Structure of the cycle map for Hilbert schemes of families of nodal curves

摘要

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 determine the structure of certain projective bundles called node scrolls which play an important role in the geometry of Hilbert schemes.