On generated coherent systems and a conjecture of D. C. Butler

TL;DR

This paper investigates generated coherent systems on algebraic curves, proving Butler's Conjecture in certain cases, especially for type (2,d,4), which was previously unresolved.

Contribution

It establishes new existence results for generated coherent systems and confirms Butler's Conjecture for specific types, including the first unknown case.

Findings

Proved Butler's Conjecture for type (2,d,4)

Established existence of generated coherent systems in new cases

Connected semistability of $E$ to kernel semistability in specific settings

Abstract

Let $(E,V)$ be a general generated coherent system of type $(n,d,n+m)$ on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of $E$ to the semistability of the kernel of the evaluation map $V\otimes \mathcal{O}_X\to E$. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler's Conjecture in some cases. The strongest results are obtained for type $(2,d,4)$, which is the first previously unknown case.