The line geometry of resonance varieties

Abstract

Let R^1(A,R) be the degree-one resonance variety over a field R of a hyperplane arrangement A. We give a geometric description of R^1(A,R) in terms of projective line complexes. The projective image of R^1(A,R) is a union of ruled varieties, parametrized by neighborly partitions of subarrangements of A. The underlying line complexes are intersections of special Schubert varieties, easily described in terms of the corresponding partition. We generalize the definition and decomposition of R^1(A,R) to arbitrary commutative rings, and point out the anomalies that arise. In general the decomposition is parametrized by neighborly graphs, which need not induce neighborly partitions of subarrangements of A. We use this approach to show that the resonance variety of the Hessian arrangement over a field of characteristic three has a nonlinear component, a cubic threefold with interesting line structure. This answers a question of A. Suciu. We show that Suciu's deleted B_3 arrangement has resonance components over Z_2 that intersect nontrivially. We also exhibit resonant weights over Z_4 supported on the deleted B_3, which has no neighborly partitions. The modular resonant weights on the deleted B_3 exponentiate to points on the complex torus which lie on, and determine, the translated 1-torus in the first characteristic variety.