Batyrev–Manin Conjecture C' for Fano varieties

Let $V$ be a Fano variety with canonical bundle $\omega_V$ not effective, let $\mathcal{L}$ be a very ample line bundle, and let $U\subseteq V$ be a sufficiently small Zariski open subset. Define

$$N_U(\mathcal{L},X)=\#\{x\in U(k):h_\phi(x)\leqslant X\},$$

where $h_\phi$ is a height associated with $\mathcal{L}$. Also define

$$\alpha(\mathcal{L})=\inf\{\lambda\in\mathbf{R}:\lambda[\mathcal{L}]+[\omega_V]\in N_{\rm eff}^1(V)\},$$

and let $t(\mathcal{L})$ be the codimension of the minimal face of $\partial N_{\rm eff}^1(V)$ containing $\alpha(\mathcal{L})[\mathcal{L}]+[\omega_V]$.

Batyrev–Manin Conjecture C'. If $U$ is sufficiently small, then

$$N_U(\mathcal{L},X)\sim cX^{\alpha(\mathcal{L})}(\log X)^{t(\mathcal{L})-1}$$

as $X\to\infty$ for some positive constant $c$.

This is the expected asymptotic formula for rational points of bounded height on Fano varieties. The source presents it as a review of the Manin conjecture in relation to the automorphic counting problem; no resolution status is supplied here.