The geometry and combinatorics of cographic toric face rings

TL;DR

This paper introduces the cographic toric face ring, a new algebraic structure associated with a graph, exploring its properties and connections to graph invariants and applications in algebraic geometry.

Contribution

It defines and analyzes the cographic ring, linking its properties to graph invariants and highlighting its relevance in the study of compactified Jacobians of nodal curves.

Findings

The cographic ring is related to the cographic arrangement, Voronoi polytope, and cyclic orientations.

Properties of the cographic ring are described and connected to graph invariants.

Cographic rings are relevant in algebraic geometry, particularly in the study of nodal curves.

Abstract

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.