Clifford indices for vector bundles on curves

TL;DR

This paper extends the classical Clifford index from line bundles to semistable vector bundles on curves, exploring their properties and computing specific cases to distinguish curves of the same genus.

Contribution

It introduces new Clifford indices for vector bundles, generalizing the classical invariant and providing computations for various ranks and curve types.

Findings

Computed all Clifford indices for curves with classical index two.

Established basic properties of the new invariants.

Performed explicit calculations for small ranks and special curves.

Abstract

For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.