or if we would be looking for the degeneration (normal crossings tend to
be unstable in general).
But nevertheless normal crossing form gives a standardized combina-
torial description of a singular fiber. Although very simple singular
fibers could have rather complicated form when resolved into normal
crossings form.
As for the insistence on reduced components, many of the results
presented in this monograph could easily be generalized to the case with
multiplicities. Although such generalizations are ultimately redundant
they would have practical value in studying non-complicated singular
fibers, which would turn complicated after a semi-stable reduction
has been performed
Now to the chapter by chapter description of the monograph.
The first chapter will be devoted to a brief survey on the structure
and classification of surfaces (both algebraic and non-algebraic with a
natural emphasis on the algebraic). The main purpose of this chapter is
to define the important invariants and to show how they tie in with the
classification problem. Some well-known facts and methods are also
thrown in here for convenience and easy reference.
It is hoped that this chapter to some extent will furnish some
motivation for the subsequent material.
The second chapter plays a pivotal role. It both presents the basic
facts about degenerations and establishes the techniques employed in the
later chapters.
The basic assumption that ^ (the total space) is smooth allows us
both to use Clemens description of the topology near the singular fiber,
as expounded in [C],and to use elementary intersection theory.
Section 1 establishes (once and for all) the basic assumptions we
impose on the special fiber, and some notation.
vi
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