The Stacks project

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. $\square$

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