Rational Curves in Calabi-Yau Threefolds

3934 Johnsen and Knutsen
5.2. Some Specific K3 Surface Computations
In the following lemma we introduce a specific K3 surface with a
specific Picard lattice, which will be instrumental in proving Theorem
4.3. The element G in the lattice will correspond to a curve of bidegree
(d, a) as described in that theorem.
Lemma 5.3. Let n 4, d > 0 and a > 0 be integers satisfying d > na
Then there exists an algebraic K3 surface S with Picard group
Pic S ’ ZH ZD ZG with the following intersection matrix:
H2 D:H
H:D
D
H:G
2 D:G
G:H G:D G 2
2
4
3 2
2n
5 ¼ 4 3
3
0
d
a
3
5
d a 2
and such that the line bundle L :¼ H b
n 4
3 cD is nef.
Proof. The signature of the matrix above is (1, 2) under the given conditions.
By a result of Nikulin (1980) (see also Morrison, 1984, Theorem
2.9(i)) there exists an algebraic K3 surface S with Picard group
Pic S ¼ ZH ZD ZG and intersection matrix as indicated.
Since L 2 > 0, we can, by using Picard-Lefschetz tranformations,
assume that L is nef (see e.g., Oguiso, 1994 or Knutsen, 2002). &
Note now that
L 2 8
< 8 if n 4 mod 3;
¼ 10
:
12
if n
if n
5
6
mod 3;
mod 3:
We will from now on write L 2 ¼ 2m, for
m :¼ n 3b
3
3
a .
ð2Þ
n 4
c¼4; 5or6 ð3Þ
3
(in other words n m (mod 3)) and define
d0 :¼ G:L ¼ d
n 4
b ca > 0:
3
ð4Þ
Note that the condition d > na
3
d0 > ma
3
3
a is equivalent to
3
: ð5Þ
a