Let L=(L,∥⋅∥v) be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety V defined over a number field.
Denote by N(V,L,B) the number of rational points in V having L-height ≤B. We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for N(V,L,B) as B→∞ in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
We introduce the notions of L-primitive varieties and L-primitive fibrations. For L-primitive varieties V over F we
propose a method to define an adelic Tamagawa number τL(V) which
is a generalization of the Tamagawa number τ(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,L,B) on the choice of v-adic metrics on L.Comment: 54 pages, minor correction