special. I believe, however, that many of the techniques involved would,
especially after some process of sophistication, be successful even in
more general cases, at least I know this to be the case for degenerations
with no triple points.
I have before vaguely alluded to the combinatorial data of a special
fiber. An important ingredient is contained in the kind of double curves
that occur, and more subtly "the way" the double curves "sit" in their
respective components.
In order to isolate these phenomena it is instructive, especially
in a very preliminary, experimental version, to limit oneself to the
case of two components.
Chapter four provides estimates on the self intersection of curves
on surfaces in terms of their genus and the number of exceptional curves
meeting them. These estimates, esoteric as they may appear, turn out to
be crucial in the classification of degenerate fibers and particularly in
excluding pathological possibilities. To give some examples:
2
Let C be a rational curve in a surface V, and assume that C 0
and that C meets all exceptional divisors (an assumption that turns out
2
to be very natural). Then we have only three possibilities. V = jp and
C is either a line or a conic, or V is a rational minimal ruled sur-
face with C a section. Applying this to a component with C a
rational double curve (with positive self intersection) we find by a
contractibility criterion that the component V is contractible in all
but the middle case. (The noncontractibility of the middle case is
very annoying, and causes all sorts of additional complications that have
to be circumvented.)
Using this circle of ideas one can prove e.g. that surfaces of
general type cannot degenerate into two rational components meeting in a
rational curve. (It seems possible that certain elliptic surfaces (H = 1)
can degenerate in such a way, an example of an Enriques degeneration
xi
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