Rational Curves in Calabi-Yau Threefolds

3930 Johnsen and Knutsen
An analogue of Step (III) is: Eliminating s from the first set of equations,
we obtain:
Q1B þ Z5Q 0 1 ¼ Q2 ¼ 0:
Eliminating s from the second set of equations, we obtain:
Q1 ¼ Q2B þ Z5Q 0 2 ¼ 0:
If g is odd, we obtain in both cases a ‘‘complete intersection’’ threefold
of type
2H
g 5
g 5
F; 3H
2 2 þ e5 F :
If g is even, the first threefold is of type
2H
g 6
g 4
F; 3H
2 2 þ e5 F ;
while the second is of type
2H
g 4
g 6
F; 3H
2 2 þ e5 F ;
Since g ¼ N 1 e5, we see that in all cases we have intersection type
(2H c1F, 3H c2H), such that c1 þ c2 ¼ N 6.
The analogue of Step (IV) seems doable for g odd, but here Step (II),
as remarked, is unclear. The details of this analogue for g even are also
not quite clear to us.
5. K3 SURFACE COMPUTATIONS
The purpose of the section is to make the necessary technical preparations
to complete Step (I) of the proof of Theorem 4.3. First we will
recall some useful facts about K3 surfaces and rational normal scrolls.
In Lemma 5.3 we introduce a specific K3 surface which will be essential
in the proof of Step (I). In the last part of the section we make some
K3-theoretical computations related to the Picard lattice of this particular
K3 surface.