The essential and Cremona dimensions of a group

TL;DR

This survey explores the Cremona dimension of groups, conjecturing it is bounded above by the essential dimension, supported by numerous examples.

Contribution

It provides evidence and examples for the conjecture relating Cremona and essential dimensions of finite groups.

Findings

Cremona dimension of finite groups is conjecturally ≤ their essential dimension.

Numerous examples support the conjecture.

The paper discusses the relationship between group actions and birational transformations.

Abstract

The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.