Flops and Clusters in the Homological Minimal Model Program

Abstract

Suppose that $f\colon X\to\mathrm{Spec}\, R$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of $f$ are at most one dimensional, so by [VdB1d] there is a noncommutative ring $\Lambda$ derived equivalent to $X$. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of $\Lambda$ is finite dimensional, improving a result of [DW2]. We further show that the mutation functor of [S6][IW4] is functorially isomorphic to the inverse of the Bridgeland--Chen flop functor in the case when the factor of $\Lambda$ is finite dimensional. These results then allow us to jump between all the minimal models of $\mathrm{Spec}\, R$ in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw--Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying $R$-module generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander--McKay correspondence to dimension three.