Supersingular reduction of Kummer surfaces in residue characteristic $2$

Yuya Matsumoto Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan [email protected] [email protected]

(Date: 2023/02/19)

Abstract.

Given an abelian surface $A$ , defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\operatorname{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$ .

2010 Mathematics Subject Classification:

14J28 (Primary) 14L15, 14J17 (Secondary)

This work was supported by JSPS KAKENHI Grant Number 16K17560 and 20K14296.

1. Introduction

Let $K$ be a Henselian discrete valuation field of characteristic $0$ with valuation ring $\mathcal{O}$ , maximal ideal $\mathfrak{p}$ , and residue field $k$ assumed to be perfect. We say that a K3 surface or an abelian surface $X$ over $K$ has good reduction if there exists an algebraic space $\mathcal{X}$ over $\mathcal{O}$ , proper and smooth, whose generic fiber $\mathcal{X}\otimes_{\mathcal{O}}K$ is isomorphic to $X$ . We say that $X$ has potential good reduction if there exists a finite extension $K^{\prime}/K$ such that the base change $X_{K^{\prime}}$ has good reduction.

The Néron–Ogg–Shafarevich criterion (Serre–Tate [Serre--Tate]*Theorem 1) shows that good reduction of an abelian variety $A$ is equivalent to the unramifiedness of the Galois action on its $l$ -adic cohomology group $H_{\mathrm{\acute{e}t}}^{1}(A_{\overline{K}},\mathbb{Q}_{l})$ . We ([Matsumoto:goodreductionK3]*Theorem 1.1, [Liedtke--Matsumoto]*Theorem 1.3) showed a similar criterion for K3 surfaces (using $H_{\mathrm{\acute{e}t}}^{2}$ ), assuming the existence of semistable models with certain properties and the residue characteristic being large enough.

In this paper we focus on Kummer surfaces. For an abelian surface $A$ over a field $F$ , the Kummer surface $\operatorname{Km}(A)$ is defined to be the minimal resolution of $A/\{\pm 1\}$ . It is known that $\operatorname{Km}(A)$ is a K3 surface if and only if $\operatorname{char}F\neq 2$ or $A$ is non-supersingular (Proposition 3.1). From this geometric nature of Kummer surfaces, the following question arises naturally.

Question 1.1.

Let $A$ be an abelian surface over the discrete valuation field $K$ , and let $X=\operatorname{Km}(A)$ be the attached Kummer surface, which is a K3 surface since we are assuming $\operatorname{char}K=0$ . Assume $A$ has good reduction, with proper smooth model (Néron model) $\mathcal{A}$ over $\mathcal{O}$ .

(1)

Does $X=\operatorname{Km}(A)$ have potential good reduction?
(2)

Are smooth proper models of $X$ related to $\mathcal{A}$ in a geometric way?
(3)

Is the special fiber $\mathcal{X}_{k}$ isomorphic to $\operatorname{Km}(\mathcal{A}_{k})$ ?

Let us say that we are in the extreme case if $\mathcal{A}_{k}$ is supersingular of characteristic $2$ , and in the non-extreme case if otherwise (i.e. if $\operatorname{char}k\neq 2$ , or $\operatorname{char}k=2$ and $\mathcal{A}_{k}$ is non-supersingular). If we are in the non-extreme cases, then it is known that the all questions have essentially affirmative answers (Proposition 3.4): Roughly, one can blow-up the “singularity” of the quotient $\mathcal{A}/\{\pm 1\}$ of the Néron model $\mathcal{A}$ of $A$ to obtain a smooth proper model of $X=\operatorname{Km}(A)$ .

If we are in the extreme case, then the above construction does not give a smooth proper model of $X=\operatorname{Km}(A)$ , mainly because the singularity of $\mathcal{A}_{k}/\{\pm 1\}$ is non-rational. In this case $\operatorname{Km}(\mathcal{A}_{k})$ is a rational surface, and (3) cannot hold. What we prove in this paper is that (1) and (2) holds even in this extreme case.

Theorem 1.2.

Let $K$ and $k$ be as above. Suppose that $\operatorname{char}K=0$ and $\operatorname{char}k=2$ , that an abelian surface $A$ over $K$ has good reduction with Néron model $\mathcal{A}$ , and that the special fiber $\mathcal{A}_{k}$ is supersingular. Then $X=\operatorname{Km}(A)$ has potential good reduction.

We overview the method of our proof and its consequences. We give (in Section 4.3) a proper birational morphism $\phi\colon\tilde{\mathcal{X}}\to\mathcal{A}/\{\pm 1\}$ , defined as the composite of explicit normalized blow-ups at singular points of the special fiber, from a proper model $\tilde{\mathcal{X}}$ of $X$ that is strictly semistable in the broad sense (Definition 4.1). Then, according to Theorem 4.2, we can run a relative MMP and obtain a birational map $f\colon\tilde{\mathcal{X}}\dashrightarrow\mathcal{X}$ to a smooth proper model $\mathcal{X}$ of $X$ , which achieves good reduction.

The special fiber $\mathcal{X}_{k}$ of $\mathcal{X}$ is not isomorphic, nor even birational, to the special fiber $\mathcal{A}_{k}/\{\pm 1\}$ of $\mathcal{A}/\{\pm 1\}$ , since the latter is a rational surface. Hence the birational map $f\circ\phi^{-1}$ (resp. its inverse $\phi\circ f^{-1}$ ) contracts $\mathcal{A}_{k}/\{\pm 1\}$ (resp. $\mathcal{X}_{k}$ ). We may understand that the “K3-ness” of the special fiber $\mathcal{A}/\{\pm 1\}$ lies in the local ring at the singular point of $\mathcal{A}_{k}/\{\pm 1\}$ , not on the smooth complement of the point in $\mathcal{A}_{k}/\{\pm 1\}$ .

Moreover, our construction shows that the special fiber $\mathcal{X}_{k}$ is birational to the quotient of a certain rational surface $A^{\prime}$ by a certain group scheme of length $2$ (see Remark 4.7). We can modify $A^{\prime}$ and make it an “abelian-like” surface, by which we mean a surface sharing numerical properties with abelian surfaces. Thus $\mathcal{X}_{k}$ can be viewed as an example of an inseparable analogue of Kummer surfaces. We will study this subject in another paper.

The proof of Theorem 4.2 is essentially given in [Matsumoto:goodreductionK3]*Section 3, except that we use Takamatsu–Yoshikawa’s MMP [Takamatsu--Yoshikawa:mixed3fold]*Theorem 1.1 (which is applicable in the case of residue characteristic $p=2$ ) in place of Kawamata’s (which requires $p\geq 5$ ).

2. Preliminaries

2.1. Some elliptic double points

In this subsection, we work over an algebraically closed field $k$ of characteristic $2$ .

Definition 2.1.

For $(m,n)=(4,4),(2,8),(2,6)$ and $r=0,1$ , we say that $k[[x,y,z]]/(z^{2}+rzx^{m/2}y^{n/2}+x^{m+1}+y^{n+1})$ is $\mathrm{EDP}_{m,n}^{(r)}$ .

Remark 2.2.

The following properties of $\mathrm{EDP}_{m,n}^{(r)}$ hold.

•

They are all elliptic double points.
•
The exceptional divisor of the minimal resolution of $\mathrm{EDP}_{m,n}^{(r)}$ is independent of $r$ and consists of:
- –
  
  one cuspidal rational curve of arithmetic genus $1$ for $\mathrm{EDP}_{2,6}^{(r)}$ ;
- –
  
  smooth rational curves, with the dual graph shown in Figure 1, for $\mathrm{EDP}_{2,8}^{(r)}$ and $\mathrm{EDP}_{4,4}^{(r)}$ .
In the figures, a vertex with $-m$ inside denotes a smooth rational curve of self-intersection $-m$ , and the absence of a number denotes self-intersection $-2$ . The symbols $\raisebox{-0.9pt}{\small19}⃝_{0}$ and $\raisebox{-0.9pt}{\small4}⃝_{0,1}^{1}$ are due to Wagreich [Wagreich:ellipticsingularities]*Theorem 3.8.
•

$\mathrm{EDP}_{m,n}^{(1)}$ is a $\mathbb{Z}/2\mathbb{Z}$ -quotient singularity by [Artin:wild2]*Theorem, and $\mathrm{EDP}_{m,n}^{(0)}$ is an $\alpha_{2}$ -quotient singularity by [Matsumoto:k3alphap]*Theorem 3.8.

Remark 2.3.

We have alternative equations for $\mathrm{EDP}_{2,8}^{(r)}$ and $\mathrm{EDP}_{4,4}^{(r)}$ .

•

$\mathrm{EDP}_{2,8}^{(r)}$ is isomorphic to $k[[x,y,z]]/(z^{2}+rzx^{2}(x+y^{2})+x(x+y^{2})^{2}+x^{4}y)$ . Indeed, starting from this equation and letting $x^{\prime}=x+y^{2}$ , $z^{\prime}=z+x^{\prime}y$ , $x^{\prime}=(x^{\prime\prime}+ry^{5})(1+rz)$ , we obtain $z^{\prime 2}u_{1}+rz^{\prime}x^{\prime\prime}y^{4}u_{2}+x^{\prime\prime 3}u_{3}+y^{9}u_{4}=0$ for some units $u_{i}$ , and by replacing $x^{\prime\prime},y,z^{\prime}$ with suitable unit multiples we obtain the equation in Definition 2.1.
•

$\mathrm{EDP}_{4,4}^{(r)}$ is also isomorphic to $k[[x,y,z]]/(z^{2}+rzx^{2}y^{2}+x^{4}y+xy^{4})$ . Indeed, starting from the equation of Definition 2.1, letting $\zeta$ be a primitive $5$ -th root of $1$ and $v=x+\zeta y$ and $w=\zeta x+y$ , we have $x^{5}+y^{5}=(\zeta^{2}+\zeta^{3})(v^{4}w+vw^{4})$ , and so on.

The EDPs with $(m,n)=(4,4),(2,8)$ are relevant to us because of the following. (The remaining one with $(m,n)=(2,6)$ is used to describe possible degenerations.)

Proposition 2.4 (Katsura [Katsura:Kummer2]).

Suppose $A$ is a supersingular abelian surface in characteristic $2$ . If $A$ is superspecial (resp. non-superspecial), then the singularity of $A/\{\pm 1\}$ consists of one point of type $\mathrm{EDP}_{4,4}^{(1)}$ (resp. $\mathrm{EDP}_{2,8}^{(1)}$ ).

Here, a supersingular abelian surface is called superspecial if it is isomorphic (not only isogenous) to the product of two supersingular elliptic curves.

Proof.

In the superspecial case, this is [Katsura:Kummer2]*Proposition 8. In the non-superspecial case, [Katsura:Kummer2]*Lemma 12 gives an equation similar to the one given in Remark 2.3. ∎


$\mathrm{EDP}_{4,4}^{(r)}$ : $\raisebox{-0.9pt}{\small19}⃝_{0}$		$\mathrm{EDP}_{2,8}^{(r)}$ : $\raisebox{-0.9pt}{\small4}⃝_{0,1}^{1}$

Figure 1. Dual graphs of minimal resolutions of EDPs

2.2. Quotients by group schemes of length $2$ in mixed characteristic

We recall Tate–Oort’s classification [Tate--Oort:groupschemes]*Theorem 2 of finite group schemes $G$ of length $2$ . For our purposes it suffices to consider $G$ over a base ring $\mathcal{O}$ that is a PID (possibly a field). Then, such group schemes are of the form $G_{\alpha,\beta}$ , defined as follows, with parameters $\alpha,\beta\in\mathcal{O}$ satisfying $\alpha\beta=2$ (which are denoted by $a,b$ in [Tate--Oort:groupschemes]).

The underlying scheme is $G_{\alpha,\beta}=\operatorname{Spec}R$ , $R=\mathcal{O}[X]/(X^{2}-\alpha X)$ . The comultiplication map is given by $R\to R\otimes_{\mathcal{O}}R\colon X\mapsto X\otimes 1+1\otimes X-\beta X\otimes X$ . Actions of $G_{\alpha,\beta}$ on affine schemes $\operatorname{Spec}S$ correspond, via $S\to R\otimes_{\mathcal{O}}S\colon s\mapsto 1\otimes s+X\otimes\delta(s)$ , to $\mathcal{O}$ -linear maps $\delta\colon S\to S$ satisfying the conditions $\delta(st)=\delta(s)t+s\delta(t)+\alpha\delta(s)\delta(t)$ and $\delta(\delta(s))=-\beta\delta(s)$ . In particular, $\mathrm{id}+\alpha\delta$ is an automorphism of the $\mathcal{O}$ -algebra $S$ of order dividing $2$ .

For a unit $e\in\mathcal{O}^{*}$ , the group schemes $G_{\alpha,\beta}$ and $G_{e\alpha,e^{-1}\beta}$ are isomorphic under the obvious maps.

Example 2.5.

Three important cases of $G_{\alpha,\beta}$ are the following.

•

If $(\alpha,\beta)=(1,2)$ , then $G_{\alpha,\beta}$ is isomorphic to the constant group scheme $\mathbb{Z}/2\mathbb{Z}$ . The map $\delta$ is equal to $g-\mathrm{id}$ , where $g$ is (the action on $S$ of) the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$ .
•

If $(\alpha,\beta)=(2,1)$ , then $G_{\alpha,\beta}$ is isomorphic to $\mu_{2}$ . Actions on $\operatorname{Spec}S$ also correspond to $\mathbb{Z}/2\mathbb{Z}$ -gradings $S=\bigoplus_{i\in\mathbb{Z}/2\mathbb{Z}}S_{i}$ on the $\mathcal{O}$ -algebra $S$ , and then $\delta$ is equal to $(-1)$ times the projection to $S_{1}$ . If moreover $2=0$ in $\mathcal{O}$ , then the maps $\delta$ are precisely the derivations of multiplicative type.
•

If $(\alpha,\beta)=(0,0)$ (which implies $2=0$ in $\mathcal{O}$ ), then $G_{\alpha,\beta}$ is isomorphic to $\alpha_{2}$ . The maps $\delta$ are precisely the derivations of additive type.

If $\mathcal{O}$ is a field of characteristic $2$ , then any $G$ is isomorphic to exactly one of above.

If $2$ is invertible in $\mathcal{O}$ , then any $G$ is isomorphic to both $\mathbb{Z}/2\mathbb{Z}$ and $\mu_{2}$ .

The fixed locus $\operatorname{Fix}(G)$ of an action on $\operatorname{Spec}S$ is the closed subscheme, or the closed subset, corresponding to the ideal of $S$ generated by $\operatorname{Im}(\delta)$ . In other words, its complement is the largest open subscheme where $G$ acts freely.

Proposition 2.6.

Let $\mathcal{O}$ be a local PID (possibly a field) with maximal ideal $\mathfrak{p}$ and residue field $k$ . Let $G=G_{\alpha,\beta}$ be a group scheme of length $2$ over $\mathcal{O}$ , acting on $\mathcal{O}[[u,v]]$ , and assume $\operatorname{Fix}(G)\supset(u=v=0)$ (as subsets of $\operatorname{Spec}\mathcal{O}[[u,v]]$ ) and $\operatorname{Fix}(G_{k}\curvearrowright k[[u,v]])=(u=v=0)$ (as subsets of $\operatorname{Spec}k[[u,v]]$ ). Then, the elements

$\displaystyle x$	$\displaystyle:=u(u+\alpha\delta(u)),$	$\displaystyle a$	$\displaystyle:=\beta u+\delta(u),$
$\displaystyle y$	$\displaystyle:=v(v+\alpha\delta(v)),$	$\displaystyle b$	$\displaystyle:=\beta v+\delta(v),$
$\displaystyle z$	$\displaystyle:=\beta uv+\delta(u)v+u\delta(v),$

are $G$ -invariant, and the invariant subalgebra $\mathcal{O}[[u,v]]^{G}$ is equal to $\mathcal{O}[[x,y,z]]/(F)$ , $F=z^{2}-\alpha abz+a^{2}y+b^{2}x-\beta^{2}xy$ .

Suppose $\operatorname{char}(\operatorname{Frac}\mathcal{O})=0$ and $\operatorname{char}k=2$ . Let $\tau$ be the Tyurina number of $k[[x,y,z]]/(F)$ . If $G\otimes_{\mathcal{O}}k=\alpha_{2}$ , then $\tau=2\deg\operatorname{Fix}(G\curvearrowright k[[u,v]])=2\deg\operatorname{Fix}(G\curvearrowright(\operatorname{Frac}\mathcal{O})\otimes_{\mathcal{O}}\mathcal{O}[[u,v]])$ . If $G\otimes_{\mathcal{O}}k=\mu_{2}$ , then $\tau=2$ and $\deg\operatorname{Fix}(G\curvearrowright k[[u,v]])=\deg\operatorname{Fix}(G\curvearrowright(\operatorname{Frac}\mathcal{O})\otimes_{\mathcal{O}}\mathcal{O}[[u,v]])=1$ .

If $G_{\alpha,\beta}=G_{1,2}=\mathbb{Z}/2\mathbb{Z}$ , then the formulas simplify to $x=ug(u)$ , $y=vg(v)$ , $z=ug(v)+g(u)v$ , $a=u+g(u)$ , $b=v+g(v)$ , where $g$ is the non-trivial element of the group.

Proof.

It is straightforward to check that $x,y,z,a,b$ are invariant. Letting $z^{\prime}:=z+\alpha\delta(u)\delta(v)$ , we observe $z+z^{\prime}=\alpha ab$ and $zz^{\prime}=a^{2}y+b^{2}x-\beta^{2}xy$ , which implies $F(x,y,z)=0$ . Thus it remains to show that $x,y,z$ generates $\mathcal{O}[[u,v]]^{G}$ . Since this can be checked modulo the maximal ideals, we may assume that $\mathcal{O}=k$ is a field. Then $G$ is isomorphic to one in Example 2.5 (i.e. $\mathbb{Z}/2\mathbb{Z}$ , $\mu_{2}$ , or $\alpha_{2}$ ).

Note that in any case the ideal generated by $\operatorname{Im}(\delta)$ is generated by two elements $\delta(u),\delta(v)$ .

Suppose $G=\mu_{2}$ or $G=\alpha_{2}$ . Then $x=u^{2},y=v^{2}$ . It follows from [Matsumoto:k3alphap]*Theorem 3.8 that $k[[u,v]]^{G}$ has a $k[[x,y]]$ -basis of the form $1,w$ . Write $w=c_{00}+c_{10}u+c_{01}v+c_{11}uv$ with $c_{ij}\in k[[u^{2},v^{2}]]=k[[x,y]]$ . Write $z=d+ew$ with $d,e\in k[[x,y]]$ . It suffices to show $e\in k[[x,y]]^{*}$ . We have $z^{2}=d^{2}+e^{2}w^{2}=d^{2}+e^{2}c_{00}^{2}+e^{2}c_{10}^{2}x+e^{2}c_{01}^{2}y+e^{2}c_{11}^{2}xy$ . We also have $z^{2}=b^{2}x+a^{2}y-\beta^{2}xy$ . It suffices to show that ${a^{2},b^{2},\beta^{2}}$ have no common divisor. By the assumption on the fixed locus, the ideal $(\delta(u),\delta(v))=(a,b)$ is supported on the closed point, and so is the ideal $(a^{2},b^{2})$ .

Now suppose $G=\mathbb{Z}/2\mathbb{Z}$ . In this case $x=N(u)$ , $y=N(v)$ , and $z=\operatorname{Tr}(ug(v))$ , and it follows from Lemma 2.7 that $x,y,z$ generates $\mathcal{O}[[u,v]]^{G}$ .

Latter assertion. Suppose $G_{k}=\alpha_{2}$ . Then we have $\deg\operatorname{Fix}(G\curvearrowright k[[u,v]])=\deg k[[u,v]]/(a,b)$ , and $\tau=\dim_{k}k[[x,y,z]]/(a^{2},b^{2},z^{2})=2\dim_{k}k[[x,y]]/(a^{2},b^{2})=2\dim_{k}k[[u,v]]/(a,b)$ since $x=u^{2}$ and $y=v^{2}$ in this case.

Suppose $G_{k}=\mu_{2}$ . In this case we can linearize the action and then $\delta(u)=u$ and $\delta(v)=v$ , hence $\deg\operatorname{Fix}(G\curvearrowright k[[u,v]])=1$ . ∎

Lemma 2.7.

Suppose $k$ is a field of characteristic $2$ , and $G=\mathbb{Z}/2\mathbb{Z}=\{1,g\}$ acts on $k[[u,v]]$ , freely outside the closed point. Let $x=N(u)=ug(u)$ , $y=N(v)=vg(v)$ , and $z=\operatorname{Tr}(ug(v))=ug(v)+g(u)v$ . Then $k[[u,v]]^{G}=k[[x,y,z]]/(z^{2}+abz+a^{2}y+b^{2}x)$ , where $a=\operatorname{Tr}(u)=u+g(u)$ and $b=\operatorname{Tr}(v)=v+g(v)$ .

Proof.

It is straightforward to check $z^{2}+abz+a^{2}y+b^{2}x=0$ (cf. the first paragraph of the proof of Proposition 2.6). It remains to show that $x,y,z$ generate the maximal ideal $\mathfrak{n}^{\prime}$ of $k[[u,v]]^{G}$ .

By [Artin:wild2]*Theorem, there exist local coordinates $u^{\prime},v^{\prime}$ with the following properties: letting $x^{\prime}=N(u^{\prime})$ , $y^{\prime}=N(v^{\prime})$ , $z^{\prime}=\operatorname{Tr}(u^{\prime}g(v^{\prime}))$ , and $R_{0}=k[[x^{\prime},y^{\prime}]]$ ,

•

$k[[u,v]]$ has an $R_{0}$ -basis $1,u^{\prime},v^{\prime},u^{\prime}g(v^{\prime})$ and
•

$k[[u,v]]^{G}$ has an $R_{0}$ -basis $1,z^{\prime}$ .

Write

\begin{pmatrix}u\\ v\end{pmatrix}=\begin{pmatrix}p_{00}&p_{10}&p_{01}&p_{11}\\ q_{00}&q_{10}&q_{01}&q_{11}\end{pmatrix}\begin{pmatrix}1\\ u^{\prime}\\ v^{\prime}\\ u^{\prime}g(v^{\prime})\end{pmatrix},

where $p_{ij},q_{ij}\in R_{0}$ , $p_{00},q_{00}\in(x^{\prime},y^{\prime})$ , and $\det\begin{pmatrix}p_{10}&p_{01}\\ q_{10}&q_{01}\end{pmatrix}\in R_{0}^{*}$ . Then a straightforward computation yields

\begin{pmatrix}x\\ y\\ z\end{pmatrix}\equiv\begin{pmatrix}p_{10}^{2}&p_{01}^{2}&p_{10}p_{01}\\ q_{10}^{2}&q_{01}^{2}&q_{10}q_{01}\\ 0&0&p_{10}q_{01}+p_{01}q_{10}\end{pmatrix}\begin{pmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}\pmod{\mathfrak{n}^{\prime 2}}.

Since the matrix is invertible, $x,y,z$ generate $\mathfrak{n}^{\prime}/\mathfrak{n}^{\prime 2}$ . ∎

3. Non-extreme cases

The good reduction problem of Kummer surfaces in the non-extreme cases are solved by Ito (see [Matsumoto:SIP]*Section 4) if $\operatorname{char}k\neq 2$ , and by Lazda–Skorobogatov [Lazda--Skorobogatov:reductionofkummer] if $\operatorname{char}k=2$ and $\mathcal{A}_{k}$ is non-supersingular. For the readers’ convenience, we include a sketch of the proof for these cases.

3.1. Singularities of Kummer surfaces in the non-extreme cases

We already mentioned the result for the extreme case in Proposition 2.4. In the non-extreme cases, we have the following.

Proposition 3.1.

Let $A$ be an abelian surface over a field $F$ . If $\operatorname{char}F\neq 2$ , or $\operatorname{char}F=2$ and $A$ is non-supersingular, then the minimal resolution of $A/\{\pm 1\}$ is a K3 surface. Moreover, if all points of $A[2]$ are $F$ -rational, then $A/\{\pm 1\}$ has only RDPs as singularities, and the number and the type of RDPs are

•

$16A_{1}$ if $\operatorname{char}F\neq 2$ ,
•

$4D_{4}^{1}$ if $\operatorname{char}F=2$ and $\operatorname{\mathit{p}-rank}(A)=2$ ,
•

$2D_{8}^{2}$ if $\operatorname{char}F=2$ and $\operatorname{\mathit{p}-rank}(A)=1$ .

For the assertion on quotient singularities, see [Katsura:Kummer2]*Proposition 3 for a proof (for the determination of the coindices, see [Artin:wild2]*Examples).

3.2. Good reduction of Kummer surfaces in the non-extreme cases

Let $K,\mathcal{O},k$ be as in Introduction.

Proposition 3.2.

Suppose $\mathcal{A}\to\operatorname{Spec}\mathcal{O}$ is an abelian scheme over $\mathcal{O}$ . Then the fibers of $\mathcal{A}/\{\pm 1\}$ are isomorphic to the quotients of the fibers.

Proof.

See [Lazda--Skorobogatov:reductionofkummer]*Proposition 4.6. ∎

Proposition 3.3.

Suppose $\mathcal{X}\to\operatorname{Spec}\mathcal{O}$ is a flat morphism of finite type, with both fibers $\mathcal{X}_{k}$ and $\mathcal{X}_{K}$ normal surfaces. Suppose $\mathcal{Z}\subset\mathcal{X}$ is a section (i.e. the composite $\mathcal{Z}\to\mathcal{X}\to\operatorname{Spec}\mathcal{O}$ is an isomorphism) such that $\mathcal{Z}_{k}$ and $\mathcal{Z}_{K}$ are RDPs of the fibers. Then $\operatorname{Bl}_{\mathcal{Z}}(\mathcal{X})_{K}\cong\operatorname{Bl}_{\mathcal{Z}_{K}}(\mathcal{X}_{K})$ and $\operatorname{Bl}_{\mathcal{Z}}(\mathcal{X})_{k}\cong\operatorname{Bl}_{\mathcal{Z}_{k}}(\mathcal{X}_{k})$ .

Proof.

See [Lazda--Skorobogatov:reductionofkummer]*Proposition 4.1 or [Overkamp:Kummer]*Lemma 3.10. ∎

Proposition 3.4.

Let $K$ and $k$ be as in the introduction (with $\operatorname{char}K=0$ ). Suppose an abelian surface $A$ over $K$ has good reduction with Néron model $\mathcal{A}$ . Assume either $\operatorname{char}k\neq 2$ , or $\operatorname{char}k=2$ and $\mathcal{A}_{k}$ is non-supersingular. Then $X=\operatorname{Km}(A)$ has potential good reduction.

Proof.

Let $\mathcal{Y}:=\mathcal{A}/\{\pm 1\}$ . By Proposition 3.2, we have $\mathcal{Y}_{K}=A/\{\pm 1\}$ and $\mathcal{Y}_{k}=A_{k}/\{\pm 1\}$ . By replacing $K$ with a finite extension, we may assume that $\mathcal{A}_{K}[2]$ and $\mathcal{A}_{k}[2]$ consist respectively of $K$ -rational points and $k$ -rational points. Then, by Proposition 3.1, $\operatorname{Sing}(\mathcal{Y}_{K})$ is $16A_{1}$ and $\operatorname{Sing}(\mathcal{Y}_{k})$ is one of $16A_{1}$ , $4D_{4}^{1}$ , or $2D_{8}^{2}$ .

Choose an RDP of $\mathcal{Y}_{K}$ , take its closure $\mathcal{Z}$ in $\mathcal{Y}$ , and take the blow-up $\mathcal{Y}^{(1)}=\operatorname{Bl}_{\mathcal{Z}}\mathcal{Y}$ . Then $\mathcal{Z}_{k}$ should be a singular point of $\mathcal{Y}_{k}$ , hence one of the RDPs. By Proposition 3.3, each fiber of $\mathcal{Y}^{(1)}$ is the blow-up at an RDP. Repeating this $16$ times, we obtain a scheme $\mathcal{Y}^{(16)}$ whose fibers are $\operatorname{Km}(\mathcal{A}_{K})$ and $\operatorname{Km}(\mathcal{A}_{k})$ . ∎

4. Proof of main theorem

Let $\mathcal{O}$ be as before (i.e. a Henselian discrete valuation ring with residue field $k$ ). In this section, $k$ is of characteristic $2$ , and the maximal ideal of $\mathcal{O}$ is denoted by $\mathfrak{p}=(\varpi)$ .

We first show in Section 4.1 that, in order to prove good reduction of a K3 surface $X$ , it suffices to construct a model of $X$ satisfying a kind of semistability. Then we give an explicit construction of such a model in Sections 4.2–4.3. We summarize the proof of Theorem 1.2 in Section 4.4.

4.1. Strict semistability in the broad sense

We need the following variant of the strict semistability.

Definition 4.1.

We say that a flat scheme $\mathcal{X}$ of finite type over $\mathcal{O}$ is strictly semistable in the broad sense if it satisfies the following conditions.

•

The generic fiber $X$ of $\mathcal{X}$ is smooth.
•

Every irreducible component of $\mathcal{X}_{k}$ is smooth.
•

For each closed point $P$ of the special fiber $\mathcal{X}_{k}$ , $\mathcal{O}_{\mathcal{X},P}$ is étale over $\mathcal{O}[x_{1},\dots,x_{n}]/(x_{1}\dots x_{r}-\epsilon)$ for some $1\leq r\leq n$ and some $\epsilon\in\mathfrak{p}$ .

Recall that $\mathcal{X}$ is strictly semistable if we can take $\epsilon$ to be a uniformizer of $\mathcal{O}$ .

The existence of such a model is quite useful for the good reduction problem:

Theorem 4.2.

Let $X$ be a K3 surface over $K$ . Suppose that the Galois representation $H_{\mathrm{\acute{e}t}}^{2}(X_{\overline{K}},\mathbb{Q}_{l})$ is unramified for some auxiliary prime $l\neq p$ after replacing $K$ with a finite extension, and that $X$ admits a proper model $\mathcal{X}$ over $\mathcal{O}$ that is strictly semistable in the broad sense. Then $X$ has potential good reduction.

Here, a representation of $\operatorname{Gal}(\overline{K}/K)$ is unramified if the inertia subgroup of $\operatorname{Gal}(\overline{K}/K)$ acts trivially.

Proof.

By replacing $K$ with a finite extension (which does not affect the semistability in the broad sense), we may assume that $H_{\mathrm{\acute{e}t}}^{2}(X_{\overline{K}},\mathbb{Q}_{l})$ is unramified.

By an argument similar to [Saito:logsmooth]*Lemma 1.7, $\mathcal{X}$ is log smooth over $\mathcal{O}$ . By [Saito:logsmooth]*Theorem 1.8, there exist a finite extension $\mathcal{O}^{\prime}$ of $\mathcal{O}$ and a morphism $\mathcal{X}^{\prime}\to\mathcal{X}\otimes_{\mathcal{O}}\mathcal{O}^{\prime}$ that is isomorphic above the interior of $\mathcal{X}$ such that $\mathcal{X}^{\prime}$ is proper and strictly semistable over $\mathcal{O}^{\prime}$ . Here the interior of $\mathcal{X}$ is the complement of the union of intersections of two or more components of $\mathcal{X}_{k}$ (hence the interior contains the generic fiber). Thus we may assume that $\mathcal{X}$ is strictly semistable.

Then we can apply the arguments given in [Matsumoto:goodreductionK3]*Section 3, with the use of Kawamata’s MMP replaced with Takamatsu–Yoshikawa’s [Takamatsu--Yoshikawa:mixed3fold]*Theorem 1.1, which, unlike Kawamata’s, has no restriction on the residue characteristic. ∎

If $X=\operatorname{Km}(A)$ is the Kummer surface associated to an abelian surface $A$ with good reduction, then, by the Néron–Ogg–Shafarevich criterion, the Galois representation $H_{\mathrm{\acute{e}t}}^{1}(A_{\overline{K}},\mathbb{Q}_{l})$ is unramified, and then by the isomorphism

H_{\mathrm{\acute{e}t}}^{2}(\operatorname{Km}(A)_{\overline{K}},\mathbb{Q}_{l})\cong\bigoplus_{A[2]}\mathbb{Q}_{l}(-1)\oplus\bigwedge^{2}H_{\mathrm{\acute{e}t}}^{1}(A_{\overline{K}},\mathbb{Q}_{l}),

$H_{\mathrm{\acute{e}t}}^{2}(\operatorname{Km}(A)_{\overline{K}},\mathbb{Q}_{l})$ is also unramified after replacing $K$ with a finite extension. Thus, our goal is to construct a strictly semistable model in the broad sense of the Kummer surface $X=\operatorname{Km}(A)$ .

4.2. Overview of the blow-up construction

The main idea is to perform a (weighted) blow-up with respect to suitable coordinates.

Suppose $\mathcal{Y}$ is a smooth affine scheme over $\mathcal{O}$ of relative dimension $2$ and $G=G_{\alpha,\beta}$ is a group scheme of length $2$ (see Section 2.2). Suppose $G$ acts on $\mathcal{Y}$ , with the quotient morphism denoted by $\pi\colon\mathcal{Y}\to\mathcal{Y}/G=\mathcal{X}$ , and the following properties hold.

•

$\operatorname{Fix}(G_{k}\curvearrowright\mathcal{Y}_{k})=\{y\}$ , and the quotient singularity $\pi(y)\in\mathcal{X}_{k}$ is either $\mathrm{EDP}_{4,4}^{(r)}$ or $\mathrm{EDP}_{2,8}^{(r)}$ .
•

All points of $\operatorname{Fix}(G_{K}\curvearrowright\mathcal{Y}_{K})$ specializes to $y$ .

Let $\mathcal{Y}^{\prime}\to\mathcal{Y}$ be the weighted blow-up specified below, with center supported on $y$ . The $G$ -action extends to $\mathcal{Y}^{\prime}$ . Let $\mathcal{X}^{\prime}:=\mathcal{Y}^{\prime}/G$ . We list the properties of $\mathcal{Y}^{\prime}$ and $\mathcal{X}^{\prime}$ to be established. The construction and the proof will be given in Section 4.3.

Claim 4.3.

The following holds.

(1)

$\mathcal{X}^{\prime}$ is a normalized (weighted) blow-up of $\mathcal{X}$ .
(2)

$\mathcal{X}^{\prime}_{k}$ consists of two components: the strict transform $Z$ of $\mathcal{X}_{k}$ and the exceptional divisor $E$ .
(3)

The $G$ -action on $\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)$ induces an action of $G^{\prime}=G_{\alpha^{\prime},\beta^{\prime}}$ with isolated fixed locus, and the quotient is $\mathcal{X}^{\prime}\setminus Z$ .
(4)

The exceptional divisor $E$ is a normal projective surface and satisfies $H^{1}(\mathcal{O}_{E})=0$ and $K_{E}=0$ . In particular, if $\operatorname{Sing}(E)$ consists only of RDPs, then $E$ is an RDP K3 surface.
(5)

The image of $\operatorname{Sing}(\mathcal{X}^{\prime}_{K})$ by the specialization map is contained in $\operatorname{Sing}(E)\setminus Z$ .

Here, an RDP K3 surface is a proper surface with only rational double point singularities (if any) whose minimal resolution is a K3 surface.

Claim 4.4.

After replacing $k$ with a finite separable extension, we have the following description of $\operatorname{Sing}(Z)$ and $\operatorname{Sing}(E)$ .

(1)
If $y$ is $\mathrm{EDP}_{4,4}^{(r)}$ , then there are $P_{1},\dots,P_{5}\in Z\cap E$ and the following hold.
1. (a)
  
  $\operatorname{Sing}(Z)=\{P_{1},\dots,P_{5}\}$ , all RDPs of type $A_{1}$ .
2. (b)
  
  $\operatorname{Sing}(E)\cap Z=\{P_{1},\dots,P_{5}\}$ , all RDPs of type $A_{1}$ .
3. (c)
  
  $\operatorname{Sing}(E)\setminus Z$ is one of $16A_{1}$ , $4D_{4}^{0}$ , $2D_{8}^{0}$ , $2E_{8}^{0}$ , $1D_{16}^{0}$ , or $1\mathrm{EDP}_{2,8}^{(0)}$ .
(2)
If $y$ is $\mathrm{EDP}_{2,8}^{(r)}$ , then there are $Q_{1},\dots,Q_{4}\in Z\cap E$ and the following hold.
1. (a)
  
  $\operatorname{Sing}(Z)=\{Q_{1},\dots,Q_{4}\}$ , $Q_{1}$ is a quotient singularity of type ${(1,1)}/{3}$ , and all others are RDPs of type $A_{1}$ .
2. (b)
  
  $\operatorname{Sing}(E)\cap Z=\{Q_{1},\dots,Q_{4}\}$ , $Q_{1}$ is an RDP of type $A_{2}$ , and all others are RDPs of type $A_{1}$ .
3. (c)
  
  Every element of $\operatorname{Sing}(E)\setminus Z$ is one of the following: $A_{1}$ , $D_{4}^{0}$ , $D_{8}^{0}$ , $D_{12}^{0}$ , $D_{16}^{0}$ , $E_{8}^{0}$ , or $\mathrm{EDP}_{2,6}^{(0)}$ .
4. (d)
  
  If all singularities of $\operatorname{Sing}(E)\setminus Z$ are of the same type, then $\operatorname{Sing}(E)\setminus Z$ is one of $16A_{1}$ , $4D_{4}^{0}$ , $2D_{8}^{0}$ , $2E_{8}^{0}$ , or $1D_{16}^{0}$ .

Let $\mathcal{X}^{\prime\prime}\to\mathcal{X}^{\prime}$ be the blow-up specified below (with center supported on $\operatorname{Sing}(Z)$ ).

Claim 4.5.

$\mathcal{X}^{\prime\prime}$ is strictly semistable in the broad sense outside $\operatorname{Sing}(E)\setminus Z$ .

Remark 4.6.

For our purpose (Theorem 1.2), what we need directly are resolutions of EDPs of type $\mathrm{EDP}_{4,4}^{(r)}$ and $\mathrm{EDP}_{2,8}^{(r)}$ with $r=1$ . However we also consider the case $r=0$ because we need the resolution of $\mathrm{EDP}_{2,8}^{(0)}$ for the proof of the case of $\mathrm{EDP}_{4,4}^{(1)}$ . The arguments for $r=0$ and $r=1$ are parallel.

Remark 4.7.

Our construction also shows that the (affine) surface $E\setminus Z$ is the quotient of $\pi^{-1}(E\setminus Z)$ by $G^{\prime}\in\{\mu_{2},\alpha_{2}\}$ . Although we will not show it in this paper, this can be extended to a quotient morphism $A^{\prime}\to E$ by $G^{\prime}$ from a proper non-normal surface $A^{\prime}$ that satisfies $\omega_{A^{\prime}}\cong\mathcal{O}_{A^{\prime}}$ and $h^{i}(\mathcal{O}_{A^{\prime}})=1,2,1$ for $i=0,1,2$ . We say that such $A^{\prime}$ is an abelian-like surface and $E$ is an inseparable analogue of Kummer surface. Details will be given in another paper.

4.3. Explicit computation of the blow-up

4.3.1. Case of $\mathrm{EDP}_{4,4}^{(r)}$

First we consider the case where $y\in\mathcal{Y}_{k}/G$ is $\mathrm{EDP}_{4,4}^{(r)}$ . Let $u,v\in\mathcal{O}_{\mathcal{Y},y}$ be local coordinates satisfying $(u=v=0)\subset\operatorname{Fix}(G\curvearrowright\mathcal{Y})$ . Let $x,y,z,a,b\in\mathcal{O}_{\mathcal{Y},y}^{G}$ as in Proposition 2.6. Then, by the proposition, the maximal ideal of $\mathcal{O}_{\mathcal{Y},y}^{G}$ is equal to $(x,y,z)+\mathfrak{p}$ , and we have $F=0$ , where

F=z^{2}-\alpha zab+a^{2}y+b^{2}x-\beta^{2}xy.

Since $(u=v=0)\subset\operatorname{Fix}(G)$ , we have $a,b\in(x,y,z)$ (not only $a,b\in(x,y,z)+\mathfrak{p}$ ). By the description of the blow-up of $\mathrm{EDP}_{4,4}^{(r)}$ at the origin, we have $\overline{a},\overline{b}\in\mathfrak{m}^{2}$ (where $\mathfrak{m}=(x,y,z)$ ) and moreover

\overline{a^{2}y+b^{2}x}\equiv c_{50}x^{5}+c_{41}x^{4}y+c_{14}xy^{4}+c_{05}y^{5}\pmod{((x,y)^{6}+(z))}

for some $c_{ij}\in k$ . Here the overline denotes the reduction modulo $\mathfrak{p}$ . By a coordinate change (on $u,v$ ), and after replacing $k$ with a finite separable extension, we may assume $\overline{a}\equiv x^{2}$ , $\overline{b}\equiv y^{2}\pmod{((x,y)^{3}+(z))}$ , and furthermore

	$\displaystyle a$	$\displaystyle\equiv x^{2}+a_{10}x+a_{01}y,$
	$\displaystyle b$	$\displaystyle\equiv y^{2}+b_{10}x+b_{01}y\pmod{((x,y)^{3}+(z))}$

with $a_{ij},b_{ij}\in\mathfrak{p}$ . After replacing $\mathcal{O}$ with a finite extension if necessary, we take $\epsilon\in\mathfrak{p}$ satisfying the following condition:

\frac{a_{10}}{\epsilon^{2}},\frac{a_{01}}{\epsilon^{2}},\frac{b_{10}}{\epsilon^{2}},\frac{b_{01}}{\epsilon^{2}},\frac{\beta^{2}}{\epsilon^{6}}\in\mathcal{O},\quad\text{and moreover at least one}\in\mathcal{O}^{*}.

Let $\mathcal{Y}^{\prime}\to\mathcal{Y}$ be the blow-up at the ideal $(u,v,\epsilon)$ . Then $\mathcal{X}^{\prime}:=\mathcal{Y}^{\prime}/G$ is the normalized weighted blow-up at $(x,y,z,\epsilon)$ with respect to the weight $(2,2,5,1)$ .

We have $\mathcal{X}^{\prime}=\mathcal{X}^{\prime}_{\epsilon}\cup\mathcal{X}^{\prime}_{x}\cup\mathcal{X}^{\prime}_{y}$ , where $\mathcal{X}^{\prime}_{?}=\operatorname{Spec}A_{?}$ are the affine open subschemes of the blow-up $\mathcal{X}^{\prime}$ with the obvious meaning. We first consider $\mathcal{X}^{\prime}_{\epsilon}=\operatorname{Spec}A_{\epsilon}$ , whose special fiber is equal to $E\setminus Z$ . The $\mathcal{O}$ -algebra $A_{\epsilon}$ is étale over the subalgebra generated by

x^{\prime}:=\frac{x}{\epsilon^{2}},\quad y^{\prime}:=\frac{y}{\epsilon^{2}},\quad z^{\prime}:=\frac{z}{\epsilon^{5}},

subject to $F^{\prime}=0$ , where $F^{\prime}$ is naively “ $\epsilon^{-10}F$ ”, i.e.,

F^{\prime}=z^{\prime 2}-\epsilon^{3}\alpha a^{\prime}b^{\prime}z^{\prime}+a^{\prime 2}y^{\prime}+b^{\prime 2}x^{\prime}-(\epsilon^{-6}\beta^{2})x^{\prime}y^{\prime}

where $a^{\prime}:=\epsilon^{-4}a$ and $b^{\prime}:=\epsilon^{-4}b$ . Let $a_{ij}^{\prime}=\overline{\epsilon^{-2}a_{ij}}\in k$ and $b_{ij}^{\prime}=\overline{\epsilon^{-2}b_{ij}}\in k$ . Here, again, the overline denotes the reduction modulo $\mathfrak{p}$ . We have $A_{\epsilon}\otimes_{\mathcal{O}}k=k[x^{\prime},y^{\prime},z^{\prime}]/(\overline{F^{\prime}})$ , $\overline{F^{\prime}}=z^{\prime 2}+a^{\prime 2}y^{\prime}+b^{\prime 2}x^{\prime}-\overline{\epsilon^{-6}\beta^{2}}x^{\prime}y^{\prime}$ , $a^{\prime}=x^{\prime 2}+a^{\prime}_{10}x^{\prime}+a^{\prime}_{01}y^{\prime}$ , $b^{\prime}=y^{\prime 2}+b^{\prime}_{10}x^{\prime}+b^{\prime}_{01}y^{\prime}$ .

Suppose $\epsilon^{-6}\beta^{2}\in\mathcal{O}^{*}$ , hence its image $\overline{\epsilon^{-6}\beta^{2}}\in k$ is nonzero. Then since $\overline{F^{\prime}}_{x^{\prime}y^{\prime}}=\overline{\epsilon^{-6}\beta^{2}}\neq 0$ , every singularity of $E\setminus Z$ is of type $A_{1}$ , and considering the degrees of $\overline{F^{\prime}}_{x^{\prime}}$ and $\overline{F^{\prime}}_{y^{\prime}}$ we conclude that the singularity of $E\setminus Z$ is $16A_{1}$ .

Suppose $\epsilon^{-6}\beta^{2}\in\mathfrak{p}$ . Then we have

	$\displaystyle\overline{a^{\prime}}$	$\displaystyle=x^{\prime 2}+a^{\prime}_{10}x^{\prime}+a^{\prime}_{01}y^{\prime},$
	$\displaystyle\overline{b^{\prime}}$	$\displaystyle=y^{\prime 2}+b^{\prime}_{10}x^{\prime}+b^{\prime}_{01}y^{\prime}$

with $a^{\prime}_{ij},b^{\prime}_{ij}\in k$ , not all $0$ by the definition of $\epsilon$ . Then

\overline{F^{\prime}}=z^{\prime 2}+x^{\prime 4}y^{\prime}+x^{\prime}y^{\prime 4}+b_{10}^{\prime 2}x^{\prime 3}+a_{10}^{\prime 2}x^{\prime 2}y^{\prime}+b_{01}^{\prime 2}x^{\prime}y^{\prime 2}+a_{01}^{\prime 2}y^{\prime 3}.

To describe the singularities, we consider $c:=(b^{\prime}_{10}:a^{\prime}_{10}:b^{\prime}_{01}:a^{\prime}_{01})\in\mathbb{P}^{3}$ and the following subsets of $\mathbb{P}^{3}$ :

	$\displaystyle Q$	$\displaystyle=\{(t_{0}:t_{1}:t_{2}:t_{3})\mid t_{1}t_{2}-t_{0}t_{3}=0\},$
	$\displaystyle\Gamma_{1}$	$\displaystyle=\{(t_{0}:t_{1}:t_{2}:t_{3})\mid t_{1}t_{2}-t_{0}t_{3}=t_{1}^{2}-t_{0}t_{2}=t_{2}^{2}-t_{1}t_{3}=0\},$
	$\displaystyle\Gamma_{2}$	$\displaystyle=\{(t_{0}:t_{1}:t_{2}:t_{3})\mid t_{1}t_{2}-t_{0}t_{3}=t_{2}^{2}-t_{0}t_{1}=t_{1}^{2}-t_{2}t_{3}=0\}.$

$Q$ is the image of the Segre embedding of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ to $\mathbb{P}^{3}$ , and $\Gamma_{1}$ and $\Gamma_{2}$ are twisted cubic curves contained in $Q$ . Then it is straightforward to check that the singularity of $E\setminus Z$ is as shown in Figure 2 in the following sense: a configuration of singularities (e.g. $2D_{8}^{0}$ ) occurs if and only if $c$ belongs to the corresponding locus ( $Q$ ) but not to its subspaces ( $\Gamma_{1}$ , $\Gamma_{2}$ , and $\Gamma_{1}\cap\Gamma_{2}$ ). We also note that $\Gamma_{1}\cap\Gamma_{2}=\Gamma_{1}(\mathbb{F}_{4})=\Gamma_{2}(\mathbb{F}_{4})$ .

Figure 2.

\operatorname{Sing}(E)\setminus Z

in the case of

\mathrm{EDP}_{4,4}^{(r)}

, assuming

\epsilon^{-6}\beta^{2}\in\mathfrak{p}

Consider the assertion on the action (Claim 4.3(3)). The action of $G=G_{\alpha,\beta}$ on $\mathcal{Y}$ corresponds to the map $\delta\colon\mathcal{O}_{\mathcal{Y}}\to\mathcal{O}_{\mathcal{Y}}$ as in Section 2.2. Since $\delta(u)=a$ has weight $\geq 4=3+\operatorname{wt}(u)$ and $\delta(v)=b$ has weight $\geq 4=3+\operatorname{wt}(v)$ (i.e. $a,b\in(u,v)^{4}$ ), it extends to $\delta\colon\mathcal{O}_{\mathcal{Y}^{\prime}}\to\mathcal{O}_{\mathcal{Y}^{\prime}}(-3E_{\mathcal{Y}})$ , where $E_{\mathcal{Y}}\subset\mathcal{Y}^{\prime}$ is the exceptional divisor. On $\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)$ , the subsheaf $\mathcal{O}_{\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)}(-3E_{\mathcal{Y}})\subset\mathcal{O}_{\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)}$ is equal to $\epsilon^{3}\mathcal{O}_{\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)}$ , hence $\delta^{\prime}:=\epsilon^{-3}\delta$ defines a morphism $\mathcal{O}_{\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)}\to\mathcal{O}_{\mathcal{Y}^{\prime}\setminus\pi^{-1}(Z)}$ and it corresponds to an action of $G^{\prime}=G_{\alpha^{\prime},\beta^{\prime}}$ , $(\alpha^{\prime},\beta^{\prime})=(\epsilon^{3}\alpha,\epsilon^{-3}\beta)$ . Since $\delta^{\prime}(\epsilon^{-1}u)=\epsilon^{-4}a=a^{\prime}$ and $\delta^{\prime}(\epsilon^{-1}v)=b^{\prime}$ , the fixed locus of the action of $G^{\prime}$ on $\mathcal{Y}^{\prime}_{\epsilon}$ is $(\overline{a^{\prime}}=\overline{b^{\prime}}=0)$ . This descrpition also shows that the image under the specialization map of $\operatorname{Sing}(X_{K})$ is equal to $\operatorname{Sing}(E)\setminus Z$ .

Next we consider singularities of $Z$ and $E$ contained in $Z\cap E$ , which is contained in $\mathcal{X}^{\prime}_{x}\cup\mathcal{X}^{\prime}_{y}$ . By symmetry, it suffices to consider $\mathcal{X}^{\prime}_{x}=\operatorname{Spec}A_{x}$ . Define elements of $A_{x}$ by

y^{\prime}=\frac{y}{x},\quad z^{\prime}=\frac{z}{x^{2}},\quad z^{\prime\prime}=\frac{\epsilon z}{x^{3}},\quad q=\frac{\epsilon^{2}}{x},\quad a^{\prime}=\frac{a}{x^{2}},\quad b^{\prime}=\frac{b}{x^{2}}.

Also let

s=\frac{z^{2}}{x^{5}}=\frac{z^{\prime 2}}{x}=\alpha a^{\prime}b^{\prime}z^{\prime}x-(b^{\prime 2}+a^{\prime 2}y^{\prime}-q^{3}(\epsilon^{-6}\beta^{2})y^{\prime}).

Then the $\mathcal{O}$ -algebra $A_{x}$ is étale over the subalgebra generated by $x,y^{\prime},z^{\prime},z^{\prime\prime},q$ , subject to

\begin{pmatrix}s\\ z^{\prime}\quad z^{\prime\prime}\\ x\quad\epsilon\quad q\end{pmatrix}\in V_{2}:=\biggl{\{}\begin{pmatrix}T_{0}^{2}\\ T_{0}T_{1}\quad T_{0}T_{2}\\ T_{1}^{2}\quad T_{1}T_{2}\quad T_{2}^{2}\end{pmatrix}\biggr{\}},

where $V_{2}\subset\mathbb{A}^{6}$ is the cone over the image of the second Veronese map $\mathbb{P}^{2}\to\mathbb{P}^{5}$ . From the above description, it is straightforward to check the following assertions:

•

$\mathcal{X}_{x}\cap Z=(\varpi=q=z^{\prime\prime}=0)$ ,
•

$\mathcal{X}_{x}\cap E=(\varpi=x=z^{\prime}=0)$ ,
•

$\mathcal{X}_{x}\cap\operatorname{Sing}(Z)\cap E=\mathcal{X}_{x}\cap\operatorname{Sing}(E)\cap Z=\mathcal{X}_{x}\cap E\cap Z\cap(s=0)$ ,
•

$\operatorname{Sing}(Z)\cap E=\operatorname{Sing}(E)\cap Z$ consists of $5$ points $\{P_{1},\dots,P_{5}\}$ on which $(x:y)\in\mathbb{P}^{1}(\mathbb{F}_{4})$ ,
•

each $P_{i}$ is an RDP of type $A_{1}$ in both $Z$ and $E$ ,

Define $\mathcal{X}^{\prime\prime}\to\mathcal{X}^{\prime}$ to be the blow-up at the ideal $(s,z^{\prime\prime},x,z^{\prime},\epsilon,q)$ over $\mathcal{X}^{\prime}_{x}$ , and similar over $\mathcal{X}^{\prime}_{y}$ . Then $\mathcal{X}^{\prime\prime}$ is strictly semistable in the broad sense above each $P_{i}$ . Each exceptional divisor of $\mathcal{X}^{\prime\prime}\to\mathcal{X}^{\prime}$ is $\mathbb{P}^{2}$ , and the local equations at the triple points are, for example, $\frac{z^{\prime}}{s}\frac{z^{\prime\prime}}{s}s-\epsilon$ .

By construction, $E$ is a hypersurface of degree $10=5+2+2+1$ in $\mathbb{P}(5,2,2,1)$ , hence we obtain $H^{1}(\mathcal{O}_{E})=0$ and $K_{E}=0$ .

$E\setminus Z$ is the quotient of $E_{\mathcal{Y}}\setminus\pi^{-1}(Z)$ by $G^{\prime}_{k}=G_{\overline{\alpha^{\prime}},\overline{\beta^{\prime}}}=G_{0,\overline{\beta^{\prime}}}$ , which is isomorphic to either $\mu_{2}$ or $\alpha_{2}$ (respectively if $\overline{\beta^{\prime}}\neq 0$ or $\overline{\beta^{\prime}}=0$ ).

4.3.2. Case of $\mathrm{EDP}_{2,8}^{(r)}$

Next we consider the case where $y$ is $\mathrm{EDP}_{2,8}^{(r)}$ .

Let $u,v,x,y,z,a,b,F$ as in the previous case. According to the equation given in Definition 2.1, we may assume

	$\displaystyle\overline{a}$	$\displaystyle\equiv y^{4}\pmod{(x,y^{5},z)},$
	$\displaystyle\overline{b}$	$\displaystyle\equiv x\pmod{(x^{2},xy,y^{3},z)}.$

We can assume moreover (by coordinate changes like $u^{\prime}=u+\epsilon_{1}v$ and $v^{\prime}=v+\epsilon_{2}y$ for $\epsilon_{1},\epsilon_{2}\in\mathfrak{p}$ ) that

	$\displaystyle a$	$\displaystyle\equiv y^{4}+a_{3}y^{3}+a_{2}y^{2}+a_{1}y,$
	$\displaystyle b$	$\displaystyle\equiv x+b_{1}y\pmod{(x^{2},xy,y^{5},z)},$

$a_{1},a_{2},a_{3},b_{1}\in\mathfrak{p}$ . We take $\epsilon\in\mathfrak{p}$ satisfying the following condition (again, after replacing $\mathcal{O}$ if necessary):

\frac{a_{i}}{\epsilon^{8-2i}},\frac{b_{i}}{\epsilon^{6-2i}},\frac{\beta^{2}}{\epsilon^{10}}\in\mathcal{O},\quad\text{and moreover at least one}\in\mathcal{O}^{*}.

Let $\mathcal{Y}^{\prime}\to\mathcal{Y}$ be the weighted blow-up at the ideal $(u,v,\epsilon)$ with respect to the weight $(3,1,1)$ . Then $\mathcal{X}^{\prime}:=\mathcal{Y}^{\prime}/G$ is the normalized weighted blow-up at $(x,y,z,\epsilon)$ with respect to the weight $(6,2,9,1)$ .

We have $\mathcal{X}^{\prime}=\mathcal{X}^{\prime}_{\epsilon}\cup\mathcal{X}^{\prime}_{x}\cup\mathcal{X}^{\prime}_{y}$ . We first consider $\mathcal{X}^{\prime}_{\epsilon}=\operatorname{Spec}A_{\epsilon}$ , whose special fiber is equal to $E\setminus Z$ . The $\mathcal{O}$ -algebra $A_{\epsilon}$ is étale over the subalgebra generated by

x^{\prime}:=\frac{x}{\epsilon^{6}},\quad y^{\prime}:=\frac{y}{\epsilon^{2}},\quad z^{\prime}:=\frac{z}{\epsilon^{9}},

subject to $F^{\prime}=0$ , where

F^{\prime}=z^{\prime 2}-\epsilon^{5}a^{\prime}b^{\prime}z^{\prime}+a^{\prime 2}y^{\prime}+b^{\prime 2}x^{\prime}-(\epsilon^{-10}\beta^{2})x^{\prime}y^{\prime}

is naively “ $\epsilon^{-18}F$ ”, where $a^{\prime}:=\epsilon^{-8}a$ and $b^{\prime}:=\epsilon^{-6}b$ . Let $a_{i}^{\prime}=\overline{\epsilon^{-(8-2i)}a_{i}}\in k$ and $b_{i}^{\prime}=\overline{\epsilon^{-(6-2i)}b_{i}}\in k$ . We have $A_{\epsilon}\otimes k=k[x^{\prime},y^{\prime},z^{\prime}]/(\overline{F^{\prime}})$ , $\overline{F^{\prime}}=z^{\prime 2}+a^{\prime 2}y^{\prime}+b^{\prime 2}x^{\prime}-\overline{\epsilon^{-10}\beta^{2}}x^{\prime}y^{\prime}$ , $a^{\prime}=y^{\prime 4}+a^{\prime}_{3}y^{\prime 3}+a^{\prime}_{2}y^{\prime 2}+a^{\prime}_{1}y^{\prime 1}$ , $b^{\prime}=x^{\prime}+b^{\prime}_{1}y^{\prime}$ .

If $\beta^{2}\epsilon^{-10}\in\mathcal{O}^{*}$ then, as in the case of $\mathrm{EDP}_{4,4}^{(r)}$ , we observe that $\operatorname{Sing}(E)\setminus Z=16A_{1}$ .

Suppose $\epsilon^{-10}\beta^{2}\in\mathfrak{p}$ . Then we have

	$\displaystyle\overline{a^{\prime}}$	$\displaystyle=y^{\prime 4}+a^{\prime}_{3}y^{\prime 3}+a^{\prime}_{2}y^{\prime 2}+a^{\prime}_{1}y^{\prime},$
	$\displaystyle\overline{b^{\prime}}$	$\displaystyle=x^{\prime}+b^{\prime}_{1}y^{\prime}$

with $a^{\prime}_{i},b^{\prime}_{i}\in k$ , not all $0$ by the definition of $\epsilon$ . Then

\overline{F^{\prime}}=z^{\prime 2}+x^{\prime 3}+b_{1}^{\prime 2}x^{\prime}y^{\prime 2}+a_{1}^{\prime 2}y^{\prime 3}+a_{2}^{\prime 2}y^{\prime 5}+a_{3}^{\prime 2}y^{\prime 7}+y^{\prime 9}.

Then the singularity at the origin is as in Figure 3.

Figure 3. Singularity of

E

at the origin in the case of

\mathrm{EDP}_{2,8}^{(r)}

, assuming

\epsilon^{-10}\beta^{2}\in\mathfrak{p}

Again we have $\dim k[x^{\prime},y^{\prime},z^{\prime}]/I_{\tau}=32$ , where $I_{\tau}=(a^{\prime 2},b^{\prime 2},z^{\prime 2})$ is the Tyurina ideal. If all singularities are of the same type, then we conclude that $\operatorname{Sing}(E)\setminus Z$ is one of $4D_{4}^{0},2D_{8}^{0},2E_{8}^{0},1D_{16}^{0}$ . In particular, $D_{12}^{0}$ and $\mathrm{EDP}_{2,6}^{(0)}$ do not appear under this assumption.

The assertion on the extension of the action of $G$ is checked similarly to the case of $\mathrm{EDP}_{4,4}^{(r)}$ . Since $\delta(u)=a$ has weight $\geq 8=5+\operatorname{wt}(u)$ and $\delta(v)=b$ has weight $\geq 6=5+\operatorname{wt}(v)$ , the morphism $\delta\colon\mathcal{O}_{\mathcal{Y}}\to\mathcal{O}_{\mathcal{Y}}$ extends to $\delta\colon\mathcal{O}_{\mathcal{Y}^{\prime}}\to\mathcal{O}_{\mathcal{Y}^{\prime}}(-5E)$ . We argue similarly (with $3$ replaced with $5$ ).

Next we consider singularities of $Z$ and $E$ contained in $Z\cap E$ , which is contained in $\mathcal{X}^{\prime}_{x}\cup\mathcal{X}^{\prime}_{y}$ . We check $\mathcal{X}^{\prime}_{x}=\operatorname{Spec}A_{x}$ and omit $\mathcal{X}^{\prime}_{y}$ (which does not give any more singularities). Define elements of $A_{x}$ by

	$\displaystyle t_{i}=\frac{y^{i}\epsilon^{6-2i}}{x}\;(i=0,1,2,3),\quad w_{i}=\frac{zy^{i}\epsilon^{3-2i}}{x^{2}}\;(i=0,1),\quad v=\frac{zy^{2}}{x^{2}},\quad z^{\prime}=\frac{z}{x},$
	$\displaystyle a^{\prime}=\frac{a}{x},\quad b^{\prime}=\frac{b}{x},\quad s=\frac{z^{2}}{x^{3}}=\alpha z^{\prime}a^{\prime}b^{\prime}-(b^{\prime 2}+\frac{ya^{\prime 2}}{x}-\frac{\beta^{2}}{\epsilon^{10}}t_{1}t_{0}).$

Then the $\mathcal{O}$ -algebra $A_{x}$ is étale over the subalgebra generated by

x,y,t_{0},t_{1},t_{2},t_{3},w_{0},w_{1},v,z^{\prime},

subject to

	$\displaystyle\operatorname{rank}\begin{pmatrix}w_{0}&\epsilon^{2}&t_{0}&t_{1}&t_{2}\\ w_{1}&y&t_{1}&t_{2}&t_{3}\\ \end{pmatrix}\leq 1,\operatorname{rank}\begin{pmatrix}w_{1}&\epsilon&t_{0}&t_{1}&t_{2}\\ v&y&\epsilon t_{1}&\epsilon t_{2}&\epsilon t_{3}\\ \end{pmatrix}\leq 1,$
	$\displaystyle\operatorname{rank}\begin{pmatrix}w_{0}&\epsilon^{3}&t_{0}\\ z^{\prime}&x&\epsilon^{3}\\ \end{pmatrix}\leq 1,\operatorname{rank}\begin{pmatrix}w_{1}&\epsilon y&t_{1}\\ z^{\prime}&x&\epsilon^{3}\\ \end{pmatrix}\leq 1,\operatorname{rank}\begin{pmatrix}v&y^{2}&t_{2}&t_{3}\\ z^{\prime}&x&\epsilon^{2}&y\\ \end{pmatrix}\leq 1,$

and

	$\displaystyle\begin{pmatrix}s\\ w_{0}\quad w_{1}\quad z^{\prime}\quad v\\ t_{0}\quad t_{1}\quad t_{2}\quad\epsilon^{3}\quad\epsilon t_{2}\quad y\epsilon\quad\epsilon t_{3}\quad x\quad y^{2}\quad yt_{3}\end{pmatrix}\in V_{4}$
	$\displaystyle:=\biggl{\{}\begin{pmatrix}T_{0}^{2}\\ T_{0}T_{1}\quad T_{0}T_{2}\quad T_{0}T_{3}\quad T_{0}T_{4}\\ T_{1}^{2}\quad T_{1}T_{2}\quad T_{2}^{2}\quad T_{1}T_{3}\quad T_{1}T_{4}\quad T_{2}T_{3}\quad T_{2}T_{4}\quad T_{3}^{2}\quad T_{3}T_{4}\quad T_{4}^{2}\end{pmatrix}\biggr{\}},$

where $V_{4}\subset\mathbb{A}^{15}$ is the cone over the image of the second Veronese map $\mathbb{P}^{4}\to\mathbb{P}^{14}$ .

We have $\mathcal{X}_{x}\cap Z=(\varpi=t_{0}=t_{1}=t_{2}=w_{0}=w_{1}=0)$ and $\mathcal{X}_{x}\cap E=(\varpi=x=z^{\prime}=y=v=0)$ .

Let $Q_{2},Q_{3},Q_{4}$ be the points on $\mathcal{X}_{x}\cap Z\cap E$ where $s$ ( $=1+t_{3}^{3}+\dots$ ) vanish. Around these points, $t_{3}$ is a unit. and hence the following elements can be eliminated: $x=t_{3}^{-1}y^{3}$ , $t_{1}=t_{3}^{-1}t_{2}^{2}$ , $t_{0}=t_{3}^{-2}t_{2}^{3}$ , $w_{0}=t_{3}^{-1}t_{2}w_{1}$ , $z^{\prime}=t_{3}^{-1}yv$ . Thus the maximal ideals of the local rings are generated by $v,w_{1},y,t_{2},s,\varpi$ , subject to $(s,w_{1},v,t_{2},\epsilon t_{3},yt_{3})\in V_{2}$ , where $V_{2}$ is (as in Section 4.3.1) the cone over the image of the Veronese map $\mathbb{P}^{2}\to\mathbb{P}^{5}$ . It is straightforward to check that they are RDPs of type $A_{1}$ on both components $Z=(w_{1}=\varpi=t_{2}=0)$ and $E=(v=y=\varpi=0)$ , and that the blow-up of $\mathcal{X}^{\prime}$ at $(s,v,y,w_{1},\epsilon,t_{2})$ is strictly semistable in the broad sense above these points.

Let $Q_{1}$ be the point $(t_{i}=x=y=z^{\prime}=v=w_{i}=0)$ . Around this point $s$ is a unit, hence the maximal ideal of the local ring is generated by $w_{0},w_{1},t_{3},v,z^{\prime},y,\varpi$ , subject to $(t_{3},v,ys,z^{\prime}s,w_{1},\epsilon s,w_{0}s)\in V^{\prime}$ , where

V^{\prime}=\{(T_{0}^{3},T_{0}^{2}T_{1},T_{0}T_{1}^{2},T_{1}^{3},T_{0}T_{2},T_{1}T_{2},T_{2}^{3})\}\subset\mathbb{A}^{7}

is the toric variety attached to the monoid $M\subset\mathbb{N}^{3}$ generated by the vectors

(3,0,0),(2,1,0),(1,2,0),(0,3,0),(1,0,1),(0,1,1),(0,0,3).

(In other words, the elements $(t_{3},\dots,w_{0}s)$ define a morphism $k[M]\to\mathcal{O}_{\mathcal{X}^{\prime},Q_{1}}$ from the monoid ring.) We can check that, around this point,

•

the component $Z$ is $(\varpi=w_{1}=w_{0}=0)$ and $\mathcal{O}_{Z,Q_{1}}$ is $\frac{(1,1)}{3}$ ,
•

the component $E$ is $(\varpi=v=y=z^{\prime}=0)$ and $\mathcal{O}_{E,Q_{1}}$ is an RDP of type $A_{2}$ .

Let $\mathcal{X}^{\prime\prime}\to\mathcal{X}^{\prime}$ be the blow-up at $(t_{3},v,y,z^{\prime},w_{1},\epsilon,w_{0})$ . Then $\mathcal{X}^{\prime\prime}$ is strictly semistable in the broad sense. More precisely, there are $4$ components $\tilde{Z},\tilde{E},E_{1},E_{2}$ over this point, $\tilde{Z}$ and $\tilde{E}$ are the minimal resolutions of $Z$ and $E$ at $Q_{1}$ respectively, $E_{1}\cong\mathbb{P}^{2}$ and $E_{2}\cong\mathbb{F}_{2}$ are exceptional divisors, and the local equations at triple points are $\frac{v}{t_{3}}\frac{w_{1}}{t_{3}}t_{3}-\epsilon s$ at $\tilde{E}\cap\tilde{Z}\cap E_{2}$ and $\frac{w_{0}s}{w_{1}}\frac{t_{3}}{w_{1}}\frac{\epsilon s}{w_{1}}-\epsilon s$ at $\tilde{E}\cap E_{1}\cap E_{2}$ .

The assertion on $H^{1}(\mathcal{O}_{E})$ and $K_{E}$ follows also similarly since $E$ is a hypersurface of degree $18=9+6+2+1$ in $\mathbb{P}(9,6,2,1)$ .

4.4. End of Proof of Theorem 1.2

Let $\mathcal{A}$ be the Néron model of the abelian surface $A$ . By Proposition 2.4, the singularity of $\mathcal{A}_{k}/\{\pm 1\}$ is an elliptic double point of type $\mathrm{EDP}_{4,4}^{(1)}$ or $\mathrm{EDP}_{2,8}^{(1)}$ , according to the supersingular abelian surface $\mathcal{A}_{k}$ being superspecial or not. We apply the normalized weighted blow-up $\mathcal{X}^{\prime\prime}\to\mathcal{X}^{\prime}\to\mathcal{X}=\mathcal{A}/\{\pm 1\}$ described in Sections 4.2–4.3. Since $A[2]$ acts on itself (by translation) transitively, all singularity of $\operatorname{Sing}(E)\setminus Z$ are of the same type. From the classification of possible singularities under this assumption (Claim 4.4 (1c) and (2d)), either $\operatorname{Sing}(E)\setminus Z$ consists of RDPs, or it is $1\mathrm{EDP}_{2,8}^{(0)}$ . In the latter case, since we know that the affine surface $E\setminus Z$ is the quotient by $G^{\prime}$ , we can apply the blow-up construction once more. Thus, in each case, we obtain a model $\mathcal{X}^{\prime\prime}$ that is strictly semistable in the broad sense outside RDPs on the exceptional component. In each case, moreover, again by Claim 4.4 (1c) and (2d), the configuration of the RDPs on the special fiber is one of $16A_{1},4D_{4}^{0},2D_{8}^{0},2E_{8}^{0},1D_{16}^{0}$ . We apply Proposition 3.3 (16 times) to $16A_{1}$ on the generic fiber and obtain a proper model that is strictly semistable in the broad sense.

Applying Theorem 4.2, we conclude that $X=\operatorname{Km}(A)$ has potential good reduction.

5. Examples

We give explicit examples of abelian surface for which $16A_{1}$ , $4D_{4}^{0}$ , $2D_{8}^{0}$ occur as $\operatorname{Sing}(E)\setminus Z$ . Also we show that $\operatorname{Sing}(E)\setminus Z$ may differ between isogenous abelian surfaces.

Let $\mathcal{O}$ be a discrete valuation ring as before (in particular $\operatorname{char}K=0$ and $\operatorname{char}k=2$ ). The $\mathfrak{p}$ -adic valuation is denoted (additively) by $v\colon\mathcal{O}\to\mathbb{Q}_{\geq 0}\cup\{\infty\}$ .

Elliptic curves over $K$ having good reduction can be written, after replacing $\mathcal{O}$ , in the form $Y^{2}+cXY+Y=X^{3}$ with $c\in\mathcal{O}$ . Let us write this curve $E_{c}$ . The reduction of $E_{c}$ is defined by the same equation (with coefficients considered modulo $\mathfrak{p}$ ), and is supersingular if and only if $c\in\mathfrak{p}$ . Using coordinate change $u=\frac{X}{Y}$ and $\overline{u}=-\frac{X}{Y+cX+1}$ , we obtain the form $u+\overline{u}+cu\overline{u}-(u\overline{u})^{2}=0$ , with origin at $u=\overline{u}=0$ , and the inversion map given by $u\leftrightarrow\overline{u}$ .

Let $c_{1},c_{2}\in\mathfrak{p}$ and consider $A_{c_{1},c_{2}}:=E_{c_{1}}\times E_{c_{2}}$ , $u+\overline{u}+c_{1}u\overline{u}-(u\overline{u})^{2}=v+\overline{v}+c_{2}v\overline{v}-(v\overline{v})^{2}=0$ . This abelian surface have good superspecial reduction. Using the notations of Section 4.3.1, we have $a=-c_{1}x+x^{2}$ and $b=-c_{2}y+y^{2}$ . Then $\epsilon$ is chosen so that $v(\epsilon)=\min\{\frac{1}{2}v(c_{1}),\frac{1}{2}v(c_{2}),\frac{1}{3}v(2)\}$ . We observe that $\operatorname{Sing}(E)\setminus Z$ is

•

$16A_{1}$ if $4\epsilon^{-6}\in\mathcal{O}^{*}$ , equivalently if both $v(c_{1})$ and $v(c_{2})$ are $\geq\frac{2}{3}v(2)$ ,
•

$4D_{4}^{0}$ if $4\epsilon^{-6}\in\mathfrak{p}$ and both $\epsilon^{-2}c_{1}\in\mathcal{O}^{*}$ and $\epsilon^{-2}c_{2}\in\mathcal{O}^{*}$ , equivalently if $v(c_{1})=v(c_{2})<\frac{2}{3}v(2)$ ,
•

$2D_{8}^{0}$ if $4\epsilon^{-6}\in\mathfrak{p}$ and either $\epsilon^{-2}c_{1}\in\mathfrak{p}$ or $\epsilon^{-2}c_{2}\in\mathfrak{p}$ , equivalently if $v(c_{1})$ and $v(c_{2})$ are different and at least one is $<\frac{2}{3}v(2)$ ,

Next, we will see that isogenous abelian varieties may result in different configuration of singularities on $E\setminus Z$ . Consider another elliptic curve $E_{c_{1}^{\prime}}$ that admits an isogeny to $E_{c_{1}}$ of degree $2$ (if $c_{1}$ is generic, then there exist exactly $3$ such elliptic curves up to isomorphism). Using the formula $j(E_{c})=c^{3}(c^{3}-24)^{3}(c^{3}-27)^{-1}$ and the explicit form of the modular polynomial $\Phi_{2}(X,Y)$ , we observe that

•

if $v(c_{1})<\frac{2}{3}v(2)$ , then there exists such $c_{1}^{\prime}$ with $v(c_{1}^{\prime})=\frac{1}{2}v(c_{1})$ ,
•

if $v(c_{1})\geq\frac{2}{3}v(2)$ , then there exists such $c_{1}^{\prime}$ with $v(c_{1}^{\prime})=\frac{1}{3}v(2)$ .

We conclude that, while $A_{c_{1},c_{2}}$ and $A_{c_{1}^{\prime},c_{2}}$ are isogenous, the resulting configurations of singularities on $\operatorname{Sing}(E)\setminus Z$ differ.

Acknowledgments

I thank Hiroyuki Ito, Kazuhiro Ito, Tetsushi Ito, Teruhisa Koshikawa, Ippei Nagamachi, Hisanori Ohashi, Teppei Takamatsu, and Fuetaro Yobuko for helpful comments and discussions.

Supersingular reduction of Kummer surfaces in residue characteristic 22

Abstract.

2010 Mathematics Subject Classification:

1. Introduction

Question 1.1.

Theorem 1.2.

2. Preliminaries

2.1. Some elliptic double points

Definition 2.1.

Remark 2.2.

Remark 2.3.

Proposition 2.4 (Katsura [Katsura:Kummer2]).

Proof.

2.2. Quotients by group schemes of length 22 in mixed characteristic

Example 2.5.

Proposition 2.6.

Proof.

Lemma 2.7.

Proof.

3. Non-extreme cases

3.1. Singularities of Kummer surfaces in the non-extreme cases

Proposition 3.1.

3.2. Good reduction of Kummer surfaces in the non-extreme cases

Proposition 3.2.

Proof.

Proposition 3.3.

Proof.

Proposition 3.4.

Proof.

4. Proof of main theorem

4.1. Strict semistability in the broad sense

Definition 4.1.

Theorem 4.2.

Proof.

4.2. Overview of the blow-up construction

Claim 4.3.

Claim 4.4.

Claim 4.5.

Remark 4.6.

Remark 4.7.

4.3. Explicit computation of the blow-up

4.3.1. Case of EDP4,4(r)\mathrm{EDP}_{4,4}^{(r)}

4.3.2. Case of EDP2,8(r)\mathrm{EDP}_{2,8}^{(r)}

4.4. End of Proof of Theorem 1.2

5. Examples

Acknowledgments

References

Supersingular reduction of Kummer surfaces in residue characteristic $2$

2.2. Quotients by group schemes of length $2$ in mixed characteristic

4.3.1. Case of $\mathrm{EDP}_{4,4}^{(r)}$

4.3.2. Case of $\mathrm{EDP}_{2,8}^{(r)}$