Minimal geometrically rational surfaces over finite fields and their zeta functions

Let $S$ be a geometrically rational surface over a finite field. Applying minimal model problem to $S$ one get a minimal surface $X$ such that either $X$ is a del Pezzo surface, or $X$ admits a structure of conic bundle. Moreover, the Galois group of the algebraic closure of the fields acts on the Picard group $Pic(\overline{X})$ and invariant Picard number is equal to $1$ for del Pezzo surfaces and is equal to $2$ for conic bundles. For any such Galois group action we want to show that the corresponding minimal surface appears. Also the action of the Galois group defines zeta function of a surface, which give a method to compute the cardinality of the set of points, defined over the ground field and its extensions.
At first we discuss the work "Zeta functions of conic bundles and del Pezzo surfaces of degree $4$ over finite fields" of Sergey Rybakov and show, how to construct minimal conic bundles with given number of singular fibres and given Galois group action on these fibres. Then applying these results we show how to construct minimal del Pezzo surfaces of degree $4$. Some standart ideas show us how to construct minimal del Pezzo surfaces of degree $8$, $6$ and $5$. Then we discuss the speaker's results about minimal cubic surfaces, and after this we construct minimal del Pezzo surfaces of degree $2$. At last we discuss, which considered methods can be useful for del Pezzo surfaces of degree $1$, and what results we can obtain, applying these methods.