The Weak Lefschetz Principle is False for Ample Cones

The Weak Lefschetz Principle is False for Ample Cones

Brendan Hassett

, Hui-Wen Lin

and Chin-Lung Wang

October 26, 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 well- known 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 P¹ 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: has- sett@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 Insti- tute of Mathematical Sciences of the Chinese University of Hong Kong and NSF Grant 0070537. He would like to thank the National Center for Theo- retical Sciences of Hsinchu, Taiwan, for the invitation that made this collab- oration possible.

2 A blow-up example

2.1 The construction

We construct our first counterexample (X, D). Let φ : X → P⁴ be the blow- up of P⁴ at two distinct points p1 and p2. Let l⁰ be the line spanned by p1

and p₂ and D⁰ a general smooth cubic hypersurface (threefold) containing p₁ and p₂ but not the line l⁰. The conditions satisfied by D⁰ will be made precise in 2.5. We take D to be the proper transform of D⁰ in X. We will see that D is a very ample divisor in X but i^∗Amp(X) 6= 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) NE(X) is strictly larger than i_∗NE(D) (by Kleiman’s criterion for ampleness [Hart]). Let l be the proper transform of l⁰ 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 6∈ 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 E₁ and E₂ be the exceptional divisors in X, H⁰ the general hyperplane in P⁴ containing p₁ and p₂, and H its proper transform in X. Note that l⁰ = H⁰³ and H = φ^∗H⁰ − E₁ − E₂. It is clear that Pic(X) = ZH + ZE₁ + ZE₂. Since E_i ∼= P³ and −E_i|_Ei is its hyperplane class, E_i³ 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^∗H⁰³+ ZE₁³ + ZE₂³ and l = φ^∗H⁰³− E₁³− E₂³. As we blow up at only two points, it is readily seen that NE(X) = R_≥0l + R_≥0E₁³+ R_≥0E₂³ which is already a rational closed cone. For later use, we also observe that l.E_i = 1 and E_i⁴ = (E_i|_Ei)³ = −1.

Proposition 2.1 The ample cone Amp(X) is the interior of the cone gen- erated by H + E₁ + E₂, H + E₁ and H + E₂. That is, a Q-divisor L = φ^∗H⁰− a1E1− a2E2 is ample if and only if ai > 0 and a1+ a2 < 1. Moreover, all ample divisors are automatically very ample.

Proof. By Kleiman’s criterion, L is ample if and only if L.l = 1 − a₁− a₂ > 0 and L.E_i³ = ai > 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^∗H⁰− E₁ − E₂ is very ample in X but

−K_X ∼ 5φ^∗H⁰− 3E₁− 3E₂ 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 S⁰ be a smooth cubic surface containing distinct points p₁ and p₂, l⁰ the line spanned by these points, and S the blow-up of S⁰ at p₁ and p₂. Assume that l⁰ ∩ S⁰ consists of three distinct points, and no line containing any one of these three points is contained in S⁰. Then S is a Del Pezzo surface.

Proof. This argument is inspired by some remarks of Naruki [Nar]. We write l⁰ ∩ S⁰ = {p₁, p₂, p₃}, τ⁰ : T = Bl(S⁰)_p1,p2,p3 → S⁰, 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^0∗K_S⁰ + F₁+ F₂ + F₃ = τ^∗K_S+ F₃. Projection from the line l⁰ induces a morphism µ : T → P¹; the fibers of µ are intersections of S⁰ with hyperplanes containing l⁰. In particular, the general fiber is a smooth plane cubic, i.e., µ is an elliptic fibration. Since S⁰ contains no lines containing any of the p_i, the fibers of µ are irreducible. Furthermore, the hyperplane class of P¹ pulls back to −K_T = −τ^∗K_S− F₃.

To prove that −K_S is ample we apply the Nakai-Moishezon criterion:

−K_S is ample provided that K_S² > 0 and −K_S.C > 0 for each closed ir- reducible 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 = F₃.C_T > 0 because the fibers of µ are irreducible and F₃ is a section of µ. On

the other hand, if C_T dominates P¹ then −K_T.C_T is positive. Since C_T 6= F₃,

(−τ^∗K_S).C_T is positive as well. Q.E.D.

Remark 2.4 The classification theory of surfaces implies that a cubic sur- face S⁰ is the blow-up of P² at six general points [GrHa]. Precisely, we require that no two of the points coincide, no three are collinear, and no six are con- tained in a plane conic. Implicit in our proof is a precise condition for when the blow-up of P² 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. Equiv- alently, 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 D⁰: 1.The intersection l⁰∩ D⁰ contains three distinct points.

2.There are a finite number of lines contained in D⁰ and meeting l⁰∩ D⁰. For a general point of a cubic threefold, there are six lines containing the point and contained in the threefold.

Let S⁰ = D⁰∩ H⁰ be a hyperplane section of D⁰ containing p1 and p2, a cubic surface in H⁰ ∼= P³. We choose S⁰ so that it is smooth and does not contain any lines meeting l⁰∩ D⁰ (such an S⁰ exists by our second assumption above.) Let S ⊂ H be its proper transform, the blow-up of S⁰ at p1 and p2.

Note that S = (φ^∗H⁰− E₁− E₂)|_D and K_D = (−2φ^∗H⁰+ 2E₁+ 2E₂)|_D =

−2S. Also, since S⁰ is a cubic surface inside H⁰ ∼= P³, adjunction shows that KS⁰ = (KH⁰ + S⁰)|S⁰ = (−4H⁰|H⁰ + 3H⁰|H⁰)|S⁰ = −H⁰|S⁰. By the blow-up formula, K_S = φ^∗K_S⁰ + E₁|_S+ E₂|_S = (−φ^∗H⁰ + E₁+ E₂)|_S = −H|_S. Proposition 2.6 Consider an effective one-cycle ˜l := φ^∗H⁰³− a₁E₁³− a₂E₂³ in X with a_i ≤ 1 and a₁+ a₂ > 1. Let ˜λ be the one-cycle class in D which corresponds to ˜l. Then ˜λ 6∈ NE(D). In particular λ 6∈ NE(D).

Proof. If ˜λ ∈ NE(D) then ˜λ is the limit of a sequence {C_i} of effective one- cycles on D with rational coefficients. By assumption, ˜λ.S = 1 − a₁− a₂ < 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 ˜l.(−H) = −1+a₁+a₂ >

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 D⁰ does not contain l⁰. In fact, if D⁰ contains l⁰ then S⁰ will also contain l⁰. Thus S is not a blow-up of P² 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 + E₁³, l + E₂³, E₁³ and E₂³. The image of the ample cone i_∗Amp(D) is the interior of the rational polyhedral cone generated by H, H + E1, H + E2 and H + E1+ E2 (the nef cone).















































































































l

l+ E

1 E³

2

E

3 1

2 3

3

E l+

K C=0 K C=0

D X

Figure 1: Mori cones of X and D

Proof. Recall that the class l + E₁³ = φ^∗H⁰³− E₂³ is represented by the proper transform of a line in P⁴ which contains p₂ and is distinct from `. Since D⁰ is a cubic threefold, through every point of D⁰ one may find lines [Harris].

The proper transform in D of one such line will have the class l + E₁³. The same argument applies to l + E₂³. Since the cone generated by l + E₁³, l + E₂³, E₁³ and E₂³ 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 + a₁E₁+ a₂E₂; intersecting L with l + E₁³, l + E₂³, E₁³ and E₂³, we

get 0 < a1, a2 < 1. Q.E.D.

3 A product example with P

factors

Much simpler examples can be found if X is a product. Take X = P¹ × 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^df0+ x^d−1yf1+ ... + fdy^d = 0,

where x, y are coordinates of P¹ 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 f₀ = f₁ = ... = 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 P¹× L on X = P¹× 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 N´eron-Severi rank one and dim X ≥ 4. One may generalize this in a straightforward manner to obtain:

Theorem 4.1 Let i : D → X = Q

X_j be a smooth ample divisor in a finite product of smooth projective varieties, each with dimension ≥ 2 and N´eron-Severi rank equal to one. Assume that dim X ≥ 4 and Pic(X) = Lp^∗_jPic(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^∗_jH_j. Notice the following fact: If n_j = dim X_j, then Q

h^m_j ^j is an effective cycle if and only if m_j ≤ n_j for all j and p = Q

hⁿ_j^j is a positive integer. It is then easy to see that P

a_jh_j is an ample class on X if and only if that a_j > 0 for all j:

simply intersect it with hⁿ_j^j−1.Q

k6=jhⁿ_k^k to get a_jp > 0.

Now let D be given by the ample classP

djhjwith dj > 0. Since dim X ≥ 4, a divisor on D takes the form L|_D = P

a_jh_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|ⁿ_D^j−2.Q

k6=jh_k|ⁿ_D^k on D to get 0 < hⁿ_j^j−2.Y

k6=jhⁿ_k^k.X

a_jh_j.X

d_jh_j = a_jd_jp.

It follows that aj > 0. Q.E.D.

Remark 4.2 The condition Pic(X) = L

p^∗_jPic(X_j) holds if all but one of the factors satisfy h¹(X_j, O_Xj) = 0, or more generally, if the Jacobians of the factors admit no nontrivial endomorphisms η : Jac(Xi) → Jac(Xj), i 6= j.

4.2 A partial theorem on Mori cones

The counterexample of section 2 suggests the following equality on the neg- ative 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 essen- tially contained in [Wi´s] and [Koll´ar].

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)_KD≤0 = NE(X)_KD≤0.

Proof. Since KD = KX|D + D|D, which is numerically strictly more positive than K_X, we know by Mori’s theory that NE(X)_KD≤0 is a finite polyhedral cone generated by extremal rays. Let R = RC with C ∼= P¹ 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, Wi´sniewski’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 Koll´ar (the Lemma of [Koll´ar]). Q.E.D.

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