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It is remarked in Shafarevich's Basic Algebraic Geometry 1 that Rees and Nagata constructed examples of quasiprojective varieties such that the ring of regular functions is not finitely generated, but I cannot find the source he is referring to. Can anyone give such examples here? Does that mean we can't really say anything about the ring of regular functions of a quasi-projective variety?

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up vote 6 down vote accepted

It's a theorem that a quasi-projective variety is affine if and only if it is Stein (we're working over C, say) and its ring of functions is finitely generated. So find a Stein manifold that isn't affine, and that will do it.

And, after a bit of looking, it appears that Vakil may have rediscovered the Rees and Nagata example, here.

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I don't think that a Stein manifold that isn't affine will do the trick. For instance, if we take Serre's example ( P^1 bundle over an elliptic curve obtained as the projectivization of the unique non-trivial extension 0-> O -> V -> O -> 0 minus the section determined by O -> V) is Stein, and every regular function on it is constant since the section has zero self-intersection. –  jvp Oct 23 '09 at 11:49
    
I was certain that I'd read that Stein + f.g. => Affine for varieties, but that seems like a counterexample. I must be missing a hypothesis. Asking a question to try to find out exactly what is true. Must be some nontriviality hypothesis for the ring of regular functions, I'm guessing (maybe separates points?) –  Charles Siegel Oct 23 '09 at 12:52
    
Ok, so according to Tony Pantev on my related question, what we need is a quasi-affine variety that is Stein. Then we have affine if and only if finitely generated. So it's a matter of looking for non-affine Stein manifolds which are quasi-affine. –  Charles Siegel Oct 23 '09 at 21:26
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"Does that mean we can't really say anything about the ring of regular functions of a quasi-projective variety?"

Since every variety contains an open affine, the ring of regular functions is always a subring of a finitely generated ring. (I assume that you consider varieties to be integral.) This is a nontrivial restriction. Also, the ring of regular functions will be noetherian, since any infinite ascending chain of ideals would give an infinite descending chain of subschemes.

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