The Chow Ring of the Hilbert Cube

TL;DR

This work determines the Chow ring $A^*(\mathrm{Hilb}^3(X))$ for a smooth projective variety $X$ by presenting it as an algebra over $A^*(X\times\mathrm{Sym}^2(X))$ with explicit generators and relations, and extends the result to smooth quasi-projective $X$ in characteristic zero. The analysis adapts Gottsche's cohomology approach, using the nested Hilbert scheme $\mathrm{Hilb}^{[2,3]}(X)$, a blow-up description along the universal subscheme $Z_2$, and a degree-3 correspondence $\pi_3$ whose push-pull action scales by $3$ on Chow groups, to extract $A^*(\mathrm{Hilb}^3(X))$ as the appropriate subring of $A^*(\mathrm{Hilb}^{[2,3]}(X))$. The paper develops a general blow-up Chow-ring formula, obtains a decomposition of $A^*(\mathrm{Hilb}^{[2,3]}(X))$ into a base and $f$-torsion parts, and then identifies the invariant subring under $\pi_3$ to give an explicit presentation. When a Chow–Künneth decomposition exists for $X$, a Gottsche-type description is recovered; the quasi-projective case is handled via compactifications, yielding a uniform framework for $A^*(\mathrm{Hilb}^3(X))$ beyond projective varieties.

Abstract

Given a smooth projective variety $X$ over a 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.

Theorems & Definitions (29)

• Proposition 2.1: gottsche2006hilbert, Proposition 2.5.8
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Definition 4.1
• Theorem 4.2: Moving lemma, fulton2013intersection, Section 11.4