On nice $\mathbb{G}_m$-actions arising from locally nilpotent derivations with slice

TL;DR

This paper explores how locally nilpotent derivations with slices induce explicit $ ext{G}_m$-actions on algebraic varieties, providing criteria for their linearizability and new descriptions of these actions.

Contribution

It offers a new explicit description of $ ext{G}_m$-actions from locally nilpotent derivations with slices and establishes a linearizability criterion in special cases.

Findings

Derivations with slices induce semisimple $ ext{G}_m$-actions.

A criterion for linearizability based on automorphic conjugation and affine-linearity.

Provides an explicit description of $ ext{G}_m$-actions in terms of derivations.

Abstract

Let $k$ be an algebraically closed field of characteristic zero and $B$ a finitely generated $k$-domain. Given a locally nilpotent derivation $D$ on $B$ admitting a slice $s$, the derivation $\partial=NsD$ ($N\in\mathbb{Z}\setminus\{0\}$) is semisimple and defines a regular $\mathbb{G}_m$-action on $\mathrm{Spec}(B)$. We show that this derivation provides a new explicit description of the $\mathbb{G}_m$-action introduced by Freudenburg in terms of the infinitesimal generator $\partial=NsD$. In the nice case ($D^2(x_i)=0$ for all generators), we prove a linearizability criterion: the associated $\mathbb{G}_m$-action is linearizable if and only if $D$ is automorphically conjugate to $\partial/\partial x_n$ and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.