Completions of normal affine surfaces with a trivial Makar-Limanov invariant
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.
scientific article; zbMATH DE number 2112903