In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.

I was talking about this previous question today and the following question came up - given any countably generated noetherian k-algebra R which is an integral domain and whose field of fractions has finite transcendence degree over k, where k is a field does there exist some quasi-projective variety X (by variety I mean an integral separated scheme of finite type over k) such that the ring of global sections of X is R?

It is possible one needs more hypotheses to make this work - if this is false I think it would be interesting to know the class of algebras which can occur.