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Strictly nef divisors. (English) Zbl 1154.14004
Let \(L\) be a line bundle on a projective variety \(X\). Then \(L\) is strictly nef if \(L\cdot C>0\) for any irreducible curve \(C\subset X\). By the Nakai-Moishezon criterion, \(L\) is ample if and only if \(L^s\cdot Y>0\) for any irreducible subvariety \(Y\subset X\) of dimension \(s\). Notice however that there are examples of strictly nef line bundles that are not ample. F. Serrano [J. Reine Angew. Math. 464, 187–206 (1995; Zbl 0826.14006)] conjectured that if \(L\) is strictly nef, then \(K_X+tL\) is ample for any \(t>n+1\) and he showed that this conjecture holds in dimension \(2\) and in dimension \(3\) with the following possible exceptions: 1) \(X\) is Calabi-Yau and \(L\cdot c_2(X)=0\); 2) \(X\) is uniruled with \(q(X)\leq 1\); 3) \(X\) is uniruled with \(q(X)=2\) and \(\chi (\mathcal O _X)=0\). In this paper, the authors show that Serrano’s conjecture holds in dimension \(3\) for cases 2) and 3) above (case 1) is still open) and it also holds (in any dimension) if \(\kappa (X)\geq n-2\).

MSC:
14C20 Divisors, linear systems, invertible sheaves
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