# Gale duality and homogeneous toric varieties

## TL;DR

This paper classifies maximal S-homogeneous toric varieties using algebraic data and shows that all homogeneous toric varieties are quasiprojective, proposing a conjecture about their S-homogeneity.

## Contribution

It provides a bijective classification of maximal S-homogeneous toric varieties via pairs of abelian groups and generating sets, and relates all homogeneous toric varieties to these maximal cases.

## Key findings

- Maximal S-homogeneous toric varieties correspond to specific algebraic pairs.
- Every non-degenerate homogeneous toric variety is an open subset of a maximal one.
- All homogeneous toric varieties are quasiprojective.

## Abstract

A non-degenerate toric variety $X$ is called $S$-homogeneous if the subgroup of the automorphism group $\text{Aut}(X)$ generated by root subgroups acts on $X$ transitively. We prove that maximal $S$-homogeneous toric varieties are in bijection with pairs $(P,\mathcal{A})$, where $P$ is an abelian group and $\mathcal{A}$ is a finite collection of elements in $P$ such that $\mathcal{A}$ generates the group $P$ and for every $a\in\mathcal{A}$ the element $a$ is contained in the semigroup generated by $\mathcal{A}\setminus\{a\}$. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal $S$-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is $S$-homogeneous.