Foliations by curves uniquely determined by minimal subschemes of its singularities

Abstract

It is well-known that a foliation by curves of degree greater than or equal to two, with isolated singularities, in the complex projective space of dimension greater than or equal to two, is uniquely determined by the scheme of its singular points. The main result in this paper is that the set of foliations which are uniquely determined by a subscheme (of the minimal possible degree) of its singular points, contains a nonempty Zariski-open subset. Our results hold in the projective space defined over any algebraically closed ground field.