The multi-height distribution implies the Batyrev-Manin principle

1. Problem Statement

The paper addresses a fundamental problem in arithmetic geometry: the asymptotic distribution of rational points on algebraic varieties. Specifically, it focuses on toric varieties and toric stacks (Deligne-Mumford stacks).

• The Batyrev-Manin Conjecture: This conjecture predicts the asymptotic growth rate of the number of rational points of bounded height on a Fano variety. The height is typically associated with the anticanonical line bundle $\omega^{-1}_X$.
• The Gap: While the conjecture has been proven for split toric varieties (by Batyrev-Tschinkel and Salberger) and recently extended to toric stacks (by Darda and Yasuda), the proofs often rely on specific harmonic analysis or universal torsor lifting techniques.
• The Specific Goal: The author aims to demonstrate that the Batyrev-Manin principle (the asymptotic formula for points of bounded anticanonical height) can be deduced directly from the multi-height distribution of rational points. The multi-height distribution describes the behavior of points when their heights with respect to all effective divisors are considered simultaneously, rather than just the anticanonical height.

2. Methodology

The paper employs a sophisticated combination of geometric analysis and analytic number theory, centered on a generalized hyperbola method.

A. Multi-Height Framework

The author utilizes the framework developed in previous works (by Peyre, Darda, Yasuda, and the author) where rational points $P \in X(\mathbb{Q})$ are mapped to a multi-height vector:
$$h: X(\mathbb{Q}) \to \text{Pic}(X)^\vee_{\mathbb{R}}$$
The distribution of these points in the dual of the Picard group is known to follow a specific asymptotic law involving the Tamagawa number $\tau(X)$ and a volume measure $\nu$.

B. The Hyperbola Method (Pieropan-Schindler)

The core technical tool is a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.

• Standard Hyperbola Method: Used to estimate sums of arithmetic functions over regions defined by multiplicative constraints (e.g., $xy \leq B$).
• Generalization: The author applies the Pieropan-Schindler theorem (Theorem 1.3 in the paper) to arithmetic functions $f: \mathbb{N}^\rho \to \mathbb{R}^+$ that count rational points with multi-heights in specific polyhedral regions.
• The Strategy:
  1. Restrict the multi-heights to a simplicial subcone $\Lambda$ of the dual effective cone $C_{\text{eff}}(X)^\vee$.
  2. Define arithmetic functions $f^\uparrow$ and $f^\downarrow$ counting points where the floor/ceiling of the multi-height coordinates fall within a box.
  3. Prove these functions satisfy the asymptotic hypotheses required by the Pieropan-Schindler theorem.
  4. Apply the theorem to sum these functions over the region defined by the anticanonical height constraint ($\langle \omega^{-1}_X, h(P) \rangle \leq \log B$).

C. Decomposition and Inclusion-Exclusion

Since the dual effective cone $C_{\text{eff}}(X)^\vee$ is generally not simplicial, the author decomposes it into a finite union of simplicial subcones $\{\Lambda_j\}$.

• Hypothesis 4.1: The author assumes (and verifies for split toric varieties and weighted projective stacks) that for points on the dense torus $T$, the multi-height lies within $C_{\text{eff}}(X)^\vee$.
• Negligibility of Intersections: A crucial technical step involves proving that the number of points lying in the intersection of two distinct simplicial cones ($\Lambda_i \cap \Lambda_j$) is of a lower order of magnitude ($o(B \log(B)^{\rho-1})$) compared to the main term. This allows the use of the inclusion-exclusion principle to sum the contributions of the simplicial cones without overcounting.

3. Key Contributions

1. Derivation of Batyrev-Manin from Multi-Height: The paper provides a rigorous deduction showing that the classical Batyrev-Manin asymptotic formula is a direct consequence of the known multi-height distribution results. This unifies two previously distinct approaches.
2. Identification of Constants: The work explicitly identifies the leading constant in the Batyrev-Manin formula for toric stacks with the orbifold Tamagawa number defined in the author's previous work. This resolves a question regarding the equivalence of constants derived via different methods (harmonic analysis vs. multi-height analysis).
3. Extension to Toric Stacks: The methodology is successfully adapted to the setting of Deligne-Mumford stacks, handling the complexities of orbifold Picard groups, twisted sectors, and stacky heights.
4. Refinement of the Hyperbola Method: The paper demonstrates the applicability of the Pieropan-Schindler hyperbola method to the specific geometric constraints of toric geometry, providing explicit error terms and constants.

4. Main Results

The central result is Theorem 1.4 (and its proof in Section 4.3), which establishes the asymptotic behavior for the number of rational points on a toric variety or stack $X$ with bounded anticanonical height:

$$\sharp \{ P \in T(\mathbb{Q}) \mid h(P)(\omega^{-1}_X) \leq \log B \} \sim_{B \to \infty} \alpha(X) \tau(X) B^{\langle \omega^{-1}_X, u \rangle} (\log B)^{\rho(X)-1}$$

Where:

• $\tau(X)$ is the Tamagawa number of $X$.
• $\rho(X) = \text{rank}_{\mathbb{Z}}(\text{Pic}(X))$ is the Picard rank.
• $\alpha(X)$ is a geometric constant defined by the volume of the dual effective cone:
  $$\alpha(X) = \frac{1}{(\rho-1)!} \int_{C_{\text{eff}}(X)^\vee} e^{-\langle \omega^{-1}_X, y \rangle} dy$$
• The term $B^{\langle \omega^{-1}_X, u \rangle}$ represents the growth rate determined by the anticanonical bundle.

Corollary 1.5 further clarifies that for toric stacks, the Tamagawa number $\tau(X)$ coincides with the constant derived by Darda and Yasuda, confirming the consistency of the orbifold Tamagawa measure definition.

5. Significance

• Unification of Programs: The paper bridges the gap between the "multi-height" program (initiated by Peyre) and the classical "Manin" program. It shows that studying the distribution of points in the full Picard dual space is sufficient to recover the specific distribution along the anticanonical ray.
• Robustness for Stacks: By successfully applying these methods to toric stacks, the paper validates the use of stacky geometry in arithmetic statistics. It confirms that the Batyrev-Manin principle holds for stacks when using the correct "orbifold" anticanonical height.
• Methodological Advancement: The successful application of the Pieropan-Schindler hyperbola method to this context provides a powerful new tool for future research in counting rational points on more complex varieties where harmonic analysis might be difficult to apply.
• Constant Verification: It settles the identification of the leading constant, ensuring that the "Tamagawa number" defined via adelic integration matches the geometric volume constants appearing in asymptotic formulas.

In summary, Bongiorno's paper provides a rigorous, analytic proof that the global distribution of rational points on toric objects (varieties and stacks) dictates their local behavior with respect to the anticanonical height, thereby confirming the Batyrev-Manin principle through the lens of multi-height analysis.