Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2, we let SAut(X) denote the special automorphism group of X, that is, 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 X_reg, then it is infinitely transitive on X_reg. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x∈X_regthe tangent space T_xX is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.