Singularities of moduli of curves with a universal root

TL;DR

This paper systematically analyzes the singularities of moduli spaces of curves with universal roots of line bundles, providing combinatorial tools to describe singular loci and characterizing non-canonical singularities.

Contribution

It introduces a comprehensive framework for understanding singularities in moduli spaces of curves with roots of line bundles, extending previous work to new loci and explicit descriptions.

Findings

New loci of singularities identified for certain line bundle roots.

Complete combinatorial description of the singular locus in terms of dual graphs.

Explicit characterization of non-canonical singularities for small roots.

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 $\ell\leq 6$ and $\ell\neq 5$ pluricanonical forms extend over any desingularization. This allows to compute 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 $k\not\in\ell\mathbb{Z}$ and $\ell>2$, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graph. We characterize the locus of non-canonical singularities, and for small values of $\ell$ we give an explicit description.