On quartic surfaces and sextic curves with certain singularities
type: Item
instance of: scholarly article
title: On quartic surfaces and sextic curves with certain singularities
zbMATH Open document ID: 0549.14013
author: Tohsuke Urabe
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: The classification of the singularities lying on a normal surface in \(P^ 3\) of small degree is a subject with classical roots. For instance one knows all possible configurations of singularities lying on a cubic surface. In this paper the author classifies completely the singularities of a normal quartic surface (resp. of a sextic curve in \(P^ 2)\) having one of the following prescribed singularities: a simple elliptic singularity of type \(\tilde E_ 7\) or \(\tilde E_ 6\), a cusp singularity of type \(T_{2,4,5}\) or \(T_{3,3,4}\), a unimodular exceptional singularity of type \(Z_{11}\) or \(Q_{10}\) (resp. \(\tilde E_ 7\), \(T_{2,4,5}\), or \(Z_{11})\). In an earlier paper of the same author the classification of the singularities of normal quartic surfaces and of sextic curves with one of the singularities \(\tilde E_ 8\), \(T_{2,3,7}\) or \(E_{12}\) was given [see the author, ''On quartic surfaces and sextic curves with singularities of type \(\tilde E_ 8\), \(T_{2,3,7}\), \(E_{12}\)'' (Preprint 1983)]. Finally, we note that this paper gives no details concerning the proofs of the results.
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 3875444
zbMATH Keywords: singularities of a normal quartic surface
zbMATH Keywords: sextic curve
zbMATH Keywords: simple elliptic singularity
zbMATH Keywords: cusp singularity
zbMATH Keywords: unimodular exceptional singularity
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
full work available at URL: pjaa.59.434
OpenAlex ID: W2037652748
cites work: Local normal forms of functions
cites work: Fifteen characterizations of rational double points and simple critical points
cites work: Einfach-elliptische Singularitäten
cites work: Quartic Surfaces of Elliptic Ruled Type
cites work: On quartic surfaces and sextic curves with singularities of type \(\tilde E_ 8\), \(T_{2,3,7}\), \(E_{12}\)
DOI: 10.3792/PJAA.59.434
description: scientific article; zbMATH DE number 3875444