$p$-elementary non-cyclic subgroups of the Cremona group of the plane

Abstract: We classify, up to conjugacy, the subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})<sup>r$, where $p$ is prime and $r \geq 2$, over an algebraically closed field $\mathbf{k}$ of characteristic not equal to $p$. In particular, we show that $r \leq 2$ if $p \geq 5$, $r \leq 3$ if $p=3$, and $r \leq 4$ if $p=2$. Furthermore, we give an explicit list of representatives via a set of 20 families consisting of subgroups of the de Jonquières group and subgroups of automorphisms of del Pezzo surfaces, and we study the possible conjugacies by birational maps between these families. Finally, we give some results on subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})<sup>r$, where $p$ is prime and $r$ is an integer, over an algebraically closed field $\mathbf{k}$ of characteristic equal to $p$.

• The paper provides a complete birational conjugacy classification of (ℤ/p)^r subgroups in the Cremona group, establishing tight bounds on r based on the prime p.
• It combines classical birational geometry with modern cohomological techniques to analyze actions on del Pezzo surfaces, conic bundles, and de Jonquières subgroups.
• The results demonstrate rigidity in subgroup conjugacy classes through the study of fixed loci and invariant curves, with implications for both characteristic zero and positive characteristic cases.

$p$-Elementary Non-Cyclic Subgroups of the Cremona Group of the Plane

Introduction

The paper "$p$-elementary non-cyclic subgroups of the Cremona group of the plane" (2607.09941) presents a complete birational conjugacy classification of subgroups of the plane Cremona group $\Cr = \mathrm{Bir}(\mathbb{P}^2)$ isomorphic to $(\mathbb{Z}/p)^r$ for $r \geq 2$, where $p$ is a prime and the ground field $k$ is algebraically closed of characteristic $\mathrm{char}(k) \neq p$. The approach integrates classical birational geometry, modern cohomological methods, and detailed analysis of group actions on del Pezzo surfaces and conic bundles. The results both generalize earlier work and resolve open cases regarding the structure and possible conjugacy classes of $p$-elementary abelian subgroups, particularly for non-maximal, non-cyclic configurations.

Birational Classification: Numerical Bounds and Explicit Representatives

The primary result is the explicit birational classification summarized as follows:

• If $p \geq 5$, then any abelian $p$0-elementary subgroup of $p$1 isomorphic to $p$2 must satisfy $p$3.
• For $p$4, one may have $p$5; for $p$6, $p$7.
• The only possible types of non-cyclic $p$8-elementary groups in $p$9 (for $\Cr = \mathrm{Bir}(\mathbb{P}^2)$0) are: $\Cr = \mathrm{Bir}(\mathbb{P}^2)$1, $\Cr = \mathrm{Bir}(\mathbb{P}^2)$2, $\Cr = \mathrm{Bir}(\mathbb{P}^2)$3, $\Cr = \mathrm{Bir}(\mathbb{P}^2)$4, $\Cr = \mathrm{Bir}(\mathbb{P}^2)$5, and $\Cr = \mathrm{Bir}(\mathbb{P}^2)$6 for $\Cr = \mathrm{Bir}(\mathbb{P}^2)$7.

For each group type, the paper gives an explicit, non-redundant list of representatives, grouped into three major geometric classes:

• Type $\Cr = \mathrm{Bir}(\mathbb{P}^2)$8: Subgroups of the de Jonquières group, characterized by the existence of non-trivial elements fixing non-rational curves.
• Type $\Cr = \mathrm{Bir}(\mathbb{P}^2)$9: Subgroups acting on del Pezzo surfaces with minimal invariant Picard rank (typically acting as automorphisms leaving the Picard group invariant only up to multiples of the canonical class).
• Type $(\mathbb{Z}/p)^r$0: Subgroups contained in $(\mathbb{Z}/p)^r$1, with all non-trivial elements fixing only rational curves.

Each class is characterized by the geometry of the fixed loci of group elements, the structure of invariant curves, and the possible existence or absence of invariant conic bundles or Picard lattice extensions.

Birational Conjugacy and Isomorphism Classes

A strong structural result in the paper establishes the rigidity of the conjugacy classes in most situations:

• Two subgroups in the $(\mathbb{Z}/p)^r$2 (del Pezzo) class are birationally conjugate if and only if they are birationally isomorphic as pairs.
• For the $(\mathbb{Z}/p)^r$3 (de Jonquières) class, birational conjugacy is equivalent to conjugacy by a de Jonquières birational map.
• In the case of $(\mathbb{Z}/p)^r$4 (product of projective lines), birational conjugacy coincides with equality as subgroups.

Exceptional conjugacies can only occur between certain de Jonquières subgroups of types $(\mathbb{Z}/p)^r$5, related to the ambiguous "amazing result," wherein two a priori different constructions yield conjugate (in $(\mathbb{Z}/p)^r$6) Klein four-subgroups with specific conic bundle structures, but only in particular geometric situations.

A detailed analysis, grounded in the invariance of fixed loci of non-rational curves under birational maps, and leveraging the Iskovskikh–Sarkisov program for minimal models, shows that—outside these explicitly described cases—there are no "hidden" birational equivalences between the given families.

Cohomological and Structural Methods

The proof of the classification proceeds by reducing each action to one of two geometric situations, as per a classical dichotomy: either the group acts minimally on a del Pezzo surface, or it leaves invariant a conic bundle. The authors utilize:

• A careful combinatorial and linear analysis of $(\mathbb{Z}/p)^r$7-elementary subgroups of $(\mathbb{Z}/p)^r$8 for $(\mathbb{Z}/p)^r$9 and explicit computation of their conjugacy invariants.
• Cohomological vanishing results (e.g., vanishing of $r \geq 2$0 for associated Galois modules) to describe extensions for non-cyclic group actions.
• Fundamental invariants of the subgroups, such as the structure of fixed loci, the determinant map modulo squares (for $r \geq 2$1), and the orbits of the action on the Picard group.

The Case $r \geq 2$2 and Open Directions

Section 5 relaxes the restriction on the ground field's characteristic, analyzing the behavior of $r \geq 2$3-elementary subgroups when $r \geq 2$4. In this context, the algebraic group structure permits $r \geq 2$5-elementary subgroups of unbounded rank inside the de Jonquières group—even infinite-dimensional in the Zariski topology—but the classification is only partial: any such group can be conjugated into a standard form built from triangular (unipotent) matrices. For $r \geq 2$6, a refined classification of Klein four-groups in $r \geq 2$7 is also obtained, showing considerable differences from the characteristic zero case, especially in the behavior of their fixed points and birational conjugacies.

An open problem is the full classification in positive characteristic and over more general non-closed fields; the present results are formulated for algebraically closed fields. Further conjectural directions include the investigation of $r \geq 2$8-elementary subgroups in higher-dimensional Cremona groups and the study of their dynamics.

Implications

The sharp numerical bounds and the rigidity of possible conjugacy classes for $r \geq 2$9-elementary non-cyclic subgroups of $p$0 significantly strengthen our understanding of the algebraic and geometric structure of the Cremona group and its finite subgroups. The results demonstrate that the rich landscape of $p$1-elementary abelian group actions on rational surfaces is, in fact, geometrically very constrained up to birational equivalence, except in characteristics matching $p$2.

Practically, this classification constrains what kinds of group actions and quotient surface geometries can arise in rational surface geometry and, more generally, in algebraic dynamics and invariant theory for plane transformations.

Theoretically, the results reinforce the connections between surface geometry, lattice theory, and birational group theory, suggesting that in higher dimensions similarly tight classifications may be attainable, albeit with more intricate behavior expected. The approach, blending cohomological and explicit geometric techniques, provides a template for future study of finite group actions in the Cremona group and related birational transformation groups.

Conclusion

This work completes the birational classification of $p$3-elementary non-cyclic subgroups of the plane Cremona group for $p$4, providing explicit representatives, geometric characterization, and rigidity statements for all possible cases. The extension to positive characteristic highlights new behaviors but also delineates outstanding challenges. The methods and results represent an overview of explicit algebraic group theory and the birational geometry of rational surfaces, resolving a central classification problem in the structure theory of the Cremona group and opening avenues for further exploration in both positive characteristic and higher dimensions.