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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 \(U\subset V(k)\), we denote by \(\beta_ U(L)\) the abscissa of convergence of \(\sum_{x\in U}h_ L(x)^{-s} \). We define also the function \(\alpha(L)=\inf\{\gamma \in {\mathbb R}| \quad \gamma L+K_ V\) is effective modulo Néron- Severi equivalence}.
The paper states some conjectures to the effect that \(\beta_ U(L)\) and \(\alpha(L)\) 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 in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14G05 Rational points
14J45 Fano varieties
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References:

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