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

Luis Cid

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.

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

Abstract

Let be an algebraically closed field of characteristic zero and a finitely generated -domain. Given a locally nilpotent derivation on admitting a slice , the derivation () is semisimple and defines a regular -action on . We show that this derivation provides a new explicit description of the -action introduced by Freudenburg in terms of the infinitesimal generator . In the nice case ( for all generators), we prove a linearizability criterion: the associated -action is linearizable if and only if is automorphically conjugate to and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.

Paper Structure

This paper contains 14 sections, 16 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally nilpotent and semisimple derivations
  4. Slices and the Slice Theorem
  5. Algebraic group actions
  6. The semisimple derivation associated with a slice
  7. Relation with Freudenburg's action
  8. The nice case and a linearizability criterion
  9. Further results and applications
  10. Independence of the choice of slice
  11. Geometric reformulation via the kernel
  12. Complete answer for $n\le 3$
  13. State of the problem in dimension $n\ge 4$
  14. An obstruction to linearizability via the Makar-Limanov invariant

Key Result

Theorem 2.6

If $D\in\operatorname{LND}(B)$ admits a slice $s$ then $B=\ker(D)[s]$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Slice Theorem; Fre06
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 37 more