# High order free hyperplane arrangements in 3-dimensional vector spaces

## TL;DR

This paper proves that all 3-dimensional hyperplane arrangements are m-free for sufficiently large m, including specific types like Weyl arrangements of types A and B, extending the understanding of m-freeness.

## Contribution

It confirms Holm's conjecture for 3-dimensional arrangements by showing they are m-free for all m beyond a certain bound and computes their m-exponents.

## Key findings

- All 3-arrangements are m-free for m ≥ |A|+2.
- Computed m-exponents for these arrangements.
- Weyl arrangements of types A and B are m-free for all m ≥ 0.

## Abstract

Holm introduced $m$-free $\ell$-arrangements which is a generalization of free arrangements, while he asked whether all $\ell$-arrangements are $m$-free for $m$ large enough. Recently Abe and the author verified that this question is in the negative when $\ell\geq 4$. In this paper we verify that $3$-arrangements $\mathscr{A}$ are $m$-free and compute the $m$-exponents for all $m\geq |\mathscr{A}|+2$, where $|\mathscr{A}|$ is the cardinality of $\mathscr{A}$. Hence Holm's question is in the positive when $\ell=3$. Finally we prove that $3$-dimensional Weyl arrangements of types A and B are $m$-free for all $m\geq 0$.