Anticanonical models of rational surfaces (Q789463)

Given a non-singular rational surface X over an algebraically closed field k, let -K denote an anticanonical divisor. The anti-Kodaira dimension \(\kappa^{-1}(X)\) is defined to be \(\kappa\) (-K,X). The main result is stated as follows: Let X be a non-singular rational surface with \(\kappa^{-1}(X)=2\). Then the anticanonical ring \(R^{- 1}(X)=\oplus_{m\geq 0}H^ 0(X,0(-mK))\) is finitely generated over k, and the anticanonical model \(Y=\Pr oj R^{-1}(X)\) satisfies the following properties: (i) Y has only isolated rational singularities, (ii) \(-K_ Y\) is an ample Q-Cartier divisor. Conversely, if a normal projective surface Y has the above properties (i) and (ii), then its minimal resolution X is a rational surface with \(\kappa^{-1}(X)=2.\) The proof is based on the theory of the Zariski decomposition of divisors (pseudo effective) established by \textit{O. Zariski} (Ann. Math., II. Ser. 76, 560-615 (1962; Zbl 0124.370)] and by \textit{T. Fujita} [Proc. Jap. Acad., Ser. A 55, 106-110 (1979; Zbl 0444.14026)]. It turns out that for a rational surface X, -K is pseudo effective if and only if \(\kappa^{- 1}(X)\geq 0\). If so, there exists the Zariski decomposition: \(-K=P+N\), where P is a numerically effective Q-divisor. The negative part N is related to the singularities of the anticanonical model Y (in case \(\kappa^{-1}(X)=2)\). In order to analyse it, we need some local properties of surface singularities. For this, combined with a vanishing theorem, a local version of the notion of the Zariski decomposition is useful. In addition, a dimension formula for the m-th anti-genus is considered, where it appears a term depending on the singularities of Y. Define \(d(X)=P^ 2\) and call it the degree of X. The effect of blowing ups on the degree is discussed. We remark that there are many interesting examples of rational surfaces with \(\kappa^{-1}(X)=2\). In this article, the structures of rational surfaces with \(\kappa^{-1}(X)=0\) and 1 are also described.