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 dimensio n n are the same as those of the Grassmanian of d-planes in the complex vec tor space of dimension n + d. In fact, we give a correspondence which is th e graph of a rational morph ism which induces an isomorphism, and whose mat rix is the identity. We also prove some properties of Euler-Chow series and state some open problems related to this series. (C) 2002 Elsevier Science B.V. All rights reserved.