Bridgeland Stability of Sheaves on del Pezzo Surface of Picard Rank Three

TL;DR

This paper investigates the Bridgeland stability conditions for sheaves on a del Pezzo surface obtained by blowing up two points in the projective plane, identifying destabilizing objects and stability of certain line bundles.

Contribution

It determines destabilizing objects for sheaves and proves the stability of line bundles restricted to (-1)-curves on the surface.

Findings

Destabilizing objects of line bundles are identified.

The line bundle |_E is Bridgeland stable for any (-1)-curve E.

Stability holds for any divisorial Bridgeland stability condition.

Abstract

This article discusses the Bridgeland stability of some sheaves on the blow-up of $\mathbb{P}^{2}$ at two general points. We have determined the destabilizing objects of the line bundles and have shown that $\mathscr{O}(E)|_{E}$ is Bridgeland stable for any $(-1)$-curve $E$ and any divisorial Bridgeland stability condition.