Let $\mathcal{X}$ be a regular proper 1-dimensional Artin stack with finite diagonal, with coarse space morphism $\mathcal{X} \to X$.

Question: Is $X$ regular?

Some comments:

  1. I'm happy to assume that $\mathcal{X} \to X$ is birational.
  2. I'm happy to assume that $\mathcal{X}$ is over a field and to replace regular with smooth. (But I am also interested in the case where $X$ is Spec of a Dedekind domain.
  3. If $\mathcal{X}$ is additionally a Deligne-Mumford stack, or tame, then $X$ is smooth.
  4. Jack Hall pointed out to me that we can weaken tameness a bit: it is enough to assume that $\mathcal{X} \to X$ is an adequate moduli space, in the sense of Alper.

1 Answer 1


Jack Hall pointed out to me over email that a stack with a coarse space is adequate, so this does it.


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