Flexible varieties and automorphism groups

TL;DR

This paper explores the relationship between the transitivity of the special automorphism group on affine varieties and the concept of flexibility, establishing that transitivity implies infinite transitivity and linking it to the variety's flexibility.

Contribution

It proves that transitivity of the special automorphism group on the smooth locus implies infinite transitivity and characterizes flexibility in affine varieties.

Findings

Transitivity of SAut(X) on the smooth locus implies infinite transitivity.

Transitivity is equivalent to the flexibility of the variety.

Provides variations and applications of the main results.

Abstract

Given an affine algebraic variety X of dimension at least 2, we let SAut (X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut (X) generated by all one-parameter unipotent subgroups. We show that if SAut (X) is transitive on the smooth locus of X then it is infinitely transitive on this locus. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x of X the tangent space at x is spanned by the velocity vectors of one-parameter unipotent subgroups of Aut (X). We provide also different variations and applications.