# PRIMER FOR THE ALGEBRAIC GEOMETRY OF SANDPILES

@article{Perkinson2009PRIMERFT,
title={PRIMER FOR THE ALGEBRAIC GEOMETRY OF SANDPILES},
author={David Perkinson and Jacob Perlman and John Wilmes},
journal={arXiv: Combinatorics},
year={2009}
}
• Published 28 December 2011
• Mathematics
• arXiv: Combinatorics
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…
61 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES
Trees, parking functions, syzygies, and deformations of monomial ideals
• Mathematics
• 2003
For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial
FUNCTORIALITY OF CRITICAL GROUPS
We offer a categorical presentation of the critical group of a graph. We take the lattice-theoretic approach to critical groups developed in [1], and consider the extent to which this construction is
The Critical Group of a Line Graph
• Mathematics
• 2009
AbstractThe critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of
Harmonic morphisms and hyperelliptic graphs
• Mathematics
• 2007
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula,
Cayley-Bacharach schemes and their canonical modules
• Mathematics
• 1993
A set of s points in P d is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s − 1 points has the same Hilbert function. We investigate the consequences of this «weak uniformity.» The
On the Sandpile Group of Dual Graphs
• Mathematics
Eur. J. Comb.
• 2000
It is proved that the sandpile group of planar graph is isomorphic to that of its dual, and a combinatorial point of view on the subject is developed.
Asymmetric Abelian Sandpile Models
In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system.
Gorenstein algebras and the Cayley-Bacharach theorem
• Mathematics
• 1985
This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in p2 and properties of graded Gorenstein algebras. Introduction. It is known,