Flops and clusters in the homological minimal model programme

Abstract

Suppose that f : X → 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 Van den Bergh (Duke Math J 122(3):423-455, 2004) there is a noncommutative ring 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 is finite dimensional, improving a result of Donovan and Wemyss (Contractions and deformations, arXiv:1511.00406). We further show that the mutation functor of Iyama and Wemyss (Invent Math 197(3):521-586, 2014, §6) is functorially isomorphic to the inverse of the Bridgeland-Chen flop functor in the case when the factor of is finite dimensional. These results then allow us to jump between all the minimal models of 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.

Gopakumar–Vafa Invariants Do Not Determine Flops

“…With their roots in homological algebra, and because they are an algebra as opposed to a number, this additional structure allows us to use contraction algebras to establish and control many geometric processes [DW2,W14], whilst at the same time recover the GV and other invariants [DW1,T14,HT] in a variety of natural ways. M.W.…”

Flops and mutations for crepant resolutions of polyhedral singularities

“…Geometrically, this means that there are a priori floppable (−2, 0) and (−3, 1)-curves in the fibre of origin of some crepant resolution of C 3 /G. Even though following [Wem14] we can ensure that the correspondence of both approaches also holds for types O and I, because of the different nature of this cases with respect to explicit computations (namely the presence of high rank modules in the McKay quiver and mutations of QPs at vertices with loops) we leave the treatment of this cases for a future work.…”

Derived categories of quasi-hereditary algebras and their derived composition series

“…Triangulated categories are used and studied in different areas of mathematics and theoretical physics -algebraic geometry (for example, with applications to classical problems in birational geometry, see e.g. [15,66]), representation theory (with relations to cluster algebras, starting with [16] and perverse sheaves [7] used in the proof of the Kazhdan-Lusztig conjectures), algebraic topology, string theory (via Kontsevich's Homological Mirror Symmetry conjecture [44]), ... . In general, triangulated categories are rather complicated structures and therefore techniques allowing a decomposition into more accessible pieces are important.…”