Fields Institute Communications Volume 00, 0000

Resolution of Morphisms

Steven Dale Cutkosky

Department of Mathematics University of Missouri Columbia Missouri 65211

Abstract. This paper considers the problem of monomialization of morphisms of algebraic varieties. We survey our recent results on this problem.

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 monoidal transform of X with center V is the blow up

π : Y = Proj(⊕_{n≥0}I_{V}^{n}) → X.

If p is a closed point of Y such that 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) 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 a quadratic transform.

Suppose that

Φ : X → Y  (1.1)

is a 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

\begin{matrix}X_{1} & →^{Ψ} & Y_{1} \\ ↓ & & ↓ \\ X & →^{Φ} & Y \end{matrix}  (1.2)
\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14E15, 14B25.

The author was partially supported by NSF.

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

Definition 1.1 Suppose that Φ : X → Y is a dominant morphism of nonsingular integral finite type k schemes. Φ is monomial if for every p ∈ X there exist regular parameters (y_{1}, …, y_{m}) in O_{Y,Φ(p)}, and an étale cover U of an affine neighborhood of p, uniformizing parameters (x_{1}, …, x_{n}) on U and a matrix a_{ij} such that

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

Since Φ is dominant, the matrix (a_{ij}) must have maximal rank m.

Definition 1.2 Suppose that Φ : X → Y is a dominant morphism of kvarieties. A morphism Ψ : X_{1} → Y_{1} is a monomialization of Φ if there are sequences of monoidal transforms 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} & \underrightarrow{Ψ} & Y_{1} \\ ↓ & & ↓ \\ X & \underrightarrow{Φ} & Y \end{matrix}

commutes, and Ψ is a monomial morphism.

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

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

The characteristic of k must be zero in the question. If k has characteristic p > 0, global monomializations 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 section 3, we discuss our positive answer ([16]) to a local analogue of the Question for generically finite morphisms.

In section 4, we outline proofs of the positive answer to the question in the previously known cases, a morphism to a curve and a morphism of surfaces.

In sections 5 and 6 we outline some aspects of our recent proof of monomialization of proper morphisms from 3 folds to surfaces ([18]).

Theorem 1.3 (Theorem 18.21, [18]) 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 blow ups 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 ([26], [8], Definition 19.1 [18]) 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 [18]) 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 blow ups 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 blow ups 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}  (1.3)

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, since the x_{i} = 0 are either local equations of exceptional components of X_{1} → X or are local equations of the strict transforms of irreducible components of y_{i} = 0. Thus (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_{m} = \overline{x}_{1}^{a_{m1}} ⋯ \overline{x}_{n}^{a_{mn}}

since this would imply that rank(a_{ij}) = m > 1. rank(a_{ij}) < m would imply, by Zariski’s subspace theorem [4], that Φ is not dominant.

In fact, in general it is necessary to blow up in both X and Y to construct a monomialization. For instance, if we blow up a point p on a nonsingular surface S, blow up a point on the exceptional curve E_{1}, blow up the intersection point of the new exceptional curve E_{2} with the strict transform of E_{1}, then blow up 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 parameters (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 blow ups 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 blow ups 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}  (1.4)

where a, b ∈ N and P (\overline{x}) is a polynomial of degree ≤ b. This follows since Y_{1} → Y can be factored near p_{1} by sequences of blow ups of the form

x_{i} = x_{i+1}^{a_{i+1}}, y_{i} = x_{i+1}^{b_{i+1}}(x_{i+1} + α_{i+1}).

If we perform a sequence of blow ups 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}_{n}) such that

\begin{aligned}u & = \overline{x}_{1}^{a} \\ v & = P (\overline{x}_{1}) + \overline{x}_{1}^{b}\overline{x}_{2} \end{aligned}  (1.5)

of the form of (1.4), 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 blow ups 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, which include the form of (1.5). This is accomplished if dim(X) = 3 in Theorem 17.3 of [18]. Almost the entirety of [18] 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 blow ups and blowdowns [7]. A local form of factorization along a valuation was proven earlier by us in Theorem 1.6 [16]. Our local proof of “strong factorization” in dimension 3 appears in [15].

2 Local Uniformization and Resolution of Singularities

A basic problem is to prove Local Uniformization (resolution of singularities) along an arbitrary valuation. This was first proved in characteristic zero by Zariski in 1940.

Theorem 2.1 (Local Uniformization, Zariski [37]) Suppose that K is a field of algebraic functions over a field k of characteristic zero, and V is a valuation ring of K (containing k). Then there exists a regular local ring which is essentially of finite type over k, with quotient field K, such that V dominates R (R ⊂ V and m_{V} ∩ R = m_{R}).

The proof is by a case by case analysis of the different types of valuations which can occur.

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 [21]). Quasi-complete and separated is equivalent to proper. In arbitrary dimension, Zariski proves resolution by a quasi-complete variety.

Theorem 2.2 (Zariski) If X is a proper variety of characteristic zero, then there exists a complete nonsingular variety Y, and a birational morphism Y → X.

The theorem follows from Local Uniformization, Theorem 2.1, and the compactness of the Zariski Riemann manifold [39].

Resolution of singularities in all dimensions and characteristic zero has been proven by Hironaka [21]. Resolution in dimension ≤ 3 and positive characteristic has been proven by Abhyankar [4].

Some recent papers on local uniformization and resolutions of singularities are Heinzer, Rotthaus, Wiegand [20], Kuhlmann [27], Teissier [34]. Some other proofs of resolution are Hironaka [22], Lipman [29], Hironaka [23], Spivakovsky [33], Villamayor [35], Moh [30], Bierstone and Milman [11], De Jong [24], Abromovich and De Jong [6], Bogomolov and Pantev [12]. Lipman’s article [28] is an excellent introduction to resolution of singularities.

Let K/k be an algebraic function field,

Z(K) = { The set of valuation rings V of K}.

Z(K) is a ringed space, with O_{Z(K),V} = V. We can define a map π_{X} : Z(K) → X for any model X of K by π_{X}(V) = p if the center of V on X is p. We can define a topology on Z(K) by taking as a basis for the topology the open sets π_{X}^{−1}(U) where X is a model of K, and U is an open subset of X. The maps π_{X} are morphisms of ringed spaces. If f : X → Y is a birational map of models of K, then we have f ∘ π_{X} = π_{Y}. For any valuation ring V of Z(K), we then have inclusions of local rings

O_{X,p_{1}} ⊂ V ⊃ O_{Y,p_{2}}

where p_{1} is the center of V on X, and p_{2} is the center of V on Y. This gives a local formulation of a rational map in commutative algebra.

If K has dimension 1, then Z(K) = X where X is the unique nonsingular model of K. If dim(K) > 1, then we know that K has nonnoetherian valuations ring, so Z(K) cannot be a variety. However, Z(K) is quasi-compact, ([39]; Theorem 40, Section 17, Chapter VI [41]).

The Zariski-Riemann manifold can be used to convert a birational problem of projective varieties into

1. A local analogue of the problem along an arbitrary valuation, and

2. A patching problem

The method is to first solve the problem locally along a valuation. Then for each valuation ring V of K, there exists a model X_{V} of K such that the problem has been solved at the center of V on X_{V}. Thus the problem has been solved in some open neighborhood U_{V} of the center of V in X_{V}. {π_{X_{V}}^{−1} (U_{V})} is an open cover of Z(K). Since Z(K) is quasi-compact, there exists a finite subcover, say {π_{X_{V_{1}}}^{−1} (U_{V_{1}}), ⋯, π_{X_{Vm}}^{−1} (U_{V_{m}})}.

If we can now patch V_{1}, …, V_{m} to obtain a separated algebraic variety V, then we will know (by the valuative criterion for properness) that V is proper. By construction, we will have a global solution to our problem.

There is an easy partial solution to the patching problem. We can always patch the V_{i} along the open sets where they are isomorphic. This is a finite type k scheme, with ring of rational functions K. The only problem is that it will probably not be separated! The difficulty is that the valuative criterion for separatedness could fail. However, the existence part of the valuative criterion for properness will hold, so it is quasi-complete.

3 Local Monomialization

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}  (3.1)

Theorem 3.1 (Monomialization)(Theorem 1.1 [16]) 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}.  (3.2)

Thus (since char(k) = 0) there exists an étale 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 (3.2) 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 3.1 in positive characteristic, even in dimension 2. Theorem 3.1 implies simultaneous resolution from above [17], 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].

The construction of a monomialization by quasi-complete varieties follows from Theorem 3.1.

Theorem 3.2 (Theorem 1.2 [16]) 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 blow ups 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 = β(Ψ(z)), such that β : W_{1} → W is a finite product of monoidal transforms.

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.

The proof of Theorem 3.2 follows from Theorem 3.1, by the patching argument explained in Section 2 and the following lemma.

Lemma 3.3 Suppose that Φ : X → Y is a morphism, p ∈ X is a closed point such that there exist regular parameters (y_{1}, …, y_{n}) in O_{Y,Φ(p)}, regular parameters (x_{1}, …, x_{n}) in O_{X,p} and units δ_{1}, …, δ_{n} ∈ O_{X,p} such that

y_{i} = x_{1}^{a_{i1}} ⋯ x_{n}^{a_{in}}δ_{i}

1 ≤ i ≤ n, and d = det(a_{ij}) ≠ 0. Then Φ is locally monomial at p.

Proof Let (b_{ij}) be the adjoint matrix of A = (a_{ij}). Set

\overline{x}_{i} = ∏_{j} δj^{\frac{b_{ij}}{d}} x_{i}

1 ≤ i ≤ n. Then

y_{i} = \overline{x}_{1}^{a_{i1}} ⋯ \overline{x}_{n}^{a_{in}}δ_{i}

1 ≤ i ≤ n.

Let U_{1} = spec(R) be an affine neighborhood of p on which (x_{1}, …, x_{n}) are uniformizing parameters, and δ_{1}, …, δ_{n} are units. Let

S = R[δ1^{\frac{1}{d}} , …, δn^{\frac{1}{d}}],

V_{1} = spec(S). f : V_{1} → U_{1} is an étale cover. k[\overline{x}_{1}, …, \overline{x}_{n}] ⊂ S defines a morphism g : V_{1} → A^{n}. Let a be the origin of A^{n}. q ∈ g^{−1}(a) if and only if \overline{x}_{1}, …, \overline{x}_{n} ∈ m_{q}, which holds if and only if x_{1}, …, x_{n} ∈ m_{q}. Thus g^{−1}(a) = f^{−1}(p).

Ô_{V_{1},q} = k[[x_{1}, …, x_{n}]] = k[[\overline{x}_{1}, …, \overline{x}_{n}]]

for all q ∈ g^{−1}(a). Thus g is étale at all points of g^{−1}(a). Since this is an open condition, 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 f(Z_{1}). Let V = f^{−1}(U). Then V is an étale cover of U on which \overline{x}_{1}, …, \overline{x}_{n} are uniformizing parameters.

4 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 4.1 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 q ∈ Φ(sing(Φ)), and monomialize the mapping above q.

By embedded resolution of hypersurfaces, there exists a sequence of blow ups of nonsingular subvarieties which dominate subvarieties of Φ^{−1}(q), π : X_{1} → X such that for all p ∈ (Φ ∘ π)^{−1}(q), there exist 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 [9].

In our paper [19] with Olivier 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 4.2 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}  (4.1)

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_{T,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}  (4.2)

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 (4.2) 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_{T,p} such that one of the forms (4.1) or (4.2) 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}  (4.3)

or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}  (4.4)

where (a, b) = 1 and ad − bc ≠ 0.

We first observe that by resolution of singularities and elimination of indeterminacy, 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 blow ups 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 [9].

Lemma 4.3 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}  (4.5)

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}  (4.6)

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 (4.5) holds. Then ax^{a−1}v_{y} = δx^{e} implies we have the form (4.3) (after making a change of parameters in Ô_{T_{1},p}). Now suppose that (4.6) 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 (4.4).

It is now not difficult to construct a monomialization. We must blow up points q on S_{1} over which the map is not monomial at some point over q, and blow up 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.

5 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 elimination of indeterminacy 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 blow ups 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. We will give some examples later on in this section.

A key step in the local proof of monomialization, Theorem 3.1, 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 [37]. 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. 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 blow ups.

We will give a brief overview of the proof of Theorem 1.3 on monomialization of morphisms from 3 folds to surfaces (Theorem 18.21 [18]).

Step 1. First construct a diagram

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

where the vertical maps are products of blow ups 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.

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 as the following examples show. This is in sharp contrast to the case of a morphism of surfaces (Lemma 4.3).

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 defined 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).

For another example, 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).

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′.

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.

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 the 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.

We use 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.

We can construct an example as follows.

u = xy, v = x^{2}y

has F = 1. blow up 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}.

However, ν(p) can go up by at most 1, and some other invariants get better, or at least no worse. For a local resolution, we reduce to two difficult cases (Sections 11 and 12 of [18]) which we settle by blowing up generic curves on E_{X′} through a particular point, and use a generalization of Abhyankar’s Good Point Algorithm ([5], [28]) 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 blow ups of nonsingular curves, S′′ → S′ is a product of blow ups of points and Φ′′′ is monomial.

π : S′′ → S′ is a sequence of blow ups 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}

A similar argument holds in the other cases.

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

6 Resolution of Generically Finite Morphisms of Surfaces

In this section, a surface is a nonsingular 2-dimensional integral scheme, of finite type over an algebraically closed field k of characteristic 0.

Suppose that Z and Y are surfaces and f : Z → Y is a generically finite morphism.

Definition 6.1 We will say that f is resolved if for all q ∈ Y there exist regular parameters (u, v) at q such that for all p ∈ f^{−1}(q) there exist regular parameters (x, y) in Ô_{Z,p} and a series P such that one of the following forms hold.

\begin{aligned}u & = x^{a} \\ v & = P (x) + x^{b}y^{c} \end{aligned}  (6.1)

with c > 0 or

\begin{aligned}u & = (x^{a}y^{b})^{m} \\ v & = P (x^{a}y^{b}) + x^{c}y^{d} \end{aligned}  (6.2)

where (a, b) = 1 and ad − bc ≠ 0.

We will prove the following Theorem.

Theorem 6.2 Suppose that Z and Y are surfaces and f : Z → Y is a generically finite morphism. Then there exists a sequence of quadratic transforms π : X → Z such that f ∘ π is resolved.

Theorem 6.2 is achieved without any blowing up in the base S. The “resolved form” of Theorem 6.2 is slightly weaker that that of the conclusions of Lemma 4.3. However, the conclusions of Lemma 4.3 cannot be achieved without blowing up in S. The proof of Theorem 6.2 is substantially more difficult. However, this generalized theorem is a necessary step in the proof of Theorem 1.3.

Theorem 6.2 is an ingredient in the proof of Theorems 1.3 and 1.4. The proof of Theorem 6.2 is based on a classical method for resolution of plane curve singularities (c.f. [10]). However, there are significant differences.

Let S be the (finitely many) points of Y over which f is not finite. Let B be the branch curve of Z − f^{−1}(S) → Y − S, T be the set of (finitely many) singular points of f(B). By Abhyankar’s Lemma, for each q ∈ f(B) − T there are regular parameters (u, v) in O_{Y,q} such that for all p ∈ f^{−1}(p), there exists regular parameters (x, y) in Ô_{Z,p} such that

\begin{matrix}u = x^{n} \\ v = y. \end{matrix}

At each of the finitely many points q of S∪T, fix regular parameters (u_{q}, v_{q}) = (u, v) in O_{Y,q}. We will preserve this notation and this choice through out this section.

After performing a sequence of quadratic transforms Z_{1} → Z, and replacing f with Z_{1} → Y, we can assume that u = u_{q} = 0 is a SNC divisor everywhere above q, for q ∈ S ∪ T. Suppose that f(p) = q. Then there are regular parameters (x, y) in Ô_{X,p} such that u = x\overline{a}y\overline{b}, and \overline{a} > 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}  (6.3)

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}  (6.4)

We will say that p is a 1 point.

For the rest of this section, we will assume that Λ : X → Z is a sequence of quadratic transforms, centered at points over S ∪ T. let f_{X} = f ∘ Λ.

Lemma 6.3 Suppose that (R, m) is the local ring of a point on a surface, (u, v) are regular parameters in R and g : W → Spec(R) is a sequence of quadratic transforms. Suppose that p ∈ g^{−1}(m) is a closed point. Then there exist a series P and regular parameters (u_{1}, v_{1}) in Ô_{W,p} such that

\begin{matrix}u = u_{1}^{a} \\ v = P (u_{1}) + u_{1}^{c}v_{1} \end{matrix}  (6.5)

with c > 0 or

\begin{matrix}u = (u_{1}^{a}v_{1}^{b})^{m} \\ v = P (u_{1}^{a}v_{1}^{b}) + u_{1}^{c}v_{1}^{d} \end{matrix}  (6.6)

with b > 0, (a, b) = 1, ad − bc = ±1.

Proof R → O_{W,p} can be factored as a sequence of quadratic transforms, and can thus be factored as a sequence of transformations of intermediate local rings

R = R_{0} → R_{1} → ⋯ R_{n} = O_{W,p}

where R_{i} has regular parameters (x_{i}, y_{i}) such that

x_{0} = u, y_{0} = v
x_{i−1} = x_{i}^{β_{i}}(y_{i} + c_{i})^{β′_{i}}, y_{i=1} = x_{i}^{α_{i}}(y_{i} + c_{i})^{α′_{i}}

with c_{i} ∈ k and α′_{i}β_{i} − α_{i}β′_{i} = ±1 for i ≤ n, and 0 ≠ c_{i} if i < n. We can then find regular parameters (\overline{x}_{i}, \overline{y}_{i}) in R̂_{i} such that

\overline{x}_{0} = u, \overline{y}_{0} = v
\overline{x}_{i−1} = \overline{x}_{i}^{β_{i}}, \overline{y}_{i−1} = \overline{x}_{i}^{α_{i}}(\overline{y}_{i} + \overline{c}_{i}).

with \overline{c}_{i} ≠ 0 for i ≤ n − 1 and \overline{x}_{n−1}, \overline{y}_{n−1} have this form, or

\overline{x}_{n−1} = \overline{x}_{n}^{β_{n}} \overline{y}_{n}^{β′_{n}}, \overline{y}_{n−1} = \overline{x}_{n}^{α_{n}} \overline{y}_{n}^{α′_{n}}.

Setting u_{1} = \overline{x}_{n}, v_{1} = \overline{y}_{n} we get the conclusions of the Lemma.

Lemma 6.4 f_{X} : X → Y is resolved at all but finitely many points of X.

Proof We need only show that if E is an exceptional curve of f_{X}, then f_{X} is resolved at all but finitely many points of E. Let ν be the (dimension 1) valuation of k(X) corresponding to E. ν restricts to a (dimension 1) valuation ν′ of k(Y). Let W → Y be the minimal sequence of quadratic transforms such that the center of ν′ on W is a curve E′. The rational map Φ : X → W is defined and is finite in a neighborhood of a generic point q of E. By Lemma 6.3

There exist regular parameters (u_{1}, v_{1}) at Φ(q) such that u_{1} = 0 is a local equation of E_{1} in W at Λ(q) and the form (6.5) holds at Λ(q).

By Abhyankar’s Lemma, there exist regular parameters (x_{1}, y_{1}) at q such that

u_{1} = x_{1}^{e}, v_{1} = y_{1}

for some e ∈ N. Thus the form (6.1) holds at q.

Suppose that p ∈ X, and (x, y) are regular parameters in Ô_{X,p} such that (u, v) have one of the forms (6.3) or (6.4). Set

ν(p) = \{ \begin{matrix}mult(F) − 1 & if p is a 1 point \\ mult(F) & if p is a 2 point \end{matrix}

Lemma 6.5 ν(p) is independent of the choice of regular parameters (x, y) in (6.3) or (6.4).

Proof First suppose that p is a 2 point. To express u and v in the form (6.3) 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  (6.7)

or

y = ω_{y}\overline{x}, x = ω_{x}\overline{y} where ω_{x}^{ma}ω_{y}^{mb} = 1.  (6.8)

ν does not change after a change of variables of one of these forms.

Now suppose p is a 1 point. To preserve the form (6.4), 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.  (6.9)

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}  (6.10)

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), which is nonzero since a_{ij} ≠ 0 for some j > 0.

A 2 point p is resolved if and only if F is a unit in (6.3). A 1 point p is resolved if and only if in (6.4) 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 6.6 Suppose that g : X_{1} → X is a quadratic transform, centered at a closed point p of X, and p_{1} ∈ X_{1} is a closed point such that g(p_{1}) = p. Then

ν(p_{1}) ≤ ν(p).

If p is resolved, then p_{1} is resolved.

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) = 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}  (6.11)

Resolution of Morphisms 17

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 ν(p_{1}) > 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}  (6.12)

First suppose that −λ is a natural number. Then

∑_{i=0}^{r} a_{i}(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 ∉ N. But then the y_{1}^{r+1} coefficient of β(y_{1} + α)^{−λ} is non zero, a contradiction to (6.12).

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}  (6.13)

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}  (6.14)

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 6.7 σ is independent of the choice of permissible parameters (x, y).

Proof The proof of Lemma 6.5 shows that mult(F (0, y)) is independent of the choice of permissible parameters at a 1 point.

Lemma 6.8 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. Further suppose that p is a 2 point, p_{1} is a 1 point and ν(p_{1}) = 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 (6.12),

F_{1} ≡ (y_{1} + α)^{λ} (∑_{i=0}^{r} a_{i,r−i}(y_{1} + α)^{j}) − β mod (x_{1}, y_{1}^{r+1})  (6.15)

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 6.6. 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} (6.16)

Thus mult(F_{1}(0, y_{1})) = r + 1.

Lemma 6.9 Suppose that g : X_{1} → X is a quadratic transform, centered at a 1 point p of X and p_{1} is a closed point above p such that g(p_{1}) = p.

If p_{1} is a 1 point and ν(p_{1}) = ν(p), then σ(p_{1}) = 0. If σ(p) = 0 and p_{1} is a 2 point then ν(p_{1}) = 0.

Proof First suppose that 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 6.10 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 ν(p_{1}) = ν(p), then σ(p_{1}) ≤ σ(p).

Proof This is immediate from Lemmas 6.8 and 6.9.

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 (6.3) or (6.4), 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 6.11 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).  (6.17)
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}}  (6.18)

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 (6.18) holds. We will show that δ(p) = δ(p; x, y_{1}). Suppose that δ_{1} = δ(p; x, y_{1}) < δ(p). By (6.17), 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 6.12 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 6.11.

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 6.11. Thus δ(p_{1}) = δ(p) − 1.

If p ∈ X is a 2 point, then p is resolved precisely when ν(p) = 0. If p ∈ X is a 1 point then p is resolved precisely when δ(p) = ∞. Thus p ∈ X is not resolved at p if and only if ν(p) > 0 and δ(p) < ∞.

We can define an invariant

Inv(p) = (ν(p), σ(p), δ(p))

for p ∈ X.

Theorem 6.13 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. Suppose that ν(p) > 0 and δ(p) < ∞. Then

Inv(p_{1}) < Inv(p)

in the lexicographic ordering.

Proof The Theorem follows from Theorem 6.6, and Lemmas 6.8, 6.9, 6.12.

The proof of Theorem 6.2 is immediate from Theorem 6.13.

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Steven Dale Cutkosky, Department of Mathematics, University of Missouri Columbia, MO 65211, USA dale@cutkosky.math.missouri.edu