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

Theorems & Definitions (21)

• Definition 1
• Theorem 1
• Corollary 1
• Corollary 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['tmain']}
• Remark 1
• proof : Proof of Corollary \ref{['cmain']}
• proof : Proof of Corollary \ref{['rigid']}