Geometry and combinatorics of resonant weights

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Citation (Scopus)

Abstract

Let A be an arrangement of n hyperplanes in ℂ. Let k be a field and A=⊕p=0 A p the Orlik-Solomon algebra of A over k. The p th resonance variety of A over k is the set Rp (A, k) of one-forms a∈A 1 annihilated by some b∈A p\(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of A-master functions Φa. For the most part we focus on the case p=1. We will describe the features of R 1 (A, k) for k = 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 R 1 (A, k), and consequently obtain a precise description of R 1 (A, k) as a ruled variety. We sketch the description of components of R 1 (A, C) 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, R 1 (A, k) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of R 1 (A, k) 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 ∈ R p (A, k) and the critical loci of the corresponding master functions Φa. For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Pages155-176
Number of pages22
Volume283
DOIs
StatePublished - Jan 1 2010

Publication series

NameProgress in Mathematics
Volume283
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

Fingerprint

Combinatorics
Positive Characteristic
Pencil of planes
Algebra
Local System
Single valued
Plane Curve
Intersect
Classification Problems
Hyperplane
Locus
Cohomology
Arrangement
Union
Branch
Necessary Conditions
Sufficient Conditions
Zero
Form

Keywords

  • Arrangement
  • Local system cohomology
  • Master function
  • Multinet
  • Net
  • Orlik-Solomon algebra
  • Pencil
  • Resonance variety

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Cite this

Falk, M. J. (2010). Geometry and combinatorics of resonant weights. In Progress in Mathematics (Vol. 283, pp. 155-176). (Progress in Mathematics; Vol. 283). Springer Basel. https://doi.org/10.1007/978-3-0346-0209-9_6

Geometry and combinatorics of resonant weights. / Falk, Michael J.

Progress in Mathematics. Vol. 283 Springer Basel, 2010. p. 155-176 (Progress in Mathematics; Vol. 283).

Research output: Chapter in Book/Report/Conference proceedingChapter

Falk, MJ 2010, Geometry and combinatorics of resonant weights. in Progress in Mathematics. vol. 283, Progress in Mathematics, vol. 283, Springer Basel, pp. 155-176. https://doi.org/10.1007/978-3-0346-0209-9_6
Falk MJ. Geometry and combinatorics of resonant weights. In Progress in Mathematics. Vol. 283. Springer Basel. 2010. p. 155-176. (Progress in Mathematics). https://doi.org/10.1007/978-3-0346-0209-9_6
Falk, Michael J. / Geometry and combinatorics of resonant weights. Progress in Mathematics. Vol. 283 Springer Basel, 2010. pp. 155-176 (Progress in Mathematics).
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