Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields

Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of split toric varieties with Picard rank 2 over global function fields, with respect to 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 precise analytic properties of the corresponding height zeta functions can be given, allowing for a finer inspection of the asymptotic number of rational points of bounded height on each subvariety.