Batyrev–Manin–Peyre counting conjecture for strongly saturated varieties

Let $F$ be a number field, and let $V$ be a smooth projective variety over $F$ that is strongly $\mathcal{L}$-saturated for a metrized line bundle $\mathcal{L}$. Let $H_{\mathcal{L},F}$ be the associated height, and define

$$N(V,\mathcal{L},B)=\#\{x\in V(F):H_{\mathcal{L},F}(x)\leq B\}.$$

Write $\alpha_V(\mathcal{L})$, $\beta_V(\mathcal{L})$, and $c_V(\mathcal{L})$ for the associated Batyrev–Manin–Peyre invariants and leading constant. Batyrev–Manin–Peyre conjecture. One has

$$N(V,\mathcal{L},B)=c_V(\mathcal{L})B^{\alpha_V(\mathcal{L})}(\log B)^{\beta_V(\mathcal{L})-1}(1+o(1)),$$

as $B\to\infty$. This is the expected asymptotic for rational points of bounded height on suitable varieties; the cited formulation is attributed to Batyrev and Tschinkel.