Hypersurfaces quasi-invariant by codimension one foliations

TL;DR

This paper extends classical theorems to characterize certain foliations on higher-dimensional varieties, revealing their structure and relation to rational maps, with applications to foliations on 3-folds.

Contribution

It introduces a new characterization of foliations as pull-backs of surface foliations, generalizing the Darboux-Jouanolou Theorem.

Findings

Foliations on 3-folds with infinitely many extremal rays have a specific structure.

A variant of the Darboux-Jouanolou Theorem is established.

Foliations can be described as pull-backs via rational maps.

Abstract

We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays.