Copyright © The Author 2007. Published by Oxford University Press.
Resonance Varieties over Fields of Positive Characteristic
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011-5717
Correspondence: Correspondence to be sent to: Michael J. Falk, Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011-5717. e-mail: michael.falk{at}nau.edu
Let 
be a hyperplane arrangement, and 
a field of arbitrary characteristic. We show that the projective degree-one resonance variety 
, ) of over is ruled by lines, and identify the underlying algebraic line complex 
, ) in the Grassmannian 
, n), 
. , ) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of . 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 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 comonent in 
, a cubic hypersurface in 
with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over 
that intersect nontrivially.