Thus one can ask, what is the degeneration, a problem that ties in

with the compactification of moduli. Using Mumfords stability approach

(see [M]) one can simply ask, what are the stable special fibers?

Another motivation from studying degenerations comes from studying

surfaces and their invariants. By finding explicit degenerations one

can recapture information about the original surface, from a sort of

combinatorial study of the special fiber, which in its very explicitness,

consisting of well-known components (e.g. rational surfaces) could be

much more accessible than the general fiber.

The subject of degenerations, as implied by the somewhat vague and

diffuse introductory statements above, is very large, and consequently

this monograph makes no claim on completeness in any aspect. On the

contrary I merely intend to present a wide range of methods and techniques

and illustrate their use in typical examples.

My aim has not primarily been to establish general theorems, but

rather to suggest the intricate fabric of detail involved.

Such an approach does not easily lend itself to a coherent treatment,

and as is to be seen, my monograph has in no way been able to transcend

these intrinsic limitations.

Before I present a description chapter by chapter I will comment on

two basic assumptions I will make on the special fiber.

By Hironaka (see [H]) we can assume that ^ is a divisor with smooth

components meeting normally along nonsingular curves.

Using Mumford's semi-stable reduction theorem (see [M']) we can also

assume, after a base change, that the components are reduced i.e. they

all have multiplicity one.

These extra assumptions impose heavy restrictions in the case when

we for instance, would be interested in the whole of N (say if we

wanted to study classifications of three-folds via fibrations of surfaces)

v