RESTRICTION OF SECTIONS FOR FAMILIES OF ABELIAN VARIETIES

TOM GRABER AND JASON MICHAEL STARR

Abstract. Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base.

1. Main results sec-intro

The starting point for this article is the following theorem.

Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-projective, thm-GHMS complex variety, let A → B be an Abelian scheme over B (i.e. a family of Abelian varieties over B), and let π : T → B be a torsor for A → B. Then π is a trivial torsor if and only if for every curve C ⊂ B, the restriction T_{C} → C is a trivial torsor for C ×_{B} A → C. Stated more succinctly, the following map of étale cohomology groups is injective

H_{ét}^{1}(B, A) → ∏_{C⊂B} H_{ét}^{1}(C, C ×_{B} A).

The main result of this article is an analogue for H_{ét}^{0}(B, A), together with an extension to positive characteristic.

Let k be an algebraically closed field. To simplify statements, assume k is uncountable. We remind the reader that a subset of a scheme is general, resp. very general, if it contains a dense, open subset, resp. the intersection of a countable collection of dense, open subsets. A property of points in a scheme holds at a general point, resp. a very general point, if the set of points where it holds is general, resp. very general.

thm-main

Theorem 1.2. Let B be a integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let A be an Abelian scheme over B.

(i) For every nontrivial A-torsor T over B, for C a very general triangle curve in B, resp. a very general cubic curve in B, the restriction C ×_{B} T is a nontrivial C ×_{B} A-torsor over C.

(ii) For a very general triangle curve C in B, the map

H_{ét}^{0}(B, A) → H_{ét}^{0}(C, C ×_{B} A)

is a bijection. If char(k) = 0, this also holds for a very general cubic curve C in B.

Both parts also hold with C replaced by a very general planar surface Π in B.

Date: January 18, 2007.

The terms “triangle curve”, “planar surface” and “cubic curve” depend on the choice of a quasi-finite, generically unramified morphism f : B → ℙ_{k}^{b}.

defn-tria Definition 1.3. An f-linear curve is an integral, smooth curve C in B of the form B ×_{ℙ_{k}^{b}} L for some line L ⊂ ℙ_{k}^{b}.

An f-triangle map is a pair (C, h) of

(i) a reducible, nodal k-curve C = C_{1} ∪ C_{2} ∪ C_{3} whose irreducible components each intersect pairwise in a single node, C_{i} ∩ C_{j} = {q_{i,j}},

(ii) together with a k-morphism h : C → B whose restriction to every component, h|_{C_{i}} : C_{i} → B, is a closed immersion to an f-linear curve in B.

An f-triangle curve is the image of an f-triangle map. Equivalently, it is a union of three f-linear curves in B intersecting pairwise in three distinct points (together with other, possibly concurrent, intersections). Stated a third way, it is a curve in B of the form C = B ×_{ℙ_{k}^{b}} T where T is a triangle of lines in ℙ_{k}^{b} (and satisfying the intersection condition above).

An f-cubic curve is an integral, smooth curve C in B of the form B ×_{ℙ_{k}^{b}} E where E is a smooth, plane, cubic curve in ℙ_{k}^{b}.

Finally, an f-planar surface is an integral, normal surface S in B of the form B×_{ℙ_{k}^{b}} Π where Π is a linear 2-plane in ℙ_{k}^{b}.

When there is no likelihood of confusion, we drop the prefix f.

rmk-main

Remark 1.4. There are several remarks.

(i) When k = ℂ, Theorem 1.2 (i) is simply Theorem 1.1. By a straightforward descent argument, part (i) follows from part (ii). Thus, the new content of Theorem 1.2 is in part (ii).

(ii) The group H_{ét}^{1}(B, A) is a torsion group. The subgroup

H_{ét}^{1}(B, A)′ ⊂ H_{ét}^{1}(B, A)

of elements whose order is not divisible by char(k) is countable (it is the whole group if char(k) = 0). Therefore, Theorem 1.2 (i) implies that for C a very general triangle curve, resp. a very general cubic curve, the restriction map

H_{ét}^{1}(B, A)′ → H_{ét}^{1}(C, C ×_{B} A)′

is injective. (iii) There is nothing special about triangle curves, resp. cubic curves. The

proof works for any type of curve which is at least as “complex” as triangle curves, resp. cubic curves.

(iv) The proof of (ii) uses a partial compactification of the pair (B, A) to a Néron model (B̃, Ã). A general triangle curve, resp. cubic curve, C in B̃ is projective. Thus, using the Hilbert scheme for instance, the sections of C ×B̃ Ã over C are the k-points of a naturally defined group k-scheme Σ_{C}. If char(k) = 0, this is a reduced group scheme. But if char(k) > 0, this group scheme is sometimes nonreduced, cf. [MB81, Propostion 3]. This is the reason for the char(k) = 0 hypothesis for cubic curves in part (ii). To extend our argument to positive characteristic, one would need to prove there exists a homomorphism of group schemes over B

Σ_{C} ×_{k} B̃ → Ã

splitting the restriction map. We do not know whether such a splitting always exists.

We also give some examples related to the theorem.

prop-neg

Proposition 1.5. (i) There exist B and A such that for every cubic curve C

in B the map

H_{ét}^{1}(B, A) → H_{ét}^{1}(C, A)

is not surjective. (ii) If char(k) = p is positive, there exist B and A such that for every triangle

curve C, resp. cubic curve C, the map

H_{ét}^{1}(B, A) → H_{ét}^{1}(C, A)

is not injective. More precisely, there exists an A-torsor T over B (depending on C) whose order equals p and whose restriction C ×_{B} T is a trivial C ×_{B} A-torsor over C.

(iii) There exists B and A such that every dense open subset of the parameter space of cubic curves contains a cubic curve C in B for which

H_{ét}^{0}(B, A) → H_{ét}^{0}(C, C ×_{B} A)

is not surjective.

Acknowledgments. We are very grateful to A. J. de Jong who pointed out that a purely inseparable Bertini result as in Corollary 2.2 must exist, who recommended replacing an earlier argument by an application of the Néron extension property, and who explained to us Moret-Bailly’s counterexamples from [MB81].

2. A Bertini theorem sec-Bertini

One consequence of the classical Bertini theorem is that sections of a finite, separable cover of a quasi-projective scheme B of dimension ≥ 2 are detected by the restriction of the cover to a general hyperplane section of B. In this section we recall this and extend the result to covers which may not be separable.

Theorem 2.1. [Jou83, Théorème 4.10, 6.10] Let B be an integral scheme and let thm-B f : B → ℙ_{k}^{N} be a finite type morphism. If f is generically unramified, then for a general hyperplane H, B ×_{ℙ_{k}^{N}} H is geometrically reduced. If dim(f(B)) ≥ 2, then for a general hyperplane H, B ×_{ℙ_{k}^{N}} H is geometrically irreducible.

A straightforward consequence is the separable case of the following result. We also explain the inseparable case.

cor-B

Corollary 2.2. Let B be an integral scheme of dimension ≥ 2 and let f : B → ℙ_{k}^{N} be a generically unramified, finite type morphism. Let g : X → B be a generically finite morphism. For a general hyperplane H ⊂ ℙ_{k}^{N}, the restriction map from the set of rational sections of g over B to the set of rational sections of

g_{H} : X ×_{ℙ_{k}^{N}} H

over B ×_{ℙ_{k}^{N}} H is a bijection.

Proof. If g has a rational section, its restriction to B ×_{ℙ_{k}^{N}} H is a rational section of g_{H} for general H. By Noetherian induction, it suffices to consider the case that X is irreducible. If g is not dominant, the result is clear. If g has a rational section, then g is birational and again the result is clear. Thus assume g is dominant and has no rational section. To prove the corollary, we must prove that also g_{H} has no rational sections for H a general hyperplane.

Because g is dominant, there is an extension of fraction fields

g^{∗} : k(B) → k(X).

Because g is generically finite, this is a finite field extension. Because there is no rational section, it is a nontrivial field extension, i.e., it has degree d > 1. Let E/k(B) be a nontrivial extension intermediate to k(X)/k(B) such that there is no field strictly intermediate between E and k(B). Denote by X_{E} the integral closure of B in E and denote by g_{E} : X_{E} → B the induced morphism. Since every rational section of g_{H} maps to a rational section of (g_{E})_{H}, it suffices to prove (g_{E})_{H} has no rational section for general H. Thus, assume finally that k(X)/k(B) has no intermediate subfields.

Case I. k(X)/k(B) is separable. There are two cases depending on whether or not k(X)/k(B) is separable. If k(X)/k(B) is separable, then the composition g ∘ f : X → ℙ_{k}^{N} is generically unramified. By Theorem 2.1 applied to f ∘ g, for a general H, X ×_{ℙ_{k}^{N}} H is integral. By generic flatness, the morphism

g_{H} : X ×_{ℙ_{k}^{N}} H → B ×_{ℙ_{k}^{N}} H

is generically finite and flat of degree d. Thus k(B ×_{ℙ_{k}^{N}} H) → k(X ×_{ℙ_{k}^{N}} H) is a finite field extension of degree d > 1. Since the extension is not degree 1, it has no splitting. Thus g_{H} has no rational section.

Case II. k(X)/k(B) is not separable. Let p denote the characteristic of k. Because k(X)/k(B) has no intermediate subfields, for any z ∈ k(X)\k(B), k(X) = k(B)[z] and the minimal polynomial of z is of the form z^{p} − u for some u ∈ k(B) \ k(B)^{p}. Thus k(X) = k(B)[z]/⟨z^{p} − u⟩. Replacing B and X by dense open subsets if necessary, assume B and X are affine and normal, assume f is unramified and factors through 𝔸_{k}^{N} ⊂ ℙ_{k}^{N}, assume u ∈ k[B], and assume k[X] = k[B][z]/⟨z^{p} − u⟩.

For general H, by Theorem 2.1, B ×_{𝔸_{k}^{N}} H is integral and normal. In this case X ×_{𝔸_{k}^{N}} H is integral and g_{H} has no rational section precisely if the image of u satisfies

\overline{u} ∉ k[B ×_{𝔸_{k}^{N}} H]^{p}

which is the same as the condition for X ×_{𝔸_{k}^{N}} H to be generically reduced. If X ×_{𝔸_{k}^{N}} H is generically nonreduced, then for a general codimension b − 1, linear subspace of 𝔸_{k}^{N} contained in H, Λ ⊂ H, also X ×_{𝔸_{k}^{N}} Λ is nonreduced. Inside the parameter space for subvarieties H, resp. Λ, the set of H, resp. Λ, such that X ×_{𝔸_{k}^{N}} H, resp. X ×_{𝔸_{k}^{N}} H, is generically reduced is open, [REFERENCE, Jouanolou Theorem 4.10 approx.] Thus, to prove X ×_{𝔸_{k}^{N}} H is generically nonreduced for all H in a dense open subset, it suffices to find a single Λ such that X ×_{𝔸_{k}^{N}} Λ is generically nonreduced. Equivalently, it suffices to find a single Λ such that for every minimal prime 𝔮 of k[B] ⊗_{k[𝔸_{k}^{N}]} k[Λ], the image of u in (k[B] ⊗_{k[𝔸_{k}^{N}]} k[Λ])_{𝔮} is not a p^{th} power. This is what we will actually prove.

Denote b = dim(B). For a general linear projection p : 𝔸_{k}^{N} → 𝔸_{k}^{b}, the induced morphism p ∘ f : B → 𝔸_{k}^{b} is dominant and étale. Because the inverse image in 𝔸_{k}^{N} of a linear subspace of 𝔸_{k}^{b} of codimension b − 1 is again a linear subspace of codimension b − 1, it suffices to prove the result with f replaced by p ∘ f. Thus, without loss of generality, assume N = b = dim(B).

Denote by

f^{∗} : k[y_{1}, …, y_{b}] → k[B]

the pullback map on coordinate rings associated to f. Extend this to a morphism of k[y_{1}, …, y_{b}]-algebras,

ψ : k[x, y_{1}, …, y_{b}] → k[B]

by ψ(x) = u. Denote k[A] = Image(ψ). Because A is between k[y_{1}, …, y_{b}] and k[B]

b = tr. deg_{k}(k(y_{1}, …, y_{b})) ≤ tr. deg_{k}(k(A)) ≤ tr. deg_{k}(k(B)) = dim(B) = b.

Thus dim(k[A]) = tr. deg_{k}(k(A)) = b. So Ker(ψ) is a height 1 prime ideal. Because k[x, y_{1}, …, y_{b}] is a UFD, Ker(ψ) = ⟨g⟩ for a nonconstant, irreducible element g ∈ k[x, y_{1}, …, y_{b}], i.e.,

k[A] = k[x, y_{1}, …, y_{b}]/⟨g⟩.

The k-algebra homomorphism

k[x] → k[B], x ↦ u

determines a field extension k(x) → k(B). Because u is not in k(B)^{p}, this is a separably generated field extension. Since k(A)/k(x) is a subextension, it is also separably generated. By the Jacobian criterion, there exists an integer i = 1, …, b such that the ideal ⟨g, ∂g/∂y_{i}⟩ has height 2. Permuting indices if necessary, assume i = 1.

For a (b − 1)-tuple of elements c = (c_{2}, …, c_{n}) in k, denote k[Λ] = k[y_{1}] and denote by

χ_{c} : k[y_{1}, …, y_{b}] → k[Λ]

the unique k[y_{1}]-algebra homomorphism with χ_{c}(y_{i}) = c_{i} for i = 2, …, b. The closed scheme of 𝔸_{k}^{N} associated to

Ker(χ_{c}) = ⟨y_{2} − c_{2}, …, y_{b} − c_{b}⟩

is a codimension b − 1 linear subspace. For every k[y_{1}, …, y_{b}]-algebra R, denote

R_{Λ} = R ⊗_{k[y_{1},...,y_{b}]} k[Λ].

Because ⟨g, ∂g/∂x_{1}⟩ ∈ k[x, y_{1}, …, y_{b}] has height 2, by standard results of dimension theory (cf. [Har77, Exercise II.3.22]), for a general c, the ideal

⟨g(x, y_{1}, c_{2}, …, c_{b}), ∂g/∂x_{1}(x, y_{1}, c_{2}, …, c_{b})⟩ ⊂ k[x, y_{1}]

also has height 2. Then by the Jacobian criterion, the homomorphism

k[x] ↦ k[A]_{Λ}

is generically smooth. Because k[A] → k[B] is generically étale, for general c also

k[A]_{Λ} → k[B]_{Λ}

is generically étale. Therefore the composition

k[x] ↦ k[B]_{Λ}

is generically smooth, i.e., for every minimal prime 𝔮 of k[B]_{Λ}, the field extension

k(x) → κ(𝔮) := (k[B]_{Λ})_{𝔮}

is separably generated. This precisely means that the image

\overline{u} ∈ (k[B] ×_{k[𝔸_{k}^{n}]} k[Λ])_{𝔮}

is not a p^{th} power for every minimal prime 𝔮. □

3. A triangle lemma

sec-triang

Let B be an integral, normal, quasi-projective k-scheme of dimension b ≥ 2. Let f : B ↪ ℙ_{k}^{b} be a quasi-finite, generically unramified morphism. Let D ⊂ Grass(1, ℙ_{k}^{b}) be the dense, Zariski open subset of the Grassmannian of lines in ℙ_{k}^{b} parametrizing lines L for which B ×_{ℙ_{k}^{b}} L is an integral, smooth curve. This is the parameter space for f-linear curves in B.

Also denote by C_{D} ⊂ B ×_{k} D the family of f-linear curves in B parametrized by D. Denote by ρ_{D} : C_{D} → D and h_{D} : C_{D} → D the obvious projections.

The following “triangle lemma” is essentially the same as in [GHMS05]. We need to use the result in two slightly different contexts. Rather than repeat the proof twice, we find it convenient to introduce the following terminology which encompasses both contexts.

defn-sec Definition 3.1. Let q : Spec κ → B be a point. A family of sections of π over f-linear curves containing q is a pair (τ, σ) of a locally finite type morphism

τ : M → h_{D}^{−1}(q) = C_{D} ×_{B} Spec κ

and a morphism of B-schemes

σ : M ×_{ρ_{D}∘τ,D,ρ_{D}} C_{D} → X,

where M ×_{D} C_{D} is considered as a B-scheme via

M ×_{D} C_{D} ⟶^{pr_{C_{D}}} C_{D} ⟶^{h_{D}} B

(rather than via h_{D} ∘ τ ∘ pr_{M}). The family is locally generically finite if for every quasi-compact open U, the restriction

τ|_{U} : U → h_{D}^{−1}(q)

is generically finite. The family avoids rational curves over q if for a very general curve [C, q] in h_{D}^{−1}(q), for every m in τ^{−1}([C, q]), the point σ_{m}(q) ∈ π^{−1}(q) is contained in no rational curve in π^{−1}(q).

Let M be a family of sections of π over f-linear curves containing q which avoids rational curves. Let C be a triangle such that q_{1,2} = q. A stable section of X_{C} → C is in M over C_{1} and C_{2} if for i = 1, 2 the associated section of X_{C_{i}} → C_{i} is the section s_{m} associated to a point m ∈ r^{−1}([C_{i}, q]). lem-tri Lemma 3.2. [GHMS05, Lemma 3.7] Let B be an integral, normal, quasi-projective

k-scheme of dimension b ≥ 2. Let f : B → ℙ^{b} be a quasi-finite, generically morphism. Let π : X → B be a projective morphism. Let q ∈ B be a smooth point. Let

(τ : M → h_{D}^{−1}(q), σ : M ×_{D} C_{D} → X)

be a locally generically finite family of sections of π over f-linear curves containing q which avoids rational curves over q, cf. Definition 3.1. Let C be a very general triangle curve in B with q_{1,2} = q.

(i) Every stable section of X_{C} → C in M over C_{1} and C_{2} contains a section of X_{C} → C in M over C_{1} and C_{2}.

(ii) For every section s of X_{C} → C in M over C_{1} and C_{2}, there exists a unique irreducible closed subvariety Z ⊂ X containing s(C) such that π|_{Z} : Z → B is an isomorphism over q and such that the restriction of Z over a very general linear curve L in B containing q is a section of X_{L} → L in M.

Proof. We may replace B by the integral closure B^{new} of ℙ^{b} in the fraction field of B, which contains B as a dense, Zariski open subset. And we may replace X by some projective morphism X^{new} → B^{new} whose restriction over B equals X. Thus, without loss of generality, assume that B is projective. Because g is proper, every rational section of X over a normal curve in B extends uniquely to a regular section (by the valuative criterion of properness).

The subset Ω of X consisting of the images of all sections parametrized by M is a countable union of irreducible, locally closed subvarieties Ω_{i}^{pre} of X, i ∈ ℤ. For each i, denote by Ω_{i} the normalization of the closure in X of Ω_{i}^{pre}. Observe that if Z is a subvariety as in (ii), then Z is contained in Ω_{i} for some i. Thus, to prove the uniqueness in (ii), it suffices to consider subvarieties Z contained in some Ω_{i}. And the next claim implies each such Z will equal Ω_{i}.

claim-1

Claim 3.3. For every i ∈ ℤ, the morphism

π_{Ω_{i}} : Ω_{i} → B

is generically finite. For a very general member C_{3} of D, for every i, Ω_{i}|_{C_{3}} → C_{3} admits a section only if Ω_{i} → B is birational. If Ω_{i} → B is birational, then it is an isomorphism over q and its restriction to every very general linear curve L in B is a section of X_{L} → L in M.

First of all, Ω_{i} → B is generically finite if and only if Ω_{i}^{pre} → B is generically finite. Let M_{i} be a quasi-compact locally closed subset of M mapping to Ω_{i}^{pre}. For every q′ ∉ f^{−1}(f(q)), there is a unique line Λ in ℙ_{k}^{b} containing f(q) and f(q′). Thus, for q′ general, there is a unique f-linear curve C containing q and q′. Because τ is locally generically finite, M_{i} → h_{D}^{−1}(q) is generically finite. Thus, for q′ general, there are only finitely many points m in M_{i} mapping to [C, q]. Thus there are only finitely many images s_{m}(q′) ∈ Ω_{i}^{pre} ∩ π^{−1}(q′). Therefore Ω_{i} → B is generically finite.

If Ω_{i} does not dominate B, then a general f-linear curve C is not contained in the image of Ω_{i}. Thus Ω_{i}|_{C_{3}} → C_{3} certainly admits no section. Thus assume Ω_{i} dominates B. By Theorem 2.2, the restriction of Ω_{i} over a general member C of D has a section if and only if Ω_{i} has a rational section over B, i.e., if and only if Ω_{i} → B is birational. Finally, by [Kol96, Theorem VI.1.9.3], Ω_{i} → B is an isomorphism over q since otherwise Ω_{i} ∩ π^{−1}(q) is not uniruled (in fact there are no rational curves in π^{−1}(q) intersecting Ω_{i}^{pre}). This proves Claim 3.3.

claim-2 Claim 3.4. For a very general f-linear curve C_{3}, for every pair r, s of very general points of C_{3}, for every i and j, there exists a section of X_{C_{3}} → C_{3} intersecting both Ω_{i} ∩ π^{−1}(r) and Ω_{j} ∩ π^{−1}(s) only if i = j and Ω_{i} → B is birational. In this case, the image of the section is contained in Ω_{i}.

Because every Ω_{i} → B is generically finite, for a very general point r of B the subsets Ω_{i} ∩ π^{−1}(r) are all finite and disjoint subsets of π^{−1}(r). Because M avoids rational curves in π^{−1}(q), for a very general f-linear curve C_{1} containing q, for a very general point r of C_{1}, every section in M over [C_{1}, q] maps r to a point contained in no rational curve in π^{−1}(r). In other words, Ω_{i} ∩ π^{−1}(r) intersects no rational curve in π^{−1}(r). Thus, by the rigidity lemma, for every i, every section of X_{C_{3}} → C_{3} intersecting Ω_{i} ∩ π^{−1}(r) is rigid, as a section intersecting Ω_{i} ∩ π^{−1}(r). Thus, there are at most countably many of these rigid sections.

For every section and for every j, either the section is contained in Ω_{j} or else the section intersects Ω_{j} in finitely many closed points. Since there are countably many indices i and j, there is a countable set S of closed points of X such that for every section of X_{C_{3}} → C_{3} intersecting Ω_{i} ∩ π^{−1}(r) and not contained in Ω_{j}, the intersection of the section and Ω_{j} is contained in S. The image π(S) is a countable set of closed points of C_{3}. Choosing s in the complement of this set, every section of X_{C_{3}} → C_{3} intersecting Ω_{i} ∩ π^{−1}(r) and Ω_{j} ∩ π^{−1}(s) is contained in Ω_{j}. In this case, firstly, it intersects Ω_{j} ∩π^{−1}(r). Since Ω_{i} ∩π^{−1}(r) and Ω_{j} ∩π^{−1}(r) are disjoint for i ≠ j, it follows that i equals j. Secondly, since the section of X_{C_{3}} → C_{3} is actually a section of Ω_{j} = Ω_{i} over C_{3}, Claim 3.3 implies that Ω_{i} → B is birational. Finally, the image of the section is contained in Ω_{j}, which equals Ω_{i}. This proves Claim 3.4.

Let C_{1}, resp. C_{2}, be a general f-linear curve containing both q and r, resp. both q and s. Form the f-triangle map C = C_{0} ∪ C_{1} ∪ C_{2} with vertices q, r and s.

claim-3 Claim 3.5. Every stable section of X_{C} → C in M over C_{1} and C_{2} is regular over q, r and s, thus contains a section of X_{C} → C in M over C_{1} and C_{2}. Every section of X_{C} → C in M over C_{1} and C_{2} is contained in an irreducible closed subset Ω_{i} such that Ω_{i} → B is birational.

Given a stable section of X_{C} → C, remove all vertical, rational components not contained in π^{−1}(q), π^{−1}(r) or π^{−1}(s). The remaining components give a stable section of X_{C} → C whose vertical, rational components, if any, occur over q, r and s. By hypothesis, the images of the sections over C_{1}, resp. C_{2}, intersect no rational curves in π^{−1}(q), and then, because r and s are very general, they also intersect no rational curve in π^{−1}(r), resp. in π^{−1}(s). Thus the stable section has no vertical, rational components over q, r and s, i.e., the stable section is a section. By hypothesis, the image of this section over C_{1}, resp. over C_{2}, is contained in a subset Ω_{i}, resp. Ω_{j}. The restriction of this section to C_{3} is a section of X_{C_{3}} → C_{3} intersecting both Ω_{i} ∩ π^{−1}(r) and Ω_{j} ∩ π^{−1}(s). By Claim 3.4, i = j, Ω_{i} → B is birational, and the section over C_{3} is contained in Ω_{i}. Thus the entire section is contained in Ω_{i}. This proves Claim 3.5, and thus the lemma. □

We will use Lemma 3.2 via the following two corollaries.

cor-tri1

Corollary 3.6. Let B be an integral, normal, quasi-projective k-scheme of dimension b ≥ 2. Let f : B → ℙ^{b} be a generically unramified, quasi-finite morphism. Let π : X → B be a projective morphism. Let q ∈ B be a smooth point and let r ∈ π^{−1}(q) be a point contained in no rational curve in π^{−1}(q). Let C be a very general f-triangle curve in B with q_{1,2} = q.

(i) Every stable section of X_{C} → C mapping q_{1,2} to r contains a section of X_{C} → C mapping q_{1,2} to r.

(ii) For every section of X_{C} → C mapping q_{1,2} to r, there exists a unique irreducible closed subvariety Z ⊂ X containing the image of the section and such that π|_{Z} : Z → B is an isomorphism over q.

Proof. Let (τ : M → h_{D}^{−1}(q), σ) be the family of all sections over f-linear curves C containing q which map q to r. Since r is contained in no rational curve in π^{−1}(q), this family avoids rational curves over q. And by the rigidity lemma, since there is no rational curve in π^{−1}(q) containing r, every section over C mapping q to r is rigid (as a section mapping q to r). Thus the family is locally generically finite. So the family satisfies the hypotheses of Lemma 3.2, which gives the corollary. □

cor-tri2

Corollary 3.7. Let B be an integral, normal, quasi-projective k-scheme of dimension b ≥ 2. Let f : B → ℙ^{b} be a generically unramified, quasi-finite morphism. Let π : X → B be a projective morphism. Let q ∈ B be a smooth point. Assume that for a very general f-linear curve C′ containing q, every section of X_{C′} → C′ is rigid and maps q to a point contained in no rational curve in π^{−1}(q). Let C be a very general triangle curve in B with q_{1,2} = q.

(i) Every stable section of X_{C} → C contains a section of X_{C} → C.

(ii) For every section of X_{C} → C, there exists a unique irreducible closed subvariety Z ⊂ X containing the image of the section and such that π|_{Z} : Z → B is birational.

Proof. As in the proof of Lemma 3.2, we may reduce to the case where B is projective. Then, by the existence of the Chow variety, etc., there exists a universal family (τ_{U} : U → h_{D}^{−1}(q), σ_{U}) of sections over f-linear curves containing q. Moreover, U has countably many connected components, each of which is quasi-projective. Denote by M the closed subvariety of U which is the union of all irreducible components U_{i} such that τ_{U}(U_{i}) is dense in h_{D}^{−1}(q). In particular, for a very general D-curve C′ containing q, every irreducible component of M intersects τ^{−1}([C′, q]). By hypothesis, every section of X_{C′} → C′ is rigid and avoids rational curves in π^{−1}(q). Therefore the family

(τ_{U}|_{M} : M → h_{D}^{−1}(q), σ_{U}|_{M})

is locally generically finite and avoids rational curves over q. So the family satisfies the hypotheses of Lemma 3.2. By definition of M, for a very general triangle C with q_{1,2} = q, since both C_{1} and C_{2} are very general f-linear curves containing q, every section of X_{C_{1}} → C_{1}, resp. of X_{C_{2}} → C_{2}, is in M. Thus Lemma 3.2 gives the corollary. □

4. Elementary reductions sec-red

In this section, we prove some elementary lemmas useful in the proof of Theorem 1.2. The next definition simplifies the statements of the lemmas. As always, let k be an algebraically closed field.

defn-P Definition 4.1. Let B be an integral, smooth, quasi-projective scheme over an algebraically closed field k. Let f : B → ℙ_{k}^{b} be a quasi-finite, generically unramified morphism. Let A be an Abelian scheme over B. We say property P(B, f, A) holds for triangle curves, resp. for cubic curves, if

H_{ét}^{0}(B, A) → H_{ét}^{0}(C, C ×_{B} A)

is a bijection for a very general f-triangle curve, resp. for a very general f-cubic curve.

lem-red0 Lemma 4.2. For every dense, open immersion U → B, P(U, f|_{U}, U ×_{B} A) holds if and only if P(B, f, A) holds.

Proof. (i) By Weil’s extension theorem, cf. [BLR90, Theorem 1, p.109] and [Kol96, Theorem VI.1.9.3], sections of U ×_{B} A over U extend uniquely to sections of A over B. Assuming C is smooth at every point of B \ U, also sections of U ×_{B} C ×_{B} A over U ×_{B} C extend uniquely to sections of C ×_{B} A over C. □

lem-red1 Lemma 4.3. Let B′ be an integral scheme and let h : B′ → B be a finite, étale, Galois morphism with Galois group G. Denote f′ = f ∘ h and A′ = A ×_{B} B′. If P(B′, f′, A′) holds, then P(B, f, A) holds.

Proof. Let C be a very general triangle curve in B, resp. cubic curve in B. The restriction map

H_{ét}^{0}(B′, A′) → H_{ét}^{0}(B′ ×_{B} C, (B′ ×_{B} C) ×_{B′} A′)

is a homomorphism of Galois modules. By étale descent, the induced map of Galois invariants is canonically isomorphic to the restriction map

H_{ét}^{0}(B, A) → H_{ét}^{0}(C, C ×_{B} A).

Thus if the first map is an isomorphism, also the second map is an isomorphism. □

The next reduction uses part of the theory of Chow’s K/k-trace. We review the part that we need.

defn-Chow Definition 4.4. Let B be an integral, smooth, quasi-projective scheme over an algebraically closed field k. Let A be an Abelian scheme over B. A Chow B/ktrace of A is an initial pair (Tr_{B/k}(A), u) of an Abelian k-scheme Tr_{B/k}(A) and a morphism of Abelian schemes over B,

u : B ×_{k} Tr_{B/k}(A) → A,

i.e., for every pair (A_{0}, v) of an Abelian k-scheme A_{0} and a morphism of Abelian schemes over B,

v : B ×_{k} A_{0} → A,

there exists a unique morphism of Abelian k-schemes

w : A_{0} → Tr_{B/k}(A)

such that v = u ∘ (Id_{B} × w).

The basic result concerning the Chow trace is the following.

Theorem 4.5. [Lan83, §VIII.3], [Con06] thm-Chow

(i) For every integral, smooth, quasi-projective k-scheme B and every Abelian scheme A over B, there exists a Chow B/k-trace of A.

(ii) Let E be an integral, smooth, quasi-projective k-scheme and let E → B be a dominant k-morphism such that k(B) is separably closed in k(E). The induced morphism of Abelian k-schemes

w : Tr_{B/k}(A) → Tr_{E/k}(E ×_{B} A)

is an isomorphism.

rmk-Chow

Remark 4.6. A dense open immersion U → B satisfies the condition in (ii). Taking the limit over all dense open subsets of B, the Chow trace depends only on the field extension k(B)/k and the Abelian k(B)-scheme A ⊗_{O_{B}} k(B). Usually the Chow trace is formulated for pairs (K/k, A_{K}) of a field extension K/k and an Abelian K-scheme. It is more useful for us to formulate it as above.

The Chow trace is closely related to the property of isotriviality of an Abelian scheme.

defn-noniso

Definition 4.7. Let B be an integral, smooth, quasi-projective k-scheme. An Abelian scheme A over B is non-isotrivial, resp. strongly non-isotrivial, if for the geometric generic point of B ×_{k} B,

(p, q) : Spec κ → B ×_{k} B

the Abelian κ-schemes

A_{p} := Spec κ ×_{p,B} A, A_{q} := Spec κ ×_{q,B} A

are not isomorphic, resp. there is no nonzero morphism of Abelian κ-schemes

A_{p} → A_{q}.

lem-Chow2

Lemma 4.8. Let B be an integral, smooth, quasi-projective k-scheme. An Abelian scheme A over B is strongly non-isotrivial if and only if Tr_{k(B)^{sep}/k}(A⊗_{O_{B}} k(B)^{sep}) is zero.

Proof. First of all, dual to the morphism of Abelian varieties

u : Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) ⊗_{k} k(B)^{sep} → A_{k(B)^{sep}},

there is a morphism

v : A_{k(B)^{sep}} → Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) ⊗_{k} k(B)^{sep}

such that v∘u is multiplication by some positive integer N. Thus, if Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) is nonzero, the composition

A_{p} ⟶^{p^{∗}v} Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) ⊗_{k} κ ⟶^{q^{∗}u} A_{q}

is a nonzero homomorphism.

Denote by e_{p}, e_{q} : k(B)^{sep} → κ the field monomorphism associated to projection p, resp. q. Then, conversely, every nonzero homomorphism

A_{p} → A_{q} factors through

Tr_{e_{p}}(A_{q}) ⊗_{k(B)^{sep},e_{p}} κ → A_{q}.

By Theorem 4.5, this second map is precisely

e_{q}^{∗}u : Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) ⊗_{k} κ → A_{q}.

So if there is a nonzero homomorphism A_{p} → A_{q} (or symmetrically A_{q} → A_{p}), then Tr_{k(B)^{sep}/k}(A_{k(B)^{sep}}) is nonzero. □

lem-rigid Lemma 4.9. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 1. Let Q be an Abelian scheme over B. If Tr_{B/k}(Q) = 0, then there are at most countably many sections of Q over B.

Proof. Let \overline{B} be a normal, projective scheme containing B as a dense, open subscheme. Let \overline{Q} → \overline{B} be a projective morphism whose restriction over B equals Q. There is a Chow variety parametrizing cycles in \overline{Q}. The Chow variety has countably many irreducible components, for the usual reason (countably many Hilbert polynomials, etc.). The claim is that every cycle Z_{0} ⊂ \overline{Q} for which Z → B is birational gives an isolated point of the Chow variety.

Towards this claim, let T be an irreducible, quasi-projective curve and let Z ⊂ \overline{Q} ×_{k} T be a cycle such that Z → \overline{B} ×_{k} T is birational. Replacing T by a dense open subset if necessary, assume T is smooth. Then Z ∩ B ×_{k} T is the graph of a B-rational transformation,

F : B ×_{k} T −−→ Q.

By [Kol96, Theorem VI.1.9.3], this rational transformation is regular. Fix a point t_{0} ∈ T and denote

G : B ×_{k} T → Q, F^{new}(q, t) = F (q, t) − F (q, t_{0}).

Denote by

e : T → Alb(T)

the Albanese morphism sending t_{0} to 0. Then, by the universal property of the Albanese, the morphism G factors through

Id_{B} × e : B ×_{k} T → B ×_{k} Alb(T).

Because Tr_{B/k}(Q) is trivial, by the universal property of the trace, the induced homomorphism of Abelian schemes over B,

G̃ : B ×_{k} Alb(T) → Q

is the zero homomorphism. Thus G is the zero map, i.e.,

F (q, t) = F (q, t_{0})

for every (q, t) ∈ B ×_{k} T. Thus Z ∩ (B ×_{k} T) is independent of t ∈ T. Since Z is the closure of Z ∩ (B ×_{k} T), the same holds for Z, i.e., Z = Z_{0} ×_{k} T for a cycle Z_{0} ⊂ \overline{Q}. Therefore every k-morphism from an irreducible, quasi-projective curve T to the Chow variety parametrizing rational transformations Z is constant. In other words, every cycle Z ⊂ \overline{Q} with Z → \overline{B} birational gives an isolated point of the Chow variety. □ lem-trick

Lemma 4.10. Let B be an integral, smooth, quasi-projective k-scheme and let A be an Abelian scheme over B. Let A_{0} be an Abelian k-variety and let v_{0} : A → B×_{k} A_{0} be a surjective homomorphism of Abelian schemes. Let C be a curve in B and let q be a closed point in C. To prove that every section s of C ×_{B} A over C is the restriction of a unique section S of A over B, it is necessary and sufficient to prove this for sections s with v_{0}(s(q)) = (0, q).

Proof. Necessity is obvious. To prove sufficiency, assume the result holds for sections with v_{0}(s(q)) = (0, q). By existence of polarizations, dual homomorphisms of Abelian schemes, etc., there exists a homomorphism of Abelian schemes

u_{0} : B ×_{k} A_{0} → A

such that v_{0} ∘ u_{0} is multiplication by N for some positive integer N. Let s be a section of C ×_{B} A, not necessarily satisfying the condition. Let a_{0} ∈ A_{0} be the unique closed point with v_{0}(s(q)) = (a_{0}, q). There exists an element a_{1} ∈ A_{0} such that a_{0} = N · a_{1}. Denote by ã_{1} : B → B ×_{k} A_{0} the section whose projection to A_{0} is the constant morphism with value a_{1}. Then s^{new} = s − u_{0} ∘ ã_{1}|_{C} is a section of C ×_{B} A over C such that v_{0}(s^{new}(q)) = (0, q). By hypothesis, there exists a unique section S^{new} of A over B whose restriction to C equals s^{new}. Then S = S^{new}+u_{0}∘ã_{1} is the unique section of A over B whose restriction to C is s. □

cor-Chow2

Corollary 4.11. Let B be an integral, smooth, quasi-projective k-scheme and let A be an Abelian scheme over B. There exists a dense open subscheme U ⊂ B, a finite, étale, Galois morphism U′ → U, an Abelian k-scheme A_{0}, an Abelian scheme Q over U′ and an isogeny of Abelian schemes over U′,

u = u_{0} ⊕ u_{Q} : (U′ ×_{k} A_{0}) ×_{U′} Q → U′ ×_{B} A,

with the following properties.

(i) The pair (A_{0}, u_{0}) is a Chow U′/k-trace of U′ ×_{B} A.

(ii) For every dense open subset V of U and every finite, étale, Galois morphism V′ → V ×_{U} U′, the induced map of Abelian k-schemes

A_{0} = Tr_{U′/k}(U′ ×_{B} A) → Tr_{V′/k}(V′ ×_{B} A)

is an isomorphism. (iii) The Abelian scheme Q is strongly non-isotrivial. (iv) The quotient of U′ ×_{B} A by U′ ×_{k} A_{0} is isomorphic to Q in such a way that

the composition

Q ⟶^{u_{Q}} U′ ×_{B} A ⟶^{quotient} Q

is the multiplication by N isogeny for some positive integer N.

lem-Chow3

Lemma 4.12. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let f : B → ℙ_{k}^{b} be a quasi-finite, generically unramified morphism. Let A_{0} be an Abelian k-scheme, let A and Q be Abelian schemes over B, and let

u = u_{0} ⊕ u_{Q} : (B ×_{k} A_{0}) ×_{B} Q → A

be an isogeny of Abelian schemes over B. Assuming Tr_{B/k}(Q) is zero, P(B, f, A) holds for triangle curves, resp. cubic curves, if and only if both P(B, f, B ×_{k} A_{0}) and P(B, f, Q) hold for triangle curves, resp. cubic curves.

Proof. Denote

A′ := (B ×_{k} A_{0}) ×_{B} Q.

Then P(B, f, A′) holds if and only if both P(B, f, B ×_{k} A_{0}) and P(B, f, Q) hold. There exists an isogeny

v = (v_{0}, v_{Q}) : A → A′ = (B ×_{k} A_{0}) ×_{B} Q

such that both u ∘ v and v ∘ u are multiplication by a positive integer N.

Let q be a very general k-point of B. Let C be a very general triangle curve, resp. cubic curve, containing q. Property P(B, f, A) says that every section s of C ×_{B} A over C is the restriction of a unique section S of A over B. Let S, resp. s, be a section of A over B, resp. of C ×_{B} A over C. By Lemma 4.10, it suffices to prove this for sections s with v_{0}(s(q)) = (0, q).

Consider the collection of all sections S′ of A′ over B with pr_{A_{0}}(S′(q)) = 0. This is the same as the collection of pairs (S′_{0}, S′_{Q}) of a section S′_{Q} of Q over B and a morphism S′_{0} : B → A_{0} such that S′_{0}(q) = 0. By the rigidity lemma, every morphism S′_{0} as above is rigid, and thus there are only countably many such morphisms. By Lemma 4.9, also there are only countably many sections S′_{Q} of Q over B. Altogether, there are only countably many sections S′ of A′ over B.

For each section S′ of A′ over B, consider the scheme

X = B ×_{S′,A′,v} A.

The projection X → A is a finite, flat morphism; in fact a torsor for Ker(v). By Corollary 2.2, for C a general curve, the restriction map from the set of sections of X over B to the set of sections of C ×_{B} X over C is a bijection. To spell this out a bit, inductively applying Corollary 2.2 implies the result for a general f-linear curve C_{3} in B. Then the result also holds for a triangle curve with q_{1,2} = q whose component C_{3} is a general f-linear curve as above. Finally the result follows for cubic curves since cubic curves specialize to triangle curves.

Said differently, the conclusion is that the restriction map from the

{S : B → A|S a section, v ∘ S = S′} → {s : C → C ×_{B} A|s a section, v ∘ s = S′|_{C}}

is a bijection if C is general. By the argument above, there are at most countably many sections S′ with pr_{A_{0}}(S′(q)) = 0. Therefore, for C very general, for every section S′ with pr_{A_{0}}(S′(q)) = 0, the restriction map is a bijection.

Let s be a section of C ×_{B} A over C with v_{0}(s(q)) = 0. Assuming P(B, f, A′), there exists a unique section S′ of A′ over B such that S′|_{C} = v ∘ s. In particular, for every section S of A over B with S|_{C} = s (assuming such exist), the composition v∘S must equal S′. Since v_{0}(s(q)) = 0, also pr_{A_{0}}(S′(q)) = 0. By the last paragraph there exists a unique section S of A over B such that S|_{C} = s. Thus P(B, f, A) follows from P(B, f, A′).

The argument in the reverse direction is similar. Again by Lemma 4.10, to prove P(B, f, A′), it suffices to prove every section s′ of C ×_{B} A′ over C such that pr_{A_{0}}(s′(q)) = 0 is the restriction of a unique section S′ of A′ over B. Assuming P(B, f, A), there exists a unique section S of A over B such that S|_{C} = u ∘ s′. In particular, v_{0}(S(q)) = 0. By the argument above, there are countably many sections S of A over B with v_{0}(S(q)) = 0. Thus, assuming C is very general, the argument above proves the restriction map from sections S′ of A′ over B for which u ∘ S′ = S to sections s′ of C ×_{B} A′ over C for which u ∘ s′ = S|_{C} is a bijection. Therefore there exists a unique sections S′ such that S′|_{C} = s′. □

5. Proof of the theorem sec-proof

We are ready to prove the theorem. We begin by proving the theorem in the special cases of a constant family of Abelian varieties and of a strongly non-isotrivial family of Abelian varieties.

lem-const

Lemma 5.1. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let A_{0} be an Abelian k-variety. For a very general triangle curve C in B, the map

H_{ét}^{0}(B, B ×_{k} A_{0}) → H_{ét}^{0}(C, C ×_{k} A_{0})

is a bijection.

Proof. Let q be a very general k-point of B. Let C be a very general triangle curve in B such that q_{1,2} = q. Set r = (q, 0) in B ×_{k} A_{0}. By Corollary 3.6, for every section s of C ×_{k} A_{0} over C with s(q) = r, there exists a unique section S of B ×_{k} A_{0} over B with S|_{C} = s. By Lemma 4.10, this holds for all sections s, whether or not s(q) = r. □

lem-const2

Lemma 5.2. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let A_{0} be an Abelian k-variety. For a very general cubic curve C in B, the map

H_{ét}^{0}(B, B ×_{k} A_{0}) → H_{ét}^{0}(C, C ×_{k} A_{0})

is a bijection.

Proof. Cubic curves are deformations of triangle curves. The proof of the result for cubic curves uses Lemma 5.1 and a deformation-specialization argument. There are two parts of the argument. First, sections over cubic curves specialize to sections over triangle curves. Second, sections over triangle curves deform uniquely.

Let \overline{B} be the integral closure of ℙ_{k}^{b} in the fraction field of B. Thus \overline{B} is an integral, normal, projective k-scheme containing B as a dense, open subscheme. Replace B by the smooth locus of \overline{B}. In particular, \overline{B} \B has codimension ≥ 2 in \overline{B}. Therefore every general triangle curve in \overline{B}, resp. every general cubic curve in \overline{B}, is actually contained in B.

Let q be a very general k-point of B. Let r be the point (q, 0) ∈ B ×_{k} A_{0}. Let C be a very general triangle curve in B containing q. Let (T, t_{0}) be an irreducible pointed variety and let C ⊂ T ×B be a flat family of curves in B containing T ×{q} and with C_{t_{0}} = C. By the valuative criterion for stable maps, stable sections s_{t} of C_{t} ×_{k} A_{0} over C_{t} containing r specialize to stable sections s of C ×_{k} A_{0} over C containing r as t specializes to t_{0}. Since there are no rational curves in A_{0}, stable sections are always sections. This observation is an elementary special case of [Kol96, Theorem VI.1.9.3].

Next we claim that s_{t} specializes uniquely to a section over C. This is the same as the claim that the relative Hom scheme over T, Hom_{T} (C, C ×_{k} A_{0}, q ↦ r) is unramified over T, cf. [Kol96, Definition II.1.4]. To see this, observe that the Zariski tangent space of the fiber of the relative Hom scheme over T at a section

(Id, s) : C_{t} → C_{t} ×_{k} A_{0}

equals

H^{0}(C_{t}, s^{∗}T_{A_{0}} ⊗ I)

where T_{A_{0}} is the tangent bundle of A_{0} and I is the ideal sheaf of q in C_{t}, cf. [Kol96, Theorem II.1.7]. Because A_{0} is an Abelian variety, T_{A_{0}} = T_{A_{0},0} ⊗_{k} O_{A_{0}}. Thus the Zariski tangent space of the fiber is

T_{A_{0},0} ⊗_{k} H^{0}(C_{t}, I) = T_{A_{0},0} ⊗_{k} 0 = 0,

i.e., the relative Hom scheme is unramified over T.

Now let t be a very general point of T and let s_{t} be a section of C_{t} ×_{k} A_{0} over C_{t} sending q to r. By the last paragraph, s_{t} specializes uniquely to a section s of C ×_{k} A_{0} over C sending q to r. By Lemma 5.1, each such section s is the restriction of a unique section S over all of B. Since S|_{C_{t}} is also a section over C_{t} specializing to s, uniqueness implies s_{t} = S|_{C_{t}}. Therefore, for t a very general point of T, every section of C_{t} ×_{k} A_{0} over C_{t} sending q to r is the restriction of a unique section of B ×_{k} A_{0} over B. Finally, Lemma 4.10 implies the result for all sections s_{t} of C_{t} ×_{k} A_{0} over C_{t}, whether or not they map q to r. □

lem-noniso Lemma 5.3. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let Q be an Abelian scheme over B. Assume that Q is strongly non-isotrivial, cf. Definition 4.7. For a very general triangle curve C in B, the map

H_{ét}^{0}(B, Q) → H_{ét}^{0}(C, C ×_{B} Q)

is a bijection.

Proof. Because Q is strongly non-isotrivial, for a very general pair of closed points p, q ∈ B, there is only the zero homomorphism of Abelian varieties Q_{p} → Q_{q}. Since a very general pair of closed points is contained in an f-linear curve C′, the same conclusion holds for the Abelian scheme C′ ×_{B} Q over C′. Thus C′ ×_{B} Q is strongly non-isotrivial. By Lemma 4.8, Tr_{C′/k}(C′ ×_{B} Q) = 0. Thus, by Lemma 4.9, every section of C′ ×_{B} Q over Q is rigid and there are only countably many such sections. By Corollary 3.7, for a very general triangle curve C in B, every section of C ×_{B} Q over C is contained in a unique “rational section” of Q over B. By [Kol96, Theorem VI.1.9.3], these rational sections are all regular. Therefore

H_{ét}^{0}(B, Q) → H_{ét}^{0}(C, C ×_{B} Q)

is a bijection. □

The most involved step in the proof of Theorem 1.2 is the analogue of Lemma 5.3 for very general cubic curves C in B. As with Lemma 5.2, the argument is a deformation-specialization argument. However, both steps are more involved than in Lemma 5.2. The first step proves that sections over general cubic curves specialize to sections over triangle curves. This no longer follows from Abhyankar’s lemma, [Kol96, Theorem VI.1.9.3], as the codimension 1 fibers of a compactification of Q are typically uniruled. Instead we use the Néron extension property. We recall the definition in the context which we use it.

defn-Neron Definition 5.4. Let T be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 1. A smooth, finite type, separated morphism X → T has the Néron extension property if for every triple (Y → T, U, s_{U}) of

(i) a smooth morphism Y → T, (ii) a dense, open subset U ⊂ T, (iii) and a T -morphism s_{U} : Y ×_{T} U → X,

there exists a pair (V, s_{V}) of (i) an open subset V ⊂ T containing U and all codimension 1 points of T (ii) and a T -morphism s_{V} : Y ×_{T} V → X whose restriction to Y ×_{T} U equals

s_{U}. If X → T has the Néron extension property, then it is called a Néron model, or a Néron model of its generic fiber.

Let X_{1} → T and X_{2} → T be Néron models. Let U ⊂ T be a dense, open subset. And let U ×_{T} X_{1} ≅ U ×_{T} X_{2} be a T -isomorphism. By the Néron extension property, there exists an open subset V ⊂ T containing U and all codimension 1 points and a T -isomorphism V ×_{T} X_{1} ≅ V ×_{T} X_{2} extending the isomorphism over U. Thus, Néron models are unique in codimension 1.

The basic result about existence of Néron models is the following.

Lemma 5.5. [BLR90, Theorem 3, p.19] Let W be an integral, smooth, quasi- lem-neron projective k-scheme of dimension b ≥ 1. Let B be a dense open subset of W and let A be an Abelian scheme over B. There exists an open subset B̃ of W containing B and all codimension 1 points and a Néron model Ã over B̃ whose restriction over B equals A.

Let \overline{B} be an integral, normal, projective k-scheme containing B as a dense, open subscheme. Let W be the smooth locus of \overline{B}. By Lemma 5.5, there exists an open subset B̃ of W containing B and all codimension 1 points and a Néron model Q̃ over B̃ whose restriction over B equals Q. By the uniqueness part of Néron models, or by the construction, the Néron model Q̃ can be chosen to be a smooth group scheme over B̃.

The parameter space for both cubic and triangle curves is somewhat involved and warrants a bit of notation.

notat-param

Notation 5.6. Denote by P_{1^{st}} the ℙ^{9}-bundle over the Grassmannian Grass(2, ℙ_{k}^{b}) parametrizing pairs ([Λ], [E]) of a linear 2-plane Λ ⊂ ℙ_{k}^{b}, i.e., [Λ] ∈ Grass(2, ℙ_{k}^{b}), together with a degree 3 Cartier divisor E ⊂ Λ. Denote by Δ_{1^{st}} ⊂ P_{1^{st}} the closed subscheme parametrizing pairs such that E is singular. Denote by Γ_{1^{st}} ⊂ P_{1^{st}} the locally closed subscheme parametrizing pairs such that E is a union of 3 non-concurrent lines in Λ. Denote by P_{2^{nd}} ⊂ P_{1^{st}} the maximal open subscheme parametrizing pairs such that

(i) \overline{B} ×_{ℙ_{k}^{b}} E is contained in B̃,

(ii) and the morphism \overline{B} ×_{ℙ_{k}^{b}} E → E is a finite, separable morphism of atworst-nodal curves which is étale over a neighborhood of every node of E and maps every node of \overline{B} ×_{ℙ_{k}^{b}} E to a node of E.

Denote Γ_{2^{nd}} = Γ_{1^{st}} ∩ P_{2^{nd}} and Δ_{2^{nd}} = Δ_{1^{st}} ∩ P_{2^{nd}}. Denote by P_{3^{rd}} → P_{2^{nd}} the blowing up of P_{2^{nd}} along Γ_{2^{nd}}. Denote the exceptional divisor by by Γ_{3^{rd}} and denote the strict transform of Δ_{2^{nd}} by Δ_{3^{rd}}.

Denote by P the open complement of Δ_{3^{rd}} in P_{3^{rd}} and denote Γ = Γ_{3^{rd}} ∩ P. The scheme P is an integral, smooth parameter space for projective cubic and triangle curves in B̃ and the locus Γ parametrizing triangle curves is a integral, smooth Cartier divisor in P. Denote by C ⊂ B̃ ×_{k} P the associated family of projective cubic and triangle curves in V parametrized by P.

Finally, denote by

Σ ⊂ Hom_{P} (C, C ×B̃ Q̃)

the locally closed subscheme of the relative Hom scheme over P parametrizing C-morphisms.

Because C ×B̃ B̃ is a group scheme over C, Σ is a group scheme over P. Because the restriction of Q̃ over a general curve in B̃ is strongly non-isotrivial, Lemma 4.9 implies that Σ → P is “locally generically finite”, i.e., every quasi-compact open subset of Σ is finite over a dense, open subset of P.

Let Σ_{i} be an irreducible component of Σ and let

F : C ×_{P} Σ_{i} → C ×B̃ Q̃

be the restriction of the universal C-morphism. Denote by

y_{Σ} : C ×_{P} Σ_{i} → B̃

the composition of F with projection to B̃. Because F is a C-morphism, F = (Id_{C}, G) where

G : C ×_{P} Σ_{i} → Q̃

is a B̃-morphism, i.e., prB̃ ∘ G = y_{Σ}.

lem-deform Lemma 5.7. If char(k) = 0 and if Σ_{i} dominates P, then the image of Σ_{i} in P intersects Γ in a dense subset of Γ.

Proof. Let Σ_{i} ↪ \overline{Σ}_{i} be a dense, open immersion of P -schemes with \overline{Σ}_{i} a normal, projective P -scheme. The B-morphism G gives a rational transformation of Bschemes

G : C ×_{P} \overline{Σ}_{i} −−→ Q̃.

By the universal property of Σ, Σ_{i} ⊂ \overline{Σ}_{i} is the maximal open subscheme over which G is regular. Thus to prove the lemma, it is equivalent to prove that G is regular over an open subscheme of \overline{Σ}_{i} whose image in P intersects Γ in a dense subset.

Denote by Σ̃_{i} the smooth locus of \overline{Σ}_{i}.

claim-nonisoA Claim 5.8. There exists an irreducible component D ⊂ Σ̃_{i} ×_{P} Γ such that D → Γ is dominant and generically finite. And D is a Cartier divisor in Σ̃_{i}.

Because \overline{Σ}_{i} → P is dominant and projective, it is surjective. Thus also \overline{Σ}_{i}×_{P} Γ → Γ is surjective. Thus some generic point of \overline{Σ}_{i} ×_{P} Γ dominates the generic point of Γ. Because Γ is a Cartier divisor in P, \overline{Σ}_{i} ×_{P} Γ is a Cartier divisor in \overline{Σ}_{i}. So each of its generic points is codimension 1 in \overline{Σ}_{i}. Because \overline{Σ}_{i} is normal, Σ̃_{i} contains every codimension 1 point, thus it contains every generic point of \overline{Σ}_{i} ×_{P} Γ. In particular, it contains a generic point dominating the generic point of Γ. Denote the closure of this generic point in Σ̃_{i} by D and give it the reduced, induced scheme structure. Then D dominates Γ. Because Σ̃_{i} → P is generically finite and Σ̃_{i} is integral, the morphism is quasi-finite at every codimension 1 point of Σ̃_{i}. Thus D → Γ is generically finite. Finally D is a Cartier divisor in Σ̃_{i} since Σ̃_{i} is smooth and D is an integral, codimension 1, closed subvariety. This proves Claim 5.8.

Denote by

y_{Γ} : C ×_{P} Γ ⟶^{pr_{C}} C ⟶^{prB̃} B̃

the natural morphism.

claim-nonisoB

Claim 5.9. The morphism y_{Γ} is smooth.

By chasing diagrams, y_{Γ} is the composition of an open immersion and the base change by B̃ → ℙ_{k}^{b} of the same morphism y_{Γ} in the special case that B̃ = ℙ_{k}^{b}. Thus it suffices to prove Claim 5.9 in the special case that B̃ = ℙ_{k}^{b}. In this case y_{Γ} is the fiber bundle over ℙ_{k}^{b} whose fiber over p ∈ ℙ_{k}^{b} is the variety parametrizing all triangles T of non-concurrent lines containing p as a smooth point, which is clearly smooth. This proves Claim 5.9.

Denote by

yΣ̃ : C ×_{P} Σ̃_{i} → B̃

the composition

C ×_{P} Σ̃_{i} ⟶^{pr_{C}} C ⟶^{prB̃} B̃.

For every open subset U ⊂ Σ̃, denote by

y_{U} : C ×_{P} U → B̃

the restriction of yΣ̃ over U.

claim-nonisoC

Claim 5.10. There exists an open subset U ⊂ Σ̃ intersecting D such that y_{U} is smooth.

The smooth locus of yΣ̃ is an open subscheme of C ×_{P} Σ̃. Its complement is closed. Because C → P is proper, the image of the closed complement in Σ̃ is closed. The open complement U ⊂ Σ̃ is the maximal open subscheme such that y_{U} is smooth. The claim is precisely that U intersects D.

Since char(k) = 0, the morphism D → Γ is smooth on a dense open subset U_{D} ⊂ D. Thus the product

C ×_{P} U_{D} → C ×_{P} Γ

is smooth. Denote by

y_{U_{D}} : C ×_{P} U_{D} → B̃

the composition of this product with y_{Γ}. By Claim 5.9, y_{Γ} is smooth. Thus the composition y_{U_{D}} is smooth. Chasing diagrams, y_{U_{D}} equals the restriction of yΣ̃ to C ×_{P} U_{D}. Since C ×_{P} D is a Cartier divisor in C ×_{P} Σ̃_{i}, yΣ̃ is smooth at every point of C ×_{P} U_{D}. Since the singular locus of yΣ̃ does not intersect the inverse image of U_{D}, U_{D} is contained in U. Therefore U intersects D. This proves Claim 5.10.

claim-nonisoD

Claim 5.11. There exists an open subset V ⊂ B̃ containing B and all codimension 1 points such that G is regular on the open subset y_{U}^{−1}(V) ⊂ C ×_{P} U.

Since y_{U} is smooth,

y_{U} : y_{U}^{−1}(B) → B

is also smooth. Since Q → B is an Abelian scheme, the rational transformation of smooth B-schemes

G : y_{U}^{−1}(B) −−→ Q

is a regular morphism by Weil’s extension theorem, cf. [BLR90, §4.4], [Kol96, Theorem VI.1.9.3]. Because Q̃ → B̃ is a Néron model, by the Neron extension property, there exists an open subset V of B̃ containing B and all codimension 1 points such that

G : y_{U}^{−1}(V) −−→ Q̃

is a regular morphism. This proves Claim 5.11.

Let P_{V} ⊂ P denote the open subscheme parametrizing curves in B̃ entirely contained in V. Because V contains all codimension 1 points of B̃, a general triangle curve is contained in P_{V}. Since Γ is irreducible, P_{V} ∩ Γ is dense. Since D → Γ is dominant, D×_{P} P_{V} is also dense in D. Since U ∩D is also dense in D, the two open subsets of D intersect. In other words, U ×_{P} P_{V} ⊂ Σ̃_{i} is an open subset intersecting D. Since D is integral, this subset is dense in D. Since D → Γ is dominant, the image of U ×_{P} P_{V} in P intersects Γ in a dense subset. By definition of P_{V},

C ×_{P} (U ×_{P} P_{V}) ⊂ y_{U}^{−1}(V).

Therefore G is regular over U ×_{P} P_{V} by Claim 5.11. Thus U ×_{P} P_{V} ⊂ \overline{Σ}_{i} is an open subset satisfying the conditions at the beginning of the proof. Therefore the image of Σ_{i} in P intersects Γ in a dense subset. □

lem-noniso2 Lemma 5.12. Assume char(k) = 0. Let B be an integral, smooth, quasi-projective k-scheme of dimension b ≥ 2. Let Q be an Abelian scheme over B. Assume that Q is strongly non-isotrivial, cf. Definition 4.7. For a very general cubic curve C in B, the map

H_{ét}^{0}(B, Q) → H_{ét}^{0}(C, C ×_{B} Q)

is a bijection.

Proof. The group H_{ét}^{0}(B, Q) is finitely generated by the Lang-Néron theorem. Applying the Néron extension property to each of the finitely many generators, there exists a single open subset of B̃ containing B and all codimension 1 points such that every section of Q over B extends to a section of Q̃ over this open subset. Replacing B̃ by this open subset, we may assume that every section of Q over B extends to a section of Q̃ over B̃. Then restriction to curves in B̃ determines a homomorphism of group schemes over P

restr : H_{ét}^{0}(B, Q) × P → Σ,

where the first group scheme is the finite, flat group scheme over P whose geometric fibers are all canonically identified with H_{ét}^{0}(B, Q). The lemma is equivalent to the statement that the induced homomorphism of geometric generic fibers over P is a bijection.

Denote by η ∈ P the generic point and denote by γ ∈ P the generic point of Γ. Denote R = O_{P,η_{Γ}}. Because P is smooth and Γ ⊂ P is a Cartier divisor, this is a DVR. Consider the base change of restr over Spec R. The geometric generic fiber of Σ ×_{P} Spec R over Spec R is

Σ\overline{η} = Σ ⊗_{O_{P}} \overline{κ(η)},

and the geometric closed fiber is

Σ\overline{γ} = Σ ⊗_{O_{P}} \overline{κ(γ)}.

By Lemma 4.9, Σ ×_{P} Spec R is a group scheme over Spec R which is “locally quasi-finite”, i.e., every quasi-compact component Σ_{i} ×_{P} Spec R is quasi-finite over Spec R. By Zariski’s Main Theorem, the morphism to Spec R is the composition of a dense open immersion and a finite morphism. By Lemma 5.12, points of Σ\overline{η} specialize to points of Σ\overline{γ}. Thus Σ_{i} ×_{P} Spec R → Spec R is finite. Since the characteristic is zero, all group schemes are reduced. Thus Σ ×_{P} Spec R is unramified over Spec R. In particular, the specialization homomorphism

spec : Σ\overline{η} → Σ\overline{γ}

is injective. But this is compatible with the restriction homomorphism, i.e.,

H_{ét}^{0}(B, Q) ⟶^{=} H_{ét}^{0}(B, Q)
restr\overline{η} ↓  ↓^{restr}\overline{γ}
Σ\overline{η}  ⟶^{spec}  Σ\overline{γ}

is a commutative diagram of group homomorphisms. By Lemma 5.3, the right vertical homomorphism is a bijection. Since spec is injective, also the right vertical homomorphism is a bijection, proving the lemma. □

Proof of Theorem 1.2. (ii) First we prove (ii), i.e., we prove that P(B, f, A) holds, cf. Definition 4.1. By Corollary 4.11, there exists a dense, open subscheme U ⊂ B, a finite, étale, Galois morphism U′ → U, and an isogeny of Abelian schemes over U′,

u = u_{0} ⊕ u_{Q} : (U′ ×_{k} A_{0}) ×_{U′} Q → U′ ×_{B} A

where A_{0} is an Abelian k-variety and Q is a strongly non-isotrivial Abelian scheme over U′. By the elementary reductions Lemma 4.2 and Lemma 4.3, it suffices to prove P(U′, f′, U′ ×_{B} A) holds. By the isogeny reduction, Lemma 4.12, it suffices to prove the two cases P(U′, f′, U′ ×_{k} A_{0}) and P(U′, f′, Q) hold. The first case follows from Lemma 5.1 for triangle curves, resp. Lemma 5.2 for cubic curves. The second case follows from Lemma 5.3 for triangle curves, resp. Lemma 5.12 for cubic curves in characteristic 0. This proves (ii).

(i) Next we prove (i), by reducing it to (ii). Let T be an A-torsor over B. Because this is a torsor for the étale topology, there exists a dense open subset U ⊂ B and a finite, étale, Galois morphism U′ → U with Galois group G such that U′ ×_{B} T is a trivial U′ ×_{B} A-torsor over U′. By (i), for a very general triangle curve C in B,

H_{ét}^{0}(U′, U′ ×_{B} A) → H_{ét}^{0}(U′ ×_{B} C, U′ ×_{B} C ×_{B} A)

is a bijection. This is an isomorphism of Galois modules. Thus the induced morphism of Galois invariants is an isomorphism. By étale descent, the morphism of Galois invariants is the restriction map

Hom_{U−sch}(U, U ×_{B} T) → Hom_{U×_{B}C−sch}(U ×_{B} C, U ×_{B} C ×_{B} T).

In particular, if C ×_{B} T has a section over C, then U ×_{B} T has a section over U. This section extends over all of B by [Kol96, Theorem VI.1.9.3]. Thus, if C ×_{B} T is a trivial torsor, then T is a trivial torsor.

Also, since sections of T over very general cubic curves specialize to sections of T over very general triangle curves, if C ×_{B} T has a section over C for a very general cubic curve, then is also has a section over a very general triangle curve. Therefore, by the previous paragraph, T is a trivial torsor.

Finally, consider the statement with planar surfaces in place of curves. Let C be a very general triangle curve in B satisfying (ii). Since C = B ×_{ℙ_{k}^{b}} E, where E is a plane cubic curve in a linear 2-plane Π ⊂ ℙ_{k}^{b}, C is contained in the planar surface S = B ×_{ℙ_{k}^{b}} Π. If C is very general, then S is very general. Moreover, since (ii) holds for B for one triangle curve in S, it holds for every very general triangle curve in S. But for a very general triangle curve in S, (ii) also applies with S in place of B. Therefore both

H_{ét}^{0}(B, A) → H_{ét}^{0}(C, C ×_{B} A)

and

H_{ét}^{0}(S, S ×_{B} A) → H_{ét}^{0}(C, C ×_{B} A)

are bijections. Since the first map factors through the second map by restriction

H_{ét}^{0}(B, A) → H_{ét}^{0}(S, S ×_{B} A),

this restriction is also a bijection. And (i) follows for planar surfaces in a similar way. □

6. Examples and the proof of Proposition 1.5

sec-ex

In this section we give some examples demonstrating limits to further generalization of Theorem 1.2.

Example (i). First of all, the restriction map

H_{ét}^{1}(B, A) → H_{ét}^{1}(C, C ×_{B} A)

can fail to be surjective for all triangle curves, resp. all cubic curves. For an example, let B = ℙ^{2}, let f be the identity map, and let A = B ×_{k} A_{0}, where A_{0} is a simple Abelian k-variety of dimension g ≥ 2. Let l be an integer not divisible by char(k). Associated to the multiplication map

0 ⟶ B ×_{k} A_{0}[l] ⟶ B ×_{k} A_{0} ⟶^{mult_{l}} B ×_{k} A_{0} ⟶ 0

there is a long exact sequence of cohomology groups part of which is

0 ⟶ H_{ét}^{0}(B, B ×_{k} A_{0}) ⊗ ℤ/lℤ ⟶ H_{ét}^{0}(B, B ×_{k} A_{0}[l]) ⟶ H_{ét}^{0}(B, B ×_{k} A_{0})[l] ⟶ 0.

Because Alb(P) = 0,

H_{ét}^{0}(B, B ×_{k} A_{0}) = A_{0}(k).

Because A_{0} is divisible, the exact sequence above gives

H_{ét}^{1}(B, B ×_{k} A_{0})[l] ≅ H_{ét}^{1}(B, B ×_{k} A_{0}[l]) ≅ H_{ét}^{1}(B, ℤ/lℤ)^{⊕2g}.

A similar argument applies to plane cubic curves and triangle curves in B. Of course the Albanese variety of a plane curve is not zero. But because A_{0} is a simple Abelian variety of dimension g ≥ 2, every morphism from C to A_{0} is a constant morphism. Thus

H_{ét}^{1}(C, C ×_{k} A_{0})[l] ≅ H_{ét}^{1}(C, C ×_{k} A_{0}[l]) ≅ H_{ét}^{1}(C, ℤ/lℤ)^{⊕2g}.

Thus the restriction homomorphism on l-torsion is equivalent to

H_{ét}^{1}(ℙ_{k}^{2}, ℤ/lℤ)^{⊕2g} → H_{ét}^{1}(C, ℤ/lℤ)^{⊕2g}.

Now

H_{ét}^{1}(ℙ_{k}^{2}, ℤ/lℤ) = 0

whereas

H_{ét}^{1}(C, ℤ/lℤ) = (ℤ/lℤ)^{⊕2}

for a smooth plane cubic and

H_{ét}^{1}(C, ℤ/lℤ) = ℤ/lℤ

for a triangle curve. Thus the restriction map is not surjective.

Example (ii). This example is similar. Let B = 𝔸_{k}^{2} and let f : 𝔸_{k}^{2} ↪ ℙ_{k}^{2} be the usual inclusion. Let A_{0} be an Abelian k-variety having a p-torsion k-point,

a ∈ A_{0}(k)[p],

e.g., A_{0} is an ordinary elliptic curve over k. Let A_{0} → A_{1} be the étale isogeny of Abelian varieties whose kernel is generate by a. There is an exact sequence

0 ⟶ ℤ/pℤ ⟶^{a} A_{0} ⟶ A_{1} ⟶ 0.

This gives rise to a long exact sequence. For the same reason as in Example (i), the induced map

H_{ét}^{1}(𝔸_{k}^{2}, ℤ/pℤ) → H_{ét}^{1}(𝔸_{k}^{2}, 𝔸_{k}^{2} ×_{k} A_{0})[p]

is an injection. Thus to produce an A_{0}-torsor over 𝔸_{k}^{2} of order p whose restriction to C is trivial, it suffices to produce a ℤ/pℤ-torsor T over 𝔸_{k}^{2} whose restriction to C is trivial. Let f ∈ k[x, y] be a polynomial function on 𝔸_{k}^{2} vanishing on C and whose degree d is not divisible by p.

Let T → 𝔸_{k}^{2} be the Artin-Schreier cover determined by the ring homomorphism

k[x, y] → k[x, y, t]/⟨t^{p} − t − f⟩.

Because d is not divisible by p, there is no element g ∈ k[x, y] such that g^{p}−g−f = 0. Since k[x, y] is normal and t^{p} − t − f is a monic polynomial in t, also there is no element g ∈ k(x, y) such that g^{p} − g − f = 0. Thus T is integral and so T → 𝔸_{k}^{2} is a nontrivial torsor. On the other hand, since f is zero on C,

C ×_{𝔸_{k}^{2}} T ≅ C × Spec k[t]/⟨t^{p} − t⟩,

which is the trivial ℤ/pℤ-torsor over C. Thus

H_{ét}^{1}(𝔸_{k}^{2}, 𝔸_{k}^{2} ×_{k} A_{0})[p] → H_{ét}^{1}(C, C ×_{k} A_{0})[p]

is not injective.

Example (iii). Let B = ℙ_{k}^{2} and let f be the identity map. Let A_{0} be an ordinary elliptic curve over k and let A = ℙ_{k}^{2} ×_{k} A_{0}. Then

H_{ét}^{0}(ℙ_{k}^{2}, ℙ_{k}^{2} ×_{k} A_{0}) = A_{0}(k).

But every open subset of the set of plane cubic curves in ℙ_{k}^{2} contains curves C isogenous to A_{0}. For these, the map

A_{0}(k) → H_{ét}^{0}(C, C ×_{k} A_{0})

is not surjective. Thus, it is necessary to use very general cubic curves in Theorem 1.2 – general cubic curves do not suffice.

References

[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

[Con06] Brian Conrad. Chow’s K/k-image and K/k-trace, and the Lang-Néron theorem. Enseign. Math. (2), 52(1-2):37–108, 2006.

[GHMS05] Tom Graber, Joe Harris, Barry Mazur, and Jason Starr. Rational connectivity and sections of families over curves. Ann. Sci. École Norm. Sup. (4), 38:671–692, 2005.

[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[Jou83] Jean-Pierre Jouanolou. Théorèmes de Bertini et applications, volume 42 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1983.

[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.

[Lan83] Serge Lang. Abelian varieties. Springer-Verlag, New York, 1983. Reprint of the 1959 original.

[MB81] Laurent Moret-Bailly. Familles de courbes et de variétés abéliennes sur ℙ^{1} ii. exemples. In Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, volume 86 of Astérisque, pages 125–140. Société Mathématique de France, Paris, 1981.

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125

E-mail address: graber@caltech.edu

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794

E-mail address: jstarr@math.sunysb.edu