# Fields of moduli and the arithmetic
of tame quotient singularities

Giulio Bresciani  and  Angelo Vistoli  Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa
Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety
$X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of
$\overline{k}$ of elements fixed by field automorphisms
$\gamma\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_{\gamma}$
is isomorphic to $X$. The field of moduli is contained in all subextensions
$k\subset k^{\prime}\subset\overline{k}$ such that $X$ descends to
$k^{\prime}$. In this paper we extend the formalism, and define the field of
moduli when $k$ is not perfect.

Furthermore, Dèbes and Emsalem identified a condition that ensures that a
smooth curve is defined over its field of moduli, and prove that a smooth
curve with a marked point is always defined over its field of moduli. Our main
theorem is a generalization of these results that applies to higher
dimensional varieties, and to varieties with additional structures.

In order to apply this, we study the problem of when a rational point of a
variety with quotient singularities lifts to a resolution. As a consequence,
we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$
such that $\operatorname{Aut}(X,p)$ is finite, étale and of degree prime to
$d!$ is defined over its field of moduli.

The second author was partially supported by research funds from Scuola
Normale Superiore, project SNS19_B_VISTOLI, and by PRIN project “Derived and
underived algebraic stacks and applications”. The paper is based upon work
partially supported by the Swedish Research Council under grant no. 2016-06596
while the second author was in residence at Institut Mittag-Leffler in
Djursholm

##  1\. Introduction

The concept of _field of moduli_ was introduced by Matsusaka in [Mat58], and
considerably clarified by Shimura in [Shi59]. Suppose that $k$ is a field with
algebraic closure $\overline{k}$. Let us assume for simplicity that $k$ is
perfect. An algebraic variety $X$, perhaps with additional structure, like a
polarization, or a marked point, will be defined over some intermediate field
$k\longrightarrow\ell\longrightarrow\overline{k}$ that is finite over $k$. If
$\Gamma$ is the Galois group of $\overline{k}$ over $k$, call
$\Delta\subseteq\Gamma$ the subgroup formed by elements $\gamma\in\Gamma$ such
that the twist $X_{\gamma}$ is isomorphic to $X$ as a $k$-scheme, possibly
with its additional structure. Then $\Delta$ is an open subgroup of $\Gamma$;
the field of moduli of $X$ is the fixed subfield $\overline{k}^{\Delta}$. It
is contained in every field of definition of $X$.

If $X$ has a finite group of automorphisms, and is one of a class of varieties
with finite automorphism groups parametrized by a coarse moduli space
$M\longrightarrow\operatorname{Spec}k$ (for example, smooth curves of genus at
least $2$), then the field of moduli has a natural interpretation as the
residue field of the image of the morphism
$\operatorname{Spec}\overline{k}\longrightarrow M$ corresponding to $X$.

One basic question is: when is $X$ defined over its field of moduli? This
problem has been the subject of a considerable amount of literature over the
years.

An important early example is due to Goro Shimura [Shi72]. Let $A_{g}$ be the
moduli space of abelian varieties of genus $g$ over $\mathbb{C}$, and call $K$
its field of rational functions. Let $X$ be the corresponding generic abelian
variety defined over $\overline{K}$; its field of moduli is $K$. Then Shimura
proved that $X$ is defined over $K$ if and only $g$ is odd.

In the case that $X$ is a smooth curve, an important advance is due to Pierre
Dèbes and Michel Emsalem [DE99].

Let $X$ be as above; assume that the group $\operatorname{Aut}X$ of
automorphisms of $X$ over $\overline{k}$ is finite. Consider the group
$\Delta$ above; for each $\delta\in\Delta$ we have an isomorphism
$X_{\delta}\simeq X$, well defined up to an automorphism of $X$ over
$\overline{k}$. This descend to an canonical automorphism of
$X/\operatorname{Aut}X$, defining an action of $\Delta$ on $X$, compatible
with it action on $\overline{k}$; by Galois descent this defines a scheme
$X^{\mathrm{c}}$ over $k$, a form of $X/\operatorname{Aut}X$, which we call
the _compression_ of $X$. If $X$ is a smooth curve, so is $X^{\mathrm{c}}$.

######  Theorem (Dèbes–Emsalem).

Assume that $X$ is a smooth curve. If $X^{\mathrm{c}}(k)\neq\emptyset$, then
$X$ is defined over its field of moduli.

Let us describe the content of this paper.

#### The definition of field of moduli

The definition of field of moduli for an object defined over $\overline{k}$ is
present in the literature in two particular cases, as explained above: when
$k$ is perfect, and when $X$ is in a class of object having a coarse moduli
space. This last hypothesis is not a very natural one: for example, one could
be interested in the field of moduli of non-polarized abelian varieties, or K3
surfaces. In §3 we give a somewhat more general and flexible formalism for
defining fields of moduli and residual gerbes under very weak hypotheses. In
particular we adapt it to non-perfect fields and objects with non-reduced
automorphism group schemes, using the fppf topology instead of the Galois
group.

In §4 we draw some consequences using results in the literature; this part is
certainly well known to the experts, and essentially contained in [DE99]; but
it seems useful to have a detailed explanation in our generality.

In the rest of the introduction we will assume $\operatorname{char}k=0$; this
will simplify the statements considerably. We refer to the main body of the
paper for precise statements in arbitrary characteristic.

#### The Dèbes–Emsalem theorem in arbitrary dimension

If $X$ is as in the introduction, if $X$ is singular, or higher-dimensional,
it is not true that if $X^{\mathrm{c}}(k)\neq\emptyset$, then $X$ is defined
over its field of moduli. Our main result (Theorem 5.4) is the following: let
$\widetilde{X}$ be a resolution of singularities of $X^{\mathrm{c}}$. If
$\widetilde{X}(k)\neq\emptyset$, then $X$ is defined over its field of moduli.

When $X$ is a smooth curve, we have that the compression $X^{\mathrm{c}}$ is
smooth, so this recovers the result of Dèbes–Emsalem. Dèbes and Emsalem
mention the fact that their methods can be generalized to curves with a
structure such as pointed curves [DE99, Remark 3.2.(b)], but they do not give
details on how to do it. So in dimension $1$, we are essentially clarifying
what curves with a structure are, and checking that the theorem of
Dèbes–Emsalem holds for them. More importantly, we are able to generalize
their result to arbitrary dimension.

Dèbes and Emsalem also observed that a suitable form of their theorem implies
that every pointed curve of genus $\geq 2$ is defined over its field of moduli
[DE99, Corollary 5.4]. This fails, however, in higher dimension: smooth
pointed varieties are not necessarily defined over their fields of moduli. An
example is Shimura’s result on generic abelian varieties explained in the
introduction. In order to apply the result above we need to ensure that a
rational point of the compression $X^{\mathrm{c}}$ lifts to a resolution of
singularities.

More precisely, we need conditions that ensure that a resolution of
singularities $\widetilde{X}\longrightarrow X^{\mathrm{c}}$ has a rational
point. In the case of pointed varieties this translates into the following
question. Suppose that $X$ is a variety over $\overline{k}$ and $G$ is a
finite group acting on $X$ with a smooth fixed point $\overline{p}\in
X(\overline{k})$. Let $X^{\mathrm{c}}$ be a form of $X/G$ defined over $k$,
with a rational point $p\in X^{\mathrm{c}}(k)$ lifting to $\overline{p}$. If
$\widetilde{X}\longrightarrow X^{\mathrm{c}}$ is a resolution of
singularities, under what conditions does it follow that there is a rational
point of $\widetilde{X}$ lying over $p$?

#### The arithmetic of quotient singularities

In §6 we introduce two related concepts.

One is that of _$R$ -singularity_ (Definition 6.13). An $R$-singularity is a
pair $(S,s)$, where $S$ is a variety over a field $K$ with quotient
singularities and $s\in S(K)$, such that, in particular, if $k\subseteq K$ is
a subfield, $(S^{\prime},s^{\prime})$ is a form of $(S,s)$ defined over $k$,
and $\widetilde{S}^{\prime}\longrightarrow S$ is a resolution of
singularities, then $\widetilde{S}^{\prime}$ has a $k$-rational point over
$s^{\prime}$.

The other key definition is that of $R_{d}$-group (Definition 6.12): if $d$ is
a positive integer, a finite group $G$ is $R_{d}$ if whenever it acts
faithfully on a smooth variety $X$ with a fixed rational point $x\in X(k)$,
the pair $(X/G,[x])$, where $[x]$ is the image of $x$, is an $R$-singularity.
It follows from our main theorem that if $X$ is a smooth variety over
$\overline{k}$ and $\overline{p}\in X(\overline{k})$ is a
$\overline{k}$-point, and $\operatorname{Aut}(X,\overline{p})$ is an $R_{d}$
group, then $(X,\overline{p})$ is defined over its field of moduli.

Not all finite groups are $R_{d}$: for example, a cyclic group of order $2$ is
not $R_{d}$ for any $d\geq 2$. Here we present two results, showing that this
is not an empty definition.

The first (Theorem 6.18) shows that there are infinitely many groups that are
$R_{d}$ for all $d$. The second (Theorem 6.19) says that any finite group of
order prime to $d!$ is $R_{d}$. As a consequence (Theorem 6.21), a
$d$-dimensional variety $X$ with a smooth marked point $p\in X$ such that
$\operatorname{Aut}(X,p)$ is finite of degree prime to $d!$ is defined over
its field of moduli.

There is much more that one could say about $R_{d}$-groups. The forthcoming
paper [Bre] will contain a complete classification of $R_{2}$-groups. The case
$d>2$ seems much harder; hopefully it will be the subject of further work.

Another result along these lines is Corollary 6.24, stating that if $X$ is an
odd-dimensional variety over $\overline{k}$ with a smooth marked point
$\overline{p}\in X(\overline{k})$, and the automorphism group of
$(X,\overline{p})$ is cyclic of order $2$ and has $\overline{p}$ as an
isolated fixed point, then $(X,\overline{p})$ is defined over its field of
moduli. This would be false without assuming that $\overline{p}$ is an
isolated fixed point, as the cyclic group $\mathrm{C}_{2}$ is not $R_{d}$ for
any $d\geq 2$. This vastly generalizes Shimura’s result that odd dimensional
generic principally polarized abelian varieties are defined over their field
of moduli.

For the second and third part, the fundamental tool is the Lang–Nishimura
theorem for tame stacks proved in [BV] (see Theorem 5.5).

### Acknowledgments

We are grateful to Dan Abramovich and Pierre Dèbes for some very useful
discussions, and to János Kollár for pointing out Lemma 6.22 to us. We thank
all of them for their interest in our work.

##  2\. Notations and conventions

We will follow the conventions of [Knu71] and [LMB00]; so the diagonals of
algebraic spaces and algebraic stacks will be separated and of finite type. In
particular, every algebraic space will be _decent_ , in the sense of [Sta21,
Definition 03I8].

We will follow the terminology of [AOV08]: a _tame stack_ is an algebraic
stack $X$ with finite inertia, such that its geometric points have linearly
reductive automorphism group. This is equivalent to requiring that $X$ is
étale locally over its moduli space a quotient by a finite, linearly reductive
group scheme [AOV08, Theorem 3.2].

If $k$ is a field and $G\longrightarrow\operatorname{Spec}k$ is a group
scheme, we denote by $\mathcal{B}_{k}G$ the classifying stack of $G$, whose
objects are $G$-torsors.

##  3\. Fields of moduli

Let $k$ be a field, $(\mathrm{Aff}/k)$ the category of affine $k$-schemes. All
stacks will be fppf stacks over $(\mathrm{Aff}/k)$. If $\mathcal{M}$ is such a
stack, and $R$ is a $k$-algebra, we set
$\mathcal{M}(R)\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathcal{M}(\operatorname{Spec}R)$; if $R\longrightarrow S$ is a
morphism of $k$-algebras and $\xi$ is an object of $\mathcal{M}(R)$, we denote
by $\xi_{S}$ the pullback of $\xi$ to $\mathcal{M}(S)$ via the induced
morphism $\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$.

Recall that a stack $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ is _locally
finitely presented_ if whenever $\\{A_{i}\\}_{i\in I}$ is a filtered inductive
system of $k$-algebras, the induced functor
$\varinjlim_{i}\mathcal{M}(A_{i})\longrightarrow\mathcal{M}(\varinjlim_{i}A_{i})$
is an equivalence. In particular we have the notion of a locally finitely
presented functor $(\mathrm{Aff}/k)\longrightarrow(\mathrm{Set})$.

Suppose that $K$ is an extension of $k$. We define, as usual, the
automorphisms functor

$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi\colon(\mathrm{Aff}/K)^{\mathrm{op}}\longrightarrow(\mathrm{Grp})\,,$

where $(\mathrm{Grp})$ is the category of groups, as the functor sending an
affine $K$-scheme $S$ into the group of automorphisms of the pullback
$\xi_{S}$.

If $\xi$ and $\eta$ are two objects of $\mathcal{M}(K)$, we denote by

$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\eta)\colon(\mathrm{Aff}/S)^{\mathrm{op}}\longrightarrow(\mathrm{Set})$

the functor of isomorphisms of the pullbacks of $\xi$ and $\eta$ along the two
projections $\operatorname{pr}_{1}$,
$\operatorname{pr}_{2}\colon\operatorname{Spec}K\times_{\operatorname{Spec}k}\operatorname{Spec}K=\operatorname{Spec}(K\otimes_{k}K)\longrightarrow\operatorname{Spec}K$.

If $\mathcal{M}$ is locally finitely presented, the two functors above are
locally finitely presented. Recall that, in our terminology, algebraic spaces
are quasi-separated: under this assumption, group algebraic spaces are
separated schemes [Sta21, Tag 08BH, Tag 0B8G]. In particular, the functor
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$ is an algebraic space of
finite type if and only if it is a group scheme of finite type.

######  Lemma 3.1.

Let $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ be a locally finitely
presented fppf stack, $K$ an extension of $k$, $\xi$ and $\eta$ two objects of
$\mathcal{M}(K)$. Let $K^{\prime}$ be an algebraic extension of $K$; then we
have the following two equivalences.

  1. (1)

The functor
$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\eta)$
is an algebraic space of finite type over $K$ if and only if
$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K^{\prime}\otimes_{k}K^{\prime}}(\operatorname{pr}_{1}^{*}\xi_{K^{\prime}},\operatorname{pr}_{2}^{*}\eta_{K^{\prime}})$
is an algebraic space of finite type over $K^{\prime}$.

  2. (2)

The functor $\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$ is a group
scheme of finite type if and only if
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K^{\prime}}\xi^{\prime}$ is a
group scheme of finite type.

Both part are immediate consequences of the following, applied to the cases
$R=K$ $R^{\prime}=K^{\prime}$, and $R=K\otimes_{k}K$,
$R^{\prime}=K^{\prime}\otimes_{k}K^{\prime}$.

######  Lemma 3.2.

Let $R$ be a commutative ring, $\\{R_{i}\\}_{i\in I}$ a filtered inductive
system of finitely presented $R$-algebras with faithfully flat transition
functions. Set $R^{\prime}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\varinjlim_{i}R_{i}$. Let
$F\colon(\mathrm{Aff}/R)^{\mathrm{op}}\longrightarrow(\mathrm{Set})$ be an
fppf sheaf that is locally of finite presentation, and call $F^{\prime}$ the
composite
$(\mathrm{Aff}/R^{\prime})^{\mathrm{op}}\longrightarrow(\mathrm{Aff}/R)^{\mathrm{op}}\longrightarrow(\mathrm{Set})$,
where the functor
$(\mathrm{Aff}/R^{\prime})^{\mathrm{op}}\longrightarrow(\mathrm{Aff}/R)^{\mathrm{op}}$
is given by restriction of scalars. Then $F$ is a finitely presented algebraic
space over $R$ if and only if $F^{\prime}$ is a finitely presented algebraic
space over $R^{\prime}$.

###### Proof.

If $F$ is a finitely presented algebraic space then
$F^{\prime}=\operatorname{Spec}R^{\prime}\times_{\operatorname{Spec}R}F$ is
also a finitely presented algebraic space. So, assume that $F^{\prime}$ is a
finitely presented algebraic space.

If $R^{\prime}$ is finitely presented over $R$, then the result follows from
Artin’s theorem [LMB00, Corollaire 10.4]. So, denote by $F_{i}$ the pullback
of $F$ to $\operatorname{Spec}R_{i}$; it is enough to show that $F_{i}$ is a
finitely presented algebraic space for some $i$. Since $F^{\prime}$ is
finitely presented, it will be obtained by pullback from a finitely presented
algebraic space $G$ over some $R_{i}$; we can replace $R,F$ with
$R_{i},F_{i}$, and assume that $G$ is defined over $R$. Then it is enough to
show that for some $i$, the pullback of $G$ to $\operatorname{Spec}R_{i}$ is
isomorphic to $F_{i}$.

We claim that the isomorphism $G_{R^{\prime}}\simeq F_{R^{\prime}}=F^{\prime}$
comes from a morphism $G_{R_{i}}\longrightarrow F_{i}$ for some $i$. For this,
choose an affine scheme $U$ with an étale surjective map $U\longrightarrow G$,
and an affine scheme $V$ with an étale surjective map $V\longrightarrow
U\times_{G}U$. Since $F$ is an fppf sheaf, for any $R$-algebra $S$ the set of
morphism $\operatorname{Hom}_{S}(G_{S},F_{S})$ is the equalizer of the two
maps $F(U_{S})\longrightarrow F(V_{S})$; since $F$ is locally of finite
presentation, and filtered colimits commute with equalizers, we easily get
$\operatorname{Hom}_{R^{\prime}}(G_{R^{\prime}},F_{R^{\prime}})=\varinjlim_{i}\operatorname{Hom}_{R_{i}}(G_{R_{i}},F_{i})$,
hence the thesis. So, by replacing $R$ with $R_{i}$ we may assume that there
exists a morphism $G\longrightarrow F$ that pulls back to the isomorphism
$G_{R^{\prime}}\simeq F_{R^{\prime}}$; we need to check that this is an
isomorphism. This is straightforward, and left to the reader. ∎

###  3.1. The residual gerbe

######  Notation 3.3.

In this subsection we will always use the following notation: $\mathcal{M}$
will be an fppf stack over $(\mathrm{Aff}/k)$, locally of finite presentation,
$K$ an algebraic extension of $k$, $\xi$ an object of $\mathcal{M}(K)$. We
will also interpret $\xi$ as a morphism
$\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$.

Assume that $K$ is finite over $k$. We define the _residual gerbe_
$\mathcal{G}_{\xi}$ of $\xi$ in $\mathcal{M}$ as in [LMB00, §11], that is, we
take $\mathcal{G}_{\xi}$ to be the fppf image of $\xi$ in $\mathcal{M}$. In
other words, $\mathcal{G}_{\xi}$ is the full fibered subcategory of
$\mathcal{M}$ such that an object $\eta$ in $\mathcal{M}(S)$ is in
$\mathcal{G}_{\xi}(S)$ if there exists an fppf cover
$S^{\prime}\longrightarrow S$ and a morphism of $k$-schemes
$S^{\prime}\longrightarrow\operatorname{Spec}K$ such that the pullback
$\eta_{S^{\prime}}$ and $\xi_{S^{\prime}}$ are isomorphic in
$\mathcal{M}(S^{\prime})$. It is immediate to check that $\mathcal{G}_{\xi}$
is an fppf stack.

Notice that by definition
$\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$ factors through
$\mathcal{G}_{\xi}$; in fact $\mathcal{G}_{\xi}\subseteq\mathcal{M}$ is the
smallest fppf substack of $\mathcal{M}$ through which $\xi$ factors.

######  Lemma 3.4.

Let $L$ be a finite extension of $K$. Then
$\mathcal{G}_{\xi_{L}}=\mathcal{G}_{\xi}$.

###### Proof.

This is clear from the fact that
$\operatorname{Spec}L\longrightarrow\operatorname{Spec}K$ is an fppf cover. ∎

In the general case $K$ is an _algebraic_ extension of $k$, and $\xi$ an
object of $\mathcal{M}(K)$ as before. Since $\mathcal{M}$ is locally of finite
presentation, there is a factorization
$\operatorname{Spec}K\longrightarrow\operatorname{Spec}L\xrightarrow{\xi^{\prime}}$
of $\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$ with $L$ finite
over $k$; we define $\mathcal{G}_{\xi}$ to be $\mathcal{G}_{\xi^{\prime}}$.
Because of Lemma 3.4, this is independent of the factorization. It is
immediate to show that the analogue of Lemma 3.4 holds when $K/k$ and $L/K$
are only assumed to be algebraic.

######  Proposition 3.5.

Let $L$ be an algebraic extension of $K$. Then
$\mathcal{G}_{\xi_{L}}=\mathcal{G}_{\xi}$.

###### Proof.

Let $E$ be an intermediate extension $k\subseteq E\subseteq K$, finite over
$k$, such that $\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$
factors as
$\operatorname{Spec}K\longrightarrow\operatorname{Spec}E\xrightarrow{\xi^{\prime}}\mathcal{M}$.
By definition we have
$\mathcal{G}_{\xi}=\mathcal{G}_{\xi^{\prime}}=\mathcal{G}_{\xi_{L}}$. ∎

######  Definition 3.6.

We say that $\xi$ is _algebraic_ if the residual gerbe $\mathcal{G}_{\xi}$ is
an algebraic stack.

From Proposition 3.5 we obtain the following.

######  Proposition 3.7.

Let $L$ be an algebraic extension of $K$. Then $\xi$ is algebraic if and only
if $\xi_{L}$ is algebraic.

######  Proposition 3.8.

The following conditions are equivalent.

  1. (1)

The object $\xi$ is algebraic.

  2. (2)

The functor

$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\xi)\colon(\mathrm{Aff}/K\otimes_{k}K)^{\mathrm{op}}\longrightarrow(\mathrm{Set})$

is an algebraic space of finite type.

###### Proof.

(1)${}\implies{}$(2). The functor in question is the fibered product
$\operatorname{Spec}K\times_{\mathcal{M}}\operatorname{Spec}K$; since
$\mathcal{G}_{\xi}$ is a full subcategory of $\mathcal{M}$ and
$\operatorname{Spec}K\longrightarrow\mathcal{M}$ factors through
$\mathcal{G}_{\xi}$, we have
$\operatorname{Spec}K\times_{\mathcal{M}}\operatorname{Spec}K=\operatorname{Spec}K\times_{\mathcal{G}_{\xi}}\operatorname{Spec}K$,
and the result follows.

(2)${}\implies{}$(1). Set
$S\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\operatorname{Spec}K$ and
$R\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\xi)$.
We obtain an fppf groupoid $R\mathbin{\rightrightarrows}S$; since
$S\longrightarrow\mathcal{G}_{\xi}$ is an fppf cover, we have an equivalence
of $\mathcal{G}_{\xi}$ with the quotient stack
$[R\mathbin{\rightrightarrows}S]$. Then it follows from Artin’s theorem
[LMB00, Corollaire 10.6] that $\mathcal{G}_{\xi}$ is an algebraic stack. ∎

If $\xi$ is algebraic, then the functor
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi\colon(\mathrm{Aff}/K)^{\mathrm{op}}\longrightarrow(\mathrm{Grp})$
is a group scheme of finite type, as it is the restriction of
$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\xi)$
along the diagonal
$\operatorname{Spec}K\subseteq\operatorname{Spec}K\otimes_{k}K$. We do not
know whether the converse holds in general; but it does if $K/k$ is separable.

######  Proposition 3.9.

Assume that the extension $K/k$ is separable, and that
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$ is a group scheme of
finite type. Then $\xi$ is algebraic.

###### Proof.

We can assume that $K$ is finite over $k$. Set
$G\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$ and Â
$R\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\xi)$;
there is a natural right action of $G$ on $R$, by right composition. Assume
that $G$ is representable, and let us show that $R$ is representable.

We have $K\otimes_{k}K=L_{1}\times\dots\times L_{r}$, where the each of the
$L_{i}$ is a finite separable extension of $k$. Let us show that for each $i$
the restriction $R_{L_{i}}$ of $R$ to $\operatorname{Spec}L_{i}$ is
representable by a scheme; this will prove the result. For each $i$ we have
two mutually exclusive cases.

  1. (a)

There exists an extension $L_{i}^{\prime}$ of $L_{i}$ such that the pullbacks
of $\operatorname{pr}_{1}^{*}\xi$ and $\operatorname{pr}_{2}^{*}\xi$ to
$L_{i}^{\prime}$ are isomorphic.

  2. (b)

For any extension $L^{\prime}_{i}$ of $L_{i}$, the pullbacks of
$\operatorname{pr}_{1}^{*}\xi$ and $\operatorname{pr}_{2}^{*}\xi$ to
$L_{i}^{\prime}$ are not isomorphic.

In the first case, the restriction $R_{L_{i}}$ is a $G$-torsor, and in the
second it is empty. In both cases it is representable. ∎

######  Proposition 3.10.

Assume that $\xi$ is algebraic. Then $\mathcal{G}_{\xi}$ is an fppf gerbe over
$\operatorname{Spec}k(\xi)$, where $k(\xi)$ is an intermediate extension
$k\subseteq k(\xi)\subseteq K$, with $k(\xi)$ finite over $k$. Furthermore,
there is a cartesian diagram

${\mathcal{B}_{K}\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi}$${\mathcal{G}_{\xi}}$${\operatorname{Spec}K}$${\operatorname{Spec}k(\xi)\,.}$

###### Proof.

We can assume that $K$ is finite over $k$. Since
$\operatorname{Spec}K\longrightarrow\mathcal{G}_{\xi}$ is flat and surjective,
and every morphism to $\operatorname{Spec}K$ is flat, it follows that every
morphism to $\mathcal{G}_{\xi}$ is flat. In particular, the inertia of
$\mathcal{G}_{\xi}$ is flat over $\mathcal{G}_{\xi}$; by [Sta21, Proposition
06QJ] it follows that $\mathcal{G}_{\xi}$ is an fppf gerbe over an algebraic
space $Z$ over $\operatorname{Spec}k$ (in the more general sense of [Sta21]).
Under the present conditions it is immediate to check that the diagonal of $Z$
is of finite type; hence $Z$ has dense open subset that is a scheme [Sta21,
Proposition 06NH]. Since $\operatorname{Spec}K\longrightarrow Z$ is flat and
surjective, it follows immediately that $Z$ has to be the spectrum of a field,
which proves the first statement.

For the second, the pullback
$\operatorname{Spec}K\times_{\operatorname{Spec}k(\xi)}\mathcal{G}_{\xi}$ is
an fppf gerbe over $\operatorname{Spec}K$. Since
$\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$ factors, by
definition, through $\mathcal{G}_{\xi}$, we obtain a section
$\operatorname{Spec}K\longrightarrow\operatorname{Spec}K\times_{\operatorname{Spec}k(\xi)}\mathcal{G}_{\xi}$;
the automorphism group scheme of the corresponding object is
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$. This concludes the proof.
∎

######  Definition 3.11.

The field $k(\xi)$ above is called the _field of moduli_ of $\xi$.

The field of moduli has the following interpretation. Denote by $R$ the
functor
$\mathop{\underline{\mathrm{Isom}}}\nolimits_{K\otimes_{k}K}(\operatorname{pr}_{1}^{*}\xi,\operatorname{pr}_{2}^{*}\xi)$,
which is an algebraic space of finite type over $K\otimes_{k}K$, by
Proposition 3.8.

######  Proposition 3.12.

The field of moduli $k(\xi)$ is the equalizer of the two arrows
$\operatorname{pr}_{1}^{*}$ and $\operatorname{pr}_{2}^{*}\colon
K\longrightarrow\mathcal{O}(R)$.

###### Proof.

Since $\mathcal{G}_{\xi}$ is a gerbe over $\operatorname{Spec}k(\xi)$, we have
$k(\xi)=\mathcal{O}(\mathcal{G}_{\xi})$. The morphism
$\operatorname{Spec}K\longrightarrow\mathcal{G}_{\xi}$ is an fpqc cover, and
$R=\operatorname{Spec}K\times_{\mathcal{G}_{\xi}}\operatorname{Spec}K$, hence
$\mathcal{O}(\mathcal{G}_{\xi})$ is the equalizer the two arrows in question.
∎

In the “classical” case, where $K/k$ is a (not necessarily finite) Galois
extension we obtain the following interpretation of the field of moduli.

######  Proposition 3.13.

Suppose that $K/k$ is a Galois extension with Galois group $G$. For each
$\gamma\in G$ call $\gamma^{*}\xi$ the pullback of $\xi$ along
$\gamma\colon\operatorname{Spec}K\longrightarrow\operatorname{Spec}K$; call
$\overline{K}$ the algebraic closure of $K$. Let $H\subseteq G$ the subgroup
consisting of elements $\gamma\in G$ such that
$(\gamma^{*}\xi)_{\overline{K}}$ is isomorphic to $\xi_{\overline{K}}$. Then
$k(\xi)$ is the field of invariants $K^{H}$.

###### Proof.

First, assume that $K$ is finite over $k$. In this case
$\operatorname{Spec}(K\otimes_{k}K)$ is a disjoint union $\coprod_{g\in
G}\operatorname{Spec}K$. The image of the natural morphism
$R\longrightarrow\operatorname{Spec}(K\otimes_{k}K)$ is $\coprod_{g\in
H}\operatorname{Spec}K\subseteq\coprod_{g\in G}\operatorname{Spec}K$; hence
$\operatorname{pr}_{1}^{*}K$, $\operatorname{pr}_{2}^{*}\colon
K\longrightarrow\mathcal{O}(R)$ factor through the pullback $\prod_{g\in
H}K\longrightarrow\mathcal{O}(R)$, which is injective. The conclusion follows
from Proposition 3.12.

If $K$ is not finite over $k$, choose an intermediate extension $k\subseteq
K^{\prime}\subseteq K$ such that $(X,\xi)$ descends to $K^{\prime}$. The
result for $K^{\prime}$ is easily seen to imply that for $K$. ∎

######  Definition 3.14.

The object $\xi$ is _tame_ if it is algebraic, and
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{K}\xi$ is finite and linearly
reductive.

Equivalently, the object $\xi$ is tame if $\mathcal{G}_{\xi}$ is a tame stack.

###  3.2. Residual gerbes and moduli spaces

In case $\mathcal{M}$ is an algebraic stack with finite inertia, there is
another interesting interpretation of the field of moduli of a tame object.

Assume that $\mathcal{M}$ is an algebraic stack with finite inertia, with
moduli space $\mathcal{M}\longrightarrow M$, and let $m\in M$ be a point. By
definition of a moduli space there exists an object $\xi$ over the algebraic
closure $\overline{k(m)}$; we say that the point $m$ is _tame_ if
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k(m)}}\xi$ is linearly
reductive.

######  Definition 3.15.

Assume that $m\in M$ is a tame point. The _residual gerbe of $m$_ is defined
to be

$\mathcal{G}_{m}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\bigl{(}\operatorname{Spec}k(m)\times_{M}\mathcal{M}\bigr{)}_{\mathrm{r}ed}\,.$

######  Proposition 3.16.

The residual gerbe $\mathcal{G}_{m}$ is a finite tame gerbe over $k(m)$.

###### Proof.

By [AOV08, Proposition 3.6] the tame points of $M$ form an open subspace
$M^{\prime}\subseteq M$; we can replace $M$ with $M^{\prime}$, and assume that
$\mathcal{M}$ is tame. Since formation of moduli spaces of tame stacks
commutes with base change, we have that the moduli space of
$\operatorname{Spec}k(m)\times_{M}\mathcal{M}$ is $\operatorname{Spec}k(m)$;
and from this, that the moduli space of $\mathcal{G}_{m}$ is
$\operatorname{Spec}k(m)$. From [Sta21, Proposition 06RC] it follows that
$\mathcal{G}_{m}$ is a gerbe over $k(m)$, as claimed. ∎

######  Proposition 3.17.

Assume that $\mathcal{M}$ is an algebraic stack with finite inertia locally of
finite type over $k$, with moduli space $\mathcal{M}\longrightarrow M$. Let
$\xi\colon\operatorname{Spec}K\longrightarrow\mathcal{M}$ be a tame object,
and call $m\in M$ the image of the composite
$\operatorname{Spec}K\xrightarrow{\xi}\mathcal{M}\longrightarrow M$. Then the
residual gerbe of $\xi$ is $\mathcal{G}_{m}$, and the field of moduli $k(\xi)$
is the residue field $k(m)$.

###### Proof.

We may assume that $K$ is finite over $k$. The morphism
$\operatorname{Spec}K\longrightarrow\mathcal{M}$ factors through
$\operatorname{Spec}k(m)\times_{M}\mathcal{M}$; since $\operatorname{Spec}K$
is reduced we get a factorization
$\operatorname{Spec}K\longrightarrow\mathcal{G}_{m}\subseteq\mathcal{M}$.
Since $\mathcal{G}_{m}$ is a finite gerbe, it follows that
$\operatorname{Spec}K\longrightarrow\mathcal{G}_{m}$ is flat and finite, hence
it is an fppf cover. The result follows from this. ∎

It is easy to give counterexamples to the statement of Proposition 3.17
without the tameness hypothesis. The point is that the moduli space of
$\operatorname{Spec}k(m)\times_{M}\mathcal{M}$ may be a non-trivial purely
inseparable extension $k^{\prime}$ of $k(m)$; and in this case the argument
above shows that the field of moduli of $\xi$ is $k^{\prime}$.

###  3.3. The basic question

Now assume that $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ is an fppf
locally finitely presented stack,
$\xi\colon\operatorname{Spec}\overline{k}\longrightarrow\mathcal{M}$ an
algebraic object defined over the algebraic closure of $k$,
$k(\xi)\subseteq\overline{k}$ its field of moduli. Is $\xi$ defined over its
field of moduli $k(\xi)$? This is equivalent to asking whether
$\mathcal{G}_{\xi}\bigl{(}k(\xi)\bigr{)}\neq\emptyset$.

From now on we will consider objects $\xi$ defined over the algebraic closure
$\overline{k}$ of $k$; from Proposition 3.5 it is clear that this is not a
restriction.

##  4\. Application of nonabelian cohomology

In the situation above, assume that $\xi\in\mathcal{M}(\overline{k})$ is an
algebraic object.

As an immediate corollary of the fact that every affine gerbe over a finite
field is neutral [DPTZ20, Theorem 8.1], we get the following.

######  Proposition 4.1.

Assume that $k$ is finite and
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$ is affine. Then
$\xi$ is defined over its field of moduli.

One can also apply standard results on the classification of gerbes, which
usually go under the name of Grothendieck–Giraud nonabelian cohomology
[Gir71], to get conditions ensuring that $\xi$ is defined over its field of
moduli. This works very cleanly when
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$ is finite and
reduced.

Set $G\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$, and
assume for the rest of the section that $G$ is finite and reduced; according
to our general conventions we think of $G$ as an ordinary group. Denote by
$\operatorname{Aut}G$ the group of automorphisms of $G$, and by
$\operatorname{Out}G$ its group of outer automorphisms, that is, the cokernel
of the homomorphism $G\longrightarrow\operatorname{Aut}G$ given by
conjugation.

######  Proposition 4.2.

Assume that the following conditions are satisfied:

  1. (1)

the center of $G$ is trivial, and

  2. (2)

the projection $\operatorname{Aut}G\longrightarrow\operatorname{Out}G$ is
split.

Then $\xi$ is defined over its field of moduli.

This should be compared with [DE99, Corollary 4.3(b)].

In similar spirit we get the following, which is a generalization of [DE99,
Corollary 4.3(a)], with the same proof.

######  Proposition 4.3.

Assume that the absolute Galois group of $k$ has cohomological dimension at
most $1$. Then $\xi$ is defined over its field of moduli.

These two propositions are immediate corollaries of the following standard
application of nonabelian cohomology.

######  Lemma 4.4.

Let $G$ be a finite group, $k$ a field. Let
$\mathcal{G}\longrightarrow(\mathrm{Aff}/k)$ be a gerbe such that
$\mathcal{G}_{\overline{k}}$ is isomorphic to $\mathcal{B}_{\overline{k}}G$.
Assume that either

  1. (1)

$G$ has trivial center, and
$\operatorname{Aut}G\longrightarrow\operatorname{Out}G$ is split, or

  2. (2)

$k$ has cohomological dimension $1$.

Then $\mathcal{G}$ is neutral.

###### Proof.

Denote by $k^{s}$ the separable closure of $k$ and by
$\Gamma=\operatorname{Gal}(k^{s}/k)$ be the absolute Galois group of $k$, we
have that $\mathcal{G}_{k^{s}}$ is isomorphic to $\mathcal{B}_{k^{s}}G$ by the
following Lemma 4.5. The tautological section
$\operatorname{Spec}k^{s}\to\mathcal{B}_{k^{s}}G\to\mathcal{G}$ induces a
continuous homomorphism $\Gamma\to\operatorname{Out}G$. Under both conditions
(1) and (2), we have a lifting $\Gamma\to\operatorname{Aut}G$: for (1) this is
obvious, while for (2) this follows from the fact that $G$ is projective
[Ser94, Proposition 45]. By descent theory, the homomorphism
$\Gamma\to\operatorname{Aut}G$ induces a finite étale group scheme
$\underline{G}$ which is a twisted form of $G$.

Let $L$ be the non-abelian band, in the sense of [Gir71], of $\mathcal{G}$. By
construction, $L$ is represented by $\underline{G}$. By [Gir71, Théorème
3.3.3], under both conditions (1) and (2), the gerbe $\mathcal{G}$ is the only
gerbe banded by $L$. Since the classifying stack
$\mathcal{B}_{k}\underline{G}$ is banded by $L$ by construction, we get that
$\mathcal{G}\simeq\mathcal{B}_{k}\underline{G}$, i.e. $\mathcal{G}$ is
neutral. ∎

######  Lemma 4.5.

A finite gerbe with unramified diagonal over a separably closed field is
neutral.

###### Proof.

Let $k$ be a separably closed field with algebraic closure $\overline{k}$, and
$\mathcal{G}$ a finite gerbe over $k$ with unramified diagonal. There exists a
finite extension $k^{\prime}/k$ with a section $s\in\mathcal{G}(k^{\prime})$.
The scheme
$\mathop{\underline{\mathrm{Isom}}}\nolimits(p_{1}^{*}s,p_{2}^{*}s)$ is finite
étale over the artinian local ring $k^{\prime}\otimes_{k}k^{\prime}$ and we
have a lifting
$\operatorname{Spec}k^{\prime}\to\mathop{\underline{\mathrm{Isom}}}\nolimits(p_{1}^{*}s,p_{2}^{*}s)$
of the closed point
$\operatorname{Spec}k^{\prime}\subset\operatorname{Spec}k^{\prime}\otimes_{k}k^{\prime}$,
hence we get a section
$\operatorname{Spec}k^{\prime}\otimes_{k}k^{\prime}\to\mathop{\underline{\mathrm{Isom}}}\nolimits(p_{1}^{*}s,p_{2}^{*}s)$.
By descent theory we obtain that $s$ descends to $k$. ∎

These results do not apply in many cases of great interest, for example, when
$G$ is abelian and $k$ is a number field.

Our main theorem, which is a generalization of [DE99, Corollary 4.3(c)], gives
a criterion for this to happen, when $\xi$ is tame, for an interesting class
of stacks, whose objects are algebraic spaces with additional structure.

##  5\. The main theorem

###  5.1. Categories of structured spaces

We will denote by $(\mathrm{FAS}/k)$ the fibered category over
$(\mathrm{Aff}/k)$ whose objects over an affine scheme $S$ over $k$ are flat
finitely presented morphisms $X\longrightarrow S$, where $X$ is an algebraic
space.

######  Definition 5.1.

A _category of structured spaces_ over $k$ is a locally finitely presented
fppf stack $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$, with a faithful
cartesian functor $\mathcal{M}\longrightarrow(\mathrm{FAS}/k)$.

Examples spring to mind.

######  Examples 5.2.

  1. (1)

The category $(\mathrm{FAS}/k)$ itself is a category of structured spaces.

  2. (2)

Any condition that we impose on objects $X\to S$ in $(\mathrm{FAS}/k)$ that is
stable under base change and fppf local defines a full subcategory
$\mathcal{M}\subseteq(\mathrm{FAS}/k)$ that is a category of structured
spaces. Thus, for example, the stacks $\mathcal{M}_{g}$ and
$\overline{\mathcal{M}}_{g}$ of smooth, or stable, curves of genus $g$, the
stack of smooth abelian varieties, of projective surfaces, and so on.

  3. (3)

The stacks $\mathcal{M}_{g,n}$ and $\overline{\mathcal{M}}_{g,n}$ of
$n$-pointed smooth, or stable, curves of genus $g$ are all categories of
structured spaces.

  4. (4)

The category of smooth polarized projective schemes is a category of
structured spaces.

An object $\xi$ of a category of structured spaces $\mathcal{M}$ will be
denoted by $(X\to S,\xi)$, or simply $(X,\xi)$, where $X\longrightarrow S$ is
the image of $\xi$ in $(\mathrm{FAS}/k)$; we want to think of $(X,\xi)$ as an
algebraic space with an additional structure.

A category of structured spaces $\mathcal{M}$ has a _universal family_
$\mathcal{X}\longrightarrow(\mathrm{Aff}/k)$; the objects of $\mathcal{X}$ are
triples $(X\to S,\xi,x)$, where $(X\longrightarrow S,\xi)$ is an object of
$\mathcal{M}(S)$, and $x\colon S\longrightarrow X$ is a section of the
morphism $X\longrightarrow S$. We have an obvious morphism
$\mathcal{X}\longrightarrow\mathcal{M}$ which forgets the section. If
$S\longrightarrow\mathcal{M}$ is a morphism, corresponding to an object $(X\to
S,\xi)$ of $\mathcal{M}(S)$, then the fibered product
$S\times_{\mathcal{M}}\mathcal{X}$ is equivalent to $X$; hence the morphism
$\mathcal{X}\longrightarrow\mathcal{M}$ is representable, flat and finitely
presented. The universal family is itself a category of structured spaces.

Now, let $(X,\xi)$ be an algebraic object of $\mathcal{M}(\overline{k})$, and
denote by $\mathcal{X}_{(X,\xi)}$ the fibered product
$\mathcal{G}_{(X,\xi)}\times_{\mathcal{M}}\mathcal{X}$. We have a commutative
diagram

${X}$${{[X/\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi]}}$${\mathcal{X}_{(X,\xi)}}$${\mathcal{X}}$${\operatorname{Spec}\overline{k}}$${\mathcal{B}_{\overline{k}}\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi}$${\mathcal{G}_{(X,\xi)}}$${\mathcal{M}}$${\operatorname{Spec}\overline{k}}$${\operatorname{Spec}k(X,\xi)}$${\operatorname{Spec}k}$${\square}$${\square}$${\square}$${\square}$

in which the squares marked with $\square$ are cartesian.

######  Definition 5.3.

Assume that $\mathcal{G}_{(X,\xi)}$ (and hence $\mathcal{X}_{(X,\xi)}$) has
finite inertia. The _compression_ of $(X,\xi)$ is the coarse moduli space of
$\mathcal{X}_{(X,\xi)}$, it is an algebraic space over the field of moduli
$k(X,\xi)$. We denote the compression using bold letters, for instance we
write $\mathbf{X}_{(X,\xi)}$ for the compression of $(X,\xi)$.

Since formation of moduli spaces commutes with flat base change (see [Con]) we
have

$\operatorname{Spec}\overline{k}\times_{\operatorname{Spec}k}\mathbf{X}_{(X,\xi)}=X/\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi\,.$

In other words, while $X$ does not necessarily descend to $k(X,\xi)$, the
quotient $X/\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$
always does, in a canonical fashion. This is a more general version of [DE99,
Theorem 3.1].

###  5.2. The main result

######  Theorem 5.4.

Let $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ be a category of structured
spaces, $(X,\xi)\in\mathcal{M}(\overline{k})$ a tame object with $X$ integral.

Assume that there exists a dominant rational map
$Y\dashrightarrow\mathbf{X}_{(X,\xi)}$ where $Y$ is an integral algebraic
space of finite type over $k(X,\xi)$ with a $k(X,\xi)$-rational regular point.
Then $(X,\xi)$ is defined over its field of moduli $k(X,\xi)$.

In this proof, and in the rest of the paper, a crucial role is played by the
Lang–Nishimura theorem for tame stack that we prove in [BV]. For the
convenience of the reader we recall its statement.

######  Theorem 5.5 ([BV, Theorem 4.1]).

Let $S$ be a scheme and $X\dashrightarrow Y$ a rational map of algebraic
stacks over $S$, with $X$ locally noetherian and integral and $Y$ tame and
proper over $S$. Let $k$ be a field,
$s\colon\operatorname{Spec}k\longrightarrow S$ a morphism. Assume that $s$
lifts to a regular point $\operatorname{Spec}k\longrightarrow X$; then it also
lifts to a morphism $\operatorname{Spec}k\longrightarrow Y$.

###### Proof of Theorem 5.4.

There exists an
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$-invariant open
subset $U\subset X$ such that the action of
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$ on $U$ is free:
by hypothesis the action of
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$ on $X$ is
faithful, and for each non-trivial subgroup
$G\subseteq\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}\xi$ the
locus of points of $X$ fixed by $G$ is a proper closed subset of $X$. This in
turn implies that there exists an open substack
$\mathcal{U}\subset\mathcal{X}_{(X,\xi)}$ which is an algebraic space; then
the composite
$\mathcal{U}\subseteq\mathcal{X}_{(X,\xi)}\longrightarrow\mathbf{X}_{(X,\xi)}$
is an open embedding.

Because of this, the hypothesis gives us a rational map
$Y\dashrightarrow\mathcal{U}$. We conclude by applying Theorem 5.5 to the
composite
$Y\dashrightarrow\mathcal{U}\subset\mathcal{X}_{(X,\xi)}\to\mathcal{G}_{(X,\xi)}$.
∎

Note that if $X$ is smooth of dimension $1$ over $\overline{k}$, then
$\mathbf{X}_{(X,\xi)}$ is also smooth over $k(X,\xi)$; hence in this case we
get the following.

######  Corollary 5.6.

Let $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ be a category of structured
spaces, $(X,\xi)\in\mathcal{M}(\overline{k})$ a tame object such that $X$ is
integral, smooth and $1$-dimensional. Assume that the compression
$\mathbf{X}_{(X,\xi)}$ has a $k(X,\xi)$-rational point. Then $(X,\xi)$ is
defined over its field of moduli $k(X,\xi)$.

When $(X,\xi)$ is a smooth projective curve with no additional structure, this
is [DE99, Corollary 4.3(c)].

###  5.3. The case of pointed spaces

One case in which we can ensure the existence of a rational point on
$\mathbf{X}_{(X,\xi)}$ is the case of pointed spaces. For the rest of the
paper we will use the following definition.

######  Definition 5.7.

A _pointed space_ $(X,p)$ over an affine scheme $S$ over $k$ is a flat locally
finitely presented morphism $X\longrightarrow S$ with a section $p\colon
S\longrightarrow X$ landing in the smooth locus of $f$.

Pointed spaces form a fibered category
$(\mathrm{PFAS}/k)\longrightarrow(\mathrm{Aff}/k)$; there is an obvious
representable cartesian functor
$(\mathrm{PFAS}/k)\longrightarrow(\mathrm{FAS}/k)$, which forgets the section.

######  Definition 5.8.

A _category of pointed structured spaces_ over $k$ is a locally finitely
presented fppf stack $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$, with a
faithful cartesian functor $\mathcal{M}\longrightarrow(\mathrm{PFAS}/k)$.

If $\mathcal{M}\longrightarrow(\mathrm{PFAS}/k)$ is a category of pointed
structured spaces and $S$ is a scheme, an element of $\mathcal{M}(S)$ will be
denoted by $(X,p,\xi)$, where $(X,\xi)$ is the corresponding structured space
given by the composition
$\mathcal{M}\longrightarrow(\mathrm{PFAS}/k)\longrightarrow(\mathrm{FAS}/k)$
and $p\colon S\longrightarrow X$ is the given section.

If $\mathcal{M}$ is a category of pointed structured spaces, it can be
considered as a category of structured spaces by composing
$\mathcal{M}\longrightarrow(\mathrm{PFAS}/k)$ with the forgetful morphism
$(\mathrm{PFAS}/k)\longrightarrow(\mathrm{FAS}/k)$. One can think of
categories of pointed structured spaces as a category of structured spaces, in
which the structure includes a smooth marked point.

There are many natural examples of categories of pointed structured spaces:
for example, the category of abelian varieties, or the category of $n$-pointed
stable, or smooth, curves, for $n\geq 1$.

######  Lemma 5.9.

Let $\mathcal{M}$ be a category of pointed structured spaces,
$(X,p,\xi)\in\mathcal{M}(\overline{k})$ a tame object. The compression
$\mathbf{X}_{(X,p,\xi)}$ has a rational point $\mathbf{p}$ over $k(X,p,\xi)$
such that $\mathbf{p}_{\overline{k}}$ corresponds to $p$ via the
identification
$\mathbf{X}_{(X,p,\xi),\overline{k}}=X/\mathop{\underline{\mathrm{Aut}}}\nolimits(X,p,\xi)$.

###### Proof.

Consider the universal family $\mathcal{X}\longrightarrow\mathcal{M}$, and its
smooth locus $\mathcal{X}_{\rm sm}\subseteq\mathcal{X}$ (this is the largest
open substack of $\mathcal{X}$ where the restriction of
$\mathcal{X}\longrightarrow\mathcal{M}$ is smooth). The fibered product
$\mathcal{M}\times_{(\mathrm{FAS}/k)}(\mathrm{PFAS}/k)$ is canonically
isomorphic to $\mathcal{X}_{\rm sm}$; hence the cartesian functor
$\mathcal{M}\longrightarrow(\mathrm{PFAS}/k)$ induces a section
$\mathcal{M}\longrightarrow\mathcal{X}_{\rm sm}$ of the projection
$\mathcal{X}_{\rm sm}\subseteq\mathcal{X}\longrightarrow\mathcal{M}$. By
restricting to $\mathcal{G}_{(X,p,\xi)}$ we obtain a section
$\mathcal{G}_{(X,p,\xi)}\longrightarrow\mathcal{X}_{(X,p,\xi)}$ of the
projection $\mathcal{X}_{(X,p,\xi)}\longrightarrow\mathcal{G}_{(X,p,\xi)}$,
and, passing to moduli spaces, a section
$\operatorname{Spec}k(X,p,\xi)\longrightarrow\mathbf{X}_{(X,p,\xi)}$ of the
projection
$\mathbf{X}_{(X,p,\xi)}\longrightarrow\operatorname{Spec}k(X,p,\xi)$, or, in
other words, a $k(X,p,\xi)$-rational point of $\mathbf{X}_{(X,p,\xi)}$. ∎

When $X$ is $1$-dimensional, it follows that
$\mathbf{p}\in\mathbf{X}_{(X,p,\xi)}(k(X,p,\xi))$ is smooth. Hence we get the
following.

######  Corollary 5.10.

Let $\mathcal{M}$ be a category of pointed structured spaces, $(X,p,\xi)$ a
tame object of $\mathcal{M}(\overline{k})$, such that $X$ is $1$-dimensional
and integral. Then $(X,p,\xi)$ is defined over its field of moduli.

As a consequence, we recover the following result by Dèbes–Emsalem.

######  Corollary 5.11 ([DE99, Corollary 5.4]).

Let $g\geq 1$ and $n\geq 1$ be positive integers and $k$ a field. Every smooth
$n$-pointed curve of genus $g$ over $\overline{k}$ with tame automorphism
group scheme is defined over its field of moduli.

In particular, if $\operatorname{char}k=0$, $K/k$ is any extension and
$\mathrm{M}_{g,n}$ is the coarse moduli space of smooth $n$-pointed curves of
genus $g$ over $k$, every $K$-valued point of $\mathrm{M}_{g,n}$ comes from a
smooth $n$-pointed curve of genus $g$ over $K$.

###### Proof.

Let $\mathcal{M}_{g,n}$ be the stack of $n$-pointed curves of genus $g$ over
$k$, since $n\geq 1$ we may think of it as a category of pointed structured
spaces, the first part then follows from Corollary 5.10. Now assume that
$\operatorname{char}k=0$, let $K/k$ be any extension and
$\operatorname{Spec}K\to\mathrm{M}_{g,n}$ a point. Let $X$ be an $n$-pointed
smooth curve over $\overline{K}$ corresponding to the composite
$\operatorname{Spec}\overline{K}\longrightarrow\operatorname{Spec}K\longrightarrow\mathrm{M}_{g,n}$;
by Proposition 3.17 we have that the field of moduli of $X$ is $K$, hence $X$
is defined over $K$. ∎

This fails for stable curves that are not irreducible: one can give examples
of $1$-pointed stable curves that are not defined over their field of moduli.
The issue here is that the automorphism group of the pointed curve may not act
faithfully on the component containing the marked point.

Corollary 5.10 fails in dimension higher than $1$. There are many
counterexamples. For example, fix a positive integer $g$, let $\mathrm{A}_{g}$
the moduli space of principally polarized $g$-dimensional abelian varieties
over $\mathbb{C}$, and let $k$ be its field of rational functions. Call $X$
the corresponding abelian variety over the algebraic closure $\overline{k}$,
which we can think of as $1$-pointed variety $(X,0)$. By Proposition 3.17 the
field of moduli of $(X,0)$ is $k$. As we mentioned in the introduction, G.
Shimura showed in [Shi72] that when $k=\mathbb{C}$ $(X,0)$ is defined over $k$
if and only if $g$ is odd (see [BRV11, Appendix] for a refinement of this
statement due to Najmuddin Fakhruddin). In this case the group of
automorphisms is cyclic of order $2$.

Given a positive integer $d$, we will study a natural class of discrete finite
groups over $k$, with the property that if $(X,\xi)$ is a tame object of
$\mathcal{M}$, and $\operatorname{Aut}_{\overline{k}}\xi$ is in this class,
then $\xi$ is defined over its field of moduli.

##  6\. The arithmetic of tame quotient singularities

Let $S$ be an algebraic space of finite type over a field $k$. We say that $S$
_has tame quotient singularities_ if there is an étale cover
$\\{S_{i}\longrightarrow S\\}$ and, for each $i$, a smooth algebraic space
$U_{i}$ and a finite group $G_{i}$ of order not divisible by
$\operatorname{char}k$ acting on $U_{i}$, such that $S_{i}$ is isomorphic to
$U_{i}/G$. In particular, $S$ is normal.

More generally, one could consider spaces that are étale-locally quotients of
smooth algebraic spaces by finite linearly reductive group schemes, as in
[Sat12]; but the technology to adequately deal with these in our context still
does not seem to be completely in place, which forces us to limit ourselves to
considering tame Deligne–Mumford stacks, as opposed to general tame stacks.

###  6.1. Minimal stacks

The following is known, see [Vis89, Proposition 2.8]. Our statement is
slightly different from the one in the reference, though, so we give details.

######  Proposition 6.1.

The moduli space of a smooth tame Deligne–Mumford stack with finite inertia
over $k$ has tame quotient singularities.

Conversely, if $S$ is an algebraic space with finite quotient singularities,
there exists a smooth tame Deligne–Mumford stack with finite inertia
$\widehat{S}$ with moduli space $S$, with the property that the morphism
$\widehat{S}\longrightarrow S$ is an isomorphism over the smooth locus of $S$.

Furthermore, if $V$ is a smooth integral Deligne–Mumford stack with a dominant
morphism $V\to\widehat{S}$, there exists a factorization $V\to\widehat{S}\to
S$, unique up to a unique isomorphism. In particular, $\widehat{S}$ is unique,
up to a unique isomorphism.

###### Proof.

Let $\mathcal{X}$ be a smooth, tame Deligne–Mumford stack with finite inertia,
and let $M$ be its moduli space, we want to show that $M$ has tame quotient
singularities. By [AV02, Lemma 2.2.3], we may assume that $\mathcal{X}=[U/G]$,
where $G$ is a finite group, so that $M=X/G$. If
$u_{0}\colon\operatorname{Spec}\Omega\longrightarrow U$ is a geometric point
of $U$, and $G_{u_{0}}$ is the stabilizer of $u_{0}$, then the natural
morphism $U/G_{u_{0}}\longrightarrow U/G$ is étale in neighborhood of $u_{0}$;
but $G_{u_{0}}$ has order prime to $\operatorname{char}k$, because $[U/G]$ is
tame, so the result follows.

If $S$ has tame quotient singularities, then the second half of the proof of
[Vis89, Proposition 2.8] shows the existence of a stack $\widehat{S}\to S$ as
in the statement (the reference’s assumption $\operatorname{char}k=0$ is not
used in the relevant part of the proof).

Now let $V\to\widehat{S}$ be as in the statement. Since $S$ is normal,
$\widehat{S}\to S$ is an isomorphism in codimension $1$. Let $U$ be the
normalization of $V\times_{S}\widehat{S}$, since everything is of finite type
over $k$ and $S$ is the moduli space of $\widehat{S}$, then $U\to V$ is proper
and birational. By purity of branch locus, it is étale too, hence $U\simeq V$
and we obtain the desired morphism $V\to\widehat{S}$. The uniqueness follows
from the fact that $\widehat{S}$ is separated. ∎

We call the stack $\widehat{S}$ above the _minimal stack_ of $S$ (also called
the _canonical stack_ in the literature). Clearly, if $S$ has tame quotient
singularities, and $k^{\prime}$ is an extension of $k$, the space
$S_{k^{\prime}}$ obtained by base change also has tame quotient singularities,
and the minimal stack of $S_{k^{\prime}}$ is $\widehat{S}_{k^{\prime}}$. In
other words, formation of the minimal stack commutes with extensions of the
base field. Furthermore, if $S^{\prime}\longrightarrow S$ is an étale morphism
and $S$ has tame quotient singularities, then so does $S^{\prime}$, and
$\widehat{S^{\prime}}=S^{\prime}\times_{S}\widehat{S}$.

It is known that algebraic spaces with tame quotient singularities have a
resolution of singularities, that is, there is a proper birational morphism
$\widetilde{S}\longrightarrow S$ where $\widetilde{S}$ is a smooth algebraic
space over $k$. If $k$ is perfect, this is [BR19, Theorem E]; in the general
case it is obtained by applying [BR19, Theorem B] to the minimal stack
$\widehat{S}\longrightarrow S$.

###  6.2. Singularities and fundamental gerbes

Let $k$ be a field; a _tame quotient singularity_ over $k$ is a pair $(S,s)$,
where $S$ is a an integral scheme of finite type over $k$ with tame quotient
singularities and $s\in S(k)$ is a $k$-rational point. No other kinds of
singularities will appear in this paper, so from now on a tame quotient
singularity will be called simply _a singularity_.

Two singularities $(S,s)$ and $(S^{\prime},s^{\prime})$ are _equivalent_ if
there exists a singularity $(S^{\prime\prime},s^{\prime\prime})$, together
with étale maps $S^{\prime\prime}\longrightarrow S$ and
$S^{\prime\prime}\longrightarrow S^{\prime}$ sending $s^{\prime\prime}$ into
$s$ and $s^{\prime}$ respectively. This is true if and only if the complete
local $k$-algebras $\widehat{\mathcal{O}}_{S,s}$ and
$\widehat{\mathcal{O}}_{S^{\prime},s^{\prime}}$ are isomorphic [Art69,
Corollary 2.6].

######  Definition 6.2.

Given a singularity $(S,s)$, the _fundamental gerbe_ $\mathcal{G}_{(S,s)}$ of
$(S,s)$ is the residual gerbe, as in Definition 3.15, of $\widehat{S}\to S$ at
$s$. The _fundamental group_ $G_{(S,s)}$ of $(S,s)$ is the automorphism group
of any geometric point of $\mathcal{G}_{(S,s)}$.

Thus, by definition, the fundamental group of $(S,s)$ is a finite group, or
order prime to $\operatorname{char}k$. It is well defined up to a non-
canonical isomorphism.

One can prove that $\mathcal{G}_{(S,s)}$ is the local fundamental gerbe of
$S$, in the following sense. Let
$S^{\prime}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\operatorname{Spec}\mathcal{O}_{S,s}^{\rm h}$, where
$\operatorname{Spec}\mathcal{O}_{S,s}^{\rm h}$ is the henselization of
$\mathcal{O}_{S,s}$, and let $U\subseteq S^{\prime}$ be the smooth locus of
$S$. Set
$\widehat{S^{\prime}}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}S^{\prime}\times_{S}\widehat{S}$. Then $\mathcal{G}_{(S,s)}$ is the
fundamental gerbe, in the sense of [BV15] of $U$ and also of
$\widehat{S^{\prime}}$. We do not prove this here, as it is not needed in what
follows.

Since formation of the minimal stack commutes with étale morphisms, equivalent
singularities have isomorphic fundamental gerbes.

######  Lemma 6.3.

Assume that $k$ is separably closed. Let $\mathcal{X}$ be a smooth, tame
Deligne–Mumford stack which is generically a scheme, with moduli space
$\mathcal{X}\longrightarrow S$. Let $\xi$ be an object in $\mathcal{X}(k)$ and
$s\in S(k)$ its image of $\xi$.

There exists a faithful representation
$\operatorname{Aut}\xi\subset\mathrm{GL}_{d}(k)$ such that $(S,s)$ is
equivalent to $(\mathbb{A}^{d}/\operatorname{Aut}\xi,[0])$, and the quotient
of $\operatorname{Aut}\xi$ by the subgroup generated by pseudoreflexions is
isomorphic to the fundamental group of $(S,s)$.

###### Proof.

After passing to an étale neighborhood of $s\in S$, we may assume
$\mathcal{X}\simeq[U/H]$ with $U$ smooth and $H$ finite of order prime to
$\operatorname{char}k$. Since $k$ is separably closed, the rational point
$\xi\in\mathcal{X}(k)$ lifts to a rational point $u\in U(k)$. Let
$H_{u}\subset H$ be the stabilizer of $u$, then $U/H_{u}\to U/H\simeq S$ is
étale in $[u]$, hence we may replace $\mathcal{X},H$ with
$[U/H_{u}],H_{u}=\operatorname{Aut}\xi$ and assume that
$H=\operatorname{Aut}\xi$ and that $u$ is a fixed point.

Call $V$ the tangent space of $U$ at $u$; then $\operatorname{Aut}\xi$ acts on
$V$, and by fixing a basis we get a representation
$\operatorname{Aut}\xi\longrightarrow\mathrm{GL}_{d}(k)$. By Cartan’s lemma,
after passing to an equivariant étale neighborhood of $u$ in $U$ we may assume
that there exists an étale $\operatorname{Aut}\xi$-equivariant map
$U\longrightarrow V$. This implies that the action of $\operatorname{Aut}\xi$
on $V$ is faithful, and that $(S,s)=(U/\operatorname{Aut}\xi,[u])$ is
equivalent to $(V/\operatorname{Aut}\xi,[0])$.

Denote by $P\subset\operatorname{Aut}\xi$ the subgroup generated by
pseudoreflexions, by the Chevalley–Shephard–Todd theorem $V/P$ is smooth and
$\operatorname{Aut}\xi/P$ acts on it with with no fixed points in codimension
$1$. It follows that $[(V/P)/(\operatorname{Aut}\xi/P)]$ is the minimal stack
of $V/\operatorname{Aut}\xi$, and hence $\operatorname{Aut}\xi/P$ is the
fundamental group of $(V/\operatorname{Aut}\xi,[0])\sim(S,s)$. ∎

######  Corollary 6.4.

Assume that $k$ is separably closed, and let $(S,s)$ be a tame quotient
singularity over $k$. There exists a faithful representation
$G_{(S,s)}\subset\mathrm{GL}_{d}(k)$ with no pseudoreflexions such that
$(S,s)\sim(\mathbb{A}^{d}_{k}/G_{(S,s)},[0])$.

###### Proof.

Thanks to Lemma 4.5, we may apply Lemma 6.3 to $\widehat{S}$. ∎

###  6.3. Liftable singularities

The following is immediate from Theorem 5.5.

######  Proposition 6.5.

Let $(S,s)$ be a tame quotient singularity, $\widetilde{S}\longrightarrow S$ a
resolution of singularities. The following conditions are equivalent.

  1. (1)

$\widetilde{S}$ has a $k$-rational point over $s$.

  2. (2)

The fundamental gerbe $\mathcal{G}_{(S,s)}$ of $(S,s)$ is neutral.

  3. (3)

The minimal stack $\widehat{S}\longrightarrow S$ has a $k$-rational point over
$s$.

  4. (4)

If $S^{\prime}\longrightarrow S$ is a proper birational morphism, where
$S^{\prime}$ is an integral tame Deligne–Mumford stack, then $S^{\prime}$ has
a $k$-rational point over $s$.

######  Definition 6.6.

Let $(S,s)$ be a tame quotient singularity. We say that $(S,s)$ _liftable_ if
it satisfies the equivalent conditions of Proposition 6.5.

######  Remarks 6.7.

  1. (1)

If two singularities are equivalent, then one is liftable if and only if the
other is.

  2. (2)

Since formation of $\widehat{S}$ commutes with extension of the base field, we
see that if $(S,s)$ is liftable over $k$, and $k^{\prime}$ is an extension of
$k$, then $(S,s)_{k^{\prime}}$ is also liftable.

The following construction gives a criterion for a singularity to be liftable,
which will be a fundamental tool in the rest of the paper.

###  6.4. The blowup construction

Let $K/k$ be a separable closure. Let $(S,s)$ be a tame quotient singularity
over $k$; by Corollary 6.4 $(S_{K},s_{K})$ is equivalent to
$(\mathbb{A}^{d}/G,[0])$ for some $G\subset\mathrm{GL}_{d}(K)$, such that $G$
contains no pseudoreflexions and has order prime to $\operatorname{char}k$.
Denote by $\overline{G}$ the image of $G$ in $\mathrm{PGL}_{d}(K)$, and by
$H\subseteq G$ the kernel of the composite
$G\longrightarrow\mathrm{GL}_{d}(K)\longrightarrow\mathrm{PGL}_{d}(K)$,
equivalently $H=G\cap K^{*}\subseteq\mathrm{GL}_{d}(K)$.

Denote by $\mathrm{I}_{\mathcal{G}},\mathrm{I}_{\mathcal{E}}$ the inertia
stacks of $\mathcal{G},\mathcal{E}$ respectively; we have an equivalence
$\mathrm{I}_{\mathcal{G},K}=G\times_{K}\mathcal{B}_{K}G$. Since
$\mathcal{E}=\mathbb{P}(\mathcal{N})=[\mathcal{N}\smallsetminus\\{0\\}/\mathbb{G}_{m}]$
and $\boldsymbol{\mu}_{n}=G\cap\mathbb{G}_{m}(K)\subset\mathrm{GL}_{d}(K)$,
the embedding $\boldsymbol{\mu}_{n}\subset\mathbb{G}_{m}$ gives us an
injective homomorphism

$\boldsymbol{\mu}_{n}\times_{k}\mathcal{E}\hookrightarrow\mathrm{I}_{\mathcal{E}}$

of relative groups over $\mathcal{E}$. Notice that
$\mathrm{I}_{\mathcal{G}}\to\mathcal{G}$ is a finite étale covering, and that
$\boldsymbol{\mu}_{n}\times_{k}\mathcal{E}\to\mathcal{G}$ is proper with Stein
factorization
$\boldsymbol{\mu}_{n}\times_{k}\mathcal{E}\to\boldsymbol{\mu}_{n}\times\mathcal{G}\to\mathcal{G}$.
Thus, the composite
$\boldsymbol{\mu}_{n}\times_{k}\mathcal{E}\to\mathrm{I}_{\mathcal{E}}\to\mathrm{I}_{\mathcal{G}}$
induces an injective homomorphism

$\boldsymbol{\mu}_{n}\times_{k}\mathcal{G}\hookrightarrow\mathrm{I}_{\mathcal{G}}$

of relative groups over $\mathcal{G}$ whose base change to $K$ is the natural
embedding

$\boldsymbol{\mu}_{n}\times_{K}\mathcal{B}_{K}G\hookrightarrow\mathrm{I}_{\mathcal{G},K}=G\times_{K}\mathcal{B}_{K}G\,.$

Denote by
$\overline{\mathcal{E}}=\mathcal{E}\mathbin{\\!\\!\boldsymbol{\fatslash}}\boldsymbol{\mu}_{n}$,
$\overline{\mathcal{G}}=\mathcal{G}\mathbin{\\!\\!\boldsymbol{\fatslash}}\boldsymbol{\mu}_{n}$
the _rigidifications_ (see for example [AGV08, Appendix C]). We obtain a
commutative diagram

${\mathcal{E}}$${\overline{\mathcal{E}}}$${E}$${\mathcal{G}}$${\overline{\mathcal{G}}}$${\operatorname{Spec}k}$

whose left hand square is cartesian, which pulls back to the diagram

${{[\mathbb{P}_{K}^{d-1}/G]}}$${{[\mathbb{P}_{K}^{d-1}/\overline{G}]}}$${\mathbb{P}_{K}^{d-1}/G}$${\mathcal{B}_{K}G}$${\mathcal{B}_{K}\overline{G}}$${\operatorname{Spec}K\,.}$

Since $\overline{\mathcal{E}}_{K}\simeq[\mathbb{P}_{K}^{d-1}/\overline{G}]$ is
smooth, the morphism $\overline{\mathcal{E}}\to E$ lifts to a morphism to the
minimal stack $\overline{\mathcal{E}}\to\widehat{E}$.

######  Definition 6.8.

We say that $E$ is the _associated variety_ of the singularity.

Recall that $\mathcal{B}$ is the blow up of $\mathcal{G}$ in $\widehat{S}$,
let $B\to S$ be its coarse moduli space. Since $\mathcal{B}$ is a tame stack
and formation of coarse moduli spaces commutes with base change for tame
stacks, the reduced fiber of $B\to S$ over $s$ is $E$.

######  Corollary 6.9.

There exists a rational morphism $E\dashrightarrow\mathcal{G}$.

###### Proof.

The morphism $\mathcal{B}\to B$ is birational. Since $\mathcal{B}$ is smooth,
then $B$ is normal, in particular it is regular at the generic point of $E$.
It follows that the morphism $\operatorname{Spec}k(E)\to B$ lifts to a
morphism $\operatorname{Spec}k(E)\to\mathcal{B}$ by our version of the
Lang–Nishimura theorem. Clearly, the composite
$\operatorname{Spec}k(E)\to\mathcal{B}\to\widehat{S}$ factors through
$\mathcal{G}\to\widehat{S}$. ∎

Given an irreducible algebraic space $X$ of finite type over a field $k$, we
say that a field extension $k^{\prime}/k$ _splits_ $X$ if there exists a
dominant rational map $Y\dashrightarrow X_{k^{\prime}}$ where $Y$ is an
integral scheme of finite type over $k^{\prime}$ with a smooth
$k^{\prime}$-rational point. If $k$ splits $X$, then we say that $X$ is split.
If $X$ has tame quotient singularities, using our version of the
Lang–Nishimura theorem we see that $k^{\prime}$ splits $X$ if and only if the
minimal stack $\widehat{X}$ has a $k^{\prime}$-rational point.

######  Proposition 6.10.

A field extension $k^{\prime}/k$ splits $E$ if and only if it splits
$\mathcal{G}$.

###### Proof.

If $k^{\prime}$ splits $\mathcal{G}$, by our version of the Lang–Nishimura
theorem there exists a morphism $\operatorname{Spec}k^{\prime}\to\mathcal{E}$,
the composition
$\operatorname{Spec}k^{\prime}\to\mathcal{E}\to\overline{\mathcal{E}}\to\widehat{E}$
shows that $k^{\prime}$ splits $E$. If $k^{\prime}$ splits $E$, since there
exists a rational map $E\dashrightarrow\mathcal{G}$ our version of the
Lang–Nishimura theorem implies that $k^{\prime}$ splits $\mathcal{G}$. ∎

###  6.5. $R$-singularities

If $k$ and $k^{\prime}$ are fields with the same characteristic, $(S,s)$ is a
singularity over $k$ and $(S^{\prime},s^{\prime})$ is a singularity over
$k^{\prime}$, we say that $(S,s)$ and $(S^{\prime},s^{\prime})$ are _stably
equivalent_ if there exists a common extension $k\subseteq K$ and
$k^{\prime}\subseteq K$, such that $(S,s)_{K}$ and
$(S^{\prime},s^{\prime})_{K}$ are equivalent. It is easily checked that this
is an equivalence relation on tame quotient singularities.

######  Definition 6.11.

An _$R$ -singularity_ is a tame quotient singularity, such that every
singularity that is stably equivalent to it is liftable.

From the definition, it is not clear that there are any non-trivial examples
of $R$-singularities.

###  6.6. $R_{d}$ groups

######  Definition 6.12.

Let $d$ be a positive integer, $p$ be either $0$ or a prime, and $G$ be a
finite group whose order is not divisible by $p$. We say that $G$ is an
_$R_{d}$ group in characteristic $p$_ if for every field $K$ of characteristic
$p$ and every faithful $d$-dimensional representation
$G\subseteq\mathrm{GL}_{d}(K)$, the singularity $(\mathbb{A}_{K}^{d}/G,[0])$
is an $R$-singularity.

If $G$ is $R_{d}$ in all characteristics not dividing the order of $G$, we say
that $G$ is _an $R_{d}$ group_, or simply that $G$ is $R_{d}$.

Another way of stating this is the following.

######  Definition 6.13.

Let $G$ be a finite group, $(S,s)$ a singularity over a field $k$ whose
characteristic does not divide the order of $G$. We say that $(S,s)$ is a _$G$
-singularity_ if there exists a field $K$ and a faithful $d$-dimensional
representation $G\subseteq\mathrm{GL}_{d}(K)$ such that $(S,s)$ is stably
equivalent to $\mathbb{A}^{d}_{K}/G$.

Then $G$ is an $R_{d}$ group if and only if every $G$-singularity is liftable.

######  Remarks 6.14.

  1. (1)

As a point of terminology, we notice that if $G$ is not a subgroup of
$\mathrm{GL}_{d}(K)$ for any field $K$ (for example, if $G$ contains an
abelian subgroup of rank larger than $d$), then it is vacuously an $R_{d}$
group. Such a group can not act faithfully on a $d$-dimensional variety with a
smooth fixed point, so it will not actually appear in the statement of Theorem
6.16. So, we are actually only interested in $R_{d}$ groups that are subgroups
of some $\mathrm{GL}_{d}(K)$.

  2. (2)

Every finite group is trivially $R_{1}$.

  3. (3)

In the definition of an $R_{d}$ group, we may assume that $K$ is algebraically
closed.

######  Lemma 6.15.

Let $\mathcal{X}$ be a geometrically integral tame Deligne–Mumford stack of
dimension $d$ over a field $k$ with finite inertia and moduli space
$\mathcal{X}\longrightarrow M$. Assume that $\mathcal{X}\longrightarrow M$ is
a birational isomorphism. Let $\xi\in\mathcal{X}(\overline{k})$ be a smooth
geometric point with image $p\in M$.

If the automorphism group $\operatorname{Aut}_{\overline{k}}\xi$ is an $R_{d}$
group in $\operatorname{char}k$, then $(M,p)$ is an $R$-singularity. In
particular, $p\in M$ lifts to a $k(p)$-rational point of $\mathcal{X}$.

###### Proof.

This is a direct consequence of Lemma 6.3. ∎

The point of this definition is the following result. Let us put ourselves in
the situation of subsection 5.3: $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$
is a category of pointed structured spaces,
$(X,p,\xi)\in\mathcal{M}(\overline{k})$ an algebraic object, as in Theorem
5.4.

######  Theorem 6.16.

Let $\mathcal{M}\longrightarrow(\mathrm{Aff}/k)$ be a category of pointed
structured spaces, $(X,p,\xi)\in\mathcal{M}(\overline{k})$ an algebraic object
with $X$ integral of dimension $d$. Assume that the automorphism group scheme
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}(X,p)$ is finite,
tame and reduced. If
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\overline{k}}(X,p,\xi)$ is an
$R_{d}$ group, then $(X,p,\xi)$ is defined over its field of moduli.

###### Proof.

We apply Lemma 6.15 to the stack $\mathcal{X}_{(X,p,\xi)}$, with moduli space
$\mathbf{X}_{(X,p,\xi)}$, to conclude that the space $\mathbf{X}_{(X,p,\xi)}$
has an $R$-singularity at the rational point corresponding to $p$. Then if
$Z\longrightarrow\mathbf{X}_{(X,p,\xi)}$ is a resolution of singularities we
see that $Z$ has a regular rational point, so the conclusion follows from
Theorem 5.4. ∎

We still have to give meaningful examples of $R_{d}$ groups.

A fairly trivial class of $R_{d}$ groups is the following: say that a group is
_strongly $R_{d}$_ if for any embedding $G\subseteq\mathrm{GL}_{d}(K)$, where
$K$ is an algebraically closed field of characteristic not dividing $|G|$, we
have that $G$ is generated by pseudoreflexions in $\mathrm{GL}_{d}(K)$. By the
Chevalley–Shephard–Todd theorem we have that $\mathbb{A}^{d}/G$ is smooth,
hence every strongly $R_{d}$ group is also $R_{d}$.

The following is straightforward.

######  Proposition 6.17.

  1. (1)

If $m$ is a positive integer, $\mathrm{C}_{m}^{d}$ is strongly $R_{d}$.

  2. (2)

Dihedral groups are strongly $R_{2}$.

To give examples of finite groups that are $R_{d}$ without being strongly
$R_{d}$ is more complicated, and requires much more technology. Here are two
results in this direction.

######  Theorem 6.18.

Let $G$ a finite group with the following properties.

  1. (1)

The center of $G$ is trivial.

  2. (2)

The projection $\operatorname{Aut}G\longrightarrow\operatorname{Out}G$ is
split.

  3. (3)

Either $G$ is perfect, or all proper normal subgroups of $G$ are perfect.

Then $G$ is $R_{d}$ for all $d$.

For example, these conditions are satisfied for all symmetric groups
$\mathrm{S}_{n}$ with $n\geq 3$, $n\neq 6$. Also, there are infinitely many
classes of simple groups such that the projection
$\operatorname{Aut}G\longrightarrow\operatorname{Out}G$ is split; a complete
classification is given in [LMM03].

While this result is interesting, it does not give any new examples of
applications of Theorem 6.16, because of Proposition 4.2. The next result,
however, does yield new examples.

######  Theorem 6.19.

A group of order prime to $d!$ is $R_{d}$.

The following easy result allows us to give more examples of $R_{d}$ groups.

######  Proposition 6.20.

Let $G$ be a finite group, $H\subseteq G$ a normal subgroup. If $H$ is
strongly $R_{d}$ and $G/H$ is $R_{d}$, then $G$ is $R_{d}$.

Thus, for example, a product of $d$ cyclic groups
$\mathrm{C}_{mr_{1}}\times\dots\times\mathrm{C}_{mr_{d}}$, where $r_{1}$, …,
$r_{d}$ are prime to $d!$, is an $R_{d}$ group.

Notice that subgroups and quotients of $R_{d}$ groups are not necessarily
$R_{d}$: for example, $\mathrm{C}_{2}\times\mathrm{C}_{2}$ is $R_{2}$, but
$\mathrm{C}_{2}$ is not. Furthermore, the product of two $R_{d}$ groups is not
necessarily $R_{d}$. A counterexample will appear in a forthcoming paper by
the first author [Bre].

By putting together Theorem 5.4, Lemma 5.9 and Theorem 6.19, we obtain the
following.

######  Theorem 6.21.

Let $\mathcal{M}$ be a category of pointed structured spaces,
$(X,p,\xi)\in\mathcal{M}(\overline{k})$ a tame object such that $X$ is
integral of dimension $d$. If
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X,p,\xi)$ is étale of degree prime
to $d!$, then $(X,p,\xi)$ is defined over its field of moduli.

###### Proof.

The rational point $\mathbf{p}$ of the compression $\mathbf{X}_{(X,p,\xi)}$
given by Lemma 5.9 is a tame quotient singularity whose fundamental group has
degree prime to $d!$ by hypothesis, hence
$(\mathbf{X}_{(X,p,\xi)},\mathbf{p})$ is liftable by Theorem 6.19. We conclude
by applying Theorem 5.4. ∎

###  6.7. The proofs of Theorems 6.18 and 6.19

Let $G\subseteq\mathrm{GL}_{d}(K)$, where $K$ is an algebraically closed field
of characteristic prime to $|G|$, where $G$ satisfies the hypotheses of one of
the theorems; and let $(S,s)$ a singularity over a field $k$ that is stably
equivalent to $(\mathbb{A}^{d}_{K}/G,[0])$. By extending $K$ we may assume
that $k\subseteq K$, and $(S,s)_{K}$ is equivalent to
$(\mathbb{A}^{d}_{K}/[0])$; under these hypotheses we need to show that
$(S,s)$ is liftable.

Let $P\subset G$ be the subgroup generated by pseudoreflexions. Under the
hypothesis of Theorem 6.18, $P$ is trivial, since either $P$ or $G$ is
perfect, so that the composite $P\subseteq
G\to\mathrm{GL}_{d}(K)\xrightarrow{\det}K^{*}$ is trivial. Under the
hypothesis of Theorem 6.19, by Lemma 6.3 applied to $[\mathbb{A}_{K}^{d}/G]$
we have that the fundamental group of $(S,s)$ is a quotient of $G$ and hence
it is abelian of order prime to $d!$. We may thus replace $G$ with the
fundamental group of $(S,s)$ and assume that $P$ is trivial by Corollary 6.4.
Hence, in both cases we may assume that $G$ is the fundamental group of
$(S,s)$.

Let $\mathcal{G}$ be the fundamental gerbe of $(S,s)$; we have
$\mathcal{G}_{K}\simeq\mathcal{B}_{K}G$.

#### The proof of Theorem 6.18

The gerbe $\mathcal{G}$ satisfies the hypotheses of Lemma 4.4, hence
$\mathcal{G}$ is neutral and $(S,s)$ is liftable.

#### The proof of Theorem 6.19

First of all, we may assume that $G$ is abelian, because of the following
elementary lemma, which was pointed out to us by János Kollár.

######  Lemma 6.22.

Let $K$ be a field, $d$ a positive integer, and $G\subseteq\mathrm{GL}_{d}(K)$
a finite subgroup whose order is not divisible by $\operatorname{char}K$ and
prime to $d!$. Then $G$ is abelian.

###### Proof.

We can assume that $K$ is algebraically closed. Since $\operatorname{char}K$
does not divide $|G|$ we have that $K^{d}$ decomposes as a sum of irreducible
representations. But the degree of any irreducible representation divides
$|G|$: this is standard in characteristic $0$, and follows from [Ser77, §15.5]
if $\operatorname{char}K>0$. So $K^{d}$ decomposes as a sum of $1$-dimensional
representations, and we obtain the result. ∎

Since $G$ is abelian, then $\mathcal{G}$ is associated to a cohomology class
$c\in\operatorname{H}^{2}(k,G^{\prime})$, where $G^{\prime}$ is a twisted form
of $G$ over $k$. Because of this, it is enough to prove that there exists a
finite field extension $k^{\prime}/k$ of degree prime with $|G|$ which splits
$\mathcal{G}$: this would imply that
$[k^{\prime}:k]c\in\operatorname{H}^{2}(k,G^{\prime})$ is trivial and hence
that $c$ is trivial, too.

Let us proceed by induction on $d$, starting from the case $d=1$, which is
trivial. Assume that the theorem holds in dimension $d-1$. Let $E$ be its
associated variety; since $E_{K}\simeq\mathbb{P}^{d-1}/\overline{G}$, we have
that $E$ has liftable singularities by the inductive hypothesis. Thanks to
Proposition 6.10 it is enough to find a finite field extension $k^{\prime}/k$
of degree prime with $|G|$ and such that $E(k^{\prime})\neq\emptyset$.

There exists a finite extension $k_{1}/k$ such that $\mathcal{G}_{k_{1}}$ is
isomorphic to $\mathcal{B}_{k_{1}}G$, and that the characters of $G$ are
defined over $k_{1}$. The pullback of $\mathcal{N}$ to $\mathcal{B}_{k_{1}}G$
corresponds to a $d$-dimensional representation $V$ of $G$, with an eigenspace
decomposition $V=\bigoplus_{\chi\in\widehat{G}}V_{\chi}$, where $\widehat{G}$
denotes the group of characters of $G$.

Define a functor
$\Gamma\colon(\mathrm{Aff}/k)^{\mathrm{op}}\longrightarrow(\mathrm{Set})$ as
follows. If $T$ is a $k$-scheme, then $\Gamma(T)$ is the set of subbundles
$\mathcal{M}\subseteq\mathcal{N}_{T}\longrightarrow\mathcal{G}_{T}$, with the
property that there exist an fppf cover $\\{\phi_{i}\colon
T_{i}\longrightarrow T\\}$ and morphisms $\psi_{i}\colon
T_{i}\longrightarrow\operatorname{Spec}k_{1}$, such that for each $i$ there
exists a $\chi\in\widehat{G}$ such that $V_{\chi}\neq~0$ and
$\phi_{i}^{*}\mathcal{M}=\psi_{i}^{*}V_{\chi}$ in
$\phi_{i}^{*}\mathcal{N}=\psi_{i}^{*}V$.

Clearly, $\Gamma$ is an fppf sheaf. The pullback
$\Gamma_{k_{1}}\colon(\mathrm{Aff}/k_{1})^{\mathrm{op}}\longrightarrow(\mathrm{Set})$
is easily checked to be represented by the disjoint union of copies of
$\operatorname{Spec}k_{1}$, one for each $\chi$ for which $V_{\chi}\neq 0$;
this implies that $\Gamma$ is represented by a finite étale scheme over $k$,
of degree at most $d$, because there are at most $d$ characters $\chi$ with
$V_{\chi}\neq~\emptyset$. So there exists a finite extension $k^{\prime}/k$ of
degree at most $d$, hence prime with $|G|$, such that
$\Gamma(k^{\prime})\neq\emptyset$.

After replacing $k$ with $k^{\prime}$, we can assume that there exists a non-
zero subbundle $\mathcal{M}\subseteq\mathcal{N}$ whose pullback to
$\mathcal{G}_{K}=\mathcal{B}_{K}G$ is $V_{\chi}\neq 0$. Consider the
projective subbundle
$\mathbb{P}(\mathcal{M})\subseteq\mathbb{P}(\mathcal{N})=\mathcal{E}$, which
descends to a projective subbundle
$\mathcal{P}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\mathbb{P}(\mathcal{M})\mathbin{\\!\\!\boldsymbol{\fatslash}}H_{1}\subseteq\mathbb{P}(\mathcal{N})\mathbin{\\!\\!\boldsymbol{\fatslash}}H_{1}=\overline{\mathcal{E}}$.
Calling $P$ the moduli space of $\mathcal{P}$, we have $P\subseteq E$;
extending the scalars to $k_{1}$ we see that
$P_{k_{1}}=\mathbb{P}(V_{\chi})/G=\mathbb{P}(V_{\chi})$, since the action of
$G$ on $\mathbb{P}(V_{\chi})$ is trivial. Hence $P$ is a Brauer–Severi variety
of dimension at most $d-1$, and it has index at most $d$. This means that
there exists a finite extension $k^{\prime}/k$ of degree at most $d$ such that
$P(k^{\prime})\neq~\emptyset$. Then $E(k^{\prime})\neq\emptyset$, and we
conclude.

###  6.8. Isolated $\mathrm{C}_{2}$-singularities in odd dimension

The proof of Theorem 6.19 can be adapted to prove that singularities of type
$\mathbb{A}^{d}/G$ are $R$-singularities for many cases that are not covered
in the statement of the theorem. Here we give just one example, that has an
interesting application.

######  Theorem 6.23.

Let $n$ be a positive integer which is not divisible by
$\operatorname{char}K$, and consider the standard action of
$\boldsymbol{\mu}_{n}$ on $\mathbb{A}^{d}_{K}$ by multiplication. If $n$ and
$d$ are relatively prime, then $\mathbb{A}_{K}^{d}/\boldsymbol{\mu}_{n}$ is an
$R$-singularity.

If $d$ is odd, $\operatorname{char}K\neq 2$, $G\subseteq\mathrm{GL}_{d}(K)$ is
a finite subgroup of order $2$, and $\mathbb{A}^{d}_{K}/G$ has an isolated
singularity, then $G=\boldsymbol{\mu}_{2}{\rm Id}\subset\mathrm{GL}_{d}(K)$.
So from the theorem we get the following.

######  Corollary 6.24.

An isolated, odd-dimensional $\mathrm{C}_{2}$-singularity is an
$R$-singularity.

Plugging this into our main result we get the following.

######  Corollary 6.25.

Let $\mathcal{M}$ be a category of pointed structured spaces, $(X,p,\xi)$ a
tame object of $\mathcal{M}(\overline{k})$, such that $X$ is $d$-dimensional
and integral. Assume that the automorphism group of $(X,p,\xi)$ is cyclic of
order $2$, and that $p$ is an isolated fixed point for its action on $X$. Then
if $d$ is odd, $(X,p,\xi)$ is defined over its field of moduli.

Thus, for example, we get that an odd dimensional abelian variety $A$ with
automorphism group as small as possible, that is, cyclic of order $2$, is
defined over its field of moduli, recovering in particular Shimura’s result on
odd-dimensional generic abelian varieties that has already been mentioned (see
[Shi72]).

###### Proof of Theorem 6.23.

Under the hypotheses of the Theorem, the associated variety $E$ is a
Brauer–Severi variety of dimension $d-1$, hence it is split by a finite
extension $k^{\prime}$ of $k$ of degree dividing $d$. By Proposition 6.10,
$k^{\prime}$ splits the fundamental gerbe $\mathcal{G}$. The result follows
from the fact that $\boldsymbol{\mu}_{n}$ is abelian of degree prime to
$[k^{\prime}:k]$. ∎

If $d$ is a positive integer, what finite groups have the property that if $G$
acts linearly on $\mathbb{A}^{d}_{K}$, where $K$ is algebraically closed, with
characteristic not dividing $|G|$, and $(\mathbb{A}^{d}_{K})^{G}=\\{0\\}$, we
have that $\mathbb{A}^{d}_{K}/G$ has an $R$-singularity at the origin, without
being $R_{d}$? We do not have any other example.

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