Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

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 asymptotic formulas for the number of rational points of bounded height can be given, with explicit leading constant and a power saving error term. In particular, for this particular family of algebraic varieties we go beyond the general expectation of the Manin-Peyre conjecture, and our results provide refinements of work done by Batyrev and Tschinkel for toric varieties. We include various examples to illustrate the scope of our results, including the case of Hirzebruch surfaces.