Algebra & Number Theory

On upper bounds of Manin type

Sho Tanimoto

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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

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

Received: 10 January 2019
Revised: 18 July 2019
Accepted: 13 November 2019
First available in Project Euclid: 2 July 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 14G05: Rational points
Secondary: 11G50: Heights [See also 14G40, 37P30] 14J28: $K3$ surfaces and Enriques surfaces 14J45: Fano varieties

heights counting rational points weak Manin conjecture


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

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