Projective manifolds with hyperplane sections being five-sheeted covers of projective space
type: Item
instance of: scholarly article
title: Projective manifolds with hyperplane sections being five-sheeted covers of projective space
zbMATH Open document ID: 1119.14008
author: Yasuharu Amitani
published in: Journal of the Mathematical Society of Japan
publication date: 2007-01-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1179759539
review text: Let \(X\) be a smooth complex projective variety of dimension \(n+1\) and let \(L\) be a very ample line bundle on \(X\). Suppose that \(| L| \) contains a smooth member \(A\) endowed with a branched covering \(\pi:A \to \mathbb P^n\) of degree \(d\). Classical results on surfaces with hyperelliptic curves as hyperplane sections and their revision made in the 1980's called the attention to the problem of classifying pairs \((X,L)\) as above. The problem has been considered by many authors according to the following values of \((n,d)\). For \((1,2)\) by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593--603 (1987; Zbl 0655.14001)]; for \((1,3)\) by Serrano (unpublished) and by \textit{M. L. Fania} [Manuscr. Math. 68, 17--34 (1990; Zbl 0729.14028)]; for \((n,2), (n,3)\) by \textit{M. Palleschi, A. J. Sommese} and the reviewer [Nagoya Math. J. 137, 1--32 (1995; Zbl 0820.14005); in: Classification of algebraic varieties. Conf. 1992, L'Aquila, Italy. Contemp. Math. 162, 277--292 (1994; Zbl 0841.14003)]; for \((n,d)\) with \(n > d=4,5\) by the reviewer in [Papers in honor of Giovanni Melzi. Milano: Univ. Cattolica del Sacro Cuore. Sci. Mat. 11, 231--248 (1994; Zbl 0831.14023)]. In the paper under review, the author focuses on case \((n,5)\) with \(n \geq 6\) again, providing a complete classification result. In particular, one case, described only in terms of numerical invariants and pointed out as possible by the reviewer, is settled positively. In addition to the four pairs \((X,L)\) already known, the author proves that \(X\) is a smooth weighted hypersurface \(W \subset \mathbb P(5,4,1^{n+1})\) of degree \(20\) with \(L=\mathcal O_W(5)\), which in fact turns out to be very ample. Recently the author considered also the case \((n,4)\) with \(n \geq 5\) [Proc. Japan Acad., Ser. A Math. Sci. 82, 8--13 (2006; Zbl 1106.14002)].
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14N30
zbMATH DE Number: 5119783
zbMATH Keywords: polarized variety
zbMATH Keywords: hyperplane section
zbMATH Keywords: branched covering
zbMATH Keywords: linear system
zbMATH Keywords: graded ring
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
OpenAlex ID: W2010256426
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DOI: 10.2969/JMSJ/1179759539
description: scientific article; zbMATH DE number 5119783