Degrees of Cayley-Chow varieties

The Cayley-Chow variety C_{n,k} (P^r), parametrizing cycles of dimension n and degree k in projective space P^r, comes out equipped with a natural embedding into a certain projective space F_{n,k}. The main result of the paper is a formula expressing the degree of a subvariety T \subset C_{n,k} (P^r) in terms of the type, the sequence of multiplicities of irreducible components of the general cycle of the family parametrized by T, and a collection of indices, numbers of cycles in the family which satisfy certain incidence conditions with respect to linear spaces of codimension n + 1 in P^r. The total index \overline\mu, the sum of all indices, is the number of cycles which meet m general linear spaces, where m = dim T, so it is what in enumerative geometry is called a characteristic number of the family. These indices depend both intrinsically on the family of cycles and on the projective characters of the carrier X \subset P^r, the locus filled in with cycles of the family (this is seen in the case of divisors). In particular, if the family is of simple type, i.e. the general cycle is irreducible, then deg T = \overline\mu. If moreover it is a family of divisors in a variety X, then \overline\mu = d^m \overline\nu, where d = deg X and \overline\nu is the number of divisors passing through m general points of X. As an application, a characterization of linear systems of divisors is derived.