Sections 2 and 3 describe the topology of a neighborhood of the

special fiber, whose main implication is that the special fiber determines

topologically the general fiber in a fairly explicit way. This is

utilized in section 5 to compute topological invariants,via a Mayer-

Vietoris spectral sequence, of the general fiber, given the data of the

special.

In this context it should be noted that nice and simple formulas for

the Euler-characteristics (c ) are easily obtainable (in fact they gener-

alize to any dimensions) but the actual computation of the Betti-numbers

is much more subtle, and depends in an intricate way on the configuration

of the special fiber. Explicit formulas are only given in the case of no

triple points.

The ramifications of intersection theory are outlined in section 4

of the same chapter.

First we are given necessary conditions for a special fiber K to

be inserted as a special fiber in a degeneration (with smooth total

space). It is unknown and unlikely that these conditions are sufficient.

Secondly we obtain a formula, essentially derived from Stokes

theorem, of computing the self-intersection of a divisor in a general

fiber from its specialization to the special (fiber).

This is applied to the computation of the self-intersection of the

2

canonical divisor (c,),and a simple formula in terms of the data of the

singular fiber is given. (Note: This approach could also be used to

compute the Chern-numbers of degenerations of arbitrary dimensions.)

We also arrive at criterions for components of the special fiber to

be contractible, and other ways of modifying the special fiber. They

are very important because they enable us to put the special fiber into

various normal forms more amenable to analysis, a technique repeatedly

used in the second half of chapter three and providing the backbone of

chapter five. I believe that a systematic development of a more

vii