An isomorphism between projective models of toric and hyperplane graphic arrangements

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 $A_{n-1}$ associated with the minimal building set is isomorphic to the one of the hyperplane arrangement of type $A_{n}$ associated again with the minimal building set; 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 the toric arrangement defined by a graph $\Gamma$ and the model of the hyperplane arrangement defined by the cone of $\Gamma$, provided that a suitable building set is chosen.