Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Height Functions
- 3 A Brief Introduction to Higher Dimensional Algebraic Geometry
- 4 Manin’s Conjecture
- 5 Proofs of Main Results
- 6 Further Examples
- 7 Moduli of Rational Curves
- 8 Examples of Geometric Manin’s Conjecture II
- References
- Index
4 - Manin’s Conjecture
Published online by Cambridge University Press: 03 July 2026
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Height Functions
- 3 A Brief Introduction to Higher Dimensional Algebraic Geometry
- 4 Manin’s Conjecture
- 5 Proofs of Main Results
- 6 Further Examples
- 7 Moduli of Rational Curves
- 8 Examples of Geometric Manin’s Conjecture II
- References
- Index
Summary
Manin’s program, initiated in [FMT89] and [BM90], provides a way to understand the asymptotic formula for the counting function of rational points on a Fano variety, and this program leads to Manin’s conjecture, which was further revised in a series of papers [Pey95], [BT98b], [Pey03], [Pey17], [LST22], and [LS24]. In this chapter, we discuss the formulation of Manin’s conjecture and examples that have been verified in the literature. In Manin’s conjecture it is important to remove the contribution of the exceptional set so that the asymptotic formula of the counting function reflects the global geometry of a projective variety.
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- Birational Geometry and Manin's Conjecture , pp. 39 - 70Publisher: Cambridge University PressPrint publication year: 2026
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