Multinets, resonance varieties, and pencils of plane curves

By Michael J. Falk, Sergey Yuzvinsky. Compositio Mathematica, 2006

132 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1a5d28a93e12e869ef1f0cff247be19a7d123594

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In Computational Mathematics

1. Stefan Papadima
2. Morihiko Saito
3. Alexander I. Suciu

In Computational Theory and Mathematics

1. Stefan Papadima
2. Alexander I. Suciu
3. Morihiko Saito
4. P. Kronheimer
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In Geometry and Topology

1. Alexander I. Suciu
2. Stefan Papadima
3. Morihiko Saito
4. Nero Budur

Citations in the corpus

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy (2014, methodology): “We will consider only simple matroids, defined by the condition that all subsets of size at most 2 are independent.”
• Hyperplane arrangements and Milnor fibrations (2013, background): “Given an arrangement A, find the primes p (if any) for which M(A) is not k-formal, over a field k of characteristic p.”
• Bernstein–Sato polynomials of hyperplane arrangements (2006, background): “1 ∈ Zd.) We thus get the quasi-isomorphism (2.1.1), and it is enough to show (5.2.2) H1(A• g,α,ω α∧) = 0. By the theory on the kernel of the differential ωα∧ of the Aomoto complexes (see [Fa], [LiYu], [FaYu], etc.), the assertion is then reduced to the following (see [BSY], [BDS, Section 1.5]): Lemma”