  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.

Algebraic Geometry;Combinatorics