On the topology of foliations with a first integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle generates the first homology group of a generic fiber of. In particular, we will prove that for any two Lefschetz vanishing cycles₀ and₁ in a regular compact fiber of, there exists a mondromyh such thath(₀)=±₁.Partially supported by CNPq-Brazil.