Resonance varieties over fields of positive characteristic

Let A be a hyperplane arrangement, and k a field of arbitrary characteristic. We show that the projective degree-one resonance variety R(A, k) of A over k is ruled by lines, and identify the underlying algebraic line complex L(A, k) in the Grassmannian G(2, kn), n = |A|. L(A, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of A. Each linear line complex is the intersection of a family of special Schubert varieties corresponding to a subspace arrangement determined by the partition. In case k has characteristic zero, the resulting ruled varieties are linear and pairwise disjoint, by results of A. Libgober and S. Yuzvinsky. We give examples to show that each of these properties fails in positive characteristic. The (4,3)-net structure on the Hessian arrangement gives rise to a nonlinear component in R(A, Z3), a cubic hypersurface in P 4 with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over Z2 that intersect nontrivially. 1 Resonance and characteristic varieties Arising out of the study of local system cohomology and fundamental groups, characteristic and resonance varieties of complex hyperplane arrangements have become the object of much of the current research in the field. The study of resonance varieties in particular has led to surprising connections with other areas of mathematics: generalized Cartan matrices [23], Latin squares and loops, nets [40], special pencils of plane curves [23, 13], graded resolutions and the BGG correspondence [10, 33, 34], and the Bethe Ansatz [7]. In this paper we establish a connection between resonance varieties and projective line complexes, which becomes evident only when one works over a field of positive characteristic.