Good reduction of complete intersections

In 1983, Faltings proved the famous Mordell conjecture on the finiteness of the set of rational points on curves of higher genus. Along the way he proved numerous other finiteness statements, including the Shafarevich conjecture, which states that there are only finitely many curves of higher genus over a number field which have good reduction outside any given set of prime ideals. In this talk we shall consider analogues of the Shafarevich conjecture for certain classes of Fano varieties given as complete intersections in projective space. This is joint work with Ariyan Javanpeykar.