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

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 h...

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Bibliographic Details
Published in:arXiv.org 2022-02
Main Author: Regeta, Andriy
Format: Article
Language:English
Subjects:
Automorphisms
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Summary: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.
ISSN:2331-8422