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?



One-point compactification of ample line bundle
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).


