The smooth surfaces in \({\mathbb{P}}^ 4\) without apparent triple points
type: Item
instance of: scholarly article
title: The smooth surfaces in \({\mathbb{P}}^ 4\) without apparent triple points
zbMATH Open document ID: 0687.14040
DOI: 10.1215/S0012-7094-88-05719-5
author: Alf Bjørn Aure
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The smooth surfaces in \({\mathbb{P}}^ 4\) without apparent triple points are the elliptic quintic scrolls and the surfaces contained in quadric hypersurfaces. The method to prove this theorem is to study the sectional genus of a surface. The author shows that a surface in \({\mathbb{P}}^ 4\) without apparent triple points and not of the types stated above has to be birationally ruled over a curve of genus 2 or more. Next, he shows that it is contained in a cubic hypersurface. Finally, by using a formula concerning quadrisecant lines he shows that such a surface without apparent triple points does not exist. The positive direction of the theorem is a by the way result of \textit{F. Severi}: ``Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a suo punti tripli apparenti'', Rend. Circ. Mat. Palermo, II. Ser. 15, 33-51 (1901).
reviewed by: Konrad Drechsler
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 4125571
zbMATH Keywords: triple points
zbMATH Keywords: elliptic quintic scrolls
zbMATH Keywords: surfaces contained in quadric hypersurfaces
zbMATH Keywords: sectional genus
zbMATH Keywords: quadrisecant lines
MaRDI profile type: Publication
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full work available at URL: s0012-7094-88-05719-5
OpenAlex ID: W2029757550
description: scientific article; zbMATH DE number 4125571