# On upper bounds of Manin type

## TL;DR

This paper introduces a birational invariant to establish upper bounds on rational points for algebraic varieties, providing new bounds for certain Fano threefolds and polarized K3 surfaces, advancing understanding in arithmetic geometry.

## Contribution

It presents a novel birational invariant and applies it to derive new upper bounds for rational points on specific classes of algebraic varieties.

## Key findings

- New upper bounds for 28 deformation types of smooth Fano 3-folds.
- Improved bounds for polarized K3 surfaces of Picard rank 1.
- Introduction of a birational invariant for counting rational points.

## 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 $\geq 2$ following Mori-Mukai's classification. We also find new upper bounds for polarized K3 surfaces $S$ of Picard rank $1$ using Bayer-Macr\`i's result on the nef cone of the Hilbert scheme of two points on $S$.