Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Abstract

We study rational and algebraic points of bounded height on weighted projective spaces. A weighted projective space ${\mathbb{P}}_{\mathbf{w}}^n$, with weights $\mathbf{w}=(q_0,\dots ,q_n)$, carries two natural heights: the tautological height $\tilde{h}$, attached to the tautological bundle on the associated stack, and the weighted height $h=H(\ddot{I}•(\cdot ){)}^{1/q}$, the normalized pullback of the Weil height under the Veronese morphism $\ddot{I}•:{\mathbb{P}}_{\mathbf{w}}^n\to {\mathbb{P}}^n$, where $q=\mathrm{lcm}(q_0,\dots ,q_n)$. For $\tilde{h}$ we prove a Schanuel-type theorem over any number field $k$ of degree $m$, with leading term $c_k(\mathbf{w})X^{mQ}$, where $Q=q_0+\cdots +q_n$. Our main result concerns $h$ over $\mathbb{Q}$, for arbitrary coprime weights. A point of ${\mathbb{P}}^n(\mathbb{Q})$ lifts along $\ddot{I}•$ only if its valuation vector at every prime lies in a set $M_{\mathbf{w}}$ cut out by Kummer congruences. We prove that the points with all coordinates nonzero satisfy \[Z^{\circ}_{h}(\mathbb{P}^n_{\mathbf{w}}(\mathbb{Q}), X) = X^{q\,a(\mathbf{w})} P_{\mathbf{w}}(\log X) + O(X^{q\,a(\mathbf{w}) - θ})\] for some $\hat{I}¸>0$, where $P_{\mathbf{w}}$ has exact degree ${\hat{I}}^2(\mathbf{w})$ and positive leading coefficient, and $a(\mathbf{w})$ and ${\hat{I}}^2(\mathbf{w})$ are the value and the dimension of the optimal face of a linear program over the minimal elements of $M_{\mathbf{w}}$. The exponent need not equal the projective benchmark $q(n+1)$, and can exceed $Q$. Since the coordinate strata are again weighted projective spaces, the full count follows by stratification, and a proper stratum may dominate. We also count points of fixed degree when the exponents $q/q_i$ are pairwise coprime, and conjecture the asymptotic for $h$ over an arbitrary number field.

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