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:
- I'm happy to assume that $\mathcal{X} \to X$ is birational.
- 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.
- If $\mathcal{X}$ is additionally a Deligne-Mumford stack, or tame, then $X$ is smooth.
- 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.