In this paper we study a notion of twisted stable map, from a curve to a tame Deligne-Mumford stack, which generalizes the well-known notion of stable map to a projective variety.
1. Introduction
We fix a noetherian base scheme ${\mathbb{S}}$.
1.1. The problem of moduli of families
Consider a Deligne-Mumford stack ${\mathcal{M}}$ (definition in Section 2.1) admitting a projective coarse moduli scheme ${\mathbf{M}}$. Given a curve $C$, it is often natural to consider morphisms $f:C \to {\mathcal{M}}$ (or equivalently, objects $f\in {\mathcal{M}}(C)$); in case ${\mathcal{M}}$ is the moduli stack of geometric objects, these morphisms correspond to families over $C$. For example, if ${\mathcal{M}}=\overline {{\mathcal{M}}}_{\gamma }$, the stack of stable curves of genus $\gamma$, then morphisms $f:C\to {\mathcal{M}}$ correspond to families of stable curves of genus $\gamma$ over $C$; and if ${\mathcal{M}}= {\mathcal{B}}G$, the classifying stack of a finite group $G$, we get principal $G$-bundles over $C$. It is interesting to study moduli of such objects; moreover, it is natural to study such moduli as $C$ varies and to find a natural compactification for such moduli.
One approach is suggested by Kontsevich’s moduli of stable maps.
1.2. Stable maps
First consider a projective scheme ${\mathbf{M}}\to {\mathbb{S}}$ with a fixed ample sheaf ${\mathcal{O}}_{\mathbf{M}}(1)$. Given integers $g,n,d$, it is known that there exists a proper algebraic stack, which here we denote by ${{\mathcal{K}}_{g,n}({\mathbf{M}},d)}$, of stable, $n$-pointed maps of genus $g$ and degree $d$ into ${\mathbf{M}}$. (See ReferenceKo, ReferenceB-M, ReferenceF-P, Reference$\aleph$-O, where the notation $\overline {\mathcal{M}}_{g,n}({\mathbf{M}},d)$ is used. Here we tried to avoid using ${\mathcal{M}},{\mathbf{M}}$ with two meanings.) This stack admits a projective coarse moduli space ${{\mathbf{K}}_{g,n}({\mathbf{M}},d)}$. If one avoids “small” residue characteristics in ${\mathbb{S}}$, which depend on $g,n,d$ and ${\mathbf{M}}$, then this stack is in fact a proper Deligne-Mumford stack.
1.3. Stable maps into stacks
Now fix a proper Deligne-Mumford stack ${\mathcal{M}}\to {\mathbb{S}}$ admitting a projective coarse moduli space ${\mathbf{M}}\to {\mathbb{S}}$ on which we fix an ample sheaf as above. We further assume that ${\mathcal{M}}$ is tame, that is, for any geometric point $s\colon \operatorname {Spec}\Omega \to {\mathcal{M}}$, the group ${\operatorname {Aut}}_{\operatorname {Spec}\Omega }(s)$ has order prime to the characteristic of the algebraically closed field $\Omega$.
It is tempting to mimic Kontsevich’s construction as follows: Let $C$ be a nodal projective connected curve; then a morphism $C \to {\mathcal{M}}$ is said to be a stable map of degree $d$ if the associated morphism to the coarse moduli scheme $C \to {\mathbf{M}}$ is a stable map of degree $d$.
It follows from our results below that the category of stable maps into ${\mathcal{M}}$ is a Deligne-Mumford stack. A somewhat surprising point is that it is not complete.
To see this, we fix $g=2$ and consider the specific case of ${\mathcal{B}}G$ with $G= ({\mathbb{Z}}/3{\mathbb{Z}})^4$. Any smooth curve $C$ of genus $2$ admits a connected principal $G$-bundle, corresponding to a surjection $H_1(C, {\mathbb{Z}}) \to G$, thus giving a map $C \to {\mathcal{B}}G$. If we let $C$ degenerate to a nodal curve $C_0$ of geometric genus $1$, then $H_1(C_0, {\mathbb{Z}}) \simeq {\mathbb{Z}}^3$, and since there is no surjection ${\mathbb{Z}}^3 \to G$, there is no connected principal $G$-bundle over $C_0$. This means that there can be no limiting stable map $C_0 \to {\mathcal{B}}G$ as a degeneration of $C \to {\mathcal{B}}G$.
1.4. Twisted stable maps
Our main goal here is to correct this deficiency. In order to do so, we will enlarge the category of stable maps into ${\mathcal{M}}$. The source curve ${{\mathcal{C}}}$ of a new stable map ${{\mathcal{C}}} \to {\mathcal{M}}$ will acquire an orbispace structure at its nodes. Specifically, we endow it with the structure of a Deligne-Mumford stack.
It is not hard to see how these orbispace structures come about. Let $S$ be the spectrum of a discrete valuation ring $R$ of pure characteristic 0, with quotient field $K$, and let $C_K\to \eta \in S$ be a nodal curve over the generic point, together with a map $C_K \to {\mathcal{M}}$ of degree $d$, whose associated map $C_K \to {\mathbf{M}}$ is stable. We can exploit the fact that ${{\mathcal{K}}}_{g,0}({\mathbf{M}},d)$ is complete; after a ramified base change on $S$ the induced map $C_K \to {\mathbf{M}}$ will extend to a stable map $C \to {\mathbf{M}}$ over $S$. Let $C_{\operatorname {sm}}$ be the smooth locus of the morphism $C \to S$; Abhyankar’s lemma, plus a fundamental purity lemma (see Lemma 2.4.1 below) shows that after a suitable base change we can extend the map $C_K \to {\mathcal{M}}$ to a map $C_{\operatorname {sm}}\to {\mathcal{M}}$; in fact the purity lemma fails to apply only at the “new” nodes of the central fiber, namely those which are not in the closure of nodes in the generic fiber. On the other hand, if $p\in C$ is such a node, then on an étale neighborhood $U$ of $p$, the curve $C$ looks like
$$\begin{equation*} uv = t^r, \end{equation*}$$
where $t$ is the parameter on the base. By taking $r$-th roots,
$$\begin{equation*} u = u_1^r,\ v = v_1^r, \end{equation*}$$
we have a nonsingular cover $V_0\to U$ where $V_0$ is defined by $u_1v_1 = t$. The purity lemma applies to $V_0$, so the composition ${V_0}_K \to C_K \to {\mathcal{M}}$ extends over all of $V_0$. There is a minimal intermediate cover $V_0\to V\to U$ such that the family extends already over $V$; this $V$ will be of the form $xy = t^{r/m}$, and the map $V \to U$ is given by $u = x^m$,$v = y^m$. Furthermore, there is an action of the group ${\boldsymbol {\mu }}_m$ of roots of 1, under which $\alpha \in {\boldsymbol {\mu }}_m$ sends $x$ to $\alpha x$ and $y$ to $\alpha ^{-1} y$, and $V/ {\boldsymbol {\mu }}_m = U$. This gives the orbispace structure ${{\mathcal{C}}}$ over $C$, and the map $C_K \to {\mathcal{M}}$ extends to a map ${{\mathcal{C}}} \to {\mathcal{M}}$.
This gives the flavor of our definition.
We define a category ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$, fibered over ${\mathcal{S}}ch/{\mathbb{S}}$, of twisted stable $n$-pointed maps ${\mathcal{C}}\to {\mathcal{M}}$ of genus $g$ and degree $d$. This category is given in two equivalent realizations: one as a category of stable twisted ${\mathcal{M}}$-valued objects over nodal pointed curves endowed with atlases of orbispace charts (see Definition 3.7.2); the other as a category of representable maps from pointed nodal Deligne-Mumford stacks into ${\mathcal{M}}$, such that the map on coarse moduli spaces is stable (see Definition 4.3.1). In our treatment, both realizations are used in proving our main theorem:
Theorem 1.4.1
(1)
The category ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$ is a proper algebraic stack.
(2)
The coarse moduli space ${{\mathbf{K}}_{g,n}({\mathcal{M}},d)}$ of ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$ is projective.
(3)
There is a commutative diagram$$\begin{equation*} \begin{array}{ccc} {{\mathcal{K}}_{g,n}({\mathcal{M}},d)}& \to & {{\mathcal{K}}_{g,n}({\mathbf{M}},d)}\\\downarrow & & \downarrow \\{{\mathbf{K}}_{g,n}({\mathcal{M}},d)}& \to & {{\mathbf{K}}_{g,n}({\mathbf{M}},d)}\end{array} \end{equation*}$$
where the top arrow is proper, quasifinite, relatively of Deligne-Mumford type and tame, and the bottom arrow is finite. In particular, if ${{\mathcal{K}}_{g,n}({\mathbf{M}},d)}$ is a Deligne-Mumford stack, then so is ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$.
1.5. Some applications and directions of further work
(1)
In our paper Reference$\aleph$-V2 we studied the situation where ${\mathcal{M}}= \overline {\mathcal{M}}_{\gamma }$, which gives a complete moduli for fibered surfaces. Some further results for elliptic surfaces are obtained in G. La Nave’s thesis ReferenceLa.
(2)
The case where ${\mathcal{M}}$ is the classifying stack of a finite group $G$ allows one to improve on the spaces of admissible covers, give moduli compactifications of spaces of curves with abelian and nonabelian level structures, and, with a suitable choice of the group $G$, show that there is a smooth, fine moduli space for admissible $G$-covers, which is a finite covering of $\overline {{\mathcal{M}}}_g$. This is the subject of our preprint Reference$\aleph$-C-V with Alessio Corti, with some ideas contributed by Johan de Jong. Some of these applications were indicated in our announcement Reference$\aleph$-V1. This approach to admissible covers is closely related to the work of Wewers ReferenceWe.
(3)
A similar reasoning applies to curves with $r$-spin structures, e.g. theta characteristics. This is studied in Reference$\aleph$-J.
(4)
The recursive nature of the theorem allows one to construct both minimal models and stable reduction for pluri-fibered varieties. This is related to recent work of Mochizuki ReferenceMo and deserves further study.
(5)
In ReferenceC-R, W. Chen and Y. Ruan introduce Gromov-Witten invariants of an orbifold, using a differential geometric counterpart of our stack of twisted stable maps. In our joint work Reference$\aleph$-G-V with Tom Graber we give an algebraic treatment of these Gromov-Witten invariants, and in the special case of 3-pointed genus 0 maps of degree 0, construct the Chen-Ruan product with integer coefficients. See also ReferenceF-G for an algebraic treatment of global quotients.
(6)
In this paper we verify that ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$ is a proper stack by going through the conditions one by one. It may be worthwhile to develop a theory of Grothendieck Quot-stacks and deduce our results from such a theory. It seems likely that some of our methods could be useful for developing such a theory.
While our paper was circulating, we were told in 1999 by Maxim Kontsevich that he had also discovered the stack of twisted stable maps, but had not written down the theory. His motivation was in the direction of Gromov-Witten invariants of stacks.
1.6. Acknowledgments
We would like to thank Kai Behrend, Larry Breen, Barbara Fantechi, Ofer Gabber, Johan de Jong, Maxim Kontsevich, and Rahul Pandharipande, for helpful discussions. We are grateful to Laurent Moret-Bailly for providing us with a preprint of the book ReferenceL-MB before it appeared. The first author thanks the Max Planck Institute für Mathematik in Bonn for a visiting period which helped in putting this paper together.
2. Generalities on stacks
2.1. Criteria for a Deligne-Mumford stack
We refer the reader to ReferenceAr and ReferenceL-MB for a general discussion of algebraic stacks (generalizing ReferenceD-M), and to the appendix in ReferenceVi for an introduction. We spell out the conditions here, as we follow them closely in the paper. We are given a category ${\mathcal{X}}$ along with a functor ${\mathcal{X}}\to {\mathcal{S}}ch/{\mathbb{S}}$. We assume
(1)
${\mathcal{X}}\to {\mathcal{S}}ch/{\mathbb{S}}$ is fibered in groupoids (see ReferenceAr, §1, (a) and (b), or ReferenceL-MB, Definition 2.1). This means:
(a)
for any morphism of schemes $T \to T'$ and any object $\xi '\in {\mathcal{X}}(T')$ there is an object $\xi \in {\mathcal{X}}(T)$ and an arrow $\xi \to \xi '$ over $T\to T'$; and
(b)
for any diagram of schemes$$\begin{equation*} \begin{array}{ccccc}T_1 & &\longrightarrow & & T_2 \\&\searrow &&\swarrow & \\&& T_3 && \end{array} \end{equation*}$$
and any objects $\xi _i \in {\mathcal{X}}(T_i)$ sitting in a compatible diagram$$\begin{equation*} \begin{array}{ccccc}\xi _1 & && & \xi _2 \\&\searrow &&\swarrow & \\&& \xi _3 && \end{array} \end{equation*}$$
there is a unique arrow $\xi _1 \to \xi _2$ over $T_1 \to T_2$ making the diagram commutative.
We remark that this condition is automatic for moduli problems, where ${\mathcal{X}}$ is a category of families with morphisms given by fiber diagrams.
(2)
${\mathcal{X}}\to {\mathcal{S}}ch/{\mathbb{S}}$ is a stack, namely:
(a)
the $\operatorname {Isom}$ functors are sheaves in the étale topology; and
(b)
any étale descent datum for objects of ${\mathcal{X}}$ is effective.
See ReferenceAr, 1.1, or ReferenceL-MB, Definition 3.1.
(3)
The stack ${\mathcal{X}}\to {\mathcal{S}}ch/{\mathbb{S}}$ is algebraic, namely:
(a)
the $\operatorname {Isom}$ functors are representable by separated algebraic spaces of finite type; and
(b)
there is a scheme $X$, locally of finite type, and a smooth and surjective morphism $X \to {\mathcal{X}}$.
See ReferenceL-MB, Definition 4.1. Notice that (3a) implies (2a).
These last two conditions are often the most difficult to verify. For the last one, M. Artin has devised a set of criteria for constructing $X\to {\mathcal{X}}$ by algebraization of formal deformation spaces (see ReferenceAr, Corollary 5.2). Thus, in case ${\mathbb{S}}$ is of finite type over a field or an excellent Dedekind domain, condition (3b) holds if
(A)
${\mathcal{X}}$ is limit preserving (see ReferenceAr, §1);
(B)
${\mathcal{X}}$ is compatible with formal completions (see ReferenceAr, 5.2 (3));
(C)
Schlessinger’s conditions for pro-representability of the deformation functors hold (see ReferenceAr, (2.2) and (2.5)); and
(D)
there exists an obstruction theory for ${\mathcal{X}}$ (see ReferenceAr, (2.6)) such that
(i)
the deformation and obstruction theory is compatible with étale localization (ReferenceAr, 4.1 (i));
(ii)
the deformation theory is compatible with formal completions (ReferenceAr, 4.1 (ii)); and
(iii)
the deformation and obstruction theory is constructible (ReferenceAr, 4.1 (iii)).
Furthermore, we say that ${\mathcal{X}}$ is a Deligne-Mumford stack if we can choose $X \to {\mathcal{X}}$ as in (3b) to be étale. This holds if and only if the diagonal ${\mathcal{X}}\to {\mathcal{X}}\times {\mathcal{X}}$ is unramified (ReferenceL-MB, Théorème 8.1). A morphism ${\mathcal{X}}\to {\mathcal{X}}_1$ is of Deligne-Mumford type if for any scheme $Y$ and morphism $Y \to {\mathcal{X}}_1$ the stack $Y \times _{{\mathcal{X}}_1} {\mathcal{X}}$ is a Deligne-Mumford stack.
For the notion of properness of an algebraic stack see ReferenceL-MB, Chapter 7. Thus a stack ${\mathcal{X}}\to {\mathbb{S}}$ is proper if it is separated, of finite type and universally closed. In ReferenceL-MB, Remarque 7.11.2, it is noted that the weak valuative criterion for properness using traits might be insufficient for properness. However, in case ${\mathcal{X}}$ has finite diagonal, it is shown in ReferenceE-H-K-V, Theorem 2.7, that there exists a finite surjective morphism from a scheme $Y \to {\mathcal{X}}$. In such a case the usual weak valuative criterion suffices (ReferenceL-MB, Proposition 7.12).
Let ${\mathcal{X}}$ be an algebraic stack with finite diagonal over a scheme $S$. There exists an algebraic space ${\mathbf{X}}$ and a morphism ${\mathcal{X}}\to {\mathbf{X}}$ such that
(1)
${\mathcal{X}}\to {\mathbf{X}}$ is proper and quasifinite;
(2)
if $k$ is an algebraically closed field, then ${\mathcal{X}}(k)/\operatorname {Isom}\to {\mathbf{X}}(k)$ is a bijection;
(3)
whenever $Y\to S$ is an algebraic space and ${\mathcal{X}}\to Y$ is a morphism, then the morphism factors uniquely as ${\mathcal{X}}\to {\mathbf{X}}\to Y$; more generally
(4)
whenever $S' \to S$ is a flat morphism of schemes, and whenever $Y\to S'$ is an algebraic space and ${\mathcal{X}}\times _S S' \to Y$ is a morphism, then the morphism factors uniquely as ${\mathcal{X}}\times _S S' \to {\mathbf{X}}\times _S S' \to Y$; in particular
Recall that an algebraic space ${\mathbf{X}}$ along with a morphism $\pi \colon {\mathcal{X}}\to {\mathbf{X}}$ satisfying properties (2) and (3) is called a coarse moduli space (or just moduli space). In particular, the theorem of Keel and Mori shows that coarse moduli spaces of algebraic stacks with finite diagonal exist. Moreover, from (4) and (5) above we have that the formation of a coarse moduli space behaves well under flat base change:
Lemma 2.2.2
Let ${{\mathcal{X}}} \to X$ be a proper quasifinite morphism, where ${{\mathcal{X}}}$ is a Deligne-Mumford stack and $X$ is a noetherian scheme. Let $X' \to X$ be a flat morphism of schemes, and denote ${{\mathcal{X}}}' = X' \times _X {{\mathcal{X}}}$.
(1)
If $X$ is the moduli space of ${{\mathcal{X}}}$, then $X'$ is the moduli space of ${{\mathcal{X}}}'$.
(2)
If $X' \to X$ is also surjective and $X'$ is the moduli space of ${{\mathcal{X}}}'$, then $X$ is the moduli space of ${{\mathcal{X}}}$.
Proof.
Given a proper quasifinite morphism $\pi \colon {{\mathcal{X}}} \to X$, it then exhibits $X$ as a moduli space if and only if $\pi _*{{\mathcal{O}}}_{{\mathcal{X}}} = {{\mathcal{O}}}_X$. If $R \rightrightarrows U$ is an étale presentation of ${{\mathcal{X}}}$, and $f \colon U \to X$ and $g \colon R \to X$ are the induced morphisms, then this condition is equivalent to the exactness of the sequence
The prototypical example of a moduli space is given by a group quotient: Let $V$ be a scheme and $\Gamma$ a finite group acting on $V$. Consider the stack $[V/\Gamma ]$; see ReferenceL-MB, 2.4.2. The morphism $[V/\Gamma ] \to V/\Gamma$ exhibits the quotient space $V/\Gamma$ as the moduli space of the stack $[V/\Gamma ]$. The following well-known lemma shows that étale-locally, the moduli space of any Deligne-Mumford stack is of this form.
Lemma 2.2.3
Let ${\mathcal{X}}$ be a separated Deligne-Mumford stack, and ${\mathbf{X}}$ its coarse moduli space. There is an étale covering $\{{\mathbf{X}}_\alpha \to {\mathbf{X}}\}$, such that for each $\alpha$ there is a scheme $U_\alpha$ and a finite group $\Gamma _\alpha$ acting on $U_\alpha$, with the property that the pullback ${{\mathcal{X}}}\times _{\mathbf{X}}{\mathbf{X}}_\alpha$ is isomorphic to the stack-theoretic quotient $[U_\alpha /\Gamma _\alpha ]$.
Sketch of proof.
Let $x_0$ be a geometric point of ${\mathbf{X}}$. Denote by ${\mathbf{X}}^{\operatorname {sh}}$ the spectrum of the strict henselization of ${\mathbf{X}}$ at the point $x_0$, and let ${{\mathcal{X}}}^{\operatorname {sh}}= {{\mathcal{X}}}\times _{\mathbf{X}}{\mathbf{X}}^{\operatorname {sh}}$. If $V \to {{\mathcal{X}}}$ is an étale morphism, with $V$ a scheme, having $x_0$ in its image, there is a component $U$ of the pullback $V \times _{\mathbf{X}}{\mathbf{X}}^{\operatorname {sh}}$ which is finite over ${\mathbf{X}}^{\operatorname {sh}}$. Denote $R = U \times _{{{\mathcal{X}}}^{\operatorname {sh}}}U$. We have that under the first projection $R \to U$, the scheme $R$ splits as a disjoint union of copies of $U$. Let $\Gamma$ be the set of connected components of $R$, so that $R$ is isomorphic to $U \times \Gamma$. Then the product $R \times _U R \to R$ induces a group structure on $\Gamma$, and the second projection $R \simeq U \times \Gamma \to U$ defines a group action of $\Gamma$ on $U$, such that ${{\mathcal{X}}}^{\operatorname {sh}}$ is the quotient $U/\Gamma$.
We need to descend from ${\mathcal{X}}^{\operatorname {sh}}$ to get the statement on ${\mathcal{X}}$. This follows from standard limit arguments.
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2.3. Tame stacks and their coarse moduli spaces
Definition 2.3.1
(1)
A Deligne-Mumford stack ${\mathcal{X}}$ is said to be tame if for any geometric point $s\colon \operatorname {Spec}\Omega \to {\mathcal{X}}$, the group ${\operatorname {Aut}}_{\operatorname {Spec}\Omega }(s)$ has order prime to the characteristic of the algebraically closed field $\Omega$.
(2)
A morphism ${\mathcal{X}}\to {\mathcal{X}}_1$ of algebraic stacks is said to be tame if for any scheme $Y$ and morphism $Y \to {\mathcal{X}}_1$ the stack $Y \times _{{\mathcal{X}}_1} {\mathcal{X}}$ is a tame Deligne-Mumford stack.
A closely related notion is the following:
Definition 2.3.2
An action of a finite group $\Gamma$ on a scheme $V$ is said to be tame if for any geometric point $s\colon \operatorname {Spec}\Omega \to V$, the group $\operatorname {Stab}(s)$ has order prime to the characteristic of $\Omega$.
The reader can verify that a separated Deligne-Mumford stack is tame if and only if the actions of the groups $\Gamma _\alpha$ on $V_\alpha$ in Lemma 2.2.3 are tame.
In case ${\mathcal{X}}$ is tame, the formation of coarse moduli spaces commutes with arbitrary morphisms:
Lemma 2.3.3
Let ${{\mathcal{X}}}$ be a tame Deligne-Mumford stack, ${{\mathcal{X}}} \to {\mathbf{X}}$ its moduli space. If ${\mathbf{X}}' \to {\mathbf{X}}$ is any morphism of schemes, then ${\mathbf{X}}'$ is the moduli space of the fiber product ${\mathbf{X}}' \times _{\mathbf{X}}{{\mathcal{X}}}$. Moreover, if ${\mathbf{X}}'$ is reduced, then it is also the moduli space of $({\mathbf{X}}' \times _{\mathbf{X}}{{\mathcal{X}}})_{\operatorname {red}}$.
Proof.
By Lemma 2.2.2, this is a local condition in the étale topology of ${\mathbf{X}}$, so we may assume that ${{\mathcal{X}}}$ is a quotient stack of type $[V / \Gamma ]$, where $\Gamma$ is a finite group acting on an affine scheme $V = \operatorname {Spec}R$. Moreover, since ${\mathcal{X}}$ is tame, we may assume that the order of $\Gamma$ is prime to all residue characteristics. Then ${\mathbf{X}}= \operatorname {Spec}R^ \Gamma$; if ${\mathbf{X}}' = \operatorname {Spec}S$, then the statement is equivalent to the map $S \to \left(R \otimes _{R^ \Gamma } S\right)^ \Gamma$ being an isomorphism. This (well-known) fact can be shown as follows: Recall that for any $R^\Gamma [\Gamma ]$-module$M$ the homomorphism
$$\begin{eqnarray*} M &\stackrel {\operatorname {Av}_M}{\longrightarrow }& M^ \Gamma \\ y & \mapsto & \frac {1}{|\Gamma |} \sum _{\gamma \in \Gamma } \gamma \cdot y \end{eqnarray*}$$
is a projector exhibiting $M^\Gamma$ as a direct summand in $M$. Thus the induced morphism
$$\begin{equation*} \operatorname {Av}_R\otimes \operatorname {Id}_S\colon R\otimes _{R^ \Gamma } S \to R^\Gamma \otimes _{R^\Gamma }S = S \end{equation*}$$
shows that $S \to \left(R \otimes _{R^ \Gamma } S\right)^ \Gamma$ is injective. The morphism $\operatorname {Av}_R\otimes \operatorname {Id}_S$ is a lifting of
This shows that ${\mathbf{X}}'$ is the moduli space of the fiber product ${\mathbf{X}}' \times _{\mathbf{X}}{{\mathcal{X}}}$. The statement about $({\mathbf{X}}' \times _{\mathbf{X}}{{\mathcal{X}}})_{\operatorname {red}}$ is immediate. This proves the result.
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Let ${\mathcal{X}}$ be a separated tame stack with coarse moduli scheme ${\mathbf{X}}$. Consider the projection $\pi \colon {\mathcal{X}}\to {\mathbf{X}}$. The functor $\pi _*$ carries sheaves of ${{\mathcal{O}}}_{{\mathcal{X}}}$-modules to sheaves of ${{\mathcal{O}}}_{\mathbf{X}}$-modules.
Lemma 2.3.4
The functor $\pi _*$ carries quasicoherent sheaves to quasicoherent sheaves, coherent sheaves to coherent sheaves, and is exact.
Proof.
The question is local in the étale topology on ${\mathbf{X}}$, so we may assume that ${{\mathcal{X}}}$ is of the form $[V/\Gamma ]$, where $V$ is a scheme and $\Gamma$ a finite group of order prime to all residue characteristics, in particular ${\mathbf{X}}= V/\Gamma$. Now sheaves on ${{\mathcal{X}}}$ correspond to equivariant sheaves on $V$. Denote by $q\colon V \to {\mathbf{X}}$ the projection. If ${\mathcal{E}}$ is a sheaf on ${\mathcal{X}}$ corresponding to a $\Gamma$-equivariant sheaf $\tilde{{\mathcal{E}}}$ on $V$, then $\pi _*{\mathcal{E}}= (q_*\tilde{{\mathcal{E}}})^\Gamma$, which, by the tameness assumption, is a direct summand in $q_*\tilde{{\mathcal{E}}}$. From this the statement follows.
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2.4. Purity lemma
We recall the following purity lemma from Reference$\aleph$-V2:
Lemma 2.4.1
Let ${\mathcal{M}}$ be a separated Deligne-Mumford stack, ${\mathcal{M}}\to {\mathbf{M}}$ the coarse moduli space. Let $X$ be a separated scheme of dimension $2$ satisfying Serre’s condition $S_2$. Let $P\subset X$ be a finite subset consisting of closed points, $U=X\smallsetminus P$. Assume that the local fundamental groups of $U$ around the points of $P$ are trivial.
Let $f\colon X \to {\mathbf{M}}$ be a morphism. Suppose there is a lifting $\tilde{f}_U\colon U \to {\mathcal{M}}$:
and $\tilde{f}$ is unique up to a unique isomorphism.
Proof.
By the descent axiom for ${\mathcal{M}}$ (see 2.1 (2)) the problem is local in the étale topology, so we may replace $X$ and ${\mathbf{M}}$ with the spectra of their strict henselizations at a geometric point; then we can also assume that we have a universal deformation space $V\to {\mathcal{M}}$ which is finite. Now $U$ is the complement of the closed point, $U$ maps to ${\mathcal{M}}$, and the pullback of $V$ to $U$ is finite and étale, so it has a section, because $U$ is simply connected; consider the corresponding map $U\to V$. Let $Y$ be the scheme-theoretic closure of the graph of this map in $X\times _{\mathbf{M}}V$. Then $Y\to X$ is finite and is an isomorphism on $U$. Since $X$ satisfies $S_2$, the morphism $Y\to X$ is an isomorphism.
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Remark 2.4.2
The reader can verify that the statement and proof work in higher dimension. See also related lemmas in ReferenceMo.
Corollary 2.4.3
Let $X$ be a smooth surface over a field, $p\in X$ a closed point with complement $U$. Let $X\to {\mathbf{M}}$ and $\, U\to {\mathcal{M}}$ be as in the purity lemma. Then there is a lifting $X \to {\mathcal{M}}$.
Corollary 2.4.4
Let $X$ be a normal crossings surface over a field $k$, namely a surface which is étale locally isomorphic to $\operatorname {Spec}k[u,v,t]/(uv)$. Let $p\in X$ be a closed point with complement $U$. Let $X\to {\mathbf{M}}$ and $\, U\to {\mathcal{M}}$ be as in the purity lemma. Then there is a lifting $X \to {\mathcal{M}}$.
Proof.
In both cases $X$ satisfies condition $S_2$ and the local fundamental group around $p$ is trivial, hence the purity lemma applies.
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2.5. Descent of equivariant objects
Lemma 2.5.1
Let $R$ be a local ring with residue field $k$, let $U = \operatorname {Spec}R$,$u_0 = \operatorname {Spec}k$, let ${\mathcal{M}}$ be a Deligne-Mumford stack, and let $\eta$ be an object of ${{\mathcal{M}}}(U)$. Assume we have a pair of compatible actions of a finite group $\Gamma$ on $R$ and on $\eta$, in such a way that the induced actions of $\Gamma$ on $k$ and on the pullback $\eta _0=\eta |_{u_0}$ are trivial. Then there exists an object $\eta '$ of ${\mathcal{M}}$ on the quotient $U/ \Gamma = \operatorname {Spec}(R^ \Gamma )$, and a $\Gamma$-invariant lifting $\eta \to \eta '$ of the projection $U \to U/ \Gamma$. Furthermore, if $\eta ''$ is another such object over $U/ \Gamma$, there is a unique isomorphism $\eta ' \simeq \eta ''$ over the identity of $U/ \Gamma$, which is compatible with the two arrows $\eta \to \eta '$ and $\eta \to \eta ''$.
As a consequence of the unicity statement, suppose that we have a triple $( \alpha , \beta , \gamma )$, where $\gamma \colon \Gamma \simeq \Gamma$ is a group isomorphism, and $\alpha \colon \eta \simeq \eta$ and $\beta \colon U \simeq U$ are compatible $\gamma$-equivariant isomorphisms. Then the given arrow $\eta \to \eta '$ and its composition with $\alpha$ both satisfy the conditions of the lemma, so there is an induced isomorphism $\overline \alpha \colon \eta ' \simeq \eta '$.
Corollary 2.5.2
Let $R,U,k,u_0,\eta ,\eta _0$ be as in the previous lemma. Let $G$ be a finite group acting compatibly on $R$ and on $\eta$. Let $\Gamma$ be the normal subgroup of $G$ consisting of elements acting on $k$ and $\eta _0$ as the identity. Then there exist a $G/ \Gamma$-equivariant object $\eta '$ on the quotient $U/ \Gamma$, and a $G$-equivariant arrow $\eta \to \eta '$ compatible with the projection $U \to U/ \Gamma$.
Proof of the corollary.
The action is defined as follows. If $g$ is an element of $G$, call $\alpha \colon \eta \simeq \eta$ and $\beta \colon U \simeq U$ the induced arrows, and $\gamma \colon \Gamma \simeq \Gamma$ the conjugation by $g$. Then the image of $g$ in $G/ \Gamma$ acts on $\eta '$ via the isomorphism $\overline \alpha \colon \eta ' \simeq \eta ''$ defined above. One checks easily that this defines an action with the required properties.
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Proof of the lemma.
First note that if $R^{\operatorname {sh}}$ is the strict henselization of $R$, the condition on the action of $\Gamma$ allows one to lift it to $R^{\operatorname {sh}}$. Also, the statement that we are trying to prove is local in the étale topology, so by standard limit arguments we can assume that $R$ is strictly henselian. Replacing ${\mathbf{M}}$ by the spectrum of the strict henselization of its local ring at the image of the closed point of $R$, we can assume that ${{\mathcal{M}}}$ is of the form $[V/H]$, where $V$ is a scheme and $H$ is a finite group. Then the object $\eta$ corresponds to a principal $H$-bundle$P \to U$, on which $\Gamma$ acts compatibly with the action of $\Gamma$ on $U$, and an $H$-equivariant and $\Gamma$-invariant morphism $P \to V$. Since $U$ is strictly henselian, the bundle $P \to U$ is trivial, so $P$ is a disjoint union of copies of $\operatorname {Spec}R$, and the group $\Gamma$ permutes these copies; furthermore the hypothesis on the action of $\Gamma$ on the closed fiber over the residue field insures that $\Gamma$ sends each component into itself. The thesis follows easily.
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We note that Lemma 2.5.1 can be proven for an arbitrary algebraic stack ${\mathcal{M}}$ as an application of the notion of relative moduli spaces, which we did not discuss here. Briefly, the assumptions of the lemma give a morphism $[U/\Gamma ] \to {\mathcal{M}}$. Using Lemma 2.2.2, it can be shown that there is a relative coarse moduli space, namely a representable morphism ${\mathcal{V}}\to {\mathcal{M}}$ such that ${\mathcal{V}}\times _{\mathcal{M}}Y$ gives the coarse moduli space of $[U/\Gamma ]\times _{\mathcal{M}}Y$ for any scheme $Y$ and flat morphism $Y \to {\mathcal{M}}$. The fact that $\Gamma$ acts trivially on $\eta _0$, together with Lemmas 2.2.2 and 4.4.3, imply that ${\mathcal{V}}= U/\Gamma$.
3. Twisted objects
Our goal in this section is to introduce the notion of a stable twisted object. This is a representable ${\mathcal{M}}$-valued object on a suitable atlas of orbispace charts on a nodal curve. Basic charts have the form $[\operatorname {Spec}k[\xi ,\eta ]/{\boldsymbol {\mu }}_r]$ at a node, or $[\operatorname {Spec}k[\xi ]/{\boldsymbol {\mu }}_r]$ along a marking, where ${\boldsymbol {\mu }}_r$ acts freely away from the origin; for completeness of the picture we allow more general charts. The category of stable twisted objects is a concrete, yet somewhat technical, incarnation of our stack ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$, which is convenient in many steps of our proof of Theorem 1.4.1.
3.1. Divisorially marked curves
The following definition is a local version of the standard definition of pointed curve; its advantage is that it is stable under localization in the étale topology.
Definition 3.1.1
A divisorially $n$-marked nodal curve, or simply $n$-marked curve$(U \to S, \Sigma _i)$, consists of a nodal curve $\pi \colon U \to S$, together with a sequence of $n$ pairwise disjoint closed subschemes $\Sigma _1, \mathinner {\ldotp \ldotp \ldotp }, \Sigma _n\subset U$ whose supports do not contain any of the singular points of the fibers of $\pi$, and such that the projections $\Sigma _i \to S$ are étale. (Any of the subschemes $\Sigma _i$ may be empty.)
If more than one curve is considered, we will often use the notation $\Sigma _i^U$ to specify the curve $U$. On the other hand, we will often omit the subschemes $\Sigma _i^U$ from the notation $(U \to S, \Sigma _i^U)$ if there is no risk of confusion.
A nodal $n$-pointed curve $C \to S$ is considered an $n$-marked curve by taking as the $\Sigma ^C_i$ the images of the sections $S \to C$.
Definition 3.1.2
If $(U \to S,\Sigma _i)$ is an $n$-marked nodal curve, we define the special locus of $U$, denoted by $U_{\text{sp}}$, to be the union of the $\Sigma _i$ with the singular locus of the projection $U \to S$, with its natural scheme structure (this makes the projection $U_{\text{sp}}\to S$ unramified). The complement of $U_{\text{sp}}$ will be called the general locus of $U$, and denoted by $U_{\text{gen}}$.
Definition 3.1.3
If $(U \to S,\Sigma _i^U)$ is a marked curve, and $S' \to S$ is an arbitrary morphism, we define the pullback to be $(U'\to S',\Sigma ^{U'}_i)$, where $U' = S' \times _S U$ and $\Sigma ^{U'}_i = S' \times _S \Sigma ^U_i$.
Definition 3.1.4
If $U \to S$ and $V \to S$ are $n$-marked curves, a morphism of $n$-marked curves$f \colon U \to V$ is a morphism of $S$-schemes which sends each $\Sigma ^U_i$ into $\Sigma ^V_i$.
A morphism of $n$-marked curves $f \colon U \to V$ is called strict if the support of $f^{-1}(\Sigma ^V_i)$ coincides with the support of $\Sigma ^U_i$ for all $i = 1, \mathinner {\ldotp \ldotp \ldotp }, n$, and similarly for the singular locus.
We notice that if a morphism of marked curves $U \to V$ is strict, then there is an induced morphism of curves $U_{\text{gen}}\to V_{\text{gen}}$. Furthermore, if $f \colon U \to V$ is strict and étale, then $f^{-1}(\Sigma ^V_i) = \Sigma ^U_i$ scheme-theoretically.
Definition 3.1.5
Let $(U \to S, \Sigma _i)$ be an $n$-marked curve and $\Gamma$ a finite group. An action of $\Gamma$ on $(U,\Sigma _i)$ is an action of $\Gamma$ on $U$ as an $S$-scheme, such that each element on $\Gamma$ acts via an automorphism, in the sense of Definition 3.1.4, of $U$ as a marked curve on $S$.
If $\Gamma$ is a finite group along with a tame action on a marked curve $U \to S$, then the quotient $U / \Gamma \to S$ can be given a marked curve structure by defining $\Sigma ^{U/ \Gamma }_i := \Sigma ^U_i/ \Gamma \subseteq U/ \Gamma$. The latter inclusion holds because the orders of stabilizers in $\Gamma$ are assumed to be prime to the residue characteristics, so $\Sigma ^U_i/ \Gamma$ is indeed a subscheme of $U/\Gamma$.
Given a morphism $f \colon U \to V$ of marked curves, and a tame action of a finite group $\Gamma$ on $U$, leaving $f$ invariant, then there is an induced morphism $U/ \Gamma \to V$ of marked curves.
Definition 3.1.6
Let $(U \to S, \Sigma _i)$ be an $n$-marked curve, with an action of a finite group $\Gamma$, and let ${\mathcal{M}}$ be a Deligne-Mumford stack. Given $\eta \in {\mathcal{M}}(U)$, an essential action of $\Gamma$ on $(\eta , U)$ is a pair of compatible actions of $\Gamma$ on $\eta$ and on $(U\to S, \Sigma _i)$, with the property that if $g$ is an element of $\Gamma$ different from the identity and $u_0$ is a geometric point of $U$ fixed by $g$, then the automorphism of the pullback of $\eta$ to $u_0$ induced by $g$ is not trivial.
3.2. Generic objects and charts
Definition 3.2.1
Let $C \to S$ be an $n$-pointed nodal curve. A generic object on $C$ is an object of ${\mathcal{M}}(C_{\text{gen}})$.
We will often write $(\xi , C)$ for a generic object $\xi$ on a curve $C$.
Definition 3.2.2
Let $C \to S$ be an $n$-pointed nodal curve and $\xi$ a generic object on $C$. A chart$(U, \eta , \Gamma )$ for $\xi$ consists of the following collection of data.
(1)
An $n$-marked curve $U \to S$ and a strict morphism $\phi \colon U \to C$.
(2)
An object $\eta$ of ${\mathcal{M}}(U)$.
(3)
An arrow $\eta |_{U_{\text{gen}}} \to \xi$ in ${\mathcal{M}}$ compatible with the restriction $\phi |_{U_{\text{gen}}} \colon U_{\text{gen}}\to C_{\text{gen}}$.
(4)
A finite group $\Gamma$.
(5)
A tame, essential action of $\Gamma$ on $(\eta ,U)$.
Furthermore, we require that the following conditions be satisfied.
a
The actions of $\Gamma$ leave the morphism $U \to C$ and the arrow $\eta |_{U_{\text{gen}}} \to \xi$ invariant.
b
The induced morphism $U/\Gamma \to C$ is étale.
The following gives a local description of the action of $\Gamma$.
Proposition 3.2.3
Let $(U, \eta , \Gamma )$ be a chart for a generic object $\xi$ on a pointed nodal curve $C \to S$. Then the action of $\Gamma$ on $\phi ^{-1} C_{\text{gen}}$ is free.
Furthermore, if $s_0$ is a geometric point of $S$ and $u_0$ a nodal point of the fiber $U_{s_0}$ of $U$ over $s_0$, then
(1)
the stabilizer $\Gamma '$ of $u_0$ is a cyclic group which sends each of the branches of $U_{s_0}$ to itself;
(2)
if $k$ is the order of $\Gamma '$, then a generator of $\Gamma '$ acts on the tangent space of each branch by multiplication with a primitive $k$-th root of $1$.
In particular, each nodal point of $U_{s_0}$ is sent to a nodal point of $C_{s_0}$.
Proof.
The first statement follows from the definition of an essential action and the invariance of the arrow $\eta |_{U_{\text{gen}}} \to E$.
As for (1), observe that if the stabilizer $\Gamma '$ of $u_0$ did not preserve the branches of $U_{s_0}$, then the quotient $U_{s_0}/\Gamma '$, which is étale at the point $u_0$ over the fiber $C_{s_0}$, would be smooth over $S$ at $u_0$, so $u_0$ would be in the inverse image of $C_{\text{gen}}$. From the first part of the proposition it would follow that $\Gamma '$ is trivial, a contradiction.
So $\Gamma '$ acts on each of the two branches individually. The action on each branch must be faithful because it is free on the complement of the set of nodes; this means that the representation of $\Gamma '$ in each of the tangent spaces to the branches is faithful, and this implies the final statement.
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Definition 3.2.4
A chart is called balanced if for any nodal point of any geometric fiber of $U$, the two roots of 1 describing the action of a generator of the stabilizer on the tangent spaces to each branch of $U$ are inverse to each other.
3.3. The transition scheme
Let $\xi$ be a generic object over a nodal curve $C\to S$, and $(U_1,\eta _1,\Gamma _1)$,$(U_2,\eta _2,\Gamma _2)$ two charts; call ${\text{pr}}_i\colon U_1\times _C U_2\to U_i$ the $i$-th projection. Consider the scheme
over $U_1\times _C U_2$ representing the functor of isomorphisms of the two objects ${\text{pr}}_1^*\eta _1$ and ${\text{pr}}_2^*\eta _2$.
There is a section of $I$ over the inverse image $\widetilde{U}$ of $C_{\text{gen}}$ in $U_1\times _C U_2$ which corresponds to the isomorphism ${\text{pr}}_1^*\eta _1|_{\widetilde{U}}\simeq {\text{pr}}_2^*\eta _2|_{\widetilde{U}}$ coming from the fact that both ${\text{pr}}_1^*\eta _1$ and ${\text{pr}}_2^*\eta _2$ are pullbacks to $\widetilde{U}$ of $\xi$. We will call the scheme-theoretic closure $R$ of this section in $I$ the transition scheme from $(U_1,\eta _1,\Gamma _1)$ to $(U_2,\eta _2,\Gamma _2)$; it comes equipped with two projections $R\to U_1$ and $R\to U_2$.
There is also an action of $\Gamma _1\times \Gamma _2$ on $I$, defined as follows. Let $(\gamma _1,\gamma _2)\in \Gamma _1\times \Gamma _2$, and let $\phi \colon {\text{pr}}_1^*\eta _1\simeq {\text{pr}}_2^*\eta _2$ be an isomorphism over $U_1\times _C U_2$; then define $(\gamma _1,\gamma _2)\cdot \phi = \gamma _2\circ \phi \circ \gamma _1^{-1}$. This action of $\Gamma _1\times \Gamma _2$ on $I$ is compatible with the action of $\Gamma _1\times \Gamma _2$ on $U_1\times _C U_2$, and leaves $R$ invariant. It follows from the definition of an essential action that the action of $\Gamma _1 = \Gamma _1\times \{1\}$ and $\Gamma _2 = \{1\}\times \Gamma _2$ on $I$ is free.
3.4. Compatibility of charts
Definition 3.4.1
Two charts $(U_1,\eta _1,\Gamma _1)$ and $(U_2,\eta _2,\Gamma _2)$ are compatible if their transition scheme $R$ is étale over $U_1$ and $U_2$.
Let us analyze this definition. First of all, $R$ is obviously étale over $(U_1)_{\text{gen}}$ and $(U_2)_{\text{gen}}$. Also, since the maps $U_j \to C$ are strict, it is clear that the inverse image of $\Sigma ^{U_1}_i$ in $R$ is set-theoretically equal to the inverse image of $\Sigma ^{U_2}_i$. If the two charts are compatible, this also holds scheme-theoretically.
Now, start from two charts $(U_1,\eta _1,\Gamma _1)$ and $(U_2,\eta _2,\Gamma _2)$. Fix two geometric points
mapping to the same geometric point $u_0\colon \operatorname {Spec}\Omega \to C$, and call $\Gamma '_j\subset \Gamma _j$ the stabilizer of $u_j$. Also call $U_1^{\operatorname {sh}}$,$U_2^{\operatorname {sh}}$ and $C^{\operatorname {sh}}$ the spectra of the strict henselizations of $U_1$,$U_2$ and $C$ at the points $u_1,u_2$ and $u_0$ respectively. The action of $\Gamma _j$ on $U_j$ induces an action of $\Gamma '_j$ on $U_j^{\operatorname {sh}}$. Also call $\eta _j^{\operatorname {sh}}$ the pullback of $\eta _j$ to $U_j^{\operatorname {sh}}$; there is an action of $\Gamma '_j$ on $\eta _j^{\operatorname {sh}}$ compatible with the action of $\Gamma '_j$ on $U_j$. The following essentially says that two charts are compatible if and only if for any choice of $u_1$ and $u_2$ the two charts are locally isomorphic in the étale topology.
Proposition 3.4.2
The two charts are compatible if and only if for any pair of geometric points $u_1$ and $u_2$ as above there exist an isomorphism of groups $\theta \colon \Gamma '_1\simeq \Gamma '_2$, a $\theta$-equivariant isomorphism $\phi \colon U_1^{\operatorname {sh}}\simeq U_2^{\operatorname {sh}}$ of schemes over $C^{\operatorname {sh}}$, and a compatible $\theta$-equivariant isomorphism $\psi \colon \eta _1^{\operatorname {sh}}\to \eta _2^{\operatorname {sh}}$.
Proof.
Consider the spectrum $(U_1\times _C U_2)^{\operatorname {sh}}$ of the strict henselization of $U_1\times _C U_2$ at the point $(u_1,u_2)\colon \operatorname {Spec}\Omega \to U_1\times _C U_2$, and call $R^{\operatorname {sh}}$ the pullback of $R$ to $(U_1\times _C U_2)^{\operatorname {sh}}$. Assume that the two charts are compatible. The action of $\Gamma _1\times \Gamma _2$ on $I$ described above induces an action of $\Gamma '_1\times \Gamma '_2$ on $R^{\operatorname {sh}}$, compatible with the action of $\Gamma '_1\times \Gamma '_2$ on $(U_1\times _C U_2)^{\operatorname {sh}}$. The action of $\Gamma '_1 = \Gamma '_1\times \{1\}$ on the inverse image of $C_{\text{gen}}$ in $R^{\operatorname {sh}}$ is free, and its quotient is the inverse image of $C_{\text{gen}}$ in $U_2^{\operatorname {sh}}$; but $R^{\operatorname {sh}}$ is finite and étale over $U_2^{\operatorname {sh}}$, so the action of $\Gamma '_1$ on all of $R^{\operatorname {sh}}$ is free, and $R^{\operatorname {sh}}/\Gamma '_1 = U_2$. Analogously the action of $\Gamma '_2$ on $R^{\operatorname {sh}}$ is free, and $R^{\operatorname {sh}}/\Gamma '_2 = U_1$.
Now, each of the connected components of $R^{\operatorname {sh}}$ maps isomorphically onto both $U_1^{\operatorname {sh}}$ and $U_2^{\operatorname {sh}}$, because $U_j^{\operatorname {sh}}$ is the spectrum of a strictly henselian ring and the projection $R^{\operatorname {sh}}\to U_j^{\operatorname {sh}}$ is étale; this implies in particular that the order of $\Gamma _1$ is the same as the number $k$ of connected components, and likewise for $\Gamma _2$. Fix one of these components, call it $R_0^{\operatorname {sh}}$; then we get isomorphisms $R_0^{\operatorname {sh}}\simeq U_j^{\operatorname {sh}}$, which yield an isomorphism $\phi \colon U_1^{\operatorname {sh}}\simeq U_2^{\operatorname {sh}}$.
Call $\Gamma '$ the stabilizer of the component $R_0^{\operatorname {sh}}$ inside $\Gamma '_1\times \Gamma '_2$; the order of $\Gamma '$ is at least $|\Gamma '_1\times \Gamma '_2|/k = k^2/k = k$. But the action of $\Gamma '_2$ on $R^{\operatorname {sh}}$ is free, and so $\Gamma '\cap \Gamma _2 = \{1\}$; this implies that the order of $\Gamma '$ is $k$, and the projection $\Gamma '\to \Gamma '_1$ is an isomorphism. Likewise the projection $\Gamma '\to \Gamma '_2$ is an isomorphism, so from these we get an isomorphism $\theta \colon \Gamma '_1\to \Gamma '_2$, and it is easy to check that the isomorphism of schemes $\phi \colon U_1^{\operatorname {sh}}\simeq U_2^{\operatorname {sh}}$ is $\theta$-equivariant.
There is also an isomorphism of the pullbacks of $\eta _1^{\operatorname {sh}}$ and $\eta _2^{\operatorname {sh}}$ to $R_0^{\operatorname {sh}}$, coming from the natural morphism $R_0^{\operatorname {sh}}\to I$, which induces an isomorphism $\psi \colon \eta _1^{\operatorname {sh}}\to \eta _2^{\operatorname {sh}}$. This isomorphism is compatible with $\phi$, and it is also $\theta$-equivariant.
Let us prove the converse. Suppose that there exist $\theta$,$\phi$ and $\psi$ as above. Then there is a morphism $\sigma \colon U_1^{\operatorname {sh}}\times \Gamma '_1\to I$ which sends a point $(u_1,\gamma _1)$ of $U_1^{\operatorname {sh}}\times \Gamma '_1$ into the point of $I$ lying over the point $(u_1, \phi \gamma _1u_1) = (u_1,\theta (\gamma _1)\phi u_1)$ corresponding to the isomorphism $\gamma _1\psi$ of the pullback of $\eta _1$ to $u_1$ with the pullback of $\eta _2$ to $\phi \gamma _1u_1$. The morphism $\sigma$ is an isomorphism of $U_1^{\operatorname {sh}}\times \Gamma '_1$ with $R^{\operatorname {sh}}$ in the inverse image of $C_{\text{gen}}$; also, from the fact that the action of $\Gamma '$ on $(\eta _1,U_1)$ is essential, it follows that $\sigma$ is injective. Since the inverse image of $C_{\text{gen}}$ is scheme-theoretically dense in $R^{\operatorname {sh}}$ and $U_1^{\operatorname {sh}}\times \Gamma _1$ is unramified over $U_1$ we see that $\sigma$ is an isomorphism of $U_1^{\operatorname {sh}}\times \Gamma '_1$ with $R^{\operatorname {sh}}$. It follows that $R^{\operatorname {sh}}$ is étale over $U_1^{\operatorname {sh}}$; analogously it is étale over $U_2^{\operatorname {sh}}$. So $R$ is étale over $U_1$ and $U_2$ at the points $u_1$ and $u_2$; since this holds for all $u_1$ and $u_2$ mapping to the same point of $C$ the conclusion follows.
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Compatibility of charts is stable under base change:
Proposition 3.4.3
(1)
Let $(U_1,\eta _1,\Gamma _1)$,$(U_2,\eta _2,\Gamma _2)$ be two compatible charts for a generic object $\xi$ on $C\to S$. If $S'\to S$ is an arbitrary morphism, then$$\begin{equation*} (S'\times _S U_1,\eta _1',\Gamma _1) \end{equation*}$$
where $\eta '_1$ and $\eta '_2$ are the pullbacks of $\eta _1$ and $\eta _2$ to $S'\times _S U_1$ and $S'\times _S U_2$, are compatible charts for the pullback of $\xi$ to $(S'\times _S C\to S')_{\text{gen}}$.
(2)
If $S' \to S$ is étale and surjective, then the converse holds.
Given two compatible charts $(U_1,\eta _1, \Gamma _1)$,$(U_2,\eta _2, \Gamma _2)$, set $\eta = {\text{pr}}_1^* \eta _1$ in ${\mathcal{M}}(R)$. There is an action of $\Gamma$, coming from pulling back the action of $\Gamma _1$ on $\eta _1$; also the tautological isomorphism $\alpha \colon {\text{pr}}_1^* \eta _1 \simeq \eta _2$ induces an action of $\Gamma _2$ on $\eta$. These two actions commute, and therefore define an action of $\Gamma _1\times \Gamma _2$ on $\eta$, compatible with the action of $\Gamma _1\times \Gamma _2$ on $\rho$. Also, $R$ has a structure of an $n$-marked curve, by defining $\Sigma ^R_i$ to be the inverse image of $\Sigma ^{U_1}_i$, and the map $R \to C$ is strict. Then
is a chart, called the product chart. It is compatible with both of the original charts.
3.6. Atlases and twisted objects
Definition 3.6.1
Fix two nonnegative integers $g$ and $n$. An $n$-pointed twisted object $(\xi , C \to S,{\mathcal{A}})$ of genus $g$ consists of
(1)
a proper, $n$-pointed curve $C\to S$ of finite presentation, with geometrically connected fibers of genus $g$;
(2)
a generic object $\xi$ over $C\to S$; and
(3)
a collection ${\mathcal{A}}= \{(U_\alpha , \eta _\alpha ,\Gamma _\alpha )\}$ of mutually compatible charts, such that the images of the $U_\alpha$ cover $C$.
A collection of charts ${\mathcal{A}}$ as in (3) is called an atlas.
A twisted object is called balanced if each chart in its atlas is balanced (Definition 3.2.4).
Lemma 3.6.2
If two charts for a twisted object are compatible with all the charts in an atlas, they are mutually compatible.
Furthermore, if the twisted object is balanced, then any chart which is compatible with every chart of the atlas is balanced.
Proof.
Both statements are immediate from the local characterization of compatibility in Proposition 3.4.2.
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Remark 3.6.3
The lemma above allows one to define a twisted object using a maximal atlas, if one prefers.
Definition 3.6.4
A morphism of twisted objects$(\xi , C \to S,{\mathcal{A}})$ to $(\xi ', C' \to S',{\mathcal{A}}')$ consists of a cartesian diagram
and an arrow $\xi \to \xi '$ lying over the restriction $f |_{C_{\text{gen}}} \colon C_{\text{gen}}\to C'_{\text{gen}}$, with the property that the pullback of the charts in ${\mathcal{A}}'$ are all compatible with the charts in ${\mathcal{A}}$.
The composition of morphisms of twisted objects is defined to be the one induced by composition of morphisms of curves.
Let $(\xi , C \to S,{\mathcal{A}})$ be a twisted object, and $T \to S$ a morphism. Then, using Proposition 3.4.3 one can define the pullback of $(\xi , C \to S,{\mathcal{A}})$ to $T$ in the obvious way.
3.7. Stability
Lemma 3.7.1
Let $(\xi , C \to S,{\mathcal{A}})$ be a twisted object. Then the morphism $C_{\text{gen}}\to {\mathbf{M}}$ induced by $\xi$ extends uniquely to a morphism $C\to {\mathbf{M}}$.
Proof.
The unicity is clear from the fact that ${\mathbf{M}}$ is separated and $C_{\text{gen}}$ is scheme-theoretically dense in $C$. To prove the existence of an extension is a local question in the étale topology; but if ${\mathcal{A}}= \{( U_\alpha ,\eta _\alpha ,\Gamma _\alpha )\}$, then the objects $\eta _\alpha$ induce morphisms $U_\alpha \to {\mathbf{M}}$, which are $\Gamma _\alpha$-equivariant, yielding morphisms $U_\alpha /\Gamma _\alpha \to {\mathbf{M}}$. These morphisms are extensions of the pullback to $(U_\alpha )_{\text{gen}}/\Gamma _\alpha$ of the morphism $C_{\text{gen}}\to {\mathbf{M}}$. Therefore they descend to $C$.
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We can now define the main object of this section:
Definition 3.7.2
A twisted object is stable if the associated map $C \to {\mathbf{M}}$ is Kontsevich stable.
3.8. The stack of stable twisted objects
Fix an ample line bundle ${{\mathcal{O}}_{{\mathbf{M}}}(1)}$ over ${\mathbf{M}}$. We define a category ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}= {{\mathcal{K}}_{g,n}({\mathcal{M}}/{\mathbb{S}},d)}$ as follows. The objects are stable twisted objects $(\xi , C \to S,{\mathcal{A}})$, where $C \to S$ is a nodal $n$-pointed curve of genus $g$, such that for the associated morphism $f \colon C \to {\mathbf{M}}$ the degree of the line bundle $f^* {{\mathcal{O}}_{{\mathbf{M}}}}(1)$ is $d$. The arrows are morphisms of twisted objects.
As stated in Theorem 1.4.1, this category is a proper algebraic stack which is relatively of Deligne-Mumford type over ${{\mathcal{K}}_{g,n}({\mathbf{M}},d)}$, admitting a projective coarse moduli space ${{\mathbf{K}}_{g,n}({\mathcal{M}},d)}$. The proof of the theorem will begin in Section 5.
We shall also consider the full subcategory ${{\mathcal{K}}^{{\operatorname {bal}}}_{g,n}({\mathcal{M}},d)}$ of balanced twisted objects. It will be shown in Proposition 8.1.1 that this is an open and closed substack in ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$, whose moduli space is open and closed in ${{\mathbf{K}}_{g,n}({\mathcal{M}},d)}$.
4. Twisted curves and twisted stable maps
In this section we give a stack-theoretic description of the category ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$ in terms of twisted stable maps. The language of stacks allows one to circumvent many of the technical details involved in twisted objects, and gives a convenient way of thinking about the category ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$. It is also convenient for studying deformation and obstruction theory for ${{\mathcal{K}}_{g,n}({\mathcal{M}},d)}$.
4.1. Nodal stacks
Let $S$ be a scheme over ${\mathbb{S}}$. Consider a proper, flat, tame Deligne-Mumford stack ${{\mathcal{C}}}\to S$ of finite presentation, such that its fibers are purely one-dimensional and geometrically connected, with at most nodal singularities. Call $C$ the moduli space of ${{\mathcal{C}}}$; by ReferenceK-M this exists as an algebraic space.
Proposition 4.1.1
The morphism $C \to S$ is a proper flat nodal curve of finite presentation, with geometrically connected fibers.
Proof.
First of all let us show that $C$ is flat over $S$. We may assume that $S$ is affine; let $R$ be its coordinate ring. Fix a geometric point $c_0 \to C$, and call $C^{\operatorname {sh}}$ the strict henselization of $C$ at $c_0$. Let $U$ be an étale cover of ${{\mathcal{C}}}$, and $u_0$ a geometric point of $U$ lying over $c_0$; denote by $U^{\operatorname {sh}}$ the strict henselization of $U$ at $u_0$. If $\Gamma$ is the automorphism group of the object of ${{\mathcal{C}}}$ corresponding to $u_0$, then $\Gamma$ acts on $U^{\operatorname {sh}}$, and $C^{\operatorname {sh}}$ is the quotient $U^{\operatorname {sh}}/ \Gamma$. Since ${\mathcal{C}}$ is tame, the order of $\Gamma$ is prime to the residue characteristic of $u_0$, therefore the coordinate ring of $C^{\operatorname {sh}}$ is a direct summand, as an $R$-module, of the coordinate ring of $U^{\operatorname {sh}}$, so it is flat over $R$.
The fact that the fibers are nodal follows from the fact that, over an algebraically closed field, the quotient of a nodal curve by a group action is again a nodal curve. Properness is clear; the fact that the morphism ${{\mathcal{C}}} \to C$ is surjective implies that the fibers are geometrically connected. The fact that $C$ is of finite presentation is an easy consequence of the fact that ${\mathcal{C}}$ is of finite presentation.
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Following tradition, when we speak of a “family over $S$” or “curve over $S$”, it is always assumed to be of finite presentation.
Definition 4.1.2
A twisted nodal $n$-pointed curve over $S$ is a diagram
${\mathcal{C}}$ is a tame Deligne-Mumford stack, proper and of finite presentation over $S$, and étale locally is a nodal curve over $S$;
(2)
$\Sigma _i^{{\mathcal{C}}} \subset {\mathcal{C}}$ are disjoint closed substacks in the smooth locus of ${\mathcal{C}}\to S$;
(3)
$\Sigma _i^{{\mathcal{C}}} \to S$ are étale gerbes;
(4)
the morphism ${\mathcal{C}}\to C$ exhibits $C$ as the coarse moduli scheme of ${\mathcal{C}}$; and
(5)
${\mathcal{C}}\to C$ is an isomorphism over $C_{\text{gen}}$.
Note that if we let $\Sigma _i^C$ be the coarse moduli spaces of $\Sigma _i^{{\mathcal{C}}}$, then, since ${\mathcal{C}}$ is tame, the schemes $\Sigma _i^C$ embed in $C$ (they are the images of $\Sigma _i^{{\mathcal{C}}}$), and $C$ becomes a usual nodal pointed curve. We say that ${\mathcal{C}}\to S$ is a twisted pointed curve of genus $g$ if $C \to S$ is a pointed curve of genus $g$.
4.2. Morphisms of twisted $n$-pointed nodal curves
Definition 4.2.1
Let ${\mathcal{C}}\to S$ and ${\mathcal{C}}' \to S'$ be twisted $n$-pointed nodal curves. A $1$-morphism (or just a morphism) $F \colon {\mathcal{C}}\to {\mathcal{C}}'$ is a cartesian diagram