[论文解读] The multi-height distribution implies the Batyrev-Manin principle

Nicolas Bongiorno|arXiv (Cornell University)|Mar 14, 2026

Algebraic Geometry and Number Theory被引用 0

一句话总结

tldr: 显示有界反对称范数（或在 toric stacks 上的轨道反对称范数）有理点的渐近来自多高度分布，通过广义双曲线方法实现。

ABSTRACT

We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively bounded anticanonical height), using a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.

研究动机与目标

• Motivate and formalize the link between multi-height distributions and classical height-bounded rational point counts on toric varieties and toric stacks.
• Extend multi-height results to cases where the multi-height is bounded above but not below.
• Apply Pieropan–Schindler hyperbola method to deduce asymptotics for anticanonical or orbifold anticanonical heights.
• Identify leading constants with Tamagawa numbers and orbifold Tamagawa measures.

提出的方法

• Define multi-height maps and local heights on the universal torsor for toric varieties and stacks.
• Prove asymptotics for multi-height distributions using a generalized hyperbola method (Pieropan–Schindler).
• Decompose the effective cone into simplicial subcones to reduce to a bounded polyhedral counting problem.
• Use Davenport’s geometry of numbers to count lattice points and obtain error terms.
• Relate leading constants to Tamagawa numbers and orbifold Tamagawa measures as in prior work.

实验结果

研究问题

• RQ1 How does the multi-height distribution determine the asymptotic count of rational points with bounded anticanonical (or orbifold anticanonical) height?
• RQ2 Can the hyperbola method of Pieropan and Schindler be adapted to toric stacks and to heights bounded above (not just below) the multi-height?
• RQ3 What is the precise relation between the leading constants in these asymptotics and Tamagawa numbers (including orbifold/toroidal settings)?

主要发现

• The multi-height asymptotic is of the form nu(D1) * tau(X) * B^{<omega_X^{-1}, u>} with a logarithmic factor in the toric case, recovered via the hyperbola method.
• For toric stacks the leading constant matches the orbifold Tamagawa number defined in prior work.
• A hyperbola-method framework yields explicit error bounds of the form O(min(Bi)^{-delta}) in the bounded region analysis.
• The leading constant in the asymptotics coincides with the Tamagawa measure on X, connecting Manin-type predictions to Tamagawa numbers.
• An explicit computation shows c_P = 1/(rho-1)! in the simplicial cone setup for the hyperbola method.