6 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 some 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 Ir (definition (2.2)), spaces of morphisms with marked points Σn(t), Σm(t) (after definition (2.1)). Using those basic objects, we construct the family of surfaces Bb and finally prove Clemens’s conjecture: Theorem 1.1. For each d > 0, there is no one parameter family cf ( 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 . So the only goal in this paper is to construct such a family {Bb} that satisfies all requirements in part I, definitions (1.2) and (1.5). Research partially supported by NSF grant DMS-0070409 Typeset by AMS-TEX 1 2 BIN WANG OCT, 2005 For reader’s convenience, in the following we briefly review those definitions and a theorem about them. 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). Assume that there exists a deformation cǫ of cǫ in each fǫ with c 0 ǫ = cǫ. Recall that R ⊂ ∆ ×X is the universal curves of the cǫ . So we have correspondence R π2 −−−−→ X