A remark on Abhyankar's space lines
type: Item
instance of: scholarly article
title: A remark on Abhyankar's space lines
zbMATH Open document ID: 0702.14027
author: Pier Carlo Craighero
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108129
review text: An (n,n-1,n-2)-line \(C_ n\) in \({\mathbb{A}}^ 3\) is the image of \({\mathbb{A}}^ 1\) under the map \((f_ 0(t),f_ 1(t),f_ 2(t))\) where \(\deg (f_ i)=n-i\). \(C_ n\) is said to be rectifiable if there exists an automorphism \(\Phi_ n\) of \({\mathbb{A}}^ 3\) such that \(\Phi_ n(C_ n)\) is a straight line. The author proves the rectifiability of the curve \(C_ 6: (t+t^ 6,t^ 5,t^ 4)\). The automorphism \(\Phi\) he obtains should not be tame (i.e. the product of linear and triangular automorphisms), according to a conjecture of \textit{M. Nagata} [cf. ``On automorphism group of k[X,Y]'' (Tokyo 1972; Zbl 0306.14001)]. On the other hand, the author gives an example of another (6,5,4)-line \(C'_ 6\) which is tamely rectifiable, showing that the possible existence of a tame automorphism that rectifies \(C_ 6\) cannot be excluded simply from the triple (6,5,4).
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 4150388
zbMATH Keywords: Abhyankar's space lines
zbMATH Keywords: automorphism
reviewed by: Juan C. Migliore
MaRDI profile type: Publication
cites work: About Abhyankar's conjectures on space lines
cites work: A result on m-flats in \({\mathbb{A}}^ n_ k\)
cites work: Q4740034
cites work: The extension of regular and rational embeddings
cites work: Q4063564
cites work: On Linear Planes
description: scientific article; zbMATH DE number 4150388