Let $X$ be a smooth projective rational variety over a field $k$. Let $CH^i(X)$ denote the Chow group of codimension $i$ algebraic cycles on $X$ modulo rational equivalence. What can one say about Chow groups $X$? Are they torsion-free?

## 1 Answer

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You cannot say much. Suppose $X$ is obtained by blowing up $\mathbb{P}^n$ along a smooth subvariety $V$, say of codimension $c$; then $CH^p(X)=CH^p(\mathbb{P}^n)\oplus CH^{p-1}(V)\oplus\ldots \oplus CH^{p+1-c}(V)$. Thus the Chow groups of $X$ look like those of $V$, which are arbitrary. All you can say for a rational variety $X$ of dimension $n$ is that $CH^1(X)=\mathrm{Pic}(X)$ is a free finitely generated abelian group, and $CH^n(X)=\mathbb{Z}$.