Tautological module and intersection theory on Hilbert schemes of nodal curves

Tautological Module and Intersection Theory on Hilbert Schemes of Nodal Curves

This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory: 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 of geometric cycles, called ’tautological module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the intersection act...

Geometry and Intersection Theory on Hilbert Schemes of Families of Nodal Curves

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 th...

Intersection theory on punctual Hilbert schemes and graded Hilbert schemes

The rational Chow ring A(S,Q) of the Hilbert scheme S parametrising the length n zero-dimensional subschemes of a toric surface S can be described with the help of equivariant techniques. In this paper, we explain the general method and we illustrate it through many examples. In the last section, we present results on the intersection theory of graded Hilbert schemes.

Geometry on Nodal Curves Ii: Cycle Map and Intersection Calculus

Structure of the Cycle Map for Hilbert Schemes of Families of Nodal Curves

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 study certain loci called node scrolls which play in important role in the geometry of the cy...