Prym-Tjurin Constructions on Cubic Hypersurfaces

Mingmin Shen

Received: May 13, 2012 Revised: January 14, 2014

Communicated by Gavril Farkas

Abstract. In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups of a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi mapping induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients. All the constructions are based on a natural relation among topological (resp. algebraic) cycles on X modulo homological (resp. rational) equivalence.

2010 Mathematics Subject Classification: 14F25, 14C25 Keywords and Phrases: Hodge structure, Chow group, incidence correspondence

1. Introduction

Algebraic cycles on a cubic hypersurface have been serving as an interesting but nontrivial example in the study of Chow groups. In [Sh], we obtained natural relations among 1-cycles and gave some applications of those relations. This paper is a continuation of [Sh] and generalization of the results to higher dimensional cycles on cubic hypersurfaces. Throughout the paper, we will work over the field ℂ of complex numbers. In this article, all homology and cohomology groups are modulo torsion. Let X ⊂ ℙ^{n+1} be a smooth cubic hypersurface of dimension n ≥ 3. We first establish Theorem 2.2 which gives a natural relation among cycles on X. To get an idea of what such a relation is, we first fix a smooth curve C ⊂ X of degree e. Let T ⊂ X be a closed subvariety of dimension r. Let T′ ⊂ X be the subvariety swept out by lines on X that meet both C and T. When C and T are in general position, then T′ has expected dimension r. If r ≥ 2, then the relation we get is

2e T + T′ = a h^{n−r}, in CH_{r}(X)

for some integer a > 0, where h is the class of a hyperplane section of X. If r = 1, the relation we get is

2e T + T′ + 2 deg(T)C = b h^{n−1}, in CH_{1}(X)

for some integer b > 0. The second case was first proved in [Sh]. Let F be the Fano scheme of lines on X. Let p : P (X) → F be the total family of lines on X and q : P (X) → X be the natural morphism. Then we have the natural cylinder homomorphism Ψ = q_{∗}p^{∗} and its transpose, the Abel-Jacobi homomorphism, Φ = p_{∗}q^{∗}. These are defined on the cohomology groups and the Chow groups. The following theorem is a more concise expression of the above relations; see Corollary 2.4.

Theorem 1.1. Let γ ∈ H_{n}(X, ℤ), 𝔞 ∈ CH_{1}(X) and 𝔟 ∈ CH_{r}(X). We use h to denote the class of a hyperplane in either the Chow group or the cohomology group of X. Then the following are true.

(1) 2 deg(𝔞)γ + Ψ(Φ(γ) · Φ([𝔞])) = 3 deg(γ) deg(𝔞)h^{\frac{n}{2}} in H_{n}(X, ℤ), where the right hand side is 0 if n is odd.

(2) If n − 2 ≥ r ≥ 2, then 2 deg(𝔞)𝔟 + Ψ(Φ(𝔞) · Φ(𝔟)) = 3 deg(𝔞) deg(𝔟)h^{n−r} in CH_{r}(X).

(3) If r = 1, then 2 deg(𝔞)𝔟 + 2 deg(𝔟)𝔞 + Ψ(Φ(𝔞) · Φ(𝔟)) = 3 deg(𝔞) deg(𝔟)h^{n−1} in CH_{1}(X).

Remark 1.2. If n ≥ 5, then CH_{1}(X) ≅ ℤ; see [Pa, Proposition 4.2]. In this case, 𝔞 ∈ CH_{1}(X) is the same as deg(𝔞)[l], where [l] is the class of a line. For fixed n, the groups CH_{r}(X) are expected to be trivial (isomorphic to ℤ) when r is small or large enough. Hence, the above relations are interesting when r is in the middle range.

With these relations, we will realize the Hodge structure and Chow groups of X as Prym-Tjurin constructions. A Prym variety is constructed from a curve together with an involution and they form a larger class of principally polarized abelian varieties (p.p.a.v.) than the Jacobian of curves. Mumford gives this a modern treatment in [Mu]. He also uses Prym varieties to characterize the intermediate Jacobian of a cubic threefold, see the appendix of [CG]. In [Tj], Tjurin generalizes this idea by replacing the involution with a correspondence that satisfies a quadratic equation. This gives what we now call Prym-Tjurin varieties. This was further developed and completed by S. Bloch and J.P. Murre, see [BM]. Welters has proved that all p.p.a.v.’s can be realized as Prym-Tjurin varieties, see [We]. Roughly speaking, this means that every principally polarized weight one Hodge structure can be realized via a Prym-Tjurin construction on some curve. We naturally ask whether similar constructions can be done for higher weight Hodge structures. The work of Lewis in [Lew] sheds some light on this question. Izadi gives a Prym construction for the cohomology of cubic hypersurfaces in [Iz].

Prym-Tjurin Constructions on Cubic Hypersurfaces 869

Definition 1.3. Let Λ be an abelian group and σ : Λ → Λ an endomorphism. Assume that σ satisfies the quadratic equation (σ − 1)(σ + q − 1) = 0. Then the Prym-Tjurin part of Λ is defined as

P (Λ, σ) = Im(σ − 1)

Remark 1.4. In many cases when Λ carries some extra structure such as a Hodge structure, the homomorphism σ is usually compatible with that extra structure and P (Λ, σ) carries an induced such structure. One very interesting case is when Λ = H^{∗}(Y, ℤ) together with the natural Hodge structure and σ is an action that is induced by some correspondence Γ ∈ CH^{dim Y} (Y × Y).

Let X ⊂ ℙ_{ℂ}^{n+1} be a smooth cubic hypersurface as above. We start with a general rational curve C ⊂ X of degree e ≥ 2. Let S_{C} be a natural resolution (in fact the normalization) of the space of lines on X that meet C. Let F = F (X) be the Fano scheme of lines on X, p : P → F the total family with a morphism q : P → X. Then S_{C} = q^{−1}(C). In particular, we have a morphism q_{0} : S_{C} → C. Consider the family p_{C} : P_{C} → S_{C} of lines parameterized by S_{C} with the morphism q_{C} : P_{C} → X. We have the natural cylinder homomorphisms Ψ_{C} = (q_{C})_{∗}(p_{C})^{∗} : H^{n−2}(S_{C}, ℤ) → H^{n}(X, ℤ) and Ψ_{C} = (q_{C})_{∗}(p_{C})^{∗} : CH_{r}(S_{C}) → CH_{r+1}(X). Similarly we have its transpose Φ_{C} = (p_{C})_{∗}(q_{C})^{∗} : H^{n}(X, ℤ) → H^{n−2}(S_{C}, ℤ) and Φ_{C} = (p_{C})_{∗}(q_{C})^{∗} : CH_{r+1}(X) → CH_{r}(S_{C}). On S_{C} we have the incidence correspondence which induces an action σ on the cohomology groups and the Chow groups. Note that there is a natural morphism from S_{C} to the Grassmannian G = G(2, n + 2). The Plücker embedding of G induces a ample class g on S_{C}. On S_{C}, we have another divisor class g′ = q_{0}^{∗}(pt) for some pt ∈ C. Let ℚ[g, g′] ⊂ H^{∗}(S_{C}, ℚ) be the subring generated by g and g′. Given a compact manifold Y, we can define the intersection pairing on H^{∗}(Y) by

⟨α, β⟩ = ∫_{Y} α ∪ β.

By abuse of notation, when α and β are of complementary dimension, we also write α · β for ⟨α, β⟩. With this notion, we define the primitive cohomology (or cycle) classes on S_{C} to be those which are orthogonal to the classes in ℚ[g, g′]. The primitive cohomology is denoted by H^{∗}(S_{C}, ℤ)^{∘} and the primitive Chow group is denoted by CH_{∗}(S_{C})^{∘}. The subring ℚ[g, g′] is invariant under the action of σ and hence σ acts on the primitive cohomology and also the primitive Chow groups. Our main result is the following

Theorem 1.5. Let C ⊂ X be a general rational curve of degree e ≥ 2 as above. Let σ be the action of the incidence correspondence on either the primitive cohomology or the primitive Chow group of S_{C}. Then the following are true.

(1) The action σ satisfies the following quadratic relation

(σ − 1)(σ + 2e − 1) = 0

(2) The map Φ_{C} induces an isomorphism of Hodge structures

Φ_{C} : H^{n}(X, ℤ)_{prim} → P (H^{n−2}(S_{C}, ℤ)^{∘}, σ)(−1) where the (−1) on the right hand side means shifting of degree by (1, 1). The intersection forms are related by the following identity

Φ_{C}(α) · Φ_{C}(β) = −2e α · β, for all α, β ∈ H^{n}(X, ℤ)_{prim}.

(3) The map Φ_{C} induces an isomorphism

Φ_{C} : A_{i}(X)_{ℚ} → P (CH_{i−1}(S_{C})_{ℚ}^{∘}, σ)

This is proved in section 4 (Theorem 4.3). In [Iz], Izadi proved a variant of statement (2) for C being a line. In [Sh], we proved the above theorem for cubic threefolds where (3) holds true for integral coefficients. Besides the natural relations stated at the very beginning of this section, another ingredient to carry out the Prym-Tjurin construction is the surjectivity of Ψ_{C} on primitive cohomologies.

Theorem 1.6. The natural homomorphism

Ψ_{C} : H^{n−2}(S_{C}, ℤ)^{∘} → H^{n}(X, ℤ)_{prim}

between the primitive cohomologies is surjective.

This is proved in Section 5 using the Clemens-Letizia method (see [Cl] and [Le]). Acknowledgement. The author was supported by Simons Foundation as a Simons Postdoctoral Fellow during the completion of this work. Part of this work was completed when the author was visiting Beijing International Center for Mathematical Research. He thanks E. Izadi for explaining her results of [Iz]. Notation: X ⊂ ℙ^{n+1}, smooth cubic hypersurface of dimension n ≥ 3 with h the hyperplane class; G = G(2, n + 2), the Grassmannian of lines in ℙ^{n+1}; F = F (X) ⊂ G, Fano scheme of lines on X, smooth of dimension 2n − 4; see [CG] and also [AK]. l ⊂ X, a line on X; [l] ∈ F the corresponding point on F; P = P (X), the universal family of lines on X, namely we have the following diagram

\xymatrix{P (X) \ar[r]^{q} \ar[d]_{p} & X \\ F}

I ⊂ F × F, the incidence correspondence, i.e. the closure of {([l_{1}], [l_{2}]) : l_{1} ≠ l_{2}, l_{1} ∩ l_{2} ≠ ∅}; Note that I has codimension n − 2 in F × F; Φ = p_{∗}q^{∗}, homomorphism from either the cohomology groups or the Chow groups of X to those of F; Ψ = q_{∗}p^{∗}, homomorphism from either the cohomology groups or the Chow groups of F to those of X; g, the polarization on F that comes from the Plücker embedding, g = Φ(h^{2}); F_{x} ⊂ F, the subscheme of lines passing through x ∈ X; it is a (2,3) complete intersection in ℙ(T_{X,x}) and smooth for general x. D_{x} ⊂ X, the variety swept out by all lines through x; D_{x} is a cone over F_{x}; F_{C} ⊂ F, the subscheme parameterizing lines meeting a curve C ⊂ X; S_{C} = q^{−1}(C) ⊂ P (X), note that there are natural morphism i_{C} = p|_{S_{C}} : S_{C} → F_{C} ⊂ F and q_{0} = q|_{S_{C}} : S_{C} → C; g|_{S_{C}} = (i_{C})^{∗}g, by abuse of notation, we still use g to denote this class; g′ = (q_{0})^{∗}[pt], where [pt] ∈ C is a closed point; ℚ[g, g′] ⊂ H^{∗}(S_{C}, ℚ), the subring generated by g and g′; P_{C} = P (X)|_{S_{C}}, to be more precise, we take the following fiber product

\xymatrix{P_{C} \ar[r]^{j_{C}} \ar[d]_{p_{C}} & P (X) \ar[d]^{p} \\ S_{C} \ar[r]^{i_{C}} & F}

q_{C} = q ∘ j_{C}; Φ_{C} = (p_{C})_{∗}(q_{C})^{∗}, homomorphism from either the cohomology groups or the Chow groups of X to those of S_{C}; Ψ_{C} = (q_{C})_{∗}(p_{C})^{∗}, homomorphism from either the cohomology groups or the Chow groups of S_{C} to those of X; D_{C} ⊂ X, the divisor swept out by all the lines meeting C; D_{C} is linearly equivalent to 5 deg(C)h; see Lemma 2.1; Given a polarization H on Y, we use H^{∗}(Y, ℤ)_{prim} to denote the primitive cohomology; Let (Y, H) be a polarized variety, we define A_{∗}(Y) ⊂ CH_{∗}(Y) to be the subgroup of degree 0 (with respect to H) cycles; For a vector bundle ℰ on Y, we use ℙ(ℰ) to denote the geometric projectivization which parameterizes all 1-dimensional linear subspaces of ℰ; more generally, if 1 ≤ r_{1} < ⋯ < r_{k} < rk ℰ is an increasing sequence of integers, we use G(r_{1}, …, r_{k}, ℰ) to denote the relative flag variety of subspaces of ℰ with corresponding ranks.

2. The fundamental relations

In this section we will establish a basic relation among algebraic/topological cycles on X. To do that, we need the following lemma which says that the space of lines on X meeting a given curve has the expected dimension.

Lemma 2.1. (i) dim F_{x} = n − 3 for all but finitely many points x_{i}, called Eckardt points. For each x_{i}, we have dim F_{x_{i}} = n − 2. (ii) Let C ⊂ X be a smooth curve on X. Then F_{C} if of pure expected dimension n − 2. (iii) The divisor D_{C} on X is linearly equivalent to 5 deg(C)h.

Proof. In [CS, Lemma 2.1], it is shown that dim F_{x} = n − 3 for a general point x ∈ X. By [CS, Corollary 2.2], there are at most finitely many Eckardt points x_{i}; see [CS, Definition 2.3]. This proves (i). Statement (ii) follows from (i) directly. Statement (ii) further implies that D_{C} is a divisor. Since Pic(X) = ℤh, the class of D_{C} has to be a multiple of h. Let l ⊂ X be a general line, then the intersection number of l and D_{C} is equal to the number of lines meeting both C and l. It is shown in [Sh, Lemma 3.10] that the above intersection number is 5 deg(C). □

Theorem 2.2. Let C ⊂ X be a smooth curve of degree e. We use h to denote the class of a hyperplane section on X, viewed as an element either in the Chow group or the (co)homology group. Then the following are true.

(1) Let γ be a topological cycle of real dimension n on X(ℂ). Then

2e[γ] + Ψ(Φ([γ]) · F_{C}) = 3e deg(γ)h^{\frac{n}{2}}

in H_{n}(X), where deg(γ) = γ · h^{i} if n = 2i and deg(γ) = 0 otherwise.

(2) Let γ be an algebraic cycle on X of dimension r with 2 ≤ r ≤ n − 2. Then

2eγ + Ψ(Φ(γ) · F_{C}) = 3e deg(γ)h^{n−r},

in CH_{r}(X), where deg(γ) = γ · h^{r}.

(3) Let γ be an algebraic cycle of dimension 1 on X. Then

2eγ + Ψ(Φ(γ) · F_{C}) + 2 deg(γ)C = 3e deg(γ)h^{n−1}

in CH_{1}(X).

Remark 2.3. Statement (1) holds for other dimensional topological cycles too. However, by the Lefschetz hyperplane theorem, it is only interesting when γ has dimension n.

Corollary 2.4. Let γ ∈ H_{n}(X, ℤ), 𝔞 ∈ CH_{1}(X) and 𝔟 ∈ CH_{r}(X). Then the following are true

(i) 2 deg(𝔞)γ + Ψ(Φ(γ) · Φ([𝔞])) = 3 deg(𝔞) deg(γ)h^{\frac{n}{2}} in H_{n}(X, ℤ). (ii) If n − 2 ≥ r ≥ 2, then 2 deg(𝔞)𝔟 + Ψ(Φ(𝔞) · Φ(𝔟)) = 3 deg(𝔞) deg(𝔟)h^{n−r} in CH_{r}(X). (iii) If r = 1, then 2 deg(𝔞)𝔟+2 deg(𝔟)𝔞+Ψ(Φ(𝔞)·Φ(𝔟)) = 3 deg(𝔞) deg(𝔟)h^{n−1} in CH_{1}(X).

Lemma 2.5. Let γ_{1} and γ_{2} be two disjoint irreducible topological cycles on ℙ^{n+1} of real dimensions r_{1} and r_{2}. Let L(γ_{1}, γ_{2}) be the set swept out by all complex lines meeting both γ_{1} and γ_{2}. Assume that γ_{1} and γ_{2} are in general position and L(γ_{1}, γ_{2}) has expected dimension. Then we have

L(γ_{1}, γ_{2}) = deg(γ_{1}) deg(γ_{2})h^{n− \frac{r_{1}+r_{2}}{2}}

in H_{r_{1}+r_{2}+2}(ℙ^{n+1}, ℤ). Here deg(γ) = γ · h^{\frac{r}{2}} if r = dim_{ℝ}(γ) is even and 0 otherwise. By convention h^{k+ \frac{1}{2}} = 0 for all integer k.

Lemma 2.6. Assume that γ_{1} and γ_{2} are two topological manifolds (resp. smooth projective varieties). Let f_{i} : γ_{i} → ℙ^{n+1}, i = 1, 2, be two disjoint irreducible topological (resp. algebraic) cycles on ℙ^{n+1}. Let φ : γ_{1} × γ_{2} → G(2, n + 2) be the map (resp. morphism) sending a pair of points (x, y) to the unique complex line connecting them. Let ℰ_{2} be the canonical rank 2 quotient bundle on G(2, n + 2) and ℰ′ = φ^{∗}ℰ_{2} its pullback. Let p_{i} : γ_{1} × γ_{2} → γ_{i}, i = 1, 2, be the two projections. Then

c_{1}(ℰ′) = p_{1}^{∗}(f_{1}^{∗}h) + p_{2}^{∗}(f_{2}^{∗}h)

in H^{2}(γ_{1} × γ_{2}, ℚ) (resp. CH^{1}(γ_{1} × γ_{2})), where h is the class of a hyperplane. In other words, c_{1}(ℰ′) is the pull-back of the class π_{1}^{∗}h + π_{2}^{∗}h via the natural map f_{1} × f_{2} : γ_{1} × γ_{2} → ℙ^{n+1} × ℙ^{n+1}, where π_{i} : ℙ^{n+1} × ℙ^{n+1} → ℙ^{n+1} are the projections.

Since the cohomology (resp. Chow ring) of ℙ^{n+1} is generated by the class of a hyperplane, both of the above lemmas can be reduced to the case where both γ_{1} and γ_{2} are complex linear subspaces. The proofs are left to the reader.

Proof of Theorem 2.2. To prove (1), we first reduce to the case γ = f_{∗}[M], where f : M → X is a continuous map from an n-dimensional topological manifold M to X such that (i) C does not meet f(M); (ii) f_{∗}M intersects D_{C} transversally and Δ_{0} := {t ∈ M : f(t) ∈ D_{C}} is of pure expected real dimension dim M − 2. The reason why we can do the above reduction can be seen as follows. First, γ is always of the form of linear combinations of cycles of the form f_{∗}[M]. If we can prove the statement (1) for each term in such a linear combination, then we prove (1) for γ. To get (i) and (ii), we note that C is a curve and D_{C} is a divisor and hence we can always move the cycle γ to a cycle γ′ such that (i) and (ii) hold. Now we assume the situation after reduction. For each pair of points (t, x) ∈ M × C, there is a unique (complex projective) line passing through f(t) and x. This gives a continuous map i_{0} : M × C → G(2, n + 2). We get the following diagram

(1)  \xymatrix{D_{1} ∪ D_{2} ∪ D \ar[r] \ar[d] & Y \ar[r] \ar[d] & X \ar[d]^{i_{X}} \\ P \ar[r]^{i′_{0}} \ar[d]_{π} & G(1, 2, n + 2) \ar[d]_{π̃} \ar[r]^{φ̃} & ℙ^{n+1} \\ M × C \ar[r]^{i_{0}} & G(2, n + 2)}

where all the squares are fiber products; D_{1} and D_{2} are sections of π corresponding to the points on f(M) and C respectively; D is the set of the third points of the intersection of the lines with X. Let π′ : D → M × C be the restriction of π to D. Let Δ ⊂ M × C be the closed subset of points (t, x) such that the line through f(t) and x is contained in X. By definition, Δ_{0} is the image of Δ under the projection to the first factor M and hence dim Δ = dim Δ_{0}. Then π′ is one-to-one away from Δ while over Δ it is an S^{2}-bundle with trivial Euler class (this is because, by taking the point in which the line meets C, we have a section of this bundle). Consider the following diagram

(2)  \xymatrix{ℙ^{1} = S^{2} \ar[r] & E \ar[r] \ar[d] & D \ar[d]^{π′} \\  & Δ \ar[r] & M × C}

For a general point z = (t_{0}, x_{0}) ∈ Δ, we use E_{z} to denote the fiber of E → Δ at the point z.

Claim 1: E · E_{z} = −1.

Proof of Claim 1. Let U_{0} ⊂ C be a small 2-dimensional (meaning real dimension) disc centered at the point x_{0} ∈ C. Let U_{z} = {t_{0}} × U_{0} ⊂ M × C be the corresponding small disc centered at z. By the assumption that z ∈ Δ is general, we see that U_{z} meets Δ transversally at the point z. Then we easily see that π^{′∗}U_{z} = Ũ_{z} +E_{z}, where Ũ_{z} = π^{′−1}(U_{z}\{z})∪{y_{0} ∈ E_{z}}. The existence of the point y_{0} can be seen as follows. Let f_{0} : C −−→ X be the rational map defined by sending x ∈ C ⊂ X to the third point of the intersection of X with the line connecting f(t_{0}) and x. Since C is a smooth curve, this rational map extends to a morphism f_{0} : C → X. Then one easily sees that y_{0} = f_{0}(x_{0}). This implies that Ũ_{z} · E = 1. The projection formula gives

(E_{z} + Ũ_{z}) · E = π^{′∗}U_{z} · E = U_{z} · π_{∗}E = U_{z} · 0 = 0.

It follows that E_{z} · E = −1. □

Let g : D → X and g_{i} : D_{i} → X, i = 1, 2, be the natural maps. Let h be the (co)homology class of a hyperplane.

Lemma 2.7. Let d = deg(f_{∗}[M]). The following are true.

(i) g_{∗}[D] = ed h^{\frac{n}{2} −1} in H_{n+2}(X); (ii) g_{∗}(h|_{D}) = ed h^{\frac{n}{2}} in H_{n}(X); (iii) h|_{D} + [E] = 2π^{′∗}c_{1}(ℰ), where ℰ = i_{0}^{∗}ℰ_{2} is the pull back of the canonical rank 2 quotient bundle ℰ_{2} on G(2, n + 2); (iv) c_{1}(ℰ) = p_{1}^{∗}(h|_{M}) + p_{2}^{∗}(h|_{C}), where p_{1} : M × C → M and p_{2} : M × C → C are the projections;

(v) g_{∗}(π^{′∗}c_{1}(ℰ)) = 2ed h^{\frac{n}{2}} − e f_{∗}[M].

Proof of Lemma 2.7. For (i), we first note that the class of the image of [D_{1}] + [D_{2}] + [D] on X is the restriction of [P] viewed as a class on ℙ^{n+1}. As a class on ℙ^{n+1}, [P] = ed h^{\frac{n}{2} −1} by Lemma 2.5. However, if we identify D_{i}, i = 1, 2, with M × C, then the natural map D_{i} → X contacts either the factor M or the factor C. Hence the classes of (g_{1})_{∗}[D_{1}] and (g_{2})_{∗}[D_{2}] are all zero on X. This proves (i). For (ii), we note that by the projection formula, we have g_{∗}(h|_{D}) = g_{∗}[D] · h. Hence (ii) follows from (i). The identity in (iv) follows from Lemma 2.6. To prove (iii), we first note that h|_{D} + [E] restricts to zero on the fibers E_{z} of E → Δ since h · E_{z} = 1 and E · E_{z} = −1 (Claim 1). By the Leray-Hirsch theorem, this implies that h|_{D} + [E] = π^{′∗}𝔞 where 𝔞 is a homology class on M × C. Applying π′_{∗} to the above equation, we get 𝔞 = π′_{∗}(h|_{D}). Note that on P, we have D_{1} + D_{2} + D = 3h, where, by abuse of notation, we still use h to denote the pullback of the class of a hyperplane to P. Hence

(3)  h · D_{1} + h · D_{2} + h · D = 3h^{2}

Since P is the projectivization of ℰ and h is the first Chern class of the relative O(1)-bundle, we have h^{2} − π^{∗}c_{1}(ℰ)h + π^{∗}c_{2}(ℰ) = 0. Applying π_{∗} to (3), we get

π_{∗}(h · D) + π_{∗}(h · D_{1}) + π_{∗}(h · D) = 3π_{∗}(h^{2})
= 3π_{∗}(π^{∗}c_{1}(ℰ) · h − π^{∗}c_{2}(ℰ))
= 3c_{1}(ℰ) · π_{∗}h = 3c_{1}(ℰ)

We also easily get that π_{∗}(h · D_{1}) = p_{1}^{∗}f^{∗}h and π_{∗}(h · D_{2}) = p_{2}^{∗}(h|_{C}). Combine this with (iv), we get

𝔞 = π′_{∗}(h|_{D}) = π_{∗}(h · D) = 2c_{1}(ℰ).

To prove (v), we note that for any class 𝔞 on M × C, if we pull back 𝔞 to D_{1} + D_{2} + D and then push forward to X, what we get is the same class as we pull back 𝔞 to P, then push forward to ℙ^{n+1} and then restrict to X. As a result, we always get a class coming from ℙ^{n+1}. When we take 𝔞 = c_{1}(ℰ), we get that the class of π^{∗}𝔞 on ℙ^{n+1} is equal to 2ed h^{\frac{n}{2}} . This can be seen as follows (using notation from Lemma 2.5).

φ̃_{∗}(i′_{0})_{∗}π^{∗}c_{1}(ℰ) = φ̃_{∗}(i′_{0})_{∗}π^{∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C})
= L(f_{∗}M · h, C) + L(M, C · h)
= ed h^{n− \frac{(n−2)+2}{2}} + ed h^{n− \frac{n+0}{2}}
= 2ed h^{\frac{n}{2}} .

Hence we have the following equalities.

g_{∗}(π^{′∗}c_{1}(ℰ)) = 2ed h^{\frac{n}{2}} − (g_{1}_{∗}(π^{∗}c_{1}(ℰ)|_{D_{1}}) + g_{2}_{∗}(π^{∗}c_{1}(ℰ)|_{D_{2}}))
= 2ed h^{\frac{n}{2}} − g_{1}_{∗}(π^{∗}c_{1}(ℰ)|_{D_{1}})
= 2ed h^{\frac{n}{2}} − e f_{∗}[M]

Here the second equality uses the fact that g_{2} : D_{2} → C ⊂ X contracts the factor M which has dimension n ≥ 3. This finishes the proof of the lemma. □

In the above lemma, we apply g_{∗} to (iii) and then take (ii) and (v) into account.

g_{∗}[E] = 2g_{∗}(π^{′∗}c_{1}(ℰ)) − g_{∗}(h|_{D})

= 4ed h^{\frac{n}{2}} − 2e f_{∗}[M] − ed h^{\frac{n}{2}} = 3ed h^{\frac{n}{2}} − 2e f_{∗}[M]

Note that g_{∗}[E] = Ψ(Φ(f_{∗}[M])·F_{C}). This way, we easily deduce the statement (1) of Theorem 2.2 for γ = f_{∗}[M] and the curve C. Now we start to prove (2) and (3) of Theorem 2.2. The basic strategy is the same as above. The only difference is that we need to consider the more delicate rational equivalence rather than homological equivalence. By linearity of the equalities in (2) and (3) and the Chow moving lemma, we reduce to the case γ = f_{∗}M where f : M → X is a morphism from a smooth projective variety M to X such that (i) f is birational onto its image; (ii) f(M) does not meet C and f_{∗}M intersects D_{C} transversally. Let r be the dimension of M. As before we construct the natural morphism i_{0} : M × C → G(2, n + 2). Let ℰ be the pull back of the canonical rank 2 quotient bundle to M × C. In this way, we get the following diagram as before.

(4)  \xymatrix{D_{1} ∪ D_{2} ∪ D \ar[r] \ar[d] & Y \ar[r] \ar[d] & X \ar[d] \\ P \ar[r]^{i′_{0}} \ar[d]_{π} & G(1, 2, n + 2) \ar[d]^{π̃} \ar[r]^{φ̃} & ℙ^{n+1} \\ M × C \ar[r]^{i_{0}} & G(2, n + 2)}

Here all the squares are fiber products; D_{1} and D_{2} are sections of π and contracted to f(M) and C via g_{1} = φ̃∘i′_{0}|_{D_{1}} : D_{1} → X and g_{2} = φ̃∘i′_{0}|_{D_{2}} : D_{2} → X respectively; D corresponds to the third point of the intersection of the lines with X. Let π′ : D → M × C be the restriction of π to D. Then π′ is a birational morphism. Let Δ ⊂ M × C be the subvariety that consists of points (t, x) such that the line through f(t) and x is contained in X. By the assumption that f_{∗}M intersects D_{C} transversally, we conclude that Δ is generically smooth. Let Δ^{sing} be the singular locus of Δ. Let E = (π′)^{−1}Δ ⊂ D and we have E ≅ ℙ(ℰ |_{Δ}).

Lemma 2.8. Away from Δ^{sing}, π′ is the blow-up along Δ.

Proof. Let ℰ_{2} be the canonical rank 2 quotient bundle on G(2, n + 2) and ℰ = i_{0}^{∗}ℰ_{2}. The point i_{0}(t, x) ∈ G(2, n + 2) represents the line connecting f(t) and x, which is also naturally ℙ(ℰ_{(t,x)}^{∨}). Hence we have canonical homomorphisms L_{f(t)} ↪ ℰ_{(t,x)}^{∨} and L_{x} ↪ ℰ_{(t,x)}^{∨}, where L ≅ O_{ℙ^{n+1}}(−1) is the tautological line bundle on ℙ^{n+1}. And since f(t) ≠ x, we have the natural identification

ℰ_{(t,x)}^{∨} = L_{f(t)} ⊕ L_{x}.

As (t, x) ∈ M × C varies, this identification glues and gives a canonical isomorphism

ℰ^{∨} ≅ p_{1}^{∗}f^{∗}L ⊕ p_{2}^{∗}(L|_{C}).

Assume that ϕ(X_{0}, …, X_{n+1}) ∈ H^{0}(ℙ^{n+1}, O(3)) is the homogeneous polynomial defining X. Since we have the canonical identification π̃_{∗}φ̃^{∗}O(3) = Sym^{3}(ℰ_{2}), the section ϕ induces a global section ϕ̃ of Sym^{3}(ℰ_{2}) and F ⊂ G(2, n + 2) is defined by ϕ̃ = 0; see [AK]. Let ϕ′ = i_{0}^{∗}ϕ̃ be the induced global section of Sym^{3} ℰ. It follows from the expression of ℰ^{∨} that

ℰ ≅ p_{1}^{∗}f^{∗}O_{X}(1) ⊕ p_{2}^{∗}O_{C}(1)

and hence

Sym^{3} ℰ = p_{1}^{∗}f^{∗}O_{X}(3) ⊕ ℰ ⊗ p_{1}^{∗}f^{∗}O_{X}(1) ⊗ p_{2}^{∗}O_{C}(1) ⊕ p_{2}^{∗}O_{C}(3).

Accordingly, we can write ϕ′ = ϕ′_{3,0} + ϕ′_{mid} + ϕ′_{0,3}. Since f(M) ⊂ X and ϕ vanishes on X, we conclude that ϕ′_{3,0} = 0. Similarly, we have ϕ′_{0,3} = 0. It follows that

ϕ′ = ϕ′_{mid} ∈ H^{0}(M × C, ℰ ⊗ p_{1}^{∗}f^{∗}O_{X}(1) ⊗ p_{2}^{∗}O_{C}(1)).

The vanishing ϕ′ = 0 at a point (t, x) exactly means that the line connecting f(t) and x is contained in X. Hence Δ ⊂ M × C is defined by ϕ′ = 0. In particular, Δ ⊂ M × C is a local complete intersection and

𝒩_{Δ/M×C} ≅ ℰ ⊗ p_{1}^{∗}f^{∗}O_{X}(1) ⊗ p_{2}^{∗}O_{C}(1)|_{Δ}.

The section ϕ′ can be viewed as a homomorphism

ϕ′ : p_{1}^{∗}f^{∗}O_{X}(−1) ⊗ p_{2}^{∗}O_{C}(−1) ⟶ ℰ .

This gives a section s : M ×C\Δ → P of π such that the image of s is the open subset D\E of D. Let σ : { \atop M × C} → M × C be the blow-up of M × C along Δ and E_{0} = σ^{−1}Δ. As a consequence of the above description of 𝒩_{Δ/M×C}, we

see that ϕ′ extends to { \atop M × C}\σ^{−1}Δ^{sing} and gives a rank one subbundle

ϕ̃′ : ℒ ↪ σ^{∗}ℰ

where ℒ = σ^{∗}(p_{1}^{∗}f^{∗}O_{X}(−1) ⊗ p_{2}^{∗}O_{C}(−1)) ⊗ O(E_{0}) is a line bundle on { \atop M × C}\σ^{−1}Δ^{sing}. This further gives a morphism

s̃: { \atop M × C}\σ^{−1}Δ^{sing} ↪ P

such that σ = π ∘s̃. Note that s̃ extends s. Hence the image of s̃ is contained in D since that of s is. Since Ẽ → Δ and E → Δ are all ℙ^{1}-bundles, we conclude that The image of s̃ is exactly D\(π′)^{−1}Δ^{sing}. This implies that

s̃: { \atop M × C}\σ^{−1}Δ^{sing} ⟶ D\(π′)^{−1}Δ^{sing}

is an isomorphism, since they are the same locally closed subvariety of P. □

Back to the proof of the theorem. We know from the above lemma that the possible singularities of D can only appear over Δ^{sing}. Take a resolution of singularities r′ : D̃ → D. Let E_{i} be the exceptional divisors of r′. We use E′ to denote the strict transform of E in D̃. We still use h to denote the class of a hyperplane in ℙ^{n+1}. Let g : D̃ → X be the natural morphism.

Lemma 2.9. Let d = deg(f_{∗}[M]), then the following are true.

(i) g_{∗}D̃ = ed h^{n−r−1} in CH_{r+1}(X); (ii) g_{∗}(h|D̃) = ed h^{n−r}; (iii) h|D̃ + E′ + ∑ a_{i}E_{i} = 2(π′ ∘ r′)^{∗}c_{1}(ℰ), for some a_{i} ∈ ℤ; (iv) c_{1}(ℰ) = p_{1}^{∗}(f^{∗}h) + p_{2}^{∗}(h|_{C});

(v) If dim M ≥ 2, then g_{∗}(r^{′∗}π^{′∗}c_{1}(ℰ)) = −e f_{∗}M + 2ed h^{n−r} in CH_{r}(X); (vi) If M is a curve, then g_{∗}(r^{′∗}π^{′∗}c_{1}(ℰ)) = ed h^{n−1} − e f_{∗}M − dC in CH_{1}(X).

Proof of Lemma 2.9. The proof is similar to that of Lemma 2.7. We note that the push forward of D_{1} + D_{2} + D to X is a class coming from ℙ^{n+1} because it is the restriction of the class of the image of P in ℙ^{n+1}. By Lemma 2.5, the class of P in ℙ^{n+1} is equal to ed h^{n−r−1}. Since D_{1} and D_{2} are contracted to smaller dimensions via g_{1} and g_{2} respectively, we get (i). Conclusion (ii) follows from the projection formula as before. (iv) follows from Lemma 2.6. For (iii), note that on D − π^{′−1}(Δ^{sing}), the class of h + E comes from a class 𝔞 on M × C − Δ^{sing} via pull back by π′. Since the codimension of Δ^{sing} in M × C is at least 3, we know that the divisor class group of M × C − Δ^{sing} is the same as that of M × C. Hence we can view 𝔞 as a divisor class on M × C. The equality

h + E = π^{′∗}𝔞, on D\π^{′−1}(Δ^{sing})

gives an equation

r^{′∗}(h + E) + ∑ a_{i}E_{i} = r^{′∗}π^{′∗}𝔞

for some a_{i} ∈ ℤ. By a very similar argument as before we know that the class 𝔞 is 2c_{1}(ℰ). To prove (v) and (vi), we do the following explicit calculation.

g_{∗}r^{′∗}π^{′∗}c_{1}(ℰ) = g_{∗}r^{′∗}π^{′∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C}),  by (iv)

= g̃_{∗}π^{∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C})|_{D∪D_{1}∪D_{2}} − g_{1∗}π^{∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C})|_{D_{1}}

− g_{2∗}π^{∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C}),  g̃ : D ∪ D_{1} ∪ D_{2} → X
= φ̃_{∗}(i′_{0})_{∗}π^{∗}(p_{1}^{∗}f^{∗}h + p_{2}^{∗}h|_{C})|_{X} − g_{1∗}(p_{2}^{∗}h|_{C}) − g_{2∗}(p_{1}^{∗}f^{∗}h)
(here we identify D_{1} and D_{2} with M × C)

= L(f_{∗}M · h, C)|_{X} + L(f_{∗}M, C · h)|_{X} − e f_{∗}M − g_{2∗}(p_{1}^{∗}f^{∗}h)

= 2ed h^{n−r} − e f_{∗}M − g_{2∗}(p_{1}^{∗}f^{∗}h)

Note that g_{2∗}(p_{1}^{∗}f^{∗}h) is supported on the curve C. If dim M ≥ 2, then g_{2∗}(p_{1}^{∗}f^{∗}h) = 0; if dim M = 1, then g_{2∗}(p_{1}^{∗}f^{∗}h) = deg(f_{∗}M)C. Hence (v) and (vi) follow from the above computation. □

Now we come back the the proof of the theorem. In the above lemma, we first apply g_{∗} to (iii) and note that all the E_{i}’s map to zero. We also easily see that g_{∗}E′ = Ψ(Φ(f_{∗}M) · F_{C}). Combine all these with (v) (or (vi) in the case of 1-cycles), we get the conclusion (2) and (3) for γ = f_{∗}M. By linearity, the conclusions hold for any given γ. □

3. The action of the incidence correspondence

Let C ⊂ X be a smooth rational curve of degree e on X. We also assume that C is general, meaning that it comes from a non-empty open subset of the corresponding component of the Hilbert scheme. Let S_{C} = q^{−1}C be the inverse image of C under the morphism q : P (X) → X. The points on S_{C} can be described as

S_{C} = {([l], x) ∈ F × C : x ∈ l}.

Let q_{0} = q|_{S_{C}} : S_{C} → C.

Lemma 3.1. If C is general, then S_{C} is smooth of dimension n − 2.

We will prove this lemma later. In [Sh, Lemma 3.4], we show that for a general rational curve C of degree e ≥ 2, there exist n_{e} = \frac{5e(e−3)}{2} + 6 secant lines of C, i.e. lines meeting C twice. Let L_{C,i}, i = 1, …, n_{e}, be all the secant lines of C and [L_{C,i}] ∈ F the corresponding points on the variety of lines. Above each point [L_{C,i}], we have a pair of points {([L_{C,i}], y_{i}), ([L_{C,i}], z_{i})} on S_{C}, where y_{i} and z_{i} are the two points in which C intersects L_{C,i}. Then Lemma 3.1 implies that [L_{C,i}]’s are the only singular points of F_{C} and p|_{S_{C}} : S_{C} → F_{C} is the normalization and also a desingularization since S_{C} is smooth. From now on, we make the assumption that e ≥ 2, unless otherwise stated.

Definition 3.2. We say that a correspondence Γ ⊂ Y ×Y is generically defined by y ↦ ∑ y_{i} if Γ is the closure of the graph of this multi-valued map.

For a general point [l] ∈ F_{C}, let x = C ∩ l be the intersection point. By [l] ↦ ([l], x) ∈ S_{C}, we view [l] as a point on S_{C}.

Lemma 3.3. There exist 5e − 5 lines l_{1}, l_{2}, …, l_{5e−5} meeting both l and C in points different from x.

By abuse of language, these lines are called secant lines of the pair (l, C); see [Sh, Definition 3.1]. Each point [l_{i}] can again be viewed as a point ([l_{i}], x_{i}) on S_{C}, where x_{i} = l_{i} ∩ C. A line meeting two disjoint curves C_{1}, C_{2} ⊂ X will be called an incidence line of C_{1} and C_{2} (note that they are also called secant lines in [Sh]).

Proof of Lemma 3.3. Note that if l̃ is a general line, the number of incidence lines of l̃ and C is 5e; this follows from a degree computation using (3) of Theorem 2.2 or [Sh, Lemma 3.10]. When l̃specializes to l, five of these incidence lines of l̃ and C will specialize to five lines passing through x and those five lines are not counted as the secant lines of the pair (l, C). We will describe this specialization in more details later. Hence the number of secant lines of the pair (l, C) is 5e − 5. □

Definition 3.4. Let the incidence correspondence I_{C} ⊂ S_{C} ×S_{C} be generically defined by

(5)  I_{C} : ([l], x) ↦ ∑_{i=1}^{5e−5}([l_{i}], x_{i}),

where l is a line meeting C and l_{i} are the secant lines of the pair (l, C). Let σ denote the action of I_{C} on either the cohomology groups or the Chow groups of S_{C}. On F, we have the incidence correspondence I = {([l_{1}], [l_{2}]) ∈ F × F : l_{1} ∩ l_{2} ≠ ∅} ⊂ F × F. This induces a correspondence

I′_{C} = (i_{C} × i_{C})^{∗}I ∈ CH_{n−2}(S_{C} × S_{C}),

where i_{C} = p|_{S_{C}} : S_{C} → F is the natural morphism.

Remark 3.5. Note that (i_{C}×i_{C})^{−1}I has a component, namely S_{C}×_{C}S_{C}, which has dimension larger than expected. The correspondence I_{C} ⊂ (i_{C} × i_{C})^{−1}I is a component of expected dimension. It turns out later that one key ingredient to understand the action of σ is the difference between I_{C} and I′_{C}.

Note that on F, we have a natural polarization g given by the Plücker embedding of G(2, n+2). It can also be written as g = Φ(h^{2}). We fix g|_{S_{C}} = (i_{C})^{∗}g as the polarization of S_{C}. By abuse of notation, we still use g to denote its restriction to S_{C}. Recall that S_{C} admits a natural morphism q_{0} = q|_{S_{C}} : S_{C} → C. This gives an extra class g′ = q_{0}^{∗}[pt] on S_{C}. Note that g′ is never ample. The following is the main result of this section.

Theorem 3.6. Let C ⊂ X be a general rational curve of degree e ≥ 2 and σ the action of the incidence correspondence as above. Then the following are true.

(1) Let 𝔞 be a topological cycle of odd dimension on S_{C}. Then

σ(𝔞) = Φ_{C}(Ψ_{C}(𝔞)) + 𝔞

in H^{∗}(X, ℤ).

(2) Let 𝔞 be a topological cycle of dimension 2m or an algebraic cycle of dimension m on S_{C}, then in either the cohomology H^{2n−2m−4}(X, ℤ) or Chow group CH_{m}(X), we have

σ(𝔞) = Φ_{C}(Ψ_{C}(𝔞)) + 𝔞 + const.

where the constant only depends on the intersection numbers 𝔞 · g^{m} and 𝔞 · g′g^{m−1}; the constant is zero if 𝔞 · g^{m} = 0 and 𝔞 · g′g^{m−1} = 0.

Before proving the above theorem, we give some technical constructions related to the geometry of S_{C}.

3.1. The normal bundle of a line meeting C. Recall from [CG, §6] that for any line l on X, we have either

𝒩_{l/X} ≅ O(1)^{n−3} ⊕ O^{2},

in which case l is said to be of first type, or

𝒩_{l/X} ≅ O(1)^{n−2} ⊕ O(−1),

in which case l is said to be of second type. A general line is of first type.

Definition 3.7. Let 𝒢 be a vector bundle on ℙ^{1}. Then we have a decomposition 𝒢 = ⨁_{i=1}^{rk 𝒢} O(a_{i}), a_{i} ∈ ℤ. We define the positive part of 𝒢 to be

Pos(𝒢) = ⨁_{a_{i}>0} O(a_{i}).

Similarly, we define the nonnegative part of 𝒢 to be

NN(𝒢) = ⨁_{a_{i}≥0} O(a_{i}).

Lemma 3.8. Assume that C ⊂ X is general and l is a line meeting C in a point x. Then l meets C transversally at the point x.

Proof. If l is tangent to C at the point x, then l is a secant line of C. We call a secant line of C simple if it meets C in two distinct points. Then we only need to show that the secant lines of a general curve C are simple. Note that having simple secant lines is an open condition and we only need to find a degenaration of C which has all secant lines being simple. First we note that the concept of a secant line can be naturally extended to most nodal curves; see [Sh]. For example, let C_{1} = L_{1} ∪ L_{2} ∪ L_{3} be a chain of three distinct lines with L_{1} ∩ L_{2} = x and L_{2} ∩ L_{3} = y. Assume that the plane spanned by L_{1} and L_{2} (resp. L_{2} and L_{3}) is not contained in X. Let l_{12} (resp. l_{23}) be the residue line, or equivalently the secant line, of L_{1} ∪ L_{2} (resp. L_{2} ∪ L_{3}). If we further assume that the pair (L_{1}, L_{3}) has finitely many incidence lines {E_{0} = L_{2}, E_{1}, E_{2}, E_{3}, E_{4}}. Then the set of secant lines of C_{1} is given by

{l_{12}, l_{23}, E_{1}, E_{2}, E_{3}, E_{4}}.

Let e ≥ 2 be the degree of C. We construct a chain C′ = ∪_{i}l_{i}, i = 1, …, e, of distinct lines such that: (i) l_{i} and l_{i+1} meet in a point y_{i}, i = 1, …, e − 1; (ii) there are finitely many distinct secant lines of C′, which are different from the lines in {l_{i} : i = 1, …, e}; (iii) all the lines l_{i} are of first type. Such a curve C′ can be constructed inductively. The condition (ii) says that the curve C′ has only simple secant lines. The condition (iii) implies that C′ can be deformed to a smooth degree e rational curve on X. Since the space of degree e rational curves on X are irreducible by [CS], we see that C′ is a specialization of C. Hence the lemma is proved. □

Definition 3.9. Let ([l], x) ∈ S_{C} be a point where l is a line on X and x is a point in which l meets C. Since C is general, l meets C transversally at the point x. We define 𝒩_{l/X}⟨T_{C,x}⟩ ↪ 𝒩_{l/X} to be the subsheaf of sections s such that s(x) is in the direction of T_{C,x}. Or equivalently, 𝒩_{l/X}⟨T_{C,x}⟩ fits into the following short exact sequence

[figure: 0; 𝒩_{l/X}⟨T_{C,x}⟩; 𝒩_{l/X}; \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}}; 0]

The tangent space of S_{C} at a point ([l], x) is canonically identified as

T_{S_{C},([l],x)} = H^{0}(l, 𝒩_{l/X}⟨T_{C,x}⟩)

See [Ko, §II.1]. It follows from the short exact sequence in Definition 3.9 that

χ(𝒩_{l/X}⟨T_{C,x}⟩) = χ(𝒩_{l/X}) − (n − 2) = n − 2 = dim S_{C}

where we use the fact that dim S_{C} = n − 2 which is a consequence of Lemma 2.1. Hence dim T_{S_{C},([l],x)} = dim S_{C} if and only if H^{1}(l, 𝒩_{l/X}⟨T_{C,x}⟩) = 0. Equivalently, we have the following

Lemma 3.10. The variety S_{C} is smooth at ([l], x) if and only if h^{1}(𝒩_{l/X}⟨T_{C,x}⟩) = 0.

Proposition 3.11. The splitting of 𝒩_{l/X}⟨T_{C,x}⟩ has the following possibilities.

(i) If l is of first type and the image of T_{C,x} in 𝒩_{l/X,x} is not contained in Pos(𝒩_{l/X})_{x}, then

𝒩_{l/X}⟨T_{C,x}⟩ ≅ O^{n−2} ⊕ O(−1).

(ii) If l is of first type and the image of T_{C,x} in 𝒩_{l/X,x} is contained in Pos(𝒩_{l/X})_{x}, then

𝒩_{l/X}⟨T_{C,x}⟩ ≅ O(1) ⊕ O^{n−4} ⊕ O(−1)^{2}.

(iii) If l is of second type and the image of T_{C,x} in 𝒩_{l/X,x} is not contained in Pos(𝒩_{l/X})_{x}, then

𝒩_{l/X}⟨T_{C,x}⟩ ≅ O^{n−2} ⊕ O(−1).

(iv) If l is of second type and the image of T_{C,x} in 𝒩_{l/X,x} is contained in Pos(𝒩_{l/X})_{x}, then

𝒩_{l/X}⟨T_{C,x}⟩ ≅ O(1) ⊕ O^{n−3} ⊕ O(−2).

Proof. Let r_{+} = rk(Pos(𝒩_{l/X})) and a = dim_{ℂ} Im\{Pos(𝒩_{l/X}) → \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}} \}. Consider the following commutative diagram

[figure: 0; 𝒢; 𝒬; ℂ^{n−2−a}; 0; 0; 𝒩_{l/X}⟨T_{C,x}⟩; 𝒩_{l/X}; \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}}; 0; 0; O(1)^{r_{+}−a} ⊕ O^{a}; Pos(𝒩_{l/X}); ℂ^{a}; 0]

where all columns and rows are short exact. Note that 𝒬 = O^{2} if l is of first type and 𝒬 = O(−1) if l is of second type. If l is of first type and T_{C,x} is not contained in Pos(𝒩_{l/X}), then Pos(𝒩_{l/X})_{x} → \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}} is injective and hence a = n − 3. Since 𝒬 ≅ O^{2}, the top row of the above diagram gives 𝒢 ≅ O ⊕ O(−1). It follows from the first column that 𝒩_{l/X}⟨T_{C,x}⟩ ≅ O^{n−2} ⊕ O(−1) since r_{+} = n − 3. If l is of first type and T_{C,x} is contained in Pos(𝒩_{l/X}), then Pos(𝒩_{l/X})_{x} → \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}} has a 1-dimensional kernel and hence a = n − 4. Since 𝒬 ≅ O^{2}, the top row of the above diagram gives 𝒢 ≅ O(−1)^{2}. It follows from the first column that 𝒩_{l/X}⟨T_{C,x}⟩ ≅ O(1) ⊕ O^{n−4} ⊕ O(−1)^{2} since r_{+} = n − 3. If l is of second type and T_{C,x} is not contained in Pos(𝒩_{l/X}), then Pos(𝒩_{l/X})_{x} → \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}} is injective and hence a = n − 2. Since 𝒬 ≅ O(−1), the top row of the above diagram gives 𝒢 ≅ O(−1). It follows from the first column that 𝒩_{l/X}⟨T_{C,x}⟩ ≅ O^{n−2} ⊕ O(−1) since r_{+} = n − 2. If l is of second type and T_{C,x} is contained in Pos(𝒩_{l/X}), then Pos(𝒩_{l/X})_{x} → \frac{T_{X,x}}{T_{l,x}⊕T_{C,x}} has a 1-dimensional kernel and and hence a = n − 3. Since 𝒬 ≅ O(−1), the top row of the above diagram gives 𝒢 ≅ O(−2). It follows from the first column that 𝒩_{l/X}⟨T_{C,x}⟩ ≅ O(1) ⊕ O^{n−4} ⊕ O(−2) since r_{+} = n − 2. □

Corollary 3.12. S_{C} is singular at ([l], x) if and only if l is of second type and the image of T_{C,x} in 𝒩_{l/X,x} is contained in Pos(𝒩_{l/X})_{x}.

Definition 3.13. The curve C has a bad direction at a point x ∈ C if there exists a line l of second type through x such that the image of T_{C,x} in 𝒩_{l/X,x} is contained in Pos(𝒩_{l/X})_{x}. Otherwise, we say that C has a good direction at x.

Lemma 3.14. If C is general, then C has good directions everywhere.

Proof. Given a line l ⊂ X of second type and a point x ∈ l, the positive part of 𝒩_{l/X} together with l itself determines an n − 1 dimensional linear subspace P_{l,x} of T_{X,x}. Let F_{0} ⊂ F be the closed subscheme of lines of second type on X. By [CG, Corollary 7.6], we know that dim F_{0} ≤ n − 2. Let D_{2} ⊂ X be the locus swept by all lines of second type. Then dim D_{2} ≤ n − 1. Let C be a general rational curve on X. Then C can meet D_{2} in at most finitely many points x_{i}, i = 1, …, m and through each point x_{i}, there is a unique line l_{i} of second type. The condition that C has a good direction at x_{i} is equivalent to T_{C,x_{i}} ∉ P_{l_{i},x_{i}}. We first note that it is proved in [Iz, Lemma 1.4] that a general line has good directions everywhere. It is proved in [CS] that the moduli space of degree e rational curves on X is irreducible. Hence to prove the lemma, we only need to construct one such curve C whose tangent directions are all good. This can be obtained by smoothing a chain of lines as follows. Take a chain of lines L = ∪_{j=1}^{e}L_{j} such that (i) each line L_{j} is of first type and has good directions everywhere and (ii) none of the nodes of L is on a line of second type. Since all the components L_{j} are of first type, the chain L is smoothable; see [Ko, Chapter II, Theorem 7.9]. Then a general smoothing of L has good directions everywhere since having a bad direction is a closed condition. □

Lemma 3.15. Let l ⊂ X be a line of first type and x ∈ l be a point, then F_{x} is smooth at [l].

Proof. Deformation theory gives us

T_{F_{x},[l]} = H^{0}(l, 𝒩_{l/X} ⊗ O_{l}(−x)) = H^{0}(l, O^{n−3} ⊕ O(−1)^{2}) ≅ ℂ^{n−3}.

Hence dim T_{F_{x},[l]} = n − 3 = dim F_{x}. It follows that F_{x} is smooth at [l]. □

Proof of Lemma 3.1. We first deal with the case when C is a general line. In this case, we see from [Iz, Lemma 1.4] that F_{C} is smooth away from the point [C] and hence so is S_{C} − p^{−1}[C]. Thus the singularities of S_{C} can only appear on C̃ = p^{−1}[C] ≅ C. Recall that we have the natural morphism q_{0} = q|_{S_{C}} : S_{C} → C and C̃ is a section of q_{0}. If q_{0}^{−1}y is smooth at the point ỹ = C̃ ∩ q_{0}^{−1}y for some y ∈ C, then S_{C} is smooth at ỹ. Note that if we identify q_{0}^{−1}y with F_{y}, then ỹ = [C] ∈ F_{y}. It follows that singularities of S_{C} can only appear at the points ỹ ∈ C̃ such that F_{y} is singular at the point [C], where y = q_{0}(ỹ) ∈ C. But this can never happen if C is of first type by Lemma 3.15. Assume that e ≥ 2. By Corollary 3.12, we only need to show that when C is general, it does not have a bad direction, which is established in Lemma 3.14. □

3.2. First order deformations of C.

Definition 3.16. Let C ⊂ X be a general smooth rational curve of degree e ≥ 2. Then C is free since it passes through a general point of X; see [Ko, Chapter II, Theorem 3.11]. In other words, 𝒩_{C/X} is globally generated. Let v ∈ H^{0}(C, 𝒩_{C/X}) be a general section of the normal bundle. Then v is nowhere vanishing since 𝒩_{C/X} is globally generated with rank n − 1 ≥ 2. We define the rank 2 subbundle T_{v} ⊂ T_{X}|_{C}, associated to v, to be such that

T_{v,x} = \{τ ∈ T_{X,x} : τ̄ ∈ ℂv ⊂ 𝒩_{C/X,x} = \frac{T_{X,x}}{T_{C,x}} \}

for all x ∈ C. Let

Σ_{v} = {([l], x) ∈ S_{C} : T_{l,x} ⊂ T_{v,x}}.

Remark 3.17. Note that we have natural embeddings

\xymatrix{S_{C} \ar@{^{(}->}[r]^{j_{1}} & ℙ(T_{X}|_{C}) & ℙ(T_{v}), \ar@{^{(}->}[l]_{j_{2}}}

where j_{1} is defined by ([l], x) ↦ {T_{l,x} ⊂ T_{X,x}} and j_{2} is induced by the natural subbundle structure T_{v} ⊂ T_{X}|_{C}. We know that dim S_{C} = n−2, dim ℙ(T_{X}|_{C}) = n and dim ℙ(T_{v}) = 2. Furthermore, the set Σ_{v} is precisely the intersection of ℙ(T_{v}) and S_{C}.

Lemma 3.18. If v is general, then S_{C} intersect ℙ(T_{v}) transversally in the set Σ_{v} = {([L_{i}], x_{i})}_{i=1}^{r_{0}}, where x_{i} ≠ x_{j} for i ≠ j. In particular, Σ_{v} is finite.

Proof. By [CG, Lemma 6.5], we see that S_{C} is a fiberwise (2,3) complete intersection of ℙ(T_{X}|_{C}). In other words, for each point x ∈ C, there exist

ϕ_{2,x} : Sym^{2}(T_{X,x}) → ℂ and ϕ_{3,x} : Sym^{3}(T_{X,x}) → ℂ

such that q_{0}^{−1}x = F_{x} ⊂ ℙ(T_{X,x}) is defined by ϕ_{2,x} = 0 and ϕ_{3,x} = 0. The tangent direction T_{C,x} defines a point x̃ = [T_{C,x}] ∈ ℙ(T_{v,x}) ⊂ ℙ(T_{X,x}). Lemma 3.8 implies that the point x̃ is not contained in F_{x}. Note that ℙ(T_{v,x}) can be viewed as a line in ℙ(T_{X,x}) passing through the point x̃. Since 𝒩_{C/X} is globally generated, we see that for a fixed x ∈ C the line ℙ(T_{v,x}) is a general line passing through x̃. Now since F_{x} ⊂ ℙ(T_{X,x}) has codimension 2, the line does not meet F_{x}. This shows that when v is general, there are at most finitely many points x_{i} ∈ C such that F_{x_{i}} meets ℙ(T_{v,x_{i}}). For any of such point x_{i}, since x̃_{i} ∉ F_{x}, the line ℙ(T_{v,x_{i}}) meets F_{x_{i}} in at most finitely many points. This proves that S_{C} intersects ℙ(T_{v}) transversally. We still need to rule out the situation x_{i} = x_{j} = y. Let X̃ → X be the blow-up of X along the curve C with E ≅ ℙ(𝒩_{C/X}) ⊂ X̃ being the exceptional divisor. Consider the diagram

\xymatrix{E \ar[r] \ar[d]_{π} & X̃ \ar[d] \\ C \ar[r] & X}

For a point x ∈ C and a line L passing through x, Lemma 3.8 implies that T_{L,x} defines a 1-dimensional linear space in 𝒩_{C/X,x} and hence gives a point z ∈ E_{x} = π^{−1}x. When (L, x) varies, the point z traces out a subvariety Z ⊂ E. Note that z is simply the intersection point of the strict transform L̃ of L and the exceptional divisor E. This gives rise to a morphism f_{x} : F_{x} → E_{x}. First we note that for general x ∈ C, the morphism f_{x} is generically 1-to-1. This can be seen as follows. When x is a general point, the tangent direction T_{C,x} defines a general point x̃ ∈ ℙ(T_{X,x}). Then for a general point [L] ∈ F_{x} ⊂ ℙ(T_{X,x}), the line connecting x̃ and [L] meets F_{x} in the single point [L]. Let Z_{1} ⊂ Z be the set of points having at least two preimages under f_{x} for some x ∈ C. The above discussion together with the fact dim F_{x} = n − 3 for all x ∈ C (since C does not pass any Eckardt point) implies

dim Z = n − 2, dim Z_{1} ≤ n − 3.

A section v ∈ H^{0}(C, 𝒩_{C/X}) defines a section C_{v} ⊂ E. Since 𝒩_{C/X} is globally generated, the curves {C_{v}} form a covering family of movable curves on E. Note that dim E = n − 1. It follows that for a general v, the curve C_{v} does not meet Z_{1}. If x_{i} = x_{j} = y, then we have two distinct lines L_{i} and L_{j} passing through y such that T_{L_{i},y}, T_{L_{j},y} and v(y) give the same point z ∈ ℙ(𝒩_{C/X,x}). This means that C_{v} meets Z_{1} at the pint z. But this does not happen for general v by the above discussion. □

Remark 3.19. The element v ∈ H^{0}(C, 𝒩_{C/X}) can be viewed as a first order deformation of C. Note that we have a canonical identification

T_{Hilb(X),[C]} = H^{0}(C, 𝒩_{C/X}).

Since [C] ∈ Hilb^{e}(X) is a smooth point, we can find a smooth pointed curve φ : (T, 0) ⟶ (Hilb(X), [C]) such that dφ(T_{T,0}) = ℂ v. The curve T parameterizes a 1-dimensional family of curves on X,

\xymatrix{𝒞 \ar[r] \ar[d] & X \\ T}

such that 𝒞_{0} = C. Let Σ_{t} = {L_{i,t}}, t ≠ 0, be the set of all incidence lines of 𝒞_{t} and C (meaning lines meeting both 𝒞_{t} and C). We have proved in [Sh, Lemma 3.10] that the cardinality of Σ_{t} is equal to 5 deg(𝒞_{t}) deg(C) = 5e^{2}. If we take the limit as t → 0 (see [Fu, §11.1]), then we see that Σ_{t} specializes to the set Σ_{v}. Hence we get r_{0} = |Σ_{v}| = 5e^{2}. We will write

Σ_{v} = {([L_{i}], x_{i}) : i = 1, 2, …, 5e^{2}}.

Lemma 3.20. If C is general, then L_{i} is of first type and

𝒩_{L_{i}/X}⟨T_{C,x_{i}}⟩ ≅ O^{n−2} ⊕ O(−1), 1 ≤ i ≤ 5e^{2}.

Furthermore, we have a decomposition

(6)  NN(𝒩_{L_{i}/X}⟨T_{C,x_{i}}⟩)_{x_{i}} = Pos(𝒩_{L_{i}/X})_{x_{i}} ⊕ T_{C,x_{i}},

which is canonical upto a scalar multiplication on Pos(𝒩_{L_{i}/X})_{x_{i}}.

Proof. When C and also v are general, all the lines L_{i} are of first type and T_{C,x_{i}} is not pointing to the positive part of 𝒩_{L_{i}/X}. Hence (i) of Proposition 3.11 applies. Consider the following short exact sequence

(7) \xymatrix{0 \ar[r] & Pos(𝒩_{L_{i}/X})_{x_{i}} \ar[r]^{·t_{i}} & NN(𝒩_{L_{i}/X}⟨T_{C,x_{i}}⟩)_{x_{i}} \ar[r] & \overline{T}_{C,x_{i}} \ar[r] & 0,}

where t_{i} is a local uniformizer of C at the point x_{i} and \overline{T}_{C,x_{i}} is the image of T_{C,x_{i}} in 𝒩_{L_{i}/X,x_{i}}. The natural homomorphism

T_{C,x_{i}} ↪ T_{X,x_{i}} → 𝒩_{L_{i}/X,x_{i}}

factors through NN(𝒩_{L_{i}/X}⟨T_{C,x_{i}}⟩)_{x_{i}} ⟶ 𝒩_{L_{i}/X,x_{i}} and gives a homomorphism θ_{i} : \overline{T}_{C,x_{i}} → 𝒩_{L_{i}/X}⟨T_{C,x_{i}}⟩)_{x_{i}}. This homomorphism θ_{i} canonically splits the sequence (7). After identifying \overline{T}_{C,x_{i}} with T_{C,x_{i}}, we get the decomposition. Note that the only choice made here is the local parameter t_{i}. A different choice of t_{i} induces a scalar multiplication on Pos(𝒩_{L_{i}/X})_{x_{i}}. □

Lemma 3.21. Let V ⊂ T_{X,x} be a 3-dimensional vector space containing T_{v,x}, for some x ∈ C. We view ℙ(V) as the space of lines tangent to X at x in a direction of V. Then F_{x} ∩ ℙ(V) is a zero dimensional scheme of length 6 if x ≠ x_{i}. If x = x_{i} for some 1 ≤ i ≤ 5e^{2}, then F_{x} ∩ ℙ(V) can have at most one component of dimension 1 and this component has to be a line (on ℙ(V) = ℙ^{2}) if it exists.

Proof. If we show that F_{x} ∩ ℙ(V) is zero dimensional, then it of length 6 since it is a (2,3)-complete intersection in ℙ(V) ≅ ℙ^{2}. Assume that B ⊂ ℙ(V) ∩ F_{x} is a curve. Then B ∩ ℙ(T_{v,x}) is nonempty, where ℙ(T_{v,x}) is viewed as a line on ℙ(V). Or in other words, ℙ(T_{v,x}) contains a line L of X. This exactly means that (L, x) ∈ Σ_{v}. Hence such B does not exist if x ≠ x_{i}. If x = x_{i}, we have already seen that ℙ(T_{v,x_{i}}) contains the unique line L_{i} of X. Hence B ⊂ ℙ(V) is of degree 1, namely a line, if it exists. This is because B can only meet the line ℙ(T_{v,x}) in the point [L_{i}]. □

3.3. The rational map ρ. Let ℱ be the quotient of T_{X}|_{C} by T_{v}. The natural quotient homomorphism T_{X}|_{C} → ℱ induces a rational map β : ℙ(T_{X}|_{C}) −−→ ℙ(ℱ). By definition, the indeterminacy locus of β is exactly ℙ(T_{v}). Let ρ = β∘j_{1} : S_{C} −−→ ℙ(ℱ), where j_{1} : S_{C} ↪ ℙ(T_{X}|_{C}) is the closed immersion induced by ([l], x) ↦ [T_{l,x} ⊂ T_{X,x}]; see Remark 3.17. Then the indeterminacy locus of ρ is the set Σ_{v}. This shows that we have a morphism

ρ : S_{C} − Σ_{v} → ℙ(ℱ).

Lemma 3.22. Let f : Y_{1} −−→ Y_{2} be a rational map between two smooth projective varieties whose indeterminacy locus Σ ⊂ Y_{1} is a finite set of closed points. Let Γ_{f} ⊂ (Y_{1} − Σ) × Y_{2} be the graph of f and Γ̄_{f} ⊂ Y_{1} × Y_{2} the closure of Γ_{f}. We define the pull-back f^{∗} and push-forward f_{∗} (on either the cohomology groups or the Chow groups) as

f^{∗}𝔞 = p_{1,∗}(Γ̄_{f} · p_{2}^{∗}𝔞), f_{∗}𝔞 = p_{2,∗}(Γ̄_{f} · p_{1}^{∗}𝔞),

where p_{i} : Y_{1} × Y_{2} → Y_{i} is the projection to the corresponding factor. Then we have f^{∗}(𝔞^{k}) = (f^{∗}𝔞)^{k}, for all 𝔞 ∈ Pic(Y_{2}) and k < dim Y_{1}.

Proof. Let E = Σ × Y_{2} ∩ Γ̄_{f} be the exceptional set of the morphism Γ̄_{f} → Y_{1}. Let D and D_{0} be two divisors on Y_{1} × Y_{2} such that D · Γ̄_{f} = D_{0} · Γ̄_{f} + γ_{1} where γ_{1} is a cycle supported on E. We conclude from the above that for all 1 ≤ k ≤ dim Y_{1},

(8)  D^{k} · Γ̄_{f} = D_{0}^{k} · Γ̄_{f} + γ_{k}

where γ_{k} is a cycle supported on E. This can be proved by induction. In equation (8), we apply p_{1,∗} to both sides and note that p_{1,∗}γ_{k} = 0, 1 ≤ k ≤ dim Y_{1} − 1, since p_{1} contracts E to points while dim γ_{k} ≥ 1. It follows that

p_{1,∗}(D^{k} · Γ̄_{f}) = p_{1,∗}(D_{0}^{k} · Γ̄_{f}), 1 ≤ k ≤ dim Y_{1} − 1.

For any divisor D on Y_{1}×Y_{2}, we take D_{0} = p_{1}^{∗}p_{1,∗}(D·Γ̄_{f}). Since p_{1}|Γ̄_{f} : Γ̄_{f} → Y_{1} is an isomorphism away from E, we see that D · Γ̄_{f} = D_{0} · Γ̄_{f} + γ_{1} for some cycle γ_{1} supported on E. Hence we get

p_{1,∗}(D^{k} · Γ̄_{f}) = p_{1,∗}(D_{0}^{k} · Γ̄_{f})
= p_{1,∗}(p_{1}^{∗}(D′)^{k} · Γ̄_{f}), D′ = p_{1,∗}(D · Γ̄_{f})
= (D′)^{k} · p_{1,∗}Γ̄_{f}, (the projection formula)
= (D′)^{k} = (p_{1,∗}(D · Γ̄_{f}))^{k}

for all divisors D on Y_{1} × Y_{2} and 1 ≤ k ≤ dim Y_{1} − 1. In particular, if we take D = p_{2}^{∗}𝔞 for some 𝔞 ∈ Pic(Y_{2}), then we get

f^{∗}𝔞^{k} = p_{1,∗}(Γ̄_{f} · p_{2}^{∗}𝔞^{k}) = p_{1,∗}(Γ̄_{f} · (p_{2}^{∗}𝔞)^{k}) = (p_{1,∗}(Γ̄_{f} · p_{2}^{∗}𝔞))^{k} = (f^{∗}𝔞)^{k},

where 1 ≤ k ≤ dim Y_{1} − 1. The case k = 0 is automatic. □

Let Γ_{ρ} ⊂ (S_{C} − Σ_{v}) × ℙ(ℱ) be the graph of ρ and Γ̄_{ρ} ⊂ S_{C} × ℙ(ℱ) its closure. Note that for any 3-dimensional vector space V ⊂ T_{X,x} containing T_{v,x}, the condition (([l], x), V/T_{v,x}) ∈ Γ_{ρ} implies that l ⊂ ℙ(V). We define the closed subvariety Z ⊂ S_{C} × ℙ(ℱ) as

Z = {(([l], x), V/T_{v,x}) ∈ S_{C} × ℙ(ℱ) | l ⊂ ℙ(V)}.

Lemma 3.23. Γ̄_{ρ} = Z in CH_{n−2}(S_{C} × ℙ(ℱ)).

Proof. From Lemma 3.21, Z agrees with Γ_{ρ} away from Σ_{v}, namely

Γ_{ρ} = Z|_{(S_{C}−Σ_{v})×ℙ(ℱ)} .

It follows that Γ̄_{ρ} ⊂ Z. Hence we have

Z = Γ̄_{ρ} + ∑_{i=1}^{5e^{2}}{([L_{i}], x_{i})} × ℙ(ℱ_{x_{i}}), in CH_{n−2}(S_{C} × ℙ(ℱ)).

Since dim ℙ(ℱ_{x_{i}}) = n−3, we see that {([L_{i}], x_{i})}×ℙ(ℱ_{x_{i}}) = 0 in CH_{n−2}(S_{C} × ℙ(ℱ)). Hence the lemma follows. □

Definition 3.24. Let Γ_{v} ⊂ S_{C} ×S_{C} be the closure of (S_{C}\Σ_{v})×_{ℙ(ℱ)} (S_{C}\Σ_{v}).

Remark 3.25. Let ([l], x) ∈ S_{C} be a general point. Then T_{l,x} and T_{v,x} span a linear 3-dimensional vector space V ⊂ T_{X,x} and ρ([l], x) = [V/T_{v,x}]. Then by Lemma 3.21, there are 5 other lines l_{i}, i = 1, …, 5, such that T_{l_{i},x} ⊂ V. Then Γ_{v} is the correspondence generically defined by

([l], x) ↦ ∑_{i=0}^{5}([l_{i}], x),

where l_{0} = l.

Lemma 3.26. Γ_{v} = Γ̄_{ρ}^{t} ∘ Γ̄_{ρ} in CH_{n−2}(S_{C} × S_{C}).

Proof. By Lemma 3.23, we may replace Γ̄_{ρ} by Z. By the definition of composition of correspondences (see [Fu, §16.1]), we know that

Z^{t} ∘ Z = p_{13,∗}(p_{12}^{∗}Z · p_{23}^{∗}Z^{t})
= p_{13,∗}(Z × S_{C} ∩ S_{C} × Z^{t})

It follows that Z^{t} ∘ Z is represented by the cycle

{(([l], x), ([l′], x)) ∈ S_{C} × S_{C} : [l], [l′] ∈ F_{x} ∩ ℙ(V) for some [V/T_{v,x}] ∈ ℙ(ℱ_{x})}

One directly checks that this cycle is Γ_{v}. □

Lemma 3.27. Let Γ_{v} act either on the cohomology groups or on the Chow groups of S_{C}.

(i) If 𝔞 is an odd dimensional topological cycle, then (Γ_{v})_{∗}𝔞 = 0; (ii) If 𝔞 is zero dimensional, then (Γ_{v})_{∗}𝔞 is a constant that only depends on the degree of 𝔞; (iii) Let 𝔞 = [S_{C}], then (Γ_{v})_{∗}𝔞 = 6𝔞; (iv) If 𝔞 is a topological cycle of real codimension 2m or an algebraic cycle of codimension m, then (Γ_{v})_{∗}𝔞 is a linear combination of g^{m} and g′g^{m−1} which only depends on the intersection numbers 𝔞 · g^{n−m−2} and 𝔞 · g′g^{n−m−3}.

Proof. By Lemma 3.26, we know that

(Γ_{v})_{∗} = (Γ̄_{ρ}^{t} ∘ Γ̄_{ρ})_{∗} = (Γ̄_{ρ}^{t})_{∗}(Γ̄_{ρ})_{∗} = ρ^{∗}ρ_{∗}.

Let ξ be the relative O_{ℙ(ℱ)} (1) class on ℙ(ℱ). Then the cohomology of ℙ(ℱ) is naturally given by

H^{∗}(ℙ(ℱ)) = H^{∗}(C)[ξ], ξ^{n−2} + π^{∗}c_{1}(ℱ) · ξ^{n−3} = 0.

Where π : ℙ(ℱ) → C is the natural projection. Similarly, we have the description of the Chow ring as

CH^{∗}(ℙ(ℱ)) = CH^{∗}(C)[ξ], ξ^{n−2} + π^{∗}c_{1}(ℱ) · ξ^{n−3} = 0.

Consider the following diagram

\xymatrix{S_{C} \ar[r]^{j_{1}} \ar@{-->}[dr]^{ρ} & ℙ(T_{X}|_{C}) \ar[r]^{α} \ar@{-->}[d]^{β} & C \\  & ℙ(ℱ) \ar@{-->}[ur]_{π}}

The rational map β is defined by the natural homomorphism T_{X}|_{C} → ℱ. The locus where β is not defined is exactly ℙ(T_{v}). The pull back β^{∗}ξ restricts to a hyperplane class on each fiber of α. The variety ℙ(T_{X}|_{C}) parameterizes all lines in ℙ^{n+1} which are tangent to X at some point x ∈ C (remembering the point x). Let g̃ ⊂ ℙ(T_{X}|_{C}) be the divisor corresponding to all the lines meeting a given linear ℙ^{n−1} ⊂ ℙ^{n+1} in general position. Then we see that i^{∗}g̃ = g|_{S_{C}} and this is denoted again by g for simplicity. One also sees that g̃ restricts to a hyperplane class of ℙ(T_{X,x}) = α^{−1}(x). Thus g̃ and β^{∗}ξ differ by a class coming from C. After pulling back to S_{C}, we get

ρ^{∗}ξ = g + rg′

for some integer r. By Lemma 3.22, we get the following key equality

(9)  ρ^{∗}ξ^{k} = (ρ^{∗}ξ)^{k} = (g + rg′)^{k}

Let 𝔞 ∈ H^{2m+1}(S_{C}) be a topological class of odd dimension, then ρ_{∗}𝔞 is an element in H^{2m+1}(ℙ(ℱ)) which is zero. Hence (Γ_{v})_{∗}(𝔞) = ρ^{∗}ρ_{∗}𝔞 = 0. Now let 𝔞 be an element of H^{2m}(S_{C}) (or CH^{m}(S_{C})), 0 < m < n − 2 (the cases of m = 0, n−2 are quite easy to deal with). Then ρ_{∗}𝔞 is an element in H^{2m}(ℙ(ℱ)) (or CH^{m}(ℙ(ℱ))). Then we have the following expression

ρ_{∗}𝔞 = aξ^{m} + bπ^{∗}[pt] · ξ^{m−1},

for some a, b ∈ ℤ. Apply ρ^{∗} to the above identity and use Lemma 3.22, we get

(Γ_{v})_{∗}(𝔞) = ρ^{∗}ρ_{∗}𝔞 = a(g + rg′)^{m} + bg′(g + rg′)^{m−1} = ag^{m} + (b + ma)g^{m−1}g′

Also, the numbers a and b can be determined in the following way

a = (aξ^{m} + bπ^{∗}[pt]ξ^{m−1}) · π^{∗}[pt]ξ^{n−m−3}
= ρ_{∗}𝔞 · π^{∗}[pt]ξ^{n−m−3}
= 𝔞 · ρ^{∗}(π^{∗}[pt]ξ^{n−m−3})
= 𝔞 · g′(g + rg′)^{n−m−3}, (by Lemma 3.22)
= 𝔞 · g′g^{n−m−3}

To get b, we consider

ρ_{∗}𝔞 · ξ^{n−m−2} = aξ^{n−2} + b
= a(−π^{∗}c_{1}(ℱ))ξ^{n−3} + b
= −a deg ℱ + b

Hence we have

b = a deg ℱ + 𝔞 · ρ^{∗}ξ^{n−m−2}
= a deg(ℱ) + 𝔞 · (g + rg′)^{n−m−2}

Thus b only depends on the intersection numbers 𝔞 · g^{n−m−2} and 𝔞 · g′g^{n−m−3}. This finishes the proof. □

3.4. Proof of Theorem 3.6. Let T ⊂ Hilb^{e}(X) be a general smooth curve passing through the point [C]. Let {𝒞_{t} : t ∈ T } be the corresponding 1dimensional family of rational curves on X such that the fiber 𝒞_{0} = C at the special point 0 = [C] ∈ T. Let S_{t} = S_{𝒞_{t}} and i_{t} = i_{𝒞_{t}} : S_{t} → F. Let I′_{t} ⊂ S_{t} × S_{C}, t ≠ 0, be the natural incidence correspondence, namely

I′_{t} = {(([l′], x′), ([l], x)) ∈ S_{t} × S_{C} : l ∩ l′ ≠ ∅} = (i_{t} × i_{C})^{∗}I ∈ CH_{n−2}(S_{t} × S_{C}).

Let I_{t} ⊂ S_{t} × S_{C} be the correspondence generically defined by

(10)  I_{t} : ([l′], x′) ↦ ∑_{i=1}^{5e}([l′_{i}], x′_{i}),

where ([l]′, x′) ∈ S_{t} is a general point, l′_{i} the incidence lines of l′ and C, x′_{i} = l′_{i} ∩ C.

Lemma 3.28. We have I_{t} = I′_{t} in CH_{n−2}(S_{t} × S_{C}).

Proof. Let Ĩ_{t} = (i_{t} × i_{C})^{−1}I. By definition, I_{t} is equal to the cycle class of Ĩ_{t}. Consider the projection p_{1} : Ĩ → S_{t}. Let ([l′], x′) ∈ S_{t} be any point. If l′ does not meet C, then p_{1}^{−1}([l′], x′) = ∑_{i=1}^{5e}([l_{i}], x_{i}) where l_{i} run through all the incidence lines of l′ and C; if l′ ∩ C = y, then p_{1}^{−1}([l′], x′) is the union of F_{y} = q_{0}^{−1}y and the set of all secant lines of the pair (l′, C). It follows that

Ĩ_{t} = I′_{t} ∪ ( ⋃_{j=1}^{5e^{2}}{[L_{j,t}], x_{j,t})} × F_{y_{j,t}}),

where {L_{j,t}}_{j=1}^{5e^{2}} is the set of incidence lines of 𝒞_{t} and C, x_{j,t} = L_{j,t} ∩ 𝒞_{t} and y_{j,t} = L_{j,t} ∩ C. Since dim F_{y_{j,t}} = n − 3, we see that the cycle class of Ĩ_{t} is equal to that of I′_{t}. □

Back to the proof of Theorem 3.6. In equation (10), when t specializes to 0, l′ specializes to a line l meeting C; there are 5 lines, say l′_{1}, …, l′_{5}, among all the l′_{i}’s, that specialize to five lines, E_{i}, i = 1, 2, …, 5, passing through x = l ∩ C; all the other lines of the l′_{i}’s specialize to secant lines, l_{i}, of the pair (l, C). We have the following description of the E_{i}’s. Since T is a smooth curve in the Hilbert scheme of degree e rational curves on X, the tangent space T_{T,0} gives a one dimensional subspace of T_{Hilb^{e}(X),[C]} = H^{0}(C, 𝒩_{C/X}). Let v ∈ H^{0}(C, 𝒩_{C/X}) be a generator of T_{T,0}. By choosing T general enough, we may assume that v, as a section of H^{0}(C, 𝒩_{C/X}), is general. To this v, we can associate T_{v} ⊂ T_{X}|_{C} with quotient ℱ, Σ_{v} ⊂ S_{C} and ρ : S_{C}\Σ_{v} → ℙ(ℱ) as before. We use the notation and results of [Fu, §11.1]. Let I_{0} = lim_{t→0} I_{t}. In the equation (10), we take the limit as t → 0, then we get

(11)  I_{0} : ([l], x) ↦ ∑_{i=6}^{5e}([l′_{i}], x_{i}) + ∑_{i=1}^{5}([E_{i}], x),

where {l′_{i}}_{i=6}^{5e} is the set of all secant lines of the pair (l, C) and x_{i} = l′_{i} ∩ C. Note that the rule in (11) generically defines I_{C} + Γ_{v} − Δ_{S_{C}}. We see that the difference of I_{0} and I_{C} + Γ_{v} − Δ_{S_{C}} is supported on Σ_{v} ×_{C} ℙ(ℱ). Since dim Σ_{v} ×_{C} ℙ(ℱ) = n − 3, we have

(12)  I_{0} = I_{C} + Γ_{v} − Δ_{S_{C}} , in CH_{n−2}(S_{C} × S_{C}).

Note that I_{t} = I|_{S_{t}×S_{C}} for t ≠ 0. By taking limits and applying [Fu, Proposition 11.1], we know that the class of I_{0} is equal to I restricted to S_{C} × S_{C}. Equivalently, we have I_{0} = I′_{C} in CH_{n−2}(S_{C} × S_{C}). Thus we get the following key identity

(13)  I′_{C} = I_{C} + Γ_{v} − Δ_{S_{C}} .

Since Ψ_{C} = Ψ ∘ (i_{C})_{∗}, Φ_{C} = (i_{C})^{∗} ∘ Φ and Φ ∘ Ψ = I_{∗}, we see that

Φ_{C} ∘ Ψ_{C} = (i_{C})^{∗} ∘ I_{∗} ∘ (i_{C})_{∗} = ((i_{C} × i_{C})^{∗}I)_{∗} = (I′_{C})_{∗}

Combine this with the equation (13), we obtain

Φ_{C} ∘ Ψ_{C} = σ − 1 + (Γ_{v})_{∗}

as actions on either the cohomology groups or the Chow groups of S_{C}. Then Theorem 3.6 follows easily from Lemma 3.27.

3.5. The subalgebra ℚ[g, g′]. On S_{C} we have the polarization g and the class g′ = q_{0}^{∗}[pt], where q_{0} = q|_{S_{C}} : S_{C} → C is the natural morphism.

Lemma 3.29. Ψ_{C}(g′ · Φ_{C}(h^{m})) = 2h^{m+1}.

Proof. First we note that Ψ_{C}(g′ · Φ_{C}(h^{m})) = Ψ(F_{x} · Φ(h^{m})). Let M ⊂ X be a general complete intersection of hyperplanes that represents h^{m}. Choose x ∈ X to be a general point. Similar to the proof of Theorem 2.2, we consider the following diagram

\xymatrix{D_{1} + D_{2} + D \ar[r] \ar[d] & Y \ar[r] \ar[d] & X \ar[d] \\ P \ar[r] \ar[d]^{π} & G(1, 2, n + 2) \ar[r] \ar[d] & ℙ^{n+1} \\ M \ar[r]^{φ} & G(2, n + 2)}

where all the squares are fiber products; φ(x′) is the line through x and x′, for all x′ ∈ M; D_{1} ⊂ P is a section of π and the natural morphism g_{1} : D_{1} → X contracts D_{1} to x ∈ X; D_{2} is a section of π and the natural morphism g_{2} : D_{2} → X maps D_{2} isomorphically to M ⊂ X; D corresponds to the third point in the intersection of the lines with X. Let g : D → X be the natural morphism. Let Δ ⊂ M be the intersection of M and D_{x} (recall that D_{x} is the variety swept out by lines through x). Then, by Bertini’s theorem, Δ is smooth of codimension 2 in M. As in Lemma 2.8, the natural map π′ : D → M is the blow-up along Δ. Let E ⊂ D be the exceptional divisor. As before, we have

(14)  h|_{D} + E = 2(π|_{D})^{∗}(h|_{M})

where h|_{D} = g^{∗}h. One checks that the push-forward of π^{∗}(h|_{M})|_{(D_{1}+D_{2}+D)} to X is equal to 3h^{m+1} and that

(g_{2})_{∗}(π^{∗}(h|_{M})|_{D_{2}}) = M · h = h^{m+1}
(g_{1})_{∗}(π^{∗}(h|_{M})|_{D_{1}}) = 0

It follows that

g_{∗}(π^{∗}(h|_{M})|_{D}) = 3h^{m+1} − h^{m+1} = 2h^{m+1}.

Note that the push-forward of D + D_{1} + D_{2} to X is P |_{X} = 3h^{m}. We also know that (g_{1})_{∗}D_{1} = 0 and (g_{2})_{∗}D_{2} = M = h^{m}. Hence we get g_{∗}D = 2h^{m}. As a consequence, we have g_{∗}(h|_{D}) = g_{∗}D · h = 2h^{m+1}. We apply g_{∗} to the equality (14) and note that g_{∗}E = Ψ(F_{x} · Φ(h^{m})). Then we get

Ψ(F_{x} · Φ(h^{m})) = 2g_{∗}(π^{′∗}h|_{M}) − g_{∗}(g^{∗}h) = 2(2h^{m+1}) − 2h^{m+1} = 2h^{m+1}.

Hence the lemma follows. □

Proposition 3.30. The following statements hold in H^{∗}(S_{C}, ℚ).

(1) g^{m} is a linear combination of Φ_{C}(h^{m+1}) and g′Φ_{C}(h^{m}); g′g^{m−1} is a multiple of g′Φ_{C}(h^{m}). The algebra ℚ[g, g′] is generated by cycles of the form Φ_{C}(h^{m+1}) with 0 ≤ m ≤ n − 2 and g′Φ_{C}(h^{m}) with 1 ≤ m ≤ n − 2.

(2) The restriction of H^{∗}(G) to H^{∗}(S_{C}) is equal to ℚ[g, g′g] ⊂ ℚ[g, g′].

(3) The action of σ preserves the algebra ℚ[g, g′].

(4) Under Ψ_{C}, the image of ℚ[g, g′] is ℚ[h]_{0<deg<n} where h is the class of a hyperplane on X.

Proof. Before proving the proposition, we review a bit of Schubert calculus. We refer to §14.7 of [Fu] for the details of the general theory. Schubert calculus on G = G(2, n + 2) shows that CH^{m}(G) = H^{2m}(G) is generated by Schubert varieties that are defined by flags ℙ^{a} ⊂ ℙ^{b} ⊂ ℙ^{n+1} with a < b ≤ n + 1 and a + b = 2n + 1 − m. A Schubert variety is the space of lines meeting ℙ^{a} and contained in ℙ^{b}. In our case, most of these classes restrict to 0 on S_{C} unless b = n or b = n + 1. If b = n + 1 and a = n − m, where 0 ≤ m ≤ n − 1, then the corresponding Schubert class restricts to Φ(h^{m+1}) on F and hence restricts to Φ_{C}(h^{m+1}) on S_{C}. If b = n and a = n − m + 1,where 2 ≤ m ≤ n, then the corresponding Schubert class restricts to deg(C) g′ · Φ_{C}(h^{m}) on S_{C}. Here we note that F_{y} ⊂ S_{C} represents g′ on S_{C}, for any y ∈ C. As a cycle on F, the class of F_{y} comes from the restriction of some class from G. However, viewed as a class on S_{C}, F_{y} is not the restriction of any class from G. The degree of a nonzero homogeneous element of ℚ[g, g′] means the degree of that element as a polynomial. Since g^{m} comes from G, we know that it can be written as a linear combination of Φ_{C}(h^{m+1}) and g′Φ_{C}(h^{m}). Hence we also see that g′g^{m−1} is a multiple of g′Φ_{C}(h^{m}) since (g′)^{2} = 0. This proves (1). The discussion of Schubert calculus above shows that the restriction of H^{∗}(G) to S_{C} is generated by Φ_{C}(h^{m+1}), 0 ≤ m ≤ n−2 and g′Φ_{C}(h^{m}), 2 ≤ m ≤ n−2. Combining this fact with (1), we see that the image of the restriction H^{∗}(G) → H^{∗}(S_{C}) consists of elements f ∈ ℚ[g, g′] such that the coefficient of the term g′ in f is zero. This proves that the image of H^{∗}(G) → H^{∗}(S_{C}) is ℚ[g, g′g]. To prove (3), we only need to consider the action of σ on Φ_{C}(h^{m+1}) and g′Φ_{C}(h^{m}). By Proposition 3.6, the following equations hold modulo ℚ[g, g′].

σ(Φ_{C}(h^{m+1})) =

= Φ_{C}(h^{m+1}) + Φ_{C}Ψ_{C}(Φ_{C}(h^{m+1}))

= Φ_{C} ∘ Ψ(Φ(h^{m+1}) · F_{C})

= \begin{cases}Φ_{C}(3e deg(h^{m+1})h^{m+1} − 2eh^{m+1}), & m < n − 2, \\ Φ_{C}(3e deg(h^{m+1})h^{m+1} − 2eh^{m+1} − 2 deg(h^{n−1})C), & m = n − 2 \end{cases}

= \begin{cases}0, & m < n − 2 \\ −6Φ_{C}(C), & m = n − 2 \end{cases}

Here the first equation uses Proposition 3.6; the second equation uses the fact that Φ_{C}(h^{m+1}) ∈ ℚ[g, g′]; the third equation uses Theorem 2.2; the last one uses the fact that the degree of h^{m} with respect to the polarization h is 3. In the case m = n − 2, we have that the cohomology class of Φ_{C}(C) is 5e^{2}[pt] ∈ H^{2n−4}(S_{C}, ℚ). Note that H^{2n−4}(S_{C}, ℚ) = ℚ[pt] = ℚ[g, g′]_{deg=n−2}. Hence we also have Φ_{C}(C) ∈ ℚ[g, g′]. Similarly, the following equalities hold modulo ℚ[g, g′].

σ(g′ · Φ_{C}(h^{m})) = Φ_{C} ∘ Ψ_{C}(g′ · Φ_{C}(h^{m})) + g′ · Φ_{C}(h^{m})
= 2Φ_{C}(h^{m+1}) + g′Φ_{C}(h^{m})

where we use Lemma 3.29. To prove (4), we also consider the action of Ψ_{C} on Φ_{C}(h^{m+1}) and g′Φ_{C}(h^{m}). By Theorem 2.2 and Lemma 3.29, we know that the images are multiples of h^{m+1}, 0 ≤ m ≤ n − 2 (we have this range since otherwise the cycles on S_{C} are automatically zero for dimension reasons). □

4. The quadratic relation and Prym-Tjurin construction

Let C ⊂ X be a general smooth rational curve on X with e = deg(C) ≥ 2. Let S_{C} = q^{−1}(C). As was shown in Lemma 3.1, S_{C} is smooth and it is the normalization of the variety of all lines meeting C. We will use the notation from the previous section.

Definition 4.1. We define the primitive cohomology, H^{∗}(S_{C}, ℤ)^{∘}, of S_{C}, to be the set of all elements α ∈ H^{∗}(S_{C}, ℤ) such that

⟨α, β⟩ := ∫_{S_{C}} α ∪ β = 0, ∀β ∈ ℚ[g, g′].

We define the primitive Chow group, CH^{∗}(S_{C})^{∘}, of S_{C} , to be the set of all elements α ∈ CH^{∗}(S_{C}) such that

⟨[α], β⟩ := ∫_{S_{C}} [α] ∪ β =_{num} 0, ∀β ∈ ℚ[g, g′],

where [α] means the cohomoloy class of α. An element α ∈ H^{∗}(S_{C})^{∘} is called a primitive cohomology class; an element α ∈ CH^{∗}(S_{C})^{∘} is called a primitive cycle class.

Remark 4.2. Since the restriction of H^{∗}(G, ℚ) to S_{C} is ℚ[g, gg′], we deduce that H^{n−2}(S_{C}, ℤ)^{∘} (n ≠ 4) consists of elements α such that ⟨α, β⟩ = 0 for all β coming from G. Note that the only difference between ℚ[g, g′] and ℚ[g, gg′] is the element g′ since (g′)^{2} = 0; when n ≠ 4, the equation ⟨α, g′⟩ = 0 is automatic for dimension reasons since g′ ∈ H^{2}(S_{C}) and α ∈ H^{n−2}(S_{C}) are not of complementary dimension. If n is odd, then every cohomology class in H^{n−2}(S_{C}) is primitive by definition. By abuse of notation, we will also use α · β to denote ⟨α, β⟩ for α and β of compementary dimensions.

By Proposition 3.30, the action σ induces an action, still denoted by σ, on the primitive cohomology and the primitive Chow groups of S_{C}. This is because σ is symmetric and preserves ℚ[g, g′]. Hence α · σ(β) = σ(α) · β. If α is a primitive class and β ∈ ℚ[g, g′], then the above identity shows that σ(α)·β = 0 and hence σ(α) is still primitive.

Theorem 4.3. Let C ⊂ X be a general rational curve of degree e ≥ 2 as above. Let σ be the action of the incidence correspondence on either H^{∗}(S_{C})^{∘} or CH^{∗}(S_{C})^{∘}. Then the following are true.

(1) On the primitive part of either the cohomology groups or the Chow groups, σ satisfies the following quadratic relation

(σ − 1)(σ + 2e − 1) = 0

(2) The map Φ_{C} induces an isomorphism of Hodge structures

Φ_{C} : H^{n}(X, ℤ)_{prim} → P (H^{n−2}(S_{C}, ℤ)^{∘}, σ)(−1)

The intersection forms are related by the following identity

Φ_{C}(α) · Φ_{C}(β) = −2e α · β

(3) The map Φ_{C} induces an isomorphism

Φ_{C} : A_{i}(X)_{ℚ} → P (CH_{i−1}(S_{C})_{ℚ}^{∘}, σ)

Proof. Since Γ_{v} is also symmetric, we know that it acts on the primitive cohomology and the primitive Chow groups. But by Lemma 3.27, the image of Γ_{v} is always non-primitive unless it is zero. Hence we get that (Γ_{v})_{∗} = 0 on primitive cohomology and Chow groups. Hence on the primitive part of the cohomology group and the Chow groups, we have

σ = Φ_{C} ∘ Ψ_{C} + 1.

The next fact that we need is

Lemma 4.4. If 𝔞 is a primitive class in either the cohomology group or the Chow groups of S_{C}, then Ψ_{C}(𝔞) has h-degree zero.

The proof of the above lemma is easy. We note that Ψ_{C}(𝔞)·h^{i} = 𝔞·Φ_{C}(h^{i}) = 0, since Φ_{C}(h^{i}) ∈ ℚ[g, g′]; see (1) of Proposition 3.30. Now we can prove statement (1) of the theorem. Theorem 2.2 shows that, on the primitive cohomology and the primitive Chow groups, we have

Ψ_{C}(Φ_{C}(𝔞)) + 2e𝔞 = 0.

From this we get

(σ − 1)(σ + 2e − 1)(𝔞) = Φ_{C} ∘ Ψ_{C}(Φ_{C} ∘ Ψ_{C} + 2e)(𝔞)

= Φ_{C}(Ψ_{C} ∘ Φ_{C}(Ψ_{C}(𝔞))) + 2eΦ_{C} ∘ Ψ_{C}(𝔞)

= Φ_{C}(−2eΨ_{C}(𝔞)) + 2eΦ_{C} ∘ Ψ_{C}(𝔞), (by Theorem 2.2) = 0

Now we prove (2). For simplicity, we write P for P (H^{n−2}(S_{C}, ℤ)^{∘}, σ). Since Φ_{C}∘Ψ_{C} = σ−1 and that Ψ_{C} is onto (Propositioon 4.5), we know that the image of Φ_{C} is exactly P. By Theorem 2.2, we know Ψ_{C} ∘ Φ_{C} = −2e. This implies that Φ_{C} is injective. Hence Φ_{C} : H^{n}(X, ℤ)_{prim} → P is an isomorphism. Φ_{C} respects the Hodge structures and hence is an isomorphism of Hodge structures. The intersection forms are related by

Φ_{C}(α) · Φ_{C}(β) = α · Ψ_{C}(Φ_{C}(β)) = α · (−2eβ) = −2eα · β

Statement (3) can be proved exactly in the same way. □

Proposition 4.5. (1) The homomorphism

Ψ_{C} : H^{n−2}(S_{C}, ℤ)^{∘} → H^{n}(X, ℤ)_{prim}

on primitive cohomology is surjective.

(2) The homomorphism

Ψ_{C} : CH_{m}(S_{C}, ℚ)^{∘} → A_{m+1}(X, ℚ)

on primitive Chow groups is surjective.

Proof. Statement (1) follows from Theorem 5.1 (for n ≠ 4) and Remark 5.10 (for n = 4). The proof of (2) is easy since we have ℚ coefficients. Let α ∈ CH_{m+1}(X)_{ℚ}, take 𝔞 = −\frac{1}{2e}Φ_{C}(α). Then we have α = Ψ_{C}(𝔞). Now assume that α has h-degree 0. For any 𝔟 ∈ ℚ[g, g′], we have Φ_{C}(α) · 𝔟 = α · Ψ_{C}(𝔟) = 0 since Ψ_{C}(𝔟) ∈ ℚ[h]. This means that Φ_{C}(α) is a primitive cycle class. This gives the surjectivity of Ψ_{C} on primitive Chow groups. □

Let π : 𝒞 → T be a family of curves together with a morphism f : 𝒞 → X. Then we can define the Abel-Jacobi homomorphism Φ_{T} := π_{∗}f^{∗} and the cylinder homomorphism Ψ_{T} := f_{∗}π^{∗} on the cohomology groups and the Chow groups. Let C ⊂ X be a general smooth rational curve of degree e ≥ 2 as before. Then there is a natural incidence correspondence Γ_{T,C} ⊂ T × S_{C} given by

t ↦ ∑([l_{i}], x_{i})

where l_{i} are the incidence lines of 𝒞_{t} and C (i.e. lines meeting both curves) and x_{i} = l_{i} ∩ C. This correspondence induces

(Γ_{T,C})_{∗} : CH_{r}(T) → CH_{r}(S_{C})

and

[Γ_{T,C}]^{∗} : H^{n−2}(S_{C}, ℤ) → H^{n−2}(T, ℤ)

We can define the primitive cohomology groups and the primitive Chow groups of T as the classes that are orthogonal to Φ_{T} (ℤ[h]). Hence Γ_{T,C} also induces homomorphisms between the primitive cohomology groups and the primitive Chow groups.

Proposition 4.6. Let 𝒞 → T be a family of curves on X and C ⊂ X a general rational curve of degree e as above. All the homomorphisms are restricted to primitive classes. Then the following hold.

(1) The homomorphism [Γ_{T,C}]^{∗} : H^{n−2}(S_{C}, ℤ)^{∘} → H^{n−2}(T, ℤ) factors as Φ′_{T} ∘ (σ − 1), where Φ′_{T} : P (H^{n−2}(S_{C}, ℤ)^{∘}, σ) → H^{n−2}(T, ℤ) is a homomorphism such that the following diagram is commutative.

\xymatrix{H^{n−2}(S_{C}, ℤ) \ar[d]_{σ−1} \ar[r]^{Ψ_{C}} & H^{n}(X, ℤ)_{prim} \ar[d]^{Φ_{T}} \ar[dl]_{Φ_{C}} \\ P (H^{n−2}(S_{C}, ℤ)^{∘}, σ) \ar[r]^{Φ′_{T}} & H^{n−2}(T, ℤ)}

(2) This image of (Γ_{T,C})_{∗} : CH_{r}(T, ℚ)^{∘} → CH_{r}(S_{C}, ℚ)^{∘} is contained in the subgroup P (CH_{r}(S_{C}, ℚ)^{∘}, σ), where CH_{r}(T, ℚ)^{∘} is the subgroup of primitive elements. In other words, the following diagram is commutative.

\xymatrix{ & A_{r+1}(X, ℚ) \ar[dr]^{Φ_{C}} \\ CH_{r}(T, ℚ)^{∘} \ar[rr]^{(Γ_{T,C})_{∗}} \ar[ur]^{Ψ_{T}} &  & P (CH_{r}(S_{C}, ℚ)^{∘}, σ)}

Proof. These are consequences of the identities [Γ_{T,C}]^{∗} = Φ_{T} ∘ Ψ_{C} and (Γ_{T,C})_{∗} = Φ_{C} ∘ Ψ_{T}. □

Let C_{1} and C_{2} be two general rational curves on X of degree e_{1} and e_{2} respectively. Then there is a natural incidence correspondence Γ_{12} ⊂ S_{C_{1}} × S_{C_{2}} defined by

([l], x) ↦ ∑_{i=1}^{5e_{2}}([l_{i}], x_{i})

where ([l], x) ∈ S_{C_{1}}, l_{i} are the incidence lines of l and C_{2}), x_{i} = l_{i} ∩ C_{2}. Let γ_{12} = (Γ_{12})_{∗} be the induced homomorphism on either the cohomology groups or the Chow groups. Note that by definition, we have γ_{12} = Φ_{C_{2}} ∘ Ψ_{C_{1}}. Since both Φ_{C} and Ψ_{C} respect primitive classes, the above identity implies that γ_{12} takes primitive classes to primitive classes. We still use γ_{12} to denote the action on the primitive cohomology and the primitive Chow groups.

Proposition 4.7. Let C_{1}, C_{2} ⊂ X be two general rational curves of degree at least 2 and γ_{12} be the homomorphism induced by incidence correspondence as above. Let σ_{1} and σ_{2} be the action of the self incidence correspondence on S_{C_{1}} and S_{C_{2}} respectively. Then the following are true.

(1) The image of γ_{12} : H^{n−2}(S_{C_{1}}, ℤ)^{∘} → H^{n−2}(S_{C_{2}}, ℤ)^{∘} is always in the Prym-Tjurin part. Furthermore there is an isomorphism of Hodge structures

t_{12} : P (H^{n−2}(S_{C_{1}}, ℤ)^{∘}, σ_{1}) → P (H^{n−2}(S_{C_{2}}, ℤ)^{∘}, σ_{2})

such that Φ_{C_{2}} = t_{12} ∘ Φ_{C_{1}} and γ_{12} = t_{12}(σ_{1} − 1).

(2) The same conclusions as in (1) hold for primitive Chow groups with ℚcoefficient.

Proof. For simplicity, we write Λ_{i} = H^{n−2}(S_{C_{i}}, ℤ)^{∘} for the primitive cohomology and P_{i} = Im(σ_{i} − 1) for the Prym-Tjurin part, i = 1, 2. Let Λ = H^{n}(X, ℤ)_{prim}, Φ_{i} = Φ_{C_{i}} : Λ → P_{i}. Then one easily checks that t_{12} = Φ_{2} ∘ Φ_{1}^{−1} satisfies (1). The proof of (2) is similar. □

5. Surjectivity of Ψ_{C} on primitive cohomology

In this section we supply a proof of the surjectivity of Ψ_{C} on the primitive cohomologies. To do this, it is more convenient to consider homology instead of cohomology. We define

(15)  V_{n−2}(S_{C}, ℤ) = ker{H_{n−2}(S_{C}, ℤ) → H_{n−2}(G, ℤ)},

and

(16)  V_{n}(X, ℤ) = ker{H_{n}(X, ℤ) → H_{n}(ℙ^{n+1}, ℤ)}.

Under the Poincaré duality H_{n−2}(S_{C}, ℤ) ≅ H^{n−2}(S_{C}, ℤ), the subspace V_{n−2}(S_{C}, ℤ) corresponds to H^{n−2}(S_{C}, ℤ)^{∘} if n ≠ 4. When n = 4, since the class g′ is not from G(2, 6), we see that V_{2}(S, ℤ) is larger than H^{2}(S_{C}, ℤ)^{∘}. Actually, we have

H^{2}(S_{C}, ℤ)^{∘} = {σ ∈ V_{2}(S_{C}, ℤ) : g′ · σ = 0}.

Similarly the Poincaré duality on X allows us to identify V_{n}(X, ℤ) with H^{n}(X, ℤ)_{prim}. Then the surjectivity of

Ψ_{C} : H^{n−2}(S_{C}, ℤ)^{∘} → H^{n}(X, ℤ)_{prim}

when n ≠ 4, is equivalent to the following

Theorem 5.1. The natural cylinder homomorphism

Ψ_{C} : V_{n−2}(S_{C}, ℤ) → V_{n}(X, ℤ)

is surjective.

The idea of the proof is the Clemens-Letizia method, see [Cl] and [Le]. Our presentation closely follows that of [Shi, §2,3]. Let π : V → Δ be a proper flat holomorphic map from a complex manifold V of dimension m+1 onto the unit disk Δ. This map is called a degeneration if π is smooth over the punctured disk Δ^{∗} = Δ − 0 and V_{t} := π^{−1}(t) is irreducible for t ≠ 0. Let Sing(V_{0}) denote the singular locus of V_{0}.

Definition 5.2. ([Shi, Definition 1]) A degeneration π : V → Δ is called quadratic of codimension r if Sing(V_{0}) is connected and, for every point p ∈ Sing(V_{0}), there exist local coordinates (z_{0}, …, z_{m}) of V around p such that π = z_{0}^{2} + ⋯ + z_{r}^{2}.

Proposition 5.3. For any smooth cubic hypersurface X ⊂ ℙ^{n+1}, there exist a Lefschetz pencil X_{t} ⊂ ℙ^{n+1} with t ∈ B ≅ ℙ^{1}, such that

(1) The total space, F = ∐_{t∈B} F (X_{t}), of the associated Fano schemes of lines is smooth.

(2) X_{0} = X is the cubic hypersurface we start with and X_{t} is smooth unless t ∈ {t_{1}, …, t_{N}}.

(3) For each degeneration point t_{j} ∈ B, j = 1, …, N, the family β : F → B is a quadratic degeneration of codimension n − 2 at the point t_{j}.

This Proposition can be viewed as a special case of [Shi, Proposition 1]. Note that we have no isolated singularities in F (X_{t}) since F (X_{t}) is smooth as long as X_{t} is smooth. By the results of [CG, Theorem 7.8], we easily get a description of the fibers of β. If t ∈ B − {t_{1}, …, t_{N}} then F_{t} = F (X_{t}) is smooth of dimension 2n − 4. Let x_{i} be the ordinary double point of X_{t_{i}} and Γ_{i} be the lines on X_{t_{i}} that pass through x_{i}. We know that Γ_{i} is smooth of dimension n − 2. The singular fiber F_{t_{i}} is irreducible with Γ_{i} being the singular locus which is an ordinary double subvariety. Let z ∈ Γ_{i} be a point and Y ⊂ F be an (n − 1)-dimensional complex submanifold is a small neighborhood of z. If Y meets Γ_{i} transversally in the point z, then z is a non-degenerate critical point of β|_{Y}. There is an associated vanishing cycle σ_{i} ∈ H_{n−2}(Y_{t_{i}+ϵ}, ℤ). The next observation we make is that a cubic hypersurface with an isolated ordinary double point has enough rational curves in its smooth locus. Let X ⊂ ℙ^{n+1} be a cubic hypersurface with an ordinary double point x_{0} ∈ X. Then the projection from the point x_{0} defines a morphism τ : X̃ → ℙ^{n}, where X̃ = Bl_{x_{0}}(X) is the blow-up of X at the point x_{0}. Let σ : X̃ → X be the blowup morphism. Then it is known that τ is the blow-up of ℙ^{n} along a smooth (2, 3)-complete intersection Z ⊂ ℙ^{n}; see [CG, Lemma 6.5]. Let Q ⊂ ℙ^{n} be the unique quadric hypersurface containing Z and Q̃ ⊂ X̃ be the strict transform of Q. Then σ is an isomorphism on X̃\Q̃ and it contracts Q̃ to the singular point x_{0}. Let h ∈ Pic(X) be the class of a hyperplane section and H ∈ Pic(ℙ^{n}) the class of O(1). Then on X̃, we have

σ^{∗}h = 3τ^{∗}H − E.

Let C ⊂ X\{x_{0}} be a degree e rational curve and C̃ = σ^{−1}C ⊂ X̃. Then this curve C̃ satisfies the following conditions

C̃ · (3τ^{∗}H − E) = e,  C̃ · (2τ^{∗}H − E) = C̃ · Q̃ = 0.

This implies that C̃·τ^{∗}H = e and C̃·E = 2e. Hence C′ = τ(C̃) ⊂ ℙ^{n} is a degree e rational curve which meets Z in 2e points and C̃ is the strict transform of C′. Conversely, if we start with a degree e rational curve C′ ⊂ ℙ^{n} which meets Z in 2e points. Let C̃ be the strict transform of C′. Then C = σ(C̃) ⊂ X\{x_{0}} is a rational curve of degree e.

Lemma 5.4. Let X ⊂ ℙ^{n+1} be a cubic hypersurface with an isolated ordinary double point x_{0} ∈ X. Then there exists a free rational curve C ⊂ X\{x_{0}} of degree e ≥ 1.

Proof. We only need to show that there is a rational curve of degree e through a general point of X\{x_{0}} and contained in X\{x_{0}}. We first do this for e = 1. In this case, we see from the above discussion that we only to show the following Claim: Let y ∈ ℙ^{n} be a general point, then there is a line l′ on ℙ^{n} that meets Z in two points. To prove the claim, we consider the projection, pr_{y} : ℙ^{n}\{y} → ℙ^{n−1}, from the point y. If there is no such line l′, then pr_{y}|_{Z} : Z ↪ ℙ^{n−1} is a closed immersion. But this would imply that Z is a degree 6 hypersurface in ℙ^{n−1} and hence −K_{Z} = (n − 6)H where H is the class of a hyperplane section of Z. Since Z ⊂ ℙ^{n} is a (2, 3)-complete intersection, we also know that −K_{Z} = (n + 1 − 2 − 3)H = (n − 4)H, which is a contradiction. For case of e > 1, we can take a chain of e lines on X\{x_{0}} and smooth to a degree e rational curve; see [Ko, §II.7]. Such a rational curve passes through a general point since a line does. □

Lemma 5.5. Under the situation of Proposition 5.3, there exists a contractible analytic open neighborhood D of 0 ∈ B containing {t_{1}, …, t_{N}} such that

(i) There is an analytic family of curves {C_{t} ⊂ X_{t}^{sm} : t ∈ D} with C_{0} = C, where X_{t}^{sm} is the smooth locus of X_{t}. (ii) S = ∪_{t∈D}S_{C_{t}} ⊂ F is a complex manifold and ρ : S → D is smooth away from the t_{i}’s; (iii) As sub-manifolds of F, S meets each Γ_{i} transversally at finitely many points z_{i1}, …, z_{ir}.

Proof. Let H/B be the component of relative Hilbert scheme of rational curves on X_{t}’s that contains C as an element. This is well-defined since C defines a smooth point in the relative Hilbert scheme. We claim that, after shrinking D, there exists a general analytic section s : t ↦ C_{t} ∈ H satisfies C_{t} ⊂ X_{t}^{sm}. To prove this we first note that by Lemma 5.4, there is a free rational curve C_{i} ∈ X_{t_{i}}^{sm} of the same degree e. Since C_{i} is free, it deforms to a curve C_{t} in nearby fibers. By the result of [CS, Theorem 1.1], the space of degree e rational curves on a smooth cubic hypersurface is irreducible. This implies that C_{t} is a point of H and hence so is C_{i}. So we can choose a family s : t ↦ C_{t} whose specializations does not meet the points x_{i} ∈ X_{t_{i}}. This proves (i). We also get (iii) easily by choosing s general enough. The smoothness of S follows from a deformation argument. First by shrinking D, we may assume that S_{t} is smooth for all t ≠ t_{i}. By choosing C_{i} general, we may assume that S_{C_{i}} is smooth away from {z_{i1,...,z_{ir}}}. We only need to show that S is smooth at the points z = z_{ij}. Let L be the line corresponds to z and C_{i} = C_{t_{i}}. Let v ∈ H^{0}(C_{i}, 𝒩_{C_{i}/ℙ^{n+1}}) be the section corresponding to \frac{∂}{∂t} at the point C_{i}. Let y = C_{i} ∩ L. Then v(y) ∈ 𝒩_{C_{i}/X,y} = \frac{T_{X,y}}{T_{C_{i},y}} determines a 2-dimensional subspace V_{y} ⊂ T_{X,y}. Then T_{S,z} ⊂ T_{F,z} is naturally given by all sections v′ ∈ T_{F,z} ⊂ H^{0}(L, 𝒩_{L/ℙ^{n+1}}) with v′(y) ∈ V̄_{y}, where V̄_{y} is the image of V_{y} in \frac{T_{X,y}}{T_{L,y}} = 𝒩_{L/ℙ^{n+1},y}. When v(y) is general, the above condition gives a codimension n − 2 subspace of T_{F,z}, i.e. dim S = dim T_{S,z}. This proves (ii). □

Remark 5.6. From the above proof, we see that the family t ↦ C_{t} can be made algebraic on some finite cover of B = ℙ^{1}. The points z_{ij} are exactly the critical points of ρ and all of them are non-degenerate.

Now we fix a small ϵ and let B_{i} be the closed ball of radius ϵ with center t_{i}. If ϵ is small enough, we have B_{i} ⊂ D. Pick a path l_{i} connecting 0 and t_{i} + ϵ such that ∪_{i}l_{i} is star-shaped and contractible. Let D_{i} ⊂ ℙ^{n+1} be a small open ball centered at the double point x_{i} ∈ X_{t_{i}}. Let D̃_{i} ⊂ F be the set of lines L ∈ F such that L meets D_{i}. Thus D̃_{i} is a small open neighborhood of Γ_{i}. By construction,

S ∩ D̃_{i} = U_{1} ∪ ⋯ ∪ U_{r}

where U_{j} are disjoint open balls in S. Let p_{j} : P_{j} → U_{j} be the family of lines parameterized by U_{j}. Hence we get the following commutative diagram

\xymatrix{P_{j} \ar[r]^{q_{j}} \ar[d]_{p_{j}} & Y \ar[d]^{π} \\ U_{j} \ar[r]^{ρ_{j}} & B}

where Y is the blow up of ℙ^{n+1} along the base locus of the Lefschetz pencil and ρ_{j} = ρ|_{U_{j}}. We fix a general analytic section s_{j} : U_{j} → P_{j}.

Lemma 5.7. ([Shi, Lemma 7]) There exists local analytic coordinates u_{0}, u_{1}, …, u_{n} of ℙ^{n+1} at the point x_{j} such that (i) The image of P_{j} in ℙ^{n+1} is locally given by u_{0} + \sqrt{−1}u_{1} = 0; (ii) The image q_{j} ∘ s_{j}(U_{j}) is given by u_{0} = u_{1} = 0; (iii) π = t_{j} + u_{0}^{2} + u_{1}^{2} + ⋯ + u_{n}^{2}.

Proof. We first show that q_{j} has injective tangent map. For any tangent vector v ∈ T_{U_{j},z_{j}}, it corresponds to a global section of 𝒩_{L_{j}/ℙ^{n+1}}, where L_{j} is the line corresponding to z_{j}. By the choice of C_{t}, we know that C_{t_{i}} meets the divisor swept out by lines through x_{i} transversally. Then v does not vanish at the point x_{i} ∈ L_{j}. This means that the point x_{i} ∈ L_{j} actually moves when we move L_{j} in U_{j}. Hence we get the injectivity of the tangent map of q_{j}. Then we can start with local coordinates on U_{j} and extend to P_{j} and then to ℙ^{n+1}. See the proof of [Shi, Lemma 7] for more details. □

Since we are doing local computations, by abuse of notation, we regard P_{j} as a submanifold of ℙ^{n+1}. Using the local coordinates obtained in Lemma 5.7, it is standard that a vanishing cycle associated to x_{i} is given by

Σ_{ij} = {u_{0}^{2} + ⋯ + u_{n}^{2} = ϵ, u_{k} ∈ ℝ}

See [La] for more details. Note that the index j is used to remember that the local coordinates are chosen with respect to P_{j} as in Lemma 5.7. If we fix an orientation of Σ_{ij} and this gives an element [Σ_{ij}] ∈ H_{n}(X_{t_{i}+ϵ}, ℤ). Also, by Lemma 5.7, we know that σ_{ij} = (q_{j} ∘ s_{j})^{−1}Σ_{ij}, with the induced orientation, gives a vanishing cycle for the critical point z_{ij} ∈ S. Let

[σ_{ij}] ∈ H_{n−2}(S_{t_{i}+ϵ}, ℤ)

be the corresponding homology class. We choose the orientation of Σ_{ij} such that [Σ_{ij}] = [Σ_{i}] is a fixed class. Then σ_{ij} has an induced orientation.

Proposition 5.8. ([Shi, Proposition 2]) For any 𝔞 ∈ H_{n−2}(S_{t_{i}+ϵ}, ℤ), we have

𝔞 · ([σ_{i1}] + ⋯ + [σ_{ir}]) = Ψ(𝔞) · [Σ_{i}]

where Ψ = Ψ_{C_{t_{i}+ϵ}}.

Proof. The proof goes the same as that of [Shi, Proposition 2]; we sketch it here for completeness. We still use 𝔞 to denote a topological cycle that represents the class 𝔞. We use the notations above and set 𝔞_{j} = 𝔞 ∩ U_{j}. If L ∈ 𝔞 − ∪_{j}𝔞_{j}, then L ∩ Σ_{i} = 0. We may assume that 𝔞_{j} meets σ_{ij} transversally at μ points a_{1}, …, a_{μ} ∈ σ_{ij}. Then the intersection of ∪_{L∈𝔞_{j}}L and Σ_{ij} are exactly s_{j}(a_{1}), …, s_{j}(a_{μ}). And by construction, this intersection is transversal. By chasing the orientations, we see that the contributions to the two sides of the identity are equal. □

Remark 5.9. Since ∪_{i}l_{i} is contractible, H_{n}(X_{t_{i}+ϵ}, ℤ) is naturally identified with H_{n}(X, ℤ) via (l_{i})_{∗}. We can similarly identify S_{t_{i}+ϵ} and S_{C} = S_{0}. Hence we can view [σ_{ij}] as elements in V_{n−2}(S_{C}, ℤ) and [Σ_{i}] as elements in V_{n}(X, ℤ). Under this identification, the above proposition still holds true.

Now we are ready to prove the main result of this section. The proof follows that of [Shi, Proposition 4].

Proof. (of Theorem 5.1). By Lefschetz theory, V_{n}(X, ℤ) is generated by vanishing cycles, see [La]. Since Ψ = Ψ_{C} commutes with the specialization map, we know that Ψ([σ_{i1}]) = λ[Σ_{i}] for some λ ∈ ℤ. Now we see that

[σ_{i1}] · ([σ_{i1}] + ⋯ + [σ_{ir}]) = [σ_{i1}] · [σ_{i1}] = ±2

See [La], p.40. By the above proposition, we get

Ψ([σ_{i1}]) · [Σ_{i}] = λ[Σ_{i}] · [Σ_{i}] = ±2

Comparing this with [Σ_{i}] · [Σ_{i}] = ±2, we get λ = ±1. This shows that [Σ_{i}] is in the image of Ψ. This proves the theorem since [Σ_{i}] generates V_{n}(X, ℤ). □

Remark 5.10. To complete the picture, we still need to prove the surjectivity of

Ψ_{C} : H^{n−2}(S_{C}, ℤ)^{∘} → H^{n}(X, ℤ)_{prim}

for n = 4. Note that under Poincaré duality, H^{2}(S_{C}, ℤ)^{∘} ⊂ H^{2}(S_{C}, ℤ) consists of all classes in V_{2}(S_{C}, ℤ) which is orthogonal to g′. Hence we only need to show that σ_{ij} · g′ = 0. But this is easy to see from construction. In fact, we pick a general point x ∈ C_{t_{i}+ε}, then g′ is represented by F_{x} (all lines through x). Then by construction, all lines through x avoids σ_{ij} (this is essentially due to the fact that the surface swept out by lines in through x is 2h^{2}; see Lemma 3.29). This means σ_{ij} · g′ = 0.

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Mingmin Shen

Korteweg-de Vries Institute for Mathematics University of Amsterdam P.O. Box 94248 1090GE Amsterdam Netherlands M.Shen@uva.nl 904

Documenta Mathematica 19 (2014)