In this paper, we define and study a ring associated to a graph that we call the
cographic toric face ring or simply the cographic ring. The cographic ring is the toric
face ring defined by the following equivalent combinatorial structures of a graph: the
cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally
cyclic orientations. We describe the properties of the cographic ring and, in
particular, relate the invariants of the ring to the invariants of the corresponding
graph.
Our study of the cographic ring fits into a body of work on describing rings
constructed from graphs. Among the rings that can be constructed from a graph,
cographic rings are particularly interesting because they appear in the study of
compactified Jacobians of nodal curves.
Keywords
toric face rings, graphs, totally cyclic orientations,
Voronoi polytopes, cographic arrangement of hyperplanes,
cographic fans, compactified Jacobians, nodal curves