# Properties inherited by the coarse moduli space

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.
• Is there any known additional condition that we can impose on the stack to ensure that $X$ is regular? – user43198 May 15 '18 at 8:14
• 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). – Ariyan Javanpeykar May 15 '18 at 8:24