Non-algebraicity of foliations via reduction modulo $2$

TL;DR

This paper introduces a criterion based on reduction modulo 2 to prove the non-algebraicity of certain foliations on the complex projective plane, providing new proofs and examples of foliations without algebraic invariant curves.

Contribution

It develops a novel criterion using reduction modulo 2 to establish the absence of algebraic invariant curves in foliations, including a new proof for Jouanolou's foliation.

Findings

Jouanolou foliation of odd degree has no algebraic invariant curves

New criterion effectively proves non-algebraicity of foliations

Identifies classes of foliations without algebraic invariant curves

Abstract

Motivated by the Jouanolou foliation problem, we investigate the non-algebraicity of foliations by curves on $\mathbb{P}^2_{\mathbb{C}}$. We present a criterion to show that such a foliation has no algebraic invariant curves, using a method of reduction modulo $2$. Finally, using this criterion, we give a new proof that the Jouanolou foliation of odd degree has no algebraic invariant curves. We also present other classes of foliations without algebraic invariant curves.