On linear systems and a conjecture of D. C. Butler

TL;DR

This paper proves a conjecture by D. C. Butler regarding the semistability of kernels of evaluation maps for line bundles on algebraic curves, advancing the understanding of coherent systems and stability conditions.

Contribution

It confirms Butler's conjecture on semistability and introduces new results on the stability of kernels in the context of coherent systems on curves.

Findings

Proves Butler's conjecture on semistability of kernels.

Establishes new stability results for kernels of evaluation maps.

Utilizes wall crossing techniques in the theory of coherent systems.

Abstract

Let $C$ be a smooth irreducible projective curve of genus $g$ and $L$ a line bundle of degree $d$ generated by a linear subspace $V$ of $H^0(L)$ of dimension $n+1$. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map $V\otimes{\mathcal O}_C\to L$ and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.