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

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    $\begingroup$ $M_g$ is not smooth for $g>1$, because of the presence of curves with extra automorphisms. $\endgroup$ – abx Nov 22 '19 at 7:25
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    $\begingroup$ Yes. This goes back to H. Rauch, The singularities of the modulus space, Bull. Amer. Math. Soc. 68 (1962), 390–394. $\endgroup$ – abx Nov 22 '19 at 8:23
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    $\begingroup$ No, I am talking here about $M_g$, not $\overline{M}_g$. But that's fairly trivial: take a smooth curve with an automorphism fixing a point, and add an elliptic tail. $\endgroup$ – abx Nov 22 '19 at 9:53
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    $\begingroup$ This is obvious, just take hyperelliptic curves. $\endgroup$ – abx Nov 22 '19 at 14:21
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    $\begingroup$ Sorry, this is enough for me. Look at the reference I gave. $\endgroup$ – abx Nov 22 '19 at 16:05
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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.

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  • $\begingroup$ This is a better answer than mine. Can you add the case $g=1$ to make it a complete answer? $\endgroup$ – Yuhang Chen Dec 2 '19 at 17:16
  • $\begingroup$ @YuhangChen: we have $\mathrm{M}_1\simeq\mathbf{A}^1$ (in particular there are no singularities), but I think it is most natural to restrict the discussion to $g\geqslant 2$. $\endgroup$ – Simpleton Dec 4 '19 at 19:19
  • $\begingroup$ Yes, and $\overline{M}_1 \cong \mathbb{P}^1$, right? $\endgroup$ – Yuhang Chen Dec 5 '19 at 11:54
  • $\begingroup$ @YuhangChen: Sure $\endgroup$ – Simpleton Dec 7 '19 at 17:28
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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.

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