Heights of hypersurfaces in toric varieties

TL;DR

This paper extends the combinatorial understanding of heights of hypersurfaces in toric varieties from degrees to an arithmetic setting, expressing heights via mixed integrals of roof functions and Ronkin functions.

Contribution

It introduces a new formula for the global height of codimension 1 cycles in toric varieties using mixed integrals, generalizing the degree formula to an arithmetic context.

Findings

Height expressed as mixed integrals of roof functions and Ronkin functions

Generalization of degree formula to arithmetic setting

Provides a combinatorial approach to heights in toric geometry

Abstract

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial of the cycle.