Embeddings of a family of Danielewski hypersurfaces and certain \(\mathbb C^+\)-actions on \(\mathbb C^3\)
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.
scientific article; zbMATH DE number 5145710