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.This collaboration initiated while both authors where visiting
James McKernan at UCSD, and continued during a visit of the second author to IMPA.
We are grateful to both institutions for the favorable working conditions.
The first author is partially supported by Cnpq and FAPERJ.
The second author was partially supported by NSF research grant no: 1200656 and no: 1265263.
During the final revision of this work he was supported by funding from the
European Union's Seventh Framework Programme (FP7/2007-2013)/ERC
Grant agreement no. 307119