Continuity of heights in families and complete intersections in toric varieties

TL;DR

This paper investigates how heights of algebraic cycles vary in families over number fields and globally valued fields, establishing continuity and confirming a conjecture on limit heights in toric varieties.

Contribution

It introduces the GVF analytification for schemes over globally valued fields and proves the continuity of fiber heights, confirming Gualdi's conjecture on complete intersections in toric varieties.

Findings

Heights of fibers vary continuously in flat families.

Established GVF analytification as a tool for studying heights.

Confirmed Gualdi's conjecture on limit heights 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.