  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 in order to investigate
the effective algebraic integration of foliations on the projective plane. In
particular, we describe the Zariski closure of the set of foliations on the
projective plane of degree d admitting rational first integrals with fibers
having geometric genus bounded by g.
