# Residual gerbes and coarse moduli space of a DM stack

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 $$k$$.)
• Giulio Bresciani has a series of papers where he consider topics like this (some joint with Angelo Vistoli). I'd recommend taking a look. For example this paper arxiv.org/pdf/2210.04789.pdf Commented May 9, 2023 at 10:53

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 .

• Thanks a lot for this reference! This confirms that (with my notations), $G_x = Z_{\mathrm{red}}$. But rereading my question, I don't understand anymore my claim that $Z_0$, the coarse space associated to $Z$, is the same as $k_1$, the coarse space/residue point of the gerbe $G_x$. (Because the reduced structure is universal for morphisms into $Z$ while the coarse space is universal for morphisms from $Z$). If I understand your last paragraph of section 3.2 of your article, you make the same claim (with your notations, $k'$ is my $Z_0$). Commented May 12, 2023 at 16:08
• So I guess my question is: if $Z$ has a reduced coarse moduli space $Z_0$, is it true that the coarse space associate to $Z_{\mathrm{red}}$ is also $Z_0$? Commented May 12, 2023 at 16:09

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 $$\mathcal{X}$$.