Algebraic Geometry Seminar, 1999
Kalle Karu (BU),
Semistable Reduction in Characteristic Zero.
Let f: X --> B be a family of varieties. We consider the problem of
replacing the family f with a new family f': X' --> B' such that all
fibers of f' are as nice as possible. The correct definition of "as nice
as possible" is given in terms of toric geometry, and a morphism f'
satisfying it is called semistable. The semistable reduction problem then
asks to find a generically finite proper base change B' --> B and a
proper birational morphism X' --> X \times_B B' such that the induced
morphism f': X' --> B' is semsitable. We discuss the proof of a slightly
weaker version of the problem and some interesting (unsolved) questions in
the combinatorics of polyhedral complexes that the strong semistable
reduction problem gives rise to.