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Toric face rings were introduced by Stanley as a simultaneous generalization of simplicial face rings and toric rings. In this talk, I will report on a joint work in progress with S. Casalaina-Martin and J. L. Kass, in which we study certain toric face rings associated to graphs. The structure of these rings depends on various combinatorial properties of their associated graphs: the poset of totally cyclic orientations, the cographic arrangement of hyperplanes and the Dirichelet-Voronoi polytope.
Our motivation for studying cographic toric face rings is that they show up naturally as local rings of coarse compactified Jacobians of nodal curves. However, apart from a couple of brief introductory remarks, the talk will focus on the algebraic and combinatorial aspects of the subject.
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