Stability of branched pull-back projective foliations

Abstract

We prove that, if $n\ge 3$, a singular foliation $\mathcal{F}$ on ${\mathbb{P}}^n$ which can be written as pull-back, where $\mathcal{G}$ is a foliation in ${\mathbb{P}}^2$ of degree $d\ge 2$ with one or three invariant lines in general position and $f:{\mathbb{P}}^n--->{\mathbb{P}}^2$, $deg(f)=\nu \ge 2,$ is an appropriated rational map, is stable under holomorphic deformations. As a consequence we conclude that the closure of the sets $\{\mathcal{F}=f^{\ast }(\mathcal{G})\}$ are new irreducible components of the space of holomorphic foliations of certain degrees.

arXiv admin note: substantial text overlap with arXiv:1503.07827, arXiv:1503.00715