The total coordinate ring of a smooth projective surface (Q1764831)

For smooth projective varieties \(X\) with finitely generated, free Picard group, one can define the Cox (or total coordinate) ring TC\((X)\) which, roughly speaking, looks like \[ \text{TC}(X)=\bigoplus_{D\in\text{Pic} X}\text{H}^0(X,\mathcal O(D)). \] It is factorial, but the general problem is to find out under which conditions it is finitely generated. A necessary condition for this is that the entire Mori cone is polyhedral. The authors of the paper under review investigate the case of \(\dim X=2\). Here they show that \(\text{TC}(X)\) is finitely generated if and only if \(X\) has only finitely many divisors with negative self-intersection and \(\text{TC}(X)\) contains a sufficiently large (e.g. covering the nef part), finitely generated subalgebra. This is fulfilled for surfaces with polyhedral Mori cone where, additionally, ``nef'' implies ``semiample''. Finally, this result is applied to investigate the case of rational surfaces. Encoding them by a configuration of points (to be blown up in) on \(\mathbb P^2\) or the Hirzebruch surfaces, sufficient conditions for the finite generatedness of \(\text{TC}(X)\) are given.