$X \cong \operatorname{Proj} \oplus_n H^0(X, O(n)) $

Question

Let $ X \subset \mathbb{P}^n$ be a smooth closed subvariety, and $O(1)$ is the pull-back of the line bundle of $O(1)$ on $\mathbb{P}^n$. Then it is claimed:

$$X \cong \operatorname{Proj}\left(\bigoplus_n H^0\big(X, O(n)\big)\right)\;.$$

This seems quite plausible for me, but I don't know how to show it.

score 5 · Accepted Answer · 2013-03-23T04:36:31.080

5

This follows from (EGA, II, 4.5.1, (b)) or (Stacks, 23.24.11), since $O(1)$ is ample on $X$.

edited Mar 23 '13 at 04:36

answered Feb 28 '13 at 00:17

2

Thank you! That's what I am looking for. I cannot voteup because of lack of reputation, sorry about that. Using (Stacks, 23.24.11), one know it is a open immersion, and because $X$ is projective, the morphism is proper, thus closed, which shows the image should be the whole thing-Add this trivial comments :) – Li Yutong Feb 28 '13 at 02:30

$X \cong \operatorname{Proj} \oplus_n H^0(X, O(n)) $

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