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Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then $$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq 2.$$ For instance for $g = 1, n = 2$ wa have that $\overline{M}_{1,2}$ is a rational surface with four singular points.

Does there exist any value of $g$ and $n$ for which $Sing(\overline{M}_{g,n})$ is in codimension at least $3$ ?

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No. In fact more is true: the locus of all $n$-pointed curves of genus $g-1$ with a single elliptic tail $E$, such that $\mathrm{Aut}(E)=\mathbf Z/6$, has codimension two in $\overline M_{g,n}$ and consists of noncanonical singularities. This was famously determined by Harris and Mumford in their paper on the Kodaira dimension of the moduli space of curves (they work with $n=0$ but this makes no difference).

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  • $\begingroup$ I guess that the locus of genus $g-1$ curves with an elliptic tail $E$ such that $Aut(E) = \mathbb{Z}/4$ is singular and in codimension $2$ as well. Let us assume that $g+n\geq 4$. Do you know if in this case these two loci are exactly the codimension $2$ part of $Sing(\overline{M}_{g,n})$ or if there is something else? $\endgroup$
    – Puzzled
    Commented Jan 7, 2014 at 16:05
  • $\begingroup$ That sounds about right. The point is that $\overline M_{g,n}$ is smooth away from the loci of curves with automorphism. For $(g,n)$ not "too small" the only codimension one locus of curves with automorphisms is the locus of elliptic tails -- but an elliptic tail involution does not produce a singularity -- and then again for $(g,n)$ not too small, the only codimension two loci are the ones where the elliptic tail has $j$-invariant $0$ or $1728$. There is of course something that needs to be checked by hand to see that $g+n \geq 4$ suffices. $\endgroup$
    – Dan Petersen
    Commented Jan 8, 2014 at 8:05

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