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

For a finite index sublattice $L$ of the root lattice $A_{n}$, we construct a deterministic algorithm to deform the lattice ideal $I_L$ to a nearby generic lattice ideal, answering a question posed by Miller and Sturmfels. Our algorithm is based on recent results of Perkinson, Perlman and Wilmes concerning commutative algebraic aspects of chip firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal $I_L$ by degenerating the Scarf complex of its deformation.