# Rational curves and the Hilbert Property on Jacobian Kummer varieties

Damián Gvirtz-Chen and Zhizhong Huang  Department of Mathematics, University
College London, Gower Street, LONDON WC1E 6BT, United Kingdom
<EMAIL_ADDRESS>Institute of Science and Technology Austria, Am Campus 1,
3400 KLOSTERNEUBURG, Austria<EMAIL_ADDRESS>
###### Abstract.

A conjecture by Corvaja and Zannier predicts that smooth, projective, simply
connected varieties over a number field with Zariski dense set of rational
points have the Hilbert Property; this was proved by Demeio for Kummer
surfaces which are associated to products of two elliptic curves.

In this article, over a finitely generated field of characteristic zero, we
establish the Hilbert Property for all Kummer surfaces associated to the
Jacobian of a genus $2$ curve. In general we prove that all Jacobian Kummer
varieties associated to a hyperelliptic curve of genus $\geq 2$ of odd degree
also have the Hilbert Property.

###### 1991 Mathematics Subject Classification:

14G05, 11G35, 14J28

##  1\. Introduction

The aim of this article is to introduce the use of rational curves into the
study of the Hilbert Property for non-unirational varieties.

###  1.1. Background for the Hilbert Property

Let $X$ be a quasi-projective integral variety of dimension at least one over
a field $K$. Recall after [Ser08, §3] (see also [Col20, §2]) that a subset
$A\subset X(K)$ is _thin_ if there exists a $K$-morphism $f:Y\to X$ from a
$K$-variety $Y$ to $X$ such that

  * •

$A\subset f(Y(K))$;

  * •

the fibre of $f$ over the generic point of $X$ is finite, and $f$ does not
have any rational sections.

######  Definition 1.1 ([CS87] p. 189).

We say that $X$ has the _Hilbert Property_ (or $X$ is _of Hilbert type_) if
$X(K)$ is not a thin set.

The field $K$ is called _Hilbertian_ if the projective line
${\mathbb{P}}^{1}_{K}$ satisfies the Hilbert Property. Not every base field is
Hilbertian. The field of rational numbers, and more generally all number
fields, are classically known to be Hilbertian by the Hilbert Irreducibility
Theorem [Hil92] (see [Ser08, Theorem 3.4.3]). In this article, unless
explicitly stated to the contrary, we will assume that $K$ is a finitely
generated field of characteristic $0$. These fields are known to be Hilbertian
(see e.g. [Lan83, Chapter 9] and [FJ08, §13.4]).

It should be pointed out that Definition 1.1 looks different from the one
introduced later in [CZ17, §1.1 Definition], [CDJLZ21, Definition 1.1]. We
shall provide a proof (see Proposition 2.1 and Corollary 2.2 below) that all
these definitions are actually equivalent.

It is clear from the definition that Zariski density of $X(K)$ in $X$ is a
necessary condition for the Hilbert Property, and that the Hilbert Property is
a birational invariant. Moreover, if $X$ satisfies the Hilbert Property, then
it is geometrically integral (see [Ser08, p. 20]). The Hilbert Property is a
specialisation of weak weak approximation (see [Ser08, §3.5, Corollary 3.5.4,
Definition 3.5.6] and also [Col20, §2 (AFF), Théorème 2.3]) and provides an
approach to solving the Inverse Galois Problem (see for example the book
[Ser08] and notably ibid., Theorem 3.5.9).

There is a well-known topological obstruction to the Hilbert Property which
goes back to the Chevalley–Weil Theorem. Namely, a projective variety $X$
cannot have the Hilbert Property if it is not algebraically simply connected
(or equivalently for normal $X$, if its fundamental group has a proper
subgroup of finite index), see [CZ17, Theorem 1.4]. Corvaja and Zannier
conjecture that the converse is true.

######  Conjecture 1.2 ([CZ17], §2 Question-Conjecture 1).

If $X(K)$ is Zariski dense and if $X$ has a smooth, projective model over $K$
which is algebraically simply connected, then X has the Hilbert Property.

While there have been quite a number of results confirming the weak weak
approximation property for unirational varieties (they are simply connected by
[Ser59]), as conjectured by Colliot-Thélène (see [CS87], [Ser08, Conjecture
3.5.8]), it is an overarching challenge to establish the Hilbert Property for
intermediate type varieties. In particular Corvaja and Zannier ask for a proof
that the Hilbert Property holds for Kummer surfaces with a Zariski dense set
of rational points [CZ17, Appendix 1]. (See [HS16, p. 817] for a stronger
conjecture of weak weak approximation on K3 surfaces and [HP15] for a
consequence on the rank of quadratic twists of abelian varieties.)

###  1.2. Main results and discussion

Let $H$ be a smooth, projective hyperelliptic curve of genus $g\geq 2$ over a
finitely generated field $K$ of characteristic $0$. Let
$\operatorname{Jac}(H)$ be its Jacobian variety and let
$\operatorname{Kum}(\operatorname{Jac}(H))$ be the associated Kummer quotient,
i.e. the quotient of $\operatorname{Jac}(H)$ by the natural involution. The
minimal desingularisation of $\operatorname{Kum}(\operatorname{Jac}(H))$ is a
simply connected variety of dimension $g$, called a _Jacobian Kummer variety_
(see §3.1 for details).

In dimension two, all principally polarised abelian surfaces over $K$ are
isomorphic to either the Jacobian of a genus $2$ curve or, up to a quadratic
extension, the product of two elliptic curves (see [GGR05, Theorem 3.1]). The
Jacobian case is the generic one, in the sense of forming a Zariski dense open
subvariety of the corresponding moduli space. Our main theorem confirms 1.2 in
the outstanding Jacobian case.

######  Theorem 1.3.

Let $H$ be a smooth, projective genus $2$ curve over $K$. Then the variety
$\operatorname{Kum}(\operatorname{Jac}(H))$ has the Hilbert Property.111Note
that any smooth genus $2$ curve is hyperelliptic.

The study of the Hilbert Property for K3 surfaces was first addressed by
Corvaja and Zannier in [CZ17], and they verified it for the diagonal quartic
surface $x^{4}+y^{4}=z^{4}+w^{4}$ [CZ17, Theorem 1.6] using multiple elliptic
fibrations. The same method was formalised and applied in the recent work of
Demeio [Dem21, Proposition 4.2.1]. He establishes the Hilbert Property for the
Kummer surface $\operatorname{Kum}(E_{1}\times E_{2})$ associated to the
product of two elliptic curves $E_{1},E_{2}$ over $K$, under the assumption
that rational points on $\operatorname{Kum}(E_{1}\times E_{2})$ are Zariski
dense. (Demeio’s theorem is stated over a number field and contains the
stronger assumption that the elliptic curves have positive Mordell-Weil ranks.
In fact, the proof of [Dem21, Theorem 1.0.2] works for any finitely generated
field $K$ of characteristic $0$ and only assumes that the set of rational
points is Zariski dense. This is known to hold unconditionally unless
$j(E_{1})=j(E_{2})\in\\{0,1728\\}$, thanks to the work of Kuwata–Wang [KW93].)
The second author studied in [Hua21] various Kummer surfaces associated to
torsors under the product of two elliptic curves, though the Hilbert Property
for such surfaces remains still open. More recently, the first author and
Mezzedimi [GM21] have shown that the Hilbert Property holds for Kummer
surfaces if one allows a finite field extension of the base field $K$.

The elliptic fibrations (modulo automophisms) of a generic Jacobian Kummer
surface (i.e. $\operatorname{Jac}(H)$ has no extra endomorphisms) over an
algebraically closed field have been classified by Kumar in [Kum14].
Unfortunately, in general none of these fibrations descend to $K$ and thus, it
is not possible to apply directly the method of [Dem21] over $K$. Instead, the
novelty of our proofs of Theorem 1.3 is, amongst other things, the use of
rational curves on $\operatorname{Kum}(\operatorname{Jac}(H))$. To the
authors’ knowledge, this is the first time that rational curves have been used
to prove the Hilbert Property for non-unirational varieties.

In fact, our approach allows us to settle 1.2 for a family of higher-
dimensional varieties. We are not aware of any previous results on the Hilbert
Property for non-unirational varieties of dimension $>2$, apart from naive
constructions like product varieties.

######  Theorem 1.4.

Let $H$ be a smooth, projective hyperelliptic curve over $K$ of odd degree,
i.e., $H$ is defined by an affine model $y^{2}=f(x)$ with $f$ separable of
degree $2g+1$. Then the variety $\operatorname{Kum}(\operatorname{Jac}(H))$
has the Hilbert Property.

###  1.3. Strategy and outline

In order to reduce the _a priori_ unknown Hilbert Property of a higher-
dimensional variety $X$ to the known Hilbert Property of (possibly singular)
rational subcurves on $X$, their behaviour under pullbacks along covers of $X$
has to be understood. This is the criterion of Proposition 5.1 which we
formulate in a way tailored to our application to Jacobian Kummer varieties.

Let $\operatorname{Alb}^{1}(H)$ be the (first) Albanese torsor of the
hyperelliptic curve $H$, and let
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ be the associated Kummer
quotient, i.e. the quotient of $\operatorname{Alb}^{1}(H)$ by the natural
involution. See §3.2 for details. In general
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ is a non-isomorphic Galois
twist of $\operatorname{Kum}(\operatorname{Jac}(H))$. In practice, it turns
out that working with $\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ instead
of $\operatorname{Kum}(\operatorname{Jac}(H))$ is more useful for our
purposes. The most general form of our main theorem is the following, from
which Theorems 1.3 and 1.4 will follow as special cases.

######  Theorem 1.5.

Let $H$ be a smooth, projective hyperelliptic curve over $K$. Then the variety
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ has the Hilbert Property.

The variety $\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ always possesses a
countable family of rational curves, whose union is Zariski dense. To show
that the conditions in the criterion of Proposition 5.1 are satisfied for
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$, we prove Pullback Lemmas that
deal with the three occurring types of covers of
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$:

  1. (1)

covers that factor rationally through $\operatorname{Alb}^{1}(H)$;

  2. (2)

covers that arise as Kummer quotients of étale covers of
$\operatorname{Alb}^{1}(H)$;

  3. (3)

covers that pull back to ramified covers of $\operatorname{Alb}^{1}(H)$.

While type (1) covers are relatively easy to treat, for the other types we
rely on two separate methods: Type (2) covers are dealt with through a
fundamental group argument supplemented by a _coprimality trick_. As for type
(3) covers, we make use of the recently established _weak Hilbert Property for
abelian varieties_ due to Corvaja, Demeio, Javanpeykar, Lombardo and Zannier
[CDJLZ21], which guarantees the existence of many rational points avoiding
these covers. To control the pullbacks of rational curves on
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$, we apply this result not to
$\operatorname{Alb}^{1}(H)$ itself but instead to the base change of
$\operatorname{Alb}^{1}(H)$ to the function field $L(H)$ of $H$ over some
finite extension $L$ of $K$.

Finally, for any given finite collection of these covers, we need to carefully
match these methods together, so as to extract a subfamily of rational curves
which simultaneously have no rational sections along any one of them. Thus _a
fortiori_ these curves contribute to many rational points not coming from
these covers, thereby achieving the Hilbert Property for
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$. The deduction of the Hilbert
Property for $\operatorname{Kum}(\operatorname{Jac}(H))$ in the case $g=2$ is
based on the fact (due to Cassels and Flynn, see Theorem 3.5) that
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$ and
$\operatorname{Kum}(\operatorname{Jac}(H))$ are birational to each other over
$K$.

It is worth pointing out that, while the ideas of this article are applied to
Jacobian Kummer varieties, we expect them to be far more fruitful beyond this
particular case, especially when combined with [GM21].

The article is structured as follows. After notation and background in §2, in
§3 we construct and prove properties of rational curves on
$\operatorname{Kum}(\operatorname{Jac}(H))$ and
$\operatorname{Kum}(\operatorname{Alb}^{1}(H))$. In §4 the crucial Pullback
Lemmas are formulated for covers of types (1)-(3). These are applied in §5 to
prove the main theorems.

##  2\. Notation, conventions and preliminaries

A _variety_ over $K$ is a separated reduced $K$-scheme of dimension $>0$ and
of finite type. To simplify exposition, throughout the rest of this article
all varieties are supposed to be integral. We write $K(X)$ for the function
field of $X$.

Let $\phi:Y\to X$ be a morphism of varieties and let $X^{\prime}\subseteq X$
be a subvariety. We say that $X^{\prime}$ _has a rational section along
$\phi$_, if the morphism
$\phi|_{\phi^{-1}(X^{\prime})}:\phi^{-1}(X^{\prime})\to X^{\prime}$ has a
rational section, i.e. there exists an open dense subset $U^{\prime}\subset
X^{\prime}$ and $\psi:U^{\prime}\to Y$ such that $\phi\circ\psi$ is the
identity map on $U^{\prime}$.

A $K$-variety birationally equivalent to ${\mathbb{P}}^{n}_{K}$ is called
$K$-_rational_. By a _rational curve_ on a $K$-variety $X$, we mean a closed
(possibly singular) subvariety on $X$ which is birationally equivalent to
${\mathbb{P}}^{1}_{K}$. By a _hyperelliptic curve_ over $K$, we mean a smooth,
projective curve over $K$ with a finite morphism to ${\mathbb{P}}^{1}_{K}$ of
degree $2$. (Note that a curve may be geometrically hyperelliptic but not
hyperelliptic over $K$, the image of the canonical map being a conic without a
$K$-rational point. This does not happen when $g$ is even.)

Assume that a $K$-variety $V$ is equipped with an involution $\tau:V\to V$,
usually taken to be implicit from the context. Then we define the _Kummer
quotient_ $\operatorname{Kum}(V)$ as the (possibly singular) quotient of $V$
by the ${\mathbb{Z}}/2{\mathbb{Z}}$-action for which the group generator acts
by $\tau$.

Let $(G,+)$ be an abelian group. A _coset_ ${\mathcal{C}}$ of $G$ is of the
form $g+H$ where $g\in G$ and $H\subset G$ is subgroup.

Our next proposition says that proving the Hilbert Property boils down to a
specific family of maps.

######  Proposition 2.1.

Let $X$ be a quasi-projective variety over a field $K$. Then a subset
$M\subset X(K)$ is thin if and only if there exists a finite collection
$(\phi_{i}:Y_{i}\to X)$, where $Y_{i}$ are normal and geometrically integral
varieties over $K$, and $\phi_{i}$ are finite surjective $K$-morphisms of
degree $[K(Y_{i}):K(X)]>1$, such that
$M\setminus\bigcup_{i=1}^{r}\phi_{i}(Y_{i}(K))$ is not Zariski dense.

###### Proof.

The “if” part is clear (by considering the morphism from the disjoint union of
$Y_{i}$ with a Zariski closed subset of $X$ to $X$).

We now prove the “only if” part. Namely, for every thin set $M\subset X(K)$,
we want to construct such a finite collection $\phi_{i}:Y_{i}\to X$ as above
such that $M\setminus(\bigcup_{i=1}^{r}\phi_{i}(Y_{i}(K)))$ is contained in a
proper Zariski closed subset. We may therefore assume that $M=\phi(Y(K))$,
where $Y$ is an integral $K$-variety with $\dim Y=\dim X$, and $\phi:Y\to X$
is a dominant $K$-morphism with degree $[K(Y):K(X)]>1$. Let us fix a
projective model $\overline{X}$ of $X$ in what follows.

By the Nagata compactification theorem (see [Lüt93, Theorem 3.2]), $\phi$
extends to a proper morphism $\overline{\phi}:\overline{Y}\to\overline{X}$
where $\overline{Y}$ is a proper model of $Y$. Using the Stein factorization
([Har77, III. Corollary 11.5]), we can decompose $\overline{\phi}=g_{1}\circ
g_{2}:\overline{Y}\to W\to\overline{X}$, where $g_{1}:\overline{Y}\to W$ is
proper with connected fibres and $g_{2}:W\to\overline{X}$ is finite. It
follows that $g_{1}$ is birational, and $g_{2}$ is proper by [Har77, Exercise
II 4.1]) and hence it is surjective (since $\phi$ is dominant). Let $Z$
denotes the non-normal locus of $W$ and let $h:\widehat{W}\to W$ be the
normalisation, which is finite birational. We may assume that
$\widehat{W}(K)\neq\varnothing$ and hence $\widehat{W}$ is geometrically
integral (see [Ser08, p. 20]). Otherwise $M$ is contained in the Zariski
closed subset $g_{2}(Z)$. We let $\widetilde{W}=\widehat{W}\setminus(h\circ
g_{2})^{-1}(\overline{X}\setminus X)$. We claim that the morphism $h\circ
g_{2}|_{\widetilde{W}}:\widetilde{W}\to X$ meets our need. Indeed, $h\circ
g_{2}$ is finite surjective of degree $[K(W):K(X)]=[K(Y):K(X)]>1$, and
$M\setminus((h\circ g_{2})(\widetilde{W}(K)))$ is contained in the Zariski
closed subset $g_{2}(Z)$. ∎

The proposition above motivates the following convention.

A _cover_ $\phi:Y\to X$ between $K$-varieties is a finite surjective morphism
with $Y$ being normal and geometrically integral. The _degree_ of $\phi$ is
$[K(Y):K(X)]$.

Therefore Proposition 2.1 may be rephrased as follows:

######  Corollary 2.2.

Let $X$ be a quasi-projective variety over a Hilbertian field $K$. Then $X$
has the Hilbert Property over $K$ if and only if for any finite family of
covers $\phi_{i}\colon Y_{i}\to X$, $i=1,\dots,r$ with $\deg(\phi_{i})>1$,
$X(K)\setminus\bigcup_{i=1}^{r}\phi_{i}(Y_{i}(K))$ is Zariski dense in $X$.

##  3\. Rational curves on Jacobian Kummer varieties

Let $H$ be a hyperelliptic curve of genus $g$ over $K$. Recall the following
classical facts about $H$, its Jacobian and Albanese varieties, and their
Kummer quotients.

The hyperelliptic involution $\tau_{H}:H\to H$ is defined over $K$. The Kummer
quotient $C=\operatorname{Kum}(H)$ is non-canonically isomorphic to
${\mathbb{P}}^{1}_{K}$, and we write $\pi_{H}:H\to C$ for the quotient
morphism. The set $W\subset H(\overline{K})$ of Weierstrass points of $H$,
i.e. the fixed points under $\tau_{H}$, forms a zero-dimensional reduced
$K$-subscheme of degree $2g+2$.

###  3.1. Jacobian variety and rational curves on its Kummer quotient

The Jacobian $J=\operatorname{Jac}(H)$ is a principally polarised abelian
variety over $K$, even when $H(K)=\emptyset$. See e.g. [Mil86a, Theorem 1.1,
Summary 6.11]. For $n\in{\mathbb{Z}}$, write $[n_{J}]:J\to J$ for the
multiplication-by-$n$ morphism, and for a closed point $x\in J$, write
$t_{x}:J\to J$ for the translation-by-$x$ morphism. The fixed points of the
involution $\tau_{J}=[(-1)_{J}]$ consist precisely of the $2$-torsion $J[2]$,
which is a zero-dimensional subgroup scheme of degree $2^{2g}$. Let
$X=\operatorname{Kum}(J)$ and $\pi:J\to X$ be the quotient morphism. Clearly
the morphisms $[n_{J}]$ and $\tau_{J}$ commute, so $[n_{J}]$ descends to $X$
and we denote the resulting morphism by $[n_{X}]$. The morphism $\pi$ is étale
outside the images of $J[2]$, which form the singular locus of $X$. Blowing up
these $2^{2g}$ canonical singularities gives a minimal resolution
${\widetilde{X}}\to X$ where ${\widetilde{X}}$ is known as the _Kummer
variety_ of $J$. It is a smooth, projective and simply connected variety (see
[Spa56, Theorem 1]).

For every $x_{0}\in H(K)$, there exists an Abel–Jacobi closed $K$-embedding

$\displaystyle\iota_{x_{0}}:H$ $\displaystyle\hookrightarrow J,$
$\displaystyle x$ $\displaystyle\mapsto\langle x_{0}-x\rangle,$

where $\langle\cdot\rangle$ denotes the class of a divisor. By the Riemann-
Roch theorem, the closed points on $H$ generate $J$ (see [Har77, IV Remark
4.10.9] or [Mil86a, §5]).

We assume throughout the rest of §3.1 that $H$ has a $K$-rational Weierstrass
point $\infty$. We consider $H$ as a subvariety of $J$ via the Abel-Jacobi
embedding $\iota_{\infty}$. Since the map $\tau_{J}$ restricted to $H$ is just
$\tau_{H}$, the image of $H$ under $\pi$ is a smooth rational curve, which we
denote by $C$. For every $n\in{\mathbb{Z}}_{>0}$, let $H_{n}=[n_{J}](H)$, the
image of $H$ under the map $[n_{J}]$, and let $C_{n}=\pi(H_{n})$. The curves
$C_{n}$ are rational for all $n\in{\mathbb{Z}}_{>0}$. (See e.g. [Huy16,
Chapter 13, Example 0.4].)

######  Lemma 3.1.

The family $(H_{n})_{n\in{\mathbb{N}}}$ consists of pairwise distinct curves.
Consequently, $(C_{n})_{n\in{\mathbb{N}}}$ are pairwise distinct rational
curves on $X$.

###### Proof.

Assume $H_{m}=H_{n}$ for some $0<m<n$. Then there is an automorphism $\sigma$
of $H_{n}$ such that

$[m_{J}]|_{H}=\sigma\circ[n_{J}]|_{H}.$

Since $g\geq 2$, the order of $\sigma$ is finite, so we may assume that
$[m_{J}]|_{H}=[n_{J}]|_{H}$. It follows that
$H\subset\operatorname{Ker}([(n-m)_{J}])\subset J$, which is impossible since
the closed points of $H$ generate $J$. ∎

######  Remark 3.2.

We would like to point out an oversight in [Huy16, p. 274], where it is stated
that when $g=2$, the rational curve $C_{n}$ is smooth for all $n$. In fact
this is false for $n>1$ and it is for this reason that we will need the
coprimality argument in Proposition 4.4. If $C_{n}$ were smooth, then since it
is an ample geometrically irreducible divisor of $J$, its pullback along
$[n_{J}]$ would also be geometrically irreducible by [BL04, Corollary 4.3.4].
But this is clearly not the case since it is (geometrically) a sum of
translates of $C$. See also Remark 4.3 below.

More concretely, let us take a look at the curve $C_{2}$. Since $H$ contains
$6$ points of $J[2]$, then $H_{2}=[2_{J}](H)$ passes through $0\in J$ with
multiplicity at least $6$. Hence $C_{2}$ has multiplicity as least $3$ at
$\pi(0)$. In general, the singular locus of $H_{n}$ (resp. $C_{n}$) is the
closed subscheme of all points $[n_{J}]x$ (resp. $[n_{X}]\pi(x)$), $x\in H$,
such that there exists $y\in H$ different from $x$ with $[n_{J}](x-y)=0$. In
other words, $x$ is in a _torsion package_ of size $>1$ in the sense of
[BP01]. We omit the proof since this fact will not be needed.

######  Lemma 3.3.

Let $\eta_{H}\in J_{K(H)}$ be the image of the generic point of $H$ under the
embedding $\iota_{\infty}:H\hookrightarrow J$. Then $\eta_{H}$ is non-
degenerate in $J_{K(H)}$, i.e. ${\mathbb{Z}}\eta_{H}$ is Zariski dense in
$J_{K(H)}$.

###### Proof.

Assume instead that $\eta_{H}$ is contained in a proper abelian subvariety of
$J_{K(H)}$. Then by the Poincaré irreducibility theorem, there exists a non-
zero endomorphism $e\in\operatorname{End}(J_{\overline{K(H)}})$ (actually in
$\operatorname{End}(J_{K(H)})$) such that $e\eta_{H}=0$. Because
$\operatorname{End}(J_{\overline{K(H)}})\simeq\operatorname{End}(J_{\overline{K}})$,
see [CCO14, Lemma 1.2.1.2], it follows that there exists
$e^{\prime}\in\operatorname{End}(J_{\overline{K}})$ such that
$e^{\prime}s(x)=e^{\prime}x=0$ for all $x\in H$. Thus $H$ would have to be
contained in a proper abelian subvariety of $J$, which is impossible again
since the closed points of $H$ generate $J$. ∎

######  Corollary 3.4.

For every finite index coset ${\mathcal{N}}\subset{\mathbb{Z}}$,
$\bigcup_{n\in{\mathcal{N}}}H_{n}$ is Zariski dense in $J$ and
$\bigcup_{n\in{\mathcal{N}}}C_{n}$ is Zariski dense in $X$.

###### Proof.

Write ${\mathcal{N}}=a{\mathbb{Z}}+b$ where $a,b$ are integers with $a\neq 0$.
Since the morphism $t_{b\eta_{H}}\circ[a_{J_{K(H)}}]:J_{K(H)}\to J_{K(H)}$ is
dominant, the conclusion follows directly from Lemma 3.3. ∎

###  3.2. Albanese torsor and its Kummer quotient

The zero-cycles of degree $1$ on $H$ fixed under $\tau_{H}$ form a torsor $T$
under $J[2]$, which is trivial if $H$ has a $K$-rational Weierstrass point.
Let $c\in{\mathrm{H}}^{1}(K,J[2])$ be the associated cohomology class. Then
for every fixed $\overline{K}$-Weierstrass point $\infty\in W$, $c$ is given
by the cocycle $\gamma\mapsto\langle\gamma(\infty)-\infty\rangle$. The twist
of $J$ by the image of $c$ in ${\mathrm{H}}^{1}(K,J)$, i.e. the quotient of
$J\times T$ by the diagonal action of $J[2]$, is isomorphic to the Albanese
torsor $J^{1}=\operatorname{Alb}^{1}(H)$.

In general, we denote by $J^{i}=\operatorname{Alb}^{i}(H)$ the $i$-th Albanese
torsor, which parametrises zero-cycles of degree $i$ on $H$. There is an
isomorphism $J^{i}\simeq J^{i+2}(H)$ for all $i\geq 0$, given by adding
$\langle 2\infty\rangle$, and $J^{0}\simeq J$. The variety $J^{1}$ is an
initial object within the category of torsors $V$ under $J$ with a morphism
$\iota_{V}:H\to V$ (see [Wit08, Appendix A]). The involution $\tau_{H}$
extends formally to an involution $\tau_{J^{1}}$ on $J^{1}$ in a way that is
compatible with the involution $\tau_{J}$ on $J$ and the $J$-torsor structure.
We write $\pi^{1}:J^{1}\to X^{1}=\operatorname{Kum}(J^{1})$ for the
corresponding Kummer quotient morphism.

In the case $g=2$, the surfaces $X$ and $X^{1}$ are projective duals of each
other as quartic surfaces in ${\mathbb{P}}^{3}$. (By [CF96, Chapter 4], they
are isomorphic if and only if at least one of the translates of $C\subset X$
by $J[2](\overline{K})$ – the so-called _tropes_ – is defined over $K$, which
happens exactly when the sextic polynomial $f$ defining $H$ has a linear or
cubic factor.) However, it is pointed out in [CF96, Chapter 16] (somehow
surprisingly) that

######  Theorem 3.5 (Cassels–Flynn).

The minimal desingularisations of the surfaces $X$ and $X^{1}$ are isomorphic
over $K$.

###### Proof.

We offer a more conceptional and simplified proof here. Indeed, consider the
following involutions

$\displaystyle\sigma_{1}:H\times H$ $\displaystyle\to H\times H$
$\displaystyle(x_{1},x_{2})$
$\displaystyle\mapsto(\tau_{H}(x_{2}),\tau_{H}(x_{1})),$
$\displaystyle\sigma_{2}:H\times H$ $\displaystyle\to H\times H$
$\displaystyle(x_{1},x_{2})$ $\displaystyle\mapsto(x_{2},x_{1}).$

We define $\operatorname{Inv}^{2}(H)$ (resp. $\operatorname{Sym}^{2}(H)$) be
the quotient of $H\times H$ by $\sigma_{1}$ (resp. by $\sigma_{2}$). For any
point $(x_{1},x_{2})\in H\times H$, we write $\\{x_{1},x_{2}\\}$ for the
corresponding quotient image.

We now define

$\displaystyle\beta:\operatorname{Inv}^{2}(H)$ $\displaystyle\to J^{0}=J$
$\displaystyle\\{x_{1},x_{2}\\}$ $\displaystyle\mapsto\langle
x_{1}-x_{2}\rangle,$ $\displaystyle\beta^{\prime}:\operatorname{Sym}^{2}(H)$
$\displaystyle\to J^{2}$ $\displaystyle\\{x_{1},x_{2}\\}$
$\displaystyle\mapsto\langle x_{1}+x_{2}\rangle.$

Then by the Riemann-Roch theorem (see also [BP01, Proposition 4(2)]), outside
the image of the diagonal
$(\operatorname{id}_{H}\times\operatorname{id}_{H})(H)$ (resp. the anti-
diagonal $(\operatorname{id}_{H}\times\tau_{H})(H)$), $\beta$ (resp.
$\beta^{\prime}$) is a birational isomorphism.

Let us define the rational maps $b,b^{\prime}$ over $K$ by

$\displaystyle b:X$ $\displaystyle\dashrightarrow X^{1}$
$\displaystyle\\{z,\tau_{J}(z)\\}$ $\displaystyle\mapsto\\{z+\langle
x_{2}\rangle,z+\langle\tau_{H}(x_{1})\rangle\\}$

for $z\in J\setminus J[2]$, where $(x_{1},x_{2})$ is a preimage of $2z$ under
$\beta$, and

$\displaystyle b^{\prime}:X^{1}$ $\displaystyle\dashrightarrow X$
$\displaystyle\\{z^{1},\tau_{J^{1}}(z^{1})\\}$
$\displaystyle\mapsto\\{z^{1}-\langle x\rangle,z^{1}-\langle
x^{\prime}\rangle\\}$

for $z^{1}\in J^{1}\setminus T$, where $\\{x,x^{\prime}\\}$ is the preimage of
$2z^{1}$ under $\beta^{\prime}$. Then $b$ and $b^{\prime}$ are inverse to each
other, and hence are birational morphisms. Blowing up the singular loci of $X$
and $X^{\prime}$ gives the desired isomorphism. ∎

######  Remark 3.6.

Alternatively, in terms of cocycles, it can be checked that the image of $c$
in ${\mathrm{H}}^{1}(K,\operatorname{Aut}_{\overline{K}}\overline{K}(X))$ is
the coboundary of the automorphism of $\overline{K}(X)$ which is induced by
the birational map

$\sigma:X_{\overline{K}}\dashrightarrow
X_{\overline{K}},\pi(z)\mapsto\pi(z+\langle x_{2}-\infty\rangle);$

thus $c$ vanishes in
${\mathrm{H}}^{1}(K,\operatorname{Aut}_{\overline{K}}\overline{K}(X))$.

The image $H^{1}$ of the natural embedding $\iota_{J^{1}}:H\to J^{1}$ is
stable under the involution $\tau_{J^{1}}$ and thus yields a smooth rational
subcurve $C^{1}=\pi^{1}(H^{1})\subset X^{1}$. For any _odd_ integer
$n\in{\mathbb{Z}}_{>0}$, we denote by $[n_{J^{1}}]$ the composition of the
formal multiplication-by-$n$ map (of zero-cycles of degree $1$) $J^{1}\to
J^{n}$ and the previously defined isomorphism $J^{n}\simeq J^{1}$. Because
$[n_{J_{1}}]$ commutes with involution, it descends to a morphism
$[n_{X_{1}}]$. We let $H^{1}_{n}=[n_{J^{1}}](H^{1})$, the image of $H^{1}$
under the map $[n_{J^{1}}]$, and let $C^{1}_{n}=\pi^{1}(H^{1}_{n})$. Since

$C^{1}_{n}=(\pi^{1}\circ[n_{J^{1}}])(H^{1})=([n_{X^{1}}]\circ\pi^{1})(H^{1})=[n_{X^{1}}](C^{1}),$

all the curves $C^{1}_{n}$ are rational.

If $H$ has a $K$-rational Weierstrass point $\infty$, then the $K$-morphism

$\displaystyle\alpha_{J}:J^{1}$ $\displaystyle\to J$ $\displaystyle x$
$\displaystyle\mapsto x-\langle\infty\rangle$

commutes with the involutions $\tau_{J},\tau_{J^{1}}$, so it descends to a
$K$-morphism $\alpha_{X}:X^{1}\to X$ and yields a commutative diagram

${J^{1}}$${X^{1}}$${J}$${X}$$\scriptstyle{\pi^{1}}$$\scriptstyle{\alpha_{J}}$$\scriptstyle{\alpha_{X}}$$\scriptstyle{\pi}$

where the vertical arrows are $K$-isomorphisms. It is easy to see that
$H^{1}_{n}$ maps to $H_{n}$ (as a subvariety of $J$) under $\alpha_{J}$, and
$C_{n}^{1}$ maps to $C_{n}$ under $\alpha_{X}$ for all odd
$n\in{\mathbb{Z}}_{>0}$.

##  4\. Pullbacks along covers

Throughout this section, $K$ denotes a finitely generated field of
characteristic zero and $X=\operatorname{Kum}(J)$ denotes the Kummer quotient
associated to the Jacobian $J$ of a genus $g$ hyperelliptic curve $H$ over $K$
with a $K$-rational Weierstrass point $\infty$. We continue to consider $H$
embedded into $J$ via $\iota_{\infty}$.

###  4.1. Pullback along covers factoring through the Jacobian

######  Lemma 4.1.

Let $\phi:Y\to X$ be a cover of degree $>1$. Suppose that the base change
variety $Y\times_{X}J$ is not geometrically integral. Then for every finite
index coset ${\mathcal{N}}\subset{\mathbb{Z}}$, there exists a subcoset
${\mathcal{N}}^{\prime}\subset{\mathcal{N}}$ of finite index, such that none
of the rational curves $(C_{n})_{n\in{\mathcal{N}}^{\prime}}$ have a rational
section along $\phi$ over $\overline{K}$.

###### Proof.

We may base change everything to $\overline{K}$ and we fix an algebraic
closure $\Omega$ of $\overline{K}(X)$. We have that
$Y_{\overline{K}}\times_{X_{\overline{K}}}J_{\overline{K}}$ is not
geometrically integral if and only if
$\overline{K}(Y)\otimes_{\overline{K}(X)}\overline{K}(J)$ is not a field, or
equivalently, the field extensions $\overline{K}(Y)$ and $\overline{K}(J)$ in
$\Omega$ are not linearly disjoint over $\overline{K}(X)$. Since
$[\overline{K}(Y)\cdot\overline{K}(J):\overline{K}(Y)]\leq[\overline{K}(J):\overline{K}(X)]=2$,
this happens if and only if
$[\overline{K}(Y)\cdot\overline{K}(J):\overline{K}(Y)]<2$, or equivalently
$\overline{K}(J)\subseteq\overline{K}(Y)$. It follows that, over
$\overline{K}$, $\phi:Y\to X$ factors as a rational map
$\phi_{1}:Y\dashrightarrow J$ then to $X$.

By Lemma 3.3, we fix a closed point $P\in H$ which is non-degenerate in $J$.
By [CDJLZ21, Lemma 4.6], there exists a finite index subcoset
${\mathcal{N}}^{\prime}\subset{\mathcal{N}}$ such that the set
$\\{nP,n\in{\mathcal{N}}^{\prime}\\}$ does not intersect with the
indeterminacy locus, say $Z\subset J$, of $\phi_{1}$. This implies that none
of the curves $(H_{n})_{n\in{\mathcal{N}}^{\prime}}$ is contained in $Z$. If a
rational curve $(C_{n})_{\overline{K}}\subset
X_{\overline{K}},n\in{\mathcal{N}}^{\prime}$ has a rational section along
$\phi$, say $\widetilde{\phi}$, then it has a rational section
$\phi_{1}\circ\widetilde{\phi}$ along $\pi$. This is impossible since there
are no non-constant rational maps from the projective line to abelian
varieties (see [Mil86, Corollary 3.8]). ∎

###  4.2. Pullbacks along unramified covers

######  Lemma 4.2.

Let $D$ be a smooth, projective curve of genus $\geq 1$ over a separably
closed field. Fix a point $x$ in $D$ inducing the Abel–Jacobi embedding
$\iota_{x}:D\hookrightarrow\operatorname{Jac}(D)$. Let $\pi_{1}(D,x)$ (resp.
$\pi_{1}(\operatorname{Jac}(D),0)$) be the pointed étale fundamental group of
$D$ (resp. $\operatorname{Jac}(D)$). Then the map

$(\iota_{x})_{*}:\pi_{1}(D,x)\to\pi_{1}(\operatorname{Jac}(D),0)$

induced by $\iota_{x}$ is surjective. Consequently, every pullback via
$\iota_{x}$ of a connected finite étale covering of $\operatorname{Jac}(D)$
remains connected as a finite étale covering over $D$.

###### Proof.

This is explained in [Mil86a, Proposition 9.1]. Indeed, $(\iota_{x})_{*}$
induces an isomorphism

$\pi_{1}(D,x)^{\operatorname{ab}}\to\pi_{1}(\operatorname{Jac}(D),0),$

where $\pi_{1}(D,x)^{\operatorname{ab}}$ is the maximal abelian quotient of
$\pi_{1}(D,x)$ and in particular $(\iota_{x})_{*}$ is surjective. Thus by the
theory of Galois categories, all connected finite étale coverings of
$\operatorname{Jac}(D)$ have abelian Galois group, and the surjectivity of
$(\iota_{x})_{*}$ implies that all connected finite abelian coverings of $D$
are obtained precisely by pulling back the connected finite étale coverings of
$J$. See also [Sza09, Proposition 5.5.4 (2)]. ∎

######  Remark 4.3.

If $g=2$, one may also prove Lemma 4.2 by using the fact that the pullback of
a geometrically irreducible, unibranch, ample divisor on an abelian variety
along an isogeny is again geometrically irreducible. (See [BL04, Corollary
4.3.4]. Note that the statement is missing the crucial unibranch (or at the
least smoothness) assumption.)

Lemma 4.2 applies to $H\hookrightarrow J$, while an analogous statement for
$H_{n}$, $n>1$ fails because of the non-smoothness. However, one can remedy
the situation by restricting to a subfamily:

######  Proposition 4.4 (“Coprimality trick”).

Let $\phi:J^{\prime}\to J$ be an unramified cover. Then $\phi^{-1}(H_{n})$ are
geometrically irreducible for all $n$ coprime to $\deg(\phi)$.

###### Proof.

We may base change everything to $\overline{K}$, without affecting the
geometrical irreducibility. By the Lang–Serre theorem [Ser88, Chapter I §2],
$J^{\prime}$ is an abelian variety and $\phi$ can be made into an isogeny. The
case where $\deg(\phi)=1$ is clear and so we assume from now on that
$\deg(\phi)>1$.

If $n=1$, then the claim follows from Lemma 4.2. In particular,
$\operatorname{Ker}(\phi)\subset\phi^{-1}(H_{1})$. In general, if $n$ is
coprime to $\deg(\phi)=\deg(\operatorname{Ker}(\phi))$, then
$[n_{J^{\prime}}]|_{\operatorname{Ker}\phi}$ is an isomorphism. Thus

$\displaystyle\phi^{-1}(H_{n})$ $\displaystyle=\phi^{-1}([n_{J}](H_{1}))$
$\displaystyle=[n_{J^{\prime}}](\phi^{-1}(H_{1}))+\operatorname{Ker}(\phi)$
$\displaystyle=[n_{J^{\prime}}]\left(\phi^{-1}(H_{1})+\operatorname{Ker}(\phi)\right)$
$\displaystyle=[n_{J^{\prime}}](\phi^{-1}(H_{1})),$

and hence $\phi^{-1}(H_{n})$ is geometrically irreducible. ∎

######  Corollary 4.5.

Let $\phi_{i}:Y_{i}\to X,i\in I$ be any finite collection of covers of with
$\deg(\phi_{i})>1$ such that $Y_{i}\times_{X}J$ are all geometrically
integral, and the base change morphisms
$\widetilde{\phi}_{i}:Y_{i}\times_{X}J\to J$ are all unramified. Then
$\phi_{i}^{-1}(C_{n})$ are all geometrically irreducible for all $n$ coprime
to $\prod_{i}\deg(\phi_{i})$.

###### Proof.

Since $H_{n}$ is stable under the involution, $\phi_{i}^{-1}(C_{n})$ is
geometrically irreducible if and only if $\widetilde{\phi}_{i}^{-1}(H_{n})$
is. Since $Y_{i}\times_{X}J$ are all geometrically integral, we have
$\deg(\widetilde{\phi}_{i})=\deg(\phi_{i})$. We then apply Proposition 4.4 to
$\widetilde{\phi}_{i}:Y_{i}\times_{X}J\to J$. ∎

###  4.3. Pullbacks along ramified covers

The following result is essentially [CDJLZ21, Theorem 1.3].

######  Theorem 4.6 (Corvaja–Demeio–Javanpeykar–Lombardo–Zannier).

Let $A$ be an abelian variety over a finitely generated field $K$ of
characteristic $0$ with a non-degenerate point $x\in A(K)$. Let
$f_{i}:B_{i}\to A$ be a finite collection of ramified covers. Then for any
finite index coset ${\mathcal{P}}\subset{\mathbb{Z}}x$, there exists a finite
index subcoset of ${\mathcal{P}}^{\prime}\subset{\mathcal{P}}$ such that for
every $x^{\prime}\in{\mathcal{P}}^{\prime}$,
$(f_{i}^{-1}(x^{\prime}))(K)=\emptyset$.

###### Proof.

It suffices to work with a single ramified cover $f:B\to A$ by successively
analysing the finite index coset obtained at each step. Write
${\mathcal{P}}=\Pi+bx$ where $b\in{\mathbb{Z}}$ and $\Pi$ is a finite index
subgroup of ${\mathbb{Z}}x$ and $P\in A(K)$. Define $f^{\prime}=t_{-bx}\circ
f$ where $t_{-bx}:A\to A$ is the translation-by-$(-bx)$ morphism. Then we
apply [CDJLZ21, Theorem 1.3] to $\Pi$ and get a finite index coset
$\Pi^{\prime}\subset\Pi$, for which any $K$-fibre of $f^{\prime}$ has no
$K$-point. Then we may take ${\mathcal{P}}^{\prime}=\Pi^{\prime}+bx$ to obtain
the same conclusion for $f$. ∎

######  Corollary 4.7.

Let $\phi_{i}:Y_{i}\to X,i\in I$ be any finite collection of covers such that
all the base change morphisms $\widetilde{\phi}_{i}:Y_{i}\times_{X}J\to J$ are
ramified with $Y_{i}\times_{X}J$ geometrically integral. Then for any finite
index coset ${\mathcal{N}}\subset{\mathbb{Z}}$, there exists a finite index
subcoset ${\mathcal{N}}^{\prime}\subset{\mathcal{N}}$ such that for any
$n\in{\mathcal{N}}^{\prime}$, the rational curve $C_{n}$ does not have a
rational section along $\phi_{i}$ for all $i\in I$.

###### Proof of Corollary 4.7.

Consider the base change

$\widetilde{\phi}_{i,K(H)}:(Y_{i}\times_{X}J)_{K(H)}\to J_{K(H)}$

to the function field $K(H)$ of $H$. This is still a ramified morphism with
$(Y_{i}\times_{X}J)_{K(H)}$ being geometrically integral.

By Lemma 3.3, the $K(H)$-rational point $\eta_{H}\in J_{K(H)}$ induced by
$\iota_{\infty}:H\hookrightarrow J$ is non-degenerate. By Theorem 4.6 applied
to ${\mathcal{N}}\eta_{H}$ over the finitely generated field $K(H)$, there
exists a finite index coset ${\mathcal{N}}^{\prime}\subset{\mathcal{N}}$ such
that for any $\eta^{\prime}_{H}\in{\mathcal{N}}^{\prime}\eta_{H}\subset
J_{K(H)}$,

$\eta^{\prime}_{H}\notin\bigcup_{i\in
I}\widetilde{\phi}_{i,K(H)}((Y_{i}\times_{X}J)_{K(H)}(K(H))).$

Consider the cartesian diagram

${(Y_{i}\times_{X}J)_{K(H)}}$${Y_{i,K(H)}}$${Y_{i,K(C)}}$${J_{K(H)}}$${X_{K(H)}}$${X_{K(C)}}$$\scriptstyle{\widetilde{\phi}_{i,K(H)}}$$\scriptstyle{\phi_{i,K(H)}}$$\scriptstyle{\phi_{i,K(C)}}$

where the rightmost square is the base change from $K(C)$ to $K(H)$. Let
$\eta_{C}\in X_{K(C)}$ be the image of $\eta_{H}$ under $J_{K(H)}\to
X_{K(C)}$. Then for any
$\eta^{\prime}_{C}\in{\mathcal{N}}^{\prime}\eta_{C}\subset X_{K(C)}$,

$\eta^{\prime}_{C}\notin\bigcup_{i\in I}\phi_{i,K(C)}(Y_{i,K(C)}(K(C))).$

Equivalently, none of the curves $C_{n}$, $n\in{\mathcal{N}}^{\prime}$, has a
rational section along $\phi_{i}$ for all $i\in I$. ∎

##  5\. Hilbert Property for Jacobian Kummer varieties

######  Proposition 5.1.

Let $X$ be a quasi-projective variety over a Hilbertian field $K$. Assume that
there exists an infinite set ${\mathcal{C}}$ of (not necessarily smooth)
$K$-rational subvarieties on $X$ satisfying the following condition:

For every finite family of covers $\phi_{i}:Y_{i}\to X$, $i=1,\dots,r$, with
$\deg(\phi_{i})>1$, there exists a subset
${\mathcal{C}}^{\prime}\subset{\mathcal{C}}$ such that
$\bigcup_{C\in{\mathcal{C}}^{\prime}}C$ is Zariski dense in $X$ and for all
$C\in{\mathcal{C}}^{\prime}$ and $i=1,\dots,r$, $C$ does not have a rational
section along $\phi_{i}$.

Then $X$ has the Hilbert Property.

###### Proof.

Let us fix a finite family of covers $\phi_{i}:Y_{i}\to X$, $i=1,\dots,r$ with
$\deg(\phi_{i})>1$. Let ${\mathcal{C}}^{\prime}\subset{\mathcal{C}}$ be as
stated in the proposition. Because the union of $C\in{\mathcal{C}}^{\prime}$
is Zariski dense, upon replacing ${\mathcal{C}}^{\prime}$ by a subset if
necessary, we may assume that for every subvariety
$C\in{\mathcal{C}}^{\prime}$, $C$ is not contained in the ramification locus
of any $\phi_{i}$ for $i=1,\dots,r$. In view of Proposition 2.1, to show the
Hilbert Property for $X$, it suffices to show that

$\left(\bigcup_{C\in{\mathcal{C}}^{\prime}}C(K)\right)\setminus\left(\bigcup_{i=1}^{r}\phi_{i}(Y_{i}(K))\right)$

is not contained in any proper Zariski closed subset of $X$.

Assume that there exists $Z\subset X$ proper Zariski closed containing all the
$K$-points above. We fix $C_{0}\in{\mathcal{C}}^{\prime}$ such that
$C_{0}\not\subset Z$. For every $i\in\\{1,\dots,r\\}$, let ${C_{ij}}$,
$j=1,\dots,k_{i}$, be the irreducible components of $\phi_{i}^{-1}(C_{0})$.
The maps $\phi_{i}|_{C_{ij}}:C_{ij}\to C_{0}$ are finite morphisms because
$C_{0}$ does not have a rational section along $\phi_{i}$, and since $C_{0}$
is not contained in the ramification locus of $\phi_{i}$ by assumption, if
$\dim(C_{ij})=\dim(C_{0})$ then $\phi_{i}|_{C_{ij}}$ is dominant, and hence
has degree $>1$. It thus follows from the Hilbert Property for rational
varieties (see [Ser08, p. 30 Examples]) that $C_{0}(K)$ is not thin, whence

$C_{0}(K)\setminus\left((Z\cap
C_{0})(K)\bigcup\left(\bigcup_{i=1}^{r}\bigcup_{j=1}^{k_{i}}\phi_{i}(C_{ij}(K))\right)\right)\neq\varnothing,$

which is a contradiction. This finishes the proof. ∎

We now turn to the proofs of the main theorems. We will apply Proposition 5.1
only in the case where ${\mathcal{C}}$ is a countable set of subcurves.

### Proof of Theorem 1.5

Consider the family of rational $K$-subcurves $\\{C^{1}_{n}\\}_{n\in
1+2{\mathbb{Z}}}$ on $X^{1}$. Let $\phi_{i}:Y_{i}\to X^{1}$, $i=1,\dots,r$ be
any fixed finite family of covers with $\deg(\phi_{i})>1$. We want to
construct a finite index subcoset ${\mathcal{N}}^{\prime}\subset
1+2{\mathbb{Z}}$ of odd integers such that
$\\{C^{1}_{n}\\}_{n\in{\mathcal{N}}^{\prime}}$ such that all conditions of
Proposition 5.1 are satisfied. Let us take a finite field extension $L$ of $K$
such that $H$ has an $L$-rational Weierstrass point $\infty$. We base change
everything to $L$ (and we omit the subscript $L$). In view of the diagram
$(\dagger)$, we shall replace $X^{1}$ by the isomorphic $X$ and $C^{1}_{n}$ by
$C_{n}$ in the following.

Let $\widetilde{\phi}_{i}:\widetilde{Y}_{i}\to J$ be the base change of
$\phi_{i}$ along the Kummer quotient $\pi:J\to X$. It may be assumed that for
certain $r_{1}$, all $\widetilde{Y}_{i}$, $1\leq i<r_{1}$ are not
geometrically integral and the remaining $\widetilde{Y}_{i}$, $r_{1}\leq i\leq
r$ are geometrically integral. For $r_{1}\leq i\leq r$, we may moreover assume
that for certain $r_{2}\geq r_{1}$, the morphisms
$\widetilde{\phi}_{i},r_{1}\leq i\leq r_{2}$ are all unramified, while the
remaining $\widetilde{\phi}_{i},r_{1}<i\leq r$ are all ramified.

We start with

${\mathcal{C}}=\\{C_{n}:n\in 1+2{\mathbb{Z}}\\}.$

If $r_{2}>r_{1}$, i.e. certain $\widetilde{\phi}_{i}$ are unramified, we
consider

$d=\operatorname{lcm}(\deg\phi_{i},i=r_{1},\dots,r_{2}),\quad{\mathcal{N}}^{\acute{e}t}=1+2d{\mathbb{Z}}.$

Then ${\mathcal{N}}^{\acute{e}t}$ is a finite index coset of ${\mathbb{Z}}$,
and by Corollary 4.5, any $C_{n}$ with $n\in{\mathcal{N}}^{\acute{e}t}$ does
not have a rational section along $\phi_{i}$ for all $r_{1}\leq i\leq r_{2}$.
Otherwise if there are no such unramified covers, we just take
${\mathcal{N}}^{\acute{e}t}=1+2{\mathbb{Z}}$.

By Corollary 4.7, one obtains a finite index subcoset
${\mathcal{N}}^{geo}\subset{\mathcal{N}}^{\acute{e}t}$ such that any $C_{n}$
with $n\in{\mathcal{N}}^{geo}$ does not have a rational section along
$\phi_{i}$ for all $i=r_{2}+1,\dots,r$. Then on applying Lemma 4.1, one
obtains a further finite index subcoset
${\mathcal{N}}^{\prime}\subset{\mathcal{N}}^{geo}$ such that any $C_{n}$ with
$n\in{\mathcal{N}}^{\prime}$ does not have a rational section along $\phi_{i}$
for all $i=1,\dots,r_{1}-1$ (still over the field extension $L$ of $K$).

Any such $C_{n}^{1},n\in{\mathcal{N}}^{\prime}$ cannot have a rational section
along any $\phi_{i}$ over the base field $K$ as well. Thus, by using again
Corollary 3.4, all conditions of Proposition 5.1 are satisfied for the family
of rational curves

${\mathcal{C}}^{\prime}=\\{C_{n}:n\in{\mathcal{N}}^{\prime}\\},$

from which we conclude the Hilbert Property. ∎

### Proof of Theorem 1.3

This follows from Theorem 1.5 together with Theorem 3.5 using the birational
invariance of the Hilbert Property. ∎

### Proof of Theorem 1.4

This follows immediately from Theorem 1.5 since $H$ has by assumption a
$K$-rational Weierstrass point (at infinity), and thus $X\cong X^{1}$ over $K$
in view of the diagram $(\dagger)$. ∎

## Acknowledgements

We are in debt to J. Demeio for generously sharing many key insights into the
Hilbert Property which simplify and generalise our previous arguments
considerably. Z. Huang was supported by the DFG grant DE1646/4-2 at Leibniz
Universität Hannover, and by the FWF grant No. P32428-N35 at IST Austria. This
work was partly undertaken at the workshop “Arithmetic Statistics and Local-
Global Principles” hosted by the Erwin Schrödinger International Institute for
Mathematics and Physics, Vienna and the workshop “Rational Points on Higher-
Dimensional Varieties” hosted by the International Centre for Mathematical
Sciences, Edinburgh. We thank the organisers and the institutes for the
hospitality.

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