7/3/2013 Aula Beltrami Dipartimento di Matematica, Universita' di Pavia.

14.00-15.00 Gavril Farkas (Humboldt-Universitaet zu Berlin)

Syzygies of torsion bundles and the geometry of the level l modular variety over M_g.

Abstract: In joint work with Chiodo, Eisenbud and Schreyer, we formulate, and in some cases prove, three statements concerning the purity of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed as analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11.

I will also discuss the surprising failure of the Prym-Green Conjecture for genera which are powers of 2.

15.00-16.00 Alessandro Verra (Universita` Roma Tre)

New properties of A_5 revisiting the Prym map

Abstract: In the moduli space A_g of principally polarized abelian varieties of dimension g the locus N_g, parametrizing isomorphism classes of pairs (A,T) such that the theta divisor T of the ppav A has a non ordinary quadratic singularity, is interesting for several reasons.

The seminar is about the case of N_5, which was the first unknown one. Via the Prym map in genus six, and its ramification and antiramification divisors, N_5 will be completely described as the union of two irreducible and unirational components of codimension two.

Joint work with G. Farkas, S. Grushevskih, R. Salvati Manni.

16.30-17.30 Angela Ortega (Humboldt-Universitaet zu Berlin)

The Minimal Resolution Conjecture for curves and Ulrich bundles

Abstract: The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the minimal graded free resolution of a general set of points on X is as simple as the geometry of X allows. In the case of a curve X the MRC can be reformulated in terms of some cohomological vanishing conditions of the Lazarsfeld bundle associated to the embedding of the curve. We will show that the MRC holds for a general embedding of a curve with general moduli and for almost the full expected range. In the second part of the talk we will discuss how Lazarsfeld-Mukai bundles on K3 surfaces provide examples of Ulrich bundles, giving more evidence to the Eisenbud - Schreyer expectation that every variety possesses an Ulrich bundle of some rank. This is a joint work with Gavril Farkas and Marian Aprodu.

17.30-18.30 Rita Pardini (Universita`di Pisa)

Paracanonical systems of varieties of maximal Albanese dimension

Abstract: I will report on some recent joint work with M. Mendes Lopes and G.P. Pirola.

Let X be a smooth complex projective variety of irregularity q > 0, and let H be an irreducible family of effective divisors of X that dominates a component of the group Pic(X): given a divisor D algebraically equivalent to the elements of H, we give a cohomological criterion to ensure that D belong to H. By applying this criterion to the study of the main paracanonical system of a variety of general type with generically finite Albanese map, we are able to refine results due to Beauville in the case of surfaces and to Lazarsfeld and Popa in higher dimension. In particular, if the dimension of X is > 2 we obtain an unexpected inequality between the numerical invariants of X, under the assumption that X has generically finite Albanese map and does not have fibrations of a certain type.