Non-recursive freeness and non-rigidity

In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, Terao’s open problem asking whether freeness depends only on combinatorics is true. A long standing problem whether all free arrangements are recursively free or not was settled by the second author and Hoge very recently, by giving a free but non-recursively free plane arrangement consisting of 27 planes. In this paper, we construct free but non-recursively free plane arrangements consisting of 13 and 15 planes, and show that the example with 13 planes is the smallest in the sense of the cardinality of planes. In other words, all free plane arrangements consisting of at most 12 planes are recursively free. To show this, we completely classify all free plane arrangements in terms of inductive freeness and three exceptions when the number of planes is at most 12. Several properties of the 15 plane arrangement are proved by computer programs. Also, these two examples solve negatively a problem posed by Yoshinaga on the moduli spaces, (inductive) freeness and, rigidity of free arrangements.