N ov 2 00 5 CLEMENS ’ 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 out attention to general quintic threefolds. In a universal quintic threefold X, we construct a family of quasi-regular deformation 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 [WI], we study the deformations of rational curves in quintic threefolds. The main goal is to introduce new geometric objects that associated to the family of rational curves in quintic threefolds: degenerated locus (definition (2.2)), space of deformations with marked points (after definition (2.1)). Finally we prove Clemens’ conjecture 1.1 Theorem. For each d > 0, there is no one parameter family cf ( for a small complex number ǫ) 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-deformation Bb of the rational curve cf in f that has a deviated member. Because in [WI], we have proved such a family Bb is an obstruction to the existence of the deformation of c 0 f in f . In the following we briefly review the definitions of the quasi-regular deformation Bb, 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 fǫ be a family of smooth Calabi-Yau threefolds. Let ∆ be an open set of C that contains 0. Let π: X π −−−−→ ∆ be a smooth morphism such that for each ǫ ∈ T , 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 π −−−−→ ∆ is also smooth and for each ǫ, π(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). Let S = ∪ǫ,sc s ǫ be a restricted total set of two parameter family of rational curves to a tubular neighborhood of c0. Let H be a generic divisor of X such that in a tubular neighborhood of c0, H ∩ S is the surface C at generic points of c0. Let q1, · · · , qk be the points on c0 such that π2∗(( ∂ ∂s , 0))(which is a section of the subbundle of Tc0X that corresponds to Oc0(k) portion in (1.2)) at these points, lies in TC. In the following we give a definition of a family Bb of quasi-regular deformations of c0. The definition describes the local behavior of Bb at all four types of points above and qi, which are all the points pi in the overview. It is a tedious definition because of large number of points pi. But over all they all come from the intersections Bb ∩ S and Bb ∩ C. For the intersection Bb ∩ C, we use Bb ∩H where H is GENERIC. Definition 1.3. Let B be a quasi-projective variety. Let