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

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

• $M_g$ is not smooth for $g>1$, because of the presence of curves with extra automorphisms. – abx Nov 22 '19 at 7:25
• Yes. This goes back to H. Rauch, The singularities of the modulus space, Bull. Amer. Math. Soc. 68 (1962), 390–394. – abx Nov 22 '19 at 8:23
• 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. – abx Nov 22 '19 at 9:53
• This is obvious, just take hyperelliptic curves. – abx Nov 22 '19 at 14:21
• Sorry, this is enough for me. Look at the reference I gave. – abx Nov 22 '19 at 16:05

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.

• This is a better answer than mine. Can you add the case $g=1$ to make it a complete answer? – Yuhang Chen Dec 2 '19 at 17:16
• @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 $g\geqslant 2$. – Simpleton Dec 4 '19 at 19:19
• Yes, and $\overline{M}_1 \cong \mathbb{P}^1$, right? – Yuhang Chen Dec 5 '19 at 11:54
• @YuhangChen: Sure – Simpleton Dec 7 '19 at 17:28

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.