  We study the structure of the relative Hilbert scheme for a family of nodal
(or smooth) curves via its natural cycle map to the relative symmetric product.
We show that the cycle map is the blowing up of the discriminant locus, which
consists of cycles with multiple points. We discuss some applications and
connections, notably with birational geometry and intersection theory on
Hilbert schemes of smooth surfaces. Revised version corrects some minor errors.
