Grauert and Manin showed that a non-isotrivial 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.
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Christophe Mourougane, Families of hypersurfaces of large degree. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 911–936DOI 10.4171/JEMS/322