Group Theoretical Characterizations of Rationality

TL;DR

This paper demonstrates how the structure of the birational transformation group of a variety reveals its rationality and ruledness, and characterizes Borel subgroups, resolving longstanding conjectures.

Contribution

It provides a group-theoretic criterion for rationality and ruledness of varieties and characterizes Borel subgroups, including examples that settle previous conjectures.

Findings

Group structure of Bir(X) determines rationality and ruledness.

Borel subgroups have derived length at most twice the dimension of X.

Examples of non-standard Borel subgroups in Bir(P^n) and Aut(A^n).

Abstract

Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni.