The Noether–Fano inequalities for codimension one singular holomorphic foliations

Abstract

The idea of the proof of the classical Noether-Fano inequalities can be adapted to the domain of codimension one singular holomorphic foliations of the projective space. We obtained criteria for proving that the degree of a foliation on the plane is minimal in the birational class of the foliation and for the non-existence of birational symmetries of generic foliations (except automorphisms). Moreover, we give several examples of birational symmetries of special foliations illustrating our results.