Variations of GIT quotients and dimer combinatorics for toric compound Du Val singularities

TL;DR

This paper explores how dimer models and zigzag paths influence the GIT quotient structures and resolutions of toric compound Du Val singularities, revealing the combinatorial and geometric interplay in these resolutions.

Contribution

It demonstrates that zigzag paths on dimer models determine the wall-and-chamber structure of stability parameters for toric singularities.

Findings

Sequences of zigzag paths determine the wall-and-chamber structure.

Wall-crossing variations can be tracked via zigzag paths.

Dimer models give rise to projective crepant resolutions.

Abstract

A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and for any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track the variations of stable representations under wall-crossing using the sequences of zigzag paths.