Stable rationality of higher dimensional conic bundles

# Stable rationality of higher dimensional conic bundles

###### Abstract

We prove that a very general nonsingular conic bundle embedded in a projective vector bundle of rank over is not stably rational if the anti-canonical divisor of is not ample and .

\setpapertype

\institutionDepartment of Mathematical Sciences, Loughborough University, LE11 3TU, United Kingdom
\institutionDepartment of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502 Japan
Accepted on May 21, 2018.

{prelims}\keywords

Stable Rationality; conic Bundles

\MSCclass

14E08

\languagesection

Français

Titre. Rationalité stable des fibrés en coniques de grande dimension \commentskipRésumé. Nous démontrons qu’un fibré en coniques non-singulier très général plongé dans le projectivisé d’un fibré vectoriel de rang au dessus de n’est pas stablement rationnel lorsque le diviseur anti-canonique de n’est pas ample et .

## 1 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 may become rational after being multiplied by a suitable projective space, i.e., is birational to , where , in which case we say is stably rational.

Stable non-rationality of conic bundles in dimension 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 given in [1] and [9].

Throughout this article, by a conic bundle we mean a Mori fibre space of relative dimension (see Definition 2.5 for details). The following is our main result.

###### Theorem 1.1

Let and be integers, and let be a direct sum of three invertible sheaves on . Let be a very general member of a complete linear system on , where is the tautological divisor and is the pullback of the hyperplane on . Suppose that the natural projection is a conic bundle.

• If is singular, then is rational.

• If is non-singular and is not ample, then is not stably rational.

This result covers the following varieties as a special case.

###### Corollary 1.2

Let be a very general hypersurface of bi-degree in . If , then is not stably rational.

This can be thought of as a higher dimensional generalisation of the main result of [2].

###### Corollary 1.3

Let be a double cover of branched along a very general divisor of bi-degree . If , then is not stably rational.

By a result of Sarkisov [16], a conic bundle is birational to a standard conic bundle which is by definition a nonsingular conic bundle flat over a smooth base. The following criterion for rationality in terms of the discriminant was conjectured by Shokurov [17] (see also [10, Conjecture I]). Remarkabe progress toward this conjecture has been made in [10] and [13].

###### Conjecture 1.4 ([17, Conjecture 10.3])

Let be a -dimensional standard conic bundle and the discriminant divisor. If , then is not rational.

Although the statement becomes weaker than Theorem 1.1, we can restate our main result in terms of the discriminant:

###### Corollary 1.5

With notation and assumptions as in Theorem 1.1, assume in addition that is nonsingular and let be the discriminant divisor of the conic bundle .

• If , then is not stably rational.

• If , is standard and , then is not stably rational.

This leads us to pose the following.

###### Conjecture 1.6

Let be an -dimensional standard conic bundle with . If , then is not rational. If in addition is very general in its moduli, then is not stably rational.

#### The argument of stable non-rationality.

It is known that a stably rational smooth projective variety is universally -trivial; see [5, Lemme 1.5] and [18, theorem 1.1] and references therein. Let be a flat family over a complex curve 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 is not universally -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

• admits a universally -trivial resolution such that is not universally -trivial,

• in mixed characteristic, that is, when with being a DVR of possibly mixed characteristic.

Thus it is enough to verify the existence of such a resolution over an algebraically closed field of characteristic . In view of [18, Lemma 2.2], the core of the proof of universal -nontriviality for in our case is done by showing that for some , 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.

## 2 Embedded conic bundles

### 2.1 Weighted projective space bundles

In this subsection we work over a field .

###### Definition 2.1

A toric weighted projective space bundle over is a projective simplicial toric variety with Cox ring

 Cox(P)=k[u0,…,un,x0,…,xm],

 (1⋯1λ0⋯λm0⋯0a0⋯am)

with the irrelevant ideal , where are integers and , , are positive integers. In other words, is the geometric quotient

 P=(An+m+2∖V(I))/G2m,

where the action of on is given by the above matrix.

The natural projection by the coordinates realizes as a -bundle over . In this paper, we simply call the -bundle over defined by

 ⎛⎜⎝u0⋯unx0⋯xm1⋯1|λ0⋯λm0⋯0|a0⋯am⎞⎟⎠.

In the following, let be as in Definition 2.1. Let be a point and a preimage of via the morphism . We can write , where . In this case we express as .

###### Remark 2.2

We will frequently use the following coordinate change. Consider a point and suppose for example that and for some . Then for such that , the replacement

 xl↦αλl−alλj0βaljxl−βluλl−alλj0xalj

induces an automorphism of . By considering the above coordinate change, we can transform (via an automorphism of ) into a point for which the -coordinate is zero for with .

We have the decomposition

 Cox(P)=⨁(α,β)∈Z2Cox(P)(α,β),

where consists of the homogeneous elements of bi-degree . An element is called a (homogeneous) polynomial of bi-degree .

The Weil divisor class group is naturally isomorphic to . Let and be the divisors on corresponding to and , respectively, which generate . Note that is the class of the pullback of a hyperplane on via . We denote by the rank reflexive sheaf corresponding to the divisor class of type , that is, the divisor . More generally, for a subscheme , we set . We remark that there is an isomorphism

 H0(P,OP(α,β))≅Cox(P)(α,β).
###### Definition 2.3

For integers with , we define (resp. ) to be the -bundle (resp. -bundle) over defined by the matrix

 ⎛⎜⎝u0⋯un−1xyz1⋯1|klm0⋯0|111⎞⎟⎠⎛⎜⎝resp.⎛⎜⎝u0⋯un−1xy1⋯1|kl0⋯0|11⎞⎟⎠⎞⎟⎠.
###### Remark 2.4

Let be as in Definition 2.1. When , is a -bundle over . More precisely we have an isomorphism

 P≅PPn(OPn(−λ0)⊕⋯⊕OPn(−λm)).

Here, for a vector bundle over , denotes the projective bundle of one-dimensional quotients of . Moreover, via the above isomorphism, the pullback of a hyperplane on and the tautological divisor on are identified with the divisors on corresponding to and , respectively.

### 2.2 Embedded conic bundles

In the rest of this section we work over . By a splitting vector bundle, we mean a vector bundle which is a direct sum of invertible sheaves.

###### Definition 2.5

Let be a normal projective -factorial variety of dimension . We say that a morphism is a conic bundle (over ) if it is a Mori fibre space, that is, has only terminal singularities, has connected fibres, is -ample and , where denotes the rank of the Picard group.

An embedded conic bundle is a conic bundle such that is embedded in a projective bundle as a member of for some splitting vector bundle of rank on and , and coincides with the restriction of to . Here and denote the pullback of a hyperplane on and the tautological class on , respectively.

In the following let be a splitting vector bundle of rank on and be a general member. We denote by the restriction of to . Without loss of generality we may assume that

 E≅OPn−1(−k)⊕OPn−1(−l)⊕OPn−1(−m)

for some . Then, by Remark 2.4, we have and the linear system on corresponds to . Here we do not assume that is a conic bundle. We study conditions on and that make a conic bundle.

###### Lemma 2.6

Let be integers such that . Set and let be a general member of .

• is smooth if and only if , , or .

• is not smooth and has only terminal singularities if and only if .

• is non-normal if and only if .

{proof}

Suppose that . Then is base point free and its general member is smooth. In the following we assume that .

Suppose that . Then is defined in by

 ax2+by2+fxy+gxz+hyz=0,

where . We have and . Then is singular along . The singular locus is of codimension in . Since is general, the hypersurfaces in defined by and are both nonsingular and intersect transversally. It is then straightforward to check that the blowup along the singular locus is a resolution and we have , where is the exceptional divisor. Thus has terminal singularities.

Suppose that . Then is defined in by

 ax2+by2+fxy+gxz+yz=0.

Replacing and suitably, we can eliminate the terms and , that is, is defined by

 ax2+yz=0.

It is then clear that is smooth, when is general.

Suppose that . Then is defined in by

 ax2+by2+fxy+gxz=0.

We have . Then is singular along , and the singularity is not terminal since the singular locus is of codimension in .

Suppose that . Then is defined in by

 ax2+by2+fxy+xz=0.

Replacing suitably, we may assume that is defined by

 by2+zx=0.

It is easy to see that is smooth.

Finally suppose that . Then is defined in by

 ax2+by2+fxy=0,

where . In this case is singular along the divisor . Thus 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 or . Then the variety is rational. Moreover we have unless .

{proof}

Suppose that , which implies . We claim that is defined by an equation of the form , where . This is already proved in Lemma 2.6, when . Suppose that . Then and is defined by

 ax2+y2+z2+fxy+gxz+αyz=0,

where and . Replacing and , the above equation can be transformed into and the claim is proved.

We consider the projection Note that . Then the projection is birational, hence is rational. The projection is defined outside . Let be a point. Then does not vanish at and we have

 y=yzz=−ax2z.

From this we deduce that is everywhere defined. Now we assume that either or . Then . We see that is a divisor and it is contracted by to a codimension subset of . This shows .

Next, suppose that . Note that . If in addition , then, by the proof of Lemma 2.6, the defining equation of can be written as . The statement follows from the same argument as above. If , then and we have . This case is already proved.

###### Lemma 2.8

In the same setting as in Lemma 2.6, is a nonsingular conic bundle if and only if one of the following holds:

• ,

• and , or

• .

{proof}

This follows from Lemmas 2.6 and 2.7.

###### Proposition 2.9

Let be an embedded conic bundle over . If is general in the linear system and singular, then is rational.

{proof}

We may assume that , where , for some . By Lemma 2.6, we have . Then a general member is defined by an equation of the form

 ax2+by2+fxy+gxz+hyz=0,

where . Here, note that, if for example , then we know that the term does not appear in the equation. The inequality implies that since is general. Let be the natural projection. Now we can write the defining equation as

 z(gx+hy)+ax2+by2+fxy=0,

which implies that the restriction is birational. Therefore is rational.

The following can be considered as a “normal form” of conic bundles, which describes nonsingular embedded conic bundles (see Proposition 2.11).

###### Definition 2.10

Let be a triplet of integers . We say that (or ) is of type if belongs to , where , and coincides with the restriction of to .

###### Proposition 2.11

Let be a nonsingular embedded conic bundle. Then is either of type for some such that or of type .

{proof}

We may assume that belongs to for some and . Since the family is non-singular, we have by Lemma 2.8 and is defined in by an equation of the form

 ax2+by2+cz2+fxy+gxz+hyz=0,

where . We set and . By comparing the weights, we have

 α+2k=β+2l=γ+2m=ν+k+l=μ+k+m=λ+l+m.

Now we have

 Pn(k,l,m)≅Pn(k+(ν−m),l+(ν−m),m+(ν−m))≅Pn(λ,μ,ν)=:P,

and the linear system is identified with . Thus is of type . By applying Lemma 2.8 for and , we get the desired result.

###### Remark 2.12

In the language of [1, Definition 3.1], a conic bundle of type with is a conic bundle of graded-free type over corresponding to the triplet .

## 3 Stable non-rationality

In this section we study stable (non-)rationality of nonsingular embedded conic bundles . By Proposition 2.11, such a conic bundle is of type , where either or . In case is of type , then and it is obviously rational. We consider the remaining cases and thus we assume that

 0<λ≤μ≤ν≤λ+μ

throughout this section. In addition we assume throughout.

We set , , and consider special members defined in by an equation of the form

 ax2+by2+cz2+fxy=0, (1)

where are general polynomials in variables . Recall that and , and .

###### Remark 3.1

By the assumptions on , we have , , and .

###### Lemma 3.2

If the ground field is an algebraically closed field of characteristic , then is smooth.

{proof}

The variety is a general member of the base point free sub linear system of on the smooth variety . Thus, by the Bertini theorem, a general is smooth.

We use universal -triviality to test stable rationality of varieties.

###### Definition 3.3

Let be a projective variety defined over a field . We denote by the Chow group of -cycles on . We say that is universally -trivial if for any field containing , the degree map is an isomorphism. A projective morphism defined over is universally -trivial if for any field containing , the push-forward map is an isomorphism.

In the rest of this section we work over an algebraically closed field of characteristic . Let be the -bundle over defined by

 ⎛⎜⎝u0u1⋯un−1xy¯z11⋯1|λμ2ν00⋯0|112⎞⎟⎠

and let be the hypersurface defined by

 ax2+by2+c¯z+fxy=0.

We have a natural morphism which is a (purely inseparable) double cover branched along . The image of under is the hypersurface . Let be the induced morphism, which is a double cover branched along the divisor cut out on by . We set . Then is a global section of , and over the non-singular locus of , is the double cover obtained by taking the roots of in the sense of [11, Construction 8].

In Sections 3.1 and 3.2 below we will analyse the singularities of and , and finally we will show the existence of a universally -trivial resolution such that under some conditions on . The latter implies that is not universally -trivial by [18, Lemma 2.2].

### 3.1 Singularities

Recall that the ground field is an algebraically closed field of characteristic and is a hypersurface in defined by

 ax2+by2+cz2+fxy=0

for general . Similarly is the hypersurface in defined by

 ax2+by2+c¯z+fxy=0.

We set

 Ξ=(x=y=0)⊂R, ΞZ=Ξ∩Z=(x=y=c=0),

and , .

In order to analyze singularities of , we consider standard affine charts of . For and a coordinate , we set . We have

 R∘=⋃i∈{0,…,n−1},w∈{x,y}Uui,w.

We remark that is an affine -space and that the restriction of the sections

 {u0,…,un−1,x,y,¯z}∖{ui,w}

are affine coordinates of . We only treat because the other open subsets can be understood by symmetry. We set

 ~ui=ui/u0, ~y=y/xuμ−λ0, ~z=¯z/x2uν−2λ0.

Then is an affine -space with affine coordinates . By a slight abuse of notation, the affine coordinates are simply denoted by .

###### Lemma 3.4

is smooth.

{proof}

If , then is a non-zero constant and thus . In this case is a bundle over and it is smooth.

In the following we assume that and set

 Ux=(x≠0), Uy=(y≠0)⊂R,

so that . We will show that for any point , the condition that is singular at imposes independent conditions on . Then the assertion will follow by a dimension count argument since . We note that , and by Remark 3.1.

Let . Replacing coordinates, we may assume . Then is an affine space with coordinates and is defined by

 ~a+~by2+~c¯z+~fy=0,

where we set for a polynomial . Note that corresponds to the origin. The variety is singular at if and only if vanish at and the linear part of is zero. This imposes independent conditions since and (cf. Remark 3.1).

Suppose that . Replacing coordinates, we may assume . Then is an affine space with coordinates and is defined by

 ~ax2+~b+~c¯z+~fx=0.

The variety is singular at if and only if vanish at and the linear part of is zero. The latter imposes independent conditions since and (cf. Remark 3.1), and the proof is complete.

We set .

###### Lemma 3.5

is smooth along .

{proof}

Note that . For a point , is smooth at if and only if the hypersurface is smooth at the image of under . Clearly the hypersurface is smooth since is general, and the assertion follows.

### 3.2 Analysis of critical points

We set , where we recall . By Lemma 3.4, is non-singular and by Kollár’s result [12, V.5] there exists an invertible sheaf on such that , where denotes the double dual. Let be the push-forward of the invertible sheaf via the open immersion . By Lemma 3.5, is an invertible sheaf on .

###### Definition 3.6

Let be a nonsingular variety of dimension defined over an algebraically closed field of characteristic , an invertible sheaf on and a section. Let be a point, a local generator of at and a local description of with respect to local coordinates of at . We say that has a critical point at if the linear term of is zero.

We say that has an admissible critical point at if for a suitable choice of coordinates ,

 f={α+x1x2+x3x4+⋯+xn−1xn+g,if n is even,α+βx21+x2x3+⋯+xn−1xn+g,if n is odd,

where , and, in case is odd, the coefficient of in is nonzero.

###### Lemma 3.7

The section has only admissible critical points on .

{proof}

We choose and fix a general so that the hypersurface is non-singular. Clearly does not have a critical point on . On , the section is invertible and thus the section has an admissible critical point if and only if the section

 s:=c(ax2+by2+fxy) (=c2¯z)

has an admissible critical point. It is then enough to show that the section , viewed as a section on , has only admissible critical points on for general and . We set and so that .

We first show that does not have a critical point on . Let be a point. We may assume . We work on the open subset which is the affine space with coordinates and . For , we set . Moreover we denote by the degree part of . Then the restriction of to is and the point corresponds to the origin. Then has a critical point at if and only if

 ~c0(~b1+~f0x)+~c1~b0=0.

Note that . Since , this imposes independent conditions on . Thus, for any point , conditions are imposed in order for to have a critical point at . By counting dimensions we conclude that does not have a critical point on since .

Let be a point. We may assume . We work on the open subset which is the affine space with coordinates and . We have . Let and be the linear, quadratic and cubic parts of , respectively. We have

 ℓ=~c0(~a1+~f0y)+~c1~a0.

Since , conditions are imposed in order for to have a critical point at . It remains to show the existence of a section which has an admissible critical point at . Now suppose that has a critical point at , that is, . This implies that and . Then, for the quadratic and cubic parts, we have

 q=~c0(~a2+~b0y2+~f1y)+~a0~c21~c0+~c2~a0,h=~c0(~a3+~b1y2+~f2y)+⋯.

Since and , we can choose so that

 q={yu1+u2