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

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