We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain (-3,1)-curves. We then show how our new invariant also controls the homological algebra of flops. For any flopping curve in a projective 3-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 3-fold by twisting around a universal family over the noncommutative deformation algebra, and prove that this autoequivalence is an inverse of Bridgeland's flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 3-fold, and thus the Bridgeland stability manifold.Comment: Minor changes. Final version, to appear in Duke Math Journal. 50 page

16.04.15 KB. OK to add accepted version to spiralWe introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent

We give a new proof of the 'Pfaffian-Grassmannian' derived equivalence between certain pairs of non-birational Calabi-Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three steps by passing through some intermediate categories of (global) matrix factorizations. The first step is global Knörrer periodicity, the second comes from a birational map between Landau-Ginzburg B-models, and for the third we develop some new techniques.

Abstract. We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functorsheaf is a P-functor, hence can be used to construct an autoequivalence of D b (M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkähler 2g-folds that are not birational, for every g ≥ 2.We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivalent.