Clemens' conjecture

A dimension-counting heuristic

Nearly thirty years ago, Herbert Clemens [Cl0] made the following speculation:

On a general complex quintic threefold, there are only finitely many smooth rational curves of fixed degree d ≥ 1.

The following naïve dimension count, recorded by Katz [Ka], justifies (a belief in) the conjecture. Note that hypersurfaces of any given degree \(m\) determine a projective space \(\mathbb{P}^N\), while rational curves embedded in a target \(\mathbb{P}^n\) are defined by \((n+1)\)-tuples of polynomials in the homogeneous coordinate variables of \(\mathbb{P}^1\). For each positive integer \(d\), there is an incidence scheme \(\Phi_d\) that parameterizes rational curves of degree \(d\) embedded in hypersurfaces of degree \(m\).

Now assume \(n=4\) and \(m=5\). Five-tuples of homogeneous degree-\(d\) polynomials in 2 variables give \(5d+5\) coefficients in total, and quotienting by automorphisms of \(\mathbb{P}^1\) and rescaling yields a \((5d+1)\)-dimensional smooth and irreducible space. Fix a choice of smooth rational curve \(C\) in \(\mathbb{P}^4\). Its preimage in \(\Phi_d\) is the projectivization of the kernel of the natural restriction of global sections \[ r: H^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(5)) \rightarrow H^0(C, \mathcal{O}_C(5)). \]

The dimension of the domain of \(r\) is the binomial coefficient \(\binom{5+4}{5}= 126\), while the dimension of its target is computed by Riemann–Roch to be \(5d+1\). In other words, we expect \(C\) to impose \(5d+1\) conditions on quintic hypersurfaces, which comprise a \(\mathbb{P}^{125}\). Since \(C\) itself varies in a \((5d+1)\)-dimensional family, we thus expect \(\Phi_d\) to be 125-dimensional. So it seems plausible that the projection of \(\Phi_d\) onto the space of quintics should be generically finite.

Remarks.

• The flaw in the heuristic argument is that it implicitly assumes that the restriction \(r\) of global sections is surjective, which isn't the case in general.

• Strictly speaking, a general hypersurface as above belongs to the complement of a countable union of proper subvarieties of the projective space \(\mathbb{P}^N\) of hypersurfaces, the union being indexed by the degrees \(d\) of rational curves. Elsewhere in the literature, such a hypersurface might be called very general.

Clemens' conjecture and mirror symmetry

Quintic hypersurfaces in \(\mathbb{P}^4\) give particularly natural examples of three-dimensional Calabi–Yau manifolds. The canonical bundle of any Calabi–Yau \(X\) is trivial; accordingly, the adjunction formula implies that the normal bundle \(\mathcal{N}_{C/X}\) of any curve \(C\) embedded in \(X\) is of degree \(-2\). So for a generic choice of \(C\), one would expect that the normal bundle splits as \(\mathcal{N}_{C/X}= (\mathcal{O}_C(-1))^{\oplus 2}\), which in turn would imply that \(C\) is a rigid subvariety of \(X\) in the sense of deformation theory. Indeed, Clemens used an explicit deformation-theoretic argument [Cl-1] to prove the existence of such rational curves of degree \(d \geq 1\) on a general quintic threefold \(X\) for infinitely many values of \(d\); subsequently, Katz [Ka] used an existence result of Mori's for curves on K3 surfaces to extend Clemens' argument to every positive value of \(d\). The normal bundle splitting is part of what is commonly called the (the strong form of) Clemens' conjecture for the quintic:

Conjecture 2.1.

Let \(X \subset \mathbb{P}^4\) denote a general quintic threefold. For every positive integer \(d \geq 1\), the following statements hold.

• There is a finite, positive number of irreducible rational curves \(C \subset X\) of degree \(d\).
• These curves are all disjoint and reduced.
• The only singular irreducible rational curves are 17,601,000 six-nodal plane quintics.
• The normalization \(f: \mathbb{P}^1 \rightarrow X\) of any rational curve \(C\) on \(X\) has normal bundle \(\mathcal{N}_f=\mathcal{O}(-1)^{\oplus 2}\).

In one of the first stunning enumerative applications of mirror symmetry, Candelas, de la Ossa, Green and Parkes [CdGP] computed genus-zero Gromov–Witten invariants \(n_d\) — the so-called instanton numbers — for the general quintic on the basis of physical considerations. The last twenty years have seen an increasingly-refined development of "virtual" techniques for cumputing Gromov–Witten invariants, notably involving \(T\)-equivariant localization. In the case of the quintic threefold, the instanton numbers were confirmed mathematically by Givental [G] and Lian–Liu–Yau [LLY]. However, their enumerative significance relies upon Clemens' conjecture, which remains unproved except when \(d \leq 11\) (see the next section).

Remarks.

• A weakened form of Clemens' conjecture is obtained by dropping the stipulation that every curve of degree \(d \neq 5\) be smooth.

• Vainsencher [Va] (see also [KlPi] for an argument with more details) proved the existence of 17,601,000 six-nodal plane quintics on \(F\), each of which is the intersection of \(F\) with a sixfold tangent plane. A proof that the corresponding normal bundles split as \(\mathcal{O}(-1)^{\oplus 2}\) appears in [CK], Sec. 9.2.2.

• The hypothesis that \(X\) be general is important. For an explicit study of an interesting family of special quintics containing lines, see [Mu].

• One might still speculate that the finiteness statement for irreducible rational curves (viewed as morphisms from \(\mathbb{P}^1 \rightarrow X\)) holds for a broader class of Calabi–Yau threefolds, for example those with Picard rank 1. Voisin [Vo3], Rmk.3.24, however, shows that the latter speculation is false for double covers of \(\mathbb{P}^3\) ramified along an octic surface. Indeed, the preimages in the double cover of lines that intersect the octic with multiplicities \((2,2,2,1,1)\) determine a one-parameter family of nodal curves. (Note that the incidence type is misprinted in [Vo3].)

• In the same paper, Voisin shows that Clemens' conjecture contradicts the following conjecture of Lang [Vo3], Conj. 2.13: Every variety not of general type is covered by the (union of) images of non-constant rational maps from abelian varieties. In the case of the quintic, Lang's conjecture predicts that a general quintic is swept out by elliptic curves. Note that the double covers of the preceding remark satisfy Lang's conjecture [Vo3], Rmk. 2.17: explicitly, they are swept out by the elliptic normalizations of preimages in the double cover of lines that intersect the octic with multiplicities \((2,2,1,1,1,1)\).

Rational curves of low degree on a general quintic 3-fold

Clemens' conjecture for rational curves of degree \(d \leq 7\). His method of proof was in two steps. First he used deformation theory to show that

Clemens' conjecture holds for smooth rational curves \(C\) of degree \(d\) provided the incidence scheme \(\Phi_d\) is irreducible.

He then proved that \(\Phi_d\) is irreducible for all \(d \leq 7\), by arguing that the fibers of its projection onto the space \(M_d\) of degree-\(d\) rational curves are equidimensional projective spaces. Equidimensionality follows from an effective cohomological vanishing result for ideal sheaf cohomology due to Gruson, Lazarsfeld, and Peskine [GLP].

Subsequently, Johnsen and Kleiman [JK1] extended Katz's analysis to show that the only reduced connected curves of degree \(d\) at most 9 on \(F\) with rational components are irreducible and either smooth or six-nodal plane quintics. Their basic insight is to stratify \(M_d\) by locally-closed subsets \(M_{d,i}\), where \(i:=h^1(\mathcal{I}_{C/\mathbb{P}^4}(5))\) and \(C\) is the image of \(f\). Pulling back by \(\pi_d:\Phi_d \rightarrow M_d\), they obtain a corresponding stratification of the incidence scheme \(\Phi_d\) into loci \(\Phi_{d,i}\), and they show that the projection \(\Phi_d \rightarrow M_d\) is dominant exactly over \(M_{d,0}\), where all the fibers are equidimensional. To do so they use [GLP], as well as codimension estimates for rational curves with singularities, to handle curves with positive arithmetic genus.

The next case, that of rational curves of degree 10, is interesting in its relation to mirror symmetry. In fact, Pandharipande showed [CK], Sec. 9.2.3 that the finiteness of the Hilbert scheme of degree-10 rational curves on the general quintic implies that the instanton number \(n_{10}\) is given by \[ n_{10}= 6 \times 17,601,000 +\#\{\text{smooth rational curves of degree 10 in }F\}. \]

On the other hand, mirror symmetry [CdGP] includes Vainsencher's singular quintics in its count of rational curves of degree five, but fails to count six double covers corresponding to each of these. So 10 is the first degree \(d\) for which the instanton number \(n_d\) fails to count smooth rational curves of degree \(d\) on the general quintic \(F\); moreover, the discrepancy between \(n_{10}\) and the actual number of smooth rational curves is explained by double covers of nodal plane quintics. More generally, enumerative contributions arising from multiple covers of 6-nodal plane quintics mean that the relationship between instanton numbers \(n_d\) and counts of rational curves of degree \(d\) is subtle whenever \(d\) is divisible by 5.

Recently, Cotterill [Co1,Co2] has proved Clemens' conjecture in its strong form for rational curves of degree at most 11. He extends the stratification-based analysis of Johnsen and Kleiman by obtaining new bounds on ideal sheaf cohomology obtained from a combinatorial analysis of monomial ideas associated to Groebner degenerations of embedded curves. He also uses the stratification of rational curves according to the splitting types of their restricted tangent bundles described in \cite{Ve}. The upshot is that the incidence scheme \(\Phi_d\) of rational curves in quintics is irreducible whenever \(d \leq 11\).

References

• [CdGP] P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory, Nuclear Phys. B359 (1991), 21–74.

• [Cl-1] H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated, Publ. Math. IHES 58 (1983), 19–38.

• [Cl0] H. Clemens, Curves on higher-dimensional complex projective manifolds, Proc. International Cong. Math., Berkeley, 1986, 634–640.

• [Cl1] H. Clemens, Curves on generic hypersurfaces, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 629–636.

• [Co1] E. Cotterill, Rational curves of degree 10 on a general quintic threefold, Comm. Alg. 33 (2005), no. 6, 1833-1872.

• [Co2] E. Cotterill, Rational curves of degree 11 on a general quintic 3-fold, Quart. J. Math. 63 (2012), no. 3, 539–568.

• [CK] D. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS, 1999.

• [G] A. Givental, The mirror formula for quintic threefolds, in ``Northern California Symplectic Geometry Seminar", 49–62, Amer. Math. Soc. Transl. Ser. 2 196, AMS, Providence, RI, 1999.

• [GLP] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491–506.

• [HRS] J. Harris, M. Roth, and J. Starr, Rational curves on hypersurfaces of low degree, J. reine angew. Math. 571 (2004), 73–106.

• [JK1] T. Johnsen and S. Kleiman, Rational curves of degree at most 9 on a general quintic threefold, Comm. Alg. 24 (1996), 2721–2753.

• [JK2] T. Johnsen and S. Kleiman, Towards Clemens' conjecture in degrees between 10 and 24, Serdica Math. J. 23 (1997), 131–142.

• [Ka] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151–162.

• [Ka1] S. Katz, Degenerations of quintic threefolds and their lines, Duke Math. J. 50 (1983), 1127–1135.

• [Ka2] S. Katz, Lines on complete intersection threefolds with \(K=0\), Math. Z. 191 (1986), 293–296.

• [Ka3] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151–162.

• [KlPi] S. Kleiman and R. Piene, Node polynomials for families: methods and applications, Math. Nachr. 271 (2004), 69-90.

• [LLY] B. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), 729–763.

• [Mu] A. Mustata, Degree 1 curves in the Dwork pencil and the mirror quintic, arXiv preprint \url{math.AG/0311252.pdf}.

• [Va] I. Vainsencher, Enumeration of \(n\)-fold tangent hyperplanes to a surface, J. Alg. Geom. 4 (1995), 503–526.

• [Ve] J.L. Verdier, Two-dimensional sigma-models and harmonic maps from \(S^2\) to \(S^n\), in "Group Theoretical Methods in Physics", Springer Lecture Notes in Physics 180 (1983), 136–41.

• [Vo1] C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom. 44 (1996), 200–214.

• [Vo2] C. Voisin, A correction to "A conjecture of Clemens on rational curves on hypersurfaces", J. Diff. Geom. 49 (1998), 601-611.

• [Vo3] C. Voisin, On some problems of Kobayashi and Lang; algebraic approaches, in "Current developments in mathematics, 2003", 53–125, Int. Press, Somerville, MA, 2003.