Rational pullbacks of toric foliations

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.