Foliations with isolated singularities on Hirzebruch surfaces

Abstract

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