The Stacks project

37.80 Blowing up and ample families of invertible modules

We prove a result from [Gross-thesis].

Lemma 37.80.1. Let $X$ be a scheme. Suppose given effective Cartier divisors $D_1, \ldots , D_ m$ on $X$ and invertible modules $\mathcal{L}_1, \ldots , \mathcal{L}_ m$ such that $\bigcap D_ i = \emptyset $ and $\mathcal{L}_ i|_{X \setminus D_ i}$ is ample. Then $X$ has an ample family of invertible modules.

Proof. Let $x \in X$. Choose an index $i \in \{ 1, \ldots , m\} $ such that $x \not\in D_ i$. Set $U_ i = X \setminus D_ i$. Since $\mathcal{L}_ i|_{U_ i}$ we can find an $n \geq 1$ and a section $s \in \Gamma (U_ i, \mathcal{L}_ i^{\otimes n})$ such that the locus $(U_ i)_ s$ where $s$ doesn't vanish is affine (Properties, Definition 28.26.1). Since $U_ i$ is the locus where the canonical section $1 \in \mathcal{O}_ X(D_ i)$ doesn't vanish, we see from Properties, Lemma 28.17.2 there exists an $N \geq 0$ such that $s$ extends to a section

\[ s' \in \Gamma (X, \mathcal{L}_ i^{\otimes n} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(N D_ i)) \]

After replacing $N$ by $N + 1$ we see that $s'$ vanishes at every point of $D_ i$ and hence that $X_{s'} = (U_ i)_ s$ is affine. This proves that $X$ has an ample family of invertible modules, see Morphisms, Definition 29.12.1. $\square$


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. $\square$

Proposition 37.80.3. Let $X$ be a quasi-compact and quasi-separated scheme. There exists a morphism $f : Y \to X$ which is of finite presentation, proper, and completely decomposed (Definition 37.78.1) such that the scheme $Y$ has an ample family of invertible modules.

Proof. By Limits, Proposition 32.5.4 there exists an affine morphism $X \to X_0$ where $X_0$ is a scheme of finite type over $\mathbf{Z}$. Below we produce a morphism $Y_0 \to X_0$ with all the desired properties. Then setting $Y = X \times _{X_0} Y_0$ and $f$ equal to the projection $f : Y \to X$ we conclude. To see this observe that $f$ is of finite presentation (Morphisms, Lemma 29.21.4), $f$ is proper (Morphisms, Lemma 29.41.5), $f$ is completely decomposed (Lemma 37.78.3). Finally, since $Y \to Y_0$ is affine (as the base change of $X \to X_0$) we see that $Y$ has an ample family of invertible modules by Lemma 37.79.2. This reduces us to the case discussed in the next paragraph.

Assume $X$ is of finite type over $\mathbf{Z}$. In particular $\dim (X) < \infty $. We will argue by induction on $\dim (X)$. If $\dim (X) = 0$, then $X$ is affine and has the resolution property. In general, there exists a dense open $U \subset X$ and a $U$-admissible blowing up $X' \to X$ such that $X'$ has an ample family of invertible modules, see Lemma 37.80.2. Since $f : X' \to X$ is an isomorphism over $U$ we see that every point of $U$ lifts to a point of $X'$ with the same residue field. Let $Z = X \setminus U$ with the reduced induced scheme structure. Then $\dim (Z) < \dim (X)$ as $U$ is dense in $X$ (see above). By induction we find a proper, completely decomposed morphism $W \to Z$ such that $W$ has an ample family of invertible modules. Then it follows that $Y = X' \amalg W \to X$ is the desired morphism. $\square$

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