Chow rings and gonality of general abelian varieties

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^0(A)$, $D^k$ is not a torsion cycle.

closed algebraic subsets in the symmetric product of the considered variety, so that their dimension is well-defined. Below we denote by {x} the 0-cycle of a point x ∈ A and 0 A will be the origin of A. The following results concerning orbits |Z| ⊂ A (k) for rational equivalence, and in particular the orbit |k{0 A }|, can be regarded as a Chow-theoretic version of Theorem 0.1.
(iv) If A is a very general abelian variety of dimension g ≥ 2k − 1, the orbit |k{0 A }| is countable.
In fact, Theorem 0.4, (iii) implies Theorem 0.1, because a k-gonal curve C ⊂ A, with normalization j : C → A and divisor D ∈ Pic k C with h 0 ( C, D) ≥ 2 provides a positive dimensional orbit {j * D ′ } D ′ ∈|D| in A (k) . We can assume one Weierstrass point c ∈ C of |D|, that is, a point c such that h 0 ( C, D(−2c)) = 0, is mapped to 0 A by j, which provides a positive dimensional orbit of the form |Z ′ + 2{0 A }|, with Z ′ effective and deg Z ′ ≤ k − 2.
Item (i) of Theorem 0.4 will be proved in Section 3 (cf. Theorem 3.1). The estimates in Theorems 0.1 and 0.4, (ii) can probably be strongly improved. Estimate (i) in Theorem 0.4 cannot be improved. To start with, it is optimal for g = 1 because for any degree k divisor D on an elliptic curve E we have |D| = P k−1 ⊂ E (k) . This immediately implies that the statement is optimal for any g because for abelian varieties A = E × B admitting an elliptic factor, we have E (k) ⊂ A (k) . In the case where g = 2, we observe that orbits |Z| ⊂ A (k) are contained in the generalized Kummer variety K k−1 (A) constructed by Beauville [6]. (More precisely, this is true for the open set of |Z| parameterizing cycles where all points appear with multiplicity 1 but this is secondary, cf. [16] for a discussion of cycles with multiplicities.) This variety is of dimension 2k − 2 and has an everywhere nondegenerate holomorphic 2-form for which any orbit |Z| is totally isotropic, which implies the estimate (i) in the case g = 2. Furthermore they are also orbits for rational equivalence in K k−1 (A), as proved in [11], hence they are as well constant cycles subvarieties in K k−1 (A) in the sense of Huybrechts [10]. The question whether Lagrangian (that is maximal dimension) constant cycles subvarieties exist in hyper-Kähler manifolds is posed in [17]. For a general abelian variety A, choosing a smooth curve C ⊂ A of genus g ′ , we have C (k) ⊂ A (k) for any k and C (k) contains linear systems P k−g ′ , for k ≥ g ′ . So when k tends to infinity, the estimate (i) has optimal growth in k.
Theorem 0.4, (iv), which will be proved in Section 2, has the following immediate consequence (which is a much better estimate than the one given in Theorem 0.1): Corollary 0.5. If A is a very general abelian variety of dimension g ≥ 2k − 1, and C ⊂ A is any curve with normalization C, one has h 0 (C, O C (kc)) = 1 for any point c ∈ C.
This corollary could be regarded as the right generalization of Theorem 0.3.
Remark 0.6. Pirola proves in [14] that for a very general abelian variety A of dimension g ≥ 4, any curve C ⊂ A has genus ≥ g(g−1) 2 + 1. This suggests that Theorem 0.4, (iv) is neither optimal, and that an inequality g ≥ O( √ k) should already imply the countability of |k{0 A }|.
where µ : A × A → A is the sum map and z × z ′ = pr * 1 z · pr * 2 z ′ for z, z ′ ∈ CH(A). The two rings are related via the Fourier transform, see [4]. Define to be the set of points x ∈ A such that ({x} − {0 A }) * k = 0 in CH 0 (A). We can also define A k ⊂ A to be the set of D ∈ Pic 0 (A) =: A such that D k = 0 in CH k (A). These two sets are in fact related as follows: choose a polarization θ on A, that is an ample divisor. The polarization gives an isogeny of abelian varieties Proof. This follows from Beauville's formulas in [4,Proposition 6]. We get in particular, the following equality: where Here the logarithm is taken with respect to the Pontryagin product * and the development is finite because 0-cycles of degree 0 are nilpotent for the Pontryagin product. If ({0 A } − {x}) * k = 0, then γ(x) * k = 0 and thus D k x = 0 by (2). Conversely, if D k x = 0, then γ(x) * k = 0 by (2). But then also ({0 A } − {x}) * k = 0 because {x} = exp(−γ(x)). (Again exp(−γ(x)) is a polynomial in γ(x), hence well-defined, since γ(x) is nilpotent for the * -product, see [7]).
Theorem 0.8. Let A be an abelian variety of dimension g. Then (ii) If A is very general and g ≥ 2k − 1, the sets A k and A k are countable.
Note that in both (i) and (ii), the two statements are equivalent by Lemma 0.7, using the fact that A → A is an open map between moduli spaces, so that, if A is very general, so is A.
The fact that Theorem 0.8 implies Theorem 0.4, (iv), uses the following intriguing result that does not seem to be written anywhere, although some related results are available, in particular the results of [8], [9], [15]. Proposition 0.9. Let A be an abelian variety and let x 1 , . . . , x k be k points of A such that In other words, For the proof of Theorem 0.8, we will show how the dimension estimate provided by (i) implies the non-existence theorem stated in (ii). This is obtained by establishing and applying Theorem 1.3, that we will present in Section 1. This theorem, which is obtained by a direct generalization of Pirola's arguments in [13], says that "naturally defined subsets" of abelian varieties (see Definition 1.1), assuming they are proper subsets for abelian varieties of a given dimension g, are at most countable for very general abelian varieties of dimension ≥ 2g − 1.
Thanks. This paper is deeply influenced by the reading of the beautiful Pirola paper [13]. I thank the organizers of the Barcelona Workshop on Complex Algebraic Geometry dedicated to Pirola's 60th birthday for giving me the opportunity to speak about Pirola's work, which led me to thinking to related questions.

Naturally defined subsets of abelian varieties
The proof of Theorem 0.3 by Pirola has two steps. First of all, Pirola shows that hyperelliptic curves in an abelian variety A, one of whose Weierstrass points coincides with 0 A , are rigid. Secondly he deduces from this rigidity statement the nonexistence of any hyperelliptic curve in a very general abelian variety of dimension ≥ 3 by an argument of specialization to abelian varieties isogenous to a product B × E, that we now extend to cover more situations. Definition 1.1. We will say that a subset Σ A ⊂ A is natural if it satisfies the following conditions: (0) Σ A ⊂ A is defined for any abelian variety A and is a countable union of closed algebraic subsets of A.
(i) For any morphism f : (ii) For any family A → S, there is a countable union of closed algebraic subsets Σ A ⊂ A such that the set-theoretic fibers satisfy Recall that the dimension of a countable union of closed algebraic subsets is defined as the supremum of the dimensions of its components (which are well defined since we are over the uncountable field C). Remark 1.2. By morphism of abelian varieties A, B, we mean group morphisms, that is, mapping 0 A to 0 B . Theorem 1.3. Let Σ A ⊂ A be a naturally defined subset.
(i) Assume that for dim A = g 0 , one has Σ A = A. Then for very general A of dimension ≥ 2g 0 − 1, Σ A is at most countable.
(ii) Assume that dim Σ A ≤ k for any A. Then for very general A of dimension ≥ 2k + 1, Σ A is at most countable.
(iii) Assume that dim Σ A ≤ k − 1 for a very general abelian variety A of dimension g 0 ≥ k. Then for a very general abelian variety A of dimension ≥ g 0 + k − 1, Σ A is at most countable.
Statement (ii) is a particular case of (i) where we do g 0 = k + 1. Both (i) and (iii) will follow from the following result: (b) If for a very general abelian variety B of dimension g > 0, Σ B is countable, then for A very general of dimension ≥ g, Σ A is countable.
Indeed, applying Proposition 1.4, (a), we conclude in case (i) that the dimension of Σ A is strictly decreasing with g ≥ g 0 as long as it is not equal to 0, and by assumption it is not greater than g 0 − 1 for g = g 0 . Hence the dimension of Σ A must be 0 for some g ≤ 2g 0 − 1. By Proposition 1.4, (b), we then conclude that Σ A is countable for any g ≥ 2g 0 − 1.
For case (iii), the argument is the same except that we start with dimension g 0 = k + 1 and we conclude similarly that the dimension of Σ A is strictly decreasing with g ≥ g 0 as long as it is not equal to 0. Furthermore, for g = g 0 , this dimension is equal to k − 1. Hence the dimension of Σ A must be 0 for some g ≤ g 0 + k − 1 and thus, by Proposition 1.4, (b), Σ A is countable for any g ≥ g 0 + k − 1. This proves Theorem 1.3 assuming Proposition 1.4 that we now prove along the same lines as in [13].
Proof of Proposition 1.4. Assume that dim Σ = k ′ for a very general abelian variety A of dimension g+1. From the definition of a naturally defined subset, and by standard arguments involving the properness and countability properties of relative Chow varieties, there exists, for each universal family A → S of polarized abelian varieties with given polarization type θ, a family Σ ′ A ⊂ Σ A S ′ ⊂ A S ′ , where S ′ → S is a generically finite dominant base-change morphism, A S ′ → S ′ is the base-changed family, and the morphism Σ ′ A → S ′ is flat, with irreducible fibers of relative dimension k ′ . In other words, we choose one k ′ -dimensional component of Σ A for each A, and we can do this in families, maybe after passing to a generically finite cover of a Zariski open set of the base.
The main observation is the fact that there is a dense contable union of algebraic subsets S ′ λ ⊂ S ′ along which the fiber A b is isogenous to a product B λ × E where B is a generic abelian variety of dimension g with polarization of type determined by λ and E is an elliptic curve (λ also encodes the structure of the isogeny). Along each S ′ λ , using axiom (i) of Definition 1.1, possibly after passing to a generically finite cover S ′′ λ , we have a morphism , is a nonzero multiple of θ l because the latter generates the space of degree 2l Hodge classes of a very general abelian variety with polarizing class θ. We thus conclude that , and as Σ ′ A b is irreducible by construction, it follows that p λ,Σ is generically finite on its image.
. We now concentrate on statement (a) and thus assume that Lemma 1.6. In the situation above, the set of varieties (of dimension is a proper subvariety of a very general abelian variety of dimension g with polarization of certain type, and Σ ′ A b ⊂ A b is the specialization of a subvariety (of codimension at least 2 by Lemma 1.5) of a general abelian variety of dimension g + 1 at a point b which is Zariski dense in S. In both cases, it follows that the Gauss maps , are generically finite on their images. We have the commutative diagram where all the maps are rational maps and the rational map π λ : G(k ′ , g + 1) G(k ′ , g) is induced by the linear map dp λ : We observe here that the density of the countable union of the S ′ λ in S has a stronger version, namely, the corresponding points [T E,0 ] ∈ P(T A b ,0 ) are Zariski dense in the projectivized bundle P(T A/S ). The projection π λ above is thus generic and the composition π λ • g A is generically finite as is g A and up to shrinking S ′ if necessary, its graph deforms in a flat way over the space of parameters (namely a Zariski open set of P(T A/S )). This is now finished because we first restrict to the Zariski dense open set U of P(T A S/B ,0 ) where the rational map π λ • g A is generically finite and its graph deforms in a flat way, and then there are finitely many generically finite covers of U parameterizing a factorization of the rational map π λ • g A . As the diagram (4) shows that there is a factorization of we conclude that all the maps Σ ′ ,p λ are, up to birational equivalence of the target, members of finitely many families of generically finite dominant rational maps ψ : As a corollary, we conclude using the density of the union of the sets S ′ λ that there is, up to replacing S ′ by a a generically finite cover of it, a family of k ′ -dimensional varieties Σ ′′ A S ′ , together with a dominant generically finite rational map In particular it does not depend on the elliptic curve E. Restricting to a dense Zariski open set S ′′ of S ′ is necessary, we can assume that we have desingularizations with smooth fibers over S ′′ . Letj : Σ ′ A b → A b be the natural map, and consider the morphismp * •j * : which is a group morphism defined at the general point of S ′′ . This morphism is nonzero because when b ∈ S ′′ λ for some λ, it is injective modulo torsion on Pic 0 (B b ) (which maps by the pull-back p * λ to Pic 0 (A b ) with finite kernel). Indeed, by the projection formula, denoting byj ′ : Σ ′′ A b → B the natural map, we have the equality of maps from Pic 0 (B b ) to Pic 0 ( Σ ′′ A b ): We note here that the morphismj ′ * : Pic 0 (B b ) → Pic 0 ( Σ ′′ A b ) has finite kernel because dim Imj ′ = k > 0. As the abelian variety Pic 0 (A b ) is simple at the very general point of S ′′ , the nonzero morphism (p λ,Σ ) * •j * must be injective. But then, by specializing at a point b of S ′′ λ , where λ is chosen in such a way that S ′′ λ = S ′′ ∩ S ′ λ is non-empty, we find that this morphism is injective on the component Pic 0 (E b ) of Pic 0 (A b ). We can now fix the abelian variety B b and deform the elliptic curve E b . We then get a contradiction, because we know that the variety Σ ′′ A b depends (at least birationally) only on B b and not on E b , so that its Picard variety cannot contain a variable elliptic curve E b .

Dimension estimate
Recall that for an abelian variety A and a nonnegative integer k, we denote by A k ⊂ A the set of points x ∈ A such that ({x} − {0 A }) * k = 0 in CH 0 (A). The following proves item (i) of Theorem 0.8: Proof. Let g := dim A and let Γ P ont k be the codimension g cycle of A × A such that for any where Γ i ⊂ A × A is the graph of the map m i of multiplication by i. Let us compute (Γ P ont k ) * η for any holomorphic form on A.
Lemma 2.2. One has (Γ P ont k ) * η = 0 for any holomorphic form η of degree < k on A, and (Γ P ont k ) * η = k!η for a holomorphic form of degree k on A. (7), the lemma is thus equivalent to This lemma directly implies Proposition 2.1. Indeed, by Mumford's theorem [12], one has (Γ P ont k ) * η |A k = 0 for any holomorphic form η of positive degree, and in particular for any holomorphic k-form. By Lemma 2.2, we conclude that, denoting by A k,reg ⊂ A k the regular locus of A k , η |A k = 0 for any holomorphic form η of degree k on A, that is, dim A k < k.

2.2
Proof of Theorem 0.8 The following result is almost obvious: (1) is naturally defined in the sense of Definition 1.1.
Proof. It is known that the set A k ⊂ A is a countable union of closed algebraic subsets. Using the fact that for a morphism f : A → B of abelian varieties, is compatible with the Pontryagin product, we conclude that f * (A k ) ⊂ B k . Finally, given a family π : A → S of abelian varieties, the set of points , is a countable union of closed algebraic subsets of A whose fiber over b ∈ S coincides set-theoretically with A b,k .
Proof of Theorem 0.8. The theorem follows from Proposition 2.1, Lemma 2.3, and Theorem 1.3.

Proof of Theorem 0.4, (iv)
We first prove the following Proposition (cf. Proposition 0.9).
Proposition 2.4. Let A be an abelian variety and let x 1 , . . . , x k ∈ A such that Then for i = 1, . . . , k.
Proof. Let γ l := |I|=l,I⊂{2,...,k} {x I }, where x I := i∈I x i . Then by (8), we have Furthermore, γ l = 0 for l ≥ k and the following inductive relation is obvious: that is: where by (10) for some rational nonzero coefficients α l,i . As the 0-cycles {jx 1 }, 0 ≤ j ≤ l and ({x 1 } − {0 A }) * j , 0 ≤ j ≤ l generate the same subgroup of CH 0 (A). The relation γ k = 0 thus provides a nontrivial degree k linear relation with Q-coefficients between the 0-cycles or equivalently a polynomial relation in the variable {x 1 }−{0 A } for the Pontryagin product, where the scalars are mapped to Q{0 A }. As we know by [7] The locus swept-out by the orbit |k{0 A }| is thus contained in A k . We thus deduce from Theorem 0.8 the following corollary: Corollary 2.5. (Cf. Theorem 0.4, (iv)) For any abelian variety A, the locus swept-out by the orbit |k{0 A }| has dimension ≤ k − 1. For a very general abelian variety A of dimension g ≥ 2k − 1, the orbit |k{0 A }| is countable.
In this statement, the locus swept-out by the orbit |k{0 A }| is the set of points x ∈ A such that a cycle x + Z ′ with Z ′ effective of degree k − 1 belongs to |k{0 A }|. The dimension of this locus can be much smaller than the dimension of the orbit itself, as shown by the examples of orbits contained in subvarieties C (k) ⊂ A (k) for some curve C.

Proof of Theorem 0.4, (i)
We give in this section the proof of item (i) in Theorem 0.4. We first recall the statement: Theorem 3.1. Let A be an abelian variety. The dimension of any orbit |Z| ⊂ A (k) for rational equivalence is at most k − 1.
Proof. We will rather work with the inverse image |Z| of the orbit |Z| in A k . By Mumford's theorem [12], for any holomorphic i-form α on A with i > 0, one has, along the regular locus |Z| reg of |Z|: where the pr j : A k → A are the various projections. Let x = (x 1 , . . . , x k ) ∈ |Z| reg and let V := T |Z| reg ,x ⊂ W k , where W = T A,x = T A,0A . One has dim V = dim |Z|, and (14) says that: (*) for any α ∈ i W * with i > 0, one has ( j pr * j α) |V = 0.
Theorem 3.1 thus follows from the following proposition 3.2.
Proposition 3.2. Let W be a vector space, V ⊂ W k be a vector subspace satisfying property (*). Then dim V ≤ k − 1.
, σ being the sum map. If dim W = 2, the result follows from the fact that, choosing a generator η of 2 W * , the 2-form j pr * j η is nondegenerate on W k 0 (which has dimension 2k − 2). A subspace V satisfying (*) is contained in W k 0 and totally isotropic for this 2-form, hence has dimension r ≤ k − 1.
Proof of Proposition 3.2. Note that the group Aut W acts on W k , with induced action on Grass(r, W k ) preserving the set of r-dimensional vector subspaces V ⊂ W k satisfying condition (*). Choosing a C * -action on W with finitely many fixed points e 1 , . . . , e n , n = dim W , the fixed points [V ] ∈ Grass (r, W k ) under the induced action of C * on the Grassmannian are of the form V = A 1 e 1 , . . . A n e n , where A i ⊂ (C k ) * are vector subspaces , with r = i dim A i . It suffices to prove the inequality r ≤ k − 1 at such a fixed point, which we do now. The spaces A i have to satisfy the following conditions: (**) For any ∅ = I = {i 1 , . . . , i s } ⊂ {1, . . . , n} and for any choices of λ l ∈ A i l , l = 1, . . . , s, where f j is the natural basis of C k .
A better way to phrase condition (**) is to use the (standard) pairing , on (C k ) * , given by Condition (**) when there are only two nonzero spaces A i is the following where e is the vector (1, . . . , 1) ∈ (C k ) * . Indeed, the case s = 2 in (**) provides (15) and the case s = 1 in (**) provides (16). The fact that the pairing , is nondegenerate on (C k ) * 0 := e ⊥ immediately implies that i dim A i ≤ k − 1 when only two of the spaces A i are nonzero. By the above arguments, the proof of Proposition 3.2 is finished used the following lemma: . . , n, be linear subspaces satisfying conditions (**).
Proof. We will use the following result: Lemma 3.5. Let A ⊂ C k , B ⊂ C k be vector subspaces satisfying the following conditions: Let us first show how Lemma 3.5 implies Lemma 3.4. Indeed, we can argue inductively on the number n of spaces A i . As already noticed, Lemma 3.4 is easy when n = 2. Assuming the statement is proved for n − 1, let A 1 , . . . , A n be as in Lemma 3.4

and let
Then the set of spaces A ′ 1 , . . . , A ′ n−1 satisfies conditions (**), and on the other hand Lemma 3.5 applies to the pair (A, B) = (A n−1 , A n ) as they satisfy the desired conditions by (**). Hence we have dim A ′ n−1 ≥ dim A n−1 + dim A n and by induction on n, Under the conditions (i) and (ii), the multiplication map µ : has image in the affine space C k 1 := e + C k 0 , where C k 0 = e ⊥ , and more precisely it generates the affine space e + A + B + A · B ⊂ e + C k 0 . It thus suffices to show that the dimension of the algebraic set Im µ is at least dim A + dim B. Lemma 3.5 is thus implied by the following: Claim 3.6. The map µ has finite fiber near the point (e, e) ∈ A 1 × B 1 .
The proof of the claim is as follows: Suppose µ has a positive dimensional fiber passing through (e, e). We choose an irreducible curve contained in the fiber, passing through (e, e) and with normalization C. The curve C admits rational functions σ i , i = 1, . . . , k mapping it to A 1 such that the functions 1 σi map C to B 1 . The conditions (i) and (ii) say that as a function of (s, t) for any choice of points x, y ∈ C and local coordinates s, t near x, resp. y, on C. We now do x = y and choose for x a pole (or a zero) of one of the σ l 's. We assume that the local coordinate s is centered at x, and write σ i (s) = s di f i , with f i a holomorphic function of s which is nonzero at 0. We then get where φ i (s, t) is holomorphic in s, t and takes value 1 at (x, x) = (0, 0) and ψ i (s, t) is holomorphic in s, t. Restricting to a curve D ⊂ C × C defined by the equation s = t l for some chosen l ≥ 2, the function (σ ′ i (s) 1 σi(t) ) |D has order l(d i − 1) − d i = (l − 1)d i − l and first nonzero coefficient in its Laurent development equal to d i . These orders are different for distinct d i and the vanishing i σ ′ i (s) 1 σi (t) = 0 is then clearly impossible: indeed, by pole order considerations, for the minimal negative value d of d i , hence minimal value of the numbers (l−1)d i −l, the first nonzero coefficient in the Laurent development of (σ ′ i (s) 1 σi(t) ) |D should be also 0 and it is the same as for the sum i, di=d (σ ′ i (s) 1 σi (t)) |D , which is equal to The claim is proved.
The proof of Proposition 3.2 is thus finished.
3.1 An alternative proof of Theorem 0.4, (iv) As a first application, let us give a second proof of Theorem 0.4, (iv). The general dimension estimate of Theorem 0.4, (i) implies that the locus swept-out by the orbit of |k0 A | is of dimension ≤ k − 1 for any abelian variety A. This locus is clearly naturally defined. Hence by Theorem 1.3, (ii), it is countable for a very general abelian variety of dimension ≥ 2k − 1.

Proof of Theorem 0.4, (ii) and (iii)
Theorem 0.4, (iv) has been proved in Section 2.3. We will now prove the following result by induction on l ∈ {0, . . . , k}: , and A a very general abelian variety of dimension g, any 0-cycle of the form (k − l){0 A } + Z, with Z ∈ A (l) , has countable orbit.
The case l = 0 is Theorem 0.4, (iv) and the case l = k is then Theorem 0.4, (ii). The case l = k − 2 is Theorem 0.4, (iii).
It thus only remains to prove Proposition 4.1. For clarity, let us write-up the detail of the first induction step: Let Σ 1 (A) ⊂ A be the set of points x ∈ A such that the orbit is a countable union of closed algebraic subsets of A. We would like to show that Σ 1 (A) is naturally defined in the sense of Definition 1.1, and there is a small difficulty here: suppose that p : A → B is a morphism of abelian varieties, and let |Z| ⊂ A (k) be a positive dimensional orbit for rational equivalence on A. Then p * (|Z|) ⊂ B (k) could be zero-dimensional. In the case where Z = (k − 1){0 A } + {x}, this prevents a priori proving that Σ 1 (A) satisfies axiom (ii) of Definition 1.1. This problem can be circumvented using the following lemma which has been in fact already used in the proof of Theorem 1.3. Let A → S be a generically complete family of abelian varieties of dimension g. This means that we fixed a polarization type λ and the moduli map S → A g,λ is dominant. Lemma 4.2. Let W ⊂ A be a closed algebraic subset which is flat over S of relative dimension k ′ . Then: (i) For any b ∈ S, any morphism p : (ii) Assume k ′ > 0. For any b ∈ S, any morphism p : , since for very generic b ∈ S, these are the only nonzero Hodge classes on A b . We thus have, using our assumption that dim Statement (ii) is obtained as an application of (i) in the case k ′ = 1. One first reduces to this case by taking complete intersection curves in W b in order to reduce to the case k ′ = 1.
In the following corollary, the orbits for rational equivalence of 0-cycles of X are taken in X l rather than X (l) . Proof. Indeed, by specialization, W b is a positive dimensional orbit for rational equivalence in A l b . Up to shrinking S, we can assume that the restrictions π |pri(W) : pr 1 (W) → S are flat for all i. Our assumption is that for one i, pr i (W) has positive relative dimension over S. Lemma 4.2, (ii), then implies that pr i (p l (W b )) has positive dimension, so that p l (W b ) is a positive dimensional orbit for rational equivalence of 0-cycles of B.
Proof of Proposition 4.1. Let now A be a very general abelian variety. This means that for some generically complete family π : A → S of polarized abelian varieties, A is isomorphic to the fiber over a very general point of S. As A is very general, the locus Σ 1 (A) is the specialization of the corresponding locus Σ 1 (A/S) of A, and more precisely, of the union of its components dominating S. For any fiber A b , let us define the deformable locus Σ 1 (A) def as the one which is obtained by specializing to A b the union of the dominating components of the locus of the relative locus Σ 1 (A/S). For a very general abelian variety A, Σ 1 (A) = Σ 1 (A) def by definition. Corollary 4.3 essentially says that this locus is naturally defined. This is not quite true because the definition of Σ 1 (A) def depends on choosing a family A of deformations of A (that is, a polarization on A). In the axioms of Definition 1.1, we thus should work, not with abelian varieties but with polarized abelian varieties. Axiom (i) should be replaced by its family version, where A → S is locally complete, S ′ ⊂ S is a subvariety, f : A S ′ → B is a morphism of abelian varieties over S ′ , and B → S ′ is locally complete. We leave to the reader proving that Theorem 1.3 extends to this context. Assume now g ≥ 2k − 1. Then Σ 1 (A), hence a fortiori Σ 1 (A) def , is different from A. Indeed, otherwise, for any x ∈ A, (k − 1){0 A } + {x} has positive dimensional orbit, hence taking x = 0 A , we get that k{0 A } has positive dimensional orbit, contradicting Theorem 0.4, (iv). Theorem 1.3, (i) then implies that for g ≥ 2(2k − 1) − 1, Σ 1 (A) def is countable. Hence there are only countably many positive dimensional orbits of the form |(k − 1){0 A } + {x}| and the locus they sweep-out forms by Corollary 4.3 a naturally defined locus in A, which is of dimension ≤ k − 1 by Theorem 3.1. It follows by applying Theorem 1.3, (iii), that for g ≥ 2(2k − 1) + k − 2, this locus itself is countable, that is, all the orbits |(k − 1){0 A } + {x}| are countable for A very general.
The general induction step works exactly in the same way, introducing the locus Σ l (A) ⊂ A of points x l ∈ A such that (k − l)0 A + x 1 + . . . + x l has a positive dimensional orbit for rational equivalence in A for some points x 1 , . . . , x l−1 ∈ A.

Further discussion
It would be nice to improve the estimates in our main theorems. As already mentioned in the introduction, none of them seems to be optimal. Let us introduce a naturally defined locus (or the deformation variant of that notion used in the last section) whose study should lead to a proof of Conjecture 0.2.
Definition 5.1. The locus Z A ⊂ A of positive dimensional normalized orbits of degree k is the set of points x ∈ A such that for some degree k zero-cycle Z = x + Z ′ , with Z ′ effective, one has dim |Z| > 0, σ(Z) = 0.
Here σ : A (k) → A is the sum map. It is constant along orbits under rational equivalence. This locus, or rather its deformation version, is naturally defined. Note also that by definition it is either of positive dimension or empty. The main remaining question is to estimate the dimension of this locus, at least for very general abelian varieties. Conjecture 0.2 would follow from: Conjecture 5.2. If A is a very general abelian variety, the locus Z A ⊂ A of positive dimensional normalized orbits of degree k has dimension ≤ k − 1.
Conjecture 5.2 is true for k = 2. Indeed, in this case the normalization condition reads Z = {x} + {−x} for some x ∈ A. The positive dimensional normalized orbits of degree 2 are thus also positive dimensional orbits of points in the Kummer variety K(A) = A/ ± Id of A. These orbits are rigid because on a surface in K(A) swept-out by a continuous family of such orbits, any holomorphic 2-form on K(A) should vanish while Ω 2 K(A)reg is generated by its sections.
It would be tempting to try to estimate the dimension of the locus of positive dimensional normalized orbits of degree k for any abelian variety. Unfortunately, the following example shows that this locus can be the whole of A: Example 5.3. Let A be an abelian variety which has a degree k − 1 positive dimensional orbit Z ⊂ A (k−1 ). Then for each x ∈ A, {x 1 +x}+. . .+{x k−1 +x}, {x 1 }+. . .+{x k−1 } ∈ Z is a positive dimensional orbit and thus the set {{x 1 +x}+. . .+{x k−1 +x}+{− i x i −(k−1)x} is a positive dimensional normalized orbit of degree k. In this case, the locus of positive dimensional normalized orbits of degree k of A is the whole of A.
Nevertheless, we can observe the following small evidence for Conjecture 5.2: Lemma 5.4. Let O ⊂ A k be a closed irreducible algebraic subset which is a union of positive dimensional normalized orbits of degree k. Let Z ∈ O reg and assume the positive dimensional orbit O Z passing through Z has a tangent vector (u 1 , . . . , u k ) such that the vector space u 1 , . . . , u k ⊂ T A,0A is of dimension k − 1. Then the locus swept-out by the pr i (O) ⊂ A has dimension ≤ k − 1.
Note that k−1 is the maximal possible dimension of the vector space u 1 , . . . , u k because i u i = 0. The example above is a case where the vector space u 1 , . . . , u k has dimension 1.
Applying Theorem 1.3, (ii), Conjecture 5.2 in fact implies the following Conjecture 5.5. If A is a very general abelian variety of dimension ≥ 2k − 1, the locus of positive dimensional normalized orbits of degree k of A is empty.
This is a generalization of Conjecture 0.2, because a k-gonal curvej : C → A, D ∈ W 1 k (C) can always be translated in such a way that σ(j * D) = 0, hence becomes contained in the locus of positive dimensional normalized orbits of degree k of A.
We discussed in this paper only the applications to gonality. The case of higher dimensional linea systems would be also interesting to investigate. In a similar but different vein, the following problem is intriguing: Question 5.6. Let A be a very general abelian variety. Is it true that there is no curve C ⊂ A, whose normalization is a smooth plane curve?
If the answer to the above question is affirmative, then one could get examples of surfaces of general type which are not birational to a normal surface in P 3 . Indeed, take a surface whose Albanese variety is a general abelian variety as above. If S is birational to a normal surface S ′ in P 3 , there are plenty of smooth plane curves in S ′ , which clearly map nontrivially to Alb S, which would be a contradiction.