Singularities of Moduli of Curves with a Universal Root

Abstract

In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an $\ell$-torsion line bundle. They show that for $\ell \le 6$ and $\ell \ne 5$ pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell =2$, and by Chiodo, Eisenbud, Farkas and Schreyer for $\ell =3$. Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves $C$ with a line bundle $L$ such that $L^{\otimes \ell }\cong {\omega }_C^{\otimes k}$. New loci of canonical and non-canonical singularities appear for any $k\notin \ell \mathbb{Z}$ and $\ell >2$, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graphs. We characterize the locus of non-canonical singularities, and for small values of $\ell$ we give an explicit description.