A birational journey: From plane curve singularities to the Cremona group over perfect fields

The plane Cremona group $Cr_2(k)$ is the group of birational transformations of the plane that are defined over a field $k$. This thesis consists of an introduction, preliminaries, and three main parts.

We begin by using birational transformations as a tool to study plane curve singularities. We are interested in the following question: For a given $d$, what is the maximal $k$ such that there exists a curve of degree $d$ in the complex affine plane $\mathbb{A}^2(\mathbb{C})$ that has an $A_k$-singularity? 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 $b\leq 12$. We embedd the curve in a well chosen Hirzebruch surface and resolve the singularities via elementary links.

Then we start to study the structure of the plane Cremona group. We construct a group homomorphism from the plane Cremona group to a free product of direct sums of $\mathbb{Z}/2\mathbb{Z}$. For perfect fields $k$ satisfying $[\bar k:k]>2$, we prove that the group homomorphism is surjective and obtain new normal subgroups of the plane Cremona group. The focus lies on birational maps whose base points form large Galois orbits. We follow 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 $n\geq3$. In this way we get an elementary proof of non-simplicity of the plane Cremona group over many fields, without using any modern machinery. This extends (a part of) a result of Lamy and Zimmermann to a larger class of fields.

Finally, we describe generators of the plane Cremona group over $\mathbb{F}_2$ explicitely. It 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 $\mathbb{F}_2$. Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. In the end, we show that the plane Cremona group over $\mathbb{F}_2$ is generated by involutions.