Embeddings of a family of Danielewski hypersurfaces and certain \(\mathbb C^+\)-actions on \(\mathbb C^3\)
type: Item
instance of: scholarly article
title: Embeddings of a family of Danielewski hypersurfaces and certain \(\mathbb C^+\)-actions on \(\mathbb C^3\)
zbMATH Open document ID: 1120.14056
author: Lucy Moser-Jauslin
author: Pierre-Marie Poloni
publication date: 2007-04-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10184
review text: In the present article polynomials of the form \(P=x^2y-z^2-xq(x,z)+c\) are considered. If \(c\neq 0\), then the hypersurface \(V(P)\) in \(\mathbb{C}^3\) defined to be the zero set of \(P\) is an example of a Danielewski surface. This type of surfaces is investigated actively during last two decades. The main objective of the article is the study of embeddings of Danielewski surfaces. Recall that two polynomials \(P_1\) and \(P_2\) in \(x,y,z\) are equivalent if there is an automorphism \(\phi\) of \(\mathbb{C}[x,y,z]\) such that \(\phi(P_1)=P_2\). The authors obtain a complete classification of the equivalence classes of polynomials of the form \(x^2y-z^2-xq(x,z)\) in the algebraic and analytic category. In particular, there are polynomials of this form which are algebraically inequivalent but analytically equivalent. Also there are explicit examples of inequivalent polynomials \(P_1\) and \(P_2\) such that \(V(P_1+c)\) is isomorphic to \(V(P_2+c)\) for all \(c\in\mathbb{C}\). There exist polynomials which are algebraically inequivalent, but when considered as polynomials in \(\mathbb{C}[x,y,z,w]\) become equivalent. This result answers a problem posed by \textit{L. Makar-Limanov, P. van Rossum, V. Shpilrain} and \textit{J.-T. Yu} [Comment. Math. Helv. 79, 341--349 (2004; Zbl 1063.14077)]. Finally, a complete classification of \(\mathbb{C}^+\)-actions on \(\mathbb{C}^3\) which are defined by a triangular locally nilpotent derivation of the form \(x^2\frac{\partial}{\partial z}+(2z+xq(x,z))\frac{\partial}{\partial y}\) is given.
reviewed by: Ivan V. Arzhantsev
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14R05
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14R20
zbMATH DE Number: 5145710
zbMATH Keywords: equivalence of polynomials
zbMATH Keywords: stable equivalence
zbMATH Keywords: algebraic embeddings
zbMATH Keywords: Danielewski surfaces
MaRDI profile type: Publication
OpenAlex ID: W1206568286
cites work: Danielewski-Fieseler surfaces
cites work: On complex affine surfaces with \(\mathbb{C}^ +\)-action
cites work: Embeddings of Danielewski surfaces
cites work: On the group of automorphisms of a surface \(x^ny= P(z)\)
cites work: The stable equivalence and cancellation problems
cites work: Embeddings of hypersurfaces in affine spaces.
cites work: Affine varieties with equivalent cylinders
DOI: 10.5802/AIF.2220
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5145710