Automorphisms of cubic surfaces without points

TL;DR

This paper classifies certain birational automorphism groups of Severi--Brauer surfaces and investigates automorphism groups of cubic surfaces over fields with no points, revealing their structure and bounds.

Contribution

It provides a classification of birational automorphism groups of Severi--Brauer surfaces and characterizes automorphism groups of cubic surfaces without points, including bounds on Jordan constants.

Findings

Finite groups acting birationally on Severi--Brauer surfaces are classified.

Automorphism groups of cubic surfaces without points are abelian.

Sharp bounds for Jordan constants of birational automorphism groups are established.

Abstract

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.