The commuting derivations conjecture
The main result of the paper (theorem 3.6) is an application of Kaliman's theorem [see e.g. \textit{S. Kaliman}, Pac. J. Math. 203, No. 1, 161--190 (2002; Zbl 1060.14085)] to pairs of commuting locally nilpotent derivations of \({\mathbb C}[x,y,z]\), or equivalently, to algebraic actions of the vector group \(({\mathbb C}^2,+)\) on \({\mathbb C}^3\). The quotient morphism for such an action is of the form \(\pi :{\mathbb C}^3\to {\mathbb C}^1\), which may be viewed as a regular function on \({\mathbb C}^3\). The author shows that the general fiber of \(\pi\) is a plane. In this case, Kaliman's theorem implies that \(\pi\) is a coordinate function on \({\mathbb C}^3\). Reviewer's remark: There exist algebraic actions of \({\mathbb C}^2\) on \({\mathbb C}^3\) which are not linear in any coordinate system. However, the results of this paper show that any such action has rank at most 2. From this, it is possible to classify all \({\mathbb C}^2\)-actions on \({\mathbb C}^3\), though the paper under review does not follow this course: see \textit{D. Daigle} and \textit{G. Freudenburg}, J. Algebra 204, No.2, 353-371 (1998; Zbl 0956.13007).
quotient morphism for vector group actions
scientific article; zbMATH DE number 1895979