Residual gerbes and coarse moduli space of a DM stack

Question

Let $X$ be a nice DM stack (Noetherian, separated), and $X_0$ its coarse moduli space which exist by the Keel-Mori theorem. (I like the exposition in D. Rydh. “Existence and properties of geometric quotients”, the map $X \to X_0$ is a separated universal homeomorphism).

$\newcommand{\Spec}{\mathop{Spec}}\newcommand{\red}{\mathop{red}}\newcommand{\Isom}{\mathop{Isom}}$ Let $x: \Spec k \to X$ be a point on $X$, it induces a point $\Spec k_0 \to X_0$ on $X_0$. I will call $k_0$ (the residue field of the point defined by $x_0$) the field of moduli of $x$.

Let $Z=X \times_{X_0} x_0$ denote the pullback. We can also look at the residual gerbe $G_{x} \to \Spec k_1$ associated to $x$.

Questions:

1. What is the relationship between $Z$ and $G_{x}$, $k_0$ and $k_1$?
2. The obstruction to $k_0$ being a field of definition should be given by some element in $H^2(k_0, \Isom(x,x))$, hence a gerbe. Is this gerbe related to the residual gerbe? In particular does $x$ descends to $k_0$ if and only if $G_x$ is neutral?
3. Some authors define the field of moduli $k'$ as the intersection of the fields of definition. Clearly $k_0 \subset k'$, and we need not have equality in general (cf Residual gerbe and field of moduli). On the other hand it seems folklore that $k_0=k'$ for moduli stacks of curves or abelian varieties. Is there a criterion for when $k_0=k'$?
4. (Meta): what are some good references on these topics?

Some partial answers (hoping I am not making any mistakes, this is far from my area of expertise):

Let $Z_0$ be the coarse moduli space of $Z$. The formation of coarse moduli only commute with flat base change in general, so we only have a morphism $g: Z_0 \to \Spec k_0$. Still by general properties (see for instance J. Alper. “Adequate moduli spaces and geometrically reductive group schemes”), $g$ is a universal homeomorphism so $Z_0$ is a point.

So $|Z|$ is a point, and $Z_{\red}$ is the residual gerbe at $x$ by the Stack Project: $G_{x}=Z_{\red}$. The residual point $\Spec k_1$ at the residual gerbe is the sheaf associated to $G_x$, this is a Noetherian reduced algebraic space with one point so indeed correspond to a field $k_1$. The sheaf associated to a gerbe is also its coarse moduli space, so by unicity we have that $Z_0=\Spec k_1$.

Hence we have a map $\Spec k_1 \to \Spec k_0$, which is a universal homeomorphism so $k_1/k_0$ is purely inseparable. Furthermore if $x$ is a tame point, ie $\Isom(x,x)$ has degree prime to the characteristic of $k$, then $X \to X_0$ is a tame/good moduli space étale locally around $x_0$, and good moduli spaces commute with pullback so in this case $k_0=k_1$. (References for this: D. Abramovich, M. Olsson, and A. Vistoli. “Tame stacks in positive characteristic”; J. Alper. “Good moduli spaces for Artin stacks”.)

Bonus questions:

1. Is there any example where $Z$ is not reduced, or $k_0 \ne k_1$?
2. By M. C. Olsson. “Hom-stacks and restriction of scalars”, Theorem 2.12, $X$ is étale locally around $x$ a quotient $[U/I_x]$ where $U$ is affine and $I_x=\Isom(x,x)$. By Luna's fundamental lemma (shrinking $U$ if necessary), we have an étale morphism of the coarse space $U//I_x$ to $X_0$, and $[U/I_x] \to X$ is strongly étale, ie is the pullback of the morphism $U//I_x \to X_0$. How much can we recover of the field of moduli $k_0$ and the residual gerbe $G_x \to \Spec k_1$ from the presentation $[U/I_x]$? The map $U//I_x \to X_0$ is étale but may not be Nisnevich so I guess we need the map $[U/I_x] \to X$ too to get the descent data?
3. In D. Abramovich, M. Olsson, and A. Vistoli. “Tame stacks in positive characteristic”, Appendix A, the authors show how to construct a rigidification $X/G$ of $X$ by a generic automorphism group $G$, and $X \to X/G$ is a gerbe (bound by $G$). (Think of some fine moduli stack of elliptic curves and take $G=\pm 1$.) They have the same coarse moduli (I hope!), so a point $x$ has the same field of moduli when seen in $X$ or $X/G$. Do they have the same fields of definitions? In other words, does a point $y: \Spec k \to X/G$ lift to $X$ over $k$? (It clearly lifts over an étale extension of $k$.)

Answer 1

As you say, things work smoothly for questions 1,2 if $x$ is tame. This is still true even if $X$ is not DM but it is tame in the sense of Abramovich-Olsson-Vistoli, see section $\S 3.2$ of my preprint with Vistoli https://arxiv.org/pdf/2210.04789.pdf .

For question 3, in general if you fix a base field the intersection of the fields of definition does not work well, even in super-nice settings, e.g. if the base field is $\mathbb{R}$ you can construct smooth projective curves of genus $\ge 2$ over $\mathbb{C}$ not defined over $\mathbb{R}$ but whose corresponding point in the coarse moduli space has residue field $\mathbb{R}$. Still, this is more or less the only thing that can go wrong (at least under mild assumptions).

Using your notation, assume that $k$ is the algebraic closure of $k_{0}$, and that $k_{0}$ is perfect. Furthermore, if the characteristic is $0$ assume that $\sqrt{-1}\in k_{0}$. By the Artin-Schreier theorem $\operatorname{Gal}(k/k_{0})$ is torsion free, hence it is topologically generated by closed subgroups of the form $\mathbb{Z}_{p}$ for some prime $p$. If $H\simeq \mathbb{Z}_{p}$ is such a subgroup, then $x$ is defined over the fixed field $k^{H}$ since $\mathbb{Z}_{p}$ has cohomological dimension $1$, see Proposition 4.3 of my preprint with Vistoli. It follows that $k_{0}$ is the intersection of the fields of definition.

For a comparison between the two notions of field of moduli, see Theorem 1.6.9 of Huggins' thesis https://arxiv.org/pdf/math/0610247.pdf .

Answer 2

The answer to the first bonus question is positive. To see this, let $k$ be any field of characteristic $p > 0$, and let $\mathbf{Z}/p$ act on $\mathbf{A}^2_k$ via $(x, y) \mapsto (x + y^p, y)$. I claim that the invariant ring $k[x,y]^{\mathbf{Z}/p}$ is generated by $s := xy^{p^2 - p} - x^p$ and $t := y$. To see this, note that $s$ and $t$ are clearly contained in the invariant ring, and $k[x,y]$ is free of rank $p$ over $k[s,t]$. Since $k[x,y]$ is of rank at least $p$ over $k[x,y]^{\mathbf{Z}/p}$ by Galois theory, the claim follows.

Note now that the induced map $k[s,t]/(t) \to k[x,y]/(y)$ is isomorphic to the inclusion $k[x^p] \subset k[x]$. In particular, the extension of fraction fields is purely inseparable, and the fiber over $0$ is $k[x]/(x^p)$. Consequently, the map $[\mathbf{A}^2_k/(\mathbf{Z}/p)] \to \mathbf{A}^2_k//(\mathbf{Z}/p)$ shows that $Z$ can fail to be reduced, and $k_0$ can fail to equal $k_1$.

The answer to the third bonus question is negative: just consider any nontrivial gerbe $\mathcal{X} \to \operatorname{Spec} k$ and note that $\operatorname{Spec} k$ is the space that Abramovich-Olsson-Vistoli attach to $\mathcal{X}$.