Specialization of Néron-Severi groups of K3 surfaces

We discuss several questions related to the behaviour of the Néron-Severi groups of
K3 surfaces under specialization over function fields and number fields. Such topics have some implications for the existence of rational curves, along the lines of recent papers by Bogomolov-Hassett-Tschinkel and Li-Liedtke as well as for the computation of Picard numbers of K3 surfaces over number fields.