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

cases.)

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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