Birational geometry of foliations associated to simple derivations

TL;DR

This paper investigates a special class of foliations on the projective plane induced by simple derivations, analyzing their classification, symmetry properties, and broader implications in birational geometry.

Contribution

It characterizes the birational classification of these foliations and proves the finiteness of their birational symmetries, extending results to wider classes.

Findings

Foliations have no invariant algebraic curves or singularities outside a line.

Established the position of these foliations in the birational classification.

Proved the finiteness of their birational symmetry groups.

Abstract

We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in the complement of a line. We establish the position of these foliations in the birational classification of foliations and prove the finiteness of their birational symmetries. Most of the results apply to wider classes of foliations.