# On open algebraic surfaces of general type whose log canonical maps composed
of a pencil

Hang Zhao  Hang Zhao
School of Mathematics and Statistics
Yunnan University
Kunming, Yunnan 650091, PR China<EMAIL_ADDRESS>
###### Abstract.

Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective
surface and $D$ a simple normal corssing reduced divisor on $S$. We assume
that its log canonial linear system $|K_{S}+D|$ is composed of a penciel, let
$f\colon\thinspace S\to B$ be the fiberation induced by the linear system
$|K_{S}+D|$ and $F$ be a general fiber of $f$. Let $b$ (resp. $g$) be the
genus of the base curve $B$ (resp. general fiber $F$) and $k=D\cdot F$ the
intersection number. We show that

  1. (1)

If $k>0$ and $b\geq 2$ then $2\leq g+k\leq 3$, when $g+k=3$ we have $b=2$ and
$h^{1}(S,K_{S}+D)=0$.

  2. (2)

Suppose $p_{a}(D)\leq 2(l+q(S))+1-h^{1,1}(S)$ where $l$ is the number of
irreducible components of $D$, then we have $g\leq 5$ for $p_{g}(S,D)\gg 0$.
Moreover if $p_{g}(S)=0$, then we have $g\leq 3$.

###### Key words and phrases:

Open algebraic surfaces, Log canonical linear system, composed of a pencil

###### 2010 Mathematics Subject Classification:

14J50, 14J30

The author was supported in part by the science and technology innovation fund
of Yunnan University: ST20210105.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
    1. 2.1 Nef vector bundles
    2. 2.2 Divisors of surfaces
    3. 2.3 Theory of Peeling
    4. 2.4 The Bogomolov-Miyaoka-Yau inequality to log surfaces
    5. 2.5 Linear systems on rational surfaces
    6. 2.6 Semi-positivity of $f_{*}\omega_{S/B}(D)$
  3. 3 Proofs of the Main Theorem
  4. 4 Examples

##  1\. Introduction

The pluricanonical maps of a surface with non-negative kodaira dimension play
a important role in the calssification of algebraic surfaces. Let $S$ be a
surface, and let $K_{S}$ be a canonical divisor. Suppose the $n$-genus
$P_{n}(S)=h^{0}(S,\mathscr{O}_{S}(nK_{S}))\geq 2$, we can define a rational
map $\phi_{n}\colon\thinspace S\dashrightarrow\mathbb{P}^{P_{n}(S)-1}$, which
is called the _$n$ th-pluricanonical map_ of $S$. Bombieri [Bom73] showed that
the $n$th-pluricanonical maps $\phi_{n}$ are embeddings when $n\geq 5$ and are
birational when $n\geq 3$ except for finite many cases; the bicanonical map
$\phi_{2}$ is birational if $K_{S}^{2}\geq 10$ and $p_{g}(S)\geq 6$ except for
$S$ admiting a genus 2 fiberation, in which case $\phi_{2}$ is generically a
double covering.

The image $\Sigma$ of the canonical map $\phi_{1}\colon\thinspace
S\dashrightarrow\mathbb{P}^{p_{g}(S)-1}$ is either a surface or a curve. If
$\Sigma$ is a curve, we say that the canonical linear system $|K_{S}|$ is
composed of a pencil. Let $f\colon\thinspace X\to B$ be the fiberation induced
by the rational map $\phi_{1}$, i.e., $\mu\colon\thinspace X\to S$ resolve the
base locus of $|K_{S}|$ and take $f$ to be the Stein factorization of the
composition morphism $\phi_{1}\circ\mu\colon\thinspace X\to\Sigma$. Let $g,b$
and $q$ be the genus of a general fiber of $f$, the genus of $B$ and the
irregularity $q(S):=h^{0}(S,\mathscr{O}_{S})$ of $S$. Beauville [Bea79] showed
that $2\leq g\leq 5$ if $\chi(S,\mathscr{O}_{S})\geq 21.$ Later, Xiao [Xia85]
showed that either $b=q=1$ or $b=0,q\leq 2$.

If $\Sigma$ is a surface. Beauville [Bea79] showed that either
$p_{g}(\Sigma)=0$ or $p_{g}(\Sigma)=p_{g}(S)$ in which case $\Sigma$ is of
general type. In the first case, if $\chi(S,\mathscr{O}_{S})\geq 31$, then
$\deg\phi_{1}\leq 9$; in the second case, if $\chi(S,\mathscr{O}_{S})\geq 14$,
then $\deg\phi_{1}\leq 3$. More refiment results can be find in [Xia86].

In this paper we will generalize the above result to open algebraic surfaces,
for general references here are [Iit76, Iit77]. Let $S^{0}$ be an open
algebraic surface, by Nagata and Hironaka, we can embed it into a complete
algebraic surface $S^{0}\hookrightarrow S$ such that the complement
$D=S\setminus S^{0}$ is a divisor with simple normal corssing singularities.
It is well known that cohomology groups of many vector bundles constructed
using logarithmic forms with poles algong $D$ depend only on $S^{0}$. Thus if
we want to study properties of $S^{0}$, it is natural to consider the
properties of the divisor $K_{S}+D$ in the spirit of the theory of birational
geometry. Thus it is convenient to consider a log pair $(S,D)$ consists of a
smooth projective surface $S$ and a reduced effective divisor $D$ on $S$ with
only simple normal crossing singularities, such a pair is also called _log
smooth surface_. We will consider a _minimal_ log smooth surface $(S,D)$, in
the sence of Tsunoda which he called _almost minimal_ (see for instance
[Fuj79, Tsu83]).

######  Definition 1.1.

We say that a projective log smooth surface $(S,D)$ is _minimal_ if $K_{S}+D$
has the _Zariski-Fujita decomposition_
$K_{S}+D=(K_{S}+D^{\sharp})+\mathrm{Bk}(D)$ satisfying the following
conditions:

  1. (1)

$\mathrm{Bk}(D)$ is an effecitve $\mathbb{Q}$-divisor such that
$\mathrm{Bk}(D)\leq D$;

  2. (2)

$K_{S}+D^{\sharp}$ is nef;

  3. (3)

the intersection matrix $\mathrm{Bk}(D)$ is negative defined;

  4. (4)

$(K_{S}+D^{\sharp})\cdot C=0$ for any prime component $C$ of $\mathrm{Bk}(D)$.

We can define analogous invariants of a log surface as usual surface:
$p_{g}(S,D):=h^{0}(S,K_{S}+D)$ the _log geometric genus_ ,
$q(S,D)=h^{0}(S,\Omega_{S}^{1}(D))$ the _log irregularity_ ,
$\kappa(S,D)=\kappa(K_{S}+D)$ the _log Kodaira dimension_. We say that a log
surface $(S,D)$ is of _log general type_ if $\kappa(S,D)=2$ or
$(K_{S}+D^{\sharp})^{2}>0$.

The log canonical map $\phi\colon\thinspace
S\dashrightarrow\mathbb{P}^{p_{g}(S,D)-1}$ has been study by many authors.
Sakai [Sak80] considered the case when $D$ is a stable curve and proved a
Noether type inequality and a generalization of Bogomolov-Miyaoka-Yau
inequality. Soon afterwards, Tsunoda and De-Qi Zhang [TZ92] generalized the
result to the case when $(S,D)$ is minimal log smooth and of log general type.

Our purpose is to consider minimal log surfaces $(S,D)$ of log general type
whose log canonical linear system $|K_{S}+D|$ is composed of a penciel. We can
assume, by blowing up at the base points of $|K_{S}+D|$ if necessary, that
$|K_{S}+D|$ is base point free (see Lemma 3.2). Let $\phi\colon\thinspace
S\to\Sigma\subset\mathbb{P}^{p_{g}(S,D)-1}$ be the correspongding morphism,
then $\Sigma$ is a curve. Let $f\colon\thinspace S\to B$ be the Stein
factorization of $\phi\colon\thinspace S\to\Sigma$. We will study the
fiberation $f\colon\thinspace S\to B$ induced by the linear system
$|K_{S}+D|$. Let $g$ and $b$ denote the genus of a general fiber $F$ of $f$
and the base curve $B$ respectively. It is of interest to know the
possibilities of invariants $g,b$ and the intersection numbers $k=D\cdot F$.
The main results of this paper are the following.

######  Theorem 1.2.

With the notations as above. Then we have

  1. (1)

If $k>0$ and $b\geq 2$ then $2\leq g+k\leq 3$, when $g+k=3$ we have $b=2$ and
$h^{1}(S,K_{S}+D)=0$.

  2. (2)

Suppose $p_{a}(D)\leq 2(l+q(S))+1-h^{1,1}(S)$ where $l$ is the number of
irreducible components of $D$, then we have $3\leq 2g+k\leq 9$ for
$p_{g}(S,D)\gg 0$. Moreover if $p_{g}(X)=0$, then we have $2g+k\leq 6$.

Noteworthy, our examples of $b=0$ demonstrates rather strikingly that $g,k$
can be arbitrarily large, see Example 4.4, note that $S$ are rational surfaces
and they do not satisfy the inequality $p_{a}(D)\leq 2(l+q(S))+1-h^{1,1}(S)$
of Theorem 1.2.

##  2\. Preliminary

Throughout we work over an algebraically closed field of characteristic $0$.
We set up notaion and terminology.

A _fiberation_ is a projective surjective morphism with connected fibers
between two normal varieties.

We write $D\geq 0$ for an effective $\mathbb{Q}$-divisor on a normal variety
$X$. Let $E$ be a prime divisor on $X$, we denote by $\mu_{D}(E)$ the
coefficient of $E$ in $D$.

###  2.1. Nef vector bundles

In this subsection we recall some basic results on nef vector bundles that
will be used throughtout the paper.

######  Definition 2.1.

A vector bundle $\mathscr{E}$ on a variety $X$ is said to be _nef_ if the
tautological line bundle $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$ is a nef
line bundle on the projective bundle $\mathbb{P}_{X}(\mathscr{E})$ over $X$.

######  Proposition 2.2.

Let $\mathscr{E}$ be a vector bundle on a variety $X$.

  1. (1)

If $\mathscr{E}$ is nef, then so is any quotient bundle $\mathscr{Q}$ of
$\mathscr{E}$.

  2. (2)

Let $f\colon\thinspace X\to Y$ be a finite morphism of varieties. If
$\mathscr{E}$ is a nef vector bundle on $Y$, then $\mathscr{E}$ is nef if and
only if the pull-back $f^{*}\mathscr{E}$ is a nef vector bundle on $X$.

  3. (3)

Let $\mathscr{E}$ be a vector bundle on a variety $X$, then $\mathscr{E}$ is
nef if and only if given any finite morphism $\nu\colon\thinspace C\to X$ form
a non-singular projective curve $C$ to $X$, and given any quotient line bundle
$\mathscr{L}$ of $\nu^{*}\mathscr{E}$, one has $\deg\mathscr{L}\geq 0$.

######  Corollary 2.3.

Let $\mathscr{E}$ be a vector bundle on a non-singular projective curve $C$ of
genus $g$. If $\mathscr{E}\otimes\omega_{C}^{-1}$ is nef, then
$\deg\mathscr{E}\geq 2(g-1)\mathrm{rank}\,\mathscr{E}$ and thus
$\chi(C,\mathscr{E})\geq(g-1)\mathrm{rank}\,{\mathscr{E}}$.

###### Proof.

By the Riemann-Roch theorem,
$\chi(C,\mathscr{E})=\deg\mathscr{E}-\mathrm{rank}\,\mathscr{E}(g-1)$. If
$\mathscr{E}\otimes\omega_{C}^{-1}$ is nef, then we have $\deg\mathscr{E}\geq
2(g-1)\mathrm{rank}\,\mathscr{E}$ and thus
$\chi(C,\mathscr{E})\geq(g-1)\mathrm{rank}\,\mathscr{E}$. ∎

Let $f\colon\thinspace S\to B$ be a fiberation from a smooth projective
surface to a smooth curve. By Fujita’s theorem [Fuj78], $f_{*}\omega_{S/B}$ is
a nef vector bundle on $B$. Let $D$ be a reduced divisor on $S$ which has only
simple normal crossing singularities. To show that the sheaf
$f_{*}\omega_{S/B}(D)$ is also nef we need the following.

######  Proposition 2.4.

[CG17] Let $\mathscr{E}$ be a vector bundle on a non-singular projective curve
$C$, and $\mathscr{F}$ be a subvector bundle of $\mathscr{E}$ with the same
rank of $\mathscr{E}$. If $\mathscr{F}$ is nef, then so is $\mathscr{E}$.

###  2.2. Divisors of surfaces

We review some of the standard facts on divisors of algebraic surfaces.

A reduced divisor $D=\sum C_{j}$ on a surface $S$ is also called a curve. The
_arithmetic genus_ of $D$ is $p_{a}(D)\colonequals 1-\chi(D,\mathscr{O}_{D})$.
By Riemann-Roch formula, we have

$K_{S}\cdot D+D^{2}=2p_{a}(D)-2.$

######  Lemma 2.5.

Let $D$ be a reduced divisor on a surface $S$. If we write $D=D_{1}+D_{2}$
into sum of two reduced divisors $D_{1}$ and $D_{2}$, then

$p_{a}(D)=p_{a}(D_{1})+p_{a}(D_{2})+D_{1}\cdot D_{2}-1.$

In particular, if we write $D=\sum_{i=1}^{r}D_{i}$ into sum of all its
irreducible components, then we have

$p_{a}(D)=\sum_{i=1}^{r}p_{a}(D_{i})+1+\sum_{i<j}D_{i}\cdot D_{j}-r.$

The _weighted dual graph_ $\Gamma=\Gamma(D)$ of $D$ is the following: each
curve $C_{j}$ in $D$ corresponds to a vertex $v_{j}$. Two vertices $v_{i}$ and
$v_{j}$ are connected by an edge of weight $m$ if the corresponding curves has
intersection number $C_{i}\cdot C_{j}=m$. Each vertex $v_{j}$ has a weight
$n_{j}=-C_{j}^{2}$. The branching muber, denoted by $\beta(v_{j})$ or
$\beta_{D}(C_{j})$, of a vertex $v_{j}$ is the muber of vertices which is
connected to $v_{j}$.

A reduced divisor $D$ is called _simple normal corssing_ (SNC for short) if
each irreducible component is non-singular and $D$ has at most node. In this
case we will say that the pair $(S,D)$ is _log smooth_. The dual graph
$\Gamma$ of a SNC divisor $D$ is a simple graph, i.e., it has no loops (an
edge connected one vertex to itself) and multiple edges.

######  Remark 2.6.

Let $D$ be a reduced SNC divisor on $S$ whose dual graph has $r$ vertices and
$l$ edges. By Lemma 2.5, its arithmetic genus $p_{a}(D)$ equals
$\sum_{i=1}^{r}p_{a}(D_{i})+1+l-r.$ Note that, as $D$ is SNC, the sum
$\sum_{i<j}D_{i}\cdot D_{j}$ of intersection numbers equals $l$.

A $\mathbb{Q}$-divisor is said to be _nef_ (resp. _pseudo-effective_) if
$D\cdot C\geq 0$ for any irreducible curve $C$ (resp. $D\cdot P$ for any nef
$\mathbb{Q}$-divisor $P$) on $S$.

For a pseudo-effective $\mathbb{Q}$-divisor $D$ on a smooth projective surface
$S$, we have the _Zariski-Fujita decomposition_ $D=P+N$ which satisfies and is
determined by the following numerical properties

  1. (1)

$P$ and $N$ are $\mathbb{Q}$-divisors;

  2. (2)

$N\geq 0$ and the intersection matrix $(C_{i}\cdot C_{j})$ for the prime
components $C_{i}$ of $N$ is negative defined.

  3. (3)

$P$ is nef;

  4. (4)

$P\cdot C_{i}=0$ for every prime componet $C_{i}$ of $N$.

We call $P$ (resp. $N$) the _nef part_ (resp. _negative part_) of $D$. For the
proofs we refer the reader to [Fuj79].

###  2.3. Theory of Peeling

A connected curve $E$ in $D$ is said to be _rational_ (resp. _admissible_) if
it consists only of rational curves (resp. if there are no $(-1)$-curves in
$E$ and the intersection matrix of $E$ is negative defiend). Let $L$ be an
admissible rational linear chain in $D$. We write $L=\sum_{i=1}^{r}C_{i}$ as
the sum of its prime components. Suppose $L$ satisfies the condition:
$\beta_{D}(C_{1})=1$ and $\beta_{D}(C_{i})=2$ for $2\leq i\leq r-1$. If
$\beta_{D}(C_{r})=1$, we call $L$ a _rod_ , i.e., $L$ is a connected component
of $D$. If $\beta_{D}(C_{r})=2$ and $C_{r}\cdot C_{r+1}=1$ for a prime
component $C_{r+1}$ other than $C_{r-1}$, we call $L$ a _twig_. An admissible
rational twig is called maximal, if it is not extended to an admissible
rational twig with more irreducible components of $D$, i.e.,
$\beta_{D}(C_{r+1})\geq 3$. A connected curve $F$ in $D$ is called a _fork_ if
its dual graph is that of the exceptional curves of a minimal resolution of a
log terminal singular point and is not a chain. The prime component $C$ of a
linear chain $L$ in $D$ with $\beta_{D}(C)=1$ is called a _tip_.

Let $(S,D)$ be a log smooth pair with $K_{S}+D$ is pseudo-effective. In this
subsection, we will consider the Zariski-Fujita decomposition

$K_{S}+D=(K_{S}+D)^{+}+(K_{S}+D)^{-}.$

Here and subseqently, $(K_{S}+D)^{+}$ (resp. $(K_{S}+D)^{-}$) will denote the
nef part (resp. negative part) of $K_{S}+D$.

######  Lemma 2.7.

Let $S$ be a smooth projective surface, and $D$ be a $\mathbb{Q}$-boundary on
$S$ such that the log pair $(S,D)$ is log smooth. If $E$ is an irreducible
curve on $S$ with $E^{2}\leq-2$ and $(K_{S}+D)\cdot E<0$, then
$E\subset\mathrm{Supp}\,(D)$ and $E$ is a smooth rational curve.

###### Proof.

By assumption, we have

$K_{S}\cdot E=2p_{a}(E)-2-E^{2}\geq 0$

and thus $D\cdot E<0$, $E$ is a prime component of $D$. Write
$D=aE+D^{\prime}$ with $0<a\leq 1$ and
$E\nsubseteq\mathrm{Supp}\,(D^{\prime})$, we have

$2p_{a}(E)-2=(K_{S}+E)\cdot E\leq(K_{S}+aE+D^{\prime})\cdot=(K_{S}+D)\cdot
E<0.$

Thus $p_{a}(E)=0$, so that $E$ is a smooth rational curve. ∎

Let $L$ be an admissible rational linear chain in $D$. We write
$L=\sum_{i=1}^{r}C_{i}$ as the sum of its prime components. Suppose
$\beta_{D}(C_{1})=1$, thus $C_{1}$ is the tip of $L$. By the adjunction
formula, for each $C_{j}$, we have

$(K_{S}+D)\cdot C_{j}=-2+\beta_{D}(C_{j}).$

Note that the intersection matrix $(C_{i}\cdot C_{j})$ is negative defined,
the following system of $r$ linear equations in $r$ variables has a unique
solution

(2.1) $\sum_{i=1}^{r}a_{i}C_{i}\cdot C_{j}=(K_{S}+D)\cdot C_{j}$

Since $(K_{S}+D)\cdot C_{1}=-1<0$, it follows that $0<a_{i}\leq 1$ for all
$1\leq i\leq r$. The $\mathbb{Q}$-divisor
$\mathrm{Bk}(L)=\sum_{i=1}^{r}a_{i}C_{i}$ is called _the bark_ of $L$, and the
process of substracting $\mathrm{Bk}(L)$ from $L$ is called _the peeling_ of
$L$.

Let $F$ be an admissible rational fork, write $F=\sum_{i=1}^{r}C_{i}$ as the
sum of its prime components. In the same manner we can see that the system of
linear equations 2.1 has a unique solution
$\mathrm{Bk}(F)=\sum_{i=1}^{r}a_{i}C_{i}$ with $0<a_{i}\leq 1$ for all $1\leq
i\leq r$, which is also called the Bark of $F$.

Let $\\{T_{\lambda}\\}$, $\\{R_{\mu}\\}$ and $\\{F_{\nu}\\}$ be the sets of
all admissible rational maximal twigs, all admissible rational rods and all
admissible rational forks, respectively, where no prime coponent of
$T_{\lambda}$’s belong to $R_{\mu}$’s and $F_{\nu}$’s. Since $T_{\lambda}$’s,
$R_{\mu}$’s and $F_{\nu}$’s are disjoint from each other, we can peel the
barks of $T_{\lambda}$’s, $R_{\mu}$’s and $F_{\nu}$’s independently. We set

$D^{\sharp}=D-\sum_{\lambda}\mathrm{Bk}(T_{\lambda})-\sum_{\mu}\mathrm{Bk}(R_{\mu})-\sum_{\nu}\mathrm{Bk}(F_{\nu}).$

The $\mathbb{Q}$-divisor $\mathrm{Bk}(D)=D-D^{\sharp}$ is called _the bark_ of
$D$. By the above construction, we have

  1. (1)

$D^{\sharp}=D-\mathrm{Bk}(D)\geq 0$;

  2. (2)

write $\mathrm{Bk}(D)=\sum_{i}a_{i}C_{i}$, the intersection matrix
$(C_{i}\cdot C_{j})_{1\leq i,j\leq r}$ is negative defined;

  3. (3)

$(K_{S}+D^{\sharp})\cdot C=0$ for every prime component $C$ of
$T_{\lambda}$’s, $R_{\mu}$’s and $F_{\nu}$’s;

  4. (4)

$(K_{S}+D)^{+}=(K_{S}+D^{\sharp})^{+}$ and
$(K_{S}+D)^{-}=(K_{S}+D^{\sharp})^{-}+\mathrm{Bk}(D)$.

######  Lemma 2.8.

Let $(S,D)$ be a log smooth pair of log general type with $D$ reduced. With
the notations as above. If $E$ is an irreducible curve on $S$ such that
$(K_{S}+D^{\sharp})\cdot E<0$, then $E$ is an exceptional curve of the first
kind.

###### Proof.

Consider the Zariski-Fujita decompositon $K_{S}+D^{\sharp}=P+N$, where $P$ is
the nef part and $N$ is the negative part of $K_{S}+D^{\sharp}$. Since $(S,D)$
is of log general type, we have $P^{2}>0$. By assumption,
$(K_{S}+D^{\sharp})\cdot E<0$ and note that $P\cdot E\geq 0$, it follows that
$N\cdot E<0$ and thus $E\subset\mathrm{Supp}\,(N)$. By the properties of
Zariski-Fujita decompositon, we have $P\cdot E=0$. By the Hodge index theorem,
we see that $E^{2}\leq 0$. Again by the properties of Zariski-Fujita
decompositon, the intersection matrix of supports of $N$ is negative defined,
we infer that $E^{2}<0$. It remains to prove that the case $E^{2}\leq-2$ can
not occur.

Suppose $E^{2}\leq-2$, then by Lemma 2.7, we have
$E\subset\mathrm{Supp}\,(D^{\sharp})$ and $E$ is a smooth rational curve.
Since $(K_{S}+D^{\sharp})\cdot C=0$ every prime component $C$ of
$\mathrm{Bk}(D)$, it follows that $\mu_{D^{\sharp}}(E)=1$. Let
$C_{1},\dots,C_{k}$ be the prime components of $D$ that meets $E$ where
$k=\beta_{D}(E)$. Set $a_{i}=\mu_{D^{\sharp}}(C_{i})$, then we have
$(K_{S}+D^{\sharp})\cdot E=-2+a_{1}+\cdots+a_{k}$. Note that $(K_{S}+D)\cdot
E=-2+k$, if $k\leq 1$, then $E$ is either an isolated component of $D$ or a
tip of an admissible rational maximal twig or an admissible rational rod, in
either case we have $\mu_{\mathrm{Bk}(D)}(E)>0$, which contradicts to
$\mu_{D^{\sharp}}(E)=1$. Therefore we can assume that $k\geq 2$. By
assumption, we have $a_{1}+\cdots+a_{k}<2$ and $k\geq 2$. Thus at least one
$i$ with $a_{i}<1$, say $a_{1}<1$. We can conclude that there exists an
admissible maximal rational twig $T$ of $D$ with $C_{1}\leq T$ and
$E\nsubseteq\mathrm{Supp}\,(T)$. If $k=2$, then $T+E$ is also an admissible
rational twig of $D$, which contradicts the maximality of $T$. By Lemma
[Fuj82, Lemma 6.16], $a_{i}=1-\frac{1}{d_{i}}$ with $d_{i}\geq 2$ or
$a_{i}=1$, let $l$ be the number of $i$ with $a_{i}<1$, so we have
$k-2<\frac{1}{d_{1}}+\cdots+\frac{1}{d_{l}}$ and $k\geq 3$. It follows that
$l=k=3$. Therefore $a_{i}<1$ for all $1\leq i\leq 3$, then there exists an
admissible rational fork $F$ such that $E+C_{1}+C_{2}+C_{3}\leq F$, in this
case $\mu_{\mathrm{Bk}(F)}(E)>0$, which contradicts to
$\mu_{\mathrm{Bk}(D)}(E)=0$ and the proof is complete. ∎

After a succesion of several contractions of excetional curve $E$ as in Lemma
2.8, we obtain log smooth pair $(\bar{S},\bar{D})$ and a birational morphism
$\sigma\colon\thinspace S\to\bar{S}$ where $\bar{D}=\sigma_{*}(D)$, which
satisfies the following properties:

  1. (1)

$K_{\bar{S}}+\bar{D}^{\sharp}$ is nef, thus the Zariski-Fujita decomposition
of $K_{\bar{S}}+\bar{D}$ is
$K_{\bar{S}}+\bar{D}^{\sharp}+\mathrm{Bk}(\bar{D})$, where
$(K_{\bar{S}}+\bar{D})^{+}=K_{\bar{S}}+\bar{D}^{\sharp}$ and
$(K_{\bar{S}}+\bar{D})^{-}=\mathrm{Bk}(\bar{D})$.

  2. (2)

$(K_{S}+D)^{+}=\sigma^{*}(K_{\bar{S}}+\bar{D}^{\sharp})$.

We say a log smooth pair $(S,D)$ with a reduced boundary $D$ and
$\kappa(S,D)\geq 0$ _minimal_ if $K_{S}+D^{\sharp}$ is nef. Given a log smooth
pair $(S,D)$ with a reduced boundary $D$ and $\kappa(S,D)=2$, we can run
$(K_{S}+D^{\sharp})$-MMP and terminete with a minimal model
$(\bar{S},\bar{D})$ which is minimal in the sence of Tsunoda, which is called
_almost minimal_. For more details we refer the reader to [Tsu83, Theorem
1.3].

######  Lemma 2.9.

Let $(S,D)$ be a log smooth pari with a reduced boundary $D$. Let $t$ be the
number of tips of $\mathrm{Supp}\,(\mathrm{Bk}(D))$. Then we have
$\mathrm{Bk}(D)^{2}\geq-t$.

###### Proof.

Let $L$ be an admissible rational linear chain in $D$. We write
$L=\sum_{i=1}^{r}C_{i}$ as the sum of its prime components such that
$C_{i}\cdot C_{i+1}=1$ for $1\leq i\leq r-1$ and $\beta_{D}(C_{1})=1$. Write
$\mathrm{Bk}(L)=\sum_{i=1}^{r}a_{i}C_{i}$, then we have

$\mathrm{Bk}(L)^{2}=\sum_{i=1}^{r}a_{i}\mathrm{Bk}(L)\cdot
C_{i}=\sum_{i=1}^{r}a_{i}(K_{S}+D)\cdot
C_{i}=\sum_{i=1}^{r}a_{i}(-2+\beta_{D}(C_{i})).$

So if $L$ is a rod, $\mathrm{Bk}(L)^{2}=-a_{1}$; if $L$ is a twig,
$\mathrm{Bk}(L)^{2}=-a_{1}-a_{r}$. Let $F$ be a fork of $D$, write
$F=C_{0}+T_{1}+T_{2}+T_{3}$. Let $C_{1},C_{2},C_{3}$ be the tip of
$T_{1},T_{2},T_{3}$ respectively. Set $a_{i}=\mu_{\mathrm{Bk}(F)}(C_{i})$ for
$i=0,1,2,3$. Similar arguments apply to $F$, we have
$\mathrm{Bk}(F)^{2}=a_{0}-a_{1}-a_{2}-a_{3}$. Note that in all cases,
$0<a_{i}\leq 1$. Thus we have $\mathrm{Bk}(D)^{2}\geq-t$. ∎

###  2.4. The Bogomolov-Miyaoka-Yau inequality to log surfaces

Let $(S,D)$ be a log smooth pair with a smooth projective surface $S$ and a
reduced divisor $D$ on $S$. The sheaf of logarithmic 1-forms
$\Omega_{S}^{1}(\log D)$ is locally free of rank $2$. Its chern classes
$\bar{c}_{1}=c_{1}(\Omega_{S}^{1}(\log
D)),\bar{c}_{2}=c_{2}(\Omega_{S}^{1}(\log D))$ are defined, and we regard
$c_{1}(\Omega^{1}_{S}(\log D))=K_{S}+D$ as a class in $H^{2}(S,\mathbb{Z})$.

######  Proposition 2.10.

Let $(S,D)$ be a log smooth pair. Then

$\bar{c}_{2}=e(S)+2(p_{a}(D)-1-l),$

where $e(S)$ is the topological Euler characteristic of $S$ and $l$ is the
number of edges of the dual graph of $D$ . Moreover, we get the log version of
Noether’s equality

$\bar{c}_{1}^{2}+\bar{c}_{2}+6(p_{a}(D)-1)+D^{2}+2l=12\bar{\chi},$

where $\bar{\chi}=\chi(S,K_{S}+D)$.

###### Proof.

Write $D=\sum_{i=1}^{r}D_{i}$, wherre $D_{i}$’s are the irreducible components
of $D$. For each $1\leq i\leq r$, put $W_{i}=\sum_{j=1}^{i}D_{j}$ thus
$W_{r}=D$ and $W_{1}=D$, and we define $W_{0}=0$, then there are exact
sequences

(2.2) $0\to\Omega_{S}^{1}(\log W_{i-1})\to\Omega_{S}^{1}(\log
W_{i})\to\mathscr{O}_{D_{i}}(W_{i-1}|_{D_{i}})\to 0.$

From this we see

$\displaystyle c_{2}(\Omega_{S}^{1}(\log W_{i}))$
$\displaystyle=c_{2}(\Omega_{S}^{1}(\log W_{i-1}))+(K_{S}+W_{i-1})\cdot
D_{i}+D_{i}^{2}-W_{i-1}\cdot D_{i}$ $\displaystyle=c_{2}(\Omega_{S}^{1}(\log
W_{i-1}))+2(p_{a}(D_{i})-1).$

Summing up these equalities for $1\leq i\leq r$, we have

$c_{2}(\Omega_{S}^{1}(\log
D))=c_{2}(\Omega_{S}^{1})+2\sum_{i=1}^{r}(p_{a}(D_{i})-1).$

Note that $c_{2}(\Omega_{S}^{1})=e(S)$, and
$\sum_{i=1}^{r}(p_{a}(D_{i})-1)=p_{a}(D)-1-l$ (see Remark 2.6). ∎

######  Proposition 2.11.

Let $e(S-D)$ be the topological Euler characteristic of the open algebraic
surface $S-D$, then we have $e(S-D)=c_{2}(\Omega_{S}^{1}(\log D))$.

###### Proof.

By a theorem of Deligne (cf. [Del71, Corollary 3.2.13 and Corollary 3.2.14]),
the spectral sequence

$E_{1}^{p,q}=H^{q}(X,\Omega^{p}_{S}(\log D))\implies H^{p+q}(S-D,\mathbb{C})$

degenerates at $E_{1}$ page, thus we have Hodge decomposition

$H^{n}(S-D,\mathbb{C})\cong H^{p,q}(S,\Omega_{S}^{p}(\log D)).$

From this, we can see that

$e(S-D)=\chi(S,\mathscr{O}_{S})-\chi(S,\Omega_{S}^{1}(\log
D))+\chi(S,\Omega_{S}^{2}(\log D)).$

The Riemann-Roch theorem gives

$\displaystyle\chi(S,\Omega_{S}^{1}(\log D))$
$\displaystyle=\frac{1}{2}(K_{S}\cdot D+D^{2})-c_{2}(\Omega_{S}^{1}(\log
D))+2\chi(S,\mathscr{O}_{S})$ $\displaystyle=\chi(S,\Omega_{S}^{2}(\log
D))-c_{2}(\Omega_{S}^{1}(\log D))+\chi(S,\mathscr{O}_{S}).$

Therefore, $c_{2}(\Omega_{S}^{1}(\log D))=e(S-D)$. ∎

To show Theorem 3.4, we need the following.

######  Proposition 2.12.

Let $(S,D)$ be a log smooth pair with a reduced boundary $D$. Suppose
$p_{a}(D)\leq 2(l+q(S))+1-h^{1,1}(S)$, where $l$ is the number of edges of the
dual graph of $D$. Then we have

$e(S-D)\leq 2p_{g}(S,D)+1.$

If $p_{g}(S)=0$, we have $e(S-D)\leq p_{g}(S,D)+1$.

###### Proof.

By Proposition 2.10 and Proposition 2.11, we have

(2.3) $e(S-D)=e(S)+2p_{a}(D)-2-2l.$

Note that

(2.4) $e(S)=2-4q(S)+2p_{g}(S)+h^{1,1}(S).$

Substituting (2.4) into (2.3) yields

(2.5) $e(S-D)=h^{1,1}(S)+2p_{g}(S)-4q(S)+2p_{a}(D)-2l.$

On the other hand, the Riemann-Roch theorem gives

(2.6) $\chi(S,\Omega_{S}^{1}(\log
D))=p_{a}(D)-1-e(S-D)+2\chi(S,\mathscr{O}_{S}).$

Substituting (2.5) into (2.6) we obtain

(2.7) $\chi(S,\Omega_{S}^{1}(\log D))=2(q(X)+l)+1-h^{1,1}(X)-p_{a}(D).$

By assumption we have $\chi(S,\Omega_{S}^{1}(\log D))\geq 0$. By the proof of
Proposition 2.11, we have

$e(S-D)=\chi(\mathscr{O}_{S})+\chi(K_{S}+D)-\chi(\Omega_{S}^{1}(\log D))\leq
1+p_{g}(S)+p_{g}(S,D).$

Note that $p_{g}(S)\leq p_{g}(S,D)$, which proves the proposition. ∎

######  Theorem 2.13.

[Miy84, Theorem 1.1] Let $(S,D)$ be a log smooth projective surface with a
reduced boundary $D$ such that $K_{S}+D$ is pseudo-effective. Let
$K_{S}+D=P+N$ be the Zariski-Fujita decomposition, where $P$ is the nef part
and $N$ is the negative part. Then the inequality

$\frac{1}{3}P^{2}\leq c_{2}(\Omega_{S}^{1}(\log D))-\frac{1}{4}N^{2}$

holds.

######  Corollary 2.14.

Let $(X,D)$ be a minimal log smooth surface of log general type. Then we have

$\frac{1}{3}(K_{X}+D^{\sharp})^{2}\leq c_{2}(\Omega_{X}^{1}(\log
D))-\frac{1}{4}\mathrm{Bk}(D)^{2}.$

###  2.5. Linear systems on rational surfaces

Consider a rational surface $S$ for which we have a sequence of blowing ups

(2.8) $\pi\colon\thinspace
S=S_{n}\xrightarrow{\pi_{n}}S_{n-1}\xrightarrow{\pi_{n-1}}\cdots\xrightarrow{\pi_{2}}S_{1}\xrightarrow{\pi_{1}}S_{0}=\mathbb{P}^{2},$

where $\pi_{i}\colon\thinspace S_{i}\to S_{i-1}$ is the blowing-up of
$S_{i-1}$ at a point $p_{i}$ with exceptional divisor $E_{i}$ on $S_{i}$ for
$i=1,\dots,n$. Let $\bar{E}_{i}$ be the total transfomational of $E_{i}$ on
$S$, and $H$ be the class of total transform of a line in $\mathbb{P}^{2}$,
then $H,\bar{E}_{1},\dots,\bar{E}_{n}$ forms a free basis of
$\mathrm{Pic}(S)$. The canonical class $K_{S}$ of $S$ is given in terms of
$H,\bar{E}_{1},\dots,\bar{E}_{n}$ as $-3H+\bar{E}_{1}+\cdots+\bar{E}_{n}$.

Let $p\in\mathbb{P}^{2}$ be a point. We say that $q$ is a _infinitely near
point_ to $P$ of _order_ $n$ on $\mathbb{P}^{2}$, if there exists a birational
morphism as (2.8), such that $p_{1}=p$, $\pi_{i}(p_{i+1})=p_{i}$,
$i=1,\dots,n$ and $q\in E_{n}$. We denote by $\mathfrak{B}(\mathbb{P}^{2})$
the set of infinitely near points on $\mathbb{P}^{2}$. A _cluster_
$\mathscr{K}$ on $\mathbb{P}^{2}$ is a finite subset of
$\mathfrak{B}(\mathbb{P}^{2})$.

Given a cluster $\mathscr{K}=\\{p_{1},\dots,p_{n}\\}$ on $\mathbb{P}^{2}$ , we
denote by $\pi_{\mathscr{K}}\colon\thinspace S_{\mathscr{K}}\to\mathbb{P}^{2}$
the blowing-up of all the points in $\mathscr{K}$. For a curve on
$\mathbb{P}^{2}$, we denote by $\bar{C}^{\mathscr{K}}$ (or simply $\bar{C}$ if
$\mathscr{K}$ is understood) the total transform of $C$, and
$\widetilde{C}^{\mathscr{K}}$ (or simply $\widetilde{C}$ if $\mathscr{K}$ is
understood) the strict transform of $C$. We define the _multiplicity_ of $C$
at the point $p_{i}\in\mathscr{K}$, denote by $m_{i}(C)$, as the multiplicity
of the strict transform $\widetilde{C}$ at $p$ on $S_{\mathscr{K}}$, we have

$\bar{C}^{\mathscr{K}}=\widetilde{C}^{\mathscr{K}}+\sum_{i=1}^{n}m_{i}(C)\bar{E}_{i},$

where $E_{p}$ is the exceptional divisor over $P$.

Given an effective divisor $E=\sum_{i=1}^{n}\nu_{i}\bar{E}_{i}$ on
$S_{\mathscr{K}}$. The ideal sheaf

$\mathscr{I}=\pi_{\mathscr{K}*}\mathscr{O}_{S_{\mathscr{K}}}(-E)$

defines a zero-dimensional subscheme $Z$ of $\mathbb{P}^{2}$. For any
$d\in\mathbb{N}$,
$H^{0}(\mathbb{P}^{2},\mathscr{O}_{\mathbb{P}^{2}}(d)\otimes\mathscr{I})$
defines a sublinear system $V$ of
$H^{0}(\mathbb{P}^{2},\mathscr{O}_{\mathbb{P}^{2}}(d))$ with base ideal
$\mathscr{I}$. We see that $C\in V$ if and only if $m_{i}(C)\geq\nu_{i}$ for
all $1\leq i\leq n$. Thus the linear system $V$ is called a linear system of
plane curves of degree $d$ with assigned base points $p_{1},\dots,p_{n}$ with
respective multiplicities at least $\nu_{1},\dots,\nu_{n}$, and we say it is
type $(d;\nu_{1},\dots,\nu_{n})$.

######  Lemma 2.15.

Let $\mathscr{K}=\\{p_{1},\dots,p_{n}\\}$ be a cluster on $\mathbb{P}^{2}$,
and $V$ a linear system of plane curves of degree $d$ with assigned base
points $\mathscr{K}$ of type $(d;\nu_{1},\dots,\nu_{n})$. We have

  1. (1)

$\dim V\geq\frac{1}{2}d(d+3)-\frac{1}{2}\sum_{i=1}^{n}\nu_{i}(\nu_{i}+1).$

  2. (2)

The divisor $dH-\nu_{1}\bar{E}_{1}-\cdots-\nu_{n}\bar{E}_{n}$ is big if
$d^{2}-\sum_{i=1}^{n}\nu_{i}^{2}>0.$

###### Proof.

For (1), see [AC02, Section 1.2]. For (2), since

$h^{0}(S_{\mathscr{K}},m(dH-\sum_{i=1}^{n}v_{i}\bar{E}_{i}))\geq\frac{1}{2}dm(dm+3)-\frac{1}{2}\sum_{i=1}^{n}v_{i}m(v_{i}m+1)=\frac{1}{2}(d^{2}-\sum_{i=1}^{n}v_{i}^{2})m^{2}+O(m)$

it follows that if $d^{2}-\sum_{i=1}^{n}v_{i}^{2}>0$ then
$dH-\nu_{1}\bar{E}_{1}-\cdots-\nu_{n}\bar{E}_{n}$ is big. ∎

Consider the Hirzebruch surface
$\Sigma_{e}=\mathbb{P}(\mathscr{O}\oplus\mathscr{O}(e))$ of degree $e\geq 0$.
Let $\pi\colon\thinspace\Sigma_{e}\to\mathbb{P}^{1}$ be the
$\mathbb{P}^{1}$-fiberation, let $\Delta_{0}$ and $\Gamma$ be its section with
$\Delta_{0}^{2}=-e$ and its general fiber. Set
$\Delta_{\infty}=\Delta_{0}+e\Gamma$, then $\Delta_{\infty}$ is the class of
tautological line bundle of $\Sigma_{e}$ and we have $\Delta_{\infty}^{2}=e$.

######  Lemma 2.16.

Let $\mathscr{K}=\\{p_{1},\dots,p_{n}\\}$ be a cluster on $\Sigma_{e}$,
consider the linear system $V$ on $\Sigma_{e}$ with assigned base points
$\mathscr{K}$ with repsective multiplicities $\nu_{1},\dots,\nu_{n}$. Consider
the very ample divisor $a\Delta_{\infty}+b\Gamma$ where $a$ and $b$ are
positive integers. Let $\mu\colon\thinspace S\to\Sigma_{e}$ be the blowing-up
at $\mathscr{K}$. We have

  1. (1)

$\dim
V\geq(a+1)(b+\frac{ae}{2})+a-\sum_{i=1}^{n}\frac{1}{2}\nu_{i}(\nu_{i}+1).$

  2. (2)

The divisor
$\mu^{*}(a\Delta_{\infty}+b\Gamma)-\sum_{i=1}^{n}\nu_{i}\bar{E}_{i}$ is big if
$a(b+\frac{ae}{2})-\frac{1}{2}\sum_{i=1}^{n}\nu_{i}^{2}>0.$

###### Proof.

For $a\geq 0,b\in\mathbb{Z}$ , we have

$\pi_{*}\mathscr{O}(a\Delta_{\infty}+b\Gamma)=\mathrm{Sym}^{a}(\mathscr{O}\oplus\mathscr{O}(e))\otimes\mathscr{O}(b)=\sum_{i=0}^{a}\mathscr{O}(b+ie)$

Therefore $h^{0}(\Sigma_{e},a\Delta_{\infty}+b)=(a+1)(b+\frac{ae}{2}+1)$ and
(1) holds. For (2), since

$\displaystyle
h^{0}(S,m(\mu^{*}(a\Delta_{\infty}+b\Gamma)-\sum_{i=1}^{n}\nu_{i}\bar{E}_{i}))$
$\displaystyle\geq(am+1)(bm+\frac{ame}{2})-\frac{1}{2}\sum_{i=1}^{n}v_{i}m(v_{i}m+1)$
$\displaystyle=(a(b+\frac{ae}{2})-\frac{1}{2}\sum_{i=1}^{n}\nu_{i}^{2})m^{2}+O(m)$

it follows that if $a(b+\frac{ae}{2})-\frac{1}{2}\sum_{i=1}^{n}\nu_{i}^{2}>0$
then $\mu^{*}(a\Delta_{\infty}+b\Gamma)-\sum_{i=1}^{n}\nu_{i}\bar{E}_{i}$ is
big. ∎

###  2.6. Semi-positivity of $f_{*}\omega_{S/B}(D)$

Let $f\colon\thinspace S\to B$ be a fiberation from a smooth projective
surface to a smooth curve. Let $D$ be a reduced divisor on $S$ which has only
simple normal corssing singularities. In this section we will show that
$f_{*}\omega_{S/B}(D)$ is a nef vector bundle on $B$. We need the following
positivity result of a fiberation under some monodromy conditions and some
assumption on the divisor $D$, which was proved by Fujino [Fuj04].

######  Theorem 2.17.

Let $f\colon\thinspace X\to Y$ be a surjective morphism from a non-singular
projective variety $X$ to a projective variety $Y$. Let $D=\sum D_{i}$ be a
simple normal crossing divisor on $X$. Assume the following conditions hold

  1. (1)

$D$ is strongly horizontal with respect to $f$, i.e., each irreducible
component of $D_{i_{1}}\cap D_{i_{2}}\cap\dots\cap D_{i_{k}}$ is dominant onto
$Y$.

  2. (2)

Let $\Sigma$ be a simple normal crossing divisor on $Y$ such that $f$ is
smooth and $D$ is relatively normal crossing over over
$Y_{0}=Y\setminus\Sigma$.

  3. (3)

Put $X_{0}=f^{-1}(Y_{0}),D_{0}=D\cap X_{0},f_{0}=f|_{X_{0}}$ and $d=\dim
X-\dim Y$. All the local monodromies on the local system
$R^{d+i}f_{0*}\mathbb{C}_{X_{0}\setminus D_{0}}$ around every irreducible
component of $\Sigma$ are unipotent.

Then $R^{i}f_{*}\omega_{X/Y}(D)$ is a nef vector bundle on $Y$.

######  Remark 2.18.

In this paper we only need the Theorem 2.17 for the case when $X$ is a surface
and $Y$ is a curve, for this case the above theorem is a consequence of the
work of Steenbrink and Zucker on variation of mxied Hodge structures (see
[SZ85, Section 5]). The condition (1) becomes that $D$ is a disjoint union of
non-singular curves $C_{j}$ which are horizontal with respect to $f$. The
condition (2) is automatically satisfied. The condition (3) can be achieved
after replacing $Y$ by a finite convering and taking a suitable birational
model of the base change of $X$.

######  Lemma 2.19.

Let $f\colon\thinspace X\to Y$ be a fiberation of non-singular surface $X$ to
a curve $Y$ and $D$ be a SNC divisor on $X$. Then there exists an birational
morphism $g\colon\thinspace Z\to X$ from a non-singular surface $Z$ to $X$
such that the horizontal part of $D^{\prime}$ is strongly horizontal with
respect to $h=f\circ g$ where $D^{\prime}$ is the strict transform of $D$ and
$g^{*}D$ is also SNC. Moreover, if $h_{*}\omega_{Z/Y}(D^{\prime})$ is nef,
then so is $f_{*}\omega_{X/Y}(D)$.

###### Proof.

For simplicity we may assume that $D$ is horizontal with respect to $f$. If
$D$ is not strongly horizontal with respect to $f$, then there exist two
horizontal components $D_{1}$ and $D_{2}$ of $D$ meet at one point $P$. There
are finitely many of points which are the intersection of two horizontal
components of $D$. Let $g\colon\thinspace Z\to X$ be the blowing up at these
points, it is clearly that the strict transform $D^{\prime}$ is strictly
horizontal.

Note that $g_{*}\omega_{Z}(g^{*}D)=\omega_{X}(D)$, so that
$h_{*}\omega_{Z/Y}(g^{*}D)=f_{*}\omega_{X/Y}(D)$. Since $g^{*}D-D^{\prime}$ is
effective and its support is vertical with respect to $h$, we have an
inclusion $h_{*}\omega_{Z/Y}(D^{\prime})\hookrightarrow
h_{*}\omega_{Z/Y}(g^{*}D)$. Note that
$\mathrm{rank}\,h_{*}\omega_{Z/Y}(g^{*}D)=h^{0}(F,K_{F}+g^{*}D|_{F})=h^{0}(F,K_{F}+D^{\prime}|_{F})=\mathrm{rank}\,h_{*}\omega_{Z/Y}(D^{\prime})$,
where $F$ is a general fiber of $h$. By Proposition 2.4, it follows that
$f_{*}\omega_{X/Y}(D)=h_{*}\omega_{Z/Y}(g^{*}D)$ is nef. ∎

######  Theorem 2.20.

Let $f\colon\thinspace S\to B$ be a fiberation from a smooth surface $S$ to a
non-singular curve $B$ and Let $D$ be a SNC-divisor on $S$. Then
$f_{*}\omega_{S/B}(D)$ is a nef vector bundle.

###### Proof.

By Lemma 2.19, we may assume $D$ is strongly horizontal. By Kawamata’s
uniponent reduction [Kaw81, Corollary 18], we have the following commutative
diagram

${S^{\prime}}$${S^{\nu}}$${S_{B^{\prime}}}$${S}$${B^{\prime}}$${B^{\prime}}$${B^{\prime}}$${B}$$\scriptstyle{\nu}$$\scriptstyle{\nu_{2}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\nu_{1}}$$\scriptstyle{f^{\nu}}$$\scriptstyle{p}$$\scriptstyle{\nu_{0}}$$\scriptstyle{f}$$\scriptstyle{\mu}$

where $\mu$ is a finite morphism of curves, $\nu_{1}$ is the normalization of
the fiber product $S_{B^{\prime}}:=S\times_{B}B^{\prime}$ and $\nu_{2}$ is a
desingularization of $S^{\nu}$, such that the induced fiberation
$f^{\prime}\colon\thinspace S^{\prime}\to B^{\prime}$ satisfies the conditions
(1) through (3) of Theorem 2.17. Since $\mu$ is flat, the relative dualizing
sheaves compatible with base change
$\omega_{S_{B^{\prime}}/B^{\prime}}=\nu_{0}^{*}\omega_{S/B}$. Consider the
reduced divisor $D^{\prime}=(\nu_{0}^{*}D)_{\mathrm{red}}$ on
$S_{B^{\prime}}$, we have an injective homomorphism

$p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})\hookrightarrow
p_{*}\nu_{0}^{*}\omega_{S/B}(D),$

because $D^{\prime}\leq\nu_{0}^{*}D$. By the base change theorem,
$p_{*}\nu_{0}^{*}\omega_{S/B}(D)\cong\mu^{*}f_{*}\omega_{S/B}(D)$, and so we
get a injection

$p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})\hookrightarrow\mu^{*}f_{*}\omega_{S/B}(D).$

Since $\mathrm{Supp}\,(\nu_{0}^{*}D-D^{\prime})$ is contained in the
ramification locus of $\nu_{0}$ and is vertical with respect to $p$, it
follows that the cokernal of the homomorphism
$p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})\hookrightarrow\mu^{*}f_{*}\omega_{S/B}(D)$
is a torsion sheaf on $B$, thus we can regard
$p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})$ as a subvector bundle of
$\mu^{*}f_{*}\omega_{S/B}(D)$ with the same rank. By Proposition 2.4 and
Proposition 2.2 (2), to show that $f_{*}\omega_{S/B}(D)$ is nef it is
sufficitent to show that $p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})$
is nef. Note that the relative dualizing sheaf
$\omega_{S^{\nu}/S_{B^{\prime}}}$ is determined by

$\nu_{1*}\omega_{S^{\nu}/S_{B^{\prime}}}=\mathscr{H}om_{S_{B^{\prime}}}({\nu}_{1*}\mathscr{O}_{S^{\nu}},\mathscr{O}_{S_{B^{\prime}}})$

therefore
$\nu_{1*}\omega_{S^{\nu}/B^{\prime}}=\mathscr{H}om_{S_{B^{\prime}}}(\nu_{1*}\mathscr{O}_{S^{\nu}},\omega_{S_{B^{\prime}}/B^{\prime}})$,
and hence we get an injective homomorphism
$\alpha\colon\thinspace\nu_{1*}\omega_{S^{\nu}/B^{\prime}}\hookrightarrow\omega_{S_{B^{\prime}}/B^{\prime}}$.
Set $D^{\nu}=(\nu_{1}^{*}D^{\prime})_{\mathrm{red}}$, push down by $p_{*}$, we
get injections

$f^{\nu}_{*}\omega_{S^{\nu}/B^{\prime}}(D^{\nu})\hookrightarrow
f^{\nu}_{*}\omega_{S^{\nu}/B^{\prime}}(\nu_{1}^{*}D^{\prime})\hookrightarrow
p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime}).$

For the same reason, the cokernal of
$f^{\nu}_{*}\omega_{S^{\nu}/B^{\prime}}(D^{\nu})\hookrightarrow
p_{*}\omega_{S_{B^{\prime}}/B^{\prime}}(D^{\prime})$ is a torsion sheaf on
$B^{\prime}$, so it is sufficient to show that
$f^{\nu}_{*}\omega_{S^{\nu}/B^{\prime}}(D^{\nu})$ is nef. Set
$D^{\prime\prime}=(\nu_{2}^{*}D^{\nu})_{\mathrm{red}}$, we wish to choose
$\nu_{2}$ as Lemma 2.19 so that the horizontal part of $D^{\prime\prime}$ is
strongly horizontal with respect to $f^{\prime}$. By the construction we have
$\nu_{2*}\omega_{S^{\prime}}(D^{\prime\prime})=\omega_{S^{\nu}}(D^{\nu})$,
thus

$\displaystyle f^{\prime}_{*}\omega_{S^{\prime}/B^{\prime}}(D^{\prime\prime})$
$\displaystyle=f^{\nu}_{*}\nu_{2*}\omega_{S^{\prime}}(D^{\prime\prime})\otimes\omega_{B^{\prime}}^{-1}$
$\displaystyle=f^{\nu}_{*}\omega_{S^{\nu}}(D^{\nu})\otimes\omega_{B^{\prime}}^{-1}$
$\displaystyle=f^{\nu}_{*}\omega_{S^{\nu}/B^{\prime}}(D^{\nu}).$

By Theorem 2.17,
$f^{\prime}_{*}\omega_{S^{\prime}/B^{\prime}}(D^{\prime\prime})$ is nef, this
complete the proof. ∎

##  3\. Proofs of the Main Theorem

We begin with some properties of fiberation of algebraic surface. Let
$f\colon\thinspace X\to B$ be a fiberation of a non-singular projective
surface $X$ onto a non-singular projective curve $B$ of genus $b$. Let $g$ be
the genus of a general fiber $F$ of $f$ and $q=h^{1}(X,\mathscr{O}_{X})$ be
the irregularity of $X$. Let $D$ be a reduced effective divisor on $X$.

######  Lemma 3.1.

Assume $D\cdot F>0$ where $F$ is a general fiber of $f$, then we have

$h^{1}(B,f_{*}\mathscr{O}_{X}(K_{X}+D))=h^{1}(X,K_{X}+D).$

###### Proof.

By the Leray spectral sequence, we have

$h^{1}(X,K_{X}+D)=h^{1}(B,f_{*}\mathscr{O}_{X}(K_{X}+D))+h^{0}(B,R^{1}f_{*}\mathscr{O}_{X}(K_{X}+D)).$

By the relative duality,

$\mathscr{H}om_{\mathscr{O}_{B}}(R^{1}f_{*}\mathscr{O}_{X}(K_{X}+D),\mathscr{O}_{B})\cong
f_{*}\mathscr{O}_{X}(-D)\otimes\omega_{B}^{-1}.$

It is sufficient to show that $R^{1}f_{*}\mathscr{O}_{X}(K_{X}+D)=0.$ By
assumption $D\cdot F>0$ for a general fiber $F$ of $f$. We conclude that
$R^{1}f_{*}\mathscr{O}_{X}(-D)=0$ by base change of cohomology, i.e.
$(f_{*}\mathscr{O}_{X}(-D))_{x}\cong H^{0}(F,\mathscr{O}_{F}(-D|_{F}))=0$
where $x$ is a point on $B$ and $F$ is the fiber over $x$. Therefore, we have
$R^{1}f_{*}\mathscr{O}_{X}(K_{X}+D)=0.$ ∎

Let $(S,D)$ be a minimal log surface of log general type. Assume
$p_{g}(S,D)\geq 2$ and the log canonical linear system $|K_{S}+D|$ is composed
of a pencil, let $\phi\colon\thinspace
S\to\Sigma=\phi(S)\hookrightarrow\mathbb{P}^{p_{g}(S,D)-1}$ be the rational
map defined by $|K_{S}+D|$, where $\Sigma$ is a curve. We will study the
algebraic fiberation $f\colon\thinspace X\to B$ which is obtained by the
following diagram:

${X}$${S}$${B}$${\Sigma}$${\mathbb{P}^{p_{g}(S,D)-1}}$$\scriptstyle{\mu}$$\scriptstyle{f}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$$\scriptstyle{\tau}$

where $\mu$ is a log resolution which resolve the base points of the linear
system $|K_{S}+D|$ such that the movable part of $\mu^{*}|K_{S}+D|$ is base
point free, it induces a morphism $\psi\colon\thinspace X\to\Sigma=\psi(X)$.
We can write $K_{X}+\mu_{*}^{-1}D=\mu^{*}(K_{S}+D)+E_{+}-E_{-}$ where $E_{+}$
and $E_{-}$ are effective $\mu$-exceptional reduced divisors having no commen
component. The fiberation $f$ is the Stein factorization of the morphism
$\psi.$ Put $D_{X}=\mu_{*}^{-1}D+E_{-}$, then $(X,D_{X})$ is agian log smooth.
For a log surface $(S,D)$, we are interesting their invariants
$(p_{g}(S,D),h^{1}(S,K_{S}+D),p_{a}(D))$. According the following lemma 3.2.
After replacing $(S,D)$ by $(X,D_{X})$, we may assume the movable part of
$|K_{S}+D|$ is base point free. The irregularity $q(S)$ of a normal surface
$S$ with rational singularities is a birational invariant. For log surface
$(S,D)$, this is ture for $h^{1}(S,K_{S}+D)$.

######  Lemma 3.2.

Let $(S,D)$ be a projective log smooth log surface. For any log resolution
$\mu\colon\thinspace Y\to S$ of singularities of $(S,D)$, we can write
$K_{Y}+\mu_{*}^{-1}D+E_{-}=\mu^{*}(K_{S}+D)+E_{+}$. Put
$D_{Y}=\mu_{*}^{-1}D+E_{-}$. Then we have

  1. (1)

$h^{i}(Y,K_{Y}+D_{Y})=h^{i}(S,K_{S}+D)$ for $i=0,1$;

  2. (2)

$p_{a}(D_{Y})=p_{a}(D)$;

  3. (3)

$\chi(Y,K_{Y}+D_{Y})=\chi(S,K_{S}+D).$

###### Proof.

For (1), it is clear when $i=0$, we show that the equality holds for $i=1$. By
projection formula, we have

$R^{1}\mu_{*}\mathscr{O}_{Y}(K_{Y}+D_{Y})=R^{1}\mu_{*}\mathscr{O}_{S}(\mu^{*}(K_{S}+D)+E_{+})=\mathscr{O}_{S}(K_{S}+D)\otimes
R^{1}\mu_{*}\mathscr{O}_{Y}(E_{+}).$

It is sufficitent to show that $R^{1}\mu_{*}\mathscr{O}_{Y}(E_{+})=0$. Since
$S$ is non-singular, we have $R^{1}\mu_{*}\mathscr{O}_{Y}=0$, we can assume
that $E_{+}\neq 0$. Note that $D_{Y}\cdot E_{+}=0$, then we get $K_{Y}\cdot
E_{+}=E_{+}^{2}<0$. Consider the exact sequence

${0}$${\mathscr{O}_{Y}}$${\mathscr{O}_{Y}(E_{+})}$${\mathscr{O}_{E_{+}}(E_{+})}$${0}$

It follows that
$h^{1}(E_{+},\mathscr{O}_{E_{+}}(E_{+}))=\chi(Y,\mathscr{O}_{Y})-\chi(Y,\mathscr{O}_{Y}(E_{+}))=-\frac{1}{2}E_{+}(E_{+}-K_{Y})=0$.
Push down the above exact sequence by $\mu_{*}$, we botain an exact sequence

${R^{1}\mu_{*}\mathscr{O}_{Y}}$${R^{1}\mu_{*}\mathscr{O}_{Y}(E_{+})}$${H^{1}(E_{+},\mathscr{O}_{E_{+}}(E_{+}))}$

So we have $R^{1}\mu_{*}\mathscr{O}_{Y}(E_{+})=0$.

For (2),

$\displaystyle p_{a}(D_{Y})$
$\displaystyle=\frac{1}{2}D_{Y}\cdot(K_{Y}+D_{Y})+1$
$\displaystyle=\frac{1}{2}D_{Y}\cdot(\mu^{*}(K_{S}+D)+E_{+})+1$
$\displaystyle=\frac{1}{2}D_{Y}\cdot(\mu^{*}(K_{S}+D))+1$
$\displaystyle=\frac{1}{2}D\cdot(K_{S}+D)+1=p_{a}(D)$

The second equality uses $D_{Y}\cdot E_{+}=0$ and the fourth equality uses
$\mu_{*}D_{Y}=D$.

For (3), we may assume $D\neq 0$, since
$h^{2}(Y,K_{Y}+D_{Y})=h^{0}(Y,-D_{Y})=0$ (resp.
$h^{2}(S,K_{S}+D)=h^{0}(S,-D)=0$). By (1), we have

$\chi(Y,K_{Y}+D_{Y})=h^{0}(Y,K_{Y}+D_{Y})-h^{1}(Y,K_{Y}+D_{Y})=h^{0}(S,K_{S}+D)-h^{1}(S,K_{S}+D)=\chi(S,K_{S}+D)$

∎

Now we may consider a log smooth surface $(S,D)$ satisfying the following
conditions

  1. (1)

$(S,D)$ is minimal in the sense of the Definition 1.1;

  2. (2)

the adjoint linear system $|K_{S}+D|$ is base point free and is composed of a
pencil.

Let $f\colon\thinspace S\to B$ be the fiberation induced by $|K_{S}+D|$. Then
we can write $K_{S}+D\equiv nF+Z$, where $F$ is a general fiber of $f$, $Z$ is
the fixed part of $|K_{S}+D|$, and $n=\deg(B/\Sigma)\deg\Sigma$ a positive
integer. Since $\deg\Sigma\geq p_{g}(S,D)-1$, we have $n\geq p_{g}(S,D)-1$.
Let $K_{S}+D=K_{S}+D^{\sharp}+\mathrm{Bk}(D)$ be the Fujita-Zariski
decomposition as Section 2.3, we need the following inequality which is a part
of Noether’s inequality of log surfaces of general type which was proved by
Tsunoda and De-Qi Zhang [TZ92].

######  Lemma 3.3.

$g\leq\frac{n+2}{2n^{2}}(K_{X}+D^{\sharp})^{2}+1$.

###### Proof.

Write $D^{\sharp}=\sum_{i=1}^{r}\alpha_{i}D_{i}$ where $D_{i}$’s are the
irreducible components and $0<\alpha_{i}\leq 1$. Suppose $\alpha_{i}<1$ for
$1\leq i\leq t$. Note that the intersection maxtrix $(D_{i}\cdot D_{j})_{1\leq
i,j\leq t}$ is negative defined. So we can define $\mathbb{Q}$-divisors
$F^{\sharp}$ and $Z^{\sharp}$ as follows:

$\displaystyle F^{\sharp}=F+\sum_{i=1}^{t}a_{i}D_{i}\text{, with
}F^{\sharp}\cdot D_{j}=0\text{ for all }1\leq j\leq t$ $\displaystyle
Z^{\sharp}=Z+\sum_{i=1}^{t}b_{i}D_{i}\text{, with }Z^{\sharp}\cdot
D_{j}=0\text{ for all }1\leq j\leq t.$

By [TZ92, Lemma 1.1], we have

  1. (1)

$K_{S}+D^{\sharp}\equiv nF^{\sharp}+Z^{\sharp}$;

  2. (2)

$F\leq F^{\sharp}$ and $0\leq Z^{\sharp}\leq Z$;

  3. (3)

$(F^{\sharp})^{2}\geq 0$ and $(K_{S}+D^{\sharp})^{2}\geq
n(K_{S}+D^{\sharp})\cdot F^{\sharp}\geq n^{2}(F^{\sharp})^{2}$.

Let $C_{i}$ ($1\leq i\leq p$) (resp. $E_{j}$ ($1\leq j\leq q$)) be the
components of $D^{\sharp}$ meeting $F$ with coefficients $\alpha_{i}<1$ (resp.
$\alpha_{i}=1$). Then we have

  1. (1)

$(K_{S}+D^{\sharp})\cdot F^{\sharp}\geq K_{S}\cdot
F+\sum_{i=1}^{p}(1-\frac{2}{\gamma_{i}})C_{i}\cdot F+\sum_{j=1}^{q}E_{j}\cdot
F$, where $\gamma_{i}=-C_{i}^{2}$;

  2. (2)

$(F^{\sharp})^{2}\geq\sum_{i=1}^{p}\frac{(C_{i}\cdot F)^{2}}{\gamma_{i}}$;

  3. (3)

$(K_{S}+D^{\sharp})\cdot F^{\sharp}+2(F^{\sharp})^{2}\geq K_{S}\cdot
F+\sum_{i=1}^{p}C_{i}\cdot F+\sum_{j=1}^{q}E_{j}\cdot F\geq 1$.

This implies that
$(\frac{n+2}{n^{2}})(K_{S}+D^{\sharp})^{2}\geq(K_{S}+D^{\sharp})\cdot F\geq
2g-2$, which proves the lemma.

∎

######  Theorem 3.4.

Let $(S,D)$ be a minimal log smooth surface of log general type with
$p_{g}(S,D)\geq 2$. Assume $p_{a}(D)\leq 2(l+q(S))+1-h^{1,1}(S)$. Suppose the
log canonical linear system $|K_{S}+D|$ is composed of a penciel. Let
$f\colon\thinspace S\to B$ be the induced fiberation to a smooth curve $B$
whose general fiber has genus $g$. Then $g\leq 5,$ for $p_{g}(S,D)\gg 0$.
Moreover if $p_{g}(S)=0$, then $g\leq 3$.

###### Proof.

Let $K_{S}+D=P+N$ be the Zariski-Fujita decomposition. By Theorem 2.14, we
have $\frac{1}{3}P^{2}+\frac{1}{4}N^{2}\leq c_{2}(\Omega_{X}^{1}(\log D))$. By
Proposition 2.11 and Proposition 2.12, we have

(3.1) $\frac{1}{3}P^{2}+\frac{1}{4}N^{2}\leq 2p_{g}(S,D)+1\leq 2n+3$

Substituting Lemma 3.3 into 3.1, we get

$g-1\leq\frac{3(n+2)(2n+3-\frac{1}{4}N^{2})}{2n^{2}}$

By Lemma 2.9, $-t<N^{2}\leq 0,$ where $t$ is the number of tips of $N$.
Therefore, for $n\gg t>0$, we have
$\frac{3(n+2)(2n+3-\frac{1}{4}N^{2})}{2n^{2}}\leq 4$, and thus $g\leq 5$.

If $p_{g}(X)=0,$ then $c_{2}(\Omega^{1}_{X}(\log D))\leq p_{g}(X,D)+1\leq
n+2$, thus the above inequality becomes

$g-1\leq\frac{3(n+2)(n+2-\frac{1}{4}N^{2})}{2n^{2}}\leq 2$

for $n\gg t$. ∎

######  Remark 3.5.

Let $(S,D)$ be a log smooth surface as Theorem 3.4. By the Riemann-Roch
theorem, we have

$\chi(S,\omega_{S}(D))-\chi(S,\mathscr{O}_{S})=\frac{1}{2}(K_{S}+D)\cdot D.$

By Serre duality, $h^{2}(S,\omega_{S}(D))=h^{0}(S,-D)=0$. Note that
$p_{a}(D)=\frac{1}{2}(K_{S}+D)\cdot D+1$ and
$p_{g}(S,D)=h^{0}(S,\omega_{S}(D))$, then we get

(3.2) $p_{g}(S,D)=h^{1}(S,\omega_{S}(D))-q(S)+p_{g}(S)+p_{a}(D).$

On the other hand, by the Serre duality, $h^{1}(S,K_{S}+D)=h^{1}(S,-D)$. Let
$k=\dim\mathrm{Ker}(\varphi)$, where $\varphi$ is the homomorphism
$\varphi\colon\thinspace H^{1}(S,\mathscr{O}_{S})\to H^{1}(D,\mathscr{O}_{D})$
induced by

$0\to\mathscr{O}_{S}(-D)\to\mathscr{O}_{S}\to\mathscr{O}_{D}\to 0$

So we have $h^{1}(S,K_{S}+D)=m+k-1$, where $m$ is the number of connected
components of $D$. Therefore, the invariant $h^{1}(S,K_{S}+D)$ measures both
the irregularity $q(S)$ of $S$ and the number of connected components of $D$.
In particular, if $D$ is connected, then $h^{1}(S,K_{S}+D)\leq q(S)$. If
$h^{1}(S,K_{S}+D)=0$, then $D$ is connected.

If $S$ is a rational surface and $D$ is connected, then we get
$p_{g}(S,D)=p_{a}(D)$. Since our proof of Theorem 3.4 relays on $p_{g}(S,D)\gg
0$. In this case we require that $p_{a}(D)\gg 0$ or $p_{a}(D)<2l+1$. We will
give some examples of rational surfaces which violate this assumption and show
that their $g$ have no bound.

######  Theorem 3.6.

Let $(S,D)$ be a porjective log smooth surface such that the adjoint linear
system $|K_{S}+D|$ is composed of a pencil. Suppose $f\colon\thinspace S\to B$
is the fiberation induced by the linear system $|K_{S}+D|$, let $g$ and $b$ be
the genus of a general fiber $F$ of $f$ and the genus of $B$. Assume $k=D\cdot
F>0$. Then the following cases occurs

  1. (1)

if $b\geq 2$, then $(g,k)=(1,1),(1,2)$ or $(2,1)$;

  2. (2)

if $g+k=3$, then $b\leq 2$ and $h^{1}(S,K_{S}+D)=0$;

###### Proof.

Let $\mathscr{L}$ be the subinvertible sheaf of $f_{*}\omega_{S}(D)$
generically generated by global sections of $f_{*}\omega_{S}(D)$. Then we have
$h^{0}(B,\mathscr{L})=h^{0}(B,f_{*}\omega_{S}(D))=p_{g}(S,D)$. Consider the
exact sequence of locally free sheaves

(3.3) ${0}$${\mathscr{L}}$${f_{*}\omega_{S}(D)}$${\mathscr{Q}}$${0}$

Put $k=D\cdot F$. By the base change of cohomology and the Riemann-Roch
theorem, we have
$\mathrm{rank}\,f_{*}\omega_{S}(D)=h^{0}(F,K_{F}+D|_{F})=g+k-1$ and thus
$\mathrm{rank}\,\mathscr{Q}=g+k-2$.

Since $f_{*}\omega_{S}(D)\otimes\omega_{B}^{-1}$ is nef, by Proposition 2.2
(1),so $\mathscr{Q}\otimes\omega_{B}^{-1}$ is also nef. By Corollary 2.3 we
obtain

(3.4) $\deg\mathscr{Q}\geq 2(b-1)(g+k-2)\text{ and
}h^{0}(B,\mathscr{Q})\geq\chi(B,\mathscr{Q})\geq(b-1)(g+k-2).$

Since $h^{0}(B,\mathscr{L})=h^{0}(B,f_{*}\omega_{S}(D))$, from the log exact
sequence induced from (3.3), we infer that

(3.5) $h^{0}(B,\mathscr{Q})\leq
h^{1}(B,\mathscr{L})=h^{0}(B,\omega_{B}\otimes\mathscr{L}^{-1})\leq b-1$

the second inequality follows from the fact $\deg\mathscr{L}>0$. Substituting
(3.4) into (3.5) yields

$(b-1)(g+k-2)\leq b-1.$

Thus when $b\geq 2$ we have $g+k\leq 3$. Note that
$g+k-1=\mathrm{rank}\,f_{*}\omega_{S}(D)\geq 1$, we have $g+k\in\\{2,3\\}$.

By Lemma 3.1, the log exact sequence induced from (3.3), (3.4) and (3.5), we
have

$h^{1}(S,K_{S}+D)=h^{1}(B,f_{*}\omega_{S}(D))=h^{1}(B,\mathscr{L})-\chi(B,\mathscr{Q})\leq(b-1)(3-g-k).$

Therefore, if $b=1$ or $g+k=3$, we have $h^{1}(S,K_{S}+D)=0$.

Suppose $b\geq 2$ and $g+k=3$. In this case $h^{1}(S,K_{S}+D)=0$, all
equalities of (3.4) and (3.5) hold. In particular, we have
$\mathrm{rank}\,f_{*}\omega_{S}(D)=2$ and
$h^{0}(B,\omega_{B}\otimes\mathscr{L}^{-1})=b-1$. By the Clifford inequality,

(3.6)
$b-1=h^{0}(B,\omega_{B}\otimes\mathscr{L}^{-1})\leq\frac{1}{2}\deg(\omega_{B}\otimes\mathscr{L}^{-1})+1=b-\frac{1}{2}\deg\mathscr{L},$

hence $\deg\mathscr{L}\leq 2$. On the other hand,
$\deg\mathscr{Q}=2(b-1)(g+k-2)=2(b-1)$. So $\deg
f_{*}\omega_{S}(D)=\deg\mathscr{L}+\deg\mathscr{Q}\leq 2b$. Since
$f_{*}\omega_{S}(D)\otimes\omega_{B}^{-1}$ is nef, we have $2b\geq\deg
f_{*}\omega_{S}(D)\geq 4(b-1)$. So we get $b=2$.

∎

##  4\. Examples

The following example illustrate that when $g=k=1$, the genus $b$ of the base
curve $B$ can be any value.

######  Example 4.1.

Let $f\colon\thinspace S\to B$ be a relative minimal elliptic fiberation,
where $B$ is a non-singular curve of genus $b$. Let $D$ be a section of $S$.
By canonical bundle formula $\omega_{S}=f^{*}(\omega_{B}\otimes\mathscr{L}),$
where $\mathscr{L}$ is an invertible sheaf on $B$ such that
$\deg\mathscr{L}=\chi(S,\mathscr{O}_{S})$. Suppose
$p_{g}(S)=\chi(S,\mathscr{O}_{S})-1+b\geq 2$.

We will show that $D$ is contianed in the fixed part of $|K_{S}+D|$. Let $F$
be a general fiber of $f$, then $D$ intersects $F$ at a unique point $P=D\cap
F$. For any $E\in|K_{S}+D|$. By Riemann-Roch formula
$h^{0}(F,(K_{S}+D)|_{F})=1$, we have $|(K_{S}+D)|_{F}|=P$, thus $P\in E$. It
follows that there exists a dense open set $U\subset B$, such that
$f^{-1}(U)\cap D\subset E$. Thus $D\leq E$. Therefore the linear system
$|K_{S}+D|$ is composed of a pencil, and the induced fiberation is
$f\colon\thinspace S\to B$, thus we have $g=k=1$.

The following two examples show that when $b=0$, the intersection number $k$
can be arbitrarily large.

######  Example 4.2.

Let $C_{1},C_{2}\subset\mathbb{P}^{2}$ be quadratic plane curves such that
$C_{1}\cap C_{2}=\\{p_{1},p_{2},p_{3},p_{4}\\}$. Let $p_{5},p_{6},p_{7}$ be
three distinct points on $C_{1}\setminus\\{p_{1},p_{2},p_{3},p_{4}\\}$ and
$p_{8}\in C_{2}\setminus\\{p_{1},p_{2},p_{3},p_{4}\\}$. Let $C_{3}$ be a
quadratic curve through $p_{5},p_{6},p_{7},p_{8}$. Then $C=C_{1}+C_{2}+C_{3}$
be a plane curve of degree $6$ passing through points $p_{1},\dots,p_{8}$ with
multiplicities are all $2$. Let $\pi\colon\thinspace S\to\mathbb{P}^{2}$ be
the blowing-up at $p_{1},\dots,p_{8}$. Then the strict transform
$\widetilde{C}$ belongs to $|6H-2E_{1}-\cdots-2E_{8}|$, take
$D=\widetilde{C}$. Note that $K_{S}+D\sim 3H-E_{1}-\cdots-E_{8}.$ By Lemma
2.15, $K_{S}+D$ is big. Note that $\widetilde{C}_{1}\sim 2H-E_{1}-\cdots-
E_{7}$ and $(K_{S}+D)\cdot\widetilde{C}_{1}=-1<0,$ thus $\widetilde{C}_{1}$ is
contained in the fixed part of $|K_{S}+D|$. Since $K_{S}+D-(2H-E_{1}-\cdots-
E_{7})=H-E_{8}$ and $(H-E_{8})^{2}=0$, the linear system $|K_{S}+D|$ is
composed of a pencil. Let $\pi^{\prime}\colon\thinspace
S^{\prime}\to\mathbb{P}^{2}$ be the blowing up at $p_{8}$, the pencil of lines
passing through $p_{8}$ induces a fiberation $f^{\prime}\colon\thinspace
S^{\prime}\to\mathbb{P}^{1}$ whose general fiber is $\mathbb{P}^{1}$. By
construction, the birational morphism $\pi\colon\thinspace S\to\mathbb{P}^{2}$
factors through $\pi^{\prime}\colon\thinspace S^{\prime}\to\mathbb{P}^{2}$, we
have a birational morphism $\mu\colon\thinspace S\to S^{\prime}$. The
composition $f=\mu\circ f^{\prime}\colon\thinspace S\to\mathbb{P}^{1}$ is a
fiberation which is induced by $|K_{S}+D|$, thus we have $g=b=0$. Note that
the general fiber $F$ of $f$ is numerically equivalent to $H-E_{8}$, thus
$k=D\cdot F=4$.

######  Example 4.3.

Let $a\geq 2$ be an integer, and
$\mathscr{K}=\\{p_{0},p_{1},\dots,p_{4a-4}\\}$ be a cluster on
$\mathbb{P}^{2}$, where $p_{0},p_{1},\dots,p_{4a-4}$ are points on
$\mathbb{P}^{2}$. Consider the linear system of plane curves of degree $3a$
with assigned base points $\mathscr{K}$ of type $(3a;3a-3,2^{4a-4})$. Let
$C\subset\mathbb{P}^{2}$ be an irreducible member of this linear system. Let
$\pi\colon\thinspace S\to\mathbb{P}^{2}$ be the blowing-up at $\mathscr{K}$.
Let $E_{i}$ be the exceptional divisor over $p_{i}$, $\widetilde{C}$ be the
strict transform of $C$ on $S$ and $H$ be the total transform of the class of
line on $\mathbb{P}^{2}$, we have

$\widetilde{C}\sim 3aH-(3a-3)E_{0}-2\sum_{i=1}^{4a-4}E_{i}.$

Take $D=\widetilde{C}$, consider the log surface $(S,D)$, we have

$K_{S}+D\sim(3a-3)H-(3a-4)E_{0}-\sum_{i=1}^{4a-4}E_{i}.$

By Lemma 2.15, $K_{S}+D$ is big and the linear system
$|(a-1)H-(a-2)E_{0}-\sum_{i=1}^{4a-4}E_{i}|$ is non-empty. Take
$G\in|(a-1)H-(a-2)E_{0}-\sum_{i=1}^{4a-4}E_{i}|$ we have

$(K_{S}+D)\cdot G<0,$

thus $G$ is contained in the fixed part of $|K_{S}+D|$. Sustracting
$(a-1)H-(a-2)E_{0}-\sum_{i=1}^{4a-4}E_{i}$ from $K_{S}+D$, we get
$(2a-2)(H-E_{0})$, which is a pencil. For the same reason as Example 4.2, the
induced fiberation $f\colon\thinspace S\to\mathbb{P}^{1}$ has invariants
$g=b=0$ and $k=3a$.

The following example show that if we drop the assumption $p_{a}(D)\leq
2(l+q(S))+1-h^{1,1}(S)$ of Theorem 3.4, then the invariants $k$ and $g$ can be
arbitrarily large.

######  Example 4.4.

Let $0\leq e\leq g$ be integers, consider the Hirzebruch surface
$\Sigma_{e}=\mathbb{P}(\mathscr{O}\oplus\mathscr{O}(e))$ of degree $e$. Let
$\pi\colon\thinspace\Sigma_{e}\to\mathbb{P}^{1}$ be the
$\mathbb{P}^{1}$-fiberation, let $\Delta_{0}$ and $\Gamma$ be its section with
$\Delta_{0}^{2}=-e$ and its general fiber. Set
$\Delta_{\infty}=\Delta_{0}+e\Gamma$, then $\Delta_{\infty}$ is the class of
tautological line bundle of $\Sigma_{e}$ and we have $\Delta_{\infty}^{2}=e$.
Set $a=g+1-e>0$, then $2\Delta_{\infty}+a\Gamma$ is very ample, and a general
member $D$ of the linear system $|2\Delta_{\infty}+a\Gamma|$ is a smooth
irreducible hyperelliptic curve of genus $g$. Choose
$D_{0},D_{1}\in|2\Delta_{\infty}+a\Gamma|$, consider the Lefschetz pencil
$\\{\alpha D_{0}+\beta D_{1}\\}$,
$[\alpha\colon\thinspace\beta]\in\mathbb{P}^{1}$. Since $D_{0}\cdot
D_{1}=(2\Delta_{\infty}+a\Gamma)^{2}=4e+4a=4g+4.$ Therefore we have $4g+4$
distinct base points $p_{1},\dots,p_{4g+4}$ of this pencil. Let
$\nu\colon\thinspace S\to\Sigma_{e}$ be the blowing-up at these base points,
then we obtain a fiberation $f\colon\thinspace S\to\mathbb{P}^{1}$ whose
general fiber $F$ with genus $g$. Note that

$\displaystyle F\sim\nu^{*}(2\Delta_{\infty}+a\Gamma)-E_{1}-\cdots-E_{4g+4},$
$\displaystyle
K_{S}\sim\nu^{*}(-2\Delta_{\infty}+(e-2)\Gamma)+E_{1}+\cdots+E_{4g+4},$

where $E_{i}$ is the exceptional divisor over $p_{i}$. Consider the linear
system

$|\nu^{*}(x\Delta_{\infty}+y\Gamma)-\sum_{i=1}^{4g+4}2E_{i}|.$

If

(4.1) $(x+1)(y+\frac{ex}{2})+x-3(4g+4)>0$

then it contains an irreducible member $D$, then we have

$K_{S}+D\sim\nu^{*}((x-2)\Delta_{\infty}+(y+e-2)\Gamma)-\sum_{i=1}^{4g-g}E_{i}.$

If

(4.2) $(x-2)(y+e-2+\frac{(x-2)e}{2})-\frac{1}{2}(4g+4)>0$

then $K_{S}+D$ is big. Set $M=\nu^{*}((x-4)\Delta+(y+e-a-2)\Gamma)$, if

(4.3) $(x-3)(y+e-a-1+\frac{(x-4)e}{2})>0$

then we can assume $M\geq 0$. If

(4.4) $(K_{S}+D)\cdot M=(x-4)(x-2)+(x-2)(y+e-a-2)+(x-4)(y+e-2)<0$

then $M$ is contained in the fixed part of $|K_{S}+D|$. Note that
$K_{S}+D-M\sim F$. So $|K_{S}+D|$ is composed of a pencil $|F|$. In this case
the corresponding fiberation is $f\colon\thinspace X\to\mathbb{P}^{1}$, its
general fiber has genus $g$ and $D\cdot F=x(g+1+e)+2y-8g-8$.

We find that $x=8,y=1$, the inequalities (4.1), (4.2), (4.3) and (4.4) become

$\displaystyle e>\frac{12g-5}{36}\quad(\ref{eq: D})$ $\displaystyle
e>\frac{g+4}{12}\quad(\ref{eq: big})$ $\displaystyle
e>\frac{g+1}{4}\quad(\ref{eq: effective})$ $\displaystyle
e<\frac{3g-4}{8}\quad(\ref{eq: fixed part})$

When $g\geq 8$, we have
$\frac{3g-4}{8}>\frac{12g-13}{36}>\frac{g+1}{4}>\frac{g+4}{12}$. Since $e$ is
an integer, not every $g\geq 8$ the interval
$(\frac{12g-13}{36},\frac{3g-4}{8})$ contains an integer, we find that
$g=10,e=3$ satisfies all the inequalities, and when $g\geq 27$ we have
$(\frac{12g-13}{36},\frac{3g-4}{8})\cap\mathbb{Z}\neq\varnothing$. We can see
that $D\cdot F=8e+2$, therefore the genus $g$ and the intersection number
$k=D\cdot F$ can be arbitrarily large.

## Acknowledgements

I would like to give heatly thanks my Ph.D supervisor JinXing Cai for
suggesting generalizing Xiao’s result about canonical maps of surface of log
general type to log surfaces of log general type, and for his continuous
support and encouragement, and to YiFei Chen, Lei Zhang, SongBo Ling, ZhiMin
Ling and JingShan Chen for useful discussion. Many parts of this paper was
accomplished during my Ph.D. The author was supported in part by the the
science and technology innovation fund of Yunnan University: ST20210105.

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