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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2604.24495 |
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- 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.