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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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