Some remarks on Chow varieties and Euler-Chow series

In this article we give a geometric explanation of the fact that the Betti numbers of the d-fold symmetric product of the proyective space of dimension n are the same as those of the Grassmanian of d-planes in the complex vector space of dimension n + d. In fact, we give a correspondence which is the graph of a rational morphism which induces an isomorphism, and whose matrix is the identity. We also prove some properties of Euler-Chow series and state some open problems related to this series.