Slope inequalities for irregular cyclic covering fibrations

TL;DR

This paper investigates the lower bounds of the slope for cyclic covering fibrations of fibered surfaces, especially when the relative irregularity is positive, contributing to the understanding of geometric inequalities.

Contribution

It provides new slope inequalities for irregular cyclic covering fibrations, extending previous results to cases with positive relative irregularity.

Findings

Established lower bounds for the slope in irregular cases

Extended slope inequality theory to cyclic coverings with positive irregularity

Enhanced understanding of the geometry of fibered surfaces

Abstract

Let $f:S\to B$ be a finite cyclic covering fibration of a fibered surface. We study the lower bound of slope $\lambda_{f}$ when the relative irregularity $q_{f}$ is positive.

Equations

$$K_f^2=K_S^2-8(g-1)(b-1),$$

$${\chi }_f:=\chi ({\mathcal{O}}_S)-(g-1)(b-1),$$

$$e_f:=e(S)-4(g-1)(b-1),$$

$${\lambda }_f:=\frac{K_f^2}{{\chi }_f}$$

$${\lambda }_f\ge \frac{4(g-1)}{g}.$$

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

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

$${\mathrm{mult}}_x{R}_h\le \frac{r}{2}=\frac{g}{n-1}+1$$

$$n\biggl{[}\frac{m}{n}\biggr{]}\leq\frac{r}{2}.$$

$$(\mathrm{deg}(\pi )-1)♯\Delta \ge \mathrm{deg}{R}_{\pi },$$

$$\mathrm{deg}{R}_{\pi }$$

$$=\mathrm{deg}(\pi )♯\Delta -(j_1+\cdots j_{♯\Delta })$$

$$\le (\mathrm{deg}(\pi )-1)♯\Delta ,$$

$$S_i=\mathrm{Spec}(\underset{j=0}{\overset{n-1}{⨁}}{\mathcal{O}}_{W_i}(-j{\mathfrak{d}}_i))$$

$$\text{mathaccent}869S=S_N\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K_{\tilde{f}}^2=n(K_{\tilde{\phi }}^2+2(n-1)\tilde{\mathfrak{d}}.K_{\tilde{\phi }}+(n-1{)}^2{\tilde{\mathfrak{d}}}^2),$$

$${\chi }_{\tilde{f}}=n{\chi }_{\tilde{\phi }}+\frac{1}{2}\sum _{j=0}^{n-1}j\tilde{\mathfrak{d}}(j\tilde{\mathfrak{d}}+K_{\tilde{\phi }}),$$

$$\displaystyle\tilde{\mathfrak{d}}^{2}=\mathfrak{d}^{2}-\sum_{i=1}^{N}\biggl{[}\frac{m_{i}}{n}\biggr{]}^{2},$$

$$\displaystyle\tilde{\mathfrak{d}}.K_{\tilde{\varphi}}=\mathfrak{d}.K_{\varphi}+\sum_{i=1}^{N}\biggl{[}\frac{m_{i}}{n}\biggr{]}$$

$$K_{\tilde{\phi }}^2=K_{\phi }^2-N.$$

$$N=\sum _{k\ge 1}{\alpha }_k,$$

$$\displaystyle\sum_{i=1}^{N}\biggl{[}\frac{m_{i}}{n}\biggr{]}=\sum_{k\geq 1}k\alpha_{k},$$

$$\displaystyle\sum_{i=1}^{N}\biggl{[}\frac{m_{i}}{n}\biggr{]}^{2}=\sum_{k\geq 1}k^{2}\alpha_{k}$$

$${\alpha }_0=(K_{\phi }+R)R-\sum _{k\ge 1}nk(nk-1){\alpha }_k.$$

$$K_{\tilde{f}}^2=n(K_{\phi }^2+2(n-1)\mathfrak{d}.K_{\phi }+(n-1{)}^2{\mathfrak{d}}^2)-n\sum _{k\ge 1}((n-1)k-1{)}^2{\alpha }_k$$

$${\chi }_{\tilde{f}}=n{\chi }_{\tilde{\phi }}+\frac{n(n-1)}{4}\mathfrak{d}.K_{\phi }+\frac{n(n-1)(2n-1)}{12}{\mathfrak{d}}^2-\frac{n(n-1)}{12}\sum _{k\ge 1}((2n-1)k^2-3k){\alpha }_k.$$

$$K_f^2\ge n(K_{\phi }^2+2(n-1)\mathfrak{d}.K_{\phi }+(n-1{)}^2{\mathfrak{d}}^2)-n\sum _{k\ge 1}((n-1)k-1{)}^2{\alpha }_k$$

$$\begin{gathered} {\chi }_f=n{\chi }_{\phi }+\frac{n(n-1)}{4}\mathfrak{d}.K_{\phi }+\frac{n(n-1)(2n-1)}{12}{\mathfrak{d}}^2 \\ -\frac{n(n-1)}{12}\sum _{k\ge 1}((2n-1)k^2-3k){\alpha }_k. \end{gathered}$$

$$(r-1)K_f^2$$

$$\displaystyle+\sum_{k\geq 1}^{nk\leq\frac{r}{2}}\biggl{(}\frac{n^{2}-1}{n}nk(r-1-(nk-1))-(r-1)n\biggr{)}\alpha_{k},$$

$$\displaystyle(r-1)\chi_{f}^{2}=\frac{(2r-3)n-r}{12n}(n-1)\alpha_{0}+\sum_{k\geq 1}^{nk\leq\frac{r}{2}}\biggl{(}\frac{n^{2}-1}{12n}nk(r-1-(nk-1))\biggr{)}\alpha_{k}.$$

$$R.K_{\phi }=-2M_0,$$

$$R^2=2r{M}_0.$$

$$(K_{\phi }+R).R=2(r-1)M_0.$$

$$2(r-1)M_0={\alpha }_0+\sum _{k\ge 1}^{nk\le \frac{r}{2}}nk(nk-1){\alpha }_k.$$

Full text

Introduction

Let $f:S\to B$ be a surjective morphism from a smooth projective surface $S$ to a smooth projective curve $B$ with connected fibers. We call it a fibration 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 $g\geq 2$. We denote by $K_{f}=K_{S}-f^{\ast}K_{B}$ a relative canonical divisor. We associate three relative invariants with $f$:

[TABLE]

where $b$ and $e(S)$ respectively denote the genus of the base curve $B$ and the topological Euler-Poincaré characteristic of $S$. Then the following are well-known:

• •

(Noether)  $12\chi_{f}=K_{f}^{2}+e_{f}$.

• •

(Arakelov) $K_{f}$ is nef.

• •

(Ueno) $\chi_{f}\geq 0$ and $\chi_{f}=0$ if and only if $f$ is locally trivial.

• •

(Segre) $e_{f}\geq 0$ and $e_{f}=0$ if and only if $f$ is smooth.

When $f$ is not locally trivial, we put

[TABLE]

and call it the slope of $f$ according to [5], in which Xiao succeeded in giving its effective lower bound as

[TABLE]

Another invariant we are interested in is the relative irregularity of $f$ defined by $q_{f}:=q(S)-b$, where $q(S):=\mathrm{dim}H^{1}(S,\mathcal{O}_{S})$ denotes the irregularity of $S$ as usual. When $q_{f}$ is positive, we call $f$ an irregular fibration. Xiao showed in [5] that $\lambda_{f}\geq 4$ holds for irregular fibrations. It seems a general rule that the lower bound of the slope goes up, when the relative irregular gets bigger.

In the present papaer, we consider primitive cyclic covering fibrations of type $(g,h,n)$ introduced in [2], where Enokizono gave the lower bound of the slope for them. Note that it is nothing more than a hyperelliptic fibration when $h=0$ and $n=2$. Recall that Lu and Zuo obtained the lower bound of the slope for irregular double covering fibrations in [4] and [3]. Inspired by their results, we try to generalize them to irregular primitive cyclic covering fibrations with $n\geq 3$. We give the lower bound of slope for those of type $(g,0,n)$ in Theorems 3.4 and 4.7, and for those of type $(g,h,n)$ with $h\geq 1$ in Theorem 3.8.

The key observation for the proof is Proposition 3.2. We apply it to the anti-invariant part of the Albanese map with respect to the action of the Galois group canonically associated to the cyclic covering fibration, and derive the “negativity” of the ramification divisor when $q_{f}>0$. Recall that $\chi_{f}$ and (the essential part of) $K_{f}^{2}$ can be expressed in terms of the so-called $k$-th singularity index $\alpha_{k}$ defined for each non-negative integer $k$. The negativity referred above can be used to get some non-trivial restrictions on $\alpha_{0}$ which is the most difficult one to handle with among all $\alpha_{k}$’s. Thanks to such information together with an analysis of the Albanese map, we can obtain the desired slope inequalities.

We also give a small contribution to the modified Xiao’s conjecture that $q_{f}\leq\lceil\frac{g+1}{2}\rceil$ holds, posed by Barja, González-Alonso and Naranjo in [1]. It is known to be true for fibrations of maximal Clifford index [1] and for hyperelliptic fibrations [4] among others. We show in Theorem 4.5 that $q_{f}\leq(g+1-n)/2$ holds, when $f$ is a primitive cyclic covering fibration of type $(g,0,n)$ under some additional assumptions. For the history around the conjecture, see the introduction of [1].

The author express his sincere gratitude to Professor Kazuhiro Konno for suggesting this assignment, his valuable advice and support. The author also thanks Dr. Makoto Enokizono for his precious advices, allowing him to use Proposition 3.2 freely.

1. Primitive cyclic covering fibrations

We recall the basis properties of primitive cyclic covering fibrations, most of which can be found in [2].

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 $(g,h,n)$, when there are $($not necessarily relatively minimal$)$ fibration $\tilde{\varphi}:\mathaccent 869{W}\to B$ of genus $h\geq 0$ and a classical $n$-cyclic covering

[TABLE]

branched over a smooth curve $\mathaccent 869{R}\in|n\mathaccent 869{\mathfrak{d}}|$ for some $n\geq 2$ and $\mathaccent 869{\mathfrak{d}}\in\mathrm{Pic}(\mathaccent 869{W})$ such that $f$ is the relatively minimal model of $\tilde{f}=\tilde{\varphi}\circ\tilde{\theta}$.

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,h,n)$. Let $\mathaccent 869{F}$ and $\mathaccent 869{\Gamma}$ be general fibers of $\tilde{f}$ and $\tilde{\varphi}$, respectively. Then the restriction map $\tilde{\theta}|_{\mathaccent 869{F}}:\mathaccent 869{F}\to\mathaccent 869{\Gamma}$ is a classical $n$-cyclic covering branched over $\mathaccent 869{R}\cap\mathaccent 869{\Gamma}$. By the Hurwitz formula for $\tilde{\theta}|_{\mathaccent 869{F}}$, we get

[TABLE]

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

Let $\tilde{\psi}:\mathaccent 869{W}\to W$ be the contraction morphism to a relative minimal model $W\to B$ of $\tilde{\varphi}:\mathaccent 869{W}\to B$. Since $\tilde{\psi}$ 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 $x_{i}\in W_{i-1}\;(i=1,\cdots,N)$, $W_{0}=W$ and $W_{N}=\mathaccent 869{W}$. We define a reduced curve $R_{i}$ inductively as $R_{i-1}=(\psi_{i})_{\ast}R_{i}$ starting from $R_{N}=\mathaccent 869{R}$ down to $R_{0}=R$. We also put $E_{i}=\psi_{i}^{-1}(x_{i})$ and $m_{i}=\mathrm{mult}_{x_{i}}R_{i-1}\;(i=1,\cdots,N)$.

Lemma 1.2** ([2], Lemma 1.5).**

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

$(1)$* Either $m_{i}\in n\mathbb{Z}$ or $n\mathbb{Z}+1$. Furthermore, $m_{i}\in n\mathbb{Z}$ 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 $\mathfrak{d}_{i}\in\mathrm{Pic}(W_{i})$ such that $\mathfrak{d}_{i}=\psi_{i}^{\ast}\mathfrak{d}_{i-1}-[\frac{m_{i}}{n}]E_{i}$ and $R_{i}\sim n\mathfrak{d}_{i}$, $\mathfrak{d}_{N}=\mathaccent 869{\mathfrak{d}}$.*

Remark 1.3*.*

By [2], we can assume the following for any primitive cyclic covering fibrations. Let $\tilde{\sigma}$ be a generator of the covering transformation group of $\tilde{\theta}$, and $\sigma$ the automorphism of $S$ over $B$ induced by $\tilde{\sigma}$. Then the natural morphism $\rho:\mathaccent 869{S}\to S$ is a minimal succession of blowing-ups that resolves all isolated fixed points of $\sigma$.

We must pay a special attention when $h=0$, since we have various relatively minimal models for $\tilde{\varphi}:\mathaccent 869{W}\to B$. Using elementary transformations, one can show the following.

Lemma 1.4** ([2], Lemma 3.1).**

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,0,n)$. Then there is a relatively minimal model of $\tilde{\varphi}:\mathaccent 869{W}\to B$ such that

[TABLE]

for all $x\in R_{h}$, where $R_{h}$ denotes the $\varphi$-horizontal part of $R$. Moreover if $\mathrm{mult}_{x}R>\frac{r}{2}$, then $\mathrm{mult}_{x}R\in n\mathbb{Z}+1$.

When $h=0$, we always assume that a relatively minimal model of $\tilde{\varphi}:\mathaccent 869{W}\to B$ is as in the above lemma.

Corollary 1.5**.**

Let the situation be the same as in Lemma 1.4. If $x$ is a singular point of $R$ and $m=\mathrm{mult}_{x}R$, then

[TABLE]

Proof.

When $m\in n\mathbb{Z}$, the inequality clearly holds by Lemma 1.4. If $m\in n\mathbb{Z}+1$, then $n[\frac{m}{n}]+1=m$. From Lemma 1.4, we have $m\leq\frac{r}{2}+1$. So we get $n[\frac{m}{n}]\leq\frac{r}{2}$. ∎

In closing the section, we give an easy lemma that will be usuful in the sequel.

Lemma 1.6**.**

Let $\pi:C_{1}\to C_{2}$ be a surjective morphism between smooth projective curves. Let $R_{\pi}$ and $\Delta$ be the ramification divisor and the branch locus of $\pi$, respectively. Then,

[TABLE]

where $\sharp\Delta$ denotes the cardinality of $\Delta$ as a set of points.

Proof.

We put $\Delta=\{Q_{1},\;\dots\;,Q_{\sharp\Delta}\}$. For any $Q_{i}\in\Delta$, we put $\pi^{-1}(Q_{i})=\{P_{1}^{i},\;\dots\;,P_{j_{i}}^{i}\}$. Note that $\deg(\pi)=r(P_{1}^{i})+\;\cdots\;+r(P_{j_{i}}^{i})$ for any $i=1,\;\dots\;,\sharp\Delta$, where $r(P)$ denotes the ramification index of $\pi$ around $P\in C_{1}$. Then, from the property of ramification divisor,

[TABLE]

which is what we want. ∎

2. Singularity indices and the formulae for $K_{f}^{2}$ and $\chi_{f}$.

We let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,h,n)$ and freely use the notation in the previous section. We obtain a classical $n$-cyclic covering $\theta_{i}:S_{i}\to W_{i}$ branched over $R_{i}$ by setting

[TABLE]

Since $R_{i}$ is reduced, $S_{i}$ is a normal surface. There exists a natural birational morphism $S_{i}\to S_{i-1}$. Set $S^{\prime}=S_{0}$, $\theta=\theta_{0}$, $\mathfrak{d}=\mathfrak{d}_{0}$ and $f^{\prime}=\varphi\circ\theta$. Then we have a commutative diagram:

[TABLE]

The well-known formulae for cyclic coverings give us

[TABLE]

(see e.g., [2]). From Lemma 1.2 and a simple calculation, we get

[TABLE]

and

[TABLE]

Definition 2.1*.*

Let $\Gamma_{p}$ and $F_{p}$ respectively denote fibers of $\varphi:W\to B$ and $f:S\to B$ over a point $p\in B$. For any fixed $p\in B$, we consider all singular points $($including infinitely near ones$)$ of $R$ on $\Gamma_{p}$. For any positive integer $k$, we let $\alpha_{k}(F_{p})$ be the number of singular points of multiplicity either $kn$ or $kn+1$ among them, and call it the $k$-th singularity index of $F_{p}$. We put $\alpha_{k}:=\sum_{p\in B}\alpha_{k}(F_{p})$ and call it the $k$-th singularity index of the fibration. We also put $\alpha_{0}:=(K_{\tilde{\varphi}}+\mathaccent 869{R})\mathaccent 869{R}$ and call it the ramification index of $\tilde{\varphi}|_{\mathaccent 869{R}}:\mathaccent 869{R}\to B$.

By a simple calculation, we get

[TABLE]

and

[TABLE]

Substituting (2.3) through (2.8) for (2.1) and (2.2), one gets

[TABLE]

and

[TABLE]

Since $K_{f}^{2}\geq K_{\tilde{f}}^{2}$ , $\chi_{\tilde{f}}=\chi_{f}$ and $\chi_{\tilde{\varphi}}=\chi_{\varphi}$, we obtain

[TABLE]

and

[TABLE]

We treat the cases $h=0$ and $h>0$ separately.

Proposition 2.2**.**

Let $f:S\to B$ be a primitive cyclic cover fibration of type $(g,0,n)$ and let $\alpha_{i}$ $(i\geq 0)$ be the singularity index in Definition 2.1. Then,

[TABLE]

[TABLE]

Proof.

Note that if $\alpha_{k}>0$, then $nk\leq\frac{r}{2}$ from Corollary 1.5.

We find that $R\equiv-\frac{r}{2}K_{\varphi}+M_{0}\Gamma$ for some $M_{0}\in\frac{1}{2}\mathbb{Z}$, where the symbol $\equiv$ means the numerical equivalence, since $\varphi:W\to B$ is a $\mathbb{P}^{1}$-bundle and we have $K_{W}.\Gamma=-2$ and $\mathaccent 869{R}.\mathaccent 869{\Gamma}=R.\Gamma=r$. Hence we get

[TABLE]

[TABLE]

Therefore we have

[TABLE]

From this equality and (2.9), we get

[TABLE]

On the other hand, substituting (2.14) and (2.15) for (2.10), we get

[TABLE]

Multiplying this by $r-1$ and substituting $(\ref{(1.16)})$ for it, we get (2.12). Similarly one can show (2.13). ∎

When $h>0$, we have the following:

Proposition 2.3**.**

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,h,n)$ such that $h\geq 1$ and $\alpha_{i}$ $(i\geq 0)$ the singularity index in Definition 2.1. Put

[TABLE]

[TABLE]

Then $t>0$ and $T\geq 0$. Furthermore,

[TABLE]

and

[TABLE]

hold, where

[TABLE]

[TABLE]

In $(\ref{1.17})$ and $(\ref{1.18})$, the quantity $\frac{K_{\varphi}^{2}}{(n-1)(h-1)}$ is understand to be zero, when $h=1$.

Proof.

We get $t>0$ from $r\geq 0$, $n\geq 2$ and $g\geq 2$. We shall show that $T\geq 0$. If $h=1$, by the canonical bundle formula, we have

[TABLE]

where $\{k_{i}\mid i=1,\dots,l\}$ denotes the set of multiplicities of all multiple fibers of $\varphi$, $k_{i}\geq 2$. Hence we get

[TABLE]

Since $W$ is an elliptic surface, we have $\chi(\mathcal{O}_{W})\geq 0$ and, hence, $T\geq 0$. If $h\geq 2$, we consider the intersection matrix

[TABLE]

for $\{K_{\varphi},\;\mathfrak{d},\;\Gamma\}$. Since we have $K_{\varphi}^{2}\geq 0$ by Arakelov’s theorem, it is not negative definite. Hence its determinant is non negative by the Hodge index theorem, and we get

[TABLE]

Since

[TABLE]

the inequality $(\ref{1.20})$ is equivalent to

[TABLE]

So we get

[TABLE]

and, hence, $T\geq 0$.

Now, by a direct calculation, one has

[TABLE]

and

[TABLE]

Hence we obtain from $(\ref{(1.10)})$ and $(\ref{(1.11)})$ that

[TABLE]

and

[TABLE]

From (2.9) and $(\ref{1.21})$, we get

[TABLE]

Since one sees $a_{k}=2(n-1)^{2}(g-1)k(nk-1)-nt((n-1)k-1)^{2}$, we obtain $(\ref{1.17})$. Similarly, we obtain $(\ref{1.18}).$ ∎

3. Slope inequality for irregular cyclic covering fibrations.

The purpose of this section is to show the slope inequalities for irregular cyclic covering fibrations of type $(g,h,n)$, $n\geq 3$. We start things in a more general setting.

Definition 3.1*.*

Let $\tilde{\theta}:\tilde{S}\to\tilde{W}$ be a finite Galois cover $($not necessarily primitive cyclic$)$ between smooth projective varieties with Galois group $G$. Let $\alpha:\mathaccent 869{S}\to\mathrm{Alb}(\mathaccent 869{S})$ be the Albanese map. For any $\tilde{\sigma}\in G$, we denote by $\alpha({\tilde{\sigma}}):\mathrm{Alb}(\mathaccent 869{S})\to\mathrm{Alb}(\mathaccent 869{S})$ the morphism induced from $\tilde{\sigma}:\mathaccent 869{S}\to\mathaccent 869{S}$ by the universality of the Albanese map. We put

[TABLE]

and let

[TABLE]

be the morphism defined by $\alpha_{\tilde{\sigma}}:=(\alpha(\tilde{\sigma})-1)\circ\alpha$.

The following is due to Makoto Enokizono.

Proposition 3.2**.**

Suppose that $G$ is a cyclic group generated by $\tilde{\sigma}$ in the above situation. If $q_{\mathaccent 869{\theta}}:=q(\mathaccent 869{S})-q(\mathaccent 869{W})>0$, then the following hold.

$(1)$* $\dim\mathrm{Alb}_{\tilde{\sigma}}(\mathaccent 869{S})=q_{\mathaccent 869{\theta}}$.*

$(2)$* If $\mathrm{Fix}(G):=\{x\in\mathaccent 869{S}\mid\tilde{\sigma}(x)=x\}\neq\emptyset$, then it is contracted by $\alpha_{\tilde{\sigma}}$ to the unit element $0\in\mathrm{Alb}_{\tilde{\sigma}}(\mathaccent 869{S})$.*

$(3)$* If $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve, then the geometric genus of $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is not less than $q_{\tilde{\theta}}$.*

Proof.

Firstly, we show (1). By the construction of $\alpha(\tilde{\sigma})-1:\mathrm{Alb}(\mathaccent 869{S})\to\mathrm{Alb}(\mathaccent 869{S})$, we get the following commutative diagram:

[TABLE]

Since $\alpha^{*}$ is an isomorphism, we have $\dim\mathrm{Ker}(\alpha(\tilde{\sigma})-1)^{\ast}=\dim\mathrm{Ker}(\tilde{\sigma}^{\ast}-1)$. Since $G$ is a cyclic group generated by $\tilde{\sigma}$, we see that $\mathrm{Ker}(\tilde{\sigma}^{\ast}-1)$ coincides with the $G$-invariant part $H^{0}(\mathaccent 869{S},\Omega^{1}_{\mathaccent 869{S}})^{G}$ of ${H}^{0}(\mathaccent 869{S},\Omega^{1}_{\mathaccent 869{S}})$. On the other hand, since $\tilde{\theta}^{\ast}:\mathrm{H}^{0}(\mathaccent 869{W},\Omega^{1}_{\mathaccent 869{W}})\to\mathrm{H}^{0}(\mathaccent 869{S},\Omega^{1}_{\mathaccent 869{S}})^{G}$ is an isomorphism, we have $\dim(\mathrm{Ker}(\alpha(\tilde{\sigma})-1)^{\ast})=q(\mathaccent 869{W})$ and, hence,

[TABLE]

It follows that $\mathrm{Alb}_{\tilde{\sigma}}(\mathaccent 869{S})$ is of dimension $q_{\mathaccent 869{\theta}}$.

Secondly, we show (2). We take a point $x_{0}$ in $\mathrm{Fix}(G)$ as the base point of the Albanese map $\alpha:\mathaccent 869{S}\to\mathrm{Alb}(\mathaccent 869{S}).$ Let $x\in\mathrm{Fix}(G)$. Note that we have

[TABLE]

and that $\alpha(\tilde{\sigma})(\alpha(x))-\alpha(x)$ is the function given by $\omega\mapsto\int_{x_{0}}^{x}\tilde{\sigma}^{\ast}\omega-\int_{x_{0}}^{x}\omega$ for $\omega\in\mathrm{H}^{0}(\mathaccent 869{S},\Omega^{1}_{\mathaccent 869{S}})$ modulo periods. Since $x$ and $x_{0}$ are both in $\mathrm{Fix}(G)$, we find

[TABLE]

Hence we get $\alpha_{\tilde{\sigma}}(x)=\alpha(\tilde{\sigma})(\alpha(x))-\alpha(x)=0$. Since $x$ can be taken arbitrarily in $\mathrm{Fix}(G)$, we get (2).

When $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve, one can check easily that the geometric genus of $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is not less than $q_{\tilde{\theta}}$, by (1) and the universality of the Abel-Jacobi map for the normalization of $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$. Hence (3). ∎

3.1. The slope inequality in the case of $h=0$

We consider the primitive cyclic covering fibration $f:S\to B$ of type $(g,0,n)$ with $q_{f}>0$. Since $\varphi:W\to B$ is a ruled surface, we have $q(W)=b$ and it follows $q_{\tilde{\theta}}=q_{f}$. We apply Proposition 3.2 to the cyclic covering $\tilde{\theta}:\mathaccent 869{S}\to\mathaccent 869{W}$ to find that its ramification divisor $\mathrm{Fix}(G)$ is contracted to a point by $\alpha_{\tilde{\sigma}}:\mathaccent 869{S}\to\mathrm{Alb}_{\tilde{\sigma}}(\mathaccent 869{S})$, where $\tilde{\sigma}$ denotes a generator of the Galois group $G:=\mathrm{Gal}(\mathaccent 869{S}/\mathaccent 869{W})$. So if $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a surface (resp. a curve), from Mumford’s theorem (resp. Zariski’s Lemma), the intersection form is negative definite (resp. semi-definite) on $\mathrm{Fix}(G)$, and we in particular get

[TABLE]

Hence, in any way, we have

[TABLE]

since $\tilde{\theta}^{*}\mathaccent 869{R}=n\mathrm{Fix}(G)$.

Here, we remark the following.

Lemma 3.3**.**

Let $f:S\to B$ be a primitive cyclic covering of type $(g,0,n)$. If it is not locally trivial and $q_{f}>0$, then $r\geq 2n$.

Proof.

We assume that $r<2n$ and show that this leads us to a contradiction.

Recall that $r$ is a multiple of $n$. If $r<2n$, then $r\leq n$ and $R$ has to be smooth by Lemmas 1.2 and 1.4. On the other hand, since $q_{f}>0$, we already know from (3.1) that the self-intersection number of any irreducible component $C$ of $R$ is non-positive.

Let $C_{0}$ be the minimal section with $C_{0}^{2}=-e$ and $\Gamma$ a fiber of $\varphi:W\to B$. Note that we can choose the normalized vector bundle of rank $2$ associated with $W$ so that there are no effective divisor numerically equivalent to $C_{0}-c\Gamma$ for any positive integer $c$. Put $C\equiv aC_{0}+b\Gamma$ with two integers $a$, $b$. We have $a\geq 0$. If $a=0$, then we have $b=1$, that is, $C$ is a single fiber by its irreducibility. So we may assume that $a$ is positive.

We have $C^{2}=a(2b-ae)\leq 0$. Hence $2b\leq ae$. Furthermore, since $C$ is irreducible and $a>0$, we have $(K_{\varphi}+C)C\geq 0$ by the Hurwitz formula applied for the normalization of $C$. Since $K_{\varphi}\equiv-2C_{0}-e\Gamma$, we have

[TABLE]

by $2b\leq ae$ and $a\geq 1$. Hence we get $(K_{\varphi}+C)C=0$ and, either $a=1$ or $2b=ae$. In particular, as the first equality shows, $C$ is smooth and $\varphi|_{C}:C\to B$ is unramified. Furthermore, we get $C^{2}=0$ when $a\geq 2$ by $2b=ae$. In this case, we also have $b\leq 0$, because $0\leq CC_{0}=b-ae=-b$. If $a=1$ and $2b<e$, then $b\geq 0$ and it follows from $CC_{0}=b-e<-b\leq 0$ that we have $C=C_{0}$ by the irreducibility of $C$. We remark here that we have $(a_{1}C_{0}+b_{1}\Gamma)(a_{2}C_{0}+b_{2}\Gamma)=0$ when $a_{i}>0$ and $2b_{i}=a_{i}e$ for $i=1,2$.

In summary, the only possibilities left for smooth $R$ are (i) $R$ consists of several fibers (including the case $R=0$), (ii) $R$ is the minimal section with $R^{2}<0$, and (iii) $R$ consists of several smooth curves with self-intersection numbers [math] which are unramified over $B$ (via $\varphi$). If (i) or (ii) is the case, then we have either $g=0$ or $r=1$, any of which is absurd. If (iii) is the case, then $f:S\to B$ is a locally trivial fibration, which is again inadequate. ∎

From $\mathaccent 869{R}^{2}=R^{2}-\sum_{k\geq 1}n^{2}k^{2}\alpha_{k}$, (2.15) and (3.1), we get

[TABLE]

Hence from this and (2.16), we get

[TABLE]

Theorem 3.4**.**

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,0,n)$ which is not locally trivial and $q_{f}>0$. Then

[TABLE]

Proof.

For $\lambda\in\mathbb{R}$, we put

[TABLE]

From Proposition 2.2, we get

[TABLE]

where

[TABLE]

[TABLE]

We can check that $A(\lambda_{g,n}^{1})\leq 0$ as follows. Since $r\geq 2n$ by Lemma 3.3, a calculation shows that the inequality

[TABLE]

is equivalent to

[TABLE]

and we can check easily its validity. Therefore $A(\lambda_{g,n}^{1})\leq 0$. Hence from $(\ref{(2.2)})$, we get

[TABLE]

For any integer $k$ satisfying $\frac{r}{2n}\geq k\geq 1$, we have $nk(r-nk)\geq n(r-n)$. Since we have

[TABLE]

the coefficient of $\alpha_{k}$ in the right hand side of $(\ref{2.2'})$ is not negative. Therefore, we get $K_{f}^{2}-\lambda_{g,n}^{1}\chi_{f}\geq 0$ as desired. ∎

3.2. The slope inequality in the case of $h\geq 1$.

Before showing the slope inequality when $h\geq 1$ and $n\geq 3$, we study the upper bound of $\alpha_{0}$.

Recall that we decomposed $\psi$ into a succession of blowing-ups $\psi_{i}$ as,

[TABLE]

We define the order of blowing-up $\psi^{\prime}$ appearing in $\psi$ as follows. If the center of $\psi^{\prime}$ is a point on the branch locus of multiplicity $m^{\prime}$, we put

[TABLE]

Moreover we introduce a partial order on these blowing-ups $\psi^{\prime}$ and $\psi^{\prime\prime}$ appearing in $\psi$,

[TABLE]

Lemma 3.5**.**

Assume that $n\geq 3$. Let $x_{j}$ $(\in R_{j}\subset W_{j})$ be a singular point infinitely near to $x_{i}\in R_{i}$. Then the multiplicities satisfy $m_{j}\leq m_{i}$.

Proof.

Though this can be found in [2], Lemma 3.7, when $n=0$, we shall give a proof for the convenience of readers.

Let $x_{i+1}$ be the singular point of $R_{i+1}$ infinitely near to $x_{i}\in R_{i}$. If $m_{i}\in n\mathbb{Z}$, then $R_{i+1}$ coincied with $\mathaccent 866{R_{i}}$, the proper transform of $R_{i}$ by $\psi_{i+1}$, by Lemma 1.2. Hence $m_{i+1}\leq m_{i}$ in this case. If $m_{i}\in n\mathbb{Z}$+1, then $R_{i+1}=\mathaccent 866{R_{i}}+E_{i+1}$. Hence we get $m_{i+1}\leq m_{i}+1\in n\mathbb{Z}+2$. From Lemma 1.2 and the assumption $n\geq 3$, we get $m_{i+1}\leq m_{i}$. ∎

From Lemma 3.5, we can reorder those blowing-ups appearing in $\psi$ so that $\psi_{i}\geq\psi_{j}$ holds whenever $i<j$. We put,

[TABLE]

Then we can decompose $\psi$ as

[TABLE]

in such a way that $\mathrm{ord}(\psi^{\prime})=M+1-i$ holds for any $\psi^{\prime}$ appearing in $\hat{\psi}_{i}$.

Lemma 3.6**.**

Let $\psi^{\prime}$ be a blowing-up appearing in $\hat{\psi}_{i}$ and $\mathaccent 869{D}$ the proper inverse image of the exceptional curve of $\psi^{\prime}$ on $\mathaccent 869{S}$. Then the geometric genus of $\mathaccent 869{D}$ satisfies

[TABLE]

Proof.

Let $m^{\prime}$ be the multiplicity of the singular point blown up by $\psi^{\prime}$, and $\mathaccent 869{E}$ the proper transform of the exceptional curve of $\psi^{\prime}$ on $\mathaccent 869{W}$.

When $m^{\prime}\in n\mathbb{Z}+1$, since $\mathaccent 869{E}$ is contained in $\mathaccent 869{R}$, $\mathaccent 869{D}$ is a smooth rational curve.

Assume that $m^{\prime}\in n\mathbb{Z}$. From $m^{\prime}=n(M+1-i)$, the intersection number of the exceptional curve of $\psi^{\prime}$ and the branch locus is $n(M+1-i)$. Hence the intersection number of their proper transforms on $\mathaccent 869{W}$ is at most $n(M+1-i)$. On the other hand, we consider the composite $\pi:\mathaccent 869{D^{\prime}}\to\mathaccent 869{D}\overset{\tilde{\theta}|_{\mathaccent 869{D}}}{\to}\mathaccent 869{E}$, where $\mathaccent 869{D^{\prime}}\to\mathaccent 869{D}$ is normalization of $\mathaccent 869{D}$, and let $B_{\pi}$ be the branch locus $\pi$. From the Hurwitz formula for $\pi$ and Lemma 1.6, we get

[TABLE]

Since $\tilde{\theta}|_{\tilde{D}}$ is totally ramified,

[TABLE]

Therefore we get

[TABLE]

which is what we want. ∎

Proposition 3.7**.**

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,h,n)$ such that $q_{\tilde{\theta}}>0$, $h\geq 1$, $n\geq 3$ and let $\alpha_{i}$ $(i\geq 0)$ be the singularity index in Definition 2.1. Then,

[TABLE]

where $\bar{a}_{k}$ is defined in Proposition 2.3. If the image $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve and $\nu(q_{\tilde{\theta}})\geq 1$, where

[TABLE]

then

[TABLE]

In $(\ref{2.4})$ and $(\ref{2.5})$, the quantity $\frac{K_{\varphi}^{2}}{(n-1)(h-1)}$ is understood to be zero when $h=1$.

Proof.

Firstly assume that $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve of geometric genus $g^{\prime}$. In this case, by Proposition 3.2, we have $g^{\prime}\geq q_{\tilde{\theta}}$ and see that any curve of geometric genus less than $g^{\prime}$ on $\mathaccent 869{S}$ is contracted by $\alpha_{\tilde{\sigma}}$. Hence, we know from Lemma 3.6 that for any $1\leq i\leq M$ satisfying

[TABLE]

the proper transform of the exceptional curve of $\hat{\psi}_{i}$ to $\mathaccent 869{S}$ is contracted by $\alpha_{\tilde{\sigma}}$. Then, since $q_{\tilde{\theta}}\leq g^{\prime}$, for any $1\leq i\leq M$ satisfying

[TABLE]

the same holds true. So we conclude that the total inverse image of $\mathaccent 866{R}_{M-\nu(q_{\tilde{\theta}})}$ in $\mathaccent 869{S}$ is contracted by $\alpha_{\tilde{\sigma}}$, where $\mathaccent 866{R}_{M-\nu(q_{\tilde{\theta}})}\subset\mathaccent 866{W}_{M-\nu(q_{\tilde{\theta}})}$ is the image of $\mathaccent 869{R}$. Therefore, the total inverse image of $\mathaccent 866{R}_{M-\nu(q_{\tilde{\theta}})}$ forms a negative semi-definite configuration. In particular, we have

[TABLE]

By the construction, we have

[TABLE]

where

[TABLE]

Hence from $(\ref{(1.9)})$, $(\ref{2.6})$ and the above equality, we get

[TABLE]

This is nothing more than $(\ref{2.5})$.

Even when $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is not a curve, we have $\mathaccent 869{R}^{2}\leq 0$ by Proposition 3.2. Using this instead of (3.1), we get $(\ref{2.4})$ by a similar argument as above. ∎

Theorem 3.8**.**

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,h,n)$ which is not locally trivial and such that $q_{\tilde{\theta}}$ and $h$ are both positive, $n\geq 3$. Put

[TABLE]

(i)* If $F(g,h,0)\geq 0$, then*

[TABLE]

(ii)* Assume that $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve and $\nu(q_{\tilde{\theta}})\geq 1$. If $F(g,h,\nu(q_{\tilde{\theta}}))\geq 0$, then*

[TABLE]

Proof.

Here we restrict ourselves to the case that $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve and show (ii) only, since (i) can be shown similarly. From (2.1) and (2.1), we obtain

[TABLE]

To apply $(\ref{2.5})$ to the above inequality, we have to check that $z^{\prime}-\lambda_{g,h,n,q_{\tilde{\theta}}}\bar{z}^{\prime}\leq 0$ in advance. Since

[TABLE]

it is sufficient to see that

[TABLE]

which is equivalent to

[TABLE]

the validity of which can be checked directly. Therefore we get $z^{\prime}-\lambda_{g,h,n,q_{\tilde{\theta}}}\bar{z}^{\prime}\leq 0$. Applying $(\ref{2.5})$ to $(\ref{2.7})$, we obtain

[TABLE]

We will show that \big{(}(n-1)k((2n-1)k-3)\lambda_{g,h,n,q_{\tilde{\theta}}}-12((n-1)k-1)^{2}\bigr{)}\geq 0 for $1\leq k\leq\nu(q_{\tilde{\theta}}).$ Note that

[TABLE]

Firstly we will show that $\lambda_{g,h,n,q_{\tilde{\theta}}}\geq\frac{12(n-1)}{2n-1}$. From $(\ref{2.7''})$, it is sufficient to check that

[TABLE]

A calculation shows that it is equivalent to

[TABLE]

which holds true clearly. So we have shown $\lambda_{g,h,n,q_{\tilde{\theta}}}\geq\frac{12(n-1)}{2n-1}$ and it follows that

[TABLE]

is increasing in $k$. Evaluating at $k=1$, we get

[TABLE]

by $\lambda_{g,h,n,q_{\tilde{\theta}}}\geq\frac{12(n-1)}{2n-1}$. Since

[TABLE]

holds for any $k\geq\nu(q_{\tilde{\theta}})+1$, we obtain

[TABLE]

If $F(g,h,\nu(q_{\tilde{\theta}}))\geq 0$ and $h=1$, then by $(\ref{1.19})$ we have $T=2(g-1)K_{\varphi}.R\geq\frac{4(g-1)^{2}}{n-1}\chi_{\varphi}$. Hence it follows from $(\ref{2.12})$ that

[TABLE]

which gives us (ii) for $h=1$. If $F(g,h,\nu(q_{\tilde{\theta}}))\geq 0$ and $h\geq 2$, then we can use Xiao’s slope inequality $K_{\varphi}^{2}\geq\frac{4(h-1)}{h}\chi_{\varphi}$ and $T\geq 0$, to get

[TABLE]

from $(\ref{2.12})$. Hence we have shown (ii) also for $h\geq 2$. ∎

4. Special irregular cyclic covering fibrations of ruled surfaces.

Let $f:S\to B$ be a primitive cyclic covering fibration of type $(g,0,n)$ with $q_{f}>0$ and suppose that it is not locally trivial. Let $\alpha_{\tilde{\sigma}}:\mathaccent 869{S}\to\mathrm{Alb}_{\tilde{\sigma}}(\mathaccent 869{S})$ be the morphism defined as in Definition 3.1 for the generator $\tilde{\sigma}$ of the covering transformation group $G$ of $\tilde{\theta}:\mathaccent 869{S}\to\mathaccent 869{W}$. Moreover we assume that there is a component $C$ of $\mathrm{Fix}(G)$ such that $C^{2}=0$. Note then that $\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$ is a curve by Proposition 3.2.

Proposition 4.1**.**

In the above situation, there are a fibrations $\tilde{f^{\prime}}:\mathaccent 869{S}\to B^{\prime}$, $\varphi^{\prime}:\mathaccent 869{W}\to\mathbb{P}^{1}$ and a morphism $\theta^{\prime}:B^{\prime}\to\mathbb{P}^{1}$, where $B^{\prime}$ is s smooth curve, such that $\mathaccent 869{R}$ is $\tilde{\varphi^{\prime}}$-vertical, $q_{f}\leq g(B^{\prime})$, and they fit into the commutative diagram:

[TABLE]

Proof.

We can obtain $\tilde{f^{\prime}}:\mathaccent 869{S}\to B^{\prime}$ from the Stein factorization of $\alpha_{\tilde{\sigma}}:\mathaccent 869{S}\to\alpha_{\tilde{\sigma}}(\mathaccent 869{S})$. Hence we have $g(B^{\prime})\geq q_{f}$ by Proposition 3.2, (3). We will show that the automorphism $\tilde{\sigma}:\mathaccent 869{S}\to\mathaccent 869{S}$ induces an automorphism of $B^{\prime}$. We assume that there is a fiber $F^{\prime}$ of $\tilde{f^{\prime}}$ such that $\tilde{\sigma}^{\ast}F^{\prime}$ has a $\tilde{f}^{\prime}$-horizontal component. Let $F_{C}^{\prime}$ be the fiber of $f^{\prime}$ which contains the curve $C$ with $C^{2}=0$. Then, from Zariski’s lemma, we see that $F_{C}^{\prime}=aC$ for some positive integer $a$ and it follows $F_{C}^{\prime}=\tilde{\sigma}^{\ast}F_{C}^{\prime}$, since $C$ is a component of $\mathrm{Fix}(G)$. Hence

[TABLE]

a contradiction. Therefore $\tilde{\sigma}$ maps fibers to fibers, and descends down to give an automorphism $\tilde{\sigma}_{B^{\prime}}:B^{\prime}\to B^{\prime}$. Furthermore we have the commutative diagram

[TABLE]

where $\theta^{\prime}:B^{\prime}\to D^{\prime}:=B^{\prime}/\langle\tilde{\sigma}_{B^{\prime}}\rangle$ denotes the quotient map. In order to complete the proof, it suffices to see that $D^{\prime}=\mathbb{P}^{1}$. This can be shown as follows. Any general fiber of $\tilde{\varphi}$ is $\tilde{\varphi^{\prime}}$-horizontal by $\mathaccent 869{R}.\mathaccent 869{\Gamma}>0$. Since $\tilde{\varphi}$ is ruled, we see that $\mathbb{P}^{1}$ dominates $D^{\prime}$ and it follows $D^{\prime}=\mathbb{P}^{1}$. ∎

The contraction $\varphi:\mathaccent 869{W}\to W$ is composed of several blowing-ups. We decompose it as $\psi=\check{\psi}\circ\bar{\psi}$ as follows. Let $\bar{\psi}:\mathaccent 869{W}\to\overline{W}$ be the longest succession of blowing-downs such that we still have the morphism $\bar{\varphi^{\prime}}$ satisfying $\tilde{\varphi^{\prime}}=\bar{\varphi^{\prime}}\circ\bar{\psi}$. Then we have the following commutative diagram.

[TABLE]

Let $\overline{R}:=\bar{\psi}_{\ast}\mathaccent 869{R}$ be the image of $\mathaccent 869{R}$ by $\bar{\psi}$.

Lemma 4.2**.**

The morphism $\check{\psi}:\overline{W}\to W$ is not the identity map.

Proof.

We will prove this by contradiction. Suppose that $\check{\psi}$ is the identity map. As one sees from the proof of Lemma 3.3, any irreducible curve $D$ on $W$ with $D^{2}\leq 0$ is smooth, and $\varphi|_{D}:D\to B$ is an unramified covering when $D^{2}=0$ and $D$ is not a fiber of $\varphi$. Hence, any irreducible fiber of $\bar{\varphi^{\prime}}:W\to\mathbb{P}^{1}$ has to be smooth. Suppose that there is a fiber of $\bar{\varphi^{\prime}}$ whose reduced scheme is reducible, and take an irreducible component $D_{0}$. Then $D_{0}^{2}<0$ and, from the proof of Lemma 3.3, we conclude that $D_{0}$ coincides with the minimal section. The unicity implies that we cannot have such reducible singular fibers. Therefore, a singular fiber of $\bar{\varphi^{\prime}}$, if any, is a multiple fiber whose support is a smooth irreducible curve. Since $R$ is a reduced divisor with support in fibers of $\bar{\varphi^{\prime}}$ by Proposition 4.1, we see that $R$ is smooth and $\varphi|_{R}:R\to B$ is unramified. Then $f:S\to B$ is a locally trivial fibration, which is inadequate. ∎

Assume that $\theta^{\prime}:B^{\prime}\to\mathbb{P}^{1}$ is branched over $\Delta\subset\mathbb{P}^{1}$. For any $y\in\Delta$, let $\mathaccent 869{\Gamma_{y}^{\prime}}=\sum\tilde{n}_{C}C$ be the fiber of $\tilde{\varphi^{\prime}}$ over $y$, and put

[TABLE]

Lemma 4.3**.**

In the above situation, $\mathaccent 869{R}_{r}\preceq\mathaccent 869{R}.$

Proof.

We put

[TABLE]

Since $\tilde{f^{\prime}}\circ\tau=\tau$ for any $\tau\in G_{\tilde{f^{\prime}}}$, the morphism $\tilde{f^{\prime}}$ induces the morphism $\pi:\mathaccent 869{S}/G_{\tilde{f^{\prime}}}\to\mathaccent 869{W}$ and we have the following commutative diagram.

[TABLE]

Note that the degree of $\pi$ is equal to that of $\theta^{\prime}$. We claim that $\mathrm{Fix}(G_{\tilde{f^{\prime}}})=\mathrm{Fix}(G)$. This can be see as follows. It is clear that $\mathrm{Fix}(G_{\tilde{f^{\prime}}})\supset\mathrm{Fix}(G)$. If there is a point $x\in\mathrm{Fix}(G_{\tilde{f^{\prime}}})\setminus\mathrm{Fix}(G)$, then we have $\tilde{\sigma}(x)\neq x$ for the generator $\tilde{\sigma}$ of $G$. On the other hand, since $G_{\tilde{f^{\prime}}}$ is a subgroup of $G$ of order $n/\mathrm{deg}\,\theta^{\prime}$, we have $G_{\tilde{f^{\prime}}}=\langle\tilde{\sigma}^{\mathrm{deg}\,\theta^{\prime}}\rangle$. Hence the number of $G$-orbits of $x$ is at most $n/\mathrm{deg}\,\theta^{\prime}$. This contradicts that $\tilde{\theta}:\mathaccent 869{S}\to\mathaccent 869{W}$ is totally ramified. Therefore $\mathrm{Fix}(G_{\tilde{f^{\prime}}})=\mathrm{Fix}(G)$. Hence $\mathaccent 869{S}/G_{\tilde{f^{\prime}}}$ is smooth. Let $R_{\pi}$ be the branch locus of $\pi$. Since $\tilde{\theta}$ is totally ramified, one can check easily that $R_{\pi}=\mathaccent 869{R}$. Hence it is sufficient to prove that $\mathaccent 869{R}_{r}\leq R_{\pi}$. Let $C$ be any component of $\mathaccent 869{R}_{r}$. We can take analytic local coordinates $(U_{\mathbb{P}}^{1},x)$ on $\mathbb{P}^{1}$, $(U_{\mathaccent 869{W}},y,z)$ on $\mathaccent 869{W}$ and $(U_{B^{\prime}},w)$ on $B^{\prime}$ such that $\tilde{\varphi^{\prime}}(C)$ is defined by $x=0$, $C$ is defined by $y=0$, $\theta^{\prime\ast}x=w^{\mathrm{deg}\,\theta^{\prime}}$ and $\bar{\varphi^{\prime}}^{\ast}x=y$. $U_{B^{\prime}}\times_{\mathbb{P}^{1}}U_{\mathaccent 869{W}}$ is defined by $y=w^{\mathrm{deg}\,\theta^{\prime}}$ in $U_{B^{\prime}}\times U_{\mathaccent 869{W}}$. So $U_{B^{\prime}}\times_{\mathbb{P}^{1}}U_{\mathaccent 869{W}}\to U_{\mathaccent 869{W}}$ is ramified over $C\cap U_{\mathaccent 869{W}}$ and $U_{B^{\prime}}\times_{\mathbb{P}^{1}}U_{\mathaccent 869{W}}$ is smooth. Hence the natural morphism $\mathaccent 869{S}/G_{\tilde{f^{\prime}}}\to B^{\prime}\times_{\mathbb{P}^{1}}\mathaccent 869{W}$ is an isomorphism around $U_{B^{\prime}}\times_{\mathbb{P}^{1}}U_{\mathaccent 869{W}}$. Therefore we get $C\preceq R_{\pi}=\mathaccent 869{R}$. ∎

We suppose that $\check{\psi}=\check{\psi}_{1}\circ\;\cdots\;\circ\check{\psi}_{u}$, where $\check{\psi}_{i}:\check{W_{i}}\to\check{W}_{i-1}$ is a blowing-up at $\check{x}_{i-1}\in\check{W}_{i-1}$ with exceptional curve $\check{\mathcal{E}_{i}}\subset\check{W}$, $\check{W}_{0}=W$ and $\check{W}_{u}=\overline{W}$. Let $\check{R}_{i}$ be the image of $\overline{R}$ in $\check{W}_{i}$, and let $\check{x_{i}}$ be a singular point of $\check{R_{i}}$ of multiplicity $\check{m_{i}}$.

Lemma 4.4**.**

Assume that $n\geq 3$. For $1\leq i\leq u-1$, we have $\check{m}_{i}\geq\frac{2q_{f}}{n-1}+2$. Moreover if there is $\check{m}_{i}$ such that equality sign holds, then $\frac{2q_{f}}{n-1}+2\in n\mathbb{Z}$ and $\deg\theta^{\prime}=n$.

Proof.

Let $\mathcal{E}\subset\overline{W}$ be any $(-1)$-curve contracted by $\check{\psi}$. Note that $\bar{\varphi^{\prime}}|:\mathcal{E}\to\mathbb{P}^{1}$ is surjective and $\mathcal{E}.\overline{R}\in n\mathbb{Z}$. We will show that $\mathcal{E}.\overline{R}\geq\frac{2q_{f}}{n-1}+2$. If $n\geq\frac{2q_{f}}{n-1}+2$, then the assertion is clear from Lemma 1.2, (1). Hence we may assume that $n<\frac{2q_{f}}{n-1}+2$ in the following. In particular, we have $q_{f}\geq n-1$.

By Lemma 4.3, it is enough to show that $\mathcal{E}.\overline{R}_{r}\geq\frac{2q_{f}}{n-1}+2$ where $\overline{R}_{r}$ is image of $\mathaccent 869{R}_{r}.$ Let $\mathcal{R}_{\theta^{\prime}}$ be the ramification divisor of $\theta^{\prime}:B^{\prime}\to\mathbb{P}^{1}$. Since $g(B^{\prime})\geq q_{f}>0$, we have $\deg\theta^{\prime}>1$. From the Hurwitz formula, we get

[TABLE]

By Lemma 1.6 and $(\ref{(3.4)})$, we get $(\deg\theta^{\prime}-1)\sharp\Delta\geq 2g(B^{\prime})-2+2\deg\theta^{\prime}$, that is,

[TABLE]

We put $\overline{R}_{all}:=\bar{\psi^{\prime}}^{\ast}\Delta$. For any $p\in\mathcal{E}\cap\overline{R}_{all}$, let $r_{p}:=I_{p}(\mathcal{E}.\overline{R}_{all})$ be the local intersection number. Since $\overline{R}_{all}=\bar{\psi^{\prime}}^{\ast}\Delta$ consists of $\sharp\Delta$ fibers of $\bar{\varphi^{\prime}}$, one has

[TABLE]

By definition, $r_{p}\geq 2$ for any $p\in(\mathcal{E}\cap\overline{R}_{all})\setminus(\mathcal{E}\cap\overline{R}_{r})$. On the other hand, by the Hurwitz formula for $\bar{\varphi^{\prime}}|_{\mathcal{E}}:\mathcal{E}\to\mathbb{P}^{1}$, one has

[TABLE]

Hence,

[TABLE]

where the last inequality comes from $(\ref{(3.5)})$.

Note that we have $g(B^{\prime})\geq q_{f}\geq n-1\geq\deg\theta^{\prime}-1$. Therefore

[TABLE]

For any $\check{x_{i}}$, let $\check{x}_{i+j_{i}}$ be the last infinitely near singular point blown up by $\check{\psi}$, $\mathcal{E}_{i+j_{i}}$ the exceptional curve. Then, from the above argument, we get

[TABLE]

Moreover if the equality signs hold everywhere, then we get

[TABLE]

Since $\deg\theta^{\prime}\leq n$, we are done. ∎

Theorem 4.5**.**

Let $f:S\to B$ be a locally non-trivial primitive cyclic covering fibration of type $(g,0,n)$ with $q_{f}>0$ and $n\geq 3$. Assume that there is a component C of $\mathrm{Fix}(\tilde{\sigma})$ such that $C^{2}=0$. Then,

[TABLE]

Proof.

There is a singular point $x$ of $R$ which is blown up by $\check{\psi}:\overline{W}\to W$ from Lemma 4.2. If $m:=\mathrm{mult}_{x}R\leq\frac{r}{2}$, then

[TABLE]

from Lemma 4.4. So we get $q_{f}\leq(g-n+1)/2$. If $m>\frac{r}{2}$, then we have $m\in n\mathbb{Z}+1$ from Lemma 1.4. Let $\check{x}$ be the last singular point, infinitely near to $x$, blown up by $\check{\psi}$ and $\check{m}$ its multiplicity. Then it holds that $\check{m}\in n\mathbb{Z}$. Indeed, if $\check{m}\in n\mathbb{Z}+1$, the exceptional curve arizing from $\check{x}$ is contained in branch locus. It contradicts the definition of $\check{\psi}$. So we get $\check{m}+1\leq m$. Therefore we get

[TABLE]

from Lemma 1.4 and Lemma 4.4. It follows $q_{f}\leq(g-n+1)/2$. ∎

Therefore, the Modified Xiao’s Conjecture is true in this particular case.

Now, we turn our attention to the slope. Let $\check{\alpha}_{k}$ be the number of the singular points of $R$ with multiplicity $nk$ or $nk+1$ appearing in $\check{\psi}$. Then $\check{\alpha}_{k}\geq 0$ and by Lemma 4.4 one has

[TABLE]

for any $k$ satisfying $nk+1\leq\frac{2q_{f}}{n-1}+2$. We put $\bar{\alpha}_{k}:=\alpha_{k}-\check{\alpha}_{k}$ then by $(\ref{(3.6)})$,

[TABLE]

for any $k$ satisfying $nk+1\leq\frac{2q_{f}}{n-1}+2$. By the construction of $\bar{\varphi^{\prime}}$, $\overline{R}$ is contained in fibers of $\bar{\varphi^{\prime}}$, hence we get

[TABLE]

On the other hand, we have

[TABLE]

As $R^{2}=2rM_{0}$ by (2.15), we get

[TABLE]

Hence

[TABLE]

where the first and the last inequalities above follow immediately from $\bar{\alpha}_{k}\geq 0$ and $(\ref{(3.7)})$, and the second one follows from ($\ref{(1.16)})$ and ($\ref{(3.8)})$. Hence we have shown:

Proposition 4.6**.**

Under the same assumptions as in Proposition 4.1,

[TABLE]

Using this, we will prove the following:

Theorem 4.7**.**

Let $f:S\to B$ be a locally non-trivial primitive cyclic covering fibration of type $(g,0,n)$ such that there is a component $C\subset\mathrm{Fix}(G)$ with $C^{2}=0$. If $q_{f}>0$ and $n\geq 3$, then

[TABLE]

Proof.

We first remark that, for two real numbers $x$, $y$ with $x+y\leq r$, we have $x(r-x)\geq y(r-y)$ if and only if $x\geq y$. Since we have $n+(2+2q_{f}/(n-1))\leq r$ by the proof of Theorem 4.5 and $r\geq 2n$, this observation works for $x=n$, $y=2+2q_{f}/(n-1)$.

(i) The case of $n\geq 2+\frac{2q_{f}}{n-1}$, i.e., $\frac{(n-2)(n-1)}{2}\geq q_{f}$.

Since $n\geq 2+\frac{2q_{f}}{n-1}$, we get

[TABLE]

Therefore, $\lambda_{g,n}^{1}\geq\lambda_{g,n,q_{f}}^{2}$, and $(\ref{(3.9)})$ follows from Theorem 3.4.

(ii) The case of $n<2+\frac{2q_{f}}{n-1}$.

In this case, we have

[TABLE]

and, hence, $\lambda_{g,n}^{1}\leq\lambda_{g,n,q_{f}}^{2}$. Then we have $A(\lambda_{g,n,q_{f}}^{2})\leq 0$, since the function $A(\lambda)$ defined in the proof of Proposition 3.4 is decreasing in $\lambda$ and we have already proved $A(\lambda_{g,n}^{1})\leq 0$ there.

From Proposition 2.2, we have

[TABLE]

where $a_{k}$ and $\bar{a_{k}}$ are the same as in the proof of Proposition 3.4. Applying Proposition 4.6 to (4.1), we get

[TABLE]

First, we will show

[TABLE]

for any positive integer $k$ with $nk+1\leq 2+\frac{2q_{f}}{n-1}$. By a simple calculation, we get

[TABLE]

We claim that $\lambda_{g,n,q_{f}}^{2}\geq\frac{12(n-1)}{2n-1}$. It is equivalent to

[TABLE]

From $g-n+1\geq 2q_{f}$ and $q_{f}\geq\frac{(n-2)(n-1)}{2}+1$, we easily see that it holds true. Since $\lambda_{g,n,q_{f}}^{2}\geq\frac{12(n-1)}{2n-1}$, the right hand side of (4.3), which is incleasing in $k$, is not less than

[TABLE]

Therefore we get (4.2).

Secondly, we will show

[TABLE]

for any positive integer $k$ satisfying $\frac{r}{2}\geq nk\geq 2+\frac{2q_{f}}{n-1}$. By a simple calculation, we get

[TABLE]

Since we have

[TABLE]

for any positive integer $k$ satisfying $2+\frac{2q_{f}}{n-1}\leq nk\leq\frac{r}{2}$, the right hand side of $(\ref{(3.14)})$ is not less than

[TABLE]

In sum, we have shown $K_{f}^{2}-\lambda_{g,n,q_{f}}^{2}\chi_{f}\geq 0$. ∎

5. An example.

We construct primitive cyclic covering fibrations of type $(g,0,n)$ with relative minimal irregularity $q_{f}$ satisfying $g+n-1=m(q_{f}+n-1)$ for any integer $m\geq 2$. Hence, when $m=2$, this implies that the bound of $q_{f}$ in Theorem 4.5 is sharp. Also, our examples show that the slope bound (4.1) in Theorem 4.7 is sharp.

Let $\varphi:W:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(e))\to B:=\mathbb{P}^{1}$ be the Hirzebruch surface of degree $e\geq 0$. Denote by $\Gamma$ and $C_{0}$ a fiber of $\varphi$ and the section with $C_{0}^{2}=-e$, respectively. We know that $mC_{0}+b\Gamma$ is very ample if and only if $b>me$. So we take $b_{0}$ with $b_{0}>me$.

We take two general members $D,D^{\prime}$ of $|mC_{0}+b_{0}\Gamma|$ which intersect each other transversely. Let $\Lambda$ be the pencil generated by $D$ and $D^{\prime}$. Then $\Lambda$ define the rational map $\varphi_{\Lambda}:W\cdots\to\mathbb{P}^{1}$. Let $\psi$ be a minimal succession blowing-ups which eliminates the base points of $\Lambda$. We get a relatively minimal fibration $\tilde{\varphi^{\prime}}:\mathaccent 869{W}\to\mathbb{P}^{1}$ by putting $\tilde{\varphi^{\prime}}=\varphi_{\Lambda}\circ\psi$. Denote by $\mathaccent 869{\Gamma}^{\prime}$ a general fiber of $\tilde{\varphi}^{\prime}$ and $K_{\mathaccent 869{W}}$ a canonical divisor of $\mathaccent 869{W}$. By a simple calculation, we get

[TABLE]

where $x$ is the a number of blowing-ups in $\psi$. Note that $x=(mC_{0}+b_{0}\Gamma)^{2}$.

Let $\Delta\subset\mathbb{P}^{1}$ be a set of $\frac{2q}{n-1}+2$ general points, where $q$ is an integer satisfying $\frac{2q}{n-1}+2\in n\mathbb{Z}$. Then there is a divisor $\mathfrak{d}^{\prime}$ on $\mathbb{P}^{1}$ such that $n\mathfrak{d}^{\prime}=\Delta$. Let $\mathaccent 869{R}=(\tilde{\varphi}^{\prime})^{\ast}\Delta$ be the fiber of $\tilde{\varphi}^{\prime}$ over $\Delta$. Since $\Delta$ is general, we can assume that $\mathaccent 869{R}$ is both reduced and smooth.

We consider a classical cyclic $n$-covering

[TABLE]

Since $\Delta$ is general, we can assume that the fiber product $\mathaccent 869{S}:=B\times_{\mathbb{P}^{1}}\mathaccent 869{W}$ is smooth. Noting that the morphism $\tilde{\theta}:\mathaccent 869{S}\to\mathaccent 869{W}$ induced by $\theta^{\prime}$ is nothing but the natural one

[TABLE]

one gets a commutative diagram

[TABLE]

where $\tilde{f}:=\tilde{\varphi}\circ\tilde{\theta}$.

By the construction, we get $q_{f}=q=g(B^{\prime})$. From the formulae

[TABLE]

and $(\ref{5.1})$, we get

[TABLE]

[TABLE]

Let $g$ be the genus of fibration $f:\mathaccent 869{S}\to B$. Then it is easy to see that

[TABLE]

Hence we get

[TABLE]

Therefore we get

[TABLE]

by $q=q_{f}$.

We remark that $\tilde{f}$ is relatively minimal. In fact the singular points of $R$, the image of $\mathaccent 869{R}$ in W, are all of multiplicity $\frac{2q_{f}}{n-1}+2\in n\mathbb{Z}$ and can be resolved by a single blowing-up. So there is no $\tilde{\varphi}$-vertical $(-n)$-curve in $\mathaccent 869{W}$. Therefore there is no $\tilde{f}$-vertical $(-1)$-curve in $\mathaccent 869{S}$.