Foliations with isolated singularities on Hirzebruch surfaces

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