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Geometry and Combinatorics of Resonant Weights

  • Michael Falk
Part of the Progress in Mathematics book series (PM, volume 283)

Abstract

Let \( \mathcal{A} \) be an arrangement of n hyperplanes in ℂ. Let \( \Bbbk \) be a field and A=⊕ p=0 A p the Orlik-Solomon algebra of \( \mathcal{A} \) over \( \Bbbk \). The p th resonance variety of \( \mathcal{A} \) over \( \Bbbk \) is the set \( \mathcal{R}^p \left( {\mathcal{A},\Bbbk } \right) \) of one-forms aA 1 annihilated by some bA p \(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of \( \mathcal{A} \)-master functions Φ a .

For the most part we focus on the case p=1. We will describe the features of R 1 (\( \left( {\mathcal{A},\Bbbk } \right) \)) for \( \Bbbk = \mathbb{C} \) and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element a to lie in \( \mathcal{R}^1 \left( {\mathcal{A},\Bbbk } \right) \), and consequently obtain a precise description of \( \mathcal{R}^1 \left( {\mathcal{A},\Bbbk } \right) \) as a ruled variety. We sketch the description of components of \( \mathcal{R}^1 \left( {\mathcal{A},\mathbb{C}} \right) \) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, \( \mathcal{R}^1 \left( {\mathcal{A},\Bbbk } \right) \) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of \( \mathcal{R}^1 \left( {\mathcal{A},\Bbbk } \right) \) do not intersect trivially.

We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms \( a \in \mathcal{R}^p \left( {\mathcal{A},\Bbbk } \right) \) and the critical loci of the corresponding master functions Φ a . For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.

Mathematics Subject Classification (2000)

Primary 32S22 Secondary 52C35 55N25 14C21 

Keywords

Arrangement Orlik-Solomon algebra local system cohomology resonance variety master function net multinet pencil 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Michael Falk
    • 1
  1. 1.Department of Mathematics and StatisticsNorthern Arizona UniversityFlagstaffUSA

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