Terao’s conjecture for line arrangements

Determine whether freeness of a line arrangement in the projective plane is completely determined by its intersection lattice; equivalently, prove that freeness is constant on the realization space of any fixed intersection lattice of lines in P^2.

Background

The paper studies freeness of line arrangements via a semicontinuous relaxation of Saito’s criterion and places the problem within the broader context of longstanding questions in arrangement theory. Terao’s conjecture predicts that freeness depends only on the intersection lattice (combinatorics) and not on the specific geometric realization.

The authors emphasize that, despite progress on related algebraic and combinatorial invariants, the conjecture remains unresolved even in the case of line arrangements in the projective plane, underscoring its centrality to the field and motivating their functional and computational approach.

References

A central open question in this area is Terao's conjecture, which predicts that freeness of an arrangement is determined solely by its intersection lattice. Even for line arrangements in $\mathbb{P}2$, this conjecture remains widely open, and much recent work has focused on clarifying the relation between combinatorial data and freeness-related algebraic invariants such as the module of logarithmic derivations and the minimal degree of Jacobian relations.

— A semicontinuous relaxation of Saito's criterion and freeness as angular minimization  (2604.02995 - Silva, 3 Apr 2026) in Section 1, Introduction