The distribution of Galois orbits of points of small height in toric varieties

TL;DR

This paper studies the distribution of small height points on toric varieties, establishing conditions for equidistribution and the Bogomolov property, with implications for Galois orbits and toric subvarieties.

Contribution

It introduces the concept of monocritical toric divisors and characterizes equidistribution and Bogomolov property conditions in this context.

Findings

Equidistribution occurs for small points if and only if the divisor is monocritical.

Limit measures of small points are translates of natural measures on the torus.

The Bogomolov property characterizes subvarieties with minimal height as translates of subtori.

Abstract

We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution occurs for every generic, small sequence with respect to a toric metrized divisor, for every place if and only if the divisor is monocritical. Furthermore, when this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of the ambient toric variety. We also study the $v$-adic modulus distribution of Galois orbits of small points. We characterize, in terms of the given toric semipositive metrized divisor, the cluster measures of $v$-adic valuations of Galois orbits of generic small sequences. The Bogomolov property now says that a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the toric variety with respect to a monocritical toric metrized divisor, must be a translate of a subtorus. We also give several examples, including a non-monocritical divisor for which the Bogomolov property does not hold.