  The Laplacian matrix of a graph G describes the combinatorial dynamics of the
Abelian Sandpile Model and the more general Riemann-Roch theory of G. The
lattice ideal associated to the lattice generated by the columns of the
Laplacian provides an algebraic perspective on this recently (re)emerging
field. This ideal I_G has a distinguished monomial initial ideal M_G,
characterized by the property that the standard monomials are in bijection with
the G-parking functions of the graph G. The ideal M_G was also introduced by
Postnikov and Shapiro (2004) in the context of monotone monomial ideals. We
study resolutions of M_G and show that a minimal free cellular resolution is
supported on the bounded subcomplex of a section of the graphical arrangement
of G. This generalizes constructions from Postnikov and Shapiro (for the case
of the complete graph) and connects to work of Manjunath and Sturmfels, and of
Perkinson et al. on the commutative algebra of Sandpiles. As a corollary we
verify a conjecture of Perkinson et al. regarding the Betti numbers of M_G, and
in the process provide a combinatorial characterization in terms of acyclic
orientations.
