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Foliations with isolated singularities on Hirzebruch surfaces

We study foliations $\mathcal{F}$ on Hirzebruch surfaces $S_δ$ 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 $ δ=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 $δ\neq 1$, 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 $s'=Φ(s)$, for some global endomorphism $Φ$ of the tangent bundle of $S_δ$.