Rational Curves in Calabi-Yau Threefolds

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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
Rational Curves in Calabi-Yau Threefolds 3935 Also note that Pic S ’ ZL ZD ZG, and that d :¼ jdiscðL; D; GÞj ¼ jdiscðH; D; GÞj ¼j2að3d naÞþ18j ¼j2að3d0 maÞþ18j ð6Þ divides jdisc(A, B, C)j for any A, B, C 2 Pic S. Now we will study the K3 surface defined in Lemma 5.3 in further detail. Lemma 5.4. Let S, H, D, G and L be as in Lemma 5.3. Then L is base point free and Cliff L ¼ 1. Furthermore, L is ample (whence very ample) if and only if we are not in any of the following cases: (i) ma ¼ 3d 0 and 9jma, (ii) m ¼ 4 and (d0, a)¼ (2, 2), (5, 4) or (9, 7), (iii) m ¼ 5 and (d0, a)¼ (2, 2), (6, 4) or (13, 8), (iv) m ¼ 6 and (d0, a)¼ (3, 2). Moreover, if L is ample, then jDj is a base point free pencil, L D is also base point free and h 1 (L D) ¼ h 1 (L 2D) ¼ 0. Proof. Since D.L ¼ 3, we have Cliff L 1 by Proposition 5.1. To prove the two first statements, it suffices to show (by classical results on line bundles on K3 surfaces such as in Saint-Donat (1974) and by Proposition 5.1) that there is no smooth curve E satisfying E 2 ¼ 0 and E.L ¼ 1, 2. Since E is base point free, being a smooth curve of non-negative selfintersection (see Saint-Donat, 1974), we must have E.D 0. If E.D ¼ 0, then the divisor B :¼ 3E (E.L)D satisfies B 2 ¼ 0 and B.L ¼ 0, whence by the Hodge index theorem we have 3E (E.L)D, contradicting that D is part of a basis of Pic S. IfE.D 2, the Hodge index theorem gives the contradiction 32 2L 2 ðE:DÞ ¼L 2 ðE þ DÞ 2 ðL:ðE þ DÞÞ 2 We now treat the case E.D ¼ 1. If E.L ¼ 1, we get 16 2L 2 ¼ L 2 ðE þ DÞ 2 ðL:ðE þ DÞÞ 2 ¼ 16; whence by the Hodge index theorem L 2(E þ D), which is impossible, since L is part of a basis of Pic S. So we have E.L ¼ 2. Write E xL þ yD þ zG. From E.L ¼ 2andE.D ¼ 1, we get x ¼ 1 3 a 2ma 3d0 z and y ¼ z 3 9 2ðm 3Þ : 9 25: ð7Þ
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Rational Curves in Calabi-Yau Threefolds

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