Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety

Abstract

In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants of $D$ and $X$. Particularly, if $\F$ is a foliation of degree $d$ on ${\mathbb{P}}_{\mathbb{C}}^2$, whose the number of invariants curves is greater that $\left(\genfrac{}{}{0ex}{}{k+2}{k}\right)$, we show that there exist a number $\mathcal{M}(d,k)$ such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral of degree $\le k$. Moreover, there exist a number $\mathcal{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus smaller than $\mathcal{G}(d,k)$, then it has a rational first integral of degree $\le k$.