We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

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 $\mathbb{F}_2$ 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. Finally, we prove that the plane Cremona group over $\mathbb{F}_2$ is generated by involutions.

The files are sagemath files for JupyterLab: sage-Cremona_over_F2.zip For v4 on arxiv. Work a priori over any finite field.

DPtoolkit.py:	some basic functions that will be used throughout
k-structure.py:	The information of the minimal del Pezzo surfaces in terms of k-structure.
points_on_P2.ipynb:	Compute points in general position on projective plane over any finite field.
points_on_Q.ipynb:	as above but for minimal del Pezzo surface of degree 8.
points_on_X5.ipynb:	as above but for minimal del Pezzo surface of degree 5.
points_on_X6.ipynb:	as above but for minimal del Pezzo surface of degree 6.
Map_P2_66.py:	some functions to give explicit equation for the 6:6-link on P2, and the 3:3-link on X6.
Map_P2_55.py:	as above but for the 2:2-link on X6.
involution_P2_66.ipynb:	Can determine over any field whether the 6:6-link on P2 is an involution, and if yes, find the explicit equation.
involution_X6_22.ipynb:	as above but for 2:2-link on X6.
involution_X6_33.ipynb:	as above but for 3:3-link on X6.

We prove the Sarkisov program for projective surfaces over excellent base rings, including the case of non-perfect base fields of characteristic $0$. We classify the Sarkisov links between Mori fibre spaces and their relations for regular surfaces, generalising work of Iskovskikh. As an application, we discuss rationality problems for regular surfaces and the structure of the plane Cremona group.

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 $\bar k$ satisfying $[\bar k:k]>2$, 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 $n\geq3$.

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 $b\leq12$.