Freeness and multirestriction of hyperplane arrangements

TL;DR

This paper extends Yoshinaga's result to higher dimensions, establishing conditions under which a hyperplane arrangement's freeness can be characterized by its restriction properties and characteristic polynomials.

Contribution

It generalizes the criteria for freeness of hyperplane arrangements from dimension 3 to higher dimensions under tameness conditions.

Findings

Freeness in 4-space characterized by restriction freeness and polynomial equality.

Extension of Yoshinaga's result to arbitrary dimensions with tameness assumptions.

Provides new criteria for analyzing hyperplane arrangement freeness.

Abstract

Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.