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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm009, 25 pages, doi:10.1093/imrn/rnm009 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Resonance Varieties over Fields of Positive Characteristic

Michael J. Falk

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 Formula be a hyperplane arrangement, and k a field of arbitrary characteristic. We show that the projective degree-one resonance variety Formula, k) of Formula over k is ruled by lines, and identify the underlying algebraic line complex Formula, k) in the Grassmannian Formula, kn), Formula. Formula, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of Formula. 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 Formula, a cubic hypersurface in Formula with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over Formula that intersect nontrivially.

Communicated by Andrei Zelevinsky



References

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