Decomposable Cycles and Noether-Lefschetz Loci

Kieran G. O’Grady^{1}

Received: June 10, 2015 Revised: December 14, 2015

Communicated by Gavril Farkas

Dedicato alla piccola Titti

Abstract. We prove that there exist smooth surfaces of degree d in ℙ^{3} whose group of rational equivalence classes of decomposable 0-cycles has rank at least ⌊\frac{d−1}{3}⌋.

2010 Mathematics Subject Classification: 14C15

Keywords and Phrases: Decomposable cycles, Noether-Lefschetz loci.

0. Introduction

Let X be a smooth complex surface: a rational equivalence class of 0-cycles on X is decomposable if it is the intersection of two divisor classes. Let DCH_{0}(X) ⊂ CH_{0}(X) be the subgroup generated by decomposable 0-cycles. Beaville and Voisin [1] proved that if X is a K3 surface then DCH_{0}(X) ≅ ℤ. What can be said of the group DCH_{0}(X) in general? An irregular surface X with non-zero map ⋀^{2} H^{0}(Ω_{X}^{1}) → H^{0}(Ω_{X}^{2}) provides an example with group of decomposable 0-cycles that is not finitely generated, even after tensorization with ℚ. Let us assume that X is a regular surface: then DCH_{0}(X) is finitely generated because CH^{1}(X) is finitely generated, and we may ask for its rank. Blowing up regular surfaces with non-zero geometric genus at (r − 1) very general points, one gets examples of regular surfaces with DCH_{0}(X) of rank at least r (see Example 1.3 b) of [2]). What about a less artificial class of surfaces, such as (smooth) surfaces in ℙ^{3}? If the rank of DCH_{0}(X) is to be larger than 1 then the rank of CH^{1}(X) must be larger than 1, but the latter condition is not sufficient, for example curves on X whose canonical line-bundle is a (fractional) power of the hyperplane bundle do not increase the rank of DCH_{0}(X), see Subsection 1.2. The papers [13, 4] provide examples of smooth surfaces in ℙ^{3} with Picard group of large rank and generated by lines: it follows that the group spanned by decomposable 0-cycles of such surfaces has rank 1. On the other hand Lie Fu proved that there exist degree-8 surfaces X ⊂ ℙ^{3} such that DCH_{0}(X) has rank at least 2, see 1.4 of [6]. In the present paper we will prove the result below.

1Partially supported by PRIN 2013, the Giorgio and Elena Petronio Fellowship Fund, the Giorgio and Elena Petronio Fellowship Fund II, the Fund for Mathematics of the IAS

Theorem 0.1. There exist smooth surfaces X ⊂ ℙ^{3} of degree d such that the rank of DCH_{0}(X) is at least ⌊\frac{d−1}{3}⌋.

In particular the rank of the group of decomposable 0-cycles of a smooth surface in ℙ^{3} can be arbitrarily large. Let us explain the main ideas that go into the proof of Theorem 0.1. Let C = C_{1} ∪ … ∪ C_{n} be the disjoint union of smooth irreducible curves C_{j} ⊂ ℙ^{3}. Suppose that d ≫ 0, and that the curves C_{j} are not rationally canonical, i.e. there exists e ∈ ℤ such that K_{C_{j}}^{⊗m} ≅ 𝒪_{C_{j}}(e) only for m = 0; we prove that for a very general smooth X ∈ |ℐ_{C}(d)|, the classes c_{1}(𝒪_{X}(1))^{2}, C_{1} ·C_{1}, …, C_{n} · C_{n} in CH_{0}(X) are linearly independent. We argue as follows. Assume that they are not linearly independent for X very general; then there exists a non-zero (a, r_{1}, …, r_{n}) ∈ ℤ^{n+1} such that

(0.1)  ac_{1}(𝒪_{X}(1))^{2} + r_{1}c_{1}(𝒪_{X}(C_{1}))^{2} + … + r_{n}c_{1}(𝒪_{X}(C_{n}))^{2} = 0

for all smooth X ∈ |ℐ_{C}(d)|. Now let π: W → ℙ^{3} be the blow up of C, let E be the exceptional divisor of π, and E_{j} be the component of E mapping to C_{j}. Let Λ(d) := |π^{∗}𝒪_{ℙ^{3}}(d)(−E)|, and let 𝒮 ⊂ W × Λ(d) be the universal surface parametrized by Λ(d). We let p_{W} : 𝒮 → W and p_{Λ(d)} : 𝒮 → Λ(d) be the projection maps. There is a natural identification Λ(d) = |ℐ_{C}(d)|, and the generic S ∈ Λ(d) is isomorphic to the corresponding X ∈ |ℐ_{C}(d)|. Since (0.1) holds for all smooth X, an application of the spreading principle shows that the class

(0.2) p_{W}^{∗} (aπ^{∗}c_{1}(𝒪_{ℙ}^{3}(1))^{2} +r_{1}c_{1}(𝒪_{W} (E_{1}))^{2} +…+r_{n}c_{1}(𝒪_{W} (E_{n}))^{2}) ∈ CH^{2}(𝒮)

is vertical, i.e. is represented by a linear combination of codimension-2 subvarieties Γ_{i} ⊂ 𝒮 such that

(0.3)  dim p_{Λ(d)}(Γ_{i}) < dim Γ_{i}.

We prove that if the class in (0.2) is vertical, then 0 = a = r_{1} = … = r_{n}. The key result that one needs is a Noether-Lefschetz Theorem for surfaces belonging to an integral codimension-1 closed subset A ∈ Λ(d). More precisely one needs to prove that the following hold:

(1) If the generic S ∈ A is isomorphic to π(S) ⊂ ℙ^{3}, i.e. S contains no fiber of π: W → ℙ^{3} over C, then CH^{1}(S) is generated (over ℚ) by π^{∗}c_{1}(𝒪_{ℙ^{3}}(1))|_{S}, c_{1}(𝒪_{S}(E_{1})), …, c_{1}(𝒪_{S}(E_{n})).

(2) If the generic S ∈ A contains a fiber R of π: W → ℙ^{3} over C, necessarily unique by genericity of S, then CH^{1}(S) is generated (over ℚ) by the classes listed in Item (1), together with c_{1}(𝒪_{S}(R)).

The reason why such a Noether-Lefschetz Theorem is needed is the following. Let Γ_{i} ⊂ 𝒮 be a codimension-2 subvariety such that (0.3) holds, and assume that the generic fiber of Γ_{i} → p_{Λ(d)}(Γ_{i}) has dimension 1; then A := p_{Λ(d)}(Γ_{i}) is an integral closed codimension-1 subset of Λ(d), and the restriction of Γ_{i} to the surface S_{t} parametrized by t ∈ A is a divisor on S_{t}. Thus we are lead to prove the above Noether-Lefschetz result. There is a substantial literature on Noether-Lefschetz, but we have not found a result taylor made for our needs. A criterion of K. Joshi [9] is very efficient in disposing of “most” choices of a codimension-1 closed subset A ∈ Λ(d). We deal with the remaining cases by appealing to the Griffiths-Harris approach to Noether-Lefschetz [8], as further developed by Lopez [12] and Brevik-Nollet [5]. The paper is organized as follows. In Section 1 we consider a smooth 3fold V with trivial Chow groups, an ample divisor H on V and surfaces in the linear system |ℐ_{C}(H)|, where C = C_{1} ∪ … ∪ C_{n} is the disjoint union of a fixed collection of smooth irreducible curves C_{i} ⊂ V. We prove that if the curves C_{i} are not rationally canonical, and a suitable Noether-Lefschetz Theorem holds, then the classes of C_{1}^{2}, …, C_{n}^{2} on a very general X ∈ |ℐ_{C}(H)| are linearly independent, and they span a subgroup intersecting trivially the image of CH^{2}(V) → CH^{2}(X). In Section 2 we prove the required Noether-Lefschetz Theorem for V = ℙ_{ℂ}^{3}. In Section 3 we prove Theorem 0.1 by combining the main results of Section 1 and Section 2.

Conventions and notation: We work over ℂ. Points are closed points. Let X be a variety: “If x is a generic point of X, then...” is shorthand for “There exists an open dense U ⊂ X such that if x ∈ U then...”. Similarly the expression “If x is a very general point of X, then...” is shorthand for “There exists a countable collection of closed nowhere dense Y_{i} ∈ X such that if x ∈ (X \ ⋃_{i} Y_{i}) then...”. From now on we will denote by CH(X) the group of rational equivalence classes of cycles with rational coefficients. Thus if Z_{1}, Z_{2} are cycles on X then Z_{1} ≡ Z_{2} means that for some non-zero integer ℓ the cycles ℓZ_{1}, ℓZ_{2} are integral and rationally equivalent. If Z is a cycle on X we will often use the same symbol (i.e. Z) for the rational equivalence class represented by Z.

Acknowledgements: It is a pleasure to thank Angelo Lopez for useful exchanges on Noether-Lefschetz results.

1. The family of surfaces containing given curves

1.1. Threefolds with trivial Chow groups. Throughout the paper V is an integral smooth projective threefold.

Hypothesis 1.1. The cycle class map cl: CH(V) ⟶ H(V; ℚ) is an isomorphism.

The archetypal such V is ℙ^{3}. A larger class of examples is given by 3-folds with an algebraic cellular decomposition (see Ex. 1.9.1 of [7]), and conjecturally the above assumption is equivalent to vanishing of H^{p,q}(V) for p ≠ q. An integral smooth projective threefold has trivial Chow group if Hypothesis 1.1 holds.

Claim 1.2. Let V be as above, in particular it has trivial Chow group. The natural map

(1.1)  S^{2} CH^{1}(V) ⟶ CH^{2}(V)

is surjective.

Proof. The natural map S^{2} H^{2}(V; ℚ) → H^{4}(V; ℚ) is surjective by Hard Lefscehtz. The claim follows because of Hypothesis 1.1. □

1.2. Standard relations. Let V be an integral smooth projective 3-fold with trivial Chow group. Let X ⊂ V be a closed surface, and i: X ↪ V be the inclusion map. Let ℛ^{s}(X) ⊂ CH^{s}(X) be the image of the restriction map

(1.2)  \begin{matrix}CH^{s}(V) & ⟶ & CH^{s}(X) \\ ξ & ↦ & i^{∗}ξ \end{matrix}

Notice that ℛ^{2}(X) ⊂ DCH_{0}(X) by Claim 1.2. Suppose that C ⊂ X is an integral smooth curve. We will assume that C · C makes sense in CH_{0}(X), for example that will be the case if X is ℚ-factorial. We will list elements of the kernel of the map

(1.3)  \begin{matrix}ℛ^{2}(X) ⊕ ℛ^{1}(X) ⊕ ℛ^{0}(X) & ⟶ & DCH_{0}(X) \\ (α, β, γ) & ↦ & α + C · β + γ · C · C \end{matrix}

Let j: C ↪ V be the inclusion map. By Cor. 8.1.1 of [7] the following relation holds in CH_{0}(X):

(1.4)  i^{∗}(j_{∗}[C]) = C · c_{1}(𝒩_{X/V}) = C · i^{∗}𝒪_{V} (X).

Thus

(1.5)  α_{C} − C · i^{∗}𝒪_{V} (X) = 0,

where α_{C} := i^{∗}(j_{∗}C) ∈ ℛ^{2}(X). Equation (1.5) is the first standard relation. Now suppose that there exists ξ ∈ CH^{1}(V) such that

(1.6)  c_{1}(K_{C}) = ξ|_{C}.

(Recall that Chow groups are with ℚ-coefficients, thus (1.6) means that there exists an integer n > 0 such that K_{C}^{⊗n} is the pull-back of a line-bundle on V.) By adjunction for X ⊂ V and for C ⊂ X,

(1.7)  C · C + C · (i^{∗}K_{V} + i^{∗}𝒪_{X}(X)) ≡ C · i^{∗}ξ.

Thus there exists β_{C} ∈ ℛ^{1}(X) such that

(1.8)  β_{C} · C − C · C = 0.

The above is the second standard relation (it holds assuming (1.6)).

Example 1.3. Let V = ℙ^{3}, let X ⊂ ℙ^{3} be a smooth surface of degree d, and let C ⊂ X be a smooth curve. The subgroup of CH_{0}(X) spanned by intersections of linear combinations of H := c_{1}(𝒪_{X}(1)) and C has rank at most 2. In fact the first standard relation reads dC · H = (deg C)H · H. Suppose that c_{1}(K_{C}) = mC · H, where m ∈ ℚ. With this hypothesis, the second standard relation reads C · C = (m + 4 − d)C · H, and hence C · C, C · H, H · H span a rank-1 subgroup. In particular a curve of genus 0 or 1 does not add anything to the rank of DCH_{0}(X).

1.3. Surfaces containing disjoint curves. Let V be a smooth projective 3-fold with trivial Chow group and C_{1}, …, C_{n} ⊂ V be pairwise disjoint integral smooth projective curves. Let C := C_{1}∪…∪C_{n} and let π: W → V be the blowup of C. Let E be the exceptional divisor of π, and let E_{j}, for j ∈ {1, …, n}, be the irreducible component of E mapping to C_{j}. Let H be an ample divisor on V. For j ∈ {1, …, n} we let (1.9) Σ_{j} := {S ∈ |π^{∗}H − E| | π(S) is singular at some point of C_{j}}, Σ := ∪_{j=1}^{n}Σ_{j}.

Let S ∈ |π^{∗}H − E|, and let X := π(S). Then S ∈ Σ_{j} if and only if S contains one (at least) of the fibers of E_{j} → C_{j}, or, equivalently, the map S → X given by restriction of π is not an isomorphism over C_{j}. We will always assume that (π^{∗}H − E) is very ample on W; with this hypothesis Σ_{j} is irreducible of codimension 1, or empty (compute the codimension of the loci of S ∈ |π^{∗}(H) − E| which contain one or two fixed fibers of E_{k} → C_{k}). Suppose that H is sufficiently ample: then, in addition, if S ∈ Σ_{k} is generic the surface X = π(S) is smooth except for one ODP (ordinary double point) belonging to C_{k}, and the set of reducible S ∈ |π^{∗}H −E| is of large codimension in |π^{∗}H −E|. We will assume that both of these facts hold (but we do not assume that H is “sufficiently ample”, because we want to prove effective results).

Hypothesis 1.4. Let C_{1}, …, C_{n} ⊂ V and H be as above, in particular H is ample on V, and (π^{∗}H − E) is very ample on W. Suppose that

(1) for j ∈ {1, …, n}, and S ∈ Σ_{j} generic, the surface π(S) is smooth except for one ODP (ordinary double point) belonging to C_{j}, and

(2) the set of reducible S ∈ |π^{∗}H −E| has codimension at least 3 in |π^{∗}H − E|.

Assume that Hypothesis 1.4 holds, and let S ∈ Σ_{j} be generic. Then there is a unique singular point of π(S), call it x, and the line π^{−1}(x) is contained in S.

Hypothesis 1.5. Let C_{1}, …, C_{n} ⊂ V and H be as above. Suppose that Hypothesis 1.4 holds, and that in addition the following hold:

(1) If A ⊂ |π^{∗}H − E| is an integral closed codimension-1 subset, not equal to one of Σ_{1}, …, Σ_{n}, and S ∈ A is very general, the restriction map CH^{1}(W) → CH^{1}(S) is surjective.

(2) For j ∈ {1, …, n}, S ∈ Σ_{j} very general, and x the unique singular point of π(S) (an ODP belonging to C_{j}, by Hypothesis 1.4), CH^{1}(S) is generated by the image of the restriction map CH^{1}(W) → CH^{1}(S) together with the class of π^{−1}(x).

Remark 1.6. Let V = ℙ^{3}, and fix C_{1}, …, C_{n} ⊂ ℙ^{3}. Let d ≫ 0, and H ∈ |𝒪_{ℙ^{3}}(d)|. If S ∈ Σ_{j} is generic, then π^{−1}(x) does not belong to the image of the restriction map CH^{1}(W) → CH^{1}(S).

In the present section we will prove the following result.

Proposition 1.7. Let C_{1}, …, C_{n} ⊂ V and H be as above, and assume that Hypothesis 1.5 holds. Suppose also that for j ∈ {1, …, n} there does not exist ξ ∈ CH^{1}(V) such that c_{1}(K_{C_{j}}) = ξ|_{C_{j}}. (Recall that Chow groups are with coefficients in ℚ.) Then for very general smooth X ∈ |ℐ_{C}(H)| the following hold:

(1) The map CH^{2}(V) → CH_{0}(X) is injective.

(2) Let {ζ_{1}, …, ζ_{m}} be a basis of CH^{1}(V) (as ℚ-vector space). Suppose that for very general smooth X ∈ |ℐ_{C}(H)|

0 = P (ζ_{1}|X, …, ζ_{m}|X) + r_{1}C_{1}^{2} + … + r_{n}C_{n}^{2},

where P ∈ ℚ[x_{1}, …, x_{m}]_{2} is a homogeneous quadratic polynomial. Then 0 = P (ζ_{1}, …, ζ_{m}) = r_{1} = … = r_{n}.

The proof of Proposition 1.7 will be given in Subsection 1.7. Throughout the present section we let V, C, W, E and H be as above.

1.4. The universal surface. Assume that Hypothesis 1.4 holds. Let

(1.10)  Λ := |π^{∗}(H) − E|
(1.11)  𝒮 := {(x, S) ∈ W × Λ | x ∈ S}.

Let p_{W} : 𝒮 → W and p_{Λ} : 𝒮 → Λ be the forgetful maps. Thus we have

(1.12)  \xymatrix{ &  & 𝒮 \ar[dl]_{p_{W}} \ar[dr]^{p_{Λ}} \\ V & W \ar[l]_{π} &  & Λ}

Let N := dim Λ. Since (π^{∗}(H) − E) is very ample it is globally generated and hence the map p_{W} is a ℙ^{N−1}-fibration. It follows that 𝒮 is smooth and

(1.13)  dim 𝒮 = (N + 2).

Definition 1.8. Let Vert^{q}(𝒮 /Λ) ⊂ CH^{q}(𝒮) be the subspace spanned by rational equivalence classes of codimension-q integral closed subsets Z ⊂ 𝒮 such that the dimension of p_{Λ}(Z) is strictly smaller than the dimension of Z.

The result below is an instance of the spreading principle.

Claim 1.9. Keep notation and assumptions as above, in particular Hypothesis 1.4 holds. Let Q ∈ ℚ[x_{1}, …, x_{m}, y_{1}, …, y_{n}]_{2} be a homogeneous polynomial of degree 2 and let ζ_{1}, …, ζ_{m} ∈ CH^{1}(V). Then

(1.14)  Q(ζ_{1}|_{X}, …, ζ_{m}|_{X}, c_{1}(𝒪_{X}(C_{1})), …, c_{1}(𝒪_{X}(C_{n}))) = 0

for all smooth X ∈ |ℐ_{C}(H)| if and only if

(1.15)  p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n}) ∈ Vert^{2}(𝒮 /Λ).

Proof. Suppose that (1.14) holds for all smooth X ∈ |ℐ_{C}(H)|. Let S ∈ Λ be generic, X := π(S). Then X is smooth and the restriction of π to S defines an isomorphism φ: S ⟶^{∼} X, thus by our assumption

p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n})|_{S} = 0.

Since S is generic in Λ it follows (see [3, 14]) that there exists an open dense subset 𝒰 ⊂ Λ such that

(1.16)  p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n})|_{p_{Λ}^{−1}𝒰} = 0.

(We recall that Chow groups are with rational coefficients, if we consider integer coefficients then (1.16) holds only up to torsion.) Let B := (Λ \ 𝒰 ). By the localization exact sequence

CH_{N}(p_{Λ}^{−1}B) ⟶ CH_{N}(𝒮) ⟶ CH_{N}(p_{Λ}^{−1}𝒰 ) ⟶ 0

p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n}) is represented by an N-cycle supported on p_{Λ}^{−1}B, and hence (1.15) holds because dim B < N. Next, suppose that (1.15) holds. Then, by definition, the left-hand side of (1.15) is represented by an N-cycle whose support is mapped by p_{Λ} to a proper closed subset B ⊂ Λ. Thus there exists an open dense 𝒰 ⊂ Λ such that the restriction of p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n}) to p_{Λ}^{−1}𝒰 vanishes, e.g. 𝒰 = Λ \ B. By shrinking 𝒰 we may assume that for S ∈ 𝒰 the surface X := π(S) is smooth. Let S ∈ 𝒰 : then 0 = p_{W}^{∗} Q(π^{∗}ζ_{1}, …, π^{∗}ζ_{m}, E_{1}, …, E_{n})|_{S}, and since X ≅ S it follows that (1.14) holds for X = π(S). On the other hand the locus of smooth X ∈ |ℐ_{C}(H)| such that (1.14) holds is a countable union of closed subsets of Λ_{sm} (the open dense subset of Λ parametrizing smooth surfaces); since it contains an open dense subset of Λ_{sm} it is equal to Λ_{sm}. □

1.5. The Chow groups of 𝒮 and W. Assume that Hypothesis 1.4 holds. Let ξ ∈ CH^{1}(𝒮) be the pull-back of the hyperplane class on Λ via the map p_{Λ} of (1.12). Since p_{W} is the projectivization of a rank-N vector-bundle on W and ξ restricts to the hyperplane class on each fiber of p_{W} the Chow ring CH(𝒮) is the ℚ-algebra generated by p_{W}^{∗} CH(W) and ξ, with ideal of relations generated by a single relation in codimension N. We have N ≥ 3 because(π^{∗}H − E) is very ample by Hypothesis 1.4; thus

(1.17)  \begin{matrix}ℚ ⊕ CH^{1}(W) ⊕ CH^{2}(W) & ⟶^{∼} & CH^{2}(𝒮) \\ (a_{0}, a_{1}, a_{2}) & ↦ & a_{0}ξ^{2} + p_{W}^{∗} (a_{1}) · ξ + p_{W}^{∗} (a_{2}) \end{matrix}

is an isomorphism. The Chow groups CH_{q}(W) are computed by first describing CH_{q}(E_{j}) for j ∈ {1, …, n}, and then considering the localization exact sequence

⨁_{j} CH_{q}(E_{j}) ⟶ CH_{q}(W) ⟶ CH_{q}(W \ (E_{1} ∪ … ∪ E_{n})) ⟶ 0.

One gets an isomorphism

(1.18)  \begin{matrix}CH^{1}(V) ⊕ ℚ^{n} & ⟶^{∼} & CH^{1}(W) \\ (a, t_{1}, …, t_{n}) & ↦ & π^{∗}a + ∑_{j=1}^{n} t_{j}E_{j} \end{matrix}

and an exact sequence

(1.19)  0 ⟶ CH^{2}(W)_{hom} ⟶ CH^{2}(W) ⟶^{cl} H^{4}(W; ℚ) ⟶ 0

where CH^{2}(W)_{hom} is described as follows. Let ρ_{j}: E_{j} → C_{j} be the restriction of the blow-up map π, and σ_{j}: E_{j} ↪ W be the inclusion map; then we have an Abel-Jacobi isomorphism

(1.20) AJ: CH^{2}(W)_{hom} ⟶^{∼}  ⨁_{j=1}^{n} CH_{0}(C_{j})_{hom} α  ↦ (ρ_{1,∗}(σ_{1}^{∗}α), …, ρ_{n,∗}(σ_{n}^{∗}α)

Let AJ_{j} be the j-th component of the map AJ.

Lemma 1.10. Assume that Hypothesis 1.4 holds. Let

ω := π^{∗}α + ∑_{j=1}^{n} E_{j} · π^{∗}β_{j} + ∑_{j=1}^{n} γ_{j}E_{j} · E_{j},

where α ∈ CH^{2}(V), β_{j} ∈ CH^{1}(V), and γ_{j} ∈ ℚ for j ∈ {1, …, n}. Then the following hold:

(1) The cohomology class of ω vanishes if and only if

(1.21)  α = ∑_{j=1}^{n} γ_{j}C_{j},

and for all j ∈ {1, …, n}

(1.22)  deg(β_{j} · C_{j}) = −γ_{j} deg(𝒩_{C_{j}/V}).

(2) Suppose that (1.21) and (1.22) hold. Then for j ∈ {1, …, n}

(1.23)  AJ_{j}(ω) = −γ_{j}c_{1}(𝒩_{C_{j}/V}) − c_{1}(β_{j}|_{C_{j}}).

Proof. Since the cohomology class map cl: CH^{1}(V) → H^{2}(V; ℚ) is a surjection (by hypothesis), also the cohomology class map cl: CH^{1}(W) → H^{2}(W; ℚ) is surjective. By Poincarè duality it follows that cl(ω) = 0 if and only if deg(ω · ξ) = 0 for all ξ ∈ CH^{1}(W). By (1.18) we must test ξ = π^{∗}ζ with ζ ∈ CH^{1}(V) and ξ = E_{i} for i ∈ {1, …, n}. We have

(1.24)  deg(ω · π^{∗}ζ) = deg ((α − ∑_{j=1}^{n} γ_{j}C_{j}) · ζ) .

Since the cycle map CH^{2}(V) → H^{4}(V; ℚ) is an isomorphism, it follows that deg(ω · π^{∗}ζ) = 0 for all ζ ∈ CH^{1}(V) if and only if (1.21) holds. Next, we test ξ = E_{i}. In CH_{0}(C_{i})

(1.25)  ρ_{i,∗}c_{1}(𝒪_{E_{i}}(E_{i}))^{2} = −c_{1}(𝒩_{C_{i}/V}),

and hence

(1.26)  deg(ω · E_{i}) = − deg(β_{i} · C_{i}) − γ_{i} deg(𝒩_{C_{i}/V}).

This proves Item (1). Item (2) follows from Equation (1.25). □

Remark 1.11. By Lemma 1.10 the kernel of the map

(1.27)

\begin{matrix}CH^{2}(V) ⊕ ⨁_{k=1}^{n} CH^{1}(V) ⊕ ⨁_{k=1}^{n} ℚ & ⟶ & CH^{2}(W) \\ (α, β_{1}, … β_{n}, γ_{1}, …, γ_{n}) & ↦ & π^{∗}α + ∑_{j=1}^{n} E_{j} · π^{∗}β_{j} + ∑_{j=1}^{n} γ_{j}E_{j} · E_{j} \end{matrix}

is generated over ℚ by the classes E_{j} · π^{∗}β, where β ∈ CH^{1}(V) and β|_{C_{j}} = 0, together with the classes

(1.28)  π^{∗}[C_{j}] + E_{j} · π^{∗}β + E_{j} · E_{j},

where β ∈ CH^{1}(V), deg(β · C_{j}) = − deg(𝒩_{C_{j}/V}), and

(1.29)  − c_{1}(𝒩_{C_{j}/V}) − c_{1}(β|_{C_{j}}) = 0.

Next notice that (1.29) holds if and only if c_{1}(K_{C_{j}}) is equal to the restriction of a class in CH^{1}(V) i.e. (1.6) holds. Assume that this is the case, and that X ∈ |ℐ_{C}(H)| is a surface smooth at all points of C_{j}. Let S ∈ |π^{∗}H − E| be the strict transform of S. Then S is isomorphic to X over C_{j}, and restricting to S the equation π^{∗}[C_{j}] + E_{j} · π^{∗}β + E_{j} · E_{j} = 0 we get the second standard relation (1.8).

1.6. A vertical cycle on 𝒮. According to Claim 1.9, for every codimension-2 relation that holds between 𝒪_{X}(C_{1}), …, 𝒪_{X}(C_{n}) and restrictions to X of divisors on V, where X is an arbitrary smooth member of ∈ |ℐ_{C}(H)|, there is a polynomial in classes of π^{∗} CH^{1}(V) and the classes of the exceptional divisors of π which is “responsible” for the relation, i.e. when we pullit back to 𝒮 it is a vertical class. We have shown that π^{∗}[C_{j}]+E_{j}·π^{∗}β+E_{j}·E_{j} is the class responsible for the second standard relation (1.8), see Remark 1.11, and in fact this class vanishes. In the present subsection we will write out a cycle responsible for the first standard relation (1.5), this time the pull-back to CH^{2}(𝒮) is a non-zero vertical class. We record for later use the following formulae:

(1.30)  σ_{j,∗}ρ_{j}^{∗}c_{1}(𝒩_{C_{j}/V}) = π^{∗}C_{j} + E_{j} · E_{j},
(1.31)  p_{W,∗}(ξ^{N}) = (π^{∗}H − E).

The first formula follows from the “Key formula” for π^{∗}C_{j}, see Prop. 6.7 of [7]. The second formula is immediate (recall that N = dim Λ). Let j ∈ {1, …, n}. By Hypothesis 1.4 there exists an open dense U ⊂ Σ_{j} such that, if S ∈ U, then S · E_{j} = L_{x} + Z, where x ∈ C_{j} is the unique singular point of π(S), L_{x} := π^{−1}(x), and Z is the residual divisor (whose support does not contain L_{x}). It follows that

(1.32)  E_{j} ∩ p_{Λ}^{−1}(U) = 𝒱_{j} + 𝒵_{j},

where, for every S ∈ U, the restrictions to E_{j} ∩ S of 𝒱_{j}, 𝒵_{j} are equal to L_{x} and Z, respectively. We let

(1.33)  Θ_{j} := \overline{𝒱} _{j}.

Thus p_{Λ}(Θ_{j}) = Σ_{j}, and the generic fiber of Θ_{j} → Σ_{j} is a projective line. By Hypothesis 1.4 Θ_{j} is of pure codimension 2 in 𝒮 (or empty), and hence

(1.34)  Θ_{j} ∈ Vert^{2}(𝒮 /Λ).

The result below will be instrumental in writing out the class of Θ_{j} in CH^{2}(𝒮) according to Decomposition (1.17).

Proposition 1.12. Let j ∈ {1, …, n}. Then

(1.35)  p_{W,∗}(Θ_{j} · ξ^{N−1}) = 2E_{j} · π^{∗}H − E_{j} · E_{j} − π^{∗}C_{j}.

Proof. Let α, β ∈ H^{0}(W, π^{∗}(H)−E) be generic. Then div(α|_{E_{j}}) and div(β|_{E_{j}}) are smooth divisors intersecting transversely at points p_{1}, …, p_{s}. Let q_{i} := π(p_{i}) for i ∈ {1, …, s}. Let R = ℙ(⟨α, β⟩) ⊂ Λ; thus p_{Λ}^{−1}R represents ξ^{N−1}. Given p_{i}, there exists [λ_{i}, μ_{i}] ∈ ℙ^{1} such that div(λ_{i}α + μ_{i}β) contains π^{−1}(q_{i}), and hence [λ_{i}α + μ_{i}β] ∈ R ∩ Σ_{j}. Conversely, every point of R ∩ Σ_{j} is of this type. The line R intersects transversely Σ_{j} because it is generic, and hence

(1.36)  p_{W,∗}(Θ_{j} · ξ^{N−1}) = σ_{j,∗}ρ_{j}^{∗}(q_{1} + … + q_{s}).

Thus in order to compute p_{W,∗}(Θ_{j} · ξ^{N−1}) we must determine the class of the 0-cycle q_{1} +…+q_{s}. Let ϕ: C_{j} ×R → C_{j} and ψ: C_{j} ×R → R be the projections and ℱ the rank-2 vector-bundle on C_{j} × R defined by

ℱ := ϕ^{∗}(𝒩_{C_{j}/V}^{∨} ⊗ 𝒪_{C_{j}}(H)) ⊗ ψ^{∗}𝒪_{R}(1).

The composition of the natural maps

(1.37) ⟨α, β⟩ ↪ H^{0}(W, π^{∗}H −E) ⟶ H^{0}(E_{j}, 𝒪_{E_{j}} (π^{∗}H −E)) ⟶ H^{0}(C_{j}, 𝒩_{C_{j} /V}^{∨} ⊗𝒪_{C_{j}} (H))

defines a section τ ∈ H^{0}(ℱ) whose zero-locus consists of points p′_{1}, …, p′_{s} such that π(p′_{i}) = q_{i}. Now, the zero-locus of τ represents c_{2}(ℱ), and hence

p_{W,∗}(Θ_{j} · ξ^{N−1}) = σ_{j,∗}(ρ_{j}^{∗}(ϕ_{∗}c_{2}(ℱ)))

by (1.36). The formula

c_{2}(ℱ) = ϕ^{∗}(2c_{1}(𝒪_{C}(H)) − c_{1}(𝒩_{C/ℙ^{3}})) · ψ^{∗}c_{1}(𝒪_{R}(1)).

gives

(1.38)  p_{W,∗}(Θ_{j} · ξ^{N−1}) = 2E_{j} · π^{∗}H − σ_{j,∗} (ρ_{j}^{∗}c_{1}(𝒩_{C_{j}/V}))) .

Then (1.35) follows from the above equality together with (1.30). □

Corollary 1.13. Let j ∈ {1, …, n}. Then

(1.39)  Θ_{j} = ξ · p_{W}^{∗} E_{j} + p_{W}^{∗} (E_{j} · π^{∗}H − π^{∗}C_{j}).

Proof. By (1.17) there exist β_{h} ∈ CH^{h}(W) for h = 0, 1, 2 such that

Θ_{j} = ξ^{2} · p_{W}^{∗} β_{0} + ξ · p_{W}^{∗} β_{1} + p_{W}^{∗} β_{2}.

Restricting p_{W} to Θ_{j} we get a ℙ^{N−2}-fibration Θ_{j} → E_{j}: it follows that β_{0} = 0 and β_{1} = E_{j}. By (1.31) (1.40) p_{W,∗}(Θ_{j} · ξ^{N−1}) = p_{W,∗}(ξ^{N} · p_{W}^{∗} E_{j} + ξ^{N−1} · p_{W}^{∗} β_{2}) = (E_{j} · π^{∗}H − E_{j} · E_{j} + β_{2}).

On the other hand p_{W,∗}(Θ_{j} · ξ^{N−1}) is equal to the right-hand side of (1.35): equating that expression and the right-hand side of (1.40) we get β_{2} = (E_{j} · π^{∗}H − π^{∗}C_{j}). □

Corollary 1.14. Let j ∈ {1, …, n}. Then p_{W}^{∗} (E_{j} · π^{∗}H − π^{∗}C_{j}) ∈ Vert^{2}(𝒮 /Λ).

Proof. By Corollary 1.13 we have

p_{W}^{∗} (E_{j} · π^{∗}H − π^{∗}C_{j}) = Θ_{j} − ξ · p_{W}^{∗} E_{j}.

Now Θ_{j} ∈ Vert^{2}(𝒮 /Λ) (see (1.34)) and ξ · p_{W}^{∗} E_{j} ∈ Vert^{2}(𝒮 /Λ) because it is supported on the inverse image of a hyperplane via p_{Λ}; thus p_{W}^{∗} (E_{j} · π^{∗}H − π^{∗}C_{j}) ∈ Vert^{2}(𝒮 /Λ). □

By Claim 1.9 the relation p_{W}^{∗} (E_{j} ·π^{∗}H −π^{∗}C_{j}) ∈ Vert^{2}(𝒮 /Λ) gives a relation in CH(X) for an arbitrary smooth X ∈ |ℐ_{C}(H)|. In fact it gives the first standard relation (1.5).

1.7. Proof of the main result of the section.

Lemma 1.15. Assume that Hypothesis 1.5 holds. Then the projection CH^{2}(𝒮) → CH^{2}(W) determined by (1.17) maps Vert^{2}(𝒮 /Λ) to the subspace spanned by

(1.41) (E_{1} · π^{∗}H − π^{∗}C_{1}), …, (E_{j} · π^{∗}H − π^{∗}C_{j}), …, (E_{n} · π^{∗}H − π^{∗}C_{n}).

Proof. Let Z ⊂ 𝒮 be an irreducible closed codimension-2 subset of 𝒮 such that

(1.42)  dim p_{Λ}(Z) < dim Z = N.

Since the fibers of p_{Λ} are surfaces,

(1.43)  dim p_{Λ}(Z) = \begin{cases}N − 2, & or \\ N − 1. \end{cases}

Suppose that dim p_{Λ}(Z) = N − 2. We claim that

(1.44)  Z = p_{Λ}^{−1}(p_{Λ}(Z)).

Since Z ⊂ p_{Λ}^{−1}(p_{Λ}(Z)), it will suffice to prove that p_{Λ}^{−1}(p_{Λ}(Z)) is irreducible of dimension N. First we notice that every irreducible component of p_{Λ}^{−1}(p_{Λ}(Z)) has dimension at least N. In fact, letting ι: p_{Λ}(Z) ↪ Λ be the inclusion and Δ_{Λ} ⊂ Λ × Λ the diagonal, p_{Λ}^{−1}(p_{Λ}(Z)) is identified with (ι, p_{Λ})^{−1}Δ_{Λ}, and the claim follows because Δ_{Λ} is a l.c.i. of codimension N. Since every fiber of p_{Λ} has dimension 2, it follows that every irreducible component of p_{Λ}^{−1}(p_{Λ}(Z)) dominates p_{Λ}(Z). On the other hand, since cod(p_{Λ}(Z), Λ) = 2, there exists an open dense U ⊂ p_{Λ}(Z) such that p_{Λ}^{−1}(t) is irreducible for all t ∈ U by Hypothesis 1.4, and hence p_{Λ}^{−1}(U) is irreducible of dimension N. It follows that there is a single irreducible component of p_{Λ}^{−1}(p_{Λ}(Z)) dominating p_{Λ}(Z), and hence p_{Λ}^{−1}(p_{Λ}(Z)) is irreducible (of dimension N). We have proved (1.44). Since Λ is a projective space, p_{Λ}([Z]) is a multiple of c_{1}(𝒪_{Λ}(1))^{2}. It follows that the class of Z is a multiple of ξ^{2} and hence the projection CH^{2}(𝒮) → CH^{2}(W) maps it to 0. Now assume that dim p_{Λ}(Z) = N − 1. Let Y := p_{Λ}(Z). For t ∈ Λ, we let S_{t} := p_{Λ}^{−1}(t). We distinguish between the two cases:

(1) p_{Λ}(Z) ∉ {Σ_{1}, …, Σ_{n}}.

(2) There exists j ∈ {1, …, n} such that p_{Λ}(Z) = Σ_{j}.

Suppose that (1) holds. Let Y^{sm} ⊂ Y be the subset of smooth points. If t ∈ Y^{sm}, we may intersect the cycles Z and S_{t} in p_{Λ}^{−1}(Y) (because S_{t} is a l.c.i.), and the resulting cycle class Z · S_{t} belongs to CH^{1}(S_{t}). By Hypothesis 1.5 there exists Γ ∈ CH^{1}(W) such that Γ|_{S_{t}} = Z · S_{t} for t ∈ Y^{sm}. It follows that there exists an open dense U ⊂ Y^{sm} such that

Γ|_{p_{Λ}^{−1}(U)} ≡ Z|_{p_{Λ}^{−1}(U)}.

(Recall that Chow groups are with ℚ-coefficients.) By the localization sequence applied to p_{Λ}^{−1}(U) ⊂ p_{Λ}^{−1}(Y), it follows that there exists a cycle Ξ ∈ CH_{N}(p_{Λ}^{−1}(Y \ U)) such that

[Z] = Ξ + p_{W}^{∗} (Γ) · p_{Λ}^{∗}([Y]).

Here, by abuse of notation, we mean cycle classes in CH_{N}(𝒮): thus [Z] and Ξ are actually the push-forwards of the corresponding classes in CH_{N}(p_{Λ}^{−1}(Y)) and CH_{N}(p_{Λ}^{−1}(Y \ U)) via the obvious closed embeddings. By (1.44) Ξ is represented by a linear combination of varieties p_{Λ}^{−1}(B_{i}), where B_{1}, …, B_{m} are the irreducible components of Y \ U; it follows that Ξ = aξ^{2} for some a ∈ ℚ. On the other hand [Y] ∈ CH^{1}(Λ) = ℚc_{1}(𝒪_{Λ}(1)), and hence p_{W}^{∗} (Γ) · p_{Λ}^{∗}([Y]) = bp_{W}^{∗} (Γ)ξ for some b ∈ ℚ. It follows that the projection CH^{2}(𝒮) → CH^{2}(W) maps Z to 0. Lastly suppose that Item (2) holds. Arguing as above, one shows that there exist Γ ∈ CH^{1}(W), an open dense U ⊂ Y, a cycle Ξ ∈ CH_{N}(p_{Λ}^{−1}(Y \ U)), and a ∈ ℚ such that

[Z] = Ξ + p_{W}^{∗} (Γ) · p_{Λ}^{∗}([Y]) + aΘ_{j}.

By Corollary 1.13 the projection CH^{2}(𝒮) → CH^{2}(W) maps [Z] to a(E_{j} · π^{∗}H − π^{∗}C_{j}). This proves that Vert^{2}(𝒮 /Λ) is mapped into the subspace spanned by the elements of (1.41). Since [Θ_{j}] is a vertical class and is mapped to (E_{j} · π^{∗}H − π^{∗}C_{j}), we have proved the lemma. □

Proof of Proposition 1.7. Let P ∈ ℚ[x_{1}, …, x_{m}] be homogeneous of degree 2 and r_{1}, …, r_{n} ∈ ℚ. The set of smooth X ∈ |ℐ_{C}(H)| such that

(1.45)  0 = P (ζ_{1}|X, …, ζ_{m}|X) + r_{1}C_{1}^{2} + … + r_{n}C_{n}^{2}

is a countable union of closed subsets of the open dense subset of |ℐ_{C}(H)| parametrizing smooth surfaces. It follows that if the proposition is false then there exist P and r_{1}, …, r_{n}, not all zero, such that (1.45) holds for all smooth X ∈ |ℐ_{C}(H)|. Now we argue by contradiction. By Claim 1.9

(1.46)  p_{W}^{∗} (P (π^{∗}ζ_{1}, …, π^{∗}ζ_{m}) + ∑_{j=1}^{n} r_{j}E_{j}^{2}) ∈ Vert^{2}(𝒮 /Λ).

By Lemma 1.15 it follows that there exist rationals s_{1}, …, s_{n} such that

P (π^{∗}ζ_{1}, …, π^{∗}ζ_{m}) + ∑_{j=1}^{n} r_{j}E_{j}^{2} = ∑_{j=1}^{n} s_{j}(E_{j} · π^{∗}H − π^{∗}C_{j}),

i.e.,

(1.47)  0 = π^{∗}(P (ζ_{1}, …, ζ_{m}) + ∑_{j=1}^{n} s_{j}C_{j}) − ∑_{j=1}^{n} s_{j}E_{j} · π^{∗}H + ∑_{j=1}^{n} r_{j}E_{j}^{2}.

Let ω be the right hand side of (1.47); then the homology class of ω vanishes, and also the Abel-Jacobi image AJ(ω), notation as in (1.20). Item (2) of Lemma 1.10, together with our hypothesis that there does not exist ξ ∈ CH^{1}(V) such that c_{1}(K_{C_{j}}) = ξ|_{C_{j}}, gives r_{j} = 0 for j ∈ {1, …, n}. By (1.21)

(1.48)  P (ζ_{1}, …, ζ_{m}) + ∑_{j=1}^{n} s_{j}C_{j} = 0,

and hence ∑_{j=1}^{n} s_{j}E_{j} · π^{∗}H = 0. Thus

(1.49)  0 = E_{i} · (∑_{j=1}^{n} s_{j}E_{j} · π^{∗}H) = −s_{i} deg(C_{i} · H).

for i ∈ {1, …, n}. By hypothesis H is ample, and hence s_{i} = 0 follows from (1.49). Thus P (ζ_{1}, …, ζ_{m}) = 0 by (1.48). □

2. Noether-Lefschetz loci for linear systems of surfaces in ℙ^{3} with base-locus

2.1. The main result. In the present section we let V = ℙ^{3}. Thus C_{1}, …, C_{n} ⊂ ℙ^{3}, and π: W → ℙ^{3}. We let Λ(d) := |π^{∗}𝒪_{ℙ^{3}}(d)(−E)|. For j ∈ {1, …, n} let Σ_{j}(d) ⊂ Λ(d) be the subset Σ_{j} considered in Section 1; thus Σ_{j}(d) parametrizes surfaces S ∈ Λ(d) such that π(S) is singular at some point of C_{j}. Let Σ(d) := Σ_{1}(d) ∪ … ∪ Σ_{n}(d). We denote the tangent sheaf of a smooth variety X by T_{X}. Below is the main result of the present section.

Theorem 2.1. Suppose that d ≥ 5, and that the following hold:

(1) π^{∗}𝒪_{ℙ^{3}}(d − 3)(−E) is very ample.

(2) H^{1}(C, T_{C}(d − 4)) = 0.

(3) The sheaf ℐ_{C} is (d − 2)-regular.

(4) The curves C_{1}, …, C_{n} are not planar. Then Hypothesis 1.5 holds for H ∈ |𝒪_{ℙ^{3}}(d)|.

Recall that Hypothesis 1.5 states that Hypothesis 1.4 holds, and that Items (1) and (2) (our Noether-Lefschetz hypotheses) of Hypothesis 1.5 hold. The proof that Hypothesis 1.4 holds is elementary, and will be given in Subsection 2.2. We will prove that Items (1) and (2) of Hypothesis 1.5 hold by applying Joshi’s main criterion (Prop. 3.1 of [9]), and the main idea in Griffiths-Harris’ proof of the classical Noether-Lefschetz Theorem [8], as further developed by Lopez [12] and Brevik-Nollet [5]. The proof will be outlined in Subsection 2.3, details are in the remaining subsections.

Remark 2.2. Choose disjoint integral smooth curves C_{1}, …, C_{n} ⊂ ℙ^{3} such that for each j ∈ {1, …, n} there does not exist r ∈ ℚ such that c_{1}(K_{C_{j}}) = rc_{1}(𝒪_{C_{j}}(1)). Let d ≫ 0. Then the hypotheses of Theorem 2.1 are satisfied, and hence by Proposition 1.7 the following holds: if X ∈ |ℐ_{C}(d)| is very general, then the 0-cycle classes c_{1}(𝒪_{X}(1))^{2}, c_{1}(𝒪_{X}(C_{1}))^{2}, …, c_{1}(𝒪_{X}(C_{n}))^{2} are linearly independent. Thus the group of decomposable 0-cycles of X has rank at least n + 1. The proof of Theorem 0.1 is achieved by making the above argument effective, see Section 3.

2.2. Dimension counts. We will prove that, if the hypotheses of Theorem 2.1 are satisfied, then Hypothesis 1.4 holds for H ∈ |𝒪_{ℙ^{3}}(d)|. First, H is ample on ℙ^{3}, and π^{∗}(H)−E is very ample on W because it is the tensor product of the line-bundle π^{∗}𝒪_{ℙ^{3}}(d − 3)(−E), which is very ample by hypothesis, and the base-point free line-bundle π^{∗}𝒪_{ℙ^{3}}(3). Let Δ(r) ⊂ Λ(r) be the closed subset parametrizing singular surfaces.

Proposition 2.3. Suppose that π^{∗}𝒪_{ℙ^{3}}(r − 1)(−E) is very ample. Then the following hold:

(1) Let x ∈ C. The linear system |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| has base locus equal to C, and codimension 2 in |ℐ_{C}(r)|. If X is generic in |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| then it has an ODP at x and no other singularity.

(2) Given x ∈ W \ E there exists S ∈ Δ(r) which has an ODP at x and is smooth away from x.

(3) The closed subset Δ(r) is irreducible of codimension 1 in Λ(r), and the generic S ∈ Δ(r) has a unique singular point, which is an ODP.

(4) Let j ∈ {1, …, n}. If S is a generic element of Σ_{j}(r), then π(S) has a unique singular point x, which is an ODP (notice that S is smooth).

Proof. Let q ∈ ℙ^{3} \ C. Since π^{∗}𝒪_{ℙ^{3}}(r − 1)(−E) is very ample there exists X ∈ |ℐ_{C}(r − 1)| such that q ∉ X. Let P ⊂ ℙ^{3} be a plane containing x but not q: then X +P ∈ |ℐ_{x}^{2}(r)|∩|ℐ_{C}(r)| does not pass through q, and this proves that |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| has base locus equal to C. Since π^{∗}𝒪_{ℙ^{3}}(r − 1)(−E) is very ample there exist F, G ∈ H^{0}(ℙ^{3}, ℐ_{C}(r − 1)) and q_{1}, …, q_{m} ∈ (C \ {x}) such that V (F), V (G) are smooth and transverse at each point of C \ {q_{1}, …, q_{m}}. Let P ⊂ ℙ^{3} be a plane not passing through x: the pencil in |ℐ_{C}(r)| spanned by V (F) + P and V (G) + P does not intersect |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)|, and hence |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| has codimension at least 2 in |ℐ_{C}(r)|. The codimension is equal to 2 because imposing on X ∈ |ℐ_{C}(r)| that it be singular at x ∈ C is equivalent to 2 linear equations being satisfied. In order to show that the singularities of a generic element of |ℐ_{x}^{2}(r)|∩|ℐ_{C}(r)| are as claimed we consider the embedding

(2.1)  \begin{matrix}ℙ(H^{0}(ℙ^{3}, ℐ_{x}(1)) ⊕ H^{0}(ℙ^{3}, ℐ_{x}(1))) & ⟶ & Σ_{j}(r) \\ [A, B] & ↦ & V (AF + BG) \end{matrix}

where F, G are as above. The image is a sublinear system of |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| whose base locus is C, hence the generic V (A · F + B · G) is smooth away from C by Bertini’s Theorem. A local computation shows that the projectivized tangent cone of V (AF + BG) at x is a smooth conic for generic A, B. Lastly let q ∈ C \ {x}. The set of [A, B] such that V (AF + BG) is singular at q has codimension 2 if q ∉ {q_{1}, …, q_{m}}, codimension 1 if q ∈ {q_{1}, …, q_{m}}: it follows that for generic [A, B] the surface V (AF + BG) is smooth at all points of C \{x}. This proves Item (1). The remaining items are proved similarly. □

Remark 2.4. Let x ∈ C. The proof of Proposition 2.3 shows that the projectivized tangent cone at x of the generic X ∈ |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| is the generic conic in ℙ(T_{x}ℙ^{3}) containing the point ℙ(T_{x}C).

Proposition 2.5. Suppose that π^{∗}𝒪_{ℙ^{3}}(r)(−E) is very ample and that π^{∗}𝒪_{ℙ^{3}}(r − 3)(−E) is base point free. Then the locus of non-integral surfaces S ∈ |Λ(r)| has codimension at least 4.

Proof. Let Dec(r) ⊂ Λ(r) be the (closed) subset of non-integral surfaces, and Dec(r)_{1}, …, Dec(r)_{m} be its irreducible components. Let j ∈ {1, …, m}; we will prove that the locus of non-integral surfaces S ∈ Dec(r)_{j} has codimension at least 4. Suppose first that the generic S ∈ Dec(r)_{j} contains one (at least) of the components of E, say E_{k}. Since π^{∗}𝒪_{ℙ^{3}}(r)(−E) is very ample, and E_{k} is a ℙ^{1}-bundle, the image of the restriction map

H^{0}(W, π^{∗}𝒪_{ℙ^{3}}(r)(−E)) → H^{0}(E_{k}, π^{∗}𝒪_{ℙ^{3}}(r)(−E)|_{E_{k}})

has dimension at least 4, and hence the locus of S ∈ |π^{∗}𝒪_{ℙ^{3}}(r)(−E)| which contain E_{k} has codimension at least 4. Next, suppose that the generic S ∈ Dec(r)_{j} does not contain any of the components of E. Let Dec(r)′_{j} ⊂ |ℐ_{C}(r)| be the image of Dec(r)_{j} under the natural isomorphism Λ(r) →^{∼} |ℐ_{C}(r)|. Let X ∈ Dec(r)′_{j} be generic; we claim that

(2.2)  dim(sing X \ C) ≥ 1.

In fact X = π(S), where S ∈ Dec(r)_{j} is generic, and since S is non-integral we may write S = S_{1} + S_{2} where S_{1}, S_{2} are effective non-zero divisors on W (we will identify effective divisors and pure codimension-1 subschemes of W and ℙ^{3}). Thus X = X_{1} + X_{2}, where X_{i} := π(S_{i}). Since X_{1}, X_{2} are effective nonzero divisors on ℙ^{3} (non-zero because neither S_{1} nor S_{2} contains a component of E), their intersection has dimension at least 1. Now X_{1}∩X_{2} ⊂ sing X, hence in order to prove (2.2) it suffices to show that X_{1} ∩ X_{2} is not contained in C. Suppose that X_{1} ∩ X_{2} is contained in C; then, since it has dimension at least 1, there exists k ∈ {1, …, n} such that X_{1} ∩ X_{2} contains C_{k}, and this implies that S contains E_{k}, contradicting our assumption. We have proved (2.2). Next, let p ≠ q ∈ (ℙ^{3} \ C), and let Ω_{p,q}(r) ⊂ |ℐ_{C}(r)| be the subset of divisors X which are singular at p, q, with degenerate quadratic terms. If X ∈ Dec(r)′_{j}, then by (2.2) there exists a couple of distinct p, q ∈ (X \ C) such that X is singular at p and q, with degenerate quadratic terms (in fact the set of such couples is infinite). Thus, if Item (2) holds, then

(2.3)  Dec(r)′_{j} ⊂ ⋃_{p≠q∈(ℙ^{3}\C)} Ω_{p,q}(r).

Hence it suffices to prove that the codimension of Ω_{p,q} in |ℐ_{C}(r)| is 10 (as expected) for each p ≠ q ∈ (ℙ^{3} \ C). Let Y ∈ |ℐ_{C}(r − 3)| be a surface not containing p nor q (it exists because π^{∗}𝒪_{ℙ^{3}}(r − 3)(−E) is base point free), and consider the subset

P_{Y} := {Y + Z | Z ∈ |𝒪_{ℙ^{3}}(3)|}.

An explicit computation shows that the codimension of the set of Z ∈ |𝒪_{ℙ^{3}}(3)| singular at p, q, with degenerate quadratic terms, has codimension 10: it follows that Ω_{p,q}(r) ∩ P_{Y} has codimension 10, and hence Ω_{p,q}(r) has codimension 10 in |ℐ_{C}(r)|. □

Proposition 2.3 and Proposition 2.5 prove that, if the hypotheses of Theorem 2.1 are satisfied, then Hypothesis 1.4 holds for H ∈ |𝒪_{ℙ^{3}}(d)|.

2.3. Outline of the proof that the Noether-Lefschetz hypothesis holds. Let A be an integral closed codimension-1 subset of Λ(d). Let A^{∨} ⊂ Λ(d)^{∨} be the projective dual of A, i.e. the closure of the locus of projective tangent hyperplanes T_{S}A for S a point in the smooth locus A^{sm} of A. Since π^{∗}𝒪_{ℙ^{3}}(d)(−E) is very ample we have the natural embedding W ↪ Λ(d)^{∨}, and hence it makes sense to distinguish between the following two cases:

(I) A^{∨} is not contained in W.

(II) A^{∨} is contained in W. Thus (I) holds if and only if, for the generic S ∈ A^{sm}, the projective tangent hyperplane T_{S}A is a base point free linear subsystem of Λ(d). On the other hand, examples of codimension-1 subsets of Λ(d) for which (II) holds are given by Δ(d) and by Σ_{j}(d) for j ∈ {1, …, n}. In fact Δ(d)^{∨} = W and Σ_{j}(d)^{∨} = E_{j}. The last equality holds because S ∈ Λ(d) belongs to Σ_{j}(d) if and only if it is tangent to E_{j}, thus Σ_{j}(d) = E_{j}^{∨}, and hence Σ_{j}(d)^{∨} = E_{j} by projective duality. Let NL(Λ(d)\Δ(d)) be the Noether-Lefschetz locus, i.e. the set of those smooth surfaces S ∈ Λ(d) such that the restriction map Pic(W)_{ℚ} → Pic(S)_{ℚ} is not surjective. As is well-known NL(Λ(d) \ Δ(d)) is a countable union of closed subsets of Λ(d) \ Δ(d). In Subsection 2.5 we will apply Joshi’s criterion (Proposition 3.1 of [9]) in order to prove the following result.

Proposition 2.6. Suppose that d ≥ 5 and that the following hold: (1) π^{∗}𝒪_{ℙ^{3}}(d)(−E) is ample.

(2) H^{1}(C, T_{C}(d − 4)) = 0.
(3) The sheaf ℐ_{C} (on ℙ^{3}) is (d − 2)-regular.

Let A ⊂ Λ(d) be an integral closed subset of codimension 1, and suppose that there exists S ∈ (A\Δ(d)) such that A is smooth at S, and the projective tangent space T_{S}A is a base-point free linear subsystem of Λ(d). Then A \ Δ(d) does not belong to the Noether-Lefschetz locus NL(Λ(d) \ Δ(d)).

The above result deals with codimension-1 subsets A ⊂ Λ(d) for which (I) above holds. Thus, in order to finish the proof of Theorem 2.1, it remains to deal with those A such that (II) above holds.

Definition 2.7. Given p ∈ W and F ⊂ T_{p}W a vector subspace, we let

(2.4)  Λ_{p,F} (d) := {S ∈ |ℐ_{p} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)| : F ⊂ T_{p}S}.

Let Γ(d) := |ℐ_{C}(d)|. We have a tautological identification Λ(d) ⟶^{∼} Γ(d): we let Γ_{p,F} (d) ⊂ Γ(d) be the image of Λ_{p,F} (d), and for j ∈ {1, …, n} we let Π_{j}(d) ⊂ Γ(d) be the image of Σ_{j}(d).

Notice that Λ_{p,F} (d) and Γ_{p,F} (d) are linear subsystems of Λ(d) and Γ(d) respectively. In Subsection 2.6 we will prove the result below by applying an idea of Griffiths-Harris [8] as further developed by Lopez [12] and Brevik-Nollet [5].

Proposition 2.8. Suppose that the following hold:

(1) d ≥ 4 and π^{∗}𝒪_{ℙ^{3}}(d − 3)(−E) is very ample.

(2) None of the curves C_{1}, …, C_{n} is planar.

Let X be a very general element (a) of Γ_{p,F} (d), where either p ∉ E, or else p ∈ E and

(2.5)  T_{p}(π^{−1}(π(p))) ⊄ F ⊊ T_{p}E,
(b) or of Π_{j}(d) for some j ∈ {1, …, n}.

Then the Chow group CH^{1}(X)_{ℚ} is generated by c_{1}(𝒪_{X}(1)) and the classes of C_{1}, …, C_{n}.

Granting Proposition 2.8, let us prove that the statement of Theorem 2.1 holds for A such that A^{∨} is contained in W. We will distinguish between the following two cases:

(IIa) A ∉ {Σ_{1}(d), …, Σ_{n}(d)}.

(IIb) A ∈ {Σ_{1}(d), …, Σ_{n}(d)}. Suppose that (IIa) holds. By projective duality A is the closure of

(2.6)  ⋃_{p∈(A^{∨})^{sm}} Λ_{p,T_{p}A^{∨}}

Let p ∈ (A^{∨})^{sm} be generic. We claim that Item (a) of Proposition 2.8 hold for p and F = T_{p}A^{∨}. In fact if A^{∨} ⊄ E then p ∉ E by genericity. If A^{∨} ⊂ E then A^{∨} is contained in E_{j} for a certain j ∈ {1, …, n}. We claim that A^{∨} is a proper subset of E_{j}, and it is not equal to a fiber of the restriction of π to E_{j}. In fact, if A^{∨} = E_{j}, then A = E_{j}^{∨} = Σ_{j}(d), and that contradicts the assumption that(IIa) holds. Now suppose that A^{∨} = π^{−1}(q) for a certain q ∈ C_{j}. Let S ∈ A be generic. Since A is the closure of (2.6), S is tangent to π^{−1}(q), and hence contains π^{−1}(q) because S has degree 1 on every fiber of E_{j} → C_{j}. It follows that S is tangent to E_{j}, and hence A ⊂ E_{j}^{∨} = Σ_{j}(d), contradicting the hypothesis that (IIa) holds.

Thus Item (a) of Proposition 2.8 hold for p ∈ (A^{∨})^{sm} generic and F = T_{p}A^{∨}, and hence if S ∈ Λ_{p,T_{p}A^{∨}}(d) is very general, then CH^{1}(X)_{ℚ} is generated by c_{1}(𝒪_{X}(1)) and the classes of C_{1}, …, C_{n}. On the other hand, since A ⊄ Σ(d), S intersects transversely E, and hence the restriction of π to S is an isomorphism S ⟶^{∼} X. It follows that CH^{1}(S)_{ℚ} is equal to the image of CH^{1}(W)_{ℚ} → CH^{1}(S)_{ℚ}. This proves that there exists S ∈ A such that CH^{1}(S)_{ℚ} is equal to the image of CH^{1}(W)_{ℚ} → CH^{1}(S)_{ℚ}. Actually our argument proves that there exists such an S which is smooth if A ≠ Δ(d), and that if A = Δ(d) there exists such an S whose singular set consists of a single ODP. If the former holds, then we are done because NL(A \ Δ(d)) is a countable union of closed subsets of A \ Δ(d), and we have shown that the complement is non-empty. If the latter holds, let Δ(d)^{0} ⊂ Δ(d) be the open dense subset parametrizing surfaces with an ODP and no other singular point, then the set of S ∈ Δ(d)^{0} such that CH^{1}(W) → CH^{1}(S) is not surjective is a countable union of closed subsets of Δ(d)^{0} (take a simultaneous resolution), and we are done because we have shown that the complement is non empty. Lastly suppose that (IIb) holds, i.e. A = Σ_{j}(d) for a certain j ∈ {1, …, n}. By Proposition 2.3 there exists an open dense subset Σ_{j}(d)^{0} ⊂ Σ_{j}(d) with the following property. If S ∈ Σ_{j}(d)^{0} and X = π(S), then X has a unique singular point, call it x (obviously x ∈ C_{j}), which is an ordinary node, and the restriction of π to S is the blow-up of X with center x (in particular S is smooth). Now suppose that S ∈ Σ_{j}(d)^{0} is very general. Then by Proposition 2.8 the Chow group CH^{1}(S)_{ℚ} is generated by the image of CH^{1}(W)_{ℚ} → CH^{1}(S)_{ℚ} and the class of π^{−1}(x). Now notice that the set of S ∈ Σ_{j}(d)^{0} such that CH^{1}(S) is not generated by the image of CH^{1}(W)_{ℚ} together with π^{−1}(x) is a countable union of closed subsets of Σ_{j}(d)^{0}; since the complement is not empty, we are done. □ Summing up: we have shown that in order to prove Theorem 2.1 it suffices to prove Proposition 2.6 and Proposition 2.8. The proofs are in the following subsections.

2.4. Infinitesimal Noether-Lefschetz results. We will recall a key result of K. Joshi. Let U ⊂ H^{0}(W, π^{∗}𝒪_{ℙ^{3}}(d)(−E)) be a subspace and σ ∈ U be non-zero. We let S := V (σ), and we assume that S is smooth. Let 𝔪_{σ,U} ⊂ 𝒪_{ℙ(U),[σ]} be the maximal ideal and 𝒯_{σ,U} := Spec(𝒪_{ℙ(U),[σ]}/𝔪_{σ}^{2}) be the first-order infinitesimal neighborhood of [σ] in ℙ(U). We let 𝒮_{σ,U} → 𝒯_{σ,U} be the restriction of the family 𝒮_{Λ} → Λ to 𝒯_{σ,U}. The Infinitesimal Noether Lefschetz (INL) Theorem is valid at (U, σ) (see Section 2 of [9]) if the group of line-bundles on 𝒮_{σ,U} is equal to the image of the composition

(2.7)  Pic(W) ⟶ Pic(W ×_{ℂ} 𝒯_{σ,U} ) ⟶ Pic(𝒮_{σ,U}).

Let A ⊂ Λ(d) be an integral closed subset. Let [σ] be a smooth point of A, and suppose that S = V (σ) is smooth. Let ℙ(U) be the projective tangent space to A at [σ]. If the INL Theorem holds for (U, σ) then A \ Δ(d) does not belong to the Noether-Lefschetz locus NL(Λ(d) \ Δ(d)). Joshi [9] gave a cohomological condition which suffices for the validity of the INL Theorem. Suppose that U ⊂ H^{0}(W, π^{∗}𝒪_{ℙ^{3}}(d)(−E)) generates π^{∗}𝒪_{ℙ^{3}}(d)(−E); we let M(U) be the locally-free sheaf on W fitting into the exact sequence

(2.8)  0 ⟶ M(U) ⟶ U ⊗ 𝒪_{W} ⟶ π^{∗}𝒪_{ℙ^{3}}(d)(−E) ⟶ 0.

Proposition 2.9 (K. Joshi, Prop. 3.1 of [9]). Let U ⊂ H^{0}(W, π^{∗}𝒪_{ℙ^{3}}(d)(−E)) be a subspace which generates π^{∗}𝒪_{ℙ^{3}}(d)(−E). Let 0 ≠ σ ∈ U. Suppose that S = V (σ) is smooth, and that

(a) H^{1}(W, Ω_{W}^{2} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0.

(b) H^{1}(W, M(U) ⊗ K_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0.

Then the INL Theorem holds at (U, σ).

2.5. The generic tangent space is a base-point free linear system. We will prove Proposition 2.6 by applying Proposition 2.9.

Lemma 2.10. Suppose that

(2.9)  0 = H^{1}(ℙ^{3}, ℐ_{C} ⊗ T_{ℙ^{3}}(d − 4)) = H^{1}(C, T_{C}(d − 4)).

Then H^{1}(W, Ω_{W}^{2} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0.

Proof. Since Ω_{W}^{2} ≅ T_{W} ⊗ K_{W} it is equivalent to prove that

(2.10) 0 = H^{1}(W, T_{W} ⊗ K_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = H^{1}(W, T_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d − 4)).

Let ρ: E → C be the restriction of π. Restricting the differential of π to E, one gets an exact sequence

(2.11)  0 ⟶ 𝒪_{W} (E)|_{E} ⟶ ρ^{∗}𝒩_{C/ℙ^{3}} ⟶ ξ ⟶ 0

of sheaves on E, where ξ is an invertible sheaf. Let ι: E ↪ W be the inclusion map. The differential of π gives the exact sequence

(2.12) 0 ⟶ T_{W} ⊗π^{∗}𝒪_{ℙ^{3}}(d−4) ⟶ π^{∗}T_{ℙ^{3}}(d−4) ⟶ ι_{∗}(ξ⊗ρ^{∗}𝒪_{C}(d−4)) ⟶ 0.

Below is a piece of the associated long exact sequence of cohomology:

\begin{matrix}H^{0}(W, π^{∗}T_{ℙ^{3}}(d − 4)) → H^{0}(W, ι_{∗}(ξ ⊗ ρ^{∗}𝒪_{C}(d − 4))) → \\ → H^{1}(W, T_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d − 4)) → H^{1}(W, π^{∗}T_{ℙ^{3}}(d − 4)). \end{matrix} ; (2.13)

We claim that H^{1}(W, π^{∗}T_{ℙ^{3}}(d − 4)) = 0. In fact the spectral sequence associated to π and abutting to the cohomology H^{q}(W, π^{∗}T_{ℙ^{3}}(d − 4)) gives an exact sequence

\begin{matrix}0 → H^{1}(ℙ^{3}, π_{∗}π^{∗}T_{ℙ^{3}}(d − 4)) → H^{1}(W, π^{∗}T_{ℙ^{3}}(d − 4)) → \\ → H^{0}(ℙ^{3}, R^{1}π_{∗}π^{∗}T_{ℙ^{3}}(d − 4)) → 0. \end{matrix} ; (2.14)

Now π_{∗}π^{∗}T_{ℙ^{3}}(d − 4) ≅ T_{ℙ^{3}}(d − 4) and hence H^{1}(ℙ^{3}, π_{∗}π^{∗}T_{ℙ^{3}}(d − 4)) = 0. Moreover R^{1}π_{∗}π^{∗}T_{ℙ^{3}}(d − 4) = 0 because R^{1}π_{∗}𝒪_{W} = 0, and hence H^{1}(W, π^{∗}T_{ℙ^{3}}(d − 4)) = 0. By (2.13), in order to complete the proof it suffices to show that the map

(2.15)  H^{0}(W, π^{∗}T_{ℙ^{3}}(d − 4)) → H^{0}(W, ι_{∗}(ξ ⊗ ρ^{∗}𝒪_{C}(d − 4)))

is surjective. The long exact cohomology sequence associated to (2.11) gives an isomorphism

H^{0}(C, 𝒩_{C/ℙ^{3}}(d − 4)) ⟶^{∼} H^{0}(W, ι_{∗}(ξ ⊗ ρ^{∗}𝒪_{C}(d − 4))),

and moreover the map of (2.15) is identified with the composition

(2.16) H^{0}(ℙ^{3}, T_{ℙ^{3}}(d − 4)) ⟶^{α} H^{0}(C, T_{ℙ^{3}}(d − 4)|_{C}) ⟶^{β} H^{0}(C, 𝒩_{C/ℙ^{3}}(d − 4)).

The map α is surjective by the first vanishing in (2.9), while β is surjective by the second vanishing in (2.9). □

Let U ⊂ H^{0}(ℙ^{3}, ℐ_{C}(d)) be a subspace which generates ℐ_{C}(d); we let \overline{M}(U) be the sheaf on ℙ^{3} fitting into the exact sequence

(2.17)  0 ⟶ \overline{M}(U) ⟶ U ⊗ 𝒪_{ℙ^{3}} ⟶ ℐ_{C}(d) ⟶ 0.

The curve C is a local complete intersection because C is smooth, and hence \overline{M}(U) is locally-free.

Lemma 2.11. Suppose that the hypotheses of Lemma 2.10 hold and that in addition the sheaf ℐ_{C} is d-regular. Let U ⊂ H^{0}(ℙ^{3}, ℐ_{C}(d)) be a subspace which generates ℐ_{C}(d), and let c be its codimension. Then ⋀^{p} \overline{M}(U) is (p+c)- regular.

Proof. Let \overline{M} := \overline{M}(H^{0}(ℐ_{C}(d))). Then \overline{M} is 1-regular: in fact H^{1}(ℙ^{3}, \overline{M}) = 0 because the exact sequence induced by (2.17) on H^{0} is exact by definition, and H^{i}(ℙ^{3}, \overline{M}(1−i)) = 0 for i ≥ 2 because ℐ_{C} is d-regular. It follows that ⋀^{p} \overline{M} is p-regular (Corollary 1.8.10 of [11]). Then, arguing as in the proof of the Lemma on p. 371 of [10] (see also Example 1.8.15 of [11]) one gets that ⋀^{p} \overline{M}(U) is (p + c)-regular. □

Proof of Proposition 2.6. By hypothesis there exists a smooth point [σ] of (A \ Δ(d)), such that the projective tangent space T_{S}A is a base-point free codimension-1 linear subsystem of Λ. We have T_{S}A = ℙ(U), where U ⊂ H^{0}(π^{∗}𝒪_{ℙ^{3}}(d)(−E)) is a codimension-1 subspace which generates 𝒪_{ℙ^{3}}(d)(−E). We will prove that the INL Theorem holds for (U, σ), and Proposition 2.6 will follow. By Joshi’s Proposition 2.9 it suffices to prove that the following hold:

(a) H^{1}(W, Ω_{W}^{2} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0.
(b) H^{1}(W, M(U) ⊗ K_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0.

We start by noting that, since T_{ℙ^{3}} is −1-regular, and by hypothesis ℐ_{C} is (d − 2) regular, the sheaf ℐ_{C} ⊗ T_{ℙ^{3}} is (d − 3)-regular, see Proposition 1.8.9 and Remark 1.8.11 of [11]. Thus the hypotheses of Lemma 2.10 are satisfied, and hence Item (a) holds. Let us prove that Item (b) holds. Tensoring (2.8) by K_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E) ≅ π^{∗}𝒪_{ℙ^{3}}(d − 4) and taking cohomology we get an exact sequence

(2.18) 0→H^{0}(W,M(U)⊗π^{∗}𝒪_{ℙ3} (d−4))→U⊗H^{0}(W,π^{∗}𝒪_{ℙ3} (d−4))→^{α}H^{0}(W,π^{∗}𝒪_{ℙ3} (2d−4)(−E))→

→H^{1}(W,M(U)⊗π^{∗}𝒪_{ℙ3} (d−4))→U⊗H^{1}(W,K_{W} ⊗π^{∗}𝒪_{ℙ3} (d)(−E)).

Now H^{1}(W, K_{W} ⊗ π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = 0 because by hypothesis π^{∗}𝒪_{ℙ^{3}}(d)(−E) is ample. Thus it suffices to prove that the map α is surjective. We have an identification H^{0}(W, π^{∗}𝒪_{ℙ^{3}}(d)(−E)) = H^{0}(ℙ^{3}, ℐ_{C}(d)), and hence U is identified with a codimension-1 subspace of H^{0}(ℙ^{3}, ℐ_{C}(d)) that we will denote by the same symbol. Clearly it suffices to prove that the natural map

(2.19)  U ⊗ H^{0}(ℙ^{3}, 𝒪_{ℙ^{3}}(d − 4)) ⟶ H^{0}(ℙ^{3}, ℐ_{C}(2d − 4))

is surjective. Tensorize Exact Sequence (2.17) by 𝒪_{ℙ^{3}}(d − 4) and take the associated long exact sequence of cohomology: then (2.19) appears in that exact sequence, and hence it suffices to prove that H^{1}(ℙ^{3}, \overline{M}(U)(d − 4)) = 0. By Lemma 2.11 the sheaf \overline{M}(U) is 2-regular, and by hypothesis d ≥ 5: the required vanishing follows. □

2.6. All tangent spaces at smooth points are linear systems with a base-point. We will prove Proposition 2.8. We start with an elementary result.

Lemma 2.12. Assume that π^{∗}𝒪_{ℙ^{3}}(r − 3)(−E) is very ample. Let p ∈ W and F ⊂ T_{p}W be a subspace such that one of the following holds:

(1) p ∉ E and F ≠ T_{p}W,
(2) p ∉ E and F = T_{p}W,
(3) p ∈ E, and T_{p}(π^{−1}(π(p))) ⊄ F ⊊ T_{p}E.

Let X ∈ Γ_{p,F} (r) (see Definition 2.7) be generic. Then X is smooth if Item (1) or (3) holds, while X has an ODP at q = π(p) and is smooth elsewhere if Item (2) holds.

Proof. Suppose first that (1) or (2) holds, i.e. p ∉ E, and let q := π(p). The linear system Γ_{p,F} (r) has base locus C ∪{q}. In fact, let z ∈ (ℙ^{3} \C \{q}); then there exists Y ∈ |ℐ_{C}(r −2)| not containing z because π^{∗}𝒪_{ℙ^{3}}(r −2)(−E) is very ample, and a quadric Q ∈ Γ_{p,F} (2) not containing z. Thus Y + Q is an element of Γ_{p,F} (r) which does not contain z. Hence the generic X ∈ Γ_{p,F} (r) is smooth away from C ∪ {q} by Bertini. Considering Y + Q ∈ Γ_{p,F} (r) as above we also get that the behaviour in q of the generic element of Γ_{p,F} (r) is as claimed. It remains to prove that the generic X ∈ Γ_{p,F} (r) is smooth at every point of C, i.e. that Γ_{p,F} (r) is not a subset of Σ(r). The proof that Γ_{p,F} (r) has base locus C ∪ {q} proves also that

(2.20)  dim Γ_{p,F} (r) = dim |ℐ_{C}(r)| − dim F − 1.

Thus in order to prove that Γ_{p,F} (r) is not a subset of Σ(r), it suffices to prove that for x ∈ C

(2.21)  dim |ℐ_{x}^{2}(r)| ∩ Γ_{p,F} (r) ≤ dim |ℐ_{C}(r)| − dim F − 3.

By Item (1) of Proposition 2.3, dim |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| = dim |ℐ_{C}(r)| − 2, and hence (2.21) is equivalent to

(2.22)  cod(|ℐ_{x}^{2}(r)| ∩ Γ_{p,F} (r), |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)|) = dim F + 1.

We must check that imposing to X ∈ |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| that it contains q and that dπ(p)(F) ⊂ T_{q}X, gives dim F + 1 linearly independent conditions. By Item (1) of Proposition 2.3, there exists Y ∈ |ℐ_{x}^{2}(r − 2)| ∩ |ℐ_{C}(r − 2)| not containing q. Consider the linear subsystem A ⊂ |ℐ_{x}^{2}(r)| ∩ |ℐ_{C}(r)| whose elements are Y + Q, where Q ∈ |𝒪_{ℙ^{3}}(2)|; imposing to X ∈ A that it contains q and that dπ(p)(F) ⊂ T_{q}X, gives dim F + 1 linearly independent conditions, and hence (2.22) follows. This finishes the proof that if (1) or (2) holds, then the conclusion of the lemma holds. Now suppose that (3) holds. Suppose that F = {0}, and let S ∈ Λ_{p,F} (r) = |ℐ_{p} ⊗π^{∗}𝒪_{ℙ^{3}}(r)(−E)| be generic. Then S is smooth at p because π^{∗}𝒪_{ℙ^{3}}(r)(−E) is very ample, and by Bertini’s Theorem it is smooth away from p as well. In order to prove that X = π(S) is smooth we must check that S does not contain any of the lines L_{x} := π^{−1}(x) for x ∈ C. Since π^{∗}𝒪_{ℙ^{3}}(r)(−E) is very ample, (2.23)

cod(|ℐ_{Lx} ⊗π^{∗}𝒪_{ℙ3} (r)(−E)|∩|ℐ_{p}⊗π^{∗}𝒪_{ℙ3} (r)(−E)|, |ℐ_{p}⊗π^{∗}𝒪_{ℙ3} (r)(−E)|)=\begin{cases}1 & if x = q, \\ 2 & if x ≠ q. \end{cases}

It follows that a generic S ∈ |ℐ_{p} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| does not contain any L_{x}. We are left to deal with the case of a 1-dimensional F ⊂ T_{p}E transverse to T_{p}(π^{−1}(q)). Let Z ⊂ W be the 0-dimensional scheme of length 2 supported at p, with tangent space F; thus Z ⊂ E. We must prove that a generic S ∈ |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| is smooth and contains no line L_{x} where x ∈ C. We claim that the (reduced) base-locus of |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| is p. In fact no z ∈ (L_{q} \ {p}) is in the base-locus of |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| because L_{q} is a line and there exists S ∈ |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| which is not tangent to L_{q} at p. Moreover no z ∈ (W \ L_{q}) is in the base-locus of |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| because of the following argument. There exist T ∈ |ℐ_{p} ⊗ π^{∗}𝒪_{ℙ^{3}}(r − 1)(−E)| not containing z, and a plane P ⊂ ℙ^{3} containing q and not containing π(z); then (T + P) ∈ |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| does not contain z. This proves that the (reduced) base-locus of |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| is p; it follows that the generic S ∈ |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| is smooth. We finish by showing that (2.23) holds with ℐ_{p} replaced by ℐ_{Z}. The case x = q is immediate. If x ∈ C \ {q} we get the result by considering elements (T + P) ∈ |ℐ_{Z} ⊗ π^{∗}𝒪_{ℙ^{3}}(r)(−E)| where P is a fixed plane containing q and not containing x, and T ∈ |ℐ_{p} ⊗ π^{∗}𝒪_{ℙ^{3}}(r − 1)(−E)|. □

Remark 2.13. The proof of Lemma 2.12 shows that, if Item (2) holds, the projectivized tangent cone at q of the generic X ∈ Γ_{p,F} (r) is the generic conic in ℙ(T_{q}ℙ^{3}).

Proof of Proposition 2.8. Let r ∈ {d−1, d}. Suppose that p ∈ W, F ⊂ T_{p}W, and either p ∉ E, or else p ∈ E and (2.5) holds. By Lemma 2.12 there exists an open dense subset 𝒰_{p,F} (r) ⊂ Γ_{p,F} (r) such that for X ∈ 𝒰_{p,F} (r) the following holds:

(1) X is smooth if p ∉ E and F ≠ T_{p}W, or p ∈ E.

(2) X has an ODP at q = π(p), and is smooth elsewhere, if p ∉ E and

F = T_{p}W. Similary, let j ∈ {1, …, n}, and q ∈ C_{j}. By Proposition 2.3 there exists an open dense subset 𝒰_{q,j}(r) ⊂ |ℐ_{q}^{2}(r)| ∩ Σ_{j}(r) such that every X ∈ 𝒰_{q,j}(r) has an ODP at q and is smooth elsewhere. It will suffice to prove that if X is a very general element of 𝒰_{p,F} (r) or of 𝒰_{q,j}(r), then CH^{1}(X)_{ℚ} is generated by c_{1}(𝒪_{X}(1)) and the classes of C_{1}, …, C_{n}. Notice that if X is an element of 𝒰_{p,F} (r) or of 𝒰_{q,j}(r), then X is ℚ-factorial. More precisely: if D is a Weil divisor on X then 2D is a Cartier divisor. Let NL(𝒰_{p,F} (d)) ⊂ 𝒰_{p,F} (d) be the subset of X such that Pic(X) ⊗ ℚ is not generated by 𝒪_{X}(1) and 𝒪_{X}(2C_{1}), …, 𝒪_{X}(2C_{n}), and define similarly NL(𝒰_{q,j}(d)) ⊂ 𝒰_{q,j}(d). Then NL(𝒰_{p,F} (d)) is a countable union of closed subsets of 𝒰_{p,F} (d) (there exists a simultaneous resolution if the surfaces in 𝒰_{p,F} (d) are not smooth), and similarly for NL(𝒰_{q,j}(d)). Hence it suffices to prove that 𝒰_{p,F} (d) \ NL(𝒰_{p,F})(d) and 𝒰_{q,j}(d) \ NL(𝒰_{q,j}(d)) are not empty. Let Y be an element of 𝒰_{p,F} (d − 1) or of 𝒰_{q,j}(d − 1), and let X be a generic element of 𝒰_{p,F} (d) or of 𝒰_{q,j}(d). Since π^{∗}𝒪_{ℙ^{3}}(d)(−E) is very ample and X is generic, the intersection of X and Y is reduced, and there exists an integral curve C_{0} ⊂ ℙ^{3} such that its irreducible decomposition is

(2.24)  X ∩ Y = C_{0} ∪ C_{1} ∪ … ∪ C_{n}.

Now let P ⊂ ℙ^{3} be a generic plane, in particular transverse to C_{0}∪C_{1}∪…∪C_{n}. Let X = V (f), Y = V (g) and P = V (l). Let

(2.25)  𝒵 := V (g · l + tf) ⊂ ℙ^{3} × 𝔸^{1}.

The projection 𝒵 → 𝔸^{1} is a family of degree-d surfaces, with central fiber Y + P. The 3-fold 𝒵 is singular. First 𝒵 is singular at the points (x, 0) such that x ∈ X ∩ Y ∩ P, and it has an ODP at each of these points because P is transverse to C_{0} ∪ C_{1} ∪ … ∪ C_{n}. Moreover

(I) 𝒵 has no other singularities if we are dealing with 𝒰_{p,F} (d) and F ≠

T_{p}W, (II) 𝒵 is also singular at {q} × 𝔸^{1} if we are dealing with 𝒰_{p,F} (d) and

F = T_{p}W, or if we are dealing with 𝒰_{q,j}(d). We desingularize 𝒵 as follows. The ODP’s are eliminated by a small resolution (we follow p. 35 of [8], and choose a specific small resolution among the many possible ones), while to desingularize {q} × 𝔸^{1} we blow-up that curve: let 𝒵̂ → 𝒵 be the birational morphism. Then 𝒵̂ is smooth (if p ∉ E and F = T_{p}W, or if we are dealing with 𝒰_{q,k}(d), then 𝒵̂is smooth over {q} × 𝔸^{1} by Remark 2.4 and Remark 2.13). The composition of 𝒵̂→ 𝒵 and the projection 𝒵 → 𝔸^{1} is a flat family of surfaces φ: 𝒵̂→ 𝔸^{1}. The central fiber Ẑ_{0} := φ^{−1}(0) has normal crossings, it is the union of Y and the blow-up P̃ of P at the points of X ∩ Y ∩ P, the curve Y ∩ P being glued to its strict transform in P̃. There will be an open dense B ⊂ 𝔸^{1} containing 0 such that Ẑ_{t} := φ^{−1}(t) is smooth for t ∈ B \ {0}, and it is isomorphic to Z_{t} := V (g · l + tf) in Case (I), while it is the blow-up of Z_{t} at q (an ODP) in Case (II). We replace 𝒵̂by φ^{−1}(B) but we do not give it a new name. One proves that if P is very general, then the following hold:

(I′) In Case (I), if t is very general in B \{0}, then Pic(Ẑ_{t})⊗ℚ is generated by the classes of 𝒪Ẑ_{t}(1), 𝒪Ẑ_{t}(C_{1}), …, 𝒪Ẑ_{t}(C_{n}). (Notice that Ẑ_{t} = Z_{t} because we are in case (I).)

(II′) In Case (II), if t is very general in B \ {0}, letting μ_{t}: Ẑ_{t} → Z_{t} be the blow-up of q and R_{t} ⊂ Ẑ_{t} the exceptional curve, the group Pic(Ẑ_{t}) ⊗ ℚ is generated by the classes of μ_{t}^{∗}𝒪_{Z_{t}}(1), μ_{t}^{∗}𝒪_{Z_{t}}(2C_{1}), …, μ_{t}^{∗}𝒪_{Z_{t}}(2C_{n}) and 𝒪Ẑ_{t}(R_{t}).

One does this by controlling the Picard group of the degenerate fiber Ẑ_{0}. As proved in [8, 12, 5] it suffices to show that the following hold:

(a) Let 𝒱 ⊂ |𝒪_{ℙ^{3}}(1)| be the open subset of planes intersecting transversely C_{0} ∪ … ∪ C_{n}, let I ⊂ (C_{0} ∪ … ∪ C_{n}) × 𝒱 be the incidence subset and ρ: I → 𝒱 be the natural finite map: then the mododromy of ρ acts on a fiber (D_{0}, …, D_{n}, P) as the product of the symmetric groups 𝔖_{deg C_{0}} × … × 𝔖_{deg C_{n}}.

(b) Let j ∈ {0, …, n}, let P ⊂ ℙ^{3} be a very general plane, and let a, b ∈ C_{j} ∩ P be distinct points; then the divisor class a − b on the (smooth) curve Y ∩ P is not torsion.

Now Item (a) is Proposition II.2.6 of [12]. It remains to prove that (b) holds. To this end we will show that C_{0} is not planar and we will control the set of planes P such that P ∩ Y is reducible (see the proof of Item (b) of Lemma 3.4 of [5]).

Claim 2.14. The curve C_{0} (see (2.24)) is not planar.

Proof. By hypothesis π^{∗}𝒪_{ℙ^{3}}(d − 3)(−E) is very ample, in particular it has a non-zero section, and hence there exists a non-zero τ ∈ H^{0}(ℙ^{3}, ℐ_{C}(d − 3)). Multiplying τ by sections of 𝒪_{ℙ^{3}}(3) we get that that h^{0}(ℙ^{3}, ℐ_{C}(d)) ≥ 20. Now assume that C_{0} is planar. Recall that C = C_{1} ∪ … ∪ C_{n}, and let

H^{0}(ℙ^{3}, ℐ_{C}(d)) ⟶^{α} H^{0}(Y, 𝒪_{Y} (d))

be the restriction map. Since (C + C_{0}) ∈ |𝒪_{Y} (d)|, the image of α is equal to H^{0}(Y, 𝒪_{Y} (C_{0})), and hence has dimension at most 4 because C_{0} is planar.

The kernel of α has dimension 4 because Y has degree (d − 1). It follows that h^{0}(ℙ^{3}, ℐ_{C}(d)) ≤ 8, contradicting the inequality h^{0}(ℙ^{3}, ℐ_{C}(d)) ≥ 20. □

Thus none of the curves C_{0}, C_{1}, …, C_{n} is planar.

Lemma 2.15. Let Y ⊂ ℙ^{3} be a surface which is either smooth or has ODP’s. The set of planes P such that P ∩ Y is reducible is the union of a finite set and the collection of pencils through lines of Y.

Proof. Suppose the contrary. Then there exists a 1-dimensional family of planes P such that P · Y = C_{1} + C_{2} with C_{1}, C_{2} divisors which intersect properly, supp C_{1} is irreducible, and deg C_{i} > 1. Next, we distinguish between the two cases:

(1) The generic P does not contain any singular point of Y.

(2) The generic P contains a single point a ∈ sing Y, or two points a, b ∈

sing Y. Assume that (1) holds. Let m_{i} := deg C_{i} for i = 1, 2. Then

(2.26) m_{1}m_{2} = (C_{1} · C_{2})_{P} = (C_{1} · C_{2})_{Y} = (C_{1} · (P − C_{1}))_{Y} = m_{1} − (C_{1} · C_{1})_{Y}

where (C_{1} · C_{2})_{P} is the intersection number of C_{1}, C_{2} in the plane P, and (C_{1} · C_{2})_{Y} is the intersection number of C_{1}, C_{2} in the surface Y (this makes sense because Y has ODP singularities, and hence is ℚ-Cartier). The first equality of (2.26) holds by Bèzout, the second equality is proved by a local computation of the multiplicity of intersection at each point of C_{1} ∩ C_{2} (one needs the hypothesis that Y is smooth at each such point). Thus (2.26) gives (C_{1} · C_{1})_{Y} = m_{1}(1 − m_{2}) < 0, and this contradicts the hypothesis that C_{1} moves in Y. If (2) holds one argues similarly. We go through the computations in the case that P contains two singular points. Let ℙ̃^{3} → ℙ^{3} be the blow up of {a, b}, and Ỹ, P̃ ⊂ ℙ̃^{3} be the strict transforms of Y and P respectively. By hypothesis Y has an ODP at each of its singular points and hence Ỹ is smooth, and of course P̃ is smooth. Let C̃_{i} be the strict transform of C_{i} in ℙ̃^{3}. Let r_{i} := mult_{a} C_{i}, s_{i} := mult_{b} C_{i}. Then the equality

(2.27)  (C̃_{1} · C̃_{2})P̃ = (C̃_{1} · C̃_{2})Ỹ

gives

(2.28)  (C̃_{1} · C̃_{2})Ỹ = −(m_{1}m_{2} − m_{1} − r_{1}r_{2} − s_{1}s_{2} + r_{1} + s_{1}).

Now r_{i} + s_{i} ≤ m_{i} for i = 1, 2, because otherwise the line ⟨a, b⟩ would be contained in Y ∩ C_{i}, and hence we would be considering curves residual to a line in Y, against the hypothesis. Since r_{i} + s_{i} ≤ m_{i} for i = 1, 2 the right-hand side of (2.28) is strictly negative, and this is a contradiction. □

Now we prove that Item (b) holds. Let j ∈ {0, …, n}. Let a, b ∈ C_{j} be generic, in particular they are smooth points of Y. By Lemma 2.15 every plane containing a, b intersects Y in an irreducible curve. Let Ŷ → Y be the blow-up of the base-locus of the pencil of plane sections of Y containing a, b. Then Ŷ has at most A_{n}-singularities, and hence is ℚ-factorial. Let E, F be the exceptional sets over a and b respectively, both have strictly negative selfintersection. Let i > 0 be such that iE and iF are Cartier. Let φ: Ŷ → ℙ^{1} be the regular map defined by the pencil of plane sections of Y containing a, b; for s ∈ ℙ^{1} we let D_{s} := φ^{−1}(s). It suffices to prove that, given r > 0, the set of s ∈ ℙ^{1} such that 𝒪Ŷ (riE − riF)|_{D_{s}} is trivial is finite. Assume the contrary: then, since every plane containing a, b intersects Y in an irreducible curve, there exists ℓ ∈ ℚ such that riE − riF ≡ φ^{∗}(ℓp) in Pic(Ŷ)_{ℚ}, where p ∈ ℙ^{1} (see the proof of Item (b) of Lemma 3.4 of [5]). It follows that the degrees of 𝒪Ŷ (riE − riF) on E and F are both equal to ℓ, and that is absurd because they have opposite signs. □

3. Proof of the main result

We will prove Theorem 0.1. Let Q ⊂ ℙ^{3} be a smooth quadric and choose an isomorphism φ: Q ⟶^{∼} ℙ^{1} × ℙ^{1}: we let 𝒪_{Q}(a, b) := φ^{∗}(𝒪_{ℙ^{1}}(a) ⊠ 𝒪_{ℙ^{1}}(b)).

Proposition 3.1. A curve in |𝒪_{Q}(2, 3)| is 3-regular.

Proof. Let D ∈ |𝒪_{Q}(2, 3)|. Considering the exact sequence 0 → ℐ_{D} → 𝒪_{ℙ^{3}} → 𝒪_{D} → 0 we see right away that if i = 2, 3, then H^{i}(ℙ^{3}, ℐ_{D}(3 − i)) = 0. In order to prove that H^{1}(ℙ^{3}, ℐ_{D}(2)) = 0 we must show that H^{0}(ℙ^{3}, 𝒪_{ℙ^{3}}(2)) → H^{0}(D, 𝒪_{D}(2)) is surjective. The map H^{0}(ℙ^{3}, 𝒪_{ℙ^{3}}(2)) → H^{0}(Q, 𝒪_{Q}(2, 2)) is surjective, hence it suffices to prove that H^{0}(Q, 𝒪_{Q}(2, 2)) → H^{0}(C, 𝒪_{D}(2)) is surjective. We have an exact sequence

0 ⟶ 𝒪_{Q}(0, −1) ⟶ 𝒪_{Q}(2, 2) ⟶ 𝒪_{D}(2) ⟶ 0,

and since H^{1}(Q, 𝒪_{Q}(0, −1)) = 0 the map H^{0}(Q, 𝒪_{Q}(2, 2)) → H^{0}(D, 𝒪_{D}(2)) is indeed surjective. □

Proof of Theorem 0.1. If d ≤ 6 there is nothing to prove, hence we may assume that d ≥ 7. Let n := ⌊\frac{d−4}{3}⌋. Choose disjoint smooth curves C_{1}, …, C_{n} such that each C_{j} is a (2, 3)-curve on a smooth quadric, and let C := C_{1} ∪ … ∪ C_{n}. We may assume that for j ∈ {1, …, n} the degree-0 class in CH_{0}(C_{j}) given by 5c_{1}(K_{C_{j}}) − 2c_{1}(𝒪_{C_{j}}(1)) is not zero. Let us show that the hypotheses of Theorem 2.1 are satisfied. Let j ∈ {1, …, n}. We let π_{j}: W_{j} → ℙ^{3} be the blow-up of C_{j}, and F_{j} ⊂ W_{j} be the exceptional divisor. Then π_{j}^{∗}𝒪_{ℙ^{3}}(3)(−F_{j}) is globally generated, and π_{j}^{∗}𝒪_{ℙ^{3}}(4)(−F_{j}) is very ample: since d−3 ≥ 3(n−1)+4 it follows that π^{∗}𝒪_{ℙ^{3}}(d−3)(−E) is very ample. Let j ∈ {1, …, n}: since d ≥ 7 the cohomology group H^{1}(C_{j}, T_{C_{j}}(d−4)) vanishes, and hence H^{1}(C, T_{C}(d−4)) = 0. By Proposition 3.1 and Example 1.8.32 of [11] the curve C is 3n-regular, and since 3n ≤ (d−4) the curve C is (d−2)-regular. Lastly, by construction no curve C_{j} is planar. We have shown that the hypotheses of Theorem 2.1 are satisfied, and hence Hypothesis 1.5 holds for H ∈ |𝒪_{ℙ^{3}}(d)|. Let X ∈ |ℐ_{C}(d)| be smooth and very generic: since for j ∈ {1, …, n} the class 5c_{1}(K_{C_{j}})−2c_{1}(𝒪_{C_{j}}(1)) is not zero, the decomposable classes H^{2}, C_{1}^{2}, …, , C_{n}^{2} on X are linearly independent by Proposition 1.7. Thus DCH_{0}(X) has rank at least n + 1 = ⌊\frac{d−1}{3}⌋. □

Decomposable Cycles and Noether-Lefschetz Loci 687

References

1. A. Beauville - C. Voisin, On the Chow ring of a K3 surface, J. Algebraic Geometry 13 (2004), pp. 417-426.

2. A. Beauville, On the splitting of the Bloch-Beilinson filtration, Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser. 344, CUP, Cambridge, 2007, pp. 38-53.

3. S. Bloch - V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), pp. 1235-1253.

4. S. Boissière - A. Sarti, On the Néron-Severi group of surfaces with many lines, Proc. Amer. Math. Soc. 136 (2008), pp. 3861-3867.

5. J. Brevik - S. Nollet, Noether-Lefschetz theorem with base locus, Int. Math. Res. Not. (2011), pp. 1220-1244.

6. L. Fu, Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections, Adv. Math. 244 (2013), pp. 894-924.

7. W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2. Springer-Verlag, Berlin, 1984.

8. P. Griffiths - J. Harris, On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), pp. 31-51.

9. K. Joshi, A Noether-Lefschetz Theorem and applicationss, J. Algebraic Geom. 4 (1995), pp. 105-135.

10. S.-O. Kim, Noether-Lefschetz locus for surfaces, Trans. Amer. Math. Soc. 324 (1991), pp. 369-384.

11. R. Lazarsfeld, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete, 48. Springer (2004).

12. A. Lopez, Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc. 89 (2011).

13. T. Shioda, On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), pp. 725-734.

14. C. Voisin, Théorie de Hodge et Géométrie Algébrique complexe, SMF, Cours spécialisés 10, 2002.

Kieran G. O’Grady Sapienza Università di Roma 688

Documenta Mathematica 21 (2016)