Divisionally free arrangements of hyperplanes

TL;DR

This paper introduces the concept of divisionally free arrangements of hyperplanes, establishing a division theorem that links the freeness of arrangements to the divisibility of their characteristic polynomials, and confirms Terao's conjecture within this class.

Contribution

It defines divisionally free arrangements, proves a division theorem for their freeness, and shows Terao's conjecture holds for this broader class.

Findings

Divisionally free arrangements are strictly larger than inductively free arrangements.

The division theorem confirms the freeness of arrangements based on polynomial divisibility.

Terao's conjecture is valid within the class of divisionally free arrangements.

Abstract

We consider the triple $(\mathcal{A},\mathcal{A}',\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\mathcal{A}^H$ and the division of $\chi(\mathcal{A};t)$ by $\chi(\mathcal{A}^H;t)$ confirm the freeness of $\mathcal{A}$. The key ingredient of this "division theorem" on freeness is the fact that, if $\chi(\mathcal{A}^H;t)$ divides $\chi(\mathcal{A};t)$, then the same holds for the localization at the codimension three flat in $H$. This implies the local-freeness of $\mathcal{A}$ in codimension three along $H$. Based on these results, several applications are obtained, which include a definition of "divisionally free arrangements". It is strictly larger than the set of inductively free arrangements. Also, in the set of divisionally free arrangements, the Terao's conjecture is true.