

Iskovskikh Seminar
November 19, 2015 18:00, Moscow, Steklov Mathematical Institute, room 540






Minimal geometrically rational surfaces over finite fields and their zeta
functions
A. S. Trepalin^{} ^{} Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

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Abstract:
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.

