The Stacks project

[Proposition 1.3.1, Gross-thesis]

Lemma 37.80.2. 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 37.80.1 that $X'$ has an ample family of invertible modules. $\square$

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