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arXiv · 2208.09738

Automorphism groups of rigid affine surfaces: the identity component

Alexander Perepechko·Mikhail Zaidenberg
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Abstract

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 of the field. In the latter case, ${\rm Aut}^0 (Y)$ is an algebraic torus of rank $\le 2$.

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Alexander Perepechko, Mikhail Zaidenberg. 2022-08-20. Automorphism groups of rigid affine surfaces: the identity component. https://arxiv.org/abs/2208.09738

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