Enumeration of hypersurfaces with prescribed non-isolated singular subschemes

Abstract

Let W be an irreducible subvariety of a Hilbert scheme Hilb P W (t) (P n ). We show under mild hypothesis that there are polynomial formulas for the degrees of the loci of hypersurfaces in P n with singular subschemes containing some member of the family W. The formulas are made explicit in a number of cases.