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Completely reducible hypersurfaces in a pencil

2008; Elsevier BV; Volume: 219; Issue: 2 Linguagem: Inglês

10.1016/j.aim.2008.05.014

1090-2082

Jorge Vitório Pereira, S. Yuzvinsky,

Geometric and Algebraic Topology

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.