1 Introduction
Let ${\mathcal{O}}_{K}$ be a local Henselian DVR (discrete valuation ring) of characteristic zero with field of fractions $K$ and perfect residue field $k$ , whose characteristic is $p\geqslant 0$ . For example, ${\mathcal{O}}_{K}$ could be $\mathbb{C}[[t]]$ or the ring of integers in a $p$ adic field. Given a variety $X$ that is smooth and proper over $K$ , one can ask whether $X$ has good reduction, that is, whether there exists an algebraic space
with generic fiber $X$ that is smooth and proper over ${\mathcal{O}}_{K}$ .
1.1 Good reduction and Galois representations
Let $\ell$ be a prime different from $p$ , let $G_{K}:=\operatorname{Gal}(\overline{K}/K)$ be the absolute Galois group of $K$ , and let $I_{K}$ be its inertia subgroup. Then the natural $\ell$ adic Galois representation
is called unramified if it satisfies $\unicode[STIX]{x1D70C}_{m,\ell }(I_{K})=\{\operatorname{id}\}$ . A necessary condition for $X$ to have good reduction is that for all $m\geqslant 1$ and all primes $\ell \neq p$ , the representation $\unicode[STIX]{x1D70C}_{m,\ell }$ is unramified.
1.2 Curves and Abelian varieties
By a famous theorem of Serre and Tate [Reference Serre and TateST68], which generalizes results of Néron, Ogg, and Shafarevich for elliptic curves to Abelian varieties, the $G_{K}$ representation $\unicode[STIX]{x1D70C}_{1,\ell }$ detects the reduction type of Abelian varieties.
Theorem 1.1 (Serre–Tate).
An Abelian variety $X$ over $K$ has good reduction if and only if the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified.
On the other hand, it is not too difficult to give counterexamples to such a result for curves of genus at least $2$ . Nevertheless, Oda [Reference OdaOda95] showed that good reduction can be detected by the outer $G_{K}$ representation on the étale fundamental group. We refer the interested reader to § 2.4 for references, examples, and details.
1.3 Kulikov–Nakkajima–Persson–Pinkham models
Before coming to the results of this article, we have to make one crucial assumption.
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 model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ that is a regular algebraic space with trivial canonical sheaf $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{O}}_{L}}$ , and whose geometric special fiber is a normal crossing divisor.
In equal characteristic zero, Assumption ( $\star$ ) always holds, and the special fibers of the corresponding models have been classified by Kulikov [Reference KulikovKul77], Persson [Reference PerssonPer77], and Persson and Pinkham [Reference Persson and PinkhamPP81]. In mixed characteristic, the corresponding classification (assuming the existence of such models) is due to Nakkajima [Reference NakkajimaNakk00]. If the expected results on resolution of singularities and toroidalization of morphisms were known to hold in mixed characteristic, then Assumption $(\star )$ would follow from Kawamata’s semistable minimal model program (MMP) in mixed characteristic [Reference KawamataKaw94] and Artin’s results [Reference ArtinArt74] on simultaneous resolutions of families of surface singularities. We refer to Proposition 3.1 for details. Using work of Maulik [Reference MaulikMau14] and some strengthenings due to the second named author [Reference MatsumotoMat15], we have at least the following.
Theorem 1.2. Let $X$ be a K3 surface over $K$ and assume that $p=0$ or that $X$ admits an ample invertible sheaf ${\mathcal{L}}$ with $p>{\mathcal{L}}^{2}+4$ . Then $X$ satisfies Assumption $(\star )$ .
1.4 K3 surfaces
In this article, we establish a Néron–Ogg–Shafarevich–Serre–Tate type result for K3 surfaces. Important steps were already taken by the second named author in [Reference MatsumotoMat15]. Over the complex numbers, similar results are classically known; see, for example, [Reference Kulikov and KurchanovKK98, ch. 5].
Before coming to the main result of this article, we define a K3 surface with at worst RDP (rational double point) singularities to be a proper surface over a field, which, after base change to an algebraically closed field, has at worst rational double point singularities, and whose minimal resolution of singularities is a K3 surface.
Theorem 1.3. Let $X$ be a K3 surface over $K$ that satisfies Assumption $(\star )$ . If the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for some $\ell \neq p$ , then the following hold.

(i) There exists a model of $X$ that is a projective scheme over ${\mathcal{O}}_{K}$ , whose special fiber is a K3 surface with at worst RDP singularities.

(ii) Moreover, there exists an integer $N$ , independent of $X$ and $K$ , and a finite unramified extension $L/K$ of degree at most $N$ , such that $X_{L}$ has good reduction over $L$ .
In [Reference Hassett and TschinkelHT17, Theorem 35], a similar result is obtained for K3 surfaces over $\mathbb{C}((t))$ , but their proof uses methods different from ours. As in the case of Abelian varieties in [Reference Serre and TateST68], we obtain the following independence of $\ell$ .
Corollary 1.4. Let $X$ be a K3 surface over $K$ that satisfies Assumption $(\star )$ . Then the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for one $\ell \neq p$ if and only if it is unramified for all $\ell \neq p$ .
In [Reference Serre and TateST68], Serre and Tate showed that if an Abelian variety of dimension $g$ over $K$ with $p>2g+1$ has potential good reduction, then good reduction can be achieved after a tame extension. Here, we establish the following analog for K3 surfaces.
Corollary 1.5. Let $X$ be a K3 surface over $K$ with $p\geqslant 23$ and potential good reduction. Then $X$ has good reduction after a tame extension of $K$ .
It is important to note that in part (2) of Theorem 1.3, we cannot avoid field extensions in general. More precisely, we construct the following explicit examples.
Theorem 1.6. For every prime $p\geqslant 5$ , there exists a K3 surface $X=X(p)$ over $\mathbb{Q}_{p}$ , such that:

(i) 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 $\ell \neq p$ ;

(ii) $X$ has good reduction over the unramified extension $\mathbb{Q}_{p^{2}}$ ; but

(iii) $X$ does not have good reduction over $\mathbb{Q}_{p}$ .
1.5 Kuga–Satake Abelian varieties
Let us recall that Kuga and Satake [Reference Kuga and SatakeKS67] associated to a polarized K3 surface $(X,{\mathcal{L}})$ over $\mathbb{C}$ a polarized Abelian variety $\operatorname{KS}(X,{\mathcal{L}})$ of dimension $2^{19}$ over $\mathbb{C}$ . Moreover, if $(X,{\mathcal{L}})$ is defined over an arbitrary field $k$ , then Rizov [Reference RizovRiz10] and Madapusi Pera [Reference Madapusi PeraMad15], building on work of Deligne [Reference DeligneDel72] and André [Reference AndréAnd96], established the existence of $\operatorname{KS}(X,{\mathcal{L}})$ over some finite extension of $k$ . As an application of Theorem 1.3, we can compare the reduction behavior of a polarized K3 surface to that of its associated Kuga–Satake Abelian variety.
Theorem 1.7. Assume $p\neq 2$ . Let $(X,{\mathcal{L}})$ be a polarized K3 surface over $K$ .

(i) If $X$ has good reduction, then $\operatorname{KS}(X,{\mathcal{L}})$ can be defined over an unramified extension $L/K$ , and it has good reduction over $L$ .

(ii) Assume that $X$ satisfies Assumption $(\star )$ . Let $L/K$ be a field extension such that both $\operatorname{KS}(X,{\mathcal{L}})$ and the Kuga–Satake correspondence can be defined over $L$ . If $\operatorname{KS}(X,{\mathcal{L}})$ has good reduction over $L$ , then $X$ has good reduction over an unramified extension of $L$ .
1.6 Organization
This article is organized as follows.
In § 2, we recall a couple of general facts on models and unramified Galois representations on $\ell$ adic cohomology. We also recall the classical Serre–Tate theorem for Abelian varieties and give explicit examples of curves of genus at least $2$ , where the Galois representation does not detect bad reduction.
In § 3, we review potential semistable reduction of K3 surfaces, Kawamata’s semistable MMP, the Kulikov–Nakkajima–Pinkham–Persson classification list, and the second named author’s results on potential good reduction of K3 surfaces. We also briefly discuss potential good and semistable reduction of Enriques surfaces.
In § 4, we establish existence and termination of certain flops, which we need later on to equip our models with suitable invertible sheaves. Moreover, we show that any two smooth models of a K3 surface $X$ over $K$ are related by a finite sequence of flopping contractions and their inverses.
Section 5 is the technical heart of this article: given a K3 surface $X$ over $K$ , a finite Galois extension $L/K$ with group $G$ , and a model of $X_{L}$ over ${\mathcal{O}}_{L}$ , we study extensions of the $G$ action $X_{L}$ to this model. Then we study quotients of such models by $G$ actions, where the most difficult case arises when $p$ divides the order of $G$ (wild action).
In § 6, we establish the main results of this article: a Néron–Ogg–Shafarevich type theorem for K3 surfaces, good reduction over tame extensions, as well as the connection to Kuga–Satake Abelian varieties.
Finally, in § 7, we give explicit examples of K3 surfaces over $\mathbb{Q}_{p}$ with unramified Galois representations on their $\ell$ adic cohomology groups that do not have good reduction over $\mathbb{Q}_{p}$ .
Notation and conventions
Throughout the whole article, we fix the following notation:
If $L/K$ is a field extension, and $X$ is a scheme over $K$ , we abbreviate the basechange $X\times _{\operatorname{Spec}K}\operatorname{Spec}L$ by $X_{L}$ .
2 Generalities
In this section, we recall a couple of general facts on models of varieties, unramified Galois representations on $\ell$ adic cohomology groups, and Néron–Ogg–Shafarevich type theorems.
2.1 Models
We start with the definition of various types of models.
Definition 2.1. Let $X$ be a smooth and proper variety over $K$ .

(i) A model of $X$ over ${\mathcal{O}}_{K}$ is an algebraic space that is flat and proper over $\operatorname{Spec}{\mathcal{O}}_{K}$ and whose generic fiber is isomorphic to $X$ .

(ii) We say that $X$ has good reduction if there exists a model of $X$ that is smooth over ${\mathcal{O}}_{K}$ .

(iii) We say that $X$ has semistable reduction if there exists a regular model of $X$ , whose geometric special fiber is a reduced normal crossing divisor with smooth components. (Sometimes, this notion is also called strictly semistable reduction.)

(iv) We say that $X$ has potential good (respectively semistable) reduction if there exists a finite field extension $L/K$ such that $X_{L}$ has good (respectively semistable) reduction.
Remark 2.2. Models of curves and Abelian varieties can be treated entirely within the category of schemes; see, for example, [Reference LiuLiu02, ch. 10] and [Reference Bosch, Lütkebohmert and RaynaudBLR90]. However, if $X$ is a K3 surface over $K$ with good reduction, then it may not be possible to find a smooth model in the category of schemes, and we refer to [Reference MatsumotoMat15, §5.2] for explicit examples. In particular, we are forced to work with algebraic spaces when studying models of K3 surfaces.
2.2 Inertia and monodromy
The $G_{K}$ action on $\overline{K}$ induces an action on ${\mathcal{O}}_{\overline{K}}$ and by reduction, an action on $\overline{k}$ . This gives rise to a continuous and surjective homomorphism $G_{K}\rightarrow G_{k}$ of profinite groups. Thus we obtain a short exact sequence
whose kernel $I_{K}$ is called the inertia group. In fact, $I_{K}$ is the absolute Galois group of the maximal unramified extension of $K$ . If $p\neq 0$ , then the wild inertia group $P_{K}$ is the normal subgroup of $G_{K}$ that is the absolute Galois group of the maximal tame extension of $K$ . We note that $P_{K}$ is the unique $p$ Sylow subgroup of $I_{K}$ .
Definition 2.3. Let $X$ be a smooth and proper variety over $K$ . Then the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{m}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is called unramified if $I_{K}$ acts trivially. It is called tame if $P_{K}$ acts trivially.
For an Abelian variety $X$ , it follows from results of Serre and Tate [Reference Serre and TateST68] that the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{m}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for one $\ell \neq p$ , if and only if it is so for all $\ell \neq p$ . In Corollary 6.4, we will show a similar result for K3 surfaces. In general, it is not known whether being unramified depends on the choice of $\ell$ , but it is expected not to.
A relation between good reduction and unramified Galois representations on $\ell$ adic cohomology groups is given by the following wellknown result, which follows from the proper smooth base change theorem. For schemes, it is stated in [Reference GrothendieckSGA4, Théorème XII.5.1], and in case the model is an algebraic space, we refer to [Reference Liu and ZhengLZ14, Theorem 0.1.1] or [Reference ArtinArt73, ch. VII].
Theorem 2.4. If $X$ has good reduction, then the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{m}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for all $m$ and for all $\ell \neq p$ .
In view of this theorem, it is natural to ask for the converse direction. Whenever such a converse holds for some class of varieties over $K$ , we obtain a purely representationtheoretic criterion to determine whether such a variety admits a model over ${\mathcal{O}}_{K}$ with good reduction.
2.3 Abelian varieties
A classical converse to Theorem 2.4 is the Néron–Ogg–Shafarevich criterion for elliptic curves. Later, Serre and Tate generalized it to Abelian varieties of arbitrary dimension.
Theorem 2.5 (Serre–Tate [Reference Serre and TateST68]).
An Abelian variety $A$ over $K$ has good reduction if and only if the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{1}(A_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for one (respectively all) $\ell \neq p$ .
2.4 Higher genus curves, part 1
Now, the converse to Theorem 2.4 already fails for curves of higher genus. Let $X$ be a smooth and proper curve of genus $g\geqslant 2$ over $K$ . Let $\operatorname{Jac}(X)$ be its Jacobian, which is an Abelian variety of dimension $g$ over $K$ . Then the exact sequence of étale sheaves on $X$
gives rise to $G_{K}$ equivariant isomorphisms $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\unicode[STIX]{x1D707}_{n})\cong \operatorname{Pic}(X_{\overline{K}})[n]\cong H_{\acute{\text{e}}\text{t}}^{1}(\operatorname{Jac}(X)_{\overline{K}},\unicode[STIX]{x1D707}_{n})$ , from which we obtain $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })\cong H_{\acute{\text{e}}\text{t}}^{1}(\operatorname{Jac}(X)_{\overline{K}},\mathbb{Q}_{\ell })$ by passing to the limit. Moreover, if $X$ has a $K$ rational point, then there is a natural embedding
and the above isomorphism coincides with $j^{\ast }$ (which is independent of the choice of a rational point). By the Serre–Tate theorem (Theorem 2.5), an unramified $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is equivalent to good reduction of $\operatorname{Jac}(X)$ . The following lemma gives a criterion that ensures the latter.
Lemma 2.6. Let $X$ be a smooth and proper curve over $K$ that admits a semistable scheme model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ such that the dual graph associated to the components of its special fiber ${\mathcal{X}}_{0}$ is a tree. Then $\operatorname{Jac}(X)$ has good reduction and the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified.
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.◻
Using this lemma, it is easy to produce counterexamples to Néron–Ogg–Shafarevich type results for curves of higher genus.
Proposition 2.7. If $p\neq 2$ , then there exists for infinitely many $g\geqslant 2$ a smooth and proper curve $X$ of genus $g$ over $K$ such that:

(i) 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 $\ell \neq p$ ; and

(ii) $X$ does not have good reduction over $K$ nor over any finite extension.
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 $(g1)$ meeting transversally in one point. In particular, neither $X$ nor any basechange $X_{L}$ 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 $\ell \neq p$ .◻
We stress that these results are well known to the experts, but since we were not able to find explicit references and explicit examples, we decided to include them here.
2.5 Higher genus curves, part 2
If $X$ is a smooth and proper curve of genus at least $2$ over $K$ , then one can also study the outer $G_{K}$ representation on its étale fundamental group, which turns out to detect good reduction. More precisely, there exists a short exact sequence of étale fundamental groups
For every prime $\ell$ , this exact sequence gives rise to a welldefined homomorphism from $G_{K}$ to the outer automorphism group of the pro $\ell$ completion $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\overline{K}})_{\ell }$ of the geometric étale fundamental group
In analogy to Definition 2.3, we will say that this representation is unramified if $\unicode[STIX]{x1D70C}_{\ell }(I_{K})=\{1\}$ . We note that the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{1}(X_{\overline{K}},\mathbb{Q}_{\ell })$ arises from the residual action of $\unicode[STIX]{x1D70C}_{\ell }$ on the Abelianization of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\overline{K}})_{\ell }$ . After these preparations, we have the following Néron–Ogg–Shafarevich type theorem for curves of higher genus, which is in terms of fundamental groups rather than cohomology groups.
Theorem 2.8 (Oda [Reference OdaOda95, Theorem 3.2]).
Let $X$ be a smooth and proper curve of genus at least $2$ over $K$ . Then $X$ has good reduction if and only if the outer Galois action $\unicode[STIX]{x1D70C}_{\ell }$ is unramified for one (respectively all) $\ell \neq p$ .
3 K3 surfaces and their models
In this section, we first introduce the crucial Assumption $(\star )$ , which ensures the existence of suitable models for K3 surfaces. These models have been studied by Kulikov, Nakkajima, Persson, and Pinkham. Following ideas of Maulik, we show how Assumption $(\star )$ would follow from a combination of potential semistable reduction (which is not known in mixed characteristic, but expected) and the semistable minimal model program (MMP) in mixed characteristic. Then we give some conditions under which Assumption $(\star )$ does hold. After that, we shortly review the second named author’s results on potential good reduction of K3 surfaces. Finally, we show by example that these results do not carry over to Enriques surfaces. Most of the results of this section are probably known to the experts.
3.1 Kulikov–Nakkajima–Persson–Pinkham models
We first introduce the crucial assumption that we shall make from now on.
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 semistable 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.
Here, we equip ${\mathcal{X}}$ with its standard log structure ${\mathcal{X}}^{\log }$ and define the relative canonical sheaf $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{O}}_{L}}$ to be $\bigwedge ^{2}\unicode[STIX]{x1D6FA}_{{\mathcal{X}}^{\log }/{\mathcal{O}}_{L}^{\log }}^{1}$ using log differentials. Since ${\mathcal{X}}^{\log }$ is log smooth over ${\mathcal{O}}_{L}$ , the sheaf $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{O}}_{L}}$ is invertible; see also the discussion in [Reference MatsumotoMat15, §3].
The main reason why Assumption $(\star )$ is not known to hold is that potential semistable reduction is not known: using resolution of singularities in mixed characteristic (recently announced by Cossart and Piltant [Reference Cossart and PiltantCP14]) and embedded resolution of singularities (Cossart et al. [Reference Cossart, Jannsen and SaitoCJS13]), we obtain a model ${\mathcal{X}}$ , whose special fiber ${\mathcal{X}}_{0}$ has simple normal crossing support, but whose components may have multiplicities. At the moment, it is not clear how to get rid of these multiplicities after base change, unless all of them are prime to $p$ . In case the residue characteristic is zero, these results are classically known to hold; see the discussion in [Reference Kollár and MoriKM98, §7.2] for details.
The following result, which is inspired by Maulik’s approach and ideas from [Reference MaulikMau14, §4], shows that Assumption $(\star )$ essentially holds once we assume potential semistable reduction. More precisely, we have the following.
Proposition 3.1. Assume $p\neq 2,3$ . Let $X$ be a K3 surface over $K$ and assume that there exists:

(i) a finite field extension $L^{\prime }/K$ ; and

(ii) a smooth surface $Y$ over $L^{\prime }$ that is birationally equivalent to $X_{L^{\prime }}$ ; and

(iii) a scheme model ${\mathcal{Y}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L^{\prime }}$ of $Y$ with semistable reduction.
Then $X$ satisfies Assumption $(\star )$ .
Proof. Let ${\mathcal{Y}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L^{\prime }}$ be as in the statement. Since $p\neq 2,3$ , Kawamata’s semistable 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 semistable 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 $(\star )$ .◻
As already mentioned above, the assumptions are fulfilled if $p=0$ ; see [Reference Kempf, Knudsen, Mumford and SaintDonatKKMS73, ch. 2] or the discussion in [Reference Kollár and MoriKM98, §7.2]. If $p\neq 0$ , then they are fulfilled for K3 surfaces that admit a very ample invertible sheaf ${\mathcal{L}}$ with $p>{\mathcal{L}}^{2}+4$ by a result of Maulik [Reference MaulikMau14, §4]. With some extra work, the condition ‘very ample’ can be weakened to ‘ample’ (see [Reference MatsumotoMat15, argument following Lemma 3.1]) and Theorem 1.2 follows. Thus we have the following result.
Theorem 3.2 ( $=$ Theorem 1.2).
Let $X$ be a K3 surface over $K$ and assume that $p=0$ or that $X$ admits an ample invertible sheaf ${\mathcal{L}}$ with $p>{\mathcal{L}}^{2}+4$ . Then $X$ satisfies Assumption $(\star )$ .
Over $\mathbb{C}$ , Kulikov [Reference KulikovKul77], Persson [Reference PerssonPer77], and Pinkham and Persson [Reference Persson and PinkhamPP81] classified the special fibers of the models asserted by Assumption $(\star )$ . We refer to [Reference MorrisonMor81, §1] and [Reference Kulikov and KurchanovKK98, ch. 5] for overview, and to Nakkajima’s extension [Reference NakkajimaNakk00] of these results to mixed characteristic.
3.2 Potential good reduction of K3 surfaces
Now, if $X$ is a K3 surface over $K$ that satisfies Assumption $(\star )$ , then there exists a finite field extension $L/K$ and a model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{L}$ of $X_{L}$ as asserted by Assumption $(\star )$ . If the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified, then the weight filtration on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })$ that arises from the Steenbrink–Rapoport–Zink spectral sequence (see [Reference SteenbrinkSte76], [Reference Rapoport and ZinkRZ82, Satz 2.10], and [Reference NakayamaNaka00, Proposition 1.9] for details) is trivial. Together with a result of Persson [Reference PerssonPer77, Proposition 3.3.6], this implies that the special fiber of ${\mathcal{X}}$ is smooth, that is, $X_{L}$ has good reduction. Thus we obtain the following result of the second named author and we refer to [Reference MatsumotoMat15] for details and a detailed proof.
Theorem 3.3 (Matsumoto).
Let $X$ be a K3 surface over $K$ that satisfies Assumption $(\star )$ . If the $G_{K}$ representation on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for one $\ell \neq p$ , then $X$ has potential good reduction.
3.3 Enriques surfaces
The previous theorem does not generalize to other classes of surfaces with numerically trivial canonical sheaves. For example, the $G_{K}$ representation on $\ell$ adic cohomology of an Enriques surface can neither exclude nor confirm any type in the Kulikov–Nakkajima–Persson–Pinkham list for these surfaces. More precisely, we have the following.
Lemma 3.4. Let $Y$ be an Enriques surface over $K$ . Then there exists a finite extension $L/K$ such that the $G_{L}$ representation on $H_{\acute{\text{e}}\text{t}}^{m}(Y_{\overline{K}},\mathbb{Q}_{\ell })$ is unramified for all $m$ and all $\ell \neq p$ .
Proof. We only have to show something for $m=2$ . But then the first Chern class induces a $G_{K}$ equivariant isomorphism
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.◻
Moreover, the next example shows that also the $G_{K}$ representation on the $\ell$ adic cohomology of the K3 double cover $X$ of an Enriques surface $Y$ does not detect potential good reduction of $Y$ . This phenomenon is related to flower pot degenerations of Enriques surfaces, see [Reference PerssonPer77, §3.3] and [Reference PerssonPer77, Appendix 2].
Example 3.5. Fix a prime $p\geqslant 5$ . Consider $\mathbb{P}_{\mathbb{Z}_{p}}^{5}$ with coordinates $x_{i},y_{i}$ , $i=0,1,2$ , and inside it the complete intersection of three quadrics
where $e\in \mathbb{Z}_{p}^{\times }$ satisfies $e^{2}\not \equiv 0,1,2~\text{mod}~p$ (for example, we could take $e=2$ ). Then $\imath :x_{i}\mapsto x_{i},y_{i}\mapsto y_{i}$ defines an involution on $\mathbb{P}_{\mathbb{Z}_{p}}^{5}$ , which induces an involution on ${\mathcal{X}}$ . We denote by $X$ the generic fiber of ${\mathcal{X}}$ , and by $Y:=X/\imath$ the quotient by the involution.
Theorem 3.6. Let $p\geqslant 5$ and let $X\rightarrow Y$ be as in Example 3.5. Then $Y$ is an Enriques surface over $\mathbb{Q}_{p}$ , such that:

(i) the K3 double cover $X$ of $Y$ has good reduction;

(ii) the $G_{\mathbb{Q}_{p}}$ action on $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{\ell })$ is unramified for all $\ell \neq p$ ;

(iii) $Y$ has semistable reduction of flower pot type; but

(iv) $Y$ does not have potential good reduction.
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 nonsmooth K3 surface with four RDP singularities of type $A_{1}$ located at $[0:0:0:\pm 1:\pm 1:1]$ . Then the blowup ${\mathcal{X}}_{1}^{\prime }\rightarrow {\mathcal{X}}$ 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 blowup 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 socalled Coble surface) meeting transversally four $\mathbb{P}^{2}$ ’s, that is, a semistable 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 basechange 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. ◻
4 Existence and termination of flops
Let $X$ be a smooth and proper surface over $K$ with numerically trivial canonical sheaf and assume that we have a smooth model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ . Now, if ${\mathcal{L}}$ is an ample invertible sheaf on $X$ , then its specialization ${\mathcal{L}}_{0}$ to the special fiber may not be ample, and not even be nef. In this section, we show that there exists a finite sequence of birational modifications (flops) of ${\mathcal{X}}$ , such that we eventually arrive at a smooth model ${\mathcal{X}}^{+}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ of $X$ , such that the restriction of ${\mathcal{L}}$ to the special fiber of ${\mathcal{X}}^{+}$ is big and nef. We end this section by showing that any two smooth models of $X$ over ${\mathcal{O}}_{K}$ are related by a finite sequence of flopping contractions and their inverses.
We start by adjusting [Reference Kollár and MoriKM98, Definition 3.33] and [Reference Kollár and MoriKM98, Definition 6.10] to our situation.
Definition 4.1. Let $X$ be a smooth and proper surface over $K$ with numerically trivial $\unicode[STIX]{x1D714}_{X/K}$ that admits a smooth model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ . Then we have the following.

(i) A proper and birational morphism $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ over ${\mathcal{O}}_{K}$ is called a flopping contraction if ${\mathcal{Y}}$ is normal and if the exceptional locus of $f$ is of codimension at least two.

(ii) If $D$ is a Cartier divisor on ${\mathcal{X}}$ , then a birational map ${\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{+}$ over ${\mathcal{O}}_{K}$ is called a $D$ flop if it decomposes into a flopping contraction $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ followed by (the inverse of) a flopping contraction $f^{+}:{\mathcal{X}}^{+}\rightarrow {\mathcal{Y}}$ such that $D$ is $f$ ample and $D^{+}$ is $f^{+}$ ample, where $D^{+}$ denotes the strict transform of $D$ on ${\mathcal{X}}^{+}$ . If ${\mathcal{L}}$ is an invertible sheaf on ${\mathcal{X}}$ , we similarly define an ${\mathcal{L}}$ flop.

(iii) A morphism $f^{+}$ as in (2) is also called a flop of $f$ .
In general, one also has to assume that $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{O}}_{K}}$ is numerically $f$ trivial in the definition of a flopping contraction. However, in our situation this is automatic. Also, a flop of $f$ , if it exists, does not depend on the choice of $D$ by [Reference Kollár and MoriKM98, Corollary 6.4] and [Reference Kollár and MoriKM98, Definition 6.10]. This justifies talking about flops without referring to the divisor $D$ .
4.1 Existence of flops
The following is an adaptation of Kollár’s proof [Reference KollárKol89, Proposition 2.2] of the existence of $3$ fold flops over $\mathbb{C}$ to our situation, which deals with special flops in mixed characteristic.
Proposition 4.2 (Existence of flops).
Let $X$ be a smooth and proper surface over $K$ with numerically trivial $\unicode[STIX]{x1D714}_{X/K}$ that has a smooth model ${\mathcal{X}}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ . If ${\mathcal{L}}$ is an ample invertible sheaf on $X$ and $C$ is an integral (but not necessarily geometrically integral) curve on the special fiber ${\mathcal{X}}_{0}$ with ${\mathcal{L}}_{0}\cdot C<0$ , then there exists a flopping contraction $f:{\mathcal{X}}\rightarrow {\mathcal{X}}^{\prime }$ and its ${\mathcal{L}}$ flop $f^{+}:{\mathcal{X}}^{+}\rightarrow {\mathcal{X}}^{\prime }$ with the following properties:

(i) $f$ contracts $C$ and no other curves;

(ii) ${\mathcal{X}}^{+}\rightarrow \operatorname{Spec}{\mathcal{O}}_{K}$ is a smooth model of $X$ ;

(iii) $f$ and $f^{+}$ induce isomorphisms of generic fibers;

(iv) ${\mathcal{L}}_{0}^{+}\cdot C^{+}>0$ , where ${\mathcal{L}}^{+}$ denotes the extension of ${\mathcal{L}}$ on ${\mathcal{X}}^{+}$ , and where $C^{+}$ denotes the flopped curve (that is, the exceptional locus of $f^{+}$ ).
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 pseudoeffective. Thus there exists a Zariski–Fujita decomposition on $({\mathcal{X}}_{0})_{\overline{k}}$
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 selfintersection $2$ , 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 basechange $C_{\overline{k}}\subset ({\mathcal{X}}_{0})_{\overline{k}}$ is a disjoint union of ADE curves. Since $C^{2}<0$ , Artin showed that there exists a morphism of projective surfaces over $k$
that contracts $C$ and nothing else (see [Reference BădescuBăd01, Theorem 3.9], for example). Since $C_{\overline{k}}$ is a union of ADEcurves, 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
Since $f_{0}$ is a contraction with $R^{1}f_{0\ast }{\mathcal{O}}_{{\mathcal{X}}_{0}}=0$ , there exists a blowdown $f_{n}:{\mathcal{X}}_{n}\rightarrow {\mathcal{X}}_{n}^{\prime }$ that extends $f_{0}$ , see [Reference Cynk and van StratenCvS09, Theorem 3.1]. Passing to limits, we obtain a contraction of formal schemes
By [Reference ArtinArt70, Theorem 3.1], there exists a contraction of algebraic spaces
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
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
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 $z^{2}h_{1}(x,y)zh_{0}(x,y)$ 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
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
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 justconstructed $\widehat{{\mathcal{Z}}}^{+}\rightarrow \widehat{w}$ . Thus there exists a birational and rational map
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 $C^{+}$ .◻
4.2 Termination of flops
Having established the existence of certain flops in mixed characteristic, we now show that there is no infinite sequence of them. To do so, one can adjust the proof of termination of flops from [Reference Kollár and MoriKM98, Theorem 6.17 and Corollary 6.19] over $\mathbb{C}$ to our situation. Instead, we give another argument that was kindly suggested to us by the referee.
We keep the notation and assumptions of Proposition 4.2. Then there are two isomorphisms between the $\ell$ adic cohomology groups of the special fibers ${\mathcal{X}}_{0}$ and ${\mathcal{X}}_{0}^{+}$ .

(i) The first is by composing the comparison isomorphisms relating the cohomology groups of special and generic fibers of ${\mathcal{X}}$ and ${\mathcal{X}}^{+}$
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0}^{+})_{\overline{k}},\mathbb{Q}_{\ell })(1)\cong H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{K}},\mathbb{Q}_{\ell })(1)\cong H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0})_{\overline{k}},\mathbb{Q}_{\ell })(1).\end{eqnarray}$$ 
(ii) Next, the composition $\unicode[STIX]{x1D711}:=(f^{+})^{1}\circ f:{\mathcal{X}}{\dashrightarrow}{\mathcal{X}}^{+}$ induces a birational and rational map of special fibers $\unicode[STIX]{x1D711}_{0}:{\mathcal{X}}_{0}{\dashrightarrow}{\mathcal{X}}_{0}^{+}$ , which extends to an isomorphism, since ${\mathcal{X}}_{0}$ and ${\mathcal{X}}_{0}^{+}$ are minimal surfaces of Kodaira dimension at least $0$ . Thus we obtain a second isomorphism via pullback
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}^{\ast }:H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0}^{+})_{\overline{k}},\mathbb{Q}_{\ell })(1)\cong H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0})_{\overline{k}},\mathbb{Q}_{\ell })(1).\end{eqnarray}$$
We note that both isomorphisms respect the intersection product coming from Poincaré duality, that is, they are isometries. For a $(2)$ curve $C^{\prime }\subset ({\mathcal{X}}_{0})_{\overline{k}}$ , we let $[C^{\prime }]$ be the associated cycle class in $H_{\acute{\text{e}}\text{t}}^{2}(({\mathcal{X}}_{0})_{\overline{k}},\mathbb{Q}_{\ell })(1)$ and we define the reflection in $C^{\prime }$ to be the isometry
The following lemma compares the isometries $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D711}_{0}^{\ast }$ in terms of reflections in $(2)$ curves.
Lemma 4.3. We keep the notation and assumptions as in Proposition 4.2 and denote by $C_{1},\ldots ,C_{m}$ the connected components of $C_{\overline{k}}$ . Then
where either:

(i) the connected components $C_{i}$ are disjoint $(2)$ curves and $r_{i}=r_{C_{i}}$ ; or

(ii) each $C_{i}$ is the union of two $(2)$ curves $C_{i,1}$ and $C_{i,2}$ intersecting in one point and $r_{i}=r_{C_{i,1}}r_{C_{i,2}}r_{C_{i,1}}=r_{C_{i,2}}r_{C_{i,1}}r_{C_{i,2}}$ .
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 onedimensional 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 pullback 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 onedimensional 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.◻
We define a generalized $(2)$ curve on a smooth and proper surface $X$ over a perfect field $k$ to be an integral (but not necessarily geometrically integral) curve $C\subset X$ such that $C_{\overline{k}}=C_{1}\cup \cdots \cup C_{m}$ is a disjoint union of ADE curves of type $A_{1}$ or $A_{2}$ . We note that $C^{2}=2m$ , that is, such curves are not necessarily of selfintersection $2$ . Moreover, we define the reflection $r_{C}$ in $\operatorname{NS}(X)$ or $H_{\acute{\text{e}}\text{t}}^{2}(X_{\overline{k}},\mathbb{Q}_{\ell })(1)$ to be equal to