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The Poincaré problem in the nondicritical case. (English) Zbl 0821.32026

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 \(L\).

MSC:

32S65 Singularities of holomorphic vector fields and foliations
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