Multi-height analysis of rational points of toric varieties

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.

💡 Research Summary

The paper “Multi‑height analysis of rational points of toric varieties” investigates the distribution of rational points on smooth, projective, split toric varieties over ℚ by introducing a multi‑height framework. Classical approaches to Manin’s conjecture consider a single height attached to a fixed ample line bundle, which can lead to “accumulating subsets” where points concentrate. The author follows Peyre’s suggestion to study all possible heights simultaneously, i.e. a height vector indexed by the Picard group, and proves that the expected asymptotic behavior holds without any accumulation phenomenon.

Section 2 recalls the Arakelov height formalism and defines a system of heights as a section s: Pic(V) → H(V) of the forgetful map from adelic line bundles to the Picard group. For each rational point P∈V(ℚ) the multi‑height h(P) is the linear form on Pic(V) given by h(P)(L)=log H_s(L)(P). The conjectural asymptotic (Conjecture 2.11) predicts that for a compact polytope D₁⊂Pic(V)∨⊗ℝ and a vector u in the interior of the dual effective cone, the number of points with h(P)∈D_B:=D₁+log B·u should be  ν(D₁)·β(V)·τ(V)·B^{⟨ω_V^{−1},u⟩}, where ν is a Haar measure on Pic(V)∨⊗ℝ, β(V)=#H¹(ℚ,Pic(V)), and τ(V) is the Tamagawa number.

The core of the paper is the proof of this conjecture for toric varieties (Theorem 2.15). The author works with a split toric variety X defined by a fan Σ. The anticanonical bundle ω_X^{−1} lies in the interior of the effective cone, guaranteeing the quasi‑Fano hypotheses. A universal torsor T→X (unique over ℤ because H¹(ℤ,T_NS)=1) is constructed; it is a principal T_NS‑torsor where T_NS≅G_m^t with t=rank Pic(X). The crucial observation is that the chosen system of heights lifts canonically to T: each line bundle L corresponds to a G_m‑equivariant morphism Φ_L:T→L^×, and the adelic norm on L is precisely the pull‑back of the standard sup‑norm on ℙ^N via the Cox coordinate map.

Counting rational points on X with multi‑height in D_B is reduced to counting integral points on T inside a region defined by inequalities derived from the height vector u. This is a lattice‑point problem in a rational polyhedral cone. The author applies the lattice‑point counting technique of Davenport (originally used for counting points in bounded domains) together with a careful analysis of the shape of the region, which yields the main term ν(D₁)·τ(X)·B^{⟨ω_X^{−1},u⟩} and an explicit error term O(B^{-(1−1/ℓ−ε)}·min_{ρ}⟨