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Density of rational points on some quadric bundle threefolds
Dante Bonolis1, Tim Browning2, Zhizhong Huang3
1Department of Mathematics, Duke University, Durham, NC 27708 USA.
Mathematische Annalen
|October 4, 2024
Summary
The Manin-Peyre conjecture regarding rational points on Fano threefolds is proven. This research addresses the number of points with bounded height on specific algebraic varieties.
Area of Science:
- Number Theory
- Algebraic Geometry
Background:
- The Manin-Peyre conjecture is a significant open problem in number theory.
- Fano threefolds are a class of algebraic varieties with specific geometric properties.
Purpose of the Study:
- To prove the Manin-Peyre conjecture for a specific family of Fano threefolds.
- To analyze the distribution of rational points of bounded height on these varieties.
Main Methods:
- Utilizing techniques from analytic number theory.
- Employing tools from algebraic geometry to study Fano threefolds of bidegree (1, 2).
Main Results:
- The Manin-Peyre conjecture is established for the considered family of Fano threefolds.
- A precise understanding of rational points of bounded height outside a thin subset is achieved.
Conclusions:
- The study provides a significant advancement in understanding rational points on Fano varieties.
- This work contributes to the broader field of Diophantine geometry.
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