On extending families of curves

A.J. de Jong* and F. Oort

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Extension after blowing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Stratification by topological type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. The key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6. An example (Raynaud-Ogus-Gabber) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1 Introduction

Let S be a scheme and let U ⊂ S be a dense open subscheme. This paper deals with the question whether a given family of stable curves over U extends to a family of stable curves over S. One can ask similar questions for families of (semi) abelian varieties, or any other type of moduli problem of algebraic varieties.

Let us assume that D = S \ U is a divisor with normal crossings and that C_{U} → U is a family of smooth stable curves. Our main result implies that C_{U} extends to a family of stable curves over S if and only if it extends as a stable curve to the generic points of D. The precise statement is Theorem 5.1; it works for families of stable curves whose topological type is locally constant over U. As a corollary of the theorem we prove a similar result for stable pointed curves, see 5.5.

A previous result in this direction was obtained by Moret-Bailly, who proved an extension theorem in the case that C_{U} is a family of smooth curves, and C_{U} extends to a smooth curve over the generic points of D, see [15].

A similar statement for families of abelian varieties is wrong. A counter example in characteristic p > 0 can be found in [4, page 192]. A counter example in mixed characteristic, due to Raynaud-Ogus-Gabber, is given in Section 6. It shows that [4, V Theorem 6.4] and [4, V Theorem 6.7] are not correct as they stand.

The technique for proving Theorem 5.1 is borrowed from [4, V Section 6]; by transitivity, the technique of the proof is borrowed from [15]. We first prove, in Section 2.1, a result on extension after blowup. Then we prove that this extension comes from downstairs by analyzing the resulting family on the exceptional divisor. Here the key point is Lemma 4.1 which says that a family of stable pointed curves over ℙ^{1}, with locally constant topological type over ℙ^{1} \ {0, ∞} is constant.

Acknowledgements. Both authors thank A. Ogus for generously communicating his ideas to us. Both authors thank the Department of Mathematics of Harvard University for hospitality while working on the paper. The authors thank the referee for the suggestions concerning the stratification by topological type.

* Partly supported by the Royal Netherlands Academy of Arts and Sciences.

2 Extension after blowing up

Let S be a quasi-compact, excellent, reduced, normal scheme. Suppose that a stable curve f : C_{U} → U over an open dense subscheme U ⊂ S is given. We want to see if we can extend C_{U} to a stable curve over a modification S̃ → S with center in S \ U.

Let R be a discrete valuation ring and φ a morphism Spec(R) → S with φ(η) ∈ U. Here η denotes the generic point of Spec(R). Pulling back C_{U} gives a stable curve C_{η} over η. Consider the following condition on the pair (R, φ):

The curve C_{η} extends to a stable curve over Spec(R).  (∗)

If S̃ → S is a modification with center in S \ U, then φ will lift to φ̃ : Spec(R) → S̃ by the valuative criterion of properness. Thus, if C_{U} extends to a stable curve CS̃ over S̃, we have that the curve C_{η} extends to a stable curve over Spec(R), i.e., that (∗) holds for (R, φ). We strongly suspect that the converse holds: If (∗) holds for any pair (R, φ) as above then C_{U} has a stable extension to some modification S̃ → S. An indication of this is that we can prove the result in case dim(S) = 2.

2.1 Theorem. Notations and assumptions as above. Assume that dim(S) = 2 and that (∗) holds whenever R is a valuation ring in the function field of S. There exists a normalized blowing up S̃ → S with center in S \ U such that C_{U} extends to a stable curve over S̃.

Proof. Let s be a point of codimension 1 in S. The ring R = O_{S,s} is a discrete valuation ring as S is normal, and we may apply the assumption of the theorem to R. By (∗) and the lemma 2.3, we can extend to an open subscheme V containing s. Hence we may assume that S \ U consists of closed points. Since S has dimension 2 the blowing up will have center in closed points of S. Hence we reduce to the case S local and U equal to the complement of the closed point. We use that there exists a proper surjective f : T → S such that the inverse image of C_{U} on f^{−1}(U) extends to a stable curve C_{T} over T. See [2, Lemma 1.6] which is formulated for algebraic spaces, actually the proof given there produces a scheme, see [2, Section 3]. We may assume that f is generically finite and that T is reduced and normal. There exists a blowing up S̃ → S such that the strict transform T′ of T is flat, hence finite over S̃. See [19, page 37]. In addition, we may assume that S̃ is regular, see [12]. Consider the normalization T̃ → T′ of T′. The morphism π : T̃ → S̃ is finite and is flat by [13, Theorem 81], here we use the assumption that the dimension of S is 2. Pulling back C_{T} gives a stable curve CT̃ over T̃.

Arguing as above we see that C_{U} extends to a stable curve C_{V} over an open subscheme V ⊂ S̃ containing the generic points of the special fibre of S̃ → S. Furthermore, the curves π^{∗}(C_{V}) and CT̃ are isomorphic over π^{−1}(V): they agree on the dense open subscheme π^{−1}(U) and π^{−1}(V) is normal. Consider the fibre product T̃(2) := T̃ ×S̃ T̃. Over this we have two stable curves C_{1}, C_{2} and an isomorphism between them over the inverse image of V. We note that this inverse image contains all the points of T̃(2) which have depth ≤ 1, as the morphism T̃(2) → S̃ is flat. This implies that there is an isomorphism ϵ : C_{1} → C_{2} by the lemma below. The isomorphism ϵ satisfies the cocycle condition on T̃ ×S̃ T̃ ×S̃ T̃, since this is true on the dense open part lying over V. We conclude that CT̃ descends to the sought after stable curve over S̃.

2.2 Remark. The result of the theorem also holds, mutatis mutandis, for families of stable pointed curves. In fact the result holds for any moduli problem defined by an algebraic stack proper over Spec(ℤ). Finally, the theorem is true for families of semi-abelian varieties. One can paraphrase the proof above, using Gabber’s lemma [2, Proposition 4.10].

2.3 Lemma. Let S be a locally Noetherian scheme, U ⊂ S open dense and C_{U} → U a stable curve over U.

(1) Suppose V ⊂ S is an open subscheme containing U such that for all s ∈ V, s ∉ U we have either O_{S,s} has depth ≥ 2 or has depth 1 and is geometrically unibranch. There is at most one extension of C_{U} to a family of stable curves over V.

(2) Let s ∈ S and consider Spec(O_{S,s}) → S. If C_{U} extends to a stable curve over Spec(O_{S,s}), then it extends to a stable curve over an open subscheme V ⊂ S containing s and U.

Proof. (1) Assume we have two extensions C_{1}, C_{2} over V. We consider the scheme Isom(C_{1}, C_{2}); it is finite unramified over V [3, Theorem 1.11] and it has a section over U. Define V′ to be the scheme theoretic closure of this section in Isom(C_{1}, C_{2}). For a point s ∈ V put O(s) = Spec(O_{S,s}) and O(s)′ = O(s) ×_{V} V′. We will prove that O(s)′ ≅ O(s) by induction on dim O(s). This is OK in case dim O(s) = 0, since such points s lie in U. We know by our induction assumption that p : O(s)′ → O(s) is an isomorphism outside the closed point s of O(s). Further, p is finite unramified, hence O(s)′ is semi-local. Also Γ(O(s)′, O) ⊂ Γ(O(s)′ \ p^{−1}({s}), O) by our definition of V′ as the Zariski closure. This gets us through in case depth ≥ 2, since O_{S,s} = Γ(O(s) \ {s}, O) in that case.

In case the depth equals 1 and O_{S,s} is geometrically unibranch we base change to O(s)^{sh}, the strict henselization of O(s). It follows that on any connected component T of O(s)′ ×_{O(s)} O(s)^{sh} the map T → O(s)^{sh}, which is finite unramified, is a closed immersion. The scheme O(s)^{sh} is irreducible (see [6, IV, 18.8.15]), hence the scheme O(s)′ ×_{O(s)} O(s)^{sh} has exactly one connected component T such that the morphism T → O(s)^{sh} induces an isomorphism outside the closed point of O(s)^{sh}. The kernel of Γ(O(s)^{sh}, O) → Γ(T, O) being nonzero would contradict our assumption of depth 1. Hence T ≅ O(s)^{sh}. The closed point of T maps to a closed point s′ of O(s)′ and the map O(s)′ → O(s) is a local isomorphism at s′. Again, by definition of V′ as the Zariski closure, it follows that O(s)′ ≅ O(s).

(2) We have O_{S,s} = lim_{→} Γ(W, O), where W ⊂ S runs through the affine open neighbourhoods of s. By assumption we get a stable curve C_{W} over some such W, such that over O(s)∩U it becomes isomorphic to C_{U}. Since O(s) = lim_{←} W and O(s)∩U = lim_{←} W ∩U, we get C_{W} |_{U∩W′} ≅ C_{U}|_{U∩W′} for some smaller W′. This gives an extension C_{V} over V = W′ ∪ U.

3 Stratification by topological type

Let C be a stable curve over an algebraically closed field k. The topological type τ = τ(C) of C is given by the graph of C, where each of the edges is labeled by the genus of the corresponding component of C. Suppose that C → S is a stable curve over a reduced scheme S. We say that C has locally constant topological type τ if all the geometric fibres of C → S are stable curves having topological type τ. If this is the case, then for any geometric point s̄ of S there is a natural action of π_{1}(S, s̄) on the graph τ(Cs̄). We say that C → S has constant topological type τ if in addition this monodromy action is trivial.

Let M = M_{g} (sometimes denoted \overline{M_{g}} in the literature) be the algebraic stack parametrizing stable curves of genus g over Spec(ℤ). We list a number of results concerning the stratification by topological type which can be deduced from the deformation theory of stable curves, see [3].

i. For each topological type τ of genus g there is a reduced locally closed substack S_{τ} ⊂ M characterized by the following property: Any geometric point x̄ : Spec(k) → M factors through S_{τ} if and only if the corresponding stable curve C over k has topological type τ.

ii. If C → S is a stable curve of genus g over the reduced scheme S and x : S → M is the associated 1-morphism, then C/S has locally constant topological type τ if and only if x factors through S_{τ}. This gives a way of defining locally constant topological type for families over non-reduced bases.

iii. Locally in the étale topology on M there exist divisors D_{i}, i ∈ I, which are normal crossing divisors relative to Spec(ℤ), such that the topological type of the universal curve is constant on each stratum D_{J}. Here J ⊂ I is a subset and D_{J} is the set of points lying in the divisors D_{j}, j ∈ J and not in the divisors D_{i}, i ∈ I \ J.

iv. The algebraic stacks S_{τ} are smooth over Spec(ℤ).

We can characterize the property ‘locally constant topological type’ (lctt) without specifying a type or using M. In fact we should not limit the discussion to stable curves. If C → S is a flat family of curves with only ordinary double points in the fibres, then it has lctt if and only if Y := Sing (C/S), with its natural subscheme structure, is flat (or, equivalently, étale) over S. This is because Y is unramified over S (see 4.3) and, therefore, the geometric number of singular points in the fibres can only increase by specialization, and this happens if and only if the topological type changes. So instead of the stratification by topological type, one can look at the Fitting stratification of the finite S-scheme Y. The above argument shows that, set-theoretically, Fitting strata (fixed number of singular points) are disjoint sums of ‘topological’ strata, so the natural subscheme structure of the former defines a subscheme structure on the latter, with a more natural definition than (ii).

Of course assertions (iii) and (iv) use the deformation theory of stable curves. But what we really use in the sequel is the following purity property of Y (also a key point in [15]): locally in the étale topology, the map Y → S looks like a closed immersion defined by one equation. Indeed, locally in the étale topology one can present C as a scheme étale over Spec(O_{S}[x, y]/(xy − f)), where f is a section of O_{S} and then Y ≅ V (f) ⊂ S. See [3, page 81] and [20], or [17, Section 2], or [9, 2.23]. This proves the following general statement: locally in the étale topology on S there exist closed subschemes D_{i} ⊂ S, i ∈ I defined by one equation such that the topological type of C is constant on the locally closed subschemes D_{J}. Here J ⊂ I is a subset and D_{J} = ⋂_{j∈J} D_{j} \ ⋃_{i∈I\J} D_{i}. In the case that C → S is a stable family, this also follows by pullback from the universal case (iii) above.

4 The key lemma

We use the concept of stable r-pointed curves of genus g from [11]. Such a curve over a base scheme S is given by a pair (C, Σ), where C → S is a flat, projective family of curves and Σ = {σ_{1}, …, σ_{r}} is an ordered set of disjoint sections into the smooth locus of C → S. The geometric fibres of C → S are reduced, connected and have only nodes as singularities; the stability condition means that these fibres have finitely many automorphisms (as r-pointed curves).

We remark that we can define the topological type of an r-pointed stable curve (C, Σ) over an algebraically closed field: we take the graph of C as defined before and also label the edges with the set of indices of the distinghuised points lying on the corresponding component. Just as in the case of stable curves we may define families having locally constant topological type and constant topological type.

4.1 Key Lemma. Let k be any field. Let (C, Σ) → ℙ_{k}^{1} be a stable r-pointed curve of genus g. If the topological type of (C, Σ) is locally constant over ℙ_{k}^{1} \ {0, ∞} then (C, Σ) is constant: there exists a stable pair (C_{0}, Σ_{0}) over k such that

(C, Σ) ≅ (C_{0} × ℙ_{k}^{1}, Σ_{0} × ℙ_{k}^{1}).

Proof. Suppose we know that (C, Σ) ⊗ k̄ is constant. In that case, define (C_{0}, Σ_{0}) to be the fibre of (C, Σ) over a k-rational point x in ℙ_{k}^{1} \ {0, ∞}. Consider

Isom_{ℙ_{k}^{1}}((C_{0} × ℙ_{k}^{1}, Σ_{0} × ℙ_{k}^{1}), (C, Σ)).

This is a scheme finite and unramified over ℙ_{k}^{1}. It has a k-rational point (lying over x) and over k̄ it is isomorphic to a disjoint union of copies of ℙ1 ; k̄. Thus we get that (C, Σ) is constant. These arguments reduce the problem to the case that k is algebraically closed.

From now on we assume k algebraically closed. By Lemma 4.2 below, and the fact that π_{1}(ℙ_{k}^{1}) = {1}, we see that the components of the generic fibre C_{η} are defined over k(η) = k(ℙ^{1}). This means that C = C_{1}∪…∪C_{m}, where C_{i,η} is geometrically irreducible. By Lemma 4.3 we see that the singularities of C_{η} correspond to the generic points of a closed subscheme F ⊂ C mapping finite étale to ℙ_{k}^{1}. Thus F is a union of sections of C/ℙ_{k}^{1}. We conclude that the normalization D of the surface C is a disjoint union D = D_{1} ⊔ … ⊔ D_{m}, where D_{i} is the normalization of C_{i}. The inverse image of F in D, mapping two to one étale to F, is a union of sections τ_{ji}, with j = 1, …, m and i = 1, …, n_{j}. Further, let σ_{ji}, with i = 1, …, t_{j}, be those sections of C among the elements of Σ which map into C_{j}. Each of these lifts uniquely to a section of D_{j}, also denoted σ_{ji}. By assumption of stability, each of the pairs

(D_{j}, {σ_{j1}, …, σ_{jt_{j}}, τ_{j1}, …, τ_{jn_{j}}})

is a stable pointed curve over ℙ_{k}^{1} of locally constant topological type over ℙ_{k}^{1} \ {0, ∞}. Therefore we have reduced the key lemma to Lemma 4.4 below.

4.2 Lemma. Let π : X → T be a proper morphism of schemes, with all fibres equidimensional of dimension d. Assume T is locally Noetherian, irreducible with generic point η. Suppose the locus where π is smooth is dense in any fibre of X → T. There exists a finite étale morphism W → T such that

Wη̄ ≅ {irreducible components of Xη̄} as Gal(η̄/η)-sets.

Proof. We need only prove that if T is strictly local, with closed point t, then the Galois action on the set of irreducible components of Xη̄ is trivial. Let Z_{η} ⊂ X_{η} be a reduced closed subscheme corresponding to a union of irreducible components of Xη̄. Let Z ⊂ X be the Zariski closure of Z_{η}. Since Z → T is proper and dominant, we get that Z_{t} is not empty. The dimension of this fibre is at least the dimension of Z_{η} [6, IV.13.1]. Hence Z_{t} contains an irreducible componenent of X_{t}, as dim X_{t} = dim X_{η} by assumption. Generically along this component the morphism π is smooth. Therefore we find that Z → T is smooth in some closed point of Z_{t}. It follows that a section σ : T → Z exists. Consequently, Z_{η} has a point rational over η and hence a component defined over η. Thus any orbit of Gal(η̄/η) in the set of irreducible components of Xη̄ consists of one point.

4.3 Lemma. ([15, Lemma 3] or [3]. See also discussion in Section 3.) Let f : C → T be a proper flat morphism, whose geometric fibres are reduced curves having at most nodes as singularities. The closed subscheme Sing(f) of C is finite unramified over T.

4.4 Lemma. Let (C, Σ) → ℙ_{k}^{1} have locally constant topological type over ℙ_{k}^{1} \ {0, ∞}. Assume that k = k̄ and that C_{η} is smooth. Then (C, Σ) is constant.

Proof. We deal with the cases g ≥ 2, g = 1 and g = 0 separately.

The case g ≥ 2. Let C → C′ be the contraction of C to a minimal model. This only contracts curves on C lying in the fibres over 0, ∞. By [21, p. 61 Theorem 4] we see that C′ → ℙ_{k}^{1} is isotrivial. By an argument as before, looking at Isom(C_{0} × ℙ_{k}^{1}, C′), we see that in fact C′ is trivial: C′ ≅ C_{0} × ℙ_{k}^{1}. The sections σ_{i} of C define morphisms ℙ_{k}^{1} \ {0, ∞} → C_{0}. These are constant, say σ_{i} corresponds to p_{i} ∈ C_{0}. Thus (C, Σ) is isomorphic to (C_{0}, {p_{1}, …, p_{r}}) × ℙ_{k}^{1} \ {0, ∞} over ℙ_{k}^{1} \ {0, ∞}. It follows that (C, Σ) is isomorphic to (C_{0}, {p_{1}, …, p_{r}}) × ℙ_{k}^{1}, as the stack M_{g,r} is separated.

The case g = 1. We use the section σ_{1} to view C as an elliptic curve over ℙ_{k}^{1} \ {0, ∞}. Since C has semi-stable reduction in 0, ∞ we see that in the monodromy representation on ℓ-power torsion points

ρ : π_{1}(ℙ_{k}^{1} \ {0, ∞}) ⟶ GL_{2}(ℤ_{ℓ})

the inertia groups act unipotently. Therefore ρ factors through the tame fundamental group which is abelian. We conclude that the image of ρ may be put into upper triangular form. Consequently, the elliptic curve (C, σ_{1}) has infinitely many distinct torsion sections over ℙ_{k}^{1} \ {0, ∞}. Thus it is constant as the modular curve X_{1}(ℓ^{n}) has genus > 0 for n >> 0. Hence

(C, σ_{1})|_{ℙ_{k}^{1}\{0,∞}} ≅ (E, 0) × (ℙ_{k}^{1} \ {0, ∞})

for some elliptic curve (E, 0) over k. The other sections σ_{2}, …, σ_{r} give morphisms of ℙ_{k}^{1} \ {0, ∞} into E, which are constant. Conclusion as in the case g ≥ 2.

The case g = 0. As (C, Σ) is stable, Σ consists of at least three sections. Over ℙ_{k}^{1} \ {0, ∞} there is a unique isomorphism of pointed curves

(C, {σ_{1}, σ_{2}, σ_{3}})|_{ℙ_{k}^{1}\{0,∞}} ≅ (ℙ_{k}^{1} \ {0, 1, ∞}) × (ℙ_{k}^{1} \ {0, ∞}).

As before the sections σ_{4}, …, σ_{r} give morphisms ℙ_{k}^{1} \ {0, ∞} → ℙ_{k}^{1} \ {0, 1, ∞}, which are constant. Using that M_{0,r} is separated, the result follows as in the case g ≥ 2.

5 The main theorem

We consider a scheme S and a normal crossing divisor D ⊂ S. Recall that this means that S is regular along D and locally in the étale topology, D is given as the zero set of a product t_{1} … t_{r}, where t_{1}, …, t_{r} is part of a regular system of parameters.

Suppose we are given a family of stable curves C_{U} over the complement U = S \ D of D in S. By Lemma 2.3, since S is regular along D, there exists a largest open subscheme V ⊂ S, U ⊂ V, such that C_{U} extends to a stable curve over V. We will say that the family extends to a stable curve over the point s ∈ S, if we have s ∈ V. Equivalently, in view of Lemma 2.3, this means that C_{U} gives rise to a (unique) stable curve over Spec(O_{S,s}).

5.1 Theorem. Let C_{U} be a stable curve over U of locally constant topological type. If C_{U} extends to a stable curve over the generic points of the divisor D, then C_{U} extends to a stable curve over S.

By Lemma 2.3 we reduce to the case that S = Spec(R) is local. Hence we assume S = Spec(R) is affine, R is local, Noetherian and regular and D ⊂ S is a normal crossings divisor. The special point of S is written s. Let V ⊂ S be the largest open subscheme of S over which C_{U} extends. The inclusion morphism is denoted j : V → S. Let us write f : C_{V} → V for the extension of C_{U} to V, which exists by assumption. For each n we put

F_{n} = j_{∗}f_{∗}(ω_{C_{V} /V}^{⊗n}).

Since codim_{S}(S \ V) ≥ 2 by assumption, these are coherent sheaves on S. The direct sum is a graded algebra F = ⨁_{n≥0} F_{n}. It is clear that our extension exists if and only if (i) the graded algebra F is generated by F_{1} ⊕ F_{2} ⊕ F_{3} over O_{S} and (ii) the scheme Proj(F) is a stable curve over S. (Remark that (ii) holds if and only if all F_{i} are flat (i.e., free) and Proj(F) ⊗ κ(s) is a stable curve.)

Suppose R^{∗} is the completion of R with respect to its maximal ideal. In S^{∗} = Spec(R^{∗}) we have D^{∗}, U^{∗}, V^{∗} and a stable curve C_{U^{∗}} obtained by pulling back these data over S. Note that all the assumptions of the theorem hold for the triple S^{∗}, D^{∗}, C_{U^{∗}}. Let F_{n}^{∗} be the sheaves on S^{∗} constructed in the same manner as before. We claim that F_{n}^{∗} = F_{n} ⊗ O_{S^{∗}}. This follows easily by checking the result over V^{∗}, where it is obvious. Since R → R^{∗} is faithfully flat the condition that F is generated by F_{1} ⊕ F_{2} ⊕ F_{3} over O_{S} is equivalent to the condition that F^{∗} is generated by F_{1}^{∗} ⊕ F_{2}^{∗} ⊕ F_{3}^{∗} over O_{S^{∗}}. Similar for condition (ii) above. We conclude that C_{U} extends if and only if C_{U^{∗}} extends.

We conclude that it suffices to prove the theorem in the case that S = Spec(R) with R a regular complete local ring. In the same manner we can deal with local flat extensions R → R′ and prove: the curve C_{U} extends if and only if C_{U′} extends. The strict henselization R → R^{sh} is one such and going over to Spec(R^{sh}) does not invalidate any of the assumptions of the theorem. Therefore, we may and do suppose that R is regular, complete local with separably closed residue field.

5.2 Dimension of the base is 2

In this subsection the ring R has dimension 2. Let τ denote the topological type of C_{U}. We denote by G → U the relative P ic^{0} of C_{U} over U. This is a semi-abelian scheme over U whose abelian part has constant dimension, equal to the sum of genera of the components, i.e., the sum of the numbers labeling the edges of τ. By assumption G extends to a semi-abelian scheme over S \ {s}. This means that the scheme G[n], where n ≥ 3 is prime to the residue characteristic of s, is tamely ramified along D. Hence, by Abhyankar’s lemma [5, Exposé XIII Section 5], for a given n as above there exists a finite flat covering π : S′ → S, with the following properties: 1) π is finite étale over U, 2) S′ is regular, 3) D′ = π^{−1}(D)_{red} is a divisor with normal crossings in S′ and 4) the covering G[n] becomes trivial over U′ = π^{−1}(U). The argument given above shows that it suffices to prove that C_{U′} extends to a stable curve over S′. Therefore, we may assume that the monodromy on n-torsion points of G is trivial.

This implies that the curve C_{U} satisfies the conditions of Theorem 2.1. Indeed, for any pair (R, φ) as in that theorem, we see that the inertia works unipotently on the n-power torsion points of G_{η}. Thus G_{η} extends to a semi-abelian scheme over R, hence C_{η} extends to a stable curve over R. In case C_{η} is smooth this is [3, Proposition 2.3]. In the general case, note that the action π_{1}(η̄, η) → Aut(τ(Cη̄)) (see Section 3) is trivial (compare [2, proof of 3.5]). Thus normalizing C_{η} will give stable pointed curves (D_{j,η}, τ_{ji,η}) as in the proof of the key lemma. These will extend to stable pointed curves (D_{j}, τ_{ji}) over R by the obvious extension of [3, Proposition 2.3] to the case of stable pointed curves. Gluing these gives C_{R} over R.

We apply Theorem 2.1 to get an extension after a normalized blowing up S̃ → S with center in the closed point of S. We can dominate S̃ by a sequence of simple blowing ups in points. Thus it suffices to consider the case that π : S̃ → S is equal to such a repeated blowing up. We are going to prove that the extension C̃ over S̃ is of the form π^{∗}(C). By induction on the number of simple blowings up, we reduce to the case that S̃ → S equals the blowing up in the closed point of S. (Note that at each stage the assumption of normal crossing on the locus of degeneracy holds.)

Let E ⊂ S̃ be the exceptional fibre. It is a projective line over the residue field k of R. The restriction C_{E} = C̃|_{E} is a stable curve over E. Generically, it has some topological type τ′. We claim that the type is locally constant over E \ W, where W consists of at most two points rational over k. Indeed, W is contained in the strict transform of D. This can be seen as follows. Let p ∈ E be a point where the topological type of C_{E} jumps. Locally in the étale topology on S̃ around p there exist Cartier divisors D_{1}, ..., D_{r}, with p ∈ D_{i} such that C̃ has constant topological type on the strata D_{J}, where J ⊂ {1, …, r} is a subset. This follows from the discussion in Section 3. Since the topological type of C̃ is constant over S̃ \ π^{−1}(D), we see that either p lies on the strict transform of D or every D_{i} is contained in E. In the last case we see that the topological type is constant on E around p. This contradicts our assumption that p is a jumping point.

The key lemma implies that C̃ is constant over E, i.e., C̃|_{E} ≅ C × E. The rest of the argument is easy. Let us write f̃: C̃ → S̃ for the structural morphism. We have locally free sheaves F̃_{n} = f̃_{∗}(ω_{C̃/S̃}^{⊗n}) on S̃. By the above

F̃_{n}|_{E} ≅ H^{0}(C, ω_{C/k}^{⊗n}) ⊗_{k} O_{E}.

It follows immediately that π^{∗}π_{∗}F̃_{n} ≅ F̃_{n}. This proves that C̃ = Proj(F̃) comes from a stable curve Proj(π_{∗}(F̃)) over S.

5.3 The general case

In this section we assume that S is the spectrum of a regular complete local ring R with separably closed residue field of dimension d at least 3. We assume that the extension theorem has already been proven in the cases of dimension less than d. This implies in particular that C_{U} extends to a stable curve C_{V} over the open subscheme V = S \ {s}.

The divisor D is the union D = ⋃ D_{i} of a number of regular divisors D_{i}. Let us consider the restriction of the curve C_{V} to the intersection V_{1} = V ∩ D_{1}. Its topological type is locally constant over the stratum D_{1} \ ⋃_{i≠1} D_{i}. This follows from the discussion on the stratification by topological type given in Section 3. Hence, by induction, we get a stable curve C_{1} → D_{1} isomorphic to the restriction of C_{V} to V_{1}.

Let us write Z_{1} = D_{1} and Z_{n} for the n^{th}-infinitesimal neighbourhood of D_{1} in S. Let V_{n} = V ∩ Z_{n} and write j_{n} for the inclusion morphism V_{n} → Z_{n}. By induction on n we are going to construct a stable curve C_{n} over Z_{n} such that its restriction to V_{n} equals the restriction of C_{V} to V_{n}.

Assume therefore that we are given C_{n−1} over Z_{n−1} for some n ≥ 2. Let E denote the locally free sheaf on Z_{n−1} given by the lemma below. By [10, I Proposition 2.1.2.3], the obstruction to lift C_{n−1} to a flat scheme over Z_{n}, lies in H^{1}(Z_{n−1}, E ⊗ J), where J is the ideal of the embedding Z_{n−1} → Z_{n}. This group is zero as Z_{n−1} is affine. Let us take any lift C′_{n} of C_{n−1} to a flat scheme over Z_{n}. This gives us two lifts C′_{n}|_{V_{n}} and C_{V} |_{V_{n}} of the curve we already have over V_{n−1}. Hence, by [10, I Proposition 2.1.2.3], the difference is measured by an element of H^{0}(V_{n−1}, E ⊗ J). Note that J ≅ O_{Z_{1}} and thus E ⊗ J may be considered as a free O_{Z_{1}}-module. Since the dimension of Z_{1} = D_{1} is at least two and since it is a regular scheme, we see that

H^{0}(V_{n−1}, E ⊗ J) = H^{0}(V_{1}, E ⊗ J) = H^{0}(Z_{1}, E ⊗ J) = H^{0}(Z_{n−1}, E ⊗ J).

Consequently, we may change our lift C′_{n−1} by the resulting element of H^{0}(Z_{n−1}, E ⊗ J) and in this way we achieve the induction step. (Note that any deformation of a stable curve is a stable curve, by definition of stable curves.)

Let Z^{∧} be the completion of S along D_{1}. The limit of the curves C_{n} defines a formal scheme C^{∧} over Z^{∧}. We can algebraize this to get a stable curve C over Spec(R) [6, III Proposition 5.4.5]. (Use that since R is complete, it is also complete with respect to any ideal.) Consider the scheme

Isom = Isom(C|_{V} , C_{V}).

It is finite unramified over V and by construction there is a section over V^{∧} = Z^{∧} ∩ V. Thus this section is algebraic, i.e., there is a finite étale V′ → V and a map V′ → Isom. But finite étale coverings V′ → V are trivial (Zariski-Nagata purity theorem [SGA 2, X 3.4]). Therefore, C|_{V} ≅ C_{V}. This ends the proof of Theorem 5.1.

5.4 Lemma. Let f : C → S be a stable curve. Consider the cotangent complex L_{C/S} of C over S. The complex

Rf_{∗}RHom(L_{C/S}, O_{C})

is quasi-isomorphic to a complex consisting of a locally free sheaf E of rank 3g − 3 placed in degree 1.

Proof. We need only to prove this in the universal case, hence we may assume the base S is Gorenstein. Let us write ω_{C/S} for the relative dualizing sheaf. The natural map L_{C/S} → Ω_{C/S}^{1} is a quasi-isomorphism (see [3, page 79]). By relative duality

RHom(Rf_{∗}RHom(L_{C/S}, O_{C}), O_{S})) ≅ Rf_{∗}(L_{C/S} ⊗ ω_{C/S}) ≅ Rf_{∗}(Ω_{C/S}^{1} ⊗ ω_{C/S}),

up to a shift in degrees. The complex Rf_{∗}(Ω_{C/S}^{1} ⊗ω_{C/S}) is quasi isomorphic to the complex consisting of a locally free sheaf E^{∗} of rank 3g − 3 placed in degree 0. Indeed, the sheaf Ω_{C/S}^{1} ⊗ ω_{C/S} is flat over S and on all the fibres C of C → S the cohomology group H^{1}(C, Ω_{C}^{1} ⊗ ω_{C}) vanishes. (Computation of rank left to the reader.) By duality on S, we get that Rf_{∗}RHom(L_{C/S}, O_{C}) is quasi-isomorphic to E = Hom(E^{∗}, O_{S}) placed in some degree, we leave it to the reader to check that it should be placed in degree 1.

5.5 Corollary. The extension theorem 5.1 holds for families (C_{U}, Σ_{U}) of stable r-pointed curves.

Proof. We could go through the proof above and see that it works in this case also. However, it is more amusing to deduce the result from 5.1.

It suffices to consider the case that S = Spec(R) is local. We can find an elliptic curve (E, 0) over S. Attaching E_{U} in 0_{U} to C_{U} at the marked points Σ_{U} gives a stable curve C′_{U} over U. This curve has locally constant topological type over U and extends to the generic points of the divisor D as we assumed that (C_{U}, Σ_{U}) did. We apply 5.1 to get a stable curve C′ over S. By deleting the elliptic tails (E, 0) from this extension we obtain the desired extension (C, Σ) over S.

6 An example (Raynaud-Ogus-Gabber)

In this section we give a counter example to [4, Theorem 6.7] and [4, Corollary 6.8]. A ”minimal” counter example is produced in 6.2. It leads to a family of stable curves of genus 2 over an open set of S; in 6.4 it is explained why this does not give a counter example to Theorem 5.1.

Let W = W (𝔽̄_{p}) be the ring of Witt vectors. The ring R = W [[x, y]]//((xy)^{p−1} − p) is a regular local ring of dimension 2. As W [ζ_{p}] ≅ W [t]/(t^{p−1} − p) we see there exists a primitive pth root of unity ζ_{p} in R. We take S = Spec(R). Let s denote the closed point of S and put V = S \ {s}, U_{1} = D(x) and U_{2} = D(y). Remark that V = U_{1} ∪ U_{2} and that U = U_{12} := U_{1} ∩ U_{2} = D(p) is the complement of a divisor with normal crossings. Put

G_{i} = (μ_{p} × ℤ/pℤ)_{U_{i}}.

These are finite locally free group schemes of rank p^{2}, with G_{1} over U_{1} and G_{2} over U_{2}. We let the isomorphism

φ_{12} : G_{1}|_{U_{12}} ⟶ G_{2}|_{U_{12}}

be given by the matrix

( \begin{matrix}1 & 0 \\ β & 1 \end{matrix} ) .

Here β is the isomorphism μ_{p,U_{12}} ≅ℤ/pℤ_{U_{12}} which maps the section ζ_{p} ∈ Γ(U_{12}, μ_{p}) to the section ¯1 ∈ Γ(U_{12}, ℤ/pℤ). Since the covering V = U_{1} ∪ U_{2} consists of two parts we do not need to check a cocycle condition: we get a unique group scheme G_{V} → V with isomorphisms G_{V} |_{U_{i}} ≅ G_{i}, inducing φ_{12} over U_{12}. It is clear from the formula for φ_{12} that the subgroup schemes ℤ/pℤ_{U_{i}} ⊂ G_{i} glue to a subgroup scheme ℤ/pℤ_{V} ⊂ G_{V}. In the same manner, we see that the quotient group scheme is isomorphic to μ_{p,V}. This means that G_{V} fits into an exact sequence

0 ⟶ ℤ/pℤ_{V} ⟶ G_{V} ⟶ μ_{p,V} ⟶ 0.  (∗)

We remark that this extension is nontrivial: On each U_{i} there is a unique splitting of (∗) given by (G_{U_{i}})^{∘} ≅ μ_{p,U_{i}}; hence the morphism β : μ_{p,U_{12}} → ℤ/pℤ_{U_{12}} is an invariant of (∗); as β is nontrivial we conclude that the extension (∗) is nontrivial.

Denote j : V → S the immersion morphism. Note that j_{∗} gives an equivalence of categories

finite locally free O_{V} -modules ↔ finite locally free O_{S}-modules.

However, j_{∗} does not preserve short exact sequences. Thus for any finite locally free group scheme H_{V} over V there is a unique finite locally free group scheme H over S, with a given isomorphism H_{V} ≅ H|_{V}. The construction H_{V} ↦ H is a functor. (This notation is not confusing!) In particular, our G_{V} gives G over S and, since obviously ℤ/pℤ_{V} ↦ ℤ/pℤ_{S}, we get a morphism of group schemes

γ : ℤ/pℤ_{S} ⟶ G.

It is clear that γ is not a closed immersion: If so then the group of connected components of G would be nontrivial and the projection G → G/G^{∘} would give a trivialization of the extension (∗) over V. This also implies that there does not exist a finite locally free closed subgroup scheme N ⊂ G with N_{V} = ℤ/pℤ_{V}, since uniqueness of the extension of N_{V} to S would give N ≅ ℤ/pℤ_{S}.

We use a theorem of Raynaud, see [1, Theorem 3.1.1]: There exists a closed immersion G ⟶ X of G into an abelian scheme X over S. We define

Y_{V} := X_{V} /(ℤ/pℤ_{V}),

where ℤ/pℤ_{V} is viewed as a closed subgroup scheme of X_{V} via ℤ/pℤ_{V} → G_{V} → X_{V}.

6.1 Claim. The abelian scheme Y_{V} does not extend to an abelian scheme Y over S.

Proof. If so the morphism X_{V} → Y_{V} would extend to a morphism X → Y (see [18, IX Corollary 1.4]). This would be an isogeny and hence would have a finite locally free closed subgroup scheme N ⊂ X as kernel. Since obviously N_{V} = ℤ/pℤ_{V} ⊂ G_{V} ⊂ X_{V} we would get N ⊂ G, contradicting what was said before.

Thus Y_{V} is a counter example to [4, Corollary 6.8]. We remark that the p-divisible group of Y_{V} is a counter example to [4, Theorem 6.4]. A mistake in the proof of that theorem can be found on page 182: the map p^{− dim(G)} · trace_{G[p^{n+1}]/G[p^{n}]} is not a splitting of the natural map O_{G[p^{n}]} ⊂ O_{G[p^{n+1}]}. This can be seen by an explicit computation in the case G = μ_{p^{∞}} × ℚ_{p}/ℤ_{p}. The arguments of [4, Ch. 5 Section 6] from the proof of Theorem 6.4′ onward seem correct. Thus these arguments prove [4, Theorem 6.7] and [4, Corollary 6.8] over base schemes S for which the assertion of [4, Theorem 6.4] holds. We were informed by A. Ogus that the result of [4, Theorem 6.4] is correct in the case that S = Spec(W [[\underline{x}]]) and p > 2.

We remark that the counter example shows that the definition of a canonical integral model of a Shimura variety as in [14, Definition 2.9] has to be modified: the example shows that the moduli stack of principally polarized Abelian schemes is not a canonical model in this sense. To be explicit [14, Theorem 2.10] and [14, Proposition 2.13] are not correct as they stand. (Compare [22].)

The example we recorded above was communicated to us by A. Ogus. It seems that Raynaud (around ’84) was the first who noted that an example of this kind exists, and that Gabber reconstructed the example on the basis of what Ogus remembered of Raynaud’s original example.

6.2 From now on we take p > 2. We claim there is a nondegenerate alternating pairing

e : G × G ⟶ μ_{p,S}.

To construct e we construct an antisymmetric isomorphism

e_{V} : G_{V} ⟶ G_{V}^{D} = Hom(G_{V} , 𝔾_{m}),

which will extend to e over S by the equivalence of categories above. In fact we may take for e on U_{i}

⟨(ζ, a), (η, b)⟩ = ζ^{b} · η^{−b} ∈ μ_{p}.

To see these pairings agree on U_{1} ∩ U_{2} we compute:

\begin{aligned}⟨φ_{12}(ζ, a), φ_{12}(η, b)⟩ & = ⟨(ζ, a + β(ζ)), (η, b + β(η))⟩ \\ & = ζ^{b+β(η)} · η^{−(a+β(ζ))} = ζ^{b} · η^{−a} · ζ^{β(η)} · η^{−β(ζ)} = ζ^{b} · η^{−a}. \end{aligned}

The last equality is clear if we realize that ζ = ζ_{p}^{β(ζ)} and similarly for η.

The classification of groups schemes of order p^{2} with a nondegenerate alternating pairing over s = Spec(𝔽̄_{p}) gives the possibilities

G_{s} ≅ (μ_{p} × ℤ/pℤ)_{s} or E[p]_{s} or (α_{p} × α_{p})_{s}.

The first case is excluded, since it would imply the sequence (∗) is split, argument as before. The second case, where E denotes a supersingular elliptic curve, is impossible also. We give an argument which also casts some light on the situation.

General considerations show there exists a blowing up S̃ → S with center in s such that over S̃ there exists a finite locally free closed subgroup scheme Ñ ⊂ GS̃ with Ñ_{V} = ℤ/pℤ_{V} ⊂ GS̃|_{V} = G. We may assume that S̃ → S is a sequence of simple blowings up:

S̃ = S_{n} ⟶ S_{n−1} ⟶ … ⟶ S_{1} ⟶ S.

Let E_{n} ≅ ℙ^{1} denote the exceptionel fibre of the blowing up S_{n} → S_{n−1}. The group scheme Ñ|_{E_{n}} moves inside the constant groupscheme G_{s} × E_{n} = GS̃|_{E_{n}}. In the case G_{s} = E[p]_{s}, there exists only one closed group scheme of order p, hence we would get Ñ|_{E_{n}} ⊂ G_{s} × E_{n} is constant. This would mean that Ñ = N_{n−1} ×_{S_{n−1}} S_{n} for some finite locally free closed subgroup scheme N_{n−1} ⊂ G_{S_{n−1}}. Induction on n would give us a N ⊂ G, contrary to what has gone before.

Thus we conclude that G_{s} ≅ (α_{p} × α_{p})_{s}. Up to isomorphism, there is exactly one nondegenerate alternating pairing on α_{p} × α_{p}, namely

⟨(x_{1}, x_{2}), (y_{1}, y_{2})⟩ = φ(x_{1})(y_{2}) · φ(y_{1})(x_{2})^{−1} ∈ μ_{p}.

Here φ : α_{p} → (α_{p})^{D} is an isomorphism. As above let E be a supersingular elliptic curve over 𝔽̄_{p}. There exists a polarization μ_{0} on X_{0} := E × E such that

Ker(μ_{0}) = α_{p} × α_{p} ⊂ E × E.

By the above we can choose an isomorphism of pairs (G_{s}, e_{s}) ≅ (Ker(μ_{0}), e^{μ_{0}}). For the construction of the pairing e^{μ_{0}}, see [7].

By [8] there exists a deformation (X, μ)/S of (X_{0}, μ_{0})/s which follows the given deformation G of G_{s}, i.e., such that (G, e) ≅ (Ker(μ), e^{μ}). [The result of [8] is the following. Let (X_{0}, μ_{0}) be a polarized abelian variety over 𝔽̄_{p}. Consider the deformation functors h_{1} = Def(X_{0}, μ_{0}) and h_{2} = Def(Ker(μ_{0}, e^{μ_{0}}) on the category of Artinian W -algebra’s. There is a transformation h_{1} → h_{2}, given by (X, μ) ↦ (Ker(μ), e^{μ}). The result is that this transformation is formally smooth. (Here we use characteristic not 2.)]

As before we define Y_{V} as the quotient of X_{V} by ℤ/pℤ_{V}. By construction

ℤ/pℤ_{V} ⊂ G_{V} = Ker(μ)_{V}

is a totally isotropic closed subgroup scheme, hence the 2-dimensional abelian scheme Y_{V} has a principal polarization λ_{V}.

6.3 Corollary. There is a counter example to [4, Corollary 6.8] in which the base is two dimensional of mixed characteristic and the abelian scheme has relative dimension 2 and is principally polarized.

It is interesting and amusing to consider the family of stable curves of genus 2

f : C_{V} → V

defined by the formula

(Pic^{∘}(C_{V} /V), Θ_{C_{V} /V}) ≅ (Y_{V} , λ_{V}).

The fibres of f are either smooth projective curves of genus 2, or consist of 2 elliptic curves joined in one point. Let Z ⊂ V be the reduced closed subscheme whose points are the points where the second possiblility occurs.

6.4 Claim. The closure of the divisor Z in S has at least 5p − 5 formal branches at s.

Proof. Let us consider the blowing up S̃ = S_{n} → … → S_{1} → S introduced before. We may assume the subgroup scheme Ñ|_{E_{n}} really moves inside GS̃|_{E_{n}} = α_{p} × α_{p}|_{E_{n}}. Note that over S̃ we may extend (Y_{V} , λ_{V}) to a family of abelian schemes (Ỹ , λ̃) by putting

Ỹ = XS̃/Ñ.

Let

(Y_{MB}, λ_{MB}) ⟶ ℙ_{MB}^{1}

be the standard Moret-Bailly family associated to the pair (X_{0}, μ_{0})/𝔽̄_{p}. This family has exactly 5p − 5 fibres which are of the form (E_{1} × E_{2}, λ_{1} × λ_{2}), see [16, Prop 2.5]. It is clear from the construction of Ỹ and Y_{MB} that (Ỹ , λ̃)|_{E_{n}} is the pullback of (Y_{MB}, λ_{MB}) by a nonconstant morphism

f : E_{n} ⟶ ℙ_{MB}^{1}.

This shows that (Ỹ , λ̃) (and hence (Y_{V} , λ_{V})) is generically the Jacobian of a nonsingular genus 2 curve, hence Z is a divisor. Because f is nonconstant, it follows that Z has at least 5p − 5 formal branches at s.

6.5 Remark. The claim shows that the family C_{V} → V does not satisfy the conditions of the extension theorem proved in this article. Hence, it really is not a counter example to the theorem. That it extends after blowing up illustrates the result proved in Section 2.

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A.J. de Jong

Harvard University

Department of mathematics

One Oxford Street

Cambridge, Massachusetts 02138 USA e-mail: dejong@math.harvard.edu F. Oort Mathematisch Instituut Universiteit Utrecht Budapestlaan 6 Postbus 80.010 NL - 3508 TA Utrecht The Netherlands e-mail: oort@math.ruu.nl