lemmatheorem lemma conjecturetheorem conjecture propositiontheorem proposition
corollarytheorem corollary problemtheorem problem questiontheorem question
claimtheorem claim definitiontheorem definition exampletheorem example
assumptiontheorem assumption remarktheorem remark remarkstheorem remarks

# The Noether inequalities for a foliated surface of general type

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<EMAIL_ADDRESS>
###### Abstract.

Let $(\mathcal{F},S)$ be a foliated surface of general type with reduced
singularities over the complex number. We establish the Noether type
inequalities for $(\mathcal{F},S)$. Namely, we prove that
$\operatorname*{vol}(\mathcal{F})\geq p_{g}(\mathcal{F})-2$, and that
$\operatorname*{vol}(\mathcal{F})\geq 2p_{g}(\mathcal{F})-4$ if moreover the
surface $S$ is also of general type. Examples show that both of the Noether
type inequalities are sharp.

###### Key words and phrases:

Noether inequality, foliated surface, canonical map

###### 2020 Mathematics Subject Classification:

14J29; 14D06; 32S65

##  1\. Introduction

The aim of this paper is to establish the Noether inequalities for a foliated
surface of general type with reduced singularities over the complex number
$\mathbb{C}$.

The classical Noether inequality [Noe70] asserts that

$\operatorname*{vol}(S)\geq 2p_{g}(S)-4,$ (1.1)

for every complex smooth projective surface $S$ of general type. Here we
recall that the volume $\operatorname*{vol}(S)$ and the geometric genus
$p_{g}(S)$ of $S$ are defined as follows. Let $K_{S}$ be the canonical divisor
of $S$. Then

$\displaystyle p_{g}(S)$ $\displaystyle\,=\dim H^{0}(S,K_{S});$
$\displaystyle\operatorname*{vol}(S)$
$\displaystyle\,=\limsup_{n\to+\infty}\frac{\dim H^{0}(S,nK_{S})}{n^{2}/2}.$

These are two important birational invariants of $S$. If $S$ is minimal, then
the volume $\operatorname*{vol}(S)$ is equal to the intersection number
$K_{S}^{2}$, which is also the first Chern number of $S$. The Noether
inequality (1.1) is one of the fundamental inequality in the surface theory.
Minimal surfaces of general type with the equality are usually said on the
Noether line, which has been systematically studied by Horikawa [Hor76].

In a series of nice works [Kob92, Che04, CCZ06, Che07, CC15, CH17, CCJ20b,
CCJ20a], the following sharp Noether inequality has been established for every
minimal $3$-fold $X$ of general type with $p_{g}(X)\leq 4$ or $p_{g}(X)\geq
11$.

$\operatorname*{vol}(X)\geq\frac{4}{3}\,p_{g}(X)-\frac{10}{3}.$

It is also proved in [CJ17] that the Noether type inequality holds also in
higher dimension: there exist positive numbers $a_{d}$ and $b_{d}$, depending
only on the dimension $d$ of the variety $X$, such that

$\operatorname*{vol}(X)\geq a_{d}\,p_{g}(X)-b_{d}.$

Recently, the foliation theory has attracted more and more attention in
algebraic geometry, especially in birational geometry. Miyaoka [Miy87, Miy88]
introduced the use of the foliations to study the abundance conjecture. For
instance, the foliations whose canonical divisor $K_{\mathcal{F}}$ is not
pseudoeffective are characterised in [Miy87], see also [BM16]. In the
beautiful paper [Cas21], Cascini proposed the minimal model program for
foliated varieties. It stimulates a new project in birational geometry to
generalize everything to foliated varieties.

Let $(X,\mathcal{F})$ be a foliated variety, and denote by $K_{\mathcal{F}}$
its canonical divisor. The canonical divisor $K_{\mathcal{F}}$ of a foliation
plays a similar role in many aspects as the canonical divisor $K_{X}$ of a
variety. For instance, Miyaoka’s theorem [Miy87] on foliated varieties with
non-pseudoeffective canonical divisor, the minimal model theory on foliated
surfaces due to Brunella and McQuillan etc. [Bru99, McQ08], and the
deformation invariance of the pluri-genera by Cascini-Floris [CF18]. We are
mainly interested in the following two birational invariants: the geometric
genus $p_{g}(\mathcal{F})$ and volume $\operatorname*{vol}(\mathcal{F})$,
which are defined similar to the case of algebraic varieties.

$\displaystyle p_{g}({\mathcal{F}})$ $\displaystyle\,=\dim
H^{0}(X,K_{\mathcal{F}});$ $\displaystyle\operatorname*{vol}({\mathcal{F}})$
$\displaystyle\,=\limsup_{n\to+\infty}\frac{\dim
H^{0}(X,nK_{\mathcal{F}})}{n^{d}/d!}.$

Here $d=\dim X$. By the asymptotic Riemann-Roch theorem, if
$(Y,\mathcal{F}_{Y})$ is a minimal model of $(X,\mathcal{F})$, then
$\operatorname*{vol}(X,\mathcal{F})=K_{\mathcal{F}_{Y}}^{d}$. The foliated
variety $(X,\mathcal{F})$ is called of general type if
$\operatorname*{vol}(\mathcal{F})>0$, i.e., the canonical divisor
$K_{\mathcal{F}}$ is big. Inspired by the Noether type inequalities for
algebraic varieties, it is natural to ask the following question.

######  Question .

Do there exist Noether type inequalities for a foliated variety of general
type?

In this paper, we try to settle the above question in dimension two.

######  Theorem 1.1.

Let $(S,\mathcal{F})$ be a foliated surface of general type with reduced
singularities.

  1. (i).

The following Noether inequality holds

$\operatorname*{vol}(\mathcal{F})\geq p_{g}(\mathcal{F})-2.$ (1.2)

Moreover, if the equality in (1.2) holds, then the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ defines a birational map whose image is a
surface of minimal degree $($equal to $p_{g}(\mathcal{F})-2)$ in
$\mathbb{P}^{p_{g}(\mathcal{F})-1}$.

  2. (ii).

Suppose furthermore that the surface $S$ is of general type. Then

$\operatorname*{vol}(\mathcal{F})\geq 2p_{g}(\mathcal{F})-4.$ (1.3)

Moreover, if the equality in (1.3) holds, then the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ defines a two-to-one map whose image is a
surface of minimal degree $($equal to $p_{g}(\mathcal{F})-2)$ in
$\mathbb{P}^{p_{g}(\mathcal{F})-1}$.

######  Remark .

(i). We will construct examples (cf. Section 6 and Section 6) reaching the
equalities in both (1.2) and (1.3), which show that both of the Noether type
inequalities are sharp.

(ii). For algebraic varieties of general type, there exists so-called ”the
second Noether inequality”, at least in lower dimensions. When $X$ is of
dimension two, the volume is an integer, and hence $\operatorname*{vol}(X)\geq
2p_{g}(X)-3$ if $\operatorname*{vol}(X)\neq 2p_{g}(X)-4$; when $X$ is a
threefold, Hu-Zhang [HZ22] proved that
$\operatorname*{vol}(X)\geq\frac{4}{3}p_{g}(X)-\frac{19}{6}$ if
$\operatorname*{vol}(X)\neq\frac{4}{3}p_{g}(X)-\frac{10}{3}$. However, such a
phenomenon disappears for foliated surfaces of general type. Indeed, we will
construct in Section 6 a sequence of foliated surfaces
$(S_{n},\mathcal{F}_{n})$ of general type with reduced singularities such that
$\operatorname*{vol}(\mathcal{F}_{n})>p_{g}(\mathcal{F}_{n})-2$, but the
difference
$\operatorname*{vol}(\mathcal{F}_{n})-\big{(}p_{g}(\mathcal{F}_{n})-2\big{)}$
can be arbitrarily close to zero.

(iii). The structure of a surface of minimal degree in $\mathbb{P}^{n}$ is
well-understood, see for intance [GH78, Chapter 4.3]. In a forthcoming paper,
we will characterize the geometry of foliated surfaces of general type
satisfying the Noether type equalities.

In the following, we briefly explain the strategy of our proof. As in the case
proving the classical Noether inequality (1.1), the starting point is to
analyze the canonical map $\varphi_{|K_{\mathcal{F}}|}$ defined by the
complete linear system $|K_{\mathcal{F}}|$. However, there are two main
difficulties compared to the case of $\varphi_{|K_{S}|}$:

(1). The canonical divisor $K_{\mathcal{F}}$ might not be nef for a foliation
$\mathcal{F}$ of general type with reduced singularities on a smooth surface
$S$. It makes some trouble in estimating the lower bound of
$\operatorname*{vol}(\mathcal{F})$. For instance, suppose that

$|K_{\mathcal{F}}|=|M|+Z,$ (1.4)

where $Z$ is the fixed part of $|K_{\mathcal{F}}|$, and $M$ is the moving
part. Then it is no longer true that $\operatorname*{vol}(\mathcal{F})\geq
K_{\mathcal{F}}\cdot M$, cf. Section 6. Of course, one can contract the
support of the negative part of $K_{\mathcal{F}}$ to a normal surface $S_{0}$,
such that the induced foliation $\mathcal{F}_{0}$ on $S_{0}$ has the advantage
that $K_{\mathcal{F}_{0}}$ is nef [Bru99, McQ08]. However, the surface $S_{0}$
would be singular with klt singularities and $K_{\mathcal{F}_{0}}$ is no
longer a line bundle (but a $\mathbb{Q}$-bundle).

(2). The second difficulty occurring in the case when the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ induces a fibration $f:\,S\to B$. In the case
proving the classical Noether inequality (1.1), the general fiber $F$ of $f$
is of genus at least two, and hence $K_{S}\cdot F=2g(F)-2\geq 2$. However, in
our case it is only known that $K_{\mathcal{F}}\cdot F>0$ since
$K_{\mathcal{F}}$ is big. It can happen that $K_{\mathcal{F}}\cdot F=1$, even
when the surface $S$ is also of general type (see Section 6 for such an
example). Suppose that

$K_{\mathcal{F}}=P+N,$ (1.5)

is the Zariski decomposition of $K_{\mathcal{F}}$, where $P$ is the nef part
and $N$ is the negative part of $K_{\mathcal{F}}$. It might happen that
$N\cdot F>0$, and hence $P\cdot F<K_{\mathcal{F}}\cdot F=1$ (see Section 6 for
such an example). This will cause trouble in estimating the lower bound of
$\operatorname*{vol}(\mathcal{F})=P^{2}$ along the usual way, cf. Section 3.

To overcome the above two difficulties, we need to control the negative part
in the Zariski decomposition of the canonical divisor $K_{\mathcal{F}}$, as
well as the structure of $K_{\mathcal{F}}$ when the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ induces a fibration. Let

$\varphi=\varphi_{|K_{\mathcal{F}}|}:\,S\dashrightarrow\Sigma\subseteq\mathbb{P}^{p_{g}(\mathcal{F})-1},$

be the rational map defined by $|K_{\mathcal{F}}|$. After some elementary
reduction, we are reduced to the case when the image $\Sigma$ is a curve and
the canonical map $\varphi_{|K_{\mathcal{F}}|}$ has no base point. Hence the
canonical map induces a fibration $f:\,S\to B$ by taking the normalization and
Stein factorization:

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}(\mathcal{F})-1}}$

Consider the two decompositions in (1.4) and (1.5). Since the moving part $M$
is always nef, it follows that $M\leq P$, or equivalently $N\leq Z$. Let $F$
be a general fiber of $f$. The moving part $M$ consists of several fibers,
whose cardinality is at least $p_{g}(\mathcal{F})-1$ by the Riemann-Roch
theorem. Hence

$\operatorname*{vol}(\mathcal{F})=P^{2}\geq P\cdot
M\geq\big{(}p_{g}(\mathcal{F})-1\big{)}P\cdot
F=\big{(}p_{g}(\mathcal{F})-1\big{)}\big{(}K_{\mathcal{F}}-N\big{)}\cdot F.$

To obtain the Noether type inequalities, it suffices to get a lower bound on
$\big{(}K_{\mathcal{F}}-N\big{)}\cdot F$, or equivalently, an upper bound on
$N\cdot F$. We will show in Section 3 an upper bound on the coefficients
$a_{C}$’s of $N=\sum a_{C}C$, from which we can prove the Noether type
inequalities in most cases, except the extreme case where
$K_{\mathcal{F}}\cdot F=1$. In this extreme case, we will prove in Theorem 4.1
an explicit description on the canonical divisor $K_{\mathcal{F}}$, base on
which the Noether type inequality follows.

The paper is organized as follows. In Section 2, we review some basic facts
about foliations, especially the foliations on smooth surfaces. In Section 3,
we recall the Zariski decomposition of the canonical divisor $K_{\mathcal{F}}$
for a relatively minimal foliation, and prove in Section 3 an upper bound on
the coefficients $a_{C}$’s of the negative part $N=\sum a_{C}C$. In Section 4,
we restrict ourselves to the special case when the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ induces a fibration, and prove in in Theorem 4.1
an explicit description on the canonical divisor $K_{\mathcal{F}}$ if
$K_{\mathcal{F}}\cdot F=1$ and $S$ is of general type. The Noether type
inequalities will be proved in Section 5. Finally, we will in Section 6
construct several examples to illustrate that the Noether type inequalities
obtained in Theorem 1.1 are sharp.

##  2\. Preliminaries

In this section, we recall some basic facts about foliations, especially the
foliations on smooth surfaces. For more details we refer to [Bru04, McQ08].

We work over the complex number $\mathbb{C}$. By a foliated variety
$(X,\mathcal{F})$ we mean a foliation $\mathcal{F}$ on the variety $X$.

######  Definition .

A foliation $\mathcal{F}$ of rank $r$ on a normal variety $X$ of dimension $n$
is defined by a rank $r$ coherent subsheaf $T_{\mathcal{F}}\subseteq T_{X}$
such that

  1. (i)

$T_{\mathcal{F}}$ is saturated, i.e., $T_{X}/T_{\mathcal{F}}$ is torsion free;

  2. (ii)

$T_{\mathcal{F}}$ is closed under the Lie bracket, i.e., $[\alpha,\beta]\in
T_{\mathcal{F}}$ for any $\alpha,\beta\in T_{\mathcal{F}}$.

The singular locus $\mathrm{sing}(\mathcal{F})$ of $\mathcal{F}$ is the set of
points $x\in X$ where either $x\in\mathrm{sing}(X)$ or the quotient sheaf
$N_{\mathcal{F}}:=T_{X}/T_{\mathcal{F}}$ fails to be locally free at $x$. Note
that the first condition above implies that the codimension of
$\mathrm{sing}(\mathcal{F})$ in $X$ is at least two. The canonical divisor of
$\mathcal{F}$ is a divisor $K_{\mathcal{F}}$ on $X$ such that
$\mathcal{O}_{X}(K_{\mathcal{F}})\cong\det(T_{\mathcal{F}})^{*}.$ Locally in
the Euclidean topology around a smooth point $x\in
X\setminus\mathrm{sing}(\mathcal{F})$, the foliation $\mathcal{F}$ can be
defined by a fibration; that is, there exists an analytic open neighborhood
$x\in U\subseteq X$ and a morphism $f:\,U\to\mathbb{C}^{n-r}$ such that

$T_{\mathcal{F}}|_{U}=\ker(\mathrm{d}f:\,T_{U}\to f^{*}T_{\mathbb{C}^{n-r}}).$

Conversely, any morphism $f:\,X\to Y$ with connected fibers defines a
foliation $\mathcal{F}$ by taking the saturation of
$\ker(\mathrm{d}f:\,T_{X}\to f^{*}T_{Y})$ in $T_{X}$.

We next turn to the foliation on a smooth projective surface $S$. A foliation
$\mathcal{F}$ on $S$ can be given by an exact sequence

$0\longrightarrow T_{\mathcal{F}}\longrightarrow T_{S}\longrightarrow
I_{\Delta}(N_{\mathcal{F}})\longrightarrow 0,$

where $T_{\mathcal{F}}$ and $N_{\mathcal{F}}$ are respectively the tangent
bundle and normal bundle of $\mathcal{F}$, and $I_{\Delta}$ is an ideal sheaf
supported on the singular locus of $\mathcal{F}$. Equivalently, a foliation on
$S$ is given by the data $\\{(U_{i},v_{i})\\}_{i\in I}$, where
$\\{U_{i}\\}_{i\in I}$ is an open covering of $S$, $v_{i}$ is a holomorphic
vector field on $U_{i}$ with at most isolated zeros, and there exists
$g_{ij}\in\mathcal{O}^{*}_{S}(U_{i}\cap U_{j})$ whenever $U_{i}\cap
U_{j}\neq\emptyset$ such that

$v_{i}|_{U_{i}\cap U_{j}}=g_{ij}v_{j}|_{U_{i}\cap U_{j}}.$ (2.1)

The cocycle $\\{g_{ij}\\}$ defines a line bundle which is nothing but the
canonical divisor $K_{\mathcal{F}}=T_{\mathcal{F}}^{*}$.

Alternatively, one can also define $\mathcal{F}$ using one-forms instead of
vector fields. A foliation on $S$ is given by a collection of one-forms
$\omega_{i}\in\Omega^{1}_{S}(U_{i})$ with at most isolated zeros and there
exists $f_{ij}\in\mathcal{O}^{*}_{S}(U_{i}\cap U_{j})$ whenever $U_{i}\cap
U_{j}\neq\emptyset$ such that

$\omega_{i}|_{U_{i}\cap U_{j}}=f_{ij}\omega_{j}|_{U_{i}\cap U_{j}}.$ (2.2)

The cocycle $\\{g_{ij}\\}$ defines a line bundle which is the conormal bundle
$N_{\mathcal{F}}^{*}$. One can also translate this into an exact sequence:

$0\longrightarrow
N_{\mathcal{F}}^{*}\longrightarrow\Omega_{S}^{1}\longrightarrow
I_{\Delta}(K_{\mathcal{F}})\longrightarrow 0.$

For any surjective morphism $\Pi:\,\widetilde{S}\to S$ from anther smooth
projective surface $\widetilde{S}$, there is a natural foliation
$\widetilde{\mathcal{F}}$ induced by pulling-back $\mathcal{F}$ on
$\widetilde{S}$. Suppose that $\\{(U_{i},\omega_{i})\\}_{i\in I}$ is a
collection of local one-forms defining the foliation $\mathcal{F}$. At the
first sight, one may think that $\widetilde{\mathcal{F}}$ is simply given by
$\\{(\Pi^{-1}U_{i},\Pi^{*}\omega_{i})\\}_{i\in I}$. But this is not the case
in general, since $\Pi^{*}\omega_{i}$ may admit one-dimensional zeros.
Alternatively, the foliation $\widetilde{\mathcal{F}}$ is defined by the
following data

$\\{(V_{ij},\widetilde{\omega}_{ij})\\},~{}i\in I,j\in J,$

where $V_{ij}$ is an open covering of $\widetilde{S}$ with
$\Pi(V_{ij})\subseteq U_{i}$, and
$\widetilde{\omega}_{ij}=\frac{\Pi^{*}(\omega_{i})}{h_{ij}}$ with $h_{ij}$
being some holomorphic function over $V_{ij}$ satisfying
$\text{\rm{div\,}}(h_{ij})=\text{\rm{div\,}}\big{(}\Pi^{*}(\omega_{i})|_{V_{ij}}\big{)}$.
In particular, for any blowing-up $\sigma:\,\widetilde{S}\to S$ centered at
$p\in S$, there is an induced foliation $\widetilde{\mathcal{F}}$ on
$\widetilde{S}$, such that $\widetilde{\mathcal{F}}|_{\widetilde{S}\setminus
E}\cong\mathcal{F}|_{S\setminus p}$ under the isomorphism
$\widetilde{S}\setminus E\cong S\setminus\\{p\\}$, where $E$ is the
exceptional curve.

Suppose that $p$ is a singular point of $\mathcal{F}$, and $v$ is a local
vector field defining $\mathcal{F}$. The two eigenvalues
$\lambda_{1},\lambda_{2}$ of the linear part $(Dv)(p)$ are well-defined up to
multiplication by a non-zero constant.

######  Definition .

A singularity $p$ of $\mathcal{F}$ is called a reduced singularity if at least
one of the two eigenvalues (say, $\lambda_{2}$) is not zero and the quotient
$\lambda=\frac{\lambda_{1}}{\lambda_{2}}$ is not a positive rational number.
The foliation $\mathcal{F}$ is said to be reduced if any singularity of
$\mathcal{F}$ is reduced.

Remark that the quotient $\lambda=\frac{\lambda_{1}}{\lambda_{2}}$ is
unchanged by multiplication of $v$ by a nonvanishing holomorphic function. Of
course, if $\lambda_{1}\neq 0$, we could also consider the quotient
$\lambda^{-1}=\frac{\lambda_{2}}{\lambda_{1}}$ instead of $\lambda$, but then
$\lambda\not\in\mathbb{Q}^{+}$ iff $\lambda^{-1}\not\in\mathbb{Q}^{+}$. The
complex number $\lambda=\frac{\lambda_{1}}{\lambda_{2}}$, with an inessential
abuse due to the exchange $\lambda\leftrightarrow\lambda^{-1}$, is called the
eigenvalue of $\mathcal{F}$ at $p$ following [Bru04]. A reduced singularity is
called non-degenerate if both of the two eigenvalues $\lambda_{1}$ and
$\lambda_{2}$ are non-zero; otherwise it is called a saddle node. Given any
foliation, one can obtain a reduced one by a sequence of blowing-ups:

######  Theorem 2.1 (Seidenberg, [Bru04, Theorem 1.1]).

Given any foliated surface $(S,\mathcal{F})$, there exists a sequence of
blowing-ups $\pi:\,\widetilde{S}\to S$, such that the foliation
$\pi^{*}(\mathcal{F})$ is reduced.

The birational geometry behaves well for foliated surfaces with reduced
singularities, as showed in [Bru99, Bru04, Men00]. For instance, the pluri-
genera $p_{n}(\mathcal{F})$’s keep invariant under birational maps between
reduced foliated surfaces, where the pluri-genus $p_{n}(\mathcal{F})$ is
defined as

$p_{n}(\mathcal{F})=\dim H^{0}(S,nK_{\mathcal{F}}).$

In particular, the geometric genus $p_{g}(\mathcal{F}):=p_{1}(\mathcal{F})$ as
well as the volume $\operatorname*{vol}(\mathcal{F})$ are well-defined
birational invariants for foliated surfaces with reduced singularities, where

$\operatorname*{vol}({\mathcal{F}})=\limsup_{n\to+\infty}\frac{\,p_{n}(\mathcal{F})}{n^{2}/2}=\limsup_{n\to+\infty}\frac{\,\dim
H^{0}(X,nK_{\mathcal{F}})}{n^{2}/2}.$

An important way to produce foliations comes from fibration on smooth
projective surfaces. Let $f:\,S\to B$ be a fibration of curves, i.e., $f$ is a
proper surjective morphism from $S$ onto $B$ with connected fibers. The
fibration defines a foliation $\mathcal{F}$ on $S$ by taking the saturation of
$\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$. The canonical divisor
$K_{\mathcal{F}}$ is simple:

$K_{\mathcal{F}}=K_{S/B}\otimes\mathcal{O}_{S}\Big{(}\sum(1-a_{i})C_{i}\Big{)},$
(2.3)

where $K_{S/B}=K_{S}-f^{*}(K_{B})$ is the relative canonical divisor, the sum
is taken over all components $C_{i}$’s in fibers of $f$, and $a_{i}$ is the
multiplicity of $C_{i}$ in fibers of $f$. The foliation $\mathcal{F}$ is
reduced if and only if every possible singular fiber of $f$ is normal
crossing. In particular, if the fibration is semi-stable, i.e., any possible
singular fiber of $f$ is a reduced node curve, and any possible smooth
rational component in such a singular fiber intersects other components at
least two points, then $\mathcal{F}$ is relatively minimal and
$K_{\mathcal{F}}=K_{S/B}$ by (2.3).

An irreducible curve $C\subseteq S$ is said to be $\mathcal{F}$-invariant if
the inclusion $T_{\mathcal{F}}|_{C}\hookrightarrow T_{X}|_{C}$ factors through
$T_{C}$. By a curve $C\subseteq S$ we mean a reduced and compact algebraic
curve. So it might be singular and reducible. Suppose that $C$ is not
$\mathcal{F}$-invariant, or more precisely every irreducible component of $C$
is not $\mathcal{F}$-invariant. Then one defines the tangency of $\mathcal{F}$
to $C$ as follows. Let $p\in C$ be any point. Around $p$, let $\\{f=0\\}$ be a
local equation of $C$, and $v$ be a local holomorphic vector field defining
$\mathcal{F}$ around $p$. Then the tangency of $\mathcal{F}$ to $C$ at $p$ is
defined to be

$\mathrm{tang}(\mathcal{F},C,p)=\dim_{\mathbb{C}}\frac{\mathcal{O}_{S,p}}{\langle
f,v(f)\rangle}.$

As $C$ is not $\mathcal{F}$-invariant,
$\mathrm{tang}(\mathcal{F},C,p)<+\infty$ and
$\mathrm{tang}(\mathcal{F},C,p)=0$ except for finitely many points. Hence one
defines the tangency of $\mathcal{F}$ to $C$.

$\mathrm{tang}(\mathcal{F},C)=\sum_{p\in C}\mathrm{tang}(\mathcal{F},C,p).$

######  Proposition  ([Bru04, Proposition 2.2]).

Let $C$ be a curve on $S$ which is not $\mathcal{F}$-invariant. Then

$\mathrm{tang}(\mathcal{F},C)=K_{\mathcal{F}}C+C^{2}.$

In particular,

$K_{\mathcal{F}}C+C^{2}\geq 0.$

We now suppose that $C$ is $\mathcal{F}$-invariant, or more precisely every
irreducible component of $C$ is $\mathcal{F}$-invariant. Given any point $p\in
C$, let $\\{f=0\\}$ be a local equation of $C$, and $\omega$ be a local
holomorphic one-form defining $\mathcal{F}$ around $p$. Because $C$ is
$\mathcal{F}$-invariant, we may write

$g\omega=hdf+f\eta,$

for some holomorphic one-form $\eta$ and holomorphic functions $g,h$ around
$p$, such that $h$ and $f$ are coprime. We define

$\displaystyle Z(\mathcal{F},C,p)$ $\displaystyle\,=\text{vanishing order of
$~{}\frac{h}{g}\Big{|}_{C}$ at $p$},$
$\displaystyle\mathrm{CS}(\mathcal{F},C,p)$ $\displaystyle\,=\text{residue of
$~{}-\frac{\eta}{h}\Big{|}_{C}$ at $p$}.$

By definition, both $Z(\mathcal{F},C,p)$ and $\mathrm{CS}(\mathcal{F},C,p)$
are zero if $p$ is not a singular point of $\mathcal{F}$. If $\mathcal{F}$ is
reduced, then $Z(\mathcal{F},C,p)\geq 0$ for any $p\in C$ [Bru97]. Let

$\displaystyle Z(\mathcal{F},C)$ $\displaystyle\,=\sum_{p\in
C}Z(\mathcal{F},C,p)=\sum_{p\in
C\,\cap\,\text{Sing}(\mathcal{F})}Z(\mathcal{F},C,p),$
$\displaystyle\mathrm{CS}(\mathcal{F},C)$ $\displaystyle\,=\sum_{p\in
C}\mathrm{CS}(\mathcal{F},C,p)=\sum_{p\in
C\,\cap\,\text{Sing}(\mathcal{F})}\mathrm{CS}(\mathcal{F},C,p).$

######  Proposition  ([Bru04, Proposition 2.2]).

Let $C$ be a curve on $S$ which is $\mathcal{F}$-invariant. Then

$\displaystyle Z(\mathcal{F},C)$
$\displaystyle\,=K_{\mathcal{F}}C+\chi(C),\quad\text{where~{}}\chi(C)=-K_{S}C-C^{2};$
$\displaystyle\mathrm{CS}(\mathcal{F},C)$ $\displaystyle\,=C^{2}.$

In particular, if $\mathcal{F}$ is reduced and $C$ is an
$\mathcal{F}$-invariant curve, then

$0\leq
K_{\mathcal{F}}C+\chi(C)=K_{\mathcal{F}}C-K_{S}C-C^{2}=N_{\mathcal{F}}C-C^{2}.$

##  3\. The Zariski decomposition of the canonical divisor $K_{\mathcal{F}}$

In this section, we are concerned about the Zariski decomposition of the
canonical divisor $K_{\mathcal{F}}$. We will first briefly recall the Zariski
decomposition of the canonical divisor $K_{\mathcal{F}}$ for a relatively
minimal foliation $\mathcal{F}$ with $K_{\mathcal{F}}$ being pseudo-effective
and refer to [McQ08] and [Bru04, Chapter 8] for more details. Then we give a
careful analysis on the negative part, and prove a technical result about the
coefficients appearing in the negative part, which will be helpful in proving
the Noether inequalities.

######  Definition .

Let $\mathcal{F}$ be reduced foliation on a smooth projective $S$. An
irreducible curve $C\subseteq S$ is $\mathcal{F}$-exceptional if

  1. (i).

$C$ is an exceptional curve of first kind on $S$, i.e., it is a smooth
rational curve with $C^{2}=-1$;

  2. (ii).

the contraction of $C$ to a point produces a new foliation
$(S_{0},\mathcal{F}_{0})$ which is still reduced.

######  Definition .

A foliated surface $(S,\mathcal{F})$ is called relatively minimal if

  1. (i).

the foliation $\mathcal{F}$ is reduced;

  2. (ii).

there is no $\mathcal{F}$-exceptional curve on $S$.

It is proved that any foliated surface $(S,\mathcal{F})$ has a relatively
minimal model, cf. [Bru04, Proposition 5.1]. We assume in the following that
$\mathcal{F}$ is a relatively minimal foliation on a smooth projective surface
$S$ such that $K_{\mathcal{F}}$ is pseudo-effective. In fact, the canonical
divisor $K_{\mathcal{F}}$ is pseudo-effective if and only if $\mathcal{F}$ is
not induced by a $\mathbb{P}^{1}$-fibration, cf. [Miy87]. Denote the Zariski
decomposition of $K_{\mathcal{F}}$ by

$K_{\mathcal{F}}=P+N,$ (3.1)

where $P$ is the nef part and $N$ is the negative one. By the Riemann-Roch
theorem, one sees directly that

$\operatorname*{vol}(\mathcal{F})=P^{2}.$

McQuillan proved that the support of the negative part $N$ is a disjoint union
of maximal $\mathcal{F}$-chains.

######  Definition .

Let $\mathcal{F}$ be a relatively minimal foliation on a smooth projective
surface $S$. We say a curve $C\subseteq S$ is an $\mathcal{F}$-chain if

  1. (i).

the curve $C$ is a Hirzebruch-Jung string, i.e., $C=\cup_{j=1}^{r}C_{j}$, each
$C_{j}$ is a smooth rational curve with $C_{j}^{2}\leq-2$, $C_{j}\cdot
C_{i}=1$ if $|i-j|=1$ and $0$ if $|i-j|\geq 2$;

  2. (ii).

each irreducible component $C_{j}$ is $\mathcal{F}$-invariant;

  3. (iii).

$\mathrm{Sing}(\mathcal{F})\cap C$ are all reduced and non-degenerate;

  4. (iv).

$Z(\mathcal{F},C_{1})=1$, and $Z(\mathcal{F},C_{j})=2$ for any $2\leq j\leq
r$.

Since each irreducible component $C_{j}$ is a smooth rational curve, by
Section 2 the last condition (iv) is also equivalent to

  1. (iv)′.

$K_{\mathcal{F}}C_{1}=-1$, and $K_{\mathcal{F}}C_{j}=0$ for any $2\leq j\leq
r$.

######  Theorem 3.1 ([Bru04, Theorem 8.1]).

Let $\mathcal{F}$ be a relatively minimal foliation on a smooth projective
surface $S$. Suppose that $K_{\mathcal{F}}$ is pseudo-effective with the
Zariski decomposition as in (3.1). Then the support $\mathrm{Supp}(N)$ is a
disjoint union of maximal $\mathcal{F}$-chains, and $\lfloor N\rfloor=0$.

The above theorem shows that all the coefficients in $N$ are less than $1$. In
fact, since the support $\mathrm{Supp}(N)$ is a disjoint union of maximal
$\mathcal{F}$-chains, which can be contracted to singularities of Hirzebruch-
Jung type, these coefficients can be explicitly computed out using continued
fractions [BHPV04, § III.5]. By contracting the support $\mathrm{Supp}(N)$,
one obtains a surface $S_{0}$ with finitely many singularities. Then the
negative part $N$ can be decomposed into

$N=\sum_{Q}N_{Q},$

where the sum runs over all singularities $Q$ on $S_{0}$, and
$\mathrm{Supp}(N_{Q})$ is supported on the inverse image of $Q$ in $S$.
Suppose that $Q$ is a singularity of type $A_{n,q}$, and let
$N_{Q}=\sum\limits_{j=1}^{r}b_{j}C_{j}$ with $C=\cup_{j=1}^{r}C_{j}$ being a
maximal $\mathcal{F}$-chain as above. Define $\lambda_{r+1}=0$,
$\lambda_{r}=1$, and the rest $\lambda_{j}$’s by the following recursion
formula:

$\lambda_{j-1}-e_{j}\lambda_{j}+\lambda_{j+1}=0,$ (3.2)

where $e_{j}=-C_{j}^{2}\geq 2$. Then it has been shown in [BHPV04, § III.5]
that $\lambda_{1}=q$, $\lambda_{0}=n$. In fact, it holds that

$\large\frac{n}{q}=e_{1}-\frac{1}{e_{2}-\frac{1}{\cdots\,-\,\frac{1}{e_{r}}}}.$

Moreover,

$b_{j}=\frac{\lambda_{j}}{n},\qquad\forall\,1\leq j\leq r.$ (3.3)

We provide a brief explanation of (3.3). According to the property of the
Zariski decomposition

$N_{Q}C_{1}=K_{\mathcal{F}}C_{1}=-1,\qquad
N_{Q}C_{j}=K_{\mathcal{F}}C_{j}=0,\quad\forall\,2\leq j\leq r.$

On the other hand, define
$N^{\prime}=\sum\limits_{j=1}^{r}\frac{\lambda_{j}}{n}C_{j}$. Based on (3.2),
one checks directly that

$N^{\prime}C_{1}=-1=N_{Q}C_{1},\qquad
N^{\prime}C_{j}=0=N_{Q}C_{j},\quad\forall\,2\leq j\leq r.$

Since the intersection matrix $\big{(}C_{i}C_{j}\big{)}$ is negatively
definite, it follows that $N_{Q}=N^{\prime}$, i.e., the equality (3.3) holds.

######  Proposition .

Let $N_{Q}=\sum\limits_{j=1}^{r}b_{j}C_{j}$ with $C=\cup_{j=1}^{r}C_{j}$ being
a maximal $\mathcal{F}$-chain as above. Then

$b_{j}<\left\\{\begin{aligned}
&\,\frac{1}{e_{1}-1},&~{}&\text{~{}if~{}}j=1;\\\\[5.69054pt]
&\,\frac{1}{e_{j}},&&\text{~{}if~{}}j\geq 2\text{~{}and~{}}e_{j}\geq
3,\end{aligned}\right.$ (3.4)

where $e_{j}=-C_{j}^{2}\geq 2$.

###### Proof.

Let $\lambda_{r+1}=0$, and $b_{j}=\frac{\lambda_{j}}{n}$ for $1\leq j\leq r$.
By the above arguments, $\lambda_{r}=1$, and the rest $\lambda_{j}$’s can be
computed by (3.2). As proved in [BHPV04, § III.5], $\lambda_{1}=q$ and
$\lambda_{0}=n$. With the help of (3.2), one proves inductively that

$\lambda_{j-1}>\lambda_{j},\qquad\forall\,1\leq j\leq r+1.$ (3.5)

Hence

$\lambda_{j-1}=e_{j}\lambda_{j}-\lambda_{j+1}>e_{j}\lambda_{j}-\lambda_{j}=(e_{j}-1)\lambda_{j}.$

In particular, $n=\lambda_{0}>(e_{1}-1)\lambda_{1}$. Equivalently,
$b_{1}=\frac{\lambda_{1}}{n}<\frac{1}{e_{1}-1}$ as required.

Suppose now that $j\geq 2$. Then

$n=\lambda_{0}\geq\lambda_{j-2}=e_{j-1}\lambda_{j-1}-\lambda_{j}\geq
2\lambda_{j-1}-\lambda_{j}>\big{(}2(e_{j}-1)-1\big{)}\lambda_{j}\geq
e_{j}\lambda_{j}.$

The last inequality follows from the assumption that $e_{j}\geq 3$. It follows
that $b_{j}=\frac{\lambda_{j}}{n}<\frac{1}{e_{j}}$ if $j\geq 2$ and $e_{j}\geq
3$. This completes the proof of (3.4). ∎

######  Remark .

The above bounds on the coefficients of the negative part $N_{Q}$ will be key
to estimate the volume $\operatorname*{vol}(K_{\mathcal{F}})$. One can
similarly define the volume $\operatorname*{vol}(L)$ for any big divisor $L$
on a smooth projective surface $S$ by

$\operatorname*{vol}(L)=\limsup_{n\to+\infty}\frac{\dim
H^{0}(X,nL)}{n^{2}/2}.$

The naive Noether type inequalities do NOT hold for $L$. For instance, let
$e>0$ and
$S=\mathbb{P}_{\mathbb{P}^{1}}\big{(}\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(e)\big{)}$
be the Hirzebruch surface admitting a unique section $C_{0}$ with
$C_{0}^{2}=-e<0$. Let $f:\,S\to\mathbb{P}^{1}$ be the geometrical ruling on
$S$, $F$ be a general fiber of $f$, and $L=mF+C_{0}$. Suppose that $0<m<e$.
Then one checks easily that the Zariski decomposition of $L$ is

$L=\Big{(}mF+\frac{m}{e}C_{0}\Big{)}+\frac{e-m}{e}C_{0},$

and hence

$\operatorname*{vol}(L)=\frac{m^{2}}{e},\qquad h^{0}(L)=m+1.$

Fixing $m$ and letting $e\to+\infty$, one deduces that there can not exist
positive constants $a,b$ such that $\operatorname*{vol}(L)\geq ah^{0}(L)-b$.

Of course, in this concrete example, one shows easily that $L=K_{\mathcal{F}}$
for some reduced foliation $\mathcal{F}$ on $S$ only when $m=e-1$ based on
(3.2) and (3.3). If it is indeed the case, then

$\operatorname*{vol}(\mathcal{F})=\frac{m^{2}}{e}=\frac{(e-1)^{2}}{e}=e-2+\frac{1}{e}>e-2=p_{g}(\mathcal{F})-2.$

We refer to Section 6 for a construction of such a foliation.

##  4\. The canonical map induces a fibration

The main purpose of this section is to prove Theorem 4.1, which will be key to
get the Noether inequalities when the canonical map of a foliated surface
$(S,\mathcal{F})$ induces a fibration.

Let $\mathcal{F}$ be a reduced foliation of general type on a smooth
projective surface with canonical divisor $K_{\mathcal{F}}$. We restrict
ourselves in this section to the case when the image $\varphi(S)$ is of one-
dimension, where

$\varphi=\varphi_{|K_{\mathcal{F}}|}:\,S\dashrightarrow\Sigma\subseteq\mathbb{P}^{p_{g}(\mathcal{F})-1},$

is the rational map defined by $|K_{\mathcal{F}}|$. In this case, by the Stein
factorization, we obtain a diagram as follows.

$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\scriptstyle{\sigma}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$desingularization$\textstyle{\Sigma\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}(\mathcal{F})-1}}$

Here $\pi:\,B\to Y$ is finite, and $f:\,S^{\prime}\to B$ is a fibration of
curves with connected fibers. The foliation $\mathcal{F}$ lifts to a foliation
$\mathcal{F}^{\prime}$ on $S^{\prime}$, which is also reduced. Since
$\mathcal{F}$ is reduced,

$K_{\mathcal{F}^{\prime}}=\sigma^{*}K_{\mathcal{F}}+\mathcal{E},$

where $\mathcal{E}$ is some effective divisor supported on the exceptional
curves of the birational morphism $\sigma$. Moreover,

$|K_{\mathcal{F}^{\prime}}|=\sigma^{*}|K_{\mathcal{F}}|+\mathcal{E}.$

It follows that the map $\varphi_{|K_{\mathcal{F}^{\prime}}|}$ factors through
$\varphi_{|K_{\mathcal{F}}|}$, and hence it induces the same fibration
$f:\,S^{\prime}\to B$. By replacing $(S,\mathcal{F})$ by
$(S^{\prime},\mathcal{F}^{\prime})$, we may assume that the moving part of the
linear system $|K_{\mathcal{F}}|$ is base-point-free, so that the canonical
map $\varphi_{|K_{\mathcal{F}}|}$ induces a fibration $f:\,S\to B$.

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\scriptstyle{\varphi=\varphi_{|K_{\mathcal{F}}|}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$desingularization$\textstyle{\Sigma\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}(\mathcal{F})-1}}$

After such a replacement, the foliation $\mathcal{F}$ is still reduced of
general type, but not necessarily relatively minimal any more. Let $F$ be a
general fiber of $f$. As the foliation $\mathcal{F}$ is assumed to be of
general type, it follows that

$K_{\mathcal{F}}\cdot F\geq 1.$

The main purpose in this section is to prove the following.

######  Theorem 4.1.

Let $\mathcal{F}$ be a reduced foliation of general type on a smooth
projective surface $S$, such that its canonical map
$\varphi=\varphi_{|K_{\mathcal{F}}|}$ induces a fibration $f:\,S\to B$ as
above. Suppose that $S$ is of general type and that $K_{\mathcal{F}}\cdot
F=1$. Then

$p_{g}(\mathcal{F})=g(B).$ (4.1)

More precisely, if $g(B)\geq 1$, then

$|K_{\mathcal{F}}|=|f^{*}K_{B}|+C_{0}+Z_{v},$ (4.2)

such that $|f^{*}K_{B}|$ (resp. $C_{0}+Z_{v}$) is the moving part (resp. fixed
part) of $|K_{\mathcal{F}}|$, where $C_{0}\subseteq S$ is a section of $f$,
and $Z_{v}$ is effective whose support is contained in fibers of $f$. Here we
understand that the moving part $|f^{*}K_{B}|$ is empty if $g(B)=1$.

We divide the key points of the proof of Theorem 4.1 into several propositions
as follows.

######  Proposition .

Let $\mathcal{F}$ be a reduced foliation on $S$ and $f:\,S\to B$ be a
fibration on $S$. Suppose that $g(F)\geq 2$ and $K_{\mathcal{F}}\cdot F=1$,
where $F$ is a general fiber of $f$. Then the foliation $\mathcal{F}$ is
different from the foliation $\mathcal{G}$ defined by taking the saturation of
$\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$.

######  Proposition .

Let $\mathcal{F}$ be a reduced foliation on $S$ and $f:\,S\to B$ be a
fibration on $S$. Suppose that $\mathcal{F}$ is different from the foliation
$\mathcal{G}$ defined by taking the saturation of $\ker(\mathrm{d}f:\,T_{S}\to
f^{*}T_{B})$ in $T_{S}$. Then the canonical divisor $K_{\mathcal{F}}$ has the
following form

$K_{\mathcal{F}}=f^{*}K_{B}+Z,$ (4.3)

where $Z$ is effective.

######  Proposition .

Let $\mathcal{F}$ be a reduced foliation on $S$ and $f:\,S\to B$ be a
fibration on $S$. Suppose that $g(F)\geq 2$ and $K_{\mathcal{F}}\cdot F=1$,
where $F$ is a general fiber of $f$. Let $Z$ be the effective divisor as in
(4.3). Then

$Z=C_{0}+\sum_{i=1}^{\ell}\sum_{j=1}^{r_{i}}n_{ij}C_{ij},$ (4.4)

where $C_{0}\subseteq S$ is a section of $f$, and $C_{ij}$’s are all contained
in fibers of $f$. Moreover, the coefficients $n_{ij}$’s satisfy the following
property: if $F_{i}=\sum\limits_{j=1}^{r_{i}}m_{ij}C_{ij}$ is a fiber of $f$,
then there exists at least one $1\leq j\leq r_{i}$ such that $n_{ij}<m_{ij}$.

We postpone the proofs of the above three propositions, and first prove
Theorem 4.1 based on them.

###### Proof of Theorem 4.1.

Since $S$ is of general type, the genus $g(F)\geq 2$ for a general fiber $F$
of $f:\,S\to B$, cf. [BHPV04, § V]. Thus the foliation $\mathcal{F}$ is
different from the foliation $\mathcal{G}$ defined by taking the saturation of
$\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$ by Section 4. According
to Section 4 and Section 4,

$K_{\mathcal{F}}=f^{*}K_{B}+C_{0}+Z_{v},\qquad\text{with~{}}Z_{v}=\sum_{i=1}^{\ell}\sum_{j=1}^{r_{i}}n_{ij}C_{ij},$
(4.5)

where $C_{0}\subseteq S$ is a section of $f$, and $C_{ij}$’s are all contained
in fibers of $f$. In particular,

$p_{g}(\mathcal{F})=h^{0}(S,K_{\mathcal{F}})\geq h^{0}(B,K_{B})=g(B).$

As $g(F)\geq 2$ for a general fiber $F$ of $f$, the section $C_{0}$ is
certainly contained in the fixed part of $|K_{\mathcal{F}}|$. Hence

$h^{0}(S,K_{\mathcal{F}})=h^{0}(S,f^{*}K_{B}+Z_{v}).$

According to Section 4, the coefficients $n_{ij}$’s satisfy the following
property: if $F_{i}=\sum\limits_{j=1}^{r_{i}}m_{ij}C_{ij}$ is a fiber of $f$,
then there exists at least one $1\leq j\leq r_{i}$ such that $n_{ij}<m_{ij}$.
This means that $h^{0}(S,\,Z_{v})=1$, and that

$f_{*}\mathcal{O}_{S}(Z_{v})=\mathcal{O}_{B}.$

Therefore,

$h^{0}(S,K_{\mathcal{F}})=h^{0}(S,f^{*}K_{B}+Z_{v})=h^{0}\big{(}B,K_{B}\otimes
f_{*}\mathcal{O}_{S}(Z_{v})\big{)}=h^{0}(B,K_{B})=g(B).$

This proves (4.1), from which together with (4.5) the equality (4.2) follows.
∎

We now come back to the proofs of Propositions 4-4. A key observation is the
following.

######  Lemma .

Let $\mathcal{F}$ be a reduced foliation on $S$ and $f:\,S\to B$ be a
fibration on $S$. Suppose that $g(F)\geq 2$ and $K_{\mathcal{F}}\cdot F=1$,
where $F$ is a general fiber of $f$. Let $F_{0}$ be any fiber of $f:\,S\to B$.
Then there is at least one component $C\subseteq F_{0}$ which is not
$\mathcal{F}$-invariant. In particular, any smooth fiber of $f$ is not
$\mathcal{F}$-invariant.

###### Proof.

We prove by contradiction. Suppose that there exists a fiber
$F_{0}=\sum\limits_{j=1}^{r}m_{j}C_{j}$, such that every component $C_{j}$ is
$\mathcal{F}$-invariant. Since $\mathcal{F}$ is reduced, the fiber $F_{0}$ is
normal crossing. According to Section 2, for any component $C_{j}\subseteq
F_{0}$,

$Z(\mathcal{F},C_{j})=K_{\mathcal{F}}C_{j}-K_{S}C_{j}-C_{j}^{2}.$ (4.6)

By definition, if $p\in C_{j}$ is a singularity of $\mathcal{F}$, then
$Z(\mathcal{F},C_{j},p)\geq 1$. Hence

$Z(\mathcal{F},C_{j})\geq\sum_{i\neq j}C_{i}C_{j}.$

Combining this with (4.6),

$\sum_{i\neq j}C_{i}C_{j}\leq
K_{\mathcal{F}}C_{j}-K_{S}C_{j}-C_{j}^{2},\qquad\text{for any fixed~{}}1\leq
j\leq r.$

It follows that

$\displaystyle\sum_{i=1}^{r}\sum_{j\neq
i}m_{j}C_{i}C_{j}=\sum_{j=1}^{r}\sum_{i\neq j}m_{j}C_{i}C_{j}$
$\displaystyle\,\leq\sum_{j=1}^{r}\left(m_{j}K_{\mathcal{F}}C_{j}-m_{j}K_{S}C_{j}-m_{j}C_{j}^{2}\right)$
$\displaystyle\,=(K_{\mathcal{F}}-K_{S})\cdot\sum_{j=1}^{r}m_{j}C_{j}-\sum_{j=1}^{r}m_{j}C_{j}^{2}$
$\displaystyle\,=(K_{\mathcal{F}}-K_{S})\cdot
F_{0}-\sum_{i=1}^{r}m_{i}C_{i}^{2}$
$\displaystyle\,=1-(2g(F)-2)-\sum_{i=1}^{r}m_{i}C_{i}^{2}.$

Thus

$1-(2g(F)-2)\geq\sum_{i=1}^{r}\left(\sum_{j\neq
i}m_{j}C_{i}C_{j}+m_{i}C_{i}^{2}\right)=\sum_{i=1}^{r}F_{0}\cdot C_{i}=0.$

This gives a contradiction if $g(F)\geq 2$. ∎

###### Proof of Section 4.

This follows directly from Section 4. Indeed, if $\mathcal{F}$ is the same as
the foliation $\mathcal{G}$ defined by taking the saturation of
$\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$, then every fiber of $f$
is $\mathcal{F}$-invariant, which contradicts Section 4. ∎

###### Proof of Section 4.

Let $\omega_{i}$ be any local holomorphic one-form on the base curve $B$.
Since $\mathcal{F}$ is different from the foliation $\mathcal{G}$ defined by
taking the saturation of $\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$,
the contraction of $f^{*}(\omega_{i})$ with the local field $v_{i}$ defining
$\mathcal{F}$ gives a non-zero local section of $K_{\mathcal{F}}$. Globally,
it is just the following contraction map

$\begin{aligned}
H^{0}\big{(}S,T_{S}(T_{\mathcal{F}}^{*})\big{)}&\quad\otimes&&f^{*}H^{0}(B,\Omega^{1}_{B})&\longrightarrow\quad&H^{0}(S,K_{\mathcal{F}})\\\
\big{(}v,&&&f^{*}\omega\big{)}&\mapsto\,\quad&(v,f^{*}\omega),\end{aligned}\qquad$

where $v=(U_{i},v_{i})$ is a local vector field defining $\mathcal{F}$, which
can be viewed as a section of
$H^{0}\big{(}S,T_{S}(T_{\mathcal{F}}^{*})\big{)}$ by (2.1), and
$\omega=(V_{i},\omega_{i})\in H^{0}(B,\Omega^{1}_{B})$ with $f(U_{i})\subseteq
V_{i}$. Remark that the above contraction map also makes sense if $\omega$ is
a rational one-form on $B$ (in this case, the image $(v,f^{*}\omega)$ would be
a rational section of $K_{\mathcal{F}}$ in general). To be concrete, let
$t_{i}$ be a local coordinate of $B$ on $V_{i}$. Then $(V_{i},dt_{i})$ defines
a holomorphic section of $\omega\in H^{0}(B,\Omega^{1}_{B}(-K_{B}))$, i.e.,
$\omega$ is a twist one-form on $B$. Hence

$\alpha:=(v,f^{*}\omega)=\big{\\{}\big{(}U_{i},\,(v_{i},f^{*}dt_{i})\big{)}\big{\\}}\in
H^{0}(S,K_{\mathcal{F}}\otimes f^{*}(-K_{B})).$ (4.7)

In other words,

$K_{\mathcal{F}}-f^{*}K_{B}=\text{\rm{div\,}}(\alpha).$

By construction, $\alpha$ is locally holomorphic, and hence
$Z=\text{\rm{div\,}}(\alpha)$ is effective. This proves (4.3). ∎

###### Proof of Section 4.

Since $g(F)\geq 2$ and $K_{\mathcal{F}}\cdot F=1$, by Section 4 the foliation
$\mathcal{F}$ is different from the foliation $\mathcal{G}$ defined by taking
the saturation of $\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$.
According to the proof of Section 4 above, we have to determine the divisor
$Z=\text{\rm{div\,}}(\alpha)$ in (4.7).

Since $K_{\mathcal{F}}\cdot F=1$, the horizontal part of
$\text{\rm{div\,}}(\alpha)$ consists of exactly one section of $f$, which we
denote by $C_{0}$. It remains to decide the vertical part of
$\text{\rm{div\,}}(\alpha)$. Let $F_{i}=\sum\limits_{j=1}^{r_{i}}m_{ij}C_{ij}$
be any fiber of $f$, and

$\text{\rm{div\,}}(\alpha)=\sum_{j=1}^{r_{i}}n_{ij}C_{ij}+Z^{\prime},$

where the support of $Z^{\prime}$ does not contain any component of $F_{i}$.
Then we have to show that

there exists at least one $1\leq j\leq r_{i}$ such that $n_{ij}<m_{ij}$. (4.8)

Let $p_{ij}\in C_{ij}$ be a general point of $C_{ij}$. As $p_{ij}\in C_{ij}$
is general, there exist a local coordinate $(x,y)$ around $p_{ij}$ such that
$C_{ij}=\\{x=0\\}$, and a local coordinate $t$ around $q=f(F_{i})$ such that
map $f$ is defined by $t=x^{m_{ij}}$. Suppose that
$v=h(x,y)\frac{\partial}{\partial x}+g(x,y)\frac{\partial}{\partial y}$ around
$p_{ij}$. Then around the point $p_{ij}$, $f^{*}dt=m_{ij}x^{m_{ij}-1}$, and
hence

$\alpha=\Big{(}h(x,y)\frac{\partial}{\partial
x}+g(x,y)\frac{\partial}{\partial y},f^{*}dt\Big{)}=m_{ij}h(x,y)x^{m_{ij}-1}.$

Therefore,

$n_{ij}=m_{ij}-1+\text{order}_{x}\big{(}h(x,y)\big{)},$

where

$\text{order}_{x}\big{(}h(x,y)\big{)}=\max\big{\\{}k~{}|~{}x^{k}\text{~{}divides~{}}h(x,y)\\}.$

It follows that

$n_{ij}\geq m_{ij},\quad\Longleftrightarrow\quad
x\text{~{}divides~{}}h(x,y),\quad\Longleftrightarrow\quad\text{$C_{ij}$ is
$\mathcal{F}$-invariant}.$

Hence (4.8) follows immediately from Section 4. This completes the proof. ∎

##  5\. The proof of the Noether type inequalities

In this section, we prove the Noether type inequalities for a foliated surface
of general type. As illustrated in Section 1, it relies on the canonical map
$\varphi_{|K_{\mathcal{F}}|}$, where

$\varphi_{|K_{\mathcal{F}}|}:\,S\dashrightarrow\Sigma\subseteq\mathbb{P}^{p_{g}(\mathcal{F})-1},$
(5.1)

is the rational map defined by $|K_{\mathcal{F}}|$. Theorem 1.1 will be proved
in Section 5 if $\Sigma$ is of dimension two, and in Section 5 if $\Sigma$ is
of dimension one.

Before going to the proof, let’s do some preparations. As $\mathcal{F}$ is of
general type, $\operatorname*{vol}(\mathcal{F})>0$. Hence we will always
assume in this section that

$p_{g}(\mathcal{F})=h^{0}(S,K_{\mathcal{F}})\geq 3.$

Given any foliated surface $(S,\mathcal{F})$ with reduced singularities, by
[Bru04, Proposition 5.1], one can contract $\mathcal{F}$-exceptional curves to
obtain a relatively minimal foliated surface
$(S^{\prime},\mathcal{F}^{\prime})$ such that
$\operatorname*{vol}(\mathcal{F}^{\prime})=\operatorname*{vol}(\mathcal{F})$
and $p_{g}(\mathcal{F}^{\prime})=p_{g}(\mathcal{F})$. Therefore, we may assume
that $\mathcal{F}$ is relatively minimal as well. Let

$|K_{\mathcal{F}}|=|M|+Z,$ (5.2)

be the decomposition of the complete linear system $|K_{\mathcal{F}}|$, where
$Z$ is the fixed part. Let $\rho:\,Y\to\Sigma$ be the desingularization of
$\Sigma$, where $\Sigma=\varphi_{|K_{\mathcal{F}}|}(S)$ is the image of $S$
under the canonical map as in (5.1). By a sequence of blowing-ups
$\sigma:\,\widetilde{S}\to S$ centered on the base points of $|M|$, we obtain
a well-defined morphism $\phi:\,\widetilde{S}\to Y$ with the following
diagram.

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\sigma}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi=\varphi_{|K_{\mathcal{F}}|}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$desingularization$\textstyle{\Sigma\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}(\mathcal{F})-1}}$

There are two possiblities of the dimension of the image $\Sigma$:
$\dim\Sigma=2$ or $\dim\Sigma=1$.

######  Proposition .

Let $(S,\mathcal{F})$ be a foliated surface of general type with reduced
singularities. Suppose that the image $\Sigma=\varphi_{|K_{\mathcal{F}}|}(S)$
is of dimension two. Then

$\left\\{\begin{aligned} &\operatorname*{vol}(\mathcal{F})\geq
p_{g}(\mathcal{F})-2;\\\ &\operatorname*{vol}(\mathcal{F})\geq
2p_{g}(\mathcal{F})-4,&~{}&\text{if moreover $S$ is of general
type.}\end{aligned}\right.$ (5.3)

Moreover, if the equality in any of the above two inequalities holds, then the
image $\Sigma$ is a surface of minimal degree $($equal to
$p_{g}(\mathcal{F})-2)$ in $\mathbb{P}^{p_{g}(\mathcal{F})-1}$, and
$\deg(\varphi)=1$ (resp. $\deg(\varphi)=2$) if the equality in the first
(resp. second) inequality holds.

###### Proof.

By construction, the map

$\rho\circ\phi:\,\widetilde{S}\longrightarrow\Sigma\hookrightarrow\mathbb{P}^{N},$

is defined by the complete linear system $|\widetilde{M}|$ with
$\widetilde{M}=\sigma^{*}M-\sum a_{j}\mathcal{E}_{j}$. According to [Bea79,
Lemme 1.2],

$M^{2}\geq\widetilde{M}^{2}\geq\deg(\varphi)\cdot\deg(\Sigma).$

On the other hand, let

$K_{\mathcal{F}}=P+N,$

be the Zariski decomposition of $K_{\mathcal{F}}$, where $P$ is nef and $N$ is
negative. Then

$\operatorname*{vol}(\mathcal{F})=P^{2}\geq
M^{2}\geq\widetilde{M}^{2}\geq\deg(\varphi)\cdot\deg(\Sigma)\geq
p_{g}(\mathcal{F})-2.$

The last equality follows from [Bea79, Lemme 1.4]. Moreover, if the equality
holds, then $\deg(\varphi)=1$ and $\deg(\Sigma)=p_{g}(\mathcal{F})-2$ in
$\mathbb{P}^{p_{g}(\mathcal{F})-1}$.

Suppose moreover that $S$ is of general type. If $\deg(\varphi)=1$, then
$\Sigma$ is a surface of general type birational to $S$, and hence by [Bea79,
Lemme 1.4] and [Bea79, Remarque 1.5] one obtains that

$\operatorname*{vol}(\mathcal{F})\geq\deg(\varphi)\cdot\deg(\Sigma)=\deg(\Sigma)>2\big{(}p_{g}(\mathcal{F})-2\big{)}.$

If $\deg(\varphi)\geq 2$, then

$\operatorname*{vol}(\mathcal{F})\geq\deg(\varphi)\cdot\deg(\Sigma)\geq
2\big{(}p_{g}(\mathcal{F})-2\big{)}.$

The equality holds implies that $\deg(\varphi)=2$ and
$\deg(\Sigma)=p_{g}(\mathcal{F})-2$. This completes the proof. ∎

######  Remark .

According to [Bea79, Lemme 1.4], for any non-ruled surface
$\Sigma\subseteq\mathbb{P}^{p_{g}(\mathcal{F})-1}$, it holds

$\deg(\Sigma)\geq 2p_{g}(\mathcal{F})-4.$

Hence by the above proof, we have actually proved that

$\operatorname*{vol}(\mathcal{F})\geq 2p_{g}(\mathcal{F})-4,$

for any foliation $\mathcal{F}$ of general type with reduced singularities, if
$S$ is not a ruled surface and the image
$\Sigma=\varphi_{|K_{\mathcal{F}}|}(S)$ is of dimension two. Moreover, if the
equality holds, then

  1. (i).

either $\deg(\varphi)=1$ and $\deg(\Sigma)=2p_{g}(\mathcal{F})-4$;

  2. (ii).

either $\deg(\varphi)=2$ and $\deg(\Sigma)=p_{g}(\mathcal{F})-2$.

In the rest part of this section, we will assume that $\dim\Sigma=1$. In this
case, by the Stein factorization, we obtain a diagram as follows.

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\scriptstyle{\sigma}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi=\varphi_{|K_{\mathcal{F}}|}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$desingularization$\textstyle{\Sigma}$

Here $\pi:\,B\to Y$ is finite, and $f:\,\widetilde{S}\to B$ is a family of
curves with connected fibers. By construction, the map

$\rho\circ\phi:\,\widetilde{S}\longrightarrow\Sigma\hookrightarrow\mathbb{P}^{N},$

is defined by the complete linear system $|\widetilde{M}|$, where
$|\widetilde{M}|$ is obtained by blowing-up the base points of $|M|$. Since
$|\widetilde{M}|$ is base-point-free and induces a fibration
$f:\,\widetilde{S}\to B$, it follows that

$p_{g}(\mathcal{F})=h^{0}(S,M)=h^{0}(\widetilde{S},\widetilde{M})=h^{0}\big{(}B,f_{*}\mathcal{O}_{\widetilde{S}}(\widetilde{M})\big{)}.$

According to the Riemann-Roch theorem,

$\deg(L)\geq p_{g}(\mathcal{F})-1,$

where $L=f_{*}\mathcal{O}_{\widetilde{S}}(\widetilde{M})$ is a line bundle on
$B$. Note that $\widetilde{M}=f^{*}(L)$. Hence, we have numerically,

$\widetilde{M}\equiv_{num}\deg(L)\,F,\qquad\text{with~{}}\deg(L)\geq
p_{g}(\mathcal{F})-1,$ (5.4)

where $F$ is a general fiber of $f$.

######  Lemma .

If the linear system $|M|$ has a base point, then

$\operatorname*{vol}(\mathcal{F})\geq\big{(}p_{g}(\mathcal{F})-1\big{)}^{2}>2p_{g}(\mathcal{F})-4.$

###### Proof.

Let $A=\sigma_{*}(F)$. If $|M|$ has a base point, then $A^{2}\geq 1$. Hence

$\operatorname*{vol}(\mathcal{F})\geq P^{2}\geq
M^{2}\geq\big{(}(p_{g}(\mathcal{F})-1)A\big{)}^{2}\geq\big{(}p_{g}(\mathcal{F})-1\big{)}^{2}\qed$

We assume from now on that $|M|$ is base-point-free. In other words,
$\varphi:\,S\to\Sigma$ is already a morphism. Hence one gets a commutative
diagram as follows.

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\scriptstyle{\varphi=\varphi_{|K_{\mathcal{F}}|}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$desingularization$\textstyle{\Sigma\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}(\mathcal{F})-1}}$

######  Proposition .

Let $(S,\mathcal{F})$ be a foliated surface of general type with reduced
singularities. Suppose that the canonical map $\varphi_{|K_{\mathcal{F}}|}$
induces a fibration $f:\,S\to B$ as above. Let $F$ be a general fiber of $f$.
Then $K_{\mathcal{F}}\cdot F\geq 1$.

  1. (i).

The following inequalities holds.

$\operatorname*{vol}(\mathcal{F})>\left\\{\begin{aligned}
&\,2p_{g}(\mathcal{F})-4,&&\text{if~{}}K_{\mathcal{F}}\cdot F\geq 2;\\\
&\,p_{g}(\mathcal{F})-2,&&\text{if~{}}K_{\mathcal{F}}\cdot
F=1.\end{aligned}\right.$ (5.5)

  2. (ii).

If $K_{\mathcal{F}}\cdot F=1$ and $S$ is also of general type, then

$\operatorname*{vol}(\mathcal{F})\geq 2p_{g}(\mathcal{F})-2.$ (5.6)

###### Proof.

Since $\mathcal{F}$ is of general type, i.e., $K_{\mathcal{F}}$ is big, it
follows that $K_{\mathcal{F}}\cdot F\geq 1$.

(i). As explained at the beginning of this section, we may assume that the
foliation $\mathcal{F}$ is relatively minimal. By Section 5, we may assume
that $|M|$ is base-point-free. The relation in (5.4) rephrases as

$M\equiv_{num}\deg(L)\,F,\qquad\text{with~{}}\deg(L)\geq
p_{g}(\mathcal{F})-1.$

The moving part $M$ is clear nef, from which it follows that $M\leq P$, or
equivalently $N\leq Z$, where $P$ is the nef part and $N$ is the negative part
of $K_{\mathcal{F}}$ in its Zariski decomposition, and $Z$ is the fixed part
of $|K_{\mathcal{F}}|$ as in (5.2). Hence

$\operatorname*{vol}(\mathcal{F})=P^{2}\geq P\cdot
M\geq\big{(}p_{g}(\mathcal{F})-1\big{)}P\cdot
F=\big{(}p_{g}(\mathcal{F})-1\big{)}\,(K_{\mathcal{F}}-N)\cdot F.$ (5.7)

Therefore, it suffices to prove a lower bound on $P\cdot F$, or equivalently
an upper bound on $N\cdot F$.

The fibration $f:\,S\to B$ defines a natural foliation $\mathcal{G}$ by taking
the saturation of the kernel $\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in
$T_{S}$. The given foliation $\mathcal{F}$ may be equal to or different from
$\mathcal{G}$.

Suppose that $\mathcal{F}=\mathcal{G}$. Then the genus $g(F)\geq 2$ since
$\mathcal{F}$ is assumed to be of general type. Moreover, the canonical
divisor is easy to compute:

$K_{\mathcal{F}}=K_{\mathcal{G}}=K_{S}\otimes
f^{*}K_{B}^{-1}\otimes\mathcal{O}_{S}\Big{(}\sum(1-a_{i})C_{i}\Big{)},$

where the sum is taken over all components in fibers of $f$, and $a_{i}$ is
the multiplicity of $C_{i}$ in its fiber. Note that the support of the
negative part $N$ is a sum of $\mathcal{F}$-chains (cf. Theorem 3.1), and thus
contained in fibers of $f$. Combining this with (5.7),

$\displaystyle\operatorname*{vol}(\mathcal{F})$
$\displaystyle\,\geq\big{(}p_{g}(\mathcal{F})-1\big{)}\,(K_{\mathcal{F}}-N)\cdot
F$ $\displaystyle\,=\big{(}p_{g}(\mathcal{F})-1\big{)}\,K_{\mathcal{F}}\cdot
F$ $\displaystyle\,=\big{(}2g(F)-2)\big{)}(p_{g}(\mathcal{F})-1)\geq
2(p_{g}(\mathcal{F})-1).$

In the rest part of the proof, we will always assume that $\mathcal{F}$ is
different from the foliation $\mathcal{G}$ defined by taking the saturation of
the kernel $\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{B})$ in $T_{S}$. By (5.7), it
suffices to prove that

$P\cdot F>\left\\{\begin{aligned}
&\,\frac{2\big{(}p_{g}(\mathcal{F})-2\big{)}}{p_{g}(\mathcal{F})-1},&&\text{if~{}}K_{\mathcal{F}}\cdot
F\geq 2;\\\\[2.0pt]
&\,\frac{p_{g}(\mathcal{F})-2}{p_{g}(\mathcal{F})-1},&&\text{if~{}}K_{\mathcal{F}}\cdot
F=1.\end{aligned}\right.$ (5.8)

Let $Z=Z_{h}+Z_{v}$, where $Z$ is the fixed part of $|K_{\mathcal{F}}|$ as in
(5.2), and each component in $Z_{v}$ is contained in fibers of $f$, while each
component in $Z_{h}$ maps surjectively to the base $B$. Similarly, we can
decompose the negative as $N=N_{h}+N_{v}$. Let

$\displaystyle Z_{h}=\sum a_{C}C,$ $\displaystyle Z_{v}=\sum a_{D}D,$
$\displaystyle N_{h}=\sum b_{C}C,$ $\displaystyle N_{v}=\sum b_{D}D.$

Then the coefficients $\\{a_{C},a_{D}\\}$’s are positive integers; while
$\\{b_{C},b_{D}\\}$’s belong to $[0,1)$ by Theorem 3.1, since the foliation
$\mathcal{F}$ is assumed to be relatively minimal.

Suppose first that there exists an irreducible component $C_{0}\subseteq
Z_{h}$ with $C_{0}^{2}=-2$ and $b_{C_{0}}>0$ (i.e., $C_{0}$ is contained in
the support of $N$). Then

$0=P\cdot
C_{0}\geq\Big{(}\big{(}a_{C_{0}}-b_{C_{0}}\big{)}C_{0}+\big{(}p_{g}(\mathcal{F})-1\big{)}F\Big{)}\cdot
C_{0}=-2\big{(}a_{C_{0}}-b_{C_{0}})+\big{(}p_{g}(\mathcal{F})-1\big{)}F\cdot
C_{0}.$

It follows that

$a_{C_{0}}-b_{C_{0}}\geq\frac{\big{(}p_{g}(\mathcal{F})-1\big{)}F\cdot
C_{0}}{2}.$

Then

$P\cdot F\geq
a_{C_{0}}-b_{C_{0}}\geq\frac{p_{g}(\mathcal{F})-1}{2}\geq\frac{2\big{(}p_{g}(\mathcal{F})-2\big{)}}{p_{g}(\mathcal{F})-1}.$
(5.9)

It remains to show that the above inequality is strict if
$K_{\mathcal{F}}\cdot F\geq 2$. Indeed, if $P\cdot
F=\frac{2\big{(}p_{g}(\mathcal{F})-2\big{)}}{p_{g}(\mathcal{F})-1}$, then
$p_{g}(\mathcal{F})=3$ and

$P\cdot F=a_{C_{0}}-b_{C_{0}}=F\cdot C_{0}=1.$

Note that $b_{C_{0}}<1$ by Theorem 3.1, and $b_{C_{0}}>0$ by assumption. This
gives a contradiction, since $a_{C_{0}}$ is an integer. Hence the inequality
(5.9) is strict as required.

We can now assume that for any component $C\subseteq D_{h}$, either $C$ is not
contained in the support of $N$, or $-C^{2}\geq 3$. In other words, every
possible irreducible curve $C$, which maps surjectively onto the base $C$ and
occurs in the negative part $N$, satisfies that $C^{2}\leq-3$. We will prove
under the above assumption that

$b_{C}<\frac{a_{C}}{p_{g}(\mathcal{F})-1}.$ (5.10)

To prove (5.10), we may assume that $b_{C}>0$, i.e., $C$ is contained in the
support of the negative part $N$, which is a union of maximal
$\mathcal{F}$-chains. Suppose that $C$ is contained in the maximal
$\mathcal{F}$-chain $\sum\limits_{j=1}^{r}C_{j}$. If $C$ is not the first
component $C_{1}$ in the maximal $\mathcal{F}$-chain, then

$0=K_{\mathcal{F}}\cdot C\geq
a_{C}C^{2}+\big{(}p_{g}(\mathcal{F})-1\big{)}F\cdot C\geq
a_{C}C^{2}+\big{(}p_{g}(\mathcal{F})-1\big{)}.$

Combining this with (3.4), one gets

$b_{C}<\frac{1}{-C^{2}}\leq\frac{a_{C}}{p_{g}(\mathcal{F})-1}.$

If $C$ is the first component $C_{1}$ in the maximal $\mathcal{F}$-chain, then
there are two possibilities: either the maximal $\mathcal{F}$-chain consists
of exactly one component $C_{1}=C$, or there is another curve, say
$C^{\prime}$, contained in the negative part $N$ intersecting $C_{1}=C$. If it
is the first case, then

$\left\\{\begin{aligned} &-1=N\cdot C=b_{C}C^{2},\\\ &-1=K_{\mathcal{F}}\cdot
C\geq a_{C}C^{2}+\big{(}p_{g}(\mathcal{F})-1\big{)}F\cdot C\geq
a_{C}C^{2}+\big{(}p_{g}(\mathcal{F})-1\big{)}.\end{aligned}\right.$

Hence

$b_{C}=\frac{1}{-C^{2}}\leq\frac{a_{C}}{p_{g}(\mathcal{F})}.$

If it is the second case, then

$-1=K_{\mathcal{F}}\cdot C\geq
a_{C}C^{2}+\big{(}p_{g}(\mathcal{F})-1\big{)}F\cdot C+a_{C^{\prime}}\geq
a_{C}C^{2}+p_{g}(\mathcal{F}),$

where $a_{C^{\prime}}$ is the coefficient of $C^{\prime}$ in $Z$ (as the
component $C^{\prime}$ is in the support of $N$ and $N\leq Z$, it follows that
$C^{\prime}$ is contained in the support of $Z$). Combining this together with
(3.4), one gets

$b_{C}<\frac{1}{-C^{2}}\leq\frac{a_{C}}{p_{g}(\mathcal{F})-1}.$

This completes the proof of (5.10).

Come back to the proof of (5.8). According to (5.10),

$\displaystyle P\cdot F=\sum(a_{C}-b_{C})C\cdot F$
$\displaystyle\,\geq\sum\Big{(}a_{C}-\frac{a_{C}}{p_{g}(\mathcal{F})-1}\Big{)}C\cdot
F$ $\displaystyle\,=\frac{p_{g}(\mathcal{F})-2}{p_{g}(\mathcal{F})-1}\cdot\sum
a_{C}C\cdot F$
$\displaystyle\,=\frac{p_{g}(\mathcal{F})-2}{p_{g}(\mathcal{F})-1}\cdot
K_{\mathcal{F}}\cdot F.$

This proves (5.8), and hence completes the proof of (5.7).

(ii). Since it is assumed that $p_{g}(\mathcal{F})\geq 3$, from Theorem 4.1 it
follows that $g(B)=p_{g}(\mathcal{F})\geq 3$ and the canonical linear system

$|K_{\mathcal{F}}|=|f^{*}K_{B}|+C_{0}+Z_{v},$

such that $|f^{*}K_{B}|$ (resp. $C_{0}+Z_{v}$) is the moving part (resp. fixed
part) of $|K_{\mathcal{F}}|$, where $C_{0}\subseteq S$ is a section of $f$,
and $Z_{v}$ is effective whose support is contained in fibers of $f$. Since
$g(C_{0})=g(B)\geq 3$, the curve $C_{0}$ is not contained in the support of
negative part $N$ in the Zariski decomposition $K_{\mathcal{F}}=P+N$. It
follows that

$\operatorname*{vol}(\mathcal{F})=P^{2}\geq P\cdot
f^{*}K_{B}=(K_{\mathcal{F}}-N)\cdot f^{*}K_{B}=K_{\mathcal{F}}\cdot
f^{*}K_{B}=2g(B)-2=2p_{g}(\mathcal{F})-2.$

This proves (5.6). ∎

###### Proof of Theorem 1.1.

As $\mathcal{F}$ is of general type, $\operatorname*{vol}(\mathcal{F})>0$, and
hence we may assume that $p_{g}(\mathcal{F})\geq 3$. Let
$\varphi=\varphi_{|K_{\mathcal{F}}|}:\,S\dashrightarrow\Sigma\subseteq\mathbb{P}^{p_{g}(\mathcal{F})-1}$
be the rational map defined by $|K_{\mathcal{F}}|$. If the image $\Sigma$ is
of dimension two, it follows from Section 5; if $\Sigma$ is of dimension one,
it follows from Section 5. ∎

##  6\. Examples

In this section, we will construct several examples. Section 6 and Section 6
show that the two Noether type inequalities in Theorem 1.1 are both sharp. In
Section 6 we construct a sequence of reduced foliated surfaces
$(S_{n},\mathcal{F}_{n})$ of general type, such that
$\operatorname*{vol}(\mathcal{F}_{n})>p_{g}(\mathcal{F}_{n})-2$ and that the
difference
$\operatorname*{vol}(\mathcal{F}_{n})-\big{(}p_{g}(\mathcal{F}_{n})-2\big{)}$
tends to zero. This shows a phenomenon different from the case for algebraic
varieties, where there exists so-called ”the second Noether inequality”, cf.
[HZ22]. In Section 6, we will construct a sequence of reduced foliated
surfaces $(S,\mathcal{F})$ of general type, whose canonical map
$\varphi_{|K_{\mathcal{F}}|}$ induces a fibration $f:\,S\to B$ with $g(B)\geq
2$, $g(F)\geq 2$ and $K_{\mathcal{F}}\cdot F=1$, where $F$ is a general fiber
of $f$.

######  Example .

In this example, we construct a sequence of reduced foliated surfaces
$(S_{n},\mathcal{F}_{n})$ of general type, such that the volume
$\operatorname*{vol}(\mathcal{F}_{n})$ and the geometric genus
$p_{g}(\mathcal{F}_{n})$ tend to the infinity and that the following equality
holds

$\operatorname*{vol}(\mathcal{F}_{n})=p_{g}(\mathcal{F}_{n})-2.$ (6.1)

Let $\mathcal{F}_{0}$ be a foliation of degree two on $\mathbb{P}^{2}$ with
reduced singularities. Such a foliation exists, cf. [LS20, Proposition 3.2].
In fact, any foliation of degree $d$ on $\mathbb{P}^{2}$ can be generated by a
vector of the form ([GO89, Bru04])

$v=\big{(}P(x,y)+xR(x,y)\big{)}\frac{\partial}{\partial
x}+\big{(}Q(x,y)+yR(x,y)\big{)}\frac{\partial}{\partial y},$

where $(x,y)$ is an affine coordinate of $\mathbb{P}^{2}$, $P(x,y),Q(x,y)$ are
polynomials of degree $\leq d$, and $R(x,y)$ is a homogeneous polynomial of
degree $d$ (plus some nondegeneracy conditions). Let

$v_{\alpha,\beta}=x\Big{(}-\alpha^{2}+x^{2}+y^{2}\Big{)}\frac{\partial}{\partial
x}+y\Big{(}2\alpha\beta+(2\alpha+\beta)y+x^{2}+y^{2}\Big{)}\frac{\partial}{\partial
y},$ (6.2)

where $\alpha,\beta\in\mathbb{C}\setminus\mathbb{Q}$ are general. The
foliation $\mathcal{F}_{0}$ defined by $v_{\alpha,\beta}$ admits $7$
singularities:

$\left\\{\begin{aligned}
&(0,0),\quad(0,-2\alpha),\quad(0,-\beta),\quad(\alpha,0),\quad(-\alpha,0),\\\
&\Big{(}~{}\frac{\alpha\sqrt{3(\alpha^{2}-\beta^{2})}}{2\alpha+\beta},~{}\frac{-\alpha(\alpha+2\beta)}{2\alpha+\beta}~{}\Big{)},\quad\Big{(}~{}\frac{-\alpha\sqrt{3(\alpha^{2}-\beta^{2})}}{2\alpha+\beta},~{}\frac{-\alpha(\alpha+2\beta)}{2\alpha+\beta}~{}\Big{)}\end{aligned}\right\\}.$

The eigenvalues at these $7$ singularities are respectively equal to

$\left\\{\begin{aligned}
&\lambda_{1}=\frac{-\alpha}{2\beta},\quad\lambda_{2}=\frac{4\alpha-2\beta}{3\alpha},\quad\lambda_{3}=\frac{\beta(\beta-2\alpha)}{\beta^{2}-\alpha^{2}},\quad\lambda_{4}=\lambda_{5}=\frac{\alpha+2\beta}{2\alpha},\\\
&\lambda_{6}=\lambda_{7}=\frac{(\alpha-2\beta)(2\alpha-\beta)-\sqrt{(\alpha-2\beta)^{2}(2\alpha-\beta)^{2}-24\alpha(\alpha+2\beta)(\beta^{2}-\alpha^{2})\,}~{}}{(\alpha-2\beta)(2\alpha-\beta)+\sqrt{(\alpha-2\beta)^{2}(2\alpha-\beta)^{2}-24\alpha(\alpha+2\beta)(\beta^{2}-\alpha^{2})\,}~{}}\end{aligned}\right\\}.$

If $\alpha,\beta$ are sufficiently general (for instance if
$\\{\alpha,\beta\\}\in\mathbb{C}\setminus\mathbb{Q}$ are algebraic independent
over $\mathbb{Q}$), then the eigenvalue
$\lambda_{i}\in\mathbb{C}\setminus\mathbb{Q}$ for $1\leq i\leq 7$, and hence
these are all reduced singularities. Moreover, the two lines
$L_{0}:=\\{x=0\\}$ and $L_{\infty}:=\\{y=0\\}$ are both
$\mathcal{F}_{0}$-invariant in view of (6.2). Let
$\sigma:\,S_{1}\to\mathbb{P}^{2}$ be the blowing-up centered at $(0,0)$, and
$\mathcal{F}_{1}$ be the induced foliation on $S_{1}$. Then $\mathcal{F}_{1}$
is still reduced, and

$K_{\mathcal{F}_{1}}=\sigma^{*}K_{\mathcal{F}_{0}}=\sigma^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)\sim
C_{0}+F,$

where ’$\sim$’ stands for the linear equivalence, $C_{0}=\mathcal{E}\subseteq
S_{1}$ is the unique section (also the exceptional curve of $\sigma$) with
$C_{0}^{2}=-1$, and $F_{1}$ is a general fiber of $f_{1}$. Hence

$\operatorname*{vol}(\mathcal{F}_{1})=\operatorname*{vol}(\mathcal{F}_{0})=1,\qquad
p_{g}(\mathcal{F}_{1})=p_{g}(\mathcal{F}_{0})=3.$

It follows that the foliations $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ (they
are birational to each other) satisfy (6.1). By construction, $S_{1}$ is
isomorphic to the Hirzebruch surface
$\mathbb{P}_{\mathbb{P}^{1}}\big{(}\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)\big{)}$.
Let

$f_{1}:\,S_{1}\to\mathbb{P}^{1}$

be the ruling on $S_{1}$, and $F_{0},F_{\infty}$ be the strict transforms of
the two lines $L_{0},L_{\infty}$ respectively. As both $L_{0}$ and
$L_{\infty}$ are $\mathcal{F}_{0}$-invariant, it follows that both $F_{0}$ and
$F_{\infty}$ are $\mathcal{F}_{1}$-invariant. Let
$\pi_{n}:\,\mathbb{P}^{1}\to\mathbb{P}^{1}$ be the cyclic cover of degree
$n\geq 2$ branched over $f_{1}(F_{0})$ and $f_{1}(F_{\infty})$, and
$S_{n}=S_{1}\times_{\pi_{n}}\mathbb{P}^{1}$ be the fiber product as follows.

$\textstyle{S_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{n}}$$\scriptstyle{f_{n}}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{\mathbb{P}^{1}}$

It is clear that
$S_{n}\cong\mathbb{P}_{\mathbb{P}^{1}}\big{(}\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n)\big{)}$.
Let $\mathcal{F}_{n}$ be the induced foliation on $S_{n}$. Then
$\mathcal{F}_{n}$ is a reduced foliation since the eigenvalues
$\lambda_{i}\in\mathbb{C}\setminus\mathbb{Q}$ at each of the $7$ singularities
of $\mathcal{F}_{0}$. Since both $F_{0}$ and $F_{\infty}$ are
$\mathcal{F}_{1}$-invariant, one obtains that (cf. [Bru04, § 2.3(4)]),

$K_{\mathcal{F}_{n}}=\Pi_{n}^{*}(K_{\mathcal{F}_{1}})=\Pi_{n}^{*}(C_{0}+F_{1})=\Pi_{n}^{-1}(C_{0})+nF_{n},$

where $F_{n}$ is a general fiber of $f_{n}$, and $\Pi_{n}^{-1}(C_{0})$ is the
strict transform of $C_{0}$ in $S_{n}$ satisfying
$\Pi_{n}^{-1}(C_{0})^{2}=-n$. Hence

$\operatorname*{vol}(\mathcal{F}_{n})=n,\qquad p_{g}(\mathcal{F}_{n})=n+2.$

Therefore, the equality (6.1) holds for the foliation $\mathcal{F}_{n}$. This
completes the construction.

######  Example .

In this example, we construct a sequence of reduced foliated surfaces
$(S_{n},\mathcal{F}_{n})$ of general type, such that the volume
$\operatorname*{vol}(\mathcal{F}_{n})$ and the geometric genus
$p_{g}(\mathcal{F}_{n})$ tend to the infinity and that the following equality
holds

$\operatorname*{vol}(\mathcal{F}_{n})=p_{g}(\mathcal{F}_{n})-2+\frac{1}{p_{g}(\mathcal{F}_{n})}.$
(6.3)

Let $n\geq 2$ and
$S_{n}=\mathbb{P}_{\mathbb{P}^{1}}\big{(}\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n)\big{)}$
be the Hirzebruch surface admitting a unique section $C_{0}$ with
$C_{0}^{2}=-n<0$. Let $f:\,S_{n}\to\mathbb{P}^{1}$ be the geometrical ruling
on $S_{n}$, which makes $S_{n}$ as a $\mathbb{P}^{1}$-bundle over the
projective line $\mathbb{P}^{1}$. Note that any $\mathbb{P}^{1}$-bundle over
an affine space is necessarily trivial. One can obtain the Hirzebruch surface
$S_{n}$ by gluing the two trivial $\mathbb{P}^{1}$-bundles
$\mathbb{C}\times\mathbb{P}^{1}$ by

$\mathbb{C}\times\mathbb{P}^{1}\qquad\longrightarrow\qquad\mathbb{C}\times\mathbb{P}^{1},$
$\qquad\big{(}x,\,[Y_{0},Y_{1}]\big{)}\quad\mapsto\quad\left(\frac{1}{x},\,[Y_{0},x^{n}Y_{1}]\right).$

More explicitly, one may obtain $S_{n}$ by gluing four affine spaces as
follows. Let $(x_{i},y_{i})$ be the affine coordinate on
$U_{i}\cong\mathbb{C}^{2}=\mathbb{C}\times\mathbb{C}$ for $1\leq i\leq 4$. The
transition functions on their overlaps are given by

$\left(x_{1},\,y_{1}\right)=\left(x_{2},\,\frac{1}{y_{2}}\right)=\left(\frac{1}{x_{3}},\,x_{3}^{n}y_{3}\right)=\left(\frac{1}{x_{4}},\,\frac{x_{4}^{n}}{y_{4}}\right).$

Moreover, $C_{0}\cap U_{1}=\\{y_{1}=0\\}$.

Let $\mathcal{F}_{n}$ be the foliation on $S_{n}$ defined $(U_{i},v_{i})$,
where

$\left\\{\begin{aligned} &v_{1}=h(x_{1},y_{1})\frac{\partial}{\partial
x_{1}}+y_{1}^{2}g(x_{1},y_{1})\frac{\partial}{\partial y_{1}},\\\
&v_{2}=y_{2}h(x_{2},1/y_{2})\frac{\partial}{\partial
x_{2}}-y_{2}g(x_{2},1/y_{2})\frac{\partial}{\partial y_{2}},\\\
&v_{3}=-x_{3}h(1/x_{3},x_{3}^{n}y_{3})\frac{\partial}{\partial
x_{3}}+\Big{(}ny_{3}h(1/x_{3},x_{3}^{n}y_{3})+x_{3}^{n-1}y_{3}^{2}g(1/x_{3},x_{3}^{n}y_{3})\Big{)}\frac{\partial}{\partial
y_{3}},\\\
&v_{4}=-x_{4}y_{4}h(1/x_{4},x_{4}^{n}/y_{4})\frac{\partial}{\partial
x_{4}}-\Big{(}ny_{4}^{2}h(1/x_{4},x_{4}^{n}/y_{4})+x_{4}^{n-1}y_{4}g(1/x_{4},x_{4}^{n}/y_{4})\Big{)}\frac{\partial}{\partial
y_{4}},\end{aligned}\right.$

where

$h(x_{1},y_{1})=a_{0}+y_{1}\cdot\sum_{i=1}^{n}\tilde{a}_{i}x_{1}^{i};\qquad
g(x_{1},y_{1})=\sum_{j=1}^{n-1}b_{j}x_{1}^{j}+y_{1}\cdot\sum_{j=1}^{n-1}\tilde{b}_{j}x_{1}^{j}.$

Here $a_{i},\tilde{a}_{i},b_{j},\tilde{b}_{j}$ are some complex numbers
satisfying certain non-degenerate conditions to insure these $v_{i}$’s contain
no one-dimensional zeros. Then one checks easily by (2.1) that

$K_{\mathcal{F}_{n}}=C_{0}+(n-1)F,$

where $F$ is a general fiber of the ruling $f:\,S_{n}\to\mathbb{P}^{1}$. It
follows that the Zariski decomposition of $K_{\mathcal{F}}$ is the following.

$K_{\mathcal{F}_{n}}=P+N,$

where $P=(n-1)F+\frac{n-1}{n}C_{0}$ is the nef part, and $N=\frac{1}{n}C_{0}$
is the negative part. Hence

$\operatorname*{vol}(\mathcal{F}_{n})=n-2+\frac{1}{n},\qquad
p_{g}(\mathcal{F}_{n})=n,$

from which the equality (6.3) follows immediately. To complete the
construction, we should insure that the foliation $\mathcal{F}_{n}$ is
reduced. A sufficient condition to ensure a singularity $p\in U_{i}$ of
$\mathcal{F}_{n}$ to be reduced is that both of the eigenvalues
$\lambda_{1},\lambda_{2}$ of $(Dv_{i})(p)$ are non-zero and the quotient
$\lambda_{1}/\lambda_{2}$ is not a positive rational number. This should be
satisfied for a general choice of these complex numbers
$\\{a_{i},\tilde{a}_{i},b_{j},\tilde{b}_{j}\\}$, similar to the situation on
$\mathbb{P}^{2}$, [LS20, Proposition 3.2]. For instance, one can take

$h(x_{1},y_{1})=1+y_{1}x_{1}^{n},\qquad\qquad g(x_{1},y_{1})=\alpha
x_{1}^{n-1}+y_{1},\quad\text{where~{}}\alpha\in\mathbb{C}\setminus\mathbb{Q}.$

The number of singularities of $\mathcal{F}_{n}$ is
$\\#Sing(\mathcal{F})=2n+2.$ There are $2n-1$ singularities in $U_{1}$:

$Sing(\mathcal{F})\cap
U_{1}=\\{p_{i}=(\xi_{i},-\alpha\xi_{i}^{n-1}),\,~{}i=1,\cdots,2n-1\\},\qquad\text{where~{}}\xi_{i}^{2n-1}=\frac{1}{\alpha}.$

By direct computation, the two eigenvalues of $(Dv_{1})(p_{i})$ are

$\frac{(\alpha-n)\pm\sqrt{(\alpha+n)^{2}+4(n-1)\alpha^{2}\xi_{i}^{2n-2}\,}\,}{2\xi_{i}}.$

Hence these $p_{i}$’s are reduced singularities of $\mathcal{F}_{n}$. The rest
three singularities of $\mathcal{F}_{n}$ are all on the fiber

$F_{\infty}=\\{x_{3}=0\\}\cup\\{x_{4}=0\\}.$

To be explicit, the rest three singularities are

$\Big{\\{}(0,0),\Big{(}0,\frac{-n}{n+\alpha}\Big{)}\Big{\\}}\subseteq
U_{3},\quad\text{~{}and~{}}\quad\big{\\{}(0,0)\big{\\}}\subseteq U_{4}.$

The eigenvalues of $\mathcal{F}_{n}$ at these three singularities are
respectively equal to $-n$, $\frac{n(n+\alpha)}{\alpha}$ and $n+\alpha$. Hence
the foliation $\mathcal{F}_{n}$ is reduced as required.

######  Remark .

In the above example, $|K_{\mathcal{F}_{n}}|=|(n-1)F|+C_{0}$, where
$|M|:=|(n-1)F|$ is the moving part and $C_{0}$ is the fixed part of
$|K_{\mathcal{F}}|$. It follows that

$\operatorname*{vol}(\mathcal{F})=n-1+\frac{1}{n}=P\cdot
M<K_{\mathcal{F}}\cdot M.$

######  Example .

In this example, we construct a sequence of reduced foliated surfaces
$(S,\mathcal{F})$ of general type with $S$ being a surface of general type and

$\operatorname*{vol}(\mathcal{F})=2p_{g}(\mathcal{F})-4.$ (6.4)

Let $f:\,S\to\mathbb{P}^{1}$ be a semi-stable fibration of curves of genus
$g=2$, whose slope

$\lambda_{f}:=\frac{K_{S/\mathbb{P}^{1}}^{2}}{\deg
f_{*}\mathcal{O}_{S}(K_{S/\mathbb{P}^{1}})}=2,\qquad\text{where~{}}K_{S/\mathbb{P}^{1}}=K_{S}-f^{*}K_{\mathbb{P}^{1}}.$

Such a semi-stable fibration $f:\,S\to\mathbb{P}^{1}$ exists; indeed, one can
construct such a fibration as follows. Let
$Y=\mathbb{P}_{\mathbb{P}^{1}}\big{(}\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n)\big{)}$
be the Hirzebruch surface with the ruling $h:\,Y\to\mathbb{P}^{1}$, and
$L_{m}=6C_{0}+2m\Gamma$, where $C$ is the section with $C_{0}^{2}=-n$ and
$\Gamma$ is a general fiber of $h$. Then $L_{m}$ is very ample if $m$ is
sufficiently large (in fact $m>3n$ is enough, cf. [Har77, Cor V.2.18]). By the
Bertini theorem [Har77, Thm II.8.18], a general element $R\in|L_{m}|$
satisfies that

  1. (i).

the divisor $R$ is smooth;

  2. (ii).

the restricted map $h|_{R}:\,R\to\mathbb{P}^{1}$ has only simply ramified
points, i.e., the ramification indices are all equal to $2$.

Let $\pi:\,S\to Y$ be the double cover branched over such a general divisor
$R$, and $f=h\circ\pi:\,S\to\mathbb{P}^{1}$ the induced fibration. Then $f$ is
a semi-stable fibration of curves of genus $2$. Moreover, one checks easily
that

$\displaystyle K_{S/\mathbb{P}^{1}}^{2}$
$\displaystyle\,=2(K_{Y/\mathbb{P}^{1}}+R/2)^{2}=4m-6n,$ $\displaystyle\deg
f_{*}\mathcal{O}_{S}(K_{S/\mathbb{P}^{1}})$
$\displaystyle\,=\frac{1}{4}R\cdot\big{(}K_{Y/\mathbb{P}^{1}}+R/2\big{)}=2m-3n.$

Let $\mathcal{F}$ be the foliation on $S$ defined by taking the saturation of
$\ker(\mathrm{d}f:\,T_{S}\to f^{*}T_{\mathbb{P}^{1}})$ in $T_{S}$. Then
$\mathcal{F}$ is reduced, relatively minimal, and
$K_{\mathcal{F}}=K_{S/\mathbb{P}^{1}}$. Hence

$\operatorname*{vol}(\mathcal{F})=K_{\mathcal{F}}^{2}=4m-6n.$

Morevoer, since the Hodge bundle $\deg
f_{*}\mathcal{O}_{S}(K_{S/\mathbb{P}^{1}})$ is of rank two and semi-positive,
it follows that

$p_{g}(\mathcal{F})=h^{0}(S,K_{\mathcal{F}})=h^{0}\big{(}\mathbb{P}^{1},f_{*}\mathcal{O}_{S}(K_{S/\mathbb{P}^{1}})\big{)}=2+\deg
f_{*}\mathcal{O}_{S}(K_{S/\mathbb{P}^{1}})=2m-3n+2.$

Therefore, the Noether equality (6.4) holds for $(S,\mathcal{F})$. Moreover,
if $m>\frac{3n+4}{2}$, then $K_{S}^{2}=K_{S/\mathbb{P}^{1}}^{2}-8>0$, and
hence $S$ would be a surface of general type.

######  Example .

In this example, we construct a sequence of reduced foliated surface
$(S,\mathcal{F})$ of general type, whose canonical map
$\varphi_{|K_{\mathcal{F}}|}$ induces a fibration $f:\,S\to B$ with $g(B)\geq
2$, $g(F)=g\geq 2$, and (where $F$ is a general fiber of $f$)

$\left\\{\begin{aligned} K_{\mathcal{F}}\cdot F&\,=1,\\\
\operatorname*{vol}(\mathcal{F})&\,=4p_{g}(\mathcal{F})-4=4g(B)-4.\end{aligned}\right.$
(6.5)

Let $B$ be a curve of genus $g(B)\geq 2$, and $\psi:\,B\to\mathbb{P}^{1}$ is a
finite cover of degree $2m$ with only simply ramified points, i.e., the
ramification indices are all equal to $2$. Let $\hat{Y}=\mathbb{P}^{1}\times
B$, with two projections $\hat{h}_{1},\hat{h}_{2}$. The curve $B$ can be
embedded into $\hat{Y}$ by mapping $x\in B$ to $(\psi(x),x)\in\hat{Y}$ as a
section of $\hat{h}_{2}$:

|
---|---
$\textstyle{B\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\psi}$$\textstyle{\hat{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}_{2}}$$\scriptstyle{\hat{h}_{1}}$$\textstyle{B}$$\textstyle{\mathbb{P}^{1}}$

Denote by $\hat{D}=i(B)\subseteq\hat{Y}$, and let
$\hat{\Gamma}_{i}=\hat{h}_{1}^{-1}(p_{i})$’s ($1\leq i\leq 2g+1$) be $2g+1$
general fibers of $\hat{h}_{1}$, where $g\geq 0$. Then one sees that
$\hat{R}=\hat{D}+\sum\limits_{i=1}^{2g+1}\hat{\Gamma}_{i}$ is an even divisor,
and hence one can construct a double cover $\hat{\pi}:\,\hat{S}\to\hat{Y}$
branched exactly over $\hat{R}$. The surface $\hat{S}$ is singular. In fact,
each $\hat{\Gamma}_{i}$ intersects $\hat{D}$ transversely at $2m$ points,
since $\hat{\Gamma}_{i}$ is general. One can perform a canonical resolution to
resolve the singularities on $\hat{S}$, cf. [BHPV04, § V.22]. Let
$\sigma:\,Y\to\hat{Y}$ be the birational map by blowing up these intersection
points $\hat{D}\bigcap\sum\limits_{i=1}^{2g+1}\hat{\Gamma}_{i}$, and let
$\pi:\,S\to Y$ be the induced double cover. Then $S$ would be a smooth surface
with following diagram.

|
---|---
$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f=f_{2}}$$\scriptstyle{\rho}$$\scriptstyle{f_{1}}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{h_{2}}$$\textstyle{B}$$\textstyle{\hat{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\scriptstyle{h_{1}}$$\textstyle{\hat{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{h}_{2}}$$\scriptstyle{\hat{h}_{1}}$$\textstyle{\mathbb{P}^{1}}$

The fibration $f_{1}:\,S\to\mathbb{P}^{1}$ defines a foliation $\mathcal{F}$
on $S$ by taking the saturation of $\ker(\mathrm{d}f_{1}:\,T_{S}\to
f_{1}^{*}T_{\mathbb{P}^{1}})$ in $T_{S}$. We want to check that the foliation
$\mathcal{F}$ satisfies our requirements.

Let $D$ (resp. $C$) be the strict transform of $\hat{D}$ in $Y$ (resp. $S$),
and $\Gamma_{i}$ (resp. $\Delta_{i}$) be the strict transform of
$\hat{\Gamma}_{i}$ in $Y$ (resp. $S$). Let $\mathcal{E}_{i}$ be the union of
exceptional curves intersecting $\Gamma_{i}$ ($\mathcal{E}_{i}$ consists of
$2m$ components). Then

$K_{Y}=\sigma^{*}(K_{\hat{Y}})+\sum_{i=1}^{2g+1}\mathcal{E}_{i},\qquad\Gamma_{i}=\sigma^{*}(\hat{\Gamma}_{i})-\mathcal{E}_{i},\qquad
D=\sigma^{*}(\hat{D})-\sum_{i=1}^{2g+1}\mathcal{E}_{i}.$

Hence

$K_{S}=\pi^{*}(K_{Y})+C+\sum_{i=1}^{2g+1}\Delta_{i}=(\sigma\circ\pi)^{*}\Big{(}K_{\hat{Y}}+\frac{1}{2}\big{(}\hat{D}+\sum\limits_{i=1}^{2g+1}\hat{\Gamma}_{i}\big{)}\Big{)}.$
(6.6)

By construction, any singular fiber of $f_{1}$ is normal crossing. Hence the
foliation $\mathcal{F}$ defined by taking the saturation of
$\ker(\mathrm{d}f_{1}:\,T_{S}\to f_{1}^{*}T_{\mathbb{P}^{1}})$ in $T_{S}$ is
reduced. Moreover, the curves $\Delta_{i}$’s are of multiplicity equal to two
in fibers of $f_{1}$, and all the other components in fibers of $f_{1}$ are of
multiplicity one. It follows that

$\displaystyle
K_{\mathcal{F}}=K_{S/\mathbb{P}^{1}}-\sum_{i=1}^{2g+1}\Delta_{i}=\pi^{*}(K_{Y/\mathbb{P}^{1}})+C$
$\displaystyle\,=(\sigma\circ\pi)^{*}(K_{\hat{Y}/\mathbb{P}^{1}})+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C$
$\displaystyle\,=(\hat{h}_{2}\circ\sigma\circ\pi)^{*}(K_{B})+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C$
$\displaystyle\,=f^{*}K_{B}+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C,$

where $\overline{\mathcal{E}}_{i}\subseteq S$ is the strict transform of
$\mathcal{E}_{i}$. Let $F$ be a general fiber $f=f_{2}:\,S\to B$. Then
$\pi(F)\cong\mathbb{P}^{1}$ and $F$ is double cover of $\pi(F)$ branched over
$2g+2$ points (the branched divisor is
$\big{(}D+\sum\limits_{i=1}^{2g+1}\Gamma_{i}\big{)}\cap\pi(F)$). By the
Hurwitz formula, one obtains

$g(F)=g.$

Moreover,

$K_{\mathcal{F}}\cdot
F=\Big{(}f^{*}K_{B}+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C\Big{)}\cdot
F=C\cdot F=1.$

Since $C\cdot F=1$ and $g(F)=g\geq 2$, the curve $C$ must be contained in the
fixed part of the linear system $|K_{\mathcal{F}}|$. It follows that

$p_{g}(\mathcal{F})=h^{0}(S,K_{\mathcal{F}})=h^{0}\Big{(}S,f^{*}K_{B}+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}\Big{)}=h^{0}(B,K_{B})=g(B).$

In fact, one proves moreover that

$|K_{\mathcal{F}}|=|f^{*}K_{B}|+Z,$

where $Z=\sum\limits_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C$ is the fixed
part of $|K_{\mathcal{F}}|$. In particular, the canonical map
$\varphi_{|K_{\mathcal{F}}|}$ is nothing but the same as the map
$\varphi_{|f^{*}K_{B}|}$ defined by the linear system $|f^{*}K_{B}|$. Hence
the canonical map $\varphi_{|K_{\mathcal{F}}|}$ induces the fibration
$f:\,S\to B$ as required. It remains to check the equality

$\operatorname*{vol}(\mathcal{F})=4g(B)-4.$ (6.7)

To this purpose, we first compute the Zariski decomposition of
$K_{\mathcal{F}}$. By construction, every irreducible component in
$\sum\limits_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}$ is a smooth rational
curve with self-intersection $-2$. Moreover,

$(\sigma\circ\pi)^{*}(\hat{D})=\pi^{*}\Big{(}D+\sum_{i=1}^{2g+1}\mathcal{E}_{i}\Big{)}=2C+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}.$

Hence

$K_{\mathcal{F}}=f^{*}K_{B}+\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}+C=P+N,$

where

$P=(\sigma\circ\pi)^{*}\Big{(}\hat{h}_{2}^{*}(K_{B})+\frac{\hat{D}}{2}\Big{)},\qquad
N=\frac{1}{2}\sum_{i=1}^{2g+1}\overline{\mathcal{E}}_{i}.$

One checks easily that this is the Zariski decomposition of $K_{\mathcal{F}}$.
Hence

$\operatorname*{vol}(\mathcal{F})=P^{2}=2\Big{(}\hat{h}_{2}^{*}(K_{B})+\frac{\hat{D}}{2}\Big{)}^{2}=2(2g(B)-2).$

This proves (6.7). Finally, By (6.6), one computes that

$K_{S}^{2}=4(g-1)\big{(}2g(B)-2+m\big{)}>0.$

In particular, $S$ is of general type.

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# The Observed Rate of Binary Black Hole Mergers can be Entirely Explained by
Globular Clusters

Carl L. Rodriguez McWilliams Center for Cosmology and Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USA  Kyle Kremer TAPIR,
California Institute of Technology, Pasadena, CA 91125, USA  The Observatories
of the Carnegie Institution for Science, Pasadena, CA 91101, USA  Sourav
Chatterjee Tata Institute of Fundamental Research, Homi Bhabha Road, Navy
Nagar, Colaba, Mumbai 400005, India  Giacomo Fragione Center for
Interdisciplinary Exploration & Research in Astrophysics (CIERA) and
Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA  Abraham Loeb  Astronomy Department, Harvard University, 60 Garden
Street, Cambridge, MA 02138  Frederic A. Rasio Center for Interdisciplinary
Exploration & Research in Astrophysics (CIERA) and Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA  Newlin C.
Weatherford Center for Interdisciplinary Exploration & Research in
Astrophysics (CIERA) and Department of Physics & Astronomy, Northwestern
University, Evanston, IL 60208, USA  Claire S. Ye Center for Interdisciplinary
Exploration & Research in Astrophysics (CIERA) and Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA

###### Abstract

Since the first signal in 2015, the gravitational-wave detections of merging
binary black holes (BBHs) by the LIGO and Virgo collaborations (LVC) have
completely transformed our understanding of the lives and deaths of compact
object binaries, and have motivated an enormous amount of theoretical work on
the astrophysical origin of these objects. We show that the phenomenological
fit to the redshift-dependent merger rate of BBHs from Abbott et al. (2020) is
consistent with a purely dynamical origin for these objects, and that the
current merger rate of BBHs from the LVC could be explained entirely with
globular clusters alone. While this does not prove that globular clusters are
the dominant formation channel, we emphasize that many formation scenarios
could contribute a significant fraction of the current LVC rate, and that any
analysis that assumes a single (or dominant) mechanism for producing BBH
mergers is implicitly using a specious astrophysical prior.

binary black holes — gravitational waves — mergers — globular clusters

## 1

Over the past 5 years, the observed rate of BBH mergers in the local universe
has been significantly constrained. After the detections of GW150914,
GW151012, and GW151226 in the first observing run, initial estimates put the
volumetric merger rate between 9 to $240~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at 90%
confidence at $z=0$ (Abbott et al., 2016). Merger rates $\gtrsim
100~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ were largely understood to be explicable by
isolated binary evolution (e.g., Belczynski et al., 2016), with dynamical
assembly (e.g., Rodriguez et al., 2016) only contributing a small fraction of
the population. But as more BBH mergers have been detected, the measured local
merger rate has decreased by an order of magnitude, while sufficient numbers
of higher-redshift mergers allow the slope of the rate to be observed. By
fitting the observed detections to a phenomenological model of the form
$R(z)=R_{0}(1+z)^{\kappa}$, an analysis of the latest GW transient catalog
(GWTC-2, Abbott et al., 2020) suggests an increasing merger rate with a local
value of $19^{+16}_{-9}~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z=0$.

Figure 1: The merger rate of BBHs from GCs compared to the latest LVC results.
The left panel shows the predictions from Rodriguez & Loeb (2018) and updated
predictions from Kremer et al. (2020) using the same cosmological model. The
right panel shows the fits to the models from Kremer et al. (2020) broken down
according to cluster virial radius, assuming that all clusters are born with
initial radii of 0.5, 1, 2, or 4 pc. The total prediction on the left is the
average of these four values. In blue, we show the median, 50%, and 90%
intervals of the phenomenological merger rate fit from GWTC-2 (Abbott et al.,
2020).

There have been many theoretical models of the merger rate of BBHs from GCs,
both before and after the first detection of GWs (e.g. Portegies Zwart &
Mcmillan, 2000; Rodriguez et al., 2015) with the majority predicting a
volumetric merger rate at $z=0$ of $\sim 10~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$.
Using the Cluster Monte Carlo code (CMC), several groups have studied the BBH
merger rate from GCs and the unique properties of their sources. The analysis
of the BBH merger rate from GCs initially predicted a merger rate anywhere
from 2 to $20~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z=0$, based on the luminosity
function and comoving spatial density of observed GCs in the local universe
(Rodriguez et al., 2016). Of course, this initial analysis ignored the
contributions of GCs that were disrupted before the present day, which can
significantly increase the contribution to the BBH merger rate (e.g., Fragione
& Kocsis, 2018). In Rodriguez et al. (2018), we combined a cosmological model
for GC formation (El-Badry et al., 2019) with star-by-star CMC models of GCs
to estimate the BBH merger rate as a function of cosmological redshift.

In the left panel of Figure 1, we show the predictions from Rodriguez et al.
(2018) and updated predictions using the same cosmological model and fitting
procedure111For the updated model we have added an additional correction
accounting for the formation of central-massive BHs in GCs, from Antonini et
al. (2019); see Kremer et al. (2019) for details. This decreases the BBH
merger rate from the densest GCs by $\sim 10\%$ with newer GC models (covering
a wider range of parameters) from Kremer et al. (2020). When compared to the
allowed range and cosmological evolution of the BBH merger rate from GWTC-2
(Abbott et al., 2020), it is obvious that _the entire phenomenological BBH
merger rate can potentially be explained by GCs alone_. At $z=0$ the original
fits predict a merger rate of $15~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$, while the
newer fits from Kremer et al. (2020) predict a merger rate of
$17~{}\rm{Gpc}^{-3}\rm{yr}^{-1}$. In both cases, the merger rate increases as
a function of redshift. Comparing the ratio of the rate at $z=1$ to $z=0$, we
find that $R(1)/R(0)=3.1$ for the original fits from Rodriguez et al. (2018),
and $R(1)/R(0)=3.2$ for the Kremer et al. (2020) models. Both ratios are
consistent with the phenomenological value of $R(1)/R(0)=2.5^{+7.8}_{-1.9}$
from Abbott et al. (2020).

The primary improvement in the models of Kremer et al. (2020) is the extent of
the grid, with the newer grid containing clusters with higher initial masses,
more compact initial virial radii (as small as 0.5 pc), and a wider range of
metallicities. All together, they capture nearly the complete spectrum of
present-day dense star clusters in the Milky Way. We have assumed in the left
panel of Figure 1 that clusters of different initial radii contribute equally
to the total rate. To make this more explicit, we show what the merger rate
would look like from GCs assuming all cluster were born with virial radii of
0.5, 1, 2, or 4 pc in the right panel of Figure 1. The four values clearly
span the 90% region of allowed merger rates from Abbott et al. (2020). As
pointed out in Kremer et al. (2020), initial concentrations of clusters
directly control the slope of the merger rate, with the contributions from
clusters with 0.5, 1, 2, and 4 pc having ratios of $R(1)/R(0)=4.2$, 2.8, 2.1,
and 2.6, respectively. Given that many young clusters in the local universe
are observed to have initial effective radii between 1 and 2 pc (e.g.
Scheepmaker et al., 2007), this suggests remarkably good agreement with
current LVC findings.

Of course, there are many such dynamical scenarios for forming merging BBHs ,
many of which are consistent with the 90% uncertainties from the LVC’s
phenomenological model. But while the measured merger rates can now be
explained by a single formation channel alone, there is no reason to believe
that is the case. Several studies of the current LVC BBH catalog have
suggested that multiple formation scenarios likely operate in producing BBH
mergers (e.g., Wong et al., 2020; Zevin et al., 2020) with the latter
suggesting that no single channel likely contributes more than 70% to the
total population. This note is meant to emphasize this point: given that GCs
alone can naturally explain the most up-to-date BBH merger rate from the LVC,
it is no longer reasonable to assume that dynamical processes constitute a
subdominant fraction of the full merger rate.

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