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 $\delta \ne 1$, 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 ${\mathcal{F}}^{\prime }$ given by sections $s$ and $s^{\prime }$ have the same singular scheme if and only if $s^{\prime }=\Phi (s)$, for some global endomorphism $\Phi$ of the tangent bundle of $S_{\delta }$.

References in corpus (2)

• On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations
• Foliations by Curves with Curves as Singularities