Campana points, fibrations and logarithmic geometry.

In the study of Diophantine problems, given by systems of polynomial equations with coefficients in a number field, one traditionally distinguishes two cases: the search for rational solutions, and the hunt for integral solutions. Geometrically, these two cases correspond to the study of rational points on projective varieties and of integral points on quasi‐projective varieties respectively. While this classical dichotomy is both natural and useful, it also makes a lot of sense to try to interpolate between these two classical notions. A geometric framework in which this unification becomes possible is provided by Campana's theory of 'orbifold pairs', leading to the intermediate notion of 'Campana point', which is still relatively poorly explored. The goal of this project is to build some of the foundations for new applications of this framework, in particular the notion of 'orbifold base' of a fibration. We will then apply this framework to two classical Diophantine problems: Serre's problem, on the number of varieties in family which are everywhere locally solvable, and generalisations of Ax-Kochen type statements (following work of Colliot-Thélène, Denef and others).