The Weak Lefschetz Principle is False for Ample Cones

Brendan Hassett^{∗}, Hui-Wen Lin^{†} and Chin-Lung Wang^{‡}

September, 2001

1 Introduction

Let X be an (n + 1)-dimensional smooth complex projective variety and let D be a smooth ample divisor of X with inclusion map i : D → X. The wellknown Weak Lefschetz Theorem (see [GrHa]) asserts that the restriction map i^{∗} : H^{k}(X; ℤ) → H^{k}(D; ℤ) is an isomorphism for k ≤ n−1 and is compatible with the Hodge decomposition. For n ≥ 3 one deduces from these results that i^{∗} : Pic(X) → Pic(D) is also an isomorphism. Grothendieck has shown that this statement is true over any algebraically closed field [Hart]. While an ample line bundle on X always restricts to an ample line bundle on D, it is not at all clear whether Amp(D) ⊂ i^{∗}Amp(X), i.e., whether the Weak Lefschetz Principle holds for the ample cone.

In this note we provide two examples showing that the Weak Lefschetz Principle for the ample cone fails in general. One is obtained by blowing up (§2) and the other is a product with a ℙ^{1} factor (§3). We also provide some partial positive results (§4). However, a complete picture of how the ample cone behaves under the Weak Lefschetz isomorphism remains elusive.

∗Mathematics Department, Rice University, Houston, USA. Email address: hassett@math.rice.edu

†Department of Mathematics, National Central University, Chungli, Taiwan. Email address: linhw@math.ncu.edu.tw

‡Department of Mathematics, National Tsing-Hua University, Hsinchu, Taiwan. Email address: dragon@math2.math.nthu.edu.tw

Acknowledgments The first author was partially supported by the Institute of Mathematical Sciences of the Chinese University of Hong Kong and NSF Grant 0070537. He would like to thank the National Center for Theoretical Sciences of Hsinchu, Taiwan, for the invitation that made this collaboration possible.

2 A blow-up example

2.1 The construction

We construct our first counterexample (X, D). Let ϕ : X → ℙ^{4} be the blowup of ℙ^{4} at two distinct points p_{1} and p_{2}. Let l′ be the line spanned by p_{1} and p_{2} and D′ a general smooth cubic hypersurface (threefold) containing p_{1} and p_{2} but not the line l′. The conditions satisfied by D′ will be made precise in 2.5. We take D to be the proper transform of D′ in X. We will see that D is a very ample divisor in X but i^{∗}Amp(X) ≠ Amp(D). More precisely, Amp(D) is strictly larger than i^{∗}Amp(X) if and only if the Mori cone (the closure of the real cone generated by numerically equivalent classes of effective one-cycles) \overline{NE}(X) is strictly larger than i_{∗}\overline{NE}(D) (by Kleiman’s criterion for ampleness [Hart]). Let l be the proper transform of l′ in X, which is an effective one-cycle in X. By Weak Lefschetz, l = i_{∗}λ for some one-cycle λ on D. Our main task is to show that λ ∉ \overline{NE}(D). In fact we will determine both the ample cones and the Mori cones of X and D.

2.2 Ample and Mori cones of X

We first set up some notation; we shall often use a single letter to denote a subvariety and its homology class. Let E_{1} and E_{2} be the exceptional divisors in X, H′ the general hyperplane in ℙ^{4} containing p_{1} and p_{2}, and H its proper transform in X. Note that l′ = H′^{3} and H = ϕ^{∗}H′ − E_{1} − E_{2}. It is clear that Pic(X) = ℤH + ℤE_{1} + ℤE_{2}. Since E_{i} ≅ ℙ^{3} and −E_{i}|_{E_{i}} is its hyperplane class, E_{i}^{3} corresponds to the class of a line in E_{i}. The group of one-cycles (modulo rational or numerical equivalence) is then given by N(X) = ℤϕ^{∗}H′^{3} + ℤE_{1}^{3} + ℤE_{2}^{3} and l = ϕ^{∗}H′^{3} − E_{1}^{3} − E_{2}^{3}. As we blow up at only two points, it is readily seen that NE(X) = ℝ_{≥0}l + ℝ_{≥0}E_{1}^{3} + ℝ_{≥0}E_{2}^{3} which is already a rational closed cone. For later use, we also observe that l.E_{i} = 1 and E_{i}^{4} = (E_{i}|_{E_{i}})^{3} = −1.

Proposition 2.1 The ample cone Amp(X) is the interior of the cone generated by H + E_{1} + E_{2}, H + E_{1} and H + E_{2}. That is, a ℚ-divisor L = ϕ^{∗}H′ −a_{1}E_{1} −a_{2}E_{2} is ample if and only if a_{i} > 0 and a_{1} +a_{2} < 1. Moreover, all ample divisors are automatically very ample.

Proof. By Kleiman’s criterion, L is ample if and only if L.l = 1 − a_{1} − a_{2} > 0 and L.E_{i}^{3} = a_{i} > 0.

For the last statement, one observes that X is naturally a smooth toric variety and for smooth toric varieties, ample divisors are automatically very ample (Theorem of Demazure, see e.g. [Oda]). Q.E.D.

Corollary 2.2 The divisor D ∼ 3ϕ^{∗}H′ − E_{1} − E_{2} is very ample in X but −K_{X} ∼ 5ϕ^{∗}H′ − 3E_{1} − 3E_{2} is not even ample. In particular, X is not Fano.

2.3 The main argument

We recall the following criterion for when the blow-up of a cubic surface at two points is a Del Pezzo surface:

Proposition 2.3 Let S′ be a smooth cubic surface containing distinct points p_{1} and p_{2}, l′ the line spanned by these points, and S the blow-up of S′ at p_{1} and p_{2}. Assume that l′ ∩ S′ consists of three distinct points, and no line containing any one of these three points is contained in S′. Then S is a Del Pezzo surface.

Proof. This argument is inspired by some remarks of Naruki [Nar]. We write l′ ∩ S′ = {p_{1}, p_{2}, p_{3}}, τ′ : T = Bl(S′)_{p_{1},p_{2},p_{3}} → S′, and τ : T → S for the induced map. Let F_{i} ⊂ T denote the exceptional curve over the point p_{i}, so that K_{T} = τ′^{∗}K_{S′} + F_{1} + F_{2} + F_{3} = τ^{∗}K_{S} + F_{3}. Projection from the line l′ induces a morphism μ : T → ℙ^{1}; the fibers of μ are intersections of S′ with hyperplanes containing l′. In particular, the general fiber is a smooth plane cubic, i.e., μ is an elliptic fibration. Since S′ contains no lines containing any of the p_{i}, the fibers of μ are irreducible. Furthermore, the hyperplane class of ℙ^{1} pulls back to −K_{T} = −τ^{∗}K_{S} − F_{3}.

To prove that −K_{S} is ample we apply the Nakai-Moishezon criterion: −K_{S} is ample provided that K_{S}^{2} > 0 and −K_{S}.C > 0 for each closed irreducible curve C ⊂ S. It suffices to check the proper transform C_{T} of C intersects −τ^{∗}K_{S} positively. If μ contracts C_{T} to a point then (−τ^{∗}K_{S}).C_{T} = F_{3}.C_{T} > 0 because the fibers of μ are irreducible and F_{3} is a section of μ. On the other hand, if C_{T} dominates ℙ^{1} then −K_{T} .C_{T} is positive. Since C_{T} ≠ F_{3}, (−τ^{∗}K_{S}).C_{T} is positive as well. Q.E.D.

Remark 2.4 The classification theory of surfaces implies that a cubic surface S′ is the blow-up of ℙ^{2} at six general points [GrHa]. Precisely, we require that no two of the points coincide, no three are collinear, and no six are contained in a plane conic. Implicit in our proof is a precise condition for when the blow-up of ℙ^{2} at eight general points is a Del Pezzo surface. In addition to the conditions listed above, we require that there exists no plane cubic passing through all eight points and singular at one of the eight. Equivalently, the pencil of cubic curves passing through the eight points should have reduced base locus. This criterion was suggested without proof in §26 of [Manin].

Assumptions 2.5 We make the following generality assumptions on D′: 1.The intersection l′ ∩ D′ contains three distinct points. 2.There are a finite number of lines contained in D′ and meeting l′ ∩ D′.

For a general point of a cubic threefold, there are six lines containing the point and contained in the threefold.

Let S′ = D′ ∩ H′ be a hyperplane section of D′ containing p_{1} and p_{2}, a cubic surface in H′ ≅ ℙ^{3}. We choose S′ so that it is smooth and does not contain any lines meeting l′ ∩D′ (such an S′ exists by our second assumption above.) Let S ⊂ H be its proper transform, the blow-up of S′ at p_{1} and p_{2}.

Note that S = (ϕ^{∗}H′ − E_{1} − E_{2})|_{D} and K_{D} = (−2ϕ^{∗}H′ + 2E_{1} + 2E_{2})|_{D} = −2S. Also, since S′ is a cubic surface inside H′ ≅ ℙ^{3}, adjunction shows that K_{S′} = (K_{H′} + S′)|_{S′} = (−4H′|_{H′} + 3H′|_{H′})|_{S′} = −H′|_{S′}. By the blow-up formula, K_{S} = ϕ^{∗}K_{S′} + E_{1}|_{S} + E_{2}|_{S} = (−ϕ^{∗}H′ + E_{1} + E_{2})|_{S} = −H|_{S}.

Proposition 2.6 Consider an effective one-cycle l̃:= ϕ^{∗}H^{′3} − a_{1}E_{1}^{3} − a_{2}E_{2}^{3} in X with a_{i} ≤ 1 and a_{1} + a_{2} > 1. Let λ̃ be the one-cycle class in D which corresponds to l̃. Then λ̃ ∉ \overline{NE}(D). In particular λ ∉ \overline{NE}(D).

Proof. If λ̃ ∈ \overline{NE}(D) then λ̃ is the limit of a sequence \{C_{i}\} of effective onecycles on D with rational coefficients. By assumption, λ̃.S = 1 − a_{1} − a_{2} < 0 which implies that C_{i}.S < 0 for i large enough. In particular this shows that C_{i} ⊂ S for i large enough.

On the other hand, the sequence C_{i}.K_{S} has limit l̃.(−H) = −1+a_{1}+a_{2} > 0. This imples that C_{i}.K_{S} > 0 for i large enough, contradicting the fact that −K_{S} is an ample divisor in S. Q.E.D.

Remark 2.7 To get counterexamples to the Weak Lefschetz Principle, it is essential that the cubic threefold D′ does not contain l′. In fact, if D′ contains l′ then S′ will also contain l′. Thus S is not a blow-up of ℙ^{2} at eight ‘general’ points, −K_{S} is not ample, and no contradiction arises.

2.4 Ample and Mori cones of D

We retain the assumptions of section 2.3. Proposition 2.6 exhibits effective classes in X that are not represented by effective classes in D. We would like to show here that all other effective classes in X lie in the image i_{∗}NE(D).

Theorem 2.8 The image of the Mori cone i_{∗}NE(D) is a closed rational polyhedral cone generated by l + E_{1}^{3}, l + E_{2}^{3}, E_{1}^{3} and E_{2}^{3}. The image of the ample cone i_{∗}Amp(D) is the interior of the rational polyhedral cone generated by H, H + E_{1}, H + E_{2} and H + E_{1} + E_{2} (the nef cone).

Figure 1: Mori cones of X and D

Proof. Recall that the class l +E_{1}^{3} = ϕ^{∗}H^{′3} −E_{2}^{3} is represented by the proper transform of a line in ℙ^{4} which contains p_{2} and is distinct from ℓ. Since D′ is a cubic threefold, through every point of D′ one may find lines [Harris]. The proper transform in D of one such line will have the class l + E_{1}^{3}. The same argument applies to l + E_{2}^{3}. Since the cone generated by l + E_{1}^{3}, l + E_{2}^{3}, E_{1}^{3} and E_{2}^{3} is precisely NE(X) with the classes considered in Proposition 2.6 removed, we conclude that it is i_{∗}NE(D).

The statement on the ample cone is an application of Kleiman’s criterion to L := H + a_{1}E_{1} + a_{2}E_{2}; intersecting L with l + E_{1}^{3}, l + E_{2}^{3}, E_{1}^{3} and E_{2}^{3}, we get 0 < a_{1}, a_{2} < 1. Q.E.D.

Remark 2.9 Szendroi [Sz] has constructed examples in a similar vein, as counterexamples to conjectures of Cox and Katz on the ample cones of anticanonical hypersurfaces in toric varieties.

3 A product example with ℙ^{1} factors

Much simpler examples can be found if X is a product. Take X = ℙ^{1} × ℙ^{d} (for d ≥ 3) and let D be a divisor of type (d, b) with b ∈ ℕ. The divisor D is very ample and thus a generic member of |D| is smooth. It has defining equation

x^{d}f_{0} + x^{d−1}yf_{1} + ... + f_{d}y^{d} = 0,

where x, y are coordinates of ℙ^{1} and the f_{i}’s are polynomials of degree b in ℙ^{d}. The projection p : D → ℙ^{d} has positive dimensional fibers exactly when f_{0} = f_{1} = ... = f_{d} = 0. This has no nontrivial solutions for general f_{i}’s since there are more equations than variables. However, if p is a finite morphism then each ample divisor L on ℙ^{d} pulls back to an ample divisor on D (this follows from either the Nakai-Moishezon criterion or Kleiman’s criterion), yet this pull-back divisor is the restriction of the divisor ℙ^{1} × L on X = ℙ^{1} × ℙ^{d}, which is evidently not ample. This gives another (easier) counterexample to the Weak Lefschetz Principle.

The presence of a one-dimensional factor here is crucial (cf. Theorem 4.1).

4 Positive results

4.1 Products

It is trivial that the Weak Lefschetz Principle holds if X has Néron-Severi rank one and dim X ≥ 4. One may generalize this in a straightforward manner to obtain:

Theorem 4.1 Let i : D → X = ∏ X_{j} be a smooth ample divisor in a finite product of smooth projective varieties, each with dimension ≥ 2 and Néron-Severi rank equal to one. Assume that dim X ≥ 4 and Pic(X) = ⨁ p_{j}^{∗} Pic(X_{j}), where p_{j} : X → X_{j} is the projection map. Then i^{∗}Amp(X) = Amp(D).

Proof. Let H_{j} be an ample class on X_{j} and let h_{j} = p_{j}^{∗}H_{j}. Notice the following fact: If n_{j} = dim X_{j}, then ∏ h_{j}^{m_{j}} is an effective cycle if and only if m_{j} ≤ n_{j} for all j and p = ∏ h_{j}^{n_{j}} is a positive integer. It is then easy to see that ∑ a_{j}h_{j} is an ample class on X if and only if that a_{j} > 0 for all j: simply intersect it with h_{j}^{n_{j}−1}. ∏_{k≠j} h_{k}^{n_{k}} to get a_{j}p > 0.

Now let D be given by the ample class ∑ d_{j}h_{j} with d_{j} > 0. Since dim X ≥ 4, a divisor on D takes the form L|_{D} = ∑ a_{j}h_{j}|_{D}. We need to show that L|_{D} is ample implies that a_{j} > 0 for all j. Indeed, since n_{j} ≥ 2, we simply intersect L|_{D} with the effective cycle h_{j}|_{D}^{n_{j}−2}. ∏_{k≠j} h_{k}|_{D}^{n_{k}} on D to get

0 < h_{j}^{n_{j}−2}. ∏_{k≠j} h_{k}^{n_{k}}. ∑ a_{j}h_{j}. ∑ d_{j}h_{j} = a_{j}d_{j}p.

It follows that a_{j} > 0. Q.E.D.

Remark 4.2 The condition Pic(X) = ⨁ p_{j}^{∗} Pic(X_{j}) holds if all but one of the factors satisfy h^{1}(X_{j}, O_{X_{j}}) = 0, or more generally, if the Jacobians of the factors admit no nontrivial endomorphisms η : Jac(X_{i}) → Jac(X_{j}), i ≠ j.

4.2 A partial theorem on Mori cones

The counterexample of section 2 suggests the following equality on the negative part of Mori cones of D and X (compare Proposition 2.1 and Theorem 2.8). The proof relies on Mori’s theory of extremal rays [Mori] and is essentially contained in [Wiś] and [Kollár].

Theorem 4.3 Let i : D → X be a smooth ample divisor in a smooth variety X with dim X ≥ 4. Then i_{∗}NE(D)_{K_{D}≤0} = NE(X)_{K_{D}≤0}.

Proof. Since K_{D} = K_{X}|_{D} + D|_{D}, which is numerically strictly more positive than K_{X}, we know by Mori’s theory that NE(X)_{K_{D}≤0} is a finite polyhedral cone generated by extremal rays. Let R = ℝC with C ≅ ℙ^{1} be such a ray and ϕ : X → Y the corresponding contraction. We want to show that the class of C is also an effective class in D. If ϕ has a fiber F of dimension at least two then D ∩ F has positive dimension and contains a curve with class in R. If all fibers are one-dimensional, Wiśniewski’s theorem shows that Y is smooth and either ϕ is a blow-up of Y along a smooth codimension-two subvariety Z or ϕ is in fact a conic bundle. These cases are ruled out when dim X ≥ 4 by an argument of Kollár (the Lemma of [Kollár]). Q.E.D.

References

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