# Coble surfaces in characteristic two

Toshiyuki Katsura  Graduate School of Mathematical Sciences, The University of
Tokyo, Meguro-ku, Tokyo 153-8914, Japan<EMAIL_ADDRESS>and
Shigeyuki Kondō  Graduate School of Mathematics, Nagoya University, Nagoya,
464-8602, Japan<EMAIL_ADDRESS>
###### Abstract.

We study Coble surfaces in characteristic $2$, in particular, singularities of
their canonical coverings. As an application we classify Coble surfaces with
finite automorphism group in characteristic $2$. There are exactly 9 types of
such surfaces.

Research of the first author is partially supported by JSPS Grant-in-Aid for
Scientific Research (C) No.20K03530 and the second author by JSPS Grant-in-Aid
for Scientific Research (A) No.20H00112.

##  1\. Introduction

A Coble surface $S$ is a rational surface with $|-K_{S}|=\emptyset$ and
$|-2K_{S}|=\\{B_{1}+\cdots+B_{n}\\}$ where $B_{i}$ is a non-singular rational
curve and $B_{i}\cap B_{j}=\emptyset$ $(i\not=j)$. This is called a terminal
Coble surface of $K3$ type in Dolgachev and Zhang [5]. We call
$B_{1},\ldots,B_{n}$ the boundary components of $S$. Morrison [18] showed that
a Coble surface appears as a degeneration of Enriques surfaces. Over the
complex numbers, the moduli space of complex Enriques surfaces can be
described as an open subset of an arithmetic quotient of a 10-dimensional
bounded symmetric domain of type IV which is the complement of a Heegner
divisor. A general point of the Heegner divisor corresponds to a Coble
surface. This is a peculiar phenomenon of Enriques surfaces which does not
occur in case of $K3$ surfaces. Coble surfaces inherit many properties of
Enriques surfaces and complement the theory of Enriques surfaces. Thus it is
interesting to study Coble surfaces like as Enriques surfaces.

By definition any Coble surface has a canonical double covering $\pi:X\to S$
defined by $-K_{S}$. In case that the characteristic $p$ of the base field $k$
is not equal to 2, $\pi$ is a separable double covering branched along a non-
singular divisor $B_{1}+\cdots+B_{n}$ and hence $X$ is smooth. In particular
$X$ is a $K3$ surface. Thus one can apply the theory of $K3$ surfaces to the
case of Coble surfaces. On the other hand, in case of $p=2$, $\pi$ is a purely
inseparable $\mu_{2}$-covering, and $S$ has always singularities and might be
non-normal. This situation is similar to the case of Enriques surfaces in
characteristic 2. Recall that Enriques surfaces $Y$ in characteristic 2 are
classified into three types of classical, $\mu_{2}$\- or $\alpha_{2}$-surfaces
according to ${\rm Pic}_{Y}^{\tau}\cong{\bf Z}/2{\bf Z}$, $\mu_{2}$ or
$\alpha_{2}$, respectively (Bombieri and Mumford [1]). A Coble surface in
charcteristic 2 is an analogue of classical Enriques surfaces. The purpose of
this paper is to study the double covering $\pi$ and, as an application, to
give the classification of Coble surfaces with finite automorphism group in
characteristic 2.

In the following we assume the the ground field $k$ is an algebraically closed
field in characteristic 2. First of all, we show that the canonical covering
$X$ of a Coble surface is a $K3$-like surface, that is, the dualizing sheaf
$\omega_{X}$ is trivial and ${\rm H}^{1}(X,{\mathcal{O}}_{X})=0$ (Theorem
3.1). One can define a rational 1-form $\eta$ on $S$ naturally whose divisor
is written as

$(\eta)=-\sum_{i=1}^{n}B_{i}+B$

with $(\sum_{i}B_{i})\cap B=\emptyset$ and $B=2A$ for a divisor $A$ (Theorem
3.6). We call $A$ the conductrix of $S$. We will show the following theorem
which is an analogue of the case of Enriques surfaces.

######  Theorem 1.1.

  * $({\rm 1})$

Assume that $A=0$. Then the singularities of $X$ are all rational double
points and the minimal non-singular model of $X$ is a supersingular $K3$
surface.

  * $({\rm 2})$

Assume that $A\neq 0$. Then $X$ is a rational surface, $A^{2}=-2$ and the
isolated singularities of $X$ are all rational double points of type $A_{1}$
and the number of isolated singularities is equal to $4-n$.

We will study more details of conductrices, and can obtain the list of
possible conductrices $A$, which is the same as the one of Enriques surfaces
given in Ekedahl and Shepherd-Barron [6, Theorems 2.2, 3.1] for $A^{2}=-2$.

Next we will discuss Coble surfaces with finite automorphism group. A general
Coble surface has an infinite group of automorphisms (e.g. Dolgachev and Kondō
[4, Theorem 9.6.1]) and hence it is natural to classify Coble surfaces with
finite automorphism group as in the case of Enriques surfaces. Recall that
Katsura, Kondō and Martin [11] classified classical Enriques surfaces in
characteristic 2 with finite automorphism group: there are exactly 8 classes
of such Enriques surfaces. On the other hand, in case of characteristic $p\neq
2$, the second author [13] recently gave the classification of such Coble
surfaces.

Let $Y$ be an Enriques surface and let ${\rm Num}(Y)$ be the quotient group of
the Néron-Severi group by the torsion subgroup. Then ${\rm Num}(Y)$ is a
lattice of signature $(1,9)$ which is an important role to study Enriques
surfaces. Also any non-singular rational curve on $Y$ has the self-
intersection number $-2$. On the other hand, a non-singular rational curve on
a Coble surface $S$ has the self-intersection number $-1,-2$ or $-4$ and the
Picard number $\rho(S)$ is greater than 10. Instead of the Néron-Severi
lattice ${\rm NS}(S)$ we consider a lattice ${\rm CM}(S)$ of signature $(1,9)$
called Coble-Mukai lattice and defined by the orthogonal complement of
$B_{1},\ldots,B_{n}$ in the quadratic space generated by ${\rm NS}(S)$ and
${1\over 2}B_{1},\ldots,{1\over 2}B_{n}$. An effective class $\alpha$ in ${\rm
CM}(S)$ with $\alpha^{2}=-2$ is called an effective root, and an effective
root $\alpha$ is called irreducible if $|\alpha-\beta|=\emptyset$ for any
other effective root $\beta$. The Coble-Mukai lattice and effective
irreducible roots work like as ${\rm Num}(Y)$ for an Enriques surface $Y$ and
non-singular rational curves on $Y$.

The following is the second main theorem of this paper. In Table 1, “dual
graph” means the one of effective irreducible roots, “Type” means the type of
the dual graph, ${\rm Aut}(S)$ is the group of automorphisms of $S$, $n$ is
the number of boundary components and “dim” is the dimension of the modli
space of Coble surfaces given type.

######  Theorem 1.2.

Coble surfaces with finite automorphism group in characteristic $2$ are
classified as in the following Table 1. The moduli space of Coble surfaces
with $n$ boundary components of each type is irreducible and of dimension
given in the Table. Every Coble surface is a specialization of classical
Enriques surfaces with finite automorphism group.

Table 1. Coble surfaces with finite automorphism group ($p=2$)

We remark that a Coble surface of type VII appears in case $p=2,5$ and other
Coble surfaces in the Table 1 appear only in characteristic 2. Moreover the
minimal resolution of the canonical covering $X$ is the supersingular $K3$
surface with Artin invariant 1 if $S$ is of type VII (i.e. $A=0$) and rational
in other cases ($A\neq 0$).

The proof of classification of possible dual graphs in Table 1 is the same as
that for Enriques surfaces given in Katsura, Kondō and Martin [11] by using
the list of conductrices. We can determine the possible number $n$ of boundary
components of Coble surfaces from its dual graph. On the other hand, since
Coble surfaces are rational, one can construct Coble surfaces with finite
automorphism group by blowing up ${\bf P}^{2}$, which allow us to determine
the moduli space of such Coble surfaces. On the other hand, in [11], the
authors gave examples of families of Enriques surfaces with finite
automorphism group by taking the quotients of rational surfaces by
derivations. We show that every Coble surface in Table 1 is also obtained as a
specialization of these Enriques surfaces.

The plan of the paper is as follows. In §2 we summarize known results used in
the paper. In Section 3 we study the canonical coverings of Coble surfaces in
characteristic 2 and their singularities, and in Section 4 discuss the
properties of the conductrices of Coble surfaces and give a classification of
possible conductrices. Section 5 devotes the classification of possible dual
graphs of $(-2)$-curves on Coble surfaces with finite automorphsim and the
number of boundary components. Finally in Section 6 we give a construction of
Coble surfaces with finite automorphism group and determine their moduli
spaces..

The authors would like to thank N. Shepherd-Barron for pointing out Lemma
4.20.

##  2\. Preliminaries

A surface is a non-singular complete algebraic surface defined over an
algebraically closed field $k$ of characteristic $p\geq 0$. A non-singular
rational curve $C$ on a surface $S$ is called $(-m)$-curve if $C^{2}=-m$. We
denote by $K_{S}$ the canonical divisor of $S$.

###  2.1. Vector fields

Assume ${\rm char}(k)=p>0$ and let $S$ be a surface defined over $k$. A
rational vector field $D$ on $S$ is said to be $p$-closed if there exists a
rational function $f$ on $S$ such that $D^{p}=fD$. A vector field $D$ is of
additive type (resp. of multiplicative type) if $D^{p}=0$ (resp. $D^{p}=D$).
Let $\\{U_{i}={\rm Spec}A_{i}\\}$ be an affine open covering of $S$. We set
$A_{i}^{D}=\\{\alpha\in A_{i}\mid D(\alpha)=0\\}$. The affine surfaces
$\\{U_{i}^{D}={\rm Spec}A_{i}^{D}\\}$ glue together to define a normal
quotient surface $S^{D}$.

Now, we assume that $D$ is $p$-closed. Then, the natural morphism

(2.1) $\pi:S\longrightarrow S^{D}$

is a purely inseparable morphism of degree $p$. If the affine open covering
$\\{U_{i}\\}$ of $S$ is fine enough, then taking local coordinates
$x_{i},y_{i}$ on $U_{i}$, we see that there exist $g_{i},h_{i}\in A_{i}$ and a
rational function $f_{i}$ such that the divisors defined by $g_{i}=0$ and by
$h_{i}=0$ have no common components, and such that

$D=f_{i}\left(g_{i}\frac{\partial}{\partial
x_{i}}+h_{i}\frac{\partial}{\partial y_{i}}\right)\quad\mbox{on}~{}U_{i}.$

By Rudakov and Shafarevich [19, Section 1], the divisors $(f_{i})$ on $U_{i}$
give a global divisor $(D)$ on $S$, and the zero-cycles defined by the ideal
$(g_{i},h_{i})$ on $U_{i}$ give a global zero cycle $\langle D\rangle$ on $S$.
A point contained in the support of $\langle D\rangle$ is called an isolated
singular point of $D$. If $D$ has no isolated singular point, $D$ is said to
be divisorial. Rudakov and Shafarevich [19, Theorem 1, Corollary] showed that
$S^{D}$ is non-singular if $\langle D\rangle=0$, i.e. $D$ is divisorial. When
$S^{D}$ is non-singular, they also showed a canonical divisor formula

(2.2) $K_{S}\sim\pi^{*}K_{S^{D}}+(p-1)(D),$

where $\sim$ means linear equivalence. As for the Euler number $c_{2}(S)$ of
$S$, we have a formula

(2.3) $c_{2}(S)={\text{deg}}\langle D\rangle-K_{S}\cdot(D)-(D)^{2}$

(cf. Katsura and Takeda [12, Proposition 2.1]).

Now we consider an irreducible curve $C$ on $S$ and we set
$C^{\prime}=\pi(C)$. Take an affine open set $U_{i}$ above such that $C\cap
U_{i}$ is non-empty. The curve $C$ is said to be integral with respect to the
vector field $D$ if $g_{i}\frac{\partial}{\partial
x_{i}}+h_{i}\frac{\partial}{\partial y_{i}}$ is tangent to $C$ at a general
point of $C\cap U_{i}$. Then, Rudakov-Shafarevich [19, Proposition 1] showed
the following proposition:

######  Proposition 2.1.

  * $({\rm i})$

If $C$ is integral, then $C=\pi^{*}(C^{\prime})$ and $C^{2}=pC^{\prime 2}$.

  * $({\rm ii})$

If $C$ is not integral, then $pC=\pi^{*}(C^{\prime})$ and $pC^{2}=C^{\prime
2}$.

###  2.2. Genus 1 fibrations

We recall elliptic and quasi-elliptic fibrations on a rational surface. For
simplicity, we call an elliptic or a quasi-elliptic fibration a genus 1
fibration. A genus 1 fibration is called relatively minimal if any fiber
contains no $(-1)$-curves. Let $f:S\to{\bf P}^{1}$ be a genus 1 fibration on a
rational surface and let $F$ be a general fiber. A non-singular rational curve
$C$ on $S$ is called a section (resp. a 2-section) if $C\cdot F=1$ (resp.
$C\cdot F=2$).

We use the symbols ${\text{I}}_{n}$ $(n\geq 1)$, ${\text{I}}_{n}^{*}$ $(n\geq
0)$, II, III, IV, ${\text{II}}^{*}$, ${\text{III}}^{*}$, ${\text{IV}}^{*}$ of
relatively minimal singular fibers of $f$ in the sense of Kodaira. The dual
graph of $(-2)$-curves in a singular fiber of type ${\text{I}}_{n}$ $(n\geq
2)$, ${\text{I}}_{n}^{*}$ $(n\geq 0)$, III, IV, ${\text{II}}^{*}$,
${\text{III}}^{*}$ or ${\text{IV}}^{*}$ is an extended Dynkin diagram of type
$\tilde{A}_{n-1}$, $\tilde{D}_{n+4}$, $\tilde{A_{1}}$, $\tilde{A_{2}}$,
$\tilde{E_{8}}$, $\tilde{E_{7}}$ or $\tilde{E_{6}}$, respectively. A fiber
with multiplicity 2 is called a multiple fiber. For a multiple singular fiber
of type $F$, we write $2F$. If, for example, $f$ has a multiple singular fiber
of type III and a singular fiber of type ${\text{IV}}^{*}$, then we say that
$f$ has singular fibers of type $(2{\text{III}},{\text{IV}}^{*})$ or of type
$(2\tilde{A}_{2},\tilde{E}_{6})$. If $f$ has a section and its Mordell-Weil
group is torsion, then $f$ is called extremal. We use the following
classifications of extremal rational elliptic and rational quasi-elliptic
fibrations.

######  Proposition 2.2.

(Lang [15], [16]) The following are the singular fibers of relatively minimal
extremal elliptic fibrations on rational surfaces in characteristic $2:$

$({\rm II}^{*}),\ ({\rm II}^{*},{\rm I}_{1}),\ ({\rm III}^{*},{\rm I}_{2}),\
({\rm IV}^{*},{\rm IV}),\ ({\rm IV}^{*},{\rm I}_{3},{\rm I}_{1}),\ ({\rm
I}_{4}^{*}),\ ({\rm I}_{1}^{*},{\rm I}_{4}),$ $({\rm I}_{9},{\rm I}_{1},{\rm
I}_{1},{\rm I}_{1}),\ ({\rm I}_{8},{\rm III}),\ ({\rm I}_{6},{\rm IV},{\rm
I}_{2}),\ ({\rm I}_{5},{\rm I}_{5},{\rm I}_{1},{\rm I}_{1}),\ ({\rm
I}_{3},{\rm I}_{3},{\rm I}_{3},{\rm I}_{3}).$

######  Proposition 2.3.

(Ito [9]) The following are the singular fibers of relatively minimal quasi-
elliptic fibrations on rational surfaces in characteristic $2:$

$({\rm II}^{*}),\ ({\rm III}^{*},{\rm III}),\ ({\rm I}_{4}^{*}),\ ({\rm
I}_{2}^{*},{\rm III},{\rm III}),\ ({\rm I}_{0}^{*},{\rm I}_{0}^{*}),$ $({\rm
I}_{0}^{*},{\rm III},{\rm III},{\rm III},{\rm III}),\ ({\rm III},{\rm
III},{\rm III},{\rm III},{\rm III},{\rm III},{\rm III},{\rm III}).$

Note that any quasi-elliptic fibration is extremal. Assume that $f$ is quasi-
elliptic. Then the closure of the set of singular points of irreducible fibers
(of type II) of $f$ is called the curve of cusps which is a 2-section of $f$.

###  2.3. Coble surfaces

Let $S$ be a Coble surface with boundary components $B_{1},\ldots,B_{n}$. By
$|-2K_{S}|=\\{B_{1}+\cdots+B_{n}\\}$ and the adjunction formula,
$B_{1}^{2}=\cdots=B_{n}^{2}=-4$ and $K_{S}^{2}=-n$. Since
$|-2K_{S}|=\\{B_{1}+\cdots+B_{n}\\}$, the only curves with negative self-
intersection on $S$ are $B_{1},\ldots,B_{n}$, $(-2)$-curves and $(-1)$-curves.
Note that $(-2)$-curves do not intersect any boundary component $B_{i}$.

######  Proposition 2.4.

(Dolgachev and Kondō [4, Proposition 9.1.2]) Any Coble surface $S$ is a basic
type, that is, there exists a birational morphism from $S$ to ${\bf P}^{2}$.
Moreover $S$ admits a birational morphism $f:S\to{\bf P}^{2}$ such that the
image of $B_{1}+\cdots+B_{n}$ is a curve $W$ of degree $6$. The fundamental
points of $f^{-1}:{\bf P}^{2}\to S$ are among singular points of $W$
(including infinitely near points).

Consider a pencil $|3mh-m(p_{1}+\cdots+p_{9})|$ of curves of degree $3m$ on
${\bf P}^{2}$ with points $p_{i}$ of multiplicity $\geq m$ $(m=1$ or $2)$
where $h$ is a line on ${\bf P}^{2}$. Blowing up its base points we obtain a
rational genus 1 fibration called a Halphen surface of index $m$. A Halphen
surface of index 1 is nothing but a Jacobian fibration. Moreover we have the
following.

######  Proposition 2.5.

(Dolgachev and Kondō [4, Proposition 9.1.4]) Let $f:S\to{\bf P}^{1}$ be a
genus $1$ fibration on a Coble surface $S$. Then $S$ is obtained from a
Halphen surface of index $2$ by blowing up all singular points (and their
infinitely near points in the case of type III, IV ) of one reduced singular
fiber or from a Halphen surface of index $1$, i.e. a Jacobian fibration, by
blowing up all singular points (and their infinitely near points in the case
of type III, IV ) of two reduced singular fibers.

######  Remark 2.6.

It follows from Proposition 2.5 that a fiber of type II, III, IV, or ${\rm
I}_{n}$ is blown up. The configurations of fibers blown up are given in Figure
1. In the figure, the dotted lines are $(-1)$-curve, and $B_{i}$ means
$(-4)$-curves. In case of type ${\rm I}_{n}$, the curves $E_{i}$
($i=1,\cdots,n$) are $(-1)$-curves and $B_{i}$ ($i=1,\cdots,n$) are the
$(-4)$-curves. The fiber is given by
$\sum_{i=1}^{n}B_{i}+2\sum_{i=1}^{n}E_{i}$. In case of type ${\rm II}$, the
fiber is given by $B+2E$. In case of type ${\rm III}$, $E_{1}$ is a
$(-2)$-curve and the fiber is given by $B_{1}+B_{2}+2(E_{1}+2E_{2})$. In case
of type ${\rm IV}$, the fiber is given by
$B_{1}+B_{2}+B_{3}+3B_{4}+4(E_{1}+E_{2}+E_{3})$.

Figure 1. Blowing-up the singular points of a fiber

######  Proposition 2.7.

(Dolgachev and Kondō [4, Corollary 9.1.5]) Let $S$ be a Coble surface. Then
$K_{S}^{2}\geq-10$. In particular the number of boundary components of $S$ is
at most $10$.

###  2.4. Coble-Mukai lattices.

For a Coble surface $S$, instead of its Néron-Severi group, we consider a
lattice ${\rm CM}(S)$ of rank 10, called Coble-Mukai lattice, which is defined
by the orthogonal complement of the boundary components $B_{1},\ldots,B_{n}$
in the quadratic space $\widetilde{\rm Pic}(S)$ generated by ${\rm Pic}(S)$
and ${1\over 2}B_{1},\ldots,{1\over 2}B_{n}$. Then ${\rm CM}(S)$ together with
the intersection form is a lattice (Dolgachev and Kondō [4, §9.2]).

We call an effective class $\alpha\in{\rm CM}(S)$ with $\alpha^{2}=-2$ an
effective root. We say that an effective root $\alpha$ is irreducible if
$|\alpha-\beta|=\emptyset$ for any other effective root $\beta$.

######  Lemma 2.8.

(Dolgachev and Kondō [4, Lemma 9.2.1]) Let $\alpha$ be an effective
irreducible root. Then $\alpha$ is either the divisor class of a $(-2)$-curve
or the ${\bf Q}$-divisor class of an effective root of the form $2E+{1\over
2}B_{j}+{1\over 2}B_{k}$, where $E$ is a $(-1)$-curve intersecting two
different boundary components $B_{j},B_{k}$.

We call an effective irreducible root $\alpha$ a $(-2)$-root if it is
represented by a $(-2)$-curve and a $(-1)$-root if it is represented by
$2E+{1\over 2}B_{j}+{1\over 2}B_{k}$. An easy calculation shows the following
Lemma.

######  Lemma 2.9.

(Kondō [13, Lemma 3.3])

  * (1)

Two $(-1)$-roots have intersection number $1$ if and only if the associated
$(-1)$-curves do not meet and the roots share precisely one boundary
component.

  * (2)

The intersection number of a $(-1)$-root and a $(-2)$-root is always even.

Let ${\rm CM}(S)_{\bf R}^{+}$ be the connected component of $\\{x\in{\rm
CM}(S)\otimes{\bf R}\ :\ x^{2}>0\\}$ containing a nef class. Denote by
$\overline{{\rm CM}(S)}_{\bf R}^{+}$ the closure of ${\rm CM}(S)_{\bf R}^{+}$
in ${\rm CM}(S)\otimes{\bf R}$. Let $W(S)$ be the subgroup of ${\rm O}({\rm
CM}(S))$ generated by all reflections $s_{\alpha}$, where $\alpha$ is an
effective irreducible root. The group $W(S)$ naturally acts on $\overline{{\rm
CM}(S)}_{\bf R}^{+}$. We denote by $D(S)$ the domain defined by

$D(S)=\\{x\in\overline{{\rm CM}(S)}_{\bf R}^{+}\ :\ x\cdot\alpha\geq 0\ {\rm
for\ any\ effective\ irreducible\ root}\ \alpha\\}.$

Obviously, the automorphism group ${\rm Aut}(S)$ leaves the set of curves
$\\{B_{1},\ldots,B_{n}\\}$ invariant and hence acts on ${\rm CM}(S)$ and on
$D(S)$. The following proposition holds.

######  Proposition 2.10.

(Dolgachev and Kondō [4, Proposition 9.2.2]) Let $x\in\overline{{\rm
CM}(S)}_{\bf R}^{+}\cap{\rm CM}(S)$. Then $x$ is nef if and only if $x\in
D(S)$. In other word, the intersection of the nef cone of $S$ with
$\overline{{\rm CM}(S)}_{\bf R}^{+}$ is a fundamental domain of $W(S)$.

A vector $f\in D(S)\cap{\rm CM}(S)$ is called isotropic if $f^{2}=0$ and
primitive if $f=mf^{\prime}$, $m\in{\bf Z}$, $f^{\prime}\in{\rm CM}(S)$
implies $m=\pm 1$.

######  Lemma 2.11.

The set of genus $1$ fibration on a Coble surface $S$ bijectively corresponds
to the set of primitive isotropic vectors in $D(S)\cap{\rm CM}(S)$.

###### Proof.

By the assumption, $f\cdot K_{S}=0$ and hence the proof of Cossec, Dolgachev
and Liedtke [3, Proposition 2.2.8] works well in this case, too. ∎

######  Remark 2.12.

Let $f:S\to{\bf P}^{1}$ be a genus 1 fibration obtained from a Halphen surface
$h:H\to{\bf P}^{1}$. When we consider $f$ in ${\rm CM}(S)$, we can say that
the type of the extended Dynkin diagram of reducible singular fibers of $f$ is
the same as those of $h$. Let $F$ be the fiber of $f$ corresponding to a
reducible singular fiber $F_{0}$ of $h$. If $F_{0}$ is not blown up, then
$F=F_{0}$. Assume that $F_{0}$ is blown up. If $F_{0}$ is of type ${\rm
I}_{n}$ $(n\geq 2)$, then $F$ consists of $n$ $(-4)$-curves
$B_{1},\ldots,B_{n}$ and $n$ $(-1)$-curves $E_{1},\ldots,E_{n}$ with
$B_{i}\cdot E_{i}=B_{i+1}\cdot E_{i}=1$. Here we use the same notation as in
Remark 2.6. Then

$F=B_{1}+\cdots+B_{n}+2E_{1}+\cdots+2E_{n}=({1\over 2}B_{1}+{1\over
2}B_{2}+2E_{1})+\cdots+({1\over 2}B_{n}+{1\over 2}B_{1}+2E_{n}).$

Thus $F$ is the sum of $n$ $(-1)$-roots forming a dual graph of type
$\tilde{A}_{n-1}$. Similarly, if $F_{0}$ is of type ${\rm III}$ (resp. of type
${\rm IV}$), then

$F=2(({1\over 2}B_{1}+{1\over 2}B_{2}+2E_{2})+E_{1})\quad(resp.\ \
2(\sum_{i=1}^{3}{1\over 2}B_{i}+{1\over 2}B_{4}+2E_{i}))$

is the double of the sum of a $(-1)$-root and a $(-2)$-root (resp. three
$(-1)$-roots) forming a dual graph of type $\tilde{A}_{1}$ (resp.
$\tilde{A}_{2}$). Note that $F$ looks like a multiple fiber in case of type
${\rm III},{\rm IV}$.

As a corollary of Proposition 2.10 and Lemma 2.11, we obtain the following
Proposition.

######  Proposition 2.13.

(Dolgachev and Kondō [4, Theorem 9.8.1]) Let $S$ be a Coble surface. Suppose
$W(S)$ is of finite index in ${\rm O}({\rm CM}(S))$. Then ${\rm Aut}(S)$ is
finite.

Consider a genus 1 fibration on a Coble surface $f:S\to{\bf P}^{1}$. Then the
Mordell-Weil group of the Jacobian fibration of $f$ acts on $S$ faithfully as
automorphisms. This implies the following Proposition.

######  Proposition 2.14.

Assume that the automorphism group of a Coble surface $S$ is finite. Then any
genus $1$ fibration on $S$ is extremal.

An effective irreducible root $r$ is called a special 2-section of a genus 1
fibration $f:S\to{\bf P}^{1}$ if $r\cdot F=2$ where $F$ is a general fiber of
$f$. In this case the fibration $f$ is called special.

In case of Enriques surfaces, the following is known.

######  Proposition 2.15.

(Cossec [2, Theorem 4], Lang [14, Theorem A.3]) If an Enriques surface $Y$
contains a $(-2)$-curve, then there exists a special genus $1$ fibration on
$Y$.

The following lemma is the analogue of Proposition 2.15 for Coble surfaces.
For the proof we refer the reader to [13, Lemma 3.13].

######  Lemma 2.16.

Let $S$ be a Coble surface. Assume that $S$ has an irreducible effective root.
Then there exists a special genus $1$ fibration on $S$.

###  2.5. Vinberg’s criterion

Let $L$ be a lattice of signature $(1,r)$. We recall Vinberg’s criterion,
which guarantees that a group generated by a finite number of reflections is
of finite index in ${\rm O}(L)$.

Let $\Delta$ be a finite set of $(-2)$-vectors in $L$. Let $\Gamma$ be the
graph of $\Delta$, that is, $\Delta$ is the set of vertices of $\Gamma$ and
two vertices $\delta$ and $\delta^{\prime}$ are joined by $m$-tuple lines if
$\langle\delta,\delta^{\prime}\rangle=m$. We assume that the cone

$K(\Gamma)=\\{x\in L\otimes{\bf R}\ :\ \langle x,\delta_{i}\rangle\geq 0,\
\delta_{i}\in\Delta\\}$

is a strictly convex cone. Such a $\Gamma$ is called non-degenerate. A
connected parabolic subdiagram $\Gamma^{\prime}$ in $\Gamma$ is a Dynkin
diagram of type $\tilde{A}_{m}$, $\tilde{D}_{n}$ or $\tilde{E}_{k}$ (see
Vinberg [21, p. 345, Table 2]). If the number of vertices of $\Gamma^{\prime}$
is $r+1$, then $r$ is called the rank of $\Gamma^{\prime}$. A disjoint union
of connected parabolic subdiagrams is called a parabolic subdiagram of
$\Gamma$. We denote by $\tilde{K_{1}}\oplus\cdots\oplus\tilde{K_{s}}$ a
parabolic subdiagram which is a disjoint union of connected parabolic
subdiagrams of type $\tilde{K_{1}},\ldots,\tilde{K_{s}}$, where $K_{i}$ is
$A_{m}$, $D_{n}$ or $E_{k}$. The rank of a parabolic subdiagram is the sum of
the ranks of its connected components. Note that the dual graph of reducible
singular fibers of a genus 1 fibration on $X$ gives a parabolic subdiagram. We
denote by $W(\Gamma)$ the subgroup of ${\rm O}(L)$ generated by reflections
associated with $\delta\in\Gamma$.

######  Theorem 2.17.

(Vinberg [21, Theorem 2.3]) Let $\Delta$ be a set of $(-2)$-vectors in $L$ and
let $\Gamma$ be the graph of $\Delta$. Assume that $\Delta$ is a finite set,
$\Gamma$ is non-degenerate and $\Gamma$ contains no $m$-tuple lines with
$m\geq 3$. Then $W(\Gamma)$ is of finite index in ${\rm O}(L)$ if and only if
every connected parabolic subdiagram of $\Gamma$ is a connected component of
some parabolic subdiagram in $\Gamma$ of rank $r-1$ (= the maximal one).

######  Remark 2.18.

Note that $\Gamma$ as in the above proposition is automatically non-degenerate
if it contains the components of the reducible fibers of a special extremal
genus 1 fibration and a special $2$-section of this fibration. Indeed, these
curves will generate $L\otimes{\bf Q}$ and hence $K(\Gamma)$ is strictly
convex.

######  Proposition 2.19.

(Cossec, Dolgachev, Liedtke [3, Proposition 0.8.16]) Let $\Delta$ be a finite
set of irreducible effective roots on a Coble surface $S$ and let $\Gamma$ be
the graph of $\Delta$. Assume that $W(\Gamma)$ is of finite index in ${\rm
O}({\rm CM}(S))$. Then $\Delta$ is the set of all irreducible effective roots
on $S$.

##  3\. The canonical covering of a Coble surface in characteristic 2

###  3.1. The canonical covering

In this subsection, we examine the structure of the canonical covering of a
Coble surface in characteristic 2. For a Coble surface $S$, let
$\\{U_{i}\\}_{i\in I}$ be an affine open covering of $S$, and we denote by
$A_{i}$ the affine coordinate ring of $U_{i}$. Let $f_{i}$ be a defining
equation of the divisor $B_{1}+\cdots+B_{n}$. Taking an enough fine affine
open covering, we may assume that each affine open set $U_{i}$ meets at most
one component of the boundary and that $f_{i}$ is one element of local
coordinates of $U_{i}$ when the boundary meets $U_{i}$. Let $\\{f_{ij}\\}$ be
a cocycle which gives the invertible sheaf $\mathcal{O}_{S}(-K_{S})$. Then, by
definition there exist $g_{i}\in\Gamma(U_{i},{\mathcal{O}}_{S}^{*})$ ($i\in
I$) such that

(3.1) $f_{ij}^{2}=(f_{i}/f_{j})(g_{i}/g_{j}).$

We define the covering $\tilde{U}_{i}$ of $U_{i}$ by

(3.2) $z_{i}^{2}=f_{i}g_{i}$

with new variables $z_{i}$ ($i\in I$). Then, we have $z_{i}=f_{ij}z_{j}$ and
we get a finite flat covering $X$ of $S$:

$\pi:X\longrightarrow S.$

We call $X$ a canonical covering of $S$.

######  Theorem 3.1.

The canonical covering $X$ is a K3-like surface, that is,
$\omega_{X}\cong{\mathcal{O}}_{X}$ and ${\rm H}^{1}(X,{\mathcal{O}}_{X})=0$.

###### Proof.

We denote by $\omega_{S}$ the dualizing sheaf of $S$. We have
$\omega_{S}\cong{\mathcal{O}}_{S}(K_{S})$. The coordinate ring of
$\tilde{U}_{i}$ is isomorphic to

$A_{i}[z_{i}]/(z_{i}^{2}-f_{i}g_{i})\cong A_{i}\oplus A_{i}z_{i}.$

Since $z_{j}/z_{i}=f_{ij}^{-1}$, we see

(3.3) $\pi_{*}{\mathcal{O}}_{X}\cong{\mathcal{O}}_{S}\oplus\omega_{S}.$

By the duality theorem of finite flat morphism, we have

$\begin{array}[]{rl}\pi_{*}(\omega_{X})&\cong\pi_{*}(\pi^{!}\omega_{S})\cong{\mathcal{H}}om_{{\mathcal{O}}_{S}}(\pi_{*}{\mathcal{O}}_{X},\omega_{S})\cong{\mathcal{H}}om_{{\mathcal{O}}_{S}}({\mathcal{O}}_{S}\oplus\omega_{S},\omega_{S})\\\
&\cong{\mathcal{H}}om_{{\mathcal{O}}_{S}}({\mathcal{O}}_{S},\omega_{S})\oplus{\mathcal{H}}om_{{\mathcal{O}}_{S}}(\omega_{S},\omega_{S})\\\
&\cong\omega_{S}\oplus{\mathcal{O}}_{S}\cong\pi_{*}({\mathcal{O}}_{X})\end{array}.$

Therefore, we have $\omega_{X}\cong{\mathcal{O}}_{X}$. By the Serre duality
theorem, we have

$\dim{\rm H}^{2}(X,{\mathcal{O}}_{X})=\dim{\rm H}^{0}(X,\omega_{X})=\dim{\rm
H}^{0}(X,{\mathcal{O}}_{X})=1.$

Since $\pi$ is a finite morphism, we have $R^{i}\pi_{*}{\mathcal{O}}_{X}=0$
($i\geq 1$). Therefore, we have

$\chi({\mathcal{O}}_{X})=\chi(\pi_{*}{\mathcal{O}}_{X})=\chi({\mathcal{O}}_{S}\oplus\omega_{S})=2\chi({\mathcal{O}}_{S})=2.$

Hence, we have $\dim{\rm H}^{1}(X,{\mathcal{O}}_{X})=0$. ∎

Now, let $g:R\longrightarrow{\bf P}^{1}$ be a genus 1 fibration on a rational
surface $R$. Let $t$ be a local coordinate of affine line ${\bf
A}^{1}\subset{\bf P}^{1}$ and assume that over the point $P\in{\bf P}^{1}$
defined by $t=0$, we have a singular fiber of type ${\rm I}_{n}$, ${\rm II}$,
${\rm III}$ or ${\rm IV}$.

We take a rational one form $dt/t$ on ${\bf P}^{1}$ and we examine the poles
of $g^{*}(dt/t)$ under the blowing-ups to make $(-4)$-curves for a Coble
surface.

In the case of type ${\rm I}_{n}$, we blow up at the singlar points of
$g^{-1}(P)$. Then, the pull-back of $g^{*}(dt/t)$ has poles of order 1 along
the proper transforms of $g^{-1}(P)$, which are $(-4)$-curves, and it is
regular along the exceptional curves (see Figure 1).

In the case of type ${\rm II}$, we blow up at the singlar point of the curve
$g^{-1}(P)$ with a cusp. Then, the pull-back of $g^{*}(dt/t)$ has a pole of
order 1 along the proper transform of $g^{-1}(P)$, which are $(-4)$-curve, and
it is regular along the exceptional curve.

In the case of type ${\rm III}$, we blow up at the singlar point of
$g^{-1}(P)$. Then, the pull-back of $g^{*}(dt/t)$ has poles of order 1 along
the proper transforms of $g^{-1}(P)$, which are $(-3)$-curves, and it has not
a pole along the exceptional curve. We again blow up at the singular point of
the fiber (non-irreducible) curve. Then, we have two $(-4)$-curves, where the
pull back of the rational 1-form has poles of order 1, and no poles along
other exceptional curves.

In the case of type ${\rm IV}$, we blow up at the singlar point of
$g^{-1}(P)$. Then, the pull-back of $g^{*}(dt/t)$ has poles of order 1 along
the proper transforms of $g^{-1}(P)$, which are three $(-3)$-curves, and also
it has a pole of order 1 along the exceptional curve. We again blow up at the
three singular points of the fiber (non-irreducible) curve. Then, we have four
$(-4)$-curves, where the pull back of the rational 1-form has poles of order
1, and no poles along other exceptional curves.

######  Lemma 3.2.

Let $S$ be a Coble surface and let $f:S\longrightarrow{\bf P}^{1}$ be a genus
$1$ fibration. Let $t$ be a local coordinate of ${\bf A}^{1}\subset{\bf
P}^{1}$ and assume that for the point $P_{0}$ defined by $t=0$ the singular
fiber $f^{-1}(P_{0})$ contains at least one $(-4)$-curve. Then, on the fiber
$f^{-1}(P_{0})$ the rational $1$-form $f^{*}(dt/t)$ has poles of order $1$
along the $(-4)$-curves and is regular on other part of $f^{-1}(P_{0})$.
Namely, the rational $1$-form $f^{*}(dt/t)$ has poles of order $1$ along the
boundary contained in $f^{-1}(P_{0})$ and is regular on other part of
$S\setminus f^{-1}(\infty)$.

######  Proposition 3.3.

Let $S$ be a Coble surface and let $f:S\longrightarrow{\bf P}^{1}$ be a genus
$1$ fibration. Under the notation above, we assume that at the point $P_{0}$
defined by $t=0$, the fiber $f^{-1}(P_{0})$ is either a multiple fiber or a
singular fiber with at least one $(-4)$-curve, and that at the point
$P_{\infty}$ defined by $t=\infty$ the fiber $f^{-1}(P_{\infty})$ is a
singular fiber with at least one $(-4)$-curve. Then. $f^{*}(dt/t)$ has poles
of order $1$ along $(-4)$-curves and is regular on other part.

###### Proof.

Assume $f:S\longrightarrow{\bf P}^{1}$ has a multiple fiber at the point
$P_{0}$. Then, we have an expression $t=uh^{2}$. Here, $h$ is a defining
equation of the half fiber and $u$ is a unit. Therefore, we have
$f^{*}(dt/t)=du/u$ and $f^{*}(dt/t)$ is regular on the multiple fiber. Using
this fact and Lemma 3.2, we complete our proof. ∎

Now, we define

$\eta=d(f_{i}g_{i})/(f_{i}g_{i})\quad\mbox{on}~{}U_{i}.$

Then, by (3.1), $\eta$ becomes a rational 1-form on $S$. The rational 1-form
$\eta$ has the poles of order 1 along the boundary $B_{1}+\cdots+B_{n}$. When
$f_{i}$ is part of local coordinates, $g_{i}f_{i}$ is also part of local
coordinates. Therefore, $\eta$ has neither isolated zeros nor zero divisors on
the boundary $B_{1}+\cdots+B_{n}$. Therefore, over the boundary the covering
$X$ is non-singular and the divisorial part of the rational 1-form $\eta$ is
written as

(3.4) $(\eta)=-\sum_{i=1}^{n}B_{i}+B$

with an effective divisor $B$. Note $(\sum_{i=1}^{n}B_{i})\cap B=\emptyset$.
We call $B$ a bi-conductrix.

######  Theorem 3.4.

$\eta=f^{*}(dt/t)$

###### Proof.

Let a boundary component $B_{k}$ intersect with affine open set $U_{i}$. Then,
we have

$\eta=d(f_{i}g_{i})/(f_{i}g_{i})=df_{i}/f_{i}+dg_{i}/g_{i}.$

Since the boundary component $B_{k}$ is defined by $f_{i}=0$ on $U_{i}$ and is
contained in a fiber as a component with odd multiplicity, we have
$t=u_{i}f_{i}$ with a unit $u_{i}$. Therefore, we have

$f^{*}(dt/t)=d(u_{i}f_{i})/(u_{i}f_{i})=df_{i}/f_{i}+du_{i}/u_{i}.$

Therefore, on $U_{i}$ we have

$\eta-f^{*}(dt/t)=dg_{i}/g_{i}-du_{i}/u_{i},$

and $\eta-f^{*}(dt/t)$ is a regular 1-form on $S$. However, since $S$ is a
rational surface, we have no non-zero regular 1-form on $S$. Therefore, we
have $\eta=f^{*}(dt/t)$. ∎

From here on, for the 1-form $f^{*}(dt/t)$ on $S$, we sometimes write it as
$dt/t$ if no confusion occurs.

###  3.2. The conductrix

In this subsection, we examine the bi-conductrix $B$ and we show that there
exists an effective divisor $A$ on $S$ such that $B=2A$. The arguments are
parallel to the one in Ekedahl, Shepherd-Barron [6] and Cossec, Dolgachev,
Liedtke [3]. But we give here a down-to-earth proof to understand the case of
Coble surfaces precisely. The following lemma is well-known (cf. Cossec,
Dolgachev, Liedtke, loc. cit., for instance).

######  Lemma 3.5.

Let $S$, $X$ be algebraic surfaces and let $f:X\longrightarrow S$ be a finite
morphism. Assume $S$ is regular. Then, $f$ is flat if and only if $X$ is
Cohen-Macauley. Moreover, if $X$ is normal, then $X$ is Cohen-Macauley.

Let $S$ be a Coble surface and let $X$ be the canonical covering. Let
$X^{\prime}$ be the normaization of $X$:

$\begin{array}[]{ccr}X^{\prime}&\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}&X\\\
\pi\circ\nu\searrow&&\swarrow\pi\\\ &S.&\end{array}$

Since $X^{\prime}$ is normal, $X^{\prime}$ is Cohen-Macauley and $\pi\circ\nu$
is flat by Lemma 3.5. Therefore, we have an exact sequence

(3.5)
$0\longrightarrow{\mathcal{O}}_{S}\longrightarrow(\pi\circ\nu)_{*}{\mathcal{O}}_{X^{\prime}}\longrightarrow{\mathcal{L}}^{\prime}\longrightarrow
0$

with an invertible sheaf ${\mathcal{L}}^{\prime}$ on $S$. Therefore, by (3.3)
we have a commutative diagram of exact sequences

$\begin{array}[]{ccccccccc}0&\longrightarrow&{\mathcal{O}}_{S}&\longrightarrow&\pi_{*}{\mathcal{O}}_{X}&\longrightarrow&\omega_{S}&\longrightarrow&0\\\
&&\parallel&&\varphi\downarrow&&\psi\downarrow&&\\\
0&\longrightarrow&{\mathcal{O}}_{S}&\longrightarrow&(\pi\circ\nu)_{*}{\mathcal{O}}_{X^{\prime}}&\longrightarrow&{\mathcal{L}}^{\prime}&\longrightarrow&0.\end{array}$

Since $\pi\circ\nu$ is a purely inseparable morphism of degree 2, on the
affine open set $U_{i}$ we have a commutative diagram of exact sequences

$\begin{array}[]{ccccccccc}0&\longrightarrow&A_{i}&\longrightarrow&A_{i}\oplus
A_{i}z_{i}&\longrightarrow&A_{i}z_{i}&\longrightarrow&0\\\
&&\parallel&&\varphi\downarrow&&\psi\downarrow&&\\\
0&\longrightarrow&A_{i}&\longrightarrow&A_{i}\oplus
A_{i}w_{i}&\longrightarrow&A_{i}w_{i}&\longrightarrow&0.\end{array}$

Here, $w_{i}$ is an element such that $w_{i}^{2}=h_{i}\in A_{i}$. Let
$\varphi(z_{i})=b_{i}+a_{i}w_{i}$ ($a_{i},b_{i}\in A_{i}$). Since
$a_{i}w_{i}=\varphi(z_{i})-b_{i}=\varphi(z_{i}-b_{i})$, the annihilator
$Ann_{A_{i}}(A_{i}\oplus A_{i}w_{i})/\varphi(A_{i}\oplus A_{i}z_{i})$ is given
by the ideal $(a_{i})=a_{i}A_{i}$. The Cartier divisors on $U_{i}$ defined by
the ideal $(a_{i})$ ($i\in I$) glue together to give a divisor $A$ on $S$. We
call $A$ a conductrix of $S$. Since $\psi$ is given by the multiplication by
$a_{i}$ on $U_{i}$, $\psi$ induces an isomorphism
$\omega_{S}\cong{\mathcal{O}}_{S}(-A)\otimes{\mathcal{L}}^{\prime}$.
Therefore, we have

(3.6) ${\mathcal{L}}^{\prime}\cong\omega_{S}\otimes{\mathcal{O}}_{S}(A).$

######  Theorem 3.6.

$(\eta)=-\sum_{i=1}^{n}B_{i}+2A$, $B=2A$.

###### Proof.

By (3.2), we have

$g_{i}f_{i}=\varphi(g_{i}f_{i})=\varphi(z_{i}^{2})=(b_{i}+a_{i}w_{i})^{2}=b_{i}^{2}+a_{i}^{2}w_{i}^{2}=b_{i}^{2}+a_{i}^{2}h_{i}.$

Therefore, we have $a_{i}^{2}h_{i}=g_{i}f_{i}-b_{i}^{2}$, and
$g_{i}f_{i}-b_{i}^{2}$ is divisible by $a_{i}^{2}$. Since $X^{\prime}$ is
normal, $dh_{i}$ has only isolated zeros. Therefore, by
$a_{i}^{2}dh_{i}=d(g_{i}f_{i}-b_{i}^{2})=d(g_{i}f_{i})$, we see
$d(g_{i}f_{i})$ is exactly divisible by $a_{i}^{2}$. Hence, by (3.4), we see
$B=2A$. ∎

###  3.3. The conductrix of the surface by Frobenius base change

Let $F:{\bf P}^{1}\longrightarrow{\bf P}^{1}$ be the Frobenius morphism. Since
$\pi^{*}f^{*}(dt/t)=\pi^{*}(\eta)=\pi^{*}(d(g_{i}f_{i})/(g_{i}f_{i}))=\pi^{*}(dz_{i}^{2}/z_{i}^{2})=0$,
we have $\pi^{*}f^{*}(dt)=0$. Therefore, there exists an element $s\in k(X)$
such that $t=s^{2}$. Since $X^{\prime}$ is normal, we have the following
diagram:

$\begin{array}[]{ccl}\quad
X^{\prime}&\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}&X\\\
\psi_{0}\downarrow&&\downarrow\pi\\\ \quad S\times_{{\bf P}^{1}}{\bf
P}^{1}&\stackrel{{\scriptstyle pr_{1}}}{{\longrightarrow}}&S\\\
pr_{2}\downarrow&&\downarrow f\\\ \quad\quad{\bf
P}^{1}&\stackrel{{\scriptstyle F}}{{\longrightarrow}}&{\bf P}^{1}.\end{array}$

We set $S_{F}=S\times_{{\bf P}^{1}}{\bf P}^{1}$. Over $U_{i}$, if $U_{i}$
exists in the open set of $S$ given by $t\neq\infty$, the covering
$pr_{1}:pr_{1}^{-1}(U_{i})\longrightarrow U_{i}$ is given by the homomorphism

$A_{i}={\mathcal{O}}_{X}(U_{i})\longrightarrow
pr_{1*}{\mathcal{O}}_{X_{F}}(U_{i})=A_{i}[s]/(t-s^{2})=A_{i}\oplus A_{i}s.$

Around the point defined by $t=\infty$, the situation is similar. Note that
$S^{\prime}$ is the normalization of $S_{F}$. We calculate the conductrix for
$S_{F}$. We have exact sequences

$\begin{array}[]{ccccccccc}0&\longrightarrow&{\mathcal{O}}_{S}&\longrightarrow&pr_{1*}{\mathcal{O}}_{S_{F}}&\longrightarrow&\quad{\mathcal{L}}&\longrightarrow&0\\\
&&\parallel&&\varphi^{\prime}\downarrow&&\psi^{\prime}\downarrow&&\\\
0&\longrightarrow&{\mathcal{O}}_{S}&\longrightarrow&(pr_{1}\circ\psi_{0})_{*}{\mathcal{O}}_{S^{\prime}}&\longrightarrow&\quad{\mathcal{L}}^{\prime}&\longrightarrow&0\\\
&&&&\parallel&&&&\\\
&&&&(\pi\circ\nu)_{*}{\mathcal{O}}_{S^{\prime}}&&&&\end{array}$

On $U_{i}$ we have exact sequences

$\begin{array}[]{ccccccccc}0&\longrightarrow&A_{i}&\longrightarrow&A_{i}\oplus
A_{i}s&\longrightarrow&A_{i}s&\longrightarrow&0\\\
&&\parallel&&\varphi^{\prime}\downarrow&&\psi^{\prime}\downarrow&&\\\
0&\longrightarrow&A_{i}&\longrightarrow&A_{i}\oplus
A_{i}w_{i}&\longrightarrow&A_{i}w_{i}&\longrightarrow&0.\end{array}$

Let $\varphi^{\prime}(s)=b^{\prime}_{i}+a^{\prime}_{i}w_{i}$
($a^{\prime}_{i},b^{\prime}_{i}\in A_{i}$). Since
$a^{\prime}_{i}w_{i}=\varphi^{\prime}(s)-b^{\prime}_{i}=\varphi^{\prime}(s-b^{\prime}_{i})$,
the annihilator $Ann_{A_{i}}(A_{i}\oplus
A_{i}w_{i})/\varphi^{\prime}(A_{i}\oplus A_{i}s)$ is given by the ideal
$(a^{\prime}_{i})=a^{\prime}_{i}A_{i}$. The Cartier divisors on $U_{i}$
defined by the ideal $(a^{\prime}_{i})$ ($i\in I$) glue together to give a
divisor $A^{\prime}$ on $S$. We call $A^{\prime}$ a conductrix of $S_{F}$. As
before, we have

$t=\varphi^{\prime}(t)=\varphi^{\prime}(s^{2})=(b^{\prime}_{i}+a^{\prime}_{i}w_{i})^{2}={b^{\prime}_{i}}^{2}+{a^{\prime}_{i}}^{2}w_{i}^{2}={b^{\prime}_{i}}^{2}+{a^{\prime}_{i}}^{2}h_{i}.$

Therefore, we have ${a^{\prime}_{i}}^{2}h_{i}=t-{b^{\prime}_{i}}^{2}$, and
$t-{b^{\prime}_{i}}^{2}$ is divisible by ${a^{\prime}_{i}}^{2}$. Since
$S^{\prime}$ is normal, $dh_{i}$ has only isolated zeros. Therefore, by
${a^{\prime}_{i}}^{2}dh_{i}=d(t-{b^{\prime}_{i}}^{2})=dt$, we see $dt$ is
exactly divisible by ${a^{\prime}_{i}}^{2}$.

We denote by $P_{0}$ (resp. $P_{\infty}$) the point on ${\bf P}^{1}$ defined
by $t=0$ (resp. $t=\infty$). We have the following two cases:

(Case 1) Halphen surface of index 1. Both fibers $f^{-1}(P_{0})$ and
$f^{-1}(P_{\infty})$ are the fibers which contain $(-4)$-curves for the Coble
surface $S$.

(Case 2) Halphen surface of index 2. The fiber $f^{-1}(P_{0})$ is the multiple
fiber and the fiber $f^{-1}(P_{\infty})$ is the fiber which contains
$(-4)$-curves for the Coble surface $S$.

In fact, for any Coble surface, by a suitable choice of the local parameter
$t$, either (Case 1) or (Case 2) holds.

Case 1. By the structure of blowing-ups which we explain in the beginning of
Subsection 3.1, the fiber $f^{-1}(P_{0})$ is expressed as

$\sum_{i=1}^{\ell}B_{i}+2G_{0}$

with a positive interger $\ell<n$ and an effective divisor $G_{0}$. Here, we
arrange some boundary components $B_{i}$ suitably. The fiber
$f^{-1}(P_{\infty})$ is also expressed as

$\sum_{i=\ell+1}^{n}B_{i}+2G_{\infty}$

with an effective divisor $G_{\infty}$. Then, using these notation, we have

(3.7) $(f^{*}(dt/t))=2A^{\prime}-\sum_{i=1}^{n}B_{i}-2(G_{0}+G_{\infty}).$

On the other hand, we have

(3.8) $(f^{*}(dt/t))=(\eta)=2A-\sum_{i=1}^{n}B_{i}.$

Therefore, we have $A=A^{\prime}-G_{0}-G_{\infty}$.

Case 2. We denote by $G_{0}$ the half fiber of the fiber defined by $t=0$. The
fiber defined by $t=\infty$ is also expressed as

$\sum_{i=1}^{n}B_{i}+2G_{\infty}$

with an effective divisor $G_{\infty}$. Then, we have again the equalities
(3.7) and (3.8). Therefore, we have $A=A^{\prime}-G_{0}-G_{\infty}$.
Summarizing these results, we get the following theorem (cf. Remark 3.9
below).

######  Theorem 3.7 (Comparisson theorem).

Under the notation above, $A=A^{\prime}-G_{0}-G_{\infty}$ holds.

######  Remark 3.8.

In the case of an Enriques surface $Y$ in characteristic 2, we can consider a
similar situation. If $Y$ is classical, the genus 1 fibration has two multiple
fibers $2G_{0}$ and $2G_{\infty}$. In this case we have
$A=A^{\prime}-G_{0}-G_{\infty}$. If $Y$ is supersingular, then the genus 1
fibration has only one multiple fiber $2G_{0}$. In this case we have
$A=A^{\prime}-2G_{0}$. Therefore, $A$ is equal to $A^{\prime}$ modulo half
fibers and we can determine $A$ from $A^{\prime}$ uniquely.

######  Remark 3.9.

We give here an explicit form of $G_{0}$ and $G_{\infty}$ for the fiber with
$(-4)$-curves. For the non-half fiber case, we write $G_{0}$ and $G_{\infty}$
as $G$ symbolically. The configuration of fibers which contain $(-4)$-curves
are given in Figure 1. For each case, we write up $G$. In case of type ${\rm
I}_{n}$, $G=\sum_{i=1}^{n}E_{i}$. In case of type ${\rm II}$, $G=E$. In case
of type ${\rm III}$, $G=E_{1}+2E_{2}$. In case of type ${\rm IV}$,
$G=B_{4}+2E_{1}+2E_{2}+2E_{3}$.

###  3.4. The singularities of the canonical covering

We first compare the canonical covering $X$ with the surface
$S_{F}=S\times_{{\bf P}^{1}}{\bf P}^{1}$ which is obtained by the Frobenius
pull-back of $S$ as in the previous section. In this section, for the local
parameter $t$ on ${\bf A}^{1}\subset{\bf P}^{1}$ we assume either (Case 1) or
(case 2) in Subsection 3.3, if otherwise mentions.

######  Proposition 3.10.

Let $\pi:X\longrightarrow S$ be a canonical covering of a Coble surface $S$,
and let $f:S\longrightarrow{\bf P}^{1}$ be a genus $1$ fibration. Under the
assumption for the parameter $t$, $\pi$ factors through the Frobenius pull-
Back $S_{F}=S\times_{{\bf P}^{1}}{\bf P}^{1}$ and $X$ is isomorphic to $S_{F}$
except two fibers defined by $t=0$ and $t=\infty$.

###### Proof.

We have $\pi_{*}{\mathcal{O}}_{X}\cong{\mathcal{O}}_{S}\oplus\omega_{S}$.
Since $f_{*}\omega_{S}\cong{\mathcal{O}}_{{\bf P}^{1}}(-1)$, we have

$f_{*}\pi_{*}{\mathcal{O}}_{X}\cong f_{*}{\mathcal{O}}_{S}\oplus
f_{*}\omega_{S}\cong{\mathcal{O}}_{{\bf P}^{1}}\oplus{\mathcal{O}}_{{\bf
P}^{1}}(-1).$

Take an affine open subset $U$ of ${\bf P}^{1}$ and let $R$ be the coordinte
ring of $U$: $U={\rm Spec}~{}R$. Then we have

$f_{*}\pi_{*}{\mathcal{O}}_{X}(U)\cong R\oplus Rz\quad(\exists z\in
f_{*}\pi_{*}{\mathcal{O}}_{X}(U)).$

Since $\pi$ is purely inseparable of degree 2, $z$ is not separable over $R$.
Therefore we have either $z^{2}=a$ ($\exists a\in R$, $a\neq 0$) or 0. Since
$Y$ is reduced, the case $z^{2}=0$ doesn’t occur. Therefore, we have $z^{2}=a$
($\exists a\in R$, $a\neq 0$).

Put $C={\rm Spec}~{}f_{*}\pi_{*}{\mathcal{O}}_{X}$. Then, we have a
commutative diagram

$\begin{array}[]{ccc}X&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&S\\\
\downarrow&&\downarrow f\\\ C&\stackrel{{\scriptstyle
F}}{{\longrightarrow}}&{\bf P}^{1}.\end{array}$

The morphism $F$ becomes the Frobenius morphism and $f\circ\pi$ factors
through the Frobenius morphism. It is easy to see that
$F_{*}{\mathcal{O}}_{{\bf P}^{1}}\cong{\mathcal{O}}_{{\bf
P}^{1}}\oplus{\mathcal{O}}_{{\bf P}^{1}}(-1)$. Therefore, for
$pr_{1}:S_{F}\longrightarrow S$, we have

(3.9)
$pr_{1*}{\mathcal{O}}_{S_{F}}\cong{\mathcal{O}}_{S}\otimes_{{\mathcal{O}}_{{\bf
P}^{1}}}({\mathcal{O}}_{{\bf P}^{1}}\oplus{\mathcal{O}}_{{\bf
P}^{1}}(-1))\cong{\mathcal{O}}_{S}\oplus f^{*}{\mathcal{O}}_{{\bf
P}^{1}}(-1).$

Put $V=S\setminus\\{f^{-1}(P_{0})\cup f^{-1}(P_{\infty})\\}$. Then, on the
open set $V$, we have

(3.10)
$\pi_{*}{\mathcal{O}}_{X}|_{V}\cong({\mathcal{O}}_{S}\oplus\omega_{S})|_{V}\cong({\mathcal{O}}_{S}\oplus
f^{*}{\mathcal{O}}_{{\bf P}^{1}}(-1))|_{V}.$

Comparing (3.9) with (3.10), we get the proof of the latter part. ∎

We state here some conditions for the non-emptiness of the conductrix $A$.

######  Lemma 3.11.

If the genus $1$ fibration $f:S\longrightarrow{\bf P}^{1}$ is quasi-elliptic,
then the conductrix $A$ contains the curve of cusps as a simple component. In
particular, $A\neq 0$.

###### Proof.

The general fiber is of the form $t=y^{2}+x^{3}$ (cf. Bombieri and Mumford
[1]). Therefore, by $dt/t=x^{2}dx/t$ we see that divisorial part
$(\eta)=(dt/t)$ contains the curve of cusps as a double component. Therefore,
$A$ contains the curve of cusps as a simple component. ∎

######  Lemma 3.12.

Let $Z$ be a fundamental divisor of ADE configulation on a Coble surface $S$.
Then,

${\rm H}^{0}(Z,{\mathcal{O}}_{Z})=1,~{}{\rm H}^{1}(Z,{\mathcal{O}}_{Z})=0.$

###### Proof.

Since the lattice generated by the irreducible components of $Z$ in the Néron-
Severi group ${\rm NS}(S)$ is negative definite, we have ${\rm
H}^{0}(S,{\mathcal{O}}_{S}(K_{S}+Z))=0$. Therefore, by the Serre duality we
have ${\rm H}^{2}(S,{\mathcal{O}}_{S}(-Z))=0$. Since $Z$ is a fundamental
divisor of ADE configulation, we have $Z^{2}=-2$, and since any $(-2)$-curve
does not intersect the boundary divisor $\sum_{i=1}^{n}B_{i}$, we have
$K_{S}\cdot Z=0$. Therefore, by the Riemann-Roch theorem we have ${\rm
H}^{1}(S,{\mathcal{O}}_{S}(-Z))=0$. Considering the exact sequence

$0\longrightarrow{\mathcal{O}}_{S}(-Z)\longrightarrow{\mathcal{O}}_{S}\longrightarrow{\mathcal{O}}_{Z}\longrightarrow
0,$

we have a long exact sequence

$\begin{array}[]{c}0\longrightarrow k\longrightarrow{\rm
H}^{0}(Z,{\mathcal{O}}_{Z})\longrightarrow{\rm
H}^{1}(S,{\mathcal{O}}_{S}(-Z))\\\ \longrightarrow{\rm
H}^{1}(S,{\mathcal{O}}_{S})\longrightarrow{\rm
H}^{1}(Z,{\mathcal{O}}_{Z})\longrightarrow{\rm
H}^{2}(S,{\mathcal{O}}_{S}(-Z)).\end{array}$

Since $S$ is rational, we have ${\rm H}^{1}(S,{\mathcal{O}}_{S})=0$.
Therefore, we have the results. ∎

######  Proposition 3.13.

Assume $S$ contains a divisor whose the dual graph is of type $D_{4}$. Then
the central component is contained in $A$. In particular, $A\neq 0$.

###### Proof.

Let $E_{0},E_{1},E_{2},E_{3}$ be the components of $Z$ with $E_{0}\cdot
E_{i}=1$ $(i=1,2,3)$. Then the divisor $Z=2E_{0}+E_{1}+E_{2}+E_{3}$ is a
fundamental divisor of type $D_{4}$. By Lemma 3.12, it is rational type in the
sence of Schröer [20]. It now follows from Schröer [20, Proposition 4.6] that
$E_{0}\subset A$. ∎

######  Corollary 3.14.

If a genus $1$ fibration $f:S\longrightarrow{\bf P}^{1}$ has a singular fiber
of type ${\rm I}_{m}^{*}$, ${\rm II}^{*}$, ${\rm III}^{*}$ or ${\rm IV}^{*}$,
then we have the conductrix $A\neq 0$.

###### Proof.

Obviously each of these singular fibers contains a fundamental divisor of type
$D_{4}$. Thus the assertion follows from Proposition 3.13. ∎

######  Lemma 3.15.

Let $g:Y\longrightarrow{\bf P}^{1}$ be a relatively minimal rational elliptic
surface without multiple fiber, and let $F:{\bf P}^{1}\longrightarrow{\bf
P}^{1}$ be the Frobenius morphism. Then, the Frobenius pull-back
$Y_{F}=Y\times_{{\bf P}^{1}}{\bf P}^{1}$ is a K3-like surface.

###### Proof.

We consider the cartesian diagram

$\begin{array}[]{lccc}Y_{F}=&Y\times_{{\bf P}^{1}}{\bf
P}^{1}&\stackrel{{\scriptstyle pr}}{{\longrightarrow}}&Y\\\
&\downarrow&&\downarrow g\\\ &{\bf P}^{1}&\stackrel{{\scriptstyle
F}}{{\longrightarrow}}&{\bf P}^{1}.\end{array}$

Let $g^{-1}(P)$ be a fiber of $g$. It is easy to see that
$F_{*}{\mathcal{O}}_{{\bf P}^{1}}={\mathcal{O}}_{{\bf
P}^{1}}\oplus{\mathcal{O}}_{{\bf P}^{1}}(-1)$. Therefore, we have

$pr_{*}{\mathcal{O}}_{Y_{F}}\cong({\mathcal{O}}_{{\bf
P}^{1}}\oplus{\mathcal{O}}_{{\bf P}^{1}}(-1))\otimes_{{\mathcal{O}}_{{\bf
P}^{1}}}{\mathcal{O}}_{Y}={\mathcal{O}}_{Y}\oplus{\mathcal{O}}_{Y}(-g^{-1}(P)).$

Since the dualizing sheaf $\omega_{Y}$ of $Y$ is given by
${\mathcal{O}}_{Y}(-g^{-1}(P))$ and the morphism $pr$ is a finite morphism,
the dualizing sheaf $\omega_{Y_{F}}$ of $Y_{F}$ is given by

$\begin{array}[]{cl}pr_{*}(\omega_{Y_{F}})&\cong{\mathcal{H}}om_{{\mathcal{O}}_{Y}}(pr_{*}{\mathcal{O}}_{Y_{F}},\omega_{Y})\\\
&\cong{\mathcal{H}}om_{{\mathcal{O}}_{Y}}({\mathcal{O}}_{Y}\oplus{\mathcal{O}}_{Y}(-g^{-1}(P)),{\mathcal{O}}_{Y}(-g^{-1}(P)))\\\
&\cong{\mathcal{O}}_{Y}(-g^{-1}(P))\oplus{\mathcal{O}}_{Y}\cong
pr_{*}({\mathcal{O}}_{Y_{F}})\end{array}$

Therefore, we have $\omega_{Y_{F}}\cong{\mathcal{O}}_{Y_{F}}$. Since

$\chi({\mathcal{O}}_{Y_{F}})=\chi(pr_{*}({\mathcal{O}}_{Y_{F}}))=\chi({\mathcal{O}}_{Y})+\chi({\mathcal{O}}_{Y}(-g^{-1}(P)))=2\chi({\mathcal{O}}_{Y})=2,$

we have ${\rm H}^{1}(Y_{F},{\mathcal{O}}_{Y_{F}})=0$. Therefore, $Y_{F}$ is a
K3-like surface. ∎

######  Lemma 3.16.

Let $X$ be a normal K3-like surface. If $X$ has a non-rational singularity,
then $X$ is a rational surface.

###### Proof.

Let $\mu:\tilde{X}\longrightarrow X$ be a resolution of singularities. We
denote by $\cup_{i=1}^{k}E_{i}$ the support of the exceptional divisors of the
resolution $\mu$. Here, $E_{i}$’s are the irreducible components. Then, by
Hidaka, Watanabe [7, Lemma 1.1] the dualizing sheaf of $\tilde{X}$ is given by
$\omega_{\tilde{X}}\cong{\mathcal{O}}_{\tilde{X}}(-\sum_{i=1}^{k}r_{i}E_{i})$
and some $r_{i}$ is positive by the fact that one of singularities is non-
rational. Therefore, we have $\dim{\rm
H}^{0}(\tilde{X},\omega_{\tilde{X}}^{m})=0$ for $m>0$. Since ${\rm
H}^{1}(X,{\mathcal{O}}_{X})=0$, we have $q(X)=0$. Therefore, we have
$q(\tilde{X})=0$. By Castelnuovo’s criterion of rationality, we see
$\tilde{X}$ is a rational surface. ∎

######  Proposition 3.17.

Assume the conductrix $A=0$. Then, the canonical covering $X$ always has some
singular points and the singular points are all rational double points.

###### Proof.

Assume $X$ is non-singular. Then, since $\pi:X\longrightarrow S$ is purely
inseparable and finite, we have the Betti number $b_{2}(X)=b_{2}(S)\geq 11$,
$\omega_{X}\cong{\mathcal{O}}_{X}$ and ${\rm H}^{1}(X,{\mathcal{O}}_{X})=0$.
Therefore, $X$ must be a K3 surface by the classification theory of algebraic
surfaces in positive characteristic. By Proposition 2.7, we have
$K_{S}^{2}\geq-10$. Therefore, the Betti number $b_{2}(X)=b_{2}(S)\leq 20$,
which contradicts the topological invarinant of K3 surfaces.

Since the covering $X\longrightarrow S$ is locally define by (3.2), the
singular points are double points. By $A=0$, $X$ has only isolated
singularities. Therefore, $X$ is normal. Let $f:S\longrightarrow{\bf P}^{1}$
be a genus 1 fibration. If this fibration is quasi-elliptic, the conductrix
$A$ contains the curve of cusps, which contradicts $A=0$. Therefore,
$f:S\longrightarrow{\bf P}^{1}$ is an elliptic surface. We may assume that
this fibration has a 2-section.

Suppose that one of isolated singularities is non-rational. We consider the
Frobenius base change

$\begin{array}[]{ccc}S_{F}&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&S\\\
\tilde{f}\downarrow&&\downarrow f\\\ {\bf P}^{1}&\stackrel{{\scriptstyle
F}}{{\longrightarrow}}&{\bf P}^{1}.\end{array}$

Since $S_{F}$ is birationally equivalent to $X$ which has a non-rational
singular point, $S_{F}$ is a rational surface by Lemma 3.16. Now, we consider
the relative Jacobian surface $J(S)\longrightarrow{\bf P}^{1}$ and the
Frobenius base change $J(S)_{F}$. We may assume that $J(S)\longrightarrow{\bf
P}^{1}$ is relatively minimal. Since $J(S)$ is a rational surface by Lemma
3.15, $J(S)_{F}$ is a K3-like surface. Since $X$ is normal, that is, the
conductrix $A$ for $S$ is empty, by the same argument for the classical
Enriques surfaces in Matsumoto [17, Proposition 4.3] we see that $S_{F}$ is
birationally equivalent to $J(S)_{F}$ and so $J(S)_{F}$ is rational and that
$J(S)_{F}$ has non-rational singularity. Since $J(S)\longrightarrow{\bf
P}^{1}$ is a rational elliptic surface, we have a list of singular fibers in
Lang [16]. By a result of Schröer [20], the only case where a non-rational
singular point appears in $J(S)_{F}\longrightarrow{\bf P}^{1}$ is only Case 9C
in Lang [16]. In this case, the singular fiber is of type II with discriminant
12. Therefore, the rational ellptic surface $J(S)\longrightarrow{\bf P}^{1}$
has only one singular fiber of type II and all other fibers are supersingular
elliptic curves by Lang [16]. Since the types of fibers of the relatively
minimal model of $f:S\longrightarrow{\bf P}^{1}$ are same as the ones of
fibers of $J(S)\longrightarrow{\bf P}^{1}$, and since a multiple fiber of
Coble surface is tame, any supersingular elliptic curve cannot become a
multiple fiber. Therefore, we have no chance to make a Coble surface by
blowing up the relatively minimal elliptic rational surface. Hence, if $A=0$,
then the canonical covering of the Coble surface $S$ has only rational
singular points. ∎

######  Corollary 3.18.

Assume the conductrix $A=0$. Then, the minimal non-singular model $\tilde{X}$
of the canonical covering $X$ is a K3 surface.

###### Proof.

Let $\mu:\tilde{X}\longrightarrow X$ be the minimal resolution of
singularities. This corollary follows from
$\omega_{\tilde{X}}\cong\mu^{*}\omega_{X}\cong\mu^{*}{\mathcal{O}_{X}}$ and
${\rm H}^{1}(\tilde{X},{\mathcal{O}}_{\tilde{X}})\cong{\rm
H}^{0}(X,R^{1}\mu_{*}{\mathcal{O}}_{\tilde{X}})=0$. ∎

Now, we consider the case with the conductrix $A\neq 0$.

######  Proposition 3.19.

Assume the conductix $A\neq 0$. Then, the singularities of the normalization
$X^{\prime}$ of the canonical covering $X$ are all rational, and $X$ is a
rational surface.

###### Proof.

The proof is similar to the one for Enriques surfaces (cf. Cossec, Dolgachev,
Liedtke [3]). We denote by $\tilde{X}$ the minimal resolution of singularities
of $X^{\prime}$:

$\begin{array}[]{ccccc}\tilde{X}&\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}&X^{\prime}&\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}&X\\\
&&\searrow&&\swarrow\pi\\\ &&&S&\end{array}$

By Lemma 3.5, $\pi\circ\nu$ is a flat morphism. Therefore, by Cossec,
Dolgachev, Liedtke [3, Proposition 0.2.22], $X^{\prime}$ is Gorenstein and
again by Cossec, Dolgachev, Liedtke [Proposition 0.28, loc. cit.] and (3.6),
we have

(3.11)
$\omega_{X^{\prime}}\cong(\pi\circ\nu)^{*}(\omega_{S}\otimes{{\mathcal{L}}^{\prime}}^{-1})\cong(\pi\circ\nu)^{*}{\mathcal{O}}_{S}(-A).$

We denote by $\cup_{i=1}^{k}E_{i}$ the support of the exceptional divisors of
the resolution $\mu$. Here, $E_{i}$’s are the irreducible components. Then, by
Hidaka, Watanabe [7, Lemma 1.1], we have

$\begin{array}[]{cl}\omega_{\tilde{X}}&\sim\mu^{*}\omega_{X^{\prime}}-\sum_{i=1}^{k}r_{i}E_{i}\quad(r_{i}\geq
0)\\\
&\sim\mu^{*}(\pi\circ\nu)^{*}{\mathcal{O}}_{S}(-A)-\sum_{i=1}^{k}r_{i}E_{i}\end{array}$

Therefore, we have the pluri-genus $P_{m}({\tilde{X}})=0$ for $m\geq 1$. Since
$\pi\circ\nu\circ\mu$ is a purely inseparable morphism of degree 2, there
exists a rational map $S^{(1/2)}\longrightarrow\tilde{X}$. By the universality
of Albanese variety, there exists a surjective morphism from the Albanese
variety ${\rm Alb}(S^{(1/2)})$ to ${\rm Alb}(\tilde{X})$. Since $S$ is
rational, $S^{(1/2)}$ is also rational and so ${\rm Alb}(S^{(1/2)})=0$.
Therefore, we have ${\rm Alb}(\tilde{X})=0$, that is, the irregularity
$q(\tilde{X})=0$. Therefore, by Castelnuovo’s criterion of rationality we see
$\tilde{X}$ is a rational surface. In particular, we have ${\rm
H}^{1}(\tilde{X},{\mathcal{O}}_{\tilde{X}})=0$.

Now, we consider the Leray-spectral sequence for $\mu:\tilde{X}\longrightarrow
X^{\prime}$:

$E_{2}^{i,j}={\rm
H}^{j}(X^{\prime},R^{i}\mu_{*}{\mathcal{O}}_{\tilde{X}})\Longrightarrow{\rm
H}^{i+j}(\tilde{X},{\mathcal{O}}_{\tilde{X}}).$

Then, we have an exact sequence

${\rm H}^{1}(X^{\prime},{\mathcal{O}}_{X^{\prime}})\longrightarrow{\rm
H}^{1}(\tilde{X},{\mathcal{O}}_{\tilde{X}})\longrightarrow{\rm
H}^{0}(X^{\prime},R^{1}\mu_{*}{\mathcal{O}}_{\tilde{X}})\longrightarrow{\rm
H}^{2}(X^{\prime},{\mathcal{O}}_{X^{\prime}}).$

Since ${\rm H}^{1}(\tilde{X},{\mathcal{O}}_{\tilde{X}})=0$ and

${\rm H}^{2}(X^{\prime},{\mathcal{O}}_{X^{\prime}})\cong{\rm
H}^{0}(X^{\prime},\omega_{X^{\prime}})\cong{\rm
H}^{0}(X^{\prime},(\pi\circ\nu)^{*}{\mathcal{O}}_{S}(-A))=0,$

we have ${\rm H}^{0}(X^{\prime},R^{1}\mu_{*}{\mathcal{O}}_{\tilde{X}})=0$,
that is, $R^{1}\mu_{*}{\mathcal{O}}_{\tilde{X}}=0$. This means that all
singular points of $X^{\prime}$ are rational. ∎

##  4\. Properties of the conductrix

Let $S$ be a Coble surface, $\pi:X\longrightarrow S$ the canonical covering
and $f:S\longrightarrow{\bf P}^{1}$ a genus 1 fibration. We assume this
fibration satisfies the condition of (Case 1) or (Case 2) in Subsection 3.3.
In this section, we assume the conductrix $A\neq 0$, if otherwise mentions. As
for the general theory of conductrix, in particular for Enriques surfaces in
characteristic 2, we have the theory by Ekedahl and Shepherd-Barron [6]. In
the case of Coble surfaces, the situation is a bit different from their paper,
and we show the details, although arguments sometimes overlap with the ones in
Ekedahl and Shepherd-Barron, loc. cit. We use here the notation in Section 3.

######  Proposition 4.1.

The intesection number $A\cdot K_{S}=0$. In particular, the boundary
components $B_{i}$ don’t intersect the conductrix $A$, and each component of
$A$ is contained in the Coble-Mukai lattice ${\rm CM(S)}$.

###### Proof.

We have $-2K_{S}\sim\sum_{i=1}^{n}B_{i}$. Therefore, this result follows from
$B\cap\sum_{i=1}^{n}B_{i}=\emptyset$ and $B=2A$. ∎

######  Proposition 4.2.

The conductrix $A$ contains neither a fiber nor a half fiber.

###### Proof.

We consider the divisorial part $(\eta)$ of the rational 1-form $\eta$, and by
the multiplication by $\eta$, we have an exact sequence:

$0\longrightarrow{\mathcal{O}}((\eta))\stackrel{{\scriptstyle\times\eta}}{{\longrightarrow}}\Omega_{S}.$

We get an inclusion ${\rm
H}^{0}(S,{\mathcal{O}}_{S}((\eta)))\hookrightarrow{\rm
H}^{0}(S,\Omega_{S}^{1})$. Since $S$ is a rational surface, we have ${\rm
H}^{0}(S,\Omega_{S}^{1})=0$. Therefore, we have ${\rm
H}^{0}(S,{\mathcal{O}}_{S}((\eta)))=0$.

Assume $f:S\longrightarrow{\bf P}^{1}$ has a multiple fiber $f^{-1}(P_{0})$.
If $A$ contains a half fiber, $B=2A$ contains the fiber $f^{-1}(P_{0})$. In
this case $\sum_{i=1}^{n}B_{i}$ is contained in $f^{-1}(P_{\infty})$. Then,
there exists a rational function on $S$ such that it has a pole of order 1
along $f^{-1}(P_{0})$ and has a zero of order 1 along $f^{-1}(P_{\infty})$. It
is a non-zero function in ${\rm H}^{0}(S,{\mathcal{O}}_{S}((\eta)))$, a
contradiction. If $A$ contain a fiber $F_{1}$, then $B=2A$ contains $2F_{1}$.
Since $A\cap\sum_{i=1}^{n}B_{i}=\emptyset$, $F_{1}$ is different from
$f^{-1}(P_{\infty})$ and $\sum_{i=1}^{n}B_{i}$ is contained in
$f^{-1}(P_{\infty})$. Therefore, there exists a rational function such that it
has a pole of order 2 along $F_{1}$ and has a zero of order 1 along
$f^{-1}(P_{\infty})$. It is a non-zero function in ${\rm
H}^{0}(S,{\mathcal{O}}_{S}((\eta))$, a contradiction.

Assume $f:S\longrightarrow{\bf P}^{1}$ has no multiple fiber. If $A$ contains
a fiber $F_{1}$, then $B=2A$ contains $2F_{1}$. The fiber $F_{1}$ is neither
$f^{-1}(P_{0})$ nor $f^{-1}(P_{\infty})$ as above. Therefore, there exists a
rational function such that it has a pole of order 2 along $F_{1}$ and has a
zero of order 1 along $f^{-1}(P_{0})$ and $f^{-1}(P_{\infty})$, respectively.
It is a non-zero function in ${\rm H}^{0}(S,{\mathcal{O}}_{S}((\eta))$, a
contradiction. ∎

######  Lemma 4.3.

${\rm H}^{i}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A))=0$ for any $i$.

###### Proof.

By (3.5) and (3.6), we have a long exact sequence:

$0\longrightarrow{\rm H}^{0}(S,{\mathcal{O}}_{S})\longrightarrow{\rm
H}^{0}(S,(\pi\circ\nu)_{*}{\mathcal{O}}_{X^{\prime}})\longrightarrow{\rm
H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A))\longrightarrow{\rm
H}^{1}(S,{\mathcal{O}}_{S}).$

Since ${\rm H}^{0}(S,{\mathcal{O}}_{S})\cong k$, ${\rm
H}^{0}(S,(\pi\circ\nu)_{*}{\mathcal{O}}_{X^{\prime}})\cong k$ and ${\rm
H}^{1}(S,{\mathcal{O}}_{S})=0$, we have ${\rm
H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A))=0$. By the Serre duality
theorem, we have

${\rm H}^{2}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A))\cong{\rm
H}^{0}(S,{\mathcal{O}}_{S}(-A))=0.$

Therefore, by the Riemann-Roch theorem, we have

$0\geq\chi(\omega_{S}\otimes{\mathcal{O}}_{S}(A))=\frac{A^{2}}{2}+1.$

Therefore, we have $A^{2}\leq-2$.

Since $K_{S}^{2}=-n$, by Noether’s formula and Igusa’s formula (cf. Igusa
[8]), we have

$\begin{array}[]{cl}12+n&=c_{2}(S)={\text{deg}}\langle\eta\rangle+(\eta)\cdot
K_{S}-(\eta)^{2}\\\
&={\text{deg}}\langle\eta\rangle+(-\sum_{i=1}^{n}B_{i}+2A)\cdot
K_{S}-(-\sum_{i=1}^{n}B_{i}+2A)^{2}\\\
&={\text{deg}}\langle\eta\rangle+2n-4A^{2}.\end{array}$

Therefore we have

(4.1) ${\text{deg}}\langle\eta\rangle=12-n+4A^{2}.$

Since ${\text{deg}}\langle\eta\rangle\geq 0$ and $n\geq 1$, we have
$A^{2}>-3$. Hence, we conclude $A^{2}=-2$, and we have ${\rm
H}^{1}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A))=0$ ∎

By the proof of this lemma, we have the following.

######  Corollary 4.4.

$A^{2}=-2$ and ${\text{deg}}\langle\eta\rangle=4-n$.

######  Proposition 4.5.

$A$ is numerically $1$-connected.

###### Proof.

Suppose we have a decomposition $A=A_{1}+A_{2}$ with divisors $A_{1}>0$ and
$A_{2}>0$. Then, we have an inclusion

${\rm H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A_{1}))\hookrightarrow{\rm
H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A)).$

Therefore, by Lemma 4.3 we have ${\rm
H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A_{1}))=0$. By the Serre duality
theorem, we have ${\rm
H}^{2}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(A_{1}))\cong{\rm
H}^{0}(S,{\mathcal{O}}_{S}(-A_{1}))=0$. Therefore, by the Riemann-Roch
theorem, we have
$0\geq\chi(\omega_{S}\otimes{\mathcal{O}}_{S}(A_{1}))=A_{1}^{2}/2+1$.
Therefore, $A_{1}^{2}\leq-2$. Similarly, we have $A_{2}^{2}\leq-2$. By
Corollary 4.4, we have $-2=A^{2}=A_{1}^{2}+2A_{1}\cdot A_{2}+A_{2}^{2}$, we
conclude $A_{1}\cdot A_{2}\geq 1$. ∎

######  Proposition 4.6.

All isolated singularities of $X$ are rational singularities of type ${\rm
A}_{1}$ and the number of the singularities are less than or equal to $3$.

###### Proof.

By Lemma 4.4, we have ${\text{deg}}\langle\eta\rangle\leq 3$. The idea of the
proof comes from Ekedahl and Shepherd-Barron [6, Proposition 0.5]. Let
$P\in\tilde{U}_{i}\subset X$ be a singular point. The covering
$\pi_{i}:\tilde{U}_{i}\longrightarrow U_{i}$ is given by
$z_{i}^{2}=f_{i}g_{i}$. By putting the square part of $f_{i}g_{i}$ into
$z_{i}$, let the equation become $z_{i}^{2}=k_{i}$. Let $(x_{i},y_{i},z_{i})$
be local coodinates at the point $\pi_{i}(P)$. Then, since $P$ is a singular
point, $k_{i}$ has no terms of degree less than 2. If it has a term of degree
2, then we have an expression $z_{i}^{2}=x_{i}y_{i}+{\rm higher\ terms}$. This
is a rational singularity of type ${\rm A}_{1}$. If $k_{i}$ has no terms of
degree 2, then considering partial differentiations $k_{ix_{i}}$,
$k_{iy_{i}}$, we have

$\dim_{k}k[[x_{i},y_{i}]]/(k_{ix_{i}},k_{iy_{i}})\geq 4.$

Therefore, we have ${\text{deg}}\langle\eta\rangle\geq 4$, a contradiction.
Hence, all isolated singularities of $Y$ is rational singularities of type
${\rm A}_{1}$ and the number of isolated singularities is less than or equal
to 3. ∎

######  Corollary 4.7.

Under the assumptin $A\neq 0$, the sum of the number of isolated singularity
and the number of boundary components is equal to $4$. In particular, if a
Coble surface $S$ has a quasi-elliptic fibration, then the sum of the number
of isolated singularity and the number of boundary components is always equal
to $4$.

###### Proof.

The former part follows from ${\text{deg}}\langle\eta\rangle+n=4$. The latter
part follows from Lemma 3.11. ∎

######  Lemma 4.8.

$\dim{\rm H}^{0}(S,{\mathcal{O}}_{S}(A))=\dim{\rm
H}^{1}(S,{\mathcal{O}}_{S}(A))=1$ and $\dim{\rm
H}^{2}(S,{\mathcal{O}}_{S}(A))=0$.

###### Proof.

Since $A$ is a connected effective divisor such that $A_{1}^{2}\leq-2$ holds
for any effective divisor $A_{1}\subset A$, we have $\dim{\rm
H}^{0}(S,{\mathcal{O}}_{S}(A))=1$. By an exact sequence
$0\longrightarrow\omega_{S}\otimes{\mathcal{O}}_{S}(-A)\longrightarrow\omega_{S}$,
we have an exact sequence

$0\longrightarrow{\rm
H}^{0}(S,\omega_{S}\otimes{\mathcal{O}}_{S}(-A))\longrightarrow{\rm
H}^{0}(S,\omega_{S}).$

Since $S$ is rational, we have ${\rm H}^{0}(S,\omega_{S})=0$. Therefore, we
have ${\rm H}^{0}(S,\omega\otimes{\mathcal{O}}_{S}(-A))=0$. By the Serre
duality theorem we have ${\rm H}^{2}(S,{\mathcal{O}}_{S}(A))=0$. Hence, by the
Riemann-Roch theorem we have $\dim{\rm H}^{1}(S,{\mathcal{O}}_{S}(A))=1$ ∎

######  Theorem 4.9.

The irreducible components of the conductrix $A$ are all $(-2)$-curves. If
different two irreducible components intersect, then they intersect each other
at a point transeversely.

###### Proof.

Let $C$ be an irreducible curve contained in $A$. In the same way as in the
proof of Proposition 4.5 we see $C^{2}\leq-2$. On $S$, there exists no curve
with self-intersection number $-3$. The irreducible curves with self-
intersection number $-4$ are only boundary components $B_{i}$. Since $A$ does
not contain $(-4)$-curves, we have $C^{2}=-2$. Since $C\subset A$, we have
$C\cdot K_{S}=0$, and by the adjunction formula, $C$ is a non-singular
rational curve with $C^{2}=-2$.

Let $C_{1}$ and $C_{2}$ be two different irreducible curves which are
contained in $A$. If $C_{1}\cdot C_{2}\geq 3$, then we have
$(C_{1}+C_{2})^{2}\geq 2$. By the Riemann-Roch theorem, we have
$\chi({\mathcal{O}}_{S}(C_{1}+C_{2}))\geq 2$. Therefore we have $\dim{\rm
H}^{0}(S,{\mathcal{O}}_{S}(C_{1}+C_{2}))\geq 2$. On the other hand, by the
exact sequence
$0\longrightarrow{\mathcal{O}}_{S}(C_{1}+C_{2})\longrightarrow{\mathcal{O}}_{S}(A)$
we have an inclusion ${\rm
H}^{0}(S,{\mathcal{O}}_{S}(C_{1}+C_{2}))\hookrightarrow{\rm
H}^{0}(S,{\mathcal{O}}_{S}(A))$. By Lemma 4.8, we see $\dim{\rm
H}^{0}(S,{\mathcal{O}}_{S}(C_{1}+C_{2}))=1$, a contradiction.

If $C_{1}\cdot C_{2}=2$, then we have $(C_{1}+C_{2})^{2}=0$. If $C_{1}$ and
$C_{2}$ are contained in a same fiber $F_{1}$, then there exists a rational
number $\lambda$ such that $C_{1}+C_{2}=\lambda F_{1}$, which contradicts
Proposition 4.2. Since $C_{1}\cdot C_{2}=2$, they cannot be containd in
different fibers. Therefore, the genus 1 fibration is quasi-elliptic, and we
may assume that $C_{1}$ is the curve of cusps and $C_{2}$ is contained in a
fiber $F_{1}$. By the general theory of quasi-elliptic surface, we have
$C_{1}\cdot F_{1}=2$. Therefore, $C_{2}$ must be a simple irreducible
component of $F_{1}$. However, since $A$ is given by a part of the divisor
$(dt/t)$, any simple irreducible component cannot be contained in $A$. Hence,
we have $0\leq C_{1}\cdot C_{2}\leq 1$. ∎

We give here a remark for the case that the genus 1 fibration
$f:S\longrightarrow{\bf P}^{1}$ is an elliptic surface.

######  Proposition 4.10.

Let $f:S\longrightarrow{\bf P}^{1}$ be an elliptic fibration. If $A$ contains
a component of the multiple fiber $2F_{1}$, then $F_{1}\supset A$ holds. If
$A$ contains a component of a non-multiple fiber $F_{1}$, then $F_{1}\supset
A$ holds.

###### Proof.

Since $A$ is connected and does not contain any points in a general fiber, $A$
consists of some components of one fiber, say $F_{1}$ or $2F_{1}$ if it is a
multiple fiber. By Theorem 4.9 $F_{1}$ contains at least two irreducible
curves. By Proposition 4.2, $A$ does not contain $F_{1}$. Therefore, there
exist an effective divisor $A_{1}$ and a non-negative divisor $A_{2}$ such
that $A_{1}$ and $A_{2}$ have no common components and such that
$F_{1}-A=A_{1}-A_{2}$. On the one hand, we have

$(F_{1}-A)^{2}=F_{1}^{2}-2F_{1}\cdot A+A^{2}=-2.$

On the other hand, we have

$(F_{1}-A)^{2}=(A_{1}-A_{2})^{2}=A_{1}^{2}-2A_{1}\cdot A_{2}+A_{2}^{2}\leq
A_{1}^{2}+A_{2}^{2}.$

If $A_{2}$ is an effective divisor, we have $A_{1}^{2}\leq-2$ and
$A_{2}^{2}\leq-2$. Therefore, we have $(F_{1}-A)^{2}\leq-4$, a contradiction.
Therefore, we have $A_{2}=0$. ∎

For a rational number $a$, we denote by $[a]$ the integral part of $a$.

######  Theorem 4.11.

Let $f:S\longrightarrow{\bf P}^{1}$ be an elliptic fibration of a Coble
surface $S$. Assume that the conductrix $A\neq 0$ and that $A$ is contained in
a simple fiber $F_{1}=\sum_{i=1}^{\ell}m_{i}C_{i}$ with irreducible components
$C_{i}$. Then, $A=\sum_{i=1}^{\ell}[\frac{m_{i}}{2}]C_{i}$.

###### Proof.

By the assumption, the fiber is neither $f^{-1}(P_{0})$ nor
$f^{-1}(P_{\infty})$. Therefore, the zero divisor of $dt$ on the open set
defined by $t\neq{\infty}$ is equal to $2A$. By a suitable translation of $t$,
we may assume the singular fiber $F_{1}$ is given by $t=0$, and let
$A=\sum_{i=1}^{\ell}\ell_{i}C_{i}$. Then, on a general point $P$ of $C_{i}$ we
have the expression $t=uf_{i}^{m_{i}}$. Here, $f_{i}$ is a defining equation
of $C_{i}$ at $P$ and $u$ is a unit. If $m_{i}$ is odd, then we have
$dt=f_{i}^{m_{i}-1}(udf_{i}+f_{i}du)$. Since we can regard $f_{i}$ is one
element of local coodinates at $P$, the zero divisor which is given by $dt$ at
$P$ is $(m_{i}-1)C_{i}$. Therefore we have $\ell_{i}=(m_{i}-1)/2$. Therefore,
the possiblity of the types of singular fibers for $A\neq 0$ is only type
${\rm I}_{\nu}^{*}$, ${\rm II}^{*}$, ${\rm III}^{*}$ and ${\rm IV}^{*}$.

In case that $m_{i}$ is even, we have $dt=f_{i}^{m_{i}}du$. Therefore, the
order of the zero divisor which is given by $dt$ is greater than or equal to
$m_{i}$. Therefore, $\ell_{i}$ is greater than or equal to $m_{i}/2$. We will
show that if $m_{i}$ is even, then $\ell_{i}=m_{i}/2$. Now, we consider the
case of type ${\rm II}^{*}$. Then, the fiber is given by

$C_{1}+2C_{2}+3C_{3}+4C_{2}+5C_{5}+6C_{6}+4C_{7}+2C_{8}+3C_{9}.$

Here, $C_{i}^{2}=-2$ ($1\leq i\leq 9$), $C_{i}\cdot C_{i+1}=1$ ($1\leq i\leq
7$) and $C_{6}\cdot C_{9}=1$. The other intersection numbers are all 0.
Considering the argument above, we set

$A=(1+a_{2})C_{2}+C_{3}+(2+a_{4})E_{4}+2C_{5}+(3+a_{6})E_{6}+(2+a_{7})E_{7}+(1+a_{8})E_{8}+C_{9}.$

with non-negative integer $a_{i}$ ($i=2,4,6,7,8$). Since we have $A^{2}=-2$,
calculating the self-intersection numbers of both sides, we have an equation:

$a_{2}(1+a_{2})+a_{4}(1+a_{4})+a_{6}^{2}/2+a_{6}+(a_{6}-a_{7})^{2}/2+(a_{8}-a_{7})^{2}/2+a_{8}^{2}/2=0.$

Since $a_{i}\geq 0$ for all $i$, we have $a_{2}=a_{4}=a_{6}=a_{7}=a_{8}=0$.
This shows the claim for the type ${\rm II}^{*}$. The proofs of the other
cases are similar. ∎

Now, we examine the properties of $(-2)$-curves on a Coble surface $S$,
following the method in Ekedahl and Shepherd-Barron [6, Definition-Lemma 0.8].
As before, we denote by $X^{\prime}$ the normalization of the canonical
covering of $S$. We donote by $\tilde{X}$ the minimal resolution of
$X^{\prime}$. Since the singularities of $X^{\prime}$ are only rational double
points, the minimal dissolution $\tilde{S}$ gives the minimal resolution of
singularities, and we get the following commutative diagram:

$\begin{array}[]{ccccc}\tilde{C}&\subset&\tilde{X}&\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}&X^{\prime}\\\
&&\quad\downarrow\tilde{\rho}&&\quad\downarrow\rho\\\
&&\tilde{S}&\stackrel{{\scriptstyle\theta}}{{\longrightarrow}}&S\\\
&&\cup&&\cup\\\ &&C^{\prime}&\longrightarrow&C\end{array}$

Here, $C$, $C^{\prime}$ and $\tilde{C}$ are irreducible curves such that
$\tilde{\rho}({\tilde{C}})=C^{\prime}$ and $\theta(C^{\prime})=C$. The
morphism $\theta$ is a morphism given by a finite number of blowing-ups. We
denote by $r$ the number of blowing-ups on the curve $C$ which includes
blowing-ups at the infinitesimal near points. We also denote by $s$ the degree
of the morphism $\rho\circ\mu:\tilde{C}\longrightarrow C$. This number $s$ is
equal to the degree of the morphism $\tilde{\rho}:\tilde{C}\longrightarrow
C^{\prime}$ and it is either 1 or 2. By (3.11) we have
$\omega_{X^{\prime}}\cong\rho^{*}({\mathcal{O}}_{S}(-A))$. Since the
singularities of $X^{\prime}$ are rational double points, we have
$\omega_{\tilde{X}}\cong\mu^{*}\omega_{X^{\prime}}$. Therefore, we have

$\omega_{\tilde{X}}\cong\mu^{*}\rho^{*}({\mathcal{O}}_{S}(-A)).$

Since $(C^{\prime})^{2}=C^{2}-r$, we have

$\left(\frac{2}{s}\tilde{C}\right)^{2}=({\tilde{\rho}}^{-1}(C^{\prime}))^{2}={\text{deg}}\
\tilde{\rho}\cdot(C^{\prime})^{2}=2(C^{2}-r).$

Therefore, we have

$\tilde{C}^{2}=(C^{2}-r)s^{2}/2.$

By the adjunction formula, we have

$\omega_{\tilde{C}}\cong(\omega_{\tilde{X}}\otimes{\mathcal{O}}_{\tilde{X}}(\tilde{C}))|_{\tilde{C}}\cong(\mu^{*}\rho^{*}({\mathcal{O}}_{S}(-A))\otimes{\mathcal{O}}_{\tilde{X}}(\tilde{C}))|_{\tilde{C}}.$

We denote by $g(\tilde{C})$ the genus of the curve $\tilde{C}$. Taking the
degrees of both sides, we have

$\begin{array}[]{cl}2(g(\tilde{C})-1)&=\mu^{*}\rho^{*}({\mathcal{O}}_{S}(-A))\cdot\frac{s}{2}\mu^{*}\rho^{*}({\mathcal{O}}_{S}(C))+{\tilde{C}}^{2}\\\
&=2\cdot\frac{s}{2}\\{(-A)\cdot C\\}+(C^{2}-r)s^{2}/2\\\ &=-sA\cdot
C+(C^{2}-r)s^{2}/2.\end{array}$

Therefore, we have

(4.2) $g(\tilde{C})=-sA\cdot C/2+(C^{2}-r)s^{2}/4+1.$

Note that this formula holds even if $A=0$.

######  Remark 4.12.

In Ekedahl, Shepherd-Barron [6, Definition-Lemma 0.8], they give similar
results to the theorem below. But the $A$ which they treat is not the
conductrix. Therefore, we give here the complete proof of the theorem for our
conductrix $A$ to make the situation clear. However, in the case of Enriques
surfaces the difference between these two A’s is just the factor of the
dualizing sheaf, which is numerically trivial. Therefore, the numerical
results are same for these two A’s. In fact, in Katsura, Kondō and Martin [11,
Lemma 3.4], these facts are used.

######  Theorem 4.13.

Let $C$ be a $(-2)$-curve on $S$.

  * (i)

If $s=1$, then $A\cdot C=1-\frac{r}{2}$ holds.

  * (ii)

If $s=2$, then $A\cdot C=-1-r$ holds.

###### Proof.

Since $C$ is a non-singular curve of genus 0 with $C^{2}=-2$, we get the
results form (4.2). ∎

######  Corollary 4.14.

Let $C$ be a $(-2)$-curve.

  * (1)

If $A\not\supset C$, then $s=1$ and only the following cases occur.

    * (i)

$A\cdot C=0$ and $r=2$.

    * (ii)

$A\cdot C=1$ and $r=0$.

  * (2)

If $A\supset C$, then only the following cases occur.

    * (a)

Case $s=1$.

      * (i)

$A\cdot C=-2$, $A=C$ and $r=6$.

      * (ii)

$A\cdot C=-1$ and $r=4$.

      * (iii)

$A\cdot C=0$ and $r=2$.

      * (iv)

$A\cdot C=1$ and $r=0$.

    * (b)

Case $s=2$.

    * (i)

$A\cdot C=-2$, $A=C$ and $r=1$.

    * (ii)

$A\cdot C=-1$ and $r=0$.

###### Proof.

(1) By assumption $A\not\supset C$, we have $A\cdot C\geq 0$. Therefore, we
have $s=1$. Therefore, by $0\leq A\cdot C=1-\frac{r}{2}$, we have $r\leq 2$.
Since $A\cdot C$ is an integer and $r\geq 0$, we have $r=2$ or $0$, and we get
our results.

(2) If $A=C$, then we have clearly $A\cdot C=-2$ and $r=6$ if $s=1$, $r=1$ if
$s=2$. Now, we assume $A\neq C$ and $A\supset C$. Since $A$ is numerically
connected, we have $(A-C)\cdot C\geq 1$. Therefore, we have $A\cdot C\geq-1$
by $C^{2}=-2$. If $s=1$, then we have $-1\leq A\cdot C=1-\frac{r}{2}$ and we
have $r\leq 4$. Therefore, we have $r=0,2$, or $4$. If $s=2$, we have $-1\leq
A\cdot C=-1-r$. Therefore, we have $r=0$, and we get our result. ∎

######  Corollary 4.15.

Let $C$ be a $(-2)$-curve on $S$. Then, if $A\neq C$, then we have $-1\leq
A\cdot C\leq 1$. If $s=2$, then we have $C\subset A$.

###### Proof.

This follows from Corollary 4.14. ∎

The following corollary is shown in Ekedahl and Shepherd-Barron [6,
Definition-Lemma 0.8 (iii)].

######  Corollary 4.16.

$({\rm i})$ Let $C_{1}$ be a $(-2)$-curve on $S$ with $A\cdot C_{1}=1$. Then,
any $(-2)$-curve $C_{2}$ which intersects $C_{1}$ only at a point transversely
has $s=2$ with $A\cdot C_{2}=-1$ unless $A=C_{2}$. Moreover, $C_{2}$ is an
irreducible component of $A$.

$({\rm ii})$ If two $(-2)$-curves $C_{1}$, $C_{2}$ on $S$ which meet
transversely at a point have $s$-invariant $1$, then their intersection is
blown-up.

###### Proof.

We denote by $C^{\prime}_{1}$ (resp. $C^{\prime}_{2}$) the proper transform of
$C_{1}$ (resp. $C_{2}$) for the morphism $\theta$.

(i) By Corollary 4.14, for $C_{1}$ we have $r=0$ and $s=1$. This means that we
don’t blow up at the intersection point of $C_{1}$ and $C_{2}$, and that there
exists an irreducible curve $\tilde{C}_{1}$ on ${\tilde{X}}$ such that
$\tilde{\rho}^{-1}(C^{\prime}_{1})=2\tilde{C}_{1}$. Therefore, we have
$C^{\prime}_{1}\cdot C^{\prime}_{2}=C_{1}\cdot C_{2}=1$ and

$2={\text{deg}}(\tilde{\rho})(C^{\prime}_{1}\cdot
C^{\prime}_{2})=\tilde{\rho}^{-1}(C^{\prime}_{1})\cdot\tilde{\rho}^{-1}(C^{\prime}_{2})=2(\tilde{C_{1}}\cdot\tilde{\rho}^{-1}(C^{\prime}_{2})).$

Therefore, we have $\tilde{C_{1}}\cdot\tilde{\rho}^{-1}(C^{\prime}_{2})=1$.
Hence, we have $s=2$ for the curve $C_{2}$ and by Corollary 4.14 we conclude
$A\cdot C_{2}=-1$ and $C_{2}\subset A$.

(ii) Suppose that their intersection is not blown-up. Then, we have
$C^{\prime}_{1}\cdot C^{\prime}_{2}=1$. Since we have $s=1$ for both $C_{1}$
and $C_{2}$, there exist irreducible curves $\tilde{C}_{1}$ and
$\tilde{C}_{2}$ on ${\tilde{X}}$ such that
$\tilde{\rho}^{-1}(C^{\prime}_{1})=2\tilde{C}_{1}$ and
$\tilde{\rho}^{-1}(C^{\prime}_{2})=2\tilde{C}_{2}$. Therefore, we have

$2={\text{deg}}~{}\tilde{\rho}~{}(C^{\prime}_{1}\cdot
C^{\prime}_{2})=\tilde{\rho}^{-1}(C^{\prime}_{1})\cdot\tilde{\rho}^{-1}(C^{\prime}_{2})=4(\tilde{C}_{1}\cdot\tilde{C}_{2}),$

a contradiction. ∎

######  Corollary 4.17.

Let $F$ be a fiber such that $F\cap A\neq\emptyset$. Assume that $F$ is of
type ${\rm I}_{m}^{*}$, ${\rm II}^{*}$, ${\rm III}^{*}$ or ${\rm IV}^{*}$.
Then, all irreducible components except irreducible components at the end of
$F$ are contained in $A$.

###### Proof.

Let $C$ be an irreducible component of $F$ which does not exist at the end of
$F$. Suppose that $C$ is not contained in $A$. By choosing a suitable
irreducible component, we may assume that $C$ intersects $A$. Since $A$ is
connected, there exists an irreducible component $G$ of $F$ such that $G$
intersects $C$ and is not contained in $A$. Since $A\cdot C>0$, we have
$A\cdot C=1$ by Corollary 4.15. Therefore, by Corollary 4.16 we have $G\subset
A$, a contradiction. ∎

######  Corollary 4.18.

Let $f:S\longrightarrow{\bf P}^{1}$ be an elliptic fibration of a Coble
surface $S$. Assume this fibration has a $(-2)$-curve $C_{s}$ as a 2-section.
Let $2F$ be a multiple fiber such that $F\cap A\neq 0$ and $2F$ be of type
$2{\rm I}_{m}^{*}$, $2{\rm II}^{*}$, $2{\rm III}^{*}$ or $2{\rm IV}^{*}$. Then
the simple irreducible component $C$ of the half fiber $F$ which intersects
the special 2-section $C_{s}$ is contained in $A$.

###### Proof.

Suppose that $C$ is not contained in $A$. Then, we have $A\cdot C_{s}=0$. By
Corollary 4.17, we have $A\cdot C>0$. Therefore, by Corollary 4.14, we have
$A\cdot C=1$. Since $C\cdot C_{s}=1$, by Corollary 4.16 we have $A\cdot
C_{s}=-1$, a contradiction. ∎

######  Remark 4.19.

Let $f:S\longrightarrow{\bf P}^{1}$ be an elliptic fibration of a Coble
surface $S$. If a simple fiber $F$ contains the conductrix $A$, then as we see
in the proof of Theorem 4.11, the conductrix $A$ for the simple fiber $F$ does
not contain any simple irreducible component of $F$. Therefore, in the case of
elliptic fibration with a $(-2)$-curve of 2-section, we can distinguish the
type of conductrix of multiple fiber from the one of simple fiber by Corollary
4.18.

Let $A=\sum_{i}m_{i}C_{i}$ be a conductrix. The graph of $A$ is the dual graph
of irreducible curves $\\{C_{i}\\}$ with the multiplicity $m_{i}$. We use
graphs of conductrices in the following Lemma 4.20, Tables 2, 3, 4.

We learned the following lemma from Shepherd-Barron. In Ekedahl, Shepherd-
Barron [6], they condidered the weight of the conductrix $A$ which is a
function on the Néron-Severi group ${\rm NS}(S)$ defined by $w(x)=x\cdot A$.
The weight satisfies several conditions in which a vertex of weight $0$ is
adjacent to at most two other vertices of weight $0$, from which the lemma
follows. We give here a direct proof.

######  Lemma 4.20.

For the multiple fiber of type ${\rm I}_{4}^{*}$ of elliptic fibration, the
following graph with given multiplicities is not represented by a conductrix.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{1}$$\textstyle{0}$$\textstyle{0}$$\textstyle{1}$$\textstyle{2}$$\textstyle{1}$$\textstyle{2}$$\textstyle{2}$$\textstyle{1}$

###### Proof.

Assume that this graph is represented by a conductrix $A$. The graph contains
a Dynkin diagram of type $D_{4}$. The four curves $C_{i}$ ($1\leq i\leq 4$)
satisfy $C_{i}\subset A$ and $A\cdot C_{i}=0$. Therefore, by Corollary 4.14
they have all $s=1$ and $r=2$. Therefore, by Corollary 4.14 the intersection
points are blown-up. However, the midle curve in the Dynkin diagram of $D_{4}$
has 3 intersetion points and it must have $r\geq 3$, a contradiction. ∎

Now, we can classify all possible conductrices for Coble surfaces. In Ekedahl
and Shepherd-Barron [6], they already classify the conductrices up to a
multiple of the Kodaira-Néron cycle of the special fiber as a general theory.
To make clear the results for Coble surfaces, we list up the possibilities of
the conductrices for Coble surfaces, although our list is a part of the one in
Ekedahl and Shepherd-Barron [6]. Our results are not up to a multiple of the
Kodaira-Néron cycle of the special fiber, but give the precise divisors. Under
the results of Propositions 4.2, 4.5, 4.10, Lemma 3.11 and Corollary 4.18, we
can calculate the possibilities of the conductrices for Coble surfaces by
using Corollaries 4.15, 4.16 (i) and 4.17 by a tedious routine calculation.
Note that since Coble surfaces are rational, the maximun number of irreducible
components of the singular fibers is 9.

In case of quasi-elliptic fibration, there are exactly two types of
conductrices for each singular fiber of type ${\rm III}$, ${\rm I}_{0}^{*}$,
${\rm I}_{2}^{*}$, ${\rm I}_{4}^{*}$, ${\rm III}^{*}$, ${\rm II}^{*}$
according to whether the fiber is simple or multiple. Hence, we get Table 2,
where the hollow vertices are the curves of cusps.

Table 2. Conductrices for quasi-elliptic fibrations Table 3. Conductrices for
elliptic fibrations ($A\neq 0$)

In case of elliptic fibration, if the singular fiber is simple, we already
calculated the conductrix $A$ in Theorem 4.11. Namely, if the singular fiber
of type ${\rm I}_{0}^{*}$, ${\rm I}_{1}^{*}$, ${\rm I}_{2}^{*}$, ${\rm
I}_{3}^{*}$, ${\rm I}_{4}^{*}$, ${\rm IV}^{*}$, ${\rm III}^{*}$ or ${\rm
II}^{*}$ is simple, we have exactly one type of conductrix for each singular
fiber. If the singular fiber of type ${\rm I}_{0}^{*}$, ${\rm I}_{1}^{*}$,
${\rm I}_{2}^{*}$, ${\rm I}_{3}^{*}$, ${\rm IV}^{*}$, ${\rm III}^{*}$ or ${\rm
II}^{*}$ is multiple and the fibration has no $(-2)$-curve of 2-section, then
there exist two possibilities of the type of conductrix for each singular
fiber. If the singular fiber of type ${\rm I}_{4}^{*}$ is multiple, then we
have exactly one type of conductrix. Because one possibility is excluded by
Lemma 4.20. Hence, we get Table 3. For a multiple fiber of elliptic fibration,
if the fibration has a $(-2)$-curve of 2-section, then the situation is
different. If the elliptic fiberation with the multiple fiber of type type
${\rm I}_{0}^{*}$, ${\rm I}_{1}^{*}$ or ${\rm IV}^{*}$ has a $(-2)$-curve of
2-section, then we have exactly one type of conductrix for each multiple
fiber, and there exists no elliptic fibration with multiple fiber of type
${\rm I}_{3}^{*}$, ${\rm I}_{4}^{*}$, ${\rm III}^{*}$ or ${\rm II}^{*}$ such
that the fibration has a $(-2)$-curve of 2-section. This follows from Lemma
4.21 below. Hence, we get Table 4.

In Tables 2, 3 and 4, the numbers are the multiplicities of the irreducible
components of $A$, “simple” means a simple fiber and “multiple” means a
multiple fiber. We will give two examples of concrete calculations of
conductrices of multiple fibers of elliptic fiber space below.

######  Lemma 4.21.

There are no special elliptic fibrations on Coble surfaces with a multiple
fiber $2F$ of type $2{\rm II}^{*},2{\rm III}^{*}$, $2{\rm I}_{m}^{*}$
$(m=2,3,4)$.

###### Proof.

Let $C_{s}$ be a $(-2)$-curve of 2-section. Then, in Table 3, the graphs of
conductrices for simple fibers are excluded by Corollary 4.18. Let $A\supset
C$ be the curve of the fiber $2F$ which intersects $C_{s}$. The curve $C$ is
the one which exists at the end of $F$ and the multiplicity in $A$ is 1. Then,
since $A\cdot C_{s}>0$, by Corollary 4.15 we have $A\cdot C_{s}=1$. Therefore,
by Corollary 4.16 (i) we have $A\cdot C=-1$. Hence, in Table 3, we have no
possibilities for the multiple fibers of type $2{\rm II}^{*}$, $2{\rm
III}^{*}$, $2{\rm I}_{m}^{*}$ ($m=2,3,4$) ∎

Table 4. Conductrices of multiple fibers for elliptic fibrations with
2-section ($A\neq 0$)

######  Example 4.22.

We consider the multiple fiber $2F$ of type ${\rm I}_{1}^{*}$ of an elliptic
fibration $f:S\longrightarrow{\bf P}^{1}$ and calculate the conductrix in this
case. We have

$F=E_{1}+E_{2}+E_{3}+E_{4}+2E_{5}+2E_{6}$

with irreducible components $E_{i}$ ($1\leq i\leq 6$). Here, $E_{i}$ ($1\leq
i\leq 6$) are $(-2)$-curves such that $E_{1}\cdot E_{5}=E_{2}\cdot E_{5}=1$,
$E_{3}\cdot E_{6}=E_{4}\cdot E_{6}=1$ and $E_{5}\cdot E_{6}=1$. We assume
there exists a special 2-section $C$, and we may assume $C$ intersects
$E_{4}$. Let $A$ be the conductrix. Then, by Proposition 4.10 we have
$F\supset A$ and by Corollary 4.17 we have the expression

$A=a_{1}E_{1}+a_{2}E_{2}+a_{3}E_{3}+a_{4}E_{4}+(1+a_{5})E_{5}+(1+a_{6})E_{6}\quad(0\leq\forall
a_{i}\leq 1).$

Since $A\not\supset C$ and $A\cdot C>0$, we have $A\cdot C=1$ by Corollary
4.14. Therefore, we have $a_{4}=1$ and $A\cdot E_{4}=-1$ by Corollary 4.16.
Therefore, we have $(1+a_{6})-2=-1$, that is, we have $a_{6}=0$. If $a_{3}=0$,
then by $A\not\supset E_{3}$ and $A\cdot E_{3}>0$, we have $A\cdot E_{3}=1$ as
above. Therefore, we have $A\cdot E_{6}=-1$. However, $A\cdot
E_{6}=(1+a_{5})+1-2=a_{5}\geq 0$, a contradiction. Therefore, we have
$a_{3}=1$. Now, we have

$A\cdot E_{6}=(1+a_{5})+1+1-2=1+a_{5}.$

By Corollary 4.15, we have $a_{5}=0$ and $A\cdot E_{6}=1$. Therefore, again by
Corollary 4.16 we have $-1=A\cdot E_{5}=a_{1}+a_{2}+1-2$. Therefore, we have
$a_{1}+a_{2}=0$. Since $a_{1}$ and $a_{2}$ are non-negative, we have
$a_{1}=a_{2}=0$, and we conclude $A=E_{3}+E_{4}+E_{5}+E_{6}$.

######  Example 4.23.

We consider the multiple fiber $2F$ of type ${\rm IV}$ of an elliptic
fibration $f:S\longrightarrow{\bf P}^{1}$ and show that the conductrix $A$ is
empty for this fiber. We have

$F=E_{1}+E_{2}+E_{3}$

with irreducible components $E_{i}$ ($1\leq i\leq 3$). Here, $E_{i}$ ($1\leq
i\leq 3$) are $(-2)$-curves such that $E_{i}\cdot E_{j}=1$ ($i\neq j$).
Suppose $A$ is non-empty. Since $A\subset F$ and $A\neq F$, we may assume
either $A=E_{1}+E_{2}$ or $A=E_{1}$ without loss of generality. Then, since we
have $A\cdot E_{3}\leq 1$ by Corollary 4.14, (1), the first case is excluded.
Therefore, we have $A\cdot E_{i}=1$ ($i=2,3$). Therefore, again by Corollary
4.14, we have $s=1$ and $r=0$ for the curves $E_{2}$ and $E_{3}$. However, by
Corollary 4.16 (ii), the intersection point of $E_{2}$ and $E_{3}$ is blown-
up, and we have $r\geq 1$ for these 2 curves, a contradiction. Hence, we have
$A=0$.

##  5\. Possible dual graphs and the number of boundary components

In this section we will determine the dual graph of effective irreducible
roots on Coble surfaces with finite automorphism group and their numbers of
boundary components.

We begin with showing some lemmas.

######  Lemma 5.1.

Assume that a Dynkin diagram of type $D_{4}$ is represented by effective
irreducible roots. Then it can be realized by only $(-2)$-curves.

###### Proof.

By Lemma 2.9 (2), a diagram of type $D_{4}$ is generated by only $(-2)$-curves
or by only $(-1)$-roots. If $D_{4}$ is generated by only $(-1)$-roots, then
the central $(-1)$-root of $D_{4}$ should contain three boundary components by
Lemma 2.9 (1) which is a contradiction. ∎

######  Lemma 5.2.

If $A\neq 0$, then a special extremal elliptic fibration with a multiple fiber
of type $\tilde{A}_{2}$ cannot exist.

###### Proof.

It follows from Proposition 2.3 and the assumption $A\neq 0$ that the
fibration has singular fibers of type $({\rm IV},{\rm IV}^{*})$ or their blow-
ups. Obviously the first case does not occur. Hence the fiber $2F$ of type
$\tilde{A}_{2}$ is obtained by blowing-up a fiber of type ${\rm IV}$, that is,
the fiber $F$ consists of $(-1)$-roots. If a special $2$-section $\alpha$ is a
$(-2)$-curve, then $F\cdot\alpha\geq 2$ which contradicts the fact that $F$ is
a multiple fiber. If $\alpha$ is a $(-1)$-root, then $\alpha$ contains a
$(-4)$-curve $B$ not contained in $F$. Since $B$ is contained in a singular
fiber, $B$ should be a component of the fiber of type ${\rm IV}^{*}$ which is
impossible. ∎

######  Lemma 5.3.

Let $f:S\to{\bf P}^{1}$ be an elliptic fibration on a Coble surface $S$ and
$F$ a fiber of $f$. Assume that $F$ is of type ${\rm I}_{n}$ or the blowing-
ups of the singular points of a fiber of type ${\rm I}_{n}$ $(n\geq 2)$. Then
$\alpha\cdot F\geq 2$ for any effective irreducible root $\alpha$ with
$\alpha\cdot F>0$.

###### Proof.

First assume that $F$ is of type ${\rm I}_{n}$. By Lemma 2.9 (2), $\alpha\cdot
F\geq 2$ if $\alpha$ is a $(-1)$-root. If $\alpha$ is a $(-2)$-curve with
$\alpha\cdot F=1$, then $\alpha$ and any component of $F$ are not contained in
the conductrix $A$. Therefore three points on the component of $F$ meeting
$\alpha$ are blown-up by Lemma 4.16 (ii) which contradicts Corollary 4.14.
Next assume that $F$ is the blowing-ups of the singular points of a fiber of
type ${\rm I}_{n}$. Then $F$ is the sum of $(-1)$-roots (Remark 2.12) and
hence $F\cdot\alpha\geq 2$ if $\alpha$ is a $(-2)$-curve (Lemma 2.9). If
$\alpha$ is a $(-1)$-root, then the $(-1)$-curve appeared in $\alpha$ meets a
$(-4)$-curve in $F$ or a $(-1)$-curve in $F$. In both cases, $F\cdot\alpha\geq
2$. ∎

######  Theorem 5.4.

Let $S$ be a Coble surface in characteristic $2$ with a finite automorphism
group. Then the dual graph of irreducible effective roots on $S$ is one of the
dual graphs given in Theorem 1.2.

###### Proof.

Let $S$ be a Coble surface with finite automorphism group. Let $A$ be the
conductrix of $S$ including the case $A=0$ and let $n$ be the number of
boundary components. It follows from Lemma 2.16 that there exists a special
genus 1 fibration on $S$ which is extremal by Proposition 2.14. Note that
every configration of effective irreducible roots forming an extended Dynkin
diagram is a fiber of a genus 1 fibration on $S$ (Lemma 2.11). These data can
allow us to control effective irreducible roots. Now starting from a special
genus 1 fibration together with the possibilities of the conductrix $A$
(Tables 2, 3 and 4), we arrive at a contradiction or $S$ contains a dual graph
of irreducible effective roots in Theorem 1.2. We observe that all dual graphs
satisfy the condition in Theorem 2.17, and hence the automorphism group of the
Coble surface with one of these dual graphs is finite by Proposition 2.13.
Proposition 2.18 implies that these are all effective irreducible roots on the
Coble surface $S$. This is a story of the proof.

We have shown the same properties of $A$ as in the case of Enriques surfaces
holds in the previous section and have prepared Lemmas 4.21, 5.3, 5.2 which
were used in the proof of the case of Enriques surfaces. Thus the proof of
[11, Theorem 4.1] works well in this case, and hence the assertion on the dual
graphs has been proved except the non-existence of the one case (of course, we
should consider $(-1)$-roots, but all arguments are easy exercise. We leave
the details to the reader). The dual graph we want to exclude is given by the
following Figure 2.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{N}$
Figure 2.

The diagram contains a Dynkin diagram of type $D_{4}$. It follows from lemma
5.1 and 2.9 (2) that all vertices are represented by $(-2)$-curves. Note that
there exists a parabolic subdiagram of type $\tilde{D}_{4}\oplus\tilde{D}_{4}$
which corresponds to a quasi-elliptic fibration with singular fibers of type
$({\rm I}_{0}^{*},{\rm I}_{0}^{*})$ by Propositions 2.2, 2.3. Since $N$ is
represented by a $(-2)$-curve by Lemma 2.9 (2), the fibration is a Halphen
surface of index 2. This contradicts the existence of two multiple fibers of
type ${\rm I}_{0}^{*}$. Thus this case does not occur. ∎

######  Theorem 5.5.

Let $S$ be a Coble surface in characteristic $2$ with a finite automorphism
group. Then the number of boundary components of $S$ is given as in Theorem
1.2.

###### Proof.

We prove the assertion for each type of dual graphs.

(1) The case of type $\tilde{E}_{8}$. The dual graph is given by the following
Figure 3.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{E_{9}}$$\textstyle{E_{8}}$$\textstyle{E_{7}}$$\textstyle{E_{6}}$$\textstyle{E_{5}}$$\textstyle{E_{3}}$$\textstyle{E_{2}}$$\textstyle{E_{1}}$$\textstyle{E_{10}}$$\textstyle{E_{4}}$
Figure 3.

It follows from Lemma 5.1 and 2.9 (2) that all vertices are represented by
$(-2)$-curves. There exists a parabolic subdiagram of type $\tilde{E}_{8}$
which corresponds to a quasi-elliptic fibration with a singular fiber of type
${\rm II}^{*}$ (Lemmas 2.11, 4.21). If the fibration is obtained from a
Jacobian fibration by blowing up the singular points of two irreducible
fibers, then the vertex $E_{10}$ is a $(-1)$-root which contradicts Lemma 2.9
(2). Thus the fibration is induced from a Halphen surface of index 2 with a
multiple fiber of type ${\rm II}^{*}$, and hence the number of boundary
component is one.

(2) The case of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$. The dual graph is
given by the following Figure 4.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{E_{8}}$$\textstyle{E_{7}}$$\textstyle{E_{6}}$$\textstyle{E_{4}}$$\textstyle{E_{3}}$$\textstyle{E_{2}}$$\textstyle{E_{1}}$$\textstyle{E_{10}}$$\textstyle{E_{9}}$$\textstyle{E_{5}}$$\textstyle{E_{11}}$
Figure 4.

It follows from lemma 5.1 and 2.9 (2) that the vertices $E_{1},\ldots,E_{10}$
are represented by $(-2)$-curves. There exists a parabolic subdiagram of type
$\tilde{E}_{7}\oplus\tilde{A}_{1}$ which corresponds to a quasi-elliptic
fibration with two singular fibers of type ${\rm III}^{*}$ and of type ${\rm
III}$ (Lemmas 2.11, 4.21). If the fibration is obtained from a Jacobian
fibration, then the vertex $E_{9}$ is represented by a $(-1)$-root which
contradicts Lemma 2.9 (2). Therefore the fibration is induced from a Halphen
surface of index 2, $E_{9}$ is the proper transform of the curve of cusps and
the boundary components are the proper transforms of the components of the
fiber of type ${\rm III}$. Thus the number of boundary components is two.

(3) The case of type $\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$. The dual graph is
given by the following Figure 5.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{E_{8}}$$\textstyle{E_{7}}$$\textstyle{E_{6}}$$\textstyle{E_{4}}$$\textstyle{E_{3}}$$\textstyle{E_{2}}$$\textstyle{E_{1}}$$\textstyle{E_{10}}$$\textstyle{E_{9}}$$\textstyle{E_{5}}$$\textstyle{E_{11}}$
Figure 5.

It follows from lemma 5.1 and 2.9 (2) that all vertices are represented by
$(-2)$-curves. There exists a parabolic subdiagram of type
$\tilde{E}_{7}\oplus\tilde{A}_{1}$ which corresponds to a quasi-elliptic
fibration with reducible fibers of type $(2{\rm III}^{*},{\rm III})$ (Lemmas
2.11, 4.21). Lemma 2.9 (2) implies that a fiber of type ${\rm II}$ is blown
up, the fiber of type ${\rm III}$ is not and the 2-section $E_{9}$ is the
proper transform of the curve of cusps. Thus the fibration is a Halphen
surface of index 2 with a multiple fiber of type ${\rm III}^{*}$ and the
number of boundary components is one.

(4) The case of type $\tilde{E}_{6}+\tilde{A}_{2}$. The dual graph is given by
the following Figure 6.

Figure 6.

It follows from lemma 5.1 and 2.9 (2) that the vertices
$E_{1},\ldots,E_{7},E_{11},E_{12},E_{13}$ are represented by $(-2)$-curves.
There exists a parabolic subdiagram of type $\tilde{E}_{6}\oplus\tilde{A}_{2}$
which corresponds to an elliptic fibration with singular fibers of type $({\rm
IV}^{*},{\rm IV})$ or of type $({\rm IV}^{*},{\rm I}_{3},{\rm I}_{1})$ by
Propositions 2.2, 2.3. If the fibration is obtained from a Jacobian fibration,
then a 2-section $E_{11}$, $E_{12}$ or $E_{13}$ is represented by a
$(-1)$-root which contradicts Lemma 2.9 (2). Therefore the fibration is
obtained from a Halphen surface of index 2. Note that the fiber of type ${\rm
IV}$ does not occur because, otherwise, the fiber of type ${\rm III}$ is blown
up and then the $(-2)$-curve $E_{11}$ meets the boundary component which is a
contradiction. Hence the number of boundary components is one or three
according to that the fiber blown up is of type ${\rm I}_{1}$ or of type ${\rm
I}_{3}$.

(5) The case of type $\tilde{D}_{8}$. The dual graph is given by the following
Figure 7.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{E_{2}}$$\textstyle{E_{3}}$$\textstyle{E_{4}}$$\textstyle{E_{5}}$$\textstyle{E_{6}}$$\textstyle{E_{7}}$$\textstyle{E_{9}}$$\textstyle{E_{10}}$$\textstyle{E_{1}}$$\textstyle{E_{8}}$
Figure 7.

It follows from lemma 5.1 and 2.9 (2) that all vertices are represented by
$(-2)$-curves. There exists a parabolic subdiagram of type $\tilde{D}_{8}$
which corresponds to a quasi-elliptic fibration with singular fibers of type
$({\rm I}_{4}^{*})$ (Lemmas 2.11, 4.21). If the fibration is induced from a
Jacobian fibration, then the vertex $E_{10}$ is a $(-1)$-root which
contradicts Lemma 2.9 (2). Thus the fibration is obtained from a Halphen
surface of index 2 with a multiple fiber of type ${\rm I}_{4}^{*}$, $E_{10}$
is the proper transform of the curve of cusps and the number of boundary
component is one.

(6) The case of type VII. The dual graph of type VII is given by the following
Figure 8.

Figure 8. Type VII

Note that the dual graph contains five vertices forming a complete graph with
double edges and the remaining fifteen vertices meet together with a single
edge. Also the dual graph contains parabolic subdiagrams of type
$\tilde{A}_{4}\oplus\tilde{A}_{4}$ corresponding to elliptic fibrations of
type $({\rm I}_{5},{\rm I}_{5},{\rm I}_{1},{\rm I}_{1})$ by Propositions 2.2,
2.3. By Lemma 2.9 (2), all of the fifteen vertices are represented by only
$(-1)$-roots or by only $(-2)$-curves. Therefore if they are represented by
only $(-1)$-roots, then the Coble surface is obtained by blowing up all the
singular points of two fibers of type ${\rm I}_{5}$. This implies that the
number of boundary components is 10. If they are represented by only
$(-2)$-curves, then the surface is obtained by blowing up one or two fibers of
type ${\rm I}_{1}$. To get the complete graph with the five vertices and
double edges, the number of the boundary components is 2. Thus we conclude
that the number of boundary components is 2 or 10.

(7) The case of type VIII. The dual graph of type VIII is given by the
following Figure 9.

Figure 9. Type VIII

It follows from lemma 5.1 and 2.9 (2) that the vertices $E_{1},\ldots,E_{10}$
are represented by $(-2)$-curves. The dual graph contains parabolic
subdiagrams of type $\tilde{D}_{5}\oplus\tilde{A}_{3}$ which correspond to
elliptic fibrations with singular fibers of type $({\rm I}_{1}^{*},{\rm
I}_{4})$ by Propositions 2.2, 2.3. This implies that the number of boundary
components is 4.

Thus we have proved that the number of boundary components of a Coble surface
with finite automorphism group is given as in Theorem 1.2. ∎

##  6\. Existence and moduli

In this section we show the existence of Coble surfaces with finite
automorphism group in Table 1 and determine their moduli. For each type, we
give two constructions of such surfaces, that is, as blowing-ups of a pencil
of sextic curves on ${\bf P}^{2}$ (Proposition 2.5) and as a quotient of a
non-singular surface by a rational derivation (§2.1).

######  Example 6.1.

A Coble surface of type $\tilde{E}_{8}$ with one boundary component.

A Coble surface of type $\tilde{E}_{8}$ with one boundary component has a
unique quasi-elliptic fibration with a multiple fiber of type ${\rm II}^{*}$.
It is obtained from a Halphen surface of index 2 with a multiple fiber of type
${\rm II}^{*}$ by blowing up the singular point of a fiber $F$ of type ${\rm
II}$. The vertex $E_{10}$ in Figure 3 is represented by the proper transform
of the curve of cusps. Contracting the $(-1)$-curve (the curve of cusps) on
the Halphen surface and then the images of
$E_{9},\ldots,E_{5},E_{3},E_{2},E_{1}$ except $E_{4}$ successively, we obtain
${\bf P}^{2}$. The image of $F$ is a sextic curve $C$ with the unique singular
point $q$ and the image of $E_{4}$ is a line $\ell$. The curves $C$ and $\ell$
meet only at the point $q$. Let $(x:y:z)$ be homogeneous coordinates of ${\bf
P}^{2}$. We may assume that $q=(0:0:1)$ and $\ell$ is defined by $x=0$. An
elementary but long calculation shows that such curve $C$ is given by

(6.1) $C:\ y^{6}+x^{5}y+x^{2}z^{4}=0.$

Note that the pencil of lines through the point $q$ gives a purely inseparable
covering $C\to{\bf P}^{1}$ of degree 4 and hence $C$ is a rational curve.
Consider the pencil of sextic curves given by

$\mathcal{P}=\\{sx^{6}+t(y^{6}+x^{5}x+x^{2}x^{4})\\}_{(s:t)\in{\bf P}^{1}}.$

Blowing up the base points of $\mathcal{P}$, we obtain a quasi-elliptic
fibration (a Halphen surface of index 2) with a unique multiple singular fiber
of type ${\rm II}^{*}$ over the point $(s:t)=(1:0)$. Let $E$ be the
$(-1)$-curve which is the exceptional curve of the last blowing-up. Then $E$
is the curve of cusps of this fibration. Blowing up the singular point of the
fiber $F$ of type ${\rm II}$ over the point $(s:t)=(0:1)$, we obtain a Coble
surface $S$ with a boundary component $B$ where $B$ is the proper transform of
$F$. The surface $S$ contains 10 $(-2)$-curves forming the desired dual graph.
Thus we obtain a Coble surface $S$ of type $\tilde{E}_{8}$. By construction,
any automorphism of $S$ is induced from one of ${\bf P}^{2}$ preserving the
pencil of sextic curves. Let $\zeta_{5}$ be a primitive 5th root of unity.
Then

$\sigma:(x:y:z)\to(x:\zeta_{5}y:\zeta_{5}^{-1}z)$

induces an automorphism of $\mathcal{P}$ of order 5. A direct calculation
shows that $\sigma$ is the only such automorphisms.

Next we give a Coble surface of type $\tilde{E}_{8}$ as a quotient of a
rational surface by a derivation. In the following we use the same notation as
in [11, §10.2]. In [11], to construct classical Enriques surfaces of type
$\tilde{E}_{8}$ (the ones with the dual graph in Figure 3), we considered the
surface $Y$, blown up a smooth quadric surface, with the canonical divisor

$K_{Y}=-(2F_{0}+2E_{1}+E_{3}+3E_{5}+4E_{6}+3E_{7}+2E_{8}+4E_{9}+5E_{10}+6E_{11}+8E_{12}+4E_{13}+6E_{14})$

where $F_{0},E_{i}$ are given in [11, Figure 18], and the following derivation
defined by

(6.2) $D={1\over x^{3}y^{2}}\left(x^{4}y^{2}\frac{\partial}{\partial
x}+(x^{2}+ax^{4}y^{4}+y^{4})\frac{\partial}{\partial y}\right),\quad a\in
k^{*},$

where $(x,y)$ is inhomogeneous coordinates of the quadric surface.

Now consider the case $a=0$. Then we have the following Lemma (compare this
with [11, Lemmas 10.6, 10.7]:

######  Lemma 6.2.

(i) The integral curves with respect to $D$ are
$E_{1},E_{3},E_{5},E_{7},E_{8},E_{9},E_{10}$.

(ii)
$(D)=-(2F_{0}+3E_{1}-E_{2}+2E_{3}+4E_{5}+4E_{6}+3E_{7}+2E_{8}+4E_{9}+5E_{10}+6E_{11}+8E_{12}+4E_{13}+6E_{14})$.

(iii) $(D)^{2}=-12$.

(iv) $K_{Y}\cdot(D)=-4.$

Let $Y^{D}$ be the quotient surface of $Y$ by $D$ and $\pi:Y\to Y^{D}$ the
canonical map. By the formula (2.3) and Lemma 6.2 (iii), (iv), $D$ is
divisorial and hence $Y^{D}$ is smooth. It follows from the canonical divisor
formula (2.2) and Lemma 6.2 that $\pi^{*}K_{Y^{D}}=E_{1}-E_{2}+E_{3}+E_{5}$.
Denote by $\bar{E}_{i}$ the image of $E_{i}$ on $Y^{D}$. Since, by
construction, $E_{1}^{2}=E_{2}^{2}=E_{3}^{2}=E_{5}^{2}=-2$ (see [11, Figure
18]) and $E_{1},E_{3},E_{5}$ are integral and $E_{2}$ is not, we have
$\bar{E}_{1}^{2}=\bar{E}_{3}^{2}=\bar{E}_{5}^{2}=-1$ and $\bar{E}_{2}^{2}=-4$
(Proposition 2.1). Again by Proposition 2.1, we have
$2K_{Y^{D}}=2\bar{E}_{1}-\bar{E}_{2}+2\bar{E}_{3}+2\bar{E}_{5}$. By
contracting $(-1)$-curves $\bar{E}_{1},\bar{E}_{3},\bar{E}_{5}$, we obtain a
non-singular surface $S$ with $|-2K_{S}|=\\{\bar{E}_{2}\\}$. The images of the
curves $F_{0},E_{6},\ldots,E_{14}$ on $S$ are $(-2)$-curves forming the dual
graph given in Figure 3. The image of $E_{4}$ is a $(-1)$-curve meeting with
$\bar{E}_{2}$ at a point with multiplicity 2. Thus we have a Coble surface
with the dual graph given in Figure 3 and have the following theorem.

######  Theorem 6.3.

The surface $S$ is a unique Coble surface with the dual graph given in Figure
3. The automorphism group ${\rm Aut}(S)$ is isomorphic to ${\bf Z}/5{\bf Z}$.
The surface $S$ is a specialization of classical Enriques surfaces of type
$\tilde{E}_{8}$.

######  Example 6.4.

A Coble surface of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ with two boundary
components.

A Coble surface $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ with two
boundary components $B_{1}+B_{2}$ has a unique quasi-elliptic fibration $f$
with a multiple fiber of type ${\rm III}^{*}$. It is obtained from a Halphen
surface of index 2 with a multiple fiber of type ${\rm III}^{*}$ and a fiber
$F$ of type ${\rm III}$ by blowing up the singular point (and infinitely near
point) of $F$ (see Figure 1 (type III)). Then $E_{9}$ is the proper transform
of the curve of cusps of $f$ and $E_{10}$ is the proper transform of the
exceptional curve of the first blowing-up. Let $E$ be the exceptional curve of
the second blowing-up. Then $B_{1},B_{2}$ are the proper transforms of the
components of $F$ and $E_{11}=2E+{1\over 2}B_{1}+{1\over 2}B_{2}$.

On the other hand, there exists a genus 1 fibration $f^{\prime}$ on $S$ with a
singular fiber of type ${\rm II}^{*}$ given by

$2E_{2}+4E_{3}+6E_{4}+3E_{5}+5E_{6}+4E_{7}+3E_{8}+2E_{9}+E_{10}.$

This fibration is quasi-elliptic because the fiber of type ${\rm II}^{*}$ does
not contain the component $E_{1}$ of the conductrix (in fact $E_{1}$ is the
curve of cusps of $f^{\prime}$). Moreover the fibration $f^{\prime}$ is
Jacobian because $E$ is a section of $f^{\prime}$. Let $F_{i}$ be the
$(-1)$-curve such that $2F_{i}+B_{i}$ is the fiber of $f^{\prime}$ containing
$B_{i}$ $(i=1,2)$. We can easily see that $F_{1}\cdot E_{1}=F_{2}\cdot
E_{1}=1,F_{1}\cdot F_{2}=F_{1}\cdot E=F_{2}\cdot E=0$, and $F_{i}$ and $B_{i}$
meet at a point with multiplicity 2.

Now we blow down curves
$F_{1},F_{2},E,E_{10},E_{9},\ldots,E_{6},E_{4},E_{3},E_{2}$ successively. Then
we obtain ${\bf P}^{2}$. Let $(x:y:z)$ be homogeneous coordinate of ${\bf
P}^{2}$. The images of $B_{1},B_{2}$ are cubics $C_{1},C_{2}$ with a cusp and
those of $E_{5},E_{1}$ are lines $\ell_{1},\ell_{2}$, respectively. Denote by
$p_{1},p_{2}$ the cusp of $C_{1},C_{2}$ respectively. The two cubics
$C_{1},C_{2}$ meet together at a point $p_{0}$ with multiplicity 9, the line
$\ell_{1}$ meets $C_{1},C_{2}$ at $p_{0}$ with multiplicity 3, and the line
$\ell_{2}$ passes through $p_{0},p_{1},p_{2}$. By changing the coordinates we
may assume that $p_{0}=(0:0:1)$, $p_{1}=(1:0:0)$, $p_{2}=(1:0:1)$, $\ell_{1}$
(resp. $\ell_{2}$) are defined by $x=0$ (resp. $y=0$). It follows from
elementary calculations that $C_{1}$ and $C_{2}$, up to projective
transformations, are given by

$C_{1}:y^{3}+xz^{2}=0;\ C_{2}:x^{3}+y^{3}+xz^{2}=0.$

Conversely, let

$\mathcal{P}=\\{s(x^{2}y)^{2}+t(y^{3}+xz^{2})(x^{3}+y^{3}+xz^{2})\\}_{(s:t)\in{\bf
P}^{1}}$

be a pencil of sextic curves. Blowing up the base points $p_{0},p_{1},p_{2}$
successively, we obtain a quasi-elliptic fibration with a multiple fiber of
type ${\rm III}^{*}$ over the point $(s:t)=(1:0)$ and a fiber
$\tilde{C}_{1}+\tilde{C}_{2}$ of type ${\rm III}$ over the point $(s:t)=(0:1)$
where $\tilde{C}_{i}$ is the proper transform of $C_{i}$. Thus we get a Coble
surface $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ with two boundary
components. Note that the projective transformation

$(x:y:z)\to(x:y:x+z)$

preserves each line $\ell_{1},\ell_{2}$ and interchanging $C_{1}$ and $C_{2}$.
A direct calculation shows that ${\rm Aut}(S)$ is generated by this
involution.

Next we give a Coble surface of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ as a
quotient of a rational surface by a derivation. We use the same notation as in
[11, §11.2]. To construct classical Enriques surfaces with the dual graph in
Figure 4, we considered a conic bundle $f:R\to{\bf P}^{1}$ defined by

$S(aX_{0}^{2}+bX_{2}^{2})+T(X_{1}^{2}+aX_{1}X_{2}+bX_{0}X_{2})=0,\
(S:T)\in{\bf P}^{1},\ a,b\in k^{*}$

in the fiber space ${\bf P}^{2}\times{\bf P}^{1}\to{\bf P}^{1}$, and the
following derivation defined by

(6.3) $D={1\over s}\left(a(s^{2}+c)\frac{\partial}{\partial
x}+(as^{2}x^{2}+bc)\frac{\partial}{\partial y}\right),\quad b\neq a^{2}c\in
k^{*},$

where $(x,y,s)=(X_{0}/X_{2},X_{1}/X_{2},S/T)$ and $c$ is a root of the
equation $t^{2}+(b/a)t+1=0$.

The surface $R$ has two rational double points and the derivation has isolated
singularities. By blowing up $R$ successively, we obtain a non-singular
surface $Y$ with

$K_{Y}=-(F_{0}+2E_{3}+E_{6}+E_{7}+2E_{8}+2E_{9}+2E_{10}+3E_{11}+4E_{12}+4E_{13}+2E_{14})$

and a divisorial derivation. In case $b=0,c=1$, we obtained Enriques surfaces
of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ birationally isomorphic to the
quoteint of $Y$ by the derivation.

Now consider the case $b=c=0$, that is,

$D=s\left(\frac{\partial}{\partial x}+x^{2}\frac{\partial}{\partial
y}\right).$

Then we have the following Lemma.

######  Lemma 6.5.

(i) The integral curves with respect to $D$ are
$F_{0},E_{4},E_{6},E_{7},E_{8},E_{9},E_{11}$ in [11, Figure 24].

(ii)
$(D)=-(F_{0}-E_{1}-E_{2}+2E_{3}++2E_{6}+2E_{7}+2E_{8}+2E_{9}+2E_{10}+3E_{11}+4E_{12}+4E_{13}+2E_{14})$.

(iii) $(D)^{2}=-12$.

(iv) $K_{Y}\cdot(D)=-4.$

Let $Y^{D}$ be the quotient of $Y$ by $D$ and $\pi:Y\to Y^{D}$ the canonical
map. By the formula (2.3) and Lemma 6.5, $Y^{D}$ is smooth. It follows from
the canonical divisor formula (2.2) and Lemma 6.5 that
$\pi^{*}K_{Y^{D}}=-E_{1}-E_{2}+E_{6}+E_{7}$. Denote by $\bar{E}_{i}$ the image
of $E_{i}$ on $Y^{D}$. Since, by construction,
$E_{1}^{2}=E_{2}^{2}=E_{6}^{2}=E_{7}^{2}=-2$ (see [11, Figure 24]) and
$E_{6},E_{7}$ are integral and $E_{1},E_{2}$ are not,
$\bar{E}_{6}^{2}=\bar{E}_{7}^{2}=-1$ and $\bar{E}_{1}^{2}=\bar{E}_{2}^{2}=-4$
(Proposition 2.1). Again by Proposition 2.1, we have
$2K_{Y^{D}}=-\bar{E}_{1}-\bar{E}_{2}+2\bar{E}_{6}+2\bar{E}_{7}$. By
contracting $(-1)$-curves $\bar{E}_{6},\bar{E}_{7}$, we obtain a non-singular
surface $S$ with $|-2K_{S}|=\\{\bar{E}_{1}+\bar{E}_{2}\\}$. The images of the
curves $E_{3},E_{8},\ldots,E_{14}$ on $S$ are $(-2)$-curves forming the fiber
of type ${\rm III}^{*}$. The images of $F_{0},E_{5}$ are $(-2)$-curves and
that of $E_{4}$ is a $(-1)$-curve meeting with the boundary components
$\bar{E}_{1},\bar{E}_{2}$. Thus we have a Coble surface of type
$\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$. Thus we have proved the following
theorem.

######  Theorem 6.6.

A Coble surface $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$ with two
boundary components is obtained from the pencil $\mathcal{P}$ as above, and is
unique up to isomorphisms. The automorphism group ${\rm Aut}(S)$ is isomorphic
to ${\bf Z}/2{\bf Z}$. The surface $S$ is a specialization of classical
Enriques surfaces of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$.

######  Example 6.7.

A Coble surface of type $\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$ with one boundary
component.

A Coble surface $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$ with one
boundary component $B$ has a unique quasi-elliptic fibration $f$ with a
multiple fiber of type ${\rm III}^{*}$. It is obtained from a Halphen surface
of index 2 with singular fibers of type $(2{\rm III}^{*},{\rm III})$ by
blowing up the singular point of a fiber of type ${\rm II}$. The proper
transform of the fiber is the boundary component $B$.

Note that instead of blowing up the singular point of a fiber of type II of
the Halphen surface, by blowing up the singular point of the fiber of type III
we obtain a Coble surface $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(2)}$.
Thus we can use the construction of the previous Example. Note that the last
blowing-up depends on the choice of a singular fiber of type II of the Halphen
surface.

Next we give a Coble surface of type $\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$ as a
quotient of a rational surface by a derivation. We use the same notation as in
[11, §11.1]. We assume that $c=0$ and $b\neq 0$ in the equation (6.3). Note
that the conic bundle $R$ has a parameter $b$ which is the difference between
this case and the previous case. The canonical divisor of $Y$ is given by

$K_{Y}=-(F_{0}+2E_{2}+E_{7}+E_{8}+2E_{9}+2E_{10}+2E_{11}+3E_{12}+4E_{13}+4E_{14}+2E_{15}).$

Then we have the following Lemma.

######  Lemma 6.8.

(i) The integral curves with respect to $D$ are
$F_{0},E_{5},E_{7},E_{8},E_{9},E_{10},E_{12}$ in [11, Figure 21].

(ii)
$(D)=-(F_{0}-E_{1}+2E_{2}+E_{5}+2E_{7}+2E_{8}+2E_{9}+2E_{10}+2E_{11}+3E_{12}+4E_{13}+4E_{14}+2E_{15})$.

(iii) $(D)^{2}=-12$.

(iv) $K_{Y}\cdot(D)=-4.$

Let $Y^{D}$ be the quotient of $Y$ by $D$ and $\pi:Y\to Y^{D}$ the canonical
map. By the formula (2.3) and Lemma 6.5, $Y^{D}$ is smooth. It follows from
the canonical divisor formula (2.2) and Lemma 6.8 that
$\pi^{*}K_{Y^{D}}=-E_{1}+E_{5}+E_{7}+E_{8}$. Denote by $\bar{E}_{i}$ the image
of $E_{i}$ on $Y^{D}$. Since, by construction,
$E_{1}^{2}=E_{5}^{2}=E_{7}^{2}=E_{8}^{2}=-2$ (see [11, Figure 21]) and
$E_{5},E_{7},E_{8}$ are integral and $E_{1}$ is not,
$\bar{E}_{5}^{2}=\bar{E}_{7}^{2}=\bar{E}_{8}^{2}=-1$ and $\bar{E}_{1}^{2}=-4$
(Proposition 2.1). Again by Proposition 2.1, we have
$2K_{Y^{D}}=-\bar{E}_{1}+2\bar{E}_{5}+2\bar{E}_{7}+2\bar{E}_{8}$. By
contracting $(-1)$-curves $\bar{E}_{5},\bar{E}_{7},\bar{E}_{8}$, we obtain a
non-singular surface $S$ with $|-2K_{S}|=\\{\bar{E}_{1}\\}$. The images of the
curves $E_{2},E_{9},\ldots,E_{15}$ on $S$ are $(-2)$-curves forming the fiber
of type ${\rm III}^{*}$. The images of $E_{3},E_{4}$ are $(-2)$-curves forming
the fiber of type III. The image of curve $F_{0}$ is a $(-2)$-curve and that
of $E_{6}$ is a $(-1)$-curve meeting the boundary components $\bar{E}_{1}$
with multiplicity 2. Thus we have a Coble surface with the dual graph of type
$\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$. Thus we have proved the following
theorem.

######  Theorem 6.9.

Coble surfaces $S$ of type $\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$ with one
boundary component form a $1$-dimensional irreducible family. The automorphism
group ${\rm Aut}(S)$ is isomorphic to ${\bf Z}/2{\bf Z}$. The surfaces $S$ are
specializations of classical Enriques surfaces of type
$\tilde{E}_{7}+\tilde{A}_{1}^{(1)}$.

######  Example 6.10.

A Coble surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ with three boundary
components.

A Coble surface $S$ of type $\tilde{E}_{6}+\tilde{A}_{2}$ with three boundary
components $B_{1}+B_{2}+B_{3}$ has a unique elliptic fibration $f$ with a
multiple fiber of type ${\rm IV}^{*}$. It is obtained from a Halphen surface
of index 2 with singular fibers of type $(2{\rm IV}^{*},{\rm I}_{3},{\rm
I}_{1})$ by blowing up the singular points of the fiber of type ${\rm I}_{3}$.
The proper transforms of the components of the fiber of type ${\rm I}_{3}$ are
the boundary components $B_{1},B_{2},B_{3}$. Let $F_{1},F_{2},F_{3}$ be
$(-1)$-curves such that $2F_{1}+{1\over 2}B_{2}+{1\over 2}B_{3}$,
$2F_{2}+{1\over 2}B_{1}+{1\over 2}B_{3}$ or $2F_{3}+{1\over 2}B_{1}+{1\over
2}B_{2}$ gives $E_{8},E_{9}$ or $E_{10}$ in Figure 6, respectively.

On the other hand, there exist three genus 1 fibrations $f_{i}$ $(i=1,2,3)$
with singular fibers of type $(\tilde{E}_{7},2\tilde{A}_{1})$. The linear
system $|2(E_{8}+E_{11})|$ (resp. $|2(E_{9}+E_{12})|$, $|2(E_{10}+E_{13})|$)
gives $f_{1}$ (resp. $f_{2}$, $f_{3}$). By Propositions 2.2, 2.3, they are
quasi-elliptic. Each of them is obtained from a Jacobian fibration with
reducible singular fibers of type $({\rm III}^{*},{\rm III})$ by blowing up
the singular point (and the infinitely near point) of the fiber of type ${\rm
III}$ and the singular point of a fiber of type ${\rm II}$. Thus there exist
three $(-1)$-curves $F_{1}^{\prime},F_{2}^{\prime},F_{3}^{\prime}$ such that
$F_{i}^{\prime}$ meets $B_{i}$ at a point with multiplicity 2 and
$F_{i}^{\prime}\cdot E_{i}=1$ $(i=1,2,3)$. The Mordell-Weil group of $f_{i}$
acts on $S$ as automorphisms. For example, $f_{1}$ has two sections
$F_{2},F_{3}$ which define an involution switching $F_{2}^{\prime}$ and
$F_{3}^{\prime}$ and fixing $F_{1}^{\prime}$ pointwisely. This implies that
all $F_{1}^{\prime},F_{2}^{\prime},F_{3}^{\prime}$ meet at one point. Moreover
they generate a symmetric group $\mathfrak{S}_{3}$.

Now by contracting $F_{1},F_{2},F_{3}$ and then
$E_{11},E_{12},E_{13},E_{1},E_{2},E_{3},E_{5},E_{6},E_{7}$ successively, we
obtain ${\bf P}^{2}$. The images of $B_{1},B_{2},B_{3}$ are smooth conics
$Q_{1},Q_{2},Q_{3}$ and those of
$E_{4},F_{1}^{\prime},F_{2}^{\prime},F_{3}^{\prime}$ are lines denoted by
$\ell,\ell_{1},\ell_{2},\ell_{3}$ respectively. Two conics $Q_{i}$ and $Q_{j}$
meet at a point $p_{ij}$ with multiplicity 4 and $\ell$ passes through three
points $p_{12},p_{13},p_{23}$. Three lines $\ell_{1},\ell_{2},\ell_{3}$ meet
at a point $p_{0}$. The line $\ell_{i}$ tangents to $Q_{i}$ at a point $p_{i}$
and tangents to both $Q_{j}$, $Q_{k}$ at the point $p_{jk}$
$(\\{i,j,k\\}=\\{1,2,3\\})$. By changing coordinates, we may assume that

$p_{1}=(1:0:0),\ p_{2}=(0:1:0),\ p_{3}=(0:0:1),\ p_{0}=(1:1:1).$

Then we have

$\ell_{1}:y+z=0,\ \ell_{2}:x+z=0,\ \ell_{3}:x+y=0.$

Since the action of $\mathfrak{S}_{3}$ descends to the one acting on ${\bf
P}^{2}$ as permutations of coordinates and $\ell$ is invariant under this
action, we have

$\ell:x+y+z=0.$

Therefore we have

$p_{12}=(1:1:0),\ p_{13}=(1:0:1),\ p_{23}=(0:1:1).$

Then one can easily see that the conics are given by

$Q_{1}:y^{2}+z^{2}+xy+yz+zx=0,$ $Q_{2}:x^{2}+z^{2}+xy+yz+zx=0,$
$Q_{3}:x^{2}+y^{2}+xy+yz+zx=0.$

Conversely consider the pencil $\mathcal{P}$ of sextic curves

(6.4)
$\mathcal{P}=\\{C_{(s:t)}\\}=\\{s\ell^{6}+tQ_{1}Q_{2}Q_{3}\\}_{(s:t)\in{\bf
P}^{1}}.$

Blowing up the base points of $\mathcal{P}$, we have a Halphen surface with
singular fibers of type $(2{\rm IV}^{*},{\rm I}_{3},{\rm I}_{1})$, and hence
we obtain a Coble surface $X$ of type $\tilde{E}_{6}+\tilde{A}_{2}$ with three
boundary components. The three fibrations $f_{i}$ are Jacobian and their
Mordell-Weil groups generate $\mathfrak{S}_{3}$ a subgroup of ${\rm Aut}(S)$
which is also induced from permutations of the coordinates $(x:y:z)$. One can
see that there are no more projective transformations preserving the lines and
conics.

Next we give a Coble surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ as a
quotient of a rational surface by a derivation. We use the same notation as in
[11, §7.2]. To construct classical Enriques surfaces with the dual graph in
Figure 6, we considered a rational elliptic surface defined by the Weierstrass
equation

$y^{2}+xy+t^{2}y=x^{3}$

which has a singular of type ${\rm I}_{6}$ over the point $t=0$, a singular
fiber of type ${\rm I}_{2}$ over the point $t=1$ and a singular fiber of type
${\rm IV}$ over the point $t=\infty$, and the derivation $D$ defined by

(6.5) $D=(t+a)\frac{\partial}{\partial t}+(x+t^{2})\frac{\partial}{\partial
x},\quad a\in k\setminus\\{0,1\\}.$

Now consider the case $a=0$. The derivation has an isolated singularity at the
singular point of the fiber of type IV. We blow up the surface successively
and then obtain a non-singular surface $Y$ with

$K_{Y}=-(F_{\infty}+E_{\infty,1}+E_{\infty,2})-2(E_{\infty,3}+E_{\infty,4}+E_{\infty,5}+E_{\infty,6})$

where the notations are as in [11, Figure 7]. Then we have the following
Lemma.

######  Lemma 6.11.

(i) The integral curves with respect to $D$ are
$E_{1},F_{0},E_{0,3},E_{0,4},F_{\infty},E_{\infty,1}$, $E_{\infty,2}$,
$E_{\infty,3}$.

(ii)
$(D)=E_{0,1}+E_{0,2}+E_{0,5}-(E_{1}+F_{\infty}+E_{\infty,1}+E_{\infty,2})-2(E_{\infty,3}+E_{\infty,4}+E_{\infty,5}+E_{\infty,6})$.

(iii) $(D)^{2}=-12$.

(iv) $K_{Y}\cdot(D)=-4.$

Let $Y^{D}$ be the quotient of $Y$ by $D$ and $\pi:Y\to Y^{D}$ the canonical
map. By the formula (2.3) and Lemma 6.5, $Y^{D}$ is smooth. It follows from
the canonical divisor formula (2.2) and Lemma 6.11 that
$\pi^{*}K_{Y^{D}}=-E_{0,1}-E_{0,2}-E_{0,5}+E_{1}$. Denote by
$\bar{E}_{0,1},\bar{E}_{0,2},\bar{E}_{0,5},\bar{E}_{1}$ the images of
$E_{0,1},E_{0,2},E_{0,5},E_{1}$ on $Y^{D}$, respectively. Since
$E_{0,1},E_{0,2},E_{0,5},E_{1}$ are $(-2)$-curves (see [11, Figure 7]),
$E_{1}$ is integral and $E_{0,1},E_{0,2},E_{0,5}$ are not (Lemma 6.11),
$\bar{E}_{0,1},\bar{E}_{0,2},\bar{E}_{0,5}$ are $(-4)$-curves and
$\bar{E}_{1}$ is a $(-1)$-curve (Proposition 2.1). Again by Proposition 2.1,
we have $2K_{Y^{D}}=-\bar{E}_{0,1}-\bar{E}_{0,2}-\bar{E}_{0,5}+2\bar{E}_{1}$.
By contracting $\bar{E}_{1}$, we obtain a non-singular surface $S$ with
$|-2K_{S}|=\\{\bar{E}_{0,1}+\bar{E}_{0,2}+\bar{E}_{0,5}\\}$. The images of the
curves $s_{1},s_{2},s_{3},F_{\infty},E_{\infty,i}\ (i=1,\ldots,6)$ on $S$ are
$(-2)$-curves. The images of $F_{1}$ is the fiber of type ${\rm I}_{1}$. Thus
we have a Coble surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ and with 3
boundary components.

######  Theorem 6.12.

The Coble surface $S$ of type $\tilde{E}_{6}+\tilde{A}_{2}$ with three
boundary components is induced from the pencil $\mathcal{P}$ of sextic curves
and is unique up to isomorphisms. The automorphism group ${\rm Aut}(S)$ is
isomorphic to $\mathfrak{S}_{3}$. The surface $S$ is a specialization of
classical Enriques surfaces of type $\tilde{E}_{6}+\tilde{A}_{2}$.

######  Example 6.13.

A Coble surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ with one boundary
component.

We consider the same situation as in Example 6.10. Blowing up the base points
of the pencil $\mathcal{P}$ given by the equation (6.4), we have obtained a
Halphen surface with singular fibers of type $(2{\rm IV}^{*},{\rm I}_{3},{\rm
I}_{1})$. Then by blowing up the singular point of the fiber of type ${\rm
I}_{1}$, we obtain a Coble surface $S$ of type $\tilde{E}_{6}+\tilde{A}_{2}$
with one boundary component. The proper transform of $C_{(1:1)}$ in (6.4) is
the boundary component of $X$.

Conversely one can easily see that any Coble surface of type
$\tilde{E}_{6}+\tilde{A}_{2}$ with one boundary component is obtained from the
pencil $\mathcal{P}$ and hence it is unique.

Next we give a Coble surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ with one
boundary component as a quotient of a rational surface by a derivation. We use
the same notation as in the previous Example 6.10 and consider the case $a=1$.
Then we have the following Lemma.

######  Lemma 6.14.

(i) The integral curves with respect to $D$ are
$F_{1},E_{0,1},E_{0,2},E_{0,5},F_{\infty},E_{\infty,1}$, $E_{\infty,2}$,
$E_{\infty,3}$.

(ii)
$(D)=E_{1}-(E_{0,1}+E_{0,2}+E_{0,5}+F_{\infty}+E_{\infty,1}+E_{\infty,2})-2(E_{\infty,3}+E_{\infty,4}+E_{\infty,5}+E_{\infty,6})$.

(iii) $(D)^{2}=-12$.

(iv) $K_{Y}\cdot(D)=-4.$

Let $Y^{D}$ be the quotient of $Y$ by $D$ and $\pi:Y\to Y^{D}$ the canonical
map. By the formula (2.3) and Lemma 6.5, $Y^{D}$ is smooth. It follows from
the canonical divisor formula (2.2) and Lemma 6.11 that
$\pi^{*}K_{Y^{D}}=E_{0,1}+E_{0,2}+E_{0,5}-E_{1}$. Denote by
$\bar{E}_{0,1},\bar{E}_{0,2},\bar{E}_{0,5},\bar{E}_{1}$ the images of
$E_{0,1},E_{0,2},E_{0,5},E_{1}$ on $Y^{D}$, respectively. Since
$E_{0,1},E_{0,2},E_{0,5},E_{1}$ are $(-2)$-curves (see [11, Figure 7]),
$E_{0,1},E_{0,2},E_{0,5}$ are integral and $E_{1}$ is not (Lemma 6.11),
$\bar{E}_{0,1},\bar{E}_{0,2},\bar{E}_{0,5}$ are $(-1)$-curves and
$\bar{E}_{1}$ is a $(-4)$-curve (Proposition 2.1). Again by Proposition 2.1,
we have $2K_{Y^{D}}=2\bar{E}_{0,1}+2\bar{E}_{0,2}+2\bar{E}_{0,5}-\bar{E}_{1}$.
By contracting $\bar{E}_{0,1},\bar{E}_{0,2},\bar{E}_{0,5}$, we obtain a non-
singular surface $S$ with $|-2K_{S}|=\\{\bar{E}_{1}\\}$. Thus we have a Coble
surface of type $\tilde{E}_{6}+\tilde{A}_{2}$ and with one boundary component.

######  Theorem 6.15.

The Coble surface $S$ of type $\tilde{E}_{6}+\tilde{A}_{2}$ with one boundary
component is induced from the pencil $\mathcal{P}$ of sextic curves given in
$(\ref{E6pencil})$, and is unique up to isomorphisms. The automorphism group
${\rm Aut}(S)$ is isomorphic to $\mathfrak{S}_{3}$. The surface $S$ is a
specialization of classical Enriques surfaces of type
$\tilde{E}_{6}+\tilde{A}_{2}$.

######  Example 6.16.