Coherent systems and Brill-Noether theory.

Abstract

Let X be a curve of genus g. A coherent system on X consists of a pair (E; V ), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter a. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k = 1; 2; 3 and n = 2 explicitly. For small values of , the moduli spaces of coherent systems are related to the Brill-Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to nd the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill{Noether loci with k < 3.