Brunella's algebraic separatrix conjecture for foliations on projective spaces

Let $d4d5$ be a codimension one foliation on $d4d9^n$, with $n d45e 3$. An invariant algebraic hypersurface is an algebraic hypersurface preserved by $d4d5$; $d4d5$ is everywhere tangent to a foliation by codimension two algebraic subvarieties when its tangent directions are contained in those of such a foliation. Brunella's conjecture. Every codimension one foliation on $d4d9^n$ either admits an invariant algebraic hypersurface or is everywhere tangent to a foliation by codimension two algebraic subvarieties. The conjecture motivated the study of non-dicritical foliations; the paper proves the first alternative for non-dicritical foliations on $d4d9^n$ when $n d45e 4$, but does not settle the conjecture in general, including dimension three.