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New cycles in the Griffiths group of the generic abelian threefold.

Introduction. In this paper we give a method to produce infinitely many new 1-cycles homologically but not algebraically equivalent to 0 on the generic abelian threefold. The method is based on the construction of the paper "Some density results for curves with non-simple Jacobians" (cf. [CP]). In [CP] we showed in particular that, for any g, there are infinitely many curves of genus g mapping to a generic principally polarized abelian threefold A. The image of each such curve generates A as a group. Furthermore these curves are rigid (up to translation) and different in moduli.

We can use these curves to construct an infinite countable number of 1-cycles on the threefold A which are homologically but not algebraically equivalent to 0, i.e. elements of G??(A), the Griffiths group of A:

[MATHEMATICAL EXPRESSION OMITTED] The Griffiths group G??(A) contains the subgroup M generated by the 1-cycles [C] - [[C.sup.-]] for C [subset] A any curve of genus 3 with jacobian isogenous to A. Ceresa (cf. [Ce]) proved that M and hence G??(A) is different from 0, showing that the cycle [C] - [[C.sup.-]] with J(C) = A is not algebraically equivalent to 0 for A generic and Nori and Bardelli (cf. [N], [B]) proved that M is infinitely generated.

In this paper we are able to define infinitely many more 1-cycles. Moreover we show that at least one of the cycles that we construct is not contained in M, so in particular G??(A) [NOT EQUAL TO] M.

Nori in [N] shows that on G??(A) there is an action of a group G obtained as an extension of the symplectic group [S.sub.P6Q] with a Galois group (cf. [N] (A), pag.194) and that M is generated by the G orbit of [C] - [[C.sub.-]] with J(C) = A. At the end of his paper Nori recalls the question raised by Clemens if G??(A) is infinitely generated as G-module. It follows from our result that the Griffiths group, seen as a G-module, has rank bigger than 1.

To explain which new cycles we obtain, we just give an outline of our construction. In [CP] we showed that the subset [S.sub.3] of [M.sub.g] of the curves C which have a map on an abelian threefold A such that the image of the curve generates the threefold as a group locally can be described as countable number of subvarieties Y. Moreover one can define a map [G.sup.0] : Y [right arrow] [A.sub.3] sending the curve C to a principally polarized abelian threefold isogenous to A. We proved that for inifinitely many of these Y the map [G.sup.0] is generically finite (cf. Sect. 1 for a summary of these results).

In this paper, for g = 4, we give a global description of the subset [D.sub.Y] of Y in which the rank of the differential of [G.sup.0] drops and we show that for infinitely many Y the generic point p of [D.sub.Y] is a ramification point (cf. Sect. 2).

We consider then a threefold A corresponding to a generic point of the image of [G.sup.0] and on the fiber [G.sup.0-1]([A]) we take 2 points specializing to p. We show that the difference of the 2 curves of genus 4 corresponding to these points of the fiber is a 1-cycle [Z.sub.Y] on A that is homologically but not algebraically equivalent to 0. This result is proven by extending this cycle to a family of cycles and showing that the differential of the associated normal function is different from 0 (cf. Sect. 3).

Moreover if the map [G.sup.0] : Y [right arrow] [A.sub.3] has some particular fiber F of positive dimension, as in the case of the locus containing the cyclic coverings of the elliptic curves, our cycle [Z.sub.Y] is not contained in M. The idea to prove this is, roughly speaking, to use the fact that the pullback of the Ceresa cycles is constant along F while [Z.sub.Y] is not (cf. Sect. 4).

1. Background material. We recall some results from [CP], considering just the cases we are interested in now. We make the further hypothesis that the families have N-level structure (this hypothesis doesn't change anyhow the construction). Let C [right arrow] [M.sub.g](N) be the universal family of curves of genus g over the moduli space [M.sub.g](N) of smooth curves of genus g with N-level structure and let J(C) [right arrow] [M.sub.g](N) be the associated jacobian fibration. Let < > be the polarization given by the intersection form on [H.sub.1]([C.sub.t],Z) and let [F.sup.1] be the Hodge bundle associated to the universal family.

For any [t.sub.0] [element of] [M.sub.g](N), let V be a contractible open neighbourhood of [t.sub.0]. Over V we can trivialize the 2k-real grassmanian bundle: [[sigma].sub.V](2k,[R.sup.1][[psi].sub.*]R) [congruent] V X [G.sub.R](2k,H??). We considered the following diagram (see [CP] (0.4)):

[MATHEMATICAL EXPRESSION OMITTED] where [f.sub.k] is the bundle projection and [[phi].sub.k] associates to any k-dimensional complex subspace of [H.sup.1,0]([A.sub.t]) its image as 2k-dim real subspace via the canonical isomorphism between [H.sup.1,0]([A.sub.t]) and [H.sup.1]([A.sub.t],R) as 2g-dim real vector spaces.

In [CP] we computed the differential of [[phi].sub.k], here we will give a simplified version following a suggestion of M. Cornalba. Let H : V [right arrow] [H.sub.g] be the period map. The diagram factorizes in the following way (cf. [CP], (1.2)):

[MATHEMATICAL EXPRESSION OMITTED] Denote by Q the antisymmetric bilinear form on H?? induced by the polarization < >. The Siegel space [H.sub.g] can be seen as the set of the vector subspaces F [element of] [G.sub.C](g, H??) satisfing the Riemann bilinear relations: (a) Q(F, F) = 0 (b) Q(F, [F, bar above]) > 0 (1.2). For every F [element of] [H.sub.g] the R-linear canonical isomorphism used to define [[psi].sub.k] is clearly: PF : F [right arrow] H??, [nu] [bar right arrow] ([nu] + [nu], bar above])/2 Let [W.sub.R] [element of] [G.sub.R](2k,2g) be a real 2k-plane in H??. Its complexification is [W.sub.C] := [W.sub.R] [cross product] C a complex 2k-plane in H??, so 2k = dim ([W.sub.C]) [greater than or aqual] dim (F [intersection] [W.sub.C] + [F, bar above] [intersection] [W.sub.C]) = 2 dim (F [intersection] [W.sub.C]) then dim F [intersection] [W.sub.C] [less than or equal to] k for every F [element of] [H.sub.g].

Remark (1.A). p?? ([W.sub.R]) is a complex subspace if and only if dim ([W.sub.C][intersection]F) = k.

In fact p?? ([W.sub.R]) is complex if and only if, for any [nu] [element of] p?? ([W.sub.R]), [nu] and [[nu], bar above] are in [W.sub.C]. So p?? ([W.sub.R]) is complex if and only if [W.sub.C] [intersection] F = p?? ([W.sub.R]). Since in every case [W.sub.C] [intersection] F [subset] p?? ([W.sub.R]), the remark follows for reasons of dimension.

Clearly, for W [element of] G?? (k, [F.sup.1]) such that [[psi].sub.k](W) = [W.sub.R]

[MATHEMATICAL EXPRESSION OMITTED] On the other hand, since the restriction of [[psi].sub.k] to each fiber of f?? is an injection, the projection of f?? restricted to the fibers of [[psi].sub.k] defines an isomorphism on the image. Therefore, if f?? (W) = F,

[MATHEMATICAL EXPRESSION OMITTED] (Note that W = [W.sub.C] [intersection] F and in fact dim ([W.sub.C] [intersection] F) = k).

Hence, to study the kernel of d[[psi].sub.k] for all the points of the fiber [psi]?? ([W.sub.R]) we can study

[MATHEMATICAL EXPRESSION OMITTED] We want to give a description of Y' as analytic subvariety of [H.sub.g]. So as tangent space in any point we will consider the holomorphic tangent space. In particular the isomorphisms (1.3), (1.4) are seen as identifications of the real tangent space with the holomorphic one. By Rem. (1.A), we have that

[MATHEMATICAL EXPRESSION OMITTED] and that Y' can be seen as the intersection, in [G.sub.C](k,H??), of [H.sub.g], which is an analytic subvariety, and of the Schubert cycle:

[MATHEMATICAL EXPRESSION OMITTED] Therefore Y' is a closed analytic subvariety of [H.sub.g]. Moreover observe that [H.sub.g] doesn't intersect the singular locus of [sigma].

Let now F be a point of Y'. As we have seen, dim (F [intersection] [W.sub.C]) = k i.e. p?? ([W.sub.R]) = F [intersection] [W.sub.C] i.e. [W.sub.C] = F [intersection] [W.sub.C] [direct sum] [F, bar above] [intersection] [W.sub.C].

Let W?? be the orthogonal of [W.sub.C] w.r.t. Q. Since Q is non degenerate on [W.sub.C], it follows that

[MATHEMATICAL EXPRESSION OMITTED] Moreover Q is non degenerate on W?? and W?? is defined over R, in fact =

[MATHEMATICAL EXPRESSION OMITTED]

If [nu] [element of] F, PF([nu]) [element of] W?? and u [element of] [W.sub.C]. Write u = [u.sub.1] + [[u, bar above].sub.2], where [u.sub.1], [u.sub.2] [element of] F [intersection] [W.sub.C]. Then:

[MATHEMATICAL EXPRESSION OMITTED] because Q(F, F) = Q([F, bar above], [F, bar above]) = 0 and [[u, bar above].sub.2] [element of] [W.sub.C].

It follows that P?? (W??) = W?? [intersection] F and that dim

[MATHEMATICAL EXPRESSION OMITTED]

So the map:

[MATHEMATICAL EXPRESSION OMITTED] where [F.sub.1] and [F.sub.2] are k-dimensional and (g -- k)-dimensional subspaces of [W.sub.C] and W?? satisfing the relations (1.2a,b), is a holomorphic isomorphism on Y'.

We use the differential of this map to describe the tangent space to Y' in F. First of all, we recall that, for V [epsilon] [H.sub.n]:

[T.sub.v][H.sub.n] [approximate] [S.sup.2]V*, [eta] (1.7) so we have:

[T.sub.F]Y' = Im([S.sup.2](F [intersection] [W.sub.C])* [symmetry] [S.sup.2] (F [intersection] W??)* [right arrow] [S.sup.2]F*). (1.8) Equivalently, denoting by [pi]F the projection:

[pi]F : [S.sup.2]F [right arrow] (F [intersection] [W.sub.C]) [symmetry] (F [intersection] W??) we obtain:

[T.sub.F]Y' = ker [[pi].sub.F]. (1.9)

In particular Y' is smooth and

[MATHEMATICAL EXPRESSION OMITTED]

To conclude, we have proven:

PROPOSITION 1.1. The projection of any fiber of the map [[psi].sub.k] on [H.sub.g] is an isomorphism on the image. The image is a smooth analytic variety of [H.sub.g] of dimension k(k + 1)/2 + (g - k)(g - k + 1)/2. In particular d[[psi].sub.k] has always maximal rank. The formula (1.8) gives an explicit description of the tangent space to the fibers in every point.

Consider now the map [[phi].sub.k]. Again the restriction of [[phi].sub.k] to any fiber of [[functions].sup.k] is an injection. Hence the projection of [[functions].sup.k] restricted to the fiber of [[phi].sub.k] defines an isomorphism on the image, and for W [element of] [[sigma].sub.V](k, [F.sup.1]), [[functions].sub.k](W) = t, [[phi].sub.k](W) = [W.sub.R] we have:

[MATHEMATICAL EXPRESSION OMITTED] Set

[MATHEMATICAL EXPRESSION OMITTED] denoting again [[functions].sub.k]([psi]??([W.sub.R])) by Y' we get:

Y = [H.sup.-1](Y')

Therefore, also Y can be seen as an analytic variety and the real tangent space in every point can be identified with the holomorphic one. In particular dH([T.sub.t]Y) [subset] [T.sub.H(t)]Y', so, since for (1.9) [T.sub.H(t)]Y' = ker [[pi].sub.H(t)]:

[T.sub.t]Y = ker [[pi].sub.H(t)] [omicron] [d.sub.t]H To summarize:

PROPOSITION 1.2. 1) The projection of any fiber of the map [[phi].sub.k] on V is an isomorphism on the image and the image is an analytic subvariety of V. 2) For any W [element of] [[sigma].sub.V](k, [F.sup.1]), with [[functions].sub.k](W) = t

ker([d.sub.w][[phi].sub.k]) [approximate] ker([[pi].sub.H(t)] [omicron] [d.sub.t]H).

Clearly H(t) = [H.sup.1,0]([A.sub.t]) and its subspace W is W = [W.sub.C] [intersection] [H.sup.1,0]([A.sub.t]). Denote by [W.sup.[perpendicular to]] the subspace W?? [intersection] [H.sup.1,0]([A.sub.t]). Note that [W.sup.[perpendicular to]] is the orthogonal of W in [H.sup.1,0]([A.sub.t]) w.r.t. to the hermitian positive definite form Q(??). Then Prop. (1.2) says that the following sequence is exact:

W [cross product] [W.sup.[perpendicular to] [right arrow] TV* [right arrow] TY*

From Prop. (1.2) it also follows that the condition "d[[phi].sub.k] has maximal rank" can be translated the condition "p [omicron] dH has maximal randk".

If dim TV [greater than or equal] k(g - k) this is equivalent to ask that p [omicron] dH is surjective or, dualizing, that the map

qw : W [cross product] [W.sup.[perpendicular to]] ?? [S.sup.2] (W [symmetry] [W.sup.[perpendicular to]]) ?? TV* is injective. Note that this map qw is the restriction of the symmetric bilinear form:

q : [H.sup.1,0]([A.sub.t]) x [H.sup.1,0]([A.sub.t]) [right arrow] TV* which represents the infinitesimal variation of the polarized Hodge structure [F.sup.1].

We say that W [elements of] [[sigma].sub.v](k, [F.sup.1]) satisfies condition ([e.sup.k]) if the map:

([e.sub.k]) W [cross product] [W.sup.[perpendicular to] [right arrow] [S.sup.2][H.sup.1,0] [right arrow] [T.sub.t]*V = [T.sub.t]*[M.sub.g](N) [congruent] [H.sup.0](2K) is injective. In Section 1 of [CP] we showed that d[[phi].sub.k] has maximal rank at W if and only if W satisfies condition ([e.sub.k]).

For k = 1 condition ([e.sub.1]) is always fullfilled, for k = 2,3 we proved that for every t [element of] V there is a W [element of] [[functions].sup.-1](t) satisfying ([e.sub.k]) (cf. Th. 3 [CP]), so for k [less than or equal to] 3 a dimension count shows that [[phi].sub.k] is surjective.

We also considered condition ([p.sub.k]): for W [element of] [[sigma].sub.v](k, [F.sup.1]) the sequence:

([p.sub.k]) 0 [right arrow] [[conjunction].sup.2]W [right arrow] W [cross product] [H.sup.1,0] [right arrow] [T.sub.t]*V [approximate] [H.sup.0](2K) is exact. This condition is in general stronger than condition ([e.sub.k]).

Remark (l.B). Fix a 2k-plane [W.sub.R] [element of] [G.sub.R](2k, 2g) and denote by Y the image by [[functions].sub.k] of the fiber of [[phi].sub.k] on [W.sub.R] i.e. Y := [[functions].sup.k]([phi]??([W.sub.R])).

Take (t,W) [element of] [phi]??([W.sub.R]). The sequence (1.16) can be fitted in the following diagram:

[MATHEMATICAL EXPRESSION OMITTED]

If W satisfies ([p.sub.k]), then the restriction of dH* : [S.sup.2] ([H.sup.1,0])( = [S.sup.2](W [symmetry] [W.sup.[perpendicular to])) [right arrow] T*:

[S.sup.2]W [symmetry] (W [cross product] [W.sup.[perpendicular to]) [right arrow] T* is injective and, with a simple diagram-chase it can be checked that the composition:

[MATHEMATICAL EXPRESSION OMITTED] is injective.

Considering [H.sub.g] as a subvariety of [G.sub.c](g,H??) the map H is:

[MATHEMATICAL EXPRESSION OMITTED]

If t [element of] [M.sub.g](N) corresponds to a nonhyperelliptic curve the differential of H is injective since its dual, that is the cup-product

[MATHEMATICAL EXPRESSION OMITTED] is surjective.

Fix [W.sub.R] [element of] [G.sub.R] and take Y := [[functions].sup.k] (??(W.sub.R])), W := [W.sub.C] [intersection] [H.sup.0] ([K.sub.C]) and [W.sup.[perpendicular to]] := W?? [intersection] [H.sup.0]([K.sub.C][C.sub.t]

So, by (1.8) and (1.11) we have:

[MATHEMATICAL EXPRESSION OMITTED]

Consider [H.sub.3] as the set of the complex 3-planes of [W.sub.C] satisfying the Riemann bilinear conditions w.r.t. the restriction of Q. Then we have the map

G : Y [right arrow] [H.sub.3], t [bar, right arrow] [W.sub.t] := [H.sup.0] ([K.sub.C][C.sub.t]) [intersection] [W.sub.C].

The dual of the differential of G, that we indicate with L

[MATHEMATICAL EXPRESSION OMITTED] is the composition of the inclusion [S.sup.2][W.sub.t] [hoor right arrow] [S.sup.2][H.sup.0](K??) with the cup-product:

[S.sup.2][H.sup.0](??) [right arrow] [H.sup.0](K??).

By Rem. (1.B), if [W.sub.t] satisfies the condition ([p.sub.k]), then this dual is injective.

Assume now that [W.sub.R] = [pi] [element of] [G.sub.Q](2k, 2g) [subset] [G.sub.R](2k, 2g). So [phi]?? ([pi]) is given by the couples (t, [W.sub.t]) such that [B.sub.t] := [W??/([pi][intersection][H.sup.1]([C.sub.t], Z))* is an abelian subvariety of J([C.sub.t]). We can restrict the universal families of curves and jacobians of [M.sub.g](N) to Y. On the other hand, also the k-dimensional abelian varieties [B.sub.t] [subset] [A.sub.t] := [[psi].sup.-1](t) give a family B on Y:

[[psi].sub.[pi]] : B [right arrow] Y and [B.sub.t] = [psi]??(t).

For a point [t.sub.0] [element of] Y, let [H.sub.t0] [subset] [B.sub.t0] be a finite subgroup such that [B.sub.t0]/[H.sub.t0] is a PPAV. Let [Y.sup.0] [subset] Y be a neighbourhood of [t.sub.0] on which the isogeny [B.sub.t0] [right arrow] [B.sub.t0]/[H.sub.t0] can be extended to an isogeny:

[MATHEMATICAL EXPRESSION OMITTED]

Note that we can put [Y.sup.0] = Y because V is contractible. So on Y we have a family [[psi].sub.[rho]] : ??/H [right arrow] Y of PPA k-folds. We can define the map:

[MATHEMATICAL EXPRESSION OMITTED] where [A.sub.k] is a moduli space of PPA k-folds. Then [G.sup.0] can be lifted to maps

[MATHEMATICAL EXPRESSION OMITTED] Clearly L can be seen also as the dual map of dG?? and dG?? in every point.

2. The case g = 4, k = 3: further results. Keeping all the notations of Sect. 1, we restrict ourselves now to the study of the map G?? (cf. (1.21)) in the case of jacobian fibrations of families of curves of genus 4 parametrized by a open set U of [M.sub.4](N).

Remark (2.A). For g=4 condition ([e.sub.3]) is always satisfied because the condition is symmetric w.r.t. interchanging W with [W.sup.[perpendicular to]], and if W is a 3-dimensional subspace of [H.sup.0](C, [K.sub.C]) then [W.sup.[perpendicular to]] has dimension 1 and ([e.sub.1]) is always fullfilled.

In particular the differential of [[phi].sub.3] is maximal in any point so [[phi].sub.3] is a smooth [C.sup.[infinity]] map and every fiber is smooth. Since the restriction of [[functions].sub.3] to the fibers of [[phi].sub.3] is an isomorphism, also all the Y := [[functions].sub.3]([phi]??([pi])) [for all][pi] [element of] [G.sub.R] (6, 8) are smooth.

Furthermore, for [pi] [element of] [G.sub.Q](6, 8), Y parametrizes a family of 4 dimensional jacobians of curves containing an abelian subvariety of dimension 3 and both the families vary holomorphically in Y.

We want to characterize the points of Y in which the differential of G?? is not maximal.

For a non-hyperelliptic genus 4 curve C there is the following exact sequence (since U is an open subset of [M.sub.4] (N) we can restrict ourselves to non-hyperelliptic curves):

[MATHEMATICAL EXPRESSION OMITTED] where Q corresponds to the quadric containing the canonical curve.

Remark (2.B). For [pi] [element of] [G.sub.Q](6, 8), [C] [element of] Y := [[functions].sub.3]([phi]??([pi])) and W = [functions]?? ([C]) [intersection] [phi]??([pi]) the linear map [d.sub.[C]]G?? : [T.sub.[C]]Y [right arrow] T[H.sub.3] is not an isomorphism if and only if ([p.sub.3]) doesn't hold i.e., by (2.1), if and only if the following sequence is exact:

[MATHEMATICAL EXPRESSION OMITTED] In this case say that W [element of] ??v(k, [F.sup.1]) satisfies (2.2).

Remark (2.C). For g=4 families of 4 dimensional jacobians with a 3 dimensional abelian subvariety can be seen as well as families of coverings of elliptic curves.

We consider now two examples of covering of degree d = 2,3.

Example 2.1: The bielliptic curve of genus 4. Let p : C [right arrow] E a 2 to 1 covering of the elliptic curve E, with g(C) = 4. Let [W.sub.E] be the 1-dimensional subspace of [H.sup.0]([K.sub.C]) spanned by p*([[omega].sub.E]), where [[omega].sub.E] is the generator of [H.sup.0]([K.sub.E]). Let W be the orthogonal of [W.sub.E] w.r.t. the positive definite form <,> on [H.sup.0]([K.sub.C]). We want to show that:

PROPOSITION 2.1. W doesn't satisfy condition (2.2) (i.e. it satisfies ([p.sub.3])).

Proof. Let i : C [right arrow] C be the involution associated to the covering p.

Let i* : [H.sup.0]([K.sub.C]) [right arrow] [H.sup.0]([K.sub.C]) be the induced isomorphism. Note that [W.sub.E] is the invariant space for i* i.e.:

[MATHEMATICAL EXPRESSION OMITTED] On the other hand, since the endomorphism of J(C) induced by i is an automorphism of abelian varieties and in particular preserves the polarization, also W is an eigenspace for i*, so it has to be the eigenspace corresponding to --1:

[MATHEMATICAL EXPRESSION OMITTED] From the theory of double coverings it follows that W = p*[H.sub.0]([K.sub.E] [cross product] L) with L a line bundle on e such that [L.sup.2] = O(D), where D is the ramification divisor of the covering. So, in particular, since [H.sup.0]([K.sub.E] [cross product] L) has no base points on E, W has no base points on C. Moreover W defines an embedding of E because [h.sup.0](([K.sub.E] [cross product] L)(-p-q)) = 1 for every p, q [element of] E.

The eigenspaces of i* in [S.sup.2]([H.sup.0}([K.sub.C])) are:

[MATHEMATICAL EXPRESSION OMITTED]

Since i acts also on [H.sup.0](2[K.sub.C]) = [H.sup.1]([T.sub.C])* and the action is equivariant w.r.t. the map [S.sup.2][H.sup.0]([K.sub.C]) [right arrow] [H.sup.0](2[K.sub.C]) the kernel <Q> is stable for the action of i*.

[MATHEMATICAL EXPRESSION OMITTED] is the space of the first order deformations of C preserving the involution. Its dimension is given by the number of the total ramification points (6 for Riemann-Hurwitz formula), minus the dimension of the group of the automorphisms of the base curve (1 because E is elliptic) and plus the dimension of the moduli space of the base curve (dim [[micro].sub.1] = 1). So dim[H.sup.1] [([T.sub.C]).sup.i*=1] = 6. Then clearly <Q> [subset] [S.sup.2][H.sup.0] [([K.sub.C]).sup.i*=1]. So we have the following exact sequence:

[MATHEMATICAL EXPRESSION OMITTED] It remains to prove that <Q> ?? [S.sup.2]W. To do this consider the canonical embedding of the curve C:

[MATHEMATICAL EXPRESSION OMITTED] we identify P([H.sup.0]([K.sub.C])*) with [P.sup.3] choosing as a basis of global sections of [H.sup.0]([K.sub.C]) {[[omega].sub.E], [w.sub.1], [w.sub.2], [w.sub.3]} with [w.sub.i] [element of] W. We denote by ?? the hyperplane of [P.sup.3] defined by the equation [x.sub.0] = 0 and by O the point [x.sub.1] = [x.sub.2] = [x.sub.3] = 0. Let [[phi].sub.W] be the map associated to the linear system W, which, as we have already observed, has no base points. Since [[phi].sub.W](i(t)) = ([w.sub.1](i(t)) : [w.sub.2](i(t)) : [w.sub.3](i(t))) = ( - [w.sub.1](t) : -[w.sub.2](t) : -[w.sub.3](t)) = [[phi].sub.W](t) the map [[phi].sub.W] factors over the quotient C/i = E defining in this way a map that we have already seen to be an embedding: [[phi].sub.E] : E [right arrow] [P.sup.2] with [[phi].sub.E](E) = [[phi].sub.W](C).

On the other hand [[phi].sub.W] can be seen as the composite of [[phi].sub. [K.sub.C] ] with the projection from O. The curve C is embedded as a curve of genus 6 and the restriction to [[phi].sub. [K.sub.C] ](C) of the projection gives a 2 to 1 map. Then in particular [[phi].sub.W](C) has degree 3 and the map [[phi].sub.E] is an embedding of E in ??.

Let's have a look now at the quadrics in [P.sup.3]:

[MATHEMATICAL EXPRESSION OMITTED] and to their restriction to ??:

[MATHEMATICAL EXPRESSION OMITTED] clearly the kernel of p is given by [x.sub.0][H.sup.0] ([P.sup.3], [[omicron].sub.[P.sup.3]] (1)), while the image is the vector space <Q([x.sub.1], [x.sub.2], [x.sub.3]))> spanned by the quadrics not containing the variable [x.sub.0]. The decomposition of [S.sup.2]([H.sup.0]([K.sub.C])) in eigenspaces of i* can be seen, with this formalism, in the following way:

[MATHEMATICAL EXPRESSION OMITTED] So Q can be written as

[MATHEMATICAL EXPRESSION OMITTED] we want to show that a [not equal to] 0. Assume that a = 0. Then {Q = 0} is a cone with vertex O, and its projection is {Q = 0} [intersection] {[x.sub.0]}, a conic in ??. But, since [[phi].sub. [K.sub.C] ](C) [subset] {Q = 0}, we get

[MATHEMATICAL EXPRESSION OMITTED] and this is absurd because [[phi].sub.E] defines an embedding of E.

Example 2.2: The cyclic 3 to 1 covering of genus 4 of an elliptic curve. Let p : C [right arrow] E be a 3 to 1 cyclic covering of an elliptic curve E with 3 ramification points. By the Riemann-Hurwitz Formula g(C) = 4. Denote by W the orthogonal of

[MATHEMATICAL EXPRESSION OMITTED]

PROPOSITION 2.2. W satisfies condition (2.2).

Proof. Let [delta] be the automorphism of order 3 of the curve associated to the covering. We have that

[MATHEMATICAL EXPRESSION OMITTED] while on W [delta]* has eigenvalues [rho] and [[rho].sup.2], with [[rho].sup.3] = 1, [rho] [not equal to] 1. By the Holomorphic Fixed Point Formula we get that, possibly replacing [rho] with [[rho].sup.2], the eigenspace [W.sub.[rho]] [subset] W of [rho] has dimension 2 and the eigenspace [W.sub.[[rho].sup.2]] of [[rho].sup.2] has dimension 1.

We want to show that <Q> [subset] W [cross procult] [H.sup.0]([K.sub.C]). Arguing as in the other example we know that Q belongs to an eigenspace in [S.sup.2][H.sup.0]([K.sub.C]). We have just to show that it is not contained in

[MATHEMATICAL EXPRESSION OMITTED] This invariant subspace has dimension 3. The invariant subspace of [H.sup.0](2[K.sub.C]) has the same dimension as the invariant part of its dual [H.sup.1]([T.sub.C]). Since the invariant subspace of [H.sup.1]([T.sub.C]) is the space of the first order deformations of C which preserve the automorphism of order three, it has dimension 3 because p has 3 ramification points and dim [[micro].sub.1] = 1. It follows that Q, which spans the kernel of the map

[MATHEMATICAL EXPRESSION OMITTED] cannot be invariant.

COROLLARY 2.1. W is different from [H.sup.0]([K.sub.C](- q)) for any q [element of] C.

Proof. W is invariant for the action of the order 3 automorphism while [H.sup.0](C, [K.sub.C]( - q)) is never invariant but for a point q which is a branch point for the covering. On the other hand, if q is a branch point, p*[[omega].sub.E] [element of] [H.sup.0](C, [K.sub.C]( -q)), where [[omega].sub.E] is the generator of [H.sup.0](E, [K.sub.E]), while p*[[omega].sub.E] is never an element of W by construction.

We want to show that all the ?? : Y [right arrow] [[eta].sub.3] have generically finite fiber and that the subsets [D.sub.Y] [subset] Y in which ?? does not have maximal rank or is empty or it is a divisor and for a countable infinite number of such maps [D.sub.Y] [not equal to] [phi]. This fact will follow as an immediate corollary of Th. (2.1) whose statement involves the construction below.

We define a divisor ?? [element] [[sigma].sub.V](3, [F.sup.1]) which is a bundle of quadrics over V. The grassmannian bundle [[sigma].sub.V](3, [F.sup.1]) is canonically dual to the projective bundle ??, whose fiber over

[MATHEMATICAL EXPRESSION OMITTED] Denote by ??* the set of those points in ?? which, over any t [element of] V, are the points of the quadric which contains the canonical curve

[MATHEMATICAL EXPRESSION OMITTED] Then ??* is clearly a bundle of quadrics over V and we define ?? [element] [[sigma].sub.V](3, [F.sup.1]) to be its dual. By definition, W [element of] ?? if and only if W satisfies (2.2).

We denote by [[phi].sub.Q] the restriction of [[phi].sub.3] to Q:

[MATHEMATICAL EXPRESSION OMITTED] In particular the fibers of [[phi].sub.Q] are the intersections of the fibers of [[phi].sub.3] with Q.

Remark (2.D). For [pi] [element of] [G.sub.Q](6,8) and Y := [functions of 3]([phi]??([pi])) the subset [D.sub.[upsilon]] of [upsilon] where [dG.sup.0] doesn't have maximal rank is [functions of 3] ([phi]??([pi]) [intersection] Q).

LEMMA 2.1.

[MATHEMATICAL EXPRESSION OMITTED]

Proof. Note that all the fibers of [G.sup.0] : Y( := [functions of 3] ([phi]??([pi]))) [right arrow] [H.sub.3] have dimension [less than or equal to] 1. In fact the tangent to the fiber in a point is the kernel of d[G.sup.0] and we have seen that d[G.sup.0] is the dual of L : [S.sup.2]W [right arrow] T*Y, with dim ker L = dim ker

[MATHEMATICAL EXPRESSION OMITTED] and, by (2.2) this dimension can be at most 1 and it is 1 exactly when W [element of] Q.

Suppose that [phi]??([pi]) [subset] Q, thus Im[G.sup.0] is an open set of a divisor of [H.sub.3] and the fiber [G.sup.0-1]([A]) of a generic point [A] of Im[G.sup.0] has dimension 1. In particular A is a PPA threefold (with a fixed symplectic basis of [H.sub.1](A,Z)). Let S [subset] [M.sub.4](N) be a component of [G.sup.0-1]([A]) and let [phi] : S [right arrow] S be the family of genus 4 curves parametrized by S. Then each fiber of [phi] maps to A. Denote by [C.sub.s] the image in A of [[phi].sup.-1](s). Fix [s.sub.0] [element of] S and consider the primitive Abel-Jacobi associated to S:

[MATHEMATICAL EXPRESSION OMITTED]

In [P] the following has been showed (cf. Prop. (1.2) and Th. 1). Let F [right arrow] T be a non isotrivial family of curves of genus g, with generic fiber [F.sub.t], let B be an abelian variety of dimension n and let [psi] : F [right arrow] B be a morphism such that [psi]([F.sub.t]) generates B as a group. Then, if n [greater than or equal to] (g + 1)/2, for [t.sub.0] [element of] T, the map [ab.sub.primT]:

[MATHEMATICAL EXPRESSION OMITTED] is not identically zero.

From this result it follows that [ab.sub.primB] can't be zero. Therefore the image of B generates an abelian subvariety of [J.sup.2][(A).sub.prim] orthogonal to the 3.0 part of positive dimension. The contradiction follows from Lemma (2.2) below.

So [phi]??([pi]) [not subset] Q.

Remark (2.E). In particular from Lem. (2.1) it follows that for any Y the map [G.sup.0] : Y [right arrow] [H.sub.3] has a generic fiber of dimension 0.

LEMMA 2.2. Let [A] be a generic point of a divisor of [H.sub.3].

Then [J.sup.2][(A).sub.prim] has no subtori orthogonal to the 3.0 part.

Proof. Note that [J.sup.2][(A).sub.prim] has a subtorus orthogonal to the (3.0) part if and only if [H.sup.3][(A, Q).sub.prim] has a Hodge substructure. Furthermore A has to have [End.sub.Q](A) = Q. In fact there are abelian threefolds with [End.sub.Q](A) [not equal to] Q just in codimension at least 2 and exactly 2 if and only if A is isogenous to a product of an abelian surface and an elliptic curve.

We recall now the definition of the (special) Mumford-Tate group SMT(A) of an abelian variety A. The Hodge structure on [H.sup.1](A, Q) is defined by a complex structure on [H.sup.1](A, Q) [cross product] R = [H.sup.1](A, R), that is by a real linear map: J : [H.sup.1](A, R) [right arrow] [H.sup.1](A, R), with [J.sup.2] = -1. The eigenspaces of J in [H.sup.1](A, R) [cross product] C = [H.sup.1](A, C) are [H.sup.1,0] and [H.sup.0,1] corresponding to the eigenvalues i and -i of J. Let [S.sup.1] := {z [element of] C : z = 1 } and define a homomorphism (of real algebraic groups): h : [S.sup.1] [right arrow] GL([H.sup.1](A, R)), h(a + bi) := a + bJ. We define SMT(A), the special Mumford-Tate group of A to be the smallest algebraic subgroup of GL([H.sup.1](A, Q)) such that h([S.sup.1]) [subset] SMT(A)(R). (In particular, SMT(A) is defined by polynomial equations with coefficients in Q and the variables are the entries of matrices in End([H.sup.1](A, Q)). It is in fact the intersections of all subgroups G of GL([H.sup.1](A, Q)) defined in this way which satisfy h([S.sup.1]) [subset] G(R), where G(R) is the group of matrices with real entries which satisfy the defining equations of G.) Since the polarization <> is invariant under J (<x,y> = <Jx, Jy>), one has that h([S.sup.1]) [subset] Sp([H.sup.1](A, R), <>). As <> is defined on [H.sup.1](A, Q) we always have SMT(A) [subset or equal to] Sp([H.sup.1](A, Q), <>).

Using the polarization on [H.sup.1](A, Q) one can prove that SMT(A) is a reductive group, that is, any (linear, algebraic) representation of SMT(A) is a direct sum of irreducible representations (cf. [DMOS]). There is a bijection between rational Hodge substructures of [H.sup.k](A, Q) and subrepresentations of SMT(A) in [H.sup.k](A, Q) = [[conjunction].sup.k][H.sup.1](A, Q).

The fact that dim A is odd and that [End.sup.Q](A) = Q imply, by a theorem of Ribet, (cf. [R] Th. 1), that SMT(A) [congruent] Sp(6, Q), the symplectic group of 6 X 6 matrices. The representation of SMT(A) on [H.sup.1](A) corresponds to the standard representation [rho] of Sp(6). It is well known that [[conjunction].sup.3] [rho] is the direct sum of exactly two irreducible components,

[MATHEMATICAL EXPRESSION OMITTED] where <> stands for the one dimensional trivial representation of Sp(6) (in fact, the decomposition of [[congruent].sup.k][H.sup.1](A) into irreducible components for the Sp-action corresponds to the Lefschetz decomposition).

Since the representation of Sp(6) on [H.sup.3][(A, Q).sub.prim] is irreducible, there is no non-trivial sub Hodge structures in [H.sup.3](A, Q) and therefore there is no abelian subvarieties in [J.sup.2][(A).sub.prim].

LEMMA 2.3. For a contractible open neighbourhood V of a suitable point [t.sub.0]

[MATHEMATICAL EXPRESSION OMITTED]

Proof. Take as [t.sub.0] a 3 to 1 cyclic covering of Ex. (2.2) and let [pi] = [[phi].sub.3](W). Then by Prop.

[MATHEMATICAL EXPRESSION OMITTED]

Now we are ready to prove Th. (2.1):

THEOREM 2.1. For the open set V of Lem. (2.1) the image of [[phi].sub.Q] contains an open subset of [G.sub.R](6, 8).

Proof. We need just to show that there is at least one fiber of [[phi].sub.Q] of dimension exactly 5. Since Q is a divisor in [G.sub.v] (3, [F.sup.1], [for all][pi] [element of] [G.sub.R](6, 8), [phi]??([pi]) = [phi]??([pi])[intersection]Q is the empty set or it has dimension [greater than or equal to] 5.

To finish, note that from Lem. (2.1) we get that, for [pi] [element of] [G.sub.Q](6, 8), this dimension is never 6 and from Lem. (2.3) that there is at least a [pi] such that [phi]??([pi]) [not equal to] [phi].

COROLLARY 2.2. There is an infinite countable number of Y's such that [D.sub.Y] is a divisor in Y.

Proof. By Th. (2.1) and the density of [G.sub.Q](6, 8) in [G.sub.R](6, 8) we have a countable number of [phi]??([pi]) with [pi] [element of] [G.sub.Q](6, 8) and of dimension 5. So, by Rem. (2.D), we have a countable number of [D.sub.Y] with dim [D.sub.Y] = 5 and so the corollary is proven.

Remark (2.F). Note that the differential of [[phi].sub.Q] is generically surjective. It follows that, by analytic continuation, both Lem. (2.3) and Th. (2.1) hold for any contractible open set of [M.sub.4](N).

Remark (2.G). We can also construct the following filtration: C [subset] Q [subset] [G.sub.V] over V, where C is a subvariety of codimension 2 of [G.sub.V] such that its dual [C.sup.*] ([subset] [Q.sup.*] [subset] P??) is the set of those points of P?? corresponding, over t [element of] V, to the canonical curve [phi] [[K.sub.C].sub.t] ([C.sub.t]).

COROLLARY 2.3. There is an infinite countable number of fibers [phi]??([pi]) for [pi] [element of] [G.sub.Q](6, 8) such that [phi]??([pi]) [intersection] Q [not equal to] [phi] but [phi]??([pi]) [intersection] Q [not subset] C.

Proof. First of all, for W [element of] [G.sub.V], [functions of 3](W) = t = [[C.sub.t]]: W [element of] C [if an only if] W = [H.sup.0]([C.sub.t], [[K.sub.C].sub.t] (- q)), for some q [element of] [C.sub.t]. (2.4) Furthermore, the 3 to 1 cyclic covering of elliptic curves p : C [right arrow] E (of Ex. (2.2)) has the property that W( := ([p.sup.*]([H.sup.0]([K.sub.E])) [perpendicular to]) never satisfies the condition above i.e. W [not element of] C, (cf. Cor. (2.1)) while, as remarked in Lem. (2.4), W [element of] Q. Denote by [[pi].sub.W] the image [[phi].sub.3](W). Since the condition [phi]??(W) [intersection] Q [subset] C is closed, we can take a neighbourhood U of [[pi].sub.W] in [G.sub.R](6, 8) in such a way that [phi]??(U) [intersection] Q [not subset] C. The countable infinite set [pi]??(U [intersection] [G.sub.Q](6, 8) is then the set we were looking for.

THEOREM 2.2. Let [pi] [element of] [G.sub.Q](6, 8) be such that

[MATHEMATICAL EXPRESSION OMITTED] Define Y := [functions of 3]([phi]??([pi])) and [D.sub.Y] = [functions of 3]([phi]?? ([pi])) [intersection] Q.

Then, for y = [[C.sub.y]] [element of] [D.sub.Y] such that

[MATHEMATICAL EXPRESSION OMITTED] the sheaf [N.sub.C] corresponding to C [right arrow] J(C) [right arrow] [W.sup.*]/([[phi].sub.3] [(W).sup.*] [intersection] [H.sub.1](A, Z)) is locally free.

Proof. Let C be a smooth curve of genus g, A an abelian variety and [micro]: C [right arrow] A the composing map [micro] : C [right arrow] J(C) [right arrow] A where the first map is the Abel-Jacobi map and the second one is a surjective homomorphism of groups. The normal sheaf [N.sub.C] is, by definition, the cokernel of the map:

[MATHEMATICAL EXPRESSION OMITTED] The sheaf [N.sub.C] is locally free if, for any p [element of] TC, the linear map

[MATHEMATICAL EXPRESSION OMITTED] between the tangent spaces is different for 0. The differential [d.sub.p][micro] is equal to 0 if, for [t.sub.p] a generator of [T.sub.p]C,

[MATHEMATICAL EXPRESSION OMITTED] but, since T?? is trivial, hence generated by the global sections, that means that for any [omega] [element of] [H.sup.1,0](A) and [[micro].sup.*][omega] [element of] [H.sup.0]([K.sub.C]) <[t.sub.p],[([micro].sup.*][omega]).sub.p]> = 0 i.e. the differential form [[micro].sup.*][omega] is 0 in p. That implies that [[micro].sup.*]([H.sup.1,0](A)) [subset] [H.sup.0]([K.sub.C](-p)). For our choice of C this condition is never satisfied so [N.sub.C] is locally free.

We want now to characterize better

[MATHEMATICAL EXPRESSION OMITTED] and [D.sub.Y]:

THEOREM 2.4. Let [pi] [element of] [G.sub.Q](6, 8) be such that [phi]??([pi]) = [phi]??[pi]) [intersection] Q [equivalent to not] [phi] Define Y := [[functions].sub.3] [phi]?? ([pi])) and [D.sub.Y] = [[functions].sub.3]([phi]?? ([pi])).

Then

[MATHEMATICAL EXPRESSION OMITTED] : Y [right arrow] [H.sub.3] generically has finite fiber but is not birational. In fact the image

[MATHEMATICAL EXPRESSION OMITTED] of [D.sub.Y] is a divisor in

[MATHEMATICAL EXPRESSION OMITTED] and a generic point of

[MATHEMATICAL EXPRESSION OMITTED] is a ramification point of

[MATHEMATICAL EXPRESSION OMITTED]

Proof. We know that Y is a smooth analytic variety and that

[MATHEMATICAL EXPRESSION OMITTED] is a holomorphic map with generically finite fiber (cf. Rem. (2.A,E)).

If

[MATHEMATICAL EXPRESSION OMITTED] is not a divisor of

[MATHEMATICAL EXPRESSION OMITTED] then dim

[MATHEMATICAL EXPRESSION OMITTED] is exactly 4 because, as we pointed out at the beginning of Prop. (2.3), dim Ker

[MATHEMATICAL EXPRESSION OMITTED] so the rank of

[MATHEMATICAL EXPRESSION OMITTED] cannot be smaller than 4.

Suppose that

[MATHEMATICAL EXPRESSION OMITTED] is not a divisor, then for a generic point [A'] of

[MATHEMATICAL EXPRESSION OMITTED] (with A' a PPA threefold), dim

[MATHEMATICAL EXPRESSION OMITTED] = 1. So there is a component B of

[MATHEMATICAL EXPRESSION OMITTED] which parametrizes a 1-dimensional family of curves of genus 4, each mapping to A'.

Then, reasoning as in the proof of Lem. (2.1), we conclude that [[J.sup.2](A').sub.prim] has an abelian subvariety i.e. [H.sup.3][(A', Z).sub.prim] has a Hodge substructure. If A' was simple then the codimension of the locus of the abelian varieties with such a substructure is at least 3. Therefore, A' can't be simple. Since, by hypothesis,

[MATHEMATICAL EXPRESSION OMITTED] has to be isogenous to a product of an abelian surface and an elliptic curve.

But in this case all the jacobians of the genus 4 curves whose moduli correspond to points of [D.sub.Y] would be isogenous to products of 2 elliptic curves and an abelian surface. So we would have a family of dimension 5 of curves of genus 4 mapping to products of two elliptic curves and hence, by a count of moduli, a family of dimension at least 3 of such curves mapping to a product of two fixed elliptic curves. Let's consider the two cases:

1) The generic curve C of the family maps birationally into E x E'. This would contradict Proposition (2.4) of [CvdGT] which says that the image on [M.sub.g](N) of the subvariety of the Hilbert scheme of curves of genus g on a fixed abelian variety has to have dimension less or equal to g -- 2.

2) The map is not birational onto the image. So, by Riemann-Hurwitz, the image of C could just be birational to a smooth genus 2 curve [C.sub.2] whose jacobian is isogenous to E x E'. Moreover the degree of the map could be just 2 or 3. Note that it is impossible to deform J([C.sub.2]) mantaining the condition that it is isogenous to E x E', so we could just deform the map C [right arrow] [C.sub.2] with [C.sub.2] fixed.

If the degree was 3 the map would be unramified so there wouldn't be any deformation.

If the degree was 2, by Riemann-Hurwitz, the degree of the ramification locus would be 2 so the dimension of the family of genus 4 curves mapping to [C.sub.2] would be at most 2.

3. 1-cycles homologically but not algebraically equivalent to 0. We want to use our curves of genus 4 to construct a countable infinite number of 1-cycles homologically but not algebraically equivalent to 0 on a generic abelian threefold of [A.sub.3].

The first step is to consider the threefold A corresponding to a generic point in the image of

[MATHEMATICAL EXPRESSION OMITTED] : Y = [f.sub.3]([phi]??([pi])) [right arrow] [H.sub.3] for [pi] [element of] [G.sub.Q](6, 8) where the map

[MATHEMATICAL EXPRESSION OMITTED] is branched on an open set of the divisor [D.sub.Y] [subset] Y. On

[MATHEMATICAL EXPRESSION OMITTED] we take two points which specialize to a unique point of [D.sub.Y]. The difference of the two corresponding genus 4 curves, seen in the group [CH.sup.2](A) of the cod2 cycles of A, gives our 1-cycles, that we will show to be homologically but not algebraic equivalent to 0.

The second step consists in globalizing the construction (cf. Rem. (3.D)) in such a way that these cycles define cycles on the abelian threefold corresponding to the generic point of [A.sub.3].

Remark (3.A). For any point y = [[C.sub.y]] [element of] Y = [f.sub.3]([phi]??([pi])) we denote by [[phi].sub.y] the surjective homomorphism:

[MATHEMATICAL EXPRESSION OMITTED] and by [[psi].sub.y] [[psi].sub.y]; = [rho] [omicron] [[phi].sub.y] (3.2) the composition with an isogeny [rho] in such a way that [A.sub.y] := [rho]([A'.sub.y]) is principally polarized (cf. (1.2) and (1.3)). Moreover

[MATHEMATICAL EXPRESSION OMITTED] i.e. the tangent space to Y in y can be seen as the space of the first order deformations of both the 2 maps [[phi].sub.y] and [[psi].sub.y].

Remark (3.B). We recall that, over any Y, there are the family of curves of genus 4: [C.sub.Y] [right arrow] Y, the jacobian fibration J([C.sub.Y]) = [{J([C.sub.y])}.sub.y[element of]Y], the family of PPA threefolds [[alpha].sub.Y] = [{[A.sub.y]}.sub.y[element of]Y] and the map: [[psi].sub.Y] : J([C.sub.Y]) [right arrow] [[alpha].sub.Y] (3.4) that globalizes the maps [[psi].sub.y].

Denoting by [[alpha].sub.[H.sub.3]] [right arrow] [H.sub.3] the universal family of PPA threefold on [H.sub.3], we have [[alpha].sub.Y] =

[MATHEMATICAL EXPRESSION OMITTED]

We introduce now some notation.

Let A be an abelian threefold. Denote by [CH.sup.2](A) the Chow group of the cycles of codimension 2, [CH.sub.2][(A).sub.hom] the subgroup of [CH.sup.2](A) of the cycles homologically equivalent to 0, [CH.sup.2][(A).sub.alg] the subgroup of the cycles algebraically equivalent to 0 and [CH.sup.2][(A).sub.tran] the subgroup generated by the cycles: Z - [[tau].sup.*]Z [for all]Z [element of] [CH.sup.2](A) [[for all].sub.[tau]] translation in A. Note that

[MATHEMATICAL EXPRESSION OMITTED] We define the quotient groups:

[MATHEMATICAL EXPRESSION OMITTED] The elements of [T.sup.2](A) will be called cycles modulo translations.

The Griffiths group [Gr.sup.2](A) is the quotient group:

[MATHEMATICAL EXPRESSION OMITTED]

Note that, with this notation, for any y [element of] Y the 2-cycle [[psi].sub.y](Ab([C.sub.y])) [subset] [A.sub.y], seen in [T.sup.2]([A.sub.y]), doesn't depend on the choice of the Abel-Jacobi map Ab : [C.sub.y] [right arrow] J([C.sub.y]). Hence we have the family of cod 2 cycles modulo translation in [[alpha].sub.Y] denoted by [[psi].sub.Y]([C.sub.Y]).

Take now the fiber product: [[micro].sub.Y] : Y [x.sub.[H.sub.3]] Y [right arrow] [H.sub.3] and denote by [[micro].sub.1] and [[micro].sub.2] the standard projections to Y.

We recall that Y [x.sub.[H.sub.3]] Y [subset] Y x Y can be seen as the set of the points (y, y') such that

[MATHEMATICAL EXPRESSION OMITTED]

Over Y [x.sub.[H.sub.3]] Y we have the pullback of the family of PPA threefolds on [H.sub.3]: [f.sub.[micro]] : [alpha] [right arrow] Y [x.sub.[H.sub.3]] Y [alpha] := [micro]??[[alpha].sub.[H.sub.3]] and the pullbacks of the families of curves: [C.sub.1] := [micro]??([C.sub.Y]) [C.sub.2] := [micro]??([C.sub.Y]) [C.sub.i] [right arrow] Y [x.sub.[H.sub.3]] Y. The family of cycles [[psi].sub.Y]([C.sub.Y]), well defined modulo translation, induces the families [[psi].sub.i]([C.sub.Y]) [subset] [alpha] for i = 1, 2. (3.5) For x [element of] Y [x.sub.[H.sub.3]] Y we denote by [A.sub.x] the fiber of [f.sub.[micro]] over x and by [[psi].sub.i]([[C.sub.i].sub.x]) the 1-cycle of [A.sub.x] modulo translation [[psi].sub.i]([C.sub.i]) [intersection] [A.sub.x].

LEMMA 3.1. For any x [element of] Y [x.sub.[H.sub.3]] Y the cycle

[MATHEMATICAL EXPRESSION OMITTED] is homologically equivalent to 0.

Proof. Let q = [A] be a generic point of

[MATHEMATICAL EXPRESSION OMITTED] Let x := (y, y') [element of] Y [x.sub.[H.sub.3]] Y, we have:

[MATHEMATICAL EXPRESSION OMITTED] Since A is generic, the image of [CH.sup.2](A) and [T.sup.2](A) in [H.sup.2](A,Z) is the cyclic group generated by

[MATHEMATICAL EXPRESSION OMITTED] where [[theat.sub.A] is the theta divisor of A. Hence

[MATHEMATICAL EXPRESSION OMITTED] for k [element of] [Z.sub.>0]. Because Y is connected this property is preserved by deformation i.e. for any [x.sub.t] [element of] Y [x.sub.[H.sub.3]] Y

[MATHEMATICAL EXPRESSION OMITTED] Hence in particular

[MATHEMATICAL EXPRESSION OMITTED] is homologically equivalent to 0.

We recall now some other basis results. Let A be a PPA threefold. Denote by [J.sup.2](A) the intermediate jacobian of A and by L(A) the image in [J.sup.2](A) of A by the map induced by the Lefschetz map. We define [J.sup.2][(A).sub.prim] := [J.sup.2](A)/L(A) the primitive intermediate jacobian. Moreover we define [J.sup.2][(A).sup.ab] the biggest abelian variety contained in [J.sup.2](A). For A generic [J.sup.2][(A).sup.ab] = L(A). Since always L(A) [subset] [J.sup.2][(A).sup.ab] we can also consider the abelian variety [J.sup.2](A)?? := [J.sup.2][(A).sup.ab]/L(A).

We recall that the Abel-Jacobi map:

[MATHEMATICAL EXPRESSION OMITTED] Z := [delta][gamma] [bar right arrow] [[integral of].sub.[gamma]] satisfies:

[MATHEMATICAL EXPRESSION OMITTED] so we can define the map:

[MATHEMATICAL EXPRESSION OMITTED] Moreover since:

[MATHEMATICAL EXPRESSION OMITTED] we have the map induced by ab:

[MATHEMATICAL EXPRESSION OMITTED] Then the map Ab can be seen as:

[MATHEMATICAL EXPRESSION OMITTED]

Consider now the intermediate jacobian bundles [J.sup.2](A) and [J.sup.2][(A).sub.prim] over Y x [H.sub.3] Y and define the following normal function:

[MATHEMATICAL EXPRESSION OMITTED]

We want to show:

THEOREM 3.1. Let [pi] [element of] [G.sub.Q](6,8) be such that [phi] [not equal to] [phi]?? ([pi]) [intersection] Q ?? C. Define Y := [[functions].sub.3] ([phi]?? ([pi])) and [D.sup.Y] :=[[functions].sub.3] ([phi]??([pi]) [intersection] Q) the divisor on which the rank of the differential of the map [G.sup.0] : Y [right arrow] [H.sub.3] falls.

Then

[MATHEMATICAL EXPRESSION OMITTED] where 0 is the zero section of [J.sup.2][(A).sub.prim].

Proof. Since Y x [H.sub.3] Y is a fiber product its tangent space [T.sub.y,y'] := [T.sub.y,y']Y x [H.sub.3]Y in y,y' can be seen as the set of the couples of vectors ([nu],[nu]') [element of] [T.sub.y]Y [direct sum] [T.sub.y']Y satisfying the condition:

[MATHEMATICAL EXPRESSION OMITTED] So [T.sub.y,y'] is the kernel of the map:

[MATHEMATICAL EXPRESSION OMITTED]

The tangent space to the bundle [J.sup.2][(A).sub.prim], in the point (y,y',s) with s [element of] [J.sup.2][(A).sub.prim] and A := [A.sub.y] =[A.sub.y'], is given by:

[MATHEMATICAL EXPRESSION OMITTED] To prove our statement we just need to prove that the map:

[MATHEMATICAL EXPRESSION OMITTED] obtained composing [d.sub.y,y'](m) with the projection to the tangent space to the fiber of the primitive intermediate jacobian bundle, is not identically 0.

Pick a point (y,y) in the diagonal of [delta (difference)] [subset] Y x [H.sub.3] Y. Since:

[MATHEMATICAL EXPRESSION OMITTED] we can write

[MATHEMATICAL EXPRESSION OMITTED] ([T.sub.y]Y [hook right arrow] [T.sub.y,y], [nu] [bar right arrow] ([nu],[nu]') and ker ([d.sub.y][G.sup.0]) [hook right arrow] [T.sub.y,y], w [bar right arrow] (0,w)),

Note that the kernel of the map:

[MATHEMATICAL EXPRESSION OMITTED] corresponds exactly to the space of the first order deformations of [[psi].sub.y] : J([C.sub.y]) [right arrow] [A.sub.y] with [A.sub.y] fixed. By Horikawa's theory [H.sup.0]([N.sub.Cy]) parametrizes the first order deformations of [[psi].sub.y] [clegrees] Ab : [C.sub.y] [right arrow] [A.sub.y] with [A.sub.y] fixed. The infinitesimal translations in [A.sub.y] correspond to [H.sup.0]([T.sub.Ay]) so:

[MATHEMATICAL EXPRESSION OMITTED]

If y [element of] Y -- [D.sub.y], [d.sub.y][G.sup.0] is injective i.e. ker ([d.sub.y][G.sup.0]) = 0 i.e. [H.sup.0]([N.sub.Cy]) = [H.sup.0]([T.sub.Ay]).

Instead in a point y [element of] [D.sub.y] dim (ker ([d.sub.y][G.sup.0])) = dim ([H.sup.0]([N.sub.Cy])/[H.sup.0] ([T.sub.Ay])) = 1.

Consider the restriction [alpha] of the map

[MATHEMATICAL EXPRESSION OMITTED] to ker [d.sub.y][G.sup.0]):

[MATHEMATICAL EXPRESSION OMITTED]

To prove that [beta] is nonzero (and hence to prove the theorem) it is enough to show that [alpha] is nonzero.

The map [alaph] fits in the following diagram:

[MATHEMATICAL EXPRESSION OMITTED] where

[MATHEMATICAL EXPRESSION OMITTED] is the "universal" infinitesimal Abel-Jacobi map (cf.[C11]): it satisfies the following condition

[MATHEMATICAL EXPRESSION OMITTED] So [alpha] is different from 0 if and only if y [element of] [D.sub.y] and [gamma] is injective and Lem. (3.2) below states that [gamma] is injective if [N.sub.Cy] is locally free.

It follows from the hypothesis of the theorem that, since [phi]?? ([pi]) [intersection] Q [not subset] C, we can take y = [[C.sub.y]] [element of] [D.sub.y] such that [functions]?? (y) [intersection] [phi]?? ([pi]) [not on element of] C. Then, by Th. (2.3) the sheaf [N.sub.Cy] is locally free. Therefore we can apply Lem. (3.2) to [[psi].sub.y] [clegrees] Ab : [C.sub.y] [right arrow] [A.sub.y].

Thus [alpha] is diferent from 0 and the map:

[MATHEMATICAL EXPRESSION OMITTED] whose differential in (y,y), composed with a projection is [beta] can't be always 0 i.e. m(Y x [H.sub.3] Y) [not subset] 0 where 0 is the section of [J.sup.2] [(A).sub.prim].

We can now show that we have cycles non algebraically equivalent to 0:

THEOREM 3.2. For Y satisfying the hypothesis of Th. (3.1), there is a component X of Y x[H.sub.3] Y such that the image of projection map [P.sub.X] ( := [[micro].sub.Y X]) : X [right arrow] [H.sub.3] contains an open set of [H.sup.3] and [m.sub. X] [not equal to] 0.

So in particular for q = [A], a generic point of Im([P.sub.X]), there exists x := (y,y') in P?? (q), such that the cycle Z := [[C.sub.1] := [[psi].sub.y]([C.sub.y]) - [[C.sub.2] := [[psi].sub.y']([C.sub.y'])] [element of] T?? (A) is not algebrically equivalent to 0 i.e. it defines a non trivial element of G?? (A).

Proof. Since [G.sup.0] : Y [right arrow] [H.sub.3] has generic fiber of dimension 0 but it is not birational, for a generic q [element of] [G.sup.0](Y) there exist y,y' [element of] Y with y [not equal to] y' and

[MATHEMATICAL EXPRESSION OMITTED] We can choose y,y' in such a way that they specialize to the same point [y.sub.0] [element of] [D.sub.y] [subset] Y when q specializes to a generic point of [G.sub.0]([D.sub.Y]) that is it specializes to a ramification point of [G.sup.0]. Since y [not equal to] y' the tangent direction

[MATHEMATICAL EXPRESSION OMITTED] defined by (y,y') is not in

[MATHEMATICAL EXPRESSION OMITTED] so t has a nonzero component in ker ([d.sub.y0] [G.sup.0]) (cf. (3.9)).

Since

[MATHEMATICAL EXPRESSION OMITTED] (see proof of Th. (3.1)), for X a component of Y x [H.sub.3] Y containing (y,y') we have that m is different from 0 when restricted to X.

LEMMA 3.2 If [N.sub.C] is locally free, [gamma] : [H.sup.0]([N.sub.C]) [right arrow] [H.sup.2,1*] (A) is injective.

Proof. Since, by hypothesis, [N.sub.C] is locally free, Clemens' results (cf. [C11], [C12]) can be used in this case.

Consider the exact cohomology sequence:

[MATHEMATICAL EXPRESSION OMITTED] From Clemens (cf. [C12]) we have that [lambda] is the dual of [gamma], so we have to show that [lambda] is surjective (cf. also [P] and [B]). Note that [H.sup.0] ([N.sub.C]) is the Serre dual of [H.sup.1]([N.sub.C]) and, because dim [H.sup.1]([T.sub.A]) = 9 and dim [H.sup.0]([T.sub.A]) = 3, we need to prove that dim [H.sup.1]([psi]) = 6 but we already recalled that [H.sup.1]([psi]) [approx imate] [Y.sub.y]Y and dim [T.sub.y]Y = 6 (cf. Rem. (3.A)).

Remark (3.C). Note that A is a jacobian of a genus 3 curve [C.sub.A] and, since A is generic, the class in [H.sup.4](A,Z) of C, embedded with some Abel-Jacobi map, is [[[theta]??/2].sup.hom]. Moreover, for the curves [C.sub.1] and [C.sub.2] of Th. (3.2), there is a k [element of] [Z.sub.[greater than]0] such that their images in [H.sup.4] (A,Z) is k[[[theta]??/2].sup.hom]. Then the cycle Z can be written as: Z = [Z.sub.1] - [Z.sub.2] [Z.sub.i] := [[C.sub.i]] - k[C] [element of] T?? (A). (3.13) On the other hand, since Z [not element of] T?? (A) at least one of the [Z.sub.i] [not element of] T?? (A). It will be this cycle, which is not algebraic equivalent to 0, that we will compare with the Ceresa cycles.

Remark (3.D). Because of the local character of the previous construction the infinite cycles built above live each on abelian threefolds corresponding to generic points of different analytic open sets of [H.sub.3].

To show that in fact they define cycles on the generic abelian threefold of [A.sub.3] we can use the same global construction of the Section 3 of [CP], with just minor changes.

The scheme [H.sub.4] (cf. () of [CP])

[MATHEMATICAL EXPRESSION OMITTED] parametrizes the triples (C,A,[psi]) with C a curve of genus 4 with N level structure, A a PPA threefold with N level structure and [psi] :J(C) [right arrow] A a surjective homomorphism of groups. Every irreducible component H of [H.sub.4] is a quasi projective variety, [p.sub.2\H] is a dominant morphism and [p.sub.1\H] is a morphism finite to 1 ([p.sub.1] is a discret map). The image of H by [p.sub.1] is a constructible set and the maximal layer is locally described by the [Y.sub.i]. Moreover, by the embeddings [F.sub.i] : [Y.sub.i] [right arrow] [H.sub.4] (cf. [CP]) the local behaviour of the morphism [p.sub.2] is perfectly described by the applications G?? : [p.sub.2] ?? [F.sub.i] (it was in fact this the way to prove that [p.sub.2] is dominant).

On H there is a family [C.sub.H] of curves of genus 4, the associated jacobian fibration J[(C).sub.H], obtained as pullback by [p.sub.1] and a family of PPA threefolds [A.sub.H] for pullback by [p.sub.2]. Moreover there is the map: [[psi].sub.H] : J[(C).sub.H] [right arrow] [A.sub.H] that globalizes the [psi] : J(C) [right arrow] A corresponding to the point h := (C,A,[psi]). So, also on [A.sub.H] there is a family of cycles of codimension 2 modulo translation [[psi].sub.H]([C.sub.H]) corresponding, for every h := (C,A,[psi]) to the image [psi] ?? Ab(C) [subset] A for Ab an Abel-Jacobi map.

Take the fiber product [[nu].sub.H] : H [x.sub.[A.sub.3](N)] H [right arrow] [A.sub.3](N), with projections [[nu].sub.1] and [[nu].sub.2], [C.sub.H1] := [nu]?? [C.sub.H], [C.sub.H2] := [nu]?? [C.sub.H] and A the family of PPA threefold obtained as pullback by [[nu].sub.H] from the universal family on [A.sub.3](N).

Then [[psi].sub.H]([C.sub.H]) induces the families [[psi].sub.H1] and [[psi].sub.H2]([C.sub.H2]) of cycles modulo translation on A.

The crucial observation is that, because [p.sub.2] is dominant, the family [[psi].sub.H1]([C.sub.H1])-[[psi].sub.H2]([C.sub.H2]) defines a cycle homologically equivalent to 0 on the threefold corresponding to the generic point of [A.sub.3](N). On the other hand, the restriction of [[psi].sub.H1]([C.sub.H1]) - [[psi].sub.H2]([C.sub.H2]) to the inverse image of a set contained in the image of a G?? such that Y [subset] [p.sub.1](H), corresponds, on each threefold of the set, to the cycle obtained as a restriction of the family of cycles [[psi].sub.1]([C.sub.1]) - [[psi].sub.2]([C.sub.2]) on Y [x.sub.[H.sub.3]] Y (cf.(3.5)).

COROLLARY 3.1. There is a countable infinite number of cycles in [Gr.sup.2](A) for A a generic PPA threefold, constructed as in Th. (3.2) as differences of genus 4 curves on A with the tecnique described above.

4. An element in the Griffiths' group of the generic abelian threefold which is independent of the Ceresa cycles. Let A be a PPA threefold. It can be seen as the jacobian of a curve C of genus 3. Denote by [C] the image in [T.sup.2](A) of the curve imbedded in its jacobian (well defined up to translation) and by [[C.sup.-]] the image of the curve in [T.sup.2](A) by the map obtained composing the chosen embedding with the multiplication by -1 in A.

Ceresa in [Ce] proves that the cycle [C]-[[C.sup.-]] [element of] [T.sup.2][(A).sub.hom] is not algebraically equivalent to 0 for A generic. Hence, for A generic, [Gr.sup.2](A) is different from 0.

Moreover Nori [N] (see also [B] for a different approach) proves that, for A generic, [Gr.sup.2](A) is infinitely generated by showing that the cycles [[C.sub.[alpha]]] - [C??] are algebraically independent, where [C.sub.[alpha]] is the image under an isogeny [r.sub.[alpha]] : j([C'.sub.[alpha]]) [right arrow] A of some Abel-Jacobi embedded curve [C'.sub.[alpha]] in J([C'.sub.[alpha]]).

Our goal is to show that at least one of the cycles constructed in Rem. (3.C) using curves of genus 4 is independent of all the cycles [[C.sub.[alpha]] - [C??].

Let [M.sub.3] be the moduli space of the couples (C,B) where C is a smooth curve of genus 3 and B is a symplectic basis of [H.sub.1] (C,Z). The Torelli map: [pi] : [M.sub.3] [right arrow] [H.sub.3] (C,B) [hook right arrow] (J(C),B) is a 2 to 1 map ramified along the hyperelliptic locus. Let A [right arrow] [H.sub.3] [A.sub.p] := [[pi].sup.-1](p) be the universal family of PPA threefolds on [H.sub.3]. Moreover let C [right arrow] [M.sub.3] be the universal family of curves over [M.sub.3] and J(C) the associated jacobian fibration. The cycle [C] - [[C.sup.-]] [element of] [T.sup.2](J(C)) globalizes to a family of cycles: C - [C.sup.-] homologically equivalent to 0, defined modulo translation, on J(C) = [[pi].sup.*] A. Let [J.sup.2][([[pi].sup.*]A).sub.prim] be the primitive internmediate jacobian over [M.sub.3] and let

[MATHEMATICAL EXPRESSION OMITTED] be the normal function associated to the family of cycles C - [C.sup.-].

The group Sp6Q acts on [H.sub.3] defining, for any [alpha] [element of] Sp6Q, an isomorphism: [[rho].sub.[alpha]] : [H.sub.3] [right arrow] [H.sub.3]. (4.1) Associated to it there is a fiber map: [R.sub.[alpha]] : A [right arrow] [rho]??A which restricted to the fibers is an isogeny.

Consider the composition:

[MATHEMATICAL EXPRESSION OMITTED] clearly [[pi].sub.[alpha]] is again a 2 to 1 map.

The map [R.sub.[alpha]] defines a map [[psi].sub.[alpha]] := [[pi].sup.*]([R.sub.[alpha]]):

[MATHEMATICAL EXPRESSION OMITTED] In this way over [pi]??A there is the family of cycles:

[MATHEMATICAL EXPRESSION OMITTED] For a fixed PPA threefold A = [A.sub.p] corresponding to the point p [element of] [H.sub.3] the cycle [[C.sub.[alpha]] - [C??], considered by Nori, defined by the isogeny [r.sub.[alpha]] : J([C.sub.[alpha]]) [right arrow] A, corresponds, for t [element of] [M.sub.3], [[pi].sub.[alpha]](t) = p and C := [C.sub.t], to the cycle

[MATHEMATICAL EXPRESSION OMITTED] As before, for any [alpha] [element of] Sp6Q we can define the primitive intermediate jacobian bundle [J.sub.2][([pi]??A).sub.prim] and the normal functions: [n.sub.[alpha]] : [M.sub.3] [right arrow] [J.sup.2][([pi]??A).sub.prim]

[MATHEMATICAL EXPRESSION OMITTED]

Remark (4.A). Note that since [[pi].sub.[alpha]] is a finite to 1 map, for any p = [[A.sub.p]] [element of] [H.sub.3], any connected component [F.sub.[alpha]] of [pi]??(p) is at most a point, so, in particular, the restriction of [n.sub.[alpha]] to [F.sub.[alpha]], [n.sub.[alpha]\[F.sub.[alpha]]] is trivially constant. Moreover, if [A.sub.p] is simple, [pi]??(p) is not empty.

We want to use exactly the property of Rem. (4.A) to show that at least one of cycles [Z.sub.i] of Rem. (3.C) is independent of the Ceresa-Nori ones.

First of all we want now to globalize the cycles [Z.sub.i]. Take the fiber product

[MATHEMATICAL EXPRESSION OMITTED] with [p.sub.1] and [p.sub.2] the projection to the 2 factors. Then the cycle [[C.sub.y]] - k[C], for [G.sup.0](y) = [A], globalizes to the family of homologically equivalent to 0 cycles, modulo translation:

[MATHEMATICAL EXPRESSION OMITTED] Again we can define the associated normal function:

[MATHEMATICAL EXPRESSION OMITTED]

THEOREM 4.1. Let Y be as in the hypothesis of Th. (3.1) and assume moreover that the map [G.sup.0] : Y [right arrow] [H.sub.3] has a fiber F of positive dimension over a point p corresponding to a simple abelian threefold.

Then, for p = [A] [element of] [G.sup.0](Y) generic, any y [element of] [G.sup.0-1](p) and [t.sub.[alpha]] [element of] [pi]??(p), we have that the cycle [Z.sub.Y,y] is algebraically independent of the cycles [Z.sub.[alpha],t[alpha]] i.e. it is not contained in the subgroup of [Gr.sup.2](A) generated by the Ceresa cycles.

Proof. First of all note that if for a particular Y the map [G.sup.0] : Y [right arrow] [H.sub.3] has fiber F of positive dimension and [G.sup.0](F) corresponds to a simple abelian threefold then p??(F) is not empty because [[pi].sup.-1]([G.sup.0](F)) is not empty. Moreover, since [[pi].sup.-1]([G.sup.0](F)) is finite (cf. Rem. (4.A)), the restriction of [n.sub.Y] along a component F of p??(F) can't be constant (see[P]).

The proof will be by contradiction.

Suppose that there is the following relation:

[MATHEMATICAL EXPRESSION OMITTED] This relation implies that

[MATHEMATICAL EXPRESSION OMITTED] Consider the fiber product of Y with l + 1 copies of ??:

[MATHEMATICAL EXPRESSION OMITTED] with the projections:

[MATHEMATICAL EXPRESSION OMITTED]

Let [eta]s : [alpha]s [right arrow] S be the pullback of the universal family of [H.sub.3] and [D.sub.Y] := P??([D.sub.Y]), [Z.sub.a] = P??([Z.sub.[alpha]]) the pullbacks of the cycles. Note that they can be seen as cod 2 cycles modulo translation on [[Alpha].sub.s]. As usual denote by [D.sub.Y,s], [Z.sub.[alpha],s] their intersection with the fiber [A.sub.s] := [eta]??(S). They are 1-cycles on the abelian threefold [A.sub.S].

Define the maps

[MATHEMATICAL EXPRESSION OMITTED]

Since A is generic, formula (4.9) implies that [n.sub.S] = 0 on an irreducible component [S.sub.0] of S containing the inverse images of P??(y) and P??([t.sub.[alpha]i]).

In particular we choose y specializing to a point of ??. Since, by hypothesis, ??(F) corresponds to a simple abelian threefold and the [[pi].sub.[alpha]] are surjective on the subset of [H.sub.3] parametrizing simple abelian threefolds, [S.sub.0] contains ??. Moreover, since the maps [[pi].sub.[alpha]] : ?? [right arrow] [H.sub.3] are finite to 1, the restriction to ?? of the maps [nz.sub.[alpha]i] is constant. Instead [n.sub.Y] is not constant on P?? so ns can't be constantly 0 on [S.sub.0].

The next theorem gives at least one example of Y satisfying the hypothesis of Th. (4.1) and hence it shows that one of our cycles is independent of the Ceresa-Nori ones.

THEOREM 4.2. There is a sublocus Y[delta (difference)] of an Y containing the 3 to 1 cyclic coverings of an elliptic curve in which the restriction of the map ??:

[G.sub.[delta (difference)]] ( = ??) : Y [right arrow] [H.sub.3] has always fibers of dimension 1.

Moreover a generic point of [G.sub.[delta (difference)]] ([Y.sub.[delta (difference)]]) corresponds to a simple abelian threefold.

Proof. Let p : C [right arrow] E be a 3 to 1 cyclic covering of the elliptic curve E as in Ex. (3.1.1) from which we will take all the notation.

We study the subspace of the space [H.sup.1]([psi]) of the first order deformations of the map [psi] : J(C) [right arrow] A :=J(C)/[p.sup.*](E) given by the deformations preserving the automorphism of order 3 of the jacobian and of the threefold.

We will denote the parameter space of such jacobians by [Y.sub.[delta (difference)]] and of such threefolds by [X.sub.[delta (difference)].

The tangent space to [Y.sub.[delta (difference)] corresponds to the space of the first order deformations of the covering preserving the cyclicity, so it is 3 dimensional, because the total ramification points are 3 and it is the invariant part of [H.sup.1] ([T.sub.C]. Its dual is

[MATHEMATICAL EXPRESSION OMITTED]

The tangent space to [X.sub.[delta (difference)], i.e. the space of the first order deformations of A preserving the automorphism, is parametrized by the dual of

[MATHEMATICAL EXPRESSION OMITTED]

[MATHEMATICAL EXPRESSION OMITTED] (cf. Prop. (2.2)), a space of dimension 2.

The dual of the differential of the map:

[G.sub.[delta (difference)]] : [Y.sub.[delta (difference)]] [right arrow] [X.sub.[delta (difference)]] is given by the restriction of

[MATHEMATICAL EXPRESSION OMITTED]

[MATHEMATICAL EXPRESSION OMITTED] and it is injective because the kernel <Q> of

[MATHEMATICAL EXPRESSION OMITTED] [right arrow]

[MATHEMATICAL EXPRESSION OMITTED] is not contained in the invariant part of

[MATHEMATICAL EXPRESSION OMITTED] (cf. Prop. (2.2)). So in particular [G.sub.[delta (difference)]] has fibers of dimension 1.

We want to prove now that a generic point [A] of [X.sub.[delta (difference)]] corresponds to a simple abelian threefold. If this was not true, A would be isogenous to a product S x E where S is an abelian surface and E an elliptic curve. We obtain a contradiction showing that in this case the first order deformation space of [X.sub.[delta (difference)]] would have dimension at most 1.

Suppose that S is simple. Since on A an automorphism of order 3 acts, [End.sup.0](A) contains an imaginary quadratic field and so the same has to be true for S and E. In particular E is fixed. The decomposition of [T.sub.0][S.sup.*] =: [W.sub.S] [subset] W in eigenspaces of [delta (difference)] can be only [W.sub.S] =

[MATHEMATICAL EXPRESSION OMITTED] So in this case the first order deformation space of S x E preserving the automorphism and the reducibility has dimension 1 = dim

[MATHEMATICAL EXPRESSION OMITTED]

If S is not simple A is the product of 3 elliptic curves and its first order deformation space preserving the automorphism has again dimension 1: the only possibility is that 2 elliptic curves are equal and varying in the moduli and the third is fixed.

REFERENCES

[B] F. Bardelli, Curve of genus 3 on a general abelian threefold and the non finite generation of the Griffiths group, Arithmetic of Complex Manifolds, Proc. Erlangen, vol. 1399, Springer-Verlag, New York, 1988, pp. 10--26.

[Ce] C. Ceresa, C is not algebraic equivalent to [C.sup.--] in its jacobian, Ann. of Math. 117 (1983), 285--291.

[CvdGT] C. Ciliberto, G. van der Geer, and M. Teixidor, On the number of parameters of jacobians with nontrivial endomorphisms, preprint.

[Cl1] H. Clemens, Some results on Abel Jacobi mappings, Topics in Trascendental Algebraic Geometry Ann. of Math. Stud., vol. 106 (P. Griffiths, ed.), Princeton Univ. Press, 1984, pp. 289--304.

[Cl2] _____, A note on some formal properties of the infinitesimal Abel Jacobi mapping, Geometry Today Prog. Math., vol. 60 (E. Arbarello, et al., eds.), Birkhauser, Boston, 1985, pp. 69--73.

[CP] E. Colombo and G. P. Pirola, Some density results for curve with non-simple jacobians, Math. Ann. 288 (1990), 161--178.

[DMOS] P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Cycles, Motives and Shimura Varieties, Springer-Verlag, New York, 1982.

[N] M. V. Nori, Cycles on the generic abelian threefold, Proc. Indian Acad. Sci. 99 (1989), 191--196.

[P] G. P. Pirola, On a conjecture of Xiao, preprint.

[R] K. A. Ribet, Hodge classes on certain types of abelian varieties, Amer. J. Math. 105 (1983), 523--538.
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Publication:American Journal of Mathematics
Date:Jun 1, 1994
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