Effective algebraic integration in bounded genus

TL;DR

This paper develops birational invariants for foliations on projective surfaces and applies them to analyze algebraic integrability of foliations on the projective plane, especially those with bounded genus fibers.

Contribution

It introduces new birational invariants based on the adjoint linear series and uses them to study algebraic integrability of plane foliations with bounded genus fibers.

Findings

Characterization of the Zariski closure of foliations with rational first integrals

Description of conditions for algebraic integrability based on genus bounds

New tools for studying foliations on projective surfaces

Abstract

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.