High Order Free Hyperplane Arrangements in 3-Dimensional Vector Spaces

Holm introduced m-free l-arrangements which is a generalization of free arrangements, while he asked whether all l-arrangements are m-free for m large enough. Recently Abe and the author gave a negative answer to this question when l >= 4. In this paper we verify that 3-arrangements A are m-free and compute the m-exponents for all m >= |A| + 2, where |A| is the cardinality of A. Hence Holm's question has a positive answer when l = 3. Finally we prove that 3-dimensional Weyl arrangements of types A and B are m-free for all m >= 0.