Peyre's leading-constant formula for weak Fano varieties

Let $V$ be a smooth weak Fano variety over a number field $F$, let $\Gamma_F$ be the absolute Galois group, and write $\rho(V)=\operatorname{rk}((\operatorname{Pic}\overline V)^{\Gamma_F})$. Let $C_{\operatorname{eff}}^1(\overline V)$ be the effective cone in $\operatorname{Pic}\overline V\otimes_{\mathbb Z}\mathbb R$, and define

$$C_{\operatorname{eff}}^{1,\Gamma_F}(\overline V):=C_{\operatorname{eff}}^1(\overline V)\cap\left((\operatorname{Pic}\overline V)^{\Gamma_F}\otimes_{\mathbb Z}\mathbb R\right).$$

Let $C_{\operatorname{eff}}^{1,\Gamma_F}(\overline V)^\vee$ be its dual cone, let $K_{\overline V}$ be the canonical class, and let $\omega$ be the Tamagawa measure on $V(\mathbb A_F)$. Peyre's constant formula. The leading constant is

$$c_{V,F}=\alpha(V)\beta(V)\tau(V),$$

where

$$\alpha(V):=\frac{1}{(\rho(V)-1)!}\int_{C_{\operatorname{eff}}^{1,\Gamma_F}(\overline V)^\vee}e^{-\langle K_{\overline V},y\rangle}\,dy,$$ $$\beta(V):=\#H^1(F,\operatorname{Pic}\overline V),\qquad \tau(V):=\omega\bigl(V(\mathbb A_F)^{\operatorname{Br}V_F}\bigr).$$

This gives the explicit factorization predicted for the constant in the Manin–Peyre conjecture. The surrounding text attributes the formula to Peyre and notes that the beta factor appeared later in work of Batyrev–Tschinkel and Salberger.