Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities

Abstract

A dimer model is a bipartite graph described on the real two-torus, from which we can define the associated quiver. 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 a wall-and-chamber structure, and 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 variations of stable representations under wall-crossing using the sequences of zigzag paths.