Nonemptiness and smoothness of twisted Brill–Noether loci

Abstract

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that h0(C,V⊗E)≥k. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and “nontwisted” Brill–Noether loci. We describe the tangent cones to the twisted Brill–Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill–Noether loci with negative expected dimension.