Limit points and additive group actions

Abstract

We show that an effective action of the one-dimensional torus ${\mathbb{G}}_m$ on a normal affine algebraic variety $X$ can be extended to an effective action of a semi-direct product ${\mathbb{G}}_m⋌{\mathbb{G}}_a$ with the same general orbit closures if and only if there is a divisor $D$ on $X$ that consists of ${\mathbb{G}}_m$-fixed points. This result is applied to the study of orbits of the automorphism group $\text{Aut}(X)$ on $X$.

8 pages; small corrections and new references are added

References in corpus (4)

• Locally nilpotent derivations on affine surfaces with a $\C^*$-action
• Non-complete rational T-varieties of complexity one
• The automorphism group of a rigid affine variety
• On the uniqueness of ${\mathbf{C}}^{\ast }$-actions on affine surfaces

Cited by in corpus (1)

• Equivariant completions of affine spaces