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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