Families of curves and alterations

A.J. de Jong, Princeton University

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Set up and statement of theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3. Genus goes up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Equal genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1 Introduction

This paper is a continuation of or sequel to the article [1]. The main point that is in this paper and that was missed by the author while writing [1] is that one can alter any family of curves into a semi-stable family. In [1] this was done only in special cases, see [1, Theorem 5.8]. It should be mentioned that this result on families of curves was ”known to be true” to mathematiciens working on moduli of varieties.

Thus the main theorem of this paper is Theorem 2.4. The applications are derived by arguments that are similar to the arguments of [1]; the results are slightly stronger than the results of that paper. For example, the theorem implies that any proper dominant morphism X → S of integral excellent schemes may be altered into a composition of semistable curve fibrations, see Theorem 5.9, Corollary 5.10 and Remark 5.16. It gives an alternative proof of a stronger version of [1, Theorem 8.2] concerning semi-stable families over an excellent base scheme of dimension 1, see Corollary 5.1.

Another point of this paper is to deal systematically with group actions as well; the advantage of this is that one gets resolution of singularities up to quotient singularities in certain cases, see Theorem 5.13 and its corollaries. Finally, the results of [1] are extended to integral schemes of finite type over excellent base schemes of dimension ≤ 2, see Corollary 5.15.

For an indication as to how to get cohomological applications of the geometric results obtained in this paper we refer to the introduction of [1].

Acknowledgements. The author would like to thank D. Abramovich, J. Harris, B. Hasset and F. Oort for discussions on the subject of this paper. The idea to prove case (vii)(b) of Theorem 2.4 came to the author only after Abramovich’s remark that one should be able to prove such a thing using Kontsevich spaces of stable maps. (Although we do not do so, this way one could prove Theorem 2.4.) Some of the formulations of the results were found in discussions with Abramovich and Hasset. A theorem in characteristic zero very similar to Theorem 5.9 can be found in a secret preprint by Abramovich.

2 Set up and statement of theorem

Throughout this paper we adopt the notations, conventions, definitions and terminology of [1, Section 2] .

2.1 Situation. Here we have a proper morphism f : X → S of integral excellent schemes whose generic fibre X_{η} has dimension 1. We have a finite group G acting on f, i.e., actions of G on X and S such that f is G-equivariant. In addition we have a proper closed subset Z ⊂ X which is G-stable, i.e., for all g ∈ G we have g(Z) ⊂ Z. We note that the extreme cases Z = ∅ or G = {1} (or both) are allowed.

In the sequel we will consider the following conditions on (f : X → S, G, Z). (2.1.1) The generic fibre X_{η} is smooth over η and Z_{η} is étale over η. (2.1.2) The morphism f is projective. (2.1.3) All fibres of f are equidimensional of dimension 1. (2.1.4) The variety X_{η} is geometrically irreducible over η. (2.1.5) The quotient X_{η}/G is geometrically irreducible over R(S)^{G}. We note that (2.1.4)

implies this condition. (2.1.6) (Assume (2.1.3).) For any geometric point s̄ of S and any irreducible component

C of Xs̄ the normalization of C has genus at least 2.

2.2 Alterations of Situation 2.1. This refers to a commutative diagram of morphisms of schemes

\begin{matrix}X′ & ⟶^{ψ} & X \\ ↓^{f′} & & ↓^{f} \\ S′ & ⟶^{ϕ} & S, \end{matrix}

where (f′ : X′ → S′, G′, Z′) and (f : X → S, G, Z) are as in Situation 2.1. Further we demand that ψ and ϕ are alterations, ψ^{−1}(Z) = Z′ and that a surjection G′ → G is given such that ψ and ϕ are G′-equivariant morphisms.

In the sequel we will consider the following conditions on the pair (ψ, ϕ). (2.2.1) The field extension R(S)^{G} ⊂ R(S′)^{G′} is purely inseparable and the field R(X′)^{G′}

is the residue field of the local ring R(S′)^{G′} ⊗_{R(S)^{G}} R(X)^{G}. (2.2.2) The field R(X′) is a quotient of the ring R(X) ⊗_{R(S)} R(S′), i.e., the generic fibre

X′_{η′} maps birationally to an irreducible component of X_{η} ×_{S} S′. (2.2.3) The field R(X′) is separable over the image of the natural map R(X) ⊗_{R(S)}

R(S′) → R(X′). This implies that the field R(X′)^{G′} is separable over the image of the natural map R(X)^{G} ⊗_{R(S)^{G}} R(S′)^{G′} → R(X′)^{G′}.

(2.2.4) Let N ⊂ G′ be the kernel of the map G′ → G × Aut S′. The field R(X′)^{N} is a quotient of the ring R(X) ⊗_{R(S)} R(S′). This means that the curve X′_{η′}/N maps birationally onto an irreducible component of X_{η} ×_{S} S′.

(2.2.5) The field extensions R(S) ⊂ R(S′) and R(X) ⊂ R(X′) are separable. This means

that the alterations ψ and ϕ are generically étale. (2.2.6) Let H = Ker(G → Aut S) and H′ = Ker(G′ → Aut S′). Then H′ → H is

surjective. (2.2.7) The morphisms ψ and ϕ are projective.

2.3 Remarks. We make some obvious observations.

(i) The conditions above are by no means independent. For example the conditions (2.2.1), (2.2.5), (2.2.2) and (2.2.4) all imply (2.2.3). (ii) Various combinations of the conditions above have interesting consequences. E.g. if (ψ, ϕ) satisfies (2.2.4) and (2.2.6) then X′_{η′}/H′ is birational to a component of (X_{η}/H) ⊗ κ(η′). If (ψ, ϕ) satisfies (2.2.1) and (2.2.5) then we have R(S)^{G} = R(S′)^{G′} and R(X)^{G} = R(X′)^{G′}. In particular the field extensions R(S) ⊂ R(S′) and R(X) ⊂ R(X′) are Galois and certain subgroups of G′ surject onto their Galois groups. From this we see that (2.2.1) and (2.2.5) imply (2.2.4). (iii) It is clear how to define compositions (ψ, ϕ) ∘ (ψ′, ϕ′) of given alterations (ψ, ϕ) and (ψ′, ϕ′). If the alterations (ψ, ϕ) and (ψ′, ϕ′) satisfy one of (2.2.1), (2.2.2), (2.2.5), (2.2.6) or (2.2.7) then so does the composition. (iv) If, in the situation of (iii), we have that (ψ, ϕ) satisfies (2.2.4) and (ψ′, ϕ′) satisfies (2.2.6) and (2.2.4), then the composition (ψ, ϕ) ∘ (ψ′, ϕ′) satisfies (2.2.4).

(v) If X_{η} is geometrically irreducible, then the condition (2.2.2) implies that the curves X_{η} and X′_{η′} have the same geometric genus. If X_{η} is also geometrically reduced, then (2.2.2) is equivalent to the birationality of the morphism X′_{η′} → X_{η} ⊗ κ(η′).

2.4 Theorem. Let (f : X → S, G, Z) be as in Situation 2.1. There exists an alteration (f_{1} : X_{1} → S_{1}, G_{1}, Z_{1}) as in 2.2

\begin{matrix}X_{1} & ⟶^{ψ_{1}} & X \\ ↓^{f_{1}} & & ↓^{f} \\ S_{1} & ⟶^{ϕ_{1}} & S, \end{matrix}

with the following properties:

(i) The morphism f_{1} is a projective semi-stable family of curves over S_{1} with (irreducible) smooth generic fibre.

(ii) There are disjoint sections σ_{i} : S_{1} → X_{1}, i = 1, …, n of f_{1} into the smooth locus

of X_{1} over S_{1} such that Z_{1,η_{1}} = {σ_{1}(η_{1}), …, σ_{n}(η_{1})}. (iii) The alteration (ψ_{1}, ϕ_{1}) satisfies (2.2.7) and (2.2.4) (and hence (2.2.3)). (iv) If (f : X → S, G, Z) satisfies (2.1.5), then (ψ_{1}, ϕ_{1}) satisfies (2.2.1).

(v) If (f : X → S, G, Z) satisfies (2.1.4), then (ψ_{1}, ϕ_{1}) satisfies (2.2.6). (vi) If (f : X → S, G, Z) satisfies (2.1.1), then (ψ_{1}, ϕ_{1}) satisfies (2.2.5).

In addition we can choose the diagram such that (vii)(a) The n-pointed curve (X_{1}, σ_{1}, …, σ_{n}) over S_{1} is stable. Or we can choose (ψ_{1}, ϕ_{1}) such that in addition to (i)-(vi) we have (vii)(b) The generic fibre (X_{1})_{η_{1}} of f_{1} is the normalization of an irreducible component

of X_{η} ⊗_{R(S)} R(S_{1}), i.e., the alteration (ψ_{1}, ϕ_{1}) satisfies (2.2.2).

2.5 Remark. The condition on Z_{1,η_{1}} implies that

Z_{1} = ψ_{1}^{−1}(Z) ⊂ σ_{1}(S_{1}) ∪ … ∪ σ_{n}(S_{1}) ∪ f_{1}^{−1}(D)

for some proper closed subset D ⊂ S_{1} invariant under G_{1}.

2.6 Procedure of the proof of Theorem 2.4. We are going to alter (2.2) repeatedly our family f : X → S, finally reaching a situation which satisfies (vii)(a) or (vii)(b). This is allowed, because a composition of alterations as in 2.2 is another such. Of course we have to check that the final alteration has all the properties required of it in Theorem 2.4, e.g. in case (vii)(b) we will have to make sure the alterations we use all have property (2.2.2). However, the first step in 2.9 will be to reduce to the case where X_{η} is geometrically irreducible over η and (X_{η}, Z_{η}) is smooth over η; after that we will have alterations satisfying the properties (2.2.1), (2.2.5), (2.2.6) and (2.2.7) (these imply (2.2.3) and (2.2.4)). In case (vii)(b) we also require (2.2.2). It is then easily verified that the end result is what we want (compare Remarks 2.3).

2.7 Let us produce a useful alteration of (f : X → S, G, Z) in the case that S = Spec k where k is a field. Choose a finite purely inseparable extension k ⊂ k′ as in Lemma 2.8 below. In the case that X is smooth and Z étale over k we take k′ = k. Put k′′ equal to the compositum of the fields g(k′), g ∈ G inside the perfect closure of k. (Note that G acts on the perfect closure of k.) Then k′′ works as well in Lemma 2.8 and G acts on k′′. Let X_{1} be the normalization of the reduction of X ⊗_{k} k′′ and let Z_{1} be the inverse image of Z in X_{1}. The group G acts on the smooth pair (X_{1}, Z_{1}) over k′′. Let X_{1} → Spec k_{1} → Spec k′′ be the Stein factorization of the morphism X_{1} → Spec k′′. The group G acts on k_{1} by uniqueness of Stein factorization. Put S_{1} = Spec k_{1}. The triple (X_{1} → S_{1}, G, Z_{1}) has properties (2.1.4) and (2.1.1).

Note that we trivially have (2.2.4) and (2.2.2) for the alteration (X_{1}, S_{1}) → (X, S) so obtained. In case the pair (X, Z) is smooth over k, we have that k ⊂ k_{1} is separable, hence (2.2.5).

In case X/G is geometrically irreducible over k^{G}, we see that the extension k^{G} ⊂ k_{1}^{G} is purely inseparable as X_{1}/G → X/G is purely inseparable. Let Y ↪ X/G ⊗_{k^{G}} (k′′)^{G} be the reduction of X/G ⊗_{k^{G}} (k′′)^{G}. The morphism α : X ⊗_{k} k′′ → X/G ⊗_{k^{G}} (k′′)^{G} is generically étale and an open part of X_{1} is equal to an open part of α^{−1}(Y). Thus

deg(X_{1} → Y) = deg(X ⊗_{k} k′′ → X/G ⊗_{k^{G}} (k′′)^{G})
= deg(X → X/G) · [k′′ : (k′′)^{G}]/[k : k^{G}]
= deg(X → X/G).

But deg(X_{1} → X_{1}/G) = deg(X → X/G) (we did not make the group bigger), hence we conclude that X_{1}/G → Y is birational. This proves the last assertion of (2.2.1).

Finally, assume that X is geometrically irreducible over k. In that case we have k_{1} = k′′, and the property (2.2.6) is clear for the alteration (X_{1}, S_{1}) → (X, S).

2.8 Lemma. Let X be a variety of dimension 1 over a field k and let Z ⊂ X be a proper closed subset. There exists a finite purely inseparable extension k ⊂ k′ such that the normalization X′ of the reduction of X ⊗_{k} k′ is smooth over k′ and the inverse image Z′ ⊂ X′ of Z is étale over k′. Any further extension k′ ⊂ k′′ has the same property.

Proof. Omitted.

2.9 (Reduction to the case where we have (2.1.1) and (2.1.4).) Let (f : X → S, G, Z) be as in 2.1. Let (Y → Spec K, G, Z_{Y}) be the alteration of (X_{η} → η, G, Z_{η}) constructed in 2.7. We set S′ equal to the normalization of S in K, and X′ equal to the normalization of X in R(Y). The universal property of normalization assures that we get a proper morphism f′ : X′ → S′ and that G acts on f′. In other words, we have an alteration (ψ, ϕ) : (X′, S′) → (X, S) as in 2.2. The generic fibre of X′ is Y and hence X′ → S′ satisfies (2.1.1) and (2.1.4).

As was shown in 2.7, the pair (ψ, ϕ) satisfies (2.2.2) and (2.2.4) and if f satisfies one of the conditions mentioned in Theorem 2.4, then (ψ, ϕ) satisfies the corresponding condition.

The morphisms ψ and ϕ are finite hence projective. As explained in 2.6, we reduce to the case that we have (2.1.1) and (2.1.4).

2.10 Assume (2.1.1) and (2.1.4). By Chow’s lemma there exists a projective modification X′ → X such that X′ is projective over S. By averaging over G, as explained in [1, 7.6], we may assume that G acts on X′. Note that X′_{η} ≅ X_{η} as X_{η} is smooth over η. Thus we may replace X by X′ and assume that we have (2.1.2).

By [6, page 37-38] there is a canonical minimal modification S′ → S which is projective, such that the strict transform X′ of X with respect to S′ → S is flat over S′. Thus G acts on X′ → S′ and we have X′_{η′} ≅ X_{η}. The alteration (X′, S′) → (X, S) satisfies all properties listed in 2.2. Thus we may replace (X, S) by (X′, S′) and assume that we have (2.1.1), (2.1.3), (2.1.2) and (2.1.4).

In Section 3 we prove case (vii)(a) of Theorem 2.4 under these assumptions. In Section 4 we deduce case (vii)(b) from case (vii)(a) (under the same assumptions).

3 Genus goes up

3.1 Throughout this section the triple (f : X → S, G, Z) satisfies (2.1.1), (2.1.3), (2.1.2) and (2.1.4). In this section we will prove Theorem 2.4 in this case.

3.2 Suppose that R(S) ⊂ L is a finite separable extension such that R(S)^{G} ⊂ L is Galois. Then we can put S′ equal to the normalization of S in L and X′ = (X ×_{S} S′)_{red}. For the group G′ we take the fibre product G′ = G ×_{Gal(R(S)/R(S)^{G})} Gal(L/R(S)^{G}) (compare with [1, 7.12]). It is easy to see that the resulting alteration (f′ : X′ → S′, G′, Z′) of (f : X → S, G, Z) satisfies all the conditions listed in 2.2 and that (f′ : X′ → S′, G′, Z′) satisfies (2.1.1), (2.1.3), (2.1.2) and (2.1.4) as well. In this way we may enlarge R(S).

3.3 These remarks prove case (vii)(b) of Theorem 2.4 if S = Spec k is the spectrum of a field, as we can make the points of Z ⊂ X rational after enlarging k as above. Case (vii)(a) follows by choosing a suitable covering of X, to make the pair (X, Z) stable, details left to the reader. Therefore, from now on we will assume that S is not the spectrum of a field, i.e., dim S > 0.

3.4 Let H = Ker(G → Aut S). As X is projective over S, the quotient Y = X/H exists and is an integral scheme, projective over S of relative dimension 1.

For any closed point s ∈ S we will construct an affine open neighborhood U ⊂ S and a finite morphism φ : X_{U} → ℙ_{U}^{1} with the following properties:

(i) φ is H-invariant, (ii) φ is generically étale, and (iii) for some G-invariant proper closed subset T ⊂ X_{η} containing Z_{η} and the ramifi-

cation points of X_{η} → ℙ_{η}^{1}, we have φ(T) ∩ {0, 1, ∞} = ∅. To do this we may assume S affine. Choose a relatively ample line bundle L on Y = X/H over S. Take n so large that (a) the map Γ(Y, L^{⊗n}) → Γ(Y_{s}, (L|_{Y_{s}})^{⊗n}) is surjective, (b) there exist s̄_{1}, s̄_{2} ∈ Γ(Y_{s}, (L|_{Y_{s}})^{⊗n}) which define a finite morphism Y_{s} → ℙ_{s}^{1}, and (c) L_{η}^{⊗n} is very ample on Y_{η}. If s_{1}, s_{2} ∈ Γ(Y, L^{⊗n}) are arbitrary lifts of s̄_{1}, s̄_{2}, then there exists an open neighborhood U = U(s_{1}, s_{2}) ⊂ S of s such that s_{1}, s_{2} define a finite morphism ξ : Y_{U} → ℙ_{U}^{1}.

Note that by (c) there exists a nonempty Zariski open subset V ⊂ Γ(Y_{η}, L_{η}^{⊗n})^{2} such that (s_{1,η}, s_{2,η}) ∈ V implies that ξ_{η} is generically étale. However, the set of lifts (s_{1}, s_{2}) of (s̄_{1}, s̄_{2}) form a Zariski dense subset of Γ(Y_{η}, L_{η}^{⊗n})^{2}, as is easy to show (use that s ≠ η as dim S > 0).

Let φ : X_{U} → ℙ_{U}^{1} be the resulting morphism X_{U} → Y_{U} → ℙ_{U}^{1}. We have (i) and (ii) but not yet (iii); we will get (iii) by changing coordinates on ℙ_{U}^{1}. Let T ⊂ X_{η} be a G-stable closed subset as in (iii). Consider the group G = PGL_{2}(Γ(U, O_{U})). We are looking for an element g ∈ G such that g(T) ∩ {0, 1, ∞} = ∅. Again this is a Zariski open condition in PGL_{2,η}. The residue field κ(η) is infinite as dim S > 0, and it follows that G ⊂ PGL_{2}(κ(η)) is Zariski dense. Thus we can find g ∈ G having the desired properties.

3.5 Since S is Noetherian we can choose a finite covering by open affines S = ⋃_{i=1}^{m} U_{i} such that there are finite morphisms φ_{i} : X_{U_{i}} → ℙ_{U_{i}}^{1} satisfying (i), (ii) and (iii) of Subsection 3.4. Further we may assume that on each U_{i} a prime number ℓ_{i} ≥ 5 is invertible.

Suppose we do a base change by S′ → S as in 3.2. The covering S = ⋃ U_{i} pulls back to a covering S′ = ⋃ U′_{i} and φ_{i} to a morphism φ′_{i} : X′_{U′_{i}} → ℙ_{U′_{i}}^{1} having properties (i), (ii) and (iii) stated in 3.4. Thus we may assume that R(S) contains a primitive ℓ_{i}th root of unity for each i. Also S is normal.

Let C_{i} → ℙ_{U_{i}}^{1} be the finite normal covering given by the function field extension R(ℙ_{U_{i}}^{1}) ⊂ L_{i} obtained by adjoining the ℓ_{i}th roots of x and x − 1 to R(ℙ_{U_{i}}^{1}). By our assumption that ζ_{i}, ℓ_{i} ∈ Γ(U_{i}, O_{U_{i}}^{∗}) we have that C_{i} → ℙ_{U_{i}}^{1} is a (ramified) Galois covering with group (ℤ/ℓ_{i}ℤ)^{2}. More to the point, the morphism C_{i} → U_{i} is smooth, projective and all its geometric fibres are nonsingular curves of genus ≥ 2.

Let N ⊂ R(X)^{∗} be the multiplicative subgroup generated by the following elements

f_{i,g} = g(φ_{i}^{∗}(x)) and f′_{i,g} = g(φ_{i}^{∗}(x − 1)), i = 1, …, m, g ∈ G.

By our choice of φ_{i} the zero’s and poles of these functions f_{i,g}, f′_{i,g} on the curve X_{η} are simple and occur in points p ∈ X_{η} \ Z_{η} with κ(η) ⊂ κ(p) separable. Set ℓ equal to the product of the distinct prime numbers that occur among the ℓ_{i}. Let R(X) ⊂ L be the finite Galois extension obtained by adjoining the ℓth roots of the elements of N, or equivalently the ℓth roots of the f_{i,g}, f′_{i,g}. In fact the extension R(X)^{G} ⊂ L is Galois as N ⊂ R(X)^{∗} is G-stable.

Suppose the algebraic closure K of R(S) in L strictly contains R(S). Let us do a base change S′ → S as in 3.2 with R(S′) = K. This is allowed as R(S)^{G} ⊂ K is Galois. We replace φ_{i} by its pullback φ′_{i} as above. Repeating our construction of N and L in the new situation, we see that we have reached a situation where R(S) is algebraically closed in L.

At this point we let X′ be the normalization of the scheme X in L. We put

G′ = G ×_{Gal(R(X)/R(X)^{G})} Gal(L/R(X)^{G})

and we see that G′ surjects onto G and G′ acts on X′. The curve X′_{η} is smooth over η as the morphism X′_{η} → X_{η} is generically étale and has only tame ramification over points p ∈ X_{η} \ Z_{η} with κ(η) ⊂ κ(p) separable. It follows that X′_{η} is geometrically irreducible over η as κ(η) = R(S) is algebraically closed in R(X′_{η}) = L. In addition Z′_{η} is étale over η as Z′_{η} → Z_{η} is étale.

Any element h ∈ H fixes all the elements f_{i,g}, f′_{i,g}, hence its action on R(X) can be lifted uniquely to an element of Gal(L/R(X)^{G}) fixing all the ℓth roots of the functions f_{i,g}, f′_{i,g}. This gives an element h′ ∈ G′ which lies in Ker(G′ → Aut(S)) and which maps to h. (The author remarks that it actually wasn’t necessary to have the elements f_{i,g}, f′_{i,g} to be H-invariant.) Finally, it is trivial to see that R(X′)^{G′} = R(X)^{G}.

Thus we have shown that (f′ : X′ → S, G′, Z′) satisfies (2.1.1), (2.1.2), (2.1.3) and (2.1.4), and that (X′ → X, S → S) satisfies (2.2.1), (2.2.5), (2.2.6) and (2.2.7). In other words we have produced an alteration which is allowed.

We claim that (f′ : X′ → S, G′, Z′) satisfies (2.1.6). This we get from the fact that the homomorphism φ_{i}^{∗} : R(ℙ_{U_{i}}^{1}) → R(X) extends to a homomorphism L_{i} → L by construction. By the universal property of normalization we get a morphism X′_{U_{i}} → C_{i}. This morphism is finite as X_{U_{i}} → ℙ_{U_{i}}^{1} is finite. We conclude that any irreducible component C of any geometric fibre X′ ; s̄ of X′ over S has a finite morphism to the curve C_{i,s̄} if s̄ ∈ U_{i}. Thus g(C) ≥ g(C_{i,s̄}) ≥ 2.

3.6 We assume (2.1.1), (2.1.3), (2.1.2), (2.1.4) and (2.1.6). Note that a base change S′ → S as in 3.2 preserves the condition (2.1.6) as well. Hence we may assume, after extending R(S), that Z_{η} = {p_{1}, …, p_{n}} with κ(p_{i}) = κ(η). Remark that (X_{η}, p_{1}, …, p_{n}) is a smooth stable n-pointed curve over η in view of (2.1.6).

3.7 Lemma. Let S be an excellent scheme with generic point η, let (X_{η}, p_{1}, …, p_{n}) be a smooth stable n-pointed curve over η. There exists a generically étale alteration S′ → S such that (X_{η}, p_{1}, …, p_{n}) ⊗ κ(η′) extends to a stable n-pointed curve over S′.

Proof. Compare [1, 4.17] and [1, 5.13]. (In order to get S′ → S generically étale use only level structures prime to the characteristic of R(S).)

3.8 We apply the lemma to our stable n-pointed curve (X_{η}, p_{1}, …, p_{n}). After replacing S by a finite covering as in 3.2 we may assume that (X_{η}, p_{1}, …, p_{n}) extends to a stable n-pointed curve on a modification of S. We can dominate this modification by a normal modification S′ on which G acts [1, 7.6]. Replace X by X′ = (X ×_{S} S′)_{red} and S by S′; this is an allowed alteration. In this way we reach a situation where in addition to (2.1.1), (2.1.3), (2.1.2), (2.1.4) and (2.1.6) we have that S is normal and the following condition:

(*) There exists a stable n-pointed curve (C, σ_{1}, …, σ_{n}) over S with (X_{η}, Z_{η}) ≅

(C_{η}, σ_{1}(η), …, σ_{n}(η)).

We remark that the group G acts on (C, σ_{1}, …, σ_{n}) over S. This is due to the fact that I = Isom(g^{∗}(C, {σ_{1}, …, σ_{n}}), (C, {σ_{1}, …, σ_{n}})) is finite unramified over S, the scheme S is normal and we have an element of I(η) given by the action of g on (X_{η}, Z_{η}). Here (C, {σ_{1}, …, σ_{n}}) denotes the curve with an non-ordered set of sections.

3.9 Let us look at the scheme

M = Mor_{S}(C, X).

It is a disjoint union of schemes quasi-projective over S, see [5, page 22]. We have a point m_{η} of M(η) given by our isomorphism in (*). We define S′ as the closure of m_{η} in M. Thus S′ is an integral scheme, S′ → S is birational and quasi-projective.

3.10 Claim. The morphism S′ → S is proper.

3.11 If we prove the claim then we are done with the proof of (vii)(a) of Theorem 2.4. Indeed, the group G acts on C and X over its action on S, and hence induces an action of G on M. The map in (*) is equivariant for these actions, hence we see that G acts on S′. Since S′ is quasi-projective and proper over S, it is projective over S. The morphism C_{S′} → X obtained from the map S′ → M is birational, G-equivariant and projective as C_{S′} is projective over S. Thus the alteration (C_{S′} → X, S′ → S) satisfies all the conditions listed in 2.2.

3.12 (Proof of Claim 3.10.) We prove this by checking the valuative criterion of properness; we need only consider morphisms T → S, where T is a trait and the generic point ξ of T maps to η ∈ S. (So ξ lifts to S′ as S′ → S is birational.) Thus we have to prove the lemma below.

3.13 Lemma. Let S be a trait, C → S a semi-stable curve with smooth generic fibre and X → S a morphism as in 2.1 that satisfies (2.1.3) and (2.1.6). Any morphism C_{η} → X_{η} over η extends to a morphism C → X.

Proof. The case that C_{η} → X_{η} is constant is left to the reader. If not, then R(X_{η}) ⊂ R(C_{η}) is finite. We replace X by the normalization of X in R(C_{η}); the conditions (2.1.3) and (2.1.6) still hold. Thus we may assume that C_{η} → X_{η} is an isomorphism.

There exists a blow up C̃ → C of the normal surface C such that the birational morphism C_{η} → X_{η} extends to a morphism C̃ → X. As the surface C has only rational singularities (loc. eq. xy − π^{n} = 0) the exceptional curves of C̃ → C are all genus zero curves. Thus they are contracted to points in X in view of (2.1.6). It follows that the morphism C̃ → X factors as C̃ → C → X.

4 Equal genus

4.1 Assume the triple (f : X → S, G, Z) satisfies (2.1.1), (2.1.3), (2.1.2) and (2.1.4). By Section 3 there exists an alteration (f′ : X′ → S′, G′, Z′) of (f : X → S, G, Z) that solves case (vii)(a) of Theorem 2.4:

\begin{matrix}X′ & ⟶^{ψ} & X \\ ↓^{f′} & & ↓^{f} \\ S′ & ⟶^{ϕ} & S. \end{matrix}

Thus (ψ, ϕ) satisfies (2.2.1), (2.2.5), (2.2.4) (2.2.6) and (2.2.7).

Let N ⊂ G′ be the kernel of the mapping G′ → G×Aut S′. By (2.2.4) we have that the morphism X′/N → X ×_{S} S′ is birational. The proposition below shows that X_{1} = X′/N is a semistable curve over S′ = S_{1}. The sections σ_{i} give a number of disjoint sections (one for each N-orbit on {σ_{1}, …, σ_{n}}), mapping into the smooth locus by Proposition 4.2. The alteration (X_{1} → S_{1}, G′/N, Z/N) → (f : X → S, G, Z) satisfies (2.2.1), (2.2.5), (2.2.4) and (2.2.7) as follows from the corresponding properties of (ψ, ϕ). The property (2.2.2) follows from the construction. We are done with the proof of Theorem 2.4 once we have shown Proposition 4.2.

4.2 Proposition. Let C → S be a projective semi-stable curve over an excellent integral normal scheme S. Assume G ⊂ Aut_{S}(C) is a finite group of automorphisms of C over S. The quotient C/G exists and is a semi-stable curve over S. The map C → C/G is finite and maps sm(C/S) into the smooth locus of C/G → S.

Proof. The quotient exists as C is projective over S: any point of C is contained in a G-stable open affine U of C. The quotient of U by G is the spectrum of the invariants for G in Γ(U, O_{U}). The construction of the quotient commutes with flat base change. (For these assertions one can consult [5].) Thus to study the local structure of the quotient, we may assume that S = Spec R, where R is a normal Noetherian complete local domain with algebraically closed residue field. We want to show that C/G has a local description as in [1, 2.23]. Let x ∈ C_{s} be a closed point of the special fibre mapping to y ∈ C/G. Let H ⊂ G be the stabilizer of x. Then (O_{C,x})^{H} = O_{C/G,y} and (Ô_{C,x})^{H} = Ô_{C/G,y}. Thus the local description and the last assertion of the proposition follow from the lemma below.

4.3 Lemma. Let (R, 𝔪, k) be a Noetherian complete local ring which is a normal domain. Let A be an R-algebra isomorphic to either (a) R[[u]] or (b) R[[u, v]]/(uv − f) for some f ∈ 𝔪. Assume given some finite group H contained in the automorphism group of A over R. In case (a) we have A^{H} ≅ R[[x]] and in case (b) we have either A^{H} ≅ R[[x]] or A ≅ R[[x, y]]/(xy − g) for some g ∈ 𝔪 (in this case g can be taken to be a power of f).

Proof. We fix some notations: We set d = #H and Nm : A → A^{H} is the map a ↦ ∏_{σ∈H} σ(a). The symbol Φ(B) will denote the fraction field of the domain B.

Note that A^{H} is a normal complete local domain with maximal ideal A^{H} ∩ 𝔪_{A} and residue field k.

Case I: A ≅ R[[u]]. Put x = Nm(u) ∈ A^{H}. It is immediate that x ≡ u^{d} · (unit) modulo 𝔪A. Hence R[[x]] → A is a finite injective ring homomorphism of degree ≤ d. It follows that A^{H} → A and R[[x]] → A^{H} are finite. We have Φ(R[[x]]) ⊂ Φ(A^{H}) ⊂ Φ(A). By Galois theory [Φ(A) : Φ(A^{H})] = d, hence Φ(R[[x]]) = Φ(A)^{H} = Φ(A^{H}). But R[[x]] is normal, hence integrally closed in Φ(R[[x]]), hence R[[x]] = A^{H}.

Case II: A ≅ R[[u, v]]/(uv − f) for some f ∈ 𝔪. We get an action of H on the set of components of Spec A/𝔪A = Spec k[[u, v]]/(uv). Let H_{0} be the subgroup preserving the components. The index [H : H_{0}] ≤ 2. Below we will do the case H = H_{0}; this reduces us to the case that H_{0} = {1} and H = {1, τ}. In this case it is easy to see that A ≅ R[[u, v]]/(uv − f) with v = τ(u) and A^{H} = R[[x]] with x = u + τ(u).

Case IIa: Here f = 0 and H = H_{0}. This case follows from Case I as A^{H} is the fibre product

(A/uA)^{H} ×_{R} (A/vA)^{H}

and the morphisms (A/uA)^{H} → R and (A/vA)^{H} → R are surjective.

Case IIb: Here H = H_{0} and f ∈ 𝔪 is not zero. Again let x = Nm(u) and y = Nm(v). As H = H_{0} we have that x ≡ u^{d} · (unit) and y ≡ v^{d} · (unit) modulo 𝔪A. We obviously have xy = f^{d}. Consider the finite homomorphism of R-algebras

B = R[[x, y]]/(xy − f^{d}) ⟶ A = R[[u, v]]/(uv − f).

As f ≠ 0 both A and B are normal domains. Thus B → A is injective in view of the fact that dim A = dim B = dim R + 1. We want to compute [Φ(A) : Φ(B)]. For any prime ideal 𝔭 ⊂ B we have

[Φ(A) : Φ(B)] ≤ dim_{κ(𝔭)} A ⊗_{B} κ(𝔭).

We take 𝔭 = 𝔪B + yB, and we have B/𝔭 = k[[x]] and κ(𝔭) = k((x)). We compute

A/𝔭A = k[[u, v]]/(uv, y) = k[[u, v]]/(uv, v^{d}),

as y ≡ v^{d} · (unit) modulo 𝔪A. Thus we get A ⊗_{B} κ(𝔭) ≅ k((u)), as a κ(𝔭) = k((x)) module. We have [k((u)) : k((x))] = d as x ≡ u^{d} · (unit) modulo 𝔪A. Thus we see that [Φ(A) : Φ(B)] ≤ d. Since B ⊂ A^{H} we get equality of degrees by Galois theory. We get B = A^{H} from the fact that B is normal.

4.4 Remark. The idea of the proof of the proposition above goes back to [3, pp. 508-510]. In the case that S has dimension 1 a proof of the proposition was given by Bas Edixhoven; our proof above closely follows his.

5 Applications

Here is an immediate application.

5.1 Corollary. Let S be an excellent scheme of dimension 1. In this situation [1, Theorem 8.2] holds, i.e., any S-variety X can be altered into a semi-stable variety over a finite extension S_{1} of S.

Proof. This is trivial from Theorem 2.4 and the results of [1]. Indeed, the only problem in the proof of [1, 8.2] was to present X (after an alteration) as a semi-stable family of curves over an S-variety whose dimension is 1 less. However, this is immediate from Theorem 2.4.

5.2 Lemma. Let f : X → S be a proper morphism of excellent integral schemes, and let G be a finite group acting on f. Assume that the generic fibre X_{η} of f is geometrically reduced and irreducible and has dimension d ≥ 2. There exists a commutative diagram on which G acts

\begin{matrix}X′ & ⟶^{ψ} & X \\ ↓^{g} & & ↓^{f} \\ Y & ⟶^{h} & S, \end{matrix}

where ψ is a projective modification, g and h are dominant projective morphisms of integral schemes having geometrically reduced and irreducible generic fibres and such that g has a generic fibre of dimension 1.

Proof. The conditions on f imply that the field extension R(S) ⊂ R(X) is separable. Choose x_{1}, …, x_{d−1} ∈ R(X)^{G} which are algebraically independent over R(S) and such that R(X) is separable over R(S)(x_{1}, …, x_{d−1}). This defines a G-equivariant rational map X...→ℙ_{S}^{d−1}. Therefore there is a projective modification X′ → X on which G acts such that there is a G-equivariant morphism X′ → ℙ_{S}^{d−1} realizing r. We may assume that X′ is normal.

Let X′ → Y → ℙ_{S}^{d−1} be the Stein factorization. Note that G acts on Y and that X′ → Y and Y → S are G-equivariant. By construction the field extension R(Y) ⊂ R(X′) = R(X) is separable and R(Y) is algebraically closed in R(X′). Hence the generic fibre of X′ → Y is geometrically reduced and irreducible. In the same way one sees that Y → S has geometrically reduced and irreducible generic fibre.

5.3 Situation. Here S is an integral Noetherian scheme, and G is a finite group acting on S. A Galois alteration of (S, G) is a system (π : S′ → S, G′ → G, G′ × S′ → S′), where (a) π : S′ → S is an alteration, (b) G′ → G is a surjection of finite groups, (c) G′ × S′ → S′ is an action of G′ on S′ such that π becomes G′ equivariant, and (d) we have that the extension R(S)^{G} ⊂ R(S′)^{G′} is purely inseparable. Condition (d) is equivalent to the following: if H = Ker(G′ → Aut(S)) then R(S) ⊂ R(S′)^{H} is purely inseparable.

5.4 Suppose we have a commutative diagram of integral schemes

\begin{matrix}X′ & ⟶ & X \\ ↓ & & ↓ \\ S′ & ⟶ & S. \end{matrix}

Suppose that G′ → G is a surjection of finite groups and that these act on the various schemes above. Assume that (X′, G′) → (X, G) and (S′, G′) → (S, G) are Galois alterations. We consider the following condition on this diagram: (5.4.1) Let H = Ker(G → Aut(S)) and let H′ = Ker(G′ → Aut(S′)). Then H′ → H is

surjective. Suppose that X → S has a geometrically irreducible generic fibre. In this case (5.4.1) implies that the map X′ → (X ×_{S} S′)^{∼} is a Galois alteration (both with group G′). The notation (X ×_{S} S′)^{∼} refers to the irreducible component of X ×_{S} S′ dominating S′.

5.5 Lemma. Let f : X → S be a proper morphism of integral excellent schemes on which the group G acts. Assume that the generic fibre of f is geometrically irreducible. Suppose that (X′, G′) is an Galois alteration of X. There exists a diagram on which G′ acts:

X′_{1} ⟶ X_{1} ⟶ S_{1}
↓ ↓ ↓
X′ ⟶ X ⟶ S

such that the generic fibres of X′_{1} → S_{1} and X_{1} → S_{1} are geometrically irreducible and reduced, such that (S_{1}, G′) → (S, G) is a Galois alteration, such that X_{1} is the strict transform of X with respect to S_{1} → S, and such that the field extension R(X′) ⊂ R(X′_{1}) is purely inseparable. Finally, X′_{1} → X_{1} is a Galois alteration as well and the diagram 5.4 composed of X′_{1}, X, S_{1} and S satisfies (5.4.1).

Proof. Let S′ be the normalization of S in a purely inseparable field extension of R(S) on which the group G acts such that the generic fibres of (X ×_{S} S′)^{∼} and (X′ ×_{S} S′)^{∼} over S′ are geometrically reduced. Here (X′×_{S} S′)^{∼} denotes the irreducible component of X′×_{S} S′ dominating S′; this will be our X′_{1}. We may replace S by S′ and assume that X′_{η} and X_{η} are geometrically reduced. We may also assume that X′ is normal. Let X′ → S′ → S be the Stein factorization of X′ → S. Note that G′ acts on S′ and that (S′, G′) → (S, G) is a Galois alteration. Indeed, if H = Ker(G′ → G), then by assumption R(X) ⊂ R(X′)^{H} is purely inseparable, hence R(S) ⊂ R(S′)^{H} is purely inseparable. Now X′_{η′} is normal, geometrically reduced and geometrically connected, hence it is geometrically irreducible. We take S_{1} := S′ and X_{1} := (X ×_{S} S′)^{∼}. The verification of the last statement of the lemma is an exercise in Galois theory that is left to the reader.

5.6 Lemma. Let (S, G) be as in Situation 5.3, with S excellent. Let {s_{1}, …, s_{n}} ⊂ S be a finite set of points of S, and suppose we are given a finite separable field extension κ(s_{i}) ⊂ k_{i} for each i. There exists an alteration (π : S′ → S, G′ → G, G′ × S′ → S′) as in 5.3 with π finite and generically étale, and such that π^{−1}(s_{i}) = {s_{ij}} ⊂ S′ where all the field extensions κ(s_{i}) ⊂ κ(s_{ij}) contain k_{i}.

Proof. Clear.

We say that a semi-stable curve X over a field k is quasi-split if (a) its singular points are rational over k, and (b) in each of its singular points the two tangents are rational over k. This means that O_{X,x}^{∧} ≅ k[[uv]]/(uv) for any singular point x ∈ X. Another formulation: the inverse image under the normalization mapping X^{n} → X of the set of singular points of X consists of k-rational points of X^{n}. We say a semi-stable curve X → S over a scheme S is quasi-split if for all s ∈ S the fibre X_{s} is a quasi-split semi-stable curve over κ(s). Compare with [1, 2.22 and 2.23] where split semi-stable curves are defined; a split semi-stable curve is quasi-split and a quasi-split semi-stable curve is split only if all its fibres have smooth components.

5.7 Lemma. Let f : X → S be a semi-stable curve over an excellent integral scheme S of finite dimension. Assume the finite group G acts on f. There exists a Galois alteration (S′, G′) of (S, G) which is finite and generically étale such that the pullback X′ = X ×_{S} S′ is a quasi-split semi-stable curve over S′.

Proof. Let T ⊂ S be an irreducible and reduced closed subscheme, with generic point ξ and assume that X_{ξ} is quasi-split. We claim that there exists a nonempty open subscheme U ⊂ T such that X|_{U} is quasi-split over U. By [2, Section 3] the curve X|_{V} has locally constant topological type over V for some nonempty open V ⊂ T. Over V we can find a morphism X_{V}^{n} → X|_{V} such that X_{v}^{n} → X_{v} is the normalization mapping of X_{v} for all v ∈ V. Let Y ⊂ X_{V}^{n} be the inverse image of Sing(f)|_{V}. Then Y → V is finite étale and X_{v} is quasi-split if and only if all points of Y_{v} are κ(v)-rational (see above). But we assumed that X_{ξ} is quasi-split, so all points of Y_{ξ} are rational, hence the covering Y → V is trivial over a nonempty open subscheme U ⊂ V. This proves the claim.

Let Z ⊂ S denote the Zariski closure of the set of points s where X_{s} is not quasi-split. By the claim the curve X_{ξ} is not quasi-split for any maximal point ξ of Z. Let {s_{i}} be the set of all these maximal points and let κ(s_{i}) ⊂ k_{i} be finite separable extensions such that X_{s_{i}} ⊗ k_{i} is quasi-split. Apply Lemma 5.6 to this situation: we get π : S′ → S. If Z′ ⊂ S′ denotes the Zariski closure of the set of points of S′ where X′ is not quasi-split, then we see that π(Z′) ⊂ Z is nowhere dense. Since π is finite we conclude dim Z′ < dim Z. We win by induction on this dimension (which is finite as dim(S) < ∞).

5.8 Situation. Let (S, G) be a pair as in 5.3. A G-pluri nodal fibration of relative dimension d over S is given by a system:

(X_{d}⟶^{f_{d}}X_{d−1}⟶ … ⟶X_{1}⟶^{f_{1}}X_{0} = S, {σ_{ij}}, Z_{0}),

where (i) f_{i} : X_{i} → X_{i−1} is a projective and quasi-split semi-stable curve over X_{i−1}, (ii) Z_{0} ⊂ S = X_{0} is a G-stable proper closed subset of S, and (iii) σ_{ij} : X_{i−1} → X_{i}, j = 1, …, n_{i} are disjoint sections of f_{i} into the smooth locus of

X_{i} over X_{i−1}. Put

Z_{i} = ⋃_{j=1}^{n_{i}} σ_{ij}(X_{i−1}) ∪ f_{i}^{−1}(Z_{i−1}).

The data are subject to the condition that X_{i+1} → X_{i} is smooth over the locus X_{i} \ Z_{i}. Furthermore G acts on all X_{i} such that all the morphisms f_{i} are G-equivariant, and ∀i, j, g, σ_{ij} ∘ g = g ∘ σ_{ij′} for some j′ = j′(g, i).

5.9 Theorem. Let f : X → S be a proper morphism of integral excellent schemes of finite dimension. Assume that a finite group G acts on f and that a G-stable proper closed subset Z ⊂ X is given. Suppose the generic fibre X_{η} of f is geometrically irreducible of dimension d ≥ 1. In this situation there exist:

(i) A projective Galois alteration (5.3) (S_{1}, G_{1}) of (S, G).

(ii) A G_{1}-pluri nodal fibration (X_{d} → … → X_{1} → S_{1}, {σ_{ij}}, Z_{0}) over S_{1}.

(iii) A G_{1}-equivariant alteration ψ_{1} : X_{d} → X making the following diagram commu-

tative

X_{d} ⟶^{ψ_{1}} X
↓^{f_{1}∘...∘f_{d}} ↓^{f}
S_{1} ⟶ S.

These will satisfy the following conditions:

(a) The map (X_{d}, G_{1}) → (X, G) is a Galois alteration and the diagram above satisfies (5.4.1). In particular, if N = Ker(G_{1} → Aut(S_{1} ×_{S} X)), then the morphism X_{d}/N → (X ×_{S} S_{1})^{∼} induces a purely inseparable extension of functions fields.

(b) We have ψ_{1}^{−1}(Z) ⊂ Z_{d}. (See Subsection 5.8 for the definition of Z_{d} ⊂ X_{d}.)

Proof. In the case dim X_{η} = 1 this follows from Theorem 2.4 combined with Lemma 5.7 to ”split” the curve. In the general case we argue by induction on the dimension of X_{η}. First we replace S by the normalization of S in a purely inseparable extension of its function field on which G acts such that X_{η} becomes geometrically reduced. Choose X′ → Y → S as in Lemma 5.2. Then by Theorem 2.4 we find an alteration (2.2) with group G′:

X′′ ⟶^{ψ} X′
↓^{g′} ↓^{g}
Y′ ⟶ Y,

having the properties (2.2.1)-(2.2.7) except possibly (2.2.5); in particular Y′ → Y is a Galois alteration. There are sections σ_{j} of g′ adapted to the inverse image Z′ ⊂ X′ of Z ⊂ X, i.e., ψ^{−1}(Z′) ⊂ ⋃_{j} σ_{j}(Y′) ∪ (g′)^{−1}(Z′′) for some proper closed subset Z′′ ⊂ Y′. We enlarge Z′′ such that it contains the locus over which X′′ → Y′ is not smooth. By Lemma 5.7 we may in addition assume that g′ is a quasi-split semi-stable curve (this does not destroy any of the properties). We apply Lemma 5.5 to Y′ → Y → S and we get Y′_{1} → Y_{1} → S′ with an action of G′.

By induction applied to the pair Z′′_{1} ⊂ Y′_{1} over S′ with group G′ we get a diagram

X_{d−1} ⟶ Y′_{1}
↓ ↓
S_{1} ⟶ S′.

We leave it to the reader to show that by taking X_{d} = X′′ ×_{Y′} X_{d−1}, we get a G_{1}-pluri nodal fibration with all the properties as stated in the theorem. For example (a) can be seen to be true by noting that we have (a) for all the rectangles in the following diagram

X_{d} = X_{d−1} ×_{Y′} X′′ ⟶  ⟶ X′′ ⟶ X′ ⟶ X
↓ ↓ ↓ ↓
X_{d−1}  ⟶ Y′_{1} ⟶ Y′ ⟶ Y
↓  ↓  ↓  ↓
S_{1}  ⟶ S′ ⟶  ⟶ S = S.

5.10 Corollary. Let f : X → S be a dominant and proper morphism of integral excellent schemes. There exists a diagram

X_{d} ⟶ X_{d−1} ⟶ … ⟶ X_{1} ⟶ S_{1}
↓^{ψ}  ↓^{ϕ}
X  ⟶  S

where ψ and ϕ are alterations and f_{d} : X_{d} → S_{1} is a composition of projective semi-stable curve fibrations X_{i} → X_{i−1}. In addition we may assume a finite group N acts on this diagram, acting trivially on X, S and S_{1}, such that R(X_{d}/N) is a purely inseparable extension of the function field of the component of X ×_{S} S_{1} it maps onto.

Proof. First take a finite alteration S′ → S such that some irreducible component X′ of X ×_{S} S′ dominating S′ is geometrically irreducible over S′. Next, apply Theorem 5.9 to f′ : X′ → S′. (Note that the condition of finite dimensionality was used in the proof of Lemma 5.7 only. However, since we do not require the semi-stable curves to be quasi-split here, we do not need it. In fact X_{d} → S_{1} will have all the properties of an N-pluri nodal fibration over S_{1} except for being ”quasi-split”.)

5.11 Proposition. Let (S, G) be as in 5.3 with S regular, and let D ⊂ S be a G-strict normal crossings divisor in S (see [1, 7.1]). Suppose that f : X → S is a split semi-stable curve, smooth over S \ D. Furthermore, the action of G on S lifts to an action of G to X. There exists a G-equivariant projective modification X_{1} → X with the following properties:

(i) The scheme X_{1} is regular and the center of X_{1} → X is contained in Sing(X). The inverse image of D in X_{1} is a normal crossings divisor.

(ii) Let σ_{1}, …, σ_{n} be disjoint sections into the smooth locus of X → S which are permuted by the action of G. Then these lift to sections σ_{i} of f_{1} : X_{1} → S, and the divisor D_{1} := ⋃_{i} σ_{i}(S) ∪ f_{1}^{−1}(D) is a G-invariant normal crossings divisor of X_{1}. There is a canonical blow up b : X_{2} → X_{1} such that X_{2} is regular and D_{2} = b^{−1}(D_{1}) is a G-strict normal crossings divisor in X_{2}.

Proof. Consider the blow ups that occur in the proof of [1, Lemma 3.2]. Let T ⊂ X be as in [1, 3.4]. Then T ⟶^{≅}D_{1} as f is quasi-split. As D is G-strict, we have T′ = ⋃_{g∈G} g(T) is a disjoint union of irreducible closed subschemes of the type described in [1, 3.4]. Hence the blow up in the closed subscheme T′ can be described locally as in [1, 3.4]. Thus we may assume that X → S is quasi-split semi-stable and codim(Sing(X), X) ≥ 3.

The rest of our argument is similar to the arguments of [1, 4.24-4.28] and [1, 7.16]. The types of complete local rings that we have now are

A[[u, v]]/(uv − t_{1} · . . . · t_{s}),

where A is a regular complete local ring with a regular system of parameters t_{1}, …, t_{d−1}, and D is given by t_{1} · . . . · t_{r} = 0, for some 1 ≤ s ≤ r ≤ d − 1.

We give the argument that proves that the blow ups that occur in the proof of [1, 4.26 and 4.27] can be made G-equivariantly. Indeed, by the remarks at the end of [1, 3.5] we have Sing(X) = ∪E_{α}, with E_{α} regular and mapping finite étale onto an irreducible component of some D_{i} ∩ D_{j}. However, since f is quasi-split and E_{α} ⊂ Sing(f) this morphism induces trivial residue field extensions, hence E_{α} maps isomorphically onto a component of D_{i}∩D_{j}. Furthermore, if E_{α} and E_{β} have the same image in S, then E_{α} ∩ E_{β} = ∅. Thus, as D is G-strict, we have that ⋃_{g∈G} g(E_{α}) is a disjoint union of components E_{β}. Thus the last sentence of [1, 4.25] should be replaced with: Any G-orbit of a component of the singular locus of X is nonsingular.

Thus we blow up in orbits of components of the singular locus of X. The resulting scheme has a local description as above by the computations of [1, 4.27]. Hence we arrive at a regular scheme X′ with an action of G and a G-stable normal crossings divisor D′ ⊂ X′.

If we are given sections σ_{i} as in (ii) then these lift to sections σ′_{i} into X_{1} = X′ (as we modified only in singular loci) and the divisor D_{1} = D′ ⋃ σ′_{i}(S) is a normal crossings divisor as well. To get D_{1} to be G-strict we apply the canonical blow up of [1, 7.2].

5.12 We consider the following condition on a pair (S, G) as in 5.3: (5.12.1) For every Galois alteration (S′, G′) of (S, G) and proper closed subset Z′ ⊂ S′,

there exists a Galois alteration (S_{1}, G_{1}) of (S′, G′), such that S_{1} is regular and such that the inverse image of Z′ in S_{1} is contained in a G_{1}-strict normal crossings divisor.

5.13 Theorem. Let S be an integral excellent scheme of finite dimension. Let X be an integral scheme and let f : X → S be a dominant morphism on which the finite group G acts. Assume (S, G) satisfies (5.12.1). If f is of finite type, is separated and has geometrically irreducible generic fibre then the pair (X, G) satisfies (5.12.1).

Proof. This is just a combination of the results obtained so far. First, we can assume that X is proper over S by the usual arguments: first make X quasi-projective by a G-equivariant Chow’s lemma (use Chow’s lemma and argue as in [1, 7.6]). To make X projective over S, take an embedding i : X → ℙ_{S}^{n} and let X̄ be the closure of i′ : X → (ℙ^{n} × … × ℙ^{n})_{S} where i′(x) = ∏_{g∈G} i(gx); G acts on X̄. If the theorem holds for (X̄, G), then it holds for X.

Let (X′, G′) be a Galois alteration of the pair (X, G) and let Z′ ⊂ X′ be a proper closed subset. By Lemma 5.5 we reduce to a situation where X = X′ → S has geometrically irreducible (and reduced) generic fibre.

By Theorem 5.9 we may assume that X → S is a G-pluri nodal fibration of relative dimension d, i.e., X = X_{d} → X_{d−1} → … → X_{1} → S and that Z = Z_{d}. Next, by (5.12.1) for S, we may assume that S is regular and that Z_{0} ⊂ S is a G-strict normal crossings divisor. Then by Proposition 5.11 applied to X_{1} → S we may assume that X is a split Gpluri nodal fibration of relative dimension d−1 over a regular scheme X′_{1} whose associated closed subset is a G-strict normal crossings divisor Z′_{1} ⊂ X′_{1}. Apply Proposition 5.11 to the split semi-stable family of curves over X′_{1}, etc.

5.14 Corollary. Any integral scheme separated, flat and of finite type over Spec ℤ has resolution of singularities up to quotient singularities.

5.15 Corollary. Any integral scheme X separated and of finite type over an excellent scheme S with dim S ≤ 2 satisfies (5.12.1) with G = {1}. In particular the singularities of X can be resolved up to quotient singularities and a purely inseparable extension of R(X).

Proof. We can find a finite morphism S′ → S of an integral scheme S′ to S, and a finite alteration X′ → X with purely inseparable function field extension such that X′ is a scheme over S′ with geometrically irreducible geometric fibre. Since dim S′ ≤ 2, we have canonical resolution of singularities for S′ [4] and any alteration of S′, and hence we get (5.12.1) for S′. By the theorem we get (5.12.1) for X′. This implies (5.12.1) for X as R(X) ⊂ R(X′) is purely inseparable.

5.16 Remark. A word about the hypothesis of excellency that is used throughout. Undoubtedly, in several places this hypothesis is too strong and one could get away with working over Noetherian (universally) Japanese base schemes. However, in some cases one can establish a result similar to the results in the text by the techniques of EGA IV §8. For example let us do this for Theorem 5.9. Suppose f : X → S is a proper dominant morphism of finite presentation of integral schemes with geometrically irreducible generic fibre, with S affine and with a finite group G acting on f. We have f = f_{0} × id_{S} for some G-equivariant morphism S → S_{0} and some proper dominant morphism f_{0} : X_{0} → S_{0} of integral schemes of finite type over Spec ℤ with geometrically irreducible generic fibre endowed with an action of G. We apply Theorem 5.9 to f_{0}. This gives a group G_{1} surjecting onto G, and G_{1}-equivariant maps X_{d,0} → S_{1,0} → S_{0} and X_{d,0} → X_{0}. Put S_{1} = S_{1,0} ×_{S_{0}} S and X_{d} = X_{d,0}×_{S_{0}} S. Then X_{d} → S_{1} is a G_{1}-pluri nodal fibration over S_{1}. The morphisms S_{1} → S and X_{d} → X are ”alterations” in the sense that they are of finite presentation, proper and finite flat over an open dense subscheme. If S is Noetherian, then we can replace S_{1} by the reduction of S_{1} and we get the result of Theorem 5.9 in this situation.

References

[1] A.J. de Jong, Smoothness, semi-stability and alterations, To appear in Publications mathématiques I.H.E.S.

[2] A.J. de Jong and F. Oort, On extending families of curves, To appear in Journal of Algebraic Geometry.

[3] N.M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press (1985).

[4] J. Lipman, Desingularization of two-dimensional schemes, Annals of Mathematics 107, 151-207 (1978).

[5] D. Mumford and J. Fogarty, Geometric invariant theory, Second Enlarged Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer Verlag (1982).

[6] M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Techniques de ≪platification≫ d’un module, Inventiones mathematicae 13, 1-89 (1971).

A.J. de Jong, Princeton University, Fine Hall – Washington Road, Princeton, New Jersey 08540-1000, USA.