[Submitted on 30 Oct 2025 (v1), last revised 2 Jun 2026 (this version, v3)]

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Abstract:We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities whose divisor class group has rank one. More precisely, such toric NCCRs are in bijection with non-trivial upper sets in a certain quotient of the divisor class group equipped with a natural 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 one another.
We further give a dimer-model realization of this classification in the non-pyramidal case. More precisely, we construct periodic quivers with cuts on a $d$-dimensional torus, establish a cut-upper set correspondence, and prove that the resulting cut quiver with relations presents the corresponding toric NCCR. For $d=2$, this recovers the quiver-theoretic part of the usual dimer-model construction.
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 Van den Bergh's conjectural equality, for Gorenstein toric singularities with divisor class group of rank one, between the number of indecomposable direct summands of a toric NCCR and the normalized volume of the corresponding lattice polytope.

Submission history

From: Ryu Tomonaga [view email]
[v1] Thu, 30 Oct 2025 08:34:25 UTC (16 KB)
[v2] Tue, 21 Apr 2026 09:56:18 UTC (29 KB)
[v3] Tue, 2 Jun 2026 07:52:32 UTC (31 KB)

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