Sheaves associated to holomorphic first integrals

Let F : L --> TS be a foliation on a complex, smooth and irreducible projec tive surface S, assume F admits a holomorphic first integral f : S --> P-1. If h(0)(S, O-S(-nK(S))) > 0 for some n greater than or equal to 1 we prove the inequality : (2n - 1)(g - 1) less than or equal to h(1)(S, L'(-1)(-(n - 1)K-S)) + h(0)(S, L') + 1. If S is rational we prove that the direct imag e sheaves of the co-normal sheaf of F under f are locally free; and give so me information on the nature of their decomposition as direct sum of invert ible sheaves.