Free Hyperplane Arrangements with Exponents 1, 2 and at Most One 3 Are Supersolvable: A Proof of a Conjecture of Tohăneanu

Ş. O. Tohăneanu conjectured (Oberwolfach Report 5/2021, p. 284; Conjecture 4.1 of the preprint arXiv:1707.07091v3) that every central free hyperplane arrangement over a field of characteristic zero whose exponents are \(1\)'s, \(2\)'s and at most one \(3\) is supersolvable. We prove this conjecture in every rank. The preprint was withdrawn by its author on 16 January 2022; no result of it is used here, but the strategy of our proof is that of the preprint and of the Oberwolfach abstract. Two cases follow from a theorem of DiPasquale (Math. Ann. 385 (2023)) and are proved again: exponents \(1\) and \(2\) only, and one exponent \(3\) together with a codimension-two flat in four hyperplanes; rank three was also known. New, to our knowledge and relative to the literature we found, is the case without such a flat in rank at least \(4\), and therefore the conjecture in general. With two exponents \(3\) the statement is false: the non-Fano arrangement (exponents \((1,3,3)\)) and nine hyperplanes of rank four (exponents \((1,2,3,3)\)) are free and not supersolvable. No computation enters the proof; the computations reported are tests. The book of Orlik and Terao was not accessible to us and was not read. No proof and no counterexample of the conjecture was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.