On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero

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.

• The paper presents a complete classification of toric varieties admitting maximal symmetric actions, rectifying prior miscounted cases in four-dimensional settings.
• It employs Cox rings, combinatorial fan analysis, and Galois descent to rigorously analyze symmetry constraints under specified arithmetic conditions.
• The results reveal rigidity in high-dimensional projective spaces and an infinite, flexible family of toric surfaces with faithful S₄ actions.

Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero

Overview

The paper "On Maximal Symmetries of Toric Varieties over Fields of Characteristic Zero" (2604.24495) presents a comprehensive classification of complete simplicial toric varieties admitting faithful actions by symmetric groups $S_m$, particularly over fields of characteristic zero. The work corrects prior results for four-dimensional cases over the complex numbers, extends the analysis to non-algebraically closed fields (e.g., $\mathbb{Q}$, $\mathbb{R}$), and reveals rigidities and exceptional families in this broader context. The dichotomy between the algebraically closed and non-closed fields leads to striking results: in dimension $n\neq 2$, maximal symmetry actions constrain the geometry uniquely to projective spaces, while dimension $n=2$ exhibits infinite families of split and non-split toric surfaces with $S_4$ actions.

Theoretical Background

Toric varieties are defined combinatorially via fans in lattices, allowing their geometric and group symmetries to be systematically analyzed. The study leverages Cox rings, their graded automorphism groups, and the classification of symmetries via partitions of rays and the associated linear equivalence classes. Over algebraically closed fields, automorphism groups are robust and classification results exploit classic representation theory. Extending to non-closed fields introduces arithmetic obstructions that restrict the realizable group actions via Galois descent and Weil restriction, intertwining algebraic geometry, arithmetic, and group theory.

The key arithmetic condition imposed (denoted $(\star)$) is that $a^2+b^2=-1$ has no solution in $k$ (e.g., $k=\mathbb{Q},\mathbb{R}$), which ensures limitations on embedding symmetric groups in linear automorphism groups over $\mathbb{Q}$0.

Correction and Completion of Four-Dimensional Classification ($\mathbb{Q}$1)

A prior classification due to Esser, Ji, and Moraga undercounted varieties admitting $\mathbb{Q}$2 actions in dimension 4. The paper proves:

Theorem A: For a complete simplicial toric fourfold $\mathbb{Q}$3 over $\mathbb{Q}$4, $\mathbb{Q}$5 acts faithfully if and only if $\mathbb{Q}$6 is isomorphic to one of:

• $\mathbb{Q}$7;
• Projective bundle $\mathbb{Q}$8 for even $\mathbb{Q}$9;
• Weighted projective space $\mathbb{R}$0 for even $\mathbb{R}$1.

These results follow from precise analysis of linear equivalence partitions in the Cox ring, combinatorial fan structure, and spin representation theory. Faithful $\mathbb{R}$2 actions require particular congruence properties (arguments involving the parity of $\mathbb{R}$3 and $\mathbb{R}$4), tied to the structure of the graded automorphism group and its semidirect product decomposition.

Maximal Symmetric Group Actions over Non-Closed Fields

When $\mathbb{R}$5 is not algebraically closed and satisfies $\mathbb{R}$6, arithmetic constraints dramatically restrict symmetric group actions.

Theorem B: For $\mathbb{R}$7 as above, a complete simplicial toric $\mathbb{R}$8-fold $\mathbb{R}$9 admits a maximal symmetric group action $n\neq 2$0 if and only if $n\neq 2$1 for $n\neq 2$2. For $n\neq 2$3, there exists an infinite family of split and non-split toric surfaces with faithful $n\neq 2$4 actions.

Implications:

• The projective space $n\neq 2$5 is the unique rigid possibility for maximal action unless $n\neq 2$6.
• For $n\neq 2$7, classification requires running the $n\neq 2$8-equivariant Minimal Model Program (MMP) and deploying Galois descent, yielding both split (torus-invariant divisor structures and $n\neq 2$9-invariant fans) and non-split cases (via quadratic extensions and $n=2$0 involution).
• Non-split forms (including the Weil restriction $n=2$1) can also admit $n=2$2 actions, provided the arithmetic condition is satisfied in the extension.

Explicit Structure of Toric Surfaces with Faithful $n=2$3 Actions

The paper constructs infinite families in $n=2$4 through:

• Lattice representations ($n=2$5 and $n=2$6) with $n=2$7 acting by coordinate permutations.
• Complete fans invariant under $n=2$8, generating surfaces on which the Klein four-group $n=2$9 (the $S_4$0-torsion subgroup of the torus) and $S_4$1 combine as $S_4$2.
• Examples include $S_4$3, the del Pezzo surface $S_4$4, and surfaces derived from intricate $S_4$5 orbits in these lattices, some smooth, some singular.

In non-split cases, Galois descent (with the involution acting as $S_4$6 on $S_4$7) ensures variant forms over $S_4$8, including surfaces like $S_4$9 over $(\star)$0 and towers of blow-ups at $(\star)$1-orbits of torus-invariant points.

Numerical Bounds and Faithful Action Constraints

Strong numerical results include:

• For $(\star)$2, only $(\star)$3 acts faithfully on $(\star)$4 over $(\star)$5 satisfying $(\star)$6; $(\star)$7 is obstructed by arithmetic constraints.
• For $(\star)$8, $(\star)$9 acts faithfully only on $a^2+b^2=-1$0.
• For $a^2+b^2=-1$1, $a^2+b^2=-1$2 is maximal, with substantially richer variety structure.

Contradictory and bold claim: The geometry is rigid for $a^2+b^2=-1$3, with the arithmetic effectively eliminating previously possible actions, while $a^2+b^2=-1$4 is infinitely flexible.

Equivariant Minimal Model Program and Smooth Surfaces

Running the $a^2+b^2=-1$5-equivariant MMP:

• For split surfaces, contractions yield either $a^2+b^2=-1$6 or $a^2+b^2=-1$7.
• For non-split surfaces, the process terminates at $a^2+b^2=-1$8, but can be expanded by blowing up at $a^2+b^2=-1$9-orbits, leading to infinite smooth non-split surfaces with $k$0 symmetry.
• Toric blow-ups of the Weil restriction $k$1 do not admit $k$2 actions, marking a geometric end point.

Practical and Theoretical Implications

The results:

• Highlight a deep interplay between arithmetic and geometric symmetries in algebraic geometry.
• Establish rigid classification for toric varieties with maximal symmetric actions over fields with specified arithmetic properties.
• Reveal that projective spaces dominate high-dimension cases, while surfaces are uniquely flexible under $k$3, including constructions not possible over $k$4.
• Enable precise construction of varieties with prescribed automorphism group via lattice combinatorics and Galois descent.
• Provide insight into the limitation of group actions from the perspective of arithmetic representation theory.

Future directions may focus on relaxing characteristic zero to positive characteristic, broadening or modifying arithmetic conditions, or analyzing further group actions (e.g., alternating groups), exploiting similar combinatorial, arithmetic, and geometric tools. Potential applications include moduli problems, arithmetic geometry, and equivariant birational geometry.

Conclusion

This paper achieves a rigorous classification of toric varieties under maximal symmetric group actions for fields of characteristic zero with specific arithmetic constraints. Correction of previous oversight in dimension four, exhaustive classification over non-closed fields, and explicit construction of infinite families in dimension two underscore the profound influence of arithmetic on geometric symmetry. The methods combine combinatorial fan analysis, Cox ring automorphisms, and arithmetic representation theory to yield robust, practical tools for the systematic study of automorphism groups of varieties.