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The
geometry and combinatorics of cographic toric face rings
Sebastian Casalaina-Martin, Jesse Leo Kass and Filippo Viviani |
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Vol. 7 (2013), No. 8, 1781–1815
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 14H40
Secondary: 13F55, 05E40, 14K30, 05B35, 52C40
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Milestones
Received: 22 December 2011
Revised: 4 December 2012
Accepted: 5 December 2012
Published: 24 November 2013
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Authors |
| Sebastian Casalaina-Martin | |
| Department of Mathematics University of Colorado Boulder Campus Box 395 Boulder, CO 80309-0395 United States |
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| http://math.colorado.edu/~sbc21/ | |
| Jesse Leo Kass | |
| Institut für Algebraische
Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover Germany |
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| http://www2.iag.uni-hannover.de/~kass/ | |
| Filippo Viviani | |
| Dipartimento di Matematica Università degli Studi Roma Tre Largo San Leonardo Murialdo 1 00146 Roma Italy |
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| http://ricerca.mat.uniroma3.it/users/viviani/ | |
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