  We study intersection theory on the relative Hilbert scheme of a family of
nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an
additive group called 'discriminant module', generated by diagonal loci, node
scrolls, and twists thereof, and determine the action of the discriminant or
big diagonal divisor on this group by intersection. We show that this suffices
to determine arbitrary polynomials in Chern classes, in particular Chern
numbers, for the tautological vector bundles on the Hilbert schemes, which are
closely related to enumerative geometry. The latest version includes some new
explicit computations and reference to a computer program due to G. Liu
implementing our intersection calculus.
