Manin's conjecture for rational points on smooth Fano varieties

Let $k$ be a number field and let $X$ be a smooth Fano variety over $k$, so that $-K_X$ is ample. Fix an adelic metrization of the anticanonical bundle $comathcal O(-K_X)$, with associated height function $\mathsf H_{-\mathcal K_X}:X(k)\to\mathbb R_{\geq 0}$. For $Q\subset X(k)$, write

$$\mathsf N(Q,-\mathcal K_X,T)=\#\{x\in Q\mid \mathsf H_{-\mathcal K_X}(x)\leq T\}.$$

Suppose that $X(k)$ is not thin. Manin's conjecture. There exists a thin set $\mathsf Z\subset X(k)$ such that

$$\mathsf N(X(k)\setminus\mathsf Z,-\mathcal K_X,T)\sim\alpha(\mathrm{Nef}_1(X))\beta(X)\tau_{-\mathcal K_X}(X)T(\log T)^{\rho(X)-1},$$

as $T\to\infty$, where $\rho(X)$ is the Picard rank, $\alpha(\mathrm{Nef}_1(X))$ is the alpha constant of the nef cone of curves, $\beta(X)$ is the size of $\mathrm{Br}(X)/\mathrm{Br}(k)$, and $\tau_{-\mathcal K_X}(X)$ is the Tamagawa number. This is a central prediction for the distribution of rational points on Fano varieties. It is largely open even for del Pezzo surfaces, with the cited results covering several toric, quintic, and individual quartic cases but no known smooth cubic surface example.