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
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
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
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