Branched pull-back components of the space of codimension 1 foliations on ${\mathbb{P}}^n$

Abstract

Let $\mathcal{F}$ be written as $f^{\ast }\mathcal{G}$, where $\mathcal{G}$ is a foliation in ${\mathbb{P}}^2$ with three invariant lines in general position, say $(XYZ)=0$, and $f:{\mathbb{P}}^n--->{\mathbb{P}}^2$, $f=(F_0^{\alpha }:F_1^{\beta }:F_2^{\gamma })$ is a nonlinear rational map. Using local stability results of singular holomorphic foliations, we prove that: if $n\ge 3$, the foliation $\mathcal{F}$ is globally stable under holomorphic deformations. As a consequence we obtain new irreducible componentes for the space of codimension one foliations on ${\mathbb{P}}^n$. We present also a result which characterizes holomorphic foliations on ${\mathbb{P}}^n,n\ge 3$ which can be obtained as a pull back of foliations on ${\mathbb{P}}^2$ of degree $d\ge 2$ with three invariant lines in general position.

arXiv admin note: text overlap with arXiv:1503.00715