On \alpha-stability and linear stability of generated coherent systems

TL;DR

This paper explores the relationships between different notions of stability for generated coherent systems over curves, establishing implications between them and providing examples that challenge these relationships, with applications to Butler's conjecture.

Contribution

It demonstrates that -stability and linear stability imply -stability for generated coherent systems and provides counterexamples, advancing understanding of stability notions and proving a case of Butler's conjecture.

Findings

-stability and linear stability imply -stability for generated systems

Counterexamples show no other implications between the stability notions

Proves a case of Butler's conjecture for systems of type (2, d, 5)

Abstract

There is a well studied notion of GIT-stability for coherent systems over curves, which depends on a real parameter $\alpha$. For generated coherent systems, there is a further notion of stability derived from Mumford's definition of linear stability for varieties in projective space. Let $\alpha_S$ be close to zero and $\alpha_L \gg 0$. We show that a generated coherent system which is $\alpha_S$-stable and linearly stable is $\alpha_L$-stable, and give examples showing that without further assumptions, there are no other implications between these three types of stability. We observe that several of the systems constructed have stable dual span bundle, including systems which are not $\alpha$-semistable for any value of $\alpha$. We use this to prove a case of Butler's conjecture for systems of type $(2, d, 5)$.