Rational Curves in Calabi-Yau Threefolds

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3950 Johnsen and Knutsen projective space P N spanned by T. Study the incidence I ¼ Id, a ¼f([C], [F]) 2 M G j C Fg. Then one easily shows: Lemma 6.5. Every component of I has dimension at least dim G, and dim G ¼ 104 if 4e 4 (N 5) 2. Proof. Let t, u, Z1, ..., Z4 as usual be coordinates of T. Let r, s be homogeneous coordinates of the C ¼ P 1 which is mapped into T as a rational curve of bidegree (d, a). This map corresponds to some (not uniquely defined) parametrization t ¼ Tðr; sÞ; u ¼ Uðr; sÞ; Z1 ¼ S1ðr; sÞ; ...; Z4 ¼ S4ðr; sÞ: Here deg T ¼ deg U ¼ a, and deg Si ¼ d aei, for i ¼ 1, 2, 3, 4. (A set of coordinates for P n are of the form Z j i ¼ Zit j u ei j . Each coordinate is then of degree d in the variables r, s. See for example Stevens (2000, p. 3), or Reid (1997). The 6-tuples ðTðr; sÞ; Uðr; sÞ; S1ðr; sÞ; ...; S4ðr; sÞÞ depend on 2(a þ 1) þ (d ae1 þ 1) þ þ(d ae4 þ 1) ¼ 4d þ a(5 N) þ 6 variables. Let N ¼ Nd, a be the set of such 6-tuples. (An open subset of) N can be viewed as a parameter space for ‘‘Parametrized rational curves of bidegree (d, a) inT’’. The parameter space M can be viewed as a quotient of N. Likewise J ¼ J d, a is defined as the corresponding incidence in N G, and can be viewed as a quotient of I. The fibres of these quotients have dimension 5 ¼ dim PGL(2) þ 2 (We have two multiplicative factors, one for (T, U), and one for (S1, ..., S4)). Hence dim M ¼ 4d þ a(5 N) þ 1, and dim I ¼ 4d þ a(5 N) þ 1 þ dim G ( ¼ 4d þ a(5 N) þ 105 if T is of reasonably well balanced type). Study the incidence R ¼f(P, [F ]) 2 T GjP 2 F g. Then R has an equation, which is setting equal to zero a sum of monomials of type: pi1;...;i4ðt; uÞZi1 1 Z i4 4 : The dim G þ 1 coefficients of the p i1, ..., i4 can be viewed as homogeneous coordinates of G. In this equation we now insert the parametrizations T(r, s), U(r, s), S1(r, s), ..., S4(r, s). This gives an equation involving the coordinates of G and the dim N coefficients of these 6-tuples, and in addition r, s. We may view this as a homogeneous polynomial of degree 4d þ a(5 N) inr, s, since deg pi 1, ..., i4(t, u) ¼ i1e1 þ þi4e4 (N 5). The equation of the incidence J in N G is obtained by setting all the 4d þ a(5 N) þ 1 coefficients of this polynomial equal to zero. Since this
Rational Curves in Calabi-Yau Threefolds 3951 number of coefficients is equal to dim M, we get dim M equations in an ambient space N G of dimension dim M þ dim G þ 6. Hence all components of J have dimension at least dim G þ 6, and consequently all components of I have dimension at least dim G. Now we simply differentiate the second projection map p 2 : I ! G. If the kernel of the tangent space map is zero at a point of I, then the tangent map is injective, and therefore surjective, since dim I dim G at all points of I. Now this kernel is zero at ([G], [V ]), since h 0 (NG=V) ¼ 0, since G is isolated in V. Hence the tangent map is injective and surjective in an open neighborhood of ([G], [V ]) on a component of I of dimension dim G, and the conclusion about the existence of an isolated rational curve of degree d holds for a general section of 4H (N 5)F. & At this point the proof of Theorem 4.3 is complete. ACKNOWLEDGMENTS We thank the referee for helpful remarks, and the organizers of the Kleiman’s 60th Birthday Conference for making this volume possible. The second author was supported by a grant from the Research Council of Norway. REFERENCES Arbarello, E., Cornalba, M., Griffiths, P. A., Harris, J. (1985). Geometry of algebraic curves. In: Grundlehren der Mathematischen Wissenschaften. Vol. I. Berlin, Heidelberg, New York, Tokyo: Springer Verlag, 267. Batyrev, V., Ciocan-Fontanine, I., Kim, B., van Straten, D. (1998). Conifold Transitions and Mirror Symmetry for Calabi-Yau Intersections in Grassmannians. Nuclear Phys. B 514(3):640–666. Clemens, H. (1983). Homological equivalence, modulo algebraic equivalence, is not finitely generated. Publ. Math. IHES 58:19–38. Ekedahl, T., Johnsen, T., Sommervoll, D. E. (1999). Isolated rational curves on K3-fibered Calabi-Yau threefolds. Manuscripta Math. 99:111–133. Green, M., Lazarsfeld, R. (1987). Special divisors on curves on a K3 surface. Invent. Math. 89:357–370.
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