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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.24495 |
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| _version_ | 1866917440009535488 |
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| author | Naito, Yutaro |
| author_facet | Naito, Yutaro |
| contents | In this paper, we study complete simplicial toric varieties admitting faithful actions of large symmetric groups. First, we correct a recent classification result by Esser, Ji, and Moraga concerning $4$-dimensional toric varieties with $S_6$-actions over the complex numbers $\mathbb{C}$, providing the complete list of such varieties. Second, we extend the study of maximal symmetric group actions to non-closed fields $k$ of characteristic zero satisfying a certain arithmetic condition (such as $\mathbb{Q}$ or $\mathbb{R}$). Over such fields, we reveal a striking rigidity in dimensions $n \neq 2$, where the maximal symmetric action uniquely restricts the variety to the projective space $\mathbb{P}^n_k$. In sharp contrast, for dimension $n=2$, we discover and classify an infinite family of split and non-split toric surfaces admitting faithful $S_4$-actions by utilizing the equivariant Minimal Model Program and Galois descent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24495 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero Naito, Yutaro Algebraic Geometry In this paper, we study complete simplicial toric varieties admitting faithful actions of large symmetric groups. First, we correct a recent classification result by Esser, Ji, and Moraga concerning $4$-dimensional toric varieties with $S_6$-actions over the complex numbers $\mathbb{C}$, providing the complete list of such varieties. Second, we extend the study of maximal symmetric group actions to non-closed fields $k$ of characteristic zero satisfying a certain arithmetic condition (such as $\mathbb{Q}$ or $\mathbb{R}$). Over such fields, we reveal a striking rigidity in dimensions $n \neq 2$, where the maximal symmetric action uniquely restricts the variety to the projective space $\mathbb{P}^n_k$. In sharp contrast, for dimension $n=2$, we discover and classify an infinite family of split and non-split toric surfaces admitting faithful $S_4$-actions by utilizing the equivariant Minimal Model Program and Galois descent. |
| title | On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2604.24495 |