Picard group of K3 surfaces over finite fields

Abstract

The aim of this work is to provide a method to find explicitly generators for the Picard group of a K3 surface of degree four defined over a finite field. This work has been motivated primarily by the difficulty related to this problem, and by the lack of examples in the literature. Another question related to this problem is if given a K3 surface of degree 4 defined over a finite field, is it possible to determine the minimal degree of a non complete intersection curve lying over it. We could not answer completely to this question, but we were still able to find algorithms to determine whether or not it contains a conic defined over an extension of the base field. We were also able to determine an algorithm to find twisted cubics defined over the base field in case it is F2