  We show that the singular holomorphic foliations induced by dominant
quasi-homogeneous rational maps fill out irreducible components of the space
$\mathscr F_q(r, d)$ 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.
