Bridgeland stability of line bundles on surfaces

We study the Bridgeland stability of line bundles on surfaces with respect to certain Bridgeland stability conditions determined by divisors. Given a smooth projective surface S, we show that a line bundle L is always Bridgeland stable for those stability conditions if there are no curves C⊆S of negative self-intersection. When a curve C of negative self-intersection is present, L is destabilized by L(−C) for some stability conditions. We conjecture that line bundles of the form L(−C) are the only objects that can destabilize L and that torsion sheaves of the form L(C)|C are the only objects that can destabilize L[1]. We prove our conjecture in several cases, in particular for Hirzebruch surfaces.