Factorial rational varieties which admit or fail to admit an elliptic $\mathbb{G}_m$-action
| Autor: | Freudenburg, Gene, Nagamine, Takanori |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 18 pages |
| Databáze: | arXiv |
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