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:**

- What is the relationship between $Z$ and $G_{x}$, $k_0$ and $k_1$?
- 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?
- 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 $k_0=k'$?
- (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:**

- Is there any example where $Z$ is not reduced, or $k_0 \ne k_1$?
- 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?
- 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 $k$.)