Remarks on divisorial ideals arising from dimer models

TL;DR

This paper explores the relationship between dimer models, divisorial ideals, and maximal Cohen-Macaulay modules in the context of 3D Gorenstein toric singularities, focusing on isoradial models and conic divisorial ideals.

Contribution

It establishes a connection between properties of dimer models and the structure of MCM modules, especially for reflexive polygons in 3D Gorenstein toric singularities.

Findings

Relationship between dimer model properties and MCM modules clarified

Analysis of conic divisorial ideals in the context of reflexive polygons

Insights into the structure of Jacobian algebras as non-commutative crepant resolutions

Abstract

The Jacobian algebra $\mathsf{A}$ arising from a consistent dimer model is derived equivalent to crepant resolutions of a $3$-dimensional Gorenstein toric singularity $R$, and it is also called a non-commutative crepant resolution of $R$. This algebra $\mathsf{A}$ is a maximal Cohen-Macaulay (= MCM) module over $R$, and it is a finite direct sum of rank one MCM $R$-modules. In this paper, we observe a relationship between properties of a dimer model and those of MCM modules appearing in the decomposition of $\mathsf{A}$ as an $R$-module. More precisely, we take notice of isoradial dimer models and divisorial ideals which are called conic. Especially, we investigate them for the case of $3$-dimensional Gorenstein toric singularities associated with reflexive polygons.