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 ℓ-torsion line bundle. They show that for ℓ ≤ 6 and ℓ = 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 ℓ = 2, and by Chiodo, Eisenbud, Farkas and Schreyer for ℓ = 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 ⊗ℓ ∼ = ω ⊗k C . New loci of canonical and non-canonical singularities appear for any k ∈ ℓZ and ℓ > 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 ℓ we give an explicit description.