Singularities of the moduli space of level curves

TL;DR

This paper characterizes the singularities of the compactified moduli space of genus g curves with l-torsion points, identifying noncanonical singularities and extending pluricanonical forms for certain l, aiding in understanding its geometric properties.

Contribution

It generalizes previous work on singularities of moduli spaces to include all positive integers l and describes the extension of pluricanonical forms for specific l values.

Findings

Identifies the singular locus of the compactified moduli space R_{g,l}.

Describes noncanonical singularities for all positive l.

Shows pluricanonical forms extend to desingularizations for g≥4 and l=3,4,6.

Abstract

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