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Speaker: Mike Falk (Northern Arizona University)

Title: Resonant weights and Hessian pencils, resonance varieties and line complexes

Abstract: (pdf version) We will survey some interesting aspects of the theory of resonance varieties of Orlik-Solomon algebras. Loosely speaking, resonance varieties are multiplicative invariants of graded algebras, consisting of resonant weights, which are degree one elements whose annihiliators are larger than expected. The Orlik-Solomon algebra of a matroid, with coeffcients in an integral domain R, appears as the cohomology algebra of the complement of a complex hyperplane arrangement. It is given by a presentation which depends on the matroid associated with the arrangement. There is a nice combinatorial characterization of resonant weights in this setting. Even so, matroids supporting resonant weights are rare and special.

For R = C (complex numbers), a resonant weight gives rise to a pencil of plane curves whose singular elements include the hyperplanes of the original arrangement. In most cases this gives rise to a fibering of the complement by nonsingular curves, implying that the complement is aspherical. The Hessian arrangement is the primordial example.

For R = Z_p, components of resonance varieties are the carriers of nontrivial line complexes, algebraic subsets of the Grassmannian of lines in projective space. The complexes which arise are intersections of Schubert varieties in special position. For p = 3, the Hessian arrangement provides an example where the result is a cubic threefold with interesting line structure.

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