An isomorphism between models of graphic arrangements

TL;DR

This paper establishes an isomorphism between models of toric and hyperplane arrangements defined by graphs, extending previous results and revealing deep structural connections in arrangement theory.

Contribution

The paper demonstrates a new isomorphism between toric and hyperplane arrangement models for graph-defined arrangements, generalizing prior specific cases.

Findings

Isomorphism between toric and hyperplane models for graph arrangements

Conditions on building sets for the isomorphism to hold

Extension of previous type A arrangements to broader families

Abstract

This paper presents a bridge between the theories of wonderful models associated with toric arrangements and wonderful models associated with hyperplane arrangements. In a previous work, the same authors noticed that the model of the toric arrangement of type $A_{n-1}$ is isomorphic to the one of the hyperplane arrangement of type $A_{n}$; it is natural to ask if there exist similar isomorphisms between other families of arrangements. The aim of this paper is to study one such family, namely the family of arrangements defined by graphs. The main result states that there is indeed an isomorphism between the model of a toric arrangement defined by a graph $\Gamma$ and the model of a hyperplane arrangement defined by the cone of $\Gamma$, provided that a suitable building set is chosen.