On upper bounds of Manin type

Tanimoto, Sho

doi:10.2140/ant.2020.14.751

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On upper bounds of Manin type

Sho Tanimoto

Vol. 14 (2020), No. 3, 731–761

DOI: 10.2140/ant.2020.14.751

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 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 $S$ .

Keywords

heights, counting rational points, weak Manin conjecture

Mathematical Subject Classification 2010

Primary: 14G05

Secondary: 11G50, 14J28, 14J45

Milestones

Received: 10 January 2019

Revised: 18 July 2019

Accepted: 13 November 2019

Published: 1 June 2020

Authors

Sho Tanimoto
Department of Mathematics, Faculty of Science Kumamoto University Kurokami Kumamoto Japan

Vol. 14, No. 3, 2020