A remark on the rank of finitep-groups of birational automorphisms

Jinsong Xu

doi:10.5802/crmath.93

Abstract

We improve a result of Prokhorov and Shramov on the rank of finite $p$-subgroups of the birational automorphism group of a rationally connected variety. Known examples show that they are sharp in many cases.

Highlights

The goal of this note is to prove the following theorem on finite groups of birational transformations
We refer to [12, Proposition 1.7] and [11, 12] for surfaces and threefolds; for instance, in [12] it is shown that a finite p-subgroup of Bir(P3k) is abelian and of rank ≤ 3 as soon as p ≥ 17; here we improve the inequality to p ≥ 5
Let X be a projective variety over k, and G be a finite p-group acting by automorphisms on X

Summary

Introduction

The goal of this note is to prove the following theorem on finite groups of birational transformations. Let X be a rationally connected variety of dimension n over an algebraically closed field k of characteristic 0. Let p be a prime number and let G be a finite psubgroup of the group of birational transformations Bir(X ).

Results

Conclusion