Cox rings of extremal rational elliptic surfaces (Q2787956)

The authors compute the Cox rings for extremal rational elliptic surfaces.NEWLINENEWLINELet \(X\) be a complete normal variety with finitely generated class group \(\mathrm{Cl}(X)\), then we defined the Cox ring of \(X\) by NEWLINE\[NEWLINE\mathcal{R} = \bigoplus_{D \in K} H^{0}(X, \mathcal{O}_{X}(D)),NEWLINE\]NEWLINE where \(K\) is a subgroup of \(\mathrm{Div}(X)\) such that the class map \(K \rightarrow \mathrm{Cl}(X)\) is an isomorphism. If \(\mathcal{R}(X)\) is a finitely generated algebra, then \(X\) is called a Mori dream space.NEWLINENEWLINELet \(X\) be a smooth rational projective surface over the complex number and \(\pi : X \rightarrow \mathbb{P}^{1}\) be a relatively minimal Jacobian elliptic fibration on \(X\), i.e. \(\pi\) admits a section and \(\pi\) does not contract \((-1)\)-curves. An elliptic fibration with the finite Mordell-Weil group \(MW(\pi)\) is called extremal. The extremal elliptic fibrations were classified by \textit{R. Miranda} and \textit{U. Persson} [Math. Z. 193, 537--558 (1986; Zbl 0652.14003)]. It is know that the Cox ring of a rational elliptic surface \(X\) is finitely generated iff \(MW(\pi)\) is finite, or equivalently if the effective cone is rational polyhedral. Now we consider the following set of divisors:NEWLINENEWLINEa) \(-K_{X}\),NEWLINENEWLINEb) smooth irreducible rational curves \(C\) with \(C^{2} \in \{-2,-1,0,1\}\).NEWLINENEWLINEFor each such divisor consider the unique \(D\in K\) which is linearly equivalent to it, we choose a basis of \(H^{0}(X, \mathcal{O}_{X}(D))\) and let \(\Omega\) be the union of all such bases. An element \(f \in \Omega\) is called a distinguished section of \(\mathcal{R}(X)\).NEWLINENEWLINEThe main result of the article can be formulated as follows.NEWLINENEWLINETheorem. Let \(\pi : X \rightarrow \mathbb{P}^{1}\) be an extremal rational elliptic surface. Then the Cox ring \(\mathcal{R}(X)\) is generated by distinguished sections.NEWLINENEWLINEMoreover, in Section 4 the authors determine the ideal of relations \(\mathcal{I}(X)\) for each Cox ring \(\mathcal{R}(X)\), which allows to present in Section 6 a complete list of the Cox rings for extremal rational elliptic surfaces. As an application the authors show in Section 5 how to compute the Cox rings for some weak del-Pezzo surfaces, for instance Cayley cubic, using results for extremal rational elliptic surfaces.