LOG FANO VARIETIES OVER FUNCTION FIELDS OF CURVES

BRENDAN HASSETT AND YURI TSCHINKEL

Abstract. Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite set of points on the base curve.

Contents

1. Introduction 1 2. Integral models and reduction to the smooth case 3 3. Atiyah classes and free curves 6 4. Spaces of morphisms and comb constructions 7 5. Proof of the smooth case 13 References 15

1. Introduction

Let k be an algebraically closed field of characteristic zero and B a smooth projective curve over k with function field F = k(B).

Our point of departure is the following theorem, combining work of Graber-Harris-Starr and Kollár-Miyaoka-Mori [6, 14]: Let X be a smooth projective rationally connected variety over F. Then X(F) is Zariski dense in X. One central example is Fano varieties, i.e., varieties with ample anticanonical class, which are known to be rationally connected (see [13, V.2.13]). In this context, it is not necessary to pass to a field extension to get rational points.

When F is a number field, it may be necessary to pass to a finite extension to get rational points; there exist Fano varieties over ℚ without rational points. Moreover, even for Fano threefolds potential Zariski density, i.e., density after a finite extension of F, is unknown in general. For some positive results in this direction see [8], [4] and [9].

Date: January 14, 2008.

In this paper we study Zariski density of integral points. Consider pairs (X, D) consisting of a variety X and a divisor D ⊂ X, and fix integral models π : (X , D) → B (see Section 2 for the definition). An F -rational point s ∈ X \ D gives rise to a section s : B → X of π, meeting D in finitely many points. As we vary s,

s^{−1}(D) = π(s(B) ∩ D) ⊂ B

may vary as well. Fixing a finite set S ⊂ B, an S-integral point of (X , D) is an F -rational point of X such that s^{−1}(D) ⊂ S (as sets).

Theorem 1. Let F be the function field of a smooth projective curve B/k. Let (X, D) be a pair over F consisting of a smooth projective variety X and a smooth divisor D ⊂ X. Assume that

• D is rationally connected;

• the divisor class of the normal bundle c_{1}(N_{D/X}) is numerically

equivalent to a nontrivial effective divisor. Given a model π : (X , D) → B, let S be a nonempty finite set of points in B containing the images of the singularities of X and D. Then S-integral points of (X , D) are Zariski dense.

Note however that we allow points of bad reduction outside S. For example, let X = ℙ_{[x,y,z]}^{2} × ℙ_{[s,t]}^{1} and

D = {s(x^{2} + yz) + t(y^{2} + xz) = 0}.

While X and D are nonsingular the fiber D_{[s,t]} is singular when s^{3}+t^{3} = 0.

Let K_{X} denote the canonical class of X and K_{X} + D the log canonical class of (X, D). The pair (X, D) is log Fano if −(K_{X} + D) is ample. By adjunction

(K_{X} + D)|D = K_{D}

so −K_{D} is ample. Thus D is Fano hence rationally connected [14] [13, V.2.13].

Corollary 2. Let (X, D) be a log Fano variety over F with X and D smooth. Assume that the divisor class c_{1}(N_{D/X}) is numerically equivalent to a nontrivial effective divisor. For each integral model and collection of places as specified in Theorem 1, integral points are Zariski dense.

We expect that the hypothesis on N_{D/X} is not needed. For example, our argument does not apply to the case X = ℙ^{1} and D = ∞, where density of integral points is immediate. And there are log Fano varieties (X, D) with N_{D/X} negative, e.g., (X, D) = (𝔽_{n}, Σ_{0}) where 𝔽_{n} is the Hirzebruch surface admitting a section Σ_{0} with self-intersection −n < 0.

Furthermore, the condition that (X, D) be log Fano can be weakened. There are numerous examples of varieties X with trivial canonical class and Zariski-dense set of rational points over number and function fields. However, the deformation-theoretic approach here is not directly applicable. It is an open problem to characterize pairs with potentially dense integral points; Campana [5] has a conjectural description of these, at least when D = ∅.

Our result is a partial converse to the function-field version of Vojta’s conjectures: If (X, D) is defined over a number field F and K_{X} + D is ample then integral points are not Zariski dense (see [16] for the number field case, with connections to value-distribution theory). Very few density results for integral points over number fields are available, most of them in dimension two (see [15], [3], [10]).

Acknowledgments: We are grateful to Dan Abramovich for helpful conversations on the deformation theory used in this article. The constructive comments of the referees also greatly improved its exposition and organization. The first author appreciates the hospitality of the Mathematics Institute of the University of Göttingen. He was partially supported by National Science Foundation Grants 0554491 and 0134259 and an Alfred P. Sloan Research Fellowship. The second author was partially supported by National Science Foundation Grants 0554280 and 0602333.

2. Integral models and reduction to the smooth case

Definition 3. A pair (X, D) consists of a smooth projective variety and a reduced effective divisor with normal crossings.

Let B be a smooth projective curve defined over an algebraically closed field k of characteristic zero and F = k(B) its function field.

Definition 4. Let (X, D) be a pair defined over F. An integral model

π : (X , D) → B

consists of a flat proper morphism from a normal variety π_{X} : X → B with generic fiber X, and a closed subscheme D ⊂ X such that π_{D} := π_{X} |_{D} : D → B is flat and has generic fiber D. A point b ∈ B is of good reduction if the fibers X_{b} = π_{X}^{−1}(b) and D_{b} = π_{D}^{−1}(b) are smooth.

We emphasize that D has no irreducible components contained in fibers of π_{X}.

While in applications the model is often specified a priori, for each projective embedding of (X, D) we can construct a natural model: The properness of the Hilbert scheme yields extensions of X and D to schemes flat and projective over B. Locally on B, these are obtained by ‘clearing denominators’ in the defining equations of X and D. Normalizing if necessary, we obtain a model of (X, D).

Definition 5. Let S be a finite subset of B. An S-integral point of (X , D) is a section s : B → X such that s^{−1}(D) ⊂ S as sets.

Thus if D = ∅ then integral points are just sections of X → B, which are F -rational points of X.

The following proposition is straightforward:

Proposition 6. Let (X_{1}, D_{1}) and (X_{2}, D_{2}) be integral models of (X, D). Let T ⊂ B denote the set over which the birational map

(X_{1}, D_{1}) −−→ (X_{2}, D_{2})

fails to be an isomorphism. Then S-integral points of (X_{1}, D_{1}) are mapped to (S ∪ T)-integral points of (X_{2}, D_{2}). If S-integral points of (X_{1}, D_{1}) are Zariski dense then (S ∪ T)-integral points of (X_{2}, D_{2}) are Zariski dense.

In particular, if we allow ourselves to enlarge the set S then Zariskidensity of integral points is independent of the model.

We discuss how Theorem 1 can be reduced to the case of nonsingular integral models:

Definition 7. A good resolution of an integral model is a birational proper morphism from a pair

ρ : (X̃, D̃) → (X , D)

such that • X̃ is smooth and D̃ is normal crossings;

• ρ^{−1}(D) = D̃;

• ρ is an isomorphism over the open subset of (X , D) where X is smooth and D is normal crossings.

Remark 8.

(1) D̃ may very well have components contained in fibers over B, so (X̃, D̃) is not necessarily an integral model.

(2) The normality condition in Definition 4 of an integral model guarantees that for each b ∈ B and each irreducible component of X_{b}, the total space X is smooth at the generic point of that component. In particular, ρ is an isomorphism over a dense open subset of each fiber of X.

Let (X̃, D̃) → (X , D) be a good resolution, S ⊂ B a finite set containing the images of the singularities of X and D, and D̃^{∘} the union of the components of D̃ dominating B. We have:

• D̃^{∘} is normal crossings;

• images of S-integral points of (X̃, D̃^{∘}) under ρ are S-integral

points of (X , D). We have a bijection

ρ : X̃ \ D̃ → X \ D,

so S-integral points of (X , D) correspond to sections

{s̃: B → X̃ : s̃^{−1}(D̃) ⊂ S}.

Since the fibral components of D̃ lie over S, S-integral points of (X , D) are equal to S-integral points of (X̃, D̃^{∘}).

This analysis and resolution of singularities in characteristic zero reduces Theorem 1 to the following special case:

Theorem 9 (Smooth Case). Retain the assumptions of Theorem 1 and assume in addition that X and D are nonsingular. Then for any nonempty S ⊂ B the S-integral points in (X , D) are Zariski dense.

Here is a roadmap for the rest of the paper. Our main strategy is to produce ‘good’ sections in the boundary s : B → D that deform to a Zariski-dense collection of S-integral points. The ‘deformation of combs’ technique of [6, §2] yields a section s_{1} : B → D such that the normal bundle N_{s_{1}(B)/D} is globally generated with no higher cohomology. Since c_{1}(N_{D/X}) is effective and nonzero, we may assume that the teeth of our combs have positive intersection numbers with D. Thus we can construct a section s_{2} : B → D such that N_{s_{2}(B)/X} is globally generated with no higher cohomology. Finally, for any prescribed point p ∈ B, we produce a section s_{3} : B → D such that s_{3}^{∗}O_{X} (D) ≃ O_{B}(Np) for some N ≫ 0. This entails controlling the divisor class of the points over which we attach our teeth.

To obtain density of S-integral points, we control how deformations of s_{3}(B) meet D. The first step is to restrict to deformations s_{t} : D → X such that s_{t}^{∗}O_{X} (D) ≃ O_{B}(Np). The key technical tool is the ‘Atiyah extension’ classifying first-order deformations of varieties with a line bundle. Taking this into account, we introduce new notions of free and very free curves in Section 3. Section 4 deduces smoothing results for morphisms of curves respecting line bundles; these generalize the results for free and very free curves in [13, II.7]. Section 5 is devoted to the proof of Theorem 9.

3. Atiyah classes and free curves

Assume that k is algebraically closed. Let B be a smooth projective variety, ψ : Y → B a smooth surjective morphism from a quasi-projective variety, and L a line bundle on Y.

Consider the dual Atiyah extension [2, pp. 196] [11, pp. 243]

0 → Ω_{Y}^{1} → F_{Y,L} → O_{Y} → 0.

Up to sign, it is classified by the first Chern class

±c_{1}(L) ∈ H^{1}(Y, Ω_{Y}^{1}) = Ext^{1}(O_{Y} , Ω_{Y}^{1} ).

There is an induced relative Atiyah extension

0 → Ω_{Y/B}^{1} → F_{Y,L/B} → O_{Y} → 0.

These are all locally free as ψ is smooth. Writing E_{Y,L/B} for the dual to F_{Y,L/B}, we obtain

0 → O_{Y} → E_{Y,L/B} → T_{Y/B} → 0.

Definition 10. Let C be a nodal projective curve. A nonconstant morphism g : C → Y is L-free over B if for each q ∈ C

H^{1}(C, g^{∗}E_{Y,L/B} ⊗ I_{q}) = 0.

It is L-very free over B if for each subscheme Σ ⊂ C of length two

H^{1}(C, g^{∗}E_{Y,L/B} ⊗ I_{Σ}) = 0.

Proposition 11. Assume that k is of characteristic zero. Let b ∈ B and y ∈ Y_{b} = ψ^{−1}(b). Suppose Y_{b} is a proper rationally connected variety. Then there exists a morphism g : ℙ^{1} → Y_{b} that is L-free over B, with image containing y. If L|Y_{b} is numerically equivalent to a nontrivial effective divisor then we can choose g to be L-very free over B.

Proof. There exists a very free morphism g : ℙ^{1} → Y_{b} [13, IV.3.9.4], with g^{∗}T_{Y/B} is ample. Moreover, given any finite collection of points y_{1}, …, y_{m} ∈ Y, we may assume the image of g contains these points.

We have the extension

0 → O_{ℙ^{1}} → g^{∗}E_{Y,L/B} → g^{∗}T_{Y/B} → 0.

It follows that each summand of g^{∗}E_{Y,L/B} is nonnegative, which implies L-freeness.

We prove L-very freeness. Assume H is an effective nonzero divisor corresponding to L. Choose g such that its image contains y, some point y′ in the support of H, and some point y′′ not in the support of H. In particular, the image is not contained in any component of H. It follows that g^{∗}L has positive degree.

If O_{ℙ^{1}} were a summand of g^{∗}E_{Y,L/B} then we would have

g^{∗}E_{Y,L/B} ≃ O_{ℙ^{1}} ⊕ g^{∗}T_{Y/B},

i.e., the Atiyah extension would split after pull-back. The inclusion T_{ℙ^{1}} ↪ g^{∗}T_{Y/B} induces the Atiyah extension on ℙ^{1} associated with g^{∗}L:

0 → O_{ℙ^{1}} → E_{ℙ^{1},g^{∗}L} → T_{ℙ^{1}} → 0

which splits as well. However, this is classified by

±c_{1}(g^{∗}L) ∈ H^{1}(ℙ^{1}, Ω_{ℙ^{1}}^{1}) = Ext^{1}(T_{ℙ^{1}}, O_{ℙ^{1}}),

which is nontrivial because L is effective and nonzero. (Here we are using the assumption that the base field is of characteristic zero.) □

4. Spaces of morphisms and comb constructions

We retain the notation introduced in Section 3.

Let C_{0} be a nodal projective connected curve. Fix a morphism f_{0} : C_{0} → Y and write

g_{0} = ψ ∘ f_{0} : C_{0} → B.

Let (S, 0) be a smooth scheme with distinguished closed point. Consider a flat projective morphism ϖ : C → S such that ϖ^{−1}(0) ≃ C_{0} and an S-morphism

C ⟶^{g} B × S
ϖ ↘  ↙ π′_{S}
S

such that g|C_{0} = g_{0}. (Here π′_{S} is the projection to S.) Fix a line bundle M on C such that M|C_{0} ≃ f_{0}^{∗}L.

Let Mor_{B×S}(C, Y × S) denote the morphisms

C  ⟶^{h}  Y × S
g ↘  ↙ (ψ,Id_{S})
B × S

over B × S; each connected component is a quasi-projective scheme over S. Let Pic_{C/S} denote the relative Picard scheme [1, Theorem 7.3], which is locally of finite type over S. We have a morphism

\begin{matrix}Mor_{B×S}(C, Y × S) & → & Pic_{C/S} \\ h & ↦ & (π_{Y} ∘ h)^{∗}L ⊗ M^{−1} \end{matrix}

where π_{Y} : Y × S → Y is the projection. The fiber over the zero-section is a closed subscheme

Mor_{B×S}^{•}(C, Y × S) ⊂ Mor_{B×S}(C, Y × S).

Finally, we have

\begin{aligned}Mor_{B×S}((C, M), (Y × S, π_{Y}^{∗}L)) & = & \{(h, ι) : h ∈ Mor_{B×S}^{•}(C, Y × S), \\ & & ι : (π_{Y} ∘ h)^{∗}L →^{∼} M\} \end{aligned}

which is a 𝔾_{m}-torsor over Mor_{B×S}^{•}(C, Y ×S). This keeps track of a choice of identification between M and the pull-back of L.

Proposition 12. The relative tangent space of

Mor_{B×S}((C, M), (Y × S, π_{Y}^{∗}L)) (resp. Mor_{B×S}(C, Y × S))

over S at [f_{0}] is isomorphic to

Γ(C_{0}, f_{0}^{∗}E_{Y,L/B}) (resp. Γ(C_{0}, f_{0}^{∗}T_{Y/B})).

The obstruction space is contained in

H^{1}(C_{0}, f_{0}^{∗}E_{Y,L/B}) (resp. H^{1}(C_{0}, f_{0}^{∗}T_{Y/B})).

In particular, the morphism space is smooth over S at f_{0} provided the corresponding first cohomology group vanishes.

Proof. The description of the tangent and obstruction spaces is an application of the technique of [11, §2.3]. Using the smoothness criterion of [7, 17.5.1-2], it suffices to show that the schemes are flat over S with regular geometric fibers. The morphism spaces are flat over S at f_{0} provided their fibers over 0 have the expected dimension there [13, I.2.17]. When the first cohomology group vanishes, the dimension of the fiber equals the dimension of its tangent space. □

Remark 13. Instead of working with pairs consisting of varieties and line bundles relative to B, we could work with varieties relative to B × B𝔾_{m}. Here B𝔾_{m} is the classifying stack of the multiplicative group; morphisms Y → B𝔾_{m} correspond to line bundles on Y. The Atiyah extension of Y/B determined by L coincides with the relative tangent bundle of the morphism Y → B × B𝔾_{m}. (We are grateful to one of the referees for suggesting this point of view.)

Definition 14. A comb is a nodal projective curve

C_{0} = B′ ∪ T_{1} ∪ … ∪ T_{r}

where B′ is smooth and connected and each T_{i} is a smooth rational curve meeting B′ in one point b′_{i}. The T_{i} are pairwise disjoint.

Proposition 15. Let B′ ⊂ Y be a smooth projective curve satisfying H^{1}(B′, E_{Y,L/B}|B′) = 0; assume that B′ meets a proper separably rationally connected fiber of ψ. Suppose we are given

• a comb C_{0} with handle B′ and teeth T_{i};

• a morphism f_{0} : C_{0} → Y, such that (1) f_{0}|B′ extends the inclusion B′ ⊂ Y;

(2) each restriction F_{i} := f_{0}|T_{i} is L-free and contained in a

proper fiber of ψ.

Then for any • smooth scheme with distinguished closed point (𝔸, 0);

• flat proper morphism ϖ : C → 𝔸 with distinguished fiber C_{0};

• morphism g : C → B × 𝔸 over 𝔸 such that g|C_{0} = ψ ∘ f_{0};

• line bundle M on C such that M|C_{0} ≃ f_{0}^{∗}L; there exists an étale neighborhood (S, 0) of (𝔸, 0) and a morphism f : C → Y × S over B × S such that f^{∗}(π_{Y}^{∗}L) ≃ M and f|C_{0} = f_{0}.

Proof. Since the teeth are L-free, E_{Y,L/B} pulls back to a semi-positive vector bundle on each T_{i}. An inductive argument [13, II.7.5] reduces the cohomology over the comb to the cohomology of the restriction to the handle

H^{1}(C_{0}, E_{Y,L/B}|C_{0}) ≃ H^{1}(B′, E_{Y,L/B}|B′) = 0.

We then apply Proposition 12 to construct f. By construction,

f_{0} ∈ Mor_{B×𝔸}((C, M), (Y × 𝔸, π_{Y}^{∗}L))

which is smooth over 𝔸 because of the vanishing of the higher cohomology of E_{Y,L/B}|C_{0}. Thus for a suitable étale neighborhood (S, 0) of the origin in 𝔸 we have a section

[f] : S → Mor_{B×S}((C, M), (Y × S, π_{Y}^{∗}L))

with [f]_{0} = f_{0}. (We use the same notation for C and M and their restrictions over S.) The universal property of our morphism space gives f : C → Y ×S over B×S and an identification ι : (π_{Y} ∘f)^{∗}L →^{∼} M|S. □

Remark 16. Alternatively, we could have analyzed the deformations of f_{0} : C_{0} → Y using the normal sheaf N_{f_{0}}. This works best when f_{0} is an embedding, which can be achieved when the fibers of ψ have dimension ≥ 3. Then we can regard f_{0}(C_{0}) as a point in the Hilbert scheme Hilb of nodal curves in Y. Suppose in addition that B′ ⊂ Y is a section of ψ : Y → B. Then there exists a morphism

\begin{matrix}α : Hilb & → & Pic(B) \\ C_{s} & ↦ & det(ℝ^{•}ψ_{∗}(L ⊗ O_{C_{s}})), \end{matrix}

where det is the determinant of cohomology, which is defined for perfect complexes (see [12, Theorem 2]). A deformation argument similar to ours can be used to produce smooth curves in suitable fibers of α corresponding to sections σ : B → Y with σ^{∗}L = L|B′. (We are grateful to one of the referees for pointing out this approach.)

Proposition 17. Let B′ ⊂ Y be a smooth projective curve. There exists an integer n > 0 with the following property:

Suppose we are given • a comb C_{0} with handle B′ and teeth T_{1}, …, T_{q} attached at b′_{1}, …, b′_{q};

• a morphism f_{0} : C_{0} → Y, such that

(1) f_{0} is an embedding along B′ and extends the inclusion B′ ⊂ Y;

(2) each F_{i} := f_{0}|T_{i} is L-free and contained in a proper fiber of

ψ;

Then there exists a subset {i_{1}, …, i_{r}} ⊂ {1, …, q} with r ≥ q − n such that

f′_{0} := f_{0}|C′_{0} : C′_{0} → Y, C′_{0} := B′ ∪ T_{i_{1}} ∪ … ∪ T_{i_{r}} ⊂ C_{0}

deforms to a morphism f′_{t} : B′ → Y with

(f′_{t})^{∗}L ≃ L|B′ ⊗ O_{B′}(e_{i_{1}}b′_{i_{1}} + … + e_{i_{r}}b′_{i_{r}}), e_{i} = deg F_{i}^{∗}L

and ψ ∘ f′_{t} = ψ|B′. Furthermore, if the F_{i} are assumed to be L-very free then for suitable n > 0 we can take f′_{t} to be L-free over B.

Proof. We first construct a family ϖ : C → 𝔸^{r} deforming C_{0} to B′ (see [13, pp. 156]). Consider the smooth codimension-two subvariety

Z = ∪_{i=1}^{r}({b′_{i}} × {t_{i} = 0}) ⊂ B′ × 𝔸^{r}

and the blow-up

C := Bl_{Z}(B′ × 𝔸^{r}) →^{σ} B′ × 𝔸^{r}

with exceptional divisors E_{1}, …, E_{r}. The composed morphism

ϖ : C → B′ × 𝔸^{r} → 𝔸^{r}

is still flat with ϖ^{−1}(0) ≃ C_{0}; every fiber of ϖ is a comb with handle B′. We introduce a line bundle on this family: Consider

L′ = L|B′ ⊗ O_{B′}(e_{1}b′_{1} + … + e_{r}b′_{r})

and write

M = (π_{B′} ∘ σ)^{∗}L′ ⊗ O_{C}(−e_{1}E_{1} − … − e_{r}E_{r}).

Note that C_{s} ≃ B′ and M|C_{s} ≃ L|B′, i.e., our modification leaves the fibers away from the coordinate axes of 𝔸^{r} unchanged.

We apply the smoothing technique of [13, II.7.9] to the morphism space

Mor_{B×𝔸^{r}}((C, M), (Y × 𝔸^{r}, π_{Y}^{∗}L)).

This gives a curve containing the origin 0 ∈ T ⊂ 𝔸^{r}, whose generic point is contained in at most c ≤ n of the coordinate hyperplanes, such that the restricted family C_{T} := C ×_{𝔸^{r}} T → T admits a morphism

C_{T}  ⟶^{f}  Y × T
↘  ↙
B × T

restricting to f_{0} at 0 ∈ T. Let C′_{T} denote the irreducible component of C_{T} containing B′ ⊂ C_{0}; its distinguished fiber is a subcomb

C′_{0} := B′ ∪ T_{i_{1}} ∪ … ∪ T_{i_{q}} ⊂ C_{0}, q = r − c.

The restriction f′ = f|C′ is our desired morphism.

When the teeth are L-very free, the argument of [13, II.7.10] yields a smoothing that is L-free over B. □

Corollary 18. Retain the assumptions of Proposition 17 and assume k is of characteristic zero. Suppose in addition that B′ meets a proper rationally connected fiber of ψ. Then there exists a free f′_{s} : B′ → Y with ψ ∘ f′_{s} = ψ|B′.

Proof. We just need to construct the comb C_{0} and the morphism f_{0}. Using Proposition 11, we choose

• points b_{1}, …, b_{r} ∈ B with Y_{b_{i}} proper;

• distinct points b′_{1}, …, b′_{r} ∈ B′ with ψ(b′_{i}) = b_{i}; and

• L-very free curves F_{i} : T_{i} → Y_{b_{i}} containing b′_{i} as a smooth point. Let C_{0} = B′∪T_{1}∪…∪T_{r} denote the corresponding comb and f_{0} : C_{0} ↪ Y the morphism obtained by gluing the inclusion B′ ⊂ Y with the F_{i}. □

Remark 19. Again, we could also work over the Hilbert scheme and apply the smoothing techniques of [6, §2] to establish this result.

Our main application is to sections of rationally connected fibrations.

Theorem 20. Let B be a smooth projective curve, ψ : \overline{Y} → B a proper morphism from a smooth variety with rationally connected generic fiber, Y ⊂ \overline{Y} the open subset where ψ is smooth, and L an invertible sheaf on Y restricting to a nontrivial effective divisor on the generic fiber of ψ. Fix an integer N ≫ 0.

For each invertible sheaf M ∈ Pic^{N}(B), there exists a section s : B → Y such that s^{∗}L = M and s is L-free over B. In particular, the sheaves

s^{∗}E_{Y,L/B} and s^{∗}T_{Y/B} = N_{s(B)/Y}

are both globally generated with no higher cohomology.

Proof of 20. The Graber-Harris-Starr Theorem [6] gives a section s_{1} : B → Y. The exact sequence

0 → T_{Y/B} → T_{Y} → ψ^{∗}T_{B} → 0

induces

0 → s_{1}^{∗}T_{Y/B} → s_{1}^{∗}T_{Y} → T_{B} → 0

which is split by the differential ds_{1} : T_{B} → s_{1}^{∗}T_{Y}. Thus we have

s_{1}^{∗}T_{Y} = s_{1}^{∗}T_{Y/B} ⊕ T_{B}

and the first term coincides with the normal bundle N_{s_{1}(B)/Y}.

Proposition 17 gives a section s_{2} : B → Y that is L-free over B. We assume this in what follows, so in particular

H^{1}(B, s_{2}^{∗}E_{Y,L/B}) = 0.

To complete the proof, we produce smoothings f_{t} : B → Y of suitable combs

f_{0} : C_{0} = B ∪_{b_{1}} T_{1} … ∪_{b_{r}} T_{r} → Y

where f_{0}|B = s_{2} F_{i} = f_{0}|T_{i} : T_{i} → Y_{b_{i}} is L-very free. These are given by Proposition 15, but it is necessary to specify the points of attachment carefully to achieve the desired value of f_{t}^{∗}L.

Choose e sufficiently large so that for each smooth fiber Y_{b} and every point y ∈ Y_{b}, there exists an L-very free curve T → Y_{b} passing through y with L · T = e. We therefore may assume

e = T_{1} · L = T_{2} · L = … = T_{r} · L > 0.

Let C → 𝔸^{r} and M denote the families constructed in the proof of Proposition 17. Thus for generic s ∈ 𝔸^{r} we have C_{s} ≃ B and

M|C_{s} ≃ s_{2}^{∗}L(eb_{1} + … + eb_{r}).

Let U ⊂ B denote the locus over which Y_{b} is smooth and contains a L-very free curve T of degree e. To complete the proof of Theorem 20, we require the following moving lemma, which governs the precise value of N:

Lemma 21. Let B be a smooth projective curve, U ⊂ B a dense open subset, and e ∈ ℕ. Fix a line bundle Λ on B of degree ℓ, r ≥ 2g(B) + 1, and N = er + ℓ. For any M ∈ Pic^{N}(B) there exist distinct points b_{1}, …, b_{r} ∈ U so that

M ≃ Λ(e(b_{1} + … + b_{r})).

This is an elementary application of Riemann-Roch. Choose an eth root of M ⊗ Λ^{−1}, i.e., a line bundle A with A^{⊗e} ⊗ Λ = M. Any line bundle of degree r on B is very ample so consider the embedding

ϕ_{A} : B ↪ ℙ^{r−g(B)}.

The divisors with any support along B \ U form a finite union of hyperplanes in the linear system |A|. The divisors admitting points of multiplicity > 1 form a proper subvariety of Δ ⊂ |A| by the Bertini Theorem. Any divisor in the complement of the hyperplanes and Δ can be expressed in the form b_{1} + … + b_{r} with the b_{i} distinct in U.

□

5. Proof of the smooth case

In this section, we prove Theorem 9; take S = {p} for some p ∈ B. Apply Theorem 20 to \overline{Y} = D, L = O_{D}(D) = N_{D/X}, and M = O_{B}(Np)

for some suitable N ≫ 0. We obtain a section s : B → D with the following properties:

• s^{∗}D = N_{D/X} |s(B) ≃ O_{B}(Np);

• E_{D,O(D)/B}|s(B) is globally generated with no higher cohomology. Consider the following diagram:

0  0
↓  ↓
0 → O_{B} → E_{D,O(D)/B}|s(B) → N_{s(B)/D} → 0
|| ↓ ↓
0 → O_{B} → E_{X ,O(D)/B}|s(B) → N_{s(B)/X} → 0
↓  ↓
N_{D/X} |s(B) = N_{D/X} |s(B)
↓  ↓
0  0

The top two horizontal exact sequences are the Atiyah extensions defining E_{D,O(D)/B} and E_{X ,O(D)/B} restricted to s(B). We identify the relative tangent bundles of D and X over B restricted to s(B) with the normal bundles N_{s(B)/D} and N_{s(B)/X}. The right vertical exact sequence is the standard normal bundle sequence for s(B) ⊂ D ⊂ X.

The terms in the bottom row are isomorphic to O_{B}(Np), which has no higher cohomology. Since the middle term in the upper row has vanishing higher cohomology, we deduce that H^{1}(E_{X ,O(D)/B}|s(B)) = 0. Furthermore, the middle vertical exact sequence yields an exact sequence on global sections.

The inclusion D ⊂ X induces an inclusion of morphism spaces

(5.1) Mor_{B}((B, M), (D, O_{D}(D))) ↪ Mor_{B}((B, M), (X , O_{X}(D))).

These are smooth by Proposition 12. The image is precisely the indeterminacy of the rational map

\begin{matrix}G : Mor_{B}((B, M), (X , O_{X} (D))) & −−→ & ℙ(Γ(B, O_{B}(Np))) \\ s_{t} & ↦ & s_{t}^{−1}D. \end{matrix}

We may interpret S-integral points as sections s_{t} mapping to elements of Γ(B, O_{B}(D)) vanishing at p to maximal order N. Thus we are interested in elements of G^{−1}[Γ(B, O_{B})], where Γ(B, O_{B}) ⊂ Γ(B, O_{B}(Np)) corresponds to the constant functions, i.e., the image of the map on global sections induced by the inclusion of sheaves O_{B} ↪ O_{B}(Np).

The indeterminacy of G is resolved by blowing up the subscheme (5.1). The stalk of its normal bundle at s(B) is canonically isomorphic to Γ(O_{X} (D)|s(B)). In particular, the proper transform of G^{−1}[Γ(B, O_{B})] meets the exceptional fiber over s(B) at the point

[Γ(B, O_{B})] ∈ ℙ(Γ(O_{X} (D)|s(B))) ≃ ℙ(Γ(B, O_{B}(Np))).

Thus s(B) deforms to s_{t}(B) ∈ G^{−1}[Γ(B, O_{B})], corresponding to an Sintegral point.

The sections thus produced are Zariski dense in X. Indeed, our construction produces sections passing through the generic point of D that deform out of D to the generic point of X.

Remark 22. Here is an alternate approach suggested by one of the referees. Fix a section r ∈ Γ(X , O_{X} (D)) inducing the canonical inclusion O_{X} ↪ O_{X} (D). Since M = O_{B}(Np), we have a morphism

\begin{matrix}G̃ : Mor_{B}((B, M), (X , O_{X} (D))) & → & Γ(B, O_{B}(Np)) \\ (s_{t}, ι : s_{t}^{∗}O_{X} (D) →^{∼} M) & ↦ & ι(s_{t}^{∗}r) \end{matrix}

which yields G on composition by

Γ(B, O_{B}(Np)) −−→ ℙ(Γ(B, O_{B}(Np))).

One can show that G̃ is smooth via deformation theory. The preimage of the curve Γ(B, O_{B}) ⊂ Γ(B, O_{B}(Np)) gives the desired sections of X.

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Department of Mathematics, Rice University, MS 136, Houston, TX 77251-1892

E-mail address: hassett@rice.edu

Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany, and, Courant Institute, NYU, 251 Mercer Str., New York, NY 10012

E-mail address: tschinkel@cims.nyu.edu