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
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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