Automorphism groups of rigid affine surfaces: the identity component
Alexander Perepechko, Mikhail Zaidenberg
TL;DR
The paper addresses when the identity component ${\rm Aut}^{\circ}(Y)$ of the automorphism group of a normal affine surface $Y$ is an algebraic group, proving this occurs precisely when $Y$ admits no effective ${\mathbb G}_{\rm a}$-action, in which case ${\rm Aut}^{\circ}(Y)$ is an algebraic torus of rank at most $2$. The authors develop a framework based on NC-pairs $(X,D)$ and their dual weighted graphs $\Gamma(D)$, translating geometric questions about automorphisms into combinatorial questions about graph transformations. A key result is a combinatorial criterion for rigidity: $Y$ is rigid iff every extremal linear segment of $\Gamma(D)$ is admissible; equivalently, the absence of ${\mathbb G}_{\rm a}$-actions corresponds to ${\rm Bir}(X,D)={\rm Inn}(X,D)$ and ${\rm Aut}^{\circ}(Y)$ being a low-rank algebraic torus. The paper also establishes an ind-structure on ${\rm Bir}(X,D)$, relates ${\rm Aut}^{\circ}(Y)$ to ${\rm Aut}^{\circ}(X,D)$, and provides a detailed graph-theoretic toolkit (Graph Lemma, inertia indices) to study birational transformations and rigidity across NC-pairs and their dual graphs. The results unify birational geometry with automorphism group structure, yielding a robust classification framework for 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 $\le 2$.