Totally free arrangements of hyperplanes

TL;DR

This paper proves that the only totally free hyperplane arrangements are products of one- and two-dimensional arrangements, extending known results from low dimensions to higher dimensions.

Contribution

It establishes that no other totally free arrangements exist beyond the obvious products, generalizing previous low-dimensional results to all dimensions.

Findings

Only trivial products are totally free arrangements

No new totally free arrangements beyond known products

Extends low-dimensional results to higher dimensions

Abstract

A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be {\it totally free} if a multiarrangement $(\A, m)$ is free for any multiplicity $ m : \A\to \Z_{> 0}$. It has been known that $\A$ is totally free whenever $\ell \le 2$. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.