Seminar Algebra and Geometry: Alexander Perepechko (Moscow)

Automorphism groups of affine algebraic surfaces preserving an A1-fibration

It is well known that A1-fibrations are of particular interest in the study of automorphism groups of affine surfaces. That is, description of automorphism groups is based on subgroups preserving A1-fibrations, except for surfaces without such fibrations. We provide a method to directly compute these subgroups in terms of a boundary divisor of an SNC-completion. We also use the concepts of arc spaces and formal neighbourhoods. This allows us to establish the following structure of a subgroup preserving a A1-fibration. Up to a finite index, it is a semidirect product of an abelian unipotent subgroup acting by translations on fibers and of a finite-dimensional subgroup that fixes a certain section.
In particular, we derive an example of a surface with infinite discrete automorphism group.

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