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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
##### Keywords:
strictly nef divisors
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##### References:
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