Discrete Torelli theorem

Abstract

For an algebraic curve that has only simplest singularities and only rational irreducible components, the generalized Jacobian coincides with the moduli variety of topologically trivial linear bundles whose canonical compactification is a toric variety constructed from a convex integer polytope. The vertices of this polytope are the simple cycles in the one-dimensional rational homology space of the dual graph of this curve. It is proved that for three-connected graphs the simple cycle polytope uniquely determines the graph.