Time: 14h00– 15h30, Friday October 18, 2024

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: A Ga-action on an affine variety X is an algebraic action of the additive group Ga(K) of the base field K.  A variety is called flexible, if for each smooth point, its tangent space is spanned by tangent vectors to orbits of Ga-actions. We will discuss the connection of flexibility and multiple transitivity of the automorphism group action. We will also survey families of varieties known to be flexible.

In the case of an affine space A^n, there is a natural notion of a subgroup of triangular automorphisms, which is an infinite-dimensional analogue of upper-triangular matrices U(n) in the matrix group GL(n). It is well known that any unipotent subgroup of GL(n) is conjugated to a subgroup of U(n). Unfortunately, this result does not hold for the subgroup of triangular automorphisms.

We will present a generalization of a triangular automorphism subgroup for an arbitrary affine variety X that describes all maximal unipotent subgroups of Aut(X). We will also discuss its properties, construction, and connection to additive actions. In particular, any unipotent subgroup of Aut(X) happens to be closed in the Zariski topology.