Stable bundles of rank 2 with 4 sections

TL;DR

This paper investigates the existence and properties of stable rank 2 vector bundles with four sections on algebraic curves, providing geometric criteria and exploring cases where bounds are attained or not.

Contribution

It establishes a geometric criterion for the degree bound of such bundles and analyzes specific cases for general and Petri curves of genus 10.

Findings

The degree bound cannot be attained for a general genus 10 curve.

Petri curves of genus 10 can attain the bound.

Results relate to Clifford indices and coherent systems.

Abstract

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve $C$. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve $C$ of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems.