Double covers of smooth hyperquadrics as ample and very ample divisors
type: Item
instance of: scholarly article
title: Double covers of smooth hyperquadrics as ample and very ample divisors
zbMATH Open document ID: 0857.14002
DOI: 10.1007/BF02940777
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-03-06T00:00:00Z
review text: The object of this paper is the classification of pairs \((X,L)\), where \(X\) is a smooth projective variety of dimension \(n\geq 4\) and \(L\) is a very ample (or ample) divisor, such that one of the smooth elements of the linear system \(|L|\) is a double cover of a smooth quadric. This problem may be regarded as a generalization of the classical question of which smooth projective surfaces admit a hyperelliptic curve as a hyperplane section, solved by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)] and by \textit{F. Serrano} [J. Reine Angew. Math. 381, 90-109 (1987; Zbl 0618.14001)]. The author solves the problem completely, and moreover he generalizes along the same lines an earlier result of his, namely the classification of pairs \((X,L)\), with \(L\) ample, such that one of the smooth elements of \(|L|\) is a double cover of \(\mathbb{P}^n\). The proof rests on Fujita's classification of varieties according to \(\Delta\)-genus.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 706665
zbMATH Keywords: very ample divisor
zbMATH Keywords: \(\Delta\)-genus
zbMATH Keywords: double cover of a smooth quadric
author: Antonio Lanteri
reviewed by: Rita Pardini
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 706665