Rational Curves in Calabi-Yau Threefolds

More documents

Recommendations

Info

3948 Johnsen and Knutsen where deg qi(t, u) ¼ ei e4, for i ¼ 1, 2, 3, and A ¼ X ri; jðt; uÞZiZj; i; j where deg ri, j(t, u) ¼ ei þ ej (g 4 e4). For ‘‘small’’ values of s ¼ u v Eq. (33) cuts out a 3-dimensional subscroll of T, while Eq. (32) cuts out a ‘‘deformed’’ K3 surface within this subscroll. For s ¼ 0 we get our well-known situation with S and T in T. We may insert Z4 ¼ sB, obtained from Eq. (33) in Eq. (32), and then we get: Q þ sðBAðt; u; Z1; Z2; Z3; sLÞþQ1ðt; u; Z1; Z2; Z3; sBÞÞ ¼ 0: By choosing Q 1, A, B in a convenient way, we may express any Q þ sQ 0 in this way, for all Q 0 of the same form as Q (We can choose Q1 not to involve Z4 if we like). Hence we can obtain all possible deformations of the equation Q this way, that is we can obtain all possible deformations as sections of 3HT (g 4)FT on T (We move T too, but in a familiy of isomorphic rational normal scrolls, and t, u, Z1, Z2, Z3 are coordinates for all these scrolls, simultaneously). By choosing Q1 (and B and A if we like) in a convenient way, we then deform the K3 surface to one with Picard lattice generated by a pair of generators L i and D i only, all with the same intersection matrix. This is true since a zero scheme of a general section of 3HT (g 4)FT on T gives a general member of an 18-dimensional family of polarized K surfaces, all having Picard lattice generated by such Li and Di. 6.3. Proof of Step (III) If we eliminate (u, v) from the two equations above, and thus form the union of all the deformed surfaces, for varying (u, v) we obtain a threefold V with equation: ðQ þ Z4AÞB þ Z4Q1 ¼ 0: By, for example, studying the description on p. 3 in Stevens (2000), one sees that V is the zero scheme of a section of KT ¼ 4H (N 5)F, where N ¼ e1 þ þe4 þ 3 ¼ g þ e4 þ 1 is the dimension of the projective space spanned by T. Moreover, one argues as in Ekedahl et al. (1999) that for a general choice of A, B, Q1 the threefold V is only singular at the finitely many points given by Q ¼ Q 1 ¼ Z 4 ¼ B ¼ 0, and that none of
Rational Curves in Calabi-Yau Threefolds 3949 these points is contained in G. The number of points is: ð3H ðg 4ÞFÞ 2 ðH e4FÞ 2 ¼ 9H 4 ¼ 9ðg 3Þ ð2g þ 2e4 8Þ ¼7g 19 2e4 ð2g 8 2e4ÞH 3 F in the numerical ring of T. The essential result, inspired from Ekedahl et al. (1999), is the following: Lemma 6.3. Let V be a rational normal scroll, and let S ¼ Z(Q0, L0) \ V be a smooth codimension 2 subvariety of V, where Q0, and L0 are effective divisors on V, which are sections of base point free line bundles of type aH þ bF on V. Let L be the linear system fQ0L1 Q1L0g on V, where L1 varies through all elements of jL0j and Q1 varies through all elements of jQ0j. Then for a general element l of L, Sing(l) ¼ Z(Q0, Q1, L0, L1). In general, Z(Q0, Q1, L0, L1) \ V will be of numerical type Q2 0L2 0 on V. Proof. A straightforward generalization of the proof of Lemma 5.3 of Ekedahl et al. (1999). In that lemma one expresses the elements of Q0, and L 0 as hypersurfaces in the projective space where V sits. In our case this is true locally each point. Since the proof is local, it works also here. & Remark 6.4. It is also clear that Z(Q0, Q1, L0, L1) \ V ¼ Z(Q1, L1) \ S will intersect C in an empty set for general Q1, L1, since Q0, andL0 are base point free divisors. Hence Z(l) will contain C and be non-singular at all points of C for general Q1, L1. Just as in the proofs of Lemmas 2.4 and 2.7 of Oguiso (1994), or in Theorem 4.3 and part 5.1 of Ekedahl et al. (1999), we see that for an arbitrary such deformation of Q the curve G is isolated in V. The essential argument is already given on pp. 22–23 in Clemens (1983). 6.4. Proof of Step (IV) This follows as on pp. 25–26 in Clemens (1983), or as in the proof of Theorem 3.4 of Ekedahl et al. (1999). Let M ¼ Md,a ¼f[C] j C has bidegree(d, a)g, and let G be the parameter space of ‘‘hypersurfaces’’ of type 4H (N 5)F in T, where N ¼ e 1 þ þe 4 þ 3 is the dimension of the
Page 1 and 2: COMMUNICATIONS IN ALGEBRA Õ Vol. 3
Page 3 and 4: Rational Curves in Calabi-Yau Three
Page 31: Rational Curves in Calabi-Yau Three
Page 37: Rational Curves in Calabi-Yau Three

Rational Curves in Calabi-Yau Threefolds

Create successful ePaper yourself

Delete template?

Save as template?