Let $X$ be an integral scheme over a field. Let $G$ be a finite group acting on $X$ faithfully. Assume the quotient stack $[X/G]$ is separated (e.g., when $G$ acts on $X$ properly). Then $[X/G]$ is a separated Deligne-Mumford (DM) stack and there is a coarse moduli space $$\pi:[X/G] \to X/G.$$ Is $\pi$ always a birational morphism of DM stacks?

## 1 Answer

Yes. For each nontrivial element $g\in G$, the fixed points form a closed set, which must not contain the whole space as then $g$ would act trivially (by reducedness). The complementary open set thus contains the generic point.

The (nonempty, by irreducibility) intersection of these open sets over all nontrivial $g\in G$ forms an open set $U$ which is $G$-invariant. Restricted to $U$, the action of $G$ is free. Thus, the image of $U$ in the stack $[X/G]$ is the quotient $U/G$, which is an algebraic space, and so the image of $U/G$ in the algebraic space $X/G$ is again isomorphic to $U/G$.

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