Problems and results on the distribution of algebraic points on algebraic varieties
This is a survey paper on the distribution of algebraic points on algebraic varieties. In the first section, the author considers varieties with many points. He quotes the main results on this topic and produces a collection of examples, especially of Fano varieties. He defines accumulators, he introduces conjectures of Batyrev and Manin, of Franke, Manin and Tschinkel. Next he pays some special attention to the case of the cubic threefold (also Theorem 11.10.11 of [\textit{E. Bombieri} and \textit{W. Gubler}, Heights in Diophantine geometry. New Mathematical Monographs~4. Cambridge: Cambridge University Press (2006; Zbl 1115.11034)]) which has important applications in additive number theory and in the theory of trigonometric sums. The next section deals with the Northcott property, and the last one with small points, including the theorems of \textit{F.~Amoroso} and \textit{S.~David} [J. Reine Angew. Math. 513, 145--179 (1999; Zbl 1011.11045)] and of \textit{Yu.~Bilu} [Duke Math. J. 89, No. 3, 465--476 (1997; Zbl 0918.11035)].
conjecture of Franke, Manin and Tschinkel
scientific article; zbMATH DE number 5620667
scientific article