Tautological module and intersection theory on Hilbert schemes of nodal curves

TL;DR

This paper develops a new intersection theory framework for Hilbert schemes of nodal curves, introducing the discriminant module to compute Chern classes and numbers relevant to enumerative geometry.

Contribution

It introduces the discriminant module and an intersection calculus to compute Chern classes on Hilbert schemes of nodal curves, advancing enumerative geometry methods.

Findings

Defined the discriminant module generated by diagonal loci and node scrolls.

Determined the action of the discriminant divisor on the module.

Provided explicit computations and a computer implementation for intersection calculations.

Abstract

We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls, and twists thereof, and determine the action of the discriminant or big diagonal divisor on this group by intersection. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry. The latest version includes some new explicit computations and reference to a computer program due to G. Liu implementing our intersection calculus.