On orbits of the automorphism group on an affine toric variety

TL;DR

This paper demonstrates that the orbits of the connected automorphism group on an affine toric variety align exactly with the Luna strata from its canonical quotient presentation, linking automorphism dynamics with geometric stratification.

Contribution

It establishes a precise correspondence between automorphism group orbits and Luna strata for affine toric varieties, clarifying their geometric structure.

Findings

Automorphism group orbits match Luna strata

Canonical quotient presentation characterizes orbit structure

Provides a geometric understanding of automorphism actions

Abstract

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation of X as a quotient of a vector space by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.