# Bounds for the order of automorphism groups of cyclic covering fibrations of
an elliptic surface

Hiroto Akaike

###### Abstract.

We study automorphism groups of fibered surfaces for finite cyclic covering
fibrations of an elliptic surface. We estimate the order of a finite subgroup
of automorphism groups in terms of the genus of the base curve, the covering
degree and the square of the relative canonical divisor.

###### Key words and phrases:

fibration, automorphism group

###### 2020 Mathematics Subject Classification:

Primary 14J50, Secondary 14J10

## Introduction

We shall work over the complex number field $\mathbb{C}$. This is a
continuation of our previous work [1] studying the order of automorphism
groups of fibred algebraic surfaces.

The study of automorphism groups of projective varieties has a long history.
Hurwitz’s classical theorem states that if $C$ is a smooth projective curve of
genus $g(C)\geq 2$, then the order of the automorphism group is bounded by
$84(g(C)-1)$. Xiao attempted to generalize this result to minimal surfaces of
general type and gave the upper bound of the order of the automorphism groups
([10],[11]). While studies of the automorphism groups of surfaces exist in
absolute settings as above, we can consider a similar problem in relative
settings. For this purpose, we define basic notions of fibered surfaces and
its automorphisms.

Let $f:S\to B$ be a surjective morphism from a complex smooth projective
surface $S$ to a smooth projective curve $B$ with connected fibers. We call it
a fibration or a fibered surface of genus $g$ when a general fiber is a curve
of genus $g$. A fibration is called relatively minimal, when any $(-1)$-curve
is not contained in fibers. Here we call a smooth rational curve $C$ with
$C^{2}=-n$ a $(-n)$-curve. A fibration is called smooth when all fibers are
smooth, isotrivial when all of the smooth fibers are isomorphic, locally
trivial when it is smooth and isotrivial.  Assume that $f:S\to B$ is a
relatively minimal fibration of genus $g$. We denote by
$K_{f}=K_{S}-f^{\ast}K_{B}$ a relative canonical divisor.

An automorphism of the fibration $f:S\to B$ is a pair of automorphisms
$(\kappa_{S},\kappa_{B})\in\mathrm{Aut}(S)\times\mathrm{Aut}(B)$ satisfying
$f\circ\kappa_{S}=\kappa_{B}\circ f$, that is, the diagram

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{S}}$$\scriptstyle{f}$$\scriptstyle{\mathchar
13320\relax}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{B}}$$\textstyle{B}$

is commutative. We denote by $\mathrm{Aut}(f)$ the group of all automorphisms
of $f$.

Our main objects are primitive cyclic covering fibrations:

######  Definition 0.1 ([5]).

Let $f:S\to B$ be a relatively minimal fibration of genus $g\geq 2$. We call
it a primitive cyclic covering fibration of type $(g,1,n)$, when there are a
$($not necessarily relatively minimal $)$ fibration
$\tilde{\varphi}:\mathaccent 869{W}\to B$ of genus $1$ (i.e. elliptic surface)
and a classical $n$-cyclic covering

$\tilde{\theta}:\mathaccent 869{S}=\mathrm{Spec}_{\mathaccent
869{W}}\left(\bigoplus_{j=0}^{n-1}\mathcal{O}_{\mathaccent
869{W}}(-j\mathaccent 869{\mathfrak{d}})\right)\to\mathaccent 869{W}$

branched over a smooth (not necessarily irreducible) curve $\mathaccent
869{R}\in|n\mathaccent 869{\mathfrak{d}}|$ and $\mathaccent
869{\mathfrak{d}}\in\mathrm{Pic}(\mathaccent 869{W})$ such that $f$ is the
relatively minimal model of $\tilde{f}=\tilde{\varphi}\circ\tilde{\theta}$.

Though it seems restrictive, it should be noticed that this class contains
some classically important fibrations as subclasses. In fact, one sees
immediately that any hyperelliptic (resp. bielliptic) fibrations of genus $g$
are necessarily primitive cyclic covering fibrations of type $(g,0,2)$ (resp.
of type $(g,1,2)$ when $g\geq 6$). Here, a bielliptic curve is a smooth
projective curve admitting a double covering of an elliptic curve and a
fibration is called bielliptic if a general fiber is a bielliptic curve.
Bielliptic fibrations are an interesting class of fibrations. For instance, it
seemed that the lower bound of slopes of fibrations increases monotonically
with respect to its gonality. But it is not true. There exists a tetragonal
fibration and its slope is less than the lower bound of the slope of trigonal
fibrations. Such tetragonal fibrations are nothing more than bielliptic
fibrations ([3],[8]).

Arakawa [2] and later Chen [4] studied the automorphism groups for
hyperelliptic fibrations and gave the upper bound of the order of a finite
subgroup of $\mathrm{Aut}(f)$ in terms of $g$, $g(B)$ (the genus of $B$) and
$K_{f}^{2}$.  In [1], we considered the case that $h=0$, and gave an upper
bound on the order of $\mathrm{Aut}(f)$, generalizing results due to Arakawa
[2] and Chen [4] for hyperelliptic fibrations. We constructed an example that
shows our bound is almost optimal in some cases.

In this paper, we study the case that $h=1$, that is, cyclic covering
fibrations of elliptic surfaces. In order to state our results, we need some
further notation. Keeping the notation in Definition 0.1 with $h=1$, let
$\varphi:W\to B$ be the relatively minimal model of $\mathaccent
869{\varphi}:\mathaccent 869{W}\to B$, $R$ the image of $\mathaccent 869{R}$
by the natural birational map $\mathaccent 869{W}\to W$, and $\Gamma_{p}$ the
fiber of $\varphi$ over $p\in B$. If $\Gamma_{p}$ is a smooth fiber, fixing a
point $O_{p}\in\Gamma_{p}$, we can regard $\Gamma_{p}$ as an abelian group
with the unit element $O_{p}$. Put

$\delta:=\min\\{\sharp\mathrm{Aut}(\Gamma_{p},O_{p})\mid p\in B\text{ and
}\Gamma_{p}\text{ smooth }\\},$

where
$\mathrm{Aut}(\Gamma_{p},O_{p}):=\\{\kappa\in\mathrm{Aut}(\Gamma_{p})\mid\kappa(O_{p})=O_{p}\\}$.

######  Theorem 0.2 (Theorem 6.1).

Let $f:S\to B$ be a non-locally trivial primitive cyclic covering fibration of
type $(g,1,n)$ with
$g\geq\mathrm{max}\\{\frac{30n^{2}-47n+25}{n+1},\frac{7}{2}n(n-1)+1\\}$. Put

$\displaystyle\mu_{n}:=$ $\displaystyle\frac{12n^{2}\delta}{n^{2}-1}.$

Assume furthermore that when $g(B)=0$, $f$ has at least $3$ singular fibers.
Let $G$ be a finite subgroup of $\mathrm{Aut}(f)$. Then it holds

$\sharp G\leq\begin{cases}6(2g(B)-1)\mu_{n}K_{f}^{2}&(g(B)\geq 1),\\\
5\mu_{n}K_{f}^{2}&(g(B)=0).\end{cases}$

######  Theorem 0.3 (Theorem 6.2).

Let $f:S\to B$ be a non-locally trivial primitive cyclic covering fibration of
type $(g,1,n)$ with
$g\geq\mathrm{max}\\{\frac{30n^{2}-47n+25}{n+1},\frac{7}{2}n(n-1)+1\\}$. Put

$\displaystyle\mu_{n}^{\prime}:=\frac{6n^{2}\delta}{(n-1)(5n-4)}.$

Assume furthermore that the branch locus $R$ has singular points on at least
three $($resp. one$)$ fibers when $g(B)=0$ $($resp. $g(B)\geq 1$$)$. Let $G$
be a finite subgroup of $\mathrm{Aut}(f)$. Then it holds

$\sharp G\leq\begin{cases}6(2g(B)-1)\mu_{n}^{\prime}K_{f}^{2}&(g(B)\geq 1),\\\
5\mu_{n}^{\prime}K_{f}^{2}&(g(B)=0).\end{cases}$

######  Theorem 0.4 (Corollary 6.3).

Let $f:S\to B$ be a non-locally trivial bielliptic fibration with $g\geq 17$.
Assume furthermore that the branch locus $R$ has singular points on at least
three $($resp. one$)$ fibers when $g(B)=0$ $($resp. $g(B)\geq 1$$)$. Let $G$
be a finite subgroup of $\mathrm{Aut}(f)$. Then it holds

$\sharp G\leq\begin{cases}24\delta(2g(B)-1)K_{f}^{2}&(g(B)\geq 1),\\\ 20\delta
K_{f}^{2}&(g(B)=0).\end{cases}$

######  Theorem 0.5 (Corollary 6.4).

Let $f:S\to B$ be a non-locally trivial primitive cyclic covering fibration of
type $(g,1,n)$ with
$g\geq\mathrm{max}\\{\frac{30n^{2}-47n+25}{n+1},\frac{7}{2}n(n-1)+1\\}$. Put
$\mathrm{Aut}(S/B):=\\{(\kappa_{S},\mathrm{id}_{B})\in\mathrm{Aut}(f)\\}$.
Assume that the branch locus $R$ has a singular point. Then it holds

$\sharp\mathrm{Aut}(S/B)\leq\frac{6n^{2}\delta}{(n-1)(5n-4)}K_{f}^{2}.$

We state the outline of the proof. Let $G$ be a finite subgroup of
$\mathrm{Aut}(f)$. Then $G$ can be expressed as the extension of its
horizontal part $H$ by its vertical part $K$, that is, $1\to K\to G\to H\to 1$
(exact). We can study $H$ by using known results for the automorphism groups
of curves. Hence it suffices to estimate $K$. Then $K$ induces a subgroup
$\mathaccent 869{K}$ of the automorphism group of the elliptic surface
$\varphi:W\to B$ preserving the branch locus $R$ and our task is reduced to
estimating the order of $\mathaccent 869{K}$. For this purpose, we use the
localization of the invariant $K_{f}^{2}$. It is known that $K_{f}^{2}$ can be
localized to fibers of $f$ ([6]). We refine the localized $K_{f}^{2}$ further
by using the quantity defined at a point on a fiber, and obtain the new
expression of it in Proposition 2.2. Then, we estimate the order of
$\mathaccent 869{K}$ from above with the localized $K_{f}^{2}$ multiplied by
an explicit function in $g$ and $n$ (Propositions 5.3, 5.4 and 5.5).

In Section 1, we recall the basic properties of primitive cyclic covering
fibrations mainly due to [5]. In Section 2, we recast Section 5 of [5] and
consider the refined localization of $K_{f}^{2}$. In Section 3, we summarize
the basic properties of elliptic surfaces as group manifolds. In Section 4, we
descend the automorphism group to the elliptic surface $W$ and analyze its
action. In Section 5, we estimate the order of $\mathaccent 869{K}$. In
Section 6, we show our main result, Theorem 6.1, 6.2, Corollary 6.3 and 6.4.
Section 7 is devoted to constructing an example of bielliptic fibrations with
a large automorphism group.

Acknowledgememt

The author is deeply grateful to Professor Kazuhiro Konno for his valuable
advices and supports. The author is particularly grateful to Professor Osamu
Fujino and Professor Kento Fujita for their continuous supports. The author
also thanks to Doctor Makoto Enokizono for his precious advices. The research
is supported by JSPS KAKENHI No. 20J20055.

###### Contents

  1. 1 Primitive cyclic covering fibrations
  2. 2 Localization of $K_{f}^{2}$
  3. 3 Automorphism groups of elliptic surfaces
  4. 4 Automorphism of fibered surface
  5. 5 Estimation of the order of $\sharp\mathaccent 869{K}$
    1. 5.1 The proof of Proposition 5.4
    2. 5.2 The proof of Proposition 5.5
  6. 6 Upper bound of the order
  7. 7 Example

##  1\. Primitive cyclic covering fibrations

We recall the basic properties of primitive cyclic covering fibrations, most
of which can be found in [5].

######  Definition 1.1.

Let $f:S\to B$ be a relatively minimal fibration of genus $g\geq 2$. We call
it a primitive cyclic covering fibration of type $(g,1,n)$, when there are a
$($not necessarily relatively minimal $)$ fibration
$\tilde{\varphi}:\mathaccent 869{W}\to B$ of genus $1$ (i.e. elliptic surface)
and a classical $n$-cyclic covering

$\tilde{\theta}:\mathaccent 869{S}=\mathrm{Spec}_{\mathaccent
869{W}}\left(\bigoplus_{j=0}^{n-1}\mathcal{O}_{\mathaccent
869{W}}(-j\mathaccent 869{\mathfrak{d}})\right)\to\mathaccent 869{W}$

branched over a smooth (not necessarily irreducible) curve $\mathaccent
869{R}\in|n\mathaccent 869{\mathfrak{d}}|$ and $\mathaccent
869{\mathfrak{d}}\in\mathrm{Pic}(\mathaccent 869{W})$ such that $f$ is the
relatively minimal model of $\tilde{f}=\tilde{\varphi}\circ\tilde{\theta}$.

In addition, we employ the following notations. We denote by
$\Sigma=\langle\tilde{\sigma}\rangle$ the covering transformation group of
$\tilde{\theta}$ and by $\varphi:W\to B$ the relatively minimal model of
$\tilde{\varphi}:\mathaccent 869{W}\to B$ with the natural contraction map
$\tilde{\psi}:\mathaccent 869{W}\to W$. Furthermore, $\mathaccent 869{F}$,
$F$, $\mathaccent 869{\Gamma}$ and $\Gamma$ will denote general fibers of
$\tilde{f}$, $f$, $\tilde{\varphi}$ and $\varphi$, respectively. We write
fibers of $\tilde{f}$, $f$, $\tilde{\varphi}$ and $\varphi$ over $p$ as
$\mathaccent 869{F}_{p}$, $F_{p}$, $\mathaccent 869{\Gamma}_{p}$ and
$\Gamma_{p}$, respectively.

######  Remark 1.2.

We note important properties of primitive cyclic covering fibrations in the
following, which can be found in [5].

  * •

There exists an automorphism $\sigma\in\mathrm{Aut}(S)$ which satisfies the
following:

    * –

The natural morphism $\mathaccent 869{S}\to S$ is a minimal succession of
blowing-ups that resolves all isolated fixed points of $\sigma$.

    * –

The automorphism $\mathaccent 869{\sigma}$ is induced by $\sigma$.

  * •

Let $\mathaccent 869{R}_{v}$ be the sum of $\mathaccent 869{\varphi}$-vertical
components of $\mathaccent 869{R}$. The self-intersection number of each
irreducible component of $\mathaccent 869{R}_{v}$ is equals to $-an$ for some
positive integer $a$.

  * •

Any $\mathaccent 869{\varphi}$-vertical $(-1)$ curve intersects $\mathaccent
869{R}$.

From now on, we let $f:S\to B$ be a primitive cyclic covering fibration of
type $(g,1,n)$ and freely use the above notations and conventions.

Since the restriction map $\tilde{\theta}|_{\mathaccent 869{F}}:\mathaccent
869{F}\to\mathaccent 869{\Gamma}$ is a classical $n$-cyclic covering branched
over $\mathaccent 869{R}\cap\mathaccent 869{\Gamma}$, the Hurwitz formula for
$\tilde{\theta}|_{\mathaccent 869{F}}$ gives us

(1.1) $r:=\mathaccent 869{R}\mathaccent
869{\Gamma}=\frac{2\bigl{(}g-1\bigr{)}}{n-1}.$

From $\mathaccent 869{R}\in|n\mathaccent 869{\mathfrak{d}}|$, it follows that
$r$ is a multiple of $n$.

Since $\tilde{\psi}:\mathaccent 869{W}\to W$ is a composite of blowing-ups, we
can write $\tilde{\psi}=\psi_{1}\circ\cdots\psi_{N}$, where $\psi_{i}:W_{i}\to
W_{i-1}$ denotes the blowing-up at $z_{i}\in W_{i-1}\;(i=1,\cdots,N)$,
$W_{0}=W$ and $W_{N}=\mathaccent 869{W}$. We define reduced curves $R_{i}$
inductively as $R_{i-1}=(\psi_{i})_{\ast}R_{i}$ starting from
$R_{N}=\mathaccent 869{R}$ down to $R_{0}=R$. We call $R_{i}$ the branch locus
on $W_{i}$. We also put $E_{i}=\psi_{i}^{-1}(z_{i})$ and
$m_{i}=\mathrm{mult}_{z_{i}}R_{i-1}\;(i=1,\cdots,N)$.

######  Lemma 1.3 ([5], Lemma 1.5).

In the above situation, the following hold for any $i=1,\cdots,N$.

  * $(1)$

Either $m_{i}\in n\mathbb{Z}_{\geq 1}$ or $n\mathbb{Z}_{\geq 1}+1$, where
$\mathbb{Z}_{\geq 1}$ is the set of positive integers. Furthermore, $m_{i}\in
n\mathbb{Z}_{\geq 1}$ if and only if $E_{i}$ is not contained in $R_{i}$.

  * $(2)$

$R_{i}={\psi}_{i}^{\ast}R_{i-1}-n[\frac{m_{i}}{n}]E_{i}$, where $[t]$ denotes
the greatest integer not exceeding $t$.

  * $(3)$

There exists a $\mathfrak{d}_{i}\in\mathrm{Pic}(P_{i})$ such that
$\mathfrak{d}_{i}=\psi_{i}^{\ast}\mathfrak{d}_{i-1}-[\frac{m_{i}}{n}]E_{i}$
and $R_{i}\sim n\mathfrak{d}_{i}$, $\mathfrak{d}_{N}=\mathaccent
869{\mathfrak{d}}$.

We say that a singular point of $R_{i}$ is of type $n\mathbb{Z}$ (resp.
$n\mathbb{Z}+1$) if its multiplicity is in $n\mathbb{Z}_{\geq 1}$ (resp.
$n\mathbb{Z}_{\geq 1}+1$).

##  2\. Localization of $K_{f}^{2}$

We recall the argument in Section 5 of [5] and give localizations of
$K_{f}^{2}$. Let $f:S\to B$ be a primitive cyclic covering fibration of type
$(g,1,n)$.

First of all, let us recall the singularity indices introduced in [5]. For any
fixed $p\in B$, we consider all singular points of the branch loci
$R=R_{0},\cdots,R_{N-1}$ over $\Gamma_{p}$. For any positive integer $k$, we
let $\alpha_{k}(\Gamma_{p})$ be the number of singular points of multiplicity
either $kn$ or $kn+1$ among them. We put $\alpha_{k}:=\sum_{p\in
B}\alpha_{k}(\Gamma_{p})$ and call it the $k$-th singularity index of the
fibration. For an effective vertical divisor $T$ and $p\in B$, we denote by
$T(p)$ the biggest subdivisor of $T$ whose support is in the fiber over $p$.
Then $T=\sum_{p\in B}T(p)$. We sometimes write $\sharp T$ as the number of
irreducible components of $T$.

Let $j_{b,a}(\Gamma_{p})$ be the number of irreducible curves with genus $b$
and self-intersection number $-an$ contained in $\mathaccent 869{R}_{v}(p)$.

We introduce the following indices:

$j_{\bullet,a}(\Gamma_{p}):=\sum_{b\geq 0}j_{b,a}(\Gamma_{p}),\quad
j_{b,\bullet}(\Gamma_{p}):=\sum_{a\geq 0}j_{b,a}(\Gamma_{p}),\quad
j_{0,1}:=\sum_{p\in B}j_{0,1}(\Gamma_{p}).$

Let $A$ be the sum of all $\mathaccent 869{\varphi}$-vertical $(-n)$-curves
contained in $\mathaccent 869{R}$ and put $\mathaccent 869{R}_{0}:=\mathaccent
869{R}-A$. Then the $0$-th singularity index is defined by
$\alpha_{0}:=(K_{\tilde{\varphi}}+\mathaccent 869{R}_{0})\mathaccent
869{R}_{0}$. Put

$\displaystyle\alpha_{0}^{+}(\Gamma_{p})$ $\displaystyle:=\mathaccent
869{R}_{h}\mathaccent 869{\Gamma}_{p}-\sharp(\mathrm{Supp}(\mathaccent
869{R}_{h})\cap\mathrm{Supp}(\mathaccent 869{\Gamma}_{p})),$
$\displaystyle\alpha_{0}(\Gamma_{p})$
$\displaystyle:=\alpha_{0}^{+}(\Gamma_{p})-2\sum_{a\geq
2}j_{0,a}(\Gamma_{p}),$

where $\mathaccent 869{R}_{h}$ is the $\mathaccent 869{\varphi}$-horizontal
part of $\mathaccent 869{R}$. We note that $\alpha_{0}=\sum_{p\in
B}\alpha_{0}(\Gamma_{p})$. We introduce some indices depending only on
$\Gamma_{p}$.

  * $\chi_{\varphi}(\Gamma_{p}):=e(\Gamma_{p})/12$ where $e(\Gamma_{p})$ is the topological Euler number of $\Gamma_{p}$

  * $\nu(\Gamma_{p}):=1-1/l(\Gamma_{p})$ where $l(\Gamma_{p})$ is a multiplicity of $\Gamma_{p}$

  * $\nu:=\sum_{p\in B}\nu(\Gamma_{p})$

Then it holds that $e_{\varphi}=\sum_{p\in B}e(\Gamma_{p})$ and
$\chi_{\varphi}=\sum_{p\in B}\chi_{\varphi}(\Gamma_{p})$.

######  Lemma 2.1 ([6], Lemma 2.1).

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,1,n)$.
Then it holds

$\displaystyle K_{f}^{2}$ $\displaystyle=\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\alpha_{k}+\frac{(n-1)^{2}}{n}(\alpha_{0}-2j_{0,1})+\frac{n^{2}-1}{n}r(\chi_{\varphi}+\nu)+j_{0,1}.$

Hence if we put

$\displaystyle K_{f}^{2}(\Gamma_{p})=$ $\displaystyle\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\alpha_{k}(\Gamma_{p})+\frac{(n-1)^{2}}{n}(\alpha_{0}(\Gamma_{p})-2j_{0,1}(\Gamma_{p}))$
$\displaystyle+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))+j_{0,1}(\Gamma_{p}).$

for $p\in B$, then we get $K_{f}^{2}=\sum_{p\in B}K_{f}^{2}(\Gamma_{p})$.

Let $\mathaccent 866{\varphi}:\mathaccent 866{W}\to B$ be any intermediate
elliptic surface between $\mathaccent 869{W}$ and $W$, and regard $\mathaccent
869{\psi}:\mathaccent 869{W}\to W$ as the composite of the natural birational
morphisms $\mathaccent 866{\psi}:\mathaccent 869{W}\to\mathaccent 866{W}$ and
$\check{\psi}:\mathaccent 866{W}\to W$.

$\textstyle{\mathaccent
869{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\psi}}$$\scriptstyle{\mathaccent 869{\varphi}}$$\textstyle{\mathaccent
866{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\varphi}}$$\scriptstyle{\check{\psi}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{B}$

We put $\mathaccent 866{R}=\mathaccent 866{\psi}_{*}\mathaccent 869{R}$. The
fiber of $\mathaccent 866{\varphi}$ over $p\in B$ will be denoted by
$\mathaccent 866{\Gamma}_{p}$.

Let $\mathaccent 866{\alpha}_{k}(\Gamma_{p})$ and
$\check{\alpha}_{k}(\Gamma_{p})$ be the number of the singular points
contributing to $\alpha_{k}(\Gamma_{p})$ appearing in $\mathaccent 866{\psi}$
and $\check{\psi}$, respectively. We note that the number of singular points
of $\mathaccent 866{R}$ on $\mathaccent 866{\Gamma}_{p}$ is counted by
$\mathaccent 866{\alpha}_{k}(\Gamma_{p})$. Then we have
$\alpha_{k}(\Gamma_{p})=\mathaccent
866{\alpha}_{k}(\Gamma_{p})+\check{\alpha}_{k}(\Gamma_{p})$.

We note that an arbitrary irreducible component of $\mathaccent 869{R}_{v}(p)$
is a proper transform of either an exceptional curve appearing from
$\psi:\mathaccent 869{W}\to W$ or an irreducible component of $\Gamma_{p}$.
Let $\mathaccent 866{j}_{0,a}(\Gamma_{p})$ (resp.
$\check{j}_{0,a}(\Gamma_{p})$) be the number of irreducible components of
$\mathaccent 869{R}_{v}(p)$ contributing to $j_{0,a}(\Gamma_{p})$ which are
proper transforms of exceptional curves appearing from $\mathaccent 866{\psi}$
(resp. $\check{\psi}$). Let $j_{0,a}^{\prime}(\Gamma_{p})$ be the number of
irreducible components contributing to $j_{0,a}(\Gamma_{p})$ which are proper
transforms of irreducible components of $\Gamma_{p}$. Then we have
$j_{0,a}(\Gamma_{p})=\mathaccent
866{j}_{0,a}(\Gamma_{p})+\check{j}_{0,a}(\Gamma_{p})+j_{0,a}^{\prime}(\Gamma_{p})$.
Put $\mathaccent 866{j}_{0,\bullet}(\Gamma_{p}):=\sum_{a\geq 1}\mathaccent
866{j}_{0,a}(\Gamma_{p})$, $\check{j}_{0,\bullet}(\Gamma_{p}):=\sum_{a\geq
1}\check{j}_{0,a}(\Gamma_{p})$ and
$j_{0,\bullet}^{\prime}(\Gamma_{p}):=\sum_{a\geq
1}j_{0,a}^{\prime}(\Gamma_{p})$.

Choose and fix $p\in B$ and $\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}$. We consider the vertical part $\mathaccent
869{R}_{v}=\sum_{p\in B}\mathaccent 869{R}_{v}(p)$ of $\mathaccent 869{R}$
with respect to $\mathaccent 869{\varphi}:\mathaccent 869{W}\to B$. We let
$\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$ be the biggest subdivisor of
$\mathaccent 869{R}_{v}(p)$ contracted to $\mathaccent 866{z}$ by $\mathaccent
866{\psi}$. Note that we have $\mathaccent 869{R}_{v}(p)_{\mathaccent
866{z}}\neq 0$ only when there exists a singular point (of the branch loci) of
$n\mathbb{Z}+1$ type and which is infinitely near to $\mathaccent 866{z}$
(including $\mathaccent 866{z}$ itself). Note also that $\mathaccent
869{R}_{v}(p)_{\mathaccent 866{z}}$ is a disjoint union of non-singular
rational curves each of which is a $(-an)$-curve for some positive integer
$a$.

To decompose $\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$, we define a
family $\\{L_{i}\\}_{i}$ consisting of vertical irreducible curves in
$\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$ as follows.

  * (i)

Choose and fix a $(-1)$-curve $E_{1}$ over $\mathaccent 866{z}$ on an elliptic
surface between $\mathaccent 869{W}$ and $\mathaccent 866{W}$, and let $L_{1}$
be the proper transform of $E_{1}$ on $\mathaccent 869{W}$.

  * (ii)

For $i\geq 2$, $L_{i}$ is the proper transform of an exceptional $(-1)$-curve
$E_{i}$ that is contracted to a point $x_{i}$ in $E_{k}$ or its proper
transform for some $k<i$.

Put $\mathfrak{L}:=\\{\\{L_{i}\\}_{i}\mid$ $\\{L_{i}\\}_{i}$ satisfies (i) and
(ii)$\\}$. If $\\{L_{i}\\}_{i}$ and $\\{L^{\prime}_{j}\\}_{j}$
($\\{L_{i}\\}_{i}$,$\\{L^{\prime}_{j}\\}_{j}\in\mathfrak{L}$) have a common
curve, one can show the union of them $\\{L_{i},L^{\prime}_{j}\\}_{i,j}$ is in
$\mathfrak{L}$. We define a partial order
$\\{L_{i}\\}_{i}\leq\\{L^{\prime}_{j}\\}_{j}$ if
$\\{L_{i}\\}_{i}\subset\\{L^{\prime}_{j}\\}_{j}$. Then $(\mathfrak{L},\leq)$
is a partial order set. Applying Zorn’s lemma, one should check that every
chain admits the upper bound. Hence there exist maximal elements
$\\{L_{1,k}\\}_{k=1}^{k_{1}},\cdots,\\{L_{\eta_{\mathaccent
866{z}},k}\\}_{k=1}^{k_{\eta_{\mathaccent 866{z}}}}$, where $\eta_{\mathaccent
866{z}}$ is the number of maximal elements. Any two of
$\\{L_{1,k}\\}_{k=1}^{k_{1}},\cdots,\\{L_{\eta_{\mathaccent
866{z}},k}\\}_{k=1}^{k_{\eta_{\mathaccent 866{z}}}}$ have no common curves. We
put $D_{t}:=\sum_{k=1}^{k_{t}}L_{t,k}$ for $t=1,\cdots,\eta_{\mathaccent
866{z}}$.

We can describe $\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$ by using (the
above) $D_{t}$’s as follows. The divisor $\mathaccent
869{R}_{v}(p)_{\mathaccent 866{z}}$ is decomposed into a disjoint sum
consisting of such sums uniquely. We denote as

(2.1) $\displaystyle{\color[rgb]{0,0,0}\mathaccent 869{R}_{v}(p)_{\mathaccent
866{z}}=D_{1}+\cdots+D_{\eta_{\mathaccent 866{z}}},\quad
D_{t}=\sum_{k=1}^{k_{t}}L_{t,k}}.$

Let $C_{t,k}$ be the exceptional $(-1)$-curve whose proper transform on
$\mathaccent 869{W}$ is $L_{t,k}$. We let $\iota^{t}(\Gamma_{p})_{\mathaccent
866{z}}$ and $\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ denote the numbers
of singular points of the branch loci $R=R_{0},\cdots,R_{N-1}$ over
$\mathaccent 866{z}$ of types $n\mathbb{Z}$ and $n\mathbb{Z}+1$, respectively,
at which the proper transforms of two curves in $\\{C_{t,k}\\}_{k=1}^{k_{t}}$
meet, and put $\iota(\Gamma_{p})_{\mathaccent
866{z}}=\sum_{t=1}^{\eta_{\mathaccent
866{z}}}\iota^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ and
$\kappa(\Gamma_{p})_{\mathaccent 866{z}}=\sum_{t=1}^{\eta_{\mathaccent
866{z}}}\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}}$. By the definition of
$\iota^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ and
$\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}}$, we note that
$\iota(\Gamma_{p})_{\mathaccent 866{z}}=\kappa(\Gamma_{p})_{\mathaccent
866{z}}=0$ if $\mathaccent 866{R}$ is smooth at $\mathaccent 866{z}$.

Furthermore, we localize some indices to points on $\mathaccent
866{\Gamma}_{p}$. Let $\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}$ be the number of singular points contributing to $\mathaccent
866{\alpha}_{k}(\Gamma_{p})$ which is infinitely near to $\mathaccent 866{z}$.
Let $\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}$ (resp.
$\mathaccent 866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ for
$t=1,\cdots\eta_{\mathaccent 866{z}}$) be the number of irreducible components
of $\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$ (resp. $D_{t}$)
contributing to $\mathaccent 866{j}_{0,a}$. Put $\mathaccent
866{j}_{0,\bullet}(\Gamma_{p})_{\mathaccent 866{z}}:=\sum_{a\geq 1}\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}$ and $\mathaccent
866{j}_{0,\bullet}^{t}(\Gamma_{p})_{\mathaccent 866{z}}:=\sum_{a\geq
1}\mathaccent 866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ for each
$t=1,\cdots\eta_{\mathaccent 866{z}}$. Note that $\mathaccent
866{j}_{0,\bullet}(\Gamma_{p})_{\mathaccent 866{z}}=\sum_{t}\mathaccent
866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent 866{z}}$. We have $\mathaccent
866{\alpha}_{k}(\Gamma_{p})=\sum_{\mathaccent 866{z}\in\Gamma_{p}}\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$ and $\mathaccent
866{j}_{0,a}(\Gamma_{p})=\sum_{\mathaccent 866{z}\in\Gamma_{p}}\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}$. We put

(2.2) $\displaystyle\alpha_{0}^{+}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle:=\begin{cases}(\mathaccent 869{R}_{h},\mathaccent
869{\Gamma}_{p,\mathaccent 866{z}})-\sharp\left(\mathrm{Supp}(\mathaccent
869{R}_{h})\cap\mathrm{Supp}(\mathaccent 869{\Gamma}_{p,\mathaccent
866{z}})\right)&({\rm if\;}\mathaccent
866{R}{\rm\;is\;singular\;at\;}\mathaccent 866{z}\;),\\\ \\\ \left({\rm
The\;ramification\;index\;of\;}\mathaccent 866{\varphi}|_{\mathaccent
866{R}_{h}}:\mathaccent 866{R}_{h}\to B{\rm\;at\;}\mathaccent
866{z}\right)-1&({\rm if\;}\mathaccent
866{R}{\rm\;is\;smooth\;at\;}\mathaccent 866{z}\;),\\\ \\\ 0&({\rm
if\;}\mathaccent 866{z}\not\in\mathaccent 866{R}),\end{cases}$
$\displaystyle\mathaccent 866{\alpha}_{0}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle:=\alpha_{0}^{+}(\Gamma_{p})_{\mathaccent 866{z}}-2\sum_{a\geq
2}\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}},$

where $\mathaccent 869{\Gamma}_{p,\mathaccent 866{z}}$ is the biggest
subdivisor of $\mathaccent 869{\Gamma}_{p}$ which is contracted to
$\mathaccent 866{z}$ by $\mathaccent 866{\psi}$.

We put

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}:=$
$\displaystyle\sum_{k\geq 1}\left((n^{2}-1)k-n\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}+\frac{(n-1)^{2}}{n}(\mathaccent
866{\alpha}_{0}(\Gamma_{p})_{\mathaccent 866{z}}-2\mathaccent
866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}})+\mathaccent
866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}}.$

Then we get the following:

######  Proposition 2.2.

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,1,n)$. Let
$\mathaccent 869{\psi}=\check{\psi}\circ\mathaccent 866{\psi}$ be an arbitrary
decomposition of $\mathaccent 869{\psi}:\mathaccent 869{W}\to W$ so that the
commutative diagram

$\textstyle{\mathaccent
869{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\psi}}$$\scriptstyle{\mathaccent 869{\varphi}}$$\textstyle{\mathaccent
866{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\varphi}}$$\scriptstyle{\check{\psi}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{B.}$

Denote a fiber of $\mathaccent 866{\varphi}$ over $p\in B$ by $\mathaccent
866{\Gamma}_{p}$. Then it holds that

$\displaystyle K_{f}^{2}(\Gamma_{p})=$ $\displaystyle\sum_{\mathaccent
866{z}\in\mathaccent 866{\Gamma}_{p}}K_{f}^{2}(\Gamma_{p})_{\mathaccent
866{z}}+\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p})-2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p})+j_{0,\bullet}^{\prime}(\Gamma_{p})\right)$
$\displaystyle+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))+\check{j}_{0,1}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p}).$

###### Proof.

By definition, we have

(2.3) $\displaystyle\alpha_{k}(\Gamma_{p})=\sum_{\mathaccent
866{z}\in\mathaccent 866{\Gamma}_{p}}\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}+\check{\alpha}_{k}(\Gamma_{p}),\quad
j_{0,a}(\Gamma_{p})=\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent
866{z}}+\check{j}_{0,a}(\Gamma_{p})+j_{0,a}^{\prime}(\Gamma_{p}).$

By (2.3), we have

$\displaystyle K_{f}^{2}(\Gamma_{p})=$ $\displaystyle\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\alpha_{k}(\Gamma_{p})+\frac{(n-1)^{2}}{n}(\alpha_{0}(\Gamma_{p})-2j_{0,1}(\Gamma_{p}))$
$\displaystyle+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))+j_{0,1}(\Gamma_{p})$
$\displaystyle=$ $\displaystyle\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\left(\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}+\check{\alpha}_{k}(\Gamma_{p})\right)$
$\displaystyle+\frac{(n-1)^{2}}{n}\biggl{(}\sum_{\mathaccent
866{z}\in\mathaccent 866{\Gamma}_{p}}\mathaccent
866{\alpha}_{0}(\Gamma_{p})_{\mathaccent 866{z}}-2\sum_{a\geq
2}\left(\check{j}_{0,a}(\Gamma_{0})+j_{0,a}^{\prime}(\Gamma_{p})\right)\biggr{.}$
$\displaystyle\biggl{.}-2\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent
866{z}}-2\left(\check{j}_{0,1}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p})\right)\biggr{)}$
$\displaystyle+\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent
866{z}}+\check{j}_{0,1}(\Gamma_{p})+j^{\prime}_{0,1}(\Gamma_{p})+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))$
$\displaystyle=$ $\displaystyle\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}+\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p})$
$\displaystyle-2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p})+j_{0,\bullet}^{\prime}(\Gamma_{p})\right)$
$\displaystyle+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))+\check{j}_{0,1}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p})$

∎

We consider and recast Lemma 5.2 in [5].

######  Lemma 2.3.

The following hold:

$(1)$ $\iota(\Gamma_{p})_{\mathaccent 866{z}}=\mathaccent
866{j}_{0,\bullet}(\Gamma_{p})_{\mathaccent 866{z}}-\eta_{\mathaccent
866{z}}$.

$(2)$ $\alpha_{0}^{+}(\Gamma_{p})_{\mathaccent
866{z}}\geq(n-2)\left(\mathaccent 866{j}_{0,\bullet}(\Gamma_{p})_{\mathaccent
866{z}}-\eta_{\mathaccent 866{z}}+2\kappa(\Gamma_{p})_{\mathaccent
866{z}}\right)$.

$(3)$ $\displaystyle{\sum_{k\geq 1}\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}\geq\sum_{a\geq
1}(an-2)\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent
866{z}}+2\eta_{\mathaccent 866{z}}-\kappa(\Gamma_{p})_{\mathaccent 866{z}}}$.

######  Proof.

For each $t$, we consider the graph $\mathbb{G}_{t}$ corresponding to $D_{t}$
as follows. The vertex set $V(\mathbb{G}_{t})$ and the edge set
$E(\mathbb{G}_{t})$ are respectively the sets of symbols
$\\{v_{t,k}\\}_{k=1}^{\mathaccent
866{j}_{0,\bullet}^{t}(\Gamma_{p})_{\mathaccent 866{z}}}$ and
$\\{e_{x}\\}_{x}$, where $x$ runs over all the singular points contributing to
$\iota^{t}(\Gamma_{p})_{\mathaccent 866{z}}$. If $C_{t,k}$ or a proper
transform of it meets a proper transform of $C_{t,k^{\prime}}$ at the singular
point $x$ of type $n\mathbb{Z}$, the edge $e_{x}$ connects $v_{t,k}$ and
$v_{t,k^{\prime}}$. By the definition of $D_{t}$, we see that $\mathbb{G}_{t}$
is a connected tree. Counting the numbers of vertices and edges, we have
$\iota^{t}(\Gamma_{p})_{\mathaccent 866{z}}=\mathaccent
866{j}_{0,\bullet}^{t}(\Gamma_{p})_{\mathaccent 866{z}}-1$. Thus, we get (1)
by summing it up for $t$.

When the component $L_{t,k}$ of $D_{t}$ is a $(-an)$-curve, it is obtained by
blowing $C_{t,k}$ up $an-1$ times. We recall that $\mathaccent
866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ is the number of
irreducible components of $D_{t}$ with self-intersection number $-an$. By
$D_{t}:=\sum_{k=1}^{k_{t}}L_{t,k}$, we need blowing-ups $\sum_{a\geq
1}(an-1)\mathaccent 866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent 866{z}}$ times
to get $D_{t}$, disregarding overlaps. Taking into the account the duplication
and the first blowing-up creating the $(-1)$-curve for $L_{t,1}$, we see that
the number of blowing-ups to obtain $D_{t}$ is not less than

$\sum_{a\geq 1}(an-1)\mathaccent 866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent
866{z}}-\iota^{t}(\Gamma_{p})_{\mathaccent
866{z}}-\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}}+1\\\ =\sum_{a\geq
2}(an-2)\mathaccent 866{j}_{0,a}^{t}(\Gamma_{p})_{\mathaccent
866{z}}+2-\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}},$

since $\iota^{t}(\Gamma_{p})_{\mathaccent 866{z}}=\mathaccent
866{j}_{0,\bullet}^{t}(\Gamma_{p})_{\mathaccent 866{z}}-1$. By $\mathaccent
869{R}_{v}(p)_{\mathaccent 866{z}}=\sum_{t=1}^{\eta_{\mathaccent
866{z}}}D_{t}$, we need blowing-ups at least

$\displaystyle\sum_{t=1}^{\eta_{\mathaccent 866{z}}}\left(\sum_{a\geq
2}(an-2)\mathaccent 866{j}_{a}^{t}(\Gamma_{p})_{\mathaccent
866{z}}+2-\kappa^{t}(\Gamma_{p})_{\mathaccent 866{z}}\right)=\sum_{a\geq
2}(an-2)\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent
866{z}}+2\eta_{\mathaccent 866{z}}-\kappa(\Gamma_{p})_{\mathaccent 866{z}}$

times to get $\mathaccent 869{R}_{v}(p)_{\mathaccent 866{z}}$. This gives
$(3)$.

It remains to show (2). We may assume $\mathaccent 866{R}$ is singular at
$\mathaccent 866{z}$. Let $\mathaccent 869{\Gamma}_{p,\mathaccent 866{z}}=\sum
m_{i}G_{i}$ be the irreducible decomposition. Then it follows from (2.2) that

$\displaystyle\alpha_{0}^{+}(\Gamma_{p})_{\mathaccent
866{z}}=\sum_{i}m_{i}\mathaccent
869{R}_{h}G_{i}-\sharp(\mathrm{Supp}(\mathaccent
869{R}_{h})\cap\mathrm{Supp}(\cup_{i}G_{i}))\geq\sum_{i}(m_{i}-1)\mathaccent
869{R}_{h}G_{i}.$

We consider a directed graph $\mathbb{F}$ whose vertex set $V(\mathbb{F})$ is
the set of symbols $\\{v_{x}\\}$, where $x$ runs over all the singular points
which contribute to either $\iota(\Gamma_{p})_{\mathaccent 866{z}}$ or
$\kappa(\Gamma_{p})_{\mathaccent 866{z}}$. We define the directed edge from
$v_{x}$ to $v_{x^{\prime}}$ if $x^{\prime}$ is a singular point infinitely
near to $x$ and any singular point between $x$ and $x^{\prime}$ contributes to
neither $\iota(\Gamma_{p})_{\mathaccent 866{z}}$ nor
$\kappa(\Gamma_{p})_{\mathaccent 866{z}}$. Let
$\mathbb{T}_{1},\cdots,\mathbb{T}_{s}$ be connected components of the graph
$\mathbb{F}$. We note that $\mathbb{T}_{j}$ ($j=1,\cdots,s$) is a directed and
rooted tree graph. We denote the leaf set of $\mathbb{T}_{j}$ by
$L(\mathbb{T}_{j})$. By the definition of $\mathbb{F}$, any vertex in
$L(\mathbb{T}_{j})$ corresponds to a singular point contributing to
$\iota(\Gamma_{p})_{\mathaccent 866{z}}$. Let $x_{\mathrm{last}}$ be a
singular point of the branch locus over $x$ such that there exist no singular
points over $x_{\mathrm{last}}$ and let $\mathaccent 869{E}^{x}$ be the
exceptional curve of the blow-up at $x_{\mathrm{last}}$. Since $\mathaccent
869{E}^{x}$ arises from a singular point of type $n\mathbb{Z}$, we see that
$\mathaccent 869{R}\mathaccent 869{E}^{x}$ is a positive multiple of $n$, but
it may be possible that $\mathaccent 869{E}^{x}$ meets two vertical components
of $\mathaccent 869{R}$. Thus we have $\mathaccent 869{R}_{h}\mathaccent
869{E}^{x}\geq n-2$. Letting $m^{x}$ be the multiplicity along $\mathaccent
869{E}^{x}$ of $\mathaccent 869{\Gamma}_{p}$, we have

$\displaystyle\sum_{i}(m_{i}-1)\mathaccent
869{R}_{h}G_{i}\geq\sum_{j=1}^{s}\sum_{x\in
L(\mathbb{T}_{j})}(m^{x}-1)\mathaccent 869{R}_{h}\mathaccent 869{E}^{x}.$

We will show that

$\displaystyle\sum_{x\in
L(\mathbb{T}_{j})}(m^{x}-1)\geq\iota(\mathbb{T}_{j})+2\kappa(\mathbb{T}_{j}),$

where $\iota(\mathbb{T}_{j})$ (resp. $\kappa(\mathbb{T}_{j})$) denotes the
number of singular points of type $n\mathbb{Z}$ (resp. $n\mathbb{Z}+1$) in
$\mathbb{T}_{j}$. Let $P^{x}(\mathbb{T}_{j})$ be the set consisting of
vertices appearing in the path connecting the root of $\mathbb{T}_{j}$ and
$x\in L(\mathbb{T}_{j})$. We denote the number of singular points of type
$n\mathbb{Z}$ (resp. $n\mathbb{Z}+1$) in $P^{x}(\mathbb{T}_{j})$ by
$\iota(P^{x}(\mathbb{T}_{j}))$ (resp. $\kappa(P^{x}(\mathbb{T}_{j}))$). We
claim that $m^{x}\geq
2\iota(P^{x}(\mathbb{T}_{j}))+\kappa(P^{x}(\mathbb{T}_{j}))$. This can be seen
as follows. Put $P^{x}(\mathbb{T}_{j})=\\{v_{1},\cdots,v_{l}=v_{x}\\}$ and let
$m_{1},\cdots,m_{l}$ be the multiplicities of the fiber $\mathaccent
869{\Gamma}_{p}$ along the proper transforms of the exceptional curves arising
from $v_{1},\cdots,v_{l}$, respectively. If $v_{1}$ is of type $n\mathbb{Z}$,
then we have $m_{2}\geq m_{1}+2$. If $v_{1}$ is of type $n\mathbb{Z}+1$, then
we have $m_{2}\geq m_{1}+1$. So we get $m^{x}=m_{l}\geq
2\iota(P^{x}(\mathbb{T}_{j}))+\kappa(P^{x}(\mathbb{T}_{j}))$ inductively.
Then,

$\displaystyle\sum_{x\in L(\mathbb{T}_{j})}(m^{x}-1)\geq$
$\displaystyle\sum_{x\in
L(\mathbb{T}_{j})}\left(2\iota(P^{x}(\mathbb{T}_{j}))+\kappa(P^{x}(\mathbb{T}_{j}))\right)-\sharp
L(\mathbb{T}_{j})$ $\displaystyle\geq$ $\displaystyle
2\iota(\mathbb{T}_{j})+2\kappa(\mathbb{T}_{j})-\sharp L(\mathbb{T}_{j})$
$\displaystyle\geq$
$\displaystyle\iota(\mathbb{T}_{j})+2\kappa(\mathbb{T}_{j})$

as wished.

Now, we get

$\displaystyle\sum_{x\in L(\mathbb{T}_{j})}(m^{x}-1)\mathaccent
869{R}_{h}G_{i}\geq$ $\displaystyle\sum_{j=1}^{s}\sum_{x\in
L(\mathbb{T}_{j})}(m^{x}-1)\mathaccent 869{R}_{h}\mathaccent 869{E}^{x}$
$\displaystyle\geq$ $\displaystyle(n-2)\sum_{j=1}^{s}\sum_{x\in
L(\mathbb{T}_{j})}(m^{x}-1)$ $\displaystyle\geq$
$\displaystyle(n-2)(\iota(\Gamma_{p})_{\mathaccent
866{z}}+2\kappa(\Gamma_{p})_{\mathaccent 866{z}}).$

Plugging the above inequalities to (2.2), we conclude that
$\alpha_{0}^{+}(\Gamma_{p})_{\mathaccent
866{z}}\geq(n-2)(\iota(\Gamma_{p})_{\mathaccent
866{z}}+2\kappa(\Gamma_{p})_{\mathaccent 866{z}})$. Then, since
$\iota(\Gamma_{p})_{\mathaccent 866{z}}=j(\Gamma_{p})_{\mathaccent
866{z}}-\eta_{\mathaccent 866{z}}$ by (1), we get (2). ∎

We can show the following inequality similarly to Lemma 4.7 of [6]

######  Lemma 2.4.

If $n=2$, then it holds

$\displaystyle\kappa(\Gamma_{p})_{\mathaccent
866{z}}\leq\frac{2}{3}\sum_{a\geq 2}(a-1)\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}.$

######  Lemma 2.5.

Let the notation and the assumption as above. Then it holds that

$\displaystyle\mathaccent 866{\alpha}_{0}(\Gamma_{p})_{\mathaccent
866{z}}+B_{n}\sum_{k\geq 1}\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}-\begin{cases}nB_{n}\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent
866{z}}&(n\geq 3)\\\ \\\ 0&(n=2)\\\ \end{cases}$

is non-negative, where $B_{n}=2/n$ if $n\geq 4$, $B_{3}=1/2$ and $B_{2}=3/2$.

###### Proof.

If $n\geq 3$, we can show the desired inequality similarly to Lemma 2.2 of
[1].

For $n=2$, we have

$\displaystyle\mathaccent 866{\alpha}_{0}(\Gamma_{p})_{\mathaccent
866{z}}+B_{2}\sum_{k\geq 1}\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$ $\displaystyle=$
$\displaystyle\alpha^{+}_{0}(\Gamma_{p})_{\mathaccent 866{z}}-2\sum_{a\geq
2}\mathaccent 866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}+B_{2}\sum_{k\geq
1}\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle\geq$ $\displaystyle\alpha^{+}_{0}(\Gamma_{p})_{\mathaccent
866{z}}+\sum_{a\geq 2}\left(B_{2}(2a-2)-2\right)\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}+2B_{2}\eta_{\mathaccent
866{z}}-B_{2}\kappa(\Gamma_{p})_{\mathaccent 866{z}}$ $\displaystyle\geq$
$\displaystyle\alpha^{+}_{0}(\Gamma_{p})_{\mathaccent 866{z}}+\sum_{a\geq
2}\left(\frac{4}{3}(a-1)B_{2}-2\right)\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}},$

where the first inequality is by (3) of Lemma 2.3, the second inequality is by
Lemma 2.4. Since $B_{2}=3/2$, the coefficients of $\mathaccent
866{j}_{0,a}(\Gamma_{p})_{\mathaccent 866{z}}$ are non-negative for $a\geq 2$.
∎

######  Lemma 2.6.

The following holds for $p\in B$:

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}\geq\sum_{k\geq
1}\left((n^{2}-1)k-n-\frac{(n-1)^{2}}{n}B_{n}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}.$

In particular, $K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}$ is non-negative.

###### Proof.

If $n\geq 4$, it is clear from Lemma 2.5.

If $n=3$, we have

$\displaystyle\mathaccent 866{\alpha}_{0}(\Gamma_{p})_{\mathaccent
866{z}}-2\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}}\geq-
B_{3}\sum_{k\geq 1}\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}-\frac{1}{2}\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}}$

from Lemma 2.5. Hence we get

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle\geq\sum_{k\geq 1}\left(8k-3\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}-\frac{4}{3}B_{3}\sum_{k\geq
1}\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}-\frac{2}{3}\;\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent
866{z}}+\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle\geq\left(8k-3-\frac{4}{3}B_{3}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}.$

If $n=2$, we have

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle=\sum_{k\geq 1}\left(3k-2\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}+\frac{1}{2}\mathaccent
866{\alpha}_{0}(\Gamma_{p})_{\mathaccent 866{z}}.$

Hence we get

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}$
$\displaystyle\geq\sum_{k\geq 1}\left(3k-2-\frac{1}{2}B_{2}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$

from Lemma 2.5. ∎

######  Lemma 2.7.

The following holds for $p\in B$.

$\displaystyle K_{f}^{2}(\Gamma_{p})\geq$ $\displaystyle\sum_{\mathaccent
866{z}\in\mathaccent
866{\Gamma}_{p}}\frac{(n-1)^{2}}{n}\alpha^{+}_{0}(\Gamma_{p})_{\mathaccent
866{z}}+\sum_{\mathaccent 866{z}\in\mathaccent 866{\Gamma}_{p}}\sum_{k\geq
1}\left((n^{2}-1)k-n-2\frac{(n-1)^{2}}{n}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$ $\displaystyle+\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p})-2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p})+j_{0,\bullet}^{\prime}(\Gamma_{p})\right)$
$\displaystyle+\frac{n^{2}-1}{n}r(\chi_{\varphi}(\Gamma_{p})+\nu(\Gamma_{p}))+\check{j}_{0,1}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p}).$

###### Proof.

By the definition of $K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}$, we have

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent 866{z}}=$
$\displaystyle\sum_{k\geq 1}\left((n^{2}-1)k-n\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent
866{z}}+\frac{(n-1)^{2}}{n}(\mathaccent
866{\alpha}_{0}(\Gamma_{p})_{\mathaccent 866{z}}-2\mathaccent
866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}})+\mathaccent
866{j}_{0,1}(\Gamma_{p})_{\mathaccent 866{z}}$

for $\mathaccent 866{z}\in\mathaccent 866{\Gamma}_{p}$. Since irreducible
components of $\mathaccent 866{R}(p)_{\mathaccent 866{z}}$ arise from singular
points of type $n\mathbb{Z}+1$, we have $\mathaccent
866{j}_{0,\bullet}(\Gamma_{p})_{\mathaccent 866{z}}\leq\sum_{k\geq
1}\mathaccent 866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}$. Hence we get

$\displaystyle K_{f}^{2}(\Gamma_{p})_{\mathaccent
866{z}}\geq\frac{(n-1)^{2}}{n}\alpha^{+}_{0}(\Gamma_{p})_{\mathaccent
866{z}}+\sum_{k\geq
1}\left((n^{2}-1)k-n-2\frac{(n-1)^{2}}{n}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p})_{\mathaccent 866{z}}.$

Combining this with Proposition 2.2, we get the desired inequality. ∎

##  3\. Automorphism groups of elliptic surfaces

In this section, we summarize some properties of automorphism groups of
elliptic surfaces. Let $\varphi:W\to B$ be a relatively minimal elliptic
surface. We fix a point $p\in B$ and denote by $\Delta\subset B$ an analytic
open neighborhood of $p$. We suppose that $\Gamma_{p}$ is not multiple. Let
$\varphi_{\Delta}:W_{\Delta}\to\Delta$ be a restriction of $\varphi:W\to B$ to
$\Delta$:

$\textstyle{W_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B.}$

Replacing $\Delta$ by a smaller neighborhood of $p$ if necessary, we may
assume that $\varphi_{\Delta}$ has only one singular fiber $\Gamma_{p}$.

Let $W_{\Delta,\mathrm{sm}}:=W_{\Delta}\setminus\mathrm{Sing}(\Gamma_{p})$ and
$\Gamma_{p,\mathrm{sm}}:=\Gamma_{p}\setminus\mathrm{Sing}(\Gamma_{p})$, where
$\mathrm{Sing}(\Gamma_{p})$ is the set of critical points of
$\varphi_{\Delta}$ on $\Gamma_{p}$. Then there exists the natural map
$\varphi_{\Delta,\mathrm{sm}}:W_{\Delta,\mathrm{sm}}\to\Delta$. The following
theorem is due to Kodaira ([7]).

######  Theorem 3.1 (Kodaira).

There exist three holomorphic maps

$\displaystyle O_{\Delta}:\Delta\to
W_{\Delta,\mathrm{sm}},\quad\mu_{\Delta}:W_{\Delta,\mathrm{sm}}\times_{\Delta}W_{\Delta,\mathrm{sm}}\to
W_{\Delta,\mathrm{sm}},\quad\iota_{\Delta}:W_{\Delta,\mathrm{sm}}\to
W_{\Delta,\mathrm{sm}}$

over $\Delta$ which satisfy the following conditions.

  * (1)

The fiber germ $\varphi_{\Delta,\mathrm{sm}}:W_{\Delta,\mathrm{sm}}\to\Delta$
is a group manifold over $\Delta$.

  * (2)

Let $\Gamma_{q}$ $(q\in\Delta)$ be a fiber of $\varphi_{\Delta}$ and let
$O_{q}:=\Gamma_{q}\cap O_{\Delta}(\Delta)$. These three maps induce the
commutative group law on $\Gamma_{q,\mathrm{sm}}$ whose unit element is
$O_{q}$. The group $(\Gamma_{q,\mathrm{sm}},O_{q})$ is one of the followings.

$(\Gamma_{q,\mathrm{sm}},O_{q})\cong\begin{cases}\text{The group law of
elliptic curve}&(\mathrm{I}_{0}\text{-type})\\\ \mathbb{G}_{m}\times
G(\Gamma_{q})&(\mathrm{I}_{c}\text{-type})\\\ \mathbb{G}_{a}\times
G(\Gamma_{q})&(\text{other})\end{cases}$

where

$\displaystyle G(\mathrm{I}_{c})\cong\mathbb{Z}/c\mathbb{Z},$ $\displaystyle
G(\mathrm{I}_{2c}^{\ast})\cong(\mathbb{Z}/2\mathbb{Z})^{2},$ $\displaystyle
G(\mathrm{I}_{2c+1}^{\ast})\cong\mathbb{Z}/4\mathbb{Z},$ $\displaystyle
G(\mathrm{II})\cong G(\mathrm{II}^{\ast})\cong\\{0\\},$ $\displaystyle
G(\mathrm{III})\cong G(\mathrm{III}^{\ast})\cong\mathbb{Z}/2\mathbb{Z},$
$\displaystyle G(\mathrm{IV})\cong
G(\mathrm{IV}^{\ast})\cong\mathbb{Z}/3\mathbb{Z}.$

###### Proof.

Since $\Gamma_{p}$ is not multiple, there exists a projective elliptic surface
over a smooth projective curve $\varphi^{\flat}:W^{\flat}\to B^{\flat}$ which
admits a global section and contains $\varphi_{\Delta}:W_{\Delta}\to\Delta$ as
a fiber germ:

$\textstyle{W_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{W^{\flat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\flat}}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\flat}.}$

We denote by $O_{\flat}:B^{\flat}\to W^{\flat}$ a global section of
$\varphi^{\flat}$. Let $W_{\mathrm{sm}}^{\flat}$ be an open subset of
$W^{\flat}$ consisting of all regular points of $\varphi^{\flat}$. From
Theorem 9.1 of [7], there exist holomorphic maps
$\mu_{\flat}:W_{\mathrm{sm}}^{\flat}\times_{B^{\flat}}W_{\mathrm{sm}}^{\flat}\to
W_{\mathrm{sm}}^{\flat}$ and $\iota_{\flat}:W_{\mathrm{sm}}^{\flat}\to
W_{\mathrm{sm}}^{\flat}$ over $B^{\flat}$ which satisfy conditions (1) and
(2). These three maps $O_{\flat}$, $\mu_{\flat}$ and $\iota_{\flat}$ induce
$O_{\Delta}$, $\mu_{\Delta}$ and $\iota_{\Delta}$ by the universality of base
change. ∎

  * •

We denote by $W_{\mathrm{sm}}(\Delta)$ the set of sections of
$\varphi_{\Delta,\mathrm{sm}}$. Since
$\varphi_{\Delta,\mathrm{sm}}:W_{\Delta,\mathrm{sm}}\to\Delta$ is a group
manifold over $\Delta$, $W_{\mathrm{sm}}(\Delta)$ is a commutative group with
the unit element $O_{\Delta}$. The symbol $\beta_{1}\oplus\beta_{2}$ denotes
the sum of $\beta_{1},\>\beta_{2}\in W_{\mathrm{sm}}(\Delta)$ and
$\ominus\beta$ denotes the inverse element of $\beta\in
W_{\mathrm{sm}}(\Delta)$.

  * •

We have the following commutative diagram for a section $\beta\in
W_{\mathrm{sm}}(\Delta)$:

$\textstyle{W_{\Delta,\mathrm{sm}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\times_{W_{\Delta,\mathrm{sm}}}(\mathrm{pr}_{1})}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{W_{\Delta,\mathrm{sm}}\times_{\Delta}W_{\Delta,\mathrm{sm}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{2}}$$\scriptstyle{\mathrm{pr}_{1}}$$\textstyle{W_{\Delta,\mathrm{sm}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{W_{\Delta,\mathrm{sm}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{\Delta}$

Hence we have an automorphism
$\tau_{\beta}:=\mu_{\Delta}\circ\left(\beta\times_{W_{\Delta,\mathrm{sm}}}(\mathrm{pr}_{1})\right)\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$
for each $\beta\in W_{\mathrm{sm}}(\Delta)$. Thus, there exists a natural
injective homomorphism
$W_{\mathrm{sm}}(\Delta)\to\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta);\beta\mapsto\tau_{\beta}$.

  * •

We define the following groups.

$\displaystyle\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta):=\\{\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}})\mid\varphi_{\Delta,\mathrm{sm}}\circ\kappa=\varphi_{\Delta,\mathrm{sm}}\\}$
$\displaystyle\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta}):=\\{\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)\mid\kappa\circ
O_{\Delta}=O_{\Delta}\\}$ (3.1)
$\displaystyle{\color[rgb]{0,0,0}\mathrm{Aut}(\Gamma_{p,\mathrm{sm}},O_{p}):=\\{\kappa\in\mathrm{Aut}(\Gamma_{p,\mathrm{sm}})\mid\kappa(O_{p})=O_{p}\\}}$

######  Proposition 3.2.

There exists the exact sequence

$\displaystyle 1\to
W_{\mathrm{sm}}(\Delta)\to\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)\to\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})\to
1.$

In particular, it holds that

$\displaystyle\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)\cong
W_{\mathrm{sm}}(\Delta)\mathchar
9583\relax\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta}).$

###### Proof.

It is sufficient to show the following two statements.

  * (1)

There exist $\tau\in
W_{\mathrm{sm}}(\Delta)\subset\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$ and
$\varepsilon\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})$ for an
action $\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$ such that
$\kappa=\tau\circ\varepsilon$.

  * (2)

$W_{\mathrm{sm}}(\Delta)\lhd\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$.

(1) An action $\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$
determines a section $\kappa\circ O_{\Delta}\in W_{\mathrm{sm}}(\Delta)$. We
consider the action $\tau_{\ominus(\kappa\circ O_{\Delta})}$. Then we have
$\varepsilon:=\tau_{\ominus(\kappa\circ
O_{\Delta})}\circ\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})$.
Hence we get $\tau_{\ominus(\kappa\circ O_{\Delta})}^{-1}\in
W_{\mathrm{sm}}(\Delta)$ and
$\varepsilon\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})$ such
that $\kappa=\tau_{\ominus(\kappa\circ O_{\Delta})}^{-1}\circ\varepsilon$.

(2) We have to show $\kappa\circ\tau\circ\kappa^{-1}\in
W_{\mathrm{sm}}(\Delta)$ for any
$\kappa\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$ and $\tau\in
W_{\mathrm{sm}}(\Delta)$. In order to show the above statement, it is
sufficient to show $\varepsilon\circ\tau\circ\varepsilon^{-1}\in
W_{\mathrm{sm}}(\Delta)$ for any
$\varepsilon\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})$ and
$\tau\in W_{\mathrm{sm}}(\Delta)$. Suppose a section $\beta\in
W_{\mathrm{sm}}(\Delta)$ corresponds to the action
$\tau\in\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$ (namely
$\tau=\tau_{\beta}$). Let $\Gamma_{q}$ be an arbitrary smooth fiber of
$\varphi_{\Delta}$. Then it holds that
$(\varepsilon\circ\tau_{\beta}\circ\varepsilon^{-1})|_{\Gamma_{q}}=(\tau_{\varepsilon\circ\beta})|_{\Gamma_{q}}$
since $\varepsilon|_{\Gamma_{q}}$ is compatible with the group law of
$\Gamma_{q}$. Therefore we have
$(\varepsilon\circ\tau_{\beta}\circ\varepsilon^{-1})=\tau_{\varepsilon\circ\beta}\in
W_{\mathrm{sm}}(\Delta)$. ∎

##  4\. Automorphism of fibered surface

For a fibration $f:S\to B$, we define the automorphism group of $f$ as

$\mathrm{Aut}(f):=\\{(\kappa_{S},\kappa_{B})\in\mathrm{Aut}(S)\times\mathrm{Aut}(B)\mid
f\circ\kappa_{S}=\kappa_{B}\circ f\\}.$

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,1,n)$. Let
$\Sigma$ be the cyclic group of order $n$ generated by $\sigma$, where
$\sigma$ is defined in Remark 1.2. Let $G$ be an arbitrary finite subgroup of
$\mathrm{Aut}(f)$. Since
$\mathrm{Aut}(S/B):=\\{(\kappa_{S},\mathrm{id})\in\mathrm{Aut}(f)\\}$ is a
finite group, we may assume $\sigma\in G$ to estimate the order of it. We have
the exact sequence

$\displaystyle 1\to K\to G\to H\to 1,$

where $K:=\\{(\kappa_{S},{\it id})\in G\\}$ and
$H:=\\{\kappa_{B}\in\mathrm{Aut}(B)\mid(\kappa_{S},\kappa_{B})\in
G,\exists\kappa_{S}\in\mathrm{Aut}(S)\\}$.

######  Lemma 4.1.

Assume $r=\mathaccent 869{R}\mathaccent 869{\Gamma}\geq 4n$. Take a smooth
fiber $F$ of $f$ and a point $z\in F$. Let $\kappa_{F}$ be an automorphism of
$F$. Then it holds that

$\displaystyle\kappa_{F}(\Sigma\cdot z)=\Sigma\cdot\kappa_{F}(z),$

where $\Sigma\cdot z$ denotes the $\Sigma$-orbits of $z$.

###### Proof.

The subgroup $\Sigma\subset{\color[rgb]{0,0,0}\mathrm{Aut}(F)}$ induces the
quotient map $\theta:F\to\Gamma:=F/\Sigma$ of degree $n$. From the assumption
$r\geq 4n$, there exists an isomorphism $\kappa_{\Gamma}:\Gamma\to\Gamma$ such
that the diagram

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\kappa_{F}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{\Gamma}}$$\textstyle{\Gamma}$

commutes for any $\kappa_{F}\in\mathrm{Aut}(F)$ from Proposition 3.1 of [1].
Thus, we obtain $\kappa_{F}(\Sigma\cdot{z})=\Sigma\cdot{\kappa_{F}(z)}$. ∎

In what follows in section 4, we tacitly assume that $r\geq 4n$.

We can show the following by a similar argument as in Lemma 3.3 of [1].

######  Lemma 4.2.

Let $F$ be a smooth fiber of $f:S\to B$. Regard $\Sigma$ as a subgroup of
$\mathrm{Aut}(F)$. Then $\Sigma$ is a normal subgroup of $\mathrm{Aut}(F)$.

Since we have $\Sigma\lhd K$, the action of $K\subset\mathrm{Aut}(S/B)$ on $S$
can be lifted to the one on $\mathaccent 869{S}$ and we can regard $K$ as a
subgroup of $\mathrm{Aut}(\mathaccent 869{S}/B)$. If we put $\mathaccent
869{K}=K/\Sigma$, then we have the exact sequence

$\displaystyle 1\to\Sigma\to K\to\mathaccent 869{K}\to 1.$

Note that $\mathaccent 869{K}\subset\mathrm{Aut}(\mathaccent 869{W}/B)$ and
$\mathaccent 869{R}$ is $\mathaccent 869{K}$-stable (namely $\mathaccent
869{K}(\mathaccent 869{R})=\mathaccent 869{R}$).

######  Lemma 4.3.

The action of $\mathaccent 869{K}$ descends down faithfully on the relatively
minimal model $\varphi:W\to B$. Hence we can regard $\mathaccent 869{K}$ as a
subgroup of $\mathrm{Aut}(W/B)$.

###### Proof.

Let $\mathaccent 869{\Gamma}_{p}$ be an arbitrary singular fiber of
$\mathaccent 869{W}\to B$ and $E$ any $(-1)$-curve contained in $\mathaccent
869{\Gamma}_{p}$. Then its $\mathaccent 869{K}$-orbits $\mathaccent
869{K}\cdot E$ satisfies one of the following:

  * •

$\mathaccent 869{K}\cdot E$ consists of a disjoint union of $(-1)$-curves in
$\mathaccent 869{\Gamma}_{p}$.

  * •

We can find two curves in $\mathaccent 869{K}=\\{E_{1},\cdots,E_{t}\\}$
meeting a point.

In the former case, contracting $\mathaccent 869{K}\cdot E$ to points, the
action of $\mathaccent 869{K}$ descends down to the action on the new fiber
obtained by the contraction.

We will show the latter case does not occur. Since the intersection form on
$E_{1}\cup E_{2}$ is negative semi-definite, $E_{1}^{2}=E_{2}^{2}=-1$ and
$E_{1}E_{2}>0$, the only possibility is: $E_{1}E_{2}=1$ and
$(E_{1}+E_{2})^{2}=0$. So we get

$\displaystyle\mathrm{Supp}\;\Gamma_{p}=E_{1}\cup E_{2}$

by Zariski’s lemma. This is impossible by the classification of singular
fibers of elliptic surfaces. ∎

We fix a point $p\in B$ and denote by $\Delta\subset B$ an analytic open
neighborhood of $p$. We suppose that $\Gamma_{p}$ is not multiple. Let
$\varphi_{\Delta}:W_{\Delta}\to\Delta$ be a restriction of $\varphi:W\to B$ to
$\Delta$. We note that there exists a natural inclusion $\mathaccent
869{K}\subset\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)$.

######  Definition 4.4.

We call $\kappa\in\mathaccent 869{K}$ a translation of $\mathaccent 869{K}$ if
$\kappa$ satisfies either $\kappa(C_{\Delta})\neq C_{\Delta}$ or
$\kappa|_{C_{\Delta}}\neq\mathrm{id}_{C_{\Delta}}$ for any
$\varphi_{\Delta}$-horizontal local analytic branch $C_{\Delta}$ on
$W_{\Delta,\mathrm{sm}}$. Let $T(\mathaccent 869{K})_{p}$ be the set
consisting of translations of $\mathaccent 869{K}$ and
$\mathrm{id}_{W_{\Delta,\mathrm{sm}}}$.

######  Proposition 4.5.

It holds that

$\displaystyle T(\mathaccent 869{K})_{p}=\mathaccent 869{K}\cap
W_{\mathrm{sm}}(\Delta).$

In particular, $T(\mathaccent 869{K})_{p}$ is a subgroup of $\mathaccent
869{K}$.

###### Proof.

Since $W_{\mathrm{sm}}(\Delta)$ is a normal subgroup of
$\mathrm{Aut}(W_{\mathrm{sm}}/\Delta)$, we deduce that $\mathaccent 869{K}\cap
W_{\Delta,\mathrm{sm}}(\Delta)$ is a normal subgroup of $\mathaccent 869{K}$.
We have an exact commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}\cap
W_{\mathrm{sm}}(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}/(\mathaccent 869{K}\cap
W_{\mathrm{sm}}(\Delta))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\mathrm{sm}}(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

We note that the injectivity of $\mathaccent 869{K}/(\mathaccent 869{K}\cap
W_{\Delta,\mathrm{sm}}(\Delta))\to\mathrm{Aut}(W_{\Delta,\mathrm{sm}}/\Delta,O_{\Delta})$
is shown by a simple diagram chasing. Therefore we have the following exact
commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}\cap
W_{\mathrm{sm}}(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}/(\mathaccent 869{K}\cap
W_{\mathrm{sm}}(\Delta))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Gamma_{q,\mathrm{sm}},O_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(\Gamma_{q,\mathrm{sm}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(\Gamma_{q,\mathrm{sm}},O_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

for an arbitrary fiber $\Gamma_{q}$ ($q\in\Delta$) of $\varphi_{\Delta}$.
Since $\mathaccent 869{K}/(\mathaccent 869{K}\cap
W_{\Delta,\mathrm{sm}}(\Delta))\to\mathrm{Aut}(\Gamma_{q,\mathrm{sm}},O_{q})$
is injective, we deduce that

$\displaystyle\mathaccent 869{K}\cap(\Gamma_{q},O_{q})\subset\mathaccent
869{K}\cap W_{\mathrm{sm}}(\Delta)$

for an arbitrary smooth fiber $\Gamma_{q}$.

Take a non-trivial arbitrary automorphism $\kappa\in T(\mathaccent
869{K})_{p}$. Since $\kappa$ is a translation of $\mathaccent 869{K}$,
$\kappa|_{\Gamma_{q}}$ has no fixed point for general $q\in\Delta$. It implies
that $\kappa|_{\Gamma_{q}}\in\mathaccent 869{K}\cap(\Gamma_{q},O_{q})$. By
$\mathaccent 869{K}\cap(\Gamma_{q},O_{q})\subset\mathaccent 869{K}\cap
W_{\mathrm{sm}}(\Delta)$, we have $\kappa\in\mathaccent 869{K}\cap
W_{\mathrm{sm}}(\Delta)$. Therefore we obtain $T(\mathaccent
869{K})_{p}\subset\mathaccent 869{K}\cap W_{\mathrm{sm}}(\Delta)$. The
converse $\mathaccent 869{K}\cap W_{\mathrm{sm}}(\Delta)\subset T(\mathaccent
869{K})_{p}$ is trivial. ∎

######  Corollary 4.6.

There exists an exact commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(\mathaccent
869{K})_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathaccent
869{K}/T(\mathaccent
869{K})_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Gamma_{q,\mathrm{sm}},O_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(\Gamma_{q,\mathrm{sm}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}(\Gamma_{q,\mathrm{sm}},O_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

for an arbitrary fiber $\Gamma_{q}$ $(q\in\Delta)$ of $\varphi_{\Delta}$.

The following two Corollaries are mentioned in [9].

######  Corollary 4.7.

If $\varphi:W\to B$ has a singular fiber $\Gamma_{p}$ $(p\in B)$ except for
type $l\mathrm{I}_{c}$. Then it holds that $\sharp T(\mathaccent
869{K})_{p}\leq 4$.

###### Proof.

From Theorem 3.1 and Corollary 4.6, we have an injective homomorphism

$\displaystyle T(\mathaccent 869{K})_{p}\hookrightarrow\mathbb{G}_{a}\times
G(\Gamma_{p,\mathrm{sm}}).$

Since $\mathbb{G}_{a}$ is a torsion free group, we have $T(\mathaccent
869{K})_{p}\hookrightarrow G(\Gamma_{p,\mathrm{sm}})$. By the assumption that
$\Gamma_{p}$ is not of type $l\mathrm{I}_{c}$, we have $\sharp
G(\Gamma_{p,\mathrm{sm}})\leq 4$. Thus, it holds that $\sharp T(\mathaccent
869{K})_{p}\leq 4$. ∎

######  Corollary 4.8.

Let $\Gamma_{p}$ be a singular fiber of type $\mathrm{I}_{c}$. Then the only
points of $\Gamma_{p}$ with non-trivial stabilizer subgroup under the action
of $T(\mathaccent 869{K})_{p}$ are nodes of $\Gamma_{p}$.

######  Corollary 4.9.

Assume $\varphi:W\to B$ is not isotrivial. Then it holds that
$\sharp(\mathaccent 869{K}/T(\mathaccent 869{K})_{p})\leq 2$ for any $p\in B$.
In particular, it holds that $\sharp\mathaccent 869{K}\leq 2\sharp
T(\mathaccent 869{K})_{p}$.

###### Proof.

Let $\varphi_{\Delta}:W_{\Delta}\to\Delta$ be a restriction of $\varphi:W\to
B$ to $p\in\Delta$. Since $\varphi:W\to B$ is not isotrivial, there exists a
smooth fiber $\Gamma_{q}$ of $\varphi|_{\Delta}$ such that
$\sharp(\mathrm{Aut}(\Gamma_{q},O_{q}))=2$. From Corollary 4.6, we have the
injective homomorphism $\mathaccent 869{K}/T(\mathaccent
869{K})_{p}\hookrightarrow\mathrm{Aut}(\Gamma_{q},O_{q})$. ∎

We suppose that $\Gamma_{p}$ is the multiple fiber of type $l\mathrm{I}_{c}$.
Let $\pi:\Delta^{\dagger}\to\Delta$ be a $l$-th root map branched at $p$. Then
there exists a commutative diagram

(4.1)
$\textstyle{W_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta^{\dagger}}}$$\scriptstyle{\Pi}$$\textstyle{W_{\Delta}\times_{\Delta}\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Delta}$

where $W_{\Delta^{\dagger}}$ is the normalization of
$W_{\Delta}\times_{\Delta}\Delta^{\dagger}$. We note that
$W_{\Delta^{\dagger}}$ is non-singular. Since $\pi^{-1}(p)$ consists of only
one point, denote this point by $p^{\dagger}$. Let $\Gamma_{p^{\dagger}}$ be a
fiber of $\varphi_{\Delta^{\dagger}}$ over ${p^{\dagger}}$. We note that
$\Gamma_{p^{\dagger}}$ is a singular fiber of type $\mathrm{I}_{lc}$. Put
$R_{\dagger}:=\Pi^{\ast}R$. We note that $\Pi:W_{\Delta^{\dagger}}\to
W_{\Delta}$ is an unramified covering of degree $l$.

######  Proposition 4.10.

There exists an injective homomorphism
$\mathrm{Aut}(W_{\Delta}/\Delta)\to\mathrm{Aut}(W_{\Delta^{\dagger}}/{\Delta^{\dagger}})$.

###### Proof.

An automorphism $\kappa:W_{\Delta}\to
W_{\Delta}\in\mathrm{Aut}(W_{\Delta}/\Delta)$ induces the automorphism

$\displaystyle\kappa\times_{\Delta}\Delta^{\dagger}:W_{\Delta}\times_{\Delta}\Delta^{\dagger}\to
W_{\Delta}\times_{\Delta}\Delta^{\dagger}$

by the universality of the fiber product. Hence there exists the injective
homomorphism
$\mathrm{Aut}(W_{\Delta}/\Delta)\to\mathrm{Aut}(W_{\Delta}\times_{\Delta}\Delta^{\dagger}/{\Delta^{\dagger}})$.
Furthermore, the universality of normalization induces the injective
homomorphism

$\displaystyle\mathrm{Aut}(W_{\Delta}\times_{\Delta}\Delta^{\dagger}/{\Delta^{\dagger}})\hookrightarrow\mathrm{Aut}(W_{\Delta^{\dagger}}/\Delta^{\dagger}).$

Thus, we get the desired homomorphism. ∎

Hence we can regard $\mathaccent 869{K}$ as a subgroup of
$\mathrm{Aut}(W_{\Delta^{\dagger}}/{\Delta^{\dagger}})$. We denote by
$\kappa^{\dagger}\in\mathrm{Aut}(W_{\Delta^{\dagger}}/{\Delta^{\dagger}})$ the
action which corresponds to $\kappa\in\mathrm{Aut}(W_{\Delta}/\Delta)$ and let
$\mathaccent 869{K}^{\dagger}:=\\{\kappa^{\dagger}\mid\kappa\in\mathaccent
869{K}\\}$.

######  Proposition 4.11.

Let $z^{\dagger}\in\Gamma_{p^{\dagger}}$ be a point and put
$z=\Pi(z^{\dagger})\in\Gamma_{p}$. Let $\kappa\in\mathrm{Stab}_{\mathaccent
869{K}}(z)$. Then $\kappa^{\dagger}\in\mathrm{Stab}_{\mathaccent
869{K}^{\dagger}}(z^{\dagger})$. In particular, it holds $\sharp(\mathaccent
869{K}\cdot z)=\sharp(\mathaccent 869{K}^{\dagger}\cdot z^{\dagger})$

###### Proof.

(1) The case that $\Gamma_{p^{\dagger}}$ is smooth at $z^{\dagger}$.

Then $\Gamma_{p,\mathrm{red}}$ is smooth at $z$. We take analytic local
neighborhoods $(U;(x_{1},x_{2}))$ of $z$ on $W_{\Delta}$, $(\Delta;y)$ of $p$
and $(\Delta^{\dagger};x_{3})$ of $p^{\dagger}$ such that
$\varphi^{\ast}y=x_{1}^{l}$ and $\pi^{\ast}y=x_{3}^{l}$. Then
$U\times_{\Delta}\Delta^{\dagger}$ is defined by $x_{1}^{l}-x_{3}^{l}=0$ in
$U\times\Delta^{\dagger}$. Put
$H_{i}:=\\{(x_{1},x_{2},\zeta_{l}^{i}x_{1})\mid(x_{1},x_{2})\in U\\}$ for
$i=0,\cdots,l-1$, where $\zeta_{l}$ is a primitive $l$-th root of unity. Then
we have

$U\times_{\Delta}\Delta^{\dagger}=\bigcup_{i=0}^{l-1}H_{i}\subset
U\times\Delta.$

Thus, the natural projection $\sqcup_{i=0}^{l-1}H_{i}\to
U\times_{\Delta}\Delta^{\dagger}$ is nothing more than the normalization of
$U\times_{\Delta}\Delta^{\dagger}$. Let
$\mathrm{id}\neq\kappa\in\mathrm{Stab}_{\mathaccent 869{K}}(z)$. Replacing $U$
by a smaller neighborhood of $z$ if necessary, we may assume that
$\kappa(U)=U$. We write
$\kappa(x_{1},x_{2})=(\kappa_{1}(x_{1,x_{2}}),\kappa_{2}(x_{1},x_{2}))$ on U.
Then the induced automorphism
$\kappa\times_{\Delta}\Delta^{\dagger}:U\times_{\Delta}\Delta^{\dagger}\to
U\times_{\Delta}\Delta^{\dagger}$ by $\kappa$ is written by
$(\kappa\times_{\Delta}\Delta^{\dagger})(x_{1},x_{2},x_{3})=(\kappa_{1}(x_{1},x_{2}),\kappa_{2}(x_{1},x_{2}),x_{3})$.
It holds that $(\kappa\times_{\Delta}\Delta^{\dagger})(H_{i})=H_{i}$ for
$i=0,\cdots,l-1$. Furthermore, we have a commutative diagram

$\textstyle{H_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi|_{H_{i}}}$$\scriptstyle{(\kappa\times_{\Delta}\Delta^{\dagger})}$$\textstyle{H_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi|_{H_{i}}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{U}$

for $i=0,\cdots,l-1$. Thus, we have
$\kappa^{\dagger}\in\mathrm{Stab}_{\mathaccent 869{K}}(z^{\dagger})$.

(2) The case that $\Gamma_{p^{\dagger}}$ has a node at $z^{\dagger}$.

Then $\Gamma_{p.\mathrm{red}}$ has a node at $z$. We take analytic local
neighborhoods $(U;(x_{1},x_{2}))$ of $z$ on $W_{\Delta}$, $(\Delta;y)$ of $p$
and $(\Delta^{\dagger};x_{3})$ of $p^{\dagger}$ such that
$\varphi^{\ast}y=(x_{1}x_{2})^{l}$ and $\pi^{\ast}y=x_{3}^{l}$. Then
$U\times_{\Delta}\Delta^{\dagger}$ is defined by
$(x_{1}x_{2})^{l}-x_{3}^{l}=0$ in $U\times\Delta^{\dagger}$. Put
$H_{i}:=\\{(x_{1},x_{2},\zeta_{l}^{i}x_{1}x_{2})\mid(x_{1},x_{2})\in U\\}$ for
$i=0,\cdots,l-1$, where $\zeta_{l}$ is a primitive $l$-th root of unity. Then
we have

$U\times_{\Delta}\Delta^{\dagger}=\bigcup_{i=0}^{l-1}H_{i}\subset
U\times\Delta.$

Thus, the natural projection $\sqcup_{i=0}^{l-1}H_{i}\to
U\times_{\Delta}\Delta^{\dagger}$ is the normalization of
$U\times_{\Delta}\Delta^{\dagger}$. We can show
$\kappa^{\dagger}\in\mathrm{Stab}_{\mathaccent 869{K}}(z^{\dagger})$ similarly
to (1). ∎

We can define $T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}$ for $\mathaccent
869{K}^{\dagger}\subset\mathrm{Aut}(W_{\Delta^{\dagger}}/{\Delta^{\dagger}})$
since $\Gamma_{p^{\dagger}}$ is not multiple. Hence we define

$T(\mathaccent 869{K})_{p}:=\\{\kappa\in\mathaccent
869{K}\mid\kappa^{\dagger}\in T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\\}$

for a multiple fiber $\Gamma_{p}$. We note that $T(\mathaccent
869{K})_{p}\cong T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}$ and
$\mathaccent 869{K}\cong\mathaccent 869{K}^{\dagger}$.

##  5\. Estimation of the order of $\sharp\mathaccent 869{K}$

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,1,n)$ with
$r=\frac{2(g-1)}{n-1}\geq 4n$.

######  Definition 5.1.

Let $z\in R_{h}$ be a (smooth) ramification point of $\varphi|_{R_{h}}$. We
call $z$ a good ramification point when the ramification index of
$\varphi|_{R_{h}}$ at $z$ is greater than the multiplicity of the fiber of
$\varphi$ passing through $z$.

The goal in this section is to show Proposition 5.3, 5.4 and 5.5.

######  Definition 5.2.

Put

$\displaystyle\delta:=\mathrm{min}\\{\sharp\mathrm{Aut}(\Gamma_{p},O_{p})|\;\forall
p\in\Delta\textrm{ with }\Gamma_{p}\textrm{ is smooth}\\},$

where $\mathrm{Aut}(\Gamma_{p},O_{p})$ is the same notation defined in (3.1).
We note that $\delta=2,4$ or $6$.

######  Proposition 5.3.

Let $f:S\to B$ be a primitive cyclic covering fibration of $(g,1,n)$ with
$r\geq 4n$. Let $\Gamma_{p}$ be of type $l\mathrm{I}_{c}$. Assume $R$ is
smooth locally around $\Gamma_{p}$ and has no good ramification points on
$\Gamma_{p}$. Then it holds that

$\displaystyle K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq\frac{(n^{2}-1)}{12n\delta}\sharp\mathaccent 869{K}.$

######  Proof.

Since $R$ is smooth locally around $\Gamma_{p}$, we have
$j_{0,a}^{\prime}(\Gamma_{p})=0$ for any $a\geq 1$ if $n\geq 3$. Similarly, we
have $j_{0,a}^{\prime}(\Gamma_{p})=0$ for $a\geq 2$ if $n=2$. Hence we have

$\displaystyle
K_{f}^{2}(\Gamma_{p})=\frac{(n-1)^{2}}{n}\alpha_{0}^{+}(\Gamma_{p})+\frac{n^{2}-1}{n}r\left(\chi_{f}(\Gamma_{p})+1-\frac{1}{l}\right)$

from Lemma 2.1. Since the ramification index of any ramification point of
$\varphi|_{R_{h}}$ over $p$ is $l$, we have

$\alpha_{0}^{+}(\Gamma_{p})=r(1-\frac{1}{l}).$

By Corollary 4.6, we have $\delta r\geq\sharp\mathaccent 869{K}$. Thus, we
have

$\displaystyle K_{f}(\Gamma_{p})$
$\displaystyle\geq\left(\frac{2(n-1)}{\delta}\left(1-\frac{1}{l}\right)+\frac{n^{2}-1}{\delta
n}\frac{c}{12}\right)\sharp\mathaccent 869{K}$
$\displaystyle\geq\frac{(n^{2}-1)}{12n\delta}\sharp\mathaccent 869{K}.$

∎

######  Proposition 5.4.

Let $f:S\to B$ be a primitive cyclic covering fibration of $(g,1,n)$ with
$r\geq\mathrm{max}\\{60+\frac{12}{n^{2}-1}-\frac{96}{n+1},7n\\}$. Assume
either $\Gamma_{p}$ is not of type $l\mathrm{I}_{c}$ or $R$ has a singular
point on $\Gamma_{p}$. Then it holds that

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq\frac{1}{3}(n-1)(5n-4)\sharp\mathaccent 869{K}.$

######  Proposition 5.5.

Let $f:S\to B$ be a primitive cyclic covering fibration of $(g,1,n)$ with
$r\geq 4n$. Let $\Gamma_{p}$ is of type $l\mathrm{I}_{c}$. Assume $R$ is
smooth locally around $\Gamma_{p}$ and has a good ramification point on
$\Gamma_{p}$. Then it holds that

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq(n-1)^{2}\sharp\mathaccent 869{K}.$

###  5.1. The proof of Proposition 5.4

######  Lemma 5.6.

Let $f:S\to B$ be a primitive cyclic covering fibration of $(g,1,n)$ with
$r\geq 4n$. Assume $\Gamma_{p}$ is a singular fiber except for type
$l\mathrm{I}_{c}$. Then it holds that

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq\frac{n^{2}-1}{12}\left(r-12\frac{n-1}{n+1}\right)\sharp\mathaccent
869{K}.$

If $r\geq 7n$, it holds

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq\frac{1}{3}(n-1)(5n-4)\sharp\mathaccent 869{K}.$

###### Proof.

Let $\Gamma_{p}$ be a singular fiber except for type $l\mathrm{I}_{c}$. We
note that $e(\Gamma_{p})\geq\sharp\Gamma_{p}+1$ in this case. So we have
$j_{0,\bullet}^{\prime}(\Gamma_{p})\leq\sharp\Gamma_{p}$ and
$\chi_{\varphi}(\Gamma_{p})\geq(\sharp\Gamma_{p}+1)/12$. We have

$\displaystyle K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq-2\frac{(n-1)^{2}}{n}j_{0,\bullet}^{\prime}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p})+\frac{n^{2}-1}{n}r\chi_{\varphi}(\Gamma_{p})$

from Proposition 2.2 and Lemma 2.6. Hence we have

$\displaystyle K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq-2\frac{(n-1)^{2}}{n}j_{0,\bullet}^{\prime}(\Gamma_{p})+j_{0,1}^{\prime}(\Gamma_{p})+\frac{n^{2}-1}{12n}r(\sharp\Gamma_{p}+1)$
$\displaystyle=\left(\frac{n^{2}-1}{12n}r-\frac{2(n-1)^{2}}{n}\right)\sharp\Gamma_{p}+\frac{n^{2}-1}{12n}r$
$\displaystyle\geq\frac{n^{2}-1}{6n}r-\frac{2(n-1)^{2}}{n}\;(\text{by}\;\sharp\Gamma_{p}\geq
1)$ $\displaystyle=\frac{n^{2}-1}{6n}\left(r-12\frac{n-1}{n+1}\right).$

Thus, it is sufficient to show

$\displaystyle
2n\delta\frac{n^{2}-1}{6n}\left(r-12\frac{n-1}{n+1}\right)-\frac{n^{2}-1}{12}\left(r-12\frac{n-1}{n+1}\right)\sharp\mathaccent
869{K}\geq 0.$

Since $\Gamma_{p}$ is not of type $l\mathrm{I}_{c}$, we have
$\sharp\mathaccent 869{K}\leq 4\delta$ from Corollary 4.7. Hence we get the
desired inequality. ∎

We assume $\varphi:W\to B$ has at most singular fibers of type
$l\mathrm{I}_{c}$ in what follows in this subsection. Let $z$ be a singular
point of $R$ and $z\in\Gamma_{p}$ a fiber of type $l\mathrm{I}_{c}$ where $l$
and $c$ are non-negative integers. Recall the diagram (4.1).

$\textstyle{W_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta^{\dagger}}}$$\scriptstyle{\Pi}$$\textstyle{W_{\Delta}\times_{\Delta}\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Delta}$

We recall that $\pi^{-1}(p)=\\{p^{\dagger}\\}$ and
$R_{\dagger}=\Pi^{\ast}(R|_{W_{\Delta}})$. Replacing $\Delta$ by a smaller
neighborhood of $p$ if necessary, we may assume that
$\varphi_{\Delta^{\dagger}}$ has only one singular fiber
$\Gamma_{p^{\dagger}}$ and $R_{\dagger}$ has a singular point only on
$\Gamma_{p^{\dagger}}$.

Since $\Pi:W_{\Delta^{\dagger}}\to W_{\Delta}$ is an unramified covering, each
singular point of $R_{\dagger}$ is analytically equivalent to the
corresponding singular point of $R|_{W_{\Delta}}$. Hence we can consider a
formal canonical resolution of $R_{\dagger}$ as follows.

######  Definition 5.7.

Let $z^{\dagger}_{1}$ be a singular point of $R_{\dagger}$ on
$\Gamma_{p^{\dagger}}$ and
$(\psi_{\Delta^{\dagger}})_{1}:W_{\Delta^{\dagger},1}\to W_{\Delta^{\dagger}}$
the blowing-up at $z^{\dagger}_{1}$. We put

$\displaystyle
R_{\dagger,1}:=(\psi_{\Delta^{\dagger}})^{\ast}R_{\dagger}-n\left[\frac{m_{1}}{n}\right]E_{1},$

where $E_{1}:=(\psi_{\Delta^{\dagger}})^{-1}_{1}(z^{\dagger}_{1})$ and
$m_{1}:=\mathrm{mult}_{z^{\dagger}_{1}}(R_{\dagger})$. We define
$(W_{\Delta^{\dagger},i},R_{\dagger,i})$ for $i=2,\cdots,N^{\dagger}$
inductively as follows. Let $z^{\dagger}_{i}$ be a singular point of
$R_{\dagger,i-1}$ and $(\psi_{\Delta^{\dagger}})_{i}:W_{\Delta^{\dagger},i}\to
W_{\Delta^{\dagger},i-1}$ the blowing-up at $z^{\dagger}_{i}$. We put

$\displaystyle
R_{\dagger,i}:=(\psi_{\Delta^{\dagger}})_{i}^{\ast}R_{\dagger,i-1}-n\left[\frac{m_{i}}{n}\right]E_{i},$

where $E_{i}:=(\psi_{\Delta^{\dagger}})^{-1}_{i}(z^{\dagger}_{i})$ and
$m_{i}=\mathrm{mult}_{z^{\dagger}_{i}}(R_{\dagger,i-1})$. We continue this
process until $R_{\dagger,i}$ is smooth. Since $\Pi$ is an unramified
covering, this process terminates. Let
$(W_{\Delta^{\dagger},N^{\dagger}},R_{\dagger,N^{\dagger}})$ be the model
which the above process terminates and put $\mathaccent
869{\psi}_{\Delta^{\dagger}}:=(\psi_{\Delta^{\dagger}})_{1}\circ\cdots\circ(\psi_{\Delta^{\dagger}})_{N^{\dagger}}$.
We call $R_{\dagger,i}$ ($i=1,\cdots N^{\dagger}$) the branch locus on
$W_{\Delta^{\dagger},i}$. We rewrite
$(W_{\Delta^{\dagger},N^{\dagger}},R_{\dagger,N^{\dagger}})$ as $(\mathaccent
869{W}_{\Delta^{\dagger}},\mathaccent 869{R}_{\dagger})$.

######  Remark 5.8.

The following hold.

  * •

The multiplicity of an arbitrary singular point of $R_{\dagger,i}$ is in
$n\mathbb{Z}$ or $n\mathbb{Z}+1$ for $i=1,\cdots,N^{\dagger}$.

  * •

The self-intersection number of any vertical component of $\mathaccent
869{R}_{\dagger}$ which is contracted to a point by $\mathaccent
869{\psi}_{\Delta^{\dagger}}$ is divisible by $n$.

Let $\mathaccent 866{\varphi}_{\Delta^{\dagger}}:\mathaccent
866{W}_{\Delta^{\dagger}}\to\Delta^{\dagger}$ be any intermediate elliptic
fiber germ between $\mathaccent 869{W}_{\Delta^{\dagger}}$ and
$W_{\Delta^{\dagger}}$, and regard $\mathaccent
869{\psi}_{\Delta^{\dagger}}:\mathaccent 869{W}_{\Delta^{\dagger}}\to
W_{\Delta^{\dagger}}$ as the composite of the natural birational morphisms
$\mathaccent 866{\psi}_{\Delta^{\dagger}}:\mathaccent
869{W}_{\Delta^{\dagger}}\to\mathaccent 866{W}_{\Delta^{\dagger}}$ and
$\check{\psi}:\mathaccent 866{W}_{\Delta^{\dagger}}\to W_{\Delta^{\dagger}}$.

$\textstyle{\mathaccent
869{W}_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\psi}_{\Delta^{\dagger}}}$$\scriptstyle{\mathaccent
869{\varphi}_{\Delta^{\dagger}}}$$\textstyle{\mathaccent
866{W}_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\varphi}_{\Delta^{\dagger}}}$$\scriptstyle{\check{\psi}_{\Delta^{\dagger}}}$$\textstyle{W_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta^{\dagger}}}$$\textstyle{B}$

We put $\mathaccent 866{R}_{\dagger}=(\mathaccent
866{\psi}_{\Delta^{\dagger}})_{\ast}\mathaccent 869{R}_{\dagger}$. The fiber
of $\mathaccent 866{\varphi}_{\Delta^{\dagger}}$ over $p\in\Delta^{\dagger}$
will be denoted by $\mathaccent 866{\Gamma}_{p^{\dagger}}$.

For any $\mathaccent 866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}$, we introduce indices as Section 2.

$\displaystyle\begin{aligned}
\alpha_{0}^{+}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}&:=&\begin{cases}\left((\mathaccent
869{R}_{\dagger})_{h},\mathaccent 869{\Gamma}_{p^{\dagger},\mathaccent
866{z}^{\dagger}}\right)-\sharp\left(\mathrm{Supp}(\mathaccent
869{R}_{\dagger})_{h}\cap\mathrm{Supp}(\mathaccent
869{\Gamma}_{p^{\dagger},\mathaccent 866{z}^{\dagger}})\right)&({\rm
if\;}\mathaccent 866{R}_{\dagger}{\rm\;is\;singular\;at\;}\mathaccent
866{z}^{\dagger}\;),\\\ \\\ \left({\rm
The\;ramification\;index\;of\;}(\mathaccent
869{R}_{\dagger})_{h}\to\Delta^{\dagger}{\rm\;at\;}\mathaccent
866{z}\right)-1&({\rm if\;}\mathaccent
866{R}_{\dagger}{\rm\;is\;smooth\;at\;}\mathaccent 866{z}^{\dagger}\;),\\\ \\\
0&({\rm if\;}\mathaccent 866{z}^{\dagger}\not\in\mathaccent
866{R}_{\dagger}),\end{cases}\end{aligned}$

where $\mathaccent 869{\Gamma}_{p^{\dagger},\mathaccent 866{z}^{\dagger}}$ is
the biggest subdivisor of $\mathaccent 869{\Gamma}_{p^{\dagger}}$ which is
contracted to $\mathaccent 866{z}^{\dagger}$ by $\mathaccent
866{\psi}_{\Delta^{\dagger}}$.

  * •

Let $\check{\alpha}_{k}(\Gamma_{p^{\dagger}})$ be the number of singular
points of multiplicity either $kn$ or $kn+1$ among all singular points of the
branch loci appearing in $\check{\psi}_{\Delta^{\dagger}}$. If
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we
define $\check{\alpha}_{k}(\Gamma_{p^{\dagger}})=0$.

  * •

Let $\mathaccent 866{\alpha}_{k}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ be the number of singular points of multiplicity either
$kn$ or $kn+1$ among all singular points of the branch loci which are
infinitely near to $\mathaccent 866{z}^{\dagger}$. If
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we
rewrite $\mathaccent 866{\alpha}_{k}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ as $\alpha_{k}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$.

  * •

Put
$j_{0,a}^{{}^{\prime}}(\Gamma_{p^{\dagger}}):=lj_{0,a}^{{}^{\prime}}(\Gamma_{p})$
for $a\geq 1$. Put
$j_{0,\bullet}^{{}^{\prime}}(\Gamma_{p^{\dagger}}):=\sum_{a\geq
1}j_{0,a}^{{}^{\prime}}(\Gamma_{p^{\dagger}})$

  * •

Let $\check{j}_{0,a}(\Gamma_{p^{\dagger}})$ be the number of vertical
irreducible components of $\mathaccent 869{R}_{\dagger}$ that satisfy
following two conditions.

    * –

The self-intersection number is $-an$.

    * –

It is a proper transform of an exceptional curve appearing from
$\check{\psi}_{\Delta^{\dagger}}$.

Put $\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}}):=\sum_{a\geq
1}\check{j}_{0,a}(\Gamma_{p^{\dagger}})$. If
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we
define $\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})=0$.

  * •

Let $\mathaccent 866{j}_{0,a}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ be the number of irreducible curves with self-intersection
number $-an$ contained in $(\mathaccent 869{R}_{\dagger})_{v}(p^{\dagger})$
and contracted to $\mathaccent 866{z}^{\dagger}$ by $\mathaccent
866{\psi}_{\Delta^{\dagger}}$. Put $\mathaccent
866{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}:=\sum_{a\geq 1}\mathaccent
866{j}_{0,a}(\Gamma_{p^{\dagger}})_{\mathaccent 866{z}^{\dagger}}$. If
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we
rewrite $\mathaccent 866{j}_{0,a}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ as
$j_{0,a}^{{}^{\prime\prime}}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$.

  * •

Put $\mathaccent 866{\alpha}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}:=\alpha_{0}^{+}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}-2\sum_{a\geq 2}\mathaccent
866{j}_{0,a}(\Gamma_{p^{\dagger}})_{\mathaccent 866{z}^{\dagger}}$. If
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we
rewrite $\mathaccent 866{\alpha}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ as $\alpha_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$.

  * •

$K_{f}^{2}(\Gamma_{p^{\dagger}})_{\mathaccent 866{z}^{\dagger}}:=\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}+\frac{(n-1)^{2}}{n}(\mathaccent
866{\alpha}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}-2\mathaccent 866{j}_{0,1}(\Gamma_{p})_{\mathaccent
866{z}^{\dagger}})+\mathaccent 866{j}_{0,1}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$.

  * •

$\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}}):=l\chi_{\varphi_{\Delta}}(\Gamma_{p})$.

Using indices of the case
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}_{W_{\Delta^{\dagger}}}$, we put

(5.1) $\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}}):=$
$\displaystyle\sum_{z^{\dagger}\in\Gamma_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})_{z^{\dagger}}-2\frac{(n-1)^{2}}{n}j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}}).$

######  Proposition 5.9.

Let the notation and the assumption be as above. Then it holds that

$\displaystyle
K_{f}^{2}(\Gamma_{p^{\dagger}})=lK_{f}^{2}(\Gamma_{p})-2(n-1)r(l-1).$

In particular, we have

$\displaystyle lK_{f}^{2}(\Gamma_{p})\geq K_{f}^{2}(\Gamma_{p^{\dagger}}).$

###### Proof.

By simple calculations, we have

(5.2)
$\displaystyle\sum_{z^{\dagger}\in\Gamma_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})_{z^{\dagger}}=l\sum_{z\in\Gamma_{p}}K_{f}^{2}(\Gamma_{p})_{z}-\frac{(n-1)^{2}}{n}(l-1)r.$

Applying Proposition 2.2 for $\check{\psi}_{\Delta}=\mathrm{id}_{W_{\Delta}}$,
we have

$\displaystyle l\sum_{z\in\Gamma_{p}}K_{f}^{2}(\Gamma_{p})_{z}=$
$\displaystyle
lK_{f}^{2}(\Gamma_{p})+2\frac{(n-1)^{2}}{n}lj^{\prime}_{0,\bullet}(\Gamma_{p})-lj^{\prime}_{0,1}(\Gamma_{p})$
$\displaystyle-\frac{n^{2}-1}{n}rl\chi_{\varphi}(\Gamma_{p})-\frac{n^{2}-1}{n}r(l-1).$

Substituting (5.2) to the above equation, we have

(5.3)
$\displaystyle\sum_{z^{\dagger}\in\Gamma_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})=$
$\displaystyle
lK_{f}^{2}(\Gamma_{p})+2\frac{(n-1)^{2}}{n}lj^{\prime}_{0,\bullet}(\Gamma_{p})-lj^{\prime}_{0,1}(\Gamma_{p})$
$\displaystyle-$
$\displaystyle\frac{n^{2}-1}{n}rl\chi_{\varphi}(\Gamma_{p})-2r(l-1)(n-1)$

On the other hand, we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})=$
$\displaystyle\sum_{z^{\dagger}\in\Gamma_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})_{z^{\dagger}}-\frac{2(n-1)^{2}}{n}j^{\prime}_{0,\bullet}(\Gamma_{p^{\dagger}})+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+j^{\prime}_{0,1}(\Gamma_{p^{\dagger}}).$

Substituting (5.3) to the above equation, we get

$\displaystyle
K_{f}^{2}(\Gamma_{p^{\dagger}})=lK_{f}^{2}(\Gamma_{p})-2(n-1)r(l-1).$

∎

######  Proposition 5.10 (cf. Proposition 2.2).

Let $\mathaccent 866{\varphi}_{\Delta^{\dagger}}:\mathaccent
866{W}_{\Delta^{\dagger}}\to\Delta^{\dagger}$ be any intermediate elliptic
fiber germ between $\mathaccent 869{W}_{\Delta^{\dagger}}$ and
$W_{\Delta^{\dagger}}$.

$\textstyle{\mathaccent
869{W}_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\psi}_{\Delta^{\dagger}}}$$\scriptstyle{\mathaccent
869{\varphi}_{\Delta^{\dagger}}}$$\textstyle{\mathaccent
866{W}_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathaccent
866{\varphi}_{\Delta^{\dagger}}}$$\scriptstyle{\check{\psi}_{\Delta^{\dagger}}}$$\textstyle{W_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta^{\dagger}}}$$\textstyle{B}$

Then it holds that

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})=$
$\displaystyle\sum_{\mathaccent 866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}+\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p^{\dagger}})$
$\displaystyle-2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+\check{j}_{0,1}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}}).$

###### Proof.

We can check it by simple calculations. ∎

We can show the following by a similar argument as in Lemma 2.6.

######  Lemma 5.11 (cf. Lemma 2.6).

It holds that

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}\geq\sum_{k\geq
1}\left((n^{2}-1)k-n-\frac{(n-1)^{2}}{n}B_{n}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p^{\dagger}})_{\mathaccent 866{z}^{\dagger}}.$

In particular, $K_{f}^{2}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}$ is non-negative.

######  Lemma 5.12 (cf. Lemma 2.7).

It holds that

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\sum_{\mathaccent 866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}}\frac{(n-1)^{2}}{n}\alpha^{+}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}+\sum_{\mathaccent 866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}}\sum_{k\geq
1}\left((n^{2}-1)k-n-2\frac{(n-1)^{2}}{n}\right)\mathaccent
866{\alpha}_{k}(\Gamma_{p^{\dagger}})_{\mathaccent 866{z}^{\dagger}}$
$\displaystyle+\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p^{\dagger}})-2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j^{\prime}_{0,\bullet}(\Gamma_{p^{\dagger}})\right)$
$\displaystyle+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+\check{j}_{0,1}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}}).$

###### Proof.

We can check it by simple calculations. ∎

######  Lemma 5.13.

Let $z^{\dagger}\in R_{\dagger}$ be a singular point of $R_{\dagger}$ and let
$\Pi(z^{\dagger})=z$. Then there exist $l\cdot\sharp(T(\mathaccent
869{K}^{\dagger})_{p}\cdot z^{\dagger})$ singular points of $R_{\dagger}$
which are analytically equivalent to $z\in R$.

###### Proof.

We note that $\sharp\Pi^{-1}(T(\mathaccent 869{K})_{p}\cdot
z)=l\cdot\sharp(T(\mathaccent 869{K})_{p}\cdot z)$ since $\Pi:W^{\dagger}\to
W$ is an unramified covering of degree $l$. There exists an analytic
neighborhood $U$ of each $z^{\dagger}\in\Pi^{-1}(T(\mathaccent
869{K})_{p}\cdot z)$ such that $\Pi|_{U}:U\to\Pi(U)$ is a biholomorphic map.
Hence the singular point $z^{\dagger}\in R_{\dagger}$ is analytically
equivalent to $z\in R$. Thus, there exist $l\cdot\sharp(T(\mathaccent
869{K})_{p}\cdot z)$ singular points of $R_{\dagger}$ which are analytically
equivalent to $z\in R$. On the other hand, we have $\sharp(T(\mathaccent
869{K})_{p}\cdot z)=\sharp(T(\mathaccent 869{K}^{\dagger})_{p}\cdot
z^{\dagger})$ from Proposition 4.11. ∎

Let $(R_{\dagger})_{h}$ be the local analytic curve consisting of
$\varphi_{\Delta^{\dagger}}$-horizontal local analytic branches of
$R_{\dagger}$. Take a singular point $z^{\dagger}\in R_{\dagger}$ on
$\Gamma_{p^{\dagger}}$. We introduce the following notations for
$z^{\dagger}$.

  * (1)

We denote by $\mathrm{L}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}((R_{\dagger})_{h},z^{\dagger})$ the sets of local analytic
branches of $R_{\dagger}$ and $(R_{\dagger})_{h}$ at $z^{\dagger}$,
respectively.

  * (2)

Let $\mathrm{L}_{\mathrm{tr}}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}_{\mathrm{tr}}((R_{\dagger})_{h},z^{\dagger})$ be the subsets of
$\mathrm{L}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}((R_{\dagger})_{h},z^{\dagger})$ which consist of local analytic
branches that meet $\Gamma_{p^{\dagger}}$ transversally at $z^{\dagger}$,
respectively.

  * (3)

Let $\mathrm{L}_{\mathrm{ta}}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}_{\mathrm{ta}}((R_{\dagger})_{h},z^{\dagger})$ be the subsets of
$\mathrm{L}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}((R_{\dagger})_{h},z^{\dagger})$ which consist of local analytic
branches that are tangent to $\Gamma_{p^{\dagger}}$ at $z^{\dagger}$,
respectively.

  * (4)

For a local analytic branch $\Gamma_{i}$ of $\Gamma_{p^{\dagger}}$ around
$z^{\dagger}$, let
$\mathrm{L}_{\mathrm{ta}}(R_{\dagger},\Gamma_{i},z^{\dagger})$ and
$\mathrm{L}_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{i},z^{\dagger})$ be the
subsets of $\mathrm{L}_{\mathrm{ta}}(R_{\dagger},z^{\dagger})$ and
$\mathrm{L}_{\mathrm{ta}}((R_{\dagger})_{h},z^{\dagger})$ which consist of
local analytic branches that are tangent to $\Gamma_{i}$ at $z^{\dagger}$,
respectively.

If the local analytic branch $\Gamma_{i}$ of $\Gamma_{p^{\dagger}}$ is
contained in $R_{\dagger}$, we regard the local analytic branch $\Gamma_{i}$
as belonging to
$\mathrm{L}_{\mathrm{ta}}(R_{\dagger},\Gamma_{i},z^{\dagger})$. Hence we have
$\Gamma_{i}\in\mathrm{L}_{\mathrm{ta}}(R_{\dagger},\Gamma_{i},z^{\dagger})\subset\mathrm{L}_{\mathrm{ta}}(R_{\dagger},z^{\dagger})$.

To show Proposition 5.4, we consider the following four conditions for
$\Gamma_{p^{\dagger}}$ where $R_{\dagger}$ has a singular point.

  * (C1)

There exists a singular point $z^{\dagger}\in R_{\dagger}$ on
$\Gamma_{p^{\dagger}}$ such that $\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})=\\{\mathrm{id}\\}$.

  * (C2)

The condition $(C1)$ does not hold and there exist a singular point
$z^{\dagger}$ of $(R_{\dagger})$ on $\Gamma_{p^{\dagger}}$ and a local
analytic branch $\Gamma_{i}$ of $\Gamma_{p^{\dagger}}$ around $z^{\dagger}$
such that
$\mathrm{L}_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{i},z^{\dagger})\neq\emptyset$.

  * (C3)

Conditions $(C1)$ and $(C2)$ do not hold and there exists a singular point
$z^{\dagger}\in R_{\dagger}$ on $\Gamma_{p}$ such that
$\in\mathrm{L}_{\mathrm{tr}}((R_{\dagger})_{h},z^{\dagger})\neq\emptyset$.

  * (C4)

Conditions $(C1)$, $(C2)$ and $(C3)$ do not hold.

######  Proposition 5.14.

Assume $r\geq 4n$ and $(C1)$ holds. Then it holds that

$\displaystyle 2\delta nK_{f}^{2}(\Gamma_{p})\geq$ $\displaystyle
2n\left(n^{2}-1-n\right)\sharp\mathaccent 869{K}.$

###### Proof.

Let $z^{\dagger}$ be a singular point of $R_{\dagger}$ on
$\Gamma_{p^{\dagger}}$. By Lemma 5.13, there exist $\sharp T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}l$ singular points of the branch loci on
$\Gamma_{p^{\dagger}}$ whose multiplicity is at least $n$.

If there exists a singular point of the branch loci over $z^{\dagger}\in
R_{\dagger}$, there exist $2\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}})l$ singular points of $R_{\dagger}$ on
$\Gamma_{p}$ whose multiplicity is at least $n$. Applying Proposition 5.10 for
$\check{\psi}_{\Delta^{\dagger}}=\mathrm{id}$ and Lemma 5.11, we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})$ $\displaystyle\geq
2\left((n^{2}-1)-n-\frac{(n-1)^{2}}{n}B_{n}\right)\sharp T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}l$
$\displaystyle-2\frac{(n-1)^{2}}{n}j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}})$
$\displaystyle\geq 2\left((n^{2}-1)-n-\frac{(n-1)^{2}}{n}B_{n}\right)\sharp
T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}l.$

Hence it is sufficient to show

$\displaystyle 4\delta
n\left((n^{2}-1)-n-\frac{(n-1)^{2}}{n}B_{n}\right)\sharp T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}-2n\left(n^{2}-1-n\right)\sharp\mathaccent
869{K}\geq 0.$

From Corollary 4.6, we have $\sharp\mathaccent 869{K}\leq\delta\sharp
T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}$. Hence it is sufficient to show

$\displaystyle
2\left((n^{2}-1)-n-\frac{(n-1)^{2}}{n}B_{n}\right)-\left(n^{2}-1-n\right)\geq
0.$

We can check it by a simple calculation.

Suppose that there exist no singular points over $z^{\dagger}\in R_{\dagger}$.
We recall that there exist $\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}})l$ singular points of $R_{\dagger}$ on
$\Gamma_{p}$ which is analytically equivalent to $z^{\dagger}\in R_{\dagger}$.
Let $\check{\psi}_{\Delta^{\dagger}}:\mathaccent 866{W}_{\Delta^{\dagger}}\to
W_{\Delta^{\dagger}}$ be the composite of blowing-ups at those
$\sharp(T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}})l$ singular points.
Since there exist no singular points over $z^{\dagger}\in R_{\dagger}$, we
have $\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})=0$. From Proposition 5.10
for $\check{\psi}_{\Delta^{\dagger}}$, we get

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})=$
$\displaystyle\sum_{\mathaccent 866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}}K_{f}^{2}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}+\left(n^{2}-1-n\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}})l$
$\displaystyle-2\frac{(n-1)^{2}}{n}j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}}).$

Since we have $\sharp\Gamma_{p^{\dagger}}\geq
j_{0,1}^{\prime}(\Gamma_{p^{\dagger}})$ and
$\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})\geq\sharp\Gamma_{p^{\dagger}}/12$,
we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})$
$\displaystyle\geq\left(n^{2}-1-n\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}})l+\frac{n^{2}-1}{n}\left(r-24\frac{n-1}{n+1}\right)\sharp\Gamma_{p^{\dagger}}$
$\displaystyle\geq\left(n^{2}-1-n\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}})l.$

Hence we have

$\displaystyle
K_{f}^{2}(\Gamma_{p})\geq\left(n^{2}-1-n\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}).$

Thus, it is sufficient to show

$\displaystyle 2\delta n\left(n^{2}-1-n\right)\sharp T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}-2n\left(n^{2}-1-n\right)\sharp\mathaccent
869{K}\geq 0.$

From Corollary 4.6, we have $\sharp\mathaccent 869{K}\leq\delta\sharp
T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}$. Hence we get the desired
inequality. ∎

######  Remark 5.15.

If (C1) does not hold, there exist no singular points of $R_{\dagger}$ on
$\Gamma_{p,\mathrm{sm}}$ by Corollary 4.8.

######  Proposition 5.16.

Assume $r\geq\mathrm{max}\\{60+\frac{12}{n^{2}-1}-\frac{96}{n+1},4n\\}$ and
$(C2)$ holds. Put

$C_{2,n}:=\frac{1}{3}(n-1)(5n-4).$

Then it holds that

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})\geq C_{2,n}\sharp\mathaccent
869{K}.$

###### Proof.

Take $z^{\dagger}\in R_{\dagger}$ and $\Gamma_{1}\subset\Gamma_{p^{\dagger}}$
such that
$L_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{1},z^{\dagger})\neq\emptyset$.  For
a local analytic branch $D\in
L_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{1},z^{\dagger})$, let $v(D)$ be the
number of blowing-ups until the proper transform of $D$ is not tangent to
$\Gamma_{1}$. Since
$L_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{1},z^{\dagger})\neq\emptyset$ is a
finite set, there exists $D_{1}\in
L_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{1},z^{\dagger})$ such that
$v(D_{1})$ is a minimal value among
$L_{\mathrm{ta}}((R_{\dagger})_{h},\Gamma_{1},z^{\dagger})$. Let
$\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(\Gamma_{1},z^{\dagger}):=\\{\kappa\in\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})\mid\kappa(\Gamma_{1})=\Gamma_{1}\\}$
and $\mathbb{D}_{1}$ be the $\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(\Gamma_{1},z^{\dagger})$-orbit of $D_{1}$. We
introduce some notations as follows.

  * •

$m_{D_{1}}:=\mathrm{mult}_{z^{\dagger}}D_{1}$

  * •

$m_{\mathbb{D}_{1}}:=\mathrm{mult}_{z^{\dagger}}\mathbb{D}_{1}$

  * •

$m:=\mathrm{min}\\{m^{\prime}\mid m^{\prime}\geq m_{\mathbb{D}},\quad
m^{\prime}\in n\mathbb{Z}$ or $n\mathbb{Z}+1\\}$

Assume that we need blowing-ups $v$ times until the proper transform of
$D_{1}$ is not tangent to that of $\Gamma_{1}$. Since $\Gamma_{p^{\dagger}}$
has a node at $z^{\dagger}$, we have
$(\mathbb{D}_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}\leq(v+2)m$.

From Lemma 5.13, there exist $\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ local analytic curves of
$R_{\dagger}$ which is analytically equivalent to
$z^{\dagger}\in\mathbb{D}_{1}$. Let $z^{\dagger}_{j}\in\mathbb{D}_{j}$ be such
local analytic curves for $j=1,\cdots,({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$, where we define
$z^{\dagger}_{1}=z^{\dagger}$. Since $z^{\dagger}_{j}\in\mathbb{D}_{j}$ is
analytically equivalent to $z^{\dagger}_{1}\in\mathbb{D}_{1}$, the proper
transform of $\mathbb{D}_{j}$ is not tangent to the proper transform of
$\Gamma_{p^{\dagger}}$ after $v$-times blowing-ups over $z^{\dagger}_{j}$ for
each $j=1,\cdots,({T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}}\cdot
z^{\dagger})l$.

We consider two cases separately.

(1) $n\geq 3$ or $n=2$ and $\mathrm{mult}_{z^{\dagger}}R_{\dagger}\in
2\mathbb{Z}$.

Let $\check{\psi}_{\Delta^{\dagger}}:\mathaccent 866{W}_{\Delta^{\dagger}}\to
W_{\Delta^{\dagger}}$ be the composite of the above $v$-times blowing-ups over
$z^{\dagger}_{j}$ for each $j=1,\cdots,({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$. Let $\mathaccent
866{R}_{\dagger}$ be the branch locus on $\mathaccent
866{W}_{\Delta^{\dagger}}$ and $\mathaccent 866{\Gamma}_{p^{\dagger}}$ a fiber
of $\mathaccent
866{\varphi}:=\varphi_{\Delta^{\dagger}}\circ\check{\psi}_{\Delta^{\dagger}}$
over $p^{\dagger}$. We show

$\displaystyle\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})\leq\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l$

if $n\geq 3$,

$\displaystyle\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})\leq\check{j}_{0,1}(\Gamma_{p^{\dagger}})+\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l$

if $n=2$. We may assume $v\geq 2$. Let
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}$ be the number of
irreducible components contributing to
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})$ which is contracted to
$z^{\dagger}$ by $\mathaccent 869{\psi}_{\Delta^{\dagger}}$. If $n\geq 3$, it
is sufficient to show
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}\leq 2$.  Let
$(\check{\psi}_{\Delta^{\dagger}})_{z^{\dagger}}:(\mathaccent
866{W}_{\Delta^{\dagger}})_{z^{\dagger}}\to W_{\Delta^{\dagger}}$ be the
composite of blowing-ups $v$-times over $z^{\dagger}$ and
$\mathcal{E}_{z^{\dagger}}$ the set of exceptional curves of
$(\check{\psi}_{\Delta^{\dagger}})_{z^{\dagger}}$. We note that
$\sharp\mathcal{E}_{z^{\dagger}}=v$. For an exceptional curve
$E\in\mathcal{E}_{z^{\dagger}}$, let $n(E)$ be the number such that $E$ is the
exceptional curve of the $n(E)$-th blowing-up in
$(\check{\psi}_{\Delta^{\dagger}})_{z^{\dagger}}$. Let $E_{i}$ be the
exceptional curve in $\mathcal{E}_{z^{\dagger}}$ such that $n(E_{i})=i$ for
$i=1,\cdots v$. We note that $E_{1}$ and $E_{v}$ are exceptional curves of
first and last blowing-ups in
$(\check{\psi}_{\Delta^{\dagger}})_{z^{\dagger}}$, respectively. Then the
proper transform of any local analytic branch in
$\mathrm{L}((R_{\dagger})_{h},z^{\dagger})\setminus\mathrm{L}_{\mathrm{ta}}(R_{\dagger},\Gamma_{i},z^{\dagger})$
can be tangent to only $E_{1}$ among $\mathcal{E}_{z^{\dagger}}$. Furthermore,
the proper transform of any local analytic branch in
$\mathrm{L}_{\mathrm{ta}}(R_{\dagger},\Gamma_{i},z^{\dagger})$ can be tangent
to only $E_{v}$ from the choice of $D_{1}$. Thus, the self-intersection number
of $\mathaccent 866{E}_{i}(i=2,\cdots,v-1)$ is $-2$, where $\mathaccent
866{E}_{i}\subset(\mathaccent 866{W}_{\Delta})_{z^{\dagger}}$ is the proper
transform of $E_{i}$. On the other hand, the self-intersection number of each
irreducible component of $\mathaccent 869{R}_{\dagger}$ contributing
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}$ is at most $-n$.
Thus, if $n\geq 3$, a proper transform of $\mathcal{E}_{z^{\dagger}}$ which is
contained in $\mathaccent 869{R}_{\dagger}$ is either $\mathaccent 866{E}_{1}$
or $\mathaccent 866{E}_{v}$ at most. Hence we get
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}\leq 2$ for $n\geq
3$.  Furthermore, since the branch loci have no double points, we have
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}\leq 1$ by chasing a
resolution of $z^{\dagger}\in R_{\dagger}$. If $n=2$, we can show
$\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})_{z^{\dagger}}\leq\check{j}_{0,1}(\Gamma_{p^{\dagger}})_{z^{\dagger}}+1$
by chasing a resolution of $z^{\dagger}\in R_{\dagger}$. Thus, we can get the
desired inequalities for $\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})$.

Since $\Gamma_{p^{\dagger}}$ has at least $\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ nodes, we have
$\sharp\Gamma_{p^{\dagger}}\geq\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$. Hence we get

$\displaystyle\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})\leq\sharp\Gamma_{p^{\dagger}}$

if $n\geq 3$,

$\displaystyle\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})\leq\sharp\Gamma_{p^{\dagger}}+\check{j}_{0,1}(\Gamma_{p^{\dagger}})$

if $n=2$. Put $\mathbb{D}:=\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})\cdot D_{1}$ and
$\gamma:=1-1/(D_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}$. Then we have

$\displaystyle{\color[rgb]{0,0,0}\sum_{\mathaccent
866{z}^{\dagger}\in\mathaccent
866{\Gamma}_{p^{\dagger}}}\alpha^{+}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}^{\dagger}}\geq\gamma(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l.}$

Since there exist $\sharp({T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}}\cdot
z^{\dagger})vl$-singular points of the branch loci on $\Gamma_{p^{\dagger}}$
which appear in $\check{\psi}_{\Delta^{\dagger}}$, we have

$\displaystyle\sum_{k\geq
1}\left((n^{2}-1)k-n\right)\check{\alpha}_{k}(\Gamma_{p^{\dagger}})\geq\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl.$

Thus, we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle+$
$\displaystyle\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl$ $\displaystyle-$
$\displaystyle
2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+\check{j}_{0,1}(\Gamma_{p^{\dagger}})+j_{0,1}^{\prime}(\Gamma_{p^{\dagger}})$

by Lemma 5.12. By
$\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})=e(\Gamma_{p^{\dagger}})/12=\sharp\Gamma_{p^{\dagger}}/12\geq\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l/12$, we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle+$
$\displaystyle\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl$ $\displaystyle+$
$\displaystyle\left(-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l.$

By $K_{f}^{2}(\Gamma_{p})\geq K_{f}^{2}(\Gamma_{p^{\dagger}})/l$, we get

$\displaystyle K_{f}^{2}(\Gamma_{p})\geq$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})$ $\displaystyle+$
$\displaystyle\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})v$ $\displaystyle+$
$\displaystyle\left(-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger}).$

Hence it is sufficient to show

$\displaystyle 2n\delta\biggl{\\{}$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\biggr{.}+\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)v$
$\displaystyle+\biggl{.}$
$\displaystyle\left(-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)\biggr{\\}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})-C_{2,n}\sharp\mathaccent
869{K}\geq 0.$

By
$(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq\sharp\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})$, it is sufficient to show

$\displaystyle 2n\biggl{\\{}$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\biggr{.}+\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)v$
$\displaystyle+\biggl{.}$
$\displaystyle\left(-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)\biggr{\\}}-C_{2,n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq
0.$

It is equivalent to

$\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)v-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+\left(2\gamma(n-1)^{2}-C_{2,n}\right)(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq
0.$

By $2m(v+2)\geq
2(\mathbb{D}_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}$,
it suffices to show

(5.4) $\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)v-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+2m(v+2)\left(2\gamma(n-1)^{2}-C_{2,n}\right)\geq
0.$

Since it holds $(D_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq 3$, we have
$\gamma=1-\frac{1}{(D_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}}\geq 2/3$. Hence
it suffices to show

(5.5) $\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)v-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+m(v+2)\left(\frac{8}{3}(n-1)^{2}-2C_{2,n}\right)\geq
0.$

The coefficient of $v$ in (5.5)

$\displaystyle
2n\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)+\left(\frac{8}{3}(n-1)^{2}-2C_{2,n}\right)m$

is non-negative. Hence we may assume $v=1$ to show (5.5). We will show

(5.6) $\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+m\left(8(n-1)^{2}-6C_{2,n}\right)\geq
0.$

Since the coefficient of $m$ is non-negative, we may assume $m=n+1$ to show
(5.6). Hence it suffices to show

$\displaystyle 2n(n^{2}-1)+(n+1)\left(8(n-1)^{2}-6C_{2,n}\right)\geq 0,$
$\displaystyle\frac{n^{2}-1}{12n}r-4\frac{(n-1)^{2}}{n}-n\geq 0.$

We can check the former inequality by a simple calculation. The latter one
clearly holds by the assumption $r\geq 60+\frac{12}{n^{2}-1}-\frac{96}{n+1}$.

(2) $n=2$ and $\mathrm{mult}_{z^{\dagger}}R_{\dagger}\in 2\mathbb{Z}+1$.

Let $(\check{\psi}_{\Delta^{\dagger}})_{0}:(\mathaccent
866{W}_{\Delta^{\dagger}})_{0}\to W_{\Delta^{\dagger}}$ be the composite of
the above blowing-ups $v$-times over $z^{\dagger}_{j}$ for each
$j=1,\cdots,({T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}}\cdot
z^{\dagger})l$. Let $(\mathaccent 866{R}_{\dagger})_{0}$ be the branch locus
on $(\mathaccent 866{W}_{\Delta^{\dagger}})_{0}$. Then all exceptional curves
appearing from $(\check{\psi}_{\Delta^{\dagger}})_{0}$ are contained in
$(\mathaccent 866{R}_{\dagger})_{0}$ by the assumption (2). Thus, there exist
$\sharp({T(\mathaccent 869{K}^{\dagger})_{p^{\dagger}}}\cdot
z^{\dagger})(v-1)l$ double points of $(\mathaccent 866{R}_{\dagger})_{0}$. Let
$\check{\psi}_{\Delta^{\dagger}}:\mathaccent 866{W}_{\Delta^{\dagger}}\to
W_{\Delta^{\dagger}}$ be the composite of blowing-ups at those double points
and $(\check{\psi}_{\Delta^{\dagger}})_{0}$. Let $\mathaccent
866{R}_{\dagger}$ be the branch locus on $\mathaccent
866{W}_{\Delta^{\dagger}}$ and  $\mathaccent 866{\Gamma}_{p^{\dagger}}$ a
fiber of $\mathaccent
866{\varphi}:=\varphi_{\Delta^{\dagger}}\circ\check{\psi}_{\Delta^{\dagger}}$
over $p^{\dagger}$. Since an arbitrary $\mathaccent
866{\varphi}_{\Delta^{\dagger}}$-vertical component of $\mathaccent
866{R}_{\dagger}$ appears from either exceptional curves in
$(\check{\psi}_{\Delta^{\dagger}})_{0}$ or irreducible components of
$\Gamma_{p^{\dagger}}$, we have

$\displaystyle\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})\leq\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})vl.$

Put $\mathbb{D}:=\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})\cdot D_{1}$ and
$\gamma:=1-1/(D_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}$. Then we have

$\displaystyle\sum_{\mathaccent 866{z}\in\mathaccent
866{\Gamma}_{p}}\alpha^{+}_{0}(\Gamma_{p^{\dagger}})_{\mathaccent
866{z}}\geq\gamma(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l.$

Since $R_{\dagger}$ has $\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl$-singular points of
multiplicity at least $m$ and $\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})(v-1)l$ double points which
appear in $(\check{\psi}_{\Delta^{\dagger}})_{0}$, we have

$\displaystyle\sum_{k\geq
1}\left(3k-2\right)\check{\alpha}_{k}(\Gamma_{p^{\dagger}})\geq\left(3\left[\frac{m}{2}\right]-2\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl+\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})(v-1)l.$

Thus, we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\frac{\gamma}{2}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle+$
$\displaystyle\left(3\left[\frac{m}{2}\right]-2\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})vl+\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})(v-1)l$ $\displaystyle-$
$\displaystyle\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)+\frac{3}{2}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})+\check{j}_{0,1}(\Gamma_{p^{\dagger}})+\j_{0,1}^{\prime}(\Gamma_{p^{\dagger}})$

by Lemma 5.12. By
$\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})\geq\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l/12$ and
$2\sharp\Gamma_{p^{\dagger}}+(v-1)l\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot
z^{\dagger})\geq\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)$,
we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\frac{\gamma}{2}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle+$
$\displaystyle\left(3\left[\frac{m}{2}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot
z^{\dagger})vl+\left(\frac{1}{8}r-2\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l.$

Hence it is sufficient to show

$\displaystyle
4\left(\left(3\left[\frac{m}{2}\right]-2\right)v+\frac{1}{8}r-2\right)+2m(v+2)\left(2\gamma-
C_{2,2}\right)\geq 0.$

It clearly holds by (5.4). ∎

######  Proposition 5.17.

Assume $r\geq\mathrm{max}\\{60+\frac{12}{n^{2}-1}-\frac{96}{n+1},4n\\}$ and
$(C3)$ holds. Put

$C_{3,n}:=(n-1)(2n-1).$

Then it holds that

$\displaystyle 2\delta nK_{f}^{2}(\Gamma_{p})\geq C_{3,n}\sharp\mathaccent
869{K}.$

###### Proof.

Take $z^{\dagger}\in R_{\dagger}$ and $D\in
L_{\mathrm{tr}}(R_{\dagger},z^{\dagger})$. Denote by $\mathbb{D}$ the
$\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger}$)-orbit of $D$. We introduce the
following notations.

  * •

$m_{\mathbb{D}}:=\mathrm{mult}_{z^{\dagger}}\mathbb{D}$.

  * •

$m:=\mathrm{min}\\{m^{\prime}\mid m^{\prime}\geq m_{\mathbb{D}},m^{\prime}\in
n\mathbb{Z}_{>0}$ or $n\mathbb{Z}_{>0}+1\\}$.

From Lemma 5.13, there exist $\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l$ singular points of the
branch loci on $\Gamma_{p^{\dagger}}$ which is analytically equivalent to
$z^{\dagger}\in\mathbb{D}$. Let $\check{\psi}_{\Delta^{\dagger}}:\mathaccent
866{W}_{\Delta^{\dagger}}\to W_{\Delta^{\dagger}}$ be the composite of
blowing-ups at $z^{\dagger}_{j}$ ($j=1,\cdots,({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$). Let $\mathaccent
866{R}_{\dagger}$ be the branch locus on $\mathaccent
866{W}_{\Delta^{\dagger}}$ and  $\mathaccent 866{\Gamma}_{p^{\dagger}}$ a
fiber of $\mathaccent
866{\varphi}:=\varphi_{\Delta^{\dagger}}\circ\check{\psi}_{\Delta^{\dagger}}$
over $p^{\dagger}$. Put
$\gamma:=1-1/(D_{1},\Gamma_{p^{\dagger}})_{z^{\dagger}}$. Then we have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle+$
$\displaystyle\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})l$ $\displaystyle-$
$\displaystyle
2\frac{(n-1)^{2}}{n}\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)+\frac{n^{2}-1}{n}r\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}}).$

By
$\chi_{\varphi_{\Delta^{\dagger}}}(\Gamma_{p^{\dagger}})\geq\sharp\Gamma_{p^{\dagger}}/12$,
$\sharp\Gamma_{p^{\dagger}}\geq\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l$ and
$2\sharp\Gamma_{p^{\dagger}}\geq\left(\check{j}_{0,\bullet}(\Gamma_{p^{\dagger}})+j_{0,\bullet}^{\prime}(\Gamma_{p^{\dagger}})\right)$,
we have

$\displaystyle K_{f}^{2}(\Gamma_{p})\geq$
$\displaystyle\gamma\frac{(n-1)^{2}}{n}(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})$ $\displaystyle+$
$\displaystyle\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger})$ $\displaystyle+$
$\displaystyle\left(-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)\sharp({T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}\cdot z^{\dagger}).$

Thus, it is sufficient to show

$\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+\left(2\gamma(n-1)^{2}-C_{3,n}\right)(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq
0.$

Since $D$ is in $L_{\mathrm{tr}}((R_{\dagger})_{h},z^{\dagger})$ and
$\Gamma_{p^{\dagger}}$ has nodes at $z^{\dagger}$, we have $2m\geq
2m_{\mathbb{D}}=(\mathbb{D},\Gamma_{p^{\dagger}})_{z^{\dagger}}$ and
$\gamma\geq 1/2$. Hence it suffices to show

(5.7) $\displaystyle
2n\left(\left((n^{2}-1)\left[\frac{m}{n}\right]-n\right)-4\frac{(n-1)^{2}}{n}+\frac{n^{2}-1}{12n}r\right)+2m\left((n-1)^{2}-C_{3,n}\right)\geq
0.$

The coefficient of $m$ in (5.7)

$2(n^{2}-1)+2((n-1)^{2}-C_{3,n})$

is non-negative. Hence we may assume $m=n+1$ to show (5.7). Thus, it suffices
to show

$\displaystyle 2n(n^{2}-1)+2(n+1)\left((n-1)^{2}-C_{3,n}\right)\geq 0,$
$\displaystyle\frac{n^{2}-1}{12n}r-4\frac{(n-1)^{2}}{n}-n\geq 0.$

We can check it by simple calculations. ∎

######  Proposition 5.18.

One of $(C1)$, $(C2)$ and $(C3)$ holds.

###### Proof.

Supposing that (C1), (C2) and (C3) do not hold, we lead a contradiction. We
note that $\Gamma_{p^{\dagger}}\not\subset R_{\dagger}$ in this assumption.
Hence we have $\Gamma_{p}\not\subset R$. Take an arbitrary singular point
$z\in R$.  Let $z^{\dagger}\in R_{\dagger}$ be the singular point such that
$\Pi(z^{\dagger})=z$, where the morphism $\Pi$ is in (4.1). Since (C1) does
not hold, we have $\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger})\neq\\{\mathrm{id}\\}$. By
Corollary 4.8, the singular point $z^{\dagger}\in R_{\dagger}$ is on a node of
$\Gamma_{p^{\dagger}}$. Hence the singular point $z\in R$ is on a node of
$\Gamma_{p}$. Let $\Gamma_{1}$ and $\Gamma_{2}$ be local analytic branches of
$(\Gamma)_{\mathrm{red}}$ at $z$.  Since (C2) and (C3) do not hold, $R$
consists of $\Gamma_{1}$ and $\Gamma_{2}$ locally around $z$. Hence it occurs
only if $n=2$. If $\Gamma_{p}$ is of type $l\mathrm{I}_{1}$, we have
$(\Gamma_{p})_{\mathrm{red}}\subset R$. But it is a contradiction. Hence we
may assume that $\Gamma_{p}$ is of type $l\mathrm{I}_{c}$ with $c\geq 2$.
Since $R$ has the singular point $z$, we need to blow up at $z$ to get
$\mathaccent 869{R}$. Let $\overline{\Gamma}_{2}$ be the irreducible component
of $(\Gamma_{p})_{\mathrm{red}}$ which contains $\Gamma_{2}$ and $\mathaccent
869{\Gamma}_{2}$ the proper transform of $\Gamma_{2}$ in $\mathaccent 869{R}$.
By $n=2$, it holds $\mathaccent 869{\Gamma}_{2}^{2}\leq-4$. Hence we need
blowing-ups on $\overline{\Gamma}_{2}$ at least two times to get $\mathaccent
869{R}$. Furthermore, since an arbitrary singular point of $R$ is on a node of
$(\Gamma_{p})_{\mathrm{red}}$, there exists a singular point of $R$ on the
other node of $(\Gamma_{p})_{\mathrm{red}}$ consisting of
$\overline{\Gamma}_{2}$.  Let $z^{\dagger}_{23}$ be the node and
$\overline{\Gamma}_{3}$ another irreducible component of
$(\Gamma_{p})_{\mathrm{red}}$ passing through $z^{\dagger}_{23}$. We note that
the node $z^{\dagger}_{23}$ of $(\Gamma_{p})_{\mathrm{red}}$ consists of the
intersection of $\overline{\Gamma}_{2}$ and $\overline{\Gamma}_{3}$. Since
(C2) and (C3) do not hold, $R$ consists with $\overline{\Gamma}_{2}$ and
$\overline{\Gamma}_{3}$ locally around $z_{23}$. Hence we have
$\overline{\Gamma}_{3}\subset R$. Inductively, we can show
$(\Gamma_{p})_{\mathrm{red}}\subset R$. But it is a contradiction. ∎

###  5.2. The proof of Proposition 5.5

Let $\Gamma_{p}$ be of type $l\mathrm{I}_{c}$. Assume $R$ is smooth locally
around $\Gamma_{p}$ and has a good ramification point on $\Gamma_{p}$. We will
show

$\displaystyle 2n\delta K_{f}^{2}(\Gamma_{p})$
$\displaystyle\geq(n-1)^{2}\sharp\mathaccent 869{K}.$

Let $z$ be a good ramification point of $\varphi|_{R_{h}}$ and $\Gamma_{p}$
the fiber through $z$ of type $l\mathrm{I}_{c}$ where $l$ and $c$ are non-
negative integers. Recall the diagram (4.1).

$\textstyle{W_{\Delta^{\dagger}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta^{\dagger}}}$$\scriptstyle{\Pi}$$\textstyle{W_{\Delta}\times_{\Delta}\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\Delta}}$$\textstyle{\Delta^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Delta}$

Since the ramification index of $\varphi|_{R_{h}}:R_{h}\to B$ at $z$ is
greater than $l$, the restriction map
$\varphi_{\Delta{\dagger}}|_{(R_{\dagger})_{h}}:(R_{\dagger})_{h}\to\Delta^{\dagger}$
is ramified at $z^{\dagger}\in\Pi^{-1}(z)$. We have

$\displaystyle K_{f}^{2}(\Gamma_{p^{\dagger}})\geq$
$\displaystyle\frac{(n-1)^{2}}{n}\left(((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}-1\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger})l.$

Hence we have

$\displaystyle K_{f}^{2}(\Gamma_{p})\geq$
$\displaystyle\frac{(n-1)^{2}}{n}\left(((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}-1\right)\sharp(T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}\cdot z^{\dagger}).$

So it is sufficient to show

$\displaystyle
2\delta(n-1)^{2}\left(((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}-1\right)-\delta(n-1)^{2}\sharp(\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger}))\geq 0.$

From
$((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq\sharp(\mathrm{Stab}_{T(\mathaccent
869{K}^{\dagger})_{p^{\dagger}}}(z^{\dagger}))$, it suffices to show

$\displaystyle
2\delta(n-1)^{2}\left(((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}-1\right)-\delta(n-1)^{2}((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq
0.$

By the assumption $((R_{\dagger})_{h},\Gamma_{p^{\dagger}})_{z^{\dagger}}\geq
2$, we have the desired inequality.

##  6\. Upper bound of the order

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,1,n)$ over
an elliptic surface $\varphi:W\to B$. We consider the upper bound of the order
of the automorphism groups of $f$. Let $\Gamma_{p}$ be a fiber of $\varphi$
such that $K_{f}^{2}(\Gamma_{p})>0$. Put $r_{p}:=\sharp\mathrm{Stab}_{H}(p)$.
Since $K_{f}^{2}(\Gamma)\geq 0$ for an arbitrary fiber $\Gamma$, we have

$\displaystyle K_{f}^{2}\geq\sharp(H\cdot p)K_{f}^{2}(\Gamma_{p}).$

By $\sharp G=\sharp K\sharp H=n\sharp\mathaccent 869{K}\sharp H$, we have

$\displaystyle\sharp G\leq\frac{n\sharp\mathaccent
869{K}}{K_{f}^{2}(\Gamma_{p})}r_{p}K_{f}^{2}.$

Put

$\displaystyle\mu_{n}:=$ $\displaystyle\frac{12n^{2}\delta}{n^{2}-1},$
$\displaystyle\mu_{n}^{\prime}:=$
$\displaystyle\frac{6n^{2}\delta}{(n-1)(5n-4)}.$

If $K_{f}^{2}(\Gamma_{p})>0$ , we have

$\frac{n\sharp\mathaccent 869{K}}{K_{f}^{2}(\Gamma_{p})}\leq\mu_{n}.$

If $R$ has a singular point on $\Gamma_{p}$, we have

$\frac{n\sharp\mathaccent 869{K}}{K_{f}^{2}(\Gamma_{p})}\leq\mu_{n}^{\prime}$

from Proposition 5.3 and 5.4.  We recall

$\displaystyle\delta:=\mathrm{min}\\{\sharp\mathrm{Aut}(\Gamma_{p},O_{p})|\;\forall
p\in\Delta\textrm{ with }\Gamma_{p}\textrm{ is smooth}\\}.$

We note that $\delta=2,4$ or $6$.

######  Theorem 6.1.

Let $f:S\to B$ be a non-locally trivial primitive cyclic covering fibration of
type $(g,1,n)$ with
$g\geq\mathrm{max}\\{\frac{30n^{2}-47n+25}{n+1},\frac{7}{2}n(n-1)+1\\}$. Put

$\displaystyle\mu_{n}:=$ $\displaystyle\frac{12n^{2}\delta}{n^{2}-1}.$

Assume furthermore that when $g(B)=0$, $f$ has at least $3$ singular fibers.
Let $G$ be a finite subgroup of $\mathrm{Aut}(f)$. Then it holds

$\sharp G\leq\begin{cases}6(2g(B)-1)\mu_{n}K_{f}^{2}&(g(B)\geq 1),\\\
5\mu_{n}K_{f}^{2}&(g(B)=0).\end{cases}$

###### Proof.

Put $r_{p}=\sharp\mathrm{Stab}_{H}(p)$ for $p\in B$. Let $\pi:B\to B/H$ be the
quotient map of $B$ by $H$. Note that $r_{p}$ is the ramification index of
$\pi$ at $p$.

(i) The case of $g(B)\geq 2$.

We denote the genus of $B/H$ by $g(B/H)$. From the Hurwitz formula, we get

$\displaystyle 2g(B)-2=\sharp
H\left(2g(B/H)-2+\sum_{i=1}^{s}\frac{r_{i}-1}{r_{i}}\right),$

where $s$ is the number of ramification points and $r_{i}$ is the ramification
index. Put

$\displaystyle T:=2g(B/H)-2+\sum_{i=1}^{s}\frac{r_{i}-1}{r_{i}},$