Heights of complete intersections in toric varieties

TL;DR

This paper extends the convex-analytic formulas for heights from hypersurfaces to complete intersections in toric varieties, providing a limit formula for the height of intersections and confirming a conjecture about average heights.

Contribution

It introduces a limit formula for the height of intersections of two hypersurfaces in toric varieties, advancing the understanding of height distributions in this setting.

Findings

Height of intersection cycles converges to an adelic sum of mixed integrals.

Partial confirmation of a conjecture on average heights of complete intersections.

Extension of convex-analytic height formulas to codimension two cases.

Abstract

The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the $2$-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions. This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties.