A universal property of the Cayley-Chow space of algebraic cycles

For every projective scheme X and projective embedding e : X \rightarrow P^r, there is a reduced scheme C_n(X,e) parametrizing cycles of pure dimension n with support in X. The construction is carried over in terms of the given embedding, so the space of cycles might in principle depend on the embedding. This may actually happen in positive characteristics, as shown by Nagata. We therefore confine ourselves to complex schemes. The question arises of describing (without reference to the embedding) which kind of families are obtained through morphisms T \rightarrow C_n(X,e) as pullbacks of a universal family over C_n(X,e). The solution proposed by Barlet is in terms of the local properties of the structural projection of a family. We show that, working in the category of reduced projective schemes and semi-regular maps, a simpler solution exists. More precisely, we define a functor of regular relative cycles on this category, and we prove that the space of cycles C_n(X,e) represents this functor. This implies the earlier result of Andreotti-Norguet, that the semi-normalization of C_n(X,e) is essentially independent of the embedding.