Semisimple derivations, rational slice and kernels over affine domains

TL;DR

This paper investigates semisimple derivations with integer eigenvalues on affine domains over an algebraically closed field, providing explicit descriptions of their kernels using rational slices and semi-invariant generators.

Contribution

It offers a detailed analysis of kernels of semisimple derivations with integer eigenvalues, including explicit descriptions via rational slices and semi-invariant generators.

Findings

Explicit kernel descriptions using rational slices

Characterization of kernels on localizations and the original domain

Properties of semisimple derivations under conjugation

Abstract

Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after passing to the field of fractions and choosing a rational slice s with D(s) = s, we describe the kernel of D explicitly in terms of semi-invariant generators. We also obtain descriptions of the kernel on suitable localizations of B and on B itself by intersection. Several basic properties of semisimple derivations and their behavior under conjugation are also discussed