Reduced class groups grafting relative invariants

TL;DR

This paper investigates the structure of class groups in algebraic geometry, focusing on how certain group actions can be simplified through quotienting by finite subgroups, revealing new insights into invariant theory.

Contribution

It introduces a method to reduce class groups by factoring out finite subgroups in torus actions on affine varieties, advancing understanding of invariants in algebraic geometry.

Findings

Existence of a finite normal subgroup N simplifying the torus action

Reduction of the action to a 'coffee' type after quotienting

Enhanced understanding of invariant structures in affine varieties

Abstract

Consider an equidimensional faithful conical action of an algebraic torus $T$ on an affine normal conical variety $X$ over an algebraically closed field of characteristic zero. Then there exists a finite normal subgroup $N$ of $T$ such that the action of $T$ on $X/N$ is coffee.