Seminar

Mahrud Sayrafi: "Splitting of vector bundles on toric varieties"

Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.

Feiyang Lin: "Finding special line bundles on special tetragonal curves"

Abstract: There is a canonical way to associate to a degree 4 cover of P^1 two vector bundles E and F, which give rise to a stratification of the Hurwitz space H_{4,g}. It is natural to ask whether the Brill-Noether theory of tetragonal curves is controlled by this data. I will describe a procedure for producing a particular line bundle on tetragonal covers in special strata, which is expected to be special in the Hurwitz-Brill-Noether sense. The main technique is the realization of an inflation of vector bundles on P^1 as a blow-up and blow-down of the associated projective bundle.