Non-commutative crepant resolutions for some toric singularities II

TL;DR

This paper provides an alternative, dimer model-free proof that all 3D Gorenstein affine toric varieties admit toric non-commutative crepant resolutions, using standard toric techniques.

Contribution

It offers a new proof for the existence of NCCRs for certain toric singularities without relying on dimer models, expanding the toolkit for such proofs.

Findings

Constructed a tilting bundle on a crepant resolution using toric methods.

Proved the existence of NCCRs for 3D Gorenstein affine toric varieties.

Provided an alternative approach to Broomhead's result.

Abstract

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.