Successive minima of toric height functions

TL;DR

This paper derives formulas for the successive minima of toric height functions on toric varieties, linking them to arithmetic invariants and providing explicit computations for various examples.

Contribution

It introduces explicit formulas for successive minima of toric height functions and explores their relation to other arithmetic invariants.

Findings

Formulas for the essential minimum of toric height functions.

Explicit computations for weighted projective spaces and toric bundles.

Connections between successive minima, height, and arithmetic volume.

Abstract

Given a toric metrized R-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also give formulae for all the successive minima. We apply these results to the study, in the toric setting, of the relation between the successive minima and other arithmetic invariants like the height and the arithmetic volume. We also apply our formulae to compute the successive minima for several families of examples, including weighted projective spaces, toric bundles and translates of subtori.