Singularities of the moduli space of level curves

Abstract

We describe the singular locus of the compactification of the moduli space ${\mathcal{R}}_{g,\ell }$ of curves of genus $g$ paired with an $\ell$-torsion point in their Jacobian. Generalising previous work for $\ell \le 2$, we also describe the sublocus of noncanonical singularities for any positive integer $\ell$. For $g$ ≥ 4 and $\ell$ = 3, 4, 6, this allows us to provide a lifting result on pluricanonical forms playing an essential role in the computation of the Kodaira dimension of ${\mathcal{R}}_{g,\ell }$: for those values of $\ell$, every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactified moduli space.