It is known that each germ of codimension one holomorphic foliation has at least one invariant hypersurface in the non dicritical case. The proof starts with the existence of invariant curve in the two dimensional situation, it continues in dimension three based on the reduction of singularities and a construction of the invariant hypersurface as a germ, after reduction of singularities, over the so called ``partial separatrices’’. In higher dimension the result is mainly cohomological , once we know the three-dimensional case. When the foliation is dicritical, the result is not true in general as it is shown by Jouanolou’s examples. Anyway, the three dimensional arguments may be extended for the dicritical case with a notion of ‘’extended partial separatrix’’ and, in this way, we obtain an equivalent notion of the existence of invariant hypersurface. The main positive result we know is for the class of toric type foliations, very close to the Newton nondegenerate foliations. We give details of the study of this dicritical situation. Finally, when there are no invariant hypersurface, we propose that a local version of Brunella’s alternative holds and we will show that this is so for the class of foliations with relatively isolated reduction of singularities.

(Work in collaboration with Beatriz Molina-Samper)