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Given a smooth complex projective variety with an ample line bundle $L$, it seems to be folklore that one can get a one-point compactification of the total space $\mathbb{V}(L)$ of $L$ such that removing the zero section yields an affine variety.

I can see how to do it when $L$ is very ample, by basically exploiting the one point compactification of the hyperplane bundle on projective space, but how can I deal with the ample case?

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    $\begingroup$ What do you mean by one-point compactification? That's a construction in topology, not algebraic geometry... $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 24, 2023 at 12:35
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    $\begingroup$ I see, I guess I want something that resembles the "natural" compactification of $\mathcal{O}(1)$ on $\mathbb{P}^n$ given by $\mathbb{P}^{n+1}$ with linear projection. $\endgroup$
    – Oromis
    Commented May 24, 2023 at 12:41

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This is EGA 2, Prop. 8.8.2. It basically says that if $L$ is ample then one can contract the zero section of the geometric realization $\mathbb V(L)$ of $L$ to a point. The result is called the affine cone of $L$. Observe that in EGA the sections of $L$ become functions on $\mathbb V(L)$. So your zero section is the "section" at infinity in EGA.

The converse is also true and is called Grauert's criterion of ampleness (EGA 2, Thm. 8.9.1).

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