Complements and higher resonance varieties of hyperplane arrangements

TL;DR

This paper explores the algebraic and geometric properties of hyperplane arrangements, focusing on Betti numbers, resonance varieties, and their positivity and determinantal structures, revealing new connections in combinatorics and algebraic geometry.

Contribution

It demonstrates Schur positivity of Betti number combinations and establishes that all resonance varieties are determinantal, linking combinatorial invariants with geometric structures.

Findings

Certain Betti number combinations satisfy Schur positivity

All resonance varieties are determinantal

Homological algebra and vector bundle techniques are used

Abstract

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal.