Linear series on curves: stability and Clifford index

TL;DR

This paper explores the stability properties of vector bundles associated with linear series on smooth complex curves, linking stability conditions to the Clifford index and establishing the existence of theta-divisors.

Contribution

It introduces new stability criteria related to the Clifford index and demonstrates the existence of stable bundles with theta-divisors on curves.

Findings

Stability of the Dual Span Bundle is related to the curve's Clifford index.

Under certain conditions, cohomological stability is achieved.

Constructs stable vector bundles of slope 3 with theta-divisors.

Abstract

We study concepts of stabilities associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated Dual Span Bundle and linear stability. Our result implies a stability condition related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Eventually using our results we obtain stable vector bundles of integral slope 3, and prove that they admit theta-divisors.