Three-dimensional scrolls in \(\mathbb{P}^ 6\)
type: Item
instance of: scholarly article
title: Three-dimensional scrolls in \(\mathbb{P}^ 6\)
zbMATH Open document ID: 0865.14029
DOI: 10.1007/BF01198076
author: Matei Toma
published in: Archiv der Mathematik
publication date: 1996-11-24T00:00:00Z
review text: If a \(n\)-dimensional scroll \(X(=\) a \(\mathbb{P}^{n-1}\)-bundle) over the smooth curve \(C\) of genus \(g\) is embedded in \(\mathbb{P}^N\) by linearly embedded fibers then \(N\geq 2n-1\); and if \(N=2 n-1\) then, by the Barth-Lefschetz theorem, \(g=0\). If \(N=2n\), the only known examples are scrolls over curves with \(g=0,1\) and the problem is to prove this or to find a scroll with \(N=2n\) and \(g\geq 2\). For \(n=2\) the problem has been solved affirmatively by \textit{Alf Aure}. In this paper, following the idea of Aure, the author gives an affirmative answer also for \(n=3\). The proof is based on the study of the family of 3-planes \(\mathbb{P}_i\) in \(\mathbb{P}^6\) intersecting all the fibers of the scroll.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14M07
zbMATH DE Number: 937034
zbMATH Keywords: embedded 3-dimensional scroll
reviewed by: Atanas Iliev
MaRDI profile type: Publication
cites work: Q3919817
cites work: Q4748281
description: scientific article; zbMATH DE number 937034