11footnotetext:  The first author is supported by NSFC (No. 12271073); the
second author is supported by National Natural Science Foundation of China,
Fundamental Research Funds for central Universities, and Science and
Technology Commission of Shanghai Municipality (No.
22DZ2229014).22footnotetext:  2010 Mathematics Subject Classification. 14D06,
14H10, 14H30 33footnotetext:  Key words and phrases. Fundamental group,
Kodaira fibration, slope, modular invariant.

# Slopes of fibrations with trivial vertical fundamental groups

Xiao-Lei Liu Xin Lu

###### Abstract

Kodaira fibrations have non-trivial vertical fundamental groups and their
slopes are all 12. In this paper, we show that 12 is indeed the sharp upper
bound for the slopes of fibrations with trivial vertical fundamental groups.
Precisely, for each $g\geq 3$ we prove the existence of fibrations of genus
$g$ with trivial vertical fundamental groups whose slopes can be arbitrarily
close to 12. This gives a relative analogy of Roulleau-Urzúa’s work [RU15] on
the slopes of surfaces of general type with trivial fundamental groups.

##  1 Introduction

We work over the complex number $\mathbb{C}$. Let $X$ be a smooth minimal
projective surface of general type. Denote by $c_{1}^{2}(X),c_{2}(X)$ the two
Chern numbers of $X$. It is well-known that $c_{1}^{2}(X)=K_{X}^{2}$ is the
self-intersection number of the canonical divisor $K_{X}$, and that
$c_{2}(X)=\chi_{\mathrm{top}}(X)$ is the Euler number of $X$. Moreover, they
satisfy the following Noether equality:

$\displaystyle
c_{1}^{2}(X)+c_{2}(X)=K_{X}^{2}+\chi_{\mathrm{top}}(X)=12\chi(\mathcal{O}_{X}),$
(1.1)

where $\chi(\mathcal{O}_{X})$ is the Euler characteristic of the structure
sheaf $\mathcal{O}_{X}$. A fundamental problem in the study of the geography
of surfaces of general type is: for which pair of integers $(x,y)$, there
exists a smooth minimal projective surface $X$ of general type such that
$\big{(}c_{1}^{2}(X),c_{2}(X)\big{)}=(x,y)$? By (1.1), this is equivalent to
describe all the possible values
$\big{(}K_{X}^{2},\chi(\mathcal{O}_{X})\big{)}$. There is a long history to
this problem and we refer to [Pe81, Pe87] and [BHPV04, VII §8] for an
introduction. First, both $K_{X}^{2}$ and $\chi(\mathcal{O}_{X})$ are positive
integers and satisfy the following restrictions (Noether’s inequality and
Miyaoka-Yau’s inequality):

$2\chi(\mathcal{O}_{X})-6\leq K_{X}^{2}\leq 9\chi(\mathcal{O}_{X}).$

Sommese [So84] proved that every rational number between $2$ and $9$ can be
realized as the quotient $K_{X}^{2}/\chi(\mathcal{O}_{X})$ (will be called the
slope of $X$ for convenience) of some surface $X$.

The situation becomes subtle when the simple connectedness condition is
imposed on the surface $X$. Persson [Pe81] constructed a series of examples
showing that all the rational numbers between $2$ and $8$ occur as the slopes
of simply connected surfaces. According to the characterization of the
Miyaoka-Yau equality, any surface of general type with
$K_{X}^{2}=9\chi(\mathcal{O}_{X})$ is a ball quotient and hence can not be
simply connected. Simply connected surfaces of general type with nonnegative
index (i.e., its slope $\geq 8$) seemed difficult to construct. In fact, the
Watershed conjecture predicted that any simply connected surface of general
type has negative index (i.e., its slope $<8$). Nevertheless, Moishezon-
Teicher [MT87] constructed the first example of simply connected surfaces with
positive index, and Chen [Ch87] found simply connected surfaces with slopes up
to $8.75$. More recently in the beautiful work [RU15], Roulleau-Urzúa proved
that the slopes of simply connected surfaces are dense in $[8,9)$. In
particular, the slopes of simply connected surfaces of general type can be
arbitrarily close to $9$.

We are interested in the analogy of the geography problem for surface
fibrations. Let $f\colon X\to C$ be a relatively minimal surface fibration
whose general fiber is of genus $g\geq 2$. We consider the following relative
invariants:

$\displaystyle\begin{cases}K_{f}^{2}=K_{X}^{2}-8(g-1)(g(C)-1),\\\
\chi_{f}=\chi(\mathcal{O}_{X})-(g-1)(g(C)-1),\\\
e_{f}=\chi_{\mathrm{top}}(X)-4(g-1)(g(C)-1).\end{cases}$

These invariants are non-negative integers satisfying the following
properties:

  1. (1).

$K_{f}^{2}=0~{}\Longleftrightarrow~{}\chi_{f}=0$, if and only if $f$ is
locally trivial.

  2. (2).

$e_{f}=0$ if and only if $f$ is smooth.

  3. (3).

The Noether equality holds: $12\chi_{f}=K_{f}^{2}+e_{f}$.

Analogously, one may ask a similar geography problem for surface fibrations
(cf. [AK02, § 1.1]): for which pair of integers $(x,y)$, there exists a
relatively minimal surface fibration $f\colon X\to C$ such that
$\big{(}K_{f}^{2},\chi_{f}\big{)}=(x,y)$? For a locally non-trivial fibration
$f$, the slope of $f$ is defined as $\lambda_{f}=K_{f}^{2}/\chi_{f}$, and it
satisfies:

$\frac{4(g-1)}{g}\leq\lambda_{f}\leq 12.$

The first inequality is the slope inequality [Xi87, CH88], and the equality
can be reached by hyperelliptic fibrations; the second inequality follows from
the non-negativity of $e_{f}$, and the equality can be reached by Kodaira
fibrations whenever $g\geq 3$.

Xiao started to consider the above geography problem for surface fibrations
around 1980s, where he got partial answers for fibrations of genus $g=2$, cf.
[Xi85, Theorem 2.9] and [Xi92, Theorem 4.3.5]. Motivated by Xiao’s work, Chen
[Ch87] generalized it to hyperelliptic fibrations, based on which Chen
constructed many simply connected surfaces with positive index as mentioned
before. Recently, Liu-Lu [LL24] proved that any rational number
$r\in\left[4(g-1)/g,\,12\right]$ can be realized as the slope of a fibration
of genus $g$ whenever $g>3$, which gives an analogy of Sommese’s result [So84]
in the relative version.

We would like to take the fundamental group into consideration. Recall that
the fundamental group of the fibered surface $X$ can be divided into two parts
with the following exact sequence (cf. [Xi91]):

$1\longrightarrow\mathcal{V}_{f}\longrightarrow\pi_{1}(X)\longrightarrow\mathcal{H}_{f}\longrightarrow
1,$

where $\mathcal{V}_{f}$ is called the vertical fundamental group of the
fibration $f$, see Section 2.3 for a more precise definition. Xiao [Xi87]
proved that the vertical fundamental group $\mathcal{V}_{f}$ is trivial if $f$
is non-hyperelliptic with $\lambda_{f}<4$. It is not difficult [Xi91] to
construct surface fibrations with lower slope admitting a trivial vertical
fundamental group. The situation is different if the slope $\lambda_{f}$ is
large. In the extreme case when $\lambda_{f}=12$, the vertical fundamental
group $\mathcal{V}_{f}$ can never be trivial, cf. Lemma 2.3. Hence it is
natural to wonder: is $12$ the sharp upper bound for the slopes of fibrations
with trivial vertical fundamental groups? Our main purpose is to answer this
question affirmatively, which gives a relative analogy of Roulleau-Urzúa’s
work [RU15] on the slopes of surfaces of general type with trivial fundamental
groups.

######  Theorem 1.1.

For each $g\geq 3$, there exists a sequence of fibrations $f_{n}\colon
X_{n}\to C_{n}$ of genus $g$ such that $\mathcal{V}_{f_{n}}=\\{1\\}$ and
$\lim\limits_{n\to\infty}\lambda_{f_{n}}=12.$

Different from the constructions (using cyclic coverings) in [Ch87, LL24], our
construction is based on the moduli spaces and the Torelli map. As a byproduct
of our method, we can construct more examples with given slopes in the case of
$g=3$.

######  Theorem 1.2.

For each rational number $r\in[\frac{8}{3},12)$, there exists a fibration $f$
of genus $3$ such that $\mathcal{V}_{f}=\\{1\\}$ and $\lambda_{f}=r$.

We tend to believe this should be true for each $g>3$; see Remark 3.2.
Combining with the results in [LL24], we get the following.

######  Corollary 1.3.

For each $g\geq 2$ and each rational number $r\in[4(g-1)/g,\lambda_{M}(g)]$,
there exists a fibration of genus $g$ with slope $r$, where

$\lambda_{M}(g):=\begin{cases}7,&\mbox{if~{}}g=2,\\\ 12,&\mbox{if~{}}g\geq
3.\end{cases}$

######  Remark 1.4.

(1). If $f$ is hyperelliptic, it is known [Xi92] that
$\lambda_{f}\leq\lambda_{M}^{h}(g)$ and the upper bounds can be reached (see
[Mo98, LT13]), where

$\lambda_{M}^{h}(g):=\begin{cases}12-\frac{8g+4}{g^{2}},&\mbox{if~{}}g\mbox{~{}is~{}even,}\\\
12-\frac{8g+4}{g^{2}-1},&\mbox{if~{}}g\mbox{~{}is~{}odd.}\end{cases}$

When $g=2$, the fibration $f$ is always hyperelliptic, and hence
$\lambda_{f}\leq 7$ in this case. It is well-known that there exist Kodaira
fibrations of genus $g$ for every $g\geq 3$. Hence $\lambda_{M}(g)$ is indeed
the sharp upper bound of the slopes of fibrations of genus $g$.

(2). The hyperelliptic fibrations constructed in [LL24] are in fact with
trivial vertical fundamental group. In other words, these examples show that
for each genus $g\geq 2$ and each rational number
$r\in\big{(}4(g-1)/g,\lambda_{M}^{h}(g)\big{)}$, there exists a fibration $f$
of genus $g$ satisfying that $\mathcal{V}_{f}=\\{1\\}$ and $\lambda_{f}=r$,
see Remark 2.5. However, the vertical fundamental groups of non-hyperelliptic
fibrations constructed in [LL24] seem far from being trivial.

##  2 Preliminaries

###  2.1 Moduli spaces and the Torelli map

Let $\mathcal{M}_{g}$ (resp. $\mathcal{H}_{g}$) be the moduli space of smooth
projective curves (resp. hyperelliptic curves) of genus $g\geq 2$, and
$\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally polarized
abelian varieties. Recall that the Torelli morphism

$j\colon\mathcal{M}_{g}\longrightarrow\mathcal{A}_{g},$

which associates to a curve its Jacobian with the canonical principal
polarization. The Torelli morphism $j$ is injective; in fact it is an
immersion [OS79]. Let
$\overline{\mathcal{M}_{g}}=\overline{\mathcal{M}_{g}}^{DM}$ be the Deligne-
Mumford compactification of $\mathcal{M}_{g}$, and
$\overline{\mathcal{A}_{g}}=\overline{\mathcal{A}_{g}}^{S}$ be the Satake
compactification of $\mathcal{A}_{g}$. The boundary
$\overline{\mathcal{M}_{g}}\setminus\mathcal{M}_{g}$ consists of several
divisors $\Delta_{i}$ ($0\leq i\leq[g/2]$); while the boundary
$\overline{\mathcal{A}_{g}}\setminus\mathcal{A}_{g}$ is of codimension at
least two. In fact,

$\overline{\mathcal{A}_{g}}\setminus\mathcal{A}_{g}=\bigcup\limits_{0\leq
i\leq g-1}\mathcal{A}_{i}.$

It turns out that the Torelli map $j$ extends to a map from
$\overline{\mathcal{M}_{g}}$ to $\overline{\mathcal{A}_{g}}$, which is still
denoted by $j$. The extended Torelli map is no longer injective: it collapses
the boundary divisors. When $g\geq 3$, the boundary
$j(\overline{\mathcal{M}_{g}})\setminus j(\mathcal{M}_{g})$ is of codimension
at least two, cf. [DM18].

In order to assure the representability, it is necessary to take certain level
structure into consideration. Fixing $l\geq 3$ an integer, let
$\mathcal{M}_{g,[l]}$ (resp. $\mathcal{A}_{g,[l]}$ and so on) be the
corresponding moduli space with full level-$l$ structure. No specific choice
of the level $l(\geq 3)$ is made because it is only imposed to assure the
representability, which plays no essential role in our study. The Torelli map
$j$ can be similarly defined over the moduli spaces with level structure.
However, the Torelli map

$j\colon\mathcal{M}_{g,[l]}\longrightarrow\mathcal{A}_{g,[l]}$

is no longer an immersion, but a two-to-one map ramified exactly over the
locus of hyperelliptic curves
$\mathcal{H}_{g,[l]}\subseteq\mathcal{M}_{g,[l]}$, cf. [OS79]. As the level
$l\geq 3$, there is a universal family of smooth curves (resp. stable curves)
over $\mathcal{M}_{g}$ (resp. $\overline{\mathcal{M}_{g}}$) with the following
commutative diagram, cf. [Po77, Theorem 10.9].

$\textstyle{\mathcal{S}_{g,[l]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}}$$\textstyle{\overline{\mathcal{S}_{g,[l]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\mathfrak{f}}}$$\textstyle{\mathcal{M}_{g,[l]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}_{g,[l]}}}$

The following lemma can be found in [LZ19, Lemma A.1].

######  Lemma 2.1.

There exists an involution $\sigma_{g}$ (resp. $\tau_{g}$) on
$\mathcal{S}_{g,[l]}$ (resp. $\mathcal{M}_{g,[l]})$ such that the following
diagram commutes.

$\textstyle{\mathcal{S}_{g,[l]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}}$$\scriptstyle{\sigma_{g}}$$\textstyle{\mathcal{S}_{g,[l]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{f}}$$\textstyle{\mathcal{M}_{g,[l]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{g}}$$\textstyle{\mathcal{M}_{g,[l]}}$

Moreover, $j\circ\tau_{g}(x)=j(x)$ for any $x\in\mathcal{M}_{g,[l]}$, the
fixed locus of $\tau_{g}$ is exactly the hyperelliptic locus
$\mathcal{H}_{g,[l]}\subseteq\mathcal{M}_{g,[l]}$, and for
$p\in\mathcal{H}_{g,[l]}$, $\sigma_{g}|_{F_{p}}\colon F_{p}\to F_{p}$ is the
hyperelliptic involution of $F_{p}$, where $F_{p}\subseteq\mathcal{S}_{g,[l]}$
is the hyperelliptic curve over $p$.

###  2.2 Modular invariants

Let $f\colon X\to C$ be a surface fibration (or simply fibraion), i.e., $f$ is
a proper surjective morphism from a smooth projective surface onto a smooth
projective curve with connected fibers. Denote by $g$ the genus of a general
fiber of $f$. We will always assume that $f$ is relatively minimal, i.e.,
there is no $(-1)$-curve contained in fibers of $f$. If $\pi\colon\tilde{C}\to
C$ is a base change of degree $d$, then the pullback fibration
$\tilde{f}\colon\tilde{X}\to\tilde{C}$ of $f$ with respect to $\pi$ is defined
as the relatively minimal model of the desingularization of
$X\times_{C}\tilde{C}\to\tilde{C}$. Denote by $F$ the singular fiber of $f$
over $p=f(F)\in C$. If $\pi$ is totally ramified over $p$, then the fiber
$\tilde{F}$ of the pullback fibration $\tilde{f}$ over $\pi^{-1}(p)$ is called
$d$-th root model of $F$. If $\pi$ is ramified over $p$ and some non-critical
points of $f$ such that the fibers of $\tilde{f}$ over $\pi^{-1}(p)$ are all
semistable, then the Chern numbers of $F$ are defined as follows (see [Ta94,
Ta96]),

$c_{1}^{2}(F)=K_{f}^{2}-\frac{1}{d}K_{\tilde{f}}^{2},~{}~{}c_{2}(F)=e_{f}-\frac{1}{d}e_{\tilde{f}},~{}~{}\chi_{F}=\chi_{f}-\frac{1}{d}\chi_{\tilde{f}}.$

The fibration $f$ induces a moduli map $J\colon
C\to\overline{\mathcal{M}_{g}}$ from $C$ to the moduli space
$\overline{\mathcal{M}_{g}}$. Let $\lambda$ be the Hodge divisor class of
$\overline{\mathcal{M}_{g}}$, $\delta$ be the boundary divisor class, and
$\kappa=12\lambda-\delta$ be the first Morita-Mumford class. Then there are
three fundamental modular invariants of $f$ defined as follows (see [Ta10]),

$\kappa(f)=\deg J^{*}\kappa,\hskip 8.5359pt\lambda(f)=\deg J^{*}\lambda,\hskip
8.5359pt\delta(f)=\deg J^{*}\delta.$

If $f$ is semistable, then

$\kappa(f)=K_{{f}}^{2},~{}~{}\delta(f)=e_{{f}},~{}~{}\lambda(f)=\chi_{{f}}.$
(2.1)

These modular invariants satisfy the base change property, i.e., if
$\tilde{f}$ is the pullback fibration of $f$ with respect to a base change of
degree $d$, then

$\kappa(\tilde{f})=d\cdot\kappa(f),~{}~{}\delta(\tilde{f})=d\cdot\delta(f),~{}~{}\lambda(\tilde{f})=d\cdot\lambda(f).$
(2.2)

It is proved in [Ta94, Ta96] that

$\begin{cases}K_{f}^{2}=\kappa(f)+\sum_{i=1}^{s}c_{1}^{2}(F_{i}),&\\\
e_{f}=\delta(f)+\sum_{i=1}^{s}c_{2}(F_{i}),&\\\
\chi_{f}=\lambda(f)+\sum_{i=1}^{s}\chi_{F_{i}},\end{cases}$ (2.3)

where $F_{1},\cdots,F_{s}$ are all the singular fibers of $f$.

######  Example 2.2.

Let $F^{h}_{g}$ be the singular hyperelliptic fiber of genus $g$ with the
following dual graph, where $\overset{n}{\underset{-e}{\circ}}$ denotes a
smooth rational curve with self-intersection number $(-e)$ and multiplicity
$n$ in $F^{h}_{g}$. For convenience, we omit the subscript $(-e)$ whenever
$e=2$.

2$C_{0}$111${C_{2g+2}}$${C_{2}}$${C_{1}}$$-(g+1)$Figure 1: Hyperelliptic fiber
$F^{h}_{g}$.

Then $F^{h}_{g}$ is simply connected [Pe81, Lemma A], and the Chern numbers of
$F^{h}_{g}$ are as follows (see also [LL24, Example 2.4])

$\displaystyle c_{1}^{2}(F^{h}_{g})=2g-2,~{}~{}\chi_{F^{h}_{g}}=\frac{g}{2}.$
(2.4)

Let $\tilde{F}^{h}_{g}$ be the $d$-th root model of $F^{h}_{g}$. Since

$F_{g}^{h}=\sum_{i=0}^{2g+2}n_{i}C_{i}=2C_{0}+C_{1}+\cdots+C_{2g+2}$

is normal crossing, by [LuT13, Section 2.2], the multiplicity of the strict
transform of $C_{i}$ in $\tilde{F}^{h}_{g}$ is $n_{i}/\gcd(n_{i},d)$ and
$\tilde{F}^{h}_{g}=F^{h}_{g}$ when $d$ is odd.

###  2.3 The fundamental group

In this subsection, we recall some general facts about the fundamental group
of a surface fibration. Let $f\colon X\to C$ be a fibration of genus $g$, and
$F$ be any general fiber. According to [Xi91, Lemma 1], the embedding of $F$
in $X$ induces a homomorphism

$\alpha\colon\pi_{1}(F)\longrightarrow\pi_{1}(X),$

whose image is a normal subgroup of $\pi_{1}(X)$ and independent of the choice
of the general fiber $F$. The image of $\alpha$, denoted by $\mathcal{V}_{f}$,
is called the vertical fundamental group of $X$. Let
$\mathcal{H}_{f}:=\pi_{1}(X)/\mathcal{V}_{f}$ be the quotient group, then one
obtains the following exact sequence:

$1\longrightarrow\mathcal{V}_{f}\longrightarrow\pi_{1}(X)\longrightarrow\mathcal{H}_{f}\longrightarrow
1.$

The next property is more or less well-known.

######  Lemma 2.3.

Suppose that $f$ is a Kodaira fibration of genus $g\geq 3$, then the vertical
fundamental group $\mathcal{V}_{f}$ is non-trivial.

###### Proof.

In this case, $f$ admits no singular fiber, and hence it is a topological
fiber bundle. It follows that the induced homomorphism from $\pi_{1}(F)$ to
$\pi_{1}(X)$ is injective with the following exact sequence:

$1\longrightarrow\pi_{1}(F)\overset{\alpha}{\longrightarrow}\pi_{1}(X)\longrightarrow\pi_{1}(C)\to
1.$

In particular, the vertical fundamental group $\mathcal{V}_{f}$ is non-
trivial. ∎

The next lemma, due to Xiao [Xi91, Lemma 3], provides a useful way to
determine whether $\mathcal{V}_{f}=\\{1\\}$.

######  Lemma 2.4.

Let $F_{0}$ be any fiber of $f\colon X\to C$ and
$\alpha_{0}\colon\pi_{1}(F_{0})\to\pi_{1}(X)$ the induced homomorphism, then
$\mathcal{V}_{f}\subseteq\mathrm{Im}(\alpha_{0})$. In particular,
$\mathcal{V}_{f}$ is trivial if $f$ admits a simply connected fiber.

######  Remark 2.5.

It is shown in [LL24] that for each $g\geq 2$ and each rational number
$r\in\big{(}4(g-1)/g,\lambda_{M}^{h}(g)\big{)}$, there exists a hyperelliptic
fibration $f_{g,r}$ of genus $g$ with $\lambda_{f_{g,r}}=r$. By the
construction of $f_{g,r}$ in the proof of [LL24, Theorem 1.3], we know that
$f_{g,r}$ has at least one singular fiber $F^{h}_{g}$, which is simply
connected by Example 2.2. Hence the vertical fundamental group of $f_{g,r}$ is
trivial by Lemma 2.4.

##  3 The construction

In this section, we mainly aim to construct surface fibrations with trivial
vertical fundamental groups whose slopes tend to $12$.

###### Proof of Theorem 1.1.

We use the notations and terminology introduced in Section 2. Let
$L_{1},L_{2},\ldots,L_{3g-4}$ be very ample divisors on the Satake
compactification
$\overline{\mathcal{A}_{g,[l]}}=\overline{\mathcal{A}_{g,[l]}}^{S}$, where the
level $l\geq 3$. Since $\dim\mathcal{M}_{g}=3g-3$ and the boundary
$j(\overline{\mathcal{M}_{g}})\setminus j(\mathcal{M}_{g})$ is of codimension
at least two, the intersection $L_{1}L_{2}\cdots L_{3g-4}$ is a complete
smooth curve $C\subseteq j(\mathcal{M}_{g})$. Moreover, by the very ampleness
of these $L_{i}$’s, we may assume that the curve $C$ intersects
$j(\mathcal{H}_{g,[l]})$ transversely with $C\cap
j(\mathcal{H}_{g,[l]})\not=\emptyset$.

Let $B=j^{-1}(C)\subseteq\mathcal{M}_{g,[l]}$ be the inverse image of $C$.
Note that the Torelli morphism

$j\colon\mathcal{M}_{g,[l]}\longrightarrow\mathcal{A}_{g,[l]}$

is two-to-one and ramified exactly over the hyperelliptic locus
$\mathcal{H}_{g,[l]}$. As $C$ intersects $j(\mathcal{H}_{g,[l]})$ transversely
with $C\cap j(\mathcal{H}_{g,[l]})\not=\emptyset$, it follows that $B$ is
smooth and irreducible with $B\cap\mathcal{H}_{g,[l]}\not=\emptyset$. Let
$h\colon X\to B$ be the family of smooth curves induced from the universal
family $\mathfrak{f}\colon\mathcal{S}_{g,[l]}\to\mathcal{M}_{g,[l]}$.
According to Lemma 2.1, there is an involution $\sigma$ (resp. $\tau$) on $X$
(resp. $B$) such that the following diagram commutes.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\sigma}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{B}$

Moreover, the quotient $B/\langle\tau\rangle\cong C$, and
$\sigma|_{\Gamma_{p}}\colon\Gamma_{p}\to\Gamma_{p}$ is the hyperelliptic
involution of $\Gamma_{p}$ for any fixed point
$p\in\mathrm{Fix}(\tau)\subseteq B$, where $\Gamma_{p}=h^{-1}(p)$ is the fiber
over $p$.

Let $f_{0}\colon X_{0}\to C\cong B/\langle\tau\rangle$ be the relatively
minimal model of the quotient fibred surface $X/\langle\sigma\rangle\to C\cong
B/\langle\tau\rangle$. By our construction, the possible singular fibers of
$f_{0}$ are those corresponding to $\Gamma_{p}$ with
$p\in\mathrm{Fix}(\tau)\subseteq B$. As the restricted map

$\sigma|_{\Gamma_{p}}\colon\Gamma_{p}\to\Gamma_{p}$

is the hyperelliptic involution of $\Gamma_{p}$, its image $\Pi(\Gamma_{p})$
is a rational curve with $2g+2$ singularities of $X/\langle\sigma\rangle$ on
$\Pi(\Gamma_{p})$, where $\Pi\colon X\to X/\langle\sigma\rangle$ is the
quotient map. By resolving the singularities, it follows that the
corresponding fiber $F_{j(p)}=f_{0}^{-1}\big{(}j(p)\big{)}$ in $X_{0}$ is
isomorphic to $F_{g}^{h}$, which is described in Example 2.2 for any
$p\in\mathrm{Fix}(\tau)$. Let $F_{1},\cdots,F_{s}$ be all the singular fibers
of $f_{0}$, then by (2.3) and (2.4) we have that

$\displaystyle K_{f_{0}}^{2}$
$\displaystyle\,=\kappa(f_{0})+\sum_{i=1}^{s}c_{1}^{2}(F_{i})=\kappa(f_{0})+(2g-2)s,$
$\displaystyle\chi_{f_{0}}$
$\displaystyle\,=\lambda(f_{0})+\sum_{i=1}^{s}\chi_{F_{i}}=\lambda(f_{0})+\frac{1}{2}gs.$

Note that $h$ is a Kodaira fibration. By (2.1) and (2.2),

$\kappa(f_{0})=\frac{1}{2}\kappa(h)=\frac{1}{2}K_{h}^{2},\quad\text{and}\quad\lambda(f_{0})=\frac{1}{2}\lambda(h)=\frac{1}{2}\chi_{h}.$

Thus

$\frac{\kappa(f_{0})}{\lambda(f_{0})}=\lambda_{h}=12.$

Let $p_{1}=f_{0}(F_{1}),\cdots,p_{s}=f_{0}(F_{s})$. Then
$\\{p_{1},\cdots,p_{s}\\}$ are just the ramification divisor of the induced
double cover

$j|_{B}\colon B\longrightarrow C.$

In particular, $s$ is even and $s\geq 2$ since we have assume that $s\neq 0$.
Let

$\pi_{n}\colon C_{n}\longrightarrow C$

be a cyclic cover of degree $2n+1$ branched exactly over
$\\{p_{1},\cdots,p_{s}\\}$ with ramification indices being all equal to
$2n+1$. Such a cyclic cover exists. Indeed, take positive integers
$0<m_{i}<2n+1$, $1\leq i\leq s$, such that

  1. (i).

$\gcd(m_{i},2n+1)=1$ for any $1\leq i\leq s$;

  2. (ii).

$\sum\limits_{i=1}^{s}m_{i}$ is a multiple of $2n+1$.

Then the divisor $R:=\sum\limits_{i=1}^{s}m_{i}p_{i}$ is $(2n+1)$-divisible,
i.e., there exists a line bundle $L$ such that $\mathcal{O}(R)\sim
L^{\otimes(2n+1)}$, where ‘$\sim$’ stands for the linear equivalence. Then the
relation $\mathcal{O}(R)\sim L^{\otimes(2n+1)}$ defines a cyclic cover
$\pi_{n}\colon C_{n}\longrightarrow C$ satisfying our requirements. Using such
a cyclic cover $\pi_{n}$ to do the base change, let $f_{n}\colon X_{n}\to
C_{n}$ be the pullback fibration of $f_{0}$, and $F_{i,n}\subseteq X_{n}$ be
the corresponding fiber of $F_{i}$ for $1\leq i\leq s$. Then by Example 2.2,

$F_{i,n}\cong F_{i}\cong F_{g}^{h},\qquad\forall~{}1\leq i\leq s.$

Hence by (2.2), (2.3) and (2.4),

$\displaystyle K_{f_{n}}^{2}$
$\displaystyle\,=\kappa(f_{n})+\sum_{i=1}^{s}c_{1}^{2}(F_{i,n})=(2n+1)\kappa(f_{0})+(2g-2)s,$
$\displaystyle\chi_{f_{n}}$
$\displaystyle\,=\lambda(f_{n})+\sum_{i=1}^{s}\chi_{F_{i,n}}=(2n+1)\lambda(f_{0})+\frac{1}{2}gs.$

Thus the slope of $f_{n}$ is

$\lambda_{f_{n}}=\frac{K_{f_{n}}^{2}}{\chi_{f_{n}}}=\frac{(2n+1)\kappa(f_{0})+(2g-2)s}{(2n+1)\lambda(f_{0})+\frac{1}{2}gs}.$

When $n$ goes to infinity, the slope of $f_{n}$ tends to $12$ as required.
Moreover, the vertical fundamental group $\mathcal{V}_{f_{n}}$ is trivial by
Lemma 2.4, since $f_{n}$ admits a singular fiber isomorphic to $F^{h}_{g}$,
which is simply connected by Example 2.2. ∎

Applying the above construction to the case $g=3$, we can prove Theorem 1.2.

###### Proof of Theorem 1.2.

We follow the notations introduced in the proof of Theorem 1.1. Take $g=3$,
and let $h\colon X\to B$ be the Kodaira fibration of genus $g=3$ constructed
in the proof of Theorem 1.1. Let $f_{0}\colon X_{0}\to C\cong
B/\langle\tau\rangle$ be the relatively minimal model of the quotient fibred
surface $X/\langle\sigma\rangle\to C\cong B/\langle\tau\rangle$. Then it is
known that

$\displaystyle K_{f_{0}}^{2}$
$\displaystyle\,=\kappa(f_{0})+\sum_{i=1}^{s}c_{1}^{2}(F_{i})=\kappa(f_{0})+4s,$
$\displaystyle\chi_{f_{0}}$
$\displaystyle\,=\lambda(f_{0})+\sum_{i=1}^{s}\chi_{F_{i}}=\lambda(f_{0})+\frac{3}{2}s,$

and that

$\kappa(f_{0})=\frac{1}{2}\kappa(h)=\frac{1}{2}K_{h}^{2},\qquad\lambda(f_{0})=\frac{1}{2}\lambda(h)=\frac{1}{2}\chi_{h},\qquad\text{with~{}}~{}\frac{K_{h}^{2}}{\chi_{h}}=12.$

Here $s$ is the number of singular fibers of $f_{0}$. Let
$\delta_{i}~{}(i=0,1)$ be the divisor class of $\Delta_{i}$ of
$\overline{\mathcal{M}_{3}}\backslash{\mathcal{M}}_{3}$, and $\mathfrak{h}$ be
the divisor class of $\overline{\mathcal{H}_{3}}^{DM}$. By the formulas (see
[HM82])

$\kappa=\frac{1}{3}\delta_{0}+3\delta_{1}+\frac{4}{3}\mathfrak{h},~{}~{}\lambda=\frac{1}{9}\delta_{0}+\frac{1}{3}\delta_{1}+\frac{1}{9}\mathfrak{h},$

it follows that

$K_{h}^{2}=\kappa(h)=\frac{4}{3}s,\qquad\chi_{h}=\lambda(h)=\frac{1}{9}s.$

Hence

$\kappa(f_{0})=\frac{2}{3}s,\qquad\lambda(f_{0})=\frac{1}{18}s,$ (3.1)

and

$K_{f_{0}}^{2}=\kappa(f_{0})+4s=\frac{14}{3}s,\qquad\chi_{f_{0}}=\lambda(f_{0})+\frac{3}{2}s=\frac{14}{9}s.$
(3.2)

We claim that

######  Claim 3.1.

For any rational number $r\in\big{[}3,12)$, there exists a base change
$\pi\colon\tilde{C}\to C$, such that

  1. (i).

the slope of $\tilde{f}$ is $\lambda_{\tilde{f}}=r$, where
$\tilde{f}\colon\tilde{X}\to\tilde{C}$ is the pullback fibration under the
base change $\pi$;

  2. (ii).

the fibration $\tilde{f}$ admits singular fibers, and every singular fiber is
isomorphic to $F^{h}_{3}$, where $F^{h}_{3}$ is the singular hyperelliptic
fiber of genus $3$ as in Example 2.2.

We first prove Theorem 1.2 based on the above claim. By the above claim, for
any given rational number $r\in\big{[}3,12)$, one can construct a non-
hyperelliptic fibration $f_{r}$ of genus $g=3$, such that the slope
$\lambda_{f_{r}}=r$ and the vertical fundamental group is trivial by Lemma
2.4. Combining this with Remark 2.5, we prove Theorem 1.2. It remains to prove
the above claim.

Proof of Claim 3.1. The main idea is more or less the same as that in the
proof of Theorem 1.1. But we have to carefully choose the branched points of
the base change instead of what we do in the proof of Theorem 1.1, where we
simply take a base change totally ramified over the images of all singular
fibers of $f_{0}$.

First, if $r=3$, then the fibration $f_{0}$ already satisfies our requirements
by (3.2). Thus we will assume in the following that $3<r<12$. Let $m$ be any
odd integer satisfying $m>\frac{27r-72}{12-r}$, and let

$d=\frac{28m(r-3)}{9(3r-8)(m-1)}.$

By replacing $f_{0}$ by a base change unbranched over
$\\{p_{1},\cdots,p_{s}\\}$, we may assume that $ds$ is also an integer at
least $2$, where $p_{i}=f_{0}(F_{i})$ is the image of the singular fiber
$F_{i}$. Indeed, let $\pi_{k}:C_{k}\to C$ be any base change of degree $k$,
whose branch locus does not contain any $p_{i}$, then the number of singular
fibers as well as the invariants $K_{f_{k}}^{2}$ and $\chi_{f_{k}}$ are
multiplied by $k$. Note that $0<d<1$ by definition. Let

$\pi\colon\tilde{C}\to C$

be a cyclic cover of degree $m$ branched exactly over
$\\{p_{1},\cdots,p_{ds}\\}$ with ramification indices being all equal to $m$.
Such a cyclic cover exists as already showed in the proof of Theorem 1.1. Let
$\tilde{f}$ be the pullback fibration. Then the number of singular fibers of
$\tilde{f}$ is $ds+m(s-ds)$ and each singular fiber is isomorphic to
$F^{h}_{3}$ since $m$ is assumed to be odd. Based on (3.1) and the formulas in
Section 2.2, one computes the relative invariants of $\tilde{f}$ as follows

$\displaystyle K_{\tilde{f}}^{2}$
$\displaystyle\,=m\kappa(f_{0})+4\big{(}ds+m(s-ds)\big{)}=\Big{(}\frac{14}{3}m+4d(1-m)\Big{)}s,$
$\displaystyle\chi_{\tilde{f}}$
$\displaystyle\,=m\lambda(f_{0})+\frac{3}{2}\big{(}ds+m(s-ds)\big{)}=\Big{(}\frac{14}{9}m+\frac{3}{2}d(1-m)\Big{)}s.$

So the slope of $\tilde{f}$ is

$\displaystyle\lambda_{\tilde{f}}=\frac{K_{\tilde{f}}^{2}}{\chi_{\tilde{f}}}=\frac{84m+72(1-m)d}{28m+27(1-m)d}=r.$

The last equality follows from the definition of $d$ above. Moreover, by
construction $\tilde{f}$ admits singular fibers and every singular fiber is
isomorphic to $F^{h}_{3}$ as required. This proves Claim 3.1, and hence
completes the proof of Theorem 1.2. ∎

######  Remark 3.2.

Regarding our result in Theorem 1.2 for the case $g=3$, we like to conjecture
that, for each $g>3$ and each rational number $r\in[4(g-1)/g,12)$, there
exists a fibration $f$ of genus $g$ such that $\lambda_{f}=r$ and
$\mathcal{V}_{f}=\\{1\\}$.

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Xiao-Lei Liu, School of Mathematical Sciences, Dalian University of
Technology, Dalian, Liaoning Province, P. R. of China.

E-mail address<EMAIL_ADDRESS>
Xin Lu, School of Mathematical Sciences, Key Laboratory of MEA(Ministry of
Education) & Shanghai Key Laboratory of PMMP, East China Normal University,
Shanghai 200241, China

E-mail address<EMAIL_ADDRESS>