A determinant on birational maps of Severi-Brauer surfaces

TL;DR

This paper introduces a determinant on automorphisms of Severi-Brauer surfaces, extending it to birational maps, and uses it to analyze the structure and subgroups of the birational transformation group, providing new insights into geometrically rational surfaces.

Contribution

It defines a new determinant for birational maps of Severi-Brauer surfaces and determines the abelianization of their birational transformation group, a novel result for such surfaces.

Findings

Computed the abelianization of the birational transformation group.

Identified maximal subgroups of the birational transformation group.

Provided the first example of abelianization for a geometrically rational surface with non-trivial automorphisms.

Abstract

We define a determinant on the group of automorphisms of non-trivial Severi-Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in arXiv:2211.17123 we can determine the abelianization of the group of birational transformations of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the group of birational transformations of a geometrically rational surface where the automorphisms are non trivial. Using the abelianization we find maximal subgroups of the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field.