Projective Foliations with a Unique Singular Point and Maximum Algebraic Multiplicity

Abstract

The study of codimension-one holomorphic foliations on the projective plane ${\mathbb{CP}}^2$ with a unique singular point has attracted increasing interest because of its connections with several important problems. In this paper, we focus on holomorphic foliations of degree $d$ having a singular point of maximal multiplicity $d$. We describe explicitly the foliations in this class and characterize those having a unique singular point. We also state some results concerning Hilbert-Mumford stability for these foliations.