Limit points and additive group actions

TL;DR

This paper characterizes when a one-dimensional torus action on a normal affine variety can be extended to include additive group actions, linking this to the existence of a fixed divisor, and explores implications for automorphism group orbits.

Contribution

It establishes a criterion for extending torus actions to semi-direct products with additive groups based on fixed divisors, advancing understanding of automorphism group actions.

Findings

Extension of torus actions depends on fixed divisors on the variety.

Effective actions of $ ext{Aut}(X)$ are influenced by these divisors.

Provides a criterion for when automorphism groups have dense orbits.

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 $\mathbb{G}_m\rightthreetimes\mathbb{G}_a$ 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 $X$.