exhibited in the appendix, shows its possibility for K = 0 cf. Prop. 5.2.)
Chapter five contains a systematic classification of all the
possible degenerate fibers that can occur (consisting of two components)
and in particular the n-invariant conjecture is verified in all cases.
It seems extremely likely that this can be done more generally, using the
same method of self intersection estimates, in the case with no triple
points. (Note: this takes care of non-minimal * = 0 surfaces.)
In the classification of special fibers, special emphasis is put on
the case of fibers giving rise to surfaces of general type with
p = q = o. (As mentioned above, the study of such surfaces might be
facilitated by a study of their degenerations.) In the appendix (the
index of cruciality for elliptic curves in elliptic ruled surfaces) a
detailed study is presented by looking at the self intersections of
elliptic curves in elliptic ruled surfaces, of the possibilities that
would arise from a deformation of two components, one rational the other
elliptic ruled, meeting in an elliptic curve. (An example of Godeaux
degenerations show that such components are smoothable at least in some
From the above the need for good and workable criterions for smooth
deformations of two surfaces meeting normally along a nonsingular curve
seems highly desirable. If such criterions would be available a wealth
of new surfaces could be constructed more or less systematically.
The last two chapters, difficult to motivate, might possibly appear
pointless. It should be kept in mind, however, their essentially experi-
mental nature. They are intended to expose the wealth of possibilities
and intricate, at times, esoteric details, and to suggest further study.
This monograph is essentially, with minor modifications, my thesis
presented at Harvard in the spring of 1975.
I would like to thank my advisor Professor D. Mumford, for innumer-
able suggestions, helpful discussions and most of all, constant
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