Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford 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.


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? $\endgroup$ – user43198 May 15 '18 at 8:14
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    $\begingroup$ 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. $\endgroup$ – Sasha May 15 '18 at 8:23
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    $\begingroup$ If $\mathcal{X}$ is one-dimensional, 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). $\endgroup$ – Ariyan Javanpeykar May 15 '18 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? $\endgroup$ – user43198 May 15 '18 at 8:27
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    $\begingroup$ @user43198: en.wikipedia.org/wiki/… $\endgroup$ – Sasha May 15 '18 at 9:05

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