G-birationally rigid cubic threefolds

Abstract: We classify pairs $(X,G)$ consisting of a (possibly singular) cubic threefold $X\subset\mathbb{P}<sup>4$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally rigid, i.e., $X$ is a $G$-Mori fiber space (over a point), and $X$ is not $G$-birational to any $G$-Mori fibre space that is not $G$-biregular to $X$.

• The paper establishes explicit, computable criteria for when a smooth cubic threefold with a finite group action is G-birationally rigid.
• It employs a blend of algebraic methods and computational tools (Magma/GAP) to analyze group actions and invariant substructures.
• The results offer a clear classification that serves as a blueprint for extending equivariant birational rigidity techniques to broader families of Fano threefolds.

G-Birational Rigidity of Cubic Threefolds: Classification and Methods

Introduction and Background

The paper "G-birationally rigid cubic threefolds" (2604.20426) addresses the equivariant birational geometry of cubic threefolds $X \subset \mathbb{P}^4$, focusing on the classification of finite group actions $G \subset \mathrm{Aut}(X)$ for which $X$ exhibits $G$-birational rigidity. Here, $G$-birational rigidity requires $X$ to be a $G$-Mori fiber space and for any $G$-Mori fiber space birationally $G$-equivalent to $X$ to be $G \subset \mathrm{Aut}(X)$0-biregular to $G \subset \mathrm{Aut}(X)$1. The concept generalizes classical birational rigidity—an obstruction to rationality—into the equivariant framework, driven by problems on irrationality of Fano varieties over non-algebraically closed fields.

The main theorem establishes concrete, computable criteria for when a smooth cubic threefold $G \subset \mathrm{Aut}(X)$2 paired with a finite group $G \subset \mathrm{Aut}(X)$3 is $G \subset \mathrm{Aut}(X)$4-birationally rigid, and, in fact, gives a full classification. The theoretical foundation is set by the Segre-Manin theorem for $G \subset \mathrm{Aut}(X)$5-birational rigidity of cubic surfaces, expanded here to dimension three. Equivalence of three key conditions is established: the absence of $G \subset \mathrm{Aut}(X)$6-invariant planes in $G \subset \mathrm{Aut}(X)$7 (subject to certain group exclusions), $G \subset \mathrm{Aut}(X)$8-birational superrigidity, and $G \subset \mathrm{Aut}(X)$9-birational rigidity of $X$0.

Main Theorem and Classification Strategy

The main theorem asserts that for $X$1 a smooth cubic threefold and $X$2 finite, the following are equivalent:

• There is no $X$3-invariant plane in $X$4, and $X$5 is not isomorphic to any of $X$6, $X$7, or $X$8;
• $X$9 is $G$0-birationally superrigid;
• $G$1 is $G$2-birationally rigid.

The rationale is both algebraic (group representation-theoretic) and geometric (using Mori theory, log pairs, and singularity theory). In particular, the analysis is closely linked to the equivariant Minimal Model Program (MMP).

Group-theoretic analysis: Using known classifications (see [WeiYu2020]), the finite subgroups of $G$3 that can arise for a smooth cubic threefold are subgroups of one of six explicitly listed groups, notably the automorphism groups of the Fermat and Klein cubics and certain products/extensions involving symmetric and cyclic groups.

Birational geometry consideration: Central to the proof is Lemma~\ref{lemma:Pn-pencil} showing that the presence of a $G$4-invariant plane leads to the failure of $G$5-birational rigidity via the existence of a $G$6-equivariant Mori fiber structure with positive-dimensional base. The core rigidity arguments employ multiplier ideal sheaves, Nadel vanishing, canonical threshold techniques, and adjunction, leveraging the absence of small $G$7-orbits and $G$8-invariant low-degree subvarieties.

Explicit exclusion: The groups $G$9, $G$0, and $G$1 are excluded based on explicit birational constructions (via Sarkisov program) showing loss of $G$2-birational rigidity, also employing computations with Magma and GAP.

Strong Claims and Contradictory Examples

The classification provides both positive and negative results. For cubic threefolds with sufficiently large and irreducibly-acting finite symmetry groups, $G$3-birational rigidity and superrigidity hold with rare exceptions. Notably, for the excluded cases, there exist explicit $G$4-equivariant Sarkisov links to other Mori fiber spaces, violating rigidity. The numerical evidence, supported by explicit computations in the representation theory of group actions and the geometry of threefolds, is strong: small $G$5-orbits and $G$6-invariant low-degree curves are forbidden, and the locus of potential non-canonical centers for log pairs is systematically excluded.

Detailed Classification and Examples

A comprehensive classification is given (Corollary~\ref{corollary:main}) for all $G$7 pairs for which $G$8 is $G$9-birationally rigid, with explicit equations and structural descriptions for $X$0 in each case:

• Fermat cubic: $X$1 with $X$2 any subgroup as rendered in Figure 1, excluding the group $X$3 and ensuring the absence of $X$4-invariant planes;
• Klein cubic: $X$5 with $X$6 isomorphic to $X$7 or $X$8;
• Various symmetric cases: Other $X$9 with symmetry group $G$0 as detailed in the full list—including geometric conditions on the intersection with the ambient projective space and parameters (for instance, absence of $G$1-invariant planes).

For singular cubic threefolds with at most terminal singularities, an analogous $G$2-birational rigidity result holds for the Segre and certain nine-nodal cubics, with $G$3 acting transitively on the singular set.

Figure 1: Plane free groups acting on the Fermat cubic threefold, visualizing group-theoretic inclusions and constraints on $G$4 relevant to birational rigidity.

Methods and Technical Core

At the technical heart of the proof lie several innovations and strong points:

• Representation-theoretic constraints: The $G$5-actions are scrutinized using character theory and explicit Magma/GAP computations. In particular, the absence of $G$6-invariant planes is translated into the absence of $G$7-dimensional subrepresentations in the action on $G$8.
• Neighborhood of rigidity: Exceptional birational maps violating rigidity are constructed for the excluded groups and for $G$9-invariant planes, providing strong negative results.
• Equivariant log canonical thresholds: The authors establish that no mobile $G$0-invariant linear system (with possible exceptions only in non-rigid cases) admits non-canonical centers, ruling out the possibility of $G$1-equivariant Sarkisov links to other Mori fiber spaces.
• Use of the multiplier ideal techniques and Nadel vanishing: These are pivotal in confirming the non-existence of non-trivial $G$2-fixed base loci arising from non-canonical pairs; particularly, for large $G$3 the combinatorics of small orbit lengths suffices to derive strong bounds.

Numerical and Computational Results

Strong numerical claims are obtained including the following:

• For $G$4-invariant orbits of points $G$5, $G$6 in the large symmetry case, enforced by representation theory and vanishing cohomology.
• Any $G$7-invariant curve of degree $G$8 is either forbidden or can only exist in explicitly handled (hence classified) configurations, preventing the formation of mobile linear systems with non-canonical singularities.
• For the exceptional cases ($G$9, $G$0, $G$1), existence of explicit $G$2-invariant pencils (in the sense of MMP/Mori theory) leads to $G$3-equivariant links violating rigidity, supporting the necessity of their exclusion.

Broader Implications and Future Work

Pragmatically, this classification offers a blueprint for understanding equivariant birational types of Fano threefolds under group actions and gives a clear birational rigidity landscape for cubic threefolds and their automorphism groups. The approach exploits the full power of computational and theoretical algebraic geometry, particularly representation-theoretic gaps and the machinery of the equivariant MMP.

The paper suggests that similar classification techniques and computational strategies should generalize to the broader family of del Pezzo threefolds, and the authors outline how this might proceed, based on existing partial results for degrees $G$4.

Figures and Visualization

Figure 1: Visualization of the lattice of plane-free subgroups acting on the Fermat cubic threefold, indicating subgroup inclusions (edges), and highlighting the structural role of these subgroups for the birational rigidity problem.

Conclusion

The work provides a comprehensive and technically rigorous determination of $G$5-birationally rigid cubic threefolds and the corresponding automorphism subgroups. The mathematical strategy—combining explicit algebraic geometry, group theory, and computational techniques—offers a robust template for subsequent work in equivariant birational geometry, especially in the context of classifying Mori fiber spaces with large or interesting symmetry. The results have significant implications for moduli theory, rationality considerations, and the further development of equivariant MMP in higher dimensions and for more general Fano varieties.