This follows from [CLNS, Ch.6. Lemma 3.4.1] and [CLNS, Ch.6. Example 3.5.2]. ∎

######  Lemma 5.37 (cf. [CLNS, Ch.5. Proposition 2.2.6]).

Let $X$ be an affine scheme of the form
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$. Let $p$
and $m$ be non-negative integers with $2p+1\geq m\geq p$. Let $\gamma\in
X_{p}(k)$ be a jet with $\pi_{m,p}^{-1}(\gamma)\not=\emptyset$. Then we have

$\pi_{m,p}^{-1}(\gamma)\simeq\operatorname{Hom}_{k[t]/(t^{p+1})}\bigl{(}\gamma^{*}\widehat{\Omega}_{X/k[[t]]},(t^{p+1})/(t^{m+1})\bigr{)}.$

###### Proof.

We set $A:=k[x_{1},\ldots,x_{N}][[t]]/I$. For the same reason as in the proof
of Lemma 5.13, we have

$\pi_{m,p}^{-1}(\gamma)\simeq\operatorname{Der}_{k[[t]]}\left(A,(t^{p+1})/(t^{m+1})\right).$

Furthermore, by the universal property of $\widehat{\Omega}_{A/k[[t]]}$ (cf.
Remark 5.29(2)), we have

$\operatorname{Der}_{k[[t]]}\left(A,(t^{p+1})/(t^{m+1})\right)\simeq\operatorname{Hom}_{A}\bigl{(}\widehat{\Omega}_{A/k[[t]]},(t^{p+1})/(t^{m+1})\bigr{)},$

which proves the assertion. ∎

###  5.3. Codimension formulae

In this subsection, we discuss a $k[t]$-morphism $f:X\to Y$ of affine
$k[t]$-schemes in the following two cases.

  1. (a)

$X$ and $Y$ are affine schemes of the forms
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I\bigr{)}$ and
$Y=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{M}][[t]]/J\bigr{)}$.

  2. (b)

$X$ and $Y$ are affine schemes of the forms
$X=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{M}][[t]]/I\bigr{)}$ and
$Y=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{L}]]/J\bigr{)}$.
Furthermore, $f$ satisfies $(x_{1},\ldots,x_{L})\mathcal{O}_{X}\subset(t)$.

######  Lemma 5.38.

In case (a) above, the canonical map
$f^{*}\widehat{\Omega}_{Y/k[[t]]}\to\widehat{\Omega}_{X/k[[t]]}$ is induced.
In case (b), the canonical map
$f^{*}\Omega^{\prime}_{Y/k[t]}\to\widehat{\Omega}_{X/k[[t]]}$ is induced.

###### Proof.

Let $A=\mathcal{O}_{X}$ and $B=\mathcal{O}_{Y}$ be the corresponding rings,
and $g:B\to A$ the corresponding $k[t]$-ring homomorphism.

First, we deal with case (a). Since $g:B\to A$ is a $k[t]$-ring homomorphism,
$\widehat{\Omega}_{A/k[[t]]}$ is a complete $B$-module with respect to the
$(t)$-adic topology. Therefore, by the universal property of
$\widehat{\Omega}_{B/k[[t]]}$ (cf. Remark 5.29(2)), the derivation
$\widehat{d}_{A/k[[t]]}\circ g:B\to\widehat{\Omega}_{A/k[[t]]}$ factors
through $\widehat{\Omega}_{B/k[[t]]}$. We complete the proof in case (a).

Next, we deal with case (b). By the universal property of
$\Omega^{\prime}_{B/k[t]}$, it is sufficient to show that the composition
$B\xrightarrow{g}A\xrightarrow{\widehat{d}_{A/k[[t]]}}\widehat{\Omega}_{A/k[[t]]}$
is a special $B$-derivation. Note that $\widehat{\Omega}_{A/k[[t]]}$ is a
complete $A$-module with respect to the $(t)$-adic topology. Since
$g\bigl{(}(x_{1},\ldots,x_{L})\bigr{)}\subset(t)$ holds by assumption,
$\widehat{\Omega}_{A/k[[t]]}$ is a separated $B$-module with respect to the
$(x_{1},\ldots,x_{L})$-adic topology. Therefore $\widehat{d}_{A/k[[t]]}\circ
g$ is a special $B$-derivation by Lemma 2.2. ∎

We define the order of the Jacobian for a morphism.

######  Definition 5.39.

  1. (1)

Let $f:X\to Y$ be a morphism of affine $k[t]$-schemes of the form (a) above.
Then $f$ induces a homomorphism
$f^{*}\widehat{\Omega}_{Y/k[[t]]}\to\widehat{\Omega}_{X/k[[t]]}$ by Lemma
5.38. Let $\gamma\in X_{\infty}$ be a $k$-arc and let
$\gamma^{\prime}:=f_{\infty}(\gamma)$. Let $S$ be the torsion part of
$\gamma^{*}\widehat{\Omega}_{X/k[[t]]}$. Then we define the order
$\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})$ of the Jacobian of $f$
at $\gamma$ as the length of the $k[[t]]$-module

$\operatorname{Coker}\bigl{(}\gamma^{\prime*}\widehat{\Omega}_{Y/k[[t]]}\to\gamma^{*}\widehat{\Omega}_{X/k[[t]]}/S\bigr{)}.$

In particular, if
$\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})<\infty$, then we have

$\operatorname{Coker}\bigl{(}\gamma^{\prime*}\widehat{\Omega}_{Y/k[[t]]}\to\gamma^{*}\widehat{\Omega}_{X/k[[t]]}/S\bigr{)}\simeq\bigoplus_{i}k[t]/(t^{e_{i}})$

as $k[[t]]$-modules with some positive integers $e_{i}$ satisfying
$\sum_{i}e_{i}=\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})$.

  2. (2)

Let $f:X\to Y$ be a morphism of affine $k[t]$-schemes of the form (b) above.
Then $f$ induces a homomorphism
$f^{*}\Omega^{\prime}_{Y/k[t]}\to\widehat{\Omega}_{X/k[[t]]}$ by Lemma 5.38.
Let $\gamma\in X_{\infty}$ be a $k$-arc and let
$\gamma^{\prime}:=f_{\infty}(\gamma)$. Let $S$ be the torsion part of
$\gamma^{*}\widehat{\Omega}_{X/k[[t]]}$. Then we also define the order
$\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})$ of the Jacobian of $f$
at $\gamma$ as the length of the $k[[t]]$-module

$\operatorname{Coker}\bigl{(}\gamma^{\prime*}\Omega^{\prime}_{Y/k[t]}\to\gamma^{*}\widehat{\Omega}_{X/k[[t]]}/S\bigr{)}.$

  3. (3)

By abuse of notation, we define

$\operatorname{Cont}^{e}(\operatorname{jac}_{f}):=\\{\gamma\in
X_{\infty}\mid\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})=e\\}$

for $e\geq 0$.

######  Lemma 5.40.

Let $n$ be a non-negative integer. Let $f:X\to Y$ be a morphism of affine
$k[t]$-schemes of the form (a) above. Let $g:Y\to Z$ be a morphism of affine
$k[t]$-schemes of the form (b). Suppose that each irreducible component
$W_{i}$ of $X$, $Y$ and $Z$ has $\dim W_{i}\geq n+1$. Let $\gamma\in
X_{\infty}$ be a $k$-arc and let $\gamma^{\prime}:=f_{\infty}(\gamma)$.
Suppose that

$\operatorname{ord}_{\gamma}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}<\infty,\quad\operatorname{ord}_{\gamma^{\prime}}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{Y/k[[t]]}\bigr{)}\bigr{)}<\infty.$

Then we have

$\operatorname{ord}_{\gamma}(\operatorname{jac}_{g\circ
f})=\operatorname{ord}_{\gamma}(\operatorname{jac}_{f})+\operatorname{ord}_{\gamma^{\prime}}(\operatorname{jac}_{g}).$

###### Proof.

The same proof as in [NS, Lemma 2.10] works due to Lemma 5.30. ∎

######  Remark 5.41.

The same statement holds if $g$ is of the form (a).

######  Remark 5.42.

We note in this remark that all propositions (Proposition 2.29, Lemmas 2.31,
2.32 and Proposition 2.33) in Subsection 2.6 in [NS] are also true for a
$k[t]$-morphism $f$ of the form (a) and (b) by making the following
modifications:

  * •

Replacing the conditions $(\star)_{n}$ and $(\star\star)_{n}$ in [NS] on $X$
with the following condition:

    * –

“Each irreducible component $X_{i}$ of $X$ has $\dim X_{i}\geq n+1$.”

  * •

Replacing $\Omega_{\text{--}/k[t]}$ with $\widehat{\Omega}_{\text{--}/k[[t]]}$
and $\Omega^{\prime}_{\text{--}/k[t]}$, and replacing
$\operatorname{Jac}_{\text{--}/k[t]}$ with
$\operatorname{Fitt}^{n}(\widehat{\Omega}_{\text{--}/k[[t]]})$ and
$\operatorname{Fitt}^{n}(\Omega^{\prime}_{\text{--}/k[t]})$.

We note that

  * •

Proposition 2.29(2), Lemmas 2.31, 2.32, and Proposition 2.33 in [NS]

are formal consequences of Proposition 2.29(1), Lemma 2.13(1) and Proposition
2.17 in [NS]. The formal power series ring versions of Proposition 2.29(1),
Lemma 2.13(1) and Proposition 2.17 in [NS] are proved in Lemmas 5.13 and 5.37,
Lemmas 5.7 and 5.30, and Propositions 5.9 and 5.31 in this paper.

Furthermore, Lemma 2.34 in [NS] is also true in the formal power series ring
setting by replacing $A=\operatorname{Spec}k[t][x_{1},\ldots,x_{N}]$ with
$\operatorname{Spec}k[x_{1},\ldots,x_{N}][[t]]$. Indeed, the same proof of
Lemma 2.34 in [NS] works in this setting.

######  Proposition 5.43.

Let $n$ be a non-negative integer. Let $f:X\to Y$ be a morphism of affine
$k[t]$-schemes of the form (b) above. Suppose that each irreducible component
$W_{i}$ of $X$ and $Y$ has $\dim W_{i}\geq n+1$. Let
$e,e^{\prime},e^{\prime\prime}\in\mathbb{Z}_{\geq 0}$. Let $A\subset
X_{\infty}$ be a cylinder and let $B=f_{\infty}(A)$. Assume that

$A\subset\operatorname{Cont}^{e^{\prime\prime}}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\cap\operatorname{Cont}^{e}(\operatorname{jac}_{f}),\quad
B\subset\operatorname{Cont}^{e^{\prime}}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\Omega^{\prime}_{Y/k[t]}\bigr{)}\bigr{)}.$

Then, $B$ is a cylinder of $Y_{\infty}$ contained in
$\operatorname{Cont}^{\geq 1}(\mathfrak{o}_{Y})$, where
$\mathfrak{o}_{Y}\subset\mathcal{O}_{Y}$ is the ideal sheaf generated by
$x_{1},\ldots,x_{L}\in\mathcal{O}_{Y}$. Moreover, if $f_{\infty}|_{A}$ is
injective, then it follows that

$\operatorname{codim}(A)+e=\operatorname{codim}(B).$

###### Proof.

By Remark 5.42, the same proofs as in Subsection 2.6 in [NS] work by making
the following modifications:

  * •

Replacing $\Omega_{X/k[t]}$ with $\widehat{\Omega}_{X/k[[t]]}$, and
$\Omega_{Y/k[t]}$ with $\Omega^{\prime}_{Y/k[t]}$.

  * •

Replacing $\operatorname{Jac}_{X/k[t]}$ with
$\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}$, and
$\operatorname{Jac}_{Y/k[t]}$ with
$\operatorname{Fitt}^{n}\bigl{(}\Omega^{\prime}_{Y/k[t]}\bigr{)}$.

∎

######  Proposition 5.44.

Suppose that a finite group $G$ acts on the ring $k[x_{1},\ldots,x_{N}]$ over
$k$. Let $I\subset k[x_{1},\ldots,x_{N}]^{G}[[t]]$ be an ideal, and let
$I^{\prime}\subset k[x_{1},\ldots,x_{N}][[t]]$ be the ideal generated by $I$.
We denote

$X:=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/I^{\prime}\bigr{)},\quad
Y:=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}]^{G}[[t]]/I\bigr{)},$

and denote $f:X\to X/G=Y$ the quotient morphism. Suppose that each irreducible
component $W_{i}$ of $X$ and $Y$ has $\dim W_{i}\geq n+1$. Let $A\subset
X_{\infty}$ be a $G$-invariant cylinder and let $B=f_{\infty}(A)$. Let
$e,e^{\prime},e^{\prime\prime}\in\mathbb{Z}_{\geq 0}$. Assume that

$A\subset\operatorname{Cont}^{e^{\prime\prime}}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{X/k[[t]]}\bigr{)}\bigr{)}\cap\operatorname{Cont}^{e}(\operatorname{jac}_{f}),\quad
B\subset\operatorname{Cont}^{e^{\prime}}\bigl{(}\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{Y/k[[t]]}\bigr{)}\bigr{)}.$

Then $B$ is a cylinder of $Y_{\infty}$ with

$\operatorname{codim}(A)+e=\operatorname{codim}(B).$

###### Proof.

Note that $f$ is a morphism of the form (a). By Remark 5.42, the same proof of
Proposition 2.35 in [NS] works. ∎

######  Remark 5.45.

  1. (1)

Proposition 5.43 is true also for $k[t]$-morphisms of the form (a). Indeed,
this is known for morphisms $f:X\to Y$ of formal $k[[t]]$-schemes (not
necessarily affine). When $X$ is smooth over $k[[t]]$, this is proved in
[Seb04, Lemme 7.1.3] (cf. [CLNS, Ch.5. Theorem 3.2.2]). The general case is
proved in [Yas, Lemmas 10.19, 10.20]. We also note that Yasuda proves it in
the more general setting, for formal Deligne-Mumford stacks of arbitrary
characteristics.

  2. (2)

Proposition 5.43 is true also for a $k[t]$-morphism $f:X\to Y$ of
$k[t][[x_{1},\ldots,x_{N}]]$-schemes of finite type (not necessarily affine)
by making the following modification:

     * •

Replacing the condition “each irreducible component $W_{i}$ of $X$ and $Y$ has
$\dim W_{i}\geq n+1$” in Proposition 5.43 with “each irreducible component
$W_{i}$ of $X$ and $Y$ has $\dim^{\prime}W_{i}\geq n+1$”.

##  6\. Denef and Loeser’s theory for quotient singularities

In this section, we review the theory of arc spaces of quotient varieties
established by Denef and Loeser [DL02] (cf. [Yas16], [NS, Section 3]). We
explain their theory in the formal power series ring setting.

Let $d$ be a positive integer and $\xi\in k$ a primitive $d$-th root of unity.
Let $G\subset\operatorname{GL}_{N}(k)$ be a finite subgroup with order $d$
that linearly acts on
$\overline{A}:=\widehat{\mathbb{A}}^{N}_{k}:=\operatorname{Spec}k[[x_{1},\ldots,x_{N}]]$.
Let $\overline{X}\subset\overline{A}$ be a $G$-invariant subscheme. We denote
by

$A:=\overline{A}/G,\qquad X:=\overline{X}/G$

the quotient schemes. Let $Z\subset A$ be the minimum closed subset such that
$\overline{A}\to A$ is étale outside $Z$.

Fix $\gamma\in G$. Since $G$ is a finite group, $\gamma$ can be diagonalized
with some new basis $x_{1}^{(\gamma)},\ldots,x_{N}^{(\gamma)}$. Let
$\operatorname{diag}(\xi^{e_{1}},\ldots,\xi^{e_{N}})$ be the diagonal matrix
with $0<e_{i}\leq d$. Then we define a $k[t]$-ring homomorphism

$\lambda^{*}_{\gamma}:k[t][[x_{1},\ldots,x_{N}]]^{G}\to
k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]];\quad x_{i}^{(\gamma)}\mapsto
t^{e_{i}/d}x_{i}^{(\gamma)},$

where $C_{\gamma}$ is the centralizer of $\gamma$ in $G$. Let
$i:k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]\to k[x_{1},\ldots,x_{N}][[t]]$ be
the inclusion map, and let
$\overline{\lambda}^{*}_{\gamma}=i\circ\lambda^{*}_{\gamma}$ be the composite
map.

Let $I_{X}\subset k[[x_{1},\ldots,x_{N}]]^{G}$ be the defining ideal of $X$ in
$A$. Let $I^{\prime}_{X}\subset k[t][[x_{1},\ldots,x_{N}]]^{G}$ be the ideal
generated by $I_{X}$. Let

$\widetilde{I}_{X}^{(\gamma)}\subset
k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]],\quad\overline{I}_{X}^{(\gamma)}\subset
k[x_{1},\ldots,x_{N}][[t]]$

be the ideals generated by $\lambda^{*}_{\gamma}(I^{\prime}_{X})$ and
$\overline{\lambda}^{*}_{\gamma}(I^{\prime}_{X})$, respectively. Then we have
the following diagram:

$\textstyle{k[t][[x_{1},\ldots,x_{N}]]^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\gamma}^{*}}$$\scriptstyle{x_{i}^{(\gamma)}\mapsto
t^{e_{i}/d}x_{i}^{(\gamma)}}$$\scriptstyle{\overline{\lambda}_{\gamma}^{*}}$$\textstyle{k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[x_{1},\ldots,x_{N}][[t]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[t][[x_{1},\ldots,x_{N}]]^{G}/I^{\prime}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\gamma}^{*}}$$\scriptstyle{\overline{\mu}_{\gamma}^{*}}$$\textstyle{k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]/\widetilde{I}_{X}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[x_{1},\ldots,x_{N}][[t]]/\overline{I}_{X}^{(\gamma)}}$

We define $k[t]$-schemes $\overline{A}^{(\gamma)}$, $\widetilde{X}^{(\gamma)}$
and $\overline{X}^{(\gamma)}$ as follows:

$\displaystyle\overline{A}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}k[x_{1},\ldots,x_{N}][[t]],$
$\displaystyle\widetilde{X}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}\left(k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]/\widetilde{I}_{X}^{(\gamma)}\right),$
$\displaystyle\overline{X}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}\left(k[x_{1},\ldots,x_{N}][[t]]/\overline{I}_{X}^{(\gamma)}\right).$

Let $\overline{A}^{(\gamma)}_{\infty}$, $\widetilde{X}_{\infty}^{(\gamma)}$
and $\overline{X}_{\infty}^{(\gamma)}$ be their arc spaces as $k[t]$-schemes
defined in Section 5. Then we have the following diagram of arc spaces:

$\textstyle{A_{\infty}}$$\textstyle{\bigl{(}\overline{A}^{(\gamma)}/C_{\gamma}\bigr{)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\gamma}}$$\textstyle{\overline{A}^{(\gamma)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\lambda}_{\gamma}}$$\textstyle{X_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}_{\infty}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\gamma}}$$\textstyle{\overline{X}_{\infty}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mu}_{\gamma}}$

We denote by $\mu_{\gamma}$ and $\overline{\mu}_{\gamma}$ the restrictions of
$\lambda_{\gamma}$ and $\overline{\lambda}_{\gamma}$ to
$\widetilde{X}_{\infty}^{(\gamma)}$ and $\overline{X}_{\infty}^{(\gamma)}$,
respectively.

######  Remark 6.1.

  1. (1)

Here, we have used the fact that the arc spaces of

$\operatorname{Spec}k[t][[x_{1},\ldots,x_{N}]]^{G},\quad\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]^{G}/I^{\prime}_{X}\bigr{)}$

as $k[t]$-schemes (defined in Section 5) are isomorphic to the arc spaces of
$A$ and $X$ as $k$-schemes (defined in Section 4).

  2. (2)

Furthermore, the vertical arrows are closed immersions. Under these
identifications, we have

$\lambda_{\gamma}^{-1}(X_{\infty})=\widetilde{X}_{\infty}^{(\gamma)},\qquad\overline{\lambda}_{\gamma}^{-1}(X_{\infty})=\overline{X}_{\infty}^{(\gamma)}.$

######  Proposition 6.2 ([DL02, Section 2], cf. [NS, Subsections 3.1, 3.2]).

The ring homomorphism $\lambda^{*}_{\gamma}$ induces the maps
$\lambda_{\gamma}:(\overline{A}^{(\gamma)}/C_{\gamma})_{\infty}\to A_{\infty}$
and $\overline{\lambda}_{\gamma}:\overline{A}^{(\gamma)}_{\infty}\to
A_{\infty}$, and the following hold.

  1. (1)

There is a natural inclusion
$\overline{A}^{(\gamma)}_{\infty}/C_{\gamma}\hookrightarrow\bigl{(}\overline{A}^{(\gamma)}/C_{\gamma}\bigr{)}_{\infty}$.

  2. (2)

The composite map
$\overline{A}^{(\gamma)}_{\infty}/C_{\gamma}\hookrightarrow\bigl{(}\overline{A}^{(\gamma)}/C_{\gamma}\bigr{)}_{\infty}\xrightarrow{\lambda_{\gamma}}A_{\infty}$
is injective outside $Z_{\infty}$.

  3. (3)

$\bigsqcup_{\langle\gamma\rangle\in\operatorname{Conj}(G)}\bigl{(}\overline{\lambda}_{\gamma}\bigl{(}\overline{A}^{(\gamma)}_{\infty}\bigr{)}\setminus
Z_{\infty}\bigr{)}=A_{\infty}\setminus Z_{\infty}$ holds, where
$\operatorname{Conj}(G)$ denotes the set of the conjugacy classes of $G$.

###### Proof.

In [DL02] and [NS], the assertions are proved for the polynomial ring
$k[t][x_{1},\ldots,x_{N}]$, and their proofs work in the formal power series
ring setting. ∎

By Remark 6.1(2), we can deduce the same statement for $X$.

######  Proposition 6.3 (cf. [NS, Subsection 3.3]).

The ring homomorphism $\lambda^{*}_{\gamma}$ induces the maps
$\mu_{\gamma}:\widetilde{X}_{\infty}^{(\gamma)}\to X_{\infty}$ and
$\overline{\mu}_{\gamma}:\overline{X}^{(\gamma)}_{\infty}\to X_{\infty}$, and
the following hold.

  1. (1)

There is a natural inclusion
$\overline{X}_{\infty}^{(\gamma)}/C_{\gamma}\hookrightarrow\widetilde{X}_{\infty}^{(\gamma)}$.

  2. (2)

The composite map
$\overline{X}_{\infty}^{(\gamma)}/C_{\gamma}\hookrightarrow\widetilde{X}_{\infty}^{(\gamma)}\xrightarrow{\mu_{\gamma}}X_{\infty}$
is injective outside $Z_{\infty}$.

  3. (3)

$\bigsqcup_{\langle\gamma\rangle\in\operatorname{Conj}(G)}\bigl{(}\overline{\mu}_{\gamma}(\overline{X}^{(\gamma)}_{\infty})\setminus
Z_{\infty}\bigr{)}=X_{\infty}\setminus Z_{\infty}$ holds.

######  Remark 6.4.

  1. (1)

In [NS], $e_{i}$ is taken to satisfy $0\leq e_{i}\leq d-1$. Note that the ring
homomorphism $\lambda_{\gamma}^{*}$ cannot be defined in this way of taking in
our formal power series ring setting.

  2. (2)

It is also natural to define a $k[t]$-ring homomorphism

$\overline{\lambda}^{\prime*}_{\gamma}:k[t][[x_{1},\ldots,x_{N}]]^{G}\to
k[t][[x_{1},\ldots,x_{N}]];\quad x_{i}^{(\gamma)}\mapsto
t^{e_{i}/d}x_{i}^{(\gamma)},$

and schemes

$\overline{A}^{\prime(\gamma)}:=\operatorname{Spec}k[t][[x_{1},\ldots,x_{N}]],\quad\overline{X}^{\prime(\gamma)}:=\operatorname{Spec}\left(k[t][[x_{1},\ldots,x_{N}]]/\overline{I}_{X}^{\prime(\gamma)}\right),$

where $\overline{I}_{X}^{\prime(\gamma)}$ is the ideal of
$k[t][[x_{1},\ldots,x_{N}]]$ generated by
$\overline{\lambda}^{\prime*}_{\gamma}(I^{\prime}_{X})$. Then by the same
argument as in this section, $\overline{\lambda}^{\prime*}_{\gamma}$ induces
maps

$\overline{\lambda}^{\prime}_{\gamma}:\overline{A}^{\prime(\gamma)}_{\infty}\to
A_{\infty},\qquad\overline{\mu}^{\prime}_{\gamma}:\overline{X}^{\prime(\gamma)}_{\infty}\to
X_{\infty}.$

However, as we can see in the discussion below, if $\overline{A}^{(\gamma)}$
and $\overline{X}^{(\gamma)}$ are replaced with
$\overline{A}^{\prime(\gamma)}$ and $\overline{X}^{\prime(\gamma)}$, then
Propositions 6.2 and 6.3 are no longer valid.

First, we note that

$\displaystyle\overline{A}^{\prime(\gamma)}_{m}$
$\displaystyle\simeq\operatorname{Spec}\bigl{(}k\bigl{[}\bigl{[}x_{1}^{(0)},\ldots,x_{N}^{(0)}\bigr{]}\bigr{]}\bigl{[}x_{1}^{(s)},\ldots,x_{N}^{(s)}\
\big{|}\ 1\leq s\leq m\bigr{]}\bigr{)},$
$\displaystyle\overline{A}^{(\gamma)}_{m}$
$\displaystyle\simeq\operatorname{Spec}\bigl{(}k\bigl{[}x_{1}^{(s)},\ldots,x_{N}^{(s)}\
\big{|}\ 0\leq s\leq m\bigr{]}\bigr{)},$

and we have a natural morphism
$\overline{A}^{\prime(\gamma)}_{m}\to\overline{A}^{(\gamma)}_{m}$ induced by
the ring inclusion

$k\bigl{[}x_{1}^{(s)},\ldots,x_{N}^{(s)}\ \big{|}\ 0\leq s\leq
m\bigr{]}\hookrightarrow
k\bigl{[}\bigl{[}x_{1}^{(0)},\ldots,x_{N}^{(0)}\bigr{]}\bigr{]}\bigl{[}x_{1}^{(s)},\ldots,x_{N}^{(s)}\
\big{|}\ 1\leq s\leq m\bigr{]}.$

Since the morphisms
$\overline{A}^{\prime(\gamma)}_{m}\to\overline{A}^{(\gamma)}_{m}$ are
compatible with the truncation maps, they induce a map
$\overline{A}^{\prime(\gamma)}_{\infty}\to\overline{A}^{(\gamma)}_{\infty}$.
The map
$\overline{X}^{\prime(\gamma)}_{\infty}\to\overline{X}^{(\gamma)}_{\infty}$ is
also induced, and we have the following commutative diagrams:

$\textstyle{A_{\infty}}$$\textstyle{\overline{A}^{(\gamma)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\lambda}_{\gamma}}$$\textstyle{X_{\infty}}$$\textstyle{\overline{X}^{(\gamma)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mu}_{\gamma}}$$\textstyle{\overline{A}^{\prime(\gamma)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\lambda}^{\prime}_{\gamma}}$$\textstyle{\overline{X}^{\prime(\gamma)}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mu}^{\prime}_{\gamma}}$

Furthermore, by contruction,
$\overline{A}^{\prime(\gamma)}_{\infty}\to\overline{A}^{(\gamma)}_{\infty}$
induces isomorphisms

$\displaystyle\overline{A}^{\prime(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\big{(}(x_{1},\ldots,x_{N})\big{)}$
$\displaystyle\simeq\overline{A}^{(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\big{(}(x_{1},\ldots,x_{N})\big{)},$
$\displaystyle\overline{X}^{\prime(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\big{(}(x_{1},\ldots,x_{N})\big{)}$
$\displaystyle\simeq\overline{X}^{(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\big{(}(x_{1},\ldots,x_{N})\big{)}.$

We also note that
$\overline{A}^{\prime(\gamma)}_{\infty}\to\overline{A}^{(\gamma)}_{\infty}$
induces injective maps

$\overline{A}^{\prime(\gamma)}_{\infty}(k)\hookrightarrow\overline{A}^{(\gamma)}_{\infty}(k),\quad\overline{X}^{\prime(\gamma)}_{\infty}(k)\hookrightarrow\overline{X}^{(\gamma)}_{\infty}(k)$

on $k$-points. However, these two maps are not surjective in general (see (3)
below).

  3. (3)

Suppose that $N=2$ and $G=\langle\gamma\rangle$, where
$\gamma:k[[x_{1},x_{2}]]\to k[[x_{1},x_{2}]]$ is the involution defined by
$\gamma(x_{i})=-x_{i}$ for $i\in\\{1,2\\}$. Then we have $d=2$,
$e_{1}=e_{2}=1$, and
$k[[x_{1},x_{2}]]^{G}=k[[x_{1}^{2},x_{1}x_{2},x_{2}^{2}]]$. We denote by
$\alpha\in A_{\infty}$ the $k$-arc corresponding to the $k[t]$-ring
homomorphism

$\alpha^{*}:k[t][[x_{1}^{2},x_{1}x_{2},x_{2}^{2}]]\to k[[t]];\quad
x_{1}^{2}\mapsto t,\ x_{1}x_{2}\mapsto t,\ x_{2}^{2}\mapsto t.$

Then, $\alpha$ is contained in the image of
$\overline{\lambda}_{\gamma}:\overline{A}^{(\gamma)}_{\infty}\to A_{\infty}$.
Indeed, if $\beta\in\overline{A}^{(\gamma)}_{\infty}$ is the $k$-arc defined
by

$\beta^{*}:k[x_{1},x_{2}][[t]]\to k[[t]];\quad x_{1}\mapsto 1,\ x_{2}\mapsto
1,$

then we have $\alpha^{*}=\beta^{*}\circ\overline{\lambda}^{*}_{\gamma}$ and
hence $\alpha=\overline{\lambda}_{\gamma}(\beta)$.

On the other hand, $\alpha$ is not contained in the image of
$\overline{\lambda}^{\prime}_{\gamma}:\overline{A}^{\prime(\gamma)}_{\infty}\to
A_{\infty}$ because there is no $k[t]$-ring homomorphism
$\beta^{*}:k[t][[x_{1},x_{2}]]\to k[[t]]$ satisfying
$\beta^{*}(x_{1})=\beta^{*}(x_{2})=1$.

##  7\. Arc spaces of hyperquotient singularities

In this section, we prove in Theorem 7.9 that [NS, Theorem 4.8] is still valid
in the formal power series ring setting.

###  7.1. Minimal log discrepancies of hyperquotient singularities

Let $d$ be a positive integer and let $\xi\in k$ be a primitive $d$-th root of
unity. Let $G\subset\operatorname{GL}_{N}(k)$ be a finite group with order $d$
that linearly acts on
$\overline{A}:=\widehat{\mathbb{A}}^{N}_{k}=\operatorname{Spec}k[[x_{1},\ldots,x_{N}]]$.
We denote by

$A:=\overline{A}/G$

the quotient scheme. Let $Z\subset A$ be the minimum closed subset such that
$\overline{A}\to A$ is étale outside $Z$. We assume that
$\operatorname{codim}Z\geq 2$, and hence the quotient morphism
$\overline{A}\to A$ is étale in codimension one. We note that $A$ is
$\mathbb{Q}$-Gorenstein (cf. Remark 3.1). We fix a positive integer $r$ such
that $\omega_{A/k}^{\prime[r]}$ is invertible.

We fix $\gamma\in G$. Let $C_{\gamma}$ be the centralizer of $\gamma$ in $G$.
Since $G$ is a finite group, $\gamma$ can be diagonalized with a suitable
basis $x_{1},\ldots,x_{N}$. Let
$\operatorname{diag}(\xi^{e_{1}},\ldots,\xi^{e_{N}})$ be the diagonal matrix
with $0<e_{i}\leq d$. We define a $k[t]$-ring homomorphism

$\lambda_{\gamma}^{*}:k[t][[x_{1},\ldots,x_{N}]]^{G}\to
k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]];\quad x_{i}\mapsto
t^{\frac{e_{i}}{d}}x_{i},$

and define $\overline{\lambda}_{\gamma}^{*}:k[t][[x_{1},\ldots,x_{N}]]^{G}\to
k[x_{1},\ldots,x_{N}][[t]]$ as the composition of $\lambda_{\gamma}^{*}$ and
the inclusion $i:k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]\to
k[x_{1},\ldots,x_{N}][[t]]$.

Let $f_{1},\ldots,f_{c}\in k[[x_{1},\ldots,x_{N}]]^{G}$ be a regular sequence.
We set

$\displaystyle B$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[[x_{1},\ldots,x_{N}]]^{G}/(f_{1},\ldots,f_{c})\bigr{)},$
$\displaystyle\overline{B}$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\bigr{)}.$

We denote by $n:=N-c$ their dimensions.

Suppose that $B$ is normal. Then it follows that $\overline{B}\to B$ is also
étale in codimension one, and $\overline{B}$ is also normal. Indeed, since
$\operatorname{codim}_{A}Z\geq 2$, we have $A_{\rm sing}=Z$ by the purity of
the branch locus (cf. [Nag59]). Therefore we have $B\cap Z\subset B_{\rm
sing}$ since $f_{1},\ldots,f_{c}$ is a regular sequence (cf. [Sta, tag 00NU]).
Then it follows that $\operatorname{codim}_{B}(Z\cap
B)\geq\operatorname{codim}_{B}(B_{\rm sing})\geq 2$ by the normality of $B$.
Therefore, $\overline{B}\to B$ is also étale in codimension one, and hence
$\overline{B}$ is also normal by Serre’s criterion of normality.

Note that $\omega_{B/k}^{\prime[r]}$ is invertible. Indeed, we have the
adjunction formula

$\omega^{\prime}_{B/k}\simeq\operatorname{det}^{-1}(I/I^{2})\otimes_{\mathcal{O}_{A}}\omega^{\prime}_{A/k},$

where $I:=(f_{1},\ldots,f_{c})\subset k[[x_{1},\ldots,x_{N}]]^{G}$, since the
sequence

$0\to
I/I^{2}\to\Omega^{\prime}_{A/k}\otimes_{\mathcal{O}_{A}}\mathcal{O}_{B}\to\Omega^{\prime}_{B/k}\to
0$

is exact at a regular point of $B$ by Proposition 2.12.

We define ideals $I^{\prime}$, $\widetilde{I}^{(\gamma)}$ and
$\overline{I}^{(\gamma)}$ by

$\displaystyle I^{\prime}$ $\displaystyle:=(f_{1},\ldots,f_{c})\subset
k[t][[x_{1},\ldots,x_{N}]]^{G},$ $\displaystyle\widetilde{I}^{(\gamma)}$
$\displaystyle:=\bigl{(}\lambda^{*}_{\gamma}(f_{1}),\ldots,\lambda^{*}_{\gamma}(f_{c})\bigr{)}\subset
k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]],$
$\displaystyle\overline{I}^{(\gamma)}$
$\displaystyle:=\bigl{(}\overline{\lambda}^{*}_{\gamma}(f_{1}),\ldots,\overline{\lambda}^{*}_{\gamma}(f_{c})\bigr{)}\subset
k[x_{1},\ldots,x_{N}][[t]].$

Then we have the following diagram.

$\textstyle{k[t][[x_{1},\ldots,x_{N}]]^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\gamma}^{*}}$$\scriptstyle{\overline{\lambda}_{\gamma}^{*}}$$\textstyle{k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{k[x_{1},\ldots,x_{N}][[t]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[t][[x_{1},\ldots,x_{N}]]^{G}/I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\gamma}^{*}}$$\scriptstyle{\overline{\mu}_{\gamma}^{*}}$$\textstyle{k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]/\widetilde{I}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{k[x_{1},\ldots,x_{N}][[t]]/\overline{I}^{(\gamma)}}$

We define $k[t]$-schemes $A^{\prime}$, $\widetilde{A}^{(\gamma)}$,
$\overline{A}^{(\gamma)}$, $B^{\prime}$, $\widetilde{B}^{(\gamma)}$ and
$\overline{B}^{(\gamma)}$ as follows:

$\displaystyle A^{\prime}$
$\displaystyle:=\operatorname{Spec}k[t][[x_{1},\ldots,x_{N}]]^{G},$
$\displaystyle\quad B^{\prime}$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]^{G}/I^{\prime}\bigr{)},$
$\displaystyle\widetilde{A}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]],$
$\displaystyle\quad\widetilde{B}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]]/\widetilde{I}^{(\gamma)}\bigr{)},$
$\displaystyle\overline{A}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}k[x_{1},\ldots,x_{N}][[t]],\quad$
$\displaystyle\overline{B}^{(\gamma)}$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{N}][[t]]/\overline{I}^{(\gamma)}\bigr{)}.$

Then we have the following morphisms between the corresponding $k[t]$-schemes.

$\textstyle{A^{\prime}}$$\textstyle{\widetilde{A}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\gamma}}$$\textstyle{\overline{A}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\overline{\lambda}_{\gamma}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\widetilde{B}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\gamma}}$$\textstyle{\overline{B}^{(\gamma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\overline{\mu}_{\gamma}}$$\scriptstyle{\tau}$

######  Remark 7.1.

  1. (1)

Note that

     * •

$A^{\prime}$ and $B^{\prime}$ are affine schemes of the form
$\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{M}]]/J\bigr{)}$, and

     * •

$\widetilde{A}^{(\gamma)}$, $\overline{A}^{(\gamma)}$,
$\widetilde{B}^{(\gamma)}$ and $\overline{B}^{(\gamma)}$ are of the form
$\operatorname{Spec}\bigl{(}k[x_{1},\ldots,x_{M}][[t]]/J\bigr{)}$.

We will use the notion of the sheaf $\Omega^{\prime}_{X/k[t]}$ of special
differentials for $X=A^{\prime},B^{\prime}$ defined in Section 2, and use the
notion of the sheaf $\widehat{\Omega}_{X/k[[t]]}$ for
$X=\widetilde{A}^{(\gamma)},\overline{A}^{(\gamma)},\widetilde{B}^{(\gamma)},\overline{B}^{(\gamma)}$
defined in Subsection 5.2.

Since $I^{\prime}$, $\widetilde{I}^{(\gamma)}$ and $\overline{I}^{(\gamma)}$
are generated by $c$ elements, each irreducible component $W_{i}$ of
$B^{\prime}$, $\widetilde{B}^{(\gamma)}$ and $\overline{B}^{(\gamma)}$ has
$\dim W_{i}\geq n+1$. Therefore, we can apply lemmas and propositions in
Subsections 5.1 and 5.2 to their arc spaces.

  2. (2)

Lemma 7.3(3)(4) below show that $\overline{B}^{(\gamma)}$ has only one
irreducible component that is flat over $k[[t]]$. Furthermore, the component
$V$ has $\dim V=n+1$.

######  Lemma 7.2.

Let $f:C_{1}\to C_{2}$ be a flat ring homomorphism of Noetherian rings. Then
the following hold.

  1. (1)

Let $P$ be a prime ideal of $C_{1}$ such that $PC_{2}\not=C_{2}$. Then
$\operatorname{ht}P=\operatorname{ht}(PC_{2})$ holds.

  2. (2)

Suppose that $f$ is faithfully flat. If $I\subsetneq C_{1}$ is a proper ideal
of $C_{1}$, then $\operatorname{ht}I=\operatorname{ht}(IC_{2})$ holds.

###### Proof.

We shall prove (1). Let $Q$ be a minimal prime of $PC_{2}$. Then by the going-
down theorem, we have $P=Q\cap C_{1}$. Therefore, we have
$\operatorname{ht}Q=\operatorname{ht}P$ by [Mat89, Theorem 15.1], which proves
(1).

We shall prove (2). Note that $IC_{2}\not=C_{2}$ holds by the faithfully
flatness. First, the inequality
$\operatorname{ht}I\leq\operatorname{ht}(IC_{2})$ follows from the going-down
theorem. We shall prove the opposite inequality. Take a minimal prime $P$ of
$I$ such that $\operatorname{ht}I=\operatorname{ht}P$. Then by (1), it follows
that

$\operatorname{ht}I=\operatorname{ht}P=\operatorname{ht}(PC_{2})\geq\operatorname{ht}(IC_{2}),$

which completes the proof. ∎

######  Lemma 7.3.

We denote $F_{i}:=\overline{\lambda}^{*}_{\gamma}(f_{i})$ for each $1\leq
i\leq c$. Consider the following diagram of rings.

$\textstyle{S_{1}:=k[x_{1},\ldots,x_{N}][[t]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{C_{1}:=k[x_{1},\ldots,x_{N}][[t]]/(F_{1},\ldots,F_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{S_{2}:=k[x_{1},\ldots,x_{N}]((t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{C_{2}:=k[x_{1},\ldots,x_{N}]((t))/(F_{1},\ldots,F_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{S_{3}:=k[x_{1},\ldots,x_{N}]((t^{1/d}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}:=k[x_{1},\ldots,x_{N}]((t^{1/d}))/(F_{1},\ldots,F_{c})}$$\textstyle{S_{4}:=k[[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{3}}$$\scriptstyle{x_{i}\mapsto
t^{e_{i}/d}x_{i}}$$\textstyle{C_{4}:=k[[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{3}}$$\scriptstyle{x_{i}\mapsto
t^{e_{i}/d}x_{i}}$

We denote $I_{i}:=(F_{1},\ldots,F_{c})\subset S_{i}$ for $i\in\\{1,2,3\\}$,
and $I_{4}:=(f_{1},\ldots,f_{c})\subset S_{4}$. Then the following hold.

  1. (1)

$h_{1}$, $h_{2}$ and $h_{3}$ are regular, and hence so are $g_{1}$, $g_{2}$
and $g_{3}$.

  2. (2)

$h_{2}$ and $h_{3}$ are faithfully flat, and hence so are $g_{2}$ and $g_{3}$.

  3. (3)

$C_{2}$ and $C_{3}$ are normal domains. In particular, $I_{2}$ and $I_{3}$ are
prime ideals.

  4. (4)

$\operatorname{ht}(I_{2})=\operatorname{ht}(I_{3})=c$.

###### Proof.

We shall prove (1) and (2). Since $h_{1}$ is the localization by $t\in S_{1}$,
it is regular. Note that the inclusion map

$h^{\prime}_{2}:S^{\prime}_{2}:=k[x_{1},\ldots,x_{N}][t,t^{-1}]\to
S^{\prime}_{3}:=k[x_{1},\ldots,x_{N}][t^{1/d},t^{-1/d}]$

is étale and faithfully flat. Since $h_{2}$ is the base change
${\rm-}\otimes_{S^{\prime}_{2}}S_{2}$ of $h^{\prime}_{2}$, it follows that
$h_{2}$ is étale (in particular, regular) and faithfully flat.

Let $P$ be a prime ideal of $S_{4}$ and let $Q$ be a prime ideal of $S_{3}$
such that $Q\cap S_{4}=P$. To see that $h_{3}$ is regular, it is sufficient to
show the following two conditions:

  * •

$\operatorname{ht}(P(S_{3})_{Q})=\operatorname{ht}P$.

  * •

$S_{3}/PS_{3}$ is regular at $Q$.

Note here that the first condition is equivalent to the flatness by [Mat89,
Theorem 23.1] since $S_{4}$ is regular and $S_{3}$ is Cohen-Macaulay. We also
note that the inequality
$\operatorname{ht}(P(S_{3})_{Q})\leq\operatorname{ht}P$ always holds by
[Mat89, Theorem 15.1].

Let $r:=\operatorname{ht}P$. Since $S_{4}/P$ is regular at $P$, by the
Jacobian criterion of regularity, there exist
$D_{1},\ldots,D_{r}\in\operatorname{Der}_{k}(S_{4})$ and
$a_{1},\ldots,a_{r}\in P$ such that $c:=\det(D_{i}(a_{j}))_{ij}\not\in P$.
Since $\operatorname{Der}_{k}(S_{4})$ is generated by $\partial/\partial
x_{i}$’s, we may assume that $D_{i}=\partial/\partial x_{i}$ holds for $1\leq
i\leq r$ by changing the indices of $x_{i}$. We note that $\partial/\partial
x_{i}$ defines a derivation on $S_{3}$, and we denote it by
$D_{i}^{\prime}\in\operatorname{Der}_{k}(S_{3})$. Then we have

$D^{\prime}_{i}(h_{3}(a_{j}))=t^{\frac{e_{i}}{d}}h_{3}(D_{i}(a_{j})),$

and hence we have

$\det\bigl{(}D^{\prime}_{i}(h_{3}(a_{j}))\bigr{)}_{ij}=t^{\frac{e^{\prime}}{d}}h_{3}(c)$

where $e^{\prime}:=\sum_{i=1}^{r}e_{i}$. Here, we have
$t^{\frac{e^{\prime}}{d}}h_{3}(c)\not\in Q$ since $P=Q\cap S_{4}$ and
$c\not\in P$. Therefore, it follows that $\operatorname{ht}(P(S_{3})_{Q})\geq
r$ by [Mat89, Theorem 30.4(1)]. Hence, we have
$\operatorname{ht}(P(S_{3})_{Q})=r$ and $S_{3}/PS_{3}$ is regular at $Q$ by
[Mat89, Theorem 30.4(2)]. We have proved the regularity of $h_{3}$.

To see the faithfully flatness of $h_{3}$, it is sufficient to show
$\mathfrak{m}S_{3}\not=S_{3}$ for the maximal ideal
$\mathfrak{m}=(x_{1},\ldots,x_{N})$ of $S_{4}$. It is clear because
$\mathfrak{m}S_{3}\subset(x_{1},\ldots,x_{N})\not=S_{3}$.

We shall prove (3). Note that $C_{4}$ is normal by our assumption. Since the
normality is preserved under faithfully flat regular ring homomorphisms (cf.
[Mat89, Theorem 32.2]), $C_{2}$ and $C_{3}$ are normal. Therefore, it is
sufficient to show that $C_{2}$ and $C_{3}$ are domains. In what follows, we
shall only prove that $C_{2}$ is a domain since the same proof works for
$C_{3}$. Suppose the contrary that $I_{2}$ has minimal primes $P_{1}$ and
$P_{2}$ with $P_{1}\not=P_{2}$. Then by the normality of $C_{2}$, we have
$P_{1}+P_{2}=S_{2}$ (cf. [Eis95, Proposition 2.20]). Set $Q_{1}:=P_{1}\cap
S_{1}$ and $Q_{2}:=P_{2}\cap S_{1}$. Since $P_{1}+P_{2}=S_{2}$, it follows
that $t^{s}\in Q_{1}+Q_{2}$ for some $s\geq 0$. To get a contradiction, we
shall prove

  * •

$Q_{1},Q_{2}\subset(x_{1},\ldots,x_{N},t^{s+1})$.

Suppose the contrary that

  * •

there exist $g\in k[[t]]$ and $h\in(x_{1},\ldots,x_{N})$ such that $g+h\in
Q_{1}$ and $g\not\in(t^{s+1})$.

Let $0\leq a\leq s$ be the minimum $a$ such that $g\not\in(t^{a+1})$. Then
$g-bt^{a}\in(t^{a+1})$ holds for some $b\in k^{\times}$. We may assume $b=1$
by replacing $g$ and $h$ with $b^{-1}g$ and $b^{-1}h$. For $c\in k^{\times}$,
we denote by $T_{c}:S_{1}\to S_{1}$ the ring isomorphism

$T_{c}:k[x_{1},\ldots,x_{N}][[t]]\to k[x_{1},\ldots,x_{N}][[t]];\quad t\mapsto
c^{-d}t,\ x_{i}\mapsto c^{e_{i}}x_{i}.$

Since $I_{1}$ is $T_{c}$-invariant, so is its minimal prime $Q_{1}$. Since the
ideal $(t^{s+1})$ of $S_{1}$ is also $T_{c}$-invariant, $T_{c}$ induces the
ring isomorphism $T_{c}^{\prime}:S_{1}/(t^{s+1})\to S_{1}/(t^{s+1})$. Hence,
$S_{1}/(t^{s+1})$ has a graded ring structure satisfying $\deg t=-d$ and $\deg
x_{i}=e_{i}$. Then $(Q_{1}+(t^{s+1}))/(t^{s+1})$ is a homogeneous ideal.
Therefore, the term $f_{-da}$ of $f:=g+h$ with degree $-da$ is contained in
$Q_{1}+(t^{s+1})$. Since $f_{-da}-t^{a}\in(x_{1},\ldots,x_{N})$, we have
$f_{-da}-t^{a}\in(t^{a+1})$ by looking at the degrees of its terms. Therefore
$f_{-da}=t^{a}(1+f^{\prime})$ holds for some $f^{\prime}\in(t)$. Since
$1+f^{\prime}\in S_{1}^{\times}$, we have $t^{a}\in Q_{1}+(t^{s+1})$, and
hence $t^{a}\in Q_{1}$. Therefore, we have $P_{1}=S_{2}$ and we get a
contradiction.

Note that $\operatorname{ht}(I_{4})=c$ by our assumption. Then, (4) follows
from (2) and Lemma 7.2. ∎

######  Lemma 7.4.

Let $S_{i}$ and $h_{i}$ be as in Lemma 7.3. Let $\mathfrak{c}\subset
S_{4}^{G}$ be an ideal of $S_{4}^{G}$. Let $\mathfrak{c}_{1}$ be the ideal of
$S_{1}$ generated by the image of $\mathfrak{c}$ by the ring homomorphism
$S_{4}^{G}\to S_{1};\ x_{i}\mapsto t^{e_{i}/d}x_{i}$. We denote
$\mathfrak{c}_{2}:=\mathfrak{c}_{1}S_{2}$. Then we have
$\operatorname{ht}(\mathfrak{c}_{2})=\operatorname{ht}\mathfrak{c}$.

###### Proof.

We define $\mathfrak{c}_{3}:=\mathfrak{c}_{2}S_{3}$ and
$\mathfrak{c}_{4}:=\mathfrak{c}S_{4}$. Then we have
$\mathfrak{c}_{3}=\mathfrak{c}_{4}S_{3}$. Since $S_{4}^{G}\hookrightarrow
S_{4}$ is an integral extension, we have
$\operatorname{ht}\mathfrak{c}=\operatorname{ht}(\mathfrak{c}S_{4})=\operatorname{ht}(\mathfrak{c}_{4})$.
Furthermore, we have

$\operatorname{ht}(\mathfrak{c}_{3})=\operatorname{ht}(\mathfrak{c}_{4}S_{3})=\operatorname{ht}(\mathfrak{c}_{4}),\quad\operatorname{ht}(\mathfrak{c}_{3})=\operatorname{ht}(\mathfrak{c}_{2}S_{3})=\operatorname{ht}(\mathfrak{c}_{2})$

by Lemma 7.3(2) and Lemma 7.2. We complete the proof. ∎

######  Definition 7.5.

  1. (1)

We shall define sheaves $\omega^{\prime}_{A^{\prime}/k[t]}$ on $A^{\prime}$
and $\omega^{\prime}_{B^{\prime}/k[t]}$ on $B^{\prime}$ using the special
canonical sheaves $\omega^{\prime}_{A/k}$ and $\omega^{\prime}_{B/k}$ defined
in Section 2. Let $A^{\prime}\to A$ and $B^{\prime}\to B$ be the morphisms
induced by the natural ring inclusions

$\displaystyle k[[x_{1},\ldots,x_{N}]]^{G}$ $\displaystyle\hookrightarrow
k[t][[x_{1},\ldots,x_{N}]]^{G},$ $\displaystyle k[[x_{1},\ldots,x_{N}]]^{G}/I$
$\displaystyle\hookrightarrow k[t][[x_{1},\ldots,x_{N}]]^{G}/I^{\prime}.$

Then, we define

$\omega^{\prime}_{A^{\prime}/k[t]}:=\omega^{\prime}_{A/k}\otimes_{\mathcal{O}_{A}}\mathcal{O}_{A^{\prime}},\quad\omega^{\prime}_{B^{\prime}/k[t]}:=\omega^{\prime}_{B/k}\otimes_{\mathcal{O}_{B}}\mathcal{O}_{B^{\prime}}.$

Note that they satisfy

$\omega^{\prime}_{B^{\prime}/k[t]}\simeq\operatorname{det}^{-1}(I^{\prime}/I^{\prime
2})\otimes_{\mathcal{O}_{B^{\prime}}}\sigma^{*}\omega^{\prime}_{A^{\prime}/k[t]}.$

Here, $\operatorname{det}^{-1}(I^{\prime}/I^{\prime
2}):=\bigl{(}\bigwedge^{c}(I^{\prime}/I^{\prime 2})\bigr{)}^{*}$ is an
invertible sheaf on $B^{\prime}$.

  2. (2)

The canonical map

$\Omega_{B^{\prime}/k[t]}^{\prime n}\to\omega^{\prime}_{B^{\prime}/k[t]}$

is induced by $\Omega_{B/k}^{\prime n}\to\omega^{\prime}_{B/k}$ and the
isomorphism
$\Omega^{\prime}_{B^{\prime}/k[t]}\simeq\Omega^{\prime}_{B/k}\otimes_{\mathcal{O}_{B}}\mathcal{O}_{B^{\prime}}$.
The canonical map $\Omega_{A^{\prime}/k[t]}^{\prime
n}\to\omega^{\prime}_{A^{\prime}/k[t]}$ is also defined. We define an ideal
sheaf $\mathfrak{n}_{r,B^{\prime}}\subset\mathcal{O}_{B^{\prime}}$ by

$\operatorname{Im}\Bigl{(}\bigl{(}\Omega_{B^{\prime}/k[t]}^{\prime
n}\bigr{)}^{\otimes
r}\to\omega^{\prime[r]}_{B^{\prime}/k[t]}\Bigr{)}=\mathfrak{n}_{r,B^{\prime}}\otimes_{\mathcal{O}_{B^{\prime}}}\omega^{\prime[r]}_{B^{\prime}/k[t]}.$

Then it satisfies
$\mathfrak{n}_{r,B^{\prime}}=\mathfrak{n}_{r,B}\mathcal{O}_{B^{\prime}}$.

  3. (3)

We define ideal sheaves $\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}$,
$\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}$ and
$\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}$ by

$\displaystyle\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}$
$\displaystyle:=\operatorname{Fitt}^{n}\bigl{(}\Omega^{\prime}_{B^{\prime}/k[t]}\bigr{)}\subset\mathcal{O}_{B^{\prime}},$
$\displaystyle\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}$
$\displaystyle:=\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{\widetilde{B}^{(\gamma)}/k[[t]]}\bigr{)}\subset\mathcal{O}_{\widetilde{B}^{(\gamma)}},$
$\displaystyle\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}$
$\displaystyle:=\operatorname{Fitt}^{n}\bigl{(}\widehat{\Omega}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}\subset\mathcal{O}_{\overline{B}^{(\gamma)}}.$

Here, we note that $\widetilde{B}^{(\gamma)}$ and $\overline{B}^{(\gamma)}$
are not necessarily equidimensional.

  4. (4)

We define an invertible sheaf $L_{\overline{B}^{(\gamma)}}$ on
$\overline{B}^{(\gamma)}$ by

$L_{\overline{B}^{(\gamma)}}:=\overline{\mu}_{\gamma}^{*}\bigl{(}\operatorname{det}^{-1}(I^{\prime}/I^{\prime
2})\bigr{)}\otimes_{\mathcal{O}_{\overline{B}^{(\gamma)}}}\tau^{*}\widehat{\Omega}^{N}_{\overline{A}^{(\gamma)}/k[[t]]}.$

Then there exist canonical homomorphisms

$\widehat{\Omega}^{n}_{\overline{B}^{(\gamma)}/k[[t]]}\to
L_{\overline{B}^{(\gamma)}},\qquad\overline{\mu}_{\gamma}^{*}\omega^{\prime[r]}_{B^{\prime}/k[t]}\to
L^{[r]}_{\overline{B}^{(\gamma)}}$

such that the following diagram commutes (cf. [NS, Lemma 4.5(2)]).

$\textstyle{\overline{\mu}_{\gamma}^{*}(\Omega^{\prime
n}_{B^{\prime}/k[t]})^{\otimes
r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigl{(}\widehat{\Omega}^{n}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}^{\otimes
r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mu}_{\gamma}^{*}\omega^{\prime[r]}_{B^{\prime}/k[t]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{[r]}_{\overline{B}^{(\gamma)}}}$

Furthermore, by the same argument as in [NS, Lemma 4.5(1)], we have

$\operatorname{Im}\bigl{(}\widehat{\Omega}^{n}_{\overline{B}^{(\gamma)}/k[[t]]}\to
L_{\overline{B}^{(\gamma)}}\bigr{)}=\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\otimes_{\mathcal{O}_{\overline{B}^{(\gamma)}}}L_{\overline{B}^{(\gamma)}}.$

  5. (5)

We define ideal sheaves $\mathfrak{n}^{\prime}_{1,p}$ and
$\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}}$ on
$\overline{B}^{(\gamma)}$ by

$\displaystyle\operatorname{Im}\bigl{(}p^{*}\widehat{\Omega}^{n}_{\widetilde{B}^{(\gamma)}/k[[t]]}\to
L_{\overline{B}^{(\gamma)}}\bigr{)}$
$\displaystyle=\mathfrak{n}^{\prime}_{1,p}\otimes_{\mathcal{O}_{\overline{B}^{(\gamma)}}}L_{\overline{B}^{(\gamma)}},$
$\displaystyle\operatorname{Im}\bigl{(}\overline{\mu}_{\gamma}^{*}\Omega^{\prime
n}_{B^{\prime}/k[t]}\to L_{\overline{B}^{(\gamma)}}\bigr{)}$
$\displaystyle=\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}}\otimes_{\mathcal{O}_{\overline{B}^{(\gamma)}}}L_{\overline{B}^{(\gamma)}}.$

  6. (6)

We define
$\operatorname{age}^{\prime}(\gamma):=\sum_{i=1}^{N}\frac{e_{i}}{d}$. Note
that we took $e_{i}$ to satisfy $0<e_{i}\leq d$. The age of $\gamma$ is
usually defined by
$\operatorname{age}(\gamma)=\operatorname{age}^{\prime}(\gamma)-\\#\\{1\leq
i\leq N\mid e_{i}=d\\}$.

######  Lemma 7.6.

Let $\alpha\in\overline{B}^{(\gamma)}_{\infty}$ be a $k$-arc with
$\operatorname{ord}_{\alpha}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}<\infty$.
Then the following hold.

  1. (1)

$\operatorname{ord}_{\alpha}(\operatorname{jac}_{p})+\operatorname{ord}_{\alpha}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}=\operatorname{ord}_{\alpha}(\mathfrak{n}^{\prime}_{1,p})$.

  2. (2)

$\operatorname{ord}_{\alpha}(\operatorname{jac}_{\overline{\mu}_{\gamma}})+\operatorname{ord}_{\alpha}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}=\operatorname{ord}_{\alpha}(\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}})$.

###### Proof.

The same proof as in [NS, Lemma 4.6] works due to Lemma 5.30. ∎

######  Lemma 7.7.

Let $\alpha\in\overline{B}^{(\gamma)}_{\infty}$ be a $k$-arc. Set
$\alpha^{\prime}:=\overline{\mu}_{\gamma\infty}(\alpha)$. Suppose that
$\alpha^{\prime}\not\in Z_{\infty}$. Then it follows that

$\operatorname{ord}_{\alpha}(\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}})=\frac{1}{r}\operatorname{ord}_{\alpha^{\prime}}(\mathfrak{n}_{r,B^{\prime}})+\operatorname{age}^{\prime}(\gamma).$

###### Proof.

The same proof as in [NS, Lemma 4.7] works. ∎

######  Lemma 7.8.

Let $I_{Z}\subset\mathcal{O}_{A}$ be the ideal sheaf defining $Z\subset A$.
Let $J$ be one of the following ideal sheaves on $\overline{B}^{(\gamma)}$:

$\displaystyle
I_{Z}\mathcal{O}_{\overline{B}^{(\gamma)}},\quad\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]},\quad\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}\mathcal{O}_{\overline{B}^{(\gamma)}},$
$\displaystyle\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}\mathcal{O}_{\overline{B}^{(\gamma)}},\quad\mathfrak{n}^{\prime}_{1,p},\quad\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}},\quad\mathfrak{n}_{r,B^{\prime}}\mathcal{O}_{\overline{B}^{(\gamma)}}.$

Let $W\subset\overline{B}^{(\gamma)}$ be the closed subscheme defined by $J$.
Then $W_{\infty}$ is a thin subset of $\overline{B}^{(\gamma)}_{\infty}$.

###### Proof.

We set

$\displaystyle J_{1}:=I_{Z}\mathcal{O}_{\overline{B}^{(\gamma)}},\quad
J_{2}:=\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]},\quad
J_{3}:=\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}\mathcal{O}_{\overline{B}^{(\gamma)}},$
$\displaystyle
J_{4}:=\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}\mathcal{O}_{\overline{B}^{(\gamma)}},\quad
J_{5}:=\mathfrak{n}^{\prime}_{1,p},\quad
J_{6}:=\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}},\quad
J_{7}:=\mathfrak{n}_{r,B^{\prime}}\mathcal{O}_{\overline{B}^{(\gamma)}},$

and we denote by $W_{i}\subset\overline{B}^{(\gamma)}$ the closed subscheme
defined by $J_{i}$.

Since $B\cap Z\subset B_{\rm sing}$, we have $(W_{1})_{\rm
red}\subset(W_{4})_{\rm red}$. Since the map
$\eta_{r}\colon(\Omega_{B/k}^{\prime n})^{\otimes
r}\to\omega_{B/k}^{\prime[r]}$ in Definition 2.13(5) is an isomorphism on the
regular locus $B_{\rm reg}$, we have $(W_{7})_{\rm red}\subset(W_{4})_{\rm
red}$. By Lemma 7.7, we have

$(W_{6})_{\infty}\cup(W_{1})_{\infty}=(W_{7})_{\infty}\cup(W_{1})_{\infty}.$

By Lemmas 7.6 and 5.40, we have

$(W_{5})_{\infty}\subset(W_{2})_{\infty}\cup(W_{3})_{\infty}\cup(W_{6})_{\infty}.$

Therefore, it is sufficient to show the assertion for

$J_{2}=\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]},\quad
J_{3}=\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}\mathcal{O}_{\overline{B}^{(\gamma)}},\quad
J_{4}=\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}\mathcal{O}_{\overline{B}^{(\gamma)}}.$

We set

$\displaystyle S_{1}$ $\displaystyle:=k[x_{1},\ldots,x_{N}][[t]],$
$\displaystyle\quad S_{2}:=$ $\displaystyle k[x_{1},\ldots,x_{N}]((t)),$
$\displaystyle T_{1}$
$\displaystyle:=k[x_{1},\ldots,x_{N}]^{C_{\gamma}}[[t]],$ $\displaystyle\quad
T_{2}:=$ $\displaystyle k[x_{1},\ldots,x_{N}]^{C_{\gamma}}((t)).$

The ideal $J_{2}=\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}$
corresponds to the ideal

$\mathfrak{r}:=\mathcal{J}_{c}\Bigl{(}\overline{I}^{(\gamma)};\operatorname{Der}_{k[[t]]}(S_{1})\Bigr{)}+\overline{I}^{(\gamma)}\subset
S_{1}$

of $S_{1}$. To show that $(W_{2})_{\infty}$ is a thin set, it is sufficient to
show $\operatorname{ht}(\mathfrak{r}S_{2})\geq c+1$. Since
$\overline{I}^{(\gamma)}S_{2}$ is a prime ideal of height $c$ by Lemma
7.3(3)(4), it is sufficient to show that
$\mathfrak{r}S_{2}\not\subset\overline{I}^{(\gamma)}S_{2}$. Since $S_{2}$
satisfies the weak Jacobian condition (WJ)k((t)) over $k((t))$ by [Nag62,
Theorem 46.3], we have

$\mathcal{J}_{c}\Bigl{(}\overline{I}^{(\gamma)};\operatorname{Der}_{k[[t]]}(S_{1})\Bigr{)}S_{2}+\overline{I}^{(\gamma)}S_{2}=\mathcal{J}_{c}\Bigl{(}\overline{I}^{(\gamma)}S_{2};\operatorname{Der}_{k((t))}(S_{2})\Bigr{)}+\overline{I}^{(\gamma)}S_{2}\not\subset\overline{I}^{(\gamma)}S_{2}.$

Here, the first equality follows from the fact that both
$\operatorname{Der}_{k[[t]]}(S_{1})$ and $\operatorname{Der}_{k((t))}(S_{2})$
are generated by the derivations $\partial/\partial x_{i}$’s. We complete the
proof of the assertion for $J_{2}$.

Let $\mathfrak{r}\subset T_{1}$ be the ideal of $T_{1}$ corresponding to
$\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}\subset
T_{1}/\widetilde{I}^{(\gamma)}$. Then it is sufficient to show
$\operatorname{ht}(\mathfrak{r}S_{2})\geq c+1$. Note that
$\widetilde{I}^{(\gamma)}T_{2}=\overline{I}^{(\gamma)}S_{2}\cap T_{2}$ holds.
Therefore, $\widetilde{I}^{(\gamma)}T_{2}$ is a prime ideal of height $c$. By
the same argument as above, we have
$\mathfrak{r}T_{2}\not\subset\widetilde{I}^{(\gamma)}T_{2}$, and hence
$\operatorname{ht}(\mathfrak{r}T_{2})\geq c+1$. Since $T_{2}\subset S_{2}$ is
an integral extension, we have
$\operatorname{ht}(\mathfrak{r}S_{2})=\operatorname{ht}(\mathfrak{r}T_{2})\geq
c+1$, which completes the proof of the assertion for $J_{3}$.

Let $\mathfrak{r}\subset k[[x_{1},\ldots,x_{N}]]^{G}$ be the ideal
corresponding to $\operatorname{Jac}^{\prime}_{B/k}\subset
k[[x_{1},\ldots,x_{N}]]^{G}/I$. Since
$J_{4}=\mathfrak{r}\mathcal{O}_{\overline{B}^{(\gamma)}}$, it is sufficient to
show $\operatorname{ht}(\mathfrak{r}S_{2})\geq c+1$. We have
$\operatorname{ht}\mathfrak{r}\geq c+2$ by the normality of $B$. Therefore, we
have $\operatorname{ht}(\mathfrak{r}S_{2})=\operatorname{ht}\mathfrak{r}\geq
c+2$ by Lemma 7.4. We complete the proof of the assertion for $J_{4}$. ∎

######  Theorem 7.9.

Let $x=0\in B$ be the origin and let $\mathfrak{m}_{x}\subset\mathcal{O}_{B}$
be the corresponding maximal ideal. Let $\mathfrak{a}\subset\mathcal{O}_{B}$
be a non-zero ideal sheaf and $\delta$ a positive real number. Then

$\displaystyle\operatorname{mld}_{x}(B,\mathfrak{a}^{\delta})$
$\displaystyle=\inf_{w,b_{1}\in\mathbb{Z}_{\geq 0},\gamma\in
G}\bigl{\\{}\operatorname{codim}(C_{w,\gamma,b_{1}})+\operatorname{age}^{\prime}(\gamma)-b_{1}-\delta
w\bigr{\\}}$ $\displaystyle=\inf_{w,b_{1}\in\mathbb{Z}_{\geq 0},\gamma\in
G}\bigl{\\{}\operatorname{codim}(C^{\prime}_{w,\gamma,b_{1}})+\operatorname{age}^{\prime}(\gamma)-b_{1}-\delta
w\bigr{\\}}$

holds for

$\displaystyle C_{w,\gamma,b_{1}}$
$\displaystyle:=\operatorname{Cont}^{w}\bigl{(}\mathfrak{a}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{m}_{x}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{b_{1}}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)},$
$\displaystyle C^{\prime}_{w,\gamma,b_{1}}$
$\displaystyle:=\operatorname{Cont}^{\geq
w}\bigl{(}\mathfrak{a}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{m}_{x}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{b_{1}}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}.$

###### Proof.

The formula for $C_{w,\gamma,b_{1}}$ is the formal power series ring version
of [NS, Theorem 4.8]. The same proof as in [NS, Theorem 4.8] works. First,
[EM09, Theorem 7.4] plays an important role in the proof of [NS, Theorem 4.8]
and it can be substituted by Theorem 4.11. Furthermore, Propositions 2.25,
2.33, 2.35, 3.4 and 3.8, and Lemmas 2.10, 4.6 and 4.7 in [NS], which are also
the key ingredients of the proof of [NS, Theorem 4.8], are substituted by
Propositions 5.36, 5.43, 5.44 and 6.3 and Lemmas 5.40, 7.6 and 7.7 in this
paper.

We also note that Proposition 5.36 is applied to $Z_{\infty}$ and
$W_{\infty}$, where $W$ is the closed subscheme of $\overline{B}^{(\gamma)}$
corresponding to one of the following ideals:

$\displaystyle\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]},\ \
\operatorname{Jac}^{\prime}_{\widetilde{B}^{(\gamma)}/k[[t]]}\mathcal{O}_{\overline{B}^{(\gamma)}},\
\
\operatorname{Jac}^{\prime}_{B^{\prime}/k[t]}\mathcal{O}_{\overline{B}^{(\gamma)}},\
\ \mathfrak{n}^{\prime}_{1,p},\ \
\mathfrak{n}^{\prime}_{1,\overline{\mu}_{\gamma}},\ \
\mathfrak{n}_{r,B^{\prime}}\mathcal{O}_{\overline{B}^{(\gamma)}}.$

By Lemma 7.8, they are actually thin sets.

The formula for $C^{\prime}_{w,\gamma,b_{1}}$ is the formal power series ring
version of [NS, Corollary 4.9], and the same proof works. ∎

######  Remark 7.10.

Theorem 7.9 can be easily extended to $\mathbb{R}$-ideals
$\mathfrak{a}=\prod_{i=1}^{r}\mathfrak{a}_{i}^{\delta_{i}}$, where
$\mathfrak{a}_{1},\ldots,\mathfrak{a}_{r}$ are non-zero ideal sheaves on $B$
and $\delta_{1},\ldots,\delta_{r}$ are positive real numbers. In this setting,
we have

$\displaystyle\operatorname{mld}_{x}\bigl{(}B,\prod_{i=1}^{r}\mathfrak{a}_{i}^{\delta_{i}}\bigr{)}$
$\displaystyle=\inf_{w_{1},\ldots,w_{r},b_{1}\in\mathbb{Z}_{\geq 0},\gamma\in
G}\left\\{\operatorname{codim}(C_{w_{1},\ldots,w_{r},\gamma,b_{1}})+\operatorname{age}^{\prime}(\gamma)-b_{1}-\sum_{i=1}^{r}\delta_{i}w_{i}\right\\}$
$\displaystyle=\inf_{w_{1},\ldots,w_{r},b_{1}\in\mathbb{Z}_{\geq 0},\gamma\in
G}\left\\{\operatorname{codim}(C^{\prime}_{w_{1},\ldots,w_{r},\gamma,b_{1}})+\operatorname{age}^{\prime}(\gamma)-b_{1}-\sum_{i=1}^{r}\delta_{i}w_{i}\right\\}$

for

$\displaystyle C_{w_{1},\ldots,w_{r},\gamma,b_{1}}$
$\displaystyle:=\Bigl{(}\bigcap_{i=1}^{r}\operatorname{Cont}^{w_{i}}\bigl{(}\mathfrak{a}_{i}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\Bigr{)}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{m}_{x}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{b_{1}}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)},$
$\displaystyle C^{\prime}_{w_{1},\ldots,w_{r},\gamma,b_{1}}$
$\displaystyle:=\Bigl{(}\bigcap_{i=1}^{r}\operatorname{Cont}^{\geq
w_{i}}\bigl{(}\mathfrak{a}_{i}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\Bigr{)}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{m}_{x}\mathcal{O}_{\overline{B}^{(\gamma)}}\bigr{)}\cap\operatorname{Cont}^{b_{1}}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}.$

###  7.2. Properties on $\overline{B}^{\prime(\gamma)}$

In the remainder of this section, we define a scheme
$\overline{B}^{\prime(\gamma)}$ and investigate its properties, which will be
used in Section 8.

We denote by $\overline{\lambda}_{\gamma}^{\prime*}$ the $k[t]$-ring
homomorphism

$\overline{\lambda}_{\gamma}^{\prime*}:k[t][[x_{1},\ldots,x_{N}]]^{G}\to
k[t][[x_{1},\ldots,x_{N}]];\quad x_{i}\mapsto t^{\frac{e_{i}}{d}}x_{i}.$

We set

$\displaystyle\overline{I}^{\prime(\gamma)}$
$\displaystyle:=\bigl{(}\overline{\lambda}^{\prime*}_{\gamma}(f_{1}),\ldots,\overline{\lambda}^{\prime*}_{\gamma}(f_{c})\bigr{)}\subset
k[t][[x_{1},\ldots,x_{N}]],$ $\displaystyle\overline{B}^{\prime(\gamma)}$
$\displaystyle:=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]/\overline{I}^{\prime(\gamma)}\bigr{)}.$

Then $\overline{B}^{\prime(\gamma)}$ is a scheme of finite type over
$R=k[t][[x_{1},\ldots,x_{N}]]$. Let
$\Omega^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}$ be the sheaf of special
differentials defined in Definition 2.13(1) with respect to $R$ and
$R_{0}=k[t]$. We set

$\operatorname{Jac}^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}:=\operatorname{Fitt}^{n}\bigl{(}\Omega^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}\bigr{)}.$

First, we study the dimensions of the irreducible components of
$\overline{B}^{\prime(\gamma)}$.

######  Lemma 7.11.

We denote $F_{i}:=\overline{\lambda}^{\prime*}_{\gamma}(f_{i})$ for each
$1\leq i\leq c$. Consider the following diagram of rings.

$\textstyle{S_{1}:=k[t][[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{C_{1}:=k[t][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{S_{2}:=k[t,t^{-1}][[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{C_{2}:=k[t,t^{-1}][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{S_{3}:=k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}:=k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})}$$\textstyle{S_{4}:=k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{x_{i}\mapsto
t^{e_{i}/d}x_{i}}$$\textstyle{C_{4}:=k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{x_{i}\mapsto
t^{e_{i}/d}x_{i}}$$\textstyle{S_{5}:=k[[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{3}}$$\textstyle{C_{5}:=k[[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{3}}$

We denote $I_{i}:=(F_{1},\ldots,F_{c})\subset S_{i}$ for $i\in\\{1,2,3\\}$,
and $I_{i}:=(f_{1},\ldots,f_{c})\subset S_{i}$ for $i\in\\{4,5\\}$. Then the
following hold.

  1. (1)

$h_{1}$, $h_{2}$ and $h_{3}$ are regular, and hence so are $g_{1}$, $g_{2}$
and $g_{3}$.

  2. (2)

$h_{2}$ and $h_{3}$ are faithfully flat, and hence so are $g_{2}$ and $g_{3}$.

  3. (3)

$C_{2}$, $C_{3}$ and $C_{4}$ are normal domains. In particular, $I_{2}$,
$I_{3}$ and $I_{4}$ are prime ideals.

  4. (4)

$\operatorname{ht}(I_{2})=\operatorname{ht}(I_{3})=\operatorname{ht}(I_{4})=c$.

  5. (5)

There exists only one minimal prime $P$ of $I_{1}$ of the form (2) in Lemma
5.25. Furthermore, it satisfies $\operatorname{ht}P=c$ and $P=I_{2}\cap
S_{1}$.

###### Proof.

We shall prove (1). We shall only see the regularity of $h_{2}$ since the
other two can be proved more easily. Since the ring inclusion $k[t,t^{-1}]\to
k[t^{1/d},t^{-1/d}]$ is étale, so is its base change

$k[t,t^{-1}][[x_{1},\ldots,x_{N}]]\longrightarrow
k[t,t^{-1}][[x_{1},\ldots,x_{N}]][t^{1/d},t^{-1/d}].$

Furthermore, the ring inclusion

$k[t,t^{-1}][[x_{1},\ldots,x_{N}]][t^{1/d},t^{-1/d}]\longrightarrow
k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]$

is regular since it can be seen as the completion at the prime ideal
$(x_{1},\ldots,x_{N})$ and the ring on the left side is an excellent ring, in
particular a G-ring (cf. [Mat80, Theorem 79]). Therefore, their composition
$h_{2}$ turns out to be regular.

We shall prove (2) for $h_{2}$. Any maximal ideal $M$ of $S_{2}$ is of the
form $M=(t-a,x_{1},\ldots,x_{N})$, where $a\in k^{\times}$. Therefore, we have
$MS_{3}\not=S_{3}$ and hence $h_{2}$ is faithfully flat. The same proof works
for $h_{3}$.

We shall prove (3). Note that $C_{5}$ is normal by our assumption. Therefore,
the normality of $C_{2}$, $C_{3}$ and $C_{4}$ follows from (1), (2) and the
fact that the normality is preserved under faithfully flat regular ring
homomorphisms (cf. [Mat89, Theorem 32.2]). In what follows, we prove that
$C_{2}$, $C_{3}$ and $C_{4}$ are domains. Since $h_{2}$ is faithfully flat,
$g_{2}$ is injective (cf. [Mat89, Theorem 7.5]). Therefore, it is sufficient
to show that $C_{4}$ is a domain. Let $P_{1},\ldots,P_{\ell}$ be the minimal
primes of $I_{4}$. Suppose the contrary that $\ell\geq 2$. Since $C_{4}$ is
normal, we have $P_{1}+P_{2}=S_{4}$ (cf. [Eis95, Proposition 2.20]). Take
maximal ideals $M_{1}$ and $M_{2}$ of $S_{4}$ such that $P_{i}\subset M_{i}$
for each $i\in\\{1,2\\}$. We may write
$M_{i}=(t^{1/d}-a_{i},x_{1},\ldots,x_{N})$ with $a_{i}\in k^{\times}$. For
each $c\in k^{\times}$, we denote by $T_{c}$ the ring isomorphism

$T_{c}:S_{4}\to S_{4};\qquad t^{1/d}\mapsto ct^{1/d},\quad x_{i}\mapsto
x_{i}.$

Then, $I_{4}$ is $T_{c}$-invariant for any $c\in k^{\times}$. Therefore, its
minimal primes $P_{1}$ and $P_{2}$ are also $T_{c}$-invariant for any $c\in
k^{\times}$. Therefore $P_{1}\subset M_{2}$ holds, and hence
$P_{1}+P_{2}\subset M_{2}$, a contradiction.

Note that $\operatorname{ht}(I_{5})=c$ by our assumption. Therefore, (4)
follows from (2) and Lemma 7.2.

We shall prove (5). Let $P_{1},\ldots,P_{\ell}$ be the minimal primes of
$I_{1}$. Then $P_{1}^{a_{1}}\cap\cdots\cap P_{\ell}^{a_{\ell}}\subset I_{1}$
holds for some $a_{1},\ldots,a_{\ell}\geq 1$. Since $h_{1}$ is flat, we have

$I_{2}=I_{1}S_{2}\supset(P_{1}^{a_{1}}\cap\cdots\cap
P_{\ell}^{a_{\ell}})S_{2}=P_{1}^{a_{1}}S_{2}\cap\cdots\cap
P_{\ell}^{a_{\ell}}S_{2}$

by [Mat89, Theorem 7.4]. If $P_{i}$ is of the form (1) or (3) in Lemma 5.25,
then we have $P_{i}S_{2}=S_{2}$. Since $I_{2}\not=S_{2}$, some $P_{i}$ has to
be of the form (2) in Lemma 5.25.

Suppose that $P$ is a minimal prime of $I_{1}$ of the form (2) in Lemma 5.25.
Then $P+(t-a)\not=S_{1}$ holds for any $a\in k^{\times}$, and hence we have
$PS_{2}\not=S_{2}$. Since we have

$c=\operatorname{ht}I_{2}\leq\operatorname{ht}(PS_{2})=\operatorname{ht}P\leq
c$

by Lemma 7.2 and Krull’s height theorem, it follows that
$\operatorname{ht}P=c$ and $I_{2}=PS_{2}$. Since we have

$c=\operatorname{ht}P\leq\operatorname{ht}(I_{2}\cap
S_{1})\leq\operatorname{ht}I_{2}=c$

by the going-down theorem, we have $P=I_{2}\cap S_{1}$, which also shows the
uniqueness of $P$. ∎

######  Remark 7.12.

Let $\overline{B}^{\prime(\gamma)}=V_{1}\cup\cdots\cup V_{\ell}$ be the
irreducible decomposition. By Lemma 7.11(5), $\overline{B}^{\prime(\gamma)}$
has the unique irreducible component $V$ of the form (2) in Lemma 5.25 and it
satisfies $\dim V=\dim^{\prime}V=n+1$. Furthermore, any irreducible component
$V^{\prime}$ other than $V$ satisfies

$V^{\prime}_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}=\emptyset$

by Remark 5.26. Here,
$\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\subset\mathcal{O}_{\overline{B}^{\prime(\gamma)}}$
denotes the ideal sheaf generated by $x_{1},\ldots,x_{N}$.

Next, we see the relationship between $\overline{B}^{(\gamma)}_{\infty}$ and
$\overline{B}^{\prime(\gamma)}_{\infty}$.

######  Lemma 7.13.

Let
$\mathfrak{o}_{\overline{B}^{(\gamma)}}\subset\mathcal{O}_{\overline{B}^{(\gamma)}}$
and
$\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\subset\mathcal{O}_{\overline{B}^{\prime(\gamma)}}$
be the ideal sheaves generated by $x_{1},\ldots,x_{N}$. Then the following
hold.

  1. (1)

For $m\geq 0$, there exist canonical morphisms
$\overline{B}^{\prime(\gamma)}_{m}\to\overline{B}^{(\gamma)}_{m}$ which
commute with the truncation morphisms. In particular, they induce
$\overline{B}^{\prime(\gamma)}_{\infty}\to\overline{B}^{(\gamma)}_{\infty}$.

  2. (2)

The map
$\overline{B}^{\prime(\gamma)}_{\infty}\to\overline{B}^{(\gamma)}_{\infty}$
induces an isomorphism

$\overline{B}^{\prime(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}\xrightarrow{\simeq}\overline{B}^{(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{(\gamma)}}\bigr{)}.$

  3. (3)

For a $k$-arc
$\gamma\in\overline{B}^{\prime(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}$ and the
corresponding arc
$\gamma^{\prime}\in\overline{B}^{(\gamma)}_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{(\gamma)}}\bigr{)}$, it follows that

$\operatorname{ord}_{\gamma}\Bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}\Bigr{)}=\operatorname{ord}_{\gamma^{\prime}}\Bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\Bigr{)}.$

###### Proof.

(1) and (2) follow from the discussion in Remark 6.4(2). (3) follows from
Remarks 2.14 and 5.29(3). ∎

######  Lemma 7.14.

Let $W\subset\overline{B}^{\prime(\gamma)}$ be the closed subscheme defined by
$\operatorname{Jac}^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}$. Then
$W_{\infty}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}$ is a thin
subset of $\overline{B}^{\prime(\gamma)}_{\infty}$.

###### Proof.

Let $T_{c}:k[t][[x_{1},\ldots,x_{N}]]\to k[t][[x_{1},\ldots,x_{N}]]$ be the
ring isomorphism defined in Lemma 5.25.

Set $J:=\operatorname{Jac}^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}$. Let
$\overline{J}\subset k[t][[x_{1},\ldots,x_{N}]]$ be the corresponding ideal.
Since $\overline{I}^{\prime(\gamma)}$ is $T_{c}$-invariant, $\overline{J}$ is
also $T_{c}$-invariant for each $c\in k^{\times}$. Therefore by Lemma 5.25,
each minimal prime $P$ of $\overline{J}$ satisfies one of the conditions in
Lemma 5.25. By Remark 5.26, it is sufficient to show that
$\operatorname{ht}P\geq c+1=N-n+1$ for $P$ satisfying (2) in Lemma 5.25. Since
$P+(t-1)\not=k[t][[x_{1},\ldots,x_{N}]]$ in this case, it is sufficies to show
that

$\operatorname{ht}(P+(t-1))\geq N-n+2.$

Under the identification $k[t][[x_{1},\ldots,x_{N}]]/(t-1)\simeq
k[[x_{1},\ldots,x_{N}]]$, the ideal $\bigl{(}\overline{J}+(t-1)\bigr{)}/(t-1)$
corresponds to $\operatorname{Jac}^{\prime}_{\overline{B}/k}$. Since
$\overline{B}$ is normal and hence regular in codimension one, it follows that

$\operatorname{ht}\bigl{(}\overline{J}+(t-1)\bigr{)}\geq N-n+3$

by the Jacobian criterion of regularity. Therefore, we get the desired
inequality for $P$. ∎

##  8\. PIA formula for quotient singularities of non-linear action

In this section, we generalize Theorem 5.1 in [NS] to non-linear group actions
(Theorem 8.2). First, we clarify the definition of quotient singularities in
this paper.

######  Definition 8.1.

Let $X$ be a variety over $k$ and $x\in X$ a closed point. We say that $X$ has
a quotient singularity at $x$ if there exist a quasi-projective variety
$\overline{M}$ over $k$, a finite subgroup
$G\subset\operatorname{Aut}(\overline{M})$, and a smooth closed point
$\overline{y}\in\overline{M}$ such that
$\mathcal{O}_{X,x}\simeq\mathcal{O}_{M,y}$ holds, where $M:=\overline{M}/G$ is
the quotient variety and $y\in M$ is the image of $\overline{y}$.

######  Theorem 8.2.

Suppose that a variety $X$ has a quotient singularity at a closed point $x\in
X$. Let $Y$ be a subvariety of $X$ of codimension $c$ that is locally defined
by $c$ equations $h_{1},\ldots,h_{c}\in\mathfrak{m}_{X,x}$ at $x$. Suppose
that $Y$ is klt at $x$. Let $\mathfrak{a}\subset\mathcal{O}_{X}$ be an ideal
sheaf and let $\delta$ be a positive real number. Suppose that
$\mathfrak{b}:=\mathfrak{a}\mathcal{O}_{Y}\not=0$. Then it follows that

$\operatorname{mld}_{x}\bigl{(}X,(h_{1}\cdots
h_{c})\mathfrak{a}^{\delta}\bigr{)}=\operatorname{mld}_{x}(Y,\mathfrak{b}^{\delta}).$

###### Proof.

Since $X$ has a quotient singularity at $x$, there exist a variety
$\overline{M}$ with a smooth closed point $\overline{y}\in\overline{M}$ and a
finite subgroup $G^{\prime}\subset\operatorname{Aut}(\overline{M})$ such that
$\mathcal{O}_{X,x}\simeq\mathcal{O}_{M^{\prime},y^{\prime}}$ holds, where
$M^{\prime}:=\overline{M}/G^{\prime}$ and $y^{\prime}\in M^{\prime}$ is the
image of $\overline{y}$.

We denote $G:=\\{g\in G^{\prime}\mid g(\overline{y})=\overline{y}\\}$ the
stabilizer group of $\overline{y}$, $M:=\overline{M}/G$ the quotient variety,
and $y\in M$ the image of $\overline{y}$. Then we note that
$M=\overline{M}/G\to M^{\prime}=\overline{M}/G^{\prime}$ is étale at $y$ (cf.
[Kol13, 3.17]). Furthermore, $G$ acts on
$\mathfrak{m}_{\overline{M},\overline{y}}^{i}$ for each $i\geq 0$, and hence
the projection
$s:\mathfrak{m}_{\overline{M},\overline{y}}\to\mathfrak{m}_{\overline{M},\overline{y}}/\mathfrak{m}_{\overline{M},\overline{y}}^{2}$
becomes a $G$-equivariant $k$-linear map. Let $u$ be any $k$-linear section of
$s$. Then the map $u^{\prime}:=\frac{1}{|G|}\sum_{g\in G}g^{-1}\circ u\circ g$
gives a $G$-equivariant $k$-linear section
$u^{\prime}:\mathfrak{m}_{\overline{M},\overline{y}}/\mathfrak{m}_{\overline{M},\overline{y}}^{2}\to\mathfrak{m}_{\overline{M},\overline{y}}$
of $s$. Let $N:=\dim X$. Then $u^{\prime}$ induces a ring homomorphism

$k[x_{1},\ldots,x_{N}]\to\mathcal{O}_{\overline{M},\overline{y}}$

which is étale. Furthermore, $G$ acts linearly on $k[x_{1},\ldots,x_{N}]$ and
the ring homomorphism above becomes $G$-equivariant. Since the ring
homomorphism above is étale, we get an isomorphism

$k[[x_{1},\ldots,x_{N}]]\overset{\simeq}{\longrightarrow}\widehat{\mathcal{O}}_{\overline{M},\overline{y}}.$

Note that $k[[x_{1},\ldots,x_{N}]]^{G_{\rm pr}}\simeq k[[x_{1},\ldots,x_{N}]]$
holds for $G_{\rm pr}\subset G$, where $G_{\rm pr}$ is the subgroup generated
by the pseudo-reflections (cf. [Kol13, 3.18]). Hence $\overline{M}/G_{\rm pr}$
is smooth at the image of $\overline{y}$. Therefore, by replacing $G$ with
$G/G_{\rm pr}$ and $\overline{M}$ with $\overline{M}/G_{\rm pr}$, we may
assume that $G$ does not contain a pseudo-reflection. Then we have the
following diagram of rings.

$\textstyle{\mathcal{O}_{X,x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\simeq$
$\textstyle{\mathcal{O}_{M^{\prime},y^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$étale$\textstyle{\mathcal{O}_{M,y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$=$
$\subset$$\textstyle{\widehat{\mathcal{O}}_{M,y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$=$
$\textstyle{\mathcal{O}_{\overline{M},\overline{y}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$
$\subset$$\textstyle{\widehat{\mathcal{O}}_{\overline{M},\overline{y}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$ $\simeq$
$\textstyle{k[[x_{1},\ldots,x_{N}]]^{G}}$$\textstyle{\mathcal{O}_{\overline{M},\overline{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\subset$$\textstyle{\widehat{\mathcal{O}}_{\overline{M},\overline{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\simeq$
$\textstyle{k[[x_{1},\ldots,x_{N}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\subset$

We denote by $(\overline{N},\overline{y})\subset(\overline{M},\overline{y})$
the germ defined by the images of $h_{1},\ldots,h_{c}\in\mathfrak{m}_{X,x}$ in
$\mathcal{O}_{\overline{M},\overline{y}}$. Let $f_{1},\ldots,f_{c}\in
k[[x_{1},\ldots,x_{N}]]^{G}$ be the images of
$h_{1},\ldots,h_{c}\in\mathfrak{m}_{X,x}$. Then we have an isomorphism

$\widehat{\mathcal{O}}_{\overline{N},\overline{y}}\simeq
k[[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c}).$

We set

$A:=\operatorname{Spec}k[[x_{1},\ldots,x_{N}]]^{G},\quad
B:=\operatorname{Spec}\bigl{(}k[[x_{1},\ldots,x_{N}]]^{G}/(f_{1},\ldots,f_{c})\bigr{)},$

and $x^{\prime}\in A$ the origin. Let
$\mathfrak{a}^{\prime}\subset\mathcal{O}_{A}$ and
$\mathfrak{b}^{\prime}\subset\mathcal{O}_{B}$ be the ideal sheaves
corresponding to $\mathfrak{a}\subset\mathcal{O}_{X}$ and
$\mathfrak{b}\subset\mathcal{O}_{Y}$. Since

$\operatorname{mld}_{x}\bigl{(}X,(h_{1}\cdots
h_{c})\mathfrak{a}^{\delta}\bigr{)}=\operatorname{mld}_{x^{\prime}}\bigl{(}A,(f_{1}\cdots
f_{c})\mathfrak{a}^{\prime\delta}\bigr{)},\quad\operatorname{mld}_{x}(Y,\mathfrak{b}^{\delta})=\operatorname{mld}_{x^{\prime}}(B,\mathfrak{b}^{\prime\delta})$

hold (cf. Remark 3.1), it is sufficient to show that

$\operatorname{mld}_{x^{\prime}}\bigl{(}A,(f_{1}\cdots
f_{c})\mathfrak{a}^{\prime\delta}\bigr{)}=\operatorname{mld}_{x^{\prime}}(B,\mathfrak{b}^{\prime\delta}).$

Note that $G$, $A$ and $B$ satisfy all assumptions in Section 7. Therefore we
take over all notation. Since $Y$ is klt at $x$, so is $B$. Since
$\overline{B}\to B$ is étale in codimension one, it follows that the germ
$(\overline{N},\overline{y})$ is also klt. This fact will be used in the proof
of Claim 8.3.

($\diamondsuit$) is proved for $k$-varieties in [NS, Theorem 5.1]. The key
ingredients of the proof of [NS, Theorem 5.1] are Corollary 4.9, Lemma 2.34
and Claim 5.2 in [NS]. First, Corollary 4.9 in [NS] can be substituted by
Theorem 7.9 in the formal power series ring setting. Second, Lemma 2.34 in
[NS] is still true in our setting by replacing $k[t][x_{1},\ldots,x_{N}]$ with
$k[x_{1},\ldots,x_{N}][[t]]$ (cf. Remark 5.42). On the other hand, the proof
of Claim 5.2 in [NS] does not work directly because they use the result on the
rational connectedness proved by Hacon and McKernan [HM07], which is not clear
for the formal power series ring setting. We also note the lack of [NS, Lemma
2.27] in our setting. It will be substituted by Proposition 5.23.

In what follows, we shall only prove Claim 5.2 in [NS] in the formal power
series ring setting.

######  Claim 8.3 (cf. [NS, Claim 5.2]).

Let $C\subset\overline{A}^{(\gamma)}_{\infty}$ be a cylinder that is the
intersection of finitely many cylinders of the form
$\operatorname{Cont}^{\geq\ell}\bigl{(}\mathfrak{c}\mathcal{O}_{\overline{A}^{(\gamma)}}\bigr{)}$,
where $\mathfrak{c}\subset\mathcal{O}_{A}$ is an ideal sheaf on $A$ and $\ell$
is a non-negative integer. Let $C^{\prime}$ be an irreducible component of
$C$. Then $C^{\prime}\cap\overline{B}^{(\gamma)}_{\infty}$ contains a $k$-arc
$\delta$ such that
$\operatorname{ord}_{\delta}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{(\gamma)}/k[[t]]}\bigr{)}<\infty$.

###### Proof of Claim 8.3.

First, we introduce a $k$-action on the arc space
$\overline{A}^{(\gamma)}_{\infty}$ as follows. For a $k$-arc
$\alpha\in\overline{A}^{(\gamma)}_{\infty}$, we denote
$g_{i}^{\alpha}:=\alpha^{*}(x_{i})\in k[[t]]$, where
$\alpha^{*}:k[x_{1},\ldots,x_{N}][[t]]\to k[[t]]$ is the corresponding
$k[t]$-ring homomorphism. Let $\alpha\in\overline{A}^{(\gamma)}_{\infty}$ and
$a\in k$. Then we define $a\cdot\alpha\in\overline{A}^{(\gamma)}_{\infty}$ by

$g_{i}^{a\cdot\alpha}(t):=a^{e_{i}}g_{i}^{\alpha}(a^{d}t).$

Then for $f\in k[[x_{1},\ldots,x_{N}]]^{G}$, we have $v(t)=u(a^{d}t)$ for

$u(t):=\alpha^{*}\bigl{(}\overline{\lambda}^{*}_{\gamma}(f)\bigr{)},\quad
v(t):=(a\cdot\alpha)^{*}\bigl{(}\overline{\lambda}^{*}_{\gamma}(f)\bigr{)}\in
k[[t]].$

Therefore, we have

$\operatorname{ord}_{\alpha}\bigl{(}\overline{\lambda}^{*}_{\gamma}(f)\bigr{)}=\operatorname{ord}_{a\cdot\alpha}\bigl{(}\overline{\lambda}^{*}_{\gamma}(f)\bigr{)}$

if $a\in k^{\times}$. Hence, any cylinder of the form
$\operatorname{Cont}^{\geq\ell}\bigl{(}\mathfrak{c}\mathcal{O}_{\overline{A}^{(\gamma)}}\bigr{)}$
with an ideal $\mathfrak{c}\subset\mathcal{O}_{A}=k[[x_{1},\ldots,x_{N}]]^{G}$
is invariant under the $k$-action. Therefore, $C$ in the statement and its
irreducible component $C^{\prime}$ are also invariant under the $k$-action.

We denote by $\beta\in\overline{A}^{(\gamma)}_{\infty}$ the trivial arc
determined by $g_{i}^{\beta}=0$ for each $i$. Note that $\beta=0\cdot\alpha$
holds for any $k$-arc $\alpha\in\overline{A}^{(\gamma)}_{\infty}$. Therefore,
we have $\beta\in C^{\prime}$ and hence

$\beta\in C^{\prime}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{(\gamma)}}\bigr{)}\not=\emptyset.$

By Lemma 7.13(1)(2), there exists a cylinder
$D\subset\overline{B}^{\prime(\gamma)}_{\infty}$ that is isomorphic to
$C^{\prime}\cap\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{(\gamma)}}\bigr{)}$ under the map in
Lemma 7.13(2). Then by Lemma 7.13(3), it is sufficient to show that $D$
contains a $k$-arc $\delta$ such that
$\operatorname{ord}_{\delta}\bigl{(}\operatorname{Jac}^{\prime}_{\overline{B}^{\prime(\gamma)}/k[t]}\bigr{)}<\infty$.
Therefore, Claim 8.3 follows from Claim 8.4 below and Lemma 7.14. ∎

######  Claim 8.4.

Let $D\subset\overline{B}^{\prime(\gamma)}_{\infty}$ be a cylinder contained
in $\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}$. If $D$
contains the trivial arc $\beta$, then $D$ is not a thin subset of
$\overline{B}^{\prime(\gamma)}_{\infty}$.

###### Proof of Claim 8.4.

Let $T\subset\overline{B}^{\prime(\gamma)}$ be the closed subscheme defined by
$\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}$. First, we prove the following
claim.

  * ($\spadesuit$)

$\overline{B}^{\prime(\gamma)}$ has a desingularization
$r:W\to\overline{B}^{\prime(\gamma)}$ with the following conditions.

    1. (1)

For each $a\in k^{\times}$, the closed subscheme $W_{a}\subset W$ defined by
the ideal $(t-a)\mathcal{O}_{W}\subset\mathcal{O}_{W}$ is an integral regular
scheme with $\dim^{\prime}W_{a}=n$.

    2. (2)

$r|_{T^{\prime}}:T^{\prime}\to T$ has a section, where
$T^{\prime}:=r^{-1}(T)$.

We set

$F_{i}:=\overline{\lambda}^{\prime*}_{\gamma}(f_{i})=f_{i}\bigl{(}t^{\frac{e_{1}}{d}}x_{1},\ldots,t^{\frac{e_{N}}{d}}x_{N}\bigr{)}.$

Then, we have the following natural morphisms.

$\textstyle{V_{1}:=\overline{B}^{\prime(\gamma)}=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\bigr{)}}$$\textstyle{V_{2}:=\operatorname{Spec}\bigl{(}k[t,t^{-1}][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{3}:=\operatorname{Spec}\bigl{(}k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[x_{i}\mapsto
t^{-e_{i}/d}x_{i}]}$$\textstyle{V_{4}:=\operatorname{Spec}\bigl{(}\mathcal{O}_{\overline{N},\overline{y}}[t^{1/d},t^{-1/d}]\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{5}:=\operatorname{Spec}\bigl{(}\mathcal{O}_{\overline{N},\overline{y}}\bigr{)}}$

Note that these four morphisms are regular morphisms (cf. Lemma 7.11(1)).
Hence by the functorial desingularization by Temkin [Tem12, Theorem 1.2.1],
there exist desingularizations $r_{i}:W_{i}\to V_{i}$ with the following
Cartesian diagram.

$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{1}}$$\textstyle{V_{1}:=\operatorname{Spec}\bigl{(}k[t][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\bigr{)}}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{2}}$$\textstyle{V_{2}:=\operatorname{Spec}\bigl{(}k[t,t^{-1}][[x_{1},\ldots,x_{N}]]/(F_{1},\ldots,F_{c})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{3}}$$\textstyle{V_{3}:=\operatorname{Spec}\bigl{(}k[t^{1/d},t^{-1/d}][[x_{1},\ldots,x_{N}]]/(f_{1},\ldots,f_{c})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{4}\simeq
W_{5}\times(\mathbb{A}^{1}_{k}\setminus\\{0\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{4}}$$\textstyle{V_{4}:=\operatorname{Spec}\bigl{(}\mathcal{O}_{\overline{N},\overline{y}}[t^{1/d},t^{-1/d}]\bigr{)}=V_{5}\times(\mathbb{A}^{1}_{k}\setminus\\{0\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{5}}$$\textstyle{V_{5}:=\operatorname{Spec}\bigl{(}\mathcal{O}_{\overline{N},\overline{y}}\bigr{)}}$

We shall prove that $r_{1}$ satisfies the conditions (1) and (2) in
($\spadesuit$).

We shall prove (1). For each $i\in\\{1,2\\}$ and $a\in k^{\times}$, we denote
by $(W_{i})_{a}\subset W_{i}$ and $(V_{i})_{a}\subset V_{i}$ the closed
subschemes defined by $(t-a)\mathcal{O}_{W_{i}}\subset\mathcal{O}_{W_{i}}$ and
$(t-a)\mathcal{O}_{V_{i}}\subset\mathcal{O}_{V_{i}}$, respectively. Similarly,
for each $i\in\\{3,4\\}$ and $a\in k^{\times}$, we denote by
$(W_{i})_{a}\subset W_{i}$ and $(V_{i})_{a}\subset V_{i}$ the closed
subschemes defined by
$(t^{1/d}-a)\mathcal{O}_{W_{i}}\subset\mathcal{O}_{W_{i}}$ and
$(t^{1/d}-a)\mathcal{O}_{V_{i}}\subset\mathcal{O}_{V_{i}}$, respectively.
Then, the Cartesian diagram above induces the following Cartesian diagram for
each $a\in k^{\times}$.

$\textstyle{(W_{1})_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V_{1})_{a}}$$\textstyle{(W_{2})_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V_{2})_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{b^{d}=a}(W_{3})_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{b^{d}=a}(V_{3})_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{b^{d}=a}(W_{4})_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup_{b^{d}=a}(V_{4})_{b}}$

Here, by construction, we have $(V_{1})_{a}\simeq(V_{2})_{a}\simeq(V_{3})_{b}$
for any $a,b\in k^{\times}$ with $b^{d}=a$. Therefore, we also have
$(W_{1})_{a}\simeq(W_{2})_{a}\simeq(W_{3})_{b}$. Since $W_{4}\simeq
W_{5}\times(\mathbb{A}^{1}_{k}\setminus\\{0\\})$, $(W_{4})_{b}$ turns out to
be regular for any $b\in k^{\times}$. Since $(W_{3})_{b}\to(W_{4})_{b}$ is a
regular morpshism, $(W_{3})_{b}$ is also regular (cf. [Mat89, Theorem 32.2]).
Note that the morphism $(V_{3})_{b}\to(V_{4})_{b}\simeq V_{5}$ is isomorphic
to the completion map

$\overline{B}=\operatorname{Spec}\bigl{(}\widehat{\mathcal{O}}_{\overline{N},\overline{y}}\bigr{)}\to\operatorname{Spec}\bigl{(}\mathcal{O}_{\overline{N},\overline{y}}\bigr{)}=V_{5}.$

Since $W_{5}\to V_{5}$ is a birational map, so is $(W_{3})_{b}\to(V_{3})_{b}$.
Since $(V_{3})_{b}\simeq\overline{B}$ is integral, so is $(W_{3})_{b}$.
Furthermore, we have
$\dim^{\prime}(W_{3})_{b}=\dim^{\prime}(V_{3})_{b}=\dim(V_{3})_{b}=n$.
Therefore, for any $a\in k^{\times}$, $(W_{1})_{a}$ is an integral regular
scheme with $\dim^{\prime}(W_{1})_{a}=n$.

We shall prove (2). Let $T_{i}\subset V_{i}$ be the closed subschemes defined
by the ideals $(x_{1},\ldots,x_{N})\mathcal{O}_{V_{i}}$ for $i\in\\{1,2,3\\}$
and by the ideal $\mathfrak{m}_{\overline{N},\overline{y}}\mathcal{O}_{V_{4}}$
for $i=4$. Let $T_{i}^{\prime}:=r_{i}^{-1}(T_{i})$. Then, we have the
following Cartesian diagram.

$\textstyle{T^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{1}\simeq\mathbb{A}^{1}_{k}}$$\textstyle{T^{\prime}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\simeq\mathbb{A}^{1}_{k}\setminus\\{0\\}}$$\textstyle{T^{\prime}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\simeq\mathbb{A}^{1}_{k}\setminus\\{0\\}}$$\scriptstyle{\simeq}$$\textstyle{T^{\prime}_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{4}\simeq\mathbb{A}^{1}_{k}\setminus\\{0\\}}$

Since the above diagram forms a Cartesian diagram, any closed fiber of
$T^{\prime}_{2}\to T_{2}$ is isomorphic to some fiber of $T^{\prime}_{4}\to
T_{4}$. Since the germ $(\overline{N},\overline{y})$ is klt, so is $V_{4}$.
Therefore, $T^{\prime}_{4}\to T_{4}$ has rationally connected fibers by [HM07,
Corollary 1.7(1)], and so does $T^{\prime}_{2}\to T_{2}$. Therefore
$T^{\prime}_{2}\to T_{2}$ has a section by [GHS03]. Hence by the properness of
$r_{1}$, the morphism $T^{\prime}_{1}\to T_{1}$ also has a section. We have
proved the claim ($\spadesuit$).

Note that $T_{\infty}=\\{\beta\\}$ consists of only one arc $\beta$. By claim
($\spadesuit$), there exists $\beta^{\prime}\in T^{\prime}_{\infty}\subset
W_{\infty}$ such that $r_{\infty}(\beta^{\prime})=\beta$. Suppose the contrary
that $D$ is a thin subset of $\overline{B}^{\prime(\gamma)}_{\infty}$. Since
$\beta^{\prime}\in r^{-1}_{\infty}(D)$ satisfies
$\operatorname{ord}_{\beta^{\prime}}(\mathfrak{o}_{W})=\infty$, to get a
contradiction by Proposition 5.23, it is sufficient to show that
$r^{-1}_{\infty}(D)$ is also a thin subset of $W_{\infty}$.

By Lemma 7.11(5), there exists the unique irreducible component $U$ of
$\overline{B}^{\prime(\gamma)}$ of the form (2) in Lemma 5.25. We also note
that $U^{\prime}_{\infty}\cap D=\emptyset$ holds for any other irreducible
component $U^{\prime}$ of $\overline{B}^{\prime(\gamma)}$ since
$D\subset\operatorname{Cont}^{\geq
1}\bigl{(}\mathfrak{o}_{\overline{B}^{\prime(\gamma)}}\bigr{)}$ (cf. Remark
5.26). Therefore, since $D$ is a thin set, there exists a closed subscheme
$F\subset U$ such that $D\subset F_{\infty}$ and $\dim F\leq n$. Let
$W^{\prime}$ be the connected component of $W$ that dominates $U$. We note
that $\dim W^{\prime}=\dim U=n+1$ by Lemma 7.11(5). We set
$F^{\prime}:=r^{-1}(F)\cap W^{\prime}$. Then we have
$r^{-1}_{\infty}(D)\subset F^{\prime}_{\infty}$ and $\dim F^{\prime}\leq n$,
and hence $r^{-1}_{\infty}(D)$ is also a thin set. We complete the proof of
Claim 8.4. ∎

We complete the proof of Theorem 8.2. ∎

######  Remark 8.5.

Theorem 8.2 can be generalized to $\mathbb{R}$-ideals due to Remark 7.10. We
have

$\operatorname{mld}_{x}\bigl{(}X,(h_{1}\cdots
h_{c})\mathfrak{a}\bigr{)}=\operatorname{mld}_{x}(Y,\mathfrak{b}).$

for an $\mathbb{R}$-ideal $\mathfrak{a}$ on $X$ and
$\mathfrak{b}:=\mathfrak{a}\mathcal{O}_{Y}$.

##  9\. Proof of the main theorems

As a corollary of Theorem 8.2, we prove the PIA conjecture for quotients of
locally complete intersection singularities.

######  Corollary 9.1.

Suppose that a variety $X$ has a quotient singularity at a closed point $x\in
X$. Let $Y$ be a subvariety of $X$ of codimension $c$ that is locally defined
by $c$ equations at $x$. Suppose that $Y$ is klt at $x$. Let $\mathfrak{a}$ be
an $\mathbb{R}$-ideal sheaf on $Y$. Let $D$ be a prime divisor on $Y$ through
$x$ that is klt and Cartier at $x$. Suppose that $D$ is not contained in the
cosupport of the $\mathbb{R}$-ideal sheaf $\mathfrak{a}$. Then it follows that

$\operatorname{mld}_{x}\bigl{(}Y,\mathfrak{a}\mathcal{O}_{Y}(-D)\bigr{)}=\operatorname{mld}_{x}(D,\mathfrak{a}\mathcal{O}_{D}).$

###### Proof.

Take an $\mathbb{R}$-ideal sheaf $\mathfrak{b}$ on $X$ such that
$\mathfrak{a}=\mathfrak{b}\mathcal{O}_{Y}$, and take local equations
$h_{1},\ldots,h_{c}\in\mathcal{O}_{X,x}$ of $Y$ in $X$ at $x$. Furthermore,
take $g\in\mathcal{O}_{X,x}$ such that its image
$\overline{g}\in\mathcal{O}_{Y,x}$ defines $D$. Then we have

$\operatorname{mld}_{x}\bigl{(}Y,\mathfrak{a}\mathcal{O}_{Y}(-D)\bigr{)}=\operatorname{mld}_{x}\bigl{(}X,(h_{1}\cdots
h_{c}\cdot
g)\mathfrak{b}\bigr{)}=\operatorname{mld}_{x}(D,\mathfrak{a}\mathcal{O}_{D})$

by applying Theorem 8.2 twice. ∎

######  Theorem 9.2.

Let $X$ be a variety with only quotient singularities. Let $Y$ be a klt
subvariety of $X$ of codimension $c$ that is locally defined by $c$ equations
in $X$. Let $\mathfrak{a}$ be an $\mathbb{R}$-ideal sheaf on $Y$. Then the
function

$|Y|\to\mathbb{R}_{\geq 0}\cup\\{-\infty\\};\quad
x\mapsto\operatorname{mld}_{x}(Y,\mathfrak{a})$

is lower semi-continuous, where we denote by $|Y|$ the set of all closed
points of $Y$ with the Zariski topology.

###### Proof.

We take over the notation in the proof of Corollary 9.1. Then we have

$\operatorname{mld}_{x}(Y,\mathfrak{a})=\operatorname{mld}_{x}\bigl{(}X,(h_{1}\cdots
h_{c})\mathfrak{b}\bigr{)}$

by Theorem 8.2. Then the assertion follows from the fact proved in [Nak16,
Corollary 1.3] that the lower semi-continuity holds for the variety $X$ with
only quotient singularities. ∎

## References