The Stacks project

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

  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).

  2. 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.

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

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0201. Beware of the difference between the letter 'O' and the digit '0'.