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Remarks on divisorial ideals arising from dimer models

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.