On a class of Danielewski surfaces in affine 3-space

Abstract

L. Makar-Limanov computed the automorphisms groups of surfaces in ${\mathbb{C}}^3$ defined by the equations $x^{n}z-P(y)=0$, where $n\ge 1$ and $P(y)$ is a nonzero polynomial. Similar results have been obtained by A. Crachiola for surfaces defined by the equations $x^{n}z-y^2-h(x)y=0$, where $n\ge 2$ and $h(0)\ne 0$, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations $x^{n}z-Q(x,y)=0$, where $n\ge 2$ and $Q(x,y)$ is a polynomial with coefficients in an arbitrary base field $k$. Among these surfaces, we characterize the ones which are Danielewski surfaces and we compute their automorphism groups. We study closed embeddings of these surfaces in affine 3-space. We show that in general their automorphisms do not extend to the ambient space. Finally, we give explicit examples of ${\mathbb{C}}^{\ast }$-actions on a surface in ${\mathbb{C}}^3$ which can be extended holomorphically but not algebraically to a ${\mathbb{C}}^{\ast }$-action on ${\mathbb{C}}^3$.

Revised version with simplified proofs. A classification of special Danielewski surfaces admitting multiplicative group actions has been added