Smooth curves on a cone which pass through its vertex
type: Item
instance of: scholarly article
title: Smooth curves on a cone which pass through its vertex
zbMATH Open document ID: 0779.14019
DOI: 10.1007/BF02567638
author: David B. Jaffe
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155663
review text: The author studies the existence of smooth curves passing through the vertex of the cone over a (possibly singular) plane curve \(D\) of degree \(d\). Given a positive integer \(m\), he gives conditions for there to exist such curves that meet each ruling of the cone \(m\) times. In particular, they exist if \(D\) is rational -- in this case, the proof consists in looking at curves on the desingularization of the cone. If \(D\) is not rational, such curves exist under certain conditions on \(m\) and \(D\) as well as on the characteristic of the base field. This proof involves a generalization of Cayley's classical monoid construction. The results can be applied to the problem of finding smooth space curves which are set-theoretic complete intersections of a cone with some other surface.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 60110
zbMATH Keywords: cone over a plane curve
zbMATH Keywords: ruling of a cone
zbMATH Keywords: space curves
zbMATH Keywords: complete intersections
reviewed by: Ragni Piene
MaRDI profile type: Publication
cites work: Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications
cites work: On a theorem of Castelnuovo, and the equations defining space curves
cites work: Q4143433
cites work: Complete Intersections in Characteristic p > 0
cites work: Space curves which are the intersection of a cone with another surface
cites work: On set theoretic complete intersections in \({\mathbb{P}}^ 3\)
cites work: Cuspidal projections of space curves
cites work: On self-linked curves
description: scientific article; zbMATH DE number 60110