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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 UV(k), we denote by β U (L) the abscissa of convergence of xU h L (x) -s . We define also the function α(L)=inf{γ|γL+K V is effective modulo Néron- Severi equivalence}.

The paper states some conjectures to the effect that β U (L) and α(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:
14G40Arithmetic varieties and schemes; Arakelov theory; heights
11G35Varieties over global fields
11G50Heights
14G25Global ground fields
14C20Divisors, linear systems, invertible sheaves
14G05Rational points
14J45Fano varieties
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