Rational Curves in Calabi-Yau Threefolds

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3932 Johnsen and Knutsen The following is a summary of the results obtained in Johnsen and Knutsen (2001) that we will need in the following. Since we only need those results for ample L, we restrict to this case and refer the reader to Johnsen and Knutsen (2001) for the results when L is only assumed to be base point free. Proposition 5.1. Let L be an ample line bundle of sectional genus g 4on g 1 a K3 surface S and let c :¼ Cliff L. Assume that c inimal integer k 0 such that there is a line bundle D on S satisfying the numerical conditions: ðiÞ 2 2D L:D ¼ D 2 þ k þ 2 ðiiÞ 2k þ 4 with equality in (i) or (ii) if and only if L 2D and L 2 ¼ 4k þ 8. (In particular, D 2 c þ 2, with equality if and only if L 2D and L 2 ¼ 4c þ 8, and by the Hodge index theorem D 2 L 2 ðL:DÞ 2 ¼ðD 2 þ c þ 2Þ 2 :Þ Moreover, any such D satisfies (with M :¼ L D and R :¼ L 2D): (i) D.M ¼ c þ 2. (ii) D.L M.L (equivalently D 2 M 2 ). (iii) h 1 (D) ¼ h 1 (M) ¼ 0. (iv) jDj and jMj are base point free and their generic members are smooth curves. (v) h 1 (R) ¼ 0, R 2 4, and h 0 (R) > 0 if and only if R 2 2. (vi) If R R1 þ R2 is a nontrivial effective decomposition, then R1.R2 > 0. Proof. The first statement is Knutsen (2001a, Lemma 8.3). The properties (i)–(iv) are the properties (C1)–(C5) in Johnsen and Knutsen (2001, p. 9–10), under the additional condition that L is ample. The fact that h 1 (R) ¼ 0 in (v) follows from Johnsen and Knutsen (2001, Prop. 5.5) (where D ¼ 0 since L is ample), and the rest of (v) is then an immediate consequence of Riemann-Roch. Finally, (vi) follows from Johnsen and Knutsen (2001, Prop. 6.6) since L is ample. & Now denote by fL the morphism f L : S !P g
Rational Curves in Calabi-Yau Threefolds 3933 defined by jLj and pick a subpencil fDlg jDj ’P 1 2D2þ1 generated by two smooth curves (so that, in particular, fDlg is without fixed components, and with exactly D 2 base points). Each fL(Dl) will span a (h 0 (L) h 0 (L D) 1)-dimensional subspace of P g , which is called the linear span of fL(Dl) and denoted by Dl. Note that Dl ¼ P cþ1þ1 2D2 . The variety swept out by these linear spaces, T ¼ [ Dl P g ; l2P 1 is a rational normal scroll (see Schreyer, 1986) of type (e1, ..., ed), where with ei ¼ #fjjdj ig 1; ð1Þ d ¼ d0 :¼ h 0 ðLÞ h 0 ðL DÞ; d1 :¼ h 0 ðL DÞ h 0 ðL 2DÞ; . . di :¼ h 0 ðL iDÞ h 0 ðL ði þ 1ÞDÞ; . Furthermore, T has dimension dim T ¼ d0 ¼ h 0 ðLÞ h 0 ðFÞ ¼cþ 2 þ 1 2 D2 and degree deg T ¼ h 0 ðFÞ ¼g c 1 1 2 D 2 . We will need the following. Lemma 5.2. Assume that L is ample, D 2 ¼ 0, and h 1 (L iD) ¼ 0 for all i 0 such that L iD 0. Then the scroll T defined by jDj as described above is smooth and of maximally balanced scroll type. Furthermore, dim T ¼ c þ 2 and deg T ¼ g c 1. Proof. Let r :¼ maxfijL iD 0g. Then by Riemann-Roch, and our hypothesis that h 1 (L finds r ¼b iD) ¼ 0 for all i 0 such that L iD 0, one easily g cþ2c and d0 ¼ ¼dr 1 ¼ L:D ¼ c þ 2; 1 dr ¼ g þ 1 ðc þ 2Þr c þ 2; di ¼ 0 for i r þ 1; whence the scroll T is smooth and of maximally balanced scroll type. The assertions about its dimension and degree are immediate. &
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