The smallest line arrangement which is free but not recursively free

Abstract

In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, the conjecture by Terao asserting that freeness depends only on combinatorics holds true. A long standing problem whether all free arrangements are recursively free or not is settled by Cuntz and Hoge very recently, by giving a free but non-recursively free plane arrangement consisting of 27 planes. In this paper, we construct a free but non-recursively free plane arrangement consisting of 13 planes, and show that this example 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 it, we completely classify all free plane arrangements in terms of inductive freeness and three exceptions when the number of planes is at most 12.