Automorphism groups of rigid affine surfaces: the identity component

It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface \(Y\), as to when the identity component \({\...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: Perepechko, Alexander, Zaidenberg, Mikhail
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Group theory
Toruses
Online Access:Get full text
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Summary:It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface \(Y\), as to when the identity component \({\rm Aut}^0 (Y)\) of the automorphism group \({\rm Aut} (Y)\) is an algebraic group? We show that this happens if and only if \(Y\) admits no effective action of the additive group. In the latter case, \({\rm Aut}^0 (Y)\) is an algebraic torus of rank \(\le 2\).
ISSN:2331-8422