Multinets, resonance varieties, and pencils of plane curves

Michael J Falk, Sergey Yuzvinsky

Research output: Contribution to journalArticle

60 Citations (Scopus)

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 2007

Fingerprint

Pencil of planes
Plane Curve
Arrangement
Line
Multiplicity
Connectivity
Projective plane
Argand diagram
Complement
Intersection
Partition
Fiber
If and only if
Restriction
Class

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Multinets, resonance varieties, and pencils of plane curves. / Falk, Michael J; Yuzvinsky, Sergey.

In: Compositio Mathematica, Vol. 143, No. 4, 07.2007, p. 1069-1088.

Research output: Contribution to journalArticle

Falk, Michael J ; Yuzvinsky, Sergey. / Multinets, resonance varieties, and pencils of plane curves. In: Compositio Mathematica. 2007 ; Vol. 143, No. 4. pp. 1069-1088.
@article{bb3799e8a8c44d67b28776d140d4c608,
title = "Multinets, resonance varieties, and pencils of plane curves",
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.",
keywords = "Line arrangement, Matroid, Net, Orlik-Solomon algebra, Pencil, Resonance variety",
author = "Falk, {Michael J} and Sergey Yuzvinsky",
year = "2007",
month = "7",
doi = "10.1112/S0010437X07002722",
language = "English (US)",
volume = "143",
pages = "1069--1088",
journal = "Compositio Mathematica",
issn = "0010-437X",
publisher = "Cambridge University Press",
number = "4",

}

TY - JOUR

T1 - Multinets, resonance varieties, and pencils of plane curves

AU - Falk, Michael J

AU - Yuzvinsky, Sergey

PY - 2007/7

Y1 - 2007/7

N2 - 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.

AB - 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.

KW - Line arrangement

KW - Matroid

KW - Net

KW - Orlik-Solomon algebra

KW - Pencil

KW - Resonance variety

UR - http://www.scopus.com/inward/record.url?scp=34547183398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547183398&partnerID=8YFLogxK

U2 - 10.1112/S0010437X07002722

DO - 10.1112/S0010437X07002722

M3 - Article

VL - 143

SP - 1069

EP - 1088

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 4

ER -