On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero

TL;DR

This paper classifies maximal symmetric group actions on toric varieties over fields of characteristic zero, correcting previous results and revealing rigidity in higher dimensions while identifying diverse actions in dimension two.

Contribution

It corrects a recent classification of 4-dimensional toric varieties with S_6-actions and extends the analysis of symmetric actions to non-closed fields, highlighting dimension-dependent rigidity and diversity.

Findings

Complete list of 4D toric varieties with S_6-actions over C

Rigidity of maximal symmetric actions for n ≠ 2 over certain fields

Infinite family of symmetric toric surfaces in dimension 2

Abstract

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.