Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Abstract

Let ${\mathbb{WP}}_{\mathbf{q}}^n$ be a weighted projective space with weights $\mathbf{q}=(q_0,\dots ,q_n)$, $q=\mathrm{lcm}(q_i)$, and let $\varphi :{\mathbb{WP}}_{\mathbf{q}}^n\to {\mathbb{P}}^n$, $[x_i]\mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of ${\mathbb{P}}^n(\mathbb{Q})$ is the image of a rational point of ${\mathbb{WP}}_{\mathbf{q}}^n$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of ${\mathbb{WP}}_{\mathbf{q}}^n$ of bounded weighted height $\mathfrak{h}=H(\varphi (\cdot ){)}^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$Z_{\mathfrak{h}}^{\circ }({\mathbb{WP}}_{\mathbf{q}}^n(\mathbb{Q}),X)=X^{qa(\mathbf{q})}{P}_{\mathbf{q}}(\log X)+O\left(X^{qa(\mathbf{q})-\theta }\right),\theta >0,$$ with $P_{\mathbf{q}}$ of exact degree $\beta (\mathbf{q})$, where $a(\mathbf{q})$ and $\beta (\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q\le qa(\mathbf{q})\le q(n+1)$, $Q=\sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1)-qa(\mathbf{q})$ measures the sparsity of the rational points of ${\mathbb{WP}}_{\mathbf{q}}^n$ 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.

39 pages. Substantially revised; new results on the exponent bounds and the leading constant