# On the MMP for rank one foliations on threefolds

Paolo Cascini and Calum Spicer  Department of Mathematics, Imperial College
London, 180 Queen’s Gate, London SW7 2AZ, UK<EMAIL_ADDRESS>Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
<EMAIL_ADDRESS>
###### Abstract.

We prove existence of flips and the base point free theorem for log canonical
foliated pairs of rank one on a $\mathbb{Q}$-factorial projective klt
threefold. This, in particular, provides a proof of the existence of a minimal
model for a rank one foliation on a threefold for a wider range of
singularities, after McQuillan.

Moreover, we show abundance in the case of numerically trivial log canonical
foliated pairs of rank one.

###### 2010 Mathematics Subject Classification:

14E30, 37F75

###### Contents

  1. 1 Introduction
  2. 2 Preliminary Results
  3. 3 Facts about terminal singularities
  4. 4 Subadjunction result in the presence of a foliation
  5. 5 The formal neighborhood of a flipping curve
  6. 6 Threefold contractions and flips
  7. 7 Termination of flips
  8. 8 Running the MMP
  9. 9 Base point free theorem
  10. 10 Abundance for rank one foliations with $c_{1}(K_{\cal F})=0$

##  1\. Introduction

As in the classical Minimal Model Program, it is expected that every foliation
on a complex projective manifold is either uniruled or it admits a minimal
model, i.e. a good representative in its birational class. For rank one
foliations on a complex surface, this is known to be true thanks to the work
of Brunella, McQuillan and Mendes (e.g. see [Bru00, McQ08, Men00]). For
foliations of rank two on a threefold, the program was carried out in [Spi20,
CS18, SS19].

In [McQ04], McQuillan proved the existence of minimal models for foliations by
curves. More specifically, he showed that if $X$ is a projective variety with
quotient singularities and $\cal F$ is a rank one foliation on $X$ with log
canonical singularities, then $\cal F$ admits a minimal model.

The goal of this paper is to explore this result in the case of rank one
foliations on threefolds. In particular, we are interested in proving a
generalisation of McQuillan’s theorem, exploring some applications of the
theorem, such as the base point free theorem, the study of foliations with
trivial canonical class, and understanding the relationship between the
birational geometry of foliations and classical birational geometry.

###  1.1. Statement of main results

Our first main result is to show that flips exist for log canonical foliated
pairs of rank one on a $\mathbb{Q}$-factorial threefold with klt
singularities:

######  Theorem 1.1 (= Theorem 8.7).

Let $X$ be a $\mathbb{Q}$-factorial klt threefold and let $(\cal F,\Delta)$ be
a rank one foliated pair on $X$ with log canonical singularities. Let $R$ be a
$(K_{\cal F}+\Delta)$-negative extremal ray such that $\operatorname{loc}\,R$
has dimension one (cf. Section 2.13).

Then the flipping contraction $\phi\colon X\to Z$ associated to $R$ and the
$(K_{\cal F}+\Delta)$-flip exist.

The theorem in particular implies that the foliated MMP can be run with
natural assumptions on the singularities of the underlying variety, as well as
allowing for the presence of a boundary divisor:

######  Theorem 1.2 (= Theorem 8.8).

Let $X$ be a $\mathbb{Q}$-factorial projective threefold with klt
singularities and let $(\cal F,\Delta)$ be a log canonical foliated pair of
rank one on $X$. Assume that $K_{\cal F}+\Delta$ is pseudo-effective.

Then $(\cal F,\Delta)$ admits a minimal model.

Our ideas and proofs are greatly indebted to McQuillan’s strategies and
insights, however ultimately our approach to the existence of minimal models
of foliations is independent from the proof given in [McQ04] and is based on
techniques from the existence of minimal models in the case of varieties.

Similarly to the case of foliation of rank two on a threefold [CS18, Theorem
1.3], we show that existence of flips and divisorial contractions yields a
base point free theorem:

######  Theorem 1.3 (=Theorem 9.2).

Let $X$ be a normal projective threefold with $\mathbb{Q}$-factorial klt
singularities and let $(\cal F,\Delta)$ be a rank one foliated pair on $X$
with log canonical singularities. Assume that $\Delta=A+B$ where $A$ is an
ample $\mathbb{Q}$-divisor and $B\geq 0$ is a $\mathbb{Q}$-divisor. Suppose
that $K_{\cal F}+\Delta$ is nef.

Then $K_{\cal F}+\Delta$ is semi-ample.

Our next main result is a special case of foliation abundance:

######  Theorem 1.4 (= Theorem 10.1).

Let $X$ be a normal projective threefold and let $(\cal F,\Delta)$ be a log
canonical foliated pair. Suppose that $K_{\cal F}+\Delta\equiv 0$.

Then $K_{\cal F}+\Delta\sim_{\mathbb{Q}}0$.

In particular, it follows that after a quasi-étale cover that such a foliation
can be defined by a global vector field.

A very satisfying structure theorem is known for foliations of co-rank one
with trivial canonical class (see [Tou08, LPT18, Dru18, Dru19, DO19]). We hope
that this theorem provides a first step to understanding such a structure
theorem for foliations of larger co-rank.

Finally, we prove several results which relate the birational geometry of
foliations to classical birational geometry. For instance, it is a striking
feature of the canonical model theorem for foliation by curves on surfaces
that the singularities on the underlying surface of the canonical model are
never worse than log canonical (see [McQ08, Fact I.2.4 and Theorem III.3.2]).

We were interested if such a bound could be proven on threefolds without
making recourse to a canonical model theorem for foliations on threefolds,
which to our knowledge is unknown. In this direction we prove the following:

######  Theorem 1.5 (=Theorem 4.5).

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. Let $0\in X$ be an isolated singularity.

Then $X$ has log canonical singularities.

Simple examples show that this result is close to optimal in the sense that if
$0\in X$ is not an isolated singularity then there is in general no such bound
on the singularities of $X$ (see Example 4.6).

###  1.2. Sketch of the proof

We briefly explain our approach to the proof of existence of flips. Let $X$ be
a $\mathbb{Q}$-factorial projective threefold with klt singularities and let
$\cal F$ be a foliation with canonical singularities on $X$. Let $f\colon
X\rightarrow Z$ be a $K_{\cal F}$-negative flipping contraction which
contracts a single curve $C$. We first note that $C$ is necessarily $\cal
F$-invariant.

Our basic approach is to reduce the $K_{\cal F}$-flip to a $(K_{X}+D)$-flip
for some well chosen divisor $D$. If $D$ is an arbitrary divisor then there is
no reason to expect any relation between $\cal F$ and the pair $(X,D)$.
However, if every component of $D$ is invariant then much of the geometry of
$(X,D)$ is controlled by $\cal F$.

In particular, in Section 4 we show that if $(X,D)$ is log canonical and $C$
is a log canonical centre of $(X,D)$ then $(K_{X}+D)\cdot C<0$. Thus, the
challenge in producing the $K_{\cal F}$-flip becomes to produce a large number
of $\cal F$-invariant divisors containing $C$. These divisors give us the
flexibility to produce a divisor $D$ with the desired properties. This is
achieved in Section 5. The idea is to perform a careful analysis of the
singularities of the induced foliation $f_{*}\cal F$ on $Z$ at $f(C)$.
Unfortunately, as in the classical MMP, the divisor $K_{f_{*}\cal F}$ is not
$\mathbb{Q}$-Cartier and so working directly with $f_{*}\cal F$ is very
difficult. Rather, we demonstrate the existence of an auxiliary divisor $E$ on
$Z$, which is a foliated version of a complement in the classical MMP and such
that $K_{f_{*}\cal F}+E$ is $\mathbb{Q}$-Cartier and the pair $(f_{*}\cal
F,E)$ has mild singularities. An analysis of the pair is much more feasible
and in fact we are able to show that $(f_{*}\cal F,E)$ admits a particularly
simple normal form which, roughly, can be given by a vector field of the form
$\sum n_{i}x_{i}\frac{\partial}{\partial x_{i}}$ where the $n_{i}$ are non-
negative integers. Examining this normal form, we are able to produce a large
number of invariant divisors containing $C$.

It is worth spending a moment to compare this with McQuillan’s approach to the
existence of a flip. In dimension three, it is possible to show that
$C\cap\operatorname{Sing}\,\cal F$ consists of a single point $P$ and that if
$\partial$ is a vector field defining $\cal F$ near $P$ then
$\partial=-t\frac{\partial}{\partial t}+ax\frac{\partial}{\partial
x}+by\frac{\partial}{\partial y}$ where $C=\\{x=y=0\\}$ and $a,b$ are positive
integers. From this, it is possible to deduce that the normal bundle of $C$
splits as $\cal O(-a)\oplus\cal O(-b)$. By an inductive analysis of $\cal F$
along $C$, we can lift this splitting of the normal bundle to a splitting on a
formal neighbourhood of $C$ in $X$, i.e., $C$ is a complete intersection of
two formal divisors. With this description of the formal neighbourhood of $C$
in hand, it is easy to construct the flip by an explicit procedure consisting
of a single weighted blow up followed by a single weighted blow down.

###  1.3. Acknowledgements

Both the authors were partially funded by EPSRC. We would like to thank F.
Ambro, F. Bongiorno, J. McKernan and M. McQuillan for many useful discussions.

##  2\. Preliminary Results

###  2.1. Notations

We work over the field of complex numbers $\mathbb{C}$.

Given a normal variety $X$, we denote by $\Omega^{1}_{X}$ its sheaf of Kähler
differentials and, by $T_{X}:=(\Omega^{1}_{X})^{*}$ its tangent sheaf. For any
positive integer $p$, we denote
$\Omega_{X}^{[p]}\coloneqq(\Omega_{X}^{p})^{**}$. Let $A$ be a
$\mathbb{R}$-Weil divisor on $X$ and let $D$ be a prime divisor. We denote by
$\mu_{D}A$ the coefficient of $D$ in $A$. A log pair $(X,\Delta)$ is a pair of
a normal variety and a $\mathbb{Q}$-divisor $\Delta$ such that $K_{X}+\Delta$
is $\mathbb{Q}$-Cartier. We refer to [KM98] for the classical definitions of
singularities (e.g., klt, log canonical) appearing in the minimal model
program, except for the fact that in our definitions we require the pairs to
have effective boundaries. In addition, we say that a log pair $(X,\Delta)$ is
sub log canonical, or sub lc, if $a(E,X,\Delta)\geq-1$ for any geometric
valuation $E$ over $X$. A fibration $f\colon X\to Y$ is a surjective morphism
between normal varieties with connected fibres. We refer to [CS18, Section
2.5] for some of the basic notions, commonly used in the MMP.

A foliation of rank $r$ on a normal variety $X$ is a rank $r$ coherent
subsheaf $\cal F\subset T_{X}$ such that

  1. (1)

${\cal F}$ is saturated in $T_{X}$, and

  2. (2)

$\cal F$ is closed under Lie bracket.

Note that if $r=1$ then (2) is automatically satisfied. By (1), it follows
that $T_{X}/{\mathcal{F}}$ is torsion free. We denote by
$\mathcal{N}^{*}_{\cal F}=(T_{X}/{\mathcal{F}})^{*}$ the conormal sheaf of
$\cal F$. The normal sheaf $\cal N_{F}$ of $\cal F$ is the dual of the
conormal sheaf. The canonical divisor of $\cal F$ is a divisor $K_{\cal F}$
such that $\mathcal{O}_{X}(-K_{\cal F})\simeq\det T_{\mathcal{F}}$. The
foliation $\mathcal{F}$ is said to be Gorenstein (resp.
$\mathbb{Q}$-Gorenstein) if $K_{\mathcal{F}}$ is a Cartier (resp.
$\mathbb{Q}$-Cartier) divisor. More generally, a rank $r$ foliated pair $(\cal
F,\Delta)$ is a pair of a foliation $\cal F$ of rank $r$ and a
$\mathbb{Q}$-divisor $\Delta\geq 0$ such that $K_{\cal F}+\Delta$ is
$\mathbb{Q}$-Cartier.

Let $X$ be a normal variety and let $\cal F$ be a rank $r$ foliation on $X$.
We can associate to $\cal F$ a morphism

$\phi\colon\Omega^{[r]}_{X}\rightarrow\cal O_{X}(K_{\cal F})$

defined by taking the double dual of the $r$-wedge product of the map
$\Omega^{[1]}_{X}\to\cal F^{*}$, induced by the inclusion $\cal F\subset
T_{X}$. We will call $\phi$ the Pfaff field associated to $\cal F$. Following
[Dru18, Definition 5.4], we define the twisted Pfaff field as the induced map

$\phi^{\prime}\colon(\Omega^{[r]}_{X}\otimes\cal O_{X}(-K_{\cal
F}))^{**}\rightarrow\cal O_{X}$

and we define the singular locus of $\cal F$, denoted by
$\operatorname{Sing}\,\cal F$, to be the cosupport of the image of
$\phi^{\prime}$.

Let $\sigma\colon Y\dashrightarrow X$ be a dominant map between normal
varieties and let $\cal F$ be a foliation of rank $r$ on $X$. We denote by
$\sigma^{-1}\cal F$ the induced foliation on $Y$ (e.g. see [Dru18, Section
3.2]). If $\sigma\colon Y\to X$ is a birational morphism then the induced
foliation $\sigma^{-1}\cal F$ is called the pulled back foliation. If $f\colon
X\dashrightarrow X^{\prime}$ is a birational map, then the induced foliation
on $X^{\prime}$ by $f^{-1}$ is called the transformed foliation of $\cal F$ by
$f$ and we will denote it by $f_{*}\cal F$. Moreover, if $q\colon
X^{\prime}\rightarrow X$ is a quasi-étale cover and $\cal
F^{\prime}=q^{-1}\cal F$ then $K_{\cal F^{\prime}}=q^{*}K_{\cal F}$ and
[Dru18, Proposition 5.13] implies that $\cal F^{\prime}$ is non-singular if
and only if $\cal F$ is.

###  2.2. Invariant subvarieties

Let $X$ be a normal variety and let $\cal F$ be a rank $r$ foliation on $X$.
Let $S\subset X$ be a subvariety. Then $S$ is said to be $\cal F$-invariant,
or invariant by $\cal F$, if for any open subset $U\subset X$ and any section
$\partial\in H^{0}(U,\cal F)$, we have that

$\partial(\mathcal{I}_{S\cap U})\subset\mathcal{I}_{S\cap U},$

where $\mathcal{I}_{S\cap U}$ denotes the ideal sheaf of $S\cap U$. If
$D\subset X$ is a prime divisor then we define $\epsilon(D)=1$ if $D$ is not
$\cal F$-invariant and $\epsilon(D)=0$ if it is $\cal F$-invariant.

Assume now that $\mathcal{F}$ is a foliation of rank one on an affine variety
$X=\text{Spec}\,A$ such that $K_{\cal F}$ is Cartier. In this case,
$\mathcal{F}$ is defined by a vector field $\partial$. Let $S$ be a subvariety
defined by the ideal $I\subset A$. Then $S$ is $\cal F$-invariant if and only
if $I$ is $\partial$-invariant.

We will occasionally need to consider a logarithmic version of invariance. Let
$X$ be a normal variety and let $(\cal F,D)$ be a rank one foliated pair on
$X$, where $D$ is an integral divisor. We say that a subvariety $S\subset X$
is $(\cal F,D)$-invariant provided the following holds. Let $q\colon V\to X$
be a quasi-étale cover where $U:=q(V)$ is an open set intersecting $S$ and
such that $\cal L:=q^{[*]}\mathcal{O}_{X}(K_{\cal F}+D)$ is locally free. Let
$T=q^{-1}(S)$, $\cal F_{V}=q^{-1}\cal F$ and let $D^{\prime}\geq 0$ be the
integral divisor on $V$ such that $K_{\cal F_{V}}+D^{\prime}=q^{*}(K_{\cal
F}+D)$. Then there exists a vector field $\partial\in H^{0}(V,{\cal
F_{V}}(-D^{\prime}))$ which generates ${\cal F_{V}}(-D^{\prime})$ and $T$ is
$\partial$-invariant.

For smooth Gorenstein foliations, we have the following generalisation of
Frobenius theorem:

######  Lemma 2.1.

Let $X$ be an $n$-dimensional normal analytic variety and let $\cal F$ be a
rank one foliation on $X$. Let $P\in X$ be a closed point around which, $\cal
F$ is Gorenstein and smooth.

Then there exists an analytic neigbourhood $U$ of $P$, a normal analytic
variety $B$ of dimension $n-1$ and a holomorphic submersion $f\colon
U\rightarrow B$ such that the $\cal F$ is the foliation induced by $f$.

###### Proof.

Since $\cal F$ is Gorenstein, there exists an open neighbourhood $U$ of $P$
over which $\cal F$ is generated by a vector field $\partial$. Consider an
embedding $U\hookrightarrow M$ where $M$ is a manifold and let $\partial_{M}$
be a lift of $\partial$ to $M$. Denote by $\cal F_{M}$ the lifted foliation.
After possibly shrinking $M$, we may assume that $\cal F_{M}$ is also smooth.
Thus, the Frobenius theorem implies that there exists a submersion $F\colon
M\rightarrow N$ such that $\cal F_{M}$ is the foliation induced by $F$. This
gives us a submersion $f:=F|_{U}\rightarrow B$ where $B$ is the normalisation
of $f(U)$ and the claim follows. ∎

We will also need the following version of Riemann-Hurwitz formula for
foliations (e.g. see [Dru18, Lemma 3.4]):

######  Proposition 2.2.

Let $\sigma\colon Y\rightarrow X$ be a finite surjective morphism between
normal varieties, let $\cal F$ be a foliation on $X$ and let $\cal
G=\sigma^{-1}\cal F$.

Then we may write

$K_{\cal G}=\sigma^{*}K_{\cal F}+\sum\epsilon(\sigma(D))(r_{D}-1)D$

where the sum runs over all the prime divisors on $Y$ and $r_{D}$ is the
ramification index of $\sigma$ along $D$. In particular, if every ramified
divisor is $\cal G$-invariant then $K_{\cal G}=\sigma^{*}K_{\cal F}$.

######  Lemma 2.3.

Let $X=\text{Spec}\,A$ be an affine variety and let $\partial\in
H^{0}(X,T_{X})$. Let $I\subset A$ be a $\partial$-invariant ideal such that
its radical $\text{rad}\,I$ is prime.

Then $\text{rad}\,I$ is $\partial$-invariant.

###### Proof.

First, note that we may replace $A$ by its localisation at $\text{rad}\,I$.
Suppose for sake of contradiction that $\text{rad}\,I$ is not
$\partial$-invariant. Choose $f\in\text{rad}\,I$ such that $\partial
f\notin\text{rad}\,I$, i.e., $\partial f$ is a unit. Let $n$ be minimal
positive integer so that $f^{n}\in I$. Then $nf^{n-1}\partial
f=\partial(f^{n})\in I$. However, since $\partial f$ is a unit, it follows
that $f^{n-1}\in I$, a contradiction. ∎

######  Lemma 2.4.

Let $\sigma\colon Y\rightarrow X$ be a finite surjective morphism between
normal varieties, let $\cal F$ be a foliation of rank one on $X$ and let $\cal
G\coloneqq\sigma^{-1}\cal F$. Let $D\geq 0$ be an integral divisor on $X$ such
that $K_{\cal F}+D$ is $\mathbb{Q}$-Cartier and

$K_{\cal G}+E=\sigma^{*}(K_{\cal F}+D)$

where $E\geq 0$ is integral. Let $W\subset X$ be a subvariety and let
$Z=\sigma^{-1}(W)$.

Then $W$ is $(\cal F,D)$-invariant if and only if $Z$ is $(\cal
G,E)$-invariant.

###### Proof.

Without loss of generality we may assume that $X=\text{Spec}\,A$ and
$Y=\text{Spec}\,B$ are affine and $W$ and $Z$ are irreducible. Note that we
have an inclusion $A\subset B$.

Let $q\colon Y^{\prime}\rightarrow Y$ be a quasi-étale cover such that
$q^{*}(K_{\cal G}+E)$ is Cartier and let $\cal G^{\prime}\coloneqq q^{-1}\cal
G$ and $E^{\prime}\coloneqq q^{-1}(E)$. By definition, a subvariety is $(\cal
G,E)$-invariant if and only if its preimage in $Y^{\prime}$ is $(\cal
G^{\prime},E^{\prime})$-invariant. Thus, we may freely replace $Y$, $\cal G$
and $E$ by $Y^{\prime}$, $\cal G^{\prime}$ and $E^{\prime}$ respectively and
so we may assume that $K_{\cal G}+E$ is Cartier.

We first handle the case where $K_{\cal F}+D$ is Cartier, in which case we
have vector fields $\partial_{X}\in{\cal F}(-D)$ and $\partial_{Y}\in{\cal
G}(-E)$ such that $\sigma^{*}\partial_{X}=\partial_{Y}$.

We first show that $Z$ being $\partial_{Y}$-invariant implies that $W$ is
$\partial_{X}$-invariant. Let $I\subset B$ be the ideal of $Z$ and let
$J\subset A$ be the ideal of $W$ and note that $J=I\cap A$. We have

$\partial_{X}(J)=\partial_{X}(I\cap A)\subset\partial_{Y}(I)\cap A\subset
I\cap A=J,$

as required. For the opposite implication, note that $J\otimes B\subset I$ is
$\partial_{X}$-invariant and that $\text{rad}\,(J\otimes B)=I$. We may apply
Lemma 2.3 to conclude.

Now we handle the general case. Let $p\colon X^{\prime}\rightarrow X$ be a
quasi-étale morphism such that $q^{*}(K_{\cal F}+D)$ is Cartier and let
$Y^{\prime}$ be the normalization of the main component of
$Y\times_{X}X^{\prime}$, let $s\colon Y^{\prime}\rightarrow Y$ and $t\colon
Y^{\prime}\rightarrow X^{\prime}$ be the induced morphisms. Applying our
previous arguments to the maps $s\colon Y^{\prime}\rightarrow Y$ and $t\colon
Y^{\prime}\rightarrow X^{\prime}$, we may conclude. ∎

######  Lemma 2.5.

Let $X$ be a normal variety and let $(\cal F,D)$ be a rank one foliated pair
on $X$ where $D$ is an integral divisor. Let $p\colon Y\rightarrow X$ be a
proper morphism and assume that

$K_{\cal G}+E=p^{*}(K_{\cal F}+D)$

where $\cal G=p^{-1}\cal F$ and $E\geq 0$ is integral.

Then the following hold:

  1. (1)

If $W\subset Y$ is a $(\cal G,E)$-invariant subvariety then $p(W)$ is $(\cal
F,D)$-invariant.

  2. (2)

If $Z\subset X$ is a $(\cal F,D)$-invariant subvariety then $p^{-1}(Z)$ is
$(\cal G,E)$-invariant.

###### Proof.

Observe that this is a local problem on $X$, so without loss of generality we
may restrict to a neighborhood of $P\in X$ which admits a quasi-étale cover
$q\colon X^{\prime}\rightarrow X$ such that $q^{*}(K_{\cal F}+D)$ is Cartier
and write $K_{\cal F^{\prime}}+D^{\prime}=q^{*}(K_{\cal F}+D)$. Let
$Y^{\prime}$ be the normalization of the main component of
$Y\times_{X}X^{\prime}$, let $r\colon Y^{\prime}\rightarrow Y$ be the induced
map and observe that we may write $K_{\cal
G^{\prime}}+E^{\prime}=r^{*}(K_{\cal G}+E)$ where $E^{\prime}\geq 0$ is
integral.

By applying Lemma 2.4 to $q$ and $r$ we see that we may assume without loss of
generality that $K_{\cal F}+D$ and $K_{\cal G}+E$ are Cartier.

Let $Y\xrightarrow{\alpha}W\xrightarrow{\beta}X$ be the Stein factorisation of
$p$. Since $\beta$ is finite we may again apply Lemma 2.4 to replace $X$ by
$W$. Thus, without loss of generality, we may assume that $p_{*}\cal
O_{Y}=\cal O_{X}$.

We have that $T_{\cal F}(-D)$ is generated by a vector field $\partial$ which
lifts to a vector field $\tilde{\partial}$ on $Y$ which generates $T_{\cal
G}(-E)$.

We first prove (1). Let $J$ denote the ideal of $p(W)$ and let $I$ denote the
ideal sheaf of $W$. By abusing notation slightly, we have that $p_{*}I=J$. Let
$f\in J$ and notice that $p^{*}\partial f=\tilde{\partial}(p^{*}f)$. Since $W$
is $(\cal G,E)$-invariant and $p$ is proper, we have

$p^{*}\partial f=\tilde{\partial}(p^{*}f)\in H^{0}(Y,I)=J.$

Thus, $J$ is $\partial$-invariant and (1) follows.

We now prove (2). Let $I$ denote the ideal sheaf of $Z$ and let
$f_{1},...,f_{k}$ be generators of $I$. Then $p^{*}f_{1},\dots,p^{*}f_{k}$ are
generators of $p^{-1}I\cal O_{Y}$, the ideal sheaf of $p^{-1}(Z)$. Since
$\partial(f_{i})\in I$ we get that

$\tilde{\partial}(p^{*}f_{i})=p^{*}\partial f_{i}\in p^{-1}I\cal O_{Y}$

and so $p^{-1}I\cal O_{Y}$ is invariant under $\tilde{\partial}$, as required.
∎

######  Lemma 2.6.

Let $X$ be a normal variety, let $\partial$ be a vector field on $X$ and let
$W\subset X$ be a $\partial$-invariant subvariety.

Then $\operatorname{Sing}\,W$ is $\partial$-invariant and, in particular,
$\operatorname{Sing}\,X$ is $\partial$-invariant.

###### Proof.

When $X$ is smooth this is [CP06, Lemma 5.1]. We can reduce the general case
to the smooth case as follows. Everything is local on $X$ so we may freely
find an embedding $\iota\colon X\hookrightarrow\mathbb{C}^{M}$ and a lift of
$\partial$ to a vector field $\tilde{\partial}$ on $\mathbb{C}^{M}$. Observe
that $\iota(W)$ is $\tilde{\partial}$-invariant and we may apply the smooth
case to conclude. ∎

###  2.3. Foliation singularities

Let $X$ be a normal variety and let $(\cal F,\Delta)$ be a foliated pair on
$X$.

Given a birational morphism $\pi\colon\widetilde{X}\rightarrow X$, let
$\widetilde{\cal F}$ be the pulled back foliation on $\tilde{X}$ and let
$\Delta^{\prime}$ be the strict transform of $\Delta$ in $\widetilde{X}$. We
may write

$K_{\widetilde{\cal F}}+\Delta^{\prime}=\pi^{*}(K_{\cal F}+\Delta)+\sum
a(E,\cal F,\Delta)E$

where the sum runs over all the prime $\pi$-exceptional divisors of
$\tilde{X}$.

The rational number $a(E,\cal F,\Delta)$ denotes the discrepancy of $(\cal
F,\Delta)$ with respect to $E$. If $\Delta=0$, then we will simply denote
$a(E,\mathcal{F})=a(E,\mathcal{F},0)$.

######  Definition 2.7.

Let $X$ be a normal variety and let $(\cal F,\Delta)$ be a foliated pair on
$X$. We say that $(\cal F,\Delta)$ is terminal (resp. canonical, log terminal,
log canonical) if $a(E,\cal F,\Delta)>0$ (resp. $\geq 0$, $>-\epsilon(E)$,
$\geq-\epsilon(E)$), for any birational morphism $\pi\colon\tilde{X}\to X$ and
for any $\pi$-exceptional divisor $E$ on $\tilde{X}$.

Moreover, we say that the foliated pair $(\cal F,\Delta)$ is Kawamata log
terminal, or klt, if $\lfloor\Delta\rfloor=0$ and if $a(E,\cal
F,\Delta)>-\epsilon(E)$ for any birational morphism $\pi\colon\tilde{X}\to X$
and for any $\pi$-exceptional prime divisor $E$ on $\tilde{X}$.

We say that a $\mathbb{Q}$-Gorenstein foliation $\cal F$ is terminal (resp.
canonical, log canonical) if the foliated pair $(\cal F,0)$ is such.

Note that these notions are well defined, i.e., $\epsilon(E)$ and $a(E,\cal
F,\Delta)$ are independent of $\pi$. Observe also that in the case where $\cal
F=T_{X}$, no exceptional divisor $E$ over $X$ is invariant, i.e.,
$\epsilon(E)=1$ for all $E$, and so this definition recovers the usual
definitions of (log) terminal and (log) canonical.

Let $P\in X$ be a, not necessarily closed, point of $X$. We say that $(\cal
F,\Delta)$ is terminal (resp. canonical, log canonical) at $P$ if for any
birational morphism $\pi\colon\widetilde{X}\to X$ and for any
$\pi$-exceptional divisor $E$ on $\widetilde{X}$ whose centre in $X$ is the
Zariski closure $\overline{P}$ of $P$, we have that the discrepancy of $E$ is
$>0$ (resp. $\geq 0$, $\geq-\epsilon(E)$). Sometimes we will phrase this as
$P$ is a terminal (resp. canonical, log canonical) point for $(\cal
F,\Delta)$. We say that $\cal F$ is terminal near $P\in X$ if there is a
neighborhood $U$ of $P$ such that $\cal F|_{U}$ is terminal. We will see (cf.
Lemma 2.12) that being terminal at a closed point $P$ is equivalent to $\cal
F$ being smooth at $P$.

Given an irreducible subvariety $W\subset X$, we say that $(\cal F,\Delta)$ is
terminal at the generic point of $W$ if $(\cal F,\Delta)$ is terminal at the
generic point $\eta_{W}$ of $W$. We say that $(\cal F,\Delta)$ is terminal at
a general point of $W$ if $(\cal F,\Delta)$ is terminal at a general closed
point of $W$. Note that if $W$ is a canonical centre of $(\cal F,\Delta)$,
then $(\cal F,\Delta)$ is not terminal at the generic point of $W$. We also
remark that if $\cal F$ is smooth and $C\subset X$ is a $\cal F$-invariant
curve then $\cal F$ is terminal at a general point of $C$, but is not terminal
at the generic point of $C$.

Given a normal variety $X$ and a foliation $\cal F$ of rank one on $X$, we say
that $\cal F$ has dicritical singularities if there exists a birational
morphism $\pi\colon X^{\prime}\to X$ and a $\pi$-exceptional divisor $E$ which
is not $\pi^{-1}\cal F$-invariant. We say that $\cal F$ is non-dicritical, if
it is not dicritical.

######  Lemma 2.8.

Let $X$ be a normal variety and let $\cal F$ be a rank one foliation with
canonical singularities.

Then $\cal F$ is non-dicritical.

###### Proof.

This is [MP13, Corollary III.i.4]. ∎

######  Definition 2.9.

Given a normal variety $X$ and a foliated pair $(\cal F,\Delta)$ on $X$, we
say that a subvariety $W\subset X$ is a log canonical centre or, in short, lc
centre (resp. canonical centre) of $(\cal F,\Delta)$ if $(\cal F,\Delta)$ is
log canonical (resp. canonical) at the generic point of $W$ and there is a
birational morphism $\pi\colon Y\to X$ and a prime divisor $E$ on $Y$ of
discrepancy $-\epsilon(E)$ (resp. $0$) whose centre in $X$ is $W$.

A subvariety $W$ is called a non log canonical centre  of $(\cal F,\Delta)$ if
there is a birational morphism $\pi\colon Y\to X$ and a prime divisor $E$ on
$Y$ of discrepancy $<-\epsilon(E)$ whose centre in $X$ is $W$.

Note that if $\cal F$ is a non-dicritical foliation then the notions of log
canonical and canonical coincide. In this case we might still refer to
canonical centres as log canonical centres. We also remark that any $\cal
F$-invariant divisor is an lc centre and a canonical centre of $(\cal
F,\Delta)$.

We will make frequent use of the following:

######  Lemma 2.10 (Negativity lemma).

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

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

be a commutative diagram, where $Y$ is a normal variety and $f$ and
$f^{\prime}$ are proper birational morphisms. Let $(\cal F,\Delta)$ be a
foliated pair on $X$. Let $\cal F^{\prime}=\phi_{*}\cal F$ and let $(\cal
F^{\prime},\Delta^{\prime})$ be a foliated pair on $X^{\prime}$ such that
$f_{*}\Delta=f^{\prime}_{*}\Delta^{\prime}$. Assume that $-(K_{\cal
F}+\Delta)$ is $f$-ample and $K_{\cal F^{\prime}}+\Delta^{\prime}$ is
$f^{\prime}$-ample.

Then, for any valuation $E$ on $X$, we have

$a(E,\cal F,\Delta)\leq a(E,\cal F^{\prime},\Delta^{\prime}).$

Moreover, the inequality holds if $f$ or $f^{\prime}$ is not an isomorphism
above the generic point of the centre of $E$ in $Y$.

###### Proof.

The proof is the same as [KM98, Lemma 3.38]. ∎

The following is essentially [MP13, Corollary III.i.5].

######  Lemma 2.11.

Let $X$ be a normal variety and let $\cal F$ be a rank one foliation on $X$.
Let $q\colon\overline{X}\to X$ be a finite morphism and let $\overline{\cal
F}:=q^{-1}\cal F$. Let $\overline{Z}\subset\overline{X}$ be a subvariety and
let $Z:=q(\overline{Z})$. Assume that $(\cal F,\Delta)$ is a foliated pair on
$X$ and assume that $\overline{\Delta}:=q^{*}(K_{\cal
F}+\Delta)-K_{\overline{\cal F}}$ is an effective $\mathbb{Q}$-divisor.

Then $(\cal F,\Delta)$ is terminal (resp. canonical, log canonical) at the
generic point of $Z$ if and only if $(\overline{\cal F},\overline{\Delta})$ is
terminal (resp. canonical, log canonical) at the generic point of
$\overline{Z}$

###### Proof.

Let $\overline{f}\colon\overline{Y}\to\overline{X}$ be a proper birational
morphism and let $E$ be an $\overline{f}$-exceptional divisor on
$\overline{Y}$ whose centre in $\overline{X}$ is $\overline{Z}$. Then, by
[Kol96, Theorem VI.1.3] there exists a commutative diagram

${\overline{Y}}$${Y}$${\overline{X}}$${X}$$\scriptstyle{p}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$$\scriptstyle{q}$

where $f$ is birational and $p$ is finite. In particular, if
$E=p(\overline{E})$ then $E$ is $f$-exceptional and $Z$ is the centre of $E$
in $X$.

Assume now that $f\colon Y\to X$ is a proper birational morphism and let $E$
be an $f$-exceptional divisor on $Y$. Let $\overline{Y}$ be a component of the
normalisation of $\overline{X}\times_{X}Y$ which maps onto $Y$ and let
$\overline{f}\colon\overline{Y}\to\overline{X}$ and $p\colon\overline{Y}\to Y$
be the induced morphisms. Let $\overline{E}$ be a prime divisor such that
$p(\overline{E})=E$.

In both cases above, Lemma 2.5 implies that
$\epsilon(E)=\epsilon(\overline{E})$. Let $r_{E}$ be the ramification index of
$p$ along $E$. Then, as in the proof of [KM98, Proposition 5.20], Proposition
2.2 implies that

$a(\overline{E},\overline{\cal F},\overline{\Delta})=r_{E}a(E,\cal
F,\Delta)+\epsilon(E)(r_{E}-1).$

It follows easily that

$a(\overline{E},\overline{\cal F},\overline{\Delta})>0\qquad\text{(resp. $\geq
0$, $\geq-\epsilon(\overline{E})$)}$

if and only if

$a(E,\cal F,\Delta)>0\qquad\text{(resp. $\geq 0$, $\geq-\epsilon(E)$).}$

Thus, the claim follows. ∎

######  Lemma 2.12.

Let $X$ be a normal variety and let $(\cal F,D)$ be a rank one foliated pair
on $X$, where $D$ is an integral divisor. Let $Z\subset X$ be a
$(\mathcal{F},D)$-invariant subvariety.

Then $(\cal F,D)$ is not terminal at the generic point of $Z$. In particular,
if $D=0$ then $\cal F$ is not terminal at the generic points of
$\operatorname{Sing}\,\cal F$ and if $\cal F$ is Gorenstein then $\cal F$ is
not terminal at the generic points of $\operatorname{Sing}\,X$.

Moreover, if $D=0$ and $P\in X$ is a closed point then $\cal F$ is terminal at
$P$ if and only if it is smooth. In particular, if $\cal F$ is terminal at
$P$, then there exists an analytic open neighbourhood $U$ of $P$ and a quasi-
étale morphism $q\colon V\rightarrow U$ such that:

  1. (1)

there exists a holomorphic submersion $f\colon V\rightarrow B$ onto a normal
analytic variety $B$ of dimension $n-1$ such that $\cal F^{\prime}:=q^{-1}\cal
F|_{U}$ is the foliation induced by $f$; and

  2. (2)

$q(f^{-1}(z))$ is a canonical centre for all $z\in B$ and, in particular,
$\cal F$ is not terminal at the generic point of $q(\operatorname{Sing}\,V)$.

###### Proof.

Let $P\in Z$ be a general point. It is enough to prove the claim in a
neighbourhood of $P$. In particular, we may consider a quasi-étale cover
$q\colon V\to X$ such that if $\overline{\cal F}=q^{-1}\cal F$ and
$\overline{D}=q^{-1}D$ then $K_{\overline{\cal F}}+\overline{D}$ is Cartier.
Thus, we have that $\overline{Z}=\pi^{-1}(Z)$ is $(\overline{\cal
F},\overline{D})$-invariant, and therefore $(\overline{\cal F},\overline{D})$
defines a vector field $\partial$ such that if $I$ is the ideal defining
$\overline{Z}$ then $I$ is invariant with respect to $\partial$. As in the
proof of [BM16, Lemma I.1.3], it follows that if
$\overline{p}\colon\overline{Y}\to X$ is the blow-up of $X$ at $\overline{Z}$
with exceptional divisor $\overline{E}$, then $a(\overline{E},\overline{\cal
F},\overline{D})\leq 0$. Thus, Lemma 2.11 implies that $(\cal F,D)$ is not
terminal at the generic point of $Z$.

Assume now that $D=0$. Since $\operatorname{Sing}\,\cal F$ is $\cal
F$-invariant, it follows that $\cal F$ is not terminal at the generic point of
each irreducible component of $\operatorname{Sing}\,\cal F$.

Assume that $\cal F$ is Gorenstein and $\Sigma$ is an irreducible component of
$\operatorname{Sing}\,X$ that is not contained in $\operatorname{Sing}\,\cal
F$. Then Lemma 2.6 implies that $\Sigma$ is $\cal F$-invariant. Thus, $\cal F$
is not terminal at the generic point of $\Sigma$.

Now assume that $\cal F$ is terminal at a closed point $P\in X$. Let $U$ be an
open neighbourhood of $P$ which admits a quasi-étale morphism $q\colon
V\rightarrow U$ such that $q^{*}K_{\cal F}$ is a Cartier divisor. Let $\cal
F^{\prime}=q^{-1}\cal F$ and let $Q\in V$ be a closed point such that
$q(Q)=P$. Lemma 2.11 implies that $Q$ is terminal for $\cal F^{\prime}$. By
our previous claim, it follows that $\cal F^{\prime}$ is smooth at $Q$. Thus,
by Lemma 2.1, after possibly shrinking $U$ and $V$, there exists $f\colon V\to
B$, as in (1). In particular, [Dru18, Proposition 5.13] implies that $\cal F$
is smooth at $P$.

Conversely, suppose that $\cal F$ is smooth at $P$. Again, by Lemma 2.11, we
may assume that $\cal F$ is in fact Gorenstein at $P$ and, by Lemma 2.1, that
$\cal F$ is given by a submersion $U\rightarrow B$. A simple calculation shows
that $\cal F$ is terminal at $P$.

We now prove (2). We have that $f^{-1}(z)$ is $\mathcal{F}^{\prime}$-invariant
for all $z\in B$. Thus, $q(f^{-1}(z))$ is $\mathcal{F}$-invariant and, by our
previous claim, it follows that $q(f^{-1}(z))$ is a canonical centre of $\cal
F^{\prime}$ for all $z\in B$.

We have that $\operatorname{Sing}\,V=f^{-1}(\operatorname{Sing}\,B)$ and, in
particular, $q(\operatorname{Sing}\,V)$ is $\cal F$-invariant and, as above,
(2) follows. ∎

Note that the above lemma implies the well known fact that if $X$ is a surface
and $\cal F$ is a terminal rank one foliation on $X$ then $X$ has at worst
quotient singularities. One can ask more generally if there is a similar way
to control the singularities of the underlying variety in higher dimensions
and higher ranks, and if such a bound holds if $\cal F$ has only canonical
singularities. For foliations of co-rank one on a normal threefold, some of
these questions were addressed in [CS18]. We will approach some cases of this
problem in the rank one case in dimension three (cf. Section 4).

We remark that if $\cal F$ is log canonical then there is no bound on the
singularities of the underlying variety, at least from the perspective of Mori
theory, as examples in [McQ08] show.

######  Lemma 2.13.

Let $X$ be a normal variety and let $\cal F$ be a $\mathbb{Q}$-Gorenstein
foliation of rank one on $X$. Let $C$ be an $\cal F$-invariant curve which is
not contained in $\operatorname{Sing}\,\cal F$.

Then $\cal F$ is Gorenstein at the generic point of $C$.

###### Proof.

We may freely replace $X$ by an analytic neighbourhood of a general point $P$
of $C$. In particular, $\cal F$ is smooth at $P$. Let $q\colon
X^{\prime}\rightarrow X$ be the quasi-étale cover and $f\colon X^{\prime}\to
B$ be the morphism whose existence is guaranteed by Lemma 2.12 and such that
$\cal F^{\prime}:=q^{-1}\cal F$ is induced by $f$. We may assume that $q$ is a
Galois cover with Galois group $G:=\mathbb{Z}/m\mathbb{Z}$. Since the fibres
of $f$ are invariant with respect to $G$, it follows that the action descends
to $B$. After possibly shrinking further, we may assume that $X^{\prime}$ is
isomorphic to $D\times B$ where $0\in D$ is a disc with coordinate $t$, such
that $q(0\times B)=P$ for some $Q\in B$. Moreover $G$ acts on $D$ so that the
diagonal action of $G$ on $D\times S$ coincides with the action on
$X^{\prime}$. Finally, $\cal F^{\prime}$ is generated by the vector field
$\frac{\partial}{\partial t}$. Thus, it follows that, away from $0\times B$,
we have that $\frac{\partial}{\partial t}$ descends to a section of $T_{X}$,
and this implies that $\cal F$ is Gorenstein away from $q(0\times B)$. Since
$q(0\times B)\cap C=P$, our claim follows. ∎

###  2.4. Foliations on a surface

We will need the following results for foliations on a surface.

######  Lemma 2.14.

Let $X$ be a normal projective surface and let $\cal F$ be a rank one
foliation on $X$ such that $K_{\cal F}\equiv 0$ and suppose that $\cal F$ is
not algebraically integrable. Let $C_{1},\dots,C_{k}$ be all the $\cal
F$-invariant curves on $X$.

Then $X$ is covered by curves $M$ not passing through
$\operatorname{Sing}\,\cal F$ and such that

$(K_{X}+\sum_{i=1}^{k}C_{i})\cdot M\leq 0.$

###### Proof.

We first show that $\cal F$ has canonical singularities. Suppose not and let
$p\colon Y\to X$ be a resolution such that $\cal F_{Y}:=p^{-1}\cal F$ has
canonical singularities. We have $K_{\cal F_{Y}}-\sum a(E,\cal F)E\equiv 0$,
where the sum runs over all the $p$-exceptional divisors and, by assumption,
there exists a $p$-exceptional divisor $E$ such that $a(E,\cal F)<0$. In
particular, $\cal F_{Y}$ is not pseudo-effective and by Miyaoka’s theorem
(e.g. see [Bru00, Theorem 7.1]), $\cal F_{Y}$ is algebraically integrable, and
so is $\cal F$, a contradiction.

Next, observe that we may freely contract $\cal F$-invariant divisors and
replace $X$ by a quasi-étale cover. Thus, we are free to assume that $\cal F$
is one of the foliations appearing in the list [McQ08, Theorem IV.3.6]. In
particular, $X$ is obtained as an equivariant compactification of a
commutative algebraic group of dimension two and $\cal F$ is induced by a
codimension one Lie subgroup. We now check each individual case:

  1. (1)

$X$ is an abelian surface and $\cal F$ is a linear foliation. In particular,
if $\cal F$ is not algebraically integrable, there are no $\cal F$-invariant
curves on $X$ and $K_{X}\sim 0$.

  2. (2)

$X$ is a $\mathbb{P}^{1}$-bundle over an elliptic curve, with projection
$p\colon X\rightarrow S$. In this case, the $\cal F$-invariant curves are
either a single section or two disjoint sections. Thus, it is enough to choose
$M$ as a general fibre of $p$.

  3. (3)

$X$ is a $\mathbb{P}^{1}$-bundle over $\mathbb{P}^{1}$, with projection
$p\colon X\to\mathbb{P}^{1}$. In this case, the $\cal F$-invariant curves are
two vertical fibres and either a single or two disjoint sections. Again, we
can choose $M$ as a general fibre of $p$. ∎

######  Lemma 2.15.

Let $X$ be a normal projective surface and let $\cal F$ be a rank one
foliation on $X$ which is algebraically integrable. Let $\Delta,\Theta\geq 0$
be $\mathbb{Q}$-divisors on $X$ such that

  1. (1)

$\mu_{C}\Theta\leq\mu_{C}\Delta$ for any curve $C$ which is not $\cal
F$-invariant, and

  2. (2)

$(X,\Theta)$ is log canonical.

Then $X$ is covered by $\cal F$-invariant curves $M$ such that

$(K_{X}+\Theta)\cdot M\leq(K_{\cal F}+\Delta)\cdot M.$

###### Proof.

We distinguish two cases. We first assume that $\cal F$ is non-dicritical.
Then $\cal F$ is induced by a fibration $p\colon X\rightarrow B$. Let $M$ be a
general fibre of $p$. Then $K_{X}\cdot M=K_{\cal F}\cdot M$ and (1) implies
that $\Theta\cdot M\leq\Delta\cdot M$. Thus, our claim follows.

Let us assume now that $\cal F$ is dicritical. Let $p\colon
X^{\prime}\rightarrow X$ be an F-dlt modification of $\cal F$ (cf. [CS18,
Theorem 1.4]). Then we may write $K_{\cal
F^{\prime}}+p_{*}^{-1}\Delta+E=p^{*}(K_{\cal F}+\Delta)$ and
$K_{X^{\prime}}+p_{*}^{-1}\Theta+E^{\prime}=p^{*}(K_{X}+\Theta)$, where
$E,E^{\prime}$ are $p$-exceptional $\mathbb{Q}$-divisors and the coefficients
of $E$ (resp. $E^{\prime}$) are greater or equal (resp. less or equal ) to
one. Since $\cal F^{\prime}$ is algebraically integrable and non-dicritical,
it follows that $\cal F^{\prime}$ is induced by a fibration $\pi\colon
X^{\prime}\rightarrow B$. Let $F$ be a general fibre of $\pi$ and observe that

  1. (i)

$K_{\cal F^{\prime}}\cdot F=K_{X^{\prime}}\cdot F$,

  2. (ii)

$p_{*}^{-1}\Theta\cdot F\leq p_{*}^{-1}\Delta\cdot F$, and

  3. (iii)

$E-E^{\prime}\geq 0$.

Thus, if $M=p(F)$ then

$\displaystyle(K_{X}+\Theta)\cdot M$
$\displaystyle=(K_{X^{\prime}}+p_{*}^{-1}\Theta+E^{\prime})\cdot F$
$\displaystyle\leq(K_{\cal F^{\prime}}+p_{*}^{-1}\Delta+E)\cdot F=(K_{\cal
F}+\Delta)\cdot M$

and the claim follows. ∎

###  2.5. Adjunction

######  Proposition 2.16.

Let $X$ be a normal variety and $\cal F$ be a rank one $\mathbb{Q}$-Gorenstein
foliation on $X$. Let $S\subset X$ be an $\cal F$-invariant subvariety such
that $\mathcal{F}$ is Gorenstein at the generic point of $S$. Let $\nu\colon
S^{\nu}\rightarrow S$ be the normalisation.

Then

  1. (1)

there is an induced foliated pair $(\cal G,\Delta)$ of rank one on $S^{\nu}$
such that

$K_{\cal F}|_{S^{\nu}}=K_{\cal G}+\Delta;$

  2. (2)

if $(\cal G,\Delta)$ is terminal at a closed point $P\in S^{\nu}$ then $\cal
F$ is terminal at $\nu(P)$.

Assume now that $C\subset X$ is an $\mathcal{F}$-invariant curve which is not
contained in $\operatorname{Sing}\,\cal F$ and such that $\cal F$ is
Gorenstein at the generic point of $C$. If $\nu\colon C^{\nu}\rightarrow C$ is
the normalisation then $K_{\cal F}|_{C^{\nu}}=K_{C^{\nu}}+\Delta$, where
$\Delta\geq 0$, and

  1. (3)

$\operatorname{Supp}\,\lfloor\Delta\rfloor=\nu^{-1}(\operatorname{Sing}\,\cal
F\cap C)$; and

  2. (4)

if $P\in C$ is a point such that $\cal F$ is terminal but not Gorenstein at
$\nu(P)$ then $\mu_{P}\Delta=\frac{r-1}{r}$ where $r$ is the Cartier index of
$K_{\cal F}$ at $\nu(P)$.

###### Proof.

We first show (1). In order to construct the foliation $\cal G$ on $S^{\nu}$,
we may work locally around a point in $S$. Let $P\in S$ be a closed point.
Then there exists a quasi-étale cyclic cover $q\colon V\to U$ with Galois
group $G$, where $U$ is a neighbourhood of $P$, and such that $q^{*}K_{\cal
F}$ is a Cartier divisor. Let $\cal F_{V}=q^{-1}\cal F|_{U}$ be the foliation
induced on $V$. By assumption, we may assume that $q$ is étale in a
neighborhood of the general point of $S$. Thus, $K_{\cal F_{V}}=q^{*}K_{\cal
F}$ and $S_{V}:=\pi^{-1}(S)$ is $\cal F_{V}$-invariant. We have a commutative
diagram:

${\Omega^{1}_{V}|_{S_{V}}}$${\cal O_{S_{V}}(K_{\cal
F_{V}})}$${\Omega^{1}_{S_{V}}}$

Note that the maps of the diagram are $G$-invariant.

Let $S^{\nu}_{V}\to S_{V}$ be the normalisation of $S_{V}$. Then the action of
$G$ lifts to $S^{\nu}_{V}$ and the induced morphism $\rho\colon S^{\nu}_{V}\to
S^{\nu}_{U}:=\nu^{-1}(U\cap S)$ is the quotient by $G$. By [ADK08, Proposition
4.5], the map $\Omega^{1}_{S_{V}}\to\cal O_{S_{V}}(K_{\cal F_{V}})$ extends
uniquely to a $G$-invariant map $\Omega^{1}_{S^{\nu}_{V}}\to\cal
O_{S^{\nu}_{V}}(K_{\cal F_{V}})$. Let $\cal L$ be its image. Then $\cal L$
defines a $G$-invariant foliation $\cal G_{V}$ on $S^{\nu}_{V}$ so that $\cal
L=\cal O_{S^{\nu}_{V}}(K_{\cal G_{V}})$. Let $\cal L^{\prime}$ be the
$G$-invariant push-forward of $\cal L$ on $\nu^{-1}(U\cap S)$ (e.g. see
[GKKP11, Appendix A]). In particular, $\cal L^{\prime}$ is a quotient of
$\Omega^{1}_{S^{\nu}}$ and a reflexive subsheaf of the rank one sheaf
$\mathcal{O}_{S^{\nu}_{U}}(K_{\cal F})$. Thus, it induces a foliation $\cal
G_{U}$ on $S^{\nu}_{U}$ such that $\cal O_{S^{\nu}_{U}}(K_{\cal G_{U}})=\cal
L^{\prime}$. Let $\cal G$ be the induced foliation on $S^{\nu}$.

Let $m$ be the Cartier index of $K_{\cal F}$. Observe that we have a morphism
$(\Omega^{1}_{X})^{\otimes m}\rightarrow\cal O_{X}(mK_{\cal F})$, which yields
a morphism $({\Omega^{1}_{X}}|_{S})^{\otimes m}\rightarrow\cal O_{S}(mK_{\cal
F})$. By [AD14, Lemma 3.7] this lifts to a morphism
$(\Omega^{1}_{S^{\nu}})^{\otimes m}\rightarrow\cal O_{S^{\nu}}(mK_{\cal F})$
which factors through $\cal O_{S^{\nu}}(mK_{\cal G})$. Thus, there is an
effective divisor $M$ on $S^{\nu}$ such that

$mK_{\cal F}|_{S^{\nu}}=mK_{\cal G}+M.$

Let $\Delta\coloneqq\frac{1}{m}M$. Then $K_{\cal F}|_{S^{\nu}}=K_{\cal
G}+\Delta$ and (1) follows.

We now prove (2). Assume by contradiction that $(\cal G,\Delta)$ is terminal
at $P\in S^{\nu}$ and that $\cal F$ is not terminal at $\nu(P)$. Let $q\colon
V\to U$ be the quasi-étale cover in a neighbourhood $U$ of $\nu(P)$ defined
above and let $D$ be the integral divisor on $S^{\nu}_{V}$ such that $K_{\cal
G_{V}}+D=K_{\cal F_{V}}|_{S^{\nu}_{V}}$.

Let $Q\in V$ such that $q(Q)=\nu(P)$. Then, Lemma 2.11 implies that $\cal
F_{V}$ is not terminal at $Q$ and Lemma 2.12 implies that
$Q\in\operatorname{Sing}\,\cal F_{V}$. In particular, $Q$ is $\cal
F_{V}$-invariant. Thus, it follows that if $Q^{\prime}\in S^{\nu}_{V}$ is such
that $\nu^{\prime}(Q^{\prime})=Q$, where $\nu^{\prime}\colon S^{\nu}_{V}\to
S_{V}$ is the normalisation, then $Q$ is $(\cal G_{V},D)$-invariant and, by
Lemma 2.12, $(\cal G_{V},D)$ is not terminal at $Q$. Lemma 2.11 implies that
$(\cal G,\Delta)$ is not terminal at $P$, a contradiction. Thus, (2) follows.

Assume now that $C\subset X$ is a curve as in our second claim. As above,
there exists a quasi-étale cyclic cover $q\colon V\to U$ where $U$ is a
neighbourhood of $P$, and such that $q^{*}K_{\cal F}$ is a Cartier divisor.
Let $\cal F^{\prime}=q^{-1}\cal F$, let $\tilde{C}=q^{-1}C$ and let
$\nu^{\prime}\colon\tilde{C}^{\nu}\to\tilde{C}$ be the normalisation. Then
$K_{\cal F^{\prime}}=q^{*}K_{\cal F}$. Let $p\colon\tilde{C}^{\nu}\to C^{\nu}$
be the induced morphism. We may write

$p^{*}(K_{C}+\Delta)=K_{\cal
F^{\prime}}|_{\tilde{C}^{\nu}}=K_{\tilde{C}^{\nu}}+D$

for some integral divisor $D\geq 0$. Let $Q\in\tilde{C}^{\nu}$ such that
$p(Q)=P$.

We now prove (3). First note that, by (2), it follows that if
$P\in\nu^{-1}(\operatorname{Sing}\,\cal F\cap C)$ then
$P\in\operatorname{Supp}\,\lfloor\Delta\rfloor$. Suppose now that
$P\in\operatorname{Supp}\,\lfloor\Delta\rfloor$. Then, by Riemann-Hurwitz for
curves, if follows that $\mu_{Q}D\geq 1$. Thus,
$\nu^{\prime}(Q)\in\operatorname{Sing}\,\cal F^{\prime}$ and by Lemma 2.12 and
Lemma 2.11, it follows that $\nu(P)\in\operatorname{Sing}\,\cal F$. Thus, (3)
follows.

Assume now that $\cal F$ is terminal but not Gorenstein at $\nu(P)$. Then, as
above, Lemma 2.11 and Lemma 2.12 imply that $\cal F^{\prime}$ is smooth at
$\nu^{\prime}(Q)$. In particular, it follows that $\tilde{C}$ is smooth at
$\nu^{\prime}(Q)$ and $\mu_{Q}D=0$. Again, by Riemann-Hurwitz, it follows that
$\mu_{P}\Delta=\frac{r-1}{r}$ where $r$ is the ramification index of $p$ at
$Q$. Note that $r$ coincides with the Cartier index of $K_{\cal F}$ at
$\nu(P)$. Thus, (4) follows. ∎

We now explain some generalities comparing foliation adjunction and classical
adjunction on a threefold:

######  Proposition 2.17.

Let $X$ be a normal threefold and let $\cal F$ be a foliation of rank one on
$X$ with canonical singularities. Let $S\subset X$ be a reduced divisor whose
irreducible components are $\cal F$-invariant and such that $(X,S)$ is log
canonical. Let $\nu\colon S^{\nu}\to S$ be its normalisation.

We may write

$K_{\cal F}|_{S^{\nu}}=K_{\cal
G}+\Delta\quad\text{and}\quad(K_{X}+S)|_{S^{\nu}}=K_{S^{\nu}}+\Theta$

where $\cal G$ is the induced foliation and $\Delta,\Theta\geq 0$ are
$\mathbb{Q}$-divisors on $S^{\nu}$. Let $C\subset S^{\nu}$ be a curve.

Then the following hold:

  1. (1)

if $\nu(C)$ is contained in $\operatorname{Sing}\,\cal F$ and $C$ is not $\cal
G$-invariant then $\mu_{C}\Delta\geq 1$ and, in particular,
$\mu_{C}\Delta\geq\mu_{C}\Theta$;

  2. (2)

if $\cal F$ is terminal at the generic point of $\nu(C)$, then
$\mu_{C}\Delta=\mu_{C}\Theta<1$.

Note that, in the notations above, if $C$ is $\cal G$-invariant then there is
in general no natural relation between $\mu_{C}\Delta$ and $\mu_{C}\Theta$, as
shown in the following example:

######  Example 2.18.

Let $T$ be a smooth surface and let $C_{0}$ be a smooth curve. Let $X=T\times
C_{0}$ and let $\cal F$ be the foliation induced by the fibration $p\colon
X\to T$. Let $D\subset T$ be a curve with high multiplicity at a point $z\in
D$ and let $S=D\times C_{0}\subset X$. Then $S$ is $\cal F$-invariant and if
$C=\\{z_{0}\\}\times C_{0}$, we have that $\mu_{C}\Delta=0$, however
$\mu_{C}\Theta$ is arbitrarily large.

###### Proof of Proposition 2.17.

The existence of $\cal G$ and $\Delta\geq 0$ follows from Proposition 2.16.
The existence of $\Theta\geq 0$ follows from the classical adjunction. Since
$(X,S)$ is log canonical, it follows that also $(S^{\nu},\Theta)$ is such.
Note that $\Delta$ is supported on
$\nu^{-1}\operatorname{Sing}\,X\cup\nu^{-1}\operatorname{Sing}\,\cal F$ and
that $\Theta$ is supported on
$\nu^{-1}\operatorname{Sing}\,X\cup\nu^{-1}\operatorname{Sing}\,S$.

We first prove (1). Assume that $C\subset S^{\nu}$ is a curve such that
$\nu(C)\subset\operatorname{Sing}\,\cal F$. We first assume that $\cal F$ is
Gorenstein and we later prove the general case. Since $S$ is invariant, by
[AD14, Lemma 3.7], we have the commutative diagram

${\Omega^{1}_{X}|_{S^{\nu}}}$${\cal O_{S^{\nu}}(K_{\cal
F})}$${\Omega^{1}_{S^{\nu}}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

and $\mathcal{O}_{S^{\nu}}(K_{\cal G})$ is the image of $\psi$. Since $\cal F$
is singular along $\nu(C)$, it follows that the map $\phi$ is not surjective
along $C$. Thus, $\mu_{C}(\Delta)\geq 1$ and the claim follows.

We now consider the general case. Let $P\in C$ be a general point. After
possibly shrinking $X$ around $\nu(P)$, we may assume that there exists a
quasi-étale cover $q\colon X^{\prime}\to X$ such that $q^{*}K_{\cal F}$ is
Cartier. We may also assume that $C^{\prime}\coloneqq q^{-1}(\nu(C))$ is
smooth. Let $T\coloneqq q^{-1}(S)$ and $\cal F^{\prime}\coloneqq q^{-1}\cal
F$. Then $\cal F^{\prime}$ is Gorenstein and $T$ is $\cal
F^{\prime}$-invariant. Let $\nu^{\prime}\colon T^{\nu}\to T$ be the
normalisation of $T$, let $p\colon T^{\nu}\to S^{\nu}$ be the induced morphism
and let $(\cal G^{\prime},\Delta^{\prime})$ be the induced foliated pair on
$T^{\nu}$, whose existence is guaranteed by Proposition 2.16. By Lemma 2.11
and Lemma 2.12, it follows that $C^{\prime}\subset\operatorname{Sing}\,\cal
F^{\prime}$. Note that

$p^{*}(K_{\cal G}+\Delta)=K_{\cal G^{\prime}}+\Delta^{\prime},$

and by the previous case, it follows that if $C^{\prime\prime}\subset T^{\nu}$
is a curve such that $\nu^{\prime}(C^{\prime\prime})=C^{\prime}$ then
$\mu_{C^{\prime\prime}}\Delta^{\prime}\geq 1$. Note that
$p(C^{\prime\prime})=C$. Let $r$ be the ramification index of $p$ along
$C^{\prime\prime}$. Since, by assumption, $C$ is not $\cal G$-invariant, the
Riemann-Hurwitz formula (cf. Proposition 2.2) implies that

$r\mu_{C}\Delta=\mu_{C^{\prime\prime}}\Delta^{\prime}+\epsilon(C)(r-1)\geq r.$

Thus, $\mu_{C}\Delta\geq 1$. Since $(S^{\nu},\Theta)$ is log canonical, it
follows that $\mu_{C}\Theta\leq\mu_{C}\Delta$ and (1) follows.

We now prove (2). Let $C\subset S^{\nu}$ be a curve such that $\cal F$ is
terminal at the generic point of $\nu(C)$. By Lemma 2.12, it follows that
$\nu(C)$ is not $\cal F$-invariant. Let $q\colon V\to U$ be a quasi-étale
cover defined over an open subset $U\subset X$ containing the generic point of
$\nu(C)$ and such that $q^{*}K_{\cal F}$ is Cartier. Let $\cal F^{\prime}$ be
the induced foliation on $V$ and let $S^{\prime}=q^{-1}(S)$ and
$C^{\prime}=q^{-1}(\nu(C))$. Then $C^{\prime}$ is not $\cal
F^{\prime}$-invariant. After possibly shrinking $U$, we may assume that
$C^{\prime}$ is smooth. Let $P\in C^{\prime}$ be a general point. By Lemma
2.12 there exists a submersion $f\colon W\rightarrow B$ where $W$ is an
analytic neighbourhood of $P$ and such that $\cal F^{\prime}|_{W}$ is the
foliation induced by $f$. In particular, $S^{\prime}\cap
W=f^{-1}(f(C^{\prime}))$ is smooth because $f$ and $C^{\prime}$ are smooth at
$P$. Similarly, it follows that $X^{\prime}$ is smooth at $P^{\prime}$ and
therefore, $X$ has quotient singularities at the general point of $\nu(C)$.

In particular, after possibly shrinking $U$, we may assume that
$S^{\prime}\rightarrow S$ factors through $S^{\nu}$. Let $\sigma\colon
S^{\prime}\to S^{\nu}$ be the induced morphism. Let $L^{\prime}\subset
S^{\prime}$ be the germ of a leaf through a general point $Q^{\prime}\in
C^{\prime}$. Let $Q=\sigma(Q^{\prime})$, let $L=\sigma(L^{\prime})$ and let
$n$ be the order of the cover $\sigma$ at $Q$. We may assume that $L$ and
$L^{\prime}$ are smooth, $X^{\prime}$, $S^{\prime}$ and $\cal F^{\prime}$ are
smooth along $L^{\prime}$ and that $S^{\nu}$ and $\cal G$ are smooth along
$L$. Let $\cal G^{\prime}$ be the foliation induced by $\cal G$ on
$S^{\prime}$. Then $\cal G^{\prime}$ is smooth along $L^{\prime}$ and

$K_{\cal F^{\prime}}|_{L^{\prime}}=K_{\cal
G^{\prime}}|_{L^{\prime}}=K_{L^{\prime}}.$

We have $q^{*}K_{\cal F}=K_{\cal F^{\prime}}$, and $K_{\cal G}|_{L}=K_{L}$.
Thus, if $\eta\colon L^{\prime}\to L$ is the induced morphism then

$\eta^{*}(K_{L}+\frac{n-1}{n}Q)=K_{L^{\prime}}=K_{\cal
F^{\prime}}|_{L^{\prime}}=q^{*}K_{\cal F}|_{L^{\prime}}=\eta^{*}(K_{\cal
F}|_{L})=\eta^{*}(K_{L}+\mu_{C}(\Delta)Q).$

A similar computation allows us to deduce that $\mu_{C}\Theta=\frac{n-1}{n}$.
Thus, $\mu_{C}\Delta=\frac{n-1}{n}=\mu_{C}\Theta$, and (2) follows. ∎

For divisors which are transverse to the foliation, we have the following
version of adjunction:

######  Proposition 2.19.

Let $X$ be a normal threefold and let $\cal F$ be a rank one
$\mathbb{Q}$-Gorenstein foliation on $X$. Let $S\subset X$ be a
$\mathbb{Q}$-Cartier prime divisor which is not $\cal F$-invariant and let
$\nu\colon S^{\nu}\to S$ be its normalisation.

Then there exists a $\mathbb{Q}$-divisor $\Delta\geq 0$ such that

$(K_{\cal F}+S)|_{S^{\nu}}\sim_{\mathbb{Q}}\Delta.$

###### Proof.

Consider the composition of the natural maps

$\mathcal{O}_{S}(-S)\stackrel{{\scriptstyle\phi}}{{\to}}\Omega^{[1]}_{X}|_{S}\stackrel{{\scriptstyle\psi}}{{\to}}\mathcal{O}_{S}(K_{\cal
F})$

Assume that the composition $\psi\circ\phi$ is zero. Then, on the smooth locus
of $X$, we have that $\psi$ factors through $\Omega^{1}_{S}$, which implies
that $S$ is $\cal F$-invariant, a contradiction.

Thus, the existence of the non-zero map $\psi\circ\phi$ implies that there
exists a non-zero map

$\mathcal{O}_{S^{\nu}}(-S)\to\mathcal{O}_{S^{\nu}}(K_{\cal F})$

and the claim follows. ∎

###  2.6. Characterising log canonical vector fields

Let $X$ be a normal variety and let $\partial$ be a vector field which defines
a foliated pair $(\cal F,D)$ such that $K_{\cal F}+D$ is Cartier. Then we say
that $\partial$ is terminal (resp. canonical, log canonical) if the foliated
pair $(\cal F,D)$ is such.

Let $P\in Z$ be a germ of a normal variety and let $\partial\in
H^{0}(Z,T_{Z})$ be a vector field which leaves $P$ invariant. By Lemma 2.12,
$\partial$ is singular at $P$. Let $V\coloneqq\mathfrak{m}/\mathfrak{m}^{2}$
where $\mathfrak{m}$ is the maximal ideal at $P$ and observe that $\partial$
induces a linear map $\partial_{0}\colon V\rightarrow V$. Let $\cal F$ be the
foliation defined by $\partial$ so that $\partial$ is a section of ${\cal
F}(-D)$ for some divisor $D\geq 0$.

We recall the following result, see [MP13, Fact I.ii.4].

######  Proposition 2.20.

Set up as above.

Then the vector field $\partial$ is log canonical at $P$ if and only if
$\partial_{0}$ is non-nilpotent.

For convenience, we will explain a proof of this proposition. We remark that
this proposition holds with no assumptions on the singularities of $Z$ other
than normality and holds even when $\partial$ has a codimension one singular
locus.

######  Lemma 2.21.

Let $U\subset\mathbb{C}^{n}$ be a neighbourhood of the point
$0\in\mathbb{C}^{n}$ and let $\mathfrak{m}$ be its maximal ideal. Let
$\partial$ be a vector field on $U$ such that $\partial$ is singular at $0$
and the induced linear map
$\partial_{0}\colon\mathfrak{m}/\mathfrak{m}^{2}\to\mathfrak{m}/\mathfrak{m}^{2}$
is non-zero and nilpotent. Let $\pi\colon X\rightarrow\mathbb{C}^{n}$ be the
blow up at $0$. Let $\partial^{\prime}=\pi^{*}\partial$ and let
$P\in\operatorname{Sing}\,\partial^{\prime}$ and let $\partial^{\prime}_{0}$
be the induced linear map at $P$.

Then $\partial^{\prime}_{0}$ is nilpotent, and possibly equal to zero.

###### Proof.

This follows from a direct calculation. For simplicity we will perform this
calculation in the case where $n=2$ (the higher dimensional case is completely
analogous). We may choose coordinates $x,y$ on $\mathbb{C}^{2}$ so that
$\partial=y\frac{\partial}{\partial x}+\delta$ where
$\delta\in\mathfrak{m}^{2}T_{\mathbb{C}^{2}}$.

First, consider the chart in which the blow up is given by
$(s,t)\mapsto(st,t)$. We can compute that
$\pi^{*}\partial=\frac{\partial}{\partial s}+\pi^{*}\delta$ where
$\pi^{*}\delta$ vanishes along $\\{t=0\\}$. In particular, we see that
$\pi^{*}\partial$ is non-singular in this chart.

Now consider the chart given by $(s,t)\mapsto(s,st)$. In this chart we have
$\pi^{*}\partial=(st\frac{\partial}{\partial s}-t^{2}\frac{\partial}{\partial
t})+\pi^{*}\delta$. Notice moreover that if we write
$\delta=ax^{2}\frac{\partial}{\partial y}+\delta^{\prime}$ where
$a\in\mathbb{C}$ then $\pi^{*}\delta=as\frac{\partial}{\partial
t}+\pi^{*}\delta^{\prime}$ where $\pi^{*}\delta^{\prime}$ vanishes to order
two at $\\{s=t=0\\}$. It follows that $\pi^{*}\partial$ is singular only at
$\\{s=t=0\\}$ and $(\pi^{*}\partial)_{0}$ is nilpotent at that point. ∎

As a corollary we get a slight modification of a calculation done in [McQ01,
Fact I.1.10].

######  Lemma 2.22.

Let $\partial$ be a vector field on $U$ such that $\partial$ is singular at
$0$ and the induced linear map
$\partial_{0}\colon\mathfrak{m}/\mathfrak{m}^{2}\to\mathfrak{m}/\mathfrak{m}^{2}$
is non-zero and nilpotent. Let $\cal F$ be the foliation defined by $\partial$
so that $\partial$ is a section of $\cal F(-D)$ for some divisor $D\geq 0$.

Then, any sequence of blow ups centred in points in the singular locus of
$\partial$ will eventually extract a divisor $E$ of discrepancy

$a(E,\cal F,D)\leq-\epsilon(E)-1.$

###### Proof.

Again, for simplicity we will consider the case $n=2$ (the general case can be
handled in a similar manner). We choose coordinates so that
$\partial=y\frac{\partial}{\partial x}+\delta$ where
$\delta\in\mathfrak{m}^{2}T_{\mathbb{C}^{2}}$.

Consider the blow up at $0$, $\pi\colon X\rightarrow\mathbb{C}^{2}$ and recall
from our previous lemma that $\pi^{*}\partial$ has a single singular point
$P\in X$. There are two cases: either $\partial$ has a term of the form
$ax^{2}\frac{\partial}{\partial y}$ in it with $a\neq 0$ or it does not.

Consider the latter case. In this case if $\mathfrak{m}_{P}$ is the maximal
ideal at $P$ then $\pi^{*}\partial\in\mathfrak{m}_{P}^{2}T_{X}$ and so a blow
up at $P$ will extract a divisor of discrepancy $\leq-\epsilon-1$.

Now consider the former case. In this case (using the notation as in the proof
of the previous lemma) we see that $\pi^{*}\partial=s\frac{\partial}{\partial
t}+\delta^{\prime}$ with $\delta^{\prime}\in\mathfrak{m}_{P}^{2}T_{X}$. Notice
that $\delta^{\prime}$ does not contain a term of the from
$t^{2}\frac{\partial}{\partial s}$ and so we have reduced to the previous
case. Thus two more blow ups centred in the singular locus of
$\pi^{*}\partial$ will extract a divisor of discrepancy $\leq-\epsilon-1$. ∎

###### Proof of Proposition 2.20.

Consider an embedding $Z\hookrightarrow M$ where $M$ is smooth and of minimal
dimension. Let $\tilde{\partial}$ be a lift of $\partial$ and let
$\tilde{\partial}_{0}$ be its induced linear part. Observe that $\partial_{0}$
is nilpotent if and only if $\tilde{\partial}_{0}$ is.

Suppose that $\partial$ is log canonical and suppose for sake of contradiction
that $\partial_{0}$ is nilpotent. Observe that if $\partial_{0}=0$ then a
single blow up of $M$ at $P$ will extract a divisor $E$ of discrepancy
$\leq-\epsilon(E)-1$, a contradiction. So we may suppose that $\partial_{0}$
is non-zero and nilpotent, in which case the same is true of
$\tilde{\partial}_{0}$. Consider, again, the blow up of $M$ at $P$ and let
$Z^{\prime}$ denote the strict transform of $Z$. Let
$\partial^{\prime}=(\pi|_{Z^{\prime}})^{*}\partial$ and
$\tilde{\partial}^{\prime}=\pi^{*}\tilde{\partial}$. Note that we must have
$\operatorname{Sing}\,\partial^{\prime}\neq\emptyset$. By Lemma 2.21, we know
that the linear part of $\tilde{\partial}^{\prime}$ is nilpotent at all of its
singular points, and so if
$Q\in\operatorname{Sing}\,\partial^{\prime}\subset\operatorname{Sing}\,\tilde{\partial}^{\prime}$
then the linear part of $\partial^{\prime}$ at $Q$ is nilpotent. Observe
moreover that $\partial^{\prime}$ is a log canonical vector field on
$Z^{\prime}$ (note that it may have a zero in codimension one even if
$\partial$ does not).

Replacing $P\in Z$ by $Q\in Z^{\prime}$ and continuing this process we see by
Lemma 2.22 that we will eventually perform a blow up of discrepancy
$\leq-\epsilon-1$, a contradiction.

Conversely, suppose that $\partial_{0}$ is non-nilpotent. Let $\pi\colon
Z^{\prime}\rightarrow Z$ be a sequence of blow ups in $\partial$-invariant
centres and let $\partial^{\prime}=\pi^{*}\partial$. Observe that
$\partial^{\prime}$ is log canonical if and only if $\partial$ is, and observe
that $\partial^{\prime}$ is necessarily non-nilpotent at all points
$Q\in\operatorname{Sing}\,\partial^{\prime}$. Indeed, if $f$ is a function
which restricts to an eigenvector of $\partial_{0}$ then $\pi^{*}f$ gives an
eigenvector of $\partial^{\prime}_{0}$. We now show that $\pi$ only extracts
divisors of discrepancy $\geq-\epsilon$. Indeed, if $E$ is a $\pi$-exceptional
divisor, then either $\partial^{\prime}$ is smooth at the generic point of
$E$, in which case the discrepancy is zero, or $E$ is contained in the
singular locus of $\partial^{\prime}$, in which case, since
$\partial^{\prime}$ is non-nilpotent along $E$, at the generic point of $E$ we
may write $\partial^{\prime}=ut\frac{\partial}{\partial t}$ where $u$ is a
unit and $t$ is a uniformizing parameter. In this case, it follows that
$\epsilon(E)=1$ and that the discrepancy is also equal to $-1$.

We now show that $\partial$ is log canonical. Suppose for sake of
contradiction that there exists a divisor $E$ of discrepancy
$\leq-\epsilon-1$. Observe that we may extract $E$ by performing a sequence of
blow ups in $\partial$-invariant centres. However, by our above observations
we know that such an $E$ must necessarily have discrepancy $\geq-\epsilon(E)$,
a contradiction. ∎

######  Proposition 2.23 (Inversion of adjunction for vector fields).

Let $X$ be a normal variety and let $\cal F$ be a rank one
$\mathbb{Q}$-Gorenstein foliation on $X$. Let $\Delta\geq 0$ be a
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor and let $V\subset X$ be an $\cal
F$-invariant subvariety not contained in $\operatorname{Sing}\,\cal
F\cup\operatorname{Supp}\,\Delta$. Assume that $\cal F$ is Gorenstein at the
generic point of $V$. Let $(\cal F_{V},\Delta_{V})$ be the induced foliated
pair on $V$ (cf. Proposition 2.16) and assume that $(\cal F_{V},\Delta_{V})$
is log canonical.

Then $(\cal F,\Delta)$ is log canonical at the general point of $V$.

###### Proof.

Without loss of generality, we may assume that $\cal F$ is Gorenstein and
$\Delta=0$. Let $P\in V$ be a general point and let $\partial$ a vector field
defining $\cal F$ around $P$. Consider an embedding $Z\hookrightarrow M$ where
$M$ is smooth and of minimal dimension and let $\tilde{\partial}$ be a lift of
$\partial$. By Proposition 2.20, it follows that $\partial$ is log canonical
at $P$ if and only if $\tilde{\partial}$ is. Thus, after replacing $X$ by $M$
and $\partial$ by $\tilde{\partial}$, we may assume that $X$ is a germ of a
manifold about $P$. Let $\mathfrak{m}$ denote the maximal ideal of $X$ at $P$
and let $\mathfrak{n}$ denote the maximal ideal of $V$ at $P$. Let
$\delta\coloneqq\partial|_{V}$. By Proposition 2.20], the vector field
$\partial$ (resp. $\delta$) is log canonical if and only if the induced linear
map
$\partial_{0}\colon\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\mathfrak{m}/\mathfrak{m}^{2}$
(resp.
$\delta_{0}\colon\mathfrak{n}/\mathfrak{n}^{2}\rightarrow\mathfrak{n}/\mathfrak{n}^{2}$)
is non-nilpotent.

Next, observe that the natural map
$\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\mathfrak{n}/\mathfrak{n}^{2}$ is
surjective and so if $\delta_{0}$ is non-nilpotent then we see that the same
is true for $\partial_{0}$. Our result then follows. ∎

###  2.7. Jordan decomposition of a vector field

We follow the notation of [Mar81]. Let $X\coloneqq\widehat{\mathbb{C}^{m}}$ be
the completion of $\mathbb{C}^{m}$ at the origin $0\in X$ and let $\partial$
be a vector field on $X$ which leaves $W\coloneqq\\{0\\}$ invariant. Let
$\mathfrak{m}$ be the maximal ideal defining $W$ and note that, by the Leibniz
rule, the ideal $\mathfrak{m}^{n}$ is $\partial$-invariant for all positive
integer $n$. Thus, we get a linear map

$\partial_{n}\colon\mathfrak{m}/\mathfrak{m}^{n+1}\rightarrow\mathfrak{m}/\mathfrak{m}^{n+1}.$

We may write $\partial_{n}=\partial_{S,n}+\partial_{N,n}$ as the Jordan
decomposition of $\partial_{n}$ into its semi-simple and nilpotent parts. This
decomposition respects the exact sequences

$0\rightarrow\mathfrak{m}^{n}/\mathfrak{m}^{n+1}\rightarrow\mathbb{C}[[X]]/\mathfrak{m}^{n+1}\rightarrow\mathbb{C}[[X]]/\mathfrak{m}^{n}\rightarrow
0$

for each positive integer $n$ and it yields a decomposition
$\partial=\partial_{S}+\partial_{N}$.

We summarise briefly some of the key properties of this decomposition:

  1. (1)

$[\partial_{S},\partial_{N}]=0$;

  2. (2)

we may find coordinates $y_{1},...,y_{m}$ on $\widehat{\mathbb{C}^{m}}$ and
$\lambda_{1},\dots,\lambda_{m}\in\mathbb{C}$ so that
$\partial_{S}=\sum_{i}\lambda_{i}y_{i}\partial_{y_{i}}$; and

  3. (3)

if $Z\subset\widehat{\mathbb{C}^{m}}$ is $\partial$-invariant then $Z$ is both
$\partial_{S}$ and $\partial_{N}$-invariant.

We briefly explain (3). Let $I_{Z}\subset\mathbb{C}[[X]]$ be the ideal of $Z$
and let $I_{Z,n}$ denote its restriction to $\mathfrak{m}/\mathfrak{m}^{n+1}$,
for each positive integer $n$. Then
$I_{Z,n}\subset\mathfrak{m}/\mathfrak{m}^{n+1}$ is a $\partial_{n}$-invariant
subspace and, in particular, it is both $\partial_{S,n}$ and
$\partial_{N,n}$-invariant. Thus, (3) follows.

More generally, we can define the Jordan decomposition for any vector field
$\partial$ on the completion of a variety $X$ at a point $P\in X$. Indeed,
consider an embedding $\iota\colon Z\hookrightarrow\mathbb{C}^{m}$ and a lift
$\widetilde{\partial}$ of $\partial$ to a vector field on $\mathbb{C}^{m}$. We
can define $\widetilde{\partial}_{S}$ and $\widetilde{\partial}_{N}$ as above.
Then $\widetilde{\partial}_{S}$ and $\widetilde{\partial}_{N}$ leave $Z$
invariant and, therefore, they restrict to vector fields $\partial_{S}$ and
$\partial_{N}$ on $Z$. Thus, $\partial=\partial_{S}+\partial_{N}$ and this
decomposition has all the properties of the Jordan decomposition, as described
above.

###  2.8. Eigenvalues of vector fields

In this section we provide some results on the eigenvalues of a vector field
with singularities in codimension one. Let $Z$ be a normal variety and let
$\partial\in H^{0}(Z,T_{Z})$ be a vector field such that the zero locus of
$\partial$ admits an irreducible component $E\subset Z$ of codimension one. By
normality, $\cal O_{Z,E}$ is a DVR and we denote by $t$ a local parameter. Let
$P\in E$ and let $\mathfrak{m}$ denote the maximal ideal of $P$ in $Z$. Note
that both $E$ and $P$ are invariant. Let $V=\mathfrak{m}/\mathfrak{m}^{2}$ and
let $\partial_{0}\colon V\rightarrow V$ be the induced linear map. Assume that
$\partial$ is log canonical and so, by Proposition 2.20, $\partial_{0}$ is
non-nilpotent.

The following proposition is related to [MP13, Fact III.i.3].

######  Proposition 2.24.

Set up as above.

Then, after possibly rescaling and taking a change of coordinates, we have
that $\partial$ is semi-simple and its eigenvalues are all non-negative
integers.

###### Proof.

For any positive integer $p$, let
$V_{p}\coloneqq\mathfrak{m}/\mathfrak{m}^{p}$. Let $\partial_{p}$ be the map
induced by $\partial$ on $V_{p}$. It is enough to show that, for all
sufficiently large $p$, the linear map $\partial_{p}$ is semi-simple and that
its eigenvalues are all non-negative integers. Note that $V_{p}$ is filtered
by subspaces

$V_{p}^{k}\coloneqq\\{f\mod\mathfrak{m}^{p}\mid\operatorname{ord}_{E}(f)\geq
k\\}\subset V_{p}\qquad\text{for }k=0,1,\dots.$

By a local computation at the general point of $E$, it follows that for all
$f\in\mathfrak{m}$, we have

$\operatorname{ord}_{E}(f)\leq\operatorname{ord}_{E}(\partial f).$

Thus, $\partial_{p}$ preserves the subspace $V_{p}^{k}$ for any $k$. Let
$D_{p}^{k}$ be the induced linear map on $V_{p}^{k}/V_{p}^{k+1}$. It is enough
to show that, for all $p$ sufficiently large, the linear map $D^{k}_{p}$
coincides with $kI$ where $I$ is the identity map on $V^{k}_{p}/V^{k+1}_{p}$.

Since $\partial$ is log canonical, Proposition 2.20 implies that
$\partial_{p}$ is non-nilpotent. Since $E$ is contained in the zero locus of
$\partial$, we have that $\partial_{p}(V_{p})\subset V^{1}_{p}$. It follows
that the restriction map $\partial_{p}|_{V^{1}_{p}}$ is non-nilpotent for each
$p$. Thus, after possibly rescaling $\partial$ by a constant, if $p$ is
sufficiently large, then there exists a non-zero element $v_{p}\in
V^{1}_{p}/V^{2}_{p}$ so that $D^{1}_{p}(v_{p})=v_{p}$. It follows that there
exists $f_{p}\in\mathfrak{m}$ such that $\operatorname{ord}_{E}(f_{p})=1$ and

$\partial f_{p}=f_{p}+R\mod\mathfrak{m}^{p}$

where $\text{ord}_{E}R\geq 2$.

Let $0\neq g\in\mathfrak{m}$ and let $k=\operatorname{ord}_{E}(g)$. At the
generic point of $E$, we may write $g=uf_{p}^{k}\mod\mathfrak{m}^{p}$ where
$u$ is a unit. We have

$\displaystyle\partial g$
$\displaystyle=\partial(uf_{p}^{k})\mod\mathfrak{m}^{p}$
$\displaystyle=u\partial(f_{p}^{k})+f_{p}^{k}\partial u\mod\mathfrak{m}^{p}$
$\displaystyle=kuf_{p}^{k-1}\partial(f_{p})+f_{p}^{k}\partial
u\mod\mathfrak{m}^{p}$ $\displaystyle=kg+kuf_{p}^{k-1}R+f_{p}^{k}\partial
u\mod\mathfrak{m}^{p}.$

Since $\operatorname{ord}_{E}R\geq 2$ and $\operatorname{ord}_{E}\partial
u\geq 1$, it follows that

$\operatorname{ord}_{E}(kuf_{p}^{k-1}R+f_{p}^{k}\partial u)\geq k+1.$

Thus, $D_{p}^{k}=kI$, as claimed. ∎

######  Remark 2.25.

Let $X$ be a normal $n$-dimensional variety and let $\partial$ be a vector
field on $X$ whose zero locus contains an irreducible component $E\subset X$
of codimension one. Let $P\in E$. Assume that in a neighbourhood of $P$, we
may write $\partial=t\frac{\partial}{\partial t}$ for some local parameter
$t$. Then, there exists an analytic neighbourhood $U$ of $P$ which is
isomorphic to $\Delta\times S$ where $\Delta$ is a smooth germ of a curve and
$S$ is a normal variety of dimension $n-1$. Indeed, choose $m$ minimal so that
we have an embedding $X\hookrightarrow\mathbb{C}^{m}$ with coordinates on the
latter given by $(t,x_{1},...,x_{m-1})$, so that $X$ is invariant under
$t\frac{\partial}{\partial t}$. Since $m$ is minimal, it follows that $X$
cannot be contained in $\\{t=0\\}$ and so $X$ is invariant under
$\frac{\partial}{\partial t}$, and the claim follows.

In particular, if $X$ is singular and $X$ is $\mathbb{Q}$-factorial (resp. it
admits quotient singularities) at the generic point of
$\operatorname{Sing}\,X$ then $X$ is in fact $\mathbb{Q}$-factorial (resp. it
admits quotient singularities).

###  2.9. Canonical bundle formula

We recall some results on the canonical bundle formula which will be used
later (see [Amb04] for more details).

Let $(X,\Delta)$ be a sub log canonical pair and let $f\colon X\to Y$ be a
fibration. Assume that the horizontal part $\Delta^{h}$ of $\Delta$ is
effective and that there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor
$D$ on $Y$ such that

$K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D.$

If $P$ is a prime divisor on $Y$, we denote by $\eta_{P}$ its generic point
and we define the log canonical threshold  of $f^{*}P$ with respect to
$(X,\Delta)$ to be

$\operatorname{lct}(X,\Delta;f^{*}P)\coloneqq\sup\\{t\in\mathbb{R}\mid(X,\Delta+tf^{*}P)\text{
is sub log canonical over $\eta_{P}$}\\}.$

Let $b_{P}\coloneqq 1-\operatorname{lct}(X,\Delta;f^{*}P)$. Then we define the
discriminant of $f$ with respect to $\Delta$ as
$B_{Y}\coloneqq\sum_{P}b_{P}P$, where the sum runs over all the prime divisors
$P$ in $Y$. Let $r$ be the smallest positive integer such that there exists a
rational function $\phi$ on $X$ satisfying

$K_{X}+\Delta+\frac{1}{r}(\phi)=f^{*}D.$

Then there exists a $\mathbb{Q}$-divisor $M_{Y}$ such that

$K_{X}+\Delta+\frac{1}{r}(\phi)=f^{*}(K_{Y}+B_{Y}+M_{Y}).$

$M_{Y}$ is called the moduli part of $f$ with respect to $\Delta$ .

We will need the following results:

######  Lemma 2.26.

Set up as above. Assume that $X$ is a threefold, $Y$ is a surface and $M_{Y}$
is $\mathbb{Q}$-Cartier.

Then $M_{Y}$ is semi-ample.

###### Proof.

By [PS09, Theorem 8.1], there exists a birational morphism $\alpha\colon
Y^{\prime}\to Y$ such that if $X^{\prime}$ is the normalisation of the main
component of $X\times_{Y}Y^{\prime}$ with induced morphisms $f^{\prime}\colon
X^{\prime}\to Y^{\prime}$ and $\beta\colon X^{\prime}\to X$, and we write
$\beta^{*}(K_{X}+\Delta)=K_{X^{\prime}}+\Delta^{\prime}$, then the moduli part
$M_{Y^{\prime}}$ of $f^{\prime}$ with respect to $\Delta^{\prime}$ is semi-
ample. Since $M_{Y}\sim_{\mathbb{Q}}\alpha_{*}M_{Y^{\prime}}$, it follows that
the stable base locus of $M_{Y}$ has codimension at least two. Since $Y$ is a
surface and $M_{Y}$ is $\mathbb{Q}$-Cartier, Fujita Theorem [Fuj83, Theorem
1.10] implies that $M_{Y}$ is semi-ample, as claimed. ∎

######  Lemma 2.27.

Let $(X,\Delta)$ be a two dimensional log canonical pair, let $f\colon X\to Y$
be a fibration onto a curve $Y$ and let $D$ be a $\mathbb{Q}$-divisor on $Y$
such that $K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D$. Let $y\in Y$ be a closed
point and assume that there exists an open neighbourhood $U$ of $y$ such that,
if we denote

$X_{U}\coloneqq f^{-1}(U)\qquad\text{and}\qquad X_{u}\coloneqq
f^{-1}(u)\quad\text{for }u\in U$

then $(X_{U},\Delta|_{X_{U}})$ is log smooth and there exists an isomorphism

$\phi_{u}\colon X_{u}\to X_{y}\qquad\text{such
that}\qquad\phi^{*}_{u}(\Delta|_{X_{y}})=\Delta|_{X_{u}}\qquad\text{for all
}u\in U.$

Then the moduli part of $f$ is trivial, i.e. $M_{Y}\sim_{\mathbb{Q}}0$.

###### Proof.

By [Kol07, Proposition 8.4.9], we may freely perform a base change. Thus,
without loss of generality, we may freely assume that $X\rightarrow Y$ is
semi-stable and $\Delta+f^{*}P$ is a divisor with simple normal crossing for
any prime divisor $P$ on $Y$.

Let $G$ be the support of $\Delta$. By our hypotheses, after possibly
replacing $Y$ by a higher cover, we may find an open subset $V\subset Y$ so
that $X=X_{0}\times V$ and $G=G_{0}\times V$, where $X_{0}$ is a smooth curve
and $G_{0}\subset X_{0}$ is a finite set. Since $M_{Y}$ only depends on the
generic fibre we are therefore free to assume that $X=X_{0}\times Y$ and
$G=G_{0}\times Y$, in which case the result is immediate. ∎

###  2.10. A recollection on approximation theorems

We recall some approximation results proven in [CS18, Section 4].

We consider the following set up. Let $\tilde{X}=\text{Spec}\,\tilde{A}$ be an
affine variety where $\tilde{A}$ is a henselian local ring with maximal ideal
$\mathfrak{m}$ and let $W\subset\tilde{X}$ be a closed subscheme, defined by
an ideal $\tilde{I}\subset\tilde{A}$. Let
$\widehat{X}\coloneqq\text{Spec}\,\widehat{A}$ where $\widehat{A}$ is the
completion of $\tilde{A}$ along $\tilde{I}$ and let $\widehat{D}$ be a divisor
on $\widehat{X}$. Equivalently, $\widehat{D}$ is given by a reflexive sheaf
$\widehat{M}$ on $\widehat{X}$ and a choice of a section
$\hat{s}\in\widehat{M}$.

The following is a slight generalisation of Artin-Elkik approximation theorem:

######  Theorem 2.28.

Set up as above. Let $m$ be a positive integer such that $m\widehat{D}$ is
Cartier on $\widehat{X}\setminus W$.

Then, for all positive integer $n$, there exists a divisor $D^{n}$ on
$\text{Spec}\,\tilde{A}$ such that

$D^{n}=\widehat{D}\mod\tilde{I}^{n}\qquad\text{and}\qquad\cal
O(mD^{n})\otimes\widehat{A}\cong\cal O(m\widehat{D}).$

###### Proof.

See [CS18, Corollary 4.4]. ∎

We will use this theorem under the following additional constraints. Let
$X=\text{Spec}\,A$ be an affine variety and let $P\in X$ be closed point and
suppose $\tilde{A}$ is the henselisation of $A$ at $P$.

######  Corollary 2.29.

Set up as above.

Then, for all positive integer $n$, there exists an étale neighbourhood
$\sigma\colon U\rightarrow X$ of $P$ and a divisor $D^{n}_{U}$ on $U$ such
that $\tau^{*}D^{n}_{U}=D^{n}$ where $\tau\colon\tilde{X}\rightarrow U$ is the
induced morphism.

In particular, if $\tilde{I}=I\otimes\tilde{A}$ for some $I\subset A$ then
$D^{n}_{U}=\widehat{D}\mod I^{n}$.

In our applications here we will always take $W=P$ and so the additional
hypotheses of the corollary are always satisfied.

We also recall the following:

######  Lemma 2.30.

Set up as above. Suppose in addition that $(\widehat{X},\widehat{D})$ is klt
(resp. (log) terminal, resp. (log) canonical).

Then for $n$ sufficiently large $(U,D^{n}_{U})$ is klt (resp. (log) terminal,
resp. (log) canonical) in a neighborhood of $\sigma^{-1}(P)$.

###### Proof.

See [CS18, Lemma 4.7]. ∎

###  2.11. Resolution of singularities of threefold vector fields

We recall the following example from [MP13].

######  Example 2.31.

[MP13, Example III.iii.3] Consider the $\mathbb{Z}/2\mathbb{Z}$-action on
$\mathbb{C}^{3}$ given by $(x,y,z)\mapsto(y,x,-z)$. Let $X$ denote the
quotient of $\mathbb{C}^{3}$ by this action.

Consider the vector field on $\mathbb{C}^{3}$ given by

$\partial_{S}:=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}$

and

$\partial_{N}:=a(xy,z)x\frac{\partial}{\partial
x}-a(xy,-z)y\frac{\partial}{\partial y}+c(xy,z)\frac{\partial}{\partial z}$

where $a,c$ are formal functions in two variables such that $c$ is not a unit
and it satisfies $c(xy,z)=c(xy,-z)$. Let
$\partial:=\partial_{S}+\partial_{N}$. Note that $\partial\mapsto-\partial$
under the group action. Thus, $\partial$ induces a foliation $\cal F$ on $X$
with an isolated canonical singularity and such that $2K_{\cal F}$ is Cartier,
but $K_{\cal F}$ is not Cartier.

By [MP13, Possibility III.iii.3.bis], there does not exist a birational
morphism $f\colon Y\to X$ such that the induced foliation $f^{-1}\cal F$ is
both Gorenstein and canonical. Moreover, by [MP13, III.iii.3.bis], we must
also have that the curve $\\{x=y=0\\}$ is not algebraic, nor analytically
convergent.

######  Definition 2.32.

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. We say that $\cal F$ admits a simple singulary
at $P\in X$ if either

  1. (1)

${\cal F}$ is terminal and no component of $\operatorname{Sing}\,X$ through
$P$ is ${\cal F}$-invariant; or

  2. (2)

${X}$ and ${\cal F}$ are formally isomorphic to the variety and the foliation
defined in Example 2.31 at $P$; or

  3. (3)

$X$ is smooth at $P$.

######  Theorem 2.33.

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$.

Then there exists a sequence of weighted blow ups in foliation invariant
centres $p\colon\widetilde{X}\rightarrow X$ so that $\tilde{\cal F}\coloneqq
p^{-1}\cal F$ has simple singularities at all points $P\in\widetilde{X}$.

###### Proof.

See [MP13, III.iii.4]. ∎

######  Lemma 2.34.

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$.
Suppose that $\cal F$ admits a simple singularity at $P$.

Then $X$ has cyclic quotient singularities at $P$.

###### Proof.

If $X$ is smooth then there is nothing to show and if $P\in X$ is as in
Example 2.31, then we are done since $X$ is a $\mathbb{Z}/2\mathbb{Z}$
quotient singularity.

So suppose that $\cal F$ is terminal at $P$. After possibly replacing $X$ by
an analytic neighbourhood of $P$, we may assume that there exists a quasi-
étale cover $q\colon X^{\prime}\rightarrow X$ with a holomorphic submersion
$f\colon X^{\prime}\rightarrow S$ as guaranteed by Lemma 2.12. Assume by
contradiction that $X^{\prime}$ is not smooth. Then
$q(\operatorname{Sing}\,X^{\prime})\subset\operatorname{Sing}\,X$ and
$q(\operatorname{Sing}\,X^{\prime})$ is $\cal F$-invariant, a contradiction.
It follows that $X^{\prime}$ is smooth and so $X$ has at worst a cyclic
quotient singularity. ∎

######  Lemma 2.35.

Let $G$ a finite group acting on $\mathbb{C}^{3}$, let
$X\coloneqq\mathbb{C}^{3}/G$ be a quotient singularity and let
$q\colon\mathbb{C}^{3}\rightarrow X$ be the quotient map. Let $\cal F$ be a
rank one foliation on $X$ and let $C\subset X$ be a smooth $\cal F$-invariant
curve.

Then the following hold:

  1. (1)

if $\cal F$ is terminal, then $q^{-1}(C)$ is a smooth irreducible curve;

  2. (2)

if $0\in X$ is a foliation singularity as in Example 2.31, then $q^{-1}(C)$ is
a nodal curve; and

  3. (3)

If the singularity of $\cal F$ at $0$ is simple, then there is a divisor
$D\subset X$ containing $C$ and such that if $\cal F$ is terminal (resp.
canonical) then $(D,C)$ is log terminal (resp. log canonical).

###### Proof.

Let $\cal G:=q^{-1}\cal F$ and let $C^{\prime}:=q^{-1}(C)$. Then $C^{\prime}$
is $\cal G$-invariant. If $\cal F$ is terminal then Lemma 2.11 implies that
$\cal G$ is a terminal foliation on a smooth variety and, by Lemma 2.12, it is
smooth. Since $C^{\prime}$ is a connected leaf of $\cal G$, it is therefore
smooth and irreducible. Thus, (1) follows.

We now prove (2). Using the same notation as in Example 2.31, we have that
$C^{\prime}$ is necessarily $\partial_{S}$-invariant. It is easy to see that
the only $\partial_{S}$-invariant curves passing through $0\in\mathbb{C}^{3}$
are $\\{x=y=0\\},\\{x=z=0\\}$ and $\\{y=z=0\\}$. As in Example 2.31, the curve
$\\{x=y=0\\}$ is not algebraic, or not analytically convergent. Thus,
$C^{\prime}$ is either smooth or $C^{\prime}=\\{x=z=0\\}\cup\\{y=z=0\\}$ as
required. Since $C$ is the quotient of $\\{x=z=0\\}\cup\\{y=z=0\\}$ by the
$\mathbb{Z}/2\mathbb{Z}$-action we see that $C$ is a smooth irreducible curve.
Thus, (2) follows.

Let $D^{\prime}\subset\mathbb{C}^{3}$ be a general surface containing
$C^{\prime}$ and let $D=q(D^{\prime})$. By [KM98, Proposition 5.20], $(D,C)$
is log terminal (resp. log canonical) if and only if $(D^{\prime},C^{\prime})$
is log terminal (resp. log canonical). Thus, (3) follows. ∎

###  2.12. Cone theorem

The cone theorem for rank one foliations was initially proven in [BM16,
Corollary IV.2.1] when $\cal F$ is Gorenstein and in [McQ04] when $\cal F$ is
$\mathbb{Q}$-Gorenstein. We will need a version of the theorem involving a
boundary $\Delta$ and, while the proofs in [BM16] and [McQ04] can be extended
to work in this generality, we have opted to present a slightly different
proof.

######  Theorem 2.36.

Let $X$ be a normal projective $\mathbb{Q}$-factorial threefold and let
$(\mathcal{F},\Delta)$ be a rank one foliated pair on $X$.

Then there are $\mathcal{F}$-invariant rational curves $C_{1},C_{2},\dots$
such that

$0<-(K_{\cal F}+\Delta)\cdot C_{i}\leq 2$

and

$\overline{\rm NE}(X)=\overline{\rm NE}(X)_{K_{\cal F}+\Delta\geq
0}+Z_{-\infty}+\sum_{i}\mathbb{R}_{+}[C_{i}]$

where $Z_{-\infty}\subset\overline{\rm NE}(X)$ is a subset contained in the
span of the images of $\overline{\rm NE}(W)\rightarrow\overline{\rm NE}(X)$
where $W\subset X$ are the non-log canonical centres of $(\cal F,\Delta)$.

Before proceeding with the proof, we show the following:

######  Lemma 2.37.

Let $X$ be a normal $\mathbb{Q}$-factorial variety and let $(\cal F,\Delta)$
be a rank one foliated pair on $X$. Let $C\subset X$ be a curve which is not
$\cal F$-invariant.

Then there exists a birational morphism $p\colon X^{\prime}\rightarrow X$ such
that if $\cal F^{\prime}\coloneqq p^{-1}\cal F$ then the following hold:

  1. (1)

$K_{\cal F^{\prime}}+\Delta^{\prime}=p^{*}(K_{\cal F}+\Delta)$ for some
$\mathbb{Q}$-divisor $\Delta^{\prime}\geq 0$;

  2. (2)

$p^{-1}$ is an isomorphism at the general point of $C$ and if $C^{\prime}$ is
the strict transform of $C$ in $X^{\prime}$ then $\cal F^{\prime}$ is terminal
at all points $P\in C^{\prime}$; and

  3. (3)

after possibly replacing $X$ by an analytic neighbourhood of $C$, there exist
a $\cal F^{\prime}$-invariant surface $S$ containing $C^{\prime}$.

###### Proof.

Since $C$ is not $\cal F$-invariant, it follows that $C$ is not contained in
$\operatorname{Sing}\,\cal F$. By Lemma 2.12, $\cal F$ is terminal at the
generic point of $C$. Let $P_{1},...,P_{k}\in C$ be all the closed points near
which $\cal F$ is not terminal. Let $H$ be a sufficient general ample divisor
meeting $C$ transversely so that $(\cal F,H)$ is terminal at $C\cap H$. We may
find a finite Galois morphism $\sigma\colon Y\rightarrow X$ with Galois group
$G$, which is quasi-étale outside $H$ and such that $\sigma^{*}K_{\cal F}$ is
Cartier. Let $\cal G\coloneqq\sigma^{-1}\cal F$ and let
$\widetilde{C}\coloneqq\sigma^{-1}(C)$. Then $K_{\cal G}$ is Cartier and Lemma
2.11 implies that $\cal G$ is terminal at all closed points
$Q\in\widetilde{C}$, such that $\sigma(Q)\notin\\{P_{1},\dots,P_{k}\\}$.

By [BM16, Proposition I.2.4], there exists a birational morphism $q\colon
Y^{\prime}\rightarrow Y$, obtained as a sequence of $G$-equivariant blow ups
in $\cal G$-invariant centres, such that the strict transform of
$\widetilde{C}$ in $Y^{\prime}$ is disjoint from the singular locus of
$q^{-1}\cal G$. Note that [BM16, Lemma I.1.3] implies that $K_{q^{-1}\cal
G}+E^{\prime}=q^{*}K_{\cal G}$ where $E^{\prime}\geq 0$ is $q$-exceptional.

We have that $G$ acts on $Y^{\prime}$ and $q$ is $G$-invariant. Thus, if
$X^{\prime}\coloneqq Y^{\prime}/G$ then there exists a birational morphism
$p\colon X^{\prime}\rightarrow X$. It follows by Riemann-Hurwitz (cf.
Proposition 2.2) that $K_{\cal F^{\prime}}+E=p^{*}K_{\cal F}$ where $\cal
F^{\prime}=p^{-1}\cal F$ and $E\geq 0$ is $p$-exceptional. Notice moreover
that $\cal F^{\prime}$ is terminal at all points of $C^{\prime}$. It follows
that there exists $\Delta^{\prime}\geq 0$ such that $K_{\cal
F^{\prime}}+\Delta^{\prime}=p^{*}(K_{\cal F}+\Delta)$. Thus, (1) and (2)
follow.

We now prove (3). Let $P\in C^{\prime}$ be a closed point and let $U$ be a
neighbourhood of $P$ which admits a quasi-étale morphism $\tau\colon
V\rightarrow U$ such that $\tau^{*}K_{\cal F^{\prime}}$ is Cartier. By Lemma
2.12, after replacing $V$ by a sufficiently small analytic neighbourhood of
$P$, there exists a submersion $F\colon V\rightarrow T$ onto a normal surface,
such that $\tau^{-1}\cal F^{\prime}$ is the foliation induced by $F$. Let
$S_{U}=\tau(F^{-1}(F(\tau^{-1}(C^{\prime}\cap U))))$. Then $S_{U}$ is the
unique $\cal F^{\prime}$-invariant surface containing $C\cap U$. Let $S$ be
the union of $S_{U}$ as we run over an open cover of $C$. Then (3) follows. ∎

######  Lemma 2.38.

Let $X$ be a normal $\mathbb{Q}$-factorial variety and let $(\cal F,\Delta)$
be a rank one foliated pair on $X$. Let $C\subset\operatorname{Sing}\,\cal F$
be a curve and suppose that $(\cal F,\Delta)$ is log canonical at the generic
point of $C$.

Then $(K_{\cal F}+\Delta)\cdot C\geq 0$.

###### Proof.

First suppose that $\Delta=0$. When $\cal F$ is Gorenstien this proven [BM16,
§4.1], the general case where $\cal F$ is only $\mathbb{Q}$-Gorenstein is
proven in [McQ04, Fact II.d.3].

Assume now that $\Delta\neq 0$. Since $(\cal F,\Delta)$ is log canonical at
the generic point of $C$, it follows that $C$ is not contained in
$\operatorname{Supp}\,\Delta$ and so $\Delta\cdot C\geq 0$. Thus, by the
previous case, it follows that $(K_{\cal F}+\Delta)\cdot C\geq 0$. ∎

######  Lemma 2.39.

Let $X$ be a normal $\mathbb{Q}$-factorial threefold and let $(\cal F,\Delta)$
be a rank one foliated pair on $X$. Let $C$ be a curve and let $D\geq 0$ be a
divisor such that neither $D$ nor $C$ is $\cal F$-invariant. Suppose that
$D\cdot C\leq 0$ and that $(\cal F,\Delta)$ is log canonical at the generic
point of $C$.

Then $(K_{\cal F}+\Delta)\cdot C\geq 0$.

###### Proof.

Suppose for sake of contradiction that $(K_{\cal F}+\Delta)\cdot C<0$. Let
$p\colon X^{\prime}\rightarrow X$ be a birational morphism as in Lemma 2.37.
Let $C^{\prime}$ be the strict transform of $C$ in $X^{\prime}$ and let
$D^{\prime}=p^{*}D$. Let $\Delta^{\prime}\geq 0$ be the $\mathbb{Q}$-divisor
on $X^{\prime}$ such that $K_{\cal F^{\prime}}+\Delta^{\prime}=p^{*}(K_{\cal
F}+\Delta)$. After possibly shrinking $X$ and replacing $X,D,\Delta$ and $C$
by $X^{\prime},D^{\prime},\Delta^{\prime}$ and $C^{\prime}$ respectively, we
may assume that $\cal F$ is terminal at all points of $C$, and that $C\subset
S$ where $S$ is a $\cal F$-invariant surface. Since $(\cal F,\Delta)$ is log
canonical at the generic point of $C$, it follows that $S$ is not contained in
$\operatorname{Supp}\,\Delta$. Thus, if $S^{\nu}\to S$ is the normalisation,
then by Proposition 2.16 we may write $(K_{\cal F}+\Delta)|_{S^{\nu}}=K_{\cal
G}+\Theta$ where $(\cal G,\Theta)$ is the induced foliated pair on $S$.

We want to show that $(\cal G,\Theta)$ is log canonical at the generic point
of $C$, i.e. $\mu_{C}\Theta\leq 1$. This is a local calculation which is
unchanged by taking the index one cover of $K_{\cal F}$ so we may freely
assume that $\cal F$ is Gorenstein. Lemma 2.12 implies that both $S$ and $X$
are smooth at the general point of $C$. Thus,
$\mu_{C}\Theta=\mu_{C}\Delta|_{S}$. Suppose that $\mu_{C}(\Delta|_{S})>1$. In
this case a single blow up of $X$ at $C$ would extract a divisor $E$ of
foliation discrepancy $<0=\epsilon(E)$, and so $(\cal F,\Delta)$ would not be
log canonical at the generic point of $C$, a contradiction. Thus,
$\mu_{C}(\Delta|_{S})\leq 1$ and there exists $t\geq 0$ such that
$\Theta+tC=\Theta^{\prime}+C$ where $\Theta^{\prime}\geq 0$ is a
$\mathbb{Q}$-divisor whose support does not contain $C$.

On one hand, $C^{2}\leq C\cdot(D|_{S})<0$ and so

$(K_{\cal G}+\Theta^{\prime}+C)\cdot C=(K_{\cal F}+\Delta)\cdot C+tC^{2}<0.$

On the other hand, by Lemma 2.19 we have $(K_{\cal G}+\Theta^{\prime}+C)\cdot
C\geq 0$. Thus, we obtain a contradiction and our result follows. ∎

###### Proof of Theorem 2.36.

Let $C$ be a $(K_{\cal F}+\Delta)$-negative curve which is $\cal F$-invariant
and such that $(\cal F,\Delta)$ is log canonical at the generic point of $C$.
In particular, $C$ is not contained in the support of $\Delta$. Moreover,
Lemma 2.38 implies that $C$ is not contained in $\operatorname{Sing}\,\cal F$
and Lemma 2.13 implies that $\cal F$ is Gorenstein at the general point of
$C$. Thus, by Proposition 2.16, it follows that $C$ is a rational curve and
$(K_{\cal F}+\Delta)\cdot C\geq-2$.

Let $R\subset\overline{NE}(X)$ be a $(K_{\cal F}+\Delta)$-negative extremal
ray and let $H_{R}$ be a supporting hyperplane to $R$. We argue based on
$\nu(H_{R})$. If $\nu(H_{R})<3$ then we may apply [Spi20, Corollary 2.28] to
conclude that $X$ is covered by $(K_{\cal F}+\Delta)$-negative rational curves
$\Sigma$ tangent to $\cal F$ which span $R$.

So suppose that $\nu(H_{R})=3$, i.e., $H_{R}$ is big. We may write
$H_{R}\sim_{\mathbb{Q}}A+E+tD$ where $A$ is ample, $E\geq 0$ is a
$\mathbb{Q}$-divisor, $t>0$, $D$ is a prime divisor not contained in the
support of $E$ and $D$ is negative on $R$. We may assume that $R$ is not in
the image of $\overline{\rm NE}(W)\rightarrow\overline{\rm NE}(X)$ where $W$
is any non-log canonical centre

Suppose for the moment that $D$ is not $\cal F$-invariant. Then by Proposition
2.19 we may write $(K_{\cal F}+\Delta+D)|_{D^{\nu}}\sim_{\mathbb{Q}}B$ where
$\nu\colon D^{\nu}\to D$ is the normalisation of $D$ and $B\geq 0$ is a
$\mathbb{Q}$-divisor on $D^{\nu}$. It follows that $R$ is spanned by a curve
$C\subset\operatorname{Supp}\,B$. Observe that $C$ is necessarily $\cal
F$-invariant, otherwise $C$ would be a non-invariant curve with $D\cdot C<0$
and $(\cal F,\Delta)$ is log canonical at the generic point of $C$,
contradicting Lemma 2.39. If $C$ is invariant then it is necessarily rational
and we are done.

Now suppose that $D$ is $\cal F$-invariant and let $(\cal G,\Theta)$ be the
induced foliated pair on $D^{\nu}$ whose existence is guaranteed by
Proposition 2.16. Observe that $(\cal G,\Theta)$ is log canonical away from
$\nu^{-1}(\operatorname{Sing}\,\cal F)$ and $\nu^{-1}(W)$ where $W$ is the
union of the non-log canonical centres of $(\cal F,\Delta)$. Thus, we may
apply Lemma 2.38 together with the cone theorem for surfaces (cf. [Spi20,
Theorem 6.3]) to conclude that $\overline{{\rm NE}}(X)$ has the claimed
structure. ∎

In fact, the proof gives the following slightly stronger claim which we will
need later.

######  Corollary 2.40.

Let $X$ be a normal projective $\mathbb{Q}$-factorial threefold and let
$(\mathcal{F},\Delta)$ be a rank one foliated pair on $X$. Let $R$ be a
$(K_{\cal F}+\Delta)$-negative extremal ray and let $D$ be an irreducible
divisor. Suppose that $D$ is covered by curves $\Sigma$ such that $[\Sigma]\in
R$ and $D\cdot\Sigma<0$.

Then $D$ is $\cal F$-invariant.

###### Proof.

Suppose by contradiction that $D$ is not $\cal F$-invariant. Then a general
choice of $\Sigma$ will not be $\cal F$-invariant. This however is a
contradiction of Lemma 2.39. ∎

######  Lemma 2.41.

Let $X$ be a normal projective threefold and let $(\cal F,\Delta)$ be a rank
one foliated pair on $X$ with log canonical singularities. Assume that
$\Delta=A+B$ where $A$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$ is a
$\mathbb{Q}$-divisor. Assume that $K_{\cal F}+\Delta$ is not nef but there
exists a $\mathbb{Q}$-divisor $H$ such that $K_{\cal F}+\Delta+H$ is nef. Let

$\lambda=\inf\\{t>0\mid K_{\cal F}+\Delta+tH\text{ is nef }\\}.$

Then there exists a $(K_{\cal F}+\Delta)$-negative extremal ray
$R=\mathbb{R}_{+}[C]$ such that $C$ is $\cal F$-invariant and $(K_{\cal
F}+\Delta+\lambda H)\cdot C=0$.

###### Proof.

The proof is the same as in [CS18, Lemma 9.2], as a consequence of Theorem
2.36. ∎

###  2.13. Nakamaye’s theorem and the structure of extremal rays

Let $X$ be a normal projective variety and let $M$ be a $\mathbb{Q}$-Cartier
divisor on $X$. We define the exceptional locus of $M$ to be

$\operatorname{Null}\,M\coloneqq\bigcup_{M|_{V}\text{ is not big }}V$

where the union runs over all the subvarieties $V\subset X$ of positive
dimension such that $M|_{V}$ is not big. We denote by $\mathbb{B}(M)$ the
stable base locus  of $M$,

$\mathbb{B}(M)\coloneqq\bigcap\operatorname{Bs}\,|mM|$

where the intersection runs over all the sufficiently divisible positive
integers $m$. Finally, given a ray $R$ in the cone of curves $\overline{\rm
NE}(X)$, we define the locus of $R$ to be the subset

$\operatorname{loc}\,R:=\bigcup_{[C]\in R}C.$

We recall the following result originally due to Nakamaye, in the case of
smooth varieties.

######  Lemma 2.42.

Let $X$ be a normal projective variety. Let $A$ be an ample
$\mathbb{Q}$-divisor and let $M$ be a big and nef Cartier divisor on $X$.

Then $\operatorname{Null}\,M=\mathbb{B}(M-\epsilon A)$ for any sufficiently
small rational number $\epsilon>0$.

###### Proof.

See [Bir17, Theorem 1.4]. ∎

######  Proposition 2.43.

Let $X$ be a $\mathbb{Q}$-factorial normal projective variety. Let $M$ be a
big and nef Cartier divisor on $X$. Let $W=\operatorname{Null}\,M$ and suppose
that $M|_{W}\equiv 0$.

Then there exists a birational contraction $\phi\colon X\to Z$ to an algebraic
spaces $Z$, such that $\phi$ contracts $W$ to a point and which is an
isomorphism outside $W$.

###### Proof.

Consider the rational map $\phi\colon X\dashrightarrow\mathbb{P}^{N}$ defined
by the linear system $|m(M-\epsilon A)|$ where $\epsilon>0$ is a sufficiently
small rational number and $m$ is a sufficiently divisible and large positive
integer and note that $\phi$ is birational onto the closure of its image
$Y\subset\mathbb{P}^{N}$. Let $p\colon\overline{X}\to X$ and
$q\colon\overline{X}\to Y$ be birational morphisms which resolve the
indeterminancy locus of $\phi$.

By Lemma 2.42, it follows that $p(\operatorname{Exc}\,q)=W$. We may write

$p^{*}(m(M-\epsilon A))=H+F$

where $F\geq 0$ is $q$-exceptional and $H=q^{*}L$ for some very ample Cartier
divisor $L$ on $Y$. Since $X$ is $\mathbb{Q}$-factorial we may choose $G\geq
0$ to be $p$-exceptional so that $-G$ is $p$-ample. Choose $\delta>0$
sufficiently small so that $A^{\prime}\coloneqq p^{*}(m\epsilon A)-\delta G$
is ample.

We therefore have $F+\delta G=p^{*}(mM)-A^{\prime}-H$ and $F+\delta G$ is a
$\mathbb{Q}$-Cartier $q$-exceptional divisor. Since
$p(\operatorname{Exc}\,q)=W$, it follows that $p^{*}M$ restricted to
$\operatorname{Exc}\,q$ is numerically trivial. Thus, if $k$ is a sufficiently
divisible positive integer so that $k(F+\delta G)$ is a Cartier divisor, then

$-k(F+\delta G)|_{k(F+\delta G)}\equiv k(A^{\prime}+H)|_{k(F+\delta G)}.$

Since ampleness of a line bundle on a scheme is equivalent to ampleness of the
line bundle restricted to the reduction and normalisation, and since
$A^{\prime}+H$ restricted to the reduction and normalisation of each component
of $\operatorname{Exc}\,q$ is ample, we see that $-k(F+\delta G)|_{k(F+\delta
G)}$ is ample.

We may therefore apply Artin’s Theorem [Art70, Theorem 6.2] to produce a
morphism of algebraic spaces $\overline{X}\rightarrow Z$ which contracts
$F+\delta G$ to a point. By the rigidity lemma this contraction factors
through $\overline{X}\rightarrow X$ giving our desired birational contraction
$\phi\colon X\rightarrow Z$. ∎

######  Proposition 2.44.

Let $X$ be a normal projective threefold and let $R$ be an extremal ray of the
cone of curves $\overline{\rm NE}(X)$.

Then $\operatorname{loc}\,R$ is Zariski closed.

###### Proof.

Let $H_{R}$ be the supporting hyperplane to $R$. In particular, $H_{R}$ is big
and nef. By Lemma 2.42, it follows that $\operatorname{Null}\,H_{R}$ is a
proper Zariski closed subset of $X$. We have
$\operatorname{loc}\,R\subset\operatorname{Null}\,H_{R}$. If
$\operatorname{loc}\,R$ is not Zariski closed, then there exists a surface
$S\subset\operatorname{Null}\,H_{R}$ which contains infinitely many curves in
$R$ but such that $S$ is not contained in $\operatorname{loc}\,R$. Thus,
$H_{R}|_{S}$ is a nef divisor on $S$ which intersects infinitely many curves
trivially and therefore it must intersect some moving curve trivially. It
follows that $S\subset\operatorname{loc}\,R$, a contradiction. ∎

##  3\. Facts about terminal singularities

The following simple observation is a crucial technical ingredient:

######  Proposition 3.1.

Let $X$ be a normal projective variety and let $(\cal F,\Delta)$ be a rank one
foliated pair on $X$ with canonical singularities. Let $\phi\colon
X\dashrightarrow X^{+}$ be a step in the $K_{\cal F}$-MMP and let $\cal F^{+}$
be the induced foliation on $X^{+}$.

Then the following hold:

  1. (1)

If $X$ is a threefold and $\cal F$ admits simple singularities (cf. Definition
2.32), then $\cal F^{+}$ also only admits simple singularities.

  2. (2)

If $X$ admits only quotient singularities, then $X^{+}$ also admits at worst
quotient singularities.

###### Proof.

Let $Z\subset X^{+}$ be $\phi(\operatorname{Exc}\,\phi)$ if $\phi$ is a
divisorial contraction and let it be the flipped locus when $\phi$ is a flip.
In either case by Lemma 2.10 if $E$ is a divisor centred in a subvariety of
$Z$ then $a(E,\cal F^{+})>0$. Thus, $\cal F^{+}$ is terminal at all points of
$Z$, including the generic point of $Z$. In particular, if a component
$\Sigma$ of $\operatorname{Sing}\,X^{+}$ is contained in $Z$ then, by Lemma
2.12, $\Sigma$ is not $\cal F^{+}$-invariant. Thus, (1) follows.

We now prove (2). Assume that $p\in\operatorname{Sing}\,X^{+}$ is not a
quotient singularity. Let $q\colon V\to U$ be a quasi-étale morphism over an
analytic open neighbourhood $U$ of $p$ such that $q^{*}\cal K_{\cal F^{+}}$ is
Cartier. Then $q(\operatorname{Sing}\,V)$ is non-empty.

By Lemma 2.12, after possibly shrinking $U$, we may assume that there exists a
submersion $f\colon V\rightarrow B$ which induces $q^{-1}\cal F|_{U}$ and
$\cal F^{+}$ is not terminal at the generic point of
$q(\operatorname{Sing}\,V)$. Thus, $q(\operatorname{Sing}\,V)$ is not
contained in $Z$. Let $Q\in V$ such that $q(Q)=P$. Since $X^{+}\setminus Z$
has at worst quotient singularities by assumption this implies that $f(Q)\in
B$ is a quotient singularity. Thus, $V$, and hence $U$, has at worst quotient
singularities, and (2) follows. ∎

###  3.1. A version of Reeb stability

Our goal is to generalise Reeb stability theorem to foliations defined over
singular varieties.

More specifically, let $X$ be a normal variety and let $\cal F$ be a rank one
foliation on $X$ which is terminal at all closed points. Let $C\subset X$ be a
compact $\cal F$-invariant curve and let $\Sigma\subset\operatorname{Sing}\,X$
be the locus where $\cal F$ is not Gorenstein. By Lemma 2.13, the set
$\\{c_{1},...,c_{N}\\}=C\cap\Sigma$ is finite. Let
$C^{\circ}=C\setminus\\{c_{1},...,c_{N}\\}$ and let $n_{k}$ be the Cartier
index of $K_{\cal F}$ at $c_{k}$ for each $k=1,\dots,N$. We now define the
holonomy of $\cal F$ along $C^{\circ}$.

Since $C$ is compact, by Lemma 2.12, we may find open sets
$U_{1},\dots,U_{\ell}$ in $X$ such that $C$ is contained in the union $\cup
U_{i}$ and for each $i=1,\dots,\ell$, there exists a finite morphism
$q_{i}\colon V_{i}\to U_{i}$ and a fibration $f_{i}\colon V_{i}\to T_{i}$ such
that $\cal F_{i}\coloneqq q_{i}^{-1}\cal F$ is the foliation induced by
$f_{i}$, $q_{i}$ is unramified outside $\Sigma$, and if $c_{k}\in U_{i}$ for
some $k=1,\dots,N$ then the ramification index of $q_{i}$ at $c_{k}$ is
$n_{k}$. In particular, the pre-image of the curve $C$ in $V_{i}$ is mapped to
a point $z_{i}\in T_{i}$.

Pick distinct $i,j$ such that $U_{i,j}:=U_{i}\cap U_{j}$ is not empty and it
intersects $C$. After possibly shrinking $U_{i}$ or $U_{j}$, we may assume
that $U_{i,j}$ does not intersect $\Sigma$. Let
$V^{j}_{i}:=q_{i}^{-1}(U_{i,j})$ and let
$V_{i,j}=V_{i}^{j}\times_{U_{i,j}}V_{j}^{i}$. Note that the induced morphism
$q_{i,j}\colon V_{i,j}\to U_{i,j}$ is unramified and there exists a morphism
$f_{i,j}\colon V_{i,j}\to T_{i,j}$ such the pulled back foliation $\cal
F_{i,j}$ on $V_{i,j}$ is induced by $f_{i,j}$. Indeed, $f_{i,j}$ is the Stein
factorisation of the morphism $V_{i,j}\to T_{i}$. Let $\sigma_{i,j}\colon
T_{i,j}\to T_{i}$ be the induced morphism. Note that the preimage of $C$ in
$V_{i,j}$ is mapped to a point $z_{i,j}\in T_{i,j}$ such that
$\sigma_{i,j}(z_{i,j})=z_{i}$. After possibly shrinking $U_{i}$ and $U_{j}$,
we may assume that $\sigma_{i,j}$ is surjective. It follows that
$\sigma_{i,j}$ is étale. Thus, after replacing $V_{i}$ by
$V_{i}\times_{T_{i}}T_{i,j}$ we may assume that $T_{i}=T_{j}$. After repeating
this process, finitely many times, we may assume that $T\coloneqq T_{i}$ and
$z\coloneqq z_{i}\in T$ do not depend on $i=1,\dots,k$. Note that, by the
construction above, the choice of the germ $(T,z)$ is uniquely determined by
$\cal F$ and $C$.

Pick $c\in C^{\circ}$. Let $\gamma_{1},\dots,\gamma_{N}$ be loops based at $c$
around $c_{1},\dots,c_{N}$ respectively. The orbifold fundamental group of
$C^{\circ}$ with weight $n_{k}$ at $c_{k}$ is defined as the quotient

$\pi(C^{\circ},c;n_{1},\dots,n_{N})\coloneqq\pi(C^{\circ},c)/\langle\gamma_{1}^{n_{1}},\dots,\gamma_{N}^{n_{N}}\rangle.$

We now want to define the holonomy map

$\rho\colon\pi(C^{\circ},c;n_{1},\dots,n_{N})\to\operatorname{Aut}(T,z),$

where $\operatorname{Aut}(T,z)$ denotes the group of biholomorhpic
automorphisms on the germ $(T,z)$. Let $\gamma\colon[0,1]\to C^{\circ}$ be a
continuous path which is contained in $U_{i}$ for some $i=1,\dots,\ell$. Then,
since $q_{i}\colon V_{i}\to U_{i}$ is unramified outside $\Sigma$, there
exists a lifting $\tilde{\gamma}\colon[0,1]\to V_{i}$ of $\gamma$ in $V_{i}$.
Note that $f_{i}$ maps the image of $\tilde{\gamma}$ to the point $z\in T$.
Proceeding as in the construction of the classic holonomy map (e.g. see
[CN85]), we can define a homomorphism

$\rho^{\prime}\colon\pi(C^{0},c)\to\operatorname{Aut}(T,z).$

Note that if $c_{k}\in U_{i}$ for some $i$ and $k$, then the ramification
index of $q_{i}$ at any point in $q_{i}^{-1}(c_{k})$ is equal to $n_{k}$.
Thus, it follows that $\rho^{\prime}(\gamma_{k}^{n_{k}})$ is the identity
automorphism of $T$ for any $k=1,\dots,N$ and, in particular, the holonomy map

$\rho\colon\pi(C^{\circ},c;n_{1},\dots,n_{N})\to\operatorname{Aut}(T,z)$

is well defined.

We are now ready to state our singular version of Reeb stability theorem:

######  Theorem 3.2.

Set up as above. Assume that the image of the holonomy map $\rho$ is finite.

Then there exists an analytic open subset $W$ of $X$ containing $C$ such that
the leaf $C_{t}$ of $\cal F$ passing through $t\in W$ is a compact analytic
subvariety of $W$.

###### Proof.

The proof of the Theorem is an easy generalisation of the classical Reeb
stability theorem (e.g. see [CN85, Theorem IV.5]). ∎

As a direct application of Reeb stability theorem, we get the following result
(see also [McQ04, II.d.5]):

######  Proposition 3.3.

Let $X$ be a normal variety and let $\cal F$ be a rank one foliation on $X$.
Let $C\subset X$ be an $\cal F$-invariant curve and suppose that $\cal F$ is
terminal at every closed point $P\in C$. Suppose moreover that $K_{\cal
F}\cdot C<0$.

Then $C$ moves in a family of $\cal F$-invariant curves covering $X$.

###### Proof.

Lemma 2.13 implies that $\cal F$ is Gorenstein at the generic point of $C$.
Let $c_{1},...,c_{N}\in C$ be the non-Gorenstein points of $\cal F$ and let
$n_{k}$ denote the Cartier index of $K_{\cal F}$ at $c_{k}$, for
$k=1,\dots,N$. Let $C^{\circ}=C\setminus\\{c_{1},\dots,c_{N}\\}$.

It follows from foliation adjunction (cf. Proposition 2.16), that

$K_{\cal F}\cdot C=-2+\sum_{k=1}^{N}\frac{n_{k}-1}{n_{k}}.$

In particular, since $K_{\cal F}\cdot C<0$ it follows that the orbifold
fundamental group $\pi_{1}(C^{\circ},c;n_{1},\dots,n_{N})$ is finite. Thus,
Theorem 3.2 implies the claim. ∎

##  4\. Subadjunction result in the presence of a foliation

Given a log pair $(X,S)$, a minimal log canonical centre $W$ of $(X,S)$ and an
ample divisor $A$ on $X$, we may write $(K_{X}+S+A)|_{W}=K_{W}+\Theta$ for an
effective divisor $\Theta\geq 0$. We are interested in this situation in the
presence of a foliation which leaves the components of $S$ invariant. In this
case we are able to get some control on $\Theta$ in terms of the singularities
of the foliation.

###  4.1. Dlt modification

Let $X$ be a normal threefold singularity and let $\cal F$ be a rank one
foliation on $X$ with canonical singularities. Let $S_{1},\dots,S_{k}$ be
prime $\cal F$-invariant divisors. Our goal here is to control the
singularities of the pair $(X,S\coloneqq\sum a_{i}S_{i})$, where
$a_{1},\dots,a_{k}\in(0,1]\cap\mathbb{Q}$, in terms of the singularities of
$\cal F$. As the following example shows a canonical foliation singularity
will in general have worse than quotient singularities on the ambient variety
(in contrast to the surface case):

######  Example 4.1.

Let $X=\\{xy-zw=0\\}\subset\mathbb{C}^{4}$ and consider the vector field
$\partial=x\partial_{x}-y\partial_{y}+z\partial_{z}-w\partial_{w}$ on
$\mathbb{C}^{4}$. Note that $X$ is $\partial$-invariant and so $\partial$
induces a rank one foliation $\cal F$ on $X$. We claim that $\cal F$ has
canonical singularities. Indeed, $\operatorname{Sing}\,\cal F=\\{0\\}$ and if
$\mathfrak{m}$ is the maximal ideal at $0$ then the induced linear map
$\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\mathfrak{m}/\mathfrak{m}^{2}$ is
non-nilpotent, and Proposition 2.20 implies that it is log canonical. The
eigenvalues of $\partial$ are not all positive rational numbers and [MP13,
Fact III.i.3] implies that $\cal F$ has a canonical singularity at
$(0,0,0,0)$.

######  Lemma 4.2.

Let $X$ be a normal variety and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. Let $(X,\Gamma=\sum a_{i}S_{i})$ be a log pair
whose irreducible components $S_{1},\dots,S_{k}$ are $\cal F$-invariant and
$a_{1},\dots,a_{k}\in(0,1]$.

Then there exists a birational morphism $\mu\colon\overline{X}\rightarrow X$
of $(X,\Gamma)$ such that

  1. (1)

$K_{\overline{\cal F}}=\mu^{*}K_{\cal F}+F$ where $\overline{\cal F}$ is the
foliation induced on $\overline{X}$ and $F\geq 0$ is a $\mu$-exceptional
divisor whose centre in $X$ is contained in the locus where $\mathcal{F}$ is
not Gorenstein; and

  2. (2)

$(\overline{X},\overline{\Gamma}+E)$ is dlt and
$K_{\overline{X}}+\overline{\Gamma}+E$ is nef over $X$, where $E$ is the sum
of all the $\mu$-exceptional divisors and $\overline{\Gamma}$ is the strict
transform of $\Gamma$ in $\overline{X}$.

We call the morphism $\mu$ a dlt modification of $(X,\Gamma)$ with respect to
$\cal F$.

###### Proof.

Let $U\subset X$ be the Gorenstein locus of $\cal F$, i.e., the open subset of
all points $P\in X$ such that $K_{\cal F}$ is Cartier in a neighbuorhood of
$P$. In particular, $X\setminus U$, being contained in
$\operatorname{Sing}\,X$, has codimension at least two. By [BM97], there
exists a log resolution $p\colon V\to U$ of $(U,\sum S_{i}|_{U})$ obtained by
only blowing up foliation invariant centres. Let $\cal F_{V}$ be the foliation
induced on $V$. By Lemma 2.12 and since $\cal F$ admits canonical
singularities, we have that $K_{\cal F_{V}}=p^{*}(K_{\cal F}|_{U})$. Let $Y$
be a normal variety which is a compactification of $V$ such that there exists
a projective morphism $\pi\colon Y\to X$ which extends $p$. Let
$\Gamma_{Y}=\pi^{-1}_{*}\Gamma$ and let $G$ be the sum of all the
$\pi$-exceptional divisors. Let $Z\to Y$ be a log resolution of
$(Y,\Gamma_{Y}+G)$, which is an isomorphism over $V$, and let $\rho\colon Z\to
X$ be the induced morphism. In particular, if $\mathcal{F}_{Z}$ is the induced
foliation on $Z$, $W=\rho^{-1}(U)$ and $q=\rho|_{W}\colon W\to U$ is the
restriction morphism, then
$K_{\mathcal{F}_{Z}}|_{W}=q^{*}(K_{\mathcal{F}}|_{U})$.

We may construct a morphism $\mu\colon\overline{X}\to X$ satisfying (2) as the
output of an MMP over $X$ starting from $Z$ (e.g. see [Kol13, Theorem 1.34]).
Let $\overline{\mathcal{F}}$ be the foliation induced on $\overline{X}$. It
follows that, if $\overline{U}=\mu^{-1}U$, then we have that
$K_{\overline{\mathcal{F}}}|_{\overline{U}}=r^{*}(K_{\mathcal{F}}|_{U})$,
where $r=\mu|_{\overline{U}}\colon\overline{U}\to U$ is the restriction
morphism. Thus, since $\cal F$ has canonical singularities, (1) follows. ∎

###  4.2. Bounding singularities

We consider the following set up. Let $X$ be a normal threefold and let $\cal
F$ be a rank one foliation on $X$ with canonical singularities such that
$K_{\cal F}$ is Cartier. Let $(X,\Gamma=\sum a_{i}S_{i})$ be a log pair whose
irreducible components $S_{1},\dots,S_{k}$ are $\cal F$-invariant and
$a_{1},\dots,a_{k}\in(0,1]$. Let $\mu\colon\overline{X}\rightarrow X$ be a dlt
modification with respect to $\cal F$, whose existence is guaranteed by Lemma
4.2. Let $\overline{\cal F}\coloneqq\mu^{-1}\cal F$. Note that
$K_{\overline{\cal F}}=\mu^{*}K_{\cal F}$ is Cartier. Let $\overline{\Gamma}$
be the strict transform of $\Gamma$ in $\overline{X}$ and let $E$ be the sum
of all the $\mu$-exceptional divisor. Lemma 2.8 implies that $E$ is
$\overline{\cal F}$-invariant. Let $E_{0},\dots,E_{k}$ be the irreducible
components of $E$. Let $(\cal G,\Delta)$ be the induced foliated pair on
$E_{0}$, as in Proposition 2.16 and such that

$K_{\overline{\cal F}}|_{E_{0}}=K_{\cal G}+\Delta.$

By classical adjunction, there exists a $\mathbb{Q}$-divisor $\Theta\geq 0$ on
$E_{0}$ such that $(E_{0},\Theta)$ is dlt and

$(K_{\overline{X}}+\overline{\Gamma}+E)|_{E_{0}}=K_{E_{0}}+\Theta.$

######  Corollary 4.3.

Set up as above. Let $C$ be a curve contained in
$\operatorname{Sing}\,\overline{X}\cap E_{0}$.

Then $C$ is $\overline{\cal F}$-invariant.

###### Proof.

Since $\overline{\cal F}$ is Gorenstein, Lemma 2.6 implies that
$\operatorname{Sing}\,\overline{X}$ is $\overline{\cal F}$-invariant. Thus,
the claim follows. ∎

######  Lemma 4.4.

Set up as above. Let $C\subset E_{0}$ be a curve which is not $\cal
G$-invariant.

Then $\mu_{C}\Delta\geq\mu_{C}\Theta$.

###### Proof.

We may assume that $C$ is contained in the support of $\Theta$ as, otherwise,
there is nothing to prove. Since $(E_{0},\Theta)$ is dlt, the coefficients of
$\Theta$ are at most one. Note that $\Theta$ is supported on
$\operatorname{Sing}\,\overline{X}\cap E_{0}$, on $E_{0}\cap E_{i}$ for $i\neq
0$ and on $\operatorname{Supp}\,\overline{\Gamma}\cap E_{0}$. By Corollary 4.3
and since $E_{1},\dots,E_{k}$ and the support of $\overline{\Gamma}$ are
$\overline{\cal F}$-invariant, it follows that, in all three cases, $C$ is
$\overline{\cal F}$-invariant. Since, by assumption, $C$ is not $\cal
G$-invariant, we have that $C\subset\operatorname{Sing}\,\overline{\cal F}$.
Since $\overline{\cal F}$ is Gorenstein, (1) of Proposition 2.17 implies that
$\mu_{C}\Delta\geq 1$. Thus, the claim follows. ∎

######  Theorem 4.5.

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. Let $0\in X$ be a closed point and let
$(X,\Gamma=\sum a_{i}S_{i})$ be a log pair where $S_{1},\dots,S_{k}$ are
irreducible $\cal F$-invariant divisors and $a_{1},\dots,a_{k}\in(0,1]$.
Suppose that $K_{X}$ and $\Gamma$ are $\mathbb{Q}$-Cartier and that
$(X,\Gamma)$ is log canonical away from $0$. Suppose moreover that $X$ is klt
away from $0$.

Then $(X,\Gamma)$ has log canonical singularities.

###### Proof.

Observe that our hypotheses are preserved by shrinking $X$ and by taking
quasi-étale covers. Thus, we may assume without loss of generality that
$K_{\cal F}$ is Cartier.

Suppose for sake of contradiction that $(X,\Gamma)$ has a worse than log
canonical singularity at 0. We may find $0<\lambda<1$, sufficiently close to
$1$ so that $(X,\lambda\Gamma)$ is not log canonical at $0$ and is klt away
from $0$. Thus, after replacing $\Gamma$ by $\lambda\Gamma$, we may assume
that $(X,\Gamma)$ is klt away from $0$.

Let $\mu\colon\overline{X}\rightarrow X$ be a dlt modification of $(X,\Gamma)$
with respect to $\cal F$, whose existence is guaranteed by Lemma 4.2. Let
$\overline{\cal F}\coloneqq\mu^{-1}\cal F$. Then, since $\cal F$ is
Gorenstein, we have that $K_{\overline{\cal F}}=\mu^{*}K_{\cal F}$ and since
$(X,\Gamma)$ is klt away from $0$, we have that every $\mu$-exceptional
divisor is centred in $0$. Let $E=\sum_{i=1}^{q}E_{i}$ be the sum of the
$\mu$-exceptional divisors and let $\overline{\Gamma}$ be the strict transform
of $\Gamma$ in $\overline{X}$. Lemma 2.8 implies that $E$ is $\overline{\cal
F}$-invariant.

By classical adjunction and by Proposition 2.16, for each $i=1,\dots,q$, we
may write

$(K_{\overline{X}}+\overline{\Gamma}+E)|_{E_{i}}=K_{E_{i}}+\Theta_{i}\qquad\text{and}\qquad
K_{\overline{\cal F}}|_{E_{i}}=K_{\cal G_{i}}+\Delta_{i}$

for some $\mathbb{Q}$-divisors $\Delta_{i},\Theta_{i}\geq 0$ on $E_{i}$ and
where $\cal G_{i}$ is the induced foliation on $E_{i}$. In particular,
$(E_{i},\Theta_{i})$ is log canonical for all $i=1,\dots,q$.

We first prove the following:

######  Claim.

For any $i=1,\dots,q$, the surface $E_{i}$ is covered by curves $M$ such that
$(K_{E_{i}}+\Theta_{i})\cdot M\leq 0$.

###### Proof of the Claim..

Note that $K_{\cal G_{i}}+\Delta_{i}\equiv 0$. Suppose first that $\cal G_{i}$
is not algebraically integrable. If $\Delta_{i}\neq 0$, as in the proof of
Lemma 2.14 it follows that $\cal G_{i}$ is uniruled, a contradiction. Thus, we
may assume that $\Delta_{i}=0$, and so, by Lemma 4.4, $\Theta_{i}$ only
consists of $\cal G_{i}$-invariant components. Thus, since
$(E_{i},\Theta_{i})$ is log canonical, we have that $\Theta_{i}\leq\sum C_{j}$
where the sum runs over all the $\cal G_{i}$-invariant divisors, and so we may
apply Lemma 2.14 to conclude.

Now suppose that $\cal G_{i}$ is algebraically integrable. Again, by Lemma 4.4
and since $K_{\cal G_{i}}+\Delta_{i}\equiv 0$, we may apply Lemma 2.15 to
conclude. Thus, the claim follows. ∎

Let $c\colon\overline{X}\dashrightarrow X_{can}$ be the log canonical model of
$(\overline{X},\overline{\Gamma}+E)$ over $X$, let $\Gamma_{can}\coloneqq
c_{*}\Gamma$ and let $m\colon X_{can}\rightarrow X$ be the induced morphism.

By (2) of Lemma 4.2, we have that $K_{\overline{X}}+\overline{\Gamma}+E$ is
nef over $X$. Thus, the inequality of the Claim is in fact an equality and as
such, each such curve is contracted by $c$. This implies that
$X_{can}\rightarrow X$ is a small contraction. In particular,
$m^{*}(K_{X}+\Gamma)=K_{X_{can}}+\Gamma_{can}$. Our result follows, since
$(X_{can},\Gamma_{can})$ has log canonical singularities. ∎

######  Example 4.6.

Observe that the fact that our singularity is isolated in the above theorem is
necessary. Indeed, let $S$ be any normal surface and let $C$ be a smooth curve
and let $\cal F$ be the foliation on $X\coloneqq S\times C$ induced by the
projection onto the first coordinate. It is straightforward to check that
$\operatorname{Sing}\,\cal F=\emptyset$ and so $\cal F$ has canonical
singularities by [Dru19, Lemma 5.9], and moreover, is terminal at all closed
points $x\in X$.

We also need the following:

######  Proposition 4.7.

Let $X$ be a normal threefold and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. Let $(X,S:=\sum S_{i})$ be a log pair where
$S_{1},\dots,S_{k}$ are irreducible $\cal F$-invariant divisors and let
$C\subset\operatorname{Sing}\,\cal F$ be a curve.

Then $(X,S)$ is log canonical at the generic point of $C$.

###### Proof.

The following proof relies on similar, and at the same time easier, ideas as
in Theorem 4.5. Thus, we only sketch its main steps.

Observe that our hypotheses are preserved by shrinking $X$ and by taking
quasi-étale covers. Thus, we may assume without loss of generality that
$K_{\cal F}$ is Cartier.

Let $\mu\colon\overline{X}\rightarrow X$ be a dlt modification of $(X,S)$ with
respect to $\cal F$, whose existence is guaranteed by Lemma 4.2. Let
$\overline{\cal F}\coloneqq\mu^{-1}\cal F$. Then, since $\cal F$ is
Gorenstein, we have that $K_{\overline{\cal F}}=\mu^{*}K_{\cal F}$. After
possibly replacing $X$ by a neighbourhood of the generic point of $C$, we may
assume that every $\mu$-exceptional divisor is centred in $C$. Let
$E=\sum_{i=1}^{q}E_{i}$ be the sum of the $\mu$-exceptional divisors and let
$\overline{S}$ be the strict transform of $S$ in $\overline{X}$.

By classical adjunction and by Proposition 2.16, for each $i=1,\dots,q$, we
may write

$(K_{\overline{X}}+\overline{S}+E)|_{E_{i}}=K_{E_{i}}+\Theta_{i}\qquad\text{and}\qquad
K_{\overline{\cal F}}|_{E_{i}}=K_{\cal G_{i}}+\Delta_{i}$

for some $\mathbb{Q}$-divisors $\Delta_{i},\Theta_{i}\geq 0$ on $E_{i}$ and
where $\cal G_{i}$ is the induced foliation on $E_{i}$. In particular,
$(E_{i},\Theta_{i})$ is log canonical, for all $i=1,\dots,q$.

Fix $i=1,\dots,q$ and consider the induced morphism $p\colon E_{i}\to C$. Let
$\Sigma$ be the general fibre of $p$ and let $\Sigma^{\nu}\to\Sigma$ be its
normalisation. Since $C\subset\operatorname{Sing}\,\cal F$, it follows that a
general closed point of $C$ is $\cal F$-invariant. Thus, Lemma 2.5 implies
that $\Sigma$ is $\overline{\cal F}$-invariant. By classical adjunction and by
Proposition 2.16, there exist $\mathbb{Q}$-divisors
$\Gamma_{i},\Delta^{\prime}\geq 0$ on $\Sigma^{\nu}$ such that

$(K_{E_{i}}+\Theta_{i})|_{\Sigma^{\nu}}=K_{\Sigma^{\nu}}+\Gamma_{i}\qquad\text{and}\qquad
0=K_{\overline{\cal F}}|_{\Sigma^{\nu}}=K_{\Sigma^{\nu}}+\Delta^{\prime}.$

As in the proof of Lemma 4.4, it follows that the support of $\Gamma_{i}$ is
contained in the support of $\Delta^{\prime}$ and since $\Delta^{\prime}$ is
integral, whilst $(\Sigma^{\nu},\Gamma_{i})$ is log canonical, it follows that
$\deg(K_{\Sigma^{\nu}}+\Gamma_{i})\leq 0$. Thus, our results follows as in the
proof of Theorem 4.5. ∎

Note that it is easy to produce examples of a canonical foliation of rank one
on a normal variety and a collection of invariant divisors $\sum S_{i}$ so
that $(X,\sum S_{i})$ has zero-dimensional non-log canonical singularities, as
shown in the following example:

######  Example 4.8.

Let $X=\mathbb{C}^{3}$, let $\cal F$ be the foliation defined by the vector
field $x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}$ and let
$S=\\{x=0\\}+\\{y=0\\}+\\{xy-z^{2}=0\\}$. Then the support of $S$ is $\cal
F$-invariant and the origin $0\in X$ is a non-lc centre for $(X,S)$. Note that
it is not isolated: the curves $\\{x=z=0\\}$ and $\\{y=z=0\\}$ are non-lc
centres for $(X,S)$ as well.

######  Remark 4.9.

Theorem 4.5 implies that if $x\in X$ is an isolated $\mathbb{Q}$-Gorenstein
singularity and $\cal F$ is a rank one foliation with canonical singularities
then $x\in X$ is a log canonical singularity. It would be interesting to know
if we could improve this bound. E.g. is $x\in X$ log terminal?

Note that if there is a $\mathbb{Q}$-Cartier $\cal F$-invariant, possibly
formal, divisor $S$ passing through $x$ then $(X,tS)$ is log canonical for
$t>0$ sufficiently small and so $X$ is log terminal.

######  Lemma 4.10.

Let $X$ be a normal variety and let $\cal F$ be a rank one foliation on $X$
with canonical singularities. Let $(X,\Delta=\sum a_{i}S_{i})$ be a log pair
where $S_{1},\dots,S_{k}$ are irreducible $\cal F$-invariant divisors and
$a_{1},\dots,a_{k}\in(0,1]$. Let $P$ be a non-klt centre of $(X,\Delta)$.

Then $P$ is $\cal F$-invariant and , in particular, it is a centre of
canonical singularities for $\cal F$.

###### Proof.

After possibly shrinking $X$ and taking a cover, we may assume without loss of
generality that $\cal F$ is Gorenstein and generated by a vector field
$\partial$.

By [BM97], there exists a $\cal F$-equivariant log resolution $\pi\colon
X^{\prime}\to X$ of $(X,\Delta)$. By Lemma 2.12, we have $K_{\cal
F^{\prime}}=\pi^{*}K_{\cal F}$ where $\cal F^{\prime}$ is the foliation
induced on $X^{\prime}$. We may also write
$K_{X^{\prime}}+\Gamma=\pi^{*}(K_{X}+\Delta)+B$, where $B,\Gamma\geq 0$ are
$\mathbb{Q}$-divisors without common components. By assumption, there exists a
divisor $E$ centred at $P$ such that $a(E,X,\Delta)\leq 0$. Let $Z$ be the
centre of $E$ on $Y$. Then $Z$ is a stratum of the divisor
$\lfloor\Gamma\rfloor$ and, in particular, since, by Lemma 2.8, each component
of $\Gamma$ is $\cal F^{\prime}$-invariant, it follows that also $Z$ is $\cal
F^{\prime}$-invariant. Thus, Lemma 2.5 implies that $P$ is $\cal F$-invariant
and Lemma 2.12 implies the claim. ∎

###  4.3. Subadjuntion

We work in the following set up. Let $X$ be a $\mathbb{Q}$-factorial threefold
with klt singularities, let $\cal F$ be a rank one foliation on $X$ and let
$\Gamma=\sum a_{i}S_{i}$ be a $\mathbb{Q}$-divisor where $S_{1},\dots,S_{k}$
are $\cal F$-invariant prime divisors and $a_{1},\dots,a_{k}\in(0,1)$. Let
$C\subset X$ be a projective curve which is a log canonical centre of
$(X,\Gamma)$ and suppose that there are no one-dimensional non-log canonical
centres. Suppose moreover that $\cal F$ has canonical singularities and that
$\cal F$ is terminal and Gorenstein at a general point of $C$. Theorem 4.5
implies that $(X,\Gamma)$ is log canonical. By Lemma 4.10, it follows that $C$
is $\mathcal{F}$-invariant.

Thus, by Proposition 2.16 we may write

$(K_{X}+\Gamma)|_{C^{\nu}}=K_{C^{\nu}}+\Theta$

where $\nu\colon C^{\nu}\rightarrow C$ is the normalisation and $\Theta\geq 0$
is a $\mathbb{Q}$-divisor.

######  Theorem 4.11.

Set up as above. Then

  1. (1)

$(C^{\nu},\Theta)$ is log canonical;

  2. (2)

$\lfloor\Theta\rfloor$ is supported on the pre-image of centres of canonical
singularities of $\cal F$;

  3. (3)

if $\cal F$ is terminal and Gorenstein at $\nu(Q)\in C$ for some $Q\in
C^{\nu}$, then $Q\notin\operatorname{Supp}\,\Theta$;

  4. (4)

the support of $\\{\Theta\\}$ contains all points which are terminal and non-
Gorenstein for $\cal F$. Moreover, if $P$ is such a point then
$\mu_{P}\Theta=\frac{n-1}{n}$ where $n$ is the Cartier index of $K_{\cal F}$
at $P$.

In particular, we have

$(K_{X}+\Gamma)\cdot C\leq K_{\cal F}\cdot C.$

###### Proof.

Let $p\colon\overline{X}\rightarrow X$ be a dlt modification of $(X,\Gamma)$
with respect to $\cal F$ as in Lemma 4.2 and let $\overline{\Gamma}$ be the
strict transform of $\Gamma$ in $\overline{X}$. Since $(X,\Gamma)$ is log
canonical, we may write

$K_{\overline{X}}+\overline{\Gamma}+E=p^{*}(K_{X}+\Gamma)$

where $E$ is the sum of all the prime exceptional divisors of $p$. Lemma 2.8
implies that $E$ is $\overline{\cal F}$-invariant. Let $E_{0}$ be an
irreducible component of $E$ dominating $C$ and let $E_{1}=E-E_{0}$. By
adjunction we may write

$(K_{\overline{X}}+E+\overline{\Gamma})|_{E_{0}}=K_{E_{0}}+\Theta_{0}$

where $\Theta_{0}\geq 0$. Let $\overline{\cal F}$ be the induced foliation on
$\overline{X}$ and let $\cal G$ be the induced foliation on $E_{0}$, as in
Proposition 2.16.

Let $f\coloneqq p|_{E_{0}}\colon E_{0}\rightarrow C^{\nu}$ be the restriction
morphism. Then $K_{E_{0}}+\Theta_{0}$ is $f$-trivial and we may write
$K_{E_{0}}+\Theta_{0}=f^{*}(K_{C^{\nu}}+M+B)$ where $M\coloneqq M_{C^{\nu}}$
is the moduli part of $f$ and $B\coloneqq B_{C^{\nu}}\geq 0$ is the
discrepancy part of $f$, as in Section 2.9. In particular, we have
$\Theta=M+B$.

We first show that property (1),…,(4) above hold for $(C^{\nu},B)$. Since
$(E_{0},\Theta_{0})$ is dlt, it follows that $(C^{\nu},B)$ is log canonical.
If $Q\in C^{\nu}$ is such that $\mu_{Q}(B)=1$ then $\nu(Q)$ is the image of a
log canonical centre of $(\overline{X},E+\overline{\Gamma})$ and, in
particular, it is a log canonical centre of $(X,\Gamma)$. Thus, Lemma 4.10
implies that $\nu(Q)$ is a canonical centre of $\cal F$. It follows that
$\lfloor B\rfloor$ is supported on the pre-image of the isolated canonical
centres of $\cal F$ contained in $C$.

Now, let $Q\in C^{\nu}$ be a point such that $\cal F$ is terminal and
Gorenstein at $\nu(Q)$. We want to show that $Q$ is not contained in the
support of $B$. Lemma 2.12 implies that $Q$ is not $\cal F$-invariant and it
is a smooth point of $\cal F$ and, therefore, of $C$. Lemma 2.5 implies that
$f^{-1}(Q)$ does not contain any $\overline{\cal F}$-invariant curve. By (1)
of Lemma 4.2, we may assume that $\overline{\cal F}$ is Gorenstein around
$p^{-1}(Q)$ and that $p^{-1}(Q)$ does not contain any canonical centre. In
particular, Lemma 2.12 implies that $\operatorname{Sing}\,\overline{X}$ is
$\overline{\cal F}$-invariant. Let $\Sigma$ be an irreducible component of
$\Theta_{0}$. Note that $\Sigma$ is supported on the union of
$E_{0}\cap\operatorname{Supp}\,(E_{1}+\overline{\Gamma})$ and