Let $\overline{\mathcal{M}}_g$ be the moduli stack of genus $g$ stable (nodal) curves and let $\overline{M}_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of curves of given genus", Deligne and Mumford constructed and showed that $\overline{\mathcal{M}}_g$ is a smooth Deligne-Mumford stack, and proved that $\overline{M}_g$ is an irreducible projective variety. Although $\overline{\mathcal{M}}_g$ is smooth, $\overline{M}_g$ may have finite quotient singularity.

**When is $\overline{M}_g$ smooth?**

The simplest case is $\overline{M}_0$, which is a point and hence smooth by definition. My question is then equivalent to the following: for which $g>0$ is $\overline{M}_g$ smooth? Is it never smooth?

**Edit 1:**

I forgot to mention the base scheme is $\text{Spec }\mathbb{C}$.

The singularities of the modulus space, Bull. Amer. Math. Soc. 68 (1962), 390–394. $\endgroup$4more comments