Completely reducible hypersurfaces in a pencil

Abstract

We study completely reducible fibers of pencils of hypersurfaces on ${\mathbb{P}}^n$ and associated codimension one foliations of ${\mathbb{P}}^n$. Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of $n$. Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements.

15 pages, 2 figures

References in corpus (3)

• Old and new examples of k-nets in P^2
• The line geometry of resonance varieties
• Characteristic varieties and constructible sheaves

Cited by in corpus (26)

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• The Classification of Exceptional CDQL Webs on Compact Complex Surfaces
• Chen ranks and resonance
• Characteristic varieties and Betti numbers of free abelian covers
• Hypersurfaces Invariant by Pfaff Equations
• Resonance varieties and Dwyer-Fried invariants
• On the total order of reducibility of a pencil of algebraic plane curves
• On line arrangements with applications to 3-nets
• Classification of k-nets
• A new bound on the number of special fibers in a pencil of curves
• The Only Complex 4-Net Is the Hesse Configuration
• k-nets embedded in a projective plane over a field
• Group-labeled light dual multinets in the projective plane (with Appendix)
• Sigma-invariants and tropical varieties
• Efficiency and complexity of hyperplane arrangements
• On the topology of the Milnor fibration of a hyperplane arrangement
• 3-nets realizing a group in a projective plane
• Foliations with degenerate Gauss maps on ${\mathbb{P}}^4$
• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties
• Fibers of pencils of curves on smooth surfaces
• Light dual multinets of order six in the projective plane
• Resonant bands, Aomoto complex, and real 4-nets
• 3-nets realizing a diassociative loop in a projective plane