TWISTED STABLE MAPS TO TAME ARTIN STACKS

DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI

Contents

1. Introduction 1 2. Twisted curves 3 3. Interlude: Relative moduli spaces 11 4. Twisted stable maps 14 5. Reduction of Spaces of Galois admissible covers 20 6. Example: reduction of 𝒳(2) in characteristic 2 33 Appendix A. Twisted curves and log curves

by Martin Olsson 42

Appendix B. Remarks on Ext-groups and base change

By Martin Olsson 50

Appendix C. Another boundedness theorem for Hom-stacks

By Martin Olsson 54

References 66

1. Introduction

This paper is a continuation of [4], where the basic theory of tame Artin stacks is developed. Our main goal here is the construction of an appropriate analogue of Kontsevich’s space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne–Mumford stack, the theory was developed in [1], and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation. However the method of construction we use here is very different from the ad-hoc method of [1], and more natural: we rely on the second author’s general results in [29, 30, 35], as extended in the appendices.

Section 2 is devoted to a number of basic properties of twisted curves in arbitrary characteristic, where, unlike the situation of [1], they may fail to be Deligne–Mumford stacks. In particular we show in Proposition 2.3 that the notion of twisted curve is stable under small deformations and can be tested on geometric fibers.

Date: April 3, 2008.

Vistoli supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed by MIUR. Olsson partially supported by NSF grant DMS-0555827 and an Alfred P. Sloan fellowship. Abramovich support in part by NSF grants DMS-0301695 and DMS-0603284.

In Section 3 we collect together some facts about relative moduli spaces which will be used in the following section.

In Section 4 we define twisted stable maps into a tame stack X and show in Theorem 4.1 that they form an artin stack, which is proper and quasifinite over the Kontsevich space of the coarse moduli space X of X. In particular, when X is projective, the stacks of twisted stable maps of X admit projective coarse moduli spaces. The construction proceeds rather naturally from the following:

(1) the existence of the stack of twisted curves, which was shown in [29] for Deligne–Mumford twisted curves and proved in our case in Appendix A, and

(2) The existence and finiteness properties of \underline{Hom}-stacks, proved in [30]

and [35] in many cases and extended in our case in Appendix C. Some extra care is needed for proving the quasi-finite claim. Properness relies on Lemma 4.4, a suitable generalization of the Purity Lemma of [1].

In section 5 we concentrate on the case where the target stack is BG^{1}, the

1→

classifying stack of a finite flat linearly reductive group scheme G. The main result here, Theorem 5.1, is that the space of stable maps with target BG is finite and flat over the corresponding Deligne–Mumford space. This result is known when G is tame and étale (see [2]) and relatively straightforward when G is a diagonalizable group scheme (or even a twist of such). However, our argument in general takes some delicate twists and turns.

As an example for the behavior of these stacks, we consider in Section 6 two ways to compactify the moduli space X(2) of elliptic curves with full level-2 structure. The first is as a component in K_{0,4}(Bμ_{2}), the stack of totally branched μ_{2}-covers of stable 4-pointed curves of genus 0. This provides an opportunity to consider the cyclotomic inertia stacks and evaluation maps. The second is as a component in K_{1,1}(Bμ_{2}^{2}) parametrizing elliptic curves with Bμ_{2}^{2}, where X(2) meets other components in a geometrically appealing way. We also find the Katz-Mazur regular model of X(2) as the closure of a component of the generic fiber.

As already mentioned, the paper contains three appendices, written by the second author. Appendix A generalizes the main results of [29]. Appendix B contains some prepatory results needed for Appendix C which generalizes some of the results from [30] and [35] to tame stacks. The logical order of Appendix A is after section 2, whereas the Appendix C only uses results from [4].

1.1. Acknowledgements. Thanks to Johan de Jong for helpful comments. Thanks to Shaul Abramovich for help and advice with visualization.

1(Dan) changed classifying stack to B everywhere

2. Twisted curves

section4

A1 Definition 2.1. Let S be a scheme. An n-marked twisted curve over S is

a collection of data (f: C → S, {Σ_{i} ⊂ C}_{i=1}^{n}) as follows:

(i) C is a proper tame stack over S whose geometric fibers are connected of dimension 1, and such that the moduli space C of C is a nodal curve over S.

(ii) The Σ_{i} ⊂ C are closed substacks which are fppf gerbes over S, and whose images in C are contained in the smooth locus of the morphism C → S.

(iii) If U ⊂ C denotes the complement of the Σ_{i} and the singular locus of

C → S, then U → C is an open immersion. (iv) For any geometric point p̄ → C^{2} mapping to a smooth point of C,

←2

there exists an integer r such that

Spec(O_{C,p̄}) ×_{C} C ≃ [D^{sh}/μ_{r}],

where D^{sh} denotes the strict henselization of D: = Spec(O_{S,f(p̄)}[z]) at the point (𝔪_{S,f(p̄)}, z) and ζ ∈ μ_{r} acts by z ↦ ζ · z (note that r = 1 unless p̄ maps to a point in the image of some Σ_{i}). Here 𝔪_{S,f(p̄)} denote the maximal ideal in the strict henselization O_{S,f(p̄)}.

(v) If p̄ → C is a geometric point mapping to a node of C, then there exists an integer r and an element t ∈ 𝔪_{S,f(p̄)} such that

Spec(O_{C,p̄}) ×_{C} C ≃ [D^{sh}/μ_{r}],

where D^{sh} denotes the strict henselization of

D: = Spec(O_{S,f(p̄)}[z, w]/(zw − t))

at the point (𝔪_{S,f(p̄)}, z, w) and ζ ∈ μ_{r} acts by x ↦ ζ ·z and y ↦ ζ^{−1} ·y.

Remark 2.2. If C → S is a proper tame Artin stack which admits a collection of closed substacks Σ_{i} ⊂ C (i = 1, …, n for some n) such that (C, {Σ_{i}}_{i=1}^{n}) is an n-marked twisted curve, we will refer to C as a twisted curve, without reference to markings.

The following proposition allows us to detect twisted curves on fibers:

Prop:2.3 Proposition 2.3. Let S be a scheme, and let (f: C → S, {Σ_{i}}_{i=1}^{n}) be a proper flat tame stack f: C → S with a collection of closed substacks Σ_{i} ⊂ C which are S-gerbes. If for some geometric point x̄ → S the fiber (Cx̄, {Σ_{i,x̄}}) is an n-marked twisted curve, then there exists an open neighborhood of x̄ in S over which (C → S, {Σ_{i}}) is an n-marked twisted curve.

Proof. Let π: C → C be the coarse moduli space of S. Since C is a tame stack flat over S, the space C is flat over S, and for any morphism S′ → S the base change C×_{S}S′ is the coarse moduli space of C×_{S}S′, by [4, Corollary 3.3]. It follows that in some étale neighborhood of x̄ → S the space C/S is a nodal curve over S (see for example [11, §1]). Shrinking on S we may therefore assume that C is a nodal curve over S. After further shrinking on S, we can also assume that the images of the Σ_{i} in C are contained in the smooth locus of the morphism C → S and that (2.1 (iii)) holds. To prove the proposition it then suffices to verify that conditions (iv) and (v) hold for geometric points of p̄ → C over x̄.

2(Dan) DANGER!! replaced all s̄ on a curve by p̄, since it can confuse with the base S

We may without loss of generality assume that S is the spectrum of a strictly henselian local ring whose closed point is x̄. Also let Cp̄ denote the fiber product

Cp̄ := C ×_{C,p̄} Spec(O_{C,p̄}).

Consider first the case when p̄ → C has image in the smooth locus of C/S. Since the closed fiber Cx̄ is a twisted curve, we can choose an isomorphism

Cp̄ ×_{S} Spec(k(x̄)) ≃ [Spec(k(x̄)[z])^{sh}/μ_{r}],

for some integer r ≥ 1, where the μ_{r}-action is as in (2.1 (iv)). Let

P_{0} → Cp̄ ×_{S} Spec(k(x̄))

be the μ_{r}-torsor

Spec(k(x̄)[z])^{sh} → [Spec(k(x̄)[z])^{sh}/μ_{r}].

As in the proof of [4, Proposition 3.6] there exists a μ_{r}-torsor P → Cp̄ whose reduction is P_{0}. Furthermore, P is an affine scheme over S. Since Cp̄ is flat over S, the scheme P is also flat over S. Since its closed fiber is smooth over S, it follows that P is a smooth S–scheme of relative dimension 1 with μ_{r}-action. Write P = Spec(A) and fix a μ_{r}-equivariant isomorphism

A ⊗_{O_{S}} k(x̄) ≃ (k(x̄)[z])^{sh}.

Since μ_{r} is linearly reductive we can find an element z̃ ∈ A lifting z such that an element ζ ∈ μ_{r} acts on z̃ by z̃ ↦ ζ · r̃. Since A is flat over O_{S} the induced map

(O_{S}[z̃])^{sh} → A

is an isomorphism, which shows that (iv) holds.

The verification of (v) requires some deformation theory. First we give an explicit argument. We then provide an argument using the cotangent complex.

Let p̄ → C be a geometric point mapping to a node of C. Then we can write

Cp̄ = [Spec(A)/μ_{r}],

for some integer r, and A is a flat O_{S,x̄}-algebra such that there exists an isomorphism

ι: A ⊗_{O_{S,x̄}} k(x̄) ≃ (k(x̄)[z, w]/(zw))^{sh},

with μ_{r}-action as in (v). We claim that we can lift this isomorphism to a μ_{r}-equivariant isomorphism

localiso (2.3.1)  A ≃ (O_{S,x̄}[z̃, w̃]/(z̃w̃ − t))^{sh}

for some t ∈ O_{S,x̄}. For this note that A^{μ_{r}} is equal to O_{C,p̄} and that A is a finite O_{C,p̄}-algebra. By a standard application of the Artin approximation theorem, in order to find 2.3.1 it suffices to prove the analogous statements for the map on completions (with respect to 𝔪_{C,p̄})

Ô_{C,p̄} → Â.

For this in turn we inductively find an element t_{q} ∈ O_{S,x̄}/𝔪_{S,x̄}^{q} and a μ_{r}-equivariant isomorphism

ρ_{q}: ((O_{S,x̄}/𝔪_{S,x̄}^{q})[z, w]/(zw − t_{q}))^{∧} → Â/𝔪_{S,x̄}^{q}Â.

For ρ_{1} we take the isomorphism induced by ι and t_{1} = 0. For the inductive step we assume (ρ_{q}, t_{q}) has been constructed and find (ρ_{q+1}, t_{q+1}). For this choose first any liftings z̃, w̃ ∈ Â/𝔪_{S,x̄}^{q+1}Â of z and w such that ζ ∈ μ_{r} acts by

goodaction (2.3.2)  z̃ ↦ ζ · z̃, w̃ ↦ ζ^{−1} · w̃.

This is possible because μ_{r} is linearly reductive.

Since Â is flat over O_{S,x̄} we have an exact sequence

0 → 𝔪_{S,x̄}^{q}/𝔪_{S,x̄}^{q+1} ⊗_{k(x̄)} Â/𝔪_{S,x̄} → Â/𝔪_{S,x̄}^{q+1}Â → Â/𝔪_{S,x̄}^{q}Â → 0.

Choosing any lifting t_{q+1} ∈ O_{S,x̄}/𝔪_{S,x̄}^{q+1} of t_{q} and consider

z̃w̃ − t_{q+1}.

This is a μ_{r}-invariant element of

𝔪_{S,x̄}^{q}/𝔪_{S,x̄}^{q+1} ⊗_{k(x̄)} Â/𝔪_{S,x̄} ≃ 𝔪_{S,x̄}^{q}/𝔪_{S,x̄}^{q+1} ⊗_{k(x̄)} ((O_{S,x̄}/𝔪_{S,x̄})[z, w]/(zw))^{∧}.

It follows that after possibly changing our choice of lifting t_{q+1} of t_{q}, we can write

z̃w̃ = t_{q+1} + z̃^{r}g + w̃^{r}h,

where t_{q+1} ∈ O_{S,x̄}/𝔪_{S,x̄}^{q+1} and reduces to t_{q}, and g, h ∈ 𝔪_{S,x̄}^{q}Â/𝔪_{S,x̄}^{q+1}Â are μ_{r}-invariant. Replacing z̃ by z̃− w̃^{r−1}h and w̃ by w̃ − z̃^{r−1}g (note that with these new choices the action of μ_{r} is still as in 2.3.2) we obtain t_{q+1} and ρ_{q+1}. ♠

We give an alternative proof of (v) using a description of the cotangent complex of C, which might be of interest on its own. Since C is an Artin stack a few words are in order. The cotangent complex 𝕃_{X /Y} of a morphism of Artin stacks X → Y is defined in [24], Chapter 17 as an object in the derived category of quasi-coherent sheaves in the lisse-étale site of X. Unfortunately, as was observed by Behrend and Gabber, this site is not functorial, which had the potential of rendering 𝕃_{X} both incomputable and useless. However, in [28], see especially section 8, it is shown that the lisse-étale site has surrogate properties replacing functoriality which are in particular sufficient for dealing with 𝕃_{X}, as an object of a carefully constructed D′_{qcoh}(X_{lis-ét}), and for deformation theory.

Two key properties are the following:

(1) given a morphism f: X → Y of algebraic stacks over a third Z, there is a natural distinguished triangle

Lf^{∗}𝕃_{Y/Z} → 𝕃_{X /Z} → 𝕃_{X /Y} →^{[1]} .

(2) Given a fiber diagram

\xymatrix{X′ \ar[r]^{f} \ar[d] & X \ar[d] \\ Y′ \ar[r] & Y}

with the horizontal maps flat, we have f^{∗}𝕃_{X /Y} = 𝕃_{X′/Y′}. If the base scheme is S, consider S → B𝔾_{m,S}. Applying the above to the

fiber square

\xymatrix{𝔾_{m,S} \ar[r] \ar[d] & S \ar[d] \\ S \ar[r] & B𝔾_{m,S}}

and the morphisms S → B𝔾_{m,S} → S it is easy to see that 𝕃_{B𝔾_{m,S}/S} = O_{B𝔾_{m,S}}[−1]. (In fact for a general smooth group scheme G we have 𝕃_{S/BG} = Lie(G)^{∗} and therefore 𝕃_{BG/S} = Lie(G)^{∗}[−1] considered as a G-equivariant object.)

Let now S = Spec k with k separably closed, and consider a twisted curve C/S with coarse moduli space π: C → C. Let p̄ → C be a geometric point mapping to a node, and fix an isomorphism

Cp̄ := C ×_{C} p̄ ≃ [D^{sh}/μ_{r}],

where D := Spec(k[z, w]/(zw)). By a standard limit argument, we can thicken this isomorphism to a diagram

\xymatrix{ & C_{U} \ar[dl]_{α} \ar[dr]^{β} \ar[dd]^{π_{U}} \\ C \ar[dd]^{π} &  & [D/μ_{r}] \ar[dd] \\  & U \ar[dl]_{a} \ar[dr]^{b} \\ C &  & D^{μ_{r}},}

where D^{μ_{r}} denote the coarse space of [D/μ_{r}] (so D^{μ_{r}} is equal to the spectrum of the invariants in k[z, w]/(zw)), a and b are étale, U is affine, and the squares are cartesian. In this setting we can calculate a versal deformation of C_{U} as follows.

First of all we have the deformation

[Spec(k[z, w, t]/(zw − t))/μ_{r}]

of [D/μ_{r}]. Since β is étale and representable and by the invariance of the étale site under infinitesimal thickenings, this also defines a formal deformation (i.e. compatible family of deformations over the reductions)

Ĉ_{U,t} → Spf(k[[t]])

of C_{U}. We claim that this deformation is versal. Since this deformation is nontrivial modulo t^{2}, to verify that Ĉ_{U,t} is versal it suffices to show that the deformation functor of C_{U} is unobstructed and that the tangent space is 1-dimensional. For this it suffices to show that

Ext^{2}(𝕃_{C_{U}} , O_{C}) = 0

and

Ext^{1}(𝕃_{C_{U}} , O_{C}) = k.

The map C_{U} → [D/μ_{r}] induces a morphism C_{U} → Bμ_{r}. Consider the composite map C_{U} → Bμ_{r} → B𝔾_{m}. Since 𝕃_{B𝔾_{m}/k} ≃ O_{S}[−1] we then obtain a distinguished triangle

\xymatrix{O_{C}[−1] \ar[r] & 𝕃_{C_{U}} \ar[r] & 𝕃_{C_{U} /B𝔾_{m}} \ar[r]^{[1]} & .}

Considering the associated long exact sequence of Ext-groups one sees that

Ext^{i}(𝕃_{C_{U}} , O_{C}) = Ext^{i}(𝕃_{C_{U} /B𝔾_{m}}, O_{C})

for i > 0. Now to prove that

Ext^{2}(𝕃_{C_{U} /B𝔾_{m}}, O_{C}) = 0, and Ext^{1}(𝕃_{C_{U} /B𝔾_{m}}, O_{C}) = k,

it suffices to show that 𝕃_{C/B𝔾_{m}} can be represented by a two-term complex

F^{−1} → F^{0}

with the F^{i} locally free sheaves of finite rank and the cokernel of the map

Hom(F^{0}, O_{C_{U}}) → Hom(F^{−1}, O_{C_{U}})

isomorphic to the structure sheaf O_{Z} of the singular locus Z ⊂ C_{U} (with the reduced structure).

Now since β is étale, the complex 𝕃_{C_{U} /B𝔾_{m}} is isomorphic to β^{∗}𝕃_{[D/μ_{r}]/B𝔾_{m}}. Therefore to construct such a presentation F^{·} of 𝕃_{C_{U} /B𝔾_{m}}, it suffices to construct a corresponding presentation of 𝕃_{[D/μ_{r}]/B𝔾_{m}}.

For this consider the fiber diagram

\xymatrix{X \ar[r] \ar[d] & C \ar[d] \\ S \ar[r] & B𝔾_{m}}

Here X is the surface Spec(k[z′, w′, u, u^{−1}]/(z′w′)) endowed with the action of 𝔾_{m} via t·(z′, w′, u) = (tz′, t^{−1}w′, t^{r}u). Since X/S is a local complete intersection, the map C → B𝔾_{m} is also a representable local complete intersection morphism, and therefore the natural map 𝕃_{C/B𝔾_{m}} → H_{0}(𝕃_{C/B𝔾_{m}}) = Ω_{C/B𝔾_{m}}^{1} is a quasi-isomorphism. Concretely, the pullback of 𝕃_{C/B𝔾_{m}} is isomorphic to Ω_{X/S}^{1}. Now we have a standard two-term resolution

0 → O_{X} → O_{X}^{3} → Ω_{X/S}^{1} → 0

where the first map is given by (w′, z′, 0) and the second by (dz′, dw′, du/u). Our desired presentation of 𝕃_{[D/μ_{r}]/B𝔾_{m}} is then obtained by noting that this resolution clearly descends to a locally free resolution on C.

Remark 2.4. Let S be a scheme. A priori the collection of twisted n-pointed curves over S (with morphisms only isomorphisms) forms a 2-category. However, by the same argument proving [1, 4.2.2] the 2-category of twisted npointed curves over S is equivalent to a 1-category. In what follows we will therefore consider the category of twisted n-pointed curves, whose arrows are isomorphism classes of 1-arrows in the 2-category.

cohlemb Proposition 2.5. Let f: C → S be a twisted curve. Then the adjunction map O_{S} → f_{∗}O_{C} is an isomorphism, and for any quasi-coherent sheaf F on C we have R^{i}f_{∗}F = 0 for i ≥ 2.

Proof. Let π: C → C be the coarse space, and let f̄: C → S be the structure morphism. Since π_{∗} is an exact functor on the category of quasi-coherent sheaves on C, we have for any quasicoherent sheaf F on C an equality

R^{i}f_{∗}F = R^{i}f̄_{∗}(π_{∗}F).

Since f̄: C → S is a nodal curve this implies that R^{i}f_{∗}F = 0 for i ≥ 2, and the first statement follows from the fact that π_{∗}O_{C} = O_{C}. ♠

We conclude this section by recording some facts about the Picard functor of a twisted curve (these results will be used in section 5 below).

Let S be a scheme and f: C → S a twisted curve over S. Let 𝒫ic_{C/S} denote the stack over S which to any S-scheme T associates the groupoid of line bundles on the base change C_{T} := C ×_{S} T.

Proposition 2.6. The stack 𝒫ic_{C/S} is a smooth locally finitely presented^{3}

3→

algebraic stack over S, and for any object L ∈ 𝒫ic_{C/S}(T) (a line bundle on C_{T}), the group scheme of automorphisms of L is canonically isomorphic to 𝔾_{m,T}.

Proof. That 𝒫ic_{C/S} is algebraic and locally finitely presented is a standard application of Artin’s criterion for verifying that a stack is algebraic [5]. This stack can also be constructed as a \underline{Hom}-stack. See [6] for details.

To see that 𝒫ic_{C/S} is smooth we apply the infinitesimal criterion. For this it suffices to consider the case where S affine and we have a closed subscheme S_{0} ⊂ S with square-0 ideal I. Writing C_{0} for C ×_{S} S_{0}, we have an exact sequence

3(Angelo) Added “locally finitely presented”

0 → O_{C} ⊗ I → O_{C} → O_{C_{0}} → 0

giving an exact sequence of cohomology

𝒫ic_{C/S}(S) → 𝒫ic_{C/S}(S_{0}) → R^{2}f_{∗}O_{C} ⊗ I.

By Proposition 2.5 the term on the right vanishes, giving the existence of the desired lifting of 𝒫ic_{C/S}(S) → 𝒫ic_{C/S}(S_{0}).

The statement about the group of automorphisms follows from the fact that the map O_{S} → f_{∗}O_{C} is an isomorphism, and the same remains true after arbitrary base change T → S. ♠

Let Pic_{C/S} denote the rigidification of 𝒫ic_{C/S} with respect to the group scheme 𝔾_{m} [4, Appendix A].

Let π: C → C denote the coarse moduli space of C (so C is a nodal curve over S). Pulling back along π defines a fully faithful functor

π^{∗}: 𝒫ic_{C/S} → 𝒫ic_{C/S}

which induces a morphism

Picmapb (2.6.1)  π^{∗}: Pic_{C/S} → Pic_{C/S}.

This morphism is a monomorphism of group schemes over S. Indeed suppose given two S-valued points [L_{1}], [L_{2}] ∈ Pic_{C/S}(S) defined by line bundles L_{1} and L_{2} on C such that π^{∗}[L_{1}] = π^{∗}[L_{2}]. Then after possibly replacing S by an étale cover, the two line bundles π^{∗}L_{1} and π^{∗}L_{2} on C are isomorphic. Since L_{i} = π_{∗}π^{∗}L_{i} (i = 1, 2) this implies that L_{1} and L_{2} are also isomorphic.

cokernel Lemma 2.7. The cokernel W of the morphism π^{∗} in (2.6.1) is an étale group scheme over S.

Remark 2.8. Note that, in general, W is non-separated. Consider S the affine line over a field k with parameter t, and consider the blowup X of ℙ^{1} × S at the origin. The zero fiber has a singularity with local equation uv = t. There is a natural action of μ_{r} along the fiber, which near the singularity looks like (u, v) ↦ (ζ_{r}u, ζ_{r}^{−1}v). The quotient C = [X/μ_{r}] is a twisted curve, with twisted markings at the sections at 0 and ∞, and a twisted node. The coarse curve C = X/μ_{r} has an A_{r−1} singularity xy = t^{r}, where x = u^{r} and y = v^{r}. Let E ⊂ C be one component of the singular fiber. The invertible sheaf O_{C}(E) gives a generator of W, but it coincides with the trivial sheaf away from the singular fiber. In fact W is the group-scheme obtained by gluing r copies of S along the open subset t ≠ 0.

Proof. The space W is the quotient of a smooth group scheme over S by the action of a smooth group scheme, hence is smooth over S. To show that W is étale we use the infinitesimal lifting criterion. If T_{0} ↪ T is a square zero nilpotent thickening defined by a sheaf of ideals I ⊂ O_{T}, and if L is a line bundle on C_{T} whose reduction L_{0} on C_{T_{0}} is obtained by pullback from a line bundle on C_{T_{0}}, then we need to show that L is the pullback of a line bundle on C_{T}. Using the exponential sequence

0 ⟶ I ⟶ O_{T}^{×} ⟶ O_{T_{0}}^{×} ⟶ 1

this amounts to the statement that the map

H^{i}(C_{T_{0}}, IO_{C_{T}}) → H^{i}(C_{T_{0}}, IO_{C_{T}})

is an isomorphism.^{4} ♠

4→

Lemma 2.9. The cokernel W is quasi-finite over S.

Proof. ^{5} We may assume that S is affine, say S = Spec A. Then the twisted

5→

curve C is defined over some finitely generated ℤ-subalgebra A_{0}, and formation of W commutes with base change; hence we may assume that A = A_{0}. In particular S is noetherian.

We may also assume that S is reduced. After passing to a stratification of S and to étale coverings, we may assume that S is connected, that C → S is of constant topological type, that the nodes of the geometric fibers of C → S are supported along closed subschemes S_{1}, …, S_{k} of C which map isomorphically onto S, and the reduced inverse image Σ_{i} of S_{i} in C is isomorphic to the classfying stack Bμ_{m_{i}} for certain positive integers m_{1}, …, m_{k}. Given a line bundle L on C, we can view the pullback of L to Σ_{i} as a line bundle L_{i} on S_{i}, with an action of μ_{m_{i}}. Such action is given by a character μ_{m_{i}} → 𝔾_{m} of the form t ↦ t^{a_{i}} for some a_{i} ∈ ℤ/(m_{i}). By sending L into the collection (a_{i}) we obtain a morphism Pic_{C/S} → ∏_{i} ℤ/(m_{i}), whose kernel is easily seen to be Pic_{C/S}. Therefore we have a categorically injective map W → ∏_{i} ℤ/(m_{i}); since W and ∏_{i} ℤ/(m_{i}) are étale over S, this is an open embedding. Since ∏_{i} ℤ/(m_{i}) is noetherian and quasi-finite over S, so is W. ♠

If N denotes an integer annihilating W we obtain a map

×N: Pic_{C/S} → Pic_{C/S}.

Def:Pic^o Definition 2.10. Let Pic_{C/S}^{o} denote the fiber product

Pic_{C/S}^{o} = Pic_{C/S} ×_{×N,Pic_{C/S}} Pic_{C/S}^{o}.

The open and closed subfunctor Pic_{C/S}^{o} ⊂ Pic_{C/S} classifies degree 0-line bundles on C. In particular, Pic_{C/S}^{o} is independent of the choice of N in the above construction. Note also that any torsion point of Pic_{C/S} is automatically contained in Pic_{C/S}^{o} since the cokernel of

Pic_{C/S}^{o} → Pic_{C/S}

is torsion free.

Since Pic_{C/S}^{o} is a semiabelian scheme over S, we obtain the following:

4(Dan) Changed here a bit. Unless I misunderstood the exponential sequence, we need isomorphism of H^{2}.

5(Angelo) changed the proof

3.7 Corollary 2.11. The group scheme Pic_{C/S}^{o} is an extension of a quasi-finite étale group scheme over S by a semiabelian group scheme.

3. Interlude: Relative moduli spaces

interlude

In this section we gather some results on relative moduli spaces which will be used in the verification of the valuative criterion for moduli stacks of twisted stable maps in the next section.

For a morphism of algebraic stacks f: X → Y, denote by I(X /Y) = Ker (I(X) → f^{∗}I(Y)) = X ×_{X ×_{Y}X} X the relative inertia stack.

Now let f: X → Y be a morphism of algebraic stacks, locally of finite presentation, and assume the relative inertia I(X /Y) → X is finite. Recall that when Y is an algebraic space we know that the morphism factors through the coarse moduli space: X → X → Y, see [21, 10]. We claim that there is a relative construction that generalizes to our situation:

Th:rel-cms Theorem 3.1. There exists an algebraic stack X, and morphisms X →^{π}

X →^{\overline{f}} Y such that f = \overline{f} ∘ π, satisfying the following properties:

(1) \overline{f}: X → Y is representable,

(2) X →^{π} X →^{\overline{f}} Y is initial for diagrams satisfying (1), namely if X →^{π′}

X′ →^{\overline{f′}} Y has representable \overline{f′} then there is a unique h: X → X′ such that π′ = h ∘ π and \overline{f} = \overline{f}′ ∘ h,

(3) π is proper and quasifinite,

(4) O_{X} → π_{∗}O_{X} is an isomorphism, and

(5) if X′ is a stack and X′ → X is a representable flat morphism, then X ×_{X} X′ → X′ → Y satisfies properties (1)-(4) starting from X ×_{X} X′ → Y.

Definition 3.2. We call \overline{f} : X→Y the relative coarse moduli space of f: X → Y.

Proof. Consider a smooth presentation R ⇉ U of Y. Denote by X_{R} the coarse moduli space of X_{R} = X ×_{Y} R and X_{U} the coarse moduli space of X_{U} = X ×_{Y} U, both exist by the assumption of finiteness of relative inertia.

Since the formation of coarse moduli spaces commutes with flat base change, we have that X_{R} = X_{U} ×_{U} R. It is straightforward to check that X_{R} ⇉ U_{R} is a smooth groupoid. Denote its quotient stack X := [X_{R} ⇉ U_{R}].

It is again straightforward to construct the morphisms X →^{π} X →^{\overline{f}} Y from the diagram of relevant groupoids.

Now X ×_{Y} U → X_{U} is an isomorphism, U → Y is surjective and X_{U} → U is representable, hence X → Y is representable, giving (1). Similarly X_{U} → X_{U} is proper and quasifinite hence X → X is proper and quasifinite, giving (3). Also (4) follows by flat base-change to X_{U}. Part (5) follows again since formation of moduli spaces commutes with flat base change: the coarse moduli space of X_{U} ×_{X} X′ is X_{U} ×_{X} X′ and similarly the coarse moduli space of X_{R} ×_{X} X′ is X_{R} ×_{X} X′.

Now consider the situation in (2). Since X′ → Y is representable, it is presented by the groupoid X′ ×_{Y} R ⇉ X′ ×_{Y} U, which, by the universal property of coarse moduli spaces, is the target of a canonical morphism of groupoids from X_{R} ⇉ U_{R}, giving the desired morphism h: X → X′. ♠

Now we consider the tame case. We say that a morphism f: X → Y with finite relative inertia is tame if for some algebraic space U and faithfully flat U → Y we have that X ×_{Y} U is a tame algebraic stack. This notion is independent of the choice of U by [4, Theorem 3.2]. Let X → X → Y be the relative coarse moduli space.

Proposition 3.3. Assume f: X → Y is tame. If X′ is an algebraic stack and X′ → X is a representable morphism of stacks, then X′ → Y is the relative coarse moduli space of X ×_{X} X′ → Y.

This is proven as in part (5) of the theorem, using the fact that formation of coarse moduli spaces of tame stacks commutes with arbitrary base change.

Now we consider a special case which is relevant for our study of tame stacks.

Let V be a strictly henselian local scheme, with a finite linearly reductive group scheme Γ acting, fixing a geometric point s → V. Let X = [V/Γ], and consider a morphism f: X → Y, with Y also tame.

Let K ⊂ Γ be the subgroup scheme fixing the composite object s → V → Y (so K is the kernel of the map Γ → Aut_{Y}(f(s))). Consider the geometric quotient U = V/K, the quotient group scheme Q = Γ/K and let \overline{X} = [U/Q]. There is a natural projection q: X → \overline{X}.

Prop:B3 Proposition 3.4. There exists a unique morphism g: \overline{X} → Y such that f = g ∘ q, and this factorization of f identifies \overline{X} as the relative coarse moduli space of X → Y.

Proof. Consider the commutative diagram

\xymatrix{X \ar[rr]^{f} \ar[r]^{f′} \ar[dr] & Y_{X} \ar[r]^{h} \ar[d] & Y \ar[d] \\  & X \ar[r]^{\overline{f}} & Y}

where X, Y are the coarse moduli spaces of X and Y, respectively, the square is cartesian, and \overline{f} is induced from f.

Since h is fully faithful we may replace Y by Y_{X}. As the formation of coarse moduli space of tame stacks commutes with arbitrary base change, X is the coarse moduli space of Y_{X}. We may therefore assume that X = Y.

It also suffices to prove the proposition after making a flat base change on X. We may therefore assume that the residue field of V is algebraically closed. We can write Y = [P/G], where P is a finite X-scheme and G is a finite flat split linearly reductive group scheme over X. We can assume P is local strictly henselian as well, and that the action of G on P fixes the closed point p ∈ P.

The map f induces, by looking at residual gerbes, a map Bρ_{0} : BΓ_{0} → BG_{0} where Γ_{0} and G_{0} denote the corresponding reductions to the closed point of X. This is induced by a group homomorphism ρ_{0} : Γ_{0} → G_{0}, uniquely defined up to conjugation. Since Γ and G are split, the homomorphism ρ_{0} lifts to a homomorphism ρ: Γ → G.

We obtain a 2-commutative diagram

\xymatrix{BΓ_{0} \ar[r]^{Bρ_{0}} \ar[d]_{s̄} & BG_{0} \ar[d]^{p̄} \\ [V/Γ] \ar[r]^{f} & [P/G].}

Let P′ → [V/Γ] be the pullback of the G-torsor P → [P/G]

Lemma 3.5. There is an isomorphism of G-torsors over [V/Γ]

[V × G/Γ] → P′.

Proof. Note that V ×_{[V/Γ]} BΓ_{0} = s and P′ ×_{[V/Γ]} BΓ_{0} = BΓ_{0} ×_{BG_{0}} p. Then the commutativity of the diagram

\xymatrix{s \ar[r] \ar[d] & p \ar[d] \\ BΓ_{0} \ar[r] & BG_{0}}

implies that there is an isomorphism of G_{0}-torsors over BΓ_{0}

\xymatrix{([V × G/Γ] ×_{[V/Γ]} BΓ_{0} \ar[r]^{∼} & P′ ×_{[V/Γ]} BΓ_{0}.}

As in the proof of [4, 3.6], this lifts to an isomorphism as required. ♠

Now we have a morphism

\xymatrix{V \ar[r] \ar[rrr]^{f̃} & [V × G/Γ] \ar[r] & P′ \ar[r] & P}

compatible with the actions of Γ and G. By the universal property of the quotient, the map f̃ factors through U. Passing to the quotient by the respective group actions, we get morphisms

[V/Γ] → [U/Q] → [P/G].

Let X → Z → Y be the relative coarse moduli space. Since Q injects in G, the morphism [U/Q] → [P/G] is representable, and therefore we have a morphism Z → [U/Q] over Y.

To check this is an isomorphism it suffices to check after base change along the flat morphism P → Y. In other words, we need to check that [V/Γ] ×_{[P/G]} P → [U/Q] ×_{[P/G]} P is the coarse moduli space. This map can be identified with [V × G/Γ] → U ×^{Q} G. This follows by noting that

(O_{V} ⊗ O_{G})^{Γ} = ((O_{V})^{K} ⊗ O_{G})^{Q} = (O_{U} ⊗ O_{G})^{Q}.
♠

4. Twisted stable maps

Section:3

This section relies on the results of Appendix A, in which the stack of twisted curves is constructed.

Let M be a finitely presented tame algebraic stack with finite inertia proper over a scheme S. A twisted stable map from an n-marked twisted curve (f: C → S, {Σ_{i} ⊂ C}_{i=1}^{n}) over S to M is a representable morphism of S-stacks

g: C → M

such that the induced morphism on coarse spaces

\overline{g}: C → M

is a stable map (in the usual sense of Kontsevich) from the n-pointed curve (C, {p_{1}, …, p_{n}}_{i=1}^{n}) to M (where the points p_{i} are the images of the Σ_{i}).

If M is projective over a field, the target class of g is the class of \overline{g}_{∗}[C] in the chow group of 1-dimensional cycles up to numerical equivalence of M.

Th:3.1 Theorem 4.1. Let M be a finitely presented tame algebraic stack proper over a scheme S with finite inertia. Then the stack K_{g,n}(M) of twisted stable maps from n-pointed genus g curves into M is a locally finitely presented algebraic stack over S, which is proper and quasi-finite over the stack of stable maps into M. If M is projective over a field, the substack K_{g,n}(M, β) of twisted stable maps from n-pointed genus g curves into M with target class β is proper, and is open and closed in K_{g,n}(M).

Proof. The statement is local in the Zariski topology of S, therefore we may assume S affine. Since M is of finite presentation, it is obtained by base change from the spectrum of a noetherian ring, and replacing S by this spectrum we may assume M is of finite type over a noetherian scheme S.

As M is now of finite type, there is an integer N such that the degree of the automorphism group scheme of any geometric point of M is ≤ N.

Consider the stack of twisted curves 𝔐_{g,n}^{tw} defined in A.6. It contains an open substack 𝔐_{g,n}^{tw,≤N} ⊂ 𝔐_{g,n}^{tw} of twisted curves of index bounded by N by A.8. The natural morphism 𝔐_{g,n}^{tw,≤N} → 𝔐_{g,n} to the stack of prestable curves is quasi-finite by A.8.

Consider 𝔅 = 𝔐_{g,n}^{tw,≤N} ×_{Spec ℤ} S. It is an algebraic stack of finite type over B = 𝔐_{g,n} ×_{Spec ℤ} S. Denote M_{𝔅} = M ×_{S} 𝔅, M_{𝔅} = M ×_{S} 𝔅 and similarly M_{B} = M ×_{S} B. These are all algebraic stacks.

Finally consider the universal curves ℭ^{tw} → 𝔐_{g,n}^{tw} and ℭ → 𝔐_{g,n}, and their corresponding pullbacks ℭ_{𝔅}^{tw}, ℭ_{𝔅} and ℭ_{B}.

As discussed in Appendix C, there is an algebraic stack

\underline{Hom}_{𝔅}^{rep}(ℭ_{𝔅}^{tw}, M_{𝔅}),

locally of finite type over 𝔅, parametrizing representable morphisms of the universal twisted curve to M. We also have the analogous \underline{Hom}_{𝔅}(ℭ_{𝔅}, M_{𝔅}) and \underline{Hom}_{B}(ℭ_{B}, M_{B}). We have the natural morphisms

compositemap (4.1.1) \xymatrix{\underline{Hom}_{𝔅}^{rep}(ℭ_{𝔅}^{tw}, M_{𝔅}) \ar[r]^{a} & \underline{Hom}_{𝔅}(ℭ_{𝔅}, M_{𝔅}) \ar[r]^{b} & \underline{Hom}_{B}(ℭ_{B}, M_{B}).}

5.2 Proposition 4.2. The morphism a is quasi-finite.

Proof. It suffices to prove the following. Let k be an algebraically closed field with a morphism Spec(k) → S and C/k a twisted curve with coarse moduli space C. Let f: C → M be a S-morphism, and let G → C denote the pullback of M to C along the composite

\xymatrix{C \ar[r] & C \ar[r]^{f} & M.}

We then need to show that the groupoid \underline{Sec}(G/C)(k) (notation as in C.17) has only finitely many isomorphism classes of objects.

We first reduce to the case where C is smooth. Let C′ denote the normalization of C (so C′ is a disjoint union of smooth curves), and let C′ → C be the maximal reduced substack of the fiber product C ×_{C} C′. For any node x ∈ C let Σ_{x} denote maximal reduced substack of C_{x} (so Σ_{x} is isomorphic to Bμ_{r} for some r), and let Σ denote the disjoint union of the Σ_{x} over the nodes x. Then there are two natural inclusions j_{1}, j_{2} : Σ ↪ C′, and by [3, A.0.2] the functor

\underline{Hom}(C, M)(k)
↓
\underline{Hom}(C′, M)(k) ×_{j_{1}^{∗}×j_{2}^{∗},\underline{Hom}(Σ,M)(k)×\underline{Hom}(Σ,M)(k),Δ} \underline{Hom}(Σ, M)(k)

is an equivalence of categories. Since the stack \underline{Hom}(Σ, M) clearly has quasi-finite diagonal, it follows that the map

\underline{Sec}(G/C) → \underline{Sec}(G ×_{C} C′/C′)

is quasi-finite.

It follows that to prove 4.2 it suffices to consider the case when C is smooth.

Let Y denote the normalization of G. Then any section C → G factors uniquely through Y so it suffices to show that the set

\underline{Sec}(Y/C)(k)

is finite. To prove this we may without loss of generality assume that there exists a section s: C → Y.

For any smooth morphism V → C the coarse space of the fiber product G_{V} := G ×_{C} V is equal to V by [4], Corollary 3.3 (a). For a smooth morphism V → C, let

\xymatrix{Y_{V} \ar[r]^{c} & \overline{Y}_{V} \ar[r]^{d} & V}

be the factorization of Y_{V} → V provided by [35, 2.1] (rigidification of the generic stabilizer). The section s induces a section V → \overline{Y}_{V}, and hence by [30, 2.9 (ii)] the projection d: \overline{Y} → V is in fact an isomorphism. It follows that Y is a gerbe over C. If 𝔾 → C denotes the automorphism group scheme of s, then in fact Y = B_{C}𝔾.

Let Δ ⊂ 𝔾 denote the connected component of 𝔾, and let ℍ denote the étale part of 𝔾, so we have a short exact sequence

1 → Δ → 𝔾 → ℍ → 1

of group schemes over C.

Let C′ → C be a finite étale cover such that ℍ and the Cartier dual of Δ constant. If C′′ denotes the product C′ ×_{C} C′ and Y′ and Y′′ the pullbacks of Y to C′ and C′′ respectively, then one sees using a similar argument to the one used in [30, §3] that it suffices to show that

\underline{Sec}(Y′/C′)(k) and \underline{Sec}(Y′′/C′′)(k)

are finite. We may therefore assume that ℍ and the Cartier dual of Δ are constant.

The map 𝔾 → ℍ induces a projection over C

B_{C}𝔾 → B_{C}ℍ.

Giving a section z: C → B_{C}ℍ is equivalent to giving an ℍ-torsor P → C. Since C is normal and ℍ is étale, such a torsor is determined by its restriction to any dense open subspace of C. Since C contains a dense open subscheme and the étale fundamental group of such a subscheme is finitely generated it follows that

\underline{Sec}(B_{C}ℍ/C)(k)

is finite.

Fix a section z: C → B_{C}ℍ, and let Q → C be the fiber product

Q := C ×_{z,B_{C}ℍ} B_{C}𝔾.

Then Q is a gerbe over C banded by a twisted form of Δ. By replacing C by a finite étale covering as above, we may assume that in fact Q is a gerbe banded by Δ and hence isomorphic to C ×_{Spec(k)} BΔ_{0} for some finite diagonalizable group scheme Δ_{0}/k. Writing Δ_{0} as a product of μ_{r}’s, we even reduce to the case when Δ_{0} = μ_{r}. In this case, as explained in C.25, giving a section w: C → B_{C}Δ is equivalent to giving a pair (L, ι), where L is a line bundle on C and ι: O_{C} → L^{⊗r} is an isomorphism. We conclude that the set of sections C → Q is in bijection with the set

\underline{Pic}_{C/k}[r](k)

of r-torsion points in the Picard space \underline{Pic}_{C/k}. By 2.11 this is a finite group, and hence this completes the proof of 4.2. ♠

Consider again the diagram 4.1.1. The morphism a is quasi-finite by 4.2 and of finite type by C.7. The second morphism is quasi-finite and of finite type as it is obtained by base change. Therefore the composite 4.1.1 is also quasi-finite and of finite type.

The Kontsevich moduli space \overline{M}_{g,n}(M) is open in \underline{Hom}_{B}(ℭ_{B}, M_{B}). We have

K_{g,n}(M) ≃ \underline{Hom}_{𝔅}^{rep}(ℭ_{𝔅}^{tw}, M_{𝔅}) ×_{\underline{Hom}_{B}(ℭ_{B},M_{B})} \overline{M}_{g,n}(M),

hence K_{g,n}(M) → \overline{M}_{g,n}(M) is quasi-finite and of finite type.

Properness now follows from the valuative criterion, which is the content of the following Proposition 4.3. ♠

Prop:valuative Proposition 4.3. Let R be a discrete valuation ring, with spectrum T having generic point η and special point t. Fix a morphism T → S and let C_{η} → M be an n-pointed twisted stable map over the generic point of T, and C → M a stable map over T extending the coarse map C_{η} → M. Then there is a discrete valuation ring R_{1} containing R as a local subring, and corresponding morphism of spectra T_{1} → T, and an n-pointed twisted stable map C_{T_{1}} → M, such that the restriction C_{T_{1}}|_{η_{1}} → M is isomorphic to the base change C_{η} ×_{T} T_{1} → M, and the coarse map coincides with C ×_{T} T_{1} → M.

Such an extension, when it exists, is unique up to a unique isomorphism.

Before proving this Proposition, we need to extend a few basic results known in case M is a Deligne–Mumford stack.

Lem:purity Lemma 4.4 (Purity Lemma). Let M be a separated tame stack with coarse

moduli space M. Let X be a locally noetherian^{6} separated scheme of dimen-

←6

sion 2 satisfying Serre’s condition S2. Let P ⊂ X be a finite subset consisting of closed points, U = X ∖ P. Assume that the local fundamental groups of U around the points of P are trivial and that the local Picard groups of U around points of P are torsion free.

Let f: X → M be a morphism. Suppose there is a lifting f̃_{U}: U → M:

\xymatrix{ &  & M \ar[d] \\ U \ar[r] \ar[urr]^{f̃_{U}} & X \ar[r]^{f} & M}

Then the lifting extends to X:

[figure: M; U; f̃_{U}; X; f; f̃; M]

6(Angelo) Added “locally noetherian”.

The lifting f̃ is unique up to a unique isomorphism.

Remark 4.5. We will use this lemma principally in the following cases: (1) X is regular,

(2) X is normal crossings, locally Spec(R[x, y]/xy)^{sh}, with R regular, or (3) X = X_{0} × G, with X_{0} one of the first two cases and G a linearly

reductive finite group scheme.

Proof. The question is local in the étale topology, so we may replace X by its strict henselization over some point p ∈ P, and correspondingly we may replace M and M by the strict henselization at f(p).

By [4, Theorem 3.2 (d)]we can write M = [V/G], where V → M is a finite morphism and G → M a linearly reductive group scheme acting on V. By [4] Lemma 2.20 we have an exact sequence

1 ⟶ Δ ⟶ G ⟶ H ⟶ 1

of group schemes over M, where Δ is diagonalizable and H is tame and étale.

The morphism f̃_{U} is equivalent to the data of a G-torsor P_{U} → U and a G-equivariant morphism U → V. We first wish to extend P_{U} over X.

Consider the H-torsor Q_{U} =^{def} P_{U}/Δ → U. Since the local fundamental groups of U around P are trivial, this H torsor is trivial, and extends trivially to an H torsor Q → X.

Now P_{U} → Q_{U} is a Δ-torsor. We claim that it extends uniquely to a Δ-torsor P → Q. Since Δ is diagonalizable, it suffices to treat the case Δ = μ_{r}. In this case P_{U} → Q_{U} corresponds to an r-torsion line bundle with a chosen trivialization of its r-th power. The line bundle extends to X by the assumption on the local Picard groups, and the trivialization extends by the S2 assumption.

Next we need to extend the principal action of G on P_{U} to P. The closure Γ of the graph of the morphism G ×_{M} P_{U} → P_{U} inside the scheme G×_{M} P ×_{X} P is finite over G×_{M} P, and an isomorphism over G×_{M} P_{U}. As G ×_{M} P is finite and flat over the S2 scheme X, it is also S2. It follows that the morphism Γ → G ×_{M} P is an isomorphism, hence the action extends to a morphism G ×_{M} P → P. It is easy to check that this is a principal action, as required.

The same argument shows that the equivariant morphism P_{U} → V extends to an equivariant morphism P → V, as required. ♠

Proof of Proposition 4.3. Step 1: We extend C_{η} → M over the generic points of the special fiber C_{s}.

The question is local in the étale topology of C, therefore we can replace C by its strict henselization at one of the generic points of C_{s}. We may similarly replace M by its strict henselization at the image point, and hence assume it is of the form [V/G], where V → M is finite and G a locally well split group scheme, see [4], Theorem 3.2 (d) and Lemma 2.20. We again write G as an extension

1 ⟶ Δ ⟶ G ⟶ H ⟶ 1.

The morphism C_{η} = C_{η} → M is equivalent to the data of a G-torsor P_{η} → C_{η} and a G-equivariant morphism P_{η} → V. By Abhyankar’s lemma, the H-torsor Q_{η} =^{def} P_{η}/Δ → C_{η} extends uniquely to Q → C after a base change on R. We want to extend the Δ-torsor P_{η} → Q_{η} to a Δ-torsor P → Q. Factoring Δ, we may assume Δ = μ_{r}. In this case P_{η} → Q_{η} corresponds to an r-torsion line bundle with a chosen trivialization of its r-th power. This line bundle is trivial since we have localized, hence it trivially extends to Q. The extension of the trivializing section may have a zero or pole along C_{s}, but after an r-th order base change on R this zero or pole has multiplicity r·l for some integer l. Twisting the line bundle by O_{C}(lC_{s}), the trivialization extends uniquely over C_{s} as required.

We need to lift the principal action of G on P. Write again Γ for the closure of the graph of G ×_{T} P_{η} → P_{η} inside G ×_{T} P_{η} ×_{C} P_{η}. This is a Δequivariant subscheme, with respect to the action δ·(g, p_{1}, p_{2}) = (δg, p_{1}, δp_{2}). The action is free, as it is already free on the last factor. Therefore the projection Γ → Γ/Δ is a Δ-torsor. The quotient Γ/Δ includes the subscheme (Γ/Δ)_{η} = Q_{η}. We claim that this is scheme-theoretically dense in Γ/Δ, because any nonzero sheaf of ideals with support over s would pull back to a nonzero sheaf of ideals on Γ. But Γ_{η} is by definition scheme theoretically dense in Γ. Since Q is normal, it follows that Γ/Δ = Q. Now as both Γ and P are Δ-torsors over Q and Γ → P is Δ-equivariant, this morphism is an isomorphism. Hence we have a morphism G×P → P, extending the action. It is again easy to see this is a principal action with quotient C. It is also clearly unique.

Exactly the same argument shows that the morphism P_{η} → V extends uniquely to a G-equivariant P → V, as required.

Step 2: We extend the twisted stable map C → M over the general locus C_{gen}, simply by applying the purity lemma (Lemma 4.4). Uniqueness in the Purity Lemma implies that the extension is unique up to unique isomorphism.

Step 3: We extend the twisted curve C and the twisted stable map C → M along the smooth locus of C.

Consider the index r_{i} of C_{η} over a marking Σ_{i}^{C} of C. Let C_{sm} be the stack obtained by taking r_{i}-th root of Σ_{i}^{C} on C for all i. Then (C_{sm})_{η} is uniquely isomorphic to (C_{η})_{sm}.

We need to construct a representable map C_{sm} → M lifting C_{sm} → M. The problem is local in the étale topology of C_{sm}, so we may present C_{sm} at a point p on Σ_{i}^{C} ∩ C_{s} as [D/μ_{r_{i}}], where D is smooth over T, with unique fixed point q over p. Applying the purity lemma to the map D∖{q} → M, we have a unique extension D → M. The uniqueness applies also for μ_{r_{i}} × D → M, which implies that the object D → M is μ_{r_{i}}-equivariant, giving a unique morphism [D/μ_{r_{i}}] → M, as needed.

Step 4: We extend the twisted curve C and the twisted stable map C → M over the closure of the singular locus of C_{η}.

This is similar to step 3.

Step 5: Extension over isolated singular points.

Consider an isolated node p of C. Passing to an extension on R we may assume it is rational over the residue field k. Its strict henselization is isomorphic to (Spec R[u, v]/(uv − π^{l}))^{sh}, where π is a uniformizer of R, and by Remark A.9 there is a twisted curve C_{l} which is regular over p with index l. This twisted curve has a chart of type [D/μ_{l}], where the strict henselization of D looks like Spec R[x, y]/(xy − π), where μ_{l} acts via (x, y) ↦ (ζ_{l}x, ζ_{l}^{−1}y), and u = x^{l}, v = y^{l}. There is a unique point q of D over p.

To construct C_{l} → M it suffices to apply the purity lemma to the map D∖{q} → M; the resulting extension D → M is μ_{l}-equivariant by applying the Purity Lemma to μ_{l} × D → M.

We need to replace the morphism C_{l} → M, as it is not necessarily representable, and the construction of C_{l} does not commute with base change.

Consider the morphism C_{l} → C × M. This morphism is proper and quasi-finite. Let C_{l} → C → C × M be the relative moduli space. By the local computation (Proposition 3.4), C is a twisted curve, hence C → M is a twisted stable map. Further, its formation commutes with further base change since the formation of moduli space does. ♠

5. Reduction of Spaces of Galois admissible covers Let S be a scheme and G/S a locally well-split finite flat group scheme.

We can then consider the stack K_{g,n}(BG) of twisted stable maps from genus g twisted curves with n marked points to the classifying stack BG. This is the stack which to any S-scheme T associates the groupoid of data

(C, {Σ_{i}}_{i=1}^{n}, h: C → BG),

where (C, Σ_{i}}_{i=1}^{n} is an n-marked twisted curve such that the coarse space (C, {p_{i}}_{i=1}^{n}) (with the marking induced by the Σ_{i}) is a stable n-pointed curve in the sense of Deligne-Mumford-Knudsen, and h is a representable morphism of stacks over S. As in [2], this is equivalent to the data

(C, {Σ_{i}}_{i=1}^{n}, P → C),

where P → C is a principal G-bundle, with P representable. When G is tame and étale, P → C is an admissible G-cover, and the stack K_{g,n}(BG) has been studied extensively. See [2] for discussion and further references.^{7}

The main result of this section is the following:

flattheorem Theorem 5.1. The stack K_{g,n}(BG) is flat over S, with local complete intersection fibers^{8}.

reductionsection

7→ 8→

7(Dan) added discussion

8(Angelo) I have added this. Correspondingly I have had to add several bits in the proof below.

Here we say that an algebraic stack X is local complete intersection if it admits a smooth surjective morphism from an local complete intersection algebraic space. This is equivalent to the condition that locally 𝕃_{X} be a perfect complex supported in degrees [−1, 1].^{9}

←9

In particular, it follows that K_{g,n}(BG) is Cohen–Macaulay if S is Cohen– Macaulay, and Gorenstein if S is Gorenstein.

Given a scheme X of finite type over a field k and an extension k′ of k, it is well known that X is a local complete intersection if and only if X_{k′} is; hence the statement of Theorem 5.1 is local in the fpqc topology on S.^{10}

←10

To prove this theorem we in fact prove a stronger result.

Let M_{g,n}^{tw}(G) denote the stack over S which associates to any S-scheme T the groupoid of data

(C, {Σ_{i}}_{i=1}^{n}, h: C → BG),

where (C, {Σ_{i}}_{i=1}^{n}) is an n-marked twisted curve over T and h is a morphism of stacks over S (so (C, {p_{i}}_{i=1}^{n}) is not required to be stable and h is not necessarily representable).

Lemma 5.2. The natural inclusion K_{g,n}(BG) ⊂ M_{g,n}^{tw}(G) is representable by open immersions.

Proof. This is because the condition that an n-pointed nodal curve is stable is an open condition as is the condition that a morphism of stacks C → BG is representable, see Corollary C.7 and the discussion preceding it. ♠

To prove 5.1 it therefore suffices to show that M_{g,n}^{tw}(G) is flat over S. For this we in fact prove an even stronger result. Let M_{g,n}^{tw} denote the stack defined in A.6. Forgetting the map to BG defines a morphism stacks

flatmap (5.2.1)  M_{g,n}^{tw}(G) → M_{g,n}^{tw} ×_{Spec(ℤ)} S.

Since M_{g,n}^{tw} is smooth over ℤ by A.6 the following theorem implies 5.1.

flattheorembis Theorem 5.3. The morphism 5.2.1 is a flat morphism of algebraic stacks,

with local complete intersection fibers.

Equivalently: let T be a scheme and C → T a twisted curve. Then \underline{Hom}_{T} (C, BG) is flat over T, with local complete intersection fibers.

The proof of 5.3 occupies the remainder of this section.

Note first of all that M_{g,n}^{tw}(G) is an algebraic stack locally of finite presentation over M_{g,n}^{tw}. Indeed the stack M_{g,n}^{tw}(G) is equal to the relative Hom-stack

\underline{Hom}_{M_{g,n}^{tw}}(C^{univ}, BG × M_{g,n}^{tw}),

where C^{univ} → M_{g,n}^{tw} denotes the universal twisted curve. This stack is locally of finite type over M_{g,n}^{tw} by [30, 1.1]. It is also shown there that a Hom stack is algebraic whenever the base is representable. This extents to the case where the base is an algebraic stack by the following well known result:

9(Dan) added this discussion - please check!

10(Dan) on S

Lem:relative-algebraic Lemma 5.4. Let X → Y be a morphism of stacks, and assume the follow-

ing.

(1) The stack Y is algebraic.

(2) There is an algebraic space Z and a smooth surjective Z → Y such

that X_{Z} := Z ×_{Y} X is algebraic. Then X is algebraic.

Proof. First we note that the relative diagonal X → X ×_{Y} X is representable. Indeed, let U be an algebraic space and U → X ×_{Y} X a morphism. Form the pullback U_{Z} = U ×_{Y} Z. We have a morphism U_{Z} → X_{Z} ×_{Z} X_{Z}, and by assumption X_{Z} ×_{X_{Z}×_{Z}X_{Z}} U_{Z} is representable. Applying the same to Z ×_{Y} Z we obtain flat descent data for X ×_{X ×_{Y}X} U, showing that the latter is representable. Therefore the relative diagonal X → X ×_{Y} X is representable.

Now X ×_{Y} X → X × X is the pullback of the diagonal Y → Y × Y via X × X → Y × Y. Since the diagonal Y → Y × Y is representable we have that X ×_{Y} X → X × X is representable.

Composing we get that the diagonal X → X × X is representable. It remains to produce a smooth presentation for X.

There exists a scheme V with a smooth surjective V → X_{Z}. Since X_{Z} → X is smooth surjective, we have V → X smooth surjective. ^{11} ♠

11→

We prove 5.3 by first studying two special cases and then reducing the general case to these special cases. 5.5. The case when G is a tame étale group scheme. In this case we claim that 5.2.1 in fact is étale. Indeed to verify this it suffices to show that 5.2.1 is formally étale since it is a morphism locally of finite presentation. If T_{0} ↪ T is an infinitesimal thickening and C_{T} → T is a twisted curve over T, then the reduction functor from G-torsors on C_{T} to G-torsors on C_{T_{0}} is an equivalence of categories since G is étale.

5.6. The case when G is locally diagonalizable. Let T be an S-scheme, and let C → T be a twisted curve. The stack over T of morphisms C → BG is then equivalent to the stack TORS_{C/T} (G) associating to any T′ → T the groupoid of G-torsors on C ×_{T} T′. Let X denote the Cartier dual Hom(G, 𝔾_{m}) so that G = Spec_{S}O_{S}[X].

We use the theory of Picard stacks - see [7, XVIII.1.4]. Let 𝒫ic_{C/T} denote the Picard stack of line bundles on C, and let Pic_{C/T} denote the rigidification of 𝒫ic_{C/T} with respect to 𝔾_{m}, so that Pic_{C/T} is the relative Picard functor of C/T. By Lemma 2.7 we have that Pic_{C/T} is an extension of a semi-abelian group scheme by an étale group scheme.

Sec:G-tame-etale

11(Dan) added this since Barbara asked where this is proven

Let Pic_{C/T} [X] denote the group scheme of homomorphisms

X → Pic_{C/T} .

Lemma 5.7. The scheme Pic_{C/T} [X] is flat over T, with local complete intersection fibers.

Proof. The assertion is clearly local in the étale topology on S so we may assume that S is connected and G diagonalizable. We may write G = ∏ μ_{n_{i}} and X = ∏ ℤ/(n_{i}). Then Pic_{C/T} [X] = ∏_{T} Pic_{C/T} [ℤ/(n_{i})].

It then suffices to consider the case where X = ℤ/(n), in which case Pic_{C/T} [X] = Pic_{C/T} [n]. This is the fiber of the map Pic_{C/T} ⟶^{×n} Pic_{C/T} over the identity section. This is a map of smooth schemes of the same dimension. It has finite fibers since Pic_{C/T} is an extension of a semi-abelian group scheme by an étale group scheme. Therefore this map is flat with local complete intersection fibers. Hence Pic_{C/T} [n] → T is flat with local complete intersection fibers as well.^{12} ♠

←12

Let 𝒫ic_{C/T} [X] denote the Picard stack of morphisms of Picard stacks

X → 𝒫ic_{C/T} ,

where X is viewed as a discrete stack. Then 𝒫ic_{C/T} [X] is a G–gerbe over Pic_{C/T} [X], hence flat over T.

The result in the locally diagonalizable case therefore follows from the following lemma:

Lem:tors-picx Lemma 5.8. There is an equivalence of categories

TORS_{C/T} (G) → 𝒫ic_{C/T} [X],

were TORS_{C/T} (G) denotes the stack associating to any T′ → T the groupoid of G-torsors on C ×_{T} T′.

Proof. For any morphism s: C → BG, the pushforward s_{∗}O_{C} on C × BG has a natural action of G and therefore decomposes as a direct sum ⊕_{x∈X}L_{x}. Base changing to C → C × BG (the map defined by the trivial torsor) one sees that each L_{x} is a line bundle, and that the algebra structure on s_{∗}O_{C} defines isomorphisms L_{x} ⊗ L_{x′} → L_{x+x′} giving a morphism of Picard stacks

F: X → 𝒫ic_{C/T} [X].

Conversely given such a morphism F, let L_{x} denote F (x). The isomorphisms

F (x + x′) ≃ F (x) + F (x′)

define an algebra structure on

⊕_{x∈X}L_{x}.

The relative spectrum over C × BG maps isomorphically to C and therefore defines a section. ♠ Remark 5.9. With a bit more work, one can prove a more general result which may be of interest: given a twisted curve C → T and a G-gerbe G → C, the stack \underline{Sec}_{T} (G/C) is a G-gerbe over its rigidification Sec_{T} (G/C), and the T -space Sec_{T} (G/C) is a pseudo-torsor under the flat group-scheme Pic_{C/T} [X]. In particular \underline{Sec}_{T} (G/C) → T is flat.

12(Dan) a few words on lci

5.10. Observations on fixed points. Before we consider general G, we make some observations about schemes of fixed points of group actions on semi-abelian group schemes.

Let S be a scheme, and let A → S be a smooth abelian group scheme. Let A^{o} denote the connected component of the identity and assume we have an exact sequence of group schemes

Aseq (5.10.1)  0 ⟶ A^{o} ⟶ A ⟶ W ⟶ 0,

with W a finite étale group scheme over S. In what follows we will assume that W is a constant group scheme (this always hold after making an étale base change on S) and that A^{o} is a semiabelian group scheme.

Let H be a finite group of order invertible on S acting on A (by homomorphisms of group schemes). Let N denote the order of H. Since A^{o} is a semiabelian group scheme multiplication by N is surjective and étale on A^{o}. It follows that multiplication by N on A is also étale and that there is an exact sequence

A ⟶^{×N} A ⟶ W/NW ⟶ 0.

In particular, the image of ×N: A → A is an open and closed subgroup scheme A′ ⊂ A preserved by the H-action.

smoothlem Lemma 5.11. The scheme of fixed points A^{H} is a smooth group scheme over S.

Proof. ^{13} It is clear that A^{H} is a group scheme; we need to show that it is

13→

smooth. This follows from the following Lemma (which is well known, but for which we don’t have a good reference). ♠

Lemma 5.12. Let X be a smooth finitely presented scheme over a scheme S, and let G be a finite group whose order is invertible in S. Suppose that G acts on X via S-scheme automorphisms. Then the scheme of fixed points X^{G} is smooth and finitely presented over S.

Proof. The problem is local on S, so we may assume that S = Spec R is affine. Then X, together with the action, will be obtained by base change from a smooth scheme over a finitely generated subring of R. So we may assume that R is noetherian; and then the scheme X^{G}, which is by definition a closed subscheme in the |G|-th product X ×…×X is automatically finitely presented.^{14}

14→

13(Angelo) Changed the proof

14(Dan) H ↦ G

To check smoothness we are going to use Grothendieck’s infinitesimal lifting criterion. Each point of X has an affine invariant neighborhood; we may replace X by such neighborhood and assume that X is affine; say X = Spec O.^{15} Then X^{G} is the spectrum of the largest invariant ring quotient

←15

of O; a morphism Spec A → X^{G} correspond to an invariant homomorphism O → A.

Suppose that we are given a commutative diagram

\xymatrix{R \ar[r]^{ρ} \ar[d] & A \ar[d] \\ O \ar[r]^{ϕ} \ar@{.>}[ur]^{ψ} & B}

(whithout the dotted arrow), where A → B is a surjective homomorphism of artinian rings, whose kernel I is such that I^{2} = 0, while ρ and ϕ are invariant ring homomorphisms. We need to find an invariant ring homomorphism ψ: O → A that fits above. Since O is smooth over R, we can find a ring homomorphism ψ, which is not however necessarily invariant. For each σ ∈ G the homomorphism O → I defined by the rule f ↦ ψ(σf) − ψ(f) is an R-linear derivation; hence it defines a function from G to the group Der_{R}(O, I) of R-linear derivations from O to I. There is a natural right action of G on Der_{R}(O, I), defined by the rule Dσ(f) = D(σf); the function G → Der_{R}(O, I) above is a 1-cocycle with respect to such action. Since H^{1}(G, Der_{R}(O, I)) = 0, we have that there exists D ∈ Der_{R}(O, I) such that

ψ(σf) − ψ(f) = D(σf) − D(f)

for all σ ∈ G and all f ∈ O. The function O → A defined by f ↦ ψ(f) − D(f) is the desired invariant ring homomorphism. ♠

Now let X be a finitely generated ℤ/p^{n}–module with H-action, where p is prime to N. Let A[X] denote the scheme of homomorphisms X → A. The group H acts on A[X] as follows. An element h ∈ H sends a homomorphism ρ: X → A to the homomorphism

1.5.6 (5.12.1)  X ⟶^{h^{−1}} X ⟶^{ρ} A ⟶^{h} A.

The group scheme of fixed points (A[X])^{H} is defined to be the fiber product of the diagram

A[X]
↓^{Δ}
A[X] \underrightarrow{∏_{h∈H}(h-action)} ∏_{h∈H} A[X].

Note that the group scheme of fixed points (A[X])^{H} is potentially quite unrelated to A^{H}.

15(Dan) S is already the base scheme... also later fixed ϕ to ψ, g to σ and s to f

5.9 Proposition 5.13. The group scheme of fixed points (A[X])^{H} is flat over S, with local complete intersection fibers.

Proof. Choose a presentation of X as an H-representation

shortseq (5.13.1)  0 ⟶ K ⟶ F ⟶ X ⟶ 0,

where the underlying groups of K and F are free ℤ-modules of finite rank. Let A[F] (resp. A[K]) denote the space of homomorphisms F → A (resp. K → A). Applying Hom(F, −) (resp. Hom(K, −)) to 5.10.1 we see that A[F] (resp. A[K]) sits in a short exact sequence

0 ⟶ A^{o}[F] ⟶ A[F] ⟶ Hom(F, W) ⟶ 0
(resp. 0 ⟶ A^{o}[K] ⟶ A[K] ⟶ Hom(K, W) ⟶ 0),

where A^{o}[F] (resp. A^{o}[K]) is a semiabelian group scheme. Note also that the relative dimension over S of A[F] (resp. A[K]) is equal to dim(A)·rk(F) (resp. dim(A)·rk(K)). Since X is a torsion module we have rk(F) = rk(K), so A[F] and A[K] have the same dimension. The inclusion K ↪ F induces a homomorphism

map1 (5.13.2)  A[F] ⟶ A[K]

whose kernel is A[X].

Lemma 5.14. The induced map

A^{o}[F] ⟶ A^{o}[K]

is surjective.

Proof. The short exact sequence 5.13.1 induces an exact sequence of fppfsheaves on S

A^{o}[F] ⟶ A^{o}[K] ⟶ ℰxt^{1}(X, A^{o}).

Since A^{o} is divisible we have ℰxt^{1}(X, A^{o}) = 0. The surjectivity of A^{o}[F] → A^{o}[K] follows. ♠

By 5.11 the induced morphism

kermap (5.14.1)  f: (A[F])^{H} ⟶ (A[K])^{H}

is a morphism of smooth group schemes with kernel the group scheme (A[X])^{H}. This is finite since A[X] is a finite flat group scheme. Note that the connected component of the identity in (A[F])^{H} (resp. (A[K])^{H}) is equal to the connected component of the identity in A^{o}[F]^{H} (resp. A^{o}[K]^{H}).

Lemma 5.15. The morphism

A^{o}[F]^{H} ⟶ A^{o}[K]^{H}

induced by 5.14.1 is surjective.

Proof. Let k be a field and suppose given an H-invariant homomorphism f: K → A^{o} over k. We wish to show that after possibly making a field extension of k we can find an extension f̃: F → A^{o} of f which is H-invariant. Replacing S by Spec(k), for the rest of the proof we view everything as being over k. Consider the short exact sequence of abelian group schemes with H-action

0 ⟶ A^{o}[X] ⟶ A^{o}[F] ⟶ A^{o}[K] ⟶ 0.

Viewing this sequence as an exact sequence of abelian sheaves on BH_{fppf} we obtain by pushing forward to Spec(k)_{fppf} an exact sequence

A^{o}[F]^{H}(k) ⟶ A^{o}[K]^{H}(k) ⟶ H^{1}(BH_{fppf} , A^{o}[X]).

The lemma therefore follows from Lemma 5.16 below (which is stated in the generality needed later). ♠

cohlem Lemma 5.16. Let D be an abelian sheaf on BH_{T,_{fppf}} such that every local section of D is torsion of order prime to N, the order of H. Let

f: BH_{T,_{fppf}} ⟶ T_{fppf}

be the topos morphism defined by the projection. Then R^{i}f_{∗}D = 0 for all i > 0.

Proof. Let T_{•} → BH_{T} be the simplicial scheme over BH_{T} associated to the covering T → BH_{T} defined by the trivial torsor (as in [24, 12.4]). For p ≥ 0 let f_{p}: T_{p,_{fppf}} → T_{fppf} be the projection. Note that T_{p} is a finite disjoint union of copies of T, and therefore for any abelian sheaf F on T_{p,_{fppf}} we have R^{i}f_{p∗}F = 0 for i > 0. From this it follows that Rf_{∗}D is quasi-isomorphic in the derived category of abelian sheaves on T_{fppf} to the complex

f_{0∗}D|_{T_{0}} ⟶ f_{1∗}D|_{T_{1}} ⟶ · · · .

Therefore the sheaf R^{i}f_{∗}D is isomorphic to the sheaf associated to the presheaf which to any T′ → T associates the group cohomology

cohgroups (5.16.1)  H^{i}(H, D(T′)),

where D(T′) denotes the H-module obtained by evaluating D on the object T′ → BH corresponding to the trivial torsor on T′. Since D(T′) is a direct limit of groups of order prime to the order of H, it follows that the groups 5.16.1 are zero for all i > 0. ♠

Now we complete the proof of Proposition 5.13. It follows that the map on connected components of the identity

(A[F]^{H})^{o} ⟶ (A[K]^{H})^{o}

is a surjective homomorphism of smooth group schemes of the same dimension, and hence a flat morphism with local complete intersection fibers. Therefore the morphism

A[F]^{H} ⟶ A[K]^{H}

is also flat and hence its fiber over the identity section, namely A[X]^{H}, is flat over S, with local complete intersection fibers. ♠

5.17. General G: setup. We return to the proof of 5.3.

The assertion that 5.2.1 is flat is local in the fppf topology on S. We may therefore assume that G is well-split so that there is a split exact sequence

1 ⟶ D ⟶ G ⟶ H ⟶ 1

with H étale and tame and D diagonalizable. The map G → H induces a morphism over M_{g,n}^{tw} × S

flatmap2 (5.17.1)  M_{g,n}^{tw}(G) ⟶ M_{g,n}^{tw}(H).

We may apply the case of étale group scheme (Section 5.5) to the group scheme H. Therefore we know that M_{g,n}^{tw}(H) is étale over M_{g,n}^{tw} × S, so it suffices to show that the morphism 5.17.1 is flat with local complete intersection fibers.

For this in turn it suffices to show that for any morphism t: T → M_{g,n}^{tw}(H), with T the spectrum of an artinian local ring, the induced map

relflat (5.17.2)  M_{g,n}^{tw}(G) ×_{M_{g,n}^{tw}(H)} T ⟶ T

is flat with local complete intersection fibers.

Let (C → T, F: C → BH) be the n-pointed genus g twisted curve together with the morphism F: C → BH corresponding to t, and let C′ → C denote the H-torsor obtained by pulling back the tautological H-torsor over BH. So we have a diagram with cartesian square

C′ ⟶ C
↓  ↓^{F}
T ⟶ BH_{T}
↓^{f}
T.

For the rest of the argument we may replace S by T, pulling back all the objects over S to T.

The action of H on D defines a group scheme D over BH_{T} whose pullback along the projection T → BH_{T} is the group scheme D with descent data defined by the action. Let 𝔻 denote the pullback of this group scheme to C. The pullback of 𝔻 to C′ is equal to D ×_{T} C′, but 𝔻 is a twisted form of D over C. Similarly, the character group X of D has an action of the group H, giving a group scheme 𝒳 over BH_{T} which is Cartier dual to D.

Let G → C denote the stack C×_{BH}BG, and let G′ → C′ denote the pullback to C′.The stack G is a gerbe over C banded by the sheaf of groups 𝔻. To prove that the morphism 5.17.2 is flat with local complete intersection fibers we need to show that the stack \underline{Sec}_{T} (G/C) is flat with local complete intersection fibers over T. As T is local artinian, we may assume that \underline{Sec}_{T} (G/C) → T is set theoretically surjective, and replacing T by a finite flat cover we may also assume that \underline{Sec}_{T} (G/C) → T has a section over the reduction of T.

Note that the gerbe G′ → C′ is trivial: there is an isomorphism G′ ≃ C′ ×BD. This is because T → BH corresponds to the trivial torsor, and any trivialization gives an isomorphism T ×_{BH} BG ≃ BD. It follows that there is an isomorphism of T -schemes \underline{Sec}_{T} (G′/C′) ≃ TORS_{C′/T} (D). However this isomorphism is not canonical.

5.18. Reduction to C′ connected. We wish to apply previous results such as 5.8 to C′ → T, but our discussion applied only when this is connected. We reduce to the connected case as follows: suppose the H-bundle C′ → C has disconnected fiber over the residue field of T, then since T is artinian we can choose a connected component C′′ ⊂ C′. Let H′′ ⊂ H be the subgroup sending C′′ to itself. Then C′′ → C is an H′′-bundle. If we denote G′′ = G ×_{H} H′′, there is an equivalence of categories between G-bundles E → C lifting the given H-bundle C′ and G′′-bundles E′′ → C lifting the H′′ bundle C′′: one direction is by restricting E to C′′ ⊂ C′, the other direction by inducing E′′ → C from G′′ to G. It therefore suffices to consider the case when C′ → T has connected fibers.

5.19. General G: strategy. Our approach goes as follows:

(1) By Lemma 5.8 we have a precise structure, as a gerbe over a group scheme, of the stack \underline{Sec}_{T} (G′/C′).

(2) This provides an analogous precise stucture on \underline{Sec}_{BH_{T}} (G/C) and its rigidification Sec_{BH_{T}} (G/C): we have that \underline{Sec}_{BH_{T}} (G/C) is a D-gerbe over Sec_{BH_{T}} (G/C). However Sec_{BH_{T}} (G/C) is not a group-scheme but a torsor (see below).

(3) The stack

\underline{Sec}_{T} (G/C) = \underline{Sec}_{T} (\underline{Sec}_{BH_{T}} (G/C) / BH_{T} )

can be thought of as a stack theoretic version of push-forward in the fppf topology from BH_{T} to T. Indeed, for a representable morphism U → BH_{T}, the fppf sheaf associated to the space \underline{Sec}_{T} (U/BH_{T}) coincides with f_{∗}(U)_{fppf} . We analyze the structure of the push-forward of its building blocks, namely the rigidification Sec_{BH_{T}} (G/C) and the group-schemes underlying the torsor and gerbe structures.

5.20. Structure of \underline{Sec}_{T} (G′/C′) and \underline{Sec}_{BH_{T}} (G/C). Let us explicitly state the structure in (1) above:

• \underline{Sec}_{T} (G′/C′) → Sec_{T} (G′/C′) is a gerbe banded by the group-scheme

D, and

• the rigidification Sec_{T} (G′/C′) of \underline{Sec}_{T} (G′/C′) is isomorphic to the T -

group scheme Pic_{C′/T} [X].

We proceed with the structure in (2).

Lem:sec-BH-gerbe Lemma 5.21. The automorphism group of any object of \underline{Sec}_{BH_{T}} (G/C) over Z → BH is canonically isomorphic to D(Z)

Proof. This follows from the facts that G is a D-gerbe and C → BH connected. Indeed if s: C → G is a section, then an automorphism of s is given by a map C → D. This necessarily factors through BH_{T} since by connectedness F_{∗}O_{C} = O_{BH_{T}} (where F: C → BH_{T} is the structure morphism) and D is affine over BH_{T}. ♠

The stack \underline{Sec}_{BH_{T}} (G/C) is algebraic by Lemma 5.4. We can define Sec_{BH_{T}} (G/C) to be the rigidification \underline{Sec}_{BH_{T}} (G/C) D. This is also the relative coarse moduli space since Sec_{BH_{T}} (G/C) → BH is representable. We have that \underline{Sec}_{BH_{T}} (G/C) → Sec_{BH_{T}} (G/C) is a gerbe banded by D.

We also have naturally that \underline{Sec}_{T} (G′/C′) = T ×_{BH_{T}} \underline{Sec}_{BH_{T}} (G/C). The lemma above and the base change property of rigidification gives

Sec_{T} (G′/C′) = T ×_{BH_{T}} Sec_{BH_{T}} (G/C).

Let us look at the structure underlying Sec_{BH_{T}} (G/C).

The group H acts on Pic_{C′/T} and on X. We therefore also obtain a left action of H on

Pic_{C′/T} [X] = Hom(X, Pic_{C′/T}),

where h ∈ H sends a homomorphism ρ: X → Pic_{C′/T} to the homomorphism

(5.21.1)  X ⟶^{h^{−1}} X ⟶^{ρ} _{Pic_{C′/T}} ⟶^{h} Pic_{C′/T} .

This defines an fppf sheaf on BH_{T}, which is easily seen to be represented by Pic_{C/BH_{T}} [𝒳].

Note that in general the identification Sec_{T} (G′/C′) = T ×_{BH_{T}} Sec_{BH_{T}} (G/C) does not give an isomorphism of Sec_{BH_{T}} (G/C) with Pic_{C/BH_{T}} [𝒳], because the descent data may in general be different. However, we have the following:

Lemma 5.22. There is a natural action of Pic_{C/BH_{T}} [𝒳] on Sec_{BH_{T}} (G/C), which pulls back to the natural action of Pic_{C′/T} [X] on itself by translation. In particular Sec_{BH_{T}} (G/C) is a torsor under Pic_{C/BH_{T}} [𝒳].

Proof. Let Φ: X → 𝒫ic_{C/BH_{T}} be an object of 𝒫ic_{C/BH_{T}} [𝒳] over some arrow ξ: Z → BH_{T} and s: C → G a section, again over Z. Then we can define a new section s^{Φ} : C → G as follows. Let P^{Φ} denote the D-torsor corresponding to Φ (see Lemma 5.8). The section s defines an isomorphism (which we denote by the same letter)

s: C × BD → G,

and we let s^{Φ} denote the composite

C ⟶^{1×P^{Φ}} C × BD ⟶^{s} G.

By the universal property of rigidification this defines an action of the group scheme Pic_{C/BH_{T}} [𝒳] on Sec_{BH_{T}} (G/C). The fact that its pullback is identified with the action of Pic_{C′/T} [X] on itself by translation is routine.^{16} ♠

←16

5.23. The pushforward of Pic_{C/BH_{T}} [𝒳] and its torsor Sec_{BH_{T}} (G/C). We write (Pic_{C/BH_{T}} [𝒳])_{fppf} for the sheaf in the fppf topology on BH_{T} represented by Pic_{C/BH_{T}} [𝒳]. The fppf pushforward f_{∗}((Pic_{C/BH_{T}} [𝒳])_{fppf} ) is represented by the scheme of fixed points Pic_{C′/T} [X]^{H}. We have:

1.5.1 Lemma 5.24. The scheme of fixed points Pic_{C′/T} [X]^{H} is finite and flat over T, with local complete intersection fibers.

Proof. Note that in the notation of Section 2.11, since X is a torsion group we have

Pic_{C′/T} [X] = Pic_{C′/T}^{o} [X]

where Pic_{C′/T}^{o} is the group scheme of degree-0 line bundles defined in 2.10. Now by 2.11, we have that Pic_{C′/T}^{o} is an extension of a semi-abelian group scheme by an étale group scheme. The latter étale group scheme is finite since T is assumed Artinian local. The lemma therefore follows from Proposition 5.13. ♠

Now consider the torsor Sec_{BH_{T}} (G/C) and the pushforward under f of the corresponding sheaf (Sec_{BH_{T}} (G/C))_{fppf} . We have the following.

torsorlem Lemma 5.25. Let E be an abelian sheaf on BH_{T,_{fppf}} such that every local section is torsion of order prime to N, the order of H, and let P → BH_{T} be a E–torsor. Then the sheaf of sets f_{∗}P on T_{fppf} is a torsor under the sheaf f_{∗}E.

Proof. The fact that P is a torsor under E immediately implies that f_{∗}P is a pseudo-torsor under f_{∗}E. So the only issue is to show that fppf locally on T the sheaf f_{∗}P has a section. Equivalently, we need to show that after making a flat surjective base change T′ → T the torsor P itself becomes trivial. The class of the torsor is a class in H^{1}(BH_{T} , E). By 5.16 and the Leray spectral sequence this lies in H^{1}(T, f_{∗}E). Any class in this group can be killed by making a flat surjective base change T′ → T. ♠

In our situation, with E = (Pic_{C/BH_{T}} [𝒳])_{fppf} and P = Sec_{BH_{T}} (G/C)_{fppf} , we obtain that the sheaf f_{∗}(Sec_{BH_{T}} (G/C))_{fppf} is represented by a torsor under Pic_{C′/T} [X]^{H}. In particular the space representing this sheaf is flat over T with local complete intersection fibers.

We denote this T -space by the shorthand notation Sec^{H} - a complete notation would look like \underline{Sec}_{T} (Sec_{BH_{T}} (G/C)/BH_{T}).

We turn our view to \underline{Sec}_{T} (G/C).

Lemma 5.26. The automorphism group-scheme of an object of \underline{Sec}_{T} (G/C) ober a T -scheme B is the group scheme D^{H} representing f_{∗}D.

16(Dan) A little nasty here

Proof. Let s: C → G be a section over some T -scheme B. Since \underline{Sec}_{T} (G/C) = \underline{Sec}_{T} (\underline{Sec}_{BH}(G/C) / BH), an automorphism of s is a section over B × BH of the automorphism group-scheme of s viewed as an object of \underline{Sec}_{BH}(G/C) over B × BH. By Lemma 5.21 this is a section over B × BH of the group scheme D, namely a section of f_{∗}D, as required. ♠

We can define Sec_{T} (G/C) to be the rigidification. We have that the morphism \underline{Sec}_{T} (G/C) → Sec_{T} (G/C) is a gerbe banded by the group scheme D^{H} representing f_{∗}D. This is automatically flat. It has local complete intersection fibers by the following:^{17}

17→

Lemma 5.27. Let Δ be a diagonalizable group scheme over an algebraically closed field. Then BΔ is local complete intersection.

Proof. Factoring Δ = ∏ μ_{r_{i}} we have BΔ = ∏ Bμ_{r_{i}}, and it suffices to consider each factor. But Bμ_{r_{i}} = [𝔾_{m}/𝔾_{m}], with the action though the r_{i}-th power map, in other words we have a smooth morphism 𝔾_{m} → Bμ_{r_{i}} with smooth source, hence Bμ_{r_{i}} is local complete intersection. ♠

It therefore suffices to show that Sec_{T} (G/C) is flat with local complete intersection fibers. There is a natural map \underline{Sec}_{T} (G/C) → Sec^{H} inducing Sec_{T} (G/C) → Sec^{H}. The following clearly suffices:

Prop:sec-T-gerbe Proposition 5.28. The morphism \underline{Sec}_{T} (G/C) → Sec^{H} is a gerbe banded by the group scheme D^{H} representing f_{∗}D, and hence Sec_{T} (G/C) → Sec^{H} is an isomorphism.

Consider a morphism Z → Sec^{H} and the fiber product

𝒮: = Z ×_{Sec}_{H} \underline{Sec}(G/C).

Lemma 5.29. Locally in the fppf topology on Z there exists a section Z → 𝒮.

Proof. Let s̄ ∈ f_{∗}(Sec_{BH_{T}} (G/C) (Z) denote the section defined by Z → Sec^{H}. By definition it corresponds to a morphism BH_{Z} → Sec_{BH_{T}} (G/C) over the natural morphism BH_{Z} → BH_{T}. Since \underline{Sec}_{BH_{T}} (G/C) is a D-gerbe over Sec_{BH_{T}} (G/C), the obstruction to finding a section of 𝒮 over s̄ is equal to the class of the D-gerbe of liftings of s̄ : BH_{Z} → Sec_{BH_{T}} (G/C) to a morphism BH_{Z} → \underline{Sec}_{BH_{T}} (G/C). This class lies in H2_{fppf} (BH_{Z}, D). Since by Lemma 5.16 we have R^{i}f_{∗}D = 0 for i > 0, the spectral sequence puts this class in H2_{fppf} (Z, f_{∗}D). This vanishes when pulled back to an fppf cover, and the lemma follows. ♠

If s, s′ ∈ 𝒮(Z) are two sections, then we claim that after making a flat base change on Z the sections s and s′ are isomorphic. Indeed let I be the sheaf over BH_{Z,_{fppf}} of isomorphisms between the two morphisms BH_{Z} → \underline{Sec}_{BH_{T}} (G/C) corresponding to s and s′. Then I is a D-torsor over BH_{Z,_{fppf}} , and since R^{1}f_{∗}D = 0 it follows that fppf locally on Z this torsor is trivial. It follows that 𝒮 is a gerbe banded by f_{∗}D. This completes the proof of 5.28, implying 5.3. ♠

17(Dan) added this, check!!

6. Example: reduction of 𝒳(2) in characteristic 2

examplesection

6.1. 𝒳(2) as a distinguished component in K_{0,4}(Bμ_{2}). Let 𝒳(2) denote the stack over ℚ associating to any scheme T the groupoid of pairs (E, ι), where E/T is a generalized elliptic curve with a full level 2-structure ι : (ℤ/2)^{2} ≃ E[2] in the sense of Deligne-Rapoport [12].

Order the elements of (ℤ/2)^{2} in some way. For any pair (E, ι) ∈ 𝒳(2)(T), the involution P ↦ −P has fixed points the 2-torsion points E[2] ⊂ E, and hence the stack-theoretic quotient [E/ ± 1] of E by this involution comes equipped with an ordered set of gerbes Σ_{i} ⊂ [E/ ± 1] in the smooth locus. Furthermore, one checks by direct calculation on the geometric fibers of E that the coarse space of [E/ ± 1] with the resulting four marked points is a stable genus 0 curve with four marked points. In this way we obtain a morphism of stacks

𝒳(2) → K_{0,4}(Bμ_{2})_{ℚ}

sending

(E, ι) ↦ (E → [E/ ± 1], {Σ_{i} ⊂ [E/ ± 1]}).

One verifies immediately that this functor is fully faithful, and identifies 𝒳(2) with the closed substack K_{ℚ} ⊂ K_{0,4}(Bμ_{2})_{ℚ} classifying data (B → 𝒫, {Σ_{i}}) where B → 𝒫 is a μ_{2}-torsor such that the resulting map B → P to the coarse space of 𝒫 is ramified over each of the marked points of P.

We have seen that K_{0,4}(Bμ_{2}) is flat over Spec(ℤ). We want to have a nice description of the closure K of K_{ℚ}, preferably as a naturally defined moduli stack flat over ℤ. It turns out that this can be done in the best possible way: K is an open and closed substack of K_{0,4}(Bμ_{2}), defined naturally in terms of the behavior of the μ_{2}-cover at the marked points. One can describe this directly, but we find that this gives us a good pretext for introducing the rigidified cyclotomic inertia stack and evaluation maps.

6.2. Cyclotomic inertia. Recall that in [3, §3] one defined the cyclotomic inertia stacks: for fixed positive integer r ≥ 1 we denote by I_{μ_{r}}(X) → X the stack whose objects are pairs (ξ, ϕ) where ξ is an object of X and ϕ : μ_{r} → Aut(ξ) is a monomorphism, and whose arrows are commutative diagrams as usual; and

I_{μ}(X) = ∐_{r=1}^{∞} I_{μ_{r}}(X).

For a noetherian stack with finite diagonal, The argument of [3, Proposition 3.1.2] shows that the morphism I_{μ}(X) → X is representable by quasiprojective schemes. Indeed, the fiber over an object ξ ∈ X (T) is the scheme ∐_{r} Hom_{gr−sch/T} (μ_{r}, Aut_{T} (ξ)), which is quasiprojective as the relevant r is bounded.

Proposition 6.3. If X is tame, the morphism I_{μ}(X) → X is finite and finitely presented.

Proof. As the problem is local in the fppf topology of the coarse moduli space X of X, we may assume X = [U/G] with G a linearly reductive group-scheme [4, Theorem 3.2 (c)]. The pull-back I_{X} ×_{X} U is a closed subgroup scheme of the finite flat linearly reductive group scheme G_{U}. It therefore suffices to show the following Lemma.

♠

grouphom Lemma 6.4. Let G_{1} and G_{2} be two finite flat finitely presented linearly reductive group schemes over a scheme U. Then the scheme \underline{Hom}_{gs/U}(G_{1}, G_{2}) of homomorphisms of group schemes is finite, flat and finitely presented over U.

18

18→

Proof. This is a local statement in the fppf topology; hence we may assume that G_{1} and G_{2} are of the form H_{i} ⋉ Δ_{i}, where the Δ_{i} are diagonalizable group stacks whose orders are powers of the same prime p, and H_{i} are constant tame group stacks of orders not divisible by p. Let us also assume that U is connected. If V is a scheme over U, every homomorphism G_{1,V} → G_{2,V} sends Δ_{1,V} into Δ_{2,V}; hence we have an induced morphism of Uschemes

\underline{Hom}_{gs/U}(G_{1}, G_{2}) ⟶ \underline{Hom}_{gs/U}(Δ_{1}, Δ_{2}) ×_{U} \underline{Hom}_{gs/U}(H_{1}, H_{2}).

Since \underline{Hom}_{gs/U}(Δ_{1}, Δ_{2}) and \underline{Hom}_{gs/U}(H_{1}, H_{2}) are both unions of a finite number of copies of U, we may fix two homomorphisms of group schemes χ: Δ_{1} → Δ_{2}^{19} and g: H_{1} → H_{2}, and show that the open and closed sub-

19→

scheme \underline{Hom}_{g,χ}(G_{1}, G_{2}) of \underline{Hom}_{gs/U}(G_{1}, G_{2}) inducing the homomorphisms g and χ is finite, flat and finitely presented.

If the homomorphism χ: Δ_{1} → Δ_{2} is not H_{1}-equivariant, where H_{1} acts on Δ_{2} through the homomorphism f: H_{1} → H_{2}, then \underline{Hom}_{g,χ}(G_{1}, G_{2}) is empty; so we may assume that χ is H_{1}-equivariant. Then \underline{Hom}_{g,χ}(G_{1}, G_{2}) is not empty: in fact, g × χ: H_{1} ⋉ Δ_{1} → H_{2} ⋉ Δ_{2} is in \underline{Hom}_{g,χ}(G_{1}, G_{2})(U). If V is a U-scheme, an element f of \underline{Hom}_{g,χ}(G_{1}, G_{2}) is determined by a morphism of schemes γ: H_{1} → Δ_{2} so that f(ax) = g(a)ϕ(a)χ(x). By a straightforward calculation, the condition that f be a homomorphism translates into the condition

ϕ(ab) = ϕ(a)^{b}b,

where we have denoted by (x, b) ↦ x^{b} the action of H_{1} on Δ_{2}(V) acted upon H_{1}. That is, ϕ should be a 1-cocycle. Since the order of H_{1} is prime to the exponent of Δ_{2}(V), every such cocyle is coboundary, that is, it is of the form ϕ(a) = d^{a}d^{−1} for some d ∈ Δ_{2}(V). The coboundary ϕ determines d up to an element of Δ_{2}(V)^{H_{1}}; hence we obtain a bijective correspondence between Δ_{2}(V)/Δ_{2}(V)^{H_{1}} and \underline{Hom}_{g,χ}(G_{1}, G_{2})(V). Now, it is easy to see that the invariant subsheaf Δ_{2}^{H_{1}} is in fact a diagonalizable group scheme, and so is the quotient Δ_{2}/Δ_{2}^{H_{1}}, and we have just produced an isomorphism of functors of Δ_{2}/Δ_{2}^{H_{1}} with \underline{Hom}_{g,χ}(G_{1}, G_{2}). Since Δ_{2}/Δ_{2}^{H_{1}} is finite, flat and finitely presented, this finishes the proof. ♠

18(Angelo) Made some changes to the statement and to the proof.

19(Dan) Changed h to χ because confused with H

6.5. Rigidified cyclotomic inertia. Each object of I_{μ_{r}}(X) has μ_{r} sitting in the center of its automorphisms. We can thus rigidify it and obtain a stack \overline{I_{μ_{r}}}(X) with a canonical morphism I_{μ_{r}}(X) → \overline{I_{μ_{r}}}(X). Taking the disjoint union over r ≥ 1 we obtain I_{μ}(X) → \overline{I_{μ}}(X). The stack \overline{I_{μ}}(X) is called the rigidified cyclotomic inertia stack of X.

The stacks \overline{I_{μ_{r}}}(X) and \overline{I_{μ}}(X) have another interpretation, discussed in [3, Section 3.3]. Consider the 2-category whose objects consist of morphisms of stacks α : G → X, where G is a gerbe banded by μ_{r} and α is a representable morphism; morphisms and 2-arrows are defined in the usual way. Since α is representable, this 2-category is equivalent to a category. This is isomorphic to the stack \overline{I_{μ_{r}}}(X). This gives the stack \overline{I_{μ}}(X) the interpretation as the stack of cyclotomic gerbes in X, and I_{μ}(X) → \overline{I_{μ}}(X) together with I_{μ}(X) → X is the universal gerbe in X. The non-identity components of \overline{I_{μ}}(X) are known as twisted sectors.

6.6. Evaluation maps. Consider now the stack of twisted stable maps K_{g,n}(X , β). This comes with a diagram

\xymatrix{Σ_{i} \ar[r] \ar[dr] & C_{g,n}(X , β) \ar[r] \ar[d] & X \\  & K_{g,n}(X , β).}

where C_{g,n}(X , β) → X is the universal twisted stable map and Σ_{i} are the n markings. For each i, the resulting diagram

\xymatrix{Σ_{i} \ar[d] \ar[r] & X \\ K_{g,n}(X , β).}

is a cyclotomic gerbe in X parametrized by K_{g,n}(X , β). This gives rise to n evaluation maps

e_{i} : K_{g,n}(X , β) ⟶ \overline{I_{μ}}(X).

Evaluation maps (and their sisters, twisted evaluation maps) are of central importance in Gromov–Witten theory of stacks, because the natural gluing maps

\xymatrix{K_{g_{1},n_{1}+1}(X , β_{1}) ×_{\overline{I_{μ}}(X)} K_{g_{2},1+n_{2}}(X , β_{2}) \ar[r] & K_{g_{1}+g_{2},n_{1}+n_{2}}(X , β_{2})}

have the fibered product based on the evaluation map on the right and the twisted evaluation map, namely the evaluation map composed with inversion on the band, on the left.

6.7. Back to K ⊂ K_{0,4}(Bμ_{2}). The rigidified cyclotomic inertia stack of Bμ_{2} has two components - the identity component is Bμ_{2} itself, and the non-identity component - the twisted sector - is a copy of the base scheme Spec ℤ.

Since K_{ℚ} corresponds to totally ramified maps, it is precisely the locus in K_{0,4}(Bμ_{2}) where all four evaluation maps land in the twisted sector. Since I_{μ}(Bμ_{2}) is finite unramified, the inverse image of each component is open and closed. Therefore K, the closure of K_{ℚ} in K_{0,4}(Bμ_{2}) is open and closed. In particular it is flat over ℤ. Also, since the generic fiber is irreducible, K is irreducible. It is simply the gerbe banded by μ_{2} over the λ line \overline{M}_{0,4} ≃ ℙ^{1}, associated to the class in H2_{fppf} (ℙ^{1}, μ_{2}) associated to c_{1}(O_{ℙ^{1}}(1).

6.8. What does K_{𝔽_{2}} parametrize? There is one little problem with the closure K of X(2)_{ℚ}: in characteristic 2 it has nothing to do with elliptic curves.

Given a smooth rational curves with 4 marked points, say at 0, 1, ∞ and λ, there is a unique μ_{2}-bundle over the twisted curve which is degenerate over the coarse curve at all the markings, namely the scheme given in affine coordinates by the equation

y^{2} = x(x − 1)(x − λ).

In characteristic 2, this is a cuspidal curve of geometric genus 0, with its cusp given by x^{2} = λ. This well known example is possibly the simplest example of geometric interest of a reduced principal bundle over a smooth scheme with singular total space.

It is not difficult to see that a similar situation holds at the node of a singular curve of genus 0: the singularity of the μ_{2}-bundle is a tacnode.

6.9. 𝒳(2) as a distinguished component in K_{1,1}(Bμ_{2}^{2}). Let

K_{1,1}^{∘}(Bμ_{2}^{2}) ⊂ K_{1,1}(Bμ_{2}^{2})

be the open substack classifying μ_{2}^{2}-torsors over smooth elliptic curves. For any object (E/T, P → E) ∈ K_{1,1}^{∘}(Bμ_{2}^{2}) over some scheme T, there is a natural action of μ_{2}^{2} on this object given by the action on P. Let \overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) be the rigidification.

A simple, but not concrete, description of \overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) was given in Lemma 5.8: let E/M_{1,1} be the universal elliptic curve. Then as discussed before, \overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) = Pic_{E/M_{1,1}}[X] where X = (ℤ/(2))^{2}. It follows that

\overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) = Pic_{E/M_{1,1}}[2] ×_{M_{1,1}} Pic_{E/M_{1,1}}[2].

The compactification over the boundary of \overline{M}_{1,1} is not hard to describe as well. But we wish to describe this stack in terms of more classical objects.

The stack \overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) can be reinterpreted as follows.

Given an object (E/T, π : P → E) ∈ K_{1,1}^{∘}(Bμ_{2}^{2}) over a scheme T, the sheaf π_{∗}O_{P} on E has a μ_{2}^{2}-action given by the action on P, and therefore decomposes as a direct sum

π_{∗}O_{P} = ⊕_{χ∈(ℤ/2)^{2}}L_{χ}.

Furthermore, since P is a torsor over E each L_{χ} is a locally free sheaf of rank 1 on E. The sheaves L_{χ} therefore define a homomorphism

\begin{matrix}ϕ_{P} : (ℤ/2)^{2} & → & \underline{Pic}^{0}(E) ≃ E, \\ χ & ↦ & [L_{χ}]. \end{matrix}

Let ℋ(2) denote the stack over ℤ associating to any scheme T the groupoid of pairs (E, ϕ), where E/T is an elliptic curve and ϕ : (ℤ/2)^{2} → E[2] is a homomorphism of group schemes over T. The above construction gives a functor

K_{1,1}^{∘}(Bμ_{2}^{2}) → ℋ(2)

and a straightforward verification shows that this functor induces an isomorphism

\overline{K}_{1,1}^{∘}(Bμ_{2}^{2}) ≃ ℋ(2).

For any quotient of finite abelian groups π : (ℤ/2)^{2} → A, let ℋ^{A}(2) denote the stack associating to any scheme T the groupoid of pairs (E, ϕ_{A}), where E/T is an elliptic curve and ϕ_{A} : A → E[2] is an A-structure in the sense of Katz-Mazur [20, 1.5.1]. Sending such a pair (E, ϕ_{A}) to

(E, \xymatrix{(ℤ/2)^{2} \ar[r]^{π} & A \ar[r]^{ϕ_{A}} & E[2]} )

defines a morphism of stacks

KMinclusion (6.9.1)  ℋ^{A}(2) → ℋ(2).

Using [20, 1.6.2] one sees that this in fact is a closed immersion.

We can describe the above in more classical notation as follows. Define stacks 𝒴(2), 𝒴_{1}(2), and 𝒴(1) by associating to any scheme T the following groupoids: 𝒴(2) : The groupoid of pairs (E, ϕ), where E/T is an elliptic curve and

ϕ : (ℤ/2)^{2} → E[2] is a (ℤ/2)^{2}-generator; 𝒴_{1}(2) : The groupoid of pairs (E, ϕ), where E/T is an elliptic curve and

ϕ : (ℤ/2) → E[2] is a (ℤ/2)-generator; 𝒴(1) : The groupoid of elliptic curves over T. Then 6.9.1 gives a closed immersion

i^{(2)} : 𝒴(2) ↪ ℋ(2)

corresponding to the identity map (ℤ/2)^{2} → (ℤ/2)^{2}, three closed immersions

i_{K}^{(1)} : 𝒴_{1}(2) ↪ ℋ(2), j = 1, 2, 3,

corresponding to the three surjections (ℤ/2)^{2} → ℤ/2 (indexed by index 2 subgroups K ⊂ (ℤ/2)^{2}), and a closed immersion

i^{(0)} : 𝒴(1) ↪ ℋ(2)

corresponding to the unique surjection (ℤ/2)^{2} → 0. The resulting map

coprod (6.9.2)  𝒴(2) ⊔ (𝒴_{1}(2) ⊔ 𝒴_{1}(2) ⊔ 𝒴_{1}(2)) ⊔ 𝒴(1) ⟶ ℋ(2)

is then a proper surjection, which over ℤ[1/2] is an isomorphism. Note in particular that the forgetful map ℋ(2) → 𝒴(1) sending (E, ϕ) to E has degree 16, as ℋ(2) → 𝒴(1) is flat by 5.3 and over ℤ[1/2] the map clearly has degree 16.

Over 𝔽_{2}, however, the map 6.9.2 joins together the various components in an interesting way. If E/k is an ordinary elliptic curve over a field k of characteristic 2, then any homomorphism

fibermap (6.9.3)  ϕ : (ℤ/2)^{2} → E[2] ≃ ℤ/(2) × μ_{2}

has a nontrivial kernel, and the irreducible components of 𝒴(2)_{𝔽_{2}} are indexed by these kernels. Since a Z/(2)^{2}-structure is surjective on geometric points, the kernel cannot be the whole group. Therefore 𝒴(2)_{𝔽_{2}} has 3 irreducible components: for a subgroup K ⊂ (ℤ/2)^{2} of index 2 we get an irreducible component 𝒴(2)_{𝔽_{2}}^{K} corresponding to pairs (E, ϕ), where the map 6.9.3 has kernel K. Note also that for any ordinary elliptic curve E/k over a field k, the fiber product of the diagram

𝒴(2)_{𝔽_{2}}^{K}
↓
Spec(k) ⟶^{E} 𝒴(1)_{𝔽_{2}}

has length 2 over k.

Similarly the reduction 𝒴_{1}(2)_{𝔽_{2}} has two irreducible components

𝒴_{1}(2)_{𝔽_{2}} = 𝒴_{1}(2)′ ∪ 𝒴_{1}(2)^{∅},

where 𝒴_{1}(2)′ (resp. 𝒴_{1}(2)^{∅}) classifies pairs (E, ϕ : ℤ/2 → E[2]) where the map ϕ is injective (resp. the zero map). This time only the fiber of 𝒴_{1}(2)′ over a k-point of 𝒴(1)_{𝔽_{2}} has length 2.

The closed fiber ℋ(2)_{𝔽_{2}} then has four irreducible components. For a subgroup K ⊂ (ℤ/2)^{2} of index 2, we have an irreducible component Z_{K} which is set-theoretically identified with 𝒴(2)_{𝔽_{2}}^{K} as well as the component 𝒴_{1}(2)′

given by the inclusion i_{K}^{(1)}. The fourth component Z_{∅} is set-theoretically identified with 𝒴(1)_{𝔽_{2}} via i^{(0)}, and with all the components 𝒴_{1}(2)^{∅} via the inclusions i_{K}^{(1)}.

Over the point of 𝒴(1)(\overline{𝔽}_{2}) given by the supersingular elliptic curve E, there is only one homomorphism (the zero map)

ϕ : (ℤ/2)^{2} → E[2],

and hence the four irreducible components of ℋ(2)_{𝔽_{2}} all intersect at this point (and nowhere else). Note also that over the ordinary locus of 𝒴(1), each of the components Z_{K} has length 4 over 𝒴(1)_{𝔽_{2}} as does the component Z_{∅} (so ℋ(2)_{𝔽_{2}} has length 16 over 𝒴(1)_{𝔽_{2}}, as we already knew).

The reduced fiber over 𝔽_{2} looks roughly like this:

The following figure schematically describes the main component 𝒴(2):

Here is the component 𝒴_{1}(2)^{K} for K generated by (1, 1):

40 ABRAMOVICH, OLSSON, AND VISTOLI

There are of course two more corresponding to the two other K of order 2. In addition of course there is 𝒴(1):

The way the main component 𝒴(2) meets 𝒴_{1}(2)^{K} is described as follows:

TWISTED YET TAME 41

and the other two are similar, of course along the corresponding component 𝒴(2)_{𝔽_{2}}^{K}.

The main component 𝒴(2) meets 𝒴(1) only at the supersingular point, but 𝒴_{1}(2)^{K} meets 𝒴(1) along the component corresponding to ϕ = 0: The complete picture is something like this:

An important question is that of describing explicitly the “most important” part of \overline{K}_{g,n}(Bμ_{m}^{2g}), in this example 𝒳(2). In genus 1, Katz and Mazur gave an ideal solution: the moduli stack of generalized elliptic curves with (ℤ/m)^{2} structure, which is a closed substack of our \overline{K}_{1,1}(Bμ_{m}^{2}), is regular and flat over \overline{M}_{1,1}. But for higher genus a satisfactory conclusion is missing: in [20, Section 1.9] Katz and Mazur proposed a solution using norms, but it was shown to be ill behaved in [9, Appendix A]. It still may be of interest to study the closure of the ℚ-stack \overline{M}_{g}^{(m)} of level m curves in \overline{K}_{g}(Bμ_{m}^{2g}).

Appendix A. Twisted curves and log curves by Martin Olsson

AppendixA

In [29] we introduced a notion of log twisted curve, and proved that the category of log twisted curves (over some scheme S) is naturally equivalent to the category of twisted Deligne-Mumford curves over S. The definition of a log twisted curve in loc. cit. included various assumptions about certain integers being invertible on the base scheme. In this appendix we show that if we remove these assumptions in the definition of log twisted curve, we obtain a category equivalent to the category of twisted curves.

locallyfree Definition A.1 ([25, 3.1]). Let X be an Artin stack.

(i) A fine log structure M on X is called locally free if for every geometric point x̄ → X the monoid \overline{M}x̄ := Mx̄/O_{X,x̄}^{∗} is isomorphic to ℕ^{r} for some r.

(ii) A morphism M → N of locally free log structures on X is called simple if for every geometric point x̄ → X the monoids \overline{M}x̄ and \overline{N} x̄ have the same rank, the morphism φ : \overline{M}_{x} → \overline{N} x̄ is injective, and for every irreducible element f ∈ \overline{N} x̄ there exists an irreducible element g ∈ \overline{M}x̄ and a positive integer n such that φ(g) = nf.

Remark A.2. This differs from [29, 1.5] where the integer n in (A.1 (ii)) was assumed invertible in k(x̄).

Let f : C → S be a nodal curve over a scheme S. As discussed in [29, §3], there exist canonical log structures M_{C} and M_{S} on C and S respectively, and an extension of f to a log smooth morphism

(C, M̃_{C}) → (S, M_{S}).

1.5 Definition A.3. A n-pointed log twisted curve over a scheme S is a collec-

tion of data

(C/S, {σ_{i}, a_{i}}_{i=1}^{n}, ℓ : M_{S} ↪ M′_{S}),

where C/S is a nodal curve, σ_{i} : S → C are sections, the a_{i} are positive integer-valued locally constant functions on S, and ℓ : M_{S} ↪ M′_{S} is a simple morphism of log structures on S, where M_{S} denotes the canonical log structure on S mentioned above.

Let (C/S, {σ_{i}, a_{i}}, ℓ : M_{S} ↪ M′_{S}) be a log twisted curve over S, and let

logmap (A.3.1)  (C, M_{C}) ⟶ (S, M_{S})

be the morphism of log schemes obtained from the pointed curve

(C/S, {σ_{i}})

as in [29, 3.10] (note that M_{C} is not equal to M̃_{C} mentioned above, as M_{C} also takes into account the marked points). We construct a twisted n–pointed curve (C, {Σ_{i}})/S from the log twisted curve (C/S, {σ_{i}, a_{i}}, ℓ : M_{S} ↪ M′_{S}) as in [29, §4].

Define C to be the fibered category over S which to any h : T → S associated the groupoid of data consisting of a morphism s : T → C over h together with a commutative diagram of locally free log structures on T

3.15.2 (A.3.2)  \begin{matrix}h^{∗}M_{S} & ⟶^{ℓ} & h^{∗}M′_{S} \\ ↓ & & ↓^{τ} \\ s^{∗}M_{C} & ⟶^{k} & M′_{C}, \end{matrix}

where: (1.3 (i)) The map k is a simple, and for every geometric point t̄ → T, the map \overline{M}′_{S,t̄} → \overline{M}′_{C,t̄} is either an isomorphism, or of the form ℕ^{r} → ℕ^{r+1} sending e_{i} to e_{i} for i < r and e_{r} to either e_{r} or e_{r} + e_{r+1}. (1.3 (ii)) For every i and geometric point t̄ → T with image under s in σ_{i}(S) ⊂ C, the group

3.14.3 (A.3.3)  Coker(\overline{M}_{S,t̄}^{′gp} ⊕ \overline{M}_{C,t̄}^{gp} ⟶ \overline{M}_{C,t̄}^{′gp})

is a cyclic group of order a_{i}(t̄).

For every i, define Σ_{i} ⊂ C to be the substack classifying morphisms s : T → C which factor through σ_{i}(S) ⊂ C and diagrams (A.3.2) such that for every geometric point t̄→ T the image of

(M′_{C,t̄} − τ(h^{∗}M′_{S,t̄})) → O_{T,t̄}

is zero.

Proposition A.4. The data (C, {Σ_{i}}_{i=1}^{n}) is a twisted n-pointed curve.

Proof. This follows from the argument given in [29, §4]. ♠

The main result of this appendix is the following:

Bthm Theorem A.5. Let S be a scheme. The functor

thefunctor (A.5.1) (n-pointed log twisted curves) → (twisted n-pointed curves)

sending (C/S, {σ_{i}, a_{i}}, ℓ : M_{S} ↪ M′_{S}) to the twisted n-pointed curve (C, {Σ_{i}}_{i=1}^{n}) constructed above is an equivalence of categories. Moreover, this equivalence of categories is compatible with base change S′ → S.

We prove this theorem below. Before giving the proof, however, let us record the main consequences of this result that we will use.

Fix integers g and n, and let 𝒮_{g,n} denote the fibered category over ℤ which to any scheme S associates the groupoid of all (not necessarily stable) n– pointed genus g nodal curves C/S. The stack 𝒮_{g,n} is algebraic, and as explained in section [29, §5] the substack 𝒮_{g,n}^{0} ⊂ 𝒮_{g,n} classifying smooth curves defines a log structure M_{𝒮_{g,n}} on 𝒮_{g,n}. The same argument used in [29, §5] yields the following theorem:

3.19 Theorem A.6. Let M_{g,n}^{tw} denote the fibered category over ℤ which to any scheme T associates the groupoid of n–marked genus g twisted curves (C, {Σ_{i}}) over T. Then M_{g,n}^{tw} is a smooth Artin stack, and the natural map

3.19.1 (A.6.1)  π : M_{g,n}^{tw} ⟶ 𝒮_{g,n}

sending (C, {Σ_{i}}) to its coarse moduli space with the marked points induced by the Σ_{i} is representable by tame stacks. Moreover, there is a natural locally free log structure M_{M_{g,n}^{tw}} on M_{g,n}^{tw}.

Remark A.7. Consider a field k and an object (C, {Σ_{i}}) ∈ M_{g,n}^{tw}(k). Let (C, {σ_{i}}) be the coarse moduli space, and let R be a versal deformation space for the object (C, {σ_{i}}) ∈ 𝒮_{g,n}(k). Let q_{1}, …, q_{m} ∈ C be the nodes and let r_{i} be the order of the stabilizer group of a point of C lying above q_{i}. As in [11, 1.5], there is a smooth divisor D_{i} ⊂ Spec(R) classifying deformations where q_{i} remains a node. In other words, if t_{i} ∈ R is an element defining D_{i} then in an étale neighborhood of q_{i} the versal deformation C̃ → Spec(R) of (C, {σ_{i}}) is isomorphic to

Spec(R[x, y]/(xy − t_{i})).

It follows from the argument in [29, §5] that the fiber product

M_{g,n}^{tw} ×_{𝒮_{g,n}} Spec(R)

is isomorphic to the stack-theoretic quotient of

Spec(R[z_{1}, …, z_{m}]/(z_{1}^{r_{1}} − t_{1}, …, z_{m}^{r_{m}} − t_{m}))

by the action of μ_{r_{1}} × ⋯ μ_{r_{m}} for which (ζ_{1}, …, ζ_{r}) ∈ μ_{r_{1}} × ⋯ μ_{r_{m}} sends z_{i} to ζ_{i}z_{i}. In particular M_{g,n}^{tw} is flat over 𝒮_{g,n}, and hence also flat over ℤ.

We also get a generalization of [29, 1.11]:

1.7 Corollary A.8. For any integer N > 0, let M_{g,n}^{tw,≤N} denote the substack of M_{g,n}^{tw} classifying n–pointed genus g twisted curves such that the order of the stabilizer group at every point is less than or equal to N. Then M_{g,n}^{tw,≤N} is an open substack of M_{g,n}^{tw} and the map M_{g,n}^{tw,≤N} → 𝒮_{g,n} is of finite type and quasi-finite.

curveconstruct Remark A.9. Let R be a discrete valuation ring with uniformizer π and separably closed residue field, and let C/R be a nodal curve. Let p_{1}, …, p_{r} be the nodes of C in the closed fiber. Then we have a log smooth morphism

(C, M_{C}) → (Spec(R), M_{R}),

where the log structure M_{R} → R admits a chart ℕ^{r} → R such that the image of the i-th standard generator is equal to π^{l_{i}}, where l_{i} ∈ ℕ∪{∞} is an element such that in an étale neighborhood of p_{i} the curve C_{i} is isomorphic to

Spec(R[x, y]/(xy − π^{l_{i}}),

where by convention if l_{i} = ∞ we set π^{l_{i}} = 0.

Now assume some l_{i} is finite. Then we obtain a twisted curve by taking the stack corresponding to the morphism of log structures M_{R} → M′_{R}, where M′_{R} is the log structure associated to the map ℕ^{r} → R sending e_{j} to π^{l_{j}} for j ≠ i, and e_{i} to π. Proof of A.5. The proof of A.5 follows the same outline as the proof of [29, 1.9]. We review the argument here indicating the necessary changes for this more general setting.

esssems Definition A.10 ([29, 3.3]). A log smooth morphism of fine log schemes f : (X, M_{X}) → (S, M_{S}) is essentially semi-stable if for each geometric point x̄ → X the monoids (f^{−1}\overline{M}_{S})x̄ and \overline{M}_{X,x̄} are free monoids, and if for suitable isomorphisms (f^{−1}\overline{M}_{S})x̄ ≃ ℕ^{r} and \overline{M}_{X,x̄} ≃ ℕ^{r+s} the map

(f^{−1}\overline{M}_{S})x̄ → \overline{M}_{X,x̄}

is of the form

chartformula (A.10.1)  e_{i} ↦ \{ \begin{matrix}e_{i} & if i ≠ r \\ e_{r} + e_{r+1} + ⋯ + e_{r+s} & if i = r, \end{matrix}

where e_{i} denotes the i-th standard generator of ℕ^{r}.

proofofBthm

With notation as in A.10, if U → X is a smooth surjection of schemes and M_{U} denotes the pullback of M_{X}, then it follows immediately from the definition that (X, M_{X}) → (S, M_{S}) is essentially semistable if and only if the morphism (U, M_{U}) → (S, M_{S}) is essentially semistable (the property of being semistable is local in the smooth topology on X). It follows that the notion of a morphism being essentially semistable extends to Artin stacks: If (𝒳, M_{𝒳}) is an Artin stack with a fine log structure, and f : (𝒳, M_{𝒳}) → (S, M_{S}) is a morphism to a fine log scheme, then f is essentially semistable if for some smooth surjection U → 𝒳 with U a scheme the induced morphism of log schemes

testmap (A.10.2)  (U, M_{𝒳} |_{U}) → (S, M_{S})

is essentially semistable. As usual if for some smooth surjection U → 𝒳 the morphism A.10.2 is essentially semistable then for any smooth surjection V → 𝒳 the induced morphism (V, M_{𝒳} |_{V}) → (S, M_{S}) is essentially semistable.

As explained in [29, §3], if (X, M_{X}) → (S, M_{S}) is an essentially semistable morphism of log schemes, then for any geometric point s̄ → S there is a canonical induced map

s_{Xs̄}: \{singular points of Xs̄\} → Irr(\overline{M}_{S,s̄}),

where Irr(\overline{M}_{S,s̄}) denotes the set of irreducible elements in the monoid \overline{M}_{S,s̄}.

If 𝒴 is an Artin stack over a field k, let π_{0}(𝒴_{sing}) denote the set of connected components of the complement of the maximal open substack 𝒰 ⊂ 𝒴 which is smooth over k. Note that if 𝒴′ → 𝒴 is a smooth morphism of k-stacks, then there is an induced map

π_{0}(𝒴′_{sing}) → π_{0}(𝒴_{sing}).

If 𝒴′ → 𝒴 is also surjective then this map is also surjective.

Definition A.11. Let f : (𝒳, M_{𝒳}) → (S, M_{S}) be an essentially semistable morphism from a log Artin stack to a fine log scheme, and let s̄ → S be a geometric point. We say that f is special at s̄ if for any smooth surjection U → 𝒳 with U a scheme the map

s_{Us̄}: \{singular points of Us̄\} → Irr(\overline{M}_{S,s̄})

factors through the composite (which is surjective)

{singular points of Us̄} ⟶ π_{0}(U_{s̄,sing}) ⟶ π_{0}(𝒳_{s̄,sing})

to give an isomorphism

π_{0}(𝒳_{s̄,sing}) → Irr(\overline{M}_{S,s̄}).

canlogthm Theorem A.12 (Generalization of [29, 3.6]). Let (f : C → S, {Σ_{i}}) be an n-marked twisted curve. Then there exist log structures M̃_{C} and M′_{S} on C and S respectively, and a special morphism

(f, f^{b}) : (C, M̃_{C}) ⟶ (S, M′_{S}).

Moreover, the datum (M̃_{C}, M′_{S}, f^{b}) is unique up to unique isomorphism.

Proof. The proof will be in several steps (ending in paragraph following A.17).

The uniqueness statement follows as in [29, 3.6] from the argument proving the uniqueness in [31, 2.7].

Given the uniqueness, to prove existence we may work étale locally on S, and by a limit argument as in the proof of [29, 3.6] may even assume that S is the spectrum of a strictly henselian local ring. Let s̄ ∈ S be the closed point.

Let p_{1}, …, p_{n} ∈ C be the nodes of the closed fiber, and choose for each i = 1, …, n an affine open set U_{i} ⊂ C containing p_{i} and no other nodes. Let U_{i} ⊂ C denote the inverse image of U_{i}.

A14 Lemma A.13. For any quasi-coherent sheaf F on U_{i}, we have

H^{j}(U_{i}, F) = 0

for j > 0.

Proof. This follows from the same argument proving 2.5. ♠

Let t_{i} ∈ O_{S} be an element such that the fiber product

C ×_{C} Spec(O_{C,p̄_{i}})

is isomorphic to

[Spec({ \atop O_{S}[z, w]/(zw − t_{i})})/μ_{n}]

as in (2.1 (v)). Let M_{S}^{i} be the log structure on S associated to the morphism ℕ → O_{S} sending 1 to t_{i}

Definition A.14. Let f : X → S be a morphism of schemes. A t_{i}- semistable log structure on X is a pair (M_{X}, f^{b}), where M_{X} is a locally free log structure on X and f^{b} : f^{∗}M_{S}^{i} → X is a morphism of log structures, such that the following hold:

(i) The morphism of log schemes

(f, f^{b}) : (X, M_{X}) → (S, M_{S}^{i})

is log smooth; (ii) For every geometric point x̄ → X the induced map of free monoids

ℕ → \overline{M}_{S,f(x̄)}^{i} → \overline{M}_{X,x̄}

is the diagonal map.

smoothdes Remark A.15. By [29, 3.4], if X → S admits a t_{i}-semistable log structure, then étale locally on X there exists a smooth morphism

X → Spec(O_{S}[z, w]/(zw − t_{i})).

To prove A.12, it suffices by the same argument used in [29, proof of 3.6] to show that there exists a t_{i}-semistable log structure on each U_{i}. localexist Lemma A.16. Let X → U_{i} be a smooth morphism with X a scheme. Then

étale locally on X there exists a t_{i}-semistable log structure.

Proof. It suffices to prove the existence in some étale neighborhood of a geometric point x̄ → X mapping to the node p_{i} of C. Making an étale base change on C, it therefore suffices to show that if

g : X → [Spec(O_{S}[z, w]/(zw − t_{i})/μ_{n}]

is a smooth morphism then there exists a t_{i}-semistable log structure on X. For this note that there is a log structure M on [Spec(O_{S}[z, w]/(zw−t_{i})/μ_{n}] and a morphism of log stacks

stackmap (A.16.1)  ([Spec(O_{S}[z, w]/(zw − t_{i})/μ_{n}], M) → (S, M_{S})

induced by the commutative diagram

ℕ^{2} ⟶^{β} O_{S}[z, w]/(zw − t_{i})
Δ↑  ↑
ℕ ⟶^{1↦t_{i}}  O_{S}

where β sends (1, 0) to z and (0, 1) to w. By [32, 5.23], the morphism A.16.1 is log étale. It follows that the pullback g^{∗}M on X is a t_{i}-semistable log structure on X. ♠

Let SS_{t_{i}} denote the presheaf on the lisse-étale site Lis-Et(U_{i}) of U_{i} which to any smooth morphism X → U_{i} associates the set of isomorphism classes of t_{i}-semistable log structures on X. As in the proof of [31, 3.18] a t_{i}-semistable log structure admits no nontrivial automorphisms, and hence SS_{t_{i}} is in fact a sheaf. In fact, SS_{t_{i}} is a torsor under a certain sheaf of abelian groups which we now describe.

For any smooth morphism X → U_{i}, there exists by A.16 and A.15 étale locally on X a smooth morphism

ρ : X → Spec(O_{S}[z, w]/(zw − t_{i})).

As explained in [31, 3.12] the ideal J := (z, w)O_{X} ⊂ O_{X} is independent of the choice of the smooth morphism ρ. It follows that these locally defined sheaves of ideals descend to a sheaf of ideals J ⊂ O_{U_{i}}. Let D ⊂ U_{i} be the closed substack defined by this sheaf of ideals. The local description (2.1 (v)) of the stack U_{i} implies that there is an isomorphism

D ≃ Bμ_{n} ×_{Spec(ℤ)} Spec(O_{S}/(t_{i})).

Let K_{t_{i}} ⊂ O_{S} be the kernel of multiplication by t_{i} on O_{S}, and let K_{t_{i}}^{U_{i}} denote the kernel of multiplication by t_{i} on O_{U_{i}}. Note that since U_{i} is flat over S the sheaf K_{t_{i}}^{U_{i}} is equal to the pullback of K_{t_{i}}. Let Z ⊂ U be the closed substack defined by K_{t_{i}}^{U_{i}} · J. Define

G := Ker(O_{U_{i}}^{∗} → O_{Z}^{∗}),

and let G_{2} ⊂ O_{U_{i}}^{∗} denote the subsheaf of units u such that ut_{i} = t_{i}. There is a natural inclusion G ⊂ G_{2}, and as explained in the proof of [31, 3.18] the sheaf SS_{t_{i}} is naturally a torsor under G_{2}/G. To prove the existence of a t_{i}-semistable log structure on U_{i} we therefore must show that the class of this torsor

o ∈ H^{1}(U_{i}, G_{2}/G)

is zero.

A18 Lemma A.17. The map H^{1}(U_{i}, G_{2}/G) → H^{1}(D, O_{D}^{∗}) induced by the com-

posite

G_{2} ⊂ O_{U_{i}}^{∗} → O_{D}^{∗}

is injective.

Proof. This follows from the same argument proving [29, 3.7] (and using A.13). ♠

The image of the class o in H^{1}(D, O_{D}^{∗}) corresponds to an invertible sheaf L on D. As in [31, proof of 3.16] this invertible sheaf L can be described as follows. Consider the inclusion

Bμ_{n} ↪ [Spec({ \atop O_{S}[z, w]/(zw − t_{i})})/μ_{n}] ≃ U_{i} ×_{U_{i}} Spec(O_{C,p̄_{i}}).

Then L corresponds to the representation of μ_{N} with basis zẇ. By the assumptions on the μ_{n}-action in (2.1 (v)) it follows that L is trivial, and hence o = 0. ♠

As in [29, 3.10], the log structure M̃_{C} is not the “right” log structure on C as it does not take into account the markings. Exactly as in loc. cit., for each i = 1, …, n the ideal sheaf I_{i} ⊂ O_{C} defining Σ_{i} defines a log structure N_{i} on C. Set

M_{C} := M̃_{C} ⊕_{O_{C}^{∗}} (⊕_{i=1,O_{C}^{∗}}^{n}N_{i}).

The map f^{∗}M′_{S} → M̃_{C} then induces a log smooth morphism

(C, M_{C}) → (S, M′_{S}).

If g : (C, σ_{1}, …, σ_{n}) → S is the coarse moduli space of C with its n sections defined by the p_{i}, then the above construction applied to (C, σ_{1}, …, σ_{n}) yields log structures M_{C} and M_{S} on C and S respectively and a special morphism

(C, M_{C}) → (S, M_{S}).

Proposition A.18 (Generalization of [29, 4.7]). Let π : C → C be the projection. There exists canonical morphisms of log structures π^{b} : π^{∗}M_{C} → M_{C} and ℓ : M_{S} ↪ M′_{S} such that the diagram of log stacks

(C, M_{C}) ⟶^{(π,π^{b})} (C, M_{C})
f↓  ↓^{g}
(S, M′_{S}) ⟶^{(id,ℓ)} (S, M_{S})

commutes. Moreover, the map ℓ : M_{S} ↪ M′_{S} is a simple extension.

Proof. This follows from the same argument proving [29, 4.7]. The key point is the local description (2.1 (iv) and (v)) of a twisted curve which enables one to rewrite [29, 4.6] verbatim in the present situation. ♠

Finally note that for any geometric point s̄ → S the gerbe Σ_{i,s̄} is necessarily trivial and hence isomorphic to Bμ_{a_{i}(s̄)} for some integer a_{i}(s̄). One verifies immediately that the a_{i} are positive integer valued locally constant functions on S.

The association

(C, {Σ_{i}}) ↦ (C/S, {σ_{i}, a_{i}}, ℓ : M_{S} ↪ M′_{S})

therefore defines a functor

inversefunctor (A.18.1) (twisted n-pointed curves) → (n-pointed log twisted curves)

As in [29, 4.8] one sees that A.5.1 and A.18.1 are inverse functors thereby proving A.5. ♠

Appendix B. Remarks on Ext-groups and base change By Martin Olsson

AppendixC

In this appendix we gather together some fairly standard results about Ext-groups and base change, which will be used in the following appendix C. The results B.2 and B.10 are well-known to experts, but we include them here due to lack of a suitable reference.

B.1. Boundedness of cohomology.

cohfinite Theorem B.2. Let f : X → Y be a separated, representable, finite type morphism between noetherian algebraic stacks. Then there exists an integer n_{0} such that for any quasi-coherent sheaf F on X and i ≥ n_{0} we have R^{i}f_{∗}F = 0.

Proof. The assertion is fppf local on Y, so we may assume that Y is a noetherian affine scheme, and that X is an algebraic space. In this case we must show that there exists an integer n_{0} such that for any quasi-coherent sheaf F on X and i ≥ n_{0} we have H^{i}(X, F) = 0.

nillem Lemma B.3. Let Z_{0} ↪ Z be a closed immersion of noetherian algebraic spaces over Y defined by an nilpotent ideal 𝒥 ⊂ O_{Z}. Assume that B.2 holds for Z_{0} → Y and let n_{Z_{0}} be an integer such that for any quasi-coherent sheaf F_{0} on Z_{0} we have H^{i}(Z_{0}, F_{0}) = 0 for i ≥ n_{Z_{0}}. Then for any quasi-coherent sheaf F on Z we have H^{i}(Z, F) = 0 for i ≥ n_{Z_{0}}.

Proof. Let F be a quasi-coherent sheaf on Z. Let n be an integer such that 𝒥^{n} = 0, and set F_{k} = 𝒥^{k}F so that we have an increasing sequence of quasi-coherent sheaves

0 = F_{n} ⊂ F_{n−1} ⊂ ⋯ F_{0} = F

with successive quotients F_{k}/F_{k+1} supported on Z_{0}. The result then follows by descending induction on k and consideration of the short exact sequences

0 → F_{k+1} → F_{k} → F_{k}/F_{k+1} → 0

which gives rise to exact sequences

H^{i}(Z, F_{k+1}) → H^{i}(Z, F_{k}) → H^{i}(Z_{0}, F_{k}/F_{k+1}),

where the last term is zero by assumption. ♠

Now we prove B.2 by noetherian induction.

By [23, II.6.7] there exists a dense open subspace j : U ⊂ X with U an affine scheme. Let Z ⊂ X be the complement (with the reduced structure). Let n_{Z} be an integer such that for any quasi-coherent sheaf F_{Z} on Z we have H^{i}(Z, F_{Z}) = 0 for i ≥ n_{Z}.

limlem Lemma B.4. Let G be a quasi-coherent sheaf on X whose restriction to U is the zero sheaf. Then H^{i}(X, G) = 0 for i ≥ n_{Z}.

Proof. By [23, III.1.1] the sheaf G is equal to the limit G = \underrightarrow{lim} G_{i} over its coherent subsheaves G_{i}. Furthermore by [23, II.4.17] we have

H^{i}(X, G) = \underrightarrow{lim} H^{i}(X, G_{i}).

It follows that it suffices to consider the case when G is coherent. In this case there exists a nilpotent thickening Z ⊂ Z′ ⊂ X of Z in X such that the scheme-theoretic support of G is contained in Z′. The result therefore follows from B.3. ♠

Since U is affine and X is separated, the morphism j : U → X is an affine morphism. It follows that the sheaf j_{∗}j^{∗}F is a quasi-coherent sheaf on X and that

H^{i}(X, j_{∗}j^{∗}F) = H^{i}(U, j^{∗}F).

Since U is affine these groups are zero for i > 0. Set n_{X} := n_{Z} + 1.

Let K (resp. I, Q) be the kernel (resp. image, cokernel) of the adjunction map F → j_{∗}j^{∗}F. By B.4 we have H^{i}(X, K) and H^{i}(X, Q) equal to 0 for i ≥ n_{Z}. Consideration of the long exact sequences associated to the short exact sequences

0 → K → F → I → 0

and

0 → I → j_{∗}j^{∗}F → Q → 0

then shows that H^{i}(X, F) = 0 for i ≥ n_{X}. ♠ B.5. Base change for Ext. Let f : X → S be a finite type morphism between noetherian algebraic stacks, and assume that S is an integral scheme. Fix L^{·} ∈ D_{coh}^{−}(X) (the derived category of bounded above complexes of O_{X} -modules with coherent cohomology sheaves) and a coherent sheaf J ∈ Coh(X).

Lemma B.6. For every integer n, there exists a dense open subset (which depends on n) U ⊂ S such that for any cartesian diagram

X′ ⟶^{h} X
f′↓  ↓^{f}
S′ ⟶^{g} S,

where g factors through U, the natural map

h^{∗}Ext^{p}(L^{·}, J) → Ext^{p}(𝕃h^{∗}L^{·}, 𝕃h^{∗}J)

is an isomorphism for p ≤ n.

Proof. The assertion is local in the fppf topology on X, S, and S′. We may therefore assume that X = Spec(R) and S = Spec(A) are affine schemes, and that L^{·} can be represented by a bounded above complex of projective R-modules of finite type, which we again denote by L^{·}. Let J denote the Rmodule corresponding to the sheaf J, and let F^{·} denote the bounded below complex of finite type R-modules

F^{·} := Hom^{·}(L^{·}, J).

After shrinking on S we may assume that J is flat over A, in which case each F^{j} is flat over A. We need to show that after possibly replacing S by a dense affine open subset, the natural map

H^{p}(F^{·}) ⊗_{A} A′ → H^{p}(F^{·} ⊗_{A} A′)

is an isomorphism for all ring homomorphisms A → A′ and all p ≤ n. This is a standard argument and we leave it to the reader (see for example [14, IV.9.4.3] where a similar argument is made). ♠

We would like to use this result to obtain base change properties of global Ext-groups. In order for this to be possible, however, we need certain finiteness properties of cohomology.

cohboundeddef Definition B.7. Let f : X → Y be a morphism of noetherian algebraic

stacks over a scheme S.

(i) f is cohomologically bounded if for every object F ∈ D_{qcoh}^{b}(X) there exists an integer n such that R^{i}f_{∗}F = 0 for i ≥ n.

(ii) f is universally cohomologically bounded if for every morphism of noetherian algebraic stacks Y′ → Y the morphism f′ : X′ := X ×_{Y} Y′ → Y′ is cohomologically bounded.

Remark B.8. By a standard devissage, f is cohomologically bounded if and only if for every quasi-coherent sheaf F on X there exists an integer n such that R^{i}f_{∗}F = 0 for i ≥ n.

Remark B.9. Note that any separated representable morphism f : X → Y is universally cohomologically bounded by B.2.

cohboundedness Theorem B.10. Let f : X → S be a flat finite type proper morphism of noetherian algebraic stacks with S = Spec(A) an integral affine scheme, and assume that f is cohomologically bounded. Let L^{·} ∈ D_{coh}^{−}(X) and J ∈ Coh(X). Then for every integer n, there exists a dense open subset (which depends on n) U ⊂ S such that for any cartesian diagram

X′  ⟶^{h} X
f′↓  ↓^{f}
S′ = Spec(A′) ⟶^{g} S,

where g factors through U, the natural map

targetmorphism (B.10.1)  Ext^{p}(L^{·}, J) ⊗_{A} A′ → Ext^{p}(𝕃h^{∗}L^{·}, 𝕃h^{∗}J)

is an isomorphism for p ≤ n.

Proof. This follows from consideration of the local-to-global spectral sequences for Ext from B.2, and standard base change properties for cohomology of coherent sheaves (see for example [30, 5.12]).

First after shrinking on S we may assume that J is flat over S and that for any morphism g : Spec(A′) → Spec(A) the pullback map

h^{∗}Ext^{p}(L^{·}, J) → Ext^{p}(𝕃h^{∗}L^{·}, 𝕃h^{∗}J)

is an isomorphism for all p ≤ n. By standard base change properties of cohomology of coherent sheaves as in [30, 5.12] (note that this is where the cohomological boundedness is used), it follows that after shrinking some more on S we may assume that each of the terms E_{r}^{pq} for p + q ≤ n in the spectral sequence

E_{2}^{pq} = H^{p}(X , Ext^{q}(L^{·}, J)) ⟹ Ext^{p+q}(L^{·}, J)

are flat over S, and that their formation commute with arbitrary base change Spec(A′) → Spec(A). That B.10.1 is an isomorphism then follows from consideration of the morphism of spectral sequences

E_{2}^{pq} = H^{p}(X , Ext^{q}(L^{·}, J)) ⟹ Ext^{p+q}(L^{·}, J)
↓
E_{2}^{pq} = H^{p}(X′, Ext^{q}(𝕃h^{∗}L^{·}, 𝕃h^{∗}J)) ⟹ Ext^{p+q}(𝕃h^{∗}L^{·}, 𝕃h^{∗}J).
♠

Appendix C. Another boundedness theorem for Hom-stacks

By Martin Olsson

AppendixB

C.1. Statement of Theorems.

C.2. Let B be a scheme, and X and Y be Artin stacks of finite presentation over B with finite diagonals. Let X and Y denote the coarse moduli spaces of X and Y. Assume that X is flat and proper over B, and that fppf locally on B there exists a finite and finitely presented flat surjection Z → X with Z an algebraic space.

By [30, 1.1], we then have Artin stacks \underline{Hom}(X , Y) and \underline{Hom}(X , Y) locally of finite presentation over B with separated and quasi–compact diagonals.

athm Theorem C.3. Assume that Y is a tame stack. Then the natural map

\underline{Hom}(X , Y) → \underline{Hom}(X , Y)

is of finite type.

There is also a variant of C.3, where one does not assume that X admits a finite flat cover by a scheme but instead that X is tame (this is in fact the theorem we will apply to twisted stable maps):

athmbis Theorem C.4. Let B be a scheme locally of finite type over an excellent Dedekind ring, and let X and Y be tame Artin stacks of finite presentation over B with finite diagonals. Let X (resp. Y) denote the moduli space of X (resp. Y). Assume that X is flat and proper over B. Then \underline{Hom}(X , Y) is an algebraic stack locally of finite presentation over B with quasi-compact and separated diagonal, and the map

\underline{Hom}(X , Y) → \underline{Hom}(X , Y) = \underline{Hom}(X, Y)

is of finite type.

Remark C.5. Note that by [4, 3.3], for any morphism B′ → B the base change X_{B′} → X_{B′} is the moduli space of X_{B′}.

Remark C.6. The assumption that X /B is flat implies that X/B is also flat by [4, 3.3 (b)]. It follows that \underline{Hom}(X, Y) is an algebraic space locally of finite presentation over B.

In the context of either C.3 or C.4, one can also consider the substack

\underline{Hom}^{rep}(X , Y) ⊂ \underline{Hom}(X , Y)

classifying representable morphisms X → Y. As in [30, 1.6] this substack is an open substack, and therefore the following corollary follows from C.3 and C.4:

repfinite Corollary C.7. The natural map

\underline{Hom}^{rep}(X , Y) → \underline{Hom}(X , Y)

is of finite type.

The rest of this appendix is devoted to the proofs of C.3 (paragraphs C.17-C.30) and C.4 (paragraphs C.31-C.36).

A3intro C.8. We begin with the proof of C.3.

As in [35, 6.2], Theorem C.3 is equivalent to the following statement. For any morphism f : X → Y the stack \underline{Sec}(X ×_{Y} Y/X) which to any B–scheme T associates the groupoid of sections s : X → X ×_{Y} Y is of finite type. It is this latter statement that we will prove. Let G denote X ×_{f,Y} Y. The proof that \underline{Sec}(G/X) is of finite type over B follows the same outline as in [35]. First (in paragraphs C.17–C.22) we reduce the proof of C.3 to the special case when X = X and G is a μ_{n}–gerbe for some integer n. We then prove the theorem in this special case (paragraphs C.23–C.36) using some relatively straightforward generalizations to “twisted sheaves” of theorems about the Picard functor.

The main new wrinkle is to use the base change properties of Ext explained in appendix B.

C.9. Some results about modifications. We use the results of appendix B to generalize three basic results about modifications in [35, §4] to tame stacks.

Passage to the maximal reduced substack.

C.10. Let S be a noetherian scheme, and let G → X be a finite type morphism between Artin stacks of finite type over S with finite diagonals. Assume further that X is flat and proper over S and that the structure morphism X → S is universally cohomologically bounded in the sense of B.7. Assume further that fppf locally on S there exists a finite flat cover of X by an algebraic space.

finitelem Proposition C.11. Let X_{0} ↪ X be a closed immersion defined by a nilpotent ideal J ⊂ O_{X}, and assume X_{0} is flat over S. Let G_{0} denote the base change G ×_{X} X_{0}. Then the natural map

\underline{Sec}(G/X) → \underline{Sec}(G_{0}/X_{0})

is of finite type.

Proof. This is essentially the same as in [30, 5.11].

By noetherian induction, it suffices to show that the morphism is of finite type over a dense open subset of S. We may further assume that S is reduced (since if U and V are S-schemes locally of finite type and g : U → V is a morphism, then g is of finite type if and only if the base change of g to S_{red} is of finite type), and using the same argument given in [30, paragraph following proof of 5.11] that J^{2} = 0.

It suffices to show that if T → \underline{Sec}(G_{0}/X_{0}) is a morphism corresponding to a section s : X_{0} → G_{0}, then the fiber product

P := \underline{Sec}(G/X) ×_{\underline{Sec}(G_{0}/X_{0})} T

is of finite type over T. Furthermore, we may assume that T is an integral noetherian affine scheme. The stack P associates to any w : W → T the groupoid of liftings s̃: X_{W} → G over X of the composite

X_{0,W} → G_{0} → G,

where the first map is the one induced by s.

To prove that P is quasi–compact it suffices by Noetherian induction to exhibit a dense open set U ⊂ T such that P_{U} is quasi–compact.

Let L_{G_{0}/X_{0}} be the cotangent complex of G_{0}/X_{0}. By B.10 (this is where the assumption of cohomological boundedness is used), after replacing T by a dense open subscheme we may assume that the groups

Ext^{i}(s^{∗}L_{G_{0}/X_{0}}, J), i = −1, 0, 1

are projective modules on T of finite type, and that the formation of these modules commutes with arbitrary base change on T.

By [34, 1.5] there is a canonical obstruction

o ∈ Ext^{1}(s^{∗}L_{G_{0}/X_{0}}, J)

whose vanishing is necessary and sufficient for the existence of a lifting s̃ as above, and the formation of this obstruction is functorial in T. After replacing T by the closed subscheme defined by the condition that o vanishes, we may assume that o = 0. In this case the set of isomorphism classes of liftings s̃ form a torsor under

Ext^{0}(s^{∗}L_{G_{0}/X_{0}}, J)

and the group of infinitessimal automorphisms of s̃ is canonically isomorphic to

Ext^{−1}(s^{∗}L_{G_{0}/X_{0}}, J).

It follows that P is an Ext^{−1}(s^{∗}L_{G_{0}/X_{0}}, J)–gerbe over

Ext^{0}(s^{∗}L_{G_{0}/X_{0}}, J),

and in particular is quasi–compact. ♠

The case of a finite morphism.

C.12. Let S be a noetherian scheme, and X /S a proper flat Artin stack with finite diagonal. Let G → X be a finite morphism. Assume that fppf locally on S, the stack X admits a finite flat surjection Z → X with Z an algebraic space.

finitecase Proposition C.13. If X → S is universally cohomologically bounded, then the stack \underline{Sec}(G/X) is of finite type and separated over S.

Proof. To verify that \underline{Sec}(G/X) is of finite type over S, it suffices to show that its pullback to S_{red} is of finite type. We may therefore assume that S is reduced. Furthermore, using the fact that X → S is universally cohomologically bounded, we may by C.11 (applied with X_{0} = X_{red}) assume that X is also reduced.) Furthermore, by noetherian induction it suffices to exhibit a dense open subset of S over which \underline{Sec}(G/X) is of finite type.

By Chow’s lemma for Artin stacks [33, 1.1], there exists a proper surjection h : X′ → X with X′ a projective S-scheme. After shrinking on S we may also assume that X′ is flat over S (since S is reduced). Since X is reduced the map O_{X} → h_{∗}O_{X′} is injective. After shrinking on S, we may assume that the formation of h_{∗}O_{X′} commutes with arbitrary base change S′ → S, and that for any such base change the map O_{X_{S′}} → h_{∗}O_{X′_{S′}} is also injective. Let G′ → X′ denote the pullback G ×_{X} X′.

By [30, 5.10] the stack (in fact an algebraic space) \underline{Sec}(G′/X′) is of finite type over S. It therefore suffices to show that the pullback map

5.9.1 (C.13.1)  \underline{Sec}(G/X) → \underline{Sec}(G′/X′)

is of finite type.

Let A be the coherent sheaf of algebras defined by G = \underline{Spec}_{X} (A), and let B denote the sheaf of O_{X} -algebras h_{∗}O_{X′}. For a morphism T → S, let A_{T} (resp. B_{T}) denote the pullback of A (resp. B) to X_{T} := X ×_{S} T. Then \underline{Sec}(G′/X′) is equal to the functor which to any scheme T → S associates the set of morphisms of O_{X_{T}} -algebras φ : A_{T} → B_{T} over X_{T}. The stack \underline{Sec}(G/X) is the subfunctor of morphisms φ which factor through O_{X_{T}} ⊂ B_{T}.

Fix a morphism φ : A_{T} → B_{T} over a noetherian scheme T defining a T -valued point of \underline{Sec}(G′/X′), and set

P := \underline{Sec}(G/X) ×_{\underline{Sec}(G′/X′),φ} T.

To prove that P is of finite type, we may again assume that T is reduced, and by noetherian induction it suffices to show that P → T is of finite type over a dense open subset of T. Let M denote the cokernel of the map O_{X_{T}} ↪ B_{T}, and let φ̄ : A_{T} → M be the map induced by φ. Let Q be the cokernel of φ̄, and let K be the kernel of the map M → Q so that there is an exact sequence

Qseq (C.13.2)  0 → K → M → Q → 0.

After shrinking on T we may assume that Q is flat over T in which case C.13.2 remains exact after arbitrary base change T′ → T. In this case, let Z ⊂ X_{T} be the support of the coherent sheaf K, and let W ⊂ T be the image of Z (which is closed since X_{T} → T is proper). Then P is represented by the complement of W in T.

To verify that \underline{Sec}(G/X) is separated, note that we already know that the diagonal is quasi-compact and separated. Therefore it suffices to verify the valuative criterion for properness. This amounts to the following. Assume that S is the spectrum of a valuation ring and let η ∈ S be the generic point. Assume given two sections s_{1}, s_{2} : X → G whose restrictions to η are equal. Then we need to show that s_{1} = s_{2}. For this note that since G is finite over X, we have G = \underline{Spec}_{X} (A) for some coherent sheaf of O_{X} -algebras A on X, and the sections s_{1} and s_{2} are specified by two morphisms of O_{X} -algebras ρ_{1}, ρ_{2} : A → O_{X}. Let j : X_{η} ↪ X be the inclusion of the generic fiber. Since X is flat over S, the natural map O_{X} → j_{∗}O_{X_{η}} is an inclusion. Therefore it suffices to show that the composite maps

\xymatrix{A \ar[r]^{ρ_{i}} & O_{X} \ar[r] & j_{∗}O_{X_{η}}}

are equal. Equivalently that the maps on the generic fiber

ρ_{i,η} : A_{η} → O_{X_{η}}

are equal, which holds by assumption. ♠

Behavior with respect to proper modifications of X.

P:C.14 C.14. Let S be a noetherian scheme, and let X and Y be Artin stacks of finite type over S with finite diagonals. Assume that the following conditions hold:

(i) The formation of the coarse spaces π_{X} : X → X and π_{Y} : Y → Y commutes with arbitrary base change S′ → S.

(ii) X is proper and flat over S, and fppf locally on S there exists a finite

flat surjection Z → X with Z an algebraic space. (iii) The coarse space X of X is flat over S (note that X is automatically

proper over S since X is proper over S). (iv) The morphism X → S is universally cohomologically bounded.

modprop Proposition C.15. Let m : X′ → X be a proper representable surjection with X′ a proper and flat algebraic stack over S with finite diagonal. Let f : X → Y be a morphism, and let G (resp. G′) denote the pullback along f (resp. f ∘ m) of Y. Assume that \underline{Sec}(G/X) and \underline{Sec}(G′/X′) are algebraic stacks locally of finite type over S with quasi-compact diagonals. Then the pullback map

pullmap (C.15.1)  \underline{Sec}(G/X) → \underline{Sec}(G′/X′)

is of finite type.

Remark C.16. Note that since m is representable, the stack X′ also admits a finite flat surjection from an algebraic space.

Proof. This follows from the same argument proving [35, 4.13]. The cohomological boundedness assumption is necessary in order to apply C.11. ♠ C.17. Dévissage. Now we begin the proof of the statement that \underline{Sec}(G/X) in C.8 is of finite type over B. In the rest of the proof that \underline{Sec}(G/X) is of finite type over B, we work under the assumptions of C.14.

By a standard limit argument as discussed for example in [30, §2] we can without loss of generality assume that B is of finite type over ℤ.

C.18. The assertion that \underline{Sec}(G/X) is of finite type is fppf local on B, and therefore we may assume that there exists a finite flat surjection Z → X with Z an algebraic space. Let Z^{(i)} denote the i–fold fiber product of Z with itself over X. Then by the description of \underline{Sec}(G/X) in terms of the \underline{Sec}(G ×_{X} Z^{(i)}/Z^{(i)}) given in [30, 3.3] it suffices to show that for i = 1, 2, 3 the stacks \underline{Sec}(G ×_{X} Z^{(i)}/Z^{(i)}) are of finite type. We may therefore assume that X = X.

proofstart

Furthermore, we can without loss of generality assume that B is integral. By noetherian induction, it suffices to exhibit a dominant morphism B′ → B such that the restriction of \underline{Sec}(G/X) to B′ is of finite type.

C.19. By the same argument used in [35, 6.4] we then reduce the proof of C.3 to the case when X is integral and G → X has a section s : X → G. Let \underline{Aut}(s) denote the group scheme of automorphisms of s (a scheme over X). Since X is reduced there exists a dense open subset U ⊂ X such that the restriction of \underline{Aut}(s) to U is flat over U. Let H ⊂ \underline{Aut}(s) denote the scheme– theoretic closure of \underline{Aut}(s)|_{U}. By [17, Premiére partie 5.2.2] there exists a blow-up X′ → X with center a proper closed subspace of X such that the strict transform of H in \underline{Aut}(s) is flat over X′. After further shrinking on S we may assume that X′ is flat over S. Since the map

\underline{Sec}(G/X) → \underline{Sec}(G ×_{X} X′/X′)

is of finite type by C.15, we may therefore replace X by X′ and hence can assume that H is a finite flat subgroup scheme of \underline{Aut}(s). Since all the geometric fibers of H are linearly reductive being closed subgroup schemes of linearly reductive groups (see [4, 2.7]), the X-group scheme H is in fact linearly reductive. If X̃ is the normalization of X the map

\underline{Sec}(G/X) → \underline{Sec}(G ×_{X} X̃/X̃)

is also of finite type by C.15. This enables us to reduce to the following situation: X is normal, s : X → G is a section, and H ⊂ \underline{Aut}(s) is a finite closed ^{20} subgroup scheme which is flat and linearly reductive over X and

←20

over a dense open subset of X the group scheme H is equal to \underline{Aut}(s).

Lemma C.20. The morphism π : BH → G induced by s identifies BH with the normalization of G_{red}.

Proof. Etale locally on X we can write G = [W/GL_{n}] for some n, where W is an X–scheme. Indeed by [4, 3.2] we can étale locally on X write G = [Z/G] with G a linearly reductive group scheme. Choosing any embedding G ↪ GL_{n} we take W to be the quotient of Z × GL_{n} by the diagonal action of G.

Let P → X be the pullback X ×_{s,G} W. Then P is a GL_{n}–torsor over X with a GL_{n}–equivariant morphism f : P → W. Since P is smooth over X the space P is in particular normal, and since s is proper and quasi–finite the morphism f is finite. After replacing X by an étale cover we may assume given a section p ∈ P. Then \underline{Aut}(s) is the closed subgroup scheme of GL_{n,X} given by the fiber product of the diagram

20(Martin) removed normal as it is not used

W × GL_{n}

ρ↓
W × W ⟵^{Δ∘f∘p} X,

where Δ : W → W × W is the diagonal and ρ is the map sending (v, g) to (v, g(v)). In particular, we obtain an embedding H ⊂ GL_{n,X}. Let \overline{P} denote the quotient P/H. The space \overline{P} is normal. By the construction the map f induces a morphism f̄: \overline{P} → W^{∗}, where W^{∗} denotes the normalization of W_{red}. This map f̄ is finite, surjective, and birational and hence by Zariski’s Main Theorem an isomorphism. On the other hand, we have \overline{P} ≃ W ×_{G} BH, and hence if G^{∗} denotes the normalization of G_{red} we find that the base change of BH → G^{∗} to W ×_{G} G^{∗} is an isomorphism. It follows that BH → G^{∗} is an isomorphism. ♠

C.21. Let G^{∗} denote the normalization of G, so that G^{∗} is a gerbe over X. As in [35, 3.14], after shrinking on B we may assume that for every field valued point b ∈ B(k) the fiber G_{b}^{∗} → G_{b} is the normalization of G_{b,red} and that X_{b} is normal. It then follows that the map

\underline{Sec}(G^{∗}/X) → \underline{Sec}(G/X)

is surjective on field valued points and hence surjective. We may therefore replace G by G^{∗} and therefore may assume that G is a gerbe over X.

Next we consider the case of a gerbe. If the generic point of B has characteristic 0, then after shrinking on B we can assume G is Deligne– Mumford in which case the result follows from [35, 1.1]. We may therefore assume that the generic point of B has characteristic p > 0, and hence after shrinking on B may assume that B is an 𝔽_{p}–scheme. In this case, for any B–scheme T and t ∈ G(T), the automorphism group scheme G_{t} of t is canonically an extension

1 → Δ_{t} → G_{t} → H_{t} → 1,

where Δ_{t} is locally diagonalizable and H_{t} is étale over T. Indeed, the group scheme G_{t} is tame so we can take Δ_{t} to be the subfunctor of G_{t} classifying elements of G_{t} killed by some power of p. This normal subgroup Δ_{t} is functorial in the pair (T, t).

Define H to be the stack (with respect to the fppf topology) associated to the prestack whose objects are the same as the objects of G but for which a morphism between two t, t′ ∈ G(T) is defined to be the set

Δ_{t}\\underline{Hom}(t, t′) = Δ_{t}\\underline{Hom}(t, t′)/Δ_{t′} = \underline{Hom}(t, t′)/Δ_{t′}.

Then H is a Deligne–Mumford stack and there is a natural map G → H over X. By [35, 1.1] the stack \underline{Sec}(H/X) is of finite type. On the other hand, for any section s : X → H, the fiber product

\underline{Sec}(G/X) ×_{\underline{Sec}(H/X),s} B

is isomorphic to \underline{Sec}(G ×_{H,s} X/B). Now the stack G ×_{H,s} X is a gerbe over X whose stabilizer groups are all diagonalizable. This further reduces the proof to the case when the H_{t}’s in the above discussion are all trivial.

B.33 C.22. In this case, there exists a canonical locally diagonalizable group scheme Δ on X such that G is bound by Δ. Indeed fppf locally on X there exists a section s : X → G giving rise to a group scheme Δ_{s}. If s′ : X → G is a second section then locally s and s′ are isomorphic so we obtain an isomorphism Δ_{s} ≃ Δ_{s′}. This isomorphism is independent of the choice of the isomorphism between s and s′ since Δ_{s} and Δ_{s′} are abelian. It follows that the Δ_{s}’s descend to a group scheme Δ on X.

Since the Cartier dual of Δ is a locally constant sheaf of finite abelian groups on X_{et}, there exists a finite étale covering X′ → X such that the pullback of Δ to X′ is a diagonalizable group scheme. Since

\underline{Sec}(G/X) → \underline{Sec}(G ×_{X} X′/X′)

is of finite type by C.15, this reduces the proof to the case when G is a gerbe over X bound by a diagonalizable group scheme Δ. In other words, when G corresponds to a cohomology class in H^{2}(X, Δ). Write Δ = μ_{n_{1}} × ⋯ × μ_{n_{r}} so that

H^{2}(X, Δ) = ∏_{i} H^{2}(X, μ_{n_{i}}).

Using the resulting decomposition of the cohomology class of G, we see that G is isomorphic to a product G_{1} ×_{X} ⋯ ×_{X} G_{r}, where G_{i} is a μ_{n_{i}}–gerbe over X. Then

\underline{Sec}(G/X) ≃ ∏_{i} \underline{Sec}(G_{i}/X).

This therefore finally reduces the proof to the case of a μ_{n}–gerbe over X which is the case treated in the following subsection.

Gtwisted

C.23. G–twisted line bundles.

case1 C.24. Fix an integer n. Assume that X = X and that G is a μ_{n}–gerbe over X. Assume furthermore that if f : X → B denotes the structural morphism then the map O_{B} → f_{∗}O_{X} is an isomorphism, and the same remains true after arbitrary base change B′ → B. This implies that for any scheme T and object s ∈ G(T) we have an isomorphism μ_{n} ≃ \underline{Aut}(s) and these isomorphisms are functorial in the pair (T, s). If L is a line bundle on G then for any such pair (T, s) we obtain a line bundle s^{∗}L on T which also comes equipped with an action of μ_{n} = \underline{Aut}(s). The line bundle L is called a G–twisted invertible sheaf on X if this action of μ_{n} coincides with the standard action induced by the embedding μ_{n} ↪ 𝔾_{m}.

Note that if χ denotes the standard character μ_{n} ↪ 𝔾_{m} then for any line bundle L on G the action of μ_{n} = \underline{Aut}(s) on s^{∗}L is given by χ^{i} for some integer i (locally constant on T). It follows that to check that a line bundle L is G–twisted it suffices to verify that the two actions coincide for pairs (T, s) with T the spectrum of an algebraically closed field.

If L is a G–twisted sheaf on X, then L^{⊗n} descends canonically to an invertible sheaf on X since the stabilizer actions of μ_{n} are trivial. We usually write just L^{⊗n} for the sheaf on X obtained from L^{⊗n}.

B.18 Proposition C.25. There is a natural equivalence between \underline{Sec}(G/X) and the stack which to any B–scheme T associates the groupoid of pairs (L, ι), where L is a G–twisted sheaf on X and ι : L^{⊗n} ≃ O_{X} is an isomorphism of invertible sheaves on X.

Proof. If s : X → G is a section, then there is a canonical action of μ_{n} = \underline{Aut}(s) on F := s_{∗}O_{X} and hence F decomposes canonically as ⊕_{χ∈ℤ/(n)}F_{χ}.

Lemma C.26. Each F_{χ} is locally free of rank 1 on G and for any two χ, ϵ ∈ ℤ/(n) the natural map F_{χ} ⊗ F_{ϵ} → F_{χ+ϵ} is an isomorphism. If χ_{1} : μ_{n} → 𝔾_{m} denotes the standard inclusion, then F_{χ_{1}} is a G–twisted sheaf on X.

Proof. The choice of the section s identifies G with B\underline{Aut}(s) ≃ Bμ_{n} × X. The fiber product X ×_{s,Bμ_{n},s} X is equal to X × Bμ_{n} from which we see that s^{∗}F is equal to O_{X} ⊗_{ℤ} ℤ[X]/(X^{n} − 1) with the natural action of μ_{n}. From this the lemma follows. ♠

In particular, by sending a section s : X → G to F_{χ_{1}} with the isomorphism F_{χ_{1}}^{⊗n} ≃ O_{G} we obtain a functor

sectwist (C.26.1)  \underline{Sec}(G/X) → (stack of pairs (L, ι) as in C.25).

Conversely, given a G–twisted sheaf L with an isomorphism ι : L^{⊗n} ≃ O_{X} we can form the cyclic algebra A = ⊕_{i=0}^{n−1}L^{⊗i} with multiplication induced by the natural maps L^{⊗i} ⊗ L^{⊗j} → L^{⊗i+j} and the map ι. We can then consider

Spec_{G}(A) → G.

If G = Bμ_{n} ×X and X → G is the map defined by the trivial torsor, then the restriction of (L, ι) to X is an invertible sheaf L on X with an isomorphism L^{⊗n} ≃ O_{X} and the restriction of A to X is just the cyclic algebra ⊕_{i=0}^{n−1}L^{⊗i} with μ_{n} acting L^{⊗i} through the character u ↦ u^{i}. It follows that the projection Spec_{G}(A) → X is an isomorphism and hence defines a point of \underline{Sec}(G/X). We leave to the reader that this defines an inverse to C.26.1. ♠

Let 𝒫ic_{X/B}^{G} denote the stack over B which to any B–scheme T associates the groupoid of G_{T} –twisted invertible sheaves on X_{T} (where G_{T} denotes G ×_{B} T etc.).

Proposition C.27. (i) The stack 𝒫ic_{X/B}^{G} is an algebraic stack locally of finite presentation over B.

(ii) The sheaf (with respect to the fppf topology) associated to the presheaf

T ↦ {isomorphism classes in 𝒫ic_{X/B}^{G}}

is representable by a separated algebraic space locally of finite presentation over B.

Proof. This follows for example by the same argument used in [5, Appendix].

♠

C.28. There is a natural morphism

Picmap (C.28.1)  \underline{Pic}_{X/B}^{G} → \underline{Pic}_{X/B}, L ↦ L^{⊗n}.

Denote by \underline{Pic}_{X/B}^{G}[n] the inverse image of the identity e = [O_{X}] ∈ \underline{Pic}_{X/B}. Another description of this space is as follows. The map C.28.1 lifts naturally to a morphism

π^{∗} : 𝒫ic_{X/B}^{G} → 𝒫ic_{X/B}, L ↦ L^{⊗n}.

Let e : B → 𝒫ic_{X/B} be the morphism corresponding to O_{X}, and set

𝒫ic_{X/B}^{G}[n] := 𝒫ic_{X/B}^{G} ×_{𝒫ic_{X/B},e} B.

The stack 𝒫ic_{X/B}^{G}[n] classifies pairs (L, ι), where L is a G–twisted sheaf on X and ι : L^{⊗n} ≃ O_{X} is an isomorphism. The space \underline{Pic}_{X/B}^{G}[n] is the coarse moduli space of 𝒫ic_{X/B}^{G}[n] via the map sending (L, ι) to the class of L. In fact, 𝒫ic_{X/B}^{G}[n] is a μ_{n}–gerbe over \underline{Pic}_{X/S}^{G}[n]. Indeed any point P of \underline{Pic}_{X/B}^{G}[n](B) can fppf–locally on B be represented by a G–twisted sheaf L with L^{⊗n} ≃ O_{X}. Conversely for any two pairs (L, ι) and (L′, ι′) defining the same point of \underline{Pic}_{X/B}^{G}[n], the two G–twisted sheaves L and L′ become isomorphic after making an fppf base change on B. Then ι and ι′ differ by a section of f_{∗}O_{G}^{∗} ≃ 𝔾_{m}. After making another fppf base change on B so that this unit becomes an n–th power, the two pairs (L, ι) and (L′, ι′) become isomorphic. Moreover, this isomorphism is unique up to multiplication by an element of μ_{n}(B).

Proposition C.29. The algebraic space \underline{Pic}_{X/B}^{G}[n] is of finite type over B.

Proof. To prove this statement it suffices by a standard limit argument to consider the case when B is a noetherian scheme. In this case by noetherian induction it suffices to find a dominant morphism B′ → B such that the restriction of \underline{Pic}_{X/B}^{G} to B′ is of finite type. We may therefore also assume that B is a noetherian affine integral scheme. By Chow’s lemma there exists a proper surjection P → G with P an algebraic space. After shrinking on B we may assume that P is also flat over B. Then G ×_{X} P is trivial and by C.15 the pullback map

\underline{Sec}(G/X) ≃ 𝒫ic_{X/B}^{G}[n] → 𝒫ic_{G×_{X}P/P}^{G} [n] ≃ \underline{Sec}(G ×_{X} P/P)

is of finite type. Consequently the map

\underline{Pic}_{X/B}^{G}[n] → \underline{Pic}_{P/B}^{G}[n]

is also of finite type. It therefore suffices to show \underline{Pic}_{P/B}^{G}[n] is of finite type.

Now in the case when G = Bμ_{n} × X there exists a globally defined Gtwisted sheaf L with L^{⊗n} trivial. Choose one such sheaf L. If p : G → X denotes the projection, then for any invertible sheaf M on X with M^{⊗n} ≃ O_{X}, the sheaf L ⊗ p^{∗}M is also a G–twisted sheaf. Conversely, if N is a G–twisted sheaf with N^{⊗n} trivial, then L ⊗ N^{−1} descends uniquely to an n–torsion sheaf on X. Using this one sees that \underline{Pic}_{X/B}^{G}[n] is a trivial torsor under \underline{Pic}_{X/B}[n]. The proposition therefore follows from [22, 6.27 and 6.28]. ♠

proofend C.30. Since \underline{Sec}(G/X) ≃ 𝒫ic_{X/B}^{G}[n] is a μ_{n}–gerbe over \underline{Pic}_{X/B}^{G}[n] this completes the proof of C.3 in the special case of C.24 and hence also the proof in general. ♠

Proof of C.4

21

21→

C.31. Algebraicity of \underline{Hom}(X , Y). By the case of algebraic spaces, the stack

C.30

\underline{Hom}(X, Y) is an algebraic space locally of finite presentation over B. To prove C.4 it therefore suffices, as in the proof of C.3, to show that for any morphism f : X → Y the stack \underline{Sec}(G/X) is algebraic locally of finite presentation over B, where G denotes the pullback of Y along the morphism

\xymatrix{X \ar[r] & X \ar[r]^{f} & Y.}

This is shown exactly as in [30, 5.3-5.8] by verifying Artin’s conditions. The key point is Grothendieck’s existence theorem for Artin stacks which is shown in [30, A.1].

Note that this implies that if X /B is a proper flat tame stack and G → X is a morphism with G a tame stack of finite presentation over B, then \underline{Sec}(G/X) is an algebraic stack locally of finite presentation over B, as it is equal to the fiber product of the diagram

\xymatrix{ & \underline{Hom}(X , G) \ar[d] \\ S \ar[r]^{_{id_{X}}} & \underline{Hom}(X , X).}

C.32. The diagonal of \underline{Sec}(G/X) is quasi-compact and separated.

L:C.33 Lemma C.33. If X /B is a proper and flat tame stack, and I → X is a finite morphism, then \underline{Sec}(I/X) is separated and of finite type over B.

Proof. For the quasi-compactness of the diagonal, it suffices to show that given two sections s_{1}, s_{2} : X → I, the fiber product M of the diagram

\xymatrix{ & S \ar[d]^{s_{1}×s_{2}} \\ \underline{Sec}(I/X) \ar[r]^{Δ} & \underline{Sec}(I/X) ×_{S} \underline{Sec}(I/X)}

21(Martin) I reorganized the proof of C.4

is quasi-compact. For this we may assume that S is reduced, and furthermore by noetherian induction it suffices to show that the map M → S is quasi-compact over some dense open in S. Let X_{0} denote the fiber product of the diagram

\xymatrix{ & X \ar[d]^{s_{1}×s_{2}} \\ I \ar[r]^{Δ} & I ×_{X} I.}

Since I is finite over X, the diagonal of I/X is a closed immersion which implies that the projection j : X_{0} → X is also a closed immersion. The S-space M represents the functor which to any scheme T/S associates the unital set if the base change

X_{0,T} → X_{T}

is an isomorphism, and the empty set otherwise. Now to prove that M is of finite type over S, we may by shrinking if necessary also assume that X_{0} is flat over S. In this case, if Z ⊂ X denotes the support of the ideal sheaf of X_{0}, then M is represented by the complement of the closed (since X /S is proper) image of Z in S. This proves the quasi-compactness of the diagonal of \underline{Sec}(I/X).

To verify that the diagonal of \underline{Sec}(I/X) is proper, one verifies the valuative criterion using the same argument as in the proof of C.13.

The proof that \underline{Sec}(I/X) is of finite type over B now proceeds exactly as in the proof of C.13. ♠

Now let X and Y be finite type tame B-stacks with X proper and flat over B, and let X and Y be the coarse moduli space of X and Y respectively. Fix a morphism f : X → Y, and let G → X be the pullback of Y along the composite

\xymatrix{X \ar[r] & X \ar[r]^{f} & Y.}

Lemma C.34. The diagonal of \underline{Sec}(G/X) is quasi-compact and separated.

Proof. Let s_{1}, s_{2} : X → G be two sections, and let I denote the fiber product of the diagram (which is a finite stack over X since G has finite diagonal)

X

↓^{s_{1}×s_{2}}
G ⟶^{Δ} G ×_{X} G.

Then the fiber product of the diagram

\xymatrix{ & B \ar[d]^{s_{1}×s_{2}} \\ \underline{Sec}(G/X) \ar[r]^{Δ} & \underline{Sec}(G/X) ×_{B} \underline{Sec}(G/X)}

is isomorphic to \underline{Sec}(I/X) which is quasi-compact and separated by C.33. ♠

C.35. Proposition C.15 still holds. Now an examination of the proof of C.15, following [35, 4.13], shows that once we know that in the case when X is tame the stack \underline{Sec}(G/X) is locally of finite type with separated and quasi-compact diagonal, then the proof of C.15 carries over also in the case when X does not necessarily admit a finite flat surjection from an algebraic space but instead is tame.

B.45 C.36. Completion of proof. To prove that \underline{Sec}(G/X) is quasi-compact, it suffices by noetherian induction to exhibit a dense open subset of B where this is so. Furthermore, we may without loss of generality assume that B is integral. By Chow’s lemma for Artin stacks [33, 1.1] there exists a proper surjection X′ → X with X′ an algebraic space. After shrinking on B we may assume that X′ is flat over B. Applying C.15 to the stack \underline{Sec}(G/X) and using the case when X is an algebraic space we get that \underline{Sec}(G/X) is quasi-compact. This completes the proof of C.4.

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(Abramovich) Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, U.S.A.

E-mail address: abrmovic@math.brown.edu

(Olsson) Department of Mathematics #3840, University of California, Berkeley, CA 94720-3840, U.S.A.

E-mail address: molsson@math.berkeley.edu

(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

E-mail address: angelo.vistoli@sns.it