Jack Huizenga : Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles

Posted by Mihnea Popa , part of the Algebraic Geometry Seminar.

At

Dec. 5, 2012, 4 p.m.

In

SEO 427

Abstract

The Hilbert scheme of n points in the projective plane parameterizes zero-dimensional subschemes of length n. An interesting problem is to describe the birational geometry of this space, and give modular interpretations for its various birational models. A first step in this program is to determine the cone of effective divisors on the Hilbert scheme. We show the sections of many stable vector bundles satisfy a natural interpolation condition, and that these bundles always give rise to the edge of the effective cone. To do this, we give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that general ideal sheaves are always destabilized by exceptional bundles.