Flexible varieties and automorphism groups

Abstract

Given an irreducible affine algebraic variety $X$ of dimension $n\ge 2$, we let $SAut\left(X\right)$ denote the special automorphism group of $X$, that is, the subgroup of the full automorphism group $Aut\left(X\right)$ generated by all one-parameter unipotent subgroups. We show that if $SAut\left(X\right)$ is transitive on the smooth locus $X_{\mathrm{reg}}$, then it is infinitely transitive on $X_{\mathrm{reg}}$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X_{\mathrm{reg}}$ the tangent space $T_{x}X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $Aut\left(X\right)$. We also provide various modifications and applications.