The Poincaré problem in the nondicritical case (Q1841984)

Let \({\mathcal F}\) be a foliation of \(\mathbb{P}^ 2_ \mathbb{C}\) and \(C\) an algebraic curve invariant by \({\mathcal F}\) having no dicritical singularities of \({\mathcal F}\). It is proved that \(\partial^ 0 C \leq \partial^ 0 {\mathcal F} + 2\); here \(\partial^ 0 {\mathcal F}\) is the number of points of a generic line such that the leaf of \({\mathcal F}\) passing through such a point is tangent to \(L\).