Some characterizations of freeness of hyperplane arrangement

TL;DR

This paper explores various characterizations of the freeness property in hyperplane arrangements, linking local properties, polynomial factorizations, and restrictions, and applies these to prove conjectured freeness in specific cases.

Contribution

It introduces new characterizations of hyperplane arrangement freeness and confirms a conjecture by Edelman and Reiner through these methods.

Findings

Freeness characterized by polynomial factorization in 3-arrangements

Freeness linked to local properties and restrictions in higher dimensions

Proved conjectured freeness for certain arrangements

Abstract

We will consider some characterizations of freeness of a hyperplane arrangement, in terms of the following properties: locally freeness, factorization of characteristic polynomial and freeness of restricted multiarrangement. In the case of 3-arrangement, freeness is characterized by factorization of characteristic polynomial and coincidence of its roots with exponents of restricted multiarrangement. In the case of higher dimension, it is characterized by a kind of locally freeness and freeness of restricted multiarrangement. As an application, we prove the freeness of certain arrangements which is conjectured by Edelman and Reiner.