Introduction

We will be concerned with the following situation:

TT

Y

A

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*•)

iv