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

We further analyse the moduli space of stable curves with level structure provided by Chiodo and Farkas in [2]. 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 ℓ = 5 and ℓ > 7 with the only exception of level 12. We also show that there exists a component of codimension 4 for ℓ = 12