Automorphism Groups of Danielewski Surfaces

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 $deg (p) \ge 3$\mathrm{.}Itisknownthattwosuchgenericsurfaces$D_p$and$D_q$ have isomorphic automorphism groups. Moreover, $\mathrm{A}\mathrm{u}\mathrm{t}(D_p)$ is generated by algebraic subgroups and there is a natural isomorphism $\phi \colon $\mathrm{A}\mathrm{u}\mathrm{t}$(D_p) \xrightarrow{\sim} $\mathrm{A}\mathrm{u}\mathrm{t}$(D_q)$ which restricts to an isomorphism of algebraic groups $G\stackrel{\sim }{\to }\varphi (G)$ for any algebraic subgroup $G \subset $\mathrm{A}\mathrm{u}\mathrm{t}$(D_p)$\mathrm{.}Incontrast,weprovethat$$\mathrm{A}\mathrm{u}\mathrm{t}$(D_p)$and$$\mathrm{A}\mathrm{u}\mathrm{t}$(D_q)$areisomorphicasind-groupsifandonlyif$D_p \cong D_q$asavariety\mathrm{.}Moreover,weshowthatanyautomorphismoftheind-group$$\mathrm{A}\mathrm{u}\mathrm{t}$(D_p)$ is inner.