Abstracts for Seminar Algebraic Geometry (SAG)

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. 
 

As an answer to Mordell problem over function fields, Grauert and Manin showed that a non-isotrivial algebraic family of compact complex hyperbolic curves has finitely many
sections. We consider a generic moving enough family of high enough degree hypersurfaces
in a complex projective space. We show the existence of a strict closed subset of its total
space that contains the image of all its sections.