Complements and higher resonance varieties of hyperplane arrangements

Details Complements and higher resonance varieties of hyperplane arrangements Complements and higher resonance varieties of hyperplane arrangements

2011-03-21

arxiv.org/abs/1103.3930v2

Mathematics

Algebraic Geometry

v2: typos corrected, a remark and a reference added; to appear in Math. Res. Lett

Scientific paper

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal.

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