Vol. 7, No. 8, 2013

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The geometry and combinatorics of cographic toric face rings

Sebastian Casalaina-Martin, Jesse Leo Kass and Filippo Viviani

Vol. 7 (2013), No. 8, 1781–1815

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.

toric face rings, graphs, totally cyclic orientations, Voronoi polytopes, cographic arrangement of hyperplanes, cographic fans, compactified Jacobians, nodal curves
Mathematical Subject Classification 2010
Primary: 14H40
Secondary: 13F55, 05E40, 14K30, 05B35, 52C40
Received: 22 December 2011
Revised: 4 December 2012
Accepted: 5 December 2012
Published: 24 November 2013
Sebastian Casalaina-Martin
Department of Mathematics
University of Colorado Boulder
Campus Box 395
Boulder, CO
United States
Jesse Leo Kass
Institut für Algebraische Geometrie
Leibniz Universität Hannover
Welfengarten 1
30167 Hannover
Filippo Viviani
Dipartimento di Matematica
Università degli Studi Roma Tre
Largo San Leonardo Murialdo 1
00146 Roma