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?



Is the coordinate ring of a variety a finitely generated algebra?
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](http://math.stanford.edu/~vakil/files/nonfg.pdf) by Ravi Vakil.


