Foliations with isolated singularities on Hirzebruch surfaces

Abstract

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