Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

TL;DR

This paper classifies toric non-commutative crepant resolutions of certain Gorenstein singularities, showing their connections via mutations and providing explicit quiver descriptions, with applications to lattice polytope volumes.

Contribution

It provides a classification of toric NCCRs for Gorenstein toric singularities with divisor class group of rank one, linking them to upper sets and mutations, and describes their quivers explicitly.

Findings

All toric NCCRs are connected by mutations and are derived equivalent.

Explicit quivers with relations are described using higher-dimensional dimer models.

Verified Van den Bergh's conjecture relating indecomposable summands to polytope volume.

Abstract

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. This classification allows us to prove that all toric NCCRs of such toric singularities are connected by iterated Iyama-Wemyss mutations, and hence are derived equivalent to each other. We also describe explicitly the quivers with relations of these toric NCCRs in terms of higher dimensional analogue of dimer models. In the Appendix, we give an explicit formula for the volume of $d$-dimensional lattice polytopes with $d+2$ vertices. As an application, we verify a conjecture of Van den Bergh in the case of Gorenstein toric singularities with divisor class group of rank one: the number of indecomposable direct summands of a toric NCCR coincides with the normalized volume of the corresponding lattice polytope.