We will be concerned with the following situation:
where K is a smooth 3-fold/CE, A a small open disc, and T T a proper,
flat map, smooth over &* = (A-{0)).
H = T T (t) , t ^ 0, will be referred to as the general or
smooth fiber.
tfn = TT (0) will be referred to as the special or
(in general) singular fiber.
The whole setup, referred to from now on as a degeneration, should
be thought of as a family of smooth surfaces degenerating as t - 0 into
a singular surface.
This monograph will study the relationship between the general fiber
N and the special w .
There are two intimately related approaches. On one hand we can
start with N and ask what it can degenerate into. We can be even more
precise and start with N* = TT (A*)an3 see what possibilities there are
for the special fiber Nn which "relatively compactifies" N*. On the
other hand we can start with a special fiber Hn and ask if a smooth
deformation exists and if so what it will be.
To return to the first question. It is clear that given N*, its
"relative compactification" N is by no means unique. (One can blow up
points and curves on the special fiber and blow down its components
without affecting x*•)
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