Foliations on $$\mathbb {CP}^2$$ with a unique singular point without invariant algebraic curves

Abstract

We prove that a foliation on $$\mathbb {CP}^2$$ of degree d with a singular point of type saddle-node with Milnor number $$d^2+d+1$$ does not have invariant algebraic curves. We give a family of this kind of foliations. We also present a family of foliations of degree d with a unique nilpotent singularity without invariant algebraic curves for d odd greater than 1. Finally we prove that the space of foliations on $$\mathbb {CP}^2$$ of degree $$d \ge 2$$ with a unique singular point has dimension at least $$3d+2$$ .