Automorphism Groups of Danielewski Surfaces

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 $\varphi :\mathrm{Aut}(D_p)\stackrel{\sim }{\to }\mathrm{Aut}(D_q)$ which restricts to an isomorphism of algebraic groups $G\stackrel{\sim }{\to }\varphi (G)$ 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.

18 pages