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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 $k$, $h\sb L$ an exponential height associated to $L$. For a subset $U\subset V(k)$, we denote by $\beta\sb U(L)$ the abscissa of convergence of $\sum\sb{x\in U}h\sb L(x)\sp{-s} $. We define also the function $\alpha(L)=\inf\{\gamma \in {\Bbb R}\vert \quad \gamma L+K\sb V$ is effective modulo Néron- Severi equivalence}. The paper states some conjectures to the effect that $\beta\sb U(L)$ 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

14G40Arithmetic varieties and schemes; Arakelov theory; heights
11G35Varieties over global fields
14G25Global ground fields
14C20Divisors, linear systems, invertible sheaves
14G05Rational points
14J45Fano varieties
Full Text: DOI EuDML
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