Primer for the algebraic geometry of sandpiles

TL;DR

This paper explores the algebraic geometry of the Abelian Sandpile Model by connecting it with lattice ideals, providing new characterizations, constructions, and conjectures related to graph Laplacians and sandpile groups.

Contribution

It introduces novel algebraic geometric characterizations of Laplacian lattice ideals and extends duality theorems to directed multigraphs, advancing the theoretical understanding of sandpile models.

Findings

Characterization of graphs with complete intersection Laplacian ideals

Construction of arithmetically Gorenstein ideals

Relation between Betti numbers and sandpile group elements

Abstract

The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.