Manin–Peyre conjecture for smooth weak Fano varieties

Let $V$ be a smooth weak Fano variety over a number field $F$ such that $V(F)$ is Zariski dense, and let $H_F$ be a relative adelic height on the anticanonical line bundle. Manin–Peyre conjecture. There exists a thin subset ${\mathcal U} \subseteq V(F)$ and a constant $c_{V,F}$, called the Peyre constant, such that

$$N_F({\mathcal U},B) \sim c_{V,F}B(\log B)^{\operatorname{rk}(\operatorname{Pic}V)-1}.$$

Moreover, the leading constant has the form $c_{V,F}=\alpha(V)\beta(V)\tau(V)$. This is the expected asymptotic for rational points on weak Fano varieties; the paper uses it as the framework for counting points on symmetric squares and Hilbert schemes. The supplied text does not state a resolution.