Example 29.38.3 (07ZR)—The Stacks project

Example 29.38.3. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$ -algebra generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ . Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$ . In this case $\mathcal{O}_ X(1)$ is a very ample invertible sheaf on $X$ . Namely, the morphism associated to the graded $\mathcal{O}_ S$ -algebra map

$\text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A}$

is a closed immersion $X \to \mathbf{P}(\mathcal{A}_1)$ which pulls back $\mathcal{O}_{\mathbf{P}(\mathcal{A}_1)}(1)$ to $\mathcal{O}_ X(1)$ , see Constructions, Lemma 27.18.5.

Comments (1)

Comment #9552 by Erik on August 05, 2024 at 08:02

It should say relatively very ample instead of very ample in the sentence "In this case $\mathcal{O}_X(1)$ is a very ample invertible sheaf on $X$ ".

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2 comment(s) on Section 29.38: Very ample sheaves

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