# Weak weak approximation and the Hilbert property for degree-two del Pezzo
surfaces

Julian Demeio  Julian Demeio
Department of Mathematics and Computer Science
University of Basel
Spiegelgasse 1
4051 Basel
Switzerland<EMAIL_ADDRESS>,  Sam Streeter  Sam Streeter
School of Mathematics
University of Bristol
Woodland Road
Bristol
BS1 8UG
UK<EMAIL_ADDRESS>and  Rosa Winter  Rosa Winter
Department of Mathematics
King’s College London
Strand Campus
London
WC2R 2LS
UK<EMAIL_ADDRESS>
###### Abstract.

We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak
weak approximation if $k$ is a number field and the Hilbert property if $k$ is
Hilbertian of characteristic zero, provided that they contain a $k$-rational
point lying neither on any $4$ of the $56$ exceptional curves nor on the
ramification divisor of the anticanonical morphism. This builds upon results
of Manin, Salgado–Testa–Várilly-Alvarado, and Festi–van Luijk on the
unirationality of such surfaces, and upon work of the first two authors
verifying weak weak approximation under the assumption of a conic fibration.

###### 2020 Mathematics Subject Classification:

14G05, 14G12 (14G25, 14G27, 11G35).

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Two geometric procedures
  4. 4 Proof of main results

##  1\. Introduction

A central belief in arithmetic geometry is that, if a Fano variety has one
rational point, then it has many, and that they are well-distributed.
Generally, the difficulty of verifying this belief increases with dimension.
For curves, the situation is simple: a smooth projective curve is Fano if and
only if it has genus $0$, in which case it is isomorphic to $\mathbb{P}^{1}$
as soon as it has a rational point. For varieties of dimension at least three,
little is known other than results via the circle method; the two-dimensional
case, that of _del Pezzo surfaces_ , has been a major area of research and
progress in the last fifty years.

One property capturing the abundance and distribution of rational points is
_weak approximation_ , satisfied by a variety $X$ over a number field $K$ if
its set $X\mathopen{}\mathclose{{}\left(K}\right)$ of rational points is dense
in $\prod_{v\in\Omega_{K}}X\mathopen{}\mathclose{{}\left(K_{v}}\right)$, where
$\Omega_{K}$ is the set of places of $K$. Many varieties fail weak
approximation, but this failure can sometimes be explained. In 1970, Manin
[20] gave a pairing between the _Brauer group_
$\operatorname{Br}X:=H^{2}_{\text{\'{e}t}}\mathopen{}\mathclose{{}\left(X,\mathbb{G}_{m}}\right)$
and the adelic points $X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)$
(equal to $\prod_{v\in\Omega_{K}}X(K_{v})$ for $X$ projective). The kernel
$X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)^{\operatorname{Br}}$ is
closed in $X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)$ and contains
$X\mathopen{}\mathclose{{}\left(K}\right)$. Hence, if
$X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)^{\operatorname{Br}}\subsetneq
X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)$, then $X$ fails weak
approximation, and we say that there is a _Brauer–Manin obstruction to weak
approximation for $X$_.

While certain del Pezzo surfaces exhibit this obstruction (see [22, §9.4] for
examples), Colliot-Thélène has conjectured [3, Conj. 14.1.2] that it is the
only one to weak approximation for a large class of varieties including Fano
varieties.

######  Conjecture A (Colliot-Thélène).

For $X$ a geometrically rationally connected smooth projective variety over a
number field $K$, we have
$\overline{X\mathopen{}\mathclose{{}\left(K}\right)}=X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)^{\operatorname{Br}}$.

For such $X$, we have
$X\mathopen{}\mathclose{{}\left(\mathbb{A}_{K}}\right)^{\operatorname{Br}}=U\times\prod_{v\not\in
S}X\mathopen{}\mathclose{{}\left(K_{v}}\right)$ for some finite set
$S\subset\Omega_{K}$ and non-empty open subset $U\subseteq\prod_{v\in
S}X(K_{v})$ [3, (6), p. 347]. Then Colliot-Thélène’s conjecture implies that
any del Pezzo surface $X/K$ satisfies _weak weak approximation_ , meaning that
$X\mathopen{}\mathclose{{}\left(K}\right)$ is dense in $\prod_{v\not\in
S}X\mathopen{}\mathclose{{}\left(K_{v}}\right)$ for some finite $S$.

Colliot-Thélène and Sansuc [1, p. 190] raised the following question
concerning the abundance of rational points on unirational varieties over a
larger class of fields.

######  Question B (Colliot-Thélène–Sansuc).

For $X$ a projective unirational variety over a Hilbertian field $k$, is
$X(k)$ always non-thin? That is, does $X$ have the Hilbert property?

See Definition 2.2 for the definitions of Hilbertianity and thin sets, and [4,
Question-Conjecture 1] for a recently posed more extensive question when $k$
is a number field.

There are no examples of del Pezzo surfaces with a rational point which are
known to be non-unirational [22, Rem. 9.4.11]. Thus, given a del Pezzo surface
$X$ over a field $k$ with $X(k)\neq\emptyset$, proving that $X$ satisfies the
Hilbert property when $k$ is Hilbertian and weak weak approximation when $k$
is a number field supports Conjecture A and a positive answer to Question B,
and is the focus of this paper.

Del Pezzo surfaces are classified by their degree
$d:=K_{X}^{2}\in\\{1,\dots,9\\}$, where $K_{X}$ is the canonical divisor. As
dimension notionally governs arithmetic complexity for all Fano varieties, so
the degree does for del Pezzo surfaces. Weak approximation when $X$ has a
rational point holds for $d\geq 5$ [22, Thm 9.4.8(vii)], and counterexamples
are given for all $d\leq 4$ ([22, §9.4]). However, Salberger and Skorobogatov
proved [23, Thm. 0.1] that Conjecture A holds for $d=4$ when $X$ has a
rational point, thus implying weak weak approximation. Work of Swinnerton-Dyer
[28] on cubic surfaces gives weak weak approximation when $d=3$, again under
the assumption of a rational point. Examples of degree-$d$ del Pezzo surfaces
with a Brauer–Manin obstruction to weak approximation have been given by
Kresch and Tschinkel [16, §7] when $d=2$ and Várilly-Alvarado [29, Thm. 1.1]
when $d=1$. However, little is known about weak weak approximation for $d=1,2$
outside of recent work of the first two authors [6], which relies on the
surface possessing a conic fibration. By adapting the strategies and methods
of Swinnerton-Dyer used to prove unirationality, we shall prove weak weak
approximation in the case $d=2$ without the requirement of a conic fibration.

###  1.1. Results

######  Theorem 1.1.

Let $X$ be a del Pezzo surface of degree $2$ over a number field $K$. Assume
that there exists a point $P_{0}\in X\mathopen{}\mathclose{{}\left(K}\right)$
lying neither on any $4$ of the $56$ exceptional curves of $X$, nor on the
ramification divisor of the anticanonical morphism $\kappa\colon
X\rightarrow\mathbb{P}^{2}$. Then $X$ satisfies weak weak approximation.

######  Remark 1.2.

Our method does not readily give an explicit description of the “bad set” of
places $S$ implicit in the definition of weak weak approximation. In contrast,
Swinnerton-Dyer (Theorem 3.8) proved that for a (smooth) cubic surface
$W\subset\mathbb{P}^{3}$, the set $S$ can be taken to be the set of (real
archimedean places and) places $v$ such that the scheme-theoretic closure of
$W$ in $\mathbb{P}^{3}_{\mathcal{O}_{K}}$ is singular over the prime ideal
corresponding to $v$. It would be interesting to determine whether a similar
bound on $S$ could be obtained in the degree-$2$ case, as would be implied by
Conjecture A.

Our approach is similar to that of [6], in that we iteratively produce covers
of $X$ with desirable properties (including rationality of the source and
geometric integrality of the generic fiber). Since the construction of these
covers works over any field of characteristic zero (see Note 4.1), we obtain
as a byproduct the following result (see Definition 2.3).

######  Theorem 1.3.

Let $X$ be a del Pezzo surface of degree $2$ over a field $k$ of
characteristic zero. Assume that there exists a point $P_{0}\in
X\mathopen{}\mathclose{{}\left(k}\right)$ as in Theorem 1.1. Then $X$ is
Hilbert-unirational. In particular, if $k$ is Hilbertian, then $X$ satisfies
the Hilbert property.

We immediately obtain the following corollary for del Pezzo surfaces of degree
$1$.

######  Corollary 1.4.

Let $X$ be a del Pezzo surface of degree $1$ over a field $k$ of
characteristic zero. Assume that one of the $240$ exceptional curves of $X$ is
defined over $k$.

  1. (i)

If $k$ is a number field, then $X$ satisfies weak weak approximation.

  2. (ii)

If $k$ is Hilbertian, then $X$ satisfies the Hilbert property.

Indeed, blowing down the exceptional curve in Corollary 1.4, we obtain a point
$P_{0}$ on the resulting degree-$2$ del Pezzo surface satisfying the
hypotheses of Theorem 1.1.

###  1.2. Conventions and definitions

We denote by $\operatorname{Spec}R$ the spectrum of a ring $R$ with the
Zariski topology. A scheme over $R$ is a scheme $X$ equipped with a morphism
$X\rightarrow\operatorname{Spec}R$. We denote by
$X\mathopen{}\mathclose{{}\left(R}\right)$ the set of $R$-rational points of
$X$, i.e. the set of sections of $X\rightarrow\operatorname{Spec}R$. Given
morphisms of schemes $X\rightarrow Z$ and $Y\rightarrow Z$, we denote by
$X\times_{Z}Y$ their fiber product. Given a scheme $X$ over $R$ and a morphism
$\operatorname{Spec}S\rightarrow\operatorname{Spec}R$, we denote by $X_{S}$
the fiber product $X\times_{\operatorname{Spec}R}\operatorname{Spec}S$, the
base change of $X$ to $S$. A variety over a field $k$ is a geometrically
integral separated scheme of finite type over $k$. We denote by $\overline{k}$
a separable closure of $k$. When working on a variety $X$ over $k$, we use the
word “points” to refer to points in
$X\mathopen{}\mathclose{{}\left(\overline{k}}\right)$ unless specified
otherwise (and the notation $P\in X$ will accordingly mean that $P\in
X(\overline{k})$), following classical terminology as in [14, Ch. 1].

######  Definition 1.5.

For a divisor $D$ and points $P_{1},\ldots,P_{n}$ on a variety $X$, we denote
by $|D-\sum_{i=1}^{n}a_{i}P_{i}|$ the linear system of divisors in $|D|$
containing each $P_{i}$ with multiplicity at least $a_{i}$.

######  Definition 1.6.

Let $\phi\colon Y\dashrightarrow X$ be a rational map of varieties. The
_domain_ of $\phi$ is defined to be the the maximum open set $U\subset Y$ on
which $\phi$ is defined as a morphism. The _image_ $\phi(V)$ of a subset
$V\subseteq Y$ is defined to be $\phi(U\cap V)$. The _closed image_ of $\phi$
is defined to be the closure of $\phi\mathopen{}\mathclose{{}\left(U}\right)$
in $X$.

For a scheme $X$ of finite type over a number field $K$ and
$S\subset\Omega_{K}$ finite, a model for $X$ over $\mathcal{O}_{K,S}$ is the
datum of a scheme $\mathcal{X}\to\operatorname{Spec}\mathcal{O}_{K,S}$ of
finite type and an isomorphism $\iota\colon
X\xrightarrow{\sim}\mathcal{X}_{K}=\mathcal{X}\times_{\mathcal{O}_{K,S}}K$. We
will always (abusively) identify $X$ with $\mathcal{X}_{K}$ via $\iota$.

###  1.3. Acknowledgements

We thank Jean-Louis Colliot-Thélène, Daniel Loughran, Cecília Salgado and
Alexei Skorobogatov for useful discussions and feedback. Meetings to complete
this work were made possible by funding from ICMS Edinburgh and Pierre Le
Boudec’s SNSF Professorship grant. The first named author started this project
while being a guest at the Max Planck Institute of Bonn, and he thanks it for
its wonderful hospitality and optimal working conditions. The second named
author was supported by the University of Bristol and the Heilbronn Institute
for Mathematical Research. The third named author was supported by UKRI
fellowship MR/T041609/1.

##  2\. Background

###  2.1. Hilbert property

######  Definition 2.1.

Let $X$ be a quasi-projective variety over a field $k$, and take $A\subset
X\mathopen{}\mathclose{{}\left(k}\right)$.

We say that $A$ is of _type I_ if $A\subset
W\mathopen{}\mathclose{{}\left(k}\right)$ for some closed $W\subsetneq X$,
i.e. $A$ is not dense.

We say that $A$ is of _type II_ if
$A\subset\phi\mathopen{}\mathclose{{}\left(Y\mathopen{}\mathclose{{}\left(k}\right)}\right)$
for $Y$ a variety and $\phi\colon Y\rightarrow X$ a generically finite
separable dominant morphism of degree $\geq 2$.

We say that $A$ is _thin_ if it is contained in a finite union of type I and
II subsets.

######  Definition 2.2.

Let $X$ be as in the previous definition. We say that $X$ has the _Hilbert
property_ (over $k$) if $X\mathopen{}\mathclose{{}\left(k}\right)$ is not
thin.

We say that a field $k$ is _Hilbertian_ if there exists a variety $X/k$ with
the Hilbert property.

A field $k$ is Hilbertian if and only if $\mathbb{P}^{1}_{k}$ has the Hilbert
property [12, Prop. 13.5.3]. Number fields are Hilbertian by Hilbert’s
irreducibility theorem, while local fields, finite fields and algebraically
closed fields are not (see [12, Ch. 13]).

Given a dominant morphism of $k$-varieties $f\colon Y\rightarrow X$ with
geometrically integral generic fiber and a non-thin subset $A\subset
Y\mathopen{}\mathclose{{}\left(k}\right)$, the image
$f\mathopen{}\mathclose{{}\left(A}\right)\subset
X\mathopen{}\mathclose{{}\left(k}\right)$ is not thin. Indeed, the pullback to
$Y$ of a cover $\phi\colon V\rightarrow X$ as in the definition of type II
thin sets is also dominant and generically finite of degree at least $2$,
hence one may deduce this fact by contradiction. (See also [1, Prop. 7.13].)
Since rational varieties over Hilbertian fields satisfy the Hilbert property
(using [12, Thm. 13.4.2] and the fact that the Hilbert property is a
birational invariant), rationality of $Y$ implies that $X$ satisfies the
Hilbert property. This leads to the following definition, put forward by
Colliot-Thélène in a talk given for the Algebra, Geometry and Physics seminar
of HU Berlin/MPIM Bonn in 2021.

######  Definition 2.3.

We say that a variety $X$ over a field $k$ is _Hilbert-unirational_ if there
exists a rational variety $Y$ over $k$ and a dominant morphism $f\colon
Y\rightarrow X$ such that $f$ has geometrically integral generic fiber.

From the above, it follows immediately that if $X$ is Hilbert-unirational and
$k$ is Hilbertian, then $X$ has the Hilbert property.

###  2.2. Weak weak approximation

We recall the following definition.

######  Definition 2.4.

We say that a smooth variety $X$ over a number field $K$ satisfies _weak weak
approximation_ if $X(K)$ is dense in $\prod_{v\notin
S}X\mathopen{}\mathclose{{}\left(K_{v}}\right)$ for some finite set $S$ of
places of $K$.

######  Definition 2.5.

We say that a scheme $Y$ of finite type over a perfect field is split if there
exists a non-empty geometrically integral open subscheme $U\subset Y$.

Analogously to the Hilbert property, the existence of rational covers of $X$
with certain properties guarantees weak weak approximation.

######  Definition 2.6.

We say that a smooth variety $X$ over a field $k$ is _split-unirational_ if
there exists a rational variety $Y$ over $k$ and a dominant morphism $f\colon
Y\rightarrow X$ such that:

  * $(a)$

$f$ has geometrically integral generic fiber; and,

  * $(b)$

every birational modification $f^{\prime}\colon Y^{\prime}\to X^{\prime}$ of
$f$ with $Y^{\prime}$ and $X^{\prime}$ smooth has split fibers over all
codimension-$1$ points.

The second condition is equivalent to $f$ having “split reduction” at every
divisorial discrete valuation ring $R\in\operatorname{DVR}(k(X),k)$ (see [6,
§4]). Denef [8] proved that every $f$ satisfying $(a)$ and $(b)$ above, when
$k$ is a number field $K$, is _arithmetically surjective_.

######  Definition 2.7.

We say that a morphism $f\colon Y\rightarrow X$ of varieties over a number
field $K$ is _arithmetically surjective_ if there exists a finite set of
places $S$ such that we have $f(Y(K_{v}))=X(K_{v})$ for all $v\notin S$.

By Denef’s result, every split-unirational variety $X$ over a number field $K$
satisfies weak weak approximation: let $(P_{v})\in\prod_{v\notin S}X(K_{v})$
and $(Q_{v})\in\prod_{v\notin S}Y(K_{v})$ be such that $f(Q_{v})=P_{v}$;
approximating $(Q_{v})$ by $Q\in Y(K)$, the point $P=f(Q)$ approximates
$(P_{v})$.

We will use the following result, giving a sufficient condition for split-
unirationality.

######  Proposition 2.8.

Let $X$ be a smooth proper variety defined over a number field $K$, and let
$f\colon Y\to X$ be a proper morphism with $Y$ rational such that there exists
an open subscheme $U\subseteq Y$ such that the restriction $f|_{U}\colon U\to
X$ is smooth and has split fibers. Then $f$ is split-unirational, hence
arithmetically surjective.

###### Proof.

This follows directly from [6, Prop. 4.4]. ∎

######  Remark 2.9.

The specific instance of split-unirationality in Proposition 2.8 is supported
by a folklore argument (not invoking Denef’s result) combining the
Lang–Weil–Nisnevich estimates and Hensel’s lemma. A sketch can be found in [2,
pp. 42-43]. We present a detailed proof here for completeness. In the proof of
the lemma we use the terminology of schemes instead of varieties (and so
points refer to scheme-theoretic points).

######  Lemma 2.10.

Let $K$ be a number field and $f\colon V\to Y$ a smooth morphism with split
fibers onto a smooth proper $K$-variety $Y$. Then $f$ is arithmetically
surjective.

###### Proof.

We first prove that, given $\mathcal{Y}$ a reduced $\mathcal{O}_{K,S}$-scheme
of finite type and $\phi\colon\mathcal{V}\to\mathcal{Y}$ a morphism of finite
type with split fibers (we require neither smoothness of $\phi$ nor properness
of $\mathcal{Y}$), then, after possibly enlarging the finite set $S$, the
morphism $\phi\colon\mathcal{V}(\mathbb{F}_{v})\to\mathcal{Y}(\mathbb{F}_{v})$
is surjective for all $v\notin S$. We use Noetherian induction on
$\mathcal{Y}$.

Let $\eta\in\mathcal{Y}$ be a generic point, and $V_{1}$ be a non-empty
geometrically integral open subscheme of the generic fiber $\phi^{-1}(\eta)$.
We may assume that $V_{1}$ is affine, thus quasi-projective, i.e. that there
exists an $\kappa(\eta)$-isomorphism $V_{1}\cong Z_{1}\setminus
Z_{2}\subseteq\mathbb{P}^{n}_{\kappa(\eta)}$, with $Z_{1},Z_{2}$ closed in
$\mathbb{P}^{n}_{\kappa(\eta)}$ (where $\kappa(\eta)$ is the residue field of
$\eta$). We may take $Z_{1}$ to be the Zariski-closure of $V_{1}$, and thus
assume that it is geometrically integral and that $\dim Z_{2}<\dim Z_{1}$. We
may then spread out the following statement:

there exists a $\kappa(\eta)$-isomorphism $V_{1}\cong Z_{1}\setminus
Z_{2}\subseteq\mathbb{P}^{n}_{\kappa(\eta)}$, with $Z_{1},Z_{2}$ ($\dim
Z_{2}<\dim Z_{1}$) closed in $\mathbb{P}^{n}_{\kappa(\eta)}$ and flat over
$\operatorname{Spec}\kappa(\eta)$, and such that
$Z_{1}\to\operatorname{Spec}\kappa(\eta)$ has a geometrically integral fiber,

(the flatness over $\operatorname{Spec}\kappa(\eta)$ is automatic as this is
the spectrum of a field) to obtain an open $\mathcal{U}\subseteq\mathcal{Y}$
(containing $\eta=\operatorname{Spec}\kappa(\eta)$), and an open subscheme
$\mathcal{V}_{1}\subseteq\phi^{-1}(\mathcal{U})$ such that:

there exists a $\mathcal{U}$-isomorphism
$\mathcal{V}_{1}\cong\mathcal{Z}_{1}\setminus\mathcal{Z}_{2}\subseteq\mathbb{P}^{n}_{\mathcal{U}}$,
with $\mathcal{Z}_{1},\mathcal{Z}_{2}$
($\dim\mathcal{Z}_{2}<\dim\mathcal{Z}_{1}$) closed in
$\mathbb{P}^{n}_{\mathcal{U}}$ and flat over $\operatorname{Spec}\mathcal{U}$,
and such that $\mathcal{Z}_{1}\to\operatorname{Spec}\mathcal{U}$ has
geometrically integral fibers.

We refer to [22, §3.2] (and Table 1 at pp. 306-307 of loc. cit.) for the proof
that all the required properties actually spread out, thus showing the
existence of such a $\mathcal{U}$ and $\mathcal{V}_{1}$. We set $n_{i}:=\dim
Z_{i}=\dim\mathcal{Z}_{i}-\dim\mathcal{U},i=1,2.$ Note that $n_{2}<n_{1}$.

For any finite field $\mathbb{F}_{q}$, the Lang–Weil–Nisnevich theorem gives
an asymptotic estimate, for every geometrically integral closed subscheme
$Z\subseteq\mathbb{P}^{n}_{\mathbb{F}_{q}}$ [18, Thm. 1](resp. for every
reduced closed subscheme $Z\subseteq\mathbb{P}^{n}_{\mathbb{F}_{q}}$ [18, Lem.
1]) of the form

$\\#Z(\mathbb{F}_{q})=q^{\dim Z}+O_{\dim Z,n,d}(q^{\dim Z-\frac{1}{2}})$

(resp. $\\#Z(\mathbb{F}_{q})=O_{\dim Z,n,d}(q^{\dim Z})$), where the implicit
constant in the “big $O$” depends only on $\dim Z,n$ and the degree $d$ of
$Z$. Since the Hilbert polynomial (from which dimension and degree may be
inferred) remains constant in flat projective families [14, Thm. III.9.9] (in
particular, the families $\mathcal{Z}_{1}\to\mathcal{U}$ and
$\mathcal{Z}_{2}\to\mathcal{U}$), these estimates show that there exists some
constant $N_{0}>0$ such that for all non-archimedean $v$ with norm
$q_{v}>N_{0}$ (which we may assume to hold for all $v\notin S$ after enlarging
$S$), the $n_{1}$-dimensional fibers of $\mathcal{Z}_{1}\to\mathcal{U}$ have
more $\mathbb{F}_{v}$-points than the $n_{2}$-dimensional ($n_{2}<n_{1}$)
fibers of $\mathcal{Z}_{2}\to\mathcal{U}$. Since
$\mathcal{V}_{1}\cong\mathcal{Z}_{1}\setminus\mathcal{Z}_{2}$ as
$\mathcal{U}$-schemes, we thus obtain that all fibers of
$\mathcal{V}_{1}(\mathbb{F}_{v})\to\mathcal{U}(\mathbb{F}_{v})$ are non-empty,
hence that the image of
$\mathcal{V}(\mathbb{F}_{v})\to\mathcal{Y}(\mathbb{F}_{v})$ contains
$\mathcal{U}(\mathbb{F}_{v})$.

Applying the Noetherian induction hypothesis on $\phi$ restricted over the
complement $\mathcal{Y}^{\prime}$ of $\mathcal{U}$ in $\mathcal{Y}$, we may
assume (after enlarging $S$ if necessary) that the image of
$\mathcal{V}(\mathbb{F}_{v})\to\mathcal{Y}(\mathbb{F}_{v})$ contains
$\mathcal{Y}^{\prime}(\mathbb{F}_{v})=\mathcal{Y}(\mathbb{F}_{v})\setminus\mathcal{U}(\mathbb{F}_{v})$.
Combined with the above, this proves surjectivity.

We now show how to deduce the lemma. Let $S$ be a finite set of places of $K$
such that there exist $\mathcal{O}_{K,S}$-models $\mathcal{V}$ and
$\mathcal{Y}$ of $V$ and $Y$, and such that $f$ extends to a morphism
$\phi\colon\mathcal{V}\to\mathcal{Y}$. By spreading out of the property of
having split fibers (that this property spreads out follows from spreading out
geometric integrality of fibers [13, Thm. 9.7.7] after restriction to an open
intersecting the generic fiber in a geometrically integral subscheme) and
using the just proven statement, we may assume (after possibly enlarging $S$)
that $\phi(\mathcal{V}(\mathbb{F}_{v}))=\mathcal{Y}(\mathbb{F}_{v})$ for all
$v\notin S$.

After enlarging $S$ again, we may also assume by spreading out that $\phi$ is
smooth and $\mathcal{Y}$ is proper over $\mathcal{O}_{K,S}$. Let now $v$ be a
place not in $S$, and $P_{v}$ be a point in $Y(K_{v})$. By the properness of
$\mathcal{Y}$, $P_{v}$ extends to an $\mathcal{O}_{v}$-point of $\mathcal{Y}$.
The restriction of $\phi$ over this $\mathcal{O}_{v}$-point gives a smooth
model $\mathcal{Y}_{P_{v}}\to\operatorname{Spec}\mathcal{O}_{v}$ of the
$K_{v}$-variety $\phi^{-1}(P_{v}):=Y_{P_{v}}$. By the surjectivity above, we
know that this smooth $\mathcal{O}_{v}$-model has an $\mathbb{F}_{v}$-point
$\tilde{Q}$ on its special fiber. Choosing a standard smooth neighborhood (see
[30, Tags 01V5, 01V7])
$\operatorname{Spec}\mathcal{O}_{v}[x_{1},\ldots,x_{N}]/(f_{1},\ldots,f_{M}),\
M\leq N$ of $\tilde{Q}$ in the smooth $\mathcal{O}_{v}$-scheme
$\mathcal{Y}_{P_{v}}$, we may use Hensel’s lemma to lift the smooth
$\mathbb{F}_{v}$-point $\tilde{Q}:=(x_{1}(\tilde{Q}),\ldots,x_{N}(\tilde{Q}))$
to an $\mathcal{O}_{v}$-point of
$\operatorname{Spec}\mathcal{O}_{v}[x_{1},\ldots,x_{N}]/(f_{1},\ldots,f_{M})\subseteq\mathcal{Y}_{P_{v}}$,
which then induces a $K_{v}$-point on the generic fiber $Y_{P_{v}}$ of the
model $\mathcal{Y}_{P_{v}}$. Thus ${Y}_{P_{v}}(K_{v})\neq\emptyset$ and
$V(K_{v})\to Y(K_{v})$ is surjective ($P_{v}$ was arbitrary), as wished. ∎

We summarize the previous two subsections with the following proposition.

######  Proposition 2.11.

A Hilbert-unirational (resp. split-unirational) variety $X$ over a Hilbertian
field (resp. a number field) satisfies the Hilbert property (resp. weak weak
approximation).

We will prove Theorem 1.1 by constructing a split-unirational cover of the del
Pezzo surface $X$. Since a split-unirational cover is a Hilbert-unirational
cover, our construction also gives Theorem 1.3. However, a Hilbert-unirational
cover already appears as an intermediate stage in our construction, hence we
will prove Theorem 1.3 first.

###  2.3. Del Pezzo surfaces

######  Definition 2.12.

Let $X$ be a smooth projective geometrically integral variety over a field
$k$. We say that $X$ is a _Fano variety_ if the anticanonical class $-K_{X}$
is ample. A _del Pezzo surface_ is a Fano variety of dimension two (with
respect to the Zariski topology).

For the rest of this section, let $X$ be a del Pezzo surface over a field $k$.
The surface $X_{\overline{k}}$ is isomorphic either to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ or to the blowup of $\mathbb{P}^{2}$ of
at most eight points in general position (see [22, §9.4]).

######  Definition 2.13.

The _degree_ of $X$ is the integer
$d=d\mathopen{}\mathclose{{}\left(X}\right):=\mathopen{}\mathclose{{}\left(K_{X}}\right)\cdot\mathopen{}\mathclose{{}\left(K_{X}}\right)$,
where $\cdot$ denotes the intersection product on
$\operatorname{Pic}\mathopen{}\mathclose{{}\left(X}\right)$.

It can be easily shown that we have
$d\mathopen{}\mathclose{{}\left(X}\right)\in\\{1,\dots,9\\}$ [21, Lem.
24.3.1].

####  2.3.1. Anticanonical morphism

The linear system $|{-K_{X}}|$ is base-point-free if and only if $d\neq 1$
[15, Prop. III.3.4.1]. Since base-point-free linear systems induce morphisms
to projective space [14, §II.7], we may consider for any del Pezzo surface $X$
of degree $d\geq 2$ the anticanonical morphism $\kappa\colon
X\rightarrow\mathbb{P}^{d}$ (it follows from [15, Cor. III.3.2.5.2] that the
dimension of $|{-K_{X}}|$ is $d=K_{X}^{2}$).

For $d\geq 3$ the class $-K_{X}$ is very ample, hence $\kappa$ is an
embedding. In particular, when $d=3$, the image of $\kappa$ is a smooth cubic
surface in $\mathbb{P}^{3}$.

When $d=1$, the linear system $|{-K_{X}}|$ consists of curves of arithmetic
genus $1$ and has a unique, $k$-rational base point, thus inducing a rational
map to $\mathbb{P}^{1}$ away from the base point. Blowing up the base point
gives a geometrically rational elliptic surface over $\mathbb{P}^{1}$.

When $d=2$, the morphism $\kappa\colon X\rightarrow\mathbb{P}^{2}$ is finite
of degree $2$ and ramifies over a quartic curve $\mathcal{B}$. We denote by
$\mathcal{R}\subset X$ the ramification divisor of $\kappa$, which we call the
ramification divisor of $X$. The surface $X$ is of the form

$X:w^{2}+fw=g\subset\mathbb{P}\mathopen{}\mathclose{{}\left(1,1,1,2}\right)$

in the weighted projective space
$\mathbb{P}\mathopen{}\mathclose{{}\left(1,1,1,2}\right)$ with variables
$x,y,z,w$, where $f,g\in k[x,y,z]$ are homogeneous polynomials of degrees 2
and 4, respectively. The curve $\mathcal{B}\subset\mathbb{P}^{2}$ is then
defined by $f^{2}+4g=0$, and it is smooth when char $k\neq 2$. The morphism
$\kappa$ induces the _Geiser involution_ $\iota\colon X\rightarrow X$, which
swaps the two sheets of the double cover. The embedding $\kappa_{2}\colon
X\hookrightarrow\mathbb{P}^{6}$ induced by $-2K_{X}$ realizes $X$ as an octic
(degree-$8$) surface contained in the cone over the Veronese surface in the
hyperplane $x_{6}=0$ (identified with $\mathbb{P}^{5}$) with vertex
$[0,\dots,0,1]$. Denoting the 2-uple embedding by $\rho_{2}$ and the
aforementioned hyperplane projection by $\pi_{6}$, we have the following
commutative diagram:

${X}$${\mathbb{P}^{6}}$${\mathbb{P}^{2}}$${\mathbb{P}^{5}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa}$$\scriptstyle{\pi_{6}}$$\scriptstyle{\rho_{2}}$

####  2.3.2. Lines

######  Definition 2.14.

Let $X$ be a del Pezzo surface. An irreducible curve $E\subset X$ is called an
_exceptional curve_ (or _line_) on $X$ if $E\cdot E=-1$ (equiv. $E\cdot
K_{X}=-1$).

The terminology “line” comes from the study of exceptional curves on the
surfaces studied by del Pezzo himself, namely those of degree $d\geq 3$, for
which the exceptional curves on $X$ are precisely the lines on $X$ under its
anticanonical embedding.

When $d=2$, there are $56$ exceptional curves coming in $28$ Geiser-involute
pairs. Following [24], we define a bitangent line to $\mathcal{B}$ to be a
line $L$ such that $\kappa^{-1}(L)$ is geometrically reducible (when char
$k\neq 2$, these are the lines that have even intersection multiplicity
everywhere with $\mathcal{B}$). Each of the 28 pairs of exceptional curves is
the inverse image under $\kappa$ of a bitangent line to $\mathcal{B}$ [24,
Lem. 2.1]. (See [9, §6.1.1] for more on the existence and configuration of the
bitangents of a smooth plane quartic.) For these surfaces the maximal number
of concurrent exceptional curves is $4$, motivating the following definition.

######  Definition 2.15.

A point $P\in X\mathopen{}\mathclose{{}\left(\overline{k}}\right)$ on a
degree-$2$ del Pezzo surface $X$ over a field $k$ is a _generalized Eckardt
point_ if it lies on $4$ of the $56$ exceptional curves of $X$.

This etymology is rooted in the study of smooth projective cubic surfaces (del
Pezzo surfaces of degree $3$). It is a famous result (the _Cayley–Salmon
theorem_) that a smooth projective cubic surface $W\subset\mathbb{P}^{3}$
(over an algebraically closed field) contains exactly $27$ lines. Any point on
a smooth cubic surface lies on at most $3$ of the $27$ lines, and points lying
on $3$ of the lines are known as _Eckardt points_.

######  Remark 2.16.

As with Eckardt points, generalized Eckardt points are rare: on a general del
Pezzo surface of degree $2$ there are no generalized Eckardt points (see [24,
Ex. 7]).

##  3\. Two geometric procedures

In this section, we describe two geometric constructions which we will use to
build covers of a degree-$2$ del Pezzo surface with good geometric properties.

We begin with analogous procedures for cubic surfaces (which we informally
name the “point procedure” and “rational curve procedure”) and their
relationship to arithmetic.

###  3.1. Cubic surfaces

Let $W$ be a smooth cubic surface over an infinite field $k$ in
$\mathbb{P}^{3}$ with $W\mathopen{}\mathclose{{}\left(k}\right)\neq\emptyset$.
We illustrate two geometric procedures employed by Segre [26] and Manin [21]
to prove that $W$ is unirational, and explain how these procedures were
employed by Swinnerton-Dyer [28] to demonstrate that $W$ is Hilbert-
unirational. First, we give a map which propagates points (the “point
procedure”).

######  Definition 3.1.

Define $\phi_{W}\colon W\times W\dashrightarrow W$ to be the map sending two
distinct points $P,Q\in W$ to the third point of the intersection $L_{P,Q}\cap
W\subset\mathbb{P}^{3}$ (where defined), where $L_{P,Q}$ is the line in
$\mathbb{P}^{3}$ through $P$ and $Q$.

Here the intersection $L_{P,Q}\cap W$ is understood scheme-theoretically, so
that it may include a non-reduced point. Provided that $L_{P,Q}$ is not one of
the $27$ lines of $W$, the intersection $L_{P,Q}\cap W$ does indeed consist of
three points with multiplicity.

######  Remark 3.2.

Note that $\phi_{W}$ operates along anticanonical curves of $W$ in the
following sense: given general $P,Q\in W$, there exists a pencil
$|{-K_{W}-P-Q}|$ of plane sections of $W$ containing $P$ and $Q$, namely those
cut out by planes containing the line $L_{P,Q}$. The point
$\phi_{W}\mathopen{}\mathclose{{}\left(P,Q}\right)$ is the third point at
which all of these curves meet, i.e. the third and final base-point of
$|{-K_{W}-P-Q}|$.

We now describe the “rational curve procedure” that produces a rational curve
from a rational point.

######  Definition 3.3.

Let $P\in W(k)$ be a rational point not on any line. Define $C_{P}$ to be the
intersection $T_{P}W\cap W$, where $T_{P}W$ is the tangent plane to $W$ at
$P$. Then $C_{P}$ is an irreducible singular cubic curve in the plane
$T_{P}W$, hence rational. Define $\psi_{P}\colon\mathbb{P}^{1}\rightarrow W$
to be an associated non-constant birational morphism (coming from
desingularising $C_{P}$).

######  Remark 3.4.

The curve $C_{P}\subset W$ admits an alternative definition, as observed by
Manin [21, Proof of Thm. 29.4]: provided that $P$ does not lie on any of the
$27$ lines of $W$, we may blow up $W$ at $P$ to obtain a del Pezzo surface $X$
of degree $2$. Letting $E\subset X$ denote the exceptional divisor of this
blowup, $E$ is one of the $56$ lines of $X$, which come in $28$ Geiser-
conjugate pairs $\mathopen{}\mathclose{{}\left(L,-K_{X}-L}\right)$. The curve
$C_{P}$ is the image on $W$ of the Geiser conjugate $-K_{X}-E$ of $E$.

####  3.1.1. Unirationality

Combining the two procedures above leads to the following result.

######  Theorem 3.5.

[21, Thm. 29.4, Thm. 30.1] Let $W$ be a smooth cubic surface over a field $k$
which has at least $37$ elements, and suppose that $W(k)\neq\emptyset$. Then
$W$ is unirational.

Kollár [15] later extended this result to include smaller finite fields, thus
settling unirationality for all cubic hypersurfaces with a rational point. The
proof of Theorem 3.5 supplied by Manin in [21], originating from ideas of
Segre [26], comes in two parts: proving unirationality given a rational point
avoiding all $27$ lines (which is done over any field, not necessarily
infinite), and proving that one rational point is enough to deduce the
existence of a point lying on none of the lines (which is done over large
enough finite fields and infinite fields). For the first part, one simply
iterates the rational curve procedure in Definition 3.3 twice, once through
the initial rational point and once through its generic point. For the second
part, one employs the map $\phi_{W}$ in Definition 3.1 in order to produce two
points conjugate over a quadratic extension and from them two curves, picking
up infinitely many points at the cost of moving to a higher-degree field. By
then employing the map $\phi_{W}$ on conjugate pairs, one is able to find one
pair for which the output is a rational point over the original ground field.
For an explanation of these and other techniques used in the study of
unirationality for cubic surfaces and their higher-dimensional analogues, the
reader is encouraged to consult the aforementioned work of Kollár [15], who
employs and extends the techniques of Segre–Manin.

####  3.1.2. Hilbert-unirationality

With the same tools, Swinnerton-Dyer was able to prove that cubic surfaces
satisfying the hypotheses of Theorem 3.5 are Hilbert-unirational.

######  Proposition 3.6 ([28, Lem. 7]).

Let $W$ be a smooth cubic surface over a field $k$, let $P_{0}\in
W\mathopen{}\mathclose{{}\left(k}\right)$ be a rational point not on any line
of $W$, and denote by $\psi$ the morphism
$\psi_{P_{0}}\colon\mathbb{P}^{1}\rightarrow W$ as defined above. Define
$Z\subset W\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times W$ to be the set

$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(P_{1},Q_{2},Q_{3},P_{4}}\right)\in
W\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times
W:\;\begin{gathered}P_{1}\neq\psi(Q_{2}),\ \psi(Q_{3})\text{ does not lie on
any line, }\\\
\phi_{W}\mathopen{}\mathclose{{}\left(\psi\mathopen{}\mathclose{{}\left(Q_{2}}\right),P_{4}}\right)=P_{1},\text{
and }P_{4}\in
C_{\psi\mathopen{}\mathclose{{}\left(Q_{3}}\right)}.\end{gathered}}\right\\}.$

Let $\pi_{1}\colon Z\rightarrow W$ denote the restriction of projection to the
first factor of $W\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times W$. Then $Z$
is a rational variety and the generic fiber of $\pi_{1}$ is geometrically
integral. That is, $W$ is Hilbert-unirational.

We immediately obtain the following corollary.

######  Corollary 3.7.

Smooth cubic surfaces over Hilbertian fields with a rational point satisfy the
Hilbert property.

Key to the proof of Proposition 3.6 is an understanding of certain curves on
$W$ constructed using the maps $\phi_{W}$ and $\psi_{P}$. One such curve is
the image of the map $\mathbb{P}^{1}\rightarrow W$,
$Q_{2}\mapsto\phi_{W}\mathopen{}\mathclose{{}\left(\psi\mathopen{}\mathclose{{}\left(Q_{2}}\right),P_{1}}\right)$
for fixed $P_{1}\in X$. Another is the locus of points $P^{\prime}\in X$ with
$P^{\prime}\in C_{P}$ for given $P\in X$. Specifically, Swinnerton-Dyer
considers the projection of a general fiber of $\pi_{1}$ onto
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, and it is by determining the properties
of such projections (including their divisor class and singularities) that he
ultimately deduces the geometric integrality of this birational image, hence
of general fibers.

As mentioned in the introduction, Swinnerton-Dyer then pushes forward his
method to prove the following result on weak weak approximation.

######  Theorem 3.8 (Swinnerton-Dyer).

[28, Thm. 1] Let $W\subseteq\mathbb{P}^{3}_{K}$ be a smooth cubic surface over
a number field $K$. Assume that $W(K)\neq\emptyset$, and let
$S\subseteq\Omega_{K}$ be the union of all real places and all finite places
corresponding to prime ideals $\mathfrak{p}$ such that the scheme-theoretic
closure of $W$ in $\mathbb{P}^{3}_{\mathcal{O}_{K}}$ has a singular fiber over
$\mathfrak{p}$. Then $W(K)$ is dense $\prod_{v\notin
S}W\mathopen{}\mathclose{{}\left(K_{v}}\right)$.

###  3.2. Del Pezzo surfaces of degree $2$

We now present analogues of the rational curve and point procedures for cubic
surfaces in the setting of del Pezzo surfaces of degree $2$. The rational
curve procedure, first given by Manin [21] in certain cases and later
generalized by Salgado, Testa and Várilly-Alvarado [24], is key to the
associated proofs of unirationality. The point procedure we give is new,
although it bears an interesting resemblance to an alternative formulation of
the rational curve procedure observed by Festi and van Luijk [11, Prop. 4.3]
in the extended preprint of [10].

Let $X$ be a del Pezzo surface of degree 2 over a field $k$. For convenience,
we make the following definition.

######  Definition 3.9.

Let $P\in X$ be a point not lying on the ramification divisor $\mathcal{R}$ of
$X$. We say that $P$ is _general_ if it is not a generalized Eckardt point,
and we say that it is _very general_ if it does not lie on any exceptional
curve of $X$.

The hypothesis of Theorem 1.1 is that $X$ contains a general rational point.
By the following result, this is equivalent to the existence of a very general
rational point for $\operatorname{char}k=0$. Thus, we henceforth assume
without loss of generality that $X$ contains a very general point $P_{0}\in
X(k)$.

######  Theorem 3.10.

[24, Cor. 3.3] Let $X$ be a del Pezzo surface of degree $2$ over a field $k$.
If $X$ contains a general $k$-rational point, then $X$ is unirational.

####  3.2.1. Rational curve procedure

A key ingredient in the proof of Theorem 3.10 is the following construction,
given a general point $P\in X\mathopen{}\mathclose{{}\left(k}\right)$, of a
rational curve containing $P$.

######  Definition 3.11.

Let $P\in X$ be a general point, and let $\rho\colon
X^{\prime}=\text{Bl}_{P}X\rightarrow X$ be the associated blowup, with
exceptional divisor $E\subset X^{\prime}$ above $P$. Denote by
$E_{1},\dots,E_{n}$ the $n\leq 3$ exceptional curves of $X$ on which $P$ lies,
and denote by $E_{i}^{\prime}\subset X^{\prime}$ the strict transform of
$E_{i}$. The curve $C_{P}\subset X$ is defined to be the image via $\rho$ of
the unique curve in the linear system
$|{-2K_{X^{\prime}}-E-\sum_{i=1}^{n}E_{i}^{\prime}}|$, which is then the
unique curve in the system
$|{-2K_{X}-\sum_{i=1}^{n}E_{i}-\mathopen{}\mathclose{{}\left(3-n}\right)P}|$.

The existence, uniqueness, integrality and rationality of $C_{P}$ is proved in
[24, Thm. 3.1].

######  Note 3.12.

Note that, in the above definition, we have
$(-2K_{X^{\prime}}-E-\sum_{i=1}^{n}E_{i}^{\prime})\cdot E=3-n$. Thus, when $P$
lies on $n\leq 2$ exceptional curves, the curve $C_{P}$ contains $P$ with
multiplicity $3-n$. However, when $P$ lies on exactly three exceptional
curves, the curve $C_{P}$ is a $k$-rational exceptional curve on $X$ not
containing $P$. This leads to the following interesting observation: when $P$
lies on exactly three exceptional curves, one may blow down $C_{P}$ and obtain
a del Pezzo surface of degree $3$ with a general rational point.

######  Note 3.13.

When $P$ is very general, one can describe $C_{P}$ in a similar way to
Definition 3.3. Consider the octic embedding $|{-2K_{X}}|=\kappa_{2}\colon
X\hookrightarrow\mathbb{P}^{6}$, and let $H:=\\{h=0\\}$ be the unique
hyperplane that passes through $P$ and osculates $X$ at it, i.e. such that the
restriction of $h$ to $X$ is $0$ to the second order (in Taylor expansion) at
the point $P$. Then $C_{P}$ is defined as the intersection $H\cap X$. Indeed,
such a hyperplane corresponds by construction to the unique curve in the
linear system $|{-2K_{X}-3P}|$.

######  Note 3.14.

If $P$ lies on $4$ exceptional curves $E_{1},\dots,E_{4}$, then
$E_{1}+E_{2}+E_{3}+E_{4}\sim-2K_{X}$ (this is part of [10, Prop. 4.12]), thus
the system $|{-2K_{X}-\sum_{i=1}^{4}E_{i}}|$ contains no non-zero effective
divisors.

####  3.2.2. The points procedure

Having established an analogue of the Segre–Manin procedure for making
rational curves, we now come to a procedure for producing one point from two
others. We will make use of the fact that anticanonical curves on a degree-$2$
del Pezzo surface are of arithmetic genus $1$, as can be seen from the
adjunction formula [14, Exercise V.1.3]. Recall that the linear system
$|{-K_{X}}|$ induces a degree-$2$ morphism $\kappa\colon
X\rightarrow\mathbb{P}^{2}$. Given two points $P,Q\in X$ such that
$\kappa\mathopen{}\mathclose{{}\left(P}\right)\neq\kappa\mathopen{}\mathclose{{}\left(Q}\right)$,
consider the preimage $E_{P,Q}$ in $X$ of the line
$L=L_{\kappa\mathopen{}\mathclose{{}\left(P}\right),\kappa\mathopen{}\mathclose{{}\left(Q}\right)}\subset\mathbb{P}^{2}$
joining $\kappa\mathopen{}\mathclose{{}\left(P}\right)$ and
$\kappa\mathopen{}\mathclose{{}\left(Q}\right)$. This curve $E_{P,Q}$ lies in
the anticanonical system $|{-K_{X}}|$, hence it is of arithmetic genus $1$. If
$\operatorname{char}k\neq 2$, it is singular if and only if $L$ is tangent to
the (smooth) plane quartic branch curve $\mathcal{B}$ of $\kappa$ and, for any
$\operatorname{char}k$, geometrically reducible if and only if $L$ is one of
the $28$ bitangents of $\mathcal{B}$ [24, Lem. 4], in which case it is the
union of two exceptional curves.

######  Definition 3.15.

Define $\phi\colon X\times X\dashrightarrow
X,\mathopen{}\mathclose{{}\left(P,Q}\right)\mapsto R$, where $R$ is the unique
point of $E_{P,Q}$ such that
$2\mathopen{}\mathclose{{}\left(\iota\mathopen{}\mathclose{{}\left(P}\right)}\right)-\mathopen{}\mathclose{{}\left(Q}\right)\sim\mathopen{}\mathclose{{}\left(R}\right)\in\operatorname{Div}E_{P,Q}$.

######  Remark 3.16.

When $\operatorname{char}k\neq 2$, the map $\phi$ is clearly defined on all
$(P,Q)\in X\times X$ with
$\kappa\mathopen{}\mathclose{{}\left(P}\right)\neq\kappa\mathopen{}\mathclose{{}\left(Q}\right)$
(meaning $Q\not\in\\{P,\iota\mathopen{}\mathclose{{}\left(P}\right)\\}$), such
that
$L_{\kappa\mathopen{}\mathclose{{}\left(P}\right),\kappa\mathopen{}\mathclose{{}\left(Q}\right)}$
is not tangent to the branch curve $\mathcal{B}\subset\mathbb{P}^{2}$ (i.e.
such that $E_{P,Q}$ is smooth), see [27, Prop. III.3.4]. We will show in
Proposition 3.19 that the domain of $\phi$ actually includes (in any
characteristic) also the pairs of points $(P,Q),\kappa(P)\neq\kappa(Q)$ for
which $E_{P,Q}$ is irreducible and both $P$ and $Q$ are smooth points on it.
Moreover, as we will show in Remark 3.28, assuming $\operatorname{char}k\neq
2$, fixing $P\in X$, the map $\phi(P,-)\colon X\dashrightarrow X$ extends to a
regular involution on $X\setminus\\{P,\iota(P),C_{P}\\}$ (where $C_{P}$ is
only excluded if it is well-defined, i.e. when $P$ is a general point).

######  Remark 3.17.

We make two observations on the above definition and its connection to the
other geometric procedures that we have seen.

  1. (1)

One reason for using $\iota\mathopen{}\mathclose{{}\left(P}\right)$ instead of
$P$ in the above definition, as might appear more natural, is that the given
map is connected to the rational curve procedure. Festi and van Luijk show
[11, Prop. 4.3] that, writing $D_{Q}$, $Q\in\mathbb{P}^{1}$ for the pencil of
curves in $|{-K_{X}}|$ through both $P$ and
$\iota\mathopen{}\mathclose{{}\left(P}\right)$, the curve $C_{P}$ is formed by
computing on each $D_{Q}$ the unique point $R$ such that
$2\mathopen{}\mathclose{{}\left(\iota\mathopen{}\mathclose{{}\left(P}\right)}\right)-\mathopen{}\mathclose{{}\left(P}\right)\sim\mathopen{}\mathclose{{}\left(R}\right)\in\operatorname{Div}D_{Q}$.
In particular, both the rational curve and point procedures arise from
constructing degree-$1$ divisors from pairs of points on anticanonical curves.

  2. (2)

The map $\phi$ admits a geometric interpretation akin to the line map employed
by Swinnerton-Dyer: for Zariski-general $P$ and $Q$, the point
$\phi\mathopen{}\mathclose{{}\left(P,Q}\right)$ is the third base-point of
$|{-2K_{X}-2P-Q}|$, just as the point
$\phi_{W}\mathopen{}\mathclose{{}\left(P,Q}\right)$ is the third base-point of
$|{-K_{W}-P-Q}|$. In fact, note first that this linear system contains the
linear system of reducible curves
$E_{P,Q}+|-K_{X}-P|=|-K_{X}-P-Q|+|-K_{X}-P|$, whose base locus is (set-
theoretically) the curve $E_{P,Q}$, hence any base-point lies on $E_{P,Q}$.
For an element $H\in\mathopen{}\mathclose{{}\left|-2K_{X}-2P-Q}\right|$ (a
hyperplane in $\mathbb{P}^{6}$, into which $X$ embeds via the morphism coming
from $|-2K_{X}|$), writing $H\cap E_{P,Q}=\\{P,P,Q,R_{H}\\}$ as four points
with multiplicity on $E_{P,Q}$, we have
$2\mathopen{}\mathclose{{}\left(P}\right)+\mathopen{}\mathclose{{}\left(Q}\right)+\mathopen{}\mathclose{{}\left(R_{H}}\right)\sim(-2K_{X})|_{E_{P,Q}}\sim
2(-K_{X})|_{E_{P,Q}}\sim
2\mathopen{}\mathclose{{}\left(P}\right)+2\mathopen{}\mathclose{{}\left(\iota\mathopen{}\mathclose{{}\left(P}\right)}\right)$
on $E_{P,Q}$. Then $\mathopen{}\mathclose{{}\left(R_{H}}\right)\sim
2\mathopen{}\mathclose{{}\left(\iota\mathopen{}\mathclose{{}\left(P}\right)}\right)-\mathopen{}\mathclose{{}\left(Q}\right)$
is constant in $H$, and is the unique third base point of $|{-2K_{X}-2P-Q}|$.

######  Note 3.18.

It would be of great interest to know whether a tactic similar to that of
Segre and Manin could be used in order to prove unirationality in the case
where one begins with a non-general rational point, i.e. whether one can use
the $\phi$ map to deduce the existence of a general rational point from a non-
general one.

####  3.2.3. A subdomain of $\phi$

In this subsubsection we find an open subset of $X\times X$ contained in the
domain of $\phi$ (i.e. a “subdomain”) which is large enough for our purposes.

More precisely, we prove the following result.

######  Proposition 3.19.

Define

$U_{\phi}:=\mathopen{}\mathclose{{}\left\\{(P,Q)\in X\times
X\;\mathopen{}\mathclose{{}\left|\;\kappa(P)\neq\kappa(Q),\begin{gathered}L_{\kappa(P),\kappa(Q)}\text{
is not bitangent to }\mathcal{B},\\\ \text{ and }P,Q\text{ are smooth points
on }E_{P,Q}\end{gathered}}\right.}\right\\}.$

Then $U_{\phi}$ is an open subset of $X\times X$ contained in the domain of
$\phi$.

Recall that the condition that $L_{\kappa(P),\kappa(Q)}$ is not bitangent to
$\mathcal{B}$ means precisely that $E_{P,Q}$ is not the union of two
exceptional curves. Moreover, when $\operatorname{char}k\neq 2$, the second
condition is equivalent to asking that, if $L_{\kappa(P),\kappa(Q)}$ is
tangent to $\mathcal{B}$, then $P,Q\notin\mathcal{R}$. It will follow from the
proof of Proposition 3.19 that, set-wise, the morphism $\phi$ sends a point
$(P,Q)$ in $U_{\phi}$ to the unique smooth point $R$ of $E_{P,Q}$ such that
$(R)\sim 2(\iota(P))-(Q)$ on $E_{P,Q}$ (this is only relevant when $E_{P,Q}$
is singular, for otherwise it is just the definition of $\phi$).

Before coming to the proof, let us recall the definition of a _relative
arithmetic genus $1$ curve_. We give this in the general setting of schemes,
but we will only need it in the case where $S$ is a $k$-variety (see Remark
3.22).

######  Definition 3.20.

Let $S$ be a locally Noetherian scheme. A proper morphism
$f\colon\mathcal{C}\to S$ is a _relative arithmetic genus $1$ curve_ if:

  * •

$f$ is flat;

  * •

the fibers are geometrically integral curves;

  * •

the fibers have arithmetic genus $1$.

For such a morphism $f$, we denote by $\mathcal{C}^{0}\subseteq\mathcal{C}$
the (open) smooth locus of $f$.

On such a relative curve we may do operations among sections as in the
following proposition, found in [7, Prop. 2.7] and formulated in the language
of schemes.

######  Proposition 3.21 (Deligne–Rapoport).

Let $f\colon\mathcal{C}\to S$ be a relative arithmetic genus $1$ curve. Assume
that there exists a section $e\colon S\to\mathcal{C}^{0}$. There exists then a
unique operation $+\colon\mathcal{C}^{0}\times_{S}\mathcal{C}\to\mathcal{C}$
such that, on $S$ and all base changes, for $x\in\mathcal{C}^{0}(S)$ and
$y\in\mathcal{C}(S)$, we have:

$\mathcal{O}(x+y)\cong\mathcal{O}(x)\otimes\mathcal{O}(y)\otimes\mathcal{O}(e)^{\otimes-1}$

locally on $S$. This operation makes $\mathcal{C}^{0}$ into a commutative
group scheme, with unity $e$, acting on $\mathcal{C}$. When
$S=\operatorname{Spec}k$, $+$ is the usual (possibly singular) elliptic curve
operation on $\mathcal{C}^{0}$.

Here $\mathcal{O}(x)$ denotes the line bundle corresponding to the effective
Cartier divisor associated to the image of $x$ in $\mathcal{C}$.

######  Remark 3.22.

As far as we are concerned, this proposition tells us that, if we have a
relative arithmetic genus $1$ curve $f\colon\mathcal{C}\to S$ over a
$k$-variety $S$ with a zero-section $e\colon S\to\mathcal{C}$, then there is a
morphism $+\colon\mathcal{C}^{0}\times_{S}\mathcal{C}^{0}\to\mathcal{C}$ that
restricts to the standard group operation of (possibly singular) elliptic
curves on the fibers of $f$.

###### Proof of Proposition 3.19.

Let $\mathcal{W}$ be the following incidence variety:

$\mathcal{W}:=\\{(R,P,Q)\in X\times U_{\phi}\mid R\in E_{P,Q}\\}.$

The projection $f:=pr_{2}\colon\mathcal{W}\to U_{\phi}$ may be identified with
the base change along $U_{\phi}\to\operatorname{Gr}(1,2),\ (P,Q)\mapsto
L_{\kappa(P),\kappa(Q)}$ of the (flat) degree-$2$ cover
$u_{2}:\mathcal{G}\times_{\mathbb{P}^{2},\kappa}X\xrightarrow{u\circ
pr_{1}}\operatorname{Gr}(1,2)$ of the (flat) universal Grassmanian family
$u:\mathcal{G}\to\operatorname{Gr}(1,2)$ of lines in $\mathbb{P}^{2}$ and is
thus flat (since $\kappa$ is flat [19, Rem. 4.3.11], both $u_{2}$ and $f$ are
flat by preservation of flatness under base change and composition). Since its
fibers satisfy the assumptions of Definition 3.20, this projection makes
$\mathcal{W}$ into a relative curve of arithmetic genus $1$ over $U_{\phi}$.

This relative curve has various sections, among which we have

$\sigma_{\iota(P)}:=(\iota\circ pr_{1},id)\colon
U_{\phi}\to\mathcal{W}\subseteq X\times U_{\phi}\text{ and
}\sigma_{Q}:=(pr_{2},id)\colon U_{\phi}\to\mathcal{W}\subseteq X\times
U_{\phi},$

i.e. “$(P,Q)\mapsto(\iota(P),P,Q)$” and “$(P,Q)\mapsto(Q,P,Q)$”, respectively.
Note that these two sections land in the smooth locus $\mathcal{W}^{0}$ of the
fibration $f$. We may then define the section

$\sigma_{2\iota(P)-Q}\colon U_{\phi}\to\mathcal{W}^{0},\ \
(P,Q)\mapsto[2]\sigma_{\iota(P)}-\sigma_{Q}$

of $f$ via the sum operation defined in Proposition 3.21 (note that to make
sense of a linear combination of sections $\sum_{i=1}^{r}[n_{i}]\sigma_{i}$
with $\sum_{i=1}^{r}n_{i}=1$, there is no need to specify a zero-section $e$
for the genus-$1$ curve, as the result of the sum is independent of the choice
of $e$). Then the composition

$U_{\phi}\xrightarrow{\sigma_{2\iota(P)-Q}}\mathcal{W}\xrightarrow{pr_{1}}X$

extends the rational map $\phi$, as wished. ∎

####  3.2.4. Associated elliptic surface

In the remainder of Section 3, we assume char $k\neq 2$. Let $P\in X(k)$ be a
rational point on $X$. Below we describe how to obtain an associated elliptic
surface $\mathcal{E}_{P}$, and how the rational curves and point procedures
can be viewed on this elliptic surface, following [11, §4].

Recall that $\mathcal{R}$ denotes the ramification divisor of $X$.

######  Definition 3.23.

Given a rational point $P$ on $X$, we define a surface $\mathcal{E}_{P}$ and
curves on it as follows.

  * •

If $P$ is not contained in $\mathcal{R}$, we set

$\mathcal{E}_{P}:=\text{Bl}_{P,\iota\mathopen{}\mathclose{{}\left(P}\right)}X,$

and we write $E_{P}$ and $E^{\prime}_{P}=E_{\iota(P)}$ for the exceptional
divisors in $\mathcal{E}_{P}$ above $P$ and $\iota(P)$ respectively.

  * •

If $P$ is contained in $\mathcal{R}$, set $X^{\prime}:=\text{Bl}_{P}X$. The
strict transform in $X^{\prime}$ of the pencil $|{-K_{X}-P}|$ on $X$ has a
unique base point $P^{\prime}$ (lying on the exceptional curve above $P$), of
multiplicity one. We define

$\mathcal{E}_{P}:=\text{Bl}_{P^{\prime}}X^{\prime},$

and we write $E_{P}$ and $E^{\prime}_{P}$ for the strict transform in
$\mathcal{E}_{P}$ of the exceptional divisor in $X^{\prime}$ above $P$ and for
the exceptional divisor in $\mathcal{E}_{P}\to X^{\prime}$ above $P^{\prime}$,
respectively.

We denote by $b\colon\mathcal{E}_{P}\to X$ the double blow-up.

In both cases, $b$ factors through the blow-up $X^{\prime}=\text{Bl}_{P}X\to
X$. Let
$\mathfrak{L}_{\kappa\mathopen{}\mathclose{{}\left(P}\right)}\cong\mathbb{P}^{1}$
be the pencil of lines in $\mathbb{P}^{2}$ passing through
$\kappa\mathopen{}\mathclose{{}\left(P}\right)$. The morphism $\kappa$ induces
a morphism
$\mu_{1}\colon\mathcal{E}_{P}\to\text{Bl}_{\kappa(P)}\mathbb{P}^{2}$. We
identify Bl${}_{\kappa(P)}\mathbb{P}^{2}$ with the variety
$\\{(Q,L)\in\mathbb{P}^{2}\times\mathfrak{L}_{\kappa(P)}\colon Q\in L\\}$, and
denote by $\mu_{2}$ the projection of Bl${}_{\kappa(P)}\mathbb{P}^{2}$ onto
the second factor. Let
$\nu\colon\mathcal{E}_{P}\to\mathfrak{L}_{\kappa\mathopen{}\mathclose{{}\left(P}\right)}$
be the composition $\mu_{2}\circ\mu_{1}$. The maps are represented in the
following diagram.

${\mathcal{E}_{P}}$${\text{Bl}_{\kappa(P)}\mathbb{P}^{2}\subset\mathbb{P}^{2}\times\mathfrak{L}_{\kappa(P)}}$${\mathfrak{L}_{\kappa(P)}\cong\mathbb{P}^{1}}$${X}$${\mathbb{P}^{2}}$$\scriptstyle{b}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{1}}$blow-
up$\scriptstyle{\mu_{2}}$$\scriptstyle{\kappa}$

The fiber of $\nu$ above a line $L\in\mathfrak{L}_{\kappa(P)}$ equals the
strict transform under the map $b$ of the curve $\kappa^{-1}(L)$ in
$|{-K_{X}}|$, which is an element of $|{-K_{\mathcal{E}_{P}}}|$. We have

$K_{\mathcal{E}_{P}}^{2}=\begin{cases}\mathopen{}\mathclose{{}\left(b^{*}(K_{X})+E_{P}+E_{\iota(P)}}\right)^{2}=2-2=0\text{
if $P\notin\mathcal{R}$,}\\\
\mathopen{}\mathclose{{}\left(b^{*}(K_{X})+E_{P}+2E^{\prime}_{P}}\right)^{2}=2-2-4+4=0\text{
if $P\in\mathcal{R}$.}\end{cases}$

By the adjunction formula, the genus of a fiber of $\nu$ above a line
$L\in\mathfrak{L}_{\kappa(P)}$ is equal to

$g(\nu^{-1}(L))=\frac{1}{2}\nu^{-1}(L)\cdot(\nu^{-1}(L)+K_{\mathcal{E}_{P}})+1=\frac{1}{2}\mathopen{}\mathclose{{}\left(-K_{\mathcal{E}_{P}}\cdot
0}\right)+1=1,$

hence $\nu$ induces a genus-$1$ fibration on $\mathcal{E}_{P}$. Moreover, if
$P\notin\mathcal{R}$, then each fiber intersects both $E_{P}$ and
$E_{\iota(P)}$ with multiplicity 1 (since exceptional curves intersect
anticanonical divisors with multiplicity $1$), hence these define two sections
of $\nu$; indeed, both $E_{P}$ and $E_{\iota(P)}$ are mapped isomorphically by
$\mu_{1}$ to the exceptional curve above $\kappa(P)$ in
Bl${}_{\kappa(P)}\mathbb{P}^{2}$, which is isomorphic to
$\mathfrak{L}_{\kappa(P)}$. We conclude that $\mathcal{E}_{P}$ is an elliptic
surface. Still following [11, §4], in this case we choose $E_{\iota(P)}$ to be
the zero-section of the Mordell–Weil group of $\mathcal{E}_{P}$ (which is the
Mordell–Weil group of the generic fiber of $\nu$). If $P\in\mathcal{R}$, then,
similarly, $E^{\prime}_{P}$ defines a section, and we choose it to be the
zero-section of the Mordell–Weil group of $\mathcal{E}_{P}$. The surface
$\mathcal{E}_{P}$ is proper and regular, and it is a minimal elliptic surface,
i.e. no fiber contains a $\mathopen{}\mathclose{{}\left(-1}\right)$-curve: its
singular fibers may be of types I1, I2, I3, II, III, IV in Kodaira’s
classification, as shown in [17, Tables 1 and 2].

######  Remark 3.24.

The pencil of fibers of $\nu$ is by construction the strict transform of the
pencil of anticanonical curves $E_{P,Q}$ passing through $P$ and $\iota(P)$.
In particular, the elliptic operations on a smooth fiber
$(\mathcal{E}_{P})_{t}:=\nu^{-1}(t),t\in\mathbb{P}^{1}$ of the surface
$\mathcal{E}_{P}$ may be identified via $b$ with the elliptic operations on
the anticanonical curve $b((\mathcal{E}_{P})_{t})$.

######  Definition 3.25.

For an integer $n\in\mathbb{Z}$, we define the rational map
$[n]\colon\mathcal{E}_{P}\dashrightarrow\mathcal{E}_{P}$ as the map defined by
multiplication by $n$ on the smooth fibers.

For $n=-1$, the rational map $[-1]\colon\mathcal{E}_{P}\to\mathcal{E}_{P}$ is
actually a morphism by the minimality of $\mathcal{E}_{P}\to\mathbb{P}^{1}$.
More precisely, this rational map comes from an automorphism (not just an
endomorphism) of the smooth fibers, and, as $\mathcal{E}_{P}$ is a _minimal_
proper regular elliptic surface, it extends to an automorphism of
$\mathcal{E}_{P}$ (see e.g. [19, Prop. 9.3.13]) The first part of the
following result shows how one could alternatively define the curve $C_{P}$
(see Section 3.2.1) through the surface $\mathcal{E}_{P}$.

######  Proposition 3.26.

Let $P\in X(k)$ be a point. The following hold.

  * (i) If $P$ is general, the image of the section $[-1]E_{P}\cong\mathfrak{L}_{\kappa(P)}$ under $b$ is mapped to the curve $C_{P}$.

  * (ii) If $P$ is not general, the curve $[-1]E_{P}$ is mapped under $b$ to $P$.

###### Proof.

(i) For $P$ very general this follows from [11, Prop. 4.1]. More generally,
assume that $P$ is general, lying on $n$ exceptional curves
$E_{1},\ldots,E_{n}$ (note that $0\leq n\leq 3$). By construction, we have
$C_{P}\sim-2K_{X}-\sum_{i=1}^{n}E_{i}$. Since $P$ is general, the exceptional
curves $E_{i}$ do not contain $\iota(P)$, since the curves $E_{i}$ and
$\iota(E_{i})$ intersect each other in exactly two points which are both
contained in $\mathcal{R}$. It follows that we have

$\displaystyle b^{*}(C_{P})$
$\displaystyle\sim-2K_{\mathcal{E}_{P}}+2E_{P}+2E_{\iota(P)}-\sum_{i=1}^{n}b^{*}(E_{i})$
$\displaystyle=-2K_{\mathcal{E}_{P}}+2E_{P}+2E_{\iota(P)}-\sum_{i=1}^{n}(\widetilde{E}_{i}+E_{P})$
$\displaystyle=-2K_{\mathcal{E}_{P}}+(2-n)E_{P}+2E_{\iota(P)}-\sum_{i=1}^{n}\widetilde{E}_{i},$

where $\widetilde{E}_{i}$ is the strict transform of $E_{i}$ on
$\mathcal{E}_{P}$. The multiplicity of $C_{P}$ at $P$ is $3-n$ by Note 3.12,
and from [11, Prop. 4.14] it follows that $\iota(P)$ is not contained in
$C_{P}$. Therefore, the strict transform $\widetilde{C}_{P}$ of $C_{P}$ in
$\mathcal{E}_{P}$ has divisor class

$\widetilde{C}_{P}=b^{*}(C_{P})-(3-n)E_{P}=-2K_{\mathcal{E}_{P}}-E_{P}+2E_{\iota(P)}-\sum_{i=1}^{n}\widetilde{E}_{i}.$

Since the $\widetilde{E}_{i}$’s are vertical divisors on the elliptic surface
$\mathcal{E}_{P}$ (see [17, Tables 1,2]) and so is $-K_{\mathcal{E}_{P}}$, we
have $\widetilde{C}_{P}=[-1]E_{P}$, as claimed.

(ii) If $P$ is not general, then it is contained in $\mathcal{R}$ or it is a
generalized Eckardt point. In the first case, we have that $E_{P}$ is a
vertical divisor on the elliptic surface $\mathcal{E}_{P}$ (see [17, Table 2])
that intersects the zero-section $E^{\prime}_{P}$. Thus we find
$[-1]E_{P}=E_{P}$ and $b([-1]E_{P})=b(E_{P})=P$. The second case follows from
[11, Thm. 4.13]. ∎

We will now relate the elliptic surface $\mathcal{E}_{P}$ to the map $\phi$
constructed in Section 3.2.2.

Note that the involution $\iota\colon X\to X$ extends to an involution
$\mathcal{E}_{P}\to\mathcal{E}_{P}$ that commutes with $\nu$, which we still
denote by $\iota$.

######  Proposition 3.27.

Let $P\in X(k)$ be any rational point. Let $T_{P}$ be the section $E_{P}$ of
$\mathcal{E}_{P}$ when $P\notin\mathcal{R}$, and $T_{P}=0$ when
$P\in\mathcal{R}$. The following diagrams commute:

${\mathcal{E}_{P}}$${\mathcal{E}_{P}}$${X}$${X,}$$\scriptstyle{b}$$\scriptstyle{[2]\circ\iota-
T_{P}}$$\scriptstyle{b}$$\scriptstyle{\phi(-,P)}$
${\mathcal{E}_{P}}$${\mathcal{E}_{P}}$${X}$${X.}$$\scriptstyle{b}$$\scriptstyle{[-1]}$$\scriptstyle{b}$$\scriptstyle{\phi(P,-)}$

###### Proof.

We prove the commutativity of the first one as the second one is proved
analogously. It suffices to verify the commutativity for a Zariski-general
fiber $\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$ of the map
$\nu\colon\mathcal{E}_{P}\to\mathbb{P}^{1}$. This (elliptic) fiber is
identified via $b$ with a Zariski-general element of the pencil
$|{-K_{X}-P}|$, and the origin of
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$ is identified via
$b$ with the point $\iota(P)$. Note that the map
$[2]\colon\mathcal{E}_{P}\to\mathcal{E}_{P}$ restricts by definition to
multiplication by $2$ on
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$, and thus
${[2]\circ\iota-T_{P}}$ restricts to the map $Q\mapsto[2]\iota(Q)-P$ on
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$. On the other
hand, ${\phi(-,P)}$ restricts on
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$ to the map sending
a point $Q$ to the unique point $R$ of
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$ such that $(R)\sim
2(\iota(Q))-(P)$. The last condition is equivalent to $(R)-(\iota(P))\sim
2(\iota(Q))-(P)-(\iota(P))$, which is the same as $R=[2]\iota(Q)-P$ since
$\iota(P)$ is the origin of the elliptic curve
$\mathopen{}\mathclose{{}\left(\mathcal{E}_{P}}\right)_{t}$. ∎

######  Remark 3.28.

It follows from the commutativity of the second diagram in Proposition 3.27
that the domain of the rational map ${\phi(P,-)}\colon X\dashrightarrow X$
contains $X\setminus\\{P,\iota(P)\\}$, and that ${\phi(P,-)}$ induces an
involution on

$X\setminus
b(E_{P},E_{P}^{\prime},[-1]E_{P})=X\setminus\\{P,\iota(P),C_{P}\\},$

since $b([-1]E_{P})=C_{P}$ (resp. $b([-1]E_{P})=P$) when $P$ is general (resp.
otherwise) by Proposition 3.26.

######  Remark 3.29.

In Proposition 3.27 and Remark 3.28 we proved that the restriction $\phi(P,-)$
is well-defined on $X\setminus\\{P,\iota(P)\\}$. Notice on the other hand how,
through Proposition 3.19 alone, we could a priori only infer that $\phi(P,-)$
is a morphism on $U_{\phi}\cap(\\{P\\}\times X)$, which is in general smaller
than $X\setminus\\{P,\iota(P)\\}$. For instance, it does not contain any point
on any of the lines passing through $P$. This phenomenon is consistent with
the fact that the domain of the restriction of a rational map may in general
be larger than the restriction of the domain.

We denote by $\phi_{P}$ (resp. $\phi^{P}$) the rational map $\phi(P,-)$ (resp.
$\phi(-,P)$). This notation serves also the purpose of making clearer
(notationwise) the domain issue raised in Remark 3.29: when $Q$ (resp. $P$)
lies in the domain of $\phi(P,-)$ (resp. $\phi(-,Q)$) but $(P,Q)$ does not lie
in the domain of $\phi$, we will only use the notation $\phi_{P}(Q)$ (resp.
$\phi^{Q}(P)$), and not the ambiguous notation $\phi(P,Q)$.

####  3.2.5. Key proposition

In this subsubsection we give a proposition that will be repeatedly used in
the proof of Theorem 1.1. Recall that $P_{0}\in X(k)$ is fixed and very
general. Let $C_{P_{0}}$ be the corresponding curve in Definition 3.11.

######  Proposition 3.30.

Let $D$ be an irreducible curve on $X$. Then the rational map

$\psi=\phi|_{C_{P_{0}}\times D}\colon C_{P_{0}}\times D\dashrightarrow X$

is dominant.

###### Proof.

We may assume that $k$ is algebraically closed. Note that
$U_{\phi}\cap(C_{P_{0}}\times D)\neq\emptyset$, thus $\psi$ is well-defined.
Moreover, one checks that:

  * •

$U_{\phi}\cap(C_{P_{0}}\times\\{Q\\})\neq\emptyset$ for all $Q\in D$;

  * •

$U_{\phi}\cap(\\{P\\}\times D)\neq\emptyset$ for all $P\in C_{P_{0}}$ unless
$D$ is a line, and $P\in(D\cup\iota(D))$.

We denote by $L$ the (finite) set $(D\cup\iota(D))\cap C_{P_{0}}$ if $D$ is a
line, while $L:=\emptyset$ otherwise. (Note that, since $P_{0}$ is very
general, it is not contained in $L$.) Proposition 3.19 then allows us to
restrict $\psi$ (equiv. $\phi$) on $C_{P_{0}}\times\\{Q\\}$ for all $Q\in D$
and on $\\{P\\}\times D$ for all $P\in C_{P_{0}}\setminus L$, which we will
both do repeatedly throughout the proof.

Since $C_{P_{0}}\times D$ is irreducible, so is the closed image of $\psi$. To
show that $\psi$ is dominant, assume by contradiction that its closed image is
a one-dimensional curve $C$.

For a Zariski-general $Q\in D$ we have, by Proposition 3.27, a commutative
diagram:

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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.8816pt}{-22.15385pt}\pgfsys@lineto{12.48164pt}{-22.15385pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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(Recall that $\phi^{Q}\colon X\dashrightarrow X$ is the restriction
$\phi(-,Q)$.) By [17, §4], all the singularities of the singular fibers of
$\mathcal{E}_{Q}\xrightarrow{\nu}\mathbb{P}^{1}$ lie on the strict transform
$\widetilde{\mathcal{R}}$ under $b$ of the ramification curve $\mathcal{R}$.
In particular,
$\mathcal{E}_{Q}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,[2]\circ\iota-
T_{Q}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,[2]\circ\iota-
T_{Q}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,[2]\circ\iota-
T_{Q}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,[2]\circ\iota-
T_{Q}$}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{}}}\mathcal{E}_{Q}$ is well-
defined and étale outside $\widetilde{\mathcal{R}}$.

Since $P_{0}$ is very general, the curve $C_{0}:={C}_{P_{0}}$ has a (triple)
singularity at the point $P_{0}$. Since $P_{0}$ is not blown-up under $b$ ($Q$
and $\iota(Q)$ are Zariski-general on some curve, and thus different from
$P_{0}$ in particular), it follows that the strict transform $C_{1}$ of
$C_{0}$ in $\mathcal{E}_{Q}$ has a singularity at the point
$P_{1}:=b^{-1}(P_{0})$. Moreover, this point does not lie on
$\widetilde{\mathcal{R}}$ (as $P_{0}\notin\mathcal{R}$ by assumption), thus
$[2]\circ\iota-T_{Q}$ is étale at $P_{1}$. Since the image of a curve
singularity under an unramified map is still a curve singularity, it follows
that the irreducible curve $C_{2}:=([2]\circ\iota-T_{Q})(C_{1})$ has a
singularity at $P_{2}:=([2]\circ\iota-T_{Q})(P_{1})$.

Note that $b(C_{2})$ is one-dimensional, as otherwise the irreducible curve
$C_{2}$ would be contained in the exceptional divisor for $b$ and would thus
be an open subvariety of a smooth rational curve, contradicting that it has a
singularity. Thus $b(C_{2})$ is a curve in $X$ with a singularity at
$b(P_{2})$. By commutativity of the diagram above, this translates into saying
that $\phi(C_{0},Q)$ is a curve with a singularity at
$\phi^{P_{0}}(Q)=\phi(P_{0},Q)$ (since $P_{0}\notin L$, $(P_{0},Q)$ lies in
the domain of $\phi$). Since $\phi(C_{0},Q)\subseteq\phi(C_{0},D)\subseteq C$,
it follows that $\phi(C_{0},Q)$ is an open subcurve of $C$, and thus the
latter has a singularity at $\phi(P_{0},Q)$ as well.

We now divide into two cases. In both cases, we will derive a contradiction by
showing that $C$ has too many singularities.

Case 1: $D\neq C_{P_{0}}$. Note that $\phi({P_{0}},D)\subseteq C$, and since
$\phi({P_{0}},D)$ is one-dimensional by Remark 3.28, this implies that
$\phi({P_{0}},D)$ is an open subcurve of $C$. Since $C$ is singular at
$\phi(P_{0},Q)$ for all Zariski-general $Q\in D$, we deduce that $C$ is
singular at all but finitely many points of the one-dimensional open subcurve
$\phi(P_{0},D)$, which leads to the desired contradiction.

Case 2: $D=C_{P_{0}}$. In this case, $D$ has the singularity $P_{0}$. Thus one
may argue using the second diagram of Proposition 3.27 to prove that $C$ is
singular at $\phi(Q,P_{0})$ for all Zariski-general $Q\in C_{P_{0}}$.

We claim now that $\phi(C_{P_{0}},P_{0})$ is one-dimensional. In fact, assume
by contradiction that it is not, then we would have that
$\phi(C_{P_{0}},P_{0})$ is constantly equal to some point $R$ (when defined).
In other words, for all Zariski-general $Q\in C_{P_{0}}$ we would have:

  1. (1)

$R$ belongs to $E_{Q,P_{0}}$; and

  2. (2)

$2(\iota(Q))-(P_{0})\sim(R)$ on $E_{Q,P_{0}}$.

The first condition implies that $R=P_{0}$ or $R=\iota(P_{0})$. Remembering
that, since $Q$ is contained in $C_{P_{0}}$, we have $(Q)\sim
2(\iota(P_{0}))-(P_{0})$ and thus $(\iota(Q))\sim 2(P_{0})-(\iota(P_{0}))$, we
deduce with a small computation that $2(P_{0})-2(\iota(P_{0}))\sim 0$ in the
first case and that $3(P_{0})-3(\iota(P_{0}))\sim 0$ in the second. In both
cases, we deduce that the section $E_{P_{0}}$ is a non-trivial torsion section
of the elliptic surface $\mathcal{E}_{P_{0}}$. Recalling that $P_{0}$ is very
general, we have that the elliptic surface $\mathcal{E}_{P_{0}}$ is the blow-
up of the del Pezzo surface $\operatorname{Bl}_{P_{0}}X$ of degree 1 in the
base-point $\iota(P_{0})$ of its anticanonical pencil. Such elliptic surfaces
are known to have a torsion-free Mordell-Weil group [25, Theorem 7.4], leading
to the desired contradiction and proving the claim.

We may now conclude as in Case 1, since we proved that $C$ is singular at all
points of the one-dimensional (open) subcurve $\phi(C_{P_{0}},P_{0})$. ∎

##  4\. Proof of main results

Let $X$ be a del Pezzo surface of degree $2$ over a field $k$. As in [6], we
will iteratively produce covers $f_{n}\colon Z_{n}\dashrightarrow X$ with
desirable properties relative to the goal of showing that the rational points
of $X$ are abundant and well-distributed. We assume again that $P_{0}\in
X\mathopen{}\mathclose{{}\left(k}\right)$ is a very general point (Definition
3.9).

######  Note 4.1.

Henceforth, we will assume that $\operatorname{char}k=0$. Emulating our
strategy in positive characteristic, one encounters issues regarding
separability of morphisms; these obstacles appear to be surmountable, at least
when $\operatorname{char}k\neq 2$, but we do not explore this here.

###  4.1. New unirationality cover

######  Definition 4.2.

Set $Z_{1}:=\mathbb{P}^{1}_{k}$, and let $f_{1}\colon Z_{1}\rightarrow X$ be
the desingularisation $\mathbb{P}^{1}\rightarrow C_{P_{0}}$.

For $n=2,3$, define $Z_{n}:=(\mathbb{P}^{1})^{n}$, and let $f_{n}\colon
Z_{n}\dashrightarrow X$ be the rational map

$f_{n}\colon Z_{n}=Z_{1}\times
Z_{n-1}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\mathopen{}\mathclose{{}\left(f_{1},f_{n-1}}\right)$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\mathopen{}\mathclose{{}\left(f_{1},f_{n-1}}\right)$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\mathopen{}\mathclose{{}\left(f_{1},f_{n-1}}\right)$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\mathopen{}\mathclose{{}\left(f_{1},f_{n-1}}\right)$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X\times
X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X.$

######  Proposition 4.3.

$Z_{2}$ is rational, and the rational map

$f_{2}\colon
Z_{2}\xrightarrow{\mathopen{}\mathclose{{}\left(f_{1},f_{1}}\right)}C_{P_{0}}\times
C_{P_{0}}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X$

is dominant.

###### Proof.

Follows from Proposition 3.30. ∎

######  Remark 4.4.

Note that Proposition 4.3 implies that $X$ is unirational. The cover
$f_{2}\colon Z_{2}\dashrightarrow X$ differs from the one used to prove
unirationality in [24].

######  Remark 4.5.

Interestingly, the analogue of this construction for a cubic surface $W$ does
not imply unirationality, since, in that case, $C_{P_{0}}$ is the intersection
of $W$ with the tangent plane $T_{P_{0}}W$ and $\phi_{W}(P,Q)$ is defined as
the third intersection point of $L_{P,Q}\cap W$. In particular, the lines
defining the points in $\phi_{W}(C_{P_{0}},C_{P_{0}})$ are all contained in
$T_{P_{0}}W$, hence $\phi_{W}(C_{P_{0}},C_{P_{0}})\subseteq T_{P_{0}}W\cap
W=C_{P_{0}}$.

###  4.2. Hilbert-unirationality

Recall that $f_{3}\colon Z_{3}\dashrightarrow X$ is given by

$f_{3}\colon Z_{3}=Z_{1}\times
Z_{2}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{1},f_{2})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{1},f_{2})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,(f_{1},f_{2})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{1},f_{2})$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X\times
X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X.$

######  Proposition 4.6.

$Z_{3}$ is rational, and $f_{3}$ is dominant with geometrically integral
generic fiber.

Before coming to the proof, we present a rational variant of Stein
factorization.

######  Lemma 4.7 (Rational Stein Factorization).

Let $k$ be a field and $f\colon W\dashrightarrow V$ a dominant rational map of
integral $k$-varieties. Then there exists a finite integral normal cover
$\pi\colon V^{\prime}\to V$ and a decomposition of $f$ as

$f\colon
W\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,f^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}V^{\prime}\to V,$ (1)

for some dominant rational map $f^{\prime}$ with geometrically integral
generic fiber.

###### Proof.

Let $L$ be the algebraic closure of $k(V)$ in $k(W)$. This is a finite
extension of $k(V)$. Let $V^{\prime}\to V$ be the relative normalization of
$V$ in $L$ ([19, Def. 4.1.24]). Then $k(V^{\prime})=L$, the embedding
$k(V^{\prime})\subseteq k(W)$ defines a dominant rational map
$f^{\prime}\colon W\dashrightarrow V^{\prime}$, and $f$ factors as in (1) by
construction. The generic fiber of $f^{\prime}$ is a $k(V^{\prime})$-variety
with function field $k(W)$. Noting that $k(V^{\prime})$ is algebraically
closed in $k(W)$ and that a variety is geometrically integral if and only if
its base field is algebraically closed in its function field [19, Exercise
3.2.12], this proves the lemma. ∎

In the following proof, it will prove useful to talk about _branch divisors_
of generically finite rational maps. We give the following definition.

######  Definition 4.8.

Let $\psi\colon X_{1}\dashrightarrow X_{2}$ be a generically finite rational
map of $k$-varieties with $X_{2}$ smooth. Let $X^{\prime}_{1}\to X_{2}$ be the
relative normalization of $X_{2}$ in $k(X_{1})$. We define the branch locus of
$\psi$ to be the branch locus of the finite cover $X^{\prime}_{1}\to X_{2}$.

Since $X_{2}$ is smooth and $X^{\prime}_{1}$ is normal, the branch locus is
always of pure codimension $1$ by the Zariski–Nagata purity theorem [30, Tag
0BMB]. Therefore, we will also refer to the branch locus as the branch
divisor.

Note that, if
$X_{1}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{1}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{1}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\psi_{1}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{1}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}}X_{2}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\psi_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\psi_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X_{3}$ is a composition of generically finite maps to
smooth varieties, then the branch divisor of $\psi_{2}$ is contained in the
branch divisor of $\psi_{2}\circ\psi_{1}$.

###### Proof of Proposition 4.6.

The rationality of $Z_{3}=Z_{2}\times Z_{1}$ follows from that of $Z_{2}$ and
$Z_{1}$. That $f_{3}$ is dominant follows from the fact that $f_{2}$ is
dominant (Proposition 4.3) and so is (evidently) $\phi|_{C_{P_{0}}\times
X}\colon C_{P_{0}}\times X\dashrightarrow X$.

It remains to prove that the generic fiber of $f_{3}$ is geometrically
integral. To do so, we will assume without loss of generality that $k$ is
algebraically closed.

Let us assume by contradiction that the generic fiber of $f_{3}$ is not
geometrically integral. Then by Lemma 4.7, we get a factorization

$f_{3}\colon
Z_{3}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{3}^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{3}^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,f_{3}^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{3}^{\prime}$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X^{\prime}\to X$

for some rational map $f_{3}^{\prime}$ and some non-trivial finite integral
normal cover $X^{\prime}\to X$. We thus get a commutative diagram:

${Z_{3}=\mathbb{P}^{1}\times Z_{2}}$${C_{P_{0}}\times
X}$${X^{\prime}}$${X.}$$\scriptstyle{(f_{1},f_{2})}$$\scriptstyle{f_{3}^{\prime}}$$\scriptstyle{\phi}$

Let $U\subset Z_{3}$ be the domain of $f_{3}^{\prime}$, and take any point
$Q^{\prime}\in\pi(U)\subset\mathbb{P}^{1}$, where $\pi\colon
Z_{3}\longrightarrow\mathbb{P}^{1}$ is the projection on the first coordinate.
Let $\phi_{Q}\colon X\dashrightarrow X$ be the specialized rational map
$\phi(Q,-)$, where $Q=f_{1}(Q^{\prime})\in C_{P_{0}}$. We may specialize the
diagram above to obtain:

${Z_{2}}$${X}$${X^{\prime}}$${X.}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{3}^{\prime}(Q^{\prime},-)}$$\scriptstyle{\phi_{Q}}$

In other words, $X^{\prime}\to X$ is a subcover of
$Z_{2}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}}X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi_{Q}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi_{Q}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi_{Q}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi_{Q}$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X$. The former is an irreducible finite cover of
degree $\geq 2$, and is thus necessarily ramified as $X$ is algebraically
simply connected [5, Cor. 4.18] and $k$ is algebraically closed. Let
$B^{\prime}\subseteq X$ be its (non-trivial) branch divisor. Then $B^{\prime}$
must be contained in the branch divisor of $\phi_{Q}\circ
f_{2}:Z_{2}\dashrightarrow X$.

We will reach a contradiction by showing that there is no divisor on $X$
contained in the intersection of the branch divisors of $\phi_{Q}\circ
f_{2}:Z_{2}\dashrightarrow X$ for all $Q\in f_{1}(\pi(U))$. Assume by
contradiction that $D\subseteq X$ is one such divisor, which we may assume to
be irreducible.

Let $B$ be the branch divisor of
$Z_{2}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,f_{2}$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X$, and take $Q\in f_{1}(\pi(U))$. By Remark 3.28, the
map $\phi_{Q}$ is a (regular) involution on
$U_{1}=X\backslash\mathopen{}\mathclose{{}\left\\{Q,\iota(Q),C_{Q}}\right\\}$.
It follows that $\phi_{Q}\mathopen{}\mathclose{{}\left(D\cap U_{1}}\right)$ is
contained in $B\cap U_{1}$. Choosing $Q\in f_{1}(\pi(U))$ sufficiently
general, the curve $C_{Q}$ intersects both $D$ and $B$ properly. Indeed, for a
very general $Q$, the curve $C_{Q}$ has a unique singularity at the point $Q$.
Thus the map $Q\mapsto C_{Q}$ is injective on the set of such points $Q$.
Since $C_{Q}$ is irreducible (when defined) and so is $D$, we have that $D\cap
C_{Q}$ is one-dimensional if and only if $C_{Q}=D$ (same for $B$). Because of
the injectivity above, this can happen for at most one point $Q$. This implies
that $D\cap U_{1}$ and $B\cap U_{1}$ are dense in $D$ and $B$ respectively.
Then it follows that for general $Q$, we have
$\phi_{Q}\mathopen{}\mathclose{{}\left(D}\right)\subset B$.

Since $f_{1}(\pi(U))$ is open in $C_{P_{0}}$, this holds for all Zariski-
general $Q\in C_{P_{0}}$, which implies that the closed image of the
composition

$C_{P_{0}}\times D\hookrightarrow X\times
X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X$

is contained in $B$. In particular, it is one-dimensional, contradicting
Proposition 3.30.∎

By Proposition 2.11, we thus obtain the following corollary.

######  Corollary 4.9.

If $k$ is a Hilbertian field, then $X$ satisfies the Hilbert property.

###  4.3. Split-unirationality

Recall that we assume $\operatorname{char}k=0$. Define $Z_{6}:=Z_{3}\times
Z_{3}$, and let $f_{6}\colon Z_{6}\dashrightarrow X$ be the map

$f_{6}\colon Z_{6}=Z_{3}\times
Z_{3}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{3},f_{3})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{3},f_{3})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,(f_{3},f_{3})$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,(f_{3},f_{3})$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X\times
X\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptscriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color$\scriptstyle\,\phi$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X.$

We follow here the reasoning in [6]. Namely, the main result of this
subsection is the following proposition, which shows that $f_{6}$ satisfies
the assumptions of Proposition 2.8, thus showing that $X$ is split-unirational
(indeed, note that $Z_{6}=(\mathbb{P}^{1})^{6}$ is rational). By Proposition
2.11, this implies that $X$ satisfies weak weak approximation.

######  Proposition 4.10.

There is an open subset $W\subset Z_{6}$ such that

  1. (i)

$f_{6}$ restricts to a morphism on $W$;

  2. (ii)

$f_{6}|_{W}$ is smooth with split or empty fibers;

  3. (iii)

$f_{6}(W)=X$.

In particular, $X$ is split-unirational.

The reader might notice that (ii) and (iii) combined may be summarized as
“$f_{6}|_{W}:W\to X$ is smooth with split fibers”. We made this separation
because we will prove that there exists such a $W$ by guaranteeing one
condition at a time: we produce $W_{1}$, $W_{2}$, and $W=W_{3}$ that satisfy
(i), (i)-(ii), and (i)-(iii), respectively.

####  4.3.1. Condition (i).

Recall that $U_{\phi}\subseteq X\times X$ was the subdomain of $\phi$ defined
in Proposition 3.19. Letting $V_{3}\subseteq Z_{3}$ be the domain of the
rational map $f_{3}$, clearly $f_{6}$ restricts to a morphism on
$W_{1}:=(V_{3}\times V_{3})\cap(f_{3},f_{3})^{-1}(U_{\phi})$.

####  4.3.2. Conditions (i)-(ii)

We first define an open subset $U_{\mathrm{inv}}\subseteq U_{\phi}$ that is
endowed with a natural involution and on which we will prove (Corollary 4.12)
that $\phi$ is smooth with geometrically integral fibers. Let $U_{0}\subseteq
X$ be the complement of the ramification divisor $\mathcal{R}$ and the lines
on $X$. Note that the curve $C_{P}$ is well-defined for $P\in U_{0}$ and that
$\\{(P,Q)\in U_{0}\times X\mid Q\in C_{P}\\}$ is closed in $U_{0}\times X$.
Thus

$U_{\mathrm{inv}}:=\\{(P,Q)\in U_{\phi}\mid P\in U_{0},Q\notin C_{P}\\}$

is open in $U_{\phi}$, equiv. in $X\times X$.

For $i=1,2$, we denote by $pr_{i}\colon U_{\mathrm{inv}}\longrightarrow X$ the
projection on the $i^{\tiny\text{th}}$ coordinate.

######  Lemma 4.11.

The map $(pr_{1},\phi)\colon U_{\mathrm{inv}}\to X\times X$ has image equal to
$U_{\mathrm{inv}}$, and the morphism $(pr_{1},\phi)\colon U_{\mathrm{inv}}\to
U_{\mathrm{inv}}$ is an involution that makes the following diagram commute:

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}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-33.99545pt}{4.20532pt}\pgfsys@lineto{-9.35886pt}{-14.48926pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{(pr_{1},\phi)}$} }}\pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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###### Proof.

Fix $P\in U_{0}$. For all $Q\notin\\{P,\iota(P)\\}\cup C_{P}$ we have
$L_{\kappa(P),\kappa(Q)}=L_{\kappa(P),\kappa(\phi(P,Q))}$, and
$\phi^{Q}(P)\notin\\{P,\iota(P)\\}\cup C_{P}$ by Remark 3.28. When $(P,Q)\in
U_{\mathrm{inv}}$, $\phi^{Q}(P)=\phi(P,Q)$. Since $P$ was arbitrary, it
follows that $(pr_{1},\phi)(U_{\mathrm{inv}})$ is contained in
$U_{\mathrm{inv}}$. Let us now show that $(pr_{1},\phi)$ is an involution. By
definition, we have $(pr_{1},\phi)(P,Q)=(P,R)$, where $R$ is the unique smooth
point on $E_{P,Q}$ such that $(R)\sim 2(\iota(P))-(Q)$. It follows that
$(Q)\sim 2(\iota(P))-(R)$, hence $(pr_{1},\phi)(P,R)=(P,Q)$, i.e.
$(pr_{1},\phi)$ is an involution, as claimed. The commutativity of the diagram
is clear. ∎

######  Corollary 4.12.

$\phi\colon U_{\mathrm{inv}}\longrightarrow X$ is smooth with geometrically
integral fibers.

###### Proof.

By the commutativity of the diagram in Lemma 4.11, this follows directly from
the fact that $pr_{2}\colon U_{\mathrm{inv}}\subseteq X\times X\to X$
satisfies the same properties. ∎

Let now $U\subseteq X$ be an open subset over which $f_{3}|_{V_{3}}$ is smooth
with geometrically integral fibers: since $V_{3}$ is smooth and the generic
fiber of $f_{3}$ is geometrically integral (Proposition 4.6), such $U$ exists
by generic smoothness in characteristic zero [14, Cor. III.10.7] and spreading
out [13, Thm. 9.7.7].

Define $W_{2}\subseteq W_{1}$ as the open subset $(V_{3}\times
V_{3})\cap(f_{3},f_{3})^{-1}(U_{\mathrm{inv}}\cap(U\times U))$. Note that in
the composition

$f_{6}\colon W_{2}\xrightarrow{(f_{3},f_{3})}U_{\mathrm{inv}}\cap(U\times
U)\xrightarrow{\phi}X,$

the first morphism is smooth with geometrically integral fibers by
construction of $U$, while the second morphism is smooth with geometrically
integral fibers by Corollary 4.12. The following lemma then proves that
condition (ii) holds.

######  Lemma 4.13.

Let $f\colon X_{1}\xrightarrow{g}X_{2}\xrightarrow{h}X_{3}$ be a composition
of morphisms of algebraic $k$-varieties such that $g$ is flat and $g$ and $h$
have geometrically integral fibers. Then $f$ has geometrically integral
fibers. Moreover, if both $g$ and $h$ are smooth, then so is $f$.

###### Proof.

We use the language of schemes (so points are scheme-theoretic points). For
$t\in X_{3}$, we have that $g|_{f^{-1}(t)}\colon f^{-1}(t)\to h^{-1}(t)$ is a
flat morphism with geometrically integral fibers over a geometrically integral
$k(t)$-variety. Applying [19, Prop. 4.3.8] to the base change of
$g|_{f^{-1}(t)}$ to $\overline{k(t)}$, we deduce that $f^{-1}(t)$ is a
geometrically integral $k(t)$-variety. For smoothness of the composition, see
[30, Lem. 29.34.4, Tag 01VA]. ∎

####  4.3.3. Conditions (i)-(iii)

After restricting $U$ if necessary and replacing $V_{3}$ by $V_{3}\cap
f_{3}^{-1}(U)$, we may assume $f_{3}(V_{3})=U$. Let $W_{3}$ be defined as
$W_{2}$ after replacing $U$ and $V_{3}$ by their restrictions. Then the first
morphism in the following composition is surjective:

$f_{6}\colon W_{3}\xrightarrow{(f_{3},f_{3})}U_{\mathrm{inv}}\cap(U\times
U)\xrightarrow{\phi}X.$

To verify that the composition is surjective, it is then enough to prove that
$\phi\colon U_{\mathrm{inv}}\to X$ is surjective and that all fibers of
$\phi|_{U_{\mathrm{inv}}}$ intersect $U\times U$. We verify these two
properties in Corollary 4.15 and Corollary 4.17, respectively.

######  Lemma 4.14.

For every $Q\in X$, there exists a non-empty open subset $U_{Q}\subseteq X$
such that $(P,Q)$ is contained in ${U_{\mathrm{inv}}}$ for every $P\in U_{Q}$.

###### Proof.

Assume by contradiction that there exists $Q\in X$ (which we fix for the
remainder of the proof) such that we have $(P,Q)\notin{U_{\mathrm{inv}}}$ for
every $P\in X$. Note that, for a Zariski-general $P\in X$, we certainly have
that $(P,Q)\in U_{\phi}$ and $P\in U_{0}$. In particular, our contradiction
assumption then implies that $Q$ is contained in $C_{P}$ for all Zariski-
general $P\in X$. Fix such a $P$. From Remark 3.17 (1) it follows that
$(Q)\sim 2(\iota(P))-(P)$ on $E_{P,Q}$. But this also holds for $\iota(P)$,
since $P$ is Zariski-general if and only $\iota(P)$ is. Then on
$E_{P,Q}=E_{\iota(P),Q}$ we have

$2(P)-(\iota(P))\sim(Q)\sim 2(\iota(P))-(P),$

so we find $3((P)-(\iota(P))\sim 0$, hence $P=\iota(P)+T$, where $T$ is a
3-torsion point of the Jacobian of $E_{P,Q}$. Setting $Q$ as the origin of
$E_{P,Q}$, we have that $\iota(P)=[2]P$, thus $P=-T$ is 3-torsion. Since this
holds for all $P^{\prime}$ in a Zariski-open set $U$ in $X$, this gives a
Zariski-open set $U^{\prime}=U\cap E_{P,Q}$ of $E_{P,Q}$, such that all points
in $U^{\prime}$ are 3-torsion on $E_{P^{\prime},Q}=E_{P,Q}$. Thus $E_{P,Q}$
has infinitely many 3-torsion points, giving a contradiction. ∎

######  Corollary 4.15.

$\phi\colon{U_{\mathrm{inv}}}\longrightarrow X$ is surjective.

###### Proof.

Combining Lemma 4.11 and Lemma 4.14, it directly follows that the fibers of
$\phi\colon{U_{\mathrm{inv}}}\longrightarrow X$ are non-empty. ∎

The following lemma proves that no fiber of $\phi$ is contained in $(D\times
X)\cup(X\times D)$ for any proper closed subvariety $D$ of $X$ (we say that
the fibers are slanted in $X\times X$).

######  Lemma 4.16 (Slantedness).

For every proper closed subvariety $D\subsetneq X$, and $R\in X$ a point,
there exists a point $(P,Q)\in{U_{\mathrm{inv}}}\cap\phi^{-1}(R)$ with
$P,Q\notin D$.

###### Proof.

Fix $R\in X$ and $D\subsetneq X$ as in the statement. Let $U_{R}\subseteq X$
be the non-empty open set of Lemma 4.14. By Proposition 3.27, the rational map
$\phi(-,R)\colon X\dashrightarrow X,P\mapsto 2(\iota(P))-(Q)$ is dominant. In
particular, for Zariski-general $P\in U_{R}$, we have that
$Q=\phi(P,R)=2(\iota(P))-(R)\notin D$ and clearly $P\not\in D$. Since
$\phi(P,Q)$ is now equal to $R$, this finishes the proof. ∎

######  Corollary 4.17.

All fibers of $\phi\colon{U_{\mathrm{inv}}}\longrightarrow X$ intersect
$U\times U$.

###### Proof.

Take $D=X\setminus U$ in Lemma 4.16. ∎

Combining the above and applying Proposition 2.8, we deduce Proposition 4.10.

####  4.3.4. Proof of the main theorem

We already phrased the argument in the paragraph above Proposition 4.10, but
we repeat it here for the reader’s convenience.

######  Proof of Theorem 1.1.

Proposition 4.10 shows that $X$ is split-unirational. Thus, according to
Proposition 2.11, we deduce that it satisfies weak weak approximation. ∎

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