Intersection theory on punctual Hilbert schemes and graded Hilbert schemes

TL;DR

This paper develops a method using equivariant techniques to describe the rational Chow ring of Hilbert schemes of points on toric surfaces and explores intersection theory on graded Hilbert schemes.

Contribution

It introduces a general method for computing the Chow ring of Hilbert schemes using equivariant techniques and applies it to graded Hilbert schemes.

Findings

Chow ring of S[n] can be described via equivariant methods

Method is illustrated through numerous examples

Results on intersection theory of graded Hilbert schemes are presented

Abstract

The rational Chow ring A?(S[n],Q) of the Hilbert scheme S[n] 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.