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 Mori-Mukai's classification. We also find new upper bounds for polarized K3 surfaces $S$ of Picard rank $1$ using Bayer-Macrì's result on the nef cone of the Hilbert scheme of two points on $S$.

The paper has been reorganized and the exposition has been improved. 29 pages. To appear in Algebra Number Theory

References in corpus (6)

• Geometric Manin's Conjecture and rational curves
• On the geometry of thin exceptional sets in Manin's Conjecture
• Rational points of bounded height on general conic bundle surfaces
• On Fujita invariants of subvarieties of a uniruled variety
• Manin's Conjecture and the Fujita invariant of finite covers
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Cited by in corpus (2)

• Homotopy Spectra and Diophantine Equations
• Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals