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Fractionally logarithmic canonical rings of algebraic surfaces. (English) Zbl 0543.14004
Author’s summary: 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 $$\kappa(K+D,S).$$

##### MSC:
 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