On rational multiplicative group actions

TL;DR

This paper establishes a correspondence between rational multiplicative group actions on algebraic varieties and rational semisimple derivations, proving the existence of rational slices and describing the structure of the field of fractions.

Contribution

It introduces the concept of rational semisimple derivations and proves their correspondence with rational multiplicative group actions, including the existence of rational slices.

Findings

Correspondence between rational multiplicative group actions and rational semisimple derivations.

Existence of rational slices for every rational semisimple derivation.

Structural description of the field of fractions as a rational function field with invariants.

Abstract

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i\}_{i\in I}$ of $K_X$ as a field such that $\partial(a_i)=\lambda_i a_i$ with $\lambda_i \in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\partial(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{\mathbb{G}_m}(s)$ and that under this isomorphism the derivation $\partial$ is given by $\partial=s\frac{d}{ds}$. Here, $K_X^{\mathbb{G}_m}$ is the field of invariant of the $\mathbb{G}_m$-action.