# Brill-Noether loci of rank two vector bundles on a general $\nu$-gonal curve

## TL;DR

This paper investigates the structure and components of the Brill-Noether locus for rank 2 vector bundles on general $
u$-gonal curves, classifying their reduced components and describing general members via extensions of line bundles.

## Contribution

It classifies the reduced components of the Brill-Noether locus with large dimension and describes their general members through minimal extensions of line bundles.

## Key findings

- Classification of reduced components with dimension at least the Brill-Noether number
- Description of general members as extensions of line bundles with minimality properties
- Insights into the birational geometry and very-ampleness of these vector bundles

## Abstract

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general $\nu$-gonal curve. We classify its reduced components whose dimensions are at least the corresponding Brill-Noether number. We moreover describe the general member $\mathcal F$ of such components just in terms of extensions of line bundles with suitable {\em minimality properties}, providing information on the birational geometry of such components as well as on the very-ampleness of $\mathcal F$.