Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-31T03:55:27.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Multinets, resonance varieties, and pencils of plane curves

Published online by Cambridge University Press:  17 July 2007

Michael Falk  and
Sergey Yuzvinsky
Michael Falk
Affiliation:
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011-5717, USA michael.falk@nau.edu
Sergey Yuzvinsky
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 94703, USA yuz@math.uoregon.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

resonance varietynetpencilline arrangementmatroidOrlik–Solomon algebra52C3514H1005B30
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 4 , July 2007 , pp. 1069 - 1088
Copyright
Foundation Compositio Mathematica 2007