Completions of normal affine surfaces with a trivial Makar-Limanov invariant
type: Item
instance of: scholarly article
title: Completions of normal affine surfaces with a trivial Makar-Limanov invariant
zbMATH Open document ID: 1072.14075
author: Adrien Dubouloz
published in: Michigan Mathematical Journal
publication date: 2004-11-01T00:00:00Z
review text: The Makar-Limanov invariant was introduced just over 10 years ago, and has emerged as one of the more important new tools in the study of affine varieties. The paper under review studies a class of surfaces \(V\) with trivial Makar-Limanov invariant, meaning that \(V\) admits two independent \(\mathbb{C}^+\)-actions. In particular, this condition implies that \(V\) has a relatively large group of automorphisms. The paper's main result is the following Theorem. A normal affine surface \(V\) that is nonisomorphic to \(\mathbb{C}^*\times \mathbb{A}^1\) has a trivial Makar-Limanov invariant if and only if \(V\) is completable by a zigzag. This generalizes earlier work of \textit{T. Bandman} and \textit{L. Makar-Limanov} [Mich. Math. J. 49, No. 3, 567--582 (2001; Zbl 1079.14539)], \textit{J. Bertin} [J. Reine Angew. Math. 341, 32--53 (1983; Zbl 0501.14028)], and \textit{M. H. Gizatullin} [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1047--1071 (1971; Zbl 0221.14023)], and is closely related to recent results of \textit{D. Daigle} and \textit{P. Russell} [Can. J. Math. 56, No. 6, 1145--1189 (2004; Zbl 1073.14075)], and \textit{M. Miyanishi} and \textit{R. V. Gurjar} [On the Makar-Limanov invariant and fundamental group at infinity, preprint (2002)]. Overall, this is a very well-written paper, drawing on standard tools from the theory of algebraic surfaces to give a clear and largely self-contained exposition of its results.
reviewed by: Gene Freudenburg
Mathematics Subject Classification ID: 14R05
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14R25
zbMATH DE Number: 2112903
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1091112077
OpenAlex ID: W2075463317
cites work: Affine surfaces with \(AK(S)=\mathbb C\).
cites work: Pinceaux de droites et automorphismes des surfaces affines.
cites work: Rational curves and rational singularities
cites work: Q5625343
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cites work: Affine open subsets of algebraic varieties and ample divisors
cites work: Q4344786
cites work: On groups of automorphisms of a class of surfaces
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cites work: Open Algebraic Surfaces
DOI: 10.1307/MMJ/1091112077
description: scientific article; zbMATH DE number 2112903