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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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