  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.
