Weighted analogue of the Batyrev–Manin conjecture

Let $W$ be a smooth weighted variety in $\mathbb{WP}^n_{\mathbf{q}}(k)$ with weights $\mathbf{q}=(q_0,\ldots,q_n)$, let $q=\operatorname{lcm}(q_0,\ldots,q_n)$, and let $A$ be an ample divisor on $W$. Define

$$Z_{\mathfrak{h}_A}(W(k),X)=\left|\{\mathfrak{p}\in W(k):\mathfrak{h}_A(\mathfrak{p})\leq X\}\right|,$$

where $\mathfrak{h}_A(\mathfrak{p})=H_A(\phi(\mathfrak{p}))^{1/q}$ and $\phi:W\to V\subset\mathbb{P}^n(k)$. Weighted analogue of the Batyrev–Manin conjecture. As $X\to\infty$,

$$Z_{\mathfrak{h}_A}(W(k),X)\sim c_W X^{a_W}(\log X)^{b_W},$$

where $c_W$ is proportional to $\gcd(q,\varphi(m))^{-1}$ compared with the projective case, $a_W=m(\dim W+1)$, and $b_W=\operatorname{rank}\operatorname{Pic}(W)-1$, with $b_W=0$ if $\dim W>1$ or $m>1$. This adapts the classical Batyrev–Manin prediction to weighted heights and the arithmetic sparsity caused by the weighted structure; the source notes that the classical prediction is known for toric varieties, while this weighted formulation is proposed as a new conjecture.