Low Codimension Strata of the Singular Locus of Moduli of Level Curves

TL;DR

This paper investigates the structure of the singular locus in the moduli space of stable curves with various level structures, revealing codimension patterns and new components for different levels.

Contribution

It extends Harris and Mumford's analysis by identifying new codimension components in the singular locus for levels 5, 7, and beyond, including exceptional cases.

Findings

Codimension 2 singularities for levels 2, 3, 4, and 6

Existence of codimension 3 components for levels 5 and ≥7 (except 12)

Existence of a codimension 4 component at level 12

Abstract

We further analyze the moduli space of stable curves with level structure provided by Chiodo and Farkas in \cite{AA}. Their result builds upon Harris and Mumford analysis of the locus of singularities of the moduli space of curves and shows in particular that for levels 2, 3, 4, and 6 the locus of noncanonical singularities is completely analogous to the locus described by Harris and Mumford, it has codimension 2 and arises from the involution of elliptic tails carrying a trivial level structure. For the remaining levels (5, 7, and beyond), the picture also involves components of higher codimension. We show that there exists a component of codimension 3 for levels $\ell=5$ and $\ell\geqslant 7$ with the only exception of level 12. We also show that there exists a component of codimension 4 for $\ell=12$.