Automorphism groups of rigid affine surfaces: the identity component

TL;DR

This paper investigates conditions under which the identity component of the automorphism group of an affine algebraic surface is an algebraic group, revealing it occurs precisely when the surface admits no additive group action.

Contribution

It establishes a necessary and sufficient condition for the automorphism group's identity component to be algebraic, linking it to the absence of additive group actions on the surface.

Findings

${ m Aut}^0 (Y)$ is algebraic iff $Y$ admits no effective additive group action.

When algebraic, ${ m Aut}^0 (Y)$ is an algebraic torus of rank ≤ 2.

Characterizes automorphism groups of rigid affine surfaces.

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 $\le 2$.