Factorial rational varieties which admit or fail to admit an elliptic ${\mathbb{G}}_m$-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 $X\times {\mathbb{A}}_k^m{\cong }_k{\mathbb{A}}_k^{n+m}$, then $X{\cong }_k{\mathbb{A}}_k^n$ if and only if $X$ admits an elliptic ${\mathbb{G}}_m$-action.

18 pages