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The Riemann-Roch theorem on surfaces with log terminal singularities. (Russian. English summary) Zbl 1073.14018
Fundam. Prikl. Mat. 10, No. 4, 35-42 (2004); translation in J. Math. Sci., New York 140, No. 2, 200-205 (2007).
Let $$X$$ be a del Pezzo log surface, $$X$$ contain exactly one non-Duval point and this point is a cyclic factor. It is proved that in this case the linear system $$| -K_X|$$ is not empty, in particular, $$X$$ is not special and divisor $$K_X$$ possess with $$1$$-, $$2$$-, $$3$$-, $$4$$-, or $$6$$-complement. It is proved also that if in the above conditions $$X$$ is not a del Pezzo log surface, but is a rational Enriques log surface, then $$2K_X\sim 0$$. As a corollary the same results are obtained for the surfaces $$X$$ with Picar number $$\rho(X)=1$$ and exactly 5 singular points such that each non-Duvale point is a cyclic factor. The method of the prove is of some special interest and may have the applications in more general cases.
##### MSC:
 14C40 Riemann-Roch theorems 14J26 Rational and ruled surfaces
##### Keywords:
Riemann-Roch theorem; singularities; del Pezzo surfaces