Resonance Varieties over Fields of Positive Characteristic

Let $\mathcal A$ be a hyperplane arrangement, and K a field of arbitrary characteristic. We show that the projective degree-one resonance variety $\mathcal R^1(\mathcal A$, K) of $\mathcal A$ over K is ruled by lines, and identify the underlying algebraic line complex $L(\mathcal A$, K) in the Grassmannian $\mathcal G(2$, Kⁿ), $n=\vert\mathcal A\vert$. $\mathrm L(\mathcal A$, K) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of $\mathcal A$. 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 comonent in $\mathcal R^1(\mathcal A\text{,}\ {\overline{\mathbb{Z}}}_3)$, a cubic hypersurface in $\mathbb{P}^4$ with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over $\mathbb{Z}_2$ that intersect nontrivially.