Polynomial vector fields with algebraic trajectories

Abstract

It is known after Jouanolou that a general holomorphic foliation of degree $\ge 2$ in projective space has no algebraic leaf. We give formulas for the degrees of the subvarieties of the parameter space of one-dimensional foliations that correspond to foliations endowed with some invariant subvariety of degree 1 or 2 and dimension $\ge 1$.

arXiv comment16 pages, including scripts