GENERICALLY FINITE MORPHISMS

STEVEN DALE CUTKOSKY

1. introduction

Suppose that k is a field, and f : Y → X is a dominant, generically finite morphism of complete k-varieties. If Y and X are complete curves, then it is classical that f is finite. If Y and X have dimensions ≥ 2 f need not be finite. The simplest example is the blowup of a nonsingular subvariety of a nonsingular projective variety.

It is however natural to ask the following question. Given a generically finite morphism f : Y → X as above, does there exist a commutative diagram

\begin{matrix}Y_{1} & →^{f_{1}} & X_{1} \\ ↓ & & ↓ \\ Y & →^{f} & X \end{matrix}  (1)

such that f_{1} is finite, Y_{1} and X_{1} are nonsingular complete k-varieties, and the vertical arrows are birational? The answer to this question is no, as is shown by a theorem of Abhyankar (Theorem 11 [2]). This theorem, (as shown in Example 6.2 of this paper) proves that such a diagram cannot always be constructed even when f : Y → X is a G-equivariant morphism of complex projective surfaces, where the extension of function fields k(X) → k(Y) is Galois with Galois group G.

In the theory of resolution of singularities a modified version of this question is important.

Question 1. With f : Y → X as above, is it possible to construct a diagram (1) such that f_{1} is finite, Y_{1} and X_{1} are complete k-varieties such that Y_{1} is nonsingular, X_{1} is normal and the vertical arrows are birational?

This question has been posed by Abhyankar (with the further conditions that Y_{1} → Y is a sequence of blowups of nonsingular subvarieties and Y, X are projective) explicitely on page 144 of [5], where it is called the “weak simultaneous resolution global conjecture” and implicitely in the paper [2].

As positive evidence for this conjecture, Abhyankar proves a local form of this conjecture for 2 dimensional function fields over an algebraically closed field of arbitrary characteristic [1], [3], (this is the two dimensional case of the “weak simultaneous resolution local conjecture” [5]).

An important case where Question 1 has a positive answer is for generically finite morphisms f : Y → X of projective varieties, over a field k of characteristic zero, which induce a Galois extension of function fields. We give a simple proof in Theorem 6.1. We can construct (with this Galois assumption) a diagram (1) such that the conclusions of Question 1 hold, and X_{1} has normal toric singularities. This is a relative version of Theorem 7, [2].

partially supported by NSF.

In Theorem 3.1 of this paper we give a counterexample to Question 1. The example is of a generically finite morphism Y → X of nonsingular, projective surfaces, defined over an algebraically closed field k of characteristic not equal to 2. This counterexample is necessarilly then a counterexample to the “weak simultaneous resolution global conjecture”.

As the “weak simultaneous resolution local conjecture”, posed by Abhyankar on page 144 of [5] is true in characteristic 0 (we prove it in [9] as a corollary to the local monomialization theorem, Theorem 1.1 [8], and prove a stronger statement in Theorem 4.3 of this paper), Theorem 3.1 also gives a counterexample to the philosophy that a theorem which is true in valuation theory should also be true in the birational geometry of projective varieties. This is the philosophy which led to successful proofs of resolution of singularities for surfaces and 3-folds, in characteristic zero by Zariski ([26], [28]), and in positive characteristic by Abhyankar ([1],[3],[4]). Recently there has been progress on the important problem of local uniformization in positive characteristic in higher dimensions (c.f. [18], [21],[23], [24]). Ramification of morphisms of algebraic surfaces in positive characteristic is analyzed in [10] and [11].

We prove in Theorem 5.3 that Question 1, and the “weak simultaneous resolution global conjecture” are almost true, as it always possible (over fields of characteristic zero) to construct a diagram (1) where f_{1} : Y_{1} → X_{1} is a quasi-finite morphism of integral, finite type k-schemes, Y_{1} is nonsingular, X_{1} has normal toric singularities, the vertical morphisms are birational and every k-valuation of k(X) has a center on X_{1}, every k-valuation of k(Y) has a center on Y_{1}. That is, the answer to Question 1 becomes true if we weaken the condition that the vertical arrows are proper by not insisting that these morphisms be separated.

The essential technical result used in the proof of Theorem 5.3 is the Local Monomialization Theorem, Theorem 1.1 of [8]. Local monomialization is used to prove a strengthened version of the “weak simultaneous local conjecture”, Theorem 4.3 of this paper, which allows us to construct local solutions of the problem, which are patched in an arbitrary manner (this is where separatedness is lost) to construct X_{1} and Y_{1}.

We will now give an overview of the proof of the construction of Theorem 3.1, which is the counterexample to Question 1 and the “weak simultaneous global conjecture”. We will use some of the notation explained in the following section on notations.

Suppose that K^{∗} is a finite extension of an algebraic function field K, defined over an algebraically closed field k of characteristic zero, ν^{∗} is a k-valuation of K^{∗} and ν is its restriction to K. By Theorem 4.3, there exists an algebraic regular local ring S with quotient field K^{∗} dominated by ν^{∗} and an algebraic normal local ring R with quotient field K such that S lies over R (S is the localization at a maximal ideal of the integral closure of R in K^{∗}). This can be refined [11] to show that there exist S and R as above such that the quotient of value groups Γ_{ν^{∗}}/Γ_{ν} acts faithfully on the power series ring Ŝ by k-algebra automorphisms so that

R̂ ⊗_{R/m_{R}} k′ ≅ (Ŝ ⊗_{S/m_{S}} k′)^{Γ_{ν^{∗}} /Γ_{ν}}  (2)

where k′ is an algebraic closure of S/m_{S}. In some special cases, such as rational rank 2 valuations of algebraic function fields of dimension two, where [K : K^{∗}] is not divisible by char(k), this construction is stable under quadratic transforms of S. We give a direct proof in this paper.

Let k be an algebraically closed field, \overline{ν} be the rational rank 2 valuation on \overline{K} = k(u, v) and L_{1} be the q-cyclic extension of \overline{K}, where q is a prime distinct from char(k), constructed in Theorem 11 [2] of Abhyankar (the construction is recalled in Theorem 3.8 of this paper). \overline{R} = k[u, v]_{(u,v)} is dominated by \overline{ν}. The extension \overline{K} → L_{1} has the property that if S_{1} is an algebraic regular local ring with quotient field L_{1} which lies over an algebraic normal local ring R_{1} with quotient field \overline{K} such that R_{1} is dominated by \overline{ν}, and contains \overline{R}, then R_{1} is singular.

Examining this example, we see that there is a unique extension ν_{1} of \overline{ν} to L_{1}, and the quotient of value groups Γ_{ν_{1}}/Γ\overline{ν} ≅ Z_{q}. By (2) we have R̂_{1} ≅ Ŝ_{1}^{Z_{q}}. Since R̂_{1} is singular and Ŝ_{1} is a power series ring in two variables over k, the algebraic fundamental group of R̂_{1} is

π_{1}(spec(R̂_{1}) − mR̂_{1}) ≅ Z_{q}.

We now consider the extension ν_{2} of \overline{ν} to a particular p-cyclic extension L_{2} of \overline{K} where p is a prime such that p ≠ q and p ≠ char(k). We have Γ_{ν_{2}}/Γ\overline{ν} ≅ Z_{p}, and if S_{2} is an algebraic regular local ring with quotient field L_{2} which is dominated by ν_{2} which contains \overline{R}, and if there exists an algebraic normal local ring R_{2} with quotient field \overline{K} which lies below S_{2}, then R̂_{2} ≅ Ŝ_{2}^{Z_{p}} by (2). If R_{2} is singular, the algebraic fundamental group of R̂_{2} is then

π_{1}(spec(R̂_{2}) − mR̂_{2}) ≅ Z_{p}.

We then construct a morphism of projective nonsingular k-surfaces Φ : Y → X such that X has the function field \overline{K} and \overline{R} is the local ring of a point on X. Y is constructed in such a way that Y splits into two sheets over spec(\overline{R}), and there are points on these two sheets which are formally the same as extensions \overline{R} → S_{1}, \overline{R} → S_{2} into algebraic regular local rings with respective quotient fields L_{1} and L_{2} which are dominated by the respective valuations ν_{1} and ν_{2}. We then use these formal embeddings to construct extensions \overline{ν}_{1} and \overline{ν}_{2} of ν to the function field L_{0} of Y.

These extensions \overline{ν}_{i} have the property that if Y_{1} is nonsingular and Y_{1} → Y is proper birational (so that it can be factored by blowups of points) then the map Y_{1} → X is formally isomorphic at the centers of the valuations \overline{ν}_{1} and \overline{ν}_{2} to the extensions of \overline{R} by the corresponding sequences of quadratic transforms of the local rings S_{1} and S_{2} along the respective valuations ν_{1} and ν_{2}.

Now suppose that we can construct a diagram

\begin{aligned}Y_{1} & → & X_{1} \\ ↓ & & ↓ \\ Y & → & X \end{aligned}

such that Y_{1} → X_{1} is finite, Y_{1} is nonsingular, X_{1} is normal, and the vertical arrows are proper and birational. Let R_{1} be the local ring of the center of \overline{ν} on X_{1}, S(1) be the local ring of the center of \overline{ν}_{1} on Y_{1}, and let S(2) be the local ring of the center of \overline{ν}_{2} on Y_{1}. Since \overline{ν}_{1} and \overline{ν}_{2} both extend \overline{ν}, and Γ_{\overline{ν}_{1}}/Γ\overline{ν} ≅ Z_{q}, Γ_{\overline{ν}_{2}}/Γ\overline{ν} ≅ Z_{p}, we must have that

Ŝ(1)^{Z_{q}} ≅ R̂_{1} ≅ Ŝ(2)^{Z_{p}}.

We then have that R_{1} is singular, by our contruction of \overline{ν}_{1}, and thus the algebraic fundamental group π_{1}(spec(R̂_{1}) − mR̂_{1}) has simultaneously order p and order q ≠ p which is impossible.

2. notations

We will denote the maximal ideal of a local ring R by m_{R}. We will denote the quotient field of a domain R by QF (R). Suppose that R ⊂ S is an inclusion of local rings. We will say that R dominates S if m_{S} ∩ R = m_{R}. Suppose that K is an algebraic function field over a field k. We will say that a subring R of K is algebraic if R is essentially of finite type over k. Suppose that K^{∗} is a finite extension of an algebraic function field K, R is a local ring with QF (K) and S is a local ring with QF (K^{∗}). We will say that S lies over R and R lies below S if S is a localization at a maximal ideal of the integral closure of R in K^{∗}. If R is a local ring, R̂ will denote the completion of R at its maximal ideal.

Good introductions to the valuation theory which we require in this paper can be found in Chapter VI of [30] and in [3]. A valuation ν of K will be called a k-valuation if ν(k) = 0. We will denote by V_{ν} the associated valuation ring, which necessarily contains k. A valuation ring V of K will be called a k-valuation ring if k ⊂ V. The value group of a valuation ν will be denoted by Γ_{ν}. If X is an integral k-scheme with function field K, then a point p ∈ X is called a center of the valuation ν (or the valuation ring V_{ν}) if V_{ν} dominates O_{X,p}. If R is a subring of V_{ν} then the center of ν (the center of V_{ν}) on R is the prime ideal R ∩ m_{V_{ν}}.

Suppose that R is a local domain. A monoidal transform R → R_{1} is a birational extension of local domains such that R_{1} = R[\frac{P}{x}]_{m} where P is a regular prime ideal of R, 0 ≠ x ∈ P and m is a prime ideal of R[\frac{P}{x}] such that m ∩ R = m_{R}. R → R_{1} is called a quadratic transform if P = m_{R}.

If R is regular, and R → R_{1} is a monodial transform, then there exists a regular sustem of parameters (x_{1}, …, x_{n}) in R and r ≤ n such that

R_{1} = R [\frac{x_{2}}{x_{1}} , …, \frac{x_{r}}{x_{1}} ]_{m} .

Suppose that ν is a valuation of the quotient field R with valuation ring V_{ν} which dominates R. Then R → R_{1} is a monoidal transform along ν (along V_{ν}) if ν dominates R_{1}.

We follow the notation of [17]. In particular, we do not require that a scheme be separated.

3. A counterexample to global weak simultaneous resolution

In this section we construct the following example. This gives a counterexample to Question 1 stated in the introduction, as well as to the “weak simultaneous resolution global conjecture” stated by Abhyankar explicitely on page 144 [5] and implicitely in the paper [2]. As the “weak simultaneous resolution local conjecture” is true in characteristic 0 (We prove it in [9], and prove a stronger version in Theorem 4.3 of this paper), Theorem 3.1 also gives a counterexample to the philosophy that a theorem which is true in valuation theory should also be true in the birational geometry of projective varieties.

Theorem 3.1. Suppose that k is an algebraically closed field of characteristic 0 or of odd prime characteristic. Then there exists a generically finite morphism Φ : Y → X of projective nonsingular k-surfaces such that there does not exist a commutative diagram

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

where the vertical arrows are birational and proper, Y_{1} is nonsingular, X_{1} is normal, and Φ_{1} is finite.

Throughout this section we will suppose that k is an algebraically closed field.

Lemma 3.2. Suppose that L is a 2 dimensional algebraic function field over k. Suppose that R is an algebraic regular local ring with quotient field L and maximal ideal m_{R} = (u, v). Suppose that \overline{ν} is a rank 1, rational rank 2 valuation of L such that \overline{ν} dominates R, \overline{ν}(u), \overline{ν}(v) > 0 and \overline{ν}(u), \overline{ν}(v) are rationally independent. Then 1. The value group of \overline{ν} is Γ\overline{ν} = Z\overline{ν}(u) + Z\overline{ν}(v).

2. Suppose that R → R_{1} is a sequence of quadratic transforms along \overline{ν}. Then there exist regular parameters (u_{1}, v_{1}) in R_{1} and a, b, c, d ∈ N such that

\begin{aligned}u & = u_{1}^{a}v_{1}^{b} \\ v & = u_{1}^{c}v_{1}^{d} \end{aligned}

with ad − bc = ±1.

3. There exists a unique extension ν̂ of \overline{ν} to L̂ = QF (R̂) which dominates R̂. The value group of ν̂ is Γν̂ = Γ_{ν}.

4. If ν_{1} is a valuation such that ν_{1} is equivalent to \overline{ν}, (and the value groups Γ\overline{ν} and Γ_{ν_{1}} are embedded as subgroups of R) then

\frac{ν_{1}(v)}{ν_{1}(u)} = \frac{\overline{ν}(v)}{\overline{ν}(u)}.

Proof. Proof of 1. f ∈ R implies there is an expression f = ∑_{i+j=r}^{n−1} a_{ij}u^{i}v^{j} + h with a_{ij} ∈ k, r = ord(f), h ∈ (m_{R})^{n}, where n is such that

n\overline{ν}(m_{R}) > ν( ∑_{i+j=r} a_{ij}u^{i}v^{j}).

Thus since \overline{ν}(u) and \overline{ν}(v) are rationally independent,

\overline{ν}(f) = \overline{ν}( ∑_{i+j=r}^{n−1} a_{ij}u^{i}v^{j}) = min\{\overline{ν}(u^{i}v^{j}) | r ≤ i + j ≤ n − 1, a_{ij} ≠ 0\}.

Thus Γ\overline{ν} = Z\overline{ν}(u) + Z\overline{ν}(v).

Proof of 2. It suffices to prove this for a single quadratic transform. We either have that \overline{ν}(u) > \overline{ν}(v) or \overline{ν}(v) > \overline{ν}(u). In the first case have that

R_{1} = R[\frac{u}{v}, v]_{( \frac{u}{v} ,v)}

and \overline{ν}(\frac{u}{v}), \overline{ν}(v) are linearly independent over Q. In the second case we have that

R_{1} = R[u, \frac{v}{u}]_{(u, \frac{v}{u})}

and \overline{ν}(u), \overline{ν}(\frac{v}{u}) are linearly independent over Q.

Proof of 3. Define an extension ν̂ of \overline{ν} to L̂ by

ν̂(f) = min\{i\overline{ν}(u) + j\overline{ν}(v) | a_{ij} ≠ 0\}

if f ∈ R̂, and f has the expression f = ∑ a_{ij}x^{i}y^{j} with a_{ij} ∈ k. ν̂ is a valuation since for i, j, α, β ∈ N,

i\overline{ν}(u) + j\overline{ν}(v) = α\overline{ν}(u) + β\overline{ν}(v)

implies i = α, j = β. ν̂ dominates R̂ and Γν̂ = Γ_{ν}.

Suppose that ν̃ is an extension of \overline{ν} to L̂ which dominates R̂. Suppose that f ∈ R̂. Write

f = ∑_{i+j=r}^{∞} a_{ij}u^{i}v^{j},

where r = ord(f), a_{ij} ∈ k. There exists n such that

\overline{ν}( ∑_{i+j=r} a_{ij}u^{i}v^{j}) < n\overline{ν}(m_{R}).

Write

f = ∑_{i+j=r}^{n−1} a_{ij}u^{i}v^{j} + g,

with g ∈ m_{R}^{n}R̂.

ν̃( ∑_{i+j=r}^{n−1} a_{ij}u^{i}v^{j}) = \overline{ν}( ∑_{i+j=r}^{n−1} a_{ij}u^{i}v^{j}) = min\{\overline{ν}(u^{i}v^{j}) | a_{ij} ≠ 0, r ≤ i+j ≤ n−1\} < n\overline{ν}(m_{R})

and ν̃(g) ≥ n\overline{ν}(m_{R}) so that

ν̃(f) = min\{\overline{ν}(u^{i}v^{j}) | a_{ij} ≠ 0\} = ν̂(f).

Proof of 4. As on page 653 [26], we consider the convergent factions \frac{f_{p}}{g_{p}} of τ = \frac{\overline{ν}(v)}{\overline{ν}(u)}. Set

ϵ = f_{p−1}g_{p} − f_{p}g_{p−1} = ±1.

ϵ, −τ + \frac{f_{p−1}}{g_{p−1}} , τ − \frac{f_{p}}{g_{p}} have the same signs.

For arbitrary p, we can define u_{1}, v_{1} ∈ L by

u = u_{1}^{g_{p}}v_{1}^{g_{p−1}}, v = u_{1}^{f_{p}}v_{1}^{f_{p−1}}.

u_{1}^{ϵ} = \frac{u^{f_{p−1}}}{v^{g_{p−1}}} , v_{1}^{ϵ} = \frac{v^{g_{p}}}{u^{f_{p}}} .

\begin{aligned}ϵ\overline{ν}(u_{1}) & = f_{p−1}\overline{ν}(u) − g_{p−1}\overline{ν}(v) \\ & = g_{p−1}\overline{ν}(u) [\frac{f_{p−1}}{g_{p−1}} − τ] \end{aligned}

which implies that \overline{ν}(u_{1}) > 0.

\begin{aligned}ϵ\overline{ν}(v_{1}) & = g_{p}\overline{ν}(v) − f_{p}\overline{ν}(u) \\ & = \overline{ν}(u)g_{p} [τ − \frac{f_{p}}{g_{p}} ] \end{aligned}

which implies \overline{ν}(v_{1}) > 0. Thus ν_{1}(u_{1}), ν_{1}(v_{1}) > 0.

f_{p−1}ν_{1}(u) − g_{p−1}ν_{1}(v) = ϵν_{1}(u_{1}), −f_{p}ν_{1}(u) + g_{p}ν_{1}(v) = ϵν_{1}(v_{1})

imply

\frac{f_{p−1}}{g_{p−1}} > \frac{ν_{1}(v)}{ν_{1}(u)} > \frac{f_{p}}{g_{p}}

if ϵ = 1,

\frac{f_{p−1}}{g_{p−1}} < \frac{ν_{1}(v)}{ν_{1}(u)} < \frac{f_{p}}{g_{p}}

if ϵ = −1. Since this holds for all p,

\frac{ν_{1}(v)}{ν_{1}(u)} = \frac{\overline{ν}(v)}{\overline{ν}(u)}.

Lemma 3.3. Suppose that L is a 2 dimensional algebraic function field over k. Suppose that R is an algebraic regular local ring with quotient field L and maximal ideal m_{R} = (u, v). Suppose that ν_{1} is a rank 1, rational rank 2 valuation of L̂ = QF (R̂) such that ν_{1}(u), ν_{1}(v) > 0 are rationally independent and which dominates R̂. Then \overline{ν} = ν_{1} | L is a rank 1, rational rank 2 valuation such that

Γ_{ν_{1}} = Z\overline{ν}(u) + Z\overline{ν}(v) = Γ\overline{ν}.

Proof. By arguments as in the proof of 3. of Lemma 3.2, we see that if f = ∑_{i+j=r}^{∞} a_{ij}u^{i}v^{j} ∈ R̂ with a_{ij} ∈ k, then ν_{1}(f) = min{ν_{1}(u^{i}v^{j}) | a_{ij} ≠ 0}. Thus

Γ_{ν_{1}} = Z\overline{ν}(u) + Z\overline{ν}(v) = Γ\overline{ν}.

Remark 3.4. Suppose that R is an algebraic regular local ring with quotient field K. There exist many extensions of a given valuation ν of K which dominates R to QF (R̂) which do not dominate R̂. Let K = k(x), ν be the rank 1 discrete valuation with valuation ring V_{ν} = k[x]_{(x)} such that ν(x) = 1. Set R = k[x]_{(x)}.

Choose f_{1}, …, f_{n} ∈ R̂ = k[[x]] such that x, f_{1}, …, f_{n} are algebraically independent over k, and choose γ_{1}, …, γ_{n} ∈ R such that 1, γ_{1}, …, γ_{n} are linearly independent over Q. K_{1} = k(x, f_{1}, …, f_{n}) is a rational function field in n + 1 variables, so we can extend ν to a rank 1, rational rank n+1 valuation ν_{1} of K_{1} by setting ν_{1}(f_{i}) = γ_{i}, 1 ≤ i ≤ n. By Proposition 2.22 [3] or Theorem 5’, Section 4, Chapter VI [30], ν_{1} extends (up to equivalence) to a valuation ν̂ of QF (R̂) which we can normalize so that it is an extension of ν.

Write f_{i} = x^{m_{i}}λ_{i} where λ_{i} ∈ R̂ is a unit series. ν_{1}(λ_{i}) = γ_{i} − m_{i} ≠ 0. Since λ_{i} and λ_{i}^{−1} ∈ R̂, R̂ contains elements of negative ν̂ value, and thus ν̂ does not dominate R̂.

Lemma 3.5. Suppose that K → K^{∗} is a finite extension of algebraic function fields over k of dimension 2. Suppose that ν is a rank 1 rational rank 2 valuation of K, ν^{∗} is an extension of ν to K^{∗}. Suppose that R_{0} is an algebraic regular local ring with quotient field K, maximal ideal m_{R_{0}} = (u, v), S is an algebraic regular local ring with quotient field K^{∗}, maximal ideal m_{S} = (x, y) and such that S dominates R,

\begin{aligned}u & = x^{a}y^{b}δ_{1} \\ v & = x^{c}y^{d}δ_{2} \end{aligned}

for some natural numbers a, b, c, d and units δ_{1}, δ_{2} ∈ S, and such that the characteristic of k does not divide ad − bc.

Suppose that V_{ν^{∗}} dominates S and ν(u), ν(v) are rationally independent over Q. Then

Γ_{ν} = Zν(u) + Zν(v)

and ν^{∗} is a rank 1, rational rank 2 valuation of K^{∗} such that ν^{∗}(x), ν^{∗}(y) are rationally independent over Q, and

Γ_{ν^{∗}} = Zν^{∗}(x) + Zν^{∗}(y).

Suppose that S → S_{1} is a sequence of quadratic transforms along ν^{∗}. Then S_{1} has regular parameters (\overline{x}_{1}, \overline{y}_{1}) such that

\begin{aligned}x & = \overline{x}\overline{a} ; 1\overline{y}\overline{b} ; 1 \\ y & = \overline{x}\overline{c} ; 1\overline{y}\overline{d} ; 1 \end{aligned}

with \overline{a}\overline{d} − \overline{b}\overline{c} = ±1, and there exists a (unique) algebraic regular local ring R_{1} with

quotient field K which lies below S_{1}. R̂_{1} ≅ Ŝ_{1}^{Γ_{ν^{∗}} /Γ_{ν}}, where Γ_{ν^{∗}}/Γ_{ν} acts faithfully on Ŝ_{1} by k-algebra automorphisms, by multiplication of \overline{x}_{1}, \overline{y}_{1} by roots of unity in k.

Proof. ν^{∗} has rational rank 2 and rank 1 since K^{∗} is finite over K (Lemmas 1 and 2 of Section 11, Chapter VI [30]). ν^{∗}(x), ν^{∗}(y) are linearly independent over Q, so Lemma 3.2 applies to ν and to ν^{∗}. We have an expression in S_{1}

\begin{aligned}u & = \overline{x}ã ; 1\overline{y}b̃ ; 1δ̃_{1} \\ v & = \overline{x}c̃ ; 1\overline{y}d̃ ; 1δ̃_{2} \end{aligned}

with natural numbers ã, b̃, c̃, d̃ and units δ̃_{1} and δ̃_{2} in S_{1} such that the characteristic of k does not divide ãd̃− b̃c̃. Let

A = ( \begin{matrix}ã & b̃ \\ c̃ & d̃ \end{matrix} ) ,

d =| ãd̃− b̃c̃ |. There exist regular parameters x̃_{1}, ỹ_{1} in Ŝ_{1} such that

u = x̃ã ; 1ỹb̃ ; 1, v = x̃c̃ ; 1ỹd̃ ; 1.

Let ω be a d^{th} root of unity in k. Z^{2}/AZ^{2} acts faithfully on Ŝ_{1} by k-algebra automorphisms. To c ∈ Z^{2}/AZ^{2} the corresponding k-algebra automorphism σ_{c} of Ŝ_{1} is defined by

σ_{c}(x̃_{1}) = ω^{<B_{1},c>}x̃_{1} σ_{c}(ỹ_{1}) = ω^{<B_{2},c>}ỹ_{1}

where B_{i} is the i^{th} row of dA^{−1} = ±adj(A). Since k(u, v) → k(x̃_{1}, ỹ_{1}) is Galois with Galois group Z^{2}/AZ^{2}, it follows that Ŝ_{1}^{Z^{2}/AZ^{2}} is the completion of a k-algebra generated by rational monomials u^{α_{1}}v^{β_{1}}, ⋯, u^{α_{r}}v^{β_{r}} (with α_{i}, β_{i} ∈ Z for all i).

R_{0}[u^{α_{1}}v^{β_{1}}, …, u^{α_{r}}v^{β_{r}}] ⊂ Ŝ_{1} ∩ K = S_{1}.

Let R_{1} = R_{0}[u^{α_{1}}v^{β_{1}}, …, u^{α_{r}}v^{β_{r}}]_{p} where p = R_{0}[u^{α_{1}}v^{β_{1}}, …, u^{α_{r}}v^{β_{r}}] ∩ m_{S_{1}}. R̂_{1} = Ŝ_{1}^{Z^{2}/AZ^{2}} is normal, so R_{1} = R̂_{1} ∩ K is normal. Since \sqrt{m_{R_{1}}S_{1}} = m_{S_{1}}, R_{1} lies below S_{1} by Zariski’s main Theorem (10.9 [4]). Uniqueness follows since the condition R_{1} lies below S_{1} implies R_{1} = S_{1} ∩ K by Proposition 1 (iv) [1].

Remark 3.6. The conclusion R̂_{1} ≅ Ŝ_{1}^{Γ_{ν^{∗}} /Γ_{ν}} in Lemma 3.5 is a special case of a general result on ramification of valuations [11].

Lemma 3.7. Suppose that p is a prime such that p is not the characteristic of k and Z_{p} acts diagonally and faithfully on the powerseries ring k[[x, y]]. Set R = k[[x, y]]^{Z_{p}}. Then R is a normal local ring such that either

1. R is regular and the algebraic fundamental group

π_{1}(spec(R) − m_{R}) = 0

or

2. R is not regular, R → k[[x, y]] is unramified away from m_{R} and the algebraic

fundamental group

π_{1}(spec(R) − m_{R}) ≅ Z_{p}.

Proof. Let ω be a primitive p^{th} root of unity in k, σ a generator of Z_{p}. There exist integers a, b with 0 ≤ a, b < p such that

σ(x) = ω^{a}x, σ(y) = ω^{b}y.

Suppose that a = 0. Then R = k[[x, y^{p}]] is regular. If b = 0 then R = k[[x^{p}, y]] is regular. In both cases,

π_{1}(spec(R) − m_{R}) = π_{1}(spec(R)) = π_{1}(k) = 0

by the purity of the branch locus (Theorems X 3.4, X 1.1 [14]).

Suppose that a, b ≠ 0. Then for 1 ≤ i ≤ p − 1 there exists a unique j_{i} such that bj_{i} ≡ ai mod p with 0 < j_{i} < p. This implies that x^{p−i}y^{j_{i}} is an invariant. Note that there exists an invariant of the form x^{p−i_{1}}y for some 0 < i_{1} < p, so that j_{i_{1}} = 1. We will show that

R = k[[x^{p}, x^{p−1}y^{j_{1}}, …, xy^{j_{p−1}}, y^{p}]].

We must show that any invariant monomial in x and y is a product of powers of these p + 1 monomials.

Suppose that x^{i}y^{j} is invariant. Then ai+bj ≡ 0 mod p. Write i = \overline{i}+λp, j = \overline{j}+τp, with 0 ≤ \overline{i} < p, 0 ≤ \overline{j} < p.

x^{i}y^{j} = x\overline{i}y\overline{j}x^{λp}y^{τp}

b\overline{j} ≡ −a\overline{i} mod p implies \overline{i} = \overline{j} = 0 or \overline{j} = j_{p−\overline{i}}.

Consider the finite map of normal local rings

Φ : Y = spec(k[[x, y]]) → X = spec(R).

The ramification locus of Φ is defined by the 2 × 2 minors of

J(Φ) = ( \begin{matrix}\frac{∂(x^{p})}{∂x} & \frac{∂(x^{p})}{∂y} \\ \frac{∂(x^{p−1}y^{j_{1}} )}{∂x} & \frac{∂(x^{p−1}y^{j_{1}} )}{∂y} \\ ⋮ & ⋮ \\ \frac{∂(y^{p})}{∂x} & \frac{∂(y^{p})}{∂y} \end{matrix} ) .

Det ( \begin{matrix}y^{j_{p−1}} & j_{p−1}xy^{j_{p−1}−1} \\ 0 & py^{p−1} \end{matrix} ) = py^{p−1+j_{p−1}}

and

Det ( \begin{matrix}px^{p−1} & 0 \\ (p − i_{1})x^{p−i_{1}−1}y & x^{p−i_{1}} \end{matrix} ) = px^{2p−1−i_{1}}

implies \sqrt{I_{2}(J(Φ))} = (x, y). Thus Φ is unramified (and étale) away from m_{R}.

Suppose that S is a complete normal local domain such that S is finite over R, and R → S is étale away from m_{R}. Let T be the normalization of the image of S ⊗_{R} k[[x, y]] in QF (S) ⊗_{QF (R)} QF (k[[x, y,]]). k[[x, y]] → T is étale away from (x, y), so by the purity of the branch locus, and since k is algebraically closed, spec(T) is a disjoint union of copies of spec(k[x, y]]). A choice of one of these copies gives a factorization

spec(k[[x, y]]) → spec(S) → spec(R).

Thus π_{1}(X − m_{R}) ≅ Z_{p}.

Abhyankar constructs an example which shows that we cannot in general take R to be regular in general in Corollary 4.4 (and thus we cannot take R to be regular in Theorem 4.3).

Theorem 3.8. (Abhyankar) There exists a two dimensional algebraic regular local ring \overline{R} with quotient field \overline{K}, a valuation \overline{ν} of \overline{K} which dominates \overline{R}, and a finite extension L_{1} of \overline{K} such that if \overline{R}_{1} is an algebraic regular local ring with quotient field \overline{K} such that \overline{R} ⊂ \overline{R}_{1} and V\overline{ν} dominates \overline{R}_{1}, then there is a unique normal algebraic local ring \overline{S} with quotient field L_{1} lying over \overline{R}_{1}. \overline{S} is not regular.

Proof. We give an outline of the construction, refering to Theorem 11 [2] for details.

Let \overline{K} = k(u, v) be a rational function field in two variables. Let q > 3 be a prime such that q ≠ char(k). Set a = q − 4, Set

τ = a + \frac{1}{1 + _{\frac{1}{a+_{\frac{1}{1+ \frac{1}{a+···}}}}}} ∈ R − Q.

Define a rank 1, rational rank 2 valuation \overline{ν} on \overline{K} by setting \overline{ν}(u) = τ, \overline{ν}(v) = 1. Set \overline{R} = k[u, v]_{(u,v)} ⊂ V\overline{ν}. Let

L_{1} = \overline{K}[\overline{z}]/\overline{z}^{q} − uv^{2}.

Let z be the image of \overline{z} in L_{1}.

Abhyankar shows that if \overline{R}_{1} is an algebraic regular local ring with quotient field \overline{K} such that \overline{R} ⊂ \overline{R}_{1}, and V\overline{ν} dominates \overline{R}_{1}, then there exists a unique normal algebraic local ring \overline{S} with quotient field L_{1} lying over \overline{R}_{1} and \overline{S} is not regular.

By Lemma 3.2, Γ\overline{ν} = Z + τZ. We will show that there is a unique extension ν_{1} of \overline{ν} to L_{1}. First suppose that ν_{1} is a valuation of L_{1} such that V_{ν_{1}} ∩ \overline{K} = V\overline{ν}. Since ν_{1} must have rank 1 and rational rank 2 (Lemmas 1 and 2, Section 11, Chapter VI [30]), we can assume that the value group of ν_{1} is a subgroup of R. We can then assume that ν_{1} is normalized so that ν_{1}(v) = 1. Since ν_{1} | \overline{K} is equivalent to \overline{ν}, and ν_{1}(v) = 1, we have ν_{1} | \overline{K} = \overline{ν} by Lemma 3.2. Thus ν_{1} is an extension of \overline{ν}. Since ν_{1}(z) = \frac{1}{q}(2 + τ), we have that Γ_{ν_{1}}/Γ\overline{ν} ≅ Z_{q}, and ν_{1} is the unique extension of \overline{ν} to L_{1}, by corollary to Theorem 25, Section 12, Chapter VI [30] and Lemma 2.18 [3].

Let p be another prime such that p ≠ q and p ≠ char(k), and set

L_{2} = \overline{K}[\overline{w}]/\overline{w}^{p} − uv^{2}.

Let w be the image of \overline{w} in L_{2}. By the same analysis as for ν_{1}, there is a unique extension ν_{2} of \overline{ν} to L_{2}. ν_{2}(w) = \frac{1}{p}(2 + τ) and Γ_{ν_{2}}/Γ\overline{ν} ≅ Z_{p}.

We remark that

τ = q − 4 + ϵ with 0 < ϵ < 1.  (3)

Set x_{1}(1) = \frac{v}{z}, y_{1}(1) = \frac{z^{2}}{v} ∈ L_{1}.

\begin{aligned}ν_{1}(x_{1}(1)) & = ν_{1}(v) − ν_{1}(z) = \\ & = \frac{q−2−τ}{q} = \frac{2−ϵ}{q} > 0 \end{aligned}

\begin{aligned}ν_{1}(y_{1}(1)) & = 2ν_{1}(z) − ν_{1}(v) = \frac{2}{q}(2 + τ) − 1 \\ & = \frac{q−4+2ϵ}{q} > 0. \end{aligned}

Set S_{1} = k[x_{1}(1), y_{1}(1)]_{(x_{1}(1),y_{1}(1))}. QF (S_{1}) = L_{1}. \overline{R} ⊂ S_{1} ⊂ V_{ν_{1}}.

\begin{aligned}u & = x_{1}(1)^{q−4}y_{1}(1)^{q−2} \\ v & = x_{1}(1)^{2}y_{1}(1). \end{aligned}  (4)

Γ_{ν_{1}} = ν_{1}(x_{1}(1))Z + ν_{1}(y_{1}(1))Z.

We will now impose the further condition that 5 ≤ q < p < 2q − 4. For example, we could take q = 11, p = 13 or q = 17, p = 23. Set

x_{1}(2) = \frac{v}{w} , y_{1}(2) = \frac{w^{2}}{v} ∈ L_{2}.

\begin{aligned}ν_{2}(x_{1}(2)) & = ν_{2}(v) − ν_{2}(w) \\ & = \frac{p−2−τ}{p} = \frac{(p−q)+2−ϵ}{p} > 0 \end{aligned}

\begin{aligned}ν_{2}(y_{1}(2)) & = 2ν_{2}(w) − ν_{2}(v) = \frac{2}{p}(2 + τ) − 1 \\ & = \frac{2q−p−4+2ϵ}{p} > 0. \end{aligned}

Set

S_{2} = k[x_{1}(2), y_{1}(2)]_{(x_{1}(2),y_{1}(2)}.

QF (S_{2}) = L_{2}. \overline{R} ⊂ S_{2} ⊂ V_{ν_{2}}.

\begin{aligned}u & = x_{1}(2)^{p−4}y_{1}(2)^{p−2} \\ v & = x_{1}(2)^{2}y_{1}(2). \end{aligned}  (5)

Γ_{ν_{2}} = ν_{2}(x_{1}(2))Z + ν_{2}(y_{1}(2))Z.

We will now assume that char(k) ≠ 2.

Let k[x, y, z_{1}] be a polynomial ring in x, y, z_{1},

f = z_{1}^{2} − 1 + x^{m}y^{n} ∈ k[x, y, z_{1}]

with m, n odd and sufficiently large, as will be determined below. We will also assume that m, n are not divisible by char(k). Then f is irreducible. Set S_{0} = k[x, y, z_{1}]/(f). By abuse of notation, we will from now on identify x, y, z_{1} with their equivalence classes in S_{0}. S_{0} is smooth over k. Suppose that a_{1}, b_{1}, c_{1}, d_{1} ∈ N are such that a_{1}d_{1} − b_{1}c_{1} is not divisible by char(k) and a_{2}, b_{2}, c_{2}, d_{2} ∈ N are such that a_{2}d_{2} − b_{2}c_{2} is not divisible by char(k). We now impose the conditions

m > max{| a_{1} − a_{2} |, | c_{1} − c_{2} |}

and

n > max{| b_{1} − b_{2} |, | d_{1} − d_{2} |}.

Let R = k[u, v] be a polynomial ring in two variables.

We define a k-algebra homomorphism

R → S_{0}

by

\begin{matrix}u = x^{a_{1}}y^{b_{1}}(1 − z_{1}) + x^{a_{2}}y^{b_{2}}(1 + z_{1}) \\ v = x^{c_{1}}y^{d_{1}}(1 − z_{1}) + x^{c_{2}}y^{d_{2}}(1 + z_{1}) \end{matrix}

Consider the prime ideals P_{1} = (x, y, z_{1} + 1) and P_{2} = (x, y, z_{1} − 1) in S_{0}. In the local ring (S_{0})_{P_{1}} we have (P_{1})_{P_{1}} = (x, y) since

(z_{1} + 1) = −(z_{1} − 1)^{−1}x^{m}y^{n}.

In the local ring (S_{0})_{P_{2}} we have (P_{2})_{P_{2}} = (x, y).

In (S_{0})_{P_{1}} we have

u = x^{a_{1}}y^{b_{1}}(1 − z_{1} − (z_{1} − 1)^{−1}x^{a_{2}+m−a_{1}}y^{b_{2}+n−b_{1}})
= x^{a_{1}}y^{b_{1}}δ_{1}
v = x^{c_{1}}y^{d_{1}}(1 − z_{1} − (z_{1} − 1)^{−1}x^{c_{2}+m−c_{1}}y^{d_{2}+n−d_{1}})
= x^{c_{1}}y^{d_{1}}δ_{2}

where δ_{1}, δ_{2} ∈ (S_{0})_{P_{1}} are units.

Since a_{1}d_{1}−b_{1}c_{1} is not divisible by char(k), we have regular parameters x_{1}(1), y_{1}(1) ∈ { \atop (S_{0})_{P_{1}}} such that

\begin{aligned}u & = x_{1}(1)^{a_{1}}y_{1}(1)^{b_{1}} \\ v & = x_{1}(1)^{c_{1}}y_{1}(1)^{d_{1}} \end{aligned} .  (6)

This implies that R ⊂ S_{0} is an inclusion.

By a similar calculation, we have units ϵ_{1}, ϵ_{2} ∈ (S_{0})_{P_{2}} such that

\begin{aligned}u & = x^{a_{2}}y^{b_{2}}ϵ_{1} \\ v & = x^{c_{2}}y^{d_{2}}ϵ_{2} \end{aligned}

and regular parameters x_{1}(2), y_{1}(2) ∈ { \atop (S_{0})_{P_{2}}} such that

\begin{aligned}u & = x_{1}(2)^{a_{2}}y_{1}(2)^{b_{2}} \\ v & = x_{1}(2)^{c_{2}}y_{1}(2)^{d_{2}} \end{aligned} .  (7)

With the notation introduced in Theorem 3.8, we have \overline{R} = R_{(u,v)} and \overline{K} = QF (R). Set L_{0} = QF (S_{0}). L_{0} is finite over \overline{K} since \overline{K} → L_{0} is an inclusion of algebraic function fields of dimension 2.

In this construction, set

a_{1} = q − 4, b_{1} = q − 2, c_{1} = 2, d_{1} = 1

and

a_{2} = p − 4, b_{2} = p − 2, c_{2} = 2, d_{2} = 1,

where p, q are the primes chosen in Theorem 3.8, and in the paragraph following Theorem 3.8.

Let Φ : Y → X be a morphism of smooth projective surfaces over k which extends our map spec(S_{0}) → spec(R). Such a map exists by resolution of singularities of surfaces in characteristic ≥ 0 ([4], [20], [22]). For i = 1, 2 we have commutative diagrams:

\begin{aligned}\overline{R} = R_{(u,v)} & → & (S_{0})_{P_{i}} \\ ↓ & & ↓ \\ ˆ\overline{R} = k[[u, v]] & → & { \atop (S_{0})_{P_{i}}} = k[[x_{1}(i), y_{1}(i)]] \end{aligned}  (8)

with (by (6) and (7))

\begin{aligned}u & = x_{1}(i)^{a_{i}}y_{1}(i)^{b_{i}} \\ v & = x_{1}(i)^{c_{i}}y_{1}(i)^{d_{i}}. \end{aligned}

We further have commutative diagrams:

\begin{aligned}\overline{R} & → & S_{i} \\ ↓ & & ↓ \\ ˆ\overline{R} = k[[u, v]] & → & Ŝ_{i} = k[[x_{1}(i), y_{1}(i)]] \end{aligned}  (9)

with (by (4) and (5))

\begin{aligned}u & = x_{1}(i)^{a_{i}}y_{1}(i)^{b_{i}} \\ v & = x_{1}(i)^{c_{i}}y_{1}(i)^{d_{i}}. \end{aligned}

Diagrams (8) and (9) patch to give commutative diagrams for i = 1, 2

\begin{aligned}R_{(u,v)} & → & (S_{0})_{P_{i}} \\ ↓ & & ↓ \\ ˆ\overline{R} & → & Ŝ_{i} \end{aligned}  (10)

Lemma 3.2 implies that for i = 1, 2, there exists a unique extension ν̂_{i} of ν_{i} to QF (Ŝ_{i}) which dominates Ŝ_{i} and Γν̂_{i} = Γ_{ν_{i}}.

For i = 1, 2, let \overline{ν}_{i} = ν̂_{i} | L_{0} (under the inclusion L_{0} ⊂ QF (Ŝ_{i}) induced by (10)). Γ_{\overline{ν}_{i}} ≅ Γ_{ν_{i}} by Lemma 3.3. \overline{ν}_{i} | \overline{K} = \overline{ν} for i = 1, 2 where \overline{ν} is the valuation of \overline{K} introduced in Theorem 3.8.

Suppose that there exists a diagram

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

such that the vertical arrows are birational and proper, Y_{1} is nonsingular, X_{1} is normal, and Φ_{1} is finite. Then Y_{1} → Y is a sequence of blowups of points (Theorem II.1.1 [29]). There exist commutative diagrams

\begin{aligned}R_{1} & → & S(1) \\ ↑ & & ↑ λ_{1} \\ \overline{R} & → & (S_{0})_{P_{1}} \end{aligned}  (11)

and

\begin{aligned}R_{1} & → & S(2) \\ ↑ & & ↑ λ_{2} \\ \overline{R} & → & (S_{0})_{P_{2}} \end{aligned}  (12)

where R_{1} is the center of \overline{ν} on X_{1}, S(1) is the local ring of the center of \overline{ν}_{1} on Y_{1} and S(2) is the local ring of the center of \overline{ν}_{2} on Y_{1}. λ_{1} and λ_{2} are products of quadratic transforms.

By Lemma 3.5

R̂_{1} ≅ Ŝ(1)^{Γ_{\overline{ν}_{1}} /Γ\overline{ν}} .  (13)

and

R̂_{1} ≅ Ŝ(2)^{Γ_{\overline{ν}_{2}} /Γ\overline{ν}} .  (14)

By Theorem 3.8, (10) with i = 1 and (11), R_{1} is not regular.

Γ_{\overline{ν}_{1}}/Γ\overline{ν} ≅ Z_{q} and Γ_{\overline{ν}_{2}}/Γ\overline{ν} ≅ Z_{p}

by our construction.

By Lemma 3.7 and (13),

π_{1}(spec(R̂_{1}) − mR̂_{1}) ≅ Z_{q}

and by (14)

π_{1}(spec(R̂_{1}) − mR̂_{1}) ≅ Z_{p}.

But p ≠ q, so we have a contradiction.

4. Ramification of valuations in algebraic function fields

Theorem 4.1. (Monomialization; Theorem 1.1 [8]) Let k be a field of characteristic zero, K an algebraic function field over k, K^{∗} a finite algebraic extension of K, ν^{∗} a k-valuation of K^{∗}. Suppose that S^{∗} is an algebraic regular local ring with quotient field K^{∗} which is dominated by ν^{∗} and R^{∗} is an algebraic regular local ring with quotient field K which is dominated by S^{∗}. Then there exist sequences of monoidal transforms R^{∗} → R_{0} and S^{∗} → S such that ν^{∗} dominates S, S dominates R_{0} and there are regular parameters (x_{1}, ...., x_{n}) in R_{0}, (y_{1}, ..., y_{n}) in S, units δ_{1}, …, δ_{n} ∈ S and a matrix A = (a_{ij}) of nonnegative integers such that det(A) ≠ 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}.  (15)

The standard theorems on resolution of singularities allow one to easily find R_{0} and S such that (15) holds, but, in general, we will not have the essential condition det(a_{ij}) ≠ 0. The difficulty in the proof of this Theorem is to achieve the condition det(a_{ij}) ≠ 0.

Let α_{i} be the images of δ_{i} in S/m_{S} for 1 ≤ i ≤ n. Let C = (a_{ij})^{−1}, a matrix with rational coefficients. Define regular parameters (\overline{y}_{1}, …, \overline{y}_{n}) in Ŝ by

\overline{y}_{i} = (\frac{δ_{1}}{α_{1}} )^{c_{i1}} ⋯ (\frac{δ_{n}}{α_{n}} )^{c_{in}} y_{i}

for 1 ≤ i ≤ n. We thus have relations

x_{i} = α_{i}\overline{y}_{1}^{a_{i1}} ⋯ \overline{y}_{n}^{a_{in}}  (16)

with α_{i} ∈ S/m_{S} for 1 ≤ i ≤ n in

R̂_{0} = R_{0}/m_{R_{0}}[[x_{1}, …, x_{n}]] → Ŝ = S/m_{S}[[\overline{y}_{1}, …, \overline{y}_{n}]].

Remark 4.2. Suppose that k′ is a field, A = (a_{ij}) is an n × n matrix of natural numbers with det(A) ≠ 0, and we have an inclusion of lattices

N′ = Z^{n} →^{A} N = Z^{n}

with a corresponding inclusion of dual lattices

M = Hom(N, Z) → M′ = Hom(N′, Z).

Let σ be the cone generated by the rows of A in N ⊗ R, σ′ be the cone generated by N^{n} in N′ ⊗ R. Suppose that σ̂ ∩ M is generated by e_{1}, …, e_{r}. By the theory of toric varieties (c.f. page 34 [12]) we have inclusions of k′-algebras

k′[\overline{x}_{1}, …, \overline{x}_{n}] → k′[σ̂ ∩ M] = k′[\overline{x}^{e_{1}}, …, \overline{x}^{e_{r}}] → k′[σ̂′ ∩ M′] = k′[\overline{y}_{1}, …, \overline{y}_{n}]

with \overline{x}_{i} = \overline{y}_{1}^{a_{i1}} ⋯ \overline{y}_{n}^{a_{in}} for 1 ≤ i ≤ n. k′[σ̂ ∩ M] is a normal ring with quotient field k′(\overline{x}_{1}, …, \overline{x}_{n}) and k′[σ̂′ ∩ M′] is finite over k[σ̂ ∩ M].

If k′ = C, this can be expressed in a particularly nice way. N/N′ ≅ Z^{n}/AZ^{n} acts on C[\overline{y}_{1}, …, \overline{y}_{n}] by associating to c ∈ N/N′ the C-algebra automorphism σ_{c} defined by

σ_{c}(\overline{y}_{i}) = exp^{2πi<F_{i},c>}\overline{y}_{i}

for 1 ≤ i ≤ n, where

A^{−1} = ( F_{1} ; ⋮ ; F_{n} ) .

Theorem 4.3. Let k be a field of characteristic zero, K an algebraic function field over k, K^{∗} a finite algebraic extension of K, ν^{∗} a k-valuation of K^{∗}. Suppose that S^{∗} is an algebraic local ring with quotient field K^{∗} which is dominated by ν^{∗} and R^{∗} is an algebraic local ring with quotient field K which is dominated by S^{∗}. Then there exists a commutative diagram

\begin{aligned}R_{0} & → & R & → & S & ⊂ & V_{ν^{∗}} \\ ↑ & & & & ↑ \\ R^{∗} & & → & & S^{∗} \end{aligned}  (17)

where S^{∗} → S and R^{∗} → R_{0} are sequences of monodial transforms along ν^{∗} such that R_{0} → S have regular parameters of the form of the conclusions of Theorem 4.1, R is an algebraic normal local ring with toric singularities, which is the localization of the blowup of an ideal in R_{0}, and the regular local ring S is the localization at a maximal ideal of the integral closure of R in K^{∗}.

Proof. By resolution of singularities [19] (c.f. Theorem 2.6, Theorem 2.9 [8]), we first reduce to the case where R^{∗} and S^{∗} are regular, and then construct, by the local monomialization theorem, Theorem 4.1 a sequence of monodial transforms along ν^{∗}

\begin{aligned}R_{0} & → & S & ⊂ & V_{ν^{∗}} \\ ↑ & & ↑ \\ R^{∗} & → & S^{∗} \end{aligned}  (18)

so that R_{0} is a regular local ring with regular parameters (x_{1}, …, x_{n}), S is a regular local ring with regular parameters (y_{1}, …, y_{n}), there are units δ_{1}, …, δ_{n} in S, and a matrix of natural numbers A = (a_{ij}) with nonzero determinant d such that

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

for 1 ≤ i ≤ n.

Let k′ be an algebraic closure of S/m_{S}. With the notation of (16), set \overline{x}_{i} = \frac{x_{i}}{α_{i}}, so that

R_{0} ⊗_{R_{0}/m_{R_{0}}} k′ ≅ k′[[\overline{x}_{1}, …, \overline{x}_{n}]] → S ⊗_{S/m_{S}} k′ ≅ k′[[\overline{y}_{1}, …, \overline{y}_{n}]]

is defined by

\overline{x}_{i} = \overline{y}_{1}^{a_{i1}} ⋯ \overline{y}_{n}^{a_{in}},

1 ≤ i ≤ n. With the notation of Remark 4.2, set R = R_{0}[x^{e_{1}}, …, x^{e_{r}}]_{m} where m = (x^{e_{1}}, …, x^{e_{r}}).

R_{0}[x^{e_{1}}, …, x^{e_{r}}] ⊂ Ŝ ∩ K^{∗} = S

(by Lemma 2 [1]) and m ⊂ mŜ∩K^{∗} = m_{S}, so S dominates R. R̂ = R_{0}/m_{R_{0}}[[\overline{x}^{e_{1}}, …, x^{e_{r}}]] is integrally closed in its quotient field (by Remark 4.2 and Theorem 32, Section 13, Chapter VIII [27]), so R = R̂ ∩ K is integrally closed in K. After possibly reindexing the y_{i}, we may assume that d = det(A) > 0. Let (b_{ij}) be the adjoint matrix of A. Then

∏_{j=1}^{n} x_{j}^{b_{ij}} = (∏_{j=1}^{n} δ_{j}^{b_{ij}} ) y_{i}^{d} ∈ R.

Thus \sqrt{m_{R}S} = m_{S}, so R lies below S by Zariski’s Main Theorem (10.9 [4]).

As an immediate consequence, we obtain a proof in characteristic zero of the “weak simultaneous resolution local conjecture”. which is stated explicitely on page 144 of [5], and is implicit in [2]. Abhyankar proves this for algebraic function fields of dimension two and any characteristic in [1] and [3]. In the paper [9], we have given a direct proof of this result, also as a consequence of Theorem 4.1 (Theorem 1.1 [8]).

Corollary 4.4. (Corollary1)(Theorem 1.1 [9]) Let k be a field of characteristic zero, K an algebraic function field over k, K^{∗} a finite algebraic extension of K, ν^{∗} a kvaluation of K^{∗}, and S^{∗} an algebraic regular local ring with quotient field K^{∗} which is dominated by ν^{∗}. Then for some sequence of monodial transforms S^{∗} → S along ν^{∗}, there exists a normal algebraic local ring R with quotient field K, such that the regular local ring S is the localization at a maximal ideal of the integral closure of R in K^{∗}.

Proof. There exists a normal algebraic local ring R^{∗} with quotient field K such that ν^{∗} dominates R^{∗} (take R^{∗} to be the local ring of the center of ν^{∗} on a normal projective model of K). There exists a finite type k-algebra T such that the integral closure of R^{∗} in K^{∗} is a localization of T, and T is generated over k by g_{1}, …, g_{m} ∈ V^{∗} = V_{ν^{∗}} such that ν^{∗}(g_{i}) ≥ 0 for all i. There exists a sequence of monoidal transforms S^{∗} → S_{1} along ν^{∗} such that T ⊂ S_{1} (Theorem 2.7 [8]). S_{1} dominates R^{∗}. After replacing S^{∗} with S_{1}, we can assume that S^{∗} dominates R^{∗}. Theorem 4.3 applies to this situation, so we can construct a diagram of the form (17).

When K^{∗} is Galois over K, it is not difficult to construct using Galois theory and resolution of singularities a regular local ring S with quotient field K^{∗} and a normal local ring R with quotient field K such that S lies over R (Theorem 7 [2], Theorem 6.1), although even in the Galois case the full statements of Theorem 4.3 and Corollary 4.4 do not follow from these results (Theorem 7 [2], Theorem 6.1). The general case of non Galois extensions is much more subtle, and not as well behaved, as can be seen from Theorem 3.1.

5. Generically finite morphisms

Suppose that f : Y → X is a dominant, generically finite morphism of complete k-varieties. In this section we construct a commutative diagram

\begin{matrix}\overline{Y} & →^{f_{1}} & \overline{X} \\ ↓ & & ↓ \\ Y & →^{f} & X \end{matrix}

of the form of (1) such that \overline{Y} is nonsingular, f_{1} is “close” to being finite, and the vertical arrows are birational and “close” to being proper.

Let K be an algebraic function field over a field k of characteristic zero. In his work on resolution of singularities, Zariski [27] constructed for each k-valuation ring V of K a projective model X_{V} of K such that the center of V is nonsingular on X_{V}. By the quasi-compactness of the Zariski-Riemann manifold of valuations of K, there exists a finite number of the X_{V}, {X_{V_{1}}, …, X_{V_{n}}} such that every valuation ring V of K has a nonsingular center on at least one of the X_{V_{i}}.

In dimension ≤ 3, Zariski [28] was able to patch the open nonsingular locus of appropriate birational transforms of the X_{V_{i}} to produce a nonsingular projective model X of K.

The only part of Zariski’s proof of the existence of a nonsingular model which does not extend to arbitrary dimension is the final step where nonsingular open subsets U_{i} of X_{V_{i}} are patched (after appropriate birational transforms) to produce a projective variety. Hironaka observes in Chapter 0, Section 6 of [17] that we can always patch the nonsingular loci U_{i} of the X_{V_{i}} along open sets where they are isomorphic, to produce an integral finite type scheme X such that X is nonsingular and every valuation ring V of K has a center on X, but X will in general not be separated. Hironaka calls such schemes “complete”.

If such an X is separated, then the following Lemma shows that X is in fact complete in the usual sense, that is X is a proper k-scheme.

Lemma 5.1. Suppose that X is a separated, integral, finite type k-scheme, such that every k-valuation of k(X) has a center on X. Then X is a proper k-scheme.

The proof of this Lemma is immediate from (ii) of Corollary II 7.3.10 [15] (Recall that in the notation of [15], a scheme is a separated pre-scheme), or can be deduced directly from the valuative criterion of properness (Theorem II 7.3.8 [15] or Theorem II 4.7 [17]).

One may expect a scheme X which satisfies all of the conditions of the Lemma above except the separatedness condition to be universally closed. This is false, as is shown by the following example.

Example 5.2. Suppose that k is a field. There exists a (nonseparated) integral finite type k-scheme such that every k-valuation ring of k(X) has a center on X, but X is not universally closed over k.

Proof. Let ϕ be an imbedding of P^{1} in P^{3}. Let Z = ϕ(P^{1} − {∞}), x_{0} = ϕ(∞). Let π : X_{1} → P^{3} be the blowup of \overline{Z} = ϕ(P^{1}), X_{2} = P^{3} − \{x_{0}\}. We can construct an integral finite type k-scheme X by glueing X_{1} to X_{2} along the open sets X_{1} − π^{−1}(\overline{Z}) and X_{2} −Z. By construction, every k-valuation of k(X) has a center on X. ϕ induces an isomorphism of P^{1} − {∞} with the closed subset Z ⊂ X_{1}. Z is closed in X. Thus the induced morphism ϕ : spec(k(P^{1}) → X does not extend to a morphism spec(O_{P^{1},∞}) → X.

Suppose that X is universally closed over k. Let U = spec(k(P^{1})), T = spec(O_{P^{1},∞}), with natural morphism i : U → T. Let t_{1} ∈ T be the generic point, t_{0} ∈ T the special point.

Let A be the closure of ϕ × i(U) in X × T. π_{2}(A) is closed since the projection π_{2} : X ×T → T is closed by assumption. So there exists y_{0} ∈ A such that π_{2}(y_{0}) = t_{0}. By Lemma II 4.4 [17] we have an extension T → X × T of U → X × T which projects to an extension spec(T) → X of U → X, and thus an extension of ϕ to spec(O_{P^{1},∞}) → X, a contradiction.

Theorem 5.3. Suppose that f : Y → X is a dominant, generically finite morphism of complete varieties over a field k of characteristic zero. Then there exists a commutative diagram of integral, finite type k-schemes

\begin{aligned}\overline{Y} & → & \overline{X} \\ ↓ & & ↓ \\ Y & → & X \end{aligned}

such that \overline{Y} is nonsingular, \overline{X} has normal toric singularities, the vertical arrows are birational morphisms, \overline{Y} → \overline{X} is quasi-finite and every k-valuation ring of k(X) has a center on \overline{X}, every k-valuation ring of k(Y) has a center on \overline{Y}.

Remark 5.4. Theorem 3.1 (and Lemma 5.1) show that we cannot take the vertical arrows in Theorem 5.3 to be proper.

Proof. Let K = k(X) be the function field of X, K^{∗} = k(Y) be the function field of Y. By assumption, k(Y) is finite over k(X). By resolution of singularities [19], we may assume that Y and X are nonsingular. Let V^{∗} be a k-valuation ring of K^{∗} such that trdeg_{k}V^{∗}/m_{V^{∗}} = 0, V = V^{∗} ∩ K. Let p be the center of V on X, q the center of V^{∗} on Y.

By Theorem 4.3, there exists a sequence of the form (17),

\begin{aligned}R_{0} & → R → & S & ⊂ & V^{∗} \\ ↑ & & ↑ \\ O_{X,p} & → & O_{Y,q} \end{aligned} ,

such that S is regular and R has toric singularities.

Let N = Z^{n}, σ be the cone generated by the rows of A = (a_{ij}) (with the notation of (15) in N ⊗ R. Let M be the dual lattice of N, σ̂ be the dual cone of σ. By the proof of Theorem 4.3 (and Remark 4.2), if σ̂ ∩ M is generated by e_{1}, …, e_{r}, then R = R_{0}[x^{e_{1}}, …, x^{e_{r}}]_{m}. There is a natural inclusion

k[σ̂ ∩ M] = k[x^{e_{1}}, …, x^{e_{r}}] → R.

U_{σ} = spec(k[σ̂ ∩ M]) is a normal affine toric variety.

Thus there exist affine open sets U_{p} of p in X and \overline{U}_{q} of q in Y and affine rings R_{V} with quotient fields K and S_{V^{∗}} with quotient field K^{∗} with the following properties:

1. R_{V} is normal and S_{V^{∗}} is regular.

2. If p_{1} is the center of V on R_{V} and q_{1} is the center of V^{∗} on S_{V^{∗}}, then (R_{V})_{p_{1}} = R, (S_{V^{∗}})_{q_{1}} = S.

3. There is a commutative diagram

\begin{aligned}R_{V} & → & S_{V^{∗}} \\ ↑ & & ↑ \\ Γ(U_{p}, O_{X}) & → & Γ(\overline{U}_{q}, O_{Y}) \end{aligned}  (19)

such that R_{V} → S_{V^{∗}} is quasi-finite,

k[σ̂ ∩ M] = k[x^{e_{1}}, …, x^{e_{r}}] → R_{V}

is étale, so that R_{V} has normal toric singularites, and the vertical arrows are birational morphisms.

Let Z(K) be the Zariski-Riemann manifold of k-valuation rings of K, Z(K^{∗}) be the Zariski-Riemann manifold of k-valuation rings of K^{∗}. These spaces have natural topologies with respect to which they are quasi-compact (Theorem 40 Section 17, Chapter VI, [30]). There is a natural continuous map Φ : Z(K^{∗}) → Z(K) defined by Φ(V^{∗}) = V^{∗} ∩ K. For each V^{∗} ∈ Z(K^{∗}) such that trdeg_{k}V^{∗}/m_{V^{∗}} = 0, let Y_{V^{∗}} be a projective variety which contains spec(S_{V^{∗}}) as an open set, and let X_{V} be a projective variety which contains spec(R_{V}) as an open set and such that the birational rational maps Y_{V^{∗}} → X_{V}, Y_{V^{∗}} → Y and X_{V} → X are morphisms. Then there are commutative diagrams of continuous maps

\begin{aligned}Z(K^{∗}) & → & Z(K) \\ ↓ π_{V^{∗}} & & ↓ π_{V} \\ Y_{V^{∗}} & → & X_{V} \\ ↓ & & ↓ \\ Y & → & X \end{aligned} .

Let \overline{W}_{V^{∗}} = π_{V^{∗}}^{−1}(spec(S_{V^{∗}})), an open neighborhood of V^{∗} in Z(K^{∗}), W_{V} = π_{V}^{−1}(spec(R_{V})), an open neighborhood of V in Z(K). Suppose that V′ ∈ Z(K^{∗}) and trdeg_{k}V′/m_{V′} >

0. Let π : V′ → V′/m_{V′} be the residue map. By corollary 3 to Theorem 5, Section 4, Chapter VI [30], there exists a k-valuation ring V_{1} with quotient field V′/m_{V′} such that trdeg_{k}V_{1}/m_{V_{1}} = 0. Set V^{∗} = π^{−1}(V_{1}), a k-valuation ring of K^{∗} such that V^{∗} is a specialization of V′ and trdeg_{k}V^{∗}/m_{V^{∗}} = 0 (page 57 [3]). Since V^{∗} dominates a local ring S of W_{V^{∗}}, V′ must dominate a localization of S, which is the local ring of a point of W_{V^{∗}}. Thus V′ ∈ \overline{W}_{V^{∗}}. \{\overline{W}_{V^{∗}}\}_{V^{∗}∈Z(K^{∗})} is thus an open cover of Z(K^{∗}). {W_{V} }_{V ∈Z(K)} is also an open cover of Z(K).

Since Z(K^{∗}) and Z(K) are quasi-compact, there is a finite set of valuation rings V_{1}^{∗}, …, V_{m}^{∗} ∈ Z(K^{∗}) with trdeg_{k}V_{i}^{∗}/m_{V_{i}^{∗}} = 0 for all i such that if V_{i} = V_{i}^{∗} ∩ K then \{\overline{W}_{V_{1}^{∗}}, …, \overline{W}_{V_{m}^{∗}}\} is an open cover of Z(K^{∗}) and \{W_{V_{1}}, …, W_{V_{m}}\} is an open cover of Z(K).

For 1 ≤ i ≤ m we have commutative diagrams

D_{i} := spec(S_{V_{i}^{∗}}) → C_{i} := spec(R_{V_{i}}) b_{i} ↓  a_{i} ↓ \overline{U}_{q_{i}} →  U_{p_{i}} ,

where q_{i} is the center of V_{i}^{∗} on Y, p_{i} is the center of V_{i} on X. Let A_{i} ⊂ X be nontrivial open sets where a_{i} is an isomorphism and let B_{i} ⊂ f^{−1}(A_{i}) be nontrivial open sets where b_{i} is an isomorphism. Then define \overline{Y} to be the finite type k-scheme obtained by patching D_{i} to D_{j} for i ≠ j along the nontrivial open set B_{i} ∩ B_{j}, and define \overline{X} to be the finite type k-scheme obtained by patching C_{i} to C_{j} for i ≠ j along the nontrivial open set A_{i} ∩ A_{j}.

By construction, \overline{Y} and \overline{X} are integral, \overline{Y} is nonsingular, \overline{X} has normal toric singularities and \overline{f} : \overline{Y} → \overline{X} is quasi-finite.

6. Galois extensions

For Galois extensions, Question 1 of the introduction has a positive answer. Suppose that K → K^{∗} is a finite Galois extension of algebraic function fields of characteristic zero. The existence of a finite map of normal projective k-varieties Y → X, where Y is nonsingular, k(X) = K, k(Y) = K^{∗} has been proven by Abhyankar in Theorem 7 [2]. We prove a relative version of this result in Theorem 6.1.

Theorem 6.1. Suppose that Φ : Y → X is a dominant morphism of projective varieties over an algebraically closed field k of characteristic zero such that k(Y) is a finite Galois extension of k(X). Then there exists a commutative diagram

\begin{aligned}\overline{Y} & → & \overline{X} \\ ↓ & & ↓ \\ Y & → & X \end{aligned}

such that \overline{Y} → Y, \overline{X} → X are birational morphisms of projective k-varieties, \overline{Y} is nonsingular, \overline{X} is normal, \overline{Y} → \overline{X} is finite and \overline{X} has normal toric singularities.

Proof. Let X_{0} be the normalization of X in k(X), Y_{0} be the normalization of X_{0} in k(Y). Let G = Gal(k(Y)/k(X)). G acts on Y_{0} and Y_{0}/G = X_{0}. Let D_{0} be the branch locus of Y_{0} → X_{0}. Let π_{1} : X_{1} → X_{0} be a resolution of singularities so that D_{1} = π_{1}^{−1}(D_{0})_{red} is a simple normal crossings divisor. If X_{1} is the blowup of an ideal sheaf I_{0} in X_{0}, let Y_{1} be the normalization of the blowup of I_{0}O_{Y_{0}}. f_{1} : Y_{1} → X_{1} is finite, and G acts on Y_{1}. The branch locus of f_{1} which is a divisor supported on D_{1} (by the purity of the branch locus) has simple normal crossings, so by Abhyankar’s Lemma ([6], XIII 5.3 [13]) Y_{1} has normal toric singularities, and (by Lemma 7 [1]) if p ∈ Y_{1} is a closed point, the stabilizer

G^{s}(p) = {σ ∈ G | σ(p) = p}

is Abelian. Suppose that I_{1} ⊂ O_{Y_{1}} is an ideal sheaf such that the blowup of I_{1} in Y_{1} dominates Y. Let J = ∏_{g∈G} g(I_{1}), and Ỹ_{2} be the blowup of J. Then G acts on Ỹ_{2} and the rational map Ỹ_{2} → Y is a morphism. Let Y_{2} → Ỹ_{2} be a G-equivariant resolution of singularities of Ỹ_{2} ([7], [25]), with composed map π_{2} : Y_{2} → Y_{1}. G acts on Y_{2} and for q ∈ Y_{2}, G^{s}(q) < G^{s}(π_{2}(q)) so that G^{s}(q) is Abelian. Let \overline{Y} = Y_{2}, \overline{X} = Y_{2}/G, a projective k-variety (c.f. page 126, [16]), which has normal toric singularities (by Lemma 7, [1]).

Example 6.2. Even if k(Y) is Galois over k(X), with Galois group G, and Y → X is G-equivariant, we cannot take both \overline{X} and \overline{Y} to be nonsingular in Theorem 5.3 or in Theorem 6.1.

This is an immediate consequence of Abhyankar’s example, Theorem 11, [2] (restated in Theorem 3.8 of this paper.) Let k be an algebraically closed field of characteristic zero. With the notations of Theorem 3.8, L_{1} is a Galois extension of \overline{K} = k(u, v) with Galois group Z_{q}. Z_{q} acts on S = k[u, v, z]/z^{q} − uv^{2} and its invariant ring is R = k[u, v]. With the notation of Theorem 3.8, \overline{R} = R_{(u,v)}. By equivariant resolution of singularites [7], [25] (applied to the normalization of X = P^{2} in L_{1}) there exists a dominant Z_{q} equivariant morphism of nonsingular projective k-surfaces Y → X such that k(X) = \overline{K}, k(Y) = L_{1} and there exists a point p ∈ X such that O_{X,p} = \overline{R}.

Suppose that there exists a diagram

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

as in the conclusions of Theorem 5.3 such that \overline{X} is nonsingular. Let \overline{ν} be the valuation of \overline{K} constructed in Theorem 3.8. Let ν_{1} be the (unique) extension of \overline{ν} to L_{1}. Let q ∈ \overline{Y} be a center of ν_{1}, p = \overline{Φ}(q). B = O_{\overline{Y} ,q} dominates A = O_{\overline{X},p}. Since \overline{Φ} is quasi-finite, B lies over A by Zariski’s Main Theorem (10.9 [4]). Since A is regular and \overline{ν} dominates A, B is not regular by Theorem 3.8, a contradiction.

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