Stability of foliations induced by rational maps

Cukierman, F.; Pereira, J. V.; Vainsencher, I.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 4, pp. 685-715.

Nous montrons que les feuilletages holomorphes induits par les applications rationnelles quasi-homogènes remplissent les composantes irréductibles de l’espace ${ℱ}_q(r,d)$ des feuilletages de codimension $q$ et degré $d$ de l’espace projectif ${\mathbb{P}}^r$ pour tout $1\le q\le r-2$. Nous étudions la géométrie de telles composantes irréductibles. Nous montrons que ce sont des variétés rationnelles et calculons leur degré dans plusieurs cas.

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

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