When is the automorphism group of an affine variety nested?

Abstract

For an affine algebraic variety $X$, we study the subgroup ${\mathrm{Aut}}_{\text{alg}}(X)$ of the group of regular automorphisms $\mathrm{Aut}(X)$ of $X$ generated by all the connected algebraic subgroups. We prove that ${\mathrm{Aut}}_{\text{alg}}(X)$ is nested, i.e., is a direct limit of algebraic subgroups of $\mathrm{Aut}(X)$, if and only if all the ${\mathbb{G}}_a$-actions on $X$ commute. Moreover, we describe the structure of such a group ${\mathrm{Aut}}_{\text{alg}}(X)$.

10 pages; minor corrections

References in corpus (2)

• On the geometry of the automorphism groups of affine varieties
• The automorphism group of a rigid affine variety

Cited by in corpus (2)

• Algebraically generated groups and their lie algebras
• Automorphism groups of affine varieties consisting of algebraic elements