MONOMIALIZATION OF MORPHISMS FROM 3 FOLDS TO SURFACES

STEVEN DALE CUTKOSKY

1. Introduction

Suppose that X is a nonsingular variety, over an algebraically closed field k of characteristic zero.

If V ⊂ X is a nonsingular subvariety, the blowup of V is the morphism

π : Y = Proj(⊕_{n≥0}I_{V}^{n}) → X.

If p is a closed point of Y and π(p) = q, there exist regular parameters (x_{1}, …, x_{n}) at q and regular parameters (y_{1}, …, y_{n}) at p such that

x_{1} = x_{2} = ⋯ x_{r} = 0

(with r ≤ n = dim(X)) are local equations of V at q and

x_{1} = y_{1}, x_{2} = y_{1}y_{2}, ⋯, x_{r} = y_{1}y_{r}, x_{r+1} = y_{r+1}, …, x_{n} = y_{n}.

If V = q, so that r = n, π is called the blowup of a point.

Another simple example of a morphism is a monomial morphism, Φ : A^{n} → A^{m} defined by

y_{1} = x_{1}^{a_{11}} ⋯ x_{n}^{a_{1n}}
⋮
y_{m} = x_{1}^{a_{m1}} ⋯ x_{n}^{a_{mn}}

Φ is dominant if and only if rank(a_{ij}) = m. This notion of a monomial morphism is a little too restrictive, so we extend it in the following way.

Definition 1.1. (Definition 18.20) Suppose that Φ : X → Y is a dominant morphism of nonsingular k-varieties (where k is a field of characteristic zero). Φ is monomial if for all p ∈ X there exists an étale neighborhood U of p, uniformizing parameters (x_{1}, …, x_{n}) on U, regular parameters (y_{1}, …, y_{m}) in O_{Y,Φ(p)}, and a matrix (a_{ij}) of nonnegative integers (which necessarily has rank m) such that

y_{1} = x_{1}^{a_{11}} ⋯ x_{n}^{a_{1n}}
⋮
y_{m} = x_{1}^{a_{m1}} ⋯ x_{n}^{a_{mn}}

Suppose that

Φ : X → Y  (1)

is a dominant morphism of k-varieties, where k is a field of characteristic 0. The structure of Φ is extremely complicated. However, we can hope to construct a commutative diagram

partially supported by NSF.

\begin{matrix}X_{1} & →^{Ψ} & Y_{1} \\ ↓ & & ↓ \\ X & →^{Φ} & Y \end{matrix}  (2)

where the vertical maps are products of blowups of nonsingular subvarieties, to obtain a morphism Ψ : X_{1} → Y_{1} which has a relatively simple structure.

The most optimistic conclusion we can hope for is to construct a diagram (2) such that Ψ is monomial.

Definition 1.2. (Definition 18.20) Suppose that Φ : X → Y is a dominant morphism of k-varieties. A morphism Ψ : X_{1} → Y_{1} is a monomialization of Φ if there are sequences of blowups of nonsingular subvarieties α : X_{1} → X and β : Y_{1} → Y, and a morphism Ψ : X_{1} → Y_{1} such that the diagram

\begin{matrix}X_{1} & →^{Ψ} & Y_{1} \\ ↓ & & ↓ \\ X & →^{Φ} & Y \end{matrix}

commutes, and Ψ is a monomial morphism.

In many cases a monomialization or something close to a monomialization exists so it is natural to ask the following question.

Question Suppose that Φ : X → Y is a dominant morphism of k-varieties (over a field k of characteristic zero). Does there exist a monomialization of Φ?

By resolution of singularities and resolution of indeterminancy, we easily reduce to the case where X and Y are nonsingular.

The characteristic of k must be zero in the question. If char k = p > 0, a monomialization may not exist even for curves.

t = x^{p} + x^{p+1}

gives a simple example of a mapping of curves which cannot be monomialized, since \sqrt[p]{1 + x} is inseparable over k[x].

The obstruction to monomialization in positive characteristic is thus wild ramification.

In [11], we prove that a local analogue of the Question has a positive answer for generically finite morphisms. A discussion of these results is given in section 2.

In Section 3, we outline short proofs of the positive answer to the question in the previously known cases, a morphism to a curve and a morphism of surfaces ([7], [13] in characteristic p ≥ 0 when no wild ramification is present).

In this paper we give a positive answer to the question in the case of a dominant morphism from a 3 fold to a surface.

Theorem 1.3. (Theorem 18.21) Suppose that Φ : X → S is a dominant morphism from a 3 fold X to a surface S (over an algebraically closed field k of characteristic zero). Then there exist sequences of blowups of nonsingular subvarieties X_{1} → X and S_{1} → S such that the induced map Φ_{1} : X_{1} → S_{1} is a monomial morphism.

From this we deduce that it is possible to toroidalize ([18], [5], Definition 10.1) a dominant morphism from a 3 fold to a surface. A toroidal morphism X → Y is a morphism which is monomial with respect to fixed SNC divisors on X and Y.

Theorem 1.4. (Theorem 19.11) Suppose that Φ : X → S is a dominant morphism from a 3 fold X to a surface S (over an algebraically closed field k of characteristic zero) and D_{S} is a reduced 1 cycle on S such that E_{X} = Φ^{−1}(D_{S})_{red} contains sing(X) and sing(Φ). Then there exist sequences of blowups of nonsingular subvarieties π_{1} : X_{1} → X and π_{2} : S_{1} → S such that the induced morphism X_{1} → S_{1} is a toroidal morphism with respect to π_{2}^{−1}(D_{S})_{red} and π_{1}^{−1}(E_{X})_{red}.

Suppose that Φ : X → Y is a dominant morphism of nonsingular k-varieties, and dim(Y) > 1.

To begin with, we point out that monomialization is not a direct consequence of embedded resolution of singularities and principalization of ideals.

Suppose that p ∈ X is a point where Φ is not smooth, and q = Φ(p). Let (y_{1}, …, y_{m}) be regular parameters in O_{Y,q}. By standard theorems on resolution, we have a sequence of blowups of nonsingular subvarieties π : X_{1} → X such that if p_{1} ∈ π^{−1}(p), then there exist regular parameters (x_{1}, …, x_{n}) in O_{X_{1},p_{1}}, a matrix (a_{ij}) with nonnegative coefficients and units δ_{1}, …, δ_{m} ∈ O_{X_{1},p_{1}} such that

y_{1} = x_{1}^{a_{11}} ⋯ x_{n}^{a_{1n}}δ_{1} ; ⋮ ; y_{m} = x_{1}^{a_{m1}} ⋯ x_{n}^{a_{mn}}δ_{n}  (3)

In general, p_{1} will lie on a single exceptional component of π, and p_{1} will be disjoint from the strict transforms of codimension 1 subschemes of X determined by y_{i} = 0, 1 ≤ i ≤ m, on a neighborhood of Φ^{−1}(q). In this case we will have a_{ij} = 0 if j > 1 and (a_{ij}) will have rank 1.

There thus cannot exist regular parameters (\overline{x}_{1}, …, \overline{x}_{n}) in Ô_{X_{1},p_{1}} such that

y_{1} = \overline{x}_{1}^{a_{11}} ⋯ \overline{x}_{n}^{a_{1n}}
⋮
y_{n} = \overline{x}_{1}^{a_{m1}} ⋯ \overline{x}_{n}^{a_{mn}}

since this would imply that rank(a_{ij}) = m > 1.

In fact, in general it is necessary to blowup in both X and Y to construct a monomialization. For instance, if we blowup a point p on a nonsingular surface S, blowup a point on the exceptional curve E_{1}, blowup the intersection point of the new exceptional curve E_{2} with the strict transform of E_{1}, then blowup a general point on the new exceptional curve E_{3} with exceptional curve E_{4}, we get a birational map π : S_{1} → S such that if p_{1} ∈ E_{4} is a general point we have regular parameters (u, v) in O_{S,p} and regular paramaters (x, y) in Ô_{S_{1},p_{1}} such that

u = x^{2}, v = αx^{3} + x^{4}y.

π is not monomial at p_{1} and further blowups over S_{1} will produce a morphism which is further from being monomial.

Suppose that Y is a nonsingular surface. If π_{2} : Y_{1} → Y is a sequence of blowups of points over q ∈ Y, and q_{1} ∈ π_{2}^{−1}(q) is a point which only lies on a single exceptional component E of π_{2}, then there exist regular parameters (u, v) in O_{Y,q} and (\overline{x}, \overline{y}) in Ô_{Y_{1},q_{1}} such that

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{b}\overline{y} \end{aligned}  (4)

where a, b ∈ N and P (\overline{x}) is a polynomial of degree ≤ b.

If we perform a sequence of blowups of nonsingular subvarieties π_{1} : X_{1} → X, and if p_{1} ∈ (Φ ∘ π_{1})^{−1}(q) is such that Ô_{X_{1},p_{1}} has regular parameters (\overline{x}_{1}, \overline{x}_{2}, \overline{x}_{3}, ⋯, \overline{x}_{n}) such that

\begin{aligned}u & = \overline{x}_{1}^{a} \\ v & = P (\overline{x}_{1}) + \overline{x}_{1}^{b}\overline{x}_{2} \end{aligned}  (5)

of the form of (4), we will have a factorization X_{1} → S_{1} which is a morphism in a neighborhood of p_{1}, and X_{1} → S_{1} will be monomial at p_{1}.

A strategy for monomializing a dominant morphism from a nonsingular variety X to a nonsingular surface S is thus to first perform a sequence of blowups of nonsingular subvarieties π_{1} : X_{1} → X so that for all points p of X_{1}, appropriate regular parameters (u, v) in O_{S_{1},q} where q = Φ ∘ π_{1}(p) will have simple forms which we will call prepared (Definition 6.6 if dim(X) = 3) which include the form of (5). This is accomplished if dim(X) = 3 in Theorem 17.3. Almost the entirety of this paper is devoted to proving this Theorem.

An interesting case when the existence of a global monomialization is still open is for birational morphisms of nonsingular, characteristic 0 varieties of dimension ≥ 3. Such birational maps are known to have a simple structure, since they can be factored by alternating sequences of blowups and blowdowns [6]. A local form of factorization along a valuation is proven in Theorem 1.6 [11].

2. Local Monomialization

A local version of monomialization is proven in [11].

Suppose that R ⊂ S is a local homomorphism of local rings essentially of finite type over a field k and that V is a valuation ring of the quotient field K of S, such that V dominates S. Then we can ask if there are sequences of monoidal transforms R → R′ and S → S′ such that V dominates S′, S′ dominates R′, and R → R′ is a monomial mapping.

\begin{aligned}R′ & → & S′ ⊂ V \\ ↑ & & ↑ \\ R & → & S \end{aligned}  (6)

A monoidal transform of a local ring R is the local ring R′ of a point in the blowup of a nonsingular subvariety of spec(R) such that R′ dominates R. If R is a regular local ring, then R′ is a regular local ring.

Theorem 2.1. (Monomialization)(Theorem 1.1 [11]) Suppose that R ⊂ S are regular local rings, essentially of finite type over a field k of characteristic zero, such that the quotient field K of S is a finite extension of the quotient field J of R.

Let V be a valuation ring of K which dominates S. Then there exist sequences of monoidal transforms R → R′ and S → S′ such that V dominates S′, S′ dominates R′ and there are regular parameters (x_{1}, ...., x_{n}) in R′, (y_{1}, ..., y_{n}) in S′, units δ_{1}, …, δ_{n} ∈ S′ and a matrix (a_{ij}) of nonnegative integers such that Det(a_{ij}) ≠ 0 and

x_{1} = y_{1}^{a_{11}}.....y_{n}^{a_{1n}}δ_{1} ; ⋮ ; x_{n} = y_{1}^{a_{n1}}.....y_{n}^{a_{nn}}δ_{n}.  (7)

Thus (since char(k) = 0) there exists an etale extension S′ → S′′ where S′′ has regular parameters \overline{y}_{1}, …, \overline{y}_{n} such that x_{1}, …, x_{n} are pure monomials in \overline{y}_{1}, …, \overline{y}_{n}.

The standard theorems on resolution of singularities allow one to easily find R′ and S′ such that (7) holds, but, in general, we will not have the essential condition Det(a_{ij}) ≠ 0. The difficulty of the problem is to achieve this condition.

It is an interesting open problem to prove Theorem 2.1 in positive characteristic, even in dimension 2. Theorem 2.1 implies simultaneous resolution from above [12], which is a key step in a program of Abhyankar’s for proving resolution in positive characteristic. This method is completely worked out by Abhyankar in dimension 2 [1].

A quasi-complete variety over a field k is an integral finite type k-scheme which satisfies the existence part of the valuative criterion for properness (c.f. Chapter 0 [17] where the notion is called complete). Quasi-complete and separated is equivalent to proper.

The construction of a monomialization by quasi-complete varieties follows from Theorem 2.1.

Theorem 2.2. (Theorem 1.2 [11]) Let k be a field of characteristic zero, Φ : X → Y a generically finite morphism of nonsingular proper k-varieties. Then there are birational morphisms of nonsingular quasi-complete k-varieties α : X_{1} → X and β : Y_{1} → Y, and a locally monomial morphism Ψ : X_{1} → Y_{1} such that the diagram

\begin{matrix}X_{1} & \underrightarrow{Ψ} & Y_{1} \\ ↓ & & ↓ \\ X & \underrightarrow{Φ} & Y \end{matrix}

commutes and α and β are locally products of blowups of nonsingular subvarieties. That is, for every z ∈ X_{1}, there exist affine neighborhoods V_{1} of z, V of x = α(z), such that α : V_{1} → V is a finite product of monoidal transforms, and there exist affine neighborhoods W_{1} of Ψ(z), W of y = α(Ψ(z)), such that β : W_{1} → W is a finite product of monoidal transforms.

In this Theorem, a monoidal transform of a nonsingular k-scheme S is the map T → S induced by an open subset T of Proj(⊕I^{n}), where I is the ideal sheaf of a nonsingular subvariety of S.

Theorems 1.1 and 1.2 of [11] are analogues for morphisms of the Theorems on local uniformization and local resolution of singularities of varieties of Zariski [29], [30].

3. Monomialization of Morphisms in Low Dimensions

We will outline proofs of monomialization in the previously known cases. Suppose that k is an algebraically closed field of characteristic zero and Φ : X → Y is a dominant morphism of nonsingular k varieties.

Let sing(Φ) be the closed subset of X where Φ is not smooth.

If Φ is a dominant morphism from a variety to a curve, the existence of a global monomialization follows immediately from resolution of singularities. In fact, it is really a restatement of embedded resolution of hypersurface singularities.

Theorem 3.1. Suppose that Φ : X → C is a dominant morphism from a k-variety to a curve. Then Φ has a monomialization.

Proof. Suppose that Φ : X → C where C is a nonsingular curve, X is a nonsingular n fold. Φ(sing(Φ)) is a finite number of points of C, so we may fix a regular parameter t at a point in q ∈ Φ(sing(Φ)), and monomialize the mapping above q.

By embedded resolution of hypersurfaces, there exists a sequence of blowups of nonsingular subvarieties which dominate subvarieties of Φ^{−1}(q), π : X_{1} → X such that for all p ∈ (Φ ∘ π)^{−1}(q), there exists regular parameters (x_{1}, …, x_{n}) at p such that

t = ux_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}

where a_{1} > 0, u ∈ O_{X_{1},p} is a unit. If x_{1} = \overline{x}_{1}u^{− \frac{1}{a_{1}}} , we have

t = \overline{x}_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}.

If Φ : T → S is a dominant morphism of surfaces, monomialization is not a direct corollary of resolution of singularities. One proof of monomialization in this case (over C) is given by Akbulut and King in [7].

In our paper [13] with Oliver Piltant, we show that if L is a perfect field and Φ : T → S is a dominant morphism of L-surfaces, then Φ can be monomialized if Φ is unramified. That is, no wild ramification occurs with respect to any divisorial valuation of L(T) over L(S). This condition occurs, for instance, if p |̸ [K : L(S)] where K is a Galois closure of L(T) over L(S).

We will now outline a simple proof of monomialization for morphisms of surfaces (when k is algebraically closed of characteristic zero).

Theorem 3.2. Suppose that Φ : T → S is a dominant morphism of surfaces over k. Then Φ has a monomialization.

If Φ is a monomial mapping, then Φ comes from an expression

\begin{aligned}u & = x^{a}y^{b} \\ v & = x^{c}y^{d} \end{aligned}  (8)

where ad − bc ≠ 0.

sing(Φ) must be contained in xy = 0. At a point p on x = 0 we have regular parameters (x̂, ŷ) in Ô_{X,p} such that

x̂ = x, ŷ = y − α

for some α ∈ k. If a > 0 and c > 0 we have

\begin{aligned}u & = x̂^{a}(ŷ + α)^{b} = \overline{x}^{a} \\ v & = x̂^{c}(ŷ + α)^{d} = β\overline{x}^{c} + \overline{x}^{c}\overline{y} \end{aligned}  (9)

where

x̂ = \overline{x}(ŷ + α)^{− \frac{b}{a}} , \overline{y} = (ŷ + α)^{d− \frac{cb}{a}} − β

with β = α^{d− \frac{cb}{a}}.

If a = 0 or c = 0 we also obtain a form (9) with respect to regular parameters (u_{1}, v_{1}) in O_{S,Φ(p)}.

Thus Φ is monomial at a point p if and only if there exist regular parameters in Ô_{X,p} such that one of the forms (8) or (9) hold.

We will say that Φ is prepared at p ∈ T if there exist regular parameters (u, v) in O_{S,Φ(p)}, regular parameters (x, y) in Ô_{T,p}, and a power series P such that one of the following forms holds at p.

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}y \end{aligned}  (10)

or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}  (11)

where (a, b) = 1 and ad − bc ≠ 0.

We first observe that by resolution of singularities and indeterminancy, there exists a commutative diagram

\begin{matrix}T_{1} & →^{Φ_{1}} & S_{1} \\ ↓ & & ↓ \\ T & →^{Φ} & S \end{matrix}

where the vertical maps are products of blowups of points, sing(Φ_{1}) is a simple normal crossings (SNC) divisor, and for all p ∈ sing(Φ_{1}), there exist regular parameters (u, v) at Φ_{1}(p) such that u = 0 is a local equation of sing(Φ_{1}) at p.

The essential observation is that Φ_{1} is now prepared. We give a simple proof that appears in [7].

Lemma 3.3. Φ_{1} is prepared.

Proof. Suppose that p ∈ T_{1}. With our assumptions, one of the following must hold at p.

\begin{aligned}u & = x^{a} \\ u_{x}v_{y} − u_{y}v_{x} & = δx^{e} \end{aligned}  (12)

where δ is a unit or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ u_{x}v_{y} − u_{y}v_{x} & = δx^{e}y^{f} \end{aligned}  (13)

where a, b, e, f > 0, (a, b) = 1 and δ is a unit.

Write v = ∑ a_{ij}x^{i}y^{j} with a_{ij} ∈ k. First suppose that (12) holds. Then ax^{a−1}v_{y} = δx^{e} implies we have the form (10) (after making a change of parameters in Ô_{T_{1},p}). Now suppose that (13) holds.

u_{x}v_{y} − u_{y}v_{x} = ∑ m(aj − bi)a_{ij}x^{am+i−1}y^{bm+j−1} = δx^{e}y^{f}.

Thus

v = ∑_{aj−bi=0} a_{ij}x^{i}y^{j} + ϵx^{e+1−am}y^{f+1−bm}

where ϵ is a unit. After making a change of parameters, multiplying x by a unit, and multiplying y by a unit, we get the form (11).

It is now not difficult to construct a monomialization. We must blowup points q on S_{1} over which some point is not monomial, and blowup points on T_{1} to make m_{q}O_{T_{1}} principal. If we iterate this procedure, it can be shown that we construct a commutative diagram

\begin{matrix}T_{2} & →^{Φ_{2}} & S_{2} \\ ↓ & & ↓ \\ T_{1} & →^{Φ_{1}} & S_{1} \end{matrix}

such that Φ_{2} is monomial.

4. An overview of the proof of Monomialization of morphisms from 3 folds to surfaces

Suppose that k is an algebraically closed field of characteristic zero, and Φ : X → Y is a dominant morphism of nonsingular k-varieties.

A natural first step in monomializing a morphism Φ : X → Y is to use resolution of singularities and resolution of indeterminancy to construct a commutative diagram

\begin{matrix}X_{1} & →^{Φ_{1}} & Y_{1} \\ ↓ & & ↓ \\ X & →^{Φ} & Y \end{matrix}

where the vertical maps are products of blowups of nonsingular subvarieties, sing(Φ_{1}) is a simple normal crossings (SNC) divisor, and for all p ∈ sing(Φ_{1}), there exist regular parameters (u_{1}, …, u_{n}) at Φ_{1}(p) such that u_{1} = 0 is a local equation of sing(Φ_{1}) at p.

We observed that if X and Y are surfaces, then Φ_{1} is prepared. Unfortunately, even for morphisms from a 3 fold to a surface, Φ_{1} may be quite complicated (Examples 6.3, 6.4).

A key step in the local proof of monomialization, Theorem 2.1, is to define a new invariant, which measures how far the situation is from a specific form which is close to being monomial. In the local valuation theoretic proof we make use of special products of monoidal transforms defined by Zariski called Perron transforms [30]. Under appropriate application of Perron transforms our invariant does not increase, and we can in fact make the invariant decrease, by an appropriate algorithm.

An essential difficulty globally is that our invariant can increase after a permissible monoidal transform (Example 7.2). This is a significant difference from resolution of singularities, where a foundational result is that the multiplicity of an ideal does not go up under permissible blowups.

We will give a brief overview of the proof of Theorem 18.21 (Monomialization of morphisms from 3 folds to surfaces).

Step 1. First construct a diagram

\begin{matrix}X′ & →^{Φ′} & S′ \\ ↓ & & ↓ \\ X & →^{Φ} & S \end{matrix}

where the vertical maps are products of blowups of nonsingular subvarieties such that X′, S′ are nonsingular, there exist reduced SNCS divisors D_{S′} on S′, E_{X′} = (Φ′)^{−1}(D_{S′})_{red} on X′ such that sing(Φ′) ⊂ E_{X′} and components of E_{X′} on X′ dominating distinct components of D_{S′} are disjoint. Such a morphism Φ′ will be called weakly prepared (Definition 6.1 and Lemma 6.2).

For all p ∈ X′ there exist regular parameters (u, v) in O_{S′,q} (q = Φ′(p)) and regular parameters (x, y, z) in Ô_{X′,p} such that u = 0 is a local equation of E_{X′}, u = 0 (or uv = 0) is a local equation of D_{S′} and exactly one of the following cases hold:

1.

u = x^{a}, v = P (x) + x^{b}F

where x |̸ F, F has no terms which are monomials in x. 2.

u = (x^{a}y^{b})^{m}, v = P (x^{a}y^{b}) + x^{c}y^{d}F

where (a, b) = 1, x |̸ F, y |̸ F, x^{c}y^{d}F has no terms which are monomials in x^{a}y^{b}.

3.

u = (x^{a}y^{b}z^{c})^{m}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

where (a, b, c) = 1, x |̸ F, y |̸ F, z |̸ F, x^{d}y^{e}z^{f}F has no terms which are monomials in x^{a}y^{b}z^{c}.

The structure of the singularities of F can be very complicated (Examples 6.3 and 6.4). This is in sharp contrast to the case of a morphism of surfaces (Lemma 3.3).

Our main invariant is

ν(p) = mult(F).

This invariant is independent of parameters in the forms above.

S_{r}(X′) = {p ∈ X′ | ν(p) ≥ r}

is a constructible (but not Zariski closed) subset of X′ (Proposition 6.22 and Example 6.13).

Step 2. This is the difficult step. We construct a commutative diagram

\begin{matrix}X′′ \\ ↓ λ & ↘^{Φ′′} \\ X′ & \underrightarrow{Φ′} & S′ \end{matrix}

so that everywhere we have one of the forms:

1. u = x^{a}, v = P (x) + x^{b}y,

2. u = (x^{a}y^{b})^{m}, v = P (x^{a}y^{b}) + x^{c}y^{d},

3. u = (x^{a}y^{b})^{m}, v = P (x^{a}y^{b}) + x^{c}y^{d}z,

4. u = (x^{a}y^{b}z^{c})^{m}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} with

rank \{ \begin{matrix}a & b & c \\ d & e & f \end{matrix} \} = 2.

We impose that further condition that 1. - 4. are compatible with the reduced SNC divisors D_{S′} and E_{X′′} = (Φ′′)^{−1}(D_{S′})_{red}. u = 0 is a local equation of E_{X′′}, u = 0 (or uv = 0) are local equations of D_{S′} in the above forms. We will say that Φ′′ is prepared (Definition 6.6). This is accomplished in Theorem 17.2.

We use descending induction on

r = max{t | ν(p) = t for some p ∈ X}

to achieve the conclusions of the Theorem. A major difficulty is that, unlike in the case of resolution of singularities, ν(p) can go up after blowing up a point or a nonsingular curve (Example 7.2).

However, ν(p) can go up by at most 1, and some other invariants get better, or at least no worse. For a local resoution, we reduce to two difficult cases (Sections 11 and 12) which we settle by blowing up generic curves on E_{X′} through a particular point, and use a generalization of Abhynakar’s Good Point Algorithm ([4], [20]) to achieve an improvement. This depends on arithmetic information which is captured in this algorithm.

Step 3. We construct a commutative diagram

X′′′ →^{Φ′′′} S′′
↓  ↓
X′′ →^{Φ′′} S′

such that X′′′ → X′′ is a product of blowups of nonsingular curves, S′′ → S′ is a product of blowups of points and Φ′′′ is monomial. This is accomplished in Theorem 18.19.

π : S′′ → S′ is a sequence of blowups of points. If q ∈ S′ and q_{1} ∈ π^{−1}(q) then there exist regular parameters (u, v) in O_{S′,q} and (u_{1}, v_{1}) in Ô_{S′′,q_{1}} such that

\begin{aligned}u & = u_{1}^{a} \\ v & = P (u_{1}) + u_{1}^{b}v_{1} \end{aligned}

or

\begin{aligned}u & = (u_{1}^{a}v_{1}^{b})^{m} \\ v & = P (u_{1}^{a}v_{1}^{b}) + u_{1}^{c}v_{1}^{d} \end{aligned}

with ad − bc ≠ 0 and (a, b) = 1.

If p ∈ X′′ is a point of the form 1. of Step 2, then there exists \overline{π} : S_{1} → S′ and q_{1} ∈ \overline{π}^{−1}(q) with regular parameters (u_{1}, v_{1}) in O_{S_{1},q_{1}}, (\overline{x}, \overline{y}, \overline{z}) in Ô_{X′′,p} such that

\begin{aligned}u_{1} & = \overline{x}\overline{a} \\ v_{1} & = \overline{x}^{b}(α + \overline{y}). \end{aligned}

We have an essentially canonical procedure for achieving Step 3. We blowup on S′ the (finitely many) images of all non monomial points of X′′, then blowup nonsingular curves on X′′ to resolve the indeterminancy of the resulting rational map. An invariant improves. By induction we eventually construct Φ′′′.

5. Notations

We will suppose that k is an algebraically closed field of characteristic zero. By a variety we will mean a separated, integral finite type k-scheme.

Suppose that Z is a variety and p ∈ Z. Then m_{p} will denote the maximal ideal of O_{Z,p}.

Definition 5.1. A reduced divisor D on a nonsingular variety Z of dimension n is a simple normal crossing divisor (SNC divisor) if

1. All components of D are nonsingular.

2. Suppose that p ∈ X. Let D_{1}, …, D_{s} be the components of D containing p. Then s ≤ n and there exist regular parameters (x_{1}, …, x_{n}) in O_{X,p} such that x_{i} = 0 are local equations of D_{i} at p for 1 ≤ i ≤ s.

A curve is a 1 dimensional k variety. A surface is a 2 dimensional k variety. A 3 fold is a 3 dimensional k variety. A point of a variety will mean a closed point.

By a generic point or a generic curve on a variety Z, we will mean a point or a curve which satisfies a good condition which holds on an open set (in some parametrizing space) of points or curves.

Suppose that Z is a variety and p ∈ Z. the blowup of p or the quadratic transform of p will denote Z_{1} = Proj(⊕_{n≥0}m_{p}^{n}). If V ⊂ Z is a nonsingular subvariety then the blowup of V or the monodial transform of Z centered at V will denote Z_{1} = Proj(⊕_{n≥0}I_{V}^{n}).

If R is a regular local ring with maximal ideal m, then a quadratic transform of R is R_{1} = R[\frac{m}{x}]_{m_{1}} where 0 ≠ x ∈ m and m_{1} is a maximal ideal of R_{1}.

Suppose that P (x) = ∑_{i=0}^{∞} b_{i}x^{i} ∈ k[[x]] is a series. Given t ∈ N, P_{t}(x) will denote the polynomial

P_{t}(x) = ∑_{i=0}^{t} b_{i}x^{i}.

Given a series f(x_{1}, …, x_{n}) ∈ k[[x_{1}, …, x_{n}]] ν(f), mult(f) or ord(f) will denote the order of f.

If x ∈ Q, [x] = n if n ∈ N, n ≤ x < n + 1. {x} = [x] − x. The greatest common divisor of a_{1}, …, a_{n} ∈ N will be denoted by (a_{1}, …, a_{n}).

6. The invariant ν

Definition 6.1. Suppose that Φ_{X} : X → S is a dominant morphism from a nonsingular 3 fold X to a nonsingular surface S, with reduced SNC divisors D_{S} on S and E_{X} on X such that Φ_{X}^{−1}(D_{S})_{red} = E_{X}. Let sing(Φ_{X}) be the locus of singular points of Φ_{X}.

We will say that Φ_{X} is weakly prepared if

1. sing(Φ_{X}) ⊂ E_{X} and

2. If p ∈ S is a singular point of D_{S}, C_{1} and C_{2} are the components of D_{S} containing p, T_{1} is a component of E_{X} dominating C_{1} and T_{2} is a component of E_{X} dominating C_{2} then T_{1} and T_{2} are disjoint.

Lemma 6.2. Suppose that Φ : X → S is a dominant morphism from a 3 fold X to a surface S, D_{S} is a reduced Weil divisor on S such that sing(Φ) ⊂ Φ^{−1}(D_{S}) and the singular locus of X, sing(X) ⊂ Φ^{−1}(D_{S}). Then there exists a commutative diagram

X_{1}  →^{Φ_{1}} S_{1}
↓ π_{1}  ↓ π_{2}
X →^{Φ} S

such that π_{1} and π_{2} are products of blowups of nonsingular subvarieties, and if D_{S_{1}} = π_{2}^{−1}(D_{S})_{red}, E_{X_{1}} = (Φ ∘ π_{1})^{−1}(D_{S})_{red}, then ϕ_{1} is weakly prepared.

Proof. By resolution of singularities and resolution of indeterminancy of mappings [17], there exists

\begin{matrix}\overline{X} & →^{\overline{Φ}} & \overline{S} \\ ↓ & & ↓ \\ X & → & S \end{matrix}

such that \overline{X} and \overline{S} are nonsingular, π^{−1}(D_{S})_{red} = D\overline{S} and E\overline{X} = \overline{Φ}^{−1}(D\overline{S})_{red} are SNC divisors.

Suppose that E_{1} and E_{2} are components of E\overline{X} which dominate distinct components C_{1} and C_{2} of \overline{S}. If E_{1} ∩E_{2} ≠ ∅ then there exists a sequence of blowups \overline{X}_{1} → \overline{X} with nonsingular centers which map into C_{1} ∩ C_{2} with induced map \overline{Φ}_{1} : \overline{X}_{1} → \overline{S} such that the strict transform of E_{1} and E_{2} are disjoint on \overline{X}_{1}, and E_{\overline{X}_{1}} = \overline{Φ}_{1}^{∗}(D\overline{S})_{red} is a SNC divisor.

One way to construct this is to blow up the conductor of E_{1} ∪ E_{2} to separate the strict transforms of E_{1} and E_{2} (c.f. section 2 of [14]), and then resolve the singularities of the resulting variety.

Iterating this procedure, we construct a weakly prepared morphism.

Example 6.3. The structure of weakly prepared morphisms can be quite complicated. Consider the germ of maps

u = x^{a}, v = x^{c}F

with a ≥ 2, c ≥ 0 where

F = x^{r}z + h(x, y)

where h is arbitrary. The singular locus of this map germ is the variety definied by the ideal where the jacobian has rank < 2. That is, the variety with ideal J= \sqrt{(x^{a+c−1}F_{y}, x^{a+c−1}F_{z})}. Since F_{z} = x^{r}, we have that \sqrt{J} = (x).

Examples of this kind can be used to construct weakly prepared projective morphisms satisfying the assumptions of Φ_{1}, by resolving the indeterminancy of the induced rational map P^{3} → P^{2}. A reasonably easy example to calculate is

\begin{aligned}u & = x^{2} \\ v & = y^{2} + xz. \end{aligned}

Example 6.4. Another example of a weakly prepared morphism.

Consider the (formal) germ of maps

u = x^{a}, v = x^{c}F

where

F = ∑_{i>0,j≥0} \frac{i^{j}}{j!}a_{i}(x)y^{i}z^{j} + x^{r}z

where a_{i}(x) are arbitrary series, a ≥ 2, c ≥ 0. The singular locus of this map germ is defined by J = x^{a+c−1}(F_{y}, F_{z}). Since F_{z} − yF_{y} = x^{r}, \sqrt{J} = (x).

Throughout this section we will suppose that Φ_{X} : X → S is weakly prepared. We define permissible parameters (u, v) at points q ∈ D_{S} by the following rules

1. If q is a nonsingular point of D_{S}, then regular parameters (u, v) in O_{S,q} are permissible parameters at q if u = 0 is a local equation for D_{S}. Necessarily, u = 0 is a local equation for E_{X} in O_{X,p} for all p ∈ Φ_{X}^{−1}(q).

2. If q is a singular point of D_{S}, then regular parameters (u, v) in O_{S,q} are permissible parameters at q if uv = 0 is a local equation for D_{S} at q. Necessarily, uv = 0 is a local equation for E_{X} at p for all p ∈ Φ_{X}^{−1}(q) and either u = 0 or v = 0 is a local equation of E_{X} at p.

Definition 6.5. Suppose that (u, v) are permissible parameters at q ∈ D_{S}, p ∈ Φ_{X}^{−1}(q) and that u = 0 is a local equation of E_{X} at p. Regular parameters (x, y, z) in Ô_{X,p} are called permissible parameters at p for (u, v) if u = x^{a}y^{b}z^{c} with a ≥ b ≥ c ≥ 0.

If (x, y, z) are permissible parameters at p for (u, v), then one of the following forms holds for v at p.

1. p is a 1 point:

\begin{aligned}u & = x^{a} \\ v & = P_{p}(x) + x^{b}F_{p} \end{aligned}

where a > 0, x |̸ F_{p} and x^{b}F_{p} has no terms which are powers of x,

2. p is a 2 point:

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P_{p}(x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}

where a, b > 0 (a, b) = 1, x, y |̸ F_{p} and x^{c}y^{d}F_{p} has no terms which are powers of x^{a}y^{b},

3. p is a 3 point:

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P_{p}(x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F_{p} \end{aligned}

where a, b, c > 0, (a, b, c) = 1, x, y, z |̸ F_{p} and x^{d}y^{e}z^{f}F_{p} has no terms which are

powers of x^{a}y^{b}z^{c}.

We will say that (x, y, z) are permissible parameters at p and that the above expression of V is the normalized form of v with respect to these parameters. We will also say that F_{p} is normalized with respect to (x, y, z).

The leading form of F_{p} will be denoted by L_{p}.

With the notation of Definition 6.5, we see that if p is a 1 point, then

Î_{sing(Φ_{X}),p} = \sqrt{x^{a+b−1} (\frac{∂F}{∂y} , \frac{∂F}{∂z} )}.  (14)

If p is a 2 point, then

Î_{sing(Φ_{X}),p} = \sqrt{x^{ma+c−1}y^{mb+d−1} ((ad − bc)F + ay \frac{∂F}{∂y} − bx\frac{∂F}{∂x} , y \frac{∂F}{∂z} , x\frac{∂F}{∂z} )}(15)

If p is a 3 point then

Î_{sing(Φ_{X}),p} = \sqrt{x^{ma+d−1}y^{mb+e−1}z^{mc+f−1} ( \begin{matrix}(ae − bd)zF + ayz \frac{∂F}{∂y} − bxz \frac{∂F}{∂x} , \\ (af − cd)yF + ayz \frac{∂F}{∂z} − cxy \frac{∂F}{∂x} , \\ (bf − ce)xF + bxz \frac{∂F}{∂z} − cxy \frac{∂F}{∂y} \end{matrix} )} (16)

Definition 6.6. We will say that permissible parameters (u, v) for Φ_{X}(p) ∈ D_{S} are prepared at p ∈ E_{X} if u = 0 is a local equation of E_{X} at p and there exist permissible parameters (x, y, z) at p such that one of the following forms hold:

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}y \end{aligned}  (17)

or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}  (18)

with ad − bc ≠ 0

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z \end{aligned}  (19)

or

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}  (20)

with

rank ( \begin{matrix}a & b & c \\ d & e & f \end{matrix} ) = 2.

We will say that Φ_{X} is prepared with respect to D_{S} if for every p ∈ E_{X} there exist permissible parameters for Φ_{X}(p) which are prepared at p.

Lemma 6.7. Suppose that p ∈ E_{X}, (u, v) are permissible parameters at q = Φ_{X}(p) such that u = 0 is a local equation of E_{X}. Then r = ν(F_{p}) is independent of permissible parameters (x, y, z) at p for (u, v).

If p is a 1 point then ν(F (0, y, z)) is independent of permissible parameters (x, y, z) at p for (u, v). If p is a 1 point and

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k},

then

τ(F_{p}) = max{j + k | a_{ijk} ≠ 0 with i + j + k = r}

is independent of permissible parameters (x, y, z) for (u, v) at p.

If p is a 2 point, then ν(F (0, 0, z)) is independent of permissible parameters (x, y, z) at p for (u, v). If p is a 2 point and

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k},

then

τ(F_{p}) = max{k | a_{ijk} ≠ 0 with i + j + k = r}

is independent of permissible parameters (x, y, z) for (u, v) at p.

Proof. Suppose that (u, v) are permissible parameters at q such that u = 0 is a local equation of E_{X} at p, (x, y, z), (x_{1}, y_{1}, z_{1}) are permissible parameters at p for (u, v). First suppose that p is a 1 point. We have a (normalized) expression

u = x^{a}, v = P (x) + x^{b}F

Thus

\begin{aligned}x & = ωx_{1} \\ y & = y(x_{1}, y_{1}, z_{1}) = b_{21}x_{1} + b_{22}y_{1} + b_{23}z_{1} + ⋯ \\ z & = z(x_{1}, y_{1}, z_{1}) = b_{31}x_{1} + b_{32}y_{1} + b_{33}z_{1} + ⋯ \end{aligned}

where ω^{a} = 1 and b_{22}b_{33} − b_{23}b_{32} ≠ 0, and

u = x_{1}^{a}, v = P_{1}(x_{1}) + x_{1}^{b}F_{1}

where

\begin{aligned}P_{1} & = P (ωx_{1}) + x_{1}^{b}ω^{b}F (ωx_{1}, y(x_{1}, 0, 0), z(x_{1}, 0, 0)) \\ F_{1} & = ω^{b}[F (ωx_{1}, y(x_{1}, y_{1}, z_{1}), z(x_{1}, y_{1}, z_{1})) − F (ωx_{1}, y(x_{1}, 0, 0), z(x_{1}, 0, 0))] \end{aligned}

Substituting into

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}

we get that ν(F) = ν(F_{1}), ν(F (0, y, z)) = ν(F_{1}(0, y_{1}, z_{1})) so that F_{1} is normalized with respect to (x_{1}, y_{1}, z_{1}), and τ(F) = τ(F_{1}).

Now suppose that p is a 2 point. Then

u = (x^{a}y^{b})^{m}, v = P (x^{a}y^{b}) + x^{c}y^{d}F.

Set r = ν(F). We have one of the following two cases.

case 1

\begin{aligned}x & = αx_{1} \\ y & = βy_{1} \\ z & = z(x_{1}, y_{1}, z_{1}) = a_{1}x_{1} + a_{2}y_{1} + a_{3}z_{1} + ⋯ \end{aligned}

where ω = α^{a}β^{b} satisfies ω^{m} = 1 and a_{3} ≠ 0, or case 2

\begin{aligned}x & = αy_{1} \\ y & = βx_{1} \\ z & = z(x_{1}, y_{1}, z_{1}) = a_{1}x_{1} + a_{2}y_{1} + a_{3}z_{1} + ⋯ \end{aligned}

where ω = α^{a}β^{b} satisfies ω^{m} = 1, and a_{3} ≠ 0.

In case 1, set t_{0} = max\{\frac{c}{a}, \frac{d}{b}\}. For t ≥ t_{0}, set

\begin{aligned}b_{t} & = (\frac{∂^{t(a+b)−c−d}(α^{c}β^{d}F)}{∂x_{1}^{ta−c}∂y_{1}^{tb−d}}) |_{x_{1}=y_{1}=z_{1}=0} \\ F_{1} & = α^{c}β^{d}F − ∑_{t≥t_{0}} b_{t}x_{1}^{ta−c}y_{1}^{tb−d} \\ P_{1} & = P (ωx_{1}^{a}y_{1}^{b}) + ∑_{t≥t_{0}} b_{t}(x_{1}^{a}y_{1}^{b})^{t}. \end{aligned}  (21)

Then

u = (x_{1}^{a}y_{1}^{b})^{m}, v = P_{1}(x_{1}^{a}y_{1}^{b}) + x_{1}^{c}y_{1}^{d}F_{1}.

F_{1} is normalized with respect to (x_{1}, y_{1}, z_{1}) and ν(F (0, 0, z)) = ν(F_{1}(0, 0, z_{1})). Set α(0, 0, 0) = α_{0}. β(0, 0, 0) = β_{0}. Let L, L_{1} be the respective leading forms of F and F_{1}. Then

L_{1} = α_{0}β_{0}L(α_{0}x_{1}, β_{0}y_{1}, a_{1}x_{1} + a_{2}y_{2} + a_{3}z_{1})

if there does not exist natural numbers i_{0}, j_{0} such that (c + i_{0})b − (d + j_{0})a = 0 and i_{0} + j_{0} = r,

L_{1} = α_{0}β_{0}L(α_{0}x_{1}, β_{0}y_{1}, a_{1}x_{1} + a_{2}y_{2} + a_{3}z_{1}) − \overline{c}x_{1}^{i_{0}}y_{1}^{j_{0}}

for some \overline{c} ∈ k, if there exist natural numbers i_{0}, j_{0} such that (c + i_{0})b − (d + j_{0})a = 0 and i_{0} + j_{0} = r. Thus ν(F) = ν(F_{1}) and τ(F) = τ(F_{1}).

To verify Case 2, we now need only consider the effect of a substitution

x = y_{1}, y = x_{1}.

Finally, suppose that p is a 3 point. We have

u = (x^{a}y^{b}z^{c})^{m}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

There exists σ ∈ S_{3}, and unit series α, β, γ with constant terms α_{0}, β_{0}, γ_{0} respectively, such that

x = αw_{σ(1)}, y = βw_{σ(2)}, z = γw_{σ(3)}

where

w_{1} = x_{1}, w_{2} = y_{1}, w_{3} = z_{1},

and if ω = α^{a}β^{b}γ^{c}, then ω^{m} = 1. Set t_{0} = max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}. For t ≥ t_{0}, set

\begin{aligned}b_{t} & = \frac{∂^{t(a+b+c)−d−e−f} (α^{d}β^{e}γ^{f} F)}{∂w_{σ(1)}^{ta−d}∂w_{σ(2)}^{tb−e}∂w_{σ(3)}^{tc−f}} |_{w_{σ(1)}=w_{σ(2)}=w_{σ(3)}=0} \\ F_{1} & = α^{d}β^{e}γ^{f}F (αw_{σ(1)}, βw_{σ(2)}, γw_{σ(3)}) − ∑_{t≥t_{0}} b_{t}w_{σ(1)}^{ta−d}w_{σ(2)}^{tb−e}w_{σ(3)}^{tc−f} \\ P_{1} & = P (ωw_{σ(1)}^{a}w_{σ(2)}^{b}w_{σ(3)}^{c}) + ∑_{t≥t_{0}} b_{t}w_{σ(1)}^{ta}w_{σ(2)}^{tb}w_{σ(3)}^{tc}. \end{aligned} (22)

Thus

u = (w_{σ(1)}^{a}w_{σ(2)}^{b}w_{σ(3)}^{c})^{m}, v = P_{1}(w_{σ(1)}^{a}w_{σ(2)}^{b}w_{σ(3)}^{c})+w_{σ(1)}^{d}w_{σ(2)}^{e}w_{σ(3)}^{f}F_{1}(w_{σ(1)}, w_{σ(2)}, w_{σ(3)})

where the leading form of F_{1} is

L_{1} = α_{0}^{d}β_{0}^{e}γ_{0}^{f}F (α_{0}w_{σ(1)}, β_{0}w_{σ(2)}, γ_{0}w_{σ(3)})

if there does not exist natural numbers i_{0}, j_{0}, k_{0} such that

(d + i_{0})b − (e + j_{0})a = 0, (d + i_{0})c − (f + k_{0})a = 0, and i_{0} + j + 0 + k_{0} = r,
L_{1} = α_{0}^{d}β_{0}^{e}γ_{0}^{f}F (α_{0}w_{σ(1)}, β_{0}w_{σ(2)}, γ_{0}w_{σ(3)}) − \overline{c}x_{1}^{i_{0}}y_{1}^{j_{0}}z_{1}^{k_{0}}

for some \overline{c} ∈ k, if there exist natural numbers i_{0}, j_{0}, k_{0} such that

(d + i_{0})b − (e + j_{0})a = 0, (d + i_{0})c − (f + k_{0})a = 0 and i_{0} + j_{0} + k_{0} = r.

Thus F_{1} is normalized with respect to (x_{1}, y_{1}, z_{1}) and ν(L_{1}) = ν(L).

Lemma 6.8. Suppose that p ∈ E_{X}, q = Φ_{X}(p). Then r = ν(F_{p}) is independent of permissible parameters (u, v) at q such that u = 0 is a local equation of E_{X} at p.

If p is a 1 point then ν(F_{p}(0, y, z)) is independent of permissible parameters (u, v) at q such that u = 0 is a local equation of E_{X} at p. If p is a 1 point, and

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k},

then

τ(F_{p}) = max{j + k | there exists a_{ijk} ≠ 0 with i + j + k = r}

is independent of permissible parameters (x, y, z) at p for (u, v) such that u = 0 is a local equation of E_{X} at p.

If p is a 2 point, then ν(F_{p}(0, 0, z)) is independent of permissible parameters (u, v) at q such that u = 0 is a local equation of E_{X} at p. If p is a 2 point, and

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k},

then

τ(F_{p}) = max{k | there exists a_{ijk} ≠ 0 with i + j + k = r}

is independent of permissible parameters (x, y, z) at p for (u, v) such that u = 0 is a local equation of E_{X} at p.

Proof. Let m be the maximal ideal of Ô_{X,p}. Suppose that (u, v) and (u_{1}, v_{1}) are permissible parameters at q such that u = 0 is a local equation of E_{X} at p and u_{1} = 0 is a local equation of E_{X} at p. We will show that the multiplicities of the Lemma are the same for these two sets of permissible parameters.

Case 1 Suppose that p is a 1 point. Then (u_{1}, v_{1}) and (u, v) are related by a composition of changes of parameters of the types of Cases 1.1, 1.2 and 1.3 below. It thus suffices to prove the Lemma in each of these 3 cases.

Case 1.1 Suppose that v_{1} = u, u_{1} = v. We have

u = x^{a}, v = P (x) + x^{c}F

with r = ν(F) > 0. In this case we must have v = unit u in O_{X,p}. v = unit u is equivalent to p = \overline{u}(x)x^{d} where \overline{u} is a unit and 0 < d ≤ c. Set

x = \overline{x}(\overline{u}(x) + x^{c−d}F)^{\frac{−1}{d}} .

Then v = \overline{x}^{d}. Set τ = \frac{−1}{d}. Write \overline{u}(x) = a_{0} + a_{1}x + · · ·.

\begin{aligned}(\overline{u}(x) + x^{c−d}F)^{\frac{−1}{d}} & = \overline{u}(x)^{τ} + τ\overline{u}(x)^{τ−1}x^{c−d}F + \frac{τ(τ−1)}{2} \overline{u}(x)^{τ−2}x^{2(c−d)}F^{2} + ⋯ \\ & ≡ \overline{u}(x)^{τ} mod \overline{x}^{c−d}m^{r} \end{aligned}

We thus have

x ≡ \overline{x}\overline{u}(x)^{τ} mod \overline{x}^{c−d+1}m^{r}.  (23)

Now suppose that P_{0}(x, \overline{x}) is a series. By substitution of (23), we see that there exist series A_{1} and P_{1} such that

P_{0}(x, \overline{x}) ≡ A_{1}(\overline{x}) + \overline{x}P_{1}(x, \overline{x}) mod \overline{x}^{c−d+1}m^{r}

By iteration, we get that there is a polynomial \overline{P} (\overline{x}), such that \overline{P} (0) = P_{0}(0, 0),

P_{0}(x, \overline{x}) ≡ \overline{P} (\overline{x}) mod \overline{x}^{c−d+1}m^{r}.  (24)

We get from (24) that

\overline{u}(x) ≡ Q(\overline{x}) mod \overline{x}^{c−d+1}m^{r}

where Q(0) = \overline{u}(0). Set u_{0} = \overline{u}(0). We also see that

x ≡ \overline{x}Q(\overline{x})^{τ} mod \overline{x}^{c−d+1}m^{r}

Set λ = \frac{−a}{d}.

u = \overline{x}^{a}(\overline{u}(x) + x^{c−d}F)^{λ}
= \overline{x}^{a}[\overline{u}(x)^{λ} + λ\overline{u}(x)^{λ−1}x^{c−d}F + \frac{λ(λ−1)}{2} \overline{u}^{λ−2}x^{2(c−d)}F^{2} + ⋯]

≡ \overline{x}^{a}[Q(\overline{x})^{λ} + λQ(\overline{x})^{λ−1+τ(c−d)}\overline{x}^{c−d}F + \frac{λ(λ−1)}{2}Q(\overline{x})^{λ−2+2τ(c−d)}\overline{x}^{2(c−d)}F^{2} + ⋯]

mod \overline{x}^{a+c−d+1}m^{r}

Thus

\begin{matrix}v = \overline{x}^{d} \\ u = P_{1}(\overline{x}) + \overline{x}^{a+c−d}F_{1}(\overline{x}, y, z) \end{matrix}

MONOMIALIZATION OF MORPHISMS  FROM 3 FOLDS TO SURFACES  17

where ν(p_{1}) = a, and

F_{1} ≡ λu_{0}^{λ−1+τ(c−d)}F (u_{0}\overline{x}, y, z)+\frac{λ(λ − 1)}{2}u_{0}^{λ−2+2τ(c−d)}\overline{x}^{c−d}F (u_{0}\overline{x}, y, z)^{2}+⋯ mod \overline{x}m^{r}.

Thus ν(F (x, y, z)) = ν(F_{1}(\overline{x}, y, z)), ν(F (0, y, z)) = ν(F_{1}(0, y, z)) and τ(F) = τ(F_{1}).

Case 1.2 Suppose that u_{1} = αu, v_{1} = v, where α(u, v) is a unit series. We have

u = x^{a}, v = p(x) + x^{b}F

with r = ν(F) > 0. Set λ = \frac{−1}{a}. Define x = \overline{x}α^{λ}, so that u_{1} = \overline{x}^{a}. Write α = α_{0}(u_{1}) + α_{1}(u_{1})v + ⋯

\begin{aligned}α^{λ} & = α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯) \\ & +\frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯)^{2} + ⋯ \\ & ≡ α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})P (x) + α_{2}(u_{1})P (x)^{2} + ⋯) \\ & +\frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}(α_{1}(u_{1})P (x) + α_{2}(u_{1})P (x)^{2} + ⋯)^{2} + ⋯ mod \overline{x}^{b}m^{r} \end{aligned} (25)

We have an expression

α^{λ} ≡ A_{0}(\overline{x}) + \overline{x}B_{0}(α^{λ}, \overline{x}) mod \overline{x}^{b}m^{r}.  (26)

Substitute (25) into (26) to get

α^{λ} ≡ A_{0}(\overline{x}) + \overline{x}A_{1}(\overline{x}) + \overline{x}^{2}B_{1}(α^{λ}, \overline{x}) mod \overline{x}^{b}m^{r}

By iteration, we get that there is a series S(\overline{x}) with S(0) = α_{0}(0)^{λ} = \overline{α}, such that

α^{λ} ≡ S(\overline{x}) mod \overline{x}^{b}m^{r}.

and

x = α^{λ}\overline{x} ≡ S(\overline{x})\overline{x} mod \overline{x}^{b+1}m^{r}.
v = P (x) + x^{b}F ≡ P (\overline{x}S(\overline{x})) + \overline{x}^{b}S(\overline{x})^{b}F (\overline{x}S(\overline{x}), y, z)] mod \overline{x}^{b+1}m^{r}

Thus

\begin{matrix}u_{1} = \overline{x}^{a} \\ v = P_{1}(\overline{x}) + \overline{x}^{b}F_{1}(\overline{x}, y, z) \end{matrix}

where

\begin{aligned}F_{1} & ≡ S(\overline{x})^{b}F (\overline{x}S(\overline{x}), y, z) mod \overline{x}m^{r} \\ & ≡ \overline{α}^{b}F (\overline{α}\overline{x}, y, z) mod \overline{x}m^{r} \end{aligned}

Thus ν(F) = ν(F_{1}), ν(F (0, y, z)) = ν(F_{1}(0, y, z)) and τ(F) = τ(F_{1}).

Case 1.3 Suppose that u_{1} = u, v_{1} = αu + βv. Write

\begin{aligned}α & = ∑ α_{ij}u^{i}v^{j} \\ β & = ∑ β_{ij}u^{i}v^{j} \end{aligned}

with β_{00} ≠ 0.

We have

u = x^{a}, v = P (x) + x^{b}F

r = ν(F) > 0.

v_{1} = ∑ α_{ij}u^{i+1}v^{j} + ∑ β_{ij}u^{i}v^{j+1}
= ∑ α_{ij}x^{a(i+1)}(P (x) + x^{b}F)^{j} + ∑ β_{ij}x^{ai}(P (x) + x^{b}F)^{j+1}

= ∑ α_{ij}x^{a(i+1)}(P (x)^{j} + jx^{b}P (x)^{j−1}F + \frac{j(j−1)}{2}x^{2b}P (x)^{j−2}F^{2} + ⋯ + x^{bj}F^{j})

+ ∑ β_{ij}x^{ai}(P (x)^{j+1} + (j + 1)x^{b}P (x)^{j}F + \frac{(j+1)j}{2} x^{2b}P (x)^{j−1}F^{2} + ⋯ + x^{b(j+1)}F^{j+1})

= Q(x) + H(x)F + x^{2b}F^{2}G(x, y, z)

where

H(x) = ∑ α_{ij}x^{a(i+1)}jx^{b}P (x)^{j−1} + ∑ β_{ij}x^{ai}(j + 1)P (x)^{j}x^{b}.

We can further write

H(x) = x^{b}(β_{00} + xΩ(x)).

\begin{aligned}v_{1} & = Q(x) + x^{b}(β_{00} + xΩ(x))F + x^{2b}F^{2}G \\ & = P_{1}(x) + x^{b}F_{1} \end{aligned}

where P_{1}(x) = Q(x), and

F_{1} = (β_{00} + xΩ(x))F + x^{b}F^{2}G.

Thus ν(F) = ν(F_{1}), ν(F (0, y, z)) = ν(F_{1}(0, y, z)) and τ(F) = τ(F_{1}).

Case 2 Suppose that p is a 2 point. It suffices to prove the Lemma in the three subcases 2.1, 2.2 and 2.3.

Case 2.1 Suppose that u_{1} = v, v_{1} = u. We have an expression

u = (x^{a}y^{b})^{k}, v = P (x^{a}y^{b}) + x^{c}y^{d}F.

Set r = ν(F). If

r = 0, c ≤ ord(P)a and d ≤ ord(P)b  (27)

then the multiplicities of the Lemma are the same for the two sets of parameters, so suppose that (27) doesn’t hold. v = x^{e}y^{f} unit (for some e, f) implies that there exists t > 0 such that

v = (x^{a}y^{b})^{t}(\overline{u}(x^{a}y^{b}) + x^{c−at}y^{d−bt}F)

where \overline{u} is a unit power series. Set τ = \frac{−1}{at},

x = \overline{x}(\overline{u}(x^{a}y^{b}) + x^{c−at}y^{d−bt}F)^{τ} .
(\overline{u}(x^{a}y^{b}) + x^{c−at}y^{d−bt}F)^{τ} = \overline{u}(x^{a}y^{b})^{τ} + τ\overline{u}(x^{a}y^{b})^{τ−1}x^{c−at}y^{d−bt}F
+\frac{τ(τ−1)}{2} \overline{u}(x^{a}y^{b})^{τ−2}x^{2(c−at)}y^{2(d−bt)}F^{2} + ⋯
≡ \overline{u}(x^{a}y^{b})^{τ} mod \overline{x}^{c−at}y^{d−bt}m^{r}
\begin{aligned}x^{a}y^{b} & = \overline{x}^{a}y^{b}(\overline{u}(x^{a}y^{b}) + x^{c−at}y^{d−bt}F)^{aτ} \\ & ≡ \overline{x}^{a}y^{b}\overline{u}(x^{a}y^{b})^{aτ} mod \overline{x}^{c−at+a}y^{d−bt+b}m^{r}. \end{aligned}  (28)

Now suppose that P_{0}(x^{a}y^{b}, \overline{x}^{a}y^{b}) is a series. By substitution of (28), we see that

P_{0}(x^{a}y^{b}, \overline{x}^{a}y^{b}) ≡ A_{1}(\overline{x}^{a}y^{b}) + \overline{x}^{a}y^{b}P_{1}(x^{a}y^{b}, \overline{x}^{a}y^{b}) mod \overline{x}^{c−at+a}y^{d−bt+b}m^{r}

By iteration, we get that there is a polynomial Q(\overline{x}^{a}y^{b}), such that u_{0} = Q(0) = \overline{u}(0),

\overline{u}(x^{a}y^{b}) ≡ Q(\overline{x}^{a}y^{b}) mod \overline{x}^{c−at+a}y^{d−bt+b}m^{r}.  (29)

we get from (29) that

\begin{aligned}x & ≡ \overline{x}\overline{u}(x^{a}y^{b}) mod \overline{x}^{c−at+1}y^{d−bt}m^{r} \\ & ≡ \overline{x}Q(\overline{x}^{a}y^{b}) mod \overline{x}^{c−at+1}y^{d−bt}m^{r} \end{aligned}

Set λ = \frac{−k}{t}.

u = (x^{a}y^{b})^{k} = (\overline{x}^{a}y^{b})^{k}[\overline{u}(x^{a}y^{b}) + x^{c−at}y^{d−bt}F]^{λ}
= (\overline{x}^{a}y^{b})^{k}[\overline{u}(x^{a}y^{b})^{λ} + λ\overline{u}(x^{a}y^{b})^{λ−1}x^{c−at}y^{d−bt}F
+\frac{λ(λ−1)}{2} \overline{u}(x^{a}y^{b})^{λ−2}x^{2(c−at)}y^{2(d−bt)}F^{2} + ⋯]
≡ (\overline{x}^{a}y^{b})^{k}[Q(\overline{x}^{a}y^{b})^{λ} + λQ(\overline{x}^{a}y^{b})^{λ−1+c−at}\overline{x}^{c−at}y^{d−bt}F (\overline{x}Q(\overline{x}^{a}y^{b}), y, z)
+\frac{λ(λ−1)}{2}Q(\overline{x}^{a}y^{b})^{λ−2+2(c−at)}\overline{x}^{2(c−at)}y^{2(d−bt)}F (\overline{x}Q(\overline{x}^{a}y^{b}), y, z)^{2}
+ ⋯] mod \overline{x}^{ak+c−at+1}y^{bk+d−bt}m^{r}

Thus

\begin{matrix}v = (\overline{x}^{a}y^{b})^{t} \\ u = P_{1}(\overline{x}^{a}y^{b}) + \overline{x}^{ak+c−at}y^{bk+d−bt}F_{1}(\overline{x}, y, z) \end{matrix}

where

F_{1}(\overline{x}, y, z) ≡ λQ(\overline{x}^{a}y^{b})^{λ−1+c−at}F (\overline{x}Q(\overline{x}^{a}y^{b}), y, z)
+\frac{λ(λ−1)}{2}Q(\overline{x}^{a}y^{b})^{λ−2+2(c−at)}\overline{x}^{c−at}y^{d−bt}F (\overline{x}Q(\overline{x}^{a}y^{b}), y, z)^{2}
+ ⋯ mod \overline{x}m^{r}
≡ λu_{0}^{λ−1+c−at}F (\overline{x}u_{0}, y, z) + \frac{λ(λ−1)}{2}u_{0}^{λ−2+2(c−at)}\overline{x}^{c−at}y^{d−bt}F (\overline{x}u_{0}, y, z)^{2}
+ ⋯ mod \overline{x}m^{r}.

Thus ν(F) = ν(F_{1}), ν(F_{1}(0, 0, z)) = ν(F (0, 0, z)) and τ(F) = τ(F_{1}).

Case 2.2 Suppose that p is a 2 point and that u_{1} = αu, v_{1} = v. We have an expression

u = (x^{a}y^{b})^{k}, v = P (x^{a}y^{b}) + x^{c}y^{d}F

Set r = ν(F). Write

α = α_{0}(u_{1}) + α_{1}(u_{1})v + ⋯

Set λ = \frac{−1}{ak},

x = \overline{x}α^{λ}.

We have that

u_{1} = (\overline{x}^{a}y^{b})^{k}.
α^{λ} = α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯)
+\frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯)^{2} + ⋯
≡ α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})P (x^{a}y^{b})
+α_{2}(u_{1})P (x^{a}y^{b})^{2} + ⋯) + \frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}
(α_{1}(u_{1})P (x^{a}y^{b}) + α_{2}(u_{1})P (x^{a}y^{b})^{2} + ⋯) + ⋯ mod \overline{x}^{c}y^{d}m^{r}

Now suppose that P_{0}(x^{a}y^{b}, \overline{x}^{a}y^{b}) is a series. By substitution of the above equation, we see that

P_{0}(x^{a}y^{b}, \overline{x}^{a}y^{b}) ≡ A_{1}(\overline{x}^{a}y^{b}) + \overline{x}^{a}y^{b}P_{1}(x^{a}y^{b}, \overline{x}^{a}y^{b}) mod \overline{x}^{c}y^{d}m^{r}

By iteration, we get that there is a polynomial S(\overline{x}^{a}y^{b}), such that u_{0} = S(0) = \overline{u}(0),

α^{λ} ≡ S(\overline{x}^{a}y^{b}) mod \overline{x}^{c}y^{d}m^{r}.  (30)

we get from (30) that

x = α^{λ}\overline{x} ≡ S(\overline{x}^{a}y^{b})\overline{x} mod \overline{x}^{c+1}y^{d}m^{r}
v = P (x^{a}y^{b}) + x^{c}y^{d}F
≡ P (\overline{x}^{a}y^{b}S(\overline{x}^{a}y^{b})^{a}) + \overline{x}^{c}y^{d}S(\overline{x}^{a}y^{b})^{c}F (S(\overline{x}^{a}y^{b})\overline{x}, y, z))
mod \overline{x}^{c+1}y^{d}m^{r}

so that

\begin{matrix}u_{1} = (\overline{x}^{a}y^{b})^{k} \\ v = P_{1}(\overline{x}^{a}y^{b}) + \overline{x}^{c}y^{d}F_{1}(\overline{x}, y, z) \end{matrix}

where

\begin{aligned}F_{1} & ≡ S(\overline{x}^{a}y^{b})^{c}F (\overline{x}S(\overline{x}^{a}y^{b}), y, z) mod \overline{x}m^{r} \\ & ≡ u_{0}^{c}F (u_{0}\overline{x}, y, z) mod \overline{x}m^{r} \end{aligned}

Thus ν(F) = ν(F_{1}), ν(F (0, 0, z)) = ν(F_{1}(0, 0, z)), and τ(F) = τ(F_{1}).

Case 2.3 Suppose that p is a 2 point and that u_{1} = u, v_{1} = αu + βv. We have an expression

u = (x^{a}y^{b})^{k}, v = P (x^{a}y^{b}) + x^{c}y^{d}F

where r = ν(F). Write

\begin{aligned}α & = ∑ α_{ij}u^{i}v^{j} \\ β & = ∑ β_{ij}u^{i}v^{j} \end{aligned}

with β_{00} ≠ 0.

v_{1} = ∑ α_{ij}u^{i+1}v^{j} + ∑ β_{ij}u^{i}v^{j+1}

= ∑ α_{ij}(x^{a}y^{b})^{(i+1)k}(P (x^{a}y^{b}) + x^{c}y^{d}F)^{j} + ∑ β_{ij}(x^{a}y^{b})^{ik}(P (x^{a}y^{b}) + x^{c}y^{d}F)^{j+1}

= ∑ α_{ij}(x^{a}y^{b})^{(i+1)k}(P (x^{a}y^{b})^{j} + j(x^{c}y^{d})P (x^{a}y^{b})^{j−1}F + ⋯ + (x^{c}y^{d})^{j}F^{j})

+ ∑ β_{ij}(x^{a}y^{b})^{ik}(P (x^{a}y^{b})^{j+1} + (j + 1)x^{c}y^{d}P (x^{a}y^{b})^{j}F + ⋯ + (x^{c}y^{d})^{j+1}F^{j+1})

= Q(x^{a}y^{b}) + H(x, y)F + (x^{c}y^{d})^{2}F^{2}G(x, y, z)

where

H(x, y) = x^{c}y^{d}(β_{0,0} + x^{a}y^{b}Ω(x, y)).

Then

\begin{matrix}u = (x^{a}y^{b})^{k} \\ v_{1} = Q_{1}(x^{a}y^{b}) + x^{c}y^{d}F_{1} \end{matrix}

where

F_{1} ≡ β_{0,0}F mod (xym^{r} + (x^{c}y^{d})m^{2r})

Thus ν(F) = ν(F_{1}), ν(F (0, y, z)) = ν(F_{1}(0, y, z)), and τ(F) = τ(F_{1}).

Case 3 Suppose that p is a 3 point. It suffices to prove the Lemma in the three subcases 3.1, 3.2 and 3.3.

Case 3.1 Suppose that u_{1} = v, v_{1} = u_{1}.

u = (x^{a}y^{b}z^{c})^{k}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

where r = ν(F), If

r = 0, d ≤ ord(P)a, e ≤ ord(P)b and f ≤ ord(P)c  (31)

then the multiplicities of the Lemma are the same for the two sets of parameters, so suppose that (31) doesn’t hold.

In this case we must have that v = x^{α}y^{β}z^{γ} unit, so that P (s) = s^{t}\overline{u}(s) where \overline{u} is a unit power series and

v = (x^{a}y^{b}z^{c})^{t}[\overline{u}(x^{a}y^{b}z^{c}) + x^{d−ta}y^{e−bt}z^{f−ct}F].

Set τ = \frac{−1}{at},

x = \overline{x}(\overline{u}(x^{a}y^{b}z^{c}) + x^{d−ta}y^{e−bt}z^{f−ct}F)^{τ} .
(\overline{u}(x^{a}y^{b}z^{c}) + x^{d−ta}y^{e−bt}z^{f−ct}F)^{τ} = \overline{u}(x^{a}y^{b}z^{c})^{τ} + τ\overline{u}(x^{a}y^{b}z^{c})^{τ−1}x^{d−ta}y^{e−bt}z^{f−ct}F
+\frac{τ(τ−1)}{2} \overline{u}(x^{a}y^{b}z^{c})^{τ−2}x^{2(d−ta)}y^{2(e−bt)}z^{2(f−ct)}F^{2} + ⋯
≡ \overline{u}(x^{a}y^{b}z^{c})^{τ} mod \overline{x}^{d−ta}y^{e−bt}z^{f−ct}m^{r}

\begin{aligned}x^{a}y^{b}z^{c} & = \overline{x}^{a}y^{b}z^{c}(\overline{u}(x^{a}y^{b}z^{c}) + x^{d−ta}y^{e−bt}z^{f−ct}F)^{aτ} \\ & ≡ \overline{x}^{a}y^{b}z^{c}\overline{u}(x^{a}y^{b}z^{c})^{aτ} mod \overline{x}^{a+d−ta}y^{e−bt+b}z^{f−ct+c}m^{r} \end{aligned}

Now suppose that P_{0}(x^{a}y^{b}z^{c}, \overline{x}^{a}y^{b}z^{c}) is a series. By substitution of the above equation, we see that

P_{0}(x^{a}y^{b}z^{c}, \overline{x}^{a}y^{b}z^{c}) ≡ A_{1}(\overline{x}^{a}y^{b}z^{c})+\overline{x}^{a}y^{b}z^{c}P_{1}(x^{a}y^{b}z^{c}, \overline{x}^{a}y^{b}z^{c}) mod \overline{x}^{a+d−ta}y^{e−bt+b}z^{f−ct+c}m^{r}

By iteration, we get that there is a polynomial Q(\overline{x}^{a}y^{b}z^{c}), such that if u_{0} = \overline{u}(0), Q(0) = \overline{u}(0) = u_{0},

\overline{u}(x^{a}y^{b}z^{c}) ≡ Q(\overline{x}^{a}y^{b}z^{c}) mod \overline{x}^{a+d−ta}y^{e−bt+b}z^{f−ct+c}m^{r}

MONOMIALIZATION OF MORPHISMS  FROM 3 FOLDS TO SURFACES  21

Thus

\begin{aligned}x & ≡ \overline{x}\overline{u}(x^{a}y^{b}z^{c})^{τ} mod \overline{x}^{d−ta+1}y^{e−bt}z^{f−ct}m^{r} \\ & ≡ \overline{x}Q(\overline{x}^{a}y^{b}z^{c})^{τ} mod \overline{x}^{d−ta+1}y^{e−bt}z^{f−ct}m^{r} \end{aligned}

Set λ = \frac{−k}{t}.

u = (x^{a}y^{b}z^{c})^{k} = (\overline{x}^{a}y^{b}z^{c})^{k}(\overline{u}(x^{a}y^{b}z^{c}) + x^{d−ta}y^{e−bt}z^{f−ct}F)^{λ}
= (\overline{x}^{a}y^{b}z^{c})^{k}[\overline{u}(x^{a}y^{b}z^{c})^{λ} + λ\overline{u}(x^{a}y^{b}z^{c})^{λ−1}x^{d−ta}y^{e−bt}z^{f−ct}F
+\frac{λ(λ−1)}{2} \overline{u}(x^{a}y^{b}z^{c})^{λ−2}x^{2(d−ta)}y^{2(e−bt)}z^{2(f−ct)}F^{2} + ⋯]
≡ (\overline{x}^{a}y^{b}z^{c})^{k}[Q(\overline{x}^{a}y^{b}z^{c})^{a})^{λ}

+λQ(\overline{x}^{a}y^{b}z^{c})^{λ−1+τ(d−ta)}\overline{x}^{d−ta}y^{e−bt}z^{f−ct}F (\overline{x}Q(\overline{x}^{a}y^{b}z^{c})^{τ} , y, z)

+\frac{λ(λ−1)}{2}Q(\overline{x}^{a}y^{b}z^{c})^{λ−2+2τ(d−ta)}\overline{x}^{2(d−ta)}y^{2(e−bt)}z^{2(f−ct)}F (\overline{x}Q(\overline{x}^{a}y^{b}z^{c})^{τ} , y, z)^{2}

+ ⋯] mod \overline{x}^{ak+d−ta+1}y^{bk+e−bt}z^{ck+f−ct}m^{r}

\begin{matrix}v = (\overline{x}^{a}y^{b}z^{c})^{t} \\ u = P_{1}(\overline{x}^{a}y^{b}z^{c}) + \overline{x}^{ak+d−ta}y^{bk+e−bt}z^{ck+f−ct}F_{1}(\overline{x}, y, z) \end{matrix}

where

F_{1} ≡ λQ(\overline{x}^{a}y^{b}z^{c})^{λ−1+τ(d−ta)}F (\overline{x}Q(\overline{x}^{a}y^{b}z^{c})^{τ} , y, z)

+\frac{λ(λ−1)}{2}Q(\overline{x}^{a}y^{b}z^{c})^{a})^{λ−2+2τ(d−ta)}\overline{x}^{d−ta}y^{e−bt}z^{f−ct}F (\overline{x}Q(\overline{x}^{a}y^{b}z^{c})^{τ} , y, z)^{2} + ⋯ mod \overline{x}m^{r}

≡ λu_{0}^{λ−1+τ(d−ta)}F (u_{0}^{τ} \overline{x}, y, z)

+\frac{λ(λ−1)}{2}u_{0}^{λ−2+τ(d−ta)}\overline{x}^{d−ta}y^{e−bt}z^{f−ct}F (u_{0}\overline{x}, y, z)^{2} + ⋯ mod \overline{x}m^{r} Thus ν(F_{1}) = ν(F).

Case 3.2 Suppose that u_{1} = αu, v_{1} = v.

u = (x^{a}y^{b}z^{c})^{k}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

where r = ν(F). Set λ = \frac{−1}{ak}, x = \overline{x}α^{λ}. Thus

u_{1} = (\overline{x}^{a}y^{b}z^{c})^{k}.

Write

α = α_{0}(u_{1}) + α_{1}(u_{1})v + ⋯
α^{λ} ≡ α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯)
+\frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}(α_{1}(u_{1})v + α_{2}(u_{1})v^{2} + ⋯)^{2} + ⋯

≡ α_{0}(u_{1})^{λ} + λα_{0}(u_{1})^{λ−1}(α_{1}(u_{1})P (x^{a}y^{b}z^{c}) + α_{2}(u_{1})P (x^{a}y^{b}z^{c})^{2} + ⋯)

+\frac{λ(λ−1)}{2}α_{0}(u_{1})^{λ−2}(α_{1}(u_{1})P (x^{a}y^{b}z^{c}) + α_{2}(u_{1})P (x^{a}y^{b}z^{c})^{2} + ⋯)^{2} + ⋯ mod \overline{x}^{d}y^{e}z^{f}m^{r} Thus

α^{λ} ≡ A_{0}(\overline{x}^{a}y^{b}z^{c}) + \overline{x}^{a}y^{b}z^{c}B_{0}(α^{λ}, \overline{x}^{a}y^{b}z^{c}) mod \overline{x}^{d}y^{e}z^{f}m^{r}

Substitute the above equation into itself and iterate to get

α^{λ} ≡ S(\overline{x}^{a}y^{b}z^{c}) mod \overline{x}^{d}y^{e}z^{f}m^{r}

Set \overline{α} = α(0). Then S(0) = \overline{α}^{λ}.

x = α^{λ}\overline{x} ≡ S(\overline{x}^{a}y^{b}z^{c})\overline{x} mod \overline{x}^{d+1}y^{e}z^{f}m^{r}

\begin{aligned}v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F \\ & ≡ P (\overline{x}^{a}y^{b}z^{c}S(\overline{x}^{a}y^{b}z^{c})^{a}) + \overline{x}^{d}y^{e}z^{f}S(\overline{x}^{a}y^{b}z^{c})^{d}F (S(\overline{x}^{a}y^{b}z^{c})\overline{x}, y, z) mod \overline{x}^{d+1}y^{e}z^{f}m^{r} \end{aligned}

Thus

\begin{matrix}u_{1} = (\overline{x}^{a}y^{b}z^{c})^{k} \\ v = P_{1}(\overline{x}^{a}y^{b}z^{c}) + \overline{x}^{d}y^{e}z^{f}F_{1}(\overline{x}, y, z) \end{matrix}

where

\begin{aligned}F_{1} & ≡ S(\overline{x}^{a}y^{b}z^{c})^{d}F (S(\overline{x}^{a}y^{b}z^{c})\overline{x}, y, z) mod \overline{x}m^{r} \\ & ≡ \overline{α}^{λd}F (\overline{α}^{λ}\overline{x}, y, z) mod \overline{x}m^{r} \end{aligned}

Thus ν(F_{1}) = ν(F).

Case 3.3 Suppose that u_{1} = u, v_{1} = αu + βv. We have an expression

u = (x^{a}y^{b}z^{c})^{k}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

where r = ν(F).

Write

\begin{aligned}α & = ∑ α_{ij}u^{i}v^{j} \\ β & = ∑ β_{ij}u^{i}v^{j} \end{aligned}

with β_{00} ≠ 0.

v_{1} = ∑ α_{ij}u^{i+1}v^{j} + ∑ β_{ij}u^{i}v^{j+1}

= ∑ α_{ij}(x^{a}y^{b}z^{c})^{(i+1)k}(P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F)^{j} + ∑ β_{ij}(x^{a}y^{b}z^{c})^{ik}(P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F)^{j+1} = ∑ α_{ij}(x^{a}y^{b}z^{c})^{(i+1)k}(P (x^{a}y^{b}z^{c})^{j} + j(x^{d}y^{e}z^{f})P (x^{a}y^{b}z^{c})^{j−1}F + ⋯ + (x^{d}y^{e}z^{f})^{j}F^{j}) + ∑ β_{ij}(x^{a}y^{b}z^{c})^{ik}(P (x^{a}y^{b}z^{c})^{j+1} + (j + 1)x^{d}y^{e}z^{f}P (x^{a}y^{b}z^{c})^{j}F + ⋯ + (x^{d}y^{e}z^{f})^{j+1}F^{j+1})

= Q(x^{a}y^{b}z^{c}) + HF + (x^{d}y^{e}z^{f})^{2}F^{2}G

where

H = x^{d}y^{e}z^{f}(β_{0,0} + x^{a}y^{b}z^{c}Ω)

Then

\begin{matrix}u = (x^{a}y^{b}z^{c})^{k} \\ v_{1} = Q_{1}(x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F_{1} \end{matrix}

where

F_{1} ≡ β_{0,0}F mod (xyzm^{r} + x^{d}y^{e}z^{f}m^{2r})

Thus ν(F) = ν(F_{1}).

By Lemmas 6.7 and 6.8, we can make the following definitions, with the notation of Definition 6.5.

Definition 6.9. Suppose that p ∈ E_{X}, (u, v) are permissible parameters at Φ_{X}(p), (x, y, z) are permissible parameters at p for (u, v) such that u = 0 is a local equation of E_{X} at p. Thus one of the forms of Definition 6.5 holds. Define ν(p) = ν(F_{p}). If p is a 1 point, define γ(p) = mult(F_{p}(0, y, z)). If p is a 2 point, define γ(p) = mult(F_{p}(0, 0, z)).

Suppose that p ∈ X is a 1 point such that

\begin{aligned}u & = x^{a} \\ v & = P (x^{a}) + x^{b}F_{p} \\ F_{p} & = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

where ν(p) = r. Define

τ(p) = max{j + k | there exits a_{ijk} ≠ 0 with i + j + k = r}.

If p is a 1 point, we have 1 ≤ τ(p) ≤ ν(p). Suppose that p ∈ X is a 2 point such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

where ν(p) = r. Define

τ(p) = max{k | there exits a_{ijk} ≠ 0 with i + j + k = r}.

Define

S_{r}(X) = {p ∈ E_{X}|ν(p) ≥ r}.

Let \overline{S}_{r}(X) be the Zariski closure of S_{r}(X) in X.

Definition 6.10. A point p ∈ E_{X} is resolved if the following condition holds. 1. If p is a 1 point then ν(p) ≤ 1.

2. If p is a 2 point then γ(p) ≤ 1.

3. If p is a 3 point then ν(p) = 0.

Remark 6.11. If p ∈ E_{X} is resolved and (u, v) are permissible parameters at Φ_{X}(p) such that u = 0 is a local equation of E_{X} at p, then (u, v) are prepared at p.

Lemma 6.12. S_{r}(X) ⊂ sing(Φ_{X}) for r ≥ 2, and all 3 points are contained in sing(Φ_{X}). If p ∈ S_{1}(X) is a 2 point then p ∈ sing(Φ_{X}).

Proof. The Lemma is immediate from (14), (15) and (16).

Example 6.13. S_{r}(X) is in general not Zariski closed. Consider the 2 point p with local equations

\begin{aligned}u & = xy \\ v & = x^{2}y. \end{aligned}

ν(p) = 0. At 1 points q on the surface x = 0 there are regular parameters (x, y_{1}, z) with y = y_{1} + α for some 0 ≠ α ∈ k. Set \overline{x} = x(y_{1} + α). There are permissible parameters (\overline{x}, \overline{y}, z) at q such that

\begin{aligned}u & = \overline{x} \\ v & = α^{−1}\overline{x}^{2} + \overline{x}^{2}\overline{y}. \end{aligned}

Thus ν(q) = 1.

Lemma 6.14. Suppose that p ∈ E_{X} is a 1 point and that I ⊂ Ô_{X,p} is a reduced ideal such that if x = 0 is a local equation of E_{X} at p then x ∈ I. Then the condition F_{p} ∈ I^{s} (with s ∈ N) and the condition F_{p} ∈ m_{p}I^{s} (with s ∈ N) are independent of the choice of permissible parameters (u, v) at Φ_{X}(p) such that u = 0 is a local equation of E_{X} at p, and permissible parameters (x, y, z) for (u, v) at p.

Proof. If I = m_{p}Ô_{X,p}, the Lemma follows from Lemmas 6.7 and 6.8. So we assume that I = (x, f) for some series f(y, z).

If (x, y, z) and (x_{1}, y_{1}, z_{1}) are permissible parameters at p for (u, v) then with the notation of the proof of Lemma 6.7,

F_{1} = ω^{b}[F − F (ωx_{1}, y(x_{1}, 0, 0), z(x_{1}, 0, 0))].

and

x^{ν(p)} | F (ωx_{1}, y(x_{1}(0, 0), z(x_{1}, 0, 0)]

implies F_{1} ∈ I^{s} (or F_{1} ∈ mI^{s}), x ∈ I and s ≤ ν(p) (or s ≤ ν(p) − 1).

Now suppose that (u, v), (u_{1}, v_{1}) are permissible parameters at f(p). Suppose that v_{1} = u, u_{1} = v. With the notation of Case 1.1 of the proof of Lemma 6.8, F ∈ I^{s} implies (23) can be modified to

x ≡ \overline{x}\overline{u}(x)^{τ} mod \overline{x}^{c−d+1}I^{s}

and thus

x ≡ \overline{x}Q(\overline{x})^{τ} mod \overline{x}^{c−d+1}I^{s}

We thus have

F_{1} ≡ λu_{0}^{λ−1+τ(c−d)}F (u_{0}\overline{x}, y, z)+\frac{λ(λ − 1)}{2}u_{0}^{λ−2+2τ(c−d)}\overline{x}^{c−d}F (u_{0}\overline{x}, y, z)^{2}+⋯ mod \overline{x}I^{s}

since I = (x, f(y, z)) for some f, we have F_{1} ∈ I^{s}. We have a similar proof when F ∈ mI^{s}. We can replace m^{r} in the formulas of case 1.1 of Lemma 6.8 with mI^{s}.

In the proofs of cases 1.2 and 1.3, we can also replace m^{r} in all the formulas with I^{s} (or mI^{s}). Again, since x ∈ I, we get F_{1} ∈ I^{s} (or F_{1} ∈ mI^{s}).

Lemma 6.15. Suppose that C is a 2 curve and p ∈ C. Then the condition F_{p} ∈ Î_{C,p}^{s}, (with s ∈ N) is independent of permissible parameters at Φ_{X}(p) and p.

Proof. Suppose that (u, v) are permissible parameters at Φ_{X}(p). We will first show that the condition is independent of permissible parameters for (u, v) at p.

If p is a 2 point, this follows from the proof of Lemma 6.7, with the observation that, in the notation of (21), F ∈ Î_{C,p}^{s} implies

\frac{∂^{t(a+b)−c−d}(α^{c}β^{d}F)}{∂x_{1}^{ta−c}∂y_{1}^{tb−d}} ∈ Î_{C,p}^{s−t(a+b)+c+d},

so that ∑ b_{t}x_{1}^{ta−c}y_{1}^{tb−d} ∈ Î_{C,p}^{s}, and thus F_{1} ∈ Î_{C,p}^{s}.

If p is a 3 point, this also follows from the proof of Lemma 6.7. With the notation of (22), after possibly permuting the parameters (w_{σ(1)}, w_{σ(2)}, w_{σ(3)}), we have Î_{C,p} = (w_{σ(1)}, w_{σ(2)}).

If G ∈ Ô_{X,p} is a series and G ∈ Î_{C,p}^{a} for some a, we have that

\frac{∂G}{∂w_{σ(1)}} , \frac{∂G}{∂w_{σ(2)}} ∈ Î_{C,p}^{a−1}

and

\frac{∂G}{∂w_{σ(3)}} ∈ Î_{C,p}^{a}

Thus

b_{t}w_{σ(1)}^{ta−d}w_{σ(2)}^{tb−e}w_{σ(3)}^{tc−f} ∈ Î_{C,p}^{s}

for all t, and F_{1} ∈ Î_{C,p}^{s}.

The independence of the conditions from permissible parameters (u, v) at Φ_{X}(p) follows from cases 2.1 - 3.3 of Lemma 6.8, with m^{r} replaced by Î_{C,p}^{s} = (x, y)^{s} in the formulas of these cases.

Example 6.16. If p is a 2 point, the condition F_{p} ∈ I^{s} where I ⊂ Ô_{X,p} is a reduced ideal can depend on the choice of permissible parameters at p.

Proof. Consider

u = xy, v = z^{2} + xz

the Jacobian is J = (xz, y(2z + x), x(2z + x)).

x^{2} = 2xz + x^{2} − 2xz ∈ J.

\sqrt{J} = (x, yz). (x, y, z) are permissible parameters for (u, v) at p. Let I = (x, z). F ∈ I^{2}.

We have other permissible parameters (x, y, \overline{z}) at p, where \overline{z} = z − y. Then I = (\overline{z} + y, x). The normalized form of v with respect to these new parameters is

u = xy, v = xy + F

where

F = [(\overline{z} + y)^{2} + x\overline{z}] ∉ I^{2}.

Lemma 6.17. Suppose that p is a 2 point, and C is a curve, making SNCs with the 2 curve through p. Then the condition F_{p} ∈ Î_{C,p}^{s} with s ∈ N is independent of permissible parameters (u, v) at Φ_{X}(p) and permissible parameters (x, y, z) at p for (u, v) such that Î_{C,p} = (x, z).

We will call parameters as in Lemma 6.17 permissible parameters for C at p.

Proof. Suppose that (u, v) are permissible parameters at Φ_{X}(p). We will first show that this is independent of such permissible parameters at p for (u, v). Suppose that (x, y, z) and (x_{1}, y_{1}, z_{1}) are permissible parameters for (u, v) at p such that Î_{C,p} = (x, z) = (x_{1}, z_{1}) and

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

with F ∈ Î_{C,p}^{s} = (x, y)^{s}. We have

x = αx_{1}, y = βy_{1}, z = z(x_{1}, y_{1}, z_{1}) = ωz_{1} + γx_{1}

where α, β, ω are units in Ô_{X,p} and γ ∈ Ô_{X,p}. If G ∈ Ô_{X,p} is such that G ∈ (x_{1}, z_{1})^{a} then

\frac{∂G}{∂x_{1}} ∈ (x_{1}, z_{1})^{a−1}

and

\frac{∂G}{∂y_{1}} ∈ (x_{1}, z_{1})^{a}.

Thus

\frac{∂^{t(a+b)−c−d}(α^{c}β^{d}F)}{∂x_{1}^{ta−c}∂y_{1}^{tb−d}} ∈ (x_{1}, z_{1})^{s−(ta−c)}

In (21) of Lemma 6.7, we have b_{t} = 0 if s > (ta − c), so that F_{1} ∈ (x_{1}, z_{1})^{s}.

The independence of the condition F ∈ Î_{C,p}^{s} from choice of permissible parameters (u, v) at Φ_{X}(p) follows from cases 2.1-2.3 of Lemma 6.8, with m^{r} replaced by Î_{C,p}^{s} = (x, z)^{s} is the formulas of these cases.

Let B_{2}(X) be the (possibly not closed) curve of 2 points in X, B_{3}(X) = {p_{1}, …, p_{r}} the set of 3 points in X. Let \overline{B}_{2}(X) = B_{2}(X)∪B_{3}(X) be the Zariski closure of B_{2}(X) in X.

Definition 6.18. Suppose that Z ⊂ E_{X} is a reduced closed subscheme of dimension ≤ 1 and p ∈ E_{X}. We will say that Z makes SNCs with \overline{B}_{2}(X) at p if

1. All components of Z are nonsingular at p.

2. If C_{1}, …, C_{s} are the curves of Z containing p and D_{1}, …, D_{t} are the components of \overline{B}_{2}(X) containing p, then C_{1}, …, C_{s}, D_{1}, …, D_{t} have independent tangent directions at p.

We will say that Z makes SNCs with \overline{B}_{2}(X) if Z makes SNCs with \overline{B}_{2}(X) at p for all p ∈ E_{X}.

Definition 6.19. Suppose that p ∈ X, U is an affine neighborhood of p in X, and σ : V → U is an étale cover. Then we will say that V is an étale neighborhood of p. Suppose that D ⊂ X. We will write D ∩ V to denote σ^{−1}(D ∩ U).

Definition 6.20. (c.f. Chapter 3, Section 6 [23].) Suppose that V is an affine kvariety. x_{1}, …, x_{n} ∈ Γ(V, O_{V}) are uniformizing parameters on V if the natural morphism V → spec(k[x_{1}, …, x_{n}]) is étale.

Lemma 6.21. Suppose that (x, y, z) are permissible parameters at p for (u, v) such that y, z ∈ O_{X,p}. Then there exists an affine neighborhood U of p and an étale cover V of U such that (x, y, z) are uniformizing parameters on V.

Proof. With the notations of Definition 6.5, let \overline{a} = a if p is a 1 point, \overline{a} = ma if p is a 2 or 3 point. There exists a unit λ ∈ O_{X,p} and x̃ ∈ O_{X,p} such that x\overline{a} = λx̃\overline{a}. There exists an affine neighborhood U_{1} of p such that x̃, y, z, λ ∈ R = Γ(U_{1}, O_{X}) and λ is a unit in R. Set S = R[λ^{\frac{1}{\overline{a}}}], V_{1} = spec(S). f : V_{1} → U_{1} is an étale cover. k[x, y, z] → S defines a morphism g : V_{1} → A^{3}. Let a be the origin of A^{3}. q ∈ g^{−1}(a) if and only if x, y, z ∈ m_{q} which holds if and only if x̃, y, z ∈ m_{q}. Thus g^{−1}(a) = f^{−1}(p). Ô_{V_{1},q} = k[[x̃, y, z]] = k[[x, y, z]] for all q ∈ g^{−1}(a). Thus g is étale at all points of g^{−1}(a). Since this is an open condition, (Proposition 4.5 [15]) there exists a closed set Z_{1} of V_{1} which is disjoint from f^{−1}(p) such that g | (V_{1} − Z_{1}) is étale. Let U be an affine neighborhood of p in U_{1} which is disjoint from the closed set f(Z_{1}). Let V = f^{−1}(U). Then V is an étale cover of U on which x, y, z are uniformizing parameters.

Proposition 6.22. S_{r}(X)∩(X−\overline{B}_{2}(X)) is Zariski closed in X−\overline{B}_{2}(X) and S_{r}(X)∩ B_{2}(X) is Zariski closed in B_{2}(X). Thus S_{r}(X) is a constructible set.

Proof. First suppose that p is a 1 point. Then there are regular parameters x̃, y, z in O_{X,p}, permissible parameters x, y, z at p, and a unit λ ∈ O_{X,p} such that

\begin{aligned}u & = x^{a} = λx̃^{a} \\ v & = P (x) + x^{b}F_{p}(x, y, z). \end{aligned}

x̃, y, z are uniformizing parameters in an affine neighborhood U of p, and there exists an étale neighborhood σ : V = spec(S) → U of p such that (x, y, z) are uniformizing parameters on V, x = 0 is a local equation of E_{X} ∩ V in V. Let

I = (\frac{∂^{i+j+k}v}{∂x^{i}∂y^{j}∂z^{k}} | j + k > 0, i + j + k ≤ b + r − 1) ⊂ S,

Z = V (I) ⊂ V.

Suppose that p′ ∈ E_{X} ∩ V. Then if α = y(p′), β = z(p′), we have that

\begin{aligned}u & = x^{a} \\ v & = ∑ \frac{1}{i!j!k!} \frac{∂^{i+j+k}v}{∂x^{i}∂y^{j}∂z^{k}} (0, α, β)x^{i}(y − α)^{j}(z − β)^{k} \end{aligned}

and

v − v(σ(p′)) = P_{p′}(x) + x^{b}F_{p′}.

ν(p′) ≥ r if and only if p′ ∈ V (I). Let Z_{1} = σ(Z). S_{r}(X) ∩ U = Z_{1} is closed in U.

Now suppose that p is a 2 point. Then there are regular parameters x̃, y, z in O_{X,p} and permissible parameters x, y, z at p and a unit λ in O_{X,p} such that

\begin{aligned}u & = (x^{a}y^{b})^{m} = λ(x̃^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p}(x, y, z) \end{aligned}

There exists an étale neighborhood σ : V = spec(S) → U of p such that (x, y, z) are uniformizing parameters on V, xy = 0 is a local equation of E_{X} ∩ V in V. Let C be the 2 curve in X containing p. Suppose that p′ ∈ C ∩ V. Then if β = z(p′), we have that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v − v(σ(p′)) & = P_{p′}(x^{a}y^{b}) + x^{c}y^{d}F_{p′}. \end{aligned}

v = ∑ \frac{1}{i!j!k!} \frac{∂^{i+j+k}v}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, β)x^{i}y^{j}(z − β)^{k}.

Let

I = (\frac{∂^{i+j+k}v}{∂x^{i}∂y^{j}∂z^{k}} | k > 0 or k = 0 and a(d + j) − b(c + i) = 0 and i + j + k ≤ c + d + r − 1) ⊂ S,

p′ ∈ C and ν(p′) ≥ r if and only if p′ ∈ V (I) ∩ C. Z = V (I) ⊂ V. Let Z_{1} = σ(Z). S_{r}(X) ∩ C ∩ U = C ∩ Z_{1} ∩ U is closed in C ∩ U.

Lemma 6.23. Suppose that p ∈ E_{X} is a 1 point or a 2 point.

1. Suppose that (x, y, z) are permissible parameters at p, I ⊂ Ô_{x,p} is a reduced ideal and F_{p} ∈ I^{r} for some r ≥ 2. Then Î_{\overline{S_{r}(X)},p} ⊂ I.

2. Suppose that (x, y, z) are permissible parameters at p, I = (x, f(y, z)) ⊂ Ô_{x,p} is a reduced ideal and F_{p} ∈ (x) + I^{2}. Then Î_{\overline{S}_{2}(X),p} ⊂ I.

Proof. Suppose that F_{p} ∈ I^{r} for some r ≥ 2. First assume that p is a 1 point. Since x ∈ I and r ≤ ν(p), we can make a permissible change of parameters, and renormalize to get that y, z ∈ O_{X,p} and F_{p} ∈ I^{r}.

Î_{\overline{S_{r}(X)},p} = \sqrt{(\frac{∂^{i+j+k}F_{p}}{∂x^{i}∂y^{j}∂z^{k}} |i + j + k ≤ r − 1, j + k > 0)}.  (32)

F_{p} ∈ I^{r} implies

\frac{∂^{i+j+k}F_{p}}{∂x^{i}∂y^{j}∂z^{k}} ∈ I

for all i + j + k ≤ r − 1. Thus Î_{\overline{S}_{r}(X),p} ⊂ I.

Suppose that p is a 2 point.

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}

with F_{p} ∈ I^{r} and r ≥ 2, F_{p} ∈ I^{r} implies

xy ∈ Î_{sing(Φ_{X}),p} = \sqrt{x^{ma+c−1}y^{mb+d−1} ((ad − bc)F_{p} + ay \frac{∂F_{p}}{∂y} − bx\frac{∂F_{p}}{∂x} , y \frac{∂F_{p}}{∂z} , x\frac{∂F_{p}}{∂z} )} ⊂ I,

so that x ∈ I or y ∈ I. Without loss of generality, x ∈ I.

There exist permissible parameters (\overline{x}, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p}, \overline{x}^{a}\overline{y}^{b} = x^{a}y^{b} and

x = σ\overline{x}, y = τ\overline{y}, z = \overline{z} + h

for some series σ, τ, h ∈ Ô_{X,p} with

σ ≡ 1 mod m\overline{a},
τ ≡ 1 mod m\overline{a}
h ≡ 0 mod m\overline{a}

where m = m_{p}Ô_{X,p},

\overline{a} ≥ \frac{r + c}{a}(a + b) − (c + d).

We have

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}[σ^{c}τ^{d}F_{p}(σ\overline{x}, τ\overline{y}, \overline{z} + h)] \end{aligned}

σ^{c}τ^{d}F_{p}(σ\overline{x}, τ\overline{y}, \overline{z} + h) ≡ F_{p}(\overline{x}, \overline{y}, \overline{z}) mod m\overline{a} Let v = P_{1}(\overline{x}^{a}\overline{y}^{b})+\overline{x}^{c}\overline{y}^{d}F_{1} be the normalized form of v. Since F_{p}(\overline{x}, \overline{y}, \overline{z}) is normalized, we can only remove terms

(\overline{x}^{a}\overline{y}^{b})^{t}/\overline{x}^{c}\overline{y}^{d}

with t(a + b) − (c + d) ≥ \overline{a} from σ^{c}τ^{d}F_{p}(σ\overline{x}, τ\overline{y}, \overline{z} + h) to construct F_{1}. Since this condition implies

at − c ≥ r

we have F_{1} ∈ I^{r}. We can thus assume that y, z ∈ O_{X,p}.

Set

w = \frac{v − P_{t}(x^{a}y^{b})}{x^{c}y^{d}}

with t > c + d + r. Thus

w = F_{p} + x^{m}y^{m}h(x, y)

with m > r. F_{p} ∈ I^{r} implies w ∈ I^{r} which implies that

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} ∈ I

if i + j + k ≤ r − 1.

There exists an étale neighborhood σ : V → U of p such that (x, y, z) are uniformizing parameters on V, xy = 0 is a local equation of E_{X} ∩ V in V.

Suppose that

q ∈ V (x, \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} | i + j + k ≤ r − 1) ⊂ V

is a 1 point in V. q has permissible parameters (x̃, ỹ, z̃) defined by

ỹ = y − α, z̃ = z − β, x̃ = xy^{\frac{b}{a}}  (33)

for some α, β ∈ k. Thus

u = x̃^{am}
v = P_{t}(x̃^{a}) + α^{d− \frac{cb}{a}} x̃^{c}w(x̃α^{− \frac{b}{a}} , α, β)
+x̃^{c} [(ỹ + α)^{d− \frac{cb}{a}} w(x, y, z) − α^{d− \frac{cb}{a}} w(x̃α^{− \frac{b}{a}} , α, β)]

ν_{q}(w) ≥ r implies q ∈ S_{r}(V).

Suppose that

q ∈ V (x, \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} | i + j + k ≤ r − 1) ⊂ V

is a 2 point in V. q has permissible parameters (x, y, z̃) where

z̃ = z − β.

for some β ∈ k.

u = (x^{a}y^{b})^{m}
v = P_{t}(x^{a}y^{b}) + x^{c}y^{d}[∑_{(c+i)b−a(j+d)=0} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (q)x^{i}y^{j}]
+x^{c}y^{d} [w − ∑_{(c+i)b−a(j+d)=0} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (q)x^{i}y^{j}]

Again, ν_{q}(w) ≥ r implies q ∈ S_{r}(V). So

V (I) ⊂ V (x, \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} | i + j + k ≤ r − 1) ⊂ \overline{S}_{r}(V)

implies

Î_{\overline{S}_{r},p} ⊂ \sqrt{(x, \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} | i + j + k ≤ r − 1)} ⊂ I.

We now prove 2. Suppose that the assumptions of 2. hold. If p is a 1 point, then (32) implies Î_{\overline{S}_{2}(X),p} ⊂ I.

If p is a 2 point, then arguing as in the proof of 1., we set

w = \frac{v − P_{t}(x^{a}y^{b})}{x^{c}y^{d}}

and conclude that \frac{∂w}{∂y} , \frac{∂w}{∂z} ∈ I. Suppose that q ∈ V (x, \frac{∂w}{∂y} , \frac{∂w}{∂z}) is a 1 point. Then there exists \overline{c} ∈ k such that mult(w − \overline{c}x) ≥ 2, where the multiplicity is computed at q. q has permissible parameters as in (33).

x ≡ α^{− \frac{b}{a}} x̃ mod m_{q}^{2}Ô_{X,q}

implies mult(w − \overline{c}α^{− \frac{b}{a}} x̃) ≥ 2, so that q ∈ \overline{S}_{2}(X). We have a simpler argument if q ∈ V (x, \frac{∂w}{∂y} , \frac{∂w}{∂z}) is a 2 point. Thus

Î_{\overline{S}_{2}(X),p} ⊂ \sqrt{(x, \frac{∂w}{∂y} , \frac{∂w}{∂z})} ⊂ I.

Lemma 6.24. Suppose that C ⊂ X is a curve and there exists p ∈ X such that F_{p} ∈ Î_{C,p}^{r} with r ≥ 2. Then C ⊂ E_{X}.

1. Suppose that C ⊂ E_{X} is a curve and there exists p ∈ X such that F_{p} ∈ Î_{C,p}^{r} with r ≥ 1. Suppose that q ∈ C is a 1 point. Then F_{q} ∈ Î_{C,q}^{r}.

2. Suppose that C is a 2 curve and there exists p ∈ C such that F_{p} ∈ Î_{C,p}^{r} with r ≥ 1. If q ∈ C is a 2 point, then F_{q} ∈ Î_{C,q}^{r}.

Proof. We will first show that 1 or 2 hold for all but finitely many q ∈ C. First Suppose that p is a 1 point. By Lemma 6.14, we may assume that y, z ∈ O_{X,p}.

u = x^{a}, v = P (x) + x^{b}F_{p}.

In an étale neighborhood U of p, (x, y, z) are uniformizing parameters. Let I = Î_{C,p}. If F_{p} ∈ I^{r} with r ≥ 2 then

\frac{∂F_{p}}{∂y} , \frac{∂F_{p}}{∂z} ∈ I^{r−1} ⊂ I.

u ∈ Î_{sing(Φ_{X}),p} = \sqrt{x^{a−1+b}(\frac{∂F_{p}}{∂y} , \frac{∂F_{p}}{∂z})} ⊂ I.

implies x ∈ I.

Now assume that F_{p} ∈ Î_{C,p}^{r} with r ≥ 1 so that C ⊂ E_{X}, either by assumption if r = 1, or by the above argument if r ≥ 2. Thus x ∈ Î_{C,p}. Set w = \frac{v−P_{b+r}(x)}{x^{b}}.

After possibly replacing U with a smaller étale neighborhood of p, there exists a reduced ideal J = (x, f) ⊂ Γ(U, O_{U}) such that JÔ_{X,p} = I. If q ∈ V (J) ⊂ U, then q has regular parameters (x, y − α, z − β) for some α, β ∈ k. w ∈ J^{r}Ô_{X,p} implies w ∈ J^{r} (since J is a complete intersection implies J^{r} has no embedded components). Since ν(F_{q}) ≥ r, we have

F_{q} = w − ∑_{i≥r} \frac{1}{i!} \frac{∂^{i}w}{∂x^{i}} (0, α, β)x^{i} ∈ J^{r}Ô_{X,q}.

Thus for all but finitely many q ∈ C, 1. holds.

Now suppose that p is a 2 point.

u = (x^{a}y^{b})^{m}, v = P (x^{a}y^{b}) + x^{c}y^{d}F

where F = F_{p}.

Suppose that F ∈ Î_{C,p}^{r} with r ≥ 2. Then F, \frac{∂F}{∂x} , \frac{∂F}{∂y} , \frac{∂F}{∂z} ∈ Î_{C,p}. By (15),

u ∈ Î_{sing(Φ_{X}),p} ⊂ \sqrt{(F, \frac{∂F}{∂x} , \frac{∂F}{∂y} , \frac{∂F}{∂z} )} ⊂ Î_{C,p}

implies x ∈ Î_{C,p} or y ∈ Î_{C,p}.

Now assume that F_{p} ∈ Î_{C,p}^{r} with r ≥ 1. Then C ⊂ E_{X}, either by assumption if r = 1, or by the above argument if r ≥ 2. Thus we have x or y ∈ Î_{C,p}. Suppose that x ∈ Î_{C,p}. As in the proof of Lemma 6.23, we may assume that y, z ∈ O_{X,p}. Set t = c + d + r, w = \frac{v−P_{t}(x^{a}y^{b})}{x^{c}y^{d}}. There exists an étale neighborhood U of p such that u = xy = 0 is a local equation of E_{X} in U, (x, y, z) are uniformizing parameters in U, and a reduced ideal

J = (x, f) ⊂ Γ(U, O_{U})

such that JÔ_{X,p} = Î_{C,p}. w ∈ J^{r} since J is a complete intersection.

Suppose that C is not a 2 curve, so that Î_{C,p} ≠ (x, y). After possibly replacing U with a smaller étale neighborhood of p, we can assume that U ∩ C ∩ \overline{B}_{2}(X) = p.

If q ≠ p, and q ∈ V (J) ⊂ U, then q has regular parameters (x, y − α, z − β) such that α ≠ 0. Φ_{X}(q) has permissible parameters

u_{1} = u, v_{1} = v − v(Φ_{X}(q))

with permissible parameters (\overline{x}, \overline{y}, \overline{z}), defined by

x = \overline{x}(\overline{y} + α)^{\frac{−b}{a}} , \overline{y} = y − α, \overline{z} = z − β

(\overline{y} + α)^{d− \frac{bc}{a}} w ∈ Î_{C,q}^{r}, \overline{x} ∈ Î_{C,q}, and

F_{q} = (\overline{y} + α)^{d− \frac{bc}{a}} w − Ω(\overline{x})

with mult(Ω) ≥ r, which implies F_{q} ∈ Î_{C,q}^{r}.

Now suppose that C is a 2 curve, so that Î_{C,p} = (x, y). If q ∈ V (J), then (u, v − v(Φ_{X}(q))) are permissible parameters at Φ_{X}(q), and q has permissible parameters (x, y, \overline{z}) with \overline{z} = z − α. w ∈ Î_{C,q}^{r}, x ∈ Î_{C,q} and

F_{q} = w − \frac{Ω(x^{a}y^{b})}{x^{c}y^{d}}

for some Ω with mult(\frac{Ω(x^{a}y^{b})}{x^{c}y^{d}}) ≥ r. Thus F_{q} ∈ Î_{C,q}^{r}.

Now suppose that p is a 3 point.

u = (x^{a}y^{b}z^{c})^{m}, v = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F

We can assume that y, z ∈ O_{X,p}. F ∈ Î_{C,p}^{r} with r ≥ 2 implies that

F, \frac{∂F_{p}}{∂x} , \frac{∂F_{p}}{∂y} , \frac{∂F_{p}}{∂z} ∈ Î_{C,p}

By (16),

u ∈ Î_{sing(f),p} ⊂ \sqrt{(F, \frac{∂F_{p}}{∂x} , \frac{∂F_{p}}{∂y} , \frac{∂F_{p}}{∂z})} ⊂ Î_{C,p}

Thus x, y or z ∈ Î_{C,p}.

Now suppose that F ∈ Î_{C,p}^{r} with r ≥ 1. If r = 1, then x, y or z ∈ Î_{C,p} by assumption. If r ≥ 2, then x, y or z ∈ Î_{C,p} by the above argument. Suppose that x ∈ Î_{C,p}. Set t = d + e + f + r,

w = \frac{v − P_{t}(x^{a}y^{b}z^{c})}{x^{d}y^{e}z^{f}}.

There exists an étale neighborhood U of p such that (x, y, z) are uniformizing parameters in U, u = xyz = 0 is a local equation of E_{X} in U and J = Γ(U, I_{C}) = (x, f) is a complete intersection. w ∈ J^{r} since J is a complete intersection.

Suppose that C is not a 2 curve. Then we can assume that U ∩ \overline{B_{2}}(X) ∩ C = p. If q ∈ V (J) ⊂ U and q ≠ p, then Φ_{X}(q) has permissible parameters

u_{1} = u, v_{1} = v − v(Φ_{X}(q))

with permissible parameters (\overline{x}, \overline{y}, \overline{z}) at q, with

x = \overline{x}(\overline{y} + α)^{\frac{−b}{a}} (\overline{z} + β)^{− \frac{c}{a}} , y = \overline{y} + α, z = \overline{z} + β

with α, β ≠ 0. w ∈ Î_{C,q}^{r}, \overline{x} ∈ Î_{C,q} and

F_{q} = (\overline{y} + α)^{e− \frac{bd}{a}} (\overline{z} + β)^{f− \frac{cd}{a}} w − Ω(\overline{x}).

mult(Ω) ≥ r implies F_{q} ∈ Î_{C,q}^{r}.

Suppose that C is a 2 curve, Î_{C,p} = (x, y), q ∈ V (J). Then

u_{1} = u, v_{1} = v − v(Φ_{X}(q))

are permissible parameters at Φ_{X}(q), with permissible parameters (\overline{x}, y, \overline{z}) at q,

\overline{z} = z − α, x = \overline{x}(\overline{z} + α)^{\frac{−c}{a}}
u = (\overline{x}^{a}y^{b})^{m} = (\overline{x}\overline{a}\overline{y}\overline{b})\overline{m},

with (\overline{a}, \overline{b}) = 1, w ∈ I_{C,q}^{r} = (\overline{x}, y)^{r} implies

F_{q} = (z + α)^{f− \frac{dc}{a}} w − \frac{Ω(\overline{x}\overline{a}y\overline{b})}{\overline{x}^{d}y^{e}}

with mult(\frac{Ω(\overline{x}\overline{a}y\overline{b})}{\overline{x}^{d}y^{e}}) ≥ r. Thus F_{q} ∈ Î_{C,q}^{r}.

We conclude that 1. or 2. hold for all but finitely many q ∈ C.

Suppose that q ∈ C is a 1 point. We have at q,

u = x^{a}, v = P (x) + x^{b}F

with x ∈ Î_{C,q}, y, z ∈ O_{X,q}. There exists an étale neighborhood U of q such that (x, y, z) are uniformizing parameters on U, x = 0 is a local equation of E_{X}, J = Γ(U, I_{C}) = (x, f) is a complete intersection. 1. holds for all q ≠ q′ ∈ U ∩ E_{X} and

w = \frac{v − P_{b+r}(x)}{x^{b}} ∈ Γ(U, O_{U}).

For q′ ∈ V (J) ⊂ U with q′ ≠ q,

u, v_{1} = v − v(Φ_{X}(q′))

are permissible parameters at Φ_{X}(q′) and (x, y − α, z − β) are permissible parameters for (u, v_{1}) at q′.

F_{q′} = w − P_{1}(x) ∈ Î_{C,q′}^{r}

where

P_{1}(x) = ∑_{i=0}^{∞} \frac{1}{i!} \frac{∂^{i}w}{∂x^{i}} (0, α, β)x^{i} = ∑_{i=0}^{∞} a_{i}x^{i}.

Set

Λ = w − ∑_{i=0}^{r−1} a_{i}x^{i} ∈ Γ(U, O_{U}).

Λ ∈ Î_{C,q′}^{r} implies Λ ∈ J^{r}, so that Λ ∈ Î_{C,p}^{r} implies

\frac{∂^{i}Λ}{∂x^{i}}(0, 0, 0) = 0

for i < r, and

F_{p} = Λ − ∑_{i=r}^{∞} \frac{1}{r!} \frac{∂^{i}Λ}{∂x^{i}}(0, 0, 0)x^{i} ∈ Î_{C,p}^{r}.

Now suppose that C is a 2 curve. Suppose that q ∈ C is a 2 point. We have at q,

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

with Î_{C,q} = (x, y), y, z ∈ O_{X,q}. There exists an étale neighborhood U of p such that (x, y, z) are uniformizing parameters on U, xy = 0 is a local equation of E_{X} ∩ U, J = Γ(U, I_{C}) = (x, y). 2. holds for all 2 points q ≠ q′ ∈ U ∩ E_{X}.

w = \frac{v − P_{c+d+r}(x^{a}y^{b})}{x^{c}y^{d}} ∈ Γ(U, O_{U}).

For q′ ∈ V (J) ⊂ U with q′ ≠ q, there exist permissible parameters (x, y, z − β) at q′ for (u, v − v(Φ_{X}(q))).

F_{q′} = w − \frac{P_{1}(x^{a}y^{b})}{x^{c}y^{d}} ∈ Î_{C,q′}^{r}

where

P_{1}(x^{a}y^{b}) = ∑_{i=0}^{∞} a_{i}(x^{a}y^{b})^{i}.

is a series. Set Λ = w − ^{\frac{∑_{i=0}^{c+d+r−1} a_{i}(x^{a}y^{b})^{i}}{x^{c}y^{d}}} ∈ Γ(U, O_{U}). Λ ∈ Î_{C,q′}^{r} implies Λ ∈ J^{r}, so that Λ ∈ Î_{C,q}^{r} and

\frac{∂^{(a+b)i−c−d}Λ}{∂x^{ai−c}∂y^{bi−d}} (0, 0, 0) = 0

for (a + b)i − c − d < r, so that

F_{p} = Λ − ∑_{(a+b)i−c−d≥r} \frac{1}{(ai − c)!(bi − d)!} \frac{∂^{(a+b)i−c−d}Λ}{∂x^{ai−c}∂y^{bi−d}} (0, 0, 0)x^{ai−c}y^{bi−d} ∈ Î_{C,p}^{r}.

Lemma 6.25. Suppose that r ≥ 2, C ⊂ \overline{S}_{r}(X) is a nonsingular curve, and p ∈ C is a 1 point, so that there exist permissible parameters x, y, z at p such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F \end{aligned}

where Î_{C,p} = (x, z). Then

F_{p} = a_{r}(x, y) + a_{r−1}(x, y)z + ⋯ + a_{1}(x, y)z^{r−1} + g(x, y, z)z^{r}

where

x^{i} | a_{i} for 1 ≤ i ≤ r − 1,

and x^{r−1} | a_{r}.

Proof. There exist permissible parameters (x, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p} and (x, \overline{z}) = Î_{C,p}. Then there exists a, b ∈ Ô_{X,p} such that \overline{z} = ax + bz where b is a unit. Assume that the conclusions of the Lemma are true for the variables (x, \overline{y}, \overline{z}). Then substituting for x, y, z we get the conclusions of the Lemma for (x, y, z), so we may suppose that y, z ∈ O_{X,p}.

There exists an étale neighborhood U of p such that x, y, z are uniformizing parameters in U, x = 0 is a local equation of E_{X} in U, x = z = 0 are equations of C ∩U. If p′ ∈ U ∩ C, and α = y(p′), then (x, y_{1} = y − α, z) are permissible parameters at p′.

F_{p′} = ∑_{i≥0,j+k>0} \frac{1}{(c + i)!j!k!} \frac{∂^{c+i+j+k}v}{∂x^{c+i}∂y^{j}∂z^{k}} (0, α, 0)x^{i}y_{1}^{j}z^{k}.

ν(p′) ≥ r for p′ ∈ C ∩ U implies that, if j + k > 0 and i + j + k < r, then

\frac{∂^{c+i+j+k}v}{∂x^{c+i}∂y^{j}∂z^{k}} (0, α, 0) = 0

for infinitely many α, so that

\frac{∂^{c+i+j+k}v}{∂x^{c+i}∂y^{j}∂z^{k}} (0, y, 0) = 0

in U, if j + k > 0 and i + j + k < r. Thus

\frac{∂^{c+i+k}v}{∂x^{c+i}∂z^{k}} (0, y, 0) = 0

in U if i + k < r, k > 0, so that

\frac{∂^{c+i+j+k}v}{∂x^{c+i}∂y^{j}∂z^{k}} (0, y, 0) = \frac{∂^{j}}{∂y^{j}} [\frac{∂^{c+i+k}v}{∂x^{c+i}∂z^{k}} (0, y, 0)] = 0

if i + k < r, k > 0 and j ≥ 0. Thus

\frac{∂^{c+i+j+k}v}{∂x^{c+i}∂y^{j}∂z^{k}} (0, 0, 0) = 0

if k > 0, i < r − k, j ≥ 0,

\frac{∂^{c+i+1}v}{∂x^{c+i}∂y} (0, y, 0) = 0

if i < r − 1, so that

\frac{∂^{c+i+j}v}{∂x^{c+i}∂y^{j}} (0, 0, 0) = 0

if i < r − 1, j > 0, and the conclusions of the Lemma follow.

Lemma 6.26. Suppose that r ≥ 1, C ⊂ X is a 2 curve such that ν(p) ≥ r if p ∈ C is a 2 point (C ⊂ \overline{S}_{r}(X) if r ≥ 2), and p ∈ C is a 2 point, so that there exist permissible parameters x, y, z at p such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

where Î_{C,p} = (x, y). Then there exists a series τ(z) with mult τ(z) ≥ 1 such that

F = \{ \begin{matrix}τ(z)x^{i_{0}}y^{j_{0}} + ∑_{i+j≥r} a_{ij}(z)x^{i}y^{j} & if there exist nonnegative integers (i_{0}, j_{0}) such that \\ & i_{0} + j_{0} = r − 1 and a(d + j_{0}) − b(c + i_{0}) = 0 \\ ∑_{i+j≥r} a_{ij}(z)x^{i}y^{j} & otherwise \end{matrix}

Proof. There exist permissible parameters (\overline{x}, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p}. σx = \overline{x}, ωy = \overline{y}, z = \overline{z} + h, σ^{a}ω^{b} = 1 with σ, ω, h ∈ Ô_{X,p}, σ, ω ≡ 1 mod m_{p}^{2}Ô_{X,p}, h ≡ 0 mod m_{p}^{2}Ô_{X,p}. Suppose that the conclusions of the Lemma hold for (\overline{x}, \overline{y}, \overline{z}). Substituting for (x, y, z) we get the conclusions of the Lemma for (x, y, z). We may thus assume that y, z ∈ O_{X,p}.

There exists an étale neighborhood U of p such that x, y, z are uniformizing parameters in U, xy = 0 is a local equation of U ∩ E_{X}. Set

w = \frac{v − P_{t}(x^{a}y^{b})}{x^{c}y^{d}}

where t > c + d + r. We have w ∈ Γ(U, O_{U,p}) and

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P_{t}(x^{a}y^{b}) + x^{c}y^{d}w. \end{aligned}

If p′ ∈ U ∩ C, and α = z(p′), then (x, y, z_{1} = z − α) are permissible parameters at p′.

F_{p′} = ∑_{k>0, i,j≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α)x^{i}y^{j}z_{1}^{k}+ ∑_{a(i+c)−b(j+d)≠0} \frac{1}{i!j!} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, α)x^{i}y^{j}

ν(p′) ≥ r for p′ ∈ C ∩ U implies that for infinitely many α, we have

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α) = 0 if k > 0 and i + j + k < r

and

\frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, α) = 0 if a(i + c) − b(j + d) ≠ 0 and i + j < r.

Thus

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, z) = 0 if a(i + c) − b(j + d) ≠ 0 and i + j < r

and

\frac{∂^{i+j+1}w}{∂x^{i}∂y^{j}∂z} (0, 0, z) = 0 if i + j + 1 < r,

so that

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, z) = 0

if i + j < r − 1, k > 0. Setting z = 0 in the above equations, we get the statement of the Lemma.

Lemma 6.27. Suppose that C ⊂ X is a nonsingular curve containing a 1 point, p ∈ C is a 2 point such that C makes SNCs with the 2 curve through p, and (x, y, z) are permissible parameters at p such that x = z = 0 are local equations of C at p.

1. Write

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

Then if r ≥ 2 and C ⊂ \overline{S}_{r}(X),

F = x^{r−1}τ(y) + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k}

where τ is a series with mult(τ(y)) ≥ 0.

2. If there exists a 1 point q ∈ C such that F_{q} ∈ Î_{C,q}^{r} with r ≥ 1, then F_{p} ∈ Î_{C,p}^{r}.

Proof. There exist permissible parameters (\overline{x}, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p},

σx = \overline{x}, ωy = \overline{y}, σ^{a}ω^{b} = 1,

with σ, ω ∈ Ô_{X,p}, σ, ω ≡ 1 mod m_{p}Ô_{X,p}, Î_{C,p} = (\overline{x}, \overline{z}). Then \overline{z} = \overline{a}x + \overline{b}z for some \overline{a}, \overline{b} ∈ Ô_{X,p}.

Suppose that the conclusions of the Lemma hold for (\overline{x}, \overline{y}, \overline{z}). Substituting back for (x, y, z), we get the conclusions of the Lemma for (x, y, z). We may thus assume that y, z ∈ O_{X,p}

There exists an étale neighborhood U of p such that x, y, z are uniformizing parameters in U, xy = 0 is a local equation of E_{X} ∩ U, C ∩ U = V (x, z) in U. Set

w = \frac{v − P_{t}(x^{a}y^{b})}{x^{c}y^{d}}.

where t > r + c + d. We have w ∈ Γ(U, O_{U}),

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P_{t}(x^{a}y^{b}) + x^{c}y^{d}w \end{aligned}

If p′ ∈ U ∩ C, and α = y(p′), then (x, y − α, z) are regular parameters in Ô_{X,p′}. We have permissible parameters \overline{x}, \overline{y}, z at p′ ≠ p defined by

x = \overline{x}(\overline{y} + α)^{− \frac{b}{a}} , y = \overline{y} + α.

At p′, we have

\begin{aligned}u & = \overline{x}^{am} \\ v & = P_{t}(\overline{x}^{a}) + \overline{x}^{c}(\overline{y} + α)^{d− \frac{cb}{a}} w \end{aligned}  (34)

x^{c}y^{d}w = \overline{x}^{c}(\overline{y} + α)^{d− \frac{cb}{a}} w
= \overline{x}^{c}(\overline{y} + α)^{d− \frac{cb}{a}} [∑_{i,j,k≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, α, 0)\overline{x}^{i}(\overline{y} + α)^{−i \frac{b}{a}} \overline{y}^{j}z^{k}]
= ∑_{i,k≥0} [∑_{j≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, α, 0)\overline{y}^{j}] (\overline{y} + α)^{d− \frac{b(c+i)}{a}} \overline{x}^{c+i}z^{k}.

Thus

\begin{aligned}F_{p′} & = ∑_{i≥0} [∑_{j≥0} \frac{1}{i!j!} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, α, 0)\overline{y}^{j}] (\overline{y} + α)^{d− \frac{b(c+i)}{a}} \overline{x}^{i} − ∑_{i≥0} \frac{1}{i!} \frac{∂^{i}w}{∂x^{i}} (0, α, 0)α^{d− \frac{b(c+i)}{a}} \overline{x}^{i} \\ & + ∑_{i≥0,k>0} [∑_{j≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, α, 0)\overline{y}^{j}] (\overline{y} + α)^{d− \frac{b(c+i)}{a}} \overline{x}^{i}z^{k}. \end{aligned}

ν(p′) = r implies that

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, α, 0) = 0  (35)

if i + j + k < r, k > 0, and for fixed i < r

∑_{j<r−i} \frac{1}{i!j!} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, α, 0)\overline{y}^{j} ≡ c_{α}^{i}(\overline{y} + α)^{λ_{i}} mod \overline{y}^{r−i}  (36)

where λ_{i} = \frac{b(c+i)}{a} −d, and the c_{α}^{i} ∈ k depend on α and i. Since (35) holds for infinitely many α,

\frac{∂^{i+k}w}{∂x^{i}∂z^{k}} (0, y, 0) = 0

if i + k < r and k > 0. Thus

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, 0) = \frac{∂^{j}}{∂y^{j}} [\frac{∂^{i+k}w}{∂x^{i}∂z^{k}} (0, y, 0)] (0, 0, 0) = 0

if i + k < r, k > 0.

If i < r − 1 we have

\frac{1}{i!} \frac{∂^{i}w}{∂x^{i}} (0, α, 0) = c_{α}^{i}α^{λ_{i}}

and

\frac{1}{i!} \frac{∂^{i+1}w}{∂x^{i}∂y} (0, α, 0) = c_{α}^{i}λ_{i}α^{λ_{i}−1}

for infinitely many α. Thus

λ_{i} \frac{∂^{i}w}{∂x^{i}} (0, α, 0) = α\frac{∂}{∂y} \frac{∂^{i}w}{∂x^{i}} (0, α, 0)

for infinitely many α, and thus for all α. Set γ_{i}(y) = \frac{∂^{i}w}{∂x^{i}} (0, y, 0). We have

λ_{i}γ_{i}(y) = y \frac{dγ_{i}}{dy} .

There is an expansion γ_{i}(y) = ∑_{j=0}^{∞} b_{j}y^{j} with b_{j} ∈ k. \frac{dγ_{i}}{dy} = ∑_{j=1}^{∞} jb_{j}y^{j−1}.

y \frac{dγ_{i}}{dy} = ∑_{j=0}^{∞} jb_{j}y^{j}

λ_{i}γ_{i} − y \frac{dγ_{i}}{dy} = ∑_{j=0}^{∞}(λ_{i}b_{j} − jb_{j})y^{j} = 0

so that b_{j}(λ_{i} − j) = 0 for all j, which implies that γ_{i} = 0, or λ_{i} ∈ N and γ_{i} = b_{λ_{i}}y^{λ_{i}}. Suppose that λ_{i} ∈ N and γ_{i}(y) = \frac{∂^{i}w}{∂x^{i}} (0, y, 0) ≠ 0. Then

\frac{∂^{i+λ_{i}}w}{∂x^{i}∂y^{λ_{i}}} (0, 0, 0) = λ_{i}!b_{λ_{i}} ≠ 0.

But

b(c + i) − a(d + λ_{i}) = b(c + i) − a(\frac{b}{a}(c + i)) = 0

implies i > r, by our choice of t in p_{t} and the assumption that F is normalized, a contradiction. Thus

\frac{∂^{i+j}w(0, 0, 0)}{∂x^{i}∂y^{j}} = 0

if i < r − 1.

Now suppose that there exists a 1 point q′ ∈ C such that F_{q′} ∈ Î_{C,q′}^{r}. By 1. of Lemma 6.24, F_{q} ∈ Î_{C,q}^{r} at every 1 point q ∈ C. With the above notation, (trivially if r = 1)

F_{p} = x^{r−1}τ(y) + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

For p ≠ q ∈ C ∩ U there exist permissible parameters (\overline{x}, \overline{y}, z) at q such that

x = \overline{x}(\overline{y} + α)^{− \frac{b}{a}} , y = \overline{y} + α
F_{q} = \overline{x}^{r−1}Λ + Ω

with

Λ = (\overline{y} + α)^{d− \frac{b}{a} (c+r−1)}τ(\overline{y} + α) − α^{d−(c+r−1) \frac{b}{a}} τ(α),

Ω ∈ Î_{C,q}^{r}. F_{q} ∈ Î_{C,q}^{r} implies τ = 0 or d − \frac{b}{a}(c + r − 1) = 0 and τ ∈ k. But ad − b(c + r − 1) = 0 and τ ∈ k is not possible since F is normalized. Thus τ = 0.

Lemma 6.28. Suppose that r ≥ 1 C ⊂ X is a 2 curve, such that ν(q) ≥ r if q ∈ C is a 2 point (C ⊂ \overline{S}_{r}(X) if r ≥ 2) and p ∈ C is a 3 point, so that there exist permissible parameters x, y, z at p such that

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F \end{aligned}

where Î_{C,p} = (x, y). Then

F = \{ \begin{matrix}τ(z)x^{i_{0}}y^{j_{0}} + ∑_{i+j≥r} a_{ij}(z)x^{i}y^{j} & if there exist (i_{0}, j_{0}) such that \\ & i_{0} + j_{0} = r − 1 and a(e + j_{0}) − b(d + i_{0}) = 0 \\ ∑_{i+j≥r} a_{ij}(z)x^{i}y^{j} & otherwise \end{matrix}

If there exists a 2 point q ∈ C such that F_{q} ∈ Î_{C,q}^{r} and r ≥ 1, then F_{p} ∈ Î_{C,p}^{r}

Proof. There exist permissible parameters (\overline{x}, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p},

σx = \overline{x}, ωy = \overline{y}, μz = \overline{z}

for some unit series σ, ω, μ ∈ Ô_{X,p}. Suppose that the conclusions of the Lemma are true for the parameters (\overline{x}, \overline{y}, \overline{z}). Substituting back for (x, y, z) we get the conclusions of the Lemma for (x, y, z). We may thus assume that y, z ∈ O_{X,p}.

There exists an étale neighborhood U of p such that (x, y, z) are uniformizing parameters in U, xyz = 0 is a local equation of E_{X} ∩ U. Set

w = \frac{v − P_{t}(x^{a}y^{b}z^{c})}{x^{d}y^{e}z^{f}}

where t ≥ d + e + f + r. We have w ∈ Γ(U, O_{U}) and

u = (x^{a}y^{b}z^{c})^{m}, v = P_{t}(x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}w.

If p′ ∈ U ∩ C and α = z(p′), then (x, y, z − α) are regular parameters in Ô_{X,p′}. If α ≠ 0, we have permissible parameters (\overline{x}, y, \overline{z}) at p′ where x = \overline{x}(\overline{z} +α)^{\frac{−c}{a}}, \overline{z} = z −α. At p′ we have

u = (\overline{x}^{a}y^{b})^{m}, v = P_{t}(\overline{x}^{a}y^{b}) + \overline{x}^{d}y^{e}(\overline{z} + α)^{f− \frac{dc}{a}} w.

\begin{aligned}\overline{x}^{d}y^{e}(\overline{z} + α)^{f− \frac{cd}{a}} w & = \overline{x}^{d}y^{e}(\overline{z} + α)^{f− \frac{cd}{a}} [∑_{i,j,k≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α)\overline{x}^{i}(\overline{z} + α)^{\frac{−ic}{a}} y^{j}\overline{z}^{k}] \\ & = \overline{x}^{d}y^{e} [∑_{i,j≥0} (∑_{k≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α)\overline{z}^{k}) (\overline{z} + α)^{f− \frac{c(i+d)}{a}} \overline{x}^{i}y^{j}] . \end{aligned}

Thus

F_{p′} = ∑_{i,j≥0} (∑_{k≥0} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α)\overline{z}^{k}) (\overline{z} + α)^{f− \frac{c(i+d)}{a}} \overline{x}^{i}y^{j}
− ∑_{i,j such that b(i+d)−a(j+e)=0} (\frac{1}{i!j!} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, α)α^{f− \frac{c(i+d)}{a}} ) \overline{x}^{i}y^{j}

ν(p′) = r implies

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α) = 0 if i + j + k < r and b(i + d) − a(j + e) ≠ 0  (37)

and if b(i + d) − a(j + e) = 0 for fixed i, j with i + j < r,

∑_{k<r−i−j} \frac{1}{i!j!k!} \frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, α)\overline{z}^{k} ≡ c_{α}^{ij}(\overline{z} + α)^{λ_{i}} mod (\overline{z}^{r−i−j})  (38)

where λ_{i} = \frac{c(i+d)}{a} − f, c_{α}^{i,j} ∈ k depend on α, i and j.

Since (37) holds for infinitely many α,

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, z) = 0

if i + j + k < r and b(i + d) − a(j + e) ≠ 0. Thus

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, 0) = \frac{∂^{k}}{∂z^{k}} [\frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, z)] (0, 0, 0) = 0

if i + j < r, k ≥ 0 and b(i + d) − a(j + e) ≠ 0.

If b(i + d) − a(j + e) = 0 and i + j < r − 1, we have

\frac{1}{i!j!} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, α) = c_{α}^{ij}α^{λ_{i}}

and

\frac{1}{i!j!} \frac{∂^{i+j+1}w}{∂x^{i}∂y^{j}∂z} (0, 0, α) = c_{α}^{ij}λ_{i}α^{λ_{i}−1}

for infinitely many α. Thus

λ_{i} \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, α) = α\frac{∂^{i+j+1}w}{∂x^{i}∂y^{j}∂z} (0, 0, α)

for infinitely many α, and thus for all α. Set

γ_{ij}(z) = \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, z).

We have an expression

γ_{ij}(z) = ∑_{k=0}^{∞} b_{k}z^{k}

with b_{k} ∈ k.

\frac{dγ_{ij}}{dz} = ∑_{k=1}^{∞} kb_{k}z^{k−1}.

λ_{i}γ_{ij} − z \frac{dγ_{ij}}{dz} = ∑_{k=0}^{∞}(λ_{i}b_{k} − kb_{k})z^{k} = 0

implies b_{k}(λ_{i} − k) = 0 for all k, so that either γ_{ij} = 0, or λ_{i} ∈ N and γ_{ij} = b_{λ_{i}}z^{λ_{i}}. Suppose that

γ_{ij}(z) = \frac{∂^{i+j}w}{∂x^{i}∂y^{j}} (0, 0, z) ≠ 0.

Then

\frac{∂^{i+j+λ_{i}}w}{∂x^{i}∂y^{j}∂z^{λ_{i}}} (0, 0, 0) = λ_{i}!b_{λ_{i}} ≠ 0

so that we have a nontrivial x^{d+i}y^{e+j}z^{f+λ_{i}} term in x^{d}y^{e}z^{f}F. Recall that λ_{i} = \frac{c(i+d)}{a} − f. By assumption, b(i + d) − a(j + e) = 0. We further have a(f + λ_{i}) − c(d + i) = 0,

\begin{aligned}b(f + λ_{i}) − c(e + j) & = b(\frac{c(i+d)}{a}) − c(e + j) \\ & = b\frac{c}{a}(d + i) − c\frac{b}{a}(i + d) = 0 \end{aligned}

a contradiction to the assumption that F is normalized. Thus

\frac{∂^{i+j+k}w}{∂x^{i}∂y^{j}∂z^{k}} (0, 0, 0) = 0

if b(i + d) − a(j + e) = 0, i + j < r − 1, k ≥ 0.

Now suppose there exists a 2 point q′ ∈ C such that F_{q′} ∈ Î_{C,q′}^{r}. By 2. of Lemma 6.24, F_{q} ∈ Î_{C,q}^{r} for all 2 points q ∈ C. With the above notation, if F_{p} ∉ Î_{C,p}^{r}, we have

F_{p} = τ(z)x^{i_{0}}y^{j_{0}} + ∑_{i+j≥r} a_{ij}(z)x^{i}y^{j}

where i_{0} + j_{0} = r − 1 and a(e + j_{0}) − b(d + i_{0}) = 0, τ(z) ≠ 0.

For p ≠ q ∈ C ∩ U, there exist permissible parameters (\overline{x}, y, \overline{z}) at q such that

x = \overline{x}(\overline{z} + α)^{− \frac{c}{a}} , z = \overline{z} + α

\begin{aligned}u & = (\overline{x}^{a}y^{b})^{m} = (\overline{x}\overline{a}y^{b})\overline{m} \\ v & = P_{q}(\overline{x}\overline{a}y\overline{b}) + \overline{x}^{d}y^{e}F_{q} \end{aligned}

with (\overline{a}, \overline{b}) = 1.

F_{q} = \overline{x}^{i_{0}}y^{j_{0}}Λ + Ω

with Ω ∈ Î_{C,q}^{r} and

Λ = (\overline{z} + α)^{f− \frac{c(d+i_{0})}{a}} τ(\overline{z} + α) − α^{f− \frac{c(d+i_{0})}{a}} τ(α).
F_{q} ∈ Î_{C,q}^{r}

implies τ = 0, or f − \frac{c(d+i_{0})}{a} = 0 and τ ∈ k. f − \frac{c(d+i_{0})}{a} = 0 and τ ∈ k is not possible since F is normalized.

Lemma 6.29. Suppose that r ≥ 2, f_{1}, …, f_{n−1} is a regular sequence in a n dimensional regular local ring A. Let I = (f_{1}, …, f_{n−1}). Then

depth A/(I^{r} + (f_{1})^{r−1}) = 1

for all r ≥ 2.

Proof. Let m be the maximal ideal of A. There is an exact sequence of A modules

0 → I^{r−1}/(I^{r} + (f_{1})^{r−1}) → A/(I^{r} + (f_{1})^{r−1}) → A/I^{r−1} → 0

I^{r−1}/(I^{r} + (f_{1})^{r−1}) is a free A/I module, since f_{1}, …, f_{n−1} is quasi regular by Theorem 27, [21]. depth A/I^{t} = 1 for all t ≥ 1 by Proposition 16.F, [21]. Thus Hom_{A}(A/m, A/I^{r−1}) = 0 and Hom_{A}(A/m, I^{r−1}/(I^{r}+(f_{1})^{r−1})) = 0, so that Hom_{A}(A/m, A/(I^{r}+ (f_{1})^{r−1})) = 0 and depth(A/(I^{r} + (f_{1})^{r−1})) = 1.

Lemma 6.30. Suppose that r ≥ 2, p ∈ X is a 2 point and (x, y, z) are permissible parameters at p such that y, z ∈ O_{X,p},

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}

p ∈ C ⊂ \overline{S}_{r}(X) is a curve such that x ∈ Î_{C,p} and C contains a 1 point q. Then there exists a polynomial g such that one of the following cases hold

Case 1): \overline{y}^{ad−bc}F_{p} − g(x\overline{y}^{b}) ∈ (Î_{C,p}^{r} + (x)^{r−1}) k[[x, \overline{y}, z]] if ad − bc ≥ 0.

Case 2): F_{p} − g(x\overline{y}^{b})\overline{y}^{bc−ad} ∈ (Î_{C,p}^{r} + (x)^{r−1}) k[[x, \overline{y}, z]] if ad − bc ≤ 0

where y = \overline{y}^{a}.

Proof. x, y, z are uniformizing parameters in an étale neighborhood U = spec(R) of p and xy = 0 is a local equation of E_{X} ∩ U, C is a complete intersection in U. For t > c + d + r,

w = \frac{1}{x^{c}y^{d}} [v − P_{t}(x^{a}y^{b})] ∈ R.

If q ∈ C ∩ U is a 1 point, such that C is nonsingular at q, then q has permissible parameters (x_{1}, y_{1}, z_{1}) where x = x_{1}(y_{1} + α)^{− \frac{b}{a}}, y = y_{1} + α, z = z_{1} + β for some α, β ∈ k with α ≠ 0.

\begin{aligned}u & = x_{1}^{am} \\ v & = P_{q}(x_{1}) + x_{1}^{c}F_{q} \end{aligned}

where

F_{q} = (y_{1} + α)^{\frac{ad−bc}{a}} w − g(x_{1})

for some series g. F_{q} ∈ Î_{C,q}^{r} + (x_{1})^{r−1} (by Lemma 6.25) implies

w − (y_{1} + α)^{\frac{bc−ad}{a}} g(x_{1}) ∈ Î_{C,q}^{r} + (x_{1})^{r−1}.

Let y = \overline{y}^{a}, S = R[\overline{y}], λ : V = spec(S) → spec(R). Suppose that q′ ∈ λ^{−1}(q). Let h(x_{1}) = g_{r}(x_{1}).

Suppose that ad − bc ≥ 0. Then

\overline{y}^{ad−bc}w − h(x\overline{y}^{b}) ∈ Î_{C,q′}^{r} + (x)^{r−1}.

I = Γ(V, I_{C}) is a complete intersection in V, so that (by Lemma 6.29)

\overline{y}^{ad−bc}w − h(x\overline{y}^{b}) ∈ (I^{r} + (x)^{r−1})S,

and

\overline{y}^{ad−bc}w − h(x\overline{y}^{b}) ∈ (Î_{C,p}^{r} + (x)^{r−1}) k[[x, \overline{y}, z]].
\overline{y}^{ad−bc}w − h(x\overline{y}^{b}) ≡ \overline{y}^{ad−bc}F_{p} − h(x\overline{y}^{b}) mod (x^{r})

implies

\overline{y}^{ad−bc}F_{p} − h(x\overline{y}^{b}) ∈ (Î_{C,p}^{r} + (x)^{r−1}) k[[x, \overline{y}, z]].

The case when ad − bc ≤ 0 is similar.

Lemma 6.31. Suppose that p ∈ X is a 1 or 2 point, D is a generic curve through p on a component of E_{X} containing p. Then F_{q} ∉ Î_{D,q} for q ∈ D (if F_{q} is computed with respect to permissible parameters (x, y, z) at q such that x = z = 0 are local equations of D at q).

Proof. By Lemma 6.24 and Lemma 6.27, we need only check this at p. When p is a 1 point this follows from Lemma 6.14.

Suppose that p is a 2 point, (x, y, z) are permissible parameters at p such that x ∈ Î_{D,p}. Then

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}

There exists a series

\overline{z} = z − ∑_{i=1}^{∞} α_{i}y^{i}

with α_{i} ∈ k such that Î_{D,p} = (x, \overline{z}). Let

v = \overline{P} (x^{a}y^{b}) + x^{c}y^{d}\overline{F}_{p}

be the normalized form of v with respect to the permissible parameters (x, y, \overline{z}). Then

\overline{F}_{p} = F_{p} − ∑ b_{i}x^{\overline{a}^{i}}y^{\overline{b}^{i}}

with b_{i} ∈ k and

a(d + \overline{b}_{i}) − b(c + \overline{a}_{i}) = 0

for all i. Suppose that \overline{F}_{p} ∈ Î_{D,p}.

F_{p} = h(y, z) + xΩ

with h ≠ 0. We either have

\overline{z} | h(y, \overline{z} + ∑ α_{i}y^{i})

or there exists \overline{c} ∈ k, \overline{d} ∈ N such that a(d + \overline{d}) − bc = 0 and

\overline{z} | (h(y, \overline{z} + ∑ α_{i}y^{i}) − \overline{c}y\overline{d}).

Thus either h(y, ∑ α_{k}y^{k}) = 0 or h(y, ∑ α_{k}y^{k}) = \overline{c}y\overline{d}.

Since D is generic, we can suppose that α_{1}, α_{2} are independent generic points of k.

Let e = ν(h). Write h = ∑_{i+j≥e} a_{ij}y^{i}z^{j}.

\begin{aligned}h(y, ∑ α_{k}y^{k}) & = ∑_{i+j=e} a_{ij}y^{i}(α_{1}^{j}y^{j} + jα_{1}^{j−1}α_{2}y^{j+1}) + ∑_{i+j=e+1} a_{ij}y^{i}(α_{1}y)^{j} + y^{e+2}Ω \\ & = (∑_{i+j=e} a_{ij}α_{1}^{j}) y^{e} + (∑_{i+j=e} ja_{ij}α_{1}^{j−1}α_{2} + ∑_{i+j=e+1} a_{ij}α_{1}^{j}) y^{e+1} + y^{e+2}Ω \end{aligned}

We must have h(y, ∑ α_{k}y^{k}) = \overline{c}\overline{y}\overline{d} and e = \overline{d} since ∑_{i+j=e} a_{ij}α_{1}^{j} ≠ 0 as α_{1} is a generic point of k. Since F_{p} is normalized, we must have a_{e0} = 0, and ν(h) = e implies there exists a_{i_{0}j_{0}} ≠ 0 such that i_{0} + j_{0} = e and j_{0} > 0.

We must have

( ∑_{i+j=e} ja_{ij}α_{1}^{j−1}) α_{2} + ( ∑_{i+j=e+1} a_{ij}α_{1}^{j}) = 0.

which is a contradiction to the assumption that α_{1}, α_{2} are independent generic points of k.

Definition 6.32. Suppose that Φ_{X} : X → S is weakly prepared.

A monoidal transform π : Y → X is called weakly permissible if π is the blowup of a point p on E_{X}, or a nonsingular curve C on E_{X} such that C makes SNCs with \overline{B}_{2}(X).

Suppose that Φ_{X} : X → S is weakly prepared, and π : Y → X is a weakly permissible monoidal transform. Define Φ_{Y} = Φ_{X} ∘ π : Y → S, E_{Y} = π^{−1}(E_{X})_{red}. Φ_{Y} is weakly prepared.

7. The invariant ν under quadratic transforms

Throughout this section we will suppose that Φ_{X} : X → S is weakly prepared.

Theorem 7.1. Suppose that ν(p) = r, π : X_{1} → X is the blowup of p, q ∈ π^{−1}(p) with ν(q) = r_{1}.

Suppose that p is a 1 point. Then

1. If q is a 1 point then r_{1} ≤ r.

2. If q is a 2 point then r_{1} ≤ r. r_{1} = r implies τ(q) > 0. Suppose that p is a 2 point. Then

1. If q is a 1 point then r_{1} ≤ r + 1. r_{1} = r + 1 implies γ(q) = r + 1.

2. If q is a 2 point then r_{1} ≤ r.

3. If q is a 3 point then r_{1} ≤ r. Suppose that p is a 3 point. Then

1. If q is a 1 point then r_{1} ≤ r + 1. r_{1} = r + 1 implies γ(q) = r + 1.

2. If q is a 2 point then r_{1} ≤ r + 1. r_{1} = r + 1 implies τ(q) > 0. Furthermore there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{b}) + x_{1}^{c}y_{1}^{d}F_{1} \end{aligned}

and the leading form of F_{1} is

L_{1} = cy_{1}^{t}z_{1}^{r+1−t} + x_{1}Ω

where 0 ≤ t ≤ r, cb − (d + t)a = 0. In this case, the leading form of F is

L = y^{t}( ∑_{i+k=r−t} b_{ik}x^{i}z^{k})

where all b_{ik} ≠ 0, and there are regular parameters (\overline{x}_{1}, \overline{y}_{1}, \overline{z}_{1}) in Ô_{X_{1},q} such that

x = \overline{x}_{1}, y = \overline{x}_{1}\overline{y}_{1}, z = \overline{x}_{1}(\overline{z}_{1} + β)

for some 0 ≠ β ∈ k.

3. If q is a 3 point then r_{1} ≤ r. If r_{1} = r, and (x_{1}, y_{1}, z_{1}) are permissible parameters

at q with

x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1},

then the leading form of F is

L = L(y_{1}, z_{1}).

Proof. Suppose that p is a 1 point

\begin{aligned}u & = x^{k} \\ v & = P (x) + x^{c}F \end{aligned}

Write F = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

Suppose that q ∈ π^{−1}(p) is a 1 point. Then there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}(y_{1} + α) \\ z & = x_{1}(z_{1} + β) \end{aligned}

\begin{aligned}u & = x_{1}^{k} \\ v & = P (x_{1}) + x_{1}^{c+r} \frac{F}{x_{1}^{r}} \end{aligned}

\frac{F}{x_{1}^{r}} = ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} + x_{1}Ω  (39)

where a_{jk} = a_{r−i−j,j,k}. Suppose that ν(q) > ν(p). Then

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} = γ ∈ k.

and the leading form of F is

L = x_{1}^{r}( ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k}) = γx_{1}^{r} = γx^{r}

a contradiction to the assumption that F is normalized. Thus ν(q) ≤ ν(p).

Now suppose that q ∈ π^{−1}(p) is a 2 point. Then, after possibly interchanging y and z, there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}x & = x_{1}y_{1} \\ y & = y_{1} \\ z & = y_{1}(z_{1} + α) \end{aligned}

\begin{aligned}u & = x_{1}^{k}y_{1}^{k} \\ v & = P (x_{1}y_{1}) + x_{1}^{c}y_{1}^{c+r} \frac{F}{y_{1}^{r}} \end{aligned}

\frac{F}{y_{1}^{r}} = ∑_{i+k≤r} a_{ik}x_{1}^{i}(z_{1} + α)^{k} + y_{1}Ω

where a_{ik} = a_{i,r−i−k,k}. Suppose that ν(q) > ν(p). Then

∑_{i+k≤r} a_{ik}x_{1}^{i}(z_{1} + α)^{k} = ∑ γ_{i}x_{1}^{a_{i}}

with a_{i} + c = c + r for all i. a_{i} = r is the only solution to this equation, so that if L is the leading form of F,

x^{c}L = γx_{1}^{r+c}y_{1}^{r+c} = γx^{r+c}

a contradiction to the assumption that F is normalized. Thus ν(q) ≤ ν(p).

Suppose that ν(q) = r. After making a permissible change of parameters, we may assume that α = 0. We have

F_{q} = ∑_{i+k≤r} a_{ik}x_{1}^{i}z_{1}^{k} + y_{1}Σ  (40)

Thus a_{ik} = 0 if i + k < r, and since L is normalized, we must have a_{ik} ≠ 0 for some k > 0. Thus τ(q) > 0.

Suppose that p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

Write F = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

Suppose that q ∈ π^{−1}(p) is a 1 point. Then there are regular parameters (x_{1}, y_{1}, z_{1}) in Ô_{X_{1},q} such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}(y_{1} + α) \\ z & = x_{1}(z_{1} + β) \end{aligned}

with α ≠ 0.

u = x_{1}^{(a+b)k}(y_{1} + α)^{bk} = \overline{x}_{1}^{(a+b)k}

where x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}.

v = P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d+r}(y_{1} + α)^{λ} \frac{F}{x_{1}^{r}}

where λ = d − \frac{b(c+d+r)}{a+b}.

\frac{F}{x_{1}^{r}} = ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} + x_{1}Ω  (41)

where a_{jk} = a_{r−i−j,j,k}. Suppose that

(y_{1} + α)^{λ} ( ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k}) ≡ γ mod (y_{1}, z_{1})^{r+2}

for some γ ∈ k. Then

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} ≡ γ(y_{1} + α)^{−λ} mod (y_{1}, z_{1})^{r+2}  (42)

Set

f(y_{1}) = (y_{1} + α)^{−λ} = ∑_{i=0}^{∞} α_{i}y_{1}^{i}

where α_{0} = α^{−λ} and

α_{i} = \frac{−λ(−λ − 1) ⋯ (−λ − i + 1)}{i!}α^{−λ−i}

for i ≥ 1. (42) implies α_{r+1} = 0, so that −λ ∈ {0, 1, …, r}, and

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} = γ(y_{1} + α)^{−λ}

Thus

∑_{i+j+k=r} a_{ijk}x_{1}^{r}(y_{1} + α)^{j}(z_{1} + β)^{k} = γx_{1}^{r+λ}x_{1}^{−λ}(y_{1} + α)^{−λ}

which implies that the leading form of F is

L = ∑_{i+j+k=r} a_{ijk}x^{i}y^{j}z^{k} = γx^{r+λ}y^{−λ}

x^{c}y^{d}L = γx^{r+c+λ}y^{d−λ}
a(d − λ) − b(r + c + λ) = a [\frac{b(c+d+r)}{a+b}] − b [r + c + d − \frac{b(c+d+r)}{a+b}] = 0

Thus x^{r+c+λ}y^{d−λ} is a power of x^{a}y^{b}, a contradiction to the assumption that F is normalized. We conclude that ν(q) ≤ ν(p) + 1.

If ν(q) = ν(p) + 1, we must then have that

F_{1} = (y_{1} + α)^{λ} ( ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k}) − γ + \overline{x}_{1}Σ

with γ = α^{λ} ∑_{j+k≤r} a_{jk}α^{j}β^{k}. There is a nonzero degree r + 1 term in F_{1}(0, y_{1}, z_{1}), so that γ(q) = r + 1.

Now suppose that q ∈ π^{−1}(p) is a 2 point. Then after possibly interchanging x and y, there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}y_{1} \\ z & = x_{1}(z_{1} + β) \end{aligned}

with β ≠ 0.

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{k} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r}y_{1}^{d} \frac{F}{x_{1}^{r}} \end{aligned}

\frac{F}{x_{1}^{r}} = ∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} + x_{1}Ω  (43)

with a_{jk} = a_{r−j−k,j,k}. Suppose that

∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} = ∑ γ_{i}y_{1}^{t_{i}}

with γ_{i} ∈ k, (c + d + r)b − (a + b)(d + t_{i}) = 0 for all i. There is at most one natural number t = t_{i} which is a solution to this equation, which simplifies to

a(d + t) − b(c + r − t) = 0.  (44)

We have

∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} = γy_{1}^{t}

so that

∑_{i+j+k=r} a_{ijk}x_{1}^{r}y_{1}^{j}(z_{1} + β)^{k} = γx_{1}^{r}y_{1}^{t}

Thus L = γx^{r−t}y^{t}. But by (44) x^{c+r−t}y^{d+t} is a power of x^{a}y^{b}, a contradiction to the assumption that F is normalized. Thus ν(q) ≤ r.

Now suppose that q ∈ π^{−1}(p) is a 3 point. Then there are regular parameters (x_{1}, y_{1}, z_{1}) at Q such that

\begin{aligned}x & = x_{1}z_{1} \\ y & = y_{1}z_{1} \\ z & = z_{1} \end{aligned}

We have

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b}z_{1}^{a+b})^{k} \\ v & = P (x_{1}^{a}y_{1}^{b}z_{1}^{a+b}) + x_{1}^{c}y_{1}^{d}z_{1}^{c+d+r} \frac{F}{z_{1}^{r}} \end{aligned}  (45)

F_{q} = \frac{F}{z_{1}^{r}}, so that ν(q) = ν(\frac{F}{z_{1}^{r}}) ≤ r.

Suppose that p is a 3 point

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F \end{aligned}

Write F = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}. Suppose that q ∈ π^{−1}(p) is a 1 point. Then there are regular parameters (x_{1}, y_{1}, z_{1}) in Ô_{X_{1},q} such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}(y_{1} + α) \\ z & = x_{1}(z_{1} + β) \end{aligned}

with α, β ≠ 0.

u = x_{1}^{(a+b+c)k}(y_{1} + α)^{bk}(z_{1} + β)^{ck} = \overline{x}_{1}^{(a+b+c)k}.

where \overline{x}_{1} is defined by

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b+c}} (z_{1} + β)^{− \frac{c}{a+b+c}} .

\begin{aligned}v & = P (\overline{x}_{1}^{a+b+c}) + x_{1}^{d+e+f+r}(y_{1} + α)^{e}(z_{1} + β)^{f} \frac{F}{x_{1}^{r}} \\ & = P (\overline{x}_{1}^{a+b+c}) + \overline{x}_{1}^{d+e+f+r}(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} \frac{F}{x_{1}^{r}} \end{aligned}

where

\begin{aligned}λ_{1} & = e − \frac{b(d+e+f+r)}{a+b+c} \\ λ_{2} & = f − \frac{c(d+e+f+r)}{a+b+c} \end{aligned}

\frac{F}{x_{1}^{r}} = ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} + \overline{x}_{1}Ω  (46)

where a_{jk} = a_{r−j−k,j,k}. Suppose that

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} ( ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k}) ≡ γ mod (y_{1}, z_{1})^{r+2}

for some γ ∈ k. We first observe that we cannot have γ = 0, for γ = 0 implies

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} ≡ 0 mod (y_{1}, z_{1})^{r+2}

46 STEVEN DALE CUTKOSKY

which implies

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} = 0,

a contradiction. Thus γ ≠ 0.

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} ≡ γ(y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}} mod (y_{1}, z_{1})^{r+2} (47)

Set f(y_{1}, z_{1}) = (y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}},

α_{ij} = \frac{1}{i! j!} \frac{∂^{i+j}f}{∂y_{1}^{i}∂z_{1}^{j}} (0, 0)

Then

(y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}} = ∑ α_{ij}y_{1}^{i}z_{1}^{j}.

α_{ij} = \{ \begin{matrix}(\frac{−λ_{1}(−λ_{1}−1)···(−λ_{1}−i+1)}{i!}α^{−λ_{1}−i}) (\frac{−λ_{2}(−λ_{2}−1)···(−λ_{2}−j+1)}{j!}β^{−λ_{2}−j}) & if i, j > 0 \\ α^{−λ_{1}} (\frac{−λ_{2}(−λ_{2}−1)···(−λ_{2}−j+1)}{j!}β^{−λ_{2}−j}) & if i = 0, j > 0 \\ (\frac{−λ_{1}(−λ_{1}−1)···(−λ_{1}−i+1)}{i!}α^{−λ_{1}−i}) β^{−λ_{2}} & if j = 0, i > 0 \\ α^{−λ_{1}}β^{−λ_{2}} & if i = j = 0 \end{matrix}

Thus α_{ij} = 0 for i + j = r + 1 by (47), and −λ_{1} ∈ {0, 1, …, r}, −λ_{2} ∈ {0, 1, …, r} and −λ_{1} − λ_{2} ≤ r. Thus

∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k} = γ(y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}}

so that

\begin{aligned}∑_{i+j+k=r} a_{ijk}x_{1}^{r}(y_{1} + α)^{j}(z_{1} + β)^{k} & = γx_{1}^{r}(y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}} \\ & = γx_{1}^{r+λ_{1}+λ_{2}} [x_{1}^{−λ_{1}}(y_{1} + α)^{−λ_{1}}] [x_{1}^{−λ_{2}}(z_{1} + β)^{−λ_{2}}] \end{aligned}

and the leading form of F is

L = ∑_{i+j+k=r} a_{ijk}x^{i}y^{j}z^{k} = γx^{r+λ_{1}+λ_{2}}y^{−λ_{1}}z^{−λ_{2}}

x^{d}y^{e}z^{f}L = γx^{d+r+λ_{1}+λ_{2}}y^{e−λ_{1}}z^{f−λ_{2}}

Set

\begin{aligned}\underline{a} & = d + r + e − \frac{b(d+e+f+r)}{a+b+c} + f − \frac{c(d+e+f+r)}{a+b+c} \\ \underline{b} & = e − (e − \frac{b(d+e+f+r)}{a+b+c}) \\ \underline{c} & = f − (f − \frac{c(d+e+f+r)}{a+b+c}) \end{aligned}

Set τ = \frac{d+e+f+r}{a+b+c}.

\begin{aligned}\underline{a} & = \frac{(d+e+f+r)(a+b+c)−b(d+e+f+r)−c(d+e+f+r)}{a+b+c} = aτ \\ \underline{b} & = bτ \\ \underline{c} & = cτ \end{aligned}

\begin{aligned}b\underline{a} − a\underline{b} & = (ba − ab)τ = 0 \\ a\underline{c} − c\underline{a} & = (ac − ca)τ = 0 \\ c\underline{b} − b\underline{c} & = (cb − bc)τ = 0 \end{aligned}

thus γ = 0 since F is normalized. This contradiction shows that ν(q) ≤ r + 1.

We have shown that

F_{1} = (y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} ( ∑_{j+k≤r} a_{jk}(y_{1} + α)^{j}(z_{1} + β)^{k}) − γ + \overline{x}_{1}Σ

with

γ = α^{λ_{1}}β^{λ_{2}} ∑_{j+k≤r} a_{jk}α^{j}β^{k}.

Thus r_{1} = r + 1 implies there is a nonzero degree r + 1 term in F_{1}(0, y_{1}, z_{1}) so that γ(q) = r + 1.

Now suppose that q ∈ π^{−1}(p) is a 2 point. Then after possibly interchanging x, y, z, there are regular parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}y_{1} \\ z & = x_{1}(z_{1} + β) \end{aligned}

with β ≠ 0.

u = x_{1}^{(a+b+c)k}y_{1}^{bk}(z_{1} + β)^{ck}

Set

x_{1} = (z_{1} + β)^{\frac{−c}{a+b+c}} \overline{x}_{1}
u = (\overline{x}_{1}^{a+b+c}y_{1}^{b})^{k}
v = P (\overline{x}_{1}^{a+b+c}y_{1}^{b}) + x_{1}^{d+e+f+r}y_{1}^{e}(z_{1} + β)^{f} \frac{F}{x_{1}^{r}}
= P (\overline{x}_{1}^{a+b+c}y_{1}^{b}) + \overline{x}_{1}^{d+e+f+r}y_{1}^{e}(z_{1} + β)^{λ_{1}} \frac{F}{x_{1}^{r}}

where

λ_{1} = f − \frac{c(d + e + f + r)}{a + b + c}.  (48)

\frac{F}{x_{1}^{r}} = ∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} + \overline{x}_{1}Ω  (49)

where a_{jk} = a_{r−i−j,j,k}. There is at most one natural number t such that

(d + e + f + r)b − (e + t)(a + b + c) = 0.  (50)

If ν(q) > r + 1 there exists a t satisfying (50), and 0 ≠ γ ∈ k such that

(z_{1} + β)^{λ_{1}} ( ∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k}) ≡ γy_{1}^{t} mod (y_{1}, z_{1})^{r+2}.

Thus

∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} ≡ γ(z_{1} + β)^{−λ_{1}}y_{1}^{t} mod (y_{1}, z_{1})^{r+2}.

Set

τ_{j} = \frac{−λ_{1}(−λ_{1} − 1) ⋯ (−λ_{1} − j + 1)}{j!}β^{−λ_{1}−j}.

∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} ≡ γy_{1}^{t}(∑_{j=0}^{∞} τ_{j}z_{1}^{j}) mod (y_{1}, z_{1})^{r+2}.

implies

0 = τ_{r+1−t} = \frac{−λ_{1}(−λ_{1} − 1) ⋯ (−λ_{1} − r + t)}{(r + 1 − t)!}β^{−λ_{1}−(r+1−t)}

so that −λ_{1} ∈ {0, 1, …, r − t} and t ≤ r. Thus

∑_{j+k≤r} a_{jk}y_{1}^{j}(z_{1} + β)^{k} = γ(z_{1} + β)^{−λ_{1}}y_{1}^{t}

\begin{aligned}∑_{i+j+k=r} a_{ijk}x_{1}^{r}y_{1}^{j}(z_{1} + β)^{k} & = γx_{1}^{r}(z_{1} + β)^{−λ_{1}}y_{1}^{t} \\ & = γx_{1}^{r−t+λ_{1}} [x_{1}^{t}y_{1}^{t}] [x_{1}^{−λ_{1}}(z_{1} + β)^{−λ_{1}}] \end{aligned}

x^{d}y^{e}z^{f}L = γx^{r−t+λ_{1}+d}y^{t+e}z^{f−λ_{1}}

where L is the leading form of F. Set

\begin{aligned}\underline{a} & = r − t + λ_{1} + d \\ \underline{b} & = t + e \\ \underline{c} & = f − λ_{1} \end{aligned}

We have the relations (50) and (48). (50) implies

t = \frac{(d + e + f + r)b − e(a + b + c)}{a + b + c}.

\begin{aligned}\underline{a} & = d + r − t + λ_{1} = \frac{a(d+e+f+r)}{a+b+c} \\ \underline{b} & = t + e = \frac{(d+e+f+r)b}{a+b+c} \\ \underline{c} & = \frac{c(d+e+f+r)}{a+b+c} \end{aligned}

0 = b\overline{a} − a\overline{b} = c\overline{a} − a\overline{c} = c\overline{b} − b\overline{c}

Thus

x^{r−t+λ_{1}+d}y^{t+e}z^{f−λ_{1}} = (x^{a}y^{b}z^{c})^{m}

for some m ∈ N, a contradiction, since F is normalized. Thus ν(q) ≤ r + 1. Suppose that ν(q) = r + 1.

F_{q} = ∑_{j≤r} [ ∑_{k≤r−j} a_{jk}(z_{1} + α)^{k+λ_{1}} ] y_{1}^{j} − γy_{1}^{t} + \overline{x}_{1}Σ

with t ≤ r, γ ∈ k implies a_{jk} = 0 if j ≠ t. Thus

F_{q} = y_{1}^{t}( ∑_{k≤r−t} a_{tk}(z_{1} + α)^{k+λ_{1}} − γ) + \overline{x}_{1}Σ

implies

L_{q} = cy_{1}^{t}z_{1}^{r+1−t} + \overline{x}_{1}Ω

where 0 ≠ c ∈ k.

Now suppose that q ∈ π^{−1}(p) is a 3 point. After possibly permuting x, y, z, there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}y_{1} \\ z & = x_{1}z_{1} \end{aligned}

\begin{aligned}u & = (x_{1}^{a+b+c}y_{1}^{b}z_{1}^{c})^{k} \\ v & = P (x_{1}^{a+b+c}y_{1}^{b}z_{1}^{c}) + x_{1}^{d+e+f+r}y_{1}^{e}z_{1}^{f} \frac{F}{x_{1}^{r}} \end{aligned}

ν(q) = ν(\frac{F}{x_{1}^{r}}) ≤ r.

Example 7.2. ν(p) can go up by 1 after a quadratic transform. We can construct the example as follows.

u = xy, v = x^{2}y

has F = 1. blowup p and consider the point p_{1} above p with regular parameters (x_{1}, y_{1}, z_{1}) defined by x = x_{1}, y = x_{1}(y_{1} + α), α ≠ 0, z = x_{1}z_{1}. Set \overline{x}_{1} = x_{1}(y_{1} + α)^{\frac{1}{2}} , \overline{y}_{1} = (y_{1} + α)^{− \frac{1}{2}} − α^{− \frac{1}{2}} . Then

u = \overline{x}_{1}^{2}, v = α^{− \frac{1}{2}} \overline{x}_{1}^{3} + \overline{x}_{1}^{3}\overline{y}_{1},

so that F_{1} = \overline{y}_{1}.

Theorem 7.3. Suppose that ν(p) = r, π : X_{1} → X is the blowup of p, q ∈ π^{−1}(p) with r_{1} = ν(q).

If p is a 1 point then

1. If q is a 1 point, then r_{1} < r if τ(p) < r, and if r_{1} = r then τ(q) = r.

2. If q is a 1 point, then γ(q) ≤ r.

3. If q is a 2 point, and r_{1} = r then τ(p) ≤ τ(q).

4. If q is a 2 point and γ(p) = r, then γ(q) ≤ r.

If p is a 2 point and 1 ≤ τ(p) then

1. If q is a 1 point then r_{1} ≤ r and γ(q) ≤ r.

2. If q is a 2 point and r_{1} = r, then τ(p) ≤ τ(q).

3. If q is a 2 point and γ(p) = r, then γ(q) ≤ r.

4. If q is a 3 point then r_{1} ≤ r − τ(p)

Proof. Suppose that p is a 1 point with γ(p) = r. Suppose that q ∈ π^{−1}(p) is a 1 point, and r_{1} = r. After making a permissible change of parameters we can assume that x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}. We than have, with the notation of (39).

F_{1} = ∑_{j+k≤r} a_{r−j−k,j,k}y_{1}^{j}z_{1}^{k} + x_{1}Ω.

Now suppose that q ∈ π^{−1}(p) is a 2 point, and r_{1} = r. After making a permissible change of parameters we can assume that x = x_{1}y_{1}, y = y_{1}, z = y_{1}z_{1}. We than have, with the notation of (40)

F_{1} = ∑_{i+k≤r} a_{i,r−i−k,k}x_{1}^{i}z_{1}^{k} + y_{1}Σ

Suppose that p is a 2 point and 1 ≤ τ(p). Suppose that q ∈ π^{−1}(p) is a 1 point. After making a permissible change of parameters, we have x = x_{1}, y = x_{1}(y_{1} +α), z = x_{1}z_{1} with α ≠ 0. We then have, with the notation of (41),

F_{1} = ∑_{j+k≤r} a_{r−j−k,j,k}(y_{1} + α)^{j}(y_{1} + α)^{λ}z_{1}^{k} − γ + \overline{x}_{1}Ω.

There exists a_{ijk} with i + j + k = r and k = τ(p) ≥ 1 such that a_{ijk} ≠ 0. Thus r_{1} ≤ r and γ(q) ≤ r.

Now suppose that q ∈ π^{−1}(p) is a 2 point. After making a permissible change of parameters, we have x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1} We then have, with the notation of (43),

F_{1} = ∑_{j+k≤r} a_{r−j−k,j,k}y_{1}^{j}z_{1}^{k} + x_{1}Ω.

and there exist i, j, k such that i + j + k = r and a_{ijk} ≠ 0 with k = τ(p). Thus if r_{1} = r, we have τ(p) ≤ τ(q). If γ(p) = r, we have γ(q) ≤ r.

Now suppose that q ∈ π^{−1}(p) is a 3 point. Then x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1} We then have, with the notation of (45),

F_{1} = ∑_{i+j≤r} a_{i,j,r−i−j}x_{1}^{i}y_{1}^{j} + z_{1}Ω.

There exists a_{ijk} with i + j + k = r and k = τ(p) ≥ 1 such that a_{ijk} ≠ 0. Thus r_{1} ≤ r − τ(p).

Lemma 7.4. Suppose that r ≥ 2 and p ∈ X is a 1 point. Suppose that (x, y, z) are permissible parameters at p and C ⊂ \overline{S}_{r}(X) is a curve such that p ∈ C. Then F_{p} ∈ Î_{C,p}^{r} + (x^{r−1}).

Proof. x ∈ Î_{C,p} by Lemma 6.12. There exist permissible parameters (x, \overline{y}, \overline{z}) at p such that \overline{y}, \overline{z} ∈ O_{X,p}, \overline{y} = y + h_{1}, \overline{z} = z + h_{2} with h_{1}, h_{2} ∈ m^{r}.

Suppose that the conclusions of the Lemma are true for the parameters (x, \overline{y}, \overline{z}).

\begin{aligned}u & = x^{a} \\ v & = \overline{P} (x) + x^{b}\overline{F} (x, \overline{y}, \overline{z}) \end{aligned}

and \overline{F} (x, \overline{y}, \overline{z}) is normalized with respect to the permissible parameters (x, \overline{y}, \overline{z}). We have an expression

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}F (x, y, z) \end{aligned}

where F (x, y, z) is normalized with respect to the permissible parameters (x, y, z).

\overline{F} (x, \overline{y}, \overline{z}) = F (x, y, z) + Ω

with Ω a series in x. Since F (x, y, z) is normalized, ν(\overline{F}) = ν(F) = r and only powers of x of order ≥ r can be removed from \overline{F} (x, \overline{y}, \overline{z}) to normalize to obtain F (x, y, z). Thus the conclusions of the Lemma hold for (x, y, z)

We may thus assume that y, z ∈ O_{X,p}.

There exists an étale neighborhood U of p such that (x, y, z) are uniformizing parameters in U, x = 0 is a local equation of E_{X} ∩U, C ∩U is a complete intersection. Let R = Γ(U, O_{U}), I_{C} = Γ(U, I_{C}). Set

w = \frac{v − P_{t}(x)}{x^{b}}

where t > b + r. Thus

w ∈ (y, z, x^{r})O_{U}, p  (51)

and

w − F ∈ (x^{r})Ô_{X,p}.  (52)

Let q be a smooth point of C ∩U. Then there exists α, β ∈ k such that (x, y−α, z −β) are permissible parameters at q. Lemma 6.25 implies

F_{q} ∈ Î_{C,q}^{r} + (x^{r−1}).

F_{q} = w − ∑_{i=0}^{∞} \frac{∂^{i}w}{∂x^{i}} (0, α, β)x^{i}

Set

Λ = w − ∑_{i<r−1} \frac{∂^{i}w}{∂x^{i}} (0, α, β)x^{i}.

Λ ∈ (I_{C,q}^{r} + (x^{r−1}))Ô_{X,q} implies by Theorem 7.1, Chapter VIII, section 4 [31] and Lemma 6.29,

Λ ∈ (I_{C,q}^{r} + (x^{r−1})) ∩ R = (I_{C}^{r} + (x^{r−1})R.

ν(p) = r and t > b + r implies ν(w) ≥ r and \frac{∂^{i}w}{∂x^{i}} (0, α, β) = 0 if i < r − 1, so that w ∈ Î_{C,p}^{r} + (x^{r−1}) which implies that F_{p} ∈ Î_{C,p}^{r} + (x^{r−1}).

Lemma 7.5. Suppose that p ∈ X is a 1 point and ν(p) = γ(p) = r ≥ 2. Then there exists at most one curve C in \overline{S}_{r}(X) containing p. If C exists then it is nonsingular at p.

Proof. Suppose that (x, y, z) are permissible parameters at p. Write Î_{C,p} = (x, f(y, z)). By Lemma 7.4, F_{p} ∈ Î_{C,p}^{r} + (x^{r−1}). f^{r} | F_{p}(0, y, z) and γ(p) = r implies ν(f) = 1 and C is nonsingular at p.

Suppose that D ⊂ \overline{S}_{r}(X) is another curve containing p. Then D is nonsingular at p. Lemma 6.25 implies there exist permissible parameters (x, y, z) at p such that Î_{C,p} = (x, z), there exist series a, b_{ij} such that

F_{p} = x^{r−1}a + ∑_{i+j=r} b_{ij}x^{i}z^{j}.

b_{0r} is a unit implies

\frac{∂^{r−1}F_{p}}{∂z^{r−1}} = xϕ + zψ

where ψ is a unit. Since F_{p} ∈ Î_{D,p}^{r} + (x^{r−1}), we have

\frac{∂^{r−1}F_{p}}{∂z^{r−1}} ∈ Î_{D,p}

which implies z ∈ Î_{D,p}, so that C = D.

Lemma 7.6. Suppose that r ≥ 2, p is a 2 point and C ⊂ \overline{S}_{r}(X) is an irreducible curve containing a 1 point, such that p ∈ C. Then ν(p) ≥ r − 1. If τ(p) > 0, then ν(p) ≥ r.

Proof. First suppose that C is nonsingular at p and is transversal at p to the 2 curve through p. Then the result follows from Lemma 6.27. Now suppose that C does not make SNCs with the 2 curve through p. Let s = ν(p). There exists a sequence of quadratic transforms π : X_{1} → X centered at 2 and 3 points such that the strict transform C′ of C makes SNCs with \overline{B}_{2}(X_{1}) at a 2 point p_{1} = C′ ∩ π^{−1}(p). We have s_{1} = ν(p_{1}) ≤ s + 1, by Theorems 7.1 and 7.3. s_{1} = s + 1 implies τ(p_{1}) > 0 and τ(p) = 0. τ(p) > 0 implies s_{1} ≤ s and if we further have s_{1} = s, then τ(p_{1}) > 0.

First supppose that τ(p) > 0. If s_{1} = s then τ(p_{1}) > 0 so that r ≤ s_{1} = s. If s_{1} < s then s > s_{1} ≥ r − 1, which implies s ≥ r.

Now suppose that τ(p) = 0. If s_{1} ≤ s then s ≥ s_{1} ≥ r − 1. If s_{1} = s + 1 then s + 1 = s_{1} ≥ r since τ(p_{1}) > 0, so that s ≥ r − 1.

Lemma 7.7. Suppose that r ≥ 2, p is a 3 point and C ⊂ \overline{S}_{r}(X) is an irreducible curve containing a 1 point such that p ∈ C. Then ν(p) ≥ r − 1.

Proof. Let s = ν(p). There exists a sequence of quadratic transforms π : X_{1} → X centered at 2 and 3 points such that the strict transform C′ of C makes SNCs with \overline{B}_{2}(X_{1}) at the 2 point p_{1} = C′ ∩ π^{−1}(p). We have s_{1} = ν(p_{1}) ≤ s + 1 by Theorems 7.1 and 7.3.

If s_{1} = s + 1 then τ(p_{1}) > 0, so that s_{1} ≥ r by Lemma 7.6, so that s ≥ r − 1. If s_{1} ≤ s, then s ≥ s_{1} ≥ r − 1 by Lemma 7.6.

Theorem 7.8. Suppose that p ∈ X has ν(p) = r ≥ 1, and (x, y, z) are permissible parameters at p, π : X_{1} → X is the blow up of p.

Suppose that p is a 3 point

1. Suppose that the leading form L_{p} = L(x, y, z) depends on x, y and z. Then there are no curves C in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}). No 2 curves C of π^{−1}(p) satisfy F_{q} ∈ Î_{C,q}^{r} for q ∈ C.

2. Suppose that L_{p} = L(x, y) depends on x and y. Then the curves in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}) are a finite union of lines passing through a single 3 point of π^{−1}(p). No 2 curves C of π^{−1}(p) satisfy F_{q} ∈ Î_{C,q}^{r} for q ∈ C.

3. Suppose that L_{p} = L(x) depends on x. Then there are no 1 points in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}) and there is at most one curve C in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}). It is the 2 curve D which is the intersection of the strict transform of x = 0 with π^{−1}(p). D is the only 2 curve C in π^{−1}(p) such that F_{q} ∈ Î_{C,q}^{r} for q ∈ C.

Suppose that p is a 2 point. Then the curves in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}) are a finite union of lines passing through the 3 point. There are no 2 curves in π^{−1}(p)∩\overline{S}_{r+1}(X_{1}).

Proof. First suppose that p is a 3 point and L_{p} = L(x, y, z) depends on x, y and z. There are no 3 points in π^{−1}(p) ∩ \overline{S_{r+1}(X_{1})} by Lemma 7.7 and Lemma 6.28 since (by direct calculation) ν(q) ≤ r − 1 at all 3 points in π^{−1}(p). There are no 2 curves in π^{−1}(p) ∩ \overline{S_{r+1}(X_{1})} and there are no 2 curves in π^{−1}(p) such that F_{q} ∈ Î_{C,q}^{r} for q ∈ C by Lemma 6.28. We will now show that there are no curves in \overline{S}_{r+1}(X_{1}) ∩ π^{−1}(p). Suppose that there is a curve C in \overline{S_{r+1}(X_{1})} ∩ π^{−1}(p) containing a 1 point. C must contain a 2 point q. ν(q) = r or r + 1 by Lemma 7.6 and Theorem 7.1.

First suppose that ν(q) = r + 1. Then by Theorem 7.1, there exist permissible parameters (x, y, z) at p such that

L = y^{t}f(x, z)  (53)

for some t with 0 < t < r, (since L depends on x, y, and z). Write

f = ∑_{i+k=r−t} b_{ik}x^{i}z^{k}

At a 1 point of C we have (with the notation of (46)) permissible parameters (x_{1}, y_{1}, z_{1}) such that x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}(z_{1} + β) with α, β ≠ 0

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} [ ∑_{i+j+k=r} a_{ijk}(y_{1} + α)^{j}(z_{1} + β)^{k}] ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1}(54)

for some 0 ≠ c_{α,β} ∈ k. Substituting (53), we have

\begin{aligned}(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} [(y_{1} + α)^{t} ∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k}] & ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1} \\ (y_{1} + α)^{t+λ_{1}} [∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k}(z_{1} + β)^{λ_{2}}] & ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1} \end{aligned}

If t ≠ −λ_{1} this is a contradiction, since there is then a nonzero y_{1}z_{1}^{s} term for some 0 ≤ s ≤ r − t. Thus −λ_{1} = t ∈ {1, …, r − 1}. But

ν( ∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k}(z_{1} + β)^{λ_{2}} − c_{α,β}) ≤ r − t + 1 ≤ r

which is contradiction.

Now suppose that ν(q) = r. We have from (49) (in the proof of Theorem 7.1) that there exist permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

x = x_{1}, y = x_{1}y_{1}, z = x_{1}(z_{1} + β),

F_{q} = [ ∑_{j+k≤r} a_{r−j−k,j,k}y_{1}^{j}(z_{1} + β)^{k}] (z_{1} + β)^{λ_{1}} − γy_{1}^{t} + \overline{x}_{1}Ω  (55)

if there exists a natural number t such that (d + e + f + r)b − (e + t)(a + b + c) = 0, and

F_{q} = [ ∑_{j+k≤r} a_{r−j−k,j,k}y_{1}^{j}(z_{1} + β)^{k}] (z_{1} + β)^{λ_{1}} + \overline{x}_{1}Ω  (56)

otherwise. Since ν(q) = r, For fixed j ≠ t, we have

ν( ∑_{k≤r−j} a_{r−j−k,j,k}(z_{1} + β)^{k}) ≥ r − j

Thus (for fixed j ≠ t)

∑_{k≤r−j} a_{r−j−k,j,k}(z_{1} + β)^{k} = γ_{j}z_{1}^{r−j}

for some γ_{j} ∈ k and (for fixed j ≠ t)

∑_{i+k=r−j} a_{ijk}x^{i}y^{j}z^{k} = y^{j} [∑_{k≤r−j} a_{r−j−k,j,k}[x_{1}^{k}(z_{1} + β)^{k}]x_{1}^{r−j−k}]
= y^{j}x_{1}^{r−j} [∑_{k≤r−j} a_{r−j−k,j,k}(z_{1} + β)^{k}]
= γ_{j}x_{1}^{r−j}z_{1}^{r−j}y^{j}
= γ_{j}x^{r−j} (\frac{z}{x} − β)^{r−j} y^{j}
= γ_{j}(z − βx)^{r−j}y^{j}

For j = t, we have

ν( ∑_{k≤r−t} a_{r−t−k,t,k}(z_{1} + β)^{k+λ_{1}} − γ) ≥ r − t

Thus we either have

L_{p} = ∑_{j≠t} γ_{j}y^{j}(z − βx)^{r−j} + y^{t}f(x, z)

where (d + e + f + r)b − (e + t)(a + b + c) = 0, and f is homogeneous of degree r − t, or

L_{p} = ∑_{j} γ_{j}y^{j}(z − βx)^{r−j}.

Thus

L_{q} = ∑ γ_{j}y_{1}^{j}z_{1}^{r−j} + x_{1}Ω_{1}.

or

L_{q} = ∑_{j≠t} γ_{j}y_{1}^{j}z_{1}^{r−j} + γ_{t}y_{1}^{t}z_{1}^{r−t} + x_{1}Ω_{1}.

for some γ_{t} ∈ k. If some γ_{j} ≠ 0 with j ≠ r, τ(q) > 0, so that ν(q) ≥ r + 1 by Lemma 7.6, a contradiction.

The remaining case is

L_{p} = γ_{r}y^{r} + y^{t}f(x, z),  (57)

with f ≠ 0, t < r and γ_{r} ≠ 0 if t = 0, since L_{p} depends on x, y and z.

f = ∑_{i+k=r−t} b_{ik}x^{i}z^{k}

At a 1 point of C we have (with the notation of (46)) regular parameters (x_{1}, y_{1}, z_{1}) such that x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}(z_{1} + β) with α, β ≠ 0

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} [ ∑_{i+j+k=r} a_{ijk}(y_{1} + α)^{j}(z_{1} + β)^{k}] ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1}(58)

for some c_{α,β} ∈ k. Substituting (57), we have

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} [γ_{r}(y_{1} + α)^{r} + (y_{1} + α)^{t} ∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k}] ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1}

γ_{r}(y_{1} + α)^{r−t} + ∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k} ≡ (y_{1} + α)^{−λ_{1}−t}(z_{1} + β)^{−λ_{2}}c_{α,β} mod (y_{1}, z_{1})^{r+1}

The LHS of the last equation has no y_{1}z_{1} term which implies λ_{1} = −t or −λ_{2} = 0. λ_{1} = −t implies γ_{r} = 0 and t > 0,

ν( ∑_{i+k=r−t} b_{ik}(z_{1} + β)^{k} − c_{α,β}(z_{1} + β)^{−λ_{2}}) ≤ r − t + 1 ≤ r

which is contradiction. λ_{2} = 0 implies f = 0, a contradiction.

Now suppose that p is a 3 point and L_{p} = L(x, y). Suppose that q ∈ π^{−1}(p) is a 1 point and ν(q) = r + 1. Ô_{X_{1},q} has regular parameters x_{1}, y_{1}, z_{1} such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}(y_{1} + α) \\ z & = x_{1}(z_{1} + β) \end{aligned}  (59)

where α, β ≠ 0. Write

L(x, y) = ∑_{i+j=r} a_{ij}x^{i}y^{j}.

(with the notation of (46))

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} [∑ a_{ij}(y_{1} + α)^{j}] ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1}  (60)

which implies

∑_{i+j=r} a_{ij}(y_{1} + α)^{j} ≡ (y_{1} + α)^{−λ_{1}}(z_{1} + β)^{−λ_{2}}c_{α,β} mod (y_{1}, z_{1})^{r+1}  (61)

so that λ_{2} = 0, and

∑_{i+j=r} a_{ij}(y_{1} + α)^{j} ≡ (y_{1} + α)^{−λ_{1}}c_{α} mod (y_{1})^{r+1}  (62)

We will now show that there exist at most finitely many values of α such that an equation (62) holds. Set

g(t) = ∑_{i≤r} a_{i}t^{i}

where a_{i} = a_{r−i,i}.

Suppose there are infinitely many values of α such that (62) holds for some q ∈ π^{−1}(p) with value β and regular parameters x_{1}, y_{1}, z_{1} in Ô_{X_{1},q} as in (59). Define g_{α} by

g_{α}(y_{1}) = g(y_{1} + α) = g(\frac{y}{x}).

Set λ = λ_{1}. By assumption,

g_{α}(y_{1}) = ∑_{i≤r} a_{i}(y_{1} + α)^{i} ≡ c_{α}(y_{1} + α)^{−λ} mod y_{1}^{r+1}  (63)

We can expand the RHS of (63) as

c_{α}(y_{1} + α)^{−λ} = c_{α}α^{−λ} + c_{α}(−λ)α^{−λ−1}y_{1}
+c_{α} \frac{−λ(−λ−1)}{2}α^{−λ−2}y_{1}^{2} + ⋯
+c_{α} \frac{−λ(−λ−1)···(−λ−r+1)}{r!}α^{−λ−r}y_{1}^{r} + ⋯

We can expand the LHS of (63) as

\begin{aligned}g_{α}(y_{1}) & = g_{α}(0) + \frac{dg_{α}}{dy_{1}} (0)y_{1} + \frac{1}{2} \frac{d^{2}g_{α}}{dy_{1}^{2}} (0)y_{1}^{2} + ⋯ + \frac{1}{r!} \frac{d^{r}g_{α}}{dy_{1}^{r}} (0)y_{1}^{r} \\ & = g(α) + \frac{dg}{dt} (α)y_{1} + \frac{1}{2} \frac{d^{2}g}{dt^{2}} (α)y_{1}^{2} + ⋯ + \frac{1}{r!} \frac{d^{r}g}{dt^{r}} (α)y_{1}^{r} \end{aligned}

We get that

r!a_{r} = \frac{d^{r}g}{dt^{r}} (α) = c_{α}(−λ)(−λ − 1) ⋯ (−λ − r + 1)α^{−λ−r}

which implies that

c_{α} = \frac{r!a_{r}}{(−λ)(−λ − 1) ⋯ (−λ − r + 1)α^{−λ−r}}

\begin{aligned}g(α) & = c_{α}α^{−λ} = \frac{r!a_{r}α^{−λ}}{(−λ)(−λ−1)···(−λ−r+1)α^{−λ−r}} \\ & = \frac{r!a_{r}α^{r}}{(−λ)(−λ−1)···(−λ−r+1)} \end{aligned}

Since this holds for infinitely many α, and g(t) is a polynomial,

g(t) = \frac{r!a_{r}t^{r}}{−λ(−λ − 1) ⋯ (−λ − r + 1)}.

Thus

∑_{i≤r} a_{r−i,i}t^{i} = \frac{r!a_{r}t^{r}}{−λ(−λ − 1) ⋯ (−λ − r + 1)}

so that a_{r−i,i} = 0 if i < r. Thus L_{p} = a_{0r}y^{r}, a contradiction to the assumption that L depends on two variables.

Thus the only curves in \overline{S_{r+1}(X_{1})}∩π^{−1}(p) which contain a 1 point are on the strict transforms of y − αx = 0 for a finite number of nonzero α. These lines contain the 3 point of X which has permissible parameters (x_{1}, y_{1}, z_{1}) defined by x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1}.

Since L_{p} = L(x, y), there is at most one 3 point q in π^{−1}(p) with ν(q) = r. Thus there are no 2 curves in \overline{S_{r+1}(X_{1})} ∩ π^{−1}(p) by Lemma 6.28.

Now suppose that p is a 3 point and L_{p} = L(x). Suppose that q ∈ π^{−1}(p) is a 1 point and ν(q) = r + 1. Ô_{X_{1},q} has regular parameters x_{1}, y_{1}, z_{1} such that

\begin{aligned}x & = x_{1} \\ y & = x_{1}(y_{1} + α) \\ z & = x_{1}(z_{1} + β) \end{aligned}

where α, β ≠ 0.

L(x) = \overline{a}x^{r}

With the notation of (46), we have

(y_{1} + α)^{λ_{1}}(z_{1} + β)^{λ_{2}} \overline{a} ≡ c_{α,β} mod (y_{1}, z_{1})^{r+1}

for some c_{α,β} ∈ k, which implies λ_{1} = λ_{2} = 0.

From equation (46) we have

\begin{aligned}e & = \frac{b(d+e+f+r)}{a+b+c} \\ f & = \frac{c(d+e+f+r)}{a+b+c} \end{aligned}

where u = x^{a}y^{b}z^{c} and x^{d}y^{e}z^{f}L_{p} = \overline{a}x^{d+r}y^{e}z^{f}. Thus ec − fb = 0, ae − b(d + r) = 0 and af − c(d + r) = 0. It follows that F_{p} is not normalized, a contradiction.

The fact that there is at most one curve C in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}), which is the 2 curve which is the intersection of the strict transform of x = 0 with π^{−1}(p), follows from Lemma 6.28, since at the 3 point q with permissible parameters (x_{1}, y_{1}, z_{1}) defined by x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}, ν(q) = 0.

Suppose that p is a 2 point. By Theorem 7.1, there are no 2 curves in \overline{S}_{r+1}(X_{1}) ∩ π^{−1}(p). Suppose that \overline{S}_{r+1}(X_{1}) ∩ π^{−1}(p) contains a 1 point. Then τ(p) = 0 by Theorem 7.3. The leading form of F_{p} has an expression

L_{p} = ∑_{i+j=r} a_{ij}x^{i}y^{j}.

After possibly interchanging x and y, we may assume that L ≠ a_{0r}y^{r}.

Suppose that there exist infinitely many distinct values of α ∈ k such that there exists a 1 point q ∈ \overline{S}_{r+1}(X_{1}) ∩ π^{−1}(p) with regular parameters (x_{1}, y_{1}, z_{1}) in Ô_{X_{1},q} defined by

x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}(z_{1} + β)

for some α, β ∈ k with α ≠ 0, such that ν(q) = r + 1.

With the notation of (41) of Theorem 7.1, there exist c_{α} ∈ k such that

∑_{i+j=r} a_{ij}(y_{1} + α)^{j} = c_{α}(y_{1} + α)^{−λ} mod y_{1}^{r+1}

Set g(t) = ∑_{i+j=r} a_{ij}t^{j}. g(α) = c_{α}α^{−λ}.

r!a_{0r} = \frac{d^{r}g}{dt^{r}} (α) = c_{α}(−λ)(−λ − 1) ⋯ (−λ − r + 1)α^{−λ−r}

implies

g(α) = \frac{r!a_{0r}α^{r}}{(−λ)(−λ − 1) ⋯ (−λ − r + 1)}

for infinitely many α, so that

L_{p} = \frac{r!a_{0r}}{(−λ)(−λ − 1) ⋯ (−λ − r + 1)}y^{r}

a contradiction. Thus 1 curves in π^{−1}(p) ∩ \overline{S}_{r+1}(X_{1}) must be the intersection of the strict transform of y − αx = 0 and π^{−1}(p) for a finite number of 0 ≠ α ∈ k. These lines intersect in the 3 point of π^{−1}(p).

Lemma 7.9. Suppose that r ≥ 2 and p ∈ X is such that 1. ν(p) ≤ r if p is a 1 point or a 2 point.

2. If p is a 2 point and ν(p) = r, then τ(p) > 0.

3. ν(p) ≤ r − 1 if p is a 3 point and π : Y → X is the blowup of a point p ∈ X. Then C is a line for every curve C in \overline{S}_{r}(Y) ∩ π^{−1}(p) containing a 1 point. Thus C intersects a 2 curve in at most one point, and this intersection must be transversal.

If p is a 1 or 2 point with ν(p) = r then there is at most one curve C in \overline{S}_{r}(Y) ∩ π^{−1}(p) containing a 1 point.

Proof. Suppose that p is a 1 point. Suppose that q ∈ π^{−1}(p) is a 1 point with ν(q) = r. After a permissible change of parameters at p, we have permissible parameters x_{1}, y_{1}, z_{1} at q defined by

x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}.

Write

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

F_{p} has leading form

L_{p} = ∑_{i+j+k=r} a_{ijk}x^{i}y^{j}k^{k}.

Thus L_{p} = L(y, z) depends only on y and z.

Suppose that q′ ∈ π^{−1}(p) is another 1 point with ν(q′) = r, with permissible parameters (x_{1}, y_{1}, z_{1}) defined by

x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}(z_{1} + β)

for some α, β ∈ k. Then there exists a form L_{1} such that

L_{p} = L_{1}(y − αx, z − βx) + cx^{r}

for some c ∈ k. There exist α_{i}, β_{i}, γ_{i}, δ_{i} ∈ k such that

L_{p}(y, z) = ∏_{i=1}^{r}(α_{i}y − β_{i}z)

L_{1}(y, z) = ∏_{i=1}^{r}(γ_{i}y − δ_{i}z)

We can also assume that α_{i}β_{j} − α_{j}β_{i} = 0 implies α_{i} = α_{j} and β_{i} = β_{j}.

∏_{i=1}^{r}(α_{i}y − β_{i}z) = ∏_{i=1}^{r}(γ_{i}(y − αx) − δ_{i}(z − βx)) + cx^{r}.

Set x = 0 to get that, after reindexing the (γ_{i}, δ_{i}), there exist 0 ≠ ϵ_{i} ∈ k such that

(α_{i}, β_{i}) = ϵ_{i}(γ_{i}, δ_{i})

for all i, and ∏_{i=1}^{r} ϵ_{i} = 1. Thus

∏_{i=1}^{r}(α_{i}y − β_{i}z) = ∏_{i=1}^{r}(α_{i}(y − αx) − β_{i}(z − βx)) + cx^{r}.

First suppose that there exists (α_{i}, β_{i}), (α_{j}, β_{j}) such that α_{i}β_{j} − α_{j}β_{i} ≠ 0. Suppose that α_{i} ≠ 0. There exist t < r distinct values of (α_{k}, β_{k}) such that α_{i}β_{k} − α_{j}β_{k} = 0. Set y = \frac{β_{i}z}{α_{i}} to get

0 = (β_{i}β − α_{i}α)^{t}x^{t} ∏_{j|α_{i}β_{j}−α_{j}β_{i}≠0} ((\frac{α_{j}}{α_{i}} β_{i} − β_{j}) z + (ββ_{j} − αα_{j}) x) + cx^{r}

We conclude that ββ_{i} − αα_{i} = 0. If β_{i} ≠ 0, we can set z = \frac{α_{i}}{β_{i}} y to again conclude that ββ_{i} − αα_{i} = 0. Thus ββ_{i} − αα_{i} = 0 for all i, and the 1 points q′ ∈ π^{−1}(p) ∩ \overline{S}_{r}(X) must thus lie on the lines γ_{i} which are the intersection of the strict transform of β_{i}z − α_{i}y = 0 and π^{−1}(p).

Thus q′ must be in the intersection ∩γ_{i} ⊂ π^{−1}(p) ≅ P^{2}, and there is at most one point q ∈ π^{−1}(p) such that ν(q) = r.

Now suppose that α_{i}β_{j} − α_{j}β_{i} = 0 for all i, j. Then after a permissible change of parameters at p, we have L_{p} = z^{r}.

L_{p} = (z −βx)^{r} +cx^{r} and r ≥ 2 implies β = c = 0, so q′ is on the line γ ⊂ π^{−1}(q) ⊂ P^{2} which is the intersection of the strict transform of z = 0 and π^{−1}(q).

Suppose that p is a 2 point such that ν(p) = r and τ(p) > 0. Write

F_{p} = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

Suppose there exists a 1 point q ∈ π^{−1}(p) such that ν(q) = r. After a permissible change of parameters at p, q has permissible parameters (\overline{x}_{1}, y_{1}, z_{1}) at q such that, with the notation of (41) of Theorem 7.1,

x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}z_{1}
x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

F_{q} = ∑_{i+j+k=r} a_{ijk}(y_{1} + α)^{j+λ}z_{1}^{k} − ∑_{i+j=r} a_{ij0}α^{j+λ} + \overline{x}_{1}Ω.

Let

L_{p} = ∑_{i+j+k=r} a_{ijk}x^{i}y^{j}z^{k}

be the leading form of F_{p}.

F_{q} = ∑_{k>0}( ∑_{i+j=r−k} a_{ijk}(y_{1}+α)^{j})(y_{1}+α)^{λ}z_{1}^{k}+( ∑_{i+j=r} a_{ij0}(y_{1}+α)^{j+λ}− ∑_{i+j=r} a_{ij0}α^{j+λ})+\overline{x}_{1}Ω.

ν(q) = r implies, for fixed k > 0,

∑_{i+j=r−k} a_{ijk}x^{i}y^{j} = c_{k}(y − αx)^{r−k}

for some c_{k} ∈ k, thus

L_{p} = ∑_{k>0} c_{k}(y − αx)^{r−k}z^{k} + G(x, y).

τ(p) > 0 implies some c_{k} ≠ 0.

Suppose that there exists another 1 point q′ ∈ π^{−1}(p) with ν(q′) = r. Ô_{Y,q′} has regular parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, y = x_{1}(y_{1} + \overline{α}), z = x_{1}(z_{1} + \overline{β})

with \overline{α} ≠ 0. Then

L_{p} = ∑_{k>0} \overline{c}_{k}(y − \overline{α}x)^{r−k}(z − \overline{β}x)^{k} + \overline{G}(x, y).

Thus

∑_{k>0} c_{k}(y − αx)^{r−k}z^{k} = ∑_{k>0} \overline{c}_{k}(y − \overline{α}x)^{r−k}(z − \overline{β}x)^{k} + H(x, y).  (64)

Set x = 0 in (64) to get c_{k} = \overline{c}_{k} for all k. Let

k_{0} = max{k | c_{k} ≠ 0} = τ(p).

By assumption, k_{0} > 0.

c_{k_{0}}(y − αx)^{r−k_{0}}z^{k_{0}} = c_{k_{0}}(y − \overline{α}x)^{r−k_{0}}z^{k_{0}},

and if k_{0} > 1,

c_{k_{0}−1}(y−αx)^{r−k_{0}+1}z^{k_{0}−1} = c_{k_{0}}(y−\overline{α}x)^{r−k_{0}}(−\overline{β}k_{0}x)z^{k_{0}−1}+c_{k_{0}−1}(y−\overline{α}x)^{r−k_{0}+1}z^{k_{0}−1}.

If k_{0} < r, then α = \overline{α} implies all 1 points in \overline{S}_{r}(X_{1}) ∩ π^{−1}(p) are contained in the line which is the intersection of the strict transform of y − αx = 0 and π^{−1}(p). This line contains the 3 point of π^{−1}(p).

If k_{0} = r (≥ 2),

c_{r−1}(y − αx) = −c_{r}\overline{β}rx + c_{r−1}(y − \overline{α}x)

so that

−αc_{r−1} = −c_{r}\overline{β}r − \overline{α}c_{r−1}

which implies that all 1 points in \overline{S}_{r}(X_{1}) ∩ π^{−1}(p) are contained in the line which is the intersection of the strict transform of

c_{r}rz + c_{r−1}y − αc_{r−1}x = 0

and π^{−1}(p).

Suppose that p is a 2 point or a 3 point,with ν(p) = r − 1. Then by Theorem 7.8, the conclusions of the Theorem hold.

8. Permissible Monoidal Transforms Centered at Curves

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared.

Lemma 8.1. Suppose that C ⊂ X is a 2 curve. Then either F_{p} ∈ Î_{C,p} for all p ∈ C or F_{p} ∉ Î_{C,p} for all p ∈ C.

Suppose that r ≥ 2 , C ⊂ \overline{S}_{r}(X) is a 2 curve. Then either F_{p} ∈ Î_{C,p}^{r} for all p ∈ C or F_{p} ∈ Î_{C,p}^{r−1}, F_{p} ∉ Î_{C,p}^{r} for all p ∈ C

Proof. This follows from Lemmas 6.24, 6.26, 6.28.

Lemma 8.2. Suppose that r ≥ 2 and C ⊂ \overline{S}_{r}(X) is a nonsingular curve such that C contains a 1 point and C makes SNCs with \overline{B}_{2}(X). Then either F_{p} ∈ Î_{C,p}^{r} with respect to permissible parameters for C at p for all p ∈ C, or F_{p} ∈ Î_{C,p}^{r−1}, F_{p} ∉ Î_{C,p}^{r} with respect to permissible parameters for C at p for all p ∈ C.

Proof. This follows from Lemmas 6.24, 6.25, 6.27.

Definition 8.3. Suppose that r ≥ 2, p ∈ X, C ⊂ \overline{S}_{r}(X) is a curve which contains p and makes SNCs with \overline{B}_{2}(X) at p and C ⊄ \overline{S}_{r+1}(X). C is r big at p if F_{p} ∈ Î_{C,p}^{r} with respect to permissible parameters for C at p. C is r small at p if C is not r big at p.

Suppose that C is a 2 curve, ν(q) ≥ 1 if q ∈ C is a 2 point, C ⊄ \overline{S}_{2}(X) and p ∈ C. Then C is 1 big at p if F_{p} ∈ Î_{C,p}. F_{p} is 1 small at p if C is not 1 big at p.

Suppose that r ≥ 2, C ⊂ \overline{S}_{r}(X) is a curve which makes SNCs with \overline{B}_{2}(X). We will say that C is r big if C is r big at p for all p ∈ C. We will say that C is r small if C is r small at p for all p ∈ C.

Suppose that C is a 2 curve, ν(q) ≥ 1 if q ∈ C is a 2 point, C ⊄ \overline{S}_{2}(X). We will say that C is 1 big if C is 1 big for all p ∈ C. We will say that C is 1 small if C is 1 small at p for all p ∈ C.

Lemma 8.4. Suppose that C is a 2 curve on X, p ∈ C is a 2 point, D_{1} and D_{2} are curves in E_{X} containing p such that D_{1} ∪ D_{2} makes SNCs with C at p. Then there are regular parameters (x, y, z) in O_{X,p} such that

I_{C,p} = (x, y), I_{D_{1},p} = (x, z), I_{D_{2},p} = (y, z)

Proof. There exist regular parameters (x̃, ỹ, z̃) in O_{X,p}, and ϕ ∈ O_{X,p} such that

I_{C,p} = (x̃, ỹ), I_{D_{1},p} = (x̃, z̃), I_{D_{2},p} = (ỹ, ϕ)

and ϕ ≡ ax̃ + cz̃ mod m_{p}^{2}, with a, c ∈ k, c ≠ 0. In Ô_{X,p}, there exist series h, g such that

ϕ = h(x̃, ỹ, z̃)ỹ + g(x̃, z̃)

g = u(z̃ − ψ(x̃))

where u is a unit, ψ is a series.

z̃ − ψ(x̃) ∈ Î_{D_{1},p} ∩ Î_{D_{2},p} = (I_{D_{1},p} ∩ I_{D_{2},p})Ô_{X,p},

where the last equality is by Corollary 2 to Theorem 11 of Chapter VIII, section 4 [31]). Suppose that

I_{D_{1},p} ∩ I_{D_{2},p} = (f_{1}, …, f_{n}).
z̃ − ψ(x̃) = ∑ λ_{i}f_{i}

implies there exists f ∈ I_{D_{1},p} ∩ I_{D_{2},p} such that

f ≡ \overline{a}x̃ + \overline{c}z̃ mod m_{p}^{2}

where \overline{a}, \overline{c} ∈ k, \overline{c} ≠ 0. Since f ∈ (x̃, z̃), we have f = λx̃ + τz̃ where τ is a unit. Thus (x̃, z̃) = (x̃, f). Since f ∈ (ỹ, ϕ), we have f = αỹ + βϕ where β is a unit. Thus (ỹ, ϕ) = (ỹ, f). (x̃, ỹ, f) are the desired regular parameters.

Lemma 8.5. Suppose that p ∈ X is a 1 point or a 2 point with γ(p) = r ≥ 2, and (u, v) are permissible parameters at Φ_{X}(p), such that u = 0 is a local equation of E_{X} at p. Then there exist regular parameters (x̃, y, z̃) in R = O_{X,p} and permissible parameters (x, y, z) at p with x = γx̃, z = σz̃ for some series γ, σ ∈ Ô_{X,p} such that if p is a 1 point,

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F \end{aligned}  (65)

with F = τz^{r} + ∑_{i=2}^{r} a_{i}(x, y)z^{r−i}, τ a unit and some a_{i} ≠ 0.

Further suppose that \overline{S}_{r}(X) ∪ \overline{B}_{2}(X) makes SNCs at p. Then there is at most one curve D in \overline{S}_{r}(X) through p. If D exists, we can choose (x, y, z) so that x = 0, z = 0 are local equations of D at p.

If p is a 2 point,

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}  (66)

with F = τz^{r} + ∑_{i=2}^{r} a_{i}(x, y)z^{r−i}, τ a unit, and some a_{i} ≠ 0.

Further suppose that \overline{S}_{r}(X) ∪ \overline{B}_{2}(X) makes SNCs at p. Then there are at most 2 curves D_{1} and D_{2} in \overline{S}_{r}(X) through p. If D_{1} exists (or if D_{1} and D_{2} exist) then we can choose (x, y, z) so that x = 0, z = 0 are local equations of D_{1} at p (x = 0, z = 0 are local equations of D_{1} at p and y = 0, z = 0 are local equations of D_{2} at p).

Proof. There exist regular parameters x̃, y, z̃ in R, and permissible parameters (x = γx̃, y, z̃) at p such that u = x^{a} or u = (x^{a}y^{b})^{m} in R̂, and ν(F (0, 0, z̃)) = r.

If p is a 1 point, then there exists at most one curve D in \overline{S}_{r}(X) containing p by Lemma 7.5. If D exists, we may assume that x = 0, z̃ = 0 are local equations of D at p. If p is a 2 point, then there exist at most 2 curves D_{1} and D_{2} in \overline{S}_{r}(X). If D_{1} (or D_{1} and D_{2} exist) we may assume that x = 0, z̃ = 0 are local equations of D_{1} at p (or x = 0, z̃ = 0 are local equations of D_{1} at p and y = 0, z̃ = 0 are local equations of D_{2} at p by Lemma 8.4).

Set

\overline{z} = \frac{∂^{r−1}F}{∂z̃^{r−1}} = ω(z̃ − ϕ(x, y))

where ω is a unit by the formal implicit function theorem. Set z_{1} = z̃ − ϕ(x, y), G(x, y, z_{1}) = F (x, y, z̃).

Suppose that p is a 1 point and there exists a curve D ⊂ \overline{S}_{r}(X) containing p, so that D has local equations x = 0, z̃ = 0. Then F_{p} ∈ Î_{D,p}^{r} + (x^{r−1}) by Lemma 6.25, so that

\frac{∂^{r−1}F}{∂z̃^{r−1}} ∈ Î_{D,p}

and x | ϕ(x, y). Thus x = 0, z_{1} = 0 are local equations of D at p.

Suppose that p is a 2 point and there exist curves D_{1}, D_{2} ⊂ \overline{S}_{r}(X) containing p, so that D_{1} has local equations x = 0, z̃ = 0 and D_{2} has local equations y = 0, z̃ = 0. F_{p} ∈ Î_{D_{1},p}^{r} + (x^{r−1}) and F_{p} ∈ Î_{D_{2},p}^{r} + (y^{r−1}) by Lemma 6.27. Thus

\frac{∂^{r−1}F}{∂z̃^{r−1}} ∈ Î_{D_{1},p}

and \frac{∂^{r−1}F}{∂z̃^{r−1}} ∈ Î_{D_{2},p}, so that xy | ϕ(x, y), and x = 0, z_{1} = 0 are local equations of D_{1} at p and y = 0, z_{1} = 0 are local equations of D_{2} at p.

G = G(x, y, 0)+\frac{∂G}{∂z_{1}} (x, y, 0)z_{1}+⋯+\frac{1}{(r − 1)!} \frac{∂^{r−1}G}{∂z_{1}^{r−1}}(x, y, 0)z_{1}^{r−1}+\frac{1}{r!} \frac{∂^{r}G}{∂z_{1}^{r}}(x, y, 0)z_{1}^{r}+⋯

\frac{∂^{r−1}G}{∂z_{1}^{r−1}}(x, y, 0) = \frac{∂^{r−1}F}{∂z̃^{r−1}} (x, y, ϕ(x, y)) = 0

\frac{∂^{r}G}{∂z_{1}^{r}}(x, y, 0) = \frac{∂^{r}F}{∂z̃^{r}} (x, y, ϕ(x, y))

is a unit. Thus with the regular parameters (x̃, y, \overline{z}) in R and permissible parameters (x, y, z_{1}) at p, F has the desired form.

We cannot have a_{i} = 0 for all i, since r ≥ 2 and x or xy ∈ \sqrt{Î_{sing(Φ_{X}),p}}.

Lemma 8.6. Suppose that r ≥ 2, C ⊂ X is a 2 curve such that C is r − 1 big or r small, π : X_{1} → X is the blowup of C.

1. (a) If q ∈ C is a 2 point with ν(q) = r − 1 and q_{1} ∈ π^{−1}(q), then (i) If q_{1} is a 1 point then ν(q_{1}) ≤ r and γ(q_{1}) ≤ r. (ii) If q_{1} is a 2 point then ν(q_{1}) ≤ r − 1.

(b) If q ∈ C is a 2 point with ν(q) = r, τ(q) > 0 and q_{1} ∈ π^{−1}(q), then (i) If q_{1} is a 1 point then ν(q_{1}) ≤ r. ν(q_{1}) = r implies γ(q_{1}) = r. (ii) If q_{1} is a 2 point then ν(q_{1}) ≤ r. ν(q_{1}) = r implies τ(q_{1}) > 0.

(c) If q ∈ C is a 3 point with ν(q) = r − 1 and q_{1} ∈ π^{−1}(q) then (i) q_{1} a 2 point implies ν(q_{1}) ≤ r and γ(q_{1}) ≤ r. (ii) q_{1} a 3 point implies ν(q_{1}) ≤ r − 1.

2. Suppose that C ⊂ \overline{S}_{r}(X) ( so that C is r small). If q ∈ C is a 2 point with ν(q) = r, τ(q) > 0 and q_{1} ∈ π^{−1}(q), then

(a) If q_{1} is a 1 point then q_{1} is resolved.

(b) If q_{1} is a 2 point then ν(q_{1}) ≤ r. ν(q_{1}) = r implies τ(q) > 0.

Proof. Suppose that q ∈ C is a 2 point with ν(q) = r − 1, and q has permissible parameters (x, y, z) with

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \\ L_{q} & = ∑_{i+j=r−1} a_{ij}x^{i}y^{j} \end{aligned}

Suppose that q_{1} ∈ π^{−1}(q) and Ô_{Y_{1},q_{1}} has regular parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. Set x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

\begin{aligned}u & = \overline{x}_{1}^{(a+b)m} \\ v & = P_{q_{1}}(\overline{x}_{1}) + \overline{x}_{1}^{c+d+r−1}F_{q_{1}} \\ F_{q_{1}} & = ∑_{i+j=r−1} a_{ij}(y_{1} + α)^{λ+j} − ∑_{i+j=r−1} a_{ij}α^{λ+j} + \overline{x}_{1}Ω + zG, \end{aligned}  (67)

λ = d − \frac{b(c + d + r − 1)}{a + b}.

Thus ν(q_{1}) ≤ r and γ(q_{1}) ≤ r.

Suppose that q_{1} ∈ π^{−1}(q) and q_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}y_{1}.

Then

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r−1}F_{q_{1}} \\ F_{q_{1}} & = ∑_{i+j=r−1} a_{ij}y_{1}^{j} + x_{1}Ω + zG \end{aligned}

implies that ν(q_{1}) ≤ r − 1.

A similar argument holds at the point q_{1} ∈ π^{−1}(q) with permissible parameters (x_{1}, y_{1}, z) such that x = x_{1}y_{1}, y = y_{1}.

Suppose that q ∈ C is a 2 point with ν(q) = r and τ(q) > 0. Then q has permissible parameters (x, y, z) with

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \\ L_{q} & = z(∑_{i+j=r−1} a_{ij1}x^{i}y^{j}) + ∑_{i+j=r} a_{ij0}x^{i}y^{j} \end{aligned}

with some a_{ij1} ≠ 0.

Suppose that q_{1} ∈ π^{−1}(q) and Ô_{Y_{1},q_{1}} has regular parameters (x_{1}, y_{1}, z) such that x = x_{1}, y = x_{1}(y_{1} + α) with α ≠ 0. Set

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}} .

\begin{aligned}u & = \overline{x}_{1}^{(a+b)m} \\ v & = P_{q_{1}}(\overline{x}_{1}) + \overline{x}_{1}^{c+d+r−1}F_{q_{1}} \end{aligned}

with

F_{q_{1}} = z(∑_{i+j=r−1} a_{ij1}(y_{1} + α)^{j})(y_{1} + α)^{λ} + \overline{x}_{1}Ω + z^{2}G,  (68)

λ = d − \frac{b(c + d + r − 1)}{a + b}.

Thus ν(q_{1}) ≤ r and ν(q_{1}) = r implies γ(q_{1}) = r. Suppose that q_{1} ∈ π^{−1}(q) and q_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}y_{1}.

Then

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r−1}y_{1}^{d}F_{q_{1}} \\ F_{q_{1}} & = ∑_{i+j=r−1} za_{ij1}y_{1}^{j} + x_{1}Ω + z^{2}G \end{aligned}

implies ν(q_{1}) ≤ r and ν(q_{1}) = r implies τ(q_{1}) > 0.

A similar analysis holds at the point q_{1} ∈ π^{−1}(q) with permissible parameters (x_{1}, y_{1}, z) such that x = x_{1}y_{1}, y = y_{1}.

Suppose that q ∈ C is a 3 point with ν(q) = r − 1,

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F_{q} \\ F_{q} & = ∑_{i+j≥r−1,k≥0} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

some a_{ij0} ≠ 0 with i + j = r − 1.

Suppose that q_{1} ∈ π^{−1}(q) is a 2 point.

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0.

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

\begin{aligned}u & = (\overline{x}_{1}^{a+b}z^{c})^{m} = (\overline{x}\overline{a} ; 1z\overline{c})\overline{m} \\ v & = P_{q_{1}}(\overline{x}\overline{a} ; 1z\overline{c}) + \overline{x}_{1}^{d+r−1+e}z^{f}F_{q_{1}} \end{aligned}

with

λ = e − \frac{b(d + r − 1 + e)}{a + b}, (\overline{a}, \overline{c}) = 1

F_{q_{1}} = (y_{1} + α)^{λ} \frac{F_{q}}{x_{1}^{r−1}} − \frac{g(\overline{x}\overline{a} ; 1z\overline{c})}{\overline{x}_{1}^{d+r−1+e}z^{f}}  (69)

Thus

F_{q_{1}} = ∑_{i+j=r−1} a_{ij0}(y_{1} + α)^{j+λ} + zG + \overline{x}_{1}Ω,

or

F_{q_{1}} = ∑_{i+j=r−1} a_{ij0}(y_{1} + α)^{j+λ} − ∑ a_{ij0}α^{j+λ} + zG + \overline{x}_{1}Ω,

implies ν(q_{1}) ≤ r, and γ(q_{1}) ≤ r.

Suppose that q ∈ C is a 2 point with ν(q) = r and τ(q) > 0 and C ⊂ \overline{S}_{r}(X). By Lemma 6.26, q has permissible parameters (x, y, z) with

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \\ F_{q} & = ∑_{i+j≥r,k≥0} c_{ijk}x^{i}y^{j}z^{k} + \overline{c}zx^{i_{0}}y^{j_{0}} \end{aligned}  (70)

where i_{0} + j_{0} = r − 1, (c + i_{0})b − a(d + j_{0}) = 0, \overline{c} ≠ 0. Suppose that q_{1} ∈ π^{−1}(q), and Ô_{Y_{1},q_{1}} has regular parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. Set x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}. Then

\begin{aligned}u & = \overline{x}_{1}^{(a+b)m} \\ v & = P_{q_{1}}(\overline{x}_{1}) + \overline{x}_{1}^{c+d+r−1}F_{q_{1}} \\ F_{q_{1}} & = \overline{c}z(y_{1} + α)^{λ+j_{0}} + \overline{x}_{1}Ω \end{aligned}

where λ = d − \frac{b(c+d+r−1)}{a+b}. Thus q_{1} is resolved.

Suppose that q_{1} ∈ π^{−1}(q), and Ô_{Y_{1},q_{1}} has regular parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}y_{1}.

Then

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r−1}y_{1}^{d}F_{q_{1}} \\ F_{q_{1}} & = \frac{F_{q}}{x_{1}^{r−1}} = \overline{c}zy_{1}^{j_{0}} + x_{1}Ω \end{aligned}

q_{1} satisfies the conclusions of the Theorem since j_{0} ≤ r − 1.

Suppose that q_{1} ∈ π^{−1}(q), and Ô_{Y_{1},q_{1}} has regular parameters (x_{1}, y_{1}, z) such that

x = x_{1}y_{1}, y = y_{1}.

Then an argument similar to the above case shows that q_{1} satisfies the conclusions of the Theorem (since i_{0} ≤ r − 1).

Lemma 8.7. Suppose that r ≥ 2, C ⊂ X is a 2 curve such that C is r-1 big and 1. p ∈ C a 2 point implies ν(p) ≤ r, and if ν(p) = r then τ(p) > 0.

2. p ∈ C a 3 point implies ν(p) ≤ r − 1. Suppose that π : X_{1} → X is the blowup of C. Then

π^{−1}(C) ∩ \overline{S_{r}(X_{1})}

contains at most one curve. If D ⊂ π^{−1}(C) ∩ \overline{S}_{r}(X_{1}) is a curve, then D is a section over C, and D contains a 1 point.

Suppose that D ⊂ π^{−1}(C) ∩ \overline{S}_{r}(X_{1}) is a curve (which is necessarily a section over C). Supppose that q ∈ C is a 2 point such that ν(q) = r − 1. Then π^{−1}(q) ∩ D is a 1 point.

Proof. Suppose that q ∈ C is a 2 point with ν(q) = r − 1. Suppose, with the notation of (67) of Lemma 8.6, that there exists q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. Then there exist regular parameters (x_{1}, y_{1}, z) in Ô_{X_{1},q_{1}} such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0, and γ_{α} ∈ k such that

∑_{i+j=r−1} a_{ij}(y_{1} + α)^{j} ≡ γ_{α}(y_{1} + α)^{−λ} mod y_{1}^{r}.

−λ ∉ {0, …, r − 1} since F_{q} is normalized.

Set g(t) = ∑_{i+j=r−1} a_{ij}t^{j}. We have

\frac{1}{i!} \frac{d^{i}g}{dt^{i}} (α) = γ_{α} (\frac{−λ(−λ − 1) ⋯ (−λ − i + 1)}{i!}) α^{−λ−i}

for i ≤ r − 1. Thus

a_{0,r−1} = \frac{1}{(r − 1)!} \frac{d^{r−1}g}{dt^{r−1}} (α) = γ_{α} (\frac{−λ(−λ − 1) ⋯ (−λ − r + 2)}{(r − 1)!}) α^{−λ−r+1}

a_{1r−2}+(r−1)a_{0r−1}α = \frac{1}{(r − 2)!} \frac{d^{r−2}g}{dt^{r−2}} (α) = γ_{α} (\frac{−λ(−λ − 1) ⋯ (−λ − r + 3)}{(r − 2)!}) α^{−λ−r+2}

\begin{aligned}a_{1r−2} & = γ_{α} [\frac{−λ(−λ−1)···(−λ−r+3)−(−λ)(−λ−1)···(−λ−r+2)}{(r−2)!}] α^{−λ−r+2} \\ & = γ_{α}[\frac{λ(−λ−1)···(−λ−r+3)(−λ−r+1)}{(r−2)!}]α^{−λ−r+2} \end{aligned}

\frac{−λ(−λ − 1) ⋯ (−λ − r + 2)}{(r − 1)!} ≠ 0

and

\frac{λ(−λ − 1) ⋯ (−λ − r + 3)(−λ − r + 1)}{(r − 2)!} ≠ 0

since −λ ∉ {0, …, r − 1}.

If q_{2} ∈ π^{−1}(q) has ν(q_{2}) = r, and q_{2} ≠ q_{1}, then there exist α ≠ β ∈ k such that

a_{0,r−1} = γ_{β} (\frac{−λ(−λ − 1) ⋯ (−λ − r + 2)}{(r − 1)!}) β^{−λ−r+1}

a_{1r−2} = γ_{β} [\frac{−λ(−λ − 1) ⋯ (−λ − r + 3) − (−λ)(−λ − 1) ⋯ (−λ − r + 2)}{(r − 2)!}] β^{−λ−r+2}

which implies that

γ_{β} = γ_{α}(\frac{α}{β})^{−λ−r+1}

and

γ_{α}α^{−λ−r+2} = γ_{β}β^{−λ−r+2} = γ_{α}α^{−λ−r+1}β.

so that α = β.

Thus there is at most one point q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. q_{1}, if it exists, is a 1 point.

Suppose that q ∈ C is 2 point with ν(q) = r and τ(q) > 0. Suppose, with the notation of (68) of Lemma 8.6, that there exists a 1 point q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. Then there exist regular parameters (x_{1}, y_{1}, z) in Ô_{X_{1},q_{1}} such that x = x_{1}, y = x_{1}(y_{1} + α).

Set g(t) = ∑_{i+j=r−1} a_{ij1}t^{j}. By (68),

ν( ∑_{i+j=r−1} a_{ij1}(y_{1} + α)^{j}) = r − 1.

which implies g(t + α) = a_{0,r−1,1}t^{r−1} which implies g(t) = a_{0,r−1,1}(t − α)^{r−1}. Thus there is at most one 1 point q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. Suppose that q ∈ C is a 3 point. Suppose that, with the notation of (69), of Lemma

8.6, that there exists a 2 point q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. Then there exist regular parameters (x_{1}, y_{1}, z) in Ô_{X_{1},q_{1}} such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0, and γ_{α} ∈ k such that

∑_{i+j=r−1} a_{ij0}(y_{1} + α)^{j} ≡ γ_{α}(y_{1} + α)^{−λ} mod y_{1}^{r}.

As in the argument for the case when q is a 2 point with ν(q) = r−1, we can conclude that there is at most one point q_{1} ∈ π^{−1}(q) with ν(q_{1}) = r. q_{1}, if it exists, is a 2 point.

Suppose that D ⊂ π^{−1}(C) ∩ \overline{S}_{r}(X_{1}) is a curve, which is necessarily a section over C, and q ∈ C is a 2 point such that ν(q) = r − 1. Suppose there exists a 2 point q′ ∈ π^{−1}(q) such that q′ ∈ D. Then ν(q′) = r − 1 by Lemma 7.6, so that (by the proof of Lemma 8.6), there exist permissible parameters (x, y, z) at q′ such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q′} \end{aligned}

F_{q′} = y^{r−1} + xΩ + zG

and there exists an irreducible series f(y, z) such that Î_{D,q′} = (x, f(y, z)).

Case 1 or Case 2 of Lemma 6.30 must hold. Suppose that Case 1 holds. Set x = 0 in the formula of Case 1 to get that there exists a series h(\overline{y}, z) such that

\overline{y}^{ad−bc}(\overline{y}^{(r−1)a} + zG(0, \overline{y}^{a}, z)) = hf(\overline{y}^{a}, z)^{r}

\overline{y} |̸ f(\overline{y}^{a}, z) implies

a(r − 1) ≥ ν(f(\overline{y}^{a}, 0)^{r}) ≥ ar

a contradiction.

Now suppose that Case 2 of Lemma 6.30 holds. a(r − 1) ≠ bc − ad since F_{q′} is normalized. Set x = 0 in the formula of Case 2 to get that there exists a series h(\overline{y}, z) such that

\overline{y}^{a(r−1)} + zG(0, \overline{y}^{a}, z) − g(0)\overline{y}^{bc−ad} = hf(\overline{y}^{a}, z)^{r}
0 ≠ \overline{y}^{a(r−1)} − g(0)\overline{y}^{bc−ad} = h(\overline{y}, 0)f(\overline{y}^{a}, 0)^{r}

Thus

a(r − 1) ≥ ν(\overline{y}^{a(r−1)} − g(0)\overline{y}^{bc−ad}) ≥ rν(f(\overline{y}^{a}, 0)) ≥ ra

which is a contradiction.

Lemma 8.8. Suppose that r ≥ 2 and C ⊂ \overline{S}_{r}(X) is a curve containing a 1 point such that C is r big. let π : X_{1} → X be the blowup of C.

1. Suppose that p ∈ C is a 1 point with ν(p) = γ(p) = r, and q ∈ π^{−1}(p). Then

(a) If q is a 1 point then γ(q) ≤ r. There is at most one 1 point q ∈ π^{−1}(p) such that γ(q) > r − 1.

(b) If q is a 2 point then ν(q) = 0.

2. Suppose that p ∈ C is a 1 point with ν(p) = r, γ(p) ≠ r, and q ∈ π^{−1}(p). Then (a) If q is a 1 point then γ(q) < r,

(b) If q ∈ π^{−1}(p) is a 2 point then ν(q) ≤ r − 1.

3. Suppose that p ∈ C is a 2 point such that γ(p) = ν(p) = r, and q ∈ π^{−1}(p). Then

(a) If q is a 2 point then ν(q) ≤ r and γ(q) ≤ r.

(b) There is at most one 2 point q ∈ π^{−1}(p) such that γ(q) > r − 1.

(c) If q is a 3 point then ν(q) = 0.

4. Suppose that p ∈ C is a 2 point with ν(p) = r and τ(p) > 0, and q ∈ π^{−1}(p). Then

(a) If q is a 2 point then γ(q) ≤ r.

(b) If q is the 3 point then ν(q) ≤ r − τ(p).

Proof. First suppose that p ∈ C is a 1 point such that ν(p) = γ(p) = r. We have permissible parameters (x, y, z) at p such that Î_{C,p} = (x, z),

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}F_{p} \\ F_{p} & = τz^{r} + ∑_{i=2}^{r} \overline{a}_{i}(x, y)x^{i}z^{r−i} \end{aligned}  (71)

where τ is a unit by Lemma 8.5.

Suppose that q ∈ π^{−1}(p) and q is a 1 point. Then q has permissible parameters (x_{1}, y, z_{1}) such that x = x_{1}, z = x_{1}(z_{1} + α). Then ν(F_{q}(0, 0, z_{1})) ≤ r and ν(F_{q}(0, 0, z_{1})) < r if α ≠ 0.

If q ∈ π^{−1}(p) is the 2 point then q has permissible parameters (x_{1}, y, z_{1}) such that x = x_{1}z_{1}, z = z_{1}. Then F_{q} = \frac{F_{p}}{z_{1}^{r}} is a unit.

Now suppose that p ∈ C is a 1 point with ν(p) = r and γ(p) ≠ r. Suppose that q ∈ π^{−1}(p) is a 1 point. Then there exist permissible parameters (x, y, z) at p such that x = z = 0 are local equations of C at p, and permissible parameters (x_{1}, y, z_{1}) at q such that x = x_{1}, z = x_{1}z_{1}.

F_{p} = ∑_{i+j≥r} a_{ij}(y)x^{i}z^{j}

where a_{r0}(0) = 0, a_{0r}(0) = 0, and a_{ij}(0) ≠ 0 for some i, j with i + j = r.

F_{q} = \frac{F_{p}}{x_{1}^{r}} = ( ∑_{i+j=r} a_{ij}(0)z_{1}^{j}) + x_{1}Ω + yG

implies ν(F_{q}(0, 0, z_{1})) ≤ r − 1.

At the 2 point q ∈ π^{−1}(p), there exist permissible parameters (x, y, z) as above, and permissible parameters (x_{1}, y, z_{1}) at q such that x = x_{1}z_{1}, z = z_{1},

F_{q} = \frac{F_{p}}{z_{1}^{r}} = ∑_{i+j=r} a_{ij}(0)x_{1}^{i} + z_{1}Ω + yG

where a_{ij}(0) ≠ 0 for some i ≤ r − 1.

Now suppose that p ∈ C is a 2 point such that ν(p) = r and γ(p) = r.

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

Î_{C,p} = (x, z). After a permissible change of parameters, we have by Lemma 8.5

F = τz^{r} + a_{2}(x, y)z^{r−2} + ⋯ + a_{r}(x, y)  (72)

where τ is a unit and x^{i} | a_{i} for all i.

If p_{1} ∈ π^{−1}(p) has permissible parameters (x_{1}, y, z_{1}) with

x = x_{1}z_{1}, z = z_{1}

then

F_{1} = τ + x_{1}Ω

so that p_{1} is resolved. Suppose that p_{1} ∈ π^{−1}(p) has regular parameters

x = x_{1}, z = x_{1}(z_{1} + α)

\frac{F}{x_{1}^{r}} = τ(z_{1} + α)^{r} + \frac{a_{2}(x, y)}{x^{2}}(z_{1} + α)^{r−2} + ⋯ + \frac{a_{r}(x, y)}{x^{r}}

Thus ν(F_{1}(0, 0, z_{1})) ≤ r and ν(F_{1}(0, 0, z_{1})) ≤ r − 1 if α ≠ 0.

Suppose that p ∈ C is a 2 point such that ν(p) = r and τ(p) > 0.

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F \end{aligned}

where F ∈ Î_{C,p}^{r} = (x, z)^{r}.

F = ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k}.

Suppose that q ∈ π^{−1}(p) is a 2 point. After a permissible change of parameters, replacing z with z − αx, q_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, z = x_{1}z_{1}

\begin{aligned}u & = (x_{1}^{a}y^{b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{b}) + x_{1}^{c+r}y^{d}F_{q} \end{aligned}

F_{q} = \frac{F}{x_{1}^{r}} = ∑_{i+k≥r} a_{ijk}x_{1}^{i+k−r}y^{j}z_{1}^{k}

F_{q} = ∑_{i+k=r} a_{i0k}z_{1}^{k} + x_{1}Ω_{1} + yG

Thus γ(q) ≤ r.

Now suppose that q ∈ π^{−1}(p) has permissible parameters

x = x_{1}z_{1}, z = z_{1}

so that q is a 3 point.

\begin{aligned}u & = (x_{1}^{a}y^{b}z_{1}^{a})^{m} \\ v & = P (x_{1}^{a}y^{b}z_{1}^{a}) + x_{1}^{c}y^{d}z_{1}^{c+r}F_{q} \end{aligned}

F_{q} = \frac{F}{z_{1}^{r}} = ∑_{i+k≥r} a_{ijk}x_{1}^{i}y^{j}z_{1}^{i+k−r}

F_{q} = ∑_{i+k=r} a_{i0k}x_{1}^{i} + yG + z_{1}Ω

a_{r−k,0,k} ≠ 0 if k = τ(p) which implies that ν(q) ≤ r − τ(p).

Lemma 8.9. Suppose that r ≥ 2, C ⊂ \overline{S}_{r}(X) is a curve containing a 1 point such that C is r small.

1. Let π : Y → X be the monodial transform centered at C.

(a) Suppose that p ∈ C is a generic point. If q ∈ π^{−1}(p) is a 1 point then ν(q) = 1. If q ∈ π^{−1}(p) is the 2 point then ν(q) ≤ r and ν(q) = r implies τ(q) > 0.

(b) Suppose that p ∈ C is a 2 point such that ν(q) = r − 1. If q ∈ π^{−1}(p) is a 2 point then ν(q) = 0. If q ∈ π^{−1}(p) is a 3 point then ν(q) ≤ r − 1.

2. Suppose that p ∈ C is a 1 point such that ν(p) = r or a 2 point such that ν(p) = r and τ(p) > 0. Then there exists a finite sequence of quadratic transforms σ : Z → X centered at points over p such that if q ∈ σ^{−1}(p) is a 1 point then ν(q) ≤ r. ν(q) = r implies γ(q) = r. If q ∈ σ^{−1}(p) is a 2 point then ν(q) ≤ r. ν(q) = r implies τ(q) > 0. If q ∈ σ^{−1}(p) is a 3 point then ν(q) ≤ r − 1. The strict transform of C intersects σ^{−1}(p) in a 2 point p′ such that ν(p′) = r − 1

Proof. Suppose that p ∈ C is a 2 point. By Lemma 6.27, there are permissible parameters (x, y, z) at p with Î_{C,p} = (x, z) such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = x^{r−1}y^{n} + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}  (73)

with n ≥ 0. Suppose that ν(p) = r and τ(p) > 0. Then n > 0 and a_{i0k} ≠ 0 for some i, k with i + k = r and k > 0. Let π′ : X′ → X be the blowup of p. Perform n quadratic transforms, π_{1} : X_{1} → X, centered at the 2 point which is the intersection of the strict transform of C and the exceptional divisor. Then by Theorem 7.3

1. All 1 points q in π_{1}^{−1}(p) with ν(q) = r have γ(q) = r.

2. All 2 points q ∈ π^{−1}(p) with ν(q) = r have τ(q) > 0.

3. All 3 points q ∈ π^{−1}(p) have ν(q) ≤ r − 1.

If C_{1} is the strict transform of C, and q is the exceptional point on C_{1}, then there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that

x = x_{1}y_{1}^{n}, y = y_{1}, z = z_{1}y_{1}^{n}

\begin{aligned}u & = (x_{1}^{a}y_{1}^{na+b})^{m} \\ F_{q} & = x_{1}^{r−1} + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y_{1}^{n(i+k−r)+j}z_{1}^{k} \end{aligned}  (74)

where Î_{C_{1},q} = (x_{1}, z_{1}). Thus ν(q) = r − 1.

Suppose that p ∈ C is a 1 point with ν(p) = r. Then by Lemma 6.25 there are regular parameters (x, y, z) in Ô_{X,p} such that Î_{C,p} = (x, z),

\begin{aligned}u & = x^{a} \\ F_{p} & = x^{r−1}y^{n} + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}  (75)

with n ≥ 1. There are only finitely many 1 points in C such that n > 1.

Suppose that n > 1. Then a_{i0k} ≠ 0 for some a_{i0k} with i + k = r and k > 0, so that τ(p) > 0. Let λ : Z → X_{2} be the sequence of n quadratic transforms centered first at p, and then at the intersection of the strict transform of C and the exceptional fiber.

Let C′ be the strict transform of C on Z. Let q′ be the exceptional point of λ on C′. By Theorems 7.1 and 7.3, the conclusions of 2. of the Theorem hold at all points above p, except possibly at q′. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}^{n}, y = y_{1}, z = z_{1}y_{1}^{n}.

\begin{aligned}u & = (x_{1}y_{1}^{n})^{a} \\ F_{q′} & = \frac{F_{q}}{y_{1}^{nr}} = x_{1}^{r−1} + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y_{1}^{(i+k−r)n+j}z_{1}^{k} \end{aligned}

Thus ν(q′) = r − 1 and C′ has the form (73) with n = 0 at q′.

Let π : Y → X be the blowup of C. Suppose that p ∈ C, and p is a 2 point such that ν(p) = r − 1 so that (73) with n = 0 holds at p. Suppose that q ∈ π^{−1}(p), and q has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, z = x_{1}(z_{1} + α)

After making a permissible change of variables, replacing z with z − αx, we may assume that α = 0. Then F_{q} = \frac{F_{p}}{x_{1}^{r−1}} , so that ν(q) = 0.

Suppose that q ∈ π^{−1}(p), and q has permissible parameters (x_{1}, y, z_{1}) such that

x = x_{1}z_{1}, z = z_{1}

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b}z_{1}^{a})^{m} \\ F_{q} & = \frac{F_{p}}{z_{1}^{r−1}} = x_{1}^{r−1} + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y^{j}z_{1}^{i+k−(r−1)} \end{aligned}

so that ν(q) ≤ r − 1.

Now suppose that p ∈ C is a generic point, so that (75) holds with n = 1 at p. Suppose that q ∈ π^{−1}(p), and q has permissible parameters (x_{1}, y, z_{1}) such that

x = x_{1}, z = x_{1}(z_{1} + α)

After making a permissible change of variables, replacing z with z − αx, we may assume that α = 0. Then F_{q} = \frac{F_{p}}{x_{1}^{r−1}} , so that ν(q) = 1.

Suppose that q ∈ π^{−1}(p), and q has permissible parameters (x_{1}, y, z_{1}) such that

x = x_{1}z_{1}, z = z_{1}

\begin{aligned}u & = (x_{1}z_{1})^{a} \\ F_{q} & = \frac{F_{p}}{z_{1}^{r−1}} = x_{1}^{r−1}y + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y^{j}z_{1}^{i+k−(r−1)} \end{aligned}

so that ν(q) ≤ r, ν(q) = r implies τ(q) > 0.

Lemma 8.10. Suppose that r ≥ 2, C ⊂ \overline{S}_{r}(X) is a curve containing a 1 point such that C is r small and γ(q) = r for q ∈ C.

1. Let π : Y → X be the monoidal transform centered at C.

(a) Suppose that p ∈ C is a generic point. Then ν(q) ≤ 1 if q ∈ π^{−1}(p).

(b) Suppose that p ∈ C is a 2 point such that ν(p) = r − 1. Suppose that q ∈ π^{−1}(p). Then ν(q) = 0 if q is a 2 point, and ν(q) ≤ 1 if q is a 3 point.

2. Suppose that p ∈ C. Then there exists a finite sequence of quadratic transforms σ : Z → X centered at points over p such that ν(q) ≤ r, and γ(q) ≤ r if q ∈ σ^{−1}(p) is a 1 or 2 point. ν(q) = 0 if q is a 3 point, and the strict transform of C intersects σ^{−1}(p) in a 2 point p′ such that ν(p′) = r − 1 and γ(p′) = r.

Proof. Suppose that p ∈ C is a 2 point. By Lemma 6.27, there exist permissible parameters (x, y, z) at p such that x = z = 0 are local equations of C at p.

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}  (76)

F_{p} = τ′(y)x^{r−1}y^{n} + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k}

with τ′ a unit, n ≥ 0, and a_{00r} ≠ 0.

Suppose that ν(p) = r − 1. Then n = 0 and τ(p) = 0. Let π : Y → X be the monoidal transform centered at C.

If q ∈ π^{−1}(p) is a 2 point, then after a permissible change of parameters at p, we have that q has permissible parameters (x_{1}, y, z_{1}) such that x = x_{1}, z = x_{1}z_{1}.

F_{q} = \frac{F_{p}}{x_{1}^{r−1}} = τ′(y) + x_{1}Ω

so that ν(q) = 0.

If q ∈ π^{−1}(p) is the 3 point, there exist permissible parameters (x_{1}, y, z_{1}) at q such that x = x_{1}z_{1}, z = z_{1}.

F_{q} = \frac{F_{p}}{z_{1}^{r−1}} = τ′(y)x_{1}^{r−1} + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y^{j}z_{1}^{i+k−(r−1)}

a_{00r} ≠ 0 implies ν(q) ≤ 1.

Suppose that p is a 2 point and ν(p) = r. Let σ : Y → X be the quadratic transform with center p. Suppose that q ∈ σ^{−1}(p) is a 1 point or a 2 point. Then by Theorem 7.3, ν(q) ≤ r and γ(q) ≤ r. If q is a 3 point then ν(q) = 0. At the 2 point q on the strict transform of C, we have permissible parameters (x_{1}, y_{1}, z_{1}) such that x = x_{1}y_{1}, y = y_{1}, z = x_{1}y_{1}. x_{1} = z_{1} = 0 are local equations of the strict transform of C at q.

F_{q} = τ′(y)x_{1}^{r−1}y_{1}^{n−1} + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y_{1}^{i+j+k−r}z_{1}^{k}

of the form of (76) with n decreased by 1.

By induction on n, we achieve the conclusion of 2. after a finite sequence of quadratic transforms.

Suppose that p ∈ C is a 1 point. By Lemma 6.25, there exist permissible parameters (x, y, z) at p such that x = z = 0 are local equations of C at p,

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F_{p} \\ F_{p} & = x^{r−1}τ′(y)y^{n} + ∑_{i+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}  (77)

with τ′ a unit, n ≥ 1, a_{00r} ≠ 0.

Suppose that p ∈ C is a generic point so that n = 1. Let π : Y → X be the monoidal transform centered at C. If q ∈ π^{−1}(p) is a 1 point, then after making a permissible change of parameters at p, there are permissible parameters (x_{1}, y, z_{1}) at q such that x = x_{1}, z = x_{1}z_{1}.

F_{q} = \frac{F_{p}}{x_{1}^{r−1}} = yτ′(y) + x_{1}Ω

implies ν(q) = 1. If q ∈ π^{−1}(p) is the 2 point, then there are permissible parameters (x_{1}, y, z_{1}) at q such that x = x_{1}z_{1}, z = z_{1}.

F_{q} = \frac{F_{p}}{z_{1}^{r−1}} = x_{1}^{r−1}τ′(y)y + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y_{1}^{j}z_{1}^{i+k−(r−1)}

which implies that ν(q) ≤ 1 since a_{00r} ≠ 0.

Suppose that n ≥ 2 in (77). Let σ : Y → X be the quadratic transform with center p.

Suppose that q ∈ σ^{−1}(p). Then q is a 1 or 2 point, and ν(q) ≤ r, γ(q) ≤ r by Theorem 7.3.

At the 2 point q ∈ π^{−1}(p) which is contained in the strict transform of C, there are permissible parameters (x_{1}, y_{1}, z_{1}) at q such that x = x_{1}y_{1}, y = y_{1}, z = y_{1}z_{1}.

\begin{aligned}u & = (x_{1}y_{1})^{a} \\ F_{q} & = \frac{F_{p}}{y_{1}^{r}} = x_{1}^{r−1}y_{1}^{n−1}τ′(y_{1}) + ∑_{i+k≥r} a_{ijk}x_{1}^{i}y_{1}^{i+j+k−r}z_{1}^{k} \end{aligned}

The strict transform of C has local equations x_{1} = z_{1} = 0 at q. We are thus at a point of the form of (76) with n decreased by 1.

We thus achieve the conclusions of 2. after a finite number of quadratic transforms.

Lemma 8.11. Suppose that r = 2 in Lemma 8.10, C ⊂ \overline{S}_{2}(X) is a curve containing a 1 point such that C is 2 small, γ(p) = 2 if p ∈ C, ν(p) = 1 if p ∈ C is a 2 point and p is a generic point of C (n = 1 in (77)) if p ∈ C is a 1 point. Let π : Y → X be the monodial transform centered at C. Suppose that there exists a 2 point p ∈ C such that ν(p) = r − 1 = 1, and q ∈ π^{−1}(p) is a 3 point such that ν(q) = 1, or p ∈ C is a generic point of C (n = 1 in (77)) and q ∈ π^{−1}(p) is a 2 point such that ν(q) = 1. Let \overline{C} be the 2 curve through q which is a section over C. Then F_{q′} ∈ Î_{\overline{C},q′} for all q′ ∈ \overline{C}.

Suppose that there does exist a 2 curve \overline{C} which is a section over C such that F_{q′} ∈ Î_{\overline{C},q′} for q′ ∈ \overline{C}. Let π_{1} : Z → Y be the blowup of \overline{C}. Then

1. Suppose that q ∈ \overline{C} is a 2 point such that q ∈ π^{−1}(p) where p is a generic point of C (n = 1 in (77)), and q′ ∈ π_{1}^{−1}(q).

(a) If q′ is a 1 point then ν(q′) = 1.

(b) If q′ ∈ π_{1}^{−1}(q) is a 2 point then γ(q′) ≤ 1.

2. Suppose that q ∈ \overline{C} is a 3 point such that q ∈ π^{−1}(p) where p ∈ C is a 2 point such that ν(p) = 1 and q′ ∈ π^{−1}(q).

(a) If q′ is a 2 point then γ(q′) ≤ 1.

(b) If q′ is a 3 point then ν(q′) = 0.

Proof. If p ∈ C is a 2 point with ν(p) = 1 (and γ(p) = 2), then there exist permissible parameters (x, y, z) at p such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = x + z^{2} \end{aligned}

where x = z = 0 are local equations of C at p. There exist permissible parameters (x_{1}, y, z_{1}) at the 3 point q ∈ π^{−1}(p) such that x = x_{1}z_{1}, z = z_{1}.

\begin{aligned}u & = (x_{1}^{a}y^{b}z_{1}^{a})^{m} \\ v & = P (x_{1}^{a}y^{b}z_{1}^{a}) + x_{1}^{c}y^{d}z_{1}^{c+1}F_{q} \\ F_{q} & = x_{1} + z_{1} \end{aligned}  (78)

and x_{1} = z_{1} = 0 are local equations of \overline{C} at q. We have F_{q} ∈ Î_{\overline{C},q} which implies F_{q′} ∈ Î_{\overline{C},q′} if q′ ∈ \overline{C} by Lemma 8.1.

If p ∈ C is a generic point, then p is a 1 point and there exist permissible parameters (x, y, z) at p such that

F_{p} = xy + ∑_{i+k≥2} a_{ijk}x^{i}y^{j}z^{k}

with a_{002} ≠ 0 and x = z = 0 are local equations of C at p. There exist permissible parameters (x_{1}, z_{1}, y) at the 2 point q ∈ π^{−1}(p) such that x = x_{1}z_{1}, z = z_{1}.

F_{q} = x_{1}y_{1} + z_{1}( ∑_{i+k=2} a_{ijk}x_{1}^{i}y_{1}^{j}) + z_{1}^{2}Ω

and x_{1} = z_{1} = 0 are local equations of \overline{C} at q. Since a_{002} ≠ 0, there exist permissible parameters (x_{1}, \overline{z}_{1}, y_{1}) at q such that

F_{q} = x_{1}y_{1} + \overline{z}_{1}  (79)

and x_{1} = \overline{z}_{1} = 0 are local equations of \overline{C} at q.

We have F_{q} ∈ Î_{\overline{C},q} implies F_{q′} ∈ Î_{\overline{C},q′} for all q′ ∈ \overline{C} by Lemma 8.1.

1. follows from (79).

Suppose that q ∈ \overline{C} is a 3 point, with permissible parameters (x_{1}, y, z_{1}) such that (78) holds at q. Suppose that q′ ∈ π_{1}^{−1}(q). If q′ is a 3 point, then ν(q′) = 0. Suppose that q′ is a 2 point. Then there exist regular parameters (x_{2}, y, z_{2}) in Ô_{Z,q′} such that

x_{1} = x_{2}, z_{1} = x_{2}(z_{2} + α)

with α ≠ 0.

u = (x_{2}^{2a}y^{b}(z_{2} + α)^{a})^{m} = (\overline{x}_{2}^{2a}y^{b})^{m} = (\overline{x}\overline{a} ; 2y\overline{b})\overline{m}

where x_{2} = \overline{x}_{2}(z_{2} + α)^{− \frac{1}{2}} , (\overline{a}, \overline{b}) = 1.

v = P_{q′}(\overline{x}\overline{a} ; 2y\overline{b}) + \overline{x}_{2}^{2c+2}y^{d}(1 + α + z_{2})

Thus γ(q′) ≤ 1 and 2. follows.

9. Power series in 2 variables

Lemma 9.1. Suppose that R = k[[x, y]] is a power series ring in two variables and u(x, y), v(x, y) ∈ R are series. Suppose that R → R′ is a quadratic transform. Set R_{1} = R̂′. Then (u, v) are analytically independent in R_{1} if and only if u and v are analytically independent in R.

Proof. By Zariski’s Subspace Theorem (Theorem 10.6 [3]), R → R_{1} is an inclusion, and the Lemma follows.

Lemma 9.2. Suppose that R = k[[x, y]] is a power series ring in two variables over an algebraically closed field k of characteristic 0 and u(x, y), v(x, y) ∈ R are series such that either

u = x^{a}

or

u = (x^{a}y^{b})^{m}

with (a, b) = 1. Then u and v are analytically dependent if and only if there exists a series p(t) such that v = p(x) in the first case and v = p(x^{a}y^{b}) in the second case.

Proof. First suppose that u = x^{a} and v = p(x) is a series. Let ω be a primitive a-th root of unity.

0 = ∏_{i=0}^{a−1}(v − p(ω^{i}x)) ∈ k[[u, v]]

implies u and v are analytically dependent.

Now suppose that u = x^{a} and u, v are analytically dependent. Suppose that v is not a series in x. Write

v = q(x) + x^{b}F

where q(x) is a polynomial, x |̸ F and F (0, y) is a nonzero series with no constant term.

F (0, y) = y^{r}μ(y)

for some r > 0 where μ(y) is a unit series. x and x^{b}F are thus analytically dependent, and x and F are analytically dependent. There exists an irreducible series

p(s, t) = ∑ a_{ij}s^{i}t^{j}

such that

0 = ∑ a_{ij}x^{i}F^{j}

which implies that

0 = ∑ a_{0j}F (0, y)^{j} = ∑ a_{0j}y^{rj}μ(y)^{j},

a contradiction, since p(s, t) irreducible implies some a_{0j} ≠ 0. Thus v is a series in x.

Now suppose that u = (x^{a}y^{b})^{m} and v = p(x^{a}y^{b}) is a series in x^{a}y^{b}. Let ω be a primitive m-th root of unity.

0 = ∏_{i=0}^{m−1}(v − p(ω^{i}x^{a}y^{b})) ∈ k[[u, v]]

implies that u, v are analytically dependent.

Suppose that

u = (x^{a}y^{b})^{m}

and u, v are analytically dependent. Consider the quadratic transform

R → R_{1} = R[x_{1}, y_{1}]_{(x_{1},y_{1})}

where x = x_{1}, y = x_{1}(y_{1} + 1). R̂_{1} has regular parameters (\overline{x}_{1}, y_{1}) where

x_{1} = \overline{x}_{1}(y_{1} + 1)^{− \frac{b}{a+b}} .

Thus in R̂_{1}, u = \overline{x}_{1}^{(a+b)m}. Since u, v must be analytically dependent in R̂_{1}, there exists a series q(\overline{x}_{1}) such that v = q(\overline{x}_{1}), by the first part of the proof.

Suppose that v is not a series in x^{a}y^{b}. Write v = ∑ a_{ij}x^{i}y^{j}. There exists a smallest r such that there exists i_{0}, j_{0} such that i_{0} + j_{0} = r, bi_{0} − aj_{0} ≠ 0 and a_{i_{0}j_{0}} ≠ 0.

v = ∑_{i+j<r,aj−bi=0} a_{ij}\overline{x}_{1}^{i+j} + \overline{x}_{1}^{r}( ∑_{i+j=r} a_{ij}(y_{1} + α)^{j− \frac{br}{a+b}}) + \overline{x}_{1}^{r+1}Ω.

∑_{i+j=r} a_{ij}(y_{1} + α)^{j− \frac{br}{a+b}} ∈ k

implies

(y_{1} + α)^{\frac{−br}{a+b}} ( ∑_{i+j=r} a_{ij}(y_{1} + α)^{j}) = c ∈ k

so that

∑_{i+j=r} a_{ij}(y_{1} + α)^{j} = c(y_{1} + α)^{\frac{br}{a+b}} .

Thus

\frac{br}{a + b} ∈ \{0, 1, ⋯, r\}

and j_{0} = \frac{br}{a+b}. This implies that aj_{0} − bi_{0} = 0, a contradiction. Thus v is a series in x^{a}y^{b}.

Lemma 9.3. Suppose that R = k[[x, y]] is a power series in two variables over an algebraically closed field k of characteristic 0, u = x^{a} or u = (x^{a}y^{b})^{m}, and (u, v) are analytically independent. let π : X → spec(R) be the blowup of m = (x, y). Then for all but finitely many points q ∈ π^{−1}(m) there exist regular parameters (\overline{x}, \overline{y}) in Ô_{X,q} such that there is an expansion

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{b}\overline{y} \end{aligned}  (80)

Proof. First suppose that u = x^{a}. Write v = P (x) + x^{b}F where x |̸ F and F has no terms which are powers of x. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}

where r = ν(F). There exists j_{0} > 0 such that i_{0} + j_{0} = r and a_{i_{0}j_{0}} ≠ 0. For all but one point q ∈ π^{−1}(m) there are regular parameters (x_{1}, y_{1}) in Ô_{X,q} such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ∈ k.

\begin{aligned}u & = x_{1}^{a} \\ v & = P (x_{1}) = x_{1}^{b+r}(∑_{i+j=r} a_{ij}(y_{1} + α)^{j} + x_{1}Ω) \end{aligned}  (81)

v has an expansion (80) if and only if

\frac{d}{dy_{1}} (∑_{i+j=r} a_{ij}(y_{1} + α)^{j}) |_{y_{1}=0}= ∑_{j≤r} ja_{r−j,j}(−α)^{j−1} ≠ 0.  (82)

Since

∑_{i+j=r} ja_{r−j,j}(−α)^{j−1}

has at most finitely many roots, all but finitely many q ∈ π^{−1}(m) have an expansion (80).

Now suppose that u = (x^{a}y^{b})^{m}. Write

v = P (x^{a}y^{b}) + x^{c}y^{d}F

where x, y |̸ F and x^{c}y^{d}F has no terms which are powers of x^{a}y^{b}. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}

where r = ν(F).

For all but two points q ∈ π^{−1}(m) there are regular parameters (x_{1}, y_{1}) in Ô_{X,q} such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. There are regular parameters (\overline{x}_{1}, y_{1}) in Ô_{X,q} such that

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}} .

\begin{aligned}u & = \overline{x}_{1}^{(a+b)m} \\ v & = P (\overline{x}_{1}^{(a+b)m}) + \overline{x}_{1}^{c+d+r}(y_{1} + α)^{λ} \frac{F}{x_{1}^{r}} \end{aligned}

where

λ = d − \frac{b(c + d + r)}{a + b}

(y_{1} + α)^{λ} \frac{F}{x_{1}^{r}} = ∑_{i+j=r} a_{ij}(y_{1} + α)^{j+λ} + \overline{x}_{1}Ω.

v does not have an expression (80) at q if and only if there exists c_{α} ∈ k such that

∑_{j=0}^{r} a_{r−j,j}(y_{1} + α)^{j} ≡ c_{α}(y_{1} + α)^{−λ} mod (y_{1})^{2}.

Set a_{j} = a_{r−j,j}. Suppose that q does not have a form (80). Then

∑_{j=0}^{r} a_{j}α^{j} = c_{α}α^{−λ}

and

∑_{j=0}^{r} ja_{j}α^{j−1} = −c_{α}λα^{−λ−1}

implies

(−λ) ∑_{j=0}^{r} a_{j}α^{j} = ∑_{j=0}^{r} ja_{j}α^{j}.  (83)

If there are infinitely many values of α satisfying (83), then (−λ − j)a_{j} = 0 for 0 ≤ j ≤ r, which implies that −λ ∈ {0, …, r} and the leading form of F is

L = ∑_{i+j=r} a_{ij}x^{i}y^{j} = a_{r+λ,−λ}x^{r+λ}y^{−λ}.

Thus x^{c}y^{d}F has a nonzero x^{c+r+λ}y^{d−λ} term.

a(d − λ) − b(c + r + λ) = ad − b(c + r) − (a + b)λ
= ad − b(c + r) − (a + b)(d − \frac{b(c+d+r)}{a+b})
= ad − b(c + r) − (a + b)d + b(c + d + r) = 0

which is impossible since F is normalized (contains no terms which are powers of x^{a}y^{b}). Thus there are at most a finite number of points q ∈ π^{−1}(m) where the form (80) does not hold.

Theorem 9.4. Suppose that k is an algebraically closed field of characteristic zero, B is a powerseries ring in 2 variables over k. Suppose that u, v ∈ B are analytically independent, and there exist regular parameters (x, y) in B such that u = x^{a} or u = x^{a}y^{b}. Let A = spec(B). Then there exists a sequence of quadratic transforms π : X → A such that for all points q ∈ X, there exist regular parameters (\overline{x}, \overline{y}) in Ô_{X,q} such that either

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{b}\overline{y}^{c} \end{aligned}  (84)

or

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d} \end{aligned}  (85)

where (a, b) = 1 and ad − bc ≠ 0.

Theorem 9.4 will follow from Theorem 9.15. Throughout this section, we will use the notations of the statement of Theorem 9.4.

If A → spec(k[[u, v]]) is weakly prepared, then a stronger result than the conclusions of Theorem 9.4 are true in B.

Remark 9.5. With the assumptions of Theorem 9.4, further suppose that

\sqrt{(\frac{∂u}{∂x} \frac{∂v}{∂y} − \frac{∂u}{∂y} \frac{∂v}{∂x})} = \sqrt{(u)}.

Then there exist regular parameters (\overline{x}, \overline{y}) in B, and a power series P in B such that one of the following forms holds.

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{c}\overline{y} \end{aligned}  (86)

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}) + \overline{x}^{c}\overline{y}^{d} \end{aligned}  (87)

where (a, b) = 1 and ad − bc ≠ 0.

Proof. (7.4 [7]) With our assumptions, one of the following must hold.

\begin{aligned}u & = x^{a} \\ u_{x}v_{y} − u_{y}v_{x} & = δx^{e} \end{aligned}  (88)

where δ is a unit or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ u_{x}v_{y} − u_{y}v_{x} & = δx^{e}y^{f} \end{aligned}  (89)

where a, b, e, f > 0, (a, b) = 1 and δ is a unit.

Write v = ∑ a_{ij}x^{i}y^{j}. First suppose that (88) holds. Then ax^{a−1}v_{y} = δx^{e} implies we have the form (86). Now suppose that (89) holds.

u_{x}v_{y} − u_{y}v_{x} = ∑ m(aj − bi)a_{ij}x^{am+i−1}y^{bm+j−1} = δx^{e}y^{f}.

Thus

v = ∑_{aj−bi=0} a_{ij}x^{i}y^{j} + ϵx^{c}y^{d}

where ϵ is a unit. After making a change of variables, multiplying x by a unit, and multiplying y by a unit, we get the form (87).

Definition 9.6. Suppose that Φ : X → A is a product of quadratic transforms, p is a point of X. We will say that (u, v) are 1-resolved at p if there exist regular parameters (x, y) in Ô_{X,p} such that one of the forms (84) or (85) hold at p.

For the rest of this section, we will assume that

Φ : X → A

is a sequence of quadratic transforms.

Suppose that p ∈ X is a point. Then there are regular parameters (x, y) of Ô_{X,p} such that u = x\overline{a}y\overline{b}, and \overline{a} > 0, \overline{b} ≥ 0.

Suppose that \overline{b} > 0. Let m = (\overline{a}, \overline{b}), let a = \frac{\overline{a}}{m}, b = \frac{\overline{b}}{m}. There are power series P (t) and F (x, y) such that x does not divide F, y does not divide F, x^{c}y^{d}F has no nonzero terms which are powers of x^{a}y^{b} and (in Ô_{X,p})

\begin{matrix}u = (x^{a}y^{b})^{m} \\ v = P (x^{a}y^{b}) + x^{c}y^{d}F (x, y) \end{matrix}  (90)

In this case, we will say that p is a 2 point.

If \overline{b} = 0, there are power series P (t) and F (x, y) such that x does not divide F, F has no nonzero terms which are powers of x and (in Ô_{X,p})

\begin{matrix}u = x^{a} \\ v = P (x) + x^{c}F (x, y) \end{matrix}  (91)

In this case we will say that p is a 1 point.

Suppose that p ∈ X, and (x, y) are regular parameters in Ô_{X,p} such that (u, v) have one of the forms (90) or (91). Set

\overline{ν}(p) = \{ \begin{matrix}mult(F) − 1 & if p is a 1 point \\ mult(F) & if p is a 2 point \end{matrix}

Lemma 9.7. \overline{ν}(p) is independent of the choice of regular parameters (x, y) in (90) or (91).

Proof. First suppose that p is a 2 point. To express u and v in the form (90) we can only make a permissible change of variables in x and y, where a permissible change of variables is one of the following two forms:

x = ω_{x}\overline{x}, y = ω_{y}\overline{y} where ω_{x}^{ma}ω_{y}^{mb} = 1  (92)

or

y = ω_{y}\overline{x}, x = ω_{x}\overline{y} where ω_{x}^{ma}ω_{y}^{mb} = 1  (93)

where ω_{x}, ω_{y} are unit series. \overline{ν}(p) does not change after a change of variables of one of these forms.

Now suppose p is a 1 point. To preserve the form (91) we can only make a permissible change of variables, where a permissible change of variables is of the form:

x = ω_{x}\overline{x}, y = ϕ(\overline{x}, \overline{y}) where mult(ϕ(0, \overline{y})) = 1.  (94)

and ω_{x} ∈ k is an a-th root of unity. Then

ϕ(\overline{x}, \overline{y}) = \overline{ϕ}(\overline{x}, \overline{y})(\overline{y} + ψ(\overline{x}))

where \overline{ϕ} is a unit. Write

ψ(\overline{x}) = ∑ b_{i}\overline{x}^{i}.
\begin{matrix}u = \overline{x}^{a} \\ v = \overline{P} (\overline{x}) + \overline{x}^{c}\overline{F} (\overline{x}, \overline{y}) \end{matrix}  (95)

where

\overline{F} = ω_{x}^{c}(F (ω_{x}\overline{x}, ϕ(\overline{x}, \overline{y})) − F (ω_{x}\overline{x}, ϕ(\overline{x}, 0)))
\overline{P} (\overline{x}) = P (ω_{x}\overline{x}) + \overline{x}^{c}ω_{x}^{c}F (ω_{x}\overline{x}, ϕ(\overline{x}, 0))

Suppose that the leading form of F is

L = ∑_{i+j=r} a_{ij}x^{i}y^{j}.

The leading form \overline{L} of \overline{F} is then

ω_{x}^{c}( ∑_{i+j=r} a_{ij}ω_{x}\overline{x}^{i}(e(\overline{y} − b_{1}\overline{x}))^{j} − ∑_{i+j=r} a_{ij}ω_{x}\overline{x}^{i}(−eb_{1}\overline{x})^{j})

where e = \overline{ϕ}(0, 0). \overline{L} is nonzero since a_{ij} ≠ 0 for some j > 0.

(u, v) are 1-resolved at a 2 point p if and only if F is a unit. (u, v) are 1-resolved at a 1 point p if and only if F (x, y) = g(x, y)^{d} + h(x) for some series g(x, y) with mult(g(0, y)) = 1, and positive integer d.

Theorem 9.8. Suppose that g : X_{1} → X is a quadratic transform, centered at a point p of X, and p_{1} ∈ X_{1} is a point such that g(p_{1}) = p. Then

\overline{ν}(p_{1}) ≤ \overline{ν}(p).

If (u, v) are 1-resolved at p then (u, v) are 1-resolved at p_{1}.

Proof. First suppose that p is a 2 point. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}.

in Ô_{X,p}, where r = mult(F) = \overline{ν}(p). Suppose that Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}(y_{1} + α) with α ≠ 0. Define \overline{x}_{1} by

x_{1} = \overline{x}_{1}(y_{1} + α)^{\frac{−b}{a+b}} .

Then (\overline{x}_{1}, y_{1}) are regular parameters in Ô_{X_{1},p_{1}}.

u = x_{1}^{m(a+b)}(y_{1} + α)^{mb} = \overline{x}_{1}^{m(a+b)}.

v = P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d+r}(y_{1} + α)^{λ}(\frac{F}{x_{1}^{r}})

where λ = d − \frac{b(c+d+r)}{a+b}.

F = ∑_{i+j=r} a_{ij}x_{1}^{r}(y_{1} + α)^{j} + x_{1}^{r+1}Ω.

\frac{F}{x_{1}^{r}} = ∑_{j=0}^{r} a_{j}(y_{1} + α)^{j} + x_{1}Ω.

where a_{j} = a_{r−j,j}. We have

\begin{matrix}u = \overline{x}_{1}^{m(a+b)} \\ v = \overline{P} (\overline{x}_{1}) + \overline{x}_{1}^{c+d+r}\overline{F} (\overline{x}_{1}, y_{1}) \end{matrix}  (96)

where

\overline{P} = P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d+r}(α)^{λ}(\frac{F (α^{\frac{−b}{a+b}} \overline{x}_{1}, α^{\frac{a}{a+b}} \overline{x}_{1})}{α^{\frac{−rb}{a+b}} \overline{x}_{1}^{r}})

\overline{F} = (y_{1} + α)^{λ}(\frac{F ((y_{1} + α)^{\frac{−b}{a+b}} \overline{x}_{1}, (y_{1} + α)^{\frac{a}{a+b}} \overline{x}_{1})}{(y_{1} + α)^{\frac{−rb}{a+b}} \overline{x}_{1}^{r}}) − (α)^{λ}(\frac{F (α^{\frac{−b}{a+b}} \overline{x}_{1}, α^{\frac{a}{a+b}} \overline{x}_{1})}{α^{\frac{−rb}{a+b}} \overline{x}_{1}^{r}})

Set

β = (∑_{j=0}^{r} a_{j}α^{j})α^{λ}.

Suppose that \overline{ν}(p_{1}) > \overline{ν}(p) = r, so that

mult(\overline{F}) ≥ mult(F) + 2 = r + 2.

Then

(y_{1} + α)^{λ}(∑_{j=0}^{r} a_{j}(y_{1} + α)^{j}) − β ≡ 0 mod (y_{1})^{r+2}.

β ≠ 0 since ∑_{j=0}^{r} a_{j}(y_{1} + α)^{j} ≠ 0. We have

∑_{j=0}^{r} a_{j}(y_{1} + α)^{j} ≡ β(y_{1} + α)^{−λ} mod (y_{1})^{r+2}  (97)

First suppose that −λ ∈ {0, 1, …, r}. Then

∑_{j=0}^{r} a_{j}(y_{1} + α)^{j} = β(y_{1} + α)^{−λ}

where t = −λ ≤ r. Thus the leading form of F is

L = ∑_{i+j=r} a_{ij}x_{1}^{r}(y_{1} + α)^{j}
= βx_{1}^{r}(y_{1} + α)^{−λ}
= βx^{r+λ}y^{−λ}
= βx^{r−t}y^{t}

So the leading form of F is βx^{r−t}y^{t}. Thus βx^{c+r−t}y^{d+t} is a nonzero term of x^{c}y^{d}F. Since

t = \frac{b(c + d + r)}{a + b} − d,

we have

b(c + r − t) − a(d + t) = 0

so that x^{c+r−t}y^{d+t} is a power of x^{a}y^{b}, a contradiction.

We must then have −λ ∉ {0, 1 …, r}. But then the y_{1}^{r+1} coefficient of β(y_{1} +α)^{−λ} is non zero, a contradiction to (97).

Now suppose that p is a 2 point and Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}y_{1}. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}.

Then

\begin{matrix}u = x_{1}^{m(a+b)}y_{1}^{mb} \\ v = \overline{P} (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r}y_{1}^{d}\overline{F} (x_{1}, y_{1}) \end{matrix}  (98)

where \overline{P} = P, \overline{F} = \frac{F}{x_{1}^{r}}. We need only check that \frac{F}{x_{1}^{r}} has no nonzero x_{1}^{α}y_{1}^{β} terms with b(c + d + r + α) = (a + b)(d + β). We have that a_{ij} = 0 if b(c + i) − a(d + j) = 0.

\frac{F}{x_{1}^{r}} = ∑ a_{ij}x_{1}^{i+j−r}y_{1}^{j}.

Suppose that b(c + d + r + α) = (a + b)(d + β). Set i = α − β + r, j = β. Then b(c + i) − a(d + j) = 0, and a_{ij} = 0. But this is the coefficient of x_{1}^{α}y_{1}^{β} in \frac{F}{x_{1}^{r}}. We have

mult(\overline{F}) ≤ mult(F).

The above argument also works, by interchanging the variables x and y, in the case where p is a 2 point and Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}y_{1}, y = y_{1}.

Now suppose that p is a 1 point and Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}y_{1}, y = y_{1}. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}.

Then

\begin{matrix}u = x_{1}^{a}y_{1}^{a} \\ v = \overline{P} (x_{1}y_{1}) + x_{1}^{c}y_{1}^{c+r}\overline{F} (x_{1}, y_{1}) \end{matrix}  (99)

where \overline{P} = P, \overline{F} = \frac{F}{y_{1}^{r}}. We must show that \overline{F} has no nonzero terms x_{1}^{α}y_{1}^{β} terms with α = r + β. But this is impossible since F has no nonzero x^{i} terms, with i ≥ 0.

The leading form of \overline{F} is

\overline{F} = ∑_{i=0}^{r−1} a_{i,r−i}x_{1}^{i} + y_{1}Ω

since a_{r0} = 0, where some a_{ij} ≠ 0 with i+j = r, j > 0. Thus mult(\overline{F}) ≤ mult(F)−1.

Now suppose that p is a 1 point and Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}(y_{1} + α). By making if necessary a permissible change of variables at p, replacing y with y − αx, we may assume that x = x_{1}, y = x_{1}y_{1}. Write

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}.

where a_{i0} = 0 for all i.

\begin{matrix}u = x_{1}^{a} \\ v = \overline{P} (x_{1}) + x_{1}^{c+r}\overline{F} (x_{1}, y_{1}) \end{matrix}

where \overline{P} = P, \overline{F} = \frac{F}{x_{1}^{r}}. \overline{F} has no nonzero terms which are powers of x_{1}. Thus

mult(\overline{F}) ≤ mult(F).

Suppose that p ∈ X. Set

σ(p) = \{ \begin{matrix}0 & if p is a 1 point and mult(F) = mult(F (0, y)) \\ \frac{1}{2} & if p is a 2 point \\ 1 & if p is a 1 point and mult(F) < mult(F (0, y)) \end{matrix}

Lemma 9.9. σ(p) is independent of the choice of permissible parameters (x, y) at p.

Proof. The proof of Lemma 9.7 shows that mult(F (0, y)) is independent of the choice of permissible parameters at a 1 point.

Lemma 9.10. Suppose that g : X_{1} → X is a quadratic transform, centered at a point p of X, and p_{1} ∈ X_{1} is a point such that g(p_{1}) = p. Further suppose that p is a 2 point, p_{1} is a 1 point and \overline{ν}(p_{1}) = \overline{ν}(p). Then σ(p_{1}) = 0.

Proof. Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}(y_{1} + α) with α ≠ 0. Let r = mult(F). mult(F_{1}) = mult(F) + 1 = r + 1. Let

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}

As in the analysis leading to (97),

F_{1} ≡ (y_{1} + α)^{λ} (∑_{j=0}^{r} a_{r−j,j}(y_{1} + α)^{j}) − β mod (x_{1}, y_{1}^{r+1})  (100)

for some β ∈ k. If F_{1} ≡ 0 mod (x_{1}, y_{1}^{r+1}), then β ≠ 0 and −λ ∉ {0, 1, …, r}, as in the proof of Theorem 9.8. Then

\begin{aligned}F_{1} & ≡ (y_{1} + α)^{λ} (−β \frac{−λ(−λ−1)···(−λ−r)}{(r+1)!}α^{−λ−r−1}) y_{1}^{r+1} mod (x_{1}, y_{1}^{r+2}) \\ & ≡ −β \frac{−λ(−λ−1)···(−λ−r)}{(r+1)!}α^{−r−1}y_{1}^{r+1} mod (x_{1}, y_{1}^{r+2}) \end{aligned} (101)

Thus mult(F_{1}(0, y_{1}, z_{1})) = r + 1.

Lemma 9.11. Suppose that g : X_{1} → X is a quadratic transform, centered at a 1 point p of X and p_{1} is a point above p such that g(p_{1}) = p.

If p_{1} is a 1 point and \overline{ν}(p_{1}) = \overline{ν}(p), then σ(p_{1}) = 0. If σ(p) = 0 and p_{1} is a 2 point then \overline{ν}(p_{1}) = 0.

Proof. First suppose that σ(p) = 0 and p_{1} is a 2 point. Then Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}y_{1}, y = y_{1}.

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}

with a_{0r} ≠ 0.

F_{1} = ∑_{i=0}^{r−1} a_{i,r−i}x_{1}^{i} + y_{1}Ω.

is then a unit.

Now suppose that p_{1} is a 1 point and ν(p_{1}) = ν(p). After appropriate choice of permissible variables (x, y) at p, Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}y_{1}. Set r = mult(F) = mult(F_{1}). Then F_{1} = \frac{F (x_{1},x_{1}y_{1})}{x_{1}^{r}} and mult(F_{1}(0, y_{1})) = r.

Theorem 9.12. Suppose that g : X_{1} → X is a quadratic transform, centered at a point p of X, and p_{1} is a closed point such that g(p_{1}) = p. If \overline{ν}(p_{1}) = \overline{ν}(p), then σ(p_{1}) ≤ σ(p).

Proof. This is immediate from Lemmas 9.10 and 9.11.

Suppose that

F = ∑_{i+j≥r} a_{ij}x^{i}y^{j}

has multiplicity r. Define

δ(F; x, y) = min(\frac{i}{r − j} | j < r, a_{ij} ≠ 0).

δ(F; x, y) = ∞ if and only if F = y^{r}ω, where ω is a unit. If δ(F; x, y) < ∞, then δ(F; x, y) ∈ \frac{1}{r!}N.

Suppose that p ∈ X. If (x, y) are permissible parameters at p with one of the forms (90) or (91), set

δ(p; x, y) = δ(F; x, y).

Then set

δ(p) = sup(δ(p; x, y))

where the sup is over all permissible parameters at p. Note that if p is a 2 point, then

δ(p) = max(δ(p; x, y), δ(p; y, x))

if (x, y) are a particular choice of permissible parameters at p.

If p is a 2 point and ν(p) > 0, then δ(p) < ∞. If p is a 1 point and σ(p) = 1, then δ(p) = 1, since δ(p; x, y) = 1 for all permissible parameters (x, y).

Lemma 9.13. Suppose that p is a 1 point, σ(p) = 0 and (x, y) are fixed permissible parameters at p. Then there exists a power series t(x) such that

δ(p) = δ(p; x, y − t(x)).

If δ(p) < ∞, then t(x) is a polynomial. δ(p) > δ = δ(p; x, y) if and only if δ ∈ N and

∑_{i+δj=rδ} a_{ij}x^{i}y^{j} = τ(y − cx^{δ})^{r} + λx^{rδ}

for some τ, c, λ ∈ k with c ≠ 0 (so that λ = −τ(−c)^{r}).

Proof. Suppose that (\overline{x}, \overline{y}) are also permisible parameters at p. Then \overline{x} = λx, with λ^{a} = 1 and \overline{y} = \overline{ϕ}(y − t(x)) for some unit series \overline{ϕ} and series t(x).

δ(p; \overline{x}, \overline{y}) = δ(p; x, y − t(x))

Thus

δ(p) = sup(δ(p; x, y − t(x) | t(x) is a polynomial of positive order). (102)

Let δ = δ(p; x, y),

\overline{L} = ∑_{i+δj=rδ} a_{ij}x^{i}y^{j}.

so that

F = \overline{L} + ∑_{i+δj>rδ} a_{ij}x^{i}y^{j}.

Suppose that

\overline{L} = τ(y − cx^{δ})^{r} + λx^{rδ}

for some τ, c, λ ∈ k with 0 ≠ c. Set y_{1} = y − cx^{δ}. Then δ ∈ N and δ(p; x, y_{1}) > δ(p; x, y) since

F_{1} = τy_{1}^{r} + ∑_{i+δj>rδ} \overline{a}_{ij}x^{i}y_{1}^{j}.

where

v = P_{1}(x) + x_{1}^{c}F_{1}

is the normalized form of v with respect to (x, y_{1}). We can repeat this process, with y replaced by y − cx^{δ}. The process will either produce a polynomial t(x) such that if y_{1} = y − t(x), and δ_{1} = δ(p; x, y_{1}), then δ_{1} ∉ N, or δ_{1} ∈ N and

∑_{i+δ_{1}j=rδ_{1}} \overline{a}_{ij}x^{i}y_{1}^{j} ≠ τ(y_{1} − cx^{δ_{1}})^{r} + λx^{rδ_{1}}  (103)

for any τ, c, λ ∈ k with 0 ≠ c, or we will produce a series t(x) such that if y_{1} = y−t(x), then δ(p; x, y_{1}) = δ(p) = ∞, so that F_{1} = y_{1}^{r}ϕ, where ϕ is a unit series.

Suppose that we have produced y_{1} such that δ(p; x, y_{1}) ∉ N or δ(p; x, y_{1}) ∈ N and (103) holds. We will show that δ(p) = δ(p; x, y_{1}). Suppose that δ_{1} = δ(p; x, y_{1}) < δ(p). By (102), there is a polynomial

t(x) = ∑ e_{i}x^{i}

such that if y_{2} = y_{1} − t(x), then δ(p; x, y_{2}) > δ(p; x, y_{1}). Substitute y_{1} = y_{2} + t(x) into

F_{1} = ∑_{i+δ_{1}j=rδ_{1}} \overline{a}_{ij}x^{i}y_{1}^{j} + ∑_{i+δ_{1}j>rδ_{1}} \overline{a}_{ij}x^{i}y_{1}^{j},

and normalize with respect to the permissible parameters to get

v = P_{2}(x) + x^{c}F_{2}(x, y_{2}).

Let \overline{d} = ord (t(x)). x^{i}y_{1}^{j} = x^{i}(y_{2} + t(x))^{j} has nonzero x^{i+m\overline{d}}y_{2}^{j−m} terms with 0 ≤ m ≤ j, and may have other nonzero x^{i+m\overline{d}+γ}y_{2}^{j−m} terms with 0 ≤ m ≤ j, γ ≥ 0.

Suppose that \overline{d} < δ_{1} = δ(p; x, y_{1}). The expansion of y_{1}^{r} has a nontrivial x\overline{d}y_{1}^{r−1} term. Suppose that x^{i}y_{1}^{j} is such that its expansion has a nontrivial x\overline{d}y_{1}^{r−1} term. Then \overline{d} = i + m\overline{d} + γ, r − 1 = j − m with 0 ≤ m ≤ j, i, γ ≥ 0. \overline{d}(1 − m) = i + γ ≥ 0 implies m = 0 or 1. m = 0 implies j = r − 1, i ≤ \overline{d}. \overline{a}_{ij} = 0 in this case since

i + δ_{1}j ≤ \overline{d} + δ_{1}(r − 1) < δ_{1}r.

m = 1 implies i = 0, j = r. Thus there exists a nontrivial x\overline{d}y_{1}^{r−1} term in F_{2}(x, y_{1}) which implies that δ_{2} < δ_{1}, a contradiction. Thus \overline{d} ≥ δ_{1}.

We then see that if i + δ_{1}j > rδ_{1}, then all terms x^{α}y^{β} in the expansion of x^{i}y_{1}^{j} = x^{i}(y_{2} + t(x))^{j} satisfy α + δ_{1}β > rδ_{1}. Since δ_{2} < δ_{1}, we see that

∑_{i+δ_{1}j=rδ_{1}} \overline{a}_{ij}x^{i}(y_{2} + t(x))^{j} = \{ \begin{matrix}cy_{2}^{r} + terms with i + δ_{1}j > rδ_{1} if δ_{1} ∉ N, \\ cy_{2}^{r} + dx^{rδ_{1}} + terms with i + δ_{1}j > rδ_{1} if δ_{1} ∈ N. \end{matrix}

Thus mult(t) = δ_{1} and

∑_{i+δ_{1}j=rδ_{1}} \overline{a}_{ij}x^{i}y_{1}^{j} = c(y_{1} − e_{δ_{1}}x^{δ_{1}})^{r} + dx^{rδ_{1}}

a contradiction.

Lemma 9.14. Suppose that g : X_{1} → X is a quadratic transform, centered at a point p of X, and p_{1} ∈ X_{1} is a closed point above p such that g(p_{1}) = p and \overline{ν}(p_{1}) = \overline{ν}(p).

Suppose that p and p_{1} are both 2 points. Then δ(p_{1}) = δ(p) − 1.

Suppose that p and p_{1} are both 1 points, σ(p) = 0 and δ(p) < ∞. Then δ(p_{1}) = δ(p) − 1.

Proof. Suppose that r = mult(F).

First suppose that p and p_{1} are both 2 points. Then p has permissible parameters (x, y) and Ô_{X_{1},p_{1}} has permissible parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}y_{1}. Since F_{1} = \frac{F}{x_{1}^{r}}, δ(p_{1}; x_{1}, y_{1}) = δ(p; x, y) − 1. Since \overline{ν}(p_{1}) = \overline{ν}(p), we have F = ∑ a_{ij}x^{i}y^{j} with a_{ij} = 0 if i + j ≤ r and j < r. Thus a_{0r} ≠ 0, so that δ(p; y, x) = 1 and δ(p; x, y) > 1. Thus δ(p) = δ(p; x, y). Since mult(F_{1}) = r and mult(F_{1}(0, y_{1})) = r, δ(p_{1}; y_{1}, x_{1}) = 1 and δ(p_{1}; x_{1}, y_{1}) ≥ 1. Then δ(p_{1}) = δ(p_{1}; x_{1}, y_{1}) = δ(p) − 1.

Now suppose that p and p_{1} are both 1 points, σ(p) = 0 and δ(p) < ∞. We can suppose that we have permissible coordinates (x, y) at p such that δ = δ(p) = δ(F; x, y) and mult(F (0, y)) = mult(F). p_{1} has permissible parameters (x_{1}, y_{1}) such that x = x_{1}, y = x_{1}(y_{1} + γ) for some γ ∈ k.

First suppose that γ ≠ 0.

F_{1} = ∑_{i+j=r} a_{ij}(y_{1} + γ)^{j} − \overline{a} + x_{1}Ω

where

\overline{a} = ∑_{i+j=r} a_{ij}γ^{j}.

mult(F_{1}) = mult(F) implies

∑_{i+j=r} a_{ij}(y_{1} + γ)^{j} − \overline{a} = a_{0r}y_{1}^{r}.

Thus

∑_{i+j=r} a_{ij}x^{i}y^{j} = a_{0r}(y − γx)^{r} + \overline{a}x^{r}.

This is a contradiction to the assumption that δ(p; x, y) = δ(p) by Lemma 9.13.

Now suppose that γ = 0. Then F_{1} = \frac{F}{x_{1}^{r}} and δ(p_{1}; x_{1}, y_{1}) = δ(p; x, y) − 1. If δ(p_{1}; x_{1}, y_{1}) < δ(p_{1}), then we must also have δ(p; x, y) < δ(p) By Lemma 9.13. Thus δ(p_{1}) = δ(p) − 1.

If p ∈ X is a 2 point, then (u, v) are 1-resolved at p precisely when \overline{ν}(p) = 0. If p ∈ X is a 1 point then (u, v) are 1-resolved at p precisely when δ(p) = ∞. Thus (u, v) are not 1-resolved at p ∈ X if and only if \overline{ν}(p) > 0 and δ(p) < ∞.

We can define an invariant

Inv(p) = (\overline{ν}(p), σ(p), δ(p))

for p ∈ X.

Theorem 9.15. Suppose that g : X_{1} → X is a quadratic transform, centered at a point p of X, and p_{1} ∈ X_{1} is such that g(p_{1}) = p. Suppose that \overline{ν}(p) > 0 and δ(p) < ∞. Then

Inv(p_{1}) < Inv(p)

in the lexicographic ordering.

Proof. The Theorem follows from Theorem 9.8, Lemmas 9.10, 9.11, 9.14.

The proof of Theorem 9.4 is immediate from Theorem 9.15.

Lemma 9.16. Suppose that f(x, y) ∈ T_{0} = k[[x, y]] is a series. Suppose that we have an infinite sequence of quadratic transforms

T_{0} → T_{1} → ⋯ → T_{n} → ⋯

Then there exists n_{0} such that n ≥ n_{0} implies there exist regular parameters (x_{n}, y_{n}) in T_{n}, α_{n}, β_{n} ∈ N and a unit u_{n} ∈ T_{n} such that f = x_{n}^{α_{n}}y_{n}^{β_{n}}u_{n}.

Proof. This follows directly from Zariski’s proof of resolution of surface singularities along a valuation ([29]), or can be deduced easily after blowing up enough to make f = 0 a SNC divisor.

Lemma 9.17. Suppose that α_{j} + β_{j} ≥ j, x^{α_{j}}y^{β_{j}} ∈ T_{0} = k[x, y]_{(x,y)} for 2 ≤ j ≤ r (or 1 ≤ j ≤ r). Suppose that we have a sequence of quadratic transforms

T_{0} → T_{1} → ⋯ → T_{n} → ⋯

where each T_{n} has regular parameters (x_{n}, y_{n}) such that either x_{n−1} = x_{n}, y_{n−1} = x_{n}y_{n}, or x_{n−1} = x_{n}y_{n}, y_{n−1} = y_{n}. There are natural numbers α_{n,i}, β_{n,i} such that

x^{α_{j}}y^{β_{j}} = x_{n}^{α_{n,i}}y_{n}^{β_{n,i}}.

Define

δ_{n,i,j} = (\frac{α_{n,i}}{i} − \frac{α_{n,j}}{j} ) (\frac{β_{n,i}}{i} − \frac{β_{n,j}}{j} )

Then 1. δ_{n+1,i,j} ≥ δ_{n,i,j}

2. δ_{n,i,j} < 0 implies δ_{n+1,i,j} − δ_{n,i,j} ≥ \frac{1}{r^{4}} .

Proof. We will first verify 1. Suppose that x_{n} = x_{n+1}y_{n+1}, y_{n} = y_{n+1}. The proof when x_{n} = x_{n+1}, y_{n} = x_{n+1}y_{n+1} is the same. 1. is immediate from

δ_{n+1,i,j} = δ_{n,i,j} + (\frac{α_{n,i}}{i} − \frac{α_{n,j}}{j} )^{2}

Now suppose that δ_{n,i,j} < 0. Then (\frac{α_{n,i}}{i} − \frac{α_{n,j}}{j} ) and (\frac{β_{n,i}}{i} − \frac{β_{n,j}}{j} ) are nonzero. We can suppose that x_{n} = x_{n+1}y_{n+1}, y_{n} = y_{n+1}.

\begin{aligned}δ_{n+1,i,j} − δ_{n,i,j} & = (\frac{α_{n,i}}{i} − \frac{α_{n,j}}{j} )^{2} \\ & = (\frac{jα_{n,i}−iα_{n,j}}{ij} )^{2} ≥ \frac{1}{r^{4}} \end{aligned}

since i, j ≤ r implies (ij)^{2} ≤ r^{4}.

Corollary 9.18. Suppose that α_{j} + β_{j} ≥ j and x^{α_{j}}y^{β_{j}} ∈ T_{0} = k[x, y]_{(x,y)} for 2 ≤ j ≤ r (or 1 ≤ j ≤ r) and

T_{0} → T_{1} → ⋯ → T_{n} → ⋯

is a sequence of quadratic transformations as in the statement of Lemma 9.17. Then 1. There exists n_{0} and i such that n ≥ n_{0} implies

\frac{α_{n,i}}{i} ≤ \frac{α_{n,j}}{j} and \frac{β_{n,i}}{i} ≤ \frac{β_{n,j}}{j}

for 2 ≤ j ≤ r (or 1 ≤ j ≤ r).

2. There exists an n_{1} ≥ n_{0} such that

\{\frac{α_{n_{1},i}}{i} \} + \{\frac{β_{n_{1},i}}{i} \} < 1

Proof. By Lemma 9.17, there exists n_{0} such that n ≥ n_{0} implies δ_{n,i,j} ≥ 0 for all i, j. Let λ_{1} = min (\frac{α_{n,j}}{j} ). Let λ_{2} = min (\frac{β_{n,j}}{j} such that \frac{α_{n,j}}{j} = λ_{1}). Choose i such that

\frac{α_{n,i}}{i} = λ_{1}, \frac{β_{n,i}}{i} = λ_{2}. Then

\frac{α_{n,i}}{i} ≤ \frac{α_{n,j}}{j} , \frac{β_{n,i}}{i} ≤ \frac{β_{n,j}}{j}

for 2 ≤ j ≤ r (or 1 ≤ j ≤ r).

Now we will prove 2. Suppose that n ≥ n_{0}. Then

\{\frac{α_{n,i}}{i} \} + \{\frac{β_{n,i}}{i} \} ∈ \frac{1}{i}N.

Suppose that

\{\frac{α_{n,i}}{i} \} + \{\frac{β_{n,i}}{i} \} ≥ 1.

Without loss of generality,

x_{n} = x_{n+1}y_{n+1}, y_{n} = y_{n+1}.

Then

\begin{aligned}\{\frac{α_{n+1,i}}{i} \} + \{\frac{β_{n+1,i}}{i} \} & = \{\frac{α_{n,i}}{i} \} + \{\frac{α_{n,i}}{i} \} + \{\frac{β_{n,i}}{i} \} − 1 \\ & < \{\frac{α_{n,i}}{i} \} + \{\frac{β_{n,i}}{i} \} \end{aligned}

Thus there exists n_{1} ≥ n_{0} such that 2. holds.

Remark 9.19. The conditions α_{i} + β_{i} ≥ i and \{\frac{α_{i}}{i} \} + \{\frac{β_{i}}{i} \} < 1 imply either α_{i} ≥ i or β_{i} ≥ i.

Lemmas 9.20 and 9.21 are used in Abhyankar’s Good Point proof of resolution of singularities [4], [20].

Lemma 9.20. Suppose that α_{j_{0}} + β_{j_{0}} ≥ j, (α_{j_{0}}, β_{j_{0}}) are nonnegative integers for 1 ≤ j ≤ r. Suppose that we have pairs of nonnegative integers (α_{n,j}, β_{n,j}) for all positive n and 1 ≤ j ≤ r such that either

(α_{n+1,j}, β_{n+1,j}) = (α_{n,j} + β_{n,j} − j, β_{n,j})

or

(α_{n+1,j}, β_{n+1,j}) = (α_{n,j}, α_{n,j} + β_{n,j} − j).

Define

δ_{n,i,j} = (\frac{α_{n,i}}{i} − \frac{α_{n,j}}{j} ) (\frac{β_{n,i}}{i} − \frac{β_{n,j}}{j} )

Then 1. δ_{n+1,i,j} ≥ δ_{n,i,j}

2. δ_{n,i,j} < 0 implies δ_{n+1,i,j} − δ_{n,i,j} ≥ \frac{1}{r^{4}} .

Lemma 9.21. Suppose that the assumptions are as in Lemma 9.20.

Suppose that α_{j} + β_{j} ≥ j, x^{α_{j}}y^{β_{j}} ∈ T_{0} = k[x, y]_{(x,y)} for 1 ≤ j ≤ r. Suppose that we have a possibly infinite sequence of quadratic transforms

T_{0} → T_{1} → ⋯ → T_{n} → ⋯

where each T_{n} has regular parameters (x_{n}, y_{n}) such that either x_{n−1} = x_{n}, y_{n−1} = x_{n}y_{n} or x_{n−1} = x_{n}y_{n}, y_{n−1} = y_{n} and (α_{n}, β_{n}) are defined by the respective rules of Lemma 9.20. Then

1. There exists n_{0} and i such that n ≥ n_{0} implies

\frac{α_{n,i}}{i} ≤ \frac{α_{n,j}}{j} and \frac{β_{n,i}}{i} ≤ \frac{β_{n,j}}{j}

for 1 ≤ j ≤ r.

2. There exists n_{1} ≥ n_{0} such that

\{\frac{α_{n_{1},i}}{i} \} + \{\frac{β_{n_{1},i}}{i} \} < 1

10. A_{r}(X)

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared.

Definition 10.1. Suppose that r ≥ 2. \overline{A}_{r}(X) holds if 1. ν(p) ≤ r if p ∈ X is a 1 point or a 2 point.

2. If p ∈ X is a 1 point and ν(p) = r, then γ(p) = r.

3. If p ∈ X is a 2 point and ν(p) = r, then τ(p) > 0.

4. ν(p) ≤ r − 1 if p ∈ X is a 3 point

Definition 10.2. Suppose that r ≥ 2. A_{r}(X) holds if 1. \overline{A}_{r}(X) holds.

2. \overline{S}_{r}(X) is a union of nonsingular curves and isolated points.

3. \overline{S}_{r}(X) ∩ (X − \overline{B}_{2}(X)) is smooth.

4. \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X) on the open set X − B_{3}(X).

5. The curves in \overline{S}_{r}(X) passing through a 3 point q ∈ X have distinct tangent directions at q. (They are however, allowed to be tangent to a 2 curve).

Definition 10.3. Suppose that A_{r}(X) holds. A weakly permissible monoidal transform π : X_{1} → X is called permissible if π is the blowup of a point, a 2 curve or a curve C containing a 1 point such that C ∪ \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X) at all points of C.

Remark 10.4. 1. If A_{r}(X) holds and π : X_{1} → X is a permissible monodial transform, then the strict transform of \overline{S}_{r}(X) on X_{1} makes SNCs with \overline{B}_{2}(X_{1}) at 1 and 2 points, and has distinct tangent directions at 3 points.

2. If π : X_{1} → X is a quadratic transform centered at a point p ∈ X with ν(p) = r and A_{r}(X) holds, then A_{r}(X_{1}) holds.

3. If A_{r}(X) holds and all 3 points q of X satisfy ν(q) ≤ r − 2, then \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X).

The Remark follows from Lemmas 7.9 and 7.7, and the observation that the strict transforms of nonsingular curves with distinct tangent directions at a point p intersect the exceptional fiber of the blowup of p transversally in distinct points.

11. Reduction of ν in a special case

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared.

Lemma 11.1. Suppose that r ≥ 2 and A_{r}(X) holds, p ∈ X is a 1 point or a 2 point with ν(p) = γ(p) = r. Let R = O_{X,p}. Suppose that (x, y, z) are permissible parameters at p as in Lemma 8.5. Then there exists a finite sequence of permissible monodial transforms π : Y → Spec(R̂) centered at sections over C = V (x, y), such that for q ∈ π^{−1}(p), there exist permissible parameters (\overline{x}, \overline{y}, z) at q such that F_{q} has one of the following forms.

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{c}F_{q} \end{aligned}  (104)

or

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}F_{q} \end{aligned}  (105)

with

F_{q} = τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}(\overline{x}, \overline{y})\overline{x}^{α_{i}} \overline{y}^{β_{i}}z^{r−i} + ϵ\overline{x}^{α_{r}} \overline{y}^{β_{r}}

with τ a unit, \overline{a}_{i} a unit (or zero), α_{i} + β_{i} ≥ i for all i, and ϵ = 0 or 1.

Proof. We have one of the forms (65) or (66) of Lemma 8.5 at p. By Lemma 9.2 and Theorem 9.4 applied to

\overline{u} = x^{a}, \overline{v} = P (x) + x^{c}F_{p}(x, y, 0)

or

\overline{u} = (x^{a}y^{b})^{m}, \overline{v} = P (x^{a}y^{b}) + x^{c}y^{d}F_{p}(x, y, 0)

there exists a sequence of permissible blowups of sections over C such that for all q over p, there are permissible parameters (\overline{x}, \overline{y}, z) at q such that

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{c}(τz^{r} + ∑_{i=2}^{r−1} a_{i}(\overline{x}, \overline{y})z^{r−i} + ϵ\overline{x}^{e_{0}} \overline{y}^{f_{0}}) \end{aligned}  (106)

with τ a unit, ϵ = 0 or 1 and f_{0} > 0, or

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}(τz^{r} + ∑_{i=2}^{r−1} a_{i}(\overline{x}, \overline{y})z^{r−i} + ϵ\overline{x}^{e_{0}} \overline{y}^{f_{0}}) \end{aligned}  (107)

with τ a unit, ϵ = 0 or 1 and a(d + f_{0}) − b(c + e_{0}) ≠ 0.

By further permissible blowing up (of sections over C) we can make

u ∏_{2≤i≤r−1,a_{i}≠0} a_{i} = 0

a SNC divisor, while preserving the forms (106) and (107). At points q over p satisfying (107) we have then achieved the conclusions of the Lemma.

Suppose that q is a point over p satisfying (106) such that the conclusions of the Lemma do not hold. We then have ϵ = 1, f_{0} > 0 and

u ∏_{2≤i≤r,a_{i}≠0} a_{i} = 0

is not a SNC divisor. Since

u ∏_{2≤i≤r−1,a_{i}≠0} a_{i} = 0

is a SNC divisor, there exists a nonzero, nonunit series g(\overline{x}) such that

a_{i} = \overline{a}_{i}(\overline{x}, \overline{y})\overline{x}^{α_{i}}(\overline{y} − g(\overline{x}))^{β_{i}}  (108)

for 2 ≤ i ≤ r − 1, where the \overline{a}_{i} are units (or 0), and some β_{i} > 0 with \overline{a}_{i} ≠ 0. If f_{0} = 1, we can set ỹ = \overline{y} − g(\overline{x}) and renormalize with respect to (\overline{x}, ỹ, z) to get in the form of the conclusions of the Lemma.

Otherwise f_{0} > 1. Let t = ν(g(\overline{x})), so that

g(\overline{x}) = α\overline{x}^{t} + higher order terms

for some 0 ≠ α. Now blow up V (\overline{x}, \overline{y}). under \overline{x} = x_{1}y_{1}, \overline{y} = y_{1}, we have

\begin{aligned}u & = x_{1}^{a}y_{1}^{a} \\ v & = P (x_{1}y_{1}) + x_{1}^{c}y_{1}^{c}(τz^{r} + ∑_{i=2}^{r−1} ã_{i}(x_{1}, y_{1})x_{1}^{α_{i}}y_{1}^{β_{i}+α_{i}}z^{r−i} + x_{1}^{e_{0}}y_{1}^{e_{0}+f_{0}}) \end{aligned}

in the form of the conclusions of the Lemma. Under \overline{x} = x_{1}, \overline{y} = x_{1}(y_{1} + β), with β ≠ 0, we have

u = x_{1}^{a}
v = P (x_{1}) + x_{1}^{c}(τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}x_{1}^{α_{i}+β_{i}}(y_{1} + β − \frac{g(x_{1})}{x_{1}})^{β_{i}}z^{r−i} + x_{1}^{e_{0}+f_{0}}(y_{1} + β)^{f_{0}})

= P (x_{1}) + β^{f_{0}}x_{1}^{c+e_{0}+f_{0}} + x_{1}^{c}(τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}x_{1}^{α_{i}+β_{i}}((\overline{y}_{1} + β^{f_{0}})^{\frac{1}{f_{0}}} − \frac{g(x_{1})}{x_{1}})^{β_{i}}z^{r−i} + x_{1}^{e_{0}+f_{0}} \overline{y}_{1})

where \overline{y}_{1} = (y_{1} + β)^{f_{0}} − β^{f_{0}}. If we are not in the form of the conclusions of the Lemma, then

(\overline{y}_{1} + β^{f_{0}})^{\frac{1}{f_{0}}} − \frac{g(x_{1})}{x_{1}} = a(x_{1}, \overline{y}_{1})(\overline{y}_{1} − ϕ(x_{1}))

where ν(ϕ) ≥ 1. We can make a change of variable in \overline{y}_{1}, replacing \overline{y}_{1} with \overline{y}_{1} −ϕ(x_{1}), and renormalize, to get in the form of the conclusions of the Lemma.

Under \overline{x} = x_{1}, \overline{y} = x_{1}y_{1}, we have

\begin{aligned}u & = x_{1}^{a} \\ v & = P (x_{1}) + x_{1}^{c}(τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}(x, y)x_{1}^{α_{i}+β_{i}}(y_{1} − \frac{g(x_{1})}{x_{1}})^{β_{i}}z^{r−i} + x_{1}^{e_{0}+f_{0}}y_{1}^{f_{0}}) \end{aligned}

the coefficients of z^{i} are in the form of (108), but we have a reduction ν (\frac{g(x_{1})}{x_{1}}) = t−1.

If ν (\frac{g(x_{1})}{x_{1}}) = 0 we are in the form of the conclusions of the Lemma. Thus after t blowups, centered at the intersection of the strict transform of the surface y = 0 with the exceptional divisor, we achieve the conclusions of the Lemma.

Theorem 11.2. Suppose that r ≥ 2 and A_{r}(X) holds, p ∈ X is a 1 point or a 2 point with ν(p) = γ(p) = r. Let R = O_{X,p}. Suppose that (x, y, z) are permissible parameters at p as in Lemma 8.5, where z = σz̃ for some z̃ ∈ R and unit σ ∈ R̂. Then there exists a finite sequence of permissible monodial transforms π : Y → Spec(R̂) centered at sections over C = V (x, y), such that for q ∈ π^{−1}(p), q has permissible parameters (\overline{x}, \overline{y}, z) such that F_{q} has one of the following forms:

1.

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{c}F_{q} with \\ F_{q} & = τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}(\overline{x}, \overline{y})\overline{x}^{α_{i}}z^{r−i} + ϵ\overline{x}^{α_{r}} \overline{y} \end{aligned}  (109)

where τ is a unit, α_{i} ≥ i for 2 ≤ i ≤ r − 1, α_{r} ≥ r − 1, ϵ = 0 or 1, and \overline{a}_{i} are units (or 0), or

2.

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}F_{q} with \\ F_{q} & = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(\overline{x}, \overline{y})\overline{x}^{α_{j}} \overline{y}^{β_{j}}z^{r−j} + ϵ\overline{x}^{α_{r}} \overline{y}^{β_{r}} \end{aligned}  (110)

where τ is a unit, α_{j} + β_{j} ≥ j and \overline{a}_{j} are units or 0 for all j, ϵ = 0 or 1, there exists an i such that \overline{a}_{i} ≠ 0, 2 ≤ i ≤ r and

\frac{α_{i}}{i} ≤ \frac{α_{j}}{j} , \frac{β_{i}}{i} ≤ \frac{β_{j}}{j}

for 2 ≤ j ≤ r. We further have

\{\frac{α_{i}}{i} \} + \{\frac{β_{i}}{i} \} < 1

or 3.

\begin{aligned}u & = \overline{x}^{a} \\ v & = P (\overline{x}) + \overline{x}^{b}F_{q} with \\ F_{q} & = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(\overline{x}, \overline{y})\overline{x}^{α_{j}} \overline{y}^{β_{j}}z^{r−j} + ϵ\overline{x}^{α_{r}} \overline{y}^{β_{r}} \end{aligned}  (111)

where τ is a unit, α_{j} + β_{j} ≥ j and \overline{a}_{j} are units or 0 for all j, ϵ = 0 or 1, there exists an i such that \overline{a}_{i} ≠ 0, 2 ≤ i ≤ r and

\frac{α_{i}}{i} ≤ \frac{α_{j}}{j} , \frac{β_{i}}{i} ≤ \frac{β_{j}}{j}

for 2 ≤ j ≤ r. We further have

\{\frac{α_{i}}{i} \} + \{\frac{β_{i}}{i} \} < 1.

Proof. We can first construct a sequence of monoidal transforms π : Y → Spec(R̂) satisfying the conclusions of Lemma 11.1. (109) holds at all but finitely many points q ∈ π^{−1}(p).

Suppose that q ∈ π^{−1}(p) and (104) holds at q, with ϵ = 1, but F_{q} is not in the form of (109) or (111). Perform a monoidal transform π′ : Y′ → Y centered at the section over C through q with local equations \overline{x} = \overline{y} = 0 in (104). Suppose that q′ ∈ (π′)^{−1}(q). Suppose that there are permissible parameters (x_{1}, y_{1}, z) at q′ such that \overline{x} = x_{1}, \overline{y} = x_{1}(y_{1} + α) where α ≠ 0. Then

F_{q} = τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}x_{1}^{α_{i}+β_{i}}(y_{1} + α)^{β_{i}}z^{r−i} + x_{1}^{α_{r}+β_{r}}(y_{1} + α)^{β_{r}}

Set ỹ_{1} = (y_{1} + α)^{β_{r}} − α^{β_{r}}. Then

F_{q′} = τz^{r} + ∑_{i=2}^{r−1} ã_{i}(x_{1}, ỹ_{1})xα̃_{i} ; 1z^{r−i} + xα̃_{r} ; 1 ỹ_{1}

in the form of (109). Thus the only points q′ ∈ (π′)^{−1}(q) which might not satisfy the conclusions of Theorem 11.2 are the points q′ which have regular parameters (x_{1}, y_{1}, z) such that \overline{x} = x_{1}, \overline{y} = x_{1}y_{1} or \overline{x} = x_{1}y_{1}, \overline{y} = y_{1}.

The analysis of the case when (104) holds at q, with ϵ = 0, is simpler. We again conclude that the only points in the blow up of the curve with local equations \overline{x} = \overline{y} = 0 above q which may not satisfy the conclusions of Theorem 11.2 are the points which have regular parameters (x_{1}, y_{1}, z) such that \overline{x} = x_{1}, \overline{y} = x_{1}y_{1} or \overline{x} = x_{1}y_{1}, \overline{y} = y_{1}.

Suppose that q ∈ π^{−1}(p) and (105) holds at q, with ϵ = 1, but F_{q} is not in the form of (110). Perform a monoidal transform π′ : Y′ → Y centered at the section over C through q with local equations \overline{x} = \overline{y} = 0. Suppose that q′ ∈ (π′)^{−1}(q). Suppose that there are regular parameters (x_{1}, y_{1}, z) at q′ such that \overline{x} = x_{1}, \overline{y} = x_{1}(y_{1} + α) where α ≠ 0. Then

u  = \overline{x}_{1}^{(a+b)m}
(y_{1} + α)^{λ}F_{q} = τ(y_{1} + α)^{λ}z^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}(y_{1} + α)^{β_{i}+λ−(α_{i}+β_{i}) \frac{b}{a+b}} \overline{x}_{1}^{α_{i}+β_{i}}z^{r−i}
+\overline{x}_{1}^{α_{r}+β_{r}}(y_{1} + α)^{λ+β_{r}−(α_{r}+β_{r}) \frac{b}{a+b}} .

Thus

F_{q′} = τ̃z^{r} + ∑_{i=2}^{r−1} ã_{i}(\overline{x}_{1}, \overline{y}_{1})\overline{x}α̃_{i} ; 1z^{r−i} + \overline{x}α̃_{r} ; 1 \overline{y}_{1}

is in the form of (109), where x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}, \overline{y}_{1} = (y_{1} + α)^{λ_{1}} − α^{λ_{1}}, where λ = d − \frac{b(c+d)}{a+b}, λ_{1} = λ + β_{r} − (α_{r} + β_{r})\frac{b}{a+b} ≠ 0 since F_{q} is normalized implies

a(d + β_{r}) − b(c + α_{r}) ≠ 0.

Thus the only points q′ ∈ (π′)^{−1}(q) which might not satisfy the conclusions of Theorem 11.2 are the points which have permissible parameters (x_{1}, y_{1}, z) such that \overline{x} = x_{1}, \overline{y} = x_{1}y_{1} or \overline{x} = x_{1}y_{1}, \overline{y} = y_{1}.

The analysis of the case when (105) holds at q, with ϵ = 0, is simpler. We again conclude that the only points in the blow up of the curve with local equations \overline{x} = \overline{y} = 0 above q which may not satisfy the conclusions of Theorem 11.2 are the points which have regular parameters (x_{1}, y_{1}, z) such that \overline{x} = x_{1}, \overline{y} = x_{1}y_{1} or \overline{x} = x_{1}y_{1}, \overline{y} = y_{1}.

We can construct a sequence of monoidal transforms

Y_{n} → ⋯ → Y_{1} → Y

with maps π_{i} : Y_{i} → Y such that Y_{i} → Y_{i−1} are centered at sections C_{i} over C, such that π_{i}^{−1}(p)∩C_{i} does not satisfy (109), (110) or (111). By the above analysis, Lemma 9.17 and Corollary 9.18, we reach the conclusions of the theorem after a finite number of blowups.

Remark 11.3. In (111) of Theorem 11.2, we must have β_{j} < j for some j.

Proof.

u ∈ Î_{sing(Φ_{X}),q} ⊂ Î_{\overline{S_{r}(X)},q}

by Lemma 6.12. Thus \overline{x} ∈ Î_{\overline{S}_{r}(X),q}. β_{j} ≥ j for all j implies F_{q} ∈ (y, z)^{r}, so that Î_{\overline{S}_{r}(X),q} ⊂ (y, z) by Lemma 6.23, a contradiction.

Theorem 11.4. Suppose that r ≥ 2 and A_{r}(X) holds. Suppose that p ∈ X is a 1 point or a 2 point with ν(p) = γ(p) = r. Let R = O_{X,p}. Suppose that π : Y → Spec(R̂) is the sequence of monoidal transforms of sections over the curve C with local equations x = y = 0 at p of Theorem 11.2. Suppose that t > r is a positive integer. Then there exists a sequence of permissible monoidal transforms \overline{π} : \overline{Y} → spec(R) of sections over C such that for all q ∈ \overline{π}^{−1}(p), F_{q} is equivalent mod (\overline{x}, z)^{t} to a form (109) or (111) or F_{q} is equivalent mod (\overline{x}\overline{y}, z)^{t} to a form (110), where (\overline{x}, \overline{y}, z) are permissible parameters for u, v at q, and z = σz̃ for some z̃ ∈ R and unit σ ∈ R̂.

\overline{π} extends to a sequence of permissible monoidal transforms \overline{U} → U over an affine neighborhood U of p. \overline{S}_{r}(\overline{U}) is the union of the curves in \overline{π}^{−1}(p) and the strict transforms of the curves D or D_{1}, D_{2} (if they exist) in the notation of Lemma 8.5. \overline{S}_{r}(\overline{U}) makes SNCs with \overline{B}_{2}(\overline{U}).

Proof. Let m_{0} be the maximal ideal of R̂. We can after possibly replacing

x̃ with x̃ω,  (112)

where ω is a unit in R, assume that x = γx̃ with γ ≡ 1 mod m_{0}^{t}. We can factor

Y = Y_{n′} → ⋯ → Y_{2} → Y_{1} → spec(R̂) = Y_{0}

so that each map is a permissible monoidal transform. In fact, if S_{0} = spec(k[x, y]), there exists a sequence of quadratic transforms

S_{n′} → ⋯ → S_{2} → S_{1} → spec(k[x, y]) = S_{0}

centered over (x, y) such that Y_{i} = S_{i} ×_{S_{0}} Y_{0} for all i. Set \overline{S}_{0} = spec(k[x̃, y]). x̃ → x induces an isomorphism \overline{S}_{0} ≅ S_{0}. We have then a sequence of quadratic transforms

\overline{S}_{n′} → ⋯ → \overline{S}_{1} → \overline{S}_{0}

where \overline{S}_{i} = S_{i} ×_{S_{0}} \overline{S}_{0} and isomorphisms S_{i} ≅ \overline{S}_{i}. Set \overline{Y}_{0} = spec(R). We have a natural map \overline{Y}_{0} → \overline{S}_{0}. Define a sequence of permissible monoidal transforms

\overline{Y}_{n′} → ⋯ → \overline{Y}_{1} → \overline{Y}_{0}

by \overline{Y}_{i} = \overline{S}_{i} ×_{\overline{S}_{0}} \overline{Y}_{0}. We have a commutative diagram

\begin{aligned}Y_{n′} & → & Y_{n′−1} & → & ⋯ & → & Y_{1} & → & Y_{0} \\ ↓ & & ↓ & & & & ↓ & & ↓ \\ S_{n′} & → & S_{n′−1} & → & ⋯ & → & S_{1} & → & S_{0} \\ ↑ & & ↑ & & & & ↑ & & ↑ \\ \overline{S}_{n′} & → & \overline{S}_{n′−1} & → & ⋯ & → & \overline{S}_{1} & → & \overline{S}_{0} \\ ↑ & & ↑ & & & & ↑ & & ↑ \\ \overline{Y}_{n′} & → & \overline{Y}_{n′−1} & → & ⋯ & → & \overline{Y}_{1} & → & \overline{Y}_{0} \end{aligned}  (113)

The maps \overline{S}_{i} → S_{i} are isomorphisms, and we have maps S_{i} ×_{S_{0}} Ŝ_{0} → Y_{i}, \overline{S}_{i} ×_{\overline{S}_{0}} ˆ \overline{S}_{0} → \overline{Y}_{i} induced by the natural projections

k[[x, y, z]] → k[[x, y]] and k[[x̃, y, z]] → k[[x̃, y]]

so that the diagrams

\begin{aligned}& & Y_{i} \\ & ↙ \\ S_{i} & & ↑ \\ & ↖ \\ & & S_{i} ×_{S_{0}} Ŝ_{0} \end{aligned}  (114)

and

\begin{aligned}\overline{Y} & i \\ & ↙ \\ \overline{S}_{i} & & ↑ \\ & ↖ \\ & & \overline{S}_{i} ×_{\overline{S}_{0}} Ŝ_{0} \end{aligned}

commute.

Suppose that q̃ ∈ \overline{Y}_{n′} is a closed point. (113) and (114) identifies q̃ with a closed point p̃ ∈ Y_{n′}, and closed points \overline{q} ∈ \overline{S}_{n′}, \overline{p} ∈ S_{n′}. We have commutative diagrams:

\begin{aligned}Ô_{S_{n′} ,\overline{p}} & ≅ & Ô_{\overline{S}_{n′} ,\overline{q}} \\ ↑ & & ↑ \\ k[[x, y]] & ≅ & k[[x̃, y]] \end{aligned}  (115)

induced by x → x̃. This induces commutative diagrams:

\begin{aligned}Ô_{Y_{n′} ,p̃} = Ô_{S_{n′} ,\overline{p}}[[z]] & ≅ & Ô_{\overline{S}_{n′} ,\overline{q}}[[z]] = & Ô_{\overline{Y}_{n′} ,q̃} \\ ↑ & & & ↑ \\ R̂ = k[[x, y, z]] & ≅ & & k[[x̃, y, z]] \end{aligned} .

Suppose that F (x, y, z) ∈ R̂. If (\overline{x}, \overline{y}) are regular parameters in Ô_{S_{n′} ,\overline{p}}, which are identified with regular parameters x̂, ŷ in Ô_{\overline{S}_{n′} ,\overline{q}} by (115), and F (x, y, z) = G(\overline{x}, \overline{y}, z) ∈

Ô_{Y_{n′} ,p̃}, then F (x̃, y, z) = G(x̂, ŷ, z) ∈ Ô_{\overline{Y}_{n′} ,q̃}.

Since x = γx̃, γ ≡ 1 mod m_{0}^{t}, and

F (x, y, z) = F (γx̃, y, z) ≡ F (x̃, y, z) mod m_{0}^{t}

implies

F (x, y, z) ≡ G(x̂, ŷ, z) mod m_{0}^{t}Ô_{\overline{Y}_{n′} ,q̃}.

Let m be the maximal ideal of R.

Suppose that p̃ ∈ Y_{n′} is a 1 point, u = \overline{x}^{a} in Ô_{Y_{n′} ,p̃}. Then (since we can assume that Y ≠ spec(R̂)) \overline{x} | x and \overline{x} | y in Ô_{Y_{n′} ,p̃} implies m_{0}^{t} ⊂ (\overline{x}, z)^{t}, so that m^{t} ⊂ (x̂, z)^{t}O_{\overline{Y}_{n′} ,q̃}.

Suppose that p̃ ∈ Y_{n′} is a 2 point, u = (\overline{x}^{a}\overline{y}^{b})^{m} in Ô_{Y_{n′} ,p̃}. If \overline{x} = 0, \overline{y} = 0 are both local equations of components of the exceptional locus of Y_{n′} → Y_{0}, we have \overline{x}\overline{y} | x, \overline{x}\overline{y} | y which implies that m_{0}^{t} ⊂ (\overline{x}\overline{y}, z)^{t}, so that m^{t} ⊂ (x̂ŷ, z)^{t}.

If one of \overline{x} = 0, \overline{y} = 0 is not a local equation of the exceptional locus, then we have regular parameters (x′, y′) in Ô_{S_{n′} ,\overline{p}} such that x = x′(y′)\overline{b}, y = y′ (or x = x′, y = y′(x′)\overline{b}). In the first case we have

Ô_{Y_{n′} ,p̃} = k[[\frac{x}{y\overline{b}} , y, z]]
= k[[\frac{γx̃}{y\overline{b}}, y, z]]
= k[[\frac{x̃}{y\overline{b}} , y, z]] = Ô_{\overline{Y}_{n′} ,q̃}.

Thus we have Ô_{Y_{n′} ,p̃} = Ô_{\overline{Y}_{n′} ,q̃}. In the second case, we also have Ô_{Y_{n′} ,p̃} = Ô_{\overline{Y}_{n′} ,q̃}.

Let \overline{π} : \overline{Y} = \overline{Y}_{n′} → \overline{Y}_{0} be the morphism of the bottom row of (113). Suppose that p_{0} ∈ \overline{Y}_{0} is a 1 point, so that in Ô_{\overline{Y}_{0},p_{0}},

u = x^{a}, v = P_{p_{0}}(x) + x^{c_{0}}F_{p_{0}}.

Suppose that p′ ∈ \overline{π}^{−1}(p_{0}). Let q be the corresponding closed point of Y = Y_{n′}. Suppose that we have permissible parameters (\overline{x}, \overline{y}, z) in Ô_{Y,q} such that

u = \overline{x}\overline{a}, v = P_{q}(\overline{x}) + \overline{x}^{c}F_{q}(\overline{x}, \overline{y}, z)  (116)

of the form (109) or (111) with \frac{\overline{a}}{a}, \frac{c}{c_{0}} ∈ N, x = \overline{x}^{\frac{\overline{a}}{a}} = \overline{x}^{\frac{c}{c_{0}}} . Let (x̃_{∗}, ỹ_{∗}, z) be the corresponding regular parameters at p′ (by the identification (115)).

u = x^{a} = γ^{a}x̃^{a} = γ^{a}x̃\overline{a} ; ∗ = (x̃′_{∗})\overline{a}

where we define

x̃_{∗} = x̃′_{∗}γ^{− \frac{a}{\overline{a}}} ≡ x̃′_{∗} mod m_{0}^{t}Ô_{\overline{Y} ,p′}.

Thus (x̃′_{∗}, ỹ_{∗}, z) are permissible parameters for (u, v) in Ô_{\overline{Y} ,p′} There exists a series P̃_{q}(\overline{x}) such that

F_{p_{0}}(x, y, z) = F_{q}(\overline{x}, \overline{y}, z) + P̃_{q}(\overline{x}).

Thus

\begin{aligned}F_{p_{0}}(x, y, z) & ≡ F_{q}(x̃_{∗}, ỹ_{∗}, z) + P̃_{q}(x̃_{∗}) mod m_{0}^{t}Ô_{\overline{Y} ,p′} \\ & ≡ F_{q}(x̃′_{∗}, ỹ_{∗}, z) + P̃_{q}(x̃′_{∗}) mod m_{0}^{t}Ô_{\overline{Y} ,p′} \end{aligned}

P_{q}(\overline{x}) = P_{p_{0}}(\overline{x}^{\frac{c}{c_{0}}} ) + \overline{x}^{c}P̃_{q}(\overline{x})

implies

v = P_{p_{0}}(x) + x^{c_{0}}F_{p_{0}}(x, y, z)
= P_{p_{0}}((x̃′_{∗})^{\frac{c}{c_{0}}} ) + (x̃′_{∗})^{c}F_{p_{0}}(x, y, z)
= P_{q}(x̃′_{∗}) + (x̃′_{∗})^{c}(F_{q}(x̃′_{∗}, ỹ_{∗}, z) + h)

with h ∈ m_{0}^{t}O_{\overline{Y} ,p′}.

The case when p_{0} ∈ \overline{Y}_{0} is a 1 point and (110) holds in Ô_{Y,q} is a combination of the case when p_{0} is a 1 point and the form (109) or (111) holds in Ô_{Y,q}, and the following case.

Suppose that p_{0} ∈ \overline{Y}_{0} is a 2 point so that in Ô_{\overline{Y}_{0},p_{0}},

\begin{aligned}u & = (x^{a}y^{b})^{m_{0}} \\ v & = P_{p_{0}}(x^{a}y^{b}) + x^{c_{0}}y^{d_{0}}F_{p_{0}} \end{aligned}

Suppose that p′ ∈ \overline{π}^{−1}(p_{0}). Let q be the corresponding closed point in Y = Y_{n′}. Suppose that we have permissible parameters (\overline{x}, \overline{y}, z) at q such that

\begin{aligned}u & = (\overline{x}\overline{a}\overline{y}\overline{b})^{m_{1}} \\ v & = P_{q}(\overline{x}\overline{a}\overline{y}\overline{b}) + \overline{x}^{c}\overline{y}^{d}F_{q}(\overline{x}, \overline{y}, z) \end{aligned}

of the form of (110). We have

\begin{aligned}x & = \overline{x}^{a_{1}} \overline{y}^{b_{1}}γ_{1}(\overline{x}, \overline{y}) \\ y & = \overline{x}^{a_{2}} \overline{y}^{b_{2}}γ_{2}(\overline{x}, \overline{y}) \end{aligned}

where γ_{1}, γ_{2} are units in Ô_{Y,q}, such that

(aa_{1} + ba_{2})m_{0} = \overline{a}m_{1}, (ab_{1} + bb_{2})m_{0} = \overline{b}m_{1}

where m_{0} | m_{1}, c_{0}a_{1} + d_{0}a_{2} = c, c_{0}b_{1} + d_{0}b_{2} = d.

We have

x^{a}y^{b} = (\overline{x}\overline{a}\overline{y}\overline{b})^{\frac{m_{1}}{m_{0}}}

and

x^{c_{0}}y^{d_{0}} = \overline{x}^{c}\overline{y}^{d}ϕ(\overline{x}, \overline{y})

where ϕ = γ_{1}^{c_{0}}γ_{2}^{d_{0}}. There exists a series P̃_{q}(\overline{x}\overline{a}\overline{y}\overline{b}) such that

F_{q}(\overline{x}, \overline{y}, z) = ϕ(\overline{x}, \overline{y})F_{p_{0}}(x, y, z) − \frac{P̃_{q}(\overline{x}\overline{a}\overline{y}\overline{b})}{\overline{x}^{c}\overline{y}^{d}}.

v = P_{p_{0}}((\overline{x}\overline{a}\overline{y}\overline{b})^{\frac{m_{1}}{m_{0}}} ) + P̃_{q}(\overline{x}\overline{a}\overline{y}\overline{b}) + \overline{x}^{c}\overline{y}^{d}F_{q}(\overline{x}, \overline{y}, z)

implies

P_{q}(\overline{x}^{a}\overline{y}^{b}) = P_{p_{0}}((\overline{x}\overline{a}\overline{y}\overline{b})^{\frac{m_{1}}{m_{0}}} ) + P̃_{q}(\overline{x}\overline{a}\overline{y}\overline{b}).

Let (x̃_{∗}, ỹ_{∗}, z) be the corresponding regular parameters at p′ to (\overline{x}, \overline{y}, \overline{z}), by the identification of (115). Define x̃′_{∗} by

x̃_{∗} = x̃′_{∗}γ^{− \frac{m_{0}a}{m_{1}\overline{a}}} ≡ x̃′_{∗} mod m_{0}^{t}O_{\overline{Y} ,p′}.
x^{a}y^{b} = (\overline{x}\overline{a}\overline{y}\overline{b})^{\frac{m_{1}}{m_{0}}}

implies

x^{a}y^{b} = γ^{a}x̃^{a}y^{b} = γ^{a}(x̃\overline{a} ; ∗ỹ\overline{b} ; ∗)^{\frac{m_{1}}{m_{0}}} = ((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗)^{\frac{m_{1}}{m_{0}}}

Thus (x̃′_{∗}, ỹ_{∗}, z) are permissible parameters for (u, v) in Ô_{\overline{Y} ,p′}. x^{c_{0}}y^{d_{0}} = \overline{x}^{c}\overline{y}^{d}ϕ implies

x^{c_{0}}y^{d_{0}} = γ^{c_{0}}x̃^{c_{0}}y^{d_{0}} = γ^{c_{0}}x̃_{∗}^{c}ỹ_{∗}^{d}ϕ(x̃_{∗}, ỹ_{∗}) = (x̃′_{∗})^{c}ỹ_{∗}^{d}ϕ̃

with

ϕ̃ ≡ ϕ(x̃′_{∗}, ỹ_{∗}) mod m_{0}^{t}O_{\overline{Y} ,p′}.

\begin{aligned}F_{p_{0}}(x, y, z) & ≡ ϕ(x̃_{∗}, ỹ_{∗})^{−1}(F_{q}(x̃_{∗}, ỹ_{∗}, z) + \frac{P̃_{q}(x̃\overline{a} ; ∗ỹ\overline{b} ; ∗)}{x̃_{∗}^{c}ỹ_{∗}^{d}}) mod m_{0}^{t}Ô_{\overline{Y} ,p′} \\ & ≡ ϕ̃(x̃′_{∗}, ỹ_{∗})^{−1}(F_{q}(x̃′_{∗}, ỹ_{∗}, z) + \frac{P̃_{q}((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗)}{(x̃′_{∗})^{c}ỹ_{∗}^{d}}) mod m_{0}^{t}Ô_{\overline{Y} ,p′} \end{aligned}

u = ((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗)^{m_{1}}
v = P_{p_{0}}(((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗)^{\frac{m_{1}}{m_{0}}} ) + (x̃′_{∗})^{c}(ỹ_{∗}^{d})ϕ̃F_{p_{0}}(x, y, z)

= P_{p_{0}}(((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗)^{\frac{m_{1}}{m_{0}}} ) + P̃_{q}((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗) + (x̃′_{∗})^{c}(ỹ_{∗}^{d})[F_{q}(x̃′_{∗}, ỹ_{∗}, z) + h]

= P_{q}((x̃′_{∗})\overline{a}ỹ\overline{b} ; ∗) + (x̃′_{∗})^{c}(ỹ_{∗}^{d})[F_{q}(x̃′_{∗}, ỹ_{∗}, z) + h]

with h ∈ m_{0}^{t}Ô_{\overline{Y} ,p′}.

The case when p_{0} ∈ \overline{Y}_{0} is a 2 point and (109) or (111) holds in Ô_{Y,q} is similar to the case when (110) holds in Ô_{Y,q}.

Suppose that p′ is a generic point of C. If (x′, y′, z′) are permissible parameters at p′ such that x′ = y′ = 0 are local equations of C at p′, then ν(F_{p′}(0, 0, z′)) ≤ 1, so that after extending \overline{π} to a sequence of permissible blowups \overline{U} → U over a small affine neighborhood U of p, γ(p∗) ≤ 1 at all points p∗ of \overline{π}^{−1}(p′). Thus the curves in \overline{S}_{r}(\overline{U}) must be components of \overline{π}^{−1}(p), and the strict transforms of the curves D or D_{1}, D_{2} in \overline{S}_{r}(\overline{U}), (if they exist), with the notation of Lemma 8.5.

Thus a curve E in \overline{S}_{r}(\overline{U}) must have local equations as asserted by the Theorem.

Theorem 11.5. Suppose that r ≥ 3 and A_{r}(X) holds. Suppose that p ∈ X is a 1 point or a 2 point with ν(p) = γ(p) = r. Let R = O_{X,p}. Suppose that π : Y_{p} → Spec(R̂) is the sequence of monoidal transforms of sections over the curve \overline{C} with local equations x̃ = y = 0 of Theorem 11.2. For q ∈ π^{−1}(p), define

l_{q} = \{ \begin{matrix}(min_{2≤i≤r}\{[\frac{α_{i}}{i}]\} + 3) r & if F_{q} is a form (109) or (111). \\ (min_{2≤i≤r}\{[\frac{α_{i}}{i}]\} + min_{2≤i≤r}\{[\frac{β_{i}}{i}]\} + 3) r & if F_{q} is a form (110). \end{matrix}

let l = max{l_{q} | q ∈ π^{−1}(p)}.

Suppose that t ≥ l. Let \overline{π} : \overline{Y}_{p} → spec(R) be the sequence of monodial transforms of Theorem 11.4. Let

⋯ → Y_{n} → ⋯ → Y_{1} → \overline{Y}_{p}

be a sequence of permissible monoidal transforms centered at curves C in \overline{S}_{r} such that C is r big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} \overline{Y}_{p} → spec(R)

extends to a permissible sequence of monoidal transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p, in the notation of Theorem 11.4, such that \overline{S}_{r}(\overline{U}_{1}) contains no curves C such that C is r big. Let

⋯ → Z_{n} → ⋯ → V_{p}

be a permissible sequence of monoidal transforms centered at curves C in \overline{S}_{r} such that C is r small. Then there exists n_{1} < ∞ such that π_{2} : Z_{p} = Z_{n_{1}} → V_{p} extends to a permissible sequence of monoidal transforms

\overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{r}(\overline{U}_{2}) = ∅.

Finally, there exists a sequence of quadratic transforms π_{3} : W_{p} → Z_{p} which extends to a permissible sequence of monoidal transforms

\overline{U}_{3} → \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{r}(\overline{U}_{3}) = ∅, and if

q ∈ (\overline{π} ∘ π_{1} ∘ π_{2} ∘ π_{3})^{−1}(p),

1. ν(q) ≤ r − 1 if q is a 1 or 2 point.

2. If q is a 2 point and ν(q) = r − 1, then τ(q) > 0.

3. ν(q) ≤ r − 2 if q is a 3 point.

Proof. Let \overline{Y} = \overline{Y}_{p}. If q ∈ \overline{π}^{−1}(p), we have permissible parameters (x, y, z) in Ô_{\overline{Y} ,q} for (u, v) with forms obtained from those of (109), (110), (111) by modifying F_{q} by adding an appropriate series h to F_{q}.

(109) is modified by changing F_{q} to

F_{q} = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x, y)x^{α_{j}}z^{r−j} + ϵx^{α_{r}}y + h  (117)

with h ∈ (x, z)^{t}. By assumption, there exists z̃ ∈ O_{\overline{Y} ,q} and a unit σ ∈ Ô_{\overline{Y} ,q} such that z = σz̃. Then x = z = 0 defines a germ of an algebraic curve D at q. We also assume that ν(q) = r.

Given a form (117) at q, suppose that D ⊂ \overline{S}_{r}(\overline{Y}) is a curve such that q ∈ D. By assumption, \overline{S}_{r}(\overline{Y}) makes SNCs with \overline{B}_{2}(\overline{Y}). Since D is nonsingular at q, x ∈ Î_{D,q} and F_{q} ∈ Î_{D,q}^{r} + (x)^{r−1} by Lemma 6.25 implies

\frac{∂^{r−1}F_{q}}{∂z^{r−1}} ∈ Î_{D,q},

so that z ∈ Î_{D,q} and x = z = 0 are local equations of D at q.

(110) is modified by changing F_{q} to

F_{q} = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x, y)x^{α_{j}}y^{β_{j}}z^{r−j} + ϵx^{α_{r}}y^{β_{r}} + h  (118)

with h ∈ (xy, z)^{t}. By assumption, there exists z̃ ∈ O_{\overline{Y} ,q} and a unit σ ∈ Ô_{\overline{Y} ,q} such that z = σz̃. Then x = z = 0 and x = y = 0 define germs of algebraic curves D_{1} and D_{2} at q. We also assume ν(q) = r.

Given a form (118) at q, suppose that D ⊂ \overline{S}_{r}(\overline{Y}) is a curve such that q ∈ D. Since (by assumption) \overline{S}_{r}(\overline{Y}) makes SNCs with \overline{B}_{2}(\overline{Y}), either x or y ∈ Î_{D,q}, and by Lemma 6.27, there exist d_{i} ∈ k such that

F_{q} − \frac{1}{x^{c}y^{d}} ∑ d_{i}(x^{a}y^{b})^{i} ∈ Î_{D,q}^{r} + (x)^{r−1}

or

F_{q} − \frac{1}{x^{c}y^{d}} ∑ d_{i}(x^{a}y^{b})^{i} ∈ Î_{D,q}^{r} + (y)^{r−1}.

\frac{∂^{r−1}F_{q}}{∂z^{r−1}} ∈ Î_{D,q}

implies z ∈ Î_{D,q} so that either x = z = 0 or y = z = 0 are local equations of D at q. (111) is modified by changing F_{q} to

F_{q} = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x, y)x^{α_{j}}y^{β_{j}}z^{r−j} + ϵx^{α_{r}}y^{β_{r}} + h  (119)

with h ∈ (x, z)^{t}. By assumption, there exists z̃ ∈ O_{\overline{Y} ,q} and a unit σ ∈ Ô_{\overline{Y} ,q} such that z = σz̃. Then x = z = 0 defines a germ of an algebraic curve D at q. We also assume ν(q) = r.

Given a form (119) at q, suppose that D ⊂ \overline{S}_{r}(\overline{Y}) is a curve such that q ∈ D. By assumption, \overline{S}_{r}(\overline{Y}) makes SNCs with \overline{B}_{2}(\overline{Y}). As in the analysis of the case when (117) holds, we conclude that x = z = 0 are local equations of D at q.

Suppose that D ⊂ \overline{S}_{r}(\overline{Y}) is a curve such that D is r big. Let π′ : Y′ → \overline{Y} be the blowup of D. By assumption, π′ is a permissible monodial transform. \overline{S}_{r}(Y′) makes SNCs with \overline{B}_{2}(Y′) by Lemma 8.8.

First suppose that q ∈ \overline{π}^{−1}(p) ∩ D, and that u, v have the form of (117). Then x = z = 0 are local equations of D at q. F_{q} ∈ (x, z)^{r} implies α_{r} ≥ r if ϵ = 1. Suppose that q′ ∈ (π′)^{−1}(q). First suppose that q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}(z_{1} + α) for some α ≠ 0. Substituting into F_{q}, we get ν(F_{q′}(0, 0, z_{1})) ≤ r−1. Suppose that q′ has regular parameters (x_{1}, y, z_{1}) where x = x_{1}z_{1}, z = z_{1}. Then F_{q′} is a unit. The remaining case is when q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}z_{1}. Then

\begin{aligned}u & = x_{1}^{a} \\ F_{q′} & = τz_{1}^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y)x_{1}^{α′_{j}}z_{1}^{r−j} + ϵx_{1}^{α′_{r}}y + h_{1} \end{aligned}  (120)

where α′_{j} = α_{j} − j for 2 ≤ j ≤ r and h_{1} ∈ (x_{1}, z_{1})^{t−r}. We either have a reduction in multiplicity ν(q′) < r, or ν(q′) = r and we are back in the form of (117) with a reduction in the α_{j} by j, and a decrease of t by r. By Lemma 8.8, \overline{S}_{r}(Y′) ∪ \overline{B}_{2}(Y′) makes SNCs in a neighborhood of (π′)^{−1}(q). Since S_{r}(Y′) is closed in the open set of 1 points of E_{Y′}, and r ≥ 2, by Lemma 7.7, \overline{S}_{r}(Y′) ∩ (π′)^{−1}(q) = ∅ or it is the point q′ of (120), if ν(q′) = r. Suppose that ν(q′) = r in (120). Since there exists a unit series σ such that σz ∈ O_{\overline{Y} ,q}, there exists a unit series σ′ such that σ′z_{1} ∈ O_{Y′,q′}.

Now suppose that q ∈ \overline{π}^{−1}(p) ∩ D, and that u, v have the form of (119). Then we have that x = z = 0 are local equations of D at q. Since F_{q} ∈ Î_{D,q}^{r}, we have α_{j} ≥ j for all j.

Suppose that q′ ∈ (π′)^{−1}(q). First suppose that q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}(z_{1} + α) for some α ≠ 0. substituting into F_{q}, we get ν(F_{q′}(0, 0, z_{1})) ≤ r − 1. Suppose that q′ has regular parameters (x_{1}, y, z_{1}) where x = x_{1}z_{1}, z = z_{1}. Then F_{q′} is a unit. The remaining case is when q′ has regular parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}z_{1}. Then

\begin{aligned}u & = x_{1}^{a} \\ F_{q′} & = τz_{1}^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y)x_{1}^{α′_{j}}y^{β_{j}}z_{1}^{r−j} + ϵx_{1}^{α′_{r}}y^{β_{r}} + h_{1} \end{aligned}  (121)

where α′_{j} = α_{j} − j for 2 ≤ j ≤ r and h_{1} ∈ (x_{1}, z_{1})^{t−r}. We either have a reduction in multiplicity ν(q′) < r, or we are back in the form of (119) with a reduction in α_{i} by i, and a decrease of t by r. As in the analysis of (117), we either have \overline{S}_{r}(Y′)∩(π′)^{−1}(q) = ∅, or \overline{S}_{r}(Y′) ∩ (π′)^{−1}(q) is the single point q′ of (121). In this case there exists a unit series σ′ such that σ′z_{1} ∈ O_{Y′,q′}.

Now suppose that q ∈ \overline{π}^{−1}(p) ∩ D, and that u, v have the form of (118). Then either x = z = 0 or y = z = 0 are local equations of D at q. We may suppose that x = z = 0 are local equations of D at q, so that F_{q} ∈ (x, z)^{r}.

Suppose that q′ ∈ (π′)^{−1}(q). First suppose that q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}(z_{1} + α) for some α ≠ 0. substituting into F_{q}, we get u = (x_{1}^{a}y^{b})^{m} and ν(F_{q′}(0, 0, z_{1})) ≤ r − 1. Suppose that q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}z_{1}, z = z_{1}. Then F_{q′} is a unit. The remaining case is when q′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}z_{1}. Then

\begin{aligned}u & = (x_{1}^{a}y^{b})^{m} \\ v & = P (x_{1}^{a}y^{b}) + x_{1}^{c+r}y^{d}F_{q′} \\ F_{q′} & = τz_{1}^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y)x_{1}^{α′_{j}}y^{β_{j}}z_{1}^{r−j} + ϵx_{1}^{α′_{r}}y^{β_{r}} + h_{1} \end{aligned}  (122)

where α′_{j} = α_{j} − j for 2 ≤ j ≤ r and h_{1} ∈ (x_{1}y_{1}, z_{1})^{t−r}. We either have a reduction in multiplicity ν(q′) < r, or we are back in the form of (118) with a reduction of α_{i} by i and a decrease of t by r. In this case, there exists a unit series σ′ such that σ′z_{1} ∈ O_{Y′,q′}.

Suppose that q′ ∈ \overline{S}_{r}(Y′) ∩ (π′)^{−1}(q) is a 2 point with ν(q′) = r − 1, and q′ lies on a curve E in \overline{S}_{r}(Y′′). E is transversal to the 2 curve at q′ by Lemma 8.8. By Lemma 6.27, there exist b_{t} ∈ k such that

F_{q′} + \frac{1}{x_{1}^{c+r}y^{d}} ∑ b_{t}(x_{1}^{a}y^{b})^{t} ∈ Î_{E,q′}^{r} + (x_{1})^{r−1},

if x_{1} ∈ Î_{E,q′} or the series is in Î_{E,q′}^{r} + (y)^{r−1} if y ∈ Î_{E,q′}. Then \frac{∂^{r−2}F_{q′}}{∂z^{r−2}} ∈ Î_{E,q′}^{2}. Suppose that q′ has permissible parameters (x_{1}, y, z_{1}) such that x = x_{1}, z = x_{1}(z_{1}+α) with α ≠ 0.

F_{q′} = Λz_{1}^{r−1} + ∑_{i=1}^{r−1} ã_{i}(x_{1}, y)z_{1}^{r−1−i}

where Λ is a unit. x_{1}^{i} | ã_{i} (or y^{i} | ã_{i}) for 1 ≤ i ≤ r − 1 since F_{q′} ∈ Î_{E,q′}^{r−1}. Then z_{1} ∈ Î_{E,q′}^{2} + (x_{1}) which is impossible.

Thus, by Lemma 7.7, the only possible point in \overline{S}_{r}(Y′) ∩ (π′)^{−1}(q) is the point q′ of (122). If there is a curve E ⊂ \overline{S}_{r}(Y′) containing q′, then we must have z_{1} ∈ Î_{E,q′} since

\frac{∂^{r−1}F_{q′}}{∂z^{r−1}} ∈ Î_{E,q′}.

Thus E has local equations x_{1} = z_{1} = 0 or y = z_{1} = 0.

After any sequence of permissible monoidal transforms, centered at r big curves C ⊂ \overline{S}_{r}, we eventually obtain π_{1} : V_{p} → \overline{Y} where there are no r big curves C in \overline{S}_{r}(V_{p}) and \overline{S}_{r}(V_{p}) makes SNCs with \overline{B}_{2}(V_{p}).

Further, if q ∈ (\overline{π} ∘ π_{1})^{−1}(p), and either q ∈ \overline{S}_{r}(V_{p}) or one of 1. - 3. of the conclusions of W_{p} fail at q then q must satisfy one of (117), (118) or (119) (with ν(q) = r or ν(q) = r − 1).

Suppose that (118) holds at q ∈ (\overline{π} ∘ π_{1})^{−1}(p), and ν(q) = r. Then we either have α_{j} ≥ j for all j or β_{j} ≥ j for all j by Remark 9.19. If α_{j} ≥ j for all j, then F_{q} ∈ (x, z)^{r}. If β_{j} ≥ j for all j, then F_{q} ∈ (y, z)^{r}. Since x = z = 0 and y = z = 0 are local equations of curves on V_{p}, in either case we have a curve D ⊂ \overline{S}_{r}(V_{p}) such that D is r big by Lemma 8.2. Thus (118) cannot hold on V_{p} with ν(q) = r.

Suppose that (119) holds at q ∈ (\overline{π} ∘ π_{1})^{−1}(p), and ν(q) = r. Then we either have α_{j} ≥ j for all j or β_{j} ≥ j for all j by Remark 9.19. By Remark 11.3, β_{j} < j for some j. Thus F_{q} ∈ (x, z)^{r}. Since x = z = 0 are local equations of a curve in V_{p}, By Lemma 8.2, we must have a curve D ⊂ \overline{S}_{r}(V_{p}) such that D is r big. Thus (119) cannot hold on V_{p}, with ν(q) = r.

The only points on (\overline{π} ∘π_{1})^{−1}(p) where the conclusions of the Theorem do not hold are at points q′ over p where one of (123) or (124) following hold.

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F_{q} \\ F_{q} & = τz^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x, y)x^{α_{j}}z^{r−j} + ϵx^{α_{r}}y + h \end{aligned}  (123)

where ν(q′) = r, some α_{j} < j, and h ∈ (x, z)^{3r}. Further, there exists a series σ such that σz ∈ O_{V_{p},q′}. The other possiblity is that q′ has permissible parameters (x_{1}, y_{1}, z_{1}) of the form of (122), with

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{b}) + x_{1}^{c}y_{1}^{d}F_{q′} \\ F_{q′} & = τz_{1}^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y_{1})x_{1}^{α′_{j}}y_{1}^{β′_{j}}z_{1}^{r−j} + ϵx_{1}^{α′_{r}}y_{1}^{β′_{r}} + h_{1} \end{aligned}  (124)

with ν(q′) = r − 1,

h_{1} ∈ (x_{1}y_{1}, z_{1})^{3r}

and there exists i such that

\frac{α′_{i}}{i} ≤ \frac{α′_{j}}{j} , \frac{β′_{i}}{i} ≤ \frac{β′_{j}}{j}

for 2 ≤ j ≤ r and

\{\frac{α′_{i}}{i} \} + \{\frac{β′_{i}}{i} \} < 1.

Further, there exists a series σ such that σz_{1} ∈ O_{V_{p},q′}.

Suppose that D ⊂ \overline{S}_{r}(V_{p}) is a curve (which is necessarily r small). Let π′ : Z_{1} → V_{p} be the blowup of D.

Suppose that q′ ∈ D ∩ (\overline{π} ∘ π_{1})^{−1}(p). q′ can only be a point of the form of (123) or (124).

Suppose that q′ satisfies (123). Then x = z = 0 are local equations of D at q′, and by Lemma 6.25, ϵ = 1, α_{r} = r − 1 and α_{j} ≥ j if j ≠ r − 1.

Suppose that q′′ ∈ (π′)^{−1}(q′). Suppose that Ô_{Z_{1},q′′} has regular parameters (x_{1}, y, z_{1}) where x = x_{1}, z = x_{1}(z_{1} + α) for some α ∈ k.

\begin{aligned}u & = x_{1}^{a} \\ F_{q′′} & = τx_{1}(z_{1} + α)^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y)x_{1}^{α_{j}−j+1}(z_{1} + α)^{r−j} + y − g(x_{1}) + h_{1} \end{aligned}

with h_{1} ∈ (x_{1}, z_{1})^{2r} for some series g(x_{1}). q′′ is resolved, since ν(F_{q′}(0, y, 0)) = 1. If q′′ has permissible parameters (x_{1}, y, z_{1}) where x = x_{1}z_{1}, z = z_{1},

\begin{aligned}u & = x_{1}^{a}z_{1}^{a} \\ F_{q′′} & = τz_{1} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}z_{1}, y)x_{1}^{α_{j}}z_{1}^{α_{j}−j+1} + x_{1}^{r−1}y + h_{1} \end{aligned}  (125)

with h_{1} ∈ (x_{1}, z_{1})^{2r}. ν(q′′) ≤ r − 2 since ν(F_{q′}(0, 0, z_{1})) = 1 and r ≥ 3. By Lemma 7.6, (π′)^{−1}(q′) ∩ \overline{S}_{r}(Z_{1}) = ∅, and the conclusions of 1. - 3. of the Theorem hold on (π′)^{−1}(q′).

Suppose that q′ ∈ (\overline{π} ∘ π_{1})^{−1}(p) satisfies (124). Then either x_{1} = z_{1} = 0 or y_{1} = z_{1} = 0 are local equations of D at q′. Without loss of generality, assume that y_{1} = z_{1} = 0 are local equations of D at q′. Then we have β′_{j} ≥ j if 2 ≤ j ≤ r − 1 and ϵ = 1, β′_{r} = r − 1 by Lemma 6.27. ν(q′) = r − 1 implies α′_{r} = 0.

We have

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{b}) + x_{1}^{c}y_{1}^{d}F_{q′} where \\ F_{q′} & = τz_{1}^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}(x_{1}, y_{1})x_{1}^{α′_{j}}y_{1}^{β′_{j}}z_{1}^{r−j} + y_{1}^{r−1} + h_{1} \end{aligned}  (126)

with β′_{j} ≥ j for all j,

h_{1} ∈ (x_{1}y_{1}, z_{1})^{3r}

Suppose that q′′ ∈ (π′)^{−1}(q′), and q′′ has regular parameters (x_{1}, y_{2}, z_{2}) defined by

y_{1} = y_{2}, z_{1} = y_{2}(z_{2} + α)

Then

u  = (x_{1}^{a}y_{2}^{b})^{m}
v  = P_{q′′}(x_{1}^{a}y_{2}^{b}) + x_{1}^{c}y_{2}^{d+r−1}F_{q′′}
\frac{F_{q′}}{y_{2}^{r−1}} = τ(z_{2} + α)^{r}y_{2} + ∑_{j=2}^{r−1} \overline{a}_{j}x_{1}^{α′_{j}}y_{2}^{β′_{j}−j+1}(z_{2} + α)^{r−j} + 1 + h_{2}
= 1 + y_{2}Ω

since

h_{2} ∈ (y_{2})^{2r}.

a(d + r − 1) − bc ≠ 0 since F_{q′} is normalized. Thus F_{q′′} = 1 + y_{2}Ω′ is a unit. Suppose that q′′ ∈ (π′)^{−1}(q′), and q′′ has regular parameters (x_{1}, y_{2}, z_{2}) defined by

y_{1} = y_{2}z_{2}, z_{1} = z_{2}

Then

\begin{aligned}F_{q′′} = \frac{F_{q′}}{z_{2}^{r−1}} & = τz_{2} + ∑_{j=2}^{r−1} \overline{a}_{j}x_{1}^{α′_{j}}y_{2}^{β′_{j}}z_{2}^{β′_{j}−j+1} + y_{2}^{r−1} + h_{2} \\ & = τz_{2} + +y_{2} + z_{2}^{2}Ω′ \end{aligned}  (127)

since

h_{2} ∈ (z_{2})^{2r}

Thus ν(q′′) = 1. ν(q′′) ≤ r − 2 since r ≥ 3. Thus (π′)^{−1}(q′) ∩ \overline{S}_{r}(Z_{1}) = ∅ by Lemma 7.7, and the conclusions 1. - 3. of the Theorem hold on (π′)^{−1}(q′).

We thus construct a permissible sequence of monodial transforms π_{2} : Z_{p} → V_{p} centered at the strict transforms of curves C ⊂ \overline{S}_{r}(V_{p}) which are r small so that \overline{S}_{r}(Z_{p}) contains no curves. Z_{p} → \overline{Y}_{p} extends to \overline{U}_{2} → U in the notation of the Theorem.

Suppose that q′ ∈ (\overline{π}∘π_{1}∘π_{2})^{−1}(p) does not satisfy the conclusions of the Theorem. Then q must either satisfy (123) or (124).

We cannot have that (123) holds at q′, since then ν(q′) = r, which implies that α_{j} ≥ j for j ≥ 2 and α_{r−1} ≥ r − 1, so that x = z = 0 are local equations of a curve D in \overline{S}_{r}(\overline{U}_{2}). We further see that \overline{S}_{r}(\overline{U}_{2}) = ∅.

Suppose that (124) holds at q′. Then ν(q′) = r − 1 and 2. of the conclusions of the Theorem does not hold, so that τ(q′) = 0. Thus α′_{j} + β′_{j} ≥ j for j ≠ r and ϵ = 1, α′_{r} + β′_{r} = r − 1.

First suppose that (124) holds, with τ(q′) = 0, α′_{r} and β′_{r} ≠ 0. Let π′′ : W_{1} → Z_{p} be the quadratic transform with center q′. Suppose that q′′ ∈ (π′′)^{−1}(q′) and Ô_{W_{1},q′′} has regular parameters (x_{2}, y_{2}, z_{2}) such that

x_{1} = x_{2}, y_{1} = x_{2}(y_{2} + α), z_{1} = x_{2}(z_{2} + β).

with α ≠ 0. Set x_{2} = \overline{x}_{2}(y_{2} + α)^{\frac{−b}{a+b}}. there exists

h_{2} ∈ (x_{2})^{2r}

such that

\begin{aligned}\frac{F_{q′}}{x_{2}^{r−1}} & = τx_{2}(z_{2} + β)^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}x_{2}^{α′_{j}+β′_{j}−j+1}(y_{2} + α)^{β′_{j}}(z_{2} + β)^{r−j} + (y_{2} + α)^{β′_{r}} + h_{2} \\ & = (y_{2} + α)^{β′_{r}} + \overline{x}_{2}Ω \end{aligned}

Thus

\begin{aligned}u & = (\overline{x}_{2}^{a+b})^{m} \\ v & = P_{q′′}(\overline{x}_{2}) + \overline{x}_{2}^{c+d+r−1}F_{q′′} \\ F_{q′′} & = (y_{2} + α)^{λ+β′_{r}} − α^{λ+β′_{r}} + \overline{x}_{2}Ω′ \end{aligned}

where λ = d − \frac{b(c+d+r−1)}{a+b}. Since F_{q′} is normalized,

\begin{aligned}(a + b)(β′_{r} + d) − b(c + d + r − 1) & = a(d + β′_{r}) − b(c + r − 1 − β′_{r}) \\ & = a(d + β′_{r}) − b(c + α′_{r}) ≠ 0 \end{aligned}

Thus λ + β′_{r} ≠ 0. q′′ is thus a resolved point.

Suppose that q′′ ∈ (π′′)^{−1}(q′) has permissible parameters (x_{2}, y_{2}, z_{2}) such that

x_{1} = x_{2}, y_{1} = x_{2}y_{2}, z_{1} = x_{2}(z_{2} + β).

There exists

h_{2} ∈ (x_{2})^{2r}

such that

\begin{aligned}\frac{F_{q′}}{x_{2}^{r−1}} & = τx_{2}(z_{2} + β)^{r} + ∑_{j=2}^{r−1} \overline{a}_{j}x_{2}^{α′_{j}+β′_{j}−j+1}y_{2}^{β′_{j}}(z_{2} + β)^{r−j} + y_{2}^{β′_{r}} + h_{2} \\ & = y_{2}^{β′_{r}} + x_{2}Ω \end{aligned}

b(c + d + r − 1) − (a + b)(d + β′_{r}) ≠ 0

since F_{q′} is normalized.

Thus

\begin{aligned}u & = (x_{2}^{a+b}y_{2}^{b})^{m} \\ v & = P_{q′′}(x_{2}^{a+b}y_{2}^{b}) + x_{2}^{c+d+r−1}y_{2}^{d}F_{q′′} \\ F_{q′′} & = y_{2}^{β′_{r}} + x_{2}Ω′ \end{aligned}

ν(F_{q′′}) ≤ β′_{r} < r − 1.

Suppose that q′′ ∈ (π′′)^{−1}(q′) has regular parameters (x_{2}, y_{2}, z_{2}) such that

x_{1} = x_{2}y_{2}, y_{1} = y_{2}, z_{1} = y_{2}(z_{2} + β).

Then

\begin{aligned}u & = (x_{2}^{a}y_{2}^{a+b})^{m} \\ v & = P_{q′′}(x_{2}^{a}y_{2}^{a+b}) + x_{2}^{c}y_{2}^{c+d+r−1}F_{q′′} \\ F_{q′′} & = x_{2}^{α′_{r}} + y_{2}Ω′ \end{aligned}

since F_{q′} is normalized. ν(F_{q′′}) ≤ α′_{r} < r − 1.

The remaining point in q′′ ∈ (π′′)^{−1}(q′) has regular parameters (x_{2}, y_{2}, z_{2}) such that

x_{1} = x_{2}z_{2}, y_{1} = y_{2}z_{2}, z_{1} = z_{2}.

There exists

h_{2} ∈ (z_{2})^{2r}

such that

\begin{aligned}F_{q′′} & = \frac{F_{q′}}{z_{1}^{r−1}} = τz_{2} + ∑_{j=2}^{r−1} \overline{a}_{j}x_{2}^{α′_{j}}y_{2}^{β′_{j}}z_{2}^{α′_{j}+β′_{j}−j+1} + x_{1}^{α′_{r}}y_{2}^{β′_{r}} + h_{2} \\ & ≡ τz_{2} mod (x_{2}, y_{2}, z_{2}^{2}) \end{aligned}  (128)

Thus q′′ is a 3 point with ν(F_{q′′}) = 1 ≤ r − 2, since r ≥ 3.

ν(q′′) < r − 1 for q′′ ∈ (π′′)^{−1}(q′), so that (π′′)^{−1}(q′) ∩ \overline{S}_{r}(Y′′) = ∅, and the conclusions of 1.-3. of Theorem 11.5 hold on (π′′)^{−1}(q′).

Now suppose that (124) holds, with τ(q′) = 0 and α′_{r} = 0 or β′_{r} = 0. Since the 2 cases are symmetric, we may assume that α′_{r} = 0.

We thus have β′_{r} = r − 1 (and α′_{j} + β′_{j} ≥ j for j ≠ m). Suppose that i ≠ r.

Then \frac{α′_{i}}{i} ≤ \frac{α′_{r}}{r} implies α′_{i} = 0 and \frac{β′_{i}}{i} ≤ \frac{r−1}{r} < 1 implies β′_{i} < i, so that τ(q′) > 0, a contradiction. We thus have i = r.

\frac{r − 1}{r} = \frac{β′_{r}}{r} ≤ \frac{β′_{j}}{j}

for all j implies β′_{j} ≥ j − \frac{j}{r} for 2 ≤ j < r. Since β′_{j} ∈ N, we have β′_{j} ≥ j, and the curve D with local equations y_{1} = z_{1} = 0 at q′ is such that D ⊂ \overline{S}_{r}(Z_{p}), a contradiction.

Theorem 11.6. Suppose that r = 2 and A_{2}(X) holds. Suppose that p ∈ X is a 1 point or a 2 point with ν(p) = γ(p) = 2. Let R = O_{X,p}. Suppose that π : Y_{p} → Spec(R̂) is the sequence of monoidal transforms of sections over the curve \overline{C} with local equations x̃ = y = 0 of Theorem 11.2. For q ∈ π^{−1}(p), define

l_{q} = \{ \begin{matrix}([\frac{α_{2}}{2} ] + 3)2 & if F_{q} is a form (109) or (111). \\ ([\frac{α_{2}}{2} ] + [\frac{β_{2}}{2} ] + 3)2 & if F_{q} is a form (110). \end{matrix}

let l = max{l_{q} | q ∈ π^{−1}(p)}.

Suppose that t ≥ l. Let \overline{π} : \overline{Y}_{p} → spec(R) be the sequence of monodial transforms of Theorem 11.4. Let

⋯ → Y_{n} → ⋯ → Y_{1} → \overline{Y}_{p}

be a sequence of permissible monoidal transforms centered at curves C in \overline{S}_{2} such that C is 2 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} \overline{Y}_{p} → spec(R)

extends to a permissible sequence of monoidal transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p, in the notation of Theorem 11.4, such that \overline{S}_{2}(\overline{U}_{1}) contains no curves C such that C is 2 big. Let

⋯ → Z_{n} → ⋯ → V_{p}

be a permissible sequence of monoidal transforms centered at curves C in \overline{S}_{2} such that C is 2 small. Then there exists n_{1} < ∞ such that π_{2} : Z_{p} = Z_{n_{1}} → V_{p} extends to a permissible sequence of monoidal transforms

\overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{2}(\overline{U}_{2}) = ∅.

Finally, there exists a sequence of quadratic transforms and monodial transforms centered at strict transforms of 2 curves C on Z_{p} such that C is 1 big and C is a section over a 2 small curve blown up in Z_{p} → V_{p}, π_{3} : W_{p} → Z_{p} which extends to a permissible sequence of monoidal transforms

\overline{U}_{3} → \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{2}(\overline{U}_{3}) = ∅, and if

q ∈ (\overline{π} ∘ π_{1} ∘ π_{2} ∘ π_{3})^{−1}(p)

then q is resolved.

Proof. The analysis of Theorem 11.5 is valid for r = 2, except in (125), (127) and (128).

The situation of (128) cannot occur when r = 2, since this comes from the case when (124) holds, with τ(q′) = 0, α′_{r} and β′_{r} ≠ 0. Since α′_{r} + β′_{r} = r − 1 = 1, this case cannot occur.

Suppose that a case (125) occurs in

(\overline{π} ∘ π_{1} ∘ π_{2}) : Z_{p} → spec(R).

Then we have a 2 point q′′ ∈ (\overline{π} ∘ π_{1} ∘ π_{2})^{−1}(p) such that

\begin{aligned}u & = (x_{1}^{a}z_{1}^{b})^{m} \\ v & = P_{q′′}(x_{1}^{a}z_{1}^{b}) + x_{1}^{c}z_{1}^{d}F_{q′′} \\ F_{q′′} & = z_{1} + x_{1}y_{1} + h_{1} \end{aligned}

with h_{1} ∈ (x_{1}, z_{1})^{4}.

Let C be the 2 curve on Z_{p} with local equations x_{1} = z_{1} = 0. C is 1 big by Lemma 8.1.

Let π′ : W_{1} → Z_{p} be the blowup of C. Suppose that \overline{q} ∈ (π′)^{−1}(q′′) is a 1 point. Then there exist regular parameters (x_{2}, y_{1}, z_{2}) in Ô_{W_{1},\overline{q}} such that

x_{1} = x_{2}, z_{1} = x_{2}(z_{2} + α)

with α ≠ 0. Set

x_{2} = \overline{x}_{2}(z_{1} + α)^{− \frac{b}{a+b}} .

\begin{aligned}u & = \overline{x}_{2}^{(a+b)m} \\ v & = P_{q′′}(\overline{x}_{2}^{a+b}) + \overline{x}_{2}^{c+d+1}(z_{2} + α)^{d− \frac{b(c+d+1)}{a+b}} (z_{2} + α + y_{1} + x_{1}Ω) \end{aligned}

Thus ν(F\overline{q}(0, y_{1}, z_{2})) = 1 and \overline{q} is resolved.

Suppose that \overline{q} ∈ (π′)^{−1}(q′′) is the 2 point with permissible parameters

x_{1} = x_{2}, z_{1} = x_{2}z_{2}.

Then F\overline{q} = \frac{F_{q′′}}{x_{2}} = z_{2} + y_{1} + x_{1}Ω and \overline{q} is resolved.

If \overline{q} ∈ (π′)^{−1}(q′′) is the 2 point with permissible parameters

x_{1} = x_{2}z_{2}, z_{1} = z_{2}

then

F\overline{q} = \frac{F_{q′′}}{z_{2}} = 1 + x_{2}y_{1} + z_{2}Ω

and is resolved.

Suppose that a case (127) occurs in

(\overline{π} ∘ π_{1} ∘ π_{2}) : Z_{p} → spec(R).

Then we have a 3 point q′′ ∈ (\overline{π} ∘ π_{1} ∘ π_{2})^{−1}(p) such that

\begin{aligned}u & = (x_{2}^{a}y_{2}^{b}z_{2}^{c})^{m} \\ v & = P_{q′′}(x_{2}^{a}y_{2}^{b}z_{2}^{c}) + x_{2}^{d}y_{2}^{e}z_{2}^{f}F_{q′′} \\ F_{q′′} & = z_{2} + y_{2} + h_{1} \end{aligned}

with h_{1} ∈ (z_{2})^{4}.

Let C be the 2 curve on Z_{p} with local equations y_{2} = z_{2} = 0. C is 1 big by Lemma 8.1. Let π′ : W_{1} → Z_{p} be the blowup of C. Suppose that \overline{q} ∈ (π′)^{−1}(q′′) is a 2 point. Then there exist regular parameters (x_{2}, y_{3}, z_{3}) in Ô_{W_{1},\overline{q}} such that

y_{2} = y_{3}, z_{2} = y_{3}(z_{3} + α)

with α ≠ 0. Set y_{3} = \overline{y}_{3}(z_{3} + α)^{− \frac{c}{b+c}}.

\begin{aligned}u & = (x_{2}^{a}\overline{y}_{3}^{b+c})^{m} = (x\overline{a} ; 2\overline{y}\overline{b} ; 3)\overline{m} \\ v & = P_{q′′}(x_{2}^{a}\overline{y}_{3}^{b+c}) + x_{2}^{d}\overline{y}_{3}^{e+f+1}(z_{3} + α)^{f− \frac{c(e+f+1)}{b+c}} (z_{3} + α + 1 + y_{3}^{3}Ω) \end{aligned}

with (\overline{a}, \overline{b}) = 1. Thus ν(F\overline{q}(0, 0, z_{3})) = 1 and \overline{q} is resolved.

Suppose that \overline{q} ∈ (π′)^{−1}(q′′) is the 3 point with permissible parameters y_{2} = y_{3}, z_{2} = y_{3}z_{3}. Then

F\overline{q} = \frac{F_{q′′}}{y_{3}} = z_{3} + 1 + y_{3}^{3}Ω

and \overline{q} is resolved.

Suppose that \overline{q} ∈ (π′)^{−1}(q′′) is the 3 point with permissible parameters y_{2} = y_{3}z_{3}, z_{2} = z_{3}. Then

F\overline{q} = \frac{F_{q′′}}{z_{3}} = 1 + y_{3} + z_{3}^{3}Ω

and \overline{q} is resolved.

Theorem 11.7. Suppose that r ≥ 3 and A_{r}(X) holds. Suppose that p ∈ X is a 2 point such that ν(p) = r − 1, τ(p) = 0 and γ(p) = r. Let \overline{C} be the 2 curve containing p. Let R = O_{X,p}.

There exists a sequence of permissible monoidal transforms centered at sections over \overline{C}, \overline{Y}_{p} → spec(R), which extends to a sequence of permissible monoidal transforms \overline{U} → U where U is an affine neighborhood of p, with the following property.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → \overline{Y}_{p}

be a sequence of permissible monoidal transforms centered at curves C in \overline{S}_{r} such that C is r big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} \overline{Y} → spec(R)

extends to a permissible sequence of monoidal transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p, in the notation of Theorem 11.4, such that \overline{S}_{r}(\overline{U}_{1}) contains no curves C such that C is r big. Let

⋯ → Z_{n} → ⋯ → V_{p}

be a permissible sequence of monoidal transforms centered at curves C in \overline{S}_{r} such that C is r small. Then there exists n_{1} < ∞ such that π_{2} : Z_{p} = Z_{n_{0}} → V_{p} extends to a permissible sequence of monoidal transforms

\overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{r}(\overline{U}_{2}) = ∅.

Finally, there exists a sequence of quadratic transforms π_{3} : W_{p} → Z_{p} which extends to a permissible sequence of monoidal transforms

\overline{U}_{3} → \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{r}(\overline{U}_{3}) = ∅, and if

q ∈ (\overline{π} ∘ π_{1} ∘ π_{2} ∘ π_{3})^{−1}(p),

1. ν(q) ≤ r − 1 if q is a 1 or 2 point.

2. If q is a 2 point and ν(q) = r − 1, then τ(q) > 0.

3. ν(q) ≤ r − 2 if q is a 3 point.

Theorem 11.8. Suppose that r = 2 and A_{2}(X) holds Suppose that p ∈ X is a 2 point such that ν(p) = 1, τ(p) = 0 and γ(p) = 2. Let R = O_{X,p}.

There exists a sequence of permissible monoidal transforms centered at sections over \overline{C}, \overline{Y}_{p} → spec(R) which extends to a sequence of permissible monoidal transforms \overline{U} → U, where U is an affine neighborhood of p, with the following property.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → \overline{Y}_{p}

be a sequence of permissible monoidal transforms centered at curves C in \overline{S}_{2} such that C is 2 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} \overline{Y}_{p} → spec(R)

extends to a permissible sequence of monoidal transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p, in the notation of Theorem 11.4, such that \overline{S}_{2}(\overline{U}_{1}) contains no curves C such that C is 2 big. Let

⋯ → Z_{n} → ⋯ → V_{p}

be a permissible sequence of monoidal transforms centered at curves C in \overline{S}_{2} such that C is 2 small. Then there exists n_{1} < ∞ such that π_{2} : Z_{p} = Z_{n_{1}} → Y_{n_{0}} extends to a permissible sequence of monoidal transforms

\overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{2}(\overline{U}_{2}) = ∅.

Finally, there exists a sequence of quadratic transforms and monodial transforms centered at strict transforms of 2 curves C on Z_{p} such that C is 1 big and C is a section over a 2 small curve blown up in Z_{p} → V_{p}, π_{3} : W_{p} → V_{p} which extends to a permissible sequence of monoidal transforms

\overline{U}_{3} → \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p such that \overline{S}_{2}(\overline{U}_{3}) = ∅, and if

q ∈ (\overline{π} ∘ π_{1} ∘ π_{2} ∘ π_{3})^{−1}(p)

then q is resolved.

Proof. (of Theorems 11.7 and 11.8) The conclusions of Lemma 8.5 hold at p.

The conclusions of Lemma 11.1 must be modified to: α_{j} +β_{j} ≥ j for 2 ≤ j ≤ r −2, α_{r} + β_{r} ≥ r − 1.

In the conclusions of Theorem 11.2, we must add a fourth case:

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \\ F_{q} & = τz^{r} + ∑_{i=2}^{r−1} \overline{a}_{i}(x, y)x^{α_{i}}y^{β_{i}}z^{r−i} + y^{r−1} \end{aligned}  (129)

with β_{i} ≥ i for all i.

Theorem 11.4 must be modified by adding the case F_{q} equivalent mod (\overline{x}\overline{y}, z)^{t} to a form (129). The proof of Theorem 11.5 must be modified by adding an analysis of (129). Such q are not effected by blowing up r big curves, so the construction of π_{1} : V_{p} → \overline{Y}_{p} is as in the proof of Theorem 11.5. Suppose that q ∈ (\overline{π} ∘ π_{1})^{−1}(p) satisfies (129). There is a unique r small curve D ⊂ \overline{S}_{r}(V_{p}) containing q, which has local equations y = z = 0. Let π′ : Z_{1} → \overline{Y}_{p} be the blowup of D. Then if r ≥ 3, all points of (π′)^{−1}(q) satisfy 1. - 3. of the conclusions of the Theorem, and \overline{S}_{r}(Z_{1}) ∩ (π′)^{−1}(q) = ∅.

If r = 2, and q′ ∈ (π′)^{−1}(q) is the 3 point, then there exist permissible parameters (x, y_{1}, z_{1}) at q′ such that

F_{q′} = z_{1} + y_{1} + z_{1}^{2}Ω.

If C is the 2 curve with local equations y_{1} = z_{1} = 0, then C is 1 big, and if π′ : W_{1} → Z_{p} is the blowup of C, then all points of (π′)^{−1}(q′) are resolved.

12. reduction of ν in a second special case

Throughout this section, we will assume that Φ_{X} : X → S is weakly prepared.

Theorem 12.1. (Theorem27) Suppose that r ≥ 2, A_{r}(X) holds, p ∈ X is a 2 point with ν(p) = r−1, C is a generic curve through p, and there are permissible parameters (x, y, z) at p for (u, v) (with y, z ∈ O_{X,p}) such that L_{p}(x, 0, 0) ≠ 0, and C has local equations y = z = 0 at p. Let R = O_{X,p}. We have an expression at p

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = τx^{r−1} + ∑_{i=1}^{r−1} ã_{i}(y, z)x^{r−i−1} \end{aligned}  (130)

where τ is a unit and ν(ã_{i}) ≥ i for all i. Then there exists a finite sequence of permissible monodial transforms π : Y → Spec(R) centered at sections over C, such that for q ∈ π^{−1}(p), there exist permissible parameters (\overline{x}, \overline{y}, \overline{z}) at q such that F_{q} has one of the following forms.

1.

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}F_{q} with \\ F_{q} & = τ\overline{x}^{r−1} + ∑_{j=1}^{r−1} \overline{y}^{d_{j}}Λ_{j}(\overline{y}, \overline{z})\overline{x}^{r−1−j} \end{aligned}  (131)

where τ is a unit, Λ_{j}(\overline{y}, \overline{z}) = 0 or e_{j} = ν(Λ_{j}(0, \overline{z})) = 0 or 1 and d_{j} + e_{j} ≥ j for all j.

2.

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b}\overline{z}^{c})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}\overline{z}^{c}) + \overline{x}^{d}\overline{y}^{e}\overline{z}^{f}F_{q} with \\ F_{q} & = τ\overline{x}^{r−1} + ∑_{j=1}^{r−1} \overline{a}_{j}(\overline{y}, \overline{z})\overline{y}^{d_{j}} \overline{z}^{e_{j}} \overline{x}^{r−1−j} \end{aligned}  (132)

where τ is a unit, d_{j} + e_{j} ≥ j, \overline{a}_{j} are units (or zero) for all j and there exists

an i such that 1 ≤ i ≤ r − 1, \overline{a}_{i} ≠ 0 and

\frac{d_{i}}{i} ≤ \frac{d_{j}}{j} , \frac{e_{i}}{i} ≤ \frac{e_{j}}{j}

for 1 ≤ j ≤ r − 1. We further have

\{\frac{d_{i}}{i} \} + \{\frac{e_{i}}{i} \} < 1.

3.

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}F_{q} with \\ F_{q} & = τ\overline{x}^{r−1} + ∑_{j=1}^{r−1} \overline{a}_{j}(\overline{y}, \overline{z})\overline{y}^{d_{j}} \overline{z}^{e_{j}} \overline{x}^{r−1−j} \end{aligned}  (133)

where τ is a unit, d_{j} + e_{j} ≥ j, \overline{a}_{j} are units (or zero) for all j and there exists an i such that 1 ≤ i ≤ r − 1, \overline{a}_{i} ≠ 0 and

\frac{d_{i}}{i} ≤ \frac{d_{j}}{j} , \frac{e_{i}}{i} ≤ \frac{e_{j}}{j}

for 1 ≤ j ≤ r − 1. We further have

\{\frac{d_{i}}{i} \} + \{\frac{e_{i}}{i} \} < 1.

In all these cases \overline{x} = x and \overline{x} = 0 is a local equation at q of the strict transform of the component of E_{X} with local equation x = 0 at p.

There exists an affine neighborhood U of p such that Y → spec(R) extends to a sequence of permissible monoidal transforms \overline{U} → U such that A_{r}(\overline{U}) holds.

Suppose that r ≥ 3. Let D_{i} be the curves in \overline{S}_{r−1}(X) which contain p, and such that x ∈ Î_{D_{i},p}. We further have that the strict transforms \overline{D}_{i} of the D_{i} on \overline{U} are nonsingular, disjoint and make SNCs with \overline{B}_{2}(\overline{U}).

Proof. Set S_{0} = Spec(k[y, z]), Y_{0} = Spec(R). Consider the sequence of monoidal transforms centered at sections over C

⋯ → Y_{n} → Y_{n−1} → Y_{n−2} → ⋯ → Y_{1} → Y_{0}  (134)

where the sequence is obtained from a sequence of quadratic transforms

⋯ → S_{n} → S_{n−1} → ⋯ → S_{1} → S_{0}  (135)

over the closed point p_{0} with local equations y = z = 0 in S_{0}, and (134) is obtained from (135) by base change with Y_{0} → S_{0}, so that Y_{i} ≅ S_{i} ×_{S_{0}} Y_{0}.

The map Ŝ_{0} → Ŷ_{0} obtained from the natural projection

k[[x, y, z]] → k[[y, z]]

induces maps S_{i} ×_{S_{0}} Ŝ_{0} → Y_{i} ×_{Y_{0}} Ŷ_{0} such that the composed map

S_{i} ×_{S_{0}} Ŝ_{0} → Y_{i} ×_{Y_{0}} Ŷ_{0} → S_{i} ×_{S_{0}} Ŝ_{0}

is an isomorphism for all i.

We can thus identify the center of the quadratic transform S_{i+1} → S_{i} with a point p_{i} ∈ Y_{i} over p. A section over C through p_{i} is blown up in (134) only if none of the forms (131), (132) or (133) hold at p_{i}.

We will show that (134) is finite, so that there exists n such that Y_{n} satisfies the conclusions of the theorem.

Suppose that (134) is not finite. Then we may assume that there exists an infinite sequence of points p_{0}, p_{1}, p_{2}, …, p_{n}, … such that Y_{i+1} → Y_{i} is a permissible monoidal transform, centered at a section C_{i} over C, containing p_{i}, such that p_{i} maps to p_{i−1} for all i, and F_{p_{i}} does not satisfy (131), (132) or (133) for any i.

Each point p_{i} has permissible parameters (x, y_{i}, z_{i}) for (u, v), such that one of the following cases hold.

Case 1 p_{i} is a 2 point

u = (x^{a_{i}}y_{i}^{b_{i}})^{m_{i}}, v = P_{i}(x^{a_{i}}y_{i}^{b_{i}}) + x^{c}y_{i}^{d_{i}}F_{i}

with a_{i}m_{i} = am, and permissible parameters at p_{i+1} are as in one of the following cases. Case 1a

y_{i} = y_{i+1}, z_{i} = y_{i+1}(z_{i+1} + α_{i+1})

Case 1b

y_{i} = y_{i+1}z_{i+1}, z_{i} = z_{i+1}

Case 2 p_{i} is a 3 point

\begin{aligned}u & = (x^{a_{i}}ω_{i}^{k_{i}})^{m_{i}}, ω_{i} = y_{i}^{\overline{b}_{i}}z_{i}^{\overline{c}_{i}} \\ v & = P_{i}(x^{a_{i}}ω_{i}^{k_{i}}) + x^{c}ω_{i}^{d_{i}}F_{i} \end{aligned}

with a_{i}m_{i} = am, (\overline{b}_{i}, \overline{c}_{i}) = 1, (a_{i}, k_{i}) = 1, and permissible parameters at p_{i+1} are as in one of the following cases. Case 2a

y_{i} = y_{i+1}, z_{i} = y_{i+1}z_{i+1}

Case 2b

y_{i} = y_{i+1}z_{i+1}, z_{i} = z_{i+1}

Case 2c

y_{i} = y_{i+1}(z_{i+1} + α_{i+1})^{− \frac{\overline{c}_{i}}{\overline{b}_{i}+\overline{c}_{i}}} , z_{i} = y_{i+1}(z_{i+1} + α_{i+1})^{\frac{\overline{b}_{i}}{\overline{b}_{i}+\overline{c}_{i}}}

with α_{i+1} ≠ 0. In Case 2c, y_{i+1}, z_{i+1} are constructed from the monoidal transform

y_{i} = \overline{y}_{i+1}, z_{i} = \overline{y}_{i+1}(\overline{z}_{i+1} + α_{i+1}).

Then define

\overline{y}_{i+1} = y_{i+1}(z_{i+1} + α_{i+1})^{− \frac{\overline{c}_{i}}{\overline{b}_{i}+\overline{c}_{i}}} , \overline{z}_{i+1} = z_{i+1}.

If p_{i} is a 2 point, then y is a power of y_{i}, and if q_{i} is a 3 point, then y is a monomial in y_{i} and z_{i}. If p_{i} is a 2 point, then there is a series g_{i} such that

F_{i} = F_{p} − \frac{g_{i}(x^{a_{i}}y_{i}^{b_{i}})}{x^{c}y_{i}^{d_{i}}}.

If p_{i} is a 3 point, then there is a series g_{i} such that

F_{i} = F_{p} − \frac{g_{i}(x^{a_{i}}ω_{i}^{k_{i}})}{x^{c}ω_{i}^{d_{i}}}.

In either case, we have an expression

F_{i} = F_{p_{i}} = τ′x^{r−1} + ∑_{j=1}^{r−1} a′_{j}(y_{j}, z_{j})x^{r−j−1}.

We will show that τ′ is a unit. Suppose not. First suppose that p_{i} is a 2 point. Then a_{i}d_{i} − b_{i}(c + r − 1) = 0.

(x^{a}y^{b})^{m} = (x^{a_{i}}y^{b_{i}})^{m_{i}}

implies y = yi^{\frac{b_{i}m_{i}}{bm}}. x^{c+r−1}y^{d} = x^{c+r−1}y_{i}^{d_{i}} implies

d_{i} = \frac{b_{i}m_{i}d}{mb}.

Thus ad − b(c + r − 1) = 0, a contradiction to the assumption that F_{p} is normalized. Now suppose that p_{i} is a 3 point and τ′ is not a unit. Then a_{i}d_{i} −(c+r −1)k_{i} = 0.

(x^{a}y^{b})^{m} = (x^{a_{i}}ω_{i}^{k_{i}})^{m_{i}}

implies y = ωi^{\frac{k_{i}m_{i}}{mb}}. x^{c+r−1}y^{d} = x^{c+r−1}ω_{i}^{d_{i}} implies

d_{i} = \frac{dk_{i}m_{i}}{mb} .

Thus ad − (c + r − 1)b = 0, a contradiction to the assumption that F_{p} is normalized. If p_{i} is a 2 point,

a′_{j} = \{ \begin{matrix}ã_{j} − c̃y_{i}^{t_{j}(i)} & if a_{i}(d_{i} + t_{j}(i)) − b_{i}(c + r − j − 1) = 0 with t_{j}(i) ∈ N \\ & and c̃ is the coefficient of y_{i}^{t_{j}(i)} in the expansion of ã_{j} in terms of y_{i}, z_{i} \\ ã_{j} & if a_{i}(d_{i} + t) − b_{i}(c + r − j − 1) ≠ 0 for any t ∈ N. \end{matrix}

If p_{i} is a 3 point,

a′_{j} = \{ \begin{matrix}ã_{j} − c̃ω_{i}^{t_{j}(i)} & if a_{i}(d_{i} + t_{j}(i)) − k_{i}(c + r − j − 1) = 0 with t_{j}(i) ∈ N \\ & and c̃ is the coefficient of ω_{i}^{t_{j}(i)} in the expansion of ã_{j} in terms of y_{i}, z_{i} \\ ã_{j} & if a_{i}(d_{i} + t) − k_{i}(c + r − j − 1) ≠ 0 for any t ∈ N. \end{matrix}

In particular, there exists at most one value of t_{j}(i) such that a term can be removed from any ã_{j}.

Set \overline{u} = y^{b}, \overline{v}_{j} = y^{d}ã_{j}(y, z). We have

\begin{aligned}u & = (x^{a}\overline{u})^{m} \\ v & = P (x^{a}\overline{u}) + τx^{c+r−1}y^{d} + ∑_{j=1}^{r−1} \overline{v}_{j}x^{c+r−1−j}. \end{aligned}

By Theorem 9.4, Lemma 9.2 and Theorem 9.8, there exists i_{0} such that for i ≥ i_{0} in (135), one of the following forms holds at p_{i}, for 1 ≤ j ≤ r − 1.

If p_{i} is a 2 point,

\begin{aligned}\overline{u} & = y_{i}^{b_{i}m_{i}} \\ \overline{v}_{j} & = P_{ji}(y_{i}) + y_{i}^{d_{j}(i)}ψ_{ji}(y_{i}, z_{i})^{e_{j}(i)} \end{aligned}  (136)

where e_{j}(i) ≥ 1, ν(ψ_{ji}(0, z_{i})) = 1 or ψ_{ji} = 0 for 1 ≤ j ≤ r − 1.

If p_{i} is a 3 point,

\begin{aligned}\overline{u} & = (ω_{i}^{k_{i}})^{m_{i}} = (y_{i}^{\overline{b}_{i}}z_{i}^{\overline{c}_{i}})^{k_{i}m_{i}} \\ \overline{v}_{j} & = P_{ji}(ω_{i}) + y_{i}^{d_{j}(i)}z_{i}^{e_{j}(i)}ϕ_{ji}(y_{i}, z_{i}) \end{aligned}  (137)

where ϕ_{ji}(y_{i}, z_{i}) is a unit and d_{j}(i)\overline{c}_{i} − \overline{b}_{i}e_{j}(i) ≠ 0, or ϕ_{ji} = 0 for 1 ≤ j ≤ r − 1. For i sufficiently large, we have that \overline{u} \overline{v}_{j} = 0 are SNC divisors for 1 ≤ j ≤ r − 1 (by Lemma 9.16).

Suppose that (136) holds at p_{i} with e_{j}(i) = 1 or ϕ_{ji} = 0 for all j. If there exists t_{j}(i) ∈ N such that

a_{i}(d_{i} + t_{j}(i)) − b_{i}(c + r − j − 1) = 0,

then the normalized form of x^{c+r−1−j}\overline{v}_{j} at p_{i} is

x^{c+r−1−j}\overline{v}_{j} − c̃x^{c+r−1−j}y_{i}^{t_{j}(i)+d_{i}} = x^{c+r−1−j}y_{i}^{λ_{j}(i)}Λ_{ji}(y_{i}, z_{i})

where Λ_{ji} is a unit, zero, or ν(Λ_{ji}(0, z_{i})) = 1.

If there does not exist t ∈ N such that

a_{i}(d_{i} + t) − b_{i}(c + r − j − 1) = 0,

then

x^{c+r−1−j}\overline{v}_{j} = x^{c+r−1−j}y_{i}^{λ_{j}(i)}Λ_{ji}(y_{i}, z_{i})

where Λ_{ji} is a unit, 0 or ν(Λ_{ji}(0, z_{i})) = 1.

Thus if (136) holds at p_{i}, with e_{j}(i) = 1 or ϕ_{ji} = 0 for all j, (131) holds at p_{i}. If p_{i} is a 3 point, so that (137) holds, and p_{i+1} is a 2 point, then (136) holds at

p_{i+1}, with e_{j}(i) = 1 or ψ_{ji} = 0 for 1 ≤ j ≤ r − 1, so that (131) holds at p_{i+1}.

We are reduced to the 2 cases where either for all i ≥ i_{0} in (135) all monoidal transforms are of the forms 2a or 2b, or for all i ≥ i_{0} in (135), all monoidal transforms are of the form 1a with some e_{j}(i) > 1.

If all monoidal transforms are of the form 2a or 2b for i ≥ i_{0}, then F_{i} = F_{i_{0}} for all i ≥ i_{0}. If

F_{i_{0}} = τ_{0}x^{r−1} + ∑_{j=1}^{r−1} ã_{j}(y_{i_{0}}, z_{i_{0}})x^{r−1−j}

then for i >> i_{0}, ã_{j}(y_{i_{0}}, z_{i_{0}}) is a monomial in y_{i} and z_{i} times a unit for all j by Lemma 9.16. By Lemmas 9.17 and Corollary 9.18, there exists i_{1} > i_{0} such that (132) holds at p_{i_{1}}.

Suppose that all monoidal transforms are of the form 1a for i ≥ i_{0} and some e_{j}(i) > 1. There exists a permissible change of parameters (x, y_{i_{0}}, \overline{z}_{i_{0}}) such that

\overline{z}_{i_{0}} = z_{i_{0}} − p̃(y_{i_{0}})

for some series p̃, such that (x, y_{i}, \overline{z}_{i}) are permissible parameters at p_{i} for all i ≥ i_{0} with y_{i} = y_{i_{0}}, \overline{z}_{i_{0}} = y_{i}^{i−i_{0}} \overline{z}_{i}. Let

F = \overline{P} (y_{i}) + \overline{F}_{i}

be the normalized form of F with respect to the parameters (x, y_{i}, \overline{z}_{i}). Then \overline{F}_{i} = \overline{F}_{i_{0}} for i ≥ i_{0}. If

\overline{F}_{i_{0}} = τ_{0}x^{r−1} + ∑_{j=1}^{r−1} ã_{j}(y_{i_{0}}, \overline{z}_{i_{0}})x^{r−1−j}

then for i >> i_{0}, ã_{j}(y_{i_{0}}, \overline{z}_{i_{0}}) is a monomial in y_{i} and \overline{z}_{i} times a unit for all j by Lemma 9.16. By Lemma 9.17 and Corollary 9.18 there exists i_{1} > i_{0} such that (133) holds at p_{i_{1}}.

C generic implies F_{q} is resolved for q ∈ C a generic point. There exists an affine neighborhood U of p such that Y → spec(R) extends to a permissible sequence of monoidal transforms of sections over C, \overline{U} → U such that \overline{S}_{r}(\overline{U})∪\overline{B}_{2}(\overline{U}) is contained in the union of \overline{B}_{2}(\overline{U}) and the strict transform of \overline{S}_{r}(U). Thus A_{r}(\overline{U}) holds.

If r ≥ 3, we can choose i_{0} sufficiently large in obtaining the forms of (136) and (137) so that the strict transforms of the curves D_{i} in \overline{S}_{r−1}(U) such that x ∈ Î_{D_{i},p} are disjoint and make SNCs with \overline{B}_{2}(\overline{U}).

Theorem 12.2. Suppose that r ≥ 3, A_{r}(X) holds, p is a 2 point with ν(p) = r − 1, and L(x, 0, 0) ≠ 0, as in the assumptions of Theorem 12.1. Let R = O_{X,p}. Suppose that π : Y_{p} → Spec(R) is the morphism of Theorem 12.1.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → Y_{p}

be a sequence of permissible monodial transforms centered at 2 curves D such that D is r-1 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p (with the notation of Theorem 12.1) such that \overline{U}_{1} contains no 2 curves D such that D is r-1 big or r small, and for q ∈ \overline{U}_{1},

1. If q is a 1 or a 2 point then ν(q) ≤ r. ν(q) = r implies γ(q) = r.

2. If q is a 3 point then ν(q) ≤ r − 2.

3. \overline{S}_{r}(\overline{U}_{1}) makes SNCs with \overline{B}_{2}(\overline{U}_{1}). There exists a sequence of quadratic transforms W_{p} → V_{p} such that if Z_{p} → W_{p} is the sequence of monodial transforms (in any order) centered at the strict transforms of curves C in \overline{S}_{r}(X) then

Z_{p} → W_{p} → V_{p} → Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{π} : \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood of p such that \overline{U}_{2} contains no 2 curves D such that D is r-1 big or r small. \overline{S}_{r}(\overline{U}_{2}) makes SNCs with \overline{B}_{2}(\overline{U}_{2}), and if q ∈ \overline{π}^{−1}(p),

1’.: ν(q) ≤ r if q is a 1 or 2 point. ν(q) = r implies γ(q) = r.

2’.: If q is a 2 point and ν(q) = r −1, then either τ(q) > 0 or γ(q) = r or τ(q) = 0 and (133) holds at q with 0 < d_{i} < i, e_{i} = i and \overline{S}_{r−1}(Y_{1}) contains a single curve D containing q, and containing a 1 point, which has local equations x = z = 0 at q.

3’.: ν(q) ≤ r − 2 if q is a 3 point.

Proof. Suppose that there exists a 2 curve D ⊂ Y = Y_{p} such that D is r-1 big. Let π_{1} : Y_{1} → Y be the blowup of D. Then A_{r}(Y_{1}) holds by Lemmas 8.6 and 8.7, since ν(q) = r − 1 for all q ∈ D.

Suppose that q ∈ D and (131) holds at q. Then d_{j} ≥ j for all j. Suppose that q′ ∈ π_{1}^{−1}(q) and Ô_{Y_{1},q′} has regular parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}, \overline{y} = x_{1}(y_{1} + α)

with α ≠ 0. Then q′ is a 1 point, so that ν(q′) ≤ r and ν(q′) = r implies γ(q′) = r by Lemma 8.6.

Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}, \overline{y} = x_{1}y_{1}

Then

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d+r−1}y_{1}^{d}F_{q′} \\ F_{q′} & = \frac{F_{q}}{x_{1}^{r−1}} = τ + ∑_{j=1}^{r−1} y_{1}^{d_{j}}Λ_{j}(x_{1}y_{1}, \overline{z})x_{1}^{d_{j}−j} \end{aligned}

so that ν(F_{q′}) = 0.

Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}y_{1}, \overline{y} = y_{1}

Then

\begin{aligned}u & = (x_{1}^{a}y_{1}^{a+b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{a+b}) + x_{1}^{c}y_{1}^{c+d+r−1}F_{q′} \\ F_{q′} & = \frac{F_{q}}{y_{1}^{r−1}} = τx_{1}^{r−1} + ∑_{j=1}^{r−1} y_{1}^{d_{j}−j}Λ_{j}(y_{1}, \overline{z})x_{1}^{r−1−j} \end{aligned}

so that either ν(F_{q′}) < r − 1, or we are back in the form (131) but the d_{j} have decreased by j.

Suppose that q ∈ D and (132) holds at q. Without loss of generality, we may assume that D has local equations \overline{x} = \overline{z} = 0 at q. Then e_{j} ≥ j for all j. Suppose that q′ ∈ π_{1}^{−1}(q) and Ô_{Y_{1},q′} has regular parameters (x_{1}, \overline{y}, z_{1}) such that

\overline{x} = x_{1}, \overline{z} = x_{1}(z_{1} + α)

with α ≠ 0. Set

x_{1} = \overline{x}_{1}(z_{1} + α)^{− \frac{c}{a+c}}

Set λ = f + (d + f + r − 1)(\frac{−c}{a+c}),

G(x_{1}, \overline{y}, z_{1}) = \frac{(z_{1} + α)^{λ}F_{q}}{x_{1}^{r−1}} = (z_{1} + α)^{λ}τ + ∑_{j=1}^{r−1}(z_{1} + α)^{λ+e_{j}} \overline{a}_{j}(\overline{y}_{,}x_{1}(z_{1} + α))\overline{y}^{d_{j}}x_{1}^{e_{j}−j}

Then

\begin{aligned}u & = (\overline{x}_{1}^{a′} \overline{y}^{b′})^{m′} \\ v & = P ((\overline{x}_{1}^{a′} \overline{y}^{b′})^{\frac{m′}{m}}) + \overline{x}_{1}^{d+f+r−1}\overline{y}^{e}G \end{aligned}

G(0, 0, z_{1}) = (z_{1} + α)^{λ} [τ_{0} + ∑_{d_{j}=0,e_{j}=j} \overline{a}_{j}(0, 0)(z_{1} + α)^{j}]

where τ_{0} = τ(0, 0, 0), (a + c)m = a′m′, bm = b′m′, (a′, b′) = 1.

F_{q′}(0, 0, z_{1}) = \{ \begin{matrix}G(0, 0, z_{1}) & if a′e − b′(d + f + r − 1) ≠ 0 \\ G(0, 0, z_{1}) − G(0, 0, 0) & if a′e − b′(d + f + r − 1) = 0 \end{matrix}

Thus ν(F_{q′}(0, 0, z_{1})) ≤ r, except possibly if λ = 0 and a′e − b′(d + f + r − 1) = 0. Then we have

af − c(d + r − 1) = 0  (138)

and

[ae − b(d + r − 1)] + [ce − fb] = 0  (139)

with a, b, c > 0. Substituting d + r − 1 = \frac{af}{c} into (139), we get

(\frac{a}{c} + 1)(ce − bf) = 0

so that

ce − bf = 0  (140)

and

ae − b(d + r − 1) = 0  (141)

(138), (140) and (141) cannot all hold since F_{q} is normalized. Thus ν(F_{q′}(0, 0, z_{1})) ≤ r and ν(q′) ≤ r, γ(q′) ≤ r.

Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, \overline{y}, z_{1}) such that

\overline{x} = x_{1}, \overline{z} = x_{1}z_{1}

Then ν(q′) = 0.

Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}z_{1}, \overline{z} = z_{1}

Then we either have a 3 point with ν(q′) < r − 1, or we are back in the form of (132) with e_{i} decreased by i.

Suppose that q ∈ D and (133) holds at q. Then d_{j} ≥ j for all j. Suppose that q′ ∈ π_{1}^{−1}(q) and Ô_{Y_{1},q′} has regular parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}, \overline{y} = x_{1}(y_{1} + α)

with α ≠ 0. Then q′ is a 1 point so that ν(q′) ≤ r and γ(q′) ≤ r by Lemma 8.6. Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}, \overline{y} = x_{1}y_{1}

Then ν(q′) = 0.

Suppose that q′ ∈ π_{1}^{−1}(q) and q′ has permissible parameters (x_{1}, y_{1}, \overline{z}) such that

\overline{x} = x_{1}y_{1}, \overline{y} = y_{1}

Then we either have a 2 point with ν(q′) < r − 1, or we are back in the form of (133) with d_{i} decreased by i.

After a finite number of blowups of 2 curves π_{1} : Y_{n_{0}} → Y we have that there are no 2 curves D on Y_{n_{0}} such that D is r-1 big. By Lemmas 8.6, 8.7, and since all 3 points q ∈ (π ∘ π_{1})^{−1}(p) have ν(q) ≤ r − 2 (if q is a 3 point and ν(q) = r − 1 so that q satisfies (132), then either d_{j} ≥ j for all j or e_{j} ≥ j for all j), there exists a neighborhood \overline{U}_{1} with the properties asserted by the statement of the Theorem.

The only 2 points q ∈ (π ∘ π_{1})^{−1}(p) where ν(q) = r − 1 and γ(q) > r either satisfy (131) with some d_{j} = j − 1 and e_{j} = 1 (so that τ(q) ≥ 1) or satisfy (133) with ν(q) = r − 1 and d_{i} < i so that e_{i} ≥ i. Furthermore, \overline{x} = 0 is a local equation at q of the strict transform of the surface with local equation x = 0 at p.

Suppose that D ⊂ \overline{S}_{r}(X) is a curve containing p, and \overline{D} is the strict transform of D on \overline{U}_{1}. \overline{D} can only intersect (π ∘ π_{1})^{−1}(p) at points q such that q is a 2 point and ν(q) = r or ν(q) = r − 1 by Lemma 7.7 and Lemma 7.6. We must either have γ(q) ≤ r or q satisfies (131) or (133).

Suppose that q ∈ \overline{D} ∩ (π ∘ π_{1})^{−1}(p) satisfies (131) or (133) and \overline{y} ∈ Î_{\overline{D},q}. Then there exist a_{j} ∈ k such that

F_{q} − ∑ a_{j} \frac{(\overline{x}^{a}\overline{y}^{b})^{j}}{\overline{x}^{c}\overline{y}^{d}} ∈ (\overline{y})^{r−1} + (\overline{y}, f(\overline{x}, \overline{z}))^{r}

where \overline{y} = f(\overline{x}, \overline{z}) = 0 are local equations of \overline{D} at q (by Lemma 6.27). This is impossible by the form of F_{q}. Thus \overline{x} ∈ Î_{\overline{D},q}.

Suppose that q satisfies (131). We have

Î_{\overline{D},q} = (\overline{x}, \overline{z} − ϕ(\overline{y}))

for some series ϕ. There exist a_{j} ∈ k such that, when renormalizing with respect to these new parameters,

F_{q} − ∑ a_{j} \frac{(\overline{x}^{a}\overline{y}^{b})^{j}}{\overline{x}^{c}\overline{y}^{d}} ∈ (\overline{x})^{r−1} + (\overline{x}, \overline{z} − ϕ(\overline{y}))^{r}

by Lemma 6.27. Setting \overline{x} = 0 in

F_{q} − ∑ a_{j} \frac{(\overline{x}^{a}\overline{y}^{b})^{j}}{\overline{x}^{c}\overline{y}^{d}} ,

we get \overline{y}^{d_{r−1}}Λ_{r−1}(\overline{y}, \overline{z}) or \overline{y}^{d_{r−1}}Λ_{r−1}(\overline{y}, \overline{z}) + c̃\overline{y}\overline{n} for some c̃ ∈ k, \overline{n} ∈ N. Thus

(\overline{z} − ϕ(\overline{y}))^{r} | \overline{y}^{d_{r−1}}Λ_{r−1}(\overline{y}, \overline{z})

or

(\overline{z} − ϕ(\overline{y}))^{r} | \overline{y}^{d_{r−1}}Λ_{r−1}(\overline{y}, \overline{z}) + c̃\overline{y}\overline{n}

which is nonzero since \overline{x} |̸ F_{q}. As ν(Λ_{r−1}(\overline{y}, \overline{z})) ≤ 1, this is a contradiction. Thus q cannot have the form of (131).

Suppose that q satisfies (133). We have Î_{\overline{D},q} = (\overline{x}, \overline{z} − ϕ(\overline{y})) for some series ϕ. There exists a_{j} ∈ k such that

F_{q} − ∑ a_{j} \frac{(\overline{x}^{a}\overline{y}^{b})^{j}}{\overline{x}^{c}\overline{y}^{d}} ∈ (\overline{x})^{r−1} + (\overline{x}, \overline{z} − ϕ(\overline{y}))^{r}

by Lemma 6.27. Setting \overline{x} = 0 in

F_{q} − ∑ a_{j} \frac{(\overline{x}^{a}\overline{y}^{b})^{j}}{\overline{x}^{c}\overline{y}^{d}} ,

we get \overline{a}_{r−1}(\overline{y}, \overline{z})\overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}} or \overline{a}_{r−1}(\overline{y}, \overline{z})\overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}} +c̃\overline{y}\overline{n} for some c̃ ∈ k, \overline{n} ∈ N. Thus

(\overline{z} − ϕ(\overline{y}))^{r} | \overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}}

or

(\overline{z} − ϕ(\overline{y}))^{r} | \overline{a}_{r−1}(\overline{y}, \overline{z})\overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}} + c̃\overline{y}\overline{n}

which is nonzero since \overline{x} |̸ F_{q}. In either case, we have

(\overline{z} − ϕ(\overline{y}))^{r−1} | \overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}−1}

since

\frac{∂}{∂\overline{z}} (\overline{a}_{r−1}\overline{y}^{d_{r−1}} \overline{z}^{e_{r−1}} + c̃\overline{y}\overline{n}) = \overline{z}^{e_{r−1}−1}\overline{y}^{d_{r−1}}(e_{r−1}\overline{a}_{r−1} + \frac{∂\overline{a}_{r−1}}{∂\overline{z}} \overline{z}),

which implies that e_{r−1} ≥ r and ϕ(\overline{y}) = 0. Thus \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q.

Suppose that \overline{D} is such that \overline{D} is r small and q ∈ (π∘π_{1})^{−1}(p)∩\overline{D} satisfies ν(q) = r and γ(q) = r. By Lemma 8.10, there exists a sequence of quadratic transforms σ_{1} : W_{1} → Y_{n_{0}} such that the strict transform D̃ of \overline{D} intersects σ_{1}^{−1}(q) in a 2 point q′ such that ν(q′) = r − 1 and γ(q′) = r. Furthermore, there are no 2 curves C in σ_{1}^{−1}(q) such that C is r-1 big, and 1’. - 3’. of the conclusions of the Theorem hold at all points of σ_{1}^{−1}(q). Thus there exists a sequence of quadratic transforms σ : W → Y_{n_{0}}, centered at 2 points {q_{1}, …, q_{m}} such that ν(q_{i}) = r and γ(q_{i}) = r on the strict transform \overline{D} of curves D in \overline{S}_{r}(X) containing p such that if \overline{D} ⊂ \overline{S}_{r}(W) is the strict transform of a curve D ⊂ \overline{S}_{r}(X) containing p, and \overline{D} is r small then \overline{D} intersects (π ∘ π_{1} ∘ σ)^{−1}(p) in 2 points q of the form of (133), and in 2 points q such that ν(q) = r − 1 and γ(q) = r. W contains no 2 curves C such that C is r-1 big, \overline{S}_{r}(W) makes SNCs with \overline{B}_{2}(W) and γ(q) ≤ r for all exceptional 1 and 2 points of σ, ν(q) = 0 for all exceptional 3 points of σ.

Suppose that \overline{D} ⊂ W is the strict transform of a curve D in \overline{S}_{r}(X) containing p. First suppose that \overline{D} is r big. If q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p), then q must be a 2 point with ν(q) = γ(q) = r. Suppose that λ_{1} : Z_{1} → W is the blowup of \overline{D}. By Lemma 8.8, 1’. - 3’. of the conclusions of the Theorem hold on Z_{1}, and the conclusions of \overline{U}_{2} hold in a neighborhood of λ_{1}^{−1}(q).

If \overline{D} is r small, then if q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p), q must be either a 2 point where ν(q) = r − 1 and γ(q) = r or q satisfies (133) and \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q.

Let λ_{1} : Z_{1} → W be the blowup of \overline{D}. If q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p) is a 2 point such that ν(p) = r − 1 and γ(p) = r, then 1.’ - 3.’ of the conclusions of the Theorem hold and the conclusions of \overline{U}_{2} hold in a neighborhood of λ_{1}^{−1}(q) (since r ≥ 3) by Lemma 8.10.

Suppose that q ∈ \overline{D} ∩(π ∘π_{1} ∘σ)^{−1}(p) satisfies (133). \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q and d_{i} < i. Since \overline{D} ⊂ \overline{S}_{r}(W), we have e_{i} > i.

Since A_{r}(X) holds, γ(q′) = r if q′ ∈ \overline{D} is a 1 point. Then e_{r−1} = r in (133). Suppose that q′ ∈ λ_{1}^{−1}(q), q′ has permissible parameters (x_{1}, \overline{y}, z_{1}) such that

\overline{x} = x_{1}, \overline{z} = x_{1}(z_{1} + α)

q′ is a 2 point. e_{i} > i implies

\frac{F_{p}}{x_{1}^{r−1}} = τ + x_{1}Ω

so that ν(q) = 0.

Suppose that q′ ∈ λ_{1}^{−1}(q) has permissible parameters (x_{1}, \overline{y}, z_{1}) such that

\overline{x} = x_{1}z_{1}, \overline{z} = z_{1}

Then we either have a 3 point with ν(q′) < r − 1, or we are in the form of (132) with e_{i} decreased by i, d_{i} < i, and \overline{x} = \overline{z} = 0 is a local equation of a 2 curve which is a section over D. Since we have e_{r−1} = 1 in (132), we must have e_{i} < i (since r ≥ 2), d_{i} < i so that ν(q′) < r − 1 by Remark 9.19.

Thus the conclusions of \overline{U}_{2} hold in a neighborhood of λ_{1}^{−1}(q) by Lemma 8.10. Then if \overline{λ} : Z → W is the sequence of monodial transforms (in any order) centered

at the strict transforms of curves C in \overline{S}_{r}(X), the conclusions of 1’. - 3’. of the Theorem hold, except possibly at a finite number of points q of the form of (133) with d_{i} < i and e_{i} ≥ i. There are no 2 curves C ⊂ Z such that C is r-1 big.

If q ∈ (π ∘ π_{1} ∘ σ ∘ \overline{λ})^{−1}(p) does not satisfy one of 1’ - 3’ of the conclusions of the Theorem then q satisfies (133), τ(q) = 0 and d_{i} < 0 so that e_{i} ≥ i. \overline{x} = 0 is then a local equation of the surface with local equation x = 0 at p.

e_{i} ≥ i implies F_{q} ∈ (\overline{x}, \overline{z})^{r−1}. Since r ≥ 3, this implies (by Lemma 6.23) that there exists an algebraic curve \overline{D} ⊂ \overline{S}_{r−1}(Z) such that \overline{x} = \overline{z} = 0 are local equations of a formal branch of \overline{D}. \overline{D} is necessarily the strict transform of a curve D ⊂ \overline{S}_{r−1}(U), since \overline{x} ∈ Î_{\overline{D},q} and \overline{y} ∉ Î_{\overline{D},q}. \overline{D} is thus nonsingular at q, by the conclusions of Theorem 12.1. Thus \overline{x} = \overline{z} = 0 are local equations of an algebraic curve \overline{D} at q. If e_{i} > i, then \overline{D} ⊂ \overline{S}_{r}(Z) and if e_{i} = i, then \overline{D} ⊂ \overline{S}_{r−1}(Z).

Since the strict transforms of all curves in \overline{S}_{r}(X) have been blown up in the map Z → W, we must have e_{i} = i.

Thus at q, (133) holds, e_{i} = i, d_{i} < i and τ(q) = 0. Let T be the component of E_{X} with local equation x = 0 at p. τ(q) = 0 implies d_{i} > 0, which implies d_{j} > 0 for all j, so that F_{q} = τ\overline{x}^{r−1} + \overline{y}Ω. Since \overline{x} = 0 is a local equation of the strict transform T′ of T, the only curve in \overline{S}_{r−1}(Z) ∩ T′ containing q is the curve \overline{D} with local equations \overline{x} = \overline{z} = 0, since a curve in \overline{S}_{r−1}(Z) ∩ T′ containing q must be the strict transform of a curve in \overline{S}_{r−1}(U) ∩ T.

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d}F \end{aligned}

and F_{q} = τ\overline{x}^{r−1} + \overline{y}Ω, where τ is a unit.

We will show that there does not exist a curve C ⊂ \overline{S}_{r−1}(Z) containing q (and a 1 point) such that \overline{y} ∈ Î_{C,q}.

After a permissible change of parameters, we may assume that \overline{y}, \overline{z} ∈ O_{X,q} with

F_{q} = τ\overline{x}^{r−1} + \overline{y}Ω.

I_{C,q} = (\overline{y}, g(\overline{x}, \overline{z})). Set \overline{x} = x̃^{b}. By Lemma 6.30, either there exists a series f such that

x̃^{bc−ad}F_{q} − f(x̃^{a}\overline{y}) ∈ ((\overline{y}, g(x̃^{b}, \overline{z}))^{r−1} + (\overline{y})^{r−2}) k[[x̃, \overline{y}, \overline{z}]]  (142)

if bc − ad ≥ 0 or

F_{q} − f(x̃^{a}\overline{y})x̃^{ad−bc} ∈ ((\overline{y}, g(x̃^{b}, \overline{z}))^{r−1} + (\overline{y})^{r−2}) k[[x̃, \overline{y}, \overline{z}]]  (143)

if ad − bc > 0.

If (142) holds, since ν(q) > 0, we have ν(f) > 0, which implies

g(x̃^{b}, \overline{z})^{r−1} | x̃^{b(r−1)+bc−ad},

and g = \overline{x}, a contradiction since C is then a 2 curve.

Suppose that (143) holds. Let \overline{c} = f(0).

τ(\overline{x}, 0, \overline{z})x̃^{b(r−1)} − \overline{c}x̃^{ad−bc} = h(x̃, \overline{z})g(x̃^{b}, \overline{z})^{r−1}

for some series h. If ad − bc = b(r − 1) then ad − b(c + r − 1) = 0, a contradiction to the assumption that F_{q} is normalized.

Let

\overline{d} = \{ \begin{matrix}min\{b(r − 1), ad − bc\} & if \overline{c} ≠ 0 \\ b(r − 1) & if \overline{c} = 0. \end{matrix}

τ(\overline{x}, 0, \overline{z})x̃^{b(r−1)} − \overline{c}x̃^{ad−bc} = Λx̃\overline{d}

where Λ is a unit. Then g(x̃^{b}, \overline{z})^{r−1} is a power of x̃, and g = x̃, a contradiction since C is then a 2 curve.

Thus the conclusions of 1’. - 3’. of the Theorem hold and the conclusions of \overline{U}_{2} hold on Z.

Theorem 12.3. Suppose that r = 2, A_{2}(X) holds, p is a 2 point with ν(p) = r − 1 = 1, and L(x, 0, 0) ≠ 0, as in the assumptions of Theorem 12.1. Let R = O_{X,p}. Suppose that π : Y_{p} → Spec(R) is the morphism of Theorem 12.1.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → Y_{p}

be a sequence of permissible monodial transforms centered at 2 curves D such that D is 1 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p (with the notation of Theorem 12.1) such that \overline{U}_{1} contains no 2 curves D such that D is 1 big or 2 small, and for q ∈ \overline{U}_{1},

1. If q is a 1 or a 2 point then ν(q) ≤ 2. ν(q) = 2 implies γ(q) = 2.

2. If q is a 3 point then ν(q) = 0.

3. \overline{S}_{2}(\overline{U}_{1}) makes SNCs with \overline{B}_{2}(\overline{U}_{1}). There exists a sequence of quadratic transforms W_{p} → V_{p} such that if Z_{p} → W_{p} is the sequence of monodial transforms (in any order) centered at the strict transforms C′ of curves C in \overline{S}_{2}(X), followed by monoidial transforms centered at any 2 curves \overline{C} which are sections over C′ such that \overline{C} is 1 big, then

Z_{p} → W_{p} → V_{p} → Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{π} : \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood of p such that \overline{U}_{2} contains no 2 curves D such that D is 1 big or 2 small. \overline{S}_{2}(\overline{U}_{2}) makes SNCs with \overline{B}_{2}(\overline{U}_{2}), and if q ∈ \overline{π}^{−1}(p),

1’.: ν(q) ≤ 2 if q is a 1 or 2 point. ν(q) = 2 implies γ(q) = 2.

2’.: If q is a 2 point and ν(q) = 1, then either q is resolved or γ(q) = 2.

3’.: ν(q) = 0 if q is a 3 point.

Proof. We can construct

W = W_{p} →^{σ} V_{p} = Y_{n_{0}} →^{π_{1}} Y = Y_{p} →^{π} spec(R)

exactly as in the proof of Theorem 12.2.

If q ∈ (π ∘ π_{1} ∘ σ)^{−1}(p) satisfies 133), then

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ F_{q} & = \overline{x} + \overline{z}^{e_{1}}. \end{aligned}  (144)

If D ⊂ \overline{S}_{2}(X) contains p, and \overline{D} is the strict transform of D on W, and q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p), then either q satisfies (144),

F_{q} = \overline{x} + \overline{z}^{e_{1}}

with e_{1} ≥ 2 and \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q, or q is a 2 point with ν(q) = 1, γ(q) = 2, or \overline{D} is 2 big and q is a 2 point with ν(q) = γ(q) = 2.

Suppose that \overline{D} ⊂ \overline{S}_{2}(W) is the strict transform of D ⊂ \overline{S}_{2}(X) such that p ∈ D. Let λ_{1} : Z_{1} → W be the blowup of \overline{D}.

First suppose that \overline{D} is 2 big. Suppose that q ∈ \overline{D} ∩(π ∘π_{1} ∘σ)^{−1}(p). Then q is a 2 point with ν(q) = γ(q) = 2. By Lemma 8.8, 1’ - 3’ of the conclusions of the Theorem hold on Z_{1}, and the conclusions of \overline{U}_{2} hold in a neighborhood of λ_{1}^{−1}(q).

Suppose that \overline{D} is 2 small. If q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p), then either q is a 2 point with ν(q) = 1, γ(q) = 2, or q satisfies (144) with e_{1} ≥ 2, \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q.

If q ∈ \overline{D}∩(π∘π_{1} ∘σ)^{−1}(p) is a 2 point with ν(q) = 1, γ(q) = 2, then by Lemma 8.10 and Lemma 8.11, either 1’ - 3’ of the conclusions of the Theorem and the conclusions of \overline{U}_{2} hold in a neighborhood of λ_{1}^{−1}(q) or there exists a 2 curve \overline{C} which is a section over \overline{D}, such that if λ_{2} : Z_{2} → Z_{1} is the blowup of \overline{C}, then 1’ - 3’ of the conclusions of the Theorem hold and the conclusions of \overline{U}_{2} hold in a neighborhood of (λ_{1} ∘λ_{2})^{−1}(q).

Suppose that q ∈ \overline{D} ∩ (π ∘ π_{1} ∘ σ)^{−1}(p) satisfies (144) with e_{1} ≥ 2, \overline{x} = \overline{z} = 0 are local equations of \overline{D} at q. Since A_{2}(X) holds, γ(q′) = 2 if q′ ∈ \overline{D} is a 1 point, so that e_{1} = 2. If q′ ∈ λ_{1}^{−1}(q) we have ν(q′) = 0 except if q′ is the 3 point with permissible parameters (x_{1}, \overline{y}, z_{1}) such that \overline{x} = x_{1}z_{1}, \overline{z} = z_{1}. Then F_{q′} = x_{1} + z_{1}.

Let \overline{C} be the 2 curve through q′ with local equations x_{1} = z_{1} = 0 at q′. \overline{C} is a section over \overline{D}. By Lemma 8.1, F_{a} ∈ Î_{\overline{C},a} for all a ∈ \overline{C}, so that \overline{C} is 1 big.

Let λ_{2} : Z_{2} → Z_{1} be the blowup of \overline{C}. Then 1’ - 3’ of the conclusions of the Theorem, and the conclusions of \overline{U}_{2} hold in a neighborhood of (λ_{1} ∘ λ_{2})^{−1}(q).

Then if \overline{λ} : Z → W is the sequence of monodial transforms (in any order) centered at the strict transform of curves C in \overline{S}_{2}(X), followed by the monodial transforms centered at 2 curves \overline{C} which are sections over C′ such that \overline{C} is 1 big, we have that there are no 2 curves C ⊂ Z such that \overline{C} is 1 big.

The only points of Z which may not satisfy the conclusions of 1’ - 3’ of the Theorem are the 2 points q ∈ (π ∘ π_{1} ∘ σ ∘ \overline{λ})^{−1}(p) which satisfy (144) with τ(q) = 0. Then (after a permissible change of parameters)

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{m} \\ F_{q} & = \overline{x} + \overline{z}^{e_{r}} \end{aligned}

with e_{r} ≥ 2.

By Lemma 6.23, there exists an algebraic curve \overline{D} ⊂ \overline{S}_{2}(Z) such that \overline{x} = \overline{z} = 0 are local equations of D at q. Since \overline{x} = 0 is a local equation of the strict transform of the component of E_{X} with local equation x = 0, and \overline{D} is not contained in the component of E_{X} with local equation \overline{y} = 0, \overline{D} is the strict transform of a curve in \overline{S}_{r}(X) containing p, a contradiction to the construction of Z. Thus q is a resolved point.

Theorem 12.4. (Theorem30) Suppose that r ≥ 2, A_{r}(X) holds, p ∈ X is a 3 point with ν(p) = r − 1, and we have permissible parameters (x, y, z) at p for u, v (with y, z ∈ O_{X,p}) such that

\begin{aligned}u & = (x^{b}y^{a+nb}z^{a+(n+1)b})^{m} \\ v & = P (x^{b}y^{a+nb}z^{a+(n+1)b}) + x^{d}y^{c+n(d+r)}z^{c+(n+1)(d+r)}F_{p} \\ F_{p} & = τx^{r−1} + ∑_{i=1}^{r−1} a_{i}(y, z)x^{r−i−1} \end{aligned}  (145)

with n ≥ 0, a, b > 0, L(x, 0, 0) ≠ 0, so that τ is a unit, and (eq510)

a(d + r − 1) − bc = 0  (146)

or

\begin{aligned}u & = (x^{b}y^{a}z^{b})^{m} \\ v & = P (x^{b}y^{a}z^{b}) + x^{d}y^{c}z^{d+r}F_{p} \\ F_{p} & = τx^{r−1} + ∑_{i=1}^{r−1} a_{i}(y, z)x^{r−i−1} \end{aligned}  (147)

with n ≥ 0, a, b > 0, L(x, 0, 0) ≠ 0, so that τ is a unit, and

a(d + r − 1) − bc = 0  (148)

Let R = O_{X,p}. Let C be the 2 curve with local equations y = z = 0 at p. Then there exists a finite sequence of permissible monodial transforms π : Y → Spec(R) centered at sections over C = V (y, z), such that for q ∈ π^{−1}(p), F_{q} has one of the forms (131), (132) or (133) of Theorem 12.1.

In all these cases \overline{x} = x and \overline{x} = 0 is a local equation at q of the strict transform of the component of E_{X} with local equation x = 0 at p.

There exists an affine neighborhood U of p such that Y → spec(R) extends to a sequence of permissible monoidal transforms \overline{U} → U such that A_{r}(\overline{U}) holds.

γ(q) ≤ 1 at a generic point q of C, so that all points q′ on the fiber of the blowup of C over q are resolved.

Suppose that r ≥ 3. Let D_{i} be the curves in \overline{S}_{r−1}(X) which contain p, and x ∈ Î_{D_{i},p}. We further have that the strict transforms \overline{D}_{i} of the D_{i} on \overline{U} are nonsingular, disjoint, and make SNCs with \overline{B}_{2}(\overline{U}).

Proof. We modify the proof of Theorem 12.1 to prove this Theorem. In the sequence of (134) we must add a new case,

Case 0:

\begin{aligned}u & = (x^{a_{i}}(y_{i}^{\overline{b}_{i}}z_{i}^{\overline{c}_{i}})^{k_{i}})^{m_{i}} \\ v & = P_{i}(x^{a_{i}}(y_{i}^{\overline{b}_{i}}z_{i}^{\overline{c}_{i}})^{k_{i}}) + x^{c}y_{i}^{\overline{d}_{i}}z_{i}^{\overline{e}_{i}}F_{i} \end{aligned}

Case 0a:

y_{i} = y_{i+1}, z_{i} = y_{i+1}z_{i+1}

Case 0b:

y_{i} = y_{i+1}z_{i+1}, z_{i} = z_{i+1}

Case 0c:

y_{i} = y_{i+1}(z_{i+1} + α_{i+1})^{− \frac{\overline{c}_{i}}{\overline{b}_{i}+\overline{c}_{i}}} ,
z_{i} = y_{i+1}(z_{i+1} + α_{i+1})^{\frac{\overline{b}_{i}}{\overline{b}_{i}+\overline{c}_{i}}}

In the sequence (134), Y_{0} has the form Case 0 (and not Case 2). The transformations of type 0a and type 0b produce a p_{i+1} of the type of Case 0, and F_{i} = F_{i+1}.

If all transformations in (134) are of types 0a or 0b, then we eventually get a p_{i} of type (132) by Lemmas 9.16, 9.17 and Corollary 9.18.

Otherwise, we eventually reach a first p_{i} where p_{i+1} is obtained by a transformation of type 0c. We have F_{i} = F_{p}.

\begin{aligned}u & = (x^{a_{i}}y_{i+1}^{(\overline{b}_{i}+\overline{c}_{i})k_{i}})^{m_{i}} = (x^{a_{i+1}}y_{i+1}^{b_{i+1}})^{m_{i+1}} \\ v & = P_{i}(x^{a_{i}}y_{i+1}^{(b_{i}+\overline{c}_{i})k_{i}}) + x^{c}y_{i+1}^{d_{i+1}}(z_{i+1} + α_{i+1})^{λ}F_{p} \end{aligned}

λ = \frac{\overline{e}_{i}\overline{b}_{i} − \overline{d}_{i}\overline{c}_{i}}{\overline{b}_{i} + \overline{c}_{i}} ,

d_{i+1} = \overline{d}_{i} + \overline{e}_{i}. Thus p_{i+1} has the form of Case 1, with

F_{i+1} = (z_{i+1} + α_{i+1})^{λ}F_{p} − \frac{g_{i+1}(x^{a_{i+1}}y_{i+1}^{b_{i+1}})}{x^{c}y_{i+1}^{d_{i+1}}}

Thus p_{i+1} satisfies the assumptions of Theorem 12.1, provided ν(F_{i+1}(x, 0, 0)) = r−1. p_{i} has permissible parameters (x, y_{1}, z_{1}) with y_{1} = y_{i}, z_{1} = z_{i} such that

y = y_{1}^{α}z_{1}^{β}, z = y_{1}^{γ}z_{1}^{δ}

with αδ − βγ = ±1, p_{i+1} has permissible parameters (x, y_{2}, z_{2}) with y_{2} = y_{i+1}, z_{2} = z_{i+1} such that

y_{1} = y_{2}, z_{1} = y_{2}(z_{2} + \overline{α})

with \overline{α} ≠ 0.

First suppose that we are in the situation of (145) and (146). Set

λ_{1} = α(a + nb) + γ(a + (n + 1)b) + β(a + nb) + δ(a + (n + 1)b),
λ_{2} = β(α + nb) + δ(a + (n + 1)b).
y_{2} = \overline{y}_{2}(z_{2} + \overline{α})^{− \frac{λ_{2}}{λ_{1}}}
u = (x^{b}y_{1}^{α(a+nb)+γ(a+(n+1)b)}z_{1}^{β(a+nb)+δ(a+(n+1)b)})^{m}
= (x^{b}y_{2}^{λ_{1}}(z_{2} + \overline{α})^{λ_{2}})^{m}
= (x^{b}\overline{y}_{2}^{λ_{1}})^{m}
x^{d+r−1}y^{c+n(d+r)}z^{c+(n+1)(d+r)} = x^{d+r−1}y_{1}^{α(c+n(d+r))+γ(c+(n+1)(d+r))}z_{1}^{β(c+n(d+r))+δ(c+(n+1)(d+r))}
= x^{d+r−1}y_{2}^{λ_{3}}(z_{2} + \overline{α})^{λ_{4}}
= x^{d+r−1}\overline{y}_{2}^{λ_{3}}(z_{2} + \overline{α})^{λ_{4}− \frac{λ_{2}λ_{3}}{λ_{1}}}

where

λ_{3} = α(c + n(d + r)) + γ(c + (n + 1)(d + r)) + β(c + n(d + r)) + δ(c + (n + 1)(d + r)),

λ_{4} = β(c + n(d + r)) + δ(c + (n + 1)(d + r)).

bλ_{3} − (d + r − 1)λ_{1}
= b[(α + β)(c + n(d + r)) + (γ + δ)(c + (n + 1)(d + r))]
−(d + r − 1)[(α + β)(a + nb) + (γ + δ)(a + (n + 1)b)]
= (α + β + γ + δ)[bc − (d + r − 1)a] + nb(α + β + γ + δ) + b(γ + δ)
= n(α + β + γ + δ)b + (γ + δ)b > 0.

Since we cannot remove

τ_{0}\overline{α}^{λ_{4}− \frac{λ_{2}λ_{3}}{λ_{1}}} x^{d+r−1}\overline{y}_{2}^{λ_{3}}

from x^{d}y^{c+n(d+r)}z^{c+(n+1)(d+r)}F_{p}, where τ_{0} = τ(0, 0, 0), when normalizing to obtain F_{p_{i+1}}, we must have ν(F_{p_{i+1}}(x, 0, 0)) = r − 1.

Now suppose that we are in the situation of (147) and (148). Set

λ_{1} = αa + γb + βa + δb,
λ_{2} = βa + δb.
y_{2} = \overline{y}_{2}(z_{2} + \overline{α})^{− \frac{λ_{2}}{λ_{1}}}
u = (x^{b}y_{1}^{αa+γb}z_{1}^{βa+δb})^{m}
= (x^{b}y_{2}^{λ_{1}}(z_{2} + \overline{α})^{λ_{2}})^{m}
= (x^{b}\overline{y}_{2}^{λ_{1}})^{m}

Set

λ_{3} = αc + γ(d + r) + βc + δ(d + r),
λ_{4} = βc + δ(d + r).
x^{d+r−1}y^{c}z^{d+r} = x^{d+r−1}y_{1}^{αc+γ(d+r)}z_{1}^{βc+δ(d+r)}
= x^{d+r−1}y_{2}^{λ_{3}}(z_{2} + \overline{α})^{λ_{4}}
= x^{d+r−1}\overline{y}_{2}^{λ_{3}}(z_{2} + \overline{α})^{λ_{4}− \frac{λ_{2}λ_{3}}{λ_{1}}}
bλ_{3} − λ_{1}(d + r − 1) = b(αc + γ(d + r) + βc + δ(d + r)) − (αa + γb + βa + δb)(d + r − 1)
= (α + β)[bc − a(d + r − 1)] + (γ + δ)[b(d + r) − b(d + r − 1)]
= (γ + δ)b ≠ 0

Thus ν(F_{p_{i+1}}(x, 0, 0)) = r−1 in this case also. The proof now preceeds as in Theorem 12.1.

Theorem 12.5. Suppose that r ≥ 3, A_{r}(X) holds, p is a 3 point with ν(p) = r − 1, and we have permissible parameters (x, y, z) at p for u, v (with y, z ∈ O_{X,p}) such that

\begin{aligned}u & = (x^{b}y^{a+nb}z^{a+(n+1)b})^{m} \\ v & = P (x^{b}y^{a+nb}z^{a+(n+1)b}) + x^{d}y^{c+n(d+r)}z^{c+(n+1)(d+r)}F_{p} \\ F_{p} & = τx^{r−1} + ∑_{i=1}^{r−1} a_{i}(y, z)x^{r−i−1} \end{aligned}

with n ≥ 0, a, b > 0, ν(p) = r − 1, L(x, 0, 0) ≠ 0, so that τ is a unit, and

a(d + r − 1) − bc = 0

or

\begin{aligned}u & = (x^{b}y^{a}z^{b})^{m} \\ v & = P (x^{b}y^{a}z^{b}) + x^{d}y^{c}z^{d+r}F_{q} \\ F_{p} & = τx^{r−1} + ∑ a_{i}(y, z)x^{r−i−1} \end{aligned}

with n ≥ 0, a, b > 0, ν(p) = r − 1, L(x, 0, 0) ≠ 0, so that τ is a unit, and

a(d + r − 1) − bc = 0

Let R = O_{X,p}.

Suppose that π : Y_{p} → Spec(R) is the morphism of Theorem 12.4.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → Y_{p}

be a sequence of permissible monodial transforms centered at 2 curves D such that D is r-1 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} Y → spec(R)

extends to a permissible sequence of monodial transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p (with the notation of Theorem 12.4) such that \overline{U}_{1} contains no 2 curves D such that D is r-1 big or r small, and for q ∈ \overline{U}_{1},

1. If q is a 1 or a 2 point then ν(q) ≤ r. ν(q) = r implies γ(q) = r.

2. If q is a 3 point then ν(q) ≤ r − 2.

3. \overline{S}_{r}(\overline{U}_{1}) makes SNCs with \overline{B}_{2}(\overline{U}_{1}). There exists a sequence of quadratic transforms W_{p} → V_{p} such that if Z_{p} → W_{p} is the sequence of monodial transforms (in any order) centered at the strict transforms of curves C in \overline{S}_{r}(X) then

Z_{p} → W_{p} → V_{p} → Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{π} : \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood of p such that \overline{U}_{2} contains no 2 curves D such that D is r-1 big or r small. \overline{S}_{r}(\overline{U}_{2}) makes SNCs with \overline{B}_{2}(\overline{U}_{2}), and if q ∈ \overline{π}^{−1}(p),

1’.: ν(q) ≤ r if q is a 1 or 2 point. ν(q) = r implies γ(q) = r.

2’.: If q is a 2 point and ν(q) = r −1, then either τ(q) > 0 or γ(q) = r or τ(q) = 0 and (133) holds at q with 0 < d_{i} < i, e_{i} = i and \overline{S}_{r−1}(Y_{1}) contains a single curve D containing q, and containing a 1 point, which has local equations x = z = 0 at q.

3’.: ν(q) ≤ r − 2 if q is a 3 point.

If p ∉ \overline{S}_{r}(X), then Z = Y_{n_{0}}.

Proof. The proof of Theorem 12.2 applied to the conclusions of Theorem 12.4 proves this theorem.

Theorem 12.6. Suppose that r = 2, A_{2}(X) holds, p is a 3 point with ν(p) = r − 1 = 1, and we have permissible parameters (x, y, z) at p for u, v (with y, z ∈ O_{X,p}) such that

\begin{aligned}u & = (x^{b}y^{a+nb}z^{a+(n+1)b})^{m} \\ v & = P (x^{b}y^{a+nb}z^{a+(n+1)b}) + x^{d}y^{c+n(d+2)}z^{c+(n+1)(d+2)}F_{p} \\ F_{p} & = τx + a_{1}(y, z) \end{aligned}

with n ≥ 0, a, b > 0, L(x, 0, 0) ≠ 0 so that τ is a unit and

a(d + r − 1) − bc = a(d + 1) − bc = 0

or

\begin{aligned}u & = (x^{b}y^{a}z^{b})^{m} \\ v & = P (x^{b}y^{a}z^{b}) + x^{d}y^{c}z^{d+2}F_{q} \\ F_{p} & = τx + a_{1}(y, z) \end{aligned}

with n ≥ 0, a, b > 0, L(x, 0, 0) ≠ 0 so that τ is a unit and

a(d + r − 1) − bc = a(d + 1) − bc = 0

Let R = O_{X,p}.

Suppose that π : Y_{p} → Spec(R) is the morphism of Theorem 12.4.

Let

⋯ → Y_{n} → ⋯ → Y_{1} → Y_{p}

be a sequence of permissible monodial transforms centered at 2 curves D such that D is 1 big. Then there exists n_{0} < ∞ such that

V_{p} = Y_{n_{0}} →^{π_{1}} Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{U}_{1} → \overline{U} → U

over an affine neighborhood U of p (with the notation of Theorem 12.4) such that \overline{U}_{1} contains no 2 curves D such that D is 1 big or 2 small, and for q ∈ \overline{U}_{1},

1. If q is a 1 or a 2 point then ν(q) ≤ 2. ν(q) = 2 implies γ(q) = 2.

2. If q is a 3 point then ν(q) = 0.

3. \overline{S}_{2}(\overline{U}_{1}) makes SNCs with \overline{B}_{2}(\overline{U}_{1}). There exists a sequence of quadratic transforms W_{p} → V_{p} such that if Z_{p} → W_{p} is the sequence of monodial transforms (in any order) centered at the strict transforms C′ of curves C in \overline{S}_{2}(X), followed by monodial transforms centered at any 2 curves \overline{C} which are sections over C′ such that \overline{C} is 1 big, then

Z_{p} → W_{p} → V_{p} → Y_{p} → spec(R)

extends to a permissible sequence of monodial transforms

\overline{π} : \overline{U}_{2} → \overline{U}_{1} → \overline{U} → U

over an affine neighborhood of p such that \overline{U}_{2} contains no 2 curves D such that D is 1 big or 2 small. \overline{S}_{2}(\overline{U}_{2}) makes SNCs with \overline{B}_{2}(\overline{U}_{2}), and if q ∈ \overline{π}^{−1}(p),

1’.: ν(q) ≤ 2 if q is a 1 or 2 point. ν(q) = 2 implies γ(q) = 2.

2’.: If q is a 2 point and ν(q) = 1, then either q is resolved or γ(q) = 2.

3’.: ν(q) = 0 if q is a 3 point.

If p ∉ \overline{S}_{2}(X), then Z_{p} = Y_{n_{0}}.

Proof. The proof of Theorem 12.3 applied to the conclusions of Theorem 12.4 proves the Theorem.

13. Resolution 1

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared. In this chapter we will need to consider the following condition on a 2 point p ∈ X

such that ν(p) = r and τ(p) = 1. The condition is that Φ_{X}(p) has permissible parameters (u, v) such that u = 0 is a local equation of E_{X} at p and p has permissible parameters (x, y, z) for (u, v) such that(eq998)

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}  (149)

and L_{p} contains a nonzero y^{r−1}z term with a(d+r −1)−bc = 0. Up to interchanging x and y, this condition is independent of permissible parameters at p for (u, v).

Lemma 13.1. Suppose that X satisfies A_{r}(X), with r ≥ 2, and C is a 2 curve on X such that C ⊂ \overline{S}_{r}(X). Suppose that

⋯ → X_{n} → ⋯ → X_{1} → X

is a sequence of permissible monodial transforms centered at 2 curves C_{i} such that C_{i} ⊂ \overline{S}_{r}(X_{i}) are sections over C. Then this sequence is finite. That is, there exists n < ∞ such that X_{n} contains no 2 curve C_{n} with this property.

Proof. Since A_{r}(X) holds, C must be r small. Suppose that q ∈ C is a 2 point and the sequence has infinite length. Let q_{n} be the point on X_{n} which is the intersection of the fiber over q and C_{n}. With the notations of (70) in the proof of Lemma 8.6, there are permissible parameters (x, y, z) at q such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ F_{q} & = \overline{c}zy^{r−1} + Σ_{i+j≥r,k≥0}c_{ijk}x^{i}y^{j}z^{k}. \end{aligned}

For all n there are permissible parameters (x_{n}, y_{n}, z) at q_{n} such that

x = x_{n}, y = x_{n}^{n}y_{n}

F_{q_{n}} = \frac{F_{q}}{x_{1}^{n(r−1)}}

If the sequence has infinite length, then c_{ijk} = 0 if j < r − 1, so that y | F_{q}, a contradiction to the assumption that F_{q} is normalized.

Lemma 13.2. Suppose that X satisfies A_{r}(X) with r ≥ 2 and C is a 2 curve on X such that C is r-1 big. Suppose that

⋯ → X_{n} → ⋯ → X_{1} → X

is a sequence of permissible monodial transforms, centered at 2 curves C_{i} such that C_{i} is a section over C and C_{i} is r-1 big. Then the sequence is finite. That is, there exists n < ∞ such that X_{n} contains no 2 curves C_{n} with this property.

Proof. Suppose that p ∈ C is a 2 point such that ν(p) = r − 1. p has permissible parameters (x, y, z) such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = ∑_{i+j≥r−1} a_{ij}(z)x^{i}y^{j} \end{aligned}

Let π : Y → spec(O_{X,p}) be the blowup of C.

Suppose that q ∈ π^{−1}(p) is the 2 point with permissible parameters (x_{1}, y_{1}, z_{1})

x = x_{1}y_{1}, y = y_{1}

\begin{aligned}u & = (x_{1}^{a}y_{1}^{a+b})^{m} \\ v & = P (x_{1}^{a}y_{1}^{a+b}) + x_{1}^{c}y_{1}^{c+d+r−1}(∑_{i+j=r−1} a_{ij}(z)x_{1}^{i} + y_{1}Ω) \end{aligned}  (150)

ν(q) ≤ r − 2 unless L_{p} = x^{r−1}. Then ν(q) = r − 1.

Suppose that L_{p} = x^{r−1}.

F_{p} = τx^{r−1} + ∑_{i=1}^{r−1} y^{b_{i}}a_{i}(y, z)x^{r−1−i}

where τ is a unit and x |̸ a_{i}, y |̸ a_{i}, b_{i} ≥ i for all i.

Suppose that the 2 curve C_{1} ⊂ Y containing q is such that C_{1} is r-1 big. C_{1} = V (x_{1}, y_{1}) in (150).

F_{q} = τx_{1}^{r−1} + ∑_{i=1}^{r−1} y_{1}^{b_{i}−i}a_{i}(y_{1}, z)x_{1}^{r−1−i}.

By induction on b_{i}, after a finite number of blowups of 2 curves, we reach λ : Z → spec(O_{X,p}) such that if D is a 2 curve in Z which is a section over C, then D is not r-1 big.

Definition 13.3. Suppose that X satisfies A_{r}(X) with r ≥ 2. A 2 point p ∈ X contained in a 2 curve C is called bad if ν(p) = r, τ(p) = 1 and one of the following holds.

1. C ⊄ \overline{S}_{r}(X).

2. C ⊂ \overline{S}_{r}(X) is r small and there exists a sequence of monodial transforms

X_{n} → X_{n−1} → ⋯ → X_{1} → spec(O_{X,p})

and 2 curves C_{i} ⊂ X_{i} which are sections over C such that C_{i} ⊂ \overline{S}_{r}(X_{i}) is r small for i < n, C_{n} ⊄ \overline{S}_{r}(X_{i}), X_{i+1} → X_{i} is centered at C_{i} if i < n, and if p_{n} is the point on C_{n} over p then ν(p_{n}) = r.

3. There exists a curve D ⊂ \overline{S}_{r}(X) such that D contains a 1 point p ∈ D, and D is r big at p.

Suppose that r ≥ 2 and A_{r}(X) holds. Then there are only finitely many bad 2 points on X.

Lemma 13.4. Suppose that X satisfies \overline{A}_{r}(X) with r ≥ 2. Then there exists a sequence of quadratic transforms X_{1} → X such that A_{r}(X_{1}) holds.

Proof. Let π : X_{1} → X be a sequence of quadratic transforms so that the strict transform of \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X). Then \overline{A}_{r}(X_{1}) holds by Theorems 7.1 and 7.3, and A_{r}(X_{1}) holds by Lemma 7.9 and Theorem 7.8.

Lemma 13.5. Suppose that A_{r}(X) holds with r ≥ 2 and p ∈ X is a bad 2 point. Suppose that there does not exist a curve D ⊂ \overline{S}_{r}(X) such that p ∈ D, D contains a 1 point and D is r big at p. Then there exists a sequence of quadratic transforms π : X_{1} → X centered at 2 points over p such that A_{r}(X_{1}) holds and all 2 points q ∈ π^{−1}(p) are good.

Proof. There exist permissible parameters (x, y, z) at p such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

Let π_{1} : X_{1} → X be the blowup of p. By Theorems 7.1, 7.3 and Lemma 7.9, A_{r}(X_{1}) holds. Suppose that q ∈ π^{−1}(p) is a 2 point such that ν(q) = r and τ(q) = 1. After a permissible change of parameters at p, we may assume that q has permissible parameters (\overline{x}_{1}, \overline{y}_{1}, \overline{z}_{1}) such that x = \overline{x}_{1}\overline{y}_{1}, y = \overline{y}_{1}, z = \overline{y}_{1}\overline{z}_{1}. τ(q) = 1 and ν(q) = r implies that, after replacing z by a constant times z, that

L_{p} = L(x, z) = \overline{d}x^{r} + x^{r−1}z

for some \overline{d} ∈ k.

Suppose that there exists a 2 point q′ ∈ π^{−1}(p) such that ν(q′) = r and q′ has permissible parameters x′, y′, z′ such that x = x′, y = x′y′, z = x′(z′ + α) for some α ∈ k. Then there exists a form L_{1} and \overline{c} ∈ k such that

L_{p} = \{ \begin{matrix}L_{1} + \overline{c}x\overline{a}y\overline{b} & if there exists \overline{a}, \overline{b} ∈ N such that \overline{a} + \overline{b} = r \\ & and a(d + \overline{b}) − b(c + \overline{a}) = 0 \\ L_{1} & otherwise. \end{matrix}

where

L_{1} = L_{1}(y, z − αx) = ey^{r} + f(z − αx)y^{r−1}

for some e, f ∈ k, with f ≠ 0. This is not possible, since r ≥ 2. Thus all 2 points q_{1} ∈ π^{−1}(p) with ν(q_{1}) = r have permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}(z_{1} + α)

for some α ∈ k. There exist at most finitely many bad 2 points q_{1} ∈ π_{1}^{−1}(p). Consider the following sequence of quadratic transforms

⋯ → X_{n} → X_{n−1} → ⋯ → X_{1} → X

with maps λ_{n} : X_{n} → X, where π_{i} : X_{i} → X_{i−1} is the blowup of all bad 2 points in λ_{i−1}^{−1}(p). We will show that there exists n < ∞ such that λ_{n}^{−1}(p) contains no bad 2 points. Suppose not. Then there exist bad 2 points q_{i} ∈ X_{i} such that π_{i}(q_{i}) = q_{i−1} for all i.

q_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}(z_{1} + α_{1})

for some α_{1} ∈ k. ν(q_{1}) = r implies

L_{q_{1}} = \overline{d}_{1}x_{1}^{r} + x_{1}^{r−1}z_{1} + y_{1}Ω_{1}

for some \overline{d}_{1} ∈ k and series Ω_{1}. Since ν(q_{2}) = r, and because of the existence of the x_{1}^{r−1}z_{1} term in L_{q_{1}}, q_{2} must have permissible parameters (x_{2}, y_{2}, z_{2}) such that

x_{1} = x_{2}y_{2}, y_{1} = y_{2}, z_{1} = y_{2}(z_{2} + α_{2})

for some α_{2} ∈ k. ν(q_{2}) = r implies

L_{q_{2}} = \overline{d}_{2}x_{2}^{r} + x_{2}^{r−1}z_{2} + y_{2}Ω_{2}.

We see that there exists a series σ(y) = ∑_{i=1}^{∞} α_{i}y^{i} such that if we replace z with z̃ = z − σ(y), we have permissible parameters (x_{n}, y_{n}, z̃_{n}) at q_{n} such that

x = x_{n}y_{n}^{n}, y = y_{n}, z̃ = z̃_{n}y_{n}^{n}.

Then

F_{q_{n}} = \frac{F_{p}(x_{n}y_{n}^{n}, y_{n}, y_{n}^{n}z̃_{n})}{y_{n}^{rn}}

for all n, so that a_{ijk} = 0 if i + k < r and F_{p} ∈ (x, z̃)^{r}. By Lemma 6.23, Î_{\overline{S}_{r}(X),p} ⊂ (x, z̃), so that x = z̃ = 0 are local equations of a curve D ⊂ \overline{S}_{r}(X), since A_{r}(X) holds. D is r big at p by Lemma 8.2.

Theorem 13.6. Suppose that A_{r}(X) holds with r ≥ 2. Then there exists a sequence of permissible monodial transforms X_{1} → X such that the following properties hold:

1. A_{r}(X_{1}) holds.

2. All bad 2 points p ∈ X_{1} satisfy (149).

3. Suppose that D ⊂ \overline{S}_{r}(X) is a curve which is r big at a 1 point. Then there exists at most one 3 point q ∈ D and D has a tangent direction at q distinct from those of \overline{B}_{2}(X) at q. Furthermore, if D is not r big, there exists only one 2 point q ∈ D. If C is the 2 curve containing q, then C is not r-1 big or r small.

4. If C is a r small or r-1 big 2 curve containing a 2 point p such that p ∈ D where D is a curve containing a 1 point and D is r big at p, then D is r big.

Proof. By Theorems 7.1 and 7.8, there exists a sequence of quadratic transforms π_{1} : X_{1} → X centered at 3 points so that A_{r}(X_{1}) holds and if q ∈ X_{1} is a bad 2 point such that (149) doesn’t hold, and there exists a curve D ⊂ \overline{S}_{r}(X_{1}) such that q ∈ D and D is r big at q, then D is r big. All exceptional 2 points for π_{1} which are bad must satisfy (149).

Let π_{2} : X_{2} → X_{1} be the blowup of such a D. By Lemma 8.8, A_{r}(X_{2}) holds and all 2 points in π_{2}^{−1}(D) are good. We have that if q ∈ X_{2} is a bad 2 point such that (149) doesn’t hold, and there exists a curve D ⊂ \overline{S}_{r}(X_{2}) such that q ∈ D and D is r big at q, then D is r big. If such a D exists, let π_{3} : X_{3} → X_{2} be the blowup of D.

After a finite sequence of blowups, we then obtain λ_{1} : Z_{1} → X such that A_{r}(Z_{1}) holds, and if q ∈ Z_{1} is a bad 2 point which doesn’t satisfy (149), then there doesn’t exist a curve D ⊂ \overline{S}_{r}(Z_{1}) such that D is r big at q. By Lemma 13.5, there exists a sequence of quadratic transforms λ_{2} : Z_{2} → Z_{1} such that A_{r}(Z_{2}) holds, and if q ∈ Z_{2} is a bad 2 point, then (149) holds at q.

By Theorems 7.1 and 7.8, there exists a sequence of quadratic transforms λ_{3} : Z_{2} → Z_{2} centered at 3 points such that the conclusions of the Theorem hold on Z_{3}.

Theorem 13.7. Suppose that the conclusions of Theorem 13.6 hold on X. Then there exists a finite sequence of quadratic transforms centered at 3 points X_{1} → X such that

1. A_{r}(X_{1}) holds.

2. All bad 2 points p ∈ X_{1} satisfy (149).

3. Suppose that D ⊂ \overline{S}_{r}(X) is a curve which is r big at a 1 point. Then there exists at most one 3 point q ∈ D and D has a tangent direction at q distinct from those of \overline{B}_{2}(X) at q. If D is not r big, there exists only one 2 point q ∈ D. If C is the 2 curve containing q, then C is not r-1 big or r small.

4. If C is a r small or r-1 big 2 curve containing a 2 point p such that p ∈ D where D is a curve r big at p, then D is r big.

5. If q ∈ X_{1} is a 3 point with ν(q) = r − 1, then either there are permissible parameters (x, y, z) at q such that

L_{q} depends on both y and z and F_{q} ∈ (y, z)^{r−1}  (151)

or there are permissible parameters (x, y, z) at q such that

F_{q} = τy^{r−1} + ∑_{j=1}^{r−1} a_{j}(x, z)x^{α_{j}}z^{β_{j}}y^{r−1−j}  (152)

where τ is a unit, a_{j} are units (or zero), α_{j} + β_{j} ≥ j for all j, and there exists i such that

\frac{α_{i}}{i} ≤ \frac{α_{j}}{j} , \frac{β_{i}}{i} ≤ \frac{β_{j}}{j}

for all j, and

\{\frac{α_{i}}{i} \} + \{\frac{β_{i}}{i} \} < 1.

Proof. X satisfies 1. - 4. of the conclusions of the Theorem. By Theorems 7.1 and 7.8, 1. - 4. are stable under quadratic transforms centered at 3 points.

Suppose that π_{1} : X_{1} → X is the blow up of a 3 point p.

If L_{p} depends on all three variables x, y, z then ν(q) ≤ r − 2 for all 3 points q ∈ π^{−1}(p).

Suppose that L_{p} depends on both y and z. Then ν(q) ≤ r − 2 for all 3 points q ∈ π^{−1}(p), except possibly under the quadratic transform

x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}.

At this 3 point q,

F_{q} = \frac{F_{p}}{x_{1}^{r−1}} = L_{p}(y_{1}, z_{1}) + x_{1}Ω.

By a sequence of quadratic transforms centered at 3 points, we can get the Theorem to hold above p, except possibly along an infinite sequence

R = O_{X,p} → R_{1} → ⋯ → R_{n} → ⋯

where for all n R_{n} has permissible parameters (x_{n}, y_{n}, z_{n}) with

x = x_{n}, y = x_{n}^{n}y_{n}, z = x_{n}^{n}z_{n}

and ν(\frac{F_{p}(x_{n},x_{n}^{n}y_{n},x_{n}^{n}z_{n})}{x^{n(r−1)}}) = r − 1. Thus F_{p} ∈ (y, z)^{r−1}.

Now suppose that L_{p} depends only on y. Then

F_{p} = τy^{r−1} + ∑_{i=1}^{r−1} a_{i}(x, z)y^{r−1−i}  (153)

where τ is a unit.

If p_{1} ∈ π_{1}^{−1}(p) is a 3 point with ν(q) = r − 1, then p_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) of one of the following 2 forms:

x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}

or

x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1}

and

F_{p_{1}} = τy_{1}^{r−1} + ∑_{i=1}^{r−1} \frac{a_{i}(x_{1}, x_{1}z_{1})}{x_{1}^{i}}y_{1}^{r−1−i}  (154)

or

F_{p_{1}} = τy_{1}^{r−1} + ∑_{i=1}^{r−1} \frac{a_{i}(x_{1}z_{1}, z_{1})}{z_{1}^{i}}y_{1}^{r−1−i}.

Suppose that

⋯ → X_{n} → X_{n−1} → ⋯ → X_{1} → X

is a sequence of quadratic transforms, π_{i} : X_{i} → X_{i−1} with induced maps λ_{i} : X_{i} → X such that for all i, π_{i} is the blowup of a 3 point p_{i−1}, with ν(p_{i−1}) = r − 1 and π_{i−1}(p_{i−1}) = p_{i−2}.

We will show that there exists n such that p_{n} satisfies (152). Each p_{i} has permissible parameters (x_{i}, y_{i}.z_{i}) such that either

x_{i−1} = x_{i}, y_{i−1} = x_{i}y_{i}, z_{i−1} = x_{i}z_{i}

or

x_{i−1} = x_{i}z_{i}, y_{i−1} = y_{i}z_{i}, z_{i−1} = z_{i}.

By (154), and resolution of plane curve singularities, there exists n_{0} such that n ≥ n_{0} implies F_{p_{n}} has the form

F_{p_{n}} = τy_{n}^{r−1} + ∑_{i=1}^{r−1} a_{n,i}(x_{n}, z_{n})x_{n}^{α_{ni}}z_{n}^{β_{ni}}y_{n}^{r−1−i}

where a_{n,i}(x_{n}, z_{n}) are units, and either

(α_{n+1,i}, β_{n+1,i}) = (α_{n,i} + β_{n,i} − i, β_{n,i})

or

(α_{n+1,i}, β_{n+1,i}) = (α_{n,i}, α_{n,i} + β_{n,i} − i)

for all i. The proof now follows from Lemmas 9.20 and 9.21.

Theorem 13.8. Suppose that A_{r}(X) holds with r ≥ 2 and p ∈ X is a 2 point such that (149) holds. Then either

1. There exists a sequence of quadratic transforms π : Y → X centered at points over p such that

(a) If q ∈ π^{−1}(p) is a 1 point then ν(q) ≤ r. ν(q) = r implies γ(q) = r.
(b) If q ∈ π^{−1}(p) is a 2 point then ν(p) ≤ r. ν(p) = r implies τ(p) ≥ 2.

(c) q ∈ π^{−1}(p) a 3 point implies ν(q) ≤ r − 1. ν(q) = r − 1 implies q satisfies the assumptions of (145) and (146) of Theorem 12.4. If D_{q} is the 2 curve containing q, with local equations y = z = 0 at q, in the notation of Theorem 12.4, then F_{q′} is resolved for all q ≠ q′ ∈ D_{q}.

(d) A_{r}(Y) holds.

or

2. There exists a curve C ⊂ \overline{S}_{r}(X) which is r big at p. Then there exists an affine neighborhood U of p such that the blowup of C ∩ U, π : Z → U is a permissible monodial transform such that

(a) If q ∈ π^{−1}(p) is a 2 point then ν(q) = 0.

(b) If q ∈ π^{−1}(p) is the 3 point then either ν(q) ≤ r − 2 or q satisfies the assumptions of (147)and (148) of Theorem 12.4. D_{q} = π^{−1}(p) is the 2 curve with local equations y = z = 0 at q in the notation of Theorem 12.4.

(c) A_{r}(Z) holds.

Proof. We first assume that the assumption of 2. doesn’t hold. There are permissible parameters (x, y, z) at p such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p}. \end{aligned}  (155)

By assumption, L_{p} has the form

L_{p} = f(x, y) + zg(x, y)  (156)

and L_{p} contains a y^{r−1}z term with

a(d + r − 1) − bc = 0.  (157)

Let π_{1} : X_{1} → X be the blowup of p.

Suppose that there exists a 2 point q ∈ π_{1}^{−1}(p) such that ν(q) = r and q has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}(z_{1} + α)

After a permissible change of parameters, we may assume that α = 0. Then F_{q} = \frac{F_{p}}{y_{1}^{r}} and ν(F_{q}(0, 0, z_{1})) ≤ 1, so that q is resolved.

Suppose that p_{1} ∈ π_{1}^{−1}(p) is a 2 point such that ν(p_{1}) = r and p_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, y = x_{1}y_{1}, z = x_{1}(z_{1} + α).

After making a permissible change of parameters, we may assume that α = 0. Then L_{p} depends only on y and z, and

L_{p} = \overline{e}y^{r} + \overline{b}zy^{r−1}

for some \overline{b}, \overline{e} ∈ k with \overline{b} ≠ 0.

Suppose that q ∈ π_{1}^{−1}(p) is another 2 point such that ν(q) = r, and q has permissible paramters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, y = x_{1}y_{1}, z = x_{1}(z_{1} + α)

with α ≠ 0.

Then there exists a form L such that

L_{p} = \{ \begin{matrix}L(y, z − αx) or \\ L(y, z − αx) + \overline{c}x\overline{α}y\overline{β} \\ for some \overline{c} ∈ k, such that a(d + \overline{β}) − b(c + \overline{α}) = 0, \overline{α} + \overline{β} = r. \end{matrix}

L_{p} = \overline{e}y^{r} + \overline{b}αxy^{r−1} + \overline{b}(z − αx)y^{r−1}

implies L(y, z − αx) = L_{p} − \overline{b}αxy^{r−1}, but \overline{α} = 1, \overline{β} = r − 1 is not possible, since

a(d + r − 1) − b(c + 1) = −b ≠ 0  (158)

Thus p_{1} is the unique 2 point q ∈ π_{1}^{−1}(p) with ν(q) = r. There are permissible parameters (x_{1}, y_{1}, z_{1}) at p_{1} such that

x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}

and L_{p} = L(y, z) = \overline{e}y^{r} + \overline{b}y^{r−1}z, L_{p_{1}} = L(y_{1}, z_{1}) + x_{1}Ω.

If p′ ∈ π^{−1}(p) is the 3 point, then ν(p′) ≤ r − 1. If p′ ∈ π^{−1}(p) is a 1 point then ν(p′) ≤ r and ν(p′) = r implies γ(p′) = r by Theorem 7.3.

By Theorem 7.3, τ(p_{1}) ≥ 1. Suppose that ν(p_{1}) = r and τ(p_{1}) = 1. Let π_{2} : X_{2} → X_{1} be the blowup of p_{1}.

We can make an analysis of π_{2}^{−1}(p_{1}) which is similar to that of π_{1}^{−1}(p). (155) is replaced with

\begin{aligned}u & = (x^{a+b}y^{b})^{m} \\ v & = P (x^{a+b}y^{b}) + x^{c+d+r}y^{d}F \end{aligned}

τ(p_{1}) = 1 implies L_{p_{1}} has the form of (156). (157) holds at p_{1}. (158) is then modified to: \overline{α} = 1, \overline{β} = r − 1 is not possible since

\begin{matrix}(a + b)(d + r − 1) − b(c + d + r + 1) \\ = a(d + r − 1) + b(d + r) − b − bc − b(d + r) − b = −2b ≠ 0 \end{matrix}

We conclude that there is at most one 2 point p_{2} ∈ π_{2}^{−1}(p_{1}) with ν(p_{2}) = r, and after replacing z with z − αx^{2} for some α ∈ k, we have that p_{2} has permissible parameters (x_{2}, y_{2}, z_{2}) such that

x = x_{2}, y = x_{2}^{2}y_{2}, z = x_{2}^{2}z_{2}.

Suppose that we can construct an infinite sequence of quadratic transforms

⋯ → X_{n} → ⋯ → X_{1} → X

Where π_{n} : X_{n+1} → X_{n} is the blowup of a 2 point p_{n} ∈ X_{n} over p such that ν(p_{n}) = r and τ(p_{n}) = 1. Then there exists a series σ(x) such that if we make a formal change of variables, replacing z with z − σ(x), we get that there are permissible parameters (x_{n}, y_{n}, z_{n}) at p_{n} such that

x = x_{1}, y = x_{1}^{n}y_{1}, z = x_{1}^{n}z_{1}

p_{n} must be the only 2 point of π_{n}^{−1}(p_{n−1}) such that ν(p_{n}) = r and τ(p_{n}) = 1. To show this, the argument following (155) is modified by replacing (155) with

\begin{aligned}u & = (x^{a+nb}y^{b})^{m} \\ v & = P (x^{a+nb}y^{b}) + x^{c+n(d+r)}y^{d}F \end{aligned}

where (157) holds at p_{n}. (158) is modified to:

\overline{α} = 1, \overline{β} = r − 1 is not possible since

(a + nb)(d + r − 1) − b(c + n(d + r) + 1)
= a(d + r − 1) + nb(d + r − 1) − bc − nb(d + r) − b
= −nb − b = −(n + 1)b ≠ 0

ν(p_{n}) = ν(\frac{F_{p}}{x_{1}^{nr}}) = r for all n implies that F_{p} ∈ (y, z)^{r}. Î_{\overline{S}_{r}(X),p} ⊂ (y, z) by Lemma 6.23, so that since A_{r}(X) holds, y = z = 0 are local equations at p of a curve C ⊂ \overline{S}_{r}(X), which is r big at p by Lemma 8.2, a contradiction to the assumption that the assumption of 2. doesn’t hold. Thus there exists a sequence of quadratic transforms π : Y → X such that if q ∈ π^{−1}(p) is a 2 point, then ν(q) ≤ r, and ν(q) = r implies τ(q) > 1. By Theorem 7.3 and Lemma 7.9, A_{r}(Y) holds.

Suppose that q ∈ π^{−1}(p) is a 3 point such that ν(q) = r−1. There exist permissible parameters (x_{2}, y_{2}, z_{2}) at q, and there exists a 2 point p_{n} ∈ X_{n} such that ν(p_{n}) = r, τ(p_{n}) = 1 and p_{n} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, y = x_{1}^{n}y_{1}, z = x_{1}^{n}z_{1}  (159)
x_{1} = x_{2}z_{2}, y_{1} = y_{2}z_{2}, z_{1} = z_{2}
\begin{aligned}u & = (x_{1}^{a+nb}y_{1}^{b})^{m} \\ v & = P (x_{1}^{a+nb}y_{1}^{b}) + x_{1}^{c+n(d+r)}y_{1}^{d}(\overline{b}y_{1}^{r−1}z_{1} + \overline{e}y_{1}^{r} + x_{1}Ω) \end{aligned}  (160)

with \overline{b} ≠ 0.

\begin{aligned}u & = (x_{2}^{a+nb}y_{2}^{b}z_{2}^{a+(n+1)b})^{m} \\ v & = P (x_{2}^{a+nb}y_{2}^{b}z_{2}^{a+(n+1)b}) + x_{2}^{c+n(d+r)}y_{2}^{d}z_{2}^{c+(n+1)(d+r)}(\overline{b}y_{2}^{r−1} + ⋯) \end{aligned} (161)

with a(d + r − 1) − bc = 0. Thus q satisfies the assumptions of (145) and (146) of Theorem 12.4, with x = y_{2}, y = x_{2}, z = z_{2}.

On Y we have:

1. If q ∈ π^{−1}(p) is a 1 point then ν(p) ≤ r. ν(p) = r implies γ(p) = r.

2. If q ∈ π^{−1}(p) is a 2 point then ν(p) ≤ r. ν(p) = r implies τ(p) ≥ 2.

3. q ∈ π^{−1}(p) a 3 point implies ν(q) ≤ r − 1. ν(q) = r − 1 implies q satisfies the assumptions of (145) and (146) of Theorem 12.4

4. A_{r}(Y) holds.

Let T be the set of 3 points q ∈ π^{−1}(p) such that ν(q) = r − 1, so that q satisfies (145) and (146) of Theorem 12.4.

Suppose that q ∈ T. In the factorization Y → X by quadratic transforms there exists a factorization Y → X_{n} → X such that q is an exceptional point on the blowup of a 2 point p_{n} of the form of (159) on X_{n}. Let τ : Y → X_{n} be this map, D_{q} be the nonsingular curve D_{q} ⊂ τ^{−1}(p_{n}) such that Î_{D_{q},q} = (x_{2}, z_{2}). The other points q′ ∈ D_{q} have regular parameters (x′, y′, z′) with the notation of (159) such that

x_{1} = x′y′, y_{1} = y′, z_{1} = y′(z′ + α)

with α ∈ k. At such a 2 point q′, we have ν(F_{q′}(0, 0, z′)) ≤ 1 by (160). Thus q′ is resolved.

We now prove 2. There are permissible parameters (x, y, z) at p such that y = z = 0 are local equations of C at p and

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p}. \end{aligned}

There exists \overline{a} ∈ k such that

L_{p} = \overline{a}y^{r} + zy^{r−1}

Let π_{1} : Y → spec(O_{X,p}) be the blowup of C. Suppose that q ∈ π_{1}^{−1}(p) is a 2 point. q has permissible parameters (x, y_{1}, z_{1}) such that

y = y_{1}, z = y_{1}(z_{1} + α)

for some α ∈ k.

\begin{aligned}u & = (x^{a}y_{1}^{b})^{m} \\ v & = p_{1}(x^{a}y_{1}^{b}) + x^{c}y_{1}^{d+r}(z_{1} + α′ + xΩ′ + y_{1}Ω′′) \end{aligned}

with α′ ∈ k, so that q is resolved. At the 3 point q ∈ π^{−1}(p), there are permissible parameters (x, y_{1}, z_{1}) such that

y = y_{1}z_{1}, z = z_{1}.

\begin{aligned}u & = (x^{a}y_{1}^{b}z_{1}^{b})^{m} \\ v & = P (x^{a}y_{1}^{b}z_{1}^{b}) + x^{c}y_{1}^{d}z_{1}^{d+r}F_{q} \\ F_{q} & = y_{1}^{r−1} + \overline{a}y_{1}^{r} + xΩ′ + zΩ′′ \end{aligned}

with a(d + r − 1) − bc = 0. Either ν(q) ≤ r − 2 or ν(q) = r − 1 and q satisfies the assumptions of (147) and (148) of Theorem 12.4 (with x = y_{1}, y = x_{1}, z = z_{1}).

The curve C blown up in Theorem 12.4 is the fiber π^{−1}(p), which is resolved away from q. There exists an affine neighborhood U of q such that if Z → U is the blowup of C ∩ U, then A_{r}(Z) holds by Lemma 8.8.

Theorem 13.9. Suppose that A_{r}(X) holds with r ≥ 2. Then there exists a finite sequence of permissible monodial transforms X_{1} → X such that

1. A_{r}(X_{1}) holds.

2. If p ∈ X_{1} is a 3 point, then ν(p) ≤ r − 2.

3. If p ∈ X_{1} is a 2 point such that ν(p) = r and τ(p) = 1 then p has permissible parameters (x, y, z) such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \end{aligned}  (162)

and L_{p} contains a nonzero y^{r−1}z term with a(d + r − 1) − bc = 0.

4. \overline{S}_{r}(X_{1}) makes SNCs with \overline{B}_{2}(X_{1}).

5. If C is a 2 curve on X_{1}, then C is not r small or r-1 big.

Proof. We may assume that the conclusions of Theorem 13.7 hold on X. Let {D_{1}, …, D_{n}} be the curves D in X which intersect a r-1 big or r small 2 curve at a 2 point such that D is r big there. By assumption, D_{1}, …, D_{n} are r big.

Let σ_{1} : W_{1} → X be the blowup of D_{1}. By Theorem 13.8 and Lemma 8.8 A_{r}(Z_{1}) holds. σ_{1}^{−1}(D_{1}) contains no bad 2 points. If q_{1} ∈ σ_{1}^{−1}(D_{1}) is a 3 point with ν(q_{1}) = r − 1, then q_{1} ∈ σ_{1}^{−1}(q) where q is a bad 2 point. In this case, 2. (b) of Theorem 13.8 holds at q_{1}.

Let \{\overline{D}_{2}, …, \overline{D}_{n}\} be the strict transforms of \{D_{2}, …, D_{n}\} on W_{1}. These curves are all r big, and are the curves D in Z_{1} which intersect a r-1 big or r small curve at a 2 point such that D is r big there.

We can blowup successively the strict transforms of \overline{D}_{2}, …, \overline{D}_{n} by a map λ : W → X to get a W such that A_{r}(W) holds, the exceptional locus of λ contains no bad 2 points, and if q is an exceptional 3 point with ν(q) = r − 1 then q must satisfy 2. (b) of Theorem 13.8.

Furthermore, if C is an r-1 big or r small 2 curve on W, and p ∈ C is a bad 2 point, then there does not exist a curve D ⊂ \overline{S}_{r}(W) such that D is r big at p.

By Theorem 13.8, after performing a sequence of quadratic transforms X_{1} → Z over bad 2 points p ∈ X such that if C is the 2 curve containing p ∈ X then C is r-1 big or r small, we have

1’.: The conclusions of 1. - 2. of Theorem 13.7 hold.

2.’: Suppose that C is a 2 curve such that C is r-1 big or r small. If p ∈ C is a 2

point, then p is good. 3’.: If p is a 3 point such that ν(p) = r − 1, then either

(a): p satisfies the assumptions of (145) and (146) of Theorem 12.4. If D_{p} is the 2 curve containing p, with local equations y = z = 0 at p, in the notation of Theorem 12.4, then F_{q} is resolved for all p ≠ q ∈ D_{p}. or (b): p satisfies the assumptions of (147) and (148) of Theorem 12.4. If D_{p} is the 2 curve containing p, with local equations y = z = 0 at p, in the notation of Theorem 12.4, then F_{q} is resolved for all p ≠ q ∈ D_{p}. or

(c): p satisfies (151) or (152) of Theorem 13.7. Let {p_{1}, …, p_{n}} be the 3 points of X_{1} which satisfy 3’. (a) or 3’ (b). By Theorem

12.4, there exist sequences of permissible monodial transforms, over sections of D_{p_{i}} for 1 ≤ i ≤ n,

λ_{p_{i}} : Y_{p_{i}} → spec(O_{X_{1},p_{i}})

such that the conclusions of Theorem 12.4 hold.

Since D_{p_{i}} is resolved at points p_{i} ≠ q ∈ D_{p_{i}}, the only obstruction to extending λ_{p_{i}} to a permissible sequence of monodial transforms of sections over D_{p_{i}} in X_{1} is if the corresponding sections over D_{p_{i}} in X_{1} do not make SNCs with 2 curves. This difficulty can be resolved by performing quadratic transforms at the points where the section does not make SNCs with 2 curves, since these points are necessarily resolved.

We can thus extend the

Y_{p_{i}} → spec(O_{X_{1},p_{i}})

to a sequence of permissible monoidal transforms

λ : Y → X_{1}

such that Y ×_{X_{1}} spec(O_{X_{1},p_{i}}) ≅ Y_{p_{i}} for 1 ≤ i ≤ n, A_{r}(Y) holds, 1’. and 2’. hold on Y − λ^{−1}({p_{1}, …, p_{n}}), and if q ∈ Y − λ^{−1}({p_{1}, …, p_{n}}) is a 3 point such that ν(q) = r − 1, then q satisfies (151) or (152) of Theorem 13.7. Let

⋯ → Z_{n} → ⋯ → Z_{1} → Y

be a sequence of permissible monodial transforms such that Z_{n} → Z_{n−1} is the blowup of a 2 curve C such that C is r-1 big or r small.

We will show that there exists n < ∞ such that Z_{n} does not contain a 2 curve C such that C is r-1 big or r small, and that Z_{n} satisfies the conclusions of the Theorem.

By Theorem 12.5, this holds above a neighborhood of λ^{−1}({p_{1}, …, p_{n}}). We must verify this condition over \overline{Y} = Y − λ^{−1}(\{p_{1}, …, p_{n}\}).

Suppose that C is a 2 curve on \overline{Y}, such that C is r-1 big or r small and p ∈ C is a 3 point. Then all 3 points q on C ⊂ \overline{Y} have ν(q) = r − 1, and satisfy (151) or (152).

Let π : Y_{1} → \overline{Y} be the blowup of C. The assumption that all 2 points of C are good and Lemma 8.6 imply that q ∈ π^{−1}(C) a 2 point implies ν(q) ≤ r and if ν(q) = r then either γ(q) = r or τ(q) = 1, q is a good 2 point, C ⊂ \overline{S}_{r}(X) and if \overline{C} is the 2 curve containing q, then \overline{C} is a section over C such that \overline{C} ⊂ \overline{S}_{r}(X). Lemma 8.6, the assumption that all 2 points are good, and Lemma 8.7 imply A_{r}(Y_{1}) holds. Further, by Lemma 8.6, if \overline{C} ⊂ π^{−1}(C) is a 2 curve such that \overline{C} is r-1 big or r small, then \overline{C} is a section over C. All 2 points of \overline{C} are good points.

Suppose that q ∈ C is a 3 point with permissible regular parameters (x, y, z) such that y = z = 0 are local equations of C and

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}F_{q}. \end{aligned}

F_{q} ∈ (y, z)^{r−1}.

Suppose that q_{1} ∈ π^{−1}(q), and q_{1} has permissible parameters (x, y_{1}, z_{1}) such that

y = y_{1}, z = y_{1}z_{1}

If (151) holds at q,

F_{q_{1}} = \frac{F_{q}}{y_{1}^{r−1}} = L_{q}(1, z_{1}) + y_{1}Ω + xΛ

implies ν(q_{1}) ≤ r − 2. If (152) holds at q we must have β_{j} ≥ j for all j, and

F_{q′} = τ + zΩ′

so that ν(q_{1}) = 0.

Now suppose that q_{1} ∈ π^{−1}(q) and q_{1} has permissible parameters (x, y_{1}, z_{1}) such that

y = y_{1}z_{1}, z = z_{1}

If (151) holds at q,

F_{q_{1}} = \frac{F_{q}}{z_{1}^{r−1}} = L_{q}(y_{1}, 1) + z_{1}Ω + xΛ

implies ν(q_{1}) ≤ r − 2. If (152) holds at q, we must have β_{j} ≥ j for all j, and

F_{q_{1}} = τy_{1}^{r−1} + ∑_{j=1}^{r−1} a_{j}(x, z_{1})x^{α_{j}}z_{1}^{β_{j}−j}y_{1}^{r−1−j}

so that either ν(q_{1}) ≤ r − 2, or q_{1} has the form of (152) also, but with \frac{β_{i}}{i} decreased by 1.

By Lemma 13.1 and Lemma 13.2, after a finite number of blowups of 2 curves C such that C is r small or r-1 big, we reach Ỹ → \overline{Y} such that Ỹ contains no 2 curves C such that C is r small or r-1 big. Since all 3 points q of Ỹ with ν(q) = r − 1 must satisfy (151) or (152), which implies that α_{j} ≥ j for all j or β_{j} ≥ j for all j, so that there exists a 2 curve through q which is r small or r-1 big, we must have ν(q) ≤ r − 2 if q ∈ Ỹ is a 3 point.

14. Resolution 2

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared. We define a new condition on X

Definition 14.1. Suppose that r ≥ 2. We will say that C_{r}(X) holds if:

1. If p ∈ X is a 1 point then ν(p) ≤ r. If ν(p) = r then γ(p) = r.

2. If p is a 2 point then ν(p) ≤ r. If ν(p) = r then γ(p) = r. If ν(p) = r − 1 then one of the following three cases must hold:

(a) τ(p) > 0 or

(b) γ(p) = r or

(c) r ≥ 3, ν(p) = r − 1, τ(p) = 0, p ∉ \overline{S}_{r}(X), there exists a unique curve D ⊂ \overline{S}_{r−1}(X) containing a 1 point such that p ∈ D, and permissible parameters (x, y, z) at p such that x = z = 0 are local equations of D,

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = τx^{r−1} + ∑_{j=1}^{r−1} \overline{a}_{j}(y, z)y^{d_{j}}z^{e_{j}}x^{r−1−j} \end{aligned}  (163)

where τ is a unit, \overline{a}_{j} are units (or 0). There exists i such that \overline{a}_{i} ≠ 0, e_{i} = i, 0 < d_{i} < i,

\frac{d_{i}}{i} ≤ \frac{d_{j}}{j} , \frac{e_{i}}{i} ≤ \frac{e_{j}}{j}

for all j, and

\{\frac{d_{i}}{i} \} + \{\frac{e_{i}}{i} \} < 1.

3. If p is a 3 point then ν(p) ≤ r − 2.

4. \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X).

Remark 14.2. If C_{r}(X) holds then there does not exist a 2 curve C on X such that C is r small or r-1 big.

Theorem 14.3. Suppose that r ≥ 2, A_{r}(X) holds, p ∈ X is a 2 point such that ν(p) = r and 2 ≤ τ(p) < r, then either

1. There exists a sequence of quadratic transforms π : Y → X over p such that (a) A_{r}(Y) holds.

(b) If q ∈ π^{−1}(p) is a 1 point then ν(q) ≤ r. ν(q) = r implies γ(q) = r.

(c) If q ∈ π^{−1}(p) is a 2 point then ν(q) ≤ r − 1.

(d) If q ∈ π^{−1}(p) is a 3 point, then ν(q) ≤ r − 2.

(e) If D ⊂ π^{−1}(p) is a 2 curve, then D is not r small or r-1 big.

or

2. There exists a curve C ⊂ \overline{S}_{r}(X) such that p ∈ C and C is r big at p. There exists an affine neighborhood U of p such that the blowup of C ∩ U, π : Y → U is a permissible monodial transform such that

(a) A_{r}(Y) holds.

(b) If q ∈ π^{−1}(p) is a 2 point, then ν(q) ≤ r − 1.

(c) If q ∈ π^{−1}(p) is the 3 point, then ν(q) ≤ r − 2.

(d) The 2 curve D = π^{−1}(p) is not r small or r-1 big. In either case, if X satisfies the conclusions of Theorem 13.9, then Y satisfies the conclusions of Theorem 13.9.

Proof. p has permissible parameters (x, y, z) such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = ∑_{i+j+k≥r} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

Suppose that there does not exist a curve C ⊂ \overline{S}_{r}(X) such that C is r big at p. Let π : X_{1} → X be the blowup of p. We will first show that (a), (b) and (d) of

1. hold on X_{1} and if q ∈ π^{−1}(p) is a 2 point with ν(q) = r then τ(q) ≥ τ(p). This follows from Theorem 7.1, Theorem 7.3 and Lemma 7.9. All exceptional 2 curves D of π contain a 3 point q such that ν(q) ≤ r − 2. (e) thus holds by Lemmas 8.1 and 7.7.

By Lemma 8.1 there are at most finitely many 2 points q ∈ π^{−1}(p) such that ν(q) = r. Suppose that there exists a 2 point q ∈ π^{−1}(p) and ν(q) = r. After a permissible change of parameters at p, we have permissible parameters (x_{1}, y_{1}, z_{1}) at q such that x = x_{1}, y = x_{1}y_{1}, z = x_{1}z_{1}. L_{p} = L_{p}(y, z) depends on both y and z.

Supppose there also exists a 2 point q′ ∈ π^{−1}(p) such that ν(q′) = r and q′ has permissible parameters (x′, y′, z′) such that

x = x′y′, y = y′, z = y′(z′ + α)

for some α ∈ k. Then there exists a form L(x, z − αy) such that

L_{p}(y, z) = \{ \begin{matrix}L(x, z − αy) + \overline{c}x\overline{a}y\overline{b} & if there exists \overline{a}, \overline{b} ∈ N such that \\ & \overline{a} + \overline{b} = r, a(d + \overline{b}) − b(c + \overline{a}) = 0 \\ L(x, z − αy) & otherwise \end{matrix}

Thus

L_{p} = \overline{d}(z − αy)^{r} + \overline{c}y^{r}

for some \overline{d}, \overline{c} ∈ k with \overline{d} ≠ 0, a contradiction to the assumption that τ(p) < r. Let

⋯ → Y_{n} → Y_{n−1} → ⋯ → Y_{1} → X

be the sequence of quadratic transforms π_{n} : Y_{n} → Y_{n−1} constructed by blowing up all 2 points q′ on Y_{n} which lie over p and have ν(q′) = r.

Suppose that this sequence has infinite length. Then there exists q_{n} ∈ Y_{n} such that π_{n}(q_{n}) = q_{n−1} and ν(q_{n}) = r for all n. There exists a series ϕ(x) = ∑ α_{i}x^{i} such that after replacing z with z − ϕ(x), q_{n} has permissible parameters (x_{n}, y_{n}, z_{n}) such that

x = x_{n}, y = x_{n}^{n}y_{n}, z = x_{n}^{n}z_{n}

and

F_{q_{n}} = L_{q}(y_{n}, z_{n}) + x_{n}Ω_{n}.

F_{q_{n}} = \frac{F_{q}}{x_{n}^{nr}} for all n > 0 implies F_{q} ∈ (y, z)^{r}.

Î_{\overline{S}_{r}(X),p} ⊂ (y, z) by Lemma 6.23. Since \overline{S}_{r}(X) makes SNCs with \overline{B}_{2}(X) at p, y = z = 0 are local equations at p of a curve C ⊂ \overline{S}_{r}(X).

Now suppose that there exists a curve C ⊂ \overline{S}_{r}(X) such that p ∈ C and C is r big at p. There exists an affine neighborhood U of p such that C ∩ U makes SNCs with \overline{B}_{2}(U). Let π : Y → U be the blowup of C ∩ U. There exist permissible parameters (x, y, z) at p such that y = z = 0 are local equations of C at p,

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ F_{p} & = ∑_{i+j≥r} a_{ij}(x)y^{i}z^{j}. \end{aligned}

At the 3 point q ∈ π^{−1}(p), there are permissible parameters (x, y_{1}, z_{1}) such that

y = y_{1}z_{1}, z = z_{1}

F_{q} = \frac{F_{p}}{z_{1}^{r}} = ∑_{i+j=r} a_{ij}(0)y_{1}^{i} + z_{1}G + xΩ

we have ν(q) ≤ r − 2, since 2 ≤ τ(p).

At a 2 point q ∈ π^{−1}(p), after a permissible change of variables at p, there exist permissible parameters (x, y_{1}, z_{1}) at q such that

y = y_{1}, z = y_{1}z_{1}

F_{q} = \frac{F_{p}}{y_{1}^{r}} = ∑_{i+j=r} a_{ij}(0)z_{1}^{j} + y_{1}G + xΩ.

ν(q) < r and γ(q) < r since τ(p) < r. Furthermore, if D = π^{−1}(p), then F_{q} ∉ Î_{D,q}^{r−1}. There exists a possibly smaller affine neighborhood U of p such that A_{r}(Y) holds by Lemma 8.8.

Theorem 14.4. Suppose that the conclusions of Theorem 13.9 hold on X with r ≥ 2, p ∈ X is a 2 point with permissible parameters (x, y, z) such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \end{aligned}

and ν(p) = r − 1, τ(p) = 0, L_{p} = f(x, y) depends on both x and y. Then there exists a sequence of quadratic transforms π : Z → X over p such that

1. q ∈ π^{−1}(p) a 1 point or a 2 point implies that ν(q) ≤ r. ν(q) = r implies γ(q) = r.

2. q ∈ π^{−1}(p) a 2 point with ν(q) = r − 1 implies that τ(q) > 0 or γ(q) = r.

3. q ∈ π^{−1}(p) a 3 point implies ν(q) ≤ r − 2.

4. The conclusions of Theorem 13.9 hold on Z Proof. Let

π : X_{1} → X  (164)

be the blowup of p.

If p_{1} ∈ π^{−1}(p) is a 1 point then ν(p_{1}) ≤ r and ν(p_{1}) = r implies γ(p_{1}) = r by Theorem 7.1. If p_{1} ∈ π^{−1}(p) is a 2 point then we must have ν(p_{1}) ≤ r − 2, by our assumption on f. Suppose that p_{1} ∈ π^{−1}(p) is the 3 point. Then ν(p_{1}) ≤ r − 1 and p_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1}

Suppose that ν(p_{1}) = r − 1. Then

L_{p_{1}} = f(x_{1}, y_{1}) + z_{1}Ω.  (165)

Let

F_{p} = ∑_{i+j+k≥r−1} a_{ijk}x^{i}y^{j}z^{k}.

Suppose that we can construct an infinite sequence of quadratic transforms

⋯ → X_{n} → ⋯ → X_{1} → X

where X_{n+1} → X_{n} is the blowup of a 3 point p_{n} lying over p_{n−1} with ν(p_{n}) = r − 1. Then p_{n} has permissible parameters (x_{n}, y_{n}, z_{n}) such that

x = x_{n}z_{n}^{n}, y = y_{n}z_{n}^{n}, z = z_{n}

and

F_{p_{n}} = \frac{F_{p}}{z_{n}^{n(r−1)}} = ∑ a_{ijk}x_{n}^{i}y_{n}^{j}z_{n}^{(k+n(i+j−r+1))}.

Thus a_{ijk} = 0 if i + j < r − 1, which implies that F_{p} ∈ (x, y)^{r−1}, a contradiction since the conclusions of Theorem 13.9 hold.

Thus by Theorem 7.1 and Lemma 7.9 there exists a finite sequence of quadratic transforms

π : X_{m} → ⋯ → X_{1} → X

where X_{n+1} → X_{n} is the blowup of a 3 point p_{n} lying over p_{n−1} with ν(p_{n}) = r − 1, such that A_{r}(X_{m}) holds, ν(q) ≤ r − 2 if q ∈ π^{−1}(p) is a 3 point, and if q ∈ π^{−1}(p) is a 1 point then ν(q) ≤ r, ν(q) = r implies γ(q) = r. Suppose that C is a 2 curve which is exceptional for π. Then C is not r-1 big or r small since C must contain a 3 point q′ with ν(q′) ≤ r − 2. Suppose that q ∈ π^{−1}(p) is a 2 point and ν(q) ≥ r − 1. Then there exists a largest n such that q maps to a 3 point p_{n} ∈ X_{n}. The point q is then a 2 point on X_{n+1}. p_{n} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}z_{1}^{n}, y = y_{1}z_{1}^{n}, z = z_{1}

By assumption, ν(p_{n}) = r − 1. Write

f = ∑_{i+j=r−1} a_{ij}x^{i}y^{j}.

We then have

\begin{aligned}u & = (x_{1}^{a}y_{1}^{b}z_{1}^{n(a+b)})^{m} \\ v & = P (x_{1}^{a}y_{1}^{b}z_{1}^{n(a+b)}) + x_{1}^{c}y_{1}^{d}z_{1}^{n(c+d+r−1)}F_{p_{n}} \\ F_{p_{n}} & = \frac{F_{p}}{z_{1}^{n(r−1)}} = ∑_{i+j=r−1} a_{ij}x_{1}^{i}y_{1}^{j} + ∑_{i+j+k=r−1,k>0} b_{ijk}x_{1}^{i}y_{1}^{j}z_{1}^{k} + Ω \end{aligned} (166)

with ν(Ω) ≥ r.

Since q is a 2 point, Ô_{X_{n+1},q} has regular parameters (x_{2}, y_{2}, z_{2}) of one of the following forms:

x_{1} = x_{2}, y_{1} = x_{2}(y_{2} + α), z_{1} = x_{2}z_{2}  (167)

with α ≠ 0, or

x_{1} = x_{2}, y_{1} = x_{2}y_{2}, z_{1} = x_{2}(z_{2} + β)  (168)

with β ≠ 0, or

x_{1} = x_{2}y_{2}, y_{1} = y_{2}, z_{1} = y_{2}(z_{2} + β)  (169)

with β ≠ 0.

First suppose that (168) holds. (169) is symmetrical, and the analysis of that case is the same. Set

x_{2} = \overline{x}_{2}(z_{2} + β)^{− \frac{n}{n+1}} .
u = (\overline{x}_{2}^{(n+1)(a+b)}y_{2}^{b})^{m} = (\overline{x}\overline{a} ; 2y\overline{b} ; 2)\overline{m}
v = P_{q}(\overline{x}\overline{a} ; 2y\overline{b} ; 2) + \overline{x}_{2}^{(n+1)(c+d+r−1)}y_{2}^{d}F_{q}
F_{q} = [∑_{i+j=r−1} a_{ij}y_{2}^{j}
+ ∑_{k>0,i+j+k=r−1} b_{ijk}y_{2}^{j}(z_{2} + β)^{k} + \overline{x}_{2}Ω^{1}] − ∑ c_{i}\overline{x}_{2}^{\overline{a}_{i}}y_{2}^{\overline{b}_{i}}

where (\overline{a}, \overline{b}) = 1,

(n + 1)(a + b)(\overline{b}_{i} + d) − b((n + 1)(c + d + r − 1) + \overline{a}_{i}) = 0.

If some b_{ijk} ≠ 0 in (166), we have ν(q) ≤ r − 1 and ν(q) = r − 1 implies that τ(q) > 0. So suppose that all b_{ijk} = 0 in (166). If ν(q) ≥ r − 1, then we must have

∑ a_{ij}y_{2}^{j} = a_{i_{0}j_{0}}y_{2}^{j_{0}} + a_{0,r−1}y_{2}^{r−1}

where a_{0,r−1} could be zero, 0 < i_{0}, a_{i_{0},j_{0}} ≠ 0 (since f(x, y) depends on x and y) and

(n + 1)(c + d + r − 1)b − (n + 1)(a + b)(j_{0} + d) = 0.

Thus

(c + r − 1 − j_{0})b − a(d + j_{0}) = 0  (170)

We then have

x^{c}y^{d}f = a_{i_{0}j_{0}}x^{c+r−1−j_{0}}y^{d+j_{0}} + a_{0,r−1}x^{c}y^{d+r−1}

which is normalized, so that a_{i_{0},j_{0}} = 0 by (170). This contradiction shows that we must have that ν(q) ≤ r − 2 in this case.

Suppose that (167) holds. Substitute (167) into (166). Set

x_{2} = \overline{x}_{2}(y_{2} + α)^{− \frac{b}{(n+1)(a+b)}}

\begin{aligned}u & = (\overline{x}_{2}^{(n+1)}z_{2}^{n})^{m(a+b)} \\ v & = P (\overline{x}_{2}^{(n+1)(a+b)}z_{2}^{n(a+b)}) + \overline{x}_{2}^{(n+1)(c+d+r−1)}z_{2}^{n(c+d+r−1)}G \end{aligned}

where

G = (y_{2} + α)^{λ} ( ∑_{i+j=r−1} a_{ij}(y_{2} + α)^{j} + ∑_{k>0,i+j+k=r−1} b_{ijk}(y_{2} + α)^{j}z_{2}^{k} + \overline{x}_{2}Ω_{2}) ,

with

λ = −\frac{b}{a + b}(c + d + r − 1) + d.

The only term which can be removed from the first sum

(y_{2} + α)^{λ} ( ∑_{i+j=r−1} a_{ij}(y_{2} + α)^{j}))

of G in obtaining F_{q} is the constant term. Thus γ(q) ≤ r.

Theorem 14.5. Suppose that r ≥ 2 and the conclusions of Theorem 13.9 hold on X, so that if C is a 2 curve, then C is not r small or r-1 big. Suppose that p ∈ X is a 1 or a 2 point and D is a generic curve through p on a component of E_{X}. Then there exists a sequence of quadratic transforms centered over a finite number of points on the strict transform of D, but not in the fiber over p, π : X_{1} → X, such that the following conditions hold.

1. There exists a neighborhood U of D − p such that C_{r}(π^{−1}(U)) holds. The case 2. (c) of C_{r} does not occur in π^{−1}(U).

2. Let D′ be the strict transform of D on X_{1}. Suppose that q ∈ D′ −p, and (x, y, z) are permissible parameters at q such that x = z = 0 are local equations of D′ at q. If q is a 1 point then ν(F_{q}(0, y, 0)) = 1. If q is a 2 point then ν(q) = 0.

3. The conclusions of Theorem 13.9 hold on X_{1}.

Proof. Suppose that q ∈ D is a 1 point. Then we can find permissible parameters (x, y, z) at q such that x = y = 0 are local equations of D at q. The multiplicity

ϕ(q) = ν(F_{q}(0, 0, z))

is independent of such permissible parameters at q. Furthermore, the set

\{q ∈ D ∩ (X − \overline{B}_{2}(X)) | ϕ(q) ≥ 2\}

is Zariski closed in D ∩ (X − \overline{B}_{2}(X)). By Lemma 6.31, F_{q} ∉ Î_{D,q} if q ∈ D. At most 1 points q on D, ϕ(q) = 1. Thus there are at most a finite number of points q ∈ D − p such that the conclusions of the Theorem do not hold at q.

1) Suppose that q ∈ D − p and ν(q) = r. Then q is a generic point on a curve C of \overline{S}_{r}(X). q is a 1 point.

1a) Suppose that C is r big. Then there exist permissible parameters (x, y, z) at q such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}F_{q} \\ F_{q} & = ∑_{i+k≥r,j≥0} a_{ijk}x^{i}y^{j}z^{k} \end{aligned}

where x = z = 0 are local equations of C at q, x = y = 0 are local equations of D at q. γ(q) = r implies a_{00r} ≠ 0.

Let π : Y → X be the blowup of q. Then ν(q′) = 0 if q′ is the point on the intersection of the strict transform of D and π^{−1}(q). Points of π^{−1}(q) satisfy the condition of C_{r} by Theorem 7.3 and Lemma 7.9.

1b) Suppose that C is r small. By Lemma 6.25,

F_{q} = ∑_{i+j≥r} a_{ij}(y)x^{i}z^{j} + τ(y)x^{r−1}

where x = z = 0 are local equations of C at q, x = y = 0 are local equations of D at q, (with ν(τ) ≥ 1). Since q is a generic point of C, ν(τ) = 1, and after a permissible change of parameters, we have τ = y. γ(q) = r implies a_{0r}(y) is a unit. Let π : Y → X be the blowup of q. Then ν(q′) = 0 if q′ is the point on the intersection of the strict transform of D and π^{−1}(q). Points of π^{−1}(q) satisfy the condition of C_{r} by Theorem 7.3 and Lemma 7.9.

2) Suppose that q ∈ D − p, ν(q) = r − 1 and the conclusions of the Theorem do not hold at q.

2a) Suppose that q is a 1 point and r ≥ 3. Then q is a general point on a curve C in \overline{S}_{r−1}(X). There are permissible parameters (x, y, z) at q such that x = z = 0 are local equations of D at q.

Let π_{1} : X_{1} → X be the blowup of p. Theorem 7.1 implies ν(q′) ≤ r − 1 for all q′ ∈ π_{1}^{−1}(q) and q′ ∈ π_{1}^{−1}(q) a 2 point with ν(q′) = r − 1 implies τ(q′) > 0.

At the 2 point q_{1} ∈ π_{1}^{−1}(q) on the strict transform of D, there are permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}z_{1}

Suppose that ν(q_{1}) = r − 1. We must have τ(q_{1}) > 0. Let π_{1} : X_{2} → X_{1} be the blowup of q_{1}. By Theorem 7.3, if q′ ∈ π^{−1}(q_{1}), then if q′ is a 1 point ν(q′) ≤ r − 1. If q′ is a 2 point, ν(q′) ≤ r − 1, ν(q′) = r − 1 implies τ(q′) > 0. q′ a 3 point implies ν(q′) ≤ r − 2. Let q_{2} ∈ π_{1}^{−1}(q) be the 2 point on the strict transform of D. There are permissible parameters (x_{2}, y_{2}, z_{2}) at q_{2} such that

x_{1} = x_{2}y_{2}, y_{1} = y_{2}, z_{1} = y_{2}z_{2}.

If ν(q_{2}) = r − 1, then τ(q_{2}) > 0.

Suppose that we can construct an infinite sequence of quadratic transforms

⋯ → X_{n} → X_{n−1} → ⋯ → X_{1} → X

centered at the point q_{n} on the strict transform of D over q on X_{n}, where q_{n} are blown up as long as ν(q_{n}) = r − 1.

By Theorem 7.3, all points q′ on X_{n} lieing over p satisfy ν(q′) ≤ r −1, ν(q′) ≤ r −2 If q′ is a 3 point and if q′ is a 2 point with ν(q′) = r − 1. Then τ(q′) > 0.

Suppose that ν(q_{n}) = r−1 for all n. Then q_{n} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}^{n}, y = y_{1}, z = z_{1}y_{1}^{n}

F_{q} = ∑_{i+j+k≥r−1} a_{ijk}x^{i}y^{j}z^{k}

F_{q_{n}} = \frac{F_{q}}{y_{1}^{n(r−1)}} = ∑_{i+j+k≥r−1} a_{ijk}x_{1}^{i}y_{1}^{j+n(i+k−(r−1))}z_{1}^{k}

implies a_{ijk} = 0 if i + k < r − 1, so that F_{q} ∈ (x, z)^{r−1}. This is a contradiction since F_{q} ∉ Î_{D,q}.

Thus after a finite sequence of quadratic transforms, π : Z → X the strict transform of D intersects π^{−1}(q) at a 2 point q_{1} with ν(q_{1}) < r − 1, so we are in case 3) below.

2b) Suppose that q is a 2 point. Suppose that C is the 2 curve through q. By Lemma 6.26, our assumption that C is not r-1 big, and since q is a generic point of C, we have τ(q) > 0. There exist permissible parameters (x, y, z) at q such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \\ F_{q} & = ∑_{i+j≥r−1,k≥0} a_{ijk}x^{i}y^{j}z^{k} + zx^{i_{0}}y^{j_{0}} \end{aligned}  (171)

with i_{0} + j_{0} = r − 2. x = z = 0 are local equations of D at q and τ(q) > 0.

Let π : X_{1} → X be the blowup of q. Then ν(q_{1}) ≤ r − 1 at all points q_{1} ∈ π^{−1}(q), ν(q_{1}) ≤ r−2 if q_{1} ∈ π^{−1}(q) is a 3 point and all 2 points q_{1} of π^{−1}(q) with ν(q_{1}) = r−1 satisfy τ(q_{1}) > 0 by Theorem 7.3.

The strict transform of D intersects π^{−1}(q) at a 2 point q′ such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}z_{1}

F_{q_{1}} = \frac{F_{q}}{y_{1}^{r−1}} = ∑_{i+j≥r−1,k≥0} a_{ijk}x_{1}^{i}y_{1}^{(i+j+k)−(r−1)}z_{1}^{k} + z_{1}x_{1}^{i_{0}}

x_{1} = z_{1} = 0 are local equations of the strict transform of D at q′. If ν(q′) ≤ r − 2 we are in case 3). Otherwise, q_{1} is a 2 point with ν(q_{1}) = r − 1 and τ(q_{1}) > 0 (so that i_{0} = r − 2). Let

⋯ → X_{n} → X_{n−1} → ⋯ → X_{1} → X

be the sequence of quadratic transforms centered at the point q_{n} on the strict transform of D over q on X_{n} where points q_{n} are blownup as long as ν(q_{n}) = r − 1. By Theorem 7.3, all points q′ on X_{n} lieing over p satisfy ν(q′) ≤ r − 1, ν(q′) ≤ r − 2 if q′ is a 3 point, and if q′ is a 2 point with ν(q′) = r − 1, then τ(q′) > 0.

Suppose that ν(q_{n}) = r − 1 for all n. q_{n} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}^{n}, y = y_{1}, z = z_{1}y_{1}^{n}

F_{q_{n}} = \frac{F_{q}}{y_{1}^{n(r−1)}} = ∑_{i+j≥r−1,k≥0} a_{ijk}x_{1}^{i}y_{1}^{j+n(i+k−(r−1))}z_{i}^{k} + z_{1}x_{1}^{r−2}

Thus i + k − (r − 1) ≥ 0 whenever a_{ijk} ≠ 0 and F_{p} ∈ (x, z)^{r−1}, so that F_{p} ∈ Î_{D,p}, which is a contradiction.

After a finite sequence of quadratic transforms, π′ : X′ → X, the strict transform of D thus intersects (π′)^{−1}(q) at a 2 point q′ with ν(q′) ≤ r − 2, so the result follows from Case 3).

3) Suppose that q ∈ D − p, ν(q) ≤ r − 2, and the conclusions of the Theorem do not hold at q. q is a 1 point or a 2 point and D makes SNCs with the 2 curve through C. The result then follows from a similar but slightly simpler argument to that of Case 2, by Theorems 7.1 and 7.3.

Theorem 14.6. Suppose that X satisfies the conclusions of Theorem 13.9 with r ≥ 2. Then there exists a sequence of permissible monoidal transforms X_{1} → X such that C_{r}(X_{1}) holds.

Proof. Let T be the finite set of 2 points p on X such that (162) holds at p, and p ∉ D for any r big curve D which contains a 1 point.

By 1. of Theorem 13.8, there exists a sequence of quadratic transforms π_{0} : X_{0} → X centered over points p ∈ T such that

1. If p ∈ X_{0} is a 1 point then ν(p) ≤ r, ν(p) = r implies γ(p) = r.

2. If p ∈ X_{0} is a 2 point then ν(p) ≤ r. ν(p) = r implies τ(p) ≥ 2, or (162) holds at p and there exists an r big curve D ⊂ \overline{S}_{r}(X_{0}) containing p.

3. If p ∈ X_{0} is a 3 point, then ν(p) ≤ r − 1. ν(p) = r − 1 implies p satisfies the assumptions of (145) and (146) of Theorem 12.4. If D_{p} is the 2 curve containing p with local equations y = z = 0 at p in the notation of Theorem 12.4, then F_{q} is resolved for all p ≠ q ∈ D_{p}.

4. A_{r}(X_{0}) holds.

5. If C is a 2 curve on X_{0}, then C is not r small. If C is r-1 big, then ν(p) = r − 1 for all p ∈ C.

Let T_{1} be the 3 points of X_{0} which satisfy (145) and (146) of Theorem 12.4. For p ∈ T_{1}, let λ_{p} : Y_{p} → spec(O_{X_{0},p}) be the sequence of monodial transforms

centered over sections of D_{p} such that the conclusions of Theorem 12.4 hold. By Theorem 12.5 (or Theorem 12.6 if r = 2), there exists a sequence of monodial transforms V_{p} → Y_{p} centered at 2 curves C such that C is r-1 big so that V_{p} satisfies the conclusions of 1. - 3. of Theorem 12.5 (or Theorem 12.6 if r = 2).

Since D_{p} is resolved at all points p ≠ q ∈ D_{p}, the only obstruction to extending λ_{p} to a permissible sequence of monodial transforms of sections over D_{p} in X_{0} is if the corresponding sections over D_{p} do not make SNCs with 2 curves. This difficulty can be removed by performing quadratic transforms at the (resolved) points where the section does not make SNCs with the 2 curves.

By 5. above and Lemmas 8.6, 8.7 and 13.2, we can thus construct a sequence of permissible monodial transforms π′_{0} : X′_{0} → X_{0} such that

X′_{0} ×_{X_{0}} spec(O_{X_{0},p}) ≅ V_{p}

for p ∈ T_{1}, and X′_{0} satisfies:

1. p ∈ X′_{0} a 1 point implies ν(p) ≤ r. ν(p) = r implies γ(p) = r.

2. p ∈ X′_{0} a 2 point implies ν(p) ≤ r. ν(p) = r implies τ(p) ≥ 2 or (162) holds at p, and there exists a r big curve D ⊂ \overline{S}_{r}(X_{0}) containing p.

3. p ∈ X′_{0} a 3 point implies ν(p) ≤ r − 2.

4. \overline{S}_{r}(X′_{0}) makes SNCSs with \overline{B}_{2}(X′_{0}).

5. If C is a 2 curve on X_{0}, then C is not r small or r-1 big. Let γ_{1}, …, γ_{n} be the r big curves in \overline{S}_{r}(X′_{0}). Each γ_{i} necessarily contains a 1

point.

Let π : X_{1} → X′_{0} be the sequence of monodial transforms (in any order) centered at the (strict transforms of) γ_{1}, …, γ_{n}.

By Lemma 8.8, 2. of Theorem 14.3, and 2. of Theorem 13.8,

1. If p ∈ X_{1} is a 1 point then ν(p) ≤ r. ν(p) = r implies γ(p) = r.

2. If p ∈ X_{1} is a 2 point then ν(p) ≤ r. ν(p) = r implies τ(p) ≥ 2. If ν(p) = r and τ(p) < r, then p does not lie on a r big curve E in \overline{S}_{r}(X_{1}).

3. p ∈ X_{1} a 3 point implies ν(p) ≤ r − 1. ν(p) = r − 1 implies p satisfies the assumptions of (147) and (148) of Theorem 12.4. If D_{p} is the 2 curve containing p with local equations y = z = 0 at p in the notation of Theorem 12.4, then F_{q} is resolved for all p ≠ q ∈ D_{p}.

4. A_{r}(X_{1}) holds.

5. If C is a 2 curve on X_{0}, then C is not r small or r-1 big.

6. There are only finitely many 2 points p ∈ X_{1} such that ν(p) = r.

6. is a consequence of 5. Let T_{2} be the set of 3 points on X_{1} satisfying (147) and (148) of Theorem 12.4.

By Theorem 14.3, there exists a sequence of quadratic transforms π_{2} : X_{2} → X_{1} centered over the 2 points p of X_{1} with ν(p) = r and 2 ≤ τ(p) < r such that 1. If p ∈ X_{2} is a 1 or 2 point then ν(p) ≤ r. If ν(p) = r then γ(p) = r.

2. If p ∈ X_{2} is a 3 point then ν(p) ≤ r − 1. If ν(p) = r − 1, then p ∈ T_{2}.

3. A_{r}(X_{2}) holds.

4. If D ⊂ X_{2} is a 2 curve, then D is not r small or r-1 big. For p ∈ T_{2}, let

λ^{p} : Y_{p} → spec(O_{X_{2},p})  (172)

be a sequence of permissible monodial transforms over sections of D_{p} such that the conclusions of Theorem 12.4 hold.

Since D_{p} is resolved at all points p ≠ q ∈ D_{p}, the only obstruction to extending λ^{p} to a permissible sequence of monodial transforms of sections over D_{p} in X_{2} is if the corresponding sections over D_{p} do not make SNCs with 2 curves. This difficulty can be removed by performing quadratic transforms at the points where the section does not make SNCs with the 2 curves.

By Theorems 7.1 and 7.3, we can thus construct a permissible sequence of monodial transforms π_{3} : X_{3} → X_{2} such that

1. If p ∈ T_{2}, then X_{3} ×_{X_{2}} spec(O_{X_{2},p}) ≅ Y_{p}.

2. If q ∈ X_{3} − π_{3}^{−1}(T_{2}), and q is a 1 or 2 point then ν(q) ≤ r. If ν(q) = r, then γ(q) = r.

3. If q ∈ X_{3} − π_{3}^{−1}(T_{2}) and q is a 3 point then ν(q) ≤ r − 2.

4. A_{r}(X_{3}) holds.

5. If D ⊂ X_{3} − π_{3}^{−1}(T_{2}) is a 2 curve, then D is not r small or r-1 big. By 5. and Theorem 12.5 (or Theorem 12.6 if r = 2), we can perform a sequence of

permissible monodial transforms σ : \overline{Z}_{2} → X_{3} centered at r-1 big 2 curves C to get that

1. If p ∈ \overline{Z}_{2} is a 1 or 2 point, then ν(p) ≤ r. ν(p) = r implies γ(p) = r.

2. If p ∈ \overline{Z}_{2} is a 3 point, then ν(p) ≤ r − 2.

3. There are no 2 curves C in \overline{Z}_{2} which are r small or r-1 big.

4. \overline{S}_{r}(\overline{Z}_{2}) makes SNCs with \overline{B}_{2}(Z_{2}).

Since 3. holds, there are only finitely many 2 points \{q_{1}, …, q_{m}\} on \overline{Z}_{2} such that ν(q_{i}) = r − 1 and τ(q_{i}) = 0.

By Theorem 14.4, we can perform a sequence of quadratic transforms σ_{1} : W_{1} → \overline{Z}_{2} over the finitely many 2 points q_{i} in \overline{Z}_{2} such that ν(q_{i}) = r − 1, τ(q_{i}) = 0 and L_{q} depends on both x and y (where (x, y, z) are permissible parameters at q_{i}) so that

1. ν(q) ≤ r and ν(q) = r implies γ(q) = r at 1 and 2 points of W_{1}.

2. ν(q) ≤ r − 2 at 3 points of W_{1}

3. If q ∈ W_{1} is a 2 point with ν(q) = r − 1 and τ(q) = 0, then either γ(q) = r or there exist permissible parameters (x, y, z) at q such that L_{q} depends only on x.

4. There are no 2 curves C on W_{1} which are r small or r-1 big.

5. \overline{S}_{r}(W_{1}) makes SNCs with \overline{B}_{2}(W_{1}). Over the (finitely many) points {a_{1}, …, a_{n}} of W_{1} which are 2 points with ν(p) =

r − 1, γ(p) > r, τ(p) = 0, and there exist permissible parameters (x, y, z) at a_{i} such that L_{a_{i}} depends only on x, by Theorem 12.1, there exist sequences of permissible monodial transforms

Y_{a_{i}} → spec(O_{W_{1},a_{i}})

where Y_{a_{i}} → spec(O_{W_{1},a_{i}}) is a sequence of blowups of sections over a general curve through a_{i}, and satisfies the conclusions of Theorem 12.1.

By Theorem 14.5 there exists a sequence of permissible monodial transforms π̃ : W_{2} → W_{1} such that

W_{2} ×_{W_{1}} spec(O_{W_{1},a_{i}}) ≅ Y_{a_{i}}

for all i, and for q ∈ π̃^{−1}(W_{2} − {a_{1}, …, a_{n}}),

1. ν(q) ≤ r, ν(q) = r implies γ(q) ≤ r if q is a 1 or 2 point.

2. q a 2 point and ν(q) = r − 1 implies τ(q) > 0 or γ(q) = r.

3. ν(q) ≤ r − 2 if q is a 3 point.

4. There are no 2 curves C in π̃^{−1}(W_{2} − {a_{1}, …, a_{n}}) which are r small or r-1 big.

5. A_{r}(W_{2}) holds.

Since all 2 curves C ⊂ W_{2} which are r-1 big must map to some a_{i} by 4., there exists a sequence of permissible monodial transforms π_{3} : W_{3} → W_{2} centered at 2 curves C which are r-1 big such that

W_{3} − (π̃ ∘ π_{3})^{−1}({a_{,} …, a_{n}}) ≅ W_{2} − π̃^{−1}({a_{1}, …, a_{n}})

and W_{3} ×_{W_{1}} spec(O_{W_{1},a_{i}}) satisfies the conclusions of V_{a_{i}} of Theorem 12.2 (or of V_{a_{i}} of Theorem 12.3).

Then the sequence of quadratic transforms W_{a_{i}} → V_{a_{i}} of Theorem 12.2 (or of W_{a_{i}} → V_{a_{i}} of Theorem 12.3) extend to π_{4} : W_{4} → W_{3} such that

W_{4} − (π̃ ∘ π_{3} ∘ π_{4})^{−1}({a_{1}, …, a_{n}}) ≅ W_{2} − π̃^{−1}({a_{1}, …, a_{n}})

and W_{4} ×_{W_{1}} spec(O_{W_{1},a_{i}}) ≅ W_{a_{i}} for 1 ≤ i ≤ n in the notation of Theorem 12.2 (or of Theorem 12.3).

Now assume that r ≥ 3. Let \{D\} be the strict transform on W_{4} of the curves \{\overline{D}\} in \overline{S}_{r}(W_{1}) which contain some a_{i}. Each \overline{D} contains a 1 point and \overline{D} is r small since \overline{D} makes SNCs with \overline{B}_{2}(W_{1}) and ν(a_{i}) = r − 1. By Theorem 12.2 and Lemma 8.10, and since by Lemma 6.27 there does not exist a 2 point q ∈ D such that ν(q) = r − 1 and τ(q) > 0, there exists a finite sequence of quadratic transforms W_{5} → W_{4} centered at points disjoint from any fiber over some a_{i} such that if W_{6} → W_{5} is a sequence of monodial transforms centered at the strict transforms of the D then C_{r}(W_{6}) holds.

Now suppose that r = 2. Let {D} be the strict transforms on W_{4} of the curves \{\overline{D}\} in \overline{S}_{r}(W_{1}) which contain some a_{i}. Each \overline{D} contains a 1 point and \overline{D} is not r big. By Theorem 12.3, Lemmas 8.10 and 8.11 there exists a sequence of quadratic transforms W_{5} → W_{4} centered at points disjoint from any fiber over some a_{i} such that if W_{6} → W_{5} is a sequence of monodial transforms centered at the strict transforms of the D, and then followed by a sequence of monodial transforms W_{7} → W_{6} centered at the strict transforms of 2 curves C on W_{6} which are sections over one of the D blownup in W_{6} → W_{5} and such that C is 1 big, then C_{r}(W_{6}) holds.

Theorem 14.7. Suppose that C_{r}(X) holds with r ≥ 2. Then there exists a sequence of quadratic transforms π : X_{1} → X such that C_{r}(X_{1}) holds and if C is a 2 curve on X_{1} such that C contains a 2 point p with ν(p) = r and p lies on a curve D in \overline{S}_{r}(X_{1}), then for p ≠ q ∈ C − B_{3}(X), ν(q) ≤ r − 1 and ν(q) = r − 1 implies γ(q) = r − 1.

Proof. Let π : X_{1} → X be the product of quadratic transforms centered at all 2 points q ∈ X such that ν(q) = r and q is on a curve D ⊂ \overline{S}_{r}(X).

Suppose that q ∈ X is a 2 point on a 2 curve C such that q ∈ D, for a curve D ⊂ \overline{S}_{r}(X). There exist permissible parameters (x, y, z) at q such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{q} \end{aligned}

where x = z = 0 are local equations of D at q. Suppose that ν(q) = r so that γ(q) = r. By Lemma 8.5, after a permissible change of variables,

F_{q} = τz^{r} + ∑_{i=2}^{r} a_{i}(x, y)z^{r−i}

with τ a unit, ν(a_{i}) ≥ i. At a 1 point q′ ∈ π^{−1}(q) we have ν(q′) ≤ r, ν(q′) = r implies γ(q′) = r.

The 2 points q′ ∈ π^{−1}(q) have permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}(z_{1} + α)

or

x = x_{1}, y = x_{1}y_{1}, z = x_{1}(z_{1} + α).

In either case

F_{q′} = τ(z_{1} + α)^{r} + terms of order ≤ r − 2 in z_{1}

implies γ(q′) ≤ r, γ(q′) ≤ r − 1 if α ≠ 0. Thus each exceptional curve \overline{C} of π contains at most one 2 point q′ such that ν(q′) = r. At the 3 point q′ ∈ π^{−1}(q),

x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1}

and ν(q′) = 0.

Thus by Lemma 7.9, C_{r}(X_{1}) holds. Let C′ be the strict transform of C, D′ the strict transform of D on X_{1}. C′ and D′ are disjoint. C′ intersects π_{1}^{−1}(q) at the 3 point q′ with ν(q′) = 0. By Lemma 7.9 there is at most one curve E in \overline{S}_{r}(X_{1}) such that E ⊂ π^{−1}(q), and E intersects each 2 curve in at most one point. If E intersects an exceptional 2 curve \overline{C} in a point q′ such that ν(q′) = r, then for q′ ≠ q′′ ∈ \overline{C}, γ(q′′) ≤ r−1 (by the above analysis). Thus each exceptional 2 curve \overline{C} for π_{1} satisfies the conditions of the conclusions of the Theorem.

The strict transform C′ of a 2 curve C on X_{1} contains no 2 points q with ν(q) = r which are contained in a curve D in \overline{S}_{r}(X_{1}).

15. resolution 3

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared.

Lemma 15.1. Suppose that C ⊂ X is a 2 curve. Suppose that t is a natural number or ∞. Then the set

{q ∈ C| q is a 2 point and γ(q) ≥ t}

is Zariski closed in C − B_{3}(X).

Proof. Suppose that p ∈ C is a 2 point. There exist permissible parameters (x, y, z) at p such that (x, y, z) are uniformizing parameters in an étale neighborhood U of p in X. At p,

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F (x, y, z). \end{aligned}

Set

w = \frac{v − P_{λ}(x^{a}y^{b})}{x^{c}y^{d}}

with λ > c + d. w ∈ Γ(U, O_{X}). If q ∈ C ∩ U, there are permissible parameters (x, y, z_{q} = z − α) at q for some α ∈ k. There exist a_{i}(q) ∈ k such that

F_{q} = w − ∑_{i=0}^{∞} a_{i}(q)\frac{(x^{a}y^{b})^{i}}{x^{c}y^{d}} .

\{q ∈ C∩U | ν(F_{q}(0, 0, z_{q}) ≥ t\} = \{ \begin{matrix}\{q ∈ C ∩ U | \frac{∂^{i}w}{∂z^{i}} (0, 0, α) = 0, 0 ≤ i < t\} & if ad − bc ≠ 0 \\ \{q ∈ C ∩ U | \frac{∂^{i}w}{∂z^{i}} (0, 0, α) = 0, 0 < i < t\} & if ad − bc = 0 \end{matrix}

is Zariski closed.

Since U is an étale cover of an affine neighborhood V of p,

{q ∈ C | q is a 2 point and γ(q) ≥ t} ∩ V

is Zariski closed in V ∩ C.

Lemma 15.2. Suppose that C is a 2 curve and there exists p ∈ C with permissible parameters (x_{p}, y_{p}, z_{p}) at p such that x_{p} = y_{p} = 0 are local equations of C at p and ν(F_{p}(0, 0, z_{p})) < ∞. If q ∈ C then ν(F_{q}(0, 0, z_{q})) < ∞, where (x_{q}, y_{q}, z_{q}) are permissible parameters at q and x_{q} = 0, y_{q} = 0 are local equations for C at q.

Proof. If ν(F_{q}(0, 0, z_{q})) = ∞, then F_{q} ∈ Î_{\overline{C},q} so that F_{p} ∈ Î_{\overline{C},p} for all p ∈ C by Lemma 8.1. Thus ν(F_{p}(0, 0, z_{p})) = ∞ for all p ∈ C, a contradiction.

Theorem 15.3. Suppose that C_{r}(X) holds with r ≥ 2 and the conclusions of Theorem 14.7 hold on X. Then there exists a sequence of permissible monoidal transforms π : Y → X centered at r big curves C in \overline{S}_{r} such that C_{r}(Y) and the conclusions of Theorem 14.7 hold on Y and if D is a curve in \overline{S_{r}(Y)}, then D is not r big.

Proof. Suppose that the C ⊂ \overline{S}_{r}(X) is r big. C must contain a 1 point. Let π : Y → X be the blowup of C.

By Lemma 8.8, C_{r}(Y) holds and the conclusions of Theorem 14.7 hold on Y. There is at most one curve D ⊂ \overline{S}_{r}(Y) ∩ π^{−1}(C). If this curve exists it must be a section over C.

Let p ∈ C be a 1 point. As in (71) of the proof of Lemma 8.8, there exist permissible parameters (x, y, z) at p such that Î_{C,p} = (x, z),

\begin{aligned}u & = x^{a} \\ F_{p} & = τz^{r} + ∑_{i=2}^{r} a_{i}(x, y)z^{r−i} \end{aligned}  (173)

where τ is a unit, x^{i} | a_{i} for 2 ≤ i ≤ r.

As shown in the proof of Lemma 8.8, the only point q ∈ π^{−1}(p) which could be in \overline{S}_{r}(Y) is the 1 point with permissible parameters x = x_{1}, z = x_{1}z_{1}.

\begin{aligned}u & = x_{1}^{a} \\ F_{q} & = τz_{1}^{r} + ∑_{i=2}^{r} \frac{a_{i}(x_{1},y)}{x_{1}^{i}}z_{1}^{r−i} \end{aligned}  (174)

In this case, (174) has the form of (173) with

min\{\frac{j}{i} such that x^{j} | a_{i}, x^{j+1} |̸ a_{i} for 2 ≤ i ≤ r\}

decreased by 1.

By induction on

min\{\frac{j}{i} such that x^{j} | a_{i}, x^{j+1} |̸ a_{i} for 2 ≤ i ≤ r\}

we can construct a sequence of permissible blowups of r big curves in \overline{S}_{r} such that the conclusions of the Theorem hold.

Theorem 15.4. Suppose that C_{r}(X) holds with r ≥ 2, the conclusions of Theorem 14.7 hold on X and if C is a curve in \overline{S}_{r}(X), then C is not r big. Suppose that p ∈ \overline{S}_{r}(X) is a 1 point, D is a general curve through p. For a 1 point q ∈ D, define

ϵ(D, q) = ν(F_{q}(0, 0, z))

where (x, y, z) are permissible parameters at q so that Î_{D,q} = (x, y) and

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F_{q}. \end{aligned}

Then there exists a sequence of blowups of points on the strict transform of D, but not at p, λ : Z → X such that

1. C_{r}(Z) and the conclusions of Theorem 14.7 hold on Z.

2. Let D̃ be the strict transform of D on Z. Then ϵ(D̃, q) = 1 for all 1 points q ≠ p on D̃, ν(q) = 0 if q ∈ D̃ is a 2 point, and there are no 3 points on D̃.

3. Suppose that p̃ is a fundamental point of λ.

(a) If q ∈ λ^{−1}(p̃) is a 1 point then ν(q) ≤ r − 1.

(b) If q ∈ λ^{−1}(p̃) is a 2 point, then ν(q) ≤ r. If ν(q) = r then γ(q) = r. If ν(q) = r − 1, but τ(q) = 0, then γ(q) = r and q is on the strict transform of a curve in \overline{S}_{r}(X).

(c) If q ∈ λ^{−1}(p̃) is a 3 point then ν(q) ≤ r − 2.

4. There does not exist a curve C ⊂ \overline{S}_{r}(Z) such that C is r big.

Proof. The existence of Z and the validity of C_{r}(Z) and 2. follow from Theorem 14.5.

Suppose that D contains a 1 point q ≠ p such that ν(q) = r. Then D intersects a curve C in \overline{S}_{r}(X) transversally at q, and q is a generic point of C. By Lemma 6.25 and Lemma 8.5, since C is not r big, γ(q) = r and q is a generic point of C, there are thus permissible parameters (x, y, z) at q such that

\begin{aligned}u & = x^{a} \\ F_{q} & = τz^{r} + ∑_{i=2}^{r−1} a_{i}(x, y)x^{α_{i}}z^{r−i} + x^{r−1}y \end{aligned}

where α_{i} ≥ i, τ is a unit and x |̸ a_{i} for 2 ≤ i ≤ r − 1, Î_{C,q} = (x, z), Î_{D,q} = (x, y). Let π_{1} : X_{1} → X be the blowup of q. Suppose that q_{1} ∈ π_{1}^{−1}(q) and there are permissible parameters (x_{1}, y_{1}, z_{1}) at q_{1}

such that

x = x_{1}, y = x_{1}(y_{1} + α), z = x_{1}(z_{1} + β).

Then

\begin{aligned}u & = x_{1}^{a} \\ \frac{F_{q}}{x_{1}^{r}} & = τ(z_{1} + β)^{r} + ∑_{i=2}^{r−1} a_{i}x_{1}^{α_{i}−i}(z_{1} + β)^{r−i} + (y_{1} + α) \end{aligned}

Thus, after normalizing to get F_{q′}, we have that γ(q′) ≤ r − 1 if β ≠ 0, and γ(q′) = 1, if β = 0.

Suppose that q_{1} ∈ π_{1}^{−1}(q) and there are permissible parameters (x_{1}, y_{1}, z_{1}) at q_{1} so that

x = x_{1}y_{1}, y = y_{1}, z = y_{1}(z_{1} + α)

If α ≠ 0, then q′ is a 2 point with γ(q′) ≤ r − 1. If α = 0, then q′ is a 2 point on the strict transform of C, ν(q′) ≤ r − 1 and γ(q′) ≤ r.

Suppose that q_{1} ∈ π_{1}^{−1}(q) and there are regular parameters (x_{1}, y_{1}, z_{1}) in Ô_{X_{1},q_{1}} so that

x = x_{1}z_{1}, y = y_{1}z_{1}, z = z_{1}

Then q′ is a 2 point with ν(q′) = 0. q′ is the point in π_{1}^{−1}(q) on the strict transform of D. Thus 3. holds for λ^{−1}(q).

If q ∈ D is a 1 point with ν(q) ≤ r − 1 or a 2 point with ν(q) ≤ r − 2, 3. for λ^{−1}(q) follows from Theorems 7.1 and 7.3 (or the proof of Theorem 14.5).

If q ∈ D is a 2 point with ν(q) = r − 1, then q is a generic point of a 2 curve C ⊂ \overline{S}_{r−1}(X) (or such that F_{q} ∈ Î_{C,q} if r = 2). Since C_{r}(X) holds, C is r-1 small.

Since q is a generic point of C, we must have τ(q) > 0 by Lemma 6.26. 1. - 3. for λ^{−1}(q) then follow from Theorem 7.1 and Theorem 7.3.

The conclusions of Theorem 14.7 hold since this condition is stable under quadratic transforms.

Definition 15.5. Suppose that C is a 2 curve of X. Then C satisfies (E) if For q ∈ C,

1. ν(q) = 0 if q is a 3 point.

2. γ(q) ≤ 1 at all but finitely many 2 points q ∈ C, where either (a) ν(q) = γ(q) = r or

(b) ν(q) = r − 1, γ(q) = r and τ(q) = 0.

Theorem 15.6. Suppose that C_{r}(X) holds with r ≥ 2, the conclusions of Theorem 14.7 hold on X and C is a 2 curve of X containing a 2 point p such that either ν(p) = γ(p) = r, or ν(p) = r − 1, γ(p) = r and τ(q) = 0. Then there exists a sequence of quadratic transforms π : Y → X (over points in C) such that the following properties hold. Let C̃ be the strict transform of C. Suppose that q is an exceptional point of π. Then

1. If q is a 1 point, then ν(q) ≤ r − 1.

2. If q is a 2 point, then ν(q) ≤ r − 1. If ν(q) = r − 1 then τ(q) > 0.

3. If q is a 3 point then ν(q) ≤ r − 2 Furthermore, C̃ satisfies (E), C_{r}(Y) holds and the conclusions of Theorem 14.7 hold on Y.

Proof. By our assumption on p, and Lemma 8.1, F_{q′} ∉ Î_{C,q′} at all points q′ ∈ C. There thus cannot exist q′ ∈ C which satisfies (163), since then F_{q′} ∈ Î_{C,q′}.

Suppose that q ∈ C. Then there exist permissible parameters (x, y, z) at q such that Î_{C,q} = (x, y) and F_{q} ∉ Î_{C,q}.

First suppose that q is a 3 point and ν(q) > 0. Suppose that π_{1} : X_{1} → X is the blowup of q. Suppose that q′ ∈ π_{1}^{−1}(q). Since ν(q) ≤ r − 2, we have that ν(q′) ≤ r − 1 if q′ is a 1 point, ν(q′) ≤ r − 1 if q′ is a 2 point, ν(q′) = r − 1 implies τ(q′) > 0, and ν(q′) ≤ r − 2 if q′ is a 3 point by Theorem 7.1. The strict transform of C intersects π_{1}^{−1}(q) in a 3 point.

Consider the infinite sequence of blowups of points X_{n+1} → X_{n}, centered at the points q_{n} on the strict transform of C on X_{n} over q,

⋯ → X_{n} → ⋯ → X_{1} → X

q_{n} has permissible parameters (x_{n}, y_{n}, z_{n}) defined by

x = x_{n}z_{n}^{n}, y = y_{n}z_{n}^{n}, z = z_{n}

F_{q_{n}} = \frac{F_{q}}{z_{n}^{∑_{i=1}^{n} s_{i}}}

where s_{i} = ν(q_{i}). Since F_{q} ∉ (x, y) we have that ν(q_{n}) = 0 for all sufficiently large n.

Let

W = {q ∈ C|q is a 2 point with γ(q) > 1}.

W is a finite set by Lemma 15.1, and since γ(q) ≤ 1 at a generic point of C.

Suppose that q ∈ W is a 2 point. Suppose that either ν(q) ≤ r − 2 or q is such that ν(q) = r − 1 and τ(q) > 0. Then arguing as in the case when q is a 3 point, and using Theorems 7.1 and 7.3 we can produce a sequence of blowups of points π : X_{m} → X, centered at the points q_{n} on the strict transform of C on X_{n} over q, such that ν(q_{m}) = 0, and the conclusions of the Theorem hold in a neighborhood of π^{−1}(q).

Theorem 15.7. Suppose that C_{r}(X) holds with r ≥ 2 and the conclusions of Theorem 14.7 hold on X. Then there exists a permissible sequence of blowups π : Y → X such that for p ∈ Y

1. ν(p) ≤ r − 1 if p is a 1 point or a 2 point.

2. If p is a 2 point and ν(p) = r − 1, then τ(p) > 0 or r ≥ 3 and (163) holds at p.

3. ν(p) ≤ r − 2 if p is a 3 point

Proof. By Theorem 15.3, we can assume that if C is a curve in \overline{S}_{r}(X) then C is not r big. Furthermore, since C_{r}(X) holds, each curve in \overline{S}_{r}(X) contains a 1 point. There are finitely many 1 points \{p_{1}, …, p_{m}\} in X such that each p_{i} is in \overline{S}_{r}(X), and p_{i} is either an isolated point in \overline{S}_{r}(X), or is a special point of a curve in \overline{S}_{r}(X) (A special 1 point on a curve in \overline{S}_{r}(X) is a point which is not generic in the sense that the conclusions of 1. (a) of Lemma 8.10 do not hold). Let D_{p_{i}} be a general curve through p_{i} for 1 ≤ i ≤ m. By Theorem 15.4, after possibly performing a finite sequence of quadratic transforms at points ≠ p_{i} on the D_{p_{i}}, we may assume that ϵ(D_{p_{i}}, q) = 1 for all 1 points q ≠ p_{i} on D_{p_{i}}, ν(q) = 0 if q ∈ D_{p_{i}} is a 2 point, and there are no 3 points on any D_{p_{i}}.

There are no exceptional 1 points in \overline{S}_{r}(X) created by the sequence of blowups in Theorem 15.4.

For each p_{i}, let t_{p_{i}} be the number l computed in Theorem 11.5 (or Theorem 11.6 if r = 2) for p_{i}. By Theorem 11.4, for 1 ≤ i ≤ m, there exist sequences of monoidal transforms

λ^{p_{i}} : X_{1}(i) → spec(O_{X,p_{i}})

where X_{1}(i) → spec(O_{X,p_{i}}) is a sequence of permissible monodial transforms centered at sections over D_{p_{i}}, such that the conclusions of Theorem 11.4 hold on X_{1}(i) (with t ≥ t_{p_{i}}). Since D_{p_{i}} is resolved at all points p_{i} ≠ q ∈ D_{p_{i}}, the only obstruction to extending λ^{p_{i}} to a permissible sequence of monodial transforms of sections over D_{p_{i}} in X is if the corresponding sections over D_{p_{i}} in X do not make SNCs with the 2 curves. This difficulty can be resolved by performing quadratic transforms at the points where the section does not make SNCs with the 2 curves.

By Theorem 15.6, we can then perform a sequence of quadratic transforms centered at 2 points, so that if C is a 2 curve containing a point p such that either

ν(p) = γ(p) = r

or

ν(p) = r − 1, γ(p) = r and τ(p) = 0

then C satisfies (E). There are no exceptional 1 points in \overline{S}_{r}(X) in this sequence of blowups.

We can thus extend the maps X_{1}(i) → spec(O_{X,p_{i}}) to a sequence of permissible monodial transforms centered at sections over D_{p_{i}} and points, λ : Y → X. C_{r}(Y − λ^{−1}({p_{1}, …, p_{m}})) holds and the conclusions of Theorem 14.7 hold on Y − λ^{−1}(\{p_{1}, …, p_{m}\}), there are no special or isolated 1 points in \overline{S}_{r}(Y)−λ^{−1}(\{p_{1}, …, p_{m}\}),

Y ×_{X} spec(O_{X,p_{i}}) ≅ X_{1}(i)

for 1 ≤ i ≤ m and if q ∈ \overline{S}_{r}(Y) − λ^{−1}(\{p_{1}, …, p_{m}\}) is a 2 point with

ν(q) = r, γ(q) = r,

or

ν(q) = r − 1, γ(q) = r and τ(q) = 0

then the 2 curve containing q satisfies (E).

Let {q_{1}, …, q_{n}} be the 2 points of Y such that ν(q_{i}) = r and q_{i} is contained in a curve of \overline{S}_{r}(Y). Let C_{q_{i}} be the 2 curve containing q_{i} for 1 ≤ i ≤ m. Then γ(q) ≤ 1 if q_{i} ≠ q ∈ C_{q_{i}} is a 2 point, and γ(q) = 0 if q ∈ C_{q_{i}} is a 3 point since C_{q_{i}} satisfies (E) and the conclusions of Theorem 14.7 hold on Y. For each q_{i}, let t_{q_{i}} be the number l computed in Theorem 11.5 (or Theorem 11.6 if r = 2) for p_{i}. For 1 ≤ i ≤ n, let

λ^{q_{i}} : Y_{1}(i) → spec(O_{Y,q_{i}})

be a permissible sequence of monodial transforms centered at sections over C_{q_{i}} such that the conclusions of Theorem 11.4 hold on Y_{1}(i) (with t ≥ t_{q_{i}}).

Since C_{q_{i}} is resolved at all points q_{i} ≠ q ∈ C_{q_{i}}, the only obstruction to extending λ^{q_{i}} to a permissible sequence of monodial transforms of sections over C_{p_{i}} in Y is if the corresponding sections over C_{p_{i}} in Y do not make SNCs with the 2 curves. This difficulty can be resolved by performing quadratic transforms at the points where the section does not make SNCs with the 2 curves.

We can thus extend the Y_{1}(i) → spec(O_{Y,q_{i}}) to a sequence of permissible monodial transforms centered at points and sections over C_{q_{i}}, ϕ : Z → Y, so that

Z ×_{Y} spec(O_{Y,q_{i}}) ≅ Y_{1}(i)

for 1 ≤ i ≤ n, and if

Z_{0} = Z − ϕ^{−1}({q_{1}, …, q_{n}}) − (λ ∘ ϕ)^{−1}({p_{1}, …, p_{m}})

then C_{r}(Z_{0}) holds and Z_{0} satisfies the conclusions of Theorem 14.7. Z_{0} contains no special or isolated 1 points. If D ⊂ \overline{S}_{r}(Z) and D is r big then D ∩ Z_{0} = ∅. If q ∈ Z_{0} is a 2 point which does not satisfy the conclusions of the Theorem, then γ(q) = r and ν(q) = r − 1 or ν(q) = r. If C ⊂ Z is the 2 curve containing q, then C satisfies (E). If ν(q) = r, then q is not contained in a curve D ⊂ \overline{S}_{r}(Z).

By Lemma 8.10 and Theorem 11.5 if r ≥ 3 (or Lemma 8.10, Lemma 8.11 and Theorem 11.6 if r = 2) there exists a sequence of permissible monodial transforms ψ : W → Z consisting of a sequence of blowups of r big curves D ⊂ \overline{S}_{r}, followed by a sequence of blowups of r small curves D ⊂ \overline{S}_{r}, and finally followed by a sequence of quadratic transforms if r ≥ 3 (or quadratic transforms and monodial transforms centered at 2 curves C such that C is 1 big and C is a section over a 2 small curve blown up in constructing ϕ if r = 2) such that C_{r}(W) holds, \overline{S}_{r}(W) is a finite union of 2 points, and the conclusions of the Theorem hold everywhere in W, except possibly at a finite number of 2 points p. If C is a 2 curve on W containing a 2 point p where the theorem fails to hold, then (E) holds on C. In particular, there are no 2 curves in \overline{S}_{r}(W).

Suppose that C ⊂ W is a 2 curve containing a 2 point such that the conclusions of the theorem do not hold. Then C satisfies (E).

Let {q_{1}, …, q_{s}} be the two points on C such that

1. ν(q_{i}) = r, γ(q_{i}) = r or

2. ν(q_{i}) = r − 1, γ(q_{i}) = r and τ(q_{i}) = 0.

γ(q) ≤ 1 if q ∈ C − {q_{1}, …, q_{s}} is a 2 point, and ν(q) = 0 if q ∈ C is a 3 point. No q_{i} is contained in a curve in \overline{S}_{r}(W), since \overline{S}_{r}(W) is finite.

We will now show that there exists an affine neighborhood U of {q_{1}, …, q_{s}} and uniformizing parameters x̃, y, z̃ on U such that x̃ = y = 0 are local equations of C on U, and all points of C ∩ U are 2 points.

Let A_{1} and A_{2} be the components of E_{W} such that C is a connected component of A_{1}∩A_{2}. There exist very ample divisors H_{1}, H_{2}, H_{3}, H_{4} on W such that A_{1} ∼ H_{1}−H_{2}, A_{2} ∼ H_{3} − H_{4} and q_{i} ∉ H_{j} for 1 ≤ i ≤ s, 1 ≤ j ≤ 4.

Let U = W − (H_{1} ∪ H_{2} ∪ H_{3} ∪ H_{4}). U = spec(A) is affine and there exist x̃, y ∈ A such that x̃ = 0 is an equation for A_{1} ∩ U in U, y = 0 is an equation for A_{2} ∩ U in U. After possibly replacing U with a smaller affine neighborhood of {q_{1}, …, q_{s}}, we may assume that U ∩ C = U ∩ A_{1} ∩ A_{2} and E_{W} ∩ U = (A_{1} ∪ A_{2}) ∩ U.

There exists a morphism π : C → P^{1} such that π is étale over π(q_{i}), 1 ≤ i ≤ s, and π(q_{i}) ≠ ∞ for any i (We can take π to be a generic projection). Let z be a coordinate on P^{1} − {∞}. After replacing U with a possibly smaller affine neighborhood of {q_{1}, …, q_{s}} we have an inclusion π^{∗} : k[z] → A, so that U → spec(k[x̃, y, z]) is étale.

There exists a component E of D_{S} such that Φ_{W} (A_{1}) ⊂ E and Φ_{W} (A_{2}) ⊂ E (since Φ_{W} : W → S is weakly prepared). There exists an affine neighborhood \overline{V} of \{Φ_{W} (q_{1}), …, Φ_{W} (q_{s})\} in S and u ∈ Γ(\overline{V} , O_{S}) such that u = 0 is a local equation of D_{S}. Then u = 0 is a local equation of E_{W} in Φ_{W}^{−1}(\overline{V}) ∩ U. Thus if we replace U with Φ_{W}^{−1}(\overline{V}) ∩ U, u extends to a system of permissible parameters at Φ_{W} (p) for all p ∈ C ∩ Φ_{W}^{−1}(\overline{V}) ∩ U.

There exist a, b ∈ N such that u = x̃^{a}y^{b}\overline{γ} where \overline{γ} ∈ A is a unit in A. Let γ = \overline{γ}^{\frac{1}{a}}, B = A[γ], V = spec(B). Then h : V → U is étale.

Let x = γx̃. k[x, y, z] → B defines a morphism g : V → A^{3}. q ∈ g^{−1}(x = 0) if and only if x ∈ m_{q} which holds if and only if x̃ ∈ m_{q}. Thus g^{−1}(x = 0) = h^{−1}(x̃ = 0). g is étale at all points of g^{−1}(x = 0). Since this is an open condition (c.f. Prop 4.5 SGA1) there exists a Zariski closed subset Z_{1} of V which is disjoint from h^{−1}(x̃ = 0) such that g | V − Z_{1} is étale. Let U_{1} be an affine neighborhood of {q_{1}, …, q_{s}} in U which is disjoint from h(Z_{1}). Let V_{1} = h^{−1}(U_{1}).

After replacing U with U_{1} and V with V_{1}, we have that V → U is an étale cover and (x, y, z) are uniformizing parameters on V.

There exist v_{i} ∈ O_{S,Φ_{W} (q_{i})} such that (u, v_{i}) are permissible parameters at Φ_{W} (q_{i}) and u = 0 is a local equation of E_{W} at q_{i} for 1 ≤ i ≤ s. For each q_{i} there exist z_{i} such that (x, y, z_{i}) are permissible parameters at q_{i} for (u, v_{i}) for 1 ≤ i ≤ s which satisfy the conclusions of Lemma 8.5.

The morphism π_{1} : Y_{q_{1}} → spec(Ô_{W,q_{1}}) of Theorem 11.2 (or Theorem 11.7 if ν(q_{i}) = r −1) extends to a sequence of permissible monodial transforms π_{1} : Ỹ_{1} → V centered at sections over C.

Ỹ_{1} ×_{spec(O_{W,q_{2}} )} spec(Ô_{W,q_{2}}) → spec(Ô_{W,q_{2}})

extends to a sequence of permissible monodial transforms Y_{q_{2}} → spec(Ô_{W,q_{2}}) of the form of the conclusions of Theorem 11.2 (or Theorem 11.7).

Y_{q_{2}} → spec(Ô_{W,q_{2}}) extends to a sequence of permissible monodial transforms

Ỹ_{2} →^{π_{2}} Ỹ_{1} − π_{1}^{−1}(h^{−1}(q_{1})) → V − {h^{−1}(q_{1})}.

Preceeding inductively, we extend

Ỹ_{s−1} ×_{spec(O_{W,qs})} spec(Ô_{W,q_{s}}) → spec(Ô_{W,q_{s}})

to a sequence of permissible monodial transforms Y_{q_{s}} → spec(Ô_{W,q_{s}}) of the form of the conclusions of Theorem 11.2 (or Theorem 11.7).

Y_{q_{s}} → spec(Ô_{W,q_{s}}) extends to a sequence of permissible monodial transforms

Ỹ_{s} →^{π_{s}} Ỹ_{s−1} − (π_{1} ∘ ⋯ ∘ π_{s−1})^{−1}(h^{−1}(q_{s−1})) → V − {q_{1}, …, q_{s−1}}.

For 1 ≤ i ≤ s, let t_{q_{i}} be the value of l in the statement of Theorem 11.5 (or Theorem 11.7) for the point q_{i}.

Let ω_{i} ∈ A, 1 ≤ i ≤ s be such that

ω_{i} ≡ γ mod m_{q_{i}}^{t_{q_{i}}} Ô_{W,q_{i}}.

By the Chinese Remainder Theorem, there exists ω ∈ A such that after possibly replacing U with a smaller affine neighborhood of {q_{1}, …, q_{s}}, we have that (ωx̃, y, z̃) are uniformizing parameters on U and

ωx̃ ≡ x mod m_{q_{i}}^{t_{q_{i}}} Ô_{W,q_{i}}

for 1 ≤ i ≤ s.

We can thus replace x̃ with ωx̃ in (112) of Theorem 11.4 for 1 ≤ i ≤ s. With this choice of x̃, The map \overline{Y}_{q_{1}} → spec(O_{W,q_{1}}) of Theorem 11.4 (or Theorem 11.7) then satisfies the assumptions of Theorem 11.5, and extends to a permissible sequence of monodial transforms centered at sections over C

λ_{1} : Ŷ_{1} → U.

The map \overline{Y}_{q_{2}} → spec(O_{W,q_{2}}) of Theorem 11.4 (or Theorem 11.7) satisfies the assumptions of Theorem 11.5 (or Theorem 11.7) and extends to a permissible sequence of monodial transforms centered at sections over C,

Ŷ_{2} →^{λ_{2}} Ŷ_{1} − λ_{1}^{−1}(q_{1}) → U − {q_{1}}.

Preceeding inductively, the map \overline{Y}_{q_{s}} → spec(O_{W,q_{s}}) of Theorem 11.4 (or Theorem 11.7) satisfies the assumptions of Theorem 11.5, and extends to a permissible sequence of monodial transforms centered at sections over C

Ŷ_{s} →^{λ_{s}} Ŷ_{s−1} − (λ_{1} ∘ ⋯ ∘ λ_{s−1})^{−1}(q_{s−1}) → U − {q_{1}, …, q_{s−1}}.

Since all points of C − U are resolved, there exists a sequence of permissible monodial transforms λ̃_{1} : Y′_{1} → W consisting of quadratic transforms centered at points over C − U and permissible monodial transforms centered at sections of C such that all points of λ̃_{1}^{−1}(C − U) are resolved, and Y′_{1} ×_{W} U ≅ Ŷ_{1}.

By Theorems 11.5 and 11.7 (or Theorems 11.6 and Theorem 11.8 if r = 2), there exists a sequence of permissible monodial transforms Z̃_{1} → Y′_{1} (with induced maps ψ_{1} : Z̃_{1} → W) centered over points and curves which map to q_{1} such that all points of ψ_{1}^{−1}(q_{1}) satisfy the conclusions of the Theorem.

By Theorem 7.1 and 7.3,

Ŷ_{2} → Ŷ_{1} − λ_{1}^{−1}(q_{1})

then extends to a sequence of permissible monodial transforms λ̃_{2} : Y′_{2} → Z̃_{1} consisting of quadratic transforms centered at points over C − (U − {q_{1}}) and permissible monodial transforms centered at sections over C such that all points of (λ̃_{1} ∘ λ̃_{2})^{−1}(C − U) are resolved, all points of (λ̃_{1} ∘ λ̃_{2})^{−1}(q_{1}) satisfy the conclusions of the Theorem and

Y′_{2} ×_{W} (U − {q_{1}}) ≅ Ŷ_{2}.

By Theorems 11.5 and 11.7 (or Theorems 11.6 and 11.8 if r = 2), there exists a sequence of permissible monodial transforms Z̃_{2} → Y′_{2} with induced maps ψ_{2} : Z̃_{2} → W centered at points and curves that map to q_{2} such that all points of ψ_{2}^{−1}(q_{2}) satisfy the conclusions of the Theorem.

By induction on s, we can then construct a sequence of permissible monodial transforms ψ_{s} : Z̃_{s} → W centered at points and curves supported over C such that all points of ψ_{s}^{−1}(C) satisfy the conclusions of the Theorem, and all points of ψ_{s}^{−1}(C − {q_{1}, …, q_{s}}) are resolved.

By induction on the number of 2 curves C ⊂ W which contain a 2 point which does not satisfy the conclusions of the Theorem, we can construct a sequence of permissible monodial transforms \overline{W} → W such that \overline{W} satisfies the conclusions of the Theorem.

16. Resolution 4

Throughout this section we will assume that Φ_{X} : X → S is weakly prepared.

Theorem 16.1. Suppose that r ≥ 1 and for p ∈ X, 1. ν(p) ≤ r if p is a 1 point or a 2 point.

2. If p is a 2 point and ν(p) = r, then τ(p) > 0 or r ≥ 2, τ(p) = 0 and there exists a unique curve D ⊂ \overline{S}_{r}(X) (containing a 1 point) such that p ∈ D, and permissible parameters (x, y, z) at p such that x = z = 0 are local equations of D.

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}F_{p} \\ F_{p} & = τx^{r} + ∑_{j=1}^{r} \overline{a}_{j}(y, z)y^{d_{j}}z^{e_{j}}x^{r−j} \end{aligned}  (175)

where τ is a unit, \overline{a}_{j} are units (or 0), there exists i such that \overline{a}_{i} ≠ 0, e_{i} = i, 0 < d_{i} < i,

\frac{d_{i}}{i} ≤ \frac{d_{j}}{j} , \frac{e_{i}}{i} ≤ \frac{e_{j}}{j}

for all j and

\{\frac{d_{i}}{i} \} + \{\frac{e_{i}}{i} \} < 1.

3. ν(p) ≤ r − 1 if p is a 3 point Then there exists a sequence of permissible monoidal transforms π : X_{1} → X such that \overline{A}_{r}(X_{1}) holds. That is,

1. ν(p) ≤ r if p ∈ X is a 1 point or a 2 point.

2. If p ∈ X is a 1 point and ν(p) = r, then γ(p) = r.

3. If p ∈ X is a 2 point and ν(p) = r, then τ(p) > 0.

4. ν(p) ≤ r − 1 if p ∈ X is a 3 point

Proof. If p is a 1 point such that ν(p) = 1, then γ(p) = 1. Thus \overline{A}_{r}(X) holds if r = 1. For the rest of the proof we will assume that r ≥ 2.

Let

W (X) = {p ∈ 1 points of X|ν(p) = r and γ(p) > r}.

W (X) is Zariski closed in the open subset of 1 points of X. Let \overline{W} (X) be the Zariski closure of W (X) in X.

Suppose that p ∈ \overline{W} (X) is a point where \overline{W} (X) does not make SNCs with \overline{B}_{2}(X). Then p can not satisfy (175). Let π : X_{1} → X be the quadratic transform with center p. By Theorems 7.1 and 7.3, all points of π^{−1}(p) satisfy the assumptions of Theorem 16.1, and there are no points of π^{−1}(p) which satisfy (175). If p is a 1 point, x | L_{p} implies ν(q) ≤ r −1 if q ∈ π^{−1}(p) is a 1 point, so π^{−1}(p) contains no curves of \overline{W} (X_{1}). By Theorems 7.1 and 7.3, π^{−1}(p) contains no curves of \overline{W} (X_{1}) if p is a 2 or 3 point.

Thus there exists a sequence of quadratic transforms π : X_{1} → X such that \overline{W} (X_{1}) is a disjoint union of nonsingular curves and isolated points, X_{1} satisfies the assumptions of Theorem 16.1, and \overline{W} (X_{1}) makes SNCs with \overline{B}_{2}(X_{1}). By Theorems 7.1 and 7.3 and Lemma 7.9, we can further assume that \overline{S}_{r}(X_{1}) makes SNCs with \overline{B}_{2}(X_{1}), except possibly at some 3 points of X_{1}, and if C ⊂ \overline{S}_{r}(X_{1}) is a curve which contains a 2 point satisfying (175), then C contains no 3 points. We can then without loss of generality assume that X = X_{1}.

Suppose that C ⊂ \overline{W} (X) is a curve. C makes SNCs with the locus of 2 curves. We either have that C is r big or r small.

For a curve C, or isolated point p in \overline{W} (X), We will show that we can construct a sequence of monoidal transforms π : Y → X, centered at points and curves over C (or over p), such that the assumptions of the theorem hold on Y, and 2. of the conclusions of the theorem hold at points over C (over p).

We can then iterate this process to obtain Z → X such that the assumptions of the theorem hold on Z, and if p ∈ Z is a 1 point with ν(p) = r, then γ(p) = r.

Suppose that C is r small Since C is r small, (175) cannot hold at any p ∈ C. By Lemma 8.9, we can construct a sequence of monoidal transforms π : Y → X, centered at points on C and the strict transform of C, such that the assumptions of the theorem hold on Y, and the conclusions of the theorem hold at points of π^{−1}(C).

Suppose that C is r big

Let π : X_{1} → X be the blowup of C. We will show that the assumptions of the theorem and 2. of then conclusions of the theorem hold at points above C.

Suppose that p ∈ C is a 2 point with τ(p) > 0 or a 1 point. Then all points of π^{−1}(p) satisfy the conclusions of the Theorem by Lemma 8.8.

Suppose that p ∈ C is a 2 point such that (175) holds. Then x = z = 0 are local equations of C at p.

Suppose that q ∈ π^{−1}(p) is a 2 point. q has permissible parameters (x_{1}, y, z_{1}) such that x = x_{1}, z = x_{1}(z_{1} + α).

\begin{aligned}u & = (x_{1}^{a}y^{b})^{m} \\ v & = P (x_{1}^{a}y^{b}) + x_{1}^{c+r}y^{d} \frac{F_{p}}{x_{1}^{r}} . \end{aligned}

F_{p} = τx^{r} + yΩ

implies

\frac{F_{p}}{x_{1}^{r}} = τ + y \frac{Ω}{x_{1}^{r}}

ad − b(c + r) ≠ 0, since F_{p} is normalized, which implies that ν(q) = 0.

Suppose that q ∈ π^{−1}(p) is the 3 point. q has permissible parameters (x_{1}, y, z_{1}) such that

x = x_{1}z_{1}, z = z_{1}.

\begin{aligned}u & = (x_{1}^{a}y^{b}z_{1}^{a})^{m} \\ v & = P (x^{a}y^{b}z_{1}^{a}) + x_{1}^{c}y^{d}z_{1}^{c+r}F_{q} \end{aligned}

where

F_{q} = \frac{F_{p}}{z_{1}^{r}} = τx_{1}^{r} + ∑_{j=1}^{r} \overline{a}_{j}(y, z_{1})y^{d_{j}}x_{1}^{r−j}z_{1}^{e_{j}−j}

By assumption d_{i} + r − i + e_{i} − i < r. Thus ν(q) ≤ r − 1.

Suppose that p is an isolated point in \overline{W} (X)

There are permissible parameters (x, y, z) at p such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}F_{p} \\ L_{p} & = x^{t}Ω(x, y, z) \end{aligned}

with 0 < t < r, x |̸ Ω.

Let π_{1} : X_{1} → X be the blowup of p. If q ∈ π_{1}^{−1}(p) is a 2 point then ν(q) ≤ r and ν(q) = r implies τ(q) > 0 by Theorem 7.1. If q ∈ π_{1}^{−1}(p) is a 1 point then ν(q) ≤ r − t < r

We are now reduced to assuming that \overline{W} (X) = ∅, so that γ(p) = r if p ∈ X is a 1 point with ν(p) = r.

Now suppose that p ∈ X satisfies (175) so that the curve D in \overline{S}_{r}(X) that p lies on satisfies γ(q) = r if q ∈ D is a 1 point.

By our initial reduction, we may assume that D is nonsingular, and makes SNCs with \overline{B}_{2}(X). Since F_{p} ∈ Î_{C,p}^{r}, C is r big.

Let π : X_{1} → X be the blowup of D. If p ∈ D is a 2 point with τ(p) > 0 or a 1 point, then all points of π^{−1}(p) satisfy the conclusions of the Theorem by Lemma 8.8. The case when p satisfies (175) is exactly as in the case when C ⊂ \overline{W} (X) is r big.

17. Proof of the Main Theorem

Theorem 17.1. Suppose that Φ_{X} : X → S is weakly prepared, r ≥ 2 and \overline{A}_{r}(X) holds. Then there exists a permissible sequence of monodial transforms Y → X such that \overline{A}_{r−1}(Y) holds.

Proof. The Theorem follows from successive application of Lemma 13.4 and Theorems 13.9, 14.6, 14.7, 15.7 and 16.1

Theorem 17.2. Suppose that Φ_{X} : X → S is weakly prepared. Then there exists a sequence of permissible monoidal transforms Y → X such that Φ_{Y} : Y → S is prepared.

Proof. For r >> 0 \overline{A}_{r}(X) holds by Zariski’s Subspace Theorem (Theorem 10.6 [3]). The theorem then follows from successive application of Theorem 17.1, and the fact that \overline{A}_{1}(X) holds if and only if Φ_{X} : X → S is prepared.

Theorem 17.3. Suppose that Φ : X → S is a dominant morphism from a 3 fold to a surface and D_{S} ⊂ S is a reduced 1 cycle such that E_{X} = Φ^{−1}(D_{S})_{red} contains sing(X) and sing(Φ). Then there exist sequences of monoidal transforms with nonsingular centers π_{1} : S_{1} → S and π_{2} : X_{1} → X such that Φ_{X_{1}} : X_{1} → S_{1} is prepared with respect to D_{S_{1}} = π_{2}^{−1}(D_{S})_{red}.

Proof. This follows from Lemma 6.2 and Theorem 17.2.

18. Monomialization

Throughout this section we will suppose that Φ : X → S is a dominant morphism from a nonsingular 3 fold to a nonsingular surface, D_{S} is a reduced SNC divisor on S, E_{X} = Φ^{−1}(D_{S})_{red} is a SNC divisor on X.

If p ∈ E_{X} we will say that p is a 1, 2 or 3 point depending on if p is contained in 1, 2 or 3 components of E_{X}. q ∈ D_{S} will be called a 1 or 2 point depending on if q is contained in 1 or 2 components of D_{S}.

Regular parameters (u, v) in O_{X,p} with q ∈ D_{S} are permissible if:

1. u = 0 is a local equation of D_{S} if q is a 1 point or

2. uv = 0 is a local equation of D_{S} if q is a 2 point.

Definition 18.1. We will say that Φ is Strongly Prepared at p ∈ X (with respect to D_{S}) if one on the following forms hold.

1. Φ is prepared at p (as defined in Definition 6.6) or

2. There exist permissible parameters (u, v) at q and regular parameters (x, y, z) in Ô_{X,p} such that one of the following hold:

(a) p is a 2 point and

u = x^{a}, v = y^{b}.

(b) p is a 3 point and

u = x^{a}, v = y^{b}z^{c}

(with a, b, c > 0).

(c) p is a 3 point and

u = x^{a}y^{b}, v = y^{c}z^{d}

(with a, b, c, d > 0).

Suppose that p ∈ X is strongly prepared and (u, v) are permissible parameters at Φ(p). Regular parameters (x, y, z) in Ô_{X,p} are called ∗-permissible parameters at p for (u, v) if one of the forms of Definition 18.1 holds in Ô_{X,p}. We will also say that (u, v) are strongly prepared at p. If a form 1. holds at p, ∗-permissible parameters are permissible as defined in Definition 6.5.

Throughout this section we will assume that Φ : X → S is strongly prepared.

Lemma 18.2. Suppose that O_{X,p} → R is finite étale, and there exists \overline{x}, \overline{y}, \overline{z} ∈ R such that (\overline{x}, \overline{y}, \overline{z}) are regular parameters in R_{q} for all primes q ⊂ R such that q ∩ O_{X,p} = m_{p}. Then there exists an étale neighborhood U of p such that (\overline{x}, \overline{y}, \overline{z}) are uniformizing parameters on U.

Proof. There exists an affine neighborhood V_{1} = spec(A) of p ∈ X and a finite étale extension B of A such that B ⊗_{A} A_{m_{p}} ≅ R. Set U_{1} = spec(B). Let π : U_{1} → V_{1} be the natural map. There exists an open neighborhood U_{2} of π^{−1}(p) such that (\overline{x}, \overline{y}, \overline{z}) are uniformizing parameters on U_{2}. Let Z = U_{1} − U_{2}. Set U_{3} = U_{1} − π^{−1}(W). U_{3} → V_{2} = V_{1} − W is finite étale. Thus there exists an étale neighborhood U of p where (\overline{x}, \overline{y}, \overline{z}) are uniformizing parameters.

Lemma 18.3. Suppose that permissible parameters (u, v) for Φ(p) ∈ D_{S} are strongly prepared at p ∈ E_{X}. Then there exist ∗-permissible parameters (x, y, z) at p such that (x, y, z) are uniformizing parameters on an étale neighborhood of p, and one of the following forms hold:

1. p is a 1 point, u = 0 is a local equation of E_{X} and

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}y \end{aligned}

where P (x) is a polynomial of degree ≤ b.

2. p is a 2 point, u = 0 is a local equation of E_{X} and

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}

where (a, b) = 1, ad − bc ≠ 0, P (t) is a polynomial of degree ≤ [ max \{\frac{c}{a}, \frac{d}{b}\}].

3. p is a 2 point, u = 0 is a local equation of E_{X} and

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z \end{aligned}

where (a, b) = 1, P (t) is a polynomial of degree ≤ [ max \{\frac{c}{a}, \frac{d}{b}\}].

4. p is a 3 point, u = 0 is a local equation of E_{X} and

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}

where (a, b, c) = 1, P (t) is a polynomial of degree ≤ [ max \{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}].

5. p is a 2 point, uv = 0 is a local equation of E_{X} and

u = x^{a}, v = y^{b}.

6. p is a 3 point, uv = 0 is a local equation of E_{X} and

u = x^{a}y^{b}, v = z^{c}

(with a, b, c > 0).

7. p is a 3 point, uv = 0 is a local equation of E_{X} and

u = x^{a}y^{b}, v = y^{c}z^{d}

(with a, b, c, d > 0).

Proof. Suppose there exist regular parameters (x, y, z) in Ô_{X,p} such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}y. \end{aligned}

There exist α ∈ Ô_{X,p} and \overline{x} ∈ O_{X,p} such that x = α\overline{x}, and α^{a} ∈ O_{X,p}. Set R = O_{X,p}[α]. Let L be the quotient field of R. R is finite étale over O_{X,p}.

v − P_{b}(α\overline{x}) = α^{b}y + \frac{(P (α\overline{x}) − P_{b}(α\overline{x})}{\overline{x}^{b}})\overline{x}^{b}

implies

\overline{y} = \frac{v − P_{b}(α\overline{x})}{\overline{x}^{b}} ∈ ˆ (R_{q}) ∩ L = R_{q}

(by Lemma 2.1 [11]) for all maximal ideals q ⊂ R. Thus \overline{y} ∈ ∩R_{q} = R. Choose \overline{z} ∈ O_{X,p} such that

z ≡ \overline{z} mod m_{p}^{2}Ô_{X,p}.

Then m_{p}R = (\overline{x}, \overline{y}, \overline{z}). By Lemma 18.2 there exists an étale neighborhood U of p such that (\overline{x}, \overline{y}, \overline{z}) are uniformizing parameters on U.

Suppose there exist regular parameters (x, y, z) in Ô_{X,p} such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}

There exists α_{1}, α_{2} ∈ Ô_{X,p} and \overline{x}, \overline{y} ∈ O_{X,p} such that x = α_{1}\overline{x}, y = α_{2}\overline{y}. Set

e = [ max \{\frac{c}{a}, \frac{d}{b}\}] .

u = (α_{1}^{a}α_{2}^{b})^{m}(\overline{x}^{a}\overline{y}^{b})^{m}.

Set γ = α_{1}^{a}α_{2}^{b}. Let K be the quotient field of O_{X,p}.

γ^{m} = \frac{u}{(\overline{x}^{a}\overline{y}^{b})^{m}} ∈ Ô_{X,p} ∩ K = O_{X,p}.

Set R = O_{X,p}[γ]. R is finite étale over O_{X,p}. Let L be the quotient field of R. Set

ω = α_{1}^{c}α_{2}^{d} + \frac{P (α_{1}^{a}α_{2}^{b}\overline{x}^{a}\overline{y}^{b}) − P_{e}(α_{1}^{a}α_{2}^{b}\overline{x}^{a}\overline{y}^{b})}{\overline{x}^{c}\overline{y}^{d}} = \frac{v − P_{e}(α_{1}^{a}α_{2}^{b}\overline{x}^{a}\overline{y}^{b})}{\overline{x}^{c}\overline{y}^{d}} ∈ ˆ (R_{q}) ∩ L = R_{q}

for all maximal ideals q of R. Thus ω ∈ ∩R_{q} = R. Set f = ad − bc. Set

x̃ = (γ^{d}ω^{−b})^{\frac{1}{f}} \overline{x},

ỹ = (γ^{−c}ω^{a})^{\frac{1}{f}} \overline{y}.

\begin{aligned}u & = (x̃^{a}ỹ^{b})^{m} \\ v & = P_{e}(x̃^{a}ỹ^{b}) + x̃^{c}ỹ^{d}. \end{aligned}

Choose z̃ ∈ O_{X,p} such that z ≡ z̃ mod m_{p}^{2}Ô_{X,p}. Then

x̃, ỹ, z̃ ∈ R_{1} = R[(γ^{d}ω^{−b})^{\frac{1}{f}} , (γ^{−c}ω^{a})^{\frac{1}{f}}]

are regular parameters at all maximal ideals of R_{1}. By Lemma 18.2, there exists an étale neighborhood U of p such that x̃, ỹ, z̃ are uniformizing parameters on U.

Suppose there exist regular parameters (x, y, z) in Ô_{X,p} such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z. \end{aligned}

There exist α_{1}, α_{2} ∈ Ô_{X,p} and \overline{x}, \overline{y} ∈ O_{X,p} such that x = α_{1}\overline{x}, y = α_{2}\overline{y}. Set e = [ max \{\frac{c}{a}, \frac{d}{b}\}]. Let K be the quotient field of O_{X,p}.

u = (α_{1}^{a}α_{2}^{b})^{m}(\overline{x}^{a}\overline{y}^{b})^{m}.

Set γ = α_{1}^{a}α_{2}^{b}.

γ^{m} = \frac{u}{(\overline{x}^{a}\overline{y}^{b})^{m}} ∈ Ô_{X,p} ∩ K = O_{X,p}.

Set R = O_{X,p}[γ]. R is finite étale over O_{X,p}. Let L be the quotient field of R. Set

\overline{z} = \frac{v − P_{e}(α_{1}^{a}α_{2}^{b}\overline{x}^{a}\overline{y}^{b})}{\overline{x}^{c}\overline{y}^{d}} ∈ ˆ (R_{q}) ∩ L = R_{q}

for all maximal ideals q of R. Thus \overline{z} ∈ ∩R_{q} = R. Set x̃ = α_{1}α2^{\frac{b}{a}} \overline{x}. x̃, \overline{y}, \overline{z} ∈ R_{1} =

R[α_{1}α2^{\frac{b}{a}}] and (x̃, \overline{y}, \overline{z}) = m_{p}R. By Lemma 18.2 there exists an étale neighborhood U of p such that (x̃, \overline{y}, \overline{z}) are uniformizing parameters on U.

Suppose there exist regular parameters (x, y, z) in Ô_{X,p} such that

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}. \end{aligned}

There exist α_{1}, α_{2}, α_{3} ∈ Ô_{X,p} and \overline{x}, \overline{y}, \overline{z} ∈ O_{X,p} such that x = α_{1}\overline{x}, y = α_{2}\overline{y}, z = α_{3}\overline{z}. Set g = [ max \{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}].

u = (α_{1}^{a}α_{2}^{b}α_{3}^{c})^{m}(\overline{x}^{a}\overline{y}^{b}\overline{z}^{c})^{m}.

Set γ = α_{1}^{a}α_{2}^{b}α_{3}^{c}. Let K be the quotient field of O_{X,p}.

γ^{m} = \frac{u}{(\overline{x}^{a}\overline{y}^{b}\overline{z}^{c})^{m}} ∈ Ô_{X,p} ∩ K = O_{X,p}

Set R = O_{X,p}[γ]. R is finite étale over O_{X,p}. Let L be the quotient field of R. Set

ω = \frac{v − P_{g}(α_{1}^{a}α_{2}^{b}α_{3}^{c}\overline{x}^{a}\overline{y}^{b}\overline{z}^{c})}{\overline{x}^{d}\overline{y}^{d}\overline{z}^{f}} ∈ ˆ (R_{q}) ∩ L = R_{q}

for all maximal ideals q of R. After possibly permuting x, y, z, we can assume that h = ae − bd ≠ 0. Set

x̃ = (γ^{e}ω^{−b})^{\frac{1}{h}} \overline{x}, ỹ = (γ^{−d}ω^{a})^{\frac{1}{h}} \overline{y}.

Set R_{1} = R[(γ^{e}ω^{−b})^{\frac{1}{h}} , (γ^{−d}ω^{a})^{\frac{1}{h}}]. x̃, ỹ, \overline{z} ∈ R_{1}.

\begin{aligned}u & = (x̃^{a}ỹ^{b}\overline{z}^{c})^{m} \\ v & = P_{g}(x̃^{a}ỹ^{b}\overline{z}^{c}) + x̃^{d}ỹ^{e}\overline{z}^{f} \end{aligned}

(x̃, ỹ, \overline{z}) = m_{p}R_{1}. By Lemma 18.2 there exists an étale neighborhood U of p such that (x̃, ỹ, \overline{z}) are uniformizing parameters on U.

The arguments for the remaining cases 5., 6. and 7. are easier.

Remark 18.4. Suppose that p ∈ X is a prepared 3 point, so that

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f}, \end{aligned}

u = 0 is a local equation of E_{X} and

rank ( \begin{matrix}a & b & c \\ d & e & f \end{matrix} ) = 2.

Then at most one of ae − bd, af − cd, bf − ce is zero.

Proof. By assumption, a, b, c are all nonnegative. Suppose that two of these forms are zero. After permuting x, y, z, we may assume that ae − bd = 0 and af − cd = 0. Then e = \frac{bd}{a}, f = \frac{cd}{a} and bf − ce = \frac{bcd}{a} − \frac{cbd}{a} = 0, a contradiction.

Definition 18.5. Suppose that Φ : X → S is strongly prepared with respect to D_{S}. Suppose that p ∈ E_{X} We will say that p is a good point for Φ if there exist permissible parameters (u, v) at Φ(p) and ∗-permissible parameters (x, y, z) at p for (u, v) such that one of the following forms hold:

p is a 3 point, u = 0 is a local equation of E_{X} at p and

\begin{aligned}u & = x^{a}y^{b}z^{c} \\ v & = x^{d}y^{e}z^{f} \end{aligned}  (176)

with

rank ( \begin{matrix}a & b & c \\ d & e & f \end{matrix} ) = 2

p is a 3 point, uv = 0 is a local equation of E_{X} at p,

\begin{aligned}u & = x^{a}y^{b} \\ v & = z^{c} \end{aligned}  (177)

p is a 3 point, uv = 0 is a local equation of E_{X} at p

\begin{aligned}u & = x^{a}y^{b} \\ v & = y^{c}z^{d} \end{aligned}  (178)

with a, b, c, d > 0.

p is a 2 point, u = 0 is a local equation of E_{X} at p,

\begin{aligned}u & = x^{a}y^{b} \\ v & = x^{c}y^{d} \end{aligned}  (179)

with ad − bc ≠ 0 p is a 2 point, u = 0 is a local equation of E_{X} at p and there exists α ∈ k such that

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = α(x^{a}y^{b})^{t} + (x^{a}y^{b})^{t}z \end{aligned}  (180)

with (a, b) = 1.

p is a 2 point, u = 0 is a local equation of E_{X} at p

\begin{aligned}u & = x^{a}y^{b} \\ v & = x^{c}y^{d}z \end{aligned}  (181)

with ad − bc ≠ 0 p is a 2 point, uv = 0 is a local equation of E_{X} at p

\begin{aligned}u & = x^{a} \\ v & = y^{b} \end{aligned}  (182)

p is a 1 point, u = 0 is a local equation of E_{X} at p and there exists α ∈ k such that

\begin{aligned}u & = x^{a} \\ v & = αx^{c} + x^{c}y \end{aligned}  (183)

p ∈ X will be called a bad point if p is not a good point.

Remark 18.6. (Remark1072) Suppose that p ∈ X is a good point of one of the forms (176), (177), (178), (179), (180), (181), (182) or (183). Then (as in Lemma 18.3) there exist ∗-permissible parameters (x, y, z) at p such that (x, y, z) are uniformizing parameters on an étale neighborhood of p, and one of the forms (176), (177), (178), (179), (180), (181), (182) or (183) hold.

Suppose that p ∈ X is a 1 point and (u, v) are permissible parameters at Φ(p), (x, y, z) are ∗-permissible parameters at p for (u, v) such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}y. \end{aligned}

with deg (P) ≤ c. Set d = ord (P) ∈ N ∪ {∞}.

Suppose that (u_{1}, v_{1}) are also ∗-permissible parameters at Φ(p) and (x_{1}, y_{1}, z_{1}) are permissible parameters at p for (u_{1}, v_{1}) such that

\begin{aligned}u_{1} & = x_{1}^{a_{1}} \\ v_{1} & = P_{1}(x_{1}) + x_{1}^{c_{1}}y_{1}. \end{aligned}

with deg (P_{1}) ≤ c_{1}. Set d_{1} = ord (P_{1}) ∈ N ∪ {∞}.

We will compare a, c, d and a_{1}, c_{1}, d_{1}.

If (u, v) = (u_{1}, v_{1}) then there exists an a-th root of unity ω ∈ k such that x = ωx_{1}, so that c = c_{1}, and

P_{1}(x_{1}) = P (ωx_{1}).

Thus a = a_{1}, c = c_{1}, d = d_{1}.

Suppose that (u, v) and (u_{1}, v_{1}) are related by a change of parameters of the type of Case 1.1 of the proof of Lemma 6.8. This case can only occur if Φ(p) is a 2 point. We have v_{1} = u and u_{1} = v. Then d = ord(P) ≤ c. The analysis of Case 1.1 in Lemma 6.8 shows that there are ∗-permissible parameters (\overline{x}, \overline{y}, \overline{z}) for (u_{1}, v_{1}) such that

\begin{aligned}u_{1} & = v = \overline{x}^{d} \\ v_{1} & = u = \overline{P} (\overline{x}) + \overline{x}^{a+c−d}\overline{y} \end{aligned}

where ord(\overline{P}) = a. Thus a_{1} = d, c_{1} = a + c − d and d_{1} = a.

Suppose that (u, v) and (u_{1}, v_{1}) are related by a change of parameters of the type of Case 1.2 of the proof of Lemma 6.8. We have u_{1} = αu and v_{1} = v where α(u, v) is a unit series. The analysis of Case 1.2 in Lemma 6.8 shows that there are ∗-permissible parameters (\overline{x}, \overline{y}, \overline{z}) for (u_{1}, v) such that

\begin{aligned}u_{1} & = \overline{x}^{a} \\ v & = \overline{P} (\overline{x}) + \overline{x}^{c}\overline{y} \end{aligned}

where ord(\overline{P}) = d. Thus a_{1} = a, c_{1} = c and d_{1} = d.

Suppose that (u, v) and (u_{1}, v_{1}) are related by a change of parameters of the type of Case 1.3 of the proof of Lemma 6.8. We have u_{1} = u and v_{1} = αu + βv where α(u, v), β(u, v) are series, β is a unit series. If Φ(p) is a 2 point then α = 0. The analysis of Case 1.3 in Lemma 6.8 shows that there are ∗-permissible parameters (\overline{x}, \overline{y}, \overline{z}) for (u_{1}, v_{1}) such that

\begin{aligned}u_{1} & = x^{a} \\ v_{1} & = \overline{P} (x) + x^{c}\overline{y} \end{aligned}

such that

\begin{aligned}\overline{P} (x) & = ∑ α_{ij}x^{a(i+1)}P (x)^{j} + ∑ β_{ij}x^{a_{i}}P (x)^{j+1} \\ & ≡ β_{00}P (x) + ∑ α_{i0}x^{a(i+1)} mod x^{d+1}. \end{aligned}

a_{1} = a, c_{1} = c, d_{1} ≤ d if a |̸ d. If α = 0, we have a_{1} = a, c_{1} = c, d_{1} = d.

Suppose that E is a component of E_{X}, p ∈ E, f ∈ Ô_{X,p}, x = 0 is a local equation of E at p. Then define

ν_{E}(f) = max {n such that x^{n} | f}.

Definition 18.7. (Def61) Suppose that p ∈ X is a 1 point, and E is the component of E_{X} containing p. Suppose that (u, v) are permissible parameters at Φ(p) such that u = 0 is a local equation of E at p. If (x, y, z) are ∗-permissible parameters at p for (u, v), then there is an expression

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}y. \end{aligned}

For fixed (u, v), a, c and ν_{E}(v) are independent of the choice of permissible parameters (x, y, z) for (u, v). Define

A(Φ, p) = min (c − ν_{E}(v))

where the minimum is over permissible parameters (u, v) at Φ(p) such that u = 0 is a local equation of E at p.

If A(Φ, p) > 0, define

C(Φ, p) = min (c − ν_{E}(v), ν_{E}(v) + a)

where the minimum (in the lexicographic order) is over permissible parameters (u, v) at Φ(p) such that u = 0 is a local equation of E at p.

Suppose that E is a component of E_{X}, p ∈ E is a 1 point. Suppose that (u, v) are permissible parameters for Φ(p) = q such that u = 0 is a local equation of E at p, (x, y, z) are ∗-permissible parameters for (u, v) at p. There is an expression

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}y. \end{aligned}  (184)

c > 0 is equivalent to Φ(E) = q. c = 0 is equivalent to Φ(E) is a component of D_{S} with local equation u = 0 at q.

Suppose that Φ(E) = q is a 1 point on S. By the discussion before Definition 18.7, A(Φ, p) = c − ν_{E}(v) if and only if a |̸ ord(P) or c = ord(P). If P (x) = ∑ a_{i}x^{i}, we can make a permissible change of parameters at q, replacing v with v − ∑ a_{ia}u^{i} to achieve A(Φ, p) = c − ν_{E}(v).

Suppose that Φ(E) = q is a 2 point on S. By the discussion before Definition 18.7, A(Φ, p) = c − ν_{E}(v).

Suppose that Φ(E) is a component D of D_{S}. This is equivalent to c = 0 in (184). Then

0 = A(Φ, p) = c − ν_{E}(v).

In all these cases, if A(Φ, p) = c − ν_{E}(v) > 0, then we have

C(Φ, p) = (c − ν_{E}(v), a + ν_{E}(v)).

and there exists an open neighborhood U of p such that A(Φ, p′) = A(Φ, p) for all p′ ∈ E ∩ U and C(Φ, p′) = C(Φ, p) if A(Φ, p) > 0. Then A(Φ, p′) = A(Φ, p) and C(Φ, p′) = C(Φ, p) at all 1 points p′ ∈ E. We can then define

A(Φ, E) = A(Φ, p)

and

C(Φ, E) = C(Φ, p)

for p ∈ E a 1 point.

Lemma 18.8. Suppose that p ∈ X is a 2 point and E_{1}, E_{2} are the components of E_{X} containing p. Then there exist permissible parameters (u, v) at q = Φ(p) and permissible parameters (x, y, z) for (u, v) at p such that, if p satisfies (18) of Definition 6.6,

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}

or if p satisfies (19) of Definition 6.6,

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z \end{aligned}

where x = 0 is a local equation of E_{1}, y = 0 is a local equation of E_{2}, then

A(Φ, E_{1}) = c − ν_{E_{1}}(v), A(Φ, E_{2}) = d − ν_{E_{2}}(v).

If A(Φ, E_{1}) > 0 then

C(Φ, E_{1}) = (c − ν_{E_{1}}(v), ν_{E_{1}}(v) + ak).

If A(Φ, E_{2}) > 0 then

C(Φ, E_{2}) = (d − ν_{E_{2}}(v), ν_{E_{2}}(v) + bk).

Suppose that p ∈ X is a 3 point, p satisifes (20) of Definiton 6.6, and E_{1}, E_{2}, E_{3} are the components of E_{X} containing p. Then there exist permissible parameters (u, v) at q = Φ(p) and permissible parameters (x, y, z) for (u, v) at p such that

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{m} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}

where x = 0 is a local equation of E_{1}, y = 0 is a local equation of E_{2}, z = 0 is a local equation of E_{3}, and

A(ϕ, E_{1}) = d − ν_{E_{1}}(v), A(Φ, E_{2}) = e − ν_{E_{2}}(v), A(Φ, E_{3}) = f − ν_{E_{3}}(v).

If A(Φ, E_{1}) > 0, then

C(Φ, E_{1}) = (d − ν_{E_{1}}(v), ν_{E_{1}}(v) + am),

If A(Φ, E_{2}) > 0, then

C(Φ, E_{2}) = (e − ν_{E_{2}}(v), ν_{E_{2}}(v) + bm),

If A(Φ_{E_{3}}) > 0, then

C(Φ, E_{3}) = (f − ν_{E_{3}}(v), ν_{E_{3}}(v) + cm),

Proof. Suppose that p ∈ X is a 2 point satisfying (18), (u, v) are permissible parameters at q and (x, y, z) are uniformizing parameters for (u, v) at p such that

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}

and (x, y, z) are uniformizing parameters on an étale neighborhood of p. Let P (t) = ∑ a_{i}t^{i}. If q is a 1 point, then we can replace v with v − ∑_{i} a_{ik}u^{i}, so that k |̸ ord (P).

If c = 0, then 0 = A(Φ, E_{1}) = c − ν_{E_{1}}(v), and if d = 0, then 0 = A(Φ, E_{2}) = d − ν_{E_{2}}(v).

Suppose that c > 0. Then Φ(E_{1}) = q. If p′ is a 1 point on E_{1} near p then there exist ∗-permissible parameters (\overline{x}, \overline{y}, \overline{z}) at p′ such that

\begin{aligned}u & = \overline{x}^{ak} \\ v & = P_{p′}(\overline{x}) + \overline{x}^{c}\overline{y} \end{aligned}

where P_{p′}(\overline{x}) = P (\overline{x}^{a}) + α\overline{x}^{c} for some nonzero α ∈ k.

If q is a 1 point we have ak |̸ ord (P_{p′}) or c = ord (P_{p′}). By the discussion before Definition 18.7, we have that A(Φ, E_{1}) = c − ν_{E_{1}}(v), and if A(Φ, E_{1}) > 0, then

C(Φ, E_{1}) = (c − ν_{E_{1}}(v), ν_{E_{1}}(v) + ak).

A similar argument shows that A(Φ, E_{2}) = d−ν_{E_{2}}(v) if d > 0, and if A(Φ, E_{2}) > 0, then

C(Φ, E_{2}) = (d − ν_{E_{2}}(v), ν_{E_{2}}(v) + bk).

If p satisfies (19) or (20) then the proof is similar.

Remark 18.9. If p is a 1 point then A(Φ, p) = 0 if and only if p is a good point.

Set

A(Φ) = max {A(Φ, E) | E is a component of E_{X}}.

If A(Φ) > 0, define

C(Φ) = max {C(Φ, E) | E is a component of E_{X}}.

Lemma 18.10. Suppose that p ∈ X is a 1 point, (u, v) are permissible parameters at Φ(p) such that u = 0 is a local equation of E_{X} at p, (x, y, z) are ∗-permissible parameters at p for (u, v) such that

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{c}y \end{aligned}

with deg(P) ≤ c. Set d = ord(P) ∈ N ∪ {∞}.

1. Suppose that Φ(p) is a 1 point. Then p is a bad point if d < c and a |̸ d.

2. Suppose that Φ(p) is a 2 point. Then p is a bad point if d < c.

Proof. Suppose that (u_{1}, v_{1}) are permissible parameters at q = Φ(p) such that u_{1} = 0 is a local equation of E_{X} at p, and (u_{1}, v_{1}) realize p as a bad point.

If q is a 1 point then there exist series \overline{α}, \overline{β}, \overline{γ} in u, v such that

\begin{aligned}u_{1} & = \overline{α}u \\ v_{1} & = \overline{β}u + \overline{γ}v. \end{aligned}

Thus (u_{1}, v_{1}) is obtained by transformations of the form of Case 1.2 and Case 1.3 of Lemma 6.8. The conclusions of the Lemma now follow from the analysis preceeding Definition 18.7.

Suppose that q is a 2 point. Then there exist unit series \overline{α}, \overline{β} in u, v such that

\begin{aligned}u_{1} & = \overline{α}u \\ v_{1} & = \overline{β}v \end{aligned}

or

\begin{aligned}u_{1} & = \overline{α}v \\ v_{1} & = \overline{β}u. \end{aligned}

In the first case we have, with the notation preceeding Definition 18.7, that a_{1} = a, c_{1} = c and d_{1} = d so that d_{1} < c_{1}. In the second case we have a_{1} = d, c_{1} = a + c − d and d_{1} = a so that d_{1} < c_{1}. p is thus a bad point.

Theorem 18.11. Suppose that Φ : X → S is strongly prepared. Then the locus of bad points in X is a Zariski closed set of pure codimension 1, consisting of a union of components of E_{X}.

Proof. We will first show that the good points of X are a Zariski open set in E_{X}.

Suppose that p ∈ X is a good 3 point. Then there exists an open neighborhood U of p, uniformizing parameters (x, y, z) in an étale cover of U such that u = 0 is a local equation of E_{X} in U and

\begin{aligned}u & = x^{a}y^{b}z^{c} \\ v & = x^{d}y^{e}z^{f}. \end{aligned}

If q ∈ U is a 2 point, then we have (after possibly permuting x, y, z) that (x, y, z_{1}) are regular parameters at q where z = z_{1} + α (with α ≠ 0). Set x = x_{1}(z_{1} + α)^{− \frac{c}{a}}. Then (x_{1}, y, z_{1}) are permissible parameters at q, and

\begin{aligned}u & = x_{1}^{a}y^{b} \\ v & = x_{1}^{d}y^{e}(z_{1} + α)^{f− \frac{cd}{a}} . \end{aligned}

If ae − bd ≠ 0, we can make a permissible change of variables (x̃, ỹ, z̃) at q to get

\begin{aligned}u & = x̃^{a}ỹ^{b} \\ v & = x̃^{d}ỹ^{e}. \end{aligned}

If ae−bd = 0 then f − \frac{cd}{a} ≠ 0, so that we can make a permissible change of parameters to get

\begin{aligned}u & = (x_{1}^{a_{1}}y^{b_{1}})^{k} \\ v & = β(x_{1}^{a_{1}}y^{b_{1}})^{t} + (x_{1}^{a_{1}}y^{b_{1}})^{t}z_{1}. \end{aligned}

If q ∈ U is a 1 point, then we have (after possibly permuting x, y, z) that (x, y_{1}, z_{1}) are regular paramaters at q where y = y_{1} + α, z = z_{1} + β (with α, β ≠ 0). Set x = x_{1}(y_{1} + α)^{− \frac{b}{a}} (z_{1} + β)^{− \frac{c}{a}}. Then (x_{1}, y_{1}, z_{1}) are permissible parameters at q, and

\begin{aligned}u & = x_{1}^{a} \\ v & = γx_{1}^{d} + x_{1}^{d}(γ_{1}y_{1} + γ_{2}z_{1} + ⋯) \end{aligned}

where γ, γ_{1}, γ_{2} ∈ k, γ ≠ 0 and either γ_{1} ≠ 0 or γ_{2} ≠ 0 since we cannot have both e − \frac{db}{a} = 0 and f − \frac{dc}{a} = 0. Thus all points in U are good points.

Suppose that p ∈ X is a good 2 point and (180) holds at p. Then there exists an open neighborhood U of p, uniformizing parameters (x, y, z) in an étale cover of U such that u = 0 is a local equation of E_{X} in U and

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = \overline{β}(x^{a}y^{b})^{t} + (x^{a}y^{b})^{t}z \end{aligned}

If q ∈ U is a 2 point, then we have that (x, y, z_{1}) are permissible parameters at q where z = z_{1} + α and q is a good point.

If q ∈ U is a 1 point, then we have (after possibly permuting x, y) that (x, y_{1}, z_{1}) are regular parameters at q where y = y_{1} + α, z = z_{1} + β (with α ≠ 0). Set x = x_{1}(y_{1} + α)^{− \frac{b}{a}}. Then (x_{1}, y_{1}, z_{1}) are permissible parameters at q, and

\begin{aligned}u & = x_{1}^{ak} \\ v & = (\overline{β} + β)x_{1}^{at} + x_{1}^{at}z_{1} \end{aligned}

Thus all points in U are good points.

Suppose that p ∈ X is a good 2 point, and (179) holds at p. Then there exists an open neighborhood U of p, uniformizing parameters (x, y, z) in an étale cover of U such that u = 0 is a local equation of E_{X} in U and

\begin{aligned}u & = x^{a}y^{b} \\ v & = x^{c}y^{d} \end{aligned}

where ad − bc ≠ 0. If q ∈ U is a 2 point, then we have that (x, y, z_{1}) are permissible parameters at q where z = z_{1} + α, and q is a good point.

If q ∈ U is a 1 point, then we have (after possibly permuting x, y) that (x, y_{1}, z_{1}) are regular paramaters at q where y = y_{1} + α, z = z_{1} + β (with α ≠ 0). Set x = x_{1}(y_{1} + α)^{− \frac{b}{a}}. Set γ = α^{d− \frac{bc}{a}}, ỹ_{1} = (y_{1} + α)^{d− \frac{bc}{a}} − γ. Then (x_{1}, ỹ_{1}, z_{1}) are permissible parameters at q, and

\begin{aligned}u & = x_{1}^{a} \\ v & = γx_{1}^{c} + x_{1}^{c}ỹ_{1} \end{aligned}

Thus all points in U are good points.

If p is a good point satisfying (177), (178), (179), (181), (182) or (183), a similar argument shows that there is a Zariski open neighborhood U of p of good points.

We will now show that the bad points of X have pure codimension 1 in X. It suffices to show that any bad point lies on a surface of bad points.

First suppose that p is a bad 3 point. Then there exists an open neighborhood U of p, uniformizing parameters (x, y, z) in an étale cover of U such that u = 0 is a local equation of E_{X} in U and

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}

where (after possibly permuting x, y, z) we have

max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \} = \frac{f}{c}

Thus ord(P) < \frac{f}{c}, since ord (P) ≥ \frac{f}{c} implies that x^{d}y^{e}z^{f}|P (x^{a}y^{b}z^{c}), and p is thus a good point.

If Φ(p) is a 1 point, we can make a permissible change of parameters so that we have that k |̸ ord (P).

Let q ∈ U be a 1 point on the surface z = 0. c, f > 0 imply z = 0 is a local equation of a component of E_{X} which maps to Φ(p). There are regular parameters (x_{1}, y_{1}, z) at q where x = x_{1} + α, y = y_{1} + β with α, β ≠ 0. There are permissible parameters (x_{1}, y_{1}, z_{1}) at q where

z = (x_{1} + α)^{− \frac{a}{c}} (y_{1} + β)^{− \frac{b}{c}} z_{1}
u = z_{1}^{ck}
v = P (z_{1}^{c}) + (x_{1} + α)^{d− \frac{fa}{c}} (y_{1} + β)^{e− \frac{fb}{c}} z_{1}^{f}
= P (z_{1}^{c}) + α^{d− \frac{fa}{c}} β^{e− \frac{fb}{c}} z_{1}^{f} + z_{1}^{f}(γ_{1}x_{1} + γ_{2}y_{1} + ⋯)

where γ_{1}, γ_{2} ∈ k and γ_{1} or γ_{2} ≠ 0.

Suppose that Φ(p) is a 1 point. Then Φ(q) = Φ(p) is a 1 point. q is a bad point by Lemma 18.10, since ck |̸ c ord(P) and c ord(P) < f

Suppose that Φ(p) is a 2 point. Then Φ(q) = Φ(p) is a 2 point. q is a bad point by Lemma 18.10 since c ord (P) < f.

Suppose that p is a bad 2 point satisfying (19). There exists an open neighborhood U of p and uniformizing parameters (x, y, z) on an étale cover of U such that

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z. \end{aligned}

We can (after possibly permuting x, y) assume that ad−bc ≥ 0. Since p is a bad point, ord (P) < \frac{d}{b}. If Φ(p) is a 1 point we can make a permissible change of parameters so that k |̸ ord (P). Let q ∈ U be a 1 point on the surface y = 0. b, d > 0 implies y = 0 is a local equation of a component of E_{X} which maps to Φ(p). There are regular parameters (x_{1}, y_{1}, z) at q where x = x_{1} + α, z = z_{1} + β (with α ≠ 0). There are permissible parameters (x_{1}, y_{1}, z_{1}) at q where

y = (x_{1} + α)^{− \frac{a}{b}} y_{1}
u = y_{1}^{bk}
v = P (y_{1}^{b}) + (x_{1} + α)^{c− \frac{da}{b}} (z_{1} + β)y_{1}^{d}
= P (y_{1}^{b}) + α^{c− \frac{da}{b}} βy_{1}^{d} + z̃_{1}y_{1}^{d}

Φ(q) = Φ(p) so that Φ(q) is a 1 point if and only if Φ(p) is a 1 point. Since b ord (P) < d and bk |̸ b ord (P) if Φ(q) is a 1 point, q is a bad point by Lemma 18.10.

A similar argument shows that there is a surface of bad points passing through a bad point satisfying (18) or (17).

Lemma 18.12. (Lemma62) Suppose that Φ : X → S is strongly prepared, q ∈ D_{S} and p ∈ Φ^{−1}(q) is such that one of the forms 1. - 7. of Lemma 18.3 hold at p. Then m_{q}O_{X,p} is not invertible if and only if one of the following holds:

p is a 1 point

\begin{aligned}u & = x^{k} \\ v & = x^{c}y \end{aligned}  (185)

with c < k.

p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}  (186)

with a, b > 0, (a, b) = 1, ad − bc ≠ 0,

min\{\frac{c}{a}, \frac{d}{b}\} < ord (P) < max\{\frac{c}{a}, \frac{d}{b}\},

min\{\frac{c}{a}, \frac{d}{b}\} < k.

p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = x^{c}y^{d} \end{aligned}  (187)

with a, b > 0, (a, b) = 1, ad − bc ≠ 0,

min\{\frac{c}{a}, \frac{d}{b}\} < k < max\{\frac{c}{a}, \frac{d}{b}\}.

p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z \end{aligned}  (188)

with a, b > 0, (a, b) = 1, ad − bc ≠ 0,

min\{\frac{c}{a}, \frac{d}{b}\} < ord (P) < max\{\frac{c}{a}, \frac{d}{b}\},

min\{\frac{c}{a}, \frac{d}{b}\} < k.

p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = x^{c}y^{d}z \end{aligned}  (189)

with a, b > 0, (a, b) = 1, ad − bc ≠ 0,

min\{\frac{c}{a}, \frac{d}{b}\} < k.

p is a 2 point

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = (x^{a}y^{b})^{t}z \end{aligned}  (190)

with a, b > 0, (a, b) = 1, t < k.

p is a 2 point

\begin{aligned}u & = x^{a} \\ v & = y^{b} \end{aligned}  (191)

p is a 3 point

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}  (192)

with a, b, c > 0, (a, b, c) = 1,

min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \} < ord (P) < max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \},

k > min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}.

p is a 3 point

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = x^{d}y^{e}z^{f} \end{aligned}  (193)

with a, b, c > 0, (a, b, c) = 1,

min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \} < k < max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}.

p is a 3 point

\begin{aligned}u & = x^{a}y^{b} \\ v & = z^{c} \end{aligned}  (194)

with a, b, c > 0.

p is a 3 point

\begin{aligned}u & = x^{a}y^{b} \\ v & = y^{c}z^{d} \end{aligned}  (195)

with a, b, c, d > 0.

Proof. Suppose that p is a 1 point. Then (185) follows easily.

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}, \end{aligned}

P ≠ 0 and e = ord (P) < max\{\frac{c}{a}, \frac{d}{b}\}. Set λ_{2} = max\{\frac{c}{a}, \frac{d}{b}\}, λ_{1} = min\{\frac{c}{a}, \frac{d}{b}\}.

u | v if and only if e ≥ k and λ_{1} ≥ k. v | u if and only if e ≤ λ_{1} and e ≤ k. Thus (u, v)O_{X,p} is not invertible if and only if λ_{1} < k and λ_{1} < e.

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = x^{c}y^{d} \end{aligned}

Set λ_{1} = min\{\frac{c}{a}, \frac{d}{b}\}, λ_{2} = max\{\frac{c}{a}, \frac{d}{b}\}. u | v if and only if k ≤ λ_{1}, v | u if and only if k ≥ λ_{2}. So (u, v)O_{X,p} is not invertible if and only if λ_{1} < k < λ_{2}.

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d}z \end{aligned}

with ad − bc ≠ 0, e = ord (P) < max\{\frac{c}{a}, \frac{d}{b}\}. Set λ_{1} = min\{\frac{c}{a}, \frac{d}{b}\}, λ_{2} = max\{\frac{c}{a}, \frac{d}{b}\}. u | v if and only if e ≥ k and k ≤ λ_{1}. v | u if and only if e ≤ k and e ≤ λ_{1}. So (u, v)O_{X,p} is not invertible if and only if λ_{1} < k and λ_{1} < e.

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = x^{c}y^{d}z \end{aligned}

and ad − bc ≠ 0. (u, v) is invertible at p if and only if c ≥ ka and d ≥ bk. Thus (u, v) is not invertible at p if and only if k > min\{\frac{c}{a}, \frac{d}{b}\}, and we get (189).

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + (x^{a}y^{b})^{t}z \end{aligned}

with P ≠ 0 and e = ord (P) ≤ t. We will show that (u, v) is invertible at p. If k ≤ e then u|v. Suppose that k > e. There are new permissible parameters (\overline{x}, \overline{y}, \overline{z}) such that

\begin{aligned}v & = (\overline{x}^{a}\overline{y}^{b})^{e} \\ u & = \overline{P} (\overline{x}^{a}\overline{y}^{b}) + (\overline{x}^{a}\overline{y}^{b})^{t−e+k}\overline{z} \end{aligned}

with ord (\overline{P}) = k. Thus v|u.

Suppose that p is a 2 point with

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = (x^{a}y^{b})^{t}z \end{aligned}

Then (u, v) not invertible at p if and only if t < k, and we get (190).

Suppose that p is a 3 point with

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = P (x^{a}y^{b}z^{c}) + x^{d}y^{e}z^{f} \end{aligned}

with P ≠ 0 and ord (P) < max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}. Set

λ_{2} = max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}, λ_{1} = min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}

u|v is equivalent to ord (P) ≥ k, k ≤ λ_{1}. v|u is equivalent to ord (P) ≤ k and ord (P) ≤ λ_{1}. That is, (u, v) is not invertible at p if and only if ord (P) > λ_{1} and k > λ_{1}. We thus get (192).

Suppose that p is a 3 point with

\begin{aligned}u & = (x^{a}y^{b}z^{c})^{k} \\ v & = x^{d}y^{e}z^{f} \end{aligned}

Set

λ_{2} = max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}, λ_{1} = min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}

u|v is equivalent to k ≤ λ_{1}. v|u is equivalent to k ≥ λ_{2}.

Thus (u, v) is not invertible at p if and only if λ_{1} < k < λ_{2}, and we get (193).

Lemma 18.13. Suppose that Φ : X → S is strongly prepared. Let S_{1} be the blowup of S at a point q ∈ D_{S}. Let U be the largest open set of X such that the rational map X → S_{1} is a morphism Φ_{1} : U → S_{1}. Then Φ_{1} is strongly prepared.

Proof. This follows from the analysis of Lemma 18.12.

Theorem 18.14. Suppose that Φ : X → S is strongly prepared, p ∈ X is a 1 point and the rational map Φ_{1} from X to the blow up S_{1} of q = Φ(p) is a morphism in a neighborhood of p. Then A(Φ_{1}, p) ≤ A(Φ, p). If A(Φ_{1}, p) = A(Φ, p) > 0, then C(Φ_{1}, p) < C(Φ, p).

Proof. At p we have permissible parameters such that

\begin{aligned}u & = x^{k} \\ v & = P (x) + x^{c}y \end{aligned}

and C(Φ, p) = (c − ν_{E}(v), ν_{E}(v) + k).

First suppose that P ≠ 0 and e = ord (P) ≤ c. If e > k then we have permissible parameters u_{1}, v_{1} at q_{1} = Φ_{1}(p) such that

u = u_{1}, v = u_{1}v_{1}.

Then

\begin{aligned}u_{1} & = x^{k} \\ v_{1} & = \frac{P (x)}{x^{k}} + x^{c−k}y \end{aligned}

A(Φ_{1}, p) ≤ (c − k − (e − k)) = A(Φ, p) and if A(Φ_{1}, p) = A(Φ, p) then C(Φ_{1}, p) ≤ (c − k − (e − k), e − k + k) = (c − e, e) < (c − e, e + k) = C(Φ, p).

If e = k then there exists 0 ≠ α ∈ k such that P (x) = αx^{k} + · · ·. There exist permissible parameters (u_{1}, v_{1}) at q_{1} = Φ_{1}(p) such that

u = u_{1}, v = u_{1}(v_{1} + α).

\begin{aligned}u_{1} & = x^{k} \\ v_{1} & = \frac{P (x)}{x^{k}} − α + x^{c−k}y. \end{aligned}

Thus A(Φ_{1}, p) < (c − k) − (e − k) = A(Φ, p).

If e < k then we have permissible parameters u_{1}, v_{1} at q_{1} = Φ_{1}(p) such that

u = u_{1}v_{1}, v = v_{1}.

We have permissible parameters (\overline{x}, \overline{y}, z) at p such that

\begin{aligned}v & = \overline{x}^{e} \\ u & = \overline{P} (\overline{x}) + \overline{x}^{k+c−e}\overline{y} \end{aligned}

where ord (\overline{P}) = k.

Then

\begin{aligned}v_{1} & = \overline{x}^{e} \\ u_{1} & = \frac{\overline{P} (\overline{x})}{\overline{x}^{e}} + \overline{x}^{k+c−2e}\overline{y} \end{aligned}

A(Φ_{1}, p) ≤ (k + c − 2e − (k − e)) = A(Φ, p)

and if A(Φ_{1}, p) = A(Φ, p) then C(Φ_{1}, p) ≤ (k + c − 2e − (k − e), (k − e) + e) < C(Φ, p). Now suppose that P (x) = 0. Then

\begin{aligned}u & = x^{k} \\ v & = x^{c}y \end{aligned}

with c ≥ k. There exist permissible parameters (u_{1}, v_{1}) at q_{1} = Φ_{1}(p) such that

u = u_{1}, v = u_{1}v_{1}

and

A(Φ_{1}, p) = A(Φ, p) = 0.

Theorem 18.15. Suppose that Φ : X → S is strongly prepared and q ∈ S. Then the locus of points Z in X where Φ does not factor through the blowup of q is a pure codimension 2 subscheme. Z makes SNCs with \overline{B}_{2}(X) except possibly at 3 points of the form (192).

Suppose that C is a component of this locus which makes SNCs with \overline{B}_{2}(X), and π : X_{1} → X is the blowup of C, E_{1} = π^{−1}(C)_{red}, Φ_{1} = Φ ∘ π. Then Φ_{1} is strongly prepared and either A(Φ_{1}) = 0 or

A(Φ_{1}, E_{1}) < A(Φ)

Proof. Suppose that p ∈ X is a 3 point such that m_{q}O_{X,p} is not invertible and (192) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U.

After possibly interchanging x, y, z, we can assume that

\frac{d}{a} = min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}

and

\frac{f}{c} = max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}.

We will now determine the locus of points in U where m_{q}O_{U} is not invertible. First suppose that q′ is a 2 point on the curve x = z = 0. q′ has regular parameters (x, y_{1}, z) where y = y_{1} + α. Thus q′ has permissible parameters (x_{1}, y_{1}, z) where x_{1} is defined by

x = x_{1}(y_{1} + α)^{− \frac{b}{a}}

Set λ = (a, c), a_{1} = \frac{a}{λ}, c_{1} = \frac{c}{λ}.

\begin{aligned}u & = (x_{1}^{a_{1}}z^{c_{1}})^{kλ} \\ v & = P ((x_{1}^{a_{1}}z^{c_{1}})^{λ}) + x_{1}^{d}z^{f}(y_{1} + α)^{e− \frac{db}{a}} . \end{aligned}

We have a_{1}f − c_{1}d > 0 and λord (P) < \frac{f}{c_{1}} . We can make a permissible change of variables to get

\begin{aligned}u & = (\overline{x}_{1}^{a_{1}} \overline{z}^{c_{1}})^{kλ} \\ v & = P ((\overline{x}_{1}^{a_{1}} \overline{z}^{c_{1}})^{λ}) + \overline{x}_{1}^{d}\overline{z}^{f}. \end{aligned}

k > \frac{d}{a} implies λk > \frac{d}{a_{1}} and ord (P) > \frac{d}{a} implies λord (P) > \frac{d}{a_{1}} . q thus has the form of (186), and we see that (u, v) is not invertible on the curve with local equations x = z = 0.

Now suppose that q′ is a 2 point on the curve y = z = 0. q′ has regular parameters (x_{1}, y, z) where x = x_{1} + α. Thus q′ has permissible parameters (x_{1}, y_{1}, z) where y_{1} is defined by

y = y_{1}(x_{1} + α)^{− \frac{a}{b}}

Set λ = (b, c), b_{1} = \frac{b}{λ}, c_{1} = \frac{c}{λ}.

\begin{aligned}u & = (y_{1}^{b_{1}}z^{c_{1}})^{kλ} \\ v & = P ((y_{1}^{b_{1}}z^{c_{1}})^{λ}) + (x_{1} + α)^{d− \frac{ea}{b}} y_{1}^{e}z^{f} \end{aligned}

First suppose that bf − ce ≠ 0. Then bf − ce > 0, and b_{1}f − c_{1}e > 0. Since λord (P) < \frac{f}{c_{1}} , we have by (186) that (u, v) is not invertible at q′ if and only if λord (P) > \frac{e}{b_{1}} and λk > \frac{e}{b_{1}} so that (u, v) is not invertible at 2 points q′ on y = z = 0 if and only if ord (P) > \frac{e}{b} and k > \frac{e}{b}.

Now suppose that bf − ce = 0, so that b_{1}f − c_{1}e = 0. Since λord (P) < \frac{f}{c_{1}} , (190) cannot hold, and we then have that (u, v) is invertible at 2 points q′ on y = z = 0.

Now suppose that q′ is a 2 point on the curve x = y = 0. q′ has regular parameters (x, y, z_{1}) where z = z_{1} + α. Thus q′ has permissible parameters (x_{1}, y, z_{1}) where x_{1} is defined by

x = x_{1}(z_{1} + α)^{− \frac{c}{a}}

Set λ = (a, b), a_{1} = \frac{a}{λ}, b_{1} = \frac{b}{λ}.

\begin{aligned}u & = (x_{1}^{a_{1}}y^{b_{1}})^{kλ} \\ v & = P ((x_{1}^{a_{1}}y^{b_{1}})^{λ}) + x_{1}^{d}y^{e}(z_{1} + α)^{f− \frac{dc}{a}} \end{aligned}

First suppose that ae − bd ≠ 0 and ord (P) < \frac{e}{b}. Then a_{1}e − b_{1}d > 0 and λord (P) < \frac{e}{b_{1}} . By assumption λk > \frac{d}{a_{1}} and λord (P) > \frac{d}{a_{1}} . By (186), (u, v) is not invertible at 2 points q′ on x = y = 0.

Now suppose that ae−bd ≠ 0 and ord (P) ≥ \frac{e}{b}. Then a_{1}e−b_{1}d > 0 and λord (P) ≥ \frac{e}{b_{1}} , so that we can choose permissible coordinates at q′ so that

\begin{aligned}u & = (x_{1}^{a_{1}}y_{1}^{b_{1}})^{kλ} \\ v & = x_{1}^{d}y_{1}^{e} \end{aligned}

By assumption λk > \frac{d}{a_{1}} , so that by (187), (u, v) is not invertible at 2 points q′ on x = y = 0 if and only if k < \frac{e}{b}.

Suppose that ae−bd = 0 and ord (P) < \frac{e}{b}. Then a_{1}e−b_{1}d = 0 and λord (P) < \frac{e}{b_{1}} . Since (190) can then not hold at q′, we have that (u, v) are invertible at 2 points q′ on x = y = 0.

Now suppose that ae−bd = 0 and ord (P) ≥ \frac{e}{b}. Then a_{1}e−b_{1}d = 0 and λord (P) ≥ \frac{e}{b_{1}} , so that we can choose permissible coordinates at q′ so that

\begin{aligned}u & = (x_{1}^{a_{1}}y_{1}^{b_{1}})^{kλ} \\ v & = (β + α^{f− \frac{dc}{a}})(x_{1}^{a_{1}}y_{1}^{b_{1}})^{t} + (x_{1}^{a_{1}}y_{1}^{b_{1}})^{t}z_{1} \end{aligned}

where t = \frac{e}{b}λ, β ∈ k is the degree t coefficient of P. For q′ in a possibly smaller neighborhood of p_{1}, (190) can then not hold at q′, so that (u, v) are invertible at 2 points q′ on x = y = 0.

Suppose that q′ is a 1 point in U on z = 0. q′ has regular parameters (x_{1}, y_{1}, z) where x = x_{1} + α, y = y_{1} + β with α, β ≠ 0. Thus q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where z_{1} is defined by

z = (x_{1} + α)^{− \frac{a}{c}} (y_{1} + β)^{− \frac{b}{c}} z_{1}

\begin{aligned}u & = z_{1}^{ck} \\ v & = P (z_{1}^{c}) + (x_{1} + α)^{d− \frac{af}{c}} (y_{1} + β)^{e− \frac{bf}{c}} z_{1}^{f}. \end{aligned}

Since by assumption c ord(P) < f, q′ cannot be in the form of (185), so that (u, v) is invertible at 1 points on z = 0.

Suppose that q′ is a 1 point in U on y = 0. q′ has regular parameters (x_{1}, y, z_{1}) where x = x_{1} + α, z = z_{1} + β with α, β ≠ 0. Thus q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where y_{1} is defined by

y = (x_{1} + α)^{− \frac{a}{b}} (z_{1} + β)^{− \frac{c}{b}} y_{1}

\begin{aligned}u & = y_{1}^{bk} \\ v & = P (y_{1}^{b}) + (x_{1} + α)^{d− \frac{ae}{b}} (z_{1} + β)^{f− \frac{ec}{b}} y_{1}^{e} \end{aligned}

If b ord (P) < e or b ord (P) > e then q′ cannot have the form of (185), so that (u, v) is invertible at all 1 points on y = 0. If k ≤ \frac{e}{b}, then (u, v) is invertible at all 1 points on y = 0.

Suppose that b ord (P) = e and k > \frac{e}{b}. Then we can write P (t) = γt^{\frac{e}{b}} + ⋯ where γ ≠ 0. We have (u, v) is invertible at q′ on y = 0 unless

α^{d− \frac{ae}{b}} β^{f− \frac{ec}{b}} + γ = 0

which holds only if (α, β) are on the algebraic curve

β^{bf−ce} = (−γ)^{b}α^{ae−bd}.

In this case (u, v) is not invertible on the curve with local equations

y = 0, z^{bf−ec} + (−γ)^{b}x^{ae−bd} = 0.

If q′ is a 1 point in U on x = 0, then there are permissible parameters (x_{1}, y_{1}, z_{1}) at q′ such that

\begin{aligned}u & = x_{1}^{ak} \\ v & = P (x_{1}^{a}) + x_{1}^{d}(y_{1} + α)^{e− \frac{db}{a}} (z_{1} + β)^{f− \frac{dc}{a}} \end{aligned}

with α, β ≠ 0. Thus (u, v) is invertible at 1 points on x = 0 in U since a ord (P) > d.

If π : X_{1} → X is the blowup of a 2 curve through p, then Φ_{1} = Φ ∘ π is strongly prepared above p. π^{−1}(p) is a 2 curve, so there are no 1 points in π^{−1}(p).

Suppose that p ∈ X is a 3 point such that m_{q}O_{X,p} is not invertible, and (193) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U.

After possibly interchanging x, y, z, we can assume that

\frac{d}{a} = min\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}

and

\frac{f}{c} = max\{\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \}

We will determine the locus of points in U where m_{q}O_{U} is not invertible. First suppose that q′ is a 2 point on the curve x = z = 0. q′ has regular parameters (x, y_{1}, z) where y = y_{1} + α. Thus q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where x_{1} is defined by

x = x_{1}(y_{1} + α)^{− \frac{b}{a}}

Set λ = (a, c), a_{1} = \frac{a}{λ}, c_{1} = \frac{c}{λ}.

\begin{aligned}u & = (x_{1}^{a_{1}}z^{c_{1}})^{kλ} \\ v & = x_{1}^{d}z_{1}^{f}(y_{1} + α)^{e− \frac{db}{a}} \end{aligned}

We have a_{1}f − c_{1}d > 0. \frac{d}{a} < k < \frac{f}{c} implies \frac{d}{a_{1}} < kλ < \frac{f}{c_{1}} . q′ thus has the form of (187), and we see that (u, v) is not invertible on the curve with local equations x = z = 0.

Suppose that q′ is a 2 point on the curve y = z = 0. q′ has permissible parameters (x_{1}, y_{1}, z) where x = x_{1} +α, y_{1} is defined by y = y_{1}(x_{1} +α)^{− \frac{a}{b}}. Set λ = (b, c), b_{1} = \frac{b}{λ}, c_{1} = \frac{c}{λ}.

\begin{aligned}u & = (y_{1}^{b_{1}}z^{c_{1}})^{kλ} \\ v & = (x_{1} + α)^{d− \frac{ea}{b}} y_{1}^{e}z^{f}. \end{aligned}

First suppose that bf − ce ≠ 0. Then bf − ce > 0 and b_{1}f − c_{1}e > 0. Since kλ < \frac{f}{c_{1}} , we have by (187) that (u, v) is not invertible at 2 points q′ on y = z = 0 if and only if \frac{e}{b_{1}} < kλ, which holds if and only if \frac{e}{b} < k.

Now suppose that bf − ce = 0, so that b_{1}f − c_{1}e = 0. Then (u, v) is invertible at 2 points q′ on y = z = 0.

Now suppose that q′ is a 2 point on the curve x = y = 0. q′ has regular parameters (x, y, z_{1}) where z = z_{1} + α. q′ has permissible parameters (x_{1}, y, z_{1}) where x_{1} is defined by x = x_{1}(z_{1} + α)^{− \frac{c}{a}}. Set λ = (a, b), a_{1} = \frac{a}{λ}, b_{1} = \frac{b}{λ}.

\begin{aligned}u & = (x_{1}^{a_{1}}y^{b_{1}})^{kλ} \\ v & = x_{1}^{d}y^{e}(z_{1} + α)^{f− \frac{dc}{a}} \end{aligned}

First suppose that ae − bd ≠ 0. Then a_{1}e − b_{1}d > 0. By assumption \frac{d}{a_{1}} < kλ. We have by (187) that (u, v) is not invertible at 2 points q′ on x = y = 0 if and only if kλ < \frac{e}{b_{1}} which holds if and only if k < \frac{e}{b}.

Now suppose that ae − bd = 0. Then a_{1}e − b_{1}d = 0 and (u, v) is invertible at 2 points on the curve x = y = 0.

Suppose that q′ is a 1 point in U on z = 0. q′ has regular parameters (x_{1}, y_{1}, z) where x = x_{1} + α, y = y_{1} + β (with α, β ≠ 0). Thus q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where z_{1} is defined by z = (x_{1} + α)^{− \frac{a}{c}} (y_{1} + β)^{− \frac{b}{c}} z_{1}.

\begin{aligned}u & = z_{1}^{ck} \\ v & = (x_{1} + α)^{d− \frac{af}{c}} (y_{1} + β)^{e− \frac{bf}{c}} z_{1}^{f} \end{aligned}

Thus (u, v) is invertible at all 1 points of z = 0.

Similarily, (u, v) is invertible at all 1 points of x = 0 and y = 0.

If π : X_{1} → X is the blow up of a 2 curve C through p, then Φ_{X_{1}} is strongly prepared above p. π^{−1}(p) is a 2 curve, so there are no 1 points in π^{−1}(p).

Suppose that p ∈ X is a 2 point such that m_{q}O_{X,p} is not invertible, and (188) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U, and the conclusions of Lemma 18.8 hold for p. After possibly interchanging x and y we may assume that ad − bc > 0. We will determine the locus of points in U where m_{q}O_{U} is not invertible. First suppose that q′ is a 2 point on the curve x = y = 0. q′ has regular parameters (x, y, z_{1}) where z = z_{1} + α. Thus q′ has permissible parameters (\overline{x}, \overline{y}, \overline{z}) such that

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{k} \\ v & = P (\overline{x}^{a}\overline{y}^{b}) + \overline{x}^{c}\overline{y}^{d} \end{aligned}

Since \frac{c}{a} < e = ord (P) < \frac{d}{b}, and k > \frac{c}{a}, we are in the form of (186). Thus (u, v) is not invertible along the curve x = y = 0.

Suppose that q′ is a 1 point near q. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where either

x = x_{1}(y_{1} + α)^{− \frac{b}{a}} , y = y_{1} + α, z = z_{1} + β  (196)

with α ≠ 0 or

x = x_{1} + α, y = y_{1}(x_{1} + α)^{− \frac{a}{b}} , z = z_{1} + β  (197)

with α ≠ 0. If q′ has permissible parameters satisfying (196), then since ord (P) > \frac{c}{a},

\begin{aligned}u & = x_{1}^{ak} \\ v & = P (x_{1}^{a}) + x_{1}^{c}(y_{1} + α)^{d− \frac{bc}{a}} (z_{1} + β) \\ & = βα^{d− \frac{bc}{a}} x_{1}^{c} + x_{1}^{c}\overline{z} \end{aligned}  (198)

(u, v) is not invertible at q′ if and only if q′ satisfies (185). Since c < ak by assumption, this holds if and only if β = 0.

If q′ is a 1 point near p on x = 0 (so that (198) holds) then A(Φ, q′) = 0.

If q′ has permissible parameters satisfying (197), then

\begin{aligned}u & = y_{1}^{bk} \\ v & = P (y_{1}^{b}) + (x_{1} + α)^{c− \frac{da}{b}} y_{1}^{d}(z_{1} + β) \end{aligned}  (199)

(u, v) is invertible at q′, since b ord P < d by assumption, so that q′ cannot satisfy (185).

If q′ is a 1 point near p on y = 0 (so that (199) holds) then

A(Φ, q′) = d − b ord (P).

We will now consider the invariant A on the blowup of V (x, y) or V (x, z) over p. Let π_{1} : X_{1} → X be the blowup of C = V (x, y). Φ_{1} = Φ ∘ π_{1} is strongly prepared

above p. If q ∈ π_{1}^{−1}(p) is a 1 point, then q has regular parameters (x, y_{1}, z) defined by

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. There are permissible parameters (\overline{x}_{1}, y_{,}z) at q where \overline{x}_{1} is defined by

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

Thus

\begin{aligned}u & = \overline{x}_{1}^{(a+b)k} \\ v & = P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d}(y_{1} + α)^{d− \frac{(c+d)b}{a+b}} z \end{aligned}

If (a + b)ord(P) ≥ c + d, then A(Φ_{1}, q) = 0. Assume that (a + b)ord p < c + d. Since ord (P) > \frac{c}{a}, we have that

c + d − (a + b) ord(P) = (d − b ord(P)) + (c − a ord(P)) < d − b ord(P)

Thus

A(Φ_{1}, q) ≤ c + d − (a + b) ord(P) < d − b ord(P) ≤ A(Φ).

If π_{1} : X_{1} → X is the blowup of C = V (x, z), then Φ_{X_{1}} is strongly prepared above p, and there are no 1 points in π_{1}^{−1}(p).

Suppose that p ∈ X is a 2 point such that m_{q}O_{X,p} is not invertible, and (189) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. After possibly interchanging x and y, we may assume that ad − bc > 0. We will determine the locus of points in U where m_{q}O_{U} is not invertible. First suppose that q′ is a 2 point on the curve x = y = 0. q′ has regular parameters (x, y, z_{1}) where z = z_{1} + α. Thus q′ has permissible parameters (\overline{x}, \overline{y}, \overline{z}) such that

\begin{aligned}u & = (\overline{x}^{a}\overline{y}^{b})^{k} \\ v & = \overline{x}^{c}\overline{y}^{d} \end{aligned}

Since \frac{c}{a} < k, we are in the form of (187), and (u, v) is not invertible along the curve x = y = 0 if and only if k < \frac{d}{b}.

Suppose that q′ is a 1 point near p. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where either

x = x_{1}(y_{1} + α)^{− \frac{b}{a}} , y = y_{1} + α, z = z_{1} + β  (200)

with α ≠ 0 or

x = x_{1} + α, y = y_{1}(x_{1} + α)^{− \frac{a}{b}} , z = z_{1} + β  (201)

with α ≠ 0. If q′ has permissible parameters satisfying (200), then

u = x_{1}^{ak}
v = x_{1}^{c}(y_{1} + α)^{d− \frac{bc}{a}} (z_{1} + β)
= βα^{d− \frac{bc}{a}} x_{1}^{c} + x_{1}^{c}\overline{z}

(u, v) is not invertible at q′ if and only if q′ satisfies (185). Since c < ak by assumption, this holds if and only if β = 0.

If q′ is a 1 point near p on x = 0 (so that (200) holds) then A(Φ, q′) = 0.

If q′ has permissible parameters satisfying (201), then

\begin{aligned}u & = y_{1}^{bk} \\ v & = (x_{1} + α)^{c− \frac{da}{b}} y_{1}^{d}(z_{1} + β) \end{aligned}

\begin{aligned}u & = y_{1}^{bk} \\ v & = α^{c− \frac{da}{b}} βy_{1}^{d} + y_{1}^{d}\overline{z}_{1} \end{aligned}

Thus (u, v) is invertible at q′ if β ≠ 0, and if β = 0, then (u, v) is invertible at q′ if and only if d ≥ kb.

If q′ is a 1 point near p on y = 0 (so that (201) holds) then A(Φ, q′) = 0.

We will now consider the invariant A on the blowup of a curve V (x, y), V (y, z) or V (x, z) where (u, v) is not invertible on the curve.

Let π_{1} : X_{1} → X be the blowup of C = V (x, y). Φ_{1} = Φ ∘ π_{1} is strongly prepared over p. If q ∈ π^{−1}(p) is a 1 point, then q has regular parameters (x, y_{1}, z) defined by

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. There are permissible parameters (x_{1}, y_{,}z) at q where x_{1} is defined by

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

Thus

\begin{aligned}u & = \overline{x}_{1}^{(a+b)k} \\ v & = \overline{x}_{1}^{c+d}(y_{1} + α)^{d− \frac{(c+d)b}{a+b}} z \end{aligned}

and A(Φ_{1}, q) = 0.

If π_{1} : X_{1} → X is the blowup of C = V (x, z) or V (y, z), then Φ_{1} = Φ ∘ π_{1} is strongly prepared over p, and there are no 1 points in π^{−1}(p).

Suppose that p ∈ X is a 2 point such that m_{q}O_{X,p} is not invertible, and (190) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. We will determine the locus of points in U where m_{q}O_{U} is not invertible. First suppose that q′ is a 2 point on the curve x = y = 0. q′ has permissible parameters (x, y, z_{1}) where z = z_{1} + α.

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = α(x^{a}y^{b})^{t} + (x^{a}y^{b})^{t}z_{1} \end{aligned}

Thus (u, v) is invertible along the curve x = y = 0, if α ≠ 0.

Suppose that q′ is a 1 point near p. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where either

x = x_{1}(y_{1} + α)^{− \frac{b}{a}} , y = y_{1} + α, z = z_{1} + β  (202)

with α ≠ 0 or

x = x_{1} + α, y = y_{1}(x_{1} + α)^{− \frac{a}{b}} , z = z_{1} + β  (203)

with α ≠ 0. If q′ has permissible parameters satisfying (202), then

\begin{aligned}u & = x_{1}^{ak} \\ v & = x_{1}^{at}(z_{1} + β) \end{aligned}

Thus (u, v) is invertible at q′ if β ≠ 0, and q′ is not invertible along V (x, z) since t < k by assumption.

If q′ has permissible parameters satisfying (203), then

\begin{aligned}u & = y_{1}^{bk} \\ v & = y_{1}^{bt}(z_{1} + β) \end{aligned}

Thus (u, v) is invertible at q′ if β ≠ 0, and q′ is not invertible along V (y, z) since t < k by assumption.

If π_{1} : X_{1} → X is the blowup of V (x, z) or V (y, z), then Φ_{1} = Φ ∘ π_{1} is strongly prepared over p and there are no 1 points in π^{−1}(p).

Suppose that p ∈ X is a 2 point such that m_{q}O_{X,p} is not invertible, and (186) holds at p. After possibly interchanging x and y, we may assume that ad − bc > 0. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U and the conclusions of Lemma 18.8 hold for p. We will determine the locus of points in U where m_{q}O_{U} is not invertible. If q′ is a 2 point on the curve x = y = 0, then q′ has the form of (186), so that Thus (u, v) is not invertible along the curve x = y = 0.

Suppose that q′ is a 1 point near q′. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where either

x = x_{1}(y_{1} + α)^{− \frac{b}{a}} , y = y_{1} + α, z = z_{1} + β  (204)

with α ≠ 0 or

x = x_{1} + α, y = y_{1}(x_{1} + α)^{− \frac{a}{b}} , z = z_{1} + β  (205)

with α ≠ 0. If q′ has permissible parameters satisfying (204), then

u = x_{1}^{ak}
v = P (x_{1}^{a}) + x_{1}^{c}(y_{1} + α)^{d− \frac{bc}{a}}
= α^{d− \frac{bc}{a}} x_{1}^{c} + x_{1}^{c}\overline{y}

for some permissible parameters (x_{1}, \overline{y}, z), since a ord (P) > c. Thus (u, v) is invertible at q′, since we have α ≠ 0.

A(Φ, q′) = 0 at points q′ near p where (204) holds.

If q′ has permissible parameters satisfying (205), then

\begin{aligned}u & = y_{1}^{bk} \\ v & = P (y_{1}^{b}) + α^{c− \frac{da}{b}} y_{1}^{d} + y_{1}^{d}\overline{x}_{1} \end{aligned}

(u, v) is invertible at q′ Since b ord(P) < d by assumption.

At points q′ near p where (205) holds, we have A(Φ, q′) = d − b ord (P) > 0.

Let π_{1} : X_{1} → X be the blowup of C = V (x, y). Then Φ_{1} = Φ ∘ π_{1} is strongly prepared above p. If q ∈ π_{1}^{−1}(p) is a 1 point, then q has regular parameters (x, y_{1}, z) defined by

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. There are permissible parameters (\overline{x}_{1}, y_{,}z) at q where \overline{x}_{1} is defined by

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}}

Thus

u = \overline{x}_{1}^{(a+b)k}
v = P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d}(y_{1} + α)^{d− \frac{(c+d)b}{a+b}}
= P (\overline{x}_{1}^{a+b}) + \overline{x}_{1}^{c+d}α^{d− \frac{(c+d)b}{a+b}} + \overline{x}_{1}^{c+d}\overline{y}_{1}

If (a + b)ord (P) ≥ c + d then A(Φ_{1}, q) = 0. Assume that (a + b)ord(P) < c + d. Since ord(P) > \frac{c}{a}, we have that

c + d − (a + b)ord(P) = (d − b ord(P)) + (c − a ord(P)) < d − b ord(P)

Thus

A(Φ_{1}, q) ≤ c + d − (a + b) ord (P) < d − b ord (P) ≤ A(Φ)

Suppose that p ∈ X is a 2 point such that m_{q}O_{X,p} is not invertible, and (187) holds at p. After possibly interchanging x and y, we may assume that ad − bc > 0. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. We will determine the locus of points in U where m_{q}O_{U} is not invertible. If q′ is a 2 point on the curve x = y = 0, then q′ has the form of (187), so that (u, v) is not invertible along the curve x = y = 0.

Suppose that q′ is a 1 point near p. q′ has permissible parameters (x_{1}, y_{1}, z_{1}) where either

x = x_{1}(y_{1} + α)^{− \frac{b}{a}} , y = y_{1} + α, z = z_{1} + β  (206)

with α ≠ 0 or

x = x_{1} + α, y = y_{1}(x_{1} + α)^{− \frac{a}{b}} , z = z_{1} + β  (207)

with α ≠ 0. If q′ has permissible parameters satisfying (206), then

u = x_{1}^{ak}
v = x_{1}^{c}(y_{1} + α)^{d− \frac{bc}{a}}
= α^{d− \frac{bc}{a}} x_{1}^{c} + x_{1}^{c}\overline{y}_{1}

(u, v) is thus invertible at q′.

A(Φ, q′) = 0 at points q′ near p where (206) holds.

If q′ has permissible parameters satisfying (207), then

u = y_{1}^{bk}
v = (x_{1} + α)^{c− \frac{da}{b}} y_{1}^{d}
= α^{c− \frac{da}{b}} y_{1}^{d} + y_{1}^{d}\overline{x}_{1}

Thus (u, v) is invertible at q′.

A(Φ, q′) = 0 at points q′ near p where (207) holds.

The locus of points where (u, v) is not invertible near p is V (x, y).

Let π_{1} : X_{1} → X be the blowup of C = V (x, y). Φ_{1} = Φ ∘ π_{1} is strongly prepared above p. If q ∈ π_{1}^{−1}(p) is a 1 point, then q has regular parameters (x_{1}, y_{1}, z) defined by

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0. There are permissible parameters (\overline{x}_{1}, y_{,}z) at q where \overline{x}_{1} is defined by

x_{1} = \overline{x}_{1}(y_{1} + α)^{− \frac{b}{a+b}} .

Thus

u = \overline{x}_{1}^{(a+b)k}
v = \overline{x}_{1}^{c+d}(y_{1} + α)^{d− \frac{(c+d)b}{a+b}}
= \overline{x}_{1}^{c+d}α^{d− \frac{(c+d)b}{a+b}} + \overline{x}_{1}^{c+d}\overline{y}_{1}

and A(Φ_{1}, q) = 0.

Suppose that p ∈ X is a 1 point such that m_{q}O_{X,p} is not invertible, so that (185) holds at p. We may assume that here exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. We will determine the locus of points in U where m_{q}O_{U} is not invertible.

Suppose that q′ is a 1 point near p. q′ has permissible parameters (x, y_{1}, z_{1}) where

y = y_{1} + α, z = z_{1} + β

\begin{aligned}u & = x^{k} \\ v & = αx^{c} + x^{c}y_{1} \end{aligned}

(u, v) is thus only not invertible on the curve V (x, y).

Let π : X_{1} → X be the blowup of V (x, y). Φ_{1} = Φ ∘ π_{1} is strongly prepared above p. If q ∈ π^{−1}(p) is a 1 point, then q has permissible parameters (x, y_{1}, z) defined by

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0.

\begin{aligned}u & = x_{1}^{k} \\ v & = x_{1}^{c+1}(y_{1} + α) \end{aligned}

and A(Φ_{1}, q) = 0.

Suppose that p ∈ X is a 3 point such that m_{p}O_{X,p} is not invertible and (194) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. The locus of points in U where m_{q}O_{U} is not invertible is the union of the 2 curves V (x, z), V (y, z) and V (x, y).

Let π : X_{1} → X be the blowup of C = V (x, y), Φ_{1} = Φ ∘ π. If q′ ∈ π^{−1}(p) is a 2 point, then q′ has permissible parameters (x_{1}, y_{1}, z) where

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0.

\begin{aligned}u & = x_{1}^{a+b}(y_{1} + α)^{b} = \overline{x}_{1}^{a+b} \\ v & = z^{c} \end{aligned}

so that Φ_{1} is strongly prepared at q′.

Suppose that q′ ∈ π^{−1}(p) is a 3 point and q′ has permissible parameters (x_{1}, y_{1}, z) where x = x_{1}, y = x_{1}y_{1}. Then

\begin{aligned}v & = x_{1}^{a+b}y_{1}^{b} \\ u & = z^{c} \end{aligned}

so that Φ_{1} is strongly prepared at q′. Suppose that q′ ∈ π^{−1}(p) is a 3 point and q′ has permissible parameters (x_{1}, y_{1}, z) where

x = x_{1}y_{1}, y = y_{1}.

Then

\begin{aligned}u & = y_{1}^{a+b} \\ v & = z^{c} \end{aligned}

so that Φ_{1} is strongly prepared at q′. A similar analysis shows that the blowup of V (x, z) or V (y, z) composed with Φ is strongly prepared.

Suppose that p ∈ X is a 3 point such that m_{q}O_{X,p} is not invertible and (195) holds at p.

We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing on an étale cover of U. The locus of points in U where m_{q}O_{U} is not invertible is the union of the 2 curves V (x, z), V (x, y,) (if c > b) and V (y, z) (if b > c). If π : X_{1} → X is the blowup of a 2 curve through C, and Φ_{1} = Φ ∘ π, then Φ_{1} is strongly prepared at points q ∈ π^{−1}(p).

Suppose that p ∈ X is a 2 point such that m_{p}O_{X,p} is not invertible, and (191) holds at p. We may assume that there exists an open neighborhood U of p such that (x, y, z) are uniformizing parameters on an étale cover of U. The locus of points in U where m_{q}O_{U} is not invertible is the 2 curve V (x, y).

Let π : X_{1} → X be the blowup of C = V (x, y), Φ_{1} = Φ ∘ π. If q′ ∈ π^{−1}(p) is a 1 point, then q′ has permissible parameters

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0.

\begin{aligned}u & = x_{1}^{a} \\ v & = x_{1}^{b}(y_{1} + α)^{b}. \end{aligned}

Set \overline{y}_{1} = (y_{1} + α)^{b} − α^{b} to get

\begin{aligned}u & = x_{1}^{a} \\ v & = α^{b}x_{1}^{b} + x_{1}^{b}\overline{y}_{1} \end{aligned}

so that A(Φ_{1}, q′) = 0. Φ_{1} is strongly prepared at points of π^{−1}(p).

If α, β are real numbers, define

S(α, β) = max {(α, β), (β, α)}

where the maximum is in the Lexicographic ordering.

Suppose that Φ : X → S is strongly prepared. Suppose that q ∈ D_{S} and C ⊂ X is a 2 curve such that m_{q}O_{X} is not invertible along C. At a generic point p of C (186),

(187) or (191) holds.

If (186) holds, then Φ(C) is a 1 point q ∈ S. Suppose that P (t) = ∑ a_{i}t^{i}. Since q is a 1 point, we can, after possibly replacing v with v −∑ a_{ik}u^{k}, assume that k |̸ ord(P) in (186). With this restriction, define

σ(C) = \{ \begin{matrix}S(| c − a ord (P) |, | d − b ord (P) |) \\ if c − ord (P), d − ord (P) have opposite signs, \\ −∞ if they have the same sign. \end{matrix}

If (187) or (191) holds, define

σ(C) = −∞.

σ(C) is well defined (independent of choice of permissible parameters (u, v) at q with the restriction that k |̸ ord(P) in (186)). This follows from Lemma 18.8.

If m_{q}O_{X} is invertible, define

\overline{σ}(Φ) = −∞.

If m_{q}O_{X} is not invertible, define

\overline{σ}(Φ) = max \left\{{σ(C) | C ⊂ X is a 2 curve \atop such that m_{q}O_{X} is not invertible along C}\right\} .

Lemma 18.16. Suppose that X is strongly prepared, q ∈ S is such that m_{q}O_{X} is not invertible. Then there exists a sequence of blowups of 2 curves X_{1} → X such that the induced map Φ_{1} : X_{1} → S is strongly prepared, A(Φ_{1}, E) < A(Φ_{1}) = A(Φ) if E is an exceptional component of E_{X_{1}} for X_{1} → X, and the forms (186), (188) and (192) do not hold at any point p ∈ X where m_{q}O_{X_{1},p} is not invertible.

Proof. \overline{σ}(Φ) ≥ 0 if and only if there exists a point p ∈ X such that m_{q}O_{X_{1},p} is not invertible, and a form (186), (188) or (192) holds at p.

Suppose that \overline{σ}(Φ) ≥ 0. Let C be a 2 curve such that σ(C) = \overline{σ}(Φ). Let π : X_{1} → X be the blowup of C. By Theorem 18.15, we need only verify that if C_{1} ⊂ π^{−1}(C) is a 2 curve such that m_{q}O_{X_{1}} is not invertible along C_{1} then σ(C_{1}) < \overline{σ}(Φ).

First suppose that C_{1} is a section over C. Let p_{1} ∈ C_{1} be a generic point. Then p = π(p_{1}) is a generic point of C. There exist permissible parameters (x, y, z) at p such that

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}

with

min\{\frac{c}{a}, \frac{d}{b}\} < ord (P) < max \{\frac{c}{a}, \frac{d}{b}\}

and

min\{\frac{c}{a}, \frac{d}{b}\} < k, k |̸ ord(P).

We may assume, after possibly interchanging x and y that

\overline{σ}(Φ) = σ(C) = (| c − a ord (P) |, | d − b ord (P) |).

Assume that p_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}y_{1}

and x_{1} = y_{1} = 0 are local equations of C_{1} at p_{1}.

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{k} \\ v & = P (x_{1}^{a+b}y_{1}^{b}) + x_{1}^{c+d}y_{1}^{d} \end{aligned}

k |̸ ord(P) implies by Lemma 18.8 that

σ(C_{1}) ≤ S(| (c + d) − (a + b) ord (P) |, | d − b ord (P) |).

If c − a ord (P) > 0 and d − b ord (P) < 0 then

0 ≤ (c − a ord (P) + (d − b ord (P)) < c − ord (P)

so that σ(C_{1}) < σ(C).

If c − a ord (P) < 0 and d − b ord (P) > 0 then

0 ≥ (c − a ord (P) + (d − b ord (P)) > c − ord (P)

so that σ(C_{1}) < σ(C).

Assume that p_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}y_{1}, y = y_{1}

and x_{1} = y_{1} = 0 are local equations of C_{1} at p_{1}.

\begin{aligned}u & = (x_{1}^{a}y_{1}^{a+b})^{k} \\ v & = P (x_{1}^{a}y_{1}^{a+b}) + x_{1}^{c}y_{1}^{c+d} \end{aligned}

k |̸ ord(P) implies

σ(C_{1}) ≤ S(| (c + d) − (a + b) ord (P) |, | c − a ord (P) |).

If c − a ord (P) > 0 and d − b ord (P) < 0 then

0 ≤ (c − a ord (P) + (d − b ord (P)) < c − ord (P)

so that σ(C_{1}) = −∞.

If c − a ord (P) < 0 and d − b ord (P) > 0 then

0 ≥ (c − a ord (P) + (d − b ord (P)) > c − ord (P)

so that σ(C_{1}) = −∞.

Now suppose that C_{1} ⊂ π^{−1}(C) is an exceptional 2 curve. Then p = π(C_{1}) satisfies (192). Let q = Φ(p). q is a 1 point. If E_{1}, E_{2}, E_{3} are the components of E_{X} containing p, then Φ(E_{1}) = Φ(E_{2}) = Φ(E_{3}) = q. Suppose that P (t) = ∑ a_{i}t^{i}. Since q is a 1 point, we may replace v with v − ∑ a_{ik}u^{i} so that k |̸ ord(P). We may also assume, after possibly interchanging (x, y, z) that

| f − c ord (P) |≥| e − b ord (P) |≥| d − a ord (P) | .

If C has local equations x = z = 0, then

σ(C) = (| f − c ord(P) |, | d − a ord(P) |)

and a generic point of C_{1} has permissible parameters (x_{1}, y, z_{1}) where

x = x_{1}, z = x_{1}(z_{1} + α)

with α ≠ 0, and x_{1} = y = 0 are local equations of C_{1}. Set \overline{x}_{1} = x_{1}(z_{1} + α)^{− \frac{c}{a+c}}.

\begin{aligned}u & = (\overline{x}\overline{a} ; 1y\overline{b})^{λk} \\ v & = P ((\overline{x}\overline{a} ; 1y\overline{b})^{λ}) + \overline{x}_{1}^{d+f}y^{e}(z_{1} + α)^{f− \frac{(d+f)c}{a+c}} \end{aligned}

where λ = (a + c, b), a + c = \overline{a}λ, b = \overline{b}λ.

If σ(C_{1}) ≥ 0, then λk |̸ λord(P) implies

σ(C_{1}) = S(| (d + f) − (a + c) ord (P) |, | e − b ord (P) |).

Similarily, if C has local equations y = z = 0,

σ(C) = (| f − c ord(P) |, | d − a ord(P) |)

and if σ(C_{1}) ≥ 0, k |̸ ord(P) implies

σ(C_{1}) = S(| d − a ord (P) |, | (e + f) − (b + c) ord (P) |).

If C has local equations x = y = 0, then

σ(C) = (| e − b ord(P) |, | d − a ord(P) |)

and if σ(C_{1}) ≥ 0, then

σ(C_{1}) = S(| f − c ord(P) |, | (d + e) − (a + b)ord (P) |).

If one of f − c ord(P), e − b ord(P), d − a ord(P) is zero, then σ(C_{1}) = −∞.

Case 1 Suppose that f − c ord (P) > 0, e − b ord (P) > 0, d − a ord (P) < 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | d − a ord (P) |)

and x = z = 0 are local equations of C.

0 ≤ (d − a ord (P)) + (f − c ord (P)) < f − c ord (P)

implies σ(C_{1}) = −∞.

Case 2 Suppose that f − c ord (P) > 0, e − b ord (P) < 0, d − a ord (P) > 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | e − b ord (P) |)

and y = z = 0 are local equations of C.

0 ≤ (e − b ord (P)) + (f − c ord (P)) < f − c ord (P)

implies σ(C_{1}) = −∞.

Case 3 Suppose that f − c ord (P) > 0, e − b ord (P) < 0, d − a ord (P) < 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | e − b ord (P) |)

and y = z = 0 are local equations of C.

0 ≤ (e − b ord (P)) + (f − c ord (P)) < f − c ord (P)

implies σ(C_{1}) < σ(C).

Case 4 Suppose that f − c ord (P) < 0, e − b ord (P) > 0, d − a ord (P) > 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | e − b ord (P) |)

and y = z = 0 are local equations of C.

0 ≥ (e − b ord (P)) + (f − c ord (P)) > f − c ord (P)

implies σ(C_{1}) < σ(C).

Case 5 Suppose that f − c ord (P) < 0, e − b ord (P) > 0, d − a ord (P) < 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | e − b ord (P) |)

and y = z = 0 are local equations of C.

0 ≥ (e − b ord (P)) + (f − c ord (P)) > f − c ord (P)

implies σ(C_{1}) = −∞.

Case 6 Suppose that f − c ord (P) < 0, e − b ord (P) < 0, d − a ord (P) > 0. Then

\overline{σ}(Φ) = σ(C) = (| f − c ord (P) |, | d − a ord (P) |)

and x = z = 0 are local equations of C.

0 ≥ (d − a ord (P)) + (f − c ord (P)) > f − c ord (P)

implies σ(C_{1}) = −∞.

We conclude that if C_{1} ⊂ π^{−1}(C) is a 2 curve such that m_{q}O_{X_{1}} is not invertible along C_{1}, then σ(C_{1}) < \overline{σ}(Φ).

By Theorem 18.15, induction on the number of 2 curves C ⊂ X such that σ(C) = \overline{σ}(Φ), and induction on \overline{σ}(Φ), we achieve the conclusions of the Lemma.

Lemma 18.17. Suppose that Φ : X → S is strongly prepared, q ∈ S is such that m_{q}O_{X} is not invertible and the forms (186), (188) and (192) do not hold at any point p ∈ X where m_{q}O_{X,p} is not invertible.

Then there exists a sequence of blowups of nonsingular curves X_{1} → X which are not 2 curves such that the induced map Φ_{1} : X_{1} → S is strongly prepared, A(Φ_{1}, E) < A(Φ_{1}) = A(Φ) if E is an exceptional component of E_{X_{1}} for X_{1} → X, the forms (186),

(188) and (192) do not hold at any point p ∈ X_{1} where m_{q}O_{X_{1},p} is not invertible, and if C ⊂ X_{1} is a curve such that m_{q}O_{X_{1}} is not invertible along C, then C is a 2 curve.

Proof. Suppose that C is a curve such that m_{q}O_{X} is not invertible along C and C is not a 2 curve. Suppose that p ∈ C. Then one of the following holds:

1. (185) holds at p, x = y = 0 are local equations of C at p.

2. (189) holds at p and x = z = 0 with \frac{c}{a} < k or y = z = 0 with \frac{d}{b} < k are local equations of C at p.

3. (190) holds at p, x = z = 0 or y = z = 0 are local equations of C at p.

At a generic point p ∈ C (185) holds. Define

Ω(C) = k − c > 0.

Let

\overline{Ω}(Φ) = max \left\{{Ω(C) | C is not a 2 curve \atop and m_{q}O_{X} is not invertible along C.}\right\}

Suppose that Ω(C) = \overline{Ω}(Φ). Let π : X_{1} → X be the blowup of C. The forms (186), (188) and (192) cannot hold at points of X_{1}. By Theorem 18.15, we need only verify that Ω(C_{1}) < \overline{Ω}(Φ) if C_{1} is a curve in π^{−1}(C) such that m_{q}O_{X_{1}} is not invertible along C_{1} and C_{1} is not a 2 curve. We then have π(C_{1}) = C.

Let p_{1} be a generic point of C_{1}, p = π(p_{1}). (185) holds at p since p is a generic point of C. p_{1} ∈ π^{−1}(p) is a 1 point. Then p_{1} has permissible parameters (x_{1}, y_{1}, z_{1}) such that

x = x_{1}, y = x_{1}(y_{1} + α).

\begin{aligned}u & = x_{1}^{k} \\ v & = x_{1}^{c+1}(y_{1} + α). \end{aligned}

m_{q}O_{X_{1},p_{1}} is invertible if α ≠ 0. If α = 0 and m_{q}O_{X_{1},p_{1}} is not invertible, then x_{1} = y_{1} = 0 are local equations of the curve C_{1} ⊂ X_{1} through p on which m_{q}O_{X_{1}} is not invertible.

0 < Ω(C_{1}) = k − (c + 1) < Ω(C) = \overline{Ω}(Φ).

By induction on the number of curves C on X such that Ω(C) = \overline{Ω}(Φ), we achieve the conclusions of the Lemma.

Lemma 18.18. Suppose that Φ : X → S is strongly prepared, q ∈ S is such that m_{q}O_{X} is not invertible, the forms (186), (188) and (192) do not hold at any point p ∈ X where m_{q}O_{X,p} is not invertible, and if C ⊂ X is a curve such that m_{q}O_{X} is not invertible along C, then C is a 2 curve.

Then there exists a sequence of blowups of 2 curves X_{1} → X such that the induced map Φ_{1} : X_{1} → S is strongly prepared, A(Φ_{1}, E) < A(Φ_{1}) = A(Φ) if E is an exceptional component of E_{X_{1}} for X_{1} → X and m_{q}O_{X_{1}} is invertible.

Proof. Suppose that C ⊂ X is a 2 curve such that m_{q}O_{X} is not invertible along C. Suppose that p ∈ C. Then (187), (193), (191), (194) or (195) hold at p. At a generic point p ∈ C (187) or (191) holds.

If C is a 2 curve such that at a generic point of C (187) holds, define

ω(C) = \{ \begin{matrix}S(| ka − c |, | kb − d |) & if ka − c, kb − d have opposite signs \\ & and m_{q}O_{X} is not invertible along C \\ −∞ & otherwise \end{matrix}

If C is a 2 curve such that at a generic point of C (191) holds, define

ω(C) = \{ \begin{matrix}S(a, b) & if m_{q}O_{X} is not invertible along C \\ −∞ & otherwise \end{matrix}

m_{q}O_{X} is not invertible along C if and only if ω(C) > 0. Set

\overline{ω}(Φ) = max \left\{{ω(C) | C is a 2 curve such that \atop m_{q}O_{X} is not invertible along C}\right\}

Suppose that ω(C) = \overline{ω}(Φ). Let π : X_{1} → X be the blowup of C. By Theorem 18.15, we need only verify that ω(C_{1}) < \overline{ω}(Φ) if C_{1} ⊂ π^{−1}(C) is a curve such that m_{q}O_{X_{1}} is not invertible along C_{1}. We must have that C_{1} is a 2 curve.

Suppose that C_{1} is a section over C. Let p_{1} ∈ C_{1} be a generic point. Then p = π(p_{1}) is a generic point on C.

Suppose that there exist permissible parameters (x, y, z) at p such that (187) holds.

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = x^{c}y^{d} \end{aligned}

with

min\{\frac{c}{a}, \frac{d}{b}\} < k < max\{\frac{c}{a}, \frac{d}{b}\}.

After possibly interchanging x and y, we may assume that

ω(C) = (| c − ak |, | d − bk |).

Assume that p_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}y_{1}

and x_{1} = y_{1} = 0 are local equations of C_{1} at p_{1}.

\begin{aligned}u & = (x_{1}^{a+b}y_{1}^{b})^{k} \\ v & = x_{1}^{c+d}y_{1}^{d} \end{aligned}

ω(C_{1}) = \{ \begin{matrix}S(| (c + d) − (a + b)k |, | bk − d |) & if (c + d) − (a + b)k, d − bk have opposite signs \\ & and m_{q}O_{X_{1}}is not invertible along C \\ −∞ & otherwise \end{matrix}

Suppose that c − ak > 0 and d − bk < 0.

0 ≤ (c − ak) + (d − bk) < c − ak

implies ω(C_{1}) < ω(C).

Suppose that c − ak < 0 and d − bk > 0.

0 ≥ (c − ak) + (d − bk) > c − ak

implies ω(C_{1}) < ω(C).

Assume that p_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}y_{1}, y = y_{1}

and x_{1} = y_{1} = 0 are local equations of C_{1} at p_{1}.

\begin{aligned}u & = (x_{1}^{a}y_{1}^{a+b})^{k} \\ v & = x_{1}^{c}y_{1}^{c+d} \end{aligned}

ω(C_{1}) = \{ \begin{matrix}S(| (c + d) − (a + b)k |, | ak − c |) & if (c + d) − (a + b)k, c − ak have opposite signs \\ & and m_{q}O_{X_{1}}is not invertible along C \\ −∞ & otherwise \end{matrix}

Suppose that c − ak > 0 and d − bk < 0.

0 ≤ (c − ak) + (d − bk) < c − ak

implies ω(C_{1}) = −∞.

Suppose that c − ak < 0 and d − bk > 0.

0 ≥ (c − ak) + (d − bk) > c − ak

implies ω(C_{1}) = −∞.

Suppose that C_{1} is a section over C, p_{1} ∈ C_{1} is a generic point and p ∈ π(p_{1}) is a generic point on C such that p satisfies (191). Then a similar argument shows that ω(C_{1}) < ω(C).

Suppose that C_{1} ⊂ π^{−1}(C) is an exceptional 2 curve. Suppose that p = π(C_{1}) satisfies (193). Without loss of generality,

| f − ck |≥| e − bk |≥| d − ak | .

If C has local equations x = z = 0 then a generic point of C_{1} has regular parameters (x_{1}, y, z_{1}) such that

x = x_{1}, z = x_{1}(z_{1} + α)

(with α ≠ 0) and x_{1} = y = 0 are local equations of C_{1}. Set \overline{x}_{1} = x_{1}(z_{1} + α)^{− \frac{c}{a+c}}.

\begin{aligned}u & = (\overline{x}\overline{a} ; 1y\overline{b})^{λk} \\ v & = \overline{x}_{1}^{d+f}y^{e}(z_{1} + α)^{f− \frac{(d+f)c}{a+c}} \end{aligned}

where λ = (a + c, b), a + c = \overline{a}λ, b = \overline{b}λ. ω(C_{1}) ≥ 0 implies

ω(C_{1}) = S(| (d + f) − (a + c)k |, | e − bk |).

Similarily, if C_{1} has local equations y = z = 0 then ω(C_{1}) ≥ 0 implies

ω(C_{1}) = S(| (e + f) − (b + c)k |, | d − ak |).

If C_{1} has local equations x = y = 0 then ω(C_{1}) ≥ 0 implies

ω(C_{1}) = S(| f − ck |, | (d + e) − (a + b)k |).

If one of d − ak, e − bk, f − ck is zero, then ω(C_{1}) = −∞.

The analysis of Cases 1 - 6 of Lemma 18.16 (with ord (P) changed to k and σ to ω) shows that ω(C_{1}) < ω(C).

A similar argument shows that ω(C_{1}) < ω(C) if p = π(C) satisfies (194) or (195). We achieve the conclusions of the Lemma by Theorem 18.15, induction on the

number of 2 curves C ⊂ X such that ω(C) = \overline{ω}(Φ), and by induction on \overline{ω}(Φ).

Theorem 18.19. Suppose that Φ : X → S is strongly prepared with respect to D_{S}. Then there exists a finite sequence of quadratic transforms π_{1} : S_{1} → S and monoidal transforms centered at nonsingular curves π_{2} : X_{1} → X such that the induced morphism Φ_{1} : X_{1} → S_{1} is strongly prepared with respect to D_{S_{1}} = π_{1}^{−1}(D_{S})_{red}, and all points of X_{1} are good for Φ_{1}.

Proof. By Remark 18.9, A(Φ) = 0 if and only if all points of X are good. Suppose that A(Φ) > 0 and E is a component of E_{X} such that C(Φ, E) = C(Φ). A(Φ, E) > 0 implies Φ(E) is a point q.

Let π_{1} : S_{1} → S be the blowup of q. By Lemmas 18.16, 18.17, 18.18 there exists a sequence of blowups of curves X_{1} → X such that Φ_{1} : X_{1} → S is strongly prepared, C(Φ_{1}) = C(Φ), A(Φ_{1}, \overline{E}) < A(Φ_{1}) if \overline{E} is exceptional for Φ_{1} and Φ_{2} : X_{1} → S_{1} is a strongly prepared morphism.

By Theorem 18.14, C(Φ_{2}, Ẽ) < C(Φ), where Ẽ is the strict transform of E on X_{1}. By induction on the number of components E of E_{X} such that C(Φ, E) = C(Φ),

and induction on C(Φ), we get the conclusions of the Theorem.

Definition 18.20. Suppose that Φ : X → Y is a dominant morphism of k-varieties, (where k is a field of characteristic zero). Φ is a monomial morphism if for all p ∈ X there exists an étale neighborhood U of p, uniformizing parameters (x_{1}, …, x_{n}) on U, regular parameters (y_{1}, …, y_{m}) in O_{Y,Φ(p)}, and a matrix (a_{ij}) of nonnegative integers such that

y_{1} = x_{1}^{a_{11}} ⋯ x_{n}^{a_{1n}}
⋮
y_{m} = x_{1}^{a_{m1}} ⋯ x_{n}^{a_{mn}}

Theorem 18.21. Suppose that Φ : X → S is a dominant morphism from a 3 fold X to a surface S (over an algebraically closed field k of characteristic zero). Then there exist sequences of blowups of nonsingular subvarieties X_{1} → X and S_{1} → S such that the induced map Φ_{1} : X_{1} → S_{1} is a monomial morphism.

Proof. This follows from Theorem 17.3, the fact that prepared implies strongly prepared, Theorem 18.19 and Remark 18.6.

19. Toroidalization

Throughout this section we will assume that Φ : X → S is strongly prepared with respect to D_{S}, and all points of X are good.

Definition 19.1. ([18] and [5]) A normal variety X with a SNC divisor E_{X} on X is called toroidal if for every point p ∈ X there exists an affine toric variety X_{σ}, a point p′ ∈ X_{σ} and an isomorphism of k algebras

Ô_{X,p} ≅ Ô_{X_{σ},p′}

such that the ideal of E_{X} corresponds to the ideal of X_{σ} − T (where T is the torus in X_{σ}). Such a pair (X_{σ}, p′) is called a local model at p ∈ X.

A dominant morphism Φ : X → Y of toroidal varieties with SNC divisors D_{Y}, E_{X} on X, Y and Φ^{−1}(D_{Y}) ⊂ E_{X} is called toroidal at p, and we will say that p is a toroidal point of Φ, if with q = Φ(p), there exist local models (X_{σ}, p′) at p, (Y_{τ} , q′) at q and a toric morphism Ψ : X_{σ} → Y_{τ} such that the following diagram commutes

Ô_{X,p} ←^{∼} Ô_{X_{σ},p′}
Φ̂^{∗} ↑ Ψ̂^{∗} ↑
O_{Y,q} ←^{∼} Ô_{Y_{τ} ,q′}

Φ : X → Y is called toroidal (with respect to D_{Y} and E_{X}) if Φ is toroidal at all p ∈ X.

Remark 19.2. 1. If one of the forms (177), (178) or (182) holds at p ∈ X then

q = Φ(p) is a 2 point.

2. If q = Φ(p) is a 2 point, then (181) cannot hold at p, and if (180) or (183) hold at p, we must have α ≠ 0, since uv = 0 is a local equation of E_{X}.

Lemma 19.3. Suppose that Φ : X → S is a morphism from a nonsingular 3 fold X to a nonsingular surface S, D_{S} is a SNC divisor on S such that E_{X} = Φ^{−1}(D_{S}) is a SNC divisor on X. Then Φ is a toroidal morphism if and only if for all p ∈ E_{X} there exist regular parameters (x, y, z) in Ô_{X,p} (u, v) in O_{S,p} such that one of the following forms hold:

1. u = 0 is a local equation for D_{S}.

(a) xy = 0 is a local equation for E_{X} and

\begin{aligned}u & = x^{a}y^{b} \\ v & = z \end{aligned}

(b) x = 0 is a local equation for E_{X} and

\begin{aligned}u & = x^{a} \\ v & = y \end{aligned}

2. uv = 0 is a local equation for D_{S}.

(a) xyz = 0 is a local equation for E_{X} and

\begin{aligned}u & = x^{a}y^{b}z^{c} \\ v & = x^{d}y^{e}z^{f} \end{aligned}

with

rank ( \begin{matrix}a & b & c \\ d & e & f \end{matrix} ) = 2.

(b) xy = 0 is a local equation for E_{X} and

\begin{aligned}u & = x^{a}y^{b} \\ v & = x^{c}y^{d} \end{aligned}

with ad − bc ≠ 0.

(c) xy = 0 is a local equation of E_{X} and

\begin{aligned}u & = (x^{a}y^{b})^{k} \\ v & = α(x^{a}y^{b})^{t} + (x^{a}y^{b})^{t}z \end{aligned}

with a, b > 0, k, t > 0, 0 ≠ α ∈ k.

(d) x = 0 is a local equation for E_{X} and

\begin{aligned}u & = x^{a} \\ v & = x^{c}(y + α) \end{aligned}

with 0 ≠ α ∈ k.

Proof. We will first determine the toroidal forms obtainable from a monomial mapping Λ : A^{3} → A^{2} defined by

\begin{aligned}u & = x^{a}y^{b}z^{c} \\ v & = x^{d}y^{e}z^{f} \end{aligned}

with

rank ( \begin{matrix}a & b & c \\ d & e & f \end{matrix} ) = 2.

First suppose that that no column of

( \begin{matrix}a & b & c \\ d & e & f \end{matrix} )

is zero. Then Λ^{−1}(D) = E, where xyz = 0 is an equation of E, uv = 0 is an equation of D.

Suppose that p ∈ A^{3} is a 2 point on y = z = 0. Then there exists 0 ≠ β ∈ k and regular parameters (\overline{x}, y, z) at p such that

\begin{aligned}u & = (\overline{x} + β)^{a}y^{b}z^{c} \\ v & = (\overline{x} + β)^{d}y^{e}z^{f}. \end{aligned}

If

Det ( \begin{matrix}b & c \\ e & f \end{matrix} ) ≠ 0,

we can make a permissible change of parameters to get 2.(b).

Suppose that

Det ( \begin{matrix}b & c \\ e & f \end{matrix} ) = 0.

There exist natural numbers \overline{b}, \overline{c} such that \overline{b}, \overline{c} > 0, (\overline{b}, \overline{c}) = 1,

\begin{aligned}u & = (\overline{x} + α)^{a}(y\overline{b}z\overline{c})^{k} \\ v & = (\overline{x} + α)^{d}(y\overline{b}z\overline{c})^{t}. \end{aligned}

After possibly interchanging u and v, we can assume that k > 0 and t ≥ 0. If t > 0 we get the form 2.(c). If t = 0, we get the form 1.(a).

Suppose that p ∈ A^{3} is a 1 point on z = 0. Then there exist 0 ≠ α, β ∈ k, and regular parameters (\overline{x}, \overline{y}, z) at p such that

\begin{aligned}u & = (\overline{x} + α)^{a}(\overline{y} + β)^{b}z^{c} \\ v & = (\overline{x} + α)^{d}(\overline{y} + β)^{e}z^{f}. \end{aligned}

After possibly interchanging u and v we may assume that c > 0. If f > 0 we get the form 2.(d). If f = 0 we get the form 1.(b).

Now suppose that a column of

( \begin{matrix}a & b & c \\ d & e & f \end{matrix} )

is zero.

After possibly interchanging (x, y, z), we may assume that c = f = 0. Then Λ^{−1}(D) = E where xy = 0 is an equation of E, uv = 0 is an equation of D. We get the forms 2.(b), 2.(d) or 1.(b).

Conversely, suppose that the forms 1. and 2. hold at all points of E_{X} and p ∈ E_{X}. By Lemma 18.3, there exists an étale neighborhood U of p and uniformizing parameters (x, y, z) on U such that a form 1. or 2. holds at p. Working backwards through the above proof, we see that Φ is toroidal at p.

We will call a point p ∈ X a non toroidal point if Φ is not toroidal at p.

Lemma 19.4. The locus of non toroidal points is Zariski closed of pure codimension 1 in X, and is a SNC divisor. The image of the non toroidal points in S is a finite set of points.

Proof. Suppose that p ∈ X is a non toroidal point. Then q = Φ(p) is a 1 point, and thus one of the forms (176), (179), (180) with t > 0, (181) or (183) with c > 0 hold at p.

1. First suppose that p is of the form of (183). We have c > 0, and all points nearby on x = 0 are non toroidal.

2. Suppose that p has the form (179). Φ is non toroidal on the line x = y = 0.

(a) Suppose that c > 0. Consider the point with regular parameters (x, ỹ + α, z̃ + β) with α ≠ 0. Set x = \overline{x}(ỹ + α)^{− \frac{b}{a}}. Then

\begin{aligned}u & = \overline{x}^{a} \\ v & = \overline{x}^{c}(ỹ + α)^{d− \frac{cb}{a}} = \overline{x}^{c}(γ + \overline{y}) \end{aligned}

which is non toroidal. Thus Φ is non toroidal on the surface x = 0.

(b) Suppose that d > 0. Then a similar analysis shows that Φ is non toroidal

on the surface y = 0.

3. Suppose that p has the form (181). A point on x = y = 0 with regular parameters (x, y, z̃ + β) with β ≠ 0 has the form of (179), and is thus not toroidal.

(a) Suppose that a, c > 0. Consider the point with regular parameters (x, ỹ + α, z̃ + β) with α ≠ 0. Set x = \overline{x}(ỹ + α)^{− \frac{b}{a}}.

\begin{aligned}u & = \overline{x}^{a} \\ v & = \overline{x}^{c}(ỹ + α)^{d− \frac{bc}{a}} (z̃ + β) = \overline{x}^{c}(γ + \overline{y}). \end{aligned}

Thus Φ is non toroidal on the surface x = 0.

(b) Suppose that b, d > 0. Then Φ is non toroidal on y = 0. Since a, b > 0 (by assumption) one of the cases (a) or (b) must hold.

4. Suppose that p has the form (180). We have t > 0. Consider a nearby point with regular parameters (x, ỹ + \overline{α}, z̃ + \overline{β}) with \overline{α} ≠ 0. Set x = \overline{x}(y + \overline{α})^{− \frac{b}{a}}.

\begin{aligned}u & = \overline{x}^{am} \\ v & = \overline{x}^{at}(α + \overline{β} + z̃) = \overline{x}^{at}(c̃ + \overline{z}). \end{aligned}

The non toroidal locus locally contains x = 0 (and y = 0).

5. Suppose that p has the form (176). Since a, b, c > 0, after possibly interchanging (x, y, z), we may assume that a, d > 0. Suppose that (x, ỹ+α, z̃+β) are regular parameters at a nearby point (with α, β ≠ 0). Set x = \overline{x}(ỹ + α)^{− \frac{b}{a}} (z̃ + β)^{− \frac{c}{a}}.

\begin{aligned}u & = \overline{x}^{a} \\ v & = \overline{x}^{d}(ỹ + α)^{e− \frac{db}{a}} (z̃ + β)^{f− \frac{dc}{a}} = \overline{x}^{d}(γ + \overline{y}) \end{aligned}

In a similar way, we see that nearby 2 points on x = 0 are non toroidal. Thus the non toroidal locus locally contains x = 0.

Suppose that p ∈ X is a 1 point such that Φ(p) = q is a 1 point. A form (183) holds at p. c − a is independent of permissible parameters (u, v) at q and (x, y, z) at p of the form of (183) (since u = 0 must be a local equation of D_{S}). Define

I(Φ, p) = c − a.

I(Φ, p) is locally constant. Thus if E is a component of E_{X} and p_{1}, p_{2} are two 1 points in E such that Φ(p_{1}) and Φ(p_{2}) are 1 points, then I(Φ, p_{1}) = I(Φ, p_{2}). We can thus define

I(Φ, E) = I(Φ, p)

if p ∈ E is a 1 point such that Φ(p) is a 1 point. Let

B_{Φ} = {q ∈ S | q is the image of a non toroidal point by Φ}.

Define

I(Φ) = max{I(Φ, p) | p ∈ Φ^{−1}(B_{Φ}) is a 1 point}.

Remark 19.5. If p ∈ Φ^{−1}(B_{Φ}) is a toroidal point then I(Φ, p) < 0.

Lemma 19.6. Suppose that q ∈ B_{Φ}. Let π : S_{1} → S be the blowup of q. Let U be the largest open set of X such that the rational map X → S_{1} is a morphism Φ_{1} : U → S_{1}. Then Φ_{1} is strongly prepared, and all points of U are good for Φ_{1}.

Suppose that p ∈ U ∩ Φ^{−1}(q) is a 1 point. If I(Φ, p) ≤ 0, then Φ_{1} is toroidal at p. If I(Φ, p) > 0, then I(Φ_{1}, p) < I(Φ, p).

The locus of points where m_{q}O_{X} is not invertible is a union of curves which make SNCs with \overline{B}_{2}(X). These points have one of the forms (187), (193), (185), (190) or (189) of Lemma 18.12.

Proof. Φ_{1} is strongly prepared by Lemma 18.13. All points of U are good for Φ_{1}, as follows by the analysis in Lemma 18.12. The locus of points where m_{q}O_{X} is not invertible is a union of curves which make SNCs with \overline{B}_{2}(X) by Theorem 18.15.

Suppose that p ∈ X is such that m_{q}O_{X} is not invertible at p. p is a good point and Φ(p) = q a 1 point implies p has one of the forms (176), (179), (180), (181) or (183). By Lemma 18.12, p must have one of the forms (187), (193), (185), (190), or (189).

Suppose that p ∈ Φ^{−1}(q) ∩ U is a 1 point. Then

\begin{aligned}u & = x^{a} \\ v & = x^{c}(α + y) \end{aligned}

where u = 0 is a local equation of D_{S} at q, with either a ≤ c or c < a and α ≠ 0. Suppose that I(Φ, p) = c − a ≤ 0.

If c < a, α ≠ 0 and we have permissible parameters (u_{1}, v_{1}) at q_{1} = Φ_{1}(p) such that

u = u_{1}v_{1}, v = v_{1}

so that q_{1} is a 2 point. There exists regular parameters (\overline{x}, \overline{y}, z) in Ô_{X,p} and 0 ≠ \overline{α} ∈ k such that

\begin{aligned}u & = \overline{x}^{a}(\overline{α} + \overline{y}) \\ v & = \overline{x}^{c}, \end{aligned}

\begin{aligned}u_{1} & = \overline{x}^{a−c}(\overline{α} + \overline{y}) \\ v_{1} & = \overline{x}^{c} \end{aligned}

(\overline{x}, \overline{y}, z) are thus permissible parameters for (v_{1}, u_{1}) at p, and p is a toroidal point for Φ_{1}.

If c = a,

u_{1} = u, v_{1} = \frac{v}{u} − α,

(u_{1}, v_{1}) are permissible parameters for Φ_{1} at q_{1} = Φ_{1}(p), and

\begin{aligned}u_{1} & = x^{a} \\ v_{1} & = \frac{v}{u} − α = y \end{aligned}

so that p is a toroidal point for Φ_{1}.

Suppose that I(Φ, p) > 0. Then (u_{1}, v_{1}) are permissible parameters at q_{1} = Φ_{1}(p), where

u = u_{1}, v = u_{1}v_{1}.

q_{1} is a 1 point.

\begin{aligned}u_{1} & = x^{a} \\ v_{1} & = x^{c−a}(α + y). \end{aligned}

Thus I(Φ_{1}, p) = c − 2a < I(Φ, p).

Lemma 19.7. Suppose that C ⊂ X is a 2 curve such that q = Φ(C) is a 1 point, if p ∈ C then p satisfies (187) or (193) and m_{q}O_{X} is not invertible along C. Let π : X_{1} → X be the blowup of C, Φ_{1} = Φ ∘ π. Then Φ_{1} : X_{1} → S is strongly prepared, all points of X_{1} are good points for Φ_{1}, and if m_{q}O_{X_{1},p_{1}} is not invertible at a point p_{1} ∈ π^{−1}(C), then p_{1} satisfies (187) or (193). If p_{1} ∈ π^{−1}(C) is a 1 point then I(Φ_{1}, p_{1}) < I(Φ).

Proof. Suppose that p ∈ C satisfies (187). Then all points of π^{−1}(p) are strongly prepared and are good points for Φ_{1}. Let p_{1} ∈ π^{−1}(p) be a 1 point. Ô_{X_{1},p_{1}} has regular parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}(y_{1} + α)

with α ≠ 0.

\begin{aligned}u & = (x_{1}^{a+b}(y_{1} + α)^{b})^{k} = \overline{x}_{1}^{(a+b)k} \\ v & = x_{1}^{c+d}(y_{1} + α)^{d} = \overline{x}_{1}^{c+d}(\overline{α} + \overline{y}) \end{aligned}

By (187) c − ak, d − bk have opposite signs.

I(Φ_{1}, p_{1}) = (c + d) − (a + b)k
= (c − ak) + (d − bk) < max{c − ak, d − bk}
≤ I(Φ).

If p satisfies (193), then all points of π^{−1}(p) are strongly prepared good points.

Lemma 19.8. Suppose that C ⊂ X is a curve such that q = Φ(C) is a 1 point, m_{q}O_{X} is not invertible along C, and C is not a 2 curve.

Let π : X_{1} → X be the blowup of C, Φ_{1} = Φ ∘ π. Then Φ_{1} : X_{1} → S is strongly prepared and all points of X_{1} are good points for Φ_{1}. If p_{1} ∈ π^{−1}(C) is a 1 point, then

I(Φ, p) < I(Φ_{1}, p_{1}) ≤ 0.

Proof. Suppose that p ∈ C. p satisfies (185), (190) or (189). Φ_{1} is strongly prepared, and all points of X_{1} are good points for Φ_{1}. A generic point p ∈ C satisfies (185), and x = y = 0 is a local equation of C at p. Suppose that p_{1} ∈ π^{−1}(p) is a 1 point. Then p_{1} has permissible parameters (x_{1}, y_{1}, z) such that

x = x_{1}, y = x_{1}(y_{1} + α),

\begin{aligned}u & = x_{1}^{k} \\ v & = x_{1}^{c+1}(y_{1} + α). \end{aligned}

I(Φ_{1}, p_{1}) = (c + 1) − k ≤ 0 since c < k by (185).

Theorem 19.9. Suppose that Φ : X → S is strongly prepared and all points p ∈ X are good points for Φ. Then there exists a sequence of quadratic transforms S_{1} → S and monodial transforms centered at nonsingular curves, X_{1} → X, such that the induced map Φ_{1} : X_{1} → S_{1} is strongly prepared, all points of X_{1} are good for Φ_{1} and I(Φ_{1}) ≤ 0.

Proof. Suppose that I(Φ) > 0. Suppose that E is a component of E_{X} such that I(Φ, E) = I(Φ). Then Φ(E) is a single 1 point q. Let π_{1} : S_{1} → S be the blowup of q.

By Lemmas 18.17, 19.6 and 19.8, there exists a sequence of blowups of curves C (which are not 2 curves) X_{1} → X such that if Φ_{1} : X_{1} → S is the induced map, Φ_{1} is strongly prepared, all points of X_{1} are good for Φ_{1}, if m_{q}O_{X_{1},p} is not invertible then (187) or (193) holds at p, and all curves in X_{1} along which m_{q}O_{X_{1}} are not invertible are 2 curves. We further have that I(Φ_{1}, \overline{E}) ≤ 0 if \overline{E} is exceptional for X_{1} → X.

By Lemmas 18.18 and 19.7, there exists a sequence of blowups of 2 curves C, X_{2} → X_{1} such that if Φ_{2} : X_{2} → S is the induced map, Φ_{2} is strongly prepared, all points of X_{2} are good for Φ_{2}, m_{q}O_{X_{2}} is invertible, and if \overline{E} is exceptional for X_{2} → X, then I(Φ_{2}, \overline{E}) < I(Φ).

Let \overline{Φ} : X_{2} → S_{1} be the induced map. By Lemma 19.6, \overline{Φ} is strongly prepared, all points of X_{2} are good for \overline{Φ}, I(\overline{Φ}) ≤ I(Φ), and if \overline{E} is a component of E_{X_{2}} which contains a 1 point q such that \overline{Φ}(p) ∈ π_{1}^{−1}(q), then I(\overline{Φ}, \overline{E}) < I(Φ).

The Theorem now follows by induction on the number of components E of X such that I(Φ, E) = I(Φ), and induction on I(Φ).

Theorem 19.10. Suppose that Φ : X → S is strongly prepared with respect to D_{S}, E_{X} = Φ^{−1}(D_{S})_{red}, all points p ∈ X are good points for Φ and I(Φ) ≤ 0. Then there exist sequences of quadratic transforms π_{1} : S_{1} → S and monodial transforms centered at nonsingular curves π_{2} : X_{1} → X such that the induced map Φ_{1} : X_{1} → S_{1} is toroidal with respect to D_{S_{1}} = π^{−1}(D_{S})_{red} and E_{X_{1}} = π_{2}^{−1}(E_{X})_{red}.

Proof. Suppose that E is a component of E_{X} such that Φ is not toroidal along E. If p ∈ E is a generic point, then at p we have an expression

\begin{aligned}u & = x^{a} \\ v & = x^{c}(α + y) \end{aligned}

with c > 0. Thus there exists a point q ∈ S such that Φ(E) = q. q is necessarily a 1 point.

Let π : S_{1} → S be the blowup of q. By Lemmas 18.17 and 19.8, there exists a sequence of blowups of nonsingular curves (which are not 2 curves) X_{1} → X such that if Φ_{1} : X_{1} → S is the induced morphism, then Φ_{1} is strongly prepared, all points of X_{1} are good for Φ_{1}, I(Φ_{1}) ≤ 0, the locus of points p_{1} of X_{1} such that m_{q}O_{X_{1},p_{1}} is not invertible is a union of 2 curves, and if m_{q}O_{X_{1},p_{1}} is not invertible, then p_{1} satisfies (187) or (193).

Suppose that C is a 2 curve on X_{1} such that m_{q}O_{X_{1}} is not invertible along C. A generic point p of C satisfies (187). Let E_{1} be the component of E_{X_{1}} with local equation x = 0 at p, E_{2} be the component of E_{X_{1}} with local equation y = 0 at p.

I(Φ_{1}, E_{1}) = c − ak, I(Φ_{1}, E_{2}) = d − bk.

Since m_{q}O_{X_{1},p} is not invertible, either 0 < d − bk or 0 < c − ak, a contradiction since I(Φ_{1}) ≤ 0. Thus m_{q}O_{X_{1}} is invertible and Φ_{1} : X_{1} → S induces a morphism \overline{Φ} : X_{1} → S_{1}. By Lemma 19.6, \overline{Φ} is strongly prepared, all points of X_{1} are good for \overline{Φ}, and I(\overline{Φ}) ≤ 0. Further, if p ∈ X_{1} is a 1 point such that p ∈ Φ_{1}^{−1}(q), then \overline{Φ} is toroidal at p.

By induction on the number of components of E_{X} along which Φ is not toroidal, we achieve the conclusions of the Theorem.

Theorem 19.11. Suppose that Φ : X → S is a dominant morphism from a 3 fold X to a surface S (over an algebraically closed field k of characteristic 0) and D_{S} is a reduced 1 cycle on S such that E_{X} = Φ^{−1}(D_{S})_{red} contains sing(X) and sing(Φ). Then there exist sequences of blowups of nonsingular subvarieties π_{1} : X_{1} → X and π_{2} : S_{1} → S such that the induced morphism X_{1} → S_{1} is a toroidal morphism with respect to π_{2}^{−1}(D_{S})_{red} and π_{1}^{−1}(E_{X})_{red}.

Proof. This follows from Theorem 17.3, the fact that prepared implies strongly prepared and Theorems 18.19, 19.9 and 19.10.

20. Glossary of Notations and Definitions

ν(p), Definition 6.9. γ(p), Definition 6.9. τ(p), Definition 6.9. S_{r}(X), \overline{S}_{r}(X), After Definition 6.9. B_{2}(X), \overline{B}_{2}(X), B_{3}(X), Before Definition 6.18. SNCs with \overline{B}_{2}(X), Definition 6.18. r small, Definition 8.3. r big, Definition 8.3. 1 point, 2 point, 3 point, Definition 6.5 and before Definition 18.1. 1-resolved, Definition 9.6. \overline{ν}(p), After Definition 9.6. σ(p), Before Lemma 9.9. δ(p), before Lemma 9.13. Inv(p), Before Theorem 9.15. A_{r}(X), Definition 10.2. \overline{A}_{r}(X), Definition 10.1. C_{r}(X), Definition 14.1. (E), Definition 15.5. ∗-permissible parameters, After Definition 18.1. ν_{E}(f), Before Definition 18.7. A(Φ, p), Definition 18.7. C(Φ, p), Definition 18.7. A(Φ, E), After Definition 18.7. C(Φ, E), After Definition 18.7. A(Φ), Before Lemma 18.10. C(Φ), Before Lemma 18.10. I(Φ, p), I(Φ, E), I(Φ), Before Remark 19.5. B_{Φ}, Before Remark 19.5. SNC divisor, Definition 5.1. P_{t}(x), After Definition 5.1.

bad point, Definition 18.5. étale neighborhood, Definition 6.19. good point, Definition 18.5. monodial transform, After Definition 5.1. monomial mapping, Definition 18.20. non toroidal point, Before Lemma 19.4. permissible monoidal transform, Definition 10.3. permissible parameters, Before Definition 6.5 and 6.5. permissible parameters for C at p, After Lemma 6.17. prepared, Definition 6.6. resolved, Definition 6.10. strongly prepared, Definition 18.1. toroidal mapping, 19.1. toroidal point, Definition 19.1. weakly permissible monodial transform, Definition 6.32. weakly prepared, Definition 6.1.

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Steven Dale Cutkosky, Department of Mathematics, University of Missouri Columbia, MO 65211, USA cutkoskys@missouri.edu