  Given the complement of a hyperplane arrangement, let $\Gamma$ 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 $\Gamma$ in
two different-seeming ways, one due to Orlik and 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 Gr\"obner basis argument that the polynomials extracted
from the Hilbert series in these two ways agree.
