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Ample Weil divisors on K3 surfaces with Du Val singularities. (English) Zbl 0766.14030
Following previous results [B. Saint-Donat, Am. J. Math. 96, 602- 639 (1974; Zbl 0301.14011)] concerning nonsingular Fano varieties and $$K3$$ surfaces the author considers singular $$\mathbb{Q}$$-Fano varieties. Mainly he proves: If $$D$$ is an ample Weil divisor on a $$K3$$ surface with Du Val singularities and the Picard number for a general element $$S$$ in the ample anticanonical class $$-K_ X$$, is $$\rho(S)=1$$ and one of the following two inequalities is true $$D^ 2>12{25\over 42}$$ or $$h^ 0(D)>7$$, then the linear system $$| D|$$ does not have multiple base curves, i.e. multiple base components of $$| D|$$.

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14C20 Divisors, linear systems, invertible sheaves 14J45 Fano varieties 14J17 Singularities of surfaces or higher-dimensional varieties
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##### References:
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