Characterization of affine ${\mathbb{G}}_m$-surfaces of hyperbolic type

Abstract

In this note we prove that if $S$ is an affine non-toric ${\mathbb{G}}_m$-surface of hyperbolic type that admits a ${\mathbb{G}}_a$-action and $X$ is an affine irreducible variety such that $Aut(X)$ is isomorphic to $Aut(S)$ as an abstract group, then $X$ is a ${\mathbb{G}}_m$-surface of hyperbolic type. Further, we show that a smooth Danielewski surface $Dp=\{xy=p(z)\}\subset {\mathbb{A}}^3$, where $p$ has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.

8 pages; comments are welcome