 There is an alternative interpretation of the Hilbert scheme which leads to a generalization of relative Hilbert schemes parameterizing subschemes of a relative scheme.For a fixed base scheme formula_46, let formula_47 and letformula_48be the functor sending a relative scheme formula_49 to the set of isomorphism classes of the setformula_50where the equivalence relation is given by the isomorphism classes of formula_51.This construction is functorial by taking pullbacks of families.