# Affine toric varieties with an open orbit of an $SL_n$ action

## TL;DR

This paper characterizes affine toric varieties with dense $SL_n$ orbits using $SL_n imes Q$-modules, revealing bounds on class group rank for $n=3$ and unboundedness for larger $n$.

## Contribution

It provides a new characterization of such varieties via module theory and explicitly analyzes the case $n=3$, including class group rank bounds.

## Key findings

- Class group rank is at most 3 for $n=3$.
- Class group rank is unbounded for $n \\geq 4$.
- Explicit module-theoretic description for $n=3$ case.

## Abstract

We study affine toric varieties with an action of group $SL_n$ with a dense orbit. A characterisation in terms of $SL_n \times Q$-modules is given where $Q$ is a quasitorus. This characterisation is more explicitly expanded in case $n=3$. It is shown that in case $n=3$ the class group rank is not greater than $3$, however it is unbounded when $n\geq 4$.