Bridgeland stability conditions on surfaces with curves of negative self-intersection

TL;DR

This paper constructs new Bridgeland stability conditions on surfaces with negatively self-intersecting curves, explores their position in the stability manifold, and develops associated moduli spaces of semistable objects.

Contribution

It introduces novel stability conditions for surfaces with negative curves and analyzes their placement within the stability manifold, along with moduli space construction.

Findings

New stability conditions on surfaces with negative curves

Identification of stability conditions on a wall of the geometric chamber

Construction of moduli spaces of semistable objects after wall-crossing

Abstract

Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on $X$ and the geometry of the variety. We construct new stability conditions for surfaces containing a curve $C$ whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of ${\rm Stab}(X)$, the stability manifold of $X$. We then construct the moduli space $M_{\sigma}(\mathcal{O}_X)$ of $\sigma$-semistable objects of class $[\mathcal{O}_X]$ in $K_0(X)$ after wall-crossing.