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I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, and that $V \subset X$ is a smooth integral subvariety. Suppose too that $\operatorname{Hilb}(X)$ is smooth at $[V]$. There is a Hilbert-Chow morphism $\operatorname{Hilb}_d(X)^{sn} \to \operatorname{Chow}_d(X)$ (where $sn$ is the seminormalization). This is constructed in Koll\'ar's book on rational curves, but I'm having trouble digesting the more technical statements in this "good" setting.

Is the irreducible component of $\operatorname{Chow}$ containing $V$ birational to the irreducible component of $\operatorname{Hilb}$ containing $V$? (I think this is what Cor. I.6.6.1 tells me, but I'm not so sure.) What is happening away from $V$, where the map is not an isomorphism? (Failing that, is there some other reasonable connection between these spaces?)

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