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Finite subgroups of Cremona group of rank 3 over the field of rational numbers

Alexandr Zaitsev

TL;DR

The paper provides an explicit bound on the orders of finite subgroups of the Cremona group of rank 3 over $\mathbb{Q}$ by reducing the problem to automorphism groups of rational Fano threefolds and Mori fiber spaces, and then applying a combination of Minkowski-type bounds for linear groups and Reid’s Riemann–Roch framework. It develops explicit additive bounds for $\mathrm{GL}_n$ and $\mathrm{PGL}_n$ over number fields via Serre and Schur invariants, and uses these to bound finite groups acting on conic bundles and del Pezzo fibrations, as well as non-Gorenstein and Gorenstein Fano threefolds. The main result is an explicit, though not sharp, bound $|G|\le 24\,103\,053\,950\,976\,000 < 10^{17}$ for $\mathrm{Cr}_3(\mathbb{Q})$, obtained by a detailed case analysis across Fano families, smoothing arguments, and equivariant minimal model programs. The methods provide a robust framework for deriving effective bounds in higher dimensions and related birational groups.

Abstract

We give an explicit bound on orders of finite subgroups of Cremona group of rank three over $\mathbb{Q}$.

Finite subgroups of Cremona group of rank 3 over the field of rational numbers

TL;DR

The paper provides an explicit bound on the orders of finite subgroups of the Cremona group of rank 3 over by reducing the problem to automorphism groups of rational Fano threefolds and Mori fiber spaces, and then applying a combination of Minkowski-type bounds for linear groups and Reid’s Riemann–Roch framework. It develops explicit additive bounds for and over number fields via Serre and Schur invariants, and uses these to bound finite groups acting on conic bundles and del Pezzo fibrations, as well as non-Gorenstein and Gorenstein Fano threefolds. The main result is an explicit, though not sharp, bound for , obtained by a detailed case analysis across Fano families, smoothing arguments, and equivariant minimal model programs. The methods provide a robust framework for deriving effective bounds in higher dimensions and related birational groups.

Abstract

We give an explicit bound on orders of finite subgroups of Cremona group of rank three over .
Paper Structure (6 sections, 92 theorems, 237 equations, 3 tables)

This paper contains 6 sections, 92 theorems, 237 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Bounds for linear groups
  3. $G$-mori fiber spaces with base of positive dimension
  4. Non-Gorenstein Fano threefolds
  5. Gorenstein Fano threefolds
  6. Galois theory

Key Result

Theorem 1.1

Let $n \geqslant 1$ be an integer and $p$ be a prime number. Define Then: In particular, if we denote by $\mathcal{P}$ the set of all prime numbers, then for any finite subgroup $G$ of $\mathop{\rm GL}\nolimits_n(\mathbb Q)$ we have

Theorems & Definitions (175)

  • Theorem 1.1: Min
  • Theorem 1.2: see PrShr higher dim
  • Theorem 1.3: Ahmed A
  • Theorem 1.4
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: cf. Serre red
  • Remark 2.6
  • ...and 165 more