Infinite transitivity, finite generation, and Demazure roots

TL;DR

This paper investigates the automorphism groups of affine algebraic varieties, demonstrating that under certain conditions, a finite set of unipotent subgroups can achieve the same high transitivity as the entire automorphism group.

Contribution

It shows that for smooth toric affine varieties in codimension 2, a finite number of unipotent subgroups can generate a subgroup with full transitivity, extending previous results on flexibility.

Findings

Finite generation of automorphism subgroups with transitivity

Three unipotent subgroups suffice for affine space of dimension ≥ 2

Flexibility extends to certain smooth toric varieties

Abstract

An affine algebraic variety X of dimension at least 2 is called flexible if the subgroup SAut(X) in Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg(X) for any m $\ge$ 1. In the previous paper we proved that any nondegenerate toric affine variety X is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety X is smooth in codimension 2. For X=$\mathbb{A}^n$ with n$\ge$2, three such subgroups suffice.