Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2

Abstract

We prove that, for a natural class of Bridgeland stability conditions on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and the blow-up of ${\mathbb{P}}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after "rotation") equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projective.

arXiv comment40 pages, 10 figures