Terao’s freeness conjecture for hyperplane arrangements

Determine whether the freeness of a complex projective hyperplane arrangement is determined solely by its intersection poset, as asserted by Terao’s freeness conjecture.

Background

The paper concerns Terao’s freeness conjecture, which asks whether two hyperplane arrangements with the same intersection poset must either both be free or both be non-free. The intersection poset records all subspaces obtained by intersecting hyperplanes, ordered by reverse inclusion.

The paper studies Ziegler pairs—arrangements with identical combinatorial intersection data but different minimal free resolutions of their Milnor algebras—and proves that any counterexample to Terao’s freeness conjecture for line arrangements in P2 would yield counterexamples in every higher-dimensional projective space Pn with n > 2. Thus, the conjecture remains unresolved in the setting addressed by the paper, while the paper establishes a conditional propagation result rather than a counterexample or a proof of the conjecture.

References

Our paper is motivated by a long-standing open conjecture in the theory of hyperplane arrangements in the complex projective spaces, Terao’s freeness conjecture [1, 2, 6]. If A ⊂ Pn is an arrangement of hyperplanes, then Terao’s freeness conjecture predicts that the freeness of A is determined by the intersection poset L(A) of A, i.e., this is the set of all subspaces that are obtained by intersecting some of the hyperplanes of A, partially ordered by the reverse inclusion. This conjecture is very open and very challenging due to its complexity.

— If Terao's conjecture fails in $\mathbb{P}^{2}$, then it fails in $\mathbb{P}^{n > 2}$  (2502.17031 - Pokora, 24 Feb 2025) in Section 1, Introduction