Quotients of an affine variety by an action of a torus

TL;DR

This paper investigates two types of quotients of an affine T-variety, establishing a connection between the toric Chow quotient and the toric Hilbert scheme, and providing explicit descriptions in the toric case.

Contribution

It introduces a main component of the toric Hilbert scheme and relates it to the toric Chow quotient, offering explicit fan descriptions in the toric setting.

Findings

The main component H_0 parameterizes general T-orbit closures.

The birational projective morphism from U_0 to W_X is established.

Explicit fan descriptions of the Altmann-Hausen family in the toric case are provided.

Abstract

Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/_CT and the toric Hilbert scheme H. We introduce a notion of the main component H_0 of H which parameterizes general T-orbit closures in X and their flat limits. The main component U_0 of the universal family U over H is a preimage of H_0. We define an analogue of a universal family W_X over the main component of the X/_CT. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U_0 to W_X. The variety W_X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W_X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.