The multi-height distribution implies the Batyrev-Manin principle

TL;DR

This paper connects multi-height analysis on toric stacks and varieties to the asymptotic counting of rational points of bounded anticanonical height, advancing the understanding of the Batyrev-Manin principle.

Contribution

It introduces a method to derive asymptotic counts of rational points from multi-height analysis on toric structures, utilizing a generalized hyperbola method.

Findings

Established a link between multi-height analysis and rational point counting

Derived asymptotic formulas for rational points of bounded height

Extended the hyperbola method to toric stacks and varieties

Abstract

We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively bounded anticanonical height), using a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.