Foliations with isolated singularities on Hirzebruch surfaces

Abstract

We study foliations ℱ on Hirzebruch surfaces Sδ and prove that, similarly to those on the projective plane, any ℱ can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ has isolated singularities, we show that, for δ=1, the singular scheme of ℱ does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ≠1, we prove that the singular scheme of ℱ does not determine the foliation. However, we prove that, in most cases, two foliations ℱ and ℱ′ given by sections s and s′ have the same singular scheme if and only if s′=Φ(s), for some global endomorphism Φ of the tangent bundle of Sδ.