# zbMATH — the first resource for mathematics

Ample vector bundles and branched coverings, II. (English) Zbl 1071.14018
Collino, Alberto (ed.) et al., The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Torino, Italy, September 29–October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 625-645 (2004).
Let $$X$$ and $$Y$$ be projective manifolds of the same dimension and let $$f\colon X\to Y$$ be a cover, namely a finite morphism, of degree $$d$$. To $$f$$ one associates $${\mathcal E}:=(f_*\mathcal {O}_X/{\mathcal O}_Y)^*$$, which is a vector bundle of rank $$d-1$$ on $$Y$$. The authors continue here the study of the positivity properties of $${\mathcal E}$$ started in the first part [Commun. Algebra 28, No. 12, 5573–5599 (2000; Zbl 0982.14025)], with special attention to the case in which $$Y$$ is a Fano manifold. Their first result is negative: they exhibit an example of a triple cover of a Fano threefold of degree 5 for which $${\mathcal E}$$ is not ample (by results of the authors and Lazarsfeld [ibid. and appendix], $${\mathcal E}$$ is known to be nef on the general curve of $$Y$$ and to be spanned when $$Y$$ is a del Pezzo manifold of degree $$\geq 5$$.) In the second part of the paper they study topological properties of low degree covers. They find sufficient conditions for the surjectivity of the map $$H^i(Y)\to H^i(X)$$, thus generalizing a well known result of R. Lazarsfeld on covers of $${\mathbb P}^n$$ of low degree [Math. Ann. 249, 153–162 (1980; Zbl 0434.32013)], and for the ampleness of the branch divisor $$R\subset X$$. The divisor $$R$$ was already known to be ample in the case in which $$Y$$ is a projective space or a quadric [cf. A. Lanteri, M. Palleschi and A. J. Sommese, Osaka J. Math. 26, No.3, 647–664 (1989; Zbl 0715.14005) and R. Lazarsfeld, loc. cit.]
For the entire collection see [Zbl 1051.00013].

##### MSC:
 14E20 Coverings in algebraic geometry 14J45 Fano varieties 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
##### Keywords:
covering of low degree; Fano manifold; ramification divisor