Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

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 $h=H(\phi (\cdot ){)}^{1\mathrm{/}q}$, the normalized pullback of the Weil height under the Veronese morphism $\phi :{\mathbb{P}}_{\mathbf{w}}^n\to {\mathbb{P}}^n$, 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 $\phi$ 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 $$Z_h^{\circ }({\mathbb{P}}_{\mathbf{w}}^n(\mathbb{Q}),X)=X^{qa(\mathbf{w})}{P}_{\mathbf{w}}(\log X)+O(X^{qa(\mathbf{w})-\theta })$$ for some $\theta >0$, where $P_{\mathbf{w}}$ has exact degree $\beta (\mathbf{w})$ and positive leading coefficient, and $a(\mathbf{w})$ and $\beta (\mathbf{w})$ 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 $q\mathrm{/}{q}_i$ are pairwise coprime, and conjecture the asymptotic for $h$ over an arbitrary number field.