Singularities of the Moduli Space of n Unordered Points on the Riemann Sphere

TL;DR

This paper classifies the finite groups linked to singularities in the moduli space of unordered points on the Riemann sphere and provides an algorithmic realization of this classification.

Contribution

It offers a complete classification of finite groups related to these singularities and introduces an algorithm to realize this classification.

Findings

Classification of finite groups associated with singularities

Development of an algorithm for realization

Enhanced understanding of moduli space singularities

Abstract

We classify the finite groups associated to the singularities of the moduli space of $n/ge5$ unordered points on the Riemann sphere. We also realize the classification by an algorithm.

Figures

Equations

$$\mathrm{PSL}(2,\mathbb{C})=\{\text{Mo¨bius transformations}\}$$

$$K_n=\{\mathbf{\lambda }=({\lambda }_1,{\lambda }_2,\cdots {\lambda }_{n-3})\in {\mathbb{C}}^{n-3}|{\lambda }_i\ne 0,1,{\lambda }_i\ne {\lambda }_j,\forall i,j=1,2,\cdots n-3,i\ne j\}$$

$$S=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1^2+x_2^2+x_3^2=1\}$$

$$\{|\alpha |<\infty |{\mathcal{A}}_{\alpha }\simeq A_5\}=\{x+60k|x=0,12,20,30,32,42,50,62,k\in \mathbb{N},x^2+k^2\ne 0\};$$

$$\{|\beta |<\infty |{\mathcal{A}}_{\alpha }\simeq S_4\}=\{x+24k|x=0,6,8,12,14,18,20,26,k\in \mathbb{N},x^2+k^2\ne 0\}.$$

$$V_I=V\cup F\cup B(B),F_I=B(N),E_I=E\cup B(Q)\cup B(R),$$

$$B_I(X)=B(X)\cup B(g(X))\cup B(g^2(X))\cup B(g^3(X))\cup B(g^4(X)),$$

$$V_O=V\cup F,F_O=E,E_O=B(P)$$

$$B_O(X)=B(X)\cup B(g(X))$$

$$\{V,F,E\}\cup \{B(X)|X\in S\backslash (V\cup F\cup E)\};$$

$$\{V_I,F_I,E_I\}\cup \{B_I(X)|X\in S\backslash (V_I\cup F_I\cup E_I)\};$$

$$\{V_O,F_O,E_O\}\cup \{B_O(X)|X\in S\backslash (V_O\cup F_O\cup E_O)\}.$$

$$\begin{array}[]{c|c|c|c|c|c|c}\mathrm{Component\,Index}&(1,0,0)&(2,0,0)&(0,1,0)&(1,1,0)&(2,1,0)&(\nu,\epsilon,k),\>k\geq 1\\ \hline\cr\mathrm{Existence\>}(|\alpha|)&\mathrm{Yea\>}(4)&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Yea\>}(10)&\mathrm{Nay}&\mathrm{Yea\>}(4\nu+6\epsilon+12k)\\ \end{array}$$

$$\{|\alpha |<\infty |{\mathcal{A}}_{\alpha }\simeq A_4\}=\{x+12k|x=0,4,6,8,10,14,k\in {\mathbb{N}}^{+},\}\cup \{4,10\}.$$

$$G=⟨z\mapsto e^{\frac{1}{n}2\pi i}z,z\mapsto \frac{1}{z}⟩\simeq D_n.$$

$$V=\{0,\infty \},$$

$$A_n=\{e^{\frac{k}{n}2\pi i}|k\in \mathbb{Z}\},B_n=\{e^{\frac{2k+1}{2n}2\pi i}|k\in \mathbb{Z}\}$$

$$C_n(z)=\{z{e}^{\frac{k}{n}2\pi i}|k\in \mathbb{Z}\}\cup \{z^{-1}{e}^{\frac{k}{n}2\pi i}|k\in \mathbb{Z}\}$$

$${\mathcal{A}}_{\alpha }=⟨z\mapsto e^{\frac{1}{pn}2\pi i}z,z\mapsto \frac{1}{z}⟩$$

$${\rho }^p(z)=-z;{\rho }^p(z)=\frac{1}{z};{\rho }^p(z)=\frac{-1}{z}.$$

$$V,A_{2p},B_{2p},C_{2p}(z),z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{4p}2\pi i}|l\in \mathbb{Z}\};$$

$$\varphi (V,)\varphi (A_{2p}),\varphi (B_{2p}),\varphi (C_{2p}(z)),z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{4p}2\pi i}|l\in \mathbb{Z}\};$$

$$\psi (V,)\psi (A_{2p}),\psi (B_{2p}),\psi (C_{2p}(z)),z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{4p}2\pi i}|l\in \mathbb{Z}\};$$

$$\varphi (z)=\frac{z-1}{z+1},\psi (z)=\frac{z+i}{iz+1}.$$

$$\{V,A_n,B_n\}\cup \{C_n(z)|z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{2n}2\pi i}|l\in \mathbb{Z}\}\};$$

$$\{V,A_{pn},B_{pn}\}\cup \{C_{pn}(z)|z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{2np}2\pi i}|l\in \mathbb{Z}\}\},p\ge 2;$$

$$\{\varphi (V),\varphi (A_{2p}),\varphi (B_{2p})\}\cup \{\varphi (C_{2p}(z))|z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{4p}2\pi i}|l\in \mathbb{Z}\}\},p\ge 2;$$

$$\{\psi (V),\psi (A_{2p}),\psi (B_{2p})\}\cup \{\psi (C_{2p}(z))|z\in {\mathbb{C}}^{\ast }\backslash \{e^{\frac{l}{4p}2\pi i}|l\in \mathbb{Z}\}\},p\ge 2;$$

$$\begin{array}[]{c|c|c|c|c|c}\mathrm{Component\>Index}&(0,1,0)&(1,1,0)&(0,2,0)&(1,2,0)&(\nu,\epsilon,k),\>k\geq 1\\ \hline\cr n\geq 3,\;n\neq 4&\mathrm{Yea\>}(n)&\mathrm{Yea\>}(n+2)&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Yea\>}(2\nu+n(\epsilon+2k))\\ \hline\cr n=4&\mathrm{Yea\>}(n)&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Yea\>}(2\nu+n(\epsilon+2k))\end{array}$$

$$\begin{array}[]{c|c|c|c|c}\mathrm{Component\>Index}&(1,0)&(2,0)&(3,0)&(\nu,k),\>k\geq 1\\ \hline\cr n=2&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Nay}&\mathrm{Yea\>}(2\nu+4k)\\ \end{array}$$

Full text

111 † The second author is supported in part by the National Natural Science Foundation of China (Grant No. 11571330) and the Fundamental Research Funds for the Central Universities.

1. 1 Introduction

2. 2 Main Results

3. 2.1 Singularities of ${\mathfrak{M_{0,n}}}$ and Stabilizers of Sets of $n$ Points on $\widehat{\mathbb{C}}$

4. 2.2 Finite Subsets with Their Stabilizers Prescribed

5. 2.3 The Classification Algorithm

6. 3 The Moduli Space $\mathfrak{M_{0,\>n}}$

7. 3.1 The Orbifold Structure of Moduli Space $\mathfrak{M_{0,\>n}}$

8. 3.2 The Explicit Structure of the Group $G_{n}$

9. 3.3 The Singularities of Moduli Space $\mathfrak{M_{0,\>n}}$

10. 4 Possible Values for the Cardinality of $\alpha$

11. 4.1 Dihedral Group

12. 4.2 Cyclic Group

13. 4.3 Trivial Group

14. 5 The Code of the Algorithm

1 Introduction

Let $\mathfrak{M_{g,\>n}}$ denote the moduli space of isomorphism classes of compact Riemann surfaces of genus $g$ with $n$ unordered marked points. It is well-known that $\mathfrak{M_{g,\>n}}$ is both a complex orbifold and an irreducible quasi-projective variety of dimension $3g-3+n$, where $n\geq 3$ if $g=0$, $n\geq 1$ if $g=1$ and $n\geq 0$ if $g\geq 2$ [1].

In section 3, using elementary method we show that $\mathfrak{M_{0,\>n}}$ is a complex orbifold of dimension $n-3$ when $n\geq 4$. Moreover, we show that when $n\geq 5$, $\mathfrak{M_{0,\>n}}$ is the quotient space of an $S_{n}$ action upon a particular Zariski open set $K_{n}\subseteq\mathbb{C}^{n-3}$, and the stabilizers of this $S_{n}$ action correspond to all the subgroups of

[TABLE]

which are stabilizers of sets that consist of $n$ distinct points on the extended complex plane ${\widehat{\mathbb{C}}}={\mathbb{C}}\cup\{\infty\}$ (for a set $\alpha$ consists of $n$ distinct points on the extended complex plane, its stabilizer ${\mathcal{A}}_{\alpha}$ refers to the group consists of all the Möbius transformations which leave $\alpha$ invariant). Thus the classification of the orbifold singularities of $\mathfrak{M_{0,\>n}}$ when $n\geq 5$ is reduced to the following problem:

Classify all the stabilizers of sets that consist of $n$ distinct points on the extended complex plane ${\widehat{\mathbb{C}}}$.

We note that such a stabilizer is a finite subgroup of ${\rm PSL}(2,\,{\mathbb{C}})$ as $n\geq 5$. We also mention the following references for the widely known classical cases of $n=4,\,5$ and $6$ about the stabilizers: [5] and [3, Chapter 8 ] for the cases of $n=4$ and $5$, and [2] for the case of $n=6$.

It is well known that there are three types of non-trivial finite subgroups of ${\rm PSL}(2,\,{\mathbb{C}})$ up to conjugacy: the polyhedral groups, the dihedral groups and the cyclic groups. Let $G$ be one of these groups. Then $\alpha$ is decomposed into finitely many orbits of $G$ if ${\mathcal{A}}_{\alpha}$ coincides with $G$. However, we note that the converse of this assertion is not true in general. We thus introduce a notion called the component index of $\alpha$ to describe the number of orbit(s) in each type in this decomposition and its definition is left to Subsection 2.2.

To tackle our problem, first we give equivalent conditions concerning the constitution of $\alpha$ for its stabilizer ${\mathcal{A}}_{\alpha}$ to coincide with $G$. Second we discuss the range of possible values for both the component index and the cardinality of $\alpha$ when ${\mathcal{A}}_{\alpha}$ coincides with $G$. Last we summarize these results to a complete answer to the problem and the classification of the orbifold singularities of $\mathfrak{M_{0,\>n}}$ when $n\geq 5$. We express the answer by an algorithm written in pseudo-code in Section 2.3 and C programming language in Section 5. We also mention an interesting observation as follows:

For each finite subgroup $G$ of $\text{\rm PSL}(2,\>\mathbb{C})$, there exists an integer $n\geq 5$ and some orbifold singularity of $\mathfrak{M_{0,\>n}}$ with stabilizer $G$.

The organization of this manuscript is as follows. Section 2 is the main body of this paper, presenting all the major steps in tackling our problem but leaving out some proof details to the following sections. Section 3 discusses the correspondence between the orbifold stabilizers of $\mathfrak{M_{0,\>n}}$ and the finite subgroups of ${\rm PSL}(2,\,{\mathbb{C}})$ which are stabilizers for sets that consist of $n$ distinct points. It also gives an explicit representation of these stabilizers. Section 4 discusses the range of possible values for both the component index and the cardinality of $\alpha$ by giving concrete examples for the cases when ${\mathcal{A}}_{\alpha}$ is dihedral, cyclic and trivial. The results are listed in Subsection 2.2 and 2.3. Section 5 is the code for classification written in C programming language.

2 Main Results

Subsection 2.1 shows the correspondence between the orbifold stabilizers of $\mathfrak{M_{0,\>n}}$ and the finite subgroups of ${\rm PSL}(2,\,{\mathbb{C}})$ which are stabilizers for sets consisting of $n$ distinct points on the extended complex plane (proof is left to Section 3). Subsection 2.2 discusses the range of possible values for both the component index and the cardinality of $\alpha$ (some proofs are left to Section 4). Subsection 2.3 presents the final answer to our problem: the classification algorithm of the orbifold singularities of $\mathfrak{M_{0,\>n}}$ when $n\geq 5$.

2.1 Singularities of ${\mathfrak{M_{0,n}}}$ and Stabilizers of Sets of $n$ Points on $\widehat{\mathbb{C}}$

Let $C$ and $C^{\prime}$ be two compact Riemann surfaces of genus $g$ and $\{p_{1},p_{2},\cdots p_{n}\}\subseteq C$, $\{p^{\prime}_{1},p^{\prime}_{2},\cdots p^{\prime}_{n}\}\subseteq C^{\prime}.$ We say that $(C,\{p_{1},p_{2},\cdots p_{n}\})$ and $(C^{\prime},\{p^{\prime}_{1},p^{\prime}_{2},\cdots p^{\prime}_{n}\})$ are isomorphic if there exists some biholomorphic map $f:C\to C^{\prime}$ such that $f(\{p_{1},p_{2},\cdots p_{n}\})=\{p^{\prime}_{1},p^{\prime}_{2},\cdots p^{\prime}_{n}\}.$

Definition 1** ([4]).**

The moduli space $\mathfrak{M_{g,\>n}}$ is the set of isomorphism classes of compact Riemann surfaces of genus $g$ with $n$ unordered marked points.

In Section 3.1 using elementary methods we shall prove that

Theorem 1**.**

$\mathfrak{M_{0,\>n}}=K_{n}/G_{n}$* for $n\geq 4$, where $K_{n}$ is the Zariski open set*

[TABLE]

and $G_{n}$ is a finite group of birational transformations of $\mathbb{C}^{n-3}$, whose restrictions to $K_{n}$ are isomorphisms on $K_{n}$.

Corollary 2**.**

$\mathfrak{M_{0,\>n}}=K_{n}/G_{n}$* is a complex orbifold of dimension $n-3$ when $n\geq 4$.*

Remark**.**

$\mathfrak{M_{0,\>n}}$* is a single point if $n\leq 3$, and $\mathfrak{M_{0,\>4}}$ coincides with the moduli space of elliptic curves.*

In Section 3.2 we shall find that

Theorem 3**.**

$G_{n}$* is isomorphic to the symmetric group $S_{n}$ when $n\geq 5$.*

Corollary 4**.**

The moduli space $\mathfrak{M_{0,\>n}}$ is the quotient space of a Zariski open set $K_{n}$ of $\mathbb{C}^{n-3}$ by an $S_{n}$ action when $n\geq 5$.

For $\boldsymbol{\lambda}\in K_{n}$, set $[\boldsymbol{\lambda}]=\{0,1,\infty,\lambda_{1},\cdots\lambda_{n-3}\}$, $G_{\boldsymbol{\lambda}}=\{g_{\sigma}\in G_{n}\>|\>g_{\sigma}(\boldsymbol{\lambda})=\boldsymbol{\lambda}\}.$ Given a subset $\alpha$ of the extended complex plane, let $\mathcal{A}_{\alpha}$ denote its stabilizer, i.e., the group consists of all the Möbius transformations which leave $\alpha$ invariant. In Section 3.3 we shall prove that

Theorem 5**.**

$G_{\boldsymbol{\lambda}}$* is isomorphic to $\mathcal{A}_{[\boldsymbol{\lambda}]}$ when $n\geq 5$.*

Thus the classification of the orbifold singularities of $\mathfrak{M_{0,\>n}}$ when $n\geq 5$ is reduced to the classification of all the stabilizers of sets that consist of $n$ distinct points on the extended complex plane ${\widehat{\mathbb{C}}}$.

2.2 Finite Subsets with Their Stabilizers Prescribed

Given a finite subset $\alpha$ of the extended complex plane, it is obvious that its stabilizer is finite when $|\alpha|\geq 3$. There are three types of non-trivial finite groups consist of Möbius transformations: the polyhedral groups, the dihedral groups and the cyclic groups. Let $G$ be one of these groups. We will give a classification of finite subsets whose stabilizer equals $G$. We shall work on the Riemann sphere

[TABLE]

in the polyhedral cases and on $\widehat{\mathbb{C}}$ in both the dihedral and the cyclic cases.

Suppose that $G$ is isomorphic to an icosahedral group (or octahedral group). Then it fixes a regular dodecahedron (resp. cube) whose center is the origin of the Riemann sphere $S$. Let $V$, $F$ and $E$ denote its vertices, the projections of the central points of its faces on $S$ and the projections of the middle points of its edges on $S$ respectively, with the origin being the central of the projection. For any $X\in S\backslash(V\cup F\cup E)$, define $B(X)$ as the orbit of $X$ under $G$.

Definition 2**.**

For any subset $\alpha\subseteq S$ that is a finite union of the orbits of $G$ ($\simeq A_{5}$ or $S_{4}$), define its (icosahedral or octahedral) component index as $(\nu,\mu,\epsilon,k)\in\{0,1\}^{3}\times\mathbb{N}$, where $\nu,\mu,\epsilon,k$ indicate the number of orbit(s) in the $V$, $F$, $E$ and $B$ type, respectively.

If the stabilizer of an finite subset $\alpha$ equals $G$, $\alpha$ must be a finite union of the orbits of $G$. Conversely, if $\alpha$ is a finite union of the orbits of $G$, its stabilizer $\mathcal{A}_{\alpha}$ includes $G$ as a subgroup. Since $\mathcal{A}_{\alpha}$ is a finite subgroup of ${\rm PSL}(2,\,{\mathbb{C}})$, $\mathcal{A}_{\alpha}$ equals $G$. Thus we have

Theorem 6**.**

For any finite subset $\alpha\subseteq S$, its stabilizer $\mathcal{A}_{\alpha}$ equals $G$ ($\simeq A_{5}$ or $S_{4}$) if and only if $\alpha$ is a finite union of certain elements in $\{V,\>F,\>E\}\cup\{B(X)|X\in S\backslash(V\cup F\cup E)\}$.

Corollary 7**.**

For each non-zero element $(\nu,\mu,\epsilon,k)\in\{0,1\}^{3}\times\mathbb{N}$, there exist some finite subsets $\alpha,\>\beta\subseteq S$ such that $\mathcal{A}_{\alpha}\simeq A_{5}$ and $\mathcal{A}_{\beta}\simeq S_{4}$, both with the component index $(\nu,\mu,\epsilon,k)$. Furthermore, we have

[TABLE]

[TABLE]

Suppose that $G$ is isomorphic to a tetrahedral group. Then it fixes two regular tetrahedrons whose centers are the origin of the Riemann sphere $S$. Choose one of the tetrahedrons (this choice will not affect our conclusions). Let $V$, $F$ and $E$ denote its vertices, the projections of the central points of its faces on $S$ and the projections of the middle points of its edges on $S$ respectively, with the origin being the central of the projection. For any $X\in S\backslash(V\cup F\cup E)$, define $B(X)$ as the orbit of $X$ under $G$. Notice that $V$ and $F$ have the same cardinality and thus are considered to be in the same type.

Definition 3**.**

For any subset $\alpha\subseteq S$ that is a finite union of the orbits of $G$ ($\simeq A_{4}$), define its (tetrahedral) component index as $(\nu,\epsilon,k)\in\{0,1,2\}\times\{0,1\}\times\mathbb{N}$, where $\nu,\epsilon,k$ indicate the number of orbit(s) in the $V(F)$, $E$ and $B$ type, respectively.

If the stabilizer of an finite subset $\alpha$ equals $G$, $\alpha$ must be a finite union of the orbits of $G$. Conversely, if $\alpha$ is a finite union of the orbits of $G$, its stabilizer $\mathcal{A}_{\alpha}$ includes $G$ as a subgroup. Since $\mathcal{A}_{\alpha}$ is a finite subgroup of ${\rm PSL}(2,\,{\mathbb{C}})$, $\mathcal{A}_{\alpha}$ could be isomorphic to $A_{5}$, $S_{4}$ or $A_{4}$.

CASE 1: $\mathcal{A}_{\alpha}$ is Isomorphic to $A_{5}$.

If $\mathcal{A}_{\alpha}$ is isomorphic to $A_{5}$, then $\mathcal{A}_{\alpha}$ fixes some regular dodecahedron $I$ whose center is the origin. Their relative positions are shown in Figure 1.

It is easy to check that ($V_{I}$, $F_{I}$, $E_{I}$ and $B_{I}(X)$ denote the orbits under the icosahedral group here)

[TABLE]

where $N$ denotes the center of the pentagon $ABCDE$, and $Q$, $R$ denote the midpoints of $AB$, $DE$ respectively, and

[TABLE]

where $g$ denotes the rotation of order $5$ that fixes the pentagon $ABCDE$ for each $X\in S\backslash(V_{I}\cup F_{I}\cup E_{I})$.

CASE 2: $\mathcal{A}_{\alpha}$ Is isomorphic to $S_{4}$.

If $\mathcal{A}_{\alpha}$ is isomorphic to $S_{4}$, then $\mathcal{A}_{\alpha}$ fixes some regular cube $O$ whose center is the origin. Their relative positions are shown in Figure 2.

It is easy to check that ($V_{O}$, $F_{O}$, $E_{O}$ and $B_{O}(X)$ denote the orbits under the octahedral group here)

[TABLE]

where $P$ denotes the midpoint of $AB$ and

[TABLE]

where $g$ denotes the rotation of order $4$ that fixes the the square $ABCD$ for each $X\in S\backslash(V_{O}\cup F_{O}\cup E_{O})$.

Theorem 8**.**

For any finite subset $\alpha\subseteq S$, its stabilizer $\mathcal{A}_{\alpha}$ equals $G$ ($\simeq A_{4}$) if and only if all of the three claims are true:

$\alpha$* is a union of certain elements in*

[TABLE] 2. 2.

$\alpha$* is NOT a union of certain elements in*

[TABLE] 3. 3.

$\alpha$* is NOT a union of certain elements in*

[TABLE]

From the discussion above we know that for each non-zero element $(\nu,\>\epsilon,\>k)\in\{0,\>1,\>2\}\times\{0,\>1\}\times\mathbb{N}$, $(\nu,\epsilon,k)\neq(2,0,0)$, $(0,1,0)$ or $(2,1,0)$, there exists some finite subset $\alpha\subseteq S$ such that $\mathcal{A}_{\alpha}\simeq A_{4}$ with tetrahedral component index $(\nu,\epsilon,k)$. When the tetrahedral component index of $\alpha$ is $(2,0,0)$, $(0,1,0)$ or $(2,1,0)$, $\mathcal{A}_{\alpha}$ is isomorphic to $S_{4}$.

Theorem 9**.**

The existence of $\alpha$ such that $\mathcal{A}_{\alpha}\simeq S_{4}$ with arbitrary element in $\{0,1,2\}\times\{0,1\}\times\mathbb{N}$ as its tetrahedral component index is shown in the table below, along with all possible cardinality of $\alpha$.

[TABLE]

[TABLE]

Suppose that G is isomorphic to a dihedral group. In this case we go back to work on $\widehat{\mathbb{C}}$. Without loss of generality set $(n\geq 2)$

[TABLE]

There are three types of orbits:

[TABLE]

[TABLE]

($A_{n}$ and $B_{n}$ have the same cardinality and are thus considered to be in the same type), and

[TABLE]

for $z\in{\mathbb{C}}^{*}\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}$ when $n\geq 3$. When $n=2$, there are only two types of orbits since $V$, $A_{2}$ and $B_{2}$ all have the same cardinality $2$.

Definition 4**.**

For any subset $\alpha\subseteq S$ that is a finite union of the orbits of $G$ ($\simeq D_{n}$, $n\geq 3$), define its (dihedral-$n$) component index as $(\nu,\epsilon,k)\in\{0,1\}\times\{0,1,2\}\times\mathbb{N}$, where $\nu,\epsilon,k$ indicate the number of orbit(s) in the $V$, $A_{n}(B_{n})$ and $C_{n}$ type, respectively.

For any subset $\alpha\subseteq S$ that is a finite union of the orbits of $G$ ($\simeq K_{4}$), define its (dihedral-$2$) component index as $(\nu,k)\in\{0,1,2,3\}\times\mathbb{N}$, where $\nu,k$ indicate the number of orbit(s) in the $V(A_{2},\>B_{2})$ and $C_{2}$ type, respectively.

If the stabilizer of an finite subset $\alpha$ equals $G$, $\alpha$ must be a finite union of the orbits of $G$. Conversely, if $\alpha$ is a finite union of the orbits of $G$, its stabilizer $\mathcal{A}_{\alpha}$ includes $G$ as a subgroup. Since $\mathcal{A}_{\alpha}$ is itself a finite group consisting of Möbius transformations, $\mathcal{A}_{\alpha}$ could be isomorphic to $A_{5}$ (when $n=2$, $3$ or $5$), $S_{4}$ (when $n=2$, $3$ or $4$), $A_{4}$ (when $n=2$) or $D_{pn}$ for $p\in\mathbb{Z}^{+}$. Since the $A_{5},\>S_{4},\>A_{4}$ cases have been explored, we shall discuss the case when $\mathcal{A}_{\alpha}$ is isomorphic to $D_{pn},\>p\geq 2$. Set $\mathcal{A}_{\alpha}=\langle\rho,\pi|(\rho)^{pn}=(\pi)^{2}=(\rho\pi)^{2}=e\rangle\simeq D_{pn}$.

CASE 1: $G\simeq D_{n}$, $n\geq 3$.

This is simple since there must be certain $m\in\mathbb{Z}$ such that $\rho^{m}(z)=e^{\frac{1}{n}2\pi i}z$. Thus

[TABLE]

and the relations of the orbits of $G$ and $\mathcal{A}_{\alpha}$ are obvious.

CASE 2: $G\simeq K_{4}$.

This is relatively complicated for there are three possible relationships between $G$ and $\mathcal{A}_{\alpha}$, i.e.,

[TABLE]

It is easy to check that the orbits of $\mathcal{A}_{\alpha}$ are

[TABLE]

[TABLE]

[TABLE]

respectively, where

[TABLE]

Theorem 10**.**

For any finite subset $\alpha\subseteq S$, $|\alpha|\geq 3$, its stabilizer $\mathcal{A}_{\alpha}$ equals $G$ ($\simeq D_{n})$ $(n\geq 2)$ if and only if all of the five claims are true:

$\alpha$* is a union of certain elements in*

[TABLE] 2. 2.

$\alpha$* is NOT a union of certain elements in*

[TABLE] 3. 3.

when $n=2$, $\alpha$ is NOT a union of certain elements in

[TABLE] 4. 4.

when $n=2$, $\alpha$ is NOT a union of certain elements in

[TABLE] 5. 5.

$\alpha$* is NOT in the icosahedral, octahedral or tetrahedral case,*

where $\phi(z)=\frac{z-1}{z+1}$ and $\psi(z)=\frac{z+i}{iz+1}$.

The discussion of the range of possible values for the component index in this case is complicated and left in Section 4.1. The results are

Theorem 11**.**

The existence of $\alpha$ such that $\mathcal{A}_{\alpha}\simeq D_{n}$, $n\geq 2$ with arbitrary element in $\{0,1\}\times\{0,1,2\}\times\mathbb{N}$ or $\{0,1,2,3\}\times\mathbb{N}^{+}$ as its dihedral-$n$ component index is shown in the tables below, along with all possible cardinality of $\alpha$ in each circumstances.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Suppose that G is isomorphic to a cyclic group. Without loss of generality set $(n\geq 2)$

[TABLE]

There are two types of orbits:

[TABLE]

for $z\in\mathbb{C}^{*}$ ($N$ and $S$ are in the same type).

Definition 5**.**

For any subset $\alpha\subseteq S$ that is a finite union of the orbits of $G$ ($\simeq\mathbb{Z}_{n}$), define its (cyclic-$n$) component index as $(\nu,k)\in\{0,1,2\}\times\mathbb{N}$, where $\nu,k$ indicate the number of orbit(s) in the $N(S)$ and $C_{n}$ type, respectively.

If the stabilizer of an finite subset $\alpha$ equals $G$, $\alpha$ must be a finite union of the orbits of $G$. Conversely, if $\alpha$ is a finite union of the orbits of $G$, its stabilizer $\mathcal{A}_{\alpha}$ includes $G$ as a subgroup. Since $\mathcal{A}_{\alpha}$ is itself a finite group consisting of Möbius transformations, $\mathcal{A}_{\alpha}$ could be isomorphic to $A_{5}$ (when $n=2$, $3$ or $5$), $S_{4}$ (when $n=2$, $3$ or $4$), $A_{4}$ (when $n=2$ or $3$), $D_{q}$ for $q\geq 2$ (when $n=2$), $D_{pn}$, $\mathbb{Z}_{pn}$ for $p\geq 1$. Since the $A_{5},\>S_{4},\>A_{4},\>D_{pn}$ cases have been explored, we shall discuss the case when $\mathcal{A}_{\alpha}$ is isomorphic to ${\mathbb{Z}}_{pn}$, $p\geq 2$. There is only one possible relationship between $G$ and $\mathcal{A}_{\alpha}$: $\mathcal{A}_{\alpha}=\langle z\mapsto e^{\frac{2\pi i}{pn}}z\rangle$ and it is obvious that

[TABLE]

Theorem 12**.**

For any finite subset $\alpha\subseteq S$, $|\alpha|\geq 3$, its stabilizer $\mathcal{A}_{\alpha}$ equals $G$ ($\simeq Z_{n})$ $(n\geq 2)$ if and only if all of the three claims are true:

$\alpha$* is a union of certain elements in*

[TABLE] 2. 2.

$\alpha$* is NOT a union of certain elements in*

[TABLE] 3. 3.

$\alpha$* is NOT in the icosahedral, octahedral, tetrahedral or dihedral case.*

The discussion of the range of possible values for the component index in this case is complicated and left in Section 4.2. The results are

Theorem 13**.**

The existence of $\alpha$ such that $\mathcal{A}_{\alpha}\simeq\mathbb{Z}_{n}$, $n\geq 2$ with arbitrary element in $\{0,1,2\}\times\mathbb{N}$ as its cyclic-$n$ component index is shown in the table below, along with all possible sizes of $\alpha$ in each circumstances.

[TABLE]

[TABLE]

[TABLE]

2.3 The Classification Algorithm

We need another theorem (proved in Subsection4.3) to complete our classification algorithm:

Theorem 14**.**

Suppose that $\mathcal{A}_{\alpha}$ is the trivial group and $\alpha$ is finite. Then the range of possible values for $|\alpha|$ is

[TABLE]

From these results we obtain the algorithm for the classification. (The reader can find the code for this algorithm written in C programming language in Section 5). Input the cardinality of a finite subset ${\alpha}$ of the extended complex plane, the algorithm will output all possible group structure(s) of its stabilizer $\mathcal{A}_{\alpha}$, along with its component index(es). Notice that this is a classification for stabilizer of finite subsets of the extended complex plane, and only applies to the orbifold singularities of $\mathfrak{M_{0,\>n}}$ when $n\geq 5$. (See the remark following Corollary 2).

n=input(|\alpha|);

if(n<3){output{infinity}};

k=\floor(n/60), r=n-60k; if(r=0&k>0){output{A_5, (0, 0, 0, k)}}; if(r=12){output{A_5, (1, 0, 0, k)}}; if(r=20){output{A_5, (0, 1, 0, k)}}; if(r=30){output{A_5, (0, 0, 1, k)}}; if(r=32){output{A_5, (1, 1, 0, k)}}; if(r=42){output{A_5, (1, 0, 1, k)}}; if(r=50){output{A_5, (0, 1, 1, k)}}; if(r=2&k>0){output{A_5, (1, 1, 1, k-1)}};

k=\floor(n/24), r=n-24k; if(r=0&k>0){output{S_4, (0, 0, 0, k)}}; if(r=6){output{S_4, (1, 0, 0, k)}}; if(r=8){output{S_4, (0, 1, 0, k)}}; if(r=12){output{S_4, (0, 0, 1, k)}}; if(r=14){output{S_4, (1, 1, 0, k)}}; if(r=18){output{S_4, (1, 0, 1, k)}}; if(r=20){output{S_4, (0, 1, 1, k)}}; if(r=2&k>0){output{S_4, (1, 1, 1, k-1)}};

k=\floor(n/12), r=n-12k; if(r=0&k>0){output{A_4, (0, 0, k)}}; if(r=4){output{A_4, (1, 0, k)}}; if(r=8$k>0){output{A_4, (2, 0, k)}}; if(r=6&k>0){output{A_4, (0, 1, k)}}; if(r=10){output{A_4, (1, 1, k)}}; if(r=2&k>1){output{A_4, (2, 1, k-1)}};

p=n; while(p>2) begin k=\floor(n/2p), l=\floor(n/p)-2k, r=n-2pk-pl; if(k>0&r=0){output{D_p, (0, l, k)}}; if(k>1&l=0&r=0){output{D_p, (0, 2, k-1)}}; if(k>0&r=2){output{D_p, (1, l, k)}}; if(k>1&l=0&r=2){output{D_p, (1, 2, k-1)}}; if(k=0&l=1&r=0){output{D_p, (0, 1, 0)}}; if(k=0&l=1&r=2&p!=4){output{D_p, (1, 1, 0)}}; p=p-1; end

k=\floor(n/4), r=n-4k; if(k>0&r=0){output{K_4, (0, k)}}; if(k>1&r=0){output{K_4, (2, k-1)}}; if(k>0&r=2){output{K_4, (1, k)}}; if(k>1&r=2){output{K_4, (3, k-1)}};

p=n; while(p>2) begin k=\floor(n/p), r=n-pk; if(k>2&r<3){output{Z_p, (r, k)}}; if(k=2&r=1){output{Z_p, (r, k)}}; if(k=1&r=1&p!=3){output{Z_p, (r, k)}}; p=p-1; end

k=\floor(n/2), r=n-2k; if(k>2){output{Z_2, (r, k)}}; if(k>3&r=0){output{Z_2, (2, k-1)}}; if(k=2&r=1){output{Z_2, (r, k)}};

if(n>4){output{(0)}};

There are some interesting observations:

Theorem 15**.**

For any finite group $G$ of linear fractional transformations, there exists a finite subset $\alpha\subseteq\widehat{\mathbb{C}}$, $|\alpha|\geq 5$ such that $\mathcal{A}_{\alpha}=G$.

Corollary 16**.**

For each finite subgroup $G$ of $\text{PSL}(2,\>\mathbb{C})$, there exists some oribfold singularity of $\mathfrak{M_{0,\>n}}$ for some $n(\geq 5)$ whose stabilizer is $G$.

3 The Moduli Space $\mathfrak{M_{0,\>n}}$

3.1 The Orbifold Structure of Moduli Space $\mathfrak{M_{0,\>n}}$

For $n\in\mathbb{Z}_{\geq 4}$, set

[TABLE]

For any $\boldsymbol{\lambda}\in K_{n}$, set

[TABLE]

For $\boldsymbol{\lambda}\in K_{n}$ and $\sigma\in S_{n}$, define $f^{\boldsymbol{\lambda}}_{\sigma}$ as the Möbius transformation such that

[TABLE]

Now we define a mapping $g_{\sigma}$ of $K_{n}$ for each $\sigma\in S_{n}$

[TABLE]

By definition $g_{\sigma}$ is a rational map from $\mathbb{C}^{n-3}$ to $\mathbb{C}^{n-3}$ which restricts to an isomorphism of $K_{n}$. Here is a useful observation:

Remark**.**

For any $\boldsymbol{\lambda}\in K_{n}$ and $\sigma\in S_{n}$, we have

[TABLE]

holds for $k=1,\>2,\>\cdots,\>n$.

Define $G_{n}=\{g_{\sigma}|\sigma\in S_{n}\}$. Notice that $|G_{n}|\leq n!$ and

Theorem 17**.**

$G_{n}$* is a finite group acting on $K_{n}$ when $n\geq 4$.*

Proof.

It is easy to check that

[TABLE]

Thus we conclude that

[TABLE]

So $g_{\pi}\circ g_{\sigma}=g_{\pi\cdot\sigma}\in G_{n}$. Thus $G_{n}$ is a group acting on $K_{n}$. ∎

As $G_{n}$ is a finite group acting on $K_{n}$, we have another conclusion

Theorem 18**.**

$\mathfrak{M_{0,\>n}}\simeq K_{n}/G_{n}$* as orbitfolds when $n\geq 4$.*

Proof.

Given two elements

[TABLE]

set

[TABLE]

If $\boldsymbol{\mu}\in G_{n}(\boldsymbol{\lambda})$, then there exists some $\sigma\in S_{n}$ s.t. $\boldsymbol{\mu}=g_{\sigma}(\boldsymbol{\lambda}).$ So we have

[TABLE]

Thus

[TABLE]

On the other hand, if

[TABLE]

then there exists some linear fractional transformation $h$ s.t.

[TABLE]

and some $\sigma\in S_{n}$ s.t.

[TABLE]

So $h=f^{\boldsymbol{\lambda}}_{\sigma}$ and $\boldsymbol{\mu}=g_{\sigma}(\boldsymbol{\lambda})$.

∎

Corollary 19**.**

$\mathfrak{M_{0,\>n}}$* is a complex orbifold of dimension $n-3$ when $n\geq 4$.*

3.2 The Explicit Structure of the Group $G_{n}$

Now let us explore $G_{n}$. Define subgroup $V_{n}$ of the symmetric group $S_{n}$

[TABLE]

and subset $T_{n}$ of $S_{n}$

[TABLE]

Thus $T_{n}$ forms a complete list of left coset representatives of $V_{n}\subseteq S_{n}$.

Define group $H$ of linear fractional transformations

[TABLE]

It is obvious that

[TABLE]

For $p=4,\>5,\>\cdots,\>n$, set $\sigma=(1,\>p)$ and we have

[TABLE]

[TABLE]

[TABLE]

So we have

[TABLE]

and

[TABLE]

By the same method I calculated $g_{\sigma}$ for each $\sigma\in S_{n}$. The next theorem is the result.

Theorem 20**.**

$G_{n}=\tilde{T}_{n}\tilde{V}_{n}$, where

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Corollary 21**.**

$G_{n}$* is isomorphic to $S_{n}$ when ${n\geq 5}$.*

3.3 The Singularities of Moduli Space $\mathfrak{M_{0,\>n}}$

For $\boldsymbol{\lambda}\in K_{n}$, $n\geq 5$, set

[TABLE]

and $G_{\boldsymbol{\lambda}}=\{g_{\sigma}\in G_{n}\>|\>g_{\sigma}(\boldsymbol{\lambda})=\boldsymbol{\lambda}\}$ the stabilizer of $\boldsymbol{\lambda}$

Definition 6**.**

When $n\geq 5$, call $\overline{[\boldsymbol{\lambda}]}\in\mathfrak{M_{0,\>n}}$ an oribfold singularity of $\mathfrak{M_{0,\>n}}$ if $G_{\boldsymbol{\lambda}}$ is non-trivial.

Given a finite subset $\alpha=\{z_{1},z_{2},\cdots,z_{n}\}\subseteq\widehat{\mathbb{C}}$, $n\geq 4$, let $\mathcal{A}_{\alpha}$ denote the stabilizing group, i.e., the group of linear fractional transformations that fix $\alpha$. It is easy to check that

Theorem 22**.**

The mapping

[TABLE]

is a group isomorphism for any $\boldsymbol{\lambda}\in K_{n}$, $n\geq 5$.

4 Possible Values for the Cardinality of $\alpha$

4.1 Dihedral Group

First suppose that $n\geq 3$. From Theorem 10 and observation we know that there exist some finite subsets $\alpha,\>\beta\subseteq S^{2}$ such that $\mathcal{A}_{\alpha}\simeq D_{n}$ with dihedral-$n$ component index $(0,1,0)$, and $\mathcal{A}_{\beta}\simeq D_{n}$ (if $n\neq 4$) with dihedral-$n$ component index $(1,1,0)$. When the dihedral-$n$ component index of $\alpha$ is $(0,2,0)$ or $(1,2,0)$, $\mathcal{A}_{\alpha}$ is isomorphic to $D_{2n}$. When $n=4$ and the dihedral-$4$ component index of $\beta$ is $(1,1,0)$, $\mathcal{A}_{\beta}$ is isomorphic to $S_{4}$.

Then suppose that $n=2$. It is obvious that when the dihedral-$2$ component index of $\alpha$ is $(1,0)$, $(2,0)$ and $(3,0)$, $\mathcal{A}_{\alpha}$ is infinite, isomorphic to $D_{4}$ and $S_{4}$, respectively.

Now suppose that $k\geq 1$. In the proposition below, I shall give examples of $\alpha$ such that $\mathcal{A}_{\alpha}\simeq D_{n}$ with dihedral-$n$ component index $(0,0,k)$ when $n\geq 3$, and $\alpha$ such that $\mathcal{A}_{\alpha}\simeq K_{4}$ with dihedral-$2$ component index $(0,k)$. Then it will be easy to find examples of $\alpha$ with dihedral-$n$ component index $(\nu,\epsilon,k)$ or $(\nu,k)$ for each element $(\nu,\epsilon,k)\in\{0,1\}\times\{0,1,2\}\times\mathbb{N}^{+}$ and $(\nu,k)\in\{0,1,2,3\}\times\mathbb{N}^{+}$ by the same means. Thus the proof of Theorem 11 is completed.

Proposition 23**.**

The finite subset

[TABLE]

where $C_{n}(z)=\{ze^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}\cup\{z^{-1}e^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}$ and $k\geq 1$, has stabilizer $\mathcal{A}_{\alpha}\simeq D_{n}$ with dihedral-$n$ component index $(0,\>0,\>k)$ when $n\geq 3$ and stabilizer $\mathcal{A}_{\alpha}\simeq K_{4}$ with dihedral-$2$ component index $(0,\>k)$ when $n=2$.

Proof.

It is obvious that

[TABLE]

$\mathcal{A}_{\alpha}$ is not isomorphic to $A_{5}$, $S_{4}$ or $A_{4}$ as all points of $\alpha$ are concyclic. Assume that

[TABLE]

for some $2\leq p\leq 2k$.

Claim: There exists some $m\in\mathbb{Z}$ such that $\rho^{m}(z)=ze^{\frac{1}{n}2\pi i}$.

Proof of Claim: The claim is obvious when $n\geq 3$. When $n=2$, it is clear that

[TABLE]

Suppose that $\rho^{p}(z)=\frac{1}{z}$. Because $\rho$ is a Möbius transformation of order $2p\geq 4$, it is elliptic with fixed points $1$ and $-1$, and thus never fixes $\alpha$. Thus $\rho^{p}(z)\neq\frac{1}{z}$. For the same reason $\rho^{p}(z)\neq\frac{-1}{z}$. Thus $\rho^{p}(z)=-z$, and the claim is proved.

Because $\rho$ is a Möbius transformation of order $pn$, it is elliptic with fixed points [math] and $\infty$. Thus

[TABLE]

Without lose of generality assume that $n^{\prime}=1$. Then

[TABLE]

with

[TABLE]

since $2\leq p\leq 2k$. However, by the construction of $\alpha$ we know that

[TABLE]

Thus

[TABLE]

which contradicts the assumption that $p\geq 2$. ∎

4.2 Cyclic Group

Suppose that $n\geq 3$. Set

[TABLE]

where $C_{n}(z)=\{ze^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}$. From Theorem 12 and observation we know that when $k\geq 3$

[TABLE]

and when $k\geq 1$ (except for $\beta_{3,1}$),

[TABLE]

Thus we have found subsets whose stabilizers isomorphic to $\mathbb{Z}_{n}$ with cyclic-$n$ component index $(0,k)$ and $(2,k)$ for $k\geq 3$ and cyclic-$n$ component index $(1,k)$ for $k\geq 1$ (except for $(1,1)$ when $n=3$.) When the cyclic-$n$ component index of $\alpha$ is $(0,1)$, $(0,2)$, $(2,1)$ or $(2,2)$, $\mathcal{A}_{\alpha}$ is isomorphic to $D_{n}$. When $n=3$ and the cyclic-$3$ component index of $\alpha$ is $(1,1)$, $\mathcal{A}_{\alpha}$ is isomorphic to $A_{4}$.

Now suppose that $n=2$. In the following proposition we will find finite subsets whose stabilizers are isomorphic to $\mathbb{Z}_{2}$ with cyclic-$2$ component index $(1,k)$ for each $k\geq 2$.

Proposition 24**.**

The finite subset

[TABLE]

has stabilizer $\mathcal{A}_{\alpha}=\langle z\mapsto-z\rangle\simeq\mathbb{Z}_{2}$ with cyclic-$2$ component index $(1,k)$.

Proof.

$\mathcal{A}_{\alpha}$ is not isomorphic to $A_{5}$, $S_{4}$ or $A_{4}$ as all points of $\alpha$ are concyclic.

Assume that

[TABLE]

for some $p\geq 2$. Then $\rho$ is an elliptic transformation such that $\rho^{p}(z)=-z$. Thus

[TABLE]

Thus $\rho$ can never fix $\alpha$ and the assumption is wrong.

Assume that

[TABLE]

for some $n\geq 2$. Notice that all points of $\alpha$ are concyclic. Then from Theorem 10 we know that $n|2k+1$ (thus $n\geq 3$), and there exists some $\tilde{\rho}=\rho^{m}$ such that

[TABLE]

We conclude that

[TABLE]

Thus $\tilde{\rho}$ can never fix $\alpha$ and the assumption is wrong again. ∎

Thus we have found finite subsets whose stabilizers are isomorphic to $\mathbb{Z}_{2}$ with cyclic-$2$ component index $(1,k)$ for each $k\geq 2$. By the same method it is easy to find subsets whose stabilizers isomorphic to $\mathbb{Z}_{2}$ with cyclic-$2$ component index $(0,k)$ and $(2,k)$ for $k\geq 3$ (consider $\{\pm\frac{1}{2},\pm\frac{3}{2},\cdots\pm\frac{2k-1}{2},\}$ and $\{0,\infty,\pm 1,\pm 2,\cdots\pm k\}$). Notice that when the cyclic-$2$ component index is $(0,1)$, $(0,2)$, $(1,1)$, $(2,1)$ or $(2,2)$, $\mathcal{A}_{\alpha}$ is not isomorphic to $\mathbb{Z}_{2}$. Now the proof of Theorem 13 is completed.

4.3 Trivial Group

Suppose that $\mathcal{A}_{\alpha}$ is the trivial group, $\alpha$ finite, what could $|\alpha|$ be? When $|\alpha|=2$, $\mathcal{A}_{\alpha}$ is infinite. When $|\alpha|=3$, $\mathcal{A}_{\alpha}$ is isomorphic to $D_{3}$. When $|\alpha|=4$, suppose that $\alpha=\{z_{1},z_{2},z_{3},z_{4}\}$. It is easy to check that

[TABLE]

Thus $\mathcal{A}_{\alpha}$ can not be trivial.

Proposition 25**.**

The finite subset

[TABLE]

where $C_{n}(z)=\{ze^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}$ has a trivial stabilizing group when $n\geq 4$.

Proof.

For any $f\in\mathcal{A}_{\alpha}$,

[TABLE]

Thus $f$ fixes the unit circle, which implies that $f$ also fixes $C_{n}(1)$. From Theorem 10 we know that

[TABLE]

However $f$ also fixes $2$. Thus $f$ has to be the identity. ∎

Thus we have proved Theorem 14.

5 The Code of the Algorithm

#include <stdio.h>

int main() {int n, k, l, r, p; printf("Please enter the size of the subset.\n"); scanf("%d", &n);

if(n<1){printf("error");} if(n==1){printf("infinity");} if(n==2){printf("infinity");}

k=n/60; r=n-60*k; //printf("k=%d, r= %d\n",k,r); if(r==0&k>=1){printf("A_5, (0, 0, 0, %d)\n",k);} if(r==12){printf("A_5, (1, 0, 0, %d)\n",k);} if(r==20){printf("A_5, (0, 1, 0, %d)\n",k);} if(r==30){printf("A_5, (0, 0, 1, %d)\n",k);} if(r==32){printf("A_5, (1, 1, 0, %d)\n",k);} if(r==42){printf("A_5, (1, 0, 1, %d)\n",k);} if(r==50){printf("A_5, (0, 1, 1, %d)\n",k);} if(r==2&k>=1){printf("A_5, (1, 1, 1, %d)\n",k-1);}

k=n/24; r=n-24*k; //printf("k=%d, r= %d\n",k,r); if(r==0&k>=1){printf("S_4, (0, 0, 0, %d)\n",k);} if(r==6){printf("S_4, (1, 0, 0, %d)\n",k);} if(r==8){printf("S_4, (0, 1, 0, %d)\n",k);} if(r==12){printf("S_4, (0, 0, 1, %d)\n",k);} if(r==14){printf("S_4, (1, 1, 0, %d)\n",k);} if(r==18){printf("S_4, (1, 0, 1, %d)\n",k);} if(r==20){printf("S_4, (0, 1, 1, %d)\n",k);} if(r==2&k>=1){printf("S_4, (1, 1, 1, %d)\n",k-1);}

k=n/12, r=n-12*k; //printf("k=%d, r= %d\n",k,r); if(r==0&k>=1){printf("A_4, (0, 0, %d)\n",k);} if(r==4){printf("A_4, (1, 0, %d)\n",k);} if(r==8&k>=1){printf("A_4, (2, 0, %d)\n",k);} if(r==6&k>=1){printf("A_4, (0, 1, %d)\n",k);} if(r==10){printf("A_4, (1, 1, %d)\n",k);} if(r==2&k>=2){printf("A_4, (2, 1, %d)\n",k-1);}

p=n; while(p>=3) { k=n/(2p); l=n/p-2k; r=n-2pk-p*l; //printf("p=%d, k=%d, l=%d, r= %d\n",p,k,l,r); if(k>=1&r==0){printf("D_%d, (0, %d, %d)\n",p,l,k);} if(k>=2&r==0&l==0){printf("D_%d, (0, 2, %d)\n",p,k-1);} if(k>=1&r==2){printf("D_%d, (1, %d, %d)\n",p,l,k);} if(k>=2&r==2&l==0){printf("D_%d, (1, 2, %d)\n",p,k-1);} if(k==0&r==0&l==1){printf("D_%d, (0, 1, 0)\n",p);} if(k==0&r==2&l==1&p!=4){printf("D_%d, (1, 1, 0)\n",p);} p=p-1; }

k=n/4; r=n-4*k; //printf("k=%d, r= %d\n",k,r); if(k>=1&r==0){printf("K_4, (0, %d)\n",k);} if(k>=2&r==0){printf("K_4, (2, %d)\n",k-1);} if(k>=1&r==2){printf("K_4, (1, %d)\n",k);} if(k>=2&r==2){printf("K_4, (3, %d)\n",k-1);}

p=n; while(p>=3) { k=n/p; r=n-p*k; //printf("p=%d, k=%d, r= %d\n",p,k,r); if(k>=3&r<=2){printf("Z_%d, (%d, %d)\n",p,r,k);} if(k==2&r==1){printf("Z_%d, (%d, %d)\n",p,r,k);} if(k==1&r==1&p!=3){printf("Z_%d, (%d, %d)\n",p,r,k);} p=p-1; }

k=n/2; r=n-2*k; //printf("k=%d, r= %d\n",k,r); if(k>=3){printf("Z_2, (%d, %d)\n",r,k);} if(k>=4&r==0){printf("Z_2, (2, %d)\n",k-1);} if(k==2&r==1){printf("Z_2, (%d, %d)\n",r,k);}

if(n>=5){printf("(0)");}

return 0; }

Example**.**

The classification of the obifold singularities of the moduli space $\mathfrak{M_{0,\>2018}}$ is

S_4, (1, 1, 1, 83) A_4, (2, 1, 167) D_2018, (0, 1, 0) D_2016, (1, 1, 0) D_1009, (0, 0, 1) D_1008, (1, 0, 1) D_672, (1, 1, 1) D_504, (1, 0, 2) D_504, (1, 2, 1) D_336, (1, 0, 3) D_336, (1, 2, 2) D_288, (1, 1, 3) D_252, (1, 0, 4) D_252, (1, 2, 3) D_224, (1, 1, 4) D_168, (1, 0, 6) D_168, (1, 2, 5) D_144, (1, 0, 7) D_144, (1, 2, 6) D_126, (1, 0, 8) D_126, (1, 2, 7) D_112, (1, 0, 9) D_112, (1, 2, 8) D_96, (1, 1, 10) D_84, (1, 0, 12) D_84, (1, 2, 11) D_72, (1, 0, 14) D_72, (1, 2, 13) D_63, (1, 0, 16) D_63, (1, 2, 15) D_56, (1, 0, 18) D_56, (1, 2, 17) D_48, (1, 0, 21) D_48, (1, 2, 20) D_42, (1, 0, 24) D_42, (1, 2, 23) D_36, (1, 0, 28) D_36, (1, 2, 27) D_32, (1, 1, 31) D_28, (1, 0, 36) D_28, (1, 2, 35) D_24, (1, 0, 42) D_24, (1, 2, 41) D_21, (1, 0, 48) D_21, (1, 2, 47) D_18, (1, 0, 56) D_18, (1, 2, 55) D_16, (1, 0, 63) D_16, (1, 2, 62) D_14, (1, 0, 72) D_14, (1, 2, 71) D_12, (1, 0, 84) D_12, (1, 2, 83) D_9, (1, 0, 112) D_9, (1, 2, 111) D_8, (1, 0, 126) D_8, (1, 2, 125) D_7, (1, 0, 144) D_7, (1, 2, 143) D_6, (1, 0, 168) D_6, (1, 2, 167) D_4, (1, 0, 252) D_4, (1, 2, 251) D_3, (1, 0, 336) D_3, (1, 2, 335) K_4, (1, 504) K_4, (3, 503) Z_2017, (1, 1) Z_672, (2, 3) Z_504, (2, 4) Z_336, (2, 6) Z_288, (2, 7) Z_252, (2, 8) Z_224, (2, 9) Z_168, (2, 12) Z_144, (2, 14) Z_126, (2, 16) Z_112, (2, 18) Z_96, (2, 21) Z_84, (2, 24) Z_72, (2, 28) Z_63, (2, 32) Z_56, (2, 36) Z_48, (2, 42) Z_42, (2, 48) Z_36, (2, 56) Z_32, (2, 63) Z_28, (2, 72) Z_24, (2, 84) Z_21, (2, 96) Z_18, (2, 112) Z_16, (2, 126) Z_14, (2, 144) Z_12, (2, 168) Z_9, (2, 224) Z_8, (2, 252) Z_7, (2, 288) Z_6, (2, 336) Z_4, (2, 504) Z_3, (2, 672) Z_2, (0, 1009) Z_2, (2, 1008) (0) *