TITLE: Is the coordinate ring of a variety a finitely generated algebra?
QUESTION [6 upvotes]: Let $k$ be an algebraically closed field and let $X$ be a (not necessarily affine) variety over $k$. Is the coordinate ring of $X$ ($k[X]$ or $O_{X}(X)$) always a finitely generated $k$-algebra?

REPLY [12 votes]: Feed google with "variety whose ring of global sections is not finitely generated". One gets An example of a nice variety whose ring of global sections is not finitely generated by Ravi Vakil.