Nonemptiness and smoothness of twisted Brill-Noether loci

TL;DR

This paper investigates the nonemptiness and smoothness of twisted Brill--Noether loci for vector bundles over smooth curves, establishing conditions for their nonemptiness, smoothness, and describing their geometric properties.

Contribution

It proves nonemptiness and generic smoothness of twisted Brill--Noether loci under certain conditions, and describes their tangent cones and irreducibility properties.

Findings

Brill--Noether loci are nonempty under specified conditions.

Many loci have a component that is generically smooth and of expected dimension.

Examples show loci can have positive dimension with negative expected dimension.

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 $h^0 (C, V \otimes E) \ge k$. 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.