Continuity of heights in families and complete intersections in toric varieties

Abstract: We study the variation of heights of cycles in flat families over number fields or, more generally, globally valued fields. To a finite type scheme over a GVF we associate a locally compact Hausdorff space which we refer to as its GVF analytification. For a flat projective family, we prove that the height of fibres is a continuous function on the GVF analytification of the base. As an application, we prove Roberto Gualdi's conjecture on limit heights of complete intersections in toric varieties.