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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
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