Modern formulation of Manin's conjecture

Let $X$ be a geometrically integral smooth projective Fano variety over $\mathbb Q$ of dimension $n$ with Picard rank $r$, let $\mathcal X$ be a proper integral model over $\mathbb Z$, and let $H$ be an anticanonical height on $X(\mathbb Q)$. A map $f:Y\to X$ of geometrically integral smooth projective varieties is thin if it is generically finite onto its image and has degree different from $1$. Define the Tamagawa measure and the constants

$$\tau=\left(\lim_{s\to 1}(s-1)^rL(s,\operatorname{Pic}X_{\overline{\mathbb Q}})\right)\prod_vL_v(s,\operatorname{Pic}X_{\overline{\mathbb Q}})^{-1}\omega_v,$$

where $\omega_v$ is the natural measure on $X(\mathbb Q_v)$ defined by $\mathcal X$ at non-Archimedean places and by the fixed volume form at $v=\infty$, and

$$\alpha(X)=r\operatorname{vol}\{y\in((\operatorname{Pic}(X)\otimes\mathbb R)^{\mathrm{eff}})^\vee\mid K_X\cdot y\leq 1\},$$ $$\beta(X)=\left|H^1(\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q),\operatorname{Pic}X_{\overline{\mathbb Q}})\right|.$$

Manin's conjecture. There exists a finite set of thin maps $f_i:Y_i\to X_i$ such that

$$\lim_{B\to\infty}\frac{1}{B(\log B)^{r-1}}\sum_{\substack{x\in X(\mathbb Q)\H(x)<B\x\notin f_i(Y_i(\mathbb Q))\ \textrm{for any }i}}\delta_x=\alpha(X)\beta(X)\tau^{Br},$$

where the weak limit is taken as measures on $X(\mathbb A_{\mathbb Q})$, $\delta_x$ is the unit point mass at $x$, and $\tau^{Br}$ is the restriction of the Tamagawa measure to the locus where the Brauer–Manin obstruction vanishes. This formulation predicts both the asymptotic number and adelic distribution of rational points after removing finitely many thin images; the source attributes it to work of Manin, Peyre, Batyrev–Tschinkel and Salberger, but the statement itself is marked with no resolution status.