Combinatoriality of freeness and asphericity for general arrangements

Determine whether freeness and asphericity are combinatorial properties for hyperplane arrangements in general.

Background

The paper proves that connected subgraph arrangements are projectively unique, which implies that freeness and asphericity are combinatorial within this class. It contrasts this result with the unresolved general situation: the broader questions of whether freeness and asphericity depend only on the intersection lattice remain longstanding problems in arrangement theory.

References

Whether both these properties are combinatorial in general are longstanding and wide open problems, see [OT92, Conj. 4.138] and [FR00, Prob. 3.8].

— On connected subgraph arrangements  (2502.18144 - Giordani et al., 25 Feb 2025) in Section 1, immediately after Theorem 1.2, p. 2