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.
In this talk I shall discuss Fano foliations on complex projective manifolds (these are foliations whose anti-canonical class is ample). I will concentrate on the special class of Del Pezzo foliations with mild singularities. This a joint work with Carolina Araujo.
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.
The moduli spaces of initial conditions for Painlevé VI can be described fairly explicitly. We describe some aspects of the relationship between the de Rham and Hitchin moduli spaces. Then we point out that the behavior at infinity of the various correspondences can be understood, to a small extent, in light of the geometry.
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