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 $\mathbb{P}^n_{\mathbf{w}}$, with weights $\mathbf{w} = (q_0, \dots, q_n)$, carries two natural heights: the tautological height $\widetilde{h}$, attached to the tautological bundle on the associated stack, and the weighted height $h = H(ϕ(\,\cdot\,))^{1/q}$, the normalized pullback of the Weil height under the Veronese morphism $ϕ: \mathbb{P}^n_{\mathbf{w}} \to \mathbb{P}^n$, where $q = \operatorname{lcm}(q_0, \dots, q_n)$. For $\widetilde{h}$ 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 $ϕ$ 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 $θ> 0$, where $P_{\mathbf{w}}$ has exact degree $β(\mathbf{w})$ and positive leading coefficient, and $a(\mathbf{w})$ and $β(\mathbf{w})$ 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.