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

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

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