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Ample vector bundles and branched coverings. (English) Zbl 0982.14025
A well known result of R. Lazarsfeld [Math. Ann. 249, 153-162 (1980; Zbl 0547.32006)] states that if $$f: X\to Y=\mathbb{P}^n$$ is a finite map and $$X$$ is smooth then $$(f_*{\mathcal O}_X/{\mathcal O}_Y)^*(-1)$$ is generated by global sections, hence, in particular, $$E_f:=(f_*{\mathcal O}_X/{\mathcal O}_Y)^*$$ is ample. In this article, the authors consider in general the question of determining the manifolds $$Y$$ such that for every finite map $$f: X\to Y$$ with $$X$$ smooth the sheaf $$E_f$$ is globally generated, nef or ample. They give a number of partial results and counterexamples. For instance they prove:
(i) if $$Y$$ is an abelian variety then $$E_f$$ is always nef;
(ii) if $$Y$$ is a curve, then $$E_f$$ is ample iff $$f$$ does not factorize through an unramified cover of $$Y$$;
(iii) if $$Y$$ is a Del Pezzo surface with $$K^2_Y\geq 5$$ then $$E_f$$ is globally generated;
(iv) if $$Y$$ is a Hirzebruch surface $${\mathbf F}_r$$ with $$r\geq 2$$ and the negative section of $$Y$$ is not contained in the branch locus of $$f$$ then $$E_f$$ is globally generated;
(v) if $$Y$$ is a Fano manifold with Picard number equal to 1, then $$E_f$$ is ample;
(vi) if $$Y$$ is a Del Pezzo manifold such that $$-K_Y=(n-1)H$$ with $$H^n\geq 5$$, then $$E_f$$ is globally generated.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14E20 Coverings in algebraic geometry 14E22 Ramification problems in algebraic geometry
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