Rational Curves in Calabi-Yau Threefolds

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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.
Rational Curves in Calabi-Yau Threefolds 3931 5.1. Some General Facts About K3 Surfaces Recall that a K3 surface is a (reduced and irreducible) surface S with trivial canonical bundle and such that H 1 (OS) ¼ 0. In particular h 2 (OS) ¼ 1 and w(OS) ¼ 2. We will use line bundles and divisors on a K3 surface with little or no distinction, as well as the multiplicative and additive notation, and denote linear equivalence of divisors by . Before continuing, we briefly recall some useful facts and some of the main results in Johnsen and Knutsen (2001) which will be used in the proof of Theorem 4.3. Let C be a smooth irreducible curve of genus g 2 and A a line bundle on C. The Clifford index of A (introduced by Martens (1968) is the integer Cliff A ¼ deg A 2ðh 0 ðAÞ 1Þ: If g 4, then the Clifford index of C itself is defined as Cliff C ¼ minfCliff Ajh 0 ðAÞ 2; h 1 ðAÞ 2g: Clifford’s theorem then states that Cliff C 0 with equality if and only if C is hyperelliptic and Cliff C ¼ 1 if and only if C is trigonal or a smooth plane quintic. At the other extreme, we obtain from Brill-Noether theory (cf. g 1 Arbarello et al., 1985, Chapter V) that Cliff C b c. For the general curve of genus g, we have Cliff C ¼b g 1 2 c. We say that a line bundle A on C contributes to the Clifford index of C if h 0 (A), h 1 (A) 2 and that it computes the Clifford index of C if in addition Cliff C ¼ Cliff A. Note that Cliff A ¼ Cliff oC A 1 . It was shown by Green and Lazarsfeld (1987) that the Clifford index is constant for all smooth curves in a complete linear system jLj on a K3 surface. Moreover, they also showed that if Cliff C ine bundle M on S such that MC :¼ M OC computes the Clifford index of C for all smooth irreducible C 2jLj. This was investigated further in Johnsen and Knutsen (2001), where we defined the Clifford index of a base point free line bundle L on a K3 surface to be the Clifford index of all the smooth curves in jLj and denoted it by Cliff L. Similarly, if (S, L) is a polarized K3 surface, we defined the Clifford index of S, denoted by CliffL(S) tobeCliffL.
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Rational Curves in Calabi-Yau Threefolds

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