The Stacks project

Lemma 54.16.9. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of finite type. Let $E \subset X$ be an exceptional curve of the first kind which is in a fibre of $f$.

  1. If $X$ is projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is projective over $S$.

  2. If $X$ is quasi-projective over $S$, then there exists a contraction $X \to X'$ of $E$ and $X'$ is quasi-projective over $S$.

Proof. Both cases follow from Lemma 54.16.8 using standard results on ample invertible modules and (quasi-)projective morphisms.

Proof of (2). Projectivity of $f$ means that $f$ is proper and there exists an $f$-ample invertible module $\mathcal{L}$, see Morphisms, Lemma 29.43.13 and Definition 29.40.1. Let $U \subset S$ be an affine open containing the image of $E$. By Lemma 54.16.8 there exists a contraction $c : f^{-1}(U) \to V'$ of $E$ and an ample invertible module $\mathcal{N}'$ on $V'$ whose pullback to $f^{-1}(U)$ is equal to $\mathcal{L}(nE)|_{f^{-1}(U)}$. Let $v \in V'$ be the closed point such that $c$ is the blowing up of $v$. Then we can glue $V'$ and $X \setminus E$ along $f^{-1}(U) \setminus E = V' \setminus \{ v\} $ to get a scheme $X'$ over $S$. The morphisms $c$ and $\text{id}_{X \setminus E}$ glue to a morphism $b : X \to X'$ which is the contraction of $E$. The inverse image of $U$ in $X'$ is proper over $U$. On the other hand, the restriction of $X' \to S$ to the complement of the image of $v$ in $S$ is isomorphic to the restriction of $X \to S$ to that open. Hence $X' \to S$ is proper (as being proper is local on the base by Morphisms, Lemma 29.41.3). Finally, $\mathcal{N}'$ and $\mathcal{L}|_{X \setminus E}$ restrict to isomorphic invertible modules over $f^{-1}(U) \setminus E = V' \setminus \{ v\} $ and hence glue to an invertible module $\mathcal{L}'$ over $X'$. The restriction of $\mathcal{L}'$ to the inverse image of $U$ in $X'$ is ample because this is true for $\mathcal{N}'$. For affine opens of $S$ avoiding the image of $v$, we see that the same is true because it holds for $\mathcal{L}$. Thus $\mathcal{L}'$ is $(X' \to S)$-relatively ample by Morphisms, Lemma 29.37.4 and (2) is proved.

Proof of (3). We can write $X$ as an open subscheme of a scheme $\overline{X}$ projective over $S$ by Morphisms, Lemma 29.43.12. By (2) there is a contraction $b : \overline{X} \to \overline{X}'$ and $\overline{X}'$ is projective over $S$. Then we let $X' \subset \overline{X}$ be the image of $X \to \overline{X}'$; this is an open as $b$ is an isomorphism away from $E$. Then $X \to X'$ is the desired contraction. Note that $X'$ is quasi-projective over $S$ as it has an $S$-relatively ample invertible module by the construction in the proof of part (2). $\square$


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 0C2N. Beware of the difference between the letter 'O' and the digit '0'.