Foliations singular along a curve

Abstract

A general one-dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes in ℙ n , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d. We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive-dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.