Note on curves in a Jacobian

Transcription

1 Note on curves in a Jacobian Elisabetta Colombo and Bert van Geemen arxiv:alg-geom/920205v2 24 Feb 992 Introduction. For an abelian variety A over C and a cycle α CH d (A) Q we define a subspace Z α of CH d (A) Q by: Z α := n α : n Z CH d (J(C)) Q. Results of Beauville imply that Z α is finite dimensional (cf. 2.5 below). In case A = J(C), the jacobian of a curve C, Ceresa has shown that the cycle C C := C ( ) C Z C is not algebraically equivalent to zero for generic C of genus g 3, which implies that for such a curve dim Q Z C 2. In this note we investigate the subspace Z Wm of CH m (J(C)) Q, with W m the image of the m-th symmetric power of C in J(C) (so W = C). To simplify matters we will actually work modulo algebraic equivalence (rather than linear equivalence, note that translates of a cycle are algebraically equivalent)..2 Let Z α / alg be the image of Z α CH d (A) Q in CH d (A) Q / alg. A d-cycle α is Abel-Jacobi equivalent to zero, α AJ 0, if α is homologically equivalent to zero and its image in J d (A), the d-th primitive Intermediate Jacobian of A, is zero. Recall that any curve of genus g is a cover of P for some d g Theorem.. For any abelian variety A and any α CH d (A) we have: dim Q (Z α / AJ ) For any curve of genus g and n g we have: dim Q (Z Wg n / alg ) n.

2 3. For a curve C which is a d : -cover of P we have: dim Q (Z C / alg ) d. (We prove.3. in 2.5,.3.2 in 2.7 and.3.3 in 3.6.).4 Recall that Ceresa showed that the image of W m W m in J m (J(C)) is non-zero for generic C of genus g 3 and m g 2. Therefore.3. and.3.2 for n = 2 are sharp. In case C is hyperelliptic, so C is a 2: cover of P, the cycles W m and W m are however algebraically equivalent. Therefore.3.3 is sharp for hyperelliptic curves (d = 2) and generic trigonal curves (d = 3). In case C is not hyperelliptic nor trigonal (like the generic curve of genus g 5), it would be interesting to know if.3.3 is actually sharp. Note that.3. implies that one cannot use the Intermediate Jacobian anymore to derive new algebraic relations among the cycles in Z C / alg. Recently M.Nori [N] constructed cycles on complete intersections in P N which are Abel-Jacobi equivalent to zero but not algebraically equivalent to zero. There is thus the possibility that similar cycles can be found on the Jacobian of a curve of genus g 5. A cycle in Z C, for certain modular curves C, was investigated by B.H.Gross and C.Schoen, [GS], esp. section 5, see also The inequalities.3. and.3.2 are consequences of work of Beauville. The main part of the paper deals with the proof of.3.3. Recall that on a smooth surface S homological and algebraical equivalence for curves coincide. Thus if we have map Φ : S J(C) and relation a [C ]+...+a n [C n ] = 0 in H 2 (S,Q), we get a Φ C +...+a n Φ C n alg 0 in J(C). We use this remark to obtain our result, the main difficulty is of course to find suitable surfaces, curves in them and maps to J(C) and to determine Neron-Severi groups (= Im(CH (S) H 2 (S,Q))) of the surfaces involved..6 We are indebted to S.J.Edixhoven and C.Schoen for several helpful discussions. 2 General results 2. The effect of n and n on the Chow groups has been investigated by Beauville ([B], [B2]), Dehninger and Murre [DM] and Künnemann [K] using the Fourier transform on abelian varieties. Below we summarize some of their results and derive the finite dimensionality of Z α as well as Thm.3. and

3 2.2 Let A be a g-dimensional abelian variety, we will view B := A A as an abelian scheme over A using the projection on the first factor: B A = A A π A For each integer n, we have an s n B(A) and a cycle Γ n CH g (B): s n : A B, a (a, na), Γ n := s n A CH g (A A), and Γ n is, as the notation suggests, the graph of multiplication by n on A. The cycle Γ n defines for each i a map on CH i (A) which is just n : n = Γ n : CH i (A) CH i (A), α n α = π 2 (π α Γ n). Next we introduce a relative version of the Pontryagin product on the Chow groups of the A-scheme B (cf. [K], (.2)). Let m B : B A B B be the multiplication map, then: α β := m B (α A β), α, β CH (B). Since the relative dimension of Γ n over A is 0, the cycle Γ n Γ m lies in CH g (B) and one has ([K], (.3.4)): Γ n Γ m = Γ n+m CH g (A A) Q, so Γ n = Γ n, where we write α n for the n-fold Pontryagin product of a cycle α with n > 0 and we put α 0 := Γ 0. In [K].4. a generalization of a theorem of Bloch is proved, which implies: (Γ Γ 0 ) (2g+) = 0 ( CH g (B) Q ). (2.2.) Using the ring structure on CH g (B) Q with product one can thus define the following cycles π i, 0 i 2g in CH g (B) Q : with: π i := (2g i)! (log Γ ) (2g i), log Γ := (Γ Γ 0 ) 2 (Γ Γ 0 ) g (Γ Γ 0 ) (2g). 3

4 Let = Γ CH g (A A) Q the class of the diagonal, then ([K], proof of 3..): = π 0 + π π 2g, π i π j = π j π i = { πi if i = j 0 if i j, (2.2.2) here αβ, for cycles α, β CH g (A A) Q, is their product as correspondences: αβ := p 3 (p 2 α p 23 β) with the p ij : A 3 A 2 the projection to the i, j factor. Moreover: Γ n π 2g i = π 2g i Γ n = n i π 2g i, (2.2.3) (one has t Γ n π i = π i t Γ n = n i π i and t π i = π 2g i ([K], 3..), now take transposes). 2.3 Remark. We sketch how these results can be obtained from Let M CH g (B) Q be the subspace spanned by the Γ n, n Z. Then 2.2. implies that dim Q M 2g + (use Γ i (Γ Γ 0 ) (2g+) = 0 for all i Z). Using the Künneth formula, Poincaré duality, one finds that the cohomology class of Γ n in H 2g (A A,Q) = H 2g i (A,Q) H i (A,Q) = Hom(H i (A,Q), H i (A,Q)) is given by (note Γ n induces n ): [Γ n ] = (n 2g, n 2g,...,n, ) 2g Hom(H i (A,Q), H i (A,Q)) (where n 2g i in the i-th component means multiplication by n 2g i on H i (A,Q)). Therefore dim Q M/ hom = 2g + and dim Q M = 2g +. Thus we have the (surprising) result that homological and linear equivalence coincide in M. We can now define π i M by [π i ] := (0,...,0,, 0,..., 0) i Hom(H i (A,Q), H i (A,Q)) ( in i-th spot) then and follow. To express π i as combination of Γ 0,...,Γ 2g, note that the (Q-linear) ring homomorphism: Q[X] M CH g (B) Q, X i Γ i = Γ i (with product on CH g (B) Q ) gives an isomorphism Q[X]/(X ) 2g+ = M. Since Γ n Γ m = Γ nm (product as correspondences), π 2g i corresponds to a polynomial f 2g i with: f 2g i (X n ) = n i f 2g i (X) so f 2g i (X) := c i (log X) i mod (X ) 2g+ and c i Q can be determined with a little more work. 4

5 2.4 Since = Γ : CH d (A) CH d (A) is the identity, we get: CH d (A) Q = CH d (A) (i), with CH d (A) (i) := π 2g i CH d (A) Q. and each CH d (A) (i) is an eigenspace for the multiplication operators: n α = n i α n Z, α CH d (A) (i), a result which was first obtained by Beauville [B2]. Moreover, he proves: CH d (A) (i) 0 = d i d + g (2.4.) (and gives sharper bounds for some d in prop.3 of [B2], it has been conjectured that CH d (A) (i) 0 2d i d + g). 2.5 Proposition. Let A be an abelian variety and let α CH d (A) Q. Then: dim Q Z α g + and dim Q (Z α / AJ ) 2. Moreover, we have π i Z α Z α for all i. Proof. We write α as a sum of weight vectors (α (i) := π 2g i α): α = α (d) + α (d+) α (g+d), so n α (i) = n i α. Taking g distinct, non-zero, integers n j, the determinant of the matrix expressing the n i α Z α in terms of the α i is a Vandermonde determinant. Thus each α (i) = π 2g i α Z α and Z α is spanned by the α (i). Since n acts as n 2d on H 2g 2d (A,Q) and as n 2d+ on J d (A), the space Z α / AJ is spanned by α (2d) and α (2d+). 2.6 In this section, we fix an abel-jacobi map C J(C). Note that W d = d! C d (Pontryagin product on J(C)). Let Θ CH (J(C)) be a symmetric theta divisor, so Θ is a translate of W g. Let Θ d CH g d (J(C)) be the d-fold intersection of Θ. 2.7 Proposition. For any d, d g, we have Θ d Z Wg d, more precisely: π 2g i W g d 0 = 2(g d) i 2g d, and π 2d W g d = d! Θd. Moreover we have: dim Q (Z Wg d / alg ) d. 5

6 Proof. We first prove the case d = g. Since the map CH (C) Q H 2g 2 (J(C),Q) factors over π 2g 2 CH (J(C)) Q (cf. the proof of 2.5 or even better, [B2], p.650) we know that π 2g 2 C 0. Then its Fourier transform F CH (π 2g 2 C) π 2 CH g (J(C)) Q ( = NS(J(C)) Q ) (cf. [B2], prop.) is non-zero. For a generic Jacobian NS Q is one dimensional and thus π 2 CH g (J(C)) Q = QΘ ([B], prop.5 and [B2], prop.). By specializing, we have for all curves that F CH (π 2g 2 C) QΘ. From [B], prop.5 we have Θ = F CH ( (g )! Θg ), so for a nonzero constant c: F CH (π 2g 2 C) = cf CH ( (g )! Θg ) and, by using F 2 CH = ( )g Γ and comparing cohomology classes, we get: π 2g 2 C = (g )! Θg. Next we recall that π 2g CH (A) = 0 for any abelian variety A ([B2], prop.3), thus: C = C (2) C g+ with C (i) = π 2g i C, so n C (i) = n i C (i) and C (2) = (g )! Θg. Therefore W g d = (g d)! C (g d) = (g d)! C (g d) (2) + Y (2.7.) with Y a sum of cycles C (i )... C (ig d ) with all i j 2 and at least one > 2. Therefore, using n (U V ) = (n U) (n V ), we get: π 2d W g d = (g d)! C (g d) (2) and π j W g d = 0 for j > 2d. By [B], corr.2 of prop.5, Θ d lies in the subspace spanned by (Θ g ) (g d), so C (g d) (2) = cθ d for a non-zero c Q and taking cohomology classes one finds c =. Finally, using 2.4. and 2.7. we can write W g d = α (2g 2d) α (2g d), with α (i) := π 2g i W g d, and since α (2g d) alg 0 by [B2], prop.4a, we get dim Q (Z Wg d / alg ) d. 2.8 In the paper [GS] the following cycle (modulo alg ) is considered: Z := 3 C 3 2 C + 3C = (Γ Γ 0 ) 3 C. This cycle is related to Ceresa s cycle in the following way: 6

7 2.9 Proposition. We have: π 2g 2 (Z) = π 2g 2 (C C ) = 0 and π 2g 3 (Z) = 3π 2g 3 (C C ). In particular, Z is abel-jacobi equivalent to 3(C C ). Proof. As we saw before (proof of 2.7) we can write: Since n C (i) = n i C (i) we have: C = C (2) + C (3) +... with C (i) := π 2g i C. C C = 2(C (3) C (5) +...), Z = 6C (3) + 36C (4) Therefore π 2g 2 (C C ) = 3π 2g 2 (Z) = 0, so both cycles are homologically equivalent to zero, and 3π 2g 3 (C C ) = π 2g 3 (Z) = 6C (3). Since the abeljacobi map factors over π 2g 3 CH (J(C)) we have that Z AJ 3(C C ). 3 Proof of Let C be a generic d: cover of P, and denote by g d the corresponding linear series. For an integer n, 0 < n < d we define a curve G n in the n-fold C (n) =Sym n (C): G n = G n (g d) = {x + x x n C (n) : x + x x n < g d}, see [ACGH], p.342 for the definition of the scheme-structure on G n. We define a surface by: S n = S n (g d) = G n C. 3.2 Proposition. Let C be a generic d:-covering of P of genus g. Then S n is a smooth, irreducible surface and dim Q NS(S n ) Q = 3. Proof. Since C is generic, the curve G n is irreducible (consider the monodromy representation) and for a simple covering the smoothness of G n follows from a local computation. Therefore also S n is smooth and irreducible. Note that for a product of 2 curves S n = G n C we have that NS(S n ) Q = QC QG n Hom(J(C), J(G n )) Q. Using Hodge structures, the last part are just the Hodge-cycles in H (C,Q) H (G n,q). Since J(C) and Pic 0 (C (n) ) are isogeneous (their H (Q) s are 7

8 isomorphic Hodge structures), composing such an isogeny with the pull-back Pic 0 (C (n) ) J(G n ), we have a non-trivial map J(C) J(G n ). To show that dim Q Hom(J(C), J(G n )) Q we use a degeneration argument. First we recall that the Jacobian of C is simple and has in fact End(J(C)) = Z. In case g = 2 this is clear since any genus 2 curve can be obtained by deformation from a given d: cover and C is generic. In case g = 3 one reasons similarly, taking the necessary care for the hyperelliptic curves. In case g(c) = g > 3, assume first that there is an elliptic curve in J(C). Specializing C to a reducible curve with two components C and C, both of genus 2, and themselves generic d:-covers we obtain a contradiction by induction. Assume now that there is no elliptic curve in J(C), then we specialize C to E C, with E an elliptic curve and C a generic d:-covering of genus g. By induction again, J(C ) is simple, which contradicts the existence of abelian subvarieties in J(C). We conclude that J(C) is simple. Thus End Q (J(C)) is a division ring and specializing again we find it must be Q. We may assume that the gd exhibits C as a simple cover of P, then G n is a cover of P (of degree ( d n )) with only twofold ramification points. Letting two branch points coincide, we obtain a curve C with a node, the normalization C of C has genus g and is again exhibited as a generic d : cover of P. The curve G n acquires ( d 2 n ) nodes (since twice that number of branch points coincide pairwise) and the normalization of that curve, G n, is G n, the curve obtained from the g d on C. The (generalized) Jacobians of C, G n are extensions of the abelian varieties (of dim ) J(C ), J(G n ) by multiplicative groups. Since the number of simple factors of J(G n ) which are isogeneous to J(C ) is greater then or equal to the number of simple factors of J(G n ) which are isogeneous to J(C), and J(C), J(C ) have End Q = Q it follows: dim Q Hom(J(C), J(G n )) Q dim Q Hom(J(C ), J(G n)) Q. Therefore it suffices to show that for a generic elliptic curve C = E and a generic gd on E we have dim Q Hom(E, J(E n )) Q (with E n = G n (gd ) and C = E). We argue again by induction, using degeneration. Note that for any C, G (gd ) = G d (gd ) = C. Since a generic elliptic curve has Hom(E, E) = Z the statement is true for d 3 (and any 0 < n < d) and since E = E it is also true for n = (and any d 2). We fix E, a generic elliptic curve, but let the gd acquire a base point y E. Then E n (gd ), n 2, becomes a reducible curve Ēn, having two components E n = E n(gd ) and E n = E n (gd ) which meet transversally 8

9 in N = ( d 2 n ) points. Indeed, let the divisor of the g d containing y be y+y +...+y d 2, then the points y i +...+y in E n and y+y i +...+y in E n are identified. Thus J(E n ) is an extension of J(E n) J(E n ) by the multiplicative group (C ) N, in fact there is an exact sequence: C (C ) N π J(E n ) J(E n ) J(E n ) 0, where is the diagonal embedding. We will write π = (π 0, π ). We will prove that Hom(E, J(E n )) Q Hom(E, J(E n )) Q, φ π0 φ is injective. By induction we may assume that dim Hom(E, J(E n )) Q, thus also dim Q Hom(E, J(E n )) Q and since dim Q Hom(E, J(E n )) Q dim Q Hom(E, J(E n )) Q the assertion on the rank of the Neron-Severi group follows. Assume that φ 0, but π 0 φ = 0. Then φ(e) J := π (J(E n )). Pulling back this (C ) N -bundle J over J(E n ) to E along π φ, we obtain a (C ) N -bundle Ẽ over E. The map φ : E J gives a section of Ẽ E, thus Ẽ is a trivial (C ) N -bundle. We show that this gives the desired contradiction. (C ) N J J(E n ) 0 π φ (C ) N Ẽ E 0 For any distinct P, Q E n which are in E n E n, the (C ) N -bundle J over J(E n ) has a quotient C -bundle J PQ whose extension class is P Q Pic 0 (E n ) = Ext (J(E n ),C ). By induction, there is a unique map in Hom(J(E n ), E) which must thus be induced by x x n x +...+x n (n )p, E n E for some p E. Taking P = y +...+y n and Q = y y n + y n, the pull-back of P Q to E = Pic 0 (E) is y y n. Choosing the degeneration suitably we may assume that y y n is not a torsion point on E and thus J PQ has a nontrivial pull-back to E, contradicting the fact that Ẽ, the pull-back of J to E, is trivial. 3.3 We define a curve n in the surface S n = G n C: n := {(x + x x n, p) S n : p {x,...,x n }} 9

10 and for n + d, another curve H n in S n : H n = {(x + x x n, p) S n : x + x x n + p < g d }. Finally we define the map: Φ n l,k : S n J(C), (3.3.) (x + x x n, p) l(x + x x n ) + kp D nl+k, where D nl+k is some divisor of degree nl + k on C. Finally we denote by u n : C (n) J(C), D D n, with D n some divisor of degree n, the Abel-Jacobi map on C (n). We simply write C for u C. 3.4 Proposition. Let C be a generic d: cover of P of genus g. Then:. NS(S n ) Q = C, G n, n. 2. In NS(S n ) we have: H n = ( d n )C + dg n n. (3.4.) 3. The image of G n in J(C) is a combination of C, 2 C,..., n C: u n G n alg ( d n )C 2 (d n 2)2 C ( )n n ( d 0)n C. (3.4.2) Proof. The first part follows from the previous proposition and from the fact that in the proof of 3.4. we will see that H n can be uniquely expressed as a combination of C, G n and n. The proof of 2 and 3 is by induction. In the case n =, so G = C, is trivial and for 3.4. we have S = C C and we must show H = d(c + G ). Note the following intersection numbers: H C = H G = d, H = 2(g + d ) (use that the gd exhibits C as d: cover of P with 2(g + d ) simple ramification points), which imply the result. Suppose that 3.4. and are true for all k < n. Note that u n (G n ) = Φ, n (H n ) and that map Φ, n restricted to H n is n: so, using 3.4. for n, we have in J(C): G n alg n ( ( d n )C + dg n (Φ n, ) n ). (3.4.3) 0

11 where we write G k for u k G k. Next we observe that (Φ k,l ) k = (Φ k,l+ ) H k and (Φ k,l+ ) C = (l + ) C. (3.4.4) Using 3.4. in the cases n 2, n 3,..., we get: (Φ n, ) n alg (Φ n 2,2 ) H n 2 alg ( d n 2)2 C + dg n 2 (Φ,3 n 3 ) H n 3 alg ( d n 2)2 C ( d n 3)3 C + d(g n 2 G n 3 ) + (Φ,4 n 4 )(3.4.5) H n 4 alg ( d n 2)2 C ( d n 3)3 C ( ) n 2 ( d 0)n C + +d(g n 2 G n ( ) n 2 G 2 + ( ) n G ) (note that (Φ,n ) H alg ( d )(n ) C + dg ( d 0 )n C). Substitute the expression for (Φ n, ) n from into 3.4.3: G n alg n ( ( d n )C ( d n 2)2 C... + ( ) n 2 ( d )(n ) C + ( ) n ( d 0)n C + d(g n G n ( ) n 3 G 2 + ( ) n 2 G ). (3.4.6) Using the formula for G k, k =, 2,..., n and the relation lk=0 ( ) k ( d l k ) = (d l ), (cf. 4.2.) we get: alg G n G n ( ) n 2 G (3.4.7) ( d n 3)C 2 (d n 4)2 C ( ) n 3 n (d 0 )(n ) C. Substituting this in formula and using the identity: n [(d n k ) + d k (d n k+ )] = k (d n k ) we obtain To obtain 3.4., note that by (), there are a, b, c Q such that: H n = ac + bg n + c n. (3.4.8) To find them we compute the homology classes of the curves in in H 2 (J(C),Q) and the intersection numbers of H n with C and G n. For the generic curve C, the homology class [B] of a curve B in J(C) is a multiple of the class [C] of C and this multiple is Θ B. We apply this to g B = G n. Using the map u : C (n) J(C), we have u (G n ) Θ = u (G n θ) with θ the pull-back of Θ to C (n).

12 The homology class of G n in H 2 (C (n),q) is: n ( d g k k=0 n ( d g k ) k=0 G n = ) xk θ n k (n k)!, so G x k θ n k n θ = (n k)!, (3.4.9) cf. formula (3.2) on pag.342 of [ACGH] (here x is the class of the divisor C (n ) in C (n) ). From pag.343 of [ACGH] one has: u (x n i θ i ) = ( g i)i![θ g ]/g! and since [Θ g ]/g! is the positive generator of H 2g (J(C),Z), we find: Therefore: G n Θ = n ( d g k k=0 n = g )( g n k ) (n k)! (n k)! ( d g k )( g n k ) k=0 = g( d 2 n ). (3.4.0) [G k ] = ( d 2 k )[C]. (3.4.) ( Next ) we compute the homology class of n. We use for k = n, l = : Φ n, n = ( ) Φ n,2 n. By induction, similar to and using (with n replaced by n) we have: ( ) Φ n, n alg d ( ) G n G n ( ) n 3 G 2 + ( ) n 2 G (3.4.2) + +( d n )2 C ( d n 2)3 C ( ) n ( d 0)(n + ) C alg n d ( ) k k (d k= Taking the homology classes in J(C) we get: Using 4.2.3: n+ k=2 we find: n+ n k )k C + ( ) k ( d n+ k)k C. ( ) [Φ n n,( n )] = d ( ) k k( d n+ n k ) + ( ) k k 2 ( d n+ k ) [C], k= n+ ( ) k ( d n+ k )k2 = ( d n ) + ( ) k ( d n+ k )k2 = ( d n ) + (d 3 n ) (d 3 n ), and because of ( d 3 k=0 k=2 k=2 [Φ n, ( n)] = ( d( d 3 n 2 ) + (d n ) + (d 3 n ) (d 3 n ) ) [C] n ) ( d 3 n ) = d( d 3 n ) (n + )( d 2 n ), ( d 2 n ) = ( d 3 n ) + ( d 3 n 2) 2

13 we finally obtain: Applying ( Φ n, equation for a, b, c: [Φ n,( n )] = ( d( d 2 n 2) (n + )( d 2 n ) + ( d n) ) [C]. (3.4.3) ) to and taking homology classes we get the following (n + )( d 2 n ) = a + b( d 2 n ) + c ( d( d 2 n ) (n + )( d 2 n ) + ( d n) ). On the other hand, we have the following intersections numbers in S n : and C C = G n G n = 0, C G n =, C n = n, G n n = ( d n ) H n C = d n, H n G n = ( d n ). Therefore, by intersecting with C and G n respectively, we find two more equations for a, b, c: and 3.4. now follows. d n = b + cn, ( d n ) = a + c( d n ), 3.5 With these results it is easy to find many relations between the cycles n C. It is a little surprising that the ones we find are equivalent to π i C alg 0 for i < 2g d and π 2g C alg Proposition. Let C be a d: covering of P. Then: π 2g i C alg 0 = 2 i d. The cycles π 2g 2 C,..., π 2g d C span Z C / alg and thus dim Q (Z C / alg ) d. For any set {n,..., n d } of non-zero distinct integers the cycles n C,..., n d C also span Z C / alg. Proof. First of all we show that for all n Z we have: P n C := (n+) C ( d d )n C +...( ) d ( d 0 )(n d+) C +F d alg 0 (3.6.) with P n = Γ n+ ( d d )Γ n+... CH g (A A) and with a cycle F d depending on d but not on n: F d = d[ C + ( d d 3 )C 2 (d d 4 )2 C +...( ) d d 2 (d 0 )(d 2) C ]. 3

14 This relation follows from the easily verified identity (for all n Z): Indeed, the l.h.s is by 3.4.: ( Φ,n ) H alg ( Φ,n ) ( ) Φ H,n = ( ) Φ d 2,n H d 2. (dc + dg ) alg dn C + dc (n + ) C, while for the r.h.s. we use (Φ k,l ) k = (Φ k,l ) H k ): ( ) Φ d 2,n H d 2 (3.6.2) alg ( d d 2)(n ) C + dg d 2 ( ) Φ d 3,n 2 H d 3 alg ( d d 2 )(n ) C ( d d 3 )(n 2) C ( ) d 2 (n d + ) C + +d(g d 2 G d ( )d 3 G ) (cf. also the proof of 3.4.2). From this 3.6. follows by using A more convenient set of relations is obtained from 3.6. as follows: (P n P n )C (3.6.3) = (n + ) C (d + )n C + ( d+ d )(n ) C ( ) d+ (n d) C = ( Γ m+d+ ( d+ )Γ m+d + ( d+ 2 )Γ m+d... + ( ) d+ Γ m ) C (3.6.4) alg 0, where we substituted: n := m + d. Relation can be rewritten as: Γ m (Γ Γ 0 ) (d+) C alg 0 m Z. (3.6.5) Next we look at the expansion of the π i s in Γ Γ 0 : π i = (2g i)! (log Γ ) (2g i) = = (Γ Γ 0 ) (2g i) ( a 2g i (Γ Γ 0 ) a 2g (Γ Γ 0 ) i). Therefore π i C alg 0 if 2g i d + so π 2g i C alg 0 if i d +. Since π 2g = Γ 0 we also have π 2g C = 0, and from [B2], prop.3 we know C () = 0 thus: C alg C (2) + C (3) C (d) with C (i) := π 2g i C 4

15 and we conclude that dim Q (Z C / alg ) d. We can in fact obtain C () alg 0 from 3.6., because a somewhat tedious computation shows that, for some c Q: Γ ( P d 0 P d ) = cγ 0 + log Γ mod (Γ Γ 0 ) (d+) CH g (A A), here Γ Γ n = Γ n is the product as correspondences, the equality is in (a quotient of) the ring CH g (A A) with -product. Since Γ 0 acts trivially on one cycles and P n C alg 0 the statement follows. In fact one shows that both sides are equal to, for some c Q: d ( ) n n (d n)γ n + c Γ 0. n= (Comparing this expression with the one for u d G d in we see that π 2g C alg u d G d alg 0 since G d = P maps to a point in J(C), note that some care must be taken as we defined G n only for n < d). Finally we observe that since n C (i) = n i C (i), the determinant of the matrix expressing the n i C in the C (i) is a Vandermonde determinant, which is nonzero. 3.7 Proposition.. For a curve C with π i C alg 0 for i 2g 4 (for example any curve with a g 3) we have, with C (i) := π 2g i C: C (2) alg 2 C ) C (3) alg 2 C ) Furthermore: n C alg n 3 +n 2 2 C n3 n 2 2 C. 2. For a curve C with π i C alg 0 for i 2g 5 (for example any curve with a g4 ) we have: C (2) alg (2 2 C 2C 4C ) C (3) alg (C 2 C ) C (4) alg (2 2 C 6C + 2C ). Proof. We give the proof of the second statement, the first being similar but easier. By assumption we have (C (i) := π 2g i C): Therefore: C alg C (2) + C (3) + C (4) with n C (i) = n i C (i). C alg C (2) + C (3) + C (4) C alg C (2) C (3) + C (4) 2 C alg 4C (2) + 8C (3) + 6C (4). The result follows by straightforward linear algebra. 5

16 3.8 Note that if one specializes a curve C with a g4 to a trigonal curve, then actually C (4) = (2 2 C 6C + 2C ) alg 0. However, we could not decide whether for a generic 4: cover of P we have C (4) alg 0. 4 Appendix 4. We recall some facts on binomial coefficients. The binomial coefficients ( n k) are defined (also for negative n Z!) as (cf. [ACGH], VIII.3) ( n k) := n(n ) (n k + ) k(k ) (k > 0), ( n 0) :=, ( n k) := 0 (k < 0). With this definition, they are the coefficients in the expansion of ( + x) n : ( + x) n = + ( n )x (n k )xk +... = ( n k )xk. k=0 Comparing the coefficients of x k in ( + x) n ( + x) m = ( + x) n+m one finds, for all n, m Z: ( n i )(m k i ) = (n+m k ), (4..) i 4.2 Lemma. We have: ( ) i ( m l i ) = (m l ), (4.2.) ( ) i i( m l i ) = (m 2 l ), (4.2.2) ( ) i i 2 ( m l i ) = (m 3 l 2 ) (m 3 l ). (4.2.3) Proof. Using ( k ) = ( )k and 4.., the first line can be written as: ( ) i ( m l i ) = ( i )( m l i ) = (m l ). The second line follows in the same way, using ( 2 k ) = ( )k k: ( ) i i( m l i ) = ( 2 i )(m l i ) = (m 2 l ). 6

17 For the last line we use ( 3 k ) = ( )k (k + )(k + 2)/2, so: Therefore: ( ) k k 2 = 2( 3 k ) 3( 2 k ) + ( k ). ( ) i i 2 ( m l i) = 2 ( 3 i )( m l i) 3 ( 2 i )( m l i) + ( i )( m l i) = 2( m 3 l ) 3( m 2 l ) + ( m l ). Now use that: ( m l ) = ( m 2 l ) + (m 2 l ) = ( m 3 l 2 ) + 2(m 3 l ) + (m 3 l ), and ( m 2 l ) = ( m 3 l ) + (m 3 l ). References [ACGH] E.Arbarello, M.Cornalba, P.Griffiths, J.Harris, Geometry of algebraic curves I, Grundlehren 267, Springer Verlag, (985). [B] [B2] [C] [DM] [GS] [K] [N] A.Beauville, Quelques remarques sur la transformation de Fourier dans l anneau de Chow d une variété abélienne, Algebraic Geometry (Tokyo/Kyoto 982), Lect. Notes Math. 06, , Springer 983. A.Beauville, Sur l anneau de Chow d une variété abélienne, Math. Ann., 273 (986) pp G.Ceresa, C is not Algebraically Equivalent to C in its Jacobian, Annals of Math., 7 (983), pp C.Deninger and J.Murre, Motivic decomposition of abelian schemes and the Fourier transform, Preprint Münster-Leiden (99). B.H.Gross and C.Schoen, The canonical cycle on the triple product of a pointed curve, to appear. K.Künnemann, On the Motive of an Abelian Scheme, preprint Münster (99). M.V.Nori, Algebraic cycles and Hodge Theoretic Connectivity, preprint University of Chicago, (99). 7

18 Elisabetta Colombo permanent adress: Università di Milano Dipartimento di Matematica Via Saldini Milano Italia until june 992: Department of Mathematics UvA Plantage Muidergracht TV Amsterdam The Netherlands Bert van Geemen Department of Mathematics RUU P.O.Box TA Utrecht The Netherlands 8