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

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

2009-01-07

arxiv.org/abs/0901.0745v6

Mathematics

Geometric Topology

Scientific paper

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 ${k+2\choose k}$, 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 $\leq k$. Moreover, there exist a number $\mathscr{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus smaller than $\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$.

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