On abelian extensions of finite abelian subgroups of Cremona groups
Luka Filin
TL;DR
The paper investigates finite abelian subgroups of the Cremona groups for dimensions $n\le 4$, reframing the problem in terms of abelian extensions $G=H\bullet K$ and product-type classes $\mathcal{A}_n$, $\mathcal{B}_n$ with associated operations $\times$ and $\bullet$. It proves that, up to dimension $4$, abelian extensions among finite abelian Cremona subgroups largely coincide with direct products, with a single exception $$(\mathbb{Z}/4)^5$$ in dimension $4$, and establishes a parallel result in the Mori-fibration setting via the identity $\mathcal{B}_1 \bullet \mathcal{B}_3' + \mathcal{B}_2 \bullet \mathcal{B}_2 = \mathcal{B}_1 \times \mathcal{B}_3' + \mathcal{B}_2 \times \mathcal{B}_2$. The approach relies on decomposing groups into $p$-parts, encoding extensions with partitions and Littlewood–Richardson coefficients, and performing detailed case analyses. The conclusions yield a positive answer to the motivating question for $n\le 3$ unconditionally and a conditional result for $n=4$ contingent on a conjectural description of finite abelian subgroups of $\mathrm{Bir}(X)$ for rationally connected threefolds.
Abstract
In this note, we study extension properties of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rational (or rationally connected) variety of dimension at most $4$. We are guided by the following question: is it true that if a finite group $G$ faithfully acts on a rationally connected variety of dimension $n$, then $G$ can faithfully act on a terminal Fano variety of dimension $n$? Using algebraic methods, we prove that up to dimension $4$, abelian extensions of finite abelian subgroups of the Cremona group coincide with direct products of such subgroups, with one exception. This result implies a positive answer to the above question up to dimension $4$ in the case of finite abelian groups, modulo a conjectural description of finite abelian subgroups of $\mathrm{Bir}(X)$ where $X$ is a rationally connected threefold.