## Algebra & Number Theory

### On upper bounds of Manin type

Sho Tanimoto

#### 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 $≥2$ 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 $S$.

#### Article information

Source
Algebra Number Theory, Volume 14, Number 3 (2020), 731-761.

Dates
Revised: 18 July 2019
Accepted: 13 November 2019
First available in Project Euclid: 2 July 2020

https://projecteuclid.org/euclid.ant/1593655270

Digital Object Identifier
doi:10.2140/ant.2020.14.731

Mathematical Reviews number (MathSciNet)
MR4113779

#### Citation

Tanimoto, Sho. On upper bounds of Manin type. Algebra Number Theory 14 (2020), no. 3, 731--761. doi:10.2140/ant.2020.14.731. https://projecteuclid.org/euclid.ant/1593655270

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