On rational multiplicative group actions
Luis Cid, Alvaro Liendo

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 and derivations of the field of fractions of satisfying that there exists a generating set of as a field such that with for all . 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 such that . By analogy with the case of additive group actions case, we prove that and that under this isomorphism the derivation is given by . Here, is the field of invariant of the -action.
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Taxonomy
TopicsFinite Group Theory Research · Advanced Algebra and Geometry · Geometric and Algebraic Topology
