Cox rings of rational surfaces and redundant blow-ups (Q2821669)

The authors propose a systematic way to study the classification of rational surfaces with finitely generated Cox rings in terms of redundant blow ups. Let us recall some notions. Suppose that \(S\) is a smooth projective rational surface and let \(D\) be a \(\mathbb{Q}\)-divisor. The Itaka dimension of \(D\) is given by NEWLINE\[NEWLINE\kappa(D):= \max \{ \dim \phi_{|-nD|}(S): n \in \mathbb{N} \}.NEWLINE\]NEWLINE We call \(\kappa(-K_{S})\) the anticanonical Itaka dimension of \(S\). Let us also recall that a given pseudoeffective \(\mathbb{Q}\)-divisor \(D\) can be written uniquely as \(P+N\), where \(P\) is a nef \(\mathbb{Q}\)-divisor, \(N\) is \(\mathbb{Q}\)-effective, \(P.N=0\), and the intersection matrix of the irreducible components of \(N\) is negative definite provided that \(N \neq 0\). We say that \(D = P + N\) is the Zariski decomposition of \(D\). Let \(-K_{S} = P + N\) be the Zariski decomposition and let \(f : \tilde{S} \rightarrow S\) be a blow-up at a point \(p\) in \(S\) with the exceptional divisor \(E\). We say that a point \(p\) is redundant if \(\text{mult}_{p}N \geq 1\). The blow-up \(f: \tilde{S} \rightarrow S\) at a redundant point \(p\) is called a redundant blow-up, and the exceptional curve \(E\) is called a redundant curve. At last, the Cox ring of \(S\) is defined as \(\bigoplus_{L \in \mathrm{Pic}(S)} H^{0}(S,L)\).NEWLINENEWLINEThe first result of the paper gives us an interesting characterization of morphisms between some rational surfaces with finitely generated Cox rings.NEWLINENEWLINETheorem 1. Let \(f: \tilde{S} \rightarrow S\) be a birational morphism of smooth projective rational surfaces with finitely generated Cox rings. If \(\kappa(-K_{S}) = \kappa(-K_{\tilde{S}}) = 0\), then \(f\) is a sequence of redundant blow ups.NEWLINENEWLINEAnother result provides conditions on blow ups of a Mori dream rational surface (i.e., a smooth rational projective surface with the rational polyhedral effective cone and for which every nef divisor is semiample) to preserve the finite generation of Cox rings.NEWLINENEWLINETheorem 2. Let \(f: \tilde{S} \rightarrow S\) be a redundant blow-up at a point \(p\) of a Mori dream rational surface with \(\kappa(-K_{S}) \geq 0\), and let \(-K_{S} = P+N\) be the Zariski decomposition. Then the Cox ring of \(S\) is finitely generated unless \(\kappa(-K_{S}) = 0\) and \(\text{mult}_{p} N = 1\).NEWLINENEWLINEUsing the above results and redundant blow ups the authors are able to construct a series of infinitely many new examples of rational surfaces with finitely generated Cox rings for \(\kappa(-K_{S}) = 0\) and \(-\infty\).NEWLINENEWLINETheorem 3. For each \(n\geq 11\), there exist smooth projective rational surfaces \(S\) and \(\tilde{S}\) with finitely generated Cox rings such that \(\kappa(-K_{S}) = 0\) and \(\rho(S) = n\), and \(\kappa(-K_{\tilde{S}}) = -\infty\) and \(\rho(\tilde{S}) = n+1\), where \(\rho\) denotes the Picard number.NEWLINENEWLINEThe last main result is devoted to the finite generation of Cox rings for minimal resolutions of singularities of certain surfaces of Picard number one.NEWLINENEWLINETheorem 4. Let \(\bar{S}\) be a normal projective rational surfaces of Picard number one, and let \(g: S \rightarrow \bar{S}\) be its minimal resolution. Assume that \(-K_{\bar{S}}\) is nef and \(\bar{S}\) contains at worst log terminal singularities which are not canonical singularities. The the Cox ring of \(S\) is finitely generated.NEWLINENEWLINEThe paper is nicely written and contains some interesting examples.