Distribution of Rational Points on Toric Varieties: A Multi-Height Approach

Manin's conjecture for Fano varieties predicts an asymptotic formula for the number of rational points of bounded height with respect to the anti-canonical height function on a small enough Zariski open set with a dense set of rational points. In the case of toric varieties, Manin's conjecture was verified by Victor Batyrev and Yuri Tschinkel. In this thesis, we will explain a multi-height variant of the Batyrev-Tschinkel theorem proposed by Emmanuel Peyre in his paper "Beyond heights: slopes and distribution of rational points", where one considers working in "height boxes" instead of using a single height function as a way to get rid of accumulating subvarieties. We prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisors for a toric variety over a number field. The main result of my thesis is the first example of a large family of varieties along the lines of Peyre's idea.