Unirationality and existence of infinitely transitive models

TL;DR

This paper investigates unirational algebraic varieties and demonstrates that, after adding finitely many variables, their rational function fields can admit models with infinitely transitive automorphism group actions, a property potentially unique to unirational varieties.

Contribution

It establishes that certain unirational varieties have models with infinitely transitive automorphism groups after finite variable extension, highlighting a special property of this class.

Findings

Existence of infinitely transitive models for some unirational varieties.

Finite variable extension enables infinite transitivity.

Potential uniqueness of this property among rationally connected varieties.

Abstract

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.