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

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