I spent some time looking for an example of non finitely generate graded ring
$R=\oplus H^0(X,mD)$ where $D$ is a divisor on a variety $X$ of dimensison $>2$.
I know there are several such examples (e.g Zariski's), but they are all on surfaces.
I believe there must exist many examples. Do you know any?



Non finitely generated graded ring of a divisor in dimension >2
Here is a way of generating lots of examples:


Start with a variety $X$ with an effective cone which is *not* rational polyhedral (i.e., not finitely generated) and let $L\_1,\ldots,L\_r$ be a collection of line bundles on $X$ such that their span $\{L\_1^{a\_1}\otimes \cdots\otimes L\_r^{a\_r} | a\_1,\ldots,a\_r \ge 0\}\,\,$ includes the effective cone. For example, one could take $X$ to be the blow-up of projective space at sufficiently many points, or a K3 surface of maximal Picard number. 


Now consider the variety $Y=\mathbb{P}(E)$ where $E=L\_1\oplus \cdots \oplus L\_r$ and the line bundle $O(1)$ on $Y$. We have for $n\geq 0$,
$$
H^0(Y,O(n))\cong H^0(X,Sym^n(E))=\bigoplus\_{a\_1+\ldots +a\_r=n}H^0(X,L\_1^{a\_1}\otimes \cdots\otimes L\_r^{a\_r})
$$so that $R(Y,O(1))$ is isomorphic to the sum of all sections of all effective line bundles on $X$: this is usually called the Cox ring of $X$. When the effective cone of $X$ is non-rational polyhedral it is clear that this ring is infinitely generated, since it requires sections from all effective divisor classes. 


It is usually a difficult problem in birational geometry to decide when a nef and big divisor is semiample, i.e., some multiple of $D$ is base-point free. A well-known theorem of Zariski says that if $X$ is normal and projective, then $D$ is semiample if and only if the section ring $R(X,D)$ is finitely generated. So this theorem gives a way of producing nef and big divisors which are not semiample. In particular, choosing $X$ such that the nef-big cone is not rational polyhedral, $L=O(1)$ is nef and big, but not semiample, since $R(Y,L)$ is not finitely generated.


