Let $\mathcal{X}$ be a regular, seperated DeligneMumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to find any reference.
1 Answer
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Not at all. Take for instance the quotient stack of $\mathbb{A}^2$ by $\pm 1$. The coarse moduli space is the quadratic cone.

$\begingroup$ Is there any known additional condition that we can impose on the stack to ensure that $X$ is regular? $\endgroup$ May 15, 2018 at 8:14

3$\begingroup$ If it is OK for you to think of the stack as of a global quotient stack by a finite group action, the condition is that the group is generated by complex reflections. $\endgroup$– SashaMay 15, 2018 at 8:23

1$\begingroup$ If $\mathcal{X}$ is onedimensional, then $X$ is regular. If the coarse space map $\mathcal{X}\to X$ is etale, then $X$ is regular (and $\mathcal{X}\to X$ is a gerbe). If the stacky locus of $\mathcal{X}$ is pure of codimension one, then $X$ might be regular (I don't remember the precise statement). $\endgroup$ May 15, 2018 at 8:24

$\begingroup$ @AriyanJavanpeykar I am new to the language of stacks. Could you suggest some reference where I can find your answer or Sasha's? $\endgroup$ May 15, 2018 at 8:27

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