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Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then

$$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{L}^{\otimes k}) \right)$$

but I am not sure how to prove this or if this is even an equivalent definition of ampleness. Can anybody provide an explanation or a reference?

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    $\begingroup$ Proposition 28.26.13 in the Stacks project — note that since $X$ is projective, the open immersion $X\rightarrow \operatorname{Proj}(S) $ is in fact an isomorphism. $\endgroup$
    – abx
    Commented Dec 19, 2019 at 13:44
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    $\begingroup$ @abx that is tag 01Q3. The numbers are not stable; see also the tags explained page. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 19, 2019 at 16:36
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    $\begingroup$ @R. van Dobben de Bruyn: Oh, OK, thanks! I am not too familiar with the Stacks project. $\endgroup$
    – abx
    Commented Dec 19, 2019 at 21:57

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