Pull-back components of the space of holomorphic foliations on \(\mathbb{C}\mathbb{P}(n)\), \(n\geqq 3\) (Q2769831)

It is known that the set \({\mathcal F}(d;n)\) of holomorphic singular foliations of degree \(d\) on \({\mathbb C P}^n\) can be identified with the projectivization of the space \(\{\omega=\sum_{j=0}^nA_j(z)dz_j: A_j\) is a homogeneous polynomial of degree \(d+1\) on \({\mathbb C}^{n+1};\;\sum_{j=0}^nz_jA_j(z)\equiv 0;\;\omega\wedge d\omega\equiv 0;\;\text{ codim}_{\mathbb C}(S(\omega))\geq 2\}\).NEWLINENEWLINENEWLINEThe authors show that the set of foliations \({\mathcal F}\) of \({\mathbb C P}^n\) that can be written as \({\mathcal F}=F^*({\mathcal G})\), where \({\mathcal G}\) is a foliation of degree \(d\) in \({\mathbb C P}^2\) and \(F: {\mathbb C P}^n\to{\mathbb C P}^2\) is a generic rational map of degree \(\nu\), is an irreducible component of \({\mathcal F}((d+2)\nu-2;n)\).NEWLINENEWLINENEWLINEThe proof involves small deformations of complete intersections.