Cycle map on Hilbert schemes of nodal curves

TL;DR

This paper investigates the structure of the relative Hilbert scheme for families of nodal curves, revealing that its cycle map is a blow-up of the discriminant locus, with implications for birational geometry and intersection theory.

Contribution

It demonstrates that the cycle map on the Hilbert scheme of nodal curves is a blow-up of the discriminant locus, providing new insights into its geometric structure.

Findings

Cycle map is the blow-up of the discriminant locus.

Connections with birational geometry and intersection theory.

Applications to Hilbert schemes of smooth surfaces.

Abstract

We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. Revised version corrects some minor errors.