Effective algebraic integration in bounded genus

We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results to investigate the effective algebraic integration of foliations on the projective plane. In particular, we describe the Zariski closure of the set d,g of foliations on P 2 of degree d admitting rational first integrals with fibers having geometric genus bounded by g.