Foliations on the projective plane with finite group of symmetries

TL;DR

This paper classifies singular holomorphic foliations on the projective plane with finite automorphism groups, determines the maximum size of these groups for a fixed degree, and explicitly describes the foliations that achieve this maximum.

Contribution

It provides a complete classification of foliations with large finite automorphism groups and identifies the maximal automorphism group size for given degrees.

Findings

Maximal automorphism group size for fixed degree determined

Explicit examples of foliations with maximal symmetry provided

Classification of foliations with large finite automorphism groups achieved

Abstract

Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.