On orbits of the automorphism group on a complete toric variety

TL;DR

This paper characterizes the orbits of the automorphism group on complete toric varieties, showing transitivity occurs only when the variety is a product of projective spaces.

Contribution

It provides an explicit description of Aut(X)-orbits on complete toric varieties and identifies when the automorphism group acts transitively.

Findings

Aut(X) acts transitively iff X is a product of projective spaces

Explicit description of Aut(X)-orbits on X

Characterization of automorphism group action on complete toric varieties

Abstract

Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.