Hyperplane arrangements and the Gauss map of a pencil

We show that the coefficients of the characteristic polynomial of a central hyperplane arrangement $\mathcal{A}$, coincide with the multidegrees of the Gauss map of a pencil of hypersurfaces naturally associated to $\mathcal{A}$. As a consequence, we obtain a proof of the Heron-Rota-Welsh conjecture for matroids representable over a field of characteristic zero.