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arxiv: 2604.24495 · v1 · submitted 2026-04-27 · 🧮 math.AG

On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero

Yutaro Naito This is my paper

Pith reviewed 2026-05-08 02:01 UTC · model grok-4.3

classification 🧮 math.AG
keywords toric varietiessymmetric group actionsequivariant minimal model programGalois descentprojective spacecharacteristic zero fieldsalgebraic surfaces
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The pith

Over fields of characteristic zero with a suitable arithmetic property, toric varieties admitting maximal symmetric group actions are uniquely projective space except in dimension two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies complete simplicial toric varieties that admit faithful actions by large symmetric groups. It first corrects an earlier classification of four-dimensional examples with S6 actions over the complex numbers. The central extension considers base fields such as the rationals or reals. In dimensions other than two, the maximal symmetric action forces the variety to be projective space over the base field. In dimension two an infinite family of both split and non-split toric surfaces with faithful S4 actions is classified using the equivariant minimal model program and Galois descent.

Core claim

Over fields k of characteristic zero satisfying a certain arithmetic condition, for n ≠ 2 the maximal symmetric action uniquely restricts the variety to the projective space P^n_k; for n=2 there exists an infinite family of split and non-split toric surfaces admitting faithful S4-actions, classified via the equivariant Minimal Model Program and Galois descent. The paper also supplies the complete list of four-dimensional toric varieties with S6 actions over C after correcting a recent result.

What carries the argument

The equivariant Minimal Model Program combined with Galois descent, which produces the classification of toric surfaces admitting faithful S4 actions over non-closed fields.

Load-bearing premise

The base fields satisfy an arithmetic condition that makes the rigidity arguments and Galois descent apply without obstruction.

What would settle it

A non-projective toric threefold over the rationals that admits a faithful action by S5 or a larger symmetric group would falsify the uniqueness claim for dimensions other than two.

Figures

Figures reproduced from arXiv: 2604.24495 by Yutaro Naito.

Figure 1
Figure 1. Figure 1: Examples of fans in N generating toric surfaces with S4-actions. since their determinant in the basis of N1 is ±3 ̸= ±1. The resulting complete toric surface has 6 quotient singularities, demonstrating that S4 can act on infinitely many singular Fano toric surfaces. Lemma 4.10. Let L/k be an arbitrary quadratic Galois extension, and let ⟨τ ⟩ = Gal(L/k) ∼= Z/2Z. Consider a lattice N (either N1 or N2) equipp… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper corrects a prior classification of 4-dimensional complete simplicial toric varieties admitting faithful S_6-actions over ℂ. It then considers complete simplicial toric varieties over fields k of characteristic zero satisfying a certain arithmetic condition (exemplified by ℚ and ℝ). For n ≠ 2 the authors claim that any faithful action of a maximal symmetric group forces the variety to be ℙ^n_k. For n = 2 they classify an infinite family of both split and non-split toric surfaces admitting faithful S_4-actions, obtained via the equivariant Minimal Model Program together with Galois descent.

Significance. If the central claims are verified, the work establishes a dimension-dependent rigidity phenomenon for toric varieties under large symmetric-group actions over non-algebraically closed fields of characteristic zero. The explicit correction to the 4-dimensional S_6 case supplies a reliable reference list. The surface classification furnishes concrete examples and demonstrates that standard equivariant MMP and Galois-descent techniques apply effectively in the toric setting, which may serve as a template for further arithmetic questions on symmetric actions.

major comments (2)
  1. [Abstract and Introduction] The arithmetic condition on the base field k is invoked repeatedly as the hypothesis that enables both the rigidity statement for n ≠ 2 and the Galois-descent arguments for n = 2, yet it is described only as “a certain arithmetic condition (such as ℚ or ℝ)” without an explicit definition or list of equivalent characterizations. Because this hypothesis is load-bearing for the main theorems, it must be stated precisely (e.g., as a numbered definition in the introduction or §2) together with a verification that the listed examples satisfy it.
  2. [Section containing the 4-dimensional correction] The correction to the Esser–Ji–Moraga list of 4-dimensional toric varieties with S_6-actions is asserted to be complete, but the manuscript provides no explicit comparison table or enumeration of the discrepancies with the earlier list. A side-by-side statement of the previous and corrected lists (or at least an indication of which varieties were added or removed) is required to substantiate the claim that the new list is exhaustive.
minor comments (2)
  1. [Throughout] Notation for the symmetric groups (S_n versus Sym_n) and for the toric varieties (X_Σ versus X) should be fixed consistently from the first appearance onward.
  2. [Main theorem for n ≠ 2] The statement of the main rigidity theorem for n ≠ 2 should include a parenthetical reminder of the precise arithmetic condition on k, even if the condition is defined earlier.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract and Introduction] The arithmetic condition on the base field k is invoked repeatedly as the hypothesis that enables both the rigidity statement for n ≠ 2 and the Galois-descent arguments for n = 2, yet it is described only as “a certain arithmetic condition (such as ℚ or ℝ)” without an explicit definition or list of equivalent characterizations. Because this hypothesis is load-bearing for the main theorems, it must be stated precisely (e.g., as a numbered definition in the introduction or §2) together with a verification that the listed examples satisfy it.

    Authors: We agree that the arithmetic condition requires an explicit definition. In the revised manuscript we will introduce a numbered definition (e.g., Definition 1.1) in the introduction that states the condition precisely, together with a brief verification that the fields ℚ and ℝ satisfy it. revision: yes

  2. Referee: [Section containing the 4-dimensional correction] The correction to the Esser–Ji–Moraga list of 4-dimensional toric varieties with S_6-actions is asserted to be complete, but the manuscript provides no explicit comparison table or enumeration of the discrepancies with the earlier list. A side-by-side statement of the previous and corrected lists (or at least an indication of which varieties were added or removed) is required to substantiate the claim that the new list is exhaustive.

    Authors: We agree that an explicit comparison would make the correction more transparent. In the revised manuscript we will add a side-by-side table or enumerated list of discrepancies with the Esser–Ji–Moraga classification, indicating which varieties are retained, removed, or newly included. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard tools

full rationale

The paper corrects an external classification of 4-dimensional toric varieties with S6-actions (Esser-Ji-Moraga) and applies the equivariant Minimal Model Program plus Galois descent to classify S4-actions on toric surfaces over fields satisfying an arithmetic condition. These are standard external methods in algebraic geometry, not derived from or reducing to the paper's own definitions, fitted parameters, or self-citations. The rigidity claim for n≠2 (restriction to projective space) follows from the action analysis without self-referential equations or ansatzes smuggled via prior work by the same author. No load-bearing step collapses to a fitted input renamed as prediction or a uniqueness theorem imported from the author's own prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in toric geometry and the minimal model program rather than new free parameters or invented entities.

axioms (2)
  • domain assumption The equivariant Minimal Model Program applies to complete simplicial toric varieties equipped with faithful actions of finite groups.
    Invoked to classify the 2-dimensional case.
  • domain assumption Galois descent is valid for the toric surfaces obtained after running the equivariant MMP over the given fields of characteristic zero.
    Required to produce both split and non-split examples over non-closed fields.

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