TRANSCENDENTAL OBSTRUCTIONS TO WEAK APPROXIMATION ON GENERAL K3 SURFACES

BRENDAN HASSETT, ANTHONY VÁRILLY-ALVARADO, AND PATRICK VARILLY

Abstract. We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

1. Introduction

Let X be a smooth projective geometrically integral variety over a number field k. Assume that the set X(𝔸_{k}) of adelic points is nonempty; we say that X satisfies weak approximation if the natural embedding X(k) ↪ X(𝔸_{k}) is dense for the adelic topology. Following Manin, we may use the Brauer group Br(X) := H_{ét}^{2}(X, 𝔾_{m}) to construct a set X(𝔸_{k})^{Br} ⊆ X(𝔸_{k}) that contains the closure of X(k) in X(𝔸_{k}). If X(𝔸_{k}) \ X(𝔸_{k})^{Br} ≠ ∅, then we say there is a Brauer-Manin obstruction to weak approximation on X. In this way, the group Br X may be used to obtain qualitative information about the distribution of rational points on X.

Fix an algebraic closure k̄ of k, and let \overline{X} = X ×_{k} k̄. Write Br_{0}(X) for the image of Br k in Br(X), and let Br_{1}(X) = ker ( Br(X) → Br(\overline{X})). An element of Br(X) is algebraic if it lies in Br_{1}(X), transcendental otherwise. The first examples of transcendental Brauer elements in an arithmetic context are due to Harari [Har96].

If X is a curve or a rational surface, then Br(\overline{X}) = 0, and thus the Brauer group is entirely algebraic. In contrast, for K3 surfaces the group Br(X)/ Br_{1}(X) can be nontrivial, albeit finite [SZ08, Theorem 1.2]. For these surfaces, the arithmetically interesting groups Br(X)/ Br_{0}(X), Br(X)/ Br_{1}(X) and Br_{1}(X)/ Br_{0}(X) are the object of much recent research [Bri06, SZ08, ISZ09, KT09, LvL09, SZ09, Cor10]. Explicit transcendental elements of Br(X), or the lack thereof, play a central role in [Wit04, SSD05, HS05, Ier10, ISZ09]. In all cases, the K3 surfaces considered are endowed with an elliptic fibration, and this additional structure is essential to the computation of transcendental classes. A ‘general’ K3 surface (i.e., one with geometric Picard rank one), however, does not carry such a structure. With this in mind, our main result is as follows.

Theorem 1.1. Let X be the K3 surface of degree 2 given by

(1) w^{2} = det (\begin{matrix}2(2x + 3y + z) & 3x + 3y & 3x + 4y & 3y^{2} + 2z^{2} \\ 3x + 3y & 2z & 3z & 4y^{2} \\ 3x + 4y & 3z & 2(x + 3z) & 4x^{2} + 5xy + 5y^{2} \\ 3y^{2} + 2z^{2} & 4y^{2} & 4x^{2} + 5xy + 5y^{2} & 2(2x^{3} + 3x^{2}z + 3xz^{2} + 3z^{3}) \end{matrix})

in ℙ_{ℚ}(1, 1, 1, 3). Then X has geometric Picard rank one, and there is a transcendental Brauer-Manin obstruction to weak approximation on X. The obstruction arises from a quaternion Azumaya algebra 𝒜 ∈ Br (k(X)) in the image of the natural injection Br(X) ↪ Br (k(X)). Explicitly, if M_{i} denotes the i-th principal minor of the matrix in (1), then

This research was supported by National Science Foundation Grants 0554491 and 0901645.

𝒜 = (−\frac{M_{2}}{M_{1}^{2}} , −\frac{M_{3}}{M_{1}M_{2}} ) .

The pair (X, 𝒜) in Theorem 1.1 is naturally associated with the smooth cubic fourfold Y ⊆ ℙ_{ℚ}^{5} given by

\begin{aligned}2X_{1}^{2}Y_{1} & + 3X_{1}^{2}Y_{2} + X_{1}^{2}Y_{3} + 3X_{1}X_{2}Y_{1} + 3X_{1}X_{2}Y_{2} + 3X_{1}X_{3}Y_{1} + 4X_{1}X_{3}Y_{2} + 3X_{1}Y_{2}^{2} \\ & + 2X_{1}Y_{3}^{2} + X_{2}^{2}Y_{3} + 3X_{2}X_{3}Y_{3} + 4X_{2}Y_{2}^{2} + X_{3}^{2}Y_{1} + 3X_{3}^{2}Y_{3} + 4X_{3}Y_{1}^{2} + 5X_{3}Y_{1}Y_{2} \\ & + 5X_{3}Y_{2}^{2} + 2Y_{1}^{3} + 3Y_{1}^{2}Y_{3} + 3Y_{1}Y_{3}^{2} + 3Y_{3}^{3} = 0. \end{aligned} ; (2)

This fourfold contains the plane {Y_{1} = Y_{2} = Y_{3} = 0}. Quite generally, we show that, for any field k, we may construct, from a given smooth cubic fourfold Y ⊆ ℙ_{k}^{5} containing a plane P and no other plane P′ meeting P along a line, a K3 surface of degree 2 over k, together with a 2-torsion element of Br(X); see Theorem 5.1. This construction is well-known in the case k = ℂ [Voi86, vG05]; Voisin used it in her proof of the Torelli theorem for cubic fourfolds. The main result of [Has99] may be interpreted as saying that Y is rational when the 2-torsion element of Br(X) is trivial. Recent work of Macri and Stellari [MS09] recasts these constructions in the context of derived categories; for a general discussion of derived categories as they apply to cubic fourfolds, see [Kuz10].

There is no a priori guarantee that the K3 surface furnished by the construction above has geometric Picard rank one. We use a method of van Luijk, together with the work of Elsenhans and Jahnel on K3 surfaces of degree 2, to assemble a K3 surface for which we can prove that this is the case [vL07, EJ08a, EJ08b]. In essence, we find smooth cubic fourfolds ℭ_{2} and ℭ_{3} over the finite fields 𝔽_{2} and 𝔽_{3}, respectively, whose corresponding K3 surfaces have rank-two geometric Picard lattices, with discriminants in different square classes (i.e., whose quotient is not a square). Then the K3 surface arising from a cubic fourfold whose reductions at 2 and 3 coincide with ℭ_{2} and ℭ_{3}, respectively, has geometric Picard rank one. This explains why the coefficients in (2), as well as the entries of the matrix in (1), are all integers less than six.

In [vG05, §9], van Geemen offers geometric interpretations for every 2-torsion element of Br(X), where (X, f) is a polarized complex K3 surface of degree 2 with Picard rank 1. He analyzes the monodromy action on Hom(T, ℤ/2ℤ) where T = f^{⊥} ⊂ H^{2}(X, ℤ); Hom(T, ℤ/2ℤ) coincides with Br(X)[2] when Pic(X) = ℤf. There are naturally three orbits and the cubic fourfold Y above is the geometric interpretation for elements in one of these orbits. In principle, it should be possible to use the remaining geometric interpretations of elements in Br(X)[2] to construct unramified Azumaya algebras in Br(k(X)) representing other transcendental classes on X. For every d ∈ ℕ, there are a finite number of orbits of Hom(T, ℤ/dℤ) under the monodromy action; we expect a geometric construction of the Brauer elements corresponding to each such orbit.

1.1. Outline of the Paper. In §2, we explain the known relation between special cubic fourfolds over ℂ and complex projective polarized K3 surfaces furnished with particular kinds of 2-torsion Brauer classes, using Hodge theory. In §3, we generalize some well-known facts about quadratic forms over base fields to the case of arbitrary base schemes. In §4, we spell out how to explicitly construct a K3 surface from a special cubic fourfold over a field k, paying particular attention to the case where char(k) = 2. We apply the results of §3 in §5 to give geometric proofs, valid over any field, of the constructions outlined in §2. We use these constructions over fields of arithmetic interest in the rest of the paper. In §6, we use van Luijk’s method to construct a K3 surface X over ℚ, arising from a special cubic fourfold, with geometric Picard rank one. In §7, we compute the unramified quaternion algebra in Br(ℚ(X)) associated to our K3 surface by the constructions of §5, and use it to prove Theorem 1.1. Finally, in §8, we describe how we implemented our construction on a computer, and we outline a few speed-ups that made the project possible.

1.2. Notation. Throughout k denotes a field (precise hypotheses are given when needed), and k̄ is a fixed algebraic closure of k. If X and Y are S-schemes, then X_{Y} := X ×_{S} Y. If Y = Spec R, then we write X_{R} instead of X_{Spec R}; if X is a k-scheme, we write \overline{X} for Xk̄. For an integral scheme X over a field, we write k(X) for the function field of X. When X is a K3 surface over a field, we write ρ(X) for the rank of the free abelian group Pic(\overline{X}).

Given a Hodge structure H of weight n, we write H(−1) for its Tate twist, the Hodge structure with an identical filtration but of weight n + 2. The Grassmannian of (r + 1)- dimensional vector subspaces of an (n + 1)-dimensional vector space, or the r-dimensional projective subspaces of ℙ^{n}, is denoted Gr(r + 1, n + 1).

Acknowledgements. We are grateful to Andreas-Stephan Elsenhans, David Harari, Kelly McKinnie, Yuri Tschinkel and Olivier Wittenberg for discussions about this work. Many computations presented here were done with Magma [BCP97]. Figure 3 was created using the Maple software package.

2. Hodge theory and two-torsion elements of the Brauer group

Given a smooth cubic fourfold Y ⊆ ℙ_{ℂ}^{5}, containing a plane P that does not meet any other plane, we may construct a K3 surface X as follows. Let π: Ỹ → ℙ_{ℂ}^{2} be the quadric bundle induced by projection away from P. Then there is a smooth sextic curve C ⊆ ℙ_{ℂ}^{2} that parametrizes the singular fibers of π [Voi86, Lemme 2, p. 586]. The K3 surface X is the double cover of ℙ_{ℂ}^{2} ramified along C.

In this section, we explain how to go in the opposite direction in the presence of extra data. To wit, following [Voi86, vG05], we can recover a smooth cubic fourfold Y ⊆ ℙ_{ℂ}^{5} (containing a plane) from a pair ((X, f), α), consisting of a polarized complex projective K3 surface (X, f) such that Pic X = ℤf, and a particular kind of element α ∈ Br(X)[2] (the 2-torsion subgroup of the Brauer group). This construction is Hodge-theoretic.

This point of view has arithmetic implications: Given a two-torsion element of the Brauer group of a degree-two K3 surface, all defined over a number field, it is natural to seek a canonically-defined unramified Azumaya algebra, arising from a concrete geometric construction, representing the Brauer class. We produce explicit examples of this type, and apply the resulting Brauer element to weak approximation questions. While the content of this section is not logically necessary for our results, it illustrates the geometric motivations behind our approach.

Let X be a complex projective K3 surface, and let ⟨ , ⟩ denote the cup product pairing on its middle cohomology. Consider the cohomology group H_{ét}^{2}(X, μ_{2}), where μ_{2} = {±1}. We have in mind the étale topology, but standard comparison theorems [Mil80, III.3.12] allow us to identify this with H^{2}(X(ℂ), μ_{2}) ≃ H^{2}(X(ℂ), ℤ/2ℤ), sheaf cohomology for the underlying analytic space. In particular, we have

H_{ét}^{2}(X, μ_{2}) ≃ (ℤ/2ℤ)^{22}.

The cup product map

\begin{matrix}H_{ét}^{2}(X, μ_{2}) × H_{ét}^{2}(X, μ_{2}) & → & H_{ét}^{4}(X, μ_{2}^{⊗2}) ≃ ℤ/2ℤ \\ (α, β) & ↦ & ⟨α, β⟩ \end{matrix}

is nondegenerate. Since the intersection form on the middle cohomology of a K3 surface is even, we have ⟨α, α⟩ = 0 for each α ∈ H_{ét}^{2}(X, μ_{2}). Nevertheless, there exists a quadratic form

q_{⟨,⟩}: H_{ét}^{2}(X, μ_{2}) → H_{ét}^{4}(X, μ_{2}^{⊗2})

constructed as follows: Take the integral (or 2-adic) intersection quadratic form on the middle cohomology, divide this by two, and reduce modulo two. An element α ∈ H_{ét}^{2}(X, μ_{2}) is even or odd depending on the parity of q_{⟨,⟩}(α).

Consider the exact sequence

Pic(X) →^{×2} Pic(X) → H_{ét}^{2}(X, μ_{2}) → Br(X)[2] → 0,

where the first arrow is the mod 2 cycle-class map. If ρ(X) = rk(Pic(X)) then

Br(X)[2] ≃ (ℤ/2ℤ)^{22−ρ(X)}.

Suppose that X has degree 2 and is endowed with a polarization f. In particular, f is an ample divisor with ⟨f, f⟩ = 2. Consider the subset

{α ∈ H_{ét}^{2}(X, μ_{2}) : q_{⟨,⟩}(α) ≡ ⟨α, f⟩ ≡ 1 (mod 2)},

i.e., the odd non-primitive classes. Note that α is in this set if and only if α + f (mod 2) is as well, so we can consider

B(X, f) := {α ∈ H_{ét}^{2}(X, μ_{2})/ ⟨f⟩ : q_{⟨,⟩}(α) ≡ ⟨α, f⟩ ≡ 1 (mod 2)}.

The map B(X, f) → Br(X)[2] is injective if and only if Pic(X) = ℤf.

Voisin [Voi86, §1] and van Geemen [vG05, §9.7] offer a geometric construction for Severi-Brauer varieties smooth of relative dimension one over X, indexed by B(X, f). We summarize their results as follows.

Proposition 2.1. Let (X, f) be a complex projective K3 surface of degree two with Pic(X) = ℤf and let α ∈ B(X, f). Then there exists a unique cubic fourfold Y containing a plane P, with variety of lines F, and a correspondence

\xymatrix{ & W \ar[dl]_{π_{1}} \ar[dr]^{π_{2}} &  & Z \ar[dl]_{p} \ar[dr]^{q} \\ X &  & F &  & Y}

where W ⊂ F parametrizes the lines in Y incident to P, π_{1} is a smooth ℙ^{1}-bundle (in the étale topology), π_{2} is an inclusion, and

Z = {[ℓ, y] : y ∈ ℓ ⊂ Y }

is the incidence correspondence between Y and F. Moreover, [W] = α in the Brauer group of X. Precisely, the correspondence induces an isogeny of Hodge structures

H^{4}(Y, ℤ)  H^{2}(X, ℤ)(−1)
∪  ∪
j: {h^{2}, P }^{⊥} ↪  f^{⊥},

where h is the hyperplane class on Y, whose image equals

{η ∈ H^{2}(X, ℤ) : ⟨η, f⟩ = 0, ⟨η, α⟩ ≡ 0 (mod 2)}.

The cubic fourfold Y does not contain another plane P′ that meets P.

We sketch the construction of j: The incidence correspondence Z induces an Abel-Jacobi map

p_{∗}q^{∗}: H^{4}(Y, ℤ) → H^{2}(F, ℤ)(−1),

i.e., an isogeny of Hodge structures that is an isomorphism between the primitive cohomology groups [BD85]. In particular, we may regard

{h^{2}, P }^{⊥} ⊂ H^{2}(F, ℤ)(−1).

The pullback maps

π_{1}^{∗}: H^{2}(X, ℤ) → H^{2}(W, ℤ), π_{2}^{∗}: H^{2}(F, ℤ) → H^{2}(W, ℤ),

are injective; indeed, for the former we know that π_{1} is projective and for the latter we use [Voi86, Proposition 1, p. 582]. Thus we can compare

π_{1}^{∗}(f^{⊥}) and π_{2}^{∗}p_{∗}q^{∗} ({h^{2}, P }^{⊥}) (1)

inside H^{2}(W, ℤ), and find that the latter is an index-two subgroup of the former [Voi86, Lemme 5, p. 583].

Remarks 2.2.

(1) The geometric manifestation of α by W in Proposition 2.1 is the kind of presentation for Brauer elements that we emulate in arithmetic contexts; see §5.

(2) The hypothesis that Pic(X) = ℤf in Proposition 2.1 is much stronger than is necessary.

(3) The analysis above does not apply to all α ∈ Br(X)[2]; van Geemen [vG05, §9] offers geometric interpretations for the classes we have not considered.

(4) Macri and Stellari have recast this discussion in the language of α-twisted sheaves on X [MS09].

3. Quadratic forms and maximal isotropic subspaces

To generalize the constructions of §2, we first generalize a few well-known results on quadratic forms over fields to the case of arbitrary base schemes. We include proofs for lack of a suitable reference.

3.1. Quadratic forms over fields. We begin with some elementary geometric observations, which can be found in [Har92, Chapter 22], at least over fields of characteristic different from two. A characteristic-independent approach can be found in [EKM08, §§7-14, 85]. Fix n ≥ 2 and let Q ⊂ ℙ_{k}^{2n−1} and Q′ ⊂ ℙ_{k}^{2n−2} denote smooth quadric hypersurfaces over an algebraically closed field k. Then Q is isomorphic to

X_{1}X_{2} + X_{3}X_{4} + … + X_{2n−1}X_{2n} = 0,

and Q′ is isomorphic to

X_{1}X_{2} + X_{3}X_{4} + … + X_{2n−1}^{2} = 0

[EKM08, §7]. Consider the schemes

W ⊂ Gr(n, 2n), W′ ⊂ Gr(n − 1, 2n − 1)

of maximal isotropic subspaces of Q and Q′, respectively. Both schemes are smooth projective of dimension

n^{2} − \binom{n + 1}{2} = (n − 1)n − \binom{n}{2} = \binom{n}{2};

furthermore, W′ is irreducible and W = W_{1} ∪W_{2}, where each component W_{i} ≃ W′ [EKM08, §85]. Indeed, if we fix a realization of Q′ as a hyperplane section of Q

Q′ ↪ Q

then we have a restriction morphism

\begin{matrix}ρ: W & → & W′ \\ Λ & ↦ & Λ ∩ Q′ \end{matrix}

mapping each component of W isomorphically onto W′. For example, if

Λ = {X_{1} = X_{3} = ⋯ = X_{2n−1} = 0} ⊂ Q

then

ρ(Λ) = {X_{1} = X_{3} = ⋯ = X_{2n−1} = 0} ⊂ Q′

and

ρ^{−1}(ρ(Λ)) = {Λ, Λ̄ = {X_{1} = ⋯ = X_{2n−3} = X_{2n} = 0}} .

Suppose now that Q ⊂ ℙ_{k}^{2n−1} is a quadric over an arbitrary field k. Consider the Clifford algebra C(Q) over k [EKM08, §11]. It decomposes into even and odd parts:

C(Q) = C_{0}(Q) ⊕ C_{1}(Q).

When Q is smooth, we have:

• C(Q) is central simple;

• C_{0}(Q) is separable, and its center is a quadratic étale k-algebra, which is denoted by Z(Q) and is called the discriminant algebra [EKM08, §13]

• C_{0}(Q) is central simple as an algebra over Z(Q). When char(k) ≠ 2, we can diagonalize

Q = {a_{1}X_{1}^{2} + ⋯ + a_{2n}X_{2n}^{2} = 0}

and Z(Q) = k[w]/ ⟨w^{2} − c⟩ with c = (−1)^{n}a_{1} … a_{2n}. When char(k) = 2, we can write

Q = {a_{1}X_{1}^{2} + X_{1}X_{2} + b_{1}X_{2}^{2} + ⋯ + a_{n}X_{2n−1}^{2} + X_{2n−1}X_{2n} + b_{n}X_{2n}^{2} = 0}

and Z(Q) = k[w]/ ⟨w^{2} + w + c⟩ where c = a_{1}b_{1} + ⋯ + a_{n}b_{n}, the Arf invariant. Finally, if W denotes the variety of maximal isotropic subspaces of Q, then we have [EKM08, §85]:

• the geometric components of W are defined over Z(Q);

• when n = 2, each component of W over Z(Q) is the conic associated with the

quaternion algebra C_{0}(Q) over Z(Q). Assume again that k is algebraically closed, and suppose that Q_{0} ⊂ ℙ_{k}^{2n−1} is a quadric

with an isolated singularity. In suitable coordinates, it has equation

X_{1}X_{2} + X_{3}X_{4} + ⋯ + X_{2n−1}^{2} = 0.

i.e., it is a cone over Q′. The scheme W_{0} ⊂ Gr(n, 2n) of (n − 1)-dimensional isotropic projective subspaces is thus W′, set-theoretically.

There is a natural scheme structure on W_{0}. Expressing ℙ_{k}^{2n−1} = ℙ(V), over the Grassmannian we have the tautological exact sequence

0 → S → O_{Gr(n,2n)} ⊗ V → Q → 0,

inducing linear maps

Sym^{2}(V^{∨}) = Γ(O_{Gr(n,2n)} ⊗ Sym^{2}(V^{∨})) → Γ(Sym^{2}(S^{∨})).

We can express W_{0} = {σ = 0}, where σ is the section of Sym^{2}(S^{∨}) arising from the defining equation of Q_{0} ⊂ ℙ(V). Note that W_{0} is Cohen-Macaulay and thus has no embedded points: this follows from the unmixedness theorem [Eis95, Cor. 18.14], as it is the codimension-\binom{n+1}{2} locus where a section of a rank-\binom{n+1}{2} vector bundle on Gr(n, 2n) vanishes. However, W_{0} has multiplicity two at its generic point.

3.2. Quadratic forms over schemes. We shall generalize the above facts to arbitrary base schemes. Let H be the Hilbert scheme parametrizing quadratic forms in 2n variables

∑_{1≤i≤j≤2n} a_{ij}X_{i}X_{j}.

Concretely, we have

H = ℙ\binom{2n+1}{2}_{ℤ} = Proj ℤ[a_{11}, …, a_{2n 2n}].

Consider the open subsets

V ⊂ U ⊂ H,

corresponding to quadrics that are geometrically smooth and have at most one isolated singularity. In characteristics ≠ 2, U (resp. V) corresponds to quadrics whose associated symmetric matrix has rank ≥ 2n − 1 (resp. 2n).

Let Δ ⊂ H denote the discriminant divisor, defined as follows: If q: Q → H is the universal family of quadrics then the differential of q drops rank along a subscheme Σ ⊂ Q; Δ is the divisor obtained by pushing forward Σ via q. Note that the discriminant algebra ramifies over the discriminant divisor.

Example 3.1. In characteristic 2, the morphism from Σ to its set-theoretic image is purely inseparable of degree two. In particular, Δ has multiplicity 2 in characteristic 2. Indeed, given the family

X_{1}X_{2} + X_{3}^{2} + sX_{4}^{2} = 0

of non-smooth quadrics over the s-line, the singular locus is

{X_{1} = X_{2} = X_{3}^{2} + sX_{4}^{2} = 0}

which is purely inseparable over the s-line. There is no section through the singularities.

Example 3.2. When n = 1 we have

Δ = {−4a_{11}a_{22} + a_{12}^{2} = 0};

the corresponding quadratic algebra is

Z(Q) = \begin{cases}k[w]/ ⟨w^{2} − a_{12}^{2} + 4a_{11}a_{22}⟩ & char(k) ≠ 2, \\ k[w]/ ⟨w^{2} + wa_{12} + a_{11}a_{22}⟩ & char(k) = 2. \end{cases}

Note the first form specializes to the second via the substitution w ↦ 2w + a_{12} and division by four.

For n = 2, setting L = a_{12}a_{34} + a_{13}a_{24} + a_{23}a_{14}, N = a_{11}a_{22}a_{33}a_{44}, and

M = − (a_{12}a_{23}a_{34}a_{14} + a_{13}a_{23}a_{24}a_{14} + a_{12}a_{24}a_{13}a_{34})
+ (a_{11}a_{23}a_{24}a_{34} + a_{22}a_{13}a_{34}a_{14} + a_{33}a_{12}a_{24}a_{14} + a_{44}a_{12}a_{23}a_{13})
− (a_{11}a_{22}a_{34}^{2} + a_{11}a_{33}a_{24}^{2} + a_{11}a_{44}a_{23}^{2} + a_{22}a_{33}a_{14}^{2} + a_{22}a_{44}a_{13}^{2} + a_{33}a_{44}a_{12}^{2}),

we have

Δ = {L^{2} + 4M + 16N = 0}.

Note that the even coefficents are divisible by four. The corresponding quadratic algebra is

Z(Q) = \begin{cases}k[w]/ ⟨w^{2} − (L^{2} + 4M + 16N)⟩ & char(k) ≠ 2, \\ k[w]/ ⟨w^{2} + wL + M⟩ & char(k) = 2. \end{cases}

The first form specializes to the second via the substitution w ↦ 2w + L.

Let W ⊂ Gr(n, 2n) × U parametrize (n − 1)-dimensional isotropic projective subspaces in the projective hypersurfaces parametrized by U, and let r: W → U be the structure map. Since W is an open subset of a projective bundle over Gr(n, 2n), it is smooth over Spec(ℤ).

Proposition 3.3. The Stein factorization

r: W →^{π} X →^{ϕ} U

is the composition of a smooth morphism by a flat double cover branched along Δ. The geometric fibers of π are all isomorphic to W′.

Proof. The fact that the Stein factorization is a double cover branched along Δ follows from our previous discussion, i.e., that the components of the maximal isotropic subspaces are defined over the discriminant algebra, which ramifies over the discriminant divisor. The assertion on the geometric fibers of π follows from the isomorphism W_{i} ≃ W′ above, once we know that π is smooth. We therefore focus on this last statement.

We first show that ϕ is flat. The scheme W is defined as the locus where a section of a rank-\binom{n+1}{2} vector bundle on the Grassmannian vanishes, with the expected dimension. Since W is Cohen-Macaulay and U is regular, W is flat over U [Eis95, Th. 18.16]. The geometric fibers of r are homogeneous varieties whose structure sheaves have no higher cohomologies. Thus ℝ^{i}r_{∗}O_{W} = 0 for each i > 0 and r_{∗}O_{W} is flat by cohomology-and-base-change.

Consider the pull-back of the universal quadric to the discriminant cover/Stein factorization

q′′: Q′′ := Q ×_{U} X → X;

W tautologically determines a component of the variety of maximal isotropic subspaces of Q′′. We claim that there exists a smooth subscheme Σ′′ ⊂ Q′′ containing the singularities of the fibers and meeting each singular fiber in a subscheme of length one. In other words, over the reduced discriminant divisor in X there exists a section through the singularities (cf. Example 3.1).

This statement is clear except in characteristic two, where the non-smooth points may be defined over a purely inseparable extension. It suffices to exhibit the sections after étale localization at a point on the base corresponding to a quadric with isolated singularity. We may assume that singularity is [0, …, 0, 1] and the intersection of the quadric with the hyperplane X_{2n} = 0 is smooth. Thus we have a local equation

F (X_{1}, …, X_{2n−1}) + X_{2n}L(X_{1}, …, X_{2n−1}) + sX_{2n}^{2}

where F is quadratic, L is linear, and s is constant. Since {F = 0} ⊂ ℙ^{2n−2} is smooth, it has a maximal isotropic subspace on passage to a separable extension. Thus after localizing in the étale topology we may change coordinates [EKM08, 7.32,7.34] so that

F = X_{1}X_{2} + ⋯ + X_{2n−3}X_{2n−2} + X_{2n−1}^{2}.

Using coordinate changes of the form X_{j} ↦ X_{j}+c_{j}X_{2n−1} for c_{j} constant and j = 1, …, 2n−2, we may eliminate X_{1}, …, X_{2n−2} from L. Thus we are left with the normal form

X_{1}X_{2} + X_{3}X_{4} + ⋯ + X_{2n−1}^{2} + tX_{2n−1}X_{2n} + sX_{2n}^{2} = 0

for our family of quadrics.

Here, the discriminant algebra takes the form

z^{2} + tz + s = 0.

More symmetrically, we write

αβ = s and α + β = t,

so that Q′′ is locally of the form

X_{1}X_{2} + X_{3}X_{4} + ⋯ + (X_{2n−1} + αX_{2n})(X_{2n−1} + βX_{2n}) = 0

with discriminant locus α = β. Here we can write

Σ′′ = {X_{1} = X_{2} = ⋯ = X_{2n−2} = X_{2n−1} + αX_{2n} = X_{2n−1} + βX_{2n} = 0}.

The local analysis yields additional dividends: We see that Q′′ has codimension-(2n − 1) ordinary double points along Σ′′. Consider the modification

β: Q̃ := Bl_{Σ′′}Q′′ → Q′′

and the induced morphism

q̃: Q̃ → X ,

which remains flat over X; let E denote its exceptional divisor. Fix a point p in the discriminant on X, so that Q′′_{p} = q′′^{−1}(p) is singular at σ = Σ′′_{p}. Then we have

Q̃_{p} := q̃^{−1}(p) = Q ∪_{Q′} Bl_{σ}Q′′_{p},

Figure 1. First modification in the n = 2 case

where Q = E ∩ Q̃_{p} is a smooth quadric of dimension 2n − 2, Q′ ⊂ Q is a smooth hyperplane section, and Q′ ⊂ Bl_{σ}Q′′_{p} is the exceptional divisor. Note that Bl_{σ}Q′′_{p} is a projective bundle over a smooth quadric of dimension 2n − 3, with section Q′.

Regard Q′′ ⊂ X × ℙ^{2n−1} and let O_{Q′′}(1) denote the pull back of O_{ℙ^{2n−1}}(1) via projection. Consider the invertible sheaf

β^{∗}O_{Q′′}(1) ⊗ OQ̃(−E),

which is semiample and induces a contraction

\xymatrix{Q̃ \ar[dr]_{q̃} \ar[rr]^{γ} &  & Q̂ \ar[dl]^{q̂} \\  & X}

This has the following properties:

• γ is the blow up along a smooth codimension-two subvariety Y ⊂ Q̂, supported over the discriminant; Y intersects singular fibers of q̂ in smooth hyperplane sections;

• for p in the discriminant, γ|Q̃_{p} is an isomorphism on Q, and it collapses Bl_{σ}Q′′_{p} along its ruling to Q′ ⊂ Q;

• q̂ is a smooth quadric bundle;

• the variety Ŵ → X parametrizing maximal isotropic subspaces in fibers of q̂ is smooth over X with two connected components.

Example 3.4. Given the quadric

Q = {X_{1}X_{2} + ⋯ + X_{2n−1}^{2} − α^{2}X_{2n}^{2} = 0}

over the complex affine α-line, we have

Q̂ = {X_{1}X_{2} + ⋯ + X_{2n−1}^{2} − y^{2} = 0},

where y = αX_{2n}.

Lemma 3.5. Consider the part of the relative Hilbert scheme of q̃: Q̃ → X

W̃ = W̃_{1} ∪ W̃_{2} → X

Figure 2. Second modification in the n = 2 case

that, for smooth fibers of q̃, parametrizes maximal isotropic subspaces. These satisfy the following properties:

• for p in the discriminant, W̃_{p} parametrizes

{Λ : Λ ⊂ Q̃_{p} = Q ∪_{Q′} Bl_{σ}Q′′_{p}},

where Λ∩Q and Λ∩Q′ are maximal isotropic, and Λ∩Bl_{σ}Q′′_{p} is the proper transform of a maximal isotropic subspace in Q′′_{p};

• γ induces an isomorphism W̃ →^{∼} Ŵ;

• β induces an isomorphism between one of the W̃_{i} and W.

Proof. (of the Lemma) The morphism γ blows up a smooth codimension-two subvariety Y ⊂ Q̂; we claim Y contains none of the maximal isotropic subspaces in the fibers of q̂. Indeed, this is clear for the smooth fibers, which are disjoint from Y. The singular fibers meet Y in smooth hyperplane sections. However, given a smooth quadric hypersurface of even dimension and a maximal isotropic subspace, any hyperplane section containing the subspace is necessarily singular.

This analysis implies that W̃_{p} has the geometric description given in the Lemma. Note that W̃ → X is smooth, as each Λ ⊂ Q̃_{p} is a complete intersection of (linear) Cartier divisors.

Consider the mapping

\begin{matrix}W̃ & → & Ŵ \\ Λ & ↦ & γ(Λ); \end{matrix}

we will show this is an isomorphism. The key point is to demonstrate how the universal family of maximal isotropic subspaces over Ŵ induces a flat family of subschemes in Q̃, corresponding to W̃. We apply a general observation: Suppose we have a smooth variety Y embedded as a codimension-two subvariety in a smooth variety P, and a flat family {Λ_{t}}_{t∈T} of subvarieties of P over an integral base T such that Y ∩ Λ_{t} ⊂ Λ_{t} is either empty or Cartier for each t ∈ T. Then the total transforms of the {Λ_{t}}_{t∈T} in Bl_{Y}P form a flat family as well. Indeed, the ‘valuative criterion for flatness’ [Gro66, 11.8.1] reduces us to the curve case, where dimensional considerations suffice [Har77, III.9.7].

The first modification yields morphisms

\begin{matrix}W̃_{i} & → & W \\ Λ & ↦ & β(Λ) \end{matrix}

of schemes smooth over Spec(ℤ). Our description of the fibers W̃_{p} guarantees this is bijective: For each maximal isotropic subspace in a fiber of q′′, there is a unique Λ mapping to that subspace. It follows that these morphisms are isomorphisms.

□

The lemma implies the smoothness of W → X, which completes the proof of Proposition 3.3. □

3.3. The variety of maximal isotropic subspaces of a quadric. In this section, we explain how to compute explicit equations for the variety of one-dimensional isotropic (projective) subspaces over the universal quadric in 4 variables. Let H = ℙ_{ℤ}^{9} = Proj ℤ[a_{11}, …, a_{44}], and let Q ⊆ ℙ_{ℤ}^{3} × H be the universal family of quadrics, given by the vanishing of

\begin{aligned}a_{11}X_{1}^{2} & + a_{22}X_{2}^{2} + a_{33}X_{3}^{2} + a_{44}X_{4}^{2} \\ & + a_{12}X_{1}X_{2} + a_{23}X_{2}X_{3} + a_{34}X_{3}X_{4} + a_{13}X_{1}X_{3} + a_{24}X_{2}X_{4} + a_{14}X_{1}X_{4}. \end{aligned} ; (3)

We define the degree-zero graded ℤ-module homomorphism

\begin{matrix}φ: ℤ[X_{1}, X_{2}, X_{3}, X_{4}] & → & ℤ[s, t] \\ X_{i} & ↦ & p_{i}s + p′_{i}t i = 1, …, 4. \end{matrix}

Consider p_{1}, …, p′_{4} as indeterminates, subject to the restriction that the rank of the matrix

(\begin{matrix}p_{1} & p_{2} & p_{3} & p_{4} \\ p′_{1} & p′_{2} & p′_{3} & p′_{4} \end{matrix})

is maximal. Then the variety of one-dimensional isotropic subspaces L of Q is given by the vanishing of Q along the image of Sym^{2} φ. Concretely, substitute the expressions X_{i} = p_{i}s + p′_{i}t for i = 1, …, 4 in (3) and think of the result as the zero polynomial in the variables s and t with coefficients in ℤ[a_{11}, …, a_{44}, p_{1}, …, p′_{4}]. Let I be the ideal generated by these coefficients.

To give equations for L as a subscheme of H × Gr(2, 4), where we consider Gr(2, 4) ⊆ Proj ℤ[p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34}] under the Plücker embedding, let

\begin{aligned}J & := ⟨p_{12} − p_{1}p′_{2} + p_{2}p′_{1}, p_{13} − p_{1}p′_{3} + p_{3}p′_{1}, p_{14} − p_{1}p′_{4} + p_{4}p′_{1}, \\ & p_{23} − p_{2}p′_{3} + p_{3}p′_{2}, p_{24} − p_{2}p′_{4} + p_{4}p′_{2}, p_{34} − p_{3}p′_{4} + p_{4}p′_{3}⟩ \end{aligned}

be an ideal of the ring ℤ[p_{1}, …, p′_{4}, p_{12}, …, p_{34}]. Denote by I^{e} and J^{e} the extensions of I and J, respectively, to the ring

ℤ[p_{1}, …, p′_{4}, a_{11}, …, a_{44}, p_{12}, …, p_{34}].

We obtain defining equations for L by eliminating p_{1}, …, p′_{4} from the ideal I^{e} + J^{e}.

Note that L ×_{H} U = W, the maximal isotropic subspaces of the quadrics with isolated singularities. To obtain equations for W, we focus on the distinguished affine subset of U given by

\frac{1}{2} det (\begin{matrix}2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_{23} & 2a_{33} \end{matrix}) = 4a_{11}a_{22}a_{33} + a_{12}a_{13}a_{23} − (a_{11}a_{23}^{2} + a_{22}a_{13}^{2} + a_{33}a_{12}^{2}) ≠ 0.

If char k ≠ 2, then the result is as follows: in the notation of Example 3.2, let Δ = L^{2} + 4M + 16N, and write K = k(\sqrt{Δ}) for the étale k-algebra Z(Q). Letting

λ(p_{14}, p_{24}, p_{34}) = a_{11}p_{14}^{2} + a_{12}p_{14}p_{24} + a_{13}p_{14}p_{34} + a_{22}p_{24}^{2} + a_{23}p_{24}p_{34} + a_{33}p_{34}^{2},

the two components of W over the field K are given by

W_{1} = {λ = 0} ∩ {p_{ij} = l_{ij}(p_{14}, p_{24}, p_{34}) : (ij) ∈ {(12), (13), (23)}}
W_{2} = {λ = 0} ∩ {p_{ij} = l′_{ij}(p_{14}, p_{24}, p_{34}) : (ij) ∈ {(12), (13), (23)}},

where l_{ij} and l′_{ij} are linear forms with coefficients in K.

4. From cubic fourfolds to K3 surfaces

In this section we explain how to explicitly construct, over an arbitrary field, a (possibly singular) K3 surface from a cubic fourfold containing a plane.

4.1. Degree two K3 surfaces. Let (X, f) denote a polarized K3 surface X of degree 2 over a field k; assume that |f| is base-point free. The induced morphism

ϕ_{|f|}: X → ℙ_{k}^{2}

is a flat double cover and induces an exact sequence

0 → O_{ℙ^{2}} → ϕ_{|f|}_{∗}O_{X} → Q → 0,

which is split by the trace mapping unless char(k) = 2. Even when char(k) = 2, Q is still an invertible sheaf [CD89, pp. 9-10]. As H^{2}(ℙ^{2}, Q) ≃ H^{2}(X, O_{X}) ≃ k, we have Q ≃ O_{ℙ^{2}}(−3) and our exact sequence still splits.

The surface X then is embedded in ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(−3)) and disjoint from the distinguished section. Blowing this down, we realize X as a smooth sextic hypersurface in the weighted projective space ℙ_{k}(1, 1, 1, 3) = Proj k[x, y, z, w]:

(4)  w^{2} + α(x, y, z)w + β(x, y, z) = 0,

where α, β ∈ k[x, y, z] have respective degrees 3 and 6. In these coordinates, the map ϕ_{|f|} is the restriction to X of the natural projection ℙ_{k}(1, 1, 1, 3) −−→ ℙ_{k}^{2}. If char(k) ≠ 2, then completing a square we may (and do) assume that α = 0. In this case, the ramification curve is C = {β(x, y, z) = 0}. In characteristic 2, (4) yields a singular surface if α(x, y, z) = 0. (Indeed, such a surface is an inseparable double cover of ℙ^{2}, typically with 21 ordinary double points; its minimal resolution is a supersingular K3 surface [Shi04].) The ramification locus of ϕ_{|f|} is a curve C = {α(x, y, z) = 0}.

4.2. Cubic fourfolds containing a plane. From now on, assume that (Y, P) is a smooth cubic fourfold containing a plane P, defined over a field k. Projection from P induces a quadric surface bundle q: Ỹ → ℙ_{k}^{2}. Let r: W → ℙ_{k}^{2} denote the relative variety of lines, parametrizing lines in Y contained in fibers of q.

We may assume, without loss of generality, that

Proj k[X_{1}, X_{2}, X_{3}] = P ⊂ Y ⊂ ℙ_{k}^{5} = Proj k[X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, Y_{3}]

so that P = {Y_{1} = Y_{2} = Y_{3} = 0}. The defining equation for Y takes the form

(5)  L_{11}X_{1}^{2} + ⋯ + L_{33}X_{3}^{2} + Q_{14}X_{1} + Q_{24}X_{2} + Q_{34}X_{3} + C_{44} = 0,

where L_{ij}, Q_{ij} and C_{44} are homogeneous forms in k[Y_{1}, Y_{2}, Y_{3}] of respective degrees 1, 2 and 3. The blow-up of ℙ_{k}^{5} along P may be identified with the projective bundle ℙ(E), where

E ≃ O_{ℙ_{k}^{2}}^{⊕3} ⊕ O_{ℙ_{k}^{2}}(−1).

The quadric bundle q: Ỹ → ℙ_{k}^{2} factors through ℙ(E); the defining equation for Ỹ ⊂ ℙ(E) is given by some s ∈ Γ(Sym^{2} E^{∨}). If char(k) ≠ 2, then we may interpret s as a homomorphism s: E → E^{∨} with s^{∨} = s. Using our trivialization for E, we can express

(6)  s = (\begin{matrix}2L_{11} & L_{12} & L_{13} & Q_{14} \\ L_{12} & 2L_{22} & L_{23} & Q_{24} \\ L_{13} & L_{23} & 2L_{33} & Q_{34} \\ Q_{14} & Q_{24} & Q_{34} & 2C_{44} \end{matrix}) .

In other words, Ỹ is isomorphic to the subscheme of Proj k[X_{1}, X_{2}, X_{3}, X_{4}] × Proj k[x, y, z] given by

\begin{aligned}L_{11}(x, y, z)X_{1}^{2} & + ⋯ + L_{33}(x, y, z)X_{3}^{2} + C_{44}(x, y, z)X^{4} \\ & + Q_{14}(x, y, z)X_{1}X_{4} + Q_{24}(x, y, z)X_{2}X_{4} + Q_{34}(x, y, z)X_{3}X_{4} = 0. \end{aligned} ; (7)

and the bundle map q: Ỹ → ℙ_{k}^{2} is the projection onto the second factor. The discriminant curve in ℙ_{k}^{2} has equation {F = 0}, where F is the determinant of the matrix (6); here X satisfies the equation

(8)  w^{2} − F = 0

in ℙ_{k}(1, 1, 1, 3) = Proj k[x, y, z, w].

On the other hand, if char k = 2, then (7) still holds, though representation of s by (6) does not. In this case, Example 3.2 shows that

(9)  w^{2} + L(x, y, z)w + M(x, y, z) = 0

is an equation for X ⊂ ℙ_{k}(1, 1, 1, 3), where

L = L_{12}Q_{34} + L_{13}Q_{24} + L_{23}Q_{14},
M = −(L_{12}L_{23}Q_{34}Q_{14} + L_{13}L_{23}Q_{24}Q_{14} + L_{12}Q_{24}L_{13}Q_{34})
+ (L_{11}L_{23}Q_{24}Q_{34} + L_{22}L_{13}Q_{34}Q_{14} + L_{33}L_{12}Q_{24}Q_{14} + C_{44}L_{12}L_{23}L_{13})

− (L_{11}L_{22}Q_{34}^{2} + L_{11}L_{33}Q_{24}^{2} + L_{11}C_{44}L_{23}^{2} + L_{22}L_{33}Q_{14}^{2} + L_{22}C_{44}L_{13}^{2} + L_{33}C_{44}L_{12}^{2}).

We apply this construction in two examples that are instrumental in the proof Theorem 1.1.

Example 4.1. The K3 surface X ⊂ ℙ_{𝔽_{2}}(1, 1, 1, 3) arising from the smooth cubic fourfold ℭ_{2} in ℙ_{𝔽_{2}}^{5}

\begin{aligned}X_{1}^{2}Y_{2} & + X_{1}^{2}Y_{3} + X_{1}X_{2}Y_{1} + X_{1}X_{2}Y_{2} + X_{1}X_{3}Y_{1} + X_{1}Y_{2}^{2} + X_{2}^{2}Y_{3} + X_{2}X_{3}Y_{3} \\ & + X_{3}^{2}Y_{1} + X_{3}^{2}Y_{3} + X_{3}Y_{1}Y_{2} + X_{3}Y_{2}^{2} + Y_{1}^{2}Y_{3} + Y_{1}Y_{3}^{2} + Y_{3}^{3} = 0 \end{aligned}

is given by

\begin{aligned}w^{2} & + w(x^{2}y + y^{3} + y^{2}z) \\ & + x^{5}z + x^{3}y^{2}z + x^{2}y^{3}z + x^{3}yz^{2} + x^{2}y^{2}z^{2} + y^{2}z^{4} + xz^{5} + yz^{5} + z^{6} = 0. \end{aligned} ; (10)

Example 4.2. The K3 surface X ⊂ ℙ_{𝔽_{3}}(1, 1, 1, 3) arising from the smooth cubic fourfold ℭ_{3} in ℙ_{𝔽_{3}}^{5}

2X_{1}^{2}Y_{1} + X_{1}^{2}Y_{3} + X_{1}X_{3}Y_{2} + 2X_{1}Y_{3}^{2} + X_{2}^{2}Y_{3} + X_{2}Y_{2}^{2} + X_{3}^{2}Y_{1} + X_{3}Y_{1}^{2} + 2X_{3}Y_{2}Y_{2} + 2X_{3}Y_{2}^{2} + 2Y_{1}^{3}

is given by

\begin{aligned}w^{2} & = 2x^{5}z + x^{4}yz + x^{4}z^{2} + 2x^{3}yz^{2} + x^{2}y^{4} + 2x^{2}y^{3}z + x^{2}y^{2}z^{2} \\ & + 2x^{2}yz^{3} + xy^{3}z^{2} + xy^{2}z^{3} + 2xz^{5} + y^{6} + 2y^{4}z^{2} + y^{3}z^{3}. \end{aligned} ; (11)

5. Constructing unramified Azumaya algebras

The Hodge-theoretic constructions of §2 can now be carried out using geometric techniques, valid over an arbitrary field.

Theorem 5.1. Let Y be a cubic fourfold smooth over a field k. Suppose that Y contains a plane P, and let Ỹ denote the blow up of Y along P, q : Ỹ → ℙ^{2} the corresponding quadric surface bundle, and r : W → ℙ^{2} its relative variety of lines. Assume that there exists no plane P′ ⊂ Yk̄ such that P′ meets P along a line.

Then the Stein factorization

r : W →^{π_{1}} X →^{ϕ} ℙ^{2}

consists of a smooth ℙ^{1}-bundle followed by a degree-two cover of ℙ^{2}, which is a K3 surface.

Proof. Our hypothesis means that no geometric fiber of q contains a plane; thus the fibers of q have at worst isolated singularities. All the statements, with the exception that X is a K3 surface, then follow directly from Proposition 3.3. For the remaining claim, we may assume that k is algebraically closed.

We start with an elementary (no doubt well-known) result whose proof we include for completeness.

Lemma 5.2. Let B be a smooth surface over an algebraically closed field of characteristic ≠ 2. Suppose that q : Q → B is quadric surface bundle such that

• the generic fiber is smooth;

• the degenerate fibers have at worst isolated singularities. Then the discriminant curve Δ ⊂ B is smooth if and only if Q is smooth.

Proof. Let b ∈ Δ be a point in the discriminant curve; we may replace B with a formal neighborhood of b. Let s and t be local coordinates centered at b. Set Q_{b} = q^{−1}(b) with equation

x^{2} + y^{2} + z^{2} = 0.

The local equation of Q takes the form

x^{2} + y^{2} + z^{2} = a_{11}(s, t) + 2a_{12}(s, t)x + 2a_{13}(s, t)y + ⋯,

where the coefficients a_{ij}(s, t) are power series with vanishing constant term. This is singular at x = y = z = s = t = 0 precisely when the linear term of a_{11}(s, t) vanishes. We have

det ((\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) + (\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & a_{34} \\ a_{14} & a_{24} & a_{34} & a_{44} \end{matrix})) = a_{11} + higher order terms,

which implies that discriminant vanishes to first order whenever a_{11} vanishes to first order. □

Lemma 5.2 yields the following amplification of [Voi86, Lemme2, p. 582]: If char k ≠ 2 then the discriminant curve Δ ⊂ ℙ_{k}^{2} of q is a smooth plane sextic curve. Thus in this case X is a K3 surface, because it is a double cover of ℙ_{k}^{2} ramified along a smooth sextic curve.

If char k = 2, then we argue as follows: The surface X can only be singular at points on the ramification locus of ϕ. Choose local coordinates s and t at b ∈ ℙ_{k}^{2} in the branch locus of ϕ. Set Ỹ_{b} = q^{−1}(b) with equation

xy + z^{2} = 0.

Then, the local equation of Ỹ takes the form

xy + z^{2} + A(s, t) + B(s, t)z = 0,

where A(s, t) and B(s, t) are power series with vanishing constant terms. On the other hand, the local equation of X is

w^{2} + B(s, t)w + A(s, t) = 0

(see Example 3.2). It is then easy to see that X is singular at s = t = w = 0 precisely when Ỹ is singular at x = y = z = s = t = 0. However, Ỹ is smooth over k, and thus so is X. It follows that X is K3, because it is isomorphic to a smooth sextic hypersurface in ℙ_{k}(1, 1, 1, 3). □

6. Computation of the Picard group

6.1. van Luijk’s Method. To prove that the geometric Picard rank ρ(X) of a K3 surface (over ℚ) arising from a cubic fourfold is one, we use a method due to van Luijk [vL07]— see [EJa] for a recent refinement of this method. The idea is as follows: Let (X, f) be a polarized K3 surface over a number field. Suppose that X has good reduction modulo p for two distinct primes p_{1} and p_{2}, and assume that

(1) the Picard rank of each reduction is two—the rank must be even over a finite field;

(2) the discriminants of the Picard groups of the reductions are in different square class. Then ρ(X) = 1. To prove that condition (2) holds, we construct explicit full-rank sublattices of the respective Picard groups, whose discriminants are in different square classes. For condition (1), we first obtain upper bounds for the Picard rank of a reduction as follows: Let (X_{p}, f̄) be a polarized K3 surface over a finite field k of cardinality q = p^{r} (p prime). Fix ℓ ≠ p a prime. Let Φ: X_{p} → X_{p} be the r-th power absolute Frobenius and write Φ^{∗} for the automorphism of H_{ét}^{2}(X_{p} × k̄, ℚ_{ℓ}) induced by the action of Φ × 1 on X_{p} × k̄. Then the number of eigenvalues of Φ^{∗} of the form qζ, with ζ a root of unity, gives an upper bound for ρ(X_{p}) [vL07, Corollary 2.3].

We determine the eigenvalues of Φ^{∗} of the required type by computing the characteristic polynomial f_{Φ^{∗}}(t) of Φ^{∗} via Newton’s identity:

f_{Φ^{∗}}(t) = t^{22} + c_{1}t^{21} + c_{2}t^{20} + ⋯ + c_{22},

where

c_{1} = − Tr(Φ^{∗}) and − kc_{k} = Tr(Φ^{∗k}) + ∑_{i=1}^{k−1} c_{i} Tr(Φ^{∗k−i}).

The traces of powers of the Frobenius maps are in turn calculated by tallying up the points of X_{p} over finite extensions of the base field, using the Lefschetz trace formula:

#X_{p}(𝔽_{q^{n}}) = q^{2n} + Tr(Φ^{∗n}) + 1.

The Weil conjectures give the functional equation

p^{22}f_{Φ^{∗}}(t) = ±t^{22}f_{Φ^{∗}}(p^{2}/t).

Hence, to compute f_{Φ^{∗}}(t), it may suffice to compute #X_{p}(𝔽_{q^{n}}) up to some n < 22. Typically, n = 11 is sufficient (if c_{11} ≠ 0, giving a positive sign for the functional equation). Sometimes, however, it is possible to do better, and get away with n = 9: see [EJ08a, §4].

6.2. Special K3 surfaces of degree 2.

6.2.1. Characteristic 2. Let (X, f) be a polarized K3 surface of degree 2 over an algebraically closed field k with char(k) = 2, given in the form (4). Note that α ≠ 0 since X is smooth. The rational map

\begin{matrix}ℙ_{k}(1, 1, 1, 3) & −−→ & ℙ_{k}(1, 1, 1, 3) \\ [x : y : z : w] & ↦ & [x : y : z : w + α] \end{matrix}

restricts to an automorphism ψ of X.

Suppose that X contains a divisor of the form

(12)  Γ := {l(x, y, z) = w + c(x, y, z) = 0},

where l, c ∈ k[x, y, z] have respective degrees 1 and 3. Let Γ′ be the scheme-theoretic image of Γ under ψ. The intersection number (Γ, Γ′) is equal to

deg(Γ ∩ Γ′) = deg ( Proj k[x, y, z, w]/(l, α, w + c)) = deg ( Proj k[x, y, z]/(l, α)) = 3.

Since {x = 0} ∈ |f|, it follows by similar computations that

(Γ, f) = (Γ′, f) = 1,

and hence Γ and Γ′ are irreducible, since f is ample. Let V := {l(x, y, z) = 0} ⊆ ℙ_{k}^{2}. Then, on the one hand, we have ϕ^{∗}V ∈ |f|, so that (ϕ^{∗}V, f) = 2. On the other hand ϕ^{∗}V − Γ − Γ′ is an effective divisor, and

(ϕ^{∗}V − Γ − Γ′, f) = 0;

whence ϕ^{∗}V = Γ + Γ′, because f is ample. Since ψ preserves the intersection form on X, the equality (ϕ^{∗}V, ϕ^{∗}V) = 2 now shows that (Γ, Γ) = (Γ′, Γ′) = −2. We conclude that the classes of Γ and Γ′ are independent in Pic X since the determinant of the intersection matrix

|\begin{matrix}−2 & 3 \\ 3 & −2 \end{matrix}| = −5

is nonzero. In particular, ρ(X) ≥ 2.

Example 6.1. Consider the K3 surface X of Example 4.1, defined by (10). The curve

Γ := {w = z = 0}

lies on this surface, and hence so does the image Γ′ of Γ under ψ:

Γ′ = {w + x^{2}y + y^{3} + y^{2}z = z = 0}

The classes of these curves over \overline{𝔽}_{2} together span a sublattice of Pic \overline{X} of discriminant −5, and thus ρ(X) ≥ 2.

6.2.2. Characteristic ≠ 2. Let (X, f) be a polarized K3 surface of degree 2 over an algebraically closed field of k with char(k) ≠ 2, given in the form (4) with α = 0.

Suppose that there exists a smooth conic C′ ⊆ ℙ_{k}^{2} tangent at six (not necessarily distinct) points to the ramification curve C of ϕ. Arguing as in [EJ08a, §2], we have ϕ^{∗}C = C_{1} + C_{2}, where C_{i} is an irreducible divisor for i = 1, 2, and the intersection matrix of the lattice spanned by the classes of C_{1} and C_{2} in Pic X has determinant

|\begin{matrix}−2 & 6 \\ 6 & −2 \end{matrix}| = −32 ≠ 0.

Hence ρ(X) ≥ 2 in this case.

Example 6.2. The polynomial

\begin{aligned}f(x, y, z) & = 2x^{5}z + x^{4}yz + x^{4}z^{2} + 2x^{3}yz^{2} + x^{2}y^{4} + 2x^{2}y^{3}z + x^{2}y^{2}z^{2} \\ & + 2x^{2}yz^{3} + xy^{3}z^{2} + xy^{2}z^{3} + 2xz^{5} + y^{6} + 2y^{4}z^{2} + y^{3}z^{3} \end{aligned} ; (13)

defines a smooth sextic curve C ⊆ ℙ_{𝔽_{3}}^{2}; the double cover X of ℙ_{𝔽_{3}}^{2} ramified along C is the K3 surface X of Example 4.2, given by (11). The smooth conic C′ in ℙ_{𝔽_{3}}^{2} given by

2x^{2} + 2xy + xz + 2y^{2} = 0.

is tangent to C at six points: indeed, let u be a multiplicative generator for 𝔽_{3^{2}}. Then C′ is parametrized by the map ℙ_{𝔽_{3^{2}}}^{1} = Proj(𝔽_{3^{2}}[s, t]) → ℙ_{𝔽_{3^{2}}}^{2} given by

x = u^{2}t^{2}, y = u^{6}st, and z = u^{2}s^{2} + u^{6}st + u^{2}t^{2}.

Substituting these expressions into f(x, y, z) we obtain

t^{2}(s^{5} + s^{4}t + s^{3}t^{2} + s^{2}t^{3} + 2st^{4} + t^{5})^{2}

The zeroes of this polynomial over \overline{𝔽}_{3} correspond to the geometric points of C ∩ C′; they clearly all have multiplicity greater than one. We conclude that ρ(X) ≥ 2.

6.3. A K3 surface of degree 2 over ℚ with geometric Picard rank one. We collect the results of this section in a single proposition for future reference. The proposition is a simple application of van Luijk’s method.

Proposition 6.3. Let X be a K3 surface of degree 2 over ℚ. Suppose that the reductions X_{2} and X_{3} of X at p = 2 and 3 are isomorphic to the K3 surfaces (10) and (11), respectively. Then ρ(X) = 1.

Proof. We use the notation introduced in §6.1. Counting 𝔽_{p^{n}} points on X_{p} for n = 1, …, 12 and p = 2, 3, and using Newton’s identities, we may compute the first twelve coefficients c_{1}, …, c_{12} of the characteristic polynomial f_{Φ^{∗}}(t) for p = 2 and 3. We obtain:

\begin{matrix}p & c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} & c_{7} & c_{8} & c_{9} & c_{10} & c_{11} & c_{12} \\ 2 & −2 & −2 & 4 & 8 & −32 & 0 & 64 & 128 & −512 & 512 & 0 & 2 048 \\ 3 & −1 & −18 & 9 & 135 & 162 & −243 & −3 645 & −6 561 & 26 244 & 118 098 & 0 & −106 288 \end{matrix}

(see Remark 6.4 below for the actual point counts). Note that to determine the sign of the functional equation of the characteristic polynomial, we must compute #X_{p}(𝔽_{p^{12}}), since c_{11} = 0 in both cases. We now have enough information to determine the remaining coefficients of the characteristic polynomial. Let f̃_{p}(t) = p^{−22}f_{Φ^{∗}}(pt), so that the number of roots of f̃_{p}(t) that are roots of unity gives an upper bound for ρ(X_{p}). We have the following factorizations into irreducible factors

f̃_{2}(t) = \frac{1}{2}(t − 1)^{2}(2t^{20} + 2t^{19} + t^{18} + t^{17} + 2t^{16} + t^{15} + t^{12} + t^{8} + t^{5} + 2t^{4} + t^{3} + t^{2} + 2t + 2),
f̃_{3}(t) = \frac{1}{3}(t − 1)(t + 1)(3t^{20} − t^{19} − 3t^{18} + 2t^{16} + 2t^{15} + t^{14} − 3t^{13} − 2t^{12}
+ t^{11} + 4t^{10} + t^{9} − 2t^{8} − 3t^{7} + t^{6} + 2t^{5} + 2t^{4} − 3t^{2} − t + 3).

In both cases, the roots of the degree 20 factor of f̃_{p}(t) are not integral, so they are not roots of unity. We conclude that ρ(X_{p}) ≤ 2 for p = 2, 3, and hence that ρ(X_{p}) = 2, by our work in Examples 6.1 and 6.2.

Finally, the discriminants of the full sublattices of Pic(X_{p}) exhibited in Examples 6.1 and 6.2 are in different square classes, and hence ρ(X) = 1. □

Remark 6.4. We record below the number of 𝔽_{p^{n}}-points of X_{p}. Set N_{n} := #X_{p}(𝔽_{p^{n}}); the second row of the table shows the counts for p = 2; the last row shows the counts for p = 3.

\begin{matrix}N_{1} & N_{2} & N_{3} & N_{4} & N_{5} & N_{6} & N_{7} & N_{8} & N_{9} & N_{10} & N_{11} & N_{12} \\ 7 & 25 & 73 & 249 & 1137 & 4273 & 16737 & 65313 & 264385 & 1047745 & 4203393 & 16767105 \\ 11 & 119 & 758 & 6707 & 58421 & 529472 & 4784357 & 43059323 & 387449246 & 3486568169 & 31380849731 & 282429079832 \end{matrix}

Counting points over 𝔽_{3^{11}} and 𝔽_{3^{12}} is computationally time consuming. See §8 for details.

7. Proof of Theorem 1.1

7.1. Geometric Picard rank one. The K3 surface X of Theorem 1.1 arises from the cubic fourfold Y given by (2): this is a direct computation applying the construction of §4.

Remark 7.1. An a posteriori analysis shows that Y does not contain a plane incident to P along a line. If it were to, then the quadric surface bundle q: Ỹ → ℙ_{ℚ}^{2} would have at least one singular geometric fiber containing a plane, and an analysis analogous to that in the proof of Lemma 5.2 would then show that the discriminant curve in ℙ_{ℚ}^{2} is singular. However, it is easy to check that this is not the case.

Proposition 7.2. We have ρ(X) = 1.

Proof. The reductions of the cubic fourfold Y at p = 2 and 3 coincide with the cubic fourfolds ℭ_{2} and ℭ_{3} of Examples 4.1 and 4.2, respectively. Thus, X has good reduction at p = 2 and 3, and its reductions at these primes are isomorphic to the K3 surfaces (10) and (11), respectively. The claim now follows directly from Proposition 6.3. □

7.2. The quaternion algebra. Let r: W → ℙ_{ℚ}^{2} denote the relative variety of lines, parametrizing lines in Y contained in fibers of q: Ỹ → ℙ_{ℚ}^{2}. By Theorem 5.1, the Stein factorization of r is given by

W ⟶^{π_{1}} X ⟶^{ϕ} ℙ_{ℚ}^{2},

where π_{1} is a smooth ℙ^{1}-bundle over X. The generic fiber of π_{1} is a Severi-Brauer variety C over the function field k(X) of X. By our discussion in §3.3, C is explicitly given by the conic

(2x + 3y + z)p_{14}^{2} + (3x + 3y)p_{14}p_{24} + (3x + 4y)p_{14}p_{34} + zp_{24}^{2} + 3zp_{24}p_{34} + (x + 3z)p_{34}^{2} = 0

in Proj k(X)[p_{14}, p_{24}, p_{34}]. Completing squares and renormalizing we obtain

p_{14}^{2} = αp_{24}^{2} + βp_{34}^{2},

where

α = \frac{9x^{2} + 18xy − 8xz + 9y^{2} − 12yz − 4z^{2}}{4(2x + 3y + z)^{2}} , and
β = −\frac{9x^{3} + 18x^{2}y + x^{2}z + 9xy^{2} + 3xyz − 10xz^{2} + 7y^{2}z − 9yz^{2} − 3z^{3}}{(2x + 3y + z)(9x^{2} + 18xy − 8xz + 9y^{2} − 12yz − 4z^{2})} .

The conic C is associated to the quaternion algebra (α, β) ∈ Br(k(X))[2] [GS06, Corollary 5.4.8]. Theorem 5.1 can be recast as saying that (α, β) is in the image of the natural injection

Br(X) ↪ Br(k(X)).

7.3. Evaluation of the local invariants. Since the K3 surface X is projective, the natural inclusion X(𝔸_{ℚ}) ⊆ ∏_{p≤∞} X(ℚ_{p}) is a bijection. Class field theory gives rise to the constraint

X(ℚ) ⊆ X(𝔸_{ℚ})^{Br} := {(x_{p})_{p} ∈ X(𝔸_{ℚ}) | ∑_{p≤∞} inv_{p} (𝒜 (x_{p})) = 0 for every 𝒜 ∈ Br(X)},

where 𝒜 (x_{p}) := 𝒜_{x_{p}} ⊗_{𝒪_{X,xp}} ℚ_{p} and inv_{p}: Br ℚ_{p} → ℚ/ℤ is the local invariant map for each p ≤ ∞.

A naive search for points on X reveals the rational point P_{1} := [15 : 15 : 16 : 13 752] and the real point P_{2} := [1 : 0 : 1 : \sqrt{8}]. Let 𝒜 = (α, β) be the quaternion algebra above, considered as an element of Br(X). Then

𝒜 (P_{1}) = (\frac{2 276}{4 · 91^{2}} , −\frac{75 852}{91 · 2 276}) and 𝒜 (P_{2}) = (−\frac{3}{36}, −\frac{1}{3})

and thus

(14)  inv_{∞} (𝒜 (P_{1})) = 0 and inv_{∞} (𝒜 (P_{2})) = \frac{1}{2}.

Proof of Theorem 1.1. The K3 surface X has geometric Picard rank one by Proposition 7.2. Thus, the algebraic Brauer group of X is trivial, because it is isomorphic, by the Hochschild-Serre spectral sequence, to the Galois cohomology group

H^{1}(Gal(\overline{ℚ}/ℚ), Pic(\overline{X})),

Figure 3. The real component (center) of the K3 surface of Theorem 1.1 where the real invariant of A is nontrivial (view of the z = 1 affine patch)

and the Galois action is trivial on the rank-one free ℤ-module Pic(\overline{X}). Therefore, the quaternion algebra A = (α, β) of §7.2, is a transcendental element of Br(X).

Define P ∈ X(𝔸_{ℚ}) as the point that is equal to P_{1} at all finite places, and is P_{2} at the real place. By (14), it follows that

∑_{p≤∞} inv_{p} (𝒜 (P_{p})) = \frac{1}{2},

and thus P ∉ X(𝔸_{ℚ})^{Br}. This shows that X does not satisfy weak approximation. □

Remarks 7.3. (1) It is natural to ask if the methods of this paper can also yield counterexamples to the Hasse principle. Numerical experiments suggest that, in the absence of more constraints, the primes of bad reduction for the K3 surfaces we construct give rise to only mild singularities. It is possible that such singularities cannot force the evaluation maps at these bad places to be constant. Thus, more constraints on the initial cubic fourfold may be necessary to violate the Hasse principle with a single Brauer element.

We also remark that imposing conditions on a smooth cubic fourfold at two different primes tends to create rather large primes of bad reduction for the associated K3 surface. Calculating the evaluation maps at these primes can be a computational challenge. The methods of [EJb] may well obviate this difficulty.

(2) It would be interesting to figure out if \overline{X(ℚ)} = X(ℚ)^{Br} (see the introduction of [SZ08]). Such an equality holds for many del Pezzo surfaces. However, in our case, we do not even know if the set X(ℚ) is Zariski dense!

8. Computations

In this section we briefly outline the steps used to construct the example in Theorem 1.1, for the benefit of those wishing to carry out similar computations.

Step 1: Generate a list of homogeneous random polynomials in 𝔽_{p}[x, y, z] (p = 2 or 3)

{L_{11}, L_{12}, L_{13}, L_{22}, L_{23}, L_{33}, Q_{14}, Q_{24}, Q_{34}, C_{44}}

where the L_{ij} are linear, the Q_{ij} are quadratic and C_{44} is cubic. Check smoothness of the cubic fourfold (5) and the K3 surface X_{p} given by (8) (or (9) if p = 2). Start over if either one is not smooth.

Step 2: If p = 2, then test to see if the K3 surface has a divisor of the form (12); if p = 3, then use [EJ08a, Algorithm 12] to test for the existence of a conic tangent at six points to the branch curve of X_{p}. In each case, go back to Step 1 if the test fails to reveal curves of the required type.

Step 3: If p = 2, then count 𝔽_{2^{n}}-points on the K3 surface for n = 1, …, 12. Use this to compute the first 12 coefficients c_{1} …, c_{12} of the characteristic polynomial of the action of Frobenius on H_{ét}^{2}(X_{p} × k̄, ℚ_{ℓ}), as described in §6.1. If either c_{11} ≠ 0 or c_{12} ≠ 0 then compute the remaining coefficients using the functional equation for the characteristic polynomial. Use this information to obtain an upper bound ρ_{up} for the geometric Picard rank of the K3 surface. If ρ_{up} > 2, then go back to Step 1.

If p = 3, then use [EJ08a, Algorithm 15] to count 𝔽_{3^{n}}-points on the K3 surface for n = 1, …, 10. This algorithm uses the fact that X_{p} is defined over 𝔽_{p}, and counts Galois orbits of points. This saves a factor of n when counting 𝔽_{p^{n}}-points. Next, use [EJ08a, Algorithms 21 and 23] to determine (with a very high degree of certainty) an upper bound ρ_{up} for the geometric Picard rank of the K3 surface. If ρ_{up} > 2 then go back to Step 1. If ρ_{up} = 2 then count 𝔽_{3^{n}}-points on the K3 surface for n large enough to ascertain the characteristic polynomial of Frobenius with total certainty; this last step may not be necessary.

Step 4 Use the Chinese remainder theorem to construct a cubic fourfold over ℚ that reduces, mod 2 and 3, to the cubic fourfolds of Step 1. The resulting K3 surface X reduces to the corresponding K3 surfaces over 𝔽_{2} and 𝔽_{3}, and thus ρ(X) = 1. Compute the quaternion algebra 𝒜 ∈ Br(X), and perform a naive search for rational and real points to test for an obstruction to weak approximation, as in the Proof of Theorem 1.1.

For Step 3, we implemented [EJ08a, Algorithm 15] in Magma to count points on double covers of the projective plane over 𝔽_{3^{n}} for n = 1, …, 10. To count points over 𝔽_{3^{11}} and 𝔽_{3^{12}} in a reasonable amount of time, we programmed the arithmetic of these fields in C++. We obtained significant speed gains by implementing the following ideas:

• We choose an isomorphism 𝔽_{3^{n}} ≅ 𝔽_{3}[X]/(f), where f is an irreducible polynomial of degree n with a minimal number of nonzero coefficients, and store elements of 𝔽_{3^{n}} using representatives of degree less than n.

• For each element g ∈ 𝔽_{3^{n}}, we precalculate values of g^{k}, 2 ≤ k ≤ 6 (use Hörner’s algorithm to perform exponentiation). Intermediate results give polynomials in X of degree less than 2n − 1; we reduce these polynomials modulo f inline, rather than using pre-stored tables. As a by-product, we build a list of squares in 𝔽_{3^{n}}, which allows us to quickly answer the query “is g a square?”.

• To evaluate a polynomial such as (13), on the affine patch x = 1, we compute each monomial by multiplying two elements from our precalculated list, obtaining a polynomial in X of degree less than 2n − 1. We add these polynomials in X first, and then perform a single reduction mod f. If the result, for example, is a nonzero square, then we have two 𝔽_{3^{n}}-rational points on the corresponding K3 surface (11).

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Department of Mathematics, Rice University, Houston, TX 77005, USA

E-mail address: hassett@rice.edu
URL: http://www.math.rice.edu/~hassett

Department of Mathematics, Rice University, Houston, TX 77005, USA

E-mail address: varilly@rice.edu
URL: http://www.math.rice.edu/~av15

Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720

E-mail address: patvarilly@gmail.com
URL: http://gold.cchem.berkeley.edu/~pvarilly