Characteristic polynomials, $\eta$-complexes and freeness of tame arrangements

TL;DR

This paper investigates the relationship between characteristic polynomials and freeness in tame arrangements, extending known criteria and establishing new inequalities and conditions for freeness based on chamber configurations.

Contribution

It generalizes Yoshinaga's freeness criterion to tame arrangements and introduces a new comparison between characteristic polynomial coefficients and Ziegler restrictions.

Findings

Coefficient of the characteristic polynomial of an arrangement is not less than that of its Ziegler restriction in tame arrangements.

A free arrangement is shown to be a minimal chamber arrangement.

Provides a new freeness criterion based on chamber configurations in tame arrangements.

Abstract

We compare each coefficient of the reduced characteristic polynomial of a simple arrangement and that of its Ziegler restriction. As a consequence we can show that the former is not less than the latter in the category of tame arrangements. This is a generalization of Yoshinaga's freeness criterion for 3-arrangements and also the recent result by the author and Yoshinaga. As a corollary, we can prove that a free arrangement is a minimal chamber arrangement, and we can give a freeness criterion in terms of chambers in the category of tame arrangements.