Density of integral points on algebraic varieties

Brendan Hassett and Yuri Tschinkel

July 30, 2000

1 Introduction

Let K be a number field, S a finite set of valuations of K, including the archimedean valuations, and O_{S} the ring of S-integers. Let X be an algebraic variety defined over K and D a divisor on X. We will use X and D to denote models over Spec(O_{S}).

We will say that integral points on (X, D) (see Section 2 for a precise definition) are potentially dense if they are Zariski dense on some model (X , D), after a finite extension of the ground field and after enlarging S. A central problem in arithmetic geometry is to find conditions insuring potential density (or nondensity) of integral points. This question motivates many interesting and concrete problems in classical number theory, transcendence theory and algebraic geometry, some of which will be presented below.

If we think about general reasons for the density of points - the first idea would be to look for the presence of a large automorphism group. There are many beautiful examples both for rational and integral points, like K3 surfaces given by a bihomogeneous (2, 2, 2) form in ℙ^{1}×ℙ^{1}×ℙ^{1} or the classical Markov equation x^{2} + y^{2} + z^{2} = 3xyz. However, large automorphism groups are “sporadic” - they are hard to find and usually, they are not well behaved in families. There is one notable exception - namely automorphisms of algebraic groups, like tori and abelian varieties.

Thus it is not a surprise that the main geometric reason for the abundance of rational points on varieties treated in the recent papers [11], [3], [12] is the presence of elliptic or, more generally, abelian fibrations with multisections having a dense set of rational points and subject to some nondegeneracy conditions. Most of the effort goes into ensuring these conditions.

In this paper we focus on cases when D is nonempty. We give a systematic treatment of known approaches to potential density and present several new ideas for proofs. The analogs of elliptic fibrations in log geometry are conic bundles with a bisection removed. We develop the necessary techniques to translate the presence of such structures to statements about density of integral points and give a number of applications.

The paper is organized as follows: in Section 2 we introduce the main definitions and notations. Section 3 is geometrical - we introduce the relevant concepts from the log minimal model program and formulate several geometric problems inspired by questions about integral points. In Section 4, we recall the fibration method and nondegeneracy properties of multisections. We consider approximation methods in Section 5. Section 6 is devoted to the study of integral points on conic bundles with sections and bisections removed. In the final section, we survey the known results concerning potential density for integral point on log K3 surfaces.

Acknowledgements. The first author was partially supported by an NSF Postdoctoral Research Fellowship, NSF Continuing Grant 0070537, and the Institute of Mathematical Sciences of the Chinese University of Hong Kong. The second author was partially supported by the NSA. We benefitted from conversations with D. Abramovich, Y. André, F. Bogomolov, A. Chambert-Loir, J.-L. Colliot-Thélène, J. Kollár, D. McKinnon, and B. Mazur. We are grateful to P. Vojta for comments that improved the paper, especially Proposition 3.12, and to D.W. Masser for information on specialization of nondegenerate sections. Our approach in Section 6 is inspired by the work of F. Beukers (see [1] and [2]).

2 Generalities

2.1 Integral points

Let π : U → Spec(O_{S}) be a flat scheme over O_{S} with generic fiber U. An integral point on U is a section of π; the set of such points is denoted U(O_{S}).

In the sequel, U will be the complement to a reduced effective Weil divisor D in a normal proper scheme X, both generally flat over Spec(O_{S}). Hence an S-integral point P of (X , D) is a section s_{P} : Spec(O_{S}) → X of π, which does not intersect D, that is, for each prime ideal 𝔭 ∈ Spec(O_{S}) we have s_{P} (𝔭) ∉ D_{𝔭}. We denote by X (resp. D) the corresponding generic fiber. We generally assume that X is a variety (i.e., a geometrically integral scheme); frequently X is smooth and D is normal crossings. Potential density of integral points on (X , D) does not depend on the choice of S or on the choices of models over Spec(O_{S}), so we will not always specify them. Hopefully, this will not create any confusion.

If D is empty then every K-rational point of X is an S-integral point for (X , D) (on some model). Every K-rational point of X, not contained in D is S-integral on (X , D) for S large enough. Clearly, for any X and D there exists a finite extension K′/K and a finite set S′ of prime ideals in O_{K′} such that there is an S′-integral point on (X′, D′) (where X′ is the base change of X to Spec(O′_{S})).

The definition of integral points can be generalized as follows: let Z be any closed subscheme of X. An S-integral point for (X , Z) is an O_{S}-valued point of X \ Z.

2.2 Vojta’s conjecture

A pair consists of a proper normal variety X and a reduced effective Weil divisor D ⊂ X. A morphism of pairs φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) is a morphism φ : X_{1} → X_{2} such that (the support of) φ^{−1}(D_{2}) is a subset of D_{1}. In particular, φ restricts to a morphism X_{1} \ D_{1} → X_{2} \ D_{2}. A morphism of pairs is dominant if φ : X_{1} → X_{2} is dominant. If (X_{1}, D_{1}) dominates (X_{2}, D_{2}) then integral points are dense on (X_{2}, D_{2}) when they are dense on (X_{1}, D_{1}) (after choosing appropriate integral models). A morphism of pairs is proper if φ : X_{1} → X_{2} is proper and the restriction X_{1} \ D_{1} → X_{2} \ D_{2} is also proper; equivalently, we may assume that φ : X_{1} → X_{2} is proper and D_{1} is a subset of φ^{−1}(D_{2}). A resolution of the pair (X, D) is a proper morphism of pairs ρ : (X̃, D̃) → (X, D) such that ρ : X̃ → X is birational, X̃ is smooth, and D̃ is normal crossings.

Let X be a normal proper variety of dimension d. Recall that a Cartier divisor D ⊂ X is big if h^{0}(O_{X}(nD)) > Cn^{d} for some C > 0 and all n sufficiently large and divisible.

Definition 2.1 A pair (X, D) is of log general type if it admits a resolution ρ : (X̃, D̃) → (X, D) with ωX̃(D̃) big.

Let us remark that the definition does not depend on the resolution.

Conjecture 2.2 (Vojta, [30]) Let (X, D) be a pair of log general type. Then integral points on (X, D) are not potentially dense.

This conjecture is known for semiabelian varieties and their subvarieties ([9], [31], [16]). Vojta’s conjecture implies that a pair with dense integral points cannot dominate a pair of log general type.

We are interested in geometric conditions which would insure potential density of integral points. The most naive statement would be the direct converse to Vojta’s conjecture. However this can’t be true even when D = ∅. Indeed, varieties which are not of general type may dominate varieties of general type, or more generally, admit finite étale covers which dominate varieties of general type (see the examples in [7]). In the next section we will analyze other types of covers with the same arithmetic property.

3 Geometry

3.1 Morphisms of pairs

Definition 3.1 We will say that a class of dominant morphisms of pairs φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) is arithmetically continuous if the density of integral points on (X_{2}, D_{2}) implies potential density of integral points on (X_{1}, D_{1}).

For example, assume that D = ∅. Then any projective bundle in the Zariski topology ℙ → X is arithmetically continuous. In the following sections we present other examples of arithmetically continuous morphisms of pairs.

Definition 3.2 A pseudo-étale cover of pairs φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) is a proper dominant morphism of pairs such that

a) φ : X_{1} → X_{2} is generically finite, and

b) the map from the normalization X_{2}^{norm} of X_{2} (in the function field of X_{1}) onto X_{2} is étale away from D_{2}.

Remark 3.3 For every pair (X, D) there exists a birational pseudo-étale morphism φ : (X̃, D̃) → (X, D) such that X̃ is smooth and D̃ is normal crossings.

The following theorem is a formal generalization of the well-known theorem of Chevalley-Weil. It shows that potential density is stable under pseudoétale covers of pairs.

Theorem 3.4 Let φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) be a pseudo-étale cover of pairs. Then φ is arithmetically continuous.

Remark 3.5 An elliptic fibration E → X, isotrivial on X \ D, is arithmetically continuous. Indeed, it splits after a pseudo-étale morphism of pairs and we can apply Theorem 3.4.

The following example is an integral analog of the example of Skorobogatov, Colliot-Thélène and Swinnerton-Dyer ([7]) of a variety which does not dominate a variety of general type but admits an étale cover which does.

Example 3.6 Consider ℙ^{1} × ℙ^{1} with coordinates (x_{1}, y_{1}), (x_{2}, y_{2}) and involutions

j_{1}(x_{1}, y_{1}) = (−x_{1}, y_{1}) j_{2}(x_{2}, y_{2}) = (y_{2}, x_{2})

on the factors. Let j be the induced involution on the product; it has fixed points

\begin{matrix}x_{1} = 0 & x_{2} = & y_{2} \\ x_{1} = 0 & x_{2} = & −y_{2} \\ y_{1} = 0 & x_{2} = & y_{2} \\ y_{1} = 0 & x_{2} = & −y_{2} \end{matrix} .

The first projection induces a map of quotients

(ℙ^{1} × ℙ^{1})/ ⟨j⟩ → ℙ^{1}/ ⟨j_{1}⟩ .

We use X to denote the source; the target is just Proj(ℂ[x_{1}^{2}, y_{1}^{2}]) ≃ ℙ^{1}. Hence we obtain a fibration f : X → ℙ^{1}. Note that f has two nonreduced fibers, corresponding to x_{1} = 0 and y_{1} = 0 respectively. Let D be the image in X of

(x_{1} = 0) ∪ (y_{1} = 0) ∪ (x_{2} = m_{1}y_{2}) ∪ (x_{2} = m_{2}y_{2})

where m_{1} and m_{2} are distinct, m_{1}m_{2} ≠ 1, and m_{1}, m_{2} ≠ 0, 1. Since D intersects the general fiber of f in just two points, (X, D) is not of log general type.

We can represent X as a degenerate quartic Del Pezzo surface with four A1 singularities (see figure 1). If we fix invariants

a = x_{1}^{2}x_{2}y_{2}, b = x_{1}^{2}(x_{2}^{2} + y_{2}^{2}), c = x_{1}y_{1}(x_{2}^{2} − y_{2}^{2}), d = y_{1}^{2}(x_{2}^{2} + y_{2}^{2}), e = y_{1}^{2}x_{2}y_{2}

then X is given as a complete intersection of two quadrics:

ad = be, c^{2} = bd − 4ae.

Figure 1: The log surface (X, D)

The components of D satisfy the equations

D_{1} = {a = b = c = 0}
D_{2} = {c = d = e = 0}
D_{3} = {(1 + m_{1}^{2})a − m_{1}b = (1 + m_{1}^{2})e − m_{1}d = 0}
D_{4} = {(1 + m_{2}^{2})a − m_{2}b = (1 + m_{2}^{2})e − m_{2}d = 0}.

Our assumptions guarantee that D_{3} and D_{4} are distinct.

We claim that (X, D) does not admit a dominant map onto a variety of log general type and that there exists a pseudo-étale cover of (X, D) which does. Indeed, the preimage of X \ D in ℙ^{1} × ℙ^{1} is

(𝔸^{1} \ 0) × (ℙ^{1} \ {m_{1}, m_{2}, 1/m_{1}, 1/m_{2}}),

which dominates a curve of log general type, namely, ℙ^{1} minus four points. However, (X, D) itself cannot dominate a curve of log general type. Any such curve must be rational, with at least three points removed; however, the boundary D contains at most two mutually disjoint irreducible components.

The following was put forward as a possible converse to Vojta’s conjecture.

Problem 3.7 (Strong converse to Vojta’s conjecture) Assume that the pair (X_{2}, D_{2}) does not admit a pseudo-étale cover (X_{1}, D_{1}) → (X_{2}, D_{2}) such that (X_{1}, D_{1}) dominates a pair of log general type. Are integral points for (X_{2}, D_{2}) potentially dense?

3.2 Projective bundles in the étale topology

We would like to produce further classes of dominant arithmetically continuous morphisms (X_{1}, D_{1}) → (X_{2}, D_{2}).

Theorem 3.8 Let φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) be a projective morphism of pairs such that φ is a projective bundle (in the étale topology) over X_{2} \ D_{2}. We also assume that φ^{−1}(D_{2}) = D_{1}. Then φ is arithmetically continuous.

Proof. We are very grateful to Prof. Colliot-Thélène for suggesting this proof.

Choose models (X_{i}, D_{i}) (i = 1, 2) over some ring of integers O_{S}, so that the morphism φ is well-defined and a projective bundle. (We enlarge S as necessary.)

We recall basic properties of the Brauer group Br(O_{S}). Let v denote a place for the quotient field K and K_{v} the corresponding completion. Classfield theory gives the following exact sequences

0 → Br(K) → ⊕_{v}Br(K_{v}) → ℚ/ℤ → 0
0 → Br(O_{S}) → Br(K) → ⊕_{v∉S}Br(K_{v}).

The Brauer groups of the local fields corresponding to nonarchimedean valuations are isomorphic to ℚ/ℤ. Given a finite extension of K_{w}/K_{v} of degree n, the induced map on Brauer groups is multiplication by n.

Let p denote an S-integral point of (X_{2}, D_{2}). The fiber φ^{−1}(p) is a Brauer-Severi variety over O_{S}. If r − 1 denotes the relative dimension of φ then the corresponding element β(p) ∈ Br(O_{S}) has order dividing r. Integral points in φ^{−1}(p) are dense if rational points are dense, which is the case when β(p) = 0.

Our exact sequences imply that β(p) yields elements of Br(K_{v}) which are zero unless v ∈ S, and are annihilated by r otherwise. It suffices to find an extension K′/K inducing a cyclic extension of K_{v} of order divisible by r for all v ∈ S. Indeed, such an extension necessarily kills β(p) for each point p defined over O_{S}. If O_{S′} is the integral closure of O_{S} in K′ then φ^{−1}(p) has dense S′-integral points. □

Remark 3.9 Let X be a smooth simply connected projective variety which does not dominate a variety of general type. It may admit an projective bundle (in the étale topology) φ : ℙ → X, for example if X is a K3 surface. However, ℙ cannot dominate a variety of general type. Indeed, given such a dominant morphism π : ℙ → Y, the fibers of φ are mapped to points by π. In particular, π necessarily factors through φ. (We are grateful to J. Kollár for emphasizing this point.)

Problem 3.10 (Geometric counterexamples to Problem 3.7) Are there pairs which do not admit pseudo-étale covers dominating pairs of log general type but which do admit arithmetically continuous covers dominating pairs of log general type?

3.3 Punctured varieties

In Section 3.1 we have seen that potential density of integral points is preserved under pseudo-étale covers. It is not an easy task, in general, to check whether or not some given variety (like an elliptic surface) admits a (pseudo-) étale cover dominating a variety of general type. What happens if we modify the variety (or pair) without changing the fundamental group?

Problem 3.11 (Geometric puncturing problem) Let X be a projective variety with canonical singularities and Z a subvariety of codimension ≥ 2. Assume that no (pseudo-) étale cover of (X, ∅) dominates a variety of general type. Then (X, Z) admits no pseudo-étale covers dominating a pair of log general type. A weaker version would be to assume that X and Z are smooth.

By definition, a pseudo-étale cover of (X, Z) is a pseudo-étale cover of a pair (X′, D′), where X′ is proper over X and X′ \ D′ ≃ X \ Z.

Proposition 3.12 Assume X and Z are as in Problem 3.11, and that X is smooth. Then

a) No pseudo-étale covers of (X, Z) dominate a curve of log general type.

b) No pseudo-étale covers of (X, Z) dominate a variety of log general type of the same dimension.

Proof. Suppose we have a pseudo-étale cover ρ : (X_{1}, D_{1}) → (X, Z) and a dominant morphism φ : (X_{1}, D_{1}) → (X_{2}, D_{2}) to a variety of log general type. By Remark 3.3, we may take the X_{i} smooth and the D_{i} normal crossings. Since D_{1} is exceptional with respect to ρ, Iitaka’s Covering Theorem ([13] Theorem 10.5) yields an equality of Kodaira dimensions

κ(K_{X}) = κ(K_{X_{1}} + D_{1}).

Assume first that X_{2} is a curve. We claim it has genus zero or one. Let X^{norm} be the normalization of X in the function field of X_{1}. The induced morphism g : X^{norm} → X is finite, surjective, and branched only over Z, a codimension ≥ 2 subset of X. Since X is smooth, it follows that g is étale (see SGA II X §3.4 [10]). If X_{2} has genus ≥ 2 then φ : X_{1} → X_{2} is constant along the fibers of X_{1} → X^{norm}, and thus descends to a map X^{norm} → X_{2}. This would contradict our assumption that no étale cover of X dominates a variety of general type.

Choose a point p ∈ D_{2} and consider the divisor F = φ^{−1}(p). Note that 2F moves because 2p moves on X_{2}. However, 2F is supported in D_{1}, which lies in the exceptional locus for ρ, and we obtain a contradiction.

Now assume φ is generically finite. We apply the Logarithmic Ramification Formula to φ (see [13] Theorem 11.5)

K_{X_{1}} + D_{1} = φ^{∗}(K_{X_{2}} + D_{2}) + R

where R is the (effective) logarithmic ramification divisor. Applying the Covering Theorem again, we find that κ(K_{X_{1}}+D_{1}−R) = κ(K_{X_{2}}+D_{2}) = dim(X). It follows that K_{X_{1}} + D_{1} is also big, which contradicts the assumption that X is not of general type. □

Problem 3.13 (Arithmetic puncturing problem) Let X be a projective variety with canonical singularities and Z a subvariety of codimension ≥ 2. Assume that rational points on X are potentially dense. Are integral points on (X, Z) potentially dense?

For simplicity, one might first assume that X and Z are smooth.

Remark 3.14 Assume that Problem 3.13 has a positive solution. Then potential density of rational points holds for all K3 surfaces.

Indeed, if Y is a K3 surface of degree 2n then potential density of rational points holds for the symmetric product X = Y^{(n)} (see [12]). Denote by Z the large diagonal in X and by Δ the large diagonal in Y^{n} (the ordinary product). Assume that integral points on (X, Z) are potentially dense. Then, by Theorem 3.4 integral points on (Y^{n}, Δ) are potentially dense. This implies potential density for rational points on Y.

4 The fibration method and nondegenerate multisections

This section is included as motivation. Let B be an algebraic variety, defined over a number field K and π : G → B be a group scheme over B. We will be mostly interested in the case when the generic fiber is an abelian variety or a split torus 𝔾_{m}^{n}. Let s be a section of π. Shrinking the base we may assume that all fibers of G are smooth. We will say that s is nondegenerate if ∪_{n}s^{n} is Zariski dense in G.

Problem 4.1 (Specialization) Assume that G → B has a nondegenerate section s. Describe the set of b ∈ B(K) such that s(b) is nondegenerate in the fiber G_{b}.

For simple abelian varieties over a field a point of infinite order is nondegenerate. If E → B is a Jacobian elliptic fibration with a section s of infinite order then this section is automatically nondegenerate, and s(b) is nondegenerate if it is nontorsion. By a result of Néron (see [26] 11.1), the set of b ∈ B(K) such that s(b) is not of infinite order is thin; this holds true for abelian fibrations of arbitrary dimension.

For abelian fibrations A → B with higher-dimensional fibers, one must also understand how rings of endomorphisms specialize. The set of b ∈ B(K) for which the restriction

End(A) → End(A(b))

fails to be surjective is also thin; this is a result of Noot [21] Corollary 1.5. In particular, a nondegenerate section of a family of generically simple abelian varieties specializes to a nondegenerate point outside a thin set of fibers.

More generally, given an arbitrary abelian fibration A → B and a nondegenerate section s, the set of b ∈ B(K) such that s(b) is degenerate is thin in B. (We are grateful to Prof. Masser for pointing out the proof.) After replacing A by an isogenous abelian variety and taking a finite extension of the function field K(B), we obtain a family A′ → B′ with A′ ≃ A_{1}^{r_{1}} × … × A_{m}^{r_{m}}, where the A_{j} are (geometrically) simple and mutually non-isogenous. By the Theorems of Néron and Noot, the A_{j}(b′) are simple and mutually nonisogenous away from some thin subset of B′. A section s′ of A′ → B′ is nondegenerate iff its projection onto each factor A_{j}^{r_{j}} is nondegenerate; for b′ not contained in our thin subset, s′(b′) is nondegenerate iff its projection onto each A_{j}^{r_{j}}(b′) is nondegenerate. Hence we are reduced to proving the claim for each A_{j}^{r_{j}}. Since A_{j} is simple, a section s_{j} of A_{j}^{r_{j}} is nondegenerate iff its projections s_{j,1}, …, s_{j,r_{j}} are linearly independent over End(A_{j}). Away from a thin subset of B′, the same statement holds for the specializations to b′. However, Néron’s theorem implies that s_{j,1}(b′), …, s_{j,r_{j}}(b′) are linearly independent away from a thin subset.

Remark 4.2 There are more precise versions of Néron’s Theorem due to Demyanenko, Manin and Silverman (see [28], for example). Masser has proposed another notion of what it means for a subset of B(K) to be small, known as ‘sparcity’. For instance, the endomorphism ring of a family of abelian varieties changes only on a ‘sparse’ set of rational points of the base (see [18]). For an analogue to Néron’s Theorem, see [17].

Similar results hold for algebraic tori and are proved using a version of Néron’s Theorem for 𝔾_{m}^{n}-fibrations (see [26] pp. 154). A sharper result (for one-dimensional bases B) can be obtained from the following recent theorem:

Theorem 4.3 ([4]) Let C be an absolutely irreducible curve defined over a number field K and x_{1}, ..., x_{r} rational functions in K(C), multiplicatively independent modulo constants. Then the set of algebraic points p ∈ C(\overline{ℚ}) such that x_{1}(p), ..., x_{r}(p) are multiplicatively dependent has bounded height.

The main idea of the papers [11], [3], [12] can be summarized as follows. We work over a number field K and we assume that all geometric data are defined over K. Let π : E → B be a Jacobian elliptic fibration over a one dimensional base B. This means that we have a family of curves of genus one and a global zero section so that every fiber is in fact an elliptic curve. Suppose that we have another section s which is of infinite order in the Mordell-Weil group of E(K(B)). The specialization results mentioned above show that for a Zariski dense set of b ∈ B(K) the restriction s(b) is of infinite order in the corresponding fiber E_{b}. If K-rational points on B are Zariski dense then rational points on E are Zariski dense as well.

Let us consider a situation when E does not have any sections but instead has a multisection M. By definition, a multisection (resp. rational multisection) M is irreducible and the induced map M → B is finite flat (resp. generically finite) of degree deg(M). The base-changed family E ×_{B} M → M has the identity section Id (i.e., the image of the diagonal under M ×_{B} M → E ×_{B} M) and a (rational) section

τ_{M} := deg(M)Id − Tr(M ×_{B} M)

where Tr(M ×_{B} M) is obtained (over the generic point) by summing all the points of M ×_{B} M. By definition, M is nondegenerate if τ_{M} is nondegenerate.

When we are concerned only with rational points, we will ignore the distinction between multisections and rational multisections, as every rational multisection is a multisection over an open subset of the base. However, this distinction is crucial when integral points are considered.

If M is nondegenerate and if rational points on M are Zariski dense then rational points on E are Zariski dense (see [3]).

Example 4.4 ([11]) Let X be a quartic surface in ℙ^{3} containing a line L. Consider planes ℙ^{2} passing through this line. The residual curve has degree 3. Thus we obtain an elliptic fibration on X together with the trisection L. If L is ramified in a smooth fiber of this fibration then the multisection is nondegenerate and rational points are Zariski dense.

This argument generalizes to abelian fibrations π : A → B. However, we do not know of any simple geometric conditions insuring nondegeneracy of a (multi)section in this case. We do know that for any abelian variety A over K there exists a finite extension K′/K with a nondegenerate point in A(K′) (see [12]). This allows us to produce nondegenerate sections over function fields.

Proposition 4.5 Let Y be a Fano threefold of type W_{2}, that is a double cover of ℙ^{3} ramified in a smooth surface of degree 6. Then rational points on the symmetric square Y^{(2)} are potentially dense.

Proof. Observe that the symmetric square Y^{(2)} is birational to an abelian surface fibration over the Grassmannian of lines in ℙ^{3}. This fibration is visualized as follows: consider two generic points in Y. Their images in ℙ^{3} determine a line, which intersects the ramification locus in 6 points and lifts to a (hyperelliptic) genus two curve on Y. On Y^{(2)} we have an abelian surface fibration corresponding to the degree two component of the relative Picard scheme. Now we need to produce a nondegenerate multisection. Pick two generic points b_{1} and b_{2} on the branch surface. The preimages in Y of the corresponding tangent planes are K3 surfaces Σ_{1} and Σ_{2}, of degree two with ordinary double points at the points of tangency. The surfaces Σ_{1} and Σ_{2} therefore have potentially dense rational points (this was proved in [3]), as does Σ_{1} × Σ_{2}. This is our multisection; we claim it is nondegenerate for generic b_{1} and b_{2}. Indeed, it suffices to show that given a (generic) point in Y^{(2)}, there exist b_{1} and b_{2} so that Σ_{1} × Σ_{2} contains the point. Observe that through a (generic) point of ℙ^{3}, there pass many tangent planes to the branch surface. □

Remark 4.6 Combining the above Proposition with the strong form of Problem 3.13 we obtain potential density of rational points on a Fano threefold of type W_{2} - the last family of smooth Fano threefolds for which potential density is not known.

Here is a formulation of the fibration method useful for the analysis of integral points:

Proposition 4.7 Let B be a scheme over a number field K, G → B a flat group scheme, T → B an étale torsor for G, and M ⊂ T a nondegenerate multisection over B. If M has potentially dense integral points then T has potentially dense integral points.

Proof. Without loss of generality, we may assume that B is geometrically connected and smooth. The base-changed family T ×_{B} M dominates T, so it suffices to prove density for T ×_{B} M. Note that since M is finite and flat over B, τ_{M} is a well-defined section over all of M (i.e., it is not just a rational section). Hence we may reduce to the case of a group scheme G → B with a nondegenerate section τ.

We may choose models G and B over Spec(O_{S}) so that G → B is a group scheme with section τ. We may also assume that S-integral points of τ are Zariski dense. The set of multiples τ^{n} of τ, each a section of G → B, is dense in G by the nondegeneracy assumption. Since each has dense S-integral points, it follows that S-integral points are Zariski dense. □

A similar argument proves the following

Proposition 4.8 Let φ : X → ℙ^{1} be a K3 surface with elliptic fibration. Let M be a multisection over its image φ(M), nondegenerate and contained in the smooth locus of φ. Let F_{1}, …, F_{n} be fibers of φ and D a divisor supported in these fibers and disjoint from M. If M has potentially dense integral points then (X, D) has potentially dense integral points.

Proof. We emphasize that X is automatically minimal and the fibers of φ are reduced (see [3]). Our assumptions imply that M is finite and flat over φ(M).

After base-changing to M, we obtain a Jacobian elliptic fibration X′ := X ×_{ℙ^{1}} M with identity and a nondegenerate section τ_{M}. Let G ⊂ X′ be the open subset equal to the connected component of the identity. Since D′ := D ×_{ℙ^{1}} M is disjoint from the identity, it is disjoint from G. Hence it suffices to show that G has potentially dense integral points.

We assumed that M is contained in the smooth locus of φ, so τ_{M} is contained in the grouplike part of X′, and some multiple of τ_{M} is contained in G. Repeating the argument for Proposition 4.7 gives the result. □

5 Approximation techniques

In this section we prove potential density of integral points for certain pairs (X, D) using congruence conditions to control intersections with the boundary. Several of these examples are included as support for the statement of Problem 3.13.

Proposition 5.1 Let G = ∏_{j}^{N} G_{j} where G_{j} are algebraic tori 𝔾_{m} or geometrically simple abelian varieties. Let Z be a subvariety in G of codimension > μ = max_{j}(dim(G_{j})) and let U = G \ Z be the complement. Then integral points on U are potentially dense.

Proof. We are grateful to Prof. McKinnon for inspiring the following argument.

The proof proceeds by induction on the number of components N. The base case N = 1 follows from the fact that rational points on tori and abelian varieties are potentially dense, so we proceed with the inductive step. Consider the projections π′ : G → G′ = ∏_{j≠N} G_{j} and π_{N} : G → G_{N}. By assumption, generic fibers of π′ are geometrically disjoint from Z.

Choose a ring of integers O_{S} and models G_{j} over Spec(O_{S}). We assume that each G_{j} is smooth over Spec(O_{S}) and that G_{N} has a nondegenerate Sintegral point q (see [12], for example, for a proof of the existence of such points on abelian varieties).

Let T be any subscheme of G_{N} supported over a finite subset of Spec(O_{S}) such that G_{N} has an S-integral point p_{N} disjoint from T . We claim that such integral points are Zariski dense. Indeed, for some m > 0 we have

mq ≡ 0 (mod 𝔭)

for each 𝔭 ∈ Spec(O_{S}) over which T has support. Hence we may take the translations of p_{N} by multiples of mq.

After extending O_{S}, we may assume U has at least one integral point p = (p′, p_{N}) so that π′^{−1}(p′) and π_{N}^{−1}(p_{N}) intersect Z in the expected dimensions. In particular, π′^{−1}(p′) is disjoint from Z. By the inductive hypothesis, we may extend O_{S} so that

(π_{N}^{−1}(p_{N}) ≃ G′, π_{N}^{−1}(p_{N}) ∩ Z)

has dense integral points. In particular, almost all such integral points are not contained in π′(Z), a closed proper subscheme of G′. Let r be such a point, so that F_{r} = π′^{−1}(r) ≃ G_{N} intersects Z in a subscheme T supported over a finite number of primes. Since (r, p_{N}) ∈ F_{r} is disjoint from T , the previous claim implies that the integral points of F_{r} disjoint from T are Zariski dense. As r varies, we obtain a Zariski dense set of integral points on G \ Z. □

Corollary 5.2 Let X be a toric variety and Z ⊂ X a subvariety of codimension ≥ 2, defined over a number field. Then integral points on (X, Z) are potentially dense.

Another special case of the Arithmetic puncturing problem 3.13 is the following:

Problem 5.3 Are integral points on punctured simple abelian varieties of dimension n > 1 potentially dense?

Example 5.4 Potential density of integral points holds for simple abelian varieties punctured in the origin, provided that their ring of endomorphisms contains units of infinite order.

6 Conic bundles and integral points

Let K be a number field, S a finite set of places for K (including all the infinite places), O_{S} the corresponding ring of S-integers, and η ∈ Spec(O_{S}) the generic point. For each place v of K, let K_{v} be the corresponding complete field and 𝔬_{v} the discrete valuation ring (if v is nonarchimedean). As before, we use calligraphic letters (e.g., X) for schemes (usually flat) over O_{S} and roman letters (e.g., X) for the fiber over η.

6.1 Results on linear algebraic groups

Consider a linear algebraic group G/K. Choose a model G for G over O_{S}, i.e., a flat group scheme of finite type G/O_{S} restricting to G at the generic point. This may be obtained by fixing a representation G ↪ GL_{n}(K) (see also [32] §10-11). The S-rank of G (denoted rank(G, O_{S})) is defined as the rank of the abelian group of sections of G(O_{S}) over O_{S}. This does not depend on the choice of a model. Indeed, consider two models G_{1} and G_{2} with a birational map b : G_{1} −−→ G_{2}; of course, b is an isomorphism over the generic point and the proper transform of the identity section I_{1} is the identity. There is a subscheme Z ⊂ Spec(O_{S}) with finite support such that the indeterminacy of b is in the preimage of Z. It follows that the sections of G_{1} congruent to I_{1} modulo Z have proper transforms which are sections of G_{2}. Such sections form a finite-index subgroup of G_{1}(O_{S}).

Let 𝔾_{m} be the multiplicative group over ℤ, i.e., Spec(ℤ[x, y]/ ⟨xy − 1⟩); it can be defined over an arbitrary scheme by extension of scalars. There is a natural projection

𝔾_{m}(ℤ) → Spec(ℤ[x]) = 𝔸_{ℤ}^{1} ⊂ ℙ_{ℤ}^{1}

so that ℙ_{ℤ}^{1} \ 𝔾_{m}(ℤ) = {0, ∞}. A form of 𝔾_{m} over K is a group scheme G/K for which there exists a finite field extension K′/K and an isomorphism G ×_{K} K′ ≃ 𝔾_{m}(K′). These are classified as follows (see [23] for a complete account). Any group automorphism

α : 𝔾_{m}(K′) → 𝔾_{m}(K′)

is either inversion or the identity, depending on whether it exchanges 0 and ∞. The corresponding automorphism group is smooth, so we may work in the étale topology (see [19] Theorem 3.9). In particular,

K − forms of 𝔾_{m} ≃ H_{ét}^{1}(Spec(K), ℤ/2ℤ).

Each such form admits a natural open imbedding into a projective curve G ↪ X, generalizing the imbedding of 𝔾_{m} into ℙ^{1}. The complement D = X \ G consists of two points. The Galois action on D is given by the cocycle in H_{ét}^{1}(Spec(K), ℤ/2ℤ) classifying G.

There is a general formula for the rank of a torus T due to T. Ono and J.M. Shyr (see [22], Theorem 6 and [27]). Let T̂ (resp. T̂_{v}) the group of characters defined over K (resp. K_{v}), and ρ(T) (resp. ρ(T, v)) the number of independent elements of T̂ (resp. T̂_{v}). The formula takes the form

rank(T, O_{S}) = ∑_{v∈S} ρ(T, v) − ρ(T).

For forms of 𝔾_{m} this is particularly simple. For split forms

rank(𝔾_{m}, O_{S}) = #{places v ∈ S} − 1.

Now let G/K be a nonsplit form, corresponding to the quadratic extension K′/K, and S′ the places of K′ lying over the places of S. Then we have

rank(G, O_{S}) = #{places v ∈ S completely splitting in S′}.

6.2 Group actions and integral points

Throughout this subsection, X is a normal, geometrically connected scheme and X → Spec(O_{S}) a flat projective morphism. Let D ⊂ X be an effective reduced Cartier divisor. Contrary to our previous conventions, we do not assume that D is flat over O_{S}. Assume that a linear algebraic group G acts on X so that X \ D is a G-torsor.

Proposition 6.1 There exists a model G for G such that G acts on X and stabilizes D.

Proof. Choose an imbedding X ↪ ℙ_{O_{S}}^{n} and a compatible linearization G ↪ GL_{n+1}(K) (see [20], Ch. 1, Cor. 1.6 and Prop. 1.7). Let G′ ↪ GL_{n+1}(O_{S}) be the resulting integral model of G, so that G′ stabilizes the ideal of X and therefore acts on it. Furthermore, G′ evidently stabilizes the irreducible components of D dominating O_{S}. The fibral components of D are supported over a finite subset of Spec(O_{S}). We take G ⊂ G′ to be the subgroup acting trivially over this subset; it has the desired properties. □

Proposition 6.2 Assume (X , D) has an S-integral point and that G has positive O_{S}-rank. Then (X , D) has an infinite number of S-integral points.

Proof. Consider the action of G(O_{S}) on the integral point σ (which has trivial stabilizer). The orbit consists of S-integral points of (X , D), an infinite collection because G has positive rank. □

Now assume that X is a smooth rational curve. A rational section (resp. bisection) D ⊂ X is a reduced effective Cartier divisor such that the generic fiber D is reduced of degree one (resp. two). Note that the open curve X \D is geometrically isomorphic to ℙ^{1} − {∞} (resp. ℙ^{1} − {0, ∞}), and thus is a torsor for some K-form G of 𝔾_{a} (resp. 𝔾_{m}). This form is easily computed. Of course, 𝔾_{a} has no nontrivial forms. In the 𝔾_{m} case, we can regard D_{η} as an element of H_{ét}^{1}(Spec(K), ℤ/2ℤ), which gives the descent data for G.

The following result is essentially due to Beukers (see [1], Theorem 2.3):

Proposition 6.3 Let (X , D) → Spec(O_{S}) be a rational curve with rational bisection and G the corresponding form of 𝔾_{m} (as described above). Assume that (X , D) has an S-integral point and rank(G, O_{S}) > 0. Then S-integral points of (X , D) are Zariski dense.

Proof. This follows from Proposition 6.2. Given an S-integral point σ of (X , D), the orbit G(O_{S})σ is infinite and thus Zariski dense. □

Combining with the formula for the rank, we obtain the following:

Corollary 6.4 Let (X , D) → Spec(O_{S}) be a rational curve with rational bisection such that (X , D) has an S-integral point. Assume that either

a) D is reducible over Spec(K) and |S| > 1; or

b) D is irreducible over Spec(K) and at least one place in S splits completely in K(D).

Then S-integral points of (X , D) are Zariski dense.

When D is a rational section we obtain a similar result (also essentially due to Beukers [1], Theorem 2.1):

Proposition 6.5 Let (X , D) → Spec(O_{S}) be a rational curve with rational section such that (X , D) has an S-integral point. Then S-integral points of (X , D) are Zariski dense.

6.3 v-adic geometry

For each place v ∈ S, consider the projective space ℙ^{1}(K_{v}) as a manifold with respect to the topology induced by the v-adic absolute value on K_{v}.

For simplicity, this will be called the v-adic topology; we will use the same term for the induced subspace topology on ℙ^{1}(K). Given an étale morphism of curves f : U → ℙ^{1} defined over K_{v}, we will say that f(U(K_{v})) is a basic étale open subset. These are open in the v-adic topology, either by the open mapping theorem (in the archimedean case) or by Hensel’s lemma (in the nonarchimedean case).

Let

χ_{f}(B) := #{z ∈ O_{{v}} : |z|_{v} ≤ B and z ∈ f(U(K_{v}))}

where B is a positive integer and

O_{{v}} := {z ∈ K : |z|_{w} ≤ 1 for each w ≠ v}.

We would like to estimate the quantity

μ_{f} := lim inf_{B→∞} χ_{f}(B)/χ_{Id}(B)

i.e., the fraction of the integers contained in the image of the v-adic points of U.

Proposition 6.6 Let f : U → ℙ^{1} be an étale morphism defined over K_{v} and f_{1} : C → ℙ^{1} a finite morphism of smooth curves extending f. If there exists a point q ∈ f_{1}^{−1}(∞) ∩ C(K_{v}) at which f_{1} is unramified then μ_{f} = 1.

Proof. This follows from the fact that f(U(K_{v})) is open if f is étale along U. □

As an illustrative example, we take K = ℚ and K_{v} = ℝ, so that O_{{v}} = ℤ. The set f(U(ℝ)) is a finite union of open intervals (r, s) with r, s ∈ ℝ ∪ {∞}, where the (finite) endpoints are branch points. We observe that

μ_{f} = \begin{cases}0 & if f(U(ℝ)) is bounded; \\ 1/2 & if \overline{f(U(ℝ))} contains a one-sided neighborhood of ∞; \\ 1 & if \overline{f(U(ℝ))} contains a two-sided neighborhood of ∞. \end{cases}

We can read off easily which alternative occurs in terms of the local behavior at infinity. Let f_{1} : C → ℙ^{1} be a finite morphism of smooth curves extending f. If f_{1}^{−1}(∞) has no real points then μ_{f} = 0. If f_{1}^{−1}(∞) has unramified (resp. ramified) real points then μ_{f} = 1 (resp. μ_{f} > 0.)

We specialize to the case of double covers:

Proposition 6.7 Let U → ℙ^{1} be an étale morphism defined over K_{v} and f_{1} : C → ℙ^{1} a finite morphism of smooth curves extending f. Assume that f_{1} has degree two and ramifies at q ∈ f_{1}^{−1}(∞). Then μ_{f} > 0.

Proof. Of course, q is necessarily defined over K_{v}. The archimedean case follows from the previous example, so we restrict to the nonarchimedean case. Assume f_{1} is given by

y^{2} = c_{n}z^{n} + c_{n−1}z^{n−1} + … + c_{0},

where z is a coordinate for the affine line in ℙ^{1}(K_{v}), c_{n} ≠ 0, and the c_{i} ∈ 𝔬_{v}. Substituting z = 1/t and y = x/t^{⌈n/2⌉}, we obtain the equation at infinity

\begin{cases}x^{2} = c_{n} + c_{n−1}t + … + c_{0}t^{n} & for n even \\ x^{2} = c_{n}t + c_{n−1}t^{2} + … + c_{0}t^{n} & for n odd \end{cases} .

If n is even then f_{1}^{−1}(∞) consists of two non-ramified points, so we may assume n odd. Then f_{1}^{−1}(∞) consists of one ramification point q, necessarily defined over K_{v}.

Write c_{n} = u_{0}π^{α} and z = u_{1}π^{−β}, where u_{0} and u_{1} are units and π is a uniformizer in 𝔬_{v}. (We may assume that some power π^{r} is contained in O_{K}.) Our equation takes the form

y^{2}π^{nβ−α} = u_{0}u_{1}^{n} + c_{n−1}u_{1}^{n−1}π^{β−α} + … + c_{0}u_{1}π^{nβ−α}.  (1)

We review a property of the v-adic numbers, (proved in [25], Ch. XIV §4). Consider the multiplicative group

U^{(m)} := {u ∈ 𝔬_{v} : u ≡ 1 (mod π^{m})}.

Then for m sufficiently large we have U^{(m)} ⊂ K_{v}^{2}. In particular, to determine whether a unit u is a square, it suffices to consider its representative mod π^{m}.

Consequently, if β is sufficiently large and has the same parity as α, then we can solve Equation 1 for y ∈ K_{v} precisely when u_{0}u_{1} is a square. For example, choose any M ∈ O_{K} so that M ≡ u_{0}π^{(r−1)β} (mod π^{rβ}) and set z = M/π^{rβ} ∈ O_{{v}}. Hence, of the z ∈ O_{{v}} with |z|_{v} ≤ B (with B ≫ 0), the fraction satisfying our conditions is bounded from below. It follows that μ_{f} > 0. □

Now let f : U → ℙ^{1} be an étale morphism of curves defined over K. Consider the function

ω_{f,S}(B) := #{z ∈ O_{S} : |z|_{v} ≤ B for each v ∈ S and z ∈ f(U(K))}

and the quantity

lim sup_{B→∞} ω_{f,{v}}(B)/χ_{f}(B).

We expect that this is zero provided that f does not admit a rational section. We shall prove this is the case when f has degree two.

A key ingredient of our argument is a version of Hilbert’s Irreducibility Theorem:

Proposition 6.8 Let f : U → ℙ^{1} be an étale morphism of curves, defined over K and admitting no rational section. Then we have

lim sup_{B→∞} ω_{f,{v}}(B)/χ_{Id}(B) = 0.

Proof. We refer the reader to Serre’s discussion of Hilbert’s irreducibility theorem ([26], §9.6, 9.7). Essentially the same argument applies in our situation. □

Combining Propositions 6.6, 6.7, and 6.8, we obtain:

Corollary 6.9 Let f : C → ℙ^{1} be a finite morphism of smooth curves defined over K. Assume that f admits no rational section and that f^{−1}(∞) contains a K_{v}-rational point. We also assume that f has degree two. Then we have

lim sup_{B→∞} ω_{f,{v}}(B)/χ_{f}(B) = 0.

In particular, the set {z ∈ O_{{v}} : z ∈ f(C(K_{v})) \ f(C(K))} is infinite.

6.4 A density theorem for surfaces

Geometric assumptions: Let X and B be flat and projective over Spec(O_{S}) and ϕ : X → B be a morphism. Let L ⊂ X be a closed irreducible subscheme, D ⊂ X a reduced effective Cartier divisor, and 𝔮 := D ∩ L. We assume the generic fibers satisfy the following: X is a geometrically connected surface, B a smooth curve, ϕ : X → B a flat morphism such that the generic fiber is a rational curve with bisection D. We also assume L ≃ ℙ_{K}^{1}, ϕ|L is finite, and L meets D at a single point q, at which D is nonsingular. Write X′ for X ×_{B} L, D′ for D ×_{B} L, L′ for the image of the diagonal in X ×_{B} L (now a section for ϕ′ : X′ → L), and 𝔮′ for L′ ∩ D′. Finally, if C′ denotes the normalization of the union of the irreducible components of D′ dominating L, we assume that C′ → L has no rational section over K (i.e., that C′ is irreducible over K).

Arithmetic assumptions: We assume that (L, 𝔮) has an S-integral point. Furthermore, we assume that for some v ∈ S, C′ has a K_{v}-rational point lying over ϕ′(q′).

Remark 6.10 The second assumption is valid if any of the following are satisfied:

1. D → B is unramified at q.

2. D → B is finite (but perhaps ramified) at q and L → B has ramification at q of odd order.

3. D → B is finite (but ramified) at q and L → B has ramification at q of order two. Choose local uniformizers t, x, and y so that we have local analytic equations t + ax^{2} = 0 and t + by^{2} = 0 (with a, b ∈ K) for D → B and L → B. We assume that ab is a square in K_{v}.

Note that in the last case, D′ and C′ have local analytic equations ax^{2}−by^{2} = 0 and x/y = ±\sqrt{b/a} respectively.

Theorem 6.11 Under the geometric and arithmetic assumptions made above, S-integral points of (X , D) are Zariski dense.

Proof. It suffices to prove that S-integral points of (X′, D′) are Zariski dense. These map to S-integral points (X , D).

Consider first S-integral points of (L′, 𝔮′). These are dense by Proposition 6.5, and contain a finite index subgroup of 𝔾_{a}(O_{S}) ⊂ ℙ_{K}^{1}. Corollary 6.9 and our geometric assumptions imply that infinitely many of these points lie in ϕ′(C′(K_{v})) \ ϕ′(C′(K)).

Choose a generic S-integral point p of (L′, 𝔮′) as described above. Let X′_{p} = ϕ′^{−1}(p), D′_{p} = X′_{p} ∩ D′, and L′_{p} = X′_{p} ∩ L′, so that (X′_{p}, D′_{p}) is a rational curve with rational bisection and integral point L′_{p}. Combining the results of the previous paragraph with Proposition 6.3, we obtain that S-integral points of (X′_{p}, D′_{p}) are Zariski dense. As we vary p, we obtain a Zariski dense collection of integral points for (X′, D′). □

6.5 Cubic surfaces containing a line

Let X_{1} be a cubic surface in ℙ_{O_{S}}^{3}, D_{1} ⊂ X_{1} a hyperplane section, and L_{1} ⊂ X_{1} a line not contained in D_{1}, all assumed to be flat over Spec(O_{S}). Write 𝔮_{1} := D_{1} ∩ L_{1}, a rational section over Spec(O_{S}). Let ℙ_{O_{S}}^{3} −−→ B be the projection associated with L_{1}, X = Bl_{L_{1}}X_{1}, and ϕ : X → B the induced projection (of course, B = ℙ_{O_{S}}^{1} if O_{S} is a UFD). Let L ⊂ X be the proper transform of L_{1}, D ⊂ X the total transform of D_{1}, and 𝔮 = L ∩ D. We shall apply Theorem 6.11 to obtain density results for S-integral points of (X_{1}, D_{1}).

We will need to assume the following geometric conditions:

GA1 D_{1} is reduced everywhere and nonsingular at q_{1};

GA2 X_{1} has only rational double points as singularities, with at most one singularity along L_{1};

GA3 D_{1} is not the union of a line and a conic containing q_{1} (defined over K).

The first assumption and the fact that D_{1} is Cartier imply that X_{1} is nonsingular at q_{1}. We analyze the projection from the line L_{1} using the first two assumptions. This induces a morphism

ϕ : X → ℙ^{1}.

Of course, X = X_{1} if and only if L_{1} is Cartier in X_{1}, which is the case exactly when X_{1} is smooth along L_{1}. We use L and D to denote the proper transforms of L_{1} and D_{1}. Our three assumptions imply that D equals the total transform of D_{1} and has a unique irreducible component C dominating ℙ^{1}. We also have that the generic fiber of ϕ is nonsingular, intersects D in two points, and intersects L in two points (if X_{1} is smooth along L_{1}) or in one point (if X_{1} has a singularity along L_{1}). In particular, L is a bisection (resp. section) of ϕ if X_{1} is nonsingular (resp. singular) along L_{1}.

We emphasize that S-integral points of (X , D) map to S-integral points of (X_{1}, D_{1}), and all the Geometric Assumptions of Theorem 6.11 are satisfied except for the last one. The last assumption is verified if any of the following hold:

GA4a The branch loci of C → ℙ^{1} and L → ℙ^{1} do not coincide.

GA4b The curve C has genus one.

GA4c X_{1} has a singularity along L_{1}.

Clearly, either the second or the third condition implies the first.

We turn next to the Arithmetic Assumptions.

AA1 (L_{1}, 𝔮_{1}) has an S-integral point.

Note that S-integral points of (L_{1}, 𝔮_{1}) not lying in the singular locus of X_{1} → Spec(O_{S}) lift naturally to S-integral points of (L, 𝔮).

Our next task is to translate the conditions of Remark 6.10 to our situation. They are satisfied in any of the following contexts:

AA2a D_{1} is irreducible over K and q_{1} is not a flex of D_{1};

AA2b X_{1} has a singularity along L_{1};

AA2c D_{1} is irreducible over K and q_{1} is a flex of D_{1}. Let H be the hyperplane section containing L_{1} and the flex line. We assume that H ∩ X_{1} = L_{1} ∪ M, where M is a smooth conic.

AA2d D_{1} is irreducible over K but q_{1} is a flex so that the hyperplane H containing L_{1} and the flex line F intersects X_{1} in three coincident lines, i.e., H ∩X_{1} = L_{1} ∪M_{1} ∪M_{2}. Choose local coordinates x and y for H so that L_{1} = {x = 0}, F = {y = 0}, and M_{1}∪M_{2} = {ax^{2}+cxy+by^{2} = 0}. Then we assume that ab is a square in K_{v}.

AA2e D_{1} consists of a line and a conic C_{1} irreducible over K, intersecting in two distinct points, each defined over K_{v}.

In the first case, the map D → B is unramified at q. Note that in the second case L is a section for ϕ. In the third case, our assumption implies that L → B is unramified at q. In the last case, we observe that the points of L lying over ϕ(q) are defined over K_{v}, hence C′ has a K_{v}-rational point over ϕ′(q′).

It remains to show that AA2d allows us to apply case 3 of Remark 6.10. We fix projective coordinates on ℙ^{3} compatibly with the coordinates already chosen on H: y = 0 is the linear equation for the hyperplane containing D_{1}, z = 0 the equation for H, x = z = 0 the equations for L_{1}, and x = z = y = 0 the equations for q_{1}. Under our assumptions, the equations for D_{1} and X_{1} take the form

g := zw^{2} + ax^{3} + c_{1}wxz + c_{2}wz^{2} + c_{4}x^{2}z + c_{5}xz^{2} + c_{6}z^{3} = 0
f := g + cx^{2}y + bxy^{2} + yzℓ(w, x, y, z) = 0

where ℓ is linear in the variables. The conic bundle structure ϕ : X → B is obtained by making the substitution z = tx

g′ = tw^{2} + x(wc_{1}t + wc_{2}t^{2}) + x^{2}(a + c_{4}t + c_{5}t^{2} + c_{6}t^{3}) = 0
f′ = g′ + cxy + by^{2} + tyℓ(w, x, y, tx).

We analyze the local behavior of D → B at q using x as a coordinate for D. First dehomogenize

g′′ = t + x(c_{1}t + c_{2}t^{2}) + x^{2}(a + c_{4}t + c_{5}t^{2} + c_{6}t^{3}) = 0

and then take a suitable analytic change of coordinate on D to obtain t + aX^{2} = 0. To analyze L → B, we set x = 0 and use y as a coordinate

f′′ = t + by^{2} + tyℓ(1, 0, y, 0) = 0.

After a suitable analytic change of coordinate on L, we obtain t + bY^{2} = 0.

Remark 6.12 We further analyze condition AA2d when K_{v} = ℝ. Then ab is a square if and only if ab ≥ 0. This is necessarily the case if c^{2} − 4ab < 0, i.e., if the lines M_{1} and M_{2} are defined over an imaginary quadratic extension.

We summarize our discussion in the following theorem:

Theorem 6.13 Let X_{1} be a cubic surface, D_{1} ⊂ X_{1} a hyperplane section, and L_{1} ⊂ X_{1} a line not contained in D_{1}, all assumed to be flat over Spec(O_{S}). Write 𝔮_{1} := D_{1} ∩ L_{1}. Assume the following:

1. GA1,GA2,GA3, and AA1;

2. at least one of the assumptions GA4a,GA4b,or GA4c;

3. at least one of the assumptions AA2a,AA2b,AA2c,AA2d, or AA2e.

Then S-integral points of (X_{1}, D_{1}) are Zariski dense.

We recover the following result (essentially Theorem 2 of Beukers [2]):

Corollary 6.14 Let X_{1} be a cubic surface, D_{1} ⊂ X_{1} a hyperplane section, and L_{1} ⊂ X_{1} a line not contained in D_{1}, all assumed to be flat over Spec(ℤ). Write 𝔮_{1} := D_{1} ∩ L_{1}. Assume that

1. X_{1} and D_{1} are smooth;

2. there exists an ℤ-integral point of (L_{1}, 𝔮_{1});

3. if q is a flex of D_{1}, we assume that the hyperplane containing L_{1} and the flex line intersects X_{1} in a smooth conic and L_{1}.

Then ℤ-integral points of (X_{1}, D_{1}) are Zariski dense.

We also recover a weak version of Theorem 1 of [2]. (This theorem is asserted to be true but the proof is not quite complete; the problem occurs in the argument for the second part of Lemma 2.)

Corollary 6.15 Retain all the hypotheses of Corollary 6.14, except that we allow the existence of a hyperplane H intersecting X_{1} in three lines L_{1}, M_{1}, and M_{2} and containing a flex line F for D_{1} at q. Let p be a place for ℤ (either infinite or finite). Choose local coordinates x and y for H so that L_{1} = {x = 0}, F = {y = 0}, and M_{1} ∪ M_{2} = {ax^{2} + cxy + by^{2} = 0}, and assume that ab is a square in ℚ_{p}. Then ℤ[1/p]-integral points of (X_{1}, D_{1}) are Zariski dense (where ℤ[1/∞] = ℤ and ℚ_{∞} = ℝ.)

Of course, there are infinitely many primes p such that ab is a square in ℚ_{p}. When p = ∞, by Remark 6.12 it suffices to verify that M_{1} and M_{2} are defined over an imaginary quadratic extension.

We also obtain results in cases where the boundary is reducible:

Corollary 6.16 Let X_{1} be a cubic surface, D_{1} ⊂ X_{1} a hyperplane section, and L_{1} ⊂ X_{1} a line not contained in D_{1}, all assumed to be flat over Spec(ℤ). Write 𝔮_{1} := D_{1} ∩ L_{1}. Assume that

1. X_{1} is smooth;

2. there exists an S-integral point of (L_{1}, 𝔮_{1}); 3. D_{1} = E∪C, where E is a line intersecting L_{1} and C is a conic irreducible

over K;

4. C intersects E in two points, defined over K_{v} where v is some place in S;

5. there exists at most one conic in X_{1} tangent to both L_{1} and C.

Then S-integral points of (X , D) are Zariski dense.

Note that the assumption on the conics tangent to L_{1} and C is used to verify GA4a.

6.6 Other applications

Theorem 6.11 can be applied in many situations.

Theorem 6.17 Let X = ℙ_{O_{S}}^{1} × ℙ_{O_{S}}^{1}, D ⊂ X a divisor of type (2, 2), and L ⊂ X a ruling of X, all flat over O_{S}. Assume that

1. D is nonsingular;

2. L is tangent to D at q;

3. S-integral points of (L, 𝔮) are Zariski dense.

Then S-integral points of (X , D) are Zariski dense.

Proof. Let ϕ be the projection for which L is a section. Since C = D in this case, the second arithmetic assumption of Theorem 6.11 is easily satisfied. □

We also obtain the following potential density result:

Theorem 6.18 Let X be a smooth Del Pezzo surface of degree K_{X}^{2} and index one (i.e., ℤK_{X} is saturated in Pic(X)). Let D ⊂ X be a smooth anticanonical divisor. If X and D are defined over a number field K then integral points of (X, D) are potentially dense.

Proof. Applying the classification theory of surfaces (and enlarging the base field), we may represent X as a conic bundle ϕ : X → ℙ^{1}. First express X as a blow-up of ℙ^{2} in 9 − K_{X}^{2} points. The pencil of lines in ℙ^{2} containing one of the points w gives the conic bundle structure. Let L denote a (−1)- curve not contained in a fiber of ϕ, e.g., the exceptional curve lying over w. By adjunction, D is a bisection for ϕ and intersects L in one point q. Finally, the irreducible components of the normalization of D ×_{ℙ^{1}} L all have positive genus, because D has positive genus. In particular, X, D, L, and q satisfy all the geometric assumptions of Theorem 6.11. On the other hand, its arithmetic assumptions may always be satisfied after judicious extensions of O_{S}. It follows that integral points of (X, D) are potentially dense. □

7 Potential density for log K3 surfaces

We consider the following general situation:

Problem 7.1 (Integral points of log K3 surfaces) Let X be a surface and D a reduced effective Weil divisor such that (X, D) has log terminal singularities and K_{X} + D is trivial. Are integral points on (X, D) potentially dense?

Problem 7.1 has been studied when D = ∅ (see, for example, [3]). In this case density holds if X has infinite automorphisms or an elliptic fibration.

The case X = ℙ^{2} and D a plane cubic has also attracted significant attention. Silverman [29] proved potential density in the case where D is singular and raised the general case as an open question. Beukers [1] established this by considering the cubic surface X_{1} obtained as the triple cover of X totally branched over D.

Implicit in [2] is a proof of potential density when X_{1} is a smooth cubic surface and D_{1} is a smooth hyperplane section. Note that this also follows from Theorem 6.13 (cf. also Corollaries 6.14 and 6.15.) After suitable extensions of K and additions to S, there exists a line L ⊂ X defined over K and the relevant arithmetic assumptions are satisfied. Similarly, the case of X = ℙ^{1} × ℙ^{1} and D a smooth divisor of type (2, 2) follows from Theorem 6.17. Finally, Theorem 6.18 gives potential density when X is an index-one Del Pezzo surface and D is a smooth anticanonical divisor.

We summarize our results as follows:

Theorem 7.2 Let X be a smooth Del Pezzo surface and D a smooth anticanonical divisor. Then integral points for (X, D) are potentially dense.

We close this section with a list of open special cases of Problem 7.1.

1. Let X be a Del Pezzo surface and D a singular anticanonical cycle. Show that integral points for (X, D) potentially dense.

2. Let X be a Hirzebruch surface and D an anticanonical cycle. Find a smooth rational curve L, intersecting D in exactly one point p, so that the induced map φ : L → ℙ^{1} is finite surjective.

7.1 Appendix: some geometric remarks

The reader will observe that the methods employed to prove density for integral points on conic bundles (with bisection removed) are not quite analogous to the methods used for elliptic fibrations. The discrepancy can be seen in a number of ways. First, given a multisection M for a conic bundle (with bisection removed), we can pull-back the conic bundle to the multisection. The resulting fibration has two rational sections, Id and τ_{M} (see section 4). However, a priori one cannot control how τ_{M} intersects the boundary divisor (clearly, this is irrelevant if the boundary is empty). A second explanation may be found in the lack of a good theory of (finite type) Néron models for algebraic tori (see chapter 10 of [5]).

We should remark that in some special cases these difficulties can be overcome, so that integral points may be obtained by geometric methods completely analogous to those used for rational points. Consider the cubic surface

x^{3} + y^{3} + z^{3} = 1

with distinguished hyperplane at infinity. This surface contains a line with equations x + y = z − 1 = 0. Euler showed that the resulting conic bundle admits a multisection (x_{0}, y_{0}, z_{0}) = (9t^{4}, 3t − 9t^{4}, 1 − 9t^{3}), which may be reparametrized as (x_{1}, y_{1}, z_{1}) = (9t^{4}, −3t − 9t^{4}, 1 + 9t^{3}). Lehmer [15] showed that this is the first in a sequence of multisections, given recursively by

(x_{n+1}, y_{n+1}, z_{n+1}) = 2(216t^{6} − 1)(x_{n}, y_{n}, z_{n}) − (x_{n−1}, y_{n−1}, z_{n−1})
+ (−108t^{4}, −108t^{4}, 216t^{6} + 4)

This is related to the fact that the norm group scheme

u^{2} − 3(108t^{6} − 1)v^{2} = 1,

admits a section of infinite order (u, v) = (216t^{6} − 1, 12t^{3}). Remarkably, this group acts regularly on the conic bundle, i.e., the coordinate transformations are integral polynomials in t. In general, one would only expect a rational action, defined over the generic point of the t-line.

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