# Stable rationality of higher dimensional conic bundles

Hamid Ahmadinezhad and Takuzo Okada

###### Abstract

We prove that a very general nonsingular conic bundle $X\to\mathbb{P}^{n-1}$
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
$n\geq 3$.

  *     * Keywords. Stable Rationality; conic Bundles

2010 Mathematics Subject Classification. 14E08

[Français]

Titre. Rationalité stable des fibrés en coniques de grande dimension Résumé.
Nous démontrons qu’un fibré en coniques non-singulier très général
$X\to\mathbb{P}^{n-1}$ plongé dans le projectivisé d’un fibré vectoriel de
rang $3$ au dessus de $\mathbb{P}^{n-1}$ n’est pas stablement rationnel
lorsque le diviseur anti-canonique de $X$ n’est pas ample et $n\geq 3$.

###### Contents

  1. 1\. Introduction
  2. 2\. Embedded conic bundles
  3. 3\. Stable non-rationality

##  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 $X$ may become rational after being multiplied by a suitable
projective space, i.e., $X\times\mathbb{P}^{m}$ is birational to
$\mathbb{P}^{n+m}$, 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 $\mathbb{P}^{3}$ 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 $n\geq 3$ and $d$ be integers, and let $\mathcal{E}$ be a direct sum of
three invertible sheaves on $\mathbb{P}^{n-1}$. Let $X$ be a very general
member of a complete linear system $|2D+dF|$ on
$\mathbb{P}_{\mathbb{P}^{n-1}}(\mathcal{E})$, where $D$ is the tautological
divisor and $F$ is the pullback of the hyperplane on $\mathbb{P}^{n-1}$.
Suppose that the natural projection $X\to\mathbb{P}^{n-1}$ 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
$\mathbb{P}^{n-1}\times\mathbb{P}^{2}$. If $d\geq n\geq 3$, then $X$ is not
stably rational.

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

###### Corollary 1.3

Let $X$ be a double cover of $\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ branched
along a very general divisor of bi-degree $(2d,2)$. If $2d\geq n\geq 3$, then
$X$ 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 $X\to S$ be a $3$-dimensional standard conic bundle and $\Delta\subset S$
the discriminant divisor. If $|2K_{S}+\Delta|\neq\emptyset$, then $X$ 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 $X$
is nonsingular and let $\Delta\subset\mathbb{P}^{n-1}$ be the discriminant
divisor of the conic bundle $X\to\mathbb{P}^{n-1}$.

  * (1)

If $|3K_{\mathbb{P}^{n-1}}+\Delta|\neq\emptyset$, then $X$ is not stably
rational.

  * (2)

If $n\geq 7$, $\pi\colon X\to\mathbb{P}^{n-1}$ is standard and
$|2K_{\mathbb{P}^{n-1}}+\Delta|\neq\emptyset$, then $X$ is not stably
rational.

This leads us to pose the following.

###### Conjecture 1.6

Let $\pi\colon X\to S$ be an $n$-dimensional standard conic bundle with $n\geq
3$. If $\left|2K_{S}+\Delta\right|\neq\emptyset$, 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
$\operatorname{CH}_{0}$-trivial; see [5, Lemme 1.5] and [18, theorem 1.1] and
references therein. Let $\mathcal{X}\rightarrow\mathcal{B}$ be a flat family
over a complex curve $\mathcal{B}$ 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 $\operatorname{CH}_{0}$-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 $\operatorname{CH}_{0}$-trivial resolution
$\varphi\colon Y\to X_{0}$ such that $Y$ is not universally
$\operatorname{CH}_{0}$-trivial,

  * 2.

in mixed characteristic, that is, when $\mathcal{B}=\operatorname{Spec}A$ with
$A$ being a DVR of possibly mixed characteristic.

Thus it is enough to verify the existence of such a resolution $\varphi\colon
Y\to X_{0}$ over an algebraically closed field of characteristic $p>0$. In
view of [18, Lemma 2.2], the core of the proof of universal
$\operatorname{CH}_{0}$-nontriviality for $Y$ in our case is done by showing
that $H^{0}(Y,\Omega^{i})\neq 0$ 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.

##  2\. 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 $\mathbb{P}^{n}$ is a projective
simplicial toric variety with Cox ring

$\operatorname{Cox}(P)=k[u_{0},\dots,u_{n},x_{0},\dots,x_{m}],$

which is $\mathbb{Z}^{2}$-graded as

$\begin{pmatrix}1&\cdots&1&\lambda_{0}&\cdots&\lambda_{m}\\\
0&\cdots&0&a_{0}&\cdots&a_{m}\end{pmatrix}$

with the irrelevant ideal $I=(u_{0},\dots,u_{n})\cap(x_{0},\dots,x_{m})$,
where $\lambda_{0},\dots,\lambda_{m}$ are integers and $n$, $m$,
$a_{0},\dots,a_{m}$ are positive integers. In other words, $P$ is the
geometric quotient

$P=(\mathbb{A}^{n+m+2}\setminus V(I))/\mathbb{G}_{m}^{2},$

where the action of $\mathbb{G}_{m}^{2}=\mathbb{G}_{m}\times\mathbb{G}_{m}$ on
$\mathbb{A}^{n+m+2}=\operatorname{Spec}\operatorname{Cox}(P)$ is given by the
above matrix.

The natural projection $\Pi\colon P\to\mathbb{P}^{n}$ by the coordinates
$u_{0},\dots,u_{n}$ realizes $P$ as a $\mathbb{P}(a_{0},\dots,a_{m})$-bundle
over $\mathbb{P}^{n}$. In this paper, we simply call $P$ the
$\mathbb{P}(a_{0},\dots,a_{m})$-bundle over $\mathbb{P}^{n}$ defined by

$\begin{pmatrix}u_{0}&\cdots&u_{n}&&x_{0}&\cdots&x_{m}\\\
1&\cdots&1&|&\lambda_{0}&\cdots&\lambda_{m}\\\
0&\cdots&0&|&a_{0}&\cdots&a_{m}\end{pmatrix}.$

In the following, let $P$ be as in Definition 2.1. Let $\mathsf{p}\in P$ be a
point and $\mathsf{q}\in\mathbb{A}^{n+m+2}\setminus V(I)$ a preimage of
$\mathsf{p}$ via the morphism $\mathbb{A}^{n+m+2}\setminus V(I)\to P$. We can
write $\mathsf{q}=(\alpha_{0},\dots,\alpha_{n},\beta_{0},\dots,\beta_{m})$,
where $\alpha_{i},\beta_{j}\in k$. In this case we express $\mathsf{p}$ as
$(\alpha_{0}\\!:\\!\cdots\\!:\\!\alpha_{n};\beta_{0}\\!:\\!\cdots\\!:\\!\beta_{m})$.

###### Remark 2.2

We will frequently use the following coordinate change. Consider a point
$\mathsf{p}=(\alpha_{0}\\!:\\!\cdots\\!:\\!\alpha_{n};\beta_{0}\\!:\\!\cdots\\!:\\!\beta_{m})\in
P$ and suppose for example that $\alpha_{0}\neq 0,\beta_{j}\neq 0$ and
$a_{j}=1$ for some $j$. Then for $l\neq j$ such that
$\lambda_{l}/a_{l}\geq\lambda_{j}$, the replacement

$x_{l}\mapsto\alpha_{0}^{\lambda_{l}-a_{l}\lambda_{j}}\beta_{j}^{a_{l}}x_{l}-\beta_{l}u_{0}^{\lambda_{l}-a_{l}\lambda_{j}}x_{j}^{a_{l}}$

induces an automorphism of $P$. By considering the above coordinate change, we
can transform $\mathsf{p}$ (via an automorphism of $P$) into a point for which
the $x_{l}$-coordinate is zero for $l$ with
$\lambda_{l}/a_{l}\geq\lambda_{j}$.

We have the decomposition

$\operatorname{Cox}(P)=\bigoplus_{(\alpha,\beta)\in\mathbb{Z}^{2}}\operatorname{Cox}(P)_{(\alpha,\beta)},$

where $\operatorname{Cox}(P)_{(\alpha,\beta)}$ consists of the homogeneous
elements of bi-degree $(\alpha,\beta)$. An element
$f\in\operatorname{Cox}(P)_{(\alpha,\beta)}$ is called a (homogeneous)
polynomial of bi-degree $(\alpha,\beta)$. The Weil divisor class group
$\operatorname{Cl}(P)$ is naturally isomorphic to $\mathbb{Z}^{2}$. Let $F$
and $D$ be the divisors on $P$ corresponding to $(1,0)$ and $(0,1)$,
respectively, which generate $\operatorname{Cl}(P)$. Note that $F$ is the
class of the pullback of a hyperplane on $\mathbb{P}^{n}$ via $\Pi\colon
P\to\mathbb{P}^{n}$. We denote by $\mathcal{O}_{P}(\alpha,\beta)$ the rank $1$
reflexive sheaf corresponding to the divisor class of type $(\alpha,\beta)$,
that is, the divisor $\alpha F+\beta D$. More generally, for a subscheme
$Z\subset P$, we set
$\mathcal{O}_{Z}(\alpha,\beta)=\mathcal{O}_{P}(\alpha,\beta)|_{Z}$. We remark
that there is an isomorphism

$H^{0}(P,\mathcal{O}_{P}(\alpha,\beta))\cong\operatorname{Cox}(P)_{(\alpha,\beta)}.$

###### Definition 2.3

For integers $k,l,m,n$ with $n\geq 3$, we define $P_{n}(k,l,m)$ (resp.
$Q_{n}(k,l)$) to be the $\mathbb{P}^{2}$-bundle (resp.
$\mathbb{P}^{1}$-bundle) over $\mathbb{P}^{n-1}$ defined by the matrix

$\begin{pmatrix}u_{0}&\cdots&u_{n-1}&&x&y&z\\\ 1&\cdots&1&|&k&l&m\\\
0&\cdots&0&|&1&1&1\end{pmatrix}\quad\left(\text{resp.}\begin{pmatrix}u_{0}&\cdots&u_{n-1}&&x&y\\\
1&\cdots&1&|&k&l\\\ 0&\cdots&0&|&1&1\end{pmatrix}\right).$

###### Remark 2.4

Let $P$ be as in Definition 2.1. When $a_{0}=\cdots=a_{m}=1$, $P$ is a
$\mathbb{P}^{m}$-bundle over $\mathbb{P}^{n}$. More precisely we have an
isomorphism

$P\cong\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}_{\mathbb{P}^{n}}(-\lambda_{0})\oplus\cdots\oplus\mathcal{O}_{\mathbb{P}^{n}}(-\lambda_{m})).$

Here, for a vector bundle $\mathcal{E}$ over $\mathbb{P}^{n}$,
$\mathbb{P}(\mathcal{E})=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{E})$ denotes the
projective bundle of one-dimensional quotients of $\mathcal{E}$. Moreover, via
the above isomorphism, the pullback of a hyperplane on $\mathbb{P}^{n-1}$ and
the tautological divisor on $\mathbb{P}(\mathcal{E})$ 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 $\mathbb{C}$. 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 $\mathbb{Q}$-factorial variety of dimension
$n$. We say that a morphism $\pi\colon X\to\mathbb{P}^{n-1}$ is a conic bundle
(over $\mathbb{P}^{n-1}$) if it is a Mori fibre space, that is, $X$ has only
terminal singularities, $\pi$ has connected fibres, $-K_{X}$ is $\pi$-ample
and $\rho(X)=2$, where $\rho(X)$ denotes the rank of the Picard group. An
embedded conic bundle $\pi\colon X\to\mathbb{P}^{n-1}$ is a conic bundle such
that $X$ is embedded in a projective bundle
$\mathbb{P}(\mathcal{E}):=\mathbb{P}_{\mathbb{P}^{n-1}}(\mathcal{E})$ as a
member of $|dF+2D|$ for some splitting vector bundle $\mathcal{E}$ of rank $3$
on $\mathbb{P}^{n-1}$ and $d\in\mathbb{Z}$, and $\pi$ coincides with the
restriction of $\Pi\colon\mathbb{P}(\mathcal{E})\to\mathbb{P}^{n-1}$ to $X$.
Here $F$ and $D$ denote the pullback of a hyperplane on $\mathbb{P}^{n-1}$ and
the tautological class $D$ on $\mathbb{P}(\mathcal{E})$, respectively.

In the following let $\mathcal{E}$ be a splitting vector bundle of rank $3$ on
$\mathbb{P}^{n-1}$ and $X\in|dF+2D|$ be a general member. We denote by
$\pi\colon X\to\mathbb{P}^{n-1}$ the restriction of
$\Pi\colon\mathbb{P}(\mathcal{E})\to\mathbb{P}^{n-1}$ to $X$. Without loss of
generality we may assume that

$\mathcal{E}\cong\mathcal{O}_{\mathbb{P}^{n-1}}(-k)\oplus\mathcal{O}_{\mathbb{P}^{n-1}}(-l)\oplus\mathcal{O}_{\mathbb{P}^{n-1}}(-m)$

for some $k\leq l\leq m$. Then, by Remark 2.4, we have
$\mathbb{P}(\mathcal{E})\cong P_{n}(k,l,m)$ and the linear system $|dF+2D|$ on
$\mathbb{P}_{\mathbb{P}^{n-1}}(\mathcal{E})$ corresponds to
$|\mathcal{O}_{P_{n}(k,l,m)}(d,2)|$. Here we do not assume that $\pi\colon
X\to\mathbb{P}^{n-1}$ is a conic bundle. We study conditions on $k,l,m$ and
$d$ that make $\pi\colon X\to\mathbb{P}^{n-1}$ a conic bundle.

###### Lemma 2.6

Let $k,l,m,d$ be integers such that $k\leq l\leq m$. Set $P=P_{n}(k,l,m)$ and
let $X$ be a general member of $|\mathcal{O}_{P}(d,2)|$.

  * (1)

$X$ is smooth if and only if $d\geq 2m$, $d=l+m$, or $d=k+m$.

  * (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 $d\geq 2m$. Then $|\mathcal{O}_{P}(d,2)|$ is base point free and
its general member $X$ is smooth. In the following we assume that $2m>d\geq
k+m$. Suppose that $2m>d>l+m$. Then $X$ is defined in $P$ by

$ax^{2}+by^{2}+fxy+gxz+hyz=0,$

where $a,b,f,g,h\in\mathbb{C}[u]$. We have $\deg h=d-(l+m)>0$ and $\deg
g=d-(k+m)>0$. Then $X$ is singular along $(x=y=g=h=0)\neq\emptyset$. The
singular locus is of codimension $3$ in $X$. Since $X$ is general, the
hypersurfaces in $\mathbb{P}^{n-1}$ defined by $g=0$ and $h=0$ are both
nonsingular and intersect transversally. It is then straightforward to check
that the blowup $\sigma\colon X^{\prime}\to X$ along the singular locus is a
resolution and we have $K_{X^{\prime}}=\sigma^{*}K_{X}+E$, where $E$ is the
exceptional divisor. Thus $X$ has terminal singularities. Suppose that
$2m>d=l+m$. Then $X$ is defined in $P$ by

$ax^{2}+by^{2}+fxy+gxz+yz=0.$

Replacing $y$ and $z$ suitably, we can eliminate the terms $by^{2},fxy$ and
$gzx$, that is, $X$ is defined by

$ax^{2}+yz=0.$

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

$ax^{2}+by^{2}+fxy+gxz=0.$

We have $\deg g=d-(k+m)>0$. Then $X$ is singular along
$(x=y=g=0)\neq\emptyset$, 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

$ax^{2}+by^{2}+fxy+xz=0.$

Replacing $z$ suitably, we may assume that $X$ is defined by

$by^{2}+zx=0.$

It is easy to see that $X$ is smooth. Finally suppose that $k+m>d$. Then $X$
is defined in $P$ by

$ax^{2}+by^{2}+fxy=0,$

where $a,b,f\in\mathbb{C}[u]$. In this case $X$ is singular along the divisor
$(x=y=0)\subset X$. Thus $X$ is not normal. The above arguments prove (1), (2)
and (3). $\Box$

###### Lemma 2.7

In the same setting as in _Lemma 2.6_, suppose that either $d=l+m$ or $d=k+m$.
Then the variety $X$ is rational. Moreover we have $\rho(X)\geq 3$ unless
$k=l=m$.

  * Proof.

Suppose that $d=l+m$, which implies $2m\geq d=l+m$. We claim that $X$ is
defined by an equation of the form $ax^{2}+yz=0$, where $a\in\mathbb{C}[u]$.
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

$ax^{2}+y^{2}+z^{2}+fxy+gxz+\alpha yz=0,$

where $\alpha\in\mathbb{C}$ and $a,f,g\in\mathbb{C}[u]$. Replacing $y$ and
$z$, the above equation can be transformed into $ax^{2}+yz=0$ and the claim is
proved. We consider the projection $X\dashrightarrow Q:=Q_{n}(k,l)$ Note that
$Q\cong\mathbb{P}(\mathcal{O}(-k)\oplus\mathcal{O}(-l))$. Then the projection
is birational, hence $X$ is rational. The projection $X\dashrightarrow Q$ is
defined outside $(x=y=0)\subset X$. Let $\mathsf{p}\in(x=y=0)$ be a point.
Then $z$ does not vanish at $\mathsf{p}$ and we have

$y=\frac{yz}{z}=-\frac{ax^{2}}{z}.$

From this we deduce that $X\dashrightarrow Q$ is everywhere defined. Now we
assume that either $k\neq l$ or $l\neq m$. Then $\deg a=d-2k=l+m-k>0$. We see
that $(y=a=0)\subset X$ is a divisor and it is contracted by $X\to Q$ to a
codimension $2$ subset of $Q$. This shows $\rho(X)\geq 3$. Next, suppose that
$d=k+m$. Note that $l+m\geq d$. 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. $\Box$

###### Lemma 2.8

In the same setting as in _Lemma 2.6_, $\pi\colon X\to\mathbb{P}^{n-1}$ is a
nonsingular conic bundle if and only if one of the following holds:

  * (1)

$d>2m$,

  * (2)

$d=2m$ and $m>l$, or

  * (3)

$d=2m=2l=2k$.

  * Proof.

This follows from Lemmas 2.6 and 2.7. $\Box$

###### Proposition 2.9

Let $X$ be an embedded conic bundle over $\mathbb{P}^{n-1}$. If $X$ is general
$($in the linear system$)$ and singular, then $X$ is rational.

  * Proof.

We may assume that $X\in|\mathcal{O}_{P}(d,2)|$, where $P=P_{n}(k,l,m)$, for
some $k\leq l\leq m$. By Lemma 2.6, we have $2m>d\geq k+m$. Then a general
member $X$ is defined by an equation of the form

$ax^{2}+by^{2}+fxy+gxz+hyz=0,$

where $a,b,f,g,h\in\mathbb{C}[u]$. Here, note that, if for example $l+m>d$,
then we know that the term $hyz$ does not appear in the equation. The
inequality $d\geq k+m$ implies that $g\neq 0$ since $X$ is general. Let
$P\dashrightarrow Q=Q_{n}(k,l)$ be the natural projection. Now we can write
the defining equation as

$z(gx+hy)+ax^{2}+by^{2}+fxy=0,$

which implies that the restriction $X\dashrightarrow Q$ is birational.
Therefore $X$ is rational. $\Box$

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 $(\lambda,\mu,\nu)$ be a triplet of integers $\lambda,\mu,\nu$. We say
that $\pi\colon X\to\mathbb{P}^{n-1}$ (or $X$) is of type $[\lambda,\mu,\nu]$
if $X$ belongs to $|\mathcal{O}_{P}(\lambda+\mu+\nu,2)|$, where
$P=P_{n}(\lambda,\mu,\nu)$, and $\pi$ coincides with the restriction of
$P\to\mathbb{P}^{n-1}$ to $X$.

###### Proposition 2.11

Let $\pi\colon X\to\mathbb{P}^{n-1}$ be a nonsingular embedded conic bundle.
Then $X$ is either of type $[\lambda,\mu,\nu]$ for some $\lambda,\mu,\nu$ such
that $0<\lambda\leq\mu\leq\nu\leq\lambda+\mu$ or of type $[0,0,0]$.

  * Proof.

We may assume that $X$ belongs to $|\mathcal{O}_{P_{n}(k,l,m)}(d,2)|$ for some
$k\leq l\leq m$ and $d$. Since the family $X$ is non-singular, we have $d\geq
2m$ by Lemma 2.8 and $X$ is defined in $P_{n}(k,l,m)$ by an equation of the
form

$ax^{2}+by^{2}+cz^{2}+fxy+gxz+hyz=0,$

where $a,b,c,f,g,h\in\mathbb{C}[u]$. We set $\alpha=\deg a,\beta=\deg
b,\gamma=\deg c,\lambda=\deg h,\mu=\deg g$ and $\nu=\deg f$. By comparing the
weights, we have

$\alpha+2k=\beta+2l=\gamma+2m=\nu+k+l=\mu+k+m=\lambda+l+m.$

Now we have

$P_{n}(k,l,m)\cong P_{n}(k+(\nu-m),l+(\nu-m),m+(\nu-m))\cong
P_{n}(\lambda,\mu,\nu)=:P,$

and the linear system $|\mathcal{O}_{P_{n}(k,l,m)}(d,2)|$ is identified with
$|\mathcal{O}_{P}(\lambda+\mu+\nu,2)|$. Thus $X$ is of type
$[\lambda,\mu,\nu]$. By applying Lemma 2.8 for $k=\lambda,l=\mu,m=\nu$ and
$d=\lambda+\mu+\nu$, we get the desired result. $\Box$

###### Remark 2.12

In the language of [1, Definition 3.1], a conic bundle $\pi\colon
X\to\mathbb{P}^{n-1}$ of type $[\lambda,\mu,\nu]$ with
$\lambda\leq\mu\leq\nu\leq\lambda+\mu$ is a conic bundle of graded-free type
over $\mathbb{P}^{n-1}$ corresponding to the triplet
$(-\lambda+\mu+\nu,\lambda-\mu+\nu,\lambda+\mu-\nu)$.

##  3\. Stable non-rationality

In this section we study stable (non-)rationality of nonsingular embedded
conic bundles $\pi\colon X\to\mathbb{P}^{n-1}$. By Proposition 2.11, such a
conic bundle is of type $[\lambda,\mu,\nu]$, where either
$0<\lambda\leq\mu\leq\nu\leq\lambda+\mu$ or $\lambda=\mu=\nu=0$. In case $X$
is of type $[0,0,0]$, then $X\cong\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ and it
is obviously rational. We consider the remaining cases and thus we assume that

$0<\lambda\leq\mu\leq\nu\leq\lambda+\mu$

throughout this section. In addition we assume $\nu\geq 3$ throughout. We set
$P=P_{n}(\lambda,\mu,\nu)$, $\delta=\lambda+\mu+\nu$, and consider special
members $X\in|\mathcal{O}_{P}(\delta,2)|$ defined in $P$ by an equation of the
form

$ax^{2}+by^{2}+cz^{2}+fxy=0,$ (1)

where $a,b,c,f$ are general polynomials in variables $u_{0},\dots,u_{n-1}$.
Recall that $\nu=\deg f$ and $\deg a=-\lambda+\mu+\nu$, $\deg
b=\lambda-\mu+\nu$ and $\deg c=\lambda+\mu-\nu$.

###### Remark 3.1

By the assumptions on $\lambda,\mu,\nu$, we have $\deg a=-\lambda+\mu+\nu\geq
3$, $\deg b=\lambda-\mu+\nu\geq 1$, $\deg c=\lambda+\mu-\nu\geq 0$ and $\deg
f=\nu\geq 3$.

###### 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 $|\mathcal{O}_{P}(\delta,2)|$ on the smooth variety $P$. Thus, by the
Bertini theorem, a general $X$ is smooth. $\Box$

We use universal $\operatorname{CH}_{0}$-triviality to test stable rationality
of varieties.

###### Definition 3.3

Let $V$ be a projective variety defined over a field $k$. We denote by
$\operatorname{CH}_{0}(V)$ the Chow group of $0$-cycles on $V$. We say that
$V$ is universally $\operatorname{CH}_{0}$-trivial if for any field $F$
containing $k$, the degree map $\operatorname{CH}_{0}(V_{F})\to\mathbb{Z}$ is
an isomorphism. A projective morphism $\varphi\colon W\to V$ defined over $k$
is universally $\operatorname{CH}_{0}$-trivial if for any field containing
$k$, the push-forward map
$\varphi_{*}\colon\operatorname{CH}_{0}(W_{F})\to\operatorname{CH}_{0}(V_{F})$
is an isomorphism.

In the rest of this section we work over an algebraically closed field $\Bbbk$
of characteristic $2$. Let $R$ be the $\mathbb{P}(1,1,2)$-bundle over
$\mathbb{P}^{n-1}$ defined by

$\begin{pmatrix}u_{0}&u_{1}&\cdots&u_{n-1}&&x&y&\bar{z}\\\
1&1&\cdots&1&|&\lambda&\mu&2\nu\\\ 0&0&\cdots&0&|&1&1&2\end{pmatrix}$

and let $Z\subset R$ be the hypersurface defined by

$ax^{2}+by^{2}+c\bar{z}+fxy=0.$

We have a natural morphism $P\to R$ which is a (purely inseparable) double
cover branched along $(\bar{z}=0)\subset R$. The image of $X$ under $P\to R$
is the hypersurface $Z\subset R$. Let $\tau\colon X\to Z$ be the induced
morphism, which is a double cover branched along the divisor cut out on $Z$ by
$\bar{z}=0$. We set $\mathcal{L}=\mathcal{O}_{Z}(\nu,1)$. Then $\bar{z}$ is a
global section of $\mathcal{L}^{2}$, and over the non-singular locus of $Z$,
$\tau$ is the double cover obtained by taking the roots of $\bar{z}\in
H^{0}(Z,\mathcal{L}^{2})$ 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
$\operatorname{CH}_{0}$-trivial resolution $\varphi\colon Y\to X$ such that
$H^{0}(Y,\Omega_{Y}^{n-1})\neq 0$ under some conditions on $\lambda,\mu,\nu$.
The latter implies that $Y$ is not universally $\operatorname{CH}_{0}$-trivial
by [18, Lemma 2.2].

###  3.A. Singularities

Recall that the ground field $\Bbbk$ is an algebraically closed field of
characteristic $2$ and $X$ is a hypersurface in $P=P_{n}(\lambda,\mu,\nu)$
defined by

$ax^{2}+by^{2}+cz^{2}+fxy=0$

for general $a,b,c,f\in\Bbbk[u_{0},\dots,u_{n-1}]$. Similarly $Z$ is the
hypersurface in $R$ defined by

$ax^{2}+by^{2}+c\bar{z}+fxy=0.$

We set

$\Xi=(x=y=0)\subset R,\ \Xi_{Z}=\Xi\cap Z=(x=y=c=0),$

and $R^{\circ}=R\setminus\Xi$, $Z^{\circ}=Z\setminus\Xi_{Z}$. In order to
analyze singularities of $Z^{\circ}\subset R^{\circ}$, we consider standard
affine charts of $R^{\circ}$. For $i=0,\dots,n-1$ and a coordinate
$w\in\\{x,y\\}$, we set $U_{u_{i},w}=(u_{i}\neq 0)\cap(w\neq 0)\subset
R^{\circ}$. We have

$R^{\circ}=\bigcup_{i\in\\{0,\dots,n-1\\},w\in\\{x,y\\}}U_{u_{i},w}.$

We remark that $U_{u_{i},w}$ is an affine $(n+1)$-space and that the
restriction of the sections

$\\{u_{0},\dots,u_{n-1},x,y,\bar{z}\\}\setminus\\{u_{i},w\\}$

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 set

$\tilde{u}_{i}=u_{i}/u_{0},\ \tilde{y}=y/xu_{0}^{\mu-\lambda},\
\tilde{z}=\bar{z}/x^{2}u_{0}^{\nu-2\lambda}.$

Then $U_{u_{0},w}$ is an affine $(n+1)$-space with affine coordinates
$\tilde{u}_{1},\dots,\tilde{u}_{n-1},\tilde{y},\tilde{z}$. By a slight abuse
of notation, the affine coordinates
$\tilde{u}_{1},\dots,\tilde{u}_{n-1},\tilde{y},\tilde{z}$ are simply denoted
by $u_{1},\dots,u_{n-1},y,\bar{z}$.

###### Lemma 3.4

$Z^{\circ}$ is smooth.

  * Proof.

If $\deg c=0$, then $c$ is a non-zero constant and thus $\Xi_{Z}=\emptyset$.
In this case $Z=Z^{\circ}$ is a $\mathbb{P}^{1}$ bundle over
$\mathbb{P}^{n-1}$ and it is smooth. In the following we assume that $\deg
c>0$ and set

$U_{x}=(x\neq 0),\ U_{y}=(y\neq 0)\subset R,$

so that $R^{\circ}=U_{x}\cup U_{y}$. We will show that for any point
$\mathsf{q}\in R^{\circ}$, the condition that $Z^{\circ}$ is singular at
$\mathsf{q}\in Z$ imposes $n+2$ independent conditions on $a,b,c,f$. Then the
assertion will follow by a dimension count argument since $\dim
R^{\circ}=n+1$. We note that $\deg b=\lambda-\mu+\nu\geq 1$, $\deg
c=\lambda+\mu-\nu\geq\lambda\geq 3$ and $\deg f=\lambda\geq 3$ by Remark 3.1.
Let $\mathsf{q}\in U_{x}$. Replacing coordinates, we may assume
$\mathsf{q}=(1\\!:\\!0\\!:\\!\cdots\\!:\\!0;1\\!:\\!0\\!:\\!0)$. Then
$U_{u_{0},x}\subset Q$ is an affine space with coordinates
$u_{1},\dots,u_{n-1},y,\bar{z}$ and $Z\cap U_{u_{0},z}$ is defined by

$\tilde{a}+\tilde{b}y^{2}+\tilde{c}\bar{z}+\tilde{f}y=0,$

where we set $\tilde{h}=h(1,u_{1},\dots,u_{n-1})$ for a polynomial
$h(u_{0},\dots,u_{n-1})$. Note that $\mathsf{q}$ corresponds to the origin.
The variety $Z^{\circ}$ is singular at $\mathsf{q}$ if and only if
$\tilde{a},\tilde{c},\tilde{f}$ vanish at $\mathsf{q}$ and the linear part of
$\tilde{a}$ is zero. This imposes $n+2$ independent conditions since $\deg
a>0$ and $\deg c,\deg f\geq 0$ (cf. Remark 3.1). Suppose that $\mathsf{q}\in
U_{y}$. Replacing coordinates, we may assume
$\mathsf{q}=(1\\!:\\!0\\!:\\!\cdots\\!:\\!0;0\\!:\\!1\\!:\\!0)$. Then
$U_{u_{0},y}\subset Q$ is an affine space with coordinates
$u_{0},\dots,u_{n-1},x,\bar{z}$ and $Z\cap U_{u_{0},y}$ is defined by

$\tilde{a}x^{2}+\tilde{b}+\tilde{c}\bar{z}+\tilde{f}x=0.$

The variety $Z^{\circ}$ is singular at $\mathsf{q}$ if and only if
$\tilde{b},\tilde{c},\tilde{f}$ vanish at $\mathsf{q}$ and the linear part of
$\tilde{b}$ is zero. The latter imposes $n+2$ independent conditions since
$\deg b>0$ and $\deg c,\deg f\geq 0$ (cf. Remark 3.1), and the proof is
complete. $\Box$

We set $X^{\circ}=\pi^{-1}(Z^{\circ})$.

###### Lemma 3.5

$X$ is smooth along $X\setminus X^{\circ}$.

  * Proof.

Note that $X\setminus X^{\circ}=X\cap(x=y=0)$. For a point $\mathsf{p}\in
X\setminus X^{\circ}$, $X$ is smooth at $\mathsf{p}$ if and only if the
hypersurface $(c=0)\subset\mathbb{P}^{n-1}$ is smooth at the image of
$\mathsf{p}$ under $X\to\mathbb{P}^{n-1}$. Clearly the hypersurface
$(c=0)\subset\mathbb{P}^{n-1}$ is smooth since $c$ is general, and the
assertion follows. $\Box$

###  3.B. Analysis of critical points

We set $\mathcal{L}^{\circ}=\mathcal{L}|_{Z^{\circ}}$, where we recall
$\mathcal{L}=\mathcal{O}_{Z}(\nu,1)$. By Lemma 3.4, $Z^{\circ}$ is non-
singular and by Kollár’s result [12, V.5] there exists an invertible sheaf
$\mathcal{Q}^{\circ}$ on $Z^{\circ}$ such that
$\mathcal{M}^{\circ}:=\tau^{*}\mathcal{Q}^{\circ}\subset(\Omega_{X^{\circ}}^{n-1})^{\vee\vee}$,
where $\vee\vee$ denotes the double dual. Let $\mathcal{M}$ be the push-
forward of the invertible sheaf $\mathcal{M}^{\circ}$ via the open immersion
$X^{\circ}\hookrightarrow X$. By Lemma 3.5, $\mathcal{M}$ is an invertible
sheaf on $X$.

###### Definition 3.6

Let $V$ be a nonsingular variety of dimension $n$ defined over an
algebraically closed field $\Bbbk$ of characteristic $2$, $\mathcal{N}$ an
invertible sheaf on $V$ and $s\in H^{0}(V,\mathcal{N}^{2})$ a section. Let
$\mathsf{p}\in V$ be a point, $\xi$ a local generator of $\mathcal{N}$ at
$\mathsf{p}$ and $s=f(x_{1},\dots,x_{n})\xi^{2}$ a local description of $s$
with respect to local coordinates $x_{1},\dots,x_{n}$ of $V$ at $\mathsf{p}$.
We say that $s$ has a critical point at $\mathsf{p}$ if the linear term of $f$
is zero. We say that $s$ has an admissible critical point at $\mathsf{p}$ if
for a suitable choice of coordinates $x_{1},\dots,x_{n}$,

$f=\begin{cases}\alpha+x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{n-1}x_{n}+g,&\text{if
$n$ is even},\\\ \alpha+\beta
x_{1}^{2}+x_{2}x_{3}+\cdots+x_{n-1}x_{n}+g,&\text{if $n$ is odd},\end{cases}$

where $\alpha,\beta\in\Bbbk$,
$g=g(x_{1},\dots,x_{n})\in(x_{1},\dots,x_{n})^{3}$ and, in case $n$ is odd,
the coefficient of $x_{1}^{3}$ in $g$ is nonzero.

###### Lemma 3.7

The section $\bar{z}\in H^{0}(Z,\mathcal{L}^{2})$ has only admissible critical
points on $Z^{\circ}$.

  * Proof.

We choose and fix a general $c\in\Bbbk[u]$ so that the hypersurface
$(c=0)\subset\mathbb{P}^{n-1}$ is non-singular. Clearly $\bar{z}$ does not
have a critical point on $(c=0)\subset Z^{\circ}$. On $Z^{\circ}\cap(c\neq
0)$, the section $c$ is invertible and thus the section $\bar{z}$ has an
admissible critical point if and only if the section

$s:=c(ax^{2}+by^{2}+fxy)\ (=c^{2}\bar{z})$

has an admissible critical point. It is then enough to show that the section
$s$, viewed as a section on $Q=Q_{n}(\lambda,\mu)$, has only admissible
critical points on $U_{c}=(c\neq 0)\subset Q$ for general $a,b$ and $f$. We
set $U_{x}=(x\neq 0)\subset Q$ and $\Pi_{y}=(x=0)\cap(y\neq 0)\subset Q$ so
that $Q=U_{x}\cup\Pi_{y}$. We first show that $s$ does not have a critical
point on $\Pi_{y}\cap U_{c}$. Let $\mathsf{p}\in\Pi_{y}\cap U_{c}$ be a point.
We may assume $\mathsf{p}=(1\\!:\\!0\\!:\\!\cdots\\!:\\!0;0\\!:\\!1)$. We work
on the open subset $U_{u_{0},y}=(u_{0}\neq 0)\cap(y\neq 0)\subset Q$ which is
the affine space with coordinates $u_{1},\dots,u_{n-1}$ and $x$. For
$e=e(u_{0},\dots,u_{n-1})$, we set $\tilde{e}=e(1,u_{1},\dots,u_{n-1})$.
Moreover we denote by $\tilde{e}_{i}$ the degree $i$ part of $\tilde{e}$. Then
the restriction of $s$ to $U_{u_{0},y}$ is
$\tilde{c}(\tilde{a}x^{2}+\tilde{b}+\tilde{f}x)$ and the point $\mathsf{p}$
corresponds to the origin. Then $s$ has a critical point at $\mathsf{p}$ if
and only if

$\tilde{c}_{0}(\tilde{b}_{1}+\tilde{f}_{0}x)+\tilde{c}_{1}\tilde{b}_{0}=0.$

Note that $\tilde{c}_{0}\neq 0$. Since $\deg b\geq 1$, this imposes $n$
independent conditions on $a,b,f$. Thus, for any point $\mathsf{p}\in\Pi_{y}$,
$n$ conditions are imposed in order for $s$ to have a critical point at
$\mathsf{p}$. By counting dimensions we conclude that $s$ does not have a
critical point on $\Pi_{y}\cap U_{c}$ since $\dim\Pi_{y}=n-1$. Let
$\mathsf{p}\in U_{x}\cap U_{c}$ be a point. We may assume
$\mathsf{p}=(1\\!:\\!0\\!:\\!\cdots\\!:\\!0;1\\!:\\!0)$. We work on the open
subset $U_{u_{0},x}=(u_{0}\neq 0)\cap(x\neq 0)\subset R$ which is the affine
space with coordinates $u_{1},\dots,u_{n-1}$ and $y$. We have
$s|_{U_{u_{0},y}}=\tilde{c}(\tilde{a}+\tilde{b}y^{2}+\tilde{f}y)$. Let
$\ell,q$ and $h$ be the linear, quadratic and cubic parts of
$s|_{U_{u_{0},y}}$, respectively. We have

$\ell=\tilde{c}_{0}(\tilde{a}_{1}+\tilde{f}_{0}y)+\tilde{c}_{1}\tilde{a}_{0}.$

Since $\deg a\geq 1$, $n$ conditions are imposed in order for $s$ to have a
critical point at $\mathsf{p}$. It remains to show the existence of a section
$s=c(ax^{2}+by^{2}+fxy)$ which has an admissible critical point at
$\mathsf{p}$. Now suppose that $s$ has a critical point at $\mathsf{p}$, that
is, $\ell=0$. This implies that $\tilde{f}_{0}=0$ and
$\tilde{a}_{1}=\tilde{a}_{0}\tilde{c}_{1}/\tilde{c}_{0}$. Then, for the
quadratic and cubic parts, we have

$\begin{split}q&=\tilde{c}_{0}(\tilde{a}_{2}+\tilde{b}_{0}y^{2}+\tilde{f}_{1}y)+\frac{\tilde{a}_{0}\tilde{c}_{1}^{2}}{\tilde{c}_{0}}+\tilde{c}_{2}\tilde{a}_{0},\\\
h&=\tilde{c}_{0}(\tilde{a}_{3}+\tilde{b}_{1}y^{2}+\tilde{f}_{2}y)+\cdots.\end{split}$

Since $\deg a\geq 3$ and $\deg f\geq 3$, we can choose $a,b,f$ so that

$q=\begin{cases}yu_{1}+u_{2}u_{3}+u_{4}u_{5}+\cdots+u_{n-2}u_{n-1},&\text{if
$n$ is even},\\\
yu_{1}+u_{2}u_{3}+u_{4}u_{5}+\cdots+u_{n-3}u_{n-2}+u_{n-1}^{2},&\text{if $n$
is odd}.\end{cases}$

In case $n$ is even, the section $s$ has a nondegenerate critical point at
$\mathsf{p}$ and we are done. Suppose that $n$ is odd. Since $\deg a\geq 3$,
then we can choose $a,b,f$ so that $q$ is as above and the coefficient of
$u_{n-1}^{3}$ in $h$ is non-zero. For this choice of $a,b,c,f$, the section
$s$ has an admissible critical point at $\mathsf{p}$ and the proof is
completed by the dimension counting argument. $\Box$

###### Proposition 3.8

Let the notation and assumption as above. Assume in addition that $\nu\geq n$.
Then there exists a universally $\operatorname{CH}_{0}$-trivial resolution
$\varphi\colon Y\to X$ of singularities such that
$H^{0}(Y,\Omega_{Y}^{n-1})\neq 0$. In particular $Y$ is not universally
$\operatorname{CH}_{0}$-trivial.

  * Proof.

By [15, Proposition 4.1] or [6], if the singularities of $X$ correspond to
admissible critical points of the section $\bar{z}$, then there exists a
universally $\operatorname{CH}_{0}$-trivial resolution $\varphi\colon Y\to X$
such that $\varphi^{*}\mathcal{M}\hookrightarrow\Omega_{Y}^{n-1}$ (in fact,
$\varphi$ is just the composite of blowups at each (isolated) singular point).
Thus, by Lemma 3.7, $X$ admits such a resolution. The branch divisor
$(\bar{z}=0)$ is clearly reduced and, by [12, Lemma V.5.9], we have an
isomorphism

$\mathcal{M}^{\circ}\cong\tau^{*}(\omega_{Z^{\circ}}\otimes{\mathcal{L}^{\circ}}^{2})\cong\mathcal{O}_{X^{\circ}}(\nu-n,0),$

so that $\mathcal{M}\cong\mathcal{O}_{X}(\nu-n,0)$. By the assumption we have
$\nu-n\geq 0$, which implies $H^{0}(X,\mathcal{M})\neq 0$. Thus
$H^{0}(Y,\Omega_{Y}^{n-1})\neq 0$ and by [18, Lemma 2.2], $Y$ is not
universally $\operatorname{CH}_{0}$-trivial. $\Box$

###  3.C. Proof of theorems and corollaries

###### Theorem 3.9

Suppose that the ground field is $\mathbb{C}$ and let $(\lambda,\mu,\nu)$ be a
triplet of integers such that $0<\lambda\leq\mu\leq\nu\leq\lambda+\mu$. If in
addition $\nu\geq n$, then a very general embedded conic bundle $\pi\colon
X\to\mathbb{P}^{n-1}$ of type $[\lambda,\mu,\nu]$ is not stably rational.

  * Proof.

For a field (or more generally a ring) $K$, we denote by $P_{K}$ the
$\mathbb{P}^{2}$-bundle $P_{n}(\lambda,\mu,\nu)$ over $\mathbb{P}^{n-1}$
defined over $K$. Let $\Bbbk$ be an algebraically closed field of
characteristic $2$ and let $X\to\mathbb{P}^{n-1}$ be a very general
hypersurface in $P_{\Bbbk}$ defined by an equation of the form (1). We take a
mixed characteristic discrete valuation ring $A$ whose residue field is
$\Bbbk$, for example the Witt ring, and then we lift $X$ to a hypersurface
$\mathcal{X}$ of $P_{A}$ defined by an equation of the form (1). We choose and
fix an embedding of the quotient field of $A$ into $\mathbb{C}$ and set
$V=\mathcal{X}\times_{A}\mathbb{C}$. Then $V$ is a very general hypersurface
of $P_{\mathbb{C}}$ defined by an equation of the form (1). By Proposition
3.8, we can apply the specialization theorem [5, Théorème 1.14] and conclude
that $V$ is not universally $\operatorname{CH}_{0}$-trivial. Note that $V$ is
nonsingular by Lemma 3.2. Note also that $V$ is not a very general conic
bundle of type $[\lambda,\mu,\nu]$. However a very general conic bundle of
type $[\lambda,\mu,\nu]$ degenerates (over a complex curve) to $V$, hence the
assertion follows from the specialization argument [19, Theorem 2.1] (or by
[14, Theorem 4.2.10]). $\Box$

Now we can prove the main theorem and corollaries in Section 1.

  * Proof of _Theorem 1.1_.

The assertion (1) follows from Proposition 2.9. Let $\pi\colon
X\to\mathbb{P}^{n-1}$ be a non-singular embedded conic bundles over
$\mathbb{P}^{n-1}$. By Proposition 2.11, we may assume that it is of type
$[\lambda,\mu,\nu]$, where either $0<\lambda\leq\mu\leq\nu\leq\lambda+\mu$ or
$\lambda=\mu=\nu=0$. By adjunction we have
$\mathcal{O}_{X}(-K_{X})\cong\mathcal{O}_{X}(n,1)$. The complete linear system
$|\mathcal{O}_{P}(n,1)|$, where $P=P_{n}(\lambda,\mu,\nu)$, is base point free
if and only if $n\geq\nu$. This shows that $\mathcal{O}_{P}(n,1)$, and hence
$\mathcal{O}_{X}(n,1)$, is ample if $n<\nu$. Since $-K_{X}$ is not ample by
assumption, we have $n\geq\nu$. Therefore (2) follows from Theorem 3.9. $\Box$

  * Proof of _Corollaries 1.2 and 1.3_.

Let $X$ be a very general hypersurface of bi-degree $(d,2)$ in
$\mathbb{P}^{n-1}\times\mathbb{P}^{2}$. Then
$\mathcal{O}_{X}(-K_{X})\cong\mathcal{O}_{X}(n-d,1)$. By assumption $d\geq n$
and this implies that $-K_{X}$ is not ample. Thus $X$ is not stably rational
by Theorem 1.1. Let $X$ be a double cover of
$\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ branched along a very general divisor
of bi-degree $(2d,2)$. Then $X$ is a very general member of
$|\mathcal{O}_{P}(2d,2)|$, where $P=P_{n}(0,0,d)$, and hence it is of type
$[d,d,2d]$. By the assumption we have $2d\geq n$. Thus $X$ is not stably
rational by Theorem 3.9 $\Box$

  * Proof of _Corollary 1.5_.

Let $\pi\colon X\to\mathbb{P}^{n-1}$ be as in Corollary 1.5. Then we may
assume that it is of type $[\lambda,\mu,\nu]$, where
$0<\lambda\leq\mu\leq\nu\leq\lambda+\nu$ or $\lambda=\mu=\nu=0$. The
discriminant divisor $\Delta$ is a hypersurface in $\mathbb{P}^{n-1}$ of
degree $\lambda+\mu+\nu$. The condition
$|3K_{\mathbb{P}^{n-1}}+\Delta|\neq\emptyset$ is equivalent to the condition
$\lambda+\mu+\nu\geq 3n$. The latter implies $\nu\geq n$ since
$\lambda\leq\mu\leq\nu$. Thus (1) follows from Theorem 3.9. Now suppose in
addition that $n\geq 7$ and $\pi\colon X\to\mathbb{P}^{n-1}$ is standard. Note
that $X$ is defined in $P_{n}(\lambda,\mu,\nu)$ by an equation of the form

$ax^{2}+by^{2}+cz^{2}+fxy+gxz+hyz=0,$

where $a,\dots,h\in\mathbb{C}[u]$. If $\deg c=\lambda+\mu-\nu>0$, then the
system of equations $a=b=\cdots=h=0$ has a non-trivial solution on
$\mathbb{P}^{n-1}$ since $n\geq 7$. This implies that $\pi$ cannot be flat, in
particular, not standard. Thus $\nu=\lambda+\mu$ and in this case the
condition
$|2K_{\mathbb{P}^{n-1}}+\Delta|=|\mathcal{O}_{\mathbb{P}^{n-1}}(2(\nu-n))|\neq\emptyset$
is equivalent to $\nu\geq n$ which implies stable non-rationality of $X$ again
by Theorem 3.9. This proves (2). $\Box$

## References

  * [1] A. Auel, C. Böhning, H.-C. Graf von Bothmer, and A. Pirutka, _Conic bundles with nontrivial unramified Brauer group over threefolds_ , preprint, 2016. arXiv:1610.04995
  * [2] C. Böhning and H.-C. Graf von Bothmer, _On stable rationality of some conic bundles and moduli spaces of Prym curves_ , Comment. Math. Helv. 93 (2018), no. 1, 133–155. MR-3777127
  * [3] H. Clemens and P. Griffiths, _The intermediate Jacobian of the cubic threefold_ , Ann. of Math. (2) 95 (1972), 281–356. MR-0302652
  * [4] J.-L. Colliot-Thélène and D. Coray, _L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques_ , Compositio Math. 39 (1979), no. 3, 301–332. MR-0550646
  * [5] J.-L. Colliot-Thélène and A. Pirutka, _Hypersurfaces quartiques de dimension $3$: non-rationalité stable_, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 371–397. MR-3481353
  * [6] J.-L. Colliot-Thélène and A. Pirutka, _Cyclic covers that are not stably rational_ , Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 35–48 (Russian). Translation in Izv. Math. 80 (2016), no. 4, 665–677. MR-3535357
  * [7] W. Fulton, _Intersection theory_. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2, Springer-Verlag, Berlin, 1998. MR-1644323
  * [8] B. Hassett, A. Kresch, and Y. Tschinkel, _Stable rationality and conic bundles_ , Math. Ann. 365 (2016), no. 3-4, 1201–1217. MR-3521088
  * [9] B. Hassett, A. Pirutka, and Y. Tschinkel, _Stable rationality of quadric surface bundles over surfaces_ , preprint, 2016. arXiv:1603.09262
  * [10] V. A. Iskovskikh, _On a rationality criterion for conic bundles_ , Mat. Sb. 187 (1996), no. 7, 75–92 (Russian). Translation in Sb. Math. 187 (1996), no. 7, 1021–1038. MR-1404188
  * [11] J. Kollár, _Nonrational hypersurfaces_ , J. Amer. Math. Soc. 8 (1995), no. 1, 241–249. MR-1273416
  * [12] J. Kollár, _Rational curves on algebraic varieties_ , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer-Verlag, Berlin, 1996. MR-1440180
  * [13] S. Mori and Y. Prokhorov, _On $\mathbb{Q}$-conic bundles_, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 315–369. MR-2426350
  * [14] J. Nicaise and E. Shinder, _The motivic nearby fiber and degeneration of stable rationality_ , preprint, 2017. arXiv:1708.02790
  * [15] T. Okada, _Stable rationality of cyclic covers of projective spaces_ , preprint, 2016. arXiv:1604.08417
  * [16] V. G. Sarkisov, _On conic bundle structures_ , Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 371–408, 432. MR-0651652
  * [17] V. V. Shokurov, _Prym varieties: theory and applications_ , Izv. Acad. SSSR Ser. Mat. 47 (1983), no. 4, 785–855 (Russian). Tranlation in Math. USSR-Izv. 23 (1984), no. 1, 83–147. MR-0712095
  * [18] B. Totaro, _Hypersurfaces that are not stably rational_ , J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. MR-3486175
  * [19] C. Voisin, _Unirational threefolds with no universal codimension $2$ cycle_, Invent. Math. 201 (2015), no. 1, 207–237. MR-3359052