4.12 (i) to the case where $X^{\bullet}$ is irreducible. This completes
Theorem 4.12 (i) (Proposition 4.11). Moreover, by gluing certain Galois multi-
admissible coverings, Theorem 4.12 (ii) follows from Theorem 3.9, Proposition
3.11 (i), and Theorem 4.12 (i).

###  4.2. A necessary condition

####  4.2.1. Settings

We maintain the notation introduced in 2.1.2 and the assumption introduced in
2.3.3 (i.e., $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1$). Moreover,
let $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$, and let

${\rm ord}_{x}(D)=\sum_{j=0}^{t-1}d_{x,j}p^{j},\ x\in D_{X}$

be the $p$-adic expansion. For any $x\in D_{X}$, we put
$d_{x}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}{\rm ord}_{x}(D)$ and
$d_{x}^{(i)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\text{ord}_{x}(D))^{(i)}$
(Definition 2.4 (ii)).

#### 4.2.2.

First, we have the following necessary condition that the first generalized
Hasse-Witt invariants attain maximum.

######  Lemma 4.1.

We maintain the notation introduced above. Suppose that $n_{X}\geq 2$. Then
the following statements hold:

(i) Suppose that $s(D)=n_{X}-1$. Then ${\rm deg}(D^{(i)})\geq{\rm deg}(D)$ for
all $i\in\\{0,1,\dots,t-1\\}$ if and only if

$\sum_{x\in D_{X}}d_{x,j}=(n_{X}-1)(p-1),\ j\in\\{0,\dots,t-1\\}.$

Note that ${\rm deg}(D^{(i)})\geq{\rm deg}(D)=(n_{X}-1)n$ is equivalent to
${\rm deg}(D^{(i)})={\rm deg}(D)=(n_{X}-1)n$.

(ii) Suppose that there exists an element $\alpha\in{\rm Rev}_{D}^{\rm
adm}(X^{\bullet})$ such that $\gamma_{(\alpha,D)}$ attains maximum. Then we
have ${\rm deg}(D^{(i)})={\rm deg}(D)=(n_{X}-1)n$ for all
$i\in\\{0,1,\dots,t-1\\}$.

(iii) Suppose that $n>n_{X}-1$. Then there exists
$D^{\prime}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ such that
$s(D^{\prime})=n_{X}-1$, and that

${\rm deg}((D^{\prime})^{(i)})={\rm deg}(D^{\prime})=(n_{X}-1)n,\
i\in\\{0,1,\dots,t-1\\}.$

###### Proof.

(i) The “if” part of (i) is trivial. We only prove the “only if” part of (i).
Since ${\rm deg}(D^{(i)})\geq{\rm deg}(D)$ for all $i\in\\{0,\dots,t-1\\}$ and
$n|{\rm deg}(D^{(i)})$, we have

${\rm deg}(D^{(i)})=\sum_{x\in D_{X}}(\text{ord}_{x}(D))^{(i)}=\sum_{x\in
D_{X}}d_{x}^{(i)}=(n_{X}-1)n.$

Moreover, for each $i\in\\{0,\dots,t-1\\}$, we have

$d^{(i+1)}_{x}=d_{x,i}p^{t-1}+\frac{d_{x}^{(i)}-d_{x,i}}{p}=\frac{1}{p}d^{(i)}_{x}+\frac{p^{t}-1}{p}d_{x,i}=\frac{1}{p}d^{(i)}_{x}+\frac{n}{p}d_{x,i}.$

Thus, we obtain

$(n_{X}-1)n=\sum_{x\in D_{X}}d_{x}^{(i+1)}=\frac{1}{p}\sum_{x\in
D_{X}}d_{x}^{(i)}+\frac{n}{p}\sum_{x\in D_{X}}d_{x,i}$
$=\frac{1}{p}(n_{X}-1)n+\frac{n}{p}\sum_{x\in D_{X}}d_{x,i}.$

Then we have

$\sum_{x\in D_{X}}d_{x,i}=(n_{X}-1)(p-1),\ i\in\\{0,\dots,t-1\\}.$

(ii) This follows from Definition 3.10 (ii) and Remark 2.4.1.

(iii) Note that there exists
$D^{\prime\prime}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ such that
$s(D^{\prime\prime})=(n_{X}-1)n$ if $n>n_{X}-1$. Then (iii) follows from (i).
We complete the proof of the lemma. ∎

#### 4.2.3.

The following lemma will be used in the constructions of Galois admissible
coverings of degenerations of $X^{\bullet}$.

######  Lemma 4.2.

We maintain the notation introduced above. Suppose that $n_{X}\geq 3$, and
that ${\rm deg}(D^{(i)})={\rm deg}(D)=(n_{X}-1)n$ for all
$i\in\\{0,1,\dots,t-1\\}$. Moreover, we put
$D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},\dots,x_{n_{X}}\\}$
and

$a_{l,l+1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\sum_{r=l+1}^{n_{X}}d_{x_{r}}],\
b_{l,l+1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\sum_{r=1}^{l}d_{x_{r}}],\
l\in\\{2,\dots,n_{X}-2\\},$

where $[(-)]$ denotes the integer which is equal to the image of $(-)$ in
$\mathbb{Z}/n\mathbb{Z}$ when we identify $\\{0,\dots,n-1\\}$ with
$\mathbb{Z}/n\mathbb{Z}$ naturally. Then, for each $i\in\\{0,\dots,t-1\\}$, we
have

$\ a_{l,l+1}^{(i)}+b_{l,l+1}^{(i)}=n,\ l\in\\{2,\dots,n_{X}-2\\},$
$d_{x_{1}}^{(i)}+d_{x_{2}}^{(i)}+a_{2,3}^{(i)}=2n,$
$b_{n_{X}-2,n_{X}-1}^{(i)}+d_{x_{n_{X}-1}}^{(i)}+d_{x_{n_{X}}}^{(i)}=2n,$
$b_{l,l+1}^{(i)}+d_{x_{l+1}}^{(i)}+a_{l+1,l+2}^{(i)}=2n,\
l\in\\{2,\dots,n_{X}-3\\}.$

###### Proof.

First, let us treat the first equality. Let $l\in\\{2,\dots,n_{X}-2\\}$ and
$i\in\\{0,\dots,t-1\\}$. Since

$\sum_{r=l+1}^{n_{X}}d_{x_{r}}+\sum_{r=1}^{l}d_{x_{r}}=\text{deg}(D)=(n_{X}-1)n,$

we see $n|(a_{l,l+1}+b_{l,l+1})$. Note that if $a_{l,l+1}+b_{l,l+1}=0$, then
$\text{deg}(D)<(n_{X}-1)n$. Thus, $a_{l,l+1}+b_{l,l+1}\neq 0$. Moreover, since
$a_{l,l+1}+b_{l,l+1}\leq n$, we obtain that $a_{l,l+1}+b_{l,l+1}=n$, and that
$a^{(i)}_{l,l+1}+b^{(i)}_{l,l+1}$ is divided by $n$. On the other hand, since
$0<a^{(i)}_{l,l+1},b^{(i)}_{l,l+1}<n$, we have
$a_{l,l+1}^{(i)}+b_{l,l+1}^{(i)}=n$. This completes the proof of the first
equality.

Let $i\in\\{0,\dots,t-1\\}$. We denote by

$S^{(i)}_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}d^{(i)}_{x_{1}}+d^{(i)}_{x_{2}}+a^{(i)}_{2,3},$
$S^{(i)}_{l}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b^{(i)}_{l,l+1}+d^{(i)}_{x_{l+1}}+a^{(i)}_{l+1,l+2},\
l\in\\{2,\dots,n_{X}-3\\},$
$S^{(i)}_{n_{X}-2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b^{(i)}_{n_{X}-2,n_{X}-1}+d^{(i)}_{x_{n_{X}-1}}+d^{(i)}_{x_{n_{X}}}.$

We have $S_{l}^{(i)}\leq 2n$, $\ l\in\\{1,\dots,n_{X}-2\\}$. Moreover, if
$i=0$, the definitions of $a_{l,l+1}$ and $b_{l,l+1}$ imply

$S_{l}^{(0)}=2n,\ l\in\\{1,\dots,n_{X}-2\\}.$

Then we have that $S_{l}^{(i)}$ is divided by $n$ for all
$i\in\\{0,\dots,t-1\\}$. Since $s(D^{(i)})=n_{X}-1$, the first equality
implies that

$\sum_{l=1}^{n_{X}-2}S_{l}^{(i)}=n(n_{X}-3)+(n_{X}-1)n=2n(n_{X}-2).$

On the other hand, since $S_{l}^{(i)}\leq 2n$, we have

$S_{l}^{(i)}=2n,\ l\in\\{1,\dots,n_{X}-2\\},\ i\in\\{0,\dots,t-1\\}.$

This completes the proof of the lemma. ∎

#### 4.2.4.

The following lemma follows immediately from [B, Corollary 6.8].

######  Lemma 4.3.

We maintain the notation introduced above. Suppose that
$X^{\bullet}=(X,D_{X})$ is a smooth component-generic pointed stable curve of
type $(g_{X},n_{X})=(0,3)$ over $k$. Moreover, we suppose that ${\rm
deg}(D^{(i)})={\rm deg}(D)=(n_{X}-1)n=2n$ for all $i\in\\{0,1,\dots,t-1\\}.$
Then the Raynaud-Tamagawa theta divisor $\Theta_{\mathcal{E}_{D}}$ associated
to $\mathcal{E}_{D}$ exists (see Definition 2.7). Moreover, we have (see 2.3.4
for $\gamma_{([\mathcal{L}],D)}$)

$\gamma_{([\mathcal{L}],D)}={\rm dim}_{k}(H^{1}(X,\mathcal{L})),\
([\mathcal{L}],D)\in\widetilde{\mathscr{P}}_{X^{\bullet},n}.$

######  Remark 4.3.1.

Note that, if $n_{X}=3$, then we have $s(D)\in\\{0,1,2\\}$.

###  4.3. Degenerations

In this subsection, we introduce certain degenerations of $X^{\bullet}$ and
prove the first main result in the case of irreducible component-generic
pointed stable curves (Proposition 4.5).

####  4.3.1. Settings

We maintain the notation introduced in 2.1.2 and the assumption introduced in
2.3.3 (i.e., $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1$). Moreover,
we assume that
$X^{\bullet}=(X,D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},\dots,x_{n_{X}}\\})$
is an irreducible component-generic pointed stable curve over $k$.

####  4.3.2. Degeneration data

We introduce some degeneration data for $X^{\bullet}$. Let $R$ be a discrete
valuation ring with algebraically closed residue field $k_{R}$, $K_{R}$ the
quotient field of $R$, and $\overline{K}_{R}$ an algebraic closure of $K_{R}$.
Suppose that $k\subseteq K_{R}$. Let
$\mathcal{X}^{\bullet}=(\mathcal{X},D_{\mathcal{X}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{e_{1},\dots,e_{n_{X}}\\})$
be a pointed stable curve of type $(g_{X},n_{X})$ over $R$. We put

$\mathcal{X}^{\bullet}_{\eta}=(\mathcal{X}_{\eta},D_{\mathcal{X}_{\eta}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{e_{\eta,1},\dots,e_{\eta,n_{X}}\\})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathcal{X}^{\bullet}\times_{R}K_{R},$
$\mathcal{X}^{\bullet}_{\overline{\eta}}=(\mathcal{X}_{\overline{\eta}},D_{\mathcal{X}_{\overline{\eta}}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{e_{\overline{\eta},1},\dots,e_{\overline{\eta},n_{X}}\\})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathcal{X}^{\bullet}\times_{R}\overline{K}_{R},$
$\mathcal{X}^{\bullet}_{s}=(\mathcal{X}_{s},D_{\mathcal{X}_{s}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{e_{s,1},\dots,e_{s,n_{X}}\\})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathcal{X}^{\bullet}\times_{R}k_{R}.$

We shall say that $X^{\bullet}$ admits a (DEG) if $(g_{X},n_{X})\neq(1,1)$ and
there exists $\mathcal{X}^{\bullet}$ such that the following conditions hold,
where “(DEG)” means “degeneration”:

(i) We have that $\mathcal{X}^{\bullet}_{\overline{\eta}}$ is
$\overline{K}_{R}$-isomorphic to $X^{\bullet}\times_{k}\overline{K}_{R}$, and
that $\mathcal{X}_{s}^{\bullet}$ is a component-generic pointed stable curve
over $k_{R}$. Then without loss of generality, we may identify
$e_{\overline{\eta},r},\ r\in\\{1,\dots,n_{X}\\}$, with
$x_{r}\times_{k}\overline{K}_{R}$ via this isomorphism.

(iii) Let $\mathscr{T}$ be a set with cardinality $\\#(\mathscr{T})=\\#(X^{\rm
sing})$ which consists of irreducible singular projective semi-stable curves
of genus $1$ (i.e., the normalizations are $\mathbb{P}^{1}_{k_{R}}$). Let
$\mathscr{P}$ be a set which consists of smooth semi-stable curves of genus
$0$ (i.e., a set of $\mathbb{P}^{1}_{k_{R}}$). Let $C_{1}$ be either an empty
set or a smooth semi-stable curve of genus $g_{X}-\\#(X^{\rm sing})$. We have
the set of irreducible components of $\mathcal{X}_{s}$ is

$\\{T\\}_{T\in\mathscr{T}}\cup\\{C_{1}\\}\cup\\{P\\}_{P\in\mathscr{P}}.$

Moreover, one of the following conditions is satisfied (see Example 4.4 below
for examples of (a) (b) (c)):

> (a) Suppose that $n_{X}\leq 1$ and $\\#(X^{\rm sing})\geq 1$. Then the
> following conditions hold:
>
>   * •
>
> $C_{1}=\emptyset$ when $n_{X}=0$ and $\\#(X^{\rm sing})=2$; otherwise,
> $C_{1}\neq\emptyset$.
>
>   * •
>
> When $C_{1}=\emptyset$, we have
> $\mathscr{T}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{T_{1},T_{2}\\}$
> such that $\\#(T_{1}\cap T_{2})=1$.
>
>   * •
>
> When $C_{1}\neq\emptyset$, we have that $T^{\prime}\cap
> T^{\prime\prime}\not\neq\emptyset$ if and only if
> $T^{\prime}=T^{\prime\prime}$ for all
> $T^{\prime},T^{\prime\prime}\in\mathscr{T}$, that $\\#(T\cap C_{1})=1$ for
> all $T\in\mathscr{T}$, and that $D_{\mathcal{X}_{s}}\subseteq C_{1}$.
>
>   * •
>
> $\mathscr{P}=\emptyset$.
>
>

>
> (b) Suppose that $n_{X}=2$. Then the following conditions hold:
>
>   * •
>
> $T^{\prime}\cap T^{\prime\prime}\not\neq\emptyset$ if and only if
> $T^{\prime}=T^{\prime\prime}$ for all
> $T^{\prime},T^{\prime\prime}\in\mathscr{T}$.
>
>   * •
>
> $C_{1}=\emptyset$ when $g_{X}-\\#(X^{\rm sing})=0$; otherwise,
> $C_{1}\neq\emptyset$ when $g_{X}-\\#(X^{\rm sing})\geq 1$.
>
>   * •
>
> $\mathscr{P}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{P\\}$ such that
> $D_{\mathcal{X}_{s}}\subseteq P$.
>
>   * •
>
> When $C_{1}=\emptyset$, we have $\\#(P\cap T)=1$ for all $T\in\mathscr{T}$.
>
>   * •
>
> When $C_{1}\neq\emptyset$, we have that $\\#(C_{1}\cap T)=1$, that
> $\\#(C_{1}\cap P)=1$, and that $P\cap T=\emptyset$ for all
> $T\in\mathscr{T}$.
>
>

>
> (c) Suppose that $n_{X}\geq 3$. Then the following conditions hold:
>
>   * •
>
> The first and the second conditions of (b) hold.
>
>   * •
>
>
> $\mathscr{P}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{P_{v}\\}_{v\in\\{2,\dots,n_{X}-1\\}}$
> such that
> $D_{\mathcal{X}_{s}}\subseteq\bigcup_{v\in\\{2,\dots,n_{X}-1\\}}P_{v}$ .
>
>   * •
>
> When $C_{1}=\emptyset$, we have $\\#(P_{2}\cap T)=1$ and $P_{v}\cap
> T=\emptyset$ for all $T\in\mathscr{T}$ and all $v\neq 2$.
>
>   * •
>
> When $C_{1}\neq\emptyset$, we have $\\#(C_{1}\cap T)=1$, $\\#(C_{1}\cap
> P_{2})=1$, $C_{1}\cap P_{v}=\emptyset$, and $P_{v}\cap T=\emptyset$ for all
> $T\in\mathscr{T}$ and all $v\neq 2$.
>
>   * •
>
> If $n_{X}\geq 4$, for each $v\in\\{2,\dots,n_{X}-2\\}$, we have
> $\\#(P_{v}\cap P_{v+1})=1$ and $P_{v}\cap P_{v^{\prime}}=\emptyset$ when
> $v^{\prime}\not\in\\{v-1,v,v+1\\}$.
>
>   * •
>
> If $n_{X}=3$, we have $D_{\mathcal{X}_{s}}\cap
> P_{2}=\\{e_{s,1},e_{s,2},e_{s,3}\\}$.
>
>   * •
>
> If $n_{X}=4$, we have $D_{\mathcal{X}_{s}}\cap P_{2}=\\{e_{s,1},e_{s,2}\\}$
> and $D_{\mathcal{X}_{s}}\cap P_{3}=\\{e_{s,3},e_{s,4}\\}$.
>
>   * •
>
> If $n_{X}\geq 5$, we have $D_{\mathcal{X}_{s}}\cap
> P_{2}=\\{e_{s,1},e_{s,2}\\}$, $D_{\mathcal{X}_{s}}\cap
> P_{n_{X}-1}=\\{e_{s,n_{X}-1},e_{s,n_{X}}\\}$, and $D_{\mathcal{X}_{s}}\cap
> P_{v}=\\{e_{s,v}\\}$, $v\in\\{3,\dots,n_{X}-2\\}$.
>
>

Note that since generic curves admit all degeneration types, we have that
$X^{\bullet}$ admits a (DEG) when $X^{\bullet}$ is a component-generic pointed
stable curve of type $(g_{X},n_{X})\neq(1,1)$.

#### 4.3.3.

Next, we give some examples to explain the degeneration data introduced above.
For simplicity, we assume that $\\#(X^{\rm sing})=2$, $C_{1}\neq\emptyset$,
and $n_{X}\neq 0$.

######  Example 4.4.

We use the notation “$\bullet$” and “$\circ$ with a line segment” to denote a
vertex and an open edge, respectively. Moreover, we use $v_{(-)}$ to denote
the vertex corresponding to the irreducible component $(-)$.

(a) If $n_{X}=1$, then the dual semi-graph
$\Gamma_{\mathcal{X}_{s}^{\bullet}}$ of $\mathcal{X}_{s}^{\bullet}$ is as
follows (see 4.3.2 (iii)-(a)):

$v_{T_{1}}$

$v_{C_{1}}$

$v_{T_{2}}$

$\Gamma_{\mathcal{X}^{\bullet}_{s}}$:

(b) If $n_{X}=2$, then the dual semi-graph
$\Gamma_{\mathcal{X}_{s}^{\bullet}}$ of $\mathcal{X}_{s}^{\bullet}$ is as
follows (see 4.3.2 (iii)-(b)):

$v_{T_{1}}$

$v_{C_{1}}$

$v_{T_{2}}$

$v_{P}$

$\Gamma_{\mathcal{X}^{\bullet}_{s}}$:

(c) If $n_{X}=5$, then the dual semi-graph
$\Gamma_{\mathcal{X}_{s}^{\bullet}}$ of $\mathcal{X}_{s}^{\bullet}$ is as
follows (see 4.3.2 (iii)-(c)):

$v_{T_{1}}$

$v_{C_{1}}$

$v_{T_{2}}$

$v_{P_{2}}$

$v_{P_{3}}$

$v_{P_{4}}$

$\Gamma_{\mathcal{X}^{\bullet}_{s}}$:

#### 4.3.4.

The main result of the present subsection is as follows:

######  Proposition 4.5.

Let
$X^{\bullet}=(X,D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},\dots,x_{n_{X}}\\})$
be an irreducible component-generic pointed stable curve over $k$,
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$, and $\alpha\in{\rm
Rev}_{D}^{\rm adm}(X^{\bullet})\setminus\\{0\\}$. Suppose that ${\rm
deg}(D)=(n_{X}-1)n$ (i.e., $s(D)=n_{X}-1$) if $n_{X}\neq 0$, and that ${\rm
deg}(D^{(i)})={\rm deg}(D)=(n_{X}-1)n$ for all $i\in\\{0,1,\dots,t-1\\}.$ Then
$\gamma_{(\alpha,D)}$ attains maximum. Namely, the following holds:

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

###### Proof.

Let $f^{\bullet}:Y^{\bullet}=(Y,D_{Y})\rightarrow X^{\bullet}$ be the Galois
multi-admissible covering over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$
induced by $\alpha$. Firstly, we note that, to verify the proposition, we may
assume that $Y^{\bullet}$ is connected.

Suppose that $X^{\bullet}$ is smooth over $k$, and that $n_{X}\leq 1$. Then
the proposition follows immediately from Proposition 3.11 (i). Thus, to verify
the proposition, it is sufficient to assume that one of the following
conditions holds: (1) $\\#(X^{\rm sing})\geq 1$ and $n_{X}\leq 1$; (2)
$n_{X}\geq 2$.

Suppose that $X^{\bullet}$ is a singular curve of type $(1,1)$ (i.e.,
$\\#(X^{\rm sing})=1$). Since $f^{\bullet}$ is étale, the proposition follows
immediately from Lemma 3.1 (i). Thus, to verify the proposition, we may assume
$(g_{X},n_{X})\neq(1,1)$. Now, we can use the degeneration data introduced in
4.3.2.

Since $X^{\bullet}$ is a component-generic pointed stable curve, $X^{\bullet}$
admits a (DEG). We maintain the notation introduced in 4.3.2. Furthermore, we
write $Q_{\overline{\eta}}$ (resp. $Q_{s}$) for the effective divisor on
$\mathcal{X}_{\overline{\eta}}$ (resp. $\mathcal{X}_{s}$) induced by $D$ and
$\alpha_{\overline{\eta}}\in\text{Rev}^{\rm
adm}_{Q_{\overline{\eta}}}(\mathcal{X}^{\bullet}_{\overline{\eta}})$ for the
element induced by $\alpha$. Then we have
$\gamma_{(\alpha,D)}=\gamma_{(\alpha_{\overline{\eta}},Q_{\overline{\eta}})}.$
Write $\Pi_{\mathcal{X}_{\overline{\eta}}^{\bullet}}$ and
$\Pi_{\mathcal{X}_{s}^{\bullet}}$ for the admissible fundamental groups of
$\mathcal{X}^{\bullet}_{\overline{\eta}}$ and $\mathcal{X}_{s}^{\bullet}$,
respectively. Then we have a specialization surjective homomorphism

$sp_{R}:\Pi_{\mathcal{X}_{\overline{\eta}}^{\bullet}}\twoheadrightarrow\Pi_{\mathcal{X}_{s}^{\bullet}}.$

We suppose that $X^{\bullet}$ satisfies (DEG)-(iii)-(c) (4.3.2). Moreover, we
suppose that $C_{1}\neq\emptyset$ and $n_{X}\geq 5$ (see Example 4.4 (c)).

Step 1: We define certain smooth pointed stable curves associated to
irreducible components of $\mathcal{X}^{\bullet}_{s}$.

We write $y_{v,v+1},\ z_{v,v+1}$, $v\in\\{2,\dots,n_{X}-2\\}$, for the inverse
image of $P_{v}\cap P_{v+1}$ of the natural closed immersion
$P_{v}\hookrightarrow\mathcal{X}_{s}$ and the inverse image of $P_{v}\cap
P_{v+1}$ of the natural closed immersion
$P_{v+1}\hookrightarrow\mathcal{X}_{s}$, respectively. We define

$P_{2}^{\bullet}=(P_{2},D_{P_{2}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{e_{s,1},e_{s,2},y_{2,3}\\}\cup(C_{1}\cap
P_{2})),$
$P_{n_{X}-1}^{\bullet}=(P_{n_{X}-1},D_{P_{n_{X}-1}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{z_{n_{X}-2,n_{X}-1},e_{s,n_{X}-1},e_{s,n_{X}}\\}),$
$P_{v}^{\bullet}=(P_{v},D_{P_{v}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{z_{v-1,v},e_{s,v},y_{v,v+1}\\}),\
v\in\\{3,\dots,n_{X}-2\\},$

to be smooth pointed stable curves of types $(0,4)$, $(0,3)$, and $(0,3)$ over
$k_{R}$, respectively. Moreover, we define

$C_{1}^{\bullet}=(C_{1},D_{C_{1}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(C_{1}\cap
P_{2})\cup((\bigcup_{T\in\mathscr{T}}T)\cap C_{1})),$
$T^{\bullet}=(T,D_{T}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}T\cap C_{1}),\
T\in\mathscr{T},$

to be a smooth pointed stable curve of type $(g_{X}-\\#(X^{\rm
sing}),1+\\#(X^{\rm sing}))$ and a singular pointed stable curve of type
$(1,1)$ over $k_{R}$, respectively. Note that since $C_{1}$ is generic, we
have $\sigma_{C_{1}}=g_{X}-\\#(X^{\rm sing})$ (i.e., $C_{1}^{\bullet}$ is
ordinary (2.2.2)).

Step 2: We construct a Galois admissible covering of
$\mathcal{X}_{s}^{\bullet}$ by using specialization isomorphisms of maximal
prime-to-$p$ quotients of admissible fundamental groups.

Let
$f^{\bullet}_{\overline{\eta}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{\bullet}\times_{k}\overline{K}_{R}:\mathcal{Y}_{\overline{\eta}}^{\bullet}=(\mathcal{Y}_{\overline{\eta}},D_{\mathcal{Y}_{\overline{\eta}}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Y^{\bullet}\times_{k}\overline{K}_{R}\rightarrow\mathcal{X}_{\overline{\eta}}^{\bullet}$
be the Galois admissible covering over $\overline{K}_{R}$ with Galois group
$\mathbb{Z}/n\mathbb{Z}$ induced by $f^{\bullet}$, and
$\Pi_{\mathcal{Y}_{\overline{\eta}}^{\bullet}}\subseteq\Pi_{\mathcal{X}_{\overline{\eta}}^{\bullet}}\cong\Pi_{X^{\bullet}}$
the admissible fundamental group of $\mathcal{Y}_{\overline{\eta}}^{\bullet}$.
By the specialization theorem of maximal prime-to-$p$ quotients of admissible
fundamental groups ([V, Théorème 2.2 (c)]), we have

$sp^{p^{\prime}}_{R}:\Pi_{\mathcal{X}^{\bullet}_{\overline{\eta}}}^{p^{\prime}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Pi_{\mathcal{X}^{\bullet}_{s}}^{p^{\prime}}.$

Then we obtain a normal open subgroup
$\Pi_{\mathcal{Y}_{s}^{\bullet}}^{p^{\prime}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}sp^{p^{\prime}}_{R}(\Pi_{\mathcal{Y}_{\overline{\eta}}^{\bullet}}^{p^{\prime}})\subseteq\Pi_{\mathcal{X}^{\bullet}_{s}}^{p^{\prime}}$.
Write
$\Pi_{\mathcal{Y}_{s}^{\bullet}}\subseteq\Pi_{\mathcal{X}_{s}^{\bullet}}$ for
the inverse image of $\Pi_{\mathcal{Y}_{s}^{\bullet}}^{p^{\prime}}$ of the
natural surjection
$\Pi_{\mathcal{X}_{s}^{\bullet}}\twoheadrightarrow\Pi_{\mathcal{X}_{s}^{\bullet}}^{p^{\prime}}$.
Then $\Pi_{\mathcal{Y}_{s}^{\bullet}}$ determines a Galois admissible covering

$f^{\bullet}_{s}:\mathcal{Y}_{s}^{\bullet}=(\mathcal{Y}_{s},D_{\mathcal{Y}_{s}})\rightarrow\mathcal{X}_{s}^{\bullet}$

over $k_{R}$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. We denote by
$\alpha_{s}\in\text{Rev}_{Q_{s}}^{\rm adm}(\mathcal{X}_{s}^{\bullet})$ the
element induced by the composition homomorphism
$\Pi_{\mathcal{X}^{\bullet}_{s}}^{p^{\prime},\rm
ab}\overset{(sp_{R}^{p^{\prime},\rm
ab})^{-1}}{\stackrel{{\scriptstyle\sim}}{{\rightarrow}}}\Pi_{\mathcal{X}^{\bullet}_{\overline{\eta}}}^{p^{\prime},\rm
ab}\overset{\alpha_{\overline{\eta}}}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$.

Step 3: We compute the generalized Hasse-Witt invariant
$\gamma_{(\alpha_{s},D_{\alpha_{s}})}$ by applying Theorem 3.9, Lemma 4.2, and
Lemma 4.3.

The structure of $\Pi_{\mathcal{X}_{s}^{\bullet}}^{p^{\prime}}$ (2.1.7)
implies that $f_{s}$ is étale over $(\bigcup_{T\in\mathscr{T}}T)\cap C_{1}$.
Then we obtain that $f_{s}$ is étale over $C_{1}\cap P_{2}$. Thus, $f_{s}$ is
étale over $D_{C_{1}}$. Let
$Y_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{-1}_{s}(P_{v})$,
$v\in\\{2,\dots,n_{X}-1\\}$. We put

$Y_{v}^{\bullet}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(Y_{v},D_{Y_{v}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{-1}_{s}(D_{P_{v}})),\
v\in\\{2,\dots,n_{X}-1\\}.$

Then $f^{\bullet}_{s}$ induces a Galois multi-admissible covering

$f_{v}^{\bullet}:Y^{\bullet}_{v}\rightarrow P_{v}^{\bullet},\
v\in\\{2,\dots,n_{X}-1\\},$

over $k_{R}$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. We maintain the
notation introduced in Lemma 4.2. Then we see that the ramification divisor on
$P_{v}$, $v\in\\{2,\dots,n_{X}-1\\}$, induced by $f_{v}^{\bullet}$ and
$\alpha_{s}$ is as follows:

$Q_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}d_{x_{1}}e_{s,1}+d_{x_{2}}e_{s,2}+a_{2,3}y_{2,3},$
$Q_{n_{X}-1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{n_{X}-2,n_{X}-1}z_{n_{X}-2,n_{X}-1}+d_{x_{n_{X}-1}}e_{s,n_{X}-1}+d_{x_{n_{X}}}e_{s,n_{X}},$
$Q_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{v-1,v}z_{v-1,v}+d_{x_{v}}e_{s,v}+a_{v,v+1}y_{v,v+1},\
v\in\\{3,\dots,n_{X}-2\\}.$

Since $f_{s}$ is étale over $C_{1}\cap P_{2}$, we see that $f_{v}^{\bullet},\
v\in\\{2,\dots,n_{X}-1\\},$ induces a pair
$([\mathcal{L}_{v}],Q_{v})\in\widetilde{\mathscr{P}}_{P_{v}^{\bullet},n}$.
Moreover, the $k_{R}[\mu_{n}]$-module
$H^{1}_{\text{\'{e}t}}(Y_{v},\mathbb{F}_{p})\otimes k_{R}$ admits the
following canonical decomposition

$H^{1}_{\text{\'{e}t}}(Y_{v},\mathbb{F}_{p})\otimes
k_{R}=\bigoplus_{j\in\mathbb{Z}/n\mathbb{Z}}M_{Y_{v}}(j),$

where $\zeta\in\mu_{n}$ acts on $M_{Y_{v}}(j)$ as the
$\zeta^{j}$-multiplication. On the other hand, lemma 4.2 implies
$\text{deg}(Q_{v}^{(i)})=\text{deg}(Q_{v})=2n$, $i\in\\{0,\dots,t-1\\}$. Then
by applying Lemma 4.3, we have

$\gamma_{([\mathcal{L}_{v}],Q_{v})}=\text{dim}_{k_{R}}(M_{Y_{v}}(1))=\text{dim}_{k_{R}}(H^{1}(P_{v},\mathcal{L}_{v}))=1.$

Let $Z_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{-1}_{s}(C_{1})$. Then
$f^{\bullet}_{s}$ induces a Galois étale covering (not necessarily connected)

$f^{\bullet}_{C_{1}}:Z^{\bullet}_{1}=(Z_{1},D_{Z_{1}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{-1}_{s}(D_{C_{1}}))\rightarrow
C_{1}^{\bullet}$

over $k_{R}$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Write
$\alpha_{C_{1}}\in\text{Rev}_{0}^{\rm adm}(C_{1}^{\bullet})$ for the element
induced by $f_{C_{1}}^{\bullet}$ and $\alpha_{s}$. The $k_{R}[\mu_{n}]$-module
$H^{1}_{\text{\'{e}t}}(Z_{1},\mathbb{F}_{p})\otimes k_{R}$ admits the
following canonical decomposition

$H^{1}_{\text{\'{e}t}}(Z_{1},\mathbb{F}_{p})\otimes
k_{R}=\bigoplus_{j\in\mathbb{Z}/n\mathbb{Z}}M_{Z_{1}}(j),$

where $\zeta\in\mu_{n}$ acts on $M_{Z_{1}}(j)$ as the
$\zeta^{j}$-multiplication. Then Proposition 3.11 (i) implies

$\displaystyle\gamma_{(\alpha_{C_{1}},0)}=\left\\{\begin{array}[]{ll}g_{X}-\\#(X^{\rm
sing}),&\text{if}\ \alpha_{C_{1}}=0,\\\ g_{X}-\\#(X^{\rm sing})-1,&\text{if}\
\alpha_{C_{1}}\neq 0.\end{array}\right.$

Let $V_{T}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{-1}_{s}(T)$,
$T\in\mathscr{T}$, and $\widetilde{T}$ the smooth compactification of
$U_{T}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}T\setminus T^{\rm sing}$. Then
$f^{\bullet}_{s}$ induces a Galois multi-admissible covering

$f_{T}^{\bullet}:V_{T}^{\bullet}=(V_{T},D_{V_{T}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f_{s}^{-1}(D_{T}))\rightarrow
T^{\bullet}$

over $k_{R}$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Then
$f_{T}^{\bullet}$ induces a Galois multi-admissible covering

$f_{\widetilde{T}}^{\bullet}:V^{\bullet}_{\widetilde{T}}=(V_{\widetilde{T}},D_{V_{\widetilde{T}}})\rightarrow\widetilde{T}^{\bullet}$

over $k_{R}$. Write
$\alpha_{\widetilde{T}}\in\text{Rev}_{0}(\widetilde{T}^{\bullet})$ for the
element induced by $f_{\widetilde{T}}^{\bullet}$ and $\alpha_{s}$. By using
the proposition of the case where $X^{\bullet}$ is a singular curve of type
$(g_{X},n_{X})=(1,1)$, we obtain that $\gamma_{(\alpha_{\widetilde{T}},0)}=0$.
Thus, Theorem 3.9 implies that

$\gamma_{(\alpha_{s},Q_{s})}=g_{X}+n_{X}-2.$

Step 4: We compute $\gamma_{(\alpha_{\overline{\eta}},Q_{\overline{\eta}})}$
by using specialization surjective homomorphisms of admissible fundamental
groups.

The $k_{R}[\mu_{n}]$-modules
$H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{\overline{\eta}},\mathbb{F}_{p})\otimes
k_{R}$ and $H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{s},\mathbb{F}_{p})\otimes
k_{R}$ admit the following canonical decompositions

$H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{\overline{\eta}},\mathbb{F}_{p})\otimes
k_{R}=\bigoplus_{j\in\mathbb{Z}/n\mathbb{Z}}M_{\mathcal{Y}_{\overline{\eta}}}(j),$
$H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{s},\mathbb{F}_{p})\otimes
k_{R}=\bigoplus_{j\in\mathbb{Z}/n\mathbb{Z}}M_{\mathcal{Y}_{s}}(j),$

respectively. Moreover, we have an injection as $k_{R}[\mu_{n}]$-modules

$H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{s},\mathbb{F}_{p})\otimes
k_{R}\hookrightarrow
H^{1}_{\rm\text{\'{e}t}}(\mathcal{Y}_{\overline{\eta}},\mathbb{F}_{p})\otimes
k_{R}$

induced by the specialization map
$\Pi_{\mathcal{Y}^{\bullet}_{\overline{\eta}}}\twoheadrightarrow\Pi_{\mathcal{Y}_{s}^{\bullet}}$.
Thus, we have

$g_{X}+n_{X}-2=\gamma_{(\alpha_{s},Q_{s})}=\text{dim}_{k_{R}}(M_{\mathcal{Y}_{s}}(1))$
$\leq\gamma_{(\alpha_{\overline{\eta}},Q_{\overline{\eta}})}=\text{dim}_{k_{R}}(M_{\mathcal{Y}_{\overline{\eta}}}(1))\leq
g_{X}+n_{X}-2.$

This means
$\gamma_{(\alpha_{\overline{\eta}},Q_{\overline{\eta}})}=g_{X}+n_{X}-2.$ We
complete the proof of the proposition when $X^{\bullet}$ satisfies
(DEG)-(iii)-(c), $C_{1}\neq\emptyset$, and $n_{X}\geq 5$.

By applying similar arguments to the arguments given in the proof above, one
can prove the proposition when $X^{\bullet}$ satisfies (DEG)-(iii)-(c) and
either $C_{1}=\emptyset$ or $n_{X}\leq 4$ holds. Moreover, similar arguments
to the arguments given in the proof above imply that the proposition holds
when $X^{\bullet}$ satisfies either (DEG)-(iii)-(a) or (DEG)-(iii)-(b). We
complete the proof of the proposition. ∎

In the remainder of this section, under certain assumptions, we generalize
Proposition 4.5 to the case where $X^{\bullet}$ is not necessary to be
irreducible.

either (DEG)-(v) or (DEG)-(vi). We complete the proof of the proposition.

#### 4.3.5.

By applying Proposition 4.5, we solve Problem 2.11 under certain assumptions
as follows:

######  Corollary 4.6.

Let $X^{\bullet}=(X,D_{X})$ be a smooth component-generic pointed stable curve
over $k$ and $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$. Suppose that
$s(D)=n_{X}-1$ if $n_{X}\neq 0$, and that ${\rm deg}(D^{(i)})\geq{\rm
deg}(D),\ i\in\\{0,1,\dots,t-1\\}.$ Then the Raynaud-Tamagawa theta divisor
$\Theta_{\mathcal{E}_{D}}$ associated to $\mathcal{E}_{D}$ exists.

###### Proof.

Since $X^{\bullet}$ is smooth over $k$, the corollary follows immediately from
Proposition 4.5 and Remark 2.7.2. ∎

###  4.4. Minimal quasi-trees

In this subsection, we introduce the so-called minimal quasi-trees which play
an important role in the remainder of the present paper.

####  4.4.1. Settings

Let $W^{\bullet}=(W,D_{W})$ be a pointed stable curve of type $(g_{W},n_{W})$
over $k$ and $\Gamma_{W^{\bullet}}$ the dual semi-graph of $W^{\bullet}$.

#### 4.4.2.

Let $Z^{\bullet}$ be a pointed sable curve over $k$, $\Gamma_{Z^{\bullet}}$
the dual semi-graph of $Z^{\bullet}$ such that $\Gamma_{Z^{\bullet}}\setminus
e^{\rm lp}(\Gamma_{Z^{\bullet}})$ is a tree (see 2.1.1 for $e^{\rm
lp}(\Gamma_{Z^{\bullet}})$), and that $E_{Z}\subseteq e^{\rm
op}(\Gamma_{Z^{\bullet}})$ is a subset of open edges.

We put

$V_{Z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{v\in
v(\Gamma_{Z^{\bullet}})\ |\ \\#(e^{\Gamma_{Z^{\bullet}}}(v)\cap(e^{\rm
cl}(\Gamma_{Z^{\bullet}})\setminus e^{\rm lp}(\Gamma_{Z^{\bullet}})))=1,\
Z_{v}\cap E_{Z}=\emptyset\\}\subseteq v(\Gamma_{Z^{\bullet}}),$

where $Z_{v}$ denotes the irreducible component corresponding to $v$. We call
$V_{Z}$ the set of terminal vertices avoiding to $E_{Z}$.

For instance, in Example 4.9 (b) below, we have
$Z^{\bullet}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}W_{1}^{\bullet}$,
$E_{Z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{\rm
op}(\Gamma_{Z^{\bullet}})=e^{\rm op}(\Gamma_{W_{1}^{\bullet}})$, and
$V_{Z}=\\{v_{3}\\}$.

####  4.4.3. Constructions of minimal quasi-trees associated to $D_{W}$

Next, we define a minimal quasi-tree associated to $D_{W}$. Let $E\subseteq
e^{\rm cl}(\Gamma_{W^{\bullet}})\setminus e^{\rm lp}(\Gamma_{W^{\bullet}})$ be
a (possibly empty) subset of closed edges such that
$\Gamma_{W^{\bullet}}\setminus(E\cup e^{\rm lp}(\Gamma_{W^{\bullet}}))$ is a
connected tree. Note that it is easy to see that $E$ exists.

Write $N$ for the set of nodes of $W$ corresponding to the closed edges which
are contained in $E$, and write $\text{norm}_{N}:W_{N}\rightarrow W$ for the
normalization morphism of the underlying curve $W$ over $N$. We define a
pointed stable curve over $k$ to be

$W_{1}^{\bullet}=(W_{1},D_{W_{1}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}W_{N}^{\bullet}=(W_{N},D_{W_{N}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{norm}_{N}^{-1}(D_{W}\cup
N)).$

Note that the above construction implies $D_{W}\subseteq D_{W_{1}}$ (i.e.,
$e^{\rm op}(\Gamma_{W^{\bullet}})\subseteq e^{\rm
op}(\Gamma_{W_{1}^{\bullet}})$). Write $\Gamma_{W_{1}^{\bullet}}$ for the dual
semi-graph of $W_{1}^{\bullet}$. Then the construction of $W_{1}^{\bullet}$
implies that $\Gamma_{W_{1}^{\bullet}}\setminus e^{\rm
lp}(\Gamma_{W_{1}^{\bullet}})$ is a tree.

Suppose that $n_{W}\neq 0$. Then for all $i\in\mathbb{N}$, if
$W_{i}^{\bullet}$ has already been defined, we may define $W_{i+1}^{\bullet}$
as follows: Let $V_{i}\subseteq v(\Gamma_{W_{i}^{\bullet}})$ be the subset of
terminal vertices avoiding to $e^{\rm op}(\Gamma_{W^{\bullet}})\subseteq
e^{\rm op}(\Gamma_{W_{i}^{\bullet}})$. If $V_{i}=\emptyset$, then we put
$W^{\bullet}_{i+1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}W^{\bullet}_{i}$.
If $V_{i}\neq\emptyset$, we write $W_{i+1}$ for the topological closure of

$W_{i}\setminus(\bigcup_{v\in V_{i}}W_{v})$

in $W_{i}$. Note that the definition of $V_{i}$ implies that $W_{i+1}$ is
connected, and that $D_{W}$ is contained in $W_{i+1}$. Then we define a
pointed stable curve over $k$ to be

$W_{i+1}^{\bullet}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(W_{i+1},D_{W_{i+1}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(D_{W_{i}}\cap
W_{i+1})\cup((\bigcup_{v\in V_{i}}W_{v})\cap W_{i+1}))).$

Note that we have $e^{\rm op}(\Gamma_{W^{\bullet}})\subseteq e^{\rm
op}(\Gamma_{W_{i+1}^{\bullet}})$.

Let $i_{0}$ be the minimal natural number such that $V_{i_{0}}=\emptyset$.
Note that the above construction implies that
$W_{i_{0}}^{\bullet}=W_{j}^{\bullet}$ for all $j\geq i_{0}$. We put

$W^{\bullet}_{\Gamma}=(W_{\Gamma},D_{W_{\Gamma}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}W_{i_{0}}^{\bullet},\
\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{W_{i_{0}}^{\bullet}}.$

Then we have that $D_{W}\subseteq D_{W_{\Gamma}}$, and that $\Gamma\setminus
e^{\rm lp}(\Gamma)$ is a tree. Moreover, we see $v(\Gamma)\subseteq
v(\Gamma_{W^{\bullet}})$ and $e(\Gamma)\subseteq e(\Gamma_{W^{\bullet}})$,
where $e(-)$ denotes the set of edges of $(-)$ (2.1.1). Note that an open edge
of $\Gamma$ is not an open edge of $\Gamma_{W^{\bullet}}$ in general, and that
a closed edge of $\Gamma$ is a closed edge of $\Gamma_{W^{\bullet}}$.

Moreover, the above constructions imply the following:

######  Lemma 4.7.

Let $W^{\bullet}=(W,D_{W})$ be a pointed stable curve of type $(g_{W},n_{W})$
over $k$. Suppose that $n_{W}\neq 0$. We denote by $\mathscr{Q}_{W^{\bullet}}$
the set of $\Gamma$ constructed as above. Then we have
$\mathscr{Q}_{W^{\bullet}}\neq\emptyset.$

Note that the construction of $\Gamma$ depends on the choice of $E$ (i.e., a
subset of closed edges of $e^{\rm cl}(\Gamma_{W^{\bullet}})\setminus e^{\rm
lp}(\Gamma_{W^{\bullet}})$ such that $\Gamma_{W^{\bullet}}\setminus(E\cup
e^{\rm lp}(\Gamma_{W^{\bullet}}))$ is a tree).

####

######  Definition 4.8.

We maintain the notation introduced above. We shall say that a semi-graph

$\Gamma_{D_{W}}$

is a minimal quasi-tree associated to $D_{W}$ if $\Gamma_{D_{W}}=\Gamma$ for
some $\Gamma$ constructed above when $n_{W}\neq 0$, and
$\Gamma_{D_{W}}=\emptyset$ when $n_{W}=0$.

#### 4.4.4.

We maintain the notation introduced in 4.4.3. Let us give some examples to
explain the above constructions.

######  Example 4.9.

(a) Let $W^{\bullet}$ be a pointed stable curve over $k$ such that the
following conditions hold: (i) The set of irreducible components of $W$ is
$\\{W_{v_{1}},W_{v_{2}},W_{v_{3}}\\}$; (ii) $D_{W}=\\{w_{b_{1}},w_{b_{2}}\\}$;
(iii) The set of nodes is $\\{w_{c},w_{a_{1}},w_{a_{2}},w_{a_{3}}\\}$; (iv)
$W_{v_{1}}$ is a singular curve with the unique node $w_{c}$; (v)
$w_{b_{1}}\in W_{v_{1}}$ and $w_{b_{2}}\in W_{v_{2}}$; (vi)
$w_{a_{1}},w_{a_{2}}\in W_{v_{1}}\cap W_{v_{2}}$; (vii) $w_{a_{3}}\in
W_{v_{2}}\cap W_{v_{3}}$. We use the notation “$\bullet$” and “$\circ$” to
denote a node and a marked point, respectively. Then $W^{\bullet}$ is as
follows:

\begin{picture}(300.0,100.0)\par\par\par\par\put(170.0,60.0){\circle{40.0}}
\put(150.0,60.0){\circle*{4.0}} \put(150.0,60.0){\line(-5,10){10.0}}
\put(150.0,60.0){\line(-5,-10){5.0}} \put(144.0,48.0){\circle{4.0}}
\put(143.0,46.0){\line(-5,-10){5.0}} \par\put(220.0,60.0){\oval(90.0,30.0)[l]}
\put(184.0,73.5){\circle*{4.0}} \put(184.0,46.5){\circle*{4.0}}
\put(222.0,45.5){\circle{4.0}} \put(224.0,45.5){\line(5,0){7.0}}
\par\put(200.0,65.0){\line(5,5){30.0}} \put(210.0,75.0){\circle*{4.0}}
\par\put(145.0,83.0){$W_{v_{1}}$} \put(230.0,83.0){$W_{v_{3}}$}
\put(205.0,32.0){$W_{v_{2}}$} \par\par\put(133.0,60.0){$w_{c}$}
\put(125.0,48.0){$w_{b_{1}}$} \put(222.0,50.0){$w_{b_{2}}$}
\put(184.0,79.0){$w_{a_{1}}$} \put(184.0,38.0){$w_{a_{2}}$}
\put(210.0,65.0){$w_{a_{3}}$} \par\put(90.0,59.0){$W^{\bullet}$:}
\end{picture}

$W_{v_{1}}$$W_{v_{3}}$$W_{v_{2}}$

$w_{c}$$w_{b_{1}}$$w_{b_{2}}$$w_{a_{1}}$$w_{a_{2}}$$w_{a_{3}}$

$W^{\bullet}$:

The dual semi-graph $\Gamma_{W^{\bullet}}$ of $W^{\bullet}$ such that the
following conditions hold: (i)
$v(\Gamma_{W^{\bullet}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{v_{1},v_{2},v_{3}\\}$;
(ii) $e^{\rm cl}(\Gamma_{W^{\bullet}})\setminus e^{\rm
lp}(\Gamma_{W^{\bullet}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{a_{1},a_{2},a_{3}\\}$
such that $a_{1}$ and $a_{2}$ abut to $v_{1}$ and $v_{2}$, respectively, and
that $a_{3}$ abuts to $v_{2}$ and $v_{3}$; (iii) $e^{\rm
lp}(\Gamma_{W^{\bullet}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{c\\}$
and $c$ abuts to $v_{1}$; (iv) $e^{\rm
op}(\Gamma_{W^{\bullet}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{b_{1},b_{2}\\}$
such that $b_{1}$ and $b_{2}$ abut to $v_{1}$ and $v_{2}$, respectively. We
use the notation $``\bullet"$ and“ $\circ$ with a line segment” to denote a
vertex and an open edge, respectively. Then $\Gamma_{W^{\bullet}}$ is as
follows:

$v_{1}$$a_{1}$$a_{2}$$c$$b_{1}$$v_{2}$$a_{3}$$v_{3}$$b_{2}$$\Gamma_{W^{\bullet}}$:

(b) Let $E\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{a_{1}\\}$. Then we see
that the dual semi-graph $\Gamma_{W_{1}^{\bullet}}$ of $W^{\bullet}_{1}$ is as
follows:

$v_{1}$$a_{2}$$c$$b_{1}$$e_{1}$$v_{2}$$e_{2}$$a_{3}$$v_{3}$$b_{2}$$\Gamma_{W^{\bullet}_{1}}$:

Note that the set of terminal vertices avoiding to $e^{\rm
op}(\Gamma_{W^{\bullet}})$ of $W_{1}^{\bullet}$ is $\\{v_{3}\\}$.

(c) We obtain a minimal quasi-tree
$\Gamma_{D_{W}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma$ associated to
$D_{W}$ is as follows:

$v_{1}$$a_{2}$$c$$b_{1}$$e_{1}$$v_{2}$$e_{2}$$a_{3}$$b_{2}$$\Gamma_{D_{W}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma$:

On the other hand, $W_{\Gamma}^{\bullet}$ is as follows:

\begin{picture}(300.0,100.0)\par\par\par\par\put(170.0,60.0){\circle{40.0}}
\put(150.0,60.0){\circle*{4.0}} \put(150.0,60.0){\line(-5,10){10.0}}
\put(150.0,60.0){\line(-5,-10){5.0}} \put(144.0,48.0){\circle{4.0}}
\put(143.0,46.0){\line(-5,-10){5.0}} \put(190.0,60.0){\circle*{4.0}}
\par\put(185.7,75.0){\circle{4.0}} \put(188.0,75.0){$w_{e_{1}}$}
\par\put(175.0,60.0){\line(5,0){30.0}} \put(207.0,60.0){\circle{4.0}}
\par\put(209.0,60.0){\line(5,0){10.0}} \put(221.0,60.0){\circle{4.0}}
\par\put(223.0,60.0){\line(5,0){10.0}} \put(235.0,60.0){\circle{4.0}}
\put(215.0,65.0){$w_{b_{2}}$} \put(200.0,50.0){$w_{e_{2}}$}
\put(235.0,50.0){$w_{a_{3}}$} \par\put(237.0,60.0){\line(5,0){10.0}}
\par\put(145.0,83.0){$W_{v_{1}}$} \put(238.0,65.0){$W_{v_{2}}$}
\par\par\put(133.0,60.0){$w_{c}$} \put(125.0,48.0){$w_{b_{1}}$}
\par\put(90.0,59.0){$W_{\Gamma}^{\bullet}$:} \end{picture}

$w_{e_{1}}$

$w_{b_{2}}$$w_{e_{2}}$$w_{a_{3}}$

$W_{v_{1}}$$W_{v_{2}}$

$w_{c}$$w_{b_{1}}$

$W_{\Gamma}^{\bullet}$:

(d) Next, we give an example $\Gamma^{\prime}\subseteq\Gamma_{W^{\bullet}}$,
which is a tree containing all open edges of $\Gamma_{W^{\bullet}}$, and which
is not a minimal quasi-tree associated to $D_{W}$

$v_{1}$$a_{2}$

$b_{1}$

$v_{2}$

$b_{2}$$\Gamma^{\prime}$:

#### 4.4.5.

We maintain the notation introduced in 4.4.3. Suppose that $n_{W}\neq 0$, and
that $\Gamma=\Gamma_{D_{W}}$ is a minimal quasi-tree associated to $D_{W}$.

The construction of $W_{\Gamma}$ implies that there is a natural morphism

$f_{\Gamma}:W_{\Gamma}\rightarrow W$

over $k$. We denote by

$\phi_{\Gamma}:\Gamma\rightarrow\Gamma_{W^{\bullet}}$

the map of dual semi-graphs induced by $f_{\Gamma}$. Let

$D_{E_{1}},D_{E_{2}}\subseteq D_{W_{\Gamma}}\setminus D_{W}$

be the subsets of marked points of $W^{\bullet}_{\Gamma}$ such that
$D_{W_{\Gamma}}=D_{W}\cup D_{E_{1}}\cup D_{E_{2}}$, that
$f_{\Gamma}(D_{E_{1}})\cap N=\emptyset$, and that
$N_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f_{\Gamma}(D_{E_{2}})\subseteq
N$. Namely, the following conditions hold: (a) $D_{W_{\Gamma}}=D_{W}\cup
D_{E_{1}}\cup D_{E_{2}}$; (b) $f_{\Gamma}(D_{W})=D_{W}$; (c)
$f_{\Gamma}(D_{E_{1}})\subseteq W^{\rm sing}\setminus N$; (d) $N_{2}\subseteq
N$; (e) $f_{\Gamma}(W^{\rm sing}_{\Gamma})\subseteq W^{\rm sing}$.

We define a pointed stable curve over $k$ to be (for instance, see Example
4.10 below)

$W_{\Gamma^{\rm im}}^{\bullet}=(W_{\Gamma^{\rm
im}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f_{\Gamma}(W_{\Gamma}),D_{W_{\Gamma^{\rm
im}}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f_{\Gamma}(D_{W}\cup
D_{E_{1}})).$

Then we see that the dual semi-graph of $W^{\bullet}_{\Gamma^{\rm im}}$
coincides with the image $\text{Im}(\phi_{\Gamma})$. We call

$\Gamma^{\rm
im}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{W^{\bullet}_{\Gamma^{\rm
im}}}$

the image of the map $\phi_{\Gamma}$ (for instance, see Example 4.10 below).
Moreover, we denote by

$\text{norm}_{\Gamma}:W_{\Gamma}\rightarrow W_{\Gamma^{\rm im}}$

the natural morphism induced by $f_{\Gamma}$ which coincides with the
normalization morphism of $W_{\Gamma^{\rm im}}$ over the set of nodes $N_{2}$.
Then we have that $D_{W_{\Gamma}}=\text{norm}^{-1}_{\Gamma}(D_{W_{\Gamma^{\rm
im}}}\cup N_{2}).$

#### 4.4.6.

We maintain the notation introduced in Example 4.9 and give an example of
$\Gamma^{\rm im}$.

######  Example 4.10.

We see that the set of open edges of $W_{\Gamma}^{\bullet}$ corresponding to
$D_{E_{1}}$ is $\\{a_{3}\\}$ (Example 4.9 (c)), and that the set of open edges
of $W_{\Gamma}^{\bullet}$ corresponding to $D_{E_{2}}$ is $\\{e_{1},e_{2}\\}$
(Example 4.9 (c)). Then the image $\Gamma^{\rm im}$ of the map
$\phi_{\Gamma}:\Gamma\rightarrow\Gamma_{W}^{\bullet}$ is as follows:

$v_{1}$$a_{1}$$a_{2}$$c$$b_{1}$$v_{2}$$a_{3}$$b_{2}$$\Gamma^{\rm im}$:

On the other hand, $W_{\Gamma^{\rm im}}^{\bullet}$ is as follows:

\begin{picture}(300.0,100.0)\par\par\par\par\put(170.0,60.0){\circle{40.0}}
\put(150.0,60.0){\circle*{4.0}} \put(150.0,60.0){\line(-5,10){10.0}}
\put(150.0,60.0){\line(-5,-10){5.0}} \put(144.0,48.0){\circle{4.0}}
\put(143.0,46.0){\line(-5,-10){5.0}} \par\put(220.0,60.0){\oval(90.0,30.0)[l]}
\put(184.0,73.5){\circle*{4.0}} \put(184.0,46.5){\circle*{4.0}}
\put(222.0,45.5){\circle{4.0}} \put(224.0,45.5){\line(5,0){7.0}}
\par\put(222.0,75.0){\circle{4.0}} \put(224.0,75.0){\line(5,0){7.0}}
\par\par\put(145.0,83.0){$W_{v_{1}}$} \put(205.0,32.0){$W_{v_{2}}$}
\par\par\put(133.0,60.0){$w_{c}$} \put(125.0,48.0){$w_{b_{1}}$}
\put(222.0,50.0){$w_{b_{2}}$} \put(184.0,79.0){$w_{a_{1}}$}
\put(184.0,38.0){$w_{a_{2}}$} \put(220.0,81.0){$w_{a_{3}}$}
\par\put(90.0,59.0){$W_{\Gamma^{\rm im}}^{\bullet}$:} \end{picture}

$W_{v_{1}}$$W_{v_{2}}$

$w_{c}$$w_{b_{1}}$$w_{b_{2}}$$w_{a_{1}}$$w_{a_{2}}$$w_{a_{3}}$

$W_{\Gamma^{\rm im}}^{\bullet}$:

###  4.5. The first main theorem

We are going to prove the first main theorem of the present paper.

####  4.5.1. Settings

We maintain the notation introduced in 2.1.2 and the assumption introduced in
2.3.3 (i.e., $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1$). Moreover,
we assume that $X^{\bullet}$ is a component-generic pointed stable curve over
$k$.

#### 4.5.2.

We have the following key proposition.

######  Proposition 4.11.

Let $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ be an effective divisor
on $X$ such that $s(D)=n_{X}-1$ if $n_{X}\neq 0$, and that ${\rm
deg}(D^{(i)})\geq{\rm deg}(D),\ i\in\\{0,1,\dots,t-1\\}.$ Then the following
statements hold:

(i) Suppose that one of the following conditions is satisfied:

  * •

$n_{X}=0$ and $\Gamma_{X^{\bullet}}\setminus e^{\rm lp}(\Gamma_{X^{\bullet}})$
is a tree.

  * •

$n_{X}\neq 0$ and $\Gamma_{X^{\bullet}}=\Gamma_{D_{W}}$ is a minimal quasi-
tree associated to $D_{X}$ (see Definition 4.8).

Then $\gamma_{(\alpha,D)}$ attains maximum for all $\alpha\in{\rm
Rev}_{D}^{\rm adm}(X^{\bullet})\setminus\\{0\\}$. Namely, the following holds:

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

(ii) There exists an element $\beta\in{\rm Rev}^{\rm
adm}_{D}(X^{\bullet})\setminus\\{0\\}$ such that $\gamma_{(\beta,D)}$ attains
maximum. Namely, the following holds:

$\displaystyle\gamma_{(\beta,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

###### Proof.

(i) Let $f^{\bullet}:Y^{\bullet}=(Y,D_{Y})\rightarrow X^{\bullet}$ be a Galois
multi-admissible covering over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$
induced by $\alpha$. To verify (i), we may assume that $Y^{\bullet}$ is
connected.

Suppose that $n_{X}=0$. Since $\Gamma_{X^{\bullet}}\setminus e^{\rm
lp}(\Gamma_{X^{\bullet}})$ is a tree. We see immediately that $f$ is étale.
Then (i) follows from Theorem 3.9 and Proposition 4.5.

Suppose that $n_{X}\neq 0$. Let $v\in v(\Gamma_{X^{\bullet}})$ and
$\pi_{0}(\overline{X\setminus X_{v}})$ the set of connected components of
$\overline{X\setminus X_{v}}$, where $\overline{X\setminus X_{v}}$ denotes the
topological closure of ${X\setminus X_{v}}$ in $X$. We put

$D_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(D_{X}\cap
X_{v})\cup(\bigcup_{C\in\pi_{0}(\overline{X\setminus X_{v}})}(C\cap X_{v})).$

Let
$X_{v}^{\bullet}=(X_{v},D_{X_{v}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{v})$,
$v\in v(\Gamma_{X^{\bullet}}),$ be a pointed stable curve of type
$(g_{X_{v}},n_{X_{v}})$ over $k$. Then $f^{\bullet}$ induces a Galois multi-
admissible covering

$f_{v}^{\bullet}:Y_{v}^{\bullet}\rightarrow X_{v}^{\bullet},\ v\in
v(\Gamma_{X^{\bullet}}),$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$.

Note that since $\Gamma_{X^{\bullet}}$ is a minimal quasi-tree associated to
$D_{X}$, we have that $C\cap X_{v}=\\{x_{C}\\}$ is a closed point of $X$ for
all $C\in\pi_{0}(\overline{X\setminus X_{v}})$. Moreover, since
$\Gamma_{X^{\bullet}}\setminus e^{\rm lp}(\Gamma_{X^{\bullet}})$ is a tree,
the ramification divisor

$Q_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{x\in D_{X}\cap
X_{v}}\text{ord}_{x}(Q_{v})x+\sum_{C\in\pi_{0}(\overline{X\setminus
X_{v}})}\text{ord}_{x_{C}}(Q_{v})x_{C}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{v}]^{0},\
v\in v(\Gamma_{X^{\bullet}}),$

on $X_{v}$ induced by $f_{v}$ satisfies the following:

$\text{ord}_{x}(Q_{v})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x}(D),\
x\in D_{X}\cap X_{v},$
$\text{ord}_{x_{C}}(Q_{v})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\sum_{c\in
D_{X}\cap C}\text{ord}_{c}(D)],\ C\in\pi_{0}(\overline{X\setminus X_{v}}),$

where $[(-)]$ denotes the integer which is equal to the image of $(-)$ in
$\mathbb{Z}/n\mathbb{Z}$ when we identify $\\{0,\dots,n-1\\}$ with
$\mathbb{Z}/n\mathbb{Z}$ naturally. By applying similar arguments to the
arguments given in the proof of Lemma 4.2, we have

$\text{deg}(Q_{v})=(\\#(D_{v})-1)n\ \text{and}\
\text{deg}(Q_{v}^{(i)})=\text{deg}(Q_{v}),\ i\in\\{0,\dots,t-1\\}.$

Let $\Pi_{X_{v}^{\bullet}}$, $v\in v(\Gamma_{X^{\bullet}})$, be the admissible
fundamental group of $X^{\bullet}_{v}$. Write $\alpha_{v}\in\text{Rev}^{\rm
adm}_{Q_{v}}(X_{v}^{\bullet})$ for the composition of the natural
homomorphisms $\Pi_{X_{v}^{\bullet}}\rightarrow\Pi_{X^{\bullet}}^{\rm
ab}\overset{\alpha}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$. Note that
$\alpha_{v}\neq 0$ for all $v\in v(\Gamma_{X^{\bullet}})$. Then Proposition
4.5 implies

$\displaystyle\gamma_{(\alpha_{v},Q_{v})}=\left\\{\begin{array}[]{ll}g_{X_{v}}-1,&\text{if}\
\text{Supp}(Q_{v})=\emptyset,\\\ g_{X_{v}}+s(Q_{v})-2,&\text{if}\
\text{Supp}(Q_{v})\neq\emptyset.\end{array}\right.$

Thus, Theorem 3.9 implies that

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

This completes the proof of (i).

(ii) Suppose that $n_{X}\leq 1$. (ii) follows from Proposition 3.11 (ii). Then
to verify (ii), we may assume that $n_{X}\geq 2$.

Let $\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$ be a
minimal quasi-tree associated to $D_{X}$, $\Gamma^{\rm im}$ the image of the
natural morphism $\phi_{\Gamma}:\Gamma\rightarrow\Gamma_{X^{\bullet}}$, and

$X^{\bullet}_{\Gamma}=(X_{\Gamma},D_{X_{\Gamma}}),\ X_{\Gamma^{\rm
im}}^{\bullet}=(X_{\Gamma^{\rm im}},D_{X_{\Gamma^{\rm im}}})$

the pointed stable curves over $k$ associated to $\Gamma,\ \Gamma^{\rm im}$,
respectively (4.4.3, 4.4.5). Note that $D$ is also an effective divisor on
$X_{\Gamma^{\rm im}}$.

Write $D_{\Gamma}$ for the pulling back divisor $\text{norm}_{\Gamma}^{*}(D)$
(see 4.4.5 for the definition of $\text{norm}_{\Gamma}$). Let
$\alpha_{\Gamma}\in\text{Rev}^{\rm adm}_{D_{\Gamma}}(X^{\bullet}_{\Gamma})$ be
an arbitrary element such that $\alpha_{\Gamma}\neq 0$. Then similar arguments
to the arguments given in (i) imply
$\gamma_{(\alpha_{\Gamma},D_{\Gamma})}=g_{X_{\Gamma}}+n_{X}-2$, where
$g_{X_{\Gamma}}$ denotes the genus of $X_{\Gamma}$. We denote by

$g_{\Gamma}^{\bullet}:Z_{\Gamma}^{\bullet}\rightarrow X^{\bullet}_{\Gamma}$

the Galois multi-admissible covering over $k$ with Galois group
$\mathbb{Z}/n\mathbb{Z}$ induced by $\alpha_{\Gamma}$. By gluing
$Z_{\Gamma}^{\bullet}$ along
$g_{\Gamma}^{-1}(D_{X_{\Gamma}}\setminus\text{norm}_{\Gamma}^{-1}(D_{X_{\Gamma^{\rm
im}}}))$ in a way that is compatible with the gluing of $X_{\Gamma}^{\bullet}$
that gives rise to $X_{\Gamma^{\rm im}}^{\bullet}$, we obtain a pointed stable
curve $Z_{\Gamma^{\rm im}}^{\bullet}$ over $k$. Moreover,
$g_{\Gamma}^{\bullet}$ induces a Galois multi-admissible covering

$g_{\Gamma^{\rm im}}^{\bullet}:Z_{\Gamma^{\rm im}}^{\bullet}\rightarrow
X^{\bullet}_{\Gamma^{\rm im}}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Let
$\Pi_{X^{\bullet}_{\Gamma}}$, $\Pi_{X^{\bullet}_{\Gamma^{\rm im}}}$ be the
admissible fundamental groups of $X^{\bullet}_{\Gamma}$,
$X^{\bullet}_{\Gamma^{\rm im}}$, respectively. Write $\alpha_{\Gamma^{\rm
im}}$ for an element of $\text{Hom}(\Pi^{\rm ab}_{X^{\bullet}_{\Gamma^{\rm
im}}},\mathbb{Z}/n\mathbb{Z})$ induced by $g^{\bullet}_{\Gamma^{\rm im}}$ such
that the composition of the natural homomorphisms
$\Pi_{X^{\bullet}_{\Gamma}}^{\rm ab}\rightarrow\Pi_{X^{\bullet}_{\Gamma^{\rm
im}}}^{\rm ab}\overset{\alpha_{\Gamma^{\rm
im}}}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$ is equal to $\alpha_{\Gamma}$. We
put $D_{\Gamma^{\rm
im}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{\alpha_{\Gamma^{\rm im}}}$.
Then Theorem 3.9 implies that

$\gamma_{(\alpha_{\Gamma^{\rm im}},D_{\Gamma^{\rm im}})}=g_{X_{\Gamma^{\rm
im}}}+n_{X}-2,$

where $g_{X_{\Gamma^{\rm im}}}$ denotes the genus of $X_{\Gamma^{\rm im}}$.

On the other hand, we write $\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})$ for the set of connected components of $\overline{X\setminus
X_{\Gamma^{\rm im}}}$, where $\overline{X\setminus X_{\Gamma^{\rm im}}}$
denotes the topological closure of $X\setminus X_{\Gamma^{\rm im}}$ in $X$. We
define the following pointed stable curve

$C^{\bullet}=(C,D_{C}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}C\cap
X_{\Gamma^{\rm im}}),\ C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}}),$

over $k$. Note that since $X^{\bullet}$ is component-generic, we have that
$C^{\bullet}$ is also component-generic. Then the $p$-rank $\sigma_{C}$ is
equal to the genus of $C^{\bullet}$.

Let $C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}})$. We put

$Z^{\bullet}_{C}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{i\in\mathbb{Z}/n\mathbb{Z}}C_{i}^{\bullet},$

where $C_{i}^{\bullet}$ is a copy of $C^{\bullet}$, and $\sqcup$ means a
disjoint union. Then we obtain a Galois multi-admissible covering

$g_{C}^{\bullet}:Z^{\bullet}_{C}\rightarrow C^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$, where the restriction
morphism $g_{C}^{\bullet}|_{C^{\bullet}_{i}}$ is an identity, and the Galois
action is $j(C_{i})=C_{i+j}$ for all $i,j\in\mathbb{Z}/n\mathbb{Z}$. By gluing
$Z_{\Gamma^{\rm im}}^{\bullet}$ and
$\\{Z_{C}^{\bullet}\\}_{C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})}$ along $g^{-1}_{\Gamma^{\rm im}}(X_{\Gamma^{\rm
im}}\cap(\bigcup_{C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}})}C))$
and $\\{g^{-1}_{C}(X_{\Gamma^{\rm im}}\cap
C)\\}_{C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}})}$ in a way that
is compatible with the gluing of $\\{X^{\bullet}_{\Gamma^{\rm
im}}\\}\cup\\{C^{\bullet}\\}_{C\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})}$ that gives rise to $X^{\bullet}$, we obtain a Galois multi-admissible
covering

$g^{\bullet}:Z^{\bullet}\rightarrow X^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$.

Let $\Pi_{X^{\bullet}}$ be the admissible fundamental group of $X^{\bullet}$.
Moreover, we write $\beta\in\text{Rev}^{\rm adm}_{D}(X^{\bullet})$ for an
element induced by $g^{\bullet}$ such that the composition of the natural
homomorphisms $\Pi_{X^{\bullet}_{\Gamma^{\rm im}}}^{\rm
ab}\rightarrow\Pi_{X^{\bullet}}^{\rm
ab}\overset{\beta}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$ is equal to
$\alpha_{\Gamma^{\rm im}}$. By applying Theorem 3.9, we see

$\gamma_{(\beta,D)}=\gamma_{X^{\bullet}}^{\rm max}=g_{X}+n_{X}-2.$

We complete the proof of (ii). ∎

#### 4.5.3.

Now, the main result of the present section is as follows.

######  Theorem 4.12.

Let $X^{\bullet}=(X,D_{X})$ be a component-generic pointed stable curve over
$k$. Let $m\in\mathbb{N}$ be an arbitrary positive natural number prime to $p$
and $D\in(\mathbb{Z}/m\mathbb{Z})^{\sim}[D_{X}]^{0}$ (2.2.5). Let
$t\in\mathbb{N}$ be a positive natural number such that $p^{t}=1$ in
$(\mathbb{Z}/m\mathbb{Z})^{\times}$. Write $n$ for $p^{t}-1$, $m^{\prime}$ for
$n/m$, and $D^{\prime}$ for the divisor
$m^{\prime}D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ when we identify
$\mathbb{Z}/m\mathbb{Z}$ with the unique subgroup of $\mathbb{Z}/n\mathbb{Z}$
of order $m$. Then the following statements hold:

(i) Suppose that one of the following conditions is satisfied:

  * •

$n_{X}=0$ and $\Gamma_{X^{\bullet}}\setminus e^{\rm lp}(\Gamma_{X^{\bullet}})$
is a tree.

  * •

$n_{X}\neq 0$ and $\Gamma_{X^{\bullet}}=\Gamma_{D_{W}}$ is a minimal quasi-
tree associated to $D_{X}$ (4.4.3).

Then the following are equivalent:

> (a) $\gamma_{(\alpha,D)}$ attains maximum for all $\alpha\in{\rm
> Rev}_{D}^{\rm adm}(X^{\bullet})\setminus\\{0\\}$ (Definition 2.4 (i) and
> Definition 3.10 (ii)). Namely, the following holds (see Definition 3.10 (i)
> for $\gamma^{\rm max}_{X^{\bullet}}$):
>
> $\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
> max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\
> n_{X}=0,\\\ g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$
>
> (b) We have (see 2.2.5 and Definition 2.4 for $(D^{\prime})^{(i)}$)
>
> $\displaystyle s(D)=\left\\{\begin{array}[]{ll}0,&\text{if}\ n_{X}=0,\\\
> n_{X}-1,&\text{if}\ n_{X}\neq 0\end{array}\right.$
>
> and ${\rm deg}((D^{\prime})^{(i)})\geq{\rm deg}(D^{\prime}),\
> i\in\\{0,1,\dots,t-1\\}$.

(ii) The following are equivalent:

> (a) There exists an element $\beta\in{\rm Rev}^{\rm
> adm}_{D}(X^{\bullet})\setminus\\{0\\}$ such that $\gamma_{(\beta,D)}$
> attains maximum. Namely, the following holds:
>
> $\displaystyle\gamma_{(\beta,D)}=\gamma^{\rm
> max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\
> n_{X}=0,\\\ g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$
>
> (b) We have
>
> $\displaystyle s(D)=\left\\{\begin{array}[]{ll}0,&\text{if}\ n_{X}=0,\\\
> n_{X}-1,&\text{if}\ n_{X}\neq 0,\end{array}\right.$
>
> and ${\rm deg}((D^{\prime})^{(i)})\geq{\rm deg}(D^{\prime}),\
> i\in\\{0,1,\dots,t-1\\}$.

###### Proof.

(i) Write $\alpha^{\prime}\in\text{Rev}_{D^{\prime}}^{\rm adm}(X^{\bullet})$
for the element induced by $\alpha$. Then we have
$\gamma_{(\alpha,D)}=\gamma_{(\alpha^{\prime},D^{\prime})}$. The “only if”
part of (i) follows from Lemma 4.1. Moreover, the “if” part of (i) follows
immediately from Proposition 4.11 (i).

(ii) The “only if” part of (ii) follows from Lemma 4.1. We prove the “if” part
of (ii). Let $\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$
be a minimal quasi-tree associated to $D_{X}$, $\Gamma^{\rm im}$ the image of
the natural morphism $\phi_{\Gamma}:\Gamma\rightarrow\Gamma_{X^{\bullet}}$,
$X^{\bullet}_{\Gamma^{\rm im}}=(X_{\Gamma^{\rm im}},D_{X_{\Gamma^{\rm im}}})$
the pointed stable curve over $k$ associated to $\Gamma^{\rm im}$ (4.4.5), and
$\Pi_{X^{\bullet}_{\Gamma^{\rm im}}}$ the admissible fundamental group of
$X^{\bullet}_{\Gamma^{\rm im}}$. Note that $D$ is also an effective divisor on
$X_{\Gamma^{\rm im}}$. Let $\beta_{\Gamma^{\rm im}}$ be an arbitrary element
of $\text{Rev}_{D}^{\rm adm}(X^{\bullet}_{\Gamma^{\rm im}})\setminus\\{0\\}$.
Write $\beta_{\Gamma^{\rm im}}^{\prime}\in\text{Rev}_{D^{\prime}}^{\rm
adm}(X^{\bullet}_{\Gamma^{\rm im}})$ for the element induced by
$\beta_{\Gamma^{\rm im}}$.

Let $\Pi_{X^{\bullet}}$ be the admissible fundamental group of $X^{\bullet}$.
Proposition 4.11 (ii) implies that there exists
$\beta^{\prime}\in\text{Rev}_{D^{\prime}}^{\rm adm}(X^{\bullet})$ such that
the composition of the natural homomorphisms $\Pi_{X_{\Gamma^{\rm
im}}^{\bullet}}^{\rm
ab}\rightarrow\Pi_{X^{\bullet}}\overset{\beta^{\prime}}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$
is equal to $\beta_{\Gamma^{\rm im}}^{\prime}$, and that
$\gamma_{(\beta^{\prime},D^{\prime})}$ attains maximum. Moreover, the
construction of $\beta^{\prime}$ given in the proof of Proposition 4.11 (ii)
(i.e., the Galois multi-admissible covering of $\widetilde{X}^{\bullet}_{v}$,
$v\in v(\Gamma_{X^{\bullet}})\setminus v(\Gamma^{\rm im})$, induced by
$\beta^{\prime}$ is trivial) implies that $\beta^{\prime}$ can be induced by
an element of $\beta\in\text{Rev}_{D}^{\rm adm}(X^{\bullet})$. Then
$\gamma_{(\beta,D)}=\gamma_{(\beta^{\prime},D^{\prime})}$ attains maximum.
This completes the proof of (ii). ∎

###  4.6. $(m,n_{X})$-ordinary curves

#### 4.6.1.

We maintain the notation introduced in 2.1.2. Moreover, let
$\overline{\mathcal{M}}_{g_{X},n_{X}}$ be the moduli stack parameterizing
pointed stable curves of type $(g_{X},n_{X})$ (2.1.4) and
$\mathcal{M}_{g_{X},n_{X}}\subseteq\overline{\mathcal{M}}_{g_{X},n_{X}}$ the
open substack parameterizing smooth pointed stable curves of type
$(g_{X},n_{X})$. We denote by $\overline{M}_{g_{X},n_{X}}$, $M_{g_{X},n_{X}}$
the coarse moduli spaces of $\overline{\mathcal{M}}_{g_{X},n_{X}}$,
$\mathcal{M}_{g_{X},n_{X}}$, respectively.

#### 4.6.2.

Let $m\in\mathbb{N}$ be an arbitrary positive natural number prime to $p$, and
let $D\in(\mathbb{Z}/m\mathbb{Z})^{\sim}[D_{X}]^{0}$ be an effective divisor
on $X$ such that $\text{deg}(D)=(n_{X}-1)m$ if $n_{X}\neq 0$.

######  Definition 4.13.

We shall say that $X^{\bullet}$ is $(m,n_{X})$-ordinary if, for all
$\alpha\in{\rm Rev}_{D}^{\rm adm}(X^{\bullet})\setminus\\{0\\}$, we have

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

Note that, if $n_{X}=0$ and $X^{\bullet}$ is non-singular, then the definition
of $(m,n_{X})$-ordinary coincides with the definition of $m$-ordinary
introduced by Nakajima ([N, §4]).

On the other hand, let $t\in\mathbb{N}$ be a positive natural number such that
$p^{t}=1$ in $(\mathbb{Z}/m\mathbb{Z})^{\times}$. Write $n$ for $p^{t}-1$,
$m^{\prime}$ for $n/m$, and $D^{\prime}$ for the divisor
$m^{\prime}D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ when we identify
$\mathbb{Z}/m\mathbb{Z}$ with the unique subgroup of $\mathbb{Z}/n\mathbb{Z}$
of order $m$. If $n_{X}\neq 0$ and $X^{\bullet}$ is $(m,n_{X})$-ordinary, then
Lemma 4.1 (ii) implies that

${\rm deg}((D^{\prime})^{(i)})={\rm deg}(D^{\prime})=(n_{X}-1)n,\
i\in\\{0,1,\dots,t-1\\}.$

#### 4.6.3.

We denote by

$\overline{U}_{(m,n_{X})}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{q\in\overline{M}_{g_{X},n_{X}}\
|\ \text{curves corresponding to}\ q\ \text{is}\
(m,n_{X})\text{-ordinary}\\}\subseteq\overline{M}_{g_{X},n_{X}}.$

Moreover, we put
$U_{(m,n_{X})}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\overline{U}_{(m,n_{X})}\cap
M_{g_{X},n_{X}}$ (i.e., the set of points corresponding to smooth
$(m,n_{X})$-ordinary curves). Then we have the following result.

######  Proposition 4.14.

(i) The set $U_{(m,n_{X})}$ is a non-empty open subset of $M_{g_{X},n_{X}}$.

(ii) Suppose that $n_{X}\leq 1$. Then we have

$M_{g_{X},n_{X}}^{\rm cl}\cap(\bigcap_{m\in\mathbb{N}\ {\text{\rm s.t.}}\
(m,p)=1}U_{(m,n_{X})})=\emptyset,$

where $M_{g_{X},n_{X}}^{\rm cl}$ denotes the set of closed points of
$M_{g_{X},n_{X}}$.

(iii) Let $q\in M_{g_{X},n_{X}}$ be an arbitrary point. We denote by $\Pi_{q}$
the admissible fundamental group of a smooth pointed stable curve
corresponding to a geometric point over $q$. Note that the isomorphism class
of $\Pi_{q}$ (as a profinite group) does not depend on the choices of
geometric points over $q$. Suppose that $n_{X}\leq 1$. Let $U\subseteq
M_{g_{X},n_{X}}$ be an arbitrary non-empty open subset. Then there exist
closed points $q_{1},q_{2}\in U\cap M_{g_{X},n_{X}}^{\rm cl}$ such that
$\Pi_{q_{1}}\not\cong\Pi_{q_{2}}$.

###### Proof.

(i) By applying similar arguments to the arguments given in the proof of [N,
Theorem 2], (i) follows from Theorem 4.12 (i).

(ii) Suppose that $k=\overline{\mathbb{F}}_{p}$, and that $X^{\bullet}$ is a
smooth pointed stable curve of type $(g_{X},n_{X})$ over $k$. To verify (ii),
we only need to prove that, if $n_{X}\leq 1$, $X^{\bullet}$ is not
$(m,n_{X})$-ordinary for some positive natural number $m\in\mathbb{N}$ prime
to $p$.

Since $n_{X}\leq 1$, we have
$(\mathbb{Z}/m\mathbb{Z})^{\sim}[D_{X}]^{0}=\\{0\\}$. We denote by
$\Theta_{X}$ the Raynaud-Tamagawa divisor associated to $\mathcal{B}^{1}_{0}$
(2.4.2, 2.4.3) and by $\Theta^{\prime}$ an arbitrary irreducible component of
$\Theta_{X}$. Write $J_{X}^{1}$ for the pulling back of the Jacobian $J_{X}$
of $X$ by the absolute Frobenius morphism $F_{k}$ of $k$. If $X^{\bullet}$ is
$(m,n_{X})$-ordinary for all prime-to-$p$ positive natural numbers, then a
property of Raynaud-Tamagawa theta divisors concerning new-ordinary coverings
(e.g. [PS, §1.2 and §1.3]) implies

$J^{1}_{X}\\{p^{\prime}\\}\cap\Theta_{X}(k)\subseteq\\{0_{J^{1}_{X}}\\},$

where $0_{J^{1}_{X}}$ denotes the zero point of $J^{1}_{X}$, and
$J^{1}_{X}\\{p^{\prime}\\}$ denotes the set of prime-to-$p$ torsion points of
$J^{1}_{X}(k)$. Since $\text{dim}(\Theta^{\prime})>0$, we have

$\Theta^{\prime}\\{p^{\prime}\\}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J^{1}_{X}\\{p^{\prime}\\}\cap\Theta^{\prime}(k)\subseteq
J^{1}_{X}\\{p^{\prime}\\}\cap\Theta_{X}(k)$

is not dense in $\Theta^{\prime}$.

On the other hand, since $\Theta^{\prime}$ is defined over
$k=\overline{\mathbb{F}}_{p}$, by applying a result of Anderson-Indik ([T3,
§5]), we have that $\Theta^{\prime}$ is a subvariety of a translate of a
proper sub-abelian variety of $J^{1}_{X}$. But this contradicts a result of
Raynaud ([R2, Proposition 1.2.1]) which says that there exists an irreducible
component $\Theta^{\prime}$ of $\Theta_{X}$ such that $\Theta^{\prime}$ is not
contained in a translate of a proper sub-abelian variety of $J^{1}_{X}$. This
completes the proof of (ii).

(iii) Let $q$ be an arbitrary closed point of $U$ and $q^{\rm gen}$ the
generic point of $M_{g_{X},n_{X}}$. Suppose that (iii) does not hold. Thus
there exists a closed point $q\in U^{\rm cl}$ such that $\Pi_{q^{\rm
gen}}\cong\Pi_{q}$. Then there exist a discrete valuation ring $R$ and a
morphism $c_{R}:\mathrm{Spec}\,R\rightarrow M_{g_{X},n_{X}}$ such that
$c_{R}(\eta_{R})=q^{\rm gen}$ and $c_{R}(s_{R})=q$, where $\eta_{R}$ is the
generic point of $\mathrm{Spec}\,R$ and $s_{R}$ is the closed point of
$\mathrm{Spec}\,R$. By replacing $R$ by a finite extension of $R$, we have a
smooth pointed stable curve $\mathcal{X}^{\bullet}$ of type $(g_{X},n_{X})$
over $\mathrm{Spec}\,R$ determined by $c_{R}$. Moreover, we obtain a
specialization surjective homomorphism

$sp_{R}:\Pi_{q^{\rm gen}}\twoheadrightarrow\Pi_{q}.$

Since $\Pi_{q^{\rm gen}}$ and $\Pi_{q}$ are topologically finitely generated,
$sp_{R}$ is an isomorphism.

On the other hand, (ii) implies that there exist a positive integer $m$ prime
to $p$ and a connected Galois étale covering
$\mathcal{Y}^{\bullet}\rightarrow\mathcal{X}^{\bullet}$ over $R$ with Galois
group $\mathbb{Z}/m\mathbb{Z}$ such that the geometric generic fiber of
$\mathcal{Y}$ is ordinary and the geometric special fiber of $\mathcal{Y}$ is
not ordinary. Namely, $sp_{R}$ is not an isomorphism. We complete the proof of
(iii). ∎

######  Remark 4.14.1.

Proposition 4.14 (iii) gives an answer of a question of D. Harbater ([H, 4.2])
which was first solved by Pop and Sa$\rm\ddot{\i}$di ([PS, Corollary]). In
[PS], Pop and Sa$\rm\ddot{\i}$di proved a result which says that
specialization homomorphisms of geometric étale fundamental groups of smooth
projective curves in positive characteristic is not an isomorphism under
certain assumptions. Then together with a result of C-L. Chai and F. Oort, and
a result of J-P. Serre, they obtained Proposition 4.14 (iii).

#### 4.6.4.

If $n_{X}\leq 1$, then Proposition 4.14 (i) was proved by Nakajima ([N,
Theorem 2]). Then Proposition 4.14 (i) is a generalized version of Nakajima’s
result to the case of admissible coverings of smooth pointed stable curves.
Moreover, Nakajima ([N, §4 Remark]) asked whether or not

$\bigcap_{m\in\mathbb{N}\ {\text{\rm s.t.}}\ (m,p)=1}U_{(m,n_{X})}$

is a non-empty open subset of $M_{g_{X},n_{X}}$. Then Proposition 4.14 (ii)
gives a negative answer of Nakajima’s question. Furthermore, we may ask the
following question:

######  Problem 4.15.

(i) Suppose that $X^{\bullet}$ is a component-generic pointed stable curve
over $k$. Can we find a necessary and sufficient condition that $X^{\bullet}$
is $(m,n_{X})$-ordinary for all $m\in\mathbb{N}$ prime to $p$?

(ii) Does

$M_{g_{X},n_{X}}^{\rm cl}\cap(\bigcap_{m\in\mathbb{N}\ {\text{\rm s.t.}}\
(m,p)=1}U_{(m,n_{X})})=\emptyset$

hold for an arbitrary non-negative integer $n_{X}$? Moreover, does

$\overline{M}_{g_{X},n_{X}}^{\rm cl}\cap(\bigcap_{m\in\mathbb{N}\ {\text{\rm
s.t.}}\ (m,p)=1}\overline{U}_{(m,n_{X})})=\emptyset$

hold for an arbitrary non-negative integer $n_{X}$?

##  5\. Maximum generalized Hasse-Witt invariants for arbitrary curves

In the present section, we discuss the maximum generalized Hasse-Witt
invariants of cyclic admissible coverings for arbitrary pointed stable curves.
The main result of this section is Theorem 5.4.

###  5.1. Idea

We briefly explain the idea of our proof of Theorem 5.4.

####  5.1.1. Settings

We maintain the notation introduced in 2.1.2.

####  5.1.2. An easy case

Firstly, let us prove an easy case (i.e., $X^{\bullet}$ is irreducible) of the
main result of the present section.

######  Proposition 5.1.

Suppose that $X^{\bullet}$ is irreducible. Then there exist a positive natural
number $n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1\in\mathbb{N}$, an
effective divisor $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ (2.2.5) on
$X$ of degree $(n_{X}-1)n$ if $n_{X}\neq 0$ (resp. degree $0$ if $n_{X}=0$),
and an element $\alpha\in{\rm Rev}^{\rm adm}_{D}(X^{\bullet})\setminus\\{0\\}$
(Definition 2.4 (i)) such that $\gamma_{(\alpha,D)}$ attains maximum
(Definition 3.10 (ii)). Namely, the following holds:

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

###### Proof.

Since $X^{\bullet}$ is irreducible, we write $\widetilde{X}^{\bullet}$ for the
smooth pointed stable curve of type $(g_{\widetilde{X}},n_{\widetilde{X}})$
over $k$ associated to the unique vertex of the dual semi-graph
$\Gamma_{X^{\bullet}}$ of $X^{\bullet}$ (2.1.3). Note that we have

$g_{\widetilde{X}}=g_{X}-\\#(X^{\rm sing}),\
n_{\widetilde{X}}=n_{X}+2\\#(X^{\rm sing}).$

Moreover, $D_{X}\subseteq D_{\widetilde{X}}$ implies
$(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}\subseteq(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{\widetilde{X}}]^{0}$.

By applying Theorem 3.9, to verify the proposition, it is sufficient to prove
that there exist a positive natural number
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1\in\mathbb{N}$, an
effective divisor $\widetilde{D}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$
of degree $(n_{X}-1)n$ if $n_{X}\neq 0$ (resp. degree $0$ if $n_{X}=0$) on
$\widetilde{X}$, and an element $\widetilde{\alpha}\in{\rm Rev}^{\rm
adm}_{\widetilde{D}}(\widetilde{X}^{\bullet})\setminus\\{0\\}$ such that the
following holds:

$\displaystyle\gamma_{(\widetilde{\alpha},\widetilde{D})}=\left\\{\begin{array}[]{ll}g_{\widetilde{X}}-1,&\text{if}\
n_{X}=0,\\\ g_{\widetilde{X}}+n_{X}-2,&\text{if}\ n_{X}\neq
0.\end{array}\right.$

Suppose that $n_{X}\leq 1$. Then the proposition follows immediately from
Proposition 3.11 (ii).

Suppose that $n_{X}\geq 2$. Let
$D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},\dots,x_{n_{X}}\\}$
and $n_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t_{i}}-1$,
$i\in\\{1,\dots,n_{X}-1\\}$, such that the following conditions are satisfied:

  * •

$n_{i}>\text{max}\\{C(g_{X})+1,\\#(e(\Gamma_{X^{\bullet}}))\\}$.

  * •

$0<a_{i,1},a_{i,2}<n_{i}$ and $a_{i,1}+a_{i,2}=n_{i}$ for all
$i\in\\{1,\dots,n_{X}-1\\}$.

Furthermore, we put

$D_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}a_{i,1}x_{i}+a_{i,2}x_{i+1}+\sum_{x\in
D_{X}\setminus\\{x_{i},x_{i+1}\\}}n_{i}x,\ i\in\\{1,\dots,n_{X}-1\\},$

which is an effective divisor on $\widetilde{X}$ of degree $(n_{X}-1)n_{i}$.
Moreover, we put

$\widetilde{D}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i=1}^{n_{X}-1}p^{\sum_{j=0}^{i-1}t_{j}}D_{i},$
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{\sum_{i=0}^{n_{X}-1}t_{j}}-1=\sum_{i=1}^{n_{X}-1}p^{\sum_{j=0}^{i-1}t_{j}}(p^{t_{i}}-1),$

where $t_{0}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}0$. We see that
$\widetilde{D}$ is an effective divisor on $\widetilde{X}$ of degree
$(n_{X}-1)n$ such that
$\widetilde{D}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$.

Let $\mathcal{L}_{\widetilde{D}}$ be a line bundle on $\widetilde{X}$ such
that $\mathcal{L}_{\widetilde{D}}^{\otimes
n}\cong\mathcal{O}_{X}(-\widetilde{D})$, and $\mathcal{L}_{\widetilde{D},t}$
the pulling back of $\mathcal{L}_{\widetilde{D}}$ by the natural morphism
$\widetilde{X}_{t}\rightarrow\widetilde{X}$ defined in 2.4.2. Then by applying
[T2, Corollary 2.6, Lemma 2.12 (ii), and Corollary 2.13], we see that the
Raynaud-Tamagawa theta divisor associated to
$\mathcal{B}^{t}_{\widetilde{D}}\otimes\mathcal{L}_{\widetilde{D},t}$ exists
(Definition 2.7). Moreover, Proposition 2.8 implies that there exists a line
bundle $\widetilde{\mathcal{I}}$ of degree $0$ on $\widetilde{X}$ such that
$[\widetilde{\mathcal{I}}]\neq[\mathcal{O}_{\widetilde{X}}]$, that
$[\widetilde{\mathcal{I}}^{\otimes n}]=[\mathcal{O}_{\widetilde{X}}]$, and
that (see 2.3.4 for
$\gamma_{([\mathcal{L}_{\widetilde{D}}\otimes\widetilde{\mathcal{I}}],\widetilde{D})}$)

$\displaystyle\gamma_{([\mathcal{L}_{\widetilde{D}}\otimes\widetilde{\mathcal{I}}],\widetilde{D})}=\left\\{\begin{array}[]{ll}g_{\widetilde{X}}-1,&\text{if}\
n_{X}=0,\\\ g_{\widetilde{X}}+n_{X}-2,&\text{if}\ n_{X}\neq
0.\end{array}\right.$

Let $\widetilde{\alpha}\in{\rm Rev}^{\rm
adm}_{\widetilde{D}}(\widetilde{X}^{\bullet})$ be the element corresponding to
the pair
$([\mathcal{L}_{\widetilde{D}}\otimes\widetilde{\mathcal{I}}],\widetilde{D})\in\widetilde{\mathscr{P}}_{\widetilde{X}^{\bullet},n}$
(2.3.2). Then we have
$\gamma_{(\widetilde{\alpha},\widetilde{D})}=\gamma_{([\mathcal{L}_{\widetilde{D}}\otimes\widetilde{\mathcal{I}}],\widetilde{D})}$.
This completes the proof of the proposition. ∎

######  Remark 5.1.1.

We maintain the notation introduced in the proof of Proposition 5.1. By
choosing a suitable $a_{i,1}$ (or $a_{i,2}$) for each
$i\in\\{1,\dots,n_{X}-1\\}$, we obtain that the Galois multi-admissible
covering induced by $\alpha$ is connected.

####  5.1.3. Strategy of the proof of Theorem 5.4

In the remainder of this section, we will generalize Proposition 5.1 to the
case where $X^{\bullet}$ is an arbitrary pointed stable curve over $k$ (i.e.,
Theorem 5.4 below).

Let $X^{\bullet}$ be an arbitrary pointed stable curve over $k$. For
simplicity, we assume that every irreducible component of $X^{\bullet}$ is
non-singular. Moreover, by Proposition 3.11 (ii), we assume $n_{X}\geq 2$. We
maintain the notation introduced in the statement of Theorem 3.9. To verify
Theorem 5.4, by applying Theorem 3.9, it is sufficient to construct a prime-
to-$p$ cyclic Galois multi-admissible covering
$\widetilde{f}^{\bullet}_{v}:\widetilde{Y}_{v}^{\bullet}\rightarrow\widetilde{X}_{v}^{\bullet}$
with Galois group $\mathbb{Z}/n\mathbb{Z}$ for every $v\in
v(\Gamma_{X^{\bullet}})$ such that the following conditions are satisfied:

  * •

$\gamma_{(\widetilde{\alpha}_{v},D_{\widetilde{\alpha}_{v}})}$ satisfies the
conditions mentioned in the statement of Theorem 3.9.

  * •

$\\{\widetilde{f}_{v}^{\bullet}\\}_{v\in v(\Gamma_{X^{\bullet}})}$ can be
glued together. Then we obtain a prime-to-$p$ cyclic Galois multi-admissible
covering $f^{\bullet}:Y^{\bullet}\rightarrow X^{\bullet}$ with Galois group
$\mathbb{Z}/n\mathbb{Z}$.

  * •

The ramification divisor $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$
associated to $f^{\bullet}$ such that $s(D)=n_{X}-1$ (2.2.5).

If $X^{\bullet}$ is not irreducible, the constructions of desired Galois
multi-admissible coverings of $X^{\bullet}$ are very difficult in general.

#### The main difficulty

We cannot determine the ramifications over nodes of a Galois admissible
covering in a unique way when the ramifications over $D_{X}$ are fixed. Then
we have the following:

(i) Let $f^{\bullet}$ be a Galois admissible covering whose ramification
divisor satisfies the third condition mentioned above. Since the Raynaud-
Tamagawa theta divisor concerning $D_{\widetilde{\alpha}_{v}}$, $v\in
v(\Gamma_{X^{\bullet}})$, does not exist in general,
$\gamma_{(\widetilde{\alpha}_{v},D_{\widetilde{\alpha}_{v}})}$ does not
satisfy the first condition mentioned above.

(ii) Even though we can construct $\widetilde{f}^{\bullet}_{v}$ for every
$v\in v(\Gamma_{X^{\bullet}})$ (by applying Proposition 5.1) such that
$\gamma_{(\widetilde{\alpha}_{v},D_{\widetilde{\alpha}_{v}})}$ satisfies the
conditions mentioned in the statement of Theorem 3.9,
$\\{\widetilde{f}_{v}^{\bullet}\\}_{v\in v(\Gamma_{X^{\bullet}})}$ cannot be
glued together (as admissible coverings) in general.

To overcome this difficulty, we observe that, when $\Gamma_{X^{\bullet}}$ is a
minimal quasi-tree, the ramifications over nodes of a Galois admissible
covering can be uniquely determined if the ramifications over $D_{X}$ are
fixed. Then for every $v\in v(\Gamma_{X^{\bullet}})$, we may construct certain
effective divisors on $\widetilde{X}_{v}$ for every marked point of
$D_{X}\cap\widetilde{X}_{v}$ and every node of $X^{\rm
sing}\cap\widetilde{X}_{v}$. Furthermore, by similar arguments to the
arguments given in the proof of Proposition 5.1, we may construct an effective
divisor on $\widetilde{X}_{v}$ such that the Raynaud-Tamagawa theta divisor
concerning this effective divisor exists. Then we can obtain Galois multi-
admissible coverings for all $v\in v(\Gamma_{X^{\bullet}})$ whose first
generalized Hasse-Witt invariants attain maximum. On the other hand, since the
ramifications over nodes of a Galois admissible covering can be uniquely
determined by the ramifications of marked points when $\Gamma_{X^{\bullet}}$
is a minimal quasi-tree, then we also obtain an effective divisor on $X$ whose
restriction on $\widetilde{X}_{v}$, $v\in v(\Gamma_{X^{\bullet}})$, is the
effective divisor constructed above. Then we may construct a desired Galois
multi-admissible covering $f^{\bullet}$ (e.g. see Example 5.2 below).

In the general case (i.e., $\Gamma_{X^{\bullet}}$ is not a tree), we take a
minimal quasi-tree
$\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$ associated to
$D_{X}$ (4.4.3) and the pointed stable curve $X^{\bullet}_{\Gamma}$ (4.4.5)
associated to $\Gamma$. Then we may construct a desired Galois multi-
admissible covering for $X^{\bullet}_{\Gamma}$ (see Lemma 5.3). In fact, this
is the motivation of the definition of minimal quasi-trees. Moreover,
Raynaud’s theorem (i.e., [R1, Théorème 4.1.1] or Theorem 2.9) implies that we
may construct a Galois (étale) multi-admissible covering for each irreducible
component corresponding to $w\in v(\Gamma_{X^{\bullet}})\setminus v(\Gamma)$
whose first generalized Hasse-Witt invariant (2.2.4) attains maximum. Then by
gluing the Galois multi-admissible coverings together, we obtain a desired
Galois multi-admissible covering of $X^{\bullet}$.

###  5.2. A key lemma

####  5.2.1. Settings

We maintain the notation introduced in 2.1.2.

#### 5.2.2.

In order to convince the reader to follow the constructions given in the proof
of Lemma 5.3 below, we give an example for constructing effective divisors
$D_{\widetilde{\alpha}_{v}}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{\widetilde{X}_{v}}]^{0}$,
$v\in v(\Gamma_{X^{\bullet}})$, and
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$.

######  Example 5.2.

Let $X^{\bullet}$ be a pointed stable curve of type $(g_{X},3)$ over $k$.
Suppose that $X$ has two non-singular irreducible components $X_{1}$ and
$X_{2}$, that $X_{1}\cap
X_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x^{-}\\}$, that $D_{X}\cap
X_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1,1},x_{1,2}\\}$, and
that $D_{X}\cap X_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{2,1}\\}$.
Let $n_{0}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t_{0}}-1>>0$,
$D_{X_{1}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1,1},x_{1,2},x^{-}\\}$,
$D_{X_{2}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{2,1},x^{-}\\}$,
$X^{\bullet}_{1}=(X_{1},D_{X_{1}})$ a pointed stable curve of type
$(g_{X_{1}},3)$, and $X^{\bullet}_{2}=(X_{2},D_{X_{2}})$ a pointed stable
curve of type $(g_{X_{2}},2)$. Then we have the following:

$X_{2}$$X_{1}$$x_{2,1}$$x^{-}$

$x_{1,2}$

$x_{1,1}$

$X^{\bullet}$:

Step 1: We construct effective divisors of degree $2n$ (resp. $n$) on $X_{1}$
(resp. $X_{2}$) such that the Raynaud-Tamagawa theta divisors concerning the
divisors exist.

We put

$Q_{1,1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}a_{1}x_{1,1}+a_{2}x_{1,2}+n_{0}x^{-}\in\text{Div}(X_{1}),$
$Q_{1,2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}n_{0}x_{2,1}\in\text{Div}(X_{2}),$
$Q_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}a_{1}x_{1,1}+a_{2}x_{1,2}+n_{0}x_{2,1}\in\text{Div}(X),$

where $0<a_{1},a_{2}<n_{0}$ such that $a_{1}+a_{2}=n_{0}$. Moreover, we put

$Q_{2,1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}n_{0}x_{1,1}+b_{1}x_{1,2}+b_{2}x^{-}\in\text{Div}(X_{1}),$
$Q_{2,2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{1}x^{-}+b_{2}x_{2,1}\in\text{Div}(X_{2}),$
$Q_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}n_{0}x_{1,1}+b_{1}x_{1,2}+b_{2}x_{2,1}\in\text{Div}(X),$

where $0<b_{1},b_{2}<n_{0}$ such that $b_{1}+b_{2}=n_{0}$.

Since $n_{0}>>0$, by applying [T2, Corollary 2.6, Lemma 2.12 (ii), and
Corollary 2.13], the Raynaud-Tamagawa theta divisors concerning $Q_{1}$,
$Q_{2}$, $Q_{1,1}$, $Q_{1,2}$, $Q_{2,1}$, $Q_{2,2}$ exist, respectively.

Note that since $\Gamma_{X^{\bullet}}$ is a tree, $Q_{1,1}$ and $Q_{1,2}$
(resp. $Q_{2,1}$ and $Q_{2,2}$) can be completely determined by $Q_{1}$ (resp.
$Q_{2}$). On the other hand, at present, we cannot construct Galois multi-
admissible coverings whose ramification divisors are the above divisors since
$Q_{1}$, $Q_{1,1}$, $Q_{1,2}$ (resp. $Q_{2}$, $Q_{2,1}$, $Q_{2,2}$) are not
contained in $(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{1}}]^{0}$ (resp.
$(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{2}}]^{0})$.

Step 2: By using the effective divisors constructed in Step 1, we construct a
global effective divisor $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ on
$X$ and $D_{i}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{i}}]^{0}$,
$i\in\\{1,2\\}$, on $X_{i}$ such that $D_{i}$ is completely determined by $D$.

Write $v_{1}$ and $v_{2}$ for the vertices of $v(\Gamma_{X^{\bullet}})$
corresponding to $X_{1}$ and $X_{2}$, respectively. We define the following
sets of effective divisors

$\text{Div}_{v_{i}}^{\rm
irr\text{-}mp}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{Q_{1,i}\\},\
\text{Div}_{v_{i}}^{\rm
irr\text{-}nd}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{Q_{2,i}\\},$
$\text{Div}_{X}^{\rm
irr}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{i=1,2}(\text{Div}_{v_{i}}^{\rm
irr\text{-}mp}\sqcup\text{Div}_{v_{i}}^{\rm irr\text{-}nd}),\
\text{Div}_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{Q_{1},Q_{2}\\},$

where $\sqcup$ means disjoint union. Let
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t^{0}}-1+p^{t_{0}}(p^{t_{0}}-1)=p^{2t_{0}}-1$.
We obtain

$D_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{1,1}+p^{t_{0}}Q_{2,1}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{1}}]^{0},$
$D_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{1,2}+p^{t_{0}}Q_{2,2}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{2}}]^{0},$
$D\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{1}+p^{t_{0}}Q_{2}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}.$

Then we have

$D|_{\\{x_{1,1},x_{1,2}\\}}=D_{1}|_{\\{x_{1,1},x_{1,2}\\}},\
D|_{\\{x_{2,1}\\}}=D_{2}|_{\\{x_{2,1}\\}}.$

Step 3: We construct a desired Galois multi-admissible covering of
$X^{\bullet}$.

By applying [T2, Corollary 2.6, Lemma 2.12 (ii), and Corollary 2.13], the
Raynaud-Tamagawa theta divisors concerning $D_{1}$ and $D_{2}$ exist. By
Proposition 2.8, if $t_{0}>>0$, then there exists
$\alpha_{v_{i}}\in\text{Rev}_{D_{i}}^{\rm
adm}(X_{i}^{\bullet})\setminus\\{0\\}$, $i\in\\{1,2\\}$, such that
$\gamma_{(\alpha_{v_{1}},D_{1})}=g_{X_{1}}+1$ and
$\gamma_{(\alpha_{v_{1}},D_{1})}=g_{X_{2}}$. Moreover, since
$\Gamma_{X^{\bullet}}$ is a tree, we have that

$D_{1}|_{x^{-}}=[\sum_{x\in
D_{X}\setminus\\{x_{1,1},x_{1,2}\\}}\text{ord}_{x}(D)]x^{-}=\text{ord}_{x_{2,1}}(D)x^{-},$
$D_{2}|_{x^{-}}=\text{ord}_{x^{-}}(D_{2})x^{-}=(n-\text{ord}_{x_{2,1}}(D_{2}))x^{-}=[\sum_{x\in
D_{X}\setminus\\{x_{2,1}\\}}\text{ord}_{x}(D)]x^{-}$
$=(n-\text{ord}_{x_{2,1}}(D))x^{-},$

where $[(-)]$ denotes the image of $(-)$ in $\mathbb{Z}/n\mathbb{Z}$. Namely,
the Galois multi-admissible coverings induced by $\alpha_{v_{1}}$ and
$\alpha_{v_{2}}$ can be glued (as admissible coverings). Then we obtain a
desired Galois multi-admissible covering of $X^{\bullet}$.

#### 5.2.3.

Let $\mathbf{G}$ be a connected semi-graph and $v\in v(\mathbf{G})$ an
arbitrary vertex. Moreover, we suppose that $\mathbf{G}$ is a tree. For each
$v^{\prime}\in v(\mathbf{G})$, there exists a path $\delta_{v,v^{\prime}}$
connecting $v$ and $v^{\prime}$ in $\mathbf{G}$. We define

$\text{leng}(\delta_{v,v^{\prime}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\#\\{\delta_{v,v^{\prime}}\cap
v(\mathbf{G})\\}-1$

to be the length of the path $\delta_{v,v^{\prime}}$. Moreover, since
$\mathbf{G}$ is a tree, there exists a unique path connecting $v$ and
$v^{\prime}$ whose length is equal to
$\text{min}\\{\text{leng}(\delta_{v,v^{\prime}})\\}_{\delta_{v,v^{\prime}}}$.
We shall write

$\delta(\mathbf{G},v,v^{\prime})$

for this unique path connecting $v$ and $v^{\prime}$ in $\mathbf{G}$, and call
$\delta(\mathbf{G},v,v^{\prime})$ the minimal path connecting $v$ and
$v^{\prime}$ in $\mathbf{G}$.

#### 5.2.4.

Now, we are going to prove the key lemma of the present section.

######  Lemma 5.3.

Let $\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$ be a
minimal quasi-tree associated to $D_{X}$,

$X_{\Gamma}^{\bullet}=(X_{\Gamma},D_{X_{\Gamma}})$

the pointed stable curve of type $(g_{X_{\Gamma}},n_{X_{\Gamma}})$ associated
to $\Gamma$ (4.4.3), and $\Pi_{X^{\bullet}_{\Gamma}}$ the admissible
fundamental group of $X^{\bullet}_{\Gamma}$. Suppose that $n_{X}\geq 2$. Then
there exist a positive natural number
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1\in\mathbb{N}$, an
effective divisor
$D_{\Gamma}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}\subseteq(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{\Gamma}}]^{0}$
on $X_{\Gamma}$, and an element $\alpha_{\Gamma}\in{\rm Rev}^{\rm
adm}_{D_{\Gamma}}(X_{\Gamma}^{\bullet})\setminus\\{0\\}$ such that the
following holds:

$\gamma_{(\alpha_{\Gamma},D_{\Gamma})}=g_{X_{\Gamma}}+n_{X}-2.$

###### Proof.

Since $\Gamma$ is a minimal quasi-tree associated to $D_{X}$, we obtain that
$\Gamma^{\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma\setminus
e^{\rm lp}(\Gamma)$ is a tree. Then we have $v(\Gamma)=v(\Gamma^{\prime})$.
Note that $D_{X}\subseteq D_{X_{\Gamma}}$. Let $v\in v(\Gamma)$ be an
arbitrary vertex and $n_{0}=p^{t_{0}}-1\in\mathbb{N}$ a positive natural
number satisfying

$n_{0}>\text{max}\\{C(g_{X})+1,\\#(e(\Gamma_{X^{\bullet}}))\\}.$

Step 1 (mp): Let $v\in v(\Gamma)$. We construct a family of effective divisors
$\text{Div}_{v}^{\rm irr\text{-}mp}$ on the irreducible component
$X^{\bullet}_{v}$ associated to the set of marked points $D_{X}\cap X_{v}$,
and construct a family of effective divisors $\text{Div}_{v}^{\rm mp}$ on
$X_{\Gamma}$, where “mp” means “marked point”.

We put

$D^{\prime}_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{X}\cap X_{v},\
m_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\#(D^{\prime}_{v}),\
\text{and}\
D^{\prime}_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{v,1},\dots,x_{v,m_{v}}\\}\
\text{if}\ m_{v}\neq 0.$

Note that $D_{v}^{\prime}\neq D_{X_{\Gamma}}\cap X_{v}$ in general. Moreover,
we put

$D_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D^{\prime}_{v}\cup(X_{v}\cap(\overline{X_{\Gamma}\setminus
X_{v}})),$

where $\overline{X_{\Gamma}\setminus X_{v}}$ denotes the topological closure
of $X_{\Gamma}\setminus X_{v}$ in $X_{\Gamma}$. Note that since $n_{X}>0$, we
have $\\#(D_{v})>0$. Let $w\in v(\Gamma)\setminus\\{v\\}$ be an arbitrary
vertex distinct from $v$. Since $\Gamma^{\prime}$ is a tree, there exists a
unique node

$x_{v,w}^{-}\in D_{w}\subseteq X_{w}$

such that the closed edge of $\Gamma^{\prime}$ corresponding to $x_{v,w}^{-}$
is contained in the minimal path $\delta(\Gamma^{\prime},v,w)$ (i.e., the
minimal path connecting $v$ and $w$ in $\Gamma^{\prime}$ defined in 5.2.3). On
the other hand, we define a set of nodes to be

$\text{Node}_{v,w}^{+}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{X_{w}\cap
X_{w^{\prime}},\ w^{\prime}\in v(\Gamma)\ |\
\text{leng}(\delta(\Gamma^{\prime},v,w^{\prime}))=\text{leng}(\delta(\Gamma^{\prime},v,w))+1\\}.$

Note that $\text{Node}_{v,w}^{+}$ may possibly be an empty set, and that
$D_{w}=\\{x^{-}_{v,w}\\}\cup\text{Node}^{+}_{v,w}\cup D^{\prime}_{w}.$

We define two sets of effective divisors

$\text{Div}_{v}^{\text{irr-mp}},\ \text{Div}_{v}^{\text{mp}}$

associated to $v$ on $X_{v}$ and $X_{\Gamma}$, as follows. Let
$i\in\\{1,\dots,m_{v}-1\\}$ and $0<a_{v,i,1},a_{v,i,2}<n_{0}$ such that
$a_{v,i,1}+a_{v,i,2}=n_{0}$. Suppose that $m_{v}\leq 1$. Then we put

$\text{Div}_{v}^{\text{irr-
mp}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\emptyset,\
\text{Div}_{v}^{\text{mp}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\emptyset.$

Suppose that $m_{v}\geq 2$. We define

$Q_{v,v,i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}a_{v,i,1}x_{v,i}+a_{v,i,2}x_{v,i+1}+\sum_{x^{\prime}\in
D^{\prime}_{v}\setminus\\{x_{v,i},x_{v,i+1}\\}}n_{0}x^{\prime}+\sum_{x\in
D_{v}\setminus D^{\prime}_{v}}n_{0}x,\ i\in\\{1,\dots,m_{v}-1\\},$

to be an effective divisor on $X_{v}$ whose support is $D_{v}$, and whose
degree is equal to $(\\#(D_{v})-1)n_{0}$. We define

$Q_{v,w,i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{x\in
D_{w}\setminus\\{x_{v,w}^{-}\\}}n_{0}x,\ w\in v(\Gamma)\setminus\\{v\\},$

to be an effective divisor on $X_{w}$ whose support is
$D_{w}\setminus\\{x_{v,w}^{-}\\}$, and whose degree is equal to
$(\\#(D_{w})-1)n_{0}$.

Moreover, we define

$Q^{v}_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}a_{v,i,1}x_{v,i}+a_{v,i,2}x_{v,i+1}+\sum_{x\in
D_{X}\setminus\\{x_{v,i},x_{v,i+1}\\}}n_{0}x,$

to be an effective divisor on $X_{\Gamma}$ whose support is $D_{X}$, and whose
degree is $(n_{X}-1)n_{0}$. Note that
$Q_{i}^{v}|_{D_{u}^{\prime}}=Q_{v,u,i}|_{D_{u}^{\prime}}$ for all $u\in
v(\Gamma)$, and that $Q_{v,w,i}=Q_{v,w,i^{\prime}}$ for all
$i,i^{\prime}\in\\{1,\dots,m_{v}-1\\}$.

We put

$\text{Div}_{v,i}^{\text{irr-
mp}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{u\in
v(\Gamma)}\\{Q_{v,u,i}\\},\ \text{Div}_{v}^{\text{irr-
mp}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{i=1}^{m_{v}-1}\text{Div}_{v,i}^{\text{irr-
mp}},\
\text{Div}_{v}^{\text{mp}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{i=1}^{m_{v}-1}\\{Q_{i}^{v}\\},$

where $\sqcup$ means disjoint union.

Step 1 (nd): Let $v\in v(\Gamma)$. We construct a family of effective divisors
$\text{Div}_{v}^{\rm irr\text{-}nd}$ on the irreducible component
$X^{\bullet}_{v}$ associated to the set of nodes $D_{v}\setminus
D^{\prime}_{v}=X_{v}\cap(\overline{X_{\Gamma}\setminus X_{v}})$, and construct
a family of effective divisors $\text{Div}_{v}^{\rm nd}$ on $X_{\Gamma}$,
where “nd” means “node”.

We define two families of effective divisors

$\text{Div}_{v}^{\text{irr-nd}},\ \text{Div}_{v}^{\text{nd}}$

associated to $v$ on $X_{v}$ and $X_{\Gamma}$, respectively, as follows. Let
$z\in D_{X}\setminus D^{\prime}_{v}$ and $0<b_{v,z,1},b_{v,z,2}<n_{0}$ such
that $b_{v,z,1}+b_{v,z,2}=n_{0}$. Suppose that $m_{v}=0$. Then we put

$\text{Div}_{v}^{\text{irr-
nd}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\emptyset,\
\text{Div}_{v}^{\text{nd}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\emptyset.$

Suppose that $m_{v}\neq 0$. Let $w_{z}$ be the vertex such that the
irreducible component $X_{w_{z}}$ corresponding to $w_{z}$ contains $z$ (i.e.,
$z\in D^{\prime}_{w_{z}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{X}\cap
X_{w_{z}}$). Note that $w_{z}\neq v$. Moreover, let
$\delta(\Gamma^{\prime},v,w_{z})$ be the minimal path connecting $v$ and
$w_{z}$ in $\Gamma^{\prime}$ and $w\in v(\Gamma)$ an arbitrary vertex distinct
from $w_{z}$ such that $w\subseteq\delta(\Gamma^{\prime},v,w_{z})$. Since
$\Gamma^{\prime}$ is a tree, we have
$\\#(\text{Node}_{v,w}^{+}\cap\delta(\Gamma^{\prime},v,w_{z}))=1$. Then we put

$x_{v,w,w_{z}}^{+}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{Node}_{v,w}^{+}\cap\delta(\Gamma^{\prime},v,w_{z})\in
D_{w}\subseteq X_{w}.$

For $v$ and $w_{z}$, we define

$Q_{v,v,z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{v,z,1}x_{v,m_{v}}+b_{v,z,2}x_{v,v,w_{z}}^{+}+\sum_{x\in
D_{v}\setminus\\{x_{v,m_{v}},x_{v,v,w_{z}}^{+}\\}}n_{0}x,$
$Q_{v,w_{z},z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{v,z,1}x_{v,w_{z}}^{-}+b_{v,z,2}z+\sum_{x\in
D_{w_{z}}\setminus\\{x_{v,w_{z}},z\\}}n_{0}x$

to be effective divisors on $X_{v}$ and $X_{w_{z}}$ whose supports are $D_{v}$
and $D_{w_{z}}$, and whose degrees are equal to $(\\#(D_{v})-1)n_{0}$ and
$(\\#(D_{w_{z}})-1)n_{0}$, respectively.

Let $w\in v(\Gamma)\setminus\\{v,w_{z}\\}$ be an arbitrary vertex such that
$w\subseteq\delta(\Gamma^{\prime},v,w_{z})$. Then we define

$Q_{v,w,z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{v,z,1}x_{v,w}^{-}+b_{v,z,2}x_{v,w,w_{z}}^{+}+\sum_{x\in
D_{w}\setminus\\{x_{v,w}^{-},x_{v,w,w_{z}}^{+}\\}}n_{0}x$

to be an effective divisor on $X_{w}$ whose support is $D_{w}$, and whose
degree is equal to $(\\#(D_{w})-1)n_{0}$.

Let $w^{\prime}\in v(\Gamma)$ be an arbitrary vertex such that
$w^{\prime}\not\subseteq\delta(\Gamma^{\prime},v,w_{z})$. Then we define

$Q_{v,w^{\prime},z}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{x\in
D_{w^{\prime}}\setminus\\{x_{v,w^{\prime}}^{-}\\}}n_{0}x$

to be an effective divisor on $X_{w^{\prime}}$ whose support is
$D_{w^{\prime}}\setminus\\{x_{v,w^{\prime}}^{-}\\}$, and whose degree is equal
to $(\\#(D_{w^{\prime}})-1)n_{0}$. Note that, if
$w^{\prime\prime}\not\subseteq\delta(\Gamma^{\prime},v,w_{z})\cup\delta(\Gamma^{\prime},v,w_{z^{\prime}})$
for $z,z^{\prime}\in D_{X}\setminus D_{v}^{\prime}$, we have
$Q_{v,w^{\prime\prime},z}=Q_{v,w^{\prime\prime},z^{\prime}}$.

Moreover, we define

$Q_{z}^{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}b_{v,z,1}x_{v,m_{v}}+b_{v,z,2}z+\sum_{x\in
D_{X}\setminus\\{x_{v,m_{v}},z\\}}n_{0}x$

to be an effective divisor on $X_{\Gamma}$ whose support is $D_{X}$, and whose
degree is equal to $(n_{X}-1)n_{0}$. Note that
$Q_{z}^{v}|_{D_{u}^{\prime}}=Q_{v,u,z}|_{D_{u}^{\prime}}$ for all $u\in
v(\Gamma)$.

We put

$\text{Div}_{v,z}^{\text{irr-
nd}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{u\in
v(\Gamma)}\\{Q_{v,u,z}\\},\ \text{Div}_{v}^{\text{irr-
nd}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{z\in D_{X}\setminus
D^{\prime}_{v}}\text{Div}_{v,z}^{\text{irr-nd}},\
\text{Div}_{v}^{\text{nd}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{z\in
D_{X}\setminus D^{\prime}_{v}}\\{Q_{z}^{v}\\}.$

Step 2: We construct an effective divisor
$P_{u}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{u}]^{0}$, $v\in v(\Gamma)$, on
$X_{u}$ and an effective divisor
$P_{\Gamma}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{\Gamma}}]^{0}$ on
$X_{\Gamma}$ for some $n$.

We put

$\text{Div}^{\text{irr}}_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{v\in
v(\Gamma)}(\text{Div}_{v}^{\text{irr-mp}}\sqcup\text{Div}_{v}^{\text{irr-
nd}}),$
$\text{Div}_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigsqcup_{v\in
v(\Gamma)}(\text{Div}_{v}^{\text{mp}}\sqcup\text{Div}_{v}^{\text{nd}}).$

We denote by
$\text{Div}^{\text{irr}}_{X}(X_{u})\subseteq\text{Div}_{X}^{\text{irr}}$,
$u\in v(\Gamma)$, the subset whose elements are effective divisors on $X_{u}$.
Note that the above constructions imply
$d\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\#(\text{Div}_{X}^{\text{irr}}(X_{u_{1}}))=\\#(\text{Div}_{X}^{\text{irr}}(X_{u_{2}}))=\\#(\text{Div}_{X})$
for all $u_{1},u_{2}\in v(\Gamma)$. Moreover, let

$o_{u}:\\{1,\dots,d\\}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\text{Div}_{X}^{\text{irr}}(X_{u})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{P_{u,1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}o_{u}(1),\dots,P_{u,d}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}o_{u}(d)\\},\
u\in v(\Gamma),$

be a bijective (as sets) such that, for all $u_{1},u_{2}\in v(\Gamma)$ and all
$j\in\\{1,\dots,d\\}$, the following conditions are satisfied:

  * •

if $P_{u_{1},j}\in\text{Div}_{v,i}^{\text{irr-mp}}$ for some $v\in v(\Gamma)$
and some $i\in\\{1,\dots,m_{v}-1\\}$, then
$P_{u_{2},j}\in\text{Div}_{v,i}^{\text{irr-mp}}$.

  * •

If $P_{u_{1},j}\in\text{Div}_{v,z}^{\text{irr-nd}}$ for some $v\in v(\Gamma)$
and some $z\in D_{X}\setminus D_{v}^{\prime}$, then
$P_{u_{2},j}\in\text{Div}_{v,z}^{\text{irr-nd}}$.

Then, by the construction of $\text{Div}_{X}$, we obtain a bijection

$o:\\{1,\dots,d\\}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\text{Div}_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{P_{1}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}o(1),\dots,P_{d}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}o(d)\\}$

induced by $o_{u}$, $u\in v(\Gamma)$.

Let $t\stackrel{{\scriptstyle\mathrm{def}}}{{=}}dt_{0}$ and
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{d}p^{(j-1)t_{0}}(p^{t_{0}}-1)=p^{t}-1$.
We define

$P_{u}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{d}p^{(j-1)t_{0}}P_{u,j}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{u}]^{0},\
u\in v(\Gamma),$
$P_{\Gamma}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=1}^{d}p^{(j-1)t_{0}}P_{j}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$

to be effective divisors of degrees $\text{deg}(P_{u})=(\\#(D_{u})-1)n$ and
$\text{deg}(P_{\Gamma})=(n_{X}-1)n$ on $X_{u}$ and $X_{\Gamma}$, respectively.
We see that the support of $P_{u}$, $u\in v(\Gamma)$, is $D_{u}$, and that the
support of $P_{\Gamma}$ is $D_{X}$.

Let $u\in v(\Gamma)$ and $x^{\prime}\in D^{\prime}_{u}$. Then the above
constructions implies $P_{\Gamma}|_{x^{\prime}}=P_{u}|_{x^{\prime}}$.
Moreover, let $x\in D_{u}\setminus D_{u}^{\prime}$. Since
$\Gamma\setminus\\{e^{\rm lp}(\Gamma)\\}$ is a tree,
$X_{\Gamma}\setminus\\{x\\}$ has two connected components $C_{1}$, $C_{2}$.
Let $C_{2}$ be the connected component such that $X_{u}\setminus\\{x\\}$ is
not contained in $C_{2}$. We denote by
$C_{x}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\overline{\\{C_{2}\\}}$ the
topological closure of $C_{2}$ in $X_{\Gamma}$ and
$M_{x}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}C_{x}\cap D_{X}$. Then the
above constructions imply

$P_{u}|_{x}=\text{ord}_{x}(P_{u})x=[\sum_{x^{\prime}\in
M_{x}}\text{ord}_{x^{\prime}}(P_{\Gamma})]x,$

where $[(-)]$ denotes the image of $(-)$ in $\mathbb{Z}/n\mathbb{Z}$.

Step 3: We construct a desired Galois multi-admissible covering of
$X^{\bullet}$.

Let $u\in v(\Gamma)$ and $\widetilde{X}^{\bullet}_{u}$ the smooth pointed
stable curve of type $(g_{u},n_{u})$ over $k$ (2.1.3). Write
$\text{norm}_{u}:\widetilde{X}_{u}\rightarrow X_{u}$ for the normalization
morphism. Then we obtain the pulling back divisor $\text{norm}_{u}^{*}(P_{u})$
on $\widetilde{X}_{u}$. Note that since $\text{Supp}(P_{u})$ is contained in
the smooth locus of $X_{u}$, we have $\text{norm}_{u}^{*}(P_{u})=P_{u}$. Let
$\mathcal{L}_{P_{u}}$ be a line bundle on $\widetilde{X}_{u}$ such that
$\mathcal{L}_{P_{u}}^{\otimes n}\cong\mathcal{O}_{\widetilde{X}_{u}}(-P_{u})$,
and $\mathcal{L}_{P_{u},t}$ the pulling back of $\mathcal{L}_{P_{u}}$ by the
natural morphism $\widetilde{X}_{u,t}\rightarrow\widetilde{X}$ defined in
2.4.2. Then by applying [T2, Corollary 2.6, Lemma 2.12 (ii), and Corollary
2.13], we see that the Raynaud-Tamagawa theta divisor associated to
$\mathcal{B}^{t}_{P_{u}}\otimes\mathcal{L}_{P_{u},t}$ exists (Definition 2.7).
Moreover, Proposition 2.8 implies that there exists
$\widetilde{\alpha}_{u}\in\text{Rev}^{\rm
adm}_{P_{u}}(\widetilde{X}_{u}^{\bullet})$ such that

$\gamma_{(\widetilde{\alpha}_{u},P_{u})}=g_{u}+\\#(D_{u})-2.$

Then Theorem 3.9 implies that the element $\alpha_{u}\in\text{Rev}^{\rm
adm}_{P_{u}}(X_{u}^{\bullet})$ induced by $\widetilde{\alpha}_{u}$ such that
the following holds:

$\gamma_{(\alpha_{u},P_{u})}=g_{X_{u}}+\\#(D_{u})-2,$

where $g_{X_{u}}$ denotes the genus of $X_{u}$. Write

$f_{u}^{\bullet}:Y_{u}^{\bullet}\rightarrow X_{u}^{\bullet}$

for the Galois multi-admissible covering over $k$ with Galois group
$\mathbb{Z}/n\mathbb{Z}$ induced by $\alpha_{u}$.

Let $u^{\prime}\in v(\Gamma)\setminus\\{u\\}$ such that $X_{u}\cap
X_{u^{\prime}}\neq\emptyset$. We denote by
$x_{u,u^{\prime}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}X_{u}\cap
X_{u^{\prime}}$ the unique node. Then the above constructions imply

$0<\text{ord}_{x_{u,u^{\prime}}}(P_{u}),\text{ord}_{x_{u,u^{\prime}}}(P_{u^{\prime}})<n,\
\text{ord}_{x_{u,u^{\prime}}}(P_{u})+\text{ord}_{x_{u,u^{\prime}}}(P_{u^{\prime}})=n.$

This means that we may glue (as admissible coverings)
$\\{Y_{u}^{\bullet}\\}_{u\in v(\Gamma)}$ along $\\{f^{-1}_{u}(D_{u}\setminus
D^{\prime}_{u})\\}_{u\in v(\Gamma)}$ in a way that is compatible with the
gluing of $\\{X_{u}^{\bullet}\\}_{u\in v(\Gamma)}$ that gives rise to
$X_{\Gamma}^{\bullet}$. Then we obtain a Galois multi-admissible covering

$f_{\Gamma}^{\bullet}:Y_{\Gamma}^{\bullet}\rightarrow X_{\Gamma}^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Note that the
construction of $f_{\Gamma}^{\bullet}$ implies that $f_{\Gamma}$ is étale over
$D_{X_{\Gamma}}\setminus D_{X}$.

Let $\Pi_{X^{\bullet}_{u}}$, $u\in v(\Gamma)$, be the admissible fundamental
group of $X_{u}^{\bullet}$. We denote by
$\alpha_{\Gamma}\in\text{Hom}(\Pi^{\rm
ab}_{X_{\Gamma}^{\bullet}},\mathbb{Z}/n\mathbb{Z})$ an element induced by
$f_{\Gamma}^{\bullet}$ such that the composition of the natural homomorphisms
$\Pi_{X_{u}^{\bullet}}^{\rm ab}\rightarrow\Pi_{X_{\Gamma}^{\bullet}}^{\rm
ab}\overset{\alpha_{\Gamma}}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$ is equal to
$\alpha_{u}$ for all $u\in v(\Gamma)$. We put
$D_{\Gamma}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}P_{\Gamma}$. Then we see
$\alpha_{\Gamma}\in\text{Rev}^{\rm
adm}_{D_{\Gamma}}(X_{\Gamma}^{\bullet})\setminus\\{0\\}$. Theorem 3.9 implies

$\gamma_{(\alpha_{\Gamma},D_{\Gamma})}=g_{X_{\Gamma}}+s(P_{\Gamma})-1=g_{X_{\Gamma}}+n_{X}-2.$

We complete the proof of the lemma. ∎

###  5.3. The second main theorem

Now, we prove the second main result of the present paper.

######  Theorem 5.4.

Let $X^{\bullet}$ be an arbitrary pointed stable curve of type $(g_{X},n_{X})$
over an algebraically closed field $k$ of characteristic $p>0$. Then there
exist a positive natural number
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1\in\mathbb{N}$, an
effective divisor $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ (2.2.5) on
$X$ of degree $(n_{X}-1)n$ if $n_{X}\neq 0$ (resp. degree $0$ if $n_{X}=0$),
and an element $\alpha\in{\rm Rev}^{\rm adm}_{D}(X^{\bullet})\setminus\\{0\\}$
(Definition 2.4 (i)) such that $\gamma_{(\alpha,D)}$ attains maximum
(Definition 3.10 (ii)). Namely, the following holds (see Definition 3.10 (i)
for $\gamma^{\rm max}_{X^{\bullet}}$):

$\displaystyle\gamma_{(\alpha,D)}=\gamma^{\rm
max}_{X^{\bullet}}=\left\\{\begin{array}[]{ll}g_{X}-1,&\text{if}\ n_{X}=0,\\\
g_{X}+n_{X}-2,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

###### Proof.

Suppose that $n_{X}\leq 1$. Then the theorem follows from Proposition 3.11
(ii). To verify the theorem, we may assume $n_{X}\geq 2$.

Let $\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$ be a
minimal quasi-tree associated to $D_{X}$, $\Gamma^{\rm im}$ the image of the
natural morphism $\phi_{\Gamma}:\Gamma\rightarrow\Gamma_{X^{\bullet}}$,
$X^{\bullet}_{\Gamma}=(X_{\Gamma},D_{X_{\Gamma}})$, $X_{\Gamma^{\rm
im}}^{\bullet}=(X_{\Gamma^{\rm im}},D_{X_{\Gamma^{\rm im}}})$ the pointed
stable curves over $k$ of types $(g_{X_{\Gamma}},n_{X_{\Gamma}})$,
$(g_{X_{\Gamma^{\rm im}}},n_{X_{\Gamma^{\rm im}}})$ associated to $\Gamma$,
$\Gamma^{\rm im}$, respectively (see 4.4.3, 4.4.5 for the definitions of
$\Gamma^{\rm im}$, $\phi_{\Gamma}$, $X^{\bullet}_{\Gamma}$, and
$X^{\bullet}_{\Gamma^{\rm im}}$), and $\Pi_{X^{\bullet}_{\Gamma}},$
$\Pi_{X^{\bullet}_{\Gamma^{\rm im}}}$ the admissible fundamental groups of
$X^{\bullet}_{\Gamma}$, $X^{\bullet}_{\Gamma^{\rm im}}$, respectively.

Lemma 5.3 implies that there exist a natural number

$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1>\text{max}\\{C(g_{X})+1,\\#(e(\Gamma_{X^{\bullet}}))\\},$

an effective divisor
$D\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{\Gamma}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}\subseteq(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{\Gamma}}]^{0}$
on $X_{\Gamma}$ of degree $(n_{X}-1)n$, and an element
$\alpha_{\Gamma}\in\text{Rev}_{D}^{\rm
adm}(X^{\bullet}_{\Gamma})\setminus\\{0\\}$ such that the following holds:

$\gamma_{(\alpha_{\Gamma},D)}=g_{X_{\Gamma}}+n_{X}-2.$

We denote by $f_{\Gamma}^{\bullet}:Z_{\Gamma}^{\bullet}\rightarrow
X^{\bullet}_{\Gamma}$ the Galois multi-admissible covering over $k$ with
Galois group $\mathbb{Z}/n\mathbb{Z}$ induced by $\alpha_{\Gamma}$. Note that
$f_{\Gamma}$ is étale over $D_{X_{\Gamma}}\setminus D_{X}$. By gluing
$Z_{\Gamma}^{\bullet}$ along

$f_{\Gamma}^{-1}(D_{X_{\Gamma}}\setminus(D_{X}\cup\\{x_{e}\\}_{e\in\phi^{-1}_{\Gamma}(e^{\rm
op}(\Gamma^{\rm im}))}))$

in a way that is compatible with the gluing of $X_{\Gamma}^{\bullet}$ that
gives rise to $X_{\Gamma^{\rm im}}^{\bullet}$, we obtain a pointed stable
curve $Z_{\Gamma^{\rm im}}^{\bullet}$ over $k$. Moreover,
$f_{\Gamma}^{\bullet}$ induces a Galois multi-admissible covering

$f_{\Gamma^{\rm im}}^{\bullet}:Z_{\Gamma^{\rm im}}^{\bullet}\rightarrow
X^{\bullet}_{\Gamma^{\rm im}}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Write
$\alpha_{\Gamma^{\rm im}}$ for an element of $\text{Hom}(\Pi^{\rm
ab}_{X^{\bullet}_{\Gamma^{\rm im}}},\mathbb{Z}/n\mathbb{Z})$ induced by
$f^{\bullet}_{\Gamma^{\rm im}}$ such that the composition of the natural
homomorphisms $\Pi_{X^{\bullet}_{\Gamma}}^{\rm ab}\rightarrow\Pi^{\rm
ab}_{X^{\bullet}_{\Gamma^{\rm im}}}\overset{\alpha_{\Gamma^{\rm
im}}}{\rightarrow}\mathbb{Z}/n\mathbb{Z}$ is equal to $\alpha_{\Gamma}$. Note
that we have $D_{\alpha_{\Gamma^{\rm im}}}=D$. Then Theorem 3.9 implies

$\gamma_{(\alpha_{\Gamma^{\rm im}},D)}=g_{X_{\Gamma^{\rm im}}}+n_{X}-2.$

On the other hand, we write $\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})$ for the set of connected components of the topological closure
$\overline{X\setminus X_{\Gamma^{\rm im}}}$ of $X\setminus X_{\Gamma^{\rm
im}}$ in $X$. We define the following pointed stable curve

$E^{\bullet}=(E,D_{E}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}E\cap
X_{\Gamma^{\rm im}}),\ E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}}),$

over $k$. Proposition 3.11 (ii) implies that there exists a Galois étale
covering

$f^{\bullet}_{E}:Z^{\bullet}_{E}=(Z_{E},D_{Z_{E}})\rightarrow E^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$ such that the following
holds:

$\displaystyle\gamma_{(\alpha_{E},0)}=\left\\{\begin{array}[]{ll}g_{E},&\text{if}\
g_{E}=0,\\\ g_{X}-1,&\text{if}\ g_{E}\neq 0,\end{array}\right.$

where $g_{E}$ denotes the genus of $E$, and $\alpha_{E}\in\text{Rev}^{\rm
adm}_{0}(E^{\bullet})$ is an element induced by $f^{\bullet}_{E}$.

We may glue $Z_{\Gamma^{\rm im}}^{\bullet}$ and
$\\{Z_{E}^{\bullet}\\}_{E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})}$ along $f^{-1}_{\Gamma}(X_{\Gamma^{\rm
im}}\cap(\bigcup_{E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}})}E))$
and $\\{f^{-1}_{E}(X_{\Gamma^{\rm im}}\cap
E)\\}_{E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}})}$ in a way that
is compatible with the gluing of $\\{X^{\bullet}_{\Gamma^{\rm
im}}\\}\cup\\{E^{\bullet}\\}_{E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm
im}}})}$ that gives rise to $X^{\bullet}$, then we obtain a Galois multi-
admissible covering

$f^{\bullet}:Z^{\bullet}\rightarrow X^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$.

Let $\Pi_{X^{\bullet}}$, $\Pi_{E^{\bullet}}$ be the admissible fundamental
groups of $X^{\bullet}$, $E^{\bullet}$, $E\in\pi_{0}(\overline{X\setminus
X_{\Gamma^{\rm im}}})$, respectively. We write
$\alpha\in\text{Hom}(\Pi_{X^{\bullet}}^{\rm ab},\mathbb{Z}/n\mathbb{Z})$ for
an element induced by $f^{\bullet}$ such that the compositions of the natural
homomorphisms $\Pi^{\rm ab}_{X^{\bullet}_{\Gamma^{\rm im}}}\rightarrow\Pi^{\rm
ab}_{X^{\bullet}}\overset{\alpha}{\rightarrow}\mathbb{Z}/n\mathbb{Z},$
$\Pi^{\rm ab}_{E^{\bullet}}\rightarrow\Pi^{\rm
ab}_{X^{\bullet}}\overset{\alpha}{\rightarrow}\mathbb{Z}/n\mathbb{Z},\
E\in\pi_{0}(\overline{X\setminus X_{\Gamma^{\rm im}}}),$ are equal to
$\alpha_{\Gamma^{\rm im}}$ and $\alpha_{E}$, respectively. We see
$\alpha\in\text{Rev}^{\rm adm}_{D}(X^{\bullet})\setminus\\{0\\}$. By applying
Theorem 3.9, we obtain

$\gamma_{(\alpha,D)}=g_{X}+n_{X}-2.$

This completes the proof of the theorem. ∎

###  5.4. A stronger version when $n_{X}=3$

In the remainder of the present section, we prove a slightly stronger version
of Theorem 5.4 when $n_{X}=3$ (i.e., we prove Theorem 5.4 for a certain fixed
effective divisor $D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$). The
stronger version will be used in the proof of reconstructions of field
structures associated to inertia subgroups (cf. Section 6.2). The main result
of this subsection is Theorem 5.7.

####  5.4.1. Settings

We maintain the notation introduced in 2.1.2. Moreover, we suppose that
$n_{X}=3$.

#### 5.4.2.

We generalize Lemma 5.3 to the case of certain fixed ramification divisors
when $n_{X}=3$. Let
$D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},x_{2},x_{3}\\}$, and
let $\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{D_{X}}$ be a
minimal quasi-tree associated to $D_{X}$,

$X_{\Gamma}^{\bullet}=(X_{\Gamma},D_{X_{\Gamma}})$

the pointed stable curve of type $(g_{X_{\Gamma}},n_{X_{\Gamma}})$ associated
to $\Gamma$, and $\Pi_{X^{\bullet}_{\Gamma}}$ the admissible fundamental group
of $X^{\bullet}_{\Gamma}$. Let
$D_{j}\in\mathbb{Z}[D_{X}]\subseteq\mathbb{Z}[D_{X_{\Gamma}}]$,
$j\in\\{1,2,3\\}$, be an effective divisor on $X_{\Gamma}$ such that the
following conditions are satisfied:

  * •

${\rm deg}(D_{j})=2(p^{t_{j}}-1)$.

  * •

${\rm ord}_{x}(D_{j})\leq p^{t_{j}}-1$ for each $x\in D_{X}$.

  * •

$\\#\\{x\in D_{X}\ |\ {\rm ord}_{x}(D_{j})=p^{t_{j}}-1\\}\geq 1$.

Let
$n\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t}-1\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t_{1}+t_{2}+t_{3}}-1$,
and let

$D_{\Gamma}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{1}+p^{t_{1}}D_{2}+p^{t_{1}+t_{2}}D_{3}\in\mathbb{Z}[D_{X}]\subseteq\mathbb{Z}[D_{X_{\Gamma}}]$

be an effective divisor on $X_{\Gamma}$ with degree $2n$. Then we have the
following lemma.

######  Lemma 5.5.

We maintain the notation introduced above. Suppose that
$D_{\Gamma}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}\subseteq(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X_{\Gamma}}]^{0}$
(i.e., ${\rm ord}_{x}(D_{\Gamma})<n$ for all $x\in D_{X}$), and that

$n>{\rm max}\\{C(g_{X})+1,\\#(e(\Gamma_{X^{\bullet}}))\\}.$

Then there exists an element $\alpha_{\Gamma}\in{\rm Rev}^{\rm
adm}_{D_{\Gamma}}(X_{\Gamma}^{\bullet})\setminus\\{0\\}$ such that the
following holds:

$\gamma_{(\alpha_{\Gamma},D_{\Gamma})}=g_{X_{\Gamma}}+1.$

###### Proof.

Since $\Gamma$ is a minimal quasi-tree associated to $D_{X}$, we obtain that
$\Gamma^{\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma\setminus
e^{\rm lp}(\Gamma)$ is a tree. If $D_{X}\subseteq X_{v}$ for some $v\in
v(\Gamma)$, then the lemma follows from Proposition 5.1. Without loss of
generality, we may assume that one of the following conditions holds:

> (i) Let $w_{1}$, $w_{2}\in v(\Gamma)$ be vertices distinct from each other
> such that $x_{1}$, $x_{2}\in X_{w_{1}}$, and $x_{3}\in X_{w_{2}}$ (see
> Example 5.6 (i) below).
>
> (ii) Let $v_{1}$, $v_{2}$, $v_{3}\in v(\Gamma)$ be vertices distinct from
> each other such that $x_{1}\in X_{v_{1}}$, $x_{2}\in X_{v_{2}}$, and
> $x_{3}\in X_{v_{3}}$ (see Example 5.6 (ii) below).

We put

$D^{\prime}_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{X}\cap X_{v},\
D_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D^{\prime}_{v}\cup(X_{v}\cap(\overline{X_{\Gamma}\setminus
X_{v}})),\ v\in v(\Gamma)$

where $\overline{X_{\Gamma}\setminus X_{v}}$ denotes the topological closure
of $X_{\Gamma}\setminus X_{v}$ in $X_{\Gamma}$. Next, we construct an
effective divisor $P_{v}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{v}]^{0}$ on
$X_{v}$ for all $v\in v(\Gamma)$.

Suppose that (i) holds. Let $y_{w_{1},w_{2}}\in D_{w_{1}}$ be the unique node
of $X_{\Gamma}$ such that the closed edge of $\Gamma^{\prime}$ corresponding
to $y_{w_{1},w_{2}}$ is contained in the minimal path
$\delta(\Gamma^{\prime},w_{1},w_{2})$ (5.2.3) and $z_{w_{1},w_{2}}\in
D_{w_{2}}$ the unique node of $X_{\Gamma}$ such that the closed edge of
$\Gamma^{\prime}$ corresponding to $z_{w_{1},w_{2}}$ is contained in the
minimal path $\delta(\Gamma^{\prime},w_{1},w_{2})$. On the other hand, let
$w\in v(\Gamma)\setminus\\{w_{1},w_{2}\\}$ be an arbitrary vertex (possibly
empty). Note that $n_{X}=3$ implies that $w$ is contained in
$\delta(\Gamma^{\prime},w_{1},w_{2})$. Since $\Gamma^{\prime}$ is a tree,
there exist a unique node $x_{w_{1},w}\in D_{w}$ and a unique node
$x_{w_{2},w}\in D_{w}$ such that the closed edges of $\Gamma^{\prime}$
corresponding to $x_{w_{1},w}$ and $x_{w_{2},w}$ are contained in
$\delta(\Gamma^{\prime},w_{1},w)$ and $\delta(\Gamma^{\prime},w_{2},w)$,
respectively.

Let $j\in\\{1,2,3\\}$. We put

$Q_{w_{1},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{1}+\text{ord}_{x_{2}}(D_{j})x_{2}+\text{ord}_{x_{3}}(D_{j})y_{w_{1},w_{2}},$
$Q_{w_{2},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]z_{w_{1},w_{2}}+\text{ord}_{x_{3}}(D_{j})x_{3},$
$Q_{w,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]x_{w_{1},w}+\text{ord}_{x_{3}}(D_{j})x_{w_{2},w},\
w\in v(\Gamma)\setminus\\{w_{1},w_{2}\\},$

where $[(-)]$ denotes the image of $(-)$ in
$\mathbb{Z}/(p^{t_{j}}-1)\mathbb{Z}$. Then $Q_{v,j}$, $v\in v(\Gamma)$, is an
effective divisor on $X_{v}$ whose degree is equal to
$(\\#(D_{v})-1)(p^{t_{j}}-1)$. Moreover, we put

$P_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{v,1}+p^{t_{1}}Q_{v,2}+p^{t_{1}+t_{2}}Q_{v,3}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{v}]^{0},\
v\in v(\Gamma).$

Then $P_{v}$ is an effective divisor on $X_{v}$ whose degree is equal to
$(\\#(D_{v})-1)n$, and whose support is equal to $D_{v}$.

Suppose that (ii) holds. Then one of the following conditions is satisfied:

> (1) There exist $a,b,c\in\\{1,2,3\\}$ distinct from each other such that
> $\delta(\Gamma^{\prime},v_{a},v_{b})\cap\delta(\Gamma^{\prime},v_{b},v_{c})\cap
> e^{\rm cl}(\Gamma)=\emptyset$ (i.e.,
> $\delta(\Gamma^{\prime},v_{a},v_{c})=\delta(\Gamma^{\prime},v_{a},v_{b})\cup\delta(\Gamma^{\prime},v_{b},v_{c})$)
> (see Example 5.6 (ii)-(1) below).
>
> (2) For all $a,b,c\in\\{1,2,3\\}$ distinct from each other, we have that
> $\delta(\Gamma^{\prime},v_{a},v_{b})\cap\delta(\Gamma^{\prime},v_{b},v_{c})\cap
> e^{\rm cl}(\Gamma)\neq\emptyset$ (see Example 5.6 (ii)-(2) below).

Suppose that (1) holds. Without loss of generality, we may assume that $a=1$,
$b=2$, and $c=3$. Note that
$\delta(\Gamma^{\prime},v_{1},v_{3})=\delta(\Gamma^{\prime},v_{1},v_{2})\cup\delta(\Gamma^{\prime},v_{2},v_{3})$.
Write $y_{v_{1},v_{3}}\in D_{v_{1}}$ for the unique node of $X_{\Gamma}$ such
that the closed edge of $\Gamma^{\prime}$ corresponding to $y_{v_{1},v_{3}}$
is contained in $\delta(\Gamma^{\prime},v_{1},v_{3})$ and $z_{v_{1},v_{3}}\in
D_{v_{3}}$ for the unique node of $X_{\Gamma}$ such that the closed edge of
$\Gamma^{\prime}$ corresponding to $z_{v_{1},v_{3}}$ is contained in
$\delta(\Gamma^{\prime},v_{1},v_{3})$. On the other hand, let $v\in
v(\Gamma)\setminus\\{v_{1},v_{3}\\}$ be an arbitrary vertex. Since $n_{X}=3$,
we see that $v\in\delta(\Gamma^{\prime},v_{1},v_{3})$, and that either
$v\in\delta(\Gamma^{\prime},v_{1},v_{2})$ or
$v\in\delta(\Gamma^{\prime},v_{2},v_{3})$ holds. Since $\Gamma^{\prime}$ is a
tree, there exist a unique node $x_{v_{1},v}\in D_{v}$ and a unique node
$x_{v_{3},v}\in D_{v}$ such that the closed edges of $\Gamma^{\prime}$
corresponding to $x_{v_{1},v}$ and $x_{v_{3},v}$ are contained in
$\delta(\Gamma^{\prime},v_{1},v)$ and $\delta(\Gamma^{\prime},v,v_{3})$,
respectively.

Let $j\in\\{1,2,3\\}$. We put

$Q_{v_{1},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{1}+[\text{deg}(D_{j})-\text{ord}_{x_{2}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]y_{v_{1},v_{3}},$
$Q_{v_{2},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{v_{1},v_{2}}+\text{ord}_{x_{2}}(D_{j})x_{2}+\text{ord}_{x_{3}}(D_{j})x_{v_{3},v_{2}}$
$Q_{v_{3},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]z_{v_{1},v_{3}}+\text{ord}_{x_{3}}(D_{j})x_{3}$
$Q_{v,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{v_{1},v}+[\text{deg}(D_{j})-\text{ord}_{x_{2}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]x_{v_{3},v}$

for all
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{1},v_{2}))\setminus\\{v_{1},v_{2}\\}$,
and

$Q_{v,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]x_{v_{1},v}+\text{ord}_{x_{3}}(D_{j})x_{v_{3},v}$

for all
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{2},v_{3}))\setminus\\{v_{2},v_{3}\\}$.
Then $Q_{v,j}$, $v\in v(\Gamma)$, is an effective divisor on $X_{v}$ whose
degree is equal to $(\\#(D_{v})-1)(p^{t_{j}}-1)$. Moreover, we put

$P_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{v,1}+p^{t_{1}}Q_{v,2}+p^{t_{1}+t_{2}}Q_{v,3}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{v}]^{0},\
v\in v(\Gamma).$

Then $P_{v}$ is an effective divisor on $X_{v}$ whose degree is equal to
$(\\#(D_{v})-1)n$, and whose support is equal to $D_{v}$.

Suppose that (2) holds. Then we have

$\\{v_{0}\\}=v(\Gamma)\cap\delta(\Gamma^{\prime},v_{1},v_{2})\cap\delta(\Gamma^{\prime},v_{2},v_{3})\cap\delta(\Gamma^{\prime},v_{3},v_{1}).$

Let $v\in v(\Gamma)$. Since $n_{X}=3$, we obtain that either
$v\in\delta(\Gamma^{\prime},v_{1},v_{0})$ or
$v\in\delta(\Gamma^{\prime},v_{2},v_{0})$ or
$v\in\delta(\Gamma^{\prime},v_{3},v_{0})$ holds. Let $y_{v_{i},v_{0}}\in
D_{v_{i}}$, $i\in\\{1,2,3\\}$, be the unique node of $X_{\Gamma}$ such that
the closed edge of $\Gamma^{\prime}$ corresponding to $y_{v_{i},v_{0}}$ is
contained in $\delta(\Gamma^{\prime},v_{i},v_{0})$ and $z_{v_{i},v_{0}}\in
D_{v_{0}}$, $i\in\\{1,2,3\\}$, be the unique node of $X_{\Gamma}$ such that
the closed edge of $\Gamma^{\prime}$ corresponding to $z_{v_{i},v_{0}}$ is
contained in $\delta(\Gamma^{\prime},v_{i},v_{0})$. Moreover, let
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{i},v_{0}))\setminus\\{v_{i},v_{0}\\}$,
$i\in\\{1,2,3\\}$. Since $\Gamma^{\prime}$ is a tree, there exist a unique
node $x_{v_{i},v}\in D_{v}$ and a unique node $x_{v_{0},v}\in D_{v}$ such that
the closed edges of $\Gamma^{\prime}$ corresponding to $x_{v_{i},v}$ and
$x_{v_{0},v}$ are contained in $\delta(\Gamma^{\prime},v_{i},v)$ and
$\delta(\Gamma^{\prime},v_{0},v)$, respectively.

Let $j\in\\{1,2,3\\}$. We put

$Q_{v_{1},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{1}+[\text{deg}(D_{j})-\text{ord}_{x_{2}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]y_{v_{1},v_{0}},$
$Q_{v_{2},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{2}}(D_{j})x_{2}+[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]y_{v_{2},v_{0}},$
$Q_{v_{3},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{3}}(D_{j})x_{3}+[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]y_{v_{3},v_{0}},$
$Q_{v_{0},j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})z_{v_{1},v_{0}}+\text{ord}_{x_{2}}(D_{j})z_{v_{2},v_{0}}+\text{ord}_{x_{3}}(D_{j})z_{v_{3},v_{0}}$
$Q_{v,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{1}}(D_{j})x_{v_{1},v}+[\text{deg}(D_{j})-\text{ord}_{x_{2}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]x_{v_{0},v}$

for all
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{1},v_{0}))\setminus\\{v_{1},v_{0}\\}$,

$Q_{v,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{2}}(D_{j})x_{v_{2},v}+[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{3}}(D_{j})]x_{v_{0},v}$

for all
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{2},v_{0}))\setminus\\{v_{2},v_{0}\\}$,
and

$Q_{v,j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{ord}_{x_{3}}(D_{j})x_{v_{3},v}+[\text{deg}(D_{j})-\text{ord}_{x_{1}}(D_{j})-\text{ord}_{x_{2}}(D_{j})]x_{v_{0},v}$

for all
$v\in(v(\Gamma)\cap\delta(\Gamma^{\prime},v_{3},v_{0}))\setminus\\{v_{3},v_{0}\\}$.
Then $Q_{v,j}$, $v\in v(\Gamma)$, is an effective divisor on $X_{v}$ whose
degree is equal to $(\\#(D_{v})-1)(p^{t_{j}}-1)$. Moreover, we put

$P_{v}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Q_{v,1}+p^{t_{1}}Q_{v,2}+p^{t_{1}+t_{2}}Q_{v,3}\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{v}]^{0},\
v\in v(\Gamma).$

Then $P_{v}$ is an effective divisor on $X_{v}$ whose degree is equal to
$(\\#(D_{v})-1)n$, and whose support is equal to $D_{v}$.

Let $v\in v(\Gamma)$, and let $\widetilde{X}^{\bullet}_{v}$ be the smooth
pointed stable curve of type $(g_{v},n_{v})$ over $k$ associated to $v$
(2.1.3). Since $D_{X}$ is contained in the smooth locus of $X_{v}$, $P_{v}$
can be also regarded as an effective divisor on $\widetilde{X}_{v}$. By
applying similar arguments to the arguments given in the proof of Proposition
5.1, there exists $\widetilde{\alpha}_{v}\in\text{Rev}^{\rm
adm}_{P_{v}}(\widetilde{X}_{v}^{\bullet})$ such that

$\gamma_{(\widetilde{\alpha}_{v},P_{v})}=g_{v}+\\#(D_{v})-2.$

Then Theorem 3.9 implies that the element $\alpha_{v}\in\text{Rev}^{\rm
adm}_{P_{v}}(X_{v}^{\bullet})$ induced by $\widetilde{\alpha}_{v}$ such that
the following holds:

$\gamma_{(\alpha_{v},P_{v})}=g_{X_{v}}+\\#(D_{v})-2,$

where $g_{X_{v}}$ denotes the genus of $X_{v}$. Write
$f_{v}^{\bullet}:Y_{v}^{\bullet}\rightarrow X_{v}^{\bullet}$ for the Galois
multi-admissible covering over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$
induced by $\alpha_{v}$.

By applying similar arguments to the arguments given in Step 3 of the proof of
Lemma 5.3, we may glue $\\{Y_{v}^{\bullet}\\}_{v\in v(\Gamma)}$ along
$\\{f^{-1}_{v}(D_{v}\setminus D^{\prime}_{v})\\}_{v\in v(\Gamma)}$ (as
admissible coverings) in a way that is compatible with the gluing of
$\\{X_{v}^{\bullet}\\}_{v\in v(\Gamma)}$ that gives rise to
$X_{\Gamma}^{\bullet}$. Then we obtain a Galois multi-admissible covering

$f_{\Gamma}^{\bullet}:Y_{\Gamma}^{\bullet}\rightarrow X_{\Gamma}^{\bullet}$

over $k$ with Galois group $\mathbb{Z}/n\mathbb{Z}$. Note that the
construction of $f_{\Gamma}^{\bullet}$ implies that $f_{\Gamma}$ is étale over
$D_{X_{\Gamma}}\setminus D_{X}$. Moreover, there exists an element
$\alpha_{\Gamma}\in{\rm Rev}^{\rm
adm}_{D_{\Gamma}}(X_{\Gamma}^{\bullet})\setminus\\{0\\}$ induced by
$f^{\bullet}_{\Gamma}$ such that the following holds:

$\gamma_{(\alpha_{\Gamma},D_{\Gamma})}=g_{X_{\Gamma}}+1.$

We complete the proof of the lemma. ∎

######  Example 5.6.

We give some examples to explain the conditions considered in the proof of
Lemma 5.5. We maintain the notation introduced in the proof of Lemma 5.5.
Moreover, we suppose that $\Gamma_{X^{\bullet}}=\Gamma=\Gamma^{\prime}$ is a
tree.

(i) Suppose that $v(\Gamma)=\\{w_{1},w_{2}\\}$. Then we have

$w_{1}$$w_{2}$$e_{x_{1}}$$e_{x_{2}}$$e_{x_{3}}$

$\Gamma$:

(ii)-(1) Suppose that $v(\Gamma)=\\{v_{1},v_{2},v_{3}\\}$. Then we have

$v_{1}$$v_{2}$$v_{3}$$e_{x_{1}}$$e_{x_{2}}$$e_{x_{3}}$

$\Gamma$:

(ii)-(2) Suppose that $v(\Gamma)=\\{v_{0},v_{1},v_{2},v_{3}\\}$. Then we have

$v_{1}$$v_{0}$$v_{3}$$v_{2}$$e_{x_{1}}$$e_{x_{2}}$$e_{x_{3}}$

$\Gamma$:

#### 5.4.3.

We generalize Theorem 5.4 to the case of certain fixed ramification divisors
as follows:

######  Theorem 5.7.

Let $X^{\bullet}$ be an arbitrary pointed stable curve of type $(g_{X},3)$
over an algebraically closed field $k$ of characteristic $p>0$. Let
$D_{X}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{x_{1},x_{2},x_{3}\\}$, and
let $D_{j}\in\mathbb{Z}[D_{X}]$, $j\in\\{1,2,3\\}$, be an effective divisor on
$X$ such that the following conditions are satisfied:

  * •

${\rm deg}(D_{j})=2(p^{t_{j}}-1)$.

  * •

${\rm ord}_{x}(D_{j})\leq p^{t_{j}}-1$ for each $x\in D_{X}$.

  * •

$\\#\\{x\in D_{X}\ |\ {\rm ord}_{x}(D_{j})=p^{t_{j}}-1\\}\geq 1$.

Let
$n=p^{t}-1\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p^{t_{1}+t_{2}+t_{3}}-1$,
and let
$D\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{1}+p^{t_{1}}D_{2}+p^{t_{1}+t_{2}}D_{3}\in\mathbb{Z}[D_{X}]$
be an effective divisor on $X$ with degree $2n$. Moreover, suppose that
$D\in(\mathbb{Z}/n\mathbb{Z})^{\sim}[D_{X}]^{0}$ (2.2.5), and that

$n>{\rm max}\\{C(g_{X})+1,\\#(e(\Gamma_{X^{\bullet}}))\\}.$

Then there exists an element $\alpha\in{\rm Rev}^{\rm
adm}_{D}(X^{\bullet})\setminus\\{0\\}$ (Definition 2.4 (i)) such that the
following holds:

$\gamma_{(\alpha,D)}=g_{X}+1.$

###### Proof.

By applying Lemma 5.5 and similar arguments to the arguments given in the
proof of Theorem 5.4, we obtain the theorem. ∎

##  6\. Applications to anabelian geometry

In this section, we give some applications of results obtained in previous
sections. The main results of the present section are Theorem 6.3 and Theorem
6.4.

###  6.1. An anabelian formula for topological types

In this subsection, by using Theorem 5.4, we prove a group-theoretical formula
for the topological type of an arbitrary pointed stable curve over an
algebraically closed field of characteristic $p>0$.

####  6.1.1. Settings

We maintain the notation introduced in 2.1.2. Moreover, let
$\Pi_{X^{\bullet}}$ be the admissible fundamental group of $X^{\bullet}$.

#### 6.1.2.

Let $\Delta$ be an arbitrary profinite group, and let $m\in\mathbb{N}$ be a
positive natural number and $\ell$ a prime number. We denote by
$D_{\ell}(\Delta)\subseteq\Delta$ the topological closure of
$[\Delta,\Delta]\Delta^{\ell}$, where $[\Delta,\Delta]$ denotes the commutator
subgroup of $\Delta$. Moreover, we define the closed normal subgroup
$D^{(m)}_{\ell}(\Delta)$ of $\Delta$ inductively by
$D^{(1)}_{\ell}(\Delta)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{\ell}(\Delta)$
and
$D_{\ell}^{(i+1)}(\Delta)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}D_{\ell}(D^{(i)}_{\ell}(\Delta))$,
$i\in\\{1,\dots,m-1\\}$. Note that
$\\#(\Delta/D_{\ell}^{(m)}(\Delta))\leq\infty$ when $\Delta$ is topologically
finitely generated. On the other hand, we denote by $F_{r,m}^{\ell}$ the
finite group $\widehat{F}_{r}/D_{\ell}^{(m)}(\widehat{F}_{r})$, where
$\widehat{F}_{r}$ denotes the free profinite group of rank $r$.

#### 6.1.3.

Let $\Pi$ be an abstract profinite group which is isomorphic to
$\Pi_{X^{\bullet}}$ as profinite groups. Moreover, we denote by $\pi_{A}(\Pi)$
the set of finite quotients of $\Pi$. We put

$b^{1}_{\Pi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{max}\\{r\ |\
\text{there exists a prime number}\ \ell\ \text{such that}\
(\mathbb{Z}/\ell\mathbb{Z})^{\oplus r}\in\pi_{A}(\Pi)\\},$ $\displaystyle
b^{2}_{\Pi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{\begin{array}[]{ll}0,&F_{b_{\Pi}^{1},2}^{\ell}\in\pi_{A}(\Pi)\
\text{for some prime number}\ \ell,\\\ 1,&\text{otherwise}.\end{array}\right.$

Note that $b^{i}_{\Pi},\ i\in\\{1,2\\}$, is a group-theoretical invariant
associated to $\Pi$ (i.e., depends only on the isomorphism class of $\Pi$).
Firstly, we have the following lemma.

######  Lemma 6.1.

(i) We have

$b^{1}_{\Pi}=2g_{X}+n_{X}-1+b^{2}_{\Pi},$ $\displaystyle
b^{2}_{\Pi}=\left\\{\begin{array}[]{ll}1,&\text{if}\ n_{X}=0,\\\ 0,&\text{if}\
n_{X}\neq 0.\end{array}\right.$

(ii) There exists a unique prime number $p_{\Pi}$ such that
$(\mathbb{Z}/p_{\Pi}\mathbb{Z})^{\oplus b^{1}_{\Pi}}\not\in\pi_{A}(\Pi)$. In
particular, we have $p=p_{\Pi}$.

###### Proof.

(i) We put
$r_{\Pi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\text{dim}_{\mathbb{F}_{\ell}}(\Pi^{\rm
ab}\otimes\mathbb{F}_{\ell})$, where $\ell$ is an arbitrary prime number
$\mathfrak{P}\setminus\\{p\\}$, and $\mathfrak{P}$ denotes the set of prime
numbers. Then the structures of maximal prime-to-$p$ quotients of admissible
fundamental groups (2.1.7) imply that (see 2.2.2 for $\sigma_{X}$)

$\Pi^{\rm
ab}\cong\mathbb{Z}_{p}^{\sigma_{X}}\times\prod_{\ell\in\mathfrak{P}\setminus\\{p\\}}\mathbb{Z}_{\ell}^{r_{\Pi}}.$

Since $X^{\bullet}$ is a pointed stable curve, we have that
$\sigma_{X}<r_{\Pi}.$ This implies that $b^{1}_{\Pi}=r_{\Pi}$. Moreover, we
have

$\displaystyle b^{1}_{\Pi}=\left\\{\begin{array}[]{ll}2g_{X},&\text{if}\
n_{X}=0,\\\ 2g_{X}+n_{X}-1,&\text{if}\ n_{X}\neq 0.\end{array}\right.$

Suppose that $n_{X}>0$. Let $\ell_{1}\in\mathfrak{P}\setminus\\{p\\}$. The
specialization theorem of maximal pro-$\ell_{1}$ quotients of admissible
fundamental groups ([V, Théorème 2.2 (c)]) implies that the maximal
pro-$\ell_{1}$ quotient $\Pi^{\ell_{1}}$ of $\Pi$ is a free pro-$\ell_{1}$
profinite group of rank $b_{\Pi}^{1}$. Then we have
$F^{\ell_{1}}_{b^{1}_{\Pi},2}\in\pi_{A}(\Pi).$ Thus, we obtain $b_{\Pi}^{2}=0$
if $n_{X}>0$.

Conversely, we assume that $F^{\ell_{2}}_{b^{1}_{\Pi},2}\in\pi_{A}(\Pi)$ for
some prime number $\ell_{2}$. Then we have $\ell_{2}\neq p$. The Schreier
index formula ([S2, Chapter I §3 3.4 Corollary]) implies the following natural
exact sequence

$1\rightarrow(\mathbb{Z}/\ell_{2}\mathbb{Z})^{\oplus\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-1)+1}\rightarrow
F_{b_{\Pi}^{1},2}^{\ell_{2}}\rightarrow(\mathbb{Z}/\ell_{2}\mathbb{Z})^{\oplus
b_{\Pi}^{1}}\rightarrow 1.$

Let $\phi:\Pi\twoheadrightarrow F^{\ell_{2}}_{b^{1}_{\Pi},2}$ be a surjection.
We denote by $X_{\ell_{2}}^{\bullet}$ the pointed stable curve over $k$
corresponding to the kernel of the natural surjection
$\Pi_{X^{\bullet}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Pi\overset{\phi}{\twoheadrightarrow}F^{\ell_{2}}_{b^{1}_{\Pi},2}\twoheadrightarrow(\mathbb{Z}/\ell_{2}\mathbb{Z})^{\oplus
b_{\Pi}^{1}}$ and by $\Pi_{\ell_{2}}\subseteq\Pi$ the kernel of the surjection
$\Pi\overset{\phi}{\twoheadrightarrow}F^{\ell_{2}}_{b^{1}_{\Pi},2}\twoheadrightarrow(\mathbb{Z}/\ell_{2}\mathbb{Z})^{\oplus
b_{\Pi}^{1}}$. Then we have
$(\mathbb{Z}/\ell_{2}\mathbb{Z})^{\oplus\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-1)+1}\in\pi_{A}(\Pi_{\ell_{2}}).$
This implies
$b^{1}_{\Pi_{\ell_{2}}}\geq\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-1)+1$. If
$n_{X}=0$, the Riemann-Hurwitz formula implies
$g_{X_{\ell_{2}}}=\ell_{2}^{b_{\Pi}^{1}}(g_{X}-1)+1,$ where $g_{X_{\ell_{2}}}$
denotes the genus of $X_{\ell_{2}}^{\bullet}$. Then we have
$b_{\Pi_{\ell_{2}}}^{1}=2(\ell_{2}^{b_{\Pi}^{1}}(g_{X}-1)+1)=\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-2)+2.$
On the other hand,
$\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-2)+2<\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-1)+1.$
This contradicts the fact that
$b^{1}_{\Pi_{\ell_{2}}}\geq\ell_{2}^{b_{\Pi}^{1}}(b_{\Pi}^{1}-1)+1$. Then we
obtain $n_{X}>0$ if $b_{\Pi}^{2}=0$. Moreover, we see

$b^{1}_{\Pi}=2g_{X}+n_{X}-1+b^{2}_{\Pi}.$

(ii) This follows immediately from the structure of $\Pi^{\rm ab}$. This
completes the proof of the lemma. ∎

#### 6.1.4.

Let $\overline{\mathbb{F}}_{p_{\Pi}}$ be an arbitrary algebraic closure of
$\mathbb{F}_{p_{\Pi}}$. Let
$\chi\in\text{Hom}(\Pi,\overline{\mathbb{F}}_{p_{\Pi}}^{\times})$. We denote