A Gröbner basis for the graph of the reciprocal plane

Abstract

Given the complement of a hyperplane arrangement, let $\mathrm{\Gamma }$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $\mathrm{\Gamma }$ in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.