# Birational morphisms from DM stacks to their coarse moduli spaces

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?

• What if $X$ has multiple components, with different elements of $G$ acting trivially on each? Jun 26 at 15:47
• @WillSawin Thanks for the comment. I edited the question to avoid that case. Jun 26 at 16:07
• I think we're still in trouble if we just glue the components together. (Say two $\mathbb P^1$s joined at a point, and the group generated by two involutions of the separate $\mathbb P^1$s that each fix the other one, and thus that point.) Maybe you want to assume $X$ is irreducible? Jun 26 at 16:53
• @WillSawin That is still not enough. Consider the action of $\mu_\ell$ on $\text{Spec}\ k[\epsilon]/\langle \epsilon^{\ell} \rangle$ by $\zeta\bullet \epsilon = \zeta\epsilon$. This is a "faithful" action, but the map from the stack to the coarse moduli space is not birational. The correct hypothesis is that for every generic point of $X$, the induced action of $G$ on the residue field of the point is faithful. Jun 26 at 17:28
• Yes, I think. 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. The complementary open set thus contains the generic point. The 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 coarse moduli space of $X/G$ is again isomorphic to $U/G$. Jun 27 at 14:07

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 $$U/G$$.