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

Abstract

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 relevant cotangent sheaves and complexes. We determine the structure of certain projective bundles called node scrolls, which play an important role in the geometry of Hilbert schemes.

To appear in Israel J. Math. arXiv admin note: text overlap with arXiv:0803.4512

References in corpus (2)

• Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation
• Tautological module and intersection theory on Hilbert schemes of nodal curves

Cited by in corpus (4)

• Unobstructedness of filling secants and the Gruson-Peskine general projection theorem
• Tautological module and intersection theory on Hilbert schemes of nodal curves
• Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures
• Modifications of Hodge bundles and enumerative geometry : the stable hyperelliptic locus