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