Automorphism Groups of Danielewski Surfaces

TL;DR

This paper investigates the automorphism groups of Danielewski surfaces, revealing conditions under which these groups are isomorphic as algebraic groups versus as ind-groups, and demonstrating that all automorphisms of the ind-group are inner.

Contribution

It establishes that automorphism groups of Danielewski surfaces are isomorphic as ind-groups only when the surfaces are isomorphic, and proves all ind-group automorphisms are inner.

Findings

Automorphism groups are isomorphic as algebraic groups for generic surfaces.

Isomorphism of automorphism groups as ind-groups characterizes surface isomorphism.

All automorphisms of the automorphism ind-group are inner.

Abstract

In this note we study the automorphism group of a smooth Danielewski surface $D_p= \{(x,y,z) \in \mathbb{A}^3 \mid xy = p(z) \} \subset \mathbb{A}^3$, where $p \in \mathbb{C}[z]$ is a polynomial without multiple roots and $deg (p) \ge 3$. It is known that two such generic surfaces $D_p$ and $D_q$ have isomorphic automorphism groups. Moreover, $\mathrm{Aut}(D_p)$ is generated by algebraic subgroups and there is a natural isomorphism $\phi \colon \mathrm{Aut}(D_p) \xrightarrow{\sim} \mathrm{Aut}(D_q)$ which restricts to an isomorphism of algebraic groups $G \xrightarrow{\sim} \phi(G)$ for any algebraic subgroup $G \subset \mathrm{Aut}(D_p)$. In contrast, we prove that $\mathrm{Aut}(D_p)$ and $\mathrm{Aut}(D_q)$ are isomorphic as ind-groups if and only if $D_p \cong D_q$ as a variety. Moreover, we show that any automorphism of the ind-group $\mathrm{Aut}(D_p)$ is inner.