Multinets, resonance varieties, and pencils of plane curves

Michael Falk, Sergey Yuzvinsky

Research output: Contribution to journalArticle

62 Scopus citations

Abstract

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a 'multinet', a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.

Original languageEnglish (US)
Pages (from-to)1069-1088
Number of pages20
JournalCompositio Mathematica
Volume143
Issue number4
DOIs
StatePublished - Jul 1 2007

Keywords

  • Line arrangement
  • Matroid
  • Net
  • Orlik-Solomon algebra
  • Pencil
  • Resonance variety

ASJC Scopus subject areas

  • Algebra and Number Theory

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