  We study the relative Hilbert scheme of a family of nodal (or smooth) curves,
over a base of arbitrary dimension, via its (birational) cycle map, going to
the relative symmetric product. We show the cycle map is the blowing up of the
discriminant locus, which consists of cycles with multiple points. We work out
the action of the blowup or 'discriminant' polarization on some natural cycles
in the Hilbert scheme, including generalized diagonals and cycles, called 'node
scrolls', parametrizing schemes supported on singular points. We derive an
intersection calculus for Chern classes of tautological vector bundles, which
are closely related to enumerative geometry.
