PRIMER FOR THE ALGEBRAIC GEOMETRY OF SANDPILES

By D. Perkinson, Jacob Perlman, John Wilmes. 2011

68 citations recorded by the source corpus.

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 al- gebraic geometry to the Laplacian matrix, drawing out connections with the ASM. A extended summary of the ASM and of the required algebraic ge- ometry 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 congurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lat- tice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/f45ce19551cedce87c8dcb9a649ebf2a36382ed9