Proof. The proof of [3] shows that, for \(K\) a number field, given covers \(\psi_j: Z_j \rightarrow S\), \(j = 1,\dots,s\) of degrees \(\geq 2\) such that \(S(K) \setminus \bigcup_{j=1}^s\psi_j(Z_j(K))\) is not dense, there is a subset \(\{j_1,\ldots,j_t\}\subset\{1,\ldots,s\}\) such that \(\psi_{j_i}: Z_{j_i} \rightarrow S\) is contained in \(\mathcal{C}\) for all \(j_i\in\{j_1,\ldots,j_t\}\) and \(S(K) \setminus \bigcup_{i=1}^t\psi_{j_i}(Z_{j_i}(K))\) is not dense (the simply connected hypothesis in loc.cit.ensures that \(\mathcal{C} = \emptyset\) in that setting). Since this non-density would contradict our hypothesis, we deduce the result for number fields. To deduce the same for \(K\) finitely generated over \(\mathbb{Q}\), note that the Mordell–Weil theorem, Faltings’ theorem and Merel’s theorem, which are the key ingredients in the proof of loc.cit.in the setting of number fields, all admit generalizations to such \(K\): the generalization of Mordell–Weil follows from the Lang–Néron theorem (see [17]), the generalization of Faltings’ theorem follows from the work of McQuillan [18], and the generalization of Merel’s theorem is established by Colliot-Thélène in [8]. ◻

Proof of Theorem 1. Let \(\mathcal{C}\) be the set in Theorem 4, and let \(\{\psi_i:V_i \rightarrow S, i=1,\dots,s\}\) be a finite subset of \(\mathcal{C}\). We will show that \(S(K) \setminus \bigcup_{i=1}^r\psi_i(V_i(K))\) is Zariski-dense in \(S\), so that we can deduce the result from Theorem 4. For each \(i \in \{1,\dots,s\}\), denote by \(F_i\) the fiber product \(X \times_{\phi,\psi_i} V_i\). Let \(X_{\eta}\) (resp.\(F_{i,\eta}\)) be the generic fiber of the low-genus fibration \(X \rightarrow B\) (resp.\(F_i \to X \rightarrow B\)). We first show that for all \(i \in \{1,\dots,s\}\), the curve \(F_{i,\eta}\) is integral over \(K(B)\). We have the following commutative diagram. \[\begin{tikzcd} {F_{i,\eta}} \arrow[d] \arrow[r] & V_i \arrow[d, "\psi_i"] \\ X_{\eta} \arrow[r, "\phi"] & S. \end{tikzcd}\] Since \(S\) is a smooth surface and \(V_i\) is normal, the morphism \(\psi_i\) is flat by miracle flatness [19]. It follows that \(F_{i,\eta} \to X_{\eta}\) is finite flat as well (both finiteness and flatness being preserved under base change), with generic fiber \(\mathop{\mathrm{Spec}}K(X) \times_{\mathop{\mathrm{Spec}}K(S)} \mathop{\mathrm{Spec}}K(V_i) \cong \mathop{\mathrm{Spec}}(K(X) \otimes_{K(S)} K(V_i))\). Since \(\phi\) has geometrically integral generic fiber, \(K(X) \otimes_{K(S)} K(V_i)\) is a field: the extensions \(K(X)/K(S)\) and \(K(V_i)/K(S)\) are regular and algebraic respectively, hence linearly disjoint. Thus the flat morphism \(F_{i, \eta} \to X_{\eta}\) has integral generic fiber, and thus \(F_{i, \eta}\) is also integral [20].

For each \(i\in\{1,\ldots,s\}\), denote by \(D_i \subset S\) the branch locus of \(\psi_i\), which is a divisor by Nagata–Zariski purity [19]. We claim that a general element of the low-genus family intersects each \(D_i\) transversally: otherwise, there is at least one \(i_0\) such that \(D_{i_0}\) belongs to \(F\), but then the restriction of \(\psi_{i_0}\) over \(S \setminus F\) is an unramified cover of degree \(\geq 2\), contradicting that \(S \setminus F\) is algebraically simply connected. Then there is at least one smooth point on \(F_{i,\eta}\) which is ramified over \(X_{\eta}\) (cf. [21]), hence \(F_{i,\eta}\) is an integral \(K(B)\)-curve ramified over \(X_{\eta}\). We may spread out over some open \(U \subset B\) to a morphism \(\xi_i\colon F_{i,U} \to X_U\) such that, for all \(P \in U\), the curve \(F_{i,P}\) is a ramified cover of \(X_P\).

Since \(K\) is finitely generated over \(\mathbb{Q}\) and \(X(K)\) is Zariski-dense, there is a non-empty Zariski open \(V \subset X_U\) such that, for each \(v \in V(K)\), the fiber \(\pi^{-1}(\pi(v))\) has infinitely many \(K\)-rational points; see [8] for the genus-\(1\) case (the genus-\(0\) case is elementary, as any smooth curve of genus \(0\) with one rational point has infinitely many). By assumption, \(V(K)\) is Zariski-dense in \(V\), and we deduce Zariski-density in \(U\) of \[\mathcal{A} := \{u \in U(K): \#\pi^{-1}(u)(K)=\infty\}.\]

When \(g = 1\), it follows from Lang–Néron that, for each \(u \in \mathcal{A}\), only finitely many points in \(\pi^{-1}(u)(K)\) can lift to \(F_{i,U}\) for some \(i\), hence \(\pi^{-1}(u)(K) \setminus \cup_{i=1}^s \xi_i(F_{i,U}(K))\) is dense in \(\pi^{-1}(u)\); the same conclusion holds for \(g = 0\) since \(\pi^{-1}(u)\) has the Hilbert property, hence \[\mathcal{A}' :=\bigcup_{u \in \mathcal{A}} \pi^{-1}(u)(K) \setminus \cup_{i=1}^s \xi_i(F_{i,U}(K))\] is Zariski-dense in \(X_U\) in either case. Thus \(\phi(\mathcal{A}')\) is Zariski-dense in \(S\). Since \(\phi(\mathcal{A}')\) is contained in \(S(K) \setminus \cup_{i=1}^s \psi_i(V_i(K))\), this concludes the proof. ◻

Proof.

1. Let \(W^{\text{aff}} \subset \mathbb{A}^3\) and \(S^{\text{aff}} \subset \mathbb{A}^3\) be the restrictions of \(W\) and \(S\) to the affine charts \(X_3 \neq 0\) and \(w \neq 0\), respectively given by \(x_1^2 = x_0^3 + a_4(x_2)x_0 + a_6(x_2)\) and \(y^2=x^3+a_4(f(z))x+a_6(f(z))\). Note that \(S^{\text{aff}} = \phi^{-1}(W^{\text{aff}})\) for the flat map \[\phi:\mathbb{A}^3 \to \mathbb{A}^3, \quad (x,y,z) \mapsto (x,y,f(z)),\] and the induced map \(S^{\text{aff}}\to W^{\text{aff}}\) is also flat by preservation of flatness under base-change. Now, any variety \(V\) with a flat surjective morphism \(V' \to V\) from a smooth variety \(V'\) is smooth (see [22]), hence \(W^{\text{aff}}\) is smooth. Then all singularities of \(W\) lie on \(X_3 = 0\).

   When \(X_3=0\), the equation for \(W\) also gives \(X_0=0\). The point \([0:1:0:0]\) is smooth, as \((\partial F_W/\partial X_3)(0,1,0,0)= -1\), with \(F_W := X_0^3 + a_4(X_2/X_3)X_0 X_3^2 + a_6(X_2/X_3) X_3^3 - X_1^2 X_3\). The remaining points on the line \(X_0 = X_3 = 0\) are covered by the affine chart \(X_2 \neq 0\), where the equation becomes \(x_1^2 x_3= x_0^3 + a_4'(x_3)x_0 x_3 + a_6'(x_3) x_3,\) with \(a_4', a_6' \in k[t]\) defined by \(a_4'(t)=ta_4(1/t)\) and \(a_6'(t)=t^2a_6(1/t)\). We rewrite the equation as \(x_1^2x_3-cx_3=O(x_0,x_3)^2\) and deduce that the only singular points with \(x_0=x_3=0\) are \((0,\pm \sqrt{c},0)\).

2. To determine the singularity type/types, we follow [23].

   1. First assume that \(c \neq 0\).

      1. If further \(a \neq 0\), then \[\begin{align} & F_W\mathopen{}\mathclose{\left(\frac{2\sqrt{c}}{a}\mathopen{}\mathclose{\left(X_0-\frac{d}{2\sqrt{c}} - X_2}\right),\frac{1}{2}(-X_2+X_3),\frac{1}{2\sqrt{c}}(X_2 + X_3), X_1}\right) \\ & = X_0 X_1 X_3 + G_W(X_0,X_1,X_2), \end{align}\] \[\begin{align} G_W(X_0,X_1,X_2) & := \frac{8c\sqrt{c}}{a^3}X_0^3 - \frac{12cd}{a^3}X_0^2 X_1 + \frac{2a^2b\sqrt{c} + 6\sqrt{c}d^2}{a^3}X_0 X_1^2 \\ & + \frac{a^3 e - a^2 b d - d^3}{a^3}X_1^3 - \frac{24c\sqrt{c}}{a^3} X_0^2 X_2 \\ & + \frac{a^3 + 24cd}{a^3}X_0 X_1 X_2 + \frac{-2a^2b\sqrt{c} - 6\sqrt{c}d^2}{a^3}X_1^2 X_2 \\ & + \frac{24 c\sqrt{c} }{a^3} X_0 X_2^2 + \frac{-a^3 - 12 c d}{a^3}X_1 X_2^2 - \frac{8 c\sqrt{c}}{a^3} X_2^3. \end{align}\] Since \(G_W(0,0,1) \neq 0\), one of the singularities is \(A_2\) by [23], hence so is the other singularity by symmetry.

      2. If instead \(a = 0\), then \[\begin{align} & F_W\mathopen{}\mathclose{\left(X_2,\frac{1}{2}\mathopen{}\mathclose{\left(-X_0 + \frac{d}{2\sqrt{c}}X_1 + X_3}\right),\frac{1}{2\sqrt{c}}\mathopen{}\mathclose{\left(X_0 - \frac{d}{2\sqrt{c}}X_1 + X_3}\right),X_1}\right) \\ & = X_0 X_1 X_3 + G_W(X_0,X_1,X_2), \end{align}\] \[\begin{align} G_W(X_0,X_1,X_2) & := \frac{d}{2\sqrt{c}}X_0 X_1^2 + \frac{4ce - d^2}{4c}X_1^3 + bX_1^2 X_2 + X_2^3. \end{align}\] Since \(G_W(0,0,1) \neq 0\), we reach the same conclusion as in (i).

   2. Next suppose that \(c = 0\) and \(a \neq 0\). Then \[F\mathopen{}\mathclose{\left(\frac{1}{a}(X_0-dX_1),X_2,X_3,X_1}\right) = X_0X_1X_3 + G_W(X_0,X_1,X_2,X_3),\] \[\begin{align} G_W(X_0,X_1,X_2) & := \frac{1}{a^3}X_0^3 - \frac{3d}{a^3}X_0^2 X_1 + \frac{a^2b + 3d^2}{a^3}X_0X_1^2 \\ & + \frac{a^3e - a^2bd - d^3}{a^3}X_1^3 - X_1 X_2^2. \end{align}\] Since \(g_1(X_1) := G_W(0,X_1,1)\) has a zero of order \(1\) at \(X_1 = 0\) and \(g_0(X_0):=G_W(X_0,0,1)\) has a zero of order \(3\) at \(X_0 = 0\), the singularity is \(A_5\) [23].

   3. Now assume that \(c = a = 0\); by smoothness of \(S\), we find \(d \neq 0\). Then \[F\mathopen{}\mathclose{\left(X_2,X_1,X_3,\frac{1}{\sqrt{d}}X_0}\right) = X_3 X_0^2 + G_W(X_0,X_1,X_2),\] \[G_W(X_0,X_1,X_2) := X_2^3 + \frac{1}{d\sqrt{d}} X_0^2 \mathopen{}\mathclose{\left(e X_0 + b\sqrt{d}X_2}\right) - \frac{1}{\sqrt{d}} X_0 X_1^2.\] Since \(G_W(0,X_1,X_2) = X_2^3\), the singularity is \(E_6\) by [23].

3. If \(W\) were a cone, then since all of its singularities are isolated by the above, it would have to be a cone over a smooth plane cubic \(C\). Denote by \(\phi:W \dashrightarrow C\) the natural projection. The composition \(\phi \circ \theta:S \dashrightarrow C\) is a dominant rational map from \(S\) to the curve \(C\) of genus \(1\). However, since \(S\) is a geometrically rational surface, such a map cannot exist and this is a contradiction.

 ◻

Proof. The set is clearly closed, so it suffices to find one point \(R \in S\) not lying inside it.

The curve \(D_P\) is irreducible, thus so is \(B_P\), which coincides with the plane section of \(W \subset \mathbb{P}^3\) given by \(\Pi_P: w_0^3 X_2 - w_0^3f(z_0/w_0)X_3 = 0\). Let \(L \subset \Pi_P\) be a general line. This intersects \(B_P\) transversely in 3 points. Let \(\sigma: W' \rightarrow W\) be the composition of the minimal desingularization of \(W\) and the blowup in the inverse image of \(L \cap W\). Let \(g: W' \rightarrow \mathbb{P}^1\) be the genus-\(1\) fibration corresponding to the pencil of plane sections cut out in \(W\) by planes containing \(L\).

Any relatively minimal model of \(W'\) is a rational elliptic surface, which has (étale/
topological) Euler number \(12\) [24], making the Euler number \(e\) of \(W'\) at least \(12\). The Euler number of a regular genus-\(1\) fibration is the sum \(\sum_v e(F_v)\) of the Euler numbers of its singular fibers \(F_v\) (loc.cit.). Consider now the planes \(\Pi_{\pm}\) passing through \(L\) and \([0:\pm \sqrt c:1:0]\), with the convention that \(\Pi_+=\Pi_-\) in the case \(c=0\). Let \(F_P,F_{\pm}\) be the fibers corresponding to the plane sections \(\Pi_P\cap W\) and \(\Pi_{\pm} \cap W\), respectively.

The diagram below summarizes part of what we defined so far. \[\begin{tikzcd} D_P \arrow[d, phantom, sloped, "\subset"]\arrow[r, "\theta"] & B_P=\Pi_P\cap W \arrow[d, phantom, sloped, "\subset"]&\arrow[l, "\sigma" above, "\sim" below] F_P\arrow[d, phantom, sloped, "\subset"]\\ S\arrow[r, "\theta"]& W &W' \arrow[l, "\sigma" above] \arrow[d, "g"]\\ & & \mathbb{P}^1 \end{tikzcd}\] We know that \(B_P\) is irreducible. We claim that each \(\Pi_{\pm} \cap W\) is also irreducible. In fact, since \(L\) is a general line on \(\Pi_P\), it is skew with respect to any finite set of lines in \(\mathbb{P}^3\) not lying on \(\Pi_P\). There are only finitely many lines on \(W\) since it is not a cone (Lemma 1), and none of these lies on \(\Pi_P\) since \(\Pi_P \cap W=B_P\) is irreducible, hence \(L\) is skew with respect to all lines of \(W\). Then \(\Pi_{\pm} \cap W\) are plane cubics with no lines, proving the claim.

From irreducibility above, we infer that \(F_P \cong \Pi_P\cap W\) has \(1\) component, while each \(F_{\pm}\) has at most \(3\) components when \(F_+ \neq F_-\) (equiv.\(c \neq 0\)) and \(6\) or \(7\) components when \(F_+ = F_-\) (equiv.\(c = 0\)): the proper transform of \(\Pi_{\pm}\cap W\) and the exceptional components coming from the desingularization (the numbers come from Lemma 1: \(A_k\) and \(E_k\)-singularities give \(k\) exceptional components). In each case we have \(\sum_{F_v \in \{F_+,F_-\}}e(F_v) \leq 8\), thus \(\sum_{F_v \in \{F_P,F_+,F_-\}}e(F_v) \leq 10\), and there is at least one other singular fiber \(F_Q \neq F_P,F_{\pm}\).

Now \(\sigma(F_Q)\) is a plane section of \(W\) with some singular point \(R' \neq [0: \pm \sqrt c:1:0]\), hence \(\sigma(F_Q) = T_{R'} W \cap W\). Then, for \(R \in \theta^{-1}(R')\), the curve \(C_R = \theta^{-1}(\sigma(F)) \subset S\) intersects \(D_P\) in at least 3 points. Since \(C_R \cdot D_P = 3\), we are done. ◻

Proof. The fibers of \(\pi_1\) are curves in \(|{-3K_S}|\), and in particular they all have the same Hilbert polynomial (with respect to any fixed ample line bundle on \(S\)), making \(\pi_1\) flat [25]. For general \(P\), the curve \(C_P\) is geometrically integral [9]. The morphism \(\pi_1:\mathcal{C} \to S^o\) is now flat over a geometrically integral base with geometrically integral generic fiber, hence \(\mathcal{C}\) is geometrically integral [20]. ◻

Proof. The proof of [9] shows that, for infinitely many \(R \in \mathbb{P}^1(K)\), the surface \(\mathcal{C}_R:=\mathcal{C} \times_{\pi_1,S}D_R\) (i.e.the restriction of \(\mathcal{C}\) to \(P \in D_R\)) has dense \(K\)-rational points. In particular, \(\{R \in \mathbb{P}^1(K): (\pi\circ\rho^{-1}\circ\pi_1)^{-1}(R)(K) \text{ is dense in } (\pi\circ\rho^{-1}\circ\pi_1)^{-1}(R)\}\) is dense in \(\mathbb{P}^1(K)\), thus \(\mathcal{C}(K)\) is dense in \(\mathcal{C}\). ◻

Proof. Assume that \(\pi_2\) does not have a geometrically integral generic fiber; then the same holds for \(\pi^{\text{n}}_2\), and we get a relative normalization decomposition: \[\label{Eq:Stein} \pi^{\text{n}}_2:\mathcal{C}^{\text{n}} \xrightarrow{g} S' \xrightarrow{h} S,\tag{1}\] where \(S'\) is integral, \(g\) has geometrically integral fibers, and \(h\) is finite of degree at least \(2\). We are going to derive a contradiction by showing that the function field extension \(K(S')/K(S)\) is a subextension of two linearly disjoint extensions of \(K(S)\).

First extension. The map \(\pi_1\) admits a rational section \(S \dashrightarrow \mathcal{C}, \;P \mapsto (P,Q)\) with \(Q \in D_P\) such that \(Q=[-2]P\) on the elliptic curve \(D_P\) with origin \(\mathcal{O}\) (see Remark 12). For general \(P\), this point \(Q\) has multiplicity \(1\) in the intersection \(D_P \cap C_P\) and is thus smooth in \(C_P\), hence \((P,Q)\) lies in the smooth locus of \(\mathcal{C}\). In particular, this section lifts to a map \(s:S \dashrightarrow \mathcal{C}^{\text{n}}\). The composition \(\pi^{\text{n}}_2 \circ s: S \dashrightarrow S\) is the map \([-2]:S \dashrightarrow S\), and thus \(K(S')\) is contained in the degree-\(4\) extension \(K(S)_2/K(S)\) (with \(K(S)_2 \cong K(S)\) as abstract fields) given by multiplication by \([-2]\).

Second extension. For general \(P\), the restriction \(\mathcal{C}^n_P := \mathcal{C}^n \times_{\mathcal{C}} C_P \to C_P\) of the normalization morphism \(\mathcal{C}^n \to \mathcal{C}\) above \(C_P \cong \{P\} \times C_P \subset \mathcal{C}\) is the normalization morphism of \(C_P\) and \(C_P\) has a simple node at \(P\), thus the inverse image \(V\) in \(\mathcal{C}^n\) of a sufficiently small Zariski-open \(U\) of the diagonal \(\Delta:= \{(P,P) : P \in S^o\} \subset \mathcal{C}\) is an étale double cover \(V \to U\). The composition \[V \to \mathcal{C}^{\text{n}} \xrightarrow{\pi^{\text{n}}_2} S\] is a generically finite map of degree \(2\), and by 1 the field extension \(K(S')/K(S)\) of degree \(\geq 2\) embeds in the étale extension \(K(V)/K(S)\) of degree \(2\). Hence \(K(S')=K(V)\).

Linear disjointness. Since the Mordell–Weil group of \(\mathcal{E}\) is torsion-free [24], the degree-\(4\) extension \(K(S)_2/K(S)\) given by multiplication by \([-2]\) has no non-trivial proper subextensions, as any of these would have degree \(2\) and would give rise to a cyclic 2-isogeny on the elliptic surface \(\mathcal{E}\), hence to a non-trivial \(2\)-torsion section. This immediately implies linear disjointness and concludes the proof. ◻

Proof of Theorem 2. Consider the low-genus family \(\pi^{\text{n}}_2:\mathcal{C}^{\text{n}} \to S\) with underlying fibration \(\pi^{\text{n}}_1\colon\mathcal{C}^{\text{n}} \rightarrow S^o\) (this is a family with general fiber the normalization of some curve \(C_P\) and general element \(C_P\)). We let \(X\) be its inverse image in \(\mathcal{E}\), i.e.\(X=\mathcal{C}^{\text{n}} \times_{\pi_2^{\text{n}},\rho}\mathcal{E}\). The variety \(X\) has a low-genus fibration with base \(S^o\), namely the composition \(X \to \mathcal{C}^{\text{n}} \xrightarrow{\pi_1^{\text{n}}} S^o\). We denote by \(p_2\) the second projection \(X \to \mathcal{E}\). By Proposition 17, \(\pi^{\text{n}}_2\) has geometrically integral generic fiber, and thus the surjective base change \(p_2\) of \(\pi_2^{\text{n}}\) has this also.

Let \(F\) denote the union of divisors on \(\mathcal{E}\) which are vertical with respect to the elliptic fibration \(\pi\colon\mathcal{E}\rightarrow\mathbb{P}^1\) and do not intersect \(\rho^{-1}(C_P) \subset \mathcal{E}\) for a general \(P\) transversally. As the fibers of \(\pi\) are all geometrically integral (they are Weierstrass curves), the irreducible divisors in \(F\) are fibers of \(\pi\). By Proposition 13, the only such fiber that possibly intersects the curve \(C_P\) non-transversally for a general \(P\) is the fiber \(\pi^{-1}([1:0])\) at infinity. So \(\mathcal{E} \setminus F\) contains \(\pi^{-1}(\mathbb{A}^1)\cong S\setminus Z\) for \(Z\) the anticanonical divisor of \(S\) given by \(w = 0\). This is simply connected, as follows for instance by applying a result of Coccia [26] (see also [25] for the case \(a_6(0) \neq 0\), or equivalently, smooth \(Z\)).¹

The existence of a \(P\) as in the statement gives Zariski–density of \(S(K)\) by Nijgh’s work [9], and the Hilbert property for \(S\) follows from Theorem 1. ◻

Proof of Theorem 3. Let \(U \subset \mathbb{A}_K^9\) be the nonempty Zariski-open whose \(\bar K\)-points are \(9\)-tuples \((a,b,\ldots,f_3)\) corresponding to a smooth \(S\), and let \(X \subset U \times \mathbb{P}(2,3,1,1)\) be the total space associated to the parametric family of Definition 9. Then the projection \(\pi=pr_1:X \to U\) is smooth surjective and its fibers are del Pezzo surfaces of degree \(1\). Note that \(X\) is birational to the affine hypersurface \(Y\) in \(\mathbb{A}_K^{11}\) defined by the equation \[y^2 = x^3 + (af(z)+b) x + (cf(z)^2+df(z)+e).\] The hypersurface \(Y\) is rational via projection to any coordinate appearing linearly.

For \(N \in \mathbb{Z}_{\geq 1}\), let \(V_N \subset X\) be the non-empty Zariski-open corresponding to pairs \((\underline{a},P), \underline{a}=(a,\ldots,e,f_0,\ldots,f_3), P=[x_0:y_0:z_0:w_0]\) such that \(w_0 \neq 0, 3z_0+2f_2w_0/f_3 \neq 0,\) the polynomial \(f(t)-f(z_0/w_0)\) is separable, and \([N](\rho^{-1}(P)) \neq O\) on \(\mathcal{E}_P\). Choose now \(N\) to be divisible by the torsion exponents of all elliptic curves over \(K\). This is a finite number by (the generalization of) Merel’s theorem.

For any \(x \in V_N(K)\), the del Pezzo surface \(\pi^{-1}(\pi(x))\) has the Hilbert property by Theorem 2. The morphism \(\pi|_{V_N}:V_N \to U\) has geometrically integral generic fiber and thus sends non-thin sets to non-thin sets. Then, since \(V_N(K)\) is non-thin by Hilbert’s Irreducibility Theorem (recall that \(X\), and thus \(V_N\), is rational), \(\pi(V_N(K))\) is non-thin.

Finally, for \(\eta\) the generic point of \(U\), we have the following properties for \(X \to U\):

1. for every \(u \in U\), we have \(\operatorname{rk} \mathop{\mathrm{Pic}}X_u \geq \operatorname{rk} \mathop{\mathrm{Pic}}X_{\eta}\);

2. the set of \(u \in U\) such that \(\operatorname{rk} \mathop{\mathrm{Pic}}X_u > \operatorname{rk} \mathop{\mathrm{Pic}}X_{\eta}\) is thin.

The first property follows from injectivity of specialization on the Picard group (see e.g.[11]). More precisely, in (2) we mean that there exists a finite collection of irreducible covers \(\pi_i:Y_i \to U, i=1,\ldots,r\) of finite degrees \(>1\) such that \(\operatorname{rk} \mathop{\mathrm{Pic}}X_u > \operatorname{rk} \mathop{\mathrm{Pic}}X_{\eta}\) if and only if \(u \in \operatorname{im} (Y_i(\kappa(u))\to U(\kappa(u)))\). This holds by applying Hilbert’s Irreducibility Theorem to the family of degree-240 polynomials whose roots parametrize the lines in each surface in our family. The Galois orbits of lines correspond to the irreducible factors of these polynomials and determine the Picard rank. The Picard rank can only increase if an irreducible factor becomes reducible upon specialization, which occurs over a thin locus. By a result of Kloosterman [28], reproving a result of Desjardins and Naskrecki [29]), the del Pezzo surface \[y^2=x^3+Az^6+Bw^6,\] where we take \(A\) and \(B\) to be transcendental variables over \(K\), has Picard rank \(1\). This surface is a member of the family \(X \to U\): it is the specialization at the generic point \(\xi\) of the copy of \(\mathbb{A}^2_K\) in \(\mathbb{A}^9_K\) defined by \(f_3=A, e=B, c=1\), and where all the other coordinates are trivial. Then by (1) we get \(1 \leq \operatorname{rk} \mathop{\mathrm{Pic}}X_{\eta} \leq \operatorname{rk} \mathop{\mathrm{Pic}}X_\xi =1\), thus \(\operatorname{rk} \mathop{\mathrm{Pic}}X_{\eta}=1\). Hence by (2) the set \(T \subset U(K)\) corresponding to del Pezzo surfaces with Picard rank \(\geq 2\) is thin. Then \(\pi(V_N(K)) \setminus T\) furnishes the sought non-thin set of parameters. ◻