are successful in verifying the ^-conjecture in the following cases.
First when
H
= -1 or 1, and we make the assumption that ^ is
weakly projective. The method employed is the principle of "generic
stability" referred to above. Incidentally the proof does not require
N to be in normal crossing form but works for any special fiber.
In the case of
H
= -1* we have simply proved that in a degeneration
of rational or ruled surfaces, the components of the special fiber are
either rational or ruled. The example of ruled surfaces degenerating
into Hopf surfaces gives a counter example to this, when we remove the
weak projectivity condition, but the example does not contradict the
K-invariant conjecture.
Clearly new ideas are needed to attack this in the general case
(i.e. relinquishing the weak projectivity assumption).
Using entirely different methods we verify the conjecture when
H = 0, and the general fiber » is minimal, and N is in normal crossing
form and with no triple points—the last condition being the most
restrictive and least natural. Note, however, that in this case no
global conditions on K need to be imposed.
Because of the minimality condition on the general fiber, we can
assume that a multiple of the canonical divisor is concentrated on the
special fiber. This imposes severe restrictions on some of its components,
and by using contractibility conditions on the components certain of them
turn out to be contractible which enables us to apply an essentially
inductive process.
It seems that with a more sophisticated arsenal of modifications
of the special fiber, essentially the same method should work, even
considering triple points, as long as minimality of the general fiber is
assumed.
In the last chapter we study the degenerations of surfaces into two
components (meeting along a nonsingular curve). This might seem very
x
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