Stable rationality of higher dimensional conic bundles

We prove that a very general nonsingular conic bundle $X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank $3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and $n\geq 3$.


Introduction
An important question in algebraic geometry is to determine whether an algebraic variety is rational; that is, birational to projective space. Two algebraic varieties are said to be birational if they become isomorphic after removing finitely many lower-dimensional subvarieties from both sides. The closest varieties to being rational are those that admit a fibration into a projective space with all fibres rational curves; so-called conic bundles.
In this article, we study stable (non-)rationality of conic bundles over a projective space of arbitrary dimension (greater than one). A non-rational variety X may become rational after being multiplied by a suitable projective space, i.e., X × P m is birational to P n+m , where n = dim X, in which case we say X is stably rational.
Stable non-rationality of conic bundles in dimension 3 has been studied extensively in [1,2] and [8], giving a satisfactory answer. In higher dimensions almost nothing is known except for a few examples of stably non-rational conic bundles over P 3 given in [1] and [9]. Throughout this article, by a conic bundle we mean a Mori fibre space of relative dimension 1 (see Definition 2.5 for details). The following is our main result. Theorem 1.1. Let n ≥ 3 and d be integers, and let E be a direct sum of three invertible sheaves on P n−1 . Let X be a very general member of a complete linear system |2D + dF| on P P n−1 (E), where D is the tautological divisor and F is the pullback of the hyperplane on P n−1 . Suppose that the natural projection X → P n−1 is a conic bundle.
(1) If X is singular, then X is rational.
(2) If X is non-singular and −K X is not ample, then X is not stably rational.
This result covers the following varieties as a special case. Corollary 1.2. Let X be a very general hypersurface of bi-degree (d, 2) in P n−1 × P 2 . If d ≥ n ≥ 3, then X is not stably rational. Corollary 1.5. With notation and assumptions as in Theorem 1.1, assume in addition that X is nonsingular and let ∆ ⊂ P n−1 be the discriminant divisor of the conic bundle X → P n−1 .
This leads us to pose the following. Conjecture 1.6. Let π : X → S be an n-dimensional standard conic bundle with n ≥ 3. If |2K S + ∆| ∅, then X is not rational. If in addition X is very general in its moduli, then X is not stably rational.
The argument of stable non-rationality. It is known that a stably rational smooth projective variety is universally CH 0 -trivial; see [5,Lemme 1.5] and [18, theorem 1.1] and references therein. Let X → B be a flat family over a complex curve B with smooth general fibre. Then, by the specialisation theorem of Voisin [19,Theorem 2.1], the stable non-rationality of a very general fibre will follow if the special fibre X 0 is not universally CH 0 -trivial and has at worst ordinary double point singularities. This was generalised by Colliot-Thélène and Pirutka [5, Théorème 1.14] to the case where 1. X 0 admits a universally CH 0 -trivial resolution ϕ : Y → X 0 such that Y is not universally CH 0 -trivial, 2. in mixed characteristic, that is, when B = Spec A with A being a DVR of possibly mixed characteristic.
Thus it is enough to verify the existence of such a resolution ϕ : Y → X 0 over an algebraically closed field of characteristic p > 0. In view of [18,Lemma 2.2], the core of the proof of universal CH 0 -nontriviality for Y in our case is done by showing that H 0 (Y , Ω i ) 0 for some i > 0, following Kollár [11] and Totaro [18]. This is done in Section 3.

Acknowledgements.
We would like to thank Professor Jean-Louis Colliot-Thélène and Professor Vyacheslav Shokurov for pointing out two oversights in an earlier version of this article. We would also like to thank the anonymous referee for helpful comments. The second author is partially supported by JSPS KAKENHI Grant Number 26800019.

Embedded conic bundles 2.A. Weighted projective space bundles
In this subsection we work over a field k. Definition 2.1. A toric weighted projective space bundle over P n is a projective simplicial toric variety with Cox ring which is Z 2 -graded as 1 · · · 1 λ 0 · · · λ m 0 · · · 0 a 0 · · · a m with the irrelevant ideal I = (u 0 , . . . , u n ) ∩ (x 0 , . . . , x m ), where λ 0 , . . . , λ m are integers and n, m, a 0 , . . . , a m are positive integers. In other words, P is the geometric quotient where the action of G 2 m = G m × G m on A n+m+2 = Spec Cox(P ) is given by the above matrix.
The natural projection Π : P → P n by the coordinates u 0 , . . . , u n realizes P as a P(a 0 , . . . , a m )-bundle over P n . In this paper, we simply call P the P(a 0 , . . . , a m )-bundle over P n defined by In the following, let P be as in Definition 2.1. Let p ∈ P be a point and q ∈ A n+m+2 \ V (I) a preimage of p via the morphism A n+m+2 \ V (I) → P . We can write q = (α 0 , . . . , α n , β 0 , . . . , β m ), where α i , β j ∈ k. In this case we express p as (α 0 : · · · : α n ; β 0 : · · · : β m ).

Remark 2.2.
We will frequently use the following coordinate change. Consider a point p = (α 0 : · · · : α n ; β 0 : · · · : β m ) ∈ P and suppose for example that α 0 0, β j 0 and a j = 1 for some j. Then for l j such that λ l /a l ≥ λ j , the replacement induces an automorphism of P . By considering the above coordinate change, we can transform p (via an automorphism of P ) into a point for which the x l -coordinate is zero for l with λ l /a l ≥ λ j .
We have the decomposition where Cox(P ) (α,β) consists of the homogeneous elements of bi-degree (α, β). An element f ∈ Cox(P ) (α,β) is called a (homogeneous) polynomial of bi-degree (α, β). The Weil divisor class group Cl(P ) is naturally isomorphic to Z 2 . Let F and D be the divisors on P corresponding to (1, 0) and (0, 1), respectively, which generate Cl(P ). Note that F is the class of the pullback of a hyperplane on P n via Π : P → P n . We denote by O P (α, β) the rank 1 reflexive sheaf corresponding to the divisor class of type (α, β), that is, the divisor αF + βD. More generally, for a subscheme Z ⊂ P , we set O Z (α, β) = O P (α, β)| Z . We remark that there is an isomorphism H 0 (P , O P (α, β)) Cox(P ) (α,β) . Definition 2.3. For integers k, l, m, n with n ≥ 3, we define P n (k, l, m) (resp. Q n (k, l)) to be the P 2 -bundle (resp. P 1 -bundle) over P n−1 defined by the matrix Remark 2.4. Let P be as in Definition 2.1. When a 0 = · · · = a m = 1, P is a P m -bundle over P n . More precisely we have an isomorphism Here, for a vector bundle E over P n , P(E) = P P n (E) denotes the projective bundle of one-dimensional quotients of E. Moreover, via the above isomorphism, the pullback of a hyperplane on P n−1 and the tautological divisor on P(E) are identified with the divisors on P corresponding to (1, 0) and (0, 1), respectively.

2.B. Embedded conic bundles
In the rest of this section we work over C. By a splitting vector bundle, we mean a vector bundle which is a direct sum of invertible sheaves.
Definition 2.5. Let X be a normal projective Q-factorial variety of dimension n. We say that a morphism π : X → P n−1 is a conic bundle (over P n−1 ) if it is a Mori fibre space, that is, X has only terminal singularities, π has connected fibres, −K X is π-ample and ρ(X) = 2, where ρ(X) denotes the rank of the Picard group. An embedded conic bundle π : X → P n−1 is a conic bundle such that X is embedded in a projective bundle P(E) := P P n−1 (E) as a member of |dF + 2D| for some splitting vector bundle E of rank 3 on P n−1 and d ∈ Z, and π coincides with the restriction of Π : P(E) → P n−1 to X. Here F and D denote the pullback of a hyperplane on P n−1 and the tautological class D on P(E), respectively.
In the following let E be a splitting vector bundle of rank 3 on P n−1 and X ∈ |dF + 2D| be a general member. We denote by π : X → P n−1 the restriction of Π : P(E) → P n−1 to X. Without loss of generality we may assume that for some k ≤ l ≤ m. Then, by Remark 2.4, we have P(E) P n (k, l, m) and the linear system |dF + 2D| on Here we do not assume that π : X → P n−1 is a conic bundle. We study conditions on k, l, m and d that make π : X → P n−1 a conic bundle.
(2) X is not smooth and has only terminal singularities if and only if 2m > d > l + m.
(3) X is non-normal if and only if k + m > d.
Proof. Suppose that d ≥ 2m. Then |O P (d, 2)| is base point free and its general member X is smooth. In the following we assume that 2m > d ≥ k + m. Suppose that 2m > d > l + m. Then X is defined in P by The singular locus is of codimension 3 in X. Since X is general, the hypersurfaces in P n−1 defined by g = 0 and h = 0 are both nonsingular and intersect transversally. It is then straightforward to check that the blowup σ : X → X along the singular locus is a resolution and we Replacing y and z suitably, we can eliminate the terms by 2 , f xy and gzx, that is, X is defined by It is then clear that X is smooth, when a is general. Suppose that l + m > d > k + m. Then X is defined in P by Then X is singular along (x = y = g = 0) ∅, and the singularity is not terminal since the singular locus is of codimension 2 in X.
Suppose that l + m > d = k + m. Then X is defined in P by Replacing z suitably, we may assume that X is defined by It is easy to see that X is smooth. Finally suppose that k + m > d. Then X is defined in P by In this case X is singular along the divisor (x = y = 0) ⊂ X. Thus X is not normal. The above arguments prove (1), (2) and (3).

Lemma 2.7.
In the same setting as in Lemma 2.6, suppose that either . This is already proved in Lemma 2.6, when 2m > d. Suppose that 2m = d = l + m. Then l = m and X is defined by where α ∈ C and a, f , g ∈ C[u]. Replacing y and z, the above equation can be transformed into ax 2 +yz = 0 and the claim is proved. We consider the projection X Q := Q n (k, l) Note that Q P(O(−k) ⊕ O(−l)). Then the projection is birational, hence X is rational. The projection X Q is defined outside (x = y = 0) ⊂ X. Let p ∈ (x = y = 0) be a point. Then z does not vanish at p and we have From this we deduce that X Q is everywhere defined. Now we assume that either k l or l m. Then Next, suppose that d = k + m. Note that l + m ≥ d. If in addition l + m > d, then, by the proof of Lemma 2.6, the defining equation of X can be written as by 2 + xz = 0. The statement follows from the same argument as above. If l + m = d, then k = l and we have d = l + m. This case is already proved. Lemma 2.8. In the same setting as in Lemma 2.6, π : X → P n−1 is a nonsingular conic bundle if and only if one of the following holds: (2) d = 2m and m > l, or Proof. This follows from Lemmas 2.6 and 2.7. Proposition 2.9. Let X be an embedded conic bundle over P n−1 . If X is general (in the linear system) and singular, then X is rational.
Proof. We may assume that X ∈ |O P (d, 2)|, where P = P n (k, l, m), for some k ≤ l ≤ m. By Lemma 2.6, we have 2m > d ≥ k + m. Then a general member X is defined by an equation of the form Here, note that, if for example l +m > d, then we know that the term hyz does not appear in the equation. The inequality d ≥ k+m implies that g 0 since X is general. Let P Q = Q n (k, l) be the natural projection. Now we can write the defining equation as z(gx + hy) + ax 2 + by 2 + f xy = 0, which implies that the restriction X Q is birational. Therefore X is rational.

Lemma 3.2.
If the ground field is an algebraically closed field of characteristic 0, then X is smooth.
Proof. The variety X is a general member of the base point free sub linear system of |O P (δ, 2)| on the smooth variety P . Thus, by the Bertini theorem, a general X is smooth.
We use universal CH 0 -triviality to test stable rationality of varieties.

Definition 3.3.
Let V be a projective variety defined over a field k. We denote by CH 0 (V ) the Chow group of 0-cycles on V . We say that V is universally CH 0 -trivial if for any field F containing k, the degree map CH 0 (V F ) → Z is an isomorphism. A projective morphism ϕ : W → V defined over k is universally CH 0trivial if for any field containing k, the push-forward map ϕ * : CH 0 (W F ) → CH 0 (V F ) is an isomorphism.
In the rest of this section we work over an algebraically closed field k of characteristic 2. Let R be the P(1, 1, 2)-bundle over P n−1 defined by and let Z ⊂ R be the hypersurface defined by ax 2 + by 2 + cz + f xy = 0.
We have a natural morphism P → R which is a (purely inseparable) double cover branched along (z = 0) ⊂ R. The image of X under P → R is the hypersurface Z ⊂ R. Let τ : X → Z be the induced morphism, which is a double cover branched along the divisor cut out on Z byz = 0. We set L = O Z (ν, 1). Thenz is a global section of L 2 , and over the non-singular locus of Z, τ is the double cover obtained by taking the roots ofz ∈ H 0 (Z, L 2 ) in the sense of [11,Construction 8].
In Sections 3.A and 3.B below we will analyse the singularities of X and Z, and finally we will show the existence of a universally CH 0 -trivial resolution ϕ : Y → X such that H 0 (Y , Ω n−1 Y ) 0 under some conditions on λ, µ, ν. The latter implies that Y is not universally CH 0 -trivial by [

3.A. Singularities
Recall that the ground field k is an algebraically closed field of characteristic 2 and X is a hypersurface in P = P n (λ, µ, ν) defined by ax 2 + by 2 + cz 2 + f xy = 0 for general a, b, c, f ∈ k[u 0 , . . . , u n−1 ]. Similarly Z is the hypersurface in R defined by ax 2 + by 2 + cz + f xy = 0.
We set In order to analyze singularities of Z • ⊂ R • , we consider standard affine charts of R • . For i = 0, . . . , n−1 and a coordinate w ∈ {x, y}, we set U u i ,w = (u i 0) ∩ (w 0) ⊂ R • . We have We remark that U u i ,w is an affine (n + 1)-space and that the restriction of the sections {u 0 , . . . , u n−1 , x, y,z} \ {u i , w} are affine coordinates of U u i ,w . We only treat U u 0 ,x because the other open subsets can be understood by symmetry. We setũ Then U u 0 ,w is an affine (n + 1)-space with affine coordinatesũ 1 , . . . ,ũ n−1 ,ỹ,z. By a slight abuse of notation, the affine coordinatesũ 1 , . . . ,ũ n−1 ,ỹ,z are simply denoted by u 1 , . . . , u n−1 , y,z.

Lemma 3.4. Z • is smooth.
Proof. If deg c = 0, then c is a non-zero constant and thus Ξ Z = ∅. In this case Z = Z • is a P 1 bundle over P n−1 and it is smooth.
In the following we assume that deg c > 0 and set We will show that for any point q ∈ R • , the condition that Z • is singular at q ∈ Z imposes n + 2 independent conditions on a, b, c, f . Then the assertion will follow by a dimension count argument since dim R • = n + 1. We note that Let q ∈ U x . Replacing coordinates, we may assume q = (1 : 0 : · · · : 0; 1 : 0 : 0). Then U u 0 ,x ⊂ Q is an affine space with coordinates u 1 , . . . , u n−1 , y,z and Z ∩ U u 0 ,z is defined bỹ a +by 2 +cz +f y = 0, where we seth = h(1, u 1 , . . . , u n−1 ) for a polynomial h(u 0 , . . . , u n−1 ). Note that q corresponds to the origin. The variety Z • is singular at q if and only ifã,c,f vanish at q and the linear part ofã is zero. This imposes n + 2 independent conditions since deg a > 0 and deg c, deg f ≥ 0 (cf. Remark 3.1).
The variety Z • is singular at q if and only ifb,c,f vanish at q and the linear part ofb is zero. The latter imposes n + 2 independent conditions since deg b > 0 and deg c, deg f ≥ 0 (cf. Remark 3.1), and the proof is complete.

Lemma 3.5. X is smooth along
Proof. Note that X \ X • = X ∩ (x = y = 0). For a point p ∈ X \ X • , X is smooth at p if and only if the hypersurface (c = 0) ⊂ P n−1 is smooth at the image of p under X → P n−1 . Clearly the hypersurface (c = 0) ⊂ P n−1 is smooth since c is general, and the assertion follows.

3.B. Analysis of critical points
We set L • = L| Z • , where we recall L = O Z (ν, 1). By Lemma 3.4, Z • is non-singular and by Kollár's result [12, V.5] there exists an invertible sheaf Q • on Z • such that M • := τ * Q • ⊂ (Ω n−1 X • ) ∨∨ , where ∨∨ denotes the double dual. Let M be the push-forward of the invertible sheaf M • via the open immersion X • → X. By Lemma 3.5, M is an invertible sheaf on X. Definition 3.6. Let V be a nonsingular variety of dimension n defined over an algebraically closed field k of characteristic 2, N an invertible sheaf on V and s ∈ H 0 (V , N 2 ) a section. Let p ∈ V be a point, ξ a local generator of N at p and s = f (x 1 , . . . , x n )ξ 2 a local description of s with respect to local coordinates x 1 , . . . , x n of V at p. We say that s has a critical point at p if the linear term of f is zero.
We say that s has an admissible critical point at p if for a suitable choice of coordinates x 1 , . . . , x n , where α, β ∈ k, g = g(x 1 , . . . , x n ) ∈ (x 1 , . . . , x n ) 3 and, in case n is odd, the coefficient of x 3 1 in g is nonzero. Note thatc 0 0. Since deg b ≥ 1, this imposes n independent conditions on a, b, f . Thus, for any point p ∈ Π y , n conditions are imposed in order for s to have a critical point at p. By counting dimensions we conclude that s does not have a critical point on Π y ∩ U c since dim Π y = n − 1.
Since deg a ≥ 1, n conditions are imposed in order for s to have a critical point at p. It remains to show the existence of a section s = c(ax 2 + by 2 + f xy) which has an admissible critical point at p. Now suppose that s has a critical point at p, that is, = 0. This implies thatf 0 = 0 andã 1 =ã 0c1 /c 0 . Then, for the quadratic and cubic parts, we have q =c 0 (ã 2 +b 0 y 2 +f 1 y) +ã 0c 2 1 c 0 +c 2ã0 , h =c 0 (ã 3 +b 1 y 2 +f 2 y) + · · · .
In case n is even, the section s has a nondegenerate critical point at p and we are done. Suppose that n is odd. Since deg a ≥ 3, then we can choose a, b, f so that q is as above and the coefficient of u 3 n−1 in h is non-zero. For this choice of a, b, c, f , the section s has an admissible critical point at p and the proof is completed by the dimension counting argument. Proposition 3.8. Let the notation and assumption as above. Assume in addition that ν ≥ n. Then there exists a universally CH 0 -trivial resolution ϕ : Y → X of singularities such that H 0 (Y , Ω n−1 Y ) 0. In particular Y is not universally CH 0 -trivial.
Proof. By [15,Proposition 4.1] or [6], if the singularities of X correspond to admissible critical points of the sectionz, then there exists a universally CH 0 -trivial resolution ϕ : Y → X such that ϕ * M → Ω n−1 Y (in fact, ϕ is just the composite of blowups at each (isolated) singular point). Thus, by Lemma 3.7, X admits such a resolution. The branch divisor (z = 0) is clearly reduced and, by [12, Lemma V.5.9], we have an isomorphism
Proof. For a field (or more generally a ring) K, we denote by P K the P 2 -bundle P n (λ, µ, ν) over P n−1 defined over K. Let k be an algebraically closed field of characteristic 2 and let X → P n−1 be a very general hypersurface in P k defined by an equation of the form (1). We take a mixed characteristic discrete valuation ring A whose residue field is k, for example the Witt ring, and then we lift X to a hypersurface X of P A defined by an equation of the form (1). We choose and fix an embedding of the quotient field of A into C and set V = X × A C. Then V is a very general hypersurface of P C defined by an equation of the form (1). By Proposition 3.8, we can apply the specialization theorem [ Now we can prove the main theorem and corollaries in Section 1.
Proof of Corollaries 1.2 and 1.3. Let X be a very general hypersurface of bi-degree (d, 2) in P n−1 ×P 2 . Then O X (−K X ) O X (n − d, 1). By assumption d ≥ n and this implies that −K X is not ample. Thus X is not stably rational by Theorem 1.1.