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)