# Slope inequality for an arbitrary divisor

Houari Benammar Ammar

###### Abstract

Let $f:S\longrightarrow C$ be a surjective morphism with connected fibers from
a smooth complex projective surface $S$ to a smooth complex projective curve
$C$ with general fiber $F$. In this paper, we develop a more general version
of slope inequality for a data $(D,\mathcal{F})$ where $D$ is an arbitrary
relatively effective divisor on $S$ and $\mathcal{F}$ is a locally free sub-
sheaf of $f_{*}\mathcal{O}_{S}(D)$ without assuming any linear stability
condition of $|D_{|_{F}}|$, we explain how the speciality of $D$ restricted to
the general fiber play a role in the results. However, we compute some natural
examples and give applications.

##  1 Introduction

Let $f:S\to C$ be a surjective morphism from a smooth complex projective
surface $S$ to a smooth complex projective curve $C$ with connected fibers. We
call the morphism $f$ a fibration or a fibred surface. Let $D$ be a relatively
effective divisor on $S$. We consider the sheaf
$\mathcal{E}=f_{*}\mathcal{O}_{S}(D)$, which is torsion free because $C$ is a
curve. Since a torsion free sheaf on curve is always locally free sheaf,
$\mathcal{E}$ is locally free and its rank is $h^{0}(F,{D}_{|_{F}})$ where $F$
is a general fiber of $f$ of genus $g(F)=g$.

The fibration $f$ is called smooth if all its fibers are smooth, isotrivial if
all its smooth fibers are mutually isomorphic and locally trivial if it is
both smooth and isotrivial. Let $\omega_{S}$ (resp, $K_{S}$) be the canonical
sheaf (resp, a canonical divisor) of $S$, $\omega_{S/C}=\omega_{S}\otimes
f^{*}\omega_{C}^{\vee}$ (resp, $K_{S/C}=K_{S}-f^{*}K_{C})$ the relative
canonical sheaf (resp, a relative canonical divisor), where $\omega_{C}$
(resp, $K_{C}$) is the canonical sheaf of $C$ (resp, a canonical divisor). In
particular, if $D=K_{S/C}$, then $\mathcal{E}$ is a nef vector bundle, [10],
its rank is $g$ and its degree is:

$\operatorname{deg}(\mathcal{E}):=\operatorname{deg}(f_{*}\omega_{S/C})=\chi(\mathcal{O}_{S})-\chi(\mathcal{O}_{F}).\chi(\mathcal{O}_{C})$
$=\chi(\mathcal{O}_{S})-(g-1)(b-1)\text{,}$

for $b=g(C)$. By the Leray spectral sequence, we remark that:

$h^{0}(C,(f_{*}\omega_{S/C})^{\vee})=h^{0}(C,\mathcal{R}^{1}f_{*}\mathcal{O}_{S})=q(S)-b,$

where $q(S)=h^{1}(S,\mathcal{O}_{S})$ is the irregularity of the surface $S$.

In [9], Severi stated that if: $S$ is a minimal smooth complex projective
surface of maximal Albanese dimension, then $K^{2}_{S}\geq
4\chi(\mathcal{O}_{S})$. But the proof was not complete, the inequality was
posed as a conjecture by Reid, [24], and proved by Manetti, [19], under the
assumption that the surface has an ample canonical divisor. Finally, the
conjecture is completely proved by Pardini, [23] which the slope inequality
techniques play an important role in the proof. Xiao Gang, in [29], wrote a
fundamental paper on fibred surfaces over curves, he discussed the geometry of
the fibration where $f$ is relatively minimal and $g(F)\geq 2$. He proved that
if $f$ is relatively minimal and not locally trivial i.e,
$\operatorname{deg}f_{*}\omega_{S/C}\neq 0$ then:

$K_{S/C}^{2}\geq 4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}\text{.}$

We call this last result a slope inequality for relative canonical divisor.
Recall in this setting that $K_{S/C}$ is a nef divisor [22, Theorem 1.4].

Independently, Cornalba and Harris, see [8], proved the above inequality for
semistable fibrations (i.e., fibrations where all the fibers are semistable
curves in the sense of Deligne and Mumford) and later Stoppino [26] showed
that a generalization of the Cornalba-Harris approach gives a full proof of
the slope inequality in which all fibrations are treated by the same method.
Yuan Xinyi and Zhang Tong, [30], gave a new approach to prove the slope
inequality by giving a sense to the relative Noether inequality and using
Frobenius iteration techniques. Motivated by these, [29], [8], in addition to
Fujita’s fundamental papers, [10], [11], there has been interest in giving a
bound related to other geometric invariants using the first and the second
Fujita decomposition combined with a study of linear stability of the general
fiber $F$ of $f$. For instance, see [25]. Konno, in [16], described directly
$K_{S/C}^{2}$ as a sum of two parts under some strict conditions on the
fibration $f$. More precisely, the first part is related to
$\operatorname{deg}f_{*}\omega_{S/C}$ and the second one is described by the
Horikawa index [15]. Since we apply Fujita’s decompositions in this paper, see
Example 5.4. Let us recall the first and second Fujita decompositions.

######  Theorem 1.1 (First Fujita decomposition for fibred surface, [10]).

Let $f:S\to C$ be a fibration from a smooth complex projective surface $S$ to
a smooth projective curve $C$. Then:

$f_{*}\omega_{S/C}=\mathcal{O}_{C}^{q(S)-b}\oplus\mathcal{N}\text{,}$

where $\mathcal{N}$ is a nef sub-vector bundle and
$h^{0}(C,\mathcal{N}^{\vee})=0$.

We remark that in conclusion of Theorem 1.1, the trivial part comes from a
nonzero global section of the dual of $f_{*}\omega_{S/C}$ i.e, from
$H^{0}(C,\mathcal{R}^{1}f_{*}\mathcal{O}_{S})$.

######  Theorem 1.2 (Second Fujita decomposition for fibred surface, [11],
[4], [5], [6]).

Let $f:S\to C$ be a fibration as above. Then:

$f_{*}\omega_{S/C}=\mathcal{A}\oplus\mathcal{U}\text{,}$

where $\mathcal{A}$ is ample sub-vector bundle and $\mathcal{U}$ is a unitary
flat sub-vector bundle.

In the situation of Theorem 1.2, we denote by $u_{f}$ the rank of
$\mathcal{U}$, and call it the _unitary rank_ of the fibred surface $f:S\to
C$. A proof of the second Fujita decomposition is given by Catanese and
Dettweiler, [4], [5], [6]. In the Theorem 5.11 and Theorem 5.12, we discuss
the first and the second Fujita decompositions for the adjoint cases.

In this paper, let $f:S\to C$ be a fibred surface and $D$ be a relatively
effective divisor on $S$ with general fiber $F$, Consider the data
$(D,\mathcal{F})$ where $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$. The
goal of this paper is to give a lower bound of $D^{2}$ in terms of
$\operatorname{deg}\mathcal{F}$ without assuming any linear stability
condition of $|D_{|_{F}}|$ and even if $D$ is not positive, for instance not
nef, we explain how the negative part of $D$ appears in the lower bound of
$D^{2}$. The method used is due to Xiao [29].

Now, we describe briefly the process of the main results, we start by defining
the Miyaoka divisors $(N_{i})_{1\leq i\leq k}$ of a data $(D,\mathcal{F})$
where $k$ is the length of the Harder-Narasimhan filtration
$(\mathcal{F}_{i})_{1\leq i\leq k}$ of $\mathcal{F}$, see Definition 3.4. We
realize that these sequence of divisors can be seeing as two sub-sequences,
the first one is a sequence of special Miyaoka divisors and the second one is
a sequence of nonspecial Miyaoka divisors, we define an important number
$\hat{n}_{(D,\mathcal{F})}$ which tells us the index of the last divisor in
the first sequence (Proposition 3.8). A natural question appears is to look to
a derived two sequences of rational numbers of the both before, the element of
index $i$ in these new sequences is the quotient
$\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}$ where $d_{i}=N_{i}.F$. In Theorem
3.15, we give a lower bound of these elements, clearly the number $2$ is a
lower bound of the sequence $(\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1})_{1\leq
i\leq\hat{n}_{(D,\mathcal{F})}}$ by Clifford Theorem, and we prove that the
number $\beta_{D}:=1+\frac{g(F)}{h^{0}(F,{D}_{|_{F}})-1}$ is a lower bound of
the sequence
$(\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1})_{\hat{n}_{(D,\mathcal{F})}+1\leq
i\leq k}$ for a fixed $D$ and any $\mathcal{F}\subseteq
f_{*}\mathcal{O}_{S}(D)$. In addition we will see that this last finite
sequence decreases. Following the Modified Xiao Lemma (Lemma 4.1), we will see
that we need to find a constant $\alpha_{(D,\mathcal{F})}$ for a data
$(D,\mathcal{F})$ such that
$d_{i}\geq\alpha_{(D,\mathcal{F})}(h^{0}(F,{N_{i}}_{|_{F}})-1)\geq\alpha_{(D,\mathcal{F})}(r_{i}-1)$
(here $r_{i}:=\operatorname{rk}\mathcal{F}_{i}$) with the aim that
$\operatorname{deg}\mathcal{F}$ appears, thus naturally the Definition 4.6
follows. The main result of the paper is the following Theorem:

######  Theorem.

(Theorem 4.10) Let $f:S\to C$ be a fibred surface, consider the data
$(D,\mathcal{F})$, where $D$ is a relatively effective divisor and
$\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$ is a locally free sub-sheaf on
$C$. Assume that $\mathcal{F}$ is not semi-stable. Set, here:

$t_{(D,\mathcal{F})}=\left\\{\begin{array}[]{ll}\operatorname{max}\\{i|\mu_{i}\geq
0\\}&\mbox{if $\mu_{1}\geq 0$}\text{.}\\\
-\infty&\mbox{otherwise}\text{.}\par\end{array}\right.$

  * •

If $t_{(D,\mathcal{F})}=1$, then:

$D^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}_{1}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

  * •

If $1<t_{(D,\mathcal{F})}\leq k$, then:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}_{t_{(D,\mathcal{F})}}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}$
$\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$

  * •

If $t_{(D,\mathcal{F})}=-\infty$, set

$C_{(D,\mathcal{F})}=\left\\{\begin{array}[]{ll}\frac{2\alpha_{(D,\mathcal{F})}d_{k}}{-\alpha_{(D,\mathcal{F})}+2\alpha_{(D,\mathcal{F})}r_{k}-d_{k}}&\mbox{if
$\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k}\leq
0$}\text{.}\\\
3\alpha_{(D,\mathcal{F})}+2d_{k}-2\alpha_{(D,\mathcal{F})}r_{k}&\mbox{otherwise}\text{.}\par\end{array}\right.$

Then:

$D^{2}\geq
C_{(D,\mathcal{F})}.\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

In addition: if $d_{k}=\alpha_{(D,\mathcal{F})}(r_{k}-1)$. Then we have the
following inequality independently to $t_{(D,\mathcal{F})}$:

$D^{2}\geq\frac{2d_{k}}{r_{k}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

We make some remarks about the notations introduced in the previous Theorem.
The proper birational morphism $\epsilon:\hat{S}\to S$ is constructed in the
Proposition 3.3, by definition the divisor $Z_{i}$ on $\hat{S}$ is a fixed
part of $\mathcal{F}_{i},$$\forall i;1\leq i\leq k$, see Definition 3.4. The
number $\mu_{i}:=\mu(\mathcal{F}_{i}/\mathcal{F}_{i-1})$ is the slope of the
quotient $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ (Proposition 3.1).

We explain the previous result as follows, we estimate the lower bound of
$D^{2}$ by a sum of two parts, the first one comes from the degree of such
specific $\mathcal{F}_{i}$ and the geometry of the Miyaoka divisors $N_{i}$,
the second one explain the negative part of $D$ when $D$ is not positive (in
many sense). The case when $\mathcal{F}$ is semi-stable is discussed in the
Remark 4.9. As an application, we apply the previous Theorem to the data
$(D,f_{*}\mathcal{O}_{S}(D))$ when $D$ is relatively nef. Thus, we obtain the
Corollary 4.13. This last Corollary it contains a new results, the constant
$\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}$ exists and it completely defined, see
Definition 4.6, moreover the last paragraph in the Corollary 4.13 extend the
result in [27, Thoerem 5], see Example 4.15 and Theorem 4.16. The form of the
Corollary 4.18 seems that it can be compared for example with [25, Theorem
3.20], our contribution is the existence of such $\alpha_{(D,\mathcal{F})}$
against in the Theorem mentioned, the authors assumed that there exists a real
number $\alpha>0$ such that for every linear subsystem $|P|$ of
$|{N_{k}}_{|_{F}}|$ we have the following:

$\frac{\operatorname{deg}|P|}{\dim|P|}\geq\alpha\text{.}$

Recently in [7, Theorem E], the authors proved a slope inequality for a data
$(D,f_{*}\mathcal{O}_{S}(D))$ of $n$-dimensional fibred variety over curve
with $n\geq 2$, and the assumption that $D$, $f_{*}\mathcal{O}_{S}(D)$ are
both nef. If we apply their result for fibred surfaces, we easly can see that
our result is sharper. More precisely, if $D_{|_{F}}$ is nef + big and non
special, they proved in [7] that:

$D^{2}\geq
2\frac{D.F}{D.F+1}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

But, if we apply the Corollary 4.18 for $(D,f_{*}\mathcal{O}_{S}(D))$ with $D$
and $f_{*}\mathcal{O}_{S}(D)$ are both nef, and $D_{|_{F}}$ is nonspecial,
then:

$D^{2}\geq\frac{2\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}D.F}{D.F+\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

Since $\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}>1$ except maybe for $g(F)=0$, we
see that the constant
$\frac{2\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}D.F}{D.F+\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}}$
is strictly bigger than $\frac{D.F}{D.F+1}$ in general.

Acknowledgements. I am very grateful to my advisors Steven Lu and Nathan
Grieve for their constant help, invaluable advice and financial support.

In the next sections, let $f:S\to C$ be a fibred surface, from a smooth
complex projective surface $S$ to a smooth complex projective curve $C$, and
we let $D$ be a relatively effective divisor on $S$.

##  2 Rational map to a projective bundle

Let $f:S\to C$ be a fibred surface and $D$ be a relatively effective divisor
on $S$. Let $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$ be a locally free
sub-sheaf of rank $r_{\mathcal{F}}$. There exist always the following
commutative diagrams:

${S}$${\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}$${C}$$\scriptstyle{\psi}$$\scriptstyle{f}$$\scriptstyle{\pi}$
${S}$${\mathbb{P}_{C}(\mathcal{F})}$${C}$$\scriptstyle{\psi_{\mathcal{F}}}$$\scriptstyle{f}$$\scriptstyle{\pi_{\mathcal{F}}}$

In the above, $\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))$ (resp
$\mathbb{P}_{C}(\mathcal{F})$) is the projective bundle of one dimensional
quotients (Grothendieck’s notations) of $f_{*}\mathcal{O}_{S}(D)$ (resp of
$\mathcal{F}$), $\pi$ (resp $\pi_{\mathcal{F}}$) is the projective morphism
from $\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))$ to $C$ (resp from
$\mathbb{P}_{C}(\mathcal{F})$ to $C$). The maps $\psi$ and
$\psi_{\mathcal{F}}$ are rational and defined by the following evaluation
maps:

$f^{*}f_{*}\mathcal{O}_{S}(D)\longrightarrow\mathcal{O}_{S}(D)\text{,}$

resp:

$f^{*}\mathcal{F}\longrightarrow\mathcal{O}_{S}(D)\text{.}$

######  Remark 2.1.

  * •

If $D$ is $f$-globally generated. Then:

$f^{*}f_{*}\mathcal{O}_{S}(D)\longrightarrow\mathcal{O}_{S}(D)$

is surjective and $\psi$ is a morphism.

  * •

If the map:

$f^{*}\mathcal{F}\longrightarrow\mathcal{O}_{S}(D)$

is surjective, then $\psi_{\mathcal{F}}$ is a morphism.

Take $A$, a sufficiently very ample divisor on $C$ such that
$f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A)$ is a very ample vector bundle.
Then the rank of $f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A)$ is

$r=H^{0}(F,D|_{F})\text{.}$

And

$\operatorname{deg}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))=\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+r.\operatorname{deg}(A).$

######  Remark 2.2.

Since $f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A)$ is globally generated,
i.e, $\exists\hskip 2.84544ptn>0$ such that we have a surjective map

$\mathcal{O}_{C}^{\oplus n}\longrightarrow
f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A)\text{,}$

or, equivalently, for any $y\in C$ we have a surjective map giving by
evaluation of section:

$H^{0}(C,f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\longrightarrow
f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A)_{|_{y}}\text{,}$

we remark that each section in $H^{0}(F,D|_{F})$ comes from some section of
$H^{0}(C,f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$. In other words, the
following map is surjective

$H^{0}(C,f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\simeq
H^{0}(S,D+f^{*}A)\longrightarrow H^{0}(F,D_{|_{F}})\text{.}$

Now, $\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$ and
$\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))$ are isomorphic by an isomorphism
$s$, see [14, Lemma 7.9].

The rational map

$\phi:S\dashrightarrow\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\text{,}$

defined by:

$f^{*}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\longrightarrow\mathcal{O}_{S}(D)\otimes
f^{*}\mathcal{O}(A)$

is the rational map given by the linear system $|D+f^{*}A|$, and for a general
fibre $F$ of $f$, $\phi_{|_{F}}$ is the map defined by $|D_{|_{F}}|$.

The line bundle
$\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))}(1)$
on $\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$ is very
ample. Then it gives an embedding of this last projective bundle to a
projective space $\mathbb{C}\mathbb{P}^{N}$. We have the following commutative
diagram:

${S}$${\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\subseteq\mathbb{C}\mathbb{P}^{N}}$${C}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\scriptstyle{\pi_{A}}$

Where $\pi_{A}$ is the projection map, again we have $\psi=s\circ\phi$, the
rational map $\phi$ is defined by the complete linear system $|D+f^{*}A|$, if
it has no nontrivial fixed part then its image is contained in no hyperplane.

We assume that there is a fixed part $Z$ of $|D+f^{*}A|$, so the linear system
$|D-Z+f^{*}A|$ factorize the map defined by $|D+f^{*}A|$ and we have the
following properties:

  * •

The fixed part $Z$ of $|D+f^{*}A|$ restricted in $F$ is just the fixed part of
the complete linear system $|D_{|_{F}}|$.

  * •

The system $|D-Z+f^{*}A|$ has no fixed part, so it has only a finite number of
base points.

  * •

The fixed part $Z$ is a divisor such that the homomorphism:

$f^{*}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\longrightarrow\mathcal{O}_{S}(D-Z)\otimes
f^{*}\mathcal{O}(A)\text{,}$

is surjective in codimension 1.

  * •

We can assume that $Z$ has no horizontal components because $A$ is
sufficiently ample.

######  Theorem 2.3.

There exist a surface $\tilde{S}$ birational to $S$ (i.e, $\exists\hskip
2.84544pt\epsilon:\tilde{S}\to S$ which is birational) and a morphism

$\lambda_{A}:\tilde{S}\longrightarrow\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$

such that the following diagram is commutative:

${\tilde{S}}$${S}$${\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\subseteq\mathbb{C}\mathbb{P}^{N}}$${C}$$\scriptstyle{\epsilon}$$\scriptstyle{\lambda_{A}}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\scriptstyle{\pi_{A}}$

$\phi\circ\epsilon=\lambda_{A}$ and:

$(\lambda_{A})^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))}(1)=\epsilon^{*}(\mathcal{O}(D-Z)\otimes
f^{*}\mathcal{O}(A))\otimes\mathcal{O}(-E)\text{.}$

Where $E$ is the exceptional divisor of $\epsilon$.

######  Remark 2.4.

$\epsilon^{*}(\mathcal{O}(D-Z)\otimes
f^{*}\mathcal{O}(A))\otimes\mathcal{O}(-E)$ is globally generated.

###### Proof of Theorem 2.3.

If $|D-Z+f^{*}A|$ has no base point, then $\phi$ is a morphism and there is
nothing to prove. We suppose that there is a base point $x$ in $|D-Z+f^{*}A|$.
We take the blow-up in $x$ defined by $\epsilon^{1}$, so
$|(\epsilon^{1})^{*}(D-Z+f^{*}A)|$ has a fixed part $k_{1}E_{1}$ with
$k_{1}\in\mathbb{Z},k_{1}\geq 1$ and
$|D_{1}|=|(\epsilon^{1})^{*}(D-Z+f^{*}A)-k_{1}E_{1}|$ has no fixed part. Hence
it defines a rational map:
$\lambda^{1}:S_{1}\dashrightarrow\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$
which is identical to $\phi\circ\epsilon^{1}$. If $\lambda^{1}$ is a morphism,
then we are done; if not, we repeat the process. Thus, we get by induction a
sequence $\epsilon^{i}:S_{i}\longrightarrow S_{i-1}$ of blow-ups and a linear
system $|D_{i}|$ with no fixed part, where
$D_{i}=(\epsilon^{i})^{*}D_{i-1}-k_{i}E_{i}$ for $i\geq 1$. But we have:

$D_{i}^{2}=D_{i-1}^{2}-k_{i}^{2}<D_{i-1}^{2}\text{.}$

Since $D_{i}$ has no fixed part, $D_{i}^{2}\geq 0$ and so this process must
terminate. In other words, we arrive at a system $D_{n}$ with no base points,
which defines a morphism:

$\epsilon=\epsilon^{1}\circ...\circ\epsilon^{n}:\tilde{S}\longrightarrow
S\text{.}$

We conclude that $|\epsilon^{*}(D-Z+f^{*}A)-E|$ defines a morphism
$\tilde{S}\overset{\lambda_{A}}{\longrightarrow}\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$
such that:

$\epsilon^{*}(\mathcal{O}(D-Z)\otimes
f^{*}\mathcal{O}(A))\otimes\mathcal{O}(-E)=(\lambda_{A})^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))}(1)\text{.}$

Where $E=\sum_{i=1}^{i=n}k_{i}E_{i}$ is the exceptional divisor. ∎

The last proof is inspired by the proof of [3, Theorem 2.7].

######  Corollary 2.5.

There exist a morphism
$\lambda:\tilde{S}\to\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))$ such that the
following diagram is
commutative.

${\tilde{S}}$${S}$${\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}$${C}$$\scriptstyle{\lambda}$$\scriptstyle{\epsilon}$$\scriptstyle{\psi}$$\scriptstyle{f}$$\scriptstyle{\pi}$

and we have:

$(\lambda)^{*}(\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}(1))=\epsilon^{*}(\mathcal{O}(D-Z))\otimes\mathcal{O}(-E)\text{.}$

###### Proof.

By Theorem 2.3, there exist a morphism
$\lambda_{A}:\tilde{S}\longrightarrow\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))$
which has the property that:

$(\lambda_{A})^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))}(1)=\epsilon^{*}(\mathcal{O}(D-Z)\otimes
f^{*}\mathcal{O}(A))\otimes\mathcal{O}(-E).$

But $\exists
s:\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))\longrightarrow\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))$
which is an isomorphism such that:

$\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D)\otimes\mathcal{O}(A))}(1)=s^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}(1)\otimes\pi_{A}^{*}\mathcal{O}(A)$
$\Longrightarrow(s\circ\lambda_{A})^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}(1)\otimes(\pi_{A}\circ\lambda_{A})^{*}\mathcal{O}(A)$
$=\epsilon^{*}(\mathcal{O}(D-Z))\otimes(f\circ\epsilon)^{*}\mathcal{O}(A)\otimes\mathcal{O}(-E)$
$\Longrightarrow(s\circ\lambda_{A})^{*}\mathcal{O}_{\mathbb{P}_{C}(f_{*}\mathcal{O}_{S}(D))}(1)=\epsilon^{*}(\mathcal{O}(D-Z))\otimes\mathcal{O}(-E).$

We take $\lambda=s\circ\lambda_{A}$. ∎

######  Remark 2.6.

More generally, for $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$ a locally
free sub-sheaf, we take $A$ a sufficiently very ample divisor such that
$\mathcal{F}\otimes\mathcal{O}(A)$ is very ample. Let $L_{\mathcal{F}}$ be a
sub-linear system of $|D+f^{*}A|$ which correspond to a sections of
$H^{0}(\mathcal{F}\otimes\mathcal{O}(A))$. Let $Z_{\mathcal{F}}$ a fixed part
of $L_{\mathcal{F}}$, so $L_{\mathcal{F}}-Z_{\mathcal{F}}$ has no fixed part
and it corresponds to a rational map from $S$ to a projective sub-variety of
$\mathbb{P}_{C}(\mathcal{F}\otimes\mathcal{O}(A))$. By the same arguments
above: $\exists\hskip
2.84544pt\tilde{S}_{\mathcal{F}}\overset{\epsilon_{\mathcal{F}}}{\longrightarrow}S$
which is birational and
$\exists\lambda_{\mathcal{F}}:\tilde{S}_{\mathcal{F}}\longrightarrow\mathbb{P}_{C}(\mathcal{F})$
such that the following diagram is commutative:

${\tilde{S}_{\mathcal{F}}}$${S}$${\mathbb{P}_{C}(\mathcal{F})}$${C}$$\scriptstyle{\lambda_{\mathcal{F}}}$$\scriptstyle{\epsilon_{\mathcal{F}}}$$\scriptstyle{\psi_{\mathcal{F}}}$$\scriptstyle{f}$$\scriptstyle{\pi_{\mathcal{F}}}$

and

$(\lambda_{\mathcal{F}})^{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F})}(1))=\epsilon_{\mathcal{F}}^{*}(\mathcal{O}(D-Z_{\mathcal{F}}))\otimes\mathcal{O}(-E_{\mathcal{F}})\text{,}$

where $E_{\mathcal{F}}$ is the exceptional divisor of
$\epsilon_{\mathcal{F}}$.

##  3 Harder-Narasimhan filtration

In this section, we study Harder-Narasimhan filtration within the context of
fibred surfaces.

######  Proposition 3.1 ([13]).

Let $\mathcal{F}$ be a vector bundle over a smooth projective curve $C$. There
exists a unique sequence of vector sub-bundles of $\mathcal{F}$:

$0=\mathcal{F}_{0}\subsetneq\mathcal{F}_{1}\subsetneq...\subsetneq\mathcal{F}_{k-1}\subsetneq\mathcal{F}_{k}=\mathcal{F}\text{,}$

that satisfies the following conditions:

  * •

for $i=1,...,k$, $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ is a semi-stable vector
bundle.

  * •

for any $i=1,...,k$, setting
$\mu_{i}:=\mu(\mathcal{F}_{i}/\mathcal{F}_{i-1})=\frac{\operatorname{deg}(\mathcal{F}_{i}/\mathcal{F}_{i-1})}{\operatorname{rk}(\mathcal{F}_{i}/\mathcal{F}_{i-1})}$,
we have that:

$\mu_{1}>\mu_{2}>...>\mu_{k}\text{.}$

In the context of Proposition 3.1 above, the filtration is called the _Harder-
Narasimhan filtration_ of $\mathcal{F}$. We set $\mu_{f}=\mu_{k}$ and call it
the _final slope_ of $\mathcal{F}$. The following elementary lemma is
important in what follows.

######  Lemma 3.2.

Let $r_{i}$ be the rank of $\mathcal{F}_{i}$. Then:

$\operatorname{deg}\mathcal{F}=\sum_{i=1}^{k-1}r_{i}(\mu_{i}-\mu_{i+1})+r_{k}\mu_{k}\text{.}$

###### Proof.

Indeed, we consider the exact sequence:

$0\longrightarrow\mathcal{F}_{k-1}\longrightarrow\mathcal{F}_{k}\longrightarrow\mathcal{F}_{k}/\mathcal{F}_{k-1}\longrightarrow
0\text{.}$

From the additivity of degree, we have that:

$\operatorname{deg}\mathcal{F}_{k}=\operatorname{deg}\mathcal{F}_{k-1}+\operatorname{deg}\mathcal{F}_{k}/\mathcal{F}_{k-1}\text{.}$

Similarly, we have that

$\operatorname{deg}\mathcal{F}_{k-1}=\operatorname{deg}\mathcal{F}_{k-2}+\operatorname{deg}\mathcal{F}_{k-1}/\mathcal{F}_{k-2}\text{.}$

And so, by induction, we can conclude that:

$\operatorname{deg}\mathcal{F}_{k}=\sum_{i=1}^{k}\operatorname{deg}\mathcal{F}_{i}/\mathcal{F}_{i-1}\text{.}$

From the definition of slope, for every $i=1,..,k$ we have:
$\operatorname{deg}\mathcal{F}_{i}/\mathcal{F}_{i-1}=\mu_{i}(r_{i}-r_{i-1})$.
Thus, we obtain the desired formula. ∎

Consider now a fibred surface $f:S\to C$, $F$ its general fiber and let $D$ be
a relatively effective divisor on $S$, $(\mathcal{F}_{i})_{0\leq i\leq k}$ be
the Harder-Narasimhan filtration of
$\mathcal{F}\subseteq\mathcal{E}=f_{*}\mathcal{O}_{S}(D)$. By a repeated
application of the Remark 2.6 to each of $\mathcal{F}_{i}$, we have the
following proposition:

######  Proposition 3.3.

There exists a suitable smooth projective surface $\hat{S}$ and a birational
morphism $\epsilon:\hat{S}\to S$ such that the following diagram is
commutative $\forall 1\leq i\leq k$:

${\hat{S}}$${\tilde{S}_{\mathcal{F}_{i}}}$${S}$${\mathbb{P}_{C}(\mathcal{F}_{i})}$${C}$$\scriptstyle{\lambda_{i}}$$\scriptstyle{\epsilon_{i}}$$\scriptstyle{\epsilon}$$\scriptstyle{\lambda_{\mathcal{F}_{i}}}$$\scriptstyle{\epsilon_{\mathcal{F}_{i}}}$$\scriptstyle{\psi_{\mathcal{F}_{i}}}$$\scriptstyle{f}$$\scriptstyle{\pi_{i}}$

Where $\epsilon_{\mathcal{F}_{i}}$ is a blow-up morphism along a finite number
of points $\\{x_{i_{1}},\dots,x_{i_{m_{i}}}\\}$, $1\leq i\leq k$, as defined
in the proof of Theorem 2.3 and Remark 2.6.

###### Proof.

Set $\bigcup_{1\leq i\leq
k}\\{x_{i_{1}},\dots,x_{i_{m_{i}}}\\}=\\{q_{1},\dots,q_{m}\\}$ and
$\hat{S}=\mathbb{BL}_{q_{1},...,q_{m}}(S)$. Then there exists a blowing up
morphism $\epsilon_{i}$ along
$\\{q_{1},...,q_{m}\\}\backslash\\{x_{i_{1}},\dots,x_{i_{m_{i}}}\\}$ which
fits into the diagram above and
$\epsilon=\epsilon_{i}\circ\epsilon_{\mathcal{F}_{i}}:\hat{S}\to S$. As
before, we fix a sufficiently ample divisor $A$ on $C$. Define the map
$\lambda_{i}:\hat{S}\to\mathbb{P}_{C}(\mathcal{F}_{i})$ by the linear system
$|\epsilon^{*}(D-Z_{\mathcal{F}_{i}})-E|$, then we have
$\lambda_{i}=\lambda_{\mathcal{F}_{i}}\circ\epsilon_{i}.$
Moreover, for any $\mathcal{F}_{i}$ in the filtration, we have:

$\lambda_{i}^{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1))=\epsilon^{*}(\mathcal{O}(D-Z_{\mathcal{F}_{i}}))\otimes\mathcal{O}(-E)\text{.}$

Where $Z_{\mathcal{F}_{i}}$ is a fixed part of
$L_{\mathcal{F}_{i}}\subseteq|D+f^{*}A|$ which correspond to a sections of
$H^{0}(\mathcal{F}_{i}\otimes\mathcal{O}(A))$. Here $E$, is the exceptional
divisor of $\epsilon$. ∎

In what follows, we study the fibred surface
$\hat{f}:=f\circ\epsilon:\hat{S}\to C$ and we denote by $F$ its general fiber.

######  Definition 3.4 (Compare with [25, Definition 3.11]).

In this setting, just described above, $\forall i;1\leq i\leq k$ we define
these divisors on $\hat{S}$:

  * •

$Z_{i}:=\epsilon^{*}Z_{\mathcal{F}_{i}}+E$ the _fixed part_ of the vector sub-
bundle $\mathcal{F}_{i},\forall 1\leq i\leq k$.

  * •

$M_{i}:=\lambda_{i}^{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1))$ the
_moving part_ of the vector sub-bundle $\mathcal{F}_{i},\forall 1\leq i\leq
k$.

  * •

Set $N_{i}:=M_{i}-\mu_{i}F,\forall 1\leq i\leq k$. We call it the $i^{th}$
_Miyaoka divisor_.

Applying, [20], [21] and [17, Proposition 6.4.11] we prove the following known
result using the language of $\mathbb{Q}$-twisted vector bundles:

######  Lemma 3.5.

$\forall 1\leq i\leq k,\hskip 5.69046ptN_{i}$ are nef divisors on $\hat{S}$.

###### Proof.

Fix $i;1\leq i\leq k$, let us see that the $\mathbb{Q}$-twisted vector bundle
$\mathcal{F}_{i}\langle-\frac{c_{1}(\mathcal{F}_{i}/\mathcal{F}_{i-1})}{\
rk(\mathcal{F}_{i}/\mathcal{F}_{k-1})}\rangle$ is nef.
Note:

$\left\\{\begin{array}[]{ll}\mathcal{G}_{i}=\mathcal{F}_{i}/\mathcal{F}_{i-1}.\\\
\delta_{i}=\frac{c_{1}(\mathcal{G}_{i})}{\operatorname{rk}(\mathcal{G}_{i})}.\end{array}\right.$

Also, $\mathcal{G}_{i}\langle-\delta_{i}\rangle$ is nef vector bundle, [17,
Proposition 6.4.11], and $\operatorname{deg}\delta_{i}=\mu_{i}$. So:

$-\operatorname{deg}\delta_{1}<-\operatorname{deg}\delta_{2}<...<-\operatorname{deg}\delta_{k}.$

Using the following vector bundle exact sequence:

$0\longrightarrow\mathcal{F}_{i-1}\longrightarrow\mathcal{F}_{i}\longrightarrow\mathcal{G}_{i}\longrightarrow
0\text{,}$

we prove that $\mathcal{F}_{i}\langle-\delta_{i}\rangle$ is nef by induction.

For $i=1$,
$\mathcal{F}_{1}\langle-\delta_{1}\rangle=\mathcal{G}_{1}\langle-\delta_{1}\rangle$
which is nef. Now assume that $\mathcal{F}_{i-1}\langle-\delta_{i-1}\rangle$
is a nef bundle, then $\mathcal{F}_{i-1}\langle-\delta_{i}\rangle$ is
$\mathbb{Q}$-ample and using the exact sequence above, since
$\mathcal{F}_{i-1}\langle-\delta_{i}\rangle$ and
$\mathcal{G}_{i}\langle-\delta_{i}\rangle$ are nef, we see that
$\mathcal{F}_{i}\langle-\delta_{i}\rangle$ is nef bundle. Thus,
$\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i}\langle-\delta_{i}\rangle)}(1)$ is
nef line bundle, so
$\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1))\otimes\pi_{i}^{*}\mathcal{O}(-\delta_{i})$
is nef, then by the remarks that
$\lambda_{i}^{*}\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1))=M_{i}$ and
$\lambda_{i}^{*}(\pi_{i}^{*}\mathcal{O}(-\delta_{i}))=\mathcal{O}(-\mu_{i}F)$,
we conclude that $N_{i}$ is nef, $\forall i;1\leq i\leq k$. ∎

######  Lemma 3.6.

$\forall 1\leq i\leq k,\hskip
5.69046ptr_{i}=\operatorname{rk}\mathcal{F}_{i}\leq h^{0}(F,{N_{i}}_{|_{F}})$.

###### Proof.

Let $\pi_{i}$ be the projection form $\mathbb{P}_{C}(\mathcal{F}_{i})$ to $C$,
we have:

$(\pi_{i}\circ\lambda_{i})_{*}(M_{i})=(\pi_{i})_{*}((\lambda_{i})_{*}M_{i})$
$=(\pi_{i})_{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1)\otimes\lambda_{i*}\mathcal{O}_{\hat{S}})\supseteq(\pi_{i})_{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1))=\mathcal{F}_{i}$
$\Longrightarrow r_{i}\leq h^{0}(F,{M_{i}}_{|_{F}})$ $\Longrightarrow
r_{i}\leq h^{0}(F,{N_{i}}_{|_{F}}).$

∎

######  Proposition 3.7.

Let $d_{i}=\operatorname{deg}({N_{i}}_{|_{F}})=N_{i}.F$; $\forall i,1\leq
i\leq k$, we have:

$d_{k}\geq d_{k-1}\geq...\geq d_{1}\geq 0.$

###### Proof.

Since $F$ is a fibre, $F^{2}=0$. Thus:

$d_{i}=N_{i}.F=(M_{i}-\mu_{i}F).F=M_{i}.F$
$=\lambda_{i}^{*}(\mathcal{O}_{\mathbb{P}_{C}(\mathcal{F}_{i})}(1)).F=(\epsilon^{*}(D-Z_{\mathcal{F}_{i}})-E).F$
$=\epsilon^{*}(D-Z_{\mathcal{F}_{i}}).F=(D-Z_{\mathcal{F}_{i}}).F\geq 0.$

But : $Z_{\mathcal{F}_{i}}\geq Z_{\mathcal{F}_{i+1}}$. Thus,
$D-Z_{\mathcal{F}_{i+1}}=D-Z_{\mathcal{F}_{i}}+(Z_{\mathcal{F}_{i}}-Z_{\mathcal{F}_{i+1}})$

$\Longrightarrow d_{i+1}\geq d_{i}.$

∎

######  Proposition 3.8.

Continuing with the setting as above, we define the following constant for the
data $(D,\mathcal{F})$:

$\hat{n}_{(D,\mathcal{F})}:=\left\\{\begin{array}[]{ll}\operatorname{max}\\{i|{N_{i}}_{|_{F}}\text{is
special}\\}\text{.}\\\ -\infty\text{ otherwise}\text{.}\end{array}\right.$

If $\hat{n}_{(D,\mathcal{F})}\neq-\infty$ then ${N_{j}}_{|_{F}}$ is special
for $1\leq j\leq\hat{n}_{(D,\mathcal{F})}$ and ${N_{j}}_{|_{F}}$ is nonspecial
for $\hat{n}_{(D,\mathcal{F})}+1\leq j\leq k$. Otherwise, ${N_{j}}_{|_{F}}$ is
nonspecial, $\forall 1\leq j\leq k$.

###### Proof.

Recall that

$Z_{\mathcal{F}_{1}}\geq...\geq Z_{\mathcal{F}_{k}}.$

Recall also, we identified the general fiber of $S$ with the general fiber of
$\hat{S}$. Let $\hat{n}_{(D,\mathcal{F})}+1\leq j\leq k-1$ thus,

${N_{j}}_{|_{F}}=\epsilon^{*}(D-{Z_{\mathcal{F}_{j}})}_{|_{F}}=(D-{Z_{\mathcal{F}_{j}})}_{|_{F}}\leq(D-{Z_{\mathcal{F}_{j+1}})}_{|_{F}}={N_{j+1}}_{|_{F}}.$

Consider the following short exact sequence:

$0\longrightarrow{N_{j}}_{|_{F}}\longrightarrow{N_{j+1}}_{|_{F}}\longrightarrow{N_{j+1}}_{|_{F}}/{N_{j}}_{|_{F}}\longrightarrow
0$

which has the long exact cohomology sequence:

$\dots\longrightarrow
H^{0}(F,{N_{j+1}}_{|_{F}}/{N_{j}}_{|_{F}})\longrightarrow
H^{1}(F,{N_{j}}_{|_{F}})\longrightarrow
H^{1}(F,{N_{j+1}}_{|_{F}})\longrightarrow 0.$

So, if $h^{1}(F,{N_{j}}_{|_{F}})=0$, then $h^{1}(F,{N_{j+1}}_{|_{F}})=0$. ∎

To illustrate the Proposition 3.8, consider the following examples.

######  Example 3.9.

Let $D=K_{S/C},g\geq 1$ and $\mathcal{F}=f_{*}\omega_{S/C}$, then we have
${N_{k}}_{|_{F}}\simeq K_{F}$, thus $H^{1}(F,K_{F})=1$. Hence
${N_{k}}_{|_{F}}$ is special. It follows also that ${N_{i}}_{|_{F}}$ is
special for all $1\leq i\leq k$, thus
$\hat{n}_{(K_{S/C},f_{*}\omega_{S/C})}=k$. Conversely, if
$\hat{n}_{(D,\mathcal{F})}=k$, then $\operatorname{deg}D_{|_{F}}\leq 2g-2$.

######  Example 3.10.

Now, let $D=K_{S/C}+\Delta,g\geq 1$,
$\mathcal{F}=f_{*}\mathcal{O}_{S}(K_{S/C}+\Delta)$ and $\Delta$ is effective
on $S$ with $\Delta.F>0$, then ${N_{k}}_{|_{F}}\simeq K_{F}+{\Delta}_{|_{F}}$,
thus $H^{1}(F,K_{F}+{\Delta}_{|_{F}})=0$ since the degree of
$K_{F}+{\Delta}_{|_{F}}$ is strictly greater than $2g-2$. So
$\hat{n}_{(K_{S/C}+\Delta,f_{*}\mathcal{O}_{S}(K_{S/C}+\Delta))}<k$.

######  Example 3.11.

As in the Example 3.10, take $D=K_{S/C}+\Delta$ and $\Delta$ is effective on
$S$ with $\Delta.F>0$, assume that $f_{*}\mathcal{O}_{S}(D)$ is semi-stable.
Then $k=1$ and
$\hat{n}_{(K_{S/C}+\Delta,f_{*}\mathcal{O}_{S}(K_{S/C}+\Delta))}=-\infty$.

######  Example 3.12.

We give another example such that $\hat{n}_{(D,\mathcal{F})}=-\infty$. Let
$\mathcal{E}=\mathcal{O}_{\mathbb{CP}_{1}}\bigoplus\mathcal{O}_{\mathbb{CP}_{1}}(1)$
and $S=\mathbb{P}_{\mathbb{CP}_{1}}(\mathcal{E})$. Then we define a natural
fibration $f:S\to\mathbb{CP}_{1}$. $S$ has only one negative curve
$E\simeq\mathbb{CP}_{1}$ such that $E^{2}=-1$ and $E=\mathcal{O}_{S}(1)-F$ in
$\textbf{Pic}(S)$, by construction the trivial part
$\mathcal{O}_{\mathbb{CP}_{1}}$ correspond to a $0$-dimensional sub-linear
system $L_{0}$ of $|\mathcal{O}_{S}(1)|$ generated by the effective divisor
$E+F$ and $\mathcal{O}_{\mathbb{CP}_{1}}(1)$ to the $1$-dimensional sub-linear
system $L_{1}$ generated by the two effective divisor of
$|\mathcal{O}_{S}(1)|$ different to $E+F$.

Let $0\subsetneq\mathcal{O}_{\mathbb{CP}_{1}}(1)\subsetneq\mathcal{E}$ is the
Harder-Narasimhan filtration of $\mathcal{E}$, since $L_{1}$ has no fixed part
then $N_{1}=\mathcal{O}_{S}(1)-F$ and $N_{2}=\mathcal{O}_{S}(1)$, thus
${N_{1}}_{|_{F}},{N_{2}}_{|_{F}}$ are nonspecial divisors, and
$\hat{n}_{(\mathcal{O}_{1}(S),\mathcal{E})}=-\infty\text{.}$

######  Example 3.13.

More generally than Example 3.12 , every fibred surface $f:S\to C$ with
$g(F)=0$ has $\hat{n}_{(D,\mathcal{F})}=-\infty$ for every relatively
effective divisor $D$ and $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$,
since there are no special divisors on $\mathbb{CP}_{1}$.

Now, it is natural to ask about the sequence
$(\frac{d_{i}}{h^{0}(F,{N_{i}}_{|_{F}})-1})_{i\in\\{1,...k\\}}$. For instance,
is it increasing finite sequence? Is it decreasing? Is it bounded from below
by strictly positive number?

######  Lemma 3.14.

$\forall i;2\leq i\leq k$, we have $h^{0}(F,{N_{i}}_{|_{F}})>1$.

###### Proof.

We have $h^{0}(F,{N_{i}}_{|_{F}})\geq\operatorname{rk}(\mathcal{F}_{i})$. So,
if $h^{0}(F,{N_{i}}_{|_{F}})=1$ the only possibility is $i=1$ and more than
that the degree: $d_{1}=g(F)-h^{1}(F,{N_{1}}_{|_{F}}).$ ∎

Without loss of generality, we can assume that $\mathcal{E}$ has at least rank
$2$ in the next Theorem.

######  Theorem 3.15.

Let $f:S\to C$ be a fibration as above, $D$ be a relatively effective divisor
on $S$ with $h^{0}(F,D_{|_{F}})>1$. Consider the Harder-Narasimhan filtration
$(\mathcal{F}_{i})$ of $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$. Let:

$\hat{s}_{(D,\mathcal{F})}:=\left\\{\begin{array}[]{ll}1&\mbox{if
$h^{0}(F,{N_{1}}_{|_{F}})>1$}\text{,}\\\
2&\mbox{otherwise}\text{,}\end{array}\right.$

and

$\beta_{D}:=1+\frac{g(F)}{h^{0}(F,{D}_{|_{F}})-1}\text{.}$

Then, we have the following result:

  * (1).

If $\hat{n}_{(D,\mathcal{F})}=-\infty$, then ${N_{i}}_{|_{F}}$ is nonspecial
$\forall i,1\leq i\leq k$ and

$\beta_{D}\leq\frac{d_{k}}{h^{0}({N_{k}}_{|_{F}})-1}\leq...\leq\frac{d_{i+1}}{h^{0}({N_{i+1}}_{|_{F}})-1}\leq$
$\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\leq...\leq\frac{d_{\hat{s}_{(D,\mathcal{F})}}}{h^{0}({N_{\hat{s}_{(D,\mathcal{F})}}}_{|_{F}})-1}.$

  * (2).

Otherwise:

    * –

$\forall i;\hat{n}_{(D,\mathcal{F})}+1\leq i\leq k$:

$\beta_{D}\leq\frac{d_{k}}{h^{0}({N_{k}}_{|_{F}})-1}\leq...\leq\frac{d_{i+1}}{h^{0}({N_{i+1}}_{|_{F}})-1}\leq$
$\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\leq...\leq\frac{d_{\hat{n}_{(D,\mathcal{F})}+1}}{h^{0}({N_{\hat{n}_{(D,\mathcal{F})}+1}}_{|_{F}})-1}.$

    * –

$\forall i;\hat{s}_{(D,\mathcal{F})}\leq i\leq\hat{n}_{(D,\mathcal{F})}$:

$\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\geq 2\text{.}$

###### Proof.

For $(1)$, if $\hat{n}_{(D,\mathcal{F})}=-\infty$, then by definition of
$\hat{n}_{(D,\mathcal{F})}$, ${N_{i}}_{|_{F}}$ is nonspecial $\forall i;1\leq
i\leq k$. Thus by Riemann-Roch:

$h^{0}({N_{i}}_{|_{F}})=d_{i}+1-g(F)\text{.}$

Since:

$h^{0}({N_{i+1}}_{|_{F}})=h^{0}({N_{i}}_{|_{F}})+d_{i+1}-d_{i}\text{,}\hskip
8.5359pt.$

It follows that $\forall i;\hat{s}_{(D,\mathcal{F})}\leq i\leq k-1$:

$\frac{d_{i+1}}{h^{0}({N_{i+1}}_{|_{F}})-1}=\frac{d_{i}+d_{i+1}-d_{i}}{h^{0}({N_{i}}_{|_{F}})+d_{i+1}-d_{i}-1}\leq\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\text{.}$

However, $D_{|_{F}}$ is nonspecial and by Riemann-Roch formula on curves, we
have:

$h^{0}(F,D_{|_{F}})=D.F+1-g(F)\text{.}$

Thus:

$h^{0}(F,D_{|_{F}})=h^{0}(F,{N_{i}}_{|_{F}})+D.F-d_{i}\hskip 5.69046pt\forall
i;1\leq i\leq k\text{.}$

Then, we deduce the desired lower bound:

$\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\geq
1+\frac{g(F)}{h^{0}(F,{D}_{|_{F}})-1}\hskip 5.69046pt\forall
i;\hat{s}_{(D,\mathcal{F})}\leq i\leq k.$

This proves $(1)$.
For $(2)$, if $\hat{n}_{(D,\mathcal{F})}+1\leq i\leq k$, then argue as in
$(1)$. Suppose now that ${N_{i}}_{|_{F}}$ is special i.e,
$\hat{s}_{(D,\mathcal{F})}\leq i\leq\hat{n}$. Then by Clifford theorem, [1],
we have:

$d_{i}\geq 2(h^{0}({N_{i}}_{|_{F}})-1)$
$\Longrightarrow\frac{d_{i}}{h^{0}({N_{i}}_{|_{F}})-1}\geq 2.$

∎

######  Remark 3.16.

Compared the Proposition 3.7 and the Theorem 3.15, we deduce that $\forall
i;\hat{s}_{(D,\mathcal{F})}\leq i\leq k$:

$d_{k}\geq d_{k-1}\geq...\geq d_{\hat{s}}\geq 1\text{.}$

##  4 Slope inequality.

Now, we are ready to present the technical lemma in the method, we called it
the Modified Xiao Lemma. Note that it is a more general form of Xiao [29,
Lemma 2].

######  Lemma 4.1 (Modified Xiao Lemma).

Let $\hat{f}:\hat{S}\to C$ be a fibred surface with $F$ its general fiber,
$\hat{D}$ be a divisor on $\hat{S}$ and suppose that there exist a sequence of
effective divisors:

$\mathcal{Z}_{1}\geq\mathcal{Z}_{2}\geq...\geq\mathcal{Z}_{j}\text{,}$

and a sequence of rational numbers:

$\mu_{1}\geq\mu_{2}\geq...\geq\mu_{j}\text{,}$

such that for every $i\in\\{1,...,j\\}$, we have:

$\mathcal{N}_{i}:=\hat{D}-\mathcal{Z}_{i}-\mu_{i}F$

are nef $\mathbb{Q}$-divisors. Then:

$\hat{D}^{2}\geq\sum_{i=1}^{j-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})+2\hat{D}.\mathcal{Z}_{j}-\mathcal{Z}_{j}^{2}+2\mu_{j}d_{j}\text{,}$

where $d_{i}=\mathcal{N}_{i}.F$.

###### Proof.

First observe that $\mathcal{N}_{1}^{2}\geq 0$ by nefness. And we have:

$\mathcal{N}_{i}^{2}=\mathcal{N}_{i}(\mathcal{N}_{i-1}+(\mathcal{Z}_{i-1}-\mathcal{Z}_{i})+(\mu_{i-1}-\mu_{i})F)$
$\geq\mathcal{N}_{i}(\mathcal{N}_{i-1}+(\mu_{i-1}-\mu_{i})F)$
$\geq(\mathcal{N}_{i-1}+(\mathcal{Z}_{i-1}-\mathcal{Z}_{i})+(\mu_{i-1}-\mu_{i})F)(\mathcal{N}_{i-1}+(\mu_{i-1}-\mu_{i})F)$
$\geq\mathcal{N}_{i-1}^{2}+(\mu_{i-1}-\mu_{i})(2\mathcal{N}_{i-1}F+(\mathcal{Z}_{i-1}-\mathcal{Z}_{i})F)$
$=\mathcal{N}_{i-1}^{2}+(\mu_{i-1}-\mu_{i})(d_{i-1}+d_{i})\text{.}$

So, by induction, we have:

$\mathcal{N}_{j}^{2}\geq\sum_{i=1}^{j-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})\text{.}$

Hence

$(\hat{D}-\mathcal{Z}_{j}-\mu_{j}F)^{2}\geq\sum_{i=1}^{j-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})\text{.}$

But

$(\hat{D}-\mathcal{Z}_{j}-\mu_{j}F)^{2}=(\hat{D}-\mathcal{Z}_{j})^{2}-2\mu_{j}(\hat{D}-\mathcal{Z}_{j})F$
$=\hat{D}^{2}-2\hat{D}.\mathcal{Z}_{j}+\mathcal{Z}_{j}^{2}-2\mu_{j}d_{j}\text{.}$

So we have:

$\hat{D}^{2}\geq\sum_{i=1}^{j-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})+2\hat{D}.\mathcal{Z}_{j}-\mathcal{Z}_{j}^{2}+2\mu_{j}d_{j}\text{.}$

∎

######  Remark 4.2.

The part: $2\hat{D}.\mathcal{Z}_{j}-\mathcal{Z}_{j}^{2}+2\mu_{j}d_{j}$
describes the negativity of $\hat{D}$.

######  Example 4.3.

In the setting of Lemma 4.1, suppose that $j=k$, $\hat{D}$ is nef and
$\mu_{k}\geq 0$. Set $\mathcal{Z}_{k+1}=0$ and $\mu_{k+1}=0$. Then apply the
Lemma 4.1 to the sequence of effective divisors:

$\mathcal{Z}_{1}\geq\mathcal{Z}_{2}\geq...\geq\mathcal{Z}_{k+1}=0\text{,}$

and a sequence of rational numbers:

$\mu_{1}\geq\mu_{2}\geq...\geq\mu_{k+1}=0\text{.}$

So, we conclude the original result of Xiao Gang [29, Lemma 2]:

$\hat{D}^{2}\geq\sum_{i=1}^{k}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1}).$

Prior to stating the main results, we apply the Lemma 4.1 to the data
$(D,\mathcal{F})$ where $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$. Let
$(\mathcal{F}_{i})_{0\leq i\leq k}$ be the Harder-Narasimhan filtration of
$\mathcal{F}$ and $(Z_{i},M_{i},N_{i})_{1\leq i\leq k}$ be the triple of fixed
part, moving part, the Miyaoka divisor respectively; see the Definiton 3.4.
Then we deduce the following inequalities:

######  Theorem 4.4.

Let $f:S\to C$ be a fibred surface as before, consider the data
$(D,\mathcal{F})$ where $D$ is a relatively effective divisor on $S$ and
$\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$ is a locally free sub-sheaf on
$C$. Assume $h^{0}(F,D_{|_{F}})>1$, then we have the following three cases:

  * •

if $\hat{n}_{(D,\mathcal{F})}=-\infty$, then:

$D^{2}\geq
2\beta_{D}\operatorname{deg}\mathcal{F}-\beta_{D}\mu_{1}+(\beta_{D}-2\beta_{D}r_{k}+2d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

  * •

if $\hat{n}_{(D,\mathcal{F})}=k$, then:

$D^{2}\geq
4\operatorname{deg}\mathcal{F}-2\mu_{1}+2(1-2r_{k}+d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

  * •

if $1\leq\hat{n}_{(D,\mathcal{F})}<k$, then:

$D^{2}\geq
4\sum_{i=1}^{\hat{n}_{(D,\mathcal{F})}-1}r_{i}(\mu_{i}-\mu_{i+1})+2\beta_{D}\sum_{i=\hat{n}_{(D,\mathcal{F})}+1}^{k-1}r_{i}(\mu_{i}-\mu_{i+1})$
$-2(\mu_{1}-\mu_{\hat{n}_{(D,\mathcal{F})}+1})-\beta_{D}(\mu_{\hat{n}_{(D,\mathcal{F})}+1}-\mu_{k})+(\beta_{D}+2)r_{\hat{n}_{(D,\mathcal{F})}}(\mu_{\hat{n}_{(D,\mathcal{F})}}-\mu_{\hat{n}_{(D,\mathcal{F})+1}})$
$+2\hat{D}.{Z}_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{.}$

Here $\epsilon$ is constructed as in the Proposition 3.3 and $\beta_{D}$ is as
in the Theorem 3.15.

###### Proof.

Recall that $\hat{f}=f\circ\epsilon:\hat{S}\to C$ is a fibred surface with $F$
its general fiber and $\hat{D}:=\epsilon^{*}D$ is a relatively effective
divisor on $\hat{S}$. Consider $(\mathcal{F}_{i})_{0\leq i\leq k}$ the Harder-
Narasimhan filtration of $\mathcal{F}$, let $(\mu_{1},\dots,\mu_{k})$ be the
sequence of slopes and $(Z_{i},M_{i},N_{i})_{1\leq i\leq k}$ the triple of
fixed part, moving part, the Miyaoka divisor respectively, recall also that
$\hat{D}=Z_{i}+M_{i},\forall 1\leq i\leq k$ and

$Z_{1}\geq Z_{2}\geq\dots\geq Z_{k}\geq 0,$
$\mu_{1}>\mu_{2}>\dots>\mu_{k}\text{.}$

Thus, $N_{i}$ are $\mathbb{Q}$-nef divisors on $\hat{S}$, $d_{i}=N_{i}.F$. Now
by the Lemma 4.1 we have the following inequality:

$\hat{D}^{2}\geq\sum_{i=1}^{k-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})+2\hat{D}.{Z}_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{.}$
(*)

Recall that $\epsilon:\hat{S}\to S$ is a proper birational morphism, then
$\hat{D}^{2}=D^{2}$. By the Theorem 3.15 we have the following natural three
cases:

  * •

if $\hat{n}_{(D,\mathcal{F})}=-\infty$ , then ${N_{i}}_{|_{F}}$ is nonspecial
$\forall i,1\leq i\leq k$ and

$d_{i}\geq\beta_{D}(h^{0}({N_{i}}_{|_{F}})-1)\geq\beta_{D}(r_{i}-1)\text{,}$

the right hand side inequality follows from the Lemma 3.6 where
$r_{i}=\operatorname{rk}(\mathcal{F}_{i});\forall 1\leq i\leq k$, we remember
also that $r_{i+1}\geq r_{i}-1;\forall 1\leq i\leq k-1$, then:

$d_{i+1}\geq\beta_{D}r_{i}\text{.}$

We replace these informations in (* ‣ 4):

$D^{2}\geq
2\beta_{D}\sum_{i=1}^{k-1}r_{i}(\mu_{i}-\mu_{i+1})-\beta_{D}(\mu_{1}-\mu_{k})+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{.}$

By Lemma 3.2 we deduce the desired inequality

$D^{2}\geq
2\beta_{D}\operatorname{deg}\mathcal{F}-\beta_{D}\mu_{1}+(\beta_{D}-2\beta_{D}r_{k}+2d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

  * •

if $\hat{n}_{(D,\mathcal{F})}=k$, then ${N_{i}}_{|_{F}}$ is special $\forall
i,1\leq i\leq k$ and

$d_{i}\geq 2(h^{0}({N_{i}}_{|_{F}})-1)\geq 2(r_{i}-1)\text{,}$

however

$d_{i+1}\geq 2r_{i}\text{.}$

This implies:

$D^{2}\geq
4\operatorname{deg}\mathcal{F}-2\mu_{1}+2(1-2r_{k}+d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

  * •

if $1\leq\hat{n}_{(D,\mathcal{F})}<k$, then:

$\forall i;\hat{n}_{(D,\mathcal{F})}+1\leq i\leq k,\hskip
11.38092ptd_{i}\geq\beta_{D}(h^{0}({N_{i}}_{|_{F}})-1)\geq\beta_{D}(r_{i}-1)\text{,}$

and

$\forall i;1\leq i\leq\hat{n}_{(D,\mathcal{F})},\hskip 11.38092ptd_{i}\geq
2(h^{0}({N_{i}}_{|_{F}})-1)\geq 2(r_{i}-1)\text{.}$

So, we decompose (* ‣ 4) in three parts:

$D^{2}\geq\sum_{i=1}^{\hat{n}_{(D,\mathcal{F})}-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})+\sum_{i=\hat{n}_{(D,\mathcal{F})}+1}^{k-1}(d_{i}+d_{i+1})(\mu_{i}-\mu_{i+1})$
$+(d_{\hat{n}_{(D,\mathcal{F})}}+d_{\hat{n}_{(D,\mathcal{F})}+1})(\mu_{\hat{n}_{(D,\mathcal{F})}}-\mu_{\hat{n}_{(D,\mathcal{F})}+1})$
$+2\hat{D}.{Z}_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{.}$ $\Longrightarrow D^{2}\geq
4\sum_{i=1}^{\hat{n}_{(D,\mathcal{F})}-1}r_{i}(\mu_{i}-\mu_{i+1})+2\beta_{D}\sum_{i=\hat{n}_{(D,\mathcal{F})}+1}^{k-1}r_{i}(\mu_{i}-\mu_{i+1})$
$-2(\mu_{1}-\mu_{\hat{n}_{(D,\mathcal{F})}})-\beta_{D}(\mu_{\hat{n}_{(D,\mathcal{F})}+1}-\mu_{k})+(\beta_{D}+2)r_{\hat{n}_{(D,\mathcal{F})}}(\mu_{\hat{n}_{(D,\mathcal{F})}}-\mu_{\hat{n}_{(D,\mathcal{F})+1}})$
$+2\hat{D}.{Z}_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{,}$

since $d_{\hat{n}_{(D,\mathcal{F})}}\geq 2(r_{\hat{n}_{(D,\mathcal{F})}}-1)$
and
$d_{\hat{n}_{(D,\mathcal{F})}+1}\geq\beta_{D}(r_{\hat{n}_{(D,\mathcal{F})}+1}-1)\geq\beta_{D}(r_{\hat{n}_{(D,\mathcal{F})}})$
.

∎

######  Remark 4.5.

The inequality that we obtain in the Theorem 4.4 if
$1\leq\hat{n}_{(D,\mathcal{F})}<k$ is sharp following the Lemma 4.1 and the
Theorem 3.15. In general, we could give a lower bound for $D^{2}$ by
$4\operatorname{deg}\mathcal{F}_{\hat{n}_{(D,\mathcal{F})}}$,
$2\beta_{D}\operatorname{deg}\mathcal{F}/\mathcal{F}_{\hat{n}_{(D,\mathcal{F})}}$
and other terms, but to formulate the slope inequality for the data
$(D,\mathcal{F})$, first we should start by taking
$\operatorname{min}(2,\beta_{D})$ and write for instance
$d_{i}\geq\operatorname{min}(2,\beta_{D})(r_{i}-1)$, following the set-up we
will obtain in the next Corollary a lower bound for $D^{2}$ by
$2\operatorname{min}(2,\beta_{D})\operatorname{deg}\mathcal{F}$ and other
terms more positive than in the Theorem 4.4.

######  Definition 4.6.

Let $f:S\to C$ be a fibred surface, $F$ its general fiber. Consider the data
$(D,\mathcal{F})$ as before where $D$ is a relatively effective divisor on $S$
and $\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$. Then we define the
following number:

$\alpha_{(D,\mathcal{F})}=\left\\{\begin{array}[]{ll}\beta_{D}&\mbox{if}\hskip
5.69046pt\hat{n}_{(D,\mathcal{F})}=-\infty\\\ 2&\mbox{if}\hskip
5.69046pt\hat{n}_{(D,\mathcal{F})}=k\\\
\operatorname{min}(2,\beta_{D})&\mbox{if}\hskip
5.69046pt1\leq\hat{n}_{(D,\mathcal{F})}<k.\end{array}\right.$

######  Corollary 4.7.

Let $f:S\to C$ be a fibred surface as before, consider the data
$(D,\mathcal{F})$ where $D$ is a relatively effective divisor on $S$ and
$\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$ is a locally free sub-sheaf on
$C$.

  * •

If $h^{0}(F,D_{|_{F}})>1$, then:

$D^{2}\geq
2\alpha_{(D,\mathcal{F})}\operatorname{deg}\mathcal{F}-\alpha_{(D,\mathcal{F})}\mu_{1}+(\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Again, here $\epsilon$ is constructed as in the Proposition 3.3.

  * •

Else, if $h^{0}(F,D_{|_{F}})=1$ then:

$D^{2}\geq
2d_{1}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

###### Proof.

Fist assume that $h^{0}(F,D_{|_{F}})>1$, for
$\hat{n}_{(D,\mathcal{F})}=-\infty,k$ the inequality is proved in the Theorem
4.4, we will prove only the case $1\leq\hat{n}_{(D,\mathcal{F})}<k$. By the
Theorem 3.15 and Lemma 3.6 we have:

$d_{i}\geq\alpha_{(D,\mathcal{F})}(h^{0}({N_{i}}_{|_{F}})-1)\hskip
5.69046pt\forall i;1\leq i\leq k\text{,}$

Putting this last inequality in (* ‣ 4), then we have the following:

$D^{2}\geq
2\alpha_{(D,\mathcal{F})}\sum_{i=1}^{k-1}r_{i}(\mu_{i}-\mu_{i+1})-\alpha_{(D,\mathcal{F})}(\mu_{1}-\mu_{k})+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}+2\mu_{k}d_{k}\text{.}$

By Lemma 3.2 we deduce the desired inequality

$D^{2}\geq
2\alpha_{(D,\mathcal{F})}\operatorname{deg}\mathcal{F}-\alpha_{(D,\mathcal{F})}\mu_{1}+(\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k})\mu_{k}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Now, if $h^{0}(F,D_{|_{F}})=1$ then $\mathcal{F}$ is a locally free sub-sheaf
of rank $1$ and $k=1$, $\mu_{k}=\mu_{1}$, thus
$h^{0}({N_{1}}_{|_{F}})=r_{1}=1$, $d_{k}=d_{1}=g(F)-h^{1}({N_{1}}_{|_{F}})$,
By the inequality (* ‣ 4) we have:

$D^{2}\geq
2d_{1}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.{Z}_{1}-Z_{1}^{2}\text{.}$

∎

######  Remark 4.8.

We see that $\alpha_{(D,\mathcal{F})}$ can always exist and is given
explicitly in the Definition 4.6 without assuming any linear stability
conditions in fibers.

######  Remark 4.9.

If $\mathcal{F}$ is semi-stable locally free sheaf. Then by Corollary 4.7 we
have the following inequality:

$D^{2}\geq
2\frac{d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.{Z}_{1}-Z_{1}^{2}\text{.}$

Now, the goal is to discuss the first point in the Corollary 4.7 depending to
$\mu_{1}$ and $\mu_{k}$. Without loss of generality we assume that
$\mathcal{F}$ is not semi-stable locally free sheaf in the next Theorem.

######  Theorem 4.10.

Let $f:S\to C$ be a fibred surface, consider the data $(D,\mathcal{F})$, where
$D$ is relatively effective divisor and $\mathcal{F}\subseteq
f_{*}\mathcal{O}_{S}(D)$ is a locally free sub-sheaf on $C$. Assume that
$\mathcal{F}$ is not semi-stable. Set, here:

$t_{(D,\mathcal{F})}=\left\\{\begin{array}[]{ll}\operatorname{max}\\{i|\mu_{i}\geq
0\\}&\mbox{if $\mu_{1}\geq 0$}\text{.}\\\
-\infty&\mbox{otherwise}\text{.}\end{array}\right.$

  * •

If $t_{(D,\mathcal{F})}=1$, then:

$D^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}_{1}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

  * •

If $1<t_{(D,\mathcal{F})}\leq k$, then:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}_{t_{(D,\mathcal{F})}}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}$
$\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$

  * •

If $t_{(D,\mathcal{F})}=-\infty$, set

$C_{(D,\mathcal{F})}=\left\\{\begin{array}[]{ll}\frac{2\alpha_{(D,\mathcal{F})}d_{k}}{-\alpha_{(D,\mathcal{F})}+2\alpha_{(D,\mathcal{F})}r_{k}-d_{k}}&\mbox{if
$\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k}\leq
0$}\text{.}\\\
3\alpha_{(D,\mathcal{F})}+2d_{k}-2\alpha_{(D,\mathcal{F})}r_{k}&\mbox{otherwise}\text{.}\end{array}\right.$

Then:

$D^{2}\geq
C_{(D,\mathcal{F})}.\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

In addition: if $d_{k}=\alpha_{(D,\mathcal{F})}(r_{k}-1)$. Then we have the
following inequality independently to $t_{(D,\mathcal{F})}$:

$D^{2}\geq\frac{2d_{k}}{r_{k}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

###### Proof.

Recall that $(\mathcal{F}_{i})_{1\leq i\leq k}$ is the Harder-Narasimhan
filtration of $\mathcal{F}$, and if $1\leq m\leq k$ then
$(\mathcal{F}_{i})_{1\leq i\leq m}$ is the Harder-Narasimhan filtration of
$\mathcal{F}_{m}$. By the Corollary 4.7 we have the following inequality for
the data $(D,\mathcal{F}_{m})$:

$D^{2}\geq
2\alpha_{(D,\mathcal{F}_{m})}\operatorname{deg}\mathcal{F}_{m}-\alpha_{(D,\mathcal{F}_{m})}\mu_{1}+(\alpha_{(D,\mathcal{F}_{m})}-2\alpha_{(D,\mathcal{F}_{m})}r_{m}+2d_{m})\mu_{m}+2\epsilon^{*}D.Z_{m}-Z_{m}^{2}\text{.}$
(**)

  * •

If $t_{(D,\mathcal{F})}=1$, we have $\mu_{1}\geq 0$ and starting from $i=2$,
$\mu_{i}<0$. We apply the inequality (** ‣ 4) for $m=1$, then:

$D^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}_{1}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

  * •

If $t_{(D,\mathcal{F})}>1$, so $\mu_{1},\dots,\mu_{t_{(D,\mathcal{F})}}\geq
0$. By the fact that:

$\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}-2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}r_{t_{(D,\mathcal{F})}}+2d_{t_{(D,\mathcal{F})}}\geq-\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}\text{,}$

and from the inequality (** ‣ 4), we deduce the following for
$m=t_{(D,\mathcal{F})}$:

$D^{2}\geq
2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}\operatorname{deg}\mathcal{F}_{t_{(D,\mathcal{F})}}-\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}(\mu_{1}+\mu_{t_{(D,\mathcal{F})}})+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$
(***)

Now, we apply the Lemma 4.1 to the sequences
$\\{Z_{1},Z_{t_{(D,\mathcal{F})}}\\}$ and
$\\{\mu_{1},\mu_{t_{(D,\mathcal{F})}}\\}$. So:

$D^{2}\geq
d_{t_{(D,\mathcal{F})}}(\mu_{1}+\mu_{t_{(D,\mathcal{F})}})+d_{1}(\mu_{1}-\mu_{t_{(D,\mathcal{F})}})+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$
(****)

By the fact that $d_{1}(\mu_{1}-\mu_{t_{(D,\mathcal{F})}})\geq 0$, and by
Remark 3.16 we have $d_{t_{(D,\mathcal{F})}}\geq 1$ i.e, we can divide by
$d_{t_{(D,\mathcal{F})}}$. Then:

$-\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}(\mu_{1}+\mu_{t_{(D,\mathcal{F})}})\geq\frac{-\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}{d_{t_{(D,\mathcal{F})}}}D^{2}+2\frac{\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}{d_{t_{(D,\mathcal{F})}}}\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-\frac{\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}{d_{t_{(D,\mathcal{F})}}}Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$

We combine this last inequality with (*** ‣ • ‣ 4), we deduce the desired
result:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}_{t_{(D,\mathcal{F})}}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}$
$\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{t_{(D,\mathcal{F})}}-Z_{t_{(D,\mathcal{F})}}^{2}\text{.}$

  * •

If $t_{(D,\mathcal{F})}=-\infty$, so both $\mu_{1},\mu_{k}<0$ and by the fact
that:

$\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k}\geq-\alpha_{(D,\mathcal{F})}\text{,}$

then:

$(\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k})\mu_{1}\leq-\alpha_{(D,\mathcal{F})}\mu_{1}\text{,}$

We replace this last inequality in (** ‣ 4) for the data $(D,\mathcal{F})$.
So:

$D^{2}\geq
2\alpha_{(D,\mathcal{F})}\operatorname{deg}\mathcal{F}+(\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k})(\mu_{1}+\mu_{k})+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

We set $A=\alpha_{(D,\mathcal{F})}-2\alpha_{(D,\mathcal{F})}r_{k}+2d_{k}$ and
we discuss the inequality above depending to $A$.

    * –

If $A\leq 0$, then we apply the Lemma 4.1 to the sequences $\\{Z_{1},Z_{k}\\}$
and $\\{\mu_{1},\mu_{k}\\}$:

$D^{2}\geq d_{k}(\mu_{1}+\mu_{k})+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$
(*****)

By the two last inequalities above, we deduce:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F})}d_{k}}{-\alpha_{(D,\mathcal{F})}+2\alpha_{(D,\mathcal{F})}r_{k}-d_{k}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

    * –

If $A>0$, we remark that:

$\operatorname{deg}\mathcal{F}\leq\mu_{1}+\mu_{k}\text{.}$

Then:

$D^{2}\geq(3\alpha_{(D,\mathcal{F})}+2d_{k}-2\alpha_{(D,\mathcal{F})}r_{k})\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Now, if $d_{k}=\alpha_{(D,\mathcal{F})}(r_{k}-1)$, we apply the inequality (**
‣ 4) for $m=k$:

$D^{2}\geq
2\alpha_{(D,\mathcal{F})}\operatorname{deg}\mathcal{F}-\alpha_{(D,\mathcal{F})}(\mu_{1}+\mu_{k})+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Combine this last inequality with (***** ‣ – ‣ • ‣ 4), we deduce the desired
result:

$D^{2}\geq\frac{2d_{k}}{r_{k}}\operatorname{deg}\mathcal{F}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

∎

######  Proposition 4.11.

Let $f:S\to C$ be a fibred surface and $D$ is a relatively effective divisor
and relatively nef divisor on $S$, consider the data
$(D,f_{*}\mathcal{O}_{S}(D))$. Then $\epsilon^{*}D.Z_{k}\geq 0$.

###### Proof.

Recall that $Z_{k}=\epsilon^{*}Z_{\mathcal{F}_{k}}+E$ and
$Z_{\mathcal{F}_{k}}$ is the fixed part of $|D+f^{*}A|$ where $A$ is a
sufficiently very ample divisor on $C$. So $Z_{\mathcal{F}_{k}}$ is an
effective divisor supported in the fibers, then:

$\epsilon^{*}D.Z_{k}=D.Z_{\mathcal{F}_{k}}\geq 0\text{,}$

since $D$ is relatively nef. ∎

######  Proposition 4.12.

Let $f:S\to C$ be a fibred surface and $D$ is a relatively effective divisor,
consider the data $(D,f_{*}\mathcal{O}_{S}(D))$. Then $Z_{k}^{2}\leq 0$

###### Proof.

Assume that $Z_{\mathcal{F}_{k}}^{2}>0$ then by Hodge index Theorem, either
$\lim_{n\to\infty}h^{0}(nZ_{\mathcal{F}_{k}})=\infty$ or
$\lim_{n\to\infty}h^{0}(K_{S}-nZ_{\mathcal{F}_{k}})=\infty$, but
$K_{S}-nZ_{\mathcal{F}_{k}}$ is never effective if $Z_{\mathcal{F}_{k}}$ is
not zero, so it must $\lim_{n\to\infty}h^{0}(nZ_{\mathcal{F}_{k}})=\infty$.
Contradiction with the fact that $Z_{\mathcal{F}_{k}}$ is not movable. Then
$Z_{\mathcal{F}_{k}}^{2}\leq 0$, thus $Z_{k}^{2}\leq 0$. ∎

######  Corollary 4.13.

Let $f:S\to C$ be a fibred surface and $D$ is a relatively effective divisor
and relatively nef divisor on $S$, consider the data
$(D,f_{*}\mathcal{O}_{S}(D))$. Then:

  * •

If $f_{*}\mathcal{O}_{S}(D)$ is semi-stable:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,{D}_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

  * •

If $f_{*}\mathcal{O}_{S}(D)$ is not semi-stable:

    * –

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=k$: $D$ is nef and

$D^{2}\geq\frac{2\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}D.F}{D.F+\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

    * –

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=1$:

$D^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

    * –

If $1<t_{(D,f_{*}\mathcal{O}_{S}(D))}<k$:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,f_{*}\mathcal{O}_{S}(D))}})}d_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}}{d_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}+\alpha_{(D,\mathcal{F}_{t_{(D,f_{*}\mathcal{O}_{S}(D))}})}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}-Z_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}^{2}\text{.}$

    * –

If $t=-\infty$:

$D^{2}\geq
C_{(D,f_{*}\mathcal{O}_{S}(D))}.\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

In addition: if
$D.F=\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}(h^{0}(F,D_{|_{F}})-1)$. Then
independtly to $t$, we have:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,{D}_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

###### Proof.

First, if $f_{*}\mathcal{O}_{S}(D)$ is semi-stable, then by Remark 4.9 we
have:

$D^{2}\geq
2\frac{d_{1}}{r_{1}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.{Z}_{1}-Z_{1}^{2}\text{.}$

But $2\epsilon^{*}D.{Z}_{1}-Z_{1}^{2}\geq 0$ by Proposition 4.11 and
Proposition 4.12, thus $d_{1}=D.F$,
$r_{1}=\operatorname{rk}(f_{*}\mathcal{O}_{S}(D))=h^{0}(F,{D}_{|_{F}})$. So we
deduce the desired inequality:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,{D}_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

Now, if $f_{*}\mathcal{O}_{S}(D)$ is not semi-stable, we apply the Theorem
4.10. Then:

  * •

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=k$:

$D^{2}\geq\frac{2\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}d_{k}}{d_{k}+\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

  * •

If $t_{(D,f_{*}\mathcal{O}_{S}(S))}=1$:

$D^{2}\geq\frac{2d_{1}}{r_{1}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{1}-Z_{1}^{2}\text{.}$

  * •

If $1<t_{(D,f_{*}\mathcal{O}_{S}(D))}<k$:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,f_{*}\mathcal{O}_{S}(D))}})}d_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}}{d_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}+\alpha_{(D,\mathcal{F}_{t_{(D,f_{*}\mathcal{O}_{S}(D))}})}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}-Z_{t_{(D,f_{*}\mathcal{O}_{S}(D))}}^{2}\text{.}$

  * •

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=-\infty$

$D^{2}\geq
C_{(D,f_{*}\mathcal{O}_{S}(D))}.\operatorname{deg}f_{*}\mathcal{O}_{S}(D)+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Again by Proposition 4.11 and Proposition 4.12,
$2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\geq 0$, thus $d_{k}=D.F$. So we deduce the
desired result:

  * •

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=k$ i.e, $\mu_{k}\geq 0$ and we have:

$D^{2}\geq\frac{2\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}D.F}{D.F+\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

  * •

If $t_{(D,f_{*}\mathcal{O}_{S}(D))}=-\infty$ i.e, $\mu_{1}<0$:

$D^{2}\geq
C_{(D,f_{*}\mathcal{O}_{S}(D))}.\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

Also, If $\mu_{k}\geq 0$, we have $f_{*}\mathcal{O}_{S}(D)$ is nef on $C$. So
$M_{k}$ is nef on $\hat{S}$. But

$\hat{D}=Z_{k}+M_{k}\text{.}$

Since $Z_{\mathcal{F}_{k}}$ is supported in the fibers and $D$ is relatively
nef, we deduce that $D$ is nef. ∎

######  Remark 4.14.

In general, if $D$ is nef, we don’t necessarily have $\mu_{k}\geq 0$.

######  Example 4.15.

In [27, Thoerem 5], they proved that: if $D$ be a relatively nef divisor on
$f:S\to B$ such that $D_{|}{{}_{F}}$ is generated by global sections on a
general fiber $F$ of $f$, $D_{|}{{}_{F}}$ is a special divisor on $F$ and

$2h^{0}(F,D_{|_{F}})-D.F-1>0\text{.}$ (P)

Then:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,D_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

First, we remark that the condition (P) is equivalent to
$2h^{0}(F,D_{|_{F}})-D.F-1=1$, since by Clifford Theorem:

$D.F\geq 2(h^{0}(F,D_{|_{F}})-1)$

Thus, we can just assume that $D_{|{{}_{F}}}$ has a section and not
necessarily is generated by global sections since we can always eliminate the
horizental fixed part of $D$. By Corollary 4.13 we see that even
$D_{|{{}_{F}}}$ is special or nonspecial, we proved the same inequality:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,{D}_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{,}$

if the following more general condition holds:

$D.F=\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}(h^{0}(F,D_{|_{F}})-1)\text{.}$

######  Theorem 4.16.

let $f:S\to C$ be a fibred surface and $F$ its general fiber, if $D$ is
relatively effective and relatively nef divisor, $D_{|_{F}}$ is nonspecial
with $h^{0}(F,D_{|_{F}})>g$. Then:

$D^{2}\geq
2\frac{D.F}{h^{0}(F,{D}_{|_{F}})}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

###### Proof.

By assumption, $D_{|_{F}}$ is nonspecial and $h^{0}(F,D_{|_{F}})>g$, then
$\beta_{D}\leq 2$ and $\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}=\beta_{D}$. So:

$D.F=\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}(h^{0}(F,D_{|_{F}})-1)\text{.}$

The result follows from the Example 4.15. ∎

######  Proposition 4.17.

Let $f:S\to C$ be a fibred surface and $D$ is nef divisor on $S$, consider the
data $(D,\mathcal{F})$. Then if $t_{(D,\mathcal{F})}\geq 1$:

$2\epsilon^{*}D.Z_{i}-Z_{i}^{2}\geq 0,\hskip 5.69046pt\forall i;1\leq i\leq
t_{(D,\mathcal{F})}\text{.}$

###### Proof.

Recall that $\epsilon^{*}D=M_{i}+Z_{i}$, thus
$2\epsilon^{*}D.Z_{i}-Z_{i}^{2}=(\epsilon^{*}D+M_{i})Z_{i}$. Since $M_{i}$ and
$\epsilon^{*}D$ are nef $\forall i;1\leq i\leq t_{(D,\mathcal{F})}$. So, we
deduce the positivity of the number $2\epsilon^{*}D.Z_{i}-Z_{i}^{2}$ ∎

######  Corollary 4.18.

(Compare with [25, Theorem 3.20]) Let $f:S\to C$ be a fibred surface and $D$
is nef divisor on $S$, consider the data $(D,\mathcal{F})$ where
$\mathcal{F}\subseteq f_{*}\mathcal{O}_{S}(D)$. Then:

  * •

If $\mathcal{F}$ is semi-stable or $t_{(D,\mathcal{F})}=1$:

$D^{2}\geq 2\frac{d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}_{1}\geq
2\frac{d_{1}}{r_{1}}\operatorname{deg}\mathcal{F}\text{.}$

  * •

If $1<t_{(D,\mathcal{F})}\leq k$, then:

$D^{2}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}_{t_{(D,\mathcal{F})}}\geq\frac{2\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}d_{t_{(D,\mathcal{F})}}}{d_{t_{(D,\mathcal{F})}}+\alpha_{(D,\mathcal{F}_{t_{(D,\mathcal{F})}})}}\operatorname{deg}\mathcal{F}\text{.}$

###### Proof.

Apply the Proposition 4.17 and the Theorem 4.10 ∎

##  5 Examples and applications

######  Example 5.1.

Let $D=K_{S/C}$ be a relative canonical divisor on $S$, Assume that $g(F)\geq
2$, thus by [10], $f_{*}\omega_{S/C}$ is nef vector bundle on $C$,
$\hat{n}_{(K_{S/C},f_{*}\omega_{S/C})}=k$ since $D_{|_{F}}=K_{F}$ is special
divisor on $F$, this implies $\alpha_{(K_{S/C},f_{*}\omega_{S/C})}=2$. We have
also $t_{(K_{S/C},f_{*}\omega_{S/C})}=k$ where $k$ is the length of Harder-
Narasimhan filtration of $f_{*}\omega_{S/C}$, $d_{k}=2g-2$ where $g=g(F)$.
Then by Remark 4.9 and Theorem 4.10 we have:

$K_{S/C}^{2}\geq
4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}+2\epsilon^{*}D.Z_{k}-Z_{k}^{2}\text{.}$

Recall that
$2\epsilon^{*}D.Z_{k}-Z_{k}^{2}=2D.Z_{\mathcal{F}_{k}}-Z_{\mathcal{F}_{k}}^{2}-E^{2}$
where $E$ is the exceptional divisor of $\epsilon$, see Proposition 3.3. In
[2, Example 2.1], they compute
$2D.Z_{\mathcal{F}_{k}}-Z_{\mathcal{F}_{k}}^{2}-E^{2}$ for nodal fibrations,
they proved if $n\geq 1$ is the total number of disconnecting nodes contained
in the fibres, we have:

$2D.Z_{\mathcal{F}_{k}}-Z_{\mathcal{F}_{k}}^{2}-E^{2}\geq n\text{.}$

So:

$K_{S/C}^{2}\geq 4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}+n\text{.}$

It means, if $K_{S/C}^{2}=4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}$,
then $f$ is never a nodal fibration with at least $1$ disconnected node.

If we assume that $D$ is relatively nef, by Corollary 4.13, $D$ is nef thus
$2D.Z_{\mathcal{F}_{k}}-Z_{\mathcal{F}_{k}}^{2}-E^{2}\geq 0$. And to have
$K_{S/C}^{2}=4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}$, it must
$2D.Z_{\mathcal{F}_{k}}-Z_{\mathcal{F}_{k}}^{2}-E^{2}=0$. Then
$|K_{S/C}+f^{*}\mathcal{O}(A)|$ has no fixed part and it is base point free
for a sufficiently ample divisor on $C$.

In general if $D$ is relatively nef, we always have the original Xiao Gang
[29] result:

$K_{S/C}^{2}\geq 4\frac{g-1}{g}\operatorname{deg}f_{*}\omega_{S/C}\text{.}$

######  Example 5.2.

Assume $g=g(F)\geq 2$. Let $D=mK_{S/C}$ be a relative pluricanonical divisor
on $S$ for $m\geq 2$. It is very known that $f_{*}\omega_{S/C}^{\otimes m}$ is
a nef vector bundle on $C$, see [28, Theorem 1.3] for instance. If $K_{S/C}$
is relatively nef, then by Corollary 4.13 we have

  * •

If $f_{*}\omega_{S/C}^{\otimes m}$ is semi-stable:

$m^{2}K_{S/C}^{2}\geq\frac{4m}{2m-1}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{,}$

since $\operatorname{rk}(f_{*}\omega_{S/C}^{\otimes m})=(2m-1)(g-1)$. So:

$K_{S/C}^{2}\geq\frac{4}{m(2m-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{.}$

  * •

If $f_{*}\omega_{S/C}^{\otimes m}$ is not semi-stable, then using the fact
that $mK_{F}$ is nonspecial divisor, we see that

$\hat{n}_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes m})}<k\text{,}$

and

$\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes
m})}=1+\frac{g}{h^{0}(F,mK_{F})-1}\text{,}$

since $h^{0}(F,mK_{F})>g$. However

$t_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes m})}=k$

because $f_{*}\omega_{S/C}^{\otimes m}$ is nef vector bundle on $C$. Then:

$m^{2}K_{S/C}\geq\frac{2\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes
m})}d_{k}}{d_{k}+\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes
m})}}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes m}\text{.}$

Again here $d_{k}=2m(g-1)$ and $h^{0}(F,mK_{F})=(2m-1)(g-1)$, thus:

$\frac{2\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes
m})}d_{k}}{m^{2}(d_{k}+\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes
m})})}=\frac{4}{m(2m-1)}$ $\Longrightarrow
K_{S/C}^{2}\geq\frac{4}{m(2m-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{.}$

In this last computations, we see that even $f_{*}\omega_{S/C}^{\otimes m}$ is
stable or not, we have the same inequality and the reason is that:
$d_{k}=\alpha_{(mK_{S/C},f_{*}\omega_{S/C}^{\otimes m})}.(r_{k}-1)$, this
explain the last paragraph in the Corollary 4.13. But in general for a data
$(D,\mathcal{F})$ where $\mathcal{F}\subsetneq f_{*}\mathcal{O}_{S}(D)$, the
constant
$\frac{2\alpha_{(D,\mathcal{F})}d_{k}}{d_{k}+\alpha_{(D,\mathcal{F})}}$ is
different form $\frac{2d_{k}}{r_{k}}$ because the inequality
$d_{k}>\alpha_{(D,\mathcal{F})}(r_{k}-1)$ can successfully hapen.

######  Example 5.3.

Now, let $f:S\to C$ be a fibred surface and $F$ its general fiber with
$g(F)\geq 2$ as before, take $D=K_{S/C}+L$ such that $L$ is nef and relatively
big with $L.F>1$. We know that $f_{*}\mathcal{O}_{S}(D)$ is nef vector bundle
on $C$ with $\operatorname{rk}(f_{*}\mathcal{O}_{S}(D))=g-1+L.F\neq 0$. If $D$
is a relatively nef divisor, then by Corollary 4.13:

$(K_{S/C}+L)^{2}\geq
2\left(1+\frac{g-1}{g-1+L.F}\right)\operatorname{deg}f_{*}(\omega_{S/C}\otimes\mathcal{O}(L))\text{.}$

since $d_{k}=\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}(r_{k}-1)$, clearly here:

$\hat{n}_{(D,f_{*}\mathcal{O}_{S}(D))}<k\text{,}$
$\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}=1+\frac{g}{g-2+L.F}\text{,}$

and since $f_{*}(\omega_{S/C}\otimes\mathcal{O}(L))$ is nef vector bundle on
$C$, the quantity $t_{(D,f_{*}\mathcal{O}_{S}(D))}$ is maximal, that means:

$t_{(D,f_{*}\mathcal{O}_{S}(D))}=k\text{.}$

######  Example 5.4.

We give a trivial example in which we can see that
$t_{(D,f_{*}\mathcal{O}_{S}(D))}=-\infty$ can be happen. Let $f:S\to C$ be a
fibred surface and $F$ its general fiber with $g(F)=2$. Assume that $K_{S/C}$
is nef divisor and the second Fujita decomposition (Theorem 1.2) of
$f_{*}\omega_{S/C}$ is not trivial i.e, there exist an ample line bundle
$\mathcal{A}$ on $C$ and a flat line bundle $\mathcal{U}$ such that:

$f_{*}\omega_{S/C}=\mathcal{A}\oplus\mathcal{U}\text{.}$

Now, let $D=K_{S/C}+f^{*}\mathcal{M}$ for a sufficientely divisor
$\mathcal{M}$ on $C$ such that
$\operatorname{deg}(\mathcal{A}\otimes\mathcal{O}(\mathcal{M}))<0$, in this
case $D$ is a relatively nef divisor on $S$ such that $D_{|_{F}}=K_{F}$ and
$f_{*}\mathcal{O}_{S}(D)$ decompose in two parts:

$f_{*}\mathcal{O}_{S}(D)=\mathcal{A}\otimes\mathcal{M}\oplus\mathcal{U}\otimes\mathcal{M}\text{.}$

The Harder-Narasimhan filtration of $f_{*}\mathcal{O}_{S}(D)$ is the
following:

$0\subsetneq\mathcal{A}\otimes\mathcal{M}\subsetneq
f_{*}\mathcal{O}_{S}(D)\text{.}$

Consider the data $(D,f_{*}\mathcal{O}_{S}(D))$, since $D_{|_{F}}=K_{F}$, we
see that $D_{|_{F}}$ is special, $\hat{n}_{(D,f_{*}\mathcal{O}_{S}(D))}=2$ and
$\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}=2$, since $g(F)=2$, we have $d_{2}=2$.
Also clearly $t_{(D,f_{*}\mathcal{O}_{S}(D))}=-\infty$, by Corollary 4.13 we
deduce the following inequality:

$D^{2}\geq
C_{(D,f_{*}\mathcal{O}_{S}(D))}\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

But $C_{(D,f_{*}\mathcal{O}_{S}(D))}=2$. Then:

$D^{2}\geq 2\operatorname{deg}f_{*}\mathcal{O}_{S}(D)\text{.}$

Else, we remark $d_{k}=\alpha_{(D,f_{*}\mathcal{O}_{S}(D))}(r_{k}-1)$. Then
again by Corollary 4.13 or Example 4.15, we deduce the same inequality.

######  Proposition 5.5.

Let $f:S\to C$ be a fibred surface and $F$ its general fiber with $g(F)\geq
2$. Then:

$K_{S/C}^{2}=\frac{2}{m(m-1)}\left(\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}-\operatorname{deg}f_{*}\omega_{S/C}\right),\hskip 5.69046pt\forall m\geq
2\text{.}$

######  Remark 5.6.

In particular:

$K_{S/C}^{2}=\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
2}-\operatorname{deg}f_{*}\omega_{S/C}\text{.}$

###### Proof.

By a simple application of the Riemann-Roch formula for vector bundle on
curve, we have:

$h^{0}(S,\omega_{S}^{\otimes m})-h^{1}(S,\omega_{S}^{\otimes
m})+h^{0}(C,\mathcal{R}^{1}f_{*}\omega_{S}^{\otimes
m})=\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}+(2m-1)^{2}(g-1)(b-1)\text{.}$

But:

$h^{0}(S,\omega_{S}^{\otimes m})-h^{1}(S,\omega_{S}^{\otimes
m})+h^{2}(S,\omega_{S}^{\otimes
m})=\chi(\mathcal{O}_{S})+\frac{1}{2}m(m-1)K_{S}^{2}\text{.}$

Combine the last two equality, we deduce:

$\chi(\mathcal{O}_{S})+\frac{1}{2}m(m-1)K_{S}^{2}-h^{2}(S,\omega_{S}^{\otimes
m})+h^{0}(C,\mathcal{R}^{1}f_{*}\omega_{S}^{\otimes
m})=\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}+(2m-1)^{2}(g-1)(b-1)\text{.}$

Since $g(F)\geq 2$, $\mathcal{R}^{1}f_{*}\omega_{S}^{\otimes m}=0$ and
$h^{2}(S,\omega_{S}^{\otimes m})=0$ we have the following:

$K_{S}^{2}=\frac{2}{m(m-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}+\frac{2(2m-1)^{2}}{m(m-1)}(g-1)(b-1)-\frac{2}{m(m-1)}\chi(\mathcal{O}_{S})$
$\Longrightarrow
K_{S}^{2}=\frac{2}{m(m-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}+8(g-1)(b-1)+\frac{2}{m(m-1)}((g-1)(b-1)-\chi(\mathcal{O}_{S}))\text{.}$

Using the fact that:

$K_{S/C}^{2}=K_{S}^{2}-8(g-1)(b-1)$

and

$\operatorname{deg}f_{*}\omega_{S/C}=\chi(\mathcal{O}_{S})-(g-1)(b-1)\text{,}$

we prove the desired result:

$K_{S/C}^{2}=\frac{2}{m(m-1)}\left(\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}-\operatorname{deg}f_{*}\omega_{S/C}\right),\hskip 5.69046pt\forall m\geq
2\text{.}$

∎

######  Remark 5.7.

Using Noether Formula, Xiao [29, Theorem 2] remarked that:

$K_{S/C}^{2}\leq 12\operatorname{deg}f_{*}\omega_{S/C}\text{.}$

######  Proposition 5.8.

Let $f:S\to C$ be a fibred surface and $F$ its general fiber with $g(F)\geq
2$. Then:

$K_{S/C}^{2}\leq\frac{12}{6m(m-1)+1}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m},\hskip 5.69046pt\forall m\geq 2\text{.}$

###### Proof.

Combine the Proposition 5.5 and Remark 5.7. ∎

######  Remark 5.9.

If $K_{S/C}^{2}>0$ and $g(F)\geq 2$, then
$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes m}>0$, $\forall m\geq 2$ .

######  Lemma 5.10.

Let $f:S\to C$ be a fibred surface and $F$ its general fiber with $g(F)\geq 2$
and $\mathcal{F}\subseteq f_{*}\omega_{S/C}^{\otimes m}$ with
$\operatorname{deg}\mathcal{F}\geq 0$, $m\geq 2$. Then:

$\mu(\mathcal{F})\leq\frac{6m}{(6m(m-1)+1)(g-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{.}$

Where
$\mu(\mathcal{F})=\frac{\operatorname{deg}\mathcal{F}}{\operatorname{rk}\mathcal{F}}$.

###### Proof.

Consider the data $(mK_{S/C},\mathcal{F})$ and $\mathcal{F^{{}^{\prime}}}$ be
the maximal destabilizing sub-vector bundle of $\mathcal{F}$. We apply the
Lemma 4.1 to the sequences $\\{Z(\mathcal{F}^{{}^{\prime}}),0\\}$ and
$(\mu(\mathcal{F}^{{}^{\prime}}),0)$, we have:

$K_{S/C}^{2}\geq\frac{2}{m}(g-1)\mu(\mathcal{F}^{{}^{\prime}})$

Here $Z(\mathcal{F}^{{}^{\prime}})$ is the fixed part of the sub-bundle
$\mathcal{F^{{}^{\prime}}}$ and
$\mu(\mathcal{F}^{{}^{\prime}})=\frac{\operatorname{deg}\mathcal{F}^{{}^{\prime}}}{\operatorname{rk}\mathcal{F}^{{}^{\prime}}}$.
By the fact that $\mathcal{F}^{{}^{\prime}}$ is the maximal destabilizing sub-
vector budle of $\mathcal{F}$, we have:

$\mu(\mathcal{F}^{{}^{\prime}})\geq\mu(\mathcal{F})$

Combine the last two inequalities with the Proposition 5.8, we deduce the
result:

$\mu(\mathcal{F})\leq\frac{6m}{(6m(m-1)+1)(g-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{.}$

∎

Now, we state the First Fujita decomposition for the relative pluricanonical
bundle and adjoint canonical bundle in the case of fibred surface.

######  Theorem 5.11.

Let $f:S\to C$ be a fibred surface and $F$ its general fiber, let $L$ be a
semi-ample line bundle on $S$. Then:

$f_{*}\omega_{S/C}^{\otimes m}=\mathcal{N}_{m}\oplus\mathcal{O}_{C}^{\oplus
p_{m}}\text{.}$ $f_{*}(\omega_{S/C}\otimes
L)=\mathcal{N}_{L}\oplus\mathcal{O}_{C}^{\oplus p_{L}}\text{.}$

Here $H^{0}(C,\mathcal{N}_{m}^{\vee})=0$ and
$H^{0}(C,\mathcal{N}_{L}^{\vee})=0\text{.}$

###### Proof.

Note $p_{m}:=h^{0}(C,(f_{*}\omega_{S/C}^{\otimes
m})^{\vee})=h^{0}(C,\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)})$ since
$(f_{*}\omega_{S/C}^{\otimes
m})^{\vee}\simeq\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)}$. Take
$\\{s_{1},\dots,s_{p_{m}}\\}$ be the sections of
$H^{0}(C,\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)})\simeq
Hom(\mathcal{O}_{C},\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)})$. Then
$s_{1}\oplus\dots\oplus s_{p_{m}}$ defines a map:

$s_{1}\oplus\dots\oplus s_{p_{m}}:\mathcal{O}_{C}^{\oplus
p_{m}}\longrightarrow\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)}$

which yields the following exact sequence:

$0\longrightarrow\mathcal{O}_{C}^{\oplus
p_{m}}\longrightarrow\mathcal{R}^{1}f_{*}\omega_{S/C}^{\otimes(1-m)}\longrightarrow\mathcal{Q}_{m}\longrightarrow
0$

where $\mathcal{Q}_{m}$ is the quotiont bundle. By duality, we have:

$0\longrightarrow\mathcal{Q}_{m}^{\vee}\longrightarrow
f_{*}\omega_{S/C}^{\otimes m}\longrightarrow\mathcal{O}_{C}^{\oplus
p_{m}}\longrightarrow 0\text{.}$

We deduce from [12, Theorem 26.4] that the last exact sequence splits as a
short exact sequence of vector bundle since the bundle
$f_{*}\omega_{S/C}^{\otimes m}$ admits a singular hermitian metric with semi-
positive curvature and verifies the minimal extention property, see [12] for
more details. Hence, the first Fujita decomposition follows:

$f_{*}\omega_{S/C}^{\otimes m}=\mathcal{N}_{m}\oplus\mathcal{O}_{C}^{\oplus
p_{m}}$

Here $\mathcal{N}_{m}=\mathcal{Q}_{m}^{\vee}$. and
$h^{0}(C,\mathcal{Q}_{m})=0$.

For the adjoint case, $p_{L}:=h^{0}(C,(f_{*}\omega_{S/C}\otimes L)^{\vee})$,
and the proof is the same as in the relative pluricanonical case, since $L$
admits a smooth hermitian metric $h$ with semi positive curvature and trivial
multiplier ideal sheaf, thus $f_{*}(\omega_{S/C}\otimes L)$ admits a singular
hermitian metric with semi positive curvature and verifies the minimal
extention property. ∎

In [18], they proved that the vector bundles $f_{*}\omega_{S/C}^{\otimes m}$
and $f_{*}(\omega_{S/C}\otimes L)$ where $L$ is semi-ample line bundle on $S$
admit a Catanese Fujita Kawamata decomposition, $\forall m\geq 2$. We state
the result in the case of fibred surfaces.

######  Theorem 5.12.

([18, Theorem 2]) Let $f:S\to C$ be a fibred surface and $F$ its general
fiber, let $L$ be a semi-ample line bundle on $S$. Then:

$f_{*}\omega_{S/C}^{\otimes m}=\mathcal{A}_{m}\oplus\mathcal{U}_{m},\hskip
5.69046pt\forall m\geq 2\text{.}$ $f_{*}(\omega_{S/C}\otimes
L)=\mathcal{A}_{L}\oplus\mathcal{U}_{L}\text{.}$

Here $\mathcal{A}_{m}$ and $\mathcal{A}_{L}$ are ample sub-vector bundles of
$f_{*}\omega_{S/C}^{\otimes m}$ and $f_{*}(\omega_{S/C}\otimes L)$
respectively, $\mathcal{U}_{m}$ and $\mathcal{U}_{L}$ are hermitian flat sub-
vector bundles of $f_{*}\omega_{S/C}^{\otimes m}$ and
$f_{*}(\omega_{S/C}\otimes L)$ respectively.

######  Example 5.13.

If $f$ is not isotrivial, it is known that $\omega_{S/C}$ is nef and big and
$f_{*}\omega_{S/C}^{\otimes m}$ is ample if not zero for $m\geq 2$, that
implies the trivial part in the First Fujita decomposition is zero and the
flat part in the Catanese Fujita Kawamata decomposition is zero.

######  Example 5.14.

If $L$ is ample line bundle on $S$, also it is very known that
$f_{*}(\omega_{S/C}\otimes L)$ is ample vector bundle on $C$, that means the
trivial part in the First Fujita decomposition is zero and the flat part in
the Catanese Fujita Kawamata decomposition is zero.

Recall now, an important result on direct image of pluricanonical sheaf over
curve due to Viehweg [28, Proposition 4.6], we restrict ourselve to a fibred
surface case.

######  Proposition 5.15.

Let $f:S\to C$ be a fibred surface and $F$ be its general fiber. the following
conditions are equivalent

  * (1).

For all $m\geq 2$, the vector bundle $f_{*}\omega_{S/C}^{\otimes m}$ is ample
if not trivial.

  * (2).

There exist some $m\geq 1$ such that $f_{*}\omega_{S/C}^{\otimes m}$ contains
an ample subsheaf.

  * (3).

There exist some $m\geq 1$ such that
$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes m}>0$.

More of that, if all the fibers of $f$ are normal crossing, then the
conditions $(1),(2),(3)$ and $(4)$ are equivalents

  * (4).

$f$ is not isotrivial.

######  Corollary 5.16.

(compare with [18, Corollary 16]) Let $f:S\to C$ be a fibred surface, then
either $f_{*}\omega_{S/C}^{\otimes m}$ is ample $\forall m\geq 2$ when
$f_{*}\omega_{S/C}^{\otimes m}\neq 0$ or $f_{*}\omega_{S/C}^{\otimes m}$ is
hermitian flat $\forall m\geq 2$.

###### Proof.

Apply the Proposition 5.15 and the Theorem 5.12. ∎

######  Remark 5.17.

If all the fibers of $f$ are normal crossing, then $f_{*}\omega_{S/C}^{\otimes
m}$ is hermitian flat $\forall m\geq 2$ is equivalent to $f$ be isotrivial.

######  Remark 5.18.

From the Corollary 5.16, we see that even $f_{*}\omega_{S/C}^{\otimes m}$ is
ample $\forall m\geq 2$ when $f_{*}\omega_{S/C}^{\otimes m}\neq 0$ or
$f_{*}\omega_{S/C}^{\otimes m}$ is hermitian flat $\forall m\geq 2$. By
Proposition 5.15, if $f_{*}\omega_{S/C}^{\otimes m}$ is flat $\forall m\geq 2$
then $f_{*}\omega_{S/C}$ is flat i.e, the family is locally trivial.
Conversely, if $f_{*}\omega_{S/C}^{\otimes m}$ is ample if not zero $\forall
m\geq 2$, $g(F)\geq 2$ and $K_{S/C}$ is nef, then $f_{*}\omega_{S/C}$ is not
flat by the Example 5.2.

In this last paragraph, we will see how
$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes m}$ is related to
$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes(m+1)}$ in the case when
$f_{*}\omega_{S/C}^{\otimes m}$ is ample $\forall m\geq 2$.

######  Corollary 5.19.

Let $f:S\to C$ be a fibred surface and $F$ its general fiber with $g(F)\geq
2$. Then:

$\operatorname{deg}f_{*}\omega_{S/C}\leq\frac{6mg}{(6m(m-1)+1)(g-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\text{,}\hskip 5.69046pt\forall m\geq 2$

and

$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
m}\leq\frac{6(m+1)(2m-1)}{6m(m+1)+1}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes(m+1)}\text{,}\hskip
5.69046pt\forall m\geq 2\text{.}$

###### Proof.

Apply the Lemma 5.10. ∎

######  Example 5.20.

If $m=2$, then

$\operatorname{deg}f_{*}\omega_{S/C}\leq\frac{12g}{13(g-1)}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes
2}\text{,}$

and

$\operatorname{deg}f_{*}\omega_{S/C}^{\otimes 2}\leq
2\operatorname{deg}f_{*}\omega_{S/C}^{\otimes 3}\leq\dots\leq
2^{n-2}\operatorname{deg}f_{*}\omega_{S/C}^{\otimes n}$

since

$\sup\frac{6(m+1)(2m-1)}{6m(m+1)+1}=2\text{.}$

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Département de Mathématiques, UQAM, C.P. 8888, Succursale Centre-ville,
Montréal ´ (Québec), H3C 3P8, Canada

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