Clemens’s Conjecture: Part Ii

This is the part II of our series of two papers, “Clemens conjecture: part I”, “Clemens conjecture: part II”. Continuing from part I, in this paper we turn our attention to general quintic threefolds. In a universal quintic threefold X, we construct a family of quasi-regular deformations Bb such that the generic member in this family is non-deviated, but some special member is deviated. By the result from part I, this is impossible unless there is no one parameter family of smooth rational curves in a generic quintic threefold. 1. Main result and review of part I In this paper which is the continuation of [Wa], we study the deformations of rational curves in quintic threefolds. The main goal is to introduce new geometric objects associated to the family of rational curves in quintic threefolds: degenerated locus (definition (2.2)), space of morphisms with marked points (after definition (2.1)). Finally we prove Clemens conjecture: Theorem 1.1. For each d > 0, there is no one parameter family csf ( for a small complex number s) of smooth rational curves of degree d in a generic quintic threefold f . The proof is based on a construction of a family of quasi-deformations Bb of the rational curve cf in f that has both deviated and non-deviated members. Because in [Wa], we have proved such a family {Bb} is an obstruction to the existence of a deformation of cf in f . In the following we briefly review the definition of the quasi-regular deformation Bb, and a theorem about it. Research partially supported by NSF grant DMS-0070409 Typeset by AMS-TEX 1 2 BIN WANG OCT, 2005 The following is the set-up. Let X be a smooth variety. Let ∆ be an open set of C that contains 0. Let π be a smooth morphism X π −−−−→ ∆ such that for each ǫ ∈ ∆, π(ǫ), denoted by fǫ, is a smooth Calabi-Yau threefold, i.e. c1(T (fǫ)) = 0. Assume there is a surface C ⊂ X such that the restriction map C πC −−−−→ ∆ is also smooth and for each ǫ, (πC) (ǫ), denoted by cǫ, is a smooth rational curve. Furthermore we assume the normal bundle of cǫ in fǫ has the following splitting Ncǫ(fǫ) = Ocǫ(k)⊕Ocǫ(−2− k), where k ≥ 0. Hence NC(X)|cǫ is also equal to (1.2) Ocǫ(k)⊕Ocǫ(−2− k). Recall that R ⊂ ∆ ×X is the universal curves of the cǫ . So we have correspondence R π2 −−−−→ X