# On logarithmic reduction of cohomologically tame
elliptic surfaces

Otto Overkamp and Arne Smeets

###### Abstract

We give cohomological criteria for logarithmic good reduction of elliptic
surfaces up to modification. Along the way, we prove several more general
results about such surfaces in positive characteristic, as well as about log
smooth morphisms.

###### Contents

  1. 1 Introduction
  2. 2 Elliptic surfaces over separably closed fields
    1. 2.1 Minimal proper regular models
    2. 2.2 Auxiliary results on elliptic surfaces over arbitrary fields
    3. 2.3 Weierstrass models
    4. 2.4 Global Weierstrass models
  3. 3 Tameness of the discriminant locus
  4. 4 Removing horizontal log structure
  5. 5 Logarithmic good reduction

##  1 Introduction

Let $\operatorname{\mathcal{O}}_{K}$ be a complete discrete valuation ring
with residue field $\kappa$ and field of fractions $K.$ Throughout, we shall
assume that $\mathrm{char}\,\kappa\not\in\\{2,3\\},$ and that $\kappa$ is
algebraically closed. Choose an algebraic closure
$K\operatorname{{{}^{\mathrm{alg}}}}$ of $K$ and let
$K\operatorname{{{}^{\mathrm{sep}}}}$ be the separable closure of $K$ in
$K\operatorname{{{}^{\mathrm{alg}}}}$. In this paper, we shall be interested
in studying the reduction of elliptic surfaces in the realm of logarithmic
geometry. Throughout this paper, we shall use the following

###### Definition 1.1

Let $k$ be an arbitrary field. An elliptic surface over $k$ is a morphism
$f\colon X\to C,$ where $X$ is a smooth, projective, and geometrically
integral algebraic surface over $k$ and $C$ is a smooth, projective and
geometrically integral curve over $k$ with generic point $\eta$ with the
following two properties:
(i) The generic fibre $X_{\eta}$ of $f$ is smooth over $\kappa(\eta)$ with
$\dim_{\kappa(\eta)}H^{1}(X_{\eta},\operatorname{\mathcal{O}}_{X_{\eta}})=1$,
and
(ii) the morphism $f$ admits a section $\sigma\colon C\to X.$

This definition has several immediate consequences. For example, the morphism
$f$ is automatically flat, and $X_{\eta}$ is, in fact, an elliptic curve over
$\kappa(\eta).$ By passing to the limit, one can furthermore show that there
exists a non-empty open subset $U\subseteq C$ such that the base change of $f$
to $U$ is smooth. Note that many authors impose additional conditions on the
morphism $f$ when defining elliptic surfaces. For example, it is often
required that the fibration be minimal (see Definition 2.2), or that $f$
possess at least one fibre which is not smooth over the residue field. We
shall not impose any such restrictions; in particular, surfaces of the form
$E\times_{\operatorname{\operatorname{Spec}}k}C$, where $E$ is an elliptic
curve over $k$ and $C$ a curve as in the Definition, will count as elliptic
surfaces over $k,$ and our results apply to such surfaces. However, many
results (such as our main theorem) will be formulated for minimal elliptic
surfaces; this will always be stated explicitly.
Now let $f\colon X\to C$ be a minimal elliptic surface over the discretely
valued field $K$ and let $\ell$ be a prime number invertible in
$\operatorname{\mathcal{O}}_{K}.$ Then, for $i\geq 0,$ the finitely generated
$\operatorname{\mathbf{Z}}_{\ell}$-modules
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{Z}}_{\ell})$
are representations of the absolute Galois group
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$ of
$K$ in a natural way (all cohomology in this paper will be étale cohomology
unless stated otherwise). This Galois group contains the subgroup $P,$ which
is commonly called the wild inertia subgroup, defined as
$P:=\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K^{\mathrm{t}}),$
where $K^{\mathrm{t}}$ is the maximal tamely ramified subextension of
$K\subseteq K\operatorname{{{}^{\mathrm{sep}}}}.$ If $M$ is an Abelian group
on which
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$
acts, we shall call the representation $M$ tamely ramified if the induced
representation of $P$ on $M$ is trivial. Intuitively speaking, it is expected
that tamely ramified representations should, when studying the reduction of
algebraic varieties over $K$ in the realm of logarithmic geometry, play a
similar role to that played by unramified representations in the classical
case. The results of the present paper can be seen as confirming this
philosophy in the case of elliptic surfaces. Our main result is

###### Theorem 1.2

Let $f\colon X\to C$ be a minimal elliptic surface over $K.$ Suppose that one
of Condition 1 or Condition 2 (see p. 14 and p. 15, respectively) are
satisfied. Let $D$ be the formal sum of closed points $x$ of $C$ such that the
fibre of $f$ above $x$ is not smooth. Assume that there exists an effective
divisor $A$ on $C$ which is étale and tamely ramified over $K$ such that

$\deg_{K}A+\deg_{K}D+2\dim_{K}H^{1}(C,\operatorname{\mathcal{O}}_{C})>2,$

and such that the supports of $D$ and $A$ do not intersect. Then there exists
a regular scheme $\mathscr{X}$ together with a flat and projective morphism
$f\colon\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
with the following properties:
(i) The morphism
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
is log smooth with respect to the log structure given by the special fibres,
and
(ii) There exists a projective modification
$\mu\colon\mathscr{X}\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}}\operatorname{\operatorname{Spec}}K\to
X.$ Furthermore, the morphism $f$ maps the exceptional locus of $\mu$ into the
support of $D.$

See Theorem 5.2 for a more precise statement. Condition 1 is satisfied, for
example, if all singular fibres of $f$ have at least two geometric irreducible
components and the Galois representations
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{Q}}_{\ell})$
are tamely ramified. The numerical condition involving $D,$ $A,$ and
$H^{1}(C,\operatorname{\mathcal{O}}_{C})$ is only of interest if the genus of
$C$ is equal to 0 or 1, as the condition is always satisfied otherwise even if
$D=A=0.$ If the genus is equal to 1, the condition excludes base curves which
do not have points in tamely ramified extensions of $K;$ for such curves, the
question of logarithmic good reduction is much more delicate than otherwise.
If the genus of $C$ is zero (which implies
$C\cong\operatorname{\mathbf{P}}^{1}_{K}$ in our situation), the condition is
always satisfied as well. However, if $\deg D\in\\{1,2\\},$ one would have to
choose $A$ inside the open subset of $C$ above which $f$ is smooth (see
Theorem 5.3 for the case $\deg D=0$). The relationship of cohomological
tameness (i.e., tame ramification of the Galois representations
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{Z}}_{\ell})$)
and logarithmic good reduction has been investigated before in various
situations; if $X$ is a curve, see [15]; if $X$ is an Abelian variety, see
[3].

$\mathbf{Acknowledgement}.$ This paper was written while the first author was
a postdoctoral researcher at Leibniz Universität Hannover and revised while he
was working at the Mathematical Institute of the University of Oxford,
supported by a research fellowship from the German Research Foundation
(Deutsche Forschungsgemeinschaft; Geschäftszeichen OV 163/1-1, Projektnummer
442615504). He would like to express his gratitude to all three of those
institutions for their support. The second author acknowledges support of an
NWO Veni grant (639.031.756) and a KU Leuven start-up grant.

##  2 Elliptic surfaces over separably closed fields

Let $k$ be a separably closed field of characteristic $p\not\in\\{2,3\\},$ and
denote by $k{\operatorname{{{}^{\mathrm{alg}}}}}$ an algebraic closure of $k.$
Suppose $f\colon X\to C$ is an elliptic surface over $k$. By assumption, $f$
is flat and smooth away from finitely many points on $C$. Assume that the
fibration $X\to C$ admits a section $\sigma\colon C\to X.$ Let $x\in C$ be a
closed point. Since
$C_{k\operatorname{{{}^{\mathrm{alg}}}}}:=C\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}}\to
C$ is a homeomorphism, there is a unique closed point $\overline{x}$ of
$C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ lying above $x.$ The corresponding
local rings $\operatorname{\mathcal{O}}_{C,x}$ and
$\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
are excellent discrete valuation rings satisfying

$\operatorname{\mathcal{O}}_{C,x}\otimes_{k}k\operatorname{{{}^{\mathrm{alg}}}}=\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}.$

###### Lemma 2.1

Let $\kappa(x)$ denote the the residue field of $x$. The local extension of
discrete valuation rings
$\operatorname{\mathcal{O}}_{C,x}\subseteq\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
has ramification index equal to $[\kappa(x):k]$, i.e., the normalised
valuation of a uniformiser of $\operatorname{\mathcal{O}}_{C,x}$ in
$\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
is equal to $[\kappa(x):k]$.

Proof. Let $y\in
C_{\kappa(x)}:=C\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\kappa(x)$
be the unique closed point of $C_{\kappa(x)}$ lying above $x.$ Then we can
factor the extension of discrete valuation rings as
$\operatorname{\mathcal{O}}_{C,x}\subseteq\operatorname{\mathcal{O}}_{C_{\kappa(x)},y}\subseteq\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}.$
We claim that the index of ramification of the first extension is
$[\kappa(x):k],$ whereas the index of the second one is $1$. To prove the
first claim, note that we have

$\operatorname{\mathcal{O}}_{C_{\kappa(x)},y}=\operatorname{\mathcal{O}}_{C,x}\otimes_{k}\kappa(x).$

In particular, the first extension is finite, of degree $[\kappa(x):k].$ On
the other hand, the residue field extension of
$\operatorname{\mathcal{O}}_{C,x}\subseteq\operatorname{\mathcal{O}}_{C_{\kappa(x)},y}$
is trivial. Hence we obtain the first claim. For the second one, let $\xi$ be
a function defined in a Zariski neighbourhood of $X$ which cuts out $y$
scheme-theoretically. Then the image of $\xi$ in
$\operatorname{\mathcal{O}}_{C_{\kappa(x)},y}$ is a uniformiser. The subscheme
cut out by the pullback of $\xi$ to $C_{k\operatorname{{{}^{\mathrm{alg}}}}}$
is isomorphic to
$\operatorname{\operatorname{Spec}}\kappa(x)\times_{\operatorname{\operatorname{Spec}}\kappa(x)}\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}},$
which is reduced. Hence this pullback of $\xi$ is a uniformiser of
$\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
and the claim follows. $\Box$

###  2.1 Minimal proper regular models

The goal of this subsection is to show that the elliptic surface
$\overline{f}\colon X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ is minimal as well. Let us first
recall the following definition (adapted from [5, Definition 3.8]):

###### Definition 2.2

Let $S$ be a Dedekind scheme with generic point $\eta$, and let
$\mathscr{E}\to S$ be a flat, proper morphism such that $\mathscr{E}$ is
regular and such that the generic fibre $\mathscr{E}_{\eta}$ is an elliptic
curve. Then $\mathscr{E}$ is a minimal proper regular model of
$\mathscr{E}_{\eta}$ if and only if the following holds: if
$\mathscr{E}\to\mathscr{E^{\prime}}$ is a morphism from $\mathscr{E}$ to
another regular proper model $\mathscr{E}^{\prime}$ of $\mathscr{E}_{\eta}$
which becomes an isomorphism on the generic fibre, then
$\mathscr{E}\to\mathscr{E^{\prime}}$ is a global isomorphism.

If $E$ is any elliptic curve over the field of fractions of $S$, then there
exists a unique minimal proper regular model of $E$ over $S$ (see, e.g., [4,
Theorem 1.2]). The minimal proper regular model $\mathscr{E}$ of a given
elliptic curve $E$ satisfies the following universal property: For every
proper regular model $\mathscr{E}^{\prime\prime}$ of $E$ over $R$, there
exists a unique morphism $\mathscr{E}^{\prime\prime}\to\mathscr{E}$ respecting
the identification of the generic fibres (see [5, Remark 3.11]).

Let $\mathscr{E}\to S$ be a proper, flat morphism $\mathscr{E}\to S$, with
$\mathscr{E}$ regular, with generic fibre an elliptic curve $E$. Then we have
the following criterion which tells us whether $\mathscr{E}$ is minimal:

###### Proposition 2.3

(Castelnuovo’s criterion) Let $S$ be a Dedekind scheme with generic point
$\eta,$ and let $s$ be a closed point of $S.$ Let $\mathscr{E}\to S$ be a
proper and flat morphism such that $\mathscr{E}$ is regular and such that
$\mathscr{E}_{\eta}$ is an elliptic curve over
$\operatorname{\operatorname{Spec}}\kappa(\eta).$ Then the following are
equivalent:
(i) the scheme $\mathscr{E}$ is a minimal proper regular model of its generic
fibre, and
(ii) for any closed point $s\in S,$ the scheme
$\mathscr{E}\times_{S}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{S,s}$
is a minimal proper regular model of its generic fibre, and
(iii) for any closed point $s\in S,$ there does not exist an irreducible
component $Z$ of $\mathscr{E}_{s}$ (endowed with its reduced closed subscheme
structure) with the following property: Let
$k_{Z}:=\Gamma(Z,\operatorname{\mathcal{O}}_{Z});$ this is a finite field
extension of $k.$ Denote by $\iota\colon Z\to\mathscr{E}$ the canonical closed
immersion. Then

$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{\mathscr{E}}(Z)=-1$

and $Z\cong\operatorname{\mathbf{P}}^{1}_{k_{Z}}.$

Proof. This is an immediate consequence of [4], Theorems 1.2, 2.1, and 3.1.
$\Box$
Assume that $f\colon X\to C$ is a minimal elliptic surface over $k.$ We will
now prove that the irreducible components of a singular fibre are projective
lines over the appropriate residue fields, provided that the fibre has at
least two irreducible components.

###### Lemma 2.4

Let $f\colon X\to C$ be a minimal elliptic surface over $k.$ Let $x\in C$ be a
closed point such that the fibre above $x$ has at least two irreducible
components. Then each component is isomorphic to
$\operatorname{\mathbf{P}}^{1}_{\kappa(x)},$ and $\kappa(x)=k.$

Proof. We begin by showing that $\operatorname{\mathbf{P}}^{1}_{k}$ can be
characterised as follows: Let $Z$ be a proper integral curve over $k$ such
that $\Gamma(Z,\operatorname{\mathcal{O}}_{Z})=k$ and
$H^{1}(Z,\operatorname{\mathcal{O}}_{Z})=0$, and assume that $Z$ admits a line
bundle of $k$-degree not divisible by $p=\mathrm{char}\,k.$ Then
$Z\cong\operatorname{\mathbf{P}}^{1}_{k}.$

Indeed, we see straight away that $Z$ is regular: If it were not, then then
its normalisation would be isomorphic to
$\operatorname{\mathbf{P}}^{1}_{k^{\prime}}$ for some non-trivial finite
extension $k^{\prime}$ of $k$ [14, Tag 0DJB]. In particular, all line bundles
on $Z$ would have degree divisible by $[k^{\prime}:k]$; since $k$ is separably
closed, this is impossible. Therefore $Z$ is regular, and $Z$ is isomorphic to
a plane projective curve of degree $2$ over $k$ (see [14, Tag 0C6N]). Moreover
$Z$ has a $k$-rational point since $\mathrm{char}\ k>2$, which implies that
$Z\cong\operatorname{\mathbf{P}}^{1}_{k}$, as desired.

Let us now assume that $Z$ is an irreducible component of the fibre of $f$
above $x$, endowed with its reduced subscheme structure. To see that
$\Gamma(Z,\operatorname{\mathcal{O}}_{Z})=\kappa(x)=k$, let
$k_{Z}:=\Gamma(Z,\operatorname{\mathcal{O}}_{Z})$, which is a finite extension
of $\kappa(x)$, and consider the commutative diagram

${X_{k_{Z}}}$${Z}$${X.}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$$\scriptstyle{\iota_{Z}}$

Note that $X_{k_{Z}}$ is smooth over $k_{Z}.$ The same argument as in the
proof of Lemma 2.1 shows that
$\pi^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z)^{\otimes{[k_{Z}:k]}},$
which implies

$\displaystyle[k_{Z}:k]\deg_{k_{Z}}\iota_{Z}^{\ast}\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z)=\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z).$

It follows from the adjunction formula

$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=-2+2\dim_{k_{Z}}H^{1}(Z,\operatorname{\mathcal{O}}_{Z})$

(see [5, pp. 20f]; here we use minimality), together with the fact that
$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)<0$, that
$\dim_{k_{Z}}H^{1}(Z,\operatorname{\mathcal{O}}_{Z})=0$ and
$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=-2.$ We conclude
that $k_{Z}=k$ and therefore also $k_{Z}=\kappa(x)$, which finishes the proof.
$\Box$
Like the previous results in the section, the following statement is probably
well-known to experts, but we were not able to find a reference; the proof is
not complicated.

###### Proposition 2.5

Let $f\colon X\to C$ be a minimal elliptic surface over $k$. Then the induced
fibration $\overline{f}\colon X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ is minimal as well.

Proof. Let $x$ be a closed point of $C$ above which the fibre of $X$ is
singular, and let $\overline{x}$ be the unique preimage of $x$ in
$C_{k\operatorname{{{}^{\mathrm{alg}}}}}.$ We will show that no irreducible
component of $\overline{f}^{-1}(\overline{x})$ satisfies the conditions from
Castelnuovo’s criterion. If $\overline{Z}$ were such a component, then since
$f^{-1}(x)$ and $\overline{f}^{-1}(\overline{x})$ are homeomorphic, there
would be a unique component $Z$ of $f^{-1}(x)$ such that
$\overline{Z}=(Z\times_{\operatorname{\operatorname{Spec}}\kappa(x)}\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}})_{\mathrm{red}}.$
It is clear that $\overline{Z}$ cannot be the only component of
$\overline{f}^{-1}(\overline{x})$. Therefore Lemma 2.4 tells us that $Z$ is
geometrically reduced over $\kappa(x)=k,$ so that we have
$\overline{Z}=Z\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}}$
scheme-theoretically. We now consider the diagram

${\overline{Z}}$${X_{k\operatorname{{{}^{\mathrm{alg}}}}}}$${Z}$${X.}$$\scriptstyle{\overline{\iota}}$$\scriptstyle{\operatorname{\mathrm{pr}}_{Z}}$$\scriptstyle{\operatorname{\mathrm{pr}}}$$\scriptstyle{\iota}$

Then we have
$\operatorname{\mathrm{pr}}^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=\operatorname{\mathcal{O}}_{X_{k\operatorname{{{}^{\mathrm{alg}}}}}}(\overline{Z}),$
and hence

$\deg_{k\operatorname{{{}^{\mathrm{alg}}}}}\overline{\iota}^{\ast}\operatorname{\mathcal{O}}_{X_{k\operatorname{{{}^{\mathrm{alg}}}}}}(\overline{Z})=\deg_{k}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=-2,$

contradicting our initial assumption that
$\deg_{k\operatorname{{{}^{\mathrm{alg}}}}}\overline{\iota}^{\ast}\operatorname{\mathcal{O}}_{X_{k\operatorname{{{}^{\mathrm{alg}}}}}}(\overline{Z})=-1$.
$\Box$

###### Proposition 2.6

Let $k$ be an arbitrary field of characteristic not equal to $2$ or $3$. Let
$f\colon X\to C$ be a minimal elliptic surface over $k$. Then the induced
morphisms $X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ and
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\to
C_{k\operatorname{{{}^{\mathrm{sep}}}}}$ are also minimal.

Proof. By Proposition 2.5, all we have to show is that
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\to
C_{k\operatorname{{{}^{\mathrm{sep}}}}}$ is minimal. Let $x\in C$ be a closed
point and let $y\in C_{k\operatorname{{{}^{\mathrm{sep}}}}}$ be a preimage of
$x$ under the canonical projection. The local extension
$\operatorname{\mathcal{O}}_{C,x}\subseteq\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}$
has ramification index $1$. Since $X$ is smooth over $k$, we know that
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}$ is smooth over
$k\operatorname{{{}^{\mathrm{sep}}}}$, and therefore
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\times_{C_{k\operatorname{{{}^{\mathrm{sep}}}}}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}$
is a proper regular model of its generic fibre. A minimal Weierstrass equation
of
$X\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}$
also yields a Weierstrass equation of
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\times_{C_{k\operatorname{{{}^{\mathrm{sep}}}}}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}$
over $\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}$.
Tate’s algorithm (see [13, pp. 364ff] and [16, Lemma 1]) shows that the
reduction type of the special fibre of the minimal proper regular model
$\mathscr{E}$ of the generic fibre of
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\times_{C_{k\operatorname{{{}^{\mathrm{sep}}}}}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}$
is the same as the reduction type of
$X\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}.$
This implies that the canonical morphism
$X_{k\operatorname{{{}^{\mathrm{sep}}}}}\times_{C_{k\operatorname{{{}^{\mathrm{sep}}}}}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{sep}}}}},y}\to\mathscr{E}$
is an isomorphism, which proves our claim. $\Box$

###  2.2 Auxiliary results on elliptic surfaces over arbitrary fields

We now collect some auxiliary results on elliptic surfaces $f\colon X\to C$
over general base fields $k$, which we do not assume to be perfect or
separably closed. These results show that minimal elliptic surfaces are quite
“common”, in the sense that any elliptic surface can be turned into a minimal
one without losing smoothness, and that any K3 surface which admits the
structure of an elliptic surface is automatically minimal. Of course such
results are well-documented over more restrictive classes of base fields.

###### Proposition 2.7

Let $f\colon X\to C$ a (not necessarily minimal) elliptic surface over $k$,
and let $f^{\prime}\colon X^{\prime}\to C$ be the model of the generic fibre
of $f$ obtained by contracting an exceptional curve on $X$ contained in a
fibre of $f$. Then $X^{\prime}$ is smooth over $k$. In particular, the minimal
proper regular model of $f$ is smooth over $k.$

Proof. Let $Z$ be a an exceptional curve, i.e., a reduced irreducible
component of a fibre of $f$ such that, if
$k_{Z}:=\Gamma(Z,\operatorname{\mathcal{O}}_{Z})$ and $\iota\colon Z\to X$
denotes the canonical closed immersion, we have
$Z\cong\operatorname{\mathbf{P}}^{1}_{k_{Z}}$ and
$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=-1$. Let $\ell$ be
the separable closure of $k$ in $k_{Z}.$ Consider the commutative diagram

${X_{k_{Z}}}$${Z}$${X_{\ell}}$${X,}$$\scriptstyle{\pi}$$\scriptstyle{\iota_{k_{Z}}}$$\scriptstyle{\iota_{\ell}}$$\scriptstyle{\iota}$$\scriptstyle{\pi^{\prime}}$

where
$X_{\ell}:=X\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\ell$
and similarly for $k_{Z}.$ The map $Z\to
Z\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\ell$
identifies $Z$ (scheme-theoretically) with a connected component of
$Z\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\ell.$
Hence we can find an effective divisor $D$ on $X_{\ell}$ such that
$\pi^{\prime\ast}\operatorname{\mathcal{O}}_{X}(Z)\cong\operatorname{\mathcal{O}}_{X_{\ell}}(Z+D)$
and such that the support of $D$ does not intersect that of $Z.$ In
particular, we have
$\iota_{\ell}^{\ast}\pi^{\prime\ast}\operatorname{\mathcal{O}}_{X}(Z)\cong\iota_{\ell}^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z),$
whence

$-1=\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=\deg_{k_{Z}}\iota_{\ell}^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z).$

We claim that, in fact, $k_{Z}=\ell.$ Indeed, we have

$-1=\deg_{k_{Z}}\iota_{\ell}^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z)=\deg_{k_{Z}}\iota_{k_{Z}}^{\ast}\pi^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z)=[k_{Z}:\ell]\deg_{k_{Z}}\iota_{k_{Z}}^{\ast}\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z).$

The third equality can be seen as follows. Recall that $\pi$ is a
homeomorphism, let $\mathfrak{z}$ be a generic point of $Z$ in $X_{\ell}$, and
let $\mathfrak{z}^{\prime}$ be its unique preimage in $X_{k_{Z}}$. Then
$\operatorname{\mathcal{O}}_{X_{\ell},\mathfrak{z}}\subseteq\operatorname{\mathcal{O}}_{X_{k_{Z}},\mathfrak{z}^{\prime}}$
is a finite extension of discrete valuation rings with ramification index
equal to $[k_{Z}:\ell].$ This implies that
$\pi^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z)=\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z)^{\otimes[k_{Z}:\ell]}.$
Therefore we obtain that $[k_{Z}:\ell]=1.$

The assumptions imply that we can contract $Z:$ We obtain a regular scheme
$X^{\prime}\to C$ together with a contraction map $X\to X^{\prime}$. It
remains to prove that $X^{\prime}$ is smooth over $k.$

All of $Z$ is mapped to a single point $q\in X^{\prime}$ with the property
that $\kappa(q)=\ell$. Let $\tilde{\ell}$ be a Galois closure of $\ell$ over
$k$. It suffices to show that
$X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}$
is smooth over $\tilde{\ell}$. Let
$\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}$ denote the perfect closure
of $\tilde{\ell}.$ Then
$\operatorname{\operatorname{Spec}}\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}\to\operatorname{\operatorname{Spec}}\tilde{\ell}$
is a universal homeomorphism, and we want to show that
$X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}$
is regular. This is clear at all points of
$X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}$
which do _not_ lie over the image of
$\operatorname{\operatorname{Spec}}\ell\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}\to
X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell};$
this image consists of finitely many $\tilde{\ell}$-rational points of
$X\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}.$

Let $x$ be a closed point in this image, with unique preimage $y$ in the fibre
product
$X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}.$
Denoting
$X^{\prime}_{\tilde{\ell}}:=X^{\prime}\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}\tilde{\ell}$
(and similarly for $\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}$), we have

$\operatorname{\mathcal{O}}_{X^{\prime}_{\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}},y}=\operatorname{\mathcal{O}}_{X^{\prime}_{\tilde{\ell}},x}\otimes_{\tilde{\ell}}\,\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}.$

Since $\kappa(x)=\tilde{\ell}$, the maximal ideals of those local rings
satisfy

$\mathfrak{m}_{X^{\prime}_{\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}},y}=\mathfrak{m}_{X^{\prime}_{\tilde{\ell}},x}\otimes_{\tilde{\ell}}\,\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}.$

This shows in particular that
$\mathfrak{m}_{X^{\prime}_{\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}},y}$
is generated by two elements; indeed, $X^{\prime}$, and hence also
$\operatorname{\mathcal{O}}_{X^{\prime}_{\tilde{\ell}},x},$ are known to be
regular. We conclude that
$\operatorname{\mathcal{O}}_{X^{\prime}_{\tilde{\ell}\operatorname{{{}^{\mathrm{perf}}}}},y}$
is regular, as required. $\Box$
We now show that elliptic K3 surfaces are minimal. Recall that an elliptic K3
surface over $k$ is an elliptic surface $f\colon X\to C$ over $k$ such that
$\omega_{X/k}\cong\operatorname{\mathcal{O}}_{X}$ and
$H^{1}(X,\operatorname{\mathcal{O}}_{X})=0.$

###### Proposition 2.8

Let $X$ be a K3 surface over $k$ and let $k_{Z}$ be a finite extension of $k$.
Let $Z:=\operatorname{\mathbf{P}}^{1}_{k_{Z}}$ and let $\iota\colon Z\to X$ be
a closed immersion over $k$. Then
$\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)\not=-1.$

Proof. Assume the Proposition is false. We first reduce to the case where
$k_{Z}$ is purely inseparable over $k.$ As above, let $\ell$ be the separable
closure of $k$ in $k_{Z}$ and consider the commutative diagram (2.2). The same
argument then shows that
$\iota_{\ell}^{\ast}\operatorname{\mathcal{O}}_{X_{\ell}}(Z)\cong\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z).$

Therefore we may now assume that $\ell=k$, $\pi^{\prime}=\mathrm{Id}_{X}$ and
$\iota_{\ell}=\iota$. Again as above, we have
$\pi^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z)^{\otimes[k_{Z}:k]}.$
The morphism $\iota_{Z}$ commutes with the canonical projections from $Z$ and
$X_{k_{Z}}$ to $\operatorname{\operatorname{Spec}}k_{Z}$, both of which are
smooth. Now adjunction tells us that

$\omega_{Z/k_{Z}}^{\otimes[k_{Z}:k]}\cong\iota_{Z}^{\ast}\Big{(}\operatorname{\mathcal{O}}_{X_{k_{Z}}}(Z)\underset{{\operatorname{\mathcal{O}}_{X_{k_{Z}}}}}{\otimes}\omega_{X_{k_{Z}}/k_{Z}}\Big{)}^{\otimes[k_{Z}:k]}\cong\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z).$

Since $Z\cong\operatorname{\mathbf{P}}^{1}_{k_{Z}}$, we obtain

$-1=\deg_{k_{Z}}\iota^{\ast}\operatorname{\mathcal{O}}_{X}(Z)=[k_{Z}:k]\deg_{k_{Z}}\omega_{Z/k_{Z}}=-2[k_{Z}:k],$

which is clearly impossible. $\Box$

###  2.3 Weierstrass models

In this section, we once again assume that $k$ is separably closed. We will
recall (and refine) some basic results on Weierstrass models of elliptic
curves, as well as their discriminants.

###### Definition 2.9

(Compare [5, Definition 2.1]) Let $R$ be a discrete valuation ring, and let
$\eta$ be the generic point of $\mathrm{Spec}\,R$. A planar integral
Weierstrass model (or Weierstrass model for short) of an elliptic curve $E$
over $\eta$ is a pair $(W,i)$ consisting of a closed subscheme
$W\subseteq\operatorname{\mathbf{P}}^{2}_{R}$ given by an equation

$y^{2}z+a_{1}xyz+a_{3}yz^{2}-x^{3}-a_{2}x^{2}z-a_{4}xz^{2}-a_{6}z^{3}=0,$

together with an isomorphism $i\colon E\to W_{\eta}$ sending the unit section
of $E$ to $[0:1:0].$

To $(W,i)$ as above, we associate the elements

$\displaystyle b_{2}$ $\displaystyle:=$ $\displaystyle a_{1}^{2}+4a_{2}$
$\displaystyle b_{4}$ $\displaystyle:=$ $\displaystyle 2a_{4}+a_{1}a_{3}$
$\displaystyle b_{6}$ $\displaystyle:=$ $\displaystyle a_{3}^{2}+4a_{6}$
$\displaystyle b_{8}$ $\displaystyle:=$ $\displaystyle
a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^{2}$

of $R,$ as well as the discriminant

$\Delta:=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}.$

It follows from [5, Corollary 2.9] that any Weierstrass model of $E$ which is
isomorphic to $W$ as a model of $E$ yields a discriminant which is an
$R^{\times}$-multiple of $\Delta$. Hence the valuation of $\Delta$ only
depends on the pair $(W,i)$ and not on the particular equation chosen.

###### Definition 2.10

A Weierstrass model $(W,i)$ of an elliptic curve $E$ over $R$ is minimal if
its discriminant has minimal valuation among all possible Weierstrass models
of $E.$

There is only one minimal Weierstrass model of a given elliptic curve, up to
(unique) isomorphism of models; the minimal proper regular model of $E$ is
equal to the minimal regular resolution of the minimal Weierstrass model (see
[5, Corollaries 4.6 and 4.7]).

The following result is an immediate consequence of Szydło’s recent extension
of Tate’s algorithm to the case of potentially imperfect residue fields [16]:

###### Lemma 2.11

Let $R$ be a discrete valuation ring, and let $\eta$ and $s$ denote the
generic and closed points respectively of $\mathrm{Spec}\,R$. Assume that
$\kappa(s)$ is separably closed, of characteristic neither $2$ nor $3$. Let
$E$ be an elliptic curve over $\eta$ with minimal proper regular model
$\mathscr{E}\to\operatorname{\operatorname{Spec}}R.$ Let $\nu$ be the
valuation of the discriminant of a minimal Weierstrass model of $E$ over $R.$
Finally, let $m$ be the number of geometric irreducible components of
$\mathscr{E}_{s}$, counted without multiplicities, and let

$\epsilon:=\begin{cases}-1&\text{if $E$ has good reduction},\\\ 0&\text{if $E$
has multiplicative reduction},\\\ 1&\text{if $E$ has additive
reduction}.\end{cases}$

Then $m=\nu-\epsilon.$

Proof. This follows from [16, Lemma 1, Tables 1 and 3]. $\Box$
As usual, we define the terms good reduction etc. to mean the following:
Suppose $\mathscr{M}$ is the Néron model of $E$ over $R$ with identity
component $\mathscr{M}^{0}.$ Choose an algebraic closure
$\kappa(s)\operatorname{{{}^{\mathrm{alg}}}}$ of the residue field of $R.$
Then $E$ has good reduction (resp. multiplicative reduction, additive
reduction) if
$\mathscr{M}^{0}\times_{\operatorname{\operatorname{Spec}}R}\operatorname{\operatorname{Spec}}\kappa(s)\operatorname{{{}^{\mathrm{alg}}}}$
is an elliptic curve (resp. isomorphic to $\operatorname{\mathbf{G}_{m}}$,
$\mathbf{G}_{\mathrm{a}}$) over $\kappa(s)\operatorname{{{}^{\mathrm{alg}}}}$.

###### Lemma 2.12

In the setting of Lemma 2.11, we have $\nu=0$ if and only if $\mathscr{E}$ is
$R$-smooth.

Proof. Let $W$ denote a minimal Weierstrass model of $E$. By [5, Corollary
4.7], we know that there is a unique morphism of models $\mathscr{E}\to W$,
which turns $\mathscr{E}$ into the minimal regular resolution of $W$. A well-
known calculation shows that $W$ is smooth if and only if $\nu=0$. Hence if
$\nu=0$, it follows immediately that $\mathscr{E}\to W$ is an isomorphism, and
hence that $\mathscr{E}$ is smooth over $R$. Conversely, if $\mathscr{E}$ is
smooth over $R$, then [5, Theorem 5.4] implies that the canonical morphism
$\mathscr{E}\to\mathscr{M}$ from $\mathscr{E}$ to the Néron model
$\mathscr{M}$ of $E$ over $R$ is an isomorphism. In particular,
$\mathscr{M}_{s}$ is connected, and [5, Theorem 5.5] implies that $W$ is
isomorphic to $\mathscr{M}$, so that $W$ is $R$-smooth. We have already
observed that this implies $\nu=0.$ $\Box$

###### Lemma 2.13

In the setting of Lemma 2.11, let $(W,i)$ be a Weierstrass model of $E$.
Assume that there is a morphism $\mathscr{E}\to W$ of models. Then $(W,i)$ is
a minimal Weierstrass model.

Proof. The morphism $\pi\colon\mathscr{E}\to W$ is certainly proper and
birational. Furthermore, its fibres cannot contain any exceptional curves:
$\mathscr{E}$ is minimal by assumption, and therefore its special fibre does
not contain any exceptional curves at all. Hence $\pi$ turns $\mathscr{E}$
into the minimal regular resolution of $W$, by [6, Theorem 2.2.2]; here we use
that $W$ is normal, which follows from [5, Lemma 2.3]. The result now follows
from [5, Corollary 4.7]. $\Box$

We are now ready to prove the main result of this section; recall that
$\mathrm{char}\,k\not\in\\{2,3\\}$.

###### Theorem 2.14

Let $f\colon X\to C$ be a minimal elliptic surface over $k$. Let $x\in C$ be a
closed point such that the fibre of $f$ above $x$ is singular. Then
$\kappa(x)=k$.

Proof. Let $\overline{x}$ be the unique preimage of $x$ under the projection
$C_{k\operatorname{{{}^{\mathrm{alg}}}}}\to C$, and let
$\operatorname{\mathcal{O}}_{C,x}\subseteq\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
be the corresponding extension of discrete valuation rings. We shall prove
that the quantities $m$ and $\epsilon$ (see Lemma 2.11) do not change when
passing from $k$ to $k\operatorname{{{}^{\mathrm{alg}}}}$, whereas $\nu$ is
replaced by $[\kappa(x):k]\nu.$ This will force $\nu=0$ or $\kappa(x)=k$. More
precisely, let $f_{k\operatorname{{{}^{\mathrm{alg}}}}}\colon
X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ be the induced elliptic surface over
$k\operatorname{{{}^{\mathrm{alg}}}}.$ Let
$X_{\operatorname{\mathcal{O}}_{C,x}}:=\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}\times_{C}X$
and
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}:=\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}\times_{C_{k\operatorname{{{}^{\mathrm{alg}}}}}}X_{k\operatorname{{{}^{\mathrm{alg}}}}}.$
Then there is a natural isomorphism

$\displaystyle
X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}=(\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}})\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}X_{\operatorname{\mathcal{O}}_{C,x}}.$
(1)

By Proposition 2.5, we know that $X_{\operatorname{\mathcal{O}}_{C,x}}$ and
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}$
are the minimal proper regular models of their respective generic fibres. Let
$m,\epsilon,\nu$ be the quantities from Lemma 2.11 associated with
$X_{\operatorname{\mathcal{O}}_{C,x}}$ and let
$m^{\prime},\epsilon^{\prime},\nu^{\prime}$ be the analogous quantities for
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}.$
We then have $m=m^{\prime}$, since
$\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}}\to\operatorname{\operatorname{Spec}}\kappa(x)$
is a universal homeomorphism.

Next, we observe that
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}\to
X_{\operatorname{\mathcal{O}}_{C,x}}$ is a homeomorphism. Hence there exists a
unique open $U\subseteq X_{\operatorname{\mathcal{O}}_{X}}$ such that
$U\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
is the smooth locus of
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}$
over
$\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}.$
Because smoothness is fpqc-local on the target, it follows that $U$ is
contained in the smooth locus of $X_{\operatorname{\mathcal{O}}_{C,x}}$ over
$\operatorname{\mathcal{O}}_{C,x}.$ On the other hand, this locus is clearly
contained in $U$, so we have an equality. Together with [5, Theorem 5.4], this
implies that $\epsilon=\epsilon^{\prime}.$

Finally, let us denote by $(W,i)$ a minimal Weierstrass model of the generic
fibre of $X_{\operatorname{\mathcal{O}}_{C,x}}.$ After choosing an equation
for $W$ as in Definition 2.9, we may calculate the corresponding discriminant
$\Delta\in\operatorname{\mathcal{O}}_{C,x}$. Clearly,
$W\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
is still a Weierstrass model of the generic fibre of
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}$.
Our chosen equation gives an equation for
$W\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
as well, and the corresponding discriminant is simply the image of $\Delta$ in
$\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$.
It now follows from [5, Corollary 4.7] that there is a proper birational map
$X_{\operatorname{\mathcal{O}}_{C,x}}\to W$ which respects the identifications
at the generic fibre. Hence there also exists a map
$X_{\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}}\to
W\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
with the same properties. Lemma 2.13 now tells us that
$W\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C_{k\operatorname{{{}^{\mathrm{alg}}}}},\overline{x}}$
is a minimal Weierstrass model. Using Lemma 2.1, we conclude that
$\nu^{\prime}=[\kappa(x):k]\nu.$ This forces $\nu=0$ or $[\kappa(x):k]=1$.
Since the former would contradict the assumption that $f^{-1}(x)$ be singular
(see Lemma 2.12), we conclude that $\kappa(x)=k$. $\Box$
$\mathbf{Remark}.$ The requirement that $f\colon X\to C$ be minimal is, in
fact, redundant. Indeed, if $f$ is not minimal and $Z\subseteq X$ is an
exceptional curve contained in some fibre of $f$, then
$Z\cong\operatorname{\mathbf{P}}^{1}_{k}$. Indeed, we know that
$Z\cong\operatorname{\mathbf{P}}^{1}_{k_{Z}}$ for some finite extension
$k_{Z}$ of $k$, and the proof of Proposition 2.5 shows that $k_{Z}$ is
separable over $k$. This shows that if $\tilde{f}\colon X^{\mathrm{min}}\to C$
is the minimal proper regular model of the generic fibre of $f$ and $x\in C$
is a closed point such that the canonical morphism $X\to X^{\mathrm{min}}$
does not induce an isomorphism of fibres above $x$ then $\kappa(x)=k.$ In
particular, if $\tilde{f}^{-1}(x)$ is smooth but $f^{-1}(x)$ is not, then
$\kappa(x)=k.$

$\mathbf{Remark.}$ Let $\eta$ be the generic point of $C.$ Theorem 2.14 can be
interpreted as a statement about elliptic curves over $\kappa(\eta)$. Indeed,
if $E$ is such an elliptic curve, we can construct the minimal proper regular
model $\mathscr{E}\to C$ of $E$. If the ground field is not perfect, there is
nothing which guarantees that
$\mathscr{E}\to\operatorname{\operatorname{Spec}}k$ be smooth. The previous
result gives us a method to construct examples where this is not the case. Let
$p>3$ be the characteristic of $k$ and assume that $a\in k$ is not a $p$-th
power. Let $C=\operatorname{\mathbf{P}}^{1}_{k}$ and let $E$ be the elliptic
curve given by the minimal Weierstrass equation $y^{2}=x^{3}+(t^{p}-a)$.
Clearly, $E$ has reduction of type $II$ at the closed point $x$ of
$\mathbf{A}^{1}_{k}\subseteq\operatorname{\mathbf{P}}^{1}_{k}$ given by the
ideal $\langle t^{p}-a\rangle\subseteq k[t].$ Since the residue field
$\kappa(x)=k(a^{1/p})$ is not separable over $k$, Theorem 2.14 tells us that
$\mathscr{E}$ is not smooth over $k$.

###  2.4 Global Weierstrass models

In this section, we denote by $k$ an arbitrary field, and by $f\colon X\to C$
a minimal elliptic surface over $k$, equipped with a fixed section
$\sigma\colon C\to X$. As before, we denote by $\eta$ the generic point of
$C$. We then have the following.

###### Proposition 2.15

There exists a unique pair $(W,\pi)$, with $w:W\to C$ projective and
$\pi\colon X\to W$ birational, such that for each closed point $x\in C$, the
pair
$(W\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x},\pi_{\eta})$
is a minimal Weierstrass model of
$X\times_{C}\operatorname{\operatorname{Spec}}\kappa(\eta)$. Furthermore, the
scheme $W$ is normal, and the fibres of $W\to C$ are geometrically integral
and have at most one geometric singularity.

Proof. Let $x_{1},\dots,x_{n}$ be the closed points of $C$ above which the
fibre of $f$ is singular, and let $U:=C\setminus\\{x_{1},\dots,x_{n}\\}.$ For
each $j=1,\dots,n,$ let $w_{j}\colon
W_{j}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x_{j}}$
be the minimal Weierstrass model of $X_{\eta}$, and let $W_{0}:=f^{-1}(U)$.
The section $\sigma\colon C\to X$ induces sections
$\sigma_{j}\colon\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x_{j}}\to
W_{j}$ and $\sigma_{0}\colon U\to W_{0}$. By abuse of notation, we shall
denote the images of the sections by the same symbol as the sections
themselves.

The line bundles
$\mathscr{L}_{j}:=\operatorname{\mathcal{O}}_{W_{j}}(\sigma_{j})$ are
relatively ample; this follows from the proof of [5, Theorem 2.8] for
$j=1,\dots,n$ and is well-known for $j=0$. Let $U_{0},\dots,U_{n}$ denote the
schemes
$U,\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x_{1}},\dots,\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x_{n}}$
respectively. For each pair of integers $0\leq i,j\leq n$, denote by $U_{ij}$
the scheme $U_{i}\times_{C}U_{j}.$ As soon as $i\not=j,$ there is a canonical
isomorphism

$U_{ij}=\operatorname{\operatorname{Spec}}\kappa(\eta),$

which means that, in this case, $W_{i}\times_{C}U_{ij}$ can be canonically
identified with $X_{\eta}$, and the pullback of $\mathscr{L}_{j}$ to this
scheme is canonically isomorphic to
$\operatorname{\mathcal{O}}_{X_{\eta}}(\sigma_{\eta}).$ Hence we obtain a
descent datum of pairs $(W_{i},\mathscr{L}_{i})$ with respect to the fpqc
covering $U_{0}\sqcup U_{1}\sqcup\cdots\sqcup U_{n}\to C$. By [2, §6.1,
Theorem 7], this descent datum is effective, which settles the existence part.

For the uniqueness, we simply observe that Lemma 2.13 implies that the
$X\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}$
are the minimal Weierstrass models of their respective generic fibres.
Therefore the descent datum obtained from $W\to C$ with respect to the fpqc
covering described above is canonically isomorphic to the one we used
previously to prove existence.

To see that $W$ is normal, let $t$ be a (set-theoretic) point of $W.$ If $t$
maps to $U_{0},$ then $W$ is smooth at $t$. If $t$ maps to one of the $x_{j},$
it suffices to observe that the $W_{j}$ are normal [5, Lemma 2.3]. The last
claim follows from [5, Lemma 2.3] $\Box$
We shall refer to $W$ as the Weierstrass model of $f\colon X\to C$.

###### Proposition 2.16

Let $w\colon W\to C$ be the Weierstrass model of $X.$ Let $x_{1},\dots,x_{r}$
be the closed points of $C$ above which the fibre of $f$ has at least two
irreducible components. For $j=1,\dots,r$, there exists a unique closed point
$z_{j}$ of $W$ above $x_{j}$ where $W$ is singular, and the extension
$k\subseteq\kappa(z_{j})$ is purely inseparable. If
$\mathrm{char}\,k\not\in\\{2,3\\},$ then $\kappa(z_{j})=k$ for all $j.$

Furthermore, the map $\pi\colon X\to W$ is an isomorphism above
$W\setminus\\{z_{1},\dots,z_{r}\\}$, and $\pi^{-1}(z_{j})$ is set-
theoretically the union of the irreducible components of $f^{-1}(x_{j})$ which
do not meet $\sigma$.

Proof. We already know that $X_{\operatorname{\mathcal{O}}_{C,x}}\to
W_{x}=W\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}$
is a minimal regular resolution for all closed points $x\in C$. In particular,
if $f^{-1}(x)$ is irreducible, then $W_{x}$ must be regular. On the other
hand, one can check easily (using equations) that $w^{-1}(x_{j})$ has
precisely one point $z_{j}$ which is not smooth over $\kappa(x_{j})$. Clearly,
$W_{x_{j}}$ is regular at all other points. If $W$ were regular at $z_{j}$,
then the morphism $X_{\operatorname{\mathcal{O}}_{C,x_{j}}}\to W_{x_{j}}$
would have to be an isomorphism, which contradicts our assumption on
$f^{-1}(x_{j})$. The morphism $X\to W$, being a minimal regular resolution, is
an isomorphism above the regular locus of $W$. The remaining claims follow
from [5], more precisely from the proof of Corollary 4.7 and the discussion
after the proof of Theorem 5.5 of loc. cit. $\Box$

###### Lemma 2.17

Let $W\to C$ be the Weierstrass model of $X.$ Then
$W\times_{\operatorname{\operatorname{Spec}}k}\operatorname{\operatorname{Spec}}k\operatorname{{{}^{\mathrm{alg}}}}$
is the Weierstrass model of $\overline{f}\colon
X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}.$

Proof. Clearly $W_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{k\operatorname{{{}^{\mathrm{alg}}}}}$ still has the property that its
localisations at closed points of $C_{k\operatorname{{{}^{\mathrm{alg}}}}}$
are Weierstrass models of their generic fibres; this can be checked using
equations. Furthermore, we still have a birational morphism
$X_{k\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{k\operatorname{{{}^{\mathrm{alg}}}}}$ which commutes with the projections
to $C_{k\operatorname{{{}^{\mathrm{alg}}}}}.$ Hence Lemma 2.13 in combination
with Proposition 2.15 implies the result. $\Box$

##  3 Tameness of the discriminant locus

Now let $K$ be a complete discretely valued field with ring of integers
$\operatorname{\mathcal{O}}_{K}$ and algebraically closed residue field of
characteristic not dividing 6. Let $f\colon X\to C$ be a minimal elliptic
surface over $K.$ We keep the notation from the previous sections; in
particular, $w\colon W\to C$ will be the global Weierstrass model of $X$,
$\pi\colon X\to W$ the corresponding projection, and for a closed point $x\in
C,$ we let
$X_{\operatorname{\mathcal{O}}_{C,x}}:=X\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x},$
and
$W_{x}:=W\times_{C}\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{C,x}.$
An immediate consequence of our previous work is the following

###### Proposition 3.1

Let $x$ be a closed point of $C$ such that the fibre of $f$ above $x$ is
singular. Then the extension $K\subseteq\kappa(x)$ is separable.

Proof. This is an immediate consequence of Theorem 2.14 and Proposition 2.6.
$\Box$
We shall now introduce the cohomological conditions that will later be shown
to guarantee logarithmic good reduction up to modification. First choose an
algebraic closure $K\operatorname{{{}^{\mathrm{alg}}}},$ and let
$K\operatorname{{{}^{\mathrm{sep}}}}$ be the separable closure of $K$ in
$K\operatorname{{{}^{\mathrm{alg}}}}.$ Then the group
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$
acts on $K\operatorname{{{}^{\mathrm{alg}}}}$ in a canonical way. Furthermore,
for any $\ell$ invertible in $\operatorname{\mathcal{O}}_{K},$ the
$\operatorname{\mathbf{Q}}_{\ell}$-vector spaces
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{Q}}_{\ell})$
and
$H^{i}(X_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell})$
are canonically identified; the same is true for cohomology with coefficients
$\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}$ (this follows
from the topological invariance of étale cohomology). In particular, we obtain
representations of
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$ on
$H^{i}(X_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell})$
(resp.
$H^{i}(X_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{F}}_{\ell})$).
These coincide with the (more commonly used) representations on
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{Q}}_{\ell})$
(resp.
$H^{i}(X_{K\operatorname{{{}^{\mathrm{sep}}}}},\operatorname{\mathbf{F}}_{\ell})$),
but our definition will turn out to be more convenient for our purposes.
Let $f^{\prime}\colon X^{\prime}\to C$ be an elliptic surface over $K$ which
is birational to $X,$ and which has the property that each singular fibre of
$f^{\prime}$ has at least two geometric irreducible components. As such models
will feature repeatedly in what follows, we make the following

###### Definition 3.2

An elliptic fibration $f^{\prime}\colon X^{\prime}\to C$ over $K$ is called
big if for each closed point $x$ of $C$ such that $f^{\prime}$ is not smooth
above $x,$ the geometric fibre of $f^{\prime}$ above $x$ has at least two
components.

Note that each (not necessarily minimal) elliptic surface $f\colon X\to C$
admits a projective modification $X^{\prime}\to X$ which is big; to construct
one, simply blow up the point hit by the section in each singular fibre of $f$
which has only one geometric irreducible component. Note that the additional
component in the singular fibres constructed this way have multiplicity one
and are geometrically integral. We can now formulate the first cohomological
condition:

  * •

Condition 1: There exists a big elliptic fibration $f^{\prime}\colon
X^{\prime}\to C,$ birational (over $C$) to $X,$ such that for all $i\geq 0,$
the Galois representations
$H^{i}(X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell})$
are tamely ramified.

To formulate the second cohomological condition, we shall first study the
sheaf $\operatorname{\mathrm{R}}^{1}f_{\ast}\operatorname{\mathbf{F}}_{\ell}$.
After proving some auxiliary lemmata, we shall see that this sheaf is
canonically isomorphic to the $\ell$-torsion subsheaf of the identity
component of the Néron model $\mathscr{M}$ of the generic fibre of $f.$

###### Lemma 3.3

The functor $\operatorname{\mathrm{Pic}}^{0}_{W/C}$ is representable by a
smooth group scheme of finite presentation over $C$ with connected fibres, and
the canonical morphism
$\operatorname{\mathrm{Pic}}^{0}_{W/C}\to\mathscr{M}^{0}$ is an isomorphism.
The same statements hold if we replace $W$ by
$W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ and $C$ by
$C_{K\operatorname{{{}^{\mathrm{alg}}}}}.$

Proof. The claim regarding the representability of
$\operatorname{\mathrm{Pic}}^{0}_{C/K}$ follows from [2, Chapter 9.3, Theorem
1]. The fibres of $\operatorname{\mathrm{Pic}}^{0}_{C/K}$ are then connected
by definition. For the second claim we observe that the Néron model commutes
with localisations at closed points of $C,$ as does $W$ (by construction).
Suppose $U\subseteq C$ is the largest open subset of $C$ such that $f$ is
smooth above $U.$ Then $U$ is non-empty, and we let $x_{1},...,x_{n}$ be the
complement of $U.$ It suffices to show that the morphism
$\operatorname{\mathrm{Pic}}^{0}_{W/C}\to\mathscr{M}^{0}$ induces isomorphisms
of the respective localisations at the points $x_{1},$…, $x_{n}.$ This,
however, follows from the fact that minimal Weierstrass models are normal and
have rational singularities (see [5, Corollary 8.4]), together with [2,
Chapter 9.3, Theorem 1]. The remaining claim now follows from Lemma 2.17.
$\Box$
Now let $\pi^{\prime}\colon X^{\prime}\to W$ be the composition of the
canonical morphism $X^{\prime}\to X$ with the projection $\pi\colon X\to W.$
Recall that we have a commutative diagram

${X^{\prime}}$${W}$${C.}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{w}$

We denote by $\overline{\pi}^{\prime},$ $\overline{f}^{\prime},$ and
$\overline{w}$ the morphisms obtained from those in the diagram by base change
to $K\operatorname{{{}^{\mathrm{alg}}}}.$

###### Lemma 3.4

There is an isomorphism
$\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\operatorname{\mathrm{R}}^{1}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
which is equivariant with respect to the canonical action of
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$ on
both sheaves.

Proof. The Leray spectral sequence for the composition
$\overline{f}^{\prime}=\overline{w}\circ\overline{\pi}^{\prime}$ gives us a
short exact sequence

$0\to\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}(\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell})\to\operatorname{\mathrm{R}}^{1}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\overline{w}_{\ast}\operatorname{\mathrm{R}}^{1}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}.$

We observe that the morphism
$\operatorname{\mathbf{F}}_{\ell}\to\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
is an isomorphism, so we have at least already constructed a morphism with the
correct source and target. We also know already that this map is injective.
The morphism $\overline{\pi}^{\prime}\colon
X_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ is an isomorphism outside finitely
many points of $W_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{K\operatorname{{{}^{\mathrm{alg}}}}},$ and the reduced preimages of those
points are trees of copies of
$\operatorname{\mathbf{P}}^{1}_{K\operatorname{{{}^{\mathrm{alg}}}}}.$ Hence
the proper base change theorem tells us that the stalk of
$\operatorname{\mathrm{R}}^{1}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
is zero at those points, and it is certainly zero at all other points as well.
Hence the morphism we constructed above is indeed an isomorphism. The claim
about equivariance follows from the fact that $\overline{f}^{\prime},$
$\overline{w},$ and $\overline{\pi}^{\prime}$ are compatible with the actions
of $\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$
on the schemes $X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},$
$W_{K\operatorname{{{}^{\mathrm{alg}}}}},$ and
$C_{K\operatorname{{{}^{\mathrm{alg}}}}}.$ $\Box$

###### Lemma 3.5

There is an isomorphism
$\operatorname{\mathrm{R}}^{1}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\operatorname{\mathrm{Pic}}^{0}_{W_{K\operatorname{{{}^{\mathrm{alg}}}}}/C_{K\operatorname{{{}^{\mathrm{alg}}}}}}[\ell]$
which is
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$-equivariant.

Proof. Since the map
$\operatorname{\mathbf{G}_{m}}\to\overline{w}_{\ast}\operatorname{\mathbf{G}_{m}}$
is an isomorphism on the small étale site of
$C_{K\operatorname{{{}^{\mathrm{alg}}}}},$ we see that the Kummer sequence
gives us a short exact sequence

$0\to\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{G}_{m}}\overset{[\ell]}{\to}\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{G}_{m}}$

Since the (geometric) fibres of $\overline{w}$ are integral curves, it follows
that
$\operatorname{\mathrm{Pic}}^{0}_{W_{K\operatorname{{{}^{\mathrm{alg}}}}}/C_{K\operatorname{{{}^{\mathrm{alg}}}}}}$
is the same as the kernel of the degree map
$\operatorname{\mathrm{Pic}}_{W_{K\operatorname{{{}^{\mathrm{alg}}}}}/C_{K\operatorname{{{}^{\mathrm{alg}}}}}}\to\operatorname{\mathbf{Z}},$
which implies

$\displaystyle\operatorname{\mathrm{Pic}}^{0}_{W_{K\operatorname{{{}^{\mathrm{alg}}}}}/C_{K\operatorname{{{}^{\mathrm{alg}}}}}}[\ell]$
$\displaystyle=\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{G}_{m}}[\ell]$
$\displaystyle=\operatorname{\mathrm{R}}^{1}\overline{w}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
$\displaystyle=\operatorname{\mathrm{R}}^{1}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}$

on the small étale site of $C_{K\operatorname{{{}^{\mathrm{alg}}}}},$ where
the last equality is due to the preceding Lemma. The second claim follows from
the fact that both $\overline{w}$ and $\overline{f}^{\prime}$ are equivariant
with respect to the action of
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$ on
$W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ and
$C_{K\operatorname{{{}^{\mathrm{alg}}}}},$ together with (again) the preceding
Lemma. $\Box$
The same argument shows that there is an isomorphism

$\operatorname{\mathrm{R}}^{1}f^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\mathscr{M}^{0}[\ell];$

in fact, the morphism constructed in the Lemma is the pull-back of this map.
As a corollary to this observation, we can construct an integral regular curve
$C^{\prime}$ of finite type over $K$ together with a flat and generically
étale morphism $\epsilon\colon C^{\prime}\to C,$ such that, at the generic
point of $C^{\prime},$ the sheaf
$\operatorname{\mathrm{R}}^{1}f^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
is constant (and isomorphic to $\operatorname{\mathbf{F}}_{\ell}^{2}$). If we
denote by $\overline{\epsilon}\colon
C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{K\operatorname{{{}^{\mathrm{alg}}}}}$ the induced morphism, we are now
ready to formulate our second cohomological condition:

  * •

Condition 2: The Galois representation on
$H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{F}}_{\ell}\oplus\overline{\epsilon}_{\ast}\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{F}}_{\ell})$
is tamely ramified.

For later use, we shall prove now a Lemma about the curve $C^{\prime}:$

###### Lemma 3.6

The curve
$C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}:=C^{\prime}\times_{\operatorname{\operatorname{Spec}}K}\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{alg}}}}$
is reduced and equal to the scheme-theoretic disjoint union of its irreducible
components.

Proof. Because the map $\epsilon$ is generically smooth, we know that so is
the morphism $C^{\prime}\to\operatorname{\operatorname{Spec}}K.$ Hence there
exists a non-empty (hence dense) open subscheme $U\subseteq C^{\prime}$ which
is smooth over $K.$ For any open subscheme $V\subseteq C^{\prime}$, the map
$\operatorname{\mathcal{O}}_{C^{\prime}}(V)\to\operatorname{\mathcal{O}}_{C^{\prime}}(U\cap
V)$ is injective, and remains so after base change to
$K\operatorname{{{}^{\mathrm{alg}}}}.$ But the target is geometrically
reduced, so the first claim follows. The second claim is purely topological;
since the map
$\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{alg}}}}\to\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{sep}}}}$
is a universal homeomorphism, it suffices to prove the corresponding claim for
$C^{\prime}\times_{\operatorname{\operatorname{Spec}}K}\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{sep}}}}.$
But this scheme is regular (as $C^{\prime}$ is regular and $K\subseteq
K\operatorname{{{}^{\mathrm{sep}}}}$ is separable), so no two irreducible
components can intersect non-trivially. $\Box$

###### Theorem 3.7

Let $f\colon X\to C$ be a minimal elliptic surface over $K$ which satisfies
one of Condition 1 or Condition 2 above. Let $x\in C$ be a closed point above
which the fibre of $f$ is singular. Then the extension $K\subseteq\kappa(x)$
is tamely ramified.

Proof. The proof of this theorem will occupy the remainder of this paragraph.
Suppose first that Condition 2 is satisfied, and assume that
$K\subseteq\kappa(x)$ is wildly ramified. Suppose that $\eta$ is the generic
point of $C^{\prime},$ and let
$\gamma\colon\operatorname{\operatorname{Spec}}(\kappa(\eta)\otimes_{K}K\operatorname{{{}^{\mathrm{alg}}}})\to
C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}$ be the canonical map. We
have a Galois-equivariant exact sequence

$0\to\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\gamma_{\ast}\gamma^{\ast}\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}\to\bigoplus_{x\in
S}x_{\ast}M_{x}\to 0,$

where $S$ is the set of $K\operatorname{{{}^{\mathrm{alg}}}}$-points of
$C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}$ above which the order of
the stalk of
$\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
drops. Note that $M_{x}$ is isomorphic to either
$\operatorname{\mathbf{F}}_{\ell}$ or $\operatorname{\mathbf{F}}_{\ell}^{2}.$
Now write

$C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}=C_{1}\sqcup...\sqcup C_{d}$

for the decomposition of $C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}$
into irreducible components. We know from Lemma 3.6 that this is possible. We
now claim that, on each $C_{j},$ there exist at least three
$K\operatorname{{{}^{\mathrm{alg}}}}$-points at which the order of the stalk
of
$\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}$
drops. This follows because the corresponding statement for
$C_{K\operatorname{{{}^{\mathrm{alg}}}}}$ is true, which, in turn, follows
from the fact that the residue characteristic of $K$ is at least three and
that $K\subseteq\kappa(x)$ is wildly ramified. This allows us to deduce that
the action of
$\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$ on
the set $\\{C_{1},...,C_{d}\\}$ is tame. Indeed, the exact sequence above
shows that, for each $j=1,...,d,$ we have

$H^{1}(C_{j},\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}\mid_{C_{j}})\not=0.$

Hence we can find a torsor $\mathscr{T}$ on
$C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}$ which is non-trivial on
$C_{j}$ but trivial on all other components. As Condition 2 guarantees that
the Galois action
$H^{1}(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell})$
is tamely ramified, this means that all elements of the wild inertia subgroup
must fix the component $C_{j}.$ This observation implies that the Galois
action on
$\Gamma(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\gamma_{\ast}\gamma^{\ast}\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell})$
is tamely ramified. Now we consider the exact sequence

$\Gamma(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\gamma_{\ast}\gamma^{\ast}\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell})\to\Gamma(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\bigoplus_{x\in
S}x_{\ast}M_{x})\to
H^{1}(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\overline{\epsilon}^{\ast}\operatorname{\mathrm{R}}^{1}\overline{f}_{\ast}\operatorname{\mathbf{F}}_{\ell}).$

All groups in this sequence are $\operatorname{\mathbf{F}}_{\ell}$-vector
spaces, so taking invariants under the wild inertia group does not destroy
exactness of this sequence. This, however, means that the Galois action on
$\Gamma(C^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\bigoplus_{x\in
S}x_{\ast}M_{x})=\bigoplus_{x\in S}M_{x}$ is tamely ramified, so the same must
be true for the Galois action on $S.$ However, the set of
$K\operatorname{{{}^{\mathrm{alg}}}}$-points of $C$ above which the fibre of
$f$ is singular is exactly the image of $S$ under $\overline{\epsilon},$ so
our claim follows.
Now suppose that Condition 1 is satisfied. Some of the arguments that follow
are inspired by Paragraph 3 of [11]; we provide some details not covered in
loc. cit., particularly relating to Galois equivariance. Until the end of this
proof, we impose the condition that $\overline{f}$ not be smooth, i.e., that
the minimal elliptic fibration $\overline{f}\colon
X_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{K\operatorname{{{}^{\mathrm{alg}}}}}$ have at least one singular fibre,
which clearly causes no loss of generality. First note that we have a Galois
equivariant spectral sequence

$E_{2}^{i,j}=H^{i}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathrm{R}}^{j}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{Q}}_{\ell})\Rightarrow
H^{i+j}(X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell}).$

###### Lemma 3.8

The spectral sequence above has the property that
$E_{2}^{0,2}=E_{\infty}^{0,2}.$

Proof. It follows from general principles that $E_{3}^{0,2}=E_{\infty}^{0,2},$
and that $E_{3}^{0,2}=\ker(E_{2}^{0,2}\to E_{2}^{2,1}).$ The proof of [11,
Proposition 3.1] shows that

$E_{2}^{2,1}=(\varprojlim\operatorname{\operatorname{Hom}}(M[\ell^{n}],\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}))\otimes_{\operatorname{\mathbf{Z}}_{\ell}}\operatorname{\mathbf{Q}}_{\ell},$

where $M$ denotes the Mordell-Weil group of $\overline{f}\colon
X_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
C_{K\operatorname{{{}^{\mathrm{alg}}}}}.$ However, by [12, Theorem 1.1], $M$
is a finitely generated Abelian group (this is where we use the condition on
the existence of a singular fibre of $\overline{f}$; otherwise, the Theorem
from loc. cit. would not apply). Hence $E_{2}^{2,1}=0,$ which implies our
claim. $\Box$
We shall now prove the formula

$\displaystyle
E_{2}^{0,2}=\operatorname{\mathbf{Q}}_{\ell}\oplus\bigoplus_{\text{$Y$
exceptional}}\operatorname{\mathbf{Q}}_{\ell}\langle Y\rangle,$ (2)

where the index set of the direct sum is meant to be the set of irreducible
components of the exceptional locus of the canonical map
$X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
X_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{K\operatorname{{{}^{\mathrm{alg}}}}}.$ The action of some
$\tau\in\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$
is given by sending such a $Y$ to $\tau^{-1}(Y).$ Recall that there is a
Galois equivariant spectral sequence

$E_{2}^{i,j}=\operatorname{\mathrm{R}}^{i}\overline{w}_{\ast}\operatorname{\mathrm{R}}^{j}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}\Rightarrow\operatorname{\mathrm{R}}^{i+j}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}.$

We have the following

###### Lemma 3.9

The spectral sequence above satisfies $E_{2}^{i,j}\not=0$ only if $ij=0$ and
$i+j\leq 2.$ Moreover, the spectral sequence stabilizes at the $E_{2}$-page.

Proof. The map $\overline{\pi}^{\prime}$ is a birational morphism of surfaces,
so
$\operatorname{\mathrm{R}}^{j}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}$
is concentrated at finitely many points as soon as $j\geq 1.$ This implies the
claim that $E_{2}^{i,j}=0$ as soon as $ij\not=0.$ The claim that $E_{2}^{i,j}$
vanishes for $ij=0,$ $i+j\geq 3$ follows from the fact that both
$\overline{\pi}^{\prime}$ and $\overline{w}$ have fibres of dimension at most
one. To see that the spectral sequence stabilises at $E_{2},$ we consider the
only differential map (on any page) whose source and target are both non-zero.
This is the morphism

$\overline{w}_{\ast}\operatorname{\mathrm{R}}^{1}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}\to\operatorname{\mathrm{R}}^{2}\overline{w}_{\ast}(\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}).$

However, the source of this map vanishes by the same argument as in the proof
of Lemma 3.4. $\Box$
Continuing with our proof of Formula (2), we note that the Lemma gives us a
Galois equivariant split exact sequence

$0\to\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}\to\operatorname{\mathrm{R}}^{2}\overline{f}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}\to\overline{w}_{\ast}\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}\to
0$

(for the splitting, see the proof of Lemma 3.2 in [11]). This gives us an
isomorphism

$E_{2}^{0,2}=\operatorname{\mathbf{Q}}_{\ell}\oplus(\varprojlim\Gamma(W_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}))\otimes_{\operatorname{\mathbf{Z}}_{\ell}}\operatorname{\mathbf{Q}}_{\ell}.$

Now let $S$ be the scheme consisting of the spectra of the residue fields of
the closed points of $W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ above which the
map $\overline{\pi}^{\prime}\colon
X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ is not an isomorphism. Let $s\colon
S_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ be the canonical closed immersion.
Further let $X^{\prime}_{S}$ be the pull-back of
$X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}$ along $s.$ To ease
notation, we note that the category of étale Abelian sheaves on $S$ is
canonically equivalent to the category whose objects are Abelian groups
together with a direct sum decomposition indexed by $S$; under this
equivalence, morphisms correspond to decomposable homomorphisms of Abelian
groups. Consider the sheaf $\bigoplus_{Y}\operatorname{\mathbf{Z}}\langle
Y\rangle$ on $S$ (the decomposition comes from the map which sends a component
of the exceptional divisor to an element of $S$). There is a canonical action
on this sheaf under which an element
$\tau\in\operatorname{\operatorname{Gal}}(K\operatorname{{{}^{\mathrm{sep}}}}/K)$
acts as $Y\mapsto\tau^{-1}(Y).$ By Kummer theory, we have a canonical Galois
equivariant isomorphism

$\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}=s_{\ast}s^{\ast}\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}=s_{\ast}(\operatorname{\mathrm{Pic}}_{X_{S}/S}\otimes_{\operatorname{\mathbf{Z}}}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}).$

By [2, Chapter 9.2, Corollary 14] there is a Galois equivariant morphism

$\operatorname{\mathrm{Pic}}_{X_{S}/S}\to\bigoplus_{Y}\operatorname{\mathbf{Z}}\langle
Y\rangle$

whose kernel is $\operatorname{\mathrm{Pic}}^{0}_{X_{S}/S}$ and whose image is
a finite-index equivariant subsheaf
$\Lambda\subseteq\bigoplus_{Y}\operatorname{\mathbf{Z}}\langle Y\rangle.$
Hence we find

$\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}=s_{\ast}(\Lambda\otimes_{\operatorname{\mathbf{Z}}}\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}}).$

Taking limits and tensoring with $\operatorname{\mathbf{Q}}_{\ell},$ we see
that

$\Gamma(W_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathrm{R}}^{2}\overline{\pi}^{\prime}_{\ast}\operatorname{\mathbf{Q}}_{\ell})=\bigoplus_{\text{$Y$
exceptional}}\operatorname{\mathbf{Q}}_{\ell}\langle{Y}\rangle$

as Galois representations, hence proving Formula (2). Now suppose $\Delta$ is
the discriminant locus of the minimal elliptic fibration $f\colon X\to C,$ and
let $\Delta^{\prime}\subseteq C_{K\operatorname{{{}^{\mathrm{alg}}}}}$ be the
set of closed points above which $\overline{f}^{\prime}$ is singular. Then we
have
$\Delta\times_{\operatorname{\operatorname{Spec}}K}\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{alg}}}}\subseteq\Delta^{\prime},$
and because the fibration $f^{\prime}\colon X^{\prime}\to C$ is big, the map
$S\to\Delta^{\prime}$ is surjective. Now Condition 1 tells us, together with
Lemma 3.8, that the Galois action on the set of irreducible components of the
exceptional divisor of $X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}}\to
W_{K\operatorname{{{}^{\mathrm{alg}}}}}$ is tame. Since this set surjects onto
$S,$ we conclude that the Galois action on
$\Delta\times_{\operatorname{\operatorname{Spec}}K}\operatorname{\operatorname{Spec}}K\operatorname{{{}^{\mathrm{alg}}}}$
is tamely ramified as well. This implies our claim. $\Box$

##  4 Removing horizontal log structure

Let $g\colon X\to Y$ be a log smooth morphism of log schemes. Suppose that
both $X$ and $Y$ are (classically) integral, and that the log structures
$\mathscr{M}_{X},$ $\mathscr{M}_{Y}$ on $X$ and $Y$ are the usual
(compactifying) log structures associated with open subsets $U_{X},$ $U_{Y}$
of $X$ and $Y$ whose complements are effective Cartier divisors. We further
impose two conditions, namely that

(i) the scheme $X$ be regular in the classical sense, and the reduced
complement of $U_{X}$ be a divisor with normal crossings, and that

(ii) the log structure $\mathscr{M}_{Y}$ be fine and saturated (fs for short),
i.e., it admit charts by finitely generated saturated integral monoids locally
in the étale topology.

Note that $\mathscr{M}_{X}$ is automatically fine, and that both log
structures are fs. Suppose $D$ is the (reduced) complement of $U_{X}$ in $X.$
We can then write

$D=D_{\mathrm{hor}}+D_{\mathrm{vert}},$

with $D_{\mathrm{hor}}$ and $D_{\mathrm{vert}}$ both effective, where
$D_{\mathrm{hor}}$ is the sum of (some) irreducible components $D_{0}$ of $D$
such that $f(D_{0})\cap U_{Y}\not=\emptyset$ (of course, such a decomposition
is not unique). Note that the complement of $D_{\mathrm{vert}}$ is contained
in $g^{-1}(U_{Y}),$ and beware that $D_{\mathrm{hor}}$ need not be flat over
$Y.$ We define a new log scheme $X^{\prime},$ whose underlying scheme is given
by that of $X,$ and whose log structure $\mathscr{M}_{X}^{\prime}$ is
associated with $U_{X^{\prime}}=X\backslash D_{\mathrm{vert}}.$ Observe that
the morphism of schemes underlying $g$ induces a morphism $g\colon
X^{\prime}\to Y$ of log schemes. The proof of the following Proposition seems
to be similar, at least intuitively, to that of [1, Proposition 3.1], although
the language and methods are somewhat different. Before we prove the
Proposition, however, we shall need the following

###### Lemma 4.1

Let $\overline{x}$ be a geometric point of $X.$ Suppose that the pullback of
$D$ to
$\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{X,\overline{x}}$
is cut out by $x_{1}\cdot...\cdot x_{d}$ such that the sequence
$x_{1},...,x_{d}$ can be extended to a minimal set of generators of
$\mathfrak{m}_{X,\overline{x}}.$ Then
$\mathscr{M}_{X,\overline{x}}\subseteq\operatorname{\mathcal{O}}_{X,\overline{x}}$
is the submonoid generated by $x_{1},...,x_{d}$ and
$\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}.$ In particular,
$\mathscr{M}_{X,\overline{x}}^{\mathrm{gp}}$ is canonically isomorphic to the
subgroup of
$(\operatorname{\operatorname{Frac}}\operatorname{\mathcal{O}}_{X,\overline{x}})^{\times}$
generated by $x_{1},...,x_{d}$ and
$\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}.$

Proof. Let us denote the image of $\overline{x}$ in any scheme to which this
geometric point maps by $x.$ We may assume without loss of generality that the
pullback of $D$ to
$\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{X,x}$ is cut
out by $x_{1}^{\prime}\cdot...\cdot x^{\prime}_{d},$ such that this sequence
extends to a minimal set of generators of $\mathfrak{m}_{X,x}.$ Let
$f\in\mathscr{M}_{X,\overline{x}}.$ Then $f$ is defined (as a function) on
some étale neighbourhood $\overline{x}\to E\to X.$ Observe that both
inclusions

$\operatorname{\mathcal{O}}_{X,x}\subseteq\operatorname{\mathcal{O}}_{E,x}\subseteq\operatorname{\mathcal{O}}_{X,\overline{x}}$

have the property that the maximal ideal in the smaller local ring generates
the maximal ideal in the bigger one (for the first inclusion, this follows
because $E\to X$ is étale; for the second inclusion, because
$\operatorname{\mathcal{O}}_{X,\overline{x}}$ is the strict Henselisation of
$\operatorname{\mathcal{O}}_{E,x}$ with respect to the geometric point
$\overline{x}\to E).$ Hence $x_{1}^{\prime},...,x_{d}^{\prime}$ remain prime
elements in $\operatorname{\mathcal{O}}_{E,x}$ and
$\operatorname{\mathcal{O}}_{X,\overline{x}}.$ We can view $f$ as an element
of $\operatorname{\mathcal{O}}_{E,x},$ and write

$f=\epsilon x_{1}^{\prime\beta_{1}}\cdot...\cdot x_{d}^{\prime\beta_{d}},$

with $\epsilon\in\operatorname{\mathcal{O}}_{E,x}^{\times},$ in a unique way.
This decomposition, and its uniqueness, remain valid in
$\operatorname{\mathcal{O}}_{X,\overline{x}}.$ Up to ordering, we know that
there are
$\epsilon_{1},...,\epsilon_{d}\in\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
such that $x^{\prime}_{j}=\epsilon_{j}x_{j}$ for $j=1,...,d.$ Hence, in
$\operatorname{\mathcal{O}}_{X,\overline{x}},$ we have

$f=\tilde{\epsilon}x_{1}^{\beta_{1}}\cdot...\cdot x_{d}^{\beta_{d}}$

for some $\tilde{\epsilon}\in\operatorname{\mathcal{O}}_{X,\overline{x}},$
which proves the inclusion "$\subseteq$" from the Lemma. The other inclusion
is obvious. The second claim from the Lemma now follows easily from the fact
that $\operatorname{\mathcal{O}}_{X,\overline{x}}$ is a unique factorization
domain. $\Box$

###### Proposition 4.2

The morphism $g\colon X^{\prime}\to Y$ is log smooth.

Proof. We shall break up the proof into several steps. Let $x\in X,$ and let
$\overline{x}$ be a geometric point of $X$ above $x.$ Because of the local
nature of our statement, we may assume that $Y$ is affine and that there is a
global chart $Q\to\Gamma(Y,\operatorname{\mathcal{O}}_{Y})$ of
$\mathscr{M}_{Y}$ around $g(\overline{x}).$ Our other assumptions on $Y$ allow
us also to assume that $Q$ is a finitely generated integral monoid.
Step 1: étale-locally around $\overline{x}$, we shall construct a chart
$P\to\mathscr{M}_{X,\overline{x}}$ of $\mathscr{M}_{X}$ such that $P$ is the
direct sum of a finitely generated Abelian group and a finitely generated free
monoid, together with an injective chart $Q\to P$ of $g.$ This step is an
adaption of the proof of [8, Theorem 3.5]. Locally around $\overline{x},$ we
can choose global sections $t_{1},...,t_{r}$ of $\mathscr{M}_{X}$ such that
$d\log t_{1},...,d\log t_{r}$ are a basis of $\Omega^{1}_{X/Y}$ (we use this
symbol to stand for log differentials). Exactly as in loc. cit., we define a
morphism

$\operatorname{\mathbf{N}}^{r}\oplus Q\to\mathscr{M}_{X,\overline{x}},$

given by $Q\to\mathscr{M}_{Y,g(\overline{x})}\to\mathscr{M}_{X,\overline{x}}$
and $e_{i}\to t_{i}$ for $i=1,...,r.$ This map gives rise to a homomorphism
$\tilde{f}\colon\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}}\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}.$
The same argument as in loc. cit. can be used to show that the cokernel of the
induced map $f\colon\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}}\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
is finite and has cardinality invertible in
$\operatorname{\mathcal{O}}_{X,\overline{x}}.$ By our assumption on $X,$ we
know that
$\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
is a free Abelian group of rank $m$ for some $m\in\operatorname{\mathbf{N}};$
hence the same is true for $\operatorname{\mathrm{im}}f.$
Let $s_{1},...,s_{m}\in\mathscr{M}_{X,\overline{x}}^{\mathrm{gp}}$ be elements
such that their images
$[s_{1}],...,[s_{m}]\in\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
are a basis, and such that $\lambda_{1}[s_{1}],...,\lambda_{m}[s_{m}]$ are a
basis of $\operatorname{\mathrm{im}}f$ for suitable
$\lambda_{1},...,\lambda_{m}\in\operatorname{\mathbf{N}}.$ Observe that all
$\lambda_{j}$ are invertible in $\operatorname{\mathcal{O}}_{X,\overline{x}}.$
Further choose
$\alpha_{1},...,\alpha_{m}\in\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}}$ such that, for all $j=1,...,m,$ we have

$f(\alpha_{j})=\lambda_{j}[s_{j}].$

The choice of the $\alpha_{j}$ gives us a section
$\operatorname{\mathrm{im}}f\to\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}},$ and hence also a retraction
$\rho\colon\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}}\to\ker f.$ Combining these, we obtain a
decomposition

$\operatorname{\mathbf{Z}}^{r}\oplus Q\operatorname{{{}^{\mathrm{gp}}}}=\ker
f\oplus\operatorname{\mathrm{im}}f.$

We define

$G:=\ker
f\oplus\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}.$

Observe that $G$ is a finitely generated Abelian group. Our next aim will be
to construct a morphism $h\colon
G\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}$ extending $\tilde{f},$ such
that the composition
$G\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
is surjective. To this end, we shall need the following
Sublemma: The map
$[-]\colon\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}\to\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
admits a section $\kappa$ such that $\tilde{f}=\kappa\circ f$ on the image of
the section $\operatorname{\mathrm{im}}f\to\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}}.$
To see this, observe that, for $j=1,...,m,$ we have

$\displaystyle[\tilde{f}(\alpha_{j})]$ $\displaystyle=f(\alpha_{j})$
$\displaystyle=\lambda_{j}[s_{j}]$ $\displaystyle=[s_{j}^{\lambda_{j}}].$

This means that there exist
$\epsilon_{j}\in\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$ such
that $\tilde{f}(\alpha_{j})=\epsilon_{j}s_{j}^{\lambda_{j}}.$ As the
$\lambda_{j}$ are invertible in $\operatorname{\mathcal{O}}_{X,\overline{x}},$
this means we can write

$\tilde{f}(\alpha_{j})=(\epsilon_{j}^{1/{\lambda_{j}}}s_{j})^{\lambda_{j}}$

for all $j=1,...,m.$ Hence
$\kappa([s_{j}]):=\epsilon_{j}^{1/\lambda_{j}}s_{j}$ gives rise to a section
with the desired properties.
We can now define $h$ by putting

$h((x,y)):=\tilde{f}(x)\kappa(y)$

for $(x,y)\in G=\ker
f\oplus\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}.$
An easy check shows that $h$ does indeed extend $\tilde{f}$. Furthermore, we
observe that $G/(\operatorname{\mathbf{Z}}^{r}\oplus
Q\operatorname{{{}^{\mathrm{gp}}}})\cong\mathrm{coker}\,f,$ which has
cardinality invertible in $\operatorname{\mathcal{O}}_{X,\overline{x}}.$ As in
loc. cit. we see that the map
$h^{-1}(\mathscr{M}_{X,\overline{x}})=:P\to\mathscr{M}_{X,\overline{x}}$
extends to a chart locally around $\overline{x}.$ From now on, we replace $X$
by such a neighbourhood of $\overline{x}.$ Similarly, we see that the map
$Q\operatorname{{{}^{\mathrm{gp}}}}\to G=P\operatorname{{{}^{\mathrm{gp}}}}$
is injective, that its cokernel’s torsion part is of cardinality invertible in
$\operatorname{\mathcal{O}}_{X,\overline{x}},$ and that the map

$X\to
Y\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[Q]}\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[P]$

is étale in the classical sense.
Step 2: We shall now go on to constructing a chart $P^{\prime}$ of
$\mathscr{M}_{X}^{\prime}$ locally around $\overline{x}$ which we can use to
show that $g\colon X^{\prime}\to Y$ is log smooth. We can find elements
$x_{1},...,x_{a}\in\operatorname{\mathcal{O}}_{X,\overline{x}}$
($a\in\operatorname{\mathbf{N}}$) such that the sequence $x_{1},...,x_{a}$ can
be extended to a minimal sequence which generates the maximal ideal
$\mathfrak{m}_{X,\overline{x}}\subseteq\operatorname{\mathcal{O}}_{X,\overline{x}},$
and such that $D$ is cut out by $x_{1}x_{2}...x_{a},$ with $D_{\mathrm{vert}}$
being cut out by $x_{1}...x_{b}$ for some $0\leq b\leq a.$ Then
$[x_{1}],...,[x_{a}]$ are a basis of
$\mathscr{M}^{\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times},$
whereas $[x_{1}],...,[x_{b}]$ are a basis of
$\mathscr{M}^{\prime\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}.$
Using Lemma 4.1 as well as that $\operatorname{\mathcal{O}}_{X,\overline{x}}$
is regular, and hence a unique factorization domain, we see that the preimage
of
$\mathscr{M}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
in $\mathscr{M}_{X,\overline{x}}^{\mathrm{gp}}$ is
$\mathscr{M}_{X,\overline{x}}$ (and similarly for
$\mathscr{M}^{\prime\mathrm{gp}}_{X,\overline{x}}$). Analogously, we see that
the preimage of
$\mathscr{M}^{\prime}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
in $\mathscr{M}_{X,\overline{x}}^{\mathrm{gp}}$ is
$\mathscr{M}^{\prime}_{X,\overline{x}}.$ Let

$G^{\prime}:=h^{-1}(\mathscr{M}_{X,\overline{x}}^{\prime\mathrm{gp}}).$

Then the induced map
$G^{\prime}\to\mathscr{M}^{\prime\mathrm{gp}}_{X,\overline{x}}\to\mathscr{M}^{\prime\mathrm{gp}}_{X,\overline{x}}/\operatorname{\mathcal{O}}_{X,\overline{x}}^{\times}$
is surjective. If we now define $P^{\prime}$ to be the preimage of
$\mathscr{M}^{\prime}_{X,\overline{x}}$ in $G^{\prime}$ under $h,$ then [8,
Lemma 2.10] tells us that the map $h\colon
P^{\prime}\to\mathscr{M}^{\prime}_{X,\overline{x}}$ can be extended to a chart
in an étale neighbourhood of $\overline{x}.$ Clearly, the map $Q\to P$ from
the chart of $g$ constructed in Step 1 factors through $P^{\prime}.$ We now
replace $X$ by such a neighbourhood of $\overline{x}.$
Step 3: We shall now see that the chart $Q\to P^{\prime}$ of the morphism
$g\colon X^{\prime}\to Y$ (locally around $\overline{x}$) satisfies part (ii)
of [8, Theorem 3.5] as relaxed by Remark 3.6 of loc. cit. Indeed, we have
$P=\ker f\oplus\operatorname{\mathbf{N}}\langle[x_{1}],...,[x_{a}]\rangle,$
and $P^{\prime}=\ker
f\oplus\operatorname{\mathbf{N}}\langle[x_{1}],...,[x_{b}]\rangle.$ In
particular, we have $P\cong P^{\prime}\oplus\operatorname{\mathbf{N}}^{a-b},$
and the morphism $Q\operatorname{{{}^{\mathrm{gp}}}}\to
P\operatorname{{{}^{\mathrm{gp}}}}$ factors through the canonical inclusion
$P^{\prime}\operatorname{{{}^{\mathrm{gp}}}}\to
P\operatorname{{{}^{\mathrm{gp}}}}=P^{\prime}\operatorname{{{}^{\mathrm{gp}}}}\oplus\operatorname{\mathbf{Z}}^{a-b}.$
Hence the torsion parts of the cokernels of
$Q\operatorname{{{}^{\mathrm{gp}}}}\to
P^{\prime}\operatorname{{{}^{\mathrm{gp}}}}$ and
$Q\operatorname{{{}^{\mathrm{gp}}}}\to P\operatorname{{{}^{\mathrm{gp}}}}$ are
isomorphic, showing part (i) of [8, Theorem 3.5] for the new log structure. To
see part (ii), observe that we have a commutative diagram

${X}$${Y\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[Q]}\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[P]}$${(Y\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[Q]}\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[P^{\prime}])\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}}\operatorname{\mathbf{A}}^{a-b}_{\operatorname{\mathbf{Z}}}}$${Y\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[Q]}\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[P^{\prime}]}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{\mathrm{pr}}_{1}}$

showing that the map $X\to
Y\times_{\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[Q]}\operatorname{\operatorname{Spec}}\operatorname{\mathbf{Z}}[P^{\prime}]$
is smooth. This concludes the proof. $\Box$

##  5 Logarithmic good reduction

We shall now prove our main theorem, using the results established in the
preceding sections. We begin with a technical lemma which will be needed
later.

###### Lemma 5.1

Let $K$ be a discretely valued field of residue characteristic not equal to 2
or 3, and let $E$ be an elliptic curve defined over $K.$ We shall denote the
valuation ring of $K$ by $R,$ and the residue field (which we do not assume to
be perfect) by $k.$ Then the finite étale $K$-group scheme $E[3]$ is tamely
ramified with respect to $R.$

Proof. We may replace $R$ by its completion and assume that $R$ is complete.
We can find finite separable extensions $E_{1},...,E_{n}$ such that there is
an isomorphism

$E[3]\cong\operatorname{\operatorname{Spec}}(K\times E_{1}\times...\times
E_{n})$

as schemes, and we know that

$1+\sum_{j=1}^{n}[E_{j}:K]=\operatorname{\operatorname{ord}}_{K}E[3]=9.$

Now observe that the $K$-scheme $E[3]$ admits the non-trivial involution
$\phi:=[-1].$ Suppose $x$ is a (set-theoretic) point of $E[3]$ with residue
field $E_{j}$ for some $j=1,...,n.$ If $\phi(x)\not=x$, then the residue field
at $\phi(x)$ is $E_{i}$ for some $i\not=j.$ In particular, we obtain an
isomorphism $E_{j}\to E_{i}$ of extensions of $K,$ which implies
$[E_{j}:K]\leq 4.$ In particular, the degree of the induced extension of
residue fields must divide 3 or 4, as must the ramification index. This
implies that $E_{j}$ is a tamely ramified extension of $K.$ On the other hand,
suppose that $\phi(x)=x.$ Then $\phi^{\ast}$ induces a non-trivial
$K$-automorphism of $E_{j}$ of order 2. Indeed, if the automorphism of $E_{j}$
induced by $\phi$ were trivial, the
$K\operatorname{{{}^{\mathrm{sep}}}}$-points of $E_{j}$ would be
$K\operatorname{{{}^{\mathrm{sep}}}}$-points of $E[3]\backslash\\{0\\}$ fixed
by $[-1],$ which is impossible since
$E[3](K\operatorname{{{}^{\mathrm{sep}}}})\cong\operatorname{\mathbf{Z}}/3\operatorname{\mathbf{Z}}\times\operatorname{\mathbf{Z}}/3\operatorname{\mathbf{Z}}.$
If $E_{j}^{\phi}$ denotes the field of elements of $E_{j}$ fixed by
$\phi^{\ast},$ then $E_{j}^{\phi}\subseteq E_{j}$ is a Galois extension of
degree 2, so $[E_{j}:K]$ must be even. Since the only prime numbers which
divide even positive integers not exceeding 8 are 2 and 3, we conclude as in
the previous case. $\Box$
We are now ready to prove the main theorem. Recall that $K$ is a complete
discretely valued field with residue field $k,$ which we assume to be
algebraically closed and of characteristic different from 2 and 3. As before,
we choose an algebraic closure $K\operatorname{{{}^{\mathrm{alg}}}}$ of $K$
and let $K\operatorname{{{}^{\mathrm{sep}}}}$ be the separable closure of $K$
in $K\operatorname{{{}^{\mathrm{alg}}}}.$ Further recall that $f\colon X\to C$
is a minimal elliptic surface over $K$ which admits a section $\sigma\colon
C\to X.$ We let $D$ be the formal sum of closed points of $C$ above which the
fibre of $f$ is singular. Then $D$ is a divisor on $C$ which is étale and
tamely ramified over $K$ (see Theorem 3.7 and Proposition 3.1), provided that
Condition 1 or Condition 2 are satisfied.

###### Theorem 5.2

Assume that the minimal elliptic surface $X$ satisfies Condition 1 or
Condition 2 above. Suppose further that there is a divisor $A$ on $C$ which is
étale and tamely ramified over $K$ such that

$\deg_{K}D+\deg_{K}A+2\dim_{K}H^{1}(C,\operatorname{\mathcal{O}}_{C})>2,$

and such that the supports of $D$ and $A$ do not intersect. Then there exists
a regular scheme $\mathscr{X}$ and a projective morphism
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
with smooth generic fibre, such that
(i) the morphism
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
is log smooth with respect to the log structures defined by the special
fibres, and
(ii) there exists a projective modification $\mu\colon\mathscr{X}_{K}\to X.$
Moreover, the morphism
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
factors as
$\mathscr{X}\overset{\phi}{\to}\mathscr{C}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$,
where $\mathscr{C}$ is a regular and projective model of $C$, log smooth over
$\operatorname{\mathcal{O}}_{K}$ with respect to the special fibres, such that
the diagram

${\mathscr{X}_{K}}$${X}$${\mathscr{C}_{K}=}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C}$$\scriptstyle{\mu}$$\scriptstyle{\phi_{K}}$$\scriptstyle{f}$

commutes. The exceptional locus of $\mu$ is mapped into the support of $D$ by
$f.$

Proof. We shall begin by constructing a projective and regular model
$\mathscr{C}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
of $C,$ flat over $\operatorname{\mathcal{O}}_{K},$ with the following
property: Let $U_{\mathscr{C}}$ be the complement of the subscheme $D\subseteq
C.$ Then the reduced complement of $U_{\mathscr{C}}$ in $\mathscr{C}$ is a
divisor with normal crossings, and the morphism
$\mathscr{C}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
is log smooth with respect to the log structure on $\mathscr{C}$ induced by
$U_{\mathscr{C}}$ (unless mentioned otherwise, the log structure on
$\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$ will always
be the one induced by the special point). Let $n>1$ and let
$U_{\mathscr{C}}^{\prime}$ be the complement of $A+D$ in $C$, viewed as an
open subscheme of $\mathscr{C}.$ If Condition 1 holds, then because
$H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Z}}_{\ell})\subseteq
H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell})\subseteq
H^{1}(X^{\prime}_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Q}}_{\ell}),$
we see that
$H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Z}}_{\ell})$
is tamely ramified. If Condition 2 holds, the same conclusion follows
immediately. This implies that
$H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}})\cong\operatorname{\mathrm{Pic}}^{0}_{C_{K\operatorname{{{}^{\mathrm{alg}}}}}/K\operatorname{{{}^{\mathrm{alg}}}}}[\ell^{n}](K\operatorname{{{}^{\mathrm{alg}}}})$
is tamely ramified as well. Furthermore, $A+D$ is étale and tamely ramified
over $K$ (Theorem 3.7). Hence [10, Theorem 1], implication $(3)\Rightarrow(1)$
(with $N=\ell^{n}$) tells us that there exists a model $\mathscr{C}$ of $C$
which is projective, regular, and flat over $\operatorname{\mathcal{O}}_{K}$
which is furthermore log smooth with respect to the log structure induced by
$U^{\prime}_{\mathscr{C}}.$ However, Proposition 4.2 now tells us that we my
replace $U_{\mathscr{C}}^{\prime}$ by $U_{\mathscr{C}}$ without affecting the
conclusion.
The map $f$ induces an elliptic curve $X_{U_{\mathscr{C}}}\to
U_{\mathscr{C}},$ and the section $\sigma$ of $f$ induces a divisor on
$X_{U_{\mathscr{C}}}$ which is obviously étale, tamely ramified, and of degree
1 over $K.$ Hence [10, Theorem 1], implication $(3)\Rightarrow(1)$ (with
$N=3$) together with Lemma 5.1 tell us that we can extend
$X_{U_{\mathscr{C}}}\to U_{\mathscr{C}}$ to a regular, projective and flat
scheme $\mathscr{X}\to\mathscr{C},$ such that, with respect to the log
structure on $\mathscr{X}$ induced by the complement of the section $\sigma$
on $X_{U_{\mathscr{C}}},$ the morphism $\mathscr{X}\to\mathscr{C}$ is log
smooth. Clearly, $\mathscr{C}_{K}\to C$ is a proper regular model of the
generic fibre of $f,$ so there exists a unique modification
$\mu\colon\mathscr{X}_{K}\to X$ compatible with the morphisms down to $C.$ It
is also clear that the map
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
is log smooth with respect to the log structure on $\mathscr{X}$ induced by
$U_{\mathscr{X}}.$ By Proposition 4.2 we may replace $U_{\mathscr{X}}$ by the
generic fibre $\mathscr{X}_{K}$ of
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
without affecting the logarithmic smoothness of the morphism, thus concluding
the proof. $\Box$
In some situations, the conclusions of the last Theorem can be strengthened
further:

###### Theorem 5.3

Let $X\to C$ be an elliptic surface such that $f$ is smooth in the classical
sense (which implies in particular that the elliptic surface is minimal).
Suppose the Galois representation on
$H^{1}(X_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{F}}_{\ell})$
is tamely ramified and that there exists an effective divisor $A$ on $C$ which
is étale and tamely ramified and such that

$\deg A+2\dim_{K}H^{1}(C,\operatorname{\mathcal{O}}_{C})>2.$

Then the statement of Theorem 5.2 can be modified to include the requirement
that $\mu$ be an isomorphism. In particular, $X$ has logarithmic good
reduction.

Proof. Our assumptions imply that for $n>1,$ the Galois representation on

$H^{1}(C_{K\operatorname{{{}^{\mathrm{alg}}}}},\operatorname{\mathbf{Z}}/\ell^{n}\operatorname{\mathbf{Z}})\cong\operatorname{\mathrm{Pic}}^{0}_{C_{K\operatorname{{{}^{\mathrm{alg}}}}}/K\operatorname{{{}^{\mathrm{alg}}}}}[\ell^{n}](K\operatorname{{{}^{\mathrm{alg}}}})$

is tamely ramified (note that the condition from the Theorem follows from
Condition 1). As in the proof of Theorem 5.2 we deduce that there exists a
proper, flat, and regular model
$\mathscr{C}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
which is log smooth with respect to the log structure given by the special
fibre. Using [10, Theorem 1], implication $(3)\Rightarrow(1)$ again, we see
that $X\to C$ can be extended to a log smooth and projective map
$\mathscr{X}\to\mathscr{C},$ and the resulting morphism
$\mathscr{X}\to\operatorname{\operatorname{Spec}}\operatorname{\mathcal{O}}_{K}$
is clearly log smooth. Now we use Proposition 4.2 as above. $\Box$

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