High order free hyperplane arrangements in 3-dimensional vector spaces

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 $\ell \ge 4$. In this paper we verify that $3$-arrangements $\mathcal{A}$ are $m$-free and compute the $m$-exponents for all $m\ge |\mathcal{A}|+2$, where $|\mathcal{A}|$ is the cardinality of $\mathcal{A}$. 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 $m\ge 0$.

25 pages, 3 figures