I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$.

Let $f : X \to S$ be a proper smooth morphism with $S$ smooth, separated and connected. He claims that, by Chow's lemma and resolution of singularities, there exists a quasi-projective and smooth $X'$ and a projective birational morphism $p : X' \to X$. Then by Bertini or Sard there exists an open subset $S_0 \subset S$ such that the base change of $f' = f\circ p$ along $S_0 \hookrightarrow S$, $f'_0 : X'_0 \to X_0 \to S_0$, is smooth. Further, he claims $f'_0$ is projective.

Can someone explain the application of Chow's lemma and resolution of singularities more explicitly? According to the statement of Chow's lemma that I found, one gets a proper birational map to $X$. And from the statement of Hironaka's resolution of singularities I found, one gets another proper birational map to that. So I don't understand how Deligne can assume $p$ is projective. And I don't see why $X'$ is quasi-projective (over $\mathrm{Spec}(\mathbf{C})$, I guess he means).

More importantly, even under these assumptions I don't understand why $f'_0$ is projective. And I don't think the base change to $S_0$ has anything to do with it; it's probably true that $f' : X' \to X \to S$ is projective.

I understand these facts are obvious to algebraic geometers, but I would really appreciate an explanation for a beginner to scheme theory.



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