Singularities of Moduli of Curves with a Universal Root

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 $\ell≤ 6$ and $\ell\neq 5$ 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 $L^\xx\ell\cong\omega_C^{\xx k}$. New loci of canonical and non-canonical singularities appear for any $k\not\in\ell\Z$ 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.

2010 Mathematics Subject Classification: 14H10; 14H60; 14H20.

Keywords and Phrases:

Full text: dvi.gz 145 k, dvi 407 k, ps.gz 409 k, pdf 439 k.

Home Page of DOCUMENTA MATHEMATICA