Polynomial vector fields with algebraic trajectories

TL;DR

This paper investigates polynomial vector fields with algebraic trajectories, providing formulas for degrees of subvarieties in the parameter space related to foliations with invariant subvarieties of degree 1 or 2.

Contribution

It introduces formulas for degrees of subvarieties of the parameter space of foliations with specific invariant subvarieties, extending understanding of algebraic trajectories.

Findings

Formulas for degrees of subvarieties with invariant degree 1 or 2

Characterization of foliations with algebraic trajectories

Extension of Jouanolou's results to specific cases

Abstract

It is known after Jouanolou that a general holomorphic foliation of degree $\geq2$ in projective space has no algebraic leaf. We give formulas for the degrees of the subvarieties of the parameter space of one-dimensional foliations that correspond to foliations endowed with some invariant subvariety of degree 1 or 2 and dimension $\geq1$.