Foliations on $\mathbb{P}^2$ with only one singular point

TL;DR

This paper investigates holomorphic foliations on the complex projective plane with a single singular point, showing that if the singularity has multiplicity one, the foliation lacks invariant algebraic curves, and provides examples in degree three.

Contribution

It establishes a new result linking singularity multiplicity to the absence of invariant algebraic curves and offers explicit examples in degree three.

Findings

Foliations with a single multiplicity-one singularity have no invariant algebraic curves.

Explicit examples of such foliations are constructed in degree three.

The study advances understanding of the structure of holomorphic foliations on projective planes.

Abstract

In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples of such foliations in degree three.