Abstract. Given an irreducible affine algebraic variety X of dimension n ≥ 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 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 reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We provide also various modifications and applications.

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International audienc

Abstract. We say that a group G acts infinitely transitively on a set X if for every m ∈ N the induced diagonal action of G is transitive on the cartesian mth power X m \ ∆ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.