Restricted Lazarsfeld-Mukai bundles and canonical curves

TL;DR

This paper proves the stability of certain vector bundles on specific algebraic curves, advancing understanding of their geometric properties and implications for Mercat's conjecture in algebraic geometry.

Contribution

It establishes stability results for normal bundles of genus 7 curves and for Lazarsfeld-Mukai bundles on K3 surfaces, providing new insights into vector bundle stability.

Findings

Normal bundle of genus 7 curves with maximal Clifford index is stable.

Rank four Lazarsfeld-Mukai bundles on general K3 surface curves are stable.

Results impact Mercat's conjecture on higher rank vector bundles.

Abstract

We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank four Lazarsfeld-Mukai vector bundles on a curve that lies on a general K3 surface are stable. Both results have consequences for Mercat's conjecture on higher rank vector bundles on generic curves.