I‑AIMS Institute for Artificial Intelligence
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Arithmetic Sparsity and Cohomological Obstructions in Weighted Projective Spaces

Tanush Shaska

I-AIMS preprint sh-94 · 2025

Let 𝕎ℙⁿ_q be a weighted projective space with weights q = (q₀, …, qₙ), q = lcm(qᵢ), and let ϕ: 𝕎ℙⁿ_q → ℙⁿ, [xᵢ] ↦ [xᵢ^(q/qᵢ)], be the Veronese morphism. A point of ℙⁿ(ℚ) is the image of a rational point of 𝕎ℙⁿ_q only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of 𝕎ℙⁿ_q of bounded weighted height h = H(ϕ(·))^(1/q) and prove that, on the locus where all coordinates are nonzero, Zₕ^∘(𝕎ℙⁿ_q(ℚ), X) = X^(q a(q)) P_q(log X) + O(X^(q a(q) − θ)), θ > 0, with P_q of exact degree β(q), where a(q) and β(q) are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies Q ≤ q a(q) ≤ q(n+1), Q = ∑ qᵢ, with equality on the right if and only if the exponents q/qᵢ are pairwise coprime; the difference q(n+1) − q a(q) measures the sparsity of the rational points of 𝕎ℙⁿ_q relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights (2,2,3,3). The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

MSC classes: 11G50, 11G35, 14G05, 14M25

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