STABLE RATIONALITY OF QUADRIC SURFACE BUNDLES OVER SURFACES

BRENDAN HASSETT, ALENA PIRUTKA, AND YURI TSCHINKEL

Abstract. We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

1. Introduction

Is a deformation of a smooth rational (or irrational) projective variety still rational (or irrational)? The main goal of this paper is to show that rationality is not deformation invariant for families of smooth complex projective varieties of dimension four. Examples along these lines are known when singular fibers are allowed, e.g., smooth cubic threefolds (which are irrational) may specialize to cubic threefolds with ordinary double points (which are rational), while smooth cubic surfaces (which are rational) may specialize to cones over elliptic curves. Totaro shows that specializations of rational varieties need not be rational in higher dimensions if mild singularities are allowed [Tot16b]. However, de Fernex and Fusi [dFF13] show that the locus of rational fibers in a smooth family of projective complex threefolds is a countable union of closed subsets on the base.

Let S be a smooth projective rational surface over the complex numbers with function field K = ℂ(S). A quadric surface bundle consists of a fourfold X and a flat projective morphism π : X → S such that the generic fiber Q/K of π is a smooth quadric surface. We assume that π factors through the projectivization of a rank four vector bundle on S such that the fibers are (possibly singular) quadric surfaces; see Section 3 for relevant background.

Theorem 1. There exist smooth families of complex projective fourfolds ϕ : X → B over connected varieties B, such that for every b ∈ B the fiber X_{b} = ϕ^{−1}(b) is a quadric surface bundle over ℙ^{2}, and satisfying:

Date: May 3, 2017.

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

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

Concretely, we consider smooth hypersurfaces

X ⊂ ℙ^{2} × ℙ^{3}

of bidegree (2, 2); projection onto the first factor gives the quadric surface bundle.

Our approach has two key elements. First, we apply the technique of the decomposition of the diagonal [Voi15b, CTP16b, CTP16a, Tot16a] to show that very general X ⊂ ℙ^{2}×ℙ^{3} of bidegree (2, 2) fail to be stably rational. The point is to identify a degenerate quadric surface fibration, with non-trivial second unramified cohomology and mild singularities. The analogous degenerate conic bundles over ℙ^{2} are the Artin-Mumford examples; deforming these allows one to show that very general conic bundles over ℙ^{2} with large degeneracy divisor fail to be stably rational [HKT15]. Second, quadric surface bundles are rational over the base whenever they admit a section, indeed, whenever they admit a multisection of odd degree. If the base is rational then the total space is rational as well; this can be achieved over a dense set of the moduli space [Has99, Voi15a]. This technique also yields rationality for a dense family of cubic fourfolds containing a plane; no cubic fourfolds have been shown not to be stably rational.

Theorem 1 is proven in Section 7, which may serve as roadmap for the steps of our argument.

This paper is inspired by the approach of Voisin [Voi15a], who also considers fourfolds birational to quadric surface bundles. While our proof of rationality is similar, the analysis of unramified cohomology relies on work of Pirutka [Pir16] and Colliot-Thélène and Ojanguren [CTO89].

Acknowledgments: The first author was partially supported through NSF grant 1551514. We are grateful to Jean-Louis Colliot-Thélène and Burt Totaro for helpful comments on drafts of this manuscript. Our paper benefited greatly from the careful reading by the referees.

2. The specialization method

We recall implications of the “integral decomposition of the diagonal and specialization” method, following [Voi15b] [CTP16b], and [Pir16].

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. Examples include smooth k-rational varieties. More complicated examples arise as follows:

Example 2. [CTP16a, Lemma 2.3, Lemma 2.4] Let X = ∪_{i}X_{i} be a projective, reduced, geometrically connected variety over a field k such that:

• Each irreducible component X_{i} is geometrically irreducible and k-rational, with isolated singularities.

• Each intersection X_{i} ∩ X_{j} is either empty or has a zero-cycle of

degree 1. Then X is universally CH_{0}-trivial.

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.

Proposition 3. [CTP16b, Proposition 1.8] Let

β : X̃ → X

be a projective morphism such that for every schematic point x of X, the fiber β^{−1}(x), considered as a variety over the residue field κ(x), is universally CH_{0}-trivial. Then β is universally CH_{0}-trivial.

For example, if X is a smooth projective variety and

β : Bl_{Z}(X) → X

is a blowup of a smooth subvariety Z ⊂ X, then β is a universally CH_{0}- trivial morphism, since all fibers over (schematic) points are projective spaces. More interesting examples arise as resolutions of singularities of certain singular projective varieties.

Examples of failure of universal CH_{0}-triviality are given by smooth projective varieties X with nontrivial Brauer group Br(X), or more generally, by varieties with nontrivial higher unramified cohomology [CTP16b, Section 1]. The following specialization argument is the key to recent advances in investigations of stable rationality:

Theorem 4. [Voi15b, 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:

• the group H_{nr}^{2}(ℂ(X)/ℂ, ℤ/2) is nontrivial;

• X admits a desingularization

β : X̃ → X

such that the morphism β is universally CH_{0}-trivial. Then a very general fiber of ϕ is not stably rational.

3. Quadric surface bundles

Let S be a smooth projective variety over ℂ. Suppose that π : X → S is a quadric surface bundle, i.e., a flat projective morphism from a variety such that the generic fiber Q is a smooth quadric surface. We assume it admits a factorization

X ↪ ℙ(V) → S,

where V → S is a rank four vector bundle and the fibers of π are expressed as quadric surfaces in the fibers of ℙ(V) → S. There is a well-defined degeneracy divisor D ⊂ S corresponding to where the associated quadratic form drops rank.

Trivializing V over an open cover of S, X may be expressed using a symmetric 4 × 4 matrix (a_{ij}):

∑ a_{ij}x_{i}x_{j} = 0.

The local equation for D is the determinant det((a_{ij})). Note that D has multiplicity ≥ 2 where the rank of fibers is less than three. Indeed, the hypersurface

{det(a_{ij}) = 0} ⊂ ℙ_{(a_{ij})}^{9}

is singular precisely where all the 3 × 3 minors vanish.

3.1. Rationality of quadric bundles. It is well known that Q is rational over K = ℂ(S) if and only if Q(K) ≠ ∅, i.e., when π admits a rational section. A theorem of Springer [Spr52] implies that Q(K) ≠ ∅ provided Q(K′) ≠ ∅ for some extension K′/K of odd degree, i.e., when π admits a rational multisection of odd degree. Thus we obtain

Proposition 5. Let π : X → S be a quadric surface bundle as above, with S rational. Then X is rational provided π admits a multisection of odd degree.

Our next step is to recast this in Hodge-theoretic terms:

Proposition 6. Let π : X → S be a quadric surface bundle as above, with X smooth and S rational. Then X is rational if it admits an integral (2, 2)-class meeting the fibers of π in odd degree.

Remark 7. See [CTV12, Cor. 8.2] for results on the integral Hodge conjecture for quadric bundles over surfaces; these suffice for our application to quadric surface bundles over ℙ^{2}.

Proof. Let F_{1}(X/S) → S denote the relative variety of lines of π. Let S_{∘} ⊂ S denote the largest open subset such that S_{∘} ∩ D is smooth and X_{∘} = X ×_{S} S_{∘}. Then F_{1}(X_{∘}/S_{∘}) → S_{∘} factors

F_{1}(X_{∘}/S_{∘}) →^{p} T_{∘} → S_{∘},

where the second morphism is a double cover branched along S_{∘} ∩ D and the first morphism is an étale ℙ^{1}-bundle. In particular F_{1}(X_{∘}/S_{∘}) is non-singular. Let α ∈ Br(T_{∘})[2] denote the Brauer class arising from p.

Let F be a resolution of the closure of F_{1}(X_{∘}/S_{∘}) in F_{1}(X/S) obtained by blowing up over the complement of S_{∘}. The incidence correspondence between X and F_{1}(X/S)

Γ′ ⊂ X ×_{S} F_{1}(X/S)

induces a correspondence Γ between X and F and a homomorphism

Γ_{∗} : CH^{2}(X) → Pic(F).

Let η denote the generic point of S; there is a quadratic map

Ξ : Pic(F_{η}) → CH^{2}(X_{η})

given by

Ξ(∑ a_{i}x_{i}) = \frac{1}{2}(∑ a_{i}ℓ(x_{i}))^{2}

where ℓ(x_{i}) ⊂ X_{η} is the line which corresponds to the point x_{i} ∈ F_{η}. In geometric terms, consider Z ⊂ F_{η} a finite reduced subscheme with support on each component of F_{η}, e.g., a choice of n lines from each ruling. Take the union of the corresponding rulings in X_{η} and set Ξ(Z) ⊂ X_{η} to be the n^{2} points where the rulings cross. This is compatible with rational equivalence and yields the desired mapping. Thus a divisor with odd degree on each geometric component of F_{η} gives rise to a rational multisection of odd degree.

The correspondence Γ and mapping Ξ guarantee the following conditions are equivalent:

• α = 0;

• F admits a divisor intersecting the generic fiber F_{η} with odd degree on each component;

• X admits a rational multisection of odd degree. As the correspondence Γ also acts at the level of Hodge classes we obtain:

If X admits an integral (2, 2)-class intersecting the fibers of π with odd degree then F admits an integral (1, 1)- class intersecting the generic fiber F_{η} with odd degree on each component.

Applying the Lefschetz (1, 1) Theorem to F and Proposition 5 we obtain the result. □

3.2. A key example. The generic fiber of π is a quadric surface, hence admits a diagonal form

(3.1)  Q =< 1, a, b, abd >,

i.e., is given by the equation

s^{2} + at^{2} + bu^{2} + abdv^{2} = 0

where a, b, d ∈ K^{×} and (s, t, u, v) are homogeneous coordinates in ℙ^{3}. Note that since k := ℂ ⊂ K, this form is equivalent to the form < 1, −a, −b, abd >.

Theorem 3.17 in [Pir16] gives a general formula for the unramified H^{2} of the field K(Q), in terms of the divisor of rational functions a, b, d ∈ K^{×}, under the assumption that d is not a square.

In Section 4 we will analyze the following special case:

Example 8. Consider the fourfold X ⊂ ℙ^{2} × ℙ^{3} given by

(3.2)  yzs^{2} + xzt^{2} + xyu^{2} + F (x, y, z)v^{2} = 0,

where

F (x, y, z) = x^{2} + y^{2} + z^{2} − 2(xy + xz + yz).

Dehomogenize by setting z = 1 to obtain a quadric surface over k(ℙ^{2}):

ys^{2} + xt^{2} + xyu^{2} + F (x, y, 1)v^{2} = 0.

Multiplying through by xy and absorbing squares into the variables yields

xS^{2} + yT^{2} + U^{2} + xyF (x, y, 1)V^{2} = 0,

which is of the form (3.1).

We compute the divisor D ⊂ ℙ^{2} parametrizing singular fibers of π : X → ℙ^{2}. This is reducible, consisting of the coordinate lines (with multiplicty two) and a conic tangent to each of the lines:

D = {x^{2}y^{2}z^{2}(x^{2} + y^{2} + z^{2} − 2(xy + xz + yz)) = 0}.

Remark 9. Hypersurfaces of bidegree (2, 2) in ℙ^{2} × ℙ^{3} may also be regarded as conic bundles over the second factor. The degeneracy surface in ℙ^{3} has degree six and at least eight nodes, corresponding to rank-one fibers. As a byproduct of the proof of Theorem 1, we obtain failure of stable rationality for very general conic bundles of this type.

4. The Brauer group of the special fiber

We refer the reader to [CTO89, Sect. 1] and [CT95] for basic properties of unramified cohomology.

Let K be a field. We write

H^{n}(K) = H^{n}(K, ℤ/2)

for its n-th Galois cohomology with constant coefficients ℤ/2. Let K = k(X) be the function field of an algebraic variety X over k = ℂ, and let ν be a rank one discrete valuation of K, trivial on k. For n ≥ 1, we have a natural homomorphism

∂_{ν}^{n} : H^{n}(K) → H^{n−1}(κ(ν)),

where κ(ν) is the residue field of ν. The group

H_{nr}^{n}(K/k) := ∩_{ν} Ker(∂_{ν}^{n})

is called the n-th unramified cohomology of K. It is a stable birational invariant [CTO89, Prop. 1.2] and vanishes if X is stably rational [CTO89, Cor. 1.2.1]. Recall that for a smooth projective X we have

Br(X)[2] = H_{nr}^{2}(k(X)/k).

The following proposition is similar to the examples in [Pir16, Section 3.5].

Proposition 10. Let K = k(x, y) = k(ℙ^{2}), X → ℙ^{2} the quadric surface bundle defined in Example 8,

α = (x, y) ∈ Br(K)[2],

and α′ its image in H^{2}(k(X)). Then α′ is contained in H_{nr}^{2}(k(X)/k) and is nontrivial; in particular,

H_{nr}^{2}(k(X)/k) ≠ 0.

Proof. Let Q be the generic fiber of the natural projection π : X → ℙ^{2}. Since the discriminant of Q is not a square, the homomorphism

H^{2}(K) → H^{2}(K(Q))

is injective [Ara75, p. 469], [Kah08, 6.4.13]. Note that α ≠ 0 as the conic xS^{2} + yT^{2} = U^{2} has no rational points over k(x, y); it follows that α′ is also nontrivial. It remains to show that for every rank one discrete valuation ν on K(Q) that is trivial on k, we have ∂_{ν}(α′) = 0. (For simplicity, we write ∂_{ν} for ∂_{ν}^{2}.) We use standard coordinates x and y (resp. y and z, resp. x and z) for the open charts of the projective plane. Let us first investigate the ramification of α on ℙ^{2}; from the definition, we only have the following nontrivial residues:

• ∂_{x}(α) = y at the line L_{x} : x = 0, where we write y for its class in the residue field k(y) modulo squares;

• ∂_{y}(α) = x at the line L_{y} : y = 0;

• ∂_{z}(α) = ∂_{z}(z, zy) = y at the line L_{z} : z = 0, in coordinates y and z on ℙ^{2}.

Let 𝔬_{ν} be the valuation ring of ν in K(Q) and consider the center of ν in ℙ^{2}. If 𝔬_{ν} ⊃ K then the ∂_{ν}(α′) = 0; hence there are two cases to consider:

• The center is the generic point of a curve C_{ν} ⊂ ℙ^{2}; we denote the corresponding residue map ∂_{C_{ν}} : H^{2}(K) → H^{1}(κ(C_{ν})).

• The center is a closed point 𝔭_{ν} ∈ ℙ^{2}.

Codimension 1. The inclusion of discrete valuation rings 𝔬_{ℙ^{2},C_{ν}} ⊂ 𝔬_{ν} induces a commutative diagram [CTO89, p. 143]:

(4.1)  \xymatrix{H^{2}(K(Q)) \ar[r]^{∂_{ν}} & H^{1}(κ(ν)) \\ H^{2}(K) \ar[r]^{∂_{Cν}} \ar[u] & H^{1}(κ(C_{ν})) \ar[u]}

The right vertical arrow need not be the functorial homomorphism induced by inclusion of the residue fields when there is ramification.

Hence we have the following cases:

(1) C_{ν} is different from L_{x}, L_{y}, or L_{z}. Then ∂_{C_{ν}}(α) = 0, so that ∂_{ν}(α′) is zero from the diagram above.

(2) C_{ν} is one of the lines L_{x}, L_{y} or L_{z}. Note that modulo the equation of C_{ν}, the element d := F (x, y, z) is a nonzero square, so that [Pir16, Cor. 3.12] gives ∂_{ν}(α′) = 0.

We deduce that for any valuation ν of K(Q) with center a codimension 1 point in ℙ_{ℂ}^{2} the residue ∂_{ν}(α′) vanishes.

Codimension 2. Let 𝔭_{ν} be the center of ν on ℙ^{2}. We have an inclusion of local rings 𝔬_{ℙ^{2},𝔭_{ν}} ⊂ 𝔬_{ν} inducing the inclusion of corresponding completions { \atop O_{ℙ^{2},𝔭_{ν}}} ⊂ 𝔬̂_{ν} with quotient fields K_{𝔭_{ν}} ⊂ K(Q)_{ν} respectively. We have three possibilities:

(1) If 𝔭_{ν} ∉ L_{x} ∪ L_{y} ∪ L_{z}, then α is a cup product of units in O_{ℙ^{2},𝔭_{ν}}, hence units in 𝔬_{v}, so that ∂_{ν}(α′) = 0.

(2) If 𝔭_{ν} lies on one curve, e.g., 𝔭_{ν} ∈ L_{x} \ (𝔭_{y} ⊔ 𝔭_{z}), where 𝔭_{y} = (0, 1, 0) and 𝔭_{z} = (0, 0, 1), then the image of y in κ(𝔭_{ν}) is a nonzero complex number, hence a square in { \atop O_{ℙ^{2},𝔭_{ν}}}, and y is also a square in 𝔬̂_{ν}. (We are using Hensel’s Lemma.) Thus α′ = (x, y) = 0 in H^{2}(K(Q)_{ν}, ℤ/2), and ∂_{ν}(α′) = 0.

(3) If 𝔭_{ν} lies on two curves, e.g., 𝔭_{ν} = L_{x} ∩ L_{y}, then the image of F (x, y, 1) in κ(𝔭_{ν}) is a nonzero complex number, hence a square. By [Pir16, Corollary 3.12], we have ∂_{ν}(α′) = 0.

□

5. Singularities of the special fiber

In this section we analyze the singularities of the fourfold introduced in Example 8 and studied in Section 4. Our main result is:

Proposition 11. The fourfold X ⊂ ℙ^{2} × ℙ^{3}, with coordinates (x, y, z) and (s, t, u, v), respectively, given by

(5.1)  yzs^{2} + xzt^{2} + xyu^{2} + F (x, y, z)v^{2} = 0,

with

(5.2)  F (x, y, z) = x^{2} + y^{2} + z^{2} − 2(xy + xz + yz),

admits a universally CH_{0}-trivial resolution of singularities.

We proceed as follows:

• identify the singular locus of X;

• construct a resolution of singularities β : X̃ → X;

• verify universal CH_{0}-triviality of β.

5.1. The singular locus. Here we describe the singularities explicitly using affine charts on ℙ^{2} × ℙ^{3}. The equations (5.1) and (5.2) are symmetric with respect to compatible permutations of {x, y, z} and {s, t, u}. In addition, there is the symmetry

(s, t, u, v) → (±s, ±t, ±u, v)

so altogether we have an action by the semidirect product (ℤ/2ℤ)^{3}⋊𝔖_{3}.

Analysis in local charts. Let L_{x}, L_{y}, L_{z} ⊂ ℙ^{2} be the coordinate lines given by

x = 0, y = 0, z = 0,

respectively, and

𝔭_{x} := (1, 0, 0), 𝔭_{y} := (0, 1, 0), 𝔭_{z} := (0, 0, 1)

their intersections.

The quadrics in the family (5.1) drop rank over coordinate lines L_{x}, L_{y}, L_{z} and over the conic C ⊂ ℙ^{2}, with equation (5.2)

F (x, y, z) = 0.

This conic is tangent to the coordinate lines in the points

𝔯_{x} := (0, 1, 1), 𝔯_{y} := (1, 0, 1), 𝔯_{z} := (1, 1, 0),

respectively.

By symmetry, and since no singular point satisfies s = t = u = 0, it suffices to consider just two affine charts:

Chart 1: z = u = 1. Equation (5.1) takes the form

(5.3)  ys^{2} + xt^{2} + xy + F (x, y, 1)v^{2} = 0.

Derivatives with respect to s, t, v give

(5.4)  ys = 0, xt = 0, vF (x, y, 1) = 0.

Hence xy = 0, from (5.3). Derivatives with respect to y, x give

(5.5) s^{2} + x + (2y − 2x − 2)v^{2} = 0, t^{2} + y + (2x − 2y − 2)v^{2} = 0.

Since xy = 0, we have two cases, modulo symmetries: Case 1: y = 0; Case 2: x = 0, y ≠ 0.

We analyze each of these cases: Case 1: y = 0. Then vF (x, y, 1) = 0 (from (5.4)) implies

v(x − 1)^{2} = 0.

So either v = 0 or x = 1. If v = 0, from (5.5) we obtain s^{2} + x = t = 0. Hence we obtain the following equations for the singular locus:

(5.6)  y = v = t = s^{2} + x = 0.

If x = 1 then (5.4) implies t = 0, and the remaining equation from (5.5) gives s^{2} + 1 − 4v^{2} = 0. Hence we obtain the following equations:

(5.7)  x − 1 = y = t = s^{2} + 1 − 4v^{2} = 0.

Case 2: x = 0, y ≠ 0. From (5.4) the condition ys = 0 implies s = 0. There are two more cases: v = 0 or v ≠ 0. If v = 0 the remaining equation (5.5) gives t^{2} + y = 0. Hence we obtain equations for the singularity:

(5.8)  x = v = s = t^{2} + y = 0.

If v ≠ 0, then F (0, y, 1) = (y − 1)^{2} = 0 from (5.4), hence y = 1. The remaining equation from (5.5) gives

t^{2} + y + (2x − 2y − 2)v^{2} = t^{2} + 1 − 4v^{2} = 0.

So we obtain equations for the singular locus:

(5.9)  x = y − 1 = s = t^{2} + 1 − 4v^{2} = 0.

Chart 2: z = s = 1. The equation of the quadric bundle is

y + xt^{2} + xyu^{2} + F (x, y, 1)v^{2} = 0.

As above, derivatives with respect to t, u, v give

(5.10)  xt = 0, xyu = 0, vF (x, y, 1) = 0.

Thus y = 0 from the equation. The conditions above and derivatives with respect to y and x yield

(5.11) xt = v(x − 1)^{2} = 1 + xu^{2} + (−2x − 2)v^{2} = t^{2} + (2x − 2)v^{2} = 0.

The second equation implies that either x = 1 or v = 0.

If x = 1, we obtain:

(5.12)  x − 1 = y = t = 1 + u^{2} − 4v^{2} = 0.

If v = 0, we obtain:

(5.13)  y = t = v = 1 + xu^{2} = 0.

Collecting these computations, we obtain the following singularities:

(1) In Chart 1:

y = v = t = s^{2} + x = 0
x − 1 = y = t = s^{2} + 1 − 4v^{2} = 0
x = v = s = t^{2} + y = 0
x = y − 1 = s = t^{2} + 1 − 4v^{2} = 0

(2) In Chart 2:

x − 1 = y = t = 1 + u^{2} − 4v^{2} = 0
y = t = v = 1 + xu^{2} = 0

Enumeration of strata. Using the symmetries, we deduce that the singular locus of X is a union of 6 conics. We distinguish between

• Horizontal conics C_{x}, C_{y}, C_{z} ⊂ X: these project onto the coordinate lines L_{x}, L_{y}, L_{z} ⊂ ℙ^{2}. We express them using our standard coordinates on ℙ^{2} × ℙ^{3}:

C_{y} ={y = t = v = 0, zs^{2} + xu^{2} = 0}
C_{x} ={x = s = v = 0, zt^{2} + yu^{2} = 0}
C_{z} ={z = u = v = 0, xt^{2} + ys^{2} = 0}

The conics intersect transversally over 𝔭_{z}, 𝔭_{x}, 𝔭_{y} ∈ ℙ^{2}, respectively:

C_{x} ∩ C_{y} =𝔮_{z} := (0, 0, 1) × (0, 0, 1, 0), π(𝔮_{z}) = 𝔭_{z}
C_{y} ∩ C_{z} =𝔮_{x} := (1, 0, 0) × (1, 0, 0, 0), π(𝔮_{x}) = 𝔭_{x}
C_{x} ∩ C_{z} =𝔮_{y} := (0, 1, 0) × (0, 1, 0, 0), π(𝔮_{y}) = 𝔭_{y}

• Vertical conics R_{y}, R_{x}, R_{z} ⊂ X: these project to the points 𝔯_{y}, 𝔯_{x}, 𝔯_{z} ∈ ℙ^{2}:

R_{y} ={x − z = y = t = 0, s^{2} + u^{2} − 4v^{2} = 0}
R_{x} ={y − z = x = s = 0, t^{2} + u^{2} − 4v^{2} = 0}
R_{z} ={x − y = z = u = 0, s^{2} + t^{2} − 4v^{2} = 0}

Vertical conics intersect the corresponding horizontal conics transversally in two points:

R_{y} ∩ C_{y} ={𝔯_{y+}, 𝔯_{y−}} = (1, 0, 1) × (±i, 0, 1, 0)
R_{x} ∩ C_{x} ={𝔯_{x+}, 𝔯_{x−}} = (0, 1, 1) × (0, 1, ±i, 0)
R_{z} ∩ C_{z} ={𝔯_{z+}, 𝔯_{z−}} = (1, 1, 0) × (1, ±i, 0, 0)

Local étale description of the singularities. The structural properties of the resolution become clearer after identifying étale normal forms for the singularities.

We now provide a local-étale description of the neighborhood of 𝔮_{z}. Equation (5.3) takes the form

ys^{2} + xt^{2} + xy + F (x, y, 1)v^{2} = 0.

At 𝔮_{z} we have F (x, y, 1) ≠ 0, so we can set

v_{0} = \sqrt{F (x, y, 1)}v

to obtain

ys^{2} + xt^{2} + xy + v_{0}^{2} = 0.

Set x = m − n and y = m + n to get

(m + n)s^{2} + (m − n)t^{2} + m^{2} − n^{2} + v_{0}^{2} = 0

or

m(s^{2} + t^{2}) + n(s^{2} − t^{2}) + m^{2} − n^{2} + v_{0}^{2} = 0.

Then let

m = m_{0} − (s^{2} + t^{2})/2 and n = n_{0} + (s^{2} − t^{2})/2

to obtain

(5.14)  m_{0}^{2} − n_{0}^{2} + v_{0}^{2} = ((s^{2} + t^{2})^{2} − (s^{2} − t^{2})^{2})/4 = s^{2}t^{2}.

We do a similar analysis in an étale-local neighborhood of either of the points 𝔯_{y±}. The singular strata for C_{y} and R_{y} are given in (5.6) and (5.7):

{y = t = v = s^{2} + x = 0}, {y = t = x − 1 = s^{2} + 1 − 4v^{2} = 0}.

We first introduce a new coordinate w = x − 1. Thus the singular stratum is the intersection of the monomial equations y = t = vw = 0 and the hypersurface

s^{2} + w + 1 − 4v^{2}.

We regard this as a local coordinate near 𝔯_{y±}. Equation (5.3) transforms to

ys^{2} + wt^{2} + t^{2} + wy + y + v^{2}(−4y + (w − y)^{2}) = 0.

Regroup terms to obtain

y(s^{2} + w + 1 − 4v^{2}) + t^{2}(1 + w) = −v^{2}(w − y)^{2}.

Note that w ≠ −1 because x ≠ 0 near 𝔯_{y±}. Let t_{0} = t\sqrt{1 + w}, s_{0} = s^{2} + w + 1 − 4v^{2}, and w_{0} = w − y we obtain

(5.15)  ys_{0} + t_{0}^{2} = −v^{2}w_{0}^{2}.

The normal forms (5.14) and (5.15) are both equivalent to

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = (b_{1}b_{2})^{2},

with ordinary threefold double points along the lines

ℓ_{1} = {a_{1} = a_{2} = a_{3} = b_{1} = 0}, ℓ_{2} = {a_{1} = a_{2} = a_{3} = b_{2} = 0}.

A direct computation – which will be presented in Section 5.2 – shows this is resolved by blowing up ℓ_{1} and ℓ_{2} in either order. The exceptional fibers over the generic points of ℓ_{1} and ℓ_{2} are smooth quadric surfaces, isomorphic to the Hirzebruch surface 𝔽_{0} ≃ ℙ^{1} × ℙ^{1}. Over the origin, we obtain a copy

𝔽_{0} ∪_{Σ} 𝔽_{2}

where Σ ≃ ℙ^{1} is the (−2)-curve on 𝔽_{2} and has self-intersection 2 on 𝔽_{0}.

By symmetry, this analysis is valid at all nine special points

𝔮_{x}, 𝔮_{y}, 𝔮_{z}, 𝔯_{x±}, 𝔯_{y±}, 𝔯_{z±}

where components of the singular locus (the horizontal and vertical conics) intersect. This explains why we can blow these conics up in any order.

5.2. Resolution of singularities.

What we need to compute. We propose blowing up as follows: (1) blow up C_{y};

(2) blow up the proper transform of C_{x};

(3) blow up the proper transform of C_{z};

(4) blow up the union of the proper transforms of R_{x}, R_{y}, and R_{z},

which are disjoint. Taking into account the symmetry, after the first step we must understand:

• What are the singularities along the proper transform of C_{x}?

• What are the singularities along the proper transform of R_{y}? Of course, answering the first questions clarifies the behavior along the proper transform of C_{z}. And R_{x} and R_{z} behave the exactly the same as R_{y}.

Let X_{1} denote the blow up of C_{y} and E_{1,y} the resulting exceptional divisor. We shall see that

• E_{1,y} is smooth, except where it meets the proper transforms of C_{x}, C_{z}, R_{y}.

• Since E_{1,y} ⊂ X_{1} is Cartier, X_{1} is also smooth at any point of E_{1,y}, except where E_{1,y} meets the proper transforms of C_{x}, C_{z}, R_{y}.

• The fibers of E_{1,y} → C_{y} are smooth quadric surfaces away from

𝔮_{x}, 𝔮_{z}, 𝔯_{y±}, over which the fibers are quadric cones. Since the quadric bundle E_{1,y} → C_{y} admits sections, E_{1,y} is rational over the function field of C_{y} and all fibers of E_{1,y} → C_{y} are rational as well.

First blow up—local charts. We describe the blow up of C_{y} in charts. We start in Chart 1, where z = u = 1. Local equations for the center are given in (5.6) and we have a local chart for each defining equation.

• Chart associated with y: Equations for the blow up of the ambient space take the form

v = yv_{1}, t = yt_{1}, s^{2} + x = yw_{1}.

The equation of the proper transform of the quadric bundle is

w_{1} + xt_{1}^{2} + F (x, y, 1)v_{1}^{2} = 0, s^{2} + x = yw_{1}.

The exceptional divisor E_{1,y} is given by y = 0, i.e.,

w_{1} + xt_{1}^{2} + (x − 1)^{2}v_{1}^{2} = 0, s^{2} + x = 0.

The blow up is smooth at any point of the exceptional divisor in this chart, as the derivative of the first equation with respect to w_{1} is 1 and the derivative of the second equation with respect to w_{1} (resp. x) is 0 (resp. 1). (The proper transforms of R_{y} and C_{x} do not appear in this chart.) We analyze E_{1,y} → C_{y}: for any field κ/ℂ and a ∈ κ, the fiber above s = a, x = −a^{2}, y = v = t = 0, is given by

(5.16)  w_{1} − a^{2}t_{1}^{2} + (1 + a^{2})^{2}v_{1}^{2} = 0,

which is smooth in this chart. Equation (5.16) makes clear that the exceptional divisor is rational and admits a section over the center.

• Chart associated with s^{2} + x: Equations for the blow up of the ambient space take the form

y = (s^{2} + x)y_{1}, v = (s^{2} + x)v_{1}, t = (s^{2} + x)t_{1}.

The proper transform of the quadric bundle has equation

y_{1} + xt_{1}^{2} + F (x, (s^{2} + x)y_{1}, 1)v_{1}^{2} = 0.

The exceptional divisor E_{1,y} satisfies

y_{1} + xt_{1}^{2} + (x − 1)^{2}v_{1}^{2} = 0, s^{2} + x = 0.

The blow up is smooth at any point of the exceptional divisor in this chart. (Again, the proper transforms of R_{y} and C_{x} do not appear in this chart.) The fiber above s = a, x = −a^{2}, y = v = t = 0, is given by

(5.17)  y_{1} − a^{2}t_{1}^{2} + (1 + a^{2})^{2}v_{1}^{2} = 0,

which is smooth and rational in this chart.

• Chart associated with t: Equations for the blow up of the ambient space are

y = ty_{1}, v = tv_{1}, s^{2} + x = tw_{1}

and the proper transform of the quadric bundle satisfies

y_{1}w_{1} + x + F (x, ty_{1}, 1)v_{1}^{2} = 0, s^{2} + x = tw_{1}.

The exceptional divisor is given by t = 0, i.e.

y_{1}w_{1} + x + (x − 1)^{2}v_{1}^{2} = 0, s^{2} + x = 0.

The blow up is smooth along the exceptional divisor, except at the point

t = v_{1} = y_{1} = s = w_{1} = x = 0,

which lies over the point 𝔮_{z}. Thus the only singularity is along the proper transform of C_{x}. The fiber above s = a, x = −a^{2}, y = v = t = 0, is given by

(5.18)  y_{1}w_{1} − a^{2} + (1 + a^{2})^{2}v^{2} = 0,

which is smooth in this chart unless a = 0.

• Chart associated with v: The equations are

y = vy_{1}, t = vt_{1}, s^{2} + x = vw_{1}

and

y_{1}w_{1} + xt_{1}^{2} + F (x, vy_{1}, 1) = 0, s^{2} + x = vw_{1}.

The exceptional divisor is given by v = 0, i.e.

y_{1}w_{1} + xt_{1}^{2} + (x − 1)^{2} = 0, s^{2} + x = 0.

The blow up is smooth at any point of the exceptional divisor except for

y_{1} = v = w_{1} = t_{1} = 0, x = 1, s = ±i.

Thus the only singularities are along the proper transform of 𝔯_{y}. The fiber above s = a, x = −a^{2}, y = v = t = 0, is given by

(5.19)  y_{1}w_{1} − a^{2}t_{1}^{2} + (1 + a^{2})^{2} = 0,

which is smooth in this chart unless a = ±i. What is missed on restricting to Chart 1? For C_{y}, we omit only

(1, 0, 0) × (1, 0, 0, 0) = 𝔮_{x} = C_{y} ∩ C_{z}

but the symmetry exchanging x and z (and s and u) takes this to 𝔮_{z}, which lies over Chart 1. For R_{y}, we omit the locus

x − z = y = t = u = s^{2} − 4v^{2} = 0

which equals (1, 0, 1)×(±2, 0, 0, 1). However, the same symmetry takes these to (1, 0, 1) × (0, 0, ±2, 1), which is over Chart 1. Thus modulo symmetries computations over Chart 1 cover these points as well.

Singularities above 𝔭_{z}. Our goal is to show explicitly that the singularity of the blow up in the exceptional divisor E_{1,y} over (x, y, z) = (0, 0, 1) = 𝔭_{z} is resolved on blowing up the proper transform of C_{x}. It suffices to examine the chart associated with t, where we have equations

y_{1}w_{1} + x + F (x, ty_{1}, 1)v_{1}^{2} = 0, s^{2} + x = tw_{1},

i.e.,

(5.20) (y_{1} + t)w_{1} − s^{2} + F (−s^{2} + tw_{1}, ty_{1}, 1)v_{1}^{2} = 0, s^{2} + x = tw_{1},

and the proper transform of C_{x} satisfies

y_{1} + t = 0, w_{1} = s = v_{1} = 0.

If we compute the singular locus for the equation (5.20) above, at the points of the exceptional divisor t = 0 and above x = 0, we recover the equations for the proper transform of C_{x} in this chart.

We analyze X_{2}, the blowup along the proper transform of C_{x}. In any chart above y_{1} = t = 0 we have F = 1 so étale locally we can introduce a new variable v_{2} = \sqrt{F} v_{1} to obtain

(y_{1} + t)w_{1} − s^{2} + v_{2}^{2} = 0.

After the change of variables y_{2} = y_{1} + t:

y_{2}w_{1} − s^{2} + v_{1}^{2} = 0,

the singular locus is y_{2} = s = w_{1} = v_{2} = 0. Here t is a free variable corresponding to an 𝔸^{1}-factor. This is the product of an ordinary threefold double point with a curve, thus is resolved on blowing up the singular locus. Note the exceptional divisor is a smooth quadric surface bundle over the proper transform of C_{x}, over this chart. (There is a singular fiber over the point where it meets the proper transform of C_{z}.)

Singularities above 𝔯_{y} = (1, 0, 1) ∈ ℙ^{2}. By the analysis above, we have only to consider the chart of the first blowup associated with v. Recall that it is obtained by setting

y = vy_{1}, t = vt_{1}, s^{2} + x = vw_{1}

with equation

y_{1}w_{1} + xt_{1}^{2} + F (x, vy_{1}, 1) = 0.

The exceptional divisor is given by v = 0. The proper transform R′_{y} of the conic

R_{y} : x − 1 = y = t = 0, s^{2} + 1 − 4v^{2} = 0

is then

(5.21)  x − 1 = y_{1} = t_{1} = 0, w_{1} − 4v = 0, s^{2} + 1 − 4v^{2} = 0.

We obtain these equations by inverting the local equation for the exceptional divisor. Eliminating x from the equation of X_{1} yields an equation that can be put in the form

y_{1}(w_{1} − 4v) + (−s^{2} + vw_{1})t_{1}^{2} + (s^{2} − vw_{1} + vy_{1} + 1)^{2} = 0.

Writing w_{2} = w_{1} − 4v we obtain

y_{1}w_{2} + (−s^{2} + vw_{2} + 4v^{2})t_{1}^{2} + (s^{2} − vw_{2} − 4v^{2} + vy_{1} + 1)^{2} = 0.

The curve R′_{y} may be expressed as a complete intersection

y_{1} = w_{2} = t_{1} = 0, σ := (s^{2} − 4v^{2} + 1) + v(y_{1} − w_{2}) = 0;

the coefficient

c := −s^{2} + vw_{2} + 4v^{2}

is non-vanishing along R′_{y} in this chart so we may introduce an étale local coordinate t_{2} = \sqrt{c}t_{1}. Then our equation takes the form

y_{1}w_{2} + t_{2}^{2} + σ^{2} = 0.

In other words, we have ordinary threefold double points along each point of R′_{y}. Blowing up R′_{y} resolves the singularity, and the exceptional divisor over R′_{y} is fibered in smooth quadric surfaces.

5.3. CH_{0}-triviality of the resolution. Let E_{1,y} denote the exceptional divisor after blowing up C_{y}. We’ve seen that the projection E_{1,y} → C_{y} is a quadric surface bundle. The fibers are smooth away from 𝔮_{x}, 𝔮_{z}, and 𝔯_{y±}; over these points the fibers are quadric cones.

Let E_{1,x} denote the exceptional divisor after blowing up the proper transform C′_{x} of C_{x}. The fibration E_{1,x} → C′_{x} is also a quadric surface bundle. The fibers are smooth away from 𝔮_{y} and 𝔯_{x±}, where the fibers are quadric cones.

Let E_{1,z} denote the exceptional divisor on blowing up the proper transform C′_{z} of C_{z}, after the first two blow ups. Again E_{1,z} → C′_{z} is a quadric surface bundle, smooth away from 𝔯_{z±}; the fibers over these points are quadric cones.

Finally, we blow up the proper transforms R′_{x}, R′_{y}, R′_{z} of the disjoint vertical conics. The local computations above show that the resulting fourfold X̃ is smooth and the exceptional divisors

E_{2,x} → R′_{x}, E_{2,y} → R′_{y}, E_{2,z} → R′_{z},

are smooth quadric surface bundles with sections.

To summarize, fibers of β : X̃ → X are one of the following: • if x is not contained in any of the conics, β^{−1}(x) is a point;

• if x is contained in exactly one of the conics, β^{−1}(x) is a smooth quadric surface isomorphic to 𝔽_{0}; when x is a generic point of one of the conics, then β^{−1}(x) is rational over the residue field of x;

• if x is contained in two of the conics, β^{−1}(x) = 𝔽_{0} ∪_{Σ} 𝔽_{2}, where 𝔽_{2} is the proper transform of a quadric cone appearing as a degenerate fiber, Σ ⊂ 𝔽_{2} is the (−2) curve, and Σ ⊂ 𝔽_{0} has self-intersection 2.

By Proposition 3 and Example 2, we conclude that β is universally CH_{0}-trivial.

6. Analysis of Hodge classes

Our approach follows Section 2 of [Voi15a]. As explained in Proposition 6, a quadric surface bundle over a rational surface π : X → S is rational provided X admits an integral class of type (2, 2) meeting the fibers of π in odd degree. Here we analyze how these classes occur.

We start by reviewing the Hodge-theoretic inputs. Let Y → B be the family of all smooth hypersurfaces in ℙ^{2} ×ℙ^{3} of bidegree (2, 2), i.e., B is the complement of the discriminant in ℙ(Γ(O_{ℙ^{2}×ℙ^{3}}(2, 2))). For each b ∈ B, let Y_{b} denote the fiber over b. The Lefschetz hyperplane theorem gives Betti/Hodge numbers

• b_{2i+1}(Y_{b}) = 0

• b_{2}(Y_{b}) = h^{1,1}(Y_{b}) = 2, b_{6}(Y_{b}) = h^{3,3}(Y_{b}) = 2.

We compute b_{4}(Y_{b}) by analyzing Y_{b} → ℙ^{2}; its degeneracy divisor is an octic plane curve D_{b}, of genus 21. Indeed, the fibers away from D_{b} are smooth quadric surfaces and the fibers over D_{b} are quadric cones, so we have

χ(Y_{b}) = χ(ℙ^{1} × ℙ^{1})χ(ℙ^{2} \ D_{b}) + χ(quadric cone)χ(D_{b})
= 4 · (3 − (−40)) + 3 · (−40) = 52.

It follows that b_{4}(Y_{b}) = 46.

We extract the remaining Hodge numbers using techniques of Griffiths for hypersurfaces in projective space, extended to the toric case by Batyrev and Cox. Let F be the defining equation of bidegree (2, 2) and consider the bigraded Jacobian ring:

Jac(F) = ℂ[x, y, z; s, t, u, v]/I(F),

where I(F) is the ideal of partials of F. Note the partials satisfy the Euler relations:

(6.1) x\frac{∂F}{∂x} + y \frac{∂F}{∂y} + z \frac{∂F}{∂z} = 2F = s\frac{∂F}{∂s} + t\frac{∂F}{∂t} + u\frac{∂F}{∂u} + v \frac{∂F}{∂v} .

Consider the vanishing cohomology

H^{4}(Y_{b})_{van} := H^{4}(Y_{b})/H^{4}(ℙ^{2} × ℙ^{3}),

i.e., we quotient by ⟨h_{1}^{2}, h_{1}h_{2}, h_{2}^{2}⟩ where h_{1} and h_{2} are pull-backs of the hyperplane classes of ℙ^{2} and ℙ^{3} respectively. Then we have [BC94, Theorem 10.13]:

• H^{4,0}(Y_{b}) = H^{4,0}(Y_{b})_{van} = Jac(F)_{(−1,−2)} = 0

• H^{3,1}(Y_{b}) = H^{3,1}(Y_{b})_{van} ≃ Jac(F)_{(1,0)} = ℂ[x, y, z]_{1} ≃ ℂ^{3}
• H^{2,2}(Y_{b})_{van} ≃ Jac(F)_{(3,2)} ≃ ℂ^{37}
• H^{1,3}(Y_{b}) = H_{1,3}(Y_{b})_{van} ≃ Jac(F)_{(5,4)} ≃ ℂ^{3}.

The first two dimension computations imply the others by the formula

b_{4}(Y_{b}) = ∑_{p+q=4} h^{p,q}(Y_{b});

or one may compute the Hilbert function of an ideal generated by three forms of degree (1, 2) and four forms of degree (2, 1), subject to the relations (6.1) but otherwise generic.

We recall the technique of Green [CHM88, Sect. 5] and Voisin [Voi07, proof of 5.3.4], which applies as our variation of Hodge structures is effective of weight two after a suitable Tate twist.

Proposition 12. Suppose there exists a b_{0} ∈ B and γ ∈ H^{2,2}(Y_{b_{0}})_{van} such that the infinitesimal period map evaluated at γ

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

is surjective. Then there exists a Zariski-dense set of b ∈ B such that for any simply connected Euclidean neighborhood B′ of b, the image of the natural map (composition of inclusion with local trivialization):

τ_{b} : H_{ℝ}^{2,2} → H^{4}(Y_{b}, ℝ)_{van}

contains an open subset V_{b} ⊂ H^{4}(Y_{b}, ℝ)_{van}. Here H_{ℝ}^{2,2} is a vector bundle over B′ with fiber

H_{ℝ,u}^{2,2} = H^{4}(Y_{u}, ℝ)_{van} ∩ F^{2}H^{4}(Y_{u}, ℂ)_{van}

over u ∈ B′.

Note that the image is the set of real degree four vanishing classes that are of type (2, 2) for some b′ ∈ B′.

The infinitesimal condition is easy to check here. Since

B ⊂ ℙ(Γ(O_{ℙ^{2}×ℙ^{3}}(2, 2)))

we may identify

T_{B,b_{0}} = (ℂ[x, y, z; s, t, u, v]/ ⟨F_{0}⟩)_{(2,2)},

where F_{0} is the defining equation of Y_{b_{0}}. The infinitesimal period map

T_{B,b_{0}} → Hom(H^{2,2}(Y_{b}), H^{1,3}(Y_{b}))

was interpreted by Carlson and Griffiths as a multiplication map

(ℂ[x, y, z; s, t, u, v]/ ⟨F_{0}⟩)_{(2,2)} × Jac(F_{0})_{(3,2)} → Jac(F_{0})_{(5,4)}.

For fixed γ ∈ Jac(F_{0})_{(3,2)}, the differential in Voisin’s hypothesis is computed by multiplying γ with the elements of bidegree (2, 2) [Voi07, Theorem 6.13].

Example 13. Consider the hypersurface Y_{b_{0}} ⊂ ℙ^{2} × ℙ^{3} with equation

F_{0} = (u^{2} + uv + ts)x^{2} + (−t^{2} + u^{2} − v^{2} − s^{2})xy + (t^{2} + uv + ts)y^{2}
+(−t^{2} + u^{2} − v^{2} − s^{2})xz + (t^{2} − 16tu − u^{2} + v^{2} + s^{2})yz
+(−3uv − 3ts + s^{2})z^{2}.

We computed the Jacobian ring using Macaulay2 [GS]. In particular, we verified that

• Jac(F_{0})_{(m_{1},m_{2})} = 0 for

(m_{1}, m_{2}) ≥ (13, 2), (7, 3), (3, 5),

so in particular Y_{b_{0}} is smooth;

• the monomials {xz^{4}v^{4}, yz^{4}v^{4}, z^{5}v^{4}} form a basis for Jac(F_{0})_{(5,4)}. Setting γ = z^{3}v^{2}, the multiples of γ generate Jac(F_{0})_{(5,4)}. Hence this example satisfies Voisin’s hypothesis on the differential of the period map.

Proposition 14. Consider the Noether-Lefschetz loci

{b ∈ B : Y_{b} admits an integral class of type (2, 2) meeting the fibers of Y_{b} → ℙ^{2} in odd degree}.

These are dense in the Euclidean topology on B.

Proof. We retain the set-up of Proposition 12. The intersection of the Noether-Lefschetz loci with B′ may be expressed as

{u ∈ B′ : H_{ℝ,u}^{2,2} ∩ τ_{b}^{−1}H^{4}(Y_{b}, ℚ)_{van} ≠ 0}.

The density of the Noether-Lefschetz loci reflects the fact that

H^{4}(Y_{b}, ℚ)_{van} ⊂ H^{4}(Y_{b}, ℝ)_{van}

is dense.

However, we are interested in vectors of H^{4}(Y_{b}, ℚ)_{van} that are rational multiples of those associated with odd degree multisections M of Y_{b} → ℙ^{2}. Such multisections exist because we can write a bidegree (2, 2) hypersurface containing a constant section of ℙ^{2} × ℙ^{3} → ℙ^{2}. The parity condition corresponds to a congruence on the image of M in H^{4}(Y_{b}, ℤ)_{van}: Indeed, write Λ = H^{4}(Y_{b}, ℤ) and consider the natural inclusions and homomorphisms

Λ ⊃ ⟨h_{1}^{2}, h_{1}h_{2}, h_{2}^{2}⟩^{⊥} ↪ Λ/ ⟨h_{1}^{2}, h_{1}h_{2}, h_{2}^{2}⟩ = H^{4}(Y_{b}, ℤ)_{van};

the cokernel of the middle arrow is the discriminant group of the lattice ⟨h_{1}^{2}, h_{1}h_{2}, h_{2}^{2}⟩^{⊥}, a finite abelian group. The class M yields an element of this group and the parity condition translates into

M · h_{1}^{2} ≡ 1 mod 2.

Rational multiples of the elements satisfying this condition remain dense in H^{4}(Y_{b}, ℝ)_{van}, so Proposition 12 gives the desired result. □

The Noether-Lefschetz loci produced by this argument have codimension at most three in moduli; each is an algebraic subvariety of B ⊂ ℙ(Γ(O_{ℙ^{2}×ℙ^{3}}(2, 2))) ≃ ℙ^{59} [CDK95]. Any projective threefold in ℙ^{59} will meet the closures of infinitely many of these loci.

7. Proof of Theorem 1

We assemble the various ingredients developed above:

(1) Theorem 4 guarantees that a very general hypersurface of bidegree (2, 2) in ℙ^{2} ×ℙ^{3} fails to be stably rational, provided we can find a special X satisfying its hypotheses.

(2) The candidate example is introduced in Example 8.

(3) In Section 4, we show that X has non-trivial unramified second cohomology. This verifies the first hypothesis of Theorem 4.

(4) In Section 5, we analyze the singularities of X, checking that it admits a resolution with universally CH_{0}-trivial fibers.

(5) Proposition 6 gives a cohomological sufficient condition for rationality of (2, 2) hypersurfaces in ℙ^{2}×ℙ^{3}; Proposition 14 shows this condition is satisfied over a dense subset of the moduli space.

Consider a family ϕ : X → B of smooth (2, 2) hypersurfaces in ℙ^{2} × ℙ^{3} over a connected base B. If the base meets both the locus parametrizing non-stably rational varieties and the Noether-Lefschetz loci then ϕ has both rational and irrational fibers.

Remark 15. Concrete examples of rational hypersurfaces X_{rat} ⊂ ℙ^{2} × ℙ^{3} of bidegree (2, 2) are easy to produce, e.g., those containing a constant section of the first projection. Any very general pencil containing X_{rat} will have both rational and irrational fibers.

References

[Ara75] Jón Kr. Arason. Cohomologische invarianten quadratischer Formen. J. Algebra, 36(3):448–491, 1975.

[BC94] Victor V. Batyrev and David A. Cox. On the Hodge structure of projective hypersurfaces in toric varieties. Duke Math. J., 75(2):293–338, 1994.

[CDK95] Eduardo Cattani, Pierre Deligne, and Aroldo Kaplan. On the locus of Hodge classes. J. Amer. Math. Soc., 8(2):483–506, 1995.

[CHM88] Ciro Ciliberto, Joe Harris, and Rick Miranda. General components of the Noether-Lefschetz locus and their density in the space of all surfaces. Math. Ann., 282(4):667–680, 1988.

[CT95] Jean-Louis Colliot-Thélène. Birational invariants, purity and the Gersten conjecture. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 1–64. Amer. Math. Soc., Providence, RI, 1995.

[CTO89] Jean-Louis Colliot-Thélène and Manuel Ojanguren. Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. Math., 97(1):141–158, 1989.

[CTP16a] Jean-Louis Colliot-Thélène and Alena Pirutka. Cyclic covers that are not stably rational. Izvestiya RAN, Ser. Math., 80(4):35–47, 2016.

[CTP16b] Jean-Louis Colliot-Thélène and Alena Pirutka. Hypersurfaces quartiques de dimension 3 : non rationalité stable. Ann. Sci. Éc. Norm. Supér. (4), 49(2):371–397, 2016.

[CTV12] Jean-Louis Colliot-Thélène and Claire Voisin. Cohomologie non ramifiée et conjecture de Hodge entière. Duke Math. J., 161(5):735–801, 2012.

[dFF13] Tommaso de Fernex and Davide Fusi. Rationality in families of threefolds. Rend. Circ. Mat. Palermo (2), 62(1):127–135, 2013.

[GS] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at

http://www.math.uiuc.edu/Macaulay2/.

[Has99] Brendan Hassett. Some rational cubic fourfolds. J. Algebraic Geom., 8(1):103–114, 1999.

[HKT15] Brendan Hassett, Andrew Kresch, and Yuri Tschinkel. Stable rationality and conic bundles. Math. Annalen, to appear, 2015. arXiv:1503.08497.

[Kah08] Bruno Kahn. Formes quadratiques sur un corps, volume 15 of Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris, 2008.

[Pir16] Alena Pirutka. Varieties that are not stably rational, zero-cycles and unramified cohomology, 2016. arXiv:1603.09261.

[Spr52] Tonny Albert Springer. Sur les formes quadratiques d’indice zéro. C. R. Acad. Sci. Paris, 234:1517–1519, 1952.

[Tot16a] Burt Totaro. Hypersurfaces that are not stably rational. J. Amer. Math. Soc., 29(3):883–891, 2016.

[Tot16b] Burt Totaro. Rationality does not specialize among terminal varieties. Math. Proc. Cambridge Philos. Soc., 161(1):13–15, 2016.

[Voi07] Claire Voisin. Hodge theory and complex algebraic geometry. II, volume 77 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, 2007. Translated from the French by Leila Schneps.

[Voi15a] Claire Voisin. (Stable) rationality is not deformation invariant, 2015.

arXiv:1511.03591.

[Voi15b] Claire Voisin. Unirational threefolds with no universal codimension 2 cycle. Invent. Math., 201(1):207–237, 2015.

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