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Jordan bounds for volume-preserving Cremona groups

This paper establishes that the Jordan constant for the volume-preserving plane Cremona group is 12, provides a Jordan bound of 144 for its three-dimensional counterpart, and derives a weak geometric Jordan bound of $2^{11} \cdot 3^2$ for the standard plane Cremona group.

Original authors: Jiahe Wang

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiahe Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect working in a magical, infinite city called Projective Space. In this city, you have a special set of tools that allow you to reshape buildings, stretch walls, and even tear holes in reality, as long as you can glue everything back together perfectly. This group of all possible reshaping tools is called the Cremona Group.

Now, imagine there's a strict rule in this city: You must preserve the "volume" of the air inside every room. If you stretch a room, you must shrink another part to keep the total amount of air exactly the same. This is the Volume-Preserving Cremona Group.

The paper by Jiahe Wang is essentially a detective story about finding the maximum chaos allowed in this city before things become too messy to control.

The Core Concept: The "Jordan Property"

Think of any group of transformations (like a team of architects) as a giant, chaotic dance party.

The Problem: Sometimes, the dancers are so wild and uncoordinated that you can't find a single, calm, orderly line of dancers (an "abelian normal subgroup") that everyone else can follow.
The Solution (Jordan Property): The mathematician Camille Jordan discovered that for many groups, no matter how chaotic the party gets, you can always find a small, calm group of dancers (the "abelian" part) that everyone else is just a few steps away from.
The "Jordan Constant": This is the maximum number of steps you might have to take to reach that calm group. If the constant is 12, it means "No matter how crazy the party gets, you are never more than 12 steps away from a calm, orderly line of dancers."

The Main Discovery: The 2D City (Plane)

The author focuses on the 2D version of this city (the plane).

The Old Rule: For the general reshaping group (without the volume rule), the maximum chaos was known to be huge (7,200 steps). That's a very wild party.
The New Rule: When you add the volume-preserving rule (keeping the air constant), the chaos drops dramatically.
The Result: Wang proves that for the volume-preserving group, the maximum chaos is only 12.
- Analogy: Imagine a room full of people trying to rearrange furniture. Without rules, they might need 7,200 different moves to get everyone to agree on a plan. But if they have to keep the total volume of furniture constant, they only need 12 moves to find a calm, orderly plan.

How did he prove this?
He realized that any chaotic reshaping team can be "regularized" (smoothed out) onto a specific type of geometric surface (like a Del Pezzo surface). On these surfaces, the volume rule forces the "pole divisors" (the edges of the rooms) to form a specific shape: a cycle of lines (like a hexagon or a triangle).
The group of transformations essentially acts like a rotation or reflection of this shape. The most complex shape they can form is a hexagon (6 sides), and the group of symmetries for a hexagon is small (12). This limits the chaos to 12.

The 3D Extension

The author then asked: "What if we go to 3D?"

In 3D, the "edges" of the rooms form a more complex shape (like a sphere made of triangles).
The symmetries of a sphere are more complex than a hexagon.
The Result: The chaos limit goes up to 144. It's still much smaller than the general 3D group (which is in the millions), but it's bigger than the 2D limit.

The "Geometric" Twist

Finally, the paper looks at a stricter version of the rule: Geometric Jordan Property.

The Analogy: In the previous version, the "calm group" just had to be orderly. In this version, the calm group must be a specific type of orderly group: one that looks like a torus (a donut shape).
The Challenge: Can we always find a calm, donut-shaped group within the chaos?
The Result: Yes! The author proves that even with this stricter rule, the 2D group is controllable. The number of steps to find this "donut group" is calculated to be $2^{11} \cdot 3^2$ (which is 2,073,600). While this number is big, the fact that it exists is the mathematical breakthrough.

Summary in a Nutshell

The Setting: A magical city where you can reshape space, but you must keep the "volume" constant.
The Question: How chaotic can the reshapers get before we can't find a calm, orderly subgroup?
The Answer (2D): Surprisingly low! Only 12. The volume rule acts like a strict bouncer, keeping the party tame.
The Answer (3D): A bit higher at 144, but still very controlled compared to the wild, unrestricted version.
The Takeaway: By adding a simple physical constraint (volume preservation), the mathematical structure of these infinite transformation groups becomes much simpler and more predictable.

The paper is a triumph of finding order in chaos by realizing that conservation laws (like volume) act as a leash on mathematical wildness.

1. Problem Statement

The paper addresses the Jordan property for groups of birational automorphisms, specifically focusing on volume-preserving subgroups.

The Jordan Property: A group $G$ has the Jordan property if there exists a constant $J$ (the Jordan constant) such that every finite subgroup $G' \leq G$ contains an abelian normal subgroup $A \unlhd G'$ with index $[G':A] \leq J$ .
Context: While the Jordan property for the full Cremona group $\text{Bir}(\mathbb{P}^n)$ is known (established by Prokhorov and Shramov), the specific constants for volume-preserving subgroups, denoted $\text{Bir}(\mathbb{P}^n, \Delta)$ , were not explicitly determined. Here, $\Delta$ represents the pole divisor of a standard torus-invariant volume form $\Omega$ .
Specific Questions:
1. What is the exact Jordan constant for the plane volume-preserving Cremona group $\text{Bir}(\mathbb{P}^2, \Delta)$ ?
2. What is a Jordan bound for the three-dimensional case $\text{Bir}(\mathbb{P}^3, \Delta)$ ?
3. Does $\text{Bir}(\mathbb{P}^2)$ satisfy the geometric Jordan property (where the abelian subgroup must be contained in an algebraic torus), and what is the bound?

2. Methodology

The author employs a combination of birational geometry, minimal model programs (MMP), and group theory to analyze finite subgroups.

A. Regularization and Minimal Models

Following the work of Dolgachev and Iskovskikh, the paper utilizes the fact that any finite subgroup $G \leq \text{Bir}(\mathbb{P}^n)$ can be "regularized" on a rational variety $X$ such that $G \leq \text{Aut}(X)$ .

For $\text{Bir}(\mathbb{P}^2, \Delta)$ , the author proves that $G$ can be regularized on a minimal $G$ -surface $X$ (either a del Pezzo surface or a conic bundle) equipped with a volume form $\omega_X$ such that the pair $(X, D_X)$ is a Calabi-Yau pair (i.e., $K_X + D_X \sim 0$ and $D_X$ is effective).
Crucially, the author shows that for volume-preserving maps, the regularization can be chosen such that the pullback volume form has no zeros, ensuring $D_X$ is effective.

B. Action on the Dual Complex

The core structural tool is the dual complex $\mathcal{D}(D_X)$ of the boundary divisor $D_X$ .

Structure: For the 2D case, $\mathcal{D}(D_X)$ is shown to be homeomorphic to a circle ( $S^1$ ). For the 3D case, it is homeomorphic to a sphere ( $S^2$ ).
Exact Sequence: The analysis yields an exact sequence:
$1 \to A \to G \to G_D \to 1$
where $A$ is the kernel (acting trivially on the dual complex) and $G_D$ is the group acting on the dual complex.
Abelian Kernel: Using results from Loginov, Przyjalkowski, and Trepalin, the author establishes that $A$ is an abelian group of rank at most $n$ (where $n$ is the dimension). Thus, bounding $G$ reduces to bounding $G_D$ .

C. Classification of Actions

2D Case: The action on the circle $\mathcal{D}(D_X) \cong S^1$ is restricted to cyclic or dihedral groups. The author classifies these actions based on the degree of the del Pezzo surface or the type of conic bundle.
3D Case: The action on $S^2$ involves finite subgroups of $\text{SO}(3)$ (icosahedral, octahedral, tetrahedral, dihedral, cyclic). The bound is derived by reducing the 3D problem to the 2D problem via adjunction on boundary strata.

D. Geometric Jordan Property

To address the geometric property, the paper utilizes the concept of torsion index for linear algebraic groups.

The strategy involves contracting disjoint $(-1)$ -curves equivariantly to reduce the surface to a simpler model (e.g., $\mathbb{P}^2$ , $\mathbb{P}^1 \times \mathbb{P}^1$ , or Hirzebruch surfaces $\mathbb{F}_n$ ).
The finite abelian subgroups are then analyzed within the automorphism groups of these simpler surfaces (which are linear algebraic groups like $\text{PGL}_3(\mathbb{C})$ ).
The torsion index $t(G)$ determines the index required to embed a finite abelian subgroup into a maximal torus.

3. Key Contributions and Results

Result 1: Exact Jordan Constant for $\text{Bir}(\mathbb{P}^2, \Delta)$

Theorem: $J(\text{Bir}(\mathbb{P}^2, \Delta)) = 12$ .
Derivation:
- The kernel $A$ is abelian.
- The quotient $G_D$ acts on the dual complex (a circle). The maximal finite group acting on the components of the boundary divisor for a degree 6 del Pezzo surface is the dihedral group $D_6$ of order 12.
- Minimality: The author constructs a specific subgroup on a degree 6 del Pezzo surface (isomorphic to $(\mathbb{Z}/n\mathbb{Z})^2 \rtimes S_3$ extended by an involution) where the induced action on the dual complex is $D_6$ . It is proven that this group has no abelian normal subgroup of index less than 12.
Significance: This is significantly smaller than the Jordan constant for the full plane Cremona group $\text{Bir}(\mathbb{P}^2)$ , which is 7200.

Result 2: Jordan Bound for $\text{Bir}(\mathbb{P}^3, \Delta)$

Theorem: $J(\text{Bir}(\mathbb{P}^3, \Delta)) \leq 144$ .
Derivation:
- The kernel $K$ (acting trivially on the dual complex $S^2$ ) is abelian of rank at most 3.
- The quotient $G/K$ acts on $S^2$ . By passing to a subgroup of index at most 2, the action fixes a point on $S^2$ .
- This reduces the problem to the stabilizer of a stratum, which is a volume-preserving group in dimension 2 ( $\text{Bir}(\mathbb{P}^2, \Delta)$ ).
- The bound is calculated as: $2 \times (\text{index for fixing point}) \times J(\text{Bir}(\mathbb{P}^2, \Delta)) = 2 \times 3 \times 12 \times 2 = 144$ . (Note: The factor of 2 accounts for the potential index 2 subgroup, and the factor of 3 for fixing a component).

Result 3: Geometric Jordan Property for $\text{Bir}(\mathbb{P}^2)$

Theorem: $\text{Bir}(\mathbb{P}^2)$ satisfies the weak geometric Jordan property.
Bound: A weak geometric Jordan bound is $2^{11} \cdot 3^2 = 20736$ .
Significance: This confirms a conjecture by Moraga regarding cluster-type varieties. The proof relies on showing that finite abelian subgroups can be embedded into algebraic tori of $\text{PGL}_3$ , $\text{PGL}_2 \times \text{PGL}_2$ , or $\text{Aut}(\mathbb{F}_n)$ with controlled indices derived from torsion indices ( $t=3, 4, 2$ ).

4. Significance and Impact

Precision in Volume-Preserving Groups: The paper provides the exact Jordan constant for the 2D volume-preserving Cremona group, a rare achievement in birational geometry where bounds are often loose.
Dimensional Induction: The methodology successfully extends the 2D result to 3D by leveraging the geometry of the dual complex and adjunction, offering a blueprint for higher-dimensional analysis.
Geometric Refinement: By establishing the geometric Jordan property, the paper connects the abstract group-theoretic Jordan property with the algebraic structure of the groups (embedding into tori), which is crucial for understanding the moduli of these groups.
Minimal Counterexamples: The construction of the degree 6 del Pezzo surface example serves as a sharp test case, proving that the bound 12 is not just an upper limit but the precise threshold.

In summary, Jiahe Wang's paper refines our understanding of the structure of volume-preserving birational maps, moving from general existence proofs to precise numerical bounds and geometric characterizations.