When is the coarse moduli space of genus $g$ stable curves smooth?

Question

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 $\text{Spec }\mathbb{C}$.

Answer 1

A more detailed description of the singular locus of $\mathrm{M}_g$ is as follows.

Theorem. Let $\mathrm{C}$ be a smooth curve of genus $g$.

If $g=2$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}_2$ if and only if $\mathrm{C}$ is given by $y^2=x^6-x$.

If $g=3$ and $\mathrm{C}$ is not hyperelliptic (resp. hyperelliptic), then $[\mathrm{C}]$ is a singular point of $\mathrm{M}_3$ if and only if $\mathrm{Aut(C)}$ is nontrivial (resp. $\mathrm{Aut(C)}$ is not $\mathbf{Z}/2\mathbf{Z}$).

If $g\geqslant 4$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}_g$ if and only if $\mathrm{Aut(C)}$ is nontrivial.

An algebro-geometric reference for this result is this paper by H. Popp.

Answer 2

The question is mostly answered by abx's comments. Here I'm posting a self-contained answer so that it will be helpful for the others.

Case 1: $g=0$

$M_0 = \overline{M}_0$ is just a point and is smooth, as already mentioned in the question.

Case 2: $g=1$
Now there is a subtlety. Actually $\overline{\mathcal{M}}_1$ is not a Deligne-Mumford stack because every elliptic curve has an infinite group of automorphisms, i.e., translations. It's more meaningful to talk about $\mathcal{M}_{1,1}$, the moduli stack of curves with genus $g=1$ and one marked point (we don't want to forget the group structure of elliptic curves, which has the marked point as the identity) and its compactification $\overline{\mathcal{M}}_{1,1}$. We know the coarse moduli space $\overline{M}_{1,1}$ is isomorphic to the weighted projective line $\mathbb{P}(4,6)$ and hence is smooth.

Case 3: $g\geq2$

By Theorem 1 in the paper "The singularities of the modulus space" by Rauch in 1962, we know $M_g$ must have some point of singularity and hence its compactification $\overline{M}_g$ is singular.