# Generalized canonical models of foliated surfaces

Yen-An Chen  Department of Mathematics, University of Utah, Salt Lake City, UT
84112, USA<EMAIL_ADDRESS>
###### Abstract.

We study generalized canonical models of foliated surfaces of general type. In
particular, we show that generalized canonical models of general type and
their minimal partial du Val resolutions are bounded. Moreover, we show the
valuative criterion of separatedness and properness and local-closedness for
${\mathcal{S}}_{P}$-families.

On the way, we also show a result on the invariance of plurigenera.

###### 2010 Mathematics Subject Classification:

Primary 32M25, Secondary 14C20, 14E99, 14D20.

The author was partially supported by NSF research grants no: DMS-1801851,
DMS-1840190 and by a grant from the Simons Foundation; Award Number: 256202.

## Introduction

The study of foliated surfaces of general type has seen significant progress
in recent years. By work of McQuillan and Brunella (see [McQ08] and [Bru15]),
it is known that foliated surfaces of general type with only canonical
foliation singularities admit a unique canonical model. Thus it is natural to
study the moduli theory for these canonical models, for instance, the
existence of the moduli functor.

A first step toward this was achieved in [HL21] where it is shown that
canonical models of foliated surfaces of general type with fixed Hilbert
function are birationally bounded. Notice that, by a result of McQuillan,
$K_{\mathscr{F}}$ is not ${\mathbb{Q}}$-Cartier for any canonical model
$(X,{\mathscr{F}})$ with cusp singularities. Thus, the moduli functor is
expected not to be algebraic. However, if we consider the moduli functor for
some natural birational modifications of these canonical models, then it is
better behaved. For instance, in [Che19], it is shown that minimal partial du
Val resolutions (see Definition 2.5) of these canonical models are bounded. As
a consequence, the boundedness of these canonical models also follows.

The next natural property to investigate is the local-closedness. We recall
the following example.

######  Example 0.1 ([Bru01, page 114]).

Consider
$\pi:{\mathbb{C}}^{2}_{x,y}\times({\mathbb{C}}_{\lambda}\backslash\\{0,1\\})\rightarrow{\mathbb{C}}_{\lambda}\backslash\\{0,1\\}$
and a family of foliations given by

$x\frac{\partial}{\partial x}+\lambda y\frac{\partial}{\partial y}.$

We can extend this family to a family
${\mathbb{P}}^{2}\times({\mathbb{C}}\backslash\\{0,1\\})\rightarrow{\mathbb{C}}\backslash\\{0,1\\}$.
Note that the singularity is reduced (see Definition 1.15) if and only if
$\lambda\not\in{\mathbb{Q}}^{+}$.

In light of this example, the moduli functor for canonical models (or their
minimal partial du Val resolutions) is _not_ locally closed. To solve this
issue, we allow some special _log canonical_ foliation singularities and
consider _generalized_ canonical models (see Definition 1.27). As in [HL21]
and [Che19], we first show the following boundedness for minimal partial du
Val resolutions of generalized canonical models. (See Theorem 2.10.)

######  Theorem 0.2.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Then the set
${\mathcal{S}}_{P}$ consisting of minimal partial du Val resolutions
$(Y,{\mathscr{G}},D)$ (see Definition 2.5) of GENERALIZED CANONICAL models
$(X,{\mathscr{F}})$ of general type with Hilbert function
$\chi(mK_{\mathscr{F}})=P(m)$ is bounded. More precisely, there exists a
projective morphism $\mu:{\mathcal{Y}}\rightarrow T$ and a family of pre-
foliations ${\mathcal{F}}$ on ${\mathcal{Y}}$ with respect to $\mu$ (see
Definition 2.9) where ${\mathcal{Y}}$ and $T$ are quasi-projective varieties
of finite type such that, for any $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$,
there is a $t\in T$ and an isomorphism $\rho:Y\rightarrow{\mathcal{Y}}_{t}$
such that
${\mathscr{G}}(-D)\cong\rho^{*}({\mathcal{F}}|_{{\mathcal{Y}}_{t}})$.

Using the techniques in [CF18], we also show the following invariance of
plurigenera for families of minimal partial du Val resolutions of
_generalized_ canonical models of general type with fixed Hilbert function.
(See Theorem 3.2.) Note that condition
$({\mathcal{X}},{\mathcal{F}},\Delta)_{t}\in{\mathcal{S}}_{P}$ puts a very
strong restriction on $\Delta$.

######  Theorem 0.3.

Given any family of foliated triples
$f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ where $T$ is connected
and $({\mathcal{X}},{\mathcal{F}},\Delta)_{t}\in{\mathcal{S}}_{P}$ for all
$t\in T$ (see Definition 2.17 for families of foliated triples), there exists
an integer $m_{f}$ depending only on this family $f$ such that

$h^{0}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}}+\Delta)|_{{\mathcal{X}}_{t}})$

is independent of $t$ for $m\geq m_{f}$ and divisible by $i(P)$. (See Lemma
3.1.)

Next, for a fixed function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$,
we use the techniques from [Spi20] and [CS18], to establish the semi-stable
$(K_{\mathcal{F}}+\Delta)$-minimal model program for a special type of
families of foliated surfaces. (See Section 4 and Theorem 4.25.)

######  Theorem 0.4.

Let $f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ be a family of
foliated triples which is a semi-stable reduction of
${\mathcal{S}}_{P}$-families. (See subsection 4.2.) Then the semi-stable
$(K_{\mathcal{F}}+\Delta)$-minimal model program exists and stops with
$f^{m}:({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})\rightarrow T$ such
that $(K_{{\mathcal{F}}^{m}}+\Delta^{m})$ is nef over $T$,
$f^{m}:{\mathcal{X}}^{m}\rightarrow T$ is dlt (see Definition 4.6), and
$({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})$ is ${\mathcal{F}}^{m}$-dlt
(see Definition 4.8).

Utilizing this semi-stable $(K_{\mathcal{F}}+\Delta)$-minimal model program,
we show that ${\mathcal{S}}_{P}$-family (see Definition 2.17) satisfies the
valuative criterions for separatedness and properness (after a finite base
change). (See Theorem 5.1 and Theorem 4.26.) This is the first step toward the
separatedness and properness of a suitable moduli functor parametrizing
${\mathcal{S}}_{P}$.

######  Theorem 0.5 (Separatedness).

Let $T$ be a smooth curve, $T^{\circ}=T\backslash\\{0\\}$ where $0\in T$, and
$({\mathcal{X}}_{i},{\mathcal{F}}_{i},\Delta_{i})$ for $i=1,2$ be two
${\mathcal{S}}_{P}$-families (see Definition 2.17) over $T$. Then any
isomorphism
$\varphi^{\circ}:({\mathcal{X}}_{1},{\mathcal{F}}_{1},\Delta_{1})_{T^{\circ}}\rightarrow({\mathcal{X}}_{2},{\mathcal{F}}_{2},\Delta_{2})_{T^{\circ}}$
over $T^{\circ}$ extends uniquely to an isomorphism
$\varphi:({\mathcal{X}}_{1},{\mathcal{F}}_{1},\Delta_{1})\rightarrow({\mathcal{X}}_{2},{\mathcal{F}}_{2},\Delta_{2})$
over $T$.

######  Theorem 0.6 (Properness).

Let $T$ be a smooth curve and
$f^{\circ}:({\mathcal{X}}^{\circ},{\mathcal{F}}^{\circ},\Delta^{\circ})\rightarrow
T^{\circ}$ be an ${\mathcal{S}}_{P}$-family (see Definition 2.17) over
$T^{\circ}$ where $T^{\circ}=T\backslash\\{0\\}$ and $0\in T$. Then there
exists a smooth curve $C$ and a finite morphism $C\rightarrow T$ such that,
after the base change via $C\rightarrow T$, the pullback
${\mathcal{S}}_{P}$-family
$({\mathcal{X}}_{C}^{\circ},{\mathcal{F}}_{C}^{\circ},\Delta_{C}^{\circ})$
extends uniquely to an ${\mathcal{S}}_{P}$-family
$({\mathcal{X}}_{C},{\mathcal{F}}_{C},\Delta_{C})$ over $C$ with the special
fiber ${\mathcal{S}}_{P}$-smoothable. (See Definition 2.19.)

Finally, by a Seidenberg-type Theorem [PS19, Proposition 7.4] for a family of
foliated surfaces, we show the following theorem (see also Theorem 6.4), which
gives us a hint of the local-closedness of a suitable moduli functor
parametrizing smoothable foliated surfaces.

######  Theorem 0.7.

Given a family of foliated triples
$f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$. ($T$ is not necessarily
a curve.) Assume that there is a dense subset $T^{\prime}\subset T$ such that
$({\mathcal{X}},{\mathcal{F}},\Delta)_{t}\in{\mathcal{S}}_{P}$ for all $t\in
T^{\prime}$. (See Definition 2.8 and Definition 2.17.) Then there exists an
open subset $U\subset T$ such that all fibers of $f$ over $U$ are in
${\mathcal{S}}_{P}$. Moreover, there is an open subset $U^{\prime}$ with
$U\subset U^{\prime}\subset T$ such that all fibers of $f$ over $U^{\prime}$
are exactly ${\mathcal{S}}_{P}$-smoothable fibers. (See Definition 2.19.)

######  Acknowledgements.

The author would like to thank Christopher D. Hacon and Fabio Bernasconi for
invaluable discussions, precious suggestions, and encouragements.

##  1\. Preliminaries

In this paper, we always work over ${\mathbb{C}}$. In this section, we recall
several definitions and results which will be used later.

###  1.1. Intersection theory on normal complete surfaces

We will use Mumford’s intersection pairing on normal surfaces (see, e.g.,
[Sak84]) and some of its properties.

Let $X$ be a normal complete surface and $f:Y\rightarrow X$ be a proper
birational morphism from a smooth surface $Y$, for instance, the minimal
resolution of singularities. Let $E=\cup_{j}E_{j}$ be the exceptional divisor
of $f$. Since the intersection matrix $(E_{i}\cdot E_{j})$ is negative
definite, for any Weil ${\mathbb{R}}$-divisor $D$ on $X$, we can define
$f^{*}D$ as $\widetilde{D}+\sum_{j}a_{j}E_{j}$ where $\widetilde{D}$ is the
proper transform of $D$ and the $a_{j}$’s are real numbers uniquely determined
by the property that $(\widetilde{D}+\sum_{j}a_{j}E_{j})\cdot E_{i}=0$ for
every $i$.

For any two Weil ${\mathbb{R}}$-divisors $D_{1}$ and $D_{2}$ on $X$, we define
the intersection number $D_{1}\cdot D_{2}:=(f^{*}D_{1})\cdot(f^{*}D_{2})$.
Note that if $D_{1}$ is Cartier and $D_{2}$ is an irreducible curve, then
$D_{1}\cdot D_{2}$ agrees with the degree of ${\mathcal{O}}_{D_{2}}(D_{1})$.

###  1.2. A criterion for very-ampleness

######  Theorem 1.1 ([Lan01, Theorem 0.2]).

Let $X$ be a normal projective surface and $M$ be a ${\mathbb{Q}}$-divisor on
$X$ such that $K_{X}+\lceil M\rceil$ is Cartier. Let $\zeta$ be a
0-dimensional subscheme of $X$ and $\delta_{\zeta}>0$ be the number defined in
[Lan01, Definition 1.1.2] associated to $\zeta$. If

  1. (1)

$M^{2}>\delta_{\zeta}$ and

  2. (2)

$M\cdot C\geq\frac{1}{2}\delta_{\zeta}$ for every curve $C$ on $X$,

then $H^{0}(X,{\mathcal{O}}_{X}(K_{X}+\lceil
M\rceil))\rightarrow{\mathcal{O}}_{\zeta}(K_{X}+\lceil M\rceil)$ is
surjective.

######  Remark 1.2.

For the case when $|\zeta|=2$, we have following remarks.

  1. (1)

If the support of $\zeta=\\{x_{1},x_{2}\\}$ is two distinct points, then
$\delta_{\zeta}=\delta_{x_{1}}+\delta_{x_{2}}$. ([Lan01, 1.1.3])

  2. (2)

$\delta_{x}\leq 4$ if $x$ is a smooth point or a du Val singularity. ([Lan01,
0.3.2])

  3. (3)

If $\zeta$ is supported at only one point $x$, then

    1. (a)

$\delta_{\zeta}\leq 8$ if $x$ is a smooth point or a du Val singularity.
([Lan01, 0.3.2])

    2. (b)

$\delta_{\zeta}\leq 4(\mbox{edim}_{x}X-1)$ if $x$ is a rational (non-smooth)
singularity where $\mbox{edim}_{x}X$ is the embedding dimension of $X$ at $x$.
([Lan01, 0.3.4])

######  Proposition 1.3 ([Har77, Proposition II.7.3]).

Let $X$ be a projective scheme over $k$ and ${\mathscr{L}}$ be a line bundle
on $X$. Then ${\mathscr{L}}$ is very ample if and only if it separates points
and tangents.

###  1.3. Foliations on schemes

Most definitions in this subsection follow from [Ayo, Section 6] with some
generalizations.

######  Definition 1.4 ([Ayo, Définition 6.1.1]).

Let $X$ be a scheme over ${\mathbb{C}}$. A _pre-foliation_ ${\mathscr{F}}$ on
$X$ is a pair $(\Omega_{{\mathscr{F}}},d_{\mathscr{F}})$ consisting of a
quasi-coherent ${\mathcal{O}}_{X}$-module $\Omega_{{\mathscr{F}}}$ and of a
${\mathbb{C}}$-linear derivation
$d_{\mathscr{F}}:{\mathcal{O}}_{X}\rightarrow\Omega_{{\mathscr{F}}}$ such that
the ${\mathcal{O}}_{X}$-linear morphism
$\rho_{\mathscr{F}}:\Omega_{X}\rightarrow\Omega_{{\mathscr{F}}}$ associated
with $d_{\mathscr{F}}$ satisfies the following conditions:

  1. (1)

it is surjective;

  2. (2)

its kernel is contained in the kernel of the composition of

$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge^{2}\rho_{\mathscr{F}}}$$\textstyle{\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{{\mathscr{F}}}}$

The rank of ${\mathscr{F}}$ is defined to be the generic rank of
$\Omega_{\mathscr{F}}$. A pre-foliation ${\mathscr{F}}$ is called a
_foliation_ if $\Omega_{\mathscr{F}}$ is _pure_ , that is, the closure of each
associated point of $\Omega_{\mathscr{F}}$ has dimension $\dim X$. A _foliated
scheme_ is a triple $(X,\Omega_{{\mathscr{F}}},d_{\mathscr{F}})$ where X is a
scheme and $(\Omega_{{\mathscr{F}}},d_{\mathscr{F}})$ is a foliation on $X$.
We will denote a foliated scheme as $(X,{\mathscr{F}})$ if there is no
confusion.

######  Remark 1.5 ([Ayo, Remarque 6.1.2]).

In Definition 1.4, the condition (2) is, under the condition (1), equivalent
to the existence of a ${\mathbb{C}}$-linear morphism
$d_{\mathscr{F}}:\Omega_{{\mathscr{F}}}\rightarrow\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{{\mathscr{F}}}$
making the following square commutative

$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{\rho_{\mathscr{F}}}$$\textstyle{\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge^{2}\rho_{\mathscr{F}}}$$\textstyle{\Omega_{{\mathscr{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{\mathscr{F}}}$$\textstyle{\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{{\mathscr{F}}}.}$

######  Remark 1.6.

We have the following remarks:

  1. (1)

Given a pre-foliation ${\mathscr{F}}$ whose support has dimension $\dim X$.
The _saturation_ ${\mathscr{F}}^{\textnormal{sat}}$ of ${\mathscr{F}}$ is
defined to be

$(\textnormal{pure}(\Omega_{\mathscr{F}}):=\Omega_{\mathscr{F}}/\textnormal{tors}(\Omega_{\mathscr{F}}),\pi\circ
d_{\mathscr{F}})$

where $\textnormal{tors}(\Omega_{\mathscr{F}})\subset\Omega_{\mathscr{F}}$ is
the subsheaf of all local sections whose supports have dimension strictly less
than $\dim X$,
and$\pi:\Omega_{\mathscr{F}}\rightarrow\textnormal{pure}(\Omega_{\mathscr{F}})$
is the canonical quotient map. Note that ${\mathscr{F}}^{\textnormal{sat}}$ is
a foliation.

  2. (2)

Suppose $X$ is smooth, the condition (2) in Definition 1.4 is equivalent to
the integrability condition of $\ker(\rho_{\mathscr{F}})$.

  3. (3)

If the rank of ${\mathscr{F}}$ is one, then the condition (2) in Definition
1.4 is automatically satisfied since
$\wedge^{2}_{{\mathcal{O}}_{X}}\Omega_{\mathscr{F}}=0$ .

######  Lemma 1.7.

If $X$ is normal and of finite type, then a foliation ${\mathscr{F}}$ of rank
one on $X$ is equivalent to a saturated rank-one coherent subsheaf
$T_{\mathscr{F}}$ of the tangent sheaf $T_{X}$. Here a subsheaf
$T_{\mathscr{F}}$ of $T_{X}$ is called saturated if $T_{X}/T_{\mathscr{F}}$ is
torsion-free.

######  Proof.

Given a foliation ${\mathscr{F}}$ on $X$, we have an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\mathscr{F}}}$$\textstyle{\Omega_{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $K$ is the kernel of $\rho_{\mathscr{F}}$. Dualizing this sequence, we
get

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{\mathscr{F}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\mathscr{F}}^{\vee}}$$\textstyle{T_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $A$ is the cokernel of $\rho_{\mathscr{F}}^{\vee}$. Note that $A$ is a
subsheaf of $K^{\vee}$ and $K^{\vee}$ is torsion-free. So $A$ is also torsion-
free. Thus, we have a rank-one coherent subsheaf $\Omega_{\mathscr{F}}^{\vee}$
of $T_{X}$. Hence, we have a function $F$ sending a foliation ${\mathscr{F}}$
of rank one on $X$ to a saturated rank-one coherent subsheaf of the tangent
sheaf $T_{X}$.

Conversely, given a saturated coherent subsheaf $T_{\mathscr{F}}$ of $T_{X}$,
that is, we have an exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
70.38821pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
70.38821pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
95.70465pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 112.69974pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
112.69974pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{X}/T_{\mathscr{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
173.12167pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
173.12167pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since $\Omega_{X}$ is reflexive, we get, after dualizing the sequence,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(T_{X}/T_{\mathscr{F}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\vee}}$$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $B$ is the cokernel of $\pi^{\vee}$. Notice that $B$ is a subsheaf of
$T_{\mathscr{F}}^{\vee}$ and $T_{\mathscr{F}}^{\vee}$ is torsion-free. Thus,
$B$ is also torsion-free and $(B,\delta)$ gives a foliation on $X$. Hence, we
have a function $G$ sending a saturated rank-one coherent subsheaf
$T_{\mathscr{F}}$ of the tangent sheaf $T_{X}$ to a foliation of rank one on
$X$.

Therefore, by [HL21, Lemma 1.8], $G\circ F$ and $F\circ G$ are identity
functions up to isomorphism. ∎

######  Definition 1.8 ([Ayo, Définition 6.1.6]).

Given two pre-foliations ${\mathscr{G}}$ on $Y$ and ${\mathscr{F}}$ on $X$. A
_morphism of pre-foliations_ ${\mathscr{G}}\rightarrow{\mathscr{F}}$ is a
morphism of schemes $f:Y\rightarrow X$ such that the kernel of
$\rho_{\mathscr{F}}:\Omega_{X}\rightarrow\Omega_{{\mathscr{F}}}$ is contained
in the kernel of the composition of

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.15056pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.14362pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{-\circ f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.15056pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.15056pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{f_{*}\Omega_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
62.17134pt\raise 6.11389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{f_{*}\rho_{\mathscr{G}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.64061pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.64061pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{f_{*}\Omega_{{\mathscr{G}}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

That is, we have the following commutative square.

$\textstyle{\Omega_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ
f}$$\scriptstyle{\rho_{\mathscr{F}}}$$\textstyle{f_{*}\Omega_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}\rho_{{\mathscr{G}}}}$$\textstyle{\Omega_{{\mathscr{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\circ
f}$$\textstyle{f_{*}\Omega_{{\mathscr{G}}}.}$

######  Lemma 1.9 ([Ayo, Lemme 6.1.10]).

Let ${\mathscr{F}}$ be a pre-foliation on $X$ and $f:Y\rightarrow X$ a
morphism of schemes. There then exists a pre-foliation $f^{*}{\mathscr{F}}$ on
$Y$, unique up to a unique isomorphism, such that the following properties are
satisfied.

  1. (1)

The morphism $f$ defines a morphism of pre-foliations
$f:f^{*}{\mathscr{F}}\rightarrow{\mathscr{F}}$.

  2. (2)

(Universal property) For any morphism of pre-foliations
${\mathscr{H}}\rightarrow{\mathscr{F}}$ where
$(\Omega_{\mathscr{H}},d_{\mathscr{H}})$ is a pre-foliation on $Z$, the
forgetful functor induces a bijection

$\textnormal{Hom}_{{\mathscr{F}}}({\mathscr{H}},f^{*}{\mathscr{F}})\cong\textnormal{Hom}_{X}(Z,Y).$

We say that the pre-foliation $f^{*}{\mathscr{F}}$ is the pullback of the pre-
foliation ${\mathscr{F}}$ via the morphism $f:Y\rightarrow X$.

###  1.4. Foliations on normal varieties

Let $X$ be a normal variety. A _foliation_ ${\mathscr{F}}$ on $X$ is a rank-
one saturated subsheaf ${\mathscr{F}}$ of the tangent sheaf $T_{X}$ of $X$. So
we have the following short exact sequence:

$0\rightarrow T_{\mathscr{F}}\rightarrow T_{X}\rightarrow
T_{X}/{\mathscr{F}}\rightarrow 0$

with $T_{X}/{\mathscr{F}}$ torsion free. By Lemma 1.7, this is equivalent to
Definition 1.4.

A point $p$ on $X$ is called a singular point of the foliation if either it is
a singular point of $X$ or a point at which the quotient $T_{X}/{\mathscr{F}}$
is _not_ locally free.

######  Definition 1.10.

A foliated variety is a pair $(X,{\mathscr{F}})$ consisting of a _normal_
variety $X$ and a foliation ${\mathscr{F}}$ on $X$.

Since $T_{X}\cong\mbox{Hom}_{{\mathcal{O}}_{X}}(\Omega_{X},{\mathcal{O}}_{X})$
is reflexive, we have that ${\mathscr{F}}$ is also reflexive because it is a
saturated subsheaf of a reflexive sheaf $T_{X}$. Hence, we can define the
canonical divisor $K_{\mathscr{F}}$ of the foliation as a Weil divisor on $X$
with ${\mathcal{O}}_{X}(-K_{\mathscr{F}})\cong{\mathscr{F}}$.

######  Definition 1.11.

Let $(X,{\mathscr{F}})$ be a foliated variety. Given any proper birational
morphism of normal varieties $f:Y\rightarrow X$. We define the _pullback
foliation_ $f^{*}{\mathscr{F}}$ as follows:

Let $U$ be the largest open subset such that $V:=f^{-1}(U)\rightarrow U$ is an
isomorphism. Note that ${\mathscr{F}}|_{U}\subset T_{U}\cong T_{V}$. By
[Har77, Exercise II.5.15], we have a coherent subsheaf ${\mathscr{G}}$ of
$T_{Y}$ such that ${\mathscr{G}}|_{V}={\mathscr{F}}|_{U}\subset T_{V}$. Then
the pullback foliation $f^{*}{\mathscr{F}}$ is defined to be the saturation of
${\mathscr{G}}$. By [HL21, Lemma 1.8], this definition is well-defined.

Also, we can define the pullback foliation $f^{*}{\mathscr{F}}$ as the
saturation (see Remark 1.6(1)) of pullback pre-foliation in Lemma 1.9. Both
definitions are equivalent by [HL21, Lemma 1.8].

Moreover if ${\mathscr{G}}$ is a foliation on $Y$, then we can define the
_pushforward foliation_ $f_{*}{\mathscr{G}}$ by taking the saturation of the
image of the composition

$f_{*}T_{\mathscr{G}}\rightarrow
f_{*}T_{Y}\rightarrow(f_{*}T_{Y})^{**}=T_{X}.$

######  Definition 1.12.

We define the Kodaira dimension of ${\mathscr{F}}$ as

$\kappa({\mathscr{F}})=\max\\{\dim\phi_{mK_{\mathscr{F}}}(X)|\,m\in{\mathbb{N}}_{>0}\mbox{
and }h^{0}(X,mK_{\mathscr{F}})>0\\}$

where by convention $\kappa({\mathscr{F}})=-\infty$ if the set above is empty,
that is, $h^{0}(X,mK_{\mathscr{F}})=0$ for all $m\in{\mathbb{N}}_{>0}$. We say
that ${\mathscr{F}}$ is _of general type_ if $\kappa(K_{\mathscr{F}})=\dim X$.

######  Definition 1.13.

Let $(X,{\mathscr{F}})$ be a foliated variety and let $f:Y\rightarrow X$ be a
proper birational morphism. For any divisor $E$ on $Y$, we define the
_foliated discrepancy_ of ${\mathscr{F}}$ along $E$ to be
$a_{E}({\mathscr{F}})=\textnormal{ord}_{E}(K_{f^{*}{\mathscr{F}}}-f^{*}K_{\mathscr{F}})$.
We say

$x\textnormal{ is
}\left\\{\begin{array}[]{ll}\textnormal{terminal}&\textnormal{ if
}a_{E}({\mathscr{F}})>0\\\ \textnormal{canonical}&\textnormal{ if
}a_{E}({\mathscr{F}})\geq 0\\\ \textnormal{log terminal}&\textnormal{ if
}a_{E}({\mathscr{F}})>-\varepsilon(E)\\\ \textnormal{log
canonical}&\textnormal{ if
}a_{E}({\mathscr{F}})\geq-\varepsilon(E)\end{array}\right.\textnormal{ for
every divisor $E$ over $x$}$

where $\varepsilon(E)=0$ if $E$ is $f^{*}{\mathscr{F}}$-invariant and
$\varepsilon(E)=1$ if $E$ is not $f^{*}{\mathscr{F}}$-invariant.

######  Definition 1.14.

Let $(X,{\mathscr{F}})$ be a foliated variety. We have the following
definitions for indices.

  1. (1)

The index $i(X)$ of $X$ is the smallest positive integer $m$ such that $mD$ is
Cartier for every Weil divisor $D$ on $X$. (We set $i(X)=\infty$ if there is
no such $m$.)

  2. (2)

The index $i(K_{X})$ of $K_{X}$ is the smallest positive integer $m$ such that
$mK_{X}$ is Cartier. (We set $i(K_{X})=\infty$ if $K_{X}$ is not
${\mathbb{Q}}$-Cartier.)

  3. (3)

The index $i({\mathscr{F}})$ of a foliation ${\mathscr{F}}$ on $X$ is the
smallest positive integer $m$ such that $mK_{\mathscr{F}}$ is Cartier. (We set
$i({\mathscr{F}})=\infty$ if $K_{\mathscr{F}}$ is not ${\mathbb{Q}}$-Cartier.)

  4. (4)

The ${\mathbb{Q}}$-index $i_{\mathbb{Q}}({\mathscr{F}})$ of a foliation
${\mathscr{F}}$ on $X$ is the smallest positive integer $m$ such that
$mK_{\mathscr{F}}$ is Cartier for all ${\mathbb{Q}}$-Gorenstein points of the
foliation.

###  1.5. Invariants on foliated normal surfaces

In this subsection, we will focus on some indices on foliated normal surfaces.
Most definitions follow from [Bru15] with some generalizations. Let
$(X,{\mathscr{F}})$ be a foliated normal surface.

Given any
$p\in\textnormal{Sing}({\mathscr{F}})\backslash\textnormal{Sing}(X)$. That is,
$p$ is a smooth point on $X$ but a singular point of the foliation
${\mathscr{F}}$. Let $v$ be the vector field around $p$ generating
${\mathscr{F}}$. Since $p\in\textnormal{Sing}({\mathscr{F}})$, we have
$v(p)=0$. Then we can consider the eigenvalues $\lambda_{1},\lambda_{2}$ of
$(Dv)|_{p}$, which do not depend on the choices of $v$.

######  Definition 1.15.

If one of the eigenvalues is non-zero, say $\lambda_{2}$, then we say $p$ is
_semi-reduced_ and define the eigenvalue of the foliation ${\mathscr{F}}$ at
$p$ to be

$\lambda:=\frac{\lambda_{1}}{\lambda_{2}}.$

For $\lambda\neq 0$, this definition is unique up to reciprocal
$\lambda\sim\frac{1}{\lambda}$.

If $\lambda=0$, then $p$ is called a _saddle-node_ ; otherwise, we say $p$ is
_non-degenerate_. If $\lambda\not\in{\mathbb{Q}}^{+}$, then $p$ is called a
_reduced_ singularity of ${\mathscr{F}}$.

######  Theorem 1.16 (Seidenberg’s theorem).

Given any foliated surface $(X,{\mathscr{F}})$ with $X$ smooth. Then there is
a sequence of blowups $\pi:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}})$ such
that $(Y,{\mathscr{G}})$ has only reduced singularities.

######  Proof.

See [Sei68], [MM80, Appendix], or [Bru15, Theorem 1.1]. ∎

######  Definition 1.17 (Invariant curves).

Let $(X,{\mathscr{F}})$ be a foliated surface. A curve $C$ on $X$ is called
${\mathscr{F}}$-invariant if the inclusion map

$T_{{\mathscr{F}}}|_{C}\rightarrow T_{X}|_{C}$

factors through $T_{C}$ on the smooth locus of $X$.

######  Definition 1.18 (Separatrices).

Let $p$ be a singular point of ${\mathscr{F}}$. A _separatrix_ of
${\mathscr{F}}$ at $p$ is a holomorphic (possibly singular) irreducible
${\mathscr{F}}$-invariant curve $C$ on a _neighborhood_ of $p$ which passes
through $p$.

######  Theorem 1.19.

Let ${\mathscr{F}}$ be a foliation on a normal surface $X$ and $C\subset X$ be
a connected, compact ${\mathscr{F}}$-invariant curve such that the followings
hold.

  1. (1)

All the singularities of ${\mathscr{F}}$ on $C$ are reduced and on the non-
singular locus of $X$.

  2. (2)

The intersection matrix of $C$ is negative definite, and the dual graph is a
tree.

Then there exists at least one point $p\in
C\cap\textnormal{Sing}({\mathscr{F}})$ and a separatrix through $p$ not
contained in $C$.

######  Proof.

See [Cam88], [Seb97], or [Bru15, Theorem 3.4]. ∎

####  1.5.1. Non-invariant curves

We first consider the non-invariant curves and define the tangency order for
them.

######  Definition 1.20 (Tangency order).

Let $(X,{\mathscr{F}})$ be a foliated normal surface and
$p\in\textnormal{Sing}({\mathscr{F}})\backslash\textnormal{Sing}(X)$. Let $v$
be the vector field generating ${\mathscr{F}}$ around $p$ and $C$ be a _non-
invariant_ curve through $p$. Let $f$ be the local defining function of $C$ at
$p$. We define the _tangency order_ of ${\mathscr{F}}$ along $C$ at $p$ to be

$\mbox{tang}({\mathscr{F}},C,p):=\dim_{\mathbb{C}}\frac{{\mathcal{O}}_{X,p}}{\langle
f,v(f)\rangle}.$

Note that $\mbox{tang}({\mathscr{F}},C,p)\geq 0$ and is independent of the
choices of $v$ and $f$. Moreover, if ${\mathscr{F}}$ is transverse to $C$ at
$p$, then $\mbox{tang}({\mathscr{F}},C,p)=0$. Therefore, if $C$ is compact,
then we can define

$\mbox{tang}({\mathscr{F}},C):=\sum_{p\in C}\mbox{tang}({\mathscr{F}},C,p).$

######  Proposition 1.21 ([Bru97],[Bru15, Proposition 2.2]).

Let ${\mathscr{F}}$ be a foliation on a smooth projective surface $X$, and $C$
be a non-invariant curve on $X$. Then we have

$K_{\mathscr{F}}\cdot C=\mbox{{\rm tang}}({\mathscr{F}},C)-C^{2}.$

######  Corollary 1.22.

Let $C$ be a non-invariant curve on a foliated surface $(X,{\mathscr{F}})$. If
$C$ is contained in the smooth locus of $X$, then we have
$(K_{\mathscr{F}}+C)\cdot C\geq 0$.

####  1.5.2. Invariant curves

Now we consider the invariant curves.

######  Definition 1.23.

Let $(X,{\mathscr{F}})$ be a foliated surface and
$p\in\textnormal{Sing}({\mathscr{F}})\backslash\textnormal{Sing}(X)$. Let
$\omega$ be a $1$-form generating ${\mathscr{F}}$ around $p$. If $C$ is an
invariant curve and $f$ is the local defining function at $p$, then we can
write

$g\omega=hdf+f\eta$

where $g$ and $h$ are holomorphic functions, $\eta$ is a holomorphic $1$-form,
and $h,f$ are relatively prime functions.

We define the index $\textnormal{Z}({\mathscr{F}},C,p)$ to be the vanishing
order of $\frac{h}{g}|_{C}$ at $p$. Also we define Camacho-Sad index
$\textnormal{CS}({\mathscr{F}},C,p)$ to be the residue of
$\frac{-1}{h}\eta|_{C}$ at $p$. These two definitions are independent of the
choices of $f,g,h,\omega,\eta$. (For a reference, see [Bru15, page 15 in
Chapter 2 and page 27 in Chapter 3].)

Note that if $p\not\in\textnormal{Sing}({\mathscr{F}})$, then
$\textnormal{Z}({\mathscr{F}},C,p)=0=\textnormal{CS}({\mathscr{F}},C,p)$.
Therefore, if $C$ is compact, then we can define

$\displaystyle\textnormal{Z}({\mathscr{F}},C)$ $\displaystyle:=$
$\displaystyle\sum_{p\in C}\textnormal{Z}({\mathscr{F}},C,p)$
$\displaystyle\textnormal{CS}({\mathscr{F}},C)$ $\displaystyle:=$
$\displaystyle\sum_{p\in C}\textnormal{CS}({\mathscr{F}},C,p)$

######  Definition 1.24.

We define the virtual Euler characteristic $\chi(C)$ of $C$ on a normal
surface $X$ to be $\chi(C)=-K_{X}\cdot C-C^{2}$.

######  Theorem 1.25 ([Bru97],[Bru15, Proposition 2.3 and Theorem 3.2]).

Let ${\mathscr{F}}$ be a foliation on a smooth projective surface $X$, and $C$
be an invariant curve on $X$. Then we have

  1. (1)

$K_{\mathscr{F}}\cdot C=\textnormal{Z}({\mathscr{F}},C)-\chi(C)$.

  2. (2)

$C^{2}=\textnormal{CS}({\mathscr{F}},C)$. This identity is called Camacho-Sad
formula.

######  Lemma 1.26.

Given a foliated normal surface $(X,{\mathscr{F}})$ and $p$ a semi-reduced
singularity.

  1. (1)

If $p$ is reduced and non-degenerate, assume $\omega=\lambda
y(1+o(1))dx-x(1+o(1))dy$ generates ${\mathscr{F}}$ around $p$. Then

$\textnormal{CS}({\mathscr{F}},x=0,p)=\frac{1}{\lambda},\textnormal{CS}({\mathscr{F}},y=0,p)=\lambda\mbox{,
and
}\textnormal{Z}({\mathscr{F}},x=0,p)=1=\textnormal{Z}({\mathscr{F}},y=0,p).$

  2. (2)

If $p$ is a saddle-node, assume $\omega=y^{k+1}dx-(x(1+\nu y^{k})+yo(k))dy$
generates ${\mathscr{F}}$ around $p$. Then

$\textnormal{CS}({\mathscr{F}},y=0,p)=0\mbox{ and
}\textnormal{Z}({\mathscr{F}},y=0,p)=1.$

Suppose there exists a weak separatrix, then

$\textnormal{CS}({\mathscr{F}},x=0,p)=\nu\mbox{ and
}\textnormal{Z}({\mathscr{F}},x=0,p)=k+1.$

  3. (3)

If $p$ is not reduced, assume its eigenvalue
$\lambda=\frac{m}{n}\in{\mathbb{Q}}^{+}$ in the reduced form. Assume
$\omega=\lambda y(1+o(1))dx-x(1+o(1))dy$ generates ${\mathscr{F}}$ around $p$.
Then

$\textnormal{CS}({\mathscr{F}},x=0,p)=\frac{1}{\lambda},\textnormal{CS}({\mathscr{F}},y=0,p)=\lambda,\mbox{
and
}\textnormal{Z}({\mathscr{F}},x=0,p)=1=\textnormal{Z}({\mathscr{F}},y=0,p).$

Moreover, for the invariant curve $C$ through $p=(0,0)$ which is locally
defined by $ax^{m}(1+o(1))-y^{n}(1+o(1))=0$, we have

$\textnormal{CS}({\mathscr{F}},C,p)=mn\mbox{ and
}\textnormal{Z}({\mathscr{F}},C,p)=1-(m-1)(n-1).$

######  Proof.

We will only show the case $(3)$. For the first two cases, the computations
are similar. (See also [Bru15, page 30-31 in Chapter 3].)

For the case $f=x$, note that $1\cdot\omega=(\lambda
y(1+o(1)))dx+x(-(1+o(1))dy)$. Then we take $g=1$, $h=\lambda y(1+o(1))$, and
$\eta=-(1+o(1))dy$. Thus we have

$\textnormal{CS}({\mathscr{F}},x=0,p)=\textnormal{res}_{p}\frac{-1}{h}\eta|_{f=0}=\textnormal{res}_{p}\frac{1}{\lambda
y(1+o(1))}(1+o(1))dy|_{x=0}=\frac{1}{\lambda}.$ $\displaystyle
Z({\mathscr{F}},x=0,p)$ $\displaystyle=$ $\displaystyle\textnormal{vanishing
order of }\frac{h}{g}|_{f=0}$ $\displaystyle=$
$\displaystyle\textnormal{vanishing order of }\frac{\lambda
y(1+o(1))}{1}|_{x=0}$ $\displaystyle=$ $\displaystyle 1.$

For the case $f=y$, it is similar.

For the case $C=(ax^{m}(1+o(1))-y^{n}(1+o(1))=0)$, we assume
$C=(f=ax^{m}-y^{n}=0)$ and $\omega=mydx-nxdy$ for simplicity. Note that

$\displaystyle ax^{m-1}\omega$ $\displaystyle=$ $\displaystyle amx^{m-1}ydx-
anx^{m}dy$ $\displaystyle=$ $\displaystyle amx^{m-1}ydx-ny^{n}dy-nfdy$
$\displaystyle=$ $\displaystyle ydf-nfdy.$

So we take $g=ax^{m-1}$, $h=y$, and $\eta=-ndy$. Thus,

$\textnormal{CS}({\mathscr{F}},C,p)=\textnormal{res}_{p}\frac{-1}{h}\eta|_{f=0}=\textnormal{res}_{p}\,\frac{1}{y}ndy|_{f=0}=mn.$
$\displaystyle\textnormal{Z}({\mathscr{F}},C,p)$ $\displaystyle=$
$\displaystyle\textnormal{vanishing order of }\frac{h}{g}|_{f=0}$
$\displaystyle=$ $\displaystyle\textnormal{vanishing order of
}\frac{y}{ax^{m-1}}|_{f=0}$ $\displaystyle=$ $\displaystyle 1-(n-1)(m-1).$

∎

###  1.6. Generalized canonical models

In light of the Example 0.1, we would like to allow some mild singularities in
order to get local-closedness in a family.

######  Definition 1.27.

A foliated normal surface $(X,{\mathscr{F}})$ is called a _generalized
canonical model_ if the following conditions hold:

  1. (1)

${\mathscr{F}}$ is a foliation on a normal proper algebraic space $X$ of
dimension two with only log canonical foliation singularities such that all
points at which ${\mathscr{F}}$ is _not_ canonical are either smooth points of
$X$ or cyclic quotient singularities of $X$.

  2. (2)

$K_{\mathscr{F}}$ is nef.

  3. (3)

$K_{\mathscr{F}}\cdot C=0$ implies $C^{2}\geq 0$ for any irreducible curve
$C$.

Moreover, we say $(X,{\mathscr{F}})$ has _mild_ log canonical foliation
singularities if the first condition above holds.

In Lemma 1.33 and Remark 1.34, we study more properties of mild log canonical
foliation singularities. Actually, there are log canonical foliation
singularities which are not mild.

######  Example 1.28 ([McQ08, Example I.2.5]).

On the normal surface $(x^{n+2}+y^{n+1}+xz^{n}=0)$ for $n\geq 2$, it is not a
rational singularity but the foliation given by

$\partial=n(n+1)x\frac{\partial}{\partial x}+n(n+2)y\frac{\partial}{\partial
y}+(n+1)^{2}z\frac{\partial}{\partial z}$

is log canonical.

######  Definition 1.29.

We say a foliated normal surface $(X,{\mathscr{F}})$ is a _canonical model_ if
it is a generalized canonical model with only canonical foliation
singularities.

######  Lemma 1.30.

Let $(X,{\mathscr{F}})$ be a generalized canonical model. If the foliation
${\mathscr{F}}$ is of general type, then $K_{{\mathscr{F}}}$ is numerically
ample, that is $K_{{\mathscr{F}}}^{2}>0$ and $K_{{\mathscr{F}}}\cdot C>0$ for
any irreducible curve $C$ on $X$.

######  Proof.

The proof follows closely from the proof of [HL21, Lemma 1.11]. ∎

To describe the resolutions of canonical foliation singularities and mild log
canonical foliation singularities, we recall the following definitions. (See
also [Bru15, Definition 5.1 and 8.1] and [McQ08, Definition III.0.2 and
III.2.3].)

######  Definition 1.31.

Given a foliated surface $(X,{\mathscr{F}})$ with $X$ smooth.

  1. (1)

$E=\bigcup_{i=1}^{s}E_{i}$ is called a string if

    1. (a)

each $E_{i}$ is a smooth rational curve and

    2. (b)

$E_{i}\cdot E_{j}=1$ if $|i-j|=1$ and $0$ if $|i-j|\geq 2$.

  2. (2)

If, moreover, $E_{i}^{2}\leq-2$ for all $i$, then we call $E$ a Hirzebruch-
Jung string.

  3. (3)

A curve $C\subset X$ is called ${\mathscr{F}}$-exceptional if

    1. (a)

$C$ is a $(-1)$-curve and

    2. (b)

the contraction of $C$ introduces $(X^{\prime},{\mathscr{F}}^{\prime})$ with
only _reduced_ singularities.

  4. (4)

An ${\mathscr{F}}$-invariant curve $C$ is called a
$(-1)$-${\mathscr{F}}$-curve (resp. $(-2)$-${\mathscr{F}}$-curve) if

    1. (a)

$C$ is a smooth rational curve and

    2. (b)

$\textnormal{Z}({\mathscr{F}},C)=1$ (resp.
$\textnormal{Z}({\mathscr{F}},C)=2$).

  5. (5)

We say $C=\bigcup_{i=1}^{s}C_{i}$ is an ${\mathscr{F}}$-chain if

    1. (a)

$C$ is a Hirzebruch-Jung string,

    2. (b)

each $C_{i}$ is ${\mathscr{F}}$-invariant,

    3. (c)

$\textnormal{Sing}{{\mathscr{F}}}\cap C$ are all reduced and non-degenerate,
and

    4. (d)

$\textnormal{Z}({\mathscr{F}},C_{1})=1$ and
$\textnormal{Z}({\mathscr{F}},C_{i})=2$ for all $i\geq 2$.

  6. (6)

If an ${\mathscr{F}}$-invariant curve $E$ is not contained in an
${\mathscr{F}}$-chain $C$ but meets the chain, then we call $E$ the tail of
the chain $C$.

  7. (7)

$C$ is called a bad tail if

    1. (a)

$C$ is a smooth rational curve with $\textnormal{Z}({\mathscr{F}},C)=3$ and
$C^{2}\leq-2$ and

    2. (b)

$C$ intersects two $(-1)$-${\mathscr{F}}$-curves whose self-intersections are
both $-2$.

######  Theorem 1.32 ([McQ08, Theorem III.3.2]).

Given any canonical model $(X,{\mathscr{F}})$. Let $\pi:Y\rightarrow X$ be the
minimal resolution at points at which ${\mathscr{F}}$ is singular. Let
${\mathscr{G}}$ be the pullback foliation on $Y$. Then all connected
components of exceptional divisors of $\pi$ are one of the following types:

  1. (1)

($A_{n}$ type) A string of smooth rational curves. More precisely, it consists
of either a ${\mathscr{G}}$-chain, two $(-1)$-${\mathscr{G}}$-curves of self-
intersections $-2$ joined by a bad tail, or a chain of
$(-2)$-${\mathscr{G}}$-curves.

  2. (2)

($D_{n}$ type) Two $(-1)$-${\mathscr{G}}$-curves of self-intersections $-2$
joined by a bad tail which itself connects to a chain of
$(-2)$-${\mathscr{G}}$-curves.

  3. (3)

(e.g.l.) Elliptic Gorenstein leaves. That is, either a cycle of
$(-2)$-${\mathscr{G}}$-curves or a rational curve with only one node.

Now we study _log canonical_ foliation singularities on _smooth_ points.

######  Lemma 1.33.

Let $(X,{\mathscr{F}},p)$ be a germ of a foliated surface. Suppose $p$ is a
semi-reduced singularity with positive rational eigenvalue, and $p\in X$ is a
smooth point. Then there is a sequence of blowups
$\pi:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}},p)$ with ${\mathscr{G}}$
having only reduced singularities on the exceptional divisor $E$. Thus we have
the following:

  1. (1)

The support of the exceptional divisor $\bigcup_{i=1}^{s}E_{i}$ is a string,

  2. (2)

All $E_{i}$’s except for one $j\in\\{1,\ldots,s\\}$ are invariant. Moreover,
this $E_{j}$ has $\textnormal{tang}({\mathscr{G}},E_{j})=0$.

  3. (3)

The string $\bigcup_{i=1}^{j-1}E_{i}$ (resp. $\bigcup_{i=j+1}^{s}E_{i}$) is a
${\mathscr{G}}$-chain with the first curve $E_{j-1}$ (resp. $E_{j+1}$).

  4. (4)

$\pi^{*}K_{\mathscr{F}}=K_{\mathscr{G}}+E_{j}$.

######  Proof.

Since the eigenvalue is defined up to reciprocal, we may assume that the
eigenvalue is $\frac{m}{n}\in{\mathbb{Q}}^{+}$ with $\textnormal{gcd}(m,n)=1$
and $m\geq n$. We proceed by induction on the pair
$(m,n)\in{\mathbb{N}}\times{\mathbb{N}}$ (by lexicographic ordering).

  1. (1)

We may assume that $p=(0,0)$ and that ${\mathscr{F}}$ is generated by

$v=nx\frac{\partial}{\partial x}+my\frac{\partial}{\partial y}$

around $p$. Let $\varphi:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}})$ be the
blowup at $p$ and $E_{0}$ be the exceptional divisor. Notice that there are
two singularities on the exceptional curve locally generated by

$v_{1}=nx^{\prime}\frac{\partial}{\partial
x^{\prime}}+(m-n)y^{\prime}\frac{\partial}{\partial y^{\prime}}$

and

$v_{2}=(n-m)x^{\prime}\frac{\partial}{\partial
x^{\prime}}+my^{\prime}\frac{\partial}{\partial y^{\prime}}.$

  2. (2)

For the base case $(m,n)=(1,1)$, by the computation above, the exceptional
curve $E_{0}$ has no foliation singularity and is not invariant with
$\mbox{tang}({\mathscr{G}},E_{0})=0$. Moreover, we have
$\pi^{*}K_{\mathscr{F}}=K_{\mathscr{G}}+E_{0}$. Thus, we complete the proof of
the base case.

  3. (3)

Assume $(m,n)\neq(1,1)$ and all pairs (strictly) less than $(m,n)$ have the
desired properties. Then $E_{0}$ is invariant and
$\pi^{*}K_{\mathscr{F}}=K_{\mathscr{G}}$. Notice that $v_{1}$ is still semi-
reduced with corresponding pair $(m-n,n)$ or $(n,m-n)$ while $v_{2}$ is
reduced.

By induction hypothesis, $v_{1}$ has a sequence of blowups
$\psi:(Z,{\mathscr{H}})\rightarrow(Y,{\mathscr{G}})$ with exceptional divisor
$E=\bigcup_{i=1}^{s}E_{i}$. Since the proper transform $\widetilde{E}_{0}$ of
$E_{0}$ on $Z$ meets $E$, $\widetilde{E}_{0}$ can only meet $E$ as a tail of
an ${\mathscr{H}}$-chain. Note that $\widetilde{E}_{0}^{2}\leq
E_{0}^{2}-1=-2$. Thus, $\widetilde{E}_{0}$ is a member of that
${\mathscr{H}}$-chain. Also
$\psi^{*}\pi^{*}K_{\mathscr{F}}=\psi^{*}K_{\mathscr{G}}=K_{\mathscr{H}}+E_{i}$
for some $i$. ∎

######  Remark 1.34.

By [McQ08, Fact I.2.4 and its remarks], a mild log canonical foliation
singularity $p$ (see Definition 1.27) is formally locally of the form
$\partial=x\frac{\partial}{\partial x}+\lambda y\frac{\partial}{\partial y}$
quotient by ${\mathbb{Z}}/n$ for some $n\in{\mathbb{Z}}$ and
$\lambda\in{\mathbb{Q}}^{+}$. Also $K_{\mathscr{F}}$ is Cartier around $p$.
Let $\pi:Y\rightarrow X$ be the _minimal_ resolution of $X$ over $p$ and
${\mathscr{G}}$ be the pullback foliation on $Y$. Then, after some
computations, we have the following:

  1. (1)

The support of the exceptional divisor $E=\bigcup_{i=1}^{s}E_{i}$ is a string.

  2. (2)

For general $\lambda\in{\mathbb{Q}}^{+}$, all $E_{i}$’s are
$(-2)$-${\mathscr{G}}$-curves and exactly one foliation singularity on $E$ is
not reduced but semi-reduced.

  3. (3)

Fix the action of ${\mathbb{Z}}/n$ on $\partial$, for any
$j\in\\{1,\ldots,s\\}$, there is a $\lambda\in{\mathbb{Q}}^{+}$ such that
$E_{j}$ is not ${\mathscr{G}}$-invariant with
$\textnormal{tang}({\mathscr{G}},E_{j})=0$, the string
$\bigcup_{i=1}^{j-1}E_{i}$ (resp. $\bigcup_{i=j+1}^{s}E_{i}$) is a
${\mathscr{G}}$-chain with the first curve $E_{j-1}$ (resp. $E_{j+1}$), and
$\pi^{*}K_{\mathscr{F}}=K_{\mathscr{G}}+E_{j}$.

######  Definition 1.35.

We call the string appearing as (3) in Remark 1.34 a _generalized
${\mathscr{G}}$-chain_.

######  Example 1.36.

Consider the foliation of the form $\partial=x\frac{\partial}{\partial
x}+\lambda y\frac{\partial}{\partial y}$ on the germ
$({\mathbb{C}}^{2}/\mu_{7},0)$ of the type $\frac{1}{7}(1,4)$. Let
$Y\rightarrow X$ be the minimal resolution of singularities. Then we have two
irreducible exceptional divisors $E_{1}$ and $E_{2}$, and we need three copies
of ${\mathbb{C}}^{2}$ to cover $Y$. More explicitly,
$Y={\mathbb{C}}^{2}_{\xi_{0},\eta_{0}}\cup{\mathbb{C}}^{2}_{\xi_{1},\eta_{1}}\cup{\mathbb{C}}^{2}_{\xi_{2},\eta_{2}}={\mathbb{A}}^{2}_{0}\cup{\mathbb{A}}^{2}_{1}\cup{\mathbb{A}}^{2}_{2}$
where

$\begin{array}[]{lll}\xi_{0}=x^{-4}y,&\xi_{1}=x^{-1}y^{2},&\xi_{2}=y^{7}\\\
\eta_{0}=x^{7},&\eta_{1}=x^{4}y^{-1},&\eta_{2}=xy^{-2}\end{array}$

Thus, on ${\mathbb{C}}^{2}_{\xi_{0},\eta_{0}}$, we have

$\partial_{0}=(\lambda-4)\xi_{0}\frac{\partial}{\partial\xi_{0}}+7\eta_{0}\frac{\partial}{\partial\eta_{0}}.$

On ${\mathbb{C}}^{2}_{\xi_{1},\eta_{1}}$, we have

$\partial_{1}=(2\lambda-1)\xi_{1}\frac{\partial}{\partial\xi_{1}}+(4-\lambda)\eta_{1}\frac{\partial}{\partial\eta_{1}}.$

And on ${\mathbb{C}}^{2}_{\xi_{2},\eta_{2}}$, we have

$\partial_{2}=7\lambda\xi_{2}\frac{\partial}{\partial\xi_{2}}+(1-2\lambda)\eta_{2}\frac{\partial}{\partial\eta_{2}}.$

Assume that $E_{1}$ is on ${\mathbb{A}}^{2}_{0}\cup{\mathbb{A}}^{2}_{1}$ and
thus defined by $\eta_{0}=0$ on ${\mathbb{A}}^{2}_{0}$ and by $\xi_{1}=0$ on
${\mathbb{A}}^{2}_{1}$. So $E_{2}$ is on
${\mathbb{A}}^{2}_{1}\cup{\mathbb{A}}^{2}_{2}$ and defined by $\eta_{1}=0$ on
${\mathbb{A}}^{2}_{1}$ and by $\xi_{2}=0$ on ${\mathbb{A}}^{2}_{2}$.

Therefore, if $\lambda=4$, then $E_{1}$ is non-invariant. Also if
$\lambda=\frac{1}{2}$, then $E_{2}$ is non-invariant. For other
$\lambda\in{\mathbb{Q}}^{+}$, the exceptional curves $E_{1}$ and $E_{2}$ are
both invariant, and precisely one of three foliation singularities is not
reduced but still semi-reduced.

Now we summarize the descriptions of the singularities on a generalized
canonical model.

######  Theorem 1.37.

Let $(X,{\mathscr{F}})$ be a generalized canonical model. Let
$\pi:Y\rightarrow X$ be the minimal resolution at points at which
${\mathscr{F}}$ is singular. Let ${\mathscr{G}}$ be the pullback foliation on
$Y$. Write $K_{\mathscr{G}}+D=\pi^{*}K_{\mathscr{F}}+E$ where $D$ and $E$ are
effective and have no common components. Then ${\mathscr{G}}$ has at worst
semi-reduced singularities, $D$ is reduced, the support of $E$ is the union of
${\mathscr{G}}$-chains which are disjoint from $D$, and all irreducible
components of $D$ are pairwise disjoint with tangency order zero. Moreover,
the connected components of $\pi$-exceptional divisors belong to one of the
following types:

  1. (1)

($A_{n}$ type) A chain of rational invariant curves consisting of an
${\mathscr{G}}$-chain, two $(-1)$-${\mathscr{G}}$-curves of self-intersections
$-2$ joined by a bad tail, a chain of $(-2)$-${\mathscr{G}}$-curves, a
generalized ${\mathscr{G}}$-chain (see Definition 1.35), or a non-invariant
smooth rational curve with tangency order zero.

  2. (2)

($D_{n}$ type) Two $(-1)$-${\mathscr{G}}$-curves of self-intersections $-2$
joined by a bad tail which itself connects to a chain of
$(-2)$-${\mathscr{G}}$-curves.

  3. (3)

(e.g.l.) Elliptic Gorenstein leaves. That is, either a cycle of
$(-2)$-${\mathscr{G}}$-curves or a rational curve with only one node.

######  Remark 1.38.

Only the eigenvalues of foliation singularities on the chain of
$(-2)$-${\mathscr{G}}$-curves can vary in a family.

###  1.7. $\varepsilon$-canonical singularities

From Example 0.1, we know that there is no Seidenberg theorem for a family.
Nevertheless, in [PS19], we can still have a Seidenberg-type theorem for a
family by allowing $\varepsilon$-canonical singularities, which will be
defined as follows.

######  Definition 1.39 ([PS19, Definition 4.1]).

Let $(X,{\mathscr{F}})$ be a foliated surface with $X$ smooth and let
$\pi:Y\rightarrow X$ be a birational morphism. Denote ${\mathscr{G}}$ be the
pullback foliation of ${\mathscr{F}}$ via $\pi$. If $E$ is a prime exceptional
divisor of $\pi$, then we define the _adjoint discrepancy_ of ${\mathscr{F}}$
along $E$ to be the function

$a({\mathscr{F}},E):[0,\infty)\rightarrow{\mathbb{R}}$

which sends $t$ to $\mbox{\rm
ord}_{E}\big{(}(1-t)(K_{\mathscr{G}}-\pi^{*}K_{\mathscr{F}})+t(K_{Y}-\pi^{*}K_{X})\big{)}$.

######  Definition 1.40 ([PS19, Definition 4.2]).

A point $x$ on a foliated surface $(X,{\mathscr{F}})$ with $X$ smooth is
called $\varepsilon$-canonical if the adjoint discrepancy of ${\mathscr{F}}$
along any divisor $E$ over $x$ satisfies $a({\mathscr{F}},E)(t)\geq 0$ for
every $t\geq\varepsilon$.

######  Lemma 1.41 ([PS19, Lemma 7.3]).

Let $({\mathcal{X}},{\mathcal{F}})$ be a smooth family over an algebraic
variety $T$ such that all fibers are smooth projective surfaces. If
$0<\varepsilon<\frac{1}{4}$, then the subset of $T$ corresponding to
foliations with isolated and $\varepsilon$-canonical singularities is a
Zariski open subset of $T$.

######  Proposition 1.42 ([PS19, Proposition 7.4]).

Given a smooth family $({\mathcal{X}},{\mathcal{F}})$ over $T$ such that all
fibers are smooth projective surfaces and a real number $\varepsilon$ with
$0<\varepsilon<\frac{1}{4}$. Then there exists a non-empty Zariski open subset
$U\subset T$ and a family $({\mathcal{Y}},{\mathcal{G}})$ over $U$ obtained
from ${\mathcal{F}}|_{U}$ by a finite composition of blowups over
(multi)-sections such that, for all $t\in U$, the foliation
${\mathcal{G}}_{t}$ has at worst $\varepsilon$-canonical singularities. In
fact, ${\mathcal{G}}_{t}$ has at worst semi-reduced singularities for all
$t\in U$.

###  1.8. Quot scheme

In this section, we recall some properties of Quot schemes.

######  Definition 1.43.

Let $X\rightarrow S$ be a scheme of finite type over a noetherian base scheme
$S$ and ${\mathcal{E}}$ be a coherent sheaf on $X$. We consider the functor

${\mathcal{Q}}uot_{{\mathcal{E}}/X/S}:(\textnormal{Sch}/S)^{\textnormal{op}}\rightarrow(\textnormal{Sets})$

sending $T\rightarrow S$ to the set

${\mathcal{Q}}uot_{{\mathcal{E}}/X/S}(T)=\left\\{({\mathcal{F}},q)\Big{|}\begin{array}[]{l}{\mathcal{F}}\in\textnormal{Coh}(X_{T}),\mbox{{\rm
Supp}}{{\mathcal{F}}}\mbox{ is proper over }T\\\ {\mathcal{F}}\mbox{ is flat
over }T,q:{\mathcal{E}}_{T}\rightarrow{\mathcal{F}}\mbox{ is
surjective.}\end{array}\right\\}\Big{/}\sim$

where $X_{T}=X\times_{S}T$,
${\mathcal{E}}_{T}=\textnormal{pr}_{X}^{*}{\mathcal{E}}$ under the projection
$\textnormal{pr}_{X}:X_{T}\rightarrow X$, and
$({\mathcal{F}},q)\sim({\mathcal{F}}^{\prime},q^{\prime})$ if $\ker q=\ker
q^{\prime}$.

######  Definition 1.44.

Let $X\rightarrow S$ be a scheme of a finite type over a noetherian base
scheme $S$ and ${\mathcal{A}}$ be a relatively ample line bundle. For any
coherent sheaf ${\mathcal{F}}$ on $X$ flat over $S$, we define the Hilbert
polynomial $\Phi_{\mathcal{F}}$ with respect to ${\mathcal{A}}$ as

$\Phi_{\mathcal{F}}(m)=\chi({\mathcal{F}}_{s}(m))=\sum_{i=0}^{\dim
X}(-1)^{i}h^{i}(X_{s},{\mathcal{F}}_{s}\otimes{\mathcal{A}}_{s}^{\otimes m})$

for any fixed $s\in S$. Note that this definition is independent of the choice
of $s\in S$.

######  Proposition 1.45.

Fix any relatively ample line bundle ${\mathcal{A}}$ on $X$. There is a
natural stratification:

${\mathcal{Q}}uot_{{\mathcal{E}}/X/S}=\coprod_{\Phi\in{\mathbb{Q}}[\lambda]}{\mathcal{Q}}uot_{{\mathcal{E}}/X/S}^{\Phi,{\mathcal{A}}}$

where
${\mathcal{Q}}uot_{{\mathcal{E}}/X/S}^{\Phi,{\mathcal{A}}}(T)=\\{({\mathcal{F}},q)\in{\mathcal{Q}}uot_{{\mathcal{E}}/X/S}(T)\,|\,\Phi_{\mathcal{F}}=\Phi\\}$.

######  Theorem 1.46 (Grothendieck).

The functor ${\mathcal{Q}}uot_{{\mathcal{E}}/X/S}^{\Phi,{\mathcal{A}}}$ is
representable by a projective scheme
$\textnormal{Quot}_{{\mathcal{E}}/X/S}^{\Phi,{\mathcal{A}}}$ over $S$.

##  2\. Boundedness of generalized canonical models

###  2.1. Local contributions of generalized canonical models

From now on, we assume that the foliation ${\mathscr{F}}$ is of general type,
that is, $K_{\mathscr{F}}$ is big. Fix $\chi(mK_{\mathscr{F}})=P(m)$ where
$P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$ is an integral valued
function. As in [HL21], we would like to study the following singular Riemann-
Roch theorem and get some boundedness results.

######  Theorem 2.1 ([Rei87],[Lan00, Section 3]).

Let $X$ be a normal complete surface and $D$ be a Weil divisor on $X$. Then we
have

$\chi(D)=\frac{1}{2}D\cdot(D-K_{X})+\chi({\mathcal{O}}_{X})+\sum_{x\in\textnormal{Sing}(X)}a(x,D)$

where $a(x,D)$ is a local contribution of ${\mathcal{O}}_{X}(D)$ at $x$
depending only on the local isomorphism class of the reflexive sheaf
${\mathcal{O}}_{X}(D)$ at $x$.

If $X$ has only quotient singularities, then the theorem follows from [Rei87].
In general, the theorem follows from [Lan00, Section 3].

######  Proposition 2.2.

Let $m$ be a non-negative integer.

  1. (1)

If $D$ is Cartier at $x$, then $a(x,D)=0$.

  2. (2)

If $(X,{\mathscr{F}},x)$ is a mild log canonical non-canonical foliation
singularity, then

$a(x,mK_{\mathscr{F}})=0\mbox{ for all }m.$

  3. (3)

If $(X,{\mathscr{F}},x)$ is a canonical foliation singularity, which is
${\mathbb{Q}}$-Gorentein but it is not terminal. Then we have two cases:

    1. (a)

${\mathscr{F}}$ is Gorenstein and $a(x,mK_{\mathscr{F}})=0$ for all $m$.

    2. (b)

${\mathscr{F}}$ is $2$-Gorenstein and

$a(x,mK_{\mathscr{F}})=\left\\{\begin{array}[]{ll}0&\mbox{ for $m$ even.}\\\
\frac{-1}{2}&\mbox{ for $m$ odd.}\end{array}\right.$

  4. (4)

If $(X,{\mathscr{F}},x)$ be a canonical foliation singularity and assume
${\mathscr{F}}$ is not ${\mathbb{Q}}$-Gorenstein at $x$, then we have

$a(x,mK_{\mathscr{F}})=\left\\{\begin{array}[]{ll}0&\mbox{ for $m=0$.}\\\
-1&\mbox{ for positive $m$.}\end{array}\right.$

  5. (5)

If $(X,{\mathscr{F}},x)$ is a terminal foliation singularity, then
$a(x,K_{\mathscr{F}})=\frac{1-n}{2n}$ where the surface singularity at $x$ is
of the type $\frac{1}{n}(1,q)$ for some $n$ and $q$.

######  Proof.

(1) follows from the definition. (2) follows from (1) since $K_{\mathscr{F}}$
is Cartier at these foliation singularities. (3) is [HL21, Proposition 2.4]
and (4) is [HL21, Proposition 2.6]. (5) follows from [HL21, Corollary 2.2] and
[Lan00, Example 5.6]. ∎

######  Remark 2.3.

In addition to any singularities on canonical models, we only allow _mild_ log
canonical foliation singularities, which have no local contributions by
Proposition 2.2. Thus, we have the following similar proposition as [HL21,
Proposition 4.1] for generalized canonical models.

######  Proposition 2.4.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. There exist
constants $B_{1}$, $B_{2}$, $B_{3}$, and $B_{4}$ such that if
$(X,{\mathscr{F}})$ is a generalized canonical model with Hilbert function
$\chi(mK_{\mathscr{F}})=P(m)$ for all $m\in{\mathbb{Z}}_{\geq 0}$, then
$K_{\mathscr{F}}^{2}=B_{1}$, $K_{\mathscr{F}}\cdot K_{X}=B_{2}$,
$\chi({\mathcal{O}}_{X})=B_{3}$, and the number of cusp singularities is equal
to $B_{4}$. Moreover, there exists some constants $C_{1}$ and $C_{2}$ such
that the number of terminal and dihedral singularities is at most $C_{1}$ and
the index at any terminal foliation singularity is at most $C_{2}$.

######  Proof.

The proof follows closely from the proof of [HL21, Proposition 4.1]. For the
reader’s convenience, we include the proof.

By Theorem 2.1, we have

$P(m)=\chi(mK_{\mathscr{F}})=\frac{1}{2}mK_{\mathscr{F}}\cdot(mK_{\mathscr{F}}-K_{X})+\chi({\mathcal{O}}_{X})+\sum_{x\in\textnormal{Sing}(X)}a(x,mK_{\mathscr{F}})$

Since $K_{\mathscr{F}}^{2}$ is the quadratic term of $P$ and
$K_{\mathscr{F}}\cdot K_{X}$ is the linear term of $P$, both of them are
fixed. Also $\chi({\mathcal{O}}_{X})=P(0)$ is also fixed. If we put $m=1$,
then we get the number

$\sum_{x\in\textnormal{Sing}(X)}a(x,K_{\mathscr{F}})$

is fixed.

Let $\Sigma_{1}$ be the set of terminal foliation singularities, $\Sigma_{2}$
be the set of dihedral quotient singularities, and $\Sigma_{3}$ be the set of
cusp singularities. Also let $\Sigma=\Sigma_{1}\cup\Sigma_{2}\cup\Sigma_{3}$.
Then by Proposition 2.2, we have

$-\sum_{x\in\textnormal{Sing}(X)}a(x,K_{\mathscr{F}})=\sum_{x\in\Sigma_{1}}\frac{n_{x}-1}{2n_{x}}+\sum_{x\in\Sigma_{2}}\frac{1}{2}+\sum_{x\in\Sigma_{3}}1\geq\frac{1}{4}|\Sigma|.$

So the number of singularities is bounded and hence

$\sum_{x\in\Sigma_{1}}\frac{1}{n_{x}}=|\Sigma|+|\Sigma_{3}|+2\sum_{x\in\textnormal{Sing}(X)}a(x,K_{\mathscr{F}})$

assumes finitely many values. By [HL21, Lemma 3.4], the indices at the
terminal foliation singularities are bounded by $C_{2}$.

Finally, let $m=(2C_{2})!$. Then by Theorem 2.1 and Proposition 2.2, we have

$P(m)=\chi(mK_{\mathscr{F}})=\frac{1}{2}mK_{\mathscr{F}}\cdot(mK_{\mathscr{F}}-K_{X})+\chi({\mathcal{O}}_{X})-|\Sigma_{3}|.$

Hence, the number of cusps is fixed. ∎

###  2.2. Minimal partial du Val resolution

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Given any
generalized canonical model $(X,{\mathscr{F}})$ of general type with
$\chi(mK_{\mathscr{F}})=P(m)$ for all $m\in{\mathbb{Z}}_{\geq 0}$. We first
consider the minimal resolution $g:X^{m}\rightarrow X$ at points at which
${\mathscr{F}}$ is _not_ terminal. Let ${\mathscr{F}}^{m}$ be the pullback
foliation on $X^{m}$. Then let $h:X^{m}\rightarrow Y$ be the (traditional)
relative canonical model of $K_{X^{m}}$ over $X$ which is given by a sequence
of contractions of smooth rational curves $C$ in the fiber of $g$ with
$C^{2}=-2$. Put ${\mathscr{G}}$ be the pushforward foliation on $Y$. Thus we
have $f:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}})$ a morphism of foliated
surfaces such that $g=f\circ h$.

By Theorem 1.37, we have $K_{{\mathscr{F}}^{m}}+D^{m}=g^{*}K_{\mathscr{F}}+E$
where each irreducible component of $E$ is an $(-1)$-${\mathscr{F}}^{m}$-curve
with self-intersection $-2$ and $D^{m}=\sum D^{m}_{i}$ is an effective reduced
divisor such that all irreducible components $D^{m}_{i}$’s are pairwise
disjoint and non-invariant with
$\textnormal{tang}({\mathscr{F}}^{m},D^{m}_{i})=0$. (See also Remark 1.34.)
Moreover, $E$ is contracted by $h$ since each irreducible component is
$(-1)$-${\mathscr{F}}^{m}$-curve with self-intersection $-2$ and any two
$(-1)$-${\mathscr{F}}^{m}$-curves are disjoint. Thus, we have
$K_{\mathscr{G}}+D=f^{*}K_{\mathscr{F}}$ where $D=h_{*}D^{m}$.

$\textstyle{(X^{m},{\mathscr{F}}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{g}$$\textstyle{(Y,{\mathscr{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(X,{\mathscr{F}})}$

######  Definition 2.5.

We call $f:(Y,{\mathscr{G}},D)\rightarrow(X,{\mathscr{F}})$ as above the
_minimal partial du Val resolution_ of the given generalized canonical model
where $(Y,{\mathscr{G}},D)$ is a foliated triple (see Definition 2.6).

######  Definition 2.6 (Foliated triple).

A _foliated triple_ is a triple $(X,{\mathscr{F}},D)$ where
$(X,{\mathscr{F}})$ is a foliated surface (see Definition 1.10) and $D$ is a
reduced divisor.

In addition, we may view a foliated surface
$(X,{\mathscr{F}})=(X,{\mathscr{F}},\emptyset)$ as a foliated triple.

######  Remark 2.7.

If $(Y,{\mathscr{G}},D)$ is the minimal partial du Val resolution of some
generalized canonical models $(X,{\mathscr{F}})$ of general type, then
$(X,{\mathscr{F}})$ is unique. More precisely, if $(Y,{\mathscr{G}},D)$ is the
minimal partial du Val resolution of generalized canonical models
$(X_{1},{\mathscr{F}}_{1})$ and $(X_{2},{\mathscr{F}}_{2})$. We have
$f_{1}^{*}K_{{\mathscr{F}}_{1}}=K_{\mathscr{G}}+D=f_{2}^{*}K_{{\mathscr{F}}_{2}}$.
Let $C$ be a curve contracted by $f_{1}$. So we have
$f_{2}^{*}K_{{\mathscr{F}}_{2}}\cdot C=f_{1}^{*}K_{{\mathscr{F}}_{1}}\cdot
C=0$. Since $K_{{\mathscr{F}}_{2}}$ is numerically ample by Lemma 1.30, $C$ is
also contracted by $f_{2}$.

######  Definition 2.8.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Let
${\mathcal{S}}_{P}$ be the set of foliated triples $(Y,{\mathscr{G}},D)$ which
are minimal partial du Val resolutions of GENERALIZED CANONICAL models
$(X,{\mathscr{F}})$ of general type with $\chi(mK_{\mathscr{F}})=P(m)$. Also
let ${\mathcal{S}}_{P}^{C}\subset{\mathcal{S}}_{P}$ be the subset consisting
of foliated triples $(Y,{\mathscr{G}},\emptyset)$ which are minimal partial du
Val resolutions of CANONICAL models $(X,{\mathscr{F}})$ of general type with
$\chi(mK_{\mathscr{F}})=P(m)$.

######  Definition 2.9.

Let $\mu:{\mathcal{Y}}\rightarrow T$ be a projective morphism between two
quasi-projective varieties of finite type. We say a (pre-)foliation
${\mathcal{F}}$ on ${\mathcal{Y}}$ is a family of (pre-)foliations with
respect to $\mu$ if ${\mathcal{F}}\subset T_{X}$ factors through the
relatively tangent sheaf of $\mu$.

Now we would like to prove the following boundedness for the set
${\mathcal{S}}_{P}$.

######  Theorem 2.10.

The set ${\mathcal{S}}_{P}$ is bounded. More precisely, there exists a
projective morphism $\mu:{\mathcal{Y}}\rightarrow T$ and a family of pre-
foliations ${\mathcal{F}}$ on ${\mathcal{Y}}$ with respect to $\mu$ where
${\mathcal{Y}}$ and $T$ are quasi-projective varieties of finite type such
that, for any $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, there is a $t\in T$
and an isomorphism $\rho:Y\rightarrow{\mathcal{Y}}_{t}$ such that
${\mathscr{G}}(-D)\cong\rho^{*}({\mathcal{F}}|_{{\mathcal{Y}}_{t}})$.

######  Proof.

The proof follows closely from the proof of [Che19, Theorem 0.1] for canonical
models. For the reader’s convenience, we include the proof. The difference is
that now we possibly have non-invariant exceptional divisors $D=\sum D_{j}$.

Let $(X,{\mathscr{F}})$ be the generalized canonical model of
$(Y,{\mathscr{G}},D)$ (see Remark 2.7) and
$f:(Y,{\mathscr{G}},D)\rightarrow(X,{\mathscr{F}})$ be the associated
morphism. Notice that $f^{*}K_{\mathscr{F}}=K_{\mathscr{G}}+D$. Let $E=\cup
E_{i}$ be the exceptional divisor of $f:Y\rightarrow X$. We divide the proof
into several steps.

  1. (1)

Claim. $K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D)$ is nef (resp. ample) when
$m\geq 3$ (resp. $m\geq 4$).

######  Proof.

Given any irreducible curve $C$ on $Y$.

    1. (a)

If $C=E_{i}$ for some $i$, then

$(K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D))\cdot C=K_{Y}\cdot C>0$

since $K_{Y}$ is ample over $X$.

    2. (b)

If $C$ is not contained in the exceptional divisor, then we have two cases:

      1. (i)

If $K_{Y}\cdot C\geq 0$, then

$\displaystyle(K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D))\cdot C$
$\displaystyle\geq$ $\displaystyle mi({\mathscr{G}})(K_{\mathscr{G}}+D)\cdot
C$ $\displaystyle=$ $\displaystyle mi({\mathscr{G}})K_{{\mathscr{F}}}\cdot
f_{*}C>0$

since $K_{\mathscr{F}}$ is numerically ample (see Lemma 1.30).

      2. (ii)

If $K_{Y}\cdot C<0$, by [Fuj12, Proposition 3.8], every $K_{Y}$ negative
extremal ray is spanned by a rational curve $C$ with $0<-K_{Y}\cdot C\leq 3$.
Thus, if we write $C\equiv\sum a_{i}C_{i}+B$ where $B\cdot K_{Y}\geq 0$, the
$C_{i}$’s are $K_{Y}$ negative extremal rays, the $a_{i}$’s are non-negative,
and at least one of $a_{i}$ is positive. By what we have seen above, we have

$(K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D))\cdot B\geq 0.$

Thus, we have

$\begin{array}[]{l}(K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D))\cdot C\\\
\geq\sum_{i}a_{i}\Big{(}K_{Y}\cdot
C_{i}+mi({\mathscr{G}})(K_{\mathscr{G}}+D)\cdot C_{i}\Big{)}\\\
\geq\sum_{i}a_{i}(-3+m)\geq 0\mbox{ (resp. $>0$ )}.\end{array}$

When $m\geq 4$, we notice that

$K_{Y}+mi({\mathscr{G}})(K_{\mathscr{G}}+D)=K_{Y}+3i({\mathscr{G}})(K_{\mathscr{G}}+D)+(m-3)i({\mathscr{G}})(K_{\mathscr{G}}+D)$

is big since it is the sum of a nef divisor and a big divisor.

  2. (2)

Claim. $i(K_{Y})$ and $i({\mathscr{G}})$ are bounded at foliation
singularities.

######  Proof.

By Proposition 2.4, $i(K_{Y})$ and $i({\mathscr{G}})$ are bounded in terms of
$\chi(mK_{\mathscr{F}})$ at terminal foliation singularities. Other foliation
singularities are at du Val singularities at which ${\mathscr{G}}$ is
canonical. Then we have $i(K_{Y})=1$ and $i({\mathscr{G}})\leq 2$ by
Proposition 2.2(3).

  3. (3)

Claim. The embedding dimension at each point at which ${\mathscr{G}}$ is
singular is bounded.

######  Proof.

By Proposition 2.4, the index at each point at which ${\mathscr{G}}$ is
terminal is bounded in terms of $\chi(mK_{\mathscr{F}})$. Actually, it is
shown that the order of each cyclic group acting at each point at which
${\mathscr{G}}$ is terminal is bounded. Thus, the embedding dimension at each
point at which ${\mathscr{G}}$ is terminal is bounded. See for example [Rei,
Corollary 2.5]. If the action is $\frac{1}{n}(1,q)$ and
$\frac{n}{n-q}=[a_{1},\ldots,a_{k}]$, then the embedding dimension is at most
$k+2$.

Other singularities on $Y$ are du Val singularities, at which the embedding
dimensions are $3$.

Thus, we have a uniform bound $\delta$ such that $\delta_{\zeta}\leq\delta$
for $|\zeta|=2$. See Remark 1.2.

  4. (4)

We consider the ${\mathbb{Q}}$-Cartier divisor

$\displaystyle L$ $\displaystyle=$ $\displaystyle\alpha
i(K_{Y})(K_{Y}+4i({\mathscr{G}})(K_{\mathscr{G}}+D))-K_{Y}$ $\displaystyle=$
$\displaystyle(\alpha
i(K_{Y})-1)(K_{Y}+3i({\mathscr{G}})(K_{\mathscr{G}}+D))+(\alpha
i(K_{Y})+3)i({\mathscr{G}})(K_{\mathscr{G}}+D).$

Since $i({\mathscr{G}})(K_{\mathscr{G}}+D)$ is Cartier, we can fix an
$\alpha$, which only depends on the Hilbert function $\chi(mK_{\mathscr{F}})$,
such that for any irreducible curve $C$, we have

$\displaystyle L^{2}$ $\displaystyle\geq$ $\displaystyle(\alpha
i(K_{Y})+3)^{2}>\delta$ $\displaystyle L\cdot C$ $\displaystyle\geq$
$\displaystyle\alpha i(K_{Y})+3>\delta/2.$

By Theorem 1.1, $K_{Y}+\lceil L\rceil=\alpha
i(K_{Y})(K_{Y}+4i({\mathscr{G}})(K_{\mathscr{G}}+D))$ separates points and
tangents. Hence by Theorem 1.3, it is very ample.

  5. (5)

Claim. The set of surfaces $X$ where $(X,{\mathscr{F}})\in{\mathcal{S}}_{P}$
forms a bounded family.

######  Proof.

Let $\varphi:Y\rightarrow{\mathbb{P}}^{N}$ be an embedding defined via
$|K_{Y}+L|$. We may assume that $N$ is minimal in the sense that $\varphi(Y)$
is not contained in any hyperplane of ${\mathbb{P}}^{N}$ . In this case, it is
known that

$\deg\varphi(Y)\geq 1+(N-\dim\varphi(Y))=N-1.$

By [HL21, Lemma 3.5], we know that $\gamma K_{{\mathscr{F}}}-K_{X}$ is pseudo-
effective for

$\gamma=\max\left\\{\frac{2K_{{\mathscr{F}}}\cdot
K_{X}}{K_{{\mathscr{F}}}^{2}}+3i_{\mathbb{Q}}({\mathscr{F}}),0\right\\}$

which is bounded in terms of $\chi(mK_{\mathscr{F}})$. Thus,

$\displaystyle\deg\varphi(Y)$ $\displaystyle=$ $\displaystyle\mbox{{\rm
vol}}(K_{Y}+L)$ $\displaystyle=$ $\displaystyle\mbox{{\rm vol}}\Big{(}\alpha
i(K_{Y})(K_{Y}+4i({\mathscr{G}})(K_{\mathscr{G}}+D))\Big{)}$
$\displaystyle\leq$ $\displaystyle\mbox{{\rm vol}}\Big{(}\alpha
i(K_{Y})(K_{X}+4i({\mathscr{G}})K_{{\mathscr{F}}})\Big{)}$ $\displaystyle\leq$
$\displaystyle\mbox{{\rm vol}}\Big{(}\alpha
i(K_{Y})(K_{X}+4i({\mathscr{G}})K_{{\mathscr{F}}})+\alpha i(K_{Y})(\gamma
K_{{\mathscr{F}}}-K_{X})\Big{)}$ $\displaystyle=$ $\displaystyle\mbox{{\rm
vol}}\Big{(}\alpha i(K_{Y})(\gamma+4i({\mathscr{G}}))K_{{\mathscr{F}}}\Big{)}$
$\displaystyle=$ $\displaystyle\Big{(}\alpha
i(K_{Y})(\gamma+4i({\mathscr{G}}))K_{{\mathscr{F}}}\Big{)}^{2}$

which is bounded in terms of $\chi(mK_{\mathscr{F}})$. Therefore $N$ is also
bounded.

Hence, by the boundedness result about Chow varieties, we have the boundedness
of surfaces $Y$ where $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$.

  6. (6)

Now, we would like to show the boundedness of the pre-foliations
${\mathscr{G}}(-D)$.

We have shown that there exists a projective morphism
$\mu:{\mathcal{Y}}\rightarrow T$ where ${\mathcal{Y}}$ and $T$ are quasi-
projective varieties of finite type such that for any
$(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, there is a $t\in T$ such that
${\mathcal{Y}}_{t}\cong Y$ and by construction we have

$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{{\mathbb{P}}^{N}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

and
${\mathcal{O}}_{Y}(m(K_{\mathscr{G}}+D)+nK_{Y})={\mathcal{O}}_{{\mathcal{Y}}_{t}}(1)$
where $m=4\alpha i(K_{Y})i({\mathscr{G}})$ and $n=\alpha i(K_{Y})$. Note that
$\alpha,i(K_{Y})$, and $i({\mathscr{G}})$ are fixed and determined by the
function $P$. Without loss of generality, we may assume that $T$ is the
closure of $T^{\prime}$ where

$T^{\prime}=\\{t\,|\,{\mathcal{Y}}_{t}\cong Y\mbox{ for some
}(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}\\}.$

  7. (7)

We may assume that $T$ is reduced.

######  Claim.

We may assume that $T$ is irreducible.

######  Proof.

Let $T=\bigcup_{i}T_{i}$ be the decomposition of $T$ into irreducible
components. If there is a family of pre-foliations ${\mathcal{F}}_{i}$ on each
${\mathcal{Y}}_{i}={\mathcal{Y}}\times_{T}T_{i}$ such that for any
$(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, there exist an $i$ and $t\in T_{i}$
with $(Y,{\mathscr{G}}(-D))\cong({\mathcal{Y}}_{i},{\mathcal{F}}_{i})_{t}$,
then the disjoint union of these families gives the required family.

######  Claim.

We may assume that $T$ is smooth.

######  Proof.

This is achieved by the existence of a finite stratification of $T$ by smooth
locally closed subsets.

  8. (8)

Recall that, by Serre’s criterion, ${\mathcal{Y}}$ is normal if and only if it
is $R_{1}$ and $S_{2}$.

######  Claim.

We may assume that ${\mathcal{Y}}$ is normal.

######  Proof.

Suppose that ${\mathcal{Y}}_{t}$ is $R_{1}$ (resp. $S_{2}$), then there exists
an open subset $U\subset T$ containing $t$ such that
${\mathcal{Y}}_{U}:={\mathcal{Y}}\times_{T}U$ is $R_{1}$ (resp. $S_{2}$) and
every fiber ${\mathcal{Y}}_{t}$ is also $R_{1}$ (resp. $S_{2}$) for $t\in U$.
(For a reference, see [Gro66, Théorème 12.2.4].) It follows that there is an
open subset $V\subset T$ such that
${\mathcal{Y}}_{V}:={\mathcal{Y}}\times_{T}V$ is normal, every fiber
${\mathcal{Y}}_{t}$ is normal for $t\in V$, and for all $Y\in{\mathcal{S}}$,
there exists a $t\in V$ such that $Y\cong{\mathcal{Y}}_{t}$. Hence, by
shrinking $T$, we may assume that ${\mathcal{Y}}$ is normal.

  9. (9)

Claim. We may assume that $K_{\mathcal{Y}}$ is ${\mathbb{Q}}$-Cartierness.

######  Proof.

This is well-known; see, for example, [Sim94, Lemma 1.27]. We include proof
for the convenience of the reader.

Since ${\mathcal{Y}}$ is normal, the canonical sheaf $K_{\mathcal{Y}}$ is
defined as a divisorial sheaf. By assumption, $i(K_{Y})K_{{\mathcal{Y}}_{t}}$
is Cartier for

$t\in T^{\circ}:=\\{t\in T\,|\,\mbox{there exists
}(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}\mbox{ such that
}{\mathcal{Y}}_{t}\cong Y\\}.$

Note that $T^{\circ}$ is a dense subset of $T$. For any $t\in T^{\circ}$, we
fix a $d$ such that $i(K_{Y})K_{{\mathcal{Y}}_{t}}(d)$ is generated by global
sections. Then by replacing $T$ by an open subset, we may assume that the map

$\mu_{*}{\mathcal{O}}_{\mathcal{Y}}(i(K_{Y})K_{\mathcal{Y}}(d))\otimes
k(t)\rightarrow H^{0}({\mathcal{Y}}_{t},i(K_{Y})K_{{\mathcal{Y}}_{t}}(d))$

is surjective by [Har77, Theorem 12.8 and Corollary 12.9]. Since
$i(K_{Y})K_{{\mathcal{Y}}_{t}}(d)$ is locally generated by some section of
$H^{0}({\mathcal{Y}}_{t},i(K_{Y})K_{{\mathcal{Y}}_{t}}(d))$, by lifting this
section to a section of
$H^{0}({\mathcal{Y}},i(K_{\mathcal{Y}})K_{\mathcal{Y}}(d))$ using the
surjection above and by Nakayama’s lemma, $i(K_{Y})K_{\mathcal{Y}}(d)$ is
locally generated by one section on a neighborhood of ${\mathcal{Y}}_{t}$.
Therefore, $i(K_{Y})K_{\mathcal{Y}}$ is Cartier on a neighborhood of
${\mathcal{Y}}_{t}$. Shrinking $T$, we may assume that
$i(K_{Y})K_{\mathcal{Y}}$ is Cartier.

  10. (10)

Now we consider the function

$\displaystyle\Phi(\ell)$ $\displaystyle=$
$\displaystyle\chi(Y,{\mathcal{O}}_{X}(K_{\mathscr{G}}+D+\ell(m(K_{\mathscr{G}}+D)+nK_{Y})))$
$\displaystyle=$
$\displaystyle\chi(X,{\mathcal{O}}_{X}((m\ell+1)(K_{\mathscr{G}}+D)+n\ell
K_{Y}))$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}(m\ell+1)(K_{\mathscr{G}}+D)+n\ell
K_{Y}\Big{)}\cdot\Big{(}(m\ell+1)(K_{\mathscr{G}}+D)+(n\ell-1)K_{Y}\Big{)}$
$\displaystyle\quad+\chi({\mathcal{O}}_{Y})+\sum_{y\in\textnormal{Sing}(Y)}a\big{(}y,(m\ell+1)(K_{\mathscr{F}}+D)+n\ell
K_{Y}\big{)}$

Note that $m(K_{\mathscr{G}}+D)$ and $nK_{Y}$ are Cartier, we have that

$a\big{(}y,(m\ell+1)(K_{\mathscr{G}}+D)+n\ell
K_{Y}\big{)}=a(y,K_{\mathscr{G}}+D)$

for any singularity $y\in Y$. Also we have

$\displaystyle\chi({\mathcal{O}}_{Y})+\sum_{y\in\textnormal{Sing}(Y)}a(y,K_{\mathscr{G}}+D)$
$\displaystyle=$
$\displaystyle\chi(K_{\mathscr{G}}+D)-\frac{1}{2}(K_{\mathscr{G}}+D)\cdot(K_{\mathscr{G}}+D-K_{Y})$
$\displaystyle=$ $\displaystyle
P(1)-\frac{1}{2}(K_{\mathscr{G}}+D)\cdot(K_{\mathscr{G}}+D-K_{Y}).$

Thus, we have

$\displaystyle\Phi(\ell)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}(m\ell+1)(K_{\mathscr{G}}+D)+n\ell
K_{Y}\Big{)}\cdot\Big{(}(m\ell+1)(K_{\mathscr{G}}+D)+(n\ell-1)K_{Y}\Big{)}$
$\displaystyle\quad+P(1)-\frac{1}{2}(K_{\mathscr{G}}+D)\cdot(K_{\mathscr{G}}+D-K_{Y})$

is a polynomial with coefficients in terms of $P(1)$,
$(K_{\mathscr{G}}+D)\cdot K_{Y}=K_{\mathscr{F}}\cdot K_{X}$,
$(K_{\mathscr{G}}+D)^{2}=K_{\mathscr{F}}^{2}$, and $K_{Y}^{2}$. Since the
function $P$ is fixed, we have $K_{\mathscr{F}}^{2}$ and $K_{\mathscr{F}}\cdot
K_{X}$ are fixed by Proposition 2.4. Moreover, $K_{Y}^{2}$ has only finitely
many possibilities since the surfaces $Y$ are in a bounded family. Therefore,
we have finitely many possible polynomials $\Phi(\ell)$.

Notice that $\frac{m+1}{n-1}>\frac{m}{n}=4i({\mathscr{G}})\geq 4$. So we have
$(m+1)(K_{\mathscr{G}}+D)+(n-1)K_{Y}$ is ample by step (1) and thus
$\Phi(1)=h^{0}(Y,{\mathcal{O}}_{Y}((m+1)(K_{\mathscr{G}}+D)+nK_{Y}))$. Let
$\varphi$ be the maximum of $\Phi(1)$ among all possible polynomials
$\Phi(\ell)$.

  11. (11)

For any polynomial $\Psi(\ell):=\Phi(\ell)-s$ where $0\leq s\leq\varphi$, we
consider the Quot scheme

$Q:=\textnormal{Quot}_{\Omega_{{\mathcal{Y}}/T}/{\mathcal{Y}}/T}^{\Psi(\ell),{\mathcal{O}}_{\mathcal{Y}}(1)},$

which is projective over $T$. Let ${\mathcal{U}}:={\mathcal{Y}}\times_{T}Q$.
By a flattening stratification on $Q$, we may assume that the projection
morphism ${\mathcal{U}}\rightarrow Q$ is flat. Then by the universal property
of Quot schemes, there is a (universal) quotient sheaf ${\mathcal{L}}$ of
$\Omega_{{\mathcal{U}}/Q}$ on ${\mathcal{U}}$. By [Gro66, Théorème 12.2.1
(v)], the locus

$\\{q\in Q\,:\,{\mathcal{L}}^{\vee}|_{{\mathcal{U}}_{q}}\mbox{ is
reflexive}\\}$

is open. After shrinking $Q$, we may assume that ${\mathcal{L}}^{\vee}$ is
reflexive over any point of $Q$ and ${\mathcal{L}}^{\vee}$ is reflexive. Then
${\mathcal{L}}^{\vee}\subset T_{{\mathcal{U}}/Q}\subset T_{\mathcal{U}}$ gives
a family of pre-foliations on ${\mathcal{U}}$ with respect to
${\mathcal{U}}\rightarrow Q$.

Now for any $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, we dualize the
following short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{T_{Y}/{\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and get the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(T_{Y}/{\mathscr{G}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\vee}}$$\textstyle{\Omega_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{coker}(\tau^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{coker}(\tau^{\vee})\otimes{\mathcal{O}}_{Y}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Y}(K_{\mathscr{G}}+D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}\otimes{\mathcal{O}}_{Y}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where ${\mathcal{Z}}$ is the sheaf supported at a $0$-dimensional subscheme
whose length is at most $\varphi$. Then we have

$\chi(Y,{\mathcal{O}}_{Y}((K_{\mathscr{G}}+D)+\ell(m(K_{\mathscr{G}}+D)+nK_{Y})))-\chi(Y,\textnormal{coker}(\tau^{\vee})\otimes{\mathcal{O}}_{Y}(D+\ell(m(K_{\mathscr{G}}+D)+nK_{Y})))$

is a non-negative constant $\leq\varphi$. Thus, the function

$\chi(Y,\textnormal{coker}(\tau^{\vee})\otimes{\mathcal{O}}_{Y}(D+\ell(m(K_{\mathscr{G}}+D)+nK_{Y})))$

has the form $\Phi(\ell)-s$. So by the construction of $Q$, there is a $q\in
Q$ such that $\rho:Y\rightarrow{\mathcal{U}}_{q}$ is an isomorphism and
$\textnormal{coker}(\tau^{\vee})\otimes{\mathcal{O}}_{Y}(D)\cong\rho^{*}({\mathcal{L}}|_{{\mathcal{U}}_{q}})$
and thus

${\mathscr{G}}(-D)\cong\textnormal{coker}(\tau^{\vee})^{\vee}\otimes{\mathcal{O}}_{Y}(-D)\cong(\textnormal{coker}(\tau^{\vee})\otimes{\mathcal{O}}_{Y}(D))^{\vee}\cong\rho^{*}({\mathcal{L}}|_{{\mathcal{U}}_{q}})^{\vee}.$

Notice that $\rho^{*}({\mathcal{L}}|_{{\mathcal{U}}_{q}})^{\vee}$ and
$\rho^{*}({\mathcal{L}}^{\vee}|_{{\mathcal{U}}_{q}})$ are isomorphic at the
generic point of ${\mathcal{U}}_{q}$. By [HL21, Lemma 1.8], they are indeed
isomorphic. Hence, ${\mathcal{L}}^{\vee}$ gives the desired family of pre-
foliations.

∎

######  Remark 2.11.

After some stratification on ${\mathcal{Q}}$, we have
$({\mathcal{U}},{\mathcal{F}},{\mathcal{D}})\rightarrow{\mathcal{Q}}$ is a
family of foliated triples (see Defition 2.17) for some reduced divisor
${\mathcal{D}}$.

###  2.3. Boundedness of non-cusp singularities

One application of Theorem 2.10 is the boundedness of non-cusp singularities.
Before we show the boundedness of non-cusp singularities, we need the
following lemma.

######  Lemma 2.12.

Let ${\mathcal{X}}\rightarrow T$ be a flat family of projective varieties of
dimension $n$ and $D$ is a ${\mathbb{Q}}$-Cartier divisor on ${\mathcal{X}}$
and flat over $T$. If there is a $t_{0}\in T$ such that $D_{t_{0}}$ is nef
(resp. strictly nef) and big, then there exists an open subset $U\subset T$
containing $t_{0}$ such that $D_{t}$ is nef (resp. strictly nef) and big for
all $t\in U$. Moreover, there exists an $N$ such that
$h^{0}({\mathcal{X}}_{t},ND_{t})\geq 1$ for all $t\in T$.

######  Proof.

We may assume that $T$ is irreducible. Indeed, suppose the lemma holds
whenever $T$ is irreducible. In general, we write $T=\bigcup_{j=1}^{m}T^{j}$
as the decomposition of irreducible components $T^{j}$. Since $t_{0}\in T$, we
have $t_{0}\in T^{j}$ for some $j$. Let’s say $t_{0}\in T^{1}$. By the
assumption, we have an open subset $U^{1}\subset T^{1}$ containing $t_{0}$
such that $D_{t}$ is nef (resp. strictly nef) and big for all $t\in U^{1}$. If
$U^{1}$ is contained in $T\backslash\bigcup_{j=2}^{m}T^{j}$ which is an open
subset of $T$, then $U^{1}$ is an open subset of $T$ and we are done. If not,
there is a point $t_{1}\in
U^{1}\cap\bigcup_{j=2}^{m}T^{j}\subset\bigcup_{j=2}^{m}T^{j}$. By induction on
the number of the irreducible components of $T$, we have an open subset
$U^{2}\subset\bigcup_{j=2}^{m}T^{j}$ containing $t_{1}$ such that $D_{t}$ is
nef (resp. strictly nef) and big for all $t\in U^{2}$. Note that
$C^{1}:=T^{1}\backslash U^{1}$ and $C^{2}:=\bigcup_{j=2}^{m}T^{j}\backslash
U^{2}$ are closed subsets of $T$. Thus $U:=T\backslash(C^{1}\cup C^{2})$ is an
open subset of $T$ containing $t_{0}$ such that $D_{t}$ is nef (resp. strictly
nef) and big for all $t\in U$.

So from now on, we assume that $T$ is irreducible. Since $D_{t_{0}}$ is nef,
we have $D_{t}$ is nef for very general $t\in
T^{*}=T\backslash\bigcup_{i=1}^{\infty}T_{i}$ where $T_{i}$’s are all proper
closed subsets of $T$. (For a reference, see [Laz04, Proposition 1.4.14].)
Note that, for all $t\in T^{*}$, we have $D_{t}^{n}=D_{t_{0}}^{n}>0$ since
$D_{t_{0}}$ is nef and big. Hence, $D_{t}$ is also nef and big for all $t\in
T^{*}$.

Let $S_{N}:=\\{t\in T|\,h^{0}({\mathcal{X}}_{t},ND_{t})\geq 1\\}$ which is a
closed subset in $T$ by the upper semi-continuity. Note that we have

$T({\mathbb{C}})=\left(\bigcup_{N}S_{N}({\mathbb{C}})\right)\cup\left(\bigcup_{i}T_{i}({\mathbb{C}})\right).$

By the Baire category theorem, there is an $N$ such that $S_{N}=T$. Thus, the
relative base locus of $D$ is proper. Let $U$ be an open subset of $T$ such
that all irreducible components of the relative base locus of $D$ are flat
over $U$. Notice that it is possible that $t_{0}$ is _not_ in $U$.

Now for any $u\in U$, if there is a curve $C_{u}$ such that $D_{u}\cdot
C_{u}<0$ (resp. $\leq 0$), then $C_{u}$ is in the relative base locus of $D$
and moves in the family over $U$. So for any $t\in T^{*}\cap U$, we have
$D_{t}\cdot C_{t}=D_{u}\cdot C_{u}<0$ (resp. $\leq 0$), which is a
contradiction. Therefore, $D_{u}$ is nef (resp. strictly nef) for all $u\in
U$. Moreover, $D_{u}^{n}=D_{t}^{n}>0$ where $t\in T^{*}\cap U$, then $D_{u}$
is also big for $u\in U$.

If $t_{0}\in U$, then we are done. If not, that is $t_{0}\in T\backslash U$,
then by induction on $\dim T$, there exists an open subset $V\subset
T\backslash U$ containing $t_{0}$ such that $D_{t}$ is nef (resp. strictly
nef) and big for all $t\in V$. So $(T\backslash U)\backslash V$ is a closed
subset of $T\backslash U$, which is a closed subset of $T$. Thus,
$(T\backslash U)\backslash V$ is a closed subset of $T$. Therefore $V\cup
U=T\backslash\big{(}(T\backslash U)\backslash V\big{)}\subset T$ is an open
subset containing $t_{0}$ such that $D_{t}$ nef (resp. strictly nef) and big
for all $t\in V\cup U$. ∎

This proof also gives the following Corollary.

######  Corollary 2.13.

Every $(K_{\mathscr{G}}+D)$-trivial curve moves in the family over $U$.

######  Proposition 2.14.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. There exist
constants $C_{1}$, $C_{2}$, and $C_{3}$ such that, for any generalized
canonical model $(X,{\mathscr{F}})$ with Hilbert function
$\chi(X,mK_{{\mathscr{F}}})=P(m)$, the index (resp. embedding dimension) at
the dihedral singularities is at most $C_{1}$ (resp. $C_{2}$) and the number
of dihedral singularities is at most $C_{3}$. Moreover, there exist constants
$B_{1}$, $B_{2}$, and $B_{3}$ such that the index (resp. embedding dimension)
at the non-terminal foliation singularities which are cyclic quotient
singularities is at most $B_{1}$ (resp. $B_{2}$) and the number of non-
terminal foliation singularities which are cyclic quotient singularities is at
most $B_{3}$.

######  Proof.

The proof follows closely from the proof of [Che19, Proposition 3.1] for
canonical models. For the reader’s convenience, we include the proof.

By Theorem 2.10, there exists a projective morphism
$\mu:{\mathcal{Y}}\rightarrow T$ of quasi-projective varieties of finite type
and a family of pre-foliations ${\mathcal{F}}$ on ${\mathcal{Y}}$ such that
for any $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, there exists a $t\in T$ and
an isomorphism $\rho:Y\rightarrow{\mathcal{Y}}_{t}$ such that
${\mathscr{G}}(-D)\cong\rho^{*}({\mathcal{F}}|_{{\mathcal{Y}}_{t}})$. We may
assume that

$T^{*}:=\\{t\in
T\,|\,({\mathcal{Y}}_{t},{\mathcal{F}}_{t})\cong(Y,{\mathscr{G}}(-D))\mbox{
for some }(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}\\}$

is dense in $T$.

Notice that $K_{\mathcal{F}}$ is relatively big. Then
$K_{\mathcal{F}}\sim_{\mathbb{Q}}{\mathcal{A}}+{\mathcal{E}}$ where
${\mathcal{A}}$ is a relative ample divisor and ${\mathcal{E}}$ is an
effective divisor. Note that, for general $t$ and for any irreducible curve
$E\subset{\mathcal{Y}}_{t}$ over a non-terminal foliation singularity, we have
$K_{\mathcal{F}}\cdot E=(K_{\mathscr{G}}+D)\cdot E=0$. Thus,
${\mathcal{E}}\cdot E<0$. That is, $E$ is in the support of ${\mathcal{E}}$.
By shrinking $T$ if necessary, we may assume that all irreducible components
of ${\mathcal{E}}$ are flat over $T$. Let ${\mathcal{E}}_{j}$’s be those
irreducible components of ${\mathcal{E}}$ of codimension 1 in ${\mathcal{Y}}$.

Let $\pi:\widetilde{{\mathcal{Y}}}\rightarrow{\mathcal{Y}}$ be the log
resolution of $({\mathcal{Y}},\sum_{j}{\mathcal{E}}_{j})$ with the exceptional
divisors $\sum{\mathcal{F}}_{\ell}$. By shrinking $T$ further, we may assume
that $\nu:=\mu\circ\pi$ is smooth and ${\mathcal{F}}_{\ell}$ is flat over $T$.
Let $\widetilde{{\mathcal{E}}_{j}}$ be the proper transform of
${\mathcal{E}}_{j}$ via $\pi$.

Note that $\Lambda:=\\{\widetilde{{\mathcal{E}}_{j}},{\mathcal{F}}_{\ell}\\}$
is a finite set. So there are only finitely many possible connected subsets of
$\Lambda$ with dual graphs of $D_{n}$ type or $A_{n}$ type.

By [Bri68, Satz 2.9 and Satz 2.11], the intersection matrix of a dihedral
singularity determines the embedding dimension and the group acting on it.
Since the intersection matrix remains the same in the family, we have the
boundedness of the embedding dimension and the order of the group acting on
these singularities. Hence the index of $X$ at the dihedral singularities is
also bounded. Similarly, by [Rei], the embedding dimension and the index of a
cyclic quotient singularity is determined by the intersection matrix.
Therefore, we have the boundedness of the embedding dimension and of the index
at the cyclic quotient singularities. ∎

######  Remark 2.15.

  1. (1)

It is possible that some connected subset of $\Lambda$ of $D_{n}$ type or of
$A_{n}$ type is redundant.

  2. (2)

By the same argument as above, we also show the boundedness of the cusps.
Precisely, the number of exceptional divisors and the intersection matrix over
any cusp singularity are bounded.

We also have the following theorem.

######  Theorem 2.16.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. There exists
an integer $m_{P}$ such that if $(X,{\mathscr{F}})$ is a generalized canonical
model of general type and $\chi(mK_{\mathscr{F}})=P(m)$ for all $m\geq 0$,
then for any $m>0$ divisible by $m_{P}$, $|mK_{\mathscr{F}}|$ defines a
birational map which is an isomorphism on the complement of the cusp
singularities.

######  Proof.

This follows closely from [HL21, Theorem 4.3]. See also [Che19, Theorem 3.3].
For the reader’s convenience, we include the proof.

Let $g:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}})$ be the minimal
resolution of cusps. Then $K_{\mathscr{G}}=g^{*}K_{\mathscr{F}}$ and
$i({\mathscr{G}})=i_{\mathbb{Q}}({\mathscr{F}})$. (See Definition 1.14.)

######  Claim.

$K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}}$ is nef.

######  Proof (Claim).

Given any irreducible curve $C$ on $Y$.

  1. (1)

If $C$ is contracted by $g$, then

$\displaystyle(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot C$
$\displaystyle=$ $\displaystyle K_{Y}\cdot C=-2-C^{2}\geq 0$

since $C^{2}\leq-2$ by the minimality of $g$.

  2. (2)

If $C$ is not contracted by $g$,

    1. (a)

If $K_{Y}\cdot C\geq 0$, then

$(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot C\geq
3i({\mathscr{G}})K_{{\mathscr{F}}}\cdot g_{*}C>0$

by Lemma 1.30.

    2. (b)

If $K_{Y}\cdot C<0$, by [Fuj12, Proposition 3.8], every $K_{Y}$ negative
extremal ray is spanned by a rational curve $C$ with $0<-K_{Y}\cdot C\leq 3$.
Thus, if we write $C\equiv\sum a_{i}C_{i}+D$ where $D\cdot K_{Y}\geq 0$, the
$C_{i}$’s are $K_{Y}$ negative extremal rays, the $a_{i}$’s are non-negative,
and at least one of $a_{i}$ is positive. By what we have seen above,
$(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot D\geq 0$. Thus,

$\displaystyle(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot C$
$\displaystyle=$
$\displaystyle\sum_{i}a_{i}\Big{(}(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot
C_{i}\Big{)}+(K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}})\cdot D$
$\displaystyle\geq$ $\displaystyle\sum_{i}a_{i}(-3+3)+0=0$

This completes the proof of the nefness of
$K_{Y}+3i({\mathscr{G}})K_{\mathscr{G}}$.

Now, by [HL21, Lemma 3.5], we know that $\gamma K_{\mathscr{G}}-K_{Y}$ is big
for any fixed integer $\gamma$ with

$\gamma>\left\lceil\max\left\\{\frac{2K_{{\mathscr{F}}}\cdot
K_{X}}{K_{{\mathscr{F}}}^{2}}+3i_{\mathbb{Q}}({\mathscr{F}}),0\right\\}\right\rceil.$

Note that $L:=(\beta-\gamma)K_{\mathscr{G}}+(\gamma K_{\mathscr{G}}-K_{Y})$ is
pseudo-effective for any $\beta>\gamma$. Let $L=P+N$ be the Zariski
decomposition. Then we have

$P^{2}\geq(\beta-\gamma)^{2}K_{\mathscr{G}}^{2}\geq\left(\frac{\beta-\gamma}{i_{\mathbb{Q}}({\mathscr{F}})}\right)^{2}$

And if $C$ is not in the exceptional divisor of $g$, then

$P\cdot C\geq(\beta-\gamma)K_{\mathscr{G}}\cdot
C\geq\frac{\beta-\gamma}{i_{\mathbb{Q}}({\mathscr{F}})}.$

By Proposition 2.14 and Proposition 2.4, we have the boundedness of embedding
dimension of each rational singularity on $Y$. Thus, we have a uniform bound
$\delta$ such that $\delta_{\zeta}\leq\delta$ for $|\zeta|=2$. Now we fix a
$\beta$ divisible by $i_{\mathbb{Q}}({\mathscr{F}})$ such that $P^{2}>\delta$
and $P\cdot C>\delta/2$ for any irreducible curve $C$. By [Lan01, Theorem 0.1]
and Proposition 1.3, we have that $K_{Y}+L=\beta K_{\mathscr{G}}$ is very
ample on the complement of the exceptional divisor of $g$. By [Sak84, Theorem
6.2], $g_{*}{\mathcal{O}}_{Y}(\beta K_{\mathscr{G}})={\mathcal{O}}_{X}(\beta
K_{\mathscr{F}})$. Hence, $\beta K_{\mathscr{F}}$ is very ample on the
complement of the cusps. ∎

###  2.4. Families of foliations

######  Definition 2.17.

Fix $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$.

  1. (1)

We call $\pi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow B$ _a family of
foliated triples_ (see also Definition 2.6) if

    1. (a)

$\pi:{\mathcal{X}}\rightarrow B$ is a projective morphism between normal
varieties such that all fibers are surfaces with at worst semi-log canonical
singularities. (For a reference, see [HK10, Definition 3.13.5].)

    2. (b)

${\mathcal{F}}$ is a foliation of rank 1 on normal variety ${\mathcal{X}}$
such that ${\mathcal{F}}$ is tangent to the fibers of $\pi$. More precisely,
the map $d_{\mathcal{F}}:\Omega_{\mathcal{X}}\rightarrow\Omega_{\mathcal{F}}$
factors through $\Omega_{{\mathcal{X}}/B}$.

    3. (c)

$\Delta=\sum\Delta_{i}$ is a reduced divisor with all $\Delta_{i}$’s flat over
$B$.

    4. (d)

For general $b\in B$ with ${\mathcal{X}}_{b}$ normal,
$({\mathcal{X}},{\mathcal{F}},\Delta)_{b}:=({\mathcal{X}}_{b},{\mathcal{F}}_{b}^{sat},\Delta_{b}+D)$
is a foliated triple where $\Delta_{b}=\Delta|_{{\mathcal{X}}_{b}}$ and
${\mathcal{F}}^{sat}_{b}$ is given by the saturation of the pre-foliation
$\Omega_{{\mathcal{X}}_{b}}\rightarrow\Omega_{{\mathcal{F}}}|_{{\mathcal{X}}_{b}}$
and $D=K_{\mathcal{F}}|_{{\mathcal{X}}_{b}}-K_{{\mathcal{F}}_{b}^{sat}}\geq
0$. (See Remark 2.18.)

  2. (2)

We say $\pi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ a
${\mathcal{S}}_{P}$-family if $\pi$ is a family of foliated triples and, for
_general_ $t\in T$, we have
$({\mathcal{X}},{\mathcal{F}},\Delta)_{t}\in{\mathcal{S}}_{P}$. (See
Definition 2.8.)

######  Remark 2.18.

  1. (1)

If $\Delta=\emptyset$, then we call
$\pi:(X,{\mathcal{F}},\emptyset)\rightarrow B$ a family of foliations, which
agrees with Definition 2.9.

  2. (2)

The restriction of a family of foliated triples
$\pi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ to a fiber
${\mathcal{X}}_{t}$ is _not_
$({\mathcal{X}}_{t},{\mathcal{F}}|_{{\mathcal{X}}_{t}},\Delta|_{{\mathcal{X}}_{t}})$.
This is because ${\mathcal{F}}|_{{\mathcal{X}}_{t}}$ may not be saturated.

  3. (3)

For general $b\in B$ with ${\mathcal{X}}_{b}$ normal, we put
$U\subset{\mathcal{X}}$ be an open subset in the smooth locus of
${\mathcal{X}}$ such that $U_{b}:=U|_{{\mathcal{X}}_{b}}$ is in the smooth
locus of ${\mathcal{X}}_{b}$. From the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{{\mathcal{X}}/B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\Omega_{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

we have the following exact sequences

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{\mathcal{F}}^{\vee}|_{U_{b}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{1}}$$\textstyle{T_{{\mathcal{X}}/B}|_{U_{b}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mbox{coker}(\rho^{\vee})|_{U_{b}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Omega_{\mathcal{F}}|_{U_{b}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{{\mathcal{X}}_{b}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(K|_{U_{b}})^{\vee}}$

where the upper horizontal one comes from taking the dual and then the
restriction, the lower horizontal one comes from taking the restriction and
then the dual, $\iota_{1}$ is the natural restriction, and $\iota_{2}$ comes
from the diagram chasing.

Note that $\iota_{1}$ is injective. Thus we have
$c_{1}(\Omega_{\mathcal{F}}^{\vee}|_{U_{b}})\leq
c_{1}((\Omega_{\mathcal{F}}|_{U_{b}})^{\vee})$ and therefore
$-K_{\mathcal{F}}|_{{\mathcal{X}}_{b}}\leq-K_{{\mathcal{F}}_{b}^{sat}}$.

######  Definition 2.19.

Fix an integral-valued function $P:{\mathbb{Z}}_{\geq
0}\rightarrow{\mathbb{Z}}$. A triple $(X,{\mathscr{F}},D)$ is called
_${\mathcal{S}}_{P}$ -smoothable_ if the followings hold:

  1. (1)

$X$ is a surface with at worst semi-log canonical singularities. (For a
reference, see [HK10, Definition 3.13.5].)

  2. (2)

${\mathscr{F}}$ is a pre-foliation on $X$.

  3. (3)

There exists a flat projective ${\mathbb{Q}}$-Gorenstein (that is, both
$K_{\mathcal{X}}$ and $K_{\mathcal{F}}$ are ${\mathbb{Q}}$-Cartier) one-
parameter ${\mathcal{S}}_{P}$-family
$\pi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow C$ (see Definition 2.17)
such that the restriction on the special fiber ${\mathcal{X}}_{0}$ of $\pi$ is
$(X,{\mathscr{F}},D)$, that is, $X\cong{\mathcal{X}}_{0}$,
${\mathcal{F}}|_{{\mathcal{X}}_{0}}\cong{\mathscr{F}}$, and
$\Delta|_{{\mathcal{X}}_{0}}=D$.

  4. (4)

$(K_{\mathcal{F}}+\Delta)|_{X}$ is nef and
$K_{X}+6i(P)(K_{\mathcal{F}}+\Delta)|_{X}$ is ample. (See Lemma 3.1 for the
definition of $i(P)$)

##  3\. Invariance of plurigenera

In this section, we would like to show the invariance of plurigenera for the
family of minimal partial du Val resolutions of generalized canonical models
of general type with fixed Hilbert function.

######  Lemma 3.1.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Then there
exists an $i(P)\in{\mathbb{N}}$ depending only on $P$ such that
$i(P)(K_{\mathscr{G}}+D)$ is Cartier for all
$(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$.

######  Proof.

We can take $i(P)=C_{2}!$ where $C_{2}$ comes from Proposition 2.4 and depends
only on $P$. ∎

######  Theorem 3.2.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Given any
family of foliated triples $f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow
T$ where $T$ is connected and
$({\mathcal{X}},{\mathcal{F}},\Delta)_{t}\in{\mathcal{S}}_{P}$ for all $t\in
T$ (see Definition 2.17 for families of foliated triples), there exists an
$m_{f}$ depending only on this family $f$ such that

$h^{0}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}}+\Delta)|_{{\mathcal{X}}_{t}})$

is independent of $t$ for $m\geq m_{f}$ and divisible by $i(P)$. (See Lemma
3.1.)

######  Proof.

The idea of the proof is from the proof of [CF18, Proposition 3.7]. We divide
the proof into the following steps. Let $m\in{\mathbb{N}}$ be a positive
integer divisible by $i(P)$.

  1. (1)

Fix a point $0\in T$ and let
$(X,{\mathscr{F}},D):=({\mathcal{X}},{\mathcal{F}},\Delta)|_{{\mathcal{X}}_{0}}\in{\mathcal{S}}_{P}$.
Let $(X_{c},{\mathscr{F}}_{c})$ be the generalized canonical model of
$(X,{\mathscr{F}},D)$, $\rho:X^{m}\rightarrow X_{c}$ be the minimal resolution
of $X_{c}$, and ${\mathscr{F}}^{m}$ be the pullback foliation. Contracting
down all ${\mathscr{F}}^{m}$-chains to get $(Y,{\mathscr{G}})$, we have a
morphism $\pi:(Y,{\mathscr{G}})\rightarrow(X,{\mathscr{F}})$. Let $E$ be the
proper transform of $D$. Thus, we have
$K_{\mathscr{G}}+E=\pi^{*}(K_{\mathscr{F}}+D)$ is nef and big.

Let $\Gamma_{1},\ldots,\Gamma_{p}$ be all elliptic Gorenstein leaves on $Y$
and $C_{1},\ldots,C_{q}$ be all disconnected chains of rational curves
_exceptional_ over its generalized canonical model such that
$(K_{\mathscr{G}}+E)\cdot C_{j}^{(k)}=0$ where
$C_{j}=\sum_{k=1}^{r_{j}}C_{j}^{(k)}$ is the sum of its irreducible
components. Also let $Z=\sum_{i=1}^{p}\Gamma_{i}+\sum_{j=1}^{q}C_{j}$.

  2. (2)

Claim. $h^{1}(\Gamma_{i},m(K_{{\mathscr{G}}}+E))=0$ for all $i$.

######  Proof.

Notice that $\Gamma_{i}$ is Cohen-Macaulay with the trivial dualizing sheaf.
Since each irreducible component of $E$ is either over a log canonical non-
canonical foliation singularity or over a dihedral singularity, we have that
$E$ does not meet $\Gamma_{i}$. So we have

$h^{1}(\Gamma_{i},m(K_{{\mathscr{G}}}+E))=h^{0}(\Gamma_{i},-m(K_{{\mathscr{G}}}+E))=h^{0}(\Gamma_{i},-mK_{{\mathscr{G}}})=0$

where the last equality is because
${\mathcal{O}}_{\Gamma_{i}}(mK_{{\mathscr{G}}})$ is not torsion and has degree
0 by [McQ08, FactIII.0.4 and Theorem IV.2.2].

  3. (3)

Claim. $h^{1}(C_{j},m(K_{{\mathscr{G}}}+E-G))=0$ for all $j$.

######  Proof.

Note that we have

$(K_{\mathscr{G}}+E)\cdot C_{j}^{(k)}=0$

for all $j$ and $k$. Since $C_{j}^{(k)}$’s are smooth rational curves, we have

(1) $h^{1}(C_{j}^{(k)},m(K_{\mathscr{G}}+E))=0$

for all $j$ and $k$. Now, for any fixed $j$, we consider the following short
exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{C_{j}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{k=1}^{r_{j}}{\mathcal{O}}_{C_{j}^{(k)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{k=1}^{r_{j}-1}{\mathcal{O}}_{C_{j}^{(k)}\cap
C_{j}^{(k)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Tensoring with ${\mathcal{O}}_{C_{j}}(m(K_{\mathscr{G}}+E))$ and taking the
induced long exact sequence, we notice that the map

$\textstyle{\displaystyle\bigoplus_{k=1}^{r_{j}}H^{0}\big{(}{C_{j}^{(k)}},m(K_{\mathscr{G}}+E)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{k=1}^{r_{j}-1}H^{0}\big{(}{C_{j}^{(k)}\cap
C_{j}^{(k+1)}},m(K_{\mathscr{G}}+E)\big{)}}$

is surjective and thus, we have that $h^{1}(C_{j},m(K_{{\mathscr{G}}}+E))=0$
for all $j$ by equation (1).

  4. (4)

Claim. $h^{j}(Y,m(K_{{\mathscr{G}}}+E)-Z)=0$ for $j>0$ and for all $m\geq
m_{0}$ for some $m_{0}$.

######  Proof.

For any irreducible curve $C$ in the support of $Z$, we have that

$\big{(}m(K_{{\mathscr{G}}}+E)-(K_{Y}+Z)\big{)}\cdot C=-(K_{Y}+Z)\cdot C\geq
0.$

Note that $K_{\mathscr{G}}+E=\pi^{*}(K_{\mathscr{F}}+D)$ is nef and big, and
the $(K_{\mathscr{G}}+E)$-trivial curves are exactly the curves in the support
of $Z$. Then there exists an $m_{0}$ depending only on $(X,{\mathscr{F}},D)$
such that

$m(K_{{\mathscr{G}}}+E)-(K_{Y}+Z)$

is nef and big for $m\geq m_{0}$. Since $Y$ is klt, we have

$h^{j}(Y,m(K_{{\mathscr{G}}}+E)-Z)=h^{j}(Y,K_{Y}+m(K_{{\mathscr{G}}}+E)-(K_{Y}+Z))=0$

by Kawamata-Viehweg vanishing theorem.

  5. (5)

Claim. $h^{j}(Y,m(K_{{\mathscr{G}}}+E))=0$ for $j>0$ and for all $m\geq
m_{0}$.

######  Proof.

By steps (2) and (3), we have that $h^{1}(Z,m(K_{\mathscr{G}}+E))=0$. Now we
consider the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Y}(-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Tensoring with ${\mathcal{O}}_{Y}(m(K_{\mathscr{G}}+E))$ and taking the
induced long exact sequence, we have the map

$H^{1}(Y,m(K_{{\mathscr{G}}}+E)-Z)\rightarrow H^{1}(Y,m(K_{{\mathscr{G}}}+E))$

is surjective and $H^{2}(Y,m(K_{{\mathscr{G}}}+E)-Z)\cong
H^{2}(Y,m(K_{{\mathscr{G}}}+E))$. Thus, we have
$h^{j}(Y,m(K_{{\mathscr{G}}}+E))=0$ for $j>0$ and for all $m\geq m_{0}$ by
step (4).

  6. (6)

Claim. $h^{1}(X,m(K_{\mathscr{F}}+D))=0$ for all $m\geq m_{0}$.

######  Proof.

Note that $\pi_{*}(m(K_{\mathscr{G}}+E))=m(K_{\mathscr{F}}+D)$ by the
projection formula. Since $h^{1}(Y,m(K_{{\mathscr{G}}}+E))=0$ for all $m\geq
m_{0}$ by step (5), we have that

$h^{1}(X,m(K_{{\mathscr{F}}}+D))=0$

for all $m\geq m_{0}$ by Leray spectral sequence.

  7. (7)

Claim. $h^{2}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))=0$ for
all $t\in T$ and for $m\geq m_{P}$ for some $m_{P}$.

######  Proof.

Note that $K_{{\mathcal{F}}_{t}}+\Delta_{t}$ is nef and big since
$({\mathcal{X}}_{t},{\mathcal{F}}_{t},\Delta_{t})\in{\mathcal{S}}_{P}$. In
addition, we have already seen, in the proof of Theorem 2.10, that
$K_{{\mathcal{X}}_{t}}+4i(P)(K_{{\mathcal{F}}_{t}}+\Delta_{t})$ is ample.
Moreover, by [HL21, Lemma 3.5], we have that
$m(K_{{\mathcal{F}}_{t}}+\Delta_{t})-K_{{\mathcal{X}}_{t}}$ is big for $m\geq
m_{P}$ where

$m_{P}:=\max\left\\{\frac{2(K_{{\mathcal{F}}_{t}}+\Delta_{t})\cdot
K_{{\mathcal{X}}_{t}}}{(K_{{\mathcal{F}}_{t}}+\Delta_{t})^{2}}+4i(P)+1,1\right\\}.$

By Proposition 2.4, we have that $m_{P}$ depends only on $P$.

Now by Serre duality, we have, for $m\geq m_{P}$, that

$h^{2}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))=h^{0}({\mathcal{X}}_{t},K_{{\mathcal{X}}_{t}}-m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))=0$

since there is no section for an anti-big divisor.

  8. (8)

Take $m_{f}:=\max\\{m_{0},m_{P}\\}$ which only depends on the family $f$. Now
we have, for all $t\in T$ and for all $m\geq m_{f}$, that

$\displaystyle h^{0}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))$
$\displaystyle\geq$
$\displaystyle\chi({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))$
$\displaystyle=$
$\displaystyle\chi({\mathcal{X}}_{0},m(K_{{\mathcal{F}}_{0}}+\Delta_{0}))$
$\displaystyle=$ $\displaystyle
h^{0}({\mathcal{X}}_{0},m(K_{{\mathcal{F}}_{0}}+\Delta_{0}))$

where the first inequality comes from step (7) and the last equality comes
from steps (6) and (7).

On the other hand, by upper semi-continuity, we have

$h^{0}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))\leq
h^{0}({\mathcal{X}}_{0},m(K_{{\mathcal{F}}_{0}}+\Delta_{0}))$

Therefore, we get the desired equality.

∎

######  Remark 3.3.

The arguments above also show that

$h^{1}({\mathcal{X}}_{t},m(K_{{\mathcal{F}}_{t}}+\Delta_{t}))=0$

for $m\geq m_{f}$ and divisible by $i(P)$.

######  Corollary 3.4.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. There is an
$m_{P}$ depending only on $P$ such that, for all $m\geq m_{P}$ and divisible
by $i(P)$, we have that

$h^{1}(X,m(K_{\mathscr{F}}+D))=h^{2}(X,m(K_{\mathscr{F}}+D))=0$

and $h^{0}(X,m(K_{\mathscr{F}}+D))$ is constant for all
$(X,{\mathscr{F}},D)\in{\mathcal{S}}_{P}$.

######  Proof.

By Theorem 2.10 and 6.4, we have finitely many ${\mathcal{S}}_{P}$-families.
Taking the maximal $m_{f}$ in Theorem 3.2, we have

$h^{1}(X,m(K_{\mathscr{F}}+D))=h^{2}(X,m(K_{\mathscr{F}}+D))=0$

for all $(X,{\mathscr{F}},D)\in{\mathcal{S}}_{P}$. Since
$\chi(X,m(K_{\mathscr{F}}+D))$ is constant for all
$(X,{\mathscr{F}},D)\in{\mathcal{S}}_{P}$, so is
$h^{0}(X,m(K_{\mathscr{F}}+D))$. ∎

##  4\. Semi-stable minimal model program

In this section, we establish the semi-stable
$(K_{\mathcal{F}}+\Delta)$-minimal model program for a particular type of
families of foliated surfaces and show the valuative criterion for properness
for ${\mathcal{S}}_{P}$-families (see Theorem 4.26).

###  4.1. Preparations

We first recall some definitions and results that will be used in this
section.

######  Lemma 4.1.

Let $\phi:X\dashrightarrow X^{\prime}$ be a birational map between normal
varieties and

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Y}$

be a commutative diagram where $Y$ is a normal variety and $f,f^{\prime}$ are
proper birational morphisms. Let ${\mathcal{F}}$ be a foliation on $X$ and
${\mathcal{F}}^{\prime}$ be the induced foliation on $X^{\prime}$. Assume that

  1. (1)

$-K_{\mathcal{F}}$ is ${\mathbb{Q}}$-Cartier and $f$-nef;

  2. (2)

$K_{{\mathcal{F}}^{\prime}}$ is ${\mathbb{Q}}$-Cartier and $f^{\prime}$-nef.

Then for an exceptional divisor $E$ over $Y$, we have
$a_{E}({\mathcal{F}})\leq a_{E}({\mathcal{F}}^{\prime})$. Moreover, the strict
inequality holds if either

  1. (a)

$-K_{\mathcal{F}}$ is $f$-ample and $f$ is not an isomorphism above the
generic point of the center of $E$ on $Y$, or

  2. (b)

$K_{{\mathcal{F}}^{\prime}}$ is $f^{\prime}$-ample and $f^{\prime}$ is not an
isomorphism above the generic point of the center of $E$ on $Y$.

######  Proof.

This is the consequence of negativity lemma. The proof is the same as [KM98,
Lemma 3.38]. For the reader’s convenience, we include the proof.

Let $W$ be a normal variety with birational morphisms $g:W\rightarrow X$ and
$W\rightarrow X^{\prime}$ such that the center of $E$ on $W$ is a divisor.
Note that ${\mathcal{F}}_{W}:=g^{*}{\mathcal{F}}\cong
g^{\prime*}{\mathcal{F}}^{\prime}$ since they are saturated and isomorphic on
an open dense subset. Put $h=f\circ g=f^{\prime}\circ g^{\prime}$. Let $m>0$
be a sufficiently divisible integer. Then we have

$\displaystyle-m\left(K_{{\mathcal{F}}_{W}}-\sum
a_{E_{i}}({\mathcal{F}})E_{i}\right)$ $\displaystyle\sim$ $\displaystyle-
mg^{*}K_{\mathcal{F}}$ $\displaystyle m\left(K_{{\mathcal{F}}_{W}}-\sum
a_{E_{i}}({\mathcal{F}}^{\prime})E_{i}\right)$ $\displaystyle\sim$
$\displaystyle mg^{\prime*}K_{{\mathcal{F}}^{\prime}}$

Adding these two, we obtain that
$H:=\sum\left(a_{E_{i}}({\mathcal{F}})-a_{E_{i}}({\mathcal{F}}^{\prime})\right)E_{i}$
is $h$-nef and $h$-exceptional. By negativity lemma, all coefficients of $H$
are non-positive. Moreover, if $H$ is not numerically $h$-trivial over the
generic point of the center of $E$ on $W$, then the coefficient of $E$ in $H$
is negative. ∎

######  Lemma 4.2 ([Spi20, Corollary 2.28, foliated Bend and Break]).

Let $X$ be a normal projective variety of dimension $n$. Let ${\mathscr{F}}$
be a foliation of rank $r$ on $X$ and $\Delta$ be an effective divisor on $X$.
Let $M$ be any nef divisor. Suppose that there are nef ${\mathbb{R}}$-divisors
$D_{1},\ldots,D_{n}$ such that

  1. (1)

$D_{1}\cdot\ldots\cdot D_{n}=0$ and

  2. (2)

$-(K_{\mathscr{F}}+\Delta)\cdot D_{2}\cdot\ldots\cdot D_{n}>0$.

Then, through a general point of $X$, there is a rational curve $\Sigma$ with
$D_{1}\cdot\Sigma=0$,

$M\cdot\Sigma\leq 2n\frac{M\cdot D_{2}\cdot\ldots\cdot
D_{n}}{-K_{\mathscr{F}}\cdot D_{2}\cdot\ldots\cdot D_{n}},$

and $\Sigma$ is tangent to ${\mathscr{F}}$.

######  Definition 4.3.

Let $X$ be a smooth variety, ${\mathscr{F}}$ be a foliation of rank 1 on $X$,
and $p$ be a singularity of ${\mathscr{F}}$. Let $v$ be a germ of a vector
field around $p$ associated to ${\mathscr{F}}$. Then $p$ is called
_elementary_ if $v$ has a non-zero non-nilpotent linear part.

######  Remark 4.4.

Elementary singularities are log canonical.

By the Introduction and Theorem 1 in [CR14], we have the following.

######  Theorem 4.5 ([CR14, Introduction and Theorem 1]).

Any germ of vector fields on $({\mathbb{C}}^{3},0)$ tangent to a codimension
one foliation can be desingularized. That is, there exists a finite sequence
of blowups such that the pullback foliation has only elementary singularities.

######  Definition 4.6 ([KM98, Definition 7.1]).

Let $X$ be a normal variety, $B$ an effective ${\mathbb{Q}}$-divisor, and
$f:X\rightarrow C$ a non-constant morphism to a smooth curve $C$. We say that
$f:(X,B)\rightarrow C$ is a _divisorial log terminal_ (dlt) if
$(X,B+f^{-1}(c))$ is dlt for every closed point $c\in C$.

######  Definition 4.7.

Let $X$ be a normal threefold and ${\mathscr{F}}$ be a rank one foliation on
$X$. We say $(X,{\mathscr{F}},\Delta)$ is _foliated log smooth_ if the
following holds:

  1. (1)

$(X,\Delta)$ is log smooth.

  2. (2)

${\mathscr{F}}$ has only elementary singularities.

Given a normal threefold $X$, a rank one foliation ${\mathscr{F}}$ on $X$, and
an effective divisor $\Delta=\sum a_{i}D_{i}$ where $0\leq a_{i}\leq 1$ for
all $i$, a _foliated log resolution_ of $(X,{\mathscr{F}},\Delta)$ is a proper
birational morphism $\pi:Y\rightarrow X$ so that $\textnormal{Exc}(\pi)$ is a
divisor and $({\mathscr{G}},\pi_{*}^{-1}\Delta+\sum_{i}E_{i})$ is foliated log
smooth where ${\mathscr{G}}$ is the pullback foliation on $Y$ and $E_{i}$’s
are all $\pi$-exceptional divisors.

######  Definition 4.8.

Let $X$ be a normal threefold and ${\mathscr{F}}$ be a rank one foliation on
$X$. Suppose $K_{\mathscr{F}}+\Delta$ is ${\mathbb{Q}}$-Cartier. We say
$(X,{\mathscr{F}},\Delta)$ is _foliated divisorial log terminal_
(${\mathscr{F}}$-dlt) if the following holds:

  1. (1)

Each irreducible component of $\Delta$ is transverse to ${\mathscr{F}}$ and
has coefficient at most one.

  2. (2)

There exists a foliated log resolution $Y\rightarrow X$ of
$(X,{\mathscr{F}},\Delta)$ which only extracts divisors $E$ of discrepancy
$>-\varepsilon(E)$ where $\varepsilon(E)=0$ if $E$ is invariant and
$\varepsilon(E)=1$ if $E$ is _not_ invariant.

######  Lemma 4.9 (Adjunction).

Let $X$ be a smooth variety and ${\mathscr{F}}$ be a foliation on $X$, which
is given by a coherent saturated subsheaf $T_{\mathscr{F}}\subset T_{X}$ of
rank 1. For any prime divisor $D$ of $X$ with normalization $D^{\nu}$, we have
the following:

  1. (1)

If $D$ is invariant, then
$K_{{\mathscr{F}}}|_{D^{\nu}}=K_{{\mathscr{F}}_{D^{\nu}}}+\Delta_{D^{\nu}}$
where $\Delta_{D^{\nu}}\geq 0$.

  2. (2)

If $D$ is not invariant, then $(K_{{\mathscr{F}}}+D)|_{D^{\nu}}=\Theta$ for
some $\Theta\geq 0$.

######  Proof.

If $D$ is invariant, then we have a morphism
$T_{\mathscr{F}}|_{D^{\nu}}\rightarrow T_{D^{\nu}}$. Thus
$(T_{\mathscr{F}}|_{D^{\nu}})^{**}\rightarrow(T_{D^{\nu}})^{**}=T_{D^{\nu}}$
gives a foliation on $D^{\nu}$ and
$K_{\mathscr{F}}|_{D^{\nu}}=K_{{\mathscr{F}}_{D^{\nu}}}+\Delta_{D^{\nu}}$
where $\Delta_{D^{\nu}}\geq 0$ since it comes from the saturation.

If $D$ is not invariant, then we consider the following composition:

$\psi:N^{*}_{\mathscr{F}}|_{D^{\nu}}\rightarrow\Omega^{1}_{X}|_{D^{\nu}}\rightarrow\Omega^{1}_{D^{\nu}}$

where $N^{*}_{\mathscr{F}}$ is the conormal bundle of the foliation. Thus,
$\textnormal{im}\,\psi$ defines a foliation ${\mathscr{F}}_{D^{\nu}}$ on
$D^{\nu}$. Since $N^{*}_{{\mathscr{F}}_{D^{\nu}}}$ is the saturation of
$N^{*}_{\mathscr{F}}|_{D^{\nu}}$ in $\Omega^{1}_{D^{\nu}}$, we have
$N^{*}_{{\mathscr{F}}_{D^{\nu}}}=N^{*}_{\mathscr{F}}|_{D^{\nu}}+\Theta$ for
some effective divisor $\Theta$. Therefore, we have

$K_{D^{\nu}}-K_{{\mathscr{F}}_{D^{\nu}}}=(K_{X}-K_{\mathscr{F}})|_{D^{\nu}}+\Theta.$

Hence, by adjunction $(K_{X}+D^{\nu})|_{D^{\nu}}=K_{D^{\nu}}$, we have that
$(K_{\mathscr{F}}+D)|_{D^{\nu}}=K_{{\mathscr{F}}_{D^{\nu}}}+\Theta$. Notice
that ${\mathscr{F}}_{D^{\nu}}$ is a foliation of rank 0 on $D^{\nu}$, then
$K_{{\mathscr{F}}_{D^{\nu}}}=0$. ∎

###  4.2. Semi-stable reduction

We are specifically interested in starting from the following two types of
families.

  * (a)

The ${\mathcal{S}}_{P}$-families
$(\overline{f}:(\overline{{\mathcal{X}}},\overline{{\mathcal{F}}},\overline{\Delta})\rightarrow
C)$ over $C$ where $\overline{{\mathcal{X}}}$ is normal and $C$ is a smooth
curve. (See Definition 2.8 and Definition 2.17.)

  * (b)

The ${\mathcal{S}}_{P}$-families
$(f^{\circ}:({\mathcal{X}}^{\circ},{\mathcal{F}}^{\circ},\Delta^{\circ})\rightarrow
C^{\circ})$ over $C^{\circ}$ where ${\mathcal{X}}^{\circ}$ is normal,
$C^{\circ}=C\backslash\\{0\\}$, and $C$ is a smooth curve containing $0$. (See
Definition 2.8 and Definition 2.17.)

We first compactify the family of type (b) to get a family
$\overline{f}:(\overline{{\mathcal{X}}},\overline{{\mathcal{F}}},\overline{\Delta})\rightarrow
C$ over $C$. Then for both types of families, after possibly taking a base
change, we have a semi-stable reduction for the family of surfaces
$(\overline{{\mathcal{X}}},\overline{\Delta})$ over the curve $C$ by [KKMSD73]
(see also [KM98, Theorem 7.17]) and we denote this family as
$f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$.

Notice that now $({\mathcal{X}},\Delta)$ is log smooth, and ${\mathcal{F}}$ is
tangent to the fibration, which is a codimension one foliation. By Theorem
4.5, we may assume, after a finite sequence of blowups, that ${\mathcal{F}}$
has only elementary singularities. Moreover, by taking base changes and
resolutions, we may assume that the central fiber of $f$ is reduced and
${\mathcal{F}}$ has only elementary singularities. We call
$f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ be a semi-stable
reduction of the ${\mathcal{S}}_{P}$-family
$\overline{f}:(\overline{{\mathcal{X}}},\overline{{\mathcal{F}}},\overline{\Delta})\rightarrow
C$ over $C$. Moreover, we call the exceptional divisors of the map
${\mathcal{X}}\rightarrow\overline{{\mathcal{X}}}_{T}$ the exceptional
divisors for the semi-stable reduction
$f:({\mathcal{X}},{\mathcal{F}},\Delta)$ of the ${\mathcal{S}}_{P}$-family
$\overline{f}:(\overline{{\mathcal{X}}},\overline{{\mathcal{F}}},\overline{\Delta})\rightarrow
C$ over $C$.

Thus, we have that $({\mathcal{X}},{\mathcal{F}},\Delta)$ is foliated log
smooth. In particular, $({\mathcal{X}},{\mathcal{F}},\Delta)$ is
${\mathcal{F}}$-dlt (see Definition 4.8) and $f:{\mathcal{X}}\rightarrow T$ is
dlt (see Definition 4.6.)

###  4.3. Foliation on the central fiber of semi-stable reduction

Let ${\mathcal{X}}_{0}=E_{0}\cup\bigcup_{i=1}^{r}E_{i}$ where $E_{0}$ and
$E_{i}$’s are distinct irreducible components. From subsection 4.2, we know
that ${\mathcal{X}}$ is smooth and ${\mathcal{F}}$ has elementary
singularities. Thus, we have three types of local expressions around a fixed
point $p\in{\mathcal{X}}_{0}$.

  1. (Type 1)

Around $E_{0}\backslash\bigcup_{i=1}^{r}E_{i}$, we may assume that
${\mathcal{F}}$ is induced by $f(x,y,z)\frac{\partial}{\partial
x}+g(x,y,z)\frac{\partial}{\partial y}$ where $E_{0}=(z=0)$,
$f(x,y,z),g(x,y,z)\in{\mathbb{C}}[\\![x,y,z]\\!]$, and one of
$f(x,y,z),g(x,y,z)$ has order $\leq 1$.

  2. (Type 2)

Around $(E_{0}\cap E_{i})\backslash\bigcup_{0<j\neq i}E_{j}$ (if non-empty)
for some $i$, we may assume that ${\mathcal{F}}$ is induced by

    1. (Type 2a)

$xf(x,y,z)\frac{\partial}{\partial x}-yf(x,y,z)\frac{\partial}{\partial
y}+z\frac{\partial}{\partial z}$ or

    2. (Type 2b)

$x\frac{\partial}{\partial x}-y\frac{\partial}{\partial
y}+g(x,y,z)\frac{\partial}{\partial z}$

where $E_{0}=(x=0)$, $E_{i}=(y=0)$,
$f(x,y,z),g(x,y,z)\in{\mathbb{C}}[\\![x,y,z]\\!]$.

  3. (Type 3)

Around $E_{0}\cap E_{i}\cap E_{j}$ (if non-empty) for some non-zero distinct
$i$ and $j$, we may assume that ${\mathcal{F}}$ is induced by
$ax\frac{\partial}{\partial x}+by\frac{\partial}{\partial
y}+cz\frac{\partial}{\partial z}$ where $E_{0}=(x=0)$, $E_{i}=(y=0)$,
$E_{j}=(z=0)$ and $a+b+c=0$

######  Lemma 4.10.

Let ${\mathcal{X}}_{0}=E_{0}\cup\bigcup_{i=1}^{r}E_{i}$ where $E_{0}$ and
$E_{i}$’s are distinct irreducible components. We have that
$K_{\mathcal{F}}|_{E_{0}}-K_{{\mathcal{F}}_{E_{0}}}=\sum_{j}C_{j}$ is a
reduced divisor with pairwise disjoint irreducible components $C_{j}$’s.
Moreover, for any irreducible component $C_{j}$, we have

  1. (1)

either $C_{j}\cap\bigcup_{i=1}^{r}E_{i}=\emptyset$

  2. (2)

or $C_{j}$ is a strata of ${\mathcal{X}}_{0}=E_{0}\cup\bigcup_{i=1}^{r}E_{i}$
of dimension one with zero tangency order.

######  Proof.

Note that $K_{\mathcal{F}}|_{E_{0}}-K_{{\mathcal{F}}_{E_{0}}}$ is supported on
components of the singular locus of ${\mathcal{F}}$ whose restrictions on
$E_{0}$ are of codimension one. Since ${\mathcal{F}}$ has only elementary
singularities, we have that
$K_{\mathcal{F}}|_{E_{0}}-K_{{\mathcal{F}}_{E_{0}}}=\sum_{j}C_{j}$ is a
reduced divisor with pairwise disjoint irreducible components $C_{j}$’s. If
$C_{j}$ meets one of the stratum of ${\mathcal{X}}_{0}$, then it is of type 2
or 3 around the intersection points.

  1. (2a)

For the type $2a$, it is impossible.

  2. (2b)

For the type $2b$, the possible irreducible component for $C_{j}$ is
$(x=y=0)$, which is some strata of ${\mathcal{X}}_{0}$ of dimension one with
zero tangency order.

  3. (3)

For the type $3$, the possible irreducible component for $C_{j}$ is either
$(x=y=0)$ or $(x=z=0)$, both of which are some stratum of ${\mathcal{X}}_{0}$
of dimension one with zero tangency order.

∎

###  4.4. Cone theorem

In proving the cone theorem, we use the following result from convex geometry.

######  Lemma 4.11.

[Roc70] If $K$ is a closed convex cone containing no lines, then $K$ is the
closure of the subcone generated by the extremal rays.

Then it is natural to investigate a $(K_{\mathcal{F}}+\Delta)$-negative
extremal ray $R$ of $\overline{{\rm NE}}({\mathcal{X}}/T)$. Let $R$ be a
relative $(K_{\mathcal{F}}+\Delta)$-negative extremal ray over $T$.

######  Definition 4.12.

We define loc$(R)$ to be all those points $x\in{\mathcal{X}}$ such that there
exists a curve $C$ with $x\in C$ and $[C]\in R$.

######  Remark 4.13.

At this point, it is not apparent that there is a curve $C$ with $[C]\in R$.
However, the proof of the following lemma will give the existence.

######  Lemma 4.14.

$\textnormal{loc}(R)$ is non-empty and closed. Moreover, if
$\textnormal{loc}(R)$ is two-dimensional, then it is irreducible and
invariant.

######  Proof.

The proof is inspired by [Spi20, Lemma 8.4].

We may pick a suitable ample divisor $H$ such that
$H_{R}=(K_{\mathcal{F}}+\Delta)+tH$ is a supporting hyperplane to $R$, that is
$H_{R}$ is nef and exactly zero on $R$.

  1. (1)

Suppose first that $\nu(H_{R})<3$. Put $D_{i}=H_{R}$ for $i\leq\nu(H_{R})+1$
and $D_{i}=H$ otherwise. Then $D_{1}\cdot D_{2}\cdot D_{3}=0$ and
$(K_{\mathcal{F}}+\Delta)\cdot D_{2}\cdot D_{3}=-tH\cdot D_{2}\cdot D_{3}<0$.
Thus, by Lemma 4.2, through a general point, there is a rational curve
$\Sigma$ with $H_{R}\cdot\Sigma=D_{1}\cdot\Sigma=0$, and hence $[\Sigma]\in
R$. Therefore, we have that $R$ is spanned by a family of rational
${\mathcal{F}}$-invariant curves passing through general points of
${\mathcal{X}}$.

Let $B$ be the closed subscheme of Hilbert scheme of ${\mathcal{X}}$
parametrizing those subvarieites tangent to ${\mathcal{F}}$. Now we consider
the following diagram

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{p}$$\textstyle{\mathcal{X}}$$\textstyle{Z}$

where $Z$ is the closed subscheme of $B$ whose general element parametrizes a
rational curve tangent to ${\mathcal{F}}$ and spanning $R$. Note that $F$ is
surjective. So if $\Sigma$ is any curve contained in the fiber of $p$, then
$[F(\Sigma)]\in R$. Thus, $\textnormal{loc}(R)={\mathcal{X}}$.

  2. (2)

Otherwise $H_{R}$ is nef and big. So we may write $H_{R}$ as the sum of an
ample divisor and an effective divisor. Thus, there is an irreducible
effective divisor $D$ with $D\cdot R<0$. In particular, we have
$\textnormal{loc}(R)\subset D$. Let $D^{\nu}$ be the normalization of $D$. If
$D\subset\Delta$, then $\varepsilon(D)=1$ and $G_{1}:=H_{R}|_{D^{\nu}}$ is
effective by Lemma 4.9. If not, we can take
$H^{\prime}_{R}=(K_{\mathcal{F}}+\Delta)+\varepsilon(D)D+A$, which is also a
supporting hyperplane to $R$ where $A$ is an ample divisor. Let
$G_{2}=H^{\prime}_{R}|_{D^{\nu}}$. Note that
$G_{2}=K_{{\mathcal{F}}_{D^{\nu}}}+\Theta$ where $\Theta\geq 0$ or $G$ itself
is effective by Lemma 4.9.

Using the similar argument as above, we have the following.

    1. (a)

If $\nu(G_{i})=2$ for $i=1$ or $2$, then $\textnormal{loc}(R)$ is a finite
union of curves.

    2. (b)

Otherwise, by bend and break, we have $\textnormal{loc}(R)=D$ and $D$ is
invariant.

∎

######  Remark 4.15.

The proof above also shows that $R={\mathbb{R}}_{\geq 0}[C]$ for some
irreducible curve $C$. If $\mbox{loc}(R)$ is not one-dimensional, then we may
take $C$ to be invariant and rational.

We will prove the following cone theorem.

######  Theorem 4.16 (Cone theorem).

Let $f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ be a family of
foliated triples which is a semi-stable reduction of
${\mathcal{S}}_{P}$-families. (See subsection 4.2.) Moreover, we put
${\mathcal{E}}_{1},\ldots,{\mathcal{E}}_{v}$ be all exceptional prime divisors
flat over $T$ for the semi-stable reduction where
${\mathcal{E}}_{1},\ldots,{\mathcal{E}}_{u}$ are invariant with
${\mathcal{E}}_{i,t}:={\mathcal{E}}_{i}|_{{\mathcal{X}}_{t}}$ which is over a
canonical foliation surface singularity for $i=1,\ldots,u$ and for general
$t\in T$. Let $a_{1},\ldots,a_{u}$ be any numbers with $0\leq a_{i}<1$ for
$i=1,\ldots,u$ such that

$\left(K_{\mathcal{X}}+\sum_{i=1}^{u}a_{i}{\mathcal{E}}_{i}\right)\cdot{\mathcal{E}}_{j,t}<0$

for all $j=1,\ldots,u$ and for general $t\in T$ where
${\mathcal{E}}_{j,t}={\mathcal{E}}_{j}|_{{\mathcal{X}}_{t}}$. Let
${\mathcal{D}}=\sum a_{i}{\mathcal{E}}_{i}$. Suppose
$\phi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow({\mathcal{Y}},{\mathcal{G}}:=\phi_{*}{\mathcal{F}},\Theta:=\phi_{*}\Delta)$
be the contraction of some $(K_{\mathcal{F}}+\Delta)$-negative extremal rays
which is given by the $(K_{\mathcal{X}}+{\mathcal{D}})$-MMP over $T$. Then:

  1. (1)

There are finitely many rational curves $C_{j}\subset{\mathcal{Y}}$ such that

$\overline{\textnormal{NE}}({\mathcal{Y}}/T)=\overline{\textnormal{NE}}({\mathcal{Y}}/T)_{K_{\mathcal{G}}+\Theta\geq
0}+\sum{\mathbb{R}}_{\geq 0}[C_{j}]$

and ${\mathbb{R}}_{\geq 0}[C_{j}]$’s are extremal rays of
$\overline{\textnormal{NE}}({\mathcal{Y}}/T)$ with two dimensional loci, which
are corresponding to
$\phi_{*}{\mathcal{E}}_{1},\ldots,\phi_{*}{\mathcal{E}}_{u}$, or irreducible
components of special fibers.

  2. (2)

$(K_{\mathcal{Y}}+\phi_{*}{\mathcal{D}})\cdot C_{j}<0$ for all $j$. Moreover,
by the $(K_{\mathcal{Y}}+\phi_{*}{\mathcal{D}})$-MMP over $T$, there is, for
each $j$, a contraction
$\textnormal{cont}_{j}:{\mathcal{Y}}\rightarrow{\mathcal{Z}}$ over $T$ such
that an irreducible curve $C\subset{\mathcal{Y}}$ is mapped to a point if and
only if $[C]\in{\mathbb{R}}_{\geq 0}[C_{j}]$.

  3. (3)

Let ${\mathcal{X}}_{0}=\sum_{i=0}^{r}E_{i}$. For any $i$ and $j$, if
$\phi_{*}E_{i}\neq 0$, then $(\textnormal{cont}_{j}\circ\phi)|_{E_{i}}$ is
either the identity or the contraction of some
${\mathcal{F}}_{E_{i}}$-exceptional curves or some
${\mathcal{F}}_{E_{i}}$-chains.

######  Remark 4.17.

We have the following remarks.

  1. (1)

By MMP on foliated surfaces, we can always find $a_{i}$’s with $0\leq a_{i}<1$
such that $\big{(}K_{\mathcal{X}}+\sum
a_{i}{\mathcal{E}}_{i}\big{)}\cdot{\mathcal{E}}_{i,t}<0$ for general $t\in T$.
Moreover, $f^{-1}(t)\cup\bigcup_{i}{\mathcal{E}}_{i}$ has simple normal
crossing support since $f$ is a semi-stable reduction. Thus,
$({\mathcal{X}},{\mathcal{D}})$ is klt and
$({\mathcal{X}},{\mathcal{D}}+f^{-1}(t))$ is dlt for any $t\in T$. Therefore,
the $(K_{\mathcal{X}}+{\mathcal{D}}+f^{-1}(t))$-MMP over $T$ exists for any
$t\in T$.

  2. (2)

Notice that $(K_{\mathcal{X}}+{\mathcal{D}})$-MMP over $T$ is equivalent to
$(K_{\mathcal{X}}+{\mathcal{D}}+f^{-1}(t))$-MMP over $T$ since
$f^{-1}(t)\equiv_{f}0$ for any $t\in T$.

  3. (3)

For any $i$ with $\phi_{*}E_{i}\neq 0$, $\phi|_{E_{i}}$ is either the identity
or the contraction of some ${\mathcal{F}}_{E_{i}}$-exceptional curves or
${\mathcal{F}}_{E_{i}}$-chains.

  4. (4)

For any ${\mathcal{E}}_{j}$ with $j>u$, we have that
$(K_{\mathcal{F}}+\Delta)\cdot{\mathcal{E}}_{j,t}=0$

Before we prove the cone theorem, we need the following lemmas.

######  Lemma 4.18.

Assumptions as in Theorem 4.16, the locus of a
$(K_{\mathcal{G}}+\Theta)$-negative extremal ray $R$ is not ${\mathcal{Y}}$.

######  Proof.

Since the general fibers of the original family of
$({\mathcal{Y}},{\mathcal{G}})\rightarrow T$ are in ${\mathcal{S}}_{P}$, the
locus $\mbox{{\rm loc}}(R)$ is not the whole ${\mathcal{Y}}$. ∎

######  Lemma 4.19.

Assumptions as in Theorem 4.16. Then the locus of a
$(K_{\mathcal{G}}+\Theta)$-negative extremal ray $R$ has dimension two.

######  Proof.

By Lemma 4.18, it suffices to show that any locus is not one-dimensional. Now
suppose $R={\mathbb{R}}_{\geq 0}[C]$ has one dimensional locus and $C\subset
F_{0}$ where ${\mathcal{Y}}_{0}=\bigcup_{i=0}^{s}F_{i}$. We divide the proof
into several steps.

  1. (1)

Claim. $\Theta\cdot C\geq 0$.

######  Proof.

Since the locus of $C$ is one-dimensional, we have $C\not\subset\Theta$. Thus
$\Theta\cdot C\geq 0$.

  2. (2)

Claim. The curve $C$ is invariant with $C^{2}<0$ on $F_{0}$.

######  Proof.

If $C^{2}\geq 0$ on $F_{0}$, then by Lemma 4.10, we have

$K_{{\mathcal{G}}_{F_{0}}}\cdot C\leq K_{\mathcal{G}}\cdot
C\leq(K_{\mathcal{G}}+\Theta)\cdot C<0.$

Then by Theorem 4.2, the locus of $R$ contains $F_{0}$, which is a
contradiction. Thus $C^{2}<0$ on $E_{0}$. If $C$ is not invariant, then

$\displaystyle 0$ $\displaystyle>$ $\displaystyle(K_{\mathcal{G}}+\Theta)\cdot
C$ $\displaystyle\geq$ $\displaystyle K_{\mathcal{G}}\cdot C$
$\displaystyle\geq$ $\displaystyle(K_{{\mathcal{G}}_{F_{0}}}+C)\cdot C$
$\displaystyle\geq$ $\displaystyle 0$

where the second inequality holds by step (1), the third inequality holds by
Lemma 4.10, and the last inequality holds by the first part of the proof of