Geometry and intersection theory on Hilbert schemes of families of nodal curves

Details Geometry and intersection theory on Hilbert schemes of families of nodal curves Geometry and intersection theory on Hilbert schemes of families of nodal curves

2008-03-31

arxiv.org/abs/0803.4512v1

Mathematics

Algebraic Geometry

Supersedes and extends arXiv:math/0410120

Scientific paper

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.

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