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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?

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    $\begingroup$ I would not call this ring the coordinate ring except in the affine case. $\endgroup$
    – Laurent Moret-Bailly
    Commented Oct 22, 2016 at 7:39

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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.

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    $\begingroup$ Note that Vakil's example is minimal: their is no such example in dimension $2$ e.g. by Cor 6.3 of this paper (reh.math.uni-duesseldorf.de/%7Eschroeer/publications_pdf/…) of Schroer. $\endgroup$
    – pinaki
    Commented Oct 21, 2016 at 20:53
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    $\begingroup$ Please stop the upvoting. Everyone can google this, this was my message sort of ... $\endgroup$
    – HeinrichD
    Commented Oct 22, 2016 at 1:47
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    $\begingroup$ Alternate link: lmgtfy.com/… $\endgroup$
    – Allen Knutson
    Commented Oct 22, 2016 at 16:36

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