Diophantine problems and algebraic geometry: new connections.

The core idea of this research proposal is that there is a lot to be gained from an intensified interaction between different subfields of arithmetic/algebraic geometry. The community of people studying rational points on algebraic varieties (to which I belong) is now very big and mature, but it would benefit enormously from more interaction with other, rapidly developing subfields of algebraic geometry, such as birational and logarithmic geometry.

The proposal consists of three broad and complementary research lines:

1) Reinventing rational points: Campana's orbifolds

2) Families of varieties over one-dimensional bases: arithmetic and geometric aspects

3) Families of varieties over higher-dimensional bases: geometric aspects