Abstract

The objective of this thesis is to understand F.Campana’s conjectures about density of integral points on varieties defined over a number field. In the case of curves this is solved by Siegel and Faltings’ theorems: the degree of the canonical divisor of a curve determines whether it has finitely or infinitely many rational points (possibly after a finite extension of the base field). In higher dimension, varieties can be neither potentially dense (the set of rational points becomes dense after at most a finite extension of the base field), nor mordellic (rational points are always finitely many in an open dense subset). The conjectures studied in the thesis address the problems of characterizing potential density and mordellicity of a variety and of constructing a (unique) fibration which splits each variety in its mordellic part, the base of the fibration, and potentially dense part, the fibers. To deal with this problem, F.Campana introduces the notion of orbifold pair: to each variety, one can attach an orbifold divisor, which allows keeping track of multiple fibers of fibrations. Using this tool, he was able to construct a fibration, the core map, which has (orbifold) special general fiber, conjectured to be the potentially dense varieties, and base of (orbifold) general type, conjectured to be the mordellic ones.