Cohomology of wheels on toric varieties

TL;DR

This paper explicitly computes the cohomology of certain sheaf diagrams called wheels on toric varieties, linking algebraic and combinatorial methods related to toric singularities and dimer models.

Contribution

It introduces a new explicit description of cohomology for wheel diagrams on toric varieties using combinatorial graph walks, connecting algebraic geometry and graph theory.

Findings

Explicit cohomology formulas for wheel diagrams on toric varieties

Connection between syzygy modules and graph walks

Application to toric singularities from dimer models

Abstract

We describe explicitly the cohomology of the total complex of certain diagrams of invertible sheaves on normal toric varieties. These diagrams, called wheels, arise in the study of toric singularities associated to dimer models. Our main tool describes the generators in a family of syzygy modules associated to the wheel in terms of walks in a family of graphs.