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Sheaves associated to holomorphic first integrals

Alexis García Zamora — 2000

Annales de l'institut Fourier

Let : L T S be a foliation on a complex, smooth and irreducible projective surface S , assume admits a holomorphic first integral f : S 1 . If h 0 ( S , 𝒪 S ( - n 𝒦 S ) ) > 0 for some n 1 we prove the inequality: ( 2 n - 1 ) ( g - 1 ) h 1 ( S , ' - 1 ( - ( n - 1 ) K S ) ) + h 0 ( S , ' ) + 1 . If S is rational we prove that the direct image sheaves of the co-normal sheaf of under f are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.

Foliations in algebraic surfaces having a rational first integral.

Alexis García Zamora — 1997

Publicacions Matemàtiques

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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