The Chow Ring of the Hilbert Cube

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.

💡 Research Summary

The paper provides a complete description of the Chow ring of the Hilbert scheme of three points on a smooth projective variety X, denoted Hilb³(X). The author works over an algebraically closed field k, assuming that the characteristic does not divide 2 or 3, and later extends the result to the quasi‑projective case when char(k)=0.

The main strategy mirrors the classical cohomological computation of FG93. The key geometric observation is that the rational map
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