Marta Pieropan

In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr), but it is still open for the maximal unramified extensions of p-adic fields. I use birational geometry in characteristic 0 to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities.