Higher Dimensional Diophantine Problems

1. Rational points. A classical conjecture of Mordell states that a curve of genus ^ 2 over the rational numbers has only a finite number of rational points. Let K be a finitely generated field over the rational numbers. Then the same statement should hold for a curve defined over K, and a specialization argument due to Néron shows in fact that this latter statement is implied by the corresponding statement over number fields (cf. [L 1, Chapter VII, §6]). Let F be a variety in projective space, defined over the complex numbers, and therefore over some finitely generated field over the rationals. We shall say that V has the Mordell property if it has only a finite number of rational points in any finitely generated field over Q. One possibility to extend Mordell's conjecture to higher dimensional varieties is as follows.