On an example of Mukai

TL;DR

This paper uses Mukai's example to construct specific semistable bundles on curves, demonstrating new properties related to Clifford indices and Brill-Noether loci with negative expected dimensions.

Contribution

It introduces explicit constructions of semistable rank 3 bundles with 6 sections on certain curves, revealing cases where Clifford indices are strictly less than the curve's Clifford index.

Findings

Existence of semistable bundles with specified properties on curves of genus 9 and 11

Demonstration of non-emptiness of Brill-Noether loci with negative expected dimension

Examples showing Clifford index inequalities on algebraic curves

Abstract

In this note we use an example of Mukai to construct semistable bundles of rank 3 with 6 independent sections on a curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill-Noether loci with negative expected dimension.