ABSTRACT. We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wallcrossing behavior" for objects with the same invariants as O C (H) when H generates Pic(S) and C ∈ |H|. If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgelandstable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of n-points on P 2 . We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

Abstract.A new tautological relation of M 3,1 in codimension 3 is derived and proved, using an invariance constraint from our previous work.

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We prove that, for a natural class of Bridgeland stability conditions on P 1 × P 1 and the blow-up of 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 [ABCH13]) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projective.