Jordan constant for Cremona group of rank 2 over a finite field

TL;DR

This paper determines the exact Jordan constant for the Cremona group of rank 2 over finite fields and constructs a unique cubic surface with a maximal automorphism group over 2.

Contribution

It provides the precise Jordan constant for the Cremona group over all finite fields and constructs a unique cubic surface with maximal automorphism group over 2.

Findings

Exact Jordan constant for Cremona group over finite fields

Construction of a cubic surface over 2 with 6 automorphism group

Proof of uniqueness of the cubic surface up to isomorphism

Abstract

In this paper we find the exact value of the Jordan constant for Cremona group of rank $2$ over all finite fields. During the proof we construct a cubic surface over $\mathbb{F}_2$ with a regular action of the group $\mathrm{S}_6$ which is the maximal automorphism group of cubic surfaces over $\mathbb{F}_2.$ Moreover, we prove the uniqueness up to isomorphism of such a cubic surface.