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On the number of rational points of bounded height on algebraic varieties. (Sur le nombre des points rationnels de hauteur borné des variétés algébriques.) (French) Zbl 0679.14008

Let $k$ be a global field, $V$ a projective variety defined over $k$, ${h}_{L}$ an exponential height associated to $L$. For a subset $U\subset V\left(k\right)$, we denote by ${\beta }_{U}\left(L\right)$ the abscissa of convergence of ${\sum }_{x\in U}{h}_{L}{\left(x\right)}^{-s}$. We define also the function $\alpha \left(L\right)=inf\left\{\gamma \in ℝ|\phantom{\rule{1.em}{0ex}}\gamma L+{K}_{V}$ is effective modulo Néron- Severi equivalence}.

The paper states some conjectures to the effect that ${\beta }_{U}\left(L\right)$ and $\alpha \left(L\right)$ are comparable (sometimes equal) if one stabilizes the situation taking $k$ sufficiently large and $U$ sufficiently small and Zariski-open. These conjectures are proved for homogeneous Fano varieties and some del Pezzo surfaces.

Reviewer: Yu. I. Manin

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G35 Varieties over global fields 11G50 Heights 14G25 Global ground fields 14C20 Divisors, linear systems, invertible sheaves 14G05 Rational points 14J45 Fano varieties
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