The Chow Ring of the Hilbert Cube

TL;DR

This paper computes the Chow ring of the Hilbert scheme of three points on a smooth projective variety, extending to quasi-projective cases in characteristic zero, providing explicit algebraic descriptions.

Contribution

It provides the first explicit computation of the Chow ring of erb3(X) as an algebra with generators and relations, generalizing previous results.

Findings

Chow ring of erb3(X) is computed explicitly.

Extension of the computation to quasi-projective varieties in characteristic zero.

Provides algebraic generators and relations for the Chow ring.

Abstract

Given a smooth projective variety $X$ over an algebraically closed field $k$, we compute the Chow ring of the Hilbert scheme of three points on $X$, $\operatorname{Hilb}^3(X)$, as an algebra with generators and relations over the Chow ring of $X\times\operatorname{Sym}^2(X)$. If in addition the characteristic of $k$ is zero, we extend the computation to the quasi-projective case.