Good reduction of K3 surfaces

Let $K$ be the field of fractions of a local Henselian discrete valuation ring ${\mathcal{O}}_{K}$ of characteristic zero with perfect residue field $k$ . Assuming potential semi-stable reduction, we show that an unramified Galois action on the second $\ell$ -adic cohomology group of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.


Introduction
Let K be a finite extension of the field Q p of p-adic numbers, and let O K be its ring of integers.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 X → Spec O K with generic fiber X that is smooth and proper over O K .
Good reduction and Galois-actions.A necessary condition for good reduction of X is that the Galois-actions on all ℓ-adic cohomology groups are unramified.This means the following: let G K := Gal(K/K) be the absolute Galois group of K, let I K be its inertia subgroup, and fix a prime ℓ = p.Being unramified means that for all m, the natural representation Curves and Abelian varieties.By a famous result of Serre and Tate [ST68], generalizing results of Néron, Ogg, and Shafarevich for elliptic curves, the Galois-action detects the reduction type of Abelian varieties: Theorem (Serre-Tate).Let X be an Abelian variety over K.Then, X has good reduction if and only if the G K -action on H 1 ét (X K , Q ℓ ) is unramified.On the other hand, it is not too difficult to give counter-examples to such a result for curves of genus ≥ 2. However, Oda [Od95] showed that good reduction can be detected by the outer Galois-action on the étale fundamental group.We refer the interested reader to Section 1.4 for references, examples, and details.
Kulikov-Nakkajima-Persson-Pinkham models.Before coming to the results of this article, we have to make one crucial assumption.
Assumption (⋆).A K3 surface X over K satisfies (⋆) if there exists a finite field extension L/K such that X L admits a model X → Spec O L that is a regular algebraic space with trivial ω X /O L , and whose geometric special fiber is a normal crossing divisor.
For a family of complex K3 surfaces over a pointed disk, the analogous assumption always holds, and the special fibers of the corresponding models have been classified by Kulikov [Ku77], Persson [P77], and Persson-Pinkham [PP81].The corresponding classification in mixed characteristic (assuming the existence of such models) is due to Nakkajima [Na00].Assumption (⋆) also holds for Abelian varieties by the theorem on potential semi-Abelian reduction, see [DM69] and [SGA7].If the expected results on resolution of singularities were known to hold in mixed characteristic, then potential semi-stable reduction would hold for K3 surfaces, and then, Kawamata's semistable MMP in mixed characteristic [Ka94] together with Artin's results [Ar74] on simultaneous resolution of singularities would imply (⋆).Using work of Maulik [Mau14], we have at least the following, and refer to Section 2.1 for details.
Theorem.Let X be a K3 surface over K such that (1) it admits a very ample invertible sheaf L with p > L 2 + 4, or (2) p ≥ 11 and X is a smooth quartic in P 3 K , or (3) p ≥ 7 and X → P 2 K is a double cover branched over a smooth sextic.Then, X satisfies (⋆).K3 surfaces.In this article, we establish a Néron-Ogg-Shafarevich type result for K3 surfaces.Important steps were already taken by the second named author in [Mat14].The main result of this article is the following.
Theorem.Let X be a K3 surface over K with p = 2 that satisfies (⋆).If the G K -action on H 2 ét (X K , Q ℓ ) is unramified for some ℓ = p, then (1) There exists a model of X that is a projective scheme over O K , whose geometric special fiber is a K3 surface with at worst RDP (rational double point) singularities.
(2) There exists a finite and unramified extension L/K such that X L has good reduction over L.
For K3 surfaces over C((t)), a similar result will appear in forthcoming work of Hassett and Tschinkel [HT].As in the case of Abelian varieties in [ST68], we obtain the following independence of ℓ.
Corollary.Let X be a K3 surface over K with p = 2 that satisfies (⋆).Then, the G K -action on H 2 ét (X K , Q ℓ ) is unramified for one ℓ = p if and only if it so for all ℓ = p.
In [ST68], Serre and Tate showed that if an Abelian variety over K of dimension g with p > 2g + 1 has potential good reduction, then good reduction can actually be achieved after a tame extension.Here, we establish the following analog for K3 surfaces.
Corollary.Let X be a K3 surface over K with p ≥ 23 and potential good reduction.Then, good reduction can be achieved after a tame extension.
It is important to note that the conclusion of the theorem cannot be strengthened, as we will show with explicit examples.
Theorem.For every prime p ≥ 5, there exists a K3 surface X = X(p) over Q p , such that (1) the G Qp -action on H 2 ét (X Q p , Q ℓ ) is unramified for all ℓ = p, (2) X has good reduction over the unramified extension Q p 2 , but (3) X does not have good reduction over Q p .
Kuga-Satake Abelian varieties.As an application, we compare the reduction behavior of a K3 surface to that of its Kuga-Satake Abelian variety.Let us recall that Kuga and Satake associated in [KS67] to a polarized K3 surface (X, L) over C a polarized Abelian variety KS(X, L) over C of dimension 2 19 .In case (X, L) is defined over some field k, Rizov [Ri10] and Madapusi Pera [M13] established the existence of KS(X, L) over some finite extension of k.This recalled, we have the following result.
Theorem.Let (X, L) be a polarized K3 surface over K with p = 2.
(1) If X has good reduction, then KS(X, L) can be defined over an unramified extension of K, and it has good reduction.(2) Assume that X satisfies (⋆).Let L/K be a field extension such that KS(X, L) can be defined over L. If KS(X, L) has good reduction over L, then X has good reduction over an unramified extension of L.
Organization.This article is organized as follows: In Section 1, we recall a couple of general facts on models and unramified Galois actions on ℓ-adic cohomology.We also recall the classical Serre-Tate theorem for Abelian varieties, and give explicit examples of curves of genus ≥ 2, where the Galois representation does not detect bad reduction.
In Section 2, we review semi-stable reduction of K3 surfaces, Kawamata's semi-stable MMP in mixed characteristic, the Kulikov-Nakkajima-Pinkham-Persson list, and the the second named author's results on potential good reduction of K3 surfaces in terms of unramified Galois actions.We also discuss shortly the case of Enriques surfaces.
In Section 3, we establish existence and termination of certain semi-stable flops, which we need later on to equip our models with suitable invertible sheaves.
Section 4 is the technical heart of this article: given a finite Galois extension L/K with group G, and a model of X L over O L , we study extensions of the G-action X L to this model.Then, we study quotients of models by G-actions, where the most difficult case arises when p divides the order of G (wild ramification).
In Section 5 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 Section 6, we give explicit examples of K3 surfaces over Q p with unramified Galois-actions on ℓ-adic cohomology that do not have good reduction over Q p , but only after an unramified extension.
Acknowledgements.It is a pleasure for us to thank François Charles, Christopher Hacon, Annabelle Hartmann, Keerthi Madapusi Pera, as well as Davesh Maulik for discussions and comments.The second named author thanks the department of mathematics at the TU München for kind hospitality while visiting the first named author there.The second named author is supported by JSPS KAKENHI Grant Number 26247002.

Notations and Conventions
Throughout the whole article, we fix the following notations K a non-Archimedean local field of characteristic zero O K the ring of integers of K k the residue field, which is finite of characteristic p G K , G k the absolute Galois groups Gal(K/K), Gal(k/k) ℓ a prime different from p If L/K is a field extension, and X is a scheme over K, we abbreviate the base-change X × Spec K Spec L by X L .

Generalities
In this section, we recall a couple of general facts on models, Galois actions on ℓ-adic cohomology, and Néron-Ogg-Shafarevich type theorems.
1.1.Models.We start with the definition of various types of models.
Definition 1.1.Let X be a smooth an proper variety over K.
(1) A model of X over O K is an algebraic space that is flat and proper over Spec O K and whose generic fiber is isomorphic to X.
(2) We say that X has good reduction if there exists a model of X that is smooth over O K .(3) We say that X has semi-stable 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 semi-stable reduction.) (4) We say that X has potential good (resp.semi-stable) reduction if there exists a finite field extension L/K such that X L has good (resp.semi-stable) reduction.
Remark 1.2.Models of curves and Abelian varieties can be treated entirely within the category of schemes, see, for example, [Liu02, Chapter 10] and [BLR].However, even 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 [Mat14, Section 5.2] for explicit examples.In particular, we are forced to work with algebraic spaces when studying models of K3 surfaces.
1.2.Inertia and monodromy.The G K -action on K induces an action on O K and by reduction, an action on k.This gives rise to a continuous and surjective homomorphism G K → 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.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 G K , and that it sits in a short exact sequence (1) More precisely, let π be a uniformizer of O K , and let µ ℓ n be the group of ℓ n .throots of unity.Then, the projection of I K onto its maximal pro-ℓ-quotient Z ℓ (1) induced by (1) can be written down explicitly as follows Next, let X be a smooth and proper variety over K.By functoriality, G K acts on all ℓ-adic cohomology groups H m ét := H m ét (X K , Q ℓ ).More precisely, these actions are quasi-unipotent by Grothendieck's monodromy theorem [SGA7, Exposé I, Théorème 1.2].Thus, there exists a finite field extension L/K such that the induced I L -action on H m ét is unipotent and factors through t ℓ .In particular, there exists a unique nilpotent map N : Then, the kernels of the powers of N give rise to a filtration on H m ét , and this filtration is G K -invariant.
(2) The nilpotent map N is called the monodromy operator, and the induced filtration on H m ét (X K , Q ℓ ) is called the monodromy filtration.
The connection between these two notions is as follows: if the G K -action is unramified, then the monodromy filtration is trivial.Conversely, if the monodromy filtration is trivial, then there exists a finite field extension L/K such that the induced G L -action on H m ét (X K , Q ℓ ) is unramified.For an Abelian variety X, it follows from the results of Serre and Tate [ST68] that the G K -action on H m ét (X K , Q ℓ ) is unramified for one ℓ = p, if and only if it is so for all ℓ = p.In Corollary 5.4, we will show a similar result for K3 surfaces.In general, it is not known whether being unramified depends on the choice of ℓ, but it is expected not to.In any case, a relation between good reduction and unramified Galois-actions on ℓ-adic cohomology is given by the following well-known result, which follows from the proper smooth base change theorem in [SGA4].
is unramified for all m and for all ℓ = p.
In view of this, 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 representation-theoretic criterion to determine whether such a variety admits a model over O K with good reduction.

Abelian varieties.
A classical converse to Theorem 1.4 is the Néron-Ogg-Shafarevich criterion for elliptic curves.Later, Serre and Tate generalized it to Abelian varieties of arbitrary dimension.
Theorem 1.5 (Serre-Tate [ST68]).An Abelian variety A over K has good reduction if and only if the G K -action on H 1 ét (A K , Q ℓ ) is unramified for one (resp.all) ℓ = p.
1.4.Higher genus curves, part 1.Now, the converse to Theorem 1.4 already fails for curves of higher genus.Let X be a curve of genus g ≥ 2 over K.If X has a K-rational point, then there is a natural embedding j : X → Jac(X) of X into its Jacobian, and then, . By the theorem of Serre and Tate, an unramified G K -action on H m ét (X K , Q ℓ ) for all m is equivalent to the good reduction of Jac(X).The following lemma gives a criterion that ensures the latter.
Lemma 1.6.Let X be a curve over K with a K-rational point and assume that it admits a semi-stable scheme model X → O K such that the dual graph associated to the components of its special fiber X 0 is a tree.Then, Jac(X) has good reduction and the G K -action on H m ét (X K , Q ℓ ) is unramified for all m.
Proof.By [BLR, Section 9.2, Example 8], Pic 0 X 0 /k is an Abelian variety, which implies that Jac(X) has good reduction.Since we assumed X to have a K-rational point, the G K -action on H m ét (X K , Q ℓ ) is unramified for all m.
Using this lemma it is easy to produce counter-examples to Néron-Ogg-Shafarevich type results for higher genus curves.
Proposition 1.7.If p = 2, then for every g ≥ 2, there exists a curve X of genus g over K with unramified G K -action on H m ét (X K , Q ℓ ) for all m and all ℓ = p, which does not allow a model with good reduction over any finite extension of K.
Proof.Let X be a hyperelliptic curve of genus g over K, which is one of the examples of [Liu02, Example 10.1.30]with the extra assumptions of [Liu02,Example 10.3.46].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 1.6 are fulfilled, the 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.1.5.Higher genus curves, part 2. If X is a curve of genus ≥ 2, then one can also study the outer Galois-action on its étale fundamental group, which turns out to be rich enough to detect good reduction.More precisely, there exists a short exact sequence of étale fundamental groups For every prime ℓ, this exact sequence gives rise to a well-defined homomorphism from G K to the pro-ℓ-completion of the outer automorphism group of In analogy to Definition 1.3, we will say that this action is unramified if ρ ℓ (I K ) = {1}.We note that the G K -action on H 1 ét (X K , Q ℓ ) arises from the residual action of ρ ℓ on the Abelianization of π ét 1 (X K ).After these preparations, we have the following Néron-Ogg-Shafarevich type theorem for curves of higher genus, which is in terms of étale fundamental groups rather than ℓ-adic cohomology groups.
Theorem 1.8 (Oda [Od95]).Let X be a curve of genus ≥ 2 over K.Then, X has good reduction if and only if the outer Galois-action ρ ℓ is unramified for one (resp.all) ℓ = p.

K3 surfaces and their models
In this section, we first introduce the crucial Assumption (⋆), which ensures the existence of suitable models for K3 surfaces.These models were studied by Kulikov, Nakkajima, Persson, and Pinkham.Then, we show how (⋆) would follow from a combination of potential semi-stable reduction (which is not known in mixed characteristic, but expected) and the semistable MMP in mixed characteristic, and we give some conditions under which (⋆) does hold.After that, we discuss the monodromy filtration in the cohomology of these models and 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 should be known to the experts.
2.1.Kulikov-Nakkajima-Persson-Pinkham models.We first introduce the crucial assumption that we shall make from now on.
Assumption (⋆).A K3 surface X over K satisfies (⋆) if there exists a finite field extension L/K such that X L admits a semi-stable model X → Spec O L (in the sense of Definition 1.1) such that ω X /O L is trivial.
For a family of complex algebraic K3 surfaces over a pointed disk, the analogous assumption always holds, and there, the special fibers of the corresponding models have been classified by Kulikov [Ku77], Persson [P77], and Persson-Pinkham [PP81].The main point why (⋆) is not known to hold is that semi-stable reduction is not known, which is the case since resolution of singularities in mixed characteristic is not known.More precisely, we have the following.
Proposition 2.1.Let X be a K3 surface over K with p ≥ 5. Assume that there exists (1) some finite field extension L ′ /K, and (2) some smooth surface Y over L ′ that is birational to X L ′ , and (3) some scheme model Y → Spec O L ′ of Y with semi-stable reduction.Then, X satisfies (⋆), where we can choose L ⊇ L ′ .
Proof.Let Y → Spec O L ′ be as in the statement.Since p ≥ 5, we apply Kawamata's semistable MMP from [Ka94] that produces a scheme Z → Spec O L ′ with trivial ω Z/O L ′ that is a model of X L ′ , and such that Z is regular outside a finite set Σ of terminal singularities.Outside Σ, this model is already a semi-stable model.From the classification of terminal singularities in [Ka94,Theorem 4.4], it follows that the geometric special fiber (Z 0 ) k is irreducible around points of Σ, and that it acquires RDP singularities in these points.Thus, after some finite field extension L/L ′ , there exists a simultaneous resolution X → Spec O L of these singularities by [Ar74].This X may exist only as an algebraic space, and it satisfies (⋆).
The assumptions of the previous proposition are fulfilled for K3 surfaces over K admitting very ample invertible sheaves L with p > L 2 + 4 by a result of Maulik [Mau14,Section 4].With some extra work the bounds on p can be slightly improved for double sextics, see [Mat14].Thus, we obtain the following corollary.
Corollary 2.2.Let X be a K3 surface over K such that (1) it admits a very ample invertible sheaf L with p > L 2 + 4, or (2) p ≥ 11 and X is a smooth quartic in P 3 K , or (3) p ≥ 7 and X → P 2 K is a double cover branched over a smooth sextic.Then, X satisfies (⋆).
Over C, Kulikov [Ku77], Persson [P77], and Pinkham-Persson [PP81] classified the special fibers of the models asserted by (⋆).We refer to [Mo81, Section 1] for an overview, and to Nakkajima's extension [Na00] of these results to mixed characteristic.For K3 surfaces, the classification is as follows.
Theorem 2.3 (Kulikov, Nakkajima, Persson, Pinkham).Let X be a K3 surface over K, assume that it satisfies (⋆), and let X → Spec O L be a model as asserted by (⋆).Then, the geometric special fiber X 0 := (X 0 ) k is one of the following: (I) X 0 is smooth.(II) X 0 is singular but has no triple points.In this case, its components form a chain, whose interior components are elliptic ruled surfaces and the end components are rational surfaces.The double curves are elliptic.(III) X 0 has triple points.In this case, its components are rational surfaces, and the associated dual graph Γ corresponds to a triangulation of the two-sphere S 2 .

2.2.
A criterion for potential good reduction.Let X be a smooth and proper surface over K, and assume that there exists a finite extension L/K such that X L admits a semi-stable model X → Spec O L .We denote by Y i the components of the geometric special fiber X 0 , and set We denote by Γ the dual graph of the configuration formed by the Y i .By [RZ82, Satz 2.10] and [RZ82, Satz 2.23], there exists a spectral sequence is called the weight filtration.(In case X is merely an algebraic space and not a scheme, the existence of this spectral sequence and its E 2 -degeneration is shown in [Mat14, Proposition 2.2].)For example, W m 0 is isomorphic to the singular cohomology group H m sing (Γ, Z) tensored with Q ℓ , and we refer to [P77, Observation 2.7.1] and [P77, Lemma 2.7.2] for the computation of some more filtration quotients.For example, computing the weight filtration in the three cases in Theorem 2.3, we obtain the following.
Corollary 2.4.Let X be a K3 surface over K, assume that it satisfies (⋆), and let X → Spec O L be a model as asserted by (⋆).Then, the weight filtration on In Section 1.2, we introduced the monodromy filtration on H m ét (X K , Q ℓ ), which arose from the monodromy operator N , which in turn arose from the G K -action.Since the residue field of O K is finite, it follows from the Weil conjectures and Deligne's theory of weights that these two filtrations agree up to a shift (see [It04, Remark 3.2], for example).Now, if X is a K3 surface over K satisfying (⋆), then there exists a finite field extension L/K and a model X → Spec O L of X L as asserted by (⋆).If the G K -action on H 2 ét (X K , Q ℓ ) was unramified to start with, then the weight filtration of X will be trivial, and thus, by Corollary 2.4, the special fiber X 0 will be smooth, which implies that X L has good reduction.Thus, we obtain the following result of the second named author and we refer the interested reader to [Mat14] for details and a complete proof.
Theorem 2.5 (Matsumoto).Let X be a K3 surface over K that satisfies (⋆).If the G K -action on H 2 ét (X K , Q ℓ ) is unramified, then X has potential good reduction.

Enriques surfaces.
The previous theorem does not generalize to other types of surfaces with numerically trivial canonical sheaves.For example, the G K -action on ℓ-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 2.6.Let Y be an Enriques surface over K.Then, there exists a finite extension L/K such that the G L -action on 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 NS(Y L ) = NS(Y K ).But then, the G L -action on NS(Y L ) is trivial, and in particular, unramified on H 2 ét .Moreover, the next example shows that the G K -action on the cohomology of the K3 double cover does not detect potential good reduction, which is related to flower pot degenerations of Enriques surfaces, see [P77, Section 3.3] and [P77, Appendix 2].
Example 2.7.Fix a prime p ≥ 5. Consider P 5 Zp with coordinates x i , y i , i = 0, 1, 2, and inside it the complete intersection of 3 quadrics p satisfies e 2 ≡ 0, 1, 2 mod p (for example, we could take e = 2).Moreover, ı : x i → x i , y i → −y i defines an involution on P 5 Zp , which induces an involution on X .We denote by X the generic fiber of X , and by Y := X/ı the quotient by the involution.
Theorem 2.8.Let p ≥ 5 and let X → Y be as in Example 2.7.Then, Y is an Enriques surface over Q p , such that (1) the K3 double cover X of Y has good reduction, Y has semi-stable reduction of flower pot type, but (4) Y does not have potential good reduction.
Proof.A straight forward computation shows that X is smooth over Q p , and that ı acts without fixed points on X.Thus, X is a K3 surface and Y is an Enriques surface over Q p .The special fiber of X is a non-smooth K3 surface with 4 singularities of type A 1 located at [0 : 0 : 0 : ±1 : ±1 : 1].Then, the blow-up X ′ 1 → X of the Weil divisor (x 0 − ex 1 = x 2 − py 2 = 0) defines a simultaneous resolution of the singularities of X → SpecZ p , and we obtain a smooth model of X over Z p .In particular, X has good reduction over Q p and the G Qp -action on H 2 ét (X Q p , Q ℓ ) is unramified for all ℓ = p.Next, let X ′ 2 → X be the blow-up of the 4 singular points of the special fiber.Then, ı extends to X ′ 2 , and the special fiber is the union of 4 divisors E i with the minimal desingularization X ′ p of the special fiber of X .The fixed locus of ı on X ′ p is the union of the four (−2)-curves of the resolution.Moreover, we can find isomorphisms E i ∼ = P 1 × P 1 such that ı acts by interchanging the two factors.Thus, the quotient X ′ 2 /ı is a model of Y over Z p , whose special fiber is a rational surface X ′ p /ı (a so-called Coble surface) meeting transversally four P 2 's, that is, a semi-stable degeneration of flower pot type (see, [P77, Section 3.3]).
Seeking a contradiction, we assume that Y has potential good reduction.Then, there exists a finite extension L/Q p and a smooth model Y → SpecO L of Y L .Let X 3 → Y → Spec O L be its canonical 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 X 3 .Since X 3 and the base-change of X ′ 1 to O L both are smooth models of X L , they are isomorphic outside a finite number of curves inside their special fibers (this is well-known, and we actually show much more in Proposition 3.3 below).The special fibers are birational, and thus, isomorphic.Moreover, the two specialized involutions agree outside a finite number of curves, and thus, agree in total.However, in one special fiber the involution acts without fixed points, whereas it has four fixed curves in the other, a contradiction.

Existence and termination of flops
Let X be a smooth and proper surface over K with numerically trivial canonical sheaf ω X , and assume that we have a smooth model X → SpecO K .Now, if L is an ample invertible sheaf on X, then its specialization L 0 to the special fiber may not be ample, and not even be big and nef.In this section, we show that there exists a finite sequence of birational modifications (semistable flops) of X , such that we eventually arrive at a smooth model X + → Spec O K of X, such that the restriction of L to the special fiber of X + is big and nef.
3.1.Existence of flops.The following is an adaptation of Kollár's proof [Kol89, Proposition 2.2] of the existence of 3-fold flops over C to our situation, which deals with special semi-stable flops in mixed characteristic.
Proposition 3.1 (Existence of flops).Let X be a smooth and proper surface over K with numerically trivial ω X that has a smooth model X → Spec O K .Assume p = 2.If L is an ample invertible sheaf on X, and C is an integral (not necessarily geometrically integral) curve on the special fiber X 0 with L 0 • C < 0, then there exists a birational and rational map ϕ induces an isomorphism of generic fibers, (3) L + 0 •C + > 0, where L + denotes the extension of L on X + , and where C + denotes the flopped curve.
Proof.Since L is ample, L ⊗n is effective for n ≫ 0, and then, also its specialization L ⊗n 0 to the special fiber X 0 is effective.In particular, L 0 has positive intersection with every ample divisor on X 0 , that is, L 0 is pseudoeffective.Thus, there exists a Zariski-Fujita decomposition on (X 0 ) k where P is nef, and where N is a sum of effective divisors, whose intersection matrix is negative definite, see for example, [Ba01, Theorem 14.14].Since ω X 0 is numerically trivial, the adjunction formula shows that every irreducible curve in N is a 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 X 0 .After these preparations, let C be as in the statement, that is, L 0 • C < 0. In particular, C is contained in the support of N , and the base-change C k ⊂ (X 0 ) k is a disjoint union of ADE curves.Since C 2 < 0, Artin showed that there exists a morphism of projective surfaces over k f 0 : X 0 → X ′ 0 that contracts C and nothing else (see [Ba01, Theorem 3.9], for example).Since C k is a union of ADE-curves, it follows that (X ′ 0 ) k has RDP singularities, which are rational and Gorenstein.Thus, also X ′ 0 has rational Gorenstein singularities.
For all n ≥ 0, we define Since f 0 is a contraction with R 1 f 0, * O X 0 = 0, there exists as blow-down Passing to limits, we obtain a contraction of formal schemes Let w ∈ X ′ 0 be a closed and singular point, and let k ′ be its residue field, which is a finite extension of k.The pre-image of w in (X ′ 0 ) k ′ is a union of closed and Galois-conjugate points, and let w be one of them.On (X ′ 0 ) k , the point w defines a RDP singularity, which is a rational singularity of multiplicity 2. Thus, also w defines a rational singularity of multiplicity 2, which implies that the completion of the local ring is of the form [Li69,Lemma 23.4].Using p = char(k ′ ) = 2 and Hensel's lemma, we may assume after a change of coordinates that the power series h(x, y, z) is of the form z 2 − g(x, y) for some polynomial g(x, y).Therefore, a formal neighborhood of w in X ′ k ′ can be viewed as a deformation of the singularity (2).More precisely, if K ′ /K denotes the unramified extension corresponding to k ′ /k, then a formal neighborhood of w is isomorphic to where G(x, y) is congruent to g(x, y) modulo the maximal ideal of O K ′ (here, we have used again p = 2 to obtain (3) from the the versal deformation of (2)), see also [Ka94,Theorem 4.4].We denote by t ′ the involution induced by z → −z.It is not difficult to see that t ′ induces −id on local Picard groups, see, for example, [Kol89, Example 2.3].Next, we note that K ′ /K is a Galois extension with Gal(K ′ /K) = Gal(k ′ /k), and we use this group to extend the involution t ′ to a Gal(K ′ /K)-invariant involution on a formal neighborhood of the pre-image of w on X ′ K ′ .Being compatible with the Galois-action, we descend t ′ to an involution t on a formal neighborhood of w in X ′ .We note that also t induces −id on local Picard groups.
Let C ⊂ X be the inverse image f −1 (Spf O X ′ ,w ).By formal GAGA, C ⊂ X is algebraizable to a closed subscheme C of X .Now, t induces an involution on C − C, and, as explained in [Kol89, Proposition 2.2], we can glue C and X − C to an algebraic space X + by using t on the overlap.By construction, there exists a birational and rational map which is an isomorphism outside C. From the glueing construction it is clear that X + is a smooth model of X over O K .
Since ϕ is an isomorphism on generic fibers, the transform L + of L on X + is ample on the generic fiber of X + .Next, ϕ is an isomorphism outside C, and since the induced map on local Picard groups around C is −id, it follows that the restriction of L + to X + 0 has positive intersection with the flopped curve C + .

Termination of flops.
Having established the existence of certain semi-stable flops in mixed characteristic, we now show that there is no infinite sequence of them.To do so, we adjust the proof of termination of flops from [KM98, Theorem 6.17] and [KM98, Corollary 6.19] over C to our situation.
Proposition 3.2 (Termination of flops).Let (X, L) and X → Spec O K be as in Proposition 3.1.Let
To prove d(Y, D) < ∞, Kollár and Mori use log resolutions in [KM98], which are not (yet) available in mixed characteristic.Now, we finally come to the proof of Proposition 3.2: since L is ample on X, we may assume, after possibly replacing L by a sufficiently high tensor power, that L ∼ = O X (D) for some irreducible divisor D. By abuse of notation, let us also denote its closure in X by D, which is still an irreducible divisor.
To prove d(X , D) < ∞ in our case, we note that X is regular, and then, the resolution result of Cossart-Jannsen-Saito [CJS13, Corollary 0.4] ensures the existence of a birational morphism f : Y → X such that Y is a regular algebraic space and f * (D) has SNC support.(That result is stated for schemes, but using canonicity, we can easily extend it to the case where X is an algebraic space.) Since ω X /O K is numerically trivial, D-flops and εD-flops are the same for all positive rationals ε, and hence, we can always replace D by εD if needed.Next, by [KM98, Corollary 2.32] (which is applicable since we have a log resolution f : Y → X by the above discussion) we may assume after possibly replacing D by εD for some positive rational number ε that the pair (X , D) is klt.Using the log resolution f : Y → X again, we can argue along the lines of the proof of [KM98, Proposition 2.36] to deduce that if E is an exceptional divisor over X satisfying a(E, X , D) < 1 + totaldiscrep(X , D), then the center of E on Y is a divisor.In particular, there are only finitely many such divisors, and it follows that d(X , D) < ∞.
After these preparations, we follow the proof of [KM98, Theorem 6.17] to deduce Proposition 3.2.
3.3.Flops along smooth models.As an application of existence and termination of flops, Kollár [Kol89, Theorem 4.9] showed that any two birational complex threefolds with Q-factorial terminal singularities and nef canonical classes are connected by a finite sequence of flops.We have the following analog in our situation, but since we are dealing with algebraic spaces rather than projective schemes (which is analogous to the case of analytic threefolds in loc.cit.), we have to allow slightly more general flops than those considered in Proposition 3.1: namely, we do not assume the existence of some ample invertible sheaf L globally on X but only formally locally around the flopping contraction.
Proposition 3.3.Let X be a smooth and proper surface over K with numerically trivial ω X that has good reduction.Assume p = 2.If X i → Spec O K are two smooth models of X, then (1) the special fibers of X 1 and X 2 are isomorphic, and (2) X 1 and X 2 are connected by a sequence of flops.
Proof.The special fibers of X 1 and X 2 are birational by the Matsusaka-Mumford theorem [MM64], and since they are minimal surfaces of Kodaira dimension ≥ 0, they are isomorphic.(Note that this statement also follows from the much more detailed analysis below.)Now, choose an ample invertible sheaf L on X.By Proposition 3.2, there exist finite sequences of flops X i ... Y i , i = 1, 2 such that L restricts to a big and nef invertible on the special fibers of Y i .
To simplify notation, set Z := Y i for some i = 1, 2, and let L 0 be the restriction of L to the special fiber Z 0 .Note that ω Z 0 is numerically trivial.Since L 0 is big and nef, we obtain a proper and birational morphism Base-changing to (Z 0 ) k , the induced morphism ̟ k contracts an integral curve C if and only if it has zero-intersection with 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 (that is, C ∼ = P 1 and C 2 = −2), it follows from the classification of Cartan matrices, that W k has at worst RDP singularities.Now, L ⊗n 0 is of degree 0 on contracted curves for all n, and over k, these curves are ADE curves.Thus, we find R 1 ̟ * L ⊗n i = 0 for all n, which implies ) for all n, and note that the latter term is zero for n ≫ 0 by Serre vanishing.Replacing L by some sufficiently high tensor power will not change ̟, and then, we may assume that H 1 (Z 0 , L ⊗n 0 ) = 0 for all n ≥ 1.If f : Z → Spec O K denotes the structure morphism, then semi-continuity and the previous vanishing result imply R 1 f * L ⊗n = 0 for all n ≥ 1.Thus, global sections of L ⊗n 0 extend to L ⊗n , and since the former is globally generated for n ≫ 0, so is the latter.Thus, we obtain a morphism of algebraic spaces over Spec O Since L is ample on X, π induces an isomorphism of generic fibers.Moreover, we can identify the induced map π 0 on special fibers with ̟ : Z 0 → W from above.
Coming back to our models Y i , we thus obtain birational morphisms that are flopping contractions in the sense of [KM98, Definition 6.10].Now, L is ample on Y ′ i .Moreover, the Y ′ i are normal projective schemes and birational outside a finite number of curves in their special fibers.In fact, there exists a birational rational map between them that is compatible with L. Thus, by [Kov09, Theorem 5.14], this birational map extends to an isomorphism, and then, we obtain a birational map Y 1 Y 2 that is a sequence of flops.(Here, we have to allow more general flops than before: there may exist no invertible sheaf on Y 2 that is relatively ample over Y ′ 2 .However, we can choose a relatively ample invertible sheaf N on the formal completion of Y 2 along the exceptional locus of Y 2 → Y ′ 2 .Pulling back to the corresponding completion of Y 1 , we have existence and termination of flops as in Proposition 3.1 and Proposition 3.2 in this local situation, and by [Kov09, Theorem 5.14], the outcome is isomorphic to Y 2 → Y ′ 2 .We leave the details to the reader.) Putting all these birational modifications together, we have connected X 1 and X 2 by a sequence of flops.

Group actions on models
In this section, we study group actions on models.More precisely, we are given a smooth and proper surface X over K with numerically trivial ω X , a finite field extension L/K, which is Galois with group G, and a smooth proper model X → Spec O L of X L .Then we study the following questions: (1) Does the G-action on X L extend to X ?
(2) If so, is the special fiber (X /G) 0 of the quotient equal to the quotient X 0 /G of the special fiber?It turns out, that the answer to question (1) is "yes", when allowing certain birational modifications of the model, and in question (2), it turns out that the case where p divides the order of G (wild group actions) is subtle.

4.1.
Extending group actions to a possibly singular model.Given a smooth proper surface X over K with numerically trivial ω X that admits a model X with good reduction after a finite Galois extension L/K with group G, we first show that the G-action extends to a (mild) birational modification of X .Proposition 4.1.Let X be a smooth and proper surface over K with numerically trivial ω X .Assume that there exist (1) a finite Galois extension L/K with Galois group G, as well as (2) a smooth model X → Spec O L of X L , and (3) an ample invertible sheaf L on X, whose pull-back to X L restricts to an invertible sheaf on the special fiber X 0 that is big and nef.Then, there exists a proper birational morphism π X π / / X ′ x x q q q q q q q q q q q Spec O L of algebraic spaces over O L , such that (1) The natural G-action on X L extends to X ′ , and is compatible with the G-action on O L .(2) X ′ is a projective scheme over Spec O L .
(3) The generic fibers of X and X ′ are isomorphic via π, whereas the induced morphism on special fibers π 0 : X 0 → X ′ 0 is birational and projective, such that the geometric special fiber (X ′ 0 ) k has at worst RDP singularities.(4) If π is not an isomorphism, then X ′ is not regular.
Proof.Since X is regular, the pull-back of L to X L extends to an invertible sheaf on X .By abuse of notation, we shall denote the pull-back to X L and its extension to X again by L. By assumption, the restriction L 0 of L to the special fiber X 0 is big and nef.Note that ω X 0 is numerically trivial.As explained in the proof of Proposition 3.3, we obtain a morphism of algebraic spaces over Spec O L π : X → X ′ := Proj n≥0 H 0 (X , L ⊗n ) that has all the properties asserted in claim (3) of the proposition.Clearly, X ′ is a projective scheme over SpecO L , and if π is not an isomorphism, then the exceptional locus is non-empty and of codimension 2, which implies that X ′ cannot be regular by van der Waerden purity, see, for example, [Liu02, Theorem 7.2.22].
It remains to establish the G-action: since L is a G-invariant invertible sheaf on X L , we have an induced G-action on H 0 (X L , L ⊗n ) for all n ≥ 0. Let U ⊂ X be an open and dense subspace such that the G-action extends to U .Since X is normal, we may assume that the complement X \U is of codimension ≥ 2 and that it is contained in X 0 .Thus, if s is a global section of L ⊗n over X and σ ∈ G, then σ(s| U ) is a well-defined global section of L ⊗n over U .Since L ⊗n is a reflexive sheaf on a regular algebraic space, σ(s| U ) extends uniquely to a global section of L ⊗n over X .Thus, we obtain a G-action on H 0 (X , L ⊗n ), which gives rise to a G-action on X ′ that is compatible with the G-action on O L , as well as with the natural G-action on X L .Remark 4.2.If p = 2 and if all assumptions of Proposition 4.1 except assumption (3) are satisfied, then Proposition 3.2 shows that there exists another smooth model of X over Spec O L for which all assumptions hold.4.2.Examples, where the action does not extend.In general, it is too much to ask for an extension of the G-action from X L to X (notation as in Proposition 4.1).The following example is typical.
Example 4.3 (Arithmetic 3-fold flop).Consider Q p with p = 2 and set L := Q p (̟), where ̟ 2 = p.Then, L/Q p is Galois with group G = Z/2Z and the non-trivial element of G acts as ̟ → −̟.We equip with the G-action that is the Galois-action on O L , and that is trivial on x, y, z.It is easy to see that the induced G-action on the special fiber X ′ 0 is trivial.Next, we consider the two ideal sheaves I ± := (x, y, z ± ̟) of O X ′ and their blow-ups Then, X ± are regular schemes, X ′ is singular at the closed point (x, y, z, ̟), π ± are both resolutions of singularities, and the exceptional locus is a P 1 in both cases.The ideals I ± are not G-invariant and the G-action on X ′ induces a rational map X + X − , which does not extend to a morphism.In particular, the G-action does not extend from X L to X ± .In fact, X ′ is an arithmetic version of a 3-fold ordinary double point, and the rational map X + X − is an arithmetic version of the classical Atiyah flop.
Even worse, the following example (which is a modification of Example 6.1 below, and rests on examples from [Mat14] and [vL07]) shows that if we have a G-action on a singular model X ′ as in Proposition 4.1, then there may exist resolutions of singularities to which the G-action extends, as well as resolutions to which the G-action does not extend.Moreover, our examples are models of K3 surfaces, that is, such phenomena are highly relevant for our results.Assume p = 2, let k ′ /k be the unique extension of degree 2, let K ′ /K be the corresponding unramified extension, and let G = {1, σ} be the Galois group of both extensions.Next, let X ′ → Spec O K be a proper scheme such that the geometric special fiber (X ′ 0 ) k has only RDP singularities and at least two of them.Set S := Sing X ′ 0 , and assume that all points of S are k ′rational but not all k-rational.Then, G acts non-trivially on S(k ′ ) = S(k).Let us finally assume that there exist two different resolutions of singularities ψ ± : X ± → X ′ , both of which are isomorphisms outside S, and both of which are obtained by blowing up ideal sheaves I ± defined over O K .From this setup, we can produce the announced counter-examples: (1) The Galois action on (X ′ ) O K ′ extends to (X + ) O K ′ , as well as to (X − ) O K ′ .Thus, there do exist resolutions of singularities to which the G-action extends.
(2) On the other hand, for each decomposition S(k ).We note that ψ S 1 ,S 2 is also a resolution of singularities.But now, if S 1 and S 2 are not G-stable, then the G-action on (X ′ ) O K ′ does not extend to X S 1 ,S 2 , but induces an isomorphism from X S 1 ,S 2 to X σ(S 1 ),σ(S 2 ) , where σ ∈ G is the non-trivial element.We now give explicit examples for p ≥ 5 and K Q p .Fix a prime p ≥ 5 and choose an integer d such that d is not a quadratic residue modulo p, and such that d 6 ≡ −2 −4 • 3 −3 mod p (one easily checks that such d exist).We define the polynomial Then, we choose a homogeneous polynomial f ∈ Z[x, y, z, w] of degree 3, such that the following congruences hold Next, we choose homogeneous quadratic polynomials 2g, 2h ∈ Z[x, y, z, w], such that the following congruences hold Finally, we define the quartic hypersurface Zp , and denote by X = X(p) its generic fiber.Then, X is a smooth K3 surface over Q p .The subscheme S = Sing X ′ 0 is given by Fp .Thus, we find 6 RDP singularities on (X ′ 0 ) Fp , all of which are defined over and G = Gal(F p 2 /F p ) acts non-trivially on S(k ′ ), since √ d ∈ F p .Finally, the blow-ups ψ ± : X ± → X ′ of the ideals I ± := (w = g ±ph = 0) are both resolutions of singularities.As explained in the strategy above, this setup yields the desired examples.

4.3.
Extending the inertia action to the smooth model.Despite all these discouraging examples, there are situations, in which the G-action on X L does extend to X , and not merely to a singular model X ′ (notation as in Proposition 4.1).More precisely, we have the following result.
Proposition 4.5.We keep the notations and assumptions of Proposition 4.1.We denote by I G the inertia subgroup of G.
(1) If X is an Abelian surface or a hyperelliptic surface, then the Gaction on X L extends to X .(2) If the I G -action on H 2 ét (X L , Q ℓ ) is trivial, that is, the G-action is unramified, then the I G -action on X L extends to X .
Proof.(1) It follows from the assumptions that X 0 is a smooth and proper surface with numerically trivial ω X 0 , and that it has the same ℓ-adic Betti numbers as X.Thus, by the classification of surfaces (see, for example, [BM2]), also X 0 is Abelian and (quasi-)hyperelliptic, respectively.As seen in the proof of Proposition 4.1, the (geometric) exceptional locus of π is a union of 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 [BM2] and [BM3] that they do not contain any smooth rational curves.In particular, π must be an isomorphism, which implies that the G-action extends to X .
(2) After replacing G with I G , may assume that L/K is totally ramified.To show that the G-action extends, it suffices to show that the σ-action extends for every σ ∈ G. Thus, let σ ∈ G, and after replacing G by the cyclic subgroup generated by σ, we may assume that G is cyclic, say G = Gal(L/K) ∼ = Z/nZ, and generated by σ.
Let U ⊂ X be the maximal open subspace to which the G-action on X L extends.Then, U contains the generic fiber X L , as well as an open dense subscheme of the special fiber X 0 .Let Γ ⊂ X n be the closure of the set {(x, σ(x), . . ., σ n−1 (x)) | x ∈ U } in X n .The group G = Z/nZ acts on X n by permutation of the factors, and this action restricted to Γ L coincides with the natural G-action on X L via X L ∼ = Γ L .
We thus obtain a commutative diagram of G-representations where we omit the coefficients (Tate twists of Q ℓ ) of the ℓ-adic cohomology groups from the notation.The commutativity of the cup product and the cospecialization follows from [Sa03, Lemma 2.17] in case X is a scheme, and in case X is an algebraic space, we reduce to the scheme case by taking an étale covering j : Y → X by a scheme Y and considering the cycle j * Γ.Let π : X → X ′ be as in Proposition 4.1.Let E ⊂ X 0 be the exceptional locus of π 0 : X 0 → X ′ 0 and E α be the irreducible components of E k .Each irreducible component E α is isomorphic to P 1 .Since π 0 is a resolution of singularities, the intersection matrix (E α •E β ) α,β is invertible.In particular, if we are given c α ∈ Q ℓ for α = 1, ..., m, such that m α=1 c α E α • E β = 0 for all β, then c α = 0 for all α = 1, ..., m.
Consider the irreducible components of Γ 0 := (Γ 0 ) k .First, there is the "diagonal" component, that is, the closure of the set {(x, σ(x), . . ., for some α 1 , . . ., α n .From the Künneth formula and the fact that and the remaining components are equal to a point).Hence, if we set We have c i,i,γ,δ = 0 for all i, γ, δ.
We want to show [Γ 0 ] nondiag = 0.For this, we will use the assumption that the G-action on H 2 (X L ) is trivial.Using the commutative diagram (4), we see that the map ), and thus, every element in its image is G-invariant.In particular, for all α and i, the cycle [ (the i.th component is equal to E α and the remaining components are equal to X 0 ).Now, G acts by σ : ] is independent of j for all β.In order to compute its value, we use equation ( 5) and find Since c i,i,γ,δ = 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 α • E β ) twice, we obtain c i,j,γ,δ + c j,i,δ,γ = 0 for all i, j, γ, δ.Thus, [Γ 0 ] nondiag = 0. Now, pr i : Γ → X is a proper birational morphism for all i, where X is regular, and Γ is integral.Thus, by van der Waerden purity (see, [Liu02, Theorem 7.2.22], for example, and note that this result can easily be extended to algebraic spaces), the exceptional locus of pr i is either empty or a divisor.If it was a divisor, it would give rise to a non-diagonal component of Γ 0 , which does not exist by the previous computations.Thus, pr i is an isomorphism for all i, and since the Gal(L/K)-action extends to Γ, this shows that the Gal(L/K)-action extends to X , as desired.
Remark 4.6.We stress that the reason for the extension of the G-action to X rather than X ′ in the case of Abelian and hyperelliptic surfaces is their "simple" geometry: they contain no smooth rational curves.4.4.The action on the special fiber.In the situation of Proposition 4.5, we now want to understand whether the induced G-action on the special fiber X 0 is trivial.Quite generally, if Y is a smooth and proper variety over some field k, then the natural representation is usually neither injective nor surjective.Thanks to Torelli theorems, we have the following exceptions: (1) If Y is a complex Abelian variety, then ρ 1 is injective.(Here, Aut(Y ) denotes the automorphism group as an Abelian variety -translations may act trivially on cohomology.)(2) If Y is a complex K3 surface, then ρ 2 is injective.Using appropriate extensions of these results to arbitrary characteristic, we have the following.Proposition 4.7.We keep the notations and assumptions of Proposition 4.1.Moreover, assume that either (1) X is an Abelian surface and the G-action on (2) X is a K3 surface and the G-action on H 2 ét (X L , Q ℓ ) is unramified.Then, the I G -action on X L extends to X , and the induced I G -action on the special fiber X 0 is trivial.
Proof.We have already shown the extension of the I G -action to X in Proposition 4.5.Moreover, the I G -action on X 0 is k-linear.By assumption, the I G -action on H m ét ((X 0 ) k , Q ℓ ) is trivial for m = 1, 2, respectively.If X is an Abelian surface, then the I G -action on (X 0 ) k is trivial by the injectivity of ρ 1 in arbitrary characteristic, see, for example, [Mu70, Theorem 3 in Section 19].If X is a K3 surface, then the I G -action on (X 0 ) k is trivial by the injectivity of ρ 2 in arbitrary characteristic, see [Og79, Corollary 2.5] and [Ke12, Theorem 1.4].In both cases, the I G -action on (X 0 ) k is trivial, and thus, also the original action on X 0 is trivial.4.5.Tame quotients.Now, in the situation of Proposition 4.5, it is natural to study the quotient X /H and its special fiber, where H is a subgroup of G.We start with the following easy result.
Proposition 4.8.Let X be a smooth and proper variety over K, and let L/K be a finite Galois extension with group G, such that X L admits a smooth model X → Spec O L .Moreover, assume that the natural G-action on X L extends to X .Let H be a subgroup of G such that (1) H is contained in the inertia subgroup of G, (2) H is of order prime to p, and (3) H acts trivially on the special fiber X 0 .Then, (1) the quotient X /H is smooth over Spec O H L , (2) the special fiber of X /H is isomorphic to X 0 .
Proof.First of all, the quotient X /H exists in the category of algebraic spaces [Kn71, Chapter IV.1].Next, let X 0 be the formal completion of X along the special fiber X 0 , which is a formal scheme.If x ∈ X 0 is a closed point, then O X 0 ,x is étale over the localization A := O L y 1 , y 2 m of the restricted power series ring The induced H-action on the residue ring A/(π) ∼ = κ(O L )[y 1 , y 2 ] m is trivial.Thus, replacing y i by 1 |H| σ∈H σ(y i ) for i = 1, 2 (here, we use that p does not divide the order of H) is simply a change of coordinates of A. But then, the H-action on A = O L y 1 , y 2 m is trivial on y 1 and y 2 , and hence From this local and formal description, the smoothness of X /H follows immediately, and we see that the quotient map X → X /H induces an isomorphism of special fibers.
4.6.Wild quotients.Unfortunately, Proposition 4.8 is no longer true if H is a subgroup of the inertia subgroup, whose order is divisible by p.Let us illustrate this with a very instructive example.We refer the interested reader to Wewers's article [We10] for a more thorough treatment of wild actions and their quotients.
and extend the H-action to R by requiring that σ(x) = ζ p−1 p • x.Then, we have R/(̟) ∼ = F p [x], and the induced H-action on the quotient is trivial.On the other hand, we find that is normal, but not regular -this is an arithmetic version of the RDP singularity of type A p−1 .We also find that the special fiber is not reduced.In particular, Proposition 4.8 does not extend to wild actions without extra assumptions.However, let us make two observations, whose significance will become clear in the proof of Proposition 4.10.
(1) The H-action on the special fiber R/(̟) only seems to be trivial, but in fact, it has become infinitesimal.More precisely, if r ∈ R and r denotes its residue class in R/(̟), then the H-action gives rise to a well-defined and non-trivial derivation (2) The augmentation ideal, that is, the ideal of R generated by all elements of the form σ(r) − r is not principal.In fact, it can be generated by the two elements ̟ p x and ̟ p+1 .
Despite this example, we have the following analog of Proposition 4.8 in the wildly ramified case.This main ideas of its proof are due to Király-Lütkebohmert [KL13, Theorem 2] and Wewers [We10, Proposition 3.2].
Proposition 4.10.Let X be a smooth and proper variety over K, and let L/K be a finite Galois extension with group G, such that X L admits a smooth model X → Spec O L .Moreover, assume that the natural G-action on X L extends to X .Let H be a subgroup of G such that (1) H is contained in the inertia subgroup of G, (2) H is cyclic of order p, and (3) H acts trivially on the special fiber X 0 .Then, the H-action induces a global and non-trivial derivation on X 0 or else both of the following two statements hold true (1) the quotient X /H is smooth over O H L , (2) the special fiber of X /H is isomorphic to X 0 .
Proof.First of all, the quotient X /H exists in the category of algebraic spaces [Kn71, Chapter IV.1].Next, we fix once and for all a generator σ ∈ H and a uniformizer π ∈ O L .We use these to define the following: Since O L is a DVR, the ideal J H (O L ) is principal.More precisely, this ideal is generated by y := σ(π) − π, and it is also generated by π N (O L ) .In [KL13], ideals generated by elements of the form σ(x) − x are called augmentation ideals.Also, it is not difficult to see that they do not depend on the choice of generator σ, which justifies the subscript H rather than σ.
Next, let X 0 be the formal completion of X along the special fiber X 0 , which is a formal scheme.For every point x ∈ X 0 , we define If η ∈ X 0 denotes the generic point, then we have the following where the leftmost inequality follows from the triviality of the H-action on X 0 .We distinguish two cases: Let x ∈ X 0 be an arbitrary point and set R := O X 0 ,x and N η := N (O X 0 ,η ).Then, we define a map which is easily seen to be a derivation.Since we have N η < N (O L ), we compute θ(π) = 0, and thus, θ induces a derivation θ : R/πR → R/πR.This globalizes and gives rise to a derivation on the special fiber X 0 .It follows from the definition of N η that this derivation is non-zero at the generic point η ∈ X 0 , whence non-trivial.
Case (II): N (O X 0 ,η ) = N (O L ).Let x ∈ X 0 be an arbitrary point and set R := O X 0 ,x .Then, all inequalities in (6) are equalities, which implies that all inclusions in From this description, we conclude that the natural map This local computation at completions shows that X /H × O H L O L is isomorphic to X , and that the special fiber X 0 of X is isomorphic to the special fiber of X /H.Since X is smooth over O L , X 0 is smooth over the residue field of O L , which implies that also the special fiber of X /H is smooth over the residue field of O H L .But this implies that X /H is smooth over O H L .

The Néron-Ogg-Shafarevich criterion
We now come to the main result of this article, which is a criterion for good reduction of K3 surfaces, similar to the classical Néron-Ogg-Shafarevich criterion for elliptic curves and its generalization to Abelian varieties by Serre and Tate.Then, we give a couple corollaries concerning potential good reduction, and good reduction after a tame extension.Finally, we relate the reduction behavior of a polarized K3 surface to that of its associated Kuga-Satake Abelian variety.
5.1.The criterion.Let us remind the reader of Section 2.1, where we introduced Assumption (⋆) and established it in several cases.
Theorem 5.1.Let X be a K3 surface over K with p = 2 that satisfies (⋆).If the G K -action on H 2 ét (X K , Q ℓ ) is unramified for some ℓ = p, then (1) there exists a model of X that is a projective scheme over O K , and whose geometric special fiber is a K3 surface with at worst RDP singularities.
(2) There exists a finite and unramified extension L/K such that X L has good reduction.
Proof.By Theorem 2.5, there exists a finite Galois extension M/K, say, with group G and possibly ramified, such that there exists a smooth model of X M X → Spec O M .Choose an ample invertible sheaf L on X.Then, by Proposition 3.2, we can replace X by another smooth model of X such that the pull-back of L to X M restricts to an invertible sheaf on X 0 that is big and nef.
Let I G be the inertia subgroup of G.By Proposition 4.5, the I G -action extends to X , and by Proposition 4.7, the induced I G -action on the special fiber X 0 is trivial.
From the short exact sequence (1) in Section 1.2, we obtain a short exact sequence 1 → P → I G → T → 1, where P is the unique p-Sylow subgroup of I G , and where T is cyclic of order prime to p.By definition, P is the wild inertia, and T is the tame inertia.
Being a p-group, P can be written as a successive extension of cyclic groups of order p.Now, the special fiber of X is a K3 surface, and thus, possesses no non-trivial global derivations by the Rudakov-Shafarevich theorem [RS76].Thus, applying Proposition 4.10 inductively, we obtain a smooth algebraic space X /P → Spec O P M with special fiber X 0 , and which is a model of X M P .
Applying Proposition 4.8 to the residual T -action on X /P , we obtain a smooth algebraic space which is a model of X L .Since L is a finite and unramified extension of K, this establishes claim (2).
The pull-back of L to X /I G is still ample on the generic fiber and big and nef when restricted to the special fiber.By Proposition 4.1, there exists a birational morphism over Spec O that is an isomorphism on generic fibers, such that the geometric special fiber Y 0 is a K3 surface with at worst RDP singularities, and such that the H := Gal(L/K)-action on X L extends to Y. Since L/K is unramified, the morphism Spec O L → Spec O K is étale, from which it follows that the quotient Y → Y/H is étale.Thus, Y/H is a projective scheme over O K , whose generic fiber is X and whose geometric special fiber is a K3 surface with at worst RDP singularities.This establishes claim (1).
Remark 5.2.Conclusion (2) of Theorem 5.1 cannot be strengthened: in the next section, we will give examples of K3 surfaces X over Q p with unramified G Qp -actions on H 2 ét (X Q p , Q ℓ ) that do not admit smooth models over Z p .
Remark 5.3.Unlike curves and Abelian varieties, even if a K3 surface has good reduction over L, then a smooth model of X L over O L need not be unique.However, by Proposition 3.3, the special fibers of all smooth models are isomorphic, and the models themselves are connected by finite sequences of flops.
If a smooth variety over K has good reduction over an unramified extension, then the G K -actions on H m ét (X K , Q ℓ ) are unramified for all m and for all ℓ = p by Theorem 1.4.Thus, as in the case of Abelian varieties in [ST68], we obtain the following independence of the auxiliary prime ℓ.
Corollary 5.4.Let X be a K3 surface over K with p = 2 that satisfies (⋆).Then, the G K -action on H 2 ét (X K , Q ℓ ) is unramified for one ℓ = p if and only if it is unramified for all ℓ = p.
We remark that this independence of ℓ can be also derived from the weaker criterion of [Mat14] (Theorem 2.5), combined with Ochiai's independence of traces [Oc99, Theorem B].
We leave the following easy consequence of Theorem 5.1 to the reader.
Corollary 5.5.Let X be a K3 surface over K such that the image of inertia If a g-dimensional Abelian variety over K with p > 2g + 1 has potential good reduction, then good reduction can be achieved over a tame extension of K by [ST68, Corollary 2 of Theorem 2].We have the following analog for K3 surfaces: Corollary 5.6.Let X be a K3 surface over K with potential good reduction.If p ≥ 23, then good reduction can be achieved after a tame extension.
Proof.The idea of proof is the same as for Abelian varieties in [ST68], we only adjust the arguments to our situation: since X is projective, there exists an ample invertible sheaf L defined over K, and then, its Chern class c 1 (L) gives rise to a G K -invariant class in H 2 ét (X K , Z ℓ ).Let T 2 ℓ be the orthogonal complement of c 1 (L) with respect to the Poincaré duality pairing.For ℓ = p, we let ρ ℓ : G K → GL T 2 ℓ be the induced ℓ-adic Galois representation, and denote by As usual, we denote by I K (resp., P K ) the inertia (resp., wild inertia) subgroup of G K .Since X has potential good reduction, ρ ℓ (I K ) is a finite group.Moreover, if ℓ is odd, since ker red ℓ has no non-trivial element of finite order, ρ ℓ (I K ) is isomorphic to red ℓ • ρ ℓ (I K ) via red ℓ .Now, suppose that ρ ℓ (P K ) is non-trivial.Then, the order of red ℓ • ρ ℓ (I K ) is divisible by p for all odd ℓ.In particular, if we set n := rank T 2 ℓ = 21, then p divides the order for all odd ℓ.By Dirichlet's theorem on arithmetic progressions, there exist infinitely many primes ℓ such that the residue class of ℓ modulo p generates the group F × p , which is of order p − 1. Choosing such an ℓ, we obtain the estimate p − 1 ≤ 21. (When working directly with H 2 (X K , Z ℓ ) instead of the primitive cohomology group T 2 ℓ , we only get the estimate p − 1 ≤ 22, which includes the prime p = 23.)Thus, if p ≥ 23, then ρ ℓ (P K ) is trivial.But then, also the P K -action on H 2 ét (X K , Z ℓ ) is trivial.Thus, there exists a tame extension L/K such that the G L -action on H 2 ét (X L , Q ℓ ) is unramified.By Theorem 5.1, there exists an unramified extension of L ′ /L such that X L ′ has good reduction.In particular, X has good reduction after a tame extension of K. 5.2.Kuga-Satake varieties.Given a polarized K3 surface (X, L) over C, Kuga and Satake [KS67] associated a polarized Abelian variety, the Kuga-Satake Abelian variety A := KS(X, L), which is of dimension 2 19 .Although their construction is transcendental, Rizov [Ri10] and Madapusi Pera [M13] extended the Kuga-Satake construction to arbitrary fields: namely, if (X, L) is a polarized K3 surface over some field k, then KS(X, L) exists over some finite extension of k.This recalled, we have the following relation between good reduction of (X, L) and KS(X, L).
(1) If X has good reduction, then KS(X, L) can be defined over an unramified extension of K, and it has good reduction.(2) Assume that X satisfies (⋆) and let L/K be a field extension such that KS(X, L) can be defined over L. If KS(X, L) has good reduction, then X has good reduction over an unramified extension of L. Proof.
(1) The pair (X, L) gives rise to a morphism SpecK → M • 2d , where M • 2d denotes the moduli space of primitively polarized K3 surfaces of degree 2d := L 2 .By assumption, there exists a smooth model of X over O K , and by Proposition 3.2, there even exists a smooth model X of X over O K , such that the restriction of L to the special fiber is big and nef.Thus, the morphism Spec K → M • 2d extends to a morphism Spec O K → M 2d , where M 2d denotes the moduli space of primitively quasi-polarized K3 surfaces of degree 2d.Passing to an unramified extension L/K of degree ≤ 2 if necessary, the previous classifying morphism extends to a morphism Spec O L → M 2d , see [M13, Section 4].Composing with the morphism M • 2d → S(Λ d ) from [M13, Proposition 4.7], we obtain a morphism Spec O L → S(Λ d ).We recall from [M13, Section 3] that there exists a finite and étale cover S(Λ d ) → S(Λ d ), such that the Kuga-Satake Abelian scheme is a relative Abelian scheme over S(Λ d ).Thus, after replacing L by a finite and unramified extension if necessary, we can lift the latter morphism to a morphism Spec O L → S(Λ d ).Thus, we obtain a Kuga-Satake Abelian variety KS(X, L) over L that has good reduction, and where L is an unramified extension of K.
(2) By assumption, KS(X, L) is defined over L and has good reduction over L. Thus, the G L -action on H 1 ét (KS(X, L) L , Q ℓ ) is unramified.By the usual properties of the Kuga-Satake construction, there exists a G Lequivariant embedding P 2 ℓ (X K , Q ℓ ) → End H 1 ét (KS(X, L) L , Q ℓ ) , where P 2 ℓ denotes the orthogonal complement of c 1 (L) inside H 2 ét (X K , Q ℓ ).This implies that also the G L -action on P 2 ℓ is unramified.Since L is defined over K, the G L -action on the Q ℓ -subvector space generated by c 1 (L) inside H 2 ét (X K , Q ℓ ) is trivial.From this, we conclude that the G L -action on H 2 ét (X K , Q ℓ ) is unramified.By Theorem 5.1, X has good reduction over an unramified extension of L.
Remark 5.8.If (X, L) is a polarized K3 surface with good reduction, then the previous theorem asserts that KS(X, L) can be defined over an unramified extension L of K. Thus, if KS(X, L) can be descended to some field K ′ with K ⊆ K ′ ⊆ L (so far, not much is known about fields of definition of Kuga-Satake Abelian varieties), then, since L/K ′ is unramified and since KS(X, L) has good reduction over L by the previous theorem, the descended Abelian variety will have good reduction over K ′ by [ST68].

Counter-examples
In this final section we give examples of K3 surfaces X over Q p for all p ≥ 5 with unramified G Qp -action on H 2 ét (X Q p , Q ℓ ) that do not have good reduction over Q p .In particular, the unramified extension from Theorem Theorem 6.2.Let p ≥ 5 and let X and X be as in Example 6.1.Then, X is a smooth K3 surface over Q p , such that (1) the G Qp -action on H 2 ét (X Q p , Q ℓ ) is unramified for all ℓ = p, (2) X is a projective model of X over Z p , whose geometric special fiber is a K3 surface with RDP singularities of type A 1 , (3) X has good reduction over the unramified extension Q p [ √ c], (4) X does not have good reduction over Q p .
Proof.Smoothness of X follows from considering the equations over Z, reducing modulo 2 and checking smoothness there.Claims (2) and (3) are straight-forward computations, and since X has good reduction after an unramified extension, also claim (1) follows.We refer to [Mat14, Section 5.3] for computations and details.
To show claim (4), we argue by contradiction, and assume that there exists a smooth and proper algebraic space Z → Spec Z p with generic fiber X.Since the generic fibers of X and Z are isomorphic, such an isomorphism extends to a birational, but possibly rational map α : Z X .
Next, let L be an ample invertible sheaf on X , for example, the restriction of O(1) from the ambient P 3 Zp .Restricting L to the generic fiber X η , and pulling it back via α, we obtain an ample invertible sheaf α * η (L η ) on Z η .Since Z is smooth over Spec O K , this invertible sheaf on Z η extends uniquely to an invertible sheaf on Z that we denote by M. By Proposition 3.2, there exists a rational and birational map where Z + is another model of X with good reduction, and such that the transform M + of M on Z + is ample on the generic fiber, and big and nef on the special fiber.We denote by α + : Z + X the composition α • ϕ −1 .Then, is a birational morphism that contracts precisely those curves on the special fiber Z + 0 that have zero-intersection with M + 0 , and nothing else.By construction, we have (α + η ) * L ∼ = M + and thus, by [Kov09, Theorem 5.14], there exists an isomorphism (Z + ) ′ → X over Spec Z p .
Thus, we have shown that the model X admits a simultaneous resolution α + : Z + → X of singularities over Z p .But then, let x ∈ X 0 be an F p -rational singular point, for example, the point x = w = y + z = 0.Then, let O X ,x be the strict local ring, and denote by Cl(O X ,x ) its Picard group.Then, α + induces a G Qp -equivariant surjection (R 1 α + * O * Z + ) x → Cl(O X ,x ).However, this is impossible for the following reason: (1) The group (R 1 α + * O * Z + ) x is generated by the class of the exceptional curve, which is F p -rational, and thus the G Qp -action on it is trivial.
(2) The G Qp -action on Cl(O X ,x ) is non-trivial.More precisely, if we define the following ideals of O X ,x This contradiction shows that X does not have good reduction over Q p , and establishes claim (4).
be a sequence of semi-stable flops obtained by flopping some integral curve that has negative intersection with the transform of L in every step (which is possible by Proposition 3.1).Then, this process stops after a finite number of steps with some smooth model of X (X +∞ , L +∞ ) → Spec O K , such that the specialization L +∞ 0 is big and nef.Proof.Let us first extend the notion of discrepancy for divisors to algebraic spaces: let Y be a normal and Q-factorial algebraic space over O K , and let D be a Q-divisor on Y.Given a birational morphism Z → Y from a normal algebraic space Z and a prime divisor E ⊂ Z, we define the discrepancy a(E, Y, D) to be the discrepancy a(E U , U, D U ) (in the sense of schemes [KM98, Definition 2.22]), where g : U → Y is an étale covering by a scheme, D U = g * (D), h : W → Z × Y U is a birational projective (hence surjective) morphism from a scheme (which exists by an algebraic space version of Chow's lemma [Kn71, Theorem IV.3.1]), and E U is a prime divisor of W mapping to E by (g • h) * .This so defined rational number is independent of the choices of g, h, and E U , since discrepancies are local for the étale topology.This allows us to define the (total) discrepancy of the pair (Y, D) as in [KM98, Section 2] discrep(Y, D) := inf E {a(E, Y, D) : E is an exceptional divisor over Y} totaldiscrep(Y, D) := inf E {a(E, Y, D) : E is a divisor over Y} The pair is called canonical if discrep(Y, D) ≥ 0. Also, a pair (Y, D), where D = i a i D i for prime divisors D i and rational numbers a i , is called klt (Kawamata log terminal) if a i < 1 for all i, and discrep(Y, D) > −1.Let us also recall [KM98, Definition 6.20]: if (Y, D) is a canonical pair, where D = i D i for prime divisors D i and rational numbers a i , we set a := max{a i }, S := i a i Z ≥0 ⊂ Q (if D = 0, we set a := 0 and S := {0}), and define d(Y, D) := ξ∈S,ξ≥a

Example 4. 4 .
Before giving explicit examples, let us explain the strategy: 5.1 needed to obtain good reduction, may be non-trivial.The examples in question already appeared in [Mat14, Section 5.3], and rest on examples due to van Luijk [vL07, Section 3].Example 6.1.Fix a prime p ≥ 5. We choose integers a, c such that a ≡ 0, 27 16 mod p, such that c ≡ 1 mod 8, and such that c is not a quadratic residue modulo p.Then, we choose a homogeneous polynomial f ∈ Z[x, y, z, w] of degree 3, such that the following congruences holdf ≡ φ mod 2 f ≡ x 3 + y 3 + z 3 + aw 3 mod p,where φ is as in Example 4.4.Finally, we define the quartic hypersurfaceX := X (p) := wf + pz 2 + xy + 2 = 0 ⊂ P 3 Zp ,and denote by X = X(p) its generic fiber.
where ζ p is a primitive p.th root of unity.Then, π := 1 − ζ p is a uniformizer in O K , and the residue field is F p , see [Was82, Lemma 1.4], for example.Let L be the finite extension K[̟], where ̟ := p √ π.Then, ̟ is a uniformizer in O L , and the residue field is F p , that is, L/K is totally ramified.By Kummer theory, L/K is Galois with group H ∼ = Z/pZ.More precisely, there exists a generator σ ∈ H such that σ