Algebraic Geometry

Let G be a finite, connected, weighted, directed graph with a vertex v that is accessible, i.e., each vertex besides v has a directed path to v. The homogeneous toppling ideal for G is the lattice ideal associated with the Laplacian matrix for G. The first paper on the subject is Polynomial Ideals for Sandpiles and their Grobner Bases, INRIA (2000) by Cori, Rossin, and Salvy. I have been working on this subject with several Reed College undergraduates. Our result so far:
  • The set of zeros of the ideal is isomorphic to the sandpile group for G with sink v.
  • Every lattice ideal defining a finite set of points in projective space arises as the toppling ideal of some graph.
  • Description of a Gröbner basis for the ideal (generalizing the work of Cori, et al. to directed graphs)
  • A description of the minimal free resolution of the ideal in terms of simplicial complexes associated to linear systems on the graph, in general, and in terms of certain connectivity properties in the the case of an undirected graph.
  • A characterization of complete intersection toppling ideals.
  • A method of constructing arithmetically Gorenstein toppling ideals.
An eprint of our results appears in the form of a primer. The online documentation for Sage Sandpiles contains a description of many of these results with examples.