Proof. By [Reference Bosch, Lütkebohmert and RaynaudBLR90, §9.2, Example 8], $\operatorname{Pic}_{{\mathcal{X}}_{0}/k}^{0}$ is an Abelian variety, which implies that $\operatorname{Jac}(X)$ has good reduction, and thus the $G_{K}$ -representation on $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified.◻

Proof. We give examples for $g\not \equiv 1~\text{mod}~p$ . Let $X$ be a hyperelliptic curve of genus $g$ over $K$ that is one of the examples of [Reference LiuLiu02, Example 10.1.30] with the extra assumptions of [Reference LiuLiu02, Example 10.3.46] (here, we need the assumption $g\not \equiv 1~\text{mod}~p$ ). Then $X$ has stable reduction over $K$ , as well as over every finite extension field $L/K$ . In this example, the special fiber of the stable model is the union of a curve of genus $1$ and a curve of genus $(g-1)$ meeting transversally in one point. In particular, neither $X$ nor any base-change $X_{L}$ have good reduction, but since the assumptions of Lemma 2.6 are fulfilled, the $G_{K}$ -representation on $H_{\acute{\text{e}}\text{t}}^{m}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for all $m$ and all $\ell \neq p$ .◻

Assumption $(\star )$ .

A K3 surface $X$ over $K$ satisfies Assumption ( $\star$ ) if there exists a finite field extension $L/K$ such that $X_{L}$ admits a semi-stable model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ (in the sense of Definition 2.1) such that $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{O}}_{L}}$ is trivial.

Proof. Let ${\mathcal{Y}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L^{\prime }}$ be as in the statement. Since $p\neq 2,3$ , Kawamata’s semi-stable MMP [Reference KawamataKaw94] (see also [Reference Kollár and MoriKM98, §7.1] for $p=0$ ) produces a scheme ${\mathcal{Z}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L^{\prime }}$ with nef relative canonical divisor $K_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ that is a model of a smooth proper surface birationally equivalent to $X_{L^{\prime }}$ , and such that ${\mathcal{Z}}$ is regular outside a finite set $\unicode[STIX]{x1D6F4}$ of terminal singularities. We refer to [Reference KawamataKaw94, § 1] for details and the definition of $K_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ , which is a Weil divisor. We also note that it coincides with the Weil divisor class associated to the relative canonical divisor $\unicode[STIX]{x1D714}_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ , see, for example, [Reference MatsumotoMat15, §3].

Since $X_{L^{\prime }}$ is a minimal surface and $K_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ is nef, the generic fiber of ${\mathcal{Z}}$ is actually isomorphic to $X_{L^{\prime }}$ , and it follows that $K_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ is trivial. Outside $\unicode[STIX]{x1D6F4}$ , this model is already a semi-stable model. From the classification of terminal singularities in [Reference KawamataKaw94, Theorem 4.4] and the fact that $K_{{\mathcal{Z}}/{\mathcal{O}}_{L^{\prime }}}$ is Cartier at points of $\unicode[STIX]{x1D6F4}$ (since it is trivial), it follows that the geometric special fiber $({\mathcal{Z}}_{0})_{\overline{k}}$ is irreducible around points of $\unicode[STIX]{x1D6F4}$ , and that it acquires RDP singularities in these points. Thus, after some finite field extension $L/L^{\prime }$ , there exists a simultaneous resolution ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ of these singularities by [Reference ArtinArt74, Theorem 2]. This ${\mathcal{X}}$ may exist only as an algebraic space, and it satisfies Assumption $(\star )$ .◻

Proof. We only have to show something for $m=2$ . But then the first Chern class induces a $G_{K}$ -equivariant isomorphism

$$\begin{eqnarray}\operatorname{NS}(Y_{\overline{K}})\otimes _{\mathbb{Z}}\mathbb{Q}_{\ell }\stackrel{c_{1}}{\longrightarrow }H_{\acute{\text{e}}\text{t}}^{2}(Y_{\overline{K}},\mathbb{Q}_{\ell })(1).\end{eqnarray}$$

After passing to a finite extension $L/K$ , we may assume that $\operatorname{NS}(Y_{L})=\operatorname{NS}(Y_{\overline{K}})$ . But then the $G_{L}$ -representation on $\operatorname{NS}(Y_{L})$ is trivial; hence it is also trivial on $H_{\acute{\text{e}}\text{t}}^{2}$ , and, in particular, unramified.◻

Proof. A straightforward computation shows that $X$ is smooth over $\mathbb{Q}_{p}$ , and that $\imath$ acts without fixed points on $X$ . Thus $X$ is a K3 surface and $Y$ is an Enriques surface over $\mathbb{Q}_{p}$ . The special fiber of ${\mathcal{X}}$ is a non-smooth K3 surface with four RDP singularities of type $A_{1}$ located at $[0:0:0:\pm 1:\pm 1:1]$ . Then the blow-up ${\mathcal{X}}_{1}^{\prime }\rightarrow {\mathcal{X}}$ of the Weil divisor $\{x_{0}-ex_{1}=x_{2}-py_{2}=0\}$ defines a simultaneous resolution of the singularities of ${\mathcal{X}}\rightarrow \operatorname{Spec}\mathbb{Z}_{p}$ , and we obtain a smooth model of $X$ over $\mathbb{Z}_{p}$ . In particular, $X$ has good reduction over $\mathbb{Q}_{p}$ and the $G_{\mathbb{Q}_{p}}$ -representation on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell })$ is unramified for all $\ell \neq p$ .

Next, let ${\mathcal{X}}_{2}^{\prime }\rightarrow {\mathcal{X}}$ be the blow-up of the four singular points of the special fiber. Then $\imath$ extends to ${\mathcal{X}}_{2}^{\prime }$ , and the special fiber is the union of four divisors $E_{i}$ with the minimal desingularization $X_{p}^{\prime }$ of the special fiber of ${\mathcal{X}}$ . The fixed locus of $\imath$ on $X_{p}^{\prime }$ is the union of the four $(-2)$ -curves of the resolution. Moreover, there exist isomorphisms $E_{i}\cong \mathbb{P}^{1}\times \mathbb{P}^{1}$ such that $\imath$ acts by interchanging the two factors. Thus the quotient ${\mathcal{X}}_{2}^{\prime }/\imath$ is a model of $Y$ over $\mathbb{Z}_{p}$ , whose special fiber is a rational surface $X_{p}^{\prime }/\imath$ (a so-called Coble surface) meeting transversally four $\mathbb{P}^{2}$ ’s, that is, a semi-stable degeneration of flower pot type (see, [Reference PerssonPer77, §3.3]).

Seeking a contradiction, we assume that $Y$ has potential good reduction. Then there exists a finite extension $L/\mathbb{Q}_{p}$ and a smooth model ${\mathcal{Y}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ of $Y_{L}$ . Let ${\mathcal{X}}_{3}\rightarrow {\mathcal{Y}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ be its K3 double cover, which is a family of smooth K3 surfaces with generic fiber $X_{L}$ , whose fixed point free involution specializes to a fixed point free involution in the special fiber of ${\mathcal{X}}_{3}$ .

Now, ${\mathcal{X}}_{3}$ and the base-change of ${\mathcal{X}}_{1}^{\prime }$ to ${\mathcal{O}}_{L}$ both are smooth models of $X_{L}$ . The isomorphism of generic fibers extends to a birational map of special fibers. The involution on generic fibers extends to rational involutions of the two special fibers, compatible with the just established birational map. Since both special fibers are K3 surfaces, the birational maps and rational involutions extend to isomorphisms and involutions. However, in one special fiber the involution acts without fixed points, whereas it has four fixed curves in the other, which contradicts the assumption. ◻

Proof. Since ${\mathcal{L}}$ is ample, ${\mathcal{L}}^{\otimes n}$ is effective for $n\gg 0$ , and thus also its specialization ${\mathcal{L}}_{0}^{\otimes n}$ to the special fiber ${\mathcal{X}}_{0}$ is effective. In particular, ${\mathcal{L}}_{0}$ has positive intersection with every ample divisor on ${\mathcal{X}}_{0}$ , that is, ${\mathcal{L}}_{0}$ is pseudo-effective. Thus there exists a Zariski–Fujita decomposition on $({\mathcal{X}}_{0})_{\overline{k}}$

$$\begin{eqnarray}({\mathcal{L}}_{0})_{\overline{k}}=P+N,\end{eqnarray}$$

where $P$ is nef, and where $N$ is a sum of effective divisors, whose intersection matrix is negative definite; see, for example, [Reference BădescuBăd01, Theorem 14.14]. Since $\unicode[STIX]{x1D714}_{{\mathcal{X}}_{0}/k}$ is numerically trivial, the adjunction formula shows that every reduced and irreducible curve in $N$ is a $\mathbb{P}^{1}$ with self-intersection $-2$ , that is, a $(-2)$ -curve. Moreover, negative definiteness and the classification of Cartan matrices implies that $N$ is a disjoint union of ADE curves. Next, $k$ is perfect and since the Zariski–Fujita decomposition is unique, it is stable under $G_{k}$ , and thus descends to ${\mathcal{X}}_{0}$ .

After these preparations, let $C$ be as in the statement, that is, ${\mathcal{L}}_{0}\cdot C<0$ . First, we want to show that there exists a morphism $f:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }$ of algebraic spaces that contracts $C$ . Being contained in the support of $N$ , the base-change $C_{\overline{k}}\subset ({\mathcal{X}}_{0})_{\overline{k}}$ is a disjoint union of ADE curves. Since $C^{2}<0$ , Artin showed that there exists a morphism of projective surfaces over $k$

$$\begin{eqnarray}f_{0}:{\mathcal{X}}_{0}\rightarrow {\mathcal{X}}_{0}^{\prime }\end{eqnarray}$$

that contracts $C$ and nothing else (see [Reference BădescuBăd01, Theorem 3.9], for example). Since $C_{\overline{k}}$ is a union of ADE-curves, it follows that $({\mathcal{X}}_{0}^{\prime })_{\overline{k}}$ has RDP singularities, which are rational and Gorenstein. Thus also ${\mathcal{X}}_{0}^{\prime }$ has rational Gorenstein singularities.

For all $n\geqslant 0$ , we define

$$\begin{eqnarray}{\mathcal{X}}_{n}:={\mathcal{X}}\times _{\operatorname{Spec}{\mathcal{O}}_{K}}\operatorname{Spec}({\mathcal{O}}_{K}/\mathfrak{m}^{n+1}).\end{eqnarray}$$

Since $f_{0}$ is a contraction with $R^{1}f_{0\ast }{\mathcal{O}}_{{\mathcal{X}}_{0}}=0$ , there exists a blow-down $f_{n}:{\mathcal{X}}_{n}\rightarrow {\mathcal{X}}_{n}^{\prime }$ that extends $f_{0}$ , see [Reference Cynk and van StratenCvS09, Theorem 3.1]. Passing to limits, we obtain a contraction of formal schemes

$$\begin{eqnarray}\widehat{f}:\widehat{{\mathcal{X}}}\rightarrow \widehat{{\mathcal{X}}}^{\prime }.\end{eqnarray}$$

By [Reference ArtinArt70, Theorem 3.1], there exists a contraction of algebraic spaces

$$\begin{eqnarray}f:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime },\end{eqnarray}$$

whose completion along their special fibers coincides with $\widehat{f}$ . In particular, $f$ is an isomorphism outside $C$ and contracts $C$ to a singular point $w\in {\mathcal{X}}^{\prime }$ .

Let $\widehat{w}$ be the formal completion of ${\mathcal{X}}^{\prime }$ along $w$ , and let

$$\begin{eqnarray}\widehat{{\mathcal{Z}}}\rightarrow \widehat{w}\end{eqnarray}$$

be the formal fiber over $\widehat{f}$ . Then $\widehat{w}$ is a formal affine scheme, say $\operatorname{Spf}R$ , and let $k^{\prime }$ be the residue field, which is a finite extension of $k$ . Let ${\mathcal{O}}_{K^{\prime }}$ be the unramified extension of ${\mathcal{O}}_{K}$ corresponding to the field extension $k\subseteq k^{\prime }$ . Since $k\subseteq k^{\prime }$ is separable, $k^{\prime }$ arises by adjoining a root $\unicode[STIX]{x1D6FC}$ of some monic polynomial $f$ with values $k$ . After lifting $f$ to a polynomial with values in ${\mathcal{O}}_{K}$ , and using that $R$ is Henselian, we can lift $\unicode[STIX]{x1D6FC}$ to $R$ , which shows that ${\mathcal{O}}_{K^{\prime }}$ is contained in $R$ . In particular, we can view $R$ as a local ${\mathcal{O}}_{K^{\prime }}$ -algebra without residue field extension – we denote by $\tilde{R}$ the ring $R$ considered as ${\mathcal{O}}_{K^{\prime }}$ -algebra.

Then the special fiber of $\operatorname{Spf}\tilde{R}$ is a rational singularity of multiplicity $2$ , and thus, by [Reference LipmanLip69, Lemma 23.4], the completion of the local ring of the special fiber is of the form

(1) $$\begin{eqnarray}k^{\prime }[[x,y,z]]/(h^{\prime }(x,y,z)).\end{eqnarray}$$

Using Hensel’s lemma, we may assume after a change of coordinates that the power series $h^{\prime }(x,y,z)$ is of the form $z^{2}-h_{1}(x,y)z-h_{0}(x,y)$ for some polynomials $h_{0}(x,y),h_{1}(x,y)$ . Using Hensel’s lemma again, the completion of $\tilde{R}$ is of the form

(2) $$\begin{eqnarray}\widehat{{\mathcal{O}}}_{K^{\prime }}[[x,y,z]]/(z^{2}-H_{1}(x,y)z-H_{0}(x,y))\end{eqnarray}$$

where $H_{i}(x,y)$ is congruent to $h_{i}(x,y)$ modulo the maximal ideal of $\widehat{{\mathcal{O}}}_{K^{\prime }}$ for $i=1,2$ ; see also [Reference KawamataKaw94, Theorem 4.4]. (If $p\neq 2$ , we may even assume $h_{1}=0$ and $H_{1}=0$ .) We denote by $t^{\prime }:\operatorname{Spf}\tilde{R}\rightarrow \operatorname{Spf}\tilde{R}$ the involution induced by $z\mapsto H_{1}(x,y)-z$ . It is not difficult to see that $t^{\prime }$ induces $-\!\operatorname{id}$ on local Picard groups, see, for example, [Reference KollárKol89, Example 2.3]. Since $R$ is equal to $\tilde{R}$ considered as rings, we have established an involution $t:\widehat{w}\rightarrow \widehat{w}$ that induces $-\text{id}$ on local Picard groups. We denote by

$$\begin{eqnarray}\widehat{{\mathcal{Z}}}^{+}\rightarrow \widehat{w}\end{eqnarray}$$

the composition $t\circ \widehat{f}$ . By [Reference KollárKol89, Proposition 2.2], this gives the desired flop formally.

By [Reference ArtinArt70, Theorem 3.2], there exists a dilatation $f^{+}:{\mathcal{X}}^{+}\rightarrow {\mathcal{X}}^{\prime }$ of algebraic spaces, such that the formal completion of ${\mathcal{X}}^{+}$ along the exceptional locus of $f^{+}$ is given by the just-constructed $\widehat{{\mathcal{Z}}}^{+}\rightarrow \widehat{w}$ . Thus there exists a birational and rational map

$$\begin{eqnarray}\unicode[STIX]{x1D711}:{\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{+},\end{eqnarray}$$

which is an isomorphism outside $C$ . From the glueing construction it is clear that ${\mathcal{X}}^{+}$ is a smooth model of $X$ over ${\mathcal{O}}_{K}$ . Finally, from the formal picture above, it is clear that the restriction of ${\mathcal{L}}^{+}$ to ${\mathcal{X}}_{0}^{+}$ has positive intersection with the flopped curve $C^{+}$ .◻

Proof. First, we consider the case, where $C$ is geometrically integral. Let ${\mathcal{Z}}\subset {\mathcal{X}}\times _{{\mathcal{O}}_{K}}{\mathcal{X}}^{+}$ be the closure of the diagonal $\unicode[STIX]{x1D6E5}(X)\subset X\times _{K}X$ . Then it is not difficult to see that the isomorphism $\unicode[STIX]{x1D6FC}$ is given by $x\mapsto \operatorname{pr}_{1,\ast }([{\mathcal{Z}}_{0}]\cdot \operatorname{pr}_{2}^{\ast }(x))$ (see also Lemma 5.6 below). We set ${\mathcal{U}}:={\mathcal{X}}\setminus C$ and ${\mathcal{U}}^{+}:={\mathcal{X}}^{+}\setminus C^{+}$ . Then we have a commutative diagram with exact rows

where $\unicode[STIX]{x1D6FC}^{\prime \prime }$ is also defined by $x\mapsto \operatorname{pr}_{1,\ast }([{\mathcal{Z}}_{0}]\cdot \operatorname{pr}_{2}^{\ast }(x))$ and where $\unicode[STIX]{x1D6FC}^{\prime }$ is the map induced by $\unicode[STIX]{x1D6FC}$ . By purity, the left terms are one-dimensional and generated by the classes $[C]$ and $[C^{+}]$ , respectively. Moreover, the right terms are canonically isomorphic to the orthogonal complements of the left terms. Since ${\mathcal{Z}}|_{{\mathcal{U}}\times {\mathcal{U}}^{+}}$ is the graph of the isomorphism $\unicode[STIX]{x1D711}|_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow {\mathcal{U}}^{+}$ , it follows that $\unicode[STIX]{x1D6FC}^{\prime \prime }$ coincides with the pull-back by the isomorphism $\unicode[STIX]{x1D711}_{0}|_{{\mathcal{U}}_{0}}=\unicode[STIX]{x1D711}|_{{\mathcal{U}}_{0}}:{\mathcal{U}}_{0}\rightarrow {\mathcal{U}}_{0}^{+}$ . Since $\unicode[STIX]{x1D6FC}$ is an isometry, so is $\unicode[STIX]{x1D6FC}^{\prime }$ , and it maps $[C^{+}]$ either to $[C]$ or to $-[C]$ . From $\unicode[STIX]{x1D6FC}({\mathcal{L}}_{0}^{+})\cdot \unicode[STIX]{x1D6FC}(C^{+})={\mathcal{L}}_{0}^{+}\cdot C^{+}>0$ and $\unicode[STIX]{x1D6FC}({\mathcal{L}}_{0}^{+})\cdot C={\mathcal{L}}_{0}\cdot C<0$ , we conclude $\unicode[STIX]{x1D6FC}([C^{+}])=-[C]$ . Putting these observations together, we find $\unicode[STIX]{x1D6FC}\circ (\unicode[STIX]{x1D711}_{0}^{\ast })^{-1}=r_{C}$ .

Now, we consider the general case. Since the absolute Galois group $G_{k}$ acts transitively on the $m$ connected components, they are mutually isomorphic. Since the flops in the disjoint $C_{i}$ commute, we may assume $m=1$ . As shown in the proof of Proposition 4.2, $C_{1}=C_{\overline{k}}$ is an ADE configuration of $(-2)$ -curves. Since $G_{k}$ acts on the irreducible components of $C_{i}$ transitively, it is not difficult to see from the classification of Dynkin diagrams that only configurations of type $A_{1}$ and $A_{2}$ can occur. We already treated the $A_{1}$ case above and thus we may assume an $A_{2}$ -configuration, that is, $C_{\overline{k}}=C_{1}=C_{1,1}\cup C_{1,2}$ . Passing to a finite unramified extension $K^{\prime }/K$ corresponding to an extension $k^{\prime }/k$ over which $C$ splits, we consider the following diagram of flops and models over ${\mathcal{O}}_{K^{\prime }}$ :

where $\unicode[STIX]{x1D711}_{j}$ denotes the flop at $C_{1,j}$ or at the corresponding curve on other models (note that our flops induce isomorphisms between the special fibers). A straightforward computation shows that $r_{C_{1,1}}r_{C_{1,2}}r_{C_{1,1}}$ and $r_{C_{1,2}}r_{C_{1,1}}r_{C_{1,2}}$ both act as $-\!\operatorname{id}$ on the one-dimensional subspace spanned by $[C_{1,1}]+[C_{1,2}]$ inside $H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0})_{\overline{k}},\mathbb{Q}_{\ell })(1)$ and as $\operatorname{id}$ on its orthogonal complement. Thus $\unicode[STIX]{x1D711}_{1}\unicode[STIX]{x1D711}_{2}\unicode[STIX]{x1D711}_{1}:{\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{121}$ and $\unicode[STIX]{x1D711}_{2}\unicode[STIX]{x1D711}_{1}\unicode[STIX]{x1D711}_{2}:{\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{212}$ both satisfy the conditions of the flop of the contraction $f$ . Hence, they coincide by the uniqueness of flops, and we set $\unicode[STIX]{x1D711}:=\unicode[STIX]{x1D711}_{1}\unicode[STIX]{x1D711}_{2}\unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D711}_{2}\unicode[STIX]{x1D711}_{1}\unicode[STIX]{x1D711}_{2}:{\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{+}$ . Clearly, $\unicode[STIX]{x1D711}$ descends to ${\mathcal{O}}_{K}$ and coincides with the flop in $C$ established in Proposition 4.2.◻

Proof. Using the Zariski–Fujita decomposition of $x$ (see the proof of Proposition 4.2) and Lemma 4.3, we see that if $x$ is not nef, then a generalized $(-2)$ -curve $C$ with $C\cdot x<0$ indeed exists. Since Abelian and bielliptic surfaces do not admit smooth rational curves, we may assume that $X$ is a K3 surface or an Enriques surface. Since $r_{C}(x)^{2}=x^{2}\geqslant 0$ , it follows from the Riemann–Roch theorem that either $r_{C}(x)$ or $-r_{C}(x)$ is effective. Let $C_{1},\ldots ,C_{m}$ be the connected components of $C_{\overline{k}}$ . Then we find $x\cdot r_{C}(x)=x^{2}+\sum _{i}(x\cdot C_{i})^{2}>0$ , from which it follows that $x$ and $r_{C}(x)$ belong to the same component of the cone $\{y\in \operatorname{NS}(X)_{\mathbb{R}}:y^{2}>0\}$ , and thus $r_{C}(x)$ is effective. This establishes assertion (i).

To show assertion (ii), we fix an ample class $H$ of $X$ and let $C_{1},\ldots ,C_{m}$ be the connected components of $C_{\overline{k}}$ . Since $G_{k}$ acts transitively on these components, we find $x\cdot C_{i}=(1/m)x\cdot C$ and $H\cdot C_{i}=(1/m)H\cdot C$ , from which we conclude

$$\begin{eqnarray}r_{C}(x)\cdot H=\biggl(x+\mathop{\sum }_{i=1}^{m}(x\cdot C_{i})C_{i}\biggr)\cdot H=x\cdot H+\frac{1}{m}(x\cdot C)(H\cdot C)<x\cdot H,\end{eqnarray}$$

since $x\cdot C<0$ by assumption and $H\cdot C>0$ by ampleness of $H$ . Therefore, if $\{x_{i}\}\in \operatorname{NS}(X)$ is as in assertion (ii), then $\{x_{i}\cdot H\}$ is a strictly decreasing sequence of positive integers. In particular, it must be of finite length, and its last class must be nef.◻

Proof. Let $\cdots {\dashrightarrow}({\mathcal{X}}^{+i},{\mathcal{L}}^{+i}){\dashrightarrow}\cdots \,$ be a sequence of flops in generalized $(-2)$ -curves $C^{i}\subset ({\mathcal{X}}^{+i})_{0}$ as asserted by Proposition 4.2 and Lemma 4.3. Moreover, we have $[{\mathcal{L}}_{0}^{+(i+1)}]=[r_{C^{i}}({\mathcal{L}}_{0}^{+i})]$ by Lemma 4.3. By Lemma 4.4, this sequence is finite and ${\mathcal{L}}_{0}^{+N}$ is nef.◻

Proof. Note that also $\unicode[STIX]{x1D714}_{{\mathcal{X}}_{0}/k}$ is numerically trivial. Since ${\mathcal{L}}_{0}$ is big and nef, we obtain a proper and birational morphism

$$\begin{eqnarray}\unicode[STIX]{x1D71B}:{\mathcal{X}}_{0}\rightarrow W:=\operatorname{Proj}\bigoplus _{n\geqslant 0}H^{0}({\mathcal{X}}_{0},{\mathcal{L}}_{0}^{\otimes n}).\end{eqnarray}$$

Base-changing to $({\mathcal{X}}_{0})_{\overline{k}}$ , the induced morphism $\unicode[STIX]{x1D71B}_{\overline{k}}$ contracts an integral curve $C$ if and only if it has zero-intersection with ${\mathcal{L}}_{0}$ . Since the intersection matrix formed by contracted curves is negative definite, and since an integral curve with negative self-intersection on a surface with numerically trivial canonical sheaf over an algebraically closed field is a $(-2)$ -curve, it follows from the classification of Cartan matrices, that $W_{\overline{k}}$ has at worst RDP singularities.

Now, ${\mathcal{L}}_{0}^{\otimes n}$ is of degree $0$ on contracted curves for all $n$ , and over $\overline{k}$ , these curves are ADE curves. Thus we find $R^{1}\unicode[STIX]{x1D71B}_{\ast }{\mathcal{L}}_{0}^{\otimes n}=0$ for all $n\geqslant 0$ , which implies $H^{1}({\mathcal{X}}_{0},{\mathcal{L}}_{0}^{\otimes n})=H^{1}(W,{\mathcal{O}}_{W}(n))$ for all $n\geqslant 0$ , and note that the latter term is zero for $n\gg 0$ by Serre vanishing. Replacing ${\mathcal{L}}$ by some sufficiently high tensor power will not change $\unicode[STIX]{x1D71B}$ , and then we may assume that $H^{1}({\mathcal{X}}_{0},{\mathcal{L}}_{0}^{\otimes n})=0$ for all $n\geqslant 1$ . If $f:{\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ denotes the structure morphism, then semi-continuity and the previous vanishing result imply $R^{1}f_{\ast }{\mathcal{L}}^{\otimes n}=0$ for all $n\geqslant 1$ . Thus global sections of ${\mathcal{L}}_{0}^{\otimes n}$ extend to ${\mathcal{L}}^{\otimes n}$ , and since the former is globally generated for $n\gg 0$ so is the latter. Thus we obtain a morphism of algebraic spaces over $\operatorname{Spec}{\mathcal{O}}_{K}$

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }:=\operatorname{Proj}\bigoplus _{n\geqslant 0}H^{0}({\mathcal{X}},{\mathcal{L}}^{\otimes n}).\end{eqnarray}$$

Since ${\mathcal{L}}$ is ample on $X$ , $\unicode[STIX]{x1D70B}$ induces an isomorphism of generic fibers. Moreover, we can identify the induced map $\unicode[STIX]{x1D70B}_{0}$ on special fibers with $\unicode[STIX]{x1D71B}:{\mathcal{X}}_{0}\rightarrow W$ from above.◻

Proof. The special fibers of ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are birational by the Matsusaka–Mumford theorem [Reference Matsusaka and MumfordMM64, Theorem 2], and since they are minimal surfaces of Kodaira dimension at least $0$ , they are isomorphic. (Note that this statement also follows from the much more detailed analysis below.)

Now, choose an ample invertible sheaf ${\mathcal{L}}$ on $X$ . By Proposition 4.5, there exist finite sequences of flops ${\mathcal{X}}_{i}{\dashrightarrow}\cdots {\dashrightarrow}{\mathcal{Y}}_{i}$ , $i=1,2$ , such that ${\mathcal{L}}$ restricts to big and nef invertible sheaves on the special fibers of ${\mathcal{Y}}_{i}$ .

Applying Proposition 4.6 to our models ${\mathcal{Y}}_{i}$ , we obtain flopping contractions

$$\begin{eqnarray}{\mathcal{Y}}_{i}\rightarrow {\mathcal{Y}}_{i}^{\prime }:=\operatorname{Proj}\bigoplus _{n\geqslant 0}H^{0}({\mathcal{Y}}_{i},{\mathcal{L}}^{\otimes n}).\end{eqnarray}$$

Now, ${\mathcal{L}}$ is ample on ${\mathcal{Y}}_{i}^{\prime }$ . Moreover, the ${\mathcal{Y}}_{i}^{\prime }$ are normal projective schemes and birational outside a finite number of curves in their special fibers. In fact, there exists a birational and rational map between them that is compatible with ${\mathcal{L}}$ . Thus, by [Reference KovacsKov09, Theorem 5.14], this birational map extends to an isomorphism, and then we obtain a birational map ${\mathcal{Y}}_{1}{\dashrightarrow}{\mathcal{Y}}_{2}$ with decomposition ${\mathcal{Y}}_{1}\rightarrow {\mathcal{Y}}_{1}^{\prime }\cong {\mathcal{Y}}_{2}^{\prime }\leftarrow {\mathcal{Y}}_{2}$ of the required form.

Putting all these birational modifications together, we have connected ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ by a sequence of birational maps of the required form.◻

Proof. Since ${\mathcal{X}}$ is regular, the pull-back of ${\mathcal{L}}$ to $X_{L}$ extends to an invertible sheaf on ${\mathcal{X}}$ . By abuse of notation, we shall denote the pull-back to $X_{L}$ and its extension to ${\mathcal{X}}$ again by ${\mathcal{L}}$ . By assumption, the restriction ${\mathcal{L}}_{0}$ of ${\mathcal{L}}$ to the special fiber ${\mathcal{X}}_{0}$ is big and nef. Note that $\unicode[STIX]{x1D714}_{{\mathcal{X}}_{0}/k}$ is numerically trivial. Let

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }:=\operatorname{Proj}\bigoplus _{n\geqslant 0}H^{0}({\mathcal{X}},{\mathcal{L}}^{\otimes n})\end{eqnarray}$$

be the morphism of algebraic spaces over ${\mathcal{O}}_{L}$ given by Proposition 4.6. Then $\unicode[STIX]{x1D70B}$ has all the properties asserted in claim (iii) of the proposition. Clearly, ${\mathcal{X}}^{\prime }$ is a projective scheme over $\operatorname{Spec}{\mathcal{O}}_{L}$ , and if $\unicode[STIX]{x1D70B}$ is not an isomorphism, then the exceptional locus is non-empty and of codimension $2$ , which implies that ${\mathcal{X}}^{\prime }$ cannot be regular by van der Waerden purity; see, for example, [Reference LiuLiu02, Theorem 7.2.22].

It remains to establish the $G$ -action on ${\mathcal{X}}^{\prime }$ . Since ${\mathcal{L}}$ is a $G$ -invariant invertible sheaf on $X_{L}$ , we have an induced $G$ -action on $H^{0}(X_{L},{\mathcal{L}}^{\otimes n})$ for all $n\geqslant 0$ . We will show that this extends to an action on $H^{0}({\mathcal{X}},{\mathcal{L}}^{\otimes n})$ . First, we show that there exists a closed subspace ${\mathcal{Z}}\subset {\mathcal{X}}$ of codimension at least $2$ that is contained in ${\mathcal{X}}_{0}$ , such that the $G$ -action on $X_{L}$ extends to an action on ${\mathcal{U}}:={\mathcal{X}}\setminus {\mathcal{Z}}$ . Since every birational rational map between two normal algebraic spaces is an isomorphism outside a closed subspace of codimension at least $2$ , there exists for every $g\in G$ a closed subspace ${\mathcal{Z}}_{g}\subset {\mathcal{X}}$ of codimension at least $2$ that is contained in ${\mathcal{X}}_{0}$ and such that $g:X_{L}\rightarrow X_{L}$ extends to $g:{\mathcal{X}}\setminus {\mathcal{Z}}_{g}\rightarrow {\mathcal{X}}$ . Since ${\mathcal{X}}_{0}$ is a minimal surface, the restriction $g|_{{\mathcal{X}}_{0}\setminus {\mathcal{Z}}_{g}}:{\mathcal{X}}_{0}\setminus {\mathcal{Z}}_{g}\rightarrow {\mathcal{X}}_{0}$ to the special fiber extends to an automorphism $g:{\mathcal{X}}_{0}\rightarrow {\mathcal{X}}_{0}$ . This defines a $G$ -action on ${\mathcal{X}}_{0}$ . Let ${\mathcal{Z}}^{\prime }:=\bigcup _{g\in G}{\mathcal{Z}}_{g}$ and ${\mathcal{Z}}:=\bigcup _{g\in G}g({\mathcal{Z}}^{\prime })$ , where $g({\mathcal{Z}}^{\prime })$ is the image by the action just defined. Since $G$ is a finite group, ${\mathcal{Z}}$ is closed. This ${\mathcal{Z}}$ satisfies the above condition, for otherwise there exists a $g\in G$ , such that the image of $g:{\mathcal{X}}\setminus {\mathcal{Z}}\rightarrow {\mathcal{X}}$ is not contained in ${\mathcal{X}}\setminus {\mathcal{Z}}$ , or, equivalently $g^{-1}({\mathcal{Z}})\not \subset {\mathcal{Z}}$ . However, since ${\mathcal{Z}}$ is $G$ -stable, this cannot happen. Thus we obtain a $G$ -action on ${\mathcal{U}}={\mathcal{X}}\setminus {\mathcal{Z}}$ .

If $s$ is a global section of ${\mathcal{L}}^{\otimes n}$ over ${\mathcal{X}}$ and $\unicode[STIX]{x1D70E}\in G$ , then $\unicode[STIX]{x1D70E}(s|_{{\mathcal{U}}})$ is a well-defined global section of ${\mathcal{L}}^{\otimes n}$ over ${\mathcal{U}}={\mathcal{X}}\setminus {\mathcal{Z}}$ . Since ${\mathcal{L}}^{\otimes n}$ is a reflexive sheaf on a regular algebraic space, $\unicode[STIX]{x1D70E}(s|_{{\mathcal{U}}})$ extends uniquely to a global section of ${\mathcal{L}}^{\otimes n}$ over ${\mathcal{X}}$ . Thus we obtain a $G$ -action on $H^{0}({\mathcal{X}},{\mathcal{L}}^{\otimes n})$ , which gives rise to a $G$ -action on ${\mathcal{X}}^{\prime }$ that is compatible with the $G$ -action on ${\mathcal{O}}_{L}$ , as well as with the natural $G$ -action on $X_{L}$ .◻

Proof. This is immediate if we define a cycle class $[{\mathcal{Z}}]$ in $H_{\acute{\text{e}}\text{t}}^{2c}({\mathcal{X}},\mathbb{Z}/n\mathbb{Z}(c))$ in such a way, that it is compatible with base change $S^{\prime }\rightarrow S$ of base schemes (and thus, in particular, compatible with restrictions to generic and special fibers). If ${\mathcal{X}}$ is a scheme, this is defined and shown in [Reference DeligneSGA4½, Cycle, Numéro 2.3], and thus it remains to treat the case where ${\mathcal{X}}$ is an algebraic space.

First, let us recall cohomological descent. Let $V\rightarrow {\mathcal{Y}}$ be an étale covering of schemes. Such a morphism is of cohomological descent by [Reference GrothendieckSGA4, Proposition V^bis.4.3.3], and we have a spectral sequence [Reference GrothendieckSGA4, Proposition V^bis.2.5.5]

$$\begin{eqnarray}E_{1}^{p,q}:=H_{\acute{\text{e}}\text{t}}^{q}(V_{p},a_{p}^{\ast }F)\Rightarrow H_{\acute{\text{e}}\text{t}}^{p+q}({\mathcal{Y}},F),\end{eqnarray}$$

where, for each $p\geqslant 0$ , $V_{p}$ is the $(p+1)$ -fold fibered product of $V$ over ${\mathcal{Y}}$ , and where $a_{p}:V_{p}\rightarrow {\mathcal{Y}}$ is the structure map. Next, we consider the more general case, where $V\rightarrow {\mathcal{Y}}$ is an étale covering of an algebraic space by a scheme. (Note that, then, the $V_{p}$ are schemes.) We observe that $V\rightarrow {\mathcal{Y}}$ is still of cohomological descent (proved in the same way) and we obtain the same spectral sequence.

After these preliminary remarks, let $U\rightarrow {\mathcal{X}}$ be an étale covering by a scheme $U$ . Applying the previous paragraph to the covering $U\times _{{\mathcal{X}}}{\mathcal{Z}}\rightarrow {\mathcal{Z}}$ and the sheaf $R^{2c}i^{!}\mathbb{Z}/n\mathbb{Z}(c)$ on ${\mathcal{Z}}$ (where $i:{\mathcal{Z}}\rightarrow {\mathcal{X}}$ is the closed immersion), and using the isomorphism $H_{{\mathcal{Z}}}^{2c}({\mathcal{X}},\mathbb{Z}/n\mathbb{Z}(c))\cong H^{0}({\mathcal{Z}},R^{2c}i^{!}\mathbb{Z}/n\mathbb{Z}(c))$ (this isomorphism is also proved by reducing to the scheme case), we obtain an isomorphism

$$\begin{eqnarray}\operatorname{Ker}(H_{Z_{0}}^{2c}(U_{0},\mathbb{Z}/n\mathbb{Z}(c))\rightarrow H_{Z_{1}}^{2c}(U_{1},\mathbb{Z}/n\mathbb{Z}(c)))\cong H_{{\mathcal{Z}}}^{2c}({\mathcal{X}},\mathbb{Z}/n\mathbb{Z}(c)),\end{eqnarray}$$

where $Z_{p}=U_{p}\times _{{\mathcal{X}}}{\mathcal{Z}}$ (note that this is a scheme). We define $[{\mathcal{Z}}]\in H_{{\mathcal{Z}}}^{2c}({\mathcal{X}},\mathbb{Z}/n\mathbb{Z}(c))$ to be $[Z_{0}]\in H_{Z_{0}}^{2c}(U_{0},\mathbb{Z}/n\mathbb{Z}(c))$ (this class lies indeed in the kernel). Since the cycle map (for schemes) is étale local, this construction does not depend on the choice of the étale covering $U\rightarrow {\mathcal{X}}$ .

Compatibility with change of base schemes reduces to the scheme case. ◻

Proof of Proposition 5.5.

(i) It follows from the assumptions that ${\mathcal{X}}_{0}$ is a smooth and proper surface with numerically trivial $\unicode[STIX]{x1D714}_{{\mathcal{X}}_{0}/k}$ , and that it has the same $\ell$ -adic Betti numbers as $X$ . Thus, by the classification of surfaces (see, for example, [Reference Bombieri and MumfordBM77, Theorem 6 and the following Proposition]), also ${\mathcal{X}}_{0}$ is Abelian and (quasi-)hyperelliptic, respectively. As seen in the proof of Proposition 5.1, the (geometric) exceptional locus of $\unicode[STIX]{x1D70B}$ is a union of $\mathbb{P}^{1}$ ’s with self-intersection number $(-2)$ . Now, there are no rational curves on Abelian varieties. Also, it follows from the explicit classification and description of (quasi-)hyperelliptic surfaces in [Reference Bombieri and MumfordBM77, Proposition 5] that they do not contain any smooth rational curves. In particular, $\unicode[STIX]{x1D70B}$ must be an isomorphism, which implies that the $G$ -action extends to ${\mathcal{X}}$ .

(ii) After replacing $K$ by an intermediate extension, we may assume $H=G$ . To show that the $G$ -action extends, it suffices to show that the $\unicode[STIX]{x1D70E}$ -action extends for every $\unicode[STIX]{x1D70E}\in G$ . Thus let $\unicode[STIX]{x1D70E}\in G$ , and after replacing $G$ by the cyclic subgroup generated by $\unicode[STIX]{x1D70E}$ , we may assume that $G$ is cyclic, say $G=\operatorname{Gal}(L/K)\cong \mathbb{Z}/n\mathbb{Z}$ , and generated by $\unicode[STIX]{x1D70E}$ .

Let $U\subset {\mathcal{X}}$ be the maximal open subspace to which the $G$ -action on $X_{L}$ extends. Then, as in the proof of Proposition 5.1, $U$ contains the generic fiber $X_{L}$ , as well as an open dense subscheme of the special fiber ${\mathcal{X}}_{0}$ . Let $\unicode[STIX]{x1D6E4}\subset {\mathcal{X}}^{n}$ be the closure of the set $\{(x,\unicode[STIX]{x1D70E}(x),\ldots ,\unicode[STIX]{x1D70E}^{n-1}(x))\mid x\in U\}$ in  ${\mathcal{X}}^{n}$ . The group $G=\mathbb{Z}/n\mathbb{Z}$ acts on ${\mathcal{X}}^{n}$ by permutation of the factors, and this action restricted to $\unicode[STIX]{x1D6E4}_{L}$ coincides with the natural $G$ -action on $X_{L}$ via $\operatorname{pr}_{1}:\unicode[STIX]{x1D6E4}_{L}\stackrel{{\sim}}{\rightarrow }X_{L}$ .

Consider the diagram of $G$ -representations

(3)

where we omit the coefficients (Tate twists of $\mathbb{Q}_{\ell }$ ) of the $\ell$ -adic cohomology groups from the notation, and we write $\unicode[STIX]{x1D6E4}_{\overline{0}}:=(\unicode[STIX]{x1D6E4}_{0})_{\overline{k}}$ and so on. The triangle on the right is clearly commutative. The commutativity of the square on the left follows from the fact that the classes $[\unicode[STIX]{x1D6E4}_{\overline{0}}]$ and $[\unicode[STIX]{x1D6E4}_{\overline{L}}]$ correspond via the isomorphism $H^{4n-4}({\mathcal{X}}_{\overline{0}}^{n})\cong H^{4n-4}(X_{\overline{L}}^{n})$ , as proved in Lemma 5.6. Note that all irreducible components of $\unicode[STIX]{x1D6E4}_{\overline{0}}$ are of dimension $2$ (by, for example, [Reference LiuLiu02, Proposition 4.4.16]).

Let $\unicode[STIX]{x1D70B}:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }$ be as in Proposition 5.1. Let $E\subset {\mathcal{X}}_{0}$ be the exceptional locus of $\unicode[STIX]{x1D70B}_{0}:{\mathcal{X}}_{0}\rightarrow {\mathcal{X}}_{0}^{\prime }$ and $E^{\unicode[STIX]{x1D6FC}}$ be the irreducible components of $E_{\overline{k}}$ . Each irreducible component $E^{\unicode[STIX]{x1D6FC}}$ is isomorphic to $\mathbb{P}^{1}$ . Since $\unicode[STIX]{x1D70B}_{0}$ is a resolution of singularities, the intersection matrix $(E^{\unicode[STIX]{x1D6FC}}\cdot E^{\unicode[STIX]{x1D6FD}})_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}$ is negative definite (by Hodge index theorem), hence invertible. In particular, if we are given $c_{\unicode[STIX]{x1D6FC}}\in \mathbb{Q}_{\ell }$ for $\unicode[STIX]{x1D6FC}=1,\ldots ,m$ , such that $\sum _{\unicode[STIX]{x1D6FC}=1}^{m}c_{\unicode[STIX]{x1D6FC}}E^{\unicode[STIX]{x1D6FC}}\cdot E^{\unicode[STIX]{x1D6FD}}=0$ for all $\unicode[STIX]{x1D6FD}$ , then $c_{\unicode[STIX]{x1D6FC}}=0$ for all $\unicode[STIX]{x1D6FC}=1,\ldots ,m$ .

Consider the irreducible components of $\unicode[STIX]{x1D6E4}_{\overline{0}}$ . First, there is the ‘diagonal’ component, that is, the closure of the set $\{(x,\unicode[STIX]{x1D70E}(x),\ldots ,\unicode[STIX]{x1D70E}^{n-1}(x))\mid x\in U_{0}\}$ . If $Z$ is a non-diagonal component (assuming there is one), then $Z$ is contained in $E^{\unicode[STIX]{x1D6FC}_{1}}\times \cdots \times E^{\unicode[STIX]{x1D6FC}_{n}}$ for some $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n}$ . From the Künneth formula and the fact that $H^{\ast }(\mathbb{P}^{1},\mathbb{Q}_{\ell })\cong \mathbb{Q}_{\ell }[\mathbb{P}^{1}]\oplus \mathbb{Q}_{\ell }[\operatorname{pt}]$ , it follows that the cycle class $[Z]\in H^{4n-4}({\mathcal{X}}_{\overline{0}}^{n})$ is a non-zero $\mathbb{Z}_{{\geqslant}0}$ -combination of $[E_{i}^{\unicode[STIX]{x1D6FE}}\times E_{j}^{\unicode[STIX]{x1D6FF}}\times \operatorname{pt}^{n-2}]$ with $i\neq j$ , where

$$\begin{eqnarray}E_{i}^{\unicode[STIX]{x1D6FE}}\times E_{j}^{\unicode[STIX]{x1D6FF}}\times \operatorname{pt}^{n-2}:=\cdots \times E^{\unicode[STIX]{x1D6FE}}\times \cdots \times E^{\unicode[STIX]{x1D6FF}}\times \cdots\end{eqnarray}$$

(the $i$ th component is equal to $E^{\unicode[STIX]{x1D6FE}}$ , the $j$ th component is equal to $E^{\unicode[STIX]{x1D6FF}}$ , and the remaining components are equal to a point). Hence, if we set $[\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}:=[\unicode[STIX]{x1D6E4}_{\overline{0}}]-[\operatorname{diag}]$ , then there exist $c_{i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}\in \mathbb{Z}_{{\geqslant}0}$ such that

(4) $$\begin{eqnarray}[\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}=\mathop{\sum }_{i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}c_{i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}[E_{i}^{\unicode[STIX]{x1D6FE}}\times E_{j}^{\unicode[STIX]{x1D6FF}}\times \operatorname{pt}^{n-2}]\in H^{4n-4}({\mathcal{X}}_{\overline{0}}^{n}).\end{eqnarray}$$

We have $c_{i,i,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}=0$ for all $i,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}$ .

We want to show $[\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}=0$ . For this, we will use the assumption that the $G$ -action on $H^{2}(X_{\overline{L}})$ is trivial. Using the commutative diagram (3), we see that the map $\cdot \,\,[\unicode[STIX]{x1D6E4}_{\overline{0}}]:H^{2}({\mathcal{X}}_{\overline{0}}^{n})\rightarrow H^{4n-2}({\mathcal{X}}_{\overline{0}}^{n})$ factors through $H^{2}(X_{\overline{L}})$ , and thus every element in its image is $G$ -invariant. In particular, for all $\unicode[STIX]{x1D6FC}$ and $i$ , the cycle $[E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\cdot [\unicode[STIX]{x1D6E4}_{\overline{0}}]\in H^{4n-2}({\mathcal{X}}_{\overline{0}}^{n})$ is $G$ -invariant, where

$$\begin{eqnarray}E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}:=\cdots \times E^{\unicode[STIX]{x1D6FC}}\times \cdots\end{eqnarray}$$

(the $i$ th component is equal to $E^{\unicode[STIX]{x1D6FC}}$ and the remaining components are equal to ${\mathcal{X}}_{\overline{0}}$ ). Now, $G$ acts by $\unicode[STIX]{x1D70E}:[E_{j}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\mapsto [E_{j+1}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]$ , which implies that for all $\unicode[STIX]{x1D6FD}$ , the cycle $[E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\cdot [\unicode[STIX]{x1D6E4}_{\overline{0}}]\cdot [E_{j}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\in H^{4n}({\mathcal{X}}_{\overline{0}}^{n})\cong \mathbb{Q}_{\ell }$ is independent of $j$ . Since $[E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\cdot [\operatorname{diag}]\cdot [E_{j}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]$ is equal to $E^{\unicode[STIX]{x1D6FC}}\cdot E^{\unicode[STIX]{x1D6FD}}$ , it is also independent of $j$ , and thus $[E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\cdot [\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}\cdot [E_{j}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]$ is independent of $j$ for all $\unicode[STIX]{x1D6FD}$ . In order to compute its value, we use (4) and find

$$\begin{eqnarray}[E_{i}^{\unicode[STIX]{x1D6FC}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]\cdot [\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}\cdot [E_{j}^{\unicode[STIX]{x1D6FD}}\times {\mathcal{X}}_{\overline{0}}^{n-1}]=\mathop{\sum }_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}(c_{i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}+c_{j,i,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE}})(E^{\unicode[STIX]{x1D6FE}}\cdot E^{\unicode[STIX]{x1D6FC}})(E^{\unicode[STIX]{x1D6FF}}\cdot E^{\unicode[STIX]{x1D6FD}}).\end{eqnarray}$$

Since $c_{i,i,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}=0$ for all $i$ , this sum is zero for $i=j$ . Since it is independent of $j$ , this sum is zero for all $i,j$ . Using invertibility of the matrix $(E^{\unicode[STIX]{x1D6FC}}\cdot E^{\unicode[STIX]{x1D6FD}})$ twice, we obtain $c_{i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}}+c_{j,i,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE}}=0$ for all $i,j,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FF}$ . Thus $[\unicode[STIX]{x1D6E4}_{\overline{0}}]^{\text{nondiag}}=0$ .

Now, $\operatorname{pr}_{i}:\unicode[STIX]{x1D6E4}\rightarrow {\mathcal{X}}$ is a proper birational morphism for all $i$ , where ${\mathcal{X}}$ is regular, and $\unicode[STIX]{x1D6E4}$ is integral. Thus, by van der Waerden purity (see, [Reference LiuLiu02, Theorem 7.2.22], for example, and note that this result can easily be extended to algebraic spaces), the exceptional locus of $\operatorname{pr}_{i}$ is either empty or a divisor. If it was a divisor, it would give rise to a non-diagonal component of $\unicode[STIX]{x1D6E4}_{\overline{0}}$ , which does not exist by the previous computations. Thus $\operatorname{pr}_{i}$ is an isomorphism for all $i$ , and since the $\operatorname{Gal}(L/K)$ -action extends to $\unicode[STIX]{x1D6E4}$ , this shows that the $\operatorname{Gal}(L/K)$ -action extends to ${\mathcal{X}}$ , as desired.◻

Proof. We have already shown the extension of the $I_{G}$ -action to ${\mathcal{X}}$ in Proposition 5.5. Moreover, the $I_{G}$ -action on ${\mathcal{X}}_{0}$ is $k$ -linear. By assumption, the $I_{G}$ -action on $H_{\acute{\text{e}}\text{t}}^{m}(({\mathcal{X}}_{0})_{\overline{k}},\mathbb{Q}_{\ell })$ is trivial for $m=1,2$ , respectively.

If $X$ is an Abelian surface, then the $I_{G}$ -action on $({\mathcal{X}}_{0})_{\overline{k}}$ is trivial by the injectivity of $\unicode[STIX]{x1D70C}_{1}$ in arbitrary characteristic; see, for example, [Reference MumfordMum70, Theorem 3 in §19]. If $X$ is a K3 surface, then the $I_{G}$ -action on $({\mathcal{X}}_{0})_{\overline{k}}$ is trivial by the injectivity of $\unicode[STIX]{x1D70C}_{2}$ in arbitrary characteristic; see [Reference OgusOgu79, Corollary 2.5] and [Reference KeumKeu16, Theorem 1.4] (in the case of complex and possibly non-algebraic K3 surfaces, see [Reference BeauvilleBea85, Proposition IX.6] and [Reference HuybrechtsHuy16, Proposition 15.2.1]). In both cases, the $I_{G}$ -action on $({\mathcal{X}}_{0})_{\overline{k}}$ is trivial, and thus also the original action on ${\mathcal{X}}_{0}$ is trivial.◻