We describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field K, proving in particular that if it contains a point of degree 6, then it is not generated by elements of finite order as it admits a surjective group homomorphism to Z. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface. We prove that any group of cardinality at most the one of C is a quotient of any Cremona group of rank at least 4. As a consequence, this gives a negative answer to the question of Dolgachev of whether the Cremona groups of all ranks are generated by involutions. We also prove that the 3-torsion of the Cremona group of rank at least 4 is not countable.

The plane Cremona group over the finite field 𝔽₂ is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve the pencil of conics through four points that form either one Galois orbit of size 4, or two Galois orbits of size 2. For each family, we give a generating set that is parametrized by the rational functions over 𝔽₂. Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. Finally, we prove that the plane Cremona group over 𝔽₂ is generated by involutions.

The files are sagemath files for Jupyter notebook.

Lemma 4.8 (orbits on projective plane P2)

Lemma 4.12 (orbits from quadric Q)

Lemma 4.14 and 4.15 (FrobTilde from D5)

Lemma 4.18 and Remark 4.19 (orbits from D5)

Lemma 4.24 (FrobTilde from D6)

Lemma 4.27 (orbits from D6)

Corollary (involutions from orbits of size 6 in P2)

Corollary (involutions from orbits of size 2 and 3 on D6)

We prove that over any perfect field the plane Cremona group is generated by involutions.

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.

For perfect fields k with algebraic closure L satisfying [L:k] > 2, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank n over (subfields of) the complex numbers is not simple for n ≥ 3.

We provide a tool how one can view a polynomial on the affine plane of bidegree (a,b) - by which we mean that its Newton polygon lies in the triangle spanned by (a,0), (0,b) and the origin - as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal A_k-singularities of curves of bidegree (3,b) and find the answer for b ≤ 12.