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.

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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.

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§0. Introduction. The (small) quantum cohomology QH * (X) of a complex projective manifold X can be thought of either as a deformation of the even degree cohomology V := H 2 * (X, C) or as a family of associative, commutative products on V indexed by H 2 (X, C). Indeed, if T 1 , ..., T m ∈ H 2 (X, C) are a basis, which we assume for simplicity to consist of nef divisors, (with dual basis t 1 , ..., t m ), then the small quantum product is a map:(reducing to the cup product when t i → −∞) determined by the enumerative geometry of rational curves (the genus-zero Gromov-Witten invariants) on X. If X is a Fano manifold then * is polynomial-valued in the exponentials.The complex Grassmannian G := G(r, n) has one of the most-studied and best-understood small quantum cohomology rings. Recall that the ordinary cohomology of G has presentation:where the σ i are the elementary symmetric polynomials in degree 2 variables x 1 , ..., x r (the Chern roots of the dual to the universal bundle) and the h i are the complete symmetric polynomials (sums of all monomials of degree i) inThe quantum cohomology of G, on the other hand, has presentation:Quantum cohomology has a mathematical partner that is frequently better suited for applications. Here one regards * as an O-linear product of T V -valued vector fields over H 2 (X, C). That is, if T 0 , T 1 , ..., T n is a basis for V , extending the basis of H 2 (X, C), with T 0 = 1 and T n its Poincaré dual, and if:.., e tm )∂ k , where ∂ i := ∂ ∂ti . If we reinterpret * once more as:then the O-linearity means that there is a family of connections:, and the associativity of * translates into the flatness of the ∇h, i.e. existence of flat sectionsGivental [Giv1] computed the fundamental (n+1)×(n+1) matrix of solutions in terms of Gromov-Witten invariants "with gravitational descendents." The solution is unique of the following form with suitable initial conditions:(polynomial in the t = (t 1 , ..., t m )). It is very useful to regard the columns as cohomology-valued, and the last column in particular:is very functorial and plays an important role in mirror symmetry (here {Ť i } is the basis dual to {T i } with respect to the intersection pairing on X).In the simplest case, let x be the hyperplane class in H 2 (P n−1 ). Then:Our goal here is to compute the J-function of the Grassmannian G and explore various applications that result. One of the nice functorial properties of Jfunctions guarantees that:where P = r i=1 P n−1 with hyperplane classes x i on the factors. We will give two proofs of the following additional property of J:denote the Vandermonde determinant and "Vandermonde operator", respectively. Then:i.e. the J-function of the Grassmannian is obtained by applying D ∆ to J P and then "symmetrizing" (and translating).We will see in the second proof of the conjecture, in particular, that this can be thought of as another nice functorial property of J-functions, which looks as though it ought to generalize to wider classes of geometric invariant theory quotients.We should note that...