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For a finite index sublattice $L$ of the root lattice $A_{n}$, we construct adeterministic algorithm to deform the lattice ideal $I_L$ to a nearby genericlattice ideal, answering a question posed by Miller and Sturmfels. Ouralgorithm is based on recent results of Perkinson, Perlman and Wilmesconcerning commutative algebraic aspects of chip firing on directed graphs. Asan application of our deformation algorithm, we construct a cellular resolutionof the lattice ideal $I_L$ by degenerating the Scarf complex of itsdeformation.

Explicit Deformation of Lattice Ideals via Chip Firing Games on Directed Graphs