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Foliations in algebraic surfaces having a rational first integral. (English) Zbl 0910.32039
Summary: Given a foliation \({\mathcal F}\) in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case \(S= \mathbb{P}^2\) some new counter-examples to the classical formulation of the Poincaré problem are presented. If \(S\) is a rational surface and \({\mathcal F}\) has singularities of type (1,1) or \((1,-1)\) we prove that the general solution is a nonsingular curve.

MSC:
32S65 Singularities of holomorphic vector fields and foliations
14J26 Rational and ruled surfaces
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