The Batyrev–Manin conjecture for rational points of bounded height

Let $V$ be a smooth projective variety over a number field $K$ such that the canonical class $K_V$ is non-effective. Endow $V(K)$ with a height function coming from an adelic metrization on a big line bundle $L$.

Batyrev–Manin conjecture. There exists an open subvariety $U \subseteq V$ such that

$$N_{U,K}(B) \sim c(V) B^a \log^{b-1} B,$$

for explicit constants $a$ and $b$ depending on $V$ and the adelically metrized line bundle $L$.

This conjecture predicts the order of growth of rational points of bounded height in terms of the geometry of the variety and the height line bundle. In the anticanonical case $L=\omega_V^\vee$, the constant $c(V)$ was predicted by Peyre.