An Example of Algebraic Cycles with Nontrivial Abel–Jacobi Images

Journal of Algebra 222, 129᎐145 Ž1999. Article ID jabr.1999.7976, available online at http:rrwww.idealibrary.com on

An Example of Algebraic Cycles with Nontrivial Abel᎐Jacobi Images Ken-ichiro Kimura Institute of Mathematics, Uni¨ ersity of Tsukuba, Tsukuba, Ibaraki 305, Japan E-mail: [email protected] Communicated by Walter Feit Received September 8, 1997

In this paper, we will study a family of quintic hypersurfaces and its Abel᎐Jacobi images. We will construct codimension 2 algebraic cycles on the blowing-up of certain dimension 3 quintic hypersurfaces defined over number fields, and show that they are homologically trivial but have nontrivial images under the 3-adic Abel᎐Jacobi map. 䊚 1999 Academic Press

1. INTRODUCTION In wBlx, Bloch has formulated a conjecture about relations between the orders of L-functions at integral arguments and algebraic cycles. Ž1.1. Conjecture. Let k be a number field, and let X k be a projective smooth k-variety. Then the following equality holds: dim ⺡ CH r Ž X k . hom s ord ssr L Ž H 2 ry1 Ž X k . , s . . Here CH r Ž X k . hom is the group of nullhomologous codimension r algebraic cycles on X k modulo rational equivalence, and the L-function LŽ H 2 ry1 Ž X k ., s . is defined by the Euler product L Ž H 2 ry1 Ž X k . , s . [

Ł Pᒍ Ž Nᒍys . y1 , ᒍ

where ᒍ runs over finite primes of k and Pᒍ Ž T . s detŽ1 y ␾ ᒍ T < H 2 ry 1 Ž X k , ⺡ l .. where ␾ ᒍ is the arithmetic Frobenius and H 2 ry1 Ž X k , ⺡ l . is the l-adic etale cohomology group. This product con129 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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verges for ReŽ s . ) r q 12 and should be continued analytically to the whole complex plane. In wBlx, Bloch has given an example in support of Conjecture Ž1.1.. To be more precise, Bloch has shown that a certain codimension 2 cycle on the quintic threefold constructed by Schoen wSchx is homologically trivial and that its image under the ‘‘arithmetic’’ Abel᎐Jacobi map is nonzero. From this result, Bloch has deduced that the cycle is of infinite order in the Griffiths group. The author has been looking for another example which might be used to test the validity of Conjecture Ž1.1., and his effort has culminated in the main result of this paper. We construct explicitly codimension 2 algebraic cycles on a family of threefolds, by blowing up certain quintics in ⺠ 4 . The quintics in question are the projective closure of the affine varieties of the form F Ž x, y . y F X Ž u, ¨ . s 0, where F s 0, F X s 0 define configurations of five lines in the Ž x, y .- Žresp. the Ž u, ¨ .-. plane. We choose F and F X so that F Ž0, 0. s F X Ž0, 0. with vanishing partial derivatives at Ž0, 0.. For a generic choice of F and F X , we see that these varieties have 101 ordinary double points as singularities. Let V denote such a variety. ŽThese varieties were earlier treated by Hirzenbruch in wHix and also by van Geemen and Werner in wWe-Gex.. Let V˜ be the blowing-up of V at all of the 101 double points. Let l 10 y l 20 be the difference of two rulings in the exceptional divisor over Ž0, 0, 0, 0.. Our main result, which is obtained assuming the absolute purity of the 3-adic etale cohomology, is formulated as follows: Ž1.2. THEOREM. When V satisfies certain conditions described later, the ˜ ⺪., but class of l 10 y l 20 has a 10 primary torsion cohomology class in H 4 Ž V, its image under the 3-adic Abel᎐Jacobi map is nonzero. This provides us an example of a family of smooth threefolds with the non-trivial 3-adic Abel᎐Jacobi image; Bloch’s example was for a single specific quintic threefold. For hypersurfaces of three folds of degree less than 5, the above conjecture reduces to the conjecture of Birch and Swinnerton-Dyer for H 1 of curves, so we need to consider hypersurfaces of degree at least 5. This has motivated us to calculate the image of the Abel᎐Jacobi map in the critical case of hypersurfaces of degree 5. One of the reasons for considering this specific hypersurface V with many nodes is that we hoped to get a ‘‘small’’ third Betti number, b 3 , so that the L-function LŽ H 3 Ž V˜k ., s . might be computable. However, b 3 of V˜ is still too large Žit is larger than 20. for the necessary computation of the L-function to be carried out.

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As it stands now, we are far from giving numerical evidence for Conjecture Ž1.1.. Our contribution is to produce a candidate of algebraic cycles of codimension 2 which might be a generator for CH 2 Ž V˜k . hom . Recall that the Griffiths group of codimension r is defined by CH r Ž V˜k . hom r algebraic equivalence4 . We state our expectation as a conjecture: Ž1.3. Conjecture. The class of l 10 y l 20 in the Griffiths group of codimension 2 is non-torsion. The basic idea of our proof of the theorem is along the same line as Bloch’s example in wBlx. To show that the cohomology class of l 10 y l 20 is 10-primary torsion, we need to control the torsion of H 4 Ž V˜⺓ , ⺪.. For this first we apply a degeneration argument to the second cohomology group to reduce to the case where V has large symmetry. Then we cut V by a smooth hyperplane section and look at H 2 of that section. In Bloch’s example, this H 2 was easier to control than in our case. For our V, we construct an algebraic correspondence between the section and the self product of a curve with complex multiplication. Through this algebraic correspondence we get algebraic cycles which generate the relevant part of the H 2 of the section. The proof of non-triviality of the image of l 10 y l 20 under the 3-adic Abel᎐Jacobi map goes exactly the same way as that in Bloch’s example. We take a regular model of V˜ over a strict henselian discrete valuation ring and then specialize l 10 y l 20 to a component of the special fiber. Then we make use of Proposition Ž3.2. in wBlx. We need the absolute purity of the etale cohomology for this. The paper is organized as follows: In Section 2, we construct the algebraic cycle l 10 y l 20 on a certain quintic threefold, and then in Section 3, we review some facts about the arithmetic Abel᎐Jacobi map. The main results are formulated in Section 4, and proofs are given in the subsequent Sections 5 and 6. Some auxiliary hypotheses imposed on the functions F and F X are stated in the Appendix.

2. THE CONSTRUCTION OF THE ALGEBRAIC CYCLE We will consider the configurations of five lines in the Ž x, y .-plane which meet only in pairs and which are stable under the map Ž x, y . ¬ Ž x, yy .. Let F s 0 be a defining equation for such a configuration. Then F has the form F Ž x, y . s Ž a q bx . Ž y 2 y Ž cx q d .

2

.Ž y 2 y Ž ex q f . 2 . .

We call a point critical for F if partial derivatives of F vanish there. F has 10 critical points at the intersections of the five lines and six other

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supplementary critical points. Let F Ž x, y . and F X Ž u, ¨ . be the defining equations of two such configurations. Let V denote the projective closure of an affine variety in ⺑4 defined by F Ž x, y . y F X Ž u, ¨ . s 0. We choose F and F X so that Ž0, 0, 0, 0. g V and Ž0, 0. is critical for both F and F X . Then Ž0, 0, 0, 0. is an ordinary double point of V. If a Žresp. b . is an intersection point of lines for F Žresp. F X ., then Ž a, b . g V is an ordinary double point. So for a generic choice of F and F X , the variety V has 10 = 10 q 1 s 101 ordinary double points. Let V˜ be the blowing-up of V at the 101 double points and Q be the exceptional divisor over Ž0, 0, 0, 0.. Then Q is a quadric hypersurface in ⺠ 3 and is isomorphic to ⺠ 1 = ⺠ 1. Let l 10 y l 20 be the difference of two rulings of Q. This is the algebraic cycle of our interest.

3. THE ‘‘ARITHMETIC’’ ABEL᎐JACOBI MAPS We begin by reviewing the definition of an arithmetic Abel᎐Jacobi map. Let Wk be a smooth projective geometrically irreducible variety over a field k and let l be a prime different from the characteristic of k. There is the cycle class map wGr-Dx r n cl W k :

CH r Ž Wk . ª H 2 r Ž Wk , ⺪rl n Ž r . . .

Then the image of CH r ŽWk . hom lies in the kernel of the restriction homomorphism H 2 r Ž Wk , ⺪rl n Ž r . . ª H 2 r Ž Wk , ⺪rl n Ž r . .

GalŽ krk .

.

The Hochschild᎐Serre spectral sequence E2p , q s H p Ž Gal Ž krk . , H q Ž Wk , ⺪rl n Ž r . . . « H pqq Ž Wk , ⺪rl n Ž r . . gives rise to a map r n cl W k , 0 :

CH r Ž Wk . hom ª H 1 Ž Gal Ž krk . , H 2 ry1 Ž Wk , ⺪rl n Ž r . . . .

Passing to the limit we obtain a map r cl W : CH r Ž Wk . hom ª H 1 Ž Gal Ž krk . , H 2 ry1 Ž Wk , ⺪ l Ž r . . . . k, 0

The cohomology group on the right is computed with continuous cochains, where H 2 ry1 ŽWk , ⺪ l Ž r .. has the inverse limit topology wTax.

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4. THE RESULT Let R be the strict henselization of the ring of integers of ⺡Ž'5 . at Ž'5 .. We write ␲ for '5 . Let F and F X be as defined in Section 2. Assume that all the coefficients of F and F X are in RU . Let k be the fraction field of R and fix an embedding k ¨ ⺓. Ž4.1. THEOREM. Assume the absolute purity of the 3-adic etale cohomology. Choose F and F X suitably so that they fulfill the condition in Ž A.1. in the ˜ Then the following assertions appendix. Lemma Ž5.1. below is applicable to V. hold so that Ža. The class of l 10 y l 20 in H 4 Ž V˜⺓ , ⺪. is 10 primary torsion. Žb. cl 2V˜ Ž l 10 y l 20 . g H 1 ŽGalŽ krk ., H 3 Ž V˜k , ⺪ 3 Ž2... is non-zero. k Ž4.2. Remarks. Ž1. The absolute purity of the etale cohomology is necessary in the proof of Žb. to apply Proposition 3.2 in wBlx. Ž2. Let f i and f iX Ž1 F i F 5. be the linear factors of F and F X , respectively, and let Di j [  f i Ž x, y . s 0, f jX Ž u, ¨ . s 04 be divisors on V for 1 F i, j F 5. Then from the proof of the proposition in Section 1 of wWe-Gex we see that Di j Ž1 F i F 5. generates H4 Ž V⺓ , ⺡.. From this we see that the ˜ ⺪. is a torsion element. class of l 10 y l 20 in H2 Ž V, We expect that l 10 y l 20 is not a torsion element in the Griffiths group and state it as a conjecture. Ž4.3. Conjecture. The class of l 10 y l 20 in the Griffiths group of V˜⺓ is non-torsion. We will prove Assertion Ž4.1Ža.. in the next section. Assertion Ž4.1Žb.. will be proved in Section 6.

5. PROOF OF ASSERTION Ž4.1Ža.. Let V0 ; ⺠ 4 be the projective closure of an affine variety defined by the equation F0 Ž x, y . y F0 Ž u, ¨ . s 0, where F0 Ž x, y . s Ž x q

1 2

. ž y 4 y y 2 Ž 2 x 2 y 2 x q 1. q 15 Ž x 2 q x y 1.

2

/.

Let F0 s 0 be the defining equation for a regular pentagon wWe-Gex. By Lemma Ž5.1. below, we may reduce the proof of Ž4.1Ža.. to the case V s V0 . V0 is a hypersurface with 126 ordinary double points. Let V˜0 be the blowup of all the double points and let V0 be the blowup of the 101 double points coming from the intersections of the lines and Ž0, 0, 0, 0..

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Then we can identify the two homology groups: Ž5.1. LEMMA.

For a suitable choice of F and F X , there is an isomorphism H2 Ž V˜⺓ , ⺪ . ( H2 Ž V0 ⺓ , ⺪ .

induced by the retraction map. Under this isomorphism, the class of l 10 y l 20 g H2 Ž V˜⺓ , ⺪. is mapped to the class of l 10 y l 20 in H2 Ž V0 ⺓ , ⺪.. The proof of this lemma will be included in the Appendix. Thus we are reduced to showing cl Ž l 10 y l 20 . g H2 Ž V0 ⺓ , ⺪ .  10 4 .

Ž 5.2.

Let ⺗ denote the direct product of two dihedral groups, i.e., ⺗ s ² ␫ 1 , ␫ 2 , ␴ 1 , ␴ 2 :, where ␫ 1Ž y . s yy, ␫ 2 Ž ¨ . s y¨ and ␴i for i s 1, 2 are the rotation by 25 ␲ . From the symmetry of V0 , ⺗ acts on V˜0 . Since the subspace of H 4 Ž V˜0 ⺓ , ⺪. generated by l 10 y l 20 is invariant under the action of ⺗, and the involutions ␫ 1 and ␫ 2 exchange l 10 and l 20 , ⺗ acts on this subspace via a character ␹ defined by ␹ Ž ␫ i . s y1 and ␹ Ž ␴i . s 1. Let H04 Ž V˜0 ⺓ . denote the part of H 4 Ž V˜0 ⺓ . generated by the classes of the difference of the rulings of the exceptional divisors. Let ␫ : Ž x, y, u, ¨ . ¬ Ž u, ¨ , x, y . be an involution of V0 . Since l 10 y l 20 is invariant under ␫ , we will consider the part of H04 Ž V˜0 ⺓ . on which ⺗ acts via ␹ , and is invariant under ␫ . First, we can compute the ⺓-dimension of this cohomology group: 5.3 LEMMA. dim ⺓ Ž H04 Ž V˜0 ⺓ , ⺓. ␹ . ␫ s 1. The proof of this lemma will be postponed until the Appendix. Let l 10 [ Ž x y u s 0, y q ¨ s 0 . ,

l 20 [ Ž x q u s 0, y q ¨ s 0 .

l 12 [ Ž xX y uX s 0, y q ¨ s 0 . ,

l 22 [ Ž xX q uX s 0, y q ¨ s 0 .

and

be the rulings of the exceptional divisors over Ž0, 0, 0, 0. and p 2 = p 2 , respectively. Here p 2 s Žy1, 0. is one of the supplementary critical points and we put xX s x q 1, uX s u q 1. Then l 10 y l 20 is the cycle of our interest. Since l 10 y l 20 and Ý i, j g ⺪ r5⺪ ␴ 1i␴ 2j Ž l 12 y l 22 . are both in Ž H04 Ž V˜0 ⺓ , ⺓. ␹ . ␫ , there must be a linear relation between them. By considering the intersection number with the strict transform of the plane Ž x y u s 0, y y ¨ s 0., we see that 0 s 5 Ž l 10 y l 20 . y

Ý i , jg⺪r5⺪

␴ 1i␴ 2j Ž l 12 y l 22 . g H04 Ž V˜0 ⺓ , ⺓ . .

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Now we denote by ␥ the class of the cycle 5Ž l 10 y l 20 . y Ý i, j g ⺪ r5⺪ ␴ 1i␴ 2j Ž l 12 yl 22 . in H04 Ž V˜0 ⺓ , ⺪.. Then we see that ␥ is a torsion element in H04 Ž V˜0 ⺓ , ⺪.. Moreover, the image of ␥ in H2 Ž V0 ⺓ , ⺪. is the same as that of 5Ž l 10 y l 20 ., because Ý ␴ 1i␴ 2j Ž l 12 y l 22 . is contained in the exceptional divisor. Thus, to prove Ž5.2., it suffices to show the following: ␹

cl Ž ␥ . g H 4 Ž V˜0 ⺓ , ⺪ l Ž 2 . .  10 4

Ž 5.2⬘.

for any prime l.

Since the set  V˜0 , l 10 , l 20 , Ý i, j g ⺪ r5⺪ ␴ 1i␴ 2j Ž l 12 y l 22 .4 is defined over ⺡, it follows that clŽ␥ . is in the GalŽ⺡r⺡.-invariant part of H 4 Ž V˜0 ⺓ , ⺪ l Ž2.. tor . Now H 4 Ž V˜0 ⺓ , ⺪ l Ž2.. tor is equal to the pull back of H 4 Ž V0 ⺓ , ⺪ l Ž2.. tor . So to complete the proof of Ž4.1Ža.., it remains to show the following proposition. 5.4 PROPOSITION. Ž H 4 Ž V0 ⺓ , ⺪ l Ž2.. ␹ w 101 x. GalŽ⺡ r ⺡ . is generated by the class of an algebraic cycle and has no torsion for any prime l. Proof. Let S [ Ž z s 0. ; V0 . This is a smooth hyperplane section, so we have a surjection H 2 Ž S⺓ , ⺪ l Ž 1 . . ª H 4 Ž V0 ⺓ , ⺪ l Ž 2 . . ª 0, and

Ž 5.4.1.

H 2 Ž S⺓ , ⺪ l Ž 1 . .

␹

ª H 4 Ž V0 ⺓ , ⺪ l Ž 2 . .

1 10

␹

1 10

ª 0.

Let ⺞ i be the subgroup of ⺗ generated by ␴i for i s 1, 2, and put ⺞ [ ⺞1 = ⺞ 2 . Then H 2 Ž Sr⺞ ⺓ , ⺪ l .

␹

1 10

( H 2 Ž S⺓ , ⺪ l .

␹

1 10

.

Thus we have only to compute H 2 Ž Sr⺞ ⺓ , ⺪ l . ␹ w 101 x. Let H Ž x, y . s x Ž x 4 y 10 x 2 y 2 q 5 y 4 . , I Ž x, y . s H Ž y, x . ,

and T Ž x, y . s x 2 q y 2 .

Further, put H [ H Ž u, ¨ . ,

I [ I Ž u, ¨ . ,

and T [ T Ž u, ¨ . .

They are subject to the relation H2 q I2 s T5

Ž resp. H 2 q I 2 s T 5 . .

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We need the following lemma to determine Sr⺞; we will postpone its proof to the end of this section. Ž5.4.2. LEMMA. Sr⺞ s ⺠ Ž I, I, H , T , T . r Ž I 2 q H 2 y T 5 , I 2 q H 2 y T 5 . where ⺠Ž I, I, H, T, T . denotes a weighted projecti¨ e space. Ž5.4.3. Construction of an Algebraic Correspondence Let Ci be the normalization of the closure in ⺠ 2 of the affine curve given by yi5 s x i2 q 1 for i s 1, 2. The group ⺪r5⺪ acts on yi diagonally via multiplication by the fifth roots of unity and the involution ␫ i acts on x i by ␫ k Ž x i . s yx i for i s 1, 2. We have a map q: C1 _  ⬁ 4 =⺓ C2 _  ⬁ 4 ª Sr⺞ _  H s 0 4 given in the affine coordinates as follows: I H

¬ x1 ,

I H

T5

¬ x2 ,

H2

¬ y 15 ,

T5 H2

¬ y 25 ,

TT 4 H2

¬ y 1 y 24 .

We will need the following fact about the map q. Ž5.4.4. LEMMA. q : Ž C1 _  ⬁ 4 =⺓ C2 _  ⬁ 4 . r⺪r5⺪ ª Sr⺞ _  H s 0 4 is an isomorphism. The proof is given at the end of this section. We blow up C1 =⺓ C2 at ⬁ and denote it by C1 = C2; . Then q extends to a regular morphism from C1 = C2; to Sr⺞. The exceptional fibers of q are C1 = ⬁ and ⬁ = C2 , so that the exceptional loci of q: C1 = C2;r⺪r5⺪ ª Sr⺞ are ⺠ 1 over Ž TT , II , HI . and ⺠ 1 over Ž TT , II , HI .. Since the cohomology of exceptional locus contains no submodule on which ⺗ acts via ␹ , it follows that q induces an isomorphism

Ž 5.4.5. qU : H 2 Ž Sr⺞ ⺓ , ⺪ l .

␹

1 10

( H 2 Ž C1 = C2; , ⺪ l .

⺪r5⺪ , ␹

1 10

.

Furthermore, for the right hand side of Ž5.4.5. we have the equality

Ž 5.4.6. H 2 Ž C1 = C2; , ⺪ l .

⺪r5⺪ , ␹

1 10

s H 1 Ž C1 , ⺪ l . m H 1 Ž C 2 , ⺪ l . Ž 1 .

⺪r5⺪

1 10

.

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This follows from Kunneth formula noting that H 0 Ž C . m H 2 Ž C ., H 2 Ž C . m H 0 Ž C ., and the cohomology of ⺠ 1 over ⬁ = ⬁ have no ␹ part. Now observe that JacŽ Ci . is a CM abelian variety with End ⺓ ŽJacŽ Ci .. s ⺪w ␨ 5 x and furthermore End ⺓ ŽJacŽ C1 .. ¨ w H 1 Ž C1 , ⺪. m H 1 Ž C2 , ⺪.Ž1.x⺪ r5⺪ . Moreover, this is in fact an isomorphism:

Ž 5.4.7.

End ⺓ Ž Jac Ž C1 . . ( H 1 Ž C1 , ⺪ . m H 1 Ž C2 , ⺪ . Ž 1 .

⺪r5⺪

,

This follows because both sides of Ž5.4.7. are finite ⺪-algebras with same rank and ⺪w ␨ 5 x is regular. Tensoring Ž5.4.7. with ⺪ l , we see that w H 1 Ž C1 , ⺪ l . m H 1 Ž C2 , ⺪ l .Ž1.x⺪ r5⺪ w 101 x is generated by the image of the cycle class of the graph of the action of ⺪w ␨ 5 x on C in the factor w H 1 Ž C1 , ⺪ l . m H 1 Ž C2 , ⺪ l .Ž1.xw 101 x of H 2 Ž C1 = C2 , ⺪ l .Ž1.w 101 x. Let ⌫i2 be the graphs of the action of ␨ 5i on C and let ⌫i1 be the graph of composition of the action of ␨ 5i and hyperelliptic involution on C for 0 F i F 4. Their equations are given in the affine coordinates by Ž y 1 y ␨ i y 2 , x 1 y x 2 . and Ž y 1 y ␨ i y 2 , x 1 q x 2 ., respectively. The projection to H 1 Ž C1 , ⺪ l . m H 1 Ž C2 , ⺪ l .Ž1.w 101 x of clŽ ⌫i2 . is equal to 12 clŽ ⌫i2 y ⌫i1 .. Let Dij Ž0 F i F 4, j s 1, 2. be the image under q of ⌫i j. Di1 s ŽT y i ␨ T, I q I . and Di2 s ŽT y ␨ i T, I y I .. We regard Dij as a divisor on S. From the above argument we see that H 2 Ž S⺓ , ⺪ l Ž1.. ␹ w 101 x is generated by the classes of Di1 y Di2 Ž0 F i F 4.. From this we see that GalŽ⺡Ž ␨ 5 .r⺡. acts on H 2 Ž S⺓ , ⺪ l Ž1.. ␹ w 101 x. Then by Ž5.4.1. there is a surjection H 2 Ž S⺓ , ⺪ l Ž 1 . .

␹

1 GalŽ ⺡ Ž ␨ 5 .r⺡ . 10

ª H 4 Ž V0 ⺓ , ⺪ l Ž 2 . .

␹

1 GalŽ ⺡ Ž ␨ 5 .r⺡ . 10

ª 0.

Ga lŽ⺡ Ž ␨ .r ⺡ .

5 Since H 2 Ž S⺓ , ⺪ l Ž1..w 101 x ␹ is generated by Ý4is1Ž Di1 y Di2 ., it suffices to show that its class is not torsion in H 4 Ž V0 ⺓ , ⺪ l Ž2..w 101 x.

Ž5.4.8. Completing the Proof A defining equation of V0 in terms of H, H, T, T, and z is given as follows: V0 : 15 Ž H y H . q 12 z Ž T y T . Ž Ž T q T . y z 2 . s 0. From this and the relation T 5 y T 5 s I 2 y I 2 q H 2 y H 2 , we see that D 10 and D 02 are the restriction of divisors on V0 to S. We will show that the intersection number Ž D 10 , Ý4is1Ž Di1 y Di2 .. is directly calculated to be nonzero. Note that 4

H Ž x, y . s c Ł Ž cos Ž 25 ␲ i . x y sin Ž 25 ␲ i . y . , is0

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where c s Ł 4is0 cosŽ 25 ␲ i .y1. Now we will compute Ž D 10 , Di1 . for i s 1, . . . , 4. We have T s T s 0, so y s " 'y 1 x and ¨ s " 'y 1 u. Since H Ž x, " 'y 1 x . s cx 5, I Ž x, 'y 1 x . s cŽ'y 1 x . 5, and I Ž x, y 'y 1 x . s cŽy 'y 1 x . 5, we see that

Ž x, y, u, ¨ . g D10 l Di1 m Ž x, y, u, ¨ . s Ž ␨ j t , 'y 1 ␨ j t , t , y'y 1 t .

or

s Ž ␨ j t , y'y 1 ␨ j t , t , 'y 1 t . and also Ž D 10 , Di2 . s 0. Thus we get

Ž D10 , Di1 y Di2 . s Ž D10 , Di1 . ) 0

for i s 1, . . . , 4

and clŽÝ4is1Ž Di1 y Di2 .. g H 4 Ž V0 ⺓ , ⺪ l Ž2.. is not torsion. This finishes the proof of Proposition Ž5.4.. Finally, combining Lemma Ž5.1., Lemma Ž5.3., and Proposition Ž5.4., we establish the assertion Ža. in Theorem 4.1. Ž5.5. Proofs of Lemmas Proof of Lemma Ž5.4.2.. We first show that ⺞

⺓ w x, y, u, ¨ x s ⺓ H , I, F , H , F , I .

Ž 5.4.2.1.

Since ⺓w x, y, u, ¨ x ( ⺓w x, y x m ⺓w u, ¨ x as a vector space, we need only show that ⺓ w x, y x

⺞1

s ⺓ w I, T , F x s ⺓ w I, T , F x r Ž H 2 q I 2 y T 5 . \ A.

Since A is Cohen᎐Macauley and is regular outside Ž H, I, T ., A is normal. Since ⺓w x, y x is finite over A and ⺓Ž x, y .⺞ 1 s ⺓Ž I, T, H ., Ž5.4.2.1. follows. From this we see that

Ž 5.4.2.2. ⺠ 3r⺞ s P Ž I, I, H , H , T , T . r Ž I 2 q H 2 y T 5 , I 2 q H 2 y T 5 . , where ⺠Ž I, I, H, H, T, T . denotes a weighted projective space. Since a defining equation for S is H y H s 0, it follows that as a topological space, Sr⺞ s ⺠ Ž I, I, H , T , T . r Ž I 2 q H 2 y T 5 , I 2 q H 2 y T 5 . . Since Sr⺞ is reduced, this equality also holds as that of schemes. Proof of Lemma Ž5.4.4.. Let B be the subring of ⌫ Ž C1 _  ⬁4 =⺓ C2 _  ⬁4 , O . generated by  x 1 , x 2 , y 1k y 25yk Ž0 F k F 5.4 . What we need to show is that

Ž C1 _  ⬁4 =⺓

C2 _  ⬁ 4 . r⺪r5⺪ s Spec B.

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Here again, we have only to show that B is normal. We see that B yy5 s⺓ w Y , x 1 , x 2 x r Ž Yyy 2ry 1 , Ž x 12 q1 . Y 5 y Ž x 22 q1 . . 1

Ž x 12 q1 .

y1

and we see that this is regular, so that B is regular outside the ideal generated by Ž y 1k y 25yk . Ž0 F k F 5.. At these points, Ž y 15, y 25 . is a regular sequence, so B is normal.

6. THE PROOF OF ASSERTION Ž4.1Žb.. First we state Proposition Ž3.2. in wBlx. This will be used in our proof. The absolute purity of the l-adic etale cohomology is required to apply this proposition to our case. Ž6.1. PROPOSITION wBlx. Let R be a strict henselian discrete ¨ aluation ring, and X be a regular proper scheme o¨ er Spec R. Let X␩ and Xs be generic and special fibers, respecti¨ ely. Let Y be a component of X . Then we ha¨ e the commutati¨ e diagram w CH 2 Ž Y .rIm h x m ⺪ l

6

␺

F 0 CH 2 Ž X␩ . ␾

␾

6

6

6

H 1 Ž␩ , H 3 Ž X␩ , ⺪ l Ž2...

␺

H 4 Ž Y, ⺪ l Ž2..rIm g

where ␺ ’s are specialization maps, ␾ on the left is the l-adic Abel᎐Jacobi map, and ␾ on the right is the cycle map. Moreo¨ er, we ha¨ e the map on chow groups i#

iU

h: CH 1 Ž Xs . ª CH 2 Ž X . ª CH 2 Ž Y . , where i: Xs ª X is the natural inclusion, and a map g on the cohomology groups i#

iU

g : HX4sŽ X , ⺪ l Ž 2 . . ª H 4 Ž X , ⺪ l Ž 2 . . ª H 4 Ž Y , ⺪ l Ž 2 . . . Assume further that the cycle map CH 2 Ž Y . ª H 4 Ž Y, ⺪ l . is injecti¨ e and that Im g is generated by ␾ ŽIm h.. Let z g F 0 CH 2 Ž X␩ . and that ␺ Ž z . / 0 in w CH 2 Ž Y .rIm h x m ⺪ l . Then ␾ Ž z . / 0 in H 1 Ž␩ , H 3 Ž X␩ , ⺪ l Ž2.... Ž6.2. Constructing a Regular Model for V˜ Under the assumption on F and F X , there is an affine piece of V˜ with a defining equation

␲ 2␣ q ␤ y 2 q ␥ u 2 q ␦ ¨ 2 q tx 3 ,

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KEN-ICHIRO KIMURA

where ␣ , ␤ , ␥ , ␦ , t f Ž␲ , x, y, u, ¨ .. When this equation is blown up at the ideal Ž␲ , x, y, u, ¨ ., the exceptional locus is a quadratic cone with vertex at ␲ s 0, x s 1, y s u s ¨ s 0 in the projectivized tangent cone at the maximum ideal of Rww x, y, u, ¨ xx. We introduce coordinates ␲ X s ␲rx, yX s yrx Žresp. uX s urx, ¨ X s ¨ rx .. Consider the affine pieces of the blowup with defining equation

Ž 6.2.1.

␲ X 2␣ X q ␤ X yX 2 q ␥ X uX 2 q ␦ X ¨ X 2 q tX x s 0, ␲ X x s ␲ ,

where ␣ X , ␤ X , ␥ X , tX f Ž␲ X , x, yX , uX , ¨ X .. The fiber ␲ s 0 now has two components: the strict transform of the original fiber defined by ␲ X s 0, and the exceptional quadric cone defined by

␲ X 2␣ X q ␤ X yX 2 q ␥ X uX 2 q ␦ X ¨ X 2 s 0 s x. These two meet along the singular quadric surface ␤ X yX 2 q ␥ X uX 2 q ␦ X ¨ X 2 s 0. We blow up the equation Ž6.2.1. again at the ideal Ž␲ X , x, yX , uX , ¨ X .. This is a regular point on the total scheme, so the new exceptional divisor is ⺠ ⺖3 5 . The fiber ␲ s 0 now consists of three divisors, D 1 , D 2 , and D 3 , with normal crossings. Here D 1 is the strict transform of the original fiber. The original fiber is the blowup of V Žmod ␲ . at Ž x, y, u, ¨ . and at the singularities coming form the intersections of the lines, noting that by the choice of F X , the terms of degree 2 at the intersections of the lines are non-zero Žmod ␲ .. So the other exceptional fibers besides Q are away from Ž␲ , x, y, u, ¨ . and similarly away from D 2 and D 3 . Now D 2 is the quadric cone ␲ X 2␣ X q ␤ X yX 2 q ␥ X uX 2 q ␦ X ¨ X 2 s 0 s x with the vertex blown up. It is a ⺠ 1-bundle over a smooth quadric surface S. Moreover, D 3 ( ⺠ ⺖3 5 , and D 2 l D 3 s S⬁ Žs the ⬁-section of D 2 .. Finally, D 1 l D 2 s T is the restriction of the ⺠ 1-bundle D 2 over the curve on S defined by ␲ X s 0. Ž6.3. Completing the Proof Consider the strict transform of the lines l i0 Ž i s 1, 2. under these transformations. Since l i0 ; Q: x s 0, the exceptional fiber after blowing up will lie in the hyperplane defined by x s 0. In particular, the line will not pass through the vertex lying under D 3 . Therefore the intersection of the strict transformation of l 10 y l 20 with D 2 is the difference of two rulings lying in a section S0 of the ⺠ 1-bundle not meeting S⬁ . The remaining procedure may be argued exactly in the same way as that in wBlx. We have Ž␲ . s D 1 q D 2 q 3 D 3 since D 3 is gotten by blowing up a point whose multiplication is 2 on D 1 and 1 on D 2 . Further, D 2 being a ⺠ 1-bundle over

ALGEBRAIC CYCLES WITH NONTRIVIAL ABEL ᎐ JACOBI IMAGES

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a surface, we see that CH 2 Ž D 2 . ( ⺪ ⭈ L [ f U CH 1 Ž S . ⭈ ␳ , where L ( ⺠ 1 is a fiber of D 2 and ␳ is the tautological divisor. Moreover, CH 1 Ž S . ( ⺪ ⭈ m1 [ ⺪ ⭈ m 2 , where m i ’s are rulings of S. From these facts one can deduce that CH 2 Ž D 2 . r Im Ž CH 1 Ž D 3 . ª CH 2 Ž D 2 . qIm Ž CH 1 Ž D 1 . ª CH 2 Ž D 2 . . q ND 2 r X ⭈ CH 1 Ž D 2 . ( ⺪r3⺪ and it is generated by the specialization of l 10 y l 20 . From this and Proposition Ž6.1. we conclude that 0 / cl 2V˜k Ž l 10 y l 20 . g H 1 Gal Ž krk . , H 3 Ž V˜k , ⺪ 3 . .

ž

/

This completes our proof for Theorem Ž4.1Žb...

APPENDIX Here we give proofs of Lemma Ž5.1. and Lemma Ž5.3.. We also give the condition on F and F X which is necessary to prove Ž4.1Žb.. of the theorem. ŽA.1. Proof of Lemma 5.1. We will make use of the argument of vanishing cycles as in wClx. Let Fc Ž x, y . s Ž x q

1 2

q c . y 4 y y 2 Ž 2 x 2 y 2 x q 1 . q 15 Ž x 2 q x y 1 .

ž

2

/

and let Ž ␻q, 0. be the critical point of Fc converging to Ž0, 0. as c ª 0. Then Ž0, 0. is a critical point of F˜c Ž x, y . s wŽ F0 Ž0, 0..rŽ Fc Ž ␻q, 0..x Fc Ž x q ␻q, y .. Let Vc ; ⺠ 4 be the projective closure of F0 Ž x, y . y F˜c Ž u, ¨ . s 0. For c / 0 small, Vc has 101 double points and let V˜c be the blowing-up of Vc at these points. Then the argument along the same line as in wClx shows that H2 Ž V˜c , ⺪ . ( H2 Ž V0 , ⺪ . . Fix c / 0 and choose F Ž x, y . Žresp. F X Ž u, ¨ .. so that Ž1. the coefficients of F Ž x, y . Žresp. F X Ž uX , ¨ X .. Žmod ␲ . satisfy the conditions in ŽA.3. below;

142

KEN-ICHIRO KIMURA

Ž2. the coefficients are sufficiently close Žas elements in ⺓. to those of F0 Ž x, y . Žresp. F˜c Ž u, ¨ .., and that Ž3. F Ž0, 0. y F X Ž0, 0. s 0. Let V ; ⺠ 4 be the projective closure of F Ž x, y . y F X Ž u, ¨ . s 0 and let V˜ be the blowing-up of the double points. Then both V˜ and V˜c are fibers of a smooth one parameter family. We can construct this one parameter family as follows: First, let Ft Žresp. FtX . be polynomials given by connecting the coefficients of F and Fc Žresp. F X and F˜c . by a ‘‘line’’ with the parameter t. We have F0 s Fc and F1 s F Žresp. F0X s F˜c and F1X s F X .. Let pre V be the family such that the fiber pre V t is the closure in ⺠ 4 of Ft Ž x, y . y FtX Ž u, ¨ . s 0. Then we modify the coefficient a of Ft and FtX so that Ž0, 0, 0, 0. g pre V t for each t. We also modify the coefficients b and b⬘ of Ft and FtX so that the partial derivatives of Ft and FtX vanish at Ž0, 0.. Let V˜t be the blow-up of pre V t in the intersection points of the lines and Ž0, 0, 0, 0.. This is a smooth family in a neighborhood of 0 with V˜0 s V˜ ˜ ⺪.. This and V˜1 s V˜c . Hence we have the isomorphism H2 Ž V˜c , ⺪. ( H2 Ž V, completes our proof for Lemma Ž5.1.. ŽA.2. Proof of Lemma Ž5.3.. By Proposition 1.3 of wSchx we have a canonical isomorphism H 1 Ž ⺠ 4 , ISing m ␻ ⺠ 4 Ž 2V0 . . ( H04 Ž V˜0 ⺓ , ⺓ . , where Sing is the singular set of V0 . So we have an exact sequence of ⺗ modules e

H 0 Ž ⺠ 4 , ␻ ⺠ 4 Ž 10 . . ª H 0 Ž Sing, O Sing m ␻ ⺠ 4 Ž 10 . . ª H04 Ž V˜0 ⺓ , ⺓ . ª 0. Here eŽ h. s hŽ x .r␻ z 10 evaluated at Sing and ␻ s dŽ xz . n dŽ zy . n dŽ u2 . n dŽ ¨z .. Since ⺗ acts on ␻ via ␹ , and since ␻ is invariant under ␫ , we see that a basis of Ž H 0 Ž⺠ 4 , ␻ ⺠ 4 Ž10.. ␹ . ␫ can be chosen to be the set

 ␻ Ž T q T . z 8 , ␻ Ž T 2 q T 2 . z 6 , ␻ Ž TT . z 6 , ␻ Ž H q H . z 5 , ␻ z 10 4 , where T, T, H, H are polynomials which will be defined below. The critical points of F0 on the x Žresp. u. axis are p1 [

ž

y1 y '5 2

p 0 [ Ž 0, 0 . ,

p 2 [ Ž y1, 0 . ,

/

,0 , p3 [

ž

y1 q '5 2

/

,0 .

ALGEBRAIC CYCLES WITH NONTRIVIAL ABEL ᎐ JACOBI IMAGES

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The points p1 and p 3 are intersection points of the lines, and p 2 is one of the supplementary critical points. Put Sing s O1 j O 2 j O 3 j O4 j O5 j p 0 = p 0 , where O1 s  ␴ 1i p1 = ␴ 2j p1 N i , j g ⺪ 4 ,

O 2 s  ␴ 1i p1 = ␴ 2j p 3 N i , j g ⺪ 4 ,

O 3 s  ␴ 1i p 3 = ␴ 2j p1 N i , j g ⺪ 4 ,

O4 s  ␴ 1i p 3 = ␴ 2j p 3 N i , j g ⺪ 4 ,

and O5 s  ␴ 1i p 2 = ␴ 2j p 2 N i , j g ⺪ 4 . Then a basis of Ž H 0 Ž S, OS m ␻ ⺠ 4 Ž10.. ␹ . ␫ consists of S1 s

Ý

s,

S2 s

sgO 1

S4 s

Ý

Ý

s,

sgO 2jO 3

s and

S4 s

Ý

s,

sgO4

p0 = p0 .

sgO 5

We see that the rank of eŽŽ H 0 Ž⺠ 4 , ␻ ⺠ 4 Ž10.. ␹ . ␫ . is equal to 4. This is done by computing the images of the basis under the map e at Ž p1 = p1 ., Ž p1 = p 3 ., Ž p 3 = p 3 ., Ž p 2 = p 2 ., and Ž p 0 = p 0 .. Therefore, the rank of Ž H04 Ž V˜0 ⺓ , ⺓. ␹ . ␫ is 1. This completes our proof for Lemma Ž5.3.. ŽA.3. Auxiliary Conditions on F and F X The condition on F and F X mentioned in Theorem Ž4.1. is described as follows: The condition for the partial derivatives of F and F X to vanish at Ž0, 0. is y2 a Ž de q cf . y2 aX Ž dX eX q cX f X . X bs and b s . df dX f X Then the expansion of F = df is given as follows: ad 3 f 3 q x 5 y2 ac 2 de 3 y 2 ac 3 e 2 f q x 4 y4 acd 2 e 3 y 7ac 2 de 2 f y 4 ac 3 ef 2 q x 3 y2 ad 3 e 3 y 8 acd 2 e 2 f y 8 ac 2 def 2 y2 ac 3 f 3 q Ž 2 ac 2 de q 2 ade 3 q 2 ac 3 f q 2 ace 2 f . y 2 q x 2 y3ad 3 e 2 f y 4 acd 2 ef 2 y 3ac 2 df 3 q Ž 4 acd 2 e q 3ac 2 df q 3ade 2 f q 4 acef 2 . y 2 q x Ž 2 ad 3 e q 2 acf 3 . y 2 q Ž y2 ade y 2 acf . y 4 q Ž y Ž ad 3 f . y adf 3 . y 2 q adfy 4 .

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KEN-ICHIRO KIMURA

Write ␣ s de, ␤ s cf. The coefficient of x 4 Žresp. x 3 , x 2 . ' aceŽ ␣ y ␤ . 2 Žresp. ' y2 aŽ ␣ q ␤ .Ž ␣ y ␤ . 2 , ' y3adf Ž ␣ y ␤ . 2 .Žmod ␲ 2 .. We take a ; f so that the coefficient of x 2 has the ␲-adic order 2, and the coefficients of x 3 and x 4 are also divisible by ␲ 2 , and that de q cf, d 2 q f 2 k 0Žmod ␲ .. Then this expansion takes the form ad 3 f 3 q ␲ 2␣ x 2 q ␤ y 2 q tx 5 , where ␣ , ␤ , t f Ž x, y, ␲ .. Note that we subsumed the terms xy 2 , x 2 y 2 , y 4 , and x 3 y 2 into ␤ y 2 and x 3 , x 4 into ␲ 2␣ x 2 . We take F X so that the lines meet only in pairs after mod ␲ , and the equations of the lines at the intersections are local parameters of spec ⺖5 w u, ¨ x at that point. This is necessary for V˜ Žmod ␲ . to be the blowing-up of V Žmod ␲ . in the intersection points of the lines and in Ž0, 0, 0, 0.. It is sufficient to take aX ; f X so that all of the elements dX eX y cX f X , dX 2 q f X 2 , cX " eX , dX " f X , dX f X Ž cX y eX . y 2 Ž dX eX q cX f X . Ž f X y dX . k 0 Ž mod ␲ . and so that dX f X Ž cX q eX . q 2 Ž dX eX q cX f X . Ž f X q dX . k 0 Ž mod ␲ . . ŽNote that dX f X Ž cX y eX . y 2Ž dX eX q cX f X .Ž f X y dX . is a non-zero multiple by an element in RU of aX q bX u at the intersections of ¨ y Ž cX u q dX . s 0 and ¨ y Ž eX u q f X . s 0. Similarly, dX f X Ž cX q eX . q 2Ž dX eX q cX f X .Ž f X q dX . is a non-zero multiple by an element of RU of aX q bX u at the intersections of ¨ y Ž cX u q dX . s 0 and ¨ q Ž eX u q f X . s 0.. Then the expansion of F y F X takes the form

␲ 2␣ x 2 q ␤ y 2 q ␥ u 2 q ␦ ¨ 2 q tx 5 , where ␣ , ␤ , ␥ , ␦ , t f Ž␲ , x, y, u, ¨ .. Moreover, if we take c " e k 0 Žmod ␲ ., the affine coordinates of the intersections of the lines in F are in R. If we let k be the field of definition of V˜ and the two rulings of Q, then it is a finite extension of ⺡Ž'5 . unramified at Ž'5 .. ŽA.4. Problem. The determination of the L-series of this variety should be pursued. One practical difficulty is that b 3 is rather large, e.g., G 20 and hence the actual computation of local factors becomes intractable at the moment. But the author hopes to come back to this problem.

ACKNOWLEDGMENTS I am grateful to Professor Shuji Saito for encouragement and especially to Professor Takeshi Saito for his advice in the proof of Proposition Ž5.4.. The revision of this work was

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carried out at Queen’s University in the Fall of 1996, where I have held a visiting postdoctoral fellowship. I thank the Department of Mathematics of Queen’s University for its hospitality and friendly atmosphere. Special thanks are due to Professor N. Yui for her reading through the earlier version of this paper and for her valuable suggestions for its improvement.

REFERENCES wBlx wClx wGr-Dx wHix wSchx wTax wWe-Gex

S. Bloch, Algebraic cycles and values of L-functions II, Duke. Math. J. 52Ž2. Ž1985., 379᎐397. C. H. Clemens, Double Solids, Ad¨ . Math. 47 Ž1983., 107᎐230. A. Grothendieck and P. Deligne, La class de cohomologie associee ` a un cycle, in ‘‘SGA4 12 ,’’ Lecture Notes in Mathematics, Vol. 569, pp. 129᎐153, Springer-Verlag, BerlinrNew York, 1977. F. Hirzebruch, Some examples of threefolds with trivial canonical bundle, in ‘‘Collected Papers,’’ Vol. II, pp. 757᎐770, Springer-Verlag, BerlinrNew York, 1987. C. Schoen, Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann. 270 Ž1985., 17᎐27. J. Tate, Relations between K 2 and Galois cohomology, In¨ ent. Math. 36 Ž1976., 257᎐274. J. Werner and B. van Geemen, New examples of threefolds with c1 s 0, Math. Z. 203 Ž1990., 211᎐225.