Free arrangements and coefficients of characteristic polynomials

TL;DR

This paper investigates the relationship between free arrangements and their multirestrictions, emphasizing the role of the second Betti number in characterizing freeness.

Contribution

It introduces a new criterion involving the second Betti number to determine the freeness of arrangements based on their multirestrictions.

Findings

Second Betti number is crucial for characterizing freeness.

Established new links between arrangement freeness and topological invariants.

Extended understanding of multirestriction properties in free arrangements.

Abstract

Ziegler showed that free arrangements have free restricted multiarrangements (multirestrictions). After Ziegler's work, several results concerning "reverse direction", namely characterizing freeness of an arrangement via that of multirestriction, have appeared. In this paper, we prove that the second Betti number of the arrangement plays a crucial role.