INTERSECTIONS OF THREE QUADRICS IN ℙ^{7}

BRENDAN HASSETT, ALENA PIRUTKA, AND YURI TSCHINKEL

Abstract. We study rationality properties of smooth complete intersections of three quadrics in ℙ^{7}. We exhibit a smooth family of such intersections with both rational and non-rational fibers.

1. Introduction

The specialization method, introduced by Voisin [Voi15], and developed by Colliot-Thélène–Pirutka, Totaro, and others, has led to major advances in higher-dimensional complex birational geometry. It makes it possible, for the first time, to prove failure of stable rationality of some smooth quartic threefolds [CTP16b], cyclic covers [Voi15], [Bea16], [CTP16a], [Oka16], and large degree smooth Fano hypersurfaces in projective space [Tot16].

The specialization method yields failure of stable rationality of a very general member of a family of complex algebraic varieties from the existence of a single, mildly singular, fiber with an explicit obstruction, that can be formulated in terms of integral decomposition of the diagonal or universal CH_{0}-triviality (see Section 2.1 for more details and references). A surprising aspect of applications of the method was that a priori different families of varieties admit specializations to the same ‘reference varieties’. This allows us to propagate the failure of stable rationality, by finding suitable chains of specializations. Examples of such ‘reference varieties’ are conic or quadric surface bundles over rational surfaces, with carefully chosen discriminant loci (see [Pir16]). A similar approach – via specialization to quartic del Pezzo fibrations over ℙ^{1} – may be used to essentially settle the stable rationality problem for very general smooth rationally connected threefolds [HKT16], [HT16], [KO17], with the exception of cubic threefolds, whose stable rationality remains elusive [Voi17].

New effects arise in dimension four: rationality properties can change in smooth families [HPT16a]. The relevant reference variety is Y ⊂ ℙ_{λ}^{2} × ℙ_{y}^{3}, given by the vanishing of the (2, 2) form

Date: June 5, 2017.

(1.1)  λ_{1}λ_{2}y_{0}^{2} + λ_{0}λ_{2}y_{1}^{2} + λ_{0}λ_{1}y_{2}^{2} + F (λ_{0}, λ_{1}, λ_{2})y_{3}^{2},

with

(1.2)  F (λ_{0}, λ_{1}, λ_{2}) := λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} − 2(λ_{0}λ_{1} + λ_{0}λ_{2} + λ_{1}λ_{2})

defining a conic tangent to each coordinate line. The family is the universal (2, 2) hypersurface, a Fano fourfold of Picard rank two.

The variety Y gives rise to other interesting families of fourfolds failing stable rationality: double covers [HPT16b], and conic bundles over ℙ^{3} [APBvB16]. In this note, we exhibit another natural family of smooth complex projective fourfolds X with rational and irrational fibers: Fano fourfolds of Picard rank one, obtained as intersections of three quadrics in ℙ^{7}.

Theorem 1. Let B ⊂ Gr(3, Γ(O_{ℙ^{7}}(2))) be the open subset of the Hilbert scheme parametrizing smooth complete intersections of three quadrics in ℙ^{7} and

(1.3)  ϕ : X → B

the corresponding universal family.

(1) For very general b ∈ B the fiber X_{b} is not stably rational.

(2) The set of b ∈ B such that X_{b} is rational is dense in B for the Euclidean topology.

Acknowledgments: The first author was partially supported by NSF grant 1551514, the second author by NSF grant 1601680, and the third by NSF grant 1601912. We would like to thank François Charles for helpful conversations.

2. Strategy

We follow the approach in [HPT16a]. In this section, we recall the main steps in the proof; details are provided in Section 3.

2.1. Fibers that are not stably rational. Recall that a projective variety X over a field k is universally CH_{0}- trivial if for all field extensions k′/k the natural degree homomorphism from the Chow group of zero-cycles

CH_{0}(X_{k′}) → ℤ

is an isomorphism. A projective morphism

β : X̃ → X

of k-varieties is universally CH_{0}-trivial if for all extensions k′/k the push-forward homomorphism

β_{∗} : CH_{0}(X̃_{k′}) → CH_{0}(X_{k′})

is an isomorphism.

In this paper, we apply the specialization method of Voisin in the following form.

Theorem 2. [Voi15, Theorem 2.1], [CTP16b, Theorem 2.3] Let

ϕ : X → B

be a flat projective morphism of complex varieties with smooth generic fiber. Assume that there exists a point b ∈ B such that the fiber

X := ϕ^{−1}(b)

satisfies the following conditions:

(R) X admits a desingularization

β : X̃ → X

such that the morphism β is universally CH_{0}-trivial; (O) the variety X̃ is not universally CH_{0}-trivial.

Then a very general fiber of ϕ is not universally CH_{0}-trivial; in particular, it is not stably rational.

Condition (O) holds, for instance, if the unramified cohomology group H_{nr}^{2}(ℂ(X)/ℂ, ℤ/2) is nontrivial. By [CTP16b, Proposition 1.8] and [CTP16a, Lemma 2.4] condition (R) is satisfied if for every scheme point x of X, the fiber β^{−1}(x), considered as a variety over the residue field κ(x), could be written as β^{−1}(x) = ∪_{i}X_{i}, where each component X_{i} is smooth, geometrically irreducible and κ(x)-rational and each intersection X_{i} ∩ X_{j} is either empty or has a zero-cycle of degree 1.

In [HPT16a, Propositions 11, 12], we constructed a hypersurface Y ⊂ ℙ^{2} × ℙ^{3} of bidegree (2, 2), satisfying the obstruction condition (O) and the resolution condition (R) as above (see (1.1)). The first projection Y → ℙ^{2} endows Y with a structure of a quadric surface bundle with discriminant curve of degree 8. As explained in [Bea77, Exemple 1.4.4], smooth intersections of three quadrics in ℙ^{7} are also birational to quadric surface bundles over ℙ^{2}, with discriminant curve of degree 8 (see Proposition 6 below). These two families, hypersurfaces of bidegree (2, 2) in ℙ^{2} × ℙ^{3} and intersections of three quadrics in ℙ^{7}, are genuinely different; see Section 4 for a precise statement. Both specialize (birationally) to the same reference fourfold: in Proposition 7 we provide an explicit example of a (singular) intersection of three quadrics X ⊂ ℙ^{7} such that X is birational to the variety Y above. We deduce Theorem 1, Part (1), from Theorem 2 at the end of Section 3.1.

2.2. One rational fiber. Let ϕ : X → B be the family (1.3). By Proposition 6, for any b ∈ B, the fiber X_{b} is birational to a quadric bundle over ℙ^{2}. In Section 3.2 (Proposition 9), we provide an explicit example of a fiber X_{b}, birational to a quadric bundle with a rational section. In particular, the fourfold X_{b} is rational.

2.3. Density of rational fibers. Let X ⊂ ℙ^{7} be a smooth intersection of three quadrics. As in the previous step, in order to establish that X is rational, it suffices to exhibit a quadric surface bundle π : Q → ℙ^{2} such that Q is birational to X and such that π admits a rational section. By Springer’s theorem, it suffices to show that π has a rational multisection of odd degree. For quadric bundles this can be formulated as a Hodge-theoretic condition:

Proposition 3. [CTV12, Corollaire 8.2] Let Q be a smooth projective complex algebraic variety, admitting a dominant morphism π : Q → ℙ^{2}, with generic fiber a quadric of dimension at least 1. Then the integral Hodge conjecture holds for classes of degree (2, 2) on Q.

Thus, in order to show that X is rational, it suffices to provide a (2, 2)-Hodge class intersecting the class of a fiber of π in odd degree. We achieve this by studying the infinitesimal period map. This technique is explained in [Voi07, 5.3.4].

The Hodge diamond of X is of the following form:

1

0  0
0  1  0
0  0  0  0
0  3  38  3  0
0  0  0  0
0  1  0
0  0

In particular, the degree 4 cohomology is essentially of weight 2. We can then apply the following criterion to the family X → B of Theorem 1 (cf. [Voi07, 5.3.4]):

Proposition 4. Suppose there exists a b_{0} ∈ B and γ ∈ H^{2,2}(X_{b_{0}}) such that the infinitesimal period map

(2.1)  ¯∇ : T_{B,b_{0}} → Hom(H^{2,2}(X_{b_{0}}), H^{1,3}(X_{b_{0}})),

evaluated at γ, gives a surjective map

(2.2)  ¯∇(γ) : T_{B,b_{0}} → H^{1,3}(X_{b_{0}}).

Then for any b ∈ B and any Euclidean neighborhood b ∈ B′ ⊂ B, the image of the natural map (composition of inclusion with local trivialization):

(2.3)  H_{ℝ}^{2,2} → H^{4}(X_{b}, ℝ)

contains an open subset V_{b} ⊂ H^{4}(X_{b}, ℝ). Here H_{ℝ}^{2,2} is a vector bundle over B′ with fiber over u equal to the real classes of type (2, 2) in H^{4}(X_{u}).

In order to check the infinitesimal criterion we use an explicit description of the period map:

Proposition 5. [Ter90, Corollary 2.5, Proposition 2.6] Let X ⊂ ℙ^{7} be a smooth complete intersection of three quadrics, defined by equations

Q_{i}(x_{0}, …, x_{7}) = 0, i = 0, 1, 2

and let

F = μ_{0}Q_{0} + μ_{1}Q_{1} + μ_{2}Q_{2} ∈ ℂ[μ_{0}, μ_{1}, μ_{2}, x_{0}, …, x_{7}].

Let I ⊂ ℂ[μ_{0}, μ_{1}, μ_{2}, x_{0}, …, x_{7}] be the ideal generated by

∂F/∂μ_{i}, i = 0, 1, 2 and ∂F/∂x_{i}, i = 0, …, 7.

Put

R = ℂ[μ_{0}, μ_{1}, μ_{2}, x_{0}, …, x_{7}]/I

and let R_{(a,b)} be the space of homogeneous elements of degree (a, b) in R, with respect to the grading (μ, x). Then there is an isomorphism

H_{prim}^{4−q,q}(X) ≃ R_{(q,2q−2)}

and the period map (2.1) is identified with the multiplication homomorphism

(2.4)  R_{(1,2)} ⊗ R_{(2,2)} → R_{(3,4)}.

Recall that the primitive cohomology H_{prim}^{p,q} is the cokernel of the natural map H^{p,q}(ℙ^{7}) → H^{p,q}(X).

In Section 3.3, we provide an explicit example X = X_{b_{0}} such that the period map 2.4 is surjective (Proposition 10). Theorem 1, Part (2), then follows. In fact, by Proposition 3.2, there exists a smooth intersection of three quadrics birational to a quadric bundle with a rational section. Similarly to [HPT16a, Proposition 14] the density of rational fibers follows from the infinitesimal criterion that we verify in Proposition 10.

3. Computations

We work over the complex numbers. We first recall the construction of Beauville [Bea77, Exemple 1.4.4]:

Proposition 6. Let X ⊂ ℙ^{7} be a smooth complete intersection of three quadrics. Then X is birational to a quadric bundle over ℙ^{2}, with discriminant curve of degree 8.

Concretely, let ℓ ⊂ X be a line and G_{ℓ} ≃ ℙ^{5} the space of 2-planes Π ⊂ ℙ^{7} containing ℓ. Then X is birational to a quadric surface bundle

π : Q → ℙ^{2},

where Q ⊂ ℙ^{2} × G_{ℓ} is given by

(3.1) Q = {([λ_{0} : λ_{1} : λ_{2}], Π)| {λ_{0}Q_{0} + λ_{1}Q_{1} + λ_{2}Q_{2} = 0} ⊃ Π} .

More explicitly, assume that the line is given by equations

ℓ : x_{2} = x_{3} = … = x_{7} = 0

and write, for i = 0, 1, 2,

Q_{i} = x_{0}L_{i}(x_{2}, x_{3}, …, x_{7}) + x_{1}M_{i}(x_{2}, x_{3}, …, x_{7}) + q_{i}(x_{2}, x_{3}, …, x_{7}),

where L_{i} and M_{i} are linear forms and q_{i} is quadratic. Any 2-plane Π ⊂ ℙ^{7} containing ℓ intersects the 5-plane x_{0} = x_{1} = 0 in a unique point [0 : 0 : x_{2} : ⋯: x_{7}]. This allows us to identify the space of 2-planes Π ⊂ ℙ^{7} containing ℓ with ℙ^{5}. Then the quadric bundle (3.1) is defined in ℙ^{2} × ℙ^{5} by the equations

(3.2) ∑_{i=0}^{2} λ_{i}L_{i}(x_{2}, x_{3}, …, x_{7}) = ∑_{i=0}^{2} λ_{i}M_{i}(x_{2}, x_{3}, …, x_{7}) =

= ∑_{i=0}^{2} λ_{i}q_{i}(x_{2}, x_{3}, …, x_{7}) = 0.

3.1. Fibers that are not stably rational. Let X ⊂ ℙ^{7} be the intersection of three quadrics

(3.3) Q_{0} : −x_{0}x_{5} + x_{3}^{2} + x_{4}x_{6} − 2x_{5}^{2} = 0;

Q_{1} : x_{0}x_{5} + x_{1}x_{4} + x_{2}^{2} − 2x_{5}^{2} = 0;

Q_{2} : x_{0}x_{7} − x_{1}x_{6} + x_{5}^{2} + x_{7}^{2} = 0.

Note that X contains a line ℓ : x_{2} = … = x_{7} = 0. Using equations (3.2), we obtain that X is birational to a quadric bundle Q → ℙ^{2}, defined in ℙ^{2} × ℙ^{5} as an intersection of two forms of bidegree (1, 1) and one form of bidegree (1, 2):

(3.4) (λ_{0} − λ_{1})x_{5} = λ_{2}x_{7}, λ_{1}x_{4} = λ_{2}x_{6}

λ_{1}x_{2}^{2} + λ_{0}x_{3}^{2} + λ_{0}x_{4}x_{6} + (λ_{2} − 2λ_{0} − 2λ_{1})x_{5}^{2} + λ_{2}x_{7}^{2} = 0.

In the open set λ_{2} ≠ 0 we can define X by a single equation

λ_{1}x_{2}^{2} + λ_{0}x_{3}^{2} + \frac{λ_{0}λ_{1}}{λ_{2}} x_{4}^{2} + (\frac{(λ_{0} − λ_{1})^{2}}{λ_{2}} + λ_{2} − 2λ_{0} − 2λ_{1})x_{5}^{2} = 0,

hence, X is birational to a hypersurface Y ⊂ ℙ^{2} × ℙ^{3} of bidegree (2, 2) defined by

(3.5)  λ_{1}λ_{2}x_{2}^{2} + λ_{0}λ_{2}x_{3}^{2} + λ_{0}λ_{1}x_{4}^{2} + F (λ_{0}, λ_{1}, λ_{2})x_{5}^{2} = 0,

where F (λ_{0}, λ_{1}, λ_{2}) = λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} − 2λ_{0}λ_{1} − 2λ_{0}λ_{2} − 2λ_{1}λ_{2}. This is precisely the hypersurface we considered in [HPT16a, Propositions 11, 12].

Proposition 7. Let Q ⊂ ℙ^{2} × ℙ^{5} be defined by the equations (3.4) and let Y ⊂ ℙ^{2} × ℙ^{3} be the hypersurface given by the equation (3.5). Then the birational map

(3.6) φ : Y −−→ Q,

(λ_{0} : λ_{1} : λ_{2}, x_{2} : …: x_{5}) ↦

(λ_{0} : λ_{1} : λ_{2}, λ_{2}x_{2} : λ_{2}x_{3} : λ_{2}x_{4} : λ_{2}x_{5} : λ_{1}x_{4} : (λ_{0} − λ_{1})x_{5})

extends to the following diagram

\xymatrix{ & Ỹ \ar[dl]_{ψ} \ar[dr]^{φ̃} \\ Y \ar@{-->}[rr] &  & Q}

where the morphisms ψ : Ỹ → Y and φ̃ : Ỹ → Q are birational and universally CH_{0}-trivial.

Proof. First note that φ is indeed a birational map between Y and Q. The locus Y^{nd} ⊂ Y where the map φ is not defined is a union of three components Y_{1} : λ_{2} = 0, x_{4} = x_{5} = 0; Y_{2} : λ_{1} = λ_{2} = 0, x_{5} = 0; Y_{3} : λ_{0} − λ_{1} = 0, λ_{2} = 0, x_{4} = 0. Note that Y_{1} is isomorphic to a product ℙ_{λ_{0}:λ_{1}}^{1} × ℙ_{x_{2}:x_{3}}^{1}, and similarly Y_{2} is isomorphic to a projective plane ℙ_{x_{2}:x_{3}:x_{4}}^{2} with homogeneous coordinates [x_{2} : x_{3} : x_{4}] and Y_{3} ≃ ℙ_{x_{2}:x_{3}:x_{5}}^{2}.

We construct Ỹ by successive blowups of Y_{1}, the proper transform of Y_{2} and the proper transform of Y_{3}. After each blowup we verify:

• the indeterminacy locus of φ on the blowup;

• the universal CH_{0}-triviality of fibers of the extension of φ to the blowup and of the blowup map. In each case we obtain that the corresponding fiber is either reduced to a point or projective (or affine, if we compute on open charts) spaces. We provide details for the first computations and the expressions in the coordinates for the remaining charts.

Blowup of Y_{1}. We have three charts:

(1) U_{1} : x_{4} = λ_{2}u_{4}, x_{5} = λ_{2}u_{5}, the exceptional divisor is given by λ_{2} = 0. Since we blow up the locus λ_{2} = 0, x_{4} = x_{5} = 0, we consider one of the charts λ_{0} ≠ 0 or λ_{1} ≠ 0 of ℙ^{2} and one of the charts x_{2} ≠ 0 or x_{3} ≠ 0 of ℙ^{3}. We extend φ to a birational map φ_{1} : U_{1} −−→ Q,

(λ_{0}, λ_{1}, λ_{2}, x_{2}, x_{3}, u_{4}, u_{5}) ↦ (λ_{0}, λ_{1}, λ_{2}, x_{2}, x_{3}, λ_{2}u_{4}, λ_{2}u_{5}, λ_{1}u_{4}, (λ_{0}−λ_{1})u_{5}).

Since one of coordinates λ_{0}, λ_{1} is nonzero, and one of coordinates x_{2}, x_{3} is nonzero, we have that φ_{1} is well-defined. The image of φ_{1} is contained in the closure of the image of φ, hence it is contained in Q, so that we obtain a map φ_{1} : U_{1} → Q.

The image of the exceptional divisor is the set of points

E_{1} = (λ_{0}, λ_{1}, 0, x_{2}, x_{3}, 0, 0, λ_{1}u_{4}, (λ_{0} − λ_{1})u_{5}).

Then for any field k′/ℂ and for any point P ∈ E_{1}(k′) the fiber φ_{1}^{−1}(P) is either a point or a line (if λ_{1} = 0 or λ_{0} − λ_{1} = 0), which ensures the universal CH_{0}-triviality of the map φ_{1} on this chart.

The equation defining U_{1} is

λ_{1}x_{2}^{2} + λ_{0}x_{3}^{2} + λ_{0}λ_{1}λ_{2}u_{4}^{2} + F (λ_{0}, λ_{1}, λ_{2})λ_{2}u_{5}^{2} = 0.

Let ψ_{1} : U_{1} → Y be the blowup map. Then, the image I_{1} of the exceptional divisor is given by the conditions

λ_{2} = 0, λ_{1}x_{2}^{2} + λ_{0}x_{3}^{2} = 0.

The latter condition defines a point since the coordinates λ_{0} : λ_{1} and x_{2} : x_{3} are homogeneous. Then for any field k′/ℂ and for any point P ∈ I_{1}(k′) the fiber ψ_{1}^{−1}(P) is a plane with coordinates u_{4} and u_{5}, which ensures the universal CH_{0}-triviality of the map ψ_{1} on this chart.

(2) U_{2} : • change of variables:

λ_{2} = x_{4}λ′_{2}, x_{5} = x_{4}u_{5};

• equation defining the blowup:

λ_{1}λ′_{2}x_{2}^{2} + λ_{0}λ_{2}^{2}x_{3}^{2} + λ_{0}λ_{1}x_{4} + F (λ_{0}, λ_{1}, λ′_{2}x_{4})x_{4}u_{5}^{2} = 0.

• exceptional divisor:

x_{4} = 0, λ_{1}λ′_{2}x_{2}^{2} + λ_{0}λ_{2}^{2}x_{3}^{2} = 0.

• extension of φ is given by:

(λ_{0}, λ_{1}, λ′_{2}x_{4}, λ′_{2}x_{2}, λ′_{2}x_{3}, λ′_{2}x_{4}, λ′_{2}x_{4}u_{5}, λ_{1}, (λ_{0} − λ_{1})u_{5}).

• domain, where the extension is not defined is the proper transform Y′_{2} of Y_{2}:

λ_{1} = λ′_{2} = 0, u_{5} = 0.

• the image of the exceptional divisor:

(λ_{0}, λ_{1}, 0, λ′_{2}x_{2}, λ′_{2}x_{3}, 0, 0, λ_{1}, (λ_{0} − λ_{1})u_{5}).

(3) U_{3} : • change of variables:

λ_{2} = x_{5}λ′_{2}, x_{4} = x_{5}u_{4};

• equation defining the blowup:

λ_{1}λ′_{2}x_{2}^{2} + λ_{0}λ′_{2}x_{3}^{2} + λ_{0}λ_{1}x_{5}u_{4}^{2} + F (λ_{0}, λ_{1}, λ′_{2}x_{5})x_{5} = 0; • exceptional divisor:

x_{5} = 0, λ_{1}λ′_{2}x_{2}^{2} + λ_{0}λ_{2}^{2}x_{3}^{2} = 0;

• extension of φ is given by:

(λ_{0}, λ_{1}, λ′_{2}x_{5}, λ′_{2}x_{2}, λ′_{2}x_{3}, λ′_{2}x_{5}u_{4}, λ′_{2}x_{5}, λ_{1}u_{4}, λ_{0} − λ_{1})

• domain, where the extension is not defined is the proper transform Y′_{3} of Y_{3}:

λ_{0} − λ_{1} = λ′_{2} = 0, u_{4} = 0.

• the image of the exceptional divisor:

(λ_{0}, λ_{1}, 0, λ′_{2}x_{2}, λ′_{2}x_{3}, 0, 0, λ_{1}u_{4}, λ_{0} − λ_{1}).

Blowup of the proper transforms Y′_{2} and Y′_{3} Note that Y_{2} and Y_{3}, and hence their proper transforms, do not

intersect. Hence we can use charts U_{2} and U_{3} independently for their blowups.

(1) On the chart U_{2}: (a) • change of variables:

λ_{1} = λ′_{2}λ′_{1}, u_{5} = λ′_{2}v_{5}

• exceptional divisor:

λ′_{2} = 0, λ_{0}x_{3}^{2} + λ_{0}λ′_{1}x_{4};

• extension of φ is everywhere defined:

(λ_{0}, λ′_{1}λ′_{2}, x_{4}λ′_{2}, x_{2}, x_{3}, x_{4}, λ′_{2}x_{4}v_{5}, λ′_{1}, (λ_{0} − λ′_{1}λ′_{2})v_{5});

• the image of the exceptional divisor:

(1, 0, 0, x_{2}, x_{3}, x_{4}, x_{4}v_{5}, λ′_{1}, v_{5}).

(b) • change of variables:

λ′_{2} = λ_{1}λ′′_{2}, u_{5} = λ_{1}v_{5};

• exceptional divisor:

λ_{1} = 0, λ_{0}λ′′_{2}x_{3}^{2} + λ_{0}x_{4} = 0;

• extension of φ is everywhere defined:

(λ_{0} : λ_{1} : λ_{1}λ′′_{2}, λ′′_{2}x_{2}, λ′′_{2}x_{3}, λ′′_{2}x_{4}, λ_{1}λ′′_{2}x_{4}v_{5}, 1, v_{5}(λ_{0} − λ_{1}));

• the image of the exceptional divisor:

(1, 0, 0, λ′′_{2}x_{2}, λ′′_{2}x_{3}, λ′′_{2}x_{4}, 0, 1, v_{5}).

INTERSECTIONS OF THREE QUADRICS 11

(c) • change of variables:

λ′_{2} = u_{5}λ′′_{2}, λ_{1} = u_{5}λ′′_{1};

• exceptional divisor:

u_{5} = 0, λ_{0}λ′′_{2}x_{3}^{2} + λ_{0}λ′′_{1}x_{4} = 0;

• extension of φ is everywhere defined: (λ_{0}, λ′′_{1}u_{5}, λ′′_{2}u_{5}, λ′′_{2}x_{2}, λ′′_{2}x_{3}, λ′′_{2}x_{4}, λ′′_{2}x_{4}u_{5}, λ′′_{1}, λ_{0} − λ′′_{1}u_{5});

• the image of the exceptional divisor:

(1, 0, 0, λ′′_{2}x_{2}, λ′′_{2}x_{3}, λ′′_{2}x_{4}, 0, λ′′_{1}, 1).

(2) On the chart U_{3}:

(a) • change of variables:

λ_{0} − λ_{1} = λ′_{2}λ′_{0}, u_{4} = λ′_{2}v_{4};

• exceptional divisor:

λ′_{2} = 0, λ_{1}x_{2}^{2} + λ_{1}x_{3}^{2} − 4λ_{1}x_{5} = 0;

• extension of φ is everywhere defined: (λ_{1} + λ′_{2}λ′_{0}, λ_{1}, λ′_{2}x_{5}, x_{2}, x_{3}, λ′_{2}x_{5}v_{4}, x_{5}, λ_{1}v_{4}, λ′_{0});

• the image of the exceptional divisor:

(λ_{1}, λ_{1}, 0, x_{2}, x_{3}, 0, x_{5}, λ_{1}v_{4}, λ′_{0}).

(b) • change of variables: λ′_{2} = (λ_{0} − λ_{1})λ′′_{2}, u_{4} = (λ_{0} − λ_{1})v_{4};

• exceptional divisor: (λ_{0} − λ_{1}) = 0, λ_{1}λ′′_{2}x_{2}^{2} + λ_{1}λ′′_{2}x_{3}^{2} − 4λ_{1}λ′′_{2}x_{5} = 0;

• extension of φ is everywhere defined:

(λ_{0}, λ_{1}, λ′′_{2}(λ_{0} − λ_{1})x_{5}, λ′′_{2}x_{2}, λ′′_{2}x_{3}, (λ_{0} − λ_{1})λ′′_{2}x_{5}v_{4}, λ′′_{2}x_{5}, λ_{1}v_{4}, 1); • the image of the exceptional divisor: (λ_{1}, λ_{1}, 0, λ′′_{2}x_{2}, λ′′_{2}x_{3}, 0, λ′′_{2}x_{5}, λ_{1}v_{4}, 1).

(c) • change of variables:

λ′_{2} = u_{4}λ′′_{2}, λ_{0} − λ_{1} = u_{4}λ′_{0};

• exceptional divisor: u_{4} = 0, λ_{1}λ′′_{2}x_{2}^{2} + λ_{1}λ′′_{2}x_{3}^{2} − 4λ_{1}λ′′_{2}x_{5} = 0;

• extension of φ is everywhere defined:

(λ_{1} + u_{4}λ′_{0}, λ_{1}, λ′′_{2}u_{4}x_{5}, λ′′_{2}x_{2}, λ′′_{2}x_{3}, λ′′_{2}x_{5}u_{4}, λ′′_{2}x_{5}, λ_{1}, λ′_{0});

• the image of the exceptional divisor:

(λ_{1}, λ_{1}, 0, λ′′_{2}x_{2}, λ′′_{2}x_{3}, 0, λ′′_{2}x_{5}, λ_{1}, λ′_{0}).

□

Corollary 8. Let Q ⊂ ℙ^{2} × ℙ^{5} be defined by the equations (3.4). Then Q admits a resolution of singularities β : Q̃ → Q such that

(i) the variety Q̃ is not universally CH_{0}-trivial; (ii) the map β is a universally CH_{0}-trivial morphism.

Proof. We use Proposition 7: Q is birational to a variety Y with H_{nr}^{2}(ℂ(Y)/ℂ, ℤ/2) ≠ 0 by [HPT16a, Proposition 11]. In particular, property (i) holds for any resolution Q̃ of Q.

In [HPT16a, Proposition 12] we constructed a resolution of singularities f : Z → Y such that f is a universally CH_{0}-trivial morphism. Then there is birational map f̃: Z̃ → Z with Z̃ smooth, such that the rational map Z −−→ Ỹ extends to a map Z̃ → Ỹ:

\xymatrix{Z̃ \ar[d]_{f̃} \ar[dr] \\ Z \ar[d]_{f} \ar@{-->}[r] & Ỹ \ar[dl]_{ψ} \ar[dr]^{φ̃} \\ Y \ar@{-->}[rr] &  & Q}

Note that the map f̃ is universally CH_{0}-trivial: by weak factorization, f̃ factors through blow-ups and blow-downs at smooth centers, each of these maps is universally CH_{0}-trivial. Hence, in the diagram above, the maps f̃, f, ψ, φ̃ are universally CH_{0}-trivial. We deduce from the diagram that the composite map Z̃ → Q is also universally CH_{0}- trivial, which shows (ii). □

Proof of Theorem 1, Part (1): From Theorem 2 and Corollary 8 we deduce that a very general quadric bundle defined by equations (3.2) is not universally CH_{0}-trivial. In particular, there exists a smooth intersection of three quadrics X birational to a smooth quadric bundle Q defined by an equation of type (3.2), such that Q is not universally CH_{0}-trivial. Since universal CH_{0}-triviality is a birational invariant of smooth projective varieties, we deduce that X is not universally CH_{0}-trivial. Then Theorem 1, Part (1), follows directly from Theorem 2, applied to the universal family ϕ : X → B of smooth complete intersections of three quadrics in ℙ^{7}. □

3.2. One rational fiber. Consider the quadrics

Q_{0} : x_{0}(x_{3} + x_{5} + 2x_{6} + 3x_{7}) + x_{1}(−x_{5} + 5x_{6} + 2x_{7})−

− x_{2}x_{3} − x_{2}x_{4} + x_{2}x_{5} + x_{3}^{2} − x_{4}x_{6} + x_{5}^{2} + x_{6}^{2} + x_{7}^{2} = 0;

Q_{1} : x_{0}(−x_{2} + 3x_{5} + 7x_{6} + 11x_{7}) + x_{1}(x_{4} + 9x_{5} + 4x_{6} + x_{7})+

+ x_{2}^{2} − x_{2}x_{3} + 2x_{3}x_{6} + x_{4}^{2} + 3x_{4}x_{7} + 2x_{5}^{2} + 3x_{6}^{2} + 5x_{7}^{2} = 0; Q_{2} : x_{0}(11x_{5} + 13x_{6} + 8x_{7}) + x_{1}(−x_{3} + 6x_{5} + 7x_{6} + 3x_{7})+

+ x_{2}^{2} + 5x_{2}x_{7} − x_{3}x_{4} + 9x_{3}x_{5} + 13x_{5}^{2} + 4x_{6}^{2} + 11x_{7}^{2} = 0.

Proposition 9. Let X ⊂ ℙ^{7} be the intersection

Q_{0} = Q_{1} = Q_{2} = 0

Then X is smooth and rational.

Proof. A Magma [BCP97] computation shows that X is smooth. Furthermore, X contains a line

ℓ : x_{2} = … = x_{7} = 0.

As in Proposition 6, considering the space G_{ℓ} ≃ ℙ^{5} of 2-planes Π ⊂ ℙ^{7} containing ℓ, we find that X is birational to a fibration in quadrics Q → ℙ^{2}, where Q ⊂ ℙ^{2} × G_{ℓ},

Q = {([λ_{0} : λ_{1} : λ_{2}], Π)| {λ_{0}Q_{0} + λ_{1}Q_{1} + λ_{2}Q_{2} = 0} ⊃ Π}.

The first projection Q → ℙ^{2} admits a rational section: the plane containing ℓ and the point [0 : 0 : λ_{0} : λ_{1} : λ_{2} : 0 : 0 : 0] is contained in {λ_{0}Q_{0} +λ_{1}Q_{1} +λ_{2}Q_{2} = 0}. Indeed, by (3.2), we have that Q ⊂ ℙ^{2} ×ℙ^{5} is defined by the equations:

λ_{0}(x_{3}+x_{5}+2x_{6}+3x_{7})+λ_{1}(−x_{2}+3x_{5}+7x_{6}+11x_{7})+λ_{2}(11x_{5}+13x_{6}+8x_{7}) = 0 λ_{0}(−x_{5}+5x_{6}+2x_{7})+λ_{1}(x_{4}+9x_{5}+4x_{6}+x_{7})+λ_{2}(−x_{3}+6x_{5}+7x_{6}+3x_{7}) = 0

λ_{0}(−x_{2}x_{3}−x_{2}x_{4}+x_{2}x_{5}+x_{3}^{2}−x_{4}x_{6}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2})+λ_{1}(x_{2}^{2}−x_{2}x_{3}+2x_{3}x_{6}+x_{4}^{2}+

+3x_{4}x_{7}+2x_{5}^{2}+3x_{6}^{2}+5x_{7}^{2})+λ_{2}(x_{2}^{2}+5x_{2}x_{7}−x_{3}x_{4}+9x_{3}x_{5}+13x_{5}^{2}+4x_{6}^{2}+11x_{7}^{2}) = 0

and, substituting

[x_{2} : x_{3} : …: x_{7}] = [0 : 0 : λ_{0} : λ_{1} : λ_{2} : 0 : 0 : 0],

we obtain

λ_{0}λ_{1} − λ_{0}λ_{1} = 0, λ_{1}λ_{2} − λ_{1}λ_{2} = 0,

λ_{0}(−λ_{0}λ_{1} − λ_{0}λ_{2} + λ_{1}^{2}) + λ_{1}(λ_{0}^{2} + λ_{2}^{2} − λ_{0}λ_{1}) + λ_{2}(λ_{0}^{2} − λ_{1}λ_{2}) = 0.

□

3.3. Density of rational fibers. Using the notation of Section 3.2, consider quadrics

\begin{aligned}Q′_{0} & := & Q_{0} + x_{0}^{2} + x_{5}^{2} \\ Q′_{1} & := & Q_{1} \\ Q′_{2} & := & Q_{2} + x_{1}^{2} + x_{3}^{2} \end{aligned}

Proposition 10. Let X′ ⊂ ℙ^{7} be the intersection

Q′_{0} = Q′_{1} = Q′_{2} = 0.

Then X′ is smooth and there exists a γ ∈ H^{2,2}(X′) such that the period map (2.2) is surjective.

Proof. A Magma computation shows that X′ is smooth. In order to compute the period map we use expression (2.4). We used Macaulay2 [GS] to verify that the following monomials

{μ_{0}μ_{2}^{2}x_{7}^{4}, μ_{1}μ_{2}^{2}x_{7}^{4}, μ_{2}^{3}x_{7}^{4}}

form a basis of the graded part R_{(3,4)} ≃ H^{1,3}(X′). In particular γ = μ_{2}^{2}x_{7}^{2} works. □

4. Differentiating quadric bundles

The goal of this section is to show that the quadric bundles arising from complete intersection of three quadrics in ℙ^{7} do in fact differ from the (2, 2) hypersurfaces in ℙ^{2} × ℙ^{3} considered in [HPT16a]. Note however that both families specialize to the same reference variety (1.1).

Let π : Q → ℙ^{2} be a quadric surface bundle with smooth degeneracy curve D ⊂ ℙ^{2}, i.e., Q is a smooth complex projective fourfold, π is a flat morphism with smooth (≃ 𝔽_{0}) fibers over ℙ^{2}\D, and quadric cones (≃ ℙ(1, 1, 2)) as fibers over D. Let τ : S → ℙ^{2} denote the associated double cover, simply branched along D. We may interpret S as the Stein factorization of the relative variety of lines

F_{1}(Q/ℙ^{2}) → S → ℙ^{2};

as such, S is equipped with a natural conic bundle structure and thus a class α_{Q} ∈ H^{2}(S, μ_{2}). We refer the reader to [APS15] for a close analysis of the equivalence between quadric surface bundles and Azumaya algebras over double covers.

We present a cohomological interpretation of this correspondence due to Laszlo [Las89]. Let H_{0}^{2}(S, ℤ) denote the primitive cohomology of S, i.e., the kernel of τ_{∗}. It carries the structure of a lattice with respect to the intersection form, as well as a weight two Hodge structure. Choose an embedding

\begin{aligned}Q & ↪ & ℙ(E) \\ & ↘^{π} & ↓ \\ & & ℙ^{2} \end{aligned}

where E → ℙ^{2} is a rank four vector bundle. Let H_{0}^{4}(Q, ℤ) denote kernel of the push forward homomorphism

H^{4}(Q, ℤ) → H^{6}(ℙ(E), ℤ).

This carries the structure of a lattice and a weight four Hodge structure. Let H_{0}^{4}(Q, ℤ)(1) denote its Tate twist, a weight two Hodge structure; this reverses the sign of the integral quadratic form.

Theorem 11. [Las89, Th. II.3.1] There exists an embedding of abelian groups

Φ : H_{0}^{4}(Q, ℤ)(1) ↪ H_{0}^{2}(S, ℤ)

compatible with the lattice and Hodge structures. The image has index two and is characterized as follows:

image(Φ) = Λ_{Q} := {γ ∈ H_{0}^{2}(S, ℤ) : (γ mod 2, α_{Q}) ≡ 0 mod 2}.

Now suppose we have a birational equivalence

Q_{1}  −−→^{∼}  Q_{2}
↘  ↙
ℙ^{2}

of quadric bundles over ℙ^{2}. It is clear that Q_{1} and Q_{2} must have the same degeneracy curve D ⊂ ℙ^{2} and induced double cover τ : S → ℙ^{2}. Consider the classes α_{Q_{1}}, α_{Q_{2}} ∈ Br(S)[2], obtained via the canonical surjection H^{2}(S, μ_{2}) → Br(S)[2]. Since α_{Q_{i}} generates the kernel of

H^{2}(ℂ(S), μ_{2}) → H^{2}(ℂ(Q_{i}), μ_{2})

by [Ara75, p.469], we have α_{Q_{1}} = α_{Q_{2}}.

Proposition 12. Let D ⊂ ℙ^{2} be a very general octic plane curve, Q_{1}, Q_{2} → ℙ^{2} quadric surface bundles with degeneracy curve D, where Q_{1} ⊂ ℙ^{2} × ℙ^{3} is a (2, 2) hypersurface and Q_{2} ⊂ ℙ^{2} × ℙ^{5} is a complete intersection of hypersurfaces of bidegrees (1, 1), (1, 1), (1, 2). Then Q_{1} and Q_{2} are not birational over ℙ^{2}.

The precise condition we require is that Pic(S) ≃ ℤ.

Proof. For the first example, let h_{1} and h_{2} denote the pull-backs of the hyperplane classes from each factor. Then we have [Q_{1}] = 2h_{1} + 2h_{2} and

h_{1}^{2} h_{1}h_{2} h_{2}^{2}
h_{1}^{2} 0 0 2
h_{1}h_{2} 0 2 2
h_{2}^{2} 2 2 0

For the second example, let g_{1} and g_{2} denote the hyperplace classes as above so that

[Q_{2}] = 4g_{1}^{2}g_{2} + 5g_{1}g_{2}^{2} + 2g_{2}^{3}.

Then we have

g_{1}^{2} g_{1}g_{2} g_{2}^{2}
g_{1}^{2} 0 0 2
g_{1}g_{2} 0 2 5
g_{2}^{2} 2 5 4

These two lattices are inequivalent over the 2-adics. Indeed, their ranks modulo two differ. It follows that the lattices H_{0}^{4}(Q_{1}, ℤ) and H_{0}^{4}(Q_{2}, ℤ) are also inequivalent, as a nondegenerate lattice and its orthogonal complement in a unimodular lattice have the same discriminant groups up to sign. (The discriminant groups are a way of packaging the p-adic invariants of a lattice.)

Under our assumption, Br(S)[2] = H^{2}(S, μ_{2})/ ⟨h⟩ where h is the hyperplane class pulled back from ℙ^{2}. If Q_{1} and Q_{2} were birational over ℙ^{2} then

α_{Q_{1}} = α_{Q_{2}} ∈ H^{2}(S, μ_{2})/ ⟨h⟩ ,

whence Λ_{Q_{1}} ≃ Λ_{Q_{2}}. This would contradict Theorem 11. □

Remark 13. Observe that the common reference variety (1.1) admits nontrivial 2-torsion in its unramified cohomology. It is intriguing that we differentiate the smooth members through a 2-adic computation of lattices.

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Department of Mathematics, Brown University, Box 1917 151 Thayer Street Providence, RI 02912, USA

E-mail address: bhassett@math.brown.edu

Courant Institute, New York University, New York, NY 10012, USA

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

Courant Institute, New York University, New York, NY 10012, USA

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

Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA