Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers

We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi--Brauer surfaces over the field of rational numbers are~$\mathbb{ Z}/3\mathbb{ Z}$ and $(\mathbb{ Z}/3\mathbb{ Z})^2.$ Moreover, we show that $(\mathbb{ Z}/3\mathbb{ Z})^2$ is contained in $\mathrm{ Bir}(V)$ for any Severi--Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{ Z}/3\mathbb{ Z})^3$ is contained in $\mathrm{ Bir}(V)$ for any Severi--Brauer surface~$V$ over a field of characteristic different from $2$ and $3$ which contains a non-trivial cube root of unity.

Categories: math.AG