Remark 30.18.2. In the situation of Chow's Lemma 30.18.1:

The morphism $\pi $ is actually H-projective (hence projective, see Morphisms, Lemma 29.43.3) since the morphism $X' \to \mathbf{P}^ n_ S \times _ S X = \mathbf{P}^ n_ X$ is a closed immersion (use the fact that $\pi $ is proper, see Morphisms, Lemma 29.41.7).

We may assume that $\pi ^{-1}(U)$ is scheme theoretically dense in $X'$. Namely, we can simply replace $X'$ by the scheme theoretic closure of $\pi ^{-1}(U)$. In this case we can think of $U$ as a scheme theoretically dense open subscheme of $X'$. See Morphisms, Section 29.6.

If $X$ is reduced then we may choose $X'$ reduced. This is clear from (2).

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