Stability of foliations induced by rational maps

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space ${\mathcal{F}}_q(r,d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space ${\mathbb{P}}^r$, when $1\le q\le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.