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Projective manifolds with hyperplane sections being four-sheeted covers of projective space. (English) Zbl 1106.14002

Summary: Let \(L\) be a very ample line bundle on a smooth complex projective variety \(X\) of dimension \(\geq 6\). We classify the polarized manifolds \((X, L)\) such that there exists a smooth member \(A\) of \(|L|\) endowed with a branched covering of degree four \(\pi \colon A \rightarrow \mathbb{P}^n\). The cases of \(\deg \pi =2\) and \(3\) are already studied by A. Lanteri, M. Palleschi and A. J. Sommese [Nagoya Math. J. 137, 1–32 (1995; Zbl 0820.14005); Contemp. Math. 162, 277–292 (1994; Zbl 0841.14003)]. Recently the case of \(\deg \pi =5\) is studied by the author.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H30 Coverings of curves, fundamental group
14N30 Adjunction problems
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