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

TL;DR

This paper characterizes affine hyperbolic $G_m$-surfaces with certain symmetries and shows that their automorphism groups uniquely determine the surfaces, including specific Danielewski surfaces.

Contribution

It proves that affine hyperbolic $G_m$-surfaces with a $G_a$-action are uniquely characterized by their automorphism groups, extending to Danielewski surfaces.

Findings

Affine hyperbolic $G_m$-surfaces with $G_a$-actions are characterized by automorphism groups.

Automorphism groups of certain Danielewski surfaces uniquely determine the surfaces.

Automorphism groups serve as complete invariants for these classes of surfaces.

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.