Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

TL;DR

This paper investigates the asymptotic behavior of rational points of bounded height on certain split toric varieties over number fields, providing explicit formulas and examples including Hirzebruch surfaces.

Contribution

It introduces a decomposition approach for these varieties and derives explicit asymptotic formulas for counting rational points, including new results for Hirzebruch surfaces.

Findings

Decomposition of varieties into subvarieties for counting

Explicit asymptotic formulas for rational points

Application to Hirzebruch surfaces

Abstract

We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where explicit asymptotic formulas for the number of rational points of bounded height can be given. Additionally, we present various examples, including the case of Hirzebruch surfaces.