Rational pullbacks of toric foliations

Abstract

This article is dedicated to the study of singular codimension $1$ foliations $\mathcal{F}$ on a simplicial complete toric variety $X$ and their pullbacks by dominant rational maps $\phi :{\mathbb{P}}^n⇢X$. First, we describe the singularities of $\mathcal{F}$ and ${\phi }^{\ast }\mathcal{F}$ for a generic pair $(\phi ,\mathcal{F})$. Then we show that the first order deformations of ${\phi }^{\ast }\mathcal{F}$ arising from first order unfoldings are the families of the form ${\phi }_{\epsilon }^{\ast }\mathcal{F}$, where ${\phi }_{\epsilon }$ is a perturbation of $\phi$. We also prove that the deformations of the form ${\phi }^{\ast }{\mathcal{F}}_{\epsilon }$ consist exactly of the families which are tangent to the fibers of $\phi$. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.

24 pages. Theorem 4.6 was improved. Accepted for publication in Forum Mathematicum. Manuscript DOI: 10.1515/forum-2022-0265