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Fractionally logarithmic canonical rings of algebraic surfaces. (English) Zbl 0543.14004
Main Theorem: Let \(D\) be an effective \(\mathbb Q\)-divisor on a smooth algebraic surface \(S\) defined over a field of any characteristic. Let \(K\) be the canonical bundle of \(S\) and suppose that \(K+D\) is pseudo-effective and that \(D\) is reduced, i.e., the coefficient of each prime component of \(D\) is not greater than one. Then the semipositive part of the Zariski decomposition of \(K+D\) is semiample. In particular \(\kappa(K+D,S)\geq 0\) and the graded algebra associated to \(K+D\) is finitely generated. A notion of minimality due to Sakai plays an important role in the proof, which consists of case-by-case arguments depending on the value of \(\kappa(K+D,S)\).
Reviewer: Takao Fujita

14C15 (Equivariant) Chow groups and rings; motives
14C20 Divisors, linear systems, invertible sheaves
14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
14J17 Singularities of surfaces or higher-dimensional varieties