Noncommutative deformations and flops

Abstract

We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a $3$-fold is representable, and hence, we associate to every such curve a noncommutative deformation algebra ${\mathrm{A}}_{\mathrm{con}}$. 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 ${\mathrm{A}}_{\mathrm{con}}$ 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 ${\mathrm{A}}_{\mathrm{con}}$ 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 ${\mathrm{A}}_{\mathrm{con}}$, 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.