Cycle map on Hilbert schemes of nodal curves

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.