Finite subgroups of maximal order of the Cremona group over the rationals
Ahmed Abouelsaad
TL;DR
The paper delivers a sharp, universal bound on the size of finite subgroups of the plane Cremona group over the rationals by reducing the problem to finite automorphism groups of rational surfaces via minimal $G$-surfaces. It combines conic-bundle and Del Pezzo surface analyses across degrees $1$–$6$, employing Galois actions (e.g., on hexagons) and known automorphism-classifications to produce per-degree caps, culminating in $|G|\le 432$ with explicit constructions achieving this bound (notably for degree $6$). The work furnishes a complete, degree-by-degree account of maximal finite subgroups over $\mathbb{Q}$, including concrete generators and rationality arguments, thereby significantly refining previous multiplicative bounds. This advances birational classification over number fields and provides precise structural insight into the finite Cremona subgroups realized over $\mathbb{Q}$.
Abstract
Let $\Cr_\Q(2)$ be the Cremona group of rank $2$ over rational numbers. we give a classification of large finite subgroups $G$ of $\Cr_\Q(2)$ and give a new sharp bound smaller (but not multiplicative) than $M(\Q)=120960 = 2^7\cdot3^3\cdot5\cdot7$; the one given in \cite{MR2567402}. In particular, we prove that any finite subgroup $G \subset\Cr_\Q(2)$ has order $\mid G\mid \le 432$ and Lemma \ref{lemm-17} provides a group of order $432$. We use the modern approach of minimal $G-$surfaces, given a (smooth) rational surface $S\subset\p^2$ defined over $\Q$, we study the finite subgroups $G \subset \Aut_{\Q}(S)$ of automorphisms of $S$. We give the best bound for the order of $G\subset\Aut(S)$ for surfaces with a conic bundle structure invariant by $G$. We also give the best bound for the order of $G\subset \Aut_\Q(S)$ for all rational Del Pezzo surfaces of some given degree. In addition, we give descriptions of the finite subgroups of automorphisms of conic bundles and Del Pezzo surfaces of maximal size.