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Let be a foliation on a complex, smooth and irreducible projective surface , assume admits a holomorphic first integral . If for some we prove the inequality: . If is rational we prove that the direct image sheaves of the co-normal sheaf of under are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.
Given a foliation in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P some new counter-examples to the classic formulation of the Poincaré problem are presented. If S is a rational surface and has singularities of type (1, 1) or (1,-1) we prove that the general solution is a non-singular curve.
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