# Conic bundle threefolds differing by a constant Brauer class and connections
to rationality

Sarah Frei  Department of Mathematics, Dartmouth College, 27 N. Main Street,
Hanover, NH 03755<EMAIL_ADDRESS>https://math.dartmouth.edu/~sfrei
,  Lena Ji  Department of Mathematics, University of Michigan, 530 Church
Street, Ann Arbor, MI 48109-1043<EMAIL_ADDRESS>https://www-
personal.umich.edu/~lenaji ,  Soumya Sankar  Mathematical Institute, Utrecht
University, Hans Freudenthal building, Budapest 6, 3584 CD Utrecht, The
Netherlands<EMAIL_ADDRESS>https://sites.google.com/site/soumya3sankar/ ,
Bianca Viray  University of Washington, Department of Mathematics, Box 354350,
Seattle, WA 98195, USA<EMAIL_ADDRESS>http://math.washington.edu/~bviray and
Isabel Vogt  Brown University, Department of Mathematics, Box 1917, 151 Thayer
Street, Providence, RI 02912, USA<EMAIL_ADDRESS>https://www.math.brown.edu/ivogt/

###### Abstract.

A double cover $Y$ of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ ramified over a
general $(2,2)$-divisor will have the structure of a geometrically standard
conic bundle ramified over a smooth plane quartic
$\Delta\subset{\mathbb{P}}^{2}$ via the second projection. These threefolds
are rational over algebraically closed fields, but over nonclosed fields,
including over ${\mathbb{R}}$, their rationality is an open problem. In this
paper, we characterize rationality over ${\mathbb{R}}$ when
$\Delta({\mathbb{R}})$ has at least two connected components (extending work
of M. Ji and the second author) and over local fields when all odd degree
fibers of the first projection have nonsquare discriminant.

We obtain these applications by proving general results comparing the conic
bundle structure on $Y$ with the conic bundle structure on a well-chosen
intersection of two quadrics. The difference between these two conic bundles
is encoded by a constant Brauer class, and we prove that this class measures a
certain failure of Galois descent for the codimension 2 Chow group of $Y$.

###### Key words and phrases:

Conic bundles, the Brauer group, rationality, intermediate Jacobians

###### 2020 Mathematics Subject Classification:

Primary: 14C25. Secondary: 14E08, 14G27, 14H40, 14K30.

##  1\. Introduction

For threefolds over algebraically closed fields, much is known about the
rationality problem (i.e., the property of being birational to projective
space), see e.g. [IskovskikhProkhorov]. However, the classical rationality
obstructions cannot always detect irrationality over nonclosed fields. In this
direction, Hassett–Tschinkel and Benoist–Wittenberg introduced a rationality
obstruction over nonclosed fields $k$ that considers restrictions on certain
torsors under the intermediate Jacobian [HT-intersection-quadrics, bw-ij, HT-
cycle], refining the classical Clemens–Griffiths intermediate Jacobian
obstruction [clemensgriffiths]. Subsequent works show that this refined
intermediate Jacobian torsor (IJT) obstruction is strong enough to
characterize rationality for many classes of Fano threefolds with $k$-points
[HT-intersection-quadrics, bw-ij, KP-Fano-3folds-rank1].

In an earlier paper [FJSVV], the authors studied the IJT obstruction for
geometrically standard conic bundle threefolds over ${\mathbb{P}}^{2}$ with
singular fibers over a smooth plane quartic $\Delta$, and proved that although
the IJT obstruction characterizes rationality for these degree $4$ conic
bundles over fields $k$ with $\operatorname{Br}k[2]=0$, the IJT obstruction
fails to characterize rationality over any subfield of ${\mathbb{R}}$. The
root cause of this discrepancy is that the intermediate Jacobian torsors of a
conic bundle $\pi\colon X\to{\mathbb{P}}^{2}$ depend only on the discriminant
cover $\tilde{\Delta}\to\Delta$ of $\pi$, whereas the isomorphism class of the
generic fiber $X_{\eta}$ depends on the discriminant cover _and_ the constant
$2$-torsion Brauer class given by a smooth fiber at any point
$w\in({\mathbb{P}}^{2}\smallsetminus\Delta)(k)$.

Thus, one may ask whether the vanishing of the IJT obstruction for a degree
$4$ conic bundle $X\to{\mathbb{P}}^{2}$ implies the rationality of _some_
degree $4$ conic bundle $X^{\prime}\to{\mathbb{P}}^{2}$ with the same
discriminant cover $\tilde{\Delta}/\Delta$ as $X$. We prove that this indeed
occurs when the so-called Prym curve111The intermediate Jacobian of $X$ is the
Prym variety of $\tilde{\Delta}/\Delta$, which, in the case $\deg\Delta=4$, is
isomorphic to the Jacobian of an explicit genus $2$ curve
$\Gamma_{\tilde{\Delta}/\Delta}$ with equation given in (2.4)
[FJSVV]*Definition 4.6. $\Gamma_{\tilde{\Delta}/\Delta}$ has a $k$-point (see
Corollary 3.2).

The question then becomes: what geometric information is captured by the
constant Brauer class $[X_{w}]-[X^{\prime}_{w}]$? The primary goal of this
paper is to answer this question for an important class of Fano threefolds
with a degree $4$ conic bundle structure. Let $\Delta\subset{\mathbb{P}}^{2}$
be a smooth plane quartic and let $\varpi\colon\tilde{\Delta}\to\Delta$ be a
smooth projective geometrically integral étale double cover. Then there exists
a unique double cover
$Y=Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$
branched over a $(2,2)$-divisor such that the second projection has the
structure of a (geometrically standard) conic bundle with discriminant cover
$\tilde{\Delta}\to\Delta$. The first projection $\pi_{1}$ realizes
$Y_{\tilde{\Delta}/\Delta}$ as a quadric surface fibration; in particular,
$Y_{\tilde{\Delta}/\Delta}$ is rational as soon as $\pi_{1}$ has a section.

######  Theorem 1.1.

Over a field $k$ of characteristic different from $2$, let
$\tilde{\Delta}/\Delta$, $Y_{\tilde{\Delta}/\Delta}$, and
$\Gamma_{\tilde{\Delta}/\Delta}$ be as above and set $X\colonequals
Y_{\tilde{\Delta}/\Delta}$. Assume that
$\Gamma_{\tilde{\Delta}/\Delta}(k)\neq\emptyset$ and that the IJT obstruction
vanishes. Then there is a conic bundle $\psi\colon
X^{\prime}\to{\mathbb{P}}^{2}$ with discriminant cover
$\tilde{\Delta}/\Delta$, such that

  1. (1)

The threefold $X^{\prime}$ is rational, and

  2. (2)

$[X_{\eta}]-[X^{\prime}_{\eta}]\in\operatorname{Br}k[2]$ is trivial if and
only if $\pi_{1}$ has a section.

Theorem 1.1 raises the question of whether conic bundles
$X\to{\mathbb{P}}^{2}$, $X^{\prime}\to{\mathbb{P}}^{2}$ that have the same
discriminant cover, but whose generic fibers are not isomorphic, can _both_ be
rational. In other words, can ${\mathbb{P}}^{3}$ have more than one
isomorphism class of conic bundle structure with the same discriminant cover?
If not, then Theorem 1.1 would imply that
$Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ is
rational if and only if $\pi_{1}$ has a section. In general, this question
seems difficult to answer, but we provide two contexts in which there is a
positive answer. First, in Proposition 2.1, we show that when $\tilde{\Delta}$
has index $1$, any two conic bundles with discriminant cover
$\tilde{\Delta}/\Delta$ necessarily have isomorphic generic fibers. When
$\Delta$ is a smooth plane quartic, to the best of our knowledge the condition
that $\tilde{\Delta}$ has index $1$ is the unique sufficient rationality
criteria that depends _only_ on the discriminant cover (see [FJSVV]*Section
8.1 and Corollary 2.2).

On the other hand, over ${\mathbb{R}}$, topological obstructions can shed
light on whether two conic bundles with the same discriminant cover can both
be rational. Exploiting this extra information, we extend the characterization
of rationality of real degree $4$ conic bundles in [JJ23] to cover one of the
two remaining isotopy classes of the quartic $\Delta({\mathbb{R}})$.

######  Theorem 1.2.

Let $k={\mathbb{R}}$ and and let $\tilde{\Delta}/\Delta$ be a geometrically
integral étale double cover of a smooth plane quartic. Assume that
$\Delta({\mathbb{R}})$ is not a single oval. Then $Y_{\tilde{\Delta}/\Delta}$
is rational if and only if $\pi_{1}$ has a section.

For further equivalent rationality conditions over ${\mathbb{R}}$, see Theorem
7.2 (which includes Theorem 1.2 as a special case).

The topological arguments over ${\mathbb{R}}$ do not extend to nonarchimedean
local fields. However, we can use our methods to show that if the Prym curve
$\Gamma_{\tilde{\Delta}/\Delta}$ has index $2$, then the IJT obstruction
characterizes rationality.

######  Theorem 1.3 (Corollary 5.4).

Let $k$ be a local field of characteristic different from $2$ and let
$\tilde{\Delta}/\Delta$ be a geometrically integral étale double cover of a
smooth plane quartic. If $\Gamma_{\tilde{\Delta}/\Delta}$ has index $2$, then
$Y_{\tilde{\Delta}/\Delta}$ is rational if and only if the IJT obstruction
vanishes.

###  1.1. Outline

We begin in Section 2 with a brief review of conic bundles, focusing on the
discriminant cover $\varpi\colon\tilde{\Delta}\to\Delta$ of a conic bundle and
the possible isomorphism classes of the generic fiber. In this section we
prove Proposition 2.1: if $\tilde{\Delta}$ has index $1$, then any two conic
bundles with discriminant cover $\tilde{\Delta}/\Delta$ necessarily have
isomorphic generic fibers. In Section 2.2 we review results from [FJSVV] that
rephrase the IJT obstruction for conic bundles whose discriminant cover is a
geometrically irreducible étale double cover of a smooth plane quartic in
terms of torsors under the Prym variety of $\tilde{\Delta}\to\Delta$; in doing
so, we also give equations for the Prym curve
$\Gamma_{\tilde{\Delta}/\Delta}$. In Section 3, we show that under the
assumption that the Prym curve has a rational point, there exists a conic
bundle with discriminant cover $\tilde{\Delta}\to\Delta$ that is the blow-up
along a smooth conic of an intersection of two quadrics. In Section 4, we
construct a conic bundle over one of the Prym torsors from Section 2.2.1 and
show that if the torsor is trivial, then this conic bundle is constant. This
construction is used in Section 5 to study the obstruction of a section to
$\pi_{1}\colon Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}$. These
ingredients are combined in Section 6 to prove Theorem 1.1. We end in Section
7 by specializing to the case $k={\mathbb{R}}$ and proving Theorem 1.2.

## Acknowledgements

The authors thank David Saltman for an intriguing question and helpful follow-
up discussion that motivated this project, and Asher Auel for helpful
conversations.

This material is based partially upon work supported by National Science
Foundation grant DMS-1928930 while the last two authors were in residence at
the Simons Laufer Mathematical Sciences Institute in Berkeley, California,
during the Spring 2023 semester. S.F. was supported in part by the Hausdorff
Research Institute for Mathematics, funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2047/1 – 390685813. L.J. was supported in part by
NSF MSPRF grant DMS-2202444. S.S. was supported by the Dutch Research Council
(NWO) grant OCENW.XL21.XL21.011. B.V. was supported in part by an AMS Birman
Fellowship and NSF grant DMS-2101434. I.V. was supported in part by NSF grant
DMS-2200655. For the purpose of open access, a CC BY public copyright license
is applied to any Author Accepted Manuscript version arising from this
submission.

### Notation

Throughout, $k$ denotes a field of characteristic different from $2$. A
variety over $k$ is a separated $k$-scheme of finite type. If $Y$ is an
integral $k$-variety, ${\mathbf{k}}(Y)$ denotes its function field. More
generally, if $Y$ is a finite union of integral $k$-varieties $Y_{i}$, then
${\mathbf{k}}(Y)\colonequals\prod_{i}{\mathbf{k}}(Y_{i})$ is the ring of
global sections of the sheaf of total quotient rings.

For a smooth variety $Y$ over a field $k$, we write $\operatorname{Br}Y$ for
the (cohomological) Brauer group
$\operatorname{H}^{2}_{\operatorname{et}}(Y,{\mathbb{G}}_{m})$. If
$Y=\operatorname{Spec}R$ is affine we write
$\operatorname{Br}R\colonequals\operatorname{Br}\operatorname{Spec}R$. For a
field $k$, $\operatorname{Br}k$ can also be viewed as describing the
isomorphism classes of Severi–Brauer varieties. In particular, any rank $3$
quadric defines an isomorphism class of a smooth plane conic which has an
associated class in the $2$-torsion part of the Brauer group. Given a rank $2$
quadric $Q$, we may associate the Brauer class given by the conic
$V(z^{2}-Q)\subset{\mathbb{P}}^{2}$. For a quadratic form $Q\in
k[\mathbf{x}]$, the symmetric matrix $M$ associated to $Q$ is defined to be
the matrix such that $\mathbf{x}M\mathbf{x}^{T}=Q$.

The Brauer group
$\operatorname{H}^{2}_{\operatorname{et}}(-,{\mathbb{G}}_{m})$ is a
contravariant functor, and given a morphism of smooth varieties $f\colon Y\to
Z$ we write $\alpha_{Y}\colonequals f^{*}\alpha$ for
$\alpha\in\operatorname{Br}Z$. Given any point $z\in Z$ we write $Y_{z}$ for
the fiber of $f$ over $z$.

For a curve $C$, i.e., a smooth projective variety of dimension $1$, we write
$\operatorname{{\bf Pic}}_{C}$ for the Picard scheme of $C$ and write
$\operatorname{{\bf Pic}}_{C}^{i}$ for the connected component that
parametrizes degree $i$ line bundles.

Throughout the paper, we will write $\eta$ for the generic point of
${\mathbb{P}}^{2}$.

##  2\. Conic bundles and discriminant covers

The central objects of interest in this paper are conic bundles over
${\mathbb{P}}^{2}$, i.e., the class of smooth projective varieties $X$ with a
flat morphism $\pi\colon X\to{\mathbb{P}}^{2}$ of relative dimension $1$ such
that $\omega_{X}$ is relatively anti-ample. This implies that locally over
${{\mathbb{P}}^{2}}$, $X$ is defined by a quadratic form and that the fiber of
$\pi$ over any $w\in{\mathbb{P}}^{2}$ is a conic in ${\mathbb{P}}^{2}$
(possibly of rank less than $3$ if $w$ is not the generic point).

There is a curve $\Delta\subset{\mathbb{P}}^{2}$ with the property that the
fiber over $w\in{\mathbb{P}}^{2}$ is a smooth conic if and only if
$w\not\in\Delta$. Furthermore, there is a curve $\tilde{\Delta}$ that is a
double cover of $\Delta$ and that parametrizes the irreducible components of
the fibers of $\pi\colon X_{\Delta}\to\Delta$. The curve $\tilde{\Delta}$ can
be obtained by considering the Stein factorization of the relative variety
$\mathcal{F}_{1}(X/{\mathbb{P}}^{2})\to\Delta\subset{\mathbb{P}}^{2}$ of lines
in the fibers of $\pi$. We call $\Delta$ the discriminant locus of $\pi$ and
$\varpi\colon\tilde{\Delta}\to\Delta$ the discriminant cover of
$X\to{\mathbb{P}}^{2}$. If $\Delta$ is smooth, then $\varpi$ is étale.

A conic bundle is standard if
$\operatorname{Pic}X=\pi^{*}\operatorname{Pic}{\mathbb{P}}^{2}\oplus{\mathbb{Z}}$;
this is equivalent to $\tilde{\Delta}$ and $\Delta$ having the same number of
irreducible components. Throughout the paper, we reserve $\pi\colon
X\to{\mathbb{P}}^{2}$ for a geometrically standard conic bundle, i.e., the
base change $X_{\overline{k}}\to{\mathbb{P}}^{2}_{\overline{k}}$ remains
standard.

###  2.1. Discriminant covers and isomorphism classes of conic bundles

Discriminant covers are crucial to the study of conic bundles and control much
of their geometry. Indeed, over fields with trivial $2$-torsion part of the
Brauer group, the discriminant cover determines the isomorphism class of
$X_{\eta}$. Over arbitrary fields, we study the isomorphism class of
$X_{\eta}$ via its class in
$\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})[2]$ by considering the purity
exact sequence [CTS-Brauer-book]*Theorems 3.7.3 and 6.1.3

$0\longrightarrow\operatorname{Br}k[2]\longrightarrow\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})[2]\xrightarrow{\;(\partial_{z})_{z}\;}\bigoplus_{z\in({\mathbb{P}}^{2})^{(1)}}H^{1}({\mathbf{k}}(z),{\mathbb{Z}}/2{\mathbb{Z}}).$

By definition of the residue maps $\partial_{z}$ [CTS-Brauer-book]*Definition
1.4.11, $\partial_{z}([X_{\eta}])$ is trivial for every $z$ outside of the
support of $\Delta$. Furthermore, if ${\mathbf{k}}(\tilde{\Delta})_{z}$
denotes the maximal subfield of ${\mathbf{k}}(\tilde{\Delta})$ that is an
extension of ${\mathbf{k}}(z)$, then
$\partial_{z}([X_{\eta}])={\mathbf{k}}(\tilde{\Delta})_{z}$ for every
$z\in({\mathbb{P}}^{2})^{(1)}$ in the support of $\Delta$. Thus, if $\pi\colon
X\to{\mathbb{P}}^{2}$ and $\pi^{\prime}\colon X^{\prime}\to{\mathbb{P}}^{2}$
are two standard conic bundles with the same discriminant cover, then

$[X_{\eta}]-[X^{\prime}_{\eta}]\in\operatorname{im}(\operatorname{Br}k[2]\to\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})[2]).$
(2.1)

Further, this difference agrees with the difference $[X_{w}]-[X^{\prime}_{w}]$
for any $(w\in{\mathbb{P}}^{2}\smallsetminus\Delta)(k)$. However, as the
following proposition shows, not every element of $\operatorname{Br}k[2]$ can
be realized as a difference of two standard conic bundles with a fixed
discriminant cover.

######  Proposition 2.1.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be an étale double cover of smooth,
projective, geometrically integral curves with $\Delta$ a plane curve over a
field $k$ of characteristic different from $2$. If $\tilde{\Delta}$ has index
$1$, i.e., if $\tilde{\Delta}(L)\neq\emptyset$ for some odd degree extension
$L/k$, then any conic bundles $\pi\colon X\to{\mathbb{P}}^{2}$,
$\tilde{\pi}\colon\tilde{X}\to{\mathbb{P}}^{2}$ with discriminant cover
$\tilde{\Delta}/\Delta$ must have isomorphic generic fibers. That is,
$[X_{\eta}]-[\tilde{X}_{\eta}]=0\in\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})$.

###### Proof.

Let $\pi\colon X\to{\mathbb{P}}^{2}$,
$\tilde{\pi}\colon\tilde{X}\to{\mathbb{P}}^{2}$ be standard conic bundles with
discriminant cover $\tilde{\Delta}/\Delta$. Then
$[X_{\eta}]-[\tilde{X}_{\eta}]\in\operatorname{im}(\operatorname{Br}k[2]\to\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})[2])$
by (2.1). We will show that for any nontrivial
$\alpha\in\operatorname{Br}k[2]$, $[X_{\eta}]+\alpha$ has no quadratic
splitting field, which implies that $[X_{\eta}]+\alpha$ is _not_ represented
by a conic. Thus, $[X_{\eta}]-[\tilde{X}_{\eta}]$ must equal
$0\in\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})$, as desired.

The statement that $[X_{\eta}]+\alpha$ has no quadratic splitting field is
equivalent to the statement that the index of $[X_{\eta}]+\alpha$ is $2^{n}$
for some $n>1$. Since the index of a $2$-primary Brauer class is unchanged
under odd degree extensions [GS]*Corollary 4.5.11, we may reduce to the case
that $\tilde{\Delta}(k)\neq\emptyset$.

We now return to proving that $[X_{\eta}]+\alpha$ has no quadratic splitting
field, assuming that $\tilde{\Delta}(k)\neq\emptyset$. By a result of Albert
[Lam-quadratic-forms]*Chapter III, Theorem 4.8, $[X_{\eta}]+\alpha$ has a
quadratic splitting field if and only if $\alpha$ and $[X_{\eta}]$ have a
common quadratic splitting field $L/{\mathbf{k}}({\mathbb{P}}^{2})$. Assume
that there exists such an $L$. Let $R$ denote the local ring
${\mathcal{O}}_{{\mathbb{P}}^{2},\Delta}$ and identify
${\mathbf{k}}({\mathbb{P}}^{2})=\operatorname{Frac}(R)$; let $S$ denote the
integral closure of $R$ in $L$. Since $L$ splits $X_{\eta}$ and $\pi\colon
X\to{\mathbb{P}}^{2}$ is proper, the valuative criterion for properness gives
a morphism $\operatorname{Spec}S\to X_{R}$. If $R\hookrightarrow S$ is
unramified, then the image of $\operatorname{Spec}S\to X_{R}$ cannot meet the
singular locus of the special fiber of $X_{R}\to\operatorname{Spec}R$. In
particular, since $X_{{\mathbf{k}}(\Delta)}$ is birational to
${\mathbb{P}}^{1}_{{\mathbf{k}}(\tilde{\Delta})}$ (by definition of the
discriminant cover), if $R\hookrightarrow S$ is an unramified extension, then
the extension of residue fields agrees with
${\mathbf{k}}(\tilde{\Delta})/{\mathbf{k}}(\Delta)$. Thus, we have the
following commutative diagrams:

${S}$${{\mathbf{k}}(\tilde{\Delta})}$${R}$${{\mathbf{k}}(\Delta)}$${k}$

and
${\operatorname{Br}L}$${\operatorname{Br}S}$${\operatorname{Br}{\mathbf{k}}(\tilde{\Delta})}$${\operatorname{Br}R}$${\operatorname{Br}{\mathbf{k}}(\Delta)}$${\operatorname{Br}k}$.

By assumption $L$ splits $\alpha$, so the diagram on the right shows that
$\alpha$ becomes trivial when pulled back to ${\mathbf{k}}(\tilde{\Delta})$.
The pullback map factors as
$\operatorname{Br}k\to\operatorname{Br}\tilde{\Delta}\hookrightarrow\operatorname{Br}{\mathbf{k}}(\tilde{\Delta})$.
Since $\tilde{\Delta}(k)\neq\emptyset$,
$\operatorname{Br}k\to\operatorname{Br}\tilde{\Delta}$ is injective [CTS-
Brauer-book]*Proposition 5.4.2 and Remark 5.4.3(1), so
$\alpha=0\in\operatorname{Br}k$.

In the case that $R\hookrightarrow S$ is ramified, then the residue field of
$S$ is ${\mathbf{k}}(\Delta)$. Since $\tilde{\Delta}$ is a cover of $\Delta$,
we have $\Delta(k)\neq\emptyset$ and so we can repeat the above argument with
$\Delta$ replacing $\tilde{\Delta}$. ∎

In the case that $\Delta$ has degree $4$, we prove the following strengthening
of [FJSVV]*Proposition 8.1(ii).

######  Corollary 2.2.

In the setting of Proposition 2.1, assume $\Delta$ is a smooth plane quartic
and $\tilde{\Delta}$ has index $1$. Then any conic bundle
$X\to{\mathbb{P}}^{2}$ with discriminant cover $\tilde{\Delta}/\Delta$ is
rational.

###### Proof.

By [FJSVV]*Section 6 there exists a threefold $Y$ with the structure of a
conic bundle with discriminant cover $\tilde{\Delta}/\Delta$ and the structure
of a quadric fibration $\pi_{1}\colon Y\to{\mathbb{P}}^{1}$.
[FJSVV]*Proposition 6.1(v) and the assumption on $\tilde{\Delta}$ imply that
there is an odd degree extension $L/k$ over which $({\pi_{1}})_{L}\colon
Y_{L}\to{\mathbb{P}}^{1}_{L}$ has a section. Then $\pi_{1}$ has a section over
$k$ by Springer’s theorem [Springer52], so $Y$ is rational. Next, let
$X\to{\mathbb{P}}^{2}$ be any conic bundle with discriminant cover
$\tilde{\Delta}/\Delta$. Then Proposition 2.1 implies that $X$ and $Y$ are
birationally equivalent, and therefore $X$ is rational. ∎

###  2.2. Prym varieties and the IJT obstruction to rationality

The primary conic bundles $\pi\colon X\to{\mathbb{P}}^{2}$ of interest in this
paper are those whose discriminant curve $\Delta\subset{\mathbb{P}}^{2}$ is a
smooth plane curve (often of degree $4$) and whose discriminant cover
$\varpi\colon\tilde{\Delta}\to\Delta$ is geometrically integral. The
smoothness of $X$ and $\Delta$ imply that $\varpi$ is an étale double cover.

The Prym variety of $\tilde{\Delta}/\Delta$, which we denote by $P$, is the
connected component of the identity of
$\ker(\varpi_{*}\colon\operatorname{{\bf
Pic}}_{\tilde{\Delta}/k}\to\operatorname{{\bf Pic}}_{\Delta/k})$; it is a
principally polarized abelian variety (see [mumford-prym]). The Prym variety
is the intermediate Jacobian of $X$, and can be thought of as a parameter
space for algebraically trivial curve classes on $X$ [beauville-
ij]*Proposition 3.3, [FJSVV]*Theorem 5.8. The kernel of
$\varpi_{*}\colon\operatorname{{\bf
Pic}}_{\tilde{\Delta}/k}\to\operatorname{{\bf Pic}}_{\Delta/k}$ has a unique
additional connected component, which we denote $\tilde{P}$, and it is a
torsor under $P$.

####  2.2.1. Étale double covers of smooth plane quartics

If $\Delta\subset{\mathbb{P}}^{2}$ is a smooth plane quartic, then any
geometrically integral étale double cover $\tilde{\Delta}$ is a smooth genus
$5$ curve. Bruin [bruin] proved that the Prym variety of a such a double cover
is always isomorphic to the Jacobian of a smooth genus $2$ curve
$\Gamma=\Gamma_{\tilde{\Delta}/\Delta}$. This implies that on the threefold
$X$, the Clemens–Griffiths intermediate Jacobian obstruction to rationality
[clemensgriffiths] vanishes. Defining equations of
$\Delta,\tilde{\Delta},\Gamma$ can be given in terms of 3 quadrics
$Q_{1},Q_{2},Q_{3}\in k[u,v,w]$ as follows:

$\displaystyle\Delta$ $\displaystyle=V(Q_{2}^{2}-Q_{1}Q_{3}),$ (2.2)
$\displaystyle\tilde{\Delta}$
$\displaystyle=V(Q_{1}-r^{2},Q_{2}-rs,Q_{3}-s^{2})\subset{\mathbb{P}}^{4},\textup{and}$
(2.3) $\displaystyle\Gamma$
$\displaystyle=V(y^{2}-\det(t_{0}^{2}M_{1}+2t_{0}t_{1}M_{2}+t_{1}^{2}M_{3}))\subset{\mathbb{P}}(1,1,3),$
(2.4)

(here $M_{i}$ is the symmetric matrix associated to $Q_{i}$); see, for
example, [bruin]. In addition, $Q_{1},Q_{2},Q_{3}$ are uniquely determined up
to a $\operatorname{PGL}_{2}$-action [FJSVV]*Theorem 4.5(i) which agrees with
changing coordinates of $t_{0},t_{1}$ on $\Gamma$.

The intermediate Jacobian torsor (IJT) obstruction for $X$, introduced by [HT-
intersection-quadrics] over ${\mathbb{R}}$ and [bw-ij] over arbitrary $k$ (see
also [HT-cycle] for $k\subset\mathbb{C}$), is an obstruction to rationality of
$X$ that is encoded by certain torsors under $P$. In the case where
$\Delta\subset{\mathbb{P}}^{2}$ is degree $4$, the obstruction is encoded by
$\tilde{P}$ and the following two $P$-torsors:

$\displaystyle{P}^{(1)}$ $\displaystyle\colonequals\\{D\in\operatorname{{\bf
Pic}}^{4}_{\tilde{\Delta}/k}\colon\varpi_{*}D\text{ lies on a line in
}{\mathbb{P}}^{2},\operatorname{h}^{0}(D)\equiv 0\bmod 2\\},\;\textup{and}$
(2.5) $\displaystyle\tilde{P}^{(1)}$
$\displaystyle\colonequals\\{D\in\operatorname{{\bf
Pic}}^{4}_{\tilde{\Delta}/k}\colon\varpi_{*}D\text{ lies on a line in
}{\mathbb{P}}^{2},\operatorname{h}^{0}(D)=1\\}.$ (2.6)

######  Proposition 2.3.

Let $k$ be a field of characteristic different from $2$, and let $\pi\colon
X\to{\mathbb{P}}^{2}$ be a conic bundle over $k$ with discriminant cover
$\tilde{\Delta}\to\Delta$ such that $\Delta$ is a smooth quartic and
$\tilde{\Delta}$ is geometrically integral.

  1. (1)

[bruin]*Section 5, Case 4 As $P$-torsors, ${P}^{(1)}\simeq\operatorname{{\bf
Pic}}^{1}_{\Gamma}$.

  2. (2)

[FJSVV]*Corollary of Theorem 6.4, Section 6.4 The IJT obstruction vanishes for
$X$ if and only if either $\tilde{P}(k)\neq\emptyset$ or
$\tilde{P}^{(1)}(k)\neq\emptyset$. Moreover, if $\operatorname{{\bf
Pic}}_{\Gamma}^{1}(k)\neq\emptyset$, then $\tilde{P}^{(1)}(k)\neq\emptyset$ if
and only if $\tilde{P}(k)\neq\emptyset$, so the vanishing of the IJT
obstruction for $X$ is equivalent to $\tilde{P}^{(1)}(k)\neq\emptyset$.

######  Remark 2.4.

The torsors $\tilde{P},{P}^{(1)},\tilde{P}^{(1)}$ are part of a larger family
of torsors, given by the polarized Prym scheme, which can be defined for
arbitrary conic bundles; see [FJSVV]*Section 4 for more details.

###  2.3. Relation to curve classes

The intermediate Jacobian torsors are indexed by Galois-invariant algebraic
curve classes $\gamma$ on $X$. For conic bundles, the vanishing of the IJT
obstruction implies the existence of certain Galois-invariant geometric curve
classes up to rational equivalence, as we now explain. Let
$\operatorname{CH}^{2}(X)$ denote the group of rational equivalence classes of
codimension $2$ cycles on $X$, and for a fixed class $\gamma$, let
$\operatorname{CH}^{2}(X)^{\gamma}$ denote the classes that are algebraically
equivalent to $\gamma$. In the degree 4 case,
$\tilde{P}^{(1)}(k)\cong(\operatorname{CH}^{2}(X_{\bar{k}})^{\tilde{\gamma}_{1}})^{G_{k}}$
for some algebraic curve class $\tilde{\gamma}_{1}$. Proposition 2.3 shows
that if the IJT obstruction vanishes and $\operatorname{{\bf
Pic}}_{\Gamma_{\tilde{\Delta}/\Delta}}^{1}(k)\neq\emptyset$, then there is a
cycle on $X_{\bar{k}}$ algebraically equivalent to $\tilde{\gamma}_{1}$ whose
rational equivalence class is Galois invariant. (See [FJSVV]*Theorem 6.4 for a
construction of $\tilde{\gamma}_{1}$.)

##  3\. Constructing an intersection of quadrics with a conic bundle structure

The goal of this section is to prove the following proposition.

######  Proposition 3.1.

Let $k$ be a field of characteristic different from $2$.

Given quadrics $Q_{1},Q_{2},Q_{3}\in k[u,v,w]$ such that
$V(Q_{2}^{2}-Q_{1}Q_{3})$ is a smooth quartic and such that
$\operatorname{disc}(Q_{1})$ is a square, there exist two quadrics
$q_{0},q_{\infty}\in k[x_{0},x_{1},\dots,x_{5}]$ such that

  1. (1)

The intersection $Z\colonequals V(q_{0},q_{\infty})\subset{\mathbb{P}}^{5}$ is
a smooth projective threefold;

  2. (2)

The linear section $Z\cap V(x_{3},x_{4},x_{5})$ is a smooth conic $C$; and

  3. (3)

Projection away from the plane $V(x_{3},x_{4},x_{5})$ induces a morphism
$\operatorname{Bl}_{C}Z\to{\mathbb{P}}^{2}$, endowing $\operatorname{Bl}_{C}Z$
with a geometrically standard conic bundle structure whose generic fiber
equals

$[(\operatorname{Bl}_{C}Z)_{\eta}]=(Q_{1},Q_{2}^{2}-Q_{1}Q_{3})+[Q_{1}]\in\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2}),$

where $[Q_{1}]$ denotes the pullback of the class in $\operatorname{Br}k[2]$
corresponding to the rank $2$ or $3$ quadric $Q_{1}$. In particular, the
discriminant cover of the conic bundle
$\operatorname{Bl}_{C}Z\to{\mathbb{P}}^{2}$ is $\tilde{\Delta}/\Delta$.

Using this proposition together with work of Hassett–Tschinkel and
Benoist–Wittenberg, we can deduce the assertion from the introduction.

######  Corollary 3.2 (Corollary of Proposition 3.1 & [HT-intersection-
quadrics]*Theorem 36, [bw-ij]*Theorem A).

Let $k$ be a field of characteristic different from $2$, and let $\pi\colon
X\to{\mathbb{P}}^{2}$ be a conic bundle over $k$ with discriminant cover
$\varpi\colon\tilde{\Delta}\to\Delta$ where $\Delta\subset{\mathbb{P}}^{2}$ is
a smooth plane quartic and $\tilde{\Delta}$ is geometrically irreducible.
Assume that $\Gamma_{\tilde{\Delta}/\Delta}(k)\neq\emptyset$.

There exists a conic bundle $\pi^{\prime}\colon X^{\prime}\to{\mathbb{P}}^{2}$
with discriminant cover $\tilde{\Delta}/\Delta$ (namely
$X^{\prime}=\operatorname{Bl}_{C}Z$ with $Z,C$ as in Proposition 3.1) such
that $X^{\prime}$ is rational if and only if the IJT obstruction vanishes for
$X$ (equivalently for $X^{\prime}$).

###### Proof.

The assumption that $\Gamma_{\tilde{\Delta}/\Delta}(k)\neq\emptyset$ means
that there exists some $[t_{0}:t_{1}]\in{\mathbb{P}}^{1}(k)$ such that the
discriminant of $t_{0}^{2}Q_{1}+2t_{0}t_{1}Q_{2}+t_{1}^{2}Q_{3}$ is a square.
By a change of coordinates on ${\mathbb{P}}^{1}(k)$, we may assume that this
point is $[1:0]$ so $\operatorname{disc}(Q_{1})$ is a square. Then we may
apply Proposition 3.1 to obtain an intersection of quadrics $Z$ containing a
conic $C$ such that the conic bundle morphism
$\pi^{\prime}\colon\operatorname{Bl}_{C}Z\to{\mathbb{P}}^{2}$ has discriminant
cover $\varpi\colon\tilde{\Delta}\to\Delta$. Then [HT-intersection-
quadrics]*Theorem 36, [bw-ij]*Theorem A imply that the IJT obstruction
characterizes rationality for $\operatorname{Bl}_{C}Z$ (see [FJSVV]*Corollary
8.3 for details). ∎

######  Remark 3.3.

If $k$ is algebraically closed, then the proposition is suggested by a
dimension count of the relevant moduli spaces. Over nonclosed fields $k$, we
must show that the intersection of quadrics $Z$ has the same field of
definition as the quadrics $Q_{1},Q_{2},Q_{3}$. We do so by constructing the
quadrics $q_{0},q_{\infty}$ (see (3.1) and (3.2)) and then we verify that $Z$
has the desired properties.

###### Proof of Proposition 3.1.

The assumption that $V(Q_{2}^{2}-Q_{1}Q_{3})$ is a smooth quartic implies that
the quadrics $Q_{1},Q_{3}$ both have rank at least $2$. This furthermore
implies that the curve $\tilde{\Delta}$ defined in (2.3) is geometrically
integral. We consider two cases for the rank of $Q_{1}$.

Case 1: $\operatorname{disc}(Q_{1})\neq 0$, i.e., $Q_{1}$ has rank $3$. We
change coordinates on ${\mathbb{P}}^{2}$ to assume that
$Q_{1}=au^{2}+bv^{2}-abw^{2}$ for some $a,b\in k^{\times}.$ Note that then
$[Q_{1}]=(a,b)\in\operatorname{Br}k[2]$. Consider the following $6\times 6$
symmetric matrices defined in terms of the symmetric matrices $M_{i}$
associated to $Q_{i}$:

$A_{0}\colonequals\begin{pmatrix}abM_{1}^{-1}&-M_{1}^{-1}M_{2}\\\
-M_{2}M_{1}^{-1}&\frac{-1}{ab}(M_{3}-M_{2}M_{1}^{-1}M_{2})\end{pmatrix},\qquad
A_{\infty}\colonequals\begin{pmatrix}0&I\\\ I&0\end{pmatrix}.$ (3.1)

Let $q_{0},q_{\infty}$ be the two quadratic forms defined by these matrices,
i.e., $q_{i}=\textbf{x}A_{i}\textbf{x}^{T}$. We claim that $Z\colonequals
V(q_{0},q_{\infty})\subset{\mathbb{P}}^{5}$ has the desired properties.

An intersection of two quadrics in six variables is smooth and three-
dimensional if and only if the discriminant polynomial
$\det(A_{0}-TA_{\infty})$ is separable [Reid-thesis]*Proposition 2.1. Using
the Schur complement, we compute that up to a nonzero scalar,
$\det(A_{0}-TA_{\infty})$ is equal to the determinant of the following
$3\times 3$ matrix

$\displaystyle
M_{3}-M_{2}M_{1}^{-1}M_{2}+(M_{2}M_{1}^{-1}+TI)M_{1}(M_{1}^{-1}M_{2}+TI)=M_{3}+2TM_{2}+T^{2}M_{1}.$

Thus, by (2.4), the discriminant polynomial $\det(A_{0}-TA_{\infty})$ is a
scalar multiple of the polynomial describing the Weierstrass points of
$\Gamma$, which is separable. Hence, $Z$ is smooth and three-dimensional, as
desired.

Since $q_{\infty}$ is identically zero when restricted to
$V(x_{3},x_{4},x_{5})$, the linear section of $Z$ given by
$V(x_{3},x_{4},x_{5})$ is equal to $V(q_{0},x_{3},x_{4},x_{5})$ so by (3.1) is
given by the vanishing of the rank $3$ form corresponding to $M_{1}^{-1}$.
Thus we have proved (2).

To prove (3), we must further describe the conic bundle structure on
$\operatorname{Bl}_{C}Z$. Let $\Lambda\colonequals V(x_{3},x_{4},x_{5})$, and
let $\phi\colon Z\dasharrow{\mathbb{P}}^{2}$ be the projection away from
$C\colonequals V(q_{0})\cap\Lambda$. Fix a point
$p=[u:v:w]\in{\mathbb{P}}^{2}$. We obtain the fiber of $\phi$ above $p$ by
taking the intersection $\operatorname{Span}(\Lambda,[0:0:0:u:v:w])\cap
Z\subset{\mathbb{P}}^{5}$, removing $\Lambda$, and then taking the Zariski
closure.

Note that the restriction of $q_{\infty}$ to
$\operatorname{Span}(\Lambda,[0:0:0:u:v:w])$ is a rank $2$ quadratic form, and
one can compute that the vanishing of this quadratic form is the union of
$\Lambda$ and the $3$-dimensional column span of the following matrix

$B_{p}=\begin{pmatrix}-v&-w&0&0\\\ u&0&-w&0\\\ 0&u&v&0\\\
&\textbf{0}_{3,3}&&[u,v,w]^{T}\end{pmatrix};$

we denote the column span of this matrix $\Lambda_{p}$. Thus the fiber of
$\phi$ above $p$ is $\Lambda_{p}\cap V(q_{0})$, which is given by the rank $3$
quadratic form associated to the $4\times 4$ symmetric matrix
$B_{p}^{T}A_{0}B_{p}$. Let $\mathbf{M}$ denote the bottom left $3\times 3$
submatrix of $B_{p}^{T}A_{0}B_{p}$. One can check222This, and all following
computational claims, can be verified with the Magma [Magma] code in the
Github repository. This submatrix $\mathbf{M}$ is denoted ‘bM’ in the code.
that the determinants of the top right $i\times i$ submatrices of $\mathbf{M}$
are:

$-(u^{2}-bw^{2}),\quad-w^{2}Q_{1},\quad\frac{-w^{2}}{ab}(Q_{2}^{2}-Q_{1}Q_{3}),\quad\textup{for
}i=1,2,3,\textup{ respectively}.$

Since a diagonalization of $\mathbf{M}$ has entries that are successive ratios
of these minors (where the $0^{th}$ minor is said to be $1$), the generic
fiber corresponds to the quaternion algebra

$\left(ab(u^{2}-bw^{2})Q_{1}(Q_{2}^{2}-Q_{1}Q_{3}),Q_{1}\right)=\left(Q_{2}^{2}-Q_{1}Q_{3},Q_{1}\right)+\left(-Q_{1},Q_{1}\right)+\left(-ab(u^{2}-bw^{2}),Q_{1}\right).$

Furthermore, using the Brauer relations $(f,g^{2})=1$ and
$(f,g)=(f,g(h^{2}-f))$ for all $f,g\in{\mathbf{k}}({\mathbb{P}}^{2})^{\times}$
and $h\in{\mathbf{k}}({\mathbb{P}}^{2})$ [CTS-Brauer-book]*Lemma 1.1.6 and
Proposition 1.1.8 and the identity $bQ_{1}=(bv)^{2}-(-ab(u^{2}-bw^{2}))$, we
see that $\left(-Q_{1},Q_{1}\right)$ is trivial and that

$\displaystyle(-ab(u^{2}-bw^{2}),Q_{1})$
$\displaystyle=(-ab(u^{2}-bw^{2}),b)=(-ab,b)=(a,b),$

which completes the proof.

Case 2: $Q_{1}$ has rank $2$. The proof will proceed similarly as in the rank
$3$ case with the specific equations modified slightly, so we will omit more
of the algebraic details. As in the previous case, verification of all steps
can be found in the Magma [Magma] code in the Github repository associated to
this paper.

We change coordinates on ${\mathbb{P}}^{2}$ to assume that
$Q_{1}=av^{2}+bw^{2}$ for some $a,b\in k^{\times}.$ Note that then
$[Q_{1}]=(a,b)\in\operatorname{Br}k[2]$. Since $\Delta$ is smooth,
$\lambda\colonequals Q_{2}(1,0,0)\neq 0$ and so by replacing $Q_{2}$ and
$Q_{3}$ with $-ab\lambda^{-1}Q_{2}$ and $(-ab\lambda^{-1})^{2}Q_{3}$, we may
assume that $Q_{2}(1,0,0)=-ab$. We will again consider $6\times 6$ symmetric
matrices defined in terms of the symmetric matrices $M_{i}$ associated to
$Q_{i}$, but we need some additional notation. Let $T_{2}$ denote the unique
upper triangular matrix such that $T_{2}+T_{2}^{T}=2M_{2}$. For any $2\times
2$ matrix $N$ and any $c\in k$, we write $c\oplus N$ for the $3\times 3$ block
diagonal matrix with $c$ in the top left entry and $N$ in the bottom right
entry. Let $N_{1}$ denote the $2\times 2$ diagonal matrix with entries $a,b$
so that $M_{1}=0\oplus N_{1}$. Now define the $6\times 6$ symmetric matrices

$A_{0}\colonequals\begin{pmatrix}1\oplus-abN_{1}^{-1}&-(0\oplus
N_{1}^{-1})T_{2}^{T}\\\ -T_{2}(0\oplus
N_{1}^{-1})&\frac{1}{ab}(M_{3}+T_{2}(0\oplus-
N_{1}^{-1})T_{2}^{T})\end{pmatrix},\quad
A_{\infty}\colonequals\begin{pmatrix}\mathbf{0}_{3,3}&0\oplus I_{2}\\\ 0\oplus
I_{2}&2\oplus\mathbf{0}_{2,2}\end{pmatrix}.$ (3.2)

Let $q_{0},q_{\infty}$ be the two quadratic forms defined by these matrices,
i.e., $q_{i}=\textbf{x}A_{i}\textbf{x}^{T}$; we will show that $Z\colonequals
V(q_{0},q_{\infty})\subset{\mathbb{P}}^{5}$ has the desired properties.

We again have $\det(A_{0}-TA_{\infty})=\det(M_{3}+2TM_{2}+T^{2}M_{1})$ so $Z$
is smooth and three-dimensional, as desired. Since $q_{\infty}$ is identically
zero when restricted to $V(x_{3},x_{4},x_{5})$, the linear section of $Z$
given by $V(x_{3},x_{4},x_{5})$ is equal to $V(q_{0},x_{3},x_{4},x_{5})$ and
so is given by the rank $3$ quadratic form associated to the top left $3\times
3$ submatrix of $A_{0}$. This proves (2).

To prove (3), we must further describe the conic bundle structure on
$\operatorname{Bl}_{C}Z$. We again let $\Lambda\colonequals
V(x_{3},x_{4},x_{5})$, and let $\phi\colon Z\dasharrow{\mathbb{P}}^{2}$ be the
projection away from $C\colonequals V(q_{0})\cap\Lambda$. Fix a point
$p=[u:v:w]\in{\mathbb{P}}^{2}$. As above, the fiber of $\phi$ above $p$ is
obtained by taking the Zariski closure of
$\left(\operatorname{Span}(\Lambda,[0:0:0:u:v:w])\cap
Z\right)\smallsetminus\Lambda\subset{\mathbb{P}}^{5}$. Note that the
restriction of $q_{\infty}$ to $\operatorname{Span}(\Lambda,[0:0:0:u:v:w])$ is
a rank $2$ quadratic form, and one can compute that vanishing of this
quadratic form is the union of $\Lambda$ and the $3$-dimensional column span
of the following matrix

$B_{p}=\left(\begin{array}[]{cc|cc}1&0&0&0\\\ 0&-w&-u^{2}&0\\\ 0&v&0&-u^{2}\\\
\hline\cr\lx@intercol\hfil\textbf{0}_{3,2}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil[u,v,w]^{T}[v,w]\hfil\lx@intercol\end{array}\right);$

we denote the column span of this matrix $\Lambda_{p}$. Thus the fiber of
$\phi$ above $p$ is $\Lambda_{p}\cap V(q_{0})$, which is given by the rank $3$
quadratic form associated to the $4\times 4$ symmetric matrix
$B_{p}^{T}A_{0}B_{p}$. Let $\mathbf{M}$ denote the negative of the middle
$2\times 2$ submatrix of $B_{p}^{T}A_{0}B_{p}$; $\mathbf{M}$ has rank $2$. One
can check by inspection of $B_{p}$ and $A_{0}$ that $B_{p}^{T}A_{0}B_{p}$ is
block diagonal with the top left entry $1$ and that the bottom right $3\times
3$ block has rank $2$. Thus, the Brauer class of the generic fiber is given by
the class of the rank $2$ quadratic form associated to $\mathbf{M}$. One can
also check that the top left entry of $\mathbf{M}$ is $Q_{1}$ and that the
discriminant of the associated quadratic form, which is $-\det(\mathbf{M})$,
is equal to $-(ab)^{-1}v^{2}(Q_{2}^{2}-Q_{1}Q_{3})$. Therefore,

$\displaystyle[(\operatorname{Bl}_{C}Z)_{\eta}]=(-(ab)^{-1}v^{2}(Q_{2}^{2}-Q_{1}Q_{3}),Q_{1})$
$\displaystyle=(-ab(Q_{2}^{2}-Q_{1}Q_{3}),Q_{1})$
$\displaystyle=(Q_{2}^{2}-Q_{1}Q_{3},Q_{1})+(-ab,Q_{1}).$

Furthermore, the Brauer relations show $(-ab,av^{2}+bw^{2})=(-ab,a)=(b,a)$, as
desired. ∎

##  4\. A constant Brauer class from $k$-points of $\tilde{P}^{(1)}$

Let $\pi\colon X\to{\mathbb{P}}^{2}$ be a geometrically standard conic bundle
whose discriminant curve is a smooth quartic, as in Section 2.2.1. In this
section, we pull back $\pi\colon X\to{\mathbb{P}}^{2}$ to obtain a conic
bundle over an open subset of $\tilde{P}^{(1)}$ and then show that if
$\tilde{P}^{(1)}(k)\neq\emptyset$ then this conic bundle is constant, i.e.,
its Brauer class is in the image of the map
$\operatorname{Br}k\to\operatorname{Br}{\mathbf{k}}(\tilde{P}^{(1)})$. This
construction can be viewed as a globalization of the argument in
[FJSVV]*Section 6.4, and, as such, the resulting Brauer class can be viewed as
an obstruction to descending representatives of curve classes to $k$ (see
Corollary 4.5 for details).

By Equation (2.6), a point $x\in\tilde{P}^{(1)}$ corresponds to an effective
divisor $D_{x}$ on $\tilde{\Delta}$, whose image under the pushforward by
$\varpi\colon\tilde{\Delta}\to\Delta$ is contained in a line in
${\mathbb{P}}^{2}$ (defined over ${\mathbf{k}}(x)$). We write
$\ell_{x}=\operatorname{Span}(\varpi_{*}{D_{x}})$ for this line. Let $I$ be
the universal line over $\tilde{P}^{(1)}$:

$I\colonequals\\{(p,x)\in{\mathbb{P}}^{2}\times\tilde{P}^{(1)}:p\in\ell_{x}\\}.$

This comes equipped with projections $p_{1}\colon I\to{\mathbb{P}}^{2}$ and
$p_{2}\colon I\to\tilde{P}^{(1)}$.

######  Lemma 4.1.

The second projection $p_{2}\colon I\to\tilde{P}^{(1)}$ is a
${\mathbb{P}}^{1}$-bundle, and any line $\ell\subset{\mathbb{P}}^{2}$ gives a
rational section $I_{\ell}\colonequals p_{1}^{-1}(\ell)\subset I$. Further,
$p_{2}^{*}\colon\operatorname{Br}\tilde{P}^{(1)}[2]\to\operatorname{Br}I[2]$
is an isomorphism.

###### Proof.

Fix a line $\ell\subset{\mathbb{P}}^{2}$. Over
$\tilde{P}^{(1)}\smallsetminus\varpi_{*}^{-1}(\ell\cap\Delta)$, the second
projection has an inverse morphism that sends a point
$D\in\tilde{P}^{(1)}\smallsetminus\varpi_{*}^{-1}(\ell\cap\Delta)$ to the
point $(\ell_{D}\cap\ell,D)\in I$. Thus $p_{2}\colon I\to\tilde{P}^{(1)}$ is a
${\mathbb{P}}^{1}$-bundle, and $I_{\ell}$ is a rational section. In
particular, the generic fiber of $p_{2}$ is
${\mathbb{P}}^{1}_{{\mathbf{k}}(\tilde{P}^{(1)})}$ [CTS-Brauer-
book]*Definition 11.1.1. Furthermore, $p_{2}$ is smooth, so, since we assumed
$2$ is invertible in $k$, we have an isomorphism on the $2$-primary torsion of
$\operatorname{Br}_{\mathrm{vert}}(I/S)$ and the pullback of
$\operatorname{Br}(S)$ by [CTS-Brauer-book]*Proof of Corollary 11.1.6(ii)
(noting that [CTS-Brauer-book]*Equation (11.3) holds for $p$-primary torsion
when $p\in k^{\times}$). ∎

Let $x\in\tilde{P}^{(1)}(k)$. We have the following commutative diagrams of
varieties and Brauer groups:

${I}$${I_{\ell_{x}}}$${{\mathbb{P}}^{2}}$${\ell_{x}}$${\tilde{P}^{(1)}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$${\operatorname{Br}{\mathbf{k}}(I)[2]}$${\operatorname{Br}{\mathbf{k}}(I_{\ell_{x}})[2]}$${\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2})[2]}$${\operatorname{Br}{\mathbf{k}}(\ell_{x})[2]}$${\operatorname{Br}{\mathbf{k}}(\tilde{P}^{(1)})[2]}$$\scriptstyle{p_{1}^{*}}$$\scriptstyle{p_{2}^{*}}$$\scriptstyle{\sim}$
(4.1)

######  Theorem 4.2 (Strengthening of [FJSVV]*Proposition 6.5).

Let $k$ be a field of characteristic different from $2$, and let
$x\in\tilde{P}^{(1)}(k)$ and let
$w\in(\ell_{x}\smallsetminus(\Delta\cap\ell_{x}))(k)$. Then

$[X_{w}]=[X_{\ell_{x}}]\in\operatorname{Br}{\mathbf{k}}(\ell_{x})\quad\textup{and}\quad[X_{w}]=p_{1}^{*}[X_{\eta}]\in\operatorname{Br}{\mathbf{k}}(I).$

In particular, $p_{1}^{*}[X_{\eta}]$ and $[X_{\ell_{x}}]$ are unramified and
constant.

######  Definition 4.3.

Assume that $\tilde{P}^{(1)}(k)\neq\emptyset$. Let $x\in\tilde{P}^{(1)}(k)$
and define

$\tilde{\alpha}^{(1)}\colonequals
x^{*}(p_{2}^{*})^{-1}\left(p_{1}^{*}[X_{\eta}]\right)\in\operatorname{Br}k[2].$

In particular, $\tilde{\alpha}^{(1)}=[X_{w}]$ for any
$w\in(\ell_{x}\smallsetminus(\Delta\cap\ell_{x}))(k)$.

######  Remarks 4.4.

  1. (1)

Theorem 4.2 implies that $\tilde{\alpha}^{(1)}$ is independent of the choice
of $x\in\tilde{P}^{(1)}(k)$. However, the definition of
$\tilde{\alpha}^{(1)}\in\operatorname{Br}k$ _does_ require that
$\tilde{P}^{(1)}(k)\neq\emptyset$, which motivates our choice of notation.

  2. (2)

It is possible to prove that
$p_{1}^{*}[X_{\eta}]\in\operatorname{Br}{\mathbf{k}}(I)$ is unramified without
assuming that $\tilde{P}^{(1)}(k)\neq\emptyset$. However, in the paper we will
only invoke Theorem 4.2 in situations when $\tilde{P}^{(1)}(k)\neq\emptyset$,
so we chose to present the shorter proof with this assumption.

###### Proof of Theorem 4.2.

By [FJSVV]*Proposition 6.5, the conic bundle $\pi|_{\ell_{x}}\colon
X_{\ell_{x}}\to\ell_{x}$ is unramified and so, since
$\ell_{x}\simeq{\mathbb{P}}^{1}$, $[X_{\ell_{x}}]=[X_{w}]$. Then, the
commutative diagrams (4.1) show that $p_{1}^{*}[X_{\eta}]$ is equal to the
image of $[X_{\ell_{x}}]=[X_{w}]$, as desired. ∎

###  4.1. Viewing $\tilde{\alpha}^{(1)}$ as an obstruction to the existence of
cycles

One of the primary reasons that vanishing of the IJT obstruction fails to
imply rationality for conic bundle threefolds is that it fails to guarantee
the existence of cycles defined over $k$ (see [FJSVV]*Prop. 6.5, Theorem 6.7,
and Theorem 1.4(i)). The class $\tilde{\alpha}^{(1)}$ measures this failure in
the following precise way.

As discussed in Section 2.2.1, the $k$-points of $\tilde{P}^{(1)}$ are in
bijection with
$(\operatorname{CH}^{2}(X_{\overline{k}})^{\tilde{\gamma}_{1}})^{G_{k}}.$ The
classes represented by $k$-rational codimension $2$ cycles form a subgroup
$\operatorname{CH}^{2}(X)^{\tilde{\gamma}_{1}}$ of
$(\operatorname{CH}^{2}(X_{{\overline{k}}})^{\tilde{\gamma}_{1}})^{G_{k}}$
[CT-CH2-descent]*Corollaire 4.2, and in general the containment can be strict.

######  Corollary 4.5.

Let $k$ be a field of characteristic different from $2$. Assume that
$X(k)\neq\emptyset$ and $\tilde{P}^{(1)}(k)\neq\emptyset$. If
$\tilde{\alpha}^{(1)}=0$, then
$\operatorname{CH}^{2}(X)^{\tilde{\gamma}_{1}}=(\operatorname{CH}^{2}(X_{{\overline{k}}})^{\tilde{\gamma}_{1}})^{G_{k}}$.

###### Proof.

Let $x\in\tilde{P}^{(1)}(k)$. By [FJSVV]*Theorem 6.4 the bijection
$(\operatorname{CH}^{2}(X_{\overline{k}})^{\tilde{\gamma}_{1}})^{G_{k}}\xrightarrow{\sim}\tilde{P}^{(1)}(k)$
can be realized explicitly as $p_{*}j^{*}\varepsilon^{*}$ where $p,j$ and
$\varepsilon$ are defined in [FJSVV]*(5.3). Thus, we will show that there
exists a curve $\gamma\subset X$, defined over $k$ such that
$p_{*}j^{*}\varepsilon^{*}[\gamma]=x\in\tilde{P}^{(1)}(k)$. If
$\tilde{\alpha}^{(1)}=0$ then, by Theorem 4.2, $\pi|_{\ell_{x}}\colon
X_{\ell_{x}}\to\ell_{x}$ has a section defined over $k$. In particular
$X_{\ell_{x}}(k)\neq\emptyset$, so by [FJSVV]*Proposition 6.5 there is a curve
$\gamma$, defined over $k$, such that
$p_{*}j^{*}\varepsilon^{*}[\gamma]=x\in\tilde{P}^{(1)}(k)$. ∎

##  5\. Double covers of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ and
obstructions to sections of $\pi_{1}$

Any smooth double cover of
${\mathbb{P}}^{1}_{[t_{0}:t_{1}]}\times{\mathbb{P}}^{2}_{[u:v:w]}$ branched
over a $(2,2)$-divisor has equation

$z^{2}=t_{0}^{2}Q_{1}+2t_{0}t_{1}Q_{2}+t_{1}^{2}Q_{3},$ (5.1)

for a tuple of quadratic forms $Q_{1},Q_{2},Q_{3}\in k[u,v,w]$. Further the
quadratic forms are well-defined up to an action of
$\operatorname{Aut}({\mathbb{P}}^{1})$, and this action agrees with the action
of $\operatorname{PGL}_{2}$ on the defining equations of $\tilde{\Delta}$ and
$\Delta$ described in Section 2.2, so the isomorphism class of the double
cover depends only on $\tilde{\Delta}/\Delta$. We write $\tilde{\pi}\colon
Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}_{[t_{0}:t_{1}]}\times{\mathbb{P}}^{2}_{[u:v:w]}$
for this isomorphism class. If $\tilde{\Delta}$ and $\Delta$ are smooth, then
so is $Y_{\tilde{\Delta}/\Delta}$ [FJSVV]*Theorem 6.1(a).

The second projection $\pi\colon Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{2}$
gives $Y=Y_{\tilde{\Delta}/\Delta}$ the structure of a standard conic bundle
with discriminant cover $\tilde{\Delta}/\Delta$, and the first projection
$\pi_{1}\colon Y\to{\mathbb{P}}^{1}$ gives $Y$ the structure of a quadric
surface bundle. By [FJSVV]*Theorem 4.5(ii) and Proposition 6.3(iii), the Prym
curve $\Gamma=\Gamma_{\tilde{\Delta}/\Delta}$ arises from the Stein
factorization of ${\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})\to{\mathbb{P}}^{1}$,
so $[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$ gives a Brauer class on $\Gamma$.
Furthermore, $\pi_{1}$ has a section if and only if
$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]=0\in\operatorname{Br}\Gamma$.333Any
quadric surface has a point if and only if it has lines defined over the
discriminant extension. Indeed, if the quadric surface has a point, then
slicing the surface with the tangent plane at that point will give a
geometrically reducible conic whose geometric components are defined over the
discriminant extension. Conversely, if there are lines defined over the
discriminant extension, then the intersection of a line and its Galois
conjugate will give a rational point.

######  Theorem 5.1.

Let $k$ be a field of characteristic different from $2$, let
$\varpi\colon\tilde{\Delta}\to\Delta$ be a geometrically irreducible étale
double cover of a smooth plane quartic over $k$, and let
$Y=Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ be the
double cover associated to $\tilde{\Delta}/\Delta$. Then for any
$x\in\tilde{P}^{(1)}(k)$ and any
$w\in(\ell_{x}\smallsetminus(\ell_{x}\cap\Delta))(k)$ ,

$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]=[Y_{w}]_{\Gamma}=(\tilde{\alpha}^{(1)})_{\Gamma}\in\operatorname{Br}\Gamma.$

###### Proof.

The second equality holds by Definition 4.3. For the first equality, it
suffices to show equality in $\operatorname{Br}{\mathbf{k}}(\Gamma)$. By
[Amitsur]*Corollary 9.5, if two finite-dimensional central simple algebras
over ${\mathbf{k}}(\Gamma)$ have the same splitting fields over
${\mathbf{k}}(\Gamma)$, then they generate the same subgroup in
$\operatorname{Br}{\mathbf{k}}(\Gamma)$. Since $[Y_{w}]$ and
$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$ are both $2$-torsion, showing they
have the same splitting fields will imply the desired equality.

Let $k^{\prime}$ be an extension of ${\mathbf{k}}(\Gamma)$ which is a
splitting field for $[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$. Then the
generic fiber of $Y\to{\mathbb{P}}^{1}$ contains lines over $k^{\prime}$, so,
over $k^{\prime}$, the generic fiber becomes isomorphic to
${\mathbb{P}}^{1}_{k^{\prime}}\times{\mathbb{P}}^{1}_{k^{\prime}}$. In
particular, every hyperplane section of the generic fiber contains
$k^{\prime}$ points. Note that $Y_{\ell_{x}}$ is a hyperplane section of the
generic fiber, so then $Y_{\ell_{x}}(k^{\prime})\neq\emptyset$. By Theorem
4.2, $Y_{\ell_{x}}$ is Brauer equivalent to $Y_{w}$, so
$Y_{w}(k^{\prime})\neq\emptyset.$

Now assume that $k^{\prime}$ is an extension of ${\mathbf{k}}(\Gamma)$ which
is a splitting field for $Y_{w}$. Then since $Y_{w}\subset Y_{\ell_{x}}$,
$Y_{\ell_{x}}(k^{\prime})\neq\emptyset$, so [FJSVV]*Prop. 6.5 implies that
$Y\to{\mathbb{P}}^{1}$ has a section over an odd degree extension of
$k^{\prime}$. By Springer’s theorem [Springer52], $Y\to{\mathbb{P}}^{1}$ then
has a section over $k^{\prime}$ so $[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$
is trivial in $\operatorname{Br}k^{\prime}$. ∎

######  Corollary 5.2.

Let $\tilde{\Delta}$,$\Delta$ and $Y=Y_{\tilde{\Delta}/\Delta}$ be as in
Theorem 5.1. Consider the following conditions:

  1. (1)

$\tilde{P}^{(1)}(k)\neq\emptyset$ and
$\tilde{\alpha}^{(1)}\in\ker(\operatorname{Br}k\to\operatorname{Br}\Gamma)$
(i.e., $(\tilde{\alpha}^{(1)})_{\Gamma}=0\in\operatorname{Br}\Gamma$).

  2. (2)

$\pi_{1}\colon Y\to{\mathbb{P}}^{1}$ has a section (over $k$).

  3. (3)

$Y$ is rational (over $k$).

Then (1)$\Rightarrow$(2)$\Rightarrow$(3).

###### Proof.

(1)$\Rightarrow$(2): By Theorem 5.1,
$(\tilde{\alpha}^{(1)})_{\Gamma}=[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]\in\operatorname{Br}\Gamma$,
and hence
$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]=0\in\operatorname{Br}\Gamma$.
Therefore $\pi_{1}$ has a section (see Footnote 3).

(2)$\Rightarrow$(3): The generic fiber of $\pi_{1}$ is a quadric, which is
rational as soon as it has a point. ∎

######  Remark 5.3.

By Proposition 2.3, the vanishing of the IJT obstruction for $Y$ is equivalent
to either $\tilde{P}^{(1)}(k)\neq\emptyset$ or $\tilde{P}(k)\neq\emptyset$.
Thus, the condition (1) in Corollary 5.2 is stronger than assuming only that
the IJT obstruction vanishes.

######  Corollary 5.4.

Let $k$ be a local field of characteristic different from $2$, let
$\tilde{\Delta},\Delta$ be as above, and let
$Y=Y_{\tilde{\Delta}/\Delta}\to{\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ be the
double cover associated to $\tilde{\Delta}/\Delta$. If $\Gamma$ has index $2$,
i.e., if $\Gamma(L)=\emptyset$ for all extensions $L/k$ of odd degree, then
$Y$ is rational if and only if the IJT obstruction vanishes.

###### Proof.

We will first show that the vanishing of the IJT obstruction is equivalent to
(1) in Corollary 5.2. First, $\operatorname{{\bf
Pic}}^{1}_{\Gamma}(k)\neq\emptyset$ since $k$ is a local field [Lichtenbaum]
(see Remark 7.3i for $k={\mathbb{R}}$). So by Proposition 2.3, the vanishing
of the IJT obstruction is equivalent to $\tilde{P}^{(1)}(k)\neq\emptyset$, and
in this setting we may define $\tilde{\alpha}^{(1)}\in\operatorname{Br}k[2]$
(see Definition 4.3). Furthermore, $\operatorname{Pic}^{1}(\Gamma)=\emptyset$
since $\Gamma$ has index $2$, so the map
$\operatorname{Pic}(\Gamma)\to\operatorname{{\bf Pic}}_{\Gamma}(k)$ has a
nontrivial element of order $2$ in its cokernel. Since
$\operatorname{Br}k[2]\simeq{\mathbb{Z}}/2{\mathbb{Z}}$, the low degree terms
of the Hochschild–Serre spectral sequence imply that
$\operatorname{Br}k[2]\to\operatorname{Br}\Gamma[2]$ is identically $0$. In
particular, the condition
$\tilde{\alpha}^{(1)}\in\ker(\operatorname{Br}k\to\operatorname{Br}\Gamma)$
always holds. This shows that (1) holds if and only if the IJT obstruction
vanishes.

Corollary 5.2 shows (1)$\Rightarrow$(3). Conversely, if (3) holds, then the
IJT obstruction vanishes by [bw-ij]*Theorem 3.11. ∎

##  6\. Proof of Theorem 1.1

Let $Q_{1},Q_{2},Q_{3}\in k[u,v,w]$ be homogeneous forms defining a smooth
plane quartic $\Delta$ and a geometrically integral étale double cover
$\tilde{\Delta}\to\Delta$ as in (2.2)–(2.3). By assumption, there exists
$y\in\Gamma_{\tilde{\Delta}/\Delta}(k)$ which, without loss of generality, we
assume lies over $[1:0]\in{\mathbb{P}}^{1}(k)$ so that $Q_{1}$ has square
discriminant. By Proposition 3.1, there exists an intersection of quadrics
$Z\subset{\mathbb{P}}^{5}$ containing a conic $C$ such that
$\operatorname{Bl}_{C}Z\to{\mathbb{P}}^{2}$ has discriminant cover
$\tilde{\Delta}/\Delta$. Furthermore, the generic fiber of the conic bundle
has Brauer class

$[(\operatorname{Bl}_{C}Z)_{\eta}]=(Q_{1},Q_{2}^{2}-Q_{1}Q_{3})+[Q_{1}]\in\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2}).$

Since $Y$ is defined by the equation
$z^{2}=Q_{1}(u,v,w)t_{0}^{2}+2Q_{2}(u,v,w)t_{0}t_{1}+Q_{3}(u,v,w)t_{1}^{2}$,
completing the square shows that the Brauer class of generic fiber of the
conic bundle $\pi_{2}\colon Y\to{\mathbb{P}}^{2}$ is equal to the quaternion
algebra $(Q_{1},Q_{2}^{2}-Q_{1}Q_{3})$. In particular,

$[(\operatorname{Bl}_{C}Z)_{\eta}]-[Y_{\eta}]=[Q_{1}]\in\operatorname{Br}k[2].$

On the other hand, the fiber of $\pi_{1}\colon Y\to{\mathbb{P}}^{1}$ above
$[1:0]$ is given by $z^{2}=Q_{1}$, so $[Q_{1}]$ is equal to the evaluation of
$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$ at $y$. By Theorem 5.1,
${\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})$ is a constant algebra, so it is equal
to its evaluation at $y$. Furthermore, the class of the relative Fano variety
of lines $[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]$ is exactly the obstruction
to $\pi_{1}$ having a section (see Footnote 3). Finally, we deduce that
$\operatorname{Bl}_{C}Z$ is rational by Corollary 3.2.∎

##  7\. Real topological consequences of Theorem 1.1

Throughout this section we work over ${\mathbb{R}}$, where the real topology
gives us additional tools to attack rationality. In particular, we are able to
show that a necessary condition for rationality is that
$\tilde{\alpha}^{(1)}\in\ker(\operatorname{Br}{\mathbb{R}}\to\operatorname{Br}\Gamma)$
for all but one isotopy class of $\Delta({\mathbb{R}})$.

######  Remark 7.1.

The rigid isotopy classes of smooth quartic plane curves over ${\mathbb{R}}$
are $\emptyset$, one oval, two nested ovals, two non-nested ovals, three
ovals, and four ovals [Zeuthen1874], where an oval is a homotopically trivial
simple closed curve in ${\mathbb{P}}^{2}({\mathbb{R}})$ [Mangolte-
real]*Definition 2.7.7.

######  Theorem 7.2.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be an étale double cover of a smooth
plane quartic over ${\mathbb{R}}$, and let $Y_{\tilde{\Delta}/\Delta}$ be the
double cover defined in (5.1). Assume that $\Delta({\mathbb{R}})$ is not a
single oval. Then the following are equivalent.

  1. (1)

The IJT obstruction vanishes and
$\tilde{\alpha}^{(1)}\in\ker(\operatorname{Br}{\mathbb{R}}\to\operatorname{Br}\Gamma)$.

  2. (2)

$\pi_{1}\colon Y\to{\mathbb{P}}^{1}$ has a section.

  3. (3)

$Y$ is rational.

  4. (4)

The IJT obstruction vanishes and $Y({\mathbb{R}})$ is connected.

######  Remarks 7.3.

  1. (i)

If the IJT obstruction vanishes, then the class $\tilde{\alpha}^{(1)}$ is
defined. Indeed, $\operatorname{{\bf
Pic}}^{1}_{\Gamma}\simeq\operatorname{{\bf Pic}}^{3}_{\Gamma}$ has an
${\mathbb{R}}$-point because there exists a Galois-invariant partition of the
Weierstrass points of $\Gamma$ into two sets of cardinality $3$. Then
Proposition 2.3 shows that the vanishing of the IJT obstruction is equivalent
to $\tilde{P}^{(1)}({\mathbb{R}})\neq\emptyset$.

  2. (ii)

The implications (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4) hold
without any assumptions on $\Delta({\mathbb{R}})$.

  3. (iii)

When $\Delta({\mathbb{R}})$ is not two nested ovals (nor a single oval), the
equivalences (2)$\Leftrightarrow$(3)$\Leftrightarrow$(4) were proven in
[JJ23]*Theorem 1.2.

  4. (iv)

There are examples where $Y$ is rational over ${\mathbb{R}}$ and
$\tilde{\alpha}^{(1)}\in\operatorname{Br}{\mathbb{R}}$ is the nontrivial
element; see [JJ23]*Example 4.10(1). If $\Delta({\mathbb{R}})$ is not a single
oval, such examples necessarily have that
$\operatorname{Br}{\mathbb{R}}\to\operatorname{Br}\Gamma$ is identically zero,
which is equivalent to $\Gamma({\mathbb{R}})=\emptyset$.

###### Proof.

(1)$\Rightarrow$(2)$\Rightarrow$(3) by Corollary 5.2. The implication
(3)$\Rightarrow$(4) follows because the number of connected components of
$Y({\mathbb{R}})$ is a birational invariant [DelfsKnebusch]*Theorem 13.3 and
the IJT obstruction is an obstruction to rationality (see Section 2.2.1).

(4)$\Rightarrow$(1): If $\Gamma({\mathbb{R}})=\emptyset$, then this follows
from Corollary 5.4. Now assume that $\Gamma({\mathbb{R}})\neq\emptyset$. Since
the IJT obstruction vanishes, $\tilde{P}^{(1)}({\mathbb{R}})\neq\emptyset$ and
so by Theorem 4.2, $\tilde{\alpha}^{(1)}$ is defined. In this case
$\operatorname{Br}{\mathbb{R}}\to\operatorname{Br}\Gamma$ is injective
(evaluating at a point of $\Gamma({\mathbb{R}})$ gives a section), so we must
show that $\tilde{\alpha}^{(1)}=0$. Assume not. By Theorem 1.1, there is a
rational threefold $X^{\prime}$ and a standard conic bundle $\psi\colon
X^{\prime}\to{\mathbb{P}}^{2}$ such that
$[Y_{\eta}]-[X^{\prime}_{\eta}]\in\operatorname{im}(\operatorname{Br}{\mathbb{R}}\hookrightarrow\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2}))$
is the obstruction to the existence of a section to $\pi_{1}$, thus is equal
to
$[{\mathcal{F}}_{1}(Y/{\mathbb{P}}^{1})]\in\operatorname{im}(\operatorname{Br}{\mathbb{R}}\hookrightarrow\operatorname{Br}\Gamma)$.
Further, Theorems 4.2 and 5.1 imply that
$[Y_{\eta}]-[X^{\prime}_{\eta}]=\tilde{\alpha}^{(1)}$. Since we assumed
$\tilde{\alpha}^{(1)}$ is nontrivial,
$\pi(Y({\mathbb{R}})-Y_{\Delta}({\mathbb{R}}))=\overline{{\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus(\psi(X^{\prime}({\mathbb{R}}))\cup\Delta({\mathbb{R}})})$.
Additionally, $\pi|_{Y_{\Delta}({\mathbb{R}})}\colon
Y_{\Delta}({\mathbb{R}})\to\Delta({\mathbb{R}})$ is surjective, so
$\pi(Y({\mathbb{R}}))=\overline{{\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\psi(X^{\prime}({\mathbb{R}}))}$.
Furthermore, both $\pi(Y({\mathbb{R}}))$ and $\psi(X^{\prime}({\mathbb{R}}))$
are nonempty and connected so Corollary 7.5(2) below implies that
$\Delta({\mathbb{R}})$ is a single oval and we arrive at a contradiction. ∎

###  7.1. Real points on conic bundles

######  Proposition 7.4.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be a geometrically integral étale
double cover of a smooth plane curve over ${\mathbb{R}}$, and let $\pi\colon
X\to{\mathbb{P}}^{2}_{{\mathbb{R}}}$ be a conic bundle with discriminant cover
$\tilde{\Delta}/\Delta$. Then
$\partial(\pi(X({\mathbb{R}})))=\Delta({\mathbb{R}})\smallsetminus\varpi(\tilde{\Delta}({\mathbb{R}}))$.

######  Corollary 7.5.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be a geometrically integral étale
double cover of a smooth plane curve over ${\mathbb{R}}$, and let $\pi\colon
X\to{\mathbb{P}}^{2}_{{\mathbb{R}}}$ be a conic bundle with discriminant cover
$\varpi\colon\tilde{\Delta}\to\Delta$.

  1. (1)

Then $\pi(X({\mathbb{R}}))$ and
$({\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\pi(X({\mathbb{R}})))$ are both
nonempty and connected if and only if
$\Delta({\mathbb{R}})\smallsetminus\varpi(\tilde{\Delta}({\mathbb{R}}))$ is a
simple closed curve.

  2. (2)

Then $\pi(X({\mathbb{R}}))$ and
$({\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\pi(X({\mathbb{R}})))\cup\Delta({\mathbb{R}})$
are both nonempty and connected if and only if $\Delta({\mathbb{R}})$ is a
simple closed curve.

######  Remark 7.6.

Note that we make no assumption on the degree of $\Delta$.

###### Proof of Corollary 7.5.

The loci $\pi(X({\mathbb{R}}))$ and
$({\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\pi(X({\mathbb{R}})))$ are both
nonempty and connected if and only if the boundary
$\partial(\pi(X({\mathbb{R}})))$ is connected. Thus, part (1) follows from
Proposition 7.4, the Jordan curve theorem, and the classification of the real
loci of smooth plane curves [Mangolte-real]*Section 2.7. Part (2) follows from
part (1) because
$({\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\pi(X({\mathbb{R}})))\cup\Delta({\mathbb{R}})$
is equal to
$\overline{({\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\pi(X({\mathbb{R}})))}\sqcup\varpi(\tilde{\Delta}({\mathbb{R}}))$.
∎

###### Proof of Proposition 7.4.

Fix a Zariski affine open $U=\operatorname{Spec}A\subset{\mathbb{P}}^{2}$
where $X_{U}$ can be viewed as a subvariety of ${\mathbb{P}}^{2}_{A}$, and fix
a defining quadratic equation $F$ for $X_{U}$. Since $\pi$ is smooth away from
$\Delta$, the signature of $F$ is constant on real connected components of
$U({\mathbb{R}})\smallsetminus(\Delta\cap U)({\mathbb{R}})$. Note that a
connected component $B$ of $U({\mathbb{R}})\smallsetminus(\Delta\cap
U)({\mathbb{R}})$ is in the image of $X_{U}({\mathbb{R}})$ if and only if
$F|_{B}$ is an indefinite quadratic form.

Fix a real connected component $c$ of $(\Delta\cap U)({\mathbb{R}})$ and let
$B_{1},B_{2}$ be the two distinct connected components of
$U({\mathbb{R}})\smallsetminus(\Delta\cap U)({\mathbb{R}})$ such that
$B_{1}\cup B_{2}\cup c$ becomes connected, i.e., $B_{1},B_{2}$ are the
immediate interior and immediate exterior of $c$. The proposition now follows
from the following two claims.

  1. (1)

The signatures of $F|_{B_{1}}$ and $F|_{B_{2}}$ differ by exactly one sign. In
particular, we may assume that $F|_{B_{1}}$ is indefinite.

  2. (2)

The quadratic form $F|_{B_{2}}$ is also indefinite if and only if
$c\subset\varpi(\tilde{\Delta}({\mathbb{R}}))$.

By continuity, it suffices to prove both claims in a (Euclidean) open
neighborhood around a point $x_{0}\in c\subset(\Delta\cap U)({\mathbb{R}})$.
In an open neighborhood of $x_{0}$, we may change coordinates so that $F$ is
diagonal and two of the coefficients are nowhere vanishing (see [IOOV]*Lemma
6.3). Thus, the sign of those coefficients are equal on $B_{1}$ and $B_{2}$.
Since the vanishing of the discriminant of $F$ exactly agrees with $\Delta$,
the third coefficient must vanish with multiplicity $1$ exactly along
$\Delta$. In particular, it has a different sign on $B_{1}$ than on $B_{2}$.
This completes the proof of the first claim.

To prove the second claim, we assume without loss of generality that the
signature of $F|_{B_{1}}$ is $(+,+,-)$. By the first claim, the signature of
$F|_{B_{2}}$ is either $(+,+,+)$ or $(+,-,-)$. Furthermore, the proof of the
first claim above shows that the signature of $F|_{B_{2}}$ is $(+,-,-)$ if and
only if the signature of $F|_{x_{0}}$ is $(+,0,-)$. Finally, since $\Delta$ is
smooth and $\tilde{\Delta}$ is geometrically integral, $F|_{x_{0}}$ is a rank
$2$ quadratic form, and its vanishing has a smooth ${\mathbb{R}}$-point if and
only if $x_{0}\in\varpi(\tilde{\Delta}({\mathbb{R}}))$. This completes the
proof of the second claim. ∎

The outside of $\Delta({\mathbb{R}})$ is the unique non-orientable component
of ${\mathbb{P}}^{2}({\mathbb{R}})\setminus\Delta({\mathbb{R}})$ (which exists
by [Mangolte-real]*Section 2.7).

######  Proposition 7.7.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be an étale double cover of a smooth
plane quartic over ${\mathbb{R}}$, and let $Y_{\tilde{\Delta}/\Delta}$ be the
double cover defined in (5.1). Assume that
$\tilde{P}^{(1)}({\mathbb{R}})\neq\emptyset$. Then the class
$\tilde{\alpha}^{(1)}$ is trivial if and only if the image of
$Y({\mathbb{R}})\to{\mathbb{P}}^{2}({\mathbb{R}})$ contains the outside of
$\Delta({\mathbb{R}})$.

###### Proof.

Let $x\in\tilde{P}^{(1)}({\mathbb{R}})$; write $D_{x}\in\operatorname{{\bf
Pic}}^{4}_{\tilde{\Delta}}$ for the associated effective divisor (see (2.6))
and $\ell_{x}$ for the line in ${\mathbb{P}}^{2}$ spanned by
$\varpi_{*}D_{x}$. The key observation is that
$\ell_{x}({\mathbb{R}})\smallsetminus(\ell_{x}\cap\Delta)({\mathbb{R}})$ meets
the outside of $\Delta({\mathbb{R}})$. If $\tilde{\alpha}^{(1)}=0$, then
$Y_{\ell_{x}}({\mathbb{R}})\to\ell_{x}({\mathbb{R}})$ is surjective by Theorem
4.2, and hence the image of $Y({\mathbb{R}})\to{\mathbb{P}}^{2}({\mathbb{R}})$
meets (and consequently contains) the outside of $\Delta({\mathbb{R}})$.
Conversely, if the image of $Y({\mathbb{R}})\to{\mathbb{P}}^{2}({\mathbb{R}})$
contains the outside of $\Delta({\mathbb{R}})$, then there exists
$w\in\ell_{x}({\mathbb{R}})\smallsetminus(\ell_{x}\cap\Delta)({\mathbb{R}})$
such that $Y_{w}({\mathbb{R}})\neq\emptyset$. Thus $\tilde{\alpha}^{(1)}=0$ by
Theorem 4.2. ∎

###  7.2. Cases when the IJT obstruction or topological obstruction
individually characterize rationality

Theorem 7.2 allows us to extend results of [JJ23] and classify the cases from
Theorem 7.2 when the IJT obstruction or the real topological obstruction
individually characterize rationality.

######  Proposition 7.8.

Let $\varpi\colon\tilde{\Delta}\to\Delta$ be an étale double cover of a smooth
plane quartic over ${\mathbb{R}}$, and let $Y_{\tilde{\Delta}/\Delta}$ be the
double cover defined in (5.1). Assume that $\Delta({\mathbb{R}})$ is not one
oval.

If $\Delta({\mathbb{R}})$ is two nested ovals, three ovals, or four ovals,
then the four conditions of Theorem 7.2 are also equivalent to

  1. (5)

$Y({\mathbb{R}})$ is connected.

If $\Delta({\mathbb{R}})$ is two nested ovals or four ovals then (5) is
additionally equivalent to

  1. (6)

The IJT obstruction vanishes for $Y$.

Moreover, the assumptions above are the strongest possible, i.e., if
$\Delta({\mathbb{R}})$ is empty or two non-nested ovals then connectedness is
not equivalent to rationality, and if $\Delta({\mathbb{R}})$ is empty, two
non-nested ovals, or three ovals then vanishing of the IJT obstruction is not
equivalent to rationality.

######  Remark 7.9.

The part of the proposition that is new is the case of two nested ovals; the
rest follows from prior work. If $\Delta({\mathbb{R}})$ is four ovals, then
$Y$ is always rational [JJ23]*Theorem 1.2(4) so the theorem vacuously holds.
If $\Delta({\mathbb{R}})$ is three ovals, then the result that connectedness
of $Y({\mathbb{R}})$ characterizes rationality is [JJ23]*Theorem 1.2(3). That
connectedness fails to characterize rationality if $\Delta({\mathbb{R}})$ is
empty or two non-nested ovals is [JJ23]*Examples 4.2 and 4.4. That the IJT
obstruction fails to characterize rationality if $\Delta({\mathbb{R}})$ is two
non-nested ovals is shown by [FJSVV]*Theorem 1.4(i); if $\Delta({\mathbb{R}})$
is empty or three ovals, this is shown by [JJ23]*Examples 4.1 and 4.6.

###### Proof.

Given the remark above, we restrict ourselves to the case that
$\Delta({\mathbb{R}})$ is two nested ovals. In this case, (6)$\Rightarrow$(5)
by [JJ23]*Corollary 3.9. This shows that (6) is equivalent to (4).

It remains to show that (5)$\Rightarrow$(6). Assume for the sake of
contradiction that (5) holds and that $Y$ has an IJT obstruction to
rationality over ${\mathbb{R}}$. Then $\pi_{1}$ does not have a section by
Theorem 7.2, so $\pi_{1}$ is not surjective on ${\mathbb{R}}$-points by [Witt-
quadratic-forms]*Satz 22. Thus there exists a point
$t\in{\mathbb{P}}^{1}({\mathbb{R}})$ such that
$Y_{t}({\mathbb{R}})=\emptyset$, and so is defined by a definite quadratic
form. In particular, there exists $z\in\Gamma_{t}({\mathbb{R}})$ with
${\mathcal{F}}(Y/{\mathbb{P}}^{1})_{z}=\mathbb{H}$. By Proposition 3.1 and
Corollary 3.2, there exists an irrational complete intersection
$Z\subset{\mathbb{P}}^{5}$ of two quadrics, containing a conic $C$, such that
$\psi\colon\operatorname{Bl}_{C}Z\to{\mathbb{P}}^{2}$ is a conic bundle with

$[(\operatorname{Bl}_{C}Z)_{\eta}]=[Y_{\eta}]+\mathbb{H}\in\operatorname{Br}{\mathbf{k}}({\mathbb{P}}^{2}),$

so the image $\psi((\operatorname{Bl}_{C}Z)({\mathbb{R}}))$ is equal to
$\overline{{\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\psi(Y({\mathbb{R}}))}\cup\Delta({\mathbb{R}})$.
Since $Y({\mathbb{R}})$ and $(\operatorname{Bl}_{C}Z)({\mathbb{R}})$ are both
3-manifolds, the images $\pi(Y({\mathbb{R}}))$ and
$\psi((\operatorname{Bl}_{C}Z)({\mathbb{R}}))$ do not have any connected
components that are 1-manifolds. Thus, since $Y({\mathbb{R}})$ is connected
and $\Delta({\mathbb{R}})$ is two nested ovals, the image
$\pi(Y({\mathbb{R}}))$ must be the region (diffeomorphic to an annulus)
enclosed by the two ovals of $\Delta({\mathbb{R}})$. This implies
$Z({\mathbb{R}})$ is disconnected, so Krasnov’s real isotopy classification of
3-dimensional intersections of two quadrics [krasnov-biquadrics]*Theorem 5.4
(see also [HT-intersection-quadrics]*Sections 11.2 and 11.4) implies that
$Z({\mathbb{R}})$ is diffeomorphic to $\mathbb{S}^{3}\sqcup\mathbb{S}^{3}$ and
that the Prym curve $\Gamma$ has 6 real Weierstrass points. On the other hand,
since we’ve assumed that
$\pi_{1}(Y({\mathbb{R}}))\subsetneq{\mathbb{P}}^{1}({\mathbb{R}})$ is
connected, the fact that $\Gamma$ has 6 real Weierstrass points implies that
$\pi_{1}\colon Y\to{\mathbb{P}}^{1}$ must have a smooth fiber with signature
$({+,+,-,-})$. After a coordinate change on ${\mathbb{P}}^{1}$ we may assume
that the fiber of $\pi_{1}$ over $[1:0]$ has signature $({+,+,-,-})$
[FJSVV]*Theorem 4.5, i.e. that the quadratic form $Q_{1}$ has signature
$({+,+,-})$. Then $(Q_{1}\geq
0)={\mathbb{P}}^{2}({\mathbb{R}})\smallsetminus\mathbb{D}^{2}\subset{\mathbb{P}}^{2}({\mathbb{R}})$,
and this region is contained in $\pi(Y({\mathbb{R}}))$ by [JJ23]*Lemma 2.8.
This contradicts our earlier conclusion that $Y({\mathbb{R}})$ is the region
enclosed by the two ovals of $\Delta({\mathbb{R}})$, so we have shown (5)
implies (6). ∎

## References