Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

Sebastián Herrero, Tobías Martínez, Pedro Montero

Published 2024-07-28Version 1

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.