# Elliptic ruled surfaces over arbitrary characteristic fields

Takato Togashi and Hokuto Uehara

###### Abstract

Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically
closed field of any characteristic. On the other hand, a rank $2$ vector
bundle on $E$ defines a surface $S$ with a $\mathbb{P}^{1}$-bundle structure
on $E$. We study when $S$ has an elliptic fibration according to the Atiyah’s
classification, and what kinds of singular fibers appear.

##  1 Introduction

Kodaira initiated a study of elliptic surfaces over $\mathbb{C}$ about 60
years ago, since then, we have a satisfactory theory of this subject. Bombieri
and Mumford studied in [BM76] elliptic surfaces over an algebraically closed
field $k$ of arbitrary characteristic $p$, and they introduced the notion of
wild fibers in the case $p>0$, which turns out a main difficulty to develop an
analogous theory of elliptic surfaces in the case $p>0$ to that in the case
$p=0$. Ueno and Katsura studied in [KU85] what extent the theory in the case
$p=0$ can be extended or have a nice analogy in the case $p>0$. In this
article, we study elliptic fibrations on elliptic ruled surfaces in arbitrary
$p$ with the aid of [KU85] and Atiyah’s study of vector bundles on elliptic
curves.

Let $S$ be a smooth projective surface defined over $k$. Suppose that $S$
admits a relatively minimal elliptic fibration $\pi\colon S\rightarrow C$, and

$\pi^{*}(p_{i})=m_{i}D_{i}\mbox{ ($m_{i}$ is the multiplicity, $p_{i}\in
C,i=1,2,\dots,\lambda$)}$

are the multiple fibers of $\pi$. We know that
$\mathbb{R}^{1}\pi_{*}\mathcal{O}_{S}\simeq\mathcal{L}_{\pi}\oplus\mathcal{T}_{\pi}$,
where $\mathcal{L}_{\pi}$ is an invertible sheaf and $\mathcal{T}_{\pi}$ a
torsion sheaf, which is known to be $0$ in the case $p=0$. The multiple fiber
$\pi^{*}(p_{i})$ is said to be _wild_ for a point
$p_{i}\in\operatorname{Supp}\mathcal{T}_{\pi}$, and _tame_ for a point
$p_{i}\not\in\operatorname{Supp}\mathcal{T}_{\pi}$. The canonical bundles on
$S$ and $C$ are related by _the canonical bundle formula_ , which states that
there is an isomorphism

$\omega_{S}\simeq\pi^{*}(\omega_{C}\otimes\mathcal{L}_{\pi}^{-1})\otimes\mathcal{O}_{S}(\sum_{i=1}^{\lambda}a_{i}D_{i})$

for some integer $a_{i}$ with $0\leq a_{i}\leq m_{i}-1$. If $a_{i}\neq
m_{i}-1$, then $\pi^{*}p_{i}$ is known to be a wild fiber.

If $\kappa(S)=-\infty$, $S$ is either a rational surface obtained by
$9$-points blow up of $\mathbb{P}^{2}$ or an _elliptic ruled surface_ , that
is, a surface with a $\mathbb{P}^{1}$-bundle over an elliptic curve. Harbourne
and Lang classified multiple fibers on rational elliptic surfaces in [HL88].
It turns out that rational elliptic surfaces have reducible fibers, but no
wild fibers. On the other hand, we can readily see that an elliptic fibration
on an elliptic ruled surface has no reducible fibers, and moreover all
multiple fibers are of type ${}_{m}\mathrm{I}_{0}$, namely the reductions of
all multiple fibers are smooth elliptic curves (Lemma 2.7). One of the aim of
this article is to determine when a given elliptic ruled surface has an
elliptic fibration $\pi$, and to determine how many wild and tame fibers
appear in $\pi$ (Theorem 1.1).

Let $\mathcal{E}$ be a normalized rank $2$ vector bundle on an elliptic curve
$E$ and

$f\colon S=\mathbb{P}(\mathcal{E})\rightarrow E$

be the $\mathbb{P}^{1}$-bundle on $E$, determined by $\mathcal{E}$. Let us set
$e=-\deg\mathcal{E}$. Then we see that $e=0$ or $-1$ if $S$ has an elliptic
fibration. If $e=0$ and $\mathcal{E}$ is indecomposable, it is easy to see
that the isomorphism class of such vector bundle is unique. If $e=0$ and
$\mathcal{E}$ is decomposable, we see that
$\mathcal{E}\cong\mathcal{O}_{E}\oplus\mathcal{L}$ for some
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$. On the other hand, if
$e=-1$, $\mathcal{E}$ is always indecomposable. We also note that there are
many isomorphism classes of such vector bundles $\mathcal{E}$ on $E$, but all
vector bundles give the unique isomorphism class of $\mathbb{P}(\mathcal{E})$
(see [Har77, Theorem V.2.15]).

Theorem 1.1 below is the first main result in this article. In the tables
contained therein, the symbol ∗ stands for a wild fiber. Moreover, as
mentioned above, if a multiple fiber $mD$ is tame, then $a=m-1$, hence we omit
the value of $a$ in the list. For example, $(2,0/2^{*})$ in the case (ii-3)
stands for one tame fiber of type ${}_{2}\mathrm{I}_{0}$ with $a_{1}=1$ and
one wild fiber of type ${}_{2}\mathrm{I}_{0}$ with $a_{2}=0$.

######  Theorem 1.1.

Let $\mathcal{E}$ be a normalized rank $2$ vector bundle on an elliptic curve
$E$, and $S=\mathbb{P}(\mathcal{E})$ the associated $\mathbb{P}^{1}$-bundle
over $E$ with $\mathcal{E}$.

  1. (i)

For $e=0$, we have the following:

| $\mathcal{E}$ |  $(a_{i}/m_{i})$ if $S$ has an elliptic fibration | $p$
---|---|---|---
(i-1) | $\mathcal{O}_{E}\oplus\mathcal{O}_{E}$ | no multiple fibers | $p\geq 0$
(i-2) |  $\mathcal{O}_{E}\oplus\mathcal{L}$, $\operatorname{ord}\mathcal{L}=m>1$ | $(m,m)$ | $p\geq 0$
(i-3) |  $\mathcal{O}_{E}\oplus\mathcal{L}$, $\operatorname{ord}\mathcal{L}=\infty$ | no elliptic fibrations | $p\geq 0$
(i-4) | indecomposable | no elliptic fibrations | $p=0$
(i-5) | indecomposable | $(p-2/{p}^{*})$ | $p>0$

Here $\mathcal{L}$ is an element of $\mathop{\mathrm{Pic}}\nolimits^{0}E$.

  2. (ii)

Suppose that $e=-1$. Then $S$ has an elliptic fibration. The list of singular
fibers are the following:

| $(a_{i}/m_{i})$ | E | $p$
---|---|---|---
(ii-1) | $(2,2,2)$ |  | $p\neq 2$
(ii-2) | $(1/2^{*})$ | supersingular | $p=2$
(ii-3) | $(2,0/2^{*})$ | ordinary | $p=2$

Maruyama also considered a condition when elliptic ruled surfaces have an
elliptic fibration [Mar71, Theorem 4], by terms of elementary transformations
of ruled surfaces (see Remark 2.16). Suwa also considered a similar condition
in [Suw69, Theorem 5] in the case $p=0$. In the case of $p\neq 2$, the result
in Theorem 1.1 was obtained in the first author’s master thesis [Tog].

We also notice that the elliptic fibration in the case (ii-2) have a _wild
fiber of strange type_ (see Remark 5.1).

In the proof of Theorem 1.1, we can construct a _resolution of singular
fibers_ on elliptic ruled surfaces. More precisely we have following:

######  Theorem 1.2.

Let $f\colon S\to E$ be a $\mathbb{P}^{1}$-bundle over an elliptic curve $E$
such that $S$ also has an elliptic fibration $\pi\colon S\to\mathbb{P}^{1}$.
Then there is a finite surjective morphism $\varphi\colon F\to E$ from an
elliptic curve $F$, fitting into the following diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{F\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}.}$

Here, $f^{\prime}$ and $\pi^{\prime}$ are natural projections, the left square
is the fiber product diagram and the right square is obtained by the Stein
factorization of $q\circ\pi$.

We observe in Theorem 1.2 that by taking a suitable finite cover of $S$, we
obtain an elliptic fibration $\pi^{\prime}$ having milder singular fibers (in
Theorem 1.2, $\pi^{\prime}$ actually has no singular fibers). The existence of
such resolutions was already observed under some suitable situations. See, for
example, [KU85, §6, §7] and [Kaw00, Theorem A]. It is notable that the notion
of a _wild fiber of strange type_ was introduced in [KU85] as an obstruction
of the existence of such resolutions in their constructions. However, we can
actually find a resolution even in the case (ii-2), $\pi$ has a wild fiber of
strange type.

The construction of this article is as follows. In §2 we refer and prove
several results on elliptic surfaces and vector bundles on elliptic curves. In
§3, we narrow down the candidates of singular fibers of elliptic fibrations on
elliptic ruled surfaces. We prove Theorem 1.1 in §4 and Theorem 1.2 in §5.

In [Ueh17], we apply the result in [Tog], which is the same as Theorem 1.1 in
the case $p\neq 2$, to study the set $\operatorname{FM}(S)$ of Fourier–Mukai
partners of elliptic ruled surfaces $S$ in the case $p=0$. In the forthcoming
paper [UW], we apply Theorem 1.1 to study $\operatorname{FM}(S)$ for elliptic
ruled surfaces $S$ for arbitrary $p$.

#### Notation and conventions

All varieties $X$ are defined over an algebraically closed field $k$ of
characteristic $p\geq 0$. A point $Q\in X$ always means a closed point. If we
denote

$D_{1}\sim D_{2}\quad(\mbox{resp. }D_{1}\equiv D_{2})$

for divisors $D_{1}$ and $D_{2}$ on a normal variety $X$, we mean that $D_{1}$
and $D_{2}$ are linearly equivalent (resp. numerically equivalent). We denote
the dimension of the cohomology $H^{i}(X,\mathcal{F})$ of a sheaf
$\mathcal{F}$ on $X$ by $h^{i}(X,\mathcal{F})$.

In the case $p>0$, we denote a relative Frobenius morphism by

$\operatorname{Fr}\colon X_{p}\to X.$

By an _elliptic surface_ , we will always mean a smooth projective surface $S$
together with a smooth projective curve $C$ and a relatively minimal
projective morphism $\pi\colon S\to C$ whose general fiber is an elliptic
curve.

Take a point $Q$ on an elliptic curve $E$. Since we have

$\mathop{\mathrm{Ext}}\nolimits^{1}_{E}(\mathcal{O}_{E},\mathcal{O}_{E})\cong\mathop{\mathrm{Ext}}\nolimits^{1}_{E}(\mathcal{O}_{E}(Q),\mathcal{O}_{E})\cong
k,$

there is the unique isomorphism class of rank $2$ vector bundle
$\mathcal{E}_{2,0}$ (resp. $\mathcal{E}_{Q}$) fitting into the non-split exact
sequence:

$0\to\mathcal{O}_{E}\to\mathcal{E}_{2,0}\to\mathcal{O}_{E}\to
0\quad(\mbox{resp.
}0\to\mathcal{O}_{E}\to\mathcal{E}_{Q}\to\mathcal{O}_{E}(Q)\to 0).$

#### Acknowledgments.

We would like to thank Hiroyuki Ito, Kentaro Mitsui, Shigeru Mukai, Toshiyuki
Katsura, Noboru Nakayama for invaluable suggestions. H.U. is supported by the
Grants-in-Aid for Scientific Research (No. 18K03249).

##  2 Preliminaries

###  2.1 Elliptic surfaces

In this subsection, we refer several useful results on elliptic surfaces in
[BM76] and [KU85]. Let $\pi:S\rightarrow C$ be an elliptic surface. Suppose
that

$\bigl{\\{}\pi^{*}(Q_{i})=m_{i}D_{i}\bigm{|}i=1,2,\dots,\lambda\bigr{\\}}$

is the set of multiple fibers for $Q_{i}\in C$. Then, we have a decomposition
$\mathbb{R}^{1}\pi_{*}\mathcal{O}_{S}\simeq\mathcal{L}_{\pi}\oplus\mathcal{T}_{\pi}$
where $\mathcal{L}_{\pi}$ is an invertible sheaf and $\mathcal{T}_{\pi}$ is a
torsion sheaf. It is known that $\mathcal{T}_{\pi}=0$ when $p=0$. If
$Q_{i}\notin\operatorname{Supp}\mathcal{T}_{\pi}$, $\pi^{*}(Q_{i})$ is said to
be a _tame fiber_ , and a _wild fiber_ otherwise. Note that for a point
$Q\in\operatorname{Supp}\mathcal{T}_{\pi}$, $\pi^{*}(Q)$ is a multiple fiber,
since $h^{0}(\pi^{*}(Q),\mathcal{O}_{\pi^{*}(Q)})\geq 2$.

Let us define $d:=\deg\mathcal{L}_{\pi}$ and $g:=g(C)$.

######  Proposition 2.1 (Theorem 2 in [BM76]).

Let $\pi\colon S\longrightarrow C$ be a smooth projective elliptic surface.
For the decomposition
$\mathbb{R}^{1}\pi_{*}\mathcal{O}_{S}\simeq\mathcal{L}_{\pi}\oplus\mathcal{T}_{\pi},$
we have

$\omega_{S}\simeq\pi^{*}(\omega_{C}\otimes\mathcal{L}_{\pi}^{-1})\otimes\mathcal{O}_{S}(\sum_{i=1}^{\lambda}a_{i}D_{i}).$

Moreover we have the following:

  1. (i)

$0\leq a_{i}\leq m_{i}-1$ for all $i=1,2,\dots,\lambda$.

  2. (ii)

If $m_{i}D_{i}$ is a tame fiber, then $a_{i}=m_{i}-1$.

  3. (iii)

$d=-\chi(\mathcal{O}_{S})-h^{0}(C,\mathcal{T}_{\pi})$ and $d\leq 0$.

For each $i$, $\mathcal{O}_{S}(D_{i})|_{D_{i}}$ is known to be a torsion
element in $\mathop{\mathrm{Pic}}\nolimits^{0}D_{i}$. Let us define
$\nu_{i}:=\operatorname{ord}\mathcal{O}_{S}(D_{i})|_{D_{i}}$. Then it is known
that

$m_{i}=p^{\alpha}\nu_{i}$ (1)

for some $\alpha\geq 0$ and that $\nu_{i}=m_{i}$ if and only if $m_{i}D_{i}$
is a tame fiber. If $D_{i}$ is a supersingular elliptic curve,
$\mathop{\mathrm{Pic}}\nolimits^{0}D_{i}$ has no torsion elements whose order
is divisible by $p$. Thus we have the following.

######  Lemma 2.2.

Suppose that $mD$ is a multiple fiber and $D$ is a supersingular elliptic
curve. Then $mD$ is tame if and only if $m$ is not divisible by $p$.

The following is useful.

######  Theorem 2.3 (Corollary to Proposition 4 in [BM76]).

Let $\pi\colon S\rightarrow\mathbb{P}^{1}$ be a smooth projective elliptic
surface satisfying $h^{1}(S,\mathcal{O}_{S})\leq 1$. Then we have
$a_{i}+1=m_{i}$ or $a_{i}+\nu_{i}+1=m_{i}$.

######  Definition 2.4.

In the above notation, assume furthermore that $C=\mathbb{P}^{1}$ and
$\chi(\mathcal{O}_{S})=0$. Such an elliptic surface $S$ is said to be of type
$(m_{1},\dots,m_{\lambda}|\nu_{1},\dots,\nu_{\lambda})$. Assume furthermore
that all singular fiber are tame. Then $S$ is said to be of type
$(m_{1},\dots,m_{\lambda})$.

The following is used as a necessary condition for algebraicity of elliptic
surface in [KU85].

######  Theorem 2.5 (Theorem 3.3 in [KU85]).

Let $\pi:S\rightarrow\mathbb{P}^{1}$ be an elliptic surface with
$\chi(\mathcal{O}_{S})=0$ and suppose that $S$ is of type
$(m_{1},\dots,m_{\lambda}|\nu_{1},\dots,\nu_{\lambda})$. Then for each
$i\in\\{1,2,\dots,\lambda\\}$, there are integers
$n_{1},n_{2},\dots,n_{\lambda}$ satisfying

  * •

$n_{i}\equiv 1\ (\bmod\ \nu_{i})$, and

  * •

$n_{1}/m_{1}+n_{2}/m_{2}+\dots+n_{\lambda}/m_{\lambda}\in\mathbb{Z}$.

######  Remark 2.6.

Let $S$ be an elliptic surface satisfying $\chi(\mathcal{O}_{S})=0$, and
$C=\mathbb{P}^{1}$. In the following cases, the information on $m$ and $\nu$
are easily deduced from Theorem 2.5.

  1. (i)

([KU85, Corollary 4.2]) Suppose that $S$ has a unique multiple fiber, and the
type of $S$ is $(m|\nu)$. Then the multiple fiber is wild, $m=p^{\alpha}$ for
some integer $\alpha>0$, and $\nu=1$.

  2. (ii)

([KU85, Corollary 4.3]) Suppose that $S$ is of type $(m_{1},m_{2})$. Then
there are integers $n_{2},n^{\prime}_{1}$ satisfying

$1/m_{1}+n_{2}/m_{2},\quad n^{\prime}_{1}/m_{1}+1/m_{2}\in\mathbb{Z}.$

These conditions imply the equality $m_{1}=m_{2}$.

###  2.2 Atiyah’s result

Atiyah classified indecomposable vector bundles on elliptic curves [Ati57]. We
summarize his results we need below.

Let $\mathcal{M}_{E}(r,d)=\mathcal{M}(r,d)$ be the set of isomorphism classes
of indecomposable vector bundles of rank $r$ and degree $d$ over an elliptic
curve $E$. For $\mathcal{E}\in\mathcal{M}(r,d)$, every other vector bundle in
$\mathcal{M}(r,d)$ is of the form $\mathcal{E}\otimes\mathcal{L}$ for some
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E=\mathcal{M}(1,0)$. There is
the unique element $\mathcal{E}_{r,0}$ in $\mathcal{M}(r,0)$ such that
$h^{0}(\mathcal{E}_{r,0})\neq 0$. Furthermore for
$\mathcal{E}\in\mathcal{M}(r,0)$, we have

$\displaystyle h^{0}(E,\mathcal{E})=h^{1}(E,\mathcal{E})=0\mbox{ when
$\mathcal{E}\neq\mathcal{E}_{r,0}$, and }$ $\displaystyle
h^{0}(E,\mathcal{E}_{r,0})=h^{1}(E,\mathcal{E}_{r,0})=1$

Actually we can define $\mathcal{E}_{r,0}$ by putting
$\mathcal{E}_{1,0}=\mathcal{O}_{E}$ and the unique non-trivial extension

$0\to\mathcal{E}_{r,0}\to\mathcal{E}_{r+1,0}\to\mathcal{O}_{E}\to 0$ (2)

inductively. We can also see that
$\mathcal{E}_{r,0}\cong(\mathcal{E}_{r,0})^{\vee}$.

Next, let us define the unique indecomposable vector bundle $\mathcal{E}_{P}$
of rank $2$ for a point $P\in E$, which fits into the following non-split
exact sequence

$0\to\mathcal{O}_{E}\to\mathcal{E}_{P}\to\mathcal{O}_{E}(P)\to 0.$ (3)

Although $\mathcal{E}_{P}\not\cong\mathcal{E}_{Q}$ for distinct points $P,Q\in
E$, we have $\mathbb{P}(\mathcal{E}_{P})\cong\mathbb{P}(\mathcal{E}_{Q})$
[Har77, Theorem V.2.15]. We can also see that

$\mathcal{M}(2,1)=\\{\mathcal{E}_{P}\mid P\in E\\}.$

###  2.3 Elliptic ruled surfaces

We use the terminology on ruled surface in [Har77, V.2], without specified
otherwise. Let $f\colon S\rightarrow E$ be a $\mathbb{P}^{1}$-bundle structure
over an elliptic curve $E$. Denote a general fiber of $f$ by $F_{f}$. Then
there is a rank $2$ vector bundle $\mathcal{E}$ on $E$ such that
$S\cong\mathbb{P}(\mathcal{E})$ and we assume that $\mathcal{E}$ is
normalized. Put $e=-\deg\mathcal{E}$ and take a minimal section $C_{0}$ with
${C_{0}}^{2}=-e$. Let us define a divisor $D$ which satisfies
$\mathcal{O}_{S}(D)\cong\det\mathcal{E}$. Then we have

$K_{S}\sim-2C_{0}-f^{*}D\equiv-2C_{0}-eF_{f}.$

Note that if $S$ has an elliptic fibration $\pi\colon S\to\mathbb{P}^{1}$,
then $-K_{S}$ is nef. Furthermore, we can easily see that $-K_{S}$ is nef if
and only if $e=0,-1$. We can also deduce from [Har77, V.2] that

$\mathcal{E}\cong\begin{cases}\mathcal{O}_{E}\oplus\mathcal{L}\ \mbox{ or }\
\mathcal{E}_{2,0}\quad&\mbox{in the case $e=0$}\\\
\mathcal{E}_{Q}\quad&\mbox{in the case $e=-1$}\end{cases}$ (4)

for some $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$ and $Q\in E$.

Moreover, any rational curves on $S$ cannot dominate the elliptic curve $E$ by
$f$. This fact immediately implies the following.

######  Lemma 2.7.

An elliptic ruled surface $S$ has no quasi-elliptic fibrations. Moreover, if
$S$ has an elliptic fibration, its singular fiber is of the of the type
${}_{m}\text{I}_{0}$ for $m>0$, that is, a multiple fiber whose reduction is a
smooth elliptic curve.

###  2.4 Non-existence of elliptic fibrations

In this subsection, we show the non-existence of elliptic fibrations on
certain elliptic ruled surfaces. First we prove the following two lemmas.

######  Lemma 2.8.

For $m\geq 0$, we have

$h^{0}(E,\operatorname{Sym}^{m}\mathcal{E}_{2,0})=\begin{cases}1&\text{$p=0$
or $m<p$}\\\ 2&\text{$m=p>0$}.\end{cases}$

###### Proof.

There is a natural exact sequence

$0\to\operatorname{Sym}^{r-1}\mathcal{E}_{2,0}\to\operatorname{Sym}^{r}\mathcal{E}_{2,0}\to\mathcal{O}_{E}\to
0,$ (5)

and Atiyah observed in [Ati57, Theorem 9] that in the case $p=0$ or $1\leq
r<p$, $\operatorname{Sym}^{r-1}\mathcal{E}_{2,0}\cong\mathcal{E}_{r,0}$, and
we can see that (2) is isomorphic to (5).

In the case $r=p>0$, we can see that the exact sequence (5) splits as follows.
Let us consider an affine open covering $\\{U_{i}\\}_{i}$ of $E$. Note that
$\mathcal{E}_{2,0}$ is trivialized on each $U_{i}$. Transition functions of
$\mathcal{E}_{2,0}$ on $U_{i}\cap U_{j}$ can be describes as

$\begin{pmatrix}1&f_{ij}\\\ 0&1\end{pmatrix}$

by (5), where $f_{ij}$ is a regular function on $U_{i}\cap U_{j}$. Then the
transition functions of $\operatorname{Sym}^{m}\mathcal{E}_{2,0}$ is

$\begin{pmatrix}1&mf_{ij}&{}_{m}C_{2}f_{ij}^{2}&{}_{m}C_{3}f_{ij}^{3}&\cdots&mf_{ij}^{m-1}&f_{ij}^{m}\\\
0&1&(m-1)f_{ij}&{}_{m-1}C_{2}f_{ij}^{2}&\cdots&(m-1)f_{ij}^{m-2}&f_{ij}^{m-1}\\\
0&0&1&(m-2)f_{ij}&\cdots&(m-2)f_{ij}^{m-3}&f_{ij}^{m-2}\\\
\vdots&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\ 0&0&0&0&\ldots&1&f_{ij}\\\
0&0&0&0&\ldots&0&1\end{pmatrix}.$

We call this $(m+1)\times(m+1)$ matrix $A_{ij}^{(m)}$. When $m=p$, it is the
following form

$\left(\begin{array}[]{@{\,}c|cccc@{\,}}1&0&\cdots&0&f_{ij}^{p}\\\ \hline\cr
0&&&&\\\
\vdots&\lx@intercol\hfil\raisebox{-10.0pt}[0.0pt][0.0pt]{\Huge$A_{ij}^{(p-1)}$}\hfil\lx@intercol\\\
0&&&&\\\ \end{array}\right),$

since ${}_{p}C_{k}$ is divisible by $p$. When $E$ is an ordinary (resp.
supersingular) elliptic curve, the Frobenius morphism $\operatorname{Fr}$
induces a bijection (resp. a zero map)

$\operatorname{Fr}^{*}\colon H^{1}(E_{p},\mathcal{O}_{E_{p}})\to
H^{1}(E,\mathcal{O}_{E}).$

Thus we put $\\{f_{ij}^{p}\\}=\lambda\\{f_{ij}\\}\in H^{1}(E,\mathcal{O}_{E})$
for some $\lambda\in k^{*}$ (resp. $\lambda=0$). In particular, there are
regular functions $g_{i}$ on $U_{i}$ such that $f_{ij}^{p}-\lambda
f_{ij}=g_{i}-g_{j}$.

Take an $(p+1)\times(p+1)$ matrix $P_{i}$ as

$\left(\begin{array}[]{@{\,}c|ccccc@{\,}}1&0&\cdots&0&\lambda&-g_{i}\\\
\hline\cr 0&&&&\\\
\vdots&\lx@intercol\hfil\raisebox{-10.0pt}[0.0pt][0.0pt]{\Huge$I_{p}$}\hfil\lx@intercol\\\
0&&&&\\\ \end{array}\right).$

Here $I_{p}$ is the $p\times p$ identity matrix. Then we can see that

$A_{ij}^{(p)}P_{i}=P_{j}\widetilde{A_{ij}}^{(p)},$

where we put

$\widetilde{A_{ij}}^{(p)}:=\left(\begin{array}[]{@{\,}c|cccc@{\,}}1&0&\cdots&0&0\\\
\hline\cr 0&&&\\\
\vdots&\lx@intercol\hfil\raisebox{-10.0pt}[0.0pt][0.0pt]{\Huge$A_{ij}^{(p-1)}$}\hfil\lx@intercol\\\
0&&&\\\ \end{array}\right).$

This means that the vector bundles defined by the transition functions
$\\{A_{ij}\\}$ and $\\{\widetilde{A_{ij}}\\}$ are isomorphic to each other.
This completes the proof. ∎

######  Lemma 2.9.

Take $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$ and
$m\in\mathbb{Z}_{\geq 0}$. Then we have

$h^{0}(E,\operatorname{Sym}^{m}(\mathcal{O}_{E}\oplus\mathcal{L}))=\begin{cases}1&m<\operatorname{ord}\mathcal{L}\\\
2&m=\operatorname{ord}\mathcal{L}.\end{cases}$

###### Proof.

The assertion follows from the isomorphism

$\displaystyle\operatorname{Sym}^{m}(\mathcal{O}_{E}\oplus\mathcal{L})\cong\mathcal{O}_{E}\oplus\mathcal{L}\oplus\cdots\oplus{\mathcal{L}}^{\otimes
m}.$

∎

######  Proposition 2.10.

Let $\mathcal{E}$ be a vector bundle of rank $2$ on an elliptic curve $E$. In
the following cases, $S:=\mathbb{P}_{E}(\mathcal{E})$ has no elliptic
fibrations.

  1. (i)

$p=0$ and $\mathcal{E}=\mathcal{E}_{2,0}$.

  2. (ii)

$p\geq 0$ and $\mathcal{E}=\mathcal{O}_{E}\oplus\mathcal{L}$ for
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$ with
$\operatorname{ord}\mathcal{L}=\infty$.

###### Proof.

In both cases, we have $e=0$ and hence $-K_{S}\sim 2C_{0}+f^{*}D$, where
$f\colon S\to E$ is the $\mathbb{P}^{1}$-bundle structure and $D$ is a divisor
satisfying $\mathcal{O}_{S}(D)\cong\det\mathcal{E}$. If $S$ has an elliptic
fibration, the complete linear system $|-mK_{S}|$ for some $m>0$ defines it,
and hence $h^{0}(S,\omega_{S}^{\otimes{-m}})\geq 2$. On the other hand, we
have

$h^{0}(S,\omega_{S}^{\otimes{-m}})=h^{0}(S,\mathcal{O}_{S}(2mC_{0}+mf^{*}D))=h^{0}(E,\operatorname{Sym}^{2m}(\mathcal{E})\otimes\mathcal{O}_{S}(mD)),$

which equals to $1$ in the case (i) by Lemma 2.8, and in the case (ii) by
Lemma 2.9. Thus the assertion follows. ∎

###  2.5 Existence of elliptic fibrations

In this subsection, we show the existence of an elliptic fibration on a given
elliptic ruled surface $S=\mathbb{P}(\mathcal{E})$. If $S$ has an elliptic
fibration, then $-K_{S}$ is nef, hence $\mathcal{E}$ is one of vector bundles
appeared in (4).

To achieve our purpose, first we study the structure of
$\varphi^{*}\mathcal{E}$, where $\varphi\colon F\to E$ is a finite isogeny of
elliptic curves. For a given such $\varphi$, there is a factorization

$\textstyle{\varphi\colon
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{E,}$

where $\psi$ is purely inseparable and $\phi$ is separable. It is known that
$\psi$ is a composition of Frobenius morphisms. If $\ker\phi$ is non-trivial,
it is a finite abelian group, which contains a cyclic group of a prime order.
Thus $\phi$ is decomposed into isogenies with a prime degree. Hence it is
essential to study the structure of $\varphi^{*}\mathcal{E}$ for an isogeny
$\varphi$ with a prime degree.

Suppose that $p>0$. Let $E$ be an ordinary elliptic curve. Then, $E$ contains
a subgroup $\mathbb{Z}/p\mathbb{Z}$ in a unique way, and let us call

$q_{E}\colon E\to E/(\mathbb{Z}/p\mathbb{Z})$

the quotient morphism. Recall that we have

$\widehat{q_{E}}=\operatorname{Fr}\colon E/(\mathbb{Z}/p\mathbb{Z})\cong
E_{p}\to E.$

To the contrary if $E$ is supersingular, we have

$\widehat{\operatorname{Fr}}=\operatorname{Fr}\colon E_{p}\to E.$

For an isogeny $\varphi\colon F\to E$ of degree $p$, note that

$\varphi=\begin{cases}\operatorname{Fr}\text{ if $\varphi$ is purely
inseparable},\\\ q_{F}\text{ if $\varphi$ is separable. (In this case, $E$ is
necessarily ordinary.)}\end{cases}$

We use the following Lemma to prove Lemma 2.12.

######  Lemma 2.11 (Corollaries 1.7 and 2.6 in [Oda71]).

Let $\varphi\colon F\to E$ be an isogeny of elliptic curves of degree $n$.
Then we have

$\varphi_{*}\mathcal{O}_{F}=\begin{cases}\mathcal{E}_{p,0}&\text{ if $n=p$ and
$\widehat{\varphi}$ is purely inseparable}\\\
\bigoplus_{\varphi^{*}\mathcal{L}\cong\mathcal{O}_{F}}\mathcal{L}&\text{ if
$\widehat{\varphi}$ is separable.}\end{cases}$

######  Lemma 2.12.

Let $F$ and $E$ be elliptic curves. Take an indecomposable vector bundle
$\mathcal{E}_{Q}\in\mathcal{M}_{E}(2,1)$ for some $Q\in E$.

  1. (i)

Suppose that $p>0$, and let $\varphi\colon F\to E$ be an isogeny of degree $p$
with $\widehat{\varphi}$ purely inseparable. Then we have

$\varphi^{*}\mathcal{E}_{2,0}\cong\mathcal{O}_{F}\oplus\mathcal{O}_{F}.$

  2. (ii)

Suppose that $p=2$. Then we have

$\operatorname{Fr}^{*}\mathcal{E}_{Q}\cong\mathcal{E}_{2,0}\otimes\mathcal{O}_{E_{2}}(Q).$

  3. (iii)

Suppose that $p\geq 0$. Let $\varphi\colon F\to E$ be a separable isogeny of
degree $2$. Then

$\varphi^{*}\mathcal{E}_{Q}\cong\mathcal{O}_{F}(Q_{1})\oplus\mathcal{O}_{F}(Q_{2}),$

where $Q_{i}\in F$ and $\mathcal{O}_{F}(Q_{1}-Q_{2})$ is an order $2$ element
of $\mathop{\mathrm{Pic}}\nolimits^{0}F$.

###### Proof.

Let $\varphi\colon F\to E$ be an isogeny with prime $n:=\deg\varphi$. Note
that Lemma 2.11 implies

$\mathop{\mathrm{Coker}}\nolimits\operatorname{adj}=\begin{cases}\mathcal{E}_{p-1,0}&\text{
if $n=p$ and $\widehat{\varphi}$ is purely inseparable, }\\\
\bigoplus_{\operatorname{ord}\mathcal{L}=n}\mathcal{L}&\text{ if
$\widehat{\varphi}$ is separable,}\end{cases}$

where

$\operatorname{adj}\colon\mathcal{O}_{E}\to\varphi_{*}\varphi^{*}\mathcal{O}_{E}=\varphi_{*}\mathcal{O}_{F}$

is the adjunction morphism.

(i) By the assumption, the connection morphism

$H^{0}(\mathop{\mathrm{Coker}}\nolimits\operatorname{adj})\to
H^{1}(\mathcal{O}_{E})$

is isomorphic, and hence the morphism

$\varphi^{*}=H^{1}(\operatorname{adj})\colon\mathop{\mathrm{Ext}}\nolimits_{E}^{1}(\mathcal{O}_{E},\mathcal{O}_{E})\cong
H^{1}(\mathcal{O}_{E})\to\mathop{\mathrm{Ext}}\nolimits^{1}_{F}(\varphi^{*}\mathcal{O}_{E},\varphi^{*}\mathcal{O}_{E})\cong
H^{1}(\varphi_{*}\mathcal{O}_{F})$

is zero. (Note that if $E$ is supersingular, this fact immediately follows
from the definition.) Hence the short exact sequence obtained by $\varphi^{*}$
of the sequence (2) for $r=1$ splits. Then the result follows.

(ii), (iii) In both cases, we have

$h^{0}(\mathop{\mathrm{Coker}}\nolimits(\operatorname{adj}\otimes\mathcal{O}_{E}(-Q)))=0,$

and hence the morphism

$H^{1}(\operatorname{adj}\otimes\mathcal{O}_{E}(-Q))\colon
H^{1}(\mathcal{O}_{E}(-Q))\to
H^{1}(\varphi_{*}\varphi^{*}\mathcal{O}_{E}\otimes\mathcal{O}_{E}(-Q))=H^{1}(\varphi^{*}\mathcal{O}_{E}(-Q))$

is injective. Since it coincides with the morphism

$\varphi^{*}\colon\mathop{\mathrm{Ext}}\nolimits^{1}_{E}(\mathcal{O}_{E}(Q),\mathcal{O}_{E})\to\mathop{\mathrm{Ext}}\nolimits^{1}_{F}(\varphi^{*}\mathcal{O}_{E}(Q),\varphi^{*}\mathcal{O}_{E}),$

the short exact sequence

$0\to\mathcal{O}_{F}\to\varphi^{*}\mathcal{E}_{Q}\to\varphi^{*}\mathcal{O}_{E}(Q)\to
0$ (6)

obtained by $\varphi^{*}$ of the sequence (3) does not split.

Assume that $p=2$, $\varphi=\operatorname{Fr}$ and
$\operatorname{Fr}^{*}\mathcal{E}_{Q}$ is decomposable. Since
$\operatorname{Fr}^{*}Q=2Q$, we have

$\operatorname{Fr}^{*}\mathcal{E}_{Q}=\mathcal{O}_{E_{2}}(Q)\oplus\mathcal{O}_{E_{2}}(Q).$

Then it follows from (6) that there is an exact sequence

$0\to\mathcal{O}_{E_{2}}(-Q)\to\mathcal{O}_{E_{2}}\oplus\mathcal{O}_{E_{2}}\to\mathcal{O}_{E_{2}}(Q)\to
0.$

But this is absurd by $h^{0}(\mathcal{O}_{E_{2}}(Q))=1$. Hence,
$\operatorname{Fr}^{*}\mathcal{E}_{Q}$ is indecomposable of degree $2$ with
$\det\operatorname{Fr}^{*}\mathcal{E}_{Q}\cong\mathcal{O}_{E_{2}}(2Q)$. This
completes the proof of (ii).

Next, let us give the proof of (iii). Set $\varphi^{*}Q=Q_{1}+Q_{2}$ for
distinct points $Q_{1}$ and $Q_{2}$ on $F$. Then we see that
$\det\varphi^{*}\mathcal{E}_{Q}\cong\mathcal{O}_{F}(Q_{1}+Q_{2})$ by (6). Note
that $\mathcal{O}_{F}(Q_{1}-Q_{2})$ is an order $2$ element of
$\mathop{\mathrm{Pic}}\nolimits^{0}F$, since $\varphi$ is a separable isogeny
of degree $2$. Assume that $\varphi^{*}\mathcal{E}_{Q}$ is indecomposable.
Then there is an element $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}F$
such that
$\varphi^{*}\mathcal{E}_{Q}\otimes\mathcal{O}_{F}(-Q_{2})\cong\mathcal{E}_{2,0}\otimes\mathcal{L}$,
and thus

$\mathcal{O}_{F}(Q_{1}-Q_{2})\cong\det(\mathcal{E}_{2,0}\otimes\mathcal{L})\cong\mathcal{L}^{\otimes
2}.$

This is absurd by the equality
$\operatorname{ord}\mathcal{O}_{F}(Q_{1}-Q_{2})=2$. Therefore, we obtain the
conclusion. ∎

######  Lemma 2.13.

Let $E$ be an elliptic curve, and take a line bundle
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$ with
$m:=\operatorname{ord}\mathcal{L}<\infty$. Then there is an isogeny
$\varphi\colon F\to E$ of degree $m$ such that
$\varphi^{*}\mathcal{L}\cong\mathcal{O}_{F}$.

###### Proof.

Define $\varphi$ to be the dual morphism of the quotient morphism

$E\cong\mathop{\mathrm{Pic}}\nolimits^{0}E\to
F:=\mathop{\mathrm{Pic}}\nolimits^{0}E/\left<\mathcal{L}\right>.$

Then, $\varphi$ satisfies the desired property. ∎

######  Lemma 2.14.

Let $\varphi\colon F\to E$ be a finite morphism of elliptic curves, and
$\mathcal{E}$ be a rank $2$ normalized vector bundle on $E$ with $e=-1$ or
$0$. Suppose that $S^{\prime}:=\mathbb{P}(\varphi^{*}\mathcal{E})$ has an
elliptic fibration $\pi^{\prime}$. Then we have the following.

  1. (i)

We have $q^{*}\omega_{S}\cong\omega_{S^{\prime}}$, where $q\colon
S^{\prime}\cong F\times_{E}S\to S:=\mathbb{P}(\mathcal{E})$ is the second
projection.

  2. (ii)

The elliptic ruled surface $S$ also has an elliptic fibration $\pi\colon
S\to\mathbb{P}^{1}$ fitting into the following commutative diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$
(7)

Here the left square is a fiber product diagram, and $\psi\circ\pi^{\prime}$
is the Stein factorization of $\pi\circ q$.

###### Proof.

(i) If $\varphi$ is étale, the result is obvious. Thus we may assume that
$\varphi$ is purely inseparable of degree $p$, that is
$\varphi=\operatorname{Fr}$. In this case,
$\Omega_{F/E}=\Omega_{F}=\mathcal{O}_{F}$, hence $\Omega_{S^{\prime}/S}\cong
f^{\prime*}\Omega_{F/E}=\mathcal{O}_{S}^{\prime}$. Then we obtain the result
by [Eke87, Corollary 3.4].

(ii) Take a sufficiently large integer $m$. Then we may assume that
$h^{0}(S^{\prime},\omega_{S^{\prime}}^{\otimes-m})$ is sufficiently large.
Now, we have a series of equalities

$\displaystyle h^{0}(S^{\prime},\omega_{S^{\prime}}^{\otimes-m})$
$\displaystyle=h^{0}(E,f_{*}q_{*}q^{*}\omega_{S}^{\otimes-m})=h^{0}(E,\varphi_{*}f^{\prime}_{*}q^{*}\omega_{S}^{\otimes-m})$
$\displaystyle=h^{0}(E,\varphi_{*}\varphi^{*}f_{*}\omega_{S}^{\otimes-m})=h^{0}(E,f_{*}\omega_{S}^{\otimes-m}\otimes\varphi_{*}\mathcal{O}_{F}).$

Here, the third equality comes from the flat base change theorem.

If $\widehat{\varphi}$ is separable, it follows from Lemma 2.11 that there
exists an invertible sheaf $\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$
such that $\varphi^{*}\mathcal{L}\cong\mathcal{O}_{F}$ and

$h^{0}(S,\omega_{S}^{\otimes-m}\otimes
f^{*}\mathcal{L}))=h^{0}(E,f_{*}\omega_{S}^{\otimes-m}\otimes\mathcal{L})\geq
2.$

Next, suppose that $\widehat{\varphi}$ is purely inseparable of degree $p$.
Since $\varphi_{*}\mathcal{O}_{F}\cong\mathcal{E}_{p,0}$ by Lemma 2.11 and
$\mathcal{E}_{p,0}$ has a filtration into $\mathcal{O}_{E}$ by (2), we have

$h^{0}(S,\omega_{S}^{\otimes-m})=h^{0}(E,f_{*}\omega_{S}^{\otimes-m})\geq 2.$

Since, in general, $\varphi$ is a composition of separable morphisms and
purely inseparable morphisms of degree $p$, we can find a divisor $H$ on $S$
satisfying $H^{2}=K_{S}\cdot H=0$ and $h^{0}(S,\mathcal{O}_{S}(H))\geq 2$. Let
us consider the moving part $|M|$ of the complete linear system $|H|.$ Note
that every effective divisor on $S$ has a non-negative self-intersection,
hence $M^{2}=0$ and $h^{0}(S,\mathcal{O}_{S}(M))\geq 2$. Thus we know that the
linear system $|M|$ is base point free, and it satisfies $M\cdot K_{S}=0$. Let
us take the Stein factorization of the projective morphism defined by $|M|$,
and then we obtain a morphism $\pi\colon S\to\mathbb{P}^{1}$ with connected
fibers. [Mac40, Theorem 2] forces the general fiber of $\pi$ is reduced. Lemma
2.7 kills the possibility that $\pi$ is quasi-elliptic, hence $\pi$ is
elliptic.

Taking the Stein factorization of $\pi\circ q$, we obtain a finite morphism
$\psi$ in the diagram. ∎

We apply Lemma 2.14 to show the following.

######  Proposition 2.15.

Let $\mathcal{E}$ be a vector bundle of rank $2$ on an elliptic curve $E$. In
the following cases, $\mathbb{P}(\mathcal{E})$ has an elliptic fibration.

  1. (i)

$p>0$ and $\mathcal{E}=\mathcal{E}_{2,0}$.

  2. (ii)

$p\geq 0$ and $\mathcal{E}=\mathcal{O}_{E}\oplus\mathcal{L}$ for
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$ with
$\operatorname{ord}\mathcal{L}<\infty$.

  3. (iii)

$p\geq 0$ and $\mathcal{E}=\mathcal{E}_{Q}$.

###### Proof.

(i) Take an isogeny $\varphi\colon F\to E$ of degree $p$ such that
$\widehat{\varphi}$ is purely inseparable, namely
$\varphi:=\widehat{\operatorname{Fr}}$. Then we have an isomorphism
$\mathbb{P}(\varphi^{*}\mathcal{E}_{2,0})\cong\mathbb{P}^{1}\times E$ by Lemma
2.12 (i). Then, the result follows from Lemma 2.14.

(ii) Lemma 2.13 tells us that
$\varphi^{*}(\mathcal{O}_{E}\oplus\mathcal{L})\cong\mathcal{O}_{F}\oplus\mathcal{O}_{F}$
for a suitable isogeny $\varphi\colon F\to E$. Hence the result follows from
Lemma 2.14.

(iii) Suppose that $p=2$. Then Lemma 2.12 (ii) assures the existence of
isomorphism
$\mathbb{P}(\operatorname{Fr}^{*}\mathcal{E}_{Q})\cong\mathbb{P}(\mathcal{E}_{2,0})$.
Then, combining (i) with Lemma 2.14, we obtain the conclusion.

Suppose that $p\neq 2$. Then Lemma 2.12 (iii) implies that there is an
isomorphism
$\mathbb{P}(\varphi^{*}\mathcal{E}_{Q})\cong\mathbb{P}(\mathcal{O}_{F}\oplus\mathcal{L})$,
where $\varphi\colon F\to E$ is an isogeny of degree $2$, and
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}F$ with
$\operatorname{ord}\mathcal{L}=2$. Combining (ii) with Lemma 2.14, we obtain
the conclusion. ∎

######  Remark 2.16.

Maruyama showed in [Mar71, Lemma 9] that $\mathbb{P}(\mathcal{E}_{Q})$
($\mathbf{P}_{1}$ in his notation) has a base point free linear pencil whose
generic member is an elliptic curve if $p\neq 2$. In addition to it, he also
stated in [Mar71, Remark 7] that a similar result is true in the case $p=2$
“by reduction”. The authors do not understand what it means.

Using this result, he showed which elliptic ruled surfaces have an elliptic
fibration in [Mar71, Theorem 4]. But the authors feel that the proof in the
case $p=2$ is quite unsatisfactory.

###  2.6 Ramifications and singular fibers

Let us consider the situation in Lemma 2.14 and the diagram (7). If an
ordinary elliptic curve is isogeneous to another elliptic curve, it is also
ordinary. Therefore, we see that all of $E$, $F$, general fibers of $\pi$ and
$\pi^{\prime}$, and reductions of multiple fibers of $\pi$ and $\pi^{\prime}$
are either ordinary or supersingular elliptic curves simultaneously. In this
subsection, we extract information on multiple fibers of $\pi$ from one of
$\pi^{\prime}$, and vice versa.

######  Lemma 2.17.

In the diagram (7), fix a point $Q\in\mathbb{P}^{1}$ and $\psi^{*}Q=\sum
e_{i}Q_{i}^{\prime}$, where $Q_{i}^{\prime}\neq Q_{j}^{\prime}$ for $i\neq j$.
Denote the fiber of $\pi$ over $Q$ by $mD$, where $m$ is its multiplicity, and
denote the fiber $m^{\prime}_{i}D^{\prime}_{i}$ of $\pi^{\prime}$ over the
point $Q_{i}^{\prime}$ similarly.

  1. (i)

For each $i$, $m^{\prime}_{i}e_{i}$ is divisible by $m$. Moreover we have

$\sum\frac{m_{i}^{\prime}e_{i}}{m}\leq\deg q.$

  2. (ii)

If $m=1$ holds, then we have $m_{i}^{\prime}=1$ for all $i$. Conversely if
$e_{i}=m_{i}^{\prime}=1$ holds for some $i$, then we have $m=1$.

  3. (iii)

Suppose that $\psi$ is not ramified at $Q^{\prime}_{i}$, that is $e_{i}=1$.
Then $\pi$ has a multiple fiber over $Q$ if and only if $\pi^{\prime}$ has a
multiple fiber over $Q^{\prime}_{i}$.

###### Proof.

(i) The assertions are a direct consequence of the equalities

$q^{*}(mD)=q^{*}\pi^{*}Q=\pi^{\prime*}\psi^{*}Q=\sum
m_{i}^{\prime}e_{i}D_{i}^{\prime}.$ (8)

(ii) First assume that $m=1$. Then the inequality

$\sum m_{i}^{\prime}e_{i}\leq\deg q=\deg\psi=\sum e_{i}$

forces that $m_{i}^{\prime}=1$ for all $i$. The second assertion is a direct
consequence of (i).

(iii) The assertion is a direct consequence of (ii). ∎

######  Lemma 2.18.

In the notation in Lemma 2.17, we assume furthermore that $\deg q=\deg\psi=2$.
Let us consider the morphism $q|_{D_{i}^{\prime}}\colon D_{i}^{\prime}\to D$.

  1. (i)

Suppose that $e_{1}=1$, equivalently $e_{2}=1$ holds. Then we have
$m_{1}^{\prime}=m_{2}^{\prime}=m$, and multiple fibers $mD$, $mD_{1}^{\prime}$
and $mD_{2}^{\prime}$ are either tame or wild simultaneously.

  2. (ii)

Suppose that $p=m=2$ and $e_{1}=2$, $m_{1}^{\prime}=1$, and
$q|_{D_{1}^{\prime}}$ is separable. Then $2D$ is a wild fiber.

  3. (iii)

Suppose that $p=m=2$ and $e_{1}=2$, $m_{1}^{\prime}=2$, and
$q|_{D_{1}^{\prime}}$ is isomorphic. Then $2D$ is a wild fiber.

###### Proof.

Note that
$q|_{D_{i}^{\prime}}^{*}\mathcal{O}_{D}(D)\cong\mathcal{O}_{D_{i}^{\prime}}(q^{*}D)$.

(i) Lemma 2.17 (i) tells us that $m_{1}^{\prime}=m_{2}^{\prime}=m$. We can see
that $q|_{D_{i}^{\prime}}$ is isomorphic and
$q|_{D_{i}^{\prime}}^{*}\mathcal{O}_{D}(D)\cong\mathcal{O}_{D_{i}^{\prime}}({D_{i}^{\prime}})$.
Hence,
$\operatorname{ord}\mathcal{O}_{D}(D)=\operatorname{ord}\mathcal{O}_{D_{i}^{\prime}}({D_{i}^{\prime}})$,
and the second assertion follows.

(ii) Note that $q^{*}D=D_{1}^{\prime}$ in this case. The morphism
$q|_{D_{1}^{\prime}}$ is either the quotient morphism by
$\mathbb{Z}/2\mathbb{Z}$ or isomorphic, hence the kernel
$\mathop{\mathrm{Ker}}\nolimits\widehat{q|_{D_{1}^{\prime}}}$ of the dual
morphism is $\mu_{2}$ in the former case, and is trivial in the latter case.
On the other hand, we have
$q|_{D_{1}^{\prime}}^{*}\mathcal{O}_{D}(D)\cong\mathcal{O}_{D_{1}^{\prime}}({D_{1}^{\prime}})\cong\mathcal{O}_{D_{1}^{\prime}}$,
and hence $\operatorname{ord}\mathcal{O}_{D}(D)=1$. In particular, $2D$ is a
wild fiber.

(iii) In this case, we have $q^{*}D=2D_{1}^{\prime}$, and hence we see
$q|_{D_{1}^{\prime}}^{*}\mathcal{O}_{D}(D)\cong\mathcal{O}_{D_{1}^{\prime}}(2D_{1}^{\prime})\cong\mathcal{O}_{D_{1}^{\prime}}$.
Thus $\operatorname{ord}\mathcal{O}_{D}(D)=1$, and therefore, $2D$ is a wild
fiber. ∎

##  3 Multiple fibers on elliptic ruled surfaces

We study multiple fibers of an elliptic fibration $\pi\colon
S\to\mathbb{P}^{1}$ on an elliptic ruled surface $S$ in this section. We use
the notation in §2.1.

######  Lemma 3.1.

We have

$\sum_{i=1}^{\lambda}\displaystyle\frac{a_{i}}{m_{i}}<2+d.$

###### Proof.

By Proposition 2.1, we have

$h^{0}(S,\omega_{S}^{\otimes
n})=h^{0}(S,\pi^{*}\mathcal{M}\otimes\mathcal{O}_{S}(\sum_{i=1}^{\lambda}(na_{i}-m_{i}[\frac{na_{i}}{m_{i}}])D_{i})),$
(9)

where we set

$\mathcal{M}:=\omega_{\mathbb{P}^{1}}^{\otimes
n}\otimes\mathcal{L}_{\pi}^{\otimes-n}\otimes\mathcal{O}_{\mathbb{P}^{1}}(\sum_{i=1}^{\lambda}\displaystyle[\frac{na_{i}}{m_{i}}]).$

Then since $\sum_{i=1}^{\lambda}(na_{i}-m_{i}[\frac{na_{i}}{m_{i}}])D_{i}$ is
a fixed part of the linear system $|nK_{S}|$, we have

$\displaystyle 0=h^{0}(S,\omega_{S}^{\otimes
n})=h^{0}(S,\pi^{*}\mathcal{M})=h^{0}(\mathbb{P}^{1},\mathcal{M}).$

Hence we obtain $\sum_{i=1}^{\lambda}[\frac{na_{i}}{m_{i}}]<n(d+2)$ for all
$n>0$. This is equivalent to the desired inequality

$\sum_{i=1}^{\lambda}\displaystyle\frac{a_{i}}{m_{i}}<2+d.$

∎

Now we can prove the following lemma.

######  Lemma 3.2.

The quantities $d,m_{i},a_{i}$ satisfy the following.

Case | $d$ | $m_{i}$ | $a_{i}$ | $p$
---|---|---|---|---
$(I)$ | $0$ | no multiple fibers |  | $p\geq 0$
$(II)$ | $0$ |  $(m,m)$, $m>1$ |  | $p\geq 0$
$(III)$ | $0$ | $(2,2,2)$ |  | $p\geq 0$
$(IV)$ | $-1$ |  $({p^{\alpha}}^{*})$, $\alpha>0$ | $a_{1}=p^{\alpha}-1$ | $p>0$
$(V)$ | $-1$ |  $({p^{\alpha}}^{*})$, $\alpha>0$ | $a_{1}=p^{\alpha}-2$ | $p>0$
$(VI)$ | $-1$ | $(2,2^{*})$ | $a_{2}=0$ | $p=2$

The symbol ∗ denotes a wild fiber.

###### Proof.

We can divide into the following two cases; (1) $d=0$ and (2) $d=-1$.
(Equivalently, $\sum_{i=1}^{\lambda}\displaystyle\frac{a_{i}}{m_{i}}<2$, and
$<1$ respectively by Lemma 3.1).

In the case (1), it follows from Proposition 2.1 (iii) that $\pi$ has no wild
fibers. In particular, we have $a_{i}=m_{i}-1$ for all $i$ by Proposition 2.1
(ii). Then $\pi$ has no multiple fibers, _or_ $\pi$ has multiple fibers of
type ($m$), ($m_{1},m_{2}$), ($2,2,m$) or ($2,3,m_{3}$), where $m$ and $m_{i}$
satisfy that

$2\leq m,m_{1},m_{2},\quad 3\leq m_{3}\leq 5.$

In the latter case, apply Theorem 2.5 (and Remark 2.6), and then we see that
the multiple fibers are of type ($m,m$) or ($2,2,2$).

In the case (2), Proposition 2.1 implies that
$h^{0}(\mathbb{P}^{1},\mathcal{T}_{\pi})=1,$ which means that $\pi$ has a
single wild fiber. Then the inequality $\sum_{i=1}^{\lambda}{a_{i}}/{m_{i}}<1$
implies that the multiple fiber is unique (2-1), _or_ that $\pi$ has one tame
fiber and one wild fiber (2-2).

(2-1) In this case, Remark 2.6 (i) implies that the unique multiple fiber has
the multiplicity $p^{\alpha}$ ($\alpha>0$) and $\nu=1$, and Theorem 2.3
implies that $a=m-1$ or $m-2$.

(2-2) Suppose that the type of $\pi$ is $(m_{1},m_{2}^{*})$. Then the
inequality

$\frac{m_{1}-1}{m_{1}}+\frac{a_{2}}{m_{2}}<1$

and Theorem 2.3 yield that $a_{2}=m_{2}-\nu_{2}-1$. Since $m_{2}$ is of the
form $p^{\alpha}\nu_{2}$ with $\alpha>0$, we can see that $p=2$, the type of
$\pi$ is $(2,2^{*})$ and $a_{2}=0$. ∎

##  4 Proof of Theorem 1.1

Let $S$ be an elliptic ruled surface admitting a $\mathbb{P}^{1}$-bundle
structure $f\colon S\rightarrow E$ over an elliptic curve $E$. We use the
notation in §2.3.

######  Lemma 4.1.

Suppose that $S$ has an elliptic fibration $\pi\colon
S\rightarrow\mathbb{P}^{1}$. Then the equality $e=0$ holds in the cases (I),
(II) and (V) in Lemma 3.2, and the equality $e=-1$ holds in the cases (III),
(IV) and (VI) in Lemma 3.2.

###### Proof.

First of all, we have $e=0$ or $-1$, as is mentioned in §2.2. Note that the
multiplicities of all multiple fibers of $\pi$ are common by Lemma 3.2. Hence
$D_{i}\equiv D_{j}$ for each $i,j$ in the notation in §2.1.. We denote the
common multiplicities by $m$. If $\pi$ has no multiple fibers, we put $m=1$
for simplicity.

Proposition 2.1 implies that

$K_{S}\sim(-2-d)F_{\pi}+\sum_{i}a_{i}D_{i}\equiv((-2-d)m+\sum_{i}a_{i})D,$

where $D$ is a reduction of a multiple fiber of $\pi$. By a direct
computation, we have

$(-2-d)m+\sum_{i}a_{i}=\begin{cases}-2&\text{in the cases (I),(II) and
(V),}\\\ -1&\text{in the cases (III), (IV) and (VI).}\end{cases}$

Consequently in the cases (I), (II) and (V), the integer $e=K_{S}\cdot C_{0}$
should be even, i.e. $e=0$. In the cases (III), (IV) and (VI), we have $D\cdot
F_{f}=-K_{S}\cdot F_{f}=2.$ Combining this with the equality $C_{0}\cdot
F_{f}=1$, we conclude that $C_{0}$ cannot be contained in a fiber of $\pi$,
equivalently $e=-(C_{0})^{2}\neq 0$. This means $e=-1$. ∎

######  Remark 4.2.

In the proof of Lemma 4.1, we show that

$K_{S}\equiv\begin{cases}-2D\equiv-2C_{0}&\mbox{ if }e=0,\\\
-D\equiv-2C_{0}+F_{f}&\mbox{ if }e=-1.\end{cases}$

In the diagram (7), take a point $Q\in\mathbb{P}^{1}$ and put
$Q^{\prime}:=\psi(Q)$. Denote the fiber over the point $Q^{\prime}$ by
$m^{\prime}D^{\prime}$, where $m^{\prime}$ is its multiplicity, and the fiber
over the point $Q$ by $mD$ similarly. Suppose that $e=0$ for $S^{\prime}$, and
$e=-1$ for $S$. Since $q^{*}K_{S}=K_{S^{\prime}}$ holds by Lemma 2.14 (i), we
obtain

$2D^{\prime}\equiv q^{*}D.$ (10)

We now give the proof of Theorem 1.1.

_Proof of Theorem 1.1._ (i) In the case $e=0$, we start with the following
claim.

######  Claim 4.3.

Assume that $S$ has an elliptic fibration $\pi$ admitting at least one
multiple fiber. Then $\pi$ is of type $(m,m)$ if and only if $\mathcal{E}$ is
decomposable.

###### Proof.

Recall first that $\mathcal{E}$ is decomposable if and only if there exist two
sections $C_{1}$ and $C_{2}$ of $f$ such that $C_{1}\cap C_{2}=\emptyset$
([Har77, Exercise V.2.2]). Moreover, Remark 4.2 says that in the case $e=0$,
$D\equiv C_{0}$, where $mD$ is a multiple fiber with multiplicity $m$. To the
contrary, if some irreducible curve $D$ satisfying $D\equiv C_{0}$, it turns
out that $D$ is the reduction of some multiple fiber of $\pi$, since
$-K_{S}\cdot D=0$.

Suppose that $\pi$ has two multiple fibers $mD_{1}$ and $mD_{2}$ with
multiplicities $m$. Then $D_{1}$ and $D_{2}$ are sections of $f$, and hence,
$\mathcal{E}$ is decomposable.

Next, suppose that $C_{1}$ and $C_{2}$ are two sections of $f$ satisfying
$C_{1}\cap C_{2}=\emptyset$. Then we can put $C_{1}\equiv C_{0}+b_{1}F_{f}$
and $C_{2}\equiv C_{0}+b_{2}F_{f}$ for some $b_{1},b_{2}\geq 0$. The
equalities

$0=C_{1}\cdot C_{2}=C_{0}^{2}+(b_{1}+b_{2})C_{0}\cdot F_{f}=b_{1}+b_{2}$

yield $b_{1}=b_{2}=0$. Hence $C_{1}$ and $C_{2}$ are the reductions of some
multiple fibers of $\pi$. Therefore, Lemma 4.1 says that this situation fits
into the case $\pi$ is of type $(m,m)$. ∎

Let us consider the case $\mathcal{E}$ is decomposable. Then, we can take
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}E$,
$\operatorname{ord}\mathcal{L}=m>1$ such that
$\mathcal{E}=\mathcal{O}_{E}\oplus\mathcal{L}$. In this case, since

$h^{0}(S,\mathcal{O}_{S}(nC_{0}))=h^{0}(E,\operatorname{Sym}^{n}(\mathcal{O}_{E}\oplus\mathcal{L}))$

holds, we can see by Lemma 2.9 that the elliptic fibration $\pi$ has a
multiple fiber $mC_{0}$, and thus we can apply Claim 4.3 to conclude that
$\pi$ is of type $(m,m)$.

Next we consider the case $\mathcal{E}$ indecomposable, that is,
$\mathcal{E}=\mathcal{E}_{2,0}$. Then combining Lemma 2.8 with Lemma 2.9, we
know that $S$ has an elliptic fibration $\pi$ in the case $p>0$, and no
elliptic fibrations in the case $p=0$. We know from Claim 4.3 that $\pi$ is of
type $(p^{\alpha*})$. Recall that we have the following diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{F\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{2,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$

Here, $\widehat{\varphi}\colon E\to F$ is a purely inseparable isogeny of
degree $p$. Applying Lemma 2.17 (i) for $m=p^{\alpha}$, $m_{i}^{\prime}=1$,
$e_{i}\leq p=\deg\psi$, we obtain $\alpha=1$.

(ii) In the case $e=-1$, $S$ has an elliptic fibration by Lemma 2.15 (iii).
When $p\neq 2$, we apply Lemma 2.17 (i) for the diagram

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{\mathbb{P}(\mathcal{O}_{F}\oplus\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}\quad}$$\scriptstyle{\quad\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1},}$
(11)

where $\varphi$ is a separable isogeny of degree $2$ and $\mathcal{L}$ is an
order $2$ element in $\mathop{\mathrm{Pic}}\nolimits^{0}F$. It turns out that
the case (IV) is impossible because $\pi^{\prime}$ is of type $(2,2)$. Assume
that $\pi$ has a wild fiber $2D$. Then
$\mu:=\operatorname{ord}\mathcal{O}_{D}(D)=1$, which contradicts with $p\neq
2$ and (1). Hence, we conclude that the case (VI) is also impossible, and thus
the case (III) occurs.

When $p=2$ and $E$ is supersingular, then Lemma 2.2 implies that the cases
(III) and (VI) in Lemma 3.2 do not occur. Hence, the unique possibility is the
case (IV) by Lemma 4.1, that is, $S$ is an elliptic surface of type
$(2^{\alpha})$. We have the following diagram:

$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Fr}}$$\scriptstyle{\square}$$\textstyle{\mathbb{P}(\mathcal{E}_{2,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$

Since $\mathbb{P}(\mathcal{E}_{2,0})$ has the unique multiple fiber of
multiplicity $2$, it forces that $\alpha=1$ by (10).

When $p=2$ and $E$ is ordinary, we want to show the case (VI) occurs. It
suffices to exclude the cases (III) and (IV) by Lemma 4.1. Let us consider the
diagram (11) again. To obtain a contradiction, first assume that the case (IV)
occurs, that is, $\pi$ has the unique multiple fiber $2^{\alpha}D$ over a
point $Q$, and $2^{\alpha}D$ is a wild fiber. Since $\pi^{\prime}$ has no wild
fibers, Lemma 2.18 (i) yields that $\psi$ is branched over $Q$, thus
$\psi^{*}Q=2Q^{\prime}$ for some $Q^{\prime}\in\mathbb{P}^{1}$. But in this
case, it follows from Lemma 2.17 (ii) that $\pi^{\prime}$ has no multiple
fibers over points besides $Q^{\prime}$, which induces a contradiction since
$\pi^{\prime}$ is of type $(2,2)$.

Next, assume that the case (III) occurs, that is, $\pi$ has three multiple
fibers $2D_{i}$ over points $Q_{i}$ ($i=1,2,3$). Since $\psi$ is separable,
its ramification divisor has degree $2$ by the Hurwitz formula. Because $\psi$
is wildly ramified, [Har77, Proposition IV.2.2(c)] yields that there is the
unique branch point in $\mathbb{P}^{1}$. Hence, we may assume that $Q_{1}$ and
$Q_{2}$ are not branch points of $\psi$, and then
$\mathbb{P}(\mathcal{O}_{F}\oplus\mathcal{L})$ has at least $4$ multiple
fibers by Lemma 2.17 (iii). This is absurd. ∎

######  Remark 4.4.

It seems worthwhile to summarize the connection between known examples in the
literature and the result in Theorem 1.1.

  1. (i)

When $p>0$ and a vector bundle $\mathcal{E}$ is of the form
$\mathcal{O}_{E}\oplus\mathcal{L}$ with $\operatorname{ord}\mathcal{L}=p$ on
an ordinary elliptic curve $E$, then the elliptic surface in [KU85, Example
4.9] is isomorphic to $\mathbb{P}(\mathcal{E})$.

  2. (ii)

When $p>0,e=0$ and $\mathcal{E}$ is an indecomposable vector bundle on an
ordinary (resp. supersingular) elliptic curve $E$, then the elliptic surface
in [KU85, Example 4.7] (resp. [KU85, Example 4.8]) is isomorphic to
$\mathbb{P}(\mathcal{E})$. Here we use the fact that an elliptic curve which
is isogeny to an ordinary (resp. supersingular) elliptic curve is again
ordinary (resp. supersingular).

  3. (iii)

Let $E$ be an elliptic curve, and consider the morphism

$a\colon E\times E\to E\qquad(x_{1},x_{2})\mapsto x_{1}+x_{2}$

defined by the addition, and the morphism

$i\colon E\times E\to E\qquad(x_{1},x_{2})\mapsto x_{1}-x_{2}$

defined by the subtraction. Then we obtain the following commutative diagram:

Here we denote the symmetric product $\operatorname{Sym}^{2}(E)$ by $S$, and
define $q^{\prime}$ to be the quotient morphism by the involution
$(x_{1},x_{2})\mapsto(x_{2},x_{1})$, $q$ to be the quotient morphism by the
action $x\mapsto-x$.

Then we know that $q$ is ramified at the origin $O$ of $E$, and the points of
order $2$ in $E$. We can also see that $\pi$ is an elliptic fibration, and $f$
is a $\mathbb{P}^{1}$-bundle. Furthermore the equality

$\displaystyle\bigl{\\{}p\in\mathbb{P}^{1}\mid\pi^{*}(p)\mbox{ is a multiple
fiber}\bigr{\\}}$ $\displaystyle=$
$\displaystyle\bigl{\\{}q(p^{\prime})\in\mathbb{P}^{1}\mid\mbox{
$\operatorname{ord}p^{\prime}=2$ in $E$}\bigr{\\}}$

holds. We also see that every multiple fiber has multiplicity $2$.

Suppose first that $p\neq 2$. Then there are exactly $3$ points of order $2$
in $E$, hence $S$ fits into the case in Theorem 1.1 (ii-1).

Secondly suppose that $p=2$ and $E$ is an ordinary elliptic curve. Then since
there is a unique point of order $2$ in $E$, $S$ fits into the case in Theorem
1.1 (i-5). (We can actually check that the section $C_{0}$ of $f$ is the
reduction of the multiple fiber of $\pi$.)

Finally suppose that $p=2$ and $E$ is a supersingular elliptic curve. Then
since there are no points of order $2$ in $E$, $S$ fits into the case in
Theorem 1.1 (i-1).

In [Ati57, page 451], it is mentioned that $\operatorname{Sym}^{n}(E)$ is
given as a projective bundle $\mathbb{P}(\mathcal{E})$ on $E$, where
$\mathcal{E}$ is a vector bundle in $\mathcal{M}_{E}(n,n-1)$. This statement
seems incompatible with the above results in the case $p=2$.

##  5 Proof of Theorem 1.2

In this section we prove Theorem 1.2. For this purpose, we illustrate the
diagram (7) in Lemma 2.14 when $S$ has an elliptic fibration with multiple
fibers, namely in the cases (i-2), (i-5), (ii-1), (ii-2) and (ii-3) in Theorem
1.1. We also study the ramification of $\psi$, if $\psi$ is separable.

###  5.1 Case (i-2): $\mathbb{P}(\mathcal{O}_{E}\oplus\mathcal{L})$, $p\geq
0$, $m=\operatorname{ord}\mathcal{L}<\infty$

Take the quotient morphism

$E\cong\mathop{\mathrm{Pic}}\nolimits^{0}E\to
F:=\mathop{\mathrm{Pic}}\nolimits^{0}E/\left<\mathcal{L}\right>$

and define $\varphi$ to be its dual. Then we obtain the following diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{F\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{O}_{E}\oplus\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\quad}$$\scriptstyle{\quad\pi}$$\textstyle{\mathbb{P}^{1}}$

Theorem 1.1 says that $\pi$ is of type $(m,m)$. Suppose that $\pi$ has
multiple fibers over points $Q_{1},Q_{2}$. When $m$ is not divisible by $p$,
then Lemma 2.17 (ii) implies that $\psi$ is branched over $Q_{1},Q_{2}$. When
$E$ is ordinary and $m=p^{n}$ ($n>0$), then all of $\varphi$, $q$ and $\psi$
are purely inseparable of degree $p^{n}$.

###  5.2 Case (i-5): $\mathbb{P}(\mathcal{E}_{2,0})$, $p>0$

Take $\varphi\colon F\to E$ such as $\widehat{\operatorname{Fr}}=\varphi$.
Then we have the following diagram.

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{F\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{2,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$

We see in Theorem 1.1 that $\pi$ has the unique (wild) multiple fiber $pD$
over a point $Q_{1}\in\mathbb{P}^{1}$. If $E$ is ordinary, $\varphi$, $q$ and
$\psi$ are separable of degree $p$, and we can see from Lemma 2.17 (ii) that
$\psi$ is wildly ramified and $Q_{1}$ is the unique branch point of $\psi$. If
$E$ is supersingular, all of $\varphi$, $q$ and $\psi$ are purely inseparable
of degree $p$.

###  5.3 Cases (ii-1), (ii-3): $\mathbb{P}(\mathcal{E}_{Q})$, except $p=2$ and
$F$ is supersingular

In this case, there is a separable isogeny $\varphi\colon F\to E$ of degree
$2$. Then there is an order $2$ element
$\mathcal{L}\in\mathop{\mathrm{Pic}}\nolimits^{0}F$ and the following diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\square}$$\textstyle{\mathbb{P}(\mathcal{O}_{F}\oplus\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}\quad}$$\scriptstyle{\quad\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$

When $E$ is ordinary and $p=2$, $\pi$ has one wild fiber $2D_{1}$ over a point
$Q_{1}$ and one tame multiple fiber $2D_{2}$ over a point $Q_{2}$. Since
$\psi$ is wildly ramified at a single point and $\pi^{\prime}$ is of type
$(2,2)$, Lemma 2.18 (i) (or (ii)) implies that $Q_{1}$ is the unique branch
point of $\psi$.

When $p\neq 2$, $\pi$ has $3$ multiple fibers over points
$Q_{1},Q_{2},Q_{3}\in\mathbb{P}^{1}$. It follows from Lemma 2.18 (i) that
$\psi$ is branched at two points of $Q_{i}$’s.

###  5.4 Cases (ii-2), (ii-3): $\mathbb{P}(\mathcal{E}_{Q})$, $p=2$

In this case, we have the following:

$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Fr}}$$\scriptstyle{\square}$$\textstyle{\mathbb{P}(\mathcal{E}_{2,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{q}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Fr}}$$\textstyle{E}$$\textstyle{\mathbb{P}(\mathcal{E}_{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathbb{P}^{1}}$

Recall that $\pi^{\prime}$ has the unique wild fiber over a point
$Q^{\prime}\in\mathbb{P}^{1}$.

When $E$ is supersingular, $\pi$ also has the unique wild fiber over the point
$\operatorname{Fr}(Q^{\prime})$. When $E$ is ordinary, $\pi$ has one wild
fiber $2D_{1}$ over a point $Q_{1}$ and one tame multiple fiber $2D_{2}$ over
a point $Q_{2}$. By Lemma 2.18 (iii), we can see
$\operatorname{Fr}(Q^{\prime})=Q_{1}$.

###  5.5 Proof of Theorem 1.2

_Proof of Theorem 1.2._ Define $q\colon F\times\mathbb{P}^{1}\to S$ to be a
suitable successive finite étale covers and purely inseparable morphisms of
degree $p$ in the previous subsections. Then we obtain Theorem 1.2. ∎

######  Remark 5.1.

A wild fiber $mD$ is said to be _of strange type_ if $a=m-1$. Katsura and Ueno
show a reduction of a wild multiple fiber to a tame multiple fiber by a finite
cover if no wild fibers of strange type appear in the procedure of reduction
(see [KU85, §6 and p.330]).

Recall that if $E$ is supersingular, the elliptic fibration
$\pi\colon\mathbb{P}_{E}(\mathcal{E}_{Q})\to\mathbb{P}^{1}$ has a multiple
fiber satisfying $(a/m)=(1/2^{*}),$ namely $\pi$ has a wild fiber of strange
type. In §5.4, we obtain a reduction of a wild multiple fiber of strange type
to a tame multiple fiber.

## References

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Takato Togashi

TOHO GIRLS’ JUNIOR AND SENIOR HIGH SCHOOL, 1-41-1 Wakaba-cho, Chofu, Tokyo,
182-8510, Japan

e-mail address<EMAIL_ADDRESS>

Hokuto Uehara

Department of Mathematical Sciences, Graduate School of Science, Tokyo
Metropolitan University, 1-1 Minamiohsawa, Hachioji, Tokyo, 192-0397, Japan

e-mail address<EMAIL_ADDRESS>


# On line arrangements with odd multiplicities

Marco Golla<EMAIL_ADDRESS>CNRS and Laboratoire de Mathématiques
Jean Leray, Nantes Université, France

###### Abstract.

We give restrictions on the weak combinatorics of line arrangements with
singular points of odd multiplicity using topological arguments on locally-
flat spheres in 4-manifolds. As a corollary, we show that there is no line
arrangement comprising 13 lines and with only triple points.

A classical result due to Gallai asserts that any non-trivial real line
arrangement must have at least a double point. This fact, known as the
Sylvester–Gallai theorem, was strengthened and generalised to pseudoline
arrangements by Melchior [Mel41]. By contrast, complex line arrangements _can_
lack double points: for instance, the arrangement defined by the polynomial
$(x^{3}-y^{3})(y^{3}-z^{3})(z^{3}-x^{3})$, known as the _dual Hesse
arrangement_ , comprises 9 lines meeting at 12 triple points. Using the
Bogomolov–Miyaoka–Yau inequality, Hirzebruch has proven that every non-trivial
complex line arrangement must contain at least a double or a triple point
[Hir83]. It has been speculated that the pencil of degree $3$ and the dual
Hesse arrangement are the only complex line arrangements to have only triple
points [Urz22, Question 3].

######  Theorem 1.

Let $L$ be a locally-flat line arrangement of degree $d$ with only triple
points. Then $d\equiv 1,3,9,19\pmod{24}$.

Here by _locally-flat_ we mean that each line in the arrangement is a locally-
flatly embedded sphere of self-intersection 1, and that every two lines
intersect transversely and positively exactly once.

If an arrangement of degree $d$ has only triple points, then $d\equiv
1,3\pmod{6}$ by an easy counting argument. Therefore, Theorem 1 covers half of
the possible degrees in which these line arrangements can exist, including in
particular $d=13$ (which was the smallest previously unknown case). The case
$d=7$ is the Fano plane: that it is not realised as a complex line
arrangements requires an easy elementary argument; see [RS19] for the case of
locally-flat line arrangements.

######  Corollary 2.

There is no pseudoline arrangement, symplectic or complex line arrangement of
degree $13$ with only triple points. ∎

Recently, the corollary has been proved for complex line arrangements by
Kühne, Szemberg, and Tutaj-Gasińska [KST24]. They used the enumeration of all
possible matroids with the combinatorial type of a line arrangement of 13
lines and 26 triple points, and showed that none of them is realised over _any
field_.

Theorem 1 is a direct consequence of the following more general statement.

######  Theorem 3.

Let $L\subset\mathbb{CP}\vphantom{\mathbb{P}}^{2}$ be a locally-flat line
arrangement of total degree $d$ whose singular points have only odd
multiplicities. Call $t_{m}$ the number of singular points of $L$ of
multiplicity $m$. Then

$\sum(m-1)t_{m}\equiv d-1\pmod{16}.$

## Notation and background

Homology is taken with integer coefficients. The intersection product and the
signature of a 4-manifold are denoted by $\cdot$ and $\sigma$, respectively.
We refer to [GS99] or [Kir89] for background on 4-dimensional topology.

If $X$ is a blow-up of $\mathbb{CP}\vphantom{\mathbb{P}}^{2}$ at distinct
points, then we have a preferred basis of $H_{2}(X)$ given by the homology
classes of a complex line and of the exceptional divisors. We say that a class
in $H_{2}(X)$ is _characteristic_ if all of its coefficients are odd in this
basis. Equivalently, a class $A\in H_{2}(X)$ is characteristic if $A\cdot
B\equiv B\cdot B$ for each $B\in H_{2}(X)$.

######  Lemma 4.

Let $F_{1},\dots,F_{d}\subset X$ be a collection of disjoint, locally-flat
spheres in a blow-up $X$ of $\mathbb{CP}\vphantom{\mathbb{P}}^{2}$ at $t$
points, such that $\sum_{k}[F_{k}]\subset H_{2}(X)$ is a characteristic class.
Then

$\sum_{k}F_{k}\cdot F_{k}\equiv 1-t\pmod{16}.$

###### Proof.

Tube the spheres $F_{1},\dots,F_{d}$ together, to obtain a locally-flat sphere
$F$ with

$[F]=\sum_{k}[F_{k}]\in H_{2}(X)$

and

$F\cdot F=\sum_{k}F_{k}\cdot F_{k}.$

Now, $F$ is characteristic and, since $X$ is a smooth manifold, its
Kirby–Siebenmann invariant vanishes. Since $X$ is simply-connected, by the
locally-flat version the Kervaire–Milnor theorem [KM61], due to Lee and
Wilczyński [LW90], we have

$\sum_{k}F_{k}\cdot F_{k}=F\cdot F\equiv\sigma(X)=1-t\pmod{16},$

which proves the statement. ∎

###### Proof of Theorem 3.

Suppose that $L=L_{1}\cup\dots\cup L_{d}$ is a locally-flat realisation of an
arrangement as in the statement. Since the multiplicities of all its singular
points are odd, each line intersects an even number of lines, so $d$ is odd.
Call $p_{k}$ the number of singular points on the line $L_{k}$.

Blow up $\mathbb{CP}\vphantom{\mathbb{P}}^{2}$ at all the singular points of
$L$, to obtain a configuration of surfaces in a blow-up $X$ of
$\mathbb{CP}\vphantom{\mathbb{P}}^{2}$ at $t:=t_{3}+t_{5}+\dots+t_{d-2}+t_{d}$
points.

First, we observe that the strict transform of $L$ comprises $d$ pairwise
disjoint locally-flat spheres, $F_{1},\dots,F_{d}$, of self-intersection
$1-p_{1},\dots,1-p_{d}$.

Since we have only blown up at points of odd multiplicity, the sum of the
homology classes $[F_{1}],\dots,[F_{d}]$ is characteristic. By Lemma 4, we
have:

$d-\sum_{k}p_{k}=\sum_{k}(1-p_{k})\equiv 1-t\pmod{16}.$

Recall that $t=\sum_{m}t_{m}$ and note that

$\sum_{k}p_{k}=\sum_{m}mt_{m},$

so that

$\sum_{m}(m-1)t_{m}=\sum_{m}mt_{m}-t=\sum_{k}p_{k}-t\equiv d-1\pmod{16}.\qed$

######  Remark 5.

Theorem 3 generalises to configurations of rational curves whose singularities
have only odd multiplicities. It also extends to certain classes of
collections of spheres with conical singularities that are modelled over plane
curve singularities whose multiplicity sequence only has odd entries. In turn,
this generalises an obstruction for rational cuspidal curves described in
[GK23, Proposition 4.2.4].

###### Proof of Theorem 1.

If $L$ is a locally-flat realisation of a line arrangement of degree $d$ with
only triple points, $t_{3}=\frac{d(d-1)}{6}$, which implies that $d\equiv
0,1\pmod{3}$. To prove the theorem, it suffices to show that $d\equiv
1,3\pmod{8}$.

Applying Theorem 3 we obtain:

$2\cdot\frac{d(d-1)}{6}=(3-1)\cdot t_{3}\equiv d-1\pmod{16},$

which is equivalent to asking that $(d-1)(d-3)\equiv 0\pmod{16},$ which in
turn implies $d\equiv 1,3\pmod{8}$. ∎

## Acknowledgements

I would like to thank the participants of the workshop _Complex and symplectic
curve configurations_ for drawing my interest to this question, and especially
Piotr Pokora for several stimulating conversations and for his comments on an
earlier draft. I would also like to thank Paolo Aceto, Erwan Brugallé, Lukas
Kühne, Danny Ruberman, and Giancarlo Urzúa for their input.

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