# Factorial rational varieties which admit or fail to admit an elliptic $\mathbb{G}_m$-action

## TL;DR

This paper classifies certain rational factorial varieties with positive gradings over algebraically closed fields, examines specific threefolds for elliptic actions, and characterizes affine spaces via elliptic $ extbf{G}_m$-actions.

## Contribution

It provides a classification of rational UFDs with positive gradings in dimension two and analyzes the existence of elliptic $ extbf{G}_m$-actions on specific threefolds and affine varieties.

## Key findings

- Classified rational UFDs of dimension two over algebraically closed fields with positive $ extbf{Z}$-grading.
- Showed that Russell cubic and Asanuma threefolds admit no elliptic $ extbf{G}_m$-action.
- Characterized affine spaces as those varieties admitting an elliptic $ extbf{G}_m$-action.

## Abstract

Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is affine over $k$. We also consider the Russell cubic threefold over $\mathbb{C}$, and the Asanuma threefolds over a field of positive characterstic, showing that these threefolds admit no elliptic $\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then $X\cong_k\mathbb{A}^n_k$ if and only if $X$ admits an elliptic $\mathbb{G}_m$-action.