Hyperplane arrangements and the Gauss map of a pencil

TL;DR

This paper links hyperplane arrangements to the Gauss map of a hypersurface pencil, proving the Heron-Rota-Welsh conjecture for certain matroids by connecting characteristic polynomials with multidegrees.

Contribution

It establishes a novel connection between the characteristic polynomial of hyperplane arrangements and the multidegrees of the Gauss map, providing a new proof of a long-standing conjecture.

Findings

Characteristic polynomial coefficients match multidegrees of the Gauss map

Proof of the Heron-Rota-Welsh conjecture for matroids over characteristic zero fields

New geometric interpretation of hyperplane arrangement invariants

Abstract

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.