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

2009/03/21 by Ziv Ran, Ran, Ziv · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0903.3693

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.

Cited by

• Modifications of Hodge bundles and enumerative geometry : the stable hyperelliptic locus

Related

• Cycle map on Hilbert schemes of nodal curves
• Geometry and intersection theory on Hilbert schemes of families of nodal curves
• Geometry on nodal curves II: cycle map and intersection calculus
• A note on Hilbert schemes of nodal curves
• Nodal curves and components of the Hilbert scheme of curves in \mathbb P^r with the expected number of moduli