Singularities of Moduli of Curves with a Universal Root
Mattia GaleottiInstitut de Mathematiques de Jussieu, UPMC, 4 Place Jussieu, 75005 Paris, France
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 -torsion line bundle. They show that for and 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 , and by Chiodo, Eisenbud, Farkas and Schreyer for . Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves with a line bundle such that . New loci of canonical and non-canonical singularities appear for any and , 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 we give an explicit description.
Cite this article
Mattia Galeotti, Singularities of Moduli of Curves with a Universal Root. Doc. Math. 22 (2017), pp. 1337–1373DOI 10.4171/DM/599