High Order Free Hyperplane Arrangements in 3-Dimensional Vector Spaces

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 gave a negative answer to this question when \(\ell \ge 4\). In this paper we verify that 3-arrangements \(\mathscr {A}\) are m-free and compute the m-exponents for all \(m\ge |\mathscr {A}|+2\), where \(|\mathscr {A}|\) is the cardinality of \(\mathscr {A}\). Hence Holm’s question has a positive answer when \(\ell =3\). Finally we prove that 3-dimensional Weyl arrangements of types A and B are m-free for all \(m\ge 0\).