On upper bounds of Manin type

Abstract

We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin type for $28$ deformation types of smooth Fano $3$-folds of Picard rank $\ge 2$ following the Mori–Mukai classification. We also find new upper bounds for polarized K3 surfaces $S$ of Picard rank $1$ using Bayer and Macrì’s result on the nef cone of the Hilbert scheme of two points on $S$.