Stable modification of relative curves

A theorem of de Jong states that any generically smooth family C -> B of proper curves can be modified to a semistable family C' -> B' after replacing the base with an appropriate alteration. Stable modification theorem strengthens this in few aspects: properness (and even separatedness) of the morphism C -> B is not needed, and once an appropriate B' is fixed, there exists a unique minimal semistable modification C' called the stable modification of C.

In my talk I will explain how the stable modification theorem is proved using Riemann-Zariski spaces and Berkovich non-archimedean geometry. In the end of the talk I will also discuss some recent results obtained in a joint work with L. Illusie (after Gabber). In particular, this includes the theorem that any variety X possesses an alteration f: X' -> X such that X' is smooth and [k(X'):k(X)] is coprime with a fixed prime l not equal to the characteristic.