The Stacks project

[Proposition 1.3.1, Gross-thesis]

Lemma 37.77.2. Let $X$ be a quasi-compact and quasi-separated scheme with finitely many irreducible components. There exists a quasi-compact dense open $U \subset X$ and a $U$-admissible blowing up $X' \to X$ such that the scheme $X'$ has an ample family of invertible modules.

Proof. Let $\eta _1, \ldots , \eta _ n \in X$ be the generic points of the irreducible components of $X$. By Properties, Lemma 28.29.4 and the fact that $X$ is quasi-compact we can find a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$ such that each $U_ i$ contains $\eta _1, \ldots , \eta _ n$. In particular the quasi-compact open subset $U = U_1 \cap \ldots \cap U_ m$ is dense in $X$. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be a finite type quasi-coherent ideal sheaf such that $U_ i = X \setminus Z_ i$ where $Z_ i = V(\mathcal{I}_ i)$, see Properties, Lemma 28.24.1. Let

\[ f : X' \longrightarrow X \]

be the blowing up of $X$ in the ideal sheaf $\mathcal{I} = \mathcal{I}_1 \cdots \mathcal{I}_ m$. Note that $f$ is a $U$-admissible blowing up as $V(\mathcal{I})$ is (set theoretically) the union of the $Z_ i$ which are disjoint from $U$. Also, $f$ is a projective morphism and $\mathcal{O}_{X'}(1)$ is $f$-relatively ample, see Divisors, Lemma 31.32.13. By Divisors, Lemma 31.32.12 for each $i$ the morphism $f'$ factors as $X' \to X'_ i \to X$ where $X'_ i \to X$ is the blowing up in $\mathcal{I}_ i$ and $X' \to X'_ i$ is another blowing up (namely in the pullback of the products of the ideals $\mathcal{I}_ j$ omitting $\mathcal{I}_ i$). It follows from this that $D_ i = f^{-1}(Z_ i) \subset X'$ is an effective Cartier divisor, see Divisors, Lemmas 31.32.11 and 31.32.4. We have $X' \setminus D_ i = f^{-1}(U_ i)$. As $\mathcal{O}_{X'}(1)$ is $f$-ample, the restriction of $\mathcal{O}_{X'}(1)$ to $X' \setminus D_ i$ is ample. It follows from Lemma 37.77.1 that $X'$ has an ample family of invertible modules. $\square$


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Your email address will not be published.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 0GTT. Beware of the difference between the letter 'O' and the digit '0'.