Representations of finite subgroups of Cremona groups

TL;DR

This paper investigates the minimal dimensions for faithful representations of finite subgroups of Cremona groups, providing exact values for low ranks and characteristic-dependent bounds for higher ranks.

Contribution

It determines the exact minimal dimensions for ranks 1 and 2 over all fields and establishes bounds and infinitude results for higher ranks and various fields.

Findings

Exact minimal dimension for rank 1 and 2 over all fields.

Infinite minimal dimension for positive characteristic fields with rank > 1.

Finite minimal dimension for many characteristic 0 fields, including number fields and complex numbers.

Abstract

The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful representation of that dimension over the same field. We find the exact value for rank 1 and 2 over all fields. We prove that the value is infinite for all fields of positive characteristic and rank greater than one. For many fields of characteristic 0, which include number fields and the complex field, we show that the value is finite for all ranks. Finally, for all fields of characteristic 0, we prove that the dimension is bounded below by a function that is exponential in the rank.