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2013 The geometry and combinatorics of cographic toric face rings
Sebastian Casalaina-Martin, Jesse Kass, Filippo Viviani
Algebra Number Theory 7(8): 1781-1815 (2013). DOI: 10.2140/ant.2013.7.1781


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.


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Sebastian Casalaina-Martin. Jesse Kass. Filippo Viviani. "The geometry and combinatorics of cographic toric face rings." Algebra Number Theory 7 (8) 1781 - 1815, 2013.


Received: 22 December 2011; Revised: 4 December 2012; Accepted: 5 December 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1287.13012
MathSciNet: MR3134034
Digital Object Identifier: 10.2140/ant.2013.7.1781

Primary: 14H40
Secondary: 05B35 , 05E40 , 13F55 , 14K30 , 52C40

Keywords: cographic arrangement of hyperplanes , cographic fans , compactified Jacobians , Graphs , nodal curves , toric face rings , totally cyclic orientations , Voronoi polytopes

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 8 • 2013
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