Bounds of the multiplicity of abelian quotient complete intersection singularities

TL;DR

This paper studies the multiplicities and log canonical thresholds of abelian quotient complete intersection singularities, providing bounds and insights into their algebraic properties.

Contribution

It introduces bounds for the multiplicity of abelian quotient complete intersection singularities based on their special data.

Findings

Established bounds for the multiplicity of these singularities.

Analyzed the relationship between multiplicity and log canonical thresholds.

Provided new insights into the structure of abelian quotient complete intersection singularities.

Abstract

In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.

Equations

$$e(R_{{\mathfrak{m}}^G})\le 2^{n-1},$$

$$e({\mathcal{O}}_{X,x})\le 2^{n-1}$$

$$e({\mathcal{O}}_{X,x})\le 2^{n-\lceil \mathrm{lct}({\mathfrak{m}}_x)\rceil }$$

$$e(R_{{\mathfrak{m}}^G})\le 2^{n-\lceil \mathrm{lct}({\mathfrak{m}}^G)\rceil }$$

$$R_{\mathbb{D}}:=\mathbb{C}[x_J^{w(J)}|J\in D],\mathrm{where}{x}_J=\prod _{j\in J}{x}_j,$$

$${\mathfrak{m}}_{\mathbb{D}}:=(x_J^{w(J)}|J\in D)\text{mbox}and$$

$$G_{\mathbb{D}}:=\Braket{E,\Set{(\zeta_{w},\zeta_{w}^{-1};i,j)}{\begin{array}[]{l}\text{$J_{1},J_{2},J\in D,i\in J_{1},j\in J_{2},J_{1}\sqsubset J,$}\\ \text{$J_{2}\sqsubset J$ and $w=w(J_{1})=w(J_{2})$}\end{array}}}{.}$$

$$\mathbb{C}[x_1,\dots ,x_n{]}^{TG{T}^{-1}}=R_{\mathbb{D}}\text{mbox}andTG{T}^{-1}=G_{\mathbb{D}}.$$

$$\mathrm{emb}(R_{\mathbb{D}})=|\{J\in D\}|=n+|\{J\in D||J|\ge 2\}|.$$

$$R_{\mathbb{D}}=\mathbb{C}[x_1^a,\dots ,x_n^a,x_1\cdots x_n],$$

$$G_{\mathbb{D}}=⟨({\zeta }_a,{\zeta }_a^{-1};1,2),\dots ,({\zeta }_a,{\zeta }_a^{-1};1,n)⟩$$

$$①\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$R_{\mathbb{D}}=\mathbb{C}[x_1^a,x_2^a,x_3^b,x_4^b,x_1{x}_2,x_3{x}_4],$$

$$G_{\mathbb{D}}=⟨({\zeta }_a,{\zeta }_a^{-1};1,2),({\zeta }_b,{\zeta }_b^{-1};3,4)⟩$$

$$ⓐ$$

$$w^{\prime }(J^{\prime })=\{\begin{array}{ll}w(J^{\prime })/w(J_i)& \text{if J′⊂Ji,}\\ w(J^{\prime })& \text{if J′⊈J.}\end{array}$$

$$v(J^{\prime })=\{\begin{array}{ll}w(J^{\prime })/w(J_i)& \text{if J′⊂Ji,}\\ w(J^{\prime })& \text{if J′⊈J.}\end{array}$$

$$w^a(J^{\prime })=\{\begin{array}{ll}1& \text{if J′=J,}\\ aw(J^{\prime })& \text{if J′∈D∖J.}\end{array}$$

$$v^b(J^{\prime })=\{\begin{array}{ll}1& \text{if J′=Ji,}\\ bv(J^{\prime })& \text{if J′∈D∖{J,J1,…,Jm}.}\end{array}$$

$$|G_{(\mathbb{D}\setminus J{)}^b}|=b^{n-m}|G_{\mathbb{D}\setminus J}|.$$

$$G_{\mathbb{D}}=\Braket{\Set{(\zeta_{w(J_{1})},\zeta_{w(J_{1})^{-1}};i_{1},i_{j})}{\begin{array}[]{l}\text{$j=2,\dots,m$}\end{array}},G_{({\mathbb{D}\setminus J})^{w(J_{1})}}}{\ }\mbox{and}$$

$$G_{\mathbb{D}^{a}}=\Braket{\Set{(\zeta_{aw(J_{1})},\zeta_{aw(J_{1})^{-1}};i_{1},i_{j})}{\begin{array}[]{l}\text{$j=2,\dots,m$}\end{array}},G_{({\mathbb{D}\setminus J})^{aw(J_{1})}}}{.}$$

$$|G_{{\mathbb{D}}^a}|=(aw(J_1){)}^{m-1}|G_{(\mathbb{D}\setminus J{)}^{aw(J_1)}}|=a^{n-1}w(J_1{)}^{m-1}|G_{(\mathbb{D}\setminus J{)}^{w(J_1)}}|=a^{n-1}|G_{\mathbb{D}}|.$$

$$|G_{\mathbb{D}}|=w(J^{\prime }{)}^{n-1}|G_{\mathbb{D}\setminus J}|.$$

$$v^{w(J_1)}(J^{\prime })=\{\begin{array}{ll}1& \text{if J′=Ji,}\\ w(J_1)v(J^{\prime })& \text{if J′∈D∖{J,J1,…,Jm}.}\end{array}$$

$$G_{\mathbb{D}}=\Braket{\Set{(\zeta_{w(J_{1})},\zeta_{w(J_{1})^{-1}};i_{1},i_{j})}{\begin{array}[]{l}\text{$j=2,\dots,m$}\end{array}},G_{({\mathbb{D}\setminus J})^{w(J_{1})}}}{.}$$

$$|G_{\mathbb{D}}|={w(J_1)}^{m-1}|G_{(\mathbb{D}\setminus J{)}^{w(J_1)}}|={w(J_1)}^{n-1}|G_{\mathbb{D}\setminus J}|.$$

$$K_Y-f^{\ast }{K}_X-tF=\sum _{i}a(E_i;X,{\mathfrak{a}}^t)E_i.$$

$$\mathrm{lct}(X,\mathfrak{a})=\sup \{t\in {\mathbb{R}}_{\ge 0}|\mathrm{the}\mathrm{pair}(X,{\mathfrak{a}}^t)\mathrm{is}\log \mathrm{canonical}\}.$$

$$\mathcal{J}(X,{\mathfrak{a}}^t)=f_{\ast }(\lceil \sum _{i}a(E_i;X,{\mathfrak{a}}^t)E_i\rceil ).$$

Full text

1. Introduction

In [8], Watanabe classified all abelian quotient complete intersection singularities. Watanabe defined a special datum (see Section 2 for detailed definitions) in order to classify abelian quotient complete intersection singularities. Using a special datum, he gave an upper bound of the multiplicity of abelian quotient complete intersection singularities.

Theorem 1.1**.**

$\mathrm{(}$Proposition 3.1 in [8]$\mathrm{)}$* Let $G$ be a finite abelian subgroup of $\mathrm{SL}(n,\mathbb{C})$. If $R=\mathbb{C}[x_{1},\dots,x_{n}]^{G}$ is a complete intersection, then*

[TABLE]

where $\mathfrak{m}^{G}=(x_{1},\dots,x_{n})\cap R$ and $e(R_{\mathfrak{m}^{G}})$ is the Hilbert-Samuel multiplicity of the local ring $R_{\mathfrak{m}^{G}}$.

The following conjecture was posed by Watanabe as a generalization of Theorem 1.1.

Conjecture 1.2**.**

Let $X$ be an n-dimensional variety of locally a complete intersection with canonical singularities. Then

[TABLE]

for a closed point $x$ of $X$.

In [7], this conjecture was refined as follows:

Conjecture 1.3**.**

Let $X$ be an n-dimensional variety of locally a complete intersection with log canonical singularities. Then

[TABLE]

for a closed point $x$ of $X$ and the equality holds if and only if $\mathrm{emb}(X,x)=2n-\lceil\mathrm{lct}(\mathfrak{m}_{x})\rceil$, where $\mathrm{emb}(X,x)$ is the embedding dimension of $X$ at $x$ and $\mathrm{lct}(\mathfrak{m}_{x})$ is the log canonical threshold of $\mathfrak{m}_{x}$.

In [7], the author gave upper bounds of the multiplicity by functions of the log canonical threshold for locally a complete intersection singularity. As an application, we obtained the affirmative answer to the conjecture if the dimension of a variety is less than or equal to $32$ and the variety has canonical singularities.

Theorem 1.4**.**

$\mathrm{(}$Theorem 5.6 in [7]$\mathrm{)}$* Let $X$ be an $n$-dimensional variety of locally a complete intersection with canonical singularities. If $n\leq 32$, then Conjecture 1.3 holds. In particular, if $n\leq 32$, then Conjecture 1.2 holds.*

In this paper, we study the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. We give an affirmative answer to Conjecture 1.3 for abelian quotient complete intersection singularities using Watanabe’s classification.

Theorem 1.5**.**

Let $G$ be a finite abelian subgroup of $\mathrm{SL}(n,\mathbb{C})$. If $R=\mathbb{C}[x_{1},\dots,x_{n}]^{G}$ is a complete intersection, then

[TABLE]

and the equality holds if and only if $\mathrm{emb}(R_{\mathfrak{m}^{G}})=2n-\lceil\mathrm{lct}(\mathfrak{m}^{G})\rceil$, where $\mathfrak{m}^{G}=(x_{1},\dots,x_{n})\cap R$ and $\mathrm{emb}(R_{\mathfrak{m}^{G}})$ is the embedding dimension of the local ring $R_{\mathfrak{m}^{G}}$.

Moreover we give a lower bound of the multiplicity of abelian quotient complete intersection singularities using the special datum and the log canonical threshold.

The paper is organized as follows: In Section $2$, we introduce the definition of a special datum and show some basic properties of abelian quotient complete intersection singularities in term of the special datum. In Section $3$, we prepare some useful and important propositions. They will play a crucial role in this paper. In Section $4$, we give an upper bound of the multiplicity of abelian quotient complete intersection singularities. In Section $5$, we give a lower bound of the multiplicity of abelian quotient complete intersection singularities.

2. abelian quotient complete intersection singularities

In this section, we recall the definition of a special datum and study the properties of abelian quotient complete intersection singularities in term of the special datum.

Definition 2.1**.**

Let $n\geq 1$ be an integer. An $n$-dimensional special datum $\mathbb{D}=(D,w)$ is a pair consisting of a set of non-empty subsets of $\{1,2,\dots,n\}$ (i.e. $D\subset 2^{\{1,2,\dots,n\}}\setminus\emptyset$), together with a function $w:D\to\mathbb{N}$ such that:

1. (1)

For each $i\in\{1,2,\dots,n\}$ we have $\{i\}\in D$. 2. (2)

For every pair of $J,J^{\prime}\in D$, either $J\subset J^{\prime}$, $J^{\prime}\subset J$ or $J\cap J^{\prime}=\emptyset$. 3. (3)

If $J$ is a maximal element with respect to the inclusion relation “$\subset$”, then $w(J)=1$. 4. (4)

If $J,J^{\prime}\in D$ and $J\subsetneq J^{\prime}$, then $w(J)>w(J^{\prime})$ and $w(J^{\prime})|w(J)$. 5. (5)

For $J_{1},J_{2},J\in D$ with $J_{1}\sqsubset J,J_{2}\sqsubset J$, we have $w(J_{1})=w(J_{2})$. (We write $J\sqsubset J^{\prime}$ if $J\subsetneq J^{\prime}$ and if there is no element of $D$ between $J$ and $J^{\prime}$.)

Definition 2.2**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. We define the ring $R_{\mathbb{D}}$ with the maximal ideal $\mathfrak{m}_{\mathbb{D}}$ and the group $G_{\mathbb{D}}$ by

[TABLE]

[TABLE]

[TABLE]

Here $E$ denotes the identity matrix in $\mathrm{SL}(n,\mathbb{C})$ and $\zeta_{w}$ denotes a primitive $w$-th root of unit and $(a,b;i,j)$ denotes the diagonal matrix in $\mathrm{SL}(n,\mathbb{C})$ whose $(i,i)$ component is $a$ and $(j,j)$ component is $b$ and the other diagonal components are $1$.

Proposition 2.3**.**

$\mathrm{(}$Proposition 1.7 in [8]$\mathrm{)}$* Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then $R_{\mathbb{D}}=\mathbb{C}[x_{1},\dots,x_{n}]^{G_{\mathbb{D}}}$ and $R_{\mathbb{D}}$ is a complete intersection.*

Theorem 2.4**.**

$\mathrm{(}$Main Theorem in [8]$\mathrm{)}$* Let $G$ be a finite abelian subgroup of $\mathrm{SL}(n,\mathbb{C})$ and if $\mathbb{C}[x_{1},\dots,x_{n}]^{G}$ is a complete intersection, then there exist an $n$-dimensional special datum $\mathbb{D}$ and $T\in\mathrm{GL}(n,\mathbb{C})$ such that*

[TABLE]

Notation 2.5**.**

To illustrate a special datum $\mathbb{D}$, we define the graph of $\mathbb{D}=(D,w)$ as follows;

1. (1)

We represent $J\in D$ by a circle and we write the integer $w(J)$ inside it. 2. (2)

If $J\sqsubset J^{\prime}$, we join the corresponding circles by a line segment in such a way that the circle corresponding $J^{\prime}$ lies above that of $J$.

The following proposition is immediate from Definition 2.2.

Proposition 2.6**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then

[TABLE]

Example 2.7**.**

Let $D=\{\{1,\dots,n\},\{1\},\dots,\{n\}\}$ and $w$ be the function from $D$ to $\mathbb{N}$ such that $w(\{1\})=\cdots=w(\{n\})=a$, $w(\{1,\dots,n\})=1$. Then $\mathbb{D}=(D,w)$ is a special datum. By the definitions, we have

[TABLE]

[TABLE]

and the graph of $\mathbb{D}$ is

[TABLE]

Example 2.8**.**

Let $D=\{\{1,2\},\{3,4\},\{1\},\{2\},\{3\},\{4\}\}$ and $w$ be the function from $D$ to $\mathbb{N}$ such that $w(\{1\})=w(\{2\})=a$, $w(\{3\})=w(\{4\})=b$, $w(\{1,2\})=w(\{3,4\})=1$. Then $\mathbb{D}=(D,w)$ is a special datum. By the definitions, we have

[TABLE]

[TABLE]

and the graph of $\mathbb{D}$ is

[TABLE]

Definition 2.9**.**

A special datum $\mathbb{D}=(D,w)$ is said to be connected if $D$ has the unique maximal element.

The following lemma is proved in the proof of Proposition 1.7 in [8]

Lemma 2.10**.**

Let $\mathbb{D}=(D,w)$ be a special datum and $J\in D$ be a maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Let $D^{\prime}=D\setminus J$ and $w^{\prime}:D^{\prime}\to\mathbb{N}$ be the function such that

[TABLE]

Then $\mathbb{D}^{\prime}=(D^{\prime},w^{\prime})$ is a special datum and $R_{\mathbb{D}}\cong R_{{\mathbb{D}}^{\prime}}[Y]/(Y^{w(J_{i})}-x_{J_{1}}\cdots x_{J_{m}})$.

Definition 2.11**.**

Let $\mathbb{D}=(D,w)$ be a special datum and $J\in D$ be a maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Then we define the special datum $\mathbb{D}\setminus J=(D\setminus J,v)$ by

[TABLE]

Definition 2.12**.**

Let $\mathbb{D}=(D,w)$ be a special datum and $J$ be a element of $D$. We define the $|J|$-dimensional special datum $\mathbb{D}_{J}=(D_{J},w_{J})$ by $D_{J}=\{J^{\prime}\in D|J^{\prime}\subset J\}$ and $w_{J}(J^{\prime})=\frac{w(J^{\prime})}{w(J)}$ for $J^{\prime}\in D_{J}$. We call $\mathbb{D}_{J}$ a connected component of $\mathbb{D}$ if $J$ is a maximal element of $D$.

We denote by $e(R_{\mathbb{D}})$ the Hilbert-Samuel multiplicity of the local ring $(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$ and denote by $\mathrm{emb}(R_{\mathbb{D}})$ the embedding dimension of the local ring $(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$. The following proposition is immediate from the definition of a connected component of a special datum.

Proposition 2.13**.**

Let $\mathbb{D}=(D,w)$ be a special datum and $J_{1},\dots,J_{m}$ be the maximal elements of $D$. Then

1. $\mathrm{(1)}$

$R_{\mathbb{D}}=R_{\mathbb{D}_{J_{1}}}\otimes_{\mathbb{C}}\cdots\otimes_{\mathbb{C}}R_{\mathbb{D}_{J_{m}}}.$** 2. $\mathrm{(2)}$

$|G_{\mathbb{D}}|=|G_{\mathbb{D}_{J_{1}}}|\cdots|G_{\mathbb{D}_{J_{m}}}|.$** 3. $\mathrm{(3)}$

$e(R_{\mathbb{D}})=e(R_{\mathbb{D}_{J_{1}}})\cdots e(R_{\mathbb{D}_{J_{m}}})$. 4. $\mathrm{(4)}$

$\mathrm{emb}(R_{\mathbb{D}})=\mathrm{emb}(R_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{emb}(R_{\mathbb{D}_{J_{m}}})$.

Lemma 2.14**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$. Let $a$ be a natural number and $w^{a}:D\to\mathbb{N}$ be the function such that

[TABLE]

Then $\mathbb{D}^{a}=(D,w^{a})$ is a special datum with $|G_{\mathbb{D}^{a}}|=a^{n-1}|G_{\mathbb{D}}|.$

Proof.

It is clear that $\mathbb{D}^{a}$ is a special datum. We prove $|G_{\mathbb{D}^{a}}|=a^{n-1}|G_{\mathbb{D}}|$ by induction on the dimension of $\mathbb{D}=(D,w)$. If the dimension of $R_{\mathbb{D}}$ is $1$, then $\mathbb{D}=\mathbb{D}^{a}$ and $|G_{\mathbb{D}}|=|G_{\mathbb{D}^{a}}|$. Therefore the lemma holds when $\mathrm{dim}R_{\mathbb{D}}=1$. Now suppose that $n\geq 2$ and this lemma holds for any special datum of dimension at most $n-1$.

Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Let $\mathbb{D}\setminus J=(D\setminus J,v)$, $b\in\mathbb{N}$ and $v^{b}:D\setminus J\to\mathbb{N}$ be the function such that

[TABLE]

Let $(\mathbb{D}\setminus J)^{b}=(D\setminus J,v^{b})$. Note that $(\mathbb{D}\setminus J)^{1}=\mathbb{D}\setminus J$ and the dimension of a connected component of $\mathbb{D}\setminus J$ is less than $n$. Therefore by the induction hypothesis and Proposition 2.13, $(\mathbb{D}\setminus J)^{b}$ is a special datum with

[TABLE]

Choose $i_{j}\in J_{j}$ for $j=1,\dots,m$. By the definition of $G_{\mathbb{D}}$, we have

[TABLE]

[TABLE]

Therefore

[TABLE]

∎

Lemma 2.15**.**

Let $n\geq 2$, $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum, $J\in D$ be the maximal element of $D$ and $J^{\prime}$ be a maximal element of $D\setminus J$. Then

[TABLE]

Proof.

Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Note that $w(J^{\prime})=w(J_{1})=\cdots=w(J_{m})$. Let $v^{w(J_{1})}:D\setminus J\to\mathbb{N}$ be the function such that

[TABLE]

Let $(\mathbb{D}\setminus J)^{w(J_{1})}=(D\setminus J,v^{w(J_{1})})$. By Lemma 2.14, $(\mathbb{D}\setminus J)^{w(J_{1})}$ is a special datum with $|G_{(\mathbb{D}\setminus J)^{w(J_{1})}}|={w(J_{1})}^{n-m}|G_{{\mathbb{D}\setminus J}}|.$ Choose $i_{j}\in J_{j}$ for $j=1,\dots,m$. By the definition of $G_{\mathbb{D}}$, we have

[TABLE]

Therefore

[TABLE]

∎

We recall the definitions of singularities of pairs and the log canonical threshold. Let $X$ be a $\mathbb{Q}$-Gorenstein normal variety over $\mathbb{C}$, $\mathfrak{a}\subset\mathcal{O}_{X}$ an ideal sheaf and $t\geq 0$ a real number. Let $f:Y\to X$ be a resolution of singularities such that the ideal sheaf $\mathfrak{a}\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ is invertible and $\mathrm{Supp}F\cup\mathrm{Exc}(f)$ is a simple normal crossing divisor, where $\mathrm{Exc}(f)$ is the exceptional locus of $f$. Let $K_{X}$ and $K_{Y}$ denote the canonical divisors of $X$ and $Y$, respectively. Then there are finitely many irreducible (not necessarily exceptional) divisors $E_{i}$ on $Y$ and real numbers $a(E_{i};X,\mathfrak{a}^{t})$ so that there exists an $\mathbb{Q}$-linear equivalence of $\mathbb{R}$-divisors

[TABLE]

Definition 2.16**.**

Under the notation as above:

1. (1)

We say that the pair $(X,\mathfrak{a}^{t})$ is canonical if $a(E_{i};X,\mathfrak{a}^{t})\geq 0$ for all exceptional $E_{i}$. 2. (2)

We say that the pair $(X,\mathfrak{a}^{t})$ is log canonical if $a(E_{i};X,\mathfrak{a}^{t})\geq-1$ for all $E_{i}$. 3. (3)

Suppose that $(X,\mathcal{O}_{X})$ is log canonical. Then we define the log canonical threshold of $\mathfrak{a}$ to be

[TABLE]

If there is not risk of confusion, we shall simply write $\mathrm{lct}(\mathfrak{a})$ instead of $\mathrm{lct}(X,\mathfrak{a})$ and $\mathrm{lct}(R,\mathfrak{a})$ instead of $\mathrm{lct}(\mathrm{Spec}R,\mathfrak{a})$ for a ring $R$. 4. (4)

We define the multiplier ideal of $\mathfrak{a}$ with coefficient $t$ to be

[TABLE]

Remark 2.17**.**

For a special datum $\mathbb{D}$, $(\mathrm{Spec}R_{\mathbb{D}},\mathcal{O}_{\mathrm{Spec}R_{\mathbb{D}}})$ is canonical by Proposition 5.20 and Corollary 5.24 in [6].

Definition 2.18**.**

Let $\mathfrak{a}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be a monomial ideal. The Newton Polygon $\mathrm{Newt}(\mathfrak{a})$ of $\mathfrak{a}$ is defined to be the covex hull of $\{(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}|\ x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in\mathfrak{a}\}$ in $\mathbb{R}^{n}$.

Howald gave a formula computing the multiplier ideal of a monomial ideal in [4].

Theorem 2.19**.**

$\mathrm{(}$Main Theorem in [4]$\mathrm{)}$* Let $X=\mathrm{Spec}\mathbb{C}[x_{1},\dots,x_{n}]$ and $\mathfrak{a}\subset\mathcal{O}_{X}$ be a monomial ideal. Then for $t\in\mathbb{R}_{>0}$,*

[TABLE]

where $t\mathrm{Newt}(\mathfrak{a})=\{t{\bf a}\in\mathbb{R}^{n}|\ {\bf a}\in\mathrm{Newt}(\mathfrak{a})\}$.

The following corollary is an immediate consequence of the definition of a log canonical threshold and Theorem2.19.

Corollary 2.20**.**

$\mathrm{(}$Example 5 in [4]$\mathrm{)}$* Let $X=\mathrm{Spec}\mathbb{C}[x_{1},\dots,x_{n}]$ and $\mathfrak{a}\subset\mathcal{O}_{X}$ be a monomial ideal. Then*

[TABLE]

Lemma 2.21**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Let $\mathfrak{a}_{\mathbb{D}}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D$. Then

[TABLE]

Proof.

It follows from Proposition 5.20 in [6] that for $t\in\mathbb{R}_{>0}$, the pair $(\mathrm{Spec}R_{\mathbb{D}},\mathfrak{m}_{\mathbb{D}}^{t})$ is log canonical if and only if the pair $(\mathrm{Spec}\mathbb{C}[x_{1},\dots,x_{n}],\mathfrak{a}_{\mathbb{D}}^{t})$ is log canonical. Therefore $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{a}_{{\mathbb{D}}})$. By Corollary 2.20, we have

[TABLE]

∎

Proposition 2.22**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum and $J_{1},\dots,J_{m}$ be the maximal elements of $D$. Then

[TABLE]

Proof.

Let $n_{0}=0$, $n_{i}$ be the dimension of $R_{\mathbb{D}_{J_{i}}}$ for $i=1,\dots,m$ and $N_{i}=n_{0}+\cdots+n_{i}$ for $i=0,\dots,m-1$. Then we may assume that $J_{i}=\{N_{i-1}+1,\dots,N_{i-1}+n_{i}\}$ for $i=1,\dots,m$. Let $\mathfrak{a}_{\mathbb{D}_{J_{i}}}\subset\mathbb{C}[x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D_{J_{i}}$. By Lemma 2.21,

[TABLE]

[TABLE]

Since $\mathfrak{a}_{\mathbb{D}}=\sum_{i}\mathfrak{a}_{\mathbb{D}_{J_{i}}}\mathbb{C}[x_{1},\dots,x_{n}]$, we have $\mathrm{lct}(\mathfrak{a}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{a}_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{lct}(\mathfrak{a}_{\mathbb{D}_{J_{m}}}).$ Therefore we have $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{m}}}).$

∎

Proposition 2.23**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum, $J\in D$ be the maximal element of $D$ with $|J|\geq 2$ and $J^{\prime}$ be a maximal element of $D\setminus J$. Then

[TABLE]

Proof.

Let $\mathfrak{a}_{\mathbb{D}}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D$. By Lemma 2.21, we have

[TABLE]

Let $\mathbb{D}\setminus J=(D\setminus J,v)$ and $a=w(J^{\prime})$. Let $\mathfrak{a}_{\mathbb{D}\setminus J}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D\setminus J$ and $\mathfrak{b}_{\mathbb{D}\setminus J}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{v(J)}$ for $J\in D\setminus J$. Then by Corollary 2.20 and Lemma 2.21,

[TABLE]

[TABLE]

Since $\mathrm{Newt}(\mathfrak{a}_{\mathbb{D}\setminus J})=a\mathrm{Newt}(\mathfrak{b}_{\mathbb{D}\setminus J})$, we have

[TABLE]

Since $\mathfrak{a}_{\mathbb{D}}=(x_{1}\cdots x_{n})+\mathfrak{a}_{\mathbb{D}\setminus J}$, we have

[TABLE]

Therefore we have $\mathrm{lct}(\mathfrak{a}_{{\mathbb{D}}})\geq 1$. If $(1,\dots,1)\in t\mathrm{Newt}(\mathfrak{a}_{\mathbb{D}})$ for $t>1$, then $(1,\dots,1)\in t\mathrm{Newt}(\mathfrak{a}_{\mathbb{D}\setminus J})$. This implies that if $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{a}_{{\mathbb{D}}})>1$, then

[TABLE]

Therefore

[TABLE]

∎

Corollary 2.24**.**

Let $\mathbb{D}=(D,w)$ be a special datum. Then

[TABLE]

Proof.

This follows immediately from Proposition 2.22 and Proposition 2.23. ∎

Proposition 2.25**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Then

[TABLE]

Moreover, the following are equvalent:

1. (1)

$\mathrm{emb}(R_{\mathbb{D}})=2n-\lceil\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})\rceil$. 2. (2)

$\mathrm{emb}(R_{{\mathbb{D}_{J_{i}}}})=2\mathrm{dim}R_{{\mathbb{D}_{J_{i}}}}-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{i}}}})\rceil$* for $i=1,\dots,m$ and*

$\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{1}}}})\rceil+\dots+\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{m}}}})\rceil-1=\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$.

Proof.

We prove this by induction on the dimension of $\mathbb{D}=(D,w)$. Note that if the dimension of $R_{\mathbb{D}}$ is $1$, then $R_{\mathbb{D}}=\mathbb{C}[x_{1}]$, $\mathrm{emb}(R_{\mathbb{D}})=1$ and $\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})=1$ and if the dimension of $R_{\mathbb{D}}$ is $2$, then $R_{\mathbb{D}}=\mathbb{C}[x_{1}^{w(1)},x_{2}^{w(2)},x_{1}x_{2}]$, $\mathrm{emb}(R_{\mathbb{D}})=3$ and $\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})=1$ by Proposition 2.23. Therefore the proposition holds when $\mathrm{dim}R_{\mathbb{D}}=2$. Now suppose that $n\geq 3$ and this proposition holds for any special datum of dimension at most $n-1$.

Let $a=w(J_{1})$. Note that the dimension of a connected component of $\mathbb{D}\setminus J$ is less than $n$ and $\mathrm{emb}(R_{\mathbb{D}})=\mathrm{emb}(R_{\mathbb{D}\setminus J})+1$. By the induction hypothesis

[TABLE]

Note that $a\geq 2$, $m\geq 2$, $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})+\cdots+\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{m}}})$ by Proposition 2.22 and \mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{max}\Bigl{\{}1,\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}\Bigl{\}} by Proposition 2.23.

If $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=1$, then

[TABLE]

If $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$, then

[TABLE]

Therefore the proposition holds. ∎

Remark 2.26**.**

Even if a complete intersection singularity $(R,\mathfrak{m})$ is not an invariant ring, the inequality $\mathrm{emb}(R)\leq 2n-\lceil\mathrm{lct}(\mathfrak{m})\rceil$ holds. See Corollary 4.5 and Lemma 5.1 in [7].

Lemma 2.27**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. If

[TABLE]

[TABLE]

then $w(J_{1})=2$ and $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})\rceil-\lceil\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})\rceil=1$.

Proof.

If $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil\geq 3$, then $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil<\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})\rceil+\cdots+\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{m}}})\rceil-1.$ Indeed, by Proposition 2.22

[TABLE]

Therefore we have that $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$ is $1$ or $2$.

Note that $m\geq 2$ and \mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{max}\Bigl{\{}1,\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}\Bigl{\}} by Proposition 2.23.

1. (1)

If $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil=1$, then $m=2,\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})\rceil=\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{2}}})\rceil=1$. Therefore $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{2}}})=1$. By Proposition 2.22, we have $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})=2$. Since $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, we have $w(J_{1})=2$ and $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})\rceil-\lceil\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})\rceil=1$. 2. (2)

If $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil=2$, then $m=2$ or $m=3$.

We assume that $m=2$. Then we may assume that $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})\rceil=1,\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{2}}})\rceil=2$. By Proposition 2.22, we have $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})\leq 3$. Since $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, we have $w(J_{1})=2$ and $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})\rceil-\lceil\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})\rceil=1$.

We assume that $m=3$. Then $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{1}}})\rceil=\cdots=\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}_{J_{3}}})\rceil=1$. By Proposition 2.22, we have $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})=3$. Since $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, we have $w(J_{1})=2$ and $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}\setminus J}})\rceil-\lceil\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})\rceil=1$.

Therefore this lemma holds. ∎

3. relation between $e(R_{\mathbb{D}})$ and $e(R_{\mathbb{D}\setminus J})$

In this section, we investigate the relation between $e(R_{\mathbb{D}})$ and $e(R_{\mathbb{D}\setminus J}).$

Proposition 3.1**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Then

[TABLE]

Moreover, if $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, then

[TABLE]

Proof.

First we prove that $e(R_{\mathbb{D}})\leq w(J_{1})e(R_{\mathbb{D}\setminus J}).$ Let $a=w(J_{1})$. Note that by Lemma 2.10,

[TABLE]

Let $\mathfrak{m}=(x_{J^{\prime}}^{w(J^{\prime})}|J^{\prime}\in D\setminus J)\subset\mathbb{C}[x_{J^{\prime}}^{w(J^{\prime})}|J^{\prime}\in D\setminus J]$ and $I$ be a minimal reduction of the maximal ideal of $\mathbb{C}[x_{J^{\prime}}^{w(J^{\prime})}|J^{\prime}\in D\setminus J]_{\mathfrak{m}}$. Note that $I(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$ is an ${\mathfrak{m}_{\mathbb{D}}}$-primary ideal of $(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$. Then

[TABLE]

The first equality holds by Proposition 11.1.10 in [5].

Next we prove that if $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, then $e(R_{\mathbb{D}})=w(J_{1})e(R_{\mathbb{D}\setminus J}).$ We assume that $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$. Let $\boldsymbol{e}_{i}=(0,\cdots,0,\overset{i}{\check{1}},0,\cdots,0)\in\mathbb{R}^{n}$. By Lemma 2.21, there exist $J^{\prime}_{1},\dots,J^{\prime}_{k}\in D\setminus J$ and $s,t_{1},\dots,t_{k}\in\mathbb{N}$ such that

[TABLE]

[TABLE]

where $\boldsymbol{e}_{J^{\prime}_{i}}=\sum_{j\in J^{\prime}_{i}}\boldsymbol{e}_{j}$. Therefore

[TABLE]

[TABLE]

By the definition of the integral closure of ideals, the integral closure of $I(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}=I\mathbb{C}[x_{J^{\prime}}^{w(J^{\prime})}|J^{\prime}\in D\setminus J][x_{J}]_{\mathfrak{m}_{\mathbb{D}}}$ contains $x_{J}=x_{1}\cdots x_{n}$. This implies that the integral closure of $I(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$ is the maximal ideal of $(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$. Hence $I(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$ is a minimal reduction of the maximal ideal of $(R_{\mathbb{D}})_{\mathfrak{m}_{\mathbb{D}}}$ (see for example Corollary 8.3.6 and Proposition 8.3.7 in [5]). Thus

[TABLE]

(see for example Proposition 11.2.2 in [5]). Therefore by the above discussion, we have $e(R_{\mathbb{D}})=w(J_{1})e(R_{\mathbb{D}\setminus J}).$

∎

Definition 3.2**.**

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $n$ and $I_{1},\dots,I_{n}$ be $\mathfrak{m}$-primary ideals. Let us consider the function $H:\mathbb{Z}_{\geq 0}^{n}\to\mathbb{Z}_{\geq 0}$ given by

[TABLE]

for all $(r_{1},\dots,r_{n})\in\mathbb{Z}_{\geq 0}^{n}$. Then there exists a polynomial $P\in\mathbb{Q}[x_{1},\dots,x_{n}]$ of degree $n$ such that

[TABLE]

for all sufficiently large $r_{1},\dots,r_{n}\in\mathbb{Z}_{\geq 0}$ and the coefficient of the monomial $x_{1}\cdots x_{n}$ in $P$ is an integer. The coefficient of the monomial $x_{1}\cdots x_{n}$ in $P$ is called the mixed multiplicity of $I_{1},\dots,I_{n}$ and is denoted by $e(I_{1},\dots,I_{n})$.

Bivi$\mathrm{\acute{a}}$-Ausina generalized the notion of mixed multiplicities and defined the invariant $\sigma_{R}(I_{1},\dots,I_{n})$ in [1]. We use this invariant in order to prove Proposition 3.8.

Definition 3.3**.**

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $n$ and $I_{1},\dots,I_{n}$ be ideals of $R$. Then we define

[TABLE]

when the number on right-hand side is finite. If the set

[TABLE]

is non-bounded then we set $\sigma_{R}(I_{1},\dots,I_{n})=\infty$.

Let $I_{1},\dots,I_{c}$ be ideals of a Noetherian local ring $(R,\mathfrak{m})$ with $k=R/\mathfrak{m}$ an infinite field. Let us consider a generating system $a_{i1},\dots,a_{is_{i}}$ of $I_{i}$. Let $s=s_{1}+\cdots+s_{c}$. We say that a property holds for sufficiently general elements of $I_{1}\oplus\cdots\oplus I_{c}$ if there exists a non-empty Zariski-open set $U$ in $k^{s}$ such that all elements $(g_{1},\dots,g_{c})\in I_{1}\oplus\cdots\oplus I_{c}$ satisfy the said property provided that $g_{i}=\sum_{1\leq j\leq s_{i}}u_{ij}a_{ij}$, $i=1,\dots,c$, where $(u_{11},\dots,u_{1s_{1}},\dots,u_{c1},\dots,u_{cs_{c}})\in U$.

Proposition 3.4**.**

$\mathrm{(Proposition\ 2.9\ in}$* [1], $\mathrm{See\ Proposition\ 2.2\ in}$ [2]$\mathrm{)}$ Let $I_{1},\dots,I_{n}$ be ideals of an $n$-dimensional Noetherian local ring $(R,\mathfrak{m})$ with $R/\mathfrak{m}$ an infinite field. Then $\sigma_{R}(I_{1},\dots,I_{n})<\infty$ if and only if there exist elements $g_{i}\in I_{i}$ for $i=1,\dots,n$ such that $(g_{1},\dots,g_{n})$ is an $\mathfrak{m}$-primary ideal. In this case, we have that $\sigma_{R}(I_{1},\dots,I_{n})=e(g_{1},\dots,g_{n})$ for sufficiently general elements $(g_{1},\dots,g_{n})\in I_{1}\oplus\cdots\oplus I_{n}$, where $e(g_{1},\dots,g_{n})$ is the Hilbert-Samuel multiplicity of the ideal of $R$ generated by $g_{1},\dots,g_{n}$.*

Lemma 3.5**.**

$\mathrm{(Corollary\ 2.5,Lemma\ 2.6\ in}$* [2]$\mathrm{)}$ Let $(R,\mathfrak{m})$ be an $n$-dimensional Noetherian local ring with $R/\mathfrak{m}$ an infinite field. Let $I_{1},\dots,I_{n}$ be ideals of $R$ with $\sigma_{R}(I_{1},\dots,I_{n})<\infty$. Then*

1. $\mathrm{(1)}$

Let $J_{1},\dots,J_{n}$ be ideals of $R$ such that $J_{i}\subset I_{i}$ for all $i=1,\dots,n$ and $\sigma_{R}(J_{1},\dots,J_{n})<\infty$. Then

[TABLE] 2. $\mathrm{(2)}$

For all $r_{1},\dots,r_{n}\in\mathbb{N}$,

[TABLE]

Lemma 3.6**.**

Let $(R,\mathfrak{m})$ be an $n$-dimensional Noetherian local ring with $R/\mathfrak{m}$ an infinite field. Let $I_{1},\dots,I_{n}$ be ideals of $R$ with $\sigma_{R}(I_{1},\dots,I_{n})<\infty$. Then

[TABLE]

where $\overline{I_{i}}$ is the integral closure of $I_{i}$.

Proof.

This lemma follows from $e(I_{1}+\mathfrak{m}^{r},\dots,I_{n}+\mathfrak{m}^{r})=e(\overline{I_{1}}+\mathfrak{m}^{r},\dots,\overline{I_{n}}+\mathfrak{m}^{r})$ for any $r$ (See Theorem 17.4.9 in [5]). ∎

Lemma 3.7**.**

Let $A=\mathbb{C}[y_{1},\dots,y_{n+c},Y]_{(y_{1},\dots,y_{n+c},Y)}$, $f_{1},\dots,f_{c},f$ be elements of the ideal $(y_{1},\dots,y_{n+c})$ of $A$ and $R=\mathbb{C}[y_{1},\dots,y_{n+c}]/(f_{1},\dots,f_{c})$. Let $a$ be a natural number and $S=R[Y]/(Y^{a}-f)$. We assume that $R$ and $S$ are $n$-dimensional complete intersection rings. Let $\mathfrak{n}=(y_{1},\dots,y_{n+c},Y)$ be the maximal ideal of $A$ and $\mathfrak{m}=(y_{1},\dots,y_{n+c},Y)$ be the maximal ideal of $S_{(y_{1},\dots,y_{n+c},Y)}$. Then

[TABLE]

Proof.

By Proposition 3.4,

[TABLE]

for sufficiently general elements $(sY^{a}+tf,h_{1},\dots,h_{n})\in(Y^{a},f)\oplus\mathfrak{n}\oplus\cdots\oplus\mathfrak{n}$. Fix such $s,t$. We may assume that $s$ and $t$ are nonzero. Then

[TABLE]

for sufficiently general elements $(h_{1},\dots,h_{n})\in\mathfrak{n}\oplus\cdots\oplus\mathfrak{n}$ since $(h_{1},\dots,h_{n})$ is a minimal reduction of the ideal $(y_{1},\dots,y_{n+c},Y)$ of $R[Y]/(sY^{a}+tf)_{(y_{1},\dots,y_{n+c},Y)}$ (see for example Theorem 8.6.6 in [5]). Since $S_{\mathfrak{m}}\cong R[Y]/(sY^{a}+tf)_{(y_{1},\dots,y_{n+c},Y)}$, we have

[TABLE]

∎

Proposition 3.8**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$ with $|J|\geq 2$. Let $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. If $1=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})>\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$, then

[TABLE]

Proof.

Let $a=w(J_{1})$ and $f_{1},\dots,f_{c},f$ be elements of the ideal $(y_{1},\dots,y_{n+c})$ of the polynomial ring $\mathbb{C}[y_{1},\dots,y_{n+c}]$ such that

[TABLE]

[TABLE]

Note that we can choose such $f_{1},\dots,f_{c},f$ by Lemma 2.10. Let $\mathbb{D}\setminus J=(D\setminus J,v)$ and $\boldsymbol{e}_{i}=(0,\cdots,0,\overset{i}{\check{1}},0,\cdots,0)\in\mathbb{R}^{n}$. By Lemma 2.21, there exist $J^{\prime}_{1},\dots,J^{\prime}_{k}\in D\setminus J$ and $s,t_{1},\dots,t_{k},p,q\in\mathbb{N}$ such that

[TABLE]

[TABLE]

[TABLE]

where $\boldsymbol{e}_{J^{\prime}_{i}}=\sum_{j\in J^{\prime}_{i}}\boldsymbol{e}_{j}$. Let $(a,q)=d$, $a=a^{\prime}d$ and $q=q^{\prime}d$. Then we have

[TABLE]

[TABLE]

Let $R=\mathbb{C}[x_{J^{\prime}}^{a^{\prime}pv(J^{\prime})}|J^{\prime}\in D\setminus J][x_{J}^{q^{\prime}}]$ and $\mathfrak{m}_{R}=(x_{J^{\prime}}^{a^{\prime}pv(J^{\prime})}|J^{\prime}\in D\setminus J)+(x_{J}^{q^{\prime}})\subset R$. Let $S=\mathbb{C}[x_{J^{\prime}}^{a^{\prime}pv(J^{\prime})}|J^{\prime}\in D\setminus J]$ and $\mathfrak{m}_{S}=(x_{J^{\prime}}^{a^{\prime}pv(J^{\prime})}|J^{\prime}\in D\setminus J)\subset S$. Note that

[TABLE]

[TABLE]

Let $I$ be a minimal reduction of the maximal ideal of the local ring $S_{\mathfrak{m}_{S}}$. Since ({(x_{1}\cdots x_{n})^{q^{\prime}}})^{sp}=\Bigl{(}(x_{J^{\prime}_{1}}^{a^{\prime}pv(J^{\prime}_{1})})^{t_{1}}\cdots(x_{J^{\prime}_{k}}^{a^{\prime}pv(J^{\prime}_{k})})^{t_{k}}\Bigr{)}^{q}, the integral closure of $IR_{\mathfrak{m}_{R}}$ contains $x_{J}^{q^{\prime}}=(x_{1}\cdots x_{n})^{q^{\prime}}$. This implies that the integral closure of $IR_{\mathfrak{m}_{R}}$ is the maximal ideal of $R_{\mathfrak{m}_{R}}$. Hence $IR_{\mathfrak{m}_{R}}$ is a minimal reduction of the maximal ideal of $R_{\mathfrak{m}_{R}}$. Thus

[TABLE]

Let $A=\mathbb{C}[y_{1},\dots,y_{n+c},Y]_{(y_{1},\dots,y_{n+c},Y)}$ and $\mathfrak{n}$ be the maximal ideal of $A$. By Lemma 3.5, Lemma 3.6 and Lemma 3.7,

[TABLE]

The third equality holds since the integral closure of $(Y^{a},f)^{q^{\prime}}$ is equal to the integral closure of $(Y^{aq^{\prime}},f^{q^{\prime}})$. ∎

4. Upper bound of the multiplicity

In this section, we give an upper bound of the multiplicity of abelian quotient complete intersection singularities.

Definition 4.1**.**

Let $\mathbb{D}=(D,w)$ be a special datum and $J\in D$ with $|J|\geq 2$. We define an invariant $\delta(J)$ by

[TABLE]

We define an invariant $m(D)$ by

[TABLE]

Theorem 4.2**.**

$\mathrm{(}$See proof of Proposition 3.1 in [8]$\mathrm{)}$* Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then*

[TABLE]

Corollary 4.3**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then

[TABLE]

and the equality holds if and only if $\mathrm{emb}(R_{\mathbb{D}})=2n-1$.

Proof.

The inequality follows from Theorem 4.2.

If $\mathrm{emb}(R_{\mathbb{D}})=2n-1$, then $e(R_{\mathbb{D}})\geq 2^{n-1}$ (See Example 12.4.9 in [3]). By Theorem 4.2, we have $e(R_{\mathbb{D}})=2^{n-1}$.

We assume that $e(R_{\mathbb{D}})=2^{n-1}$. By Theorem 4.2, we have $m(D)=2^{n-1}$. Note that

[TABLE]

(See proof of Proposition 3.1 in [8]). Therefore

[TABLE]

This implies that $|\{J\in D||J|\geq 2\}|=n-1$ and $\delta(J)=2$ for $J\in D$ with $|J|\geq 2$. By Proposition 2.6, we have $\mathrm{emb}(R_{\mathbb{D}})=2n-1$. ∎

In order to prove Theorem 4.5, we need the following inequality.

Lemma 4.4**.**

If $a\in\mathbb{N}$, $b\in\mathbb{R}$ and $2\leq a\leq b$, then $a\leq 2^{\lceil b\rceil-\lceil\frac{b}{a}\rceil}$ and the equality holds if and only if $a=2$ and $\lceil b\rceil-\lceil\frac{b}{a}\rceil=1$.

Proof.

Since $a-1\leq\lfloor\frac{b}{a}\rfloor(a-1)\leq\lceil b\rceil-\lceil\frac{b}{a}\rceil$, we have $a\leq 2^{a-1}\leq 2^{\lceil b\rceil-\lceil\frac{b}{a}\rceil}$. By the above inequalities, $a=2^{\lceil b\rceil-\lceil\frac{b}{a}\rceil}$ if and only if $a=2$ and $\lceil b\rceil-\lceil\frac{b}{a}\rceil=1$. ∎

Theorem 4.5**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then

[TABLE]

and the equality holds if and only if $\mathrm{emb}(R_{\mathbb{D}})=2n-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$.

Proof.

We prove this by induction on the dimension of $\mathbb{D}=(D,w)$. If the dimension of $R_{\mathbb{D}}$ is $1$, then $e(R_{\mathbb{D}})=1$ and $\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})=1$. Therefore the theorem holds when $\mathrm{dim}R_{\mathbb{D}}=1$. Now suppose that $n\geq 2$ and the theorem holds for any special datum of dimension at most $n-1$.

We assume that $\mathbb{D}$ is not connected. Note that the dimension of a connected component of $\mathbb{D}$ is less than $n$. Let $J_{1},\dots,J_{m}$ be the maximal elements of $D$ and $n_{i}=\mathrm{dim}R_{\mathbb{D}_{J_{i}}}$. Then we have $e(R_{\mathbb{D}})=e(R_{\mathbb{D}_{J_{1}}})\cdots e(R_{\mathbb{D}_{J_{m}}})$, $\mathrm{emb}(R_{\mathbb{D}})=\mathrm{emb}(R_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{emb}(R_{\mathbb{D}_{J_{m}}})$ and $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{m}}})$ by Proposition 2.13 and Proposition 2.22. By hypothesis, we have for $i=1,\dots,m$

[TABLE]

and the equality holds if and only if $\mathrm{emb}(R_{\mathbb{D}_{J_{i}}})=2n_{i}-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{i}}}})\rceil$. Therefore

[TABLE]

Note that

[TABLE]

by Proposition 2.25. The equality $e(R_{\mathbb{D}})=2^{n-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil}$ holds if and only if $\mathrm{emb}(R_{\mathbb{D}_{J_{i}}})=2n_{i}-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{i}}}})\rceil$ for $i=1,\dots,m$ and $\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{1}}}})\rceil+\dots+\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}_{J_{m}}}})\rceil=\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$ if and only if $\mathrm{emb}(R_{\mathbb{D}})=2n-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$. Therefore the theorem holds if $\mathbb{D}$ is not connected.

We assume that $\mathbb{D}=(D,w)$ is connected. Let $J$ be the maximal element of $D$ and $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Let $a=w(J_{1})$ and $n_{i}=\mathrm{dim}R_{\mathbb{D}_{J_{i}}}$. Note that $w(J_{1})=\cdots=w(J_{m})$ by the definition of a special datum and the dimension of a connected component of $\mathbb{D}\setminus J$ is less than $n$.

If $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$, then by Proposition 3.1

[TABLE]

Therefore by hypothesis, Proposition 2.13 and Lemma 4.4, we have

[TABLE]

By Proposition 2.25, Lemma 2.27 and Lemma 4.4, the equality $e(R_{\mathbb{D}})=2^{n-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil}$ holds if and only if

[TABLE]

[TABLE]

[TABLE]

if and only if $\mathrm{emb}(R_{\mathbb{D}})=2n-\lceil\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\rceil$. Therefore the theorem holds if $\mathbb{D}$ is connected and $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$.

If $1=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})>\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$, then by Corollary 4.3, we have $e(R_{\mathbb{D}})\leq 2^{n-1}$ and the equality holds if and only if $\mathrm{emb}(R_{\mathbb{D}})=2n-1$. Therefore the theorem holds if $\mathbb{D}$ is connected and $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=1$. ∎

Theorem 1.5 follows from Theorem 2.4 and Theorem 4.5.

5. Lower bound of the multiplicity

In this section, we give a lower bound of the multiplicity of abelian quotient complete intersection singularities.

Definition 5.1**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional connected special datum and $J\in D$ be the maximal element of $D$. Then we define

[TABLE]

and

[TABLE]

Proposition 5.2**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. If $R_{\mathbb{D}}$ is a hypersurface (i.e. $\mathrm{emb}(R_{\mathbb{D}})=\mathrm{dim}R_{\mathbb{D}}+1$), then

[TABLE]

Proof.

Let $J_{1},\dots,J_{m}$ be the maximal elements of $D$. By Proposition 2.6 and Proposition 2.13, we may assume that $\mathrm{emb}(R_{\mathbb{D}_{J_{1}}})=\mathrm{dim}R_{\mathbb{D}_{J_{1}}}+1$ and $\mathrm{dim}R_{\mathbb{D}_{J_{i}}}=1$ for $i\geq 2$. Let $J^{\prime}$ be a element of $D$ with $J^{\prime}\sqsubset J_{1}$, $n_{1}=\mathrm{dim}R_{\mathbb{D}_{J_{1}}}$ and $a=w(J^{\prime})$. Then

[TABLE]

by Lemma 2.10. Therefore we have $e(R_{\mathbb{D}_{1}})=\mathrm{min}\{a,n_{1}\}$. Thus $e(R_{\mathbb{D}})=\mathrm{min}\{a,n_{1}\}$ by Proposition 2.13.

On the other hand, by Proposition 2.22, we have $\mathrm{lct}(\mathfrak{m}_{{{\mathbb{D}_{J_{1}}}}\setminus J1})=n_{1}$. Note that $|J|=1$ for any element $J$ of $D$ with $J\neq J_{1}$. Hence

[TABLE]

Therefore $e(R_{\mathbb{D}})=\prod_{J\in D}\alpha(\mathbb{D}_{J}).$ ∎

Proposition 5.3**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Then

[TABLE]

and the equality $e(R_{\mathbb{D}})=\prod_{J\in D}\alpha(\mathbb{D}_{J})$ holds if $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D$.

Proof.

We will prove this by induction on the dimension of $\mathbb{D}=(D,w)$. If the dimension of $R_{\mathbb{D}}$ is $1$, then $e(R_{\mathbb{D}})=\alpha(\mathbb{D})=\beta(\mathbb{D})=1$. Therefore the proposition holds when $\mathrm{dim}R_{\mathbb{D}}=1$. Now suppose that $n\geq 2$ and the proposition holds for any special datum of dimension at most $n-1$.

We assume that $\mathbb{D}$ is not connected. Let $J_{1},\dots,J_{m}$ be the maximal elements of $D$ and $n_{i}=\mathrm{dim}R_{\mathbb{D}_{J_{i}}}$. Then we have $e(R_{\mathbb{D}})=e(R_{\mathbb{D}_{J_{1}}})\cdots e(R_{\mathbb{D}_{J_{m}}})$ by Proposition 2.13. By hypothesis, we have

[TABLE]

Therefore the inequality holds if $\mathbb{D}$ is not connected. We assume that $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D$. Then by hypothesis, we have

[TABLE]

Therefore the proposition holds if $\mathbb{D}$ is not connected.

We assume that $\mathbb{D}$ is connected. Let $J$ be the maximal element of $D$ and $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Then we have $e(R_{\mathbb{D}\setminus J})=e(R_{\mathbb{D}_{J_{1}}})\cdots e(R_{\mathbb{D}_{J_{m}}})$ by Proposition 2.13. Then by Proposition 3.1 and Proposition 3.8, we have

[TABLE]

Note that $\mathbb{D}=\mathbb{D}_{J}$. By hypothesis, we have

[TABLE]

Therefore the inequality holds if $\mathbb{D}$ is connected. We assume that $\alpha(\mathbb{D}_{K})=\beta(\mathbb{D}_{K})$ for every $K\in D$. Then since $\mathrm{lct}(\mathfrak{m}_{{{\mathbb{D}}}\setminus J})\geq{w(J_{1})}$, we have $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{w(J_{1})}$ by Proposition 2.23. By hypothesis and Proposition 3.1,

[TABLE]

Therefore the proposition holds if $\mathbb{D}$ is connected. ∎

In order to prove Proposition 5.5, we need the following inequality.

Lemma 5.4**.**

For positive real numbers $x_{1},\dots,x_{n},c_{1},\dots,c_{n}\in\mathbb{R}_{>0}$,

[TABLE]

Moreover, the equality holds if and only if $\frac{x_{1}}{c_{1}}=\cdots=\frac{x_{n}}{c_{n}}$.

Proof.

Since the logarithm function $f(x)=\mathrm{log}{x}$ is concave on its domain $(0,\infty)$, we have

[TABLE]

and the equality holds if and only if $\frac{x_{1}}{c_{1}}=\cdots=\frac{x_{n}}{c_{n}}$. This implies that

[TABLE]

and the equality holds if and only if $\frac{x_{1}}{c_{1}}=\cdots=\frac{x_{n}}{c_{n}}$. ∎

Proposition 5.5**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Let $\mathfrak{a}_{\mathbb{D}}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D$. Then

[TABLE]

and the equality \prod_{J\in D}\alpha(\mathbb{D}_{J})=\frac{1}{|G_{\mathbb{D}}|}\bigl{(}\frac{n}{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})}\bigr{)}^{n} holds if and only if there is a positive integer $q$ such that the integral closure $\overline{\mathfrak{a}_{\mathbb{D}}}$ of $\mathfrak{a}_{\mathbb{D}}$ is equal to $(x_{1},\dots,x_{n})^{q}$. Moreover, in this case

[TABLE]

and $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D$.

Proof.

We will prove this by induction on the dimension of $\mathbb{D}=(D,w)$. If the dimension of $R_{\mathbb{D}}$ is $1$, then $|G_{\mathbb{D}}|=1$, $\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})=1$, $\alpha(\mathbb{D})=\beta(\mathbb{D})=1$, $w(\{1\})=1$ and $\mathfrak{a}_{\mathbb{D}}=x_{1}$. Therefore the proposition holds when $\mathrm{dim}R_{\mathbb{D}}=1$. Now suppose that $n\geq 2$ and the proposition holds for any special datum of dimension at most $n-1$.

We assume that $\mathbb{D}$ is not connected. Let $J_{1},\dots,J_{m}$ be the maximal elements of $D$ and $n_{i}=\mathrm{dim}R_{\mathbb{D}_{J_{i}}}$. Then we have $e(R_{\mathbb{D}})=e(R_{\mathbb{D}_{J_{1}}})\cdots e(R_{\mathbb{D}_{J_{m}}})$, $|G_{\mathbb{D}}|=|G_{\mathbb{D}_{J_{1}}}|\cdots|G_{\mathbb{D}_{J_{m}}}|$ and $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{1}}})+\cdots+\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{m}}})$ by Proposition 2.13 and Proposition 2.22. By hypothesis and Lemma 5.4, we have

[TABLE]

Therefore the inequality holds if $\mathbb{D}$ is not connected.

Let $n_{0}=0$ and $N_{i}=n_{0}+\cdots+n_{i}$ for $i=0,\dots,m-1$. Then we may assume that $J_{i}=\{N_{i-1}+1,\dots,N_{i-1}+n_{i}\}$ for $i=1,\dots,m$. Let $\mathfrak{a}_{\mathbb{D}_{J_{i}}}\subset\mathbb{C}[x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D_{J_{i}}$.

We assume that

[TABLE]

Then we have for $i=1,\dots,m$

[TABLE]

since the above two inequalities are equalities. By hypothesis, we have for $i=1,\dots,m$, $\overline{\mathfrak{a}_{\mathbb{D}_{i}}}=(x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}})^{w(\{N_{i-1}+1\})},$

[TABLE]

and $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D_{J_{i}}$. Therefore we have $\overline{\mathfrak{a}_{\mathbb{D}}}=(x_{1},\dots,x_{n})^{w(\{1\})}$,

[TABLE]

and $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D$.

We assume that there is a positive integer $q$ such that $\overline{\mathfrak{a}_{\mathbb{D}}}=(x_{1},\dots,x_{n})^{q}$. Then we have $\overline{\mathfrak{a}_{\mathbb{D}_{J_{i}}}}=(x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}})^{q}$. By hypothesis, we have

[TABLE]

[TABLE]

for $i=1,\dots,m$. By Lemma 5.4, we have

[TABLE]

Therefore the proposition holds if $\mathbb{D}$ is not connected.

We assume that $\mathbb{D}$ is connected. Let $J$ be the maximal element of $D$, $J_{1},\dots,J_{m}$ be the elements of $D$ such that $J=J_{1}\cup\dots\cup J_{m}$ and $J_{i}\sqsubset J$ for $i=1,\dots,m$. Let $a=w(J_{1})$.

If $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})=\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$, then $\alpha(\mathbb{D})=a.$ By hypothesis, Lemma 2.15, Proposition 2.22 and Lemma 5.4, we have

[TABLE]

If $1=\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})>\frac{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J})}{a}$, then $\alpha(\mathbb{D})=\mathrm{lct}(\mathfrak{m}_{\mathbb{D}\setminus J}).$ By hypothesis, Lemma 2.15, Proposition 2.22 and Lemma 5.4, we have

[TABLE]

Therefore the inequality holds if $\mathbb{D}$ is connected.

Let $n_{0}=0$, $N_{i}=n_{0}+\cdots+n_{i}$ for $i=0,\dots,m-1$ and $\mathbb{D}\setminus J=(D\setminus J,v)$. Then we may assume that $J_{i}=\{N_{i-1}+1,\dots,N_{i-1}+n_{i}\}$ for $i=1,\dots,m$. Let $\mathfrak{a}_{\mathbb{D}_{J_{i}}}\subset\mathbb{C}[x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}}]$ be the ideal generated by $x_{K}^{v(K)}$ for $K\in D_{J_{i}}$. Note that $v(K)=\frac{w(K)}{a}$.

We assume that

[TABLE]

Then we have for $i=1,\dots,m$

[TABLE]

[TABLE]

by the above discussion. By hypothesis, we have for $i=1,\dots,m$,

[TABLE]

[TABLE]

and $\alpha(\mathbb{D}_{K})=\beta(\mathbb{D}_{K})$ for every $K\in D_{J_{i}}$. Therefore we have $\overline{\mathfrak{a}_{\mathbb{D}}}=(x_{1},\dots,x_{n})^{w(\{1\})}$,

[TABLE]

and $\alpha(\mathbb{D}_{K})=\beta(\mathbb{D}_{K})$ for every $K\in D$.

We assume that there is a positive integer $q$ such that $\overline{\mathfrak{a}_{\mathbb{D}}}=(x_{1},\dots,x_{n})^{q}$. Since ${\mathfrak{a}_{\mathbb{D}}}$ is a monomial ideal,

[TABLE]

(see for example Proposition 1.4.6 in [5]). Therefore we have $q=w(\{1\})=\cdots=w(\{n\})$. We have $\overline{\mathfrak{a}_{\mathbb{D}_{J_{i}}}}=(x_{N_{i-1}+1},\dots,x_{N_{i-1}+n_{i}})^{q/a}$ since $\overline{\mathfrak{a}_{\mathbb{D}}}=(x_{1},\dots,x_{n})^{q}$. By Lemma 2.21, ${\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})}=\frac{n}{q}$ and ${\mathrm{lct}(\mathfrak{m}_{\mathbb{D}_{J_{i}}})}=\frac{n_{i}a}{q}$. These imply that

[TABLE]

Therefore we have $\alpha(\mathbb{D})=a$ as $\mathrm{lct}(\mathfrak{m}_{{\mathbb{D}}})\geq 1$. By hypothesis, we have

[TABLE]

for $i=1,\dots,m$. By Lemma 2.15 and Lemma 5.4, we have

[TABLE]

Therefore the proposition holds if $\mathbb{D}$ is connected.

∎

Theorem 5.6**.**

Let $\mathbb{D}=(D,w)$ be an $n$-dimensional special datum. Let $\mathfrak{a}_{\mathbb{D}}\subset\mathbb{C}[x_{1},\dots,x_{n}]$ be the ideal generated by $x_{J}^{w(J)}$ for $J\in D$. Then

[TABLE]

Moreover,

1. (1)

the equality $e(R_{\mathbb{D}})=\prod_{J\in D}\alpha(\mathbb{D}_{J})$ holds if $\alpha(\mathbb{D}_{J})=\beta(\mathbb{D}_{J})$ for every $J\in D$. 2. (2)

The equality \prod_{J\in D}\alpha(\mathbb{D}_{J})=\frac{1}{|G_{\mathbb{D}}|}\bigl{(}\frac{n}{\mathrm{lct}(\mathfrak{m}_{\mathbb{D}})}\bigr{)}^{n} holds if and only if there is a positive integer $q$ such that the integral closure $\overline{\mathfrak{a}_{\mathbb{D}}}$ of $\mathfrak{a}_{\mathbb{D}}$ is equal to $(x_{1},\dots,x_{n})^{q}$. Furthermore, in this case

[TABLE]

[TABLE]

Proof.

This theorem follows from Proposition 5.3 and Proposition 5.5 ∎

Acknowledgments. The author would like to thank Professor Shihoko Ishii and Professor Shunsuke Takagi for valuable conversations. The author is partially supported by JSPS Grant-in-Aid for Early-Career Scientists 19K14496 and the Iwanami Fujukai Foundation.