Rational pullbacks of toric foliations

TL;DR

This paper investigates the singularities and deformation behavior of codimension 1 toric foliations and their pullbacks via rational maps, revealing how these structures deform and relate to each other.

Contribution

It provides a detailed description of singularities and deformation families of pulled-back toric foliations, introducing new insights into their unfolding and tangent deformation properties.

Findings

Characterization of singularities of foliations and their pullbacks

Identification of deformation families arising from perturbations

Connection between deformations and tangent to fibers of rational maps

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 $\varphi:\mathbb{P}^n\dashrightarrow X$. First, we describe the singularities of $\mathcal{F}$ and $\varphi^*\mathcal{F}$ for a generic pair $(\varphi,\mathcal{F})$. Then we show that the first order deformations of $\varphi^*\mathcal{F}$ arising from first order unfoldings are the families of the form $\varphi_\varepsilon^*\mathcal{F}$, where $\varphi_\varepsilon$ is a perturbation of $\varphi$. We also prove that the deformations of the form $\varphi^*\mathcal{F}_\varepsilon$ consist exactly of the families which are tangent to the fibers of $\varphi$. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.