Cox rings of cubic surfaces and Fano threefolds (Q2345564)

To a normal complete variety \(X\) with a finitely generated divisor class group \(\text{Cl}(X)\), one can associat its \( Cox\) \(ring\): \[ \mathcal{R}(X) = \bigoplus _{\text{Cl}(X)} \Gamma (X,\mathcal{O}_X(D)). \] The Cox ring is a powerful invariant of \(X\) and it has been very much studied since its knowledge carries, in many cases,the possibility to reconstruct \(X\) and to do computational geometry on it. In this paper the Cox ring for several kind of varieties is computed; in Section 3, the authors complete the list of the Cox rings for minimal desingularizations of singular cubic surfaces, and also for blow ups of \(\mathbb {P}^2 \) at non generic sets of six points. There are few cases of threefolds for which the Cox ring is known; here the computation is done for Fano threefolds whose Picard number is 1 or 2. An implementation of algorithms for the computations of Cox rings is provided in the computer algebra system Singular.