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

TL;DR

This paper proves the projectivity of Bridgeland moduli spaces on certain Del Pezzo surfaces, using quiver representations and stability conditions, and conjectures this extends to all Del Pezzo surfaces.

Contribution

It establishes projectivity for Bridgeland moduli spaces on specific Del Pezzo surfaces and proposes a general conjecture for all such surfaces.

Findings

Moduli spaces are projective for certain surfaces.

Stability conditions relate to quiver representations.

Conjecture extends results to all Del Pezzo surfaces.

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.