Foliations with isolated singularities on Hirzebruch surfaces

TL;DR

This paper investigates foliations with isolated singularities on Hirzebruch surfaces, showing how their singular schemes relate to the foliations themselves, with distinctions based on the value of .

Contribution

It extends the understanding of foliation singularities from the projective plane to Hirzebruch surfaces, characterizing when the singular scheme determines the foliation.

Findings

For , the singular scheme determines the foliation with some exceptions.

For f1, the singular scheme generally does not determine the foliation.

Two foliations have the same singular scheme iff their defining sections are related by a global endomorphism.

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 $S_\delta$.