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 The preprint was withdrawn by its author on 16 January 2022; no result of it is used here, but the strategy of our proof is that of the preprint and of the Oberwolfach abstract. The case of exponents 1 and 2 only was first proved by Tohăneanu, in the first version of the preprint (2017). This case, and the case of one exponent 3 together with a codimension-two flat in four hyperplanes, also follow from a theorem of DiPasquale (Math. Ann. 385 (2023)); both are proved again here. 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.

Version 1.1 (12 October 2026) corrects the credit for the case of exponents 1 and 2 only, which was first proved by Tohăneanu in the first version of the preprint arXiv:1707.07091 (2017); version 1.0 named only the theorem of DiPasquale for it. No statement and no proof is changed.

Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.

Corpus identifier: OWR-4990374-015 (Tohăneanu, Free arrangements with low exponents, Oberwolfach Reports 18 (2021), Report No. 5/2021, p. 284).