  Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety $V$ defined over a number field.
Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal
L}$-height $\leq B$. We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
  We introduce the notions of ${\cal L}$-primitive varieties and ${\cal
L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we
propose a method to define an adelic Tamagawa number $\tau_{\cal L}(V)$ which
is a generalization of the Tamagawa number $\tau(V)$ introduced by Peyre for
smooth Fano varieties. Our method allows us to construct Tamagawa numbers for
$Q$-Fano varieties with at worst canonical singularities. In a series of
examples of smooth polarized varieties and singular Fano varieties we show that
our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal
L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.
