Multi-height analysis of rational points of toric varieties

1. Problem Statement and Context

The Core Problem:
The paper addresses the asymptotic distribution of rational points on algebraic varieties. Specifically, it investigates the multi-height distribution of rational points on smooth, projective, and split toric varieties over $\mathbb{Q}$.

• Traditional Approach (Manin's Conjecture): Historically, the study of rational points $V(\mathbb{Q})$ has focused on a single height function $H_L$ associated with a specific line bundle $L$ (usually the anticanonical bundle $\omega^{-1}_V$). The goal is to count points with $H_L(P) \leq B$ as $B \to \infty$.
• The Multi-Height Framework: Following a proposal by Emmanuel Peyre [19], the paper shifts focus to the system of all possible heights. Instead of bounding a single height, the study considers the distribution of points where the vector of logarithmic heights lies within a specific domain $D_B$ in the dual of the Picard group, $\text{Pic}(V)^\vee_{\mathbb{R}}$.
• The Gap: While Peyre conjectured the asymptotic behavior for "quasi-Fano" varieties in this multi-height setting, previous proofs for toric varieties (e.g., Batyrev-Tschinkel) relied on harmonic analysis on adelic groups. A previous attempt using height zeta functions [11] was noted by the author to have a missing condition regarding the direction of the height vector, leading to incorrect asymptotic exponents in specific cases (e.g., $\mathbb{P}^1$).

Objective:
To prove Peyre's multi-height conjecture for smooth, proper, split toric varieties over $\mathbb{Q}$ using a method distinct from zeta functions, specifically utilizing universal torsors and descent.

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2. Methodology

The author employs a geometric and arithmetic approach centered on universal torsors and the geometry of numbers. The methodology proceeds in four main stages:

A. Lifting to Universal Torsors

The rational points of the toric variety $X$ are lifted to the universal torsor $T$.

• Construction: For a split toric variety $X$ defined by a fan $\Sigma$, the universal torsor $T$ is realized as an open subset of the affine space $\mathbb{A}^{\Sigma(1)}$ (where $\Sigma(1)$ are the rays of the fan). Specifically, $T = \mathbb{A}^{\Sigma(1)} \setminus Z(\Sigma)$, where $Z(\Sigma)$ is the "exceptional set" defined by the vanishing of monomials corresponding to non-cones in the fan.
• Descent: The map $q: T \to X$ is a torsor under the Néron-Severi torus $T_{NS}$. The author establishes that counting rational points on $X$ is equivalent to counting integral points on $T$ (specifically $T(\mathbb{Z})$) modulo the action of $T_{NS}$, adjusted by a factor of $2^t$ (where $t$ is the rank of the Picard group).

B. Lifting the Height Function

A crucial step is defining how the multi-height on $X$ lifts to the torsor $T$.

• Local Heights: The author defines local height functions on $T$ associated with the universal torsor structure.
• Global Relation: It is proven that for a rational point $P \in X(\mathbb{Q})$ and a lift $y \in T(\mathbb{Z})$, the multi-height $h(P)$ corresponds exactly to the local logarithmic height at the infinite place on the torsor, $h_{T, \infty}(y)$.
• Functoriality: The height system is shown to be functorial with respect to toric morphisms, allowing the explicit calculation of heights on $T$ using the coordinates of the affine space.

C. Geometry of Numbers and Lattice Point Counting

The problem is reduced to counting lattice points in a bounded domain.

• Domain Definition: The condition $h(P) \in D_B$ (where $D_B = D_1 + \log(B)u$) translates to the condition $y \in T(\mathbb{R})$ such that $h_{T, \infty}(y) \in D_B$. This defines a semi-algebraic domain in $\mathbb{R}^{\Sigma(1)}$.
• Davenport's Estimate: The author applies a theorem by Davenport [9] regarding the error term in counting lattice points in semi-algebraic sets. The number of integral points is approximated by the volume of the domain, with an error term dependent on the projections of the domain onto lower-dimensional subspaces.
• Volume Calculation: The volume of the domain on the torsor is computed and linked to the Tamagawa number $\tau(X)$ and the measure $\nu$ on the dual Picard group.

D. Möbius Inversion

To isolate the points on the torsor $T(\mathbb{Z})$ from the full lattice $\mathbb{Z}^{\Sigma(1)}$, the author uses Möbius inversion.

• The indicator function for $T(\mathbb{Z})$ is expressed as a sum over divisors $d \in (\mathbb{N}^*)^{\Sigma(1)}$ involving a Möbius function $\mu(d)$.
• The sum is split into two parts: small divisors (where the main term dominates) and large divisors (where the error term is controlled).
• Convergence: The convergence of the Euler product associated with the Möbius sum is established for $\text{Re}(s) > 1/l$, where $l$ is the smallest number of rays not contained in a single cone. This determines the quality of the error term.

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3. Key Contributions and Results

Main Theorem (Theorem 2.15 & 4.16)

The paper proves that for a smooth, proper, split toric variety $X$ over $\mathbb{Q}$, the number of rational points with multi-height in the domain $D_B = D_1 + \log(B)u$ (where $u$ is in the interior of the dual effective cone) satisfies:

$$\sharp \{ P \in X(\mathbb{Q}) \mid h(P) \in D_B \} \sim_{B \to \infty} \nu(D_1) \cdot \tau(X) \cdot B^{\langle \omega^{-1}_X, u \rangle}$$

Key Aspects of the Result:

1. Asymptotic Formula: The leading term matches Peyre's conjecture, involving the volume of the domain $\nu(D_1)$, the Tamagawa number $\tau(X)$, and the growth rate determined by the anticanonical bundle pairing with the direction $u$.
2. Error Term: The paper provides a precise error term:
   $$O\left( B^{\langle \omega^{-1}_X, u \rangle - (1 - \frac{1}{l} - \epsilon) \min_{\rho} \langle [D_\rho], u \rangle} \right)$$
   where $l$ is the combinatorial invariant of the fan (smallest number of rays not in a cone) and $\epsilon > 0$.
3. No Accumulating Subsets: A significant consequence is that there are no accumulating subsets for the multi-height distribution in the toric case when restricting to the interior of the dual effective cone. This contrasts with some single-height scenarios where points might accumulate on subvarieties.

Technical Corrections

• The author identifies a flaw in a previous result by [11] regarding the condition $\beta = \pi^\vee(u)$. The paper demonstrates that without this condition, the asymptotic exponent is incorrect (e.g., predicting $B^3$ instead of $B^2$ for $\mathbb{P}^1$ with specific weights). The descent method used here avoids this pitfall.

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4. Significance

1. Validation of Peyre's Conjecture: This work provides a rigorous proof of Peyre's multi-height conjecture for the important class of toric varieties, confirming that the geometric invariants (Tamagawa number, volume of the effective cone dual) correctly predict the distribution of points across the entire space of heights.
2. Methodological Advancement: By utilizing universal torsors and Möbius inversion rather than height zeta functions, the paper offers a robust alternative framework. This descent method is particularly well-suited for toric varieties due to their explicit Cox ring structure.
3. Precision in Error Terms: The derivation of the explicit error term involving the parameter $l$ (related to the geometry of the fan) provides deeper insight into how the combinatorial structure of the toric variety influences the rate of convergence of the point count.
4. Foundation for Future Work: The author notes that this result implies the classical Manin conjecture (single height) and sets the stage for analyzing non-split or non-projective cases, as well as varieties with more complex accumulation phenomena.

In summary, Bongiorno's paper successfully bridges the gap between the abstract multi-height conjecture and concrete arithmetic geometry for toric varieties, using the powerful machinery of universal torsors to derive precise asymptotic formulas with controlled error terms.