Stability of foliations induced by rational maps

TL;DR

This paper investigates the structure and geometry of singular holomorphic foliations induced by dominant quasi-homogeneous rational maps on complex projective spaces, revealing their irreducibility, rationality, and degree properties.

Contribution

It demonstrates that these foliations form irreducible components of the foliation space, proves their rationality, and computes their degrees in specific cases.

Findings

Foliations fill out irreducible components of the foliation space.

All such components are rational varieties.

Projective degrees are computed in several cases.

Abstract

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.