High order free hyperplane arrangements in 3-dimensional vector spaces

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