Del Pezzo surfaces over finite fields

Properties of algebraic varieties over finite fields are interesting for algebraic geometry and also for some branches of applied mathematics, for example coding theory. One of the main properties of an algebraic variety over finite field is the number of points on it.
One of the most important cases of algebraic surfaces are del Pezzo surfaces. The main invariant of a del Pezzo surface is degree that is an integer from 1 to 9.
For algebraically nonclosed fields the Galois group of the algebraic closure of the field acts on the Picard lattice of a del Pezzo surface, and this action defines many geometric properties of the surface. In particular, for finite fields this action allows to restore zeta function of a del Pezzo surface. Therefore one can find number of points over the main fields and all finite extensions of this field. We say that conjugacy class of the image of the Galois group in the automorphism group of the Picard lattice, preserving the intersection form, is the type of a del Pezzo surfaces.
A natural question arises: for which finite fields there exists a del Pezzo surface of certain type. In this talk we observe the main methods, allowing to obtain a complete answer to this question for del Pezzo surfaces of degree 2 or greater.

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