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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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