**Venue: **Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Sala Seminari "-1"**At:** 11:00

**Speaker:**

**Mattia Francesco Galeotti**(UniTrento - Dipartimento di Matematica)

**Abstract:**

In the recent period, a series of papers carried out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an ℓ-torsion line bundle. 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.

We can generalize these works in two directions. At first 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}.

Furthermore, we treat moduli spaces of curves with a G-cover, where G is any finite group. In particular for G=S_{3} we prove that the moduli space is of general type for odd genus >11.

To analyze canonical and non-canonical singularities, we provide a set of combinatorial tools allowing the description of the singular locus via the dual graphs of curves.

**Contact person: **Claudio Fontanari