Jordan constants for volume-preserving Cremona groups

TL;DR

This work determines optimal and weak geometric Jordan constants for volume-preserving Cremona groups. By reducing finite subgroups to actions on Calabi–Yau pairs and their dual complexes, the authors derive an exact kernel–quotient structure $1\to A\to G\to G_d\to 1$ with $A$ abelian, then bound $G_d$ via Calabi–Yau classifications and del Pezzo geometry; in dimension 3, a CY-pair analysis yields $J=60$ for $\mathrm{Bir}(\mathbb{P}^3,\Delta)$ from the $\mathcal{D}(X,D_X)$ being a 2-sphere quotient. They also establish a weak geometric Jordan constant for $\mathrm{Bir}(\mathbb{P}^2)$, namely $288\cdot 16$, by embedding abelian subgroups into tori after equivariant contractions and reducing to conic-bundle or del Pezzo cases. The results give explicit, dimension-dependent bounds and identify the degree-$6$ del Pezzo and related conic-bundle configurations as extremal cases, with potential computational checks for certain finite subgroups via Magma. Overall, the paper provides precise constants and a coherent framework connecting birational dynamics, Calabi–Yau pair geometry, and torus embeddings to quantify the Jordan property in volume-preserving Cremona groups.

Abstract

We show that the optimal Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, Δ)$ is 12. We provide a Jordan constant of $60$ for the three-dimensional volume-preserving Cremona group $\mathrm{Bir}(\mathbb P^3,Δ)$. We also provide a weak geometric Jordan constant of $288\cdot 16$ for $\mathrm{Bir}(\mathbb P^2)$.

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Theorems & Definitions (62)

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