[McQ08, Proposition III.1.2]. Thus, we get a contradiction, and hence $C$ is
invariant.

  3. (3)

Claim. $K_{\mathcal{G}}\cdot C\geq K_{{\mathcal{G}}_{F_{0}}}\cdot C$.

######  Proof.

By Lemma 4.10, we have

$K_{\mathcal{G}}\cdot C=\left(K_{{\mathcal{G}}_{F_{0}}}+\sum_{j\in
J}C_{j}\right)\cdot C\geq K_{{\mathcal{G}}_{F_{0}}}\cdot C$

where the last inequality holds since $C_{j}$’s are non-invariant and $C$ is
invariant.

  4. (4)

Claim. $\textnormal{Z}({\mathcal{G}}_{F_{0}},C,p)\geq 0$ for all $p$ on the
smooth locus of $C$.

######  Proof.

Let $p_{1},\ldots,p_{w}$ be all foliation singularities on the smooth locus of
$C$ such that $\textnormal{Z}({\mathcal{G}}_{F_{0}},C,p)<0$. By Lemma 1.26, we
may assume that the eigenvalue for $p_{i}$ is $\frac{m_{i}}{n_{i}}$ where
$m_{i}\geq n_{i}\geq 2$ and $\textnormal{gcd}(m_{i},n_{i})=1$. Thus, we
notice, by the proof of [McQ08, Proposition III.2.1], that

$\displaystyle 0$ $\displaystyle>$ $\displaystyle(K_{\mathcal{G}}+\Theta)\cdot
C$ $\displaystyle\geq$ $\displaystyle K_{{\mathcal{G}}_{F_{0}}}\cdot C$
$\displaystyle\geq$ $\displaystyle
k-2-\frac{k}{2}+\sum_{i=1}^{w}\textnormal{Z}({\mathcal{G}}_{F_{0}},C,p_{i})+(K_{F_{0}}+C)\cdot
C$ $\displaystyle\geq$
$\displaystyle\frac{k}{2}-2+\sum_{i=1}^{w}\big{(}1-(m_{i}-1)(n_{i}-1)\big{)}+\sum_{i=1}^{w}(m_{i}-1)(n_{i}-1)-2$
$\displaystyle=$ $\displaystyle\frac{k}{2}+w-4$

where $k$ is the number of singularities of $C$. Hence, $w\leq 3$ and $k\leq
7$. Therefore, by Lemmas 1.25 and 1.26, we have

$0>C^{2}>\widetilde{C}^{2}>\sum_{i=1}^{w}m_{i}n_{i}-k\geq 4w-7$

where $\widetilde{C}$ is the proper transform of $C$ on $\widetilde{F_{0}}$,
which is the minimal resolution of $F_{0}$. Thus $w=1$, $k\leq 5$, and
$\widetilde{C}^{2}>m_{1}n_{1}-k\geq-1$. Since $\widetilde{C}^{2}$ is an
integer, we have that $C^{2}>\widetilde{C}^{2}\geq 0$, which is a
contradiction.

  5. (5)

Therefore, we have that

$\displaystyle 0$ $\displaystyle>$ $\displaystyle(K_{\mathcal{G}}+\Theta)\cdot
C$ $\displaystyle\geq$ $\displaystyle K_{\mathcal{G}}\cdot C$ $\displaystyle=$
$\displaystyle K_{{\mathcal{G}}_{F_{0}}}\cdot C+\sum_{j\in J}C_{j}\cdot C$
$\displaystyle\geq$ $\displaystyle k-2-\frac{k}{2}+\sum_{j\in J}C_{j}\cdot C$

where the equality comes from Lemma 4.10. So $\sum_{j\in J}C_{j}\cdot C\leq
1$. We will study and exclude all cases. In particular, there is no flipping
contraction.

  6. (6)

If $\sum_{j\in J}C_{j}\cdot C=1$, then $k\leq 1$. Thus $\widetilde{C}$ must be
a tail of some ${\mathcal{G}}_{\widetilde{F_{0}}}$-chains with at most one
reduced singularity, which is impossible by Theorem 1.19.

  7. (7)

So $\sum_{j\in J}C_{j}\cdot C=0$, and thus we have $k<4$ and $\widetilde{C}$
is a smooth rational curve.

  8. (8)

If $k=0$. Then $C$ is in the smooth locus of $F_{0}$ and has no foliation
singularities. Thus $C^{2}=0$, which contradicts to step (2).

  9. (9)

If $k=2$ (resp. $k=3$), then $\widetilde{C}$ must meet three
${\mathcal{G}}_{\widetilde{F_{0}}}$-chains and has exactly two (resp. three)
reduced singularities, which is impossible by Theorem 1.19.

  10. (10)

If $k=1$, then $\widetilde{C}$ must meet exactly one
${\mathcal{G}}_{\widetilde{F_{0}}}$-chain $D$ and has exactly two reduced
singularities. So $\widetilde{C}\cup D$ gives a new
${\mathcal{G}}_{\widetilde{F_{0}}}$-chain. Then by Lemma 4.20, we see
$\textnormal{loc}([\widetilde{C}])$ has dimension at least two. Therefore,
$\textnormal{loc}(R)=\textnormal{loc}([C])$ also has dimension at least two,
which gives a contradiction.

∎

######  Lemma 4.20.

Let $f:({\mathcal{X}},{\mathcal{F}})\rightarrow T$ be an
${\mathcal{S}}_{P}$-family over $T$ with ${\mathcal{X}}$ smooth. For a fixed
$t\in T$ and an ${\mathcal{F}}_{t}$-chain $C_{t}=\sum_{i}C_{t,i}$, we can
deform $C^{t}$ to an ${\mathcal{F}}_{s}$-chain $C_{s}=\sum_{i}C_{s,i}$ for
$s\in T$ close to $t$.

######  Proof.

The proof is similar to the proof of [Bru01, Lemme 3]. We first deform $C_{t}$
to very general $s$ closed to $t$. Thus the eigenvalues of ${\mathcal{F}}_{s}$
at singularities on $C_{s}$ remain the same for $s$ close to $t$. Therefore,
$C_{s}$ is an ${\mathcal{F}}_{s}$-chain all $s$ close to $t$. ∎

######  Lemma 4.21.

Assumptions as in Theorem 4.16. Let $R$ be a
$(K_{\mathcal{G}}+\Theta)$-negative extremal ray and
${\mathcal{E}}=\textnormal{loc}(R)$. Then we have, for general $t\in T$, that

  1. (1)

either ${\mathcal{E}}_{t}:={\mathcal{E}}|_{{\mathcal{Y}}_{t}}$ is a
${\mathcal{G}}_{t}$-exceptional curve or the first curve of a
${\mathcal{G}}_{t}$-chain

  2. (2)

or the proper transform $(\phi_{t*})^{-1}({\mathcal{E}}_{t})$ is the tail of
some $\phi_{t}$-contracted ${\mathcal{F}}_{t}$-chain.

Moreover, ${\mathcal{E}}$ meets exactly one irreducible component of
${\mathcal{Y}}_{0}$.

######  Proof.

The first statement follows because ${\mathcal{Y}}_{t}$ is a birational model
over an element in ${\mathcal{S}}_{P}$ and ${\mathcal{E}}_{t}$ is a
$(K_{{\mathcal{G}}_{t}}+\Theta_{t})$-negative curve. Assume that
${\mathcal{E}}_{0}=\sum C_{j}$ has at least two irreducible components.

  1. (1)

If ${\mathcal{E}}_{t}$ is an ${\mathcal{G}}_{t}$-exceptional curve or the
first curve of an ${\mathcal{G}}_{t}$-chain, then

$K_{\mathcal{G}}\cdot\sum_{j}C_{j}=K_{{\mathcal{G}}}\cdot{\mathcal{E}}_{t}=-1$

for all $j$. Since $R$ is extremal, all $[C_{j}]$’s are in $R$. Thus
$K_{\mathcal{G}}\cdot C_{j}$ is a negative integer for any $j$. Therefore,
$K_{\mathcal{G}}\cdot\sum_{j}C_{j}\leq-2$, which is impossible.

  2. (2)

If the proper transform $E_{t}:=(\phi_{t*})^{-1}({\mathcal{E}}_{t})$ is the
tail of some $\phi_{t}$-contracted ${\mathcal{F}}_{t}$-chains, then
$-1<K_{\mathcal{G}}\cdot{\mathcal{E}}_{t}<0$. So $K_{\mathcal{G}}\cdot
C_{j}<0$ for all $j$. Notice that at least one $C_{j}$ is in the smooth locus
${\mathcal{Y}}$. Thus, for such $j$, we have $K_{\mathcal{G}}\cdot
C_{j}\leq-1$ and hence $K_{\mathcal{G}}\cdot\sum_{j}C_{j}<-1$, which is
impossible. ∎

######  Lemma 4.22.

Assumptions as in Theorem 4.16. Let $R={\mathbb{R}}_{\geq 0}[C_{R}]$ be a
$(K_{\mathcal{G}}+\Theta)$-negative extremal ray. Then
$(K_{\mathcal{Y}}+\phi_{*}{\mathcal{D}})\cdot C_{R}<0$.

######  Proof.

Let ${\mathcal{E}}=\textnormal{loc}(R)$. Then we have two cases:

  1. (1)

If ${\mathcal{E}}$ is flat over $T$, then
${\mathcal{E}}=\phi_{*}{\mathcal{E}}_{i}$ for some $i=1,\ldots,u$. By Lemma
4.21 and the assumption in Theorem 4.16, we have that

$\left(K_{\mathcal{Y}}+\phi_{*}{\mathcal{D}}\right)\cdot
C_{R}=\left(K_{\mathcal{X}}+{\mathcal{D}}\right)\cdot(\phi_{*})^{-1}(C_{R})<0.$

  2. (2)

If ${\mathcal{E}}=\mbox{{\rm loc}}(R)$ is one of the irreducible component of
the central fiber ${\mathcal{Y}}_{0}$ of
$g:({\mathcal{Y}},{\mathcal{G}})\rightarrow T$.

Let $C_{R}$ be an invariant rational curve by Remark 4.15. Notice that
${\mathcal{E}}$ is covered by rational curves. Also,
${\mathcal{G}}_{\mathcal{E}}$ has only finitely many foliation singularities,
all of which are ${\mathbb{Q}}$-foliated Gorenstein log canonical
singularities. By classification of canonical foliation singularities, there
are at most two separatrices passing through these singularities. Therefore we
may assume that $C_{R}$ has only non-reduced foliation singularities and is in
the smooth locus of ${\mathcal{E}}$. Let $p_{1},\ldots,p_{r}$ be those non-
reduced singularities with the eigenvalues $\frac{m_{i}}{n_{i}}$ where
$m_{i}\geq n_{i}\geq 1$ and $\textnormal{gcd}(m_{i},n_{i})=1$.

In the proof of Lemma 4.14, we have known that $C_{R}\cdot{\mathcal{E}}<0$.
Note that

$\displaystyle 0$ $\displaystyle>$ $\displaystyle(K_{\mathcal{G}}+\Theta)\cdot
C_{R}$ $\displaystyle=$ $\displaystyle K_{{\mathcal{G}}_{\mathcal{E}}}\cdot
C_{R}+\sum_{j\in J}C_{R}\cdot C_{j}$ $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{r}\textnormal{Z}({\mathcal{G}}_{\mathcal{E}},C_{R},p_{i})+(K_{\mathcal{E}}+C_{R})\cdot
C_{R}$ $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{r}\big{(}1-(m_{i}-1)(n_{i}-1)\big{)}+\sum_{i=1}^{r}(m_{i}-1)(n_{i}-1)-2$
$\displaystyle=$ $\displaystyle r-2$

Thus, we have that $r\leq 1$. Let ${\mathcal{Y}}_{0}=\sum_{i=0}^{s}F_{i}$ and
${\mathcal{E}}=F_{0}$. Then we have

$0<-C_{R}\cdot{\mathcal{E}}=C_{R}\cdot\sum_{i=1}^{s}F_{i}\leq\sum_{j\in
J}C_{R}\cdot C_{j}+r.$

    1. (a)

If $r=1$, then $\sum_{j\in J}C_{R}\cdot C_{j}=0$ and
$C_{R}\cdot{\mathcal{E}}=-1$. Thus $C_{R}^{2}=m_{1}n_{1}$. Therefore, by
adjunction, we have

$\displaystyle K_{\mathcal{Y}}\cdot C_{R}$ $\displaystyle=$ $\displaystyle
K_{\mathcal{E}}\cdot C_{R}-{\mathcal{E}}\cdot C_{R}$ $\displaystyle=$
$\displaystyle-2+(m_{1}-1)(n_{1}-1)-m_{1}n_{1}-(-1)$ $\displaystyle=$
$\displaystyle-m_{1}-n_{1}<0.$

    2. (b)

If $r=0$, then $\sum_{j\in J}C_{R}\cdot C_{j}=1$ and
$C_{R}\cdot{\mathcal{E}}=-1$. Since there is no foliation singularity on
$C_{R}$, we have $C_{R}^{2}=0$. Therefore, by adjunction, we have

$K_{\mathcal{Y}}\cdot C_{R}=K_{\mathcal{E}}\cdot C_{R}-{\mathcal{E}}\cdot
C_{R}=-2-C_{R}^{2}-(-1)=-1.$

Hence, ${\mathcal{Y}}_{0}$ is a ruled surface with smooth rational general
fibers. Since there is no reduced singularity on $C_{R}$, we have that
$\phi_{*}{\mathcal{D}}\cdot C_{R}=0$ by Lemma 4.21 and
$(K_{\mathcal{Y}}+\phi_{*}{\mathcal{D}})\cdot C_{R}<0$. ∎

######  Proof (Cone Theorem 4.16).

By Lemma 4.19, all loci ${\mathcal{E}}$’s of
$(K_{\mathcal{G}}+\Theta)$-negative extremal rays are of dimension two. Then
by Lemma 4.21 and MMP on foliated surfaces, there are finitely many
${\mathcal{E}}$ flat over $T$. For ${\mathcal{E}}$ non-flat over $T$,
${\mathcal{E}}$ is in an irreducible component of the special fiber of
$g:({\mathcal{Y}},{\mathcal{G}})\rightarrow T$. This proves (1).

(2) follows from Lemma 4.22 and (3) follows from Lemma 4.21. ∎

###  4.5. Semi-stable minimal model program

Let $f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ be a semi-stable
reduction as in subsection 4.2. Since Picard number is finite and is decreased
by one after each divisorial contraction, by Theorem 4.16, we can get, after
partial $(K_{\mathcal{X}}+{\mathcal{D}}+f^{-1}(t))$-MMP for any $t\in T$, that
$f^{m}:({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})\rightarrow T$ has no
$(K_{{\mathcal{F}}^{m}}+\Delta^{m})$-negative extremal ray over $T$, that is,
$(K_{{\mathcal{F}}^{m}}+\Delta^{m})$ is nef over $T$. Let
$\pi:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})$
be the contraction over $T$.

######  Lemma 4.23.

$f^{m}:{\mathcal{X}}^{m}\rightarrow T$ is dlt (see Definition 4.6) and
$({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})$ is ${\mathcal{F}}^{m}$-dlt
(see Definition 4.8).

######  Proof.

Note that $\left({\mathcal{X}},{\mathcal{D}}+f^{-1}(t)\right)$ is dlt for any
$t\in T$. Since $\pi$ is the composition of divisorial contractions of
$\left(K_{\mathcal{X}}+{\mathcal{D}}+f^{-1}(t)\right)$-negative extremal rays,
we have that
$\left({\mathcal{X}}^{m},\pi_{*}{\mathcal{D}}+(f^{m})^{-1}(t)\right)$ is also
dlt. (For a reference, see [KM98, Corollary 3.44].) Thus,
$({\mathcal{X}}^{m},(f^{m})^{-1}(t))$ is dlt for any $t\in T$ and hence
$f^{m}:{\mathcal{X}}^{m}\rightarrow T$ is dlt.

Also $({\mathcal{X}}^{m},{\mathcal{F}}^{m},\Delta^{m})$ is
${\mathcal{F}}^{m}$-dlt by Lemma 4.1. ∎

######  Remark 4.24.

In fact, we have $\pi_{*}{\mathcal{D}}=\emptyset$.

To sum up, we have the following theorem.

######  Theorem 4.25.

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

Moreover, we show the following valuative criterion of properness for
${\mathcal{S}}_{P}$-families.

######  Theorem 4.26.

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

######  Proof.

As in subsection 4.2, we take the closure and semi-stable resolution to get
$g:({\mathcal{Y}},{\mathcal{G}},\Theta)\rightarrow C$ where $C\rightarrow T$
is a finite morphism and $C$ is a smooth curve. Then by Theorem 4.25, we run
the semi-stable $(K_{\mathcal{G}}+\Theta)$-MMP over $C$. Thus, we get a family
$g:({\mathcal{Y}},{\mathcal{G}},\Theta)\rightarrow C$ such that
$K_{\mathcal{G}}+\Theta$ is nef over $C$. By running additional partial
$K_{\mathcal{Y}}$-MMP over $C$, we may assume that
$K_{\mathcal{Y}}+6i(P)(K_{\mathcal{G}}+\Theta)$ is nef over $C$. (See Lemma
3.1 for the definition of $i(P)$.) Note that
$\big{(}K_{\mathcal{Y}}+6i(P)(K_{\mathcal{G}}+\Theta)\big{)}-K_{\mathcal{Y}}$
is nef and big over $C$. By relative base point free theorem (see [KM98,
Theorem 3.24] for a reference),
$K_{\mathcal{Y}}+6i(P)(K_{\mathcal{G}}+\Theta)$ is semi-ample over $C$. Thus,
we have the relative canonical model
$g:({\mathcal{Y}},{\mathcal{G}},\Theta)\rightarrow C$.

Let $f^{\circ}_{C}$ be the base change of $f^{\circ}$ via the morphism
$C^{\circ}\rightarrow T^{\circ}$ where $C^{\circ}=C\backslash\\{0\\}$. Notice
that for any $c\in C\backslash\\{0\\}$, the fibers of $f^{\circ}_{C}$ and $g$
over $c$ are isomorphic. Moreover, for $c=0$, we have that
$(K_{\mathcal{G}}+\Theta)|_{{\mathcal{Y}}_{0}}$ is nef and
$K_{{\mathcal{Y}}_{0}}+6i(P)(K_{{\mathcal{G}}}+\Theta)|_{{\mathcal{Y}}_{0}}$
is ample. Hence, the fiber of $g$ over $0\in C$ is smoothable. Therefore,
$g:({\mathcal{Y}},{\mathcal{G}},\Theta)\rightarrow C$ is an
${\mathcal{S}}_{P}$-family over $C$ with the special fiber
${\mathcal{S}}_{P}$-smoothable. ∎

##  5\. Valuative criterion of separatedness for ${\mathcal{S}}_{P}$-families

In this section, we show the following theorem.

######  Theorem 5.1.

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

The proof follows the classical approach. We will need the following lemmas
that, for $(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, $Y$ is indeed a Proj of a
(finitely generated) graded ring associated with the adjoint bundle
$K_{Y}+6i(P)(K_{\mathscr{G}}+D)$ and this adjoint bundle behaves well after
taking resolutions.

######  Lemma 5.2.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. For any
$(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$, we have

$Y\cong\mbox{{\rm Proj }}\bigoplus_{m\geq
0}H^{0}(Y,m(K_{Y}+6i(P)(K_{\mathscr{G}}+D)))$

where $i(P)$ is defined in Lemma 3.1.

######  Proof.

Notice that $K_{Y}+6i(P)(K_{\mathscr{G}}+D)$ is ample by the part (1) of the
proof of Theorem 2.10. ∎

######  Lemma 5.3.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. For any
$(Y,{\mathscr{G}},D)\in{\mathcal{S}}_{P}$ and any resolution of singularities
$f:Z\rightarrow Y$, we write

$K_{Z}+6i(P)\left(K_{\mathscr{H}}+f_{*}^{-1}D+\sum\varepsilon(E_{j})E_{j}\right)=f^{*}(K_{Y}+6i(P)(K_{\mathscr{G}}+D))+\sum
a_{j}E_{j}$

where ${\mathscr{H}}$ is the pullback foliation on $Z$ and $E_{j}$’s are
exceptional divisors of $f$. Then we have $a_{j}\geq-1$ for all $j$ where
$\varepsilon(E_{j})=0$ if $E_{j}$ is ${\mathscr{H}}$-invariant and
$\varepsilon(E_{j})=1$ otherwise.

######  Proof.

We first consider that $Z=Y^{m}$ is the minimal resolution $h$ of
singularities of $Y$ and ${\mathscr{G}}^{m}$ is the pullback foliation on
$Y^{m}$. We have two cases for any singularity $p$ of $Y$:

  1. (1)

If $p$ is a terminal foliation singularity, then we have

$i(P)(K_{{\mathscr{G}}^{m}}+h_{*}^{-1}D)=h^{*}(i(P)(K_{\mathscr{G}}+D))+\sum
a^{\prime}_{j}E_{j}$

and $a^{\prime}_{j}>0$ for all $j$. Besides, $p$ is a klt singularity for $Y$.
So we have $K_{Y^{m}}=h^{*}K_{Y}+\sum a^{\prime\prime}_{j}E_{j}$ where
$a^{\prime\prime}_{j}>-1$. Thus, we have
$a_{j}=6i(P)a^{\prime}_{j}+a^{\prime\prime}_{j}>-1$.

  2. (2)

If $p$ is a non-terminal foliation singularity, then $p$ is a canonical
singularity of $Y$, $K_{Y^{m}}=h^{*}K_{Y}$, and
$K_{{\mathscr{G}}^{m}}+h_{*}^{-1}D=h^{*}(K_{\mathscr{G}}+D)$. Hence $a_{j}=0$.

Now for general $f$, $f$ factors through the minimal resolution $h$. Put
$f=h\circ g$ where $g:Z\rightarrow Y^{m}$ is a sequence of blowups. Since
$Y^{m}$ is smooth, all discrepancy contributions from $K_{Y^{m}}$ are positive
integers. Moreover, the discrepancy contributions from
$K_{{\mathscr{G}}^{m}}+h_{*}^{-1}D$ are non-negative over all singularities
except for the non-reduced singularities. However, by Lemma 1.33 and Remark
1.34, those discrepancies are at least $-1$ and the corresponding exceptional
divisors are non-invariant. Therefore, $a_{j}\geq-1$ for all $j$. ∎

######  Lemma 5.4.

Fix a function $P:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}$. Let
$f:({\mathcal{X}},{\mathcal{F}},\Delta)\rightarrow T$ be an
${\mathcal{S}}_{P}$-family where $T$ is a smooth affine curve. Suppose there
is a birational morphism $g:{\mathcal{Y}}\rightarrow{\mathcal{X}}$ such that
$f\circ g:{\mathcal{Y}}\rightarrow T$ is a semi-stable reduction of
$({\mathcal{X}},{\mathcal{F}},\Delta)$. Let ${\mathcal{G}}$ be the pullback
foliation of ${\mathcal{F}}$ and $\Theta$ be the proper transform of $\Delta$.
We may write

$K_{\mathcal{Y}}+6i(P)(K_{\mathcal{G}}+\Theta)=g^{*}\big{(}K_{\mathcal{X}}+6i(P)(K_{\mathcal{F}}+\Delta)\big{)}+\sum
a_{i}E_{i}+\sum b_{j}F_{j}$

where $E_{i}$ are all exceptional divisors mapping to $0\in T$ and $F_{j}$ are
all exceptional divisors flat over $T$. Then $a_{i}\geq 0$ for all $i$ and
$b_{j}\geq-1$ for all $j$.

######  Proof.

Restriction the equality to the general element $t\in T$, not $0$, we have

$K_{{\mathcal{Y}}_{t}}+6i(P)(K_{{\mathcal{G}}}+\Theta)|_{{\mathcal{Y}}_{t}}=g_{t}^{*}\big{(}K_{{\mathcal{X}}_{t}}+6i(P)(K_{{\mathcal{F}}_{t}}+\Delta_{t})\big{)}+\sum
b_{j}F_{j}.$

By Lemma 5.3, we have, for all $j$, that $b_{j}\geq-1$.

Moreover, let $X$ be an irreducible component of ${\mathcal{X}}_{0}$ and $Y$
be its proper transform on ${\mathcal{Y}}$. Let
${\mathcal{Y}}_{0}=\bigcup_{i=0}^{r}Y_{i}$ be the decomposition into distinct
irreducible components. By adjunction, we have

$\displaystyle
K_{Y}+6i(P)\left(K_{{\mathcal{G}}_{Y}}+\sum\varepsilon(E_{i}|_{Y})E_{i}|_{Y}+\Theta|_{Y}\right)$
$\displaystyle=$ $\displaystyle K_{Y}+6i(P)(K_{{\mathcal{G}}}+\Theta)|_{Y}$
$\displaystyle=$
$\displaystyle\big{(}K_{{\mathcal{Y}}}+Y+6i(P)(K_{{\mathcal{G}}}+\Theta)\big{)}|_{Y}$
$\displaystyle=$
$\displaystyle\Big{(}K_{{\mathcal{Y}}}-\sum_{i=1}^{r}Y_{i}+6i(P)(K_{{\mathcal{G}}}+\Theta)\Big{)}|_{Y}$
$\displaystyle=$
$\displaystyle\Big{(}g^{*}\big{(}K_{{\mathcal{X}}}+6i(P)(K_{{\mathcal{F}}}+\Delta)\big{)}-\sum_{i=1}^{r}Y_{i}+\sum
a_{i}E_{i}+\sum b_{j}F_{j}\Big{)}|_{Y}$ $\displaystyle=$ $\displaystyle
g_{0}^{*}\big{(}K_{X}+6i(P)(K_{{\mathcal{F}}_{X}}+\Delta|_{X})\big{)}+\sum(a_{i}-1)E_{i}|_{Y}+\sum
b_{j}F_{j}|_{Y}$

where $g_{0}$ is the morphism from $Y$ to $X$. By Lemma 5.3, we have
$a_{i}\geq 0$ for all $i$. ∎

Now we are ready to prove Theorem 5.1.

######  Proof (Theorem 5.1).

Let ${\mathcal{Y}}$ be a common resolution of
$({\mathcal{X}}_{1},\Delta_{1})$, $({\mathcal{X}}_{2},\Delta_{2})$, and
$\varphi^{\circ}$. Let $g_{1}$ and $g_{2}$ be the morphisms fitting into the
following commutative diagram.

$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{g_{2}}$$\textstyle{{\mathcal{X}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{\varphi^{\circ}}$$\textstyle{{\mathcal{X}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{T}$

Note that $g_{1}^{*}{\mathcal{F}}_{1}$ and $g_{2}^{*}{\mathcal{F}}_{2}$ are
isomorphic over $U$, so they are isomorphic at the generic point of
${\mathcal{Y}}$. By [HL21, Lemma 1.8], they are in fact isomorphic. Let
${\mathcal{G}}=g_{1}^{*}{\mathcal{F}}_{1}=g_{2}^{*}{\mathcal{F}}_{2}$.
Moreover, we have the proper transforms $(g_{1*})^{-1}\Delta_{1}$ and
$(g_{2*})^{-1}\Delta_{2}$ are the same. Then we put
$\Theta=(g_{1*})^{-1}\Delta_{1}=(g_{2*})^{-1}\Delta_{2}$.

As in subsection 4.2, we may assume that
$({\mathcal{Y}},{\mathcal{G}},\Theta)\rightarrow T$ is a semi-stable
reduction. Note that ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ may not be
normal after base change.

Suppose first that ${\mathcal{X}}_{1}$ is normal. We write

$K_{{\mathcal{Y}}}+6i(P)(K_{{\mathcal{G}}}+\Theta)=g_{1}^{*}\big{(}K_{{\mathcal{X}}_{1}}+6i(P)(K_{{\mathcal{F}}_{1}}+\Delta_{1})\big{)}+\sum
a_{i}E_{i}+\sum b_{j}F_{j}$

where $E_{i}$ are exceptional divisors mapping to $0\in T$ and $F_{j}$ are
flat over $T$.

Now, by Lemma 5.4, $a_{i}\geq 0$, and $b_{j}\geq-1$. Also, by adjunction
formula, we notice that

$\displaystyle(g_{1})_{*}\big{(}K_{{\mathcal{Y}}}+6i(P)(K_{{\mathcal{G}}}+\Theta)+\sum
F_{j}\big{)}$ $\displaystyle=$
$\displaystyle(g_{1})_{*}\left(g_{1}^{*}\big{(}K_{{\mathcal{X}}_{1}}+6i(P)(K_{{\mathcal{F}}_{1}}+\Delta_{1})\big{)}+\sum
a_{i}E_{i}+\sum(b_{j}+1)F_{j}\right)$ $\displaystyle=$ $\displaystyle
K_{{\mathcal{X}}_{1}}+6i(P)(K_{{\mathcal{F}}_{1}}+\Delta_{1}).$

Hence, we have

$\displaystyle{\mathcal{X}}_{1}$ $\displaystyle\cong$ $\displaystyle\mbox{{\rm
Proj \\!}}\bigoplus_{m\geq
0}H^{0}({\mathcal{X}}_{1},m(K_{{\mathcal{X}}_{1}}+6i(P)(K_{{\mathcal{F}}_{1}}+\Delta_{1})))$
$\displaystyle=$ $\displaystyle\mbox{{\rm Proj \\!}}\bigoplus_{m\geq
0}H^{0}\left({\mathcal{Y}},m\left(K_{{\mathcal{Y}}}+6i(P)(K_{{\mathcal{G}}}+\Theta)+\sum
F_{j}\right)\right).$

If ${\mathcal{X}}_{1}$ is not normal, then using the same argument as above,
we have

${\mathcal{X}}_{1}^{\nu}\cong\mbox{{\rm Proj \\!}}\bigoplus_{m\geq
0}H^{0}\left({\mathcal{Y}},m\left(K_{{\mathcal{Y}}}+6i(P)(K_{{\mathcal{G}}}+\Theta)+\sum
F_{j}\right)\right).$

Note that the conductor of ${\mathcal{X}}_{1}^{\nu}$ is in the support of
$\sum F_{j}$. Thus we have

${\mathcal{X}}_{1}^{\nu}\cong\mbox{{\rm Proj \\!}}\bigoplus_{m\geq
0}H^{0}\left({\mathcal{Y}},m\left(K_{{\mathcal{Y}}}+6i(P)(K_{{\mathcal{G}}}+\Theta)+\sum
F_{j}\right)\right)\cong{\mathcal{X}}_{2}^{\nu}.$

Therefore ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are isomorphic in
codimension 1 where ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ here are the
original one (before taking a base change). Since they both satisfy Serre’s
condition $S_{2}$, they are, in fact, isomorphic.

Now we put ${\mathcal{X}}_{1}\cong{\mathcal{X}}_{2}\cong{\mathcal{X}}$. Notice
that ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$ are isomorphic at the generic
point of ${\mathcal{X}}$. So they are indeed isomorphic. Moreover,
$\Delta_{1}$ and $\Delta_{2}$ are isomorphic. ∎

##  6\. local-closedness for ${\mathcal{S}}_{P}$-families

In this section, we will show in Theorem 6.4 the local-closedness for
${\mathcal{S}}_{P}$-families. (See Definition 2.8.)

######  Lemma 6.1.

Let $(X,{\mathscr{F}})$ be a foliated surface with only semi-reduced
singularities. Given a Hirzebruch-Jung string $\bigcup_{i=1}^{s}E_{i}$ of
$(-2)$-${\mathscr{F}}$-curves. Assume that there are at most two foliation
singularities on each $E_{i}$. Then, after re-indexing, we have that $E_{1}$
has one or two singularities and $E_{i}$ has exactly two singularities for
$i=2,\ldots,s$. Moreover, let $p_{i-1}$ and $p_{i}$ be the singularities on
$E_{i}$ for $i=2,\ldots,s$ where $p_{i}$ is the intersection of $E_{i}$ and
$E_{i+1}$ for $i=2,\ldots,s-1$. (See Figure 1.) Then we have the following two
cases:

  1. (1)

If $E_{1}$ has only one singularity $p_{1}$, then $p_{1}$ is a saddle-node
and, for $i\geq 2$, $p_{i}$ is a reduced and non-degenerate singularity with
eigenvalue uniquely determined by $E_{2}^{2},\ldots,E_{s}^{2}$.

  2. (2)

If $E_{1}$ has two singularities $p_{0}$ and $p_{1}$, then there is at most
one $i\in\\{0,\ldots,s\\}$ such that $p_{i}$ is not reduced. Moreover, if
$\textnormal{Z}({\mathscr{F}},E_{i},p_{i})$ (resp.
$\textnormal{Z}({\mathscr{F}},E_{i+1},p_{i})$) is non-positive, then $p_{i-1}$
(resp. $p_{i+1}$) is a saddle-node.

$E_{1}$$p_{0}$$p_{1}$$E_{2}$$p_{2}$$E_{3}$$p_{3}$$E_{4}$$\ldots$$E_{s-1}$$p_{s-1}$$E_{s}$$p_{s}$
Figure 1. Chain of $(-2)$-${\mathscr{F}}$-curves

######  Proof.

  1. (1)

If $E_{1}$ has only one singularity $p_{1}$, we have that

$\textnormal{Z}({\mathscr{F}},E_{1},p_{1})=\textnormal{Z}({\mathscr{F}},E_{1})=2.$

Thus by Lemma 1.26, we know that $p_{1}$ is a saddle-node,
$\textnormal{Z}({\mathscr{F}},E_{2},p_{1})=1$, and
$\textnormal{CS}({\mathscr{F}},E_{2},p_{1})=0$. By Camacho-Sad formula
(Theorem 1.25), we have that

$\lambda_{2}:=\textnormal{CS}({\mathscr{F}},E_{2},p_{2})=E_{2}^{2}-\textnormal{CS}({\mathscr{F}},E_{2},p_{1})=E_{2}^{2}\leq-2.$

Hence $p_{2}$ is reduced and non-degenerate.

######  Claim.

For all $i=2,\ldots,s$,
$\lambda_{i}:=\textnormal{CS}({\mathscr{F}},E_{i},p_{i})$ are negative
rational numbers less than $-1$ and $p_{i}$ are reduced non-degenerate
singularities.

We have seen that the claim holds for $i=2$. Now we assume that $p_{i}$’s are
reduced and non-degenerate with $\lambda_{i}<-1$ for $i=2,\ldots,k-1$. Then we
have that $\textnormal{Z}({\mathscr{F}},E_{k},p_{k-1})=1$ and
$\textnormal{Z}({\mathscr{F}},E_{k},p_{k})=1$. By Camacho-Sad formula (Theorem
1.25) and Lemma 1.26, we have that

$\displaystyle\lambda_{k}$ $\displaystyle=$ $\displaystyle
E_{k}^{2}-\textnormal{CS}({\mathscr{F}},E_{k},p_{k-1})$ $\displaystyle=$
$\displaystyle E_{k}^{2}-\frac{1}{\lambda_{k-1}}<-2+1=-1.$

Therefore, by induction, we complete the proof of the claim and of the first
case.

  2. (2)

Now we assume that all $E_{i}$’s have two foliation singularities. Suppose
there are two non-reduced singularities $p_{i}$ and $p_{j}$. As the proof of
Lemma 1.33, after a sequence of blowups, we get a string with two separatrices
at each singularity, which is impossible by Theorem 1.19.

Moreover, if $\textnormal{Z}({\mathscr{F}},E_{i},p_{i})\leq 0$, then
$\textnormal{Z}({\mathscr{F}},E_{i},p_{i-1})=2-\textnormal{Z}({\mathscr{F}},E_{i},p_{i})\geq
2$. So, by Lemma 1.26, we have that $p_{i-1}$ is a saddle-node.

∎

######  Lemma 6.2.

Let $(X,{\mathscr{F}})$ be a foliated surface with only semi-reduced
singularities. Given a cycle $E=\bigcup_{i=1}^{s}E_{i}$ of
$(-2)$-${\mathscr{F}}$-curves where $s\geq 2$. Then all singularities on $E$
are reduced and non-degenerate.

######  Proof.

  1. (1)

If there is only one non-reduced singularity $p$, then we can (virtually)
break the cycle into one Hirzebruch-Jung string of
$(-2)$-${\mathscr{F}}$-curves with 2 non-reduced singularities. Thus by Lemma
6.1, it is impossible.

  2. (2)

If there are more than two non-reduced singularities, then we can take any
Hirzebruch-Jung string of $(-2)$-${\mathscr{F}}$-curves such that both
endpoints are non-reduced. Then by Lemma 6.1, it is impossible.

  3. (3)

Now we know that all singularities on $E$ are reduced. Since all $E_{i}$’s are
$(-2)$-${\mathscr{F}}$-curves, we have that all singularities on $E$ are non-
degenerate by Lemma 1.26.

∎

######  Lemma 6.3.

Let $\pi:({\mathcal{X}},{\mathcal{F}})\rightarrow T$ be a smooth family such
that

  1. (1)

each irreducible component of foliation singularities is finite over $T$ and
pairwise disjoint, and

  2. (2)

${\mathcal{F}}_{t}$ has at worst semi-reduced singularities for all $t\in T$.

Given any irreducible component $W$ of singularities of ${\mathcal{F}}$, we
can define a holomorphic function $\lambda:W\rightarrow{\mathbb{P}}^{1}$ such
that $\lambda(w)$ is the eigenvalue of ${\mathcal{F}}_{\pi(w)}$ at $w$.

######  Proof.

For any $w\in W$, we can pick an étale open subset ${\mathcal{U}}$ of
${\mathcal{X}}$ containing $w$ such that $(x,y,t)$ are local coordinates of
${\mathcal{U}}$ with $\pi(x,y,t)=t$ and $W\cap{\mathcal{U}}=(x=y=0)$.

Then, on ${\mathcal{U}}$, the foliation is given by

$(a(t)x+b(t)y+\textnormal{h.o.t.})\frac{\partial}{\partial
x}+(c(t)x+d(t)y+\textnormal{h.o.t.})\frac{\partial}{\partial y}$

where $a$, $b$, $c$, and $d$ are holomorphic functions from $\pi(V)$ to
${\mathbb{C}}$ and h.o.t stands for higher order terms (in $x$, $y$, and $t$).
Thus we form a matrix

$A(t):=\left(\begin{matrix}a(t)&b(t)\\\ c(t)&d(t)\end{matrix}\right)$

which has two eigenvalues $\mu_{+}(t)$ and $\mu_{-}(t)$. Since all
singularities are semi-reduced, at least one of two eigenvalues is not zero.
Therefore, we can define a holomorphic map
$\lambda:W\cap{\mathcal{U}}\rightarrow{\mathbb{P}}^{1}$ by
$\lambda(w)=[\mu_{+}(\pi(w)):\mu_{-}(\pi(w))]$. Notice that this map is
independent of the open subset ${\mathcal{U}}$ and of change of coordinates.
Thus, $\lambda$ can be defined on $W$ and is holomorphic. ∎

######  Theorem 6.4.

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

######  Proof.

We divide the proof into several steps.

  1. (1)

Let $g:{\mathcal{Y}}\rightarrow{\mathcal{X}}$ be a resolution of
singularities, ${\mathcal{G}}=g^{*}{\mathcal{F}}$ be the pullback foliation,
and $\Theta$ be the proper transform of $\Delta$. Let $\pi=f\circ g$. By
Proposition 1.42, replacing ${\mathcal{Y}}$ with its higher model (via some
blowups), we may assume that ${\mathcal{G}}_{t}$ has at worst semi-reduced
singularities for all $t\in U$ for some open subset $U\subset T$.

  2. (2)

Claim. After shrinking $U$ further if necessary, the following conditions
hold:

    1. (a)

$\pi_{U}$ is smooth.

    2. (b)

Each irreducible component of singularities of ${\mathcal{G}}$ is finite over
$U$ and pairwise disjoint.

    3. (c)

Each irreducible component of exceptional divisors of $g$ is flat over $U$.

    4. (d)

For any irreducible component $C$ of singularities of ${\mathcal{G}}$ and any
irreducible component $E$ of exceptional divisors of $g$, we have $C\cap E$ is
either $C$ or empty.

    5. (e)

For each irreducible component $E$ of exceptional divisors of $g$, either
$E_{u}$ is invariant for all $u\in U$ or $E_{u}$ is not invariant for all
$u\in U$.

######  Proof.

More explicitly, (a) comes from the generic smoothness. For others, let

$S_{1}=\bigcup\pi(C)$

where the union is taken over all irreducible components $C$ of singularities
of ${\mathcal{G}}$ and of exceptional divisors of $g$, which are _not_ finite
over $U$. Let

$S_{2}=\bigcup\\{t\,|\,C_{i,t}=C_{j,t}\\}$

where the union is taken over all distinct irreducible components $C_{i}$ and
$C_{j}$ of singularities of ${\mathcal{G}}$. Let

$S_{3}=\bigcup\pi(C\cap E)$

where the union is taken over all irreducible components $C$ of singularity of
${\mathcal{G}}$ and all irreducible components $E$ of exceptional divisors of
$g$ such that $C\cap E$ is neither $C$ nor empty.

Note that $S_{1}$, $S_{2}$, and $S_{3}$ are all proper closed subsets of $U$.
So by replacing $U$ by $U\backslash(S_{1}\cup S_{2}\cup S_{3})$, conditions
(a)-(d) hold. The condition (e) can be achieved since being non-invariant is
an open condition.

  3. (3)

The conditions (a)-(d) will ensure that the number of foliation singularities
on an exceptional prime divisor of $g$ is constant over $U$. Furthermore, all
singularities of ${\mathcal{G}}$ do not collide over $U$. Also there is a
$t_{0}\in T^{\prime}\cap U$ since $T^{\prime}$ is dense in $T$.

  4. (4)

We say an irreducible component $E$ of exceptional divisors a _relative
${\mathcal{G}}$-exceptional curve_ if $E_{t_{0}}$ is a
${\mathcal{G}}_{t_{0}}$-exceptional curve. Then, for any $u\in U$, we have
that $E_{u}$ is invariant, and the contraction of $E_{u}$ introduces either a
smooth or a semi-reduced singularity.

Contracting all relative ${\mathcal{G}}$-exceptional curves and by induction
on Picard number, we may assume that there is no relative
${\mathcal{G}}$-exceptional curve.

  5. (5)

Now we say $E=\bigcup_{i=1}^{s}E_{i}$ is a _relative ${\mathcal{G}}$-chain_ if
$E_{t_{0}}=\bigcup_{i=1}^{s}E_{i,t_{0}}$ is a ${\mathcal{G}}_{t_{0}}$-chain
and $E_{i,t_{0}}\cdot\Theta_{t_{0}}=0$ for $i=1,\ldots,s$. (See Figure 2.)

$p_{1}$$p_{2}$$p_{3}$$\ldots$$p_{s-1}$$p_{s}$ Figure 2. ${\mathscr{F}}$-chain

Since, over $t_{0}$, there is only one singularity $p_{1,t_{0}}$ on
$E_{1,t_{0}}$, we have that, over any $u\in U$, there is also only one
singularity $p_{1,u}$ on $E_{1,u}$.

By Camacho-Sad formula (Theorem 1.25), we have
$\mbox{CS}({\mathcal{G}}_{u},E_{1,u},p_{1,u})=E_{1,u}^{2}\leq-2$. And by Lemma
1.26, the eigenvalue
$\lambda_{1}=\mbox{CS}({\mathcal{G}}_{u},E_{1,u},p_{1,u})$ must be a negative
integer less than $-2$. Thus, $p_{1,u}$ is a reduced non-degenerate
singularity.

Let the singularities on $E_{k,u}$ be $p_{k-1,u}$ and $p_{k,u}$ for
$k=2,\ldots,s$.

######  Claim.

For all $k=1,\ldots,s$,
$\lambda_{k}:=\textnormal{CS}({\mathcal{G}}_{u},E_{k,u},p_{k,u})$ are negative
rational numbers less than $-1$ and $p_{k,u}$ are reduced non-degenerate
singularities.

######  Proof.

We have shown that this holds when $k=1$. Assume the claim holds for $k-1$. By
Camacho-Sad formula (Theorem 1.25) and Lemma 1.26, we have that

$-2\geq
E_{k,u}^{2}=\mbox{CS}({\mathcal{G}}_{u},E_{k,u},p_{k-1,u})+\mbox{CS}({\mathcal{G}}_{u},E_{k,u},p_{k,u})=\frac{1}{\lambda_{k-1}}+\lambda_{k}.$

Thus, $\lambda_{k}\leq-2-\frac{1}{\lambda_{k-1}}<-2+1=-1$. Hence,
$\lambda_{k}$ is a negative rational number less than $-1$. Therefore,
$p_{k,u}$ is also a reduced non-degenerate singularity. This completes the
proof of the claim.

  6. (6)

We contract all relative $({\mathcal{G}},\Theta)$-chains and still using the
notation $({\mathcal{Y}},{\mathcal{G}},\Theta)$ for the resulting foliation.
Then we have a family such that

$(K_{\mathcal{G}}+\Theta)|_{{\mathcal{Y}}_{t}}$

is nef and big for all $t\in T^{\prime}\cap U$. By Lemma 2.12, after shrinking
$U$ further, we have $(K_{\mathcal{G}}+\Theta)|_{{\mathcal{Y}}_{u}}$ is nef
and big for all $u\in U$.

Next, in order to show that
$({\mathcal{Y}},{\mathcal{G}},\Theta)_{t}\in{\mathcal{S}}_{P}$, we will study
the _relative $(K_{\mathcal{G}}+\Theta)$-trivial curves_, that is, the
irreducible component $E$ of exceptional divisors such that $E_{t_{0}}$ is
$(K_{{\mathcal{G}}_{t_{0}}}+\Theta_{t_{0}})$-trivial.

  7. (7)

Let $E=E_{0,1}\cup E_{0,2}\cup\bigcup_{i=1}^{s}E_{i}$ be the exceptional
divisor over a _dihedral singularity_ where, for all $u\in U$,
$\bigcup_{i=1}^{s}E_{i,u}$ is a Hirzebruch-Jung string, and $E_{0,1,u}$ and
$E_{0,2,u}$ are disjoint, of self-intersection $-2$ and both intersect the bad
tail $E_{1,u}$. (See the Figure 3 for its dual graph.)

$E_{0,1}$$E_{1}$$E_{2}$$E_{0,2}$$\ldots$$E_{s-1}$$E_{s}$ Figure 3. Dual graph
of a $D_{n}$ type singularity

Let $p_{0,1,u},p_{0,2,u},$ and $p_{1,u}$ be three singularities on $E_{1,u}$.
Let $p_{k-1,u}$ and $p_{k,u}$ be two singularities on $E_{k,u}$ for
$k=2,\ldots,s$. (See the Figure 4.)

$p_{0}$$p_{0,1}$$p_{0,2}$$p_{1}$$p_{2}$$p_{3}$……$p_{s-1}$$p_{s}$ Figure 4.
$D_{n}$ type singularity

By Camacho-Sad formula (Theorem 1.25), we have, for any $u\in U$, that

$-2=E_{0,1,u}^{2}=\textnormal{CS}({\mathcal{G}}_{u},E_{0,1,u},p_{0,1,u})\mbox{
and }-2=E_{0,2,u}^{2}=\textnormal{CS}({\mathcal{G}}_{u},E_{0,2,u},p_{0,2,u}).$

So $p_{0,1,u}$ and $p_{0,2,u}$ are reduced. Thus,

$\displaystyle E_{1,u}^{2}$ $\displaystyle=$
$\displaystyle\textnormal{CS}({\mathcal{G}}_{u},E_{1,u},p_{1,u})+\textnormal{CS}({\mathcal{G}}_{u},E_{1,u},p_{0,1,u})+\textnormal{CS}({\mathcal{G}}_{u},E_{1,u},p_{0,2,u})$
$\displaystyle=$
$\displaystyle\textnormal{CS}({\mathcal{G}}_{u},E_{1,u},p_{1,u})-\frac{1}{2}-\frac{1}{2}.$

So
$\lambda_{1}:=\textnormal{CS}({\mathcal{G}}_{u},E_{1,u},p_{1,u})=E_{1,u}^{2}+1\leq-1$

######  Claim.

For all $k=1,\ldots,s$, $\lambda_{k}:=\mbox{{\rm
CS}}({\mathcal{G}}_{u},E_{k,u},p_{k,u})$ are negative rational numbers at most
$-1$ and $p_{k,u}$ are reduced non-degenerate singularities.

######  Proof.

We have shown that this holds when $k=1$. Assume the claim holds for $k-1$.

$-2\geq
E_{k,u}^{2}=\mbox{CS}({\mathcal{G}}_{u},E_{k,u},p_{k-1,u})+\mbox{CS}({\mathcal{G}}_{u},E_{k,u},p_{k,u})=\frac{1}{\lambda_{k-1}}+\lambda_{k}.$

Thus, $\lambda_{k}\leq-2-\frac{1}{\lambda_{k-1}}\leq-2+1=-1$. Hence,
$\lambda_{k}$ is a negative rational number at most $-1$. Therefore, $p_{k,u}$
is also a reduced non-degenerate singularity. This completes the proof of the
claim.

So the contraction of $E_{u}$ introduces a dihedral singularity.

  8. (8)

Let $E=\bigcup_{i=1}^{s}E_{i}$ be a connected component of exceptional
divisors such that $E_{t_{0}}=\bigcup_{i=1}^{s}E_{i,t_{0}}$ is an _elliptic
Gorenstein leaf_. Then $E_{u}=\bigcup_{i=1}^{s}E_{i,u}$ is also an elliptic
Gorenstein leaf for all $u\in U$. By Lemma 6.2, all singularities on $E_{u}$
are reduced and non-degenerate. Therefore, the contraction of $E_{u}$
introduces a cusp singularity.

  9. (9)

(Cyclic non-terminal canonical foliation singularities) Let
$E=\bigcup_{i=1}^{s}E_{i}$ be a connected component of exceptional divisors
such that the contraction of $E_{t_{0}}=\bigcup_{i=1}^{s}E_{i,t_{0}}$ gives a
_cyclic non-terminal_ foliation singularity such that $E_{i,t_{0}}$ are
_invariant_ for all $i$.

Then $E_{t_{0}}$ is a chain of $(-2)$-${\mathcal{G}}_{t_{0}}$-curves. Thus
$E_{u}$ is also a chain of $(-2)$-${\mathcal{G}}_{u}$-curves for $u\in U$. By
Lemma 6.1, we have the following two cases.

    1. (a)

In the first case, all singularities are reduced. So the contraction of
$E_{u}$ introduces a canonical foliation singularity.

    2. (b)

In the second case, if all $p_{i,u}$ are reduced (see Figure 5), then the
contraction of $E_{u}$ introduces a canonical foliation singularity as well.

$p_{0}$$p_{1}$$p_{2}$$p_{3}$$\ldots$$p_{s-1}$$p_{s}$ Figure 5. $A_{n}$ type
singularity

######  Claim.

If there is an $i$ such that $p_{i,u}$ is not reduced, then all $p_{j,u}$ are
non-degenerate for $j\neq i$.

######  Proof.

Suppose not, then one of $p_{i-1,u}$ and $p_{i+1,u}$ is a saddle-node. Without
loss of generality, we assume that $p_{i+1,u}$ is a saddle-node and

$\textnormal{Z}({\mathcal{G}}_{u},E_{i+1,u},p_{i,u})\leq 0.$

Let $W_{i}$ be the irreducible component of the singularities of
${\mathcal{G}}$ containing $p_{i,u}$. By Lemma 6.3, we have a holomorphic map
$\lambda_{i}:W_{i}\rightarrow{\mathbb{P}}^{1}$. Note that $W_{i}$ is connected
and locally path-connected (in the analytic topology). We have that $W_{i}$ is
path-connected. Note that the claim holds true when $\lambda$ is a constant
function. So we now assume that $\lambda$ is not a constant function.

There is a sequence $\\{w_{j}\\}_{j=1}^{\infty}$ converging to $p_{i,u}$ such
that $\lambda_{i}(w_{j})$ is not a positive rational and converges to
$\lambda(p_{i,u})$. For any $w_{j}$ close to $p_{i,u}$, there are only two
separatrices, one of which is $E_{i+1,\pi(w_{j})}$. So we may assume
$E_{i+1,\pi(w_{j})}$ converges to $E_{i+1,u}$. Thus
$\textnormal{Z}({\mathcal{G}}_{u},E_{i+1,u},p_{i,u})=1$, which gives a
contradiction.

This completes the proof of the claim.

Therefore, the contraction of $E_{u}$ introduces a mild log canonical
foliation singularity.

  10. (10)

Let $E=\bigcup_{i=1}^{s}E_{i}$ be a connected component of exceptional
divisors such that the contraction of $E_{t_{0}}=\bigcup_{i=1}^{s}E_{i,t_{0}}$
gives a _cyclic non-canonical foliation singularity_. So there is a unique
$j\in\\{1,\ldots,s\\}$ such that $E_{j,t_{0}}$ is _non-invariant_. Since being
non-invariant is an open condition, by shrinking $U$ further, we may assume
that $E_{u,t_{0}}$ is non-invariant for all $u\in U$.

By (5), when $u$ varies, we know that $\bigcup_{i=1}^{j-1}E_{i,u}$ and
$\bigcup_{i=j+1}^{s}E_{i,u}$ have the same structures, respectively.
Therefore, the contraction of $E_{u}$ introduces a _mild log canonical_
foliation singularities.

  11. (11)

Fix any $u\in U$, we may contract all $E_{u}$ where $E$ is any _relative
$(K_{\mathcal{G}}+\Theta)$-trivial curve_. It will end up with a _foliated
surface_ $(Z,{\mathscr{H}})$ with

$K_{{\mathcal{G}}_{u}}+\Theta_{u}=\pi^{*}K_{\mathscr{H}}$

where $\pi:Y\rightarrow Z$ is the contraction morphism.

######  Claim.

There is no $K_{{\mathscr{H}}}$-trivial curve $C$ with $C^{2}<0$.

######  Proof.

Suppose $C$ is a $K_{\mathscr{H}}$-trivial curve with $C^{2}<0$. Then the
proper transform $\widetilde{C}$ on $Y$ is a
$(K_{{\mathcal{G}}_{u}}+\Theta_{u})$-trivial curve. Note that
$\widetilde{C}^{2}\leq C^{2}<0$. Also, by Corollary 2.13, this $\widetilde{C}$
moves in a family $\mathscr{C}$ with $\mathscr{C}_{u}=\widetilde{C}$. However,
$\mathscr{C}_{t_{0}}^{2}=\mathscr{C}_{u}^{2}=\widetilde{C}^{2}<0$ and
$(K_{{\mathcal{G}}_{t_{0}}}+\Theta_{t_{0}})\cdot\mathscr{C}_{t_{0}}=(K_{{\mathcal{G}}_{u}}+\Theta_{u})\cdot\mathscr{C}_{u}=K_{\mathscr{H}}\cdot
C=0$. Therefore, _$\mathscr{C}$ is a relative
$(K_{\mathcal{G}}+\Theta)$-trivial curve_. Hence,
$\widetilde{C}=\mathscr{C}_{u}$ is contracted by $\pi$, which gives a
contradiction. This completes the proof of the claim.

Therefore, $(Z,{\mathscr{H}})$ is a generalized canonical model and hence
$({\mathcal{X}},{\mathcal{F}},\Delta)_{u}\in{\mathcal{S}}_{P}$ for any $u\in
U$.

∎

## References

  * [Ayo] J. Ayoub, _Topologie feuilletée et théorie de galois différentielle_ , Preprint available at http://user.math.uzh.ch/ayoub/PDF-Files/Feui-DGal.pdf, Accessed: 2021-03-15.
  * [Bri68] E. Brieskorn, _Rationale Singularitäten komplexer Flächen_ , Invent. Math. 4 (1967/68), 336–358. MR 222084
  * [Bru97] M. Brunella, _Feuilletages holomorphes sur les surfaces complexes compactes_ , Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 569–594. MR 1474805
  * [Bru01] by same author, _Invariance par déformations de la dimension de Kodaira d’un feuilletage sur une surface_ , Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 113–132. MR 1929324
  * [Bru15] by same author, _Birational geometry of foliations_ , IMPA Monographs, vol. 1, Springer, Cham, 2015. MR 3328860
  * [Cam88] C. Camacho, _Quadratic forms and holomorphic foliations on singular surfaces_ , Math. Ann. 282 (1988), no. 2, 177–184. MR 963011
  * [CF18] P. Cascini and E. Floris, _On invariance of plurigenera for foliations on surfaces_ , J. Reine Angew. Math. 744 (2018), 201–236. MR 3871444
  * [Che19] Y.-A. Chen, _Boundedness of minimal partial du Val resolutions of canonical surface foliations_ , arXiv preprint arXiv:1912.02931 (2019).
  * [CR14] F. Cano and C. Roche, _Vector fields tangent to foliations and blow-ups_ , J. Singul. 9 (2014), 43–49. MR 3249046
  * [CS18] P. Cascini and C. Spicer, _MMP for co-rank one foliation on threefolds_ , arXiv preprint arXiv:1808.02711 (2018).
  * [Fuj12] O. Fujino, _Minimal model theory for log surfaces_ , Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371. MR 2928144
  * [Gro66] A. Grothendieck, _Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III_ , Inst. Hautes Études Sci. Publ. Math. (1966), no. 28, 255. MR 217086
  * [Har77] R. Hartshorne, _Algebraic geometry_ , Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No. 52. MR 0463157
  * [HK10] C. D. Hacon and S. J. Kovács, _Classification of higher dimensional algebraic varieties_ , Oberwolfach Seminars, vol. 41, Birkhäuser Verlag, Basel, 2010. MR 2675555
  * [HL21] C. D. Hacon and A. Langer, _On birational boundedness of foliated surfaces_ , J. Reine Angew. Math. 770 (2021), 205–229. MR 4193468
  * [KKMSD73] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, _Toroidal embeddings. I_ , Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
  * [KM98] J. Kollár and S. Mori, _Birational geometry of algebraic varieties_ , Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR 1658959
  * [Lan00] A. Langer, _Chern classes of reflexive sheaves on normal surfaces_ , Math. Z. 235 (2000), no. 3, 591–614. MR 1800214
  * [Lan01] by same author, _Adjoint linear systems on normal log surfaces_ , Compositio Math. 129 (2001), no. 1, 47–66. MR 1856022
  * [Laz04] R. Lazarsfeld, _Positivity in algebraic geometry. I_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004, Classical setting: line bundles and linear series. MR 2095471
  * [McQ08] M. McQuillan, _Canonical models of foliations_ , Pure Appl. Math. Q. 4 (2008), no. 3, Special Issue: In honor of Fedor Bogomolov. Part 2, 877–1012. MR 2435846
  * [MM80] J.-F. Mattei and R. Moussu, _Holonomie et intégrales premières_ , Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 469–523. MR 608290
  * [PS19] J. V. Pereira and R. Svaldi, _Effective algebraic integration in bounded genus_ , Algebr. Geom. 6 (2019), no. 4, 454–485. MR 3957403
  * [Rei] M. Reid, _Surface cyclic quotient singularities and Hirzebruch-Jung resolutions_ , manuscript available at http://homepages.warwick.ac.uk/~masda/surf/more/cyclic.pdf, Accessed: 2021-03-15.
  * [Rei87] by same author, _Young person’s guide to canonical singularities_ , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
  * [Roc70] R. T. Rockafellar, _Convex analysis_ , Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  * [Sak84] F. Sakai, _Weil divisors on normal surfaces_ , Duke Math. J. 51 (1984), no. 4, 877–887. MR 771385
  * [Seb97] M. Sebastiani, _Sur l’existence de séparatrices locales des feuilletages des surfaces_ , An. Acad. Brasil. Ciênc. 69 (1997), no. 2, 159–162. MR 1754036
  * [Sei68] A. Seidenberg, _Reduction of singularities of the differential equation $A\,dy=B\,dx$_, Amer. J. Math. 90 (1968), 248–269. MR 220710
  * [Sim94] C. T. Simpson, _Moduli of representations of the fundamental group of a smooth projective variety. I_ , Inst. Hautes Études Sci. Publ. Math. (1994), no. 79, 47–129. MR 1307297
  * [Spi20] C. Spicer, _Higher-dimensional foliated Mori theory_ , Compos. Math. 156 (2020), no. 1, 1–38. MR 4036447


# On the maximum of the weighted
binomial sum $2^{-r}\sum_{i=0}^{r}\binom{m}{i}$

S. P. Glasby  and  G. R. Paseman  —S. P. Glasby, Centre for the Mathematics of
Symmetry and Computation, University of Western Australia, 35 Stirling
Highway, Perth 6009, Australia.
_E-mail address:_<EMAIL_ADDRESS>R. Paseman. _E-mail address:_
<EMAIL_ADDRESS>
###### Abstract.

The weighted binomial sum $f_{m}(r)=2^{-r}\sum_{i=0}^{r}\binom{m}{i}$ arises
in coding theory and information theory. We prove that, for
$m\not\in\\{0,3,6,9,12\\}$, the maximum value of $f_{m}(r)$ with $0\leqslant
r\leqslant m$ occurs when $r=\lfloor m/3\rfloor+1$. We also show this maximum
value is asymptotic to $\frac{3}{\sqrt{{\pi}m}}\left(\frac{3}{2}\right)^{m}$
as $m\to\infty$.

Acknowledgements: SPG gratefully acknowledges support from the Australian
Research Council (ARC) Discovery Project DP190100450. Keywords: Maximum, sum,
binomial coefficients.
2010 Math Subject Classification: 05A10, 11B65, 94B65. Date: .

##  1\. Introduction

Let $m$ be a non-negative integer, and let $f_{m}(r)$ be the function:

$f_{m}(r)=\frac{1}{2^{r}}\sum_{i=0}^{r}\binom{m}{i}.$

This function arises in coding theory and information theory e.g.
[Ash]*Theorem 4.5.3. It is desirable for a linear code to have large rate (to
communicate a lot of information) and large minimal distance (to correct many
errors). So for a linear code with parameters $[n,k,d]$, one wants both $k/n$
and $d/n$ to be large. The case that $kd/n$ is large is studied in [AGP]. A
Reed-Muller code $\textup{RM}(r,m)$ has $n=2^{m}$,
$k=\sum_{i=0}^{r}\binom{m}{i}$ and $d=2^{m-r}$ by [LX]*§6.2, and hence $kd/n$
equals $f_{m}(r)$. It is natural to ask which value of $r$ maximizes
$f_{m}(r)$, and what is the size of the maximum value.

######  Theorem 1.

Suppose that $m,r$ are integers where $0\leqslant r\leqslant m$. The maximum
value of $f_{m}(r)=2^{-r}\sum_{i=0}^{r}\binom{m}{i}$ occurs when
$r=\lfloor\frac{m}{3}\rfloor+1$ provided $m\not\in\\{0,3,6,9,12\\}$.

We give an optimal asymptotic bound for the maximum value of $f_{m}(r)$.

######  Theorem 2.

Suppose that $m\not\in\\{0,1,3,6,9,12\\}$ and
$r_{0}=\lfloor\frac{m}{3}\rfloor+1$. Then

(1)
$\frac{1}{2^{\lfloor\frac{m}{3}\rfloor}}\left(1-\frac{k+2}{2(r_{0}+1)}\right)\binom{m}{r_{0}}<f_{m}(r_{0})<\frac{1}{2^{\lfloor\frac{m}{3}\rfloor}}\binom{m}{r_{0}}$

where $k:=3r_{0}-m\in\\{1,2,3\\}$. Furthermore,

(2) $f_{m}(r_{0})<\frac{3}{\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{m}\qquad\textup{and}\qquad\lim_{m\to\infty}f_{m}(r_{0})\sqrt{m}\left(\frac{2}{3}\right)^{m}=\frac{3}{\sqrt{\pi}}.$

We prove that $f_{m}(r)$ increases strictly if $0\leqslant r\leqslant
r_{0}:=\lfloor\frac{m}{3}\rfloor+1$ and $m>12$ (see Theorem 6), and it
decreases strictly for $r_{0}\leqslant r\leqslant m$ (see Theorem 8).
Elementary arguments in Lemma 4(c) show that
$f_{m}(0)<f_{m}(1)<\cdots<f_{m}(r_{0}-1)$. More work is required to prove that
$f_{m}(r_{0}-1)<f_{m}(r_{0})$. Determining when $f_{m}(r)$ decreases involves
a delicate inductive proof requiring a growing amount of precision, and
inequalities with rational functions such as $X_{i}=\frac{r-i+1}{m-r+i}$, see
Lemma 5. In Section 5 we establish bounds (and asymptotic behavior) for
$f_{m}(r_{0})$ using standard methods.

Brendan McKay [Gmo] showed, using approximations for sufficiently large $m$,
that the maximum value of $f_{m}(r)$ is near $m/3$. His method may well extend
to a proof of Theorem 1. If so, it would involve very different techniques
from ours.

##  2\. Data, comparisons and strategies

The values of
$f_{m}(0),f_{m}(1),f_{m}(2),\dots,f_{m}(m-2),f_{m}(m-1),f_{m}(m)$ appear to
increase to a maximum and then decrease. For ‘large’ $m$ we see that

$1<\frac{m{+}1}{2}<\frac{m^{2}{+}m{+}2}{8}<\cdots\;?\;\cdots>8-\frac{m^{2}{+}m{+}2}{2^{m-2}}>4-\frac{m+1}{2^{m-2}}>2-\frac{1}{2^{m-1}}>1.$

Computer calculations for ‘large’ $m$ suggest that a maximum value for
$f_{m}(r)$ occurs at $r_{0}=\lfloor\frac{m}{3}\rfloor+1$, see Table 1 which
lists the _integer part_ $\lfloor f_{m}(r)\rfloor$. Computing $f_{m}(r)$
exactly shows that for $m\in\\{0,3,6,9,12\\}$ the maximum occurs at $r_{0}-1$
and not $r_{0}$, see Table 2. The maximum happens to occur for a unique $r$,
except for $m=1$.

Table 1. Maximum values of $\lfloor f_{m}(r)\rfloor$ for $0\leqslant r\leqslant m$ and $m\in\\{6,7,\dots,15\\}$. 6 |  | 1 | 3 | $5$ | $5$ | 3 | 1 | 1 |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
7 |  | 1 | 4 | 7 | $8$ | 6 | 3 | 1 | 1 |  |  |  |  |  |  |  |
8 |  | 1 | 4 | 9 | $11$ | 10 | 6 | 3 | 1 | 1 |  |  |  |  |  |  |
9 |  | 1 | 5 | 11 | $16$ | $16$ | 11 | 7 | 3 | 1 | 1 |  |  |  |  |  |
10 |  | 1 | 5 | 14 | 22 | $24$ | 19 | 13 | 7 | 3 | 1 | 1 |  |  |  |  |
11 |  | 1 | 6 | 16 | 29 | $35$ | 32 | 23 | 14 | 7 | 3 | 1 | 1 |  |  |  |
12 |  | 1 | 6 | 19 | 37 | $49$ | $49$ | 39 | 25 | 14 | 7 | 3 | 1 | 1 |  |  |
13 |  | 1 | 7 | 23 | 47 | 68 | $74$ | 64 | 45 | 27 | 15 | 7 | 3 | 1 | 1 |  |
14 |  | 1 | 7 | 26 | 58 | 91 | $108$ | 101 | 77 | 50 | 29 | 15 | 7 | 3 | 1 | 1 |
15 |  | 1 | 8 | 30 | 72 | 121 | 154 | $155$ | 128 | 89 | 54 | 30 | 15 | 7 | 3 | 1 | 1

Determining the relative sizes of $f_{m}(r)$ and $f_{m}(r+1)$ is reduced in
Lemma 3 to determining the relative sizes of $\sum_{i=0}^{r}\binom{m}{i}$ and
$\binom{m}{r+1}$.

######  Lemma 3.

Suppose that $0\leqslant r<m$. Then

  1. (a)

the inequality $f_{m}(r)<f_{m}(r+1)$ is equivalent to
$\sum_{i=0}^{r}\binom{m}{i}<\binom{m}{r+1}$,

  2. (b)

if $\sum_{i=0}^{r}\binom{m}{i}\leqslant\binom{m}{r+1}$, then
$\sum_{i=0}^{r}\binom{m+1}{i}<\binom{m+1}{r+1}$,

  3. (c)

the inequality $f_{m}(r)>f_{m}(r+1)$ is equivalent to
$\sum_{i=0}^{r}\binom{m}{i}>\binom{m}{r+1}$, and

  4. (d)

if $\sum_{i=0}^{r}\binom{m}{i}\geqslant\binom{m}{r+1}$, then
$\sum_{i=0}^{r}\binom{m-1}{i}>\binom{m-1}{r+1}$.

###### Proof.

(a,b) Clearly $f_{m}(r)<f_{m}(r+1)$ is equivalent to
$2\sum_{i=0}^{r}\binom{m}{i}<\sum_{i=0}^{r+1}\binom{m}{i}$ which is equivalent
to $\sum_{i=0}^{r}\binom{m}{i}<\binom{m}{r+1}$. If $r<m$ and
$\sum_{i=0}^{r}\binom{m}{i}\leqslant\binom{m}{r+1}$, then

$\displaystyle\sum_{i=0}^{r}\binom{m+1}{i}$
$\displaystyle=\sum_{i=0}^{r}\frac{m+1}{m-i+1}\binom{m}{i}\leqslant\frac{m+1}{m-r+1}\sum_{i=0}^{r}\binom{m}{i}$
$\displaystyle<\frac{m+1}{m-r}\binom{m}{r+1}=\binom{m+1}{r+1}.$

(c,d) Clearly $f_{m}(r)>f_{m}(r+1)$ is equivalent to
$2\sum_{i=0}^{r}\binom{m}{i}>\sum_{i=0}^{r+1}\binom{m}{i}$ which, in turn, is
equivalent to $\sum_{i=0}^{r}\binom{m}{i}>\binom{m}{r+1}$. If
$\sum_{i=0}^{r}\binom{m}{i}\geqslant\binom{m}{r+1}$, then as $m>r\geqslant 0$,

$\displaystyle\sum_{i=0}^{r}\binom{m-1}{i}$
$\displaystyle=\sum_{i=0}^{r}\frac{m-i}{m}\binom{m}{i}\geqslant\frac{m-r}{m}\sum_{i=0}^{r}\binom{m}{i}$
$\displaystyle>\frac{m-r-1}{m}\binom{m}{r+1}=\binom{m-1}{r+1}.\qed$

The following easy lemma elucidates which $r\in\\{0,\dots,m\\}$ maximize
$f_{m}(r)$.

######  Lemma 4.

Let $s_{m}(m+1)=2^{m}$, and for $0\leqslant r\leqslant m$ define

$s_{m}(r)=\sum_{i=0}^{r}\binom{m}{r},\qquad
t_{m}(r)=\frac{s_{m}(r+1)}{s_{m}(r)},\quad\textup{and}\quad
c_{m}(r)=\frac{\binom{m}{r+1}}{\binom{m}{r}}=\frac{m-r}{r+1}.$

(a) If $0\leqslant r\leqslant m$, then $c_{m}(r)<t_{m}(r)$, and if $0\leqslant
r<m$, then $t_{m}(r+1)<t_{m}(r)$.
(b) If $m\geqslant 2$, then for some $r^{*}$, $f_{m}(0)<\cdots<f_{m}(r^{*})$
and $f_{m}(r^{*}+1)>\cdots>f_{m}(m)$.
(c)
$\max\\{f_{m}(0),\dots,f_{m}(m)\\}=\max\\{f_{m}(r^{*}),f_{m}(r^{*}{+}1)\\}$
and $f_{m}(0)<\cdots<f_{m}(r_{0}{-}1)$.

###### Proof.

(a) We show $c_{m}(r)<t_{m}(r)$ via induction on $r$. This is true when $r=0$
as $c_{m}(0)=m<m+1=t_{m}(0)$. Suppose that $0\leqslant r<m$ and
$c_{m}(r)<t_{m}(r)$ holds. That is,
$\binom{m}{r+1}/\binom{m}{r}<s_{m}(r+1)/s_{m}(r)$ holds. Since
$c_{m}(r+1)=\frac{m-r-1}{r+2}<\frac{m-r}{r+1}=c_{m}(r)$ we have
$c_{m}(r+1)<c_{m}(r)<t_{m}(r)$. Using properties of mediants, it follows that

$c_{m}(r+1)=\frac{\binom{m}{r+2}}{\binom{m}{r+1}}<\frac{\binom{m}{r+2}+s_{m}(r+1)}{\binom{m}{r+1}+s_{m}(r)}<\frac{s_{m}(r+1)}{s_{m}(r)}=t_{m}(r).$

Hence $c_{m}(r+1)<t_{m}(r+1)<t_{m}(r)$ as
$s_{m}(n+1)=\binom{m}{n+1}+s_{m}(n)$. This completes the induction, and it
also proves that $t_{m}(r+1)<t_{m}(r)$, as claimed.

(b) Since $s_{m}(m+1)=2^{m}$, part (a) shows that
$1=t_{m}(m)<\cdots<t_{m}(0)=m+1$. Choose an integer $r^{*}$ such that
$t_{m}(r^{*})\leqslant 2<t_{m}(r^{*}-1)$. The following are equivalent:
$2<t_{m}(r)$; $2s_{m}(r)<s_{m}(r+1)$; $f_{m}(r)<f_{m}(r+1)$. Thus
$2<t_{m}(r^{*}-1)<\cdots<t_{m}(0)$ implies $f_{m}(0)<\cdots<f_{m}(r^{*})$.
Similarly, $t_{m}(m-1)<\cdots<t_{m}(r^{*}+1)<2$ and $t_{m}(r)<2$ implies
$f_{m}(r+1)<f_{m}(r)$. Hence $f_{m}(r^{*}+1)>\cdots>f_{m}(m)$.

(c) By part (b), $\max\\{f_{m}(r)\mid 0\leqslant r\leqslant
m\\}=\max\\{f_{m}(r^{*}),f_{m}(r^{*}+1)\\}$. If $2\leqslant
c_{m}(r)=\frac{m-r}{r+1}$, then $3r+2\leqslant m$ and
$r\leqslant\lfloor\frac{m-2}{3}\rfloor$. Hence $2\leqslant c_{m}(r)<t_{m}(r)$
by part (b), and $\lfloor\frac{m-2}{3}\rfloor\leqslant r^{*}-1$ by the
definition of $r^{*}$. Thus $r_{0}-1=\lfloor\frac{m}{3}\rfloor\leqslant r^{*}$
and it follows from part (b) that $f_{m}(0)<\cdots<f_{m}(r_{0}-1)$. ∎

Fix $m$ and $r$ where $0\leqslant r<m$. We shall use the following notation:

(3) $\displaystyle X_{i}$ $\displaystyle=\frac{r-i+1}{m-r+i}$
$\displaystyle\textup{for $0\leqslant i\leqslant r$},$ (4) $\displaystyle
S_{j}$ $\displaystyle=1+X_{j+1}+X_{j+1}X_{j+2}+\cdots+X_{j+1}X_{j+2}\cdots
X_{r}$ for $0\leqslant j<r$, (5) $\displaystyle T_{j}$
$\displaystyle=1+X_{1}+X_{1}X_{2}+\cdots+X_{1}X_{2}\cdots X_{j}$ for
$0\leqslant j\leqslant r$.

Our convention in (5) is that $T_{0}=1$ as
$T_{j}=\sum_{i=0}^{j}(\prod_{k=1}^{i}X_{k})$ equals 1 when $j=0$.

######  Lemma 5.

Fix $m,r,j$ where $0\leqslant j\leqslant r<m$. Using the above definitions,

  1. (a)

the inequality $\sum_{i=0}^{j}\binom{m}{r-i}>\binom{m}{r+1}$ is equivalent to
$T_{j}>X_{0}^{-1}$,

  2. (b)

the inequality $\sum_{i=0}^{r}\binom{m}{i}<\binom{m}{r+1}$ is equivalent to
$S_{0}<X_{0}^{-1}$.

###### Proof.

For $0\leqslant i\leqslant r$, we have $\binom{m}{r-i}=X_{i}\binom{m}{r-i+1}$
so $\binom{m}{r-i}=(\prod_{k=1}^{i}X_{k})\binom{m}{r}$ holds. Therefore
$\sum_{i=0}^{j}\binom{m}{r-i}=\binom{m}{r}\sum_{i=0}^{j}\left(\prod_{k=1}^{i}X_{k}\right)=\binom{m}{r}T_{j}$.
Since $\binom{m}{r}=X_{0}\binom{m}{r+1}$, the inequality
$\sum_{i=0}^{j}\binom{m}{r-i}>\binom{m}{r+1}$ is equivalent to
$\binom{m}{r}T_{j}>X_{0}^{-1}\binom{m}{r}$ which is equivalent to
$T_{j}>X_{0}^{-1}$. This proves part (a).

Note that
$\sum_{i=0}^{r}\binom{m}{i}=\sum_{i=0}^{r}\binom{m}{r-i}=\binom{m}{r}T_{r}=\binom{m}{r}S_{0}$
since $S_{0}=T_{r}$. Since $\binom{m}{r+1}=X_{0}^{-1}\binom{m}{r}$, the
inequality $\sum_{i=0}^{r}\binom{m}{i}<\binom{m}{r+1}$ is equivalent to
$\binom{m}{r}S_{0}<X_{0}^{-1}\binom{m}{r}$ which is equivalent to
$S_{0}<X_{0}^{-1}$. This proves part (b). ∎

##  3\. Proof that $f_{m}(r)$ is increasing for $0\leqslant r\leqslant r_{0}$

Recall that $m\geqslant 0$ and $r_{0}:=\lfloor\frac{m}{3}\rfloor+1$. We now
strengthen Lemma 4(c).

######  Theorem 6.

If $m\not\in\\{0,1,3,6,9,12\\}$, then $f_{m}(0)<f_{m}(1)<\cdots<f_{m}(r_{0})$.

###### Proof.

The statement is easy to check for $m\in\\{2,4,5\\}$. The statement follows
from Tables 1 and 2 for $m\in\\{7,8,10,11,13,14\\}$. Suppose now that
$m\geqslant 15$.

Table 2. $\square=\max\\{f_{m}(r_{0}-1),f_{m}(r_{0})\\}$ for $0\leqslant m\leqslant 12$, $r_{0}=\lfloor m/3\rfloor+1$. $m$ |  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$f_{m}(r_{0}-1)$ |  | $1$ | $1$ | $1$ | $2$ | $\frac{5}{2}$ | $3$ | $\frac{11}{2}$ | $\frac{29}{4}$ | $\frac{37}{4}$ | $\frac{65}{4}$ | $22$ | $29$ | $\frac{397}{8}$
$f_{m}(r_{0})$ |  |  $\frac{1}{2}$***Observe that $f_{0}(1)=2^{-1}\left(\binom{0}{0}{+}\binom{0}{1}\right)=2^{-1}(1{+}0)=\frac{1}{2}$. | $1$ | $\frac{3}{2}$ | $\frac{7}{4}$ | $\frac{11}{4}$ | $4$ | $\frac{21}{4}$ | $8$ | $\frac{93}{8}$ | $16$ | $\frac{193}{8}$ | $\frac{281}{8}$ | $\frac{793}{16}$

Recall that $r_{0}=\lfloor\frac{m}{3}\rfloor+1$ and
$m\in\\{3r_{0}-3,3r_{0}-2,3r_{0}-1\\}$. By Lemma 4(c) it suffices to show that
$f_{m}(r_{0}-1)<f_{m}(r_{0})$. If we prove this for $m=3r_{0}-3$, Lemma 3(b,a)
gives it for $m=3r_{0}-2$ and $m=3r_{0}-1$ as well, so for $r_{0}\geqslant 6$
we want to show $f_{3r_{0}-3}(r_{0}-1)<f_{3r_{0}-3}(r_{0})$. This is true for
$r_{0}=6$ by Table 1. We set $t:=r_{0}-1$, $m:=3t$ and we prove, using
induction on $t$, that $f_{3t}(t)<f_{3t}(t+1)$ holds for all $t\geqslant 6$.

Note that $f_{3t}(t)<f_{3t}(t+1)$ is equivalent by Lemma 3(a) to
$\sum_{i=0}^{t}\binom{3t}{i}<\binom{3t}{t+1}$, and this is equivalent to
$S_{0}<X_{0}^{-1}$ by Lemma 5(b). Putting $m=3t$ and $r=t$ in (3), gives
$X_{i}=\frac{t-i+1}{2t+i}$ and
$S_{0}=1+X_{1}+X_{1}X_{2}+\cdots+X_{1}X_{2}\cdots X_{t}$ by (4).

It follows from $0<X_{t}<\cdots<X_{5}<X_{4}$ and
$X_{4}=\frac{t-3}{2t+4}<\frac{1}{2}$ that

$S_{3}=1+X_{4}+X_{4}X_{5}+\cdots+X_{4}X_{5}\cdots
X_{t}<1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{t-3}}<\sum_{i=0}^{\infty}\frac{1}{2^{i}}=2.$

The recurrence relation $S_{j}=1+X_{j+1}S_{j+1}$ for $0\leqslant j<t$ implies
that

$\displaystyle S_{0}$
$\displaystyle=1+X_{1}\left(1+X_{2}\left(1+X_{3}S_{3}\right)\right)<1+X_{1}\left(1+X_{2}\left(1+2X_{3}\right)\right)$
$\displaystyle=1+\frac{t}{2t+1}\left(1+\frac{t-1}{2t+2}\left(1+\frac{2(t-2)}{2t+3}\right)\right).$

We aim to show that $S_{0}<X_{0}^{-1}$. It suffices to prove
$1+X_{1}\left(1+X_{2}\left(1+2X_{3}\right)\right)\leqslant X_{0}^{-1}$ where
$X_{0}=\frac{t+1}{2t}$. This amounts to proving that

$1+\frac{t}{2t+1}\left(1+\frac{t-1}{2t+2}\left(1+\frac{2(t-2)}{2t+3}\right)\right)\leqslant\frac{2t}{t+1}.$

Rearranging, and using the denominator $(2t+1)(2t+2)(2t+3)$, gives

$0\leqslant\frac{3t^{2}-17t-6}{(2t+1)(2t+2)(2t+3)}=\frac{(3t+1)(t-6)}{(2t+1)(2t+2)(2t+3)}$

This inequality is valid for all $t\geqslant 6$. This completes the proof. ∎

How might one prove a nice formula such as
$\lim_{s\to\infty}\sum_{i=0}^{s}\binom{3s}{i}/\binom{3s}{s}=2$?

######  Remark 7.

For $s>4$ set $m=3s$ and $r_{0}=s+1$. Then $f_{m}(r_{0}-1)<f_{m}(r_{0})$ by
Theorem 6. Hence
$\sum_{i=0}^{s}\binom{3s}{i}<\binom{3s}{s+1}=\frac{2s}{s+1}\binom{3s}{s}$ and
so $\lim_{s\to\infty}\sum_{i=0}^{s}\binom{3s}{i}/\binom{3s}{s}\leqslant 2$. We
show $f_{m}(r_{0})>f_{m}(r_{0}+1)$ in Section 4, and therefore
$\lim_{s\to\infty}\sum_{i=0}^{s}\binom{3s}{i}/\binom{3s}{s}\geqslant 2$.

##  4\. Proof that $f_{m}(r)$ is decreasing for $r_{0}\leqslant r\leqslant m$

Showing that $f_{m}(r)$ decreases strictly for $r_{0}\leqslant r\leqslant m$
is much harder. Recall that $\binom{r}{i}=0$ if $i<0$, and
$\binom{r}{i}=\frac{1}{i!}\prod_{j=0}^{i-1}(r-j)$ if $i\geqslant 0$. In this
section we prove:

######  Theorem 8.

If $m\geqslant 2$, then $f_{m}(\lfloor m/3\rfloor{+}1)>f_{m}(\lfloor
m/3\rfloor{+}2)>{\cdots}>f_{m}(m)\,{=}\,1$.

Our proof of Theorem 8 depends on two technical lemmas, the first of which
proves that the non-leading coefficients of a certain polynomial $A(r)$ are
all negative.

First define $B_{i}(r)=\prod_{\ell=1}^{i}(r-\ell)$. Now
$\prod_{\ell=1}^{i}(r-\ell)=r^{i}+\sum_{k=0}^{i-1}b_{i,k}r^{k}$ and the
coefficients $b_{i,k}$ alternate in sign: for $0\leqslant k\leqslant i$, we
have $b_{i,k}>0$ if $i-k$ is even and $b_{i,k}<0$ if $i-k$ is odd. Next define
polynomials $A_{i}(r)$ via:

(6) $A_{2}(r)=r^{2}-15r-10\quad\textup{and $\quad
A_{i}(r)=(2r+i)A_{i-1}(r)-B_{i}(r)\quad$ for $i\geqslant 3$.}$

Clearly $\deg(A_{i}(r))=i$ and we may write
$A_{i}(r)=r^{i}+\sum_{k=0}^{i-1}a_{i,k}r^{k}$. We use $a_{i,i}=1$.

Comparing coefficients in this recurrence and $B_{i}(r)=(r-i)B_{i-1}(r)$,
shows that

(Ra) $\displaystyle a_{2,0}=-10,$ $\displaystyle a_{2,1}=-15,\qquad$
$\displaystyle a_{i,k}=ia_{i-1,k}+2a_{i-1,k-1}-b_{i,k}\quad$ for $i\geqslant
3$, (Rb) $\displaystyle b_{2,0}=2,$ $\displaystyle b_{2,1}=-3,\hskip
9.95845pt\quad$ $\displaystyle b_{i,k}=-ib_{i-1,k}+b_{i-1,k-1}$
$\displaystyle\textup{for $i\geqslant 3$}.$

######  Lemma 9.

Let $a_{i,k},b_{i,k},A_{i}(r),B_{i}(r)$ be as above.

  1. (a)

If $i\geqslant 2$, then $b_{i,i-1}=-\binom{i+1}{2}$ and
$a_{i,i-1}=-\binom{i+4}{2}$.

  2. (b)

If $i\geqslant 2$ and $0\leqslant k\leqslant i-1$, then
$a_{i,k}\leqslant-2b_{i,k}<0$ if $i-k$ is even, and $\qquad a_{i,k}\leqslant
b_{i,k}<0$ if $i-k$ is odd.

  3. (c)

If $i\geqslant 2$, then the coefficients $a_{i,k}$ are negative for
$0\leqslant k<i$.

###### Proof.

(a) Clearly $b_{i,i-1}=-\sum_{j=1}^{i}j=-\binom{i+1}{2}$. The formula for
$a_{i,i-1}$ holds for $i=2$ and by induction using the recurrence (Ra).

(b) We use induction on $i$. For the base case $i=2$, either $i-k$ is even and
$a_{2,0}=-10\leqslant-2b_{2,0}=-4$, or $i-k$ is odd and
$a_{2,1}=-15<b_{2,1}=-3$. Thus the claims are true for $i=2$. Suppose now that
$i\geqslant 3$, and the claims are valid for $i-1$.

By part (a), $a_{i-1,i}=-\binom{i+4}{2}<-\binom{i+2}{2}=b_{i,i-1}<0$ as
claimed. It remains to consider $k$ in the range $0\leqslant k<i-1$. It is
useful to set $a_{i,-1}=b_{i,-1}=0$. Suppose first that $i-k$ is even. Using
the recurrences (Ra), (Rb) and induction gives

$\displaystyle a_{i,k}$
$\displaystyle=i(a_{i-1,k}+b_{i-1,k})+(2a_{i-1,k-1}-b_{i-1,k-1})$
$\displaystyle\leqslant i(b_{i-1,k}+b_{i-1,k})+(-4b_{i-1,k-1}-b_{i-1,k-1})$
$\displaystyle\leqslant 2ib_{i-1,k}-2b_{i-1,k-1}=-2b_{i,k}<0.$

If $i-k$ is odd, then a similar argument gives

$\displaystyle a_{i,k}$
$\displaystyle=i(a_{i-1,k}+b_{i-1,k})+(2a_{i-1,k-1}-b_{i-1,k-1})$
$\displaystyle\leqslant i(-2b_{i-1,k}+b_{i-1,k})+(2b_{i-1,k-1}-b_{i-1,k-1})$
$\displaystyle\leqslant-ib_{i-1,k}+b_{i-1,k-1}=b_{i,k}<0.$

(c) This follows immediately from part (b). ∎

######  Lemma 10.

Suppose that $j\geqslant 4$. Then
$\sum_{i=r-j}^{r}\binom{3r-1}{i}>\binom{3r-1}{r+1}$ holds for all $r$ in the
range $j\leqslant r\leqslant\binom{j+2}{2}$.

###### Proof.

We apply Lemma 5(a) with $m=3r-1$. Hence $X_{i}=\frac{r-i+1}{2r+i-1}$ by (3).
Since $\sum_{i=r-j}^{r}\binom{m}{i}=\sum_{i=0}^{j}\binom{m}{r-i}$ it suffices
by Lemma 5(a) to prove that

$T_{j}=1+X_{1}+X_{1}X_{2}+\cdots+X_{1}X_{2}\cdots X_{j}>X_{0}^{-1}.$

We prove that this inequality holds for all $r$ in the range $j\leqslant
r\leqslant\binom{j+2}{2}$. This inequality is equivalent to

(7)
$X_{j}>X_{j-1}^{-1}(\cdots(X_{2}^{-1}(X_{1}^{-1}(X_{0}^{-1}-1)-1)-1)\cdots)-1.$

The right-side of (7) is a rational function in $r$, which when $j=4$, equals

$\frac{P_{4}(r)}{Q_{4}(r)}=\frac{2r+2}{r-2}\left(\frac{2r+1}{r-1}\left(\frac{2r}{r}\left(\frac{2r-1}{r+1}-1\right)-1\right)-1\right)-1$

where the denominator is $Q_{4}(r)=(r+1)r(r-1)(r-2)$, and the numerator is
$P_{4}(r)=(r+1)r(r^{2}-15r-10)$. Since $\gcd(P_{4}(r),Q_{4}(r))=(r+1)r$, the
polynomials $A_{2}(r):=r^{2}-15r-10$ and $B_{2}(r):=(r-1)(r-2)$ are coprime.
The putative inequality (7) when $j=4$ is therefore

$\frac{r-3}{2r+3}>\frac{P_{4}(r)}{Q_{4}(r)}=\frac{A_{2}(r)}{B_{2}(r)}=\frac{r^{2}-15r-10}{(r-1)(r-2)}.$

Observe that $A_{2}(r)<r^{2}-15r\leqslant 0$ for $4\leqslant r\leqslant
15=\binom{6}{2}$. Thus for $r$ in the range $4\leqslant
r\leqslant\binom{6}{2}$, the left side of (7) is positive, and the right side
is at most 0. Thus the inequality is valid for $4\leqslant
r\leqslant\binom{6}{2}$ and the claim is true for $j=4$.

Assume now that $j>4$, and that the claim is true for $j-1$. Therefore the
inequality (7) can be written

$X_{j}=\frac{r-j+1}{2r+j-1}>\frac{P_{j}(r)}{Q_{j}(r)}\qquad\textup{where}\qquad\frac{P_{j}(r)}{Q_{j}(r)}=(X_{j-1})^{-1}\frac{P_{j-1}(r)}{Q_{j-1}(r)}-1.$

Since $(X_{j-1})^{-1}=\frac{2r+j-2}{r-j+2}$, this gives rise to the
recurrences:

$\displaystyle P_{j}(r)$ $\displaystyle=(2r+j-2)P_{j-1}(r)-(r-j+2)Q_{j-1}(r)$
for $j>4$, $\displaystyle Q_{j}(r)$ $\displaystyle=(r-j+2)Q_{j-1}(r)$ for
$j>4$.

It is clear that $Q_{j}(r)=(r+1)r(r-1)\cdots(r-j+2)=(r+1)rB_{j-2}(r)$ holds
and $B_{j-2}(r)$ has degree $j-2$. Furthermore, $(r+1)r$ divides
$\gcd(P_{j}(r),Q_{j}(r))$, so the polynomials $A_{j-2}(r)$, which are defined
by the similar recurrence (6), satisfy $P_{j}(r)=(r+1)rA_{j-2}(r)$ and also
have degree $j-2$.

By Lemma 9, $A_{i}(r)-r^{i}$ has negative coefficients and leading coefficient
$-\binom{i+4}{2}$. So for $i\geqslant 2$ and $r\leqslant\binom{i+4}{2}$, we
have $A_{i}(r)<r^{i}-\binom{i+4}{2}r^{i-1}\leqslant 0$. Further,
$B_{i}(r)=\prod_{\ell=1}^{i}(r-\ell)>0$ for $r\geqslant i+1$. Hence
$A_{i}(r)/B_{i}(r)<0$ for $r$ satisfying $i+2\leqslant
r\leqslant\binom{i+4}{2}$. Suppose that $j=i+2$, then $P_{j}(r)/Q_{j}(r)<0$
for $r$ in the interval $j\leqslant r\leqslant\binom{j+2}{2}$. Using the
definitions of $P_{j}(r),Q_{j}(r)$, the inequality (7) is the same as

$X_{j}=\frac{r-j+1}{2r+j-1}>\frac{P_{j}(r)}{Q_{j}(r)}=\frac{A_{j-2}(r)}{B_{j-2}(r)}.$

Thus for $r$ satisfying $j\leqslant r\leqslant\binom{j+2}{2}$, the left side
of (7) is positive, and the right side is negative. Thus the claim is valid
for $j\leqslant r\leqslant\binom{j+2}{2}$. ∎

###### Proof of Theorem 8.

It follows from $\sum_{i=0}^{m}\binom{m}{i}=2^{m}$ that $f_{m}(m)=1$. Since
$r_{0}:=\lfloor m/3\rfloor+1$, we have $m\in\\{3r_{0}-3,3r_{0}-2,3r_{0}-1\\}$.
If we can prove that $f_{m}(r_{0})>f_{m}(r_{0}+1)$ for $m=3r_{0}-1$, then
$f_{m}(r_{0})>f_{m}(r_{0}+1)$ holds for $m=3r_{0}-2$ and $3r_{0}-3$ by Lemma
3(d). With the notation in Lemma 4, we have $2>t_{m}(r_{0})$ and hence
$r^{*}\leqslant r_{0}$. Therefore $f_{m}(r_{0}+1)>\cdots>f_{m}(m)$ holds by
Lemma 4(b).

In summary, it remains to prove
$\sum_{i=0}^{r_{0}}\binom{3r_{0}-1}{i}>\binom{3r_{0}-1}{r_{0}+1}$ for
$r_{0}\geqslant 1$. This is true for $r_{0}=1,2,3$ since $\frac{3}{2}>1$,
$4>\frac{13}{4}$ and $\frac{93}{8}>\frac{163}{16}$. For each $r_{0}\geqslant
4$ set $j=r_{0}$. Then $j\geqslant 4$ and
$\sum_{i=0}^{r_{0}}\binom{m}{i}>\binom{m}{r_{0}+1}$ follows by Lemma 10. This
completes the proof. ∎

###### Proof of Theorem 1.

The result follows from Theorems 6 and 8. There are two equal sized maxima if
$m=1$, otherwise the maximum is unique. ∎

##  5\. Estimating $f_{m}(r_{0})$

This section is devoted to proving asymptotically optimal bounds for
$f_{m}(r_{0})$.

###### Proof of Theorem 2.

We first prove the upper bound in (1). This is true if $m=1$. For
$m\not\in\\{0,1,3,6,9,12\\}$ and $r_{0}=\lfloor m/3\rfloor+1$ it follows from
Theorem 6 that $f_{m}(r_{0}-1)<f_{m}(r_{0})$ and by Lemma 3(a) that
$\sum_{i=0}^{r_{0}-1}\binom{m}{i}<\binom{m}{r_{0}}$. Therefore
$\sum_{i=0}^{r_{0}}\binom{m}{i}<2\binom{m}{r_{0}}$ and the upper bound
follows. For the lower bound, $f_{m}(r_{0})>f_{m}(r_{0}+1)$ holds by Theorem
8, and so $\sum_{i=0}^{r_{0}}\binom{m}{i}>\binom{m}{r_{0}+1}$ by Lemma 3(c).
Hence $2^{-r_{0}}\binom{m}{r_{0}+1}<f_{m}(r_{0})$, and the lower bound of (1)
follows from
$\binom{m}{r_{0}+1}=\frac{2r_{0}-k}{r_{0}+1}\binom{m}{r_{0}}=(2-\frac{k+2}{r_{0}+1})\binom{m}{r_{0}}$.
To prove (2), we use binomial approximations.

Suppose that $0<p<1$ and $q:=1-p$. If $pn$ is an integer, then $qn=n-pn$ is an
integer, and Stirling’s approximation $n!=\sqrt{2\pi
n}(\frac{n}{e})^{n}(1+\textup{O}(\frac{1}{n}))$ gives

(8) $\binom{n}{pn}=\frac{c^{n}}{\sqrt{2\pi
pqn}}\left(1+\textup{O}\left(\frac{1}{n}\right)\right)\qquad\textup{where
$c=\frac{1}{p^{p}q^{q}}$.}$

Paraphrasing [Ash]*Lemma 4.7.1 gives the following upper and lower bounds:

(9)
$\frac{c^{n}}{\sqrt{8pqn}}\leqslant\binom{n}{pn}\leqslant\frac{c^{n}}{\sqrt{2\pi
pqn}}\qquad\qquad\textup{where $c=\frac{1}{p^{p}q^{q}}$.}$

Henceforth set $p=\frac{1}{3}$, so $q=\frac{2}{3}$ and $c=\frac{3}{2^{2/3}}$.
Therefore $c^{3}=\frac{27}{4}$ and

$\frac{c^{3r_{0}}}{2^{r_{0}}}=\frac{1}{2^{r_{0}}}\left(\frac{27}{4}\right)^{r_{0}}=\left(\frac{27}{8}\right)^{r_{0}}=\left(\frac{3}{2}\right)^{3r_{0}}\qquad\textup{and}\qquad\frac{1}{\sqrt{2pq}}=\frac{3}{2}.$

We write $m=3r_{0}-k$ where $k\in\\{1,2,3\\}$.

Consider the upper bound for $f_{m}(r_{0})$ in (2). It follows from

$\binom{m}{r_{0}}=\binom{3r_{0}-k}{r_{0}}=\frac{2r_{0}-k+1}{3r_{0}-k+1}\binom{3r_{0}-k+1}{r_{0}}\leqslant\frac{2}{3}\binom{3r_{0}-k+1}{r_{0}}$

that $\binom{m}{r_{0}}\leqslant(\frac{2}{3})^{k}\binom{3r_{0}}{r_{0}}$.
Setting $n=3r_{0}$ and $p=\frac{1}{3}$ in (9) and using $m<n$ shows

$\displaystyle\frac{2}{2^{r_{0}}}\binom{3r_{0}}{r_{0}}=\frac{2}{2^{r_{0}}}\binom{n}{pn}\leqslant\frac{2}{2^{r_{0}}}\frac{c^{3r_{0}}}{\sqrt{2\pi
pqn}}=\frac{2}{\sqrt{2\pi
pqn}}\left(\frac{3}{2}\right)^{3r_{0}}<\frac{3}{\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{3r_{0}}.$

Using $\binom{m}{r_{0}}\leqslant(\frac{3}{2})^{-k}\binom{3r_{0}}{r_{0}}$ and
$m=3r_{0}-k$ gives

$f_{m}(r_{0})\leqslant\frac{2}{2^{r_{0}}}\binom{m}{r_{0}}\leqslant\frac{2}{2^{r_{0}}}\left(\frac{3}{2}\right)^{-k}\binom{3r_{0}}{r_{0}}<\frac{3}{\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{m}.$

We now consider approximate lower bounds for $f_{m}(r_{0})$. Our argument
involves constants depending on $k$ but not $r_{0}$ whose values are not
relevant here. We have

$\binom{m}{r_{0}+1}=2\left(1+\textup{O}\left(\frac{1}{r_{0}}\right)\right)\binom{m}{r_{0}}=4\left(1+\textup{O}\left(\frac{1}{r_{0}}\right)\right)\binom{m}{r_{0}-1}.$

Further, if $k=1,2$ and $r_{0}\geqslant 1$ it follows that

$\displaystyle\binom{m}{r_{0}-1}$
$\displaystyle=\binom{3r_{0}-k}{r_{0}-1}=\frac{3r_{0}-k}{2r_{0}-k+1}\binom{3r_{0}-k-1}{r_{0}-1}$
$\displaystyle=\left(\frac{3}{2}+\frac{k-3}{2(2r_{0}-k+1)}\right)\binom{3r_{0}-k-1}{r_{0}-1}>\frac{3}{2}\binom{3r_{0}-k-1}{r_{0}-1}.$

Hence
$\binom{m}{r_{0}-1}\geqslant(\frac{3}{2})^{3-k}\binom{3r_{0}-3}{r_{0}-1}$
holds for $k\in\\{1,2,3\\}$ and $r_{0}\geqslant 1$.

Setting $n=3r_{0}-3$ and $p=\frac{1}{3}$ in (8) yields

$\displaystyle\frac{1}{2^{r_{0}}}\binom{3r_{0}-3}{r_{0}-1}$
$\displaystyle=\frac{1}{2^{r_{0}}}\binom{n}{pn}=\frac{1}{2^{r_{0}}}\frac{c^{3r_{0}-3}}{\sqrt{2pq\pi
n}}\left(1+\textup{O}\left(\frac{1}{n}\right)\right).$

However, $\frac{c^{3r_{0}}}{2^{r_{0}}}=(\frac{3}{2})^{3r_{0}}$ and
$\frac{c^{-3}}{\sqrt{2pq\pi n}}=\frac{3c^{-3}}{2\sqrt{\pi
n}}=\frac{2}{9\sqrt{\pi n}}=\frac{2}{9\sqrt{\pi
m}}(1+\textup{O}\left(\frac{1}{m}\right))$. Therefore

$\frac{1}{2^{r_{0}}}\binom{3r_{0}-3}{r_{0}-1}=\frac{2}{9\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{3r_{0}}\left(1+\textup{O}\left(\frac{1}{m}\right)\right).$

The above bounds give

$\displaystyle f_{m}(r_{0})$
$\displaystyle\geqslant\frac{1}{2^{r_{0}}}\binom{m}{r_{0}+1}\geqslant
4\left(1+\textup{O}\left(\frac{1}{r_{0}}\right)\right)\left(\frac{3}{2}\right)^{3-k}\frac{1}{2^{r_{0}}}\binom{3r_{0}-3}{r_{0}-1}$
$\displaystyle=4\left(1{+}\textup{O}\left(\frac{1}{m}\right)\right)\left(\frac{3}{2}\right)^{3-k}\frac{2}{9\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{3r_{0}}=\left(1{+}\textup{O}\left(\frac{1}{m}\right)\right)\frac{3}{\sqrt{\pi
m}}\left(\frac{3}{2}\right)^{m}.$

Finally, since $1+\textup{O}\left(\frac{1}{m}\right)\to 1$ as $m\to\infty$,
the limit in (2) follows. ∎

## References

  * [1]
AlizadehFaezehGlasbyS. P.PraegerCheryl E.Sequences of linear codes where the
rate times distance grows rapidlyIn preparation@article{AGP, author =
{Alizadeh, Faezeh}, author = {Glasby, S.\,P.}, author = {Praeger, Cheryl E.},
title = {Sequences of linear codes where the rate times distance grows
rapidly}, note = {In preparation}}

  * [3]
AshRobert B.Information theoryCorrected reprint of the 1965 originalDover
Publications, Inc., New York1990xii+339ISBN 0-486-66521-6@book{Ash, author =
{Ash, Robert B.}, title = {Information theory}, note = {Corrected reprint of
the 1965 original}, publisher = {Dover Publications, Inc., New York}, date =
{1990}, pages = {xii+339}, isbn = {0-486-66521-6}}

  * [5]
GlasbyS. P.On the maximum of the weighted binomial sum
$2^{-r}\sum_{i=0}^{r}\binom{m}{r}$MathOverflowQuestion 389857@article{Gmo,
author = {Glasby, S.\,P.}, title = {On the maximum of the weighted binomial
sum \texorpdfstring{$2^{-r}\sum_{i=0}^r\binom{m}{r}$}{}}, journal =
{MathOverflow}, note =
{\href{https://mathoverflow.net/questions/389857/}{Question 389857}}}

  * [7]
LingSanXingChaopingCoding theoryA first courseCambridge University Press,
Cambridge2004xii+222ISBN 0-521-52923-9@book{LX, author = {Ling, San}, author =
{Xing, Chaoping}, title = {Coding theory}, note = {A first course}, publisher
= {Cambridge University Press, Cambridge}, date = {2004}, pages = {xii+222},
isbn = {0-521-52923-9}}

  * [9]


* [4] Francisco Aparecido Rodrigues.  Network centrality: an introduction.  In A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems, pages 177–196. Springer, 2019.
  * [5] Sima Das.  On local and global centrality in large scale networks.  arXiv preprint arXiv:1405.5512, 2014.
  * [6] Linyuan Lü, Duanbing Chen, Xiao-Long Ren, Qian-Ming Zhang, Yi-Cheng Zhang, and Tao Zhou.  Vital nodes identification in complex networks.  Physics Reports, 650:1–63, 2016.
  * [7] Carla Sciarra, Guido Chiarotti, Francesco Laio, and Luca Ridolfi.  A change of perspective in network centrality.  Scientific Reports, 8, 12 2018.
  * [8] Zakariya Ghalmane, Mohammed El Hassouni, Chantal Cherifi, and Hocine Cherifi.  Centrality in modular networks.  EPJ Data Science, 8(1):15, 2019.
  * [9] Zakariya Ghalmane, Chantal Cherifi, Hocine Cherifi, and Mohammed El Hassouni.  Centrality in complex networks with overlapping community structure.  Scientific reports, 9(1):1–29, 2019.
  * [10] Hocine Cherifi, Gergely Palla, Boleslaw K Szymanski, and Xiaoyan Lu.  On community structure in complex networks: challenges and opportunities.  Applied Network Science, 4(1):1–35, 2019.
  * [11] Herbert A. Simon.  The architecture of complexity.  Proceedings of the American Philosophical Society, 106(6):467–482, 1962.
  * [12] Anna Zafeiris and Tamás Vicsek.  Why We Live in Hierarchies?: A Quantitative Treatise.  Springer, 2017.
  * [13] Alexander Clark, Chris Fox, and Shalom Lappin.  The handbook of computational linguistics and natural language processing.  John Wiley & Sons, 2013.
  * [14] Enys Mones, Lilla Vicsek, and Tamás Vicsek.  Hierarchy measure for complex networks.  PloS one, 7:e33799, 03 2012.
  * [15] Jianxi Luo and Christopher Magee.  Detecting evolving patterns of self-organizing networks by flow hierarchy measurement.  Complexity, 16:53–61, 07 2011.
  * [16] Dániel Czégel and Gergely Palla.  Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?  Scientific reports, 5:17994, 2015.
  * [17] Ala Trusina, Sergei Maslov, Petter Minnhagen, and Kim Sneppen.  Hierarchy measures in complex networks.  Physical review letters, 92(17):178702, 2004.
  * [18] Hongsuda Tangmunarunkit, Ramesh Govindan, Sugih Jamin, Scott Shenker, and Walter Willinger.  Network topologies, power laws, and hierarchy.  ACM SIGCOMM Computer Communication Review, 32(1):76, 2002.
  * [19] Javier García-Algarra, Juan Manuel Pastor, José María Iriondo, and Javier Galeano.  Ranking of critical species to preserve the functionality of mutualistic networks using the k-core decomposition.  PeerJ, 5:e3321, 2017.
  * [20] Jia Wang and James Cheng.  Truss decomposition in massive networks.  arXiv preprint arXiv:1205.6693, 2012.
  * [21] Fragkiskos D Malliaros, Christos Giatsidis, Apostolos N Papadopoulos, and Michalis Vazirgiannis.  The core decomposition of networks: Theory, algorithms and applications.  The VLDB Journal, 29(1):61–92, 2020.
  * [22] M. E. J. Newman.  Networks: an introduction.  Oxford University Press, Oxford; New York, 2010.
  * [23] Maksim Kitsak, Lazaros K Gallos, Shlomo Havlin, Fredrik Liljeros, Lev Muchnik, H Eugene Stanley, and Hernán A Makse.  Identification of influential spreaders in complex networks.  Nature physics, 6(11):888–893, 2010.
  * [24] Rakhi Saxena, Sharanjit Kaur, and Vasudha Bhatnagar.  Social centrality using network hierarchy and community structure.  Data Mining and Knowledge Discovery, 32(5):1421–1443, 2018.
  * [25] Yang-Yu Liu, Jean-Jacques Slotine, and Albert-László Barabási.  Control centrality and hierarchical structure in complex networks.  Plos one, 7(9):e44459, 2012.
  * [26] Yong Li, Wenguo Li, Yi Tan, Fang Liu, Yijia Cao, and Kwang Y Lee.  Hierarchical decomposition for betweenness centrality measure of complex networks.  Scientific reports, 7:46491, 2017.
  * [27] Stuart Oldham, Ben Fulcher, Linden Parkes, Aurina Arnatkeviciute, Chao Suo, and Alex Fornito.  Consistency and differences between centrality measures across distinct classes of networks.  PloS one, 14(7), 2019.
  * [28] David Lane.  Hierarchy, complexity, society.  In Hierarchy in Natural and Social Sciences, chapter 4, pages 81–119. Springer, 2006.
  * [29] Timothy FH Allen and Thomas B Starr.  Hierarchy: perspectives for ecological complexity.  University of Chicago Press, 2017.
  * [30] Steve Harenberg, Gonzalo Bello, La Gjeltema, Stephen Ranshous, Jitendra Harlalka, Ramona Seay, Kanchana Padmanabhan, and Nagiza Samatova.  Community detection in large-scale networks: a survey and empirical evaluation.  Wiley Interdisciplinary Reviews: Computational Statistics, 6(6):426–439, 2014.
  * [31] Jaewon Yang and Jure Leskovec.  Defining and evaluating network communities based on ground-truth.  Knowledge and Information Systems, 42(1):181–213, 2015.
  * [32] Josiane Mothe, Karen Mkhitaryan, and Mariam Haroutunian.  Community detection: Comparison of state of the art algorithms.  In 2017 Computer Science and Information Technologies (CSIT), pages 125–129. IEEE, 2017.
  * [33] Paul Wagenseller and Feng Wang.  Community detection algorithm evaluation using size and hashtags.  ArXiv, abs/1612.03362, 2016.
  * [34] Zakariya Ghalmane, Mohammed El Hassouni, and Hocine Cherifi.  Immunization of networks with non-overlapping community structure.  Social Network Analysis and Mining, 9(1):45, 2019.
  * [35] Ahmed Ibnoulouafi, Mohamed El Haziti, and Hocine Cherifi.  M-centrality: identifying key nodes based on global position and local degree variation.  Journal of Statistical Mechanics: Theory and Experiment, 2018(7):073407, 2018.
  * [36] Naveen Gupta, Anurag Singh, and Hocine Cherifi.  Centrality measures for networks with community structure.  Physica A: Statistical Mechanics and its Applications, 452:46–59, 2016.
  * [37] Zakariya Ghalmane, Mohammed El Hassouni, and Hocine Cherifi.  Betweenness centrality for networks with non-overlapping community structure.  In 2018 IEEE workshop on complexity in engineering (COMPENG), pages 1–5. IEEE, 2018.
  * [38] Minoo Ashtiani, Ali Salehzadeh-Yazdi, Zahra Razaghi-Moghadam, Holger Hennig, Olaf Wolkenhauer, Mehdi Mirzaie, and Mohieddin Jafari.  A systematic survey of centrality measures for protein-protein interaction networks.  BMC systems biology, 12(1):80, 2018.
  * [39] Ulrik Brandes and Daniel Fleischer.  Centrality measures based on current flow.  In Annual symposium on theoretical aspects of computer science, pages 533–544. Springer, 2005.
  * [40] William Webber, Alistair Moffat, and Justin Zobel.  A similarity measure for indefinite rankings.  ACM Transactions on Information Systems (TOIS), 28(4):1–38, 2010\.
  * [41] Markus Schulze.  A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method.  Social Choice and Welfare, 36(2):267–303, 2011.
  * [42] Ryan A. Rossi and Nesreen K. Ahmed.  The network data repository with interactive graph analytics and visualization.  In AAAI, 2015.  [Online]. Available: http://networkrepository.com.
  * [43] Vito Latora, Vincenzo Nicosia, and Giovanni Russo.  Complex networks: principles, methods and applications.  Cambridge University Press, 2017.  [Online]. Available: https://www.complex-networks.net/datasets.html.
  * [44] Wouter De Nooy, Andrej Mrvar, and Vladimir Batagelj.  Exploratory social network analysis with Pajek: Revised and expanded edition for updated software, volume 46.  Cambridge University Press, 2018.  [Online]. Available: http://vlado.fmf.uni-lj.si/pub/networks/data/esna/default.htm.
  * [45] Jérôme Kunegis.  Handbook of network analysis [konect–the koblenz network collection].  arXiv preprint arXiv:1402.5500, 2014.  [Online]. Available: http://konect.uni-koblenz.de.