A characterization of high order freeness for product arrangements and answers to Holm's questions

TL;DR

This paper characterizes m-freeness in product arrangements and shows localizations of m-free arrangements are m-free, providing answers to Holm's questions about the implications and prevalence of m-freeness.

Contribution

It offers a characterization of m-freeness for product arrangements and proves localizations of m-free arrangements are m-free, addressing Holm's open questions.

Findings

m-freeness is characterized for product arrangements

Localizations of m-free arrangements are also m-free

Answers to Holm's questions about implications and prevalence of m-freeness

Abstract

An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm's questions.