On germs of finite morphisms of smooth surfaces

TL;DR

This paper classifies four-sheeted finite morphism germs between smooth complex surfaces up to smooth deformation, analyzing their branch curve singularities and local monodromy groups.

Contribution

It provides a classification of four-sheeted finite morphism germs of smooth surfaces, including their singularities and monodromy, advancing understanding of their deformation behavior.

Findings

Classification of four-sheeted germs up to smooth deformation

Analysis of branch curve singularities

Investigation of local monodromy groups

Abstract

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$ are two connected germs of smooth complex-analytic surfaces. The singularity types of their branch curves and the local monodromy groups are investigated also.

Equations

$$u=f_1(z,w),v=f_2(z,w),$$

$$J(f):=\det\left(\begin{array}[]{cc}\frac{\partial u}{\partial z}&\frac{\partial u}{\partial w}\\ \frac{\partial v}{\partial z}&\frac{\partial v}{\partial w}\end{array}\right)=0$$

$${\mathbb{B}}_r=\{(u,v)\in {\mathbb{C}}^2\mid |u{|}^2+|v{|}^2<r^2\}$$

$${\tilde{\mathfrak{B}}}_T=\{\tilde{h}=(h,\epsilon ,p)\in {\mathfrak{B}}_T\times \mathbb{R}\times {\mathbb{C}}^2\mid h\in {\mathfrak{B}}_T,\epsilon <r(B_h),p\in {\mathbb{B}}_{\epsilon }\setminus B_h\}$$

$$\text{Def}:{\Omega }_T({\tilde{h}}_0)\to \text{Aut}({\pi }_1({\mathbb{B}}_{{\epsilon }_0}\setminus B_{h_0},p_0))=\text{Aut}({\pi }_1^{loc}(B_{h_0},o))$$

$$v^m+\sum _{i=1}^m{q}_i(u)v^{m-i}=0,$$

$${\overline{D}}_1={\overline{D}}_1({\epsilon }_1)=\{u\in \mathbb{C}\mid \mid u\mid ⩽{\epsilon }_1\},{\overline{D}}_2={\overline{D}}_2({\epsilon }_2)=\{v\in \mathbb{C}\mid \mid v\mid ⩽{\epsilon }_2\},$$

$$K_B:=K_B({\epsilon }_1)={\text{pr}}_2({\text{pr}}^{-1}(p_1))=\{v_1,\dots ,v_m\}\subset {\overline{D}}_2.$$

$$\begin{array}[]{clc}a_{i}a_{i+1}a_{i}&=&a_{i+1}a_{i}a_{i+1}\qquad\qquad 1\leqslant i\leqslant m-1,\\ a_{i}a_{k}&=&a_{k}a_{i}\qquad\qquad\qquad\,\,\mid i-k\mid\,\geqslant 2\end{array}$$

$$b_{(B,o)}=a_1^{n+1}.$$

$$v_1(t)={\epsilon }^{2k}{e}^{4kt\pi i}+\sqrt{{\epsilon }^{4k+n+1}}{e}^{(2k+\frac{n+1}{2})\pi i},v_2(t)={\epsilon }^{2k}{e}^{4kt\pi i}-\sqrt{{\epsilon }^{4k+n+1}}{e}^{(2k+\frac{n+1}{2})\pi i},v_3(t)=0$$

$$b_{(B,o)}=(a_1{a}_2{a}_1{)}^n.$$

$$v_1(t)=\sqrt{{\epsilon }^n}{e}^{\pi ni},v_2(t)=0,v_3(t)=-\sqrt{{\epsilon }^n}{e}^{\pi ni}$$

$$b_{(B,o)}={\Delta }_3^n(a_1{a}_2{)}^{\beta }.$$

$$\begin{array}[]{lcl}(\gamma_{j})a_{i}&=&\gamma_{j}\qquad\qquad\text{if}\,\,j\neq i,i+1,\\ (\gamma_{i})a_{i}&=&\gamma_{i}\gamma_{i+1}\gamma_{i}^{-1},\\ (\gamma_{i+1})a_{i}&=&\gamma_{i},\end{array}$$

$$({\gamma }_i){\Delta }_m=({\gamma }_1\dots {\gamma }_m){\gamma }_i({\gamma }_1\dots {\gamma }_m{)}^{-1}\text{for}i=1,\dots ,m.$$

$$i_{\ast }:{\pi }_1({\overline{D}}_2\setminus K_B,p_2)\to {\pi }_1({\mathbb{B}}_{\epsilon }\setminus B,p)\simeq {\pi }_1^{loc}(B,o).$$

$${\pi }_1^{loc}(B,o)=⟨{\gamma }_1,\dots ,{\gamma }_m\mid {\gamma }_i=({\gamma }_i)b_{(B,o)},i=1,\dots ,m⟩,$$

$${\pi }_1^{loc}(B,o)=⟨{\gamma }_1,{\gamma }_2\mid ({\gamma }_1{\gamma }_2{)}^k{\gamma }_1^{1-\delta }=({\gamma }_2{\gamma }_1{)}^k{\gamma }_2^{1-\delta }⟩.$$

$$\begin{array}[]{ll}\pi_{1}^{loc}(B_{n,1},o)=\langle\gamma_{1},\gamma_{2},\gamma_{3}\,\,\mid&\gamma_{1}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n+1}\gamma_{3}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(n+1)},\\ &\gamma_{2}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n}\gamma_{1}(\gamma_{1}\gamma_{2}\gamma_{3})^{-n},\\ &\gamma_{3}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n}\gamma_{2}(\gamma_{1}\gamma_{2}\gamma_{3})^{-n}\rangle\end{array}$$

$$\begin{array}[]{ll}\pi_{1}^{loc}(B_{n,2},o)=\langle\gamma_{1},\gamma_{2},\gamma_{3}\,\,\mid&\gamma_{1}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n+1}\gamma_{2}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(n+1)},\\ &\gamma_{2}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n+1}\gamma_{3}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(n+1)},\\ &\gamma_{3}=(\gamma_{1}\gamma_{2}\gamma_{3})^{n}\gamma_{1}(\gamma_{1}\gamma_{2}\gamma_{3})^{-n}\rangle.\end{array}$$

$$(i_{\ast }(\gamma ))\text{Def}(l)=i_{\ast }((\gamma )\text{def}(l))$$

$$v[(v-u^k{)}^2-e^{2\pi ti}{u}^{2k+n+1}]=0$$

$$(v+u^n)(v-\frac{1}{2}(1-e^{\pi ti})u^n)(v-\frac{1}{2}(1+e^{\pi ti})u^n)=0,$$

$$(v+\frac{1}{2}(1+e^{\pi ti})u^n)(v+\frac{1}{2}(1-e^{\pi ti})u^n)(v-u^n)=0$$

$$\begin{array}[]{ll}u=&f_{1}(z,w),\\ v=&f_{2}(z,w),\end{array}$$

$$J(f):=\det\left(\begin{array}[]{cc}\frac{\partial u}{\partial z}&\frac{\partial u}{\partial w}\\ \frac{\partial v}{\partial z}&\frac{\partial v}{\partial w}\end{array}\right)=0$$

$$F_{\ast }:{\pi }_1^{loc}(B,o)={\pi }_1(V\setminus B,p)\to {\mathbb{S}}_d,$$

$$\begin{array}[]{ll}\pi_{1}^{loc}(B,o)=\langle\gamma_{1},\gamma_{2},\gamma_{3}\mid&\gamma_{1}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4k+2}(\gamma_{1}\gamma_{2})^{n}\gamma_{1}\gamma_{2}\gamma_{1}^{-1}(\gamma_{1}\gamma_{2})^{-n}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(4k+2)},\\ &\gamma_{2}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4k+2}(\gamma_{1}\gamma_{2})^{n}\gamma_{1}(\gamma_{1}\gamma_{2})^{-n}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(4k+2)},\\ &\gamma_{3}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4k+2}\gamma_{3}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(4k+2)}\rangle\end{array}.$$

$$\begin{array}[]{ll}\pi_{1}^{loc}(B,o)=\langle\gamma_{1},\gamma_{2},\gamma_{3}\mid&\gamma_{1}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4m}(\gamma_{1}\gamma_{2})^{n}\gamma_{1}\gamma_{2}\gamma_{1}^{-1}(\gamma_{1}\gamma_{2})^{-n}(\gamma_{1}\gamma_{2}\gamma_{3})^{-4m},\\ &\gamma_{2}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4m}(\gamma_{1}\gamma_{2})^{n}\gamma_{1}(\gamma_{1}\gamma_{2})^{-n}(\gamma_{1}\gamma_{2}\gamma_{3})^{-4m},\\ &\gamma_{3}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4m}\gamma_{3}(\gamma_{1}\gamma_{2}\gamma_{3})^{-4m}\,\,\rangle.\end{array}$$

Full text

0. Introduction

Let $(V,o)=(\mathbb{B}_{\varepsilon},o)$ be a ball in $\mathbb{C}^{2}$ of small radius $\varepsilon>0$, $(U,o^{\prime})$ a connected germ of smooth complex-analytic surface, and $F:(U,o^{\prime})\to(V,o)$ a germ of finite holomorphic mapping (below, a germ of finite cover) of local degree $\deg_{o^{\prime}}F=d$, given in local coordinates $z,w$ in $(U,o^{\prime})$ and $u,v$ in $(V,o)$ by functions

[TABLE]

where $f_{i}(z,w)\in\mathbb{C}[[z,w]]$ are convergent power series. Denote by $R\subset(U,o^{\prime})$ the ramification divisor of $F$ given by equation

[TABLE]

and by $B=F(R_{red})\subset(V,o)$ the germ of branch curve of $F$.

The germ $F$ defines a homomorphism $F_{*}:\pi_{1}(V\setminus B,p)\to\mathbb{S}_{d}$ (the monodromy of the germ $F$), where $\mathbb{S}_{d}$ is the symmetric group acting on the fibre $F^{-1}(p)$. The group $G_{F}=\text{im}F_{*}$ is called the (local) monodromy group of $F$. Note that $G_{F}$ is a transitive subgroup of $\mathbb{S}_{d}$.

We say that two germs $F_{1}:(U,o^{\prime})\to(V,o)$ and $F_{2}:(U,o^{\prime})\to(V,o)$ are equivalent if they differ from each other by changes of coordinates in $(U,o^{\prime})$ and $(V,o)$.

The aim of this article is to investigate the germs of finite covers with up to deformation equivalence. In short (see Definition 3 in Subsection 1.5), two germs $F_{1}$ and $F_{2}$ are deformation equivalent if they can be included in a smooth family of the germs of finite covers preserving the singularity type of germs of branch curves.

The germs of finite covers of local degree $\deg_{o^{\prime}}F=3$ were investigated in [8]. In this article, we investigate the singularity types of the germs of branch curves of finite covers of local degree $\deg_{o^{\prime}}F=4$, describe their monodromy groups, and give a complete classification of them with up to deformation equivalence.

Note that, a priori, the monodromy group $G_{F}$ of the four-sheeted germ of a finite cover $F$ is one of the following subgroups of $\mathbb{S}_{4}$: a cyclic group $\mathbb{Z}_{4}$ generated by a cycle of length 4, the Klein four group $Kl_{4}$, a dihedral group $\mathbb{D}_{4}$, the alternating group $\mathbb{A}_{4}$, and $\mathbb{S}_{4}$.

Before to formulate the main result of the article, we introduce several notations.

• •

$T[f(u,v)=0]=T(B)$, the singularity type of a curve germ $B$ given by equation $f(u,v)=0$;

• •

$A_{n}:=T[v^{2}-u^{n+1}=0]$, $n\geqslant 0$;

• •

$D_{n}:=T[u(v^{2}-u^{n-2})=0]$, $n\geqslant 4$;

• •

$E_{6}:=T[v(v^{2}-u^{3})=0]$;

• •

$E_{7}:=T[v^{3}-u^{4}=0]$;

• •

$E_{8}:=T[v^{3}-u^{5}=0]$;

• •

$T_{3}(n,\beta):=T[v^{3}-u^{3n+\beta}=0]$, $n\geqslant 0$, $\beta=1,2$;

• •

$T_{3,2k+n,2k}:=T[v((v-u^{k})^{2}-u^{2k+n+1})=0]$, $k\geqslant 1$, $n\geqslant 0$;

• •

$T_{3,n-1,n}:=T[v(v^{2}-u^{n})=0]$, $n\geqslant 2$;

• •

$T_{4,2n_{1},2n_{2}}:=[(v^{2}-u^{2n_{1}+1})(u^{2}-v^{2n_{2}+1})=0]$, $n_{1},n_{2}\in\mathbb{N}$.

Note that in these notations there are several intersections, for example, $E_{6}=T_{3,2,3}$ and $E_{7}=T_{3}(1,1)$, $E_{8}=T_{3}(1,2)$.

The main result of the article is the following theorem in which we give a complete classification of four-sheeted germs of finite covers with up to deformation equivalence and compute their main invariants: the singularity types of branch curves and their monodromy groups.

Theorem 1**.**

A germ of finite cover $F:(U,o^{\prime})\to(V,o)$, $\deg_{o^{\prime}}F=4$, given by functions $u=f_{1}(z,w)$ and $v=f_{2}(z,w)$, either is equivalent to one of the following germs of finite covers:

• $({\bf 1}_{1})$

$F_{{\bf 1}_{1}}=\{u=z,\quad v=w^{4}\}$, $G_{F_{{\bf 1}_{1}}}=\mathbb{Z}_{4}$, $T(B_{{\bf 1}_{1}})=A_{0}$;

• $({\bf 2}_{1})$

$F_{{\bf 2}_{1}}=\{u=z^{2},\quad v=w^{2}\}$, $G_{F_{{\bf 2}_{1}}}=Kl_{4}$,    $T(B_{{\bf 2}_{1}})=A_{1}$;

• $({\bf 3}_{1})$

$F_{{\bf 3}_{1},n}=\{u=z,\quad v=w^{4}-2z^{n}w^{2}\}$, $G_{F_{{\bf 3}_{1},n}}=\mathbb{D}_{4}$, $T(B_{{\bf 3}_{1},n})=A_{4n-1}$, $n\geqslant 1$;

• $({\bf 3}_{2})$

$F_{{\bf 3}_{2},n}=\{u=z^{2},\quad v=w^{2}-z^{2n+1}\}$, $G_{F_{{\bf 3}_{2},n}}=\mathbb{D}_{4}$, $T(B_{{\bf 3}_{2},n})=D_{2n+3}$, $n\geqslant 1$;

• $({\bf 4}_{1})$

$F_{{\bf 4}_{1},n}=\{u=z,\quad v=w^{4}+w^{3}z^{n}\}$, $G_{F_{{\bf 4}_{1},n}}=\mathbb{S}_{4}$, $T(B_{{\bf 4}_{1},n})=A_{8n-1}$, $n\geqslant 1$,

or is deformation equivalent to one of the following germs of finite covers:

• $({\bf 4}_{2})$

$F_{{\bf 4}_{2},n,\beta}=\{u=z,\qquad v=w^{4}+4wz^{3n+\beta}\}$,   $G_{F_{{\bf 4}_{2},n,\beta}}=\mathbb{S}_{4}$,   $T(B_{{\bf 4}_{2},n,\beta})=\newline T_{3}(4n+\beta,\beta)$, $n\geqslant 0$, $\beta=1,2$;

• $({\bf 4}_{3})$

$F_{{\bf 4}_{3},n,m}=\{u=z,\,\,v=w^{4}-\frac{8}{3}z^{m}w^{3}+2(z^{2m}-z^{2(m+n)+1})w^{2}\}$, $G_{F_{{\bf 4}_{3},n,m}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{3},n,m}})=T_{3,8m+6n+2,8m}$, $m\geqslant 1$, $n\geqslant 0$;

• $({\bf 4}_{4})$

$F_{{\bf 4}_{4},n,m}=\{u=z,\,\,v=w^{4}+12z^{m+n+1}w^{3}-2z^{2m+1}w^{2}\}$, $G_{F_{{\bf 4}_{4},n,m}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{4},n,m}})=T_{3,8m+2n+4,8m+4}$, $m\geqslant 1$, $n\geqslant 0$;

• $({\bf 4}_{5})$

$F_{{\bf 4}_{5},k,m}=\{u=z,\,\,v=w^{4}-\frac{8}{3}z^{m}(1+2z^{k})w^{3}+2z^{2m}(1+4z^{k}+3z^{2k})w^{2}\}$, $G_{F_{{\bf 4}_{5},k,m}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{5},k,m}})=T_{3,8m+6k-1,8m}$, $m\geqslant 1$, $k\geqslant 1$;

• $({\bf 4}_{6})$

$F_{{\bf 4}_{6},n,m}=\{u=z,\,\,v=w^{4}-\frac{4}{3}z^{m}(1+3z^{n})w^{3}+4z^{2m}(1+z^{n})w^{2}\}$, $G_{F_{{\bf 4}_{6},n,m}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{6},n,m}})=T_{3,8m+2n-1,8m}$, $m\geqslant 1$, $n\geqslant 1$;

• $({\bf 4}_{7})$

$F_{{\bf 4}_{7},m}=\{u=z,\,\,v=w^{4}-\frac{16}{3}z^{m}w^{3}+6z^{2m}w^{2}\}$, $G_{F_{{\bf 4}_{7},m}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{7},m}})=T_{3,8m-1,8m}$, $m\geqslant 1$;

• $({\bf 4}_{8})$

$F_{{\bf 4}_{8},n_{1},n_{2}}=\{u=z^{2}+w^{2n_{1}+1},\quad v=w^{2}+z^{2n_{2}+1}\}$, $G_{F_{{\bf 4}_{8}n_{1},n_{2}}}=\mathbb{S}_{4}$, $T(B_{F_{{\bf 4}_{8},n_{1},n_{2}}})=T_{4,2n_{1},2n_{2}}$, $n_{1},n_{2}\geqslant 1$;

In case $({\bf 4}_{2})$, any two germs of finite covers deformation equivalent to the germ $F_{{\bf 4}_{2},0,1}$ are equivalent.

Corollary 1**.**

The set of monodromy groups $G_{F}$ of the four-sheeted germs of finite covers is $\{\mathbb{Z}_{4},Kl_{4},\mathbb{D}_{4},\mathbb{S}_{4}\}.$

The alternating group $\mathbb{A}_{4}$ is not the monodromy group of any four-sheeted germ of finite cover.

Moreover, in Proposition 14 it is proved that the alternating group $\mathbb{A}_{4}$ is not the monodromy group of any germ of finite cover, nevertheless (see Proposition 15), the groups $\mathbb{A}_{2n-1}$ for $n\geqslant 2$ are the monodromy groups of germs of finite covers.

Note that all possible deformation types of four-sheeted germs of finite covers, described in Theorem 1, with the exception of two (germs $F_{{\bf 4}_{5},k,m}$ and $F_{{\bf 4}_{6},3k,m}$), have different pair of main invariants. The monodromy groups of the germs $F_{{\bf 4}_{5},k,m}$ and $F_{{\bf 4}_{6},3k,m}$ are the same and the singularity types of their branch curves are the same also, but deformation types of these covers differ in the singularity types of their ramification divisors. Therefore, the following problem is interesting.

Problem 1**.**

To find a finite set of invariants defining completely the deformation types of the germs of finite covers.

In particular, would be interesting to know if there are deformation non-equivalent germs of finite covers of local degree $d\geqslant 5$, the singularity types of branch curves and ramification divisors of which are the same, the covers have conjugated monodromy groups, and the sets of conjugacy classes of the images of geometric generators of the local fundamental groups of the branch curves are the same also.

In Section 1, the main definitions are given and several auxiliary lemmas are proved. In this section, we consider different aspects of the relationship between the deformation equivalence of germs of finite covers, the singularity types of their branch curves, and the monodromy groups of these germs. The proof of Theorem 1 is given in Section 2. In Section 3, we give examples of finite groups that are not the monodromy groups of any germs of finite covers.

1. Definitions and preliminary facts

1.1. Equisingular deformations of curve germs.

Let

[TABLE]

be a ball of radius $r$ with center at the point $o$ and $C\subset\mathbb{B}_{r}$ a reduced curve given by equation $h(u,v)=0$, where $h(u,v)\in H^{0}(\mathbb{B}_{r},\mathcal{O}_{\mathbb{B}_{r}})$. Below, the words ”$B$ is a curve germ in $(V,o)$”  means that $V=\mathbb{B}_{\varepsilon}$, were $\varepsilon\ll r$, and $B=C\cap V$.

Let $B_{1},\dots,B_{k}$ be the irreducible components of a curve germ $B\subset V$ and $\sigma=\sigma_{1}\circ\dots\circ\sigma_{n}:V_{n}\to V$ the minimal sequence of $\sigma$-processes $\sigma_{i}:V_{i}\to V_{i-1}$ with centers at points resolving the singular point $o$ of $B$ and such that $\sigma^{-1}(B)$ is a divisor with normal crossings. Denote by $E_{i+k}\subset V_{n}$ the proper inverse image of the exceptional curve of $\sigma_{i}$ and $B_{j}^{\prime}\subset V_{n}$ the proper inverse image of $B_{j}$. Put $\delta_{i,j}:=(E_{i},E_{j})_{V_{n}}$ for $k+1\leqslant i,j\leqslant k+n$, $i\neq j$; $\delta_{i,j}:=(B^{\prime}_{i},E_{j})_{V_{n}}$ for $1\leqslant i\leqslant k$ and $k+1\leqslant j\leqslant k+n$; $\delta_{i,j}:=0$ for $1\leqslant i,j\leqslant k$, and $\delta_{i,i}:=0$ for $1\leqslant i\leqslant k+n$. Note that, by definition, $\delta_{i,j}$ takes only two values: [math] or $1$.

Let $\Gamma(B)$ be the graph of the curve germ $B$.

Definition 1**.**

The graph $\Gamma(B)$ is a weighted graph having $n+m$ vertices $v_{i}$. The vertices $v_{i}:=b_{i}$, $i=1,\dots,k$, are in one-to-one correspondence with the curve germs $B^{\prime}_{1},\dots,B^{\prime}_{m}$ and they have weights $w_{i}=0$. The vertices $v_{i+k}:=e_{i+m}$, $i=1,\dots,n$, are in one-to-one correspondence with the curves $E_{1+k},\dots,E_{n+k}$ and they have weights $w_{i+k}=(E_{i+k}^{2})_{V_{n}}$. The vertices $v_{i}$ and $v_{j}$ are connected by the edge of $\Gamma(B)$ if and only if $\delta_{i,j}=1$.

Note that the graph $\Gamma(B)$ is a tree if $(B,o)$ is not a divisor with normal crossings consisting of two components.

By definition, a family of curve germs is a triple $(\mathcal{V}=V\times D_{\varepsilon},\mathcal{B},\text{pr}_{2})$, where $V=\mathbb{B}_{r}\subset\mathbb{C}^{2}$ is a ball, $D_{\varepsilon}=\{t\in\mathbb{C}\mid|t|<\varepsilon\}$ is a disc in the complex plane, $\mathcal{B}$ is an effective reduced divisor in $\mathcal{V}$, the restriction to $\mathcal{B}$ of the projection $\text{pr}_{2}:\mathcal{V}\to D_{\varepsilon}$ is a flat holomorphic map.

Definition 2**.**

([17], see also [16])* A family $(\mathcal{V},\mathcal{B},\text{pr}_{2})$ is an equisingular deformation of curve germs if $\operatorname{Sing}\,\mathcal{B}=\{o\}\times D_{\varepsilon}$ and there exists a finite sequence of $n$ monoidal transformations (blowups) $\widetilde{\sigma}_{i}:\mathcal{V}_{i}\to\mathcal{V}_{i-1}$ (where $\mathcal{V}_{0}=\mathcal{V}$) with centers in smooth curves $\mathcal{S}_{i}\subset\operatorname{Sing}\,\mathcal{B}_{i}$, where $\mathcal{B}_{0}=\mathcal{B}$ and $\mathcal{B}_{i+1}=\widetilde{\sigma}_{i}^{-1}(\mathcal{B}_{i})$, and such that*

• $(i)$

$Sing\,\mathcal{B}_{i}$* is a disjoint union of sections of $\text{pr}_{2}\circ\widetilde{\sigma}_{1}\circ\dots\circ\widetilde{\sigma}_{i-1}$ for each $i$,*

• $(ii)$

$\mathcal{B}_{n}$* is a divisor with normal crossings in $\mathcal{V}_{n}$.*

Note that in this case the divisor $\mathcal{B}_{n}$ has only simple double points as its singular points.

We say that two curve germs are strongly equisingular equivalent if they can be imbedded in an equisingular deformation of curve germs as fibres of $\text{pr}_{2}$. Let us continue the strong equisingular equivalence to an equivalence relation and will say that two curve germs have the same singularity type iff they are equisingular equivalent.

The following two propositions are well known.

Proposition 1**.**

([16])* Two curve germs $(B_{1},o)$ and $(B_{2},o)$ are equisingular equivalent if and only if their graphs $\Gamma(B_{1})$ and $\Gamma(B_{2})$ are isomorphic as weighted graphs.*

We say that a curve germ $(B,o)\subset(V,o)$ is rigid if for any curve germ $(B_{1},o)\subset(V,o)$ equisingular equivalent to $(B,o)$, there is a biholomorphic mapping $G:(V,o)\to(V,o)$ such that $G(B_{1})=B$.

Proposition 2**.**

([1])* Curve germs $(B,o)$ having one of the following singularity types $A_{n}$, $D_{n}$, $E_{6},$, $E_{7}$, $E_{8}$ are rigid.*

The imbedding $\mathbb{B}_{r_{2}}\subset\mathbb{B}_{r_{1}}\subset\mathbb{B}_{r}$ of balls, $r_{2}\leqslant r_{1}\leqslant r$, induces a homomorphism $i_{*}:\pi_{1}(\mathbb{B}_{r_{2}}\setminus B)\to\pi_{1}(\mathbb{B}_{r_{1}}\setminus B)$ of the fundamental groups. The following theorem is well known (see, for example, [5]).

Theorem 2**.**

There is a radius $r_{0}$ such that for any $\varepsilon\leqslant r_{0}$ the homomorphism $i_{*}:\pi_{1}(\mathbb{B}_{\varepsilon}\setminus B)\to\pi_{1}(\mathbb{B}_{r_{0}}\setminus B)$, induced by ball imbedding, is an isomorphism.

The group $\pi_{1}^{loc}(B,o):=\pi_{1}(\mathbb{B}_{r_{0}}\setminus B)$ is called the local fundamental group of $B$.

We say that an equisingular deformation $(\mathbb{B}_{\varepsilon}\times D_{\delta},\mathcal{B},\text{pr}_{2})$ is strong if $\varepsilon<r(B_{\tau})$ for all $\tau\in D_{\delta}$. Let $l:[0,1]=\{0\leqslant t\leqslant 1\}\to D_{\delta}$ be a smooth path in $D_{\delta}$ and $(\mathbb{B}_{\varepsilon}\times D_{\delta},\mathcal{B},\text{pr}_{2})$ a strong equisingular deformation. In this case it is easy to show that $\text{pr}_{2}:(\mathbb{B}_{\varepsilon}\times D_{\delta}\setminus\mathcal{B})\times_{D_{\delta}}[0,1]\to[0,1]$ is a $C^{\infty}$-trivial fibration with a fibre $\mathbb{B}_{\varepsilon}\setminus B_{l(0)}$ and, in particular, the fundamental groups $\pi_{1}((\mathbb{B}_{\varepsilon}\times D_{\delta}\setminus\mathcal{B})\times_{D_{\delta}}[0,1],(p_{t},l(t))$ and $\pi_{1}(\mathbb{B}_{\varepsilon}\setminus B_{l(t)},p_{t})$ are naturally isomorphic for each $t\in[0,1]$ and $p_{t}\in\mathbb{B}_{\varepsilon}\setminus B_{l(t)}$. Below, for each smooth path $l$ in $D_{\delta}$, we fix one of $C^{\infty}$-trivializations of the fibration $\text{pr}_{2}:(\mathbb{B}_{\varepsilon}\times D_{\delta}\setminus\mathcal{B})\times_{D_{\delta}}[0,1]\to[0,1]$ and call it an equipment of $l$.

1.2. $D$-automorphisms.

Denote by $\mathfrak{B}_{T}$ the infinite dimensional subvariety of the variety of convergent power series in $\mathbb{C}[[z,w]]$, consisting of all power series $h(u,v)$ such that the germs of curves given by equations $h(u,v)=0$ in $(V,o)$, have the same type of singularity $T$ (i.e., all of these germs are equisingular deformation equivalent). Consider a variety

[TABLE]

and define in this variety smooth paths $l:[0,1]=\{0\leqslant t\leqslant 1\}\to\widetilde{\mathfrak{B}}_{T}$ of three types (below, elementary admissible paths); the smoothness of a path $l=(h_{t},\varepsilon_{t},p_{t})$ means that the coefficients of the series $h_{t}$ and the points $\varepsilon_{t}$ and $p_{t}$ smoothly depend on $t$). The first type consists of the paths $l(t)=(h_{t},\varepsilon_{t},p_{t})$ in which $h_{t}=h_{0}$ and $p_{t}=p_{0}$ for all $t\in[0,1]$. The second type consists of the paths $l(t)=(h_{t},\varepsilon_{t},p_{t})$ in which $h_{t}=h_{0}$ and $\varepsilon_{t}=\varepsilon_{0}$ for all $t\in[0,1]$. We call the paths of the first and second types the [math]-paths. The third type consists of the paths (we call them the $d$-paths) $l(t)=(h_{t},\varepsilon_{t},p_{t})$ for which

• $(i)$

$\varepsilon_{t}=\varepsilon_{0}$ for all $t\in[0,1]$,

• $(ii)$

there exist a smooth path $\widetilde{l}:[0,1]\to D_{\delta}$ and a strong equisingular deformation $(\mathbb{B}_{\varepsilon}\times D_{\delta},\mathcal{B},\text{pr}_{2})$ of the curve germ $B_{h_{0}}$ such that $B_{h_{t}}=\mathcal{B}\cap\text{pr}_{2}^{-1}(\widetilde{l}(t))$ for all $t\in[0,1]$ and $\{(p_{t},t)\mid t\in[0,1]\}$ is a constant section of the equipment of $\widetilde{l}$.

Obviously, each elementary admissible path $l$ defines an isomorphism from the group $\pi_{1}(\mathbb{B}_{\varepsilon_{l(0)}}\setminus B_{h_{l(0)}},p_{l(0)})$ to $\pi_{1}(\mathbb{B}_{\varepsilon_{l(1)}}\setminus B_{h_{l(1)}},p_{l(1)})$.

An admissible path $l$ in $\widetilde{\mathfrak{B}}_{T}$ is a finite sequence of elementary admissible paths $(l_{1},\dots,l_{n})$, in which the end of the path $l_{i}$ coincides with the beginning of the path $l_{i+1}$ for $1\leqslant i\leqslant n-1$.

We fix a point $\widetilde{h}_{0}=(h_{0}(u,v),\varepsilon_{0},p_{0})\in\widetilde{\mathfrak{B}}_{T}$ and call it (and, resp., the curve germ $(B_{h_{0}},o)$ given by $h_{0}(u,v)=0$) a base representative of singularity type $T$. Denote by $\Omega_{T}(\widetilde{h}_{0})$ the space of all admissible loops in $\widetilde{\mathfrak{B}}_{T}$ beginning at the point $\widetilde{h}_{0}$. Obviously, $\Omega_{T}(\widetilde{h}_{0})$ is a group and a natural homomorphism

[TABLE]

is correctly defined. The image $\mathfrak{D}_{T}:=\text{{\bf Def}}(\Omega_{T}(\widetilde{h}_{0}))$ is called the $D$-automorphism group. Note that $\mathfrak{D}_{T}$ contains the group of inner automorphisms of the group $\pi_{1}^{loc}(B_{h_{0}},o)$.

1.3. On the fundamental groups of the complements of curve germs.

Zariski – van Kampen Theorem (see below) gives an approach to finding representations of the local fundamental groups of curve germs. It is based on the calculation of the braid monodromy of the singularity and consists in the following (for a more detailed description, see, for example, [11]). Let a germ $(B,o)\subset(V,o)$ of a reduced holomorphic curve do not contain the germ $u=0$ and $m$ the multiplicity of the singularity of $B$ at the point $o$. Assume also that it is given by an equation

[TABLE]

where $q_{i}(u)$ are convergent power series in $V=\mathbb{B}_{\varepsilon}$, $q_{i}(0)=0$, and the polynomial $v^{m}+\sum q_{i}(u)v^{m-i}\in\mathbb{C}[[u]][v]$ has not multiple factors. Therefore, one can choose a small bidisk $\overline{D}=\overline{D}_{1}\times\overline{D}_{2}\subset\mathbb{B}_{\varepsilon}$,

[TABLE]

such that:

• $(P1)$

the projection on the $u$-factor $\text{pr}=\text{pr}_{1}:B\cap\overline{D}\to\overline{D}_{1}$ is a proper finite map of degree $m$,

• $(P2)$

$|v|<\varepsilon_{2}$ for each point $(u,v)\in\text{pr}^{-1}(\overline{D}_{1})$ and $o=(0,0)$ is the unique critical point of $\mathsf{pr}_{\mid B\cap\overline{D}}$.

Let us pick a point $(p_{1},p_{2})\in\partial\overline{D}_{1}\times\partial\overline{D}_{2}$, $p_{1}=\varepsilon_{1}$, $p_{2}=e^{3\pi i/2}\varepsilon_{2}$, and put

[TABLE]

The fundamental group $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})\simeq\mathbb{F}_{m}$ is the free group of rank $m$ and it is generated by $m$ bypasses $\gamma_{1},\dots,\gamma_{m}$ around the points $v_{1}\dots,v_{m}$. An ordered set $\{\gamma_{1},\dots,\gamma_{m}\}$ is called a good geometric base of $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})$ if the product $\gamma_{1}\cdot.\,.\,.\,\cdot\gamma_{m}$ is equal to the element in $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})$ represented by the circuit along the circle $\partial D_{2}$ in positive direction.

Note that $\text{pr}^{-1}((\varepsilon_{1}t,\varepsilon_{2}))$, $0<t\leqslant 1$, is the disjoint union of $m$ paths. Therefore, hereafter, we can identify groups $\pi_{1}(\overline{D}_{2}\setminus K_{B}(\varepsilon_{1}t),p_{2})$ for different $t$ and if we need to decrease $\varepsilon_{2}$, we will identify the groups $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})$ using the path $v(t)=e^{3\pi i/2}\varepsilon_{2}t$.

An example of a good geometric base of $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})$, where $B$ is given by equation $v((v-ku^{k})^{2}-u^{2k+n})=0$, is shown in Fig. 1 in which $\varepsilon<\varepsilon_{2}$ and $v_{1}=k\varepsilon^{k}+\sqrt{\varepsilon^{2k+n}}$, $v_{2}=k\varepsilon^{k}-\sqrt{\varepsilon^{2k+n}}$, $v_{3}=0$ if $k\geqslant 1$, and $v_{1}=\sqrt{\varepsilon^{n}}$, $v_{2}=0$, $v_{3}=-\sqrt{\varepsilon^{n}}$ if $k=0$.

\begin{picture}(400.0,100.0)\par\put(185.0,100.0){\vector(0,-1){1.0}} \put(200.0,100.0){\circle*{2.0}} \put(203.0,100.0){$v_{3}$} \put(200.0,100.0){\circle{30.0}} \put(200.0,0.0){\circle*{2.0}} \put(200.0,0.0){\line(0,1){85.0}} \put(188.0,50.0){$\gamma_{3}$} \put(188.0,0.0){$p_{2}$} \put(265.0,100.0){\vector(0,-1){1.0}} \put(280.0,100.0){\circle*{2.0}} \put(283.0,100.0){$v_{2}$} \put(280.0,100.0){\circle{30.0}} \put(200.0,0.0){\line(3,4){68.0}} \put(225.0,50.0){$\gamma_{2}$} \put(305.0,100.0){\vector(0,-1){1.0}} \put(320.0,100.0){\circle*{2.0}} \put(323.0,100.0){$v_{1}$} \put(320.0,100.0){\circle{30.0}} \put(200.0,0.0){\line(4,3){113.0}} \put(280.0,50.0){$\gamma_{1}$} \put(190.0,-30.0){$\text{Fig.}\,1$} \end{picture}

v_{3}$$\gamma_{3}$$p_{2}$$v_{2}$$\gamma_{2}$$v_{1}$$\gamma_{1}$$\text{Fig.}\,1

To recall the definition of the braid monodromy $b_{(B,o)}$ of $B$, consider the braid group $\text{Br}_{m}$. The group $\text{Br}_{m}$ has the following presentation. It generated by elements $a_{1},\dots,a_{m-1}$ being subject to relations

[TABLE]

(such generators of the braid group $\text{Br}_{m}$ are called Artin’s or standard). The element $\Delta_{m}=(a_{1}a_{2}\dots a_{m-1})^{m}$ is called the full twist and it belongs to the center of $\text{Br}_{m}$.

The loop $\partial\overline{D}_{1}$ oriented counter-clockwise and starting at $p_{1}$ lifts, via $\text{pr}^{-1}(\partial\overline{D}_{1})\cap B$, to a motion $\mathsf{pr}_{2}(\{v_{1}(t),\dots,v_{m}(t)\})$ of $m$ distinct points in $\overline{D}_{2}$ starting and ending at $K_{B}$. This motion defines a braid $b_{(B,o)}\in\text{Br}_{m}$ which is called the braid monodromy of $(B,o)$ with respect to pr.

Lemma 1**.**

([11], Lemma 4.1)* Let $(B,o)$ be the singularity of type $A_{n}$, then*

[TABLE]

Lemma 2**.**

Let $(B,o)$ be the singularity of type $T_{3,4k+n,4k}$, then $b_{(B,o)}=\Delta_{3}^{k}a_{1}^{n+1}.$

Proof.

The braid $b_{(B,o)}$ consists of three threads

[TABLE]

in the space $\overline{D}_{2}\times\{0\leqslant t\leqslant 1\}$ starting at points of $K_{B}\times\{0\}=\{v_{1},v_{2},v_{3}\}\times\{0\}$; threads $v_{1}(t)$ and $v_{2}(t)$ make parallel $2k$ full twists around the thread $v_{3}(t)$ and make twists around each other $2k+\frac{n+1}{2}$ times. It is easy to see that the braid consisting of one parallel turn of $v_{1}(t)$ and $v_{2}(t)$ around $v_{3}(t)$ equals to $a_{2}a_{1}^{2}a_{2}$ and one half-twist of $v_{1}(t)$ and $v_{2}(t)$ is the braid $a_{1}$. It is easy to check that $a_{2}a_{1}^{2}a_{2}$ and $a_{1}$ commute in $\text{Br}_{3}$ and $(a_{2}a_{1}^{2}a_{2})a_{1}^{2}=\Delta_{3}$. ∎

Lemma 3**.**

([11], Lemma 4.1)* Let $(B,o)$ be the singularity of type $T_{3,n-1,n}$, then*

[TABLE]

Proof.

The braid $b_{(B,o)}$ consists of three threads

[TABLE]

in the space $\overline{D}_{2}\times\{0\leqslant t\leqslant 1\}$ starting at points of $K_{B}\times\{0\}=\{v_{1},v_{2},v_{3}\}\times\{0\}$. Threads $v_{1}(t)$ and $v_{3}(t)$ make $n$ half-twists around the thread $v_{2}(t)$. It is easy to see that the braid consisting of one half-twist of $v_{1}(t)$ and $v_{3}(t)$ around $v_{2}(t)$ equals to $a_{1}a_{2}a_{1}=a_{2}a_{1}a_{2}$. ∎

Lemma 4**.**

Let $(B,o)$ be the singularity of type $D_{n}$, $n\geqslant 4$, then $b_{(B,o)}=\Delta_{3}a_{2}^{n-4}.$

Proof.

As a representative $(B,o)$ of the singularity type $D_{n}$, take the germ given by equation $(u-v)(v^{2}-u^{n-2})=0$. Then, as with the proof of the Lemma 2, it is easy to see that the braid $b_{(B,o)}$ consists of three strands, the first of which makes a full twist around the other two strands, and these two strands are twisted together $n/2-1$ times. ∎

Lemma 5**.**

([11], Lemma 4.1)* Let $(B,o)$ be the singularity of type $T_{3}(n,\beta)$, then*

[TABLE]

Fix a good geometric base $\gamma_{1},\dots,\gamma_{m}$ of $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})$. The braid group $\text{Br}_{m}$ acts on the free group $\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})\simeq\mathbb{F}_{m}$ as follows

[TABLE]

In particular,

[TABLE]

The imbedding $i:\{p_{1}\}\times\overline{D}_{2}\hookrightarrow\overline{D}_{1}\times\overline{D}_{2}\hookrightarrow\mathbb{B}_{\varepsilon}$ defines a homomorphism

[TABLE]

Zariski – van Kampen Theorem. The homomorphism $i_{*}:\pi_{1}(\overline{D}_{2}\setminus K_{B},p_{2})\to\pi_{1}^{loc}(B,o)$ is an epimorphism. The group $\pi_{1}^{loc}(B,o)$ has the following presentation:

[TABLE]

where $\gamma_{1},\dots,\gamma_{m}$ is a good geometric base of $\pi_{1}^{loc}(B,o)$.

The following two Lemmas are well known (see, for example, [11]) and easily follows from Lemmas 1 and 5.

Lemma 6**.**

Let $(B,o)$ has a singularity of type $A_{n}$, where $n=2k-\delta$ and $\delta=0$ or $1$. Then

[TABLE]

Lemma 7**.**

Let $(B_{n,\beta},o)$ has a singularity of type $T_{3}(n,\beta)$. If $\beta=1$ then

[TABLE]

and if $\beta=2$ then

[TABLE]

1.4. $WD$-subgroups of $D$-automorphism groups.

In this subsection, we use the notations and agreements of the previous paragraph. Denote by $\mathcal{W}_{T}$ a subspace of the space $\mathfrak{B}_{T}$ consisting of the power serious of the form (1) and, respectively, $\widetilde{\mathcal{W}}_{T}=\{\widetilde{h}=(h,\varepsilon,(\varepsilon_{1},e^{3\pi i/2}\varepsilon_{2}))\in\widetilde{\mathfrak{B}}_{T}\mid h\in\mathcal{W}_{T},\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}_{+}\}$, where $\varepsilon_{1}$ and $\varepsilon_{2}$ are such that $h(u,v)$ has properties $(P1)$ and $(P2)$ in the bidisk $\overline{D}=\overline{D}_{1}(\varepsilon_{1})\times\overline{D}_{2}(\varepsilon_{2})\subset\mathbb{B}_{\varepsilon}$.

A $d$-path $l=(h_{t},\varepsilon,(\varepsilon_{1},e^{3\pi i/2}\varepsilon_{2}))$ in $\widetilde{\mathcal{W}}_{T}$ is called a $wd$-path if the restriction of the equipment of $l$ to $\{\partial_{2}\overline{D}\times\{l(t)\}\mid t\in[0,1]\}$ is a trivial fibration $\text{pr}_{2}:\partial_{2}\overline{D}\times[0,1]\to[0,1]$, where $\partial_{2}\overline{D}=\overline{D}_{1}(\varepsilon_{1})\times\partial\overline{D}_{2}(\varepsilon_{2})$.

A loop $l=(l_{1},\dots,l_{k},l_{k+1},\dots,l_{k+n},l_{k}^{-1},\dots,l_{1}^{-1})$ in $\widetilde{\mathcal{W}}_{T}$, where $l_{1},\dots,l_{k}$ are [math]-paths and $l_{k+1},\dots,l_{k+n}$ – $wd$-paths, is called a $w$-loop.

Fix a point $\widetilde{h}_{0}=(h_{0}(u,v),\varepsilon,(\varepsilon_{1},e^{3\pi i/2}\varepsilon_{2}))$ in $\widetilde{\mathcal{W}}_{T}\subset\widetilde{\mathfrak{B}}_{T}$ as the base representative of singularity type $T$ and denote by $\Omega_{T,W}(\widetilde{h}_{0})$ the subgroup of $\Omega_{T}(\widetilde{h}_{0})$ generated by the $w$-loops in $\widetilde{\mathfrak{B}}_{T}$ beginning at the point $\widetilde{h}_{0}$.

Fix a good geometric base ofI $\pi_{1}(\overline{D}_{2}(\varepsilon_{2})\setminus K_{B_{h_{0}}},e^{3\pi i/2}\varepsilon_{2})$. Due to the identification of groups $\pi_{1}(\overline{D}_{2}(\lambda_{2}e^{3\pi i/2}\varepsilon_{2})\setminus K_{B_{r}}(\lambda_{1}\varepsilon_{1}),\lambda e^{3\pi i/2}\varepsilon_{2}))$ for all $\lambda_{1}$, $\lambda_{2}\in(0,1)$ defined in Subsection 1.3, the movements along $w$-loops define a homomorphism $\text{{\bf def}}:\Omega_{T,W}(\widetilde{h}_{0})\to\text{Br}_{m}\subset\text{Aut}(\pi_{1}(\overline{D}_{2}(\varepsilon_{2})\setminus K_{B_{h_{0}}},e^{3\pi i/2}\varepsilon_{2}))$ such that

[TABLE]

for all $l\in\Omega_{T,W}(\widetilde{h}_{0})$ and $\gamma\in\pi_{1}(\overline{D}_{2}\setminus K_{B_{h_{0}}},p_{2})$. The images $\mathfrak{d}_{T,W}:=\text{{\bf def}}(\Omega_{T,W}(\widetilde{h}_{0}))$ and $\mathfrak{D}_{T,W}:=\text{{\bf Def}}(\Omega_{T,W}(\widetilde{h}_{0}))\subset\mathfrak{D}_{T}$ are called the $WD$-automorphism groups.

Lemma 8**.**

If $v((v-u^{k})^{2}-u^{2k+n+1})=0$ is the equation of $(B_{h_{0}},o)$, $k\geqslant 1$, $n\geqslant 0$, then the braids $a_{1}$ and $a_{2}a_{1}^{2}a_{2}$ contain in $\mathfrak{d}_{T_{3,4k+n,4k},W}\subset Br_{3}$.

Proof.

It is easy to see that the loop $l=\{h_{t}\mid 0\leqslant t\leqslant 1\}$, given by

[TABLE]

in $\mathfrak{B}_{T_{3,4k+n,4k}}$ can be lifted in $\Omega_{T_{3,2k+n,2k},W}(\widetilde{h}_{0})$ and $\text{\bf def}(\widetilde{l})$ is the standard generator $a_{1}\in Br_{3}$. Similarly, the loop $l=\{h_{t}\mid 0\leqslant t\leqslant 1\}$ given by $v[(v-e^{2\pi ti}u^{k})^{2}-u^{2k+n+1}]=0$ in $\mathfrak{B}_{T_{3,2k+n,2k}}$ gives the element $a_{2}a_{1}^{2}a_{2}\in\mathfrak{d}_{T_{3,2k+n,2k}W}$. ∎

Lemma 9**.**

If $v(v^{2}-u^{2n})=0$ is the equation of $(B_{h_{0}},o)$, then $\mathfrak{d}_{T_{3,2n-1,2n},W}=Br_{3}$.

Proof.

It is easy to see that for uplifts $\widetilde{l}_{i}$ in $\Omega_{T_{3,2n-1,2n},W}(\widetilde{h}_{0})$ of the loops $l_{i}$, given by

[TABLE]

[TABLE]

in $\mathfrak{B}_{T_{3,2n-1,2n}}$, their images $\text{\bf def}(\widetilde{l}_{i})$ are the standard generators $a_{1}$ and $a_{2}$ of the braid group $Br_{3}$ acting on $\pi_{1}(\overline{D}_{2}\setminus K_{B_{h_{0}}})$. ∎

1.5. Deformation equivalence of germs of covers.

Consider the germ of a finite cover $F:(U,o^{\prime})\to(V,o)$, $\deg F=d$. Choose local coordinates $z,w$ in $(U,o^{\prime})$ and $u,v$ in $(V,o)$. The germ $F$ is given by two functions

[TABLE]

where $f_{i}(z,w)\in H^{0}(U,\mathcal{O}_{U})$. The ramification divisor $R$ in $U$ is defined by equation

[TABLE]

and let $B=F(R_{red})\subset(V,o)$ be the branch curve of the germ of finite cover $F$.

Remark 1**.**

The divisor $R\subset(U,o^{\prime})$ and the curve germ $B\subset(V,o)$ depend only on $F$ and do not depend on the choice of coordinates in $(U,o^{\prime})$ and $(V,o)$.

Definition 3**.**

Let $\mathcal{F}:(U,o^{\prime})\times D_{\delta}\to(V,o)\times D_{\delta}$ be a finite holomorphic mapping branched along a surface $\mathcal{B}\subset(V,o)\times D_{\delta}$ and such that $\text{pr}_{2}\circ\mathcal{F}=\text{Pr}_{2}$, where $\text{Pr}_{2}:(U,o^{\prime})\times D_{\delta}\to D_{\delta}$ and $\text{pr}_{2}:(V,o)\times D_{\delta}\to D_{\delta}$ are the projections to the second factor. The cover $\mathcal{F}$ is called a strong deformation of the germ of finite cover $F_{0}=\mathcal{F}_{\mid(U,o^{\prime})\times\{0\}}:(U,o^{\prime})\times\{0\}\to(V,o)\times\{0\}$ if $((V,o)\times D_{\delta},\mathcal{B},\text{pr}_{2})$ is a strong equisingular deformation of the curve germ $B_{0}=\mathcal{B}\cap\text{pr}_{2}^{-1}(0)$ and the germs of finite covers $F_{\tau}=\mathcal{F}_{\mid(U,o^{\prime})\times\{\tau\}}:(U,o^{\prime})\times\{t\}\to(V,o)\times\{\tau\}$ are called strong deformation equivalent to the germ $F_{0}$.

Let us continue the strong deformation equivalence of germs of finite covers to an equivalence relation.

The germ of finite cover $F:(U,o^{\prime})\to(V,o)$ defines a homomorphism

[TABLE]

where $\mathbb{S}_{d}$ is the symmetric group acting on the fibre $F^{-1}(p)$. Note that the homomorphism $F_{*}$ is defined uniquely only if we fix a numbering of the points of $F^{-1}(p)$ and in general case it is defined uniquely up to inner automorphism of $\mathbb{S}_{d}$. The group $G_{F}=\text{im}F_{*}$ is called the (local) monodromy group of the germ $F$. The group $G_{F}$ is a transitive subgroup of $\mathbb{S}_{d}$, since $(U,o^{\prime})$ is an irreducible germ of a smooth surface and hence $U\setminus F^{-1}(B)$ is connected. By Grauert - Remmert - Riemann - Stein Theorem ([14]), the epimorphism $F_{*}:\pi_{1}(V\setminus B)\to G_{F}\subset\mathbb{S}_{d}$ uniquely determines the germ of a finite cover $F$.

Let $(V\times D_{\delta},\mathcal{B},\text{pr}_{2})$   be a strong equisingular deformation of a germ $(B,o)=\mathcal{B}\cap\text{pr}_{2}^{-1}(0)$. By Grauert - Remmert - Riemann - Stein Theorem, the homomorphism $F_{*}:\pi_{1}^{loc}(B,o)\simeq\pi_{1}((V\times D_{\delta})\setminus\mathcal{B})\to\mathbb{S}_{d}$ defines a finite $d$-sheeted cover $\mathcal{F}:\mathcal{U}\to V\times D_{\delta}$, where in general case $\mathcal{U}$ is a normal complex-analytic variety. But, if $F_{*}$ is defined by a germ of finite cover $F:(U,o^{\prime})\to(V,o)$ branched in $(B,o)$ then we have

Proposition 3**.**

The cover $\mathcal{F}:\mathcal{U}\to V\times D_{\delta}$ is a strong deformation of the germ of finite cover $F=\mathcal{F}_{\mid\mathcal{F}^{-1}((V,o)\times\{0\})}:(U,o^{\prime})\times\{0\}\to(V,o)\times\{0\}$.

Proof.

First of all, note that $\mathcal{F}^{-1}(V\times\{0\})=(U,o^{\prime})$ and $\mathcal{F}_{\mid U}=F$.

In notations of Definition 2, the homomorphism $F_{*}$ defines a finite cover $\widetilde{\mathcal{F}}:\widetilde{\mathcal{U}}\to\mathcal{V}_{n}$ branched in $\mathcal{B}_{n}$. Since $\mathcal{B}_{n}$ is a divisor with normal crossings and all its singular points are sections of $\text{pr}_{2}\circ\widetilde{\sigma}_{1}\circ\dots\circ\widetilde{\sigma}_{n}$, then the local fundamental groups of the complement to $\mathcal{B}_{n}$ at the points of $\mathcal{B}_{n}$ are abelian and hence $Sing\,\widetilde{\mathcal{U}}$ is, first, a disjoint union $\bigsqcup\widetilde{S}_{j}$ of sections $\widetilde{S}_{j}$ of $\text{pr}_{2}\circ\widetilde{\sigma}_{1}\circ\dots\circ\widetilde{\sigma}_{n}\circ\widetilde{F}$ lying over $Sing\,\mathcal{B}_{n}$ and second, at the points of $\widetilde{S}_{j_{0}}\subset Sing\,\widetilde{\mathcal{U}}$, the variety $\widetilde{\mathcal{U}}$ locally biholomorphic to $W_{j_{0}}\times D_{\delta_{1}}$, where $W_{j_{0}}$ is a germ of two-dimensional cyclic quotient singularity depending on the local monodromy at the points of $S_{j_{0}}\subset Sing\,\mathcal{B}_{n}$ (see details in [2], III.6). The minimal resolutions of singularities $\rho_{j}:\overline{W}_{j}\to W_{j}$ defines a resolution of singularities $\rho:\overline{\mathcal{U}}\to\widetilde{\mathcal{U}}$. By Stein Factorisation Theorem, there is a holomorphic mapping $\Sigma:\overline{\mathcal{U}}\to\mathcal{U}$ contracting the divisor $(\mathcal{F}\circ\Sigma)^{-1}(\mathcal{E})$ to the section $\mathcal{F}^{-1}(\{o\}\times D_{\varepsilon})$ of projection $\mathcal{F}\circ\text{pr}_{2}$, where $\mathcal{E}$ is the exceptional divisor of the sequence of the monoidal transformations $\widetilde{\sigma}_{1}\circ\dots\circ\widetilde{\sigma}_{n}$.

Consider the restriction $\sigma:=\Sigma_{\mid\Sigma^{-1}((U,o^{\prime}))}:\overline{U}=\Sigma^{-1}((U,o^{\prime}))\to(U,o^{\prime})$. The exceptional divisor $E$ of $\sigma$ is $E=\mathcal{E}\cap\overline{U}$. By Zariski Theorem $\sigma$ is a composition of $\sigma$-processes, since $\overline{U}$ and $U$ are nonsingular and $\sigma$ is a bimeromorphic holomorphic mapping. It follows from [6] (see Chapter 2, Example 6.2.2.) and Nokano Contractibility Criterion ([13]) that $\Sigma$ is the composition of monoidal transformations (being in one-to-one correspondence with the composition of $\sigma$-processes $\sigma$) of smooth threefolds with centers in sections of the projection to $D_{\delta}$. ∎

The following proposition easily follows from the proof of Proposition 3.

Proposition 4**.**

If $F_{1}:(U,o^{\prime})\to(V,o)$ and $F_{2}:(U,o^{\prime})\to(V,o)$ are deformation equivalent covers, then their ramification divisors $R_{1,red}$ and $R_{2,red}$ are equisingular deformation equivalent.

Let $\gamma_{1},\dots,\gamma_{m}$ be a good geometric base of the fundamental group $\pi_{1}^{loc}(B,o)$ and $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{d}$ a homomorphism to the symmetric group $\mathbb{S}_{d}$ such that $G_{F}=\text{im}\,F_{*}$ is a transitive subgroup of $\mathbb{S}_{d}$. The collection $\{C_{1},\dots,C_{m}\}$ of conjugacy classes in $\mathbb{S}_{d}$, $F_{*}(\gamma_{i})\in C_{i}$ for $i=1,\dots,m$, is called the dataset of the homomorphism $F_{*}$. We say that the homomorphism $F_{*}$ is sole if it is uniquely, up to inner automorphisms of $\mathbb{S}_{d}$, defined by its dataset.

We say that two homomorphisms $F_{i*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{d}$, $i=1,2$, are equivalent if they differ from each other by an inner automorphism of $\mathbb{S}_{d}$ and they are deformation equivalent if they differ from each other on a $D$-automorphism of $\mathfrak{D}_{T}$ and an inner automorphism of $\mathbb{S}_{d}$.

From the above it follows

Corollary 2**.**

Two germs of finite covers $F_{1}:(U,o^{\prime})\to(V,o)$ and $F_{2}:(U,o^{\prime})\to(V,o)$ of degree $d$, branched along $(B_{1},o)\subset(V,o)$ and $(B_{2},o)\subset(V,o)$, are deformation equivalent if and only if the curve germs $(B_{1},o)$ and $(B_{2},o)$ are equisingular deformation equivalent and the monodromies $F_{1*}:\pi_{1}\to\mathbb{S}_{d}$ and $F_{2*}:\pi_{1}\to\mathbb{S}_{d}$ are deformation equivalent, where $\pi_{1}=\pi_{1}^{loc}(B_{1},o)=\pi_{1}^{loc}(B_{2},o)$.

Corollary 3**.**

Two deformation equivalent germs of finite covers $F_{1}:(U,o^{\prime})\to(V,o)$ and $F_{2}:(U,o^{\prime})\to(V,o)$ of degree $d$, branched along $(B_{1},o)\subset(V,o)$ and $(B_{2},o)\subset(V,o)$, are equivalent if the curve germ $(B_{1},o)$ is rigid and the monodromy $F_{1*}:\pi_{1}^{loc}(B_{1},o)\to\mathbb{S}_{d}$ is sole.

Lemma 10**.**

Let $T_{3,8k+2n+4,8k+4}$ be the singularity type of $(B,o)$ and a homomorphism $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ have the following properties:

• $(i)$

the image $F_{*}(\pi_{1}^{loc}(B,o))$ is a transitive subgroup of $\mathbb{S}_{4}$,

• $(ii)$

$F_{*}(\gamma_{i})=\tau_{i}$* are transpositions, where $\gamma_{1},\gamma_{2},\gamma_{3}$ is a good geometric base.*

Then $F_{*}$ is sole and $F_{*}(\pi_{1}^{loc}(B_{r},o))=\mathbb{S}_{4}$.

Proof.

By Lemma 2 and Zariski – van Kampen Theorem, the group $\pi_{1}^{loc}(B,o)$ has the following presentation

[TABLE]

Since $F_{*}(\pi_{1}^{loc}(B,o))$ is a transitive subgroup of $\mathbb{S}_{4}$ and $F_{*}(\gamma_{i})=\tau_{i}$ are transpositions, then $F_{*}(\pi_{1}^{loc}(B,o))=\mathbb{S}_{4}$ and $\tau_{1}\tau_{2}\tau_{3}$ is a cycle of length $4$. Therefore, with up to conjugation in $\mathbb{S}_{4}$, we can assume that $\tau_{1}\tau_{2}\tau_{3}=(1,2,3,4)$ and $\tau_{3}=(1,3)$, since $\gamma_{3}=(\gamma_{1}\gamma_{2}\gamma_{3})^{4k+2}\gamma_{3}(\gamma_{1}\gamma_{2}\gamma_{3})^{-(4k+2)}$. Consequently, $\tau_{1}\tau_{2}=[((1,2)(3,4)]$ and we can assume that $\tau_{1}=(1,2)$ and $\tau_{2}=(3,4)$ (if $\tau_{1}=(3,4)$ and $\tau_{2}=(1,2)$ then we conjugate by $[(1,3)(2,4)]$).∎

Lemma 11**.**

Let $(B,o)$ have the singularity type $T_{3,8m+2n,8m}$ and let $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ be a homomorphism such that

• $(i)$

$F_{*}(\pi_{1}^{loc}(B,o))$* is a transitive subgroup of $\mathbb{S}_{4}$,*

• $(ii)$

$F_{*}(\gamma_{i})=\tau_{i}$* are transpositions, where $\gamma_{1},\gamma_{2},\gamma_{3}$ is a good geometric base.*

Then $n=3k+1$, $k\in\mathbb{Z}_{\geqslant 0}$, $F_{*}(\pi_{1}^{loc}(B,o))=\mathbb{S}_{4}$, and $F_{*}$ is defined uniquely with up to inner automorphisms of $\mathbb{S}_{4}$ and $D$-automorphisms of $\mathfrak{D}_{T_{3,8m+6k+2,8m}}$.

Proof.

By Lemma 2 and Zariski – van Kampen Theorem, the group $\pi_{1}^{loc}(B,o)$ has the following presentation

[TABLE]

As in the proof of Lemma 10, since $F_{*}(\pi_{1}^{loc}(B,o))$ is a transitive subgroup of $\mathbb{S}_{4}$ and $F_{*}(\gamma_{i})=\tau_{i}$ are transpositions, then $F_{*}(\pi_{1}^{loc}(B,o))=\mathbb{S}_{4}$ and $\tau_{1}\tau_{2}\tau_{3}$ is a cycle of length $4$ and $\tau_{1}\neq\tau_{2}$. It follows from (11) that

[TABLE]

and hence, $\tau_{1}$ and $\tau_{2}$ are not commute, since they are conjugated in the group $\langle\tau_{1},\tau_{2}\rangle$. Therefore, with up to conjugation in $\mathbb{S}_{4}$, we can assume that $\tau_{1}=(1,2)$, $\tau_{2}=(1,3)$, and $\tau_{1}\tau_{2}=(1,2,3)$. Now, it is easy to see that equalities (12) hold iff $n=3k+1$.

There are three possibilities for $\tau_{3}$: either $\tau_{3}=(1,4)$ (denote this homomorphism by $F_{1}$), or $\tau_{3}=(2,4)$ (denote this homomorphism by $F_{2}$), or $\tau_{3}=(3,4)$ (denote this homomorphism by $F_{3}$). Note that, by Lemma 8, $a_{1}\in\mathfrak{d}_{T_{3,8m+6k+2,8m},W}$. Then the homomorphism $F_{1*}$ and the homomorphism $\widetilde{F}_{1*}$ sending $\gamma_{1}$ to $(1,3)$, $\gamma_{2}$ to $(2,3)$, and $\gamma_{3}$ to $(1,4)$ differ by the action of $D$-automorphism $i_{*}(a_{1})$. It is easy to check that the homomorphism $\widetilde{F}_{1*}$ coincides with $F_{2*}$ after conjugation by $(1,2,3)$. Similarly, the homomorphism $F_{2*}$ and the homomorphism $\widetilde{F}_{2*}$ sending $\gamma_{1}$ to $(1,3)$, $\gamma_{2}$ to $(2,3)$, and $\gamma_{3}$ to $(2,4)$ also differ by the action of $D$-automorphism $i_{*}(a_{1})$. It is easy to check that the homomorphism $\widetilde{F}_{2*}$ coincides with $F_{3*}$ after conjugation by $(1,2,3)$. ∎

Lemma 12**.**

Let $(B,o)$ have the singularity type $T_{3,8m+2n-1,8m}$, $m\geqslant 1$, $n\geqslant 0$, and $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ a homomorphism such that

• $(i)$

$F_{*}(\pi_{1}^{loc}(B,o))$* is a transitive subgroup of $\mathbb{S}_{4}$,*

• $(ii)$

$F_{*}(\gamma_{i})=\tau_{i}$* are transpositions, where $\gamma_{1},\gamma_{2},\gamma_{3}$ is a good geometric base.*

Then $F_{*}(\pi_{1}^{loc}(B,o))=\mathbb{S}_{4}$ and if either $n=3k+1$, or $n=3k+2$, $k\in\mathbb{Z}_{\geqslant 0}$, or $n=0$, then with up to deformation equivalence, there exist the unique such homomorphism $F_{*}$, and if $n=3k>0$ then there are at most two homomorphisms satisfying $(i)$ and $(ii)$.

Proof.

By Lemma 2 and Zariski – van Kampen Theorem, the group $\pi_{1}^{loc}(B,o)$ has the following presentation

[TABLE]

As in the proof of Lemma 10, since $F_{*}(\pi_{1}^{loc}(B,o))$ is a transitive subgroup of $\mathbb{S}_{4}$ and $F_{*}(\gamma_{i})=\tau_{i}$ are transpositions, then $F_{*}(\pi_{1}^{loc}(B,o))=\mathbb{S}_{4}$. Therefore $\tau_{1}\neq\tau_{2}$ and $\tau_{1}\tau_{2}\tau_{3}$ is a cycle of length $4$.

It follows from (11) that

[TABLE]

Let $n=3k+1$ or $3k+2$. Then it follows from (12) that $\tau_{1}$ and $\tau_{2}$ must commute with each other and hence, with up to conjugation, we can assume that $\tau_{1}=(1,2)$ and $\tau_{2}=(3,4)$. Then $\tau_{3}$ belongs to the set $\{(1,3),(1,4),(2,3),2,4)\}$. But again, it is easy to see that there is an inner automorphism of $\mathbb{S}_{4}$ saving fixed $\tau_{1}$ and $\tau_{2}$ and sending $\tau_{3}$ to $(1,3)$.

Since $\tau_{1}\tau_{2}\tau_{3}$ is a cycle of length $4$, we can assume that $\tau_{1}\tau_{2}\tau_{3}=(1,2,3,4)$ and Lemma 12 in case $n=0$ follows from Theorem 2.1 in [10] and Lemma 9.

Let $n=3k>0$. With up to conjugation in $\mathbb{S}_{4}$, we have only two possibilities: either $\tau_{1}=(1,2)$ and $\tau_{2}=(1,3)$, or $\tau_{1}=(1,2)$ and $\tau_{2}=(3,4)$. The case when $\tau_{1}=(1,2)$ and $\tau_{2}=(3,4)$ was considered above and the case when $\tau_{1}=(1,2)$ and $\tau_{2}=(1,3)$ was considered in the proof of Lemma 11. ∎

Lemma 13**.**

Let $A_{4n-1}$ be the singularity type of a curve germ $(B=B_{1}\cup B_{2},o)$ and a homomorphism $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ have the following properties:

• $(i)$

$G_{F}=\text{im},F_{*}$* is a transitive group of $\mathbb{S}_{4}$;*

• $(ii)$

$F_{*}(\gamma_{1})=\tau_{1}\tau_{2}$* is the product of two commuting transpositions $\tau_{1}$ and $\tau_{2}$, and $F_{*}(\gamma_{2})=\tau_{3}$ is a transposition, where $\gamma_{1}$ is a bypass around $B_{1}$, $\gamma_{2}$ is a bypass around $B_{2}$, and $\gamma_{1},\gamma_{2}$ is a good geometric base of $\pi_{1}^{loc}(B,o)$;*

Then $G_{F}=\mathbb{D}_{4}$ is a dihedral subgroup of $\mathbb{S}_{4}$ and $F_{*}$ is sole.

Proof.

With up to conjugation in $\mathbb{S}_{4}$, we can assume that $\tau_{1}=(1,2)$ and $\tau_{2}=(3,4)$. Note that $\tau_{3}\neq\tau_{i}$ for $i=1,2$, since $G_{F}$ is a transitive subgroup of $\mathbb{S}_{4}$ and $\gamma_{1}$, $\gamma_{2}$ generate the group $\pi_{1}^{loc}(B,o)$. Therefore $\tau_{3}=(i_{1},i_{2})$, where $i_{1}\in\{1,2\}$ and $i_{2}\in\{3,4\}$. Again, applying a conjugation in $\mathbb{S}_{4}$, we can assume that $\tau_{3}=(1,3)$. ∎

Lemma 14**.**

Let the singularity type of a germ $(B=B_{1}\cup B_{2},o)$ ($v-2u=0$ is an equation of the germ $B_{1}$ and $v^{2}-u^{2k+1}=0$ is the equation of the germ $B_{2}$)* be $D_{2k++3}$ and a homomorphism $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ such that*

• $(i)$

$G_{F}=\text{im}\,F_{*}$* is a transitive subgroup of $\mathbb{S}_{4}$,*

• $(ii)$

$F_{*}(\gamma_{1})$* is a product of two different commuting transpositions and $F_{*}(\gamma_{2})$, $F_{*}(\gamma_{3})$ are transpositions,*

where $\gamma_{1},\gamma_{2},\gamma_{3}$ is a good geometric base of $\pi_{1}^{loc}(B,o)$, $\gamma_{2}$ and $\gamma_{3}$ are bypasses around $B_{2}$ and $\gamma_{1}$ is a bypass around $B_{1}$. Then $G_{F}$ is a dihedral subgroup $\mathbb{D}_{4}$ of $\mathbb{S}_{4}$ and $F_{*}$ is solo.

Proof.

By Lemma 4 and Zariski – van Kampen Theorem, the group $\pi_{1}^{loc}(B,o)$ has the following presentation

[TABLE]

Note that $\nu=F_{*}(\gamma_{1})F_{*}(\gamma_{2})F_{*}(\gamma_{3})$ is an even permutation. Therefore $\nu$ is either a cycle of length $3$ or $\nu\in Kl_{4}$. But, it follows from (13) that $\nu$ can not be a cycle of length $3$, since it should commute with $F_{*}(\gamma_{1})$, and $\nu\neq 1$, since then $F_{*}(\gamma_{2})F_{*}(\gamma_{3})=F_{*}(\gamma_{1})$ and $F_{*}(\gamma_{1}),F_{*}(\gamma_{2}),F_{*}(\gamma_{3})$ can not generate a transitive subgroup of $\mathbb{S}_{4}$. Note also that $F_{*}(\gamma_{2})\neq F_{*}(\gamma_{3})$, since, otherwise, it follows from (13) that $F_{*}(\gamma_{2})=F_{*}(\gamma_{3})$ commutes with $F_{*}(\gamma_{1})$ and hence, again, $F_{*}(\gamma_{1}),F_{*}(\gamma_{2}),F_{*}(\gamma_{3})$ can not generate a transitive subgroup of $\mathbb{S}_{4}$. Consequently, $F_{*}(\gamma_{2})F_{*}(\gamma_{3})\in Kl_{4}$ and, with up to conjugation in $\mathbb{S}_{4}$, we can assume that $F_{*}(\gamma_{1})=[(1,2)(3,4)]$, $F_{*}(\gamma_{2})=(1,3)$, and $F_{*}(\gamma_{3})=(2,4)$. ∎

Lemma 15**.**

Let the singularity type of a germ $(B,o)$ be $T_{3}(4n+\beta,\beta)$ and a homomorphism $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ be such that

• $(i)$

$G_{F}=\text{im}\,F_{*}$* is a transitive subgroup of $\mathbb{S}_{4}$,*

• $(ii)$

$F_{*}(\gamma_{i})$, $i=1,2,3$, are transpositions,

where $\gamma_{1},\gamma_{2},\gamma_{3}$ is a good geometric base of $\pi_{1}^{loc}(B,o)$. Then $G_{F}=\mathbb{S}_{4}$ and $F_{*}$ is solo.

Proof.

It follows from $(i)$ and $(ii)$ that $\nu=F_{*}(\gamma_{1})F_{*}(\gamma_{2})F_{*}(\gamma_{3})$ is a cycle of length four. Without loss of generality, we can assume that $\nu=(1,2,3,4)$. Consider the case $\beta=1$ (the case $\beta=2$ is similar and therefore it will be omitted). By Lemma 7, we have

[TABLE]

The element $\nu$ acts by conjugation on the set of transpositions in $\mathbb{S}_{4}$. There are two orbits of this action:

[TABLE]

Therefore, by $(i)$ and since we can conjugate on the element $\nu$, it follows from (14) that we can put $F_{*}(\gamma_{1})=(1,2)$, $F_{*}(\gamma_{2})=(1,4)$, $F_{*}(\gamma_{3})=(3,4)$, and after that to check that $\nu=[(1,2)]\cdot[(1,4)]\cdot[(3,4)]$. ∎

2. Proof of Theorem 1

Let $L_{1}=\{u_{0}=0\}$ and $L_{2}=\{v_{0}=0\}$ be the axes of some local complex-analytic coordinates $u_{0},v_{0}$ in $(V,o)$. Then the local intersection number of the divisors $M_{1}=F^{*}(L_{1})$ and $M_{2}=F^{*}(L_{2})$ at the point $o^{\prime}$ is equal to $(M_{1},M_{2})_{o^{\prime}}=\deg_{o^{\prime}}F=4$. Therefore we have the following possibilities:

• $({\rm I})$

either $M_{1}$ or $M_{2}$ is a germ of a non-singular curve,

• $({\rm II})$

$M_{1}$ and $M_{2}$ have a singularity of multiplicity 2 at the point $o^{\prime}$.

Let $R\subset(U,o^{\prime})$ be the ramification divisor of the germ of finite cover $F$ and $B=F(R_{red})\subset(V,o)$ the branch curve.

Denote $\mathfrak{m}\subset\mathbb{C}[[z_{0},w_{0}]]$ the maximal ideal in the ring of power series $\mathbb{C}[[z_{0},w_{0}]]$.

2.1. Case $({\rm I})$.

Let $M_{1}$ be non-singular. Then we can choose local coordinates $z_{0},w_{0}$ in $(U,o^{\prime})$ such that $F^{*}(u_{0})=z_{0}$ and $F^{*}(v_{0})=v_{0}(z_{0},w_{0})=\sum_{i=0}^{\infty}a_{i}(z_{0})w_{0}^{i}$, where $a_{i}(z_{0})=\sum_{j=0}^{\infty}a_{i,j}z_{0}^{j}\in\mathbb{C}[[z_{0}]]$. Performing the coordinates change $v_{0}\leftrightarrow v_{0}-a_{0}(u_{0})$, we can assume that $a_{0}(z_{0})\equiv 0$. In addition, we have $a_{1,0}=a_{2,0}=a_{3,0}=0$ and can assume that $a_{4,0}=1$, since $(M_{1},M_{2})_{o^{\prime}}=4$.

The divisor $R$ is given by equation

[TABLE]

i.e., $R$ is given by equation

[TABLE]

Let us write equation (15) in the following form

[TABLE]

where $H_{1}(z_{0},w_{0})$ is a polynomial of degree $2$ and $H_{2}(z_{0},w_{0})\in\mathfrak{m}^{4}$. It follows from (16) that $(M_{1},R)_{o^{\prime}}=3$. Therefore there are seven possibilities:

• $({\rm I}_{1})$

$R=3R_{1}$, where $R_{1}$ is a germ of a smooth curve and $(M_{1},R_{1})_{o^{\prime}}=1$,

• $({\rm I}_{2})$

$R=2R_{1}+R_{2}$, where $R_{1}$ and $R_{2}$ are germs of smooth curves $(M_{1},R_{1})_{o^{\prime}}=(M_{1},R_{2})_{o^{\prime}}=1$,

• $({\rm I}_{3})$

$R=R_{1}$ is reduced irreducible and $(M_{1},R_{1})_{o^{\prime}}=3$,

• $({\rm I}_{4.1})$

$R=R_{1}+R_{2}$, where the germ $R_{1}$ is irreducible, $(M_{1},R_{1})_{o^{\prime}}=2$ and $\deg f_{\mid R_{1}}=1$, $R_{2}$ is a germ of smooth curve and $(M_{1},R_{2})_{o^{\prime}}=1$,

• $({\rm I}_{4.2})$

$R=R_{1}+R_{2}$, where the germ $R_{1}$ is irreducible, $(M_{1},R_{1})_{o^{\prime}}=2$, and $\deg f_{\mid R_{1}}=2$, $R_{2}$ is a germ of smooth curve and $(M_{1},R_{2})_{o^{\prime}}=1$,

• $({\rm I}_{5.1})$

$R=R_{1}+R_{2}+R_{3}$, where $R_{1}$, $R_{2}$, and $R_{3}$ are germs of smooth curves $(M_{1},R_{1})_{o^{\prime}}=(M_{1},R_{2})_{o^{\prime}}=(M_{1},R_{3})_{o^{\prime}}=1$ and the branch curve $B=F(R)$ consists of three irreducible germs,

• $({\rm I}_{5.2})$

$R=R_{1}+R_{2}+R_{3}$, where $R_{1}$, $R_{2}$, and $R_{3}$ are germs of smooth curves $(M_{1},R_{1})_{o^{\prime}}=(M_{1},R_{2})_{o^{\prime}}=(M_{1},R_{3})_{o^{\prime}}=1$, the branch curve $B=F(R)$ consists of two irreducible germs and has singularity of type $A_{2n-1}$ for some $n\in\mathbb{N}$.

Note that the case, when $R=R_{1}+R_{2}+R_{3}$, where $R_{1}$, $R_{2}$, and $R_{3}$ are germs of smooth curves , $(M_{1},R_{1})_{o^{\prime}}=(M_{1},R_{2})_{o^{\prime}}=(M_{1},R_{3})_{o^{\prime}}=1$, and the germ of braanch curve $B=F(R)$ is irreducible, is impossible, since $\deg_{o^{\prime}}F=4$.

Proposition 5**.**

In case $({\rm I}_{1})$, the germ of a finite cover $F$ is equivalent to the germ $F_{{\bf 1}_{1}}$.

Proof.

A germ $F$ is ramified along $R_{1}$ with multiplicity $4$ and the function $u=z$ is a local parameter on $R_{1}$ at the point $o^{\prime}$. Therefore $\deg F_{\mid R_{1}}=1$ and $B=F(R_{1})$ is a smooth curve germ at the point $o$. Hence, $\pi_{1}^{loc}(B,o)\simeq\mathbb{Z}$. Since the germ $F$ is ramified along $R_{1}$ with multiplicity $4$, then the monodromy group $G_{F}$ of $F$ is a cyclic group of the fourth order and $G_{F}$ is generated in $\mathbb{S}_{4}$ by a cycle of length $4$. But all such subgroups are conjugated in $\mathbb{S}_{4}$. Therefore, by Grauert - Remmert - Riemann - Stein Theorem, the germ $F$ is equivalent to the germ of finite cover given by functions $u=z$, $v=w^{4}$. ∎

Proposition 6**.**

In case $({\rm I}_{2})$, the germ of a finite cover $F$ is equivalent to the germ $F_{{\bf 4}_{1},n}$ for some $n\in\mathbb{N}$. The singularity type of the germ $(B,o)$ of the branch curve of $F_{{\bf 4}_{1},n}$ is $A_{8n-1}$ and the monodromy group $G_{F_{{\bf 4}_{1},n}}=\mathbb{S}_{4}$.

Proof.

The germ of a finite cover $F$ is ramified along $R_{1}$ with multiplicity $3$ and along $R_{2}$ with multiplicity $2$. The function $u=z$ is a local parameter on both $R_{1}$ and $R_{2}$ at the point $o^{\prime}$. Therefore $B_{1}=F(R_{1})$ and $B_{2}=F(R_{2})$ are smooth curve germs at the point $o$ and $\deg F_{\mid R_{1}}=\deg F_{\mid R_{2}}=1$. Note that $B_{1}\neq B_{2}$, since $\deg_{o^{\prime}}F=4$. Let $(B_{1},B_{2})_{o}=k$. Therefore the point $o$ is the singular point of the branch curve $B=B_{1}\cap B_{2}$ of type $A_{2k-1}$. By Lemma 6, $\pi_{1}^{loc}(B,o)=\langle\gamma_{1},\gamma_{2}\,\,\mid\,\,(\gamma_{1}\gamma_{2})^{k}=(\gamma_{2}\gamma_{1})^{k}\rangle,$ where $\gamma_{i}$ are bypasses around $B_{i}$, $i=1,2$, and the monodromy group $G_{F}$ is generated by a cycle $\tau_{1}=F_{*}(\gamma_{1})$ of length $3$ and a transposition $\tau_{2}=F_{*}(\gamma_{2})$. Since $G_{F}$ is a transitive subgroup of $\mathbb{S}_{4}$, then $G_{F}=\mathbb{S}_{4}$ and with up to renumbering, we can assume that $\tau_{1}=(1,2,3)$ and $\tau_{2}=(3,4)$. We have $\tau_{1}\tau_{2}=(1,2,4,3)$ and $\tau_{2}\tau_{1}=(1,2,3,4)$. Therefore $(\tau_{1}\tau_{2})^{k}=(\tau_{2}\tau_{1})^{k}$ if and only if $k=4n$ and, by Grauert - Remmert - Riemann - Stein Theorem, the finite cover $F$, with up to changes of coordinates, coincides with the germ of finite cover $F_{{\bf 4}_{1},n}$ given by

[TABLE]

Indeed, the ramification divisor $R$ of $F_{{\bf 4}_{1},n}$ is given by equation $4w^{3}-3z^{n}w^{2}=0$ and hence, $R_{1}$ is given by $w=0$ and $R_{2}$ is given by equation $4w=3z^{n}$. Then it is easy to see that $B_{1}$ is given by equation $v=0$ and $B_{2}$ is given by equation $4^{4}v+3^{3}u^{4n}=0$, i.e., the point $o$ is the singular point of the branch curve $B$ of type $A_{8n-1}$. To complete the proof, note that, by Proposition 2, the singularities of type $A_{m}$ are rigid. ∎

Proposition 7**.**

In case $({\rm I}_{3})$, the germ $F$ is deformation equivalent to the germ $F_{{\bf 4}_{2},n,\beta}$ for some $n\in\mathbb{Z}_{\geqslant 0}$ and $\beta\neq 0$.

The germs $F$ deformation equivalent to the germ $F_{{\bf 4}_{2},0,1}$ are equivalent.

Proof.

We need the following Lemma.

Lemma 16**.**

Let an irreducible curve germ $R_{1}\subset(U,o^{\prime})$ be given by equation

[TABLE]

where $\widetilde{h}_{4}(z_{0},w_{0})\in\mathbb{C}[[z_{0},w_{0}]]$, $\widetilde{h}_{i}(z_{0})\in\mathfrak{m}\cap\mathbb{C}[[z_{0}]]$ for $i=0,1,2,3$, and $k_{i}\in\mathbb{N}$, $b_{i}\in\mathbb{C}^{*}$ for $i=0,1,2$. Then there is a local change of coordinates in $(U,o^{\prime})$ of the form

[TABLE]

where $g(z_{0})\in\mathbb{C}[[z_{0}]]$ such that the equation of $R_{1}$ has the following form

[TABLE]

where $h_{4}(z,w)\in\mathbb{C}[[z,w]]$, $h_{i}(z)\in\mathfrak{m}\cap\mathbb{C}[[z]]$ for $i=0,1,2,3$, and $n_{i}\in\mathbb{Z}_{\geqslant 0}$, $a_{i}\in\mathbb{C}^{*}$ for $i=0,1,2$, and

[TABLE]

Remark 2**.**

Note that if we change the coordinate $w_{0}$ by $w_{1}=w_{0}+g(z_{0})$, then to calculate the Jacobian $J_{F}$ of the germ of finite cover $F$ in coordinates $(z_{0},w_{1})$, it is sufficient to substitute $w_{1}-g(z_{0})$ in the equation (17) instead of $w_{0}$.

Proof of Lemma 16. Let $k_{0}=3m_{0}+\beta_{0}$, where $\beta_{0}$ is the remainder of $k_{0}$ divided by $3$; $k_{1}=2m_{1}+\beta_{1}$, where $\beta_{1}$ is the remainder of $k_{1}$ divided by $2$; and $k_{2}:=m_{2}$.

Let us show that $m_{0}\leqslant\min(m_{1},m_{2})$, since otherwise, the curve germ $R_{1}$ is not irreducible. Indeed, let $m_{1}<m_{0}$ and $m_{1}\leqslant m_{2}$ (the case when $m_{2}<m_{0}$ and $m_{2}\leqslant m_{1}$ is similar and therefore its consideration will be omitted). Consider a sequence of $\sigma$-processes $\sigma:U_{m_{1}}\to U$ given by functions $z_{0}=z_{m_{1}}$ and $w_{0}=w_{m_{1}}z_{m_{1}}^{m_{1}}$. Then the exceptional curve $E_{m_{1}}$ of the last $\sigma$-process is given by equation $z_{m_{1}}=0$ and the proper inverse image $\sigma^{-1}(R_{1})$ of $R_{1}$ is given by equation

[TABLE]

Rewrite this equation in the following form:

[TABLE]

It follows from (20) that if $\beta_{1}=m_{2}-m_{1}=0$ then $\sigma^{-1}(R_{1})$ intersects with $E_{{}_{1}}$ at three points $(z_{m_{1}},w_{m_{1}})=(0,0)$ and $(z_{m_{1}},w_{m_{1}})=(0,\frac{-b_{1}\pm\sqrt{b_{1}^{2}-4b_{2}}}{2})$; if $\beta_{1}=1$ and $m_{2}-m_{1}=0$ then $\sigma^{-1}(R_{1})$ intersects with $E_{m_{1}}$ at two points $(z_{m_{1}},w_{m_{1}})=(0,0)$ and $(z_{m_{1}},w_{m_{1}})=(0,-b_{2})$; if $\beta_{1}=0$ and $m_{2}-m_{1}>0$ then $\sigma^{-1}(R_{1})$ intersects with $E_{m_{1}}$ at three points $(z_{m_{1}},w_{m_{1}})=(0,0)$ and $(z_{m_{1}},w_{m_{1}})=(0,\pm\sqrt{-b_{1}})$, which contradicts irreducibility of the curve germ $R_{1}$. Finally, if $\beta_{1}=1$ and $m_{2}-m_{1}>0$, then the quadratic part of the left side of equation (20) is $b_{1}z_{m_{1}}w_{m_{1}}$, which also contradicts irreducibility of the curve germ $R_{1}$.

Now, let us show that we can assume that $\beta_{0}\neq 0$. Indeed, suppose that $\beta_{0}=0$. As above, consider a sequence of $\sigma$-process $\sigma:U_{m_{0}}\to U$ given by functions $z_{0}=z_{m_{0}}$ and $w_{0}=w_{m_{0}}z_{m_{0}}^{m_{0}}$. Then the exceptional curve $E_{m_{0}}$ of the last $\sigma$-process is given by equation $z_{m_{0}}=0$ and the proper inverse image $\sigma^{-1}(R_{1})$ of $R_{1}$ is given by equation

[TABLE]

It follows from irreducibility of the curve germ $\sigma^{-1}(R_{1})$ that

[TABLE]

Therefore, if we change the coordinate $w_{0}$ by $\overline{w}_{0}=w_{0}+\sqrt[3]{b_{0}}z_{0}^{m_{0}}$, then we obtain that $R_{1}$ is given by an equation of the same type as the equation (17):

[TABLE]

in which $\overline{k}_{0}>k_{0}$. After a finite number (since the singular point $o^{\prime}$ of $R_{1}$ can be resolved by finite number of $\sigma$-processes) of such coordinate substitutions, we obtain that the germ $R_{1}$ is given by an equation of the same type as equation (18) in which $\beta_{0}=1$ or $2$ and $n_{0}\leqslant\min(n_{1},n_{2})$.

Note that the same arguments as above (the irreducibility of $R_{1}$ and possibility to carry out $n_{1}$ or $n_{2}$ $\sigma$-processes if necessary) show that $n_{2}\geqslant n_{0}+1$ and $\beta_{1}>0$ if $n_{1}=n_{0}$. ∎

In the case $({\rm I}_{3})$, the ramification divisor $R=R_{1}$ of the germ of a finite cover $F$ is given by equation (17) and by Lemma 16, we can assume that it is given by equation (18) satisfying conditions (19).

Then, by Remark 2, the germ $F$ is given by functions

[TABLE]

where $w^{5}g(z,w)=\int{w^{4}h_{4}(z,w)dw}$.

To find the singularity type of the branch locus of $F$, let us show that $R_{1}$ can be given by parametrisation

[TABLE]

where $g_{1}(\tau)\in\mathbb{C}[[\tau]]$ . Indeed, denote by $\nu:\widetilde{R}_{1}\to R_{1}$ the resolution of singularity of the germ $R_{1}$. It follows from equation (18) that $(M_{1},R_{1})_{o^{\prime}}=3$ and $(M_{2},R_{1})_{o^{\prime}}=3n_{0}+\beta_{0}$. Therefore there exist a local parameter $\tau$ at the point $\nu^{-1}(o^{\prime})$ in $\widetilde{R}_{1}$ such that $\nu^{-1}(z)=\tau^{3}$ and $\nu^{*}(w)=\tau^{3n_{0}+\beta_{0}}\sum_{i=0}^{\infty}c_{i}\tau^{i}$. If we substitute $\tau^{3}$ instead of $z$ and ${3n_{0}+\beta_{0}}\sum_{i=0}^{\infty}c_{i}\tau^{i+\beta_{0}}$ instead of $w$ in (18) then, as a result, we must obtain a power series identically equal to zero. In particular, we obtain that $c_{0}=\sqrt[3]{-a_{0}}$.

By (19), we have $n_{2}-n_{0}>0$ and $n_{1}-n_{0}+\beta_{1}>0$. Therefore, if we substitute $\tau^{3n_{0}+\beta_{0}}(\sqrt[3]{-a_{0}}+\tau g_{1}(\tau))$ instead of $w$ and $\tau^{3}$ instead of $z$ in (21), then we obtain that $B=F(R_{1})$ is given parametrically by equations of the form

[TABLE]

where $g_{2}(\tau)\in\mathbb{C}[[\tau]]$. Therefore the branch curve $B$ of $F$ has the singularity at $o$ of type $T_{3}(4n_{0}+\beta_{0},\beta_{0})$.

It is easy to see that $\deg F_{\mid R_{1}}=1$ and $F$ is ramified along $R_{1}$ with multiplicity two, therefore the monodromy group $G_{F}$ of the germ $F$ is generated by transpositions, and since $G_{F}$ is a transitive subgroup of $\mathbb{S}_{4}$, we have $G_{F}=\mathbb{S}_{4}$.

To complete the proof of Proposition 7, it suffices to consider the germs $F_{{\bf 4}_{2},n_{0},\beta_{0}}$ given by functions $u=z$, $v=w^{4}+4wz^{3n_{0}+\beta_{0}}$ and apply Lemma 15 and Corollaries 2 and 3. ∎

The following two Propositions will be proved simultaneously.

Proposition 8**.**

In case $({\rm I}_{4.1})$, the germ of finite cover $F$ is deformation equivalent either to the germ $F_{{\bf 4}_{3},k,m}$, or to the germ $F_{{\bf 4}_{4},k,m}$ for some $m\in\mathbb{N}$ and $k\in\mathbb{Z}_{\geqslant 0}$.

The singularity type of $(B_{{\bf 4}_{3},k,m},o)$ is $T_{3,8m+6k+2,8m}$ and the singularity type of $(B_{{\bf 4}_{4},k,m},o)$ is $T_{3,8m+2k+4,8m+4}$; in both cases $G_{F}=\mathbb{S}_{4}$.

Proposition 9**.**

In case $({\rm I}_{4.2})$, the germ $F$ is equivalent to the germ $F_{{\bf 3}_{1},2k+1}$ for some $k\in\mathbb{Z}_{\geqslant 0}$. The singularity type of the germ $(B_{{\bf 3}_{1},2k+1},o)$ is $A_{8k+3}$ and $G_{F_{{\bf 3}_{1},2k+1}}$ is a dihedral group $\mathbb{D}_{4}\subset\mathbb{S}_{4}$.

Proof.

By Remark 1, we can choose coordinates $(z,w)$ in $(U,o^{\prime})$ such that $w=0$ is an equation of $R_{2}$ and $u=z$, where $(u,v)$ are coordinates in $(V,o)$. Since the germ $R_{1}$ is irreducible and $(M_{1},R_{1})_{o^{\prime}}=2$, then arguments similar to that which we used in the proof of Lemma 16, imply that an equation of $R_{1}$ has the following form

[TABLE]

where $h_{i}\in\mathbb{C}[[z]]$ for $i\in\mathbb{N}$, $h_{0}\in\mathbb{C}[z]$, $\deg h_{0}\leqslant n-1$, $a_{0}\in\mathbb{C}$, $m\in\mathbb{N}$ and $n\in\mathbb{Z}_{\geqslant 0}$. Therefore

[TABLE]

and hence

[TABLE]

The germ of the ramification curve $R=R_{1}\cup R_{2}$ has the singularity type $T_{3,2m+2n,2m}$ if $a_{0}\neq 0$ and $T_{3,2m+2n,2m+2n+1}$ if $a_{0}=0$.

It follows from (25) that $B_{2}=F(R_{2})$ is given by equation $v=0$.

Consider the case when $a_{0}\neq 0$. As in the proof of Proposition 7, it is easy to show that equation (24) implies that $R_{1}$ can be given parametrically by functions of the following form:

[TABLE]

To obtain a parametrisation of $B_{1}$, we substitute $z(\tau)$ and $w(\tau)$ from (26) instead of $z$ and $w$ in (25) and obtain that $B_{1}$ has a parametrisation of the following form

[TABLE]

If $a_{i}=0$ for all odd $i$, then $\deg F_{\mid R_{1}}=2$, $F(R_{1})$ is a smooth germ, and $B$ has the singularity type $A_{8m-1}$. Therefore $F_{*}(\gamma_{1})\subset\mathbb{S}_{4}$ is a product of two commuting transpositions and $F_{*}(\gamma_{2})$ is a transposition, where $\gamma_{1},\gamma_{2}$ is a good geometric base in which $\gamma_{1}$ is a bypass around $B_{1}$ and $\gamma_{2}$ is a bypass around $B_{2}$. Then, by Lemma 13, the monodromy group $G_{F}$ is a dihedral group $\mathbb{D}_{4}\subset\mathbb{S}_{4}$. But, Remark 3 (see the end of Subsection 2.1) claims that it is impossible, since the germ $R$ consists of two irreducible germs.

Let $i_{0}$ be the smallest number for which $a_{i_{0}}\neq 0$. Then $\deg F_{\mid R_{1}}=1$ and $B$ has the singularity of type $T_{3,8m+i_{0}-1,8m}$. It follows from Lemma 11 that $i_{0}=6k+3$ for some $k\in\mathbb{Z}_{\geqslant 0}$.

Let us show that in this case the germ $F$ is deformation equivalent to the germ $F_{{\bf 4}_{3},k,m}$ given by functions

[TABLE]

Indeed, we have

[TABLE]

and hence, $R_{m,k}=R_{1}\cup R_{2}$, where an equation of $R_{2}$ is $w=0$ and an equation of $R_{1}$ is

[TABLE]

The germ $R_{1}$ has the following parametrisation

[TABLE]

and hence, the curve $R_{m,k}$ has a singularity of type $T_{3,2m+2k,2m}$. The branch curve $B_{m,k}=B_{1}\cup B_{2}$, where $B_{2}$ is given by equation $v=0$. In order to obtain a parametrisation of $B_{1}$, we substitute $z(\tau)$ and $w(\tau)$ from (29) instead of $z$ and $w$ in (28). As a result, we obtain that $B_{1}$ is given parametrically by functions

[TABLE]

Consequently, the curve germs $B_{m,k}$ has the singularity of type $T_{3,8m+6k+2,8m}$. It follows from Lemma 11 and Corollary 2 that $F$ is deformation equivalent to $F_{{\bf 4}_{3},k,m}$.

Now, consider the case $a_{0}=0$ (in this case we put $k:=m+n$). Then $R=R_{1}\cup R_{2}$ has the singularity of type $T_{3,2k,2k+1}$. Let us write equation (24) of $R_{1}$ in the following form

[TABLE]

Then it is easy to see that a parametrisation of $R_{1}$ has the following form

[TABLE]

We have

[TABLE]

and consequently, $F$ is given by functions

[TABLE]

To obtain a parametrisation of $B_{1}$, we substitute $z(\tau)$ and $w(\tau)$ from (32) instead of $z$ and $w$ in (33) and obtain that $B_{1}$ has a parametrisation of the following form

[TABLE]

There are two possibilities: either there is $m_{0}=8k+2(n+2)+1$ being the smallest odd number such that $b_{m_{0}}\neq 0$ or $b_{m}=0$ for all odd $m\geqslant 8k+5$.

Let $b_{8k+2(n+2)+1}\neq 0$. Then $B=B_{1}\cup B_{2}$ has the singularity of type $T_{3,8k+2n+4,8k+4}$, $k$ and $n\in\mathbb{Z}_{\geqslant 0}$. Let us show that in this case the germ $F$ is deformation equivalent to the germ $F_{{\bf 4}_{4},n,k}$ given by functions

[TABLE]

The ramification divisor of $F_{{\bf 4}_{4},n,k}$ is $R=R_{1}\cup R_{2}$, where $R_{2}$ is given by equation $w=0$ and $R_{1}$ is given by equation

[TABLE]

It is easy to check that $R_{1}$ is given parametrically by functions

[TABLE]

where $h(\tau)\in\mathbb{C}[[\tau]]$. To obtain a parametrisation of $B_{1}=F_{{\bf 4}_{4},n,k}(R_{1})$, let us substitute $z(\tau)$ and $w(\tau)$ from (37) in (35) and, as a result, we obtain

[TABLE]

where $h_{1}(\tau)\in\mathbb{C}[[\tau]]$. Consequently, $B$ has the singularity type $T_{3,8k+2n+4,8k+4}$ and Proposition 8 follows from Lemma 10 and Corollary 2.

Consider the case when $b_{m}=0$ for all odd $m\geqslant 8k+5$. It follows from (34) that $\deg F_{\mid R_{1}}=2$, $B_{1}=F(R_{1})$ is a smooth germ, and the singularity type of the germ $(B,o)$ is $A_{8k+3}$. Therefore $F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{4}$ has the following properties: $(*_{1})$ $G_{F}=\text{im},F_{*}$ is a transitive group of $\mathbb{S}_{4}$; $(*_{2})$ $F_{*}(\gamma_{1})$ is the product of two commuting transpositions and $F_{*}(\gamma_{2})$ is a transposition, where $\gamma_{1}$ is a bypass around $B_{1}$ and $\gamma_{2}$ is a bypass around $B_{2}$.

By Lemma 13, $G_{F}=\mathbb{D}_{4}$ is a dihedral subgroup of $\mathbb{S}_{4}$. Therefore, Corollary 2, to complete the proof of Proposition 9, it suffices to show that the branch curve $B$ of the germ of finite cover $F_{{\bf 3}_{1},2k+1}$ given by functions

[TABLE]

has the singularity of type $A_{8k+3}$ and $F_{{\bf 3}_{1},2k+1*}$ has properties $(*_{1})$ and $(*_{2})$.

The ramification divisor $R=R_{1}\cup R_{2}$ of $F_{{\bf 3}_{1},2k+1}$ is given by $w(w^{2}-z^{2k+1})=0$, where $R_{1}$ is given by $w^{2}-z^{2k+1}=0$. Therefore $R_{1}$ has the following parametrisation:

[TABLE]

Consequently, $F_{{\bf 3}_{1},2k+1\mid R_{1}}:R_{1}\to(V,o)$ is given by functions

[TABLE]

and, hence $\deg F_{{\bf 3}_{1},2k+1\mid R_{1}}=2$, $B_{1}$ is a smooth germ touching the germ $B_{2}$ given by $v=0$ with multiplicity $4k+2$. Therefore the branch curve $B$ of the germ $F_{{\bf 3}_{1},2k+1}$ has the singularity of type $A_{8k+3}$ and $F_{{\bf 3}_{1},2k+1*}$ has properties $(*_{1})$ and $(*_{2})$.∎

The following two Propositions also will be proved simultaneously.

Proposition 10**.**

In case $({\rm I}_{5.1})$, the germ $F$ is deformation equivalent either to one of the germs $F_{{\bf 4}_{5},k,m}$, $m,k\geqslant 1$, (the singularity type of $B_{F_{{\bf 4}_{5},k,m}}$ is $T_{3,8m+6k-1,8m}$), or to one of the germs $F_{{\bf 4}_{6},k,m}$, $m,k\geqslant 1$, (the singularity type of $B_{F_{{\bf 4}_{6},k,m}}$ is $T_{3,8m+2k-1,8m}$), or to one of the germs $F_{{\bf 4}_{7},m}$, $m\geqslant 1$, (the singularity type of $B_{F_{{\bf 4}_{7},m}}$ is $T_{3,8m-1,8m}$). In all cases, the monodromy groups of the germs of finite covers are $\mathbb{S}_{4}$.

Proposition 11**.**

In case $({\rm I}_{5.2})$, the germ $F$ is equivalent to the germ $F_{{\bf 3}_{1},2k}$ for some $k\in\mathbb{N}$. The singularity type of the germ of branch curve $(B_{F_{{\bf 3}_{1},2k}},o)$ is $A_{8k-1}$ and the monodromy group $G_{F_{{\bf 3}_{1},2k}}$ is the dihedral group $\mathbb{D}_{4}\subset\mathbb{S}_{4}$.

Proof.

In cases $({\rm I}_{5.1})$ and $({\rm I}_{5.2})$ the ramification divisor $R$ consists of three irreducible germs, $R=R_{1}\cup R_{2}\cup R_{3}$. Let us renumber them so that

[TABLE]

and choose coordinates $z,w$ such that $w=0$ is an equation of $R_{3}$.

Since $(M_{1},R_{i})_{o^{\prime}}=1$, then $z$ is a local parameter on each germ $R_{i}$ and we can choose equations of $R_{i}$ of the following form: $w=f_{i}(z)$, $i=1,2$, where if $n\geqslant 1$ then

[TABLE]

where $n_{2}\geqslant 0$, and if $n=0$ then

[TABLE]

where $p(z)\in\mathbb{C}[z]$, $\deg p(z)\leqslant n-1$, $p(0)\neq 0$ and $h_{i}(z)\in\mathbb{C}[[z]]$, $h_{1}(0)\neq 0$, $h_{2}(0)\neq 0$, and $h_{1}(0)\neq h_{2}(0)$ if $n_{2}=0$. If $n\geqslant 1$ then

[TABLE]

where $g_{1}\in\mathfrak{m}$. Consequently, $F$ is given by functions

[TABLE]

where $g_{i}\in\mathfrak{m}$. Similarly, if $n=0$ then $F$ is given by functions

[TABLE]

We have $v=0$ is an equation of $B_{3}=F(R_{3})$. To obtain equations of $B_{i}=F(R_{i})$, let us substitute $f_{i}(u)$ from (41) or (42), resp., in (43) or (44) instead of $w$. After substitutions, we obtain that the equation of $B_{i}$, $i=,1,2$, have the following form

[TABLE]

if $n>0$ and

[TABLE]

if $n=0$, where $\{j\}=\{1,2\}\setminus\{i\}$ and $h_{i,3}(u),h_{4}(u)\in\mathbb{C}[[u]]$.

Consider the case when $B=B_{1}\cup B_{2}\cup B_{3}$ consists of three different germs. Then in case $n>0$, it follows from (45) that $(B_{1},B_{3})_{o}=(B_{2},B_{3})_{o}=4m$, and $(B_{1},B_{2})_{o}=4m+k>4m$ for some $k$, since $p(0)\neq 0$. Therefore the singularity type of $(B,o)$ is $T_{3,8m+2k-1,8m}$.

In particular, it is easily to check that the ramification locus of the germ $F_{{\bf 4}_{5},k,m}$ given by functions

[TABLE]

is given by equation $w(w-z^{m}-z^{m+k})(w-z^{m}-3z^{m+k})=0$ (i.e., $R$ has the singularity of type $T_{3,2m+2k-1,2m}$) and the singularity type of the branch curve $B$ is $T_{3,8m+6k-1,8m}$.

Consider the case when $n=0$. Note that $2h_{2}(0)-h_{1}(0)\neq 2h_{1}(0)-h_{2}(0)$ and, in particular, it is impossible that $2h_{2}(0)-h_{1}(0)=0$ and $2h_{1}(0)-h_{2}(0)=0$ simultaneously. Therefore it follows from (46) that if $2h_{1}(0)-h_{2}(0)\neq 0$ and $2h_{2}(0)-h_{1}(0)\neq 0$, then $B$ has the singularity of type $T_{3,8m-1,8m}$, and if $2h_{1}(0)-h_{2}(0)=0$ (resp., if $2h_{2}(0)-h_{1}(0)=0)$, then $B$ has the singularity type $T_{3,8m+2k-1,8m}$ for some $k\geqslant 1$.

In particular, it is easily to check that the ramification locus of the germ $F_{{\bf 4}_{6},k,m}$ given by functions

[TABLE]

is given by equation $w(w-z^{m}-z^{m+k})(w-2z^{m})=0$ (i.e., $R$ has the singularity of type $T_{3,2m-1,2m}$) and the singular type of the branch curve $B$ is $T_{3,8m+2k-1,8m}$.

And again, it is easily to check that the ramification locus of the germ $F_{{\bf 4}_{7},m}$ given by functions

[TABLE]

is given by equation $w(w-z^{m})(w-3z^{m})=0$ (i.e., $R$ has the singularity of type $T_{3,2m-1,2m}$) and the singularity type of the branch curve $B$ is $T_{3,8m-1,8m}$.

So, in all cases considered above, the branch curves $B$ of $F$ have singularity types $T_{3,8m+2k-1,8m}$ for some $k\in\mathbb{Z}_{\geqslant 0}$. The germs $F_{{\bf 4}_{5},k,m}$ and $F_{{\bf 4}_{6},3k,m}$ have the branch curves of the same singularity types. But, by Proposition 4, the covers $F_{{\bf 4}_{5},k,m}$ and $F_{{\bf 4}_{6},3k,m}$ are not deformation equivalent, since their ramification divisors have different types of singularity. Therefore to complete the proof of Proposition 10, it suffices to apply Lemma 12 and Corollary 2.

If $F(R_{1})=F(R_{2})$ in case when $n>0$ or if $F(R_{3})$ coincides with either $F(R_{1})$ or with $F(R_{2})$ in case when $n=0$, then it follows from (45) and (46) that $B$ consists of two irreducible germs and has the singularity type $A_{8m-1}$, since $F(R_{i})$ are germs of smooth curves. Denote the irreducible components of $B$ by $B_{1}$ and $B_{2}$ and let $F^{-1}(B_{2})$ be the union of two components of the divisor $R$. Then $G_{F}$ is generated by a transposition $\tau_{1}=F_{*}(\gamma_{1})$, where $\gamma_{1}$ is a bypass around $B_{1}$ and a product $\nu=[\tau_{2}\tau_{1}]=F_{*}(\gamma_{2})$ of two commuting transpositions $\tau_{2}$ and $\tau_{3}$, where $\gamma_{2}$ is a bypass around $B_{2}$. Therefore, by Lemma 13, $G_{F}$ is a dihedral subgroup $\mathbb{D}_{4}\subset\mathbb{S}_{4}$.

Let us show that in this case $F$ is equivalent to the cover $F_{{\bf 3}_{1},2m}$ given by functions

[TABLE]

Indeed, it is easy to see that the branch locus $R$ of $F_{{\bf 3}_{1},2m}$ is given by equation $w(w-z^{m})(w+z^{m})$, the germ $B_{1}$ is given by equation $v=0$ and $B_{2}$ is given by equation $v+u^{4m}=0$. Therefore, by Lemma 13 and Corollary 3, the cover $F$ is equivalent to $F_{{\bf 3}_{1},2m}$. This completes the proof of Proposition 11. ∎

Remark 3**.**

It follows from the proof of Proposition 11 that if the branch curve $B$ of the germ of finite cover $F$, $\deg_{o^{\prime}}F=4$, has the singularity of type $A_{8m-1}$ and $G_{F}=\mathbb{D}_{4}\subset\mathbb{S}_{4}$, then the ramification locus $R$ of $F$ consists of three irreducible components.

2.2. Case $({\rm II})$.

To comlete the proof of Theorem 1, it suffices to prove

Proposition 12**.**

In case $({\rm II})$, the ramification divisor $R$ of the germ of finite cover $F$ is the union of two smooth curves, $R=R_{1}\cup R_{2}$, meeting transversally at the point $o^{\prime}$ and $F$ is ramified along $R$ with multiplicity two.

There are three possibilities:

• $({\rm II}_{1})$

$\deg F_{\mid R_{1}}=\deg F_{\mid R_{2}}=1$, the point $o\in V$ is the singular point of types $A_{n_{1}}$ and $A_{n_{2}}$ of the curves $B_{1}=F(R_{1})$ and $B_{2}=F(R_{2})$ for some $n_{1}$, $n_{2}\in\mathbb{N}$, and $(B_{1},B_{2})_{o}=4$, the germ $F$ is deformation equivalent to the germ $F_{{\bf 4},n_{1},n_{2}}$, the branch curve $B_{F_{{\bf 4}_{8}},n_{1},n_{2}}$ has the singularity of type $T_{4,2n_{1},2n_{2}}$ and $G_{F_{{\bf 4}_{8},n_{1},n_{2}}}$ is $\mathbb{S}_{4}$.

• $({\rm II}_{2})$

accurate to the numbering, $\deg F_{\mid R_{1}}=1$, $\deg F_{\mid R_{2}}=2$, the point $o\in V$ is the singular point of the curve germ $B_{1}=F(R_{1})$ of type $A_{2n}$ for some $n\geqslant 1$, the curve germ $B_{2}=F(R_{2})$ is smooth, and $(B_{1},B_{2})_{o}=2$, the germ $F$ is equivalent to the germ $F_{{\bf 3}_{2},n}$, the branch curve $B_{F_{{\bf 3}_{2},n}}$ has the singularity of type $D_{2n+3}$, the monodromy group $G_{F_{{\bf 3}_{2},n}}$ is a dihedral group $\mathbb{D}_{4}\subset\mathbb{S}_{4}$.

• $({\rm II}_{3})$

$\deg F_{\mid R_{1}}=\deg F_{\mid R_{2}}=2$, the curve germs $B_{1}=F(R_{1})$ and $B_{2}=F(R_{2})$ are smooth, and $(B_{1},B_{2})_{o}=1$, the germ $F$ is equivalent to the germ $F_{{\bf 2}_{1}}$, the branch curve $B_{F_{{\bf 2}_{1}}}$ has the singularity of type $A_{1}$, the monodromy group $G_{F_{{\bf 2}_{1}}}$ is the Klien four group $Kl_{4}\subset\mathbb{S}_{4}$.

Proof.

In case $({\rm II})$, the cover $F$ is given by functions of the following form:

[TABLE]

where $f_{i}(z_{0},w_{0})$, $h_{i}(z_{0},w_{0})$, $i=1,2$, are linear forms and $f_{\geqslant 3}(z_{0},w_{0})$, $h_{\geqslant 3}(z_{0},w_{0})\in\mathfrak{m}^{3}$. Note that the forms $f_{i}(z_{0},w_{0})$ and $h_{j}(z_{0},w_{0})$ are linear independent for each pair $(i,j)$, $i,j=1,2$, since $(M_{1},M_{2})_{o^{\prime}}=4$.

In the beginning, let us show that there are coordinates $z_{1},w_{1}$ in $U$ and $u_{1},v_{1}$ in $V$ such that $F$ is given by functions

[TABLE]

where $g_{1}(z_{1},w_{1})$ and $g_{2}(z_{1},w_{1})\in\mathfrak{m}^{3}$. Accurate to the coordinate change $u_{0}\leftrightarrow v_{0}$, we have two possibilities: 1) $f_{1}(z_{0},w_{0})$ and $f_{2}(z_{0},w_{0})$ are proportional and 2) $f_{1}(z_{0},w_{0})$ and $f_{2}(z_{0},w_{0})$ are linear independent.

In the first case we can assume (after coordinate changes in $U$ and $V$) that

[TABLE]

with some $a\in\mathbb{C}$. After the coordinate changes $z_{0}=z_{1}$, $w_{0}=\frac{1}{2}(w_{1}-az_{1})$ and $u_{0}=u_{1}$, $v_{0}=4v_{1}$, we obtain functions of the form (47).

In the second case we can assume (after changes of coordinates in $U$ and $V$) that

[TABLE]

where $c=0$ and $a\neq 0$, $b\neq 0$ (if $c\neq 0$ then the change of coordinates $\widetilde{u}_{0}=u_{0}$, $\widetilde{v}_{0}=v_{0}-cu_{0}$ ”kills”  the coefficient of $z_{0}w_{0}$). After the changes of coordinates

[TABLE]

we obtain functions of the form (47).

We have

[TABLE]

where $S(z_{1},w_{1})\in\mathfrak{m}^{3}$. Therefore

[TABLE]

in the ring $\mathbb{C}[[z_{1},w_{1}]]$, where $S_{i}(z_{1},w_{1})\in\mathfrak{m}^{2}$ and hence $F$ is ramified along $R$ with multiplicity two.

Let us make the change of coordinates $z_{2}=z_{1}+S_{1}(z_{1},w_{1})$, $w_{2}=w_{1}+S_{2}(z_{1},w_{1})$ and write functions $u$ and $v$ in the form

[TABLE]

where $\widetilde{f}_{1}(z_{2})\in\mathbb{C}[[z_{2}]]$, $\widetilde{h}_{1}(w_{2})\in\mathbb{C}[[w_{2}]]$. Note that the ramification divisor $R$ is given by equation $z_{2}w_{2}=0$. Therefore $R=R_{1}+R_{2}$, where $R_{1}=\{z_{2}=0\}$ and $R_{2}=\{w_{2}=0\}$.

Finally, if we put $z=z_{2}\sqrt{1+z_{2}\widetilde{f}_{1}(z_{2})}$ and $w=w_{2}\sqrt{1+w_{2}\widetilde{h}_{1}(w_{2})}$, we obtain that $F$ is given by functions of the following form

[TABLE]

where $\widetilde{g}_{3}(z,w)$ and $\widetilde{g}_{4}(z,w)\in\mathfrak{m}$. Note that $z=0$ is an equation of $R_{1}$ and $w=0$ is an equation of $R_{2}$. Note also that $w$ is a local parameter in $R_{1}$ at the point $o^{\prime}$ and $z$ is a local parameter in $R_{2}$.

The restrictions $F_{\mid R_{1}}$ to $R_{1}$ and $F_{\mid R_{2}}$ to $R_{2}$ of $F$ are given by functions

[TABLE]

If $\alpha_{i}=0$ for all odd $i$, then it is easy to see that $\deg F_{\mid R_{1}}=2$ and $B_{1}=F(R_{1})$ is given by equation

[TABLE]

Similarly, if $\beta_{i}=0$ for all odd $i$, then $\deg F_{\mid R_{2}}=2$ and $B_{2}=F(R_{2})$ is given by equation

[TABLE]

If $i=2n_{1}+1$ is the smallest odd index for which $\alpha_{i}\neq 0$, then it follows from (49) that $\deg F_{\mid R_{1}}=1$, the germ $B_{1}=F(R_{1})$ has the singularity of type $A_{2n_{1}}$, and the coordinate axis $\{u=0\}$ is the tangent line of $B_{1}$ at the point $o$.

Similarly, if $i=2n_{2}+1$ is the smallest odd index for which $\beta_{i}\neq 0$, then $\deg F_{\mid R_{2}}=1$, the germ $B_{2}=F(R_{2})$ has the singularity of type $A_{2n_{2}}$, and the coordinate axis $\{v=0\}$ is the tangent line of $B_{2}$ at the point $o$.

If $i=2n_{1}+1$ and $j=2n_{2}+1$ are the smallest odd indexes for which $\alpha_{i}\neq 0$ and $\beta_{j}\neq 0$, then the branch curve $B$ has the singularity of type $T_{4,2n_{1},2n_{2},2}$, the monodromy group $G_{F}$ is $\mathbb{S}_{4}$, since $G_{F}$ is a transitive subgroup of $\mathbb{S}_{4}$ generated by transpositions, and it is easy to see that the family of the germs of finite covers given by functions

[TABLE]

defines a deformation equivalence between $F$ and the gxerm $F_{{\bf 4}_{8},n_{1},n_{2}}$ given by

[TABLE]

If $i=2n+1$ is the smallest odd index for which $\beta_{i}\neq 0$ and $\alpha_{i}=0$ for all odd $i$, then the branch curve $B=B_{1}\cup B_{2}$ has the singularity of type $D_{2n+3}$ and it is easy to see that the family of the germs of finite covers given by functions

[TABLE]

defines a deformation equivalence between $F$ and the germ $F_{{\bf 3}_{2},n}$ given by

[TABLE]

The group $\pi_{1}^{loc}(B,o)$ is generated by bypasses $\gamma_{1}$ around $B_{1}$ and $\gamma_{2}$, $\gamma_{3}$ around $B_{2}$. The permutation $F_{*}(\gamma_{1})$ is a product of two commuting transpositions, since $\deg F_{\mid R_{1}}=2$, and $F_{*}(\gamma_{i})$, $i=2,3$. are transpositions, since $\deg F_{\mid R_{2}}=1$. These permutations generate a transitive group in $\mathbb{S}_{4}$. It follows from Lemma 14 that $G_{F_{{\bf 3}_{2},n}}\simeq\mathbb{D}_{4}$. Recall also that the germ of singularity $D_{n}$ is rigid. Therefore, by Lemma 14 and Corollary 3, the germ $F$ is equivalent to $F_{{\bf 3}_{2},n}$.

If $\alpha_{i}=0$ and $\beta_{i}=0$ for all odd $i$, then it follows from (50) and (51) that the singular point of $B=B_{1}\cup B_{2}$ is of type $A_{1}$. Consequently, $\pi_{1}^{loc}(B,o)=\mathbb{Z}\times\mathbb{Z}$ is generated by bypasses $\gamma_{1}$ and $\gamma_{2}$ around $B_{1}$ and $B_{2}$. The permutations $F_{*}(\gamma_{1})$ and $F_{*}(\gamma_{2})$ are products of two commuting transpositions, since $\deg F_{\mid R_{1}}=\deg F_{\mid R_{2}}=2$. These permutations generate a transitive group in $\mathbb{S}_{4}$. Therefore $G_{F}$ is the Klein four group $Kl_{4}\subset\mathbb{S}_{4}$. The singularity of type $A_{1}$ with up to change of coordinates is given by equation $uv=0$ and for the singularity $(B,o)$ of type $A_{1}$, there is the unique epimomorphism from $\pi_{1}^{loc}(B,o)$ to $Kl_{4}$. Therefore the germ $F$ is equivalent to the germ $F_{{\bf 2}_{1}}$ given by functions $u=z^{2}$ and $v=w^{2}$.

3. On the monofromy groups of germs of finite covers

3.1. On the transitive subgroups of the symmetric groups.

Let $G$ be a finite group. We say that two subgroups $H_{1}$ and $H_{2}$ of $G$ are equivalent if there is an inner automorphism $g\in Aut(G)$ such that $g(H_{1})=H_{2}$. A subgroup $H$ of $G$ is called relatively simple if there is not a proper non-trivial normal subgroup of $G$ contained in $H$. Denote by $A_{G}$ the set of representatives of equivalence classes of relatively simple subgroups of $G$ and let $I_{G}\subset N$ be the set of indices $i_{H}=(G:H)$ of subgroups $H\in A_{G}$.

We say that an imbedding $\varphi:G\to\mathbb{S}_{d}$ is transitive if $\varphi(G)$ is transitive subgroup of the symmetric group $\mathbb{S}_{d}$.

Proposition 13**.**

The set of transitive imbedding of $G$, considered up to conjugations in symmetric groups, is in one-to-one correspondence with the set $A_{G}$. For any $d\in I_{G}$ there is a transitive imbedding $\varphi:G\to\mathbb{S}_{d}$.

Proof.

Consider a symmetric group $\mathbb{S}_{d}$ as the group acting on the interval of natural numbers $\mathbb{N}_{d}=\{1,\dots,d\}$ and denote by $\mathbb{S}_{d-1}$ the subgroup of $\mathbb{S}_{d}$ consisting of the permutations $\tau\in\mathbb{S}_{d}$ leaving fixed $1$. Let $G$ be a transitive subgroup of $\mathbb{S}_{d}$. Then $H=G\cap\mathbb{S}_{d-1}$ is a subgroup of $G$ of index $(G:H)=d$, since $G$ is a transitive subgroup of $\mathbb{S}_{d}$. Let us show that $H$ is a relatively simple subgroup of $G$. Indeed, assume that a normal subgroup $N$ of $G$ is contained in $H$ and $h\in N$ is a non-trivial element. But, in this case for any $i\in\mathbb{N}_{d}$, there is an element $g_{i}\in G$ such that $g_{i}(1)=i$ and therefore $h(i)=i$ for each $i\in\mathbb{N}_{d}$, since $g_{i}^{-1}hg_{i}\in N\subset H$. As a result, we get a contradiction with the assumption that $G$ acts effectively on $\mathbb{N}_{d}$.

Conversely, let $H$ be a subgroup of $G$. Then $G$ acts on the set of left cosets of $H$ in $G$. This action defines a homomorphism $\varphi:G\to\mathbb{S}_{d}$, where $d=(G:H)$. It is easy to check that $\varphi$ is an imbedding if and only if $H$ is a relatively simple subgroup of $G$. ∎

Note that for each finite group $G$ there is at least one transitive imbedding, namely, Cayley’s imbedding $c:G\hookrightarrow\mathcal{S}_{|G|}$ corresponding to the trivial subgroup of $G$.

3.2. Germs of Galois smooth covers.

Consider a germ of finite cover $F:(U,o^{\prime})\to(V,o)$, $\deg_{o^{\prime}}F=d$, and its monodromy $F_{*}:\pi_{1}^{loc}(B,o)\to G_{F}\subset\mathbb{S}_{d}$. Let $c:G=G_{F}\hookrightarrow\mathcal{S}_{|G|}$ be Cayley’s imbedding. By Grauert - Remmert - Riemann - Stein Theorem, the homomorphism $c\circ F_{*}:\pi_{1}^{loc}(B,o)\to\mathbb{S}_{|G_{F}|}$ defines a germ of Galois cover $\widetilde{F}:(\widetilde{U},\widetilde{o})\to(V,o)$ of degree $\deg_{\widetilde{o}}\widetilde{F}=|G|$, where $\widetilde{F}$ is a holomorphic finite map and in general case $(\widetilde{U},\widetilde{o})$ is an irreducible germ of a normal complex-analytic variety. The group $G$ acts on $(\widetilde{U},\widetilde{o})$ such that the quotient variety $(\widetilde{U},\widetilde{o})/G$ is $(V,o)$ and $\widetilde{F}$ is the quotient map. By Proposition 13, the imbedding $G=G_{F}\hookrightarrow\mathbb{S}_{d}$ corresponds to a relatively simple subgroup $H$ of $G$, $(G:H)=d$, and it is well known that we can choose a subgroup $H_{F}$ of $G$ equivalent to $H$ such that the quotient variety $(\widetilde{U},\widetilde{o})/H_{F}$ is biholomorphic to $(U,o^{\prime})$ and the cover $\widetilde{F}$ is the composition of two covers, $\widetilde{F}=F\circ F_{H_{F}}$, where $F_{H_{F}}:(\widetilde{U},\widetilde{o})\to(U,o^{\prime})$ is the quotient map defined by the action of $H_{F}$ on $(\widetilde{U},\widetilde{o})$.

Before to formulate the statement describing the germs of the Galois smooth covers, let us recall the invariants of the binary linear groups given by Klein [7]. For each of the three groups (tetrahedral, octahedral, icosahedral) there are three invariant forms which we denote by $\varphi(z,w)$, $\psi(z,w)$, and $\theta(z,w)$, where $\psi(z,w)$ is the Hessian of $\varphi(z,w)$, and $\theta(z,w)$ is the Jacobian of $\varphi(z,w)$ and $\psi(z,w)$. We have

[TABLE]

in the case of tetrahedral group;

[TABLE]

in the case of octahedral group; and

[TABLE]

in the case of icosahedral group.

Theorem 3**.**

([15])* A germ of a Galois smooth cover $\widetilde{F}:(\widetilde{U},\widetilde{o})\to(V,o)$ is equivalent to one of the following rigid germs:*

• $1)$

$F_{1}=\{u=z^{2}+zw+w^{2},\,\,v=z_{1}^{2}w+zw^{2}\}$, $T(B_{1})=A_{2}$;

• $2_{1})$

$F_{2_{1},pq,q}=\{u=z^{pq}+w^{pq},\,\,v=z^{q}w^{q}\}$, $T(B_{2_{1},pq,q})=D_{p+2}$, $p>1$, $q>1$;

• $2_{2})$

$F_{2_{2},p,1}=\{u=z^{p}+w^{p},\,\,v=zw\}$, $T(B_{2_{2},p,p})=A_{p-1}$, $p>1$, $q=1$;

• $2_{3})$

$F_{2_{3},q,q}=\{u=z^{q}+w^{q},\,\,v=z^{q}w^{q}\}$, $T(B_{2_{3},q,1})=A_{3}$, $p=1$, $q>1$;

• $3)$

$F_{3,n_{1},n_{2}}=\{u=z^{n_{1}},\,\,v=w^{n_{2}}\}$, $T(B_{3,n_{1},n_{2}})=A_{1}$, $n_{1},n_{2}\in\mathbb{N}$;

• $4)$

$F_{4}=\{u=\psi_{4}(z,w),\,\,v=\theta_{6}(z,w)\}$, $T(B_{4})=A_{2}$;

• $5)$

$F_{5}=\{u=\varphi^{3}_{4}(z,w),\,\,v=\theta_{6}(z,w)\}$, $T(B_{5})=A_{3}$;

• $6)$

$F_{6}=\{u=\varphi_{4}(z,w),\,\,v=\psi_{4}^{3}(z,w)\}$, $T(B_{6})=A_{5}$;

• $7)$

$F_{7}=\{u=\varphi_{4}^{3}(z,w),\,\,v=\psi^{3}_{4}(z,w)\}$, $T(B_{7})=D_{4}$;

• $8)$

$F_{8}=\{u=\psi_{8}(z,w),\,\,v=\theta_{12}(z,w)\}$, $T(B_{8})=A_{2}$;

• $9)$

$F_{9}=\{u=\psi_{8}(z,w),\,\,v=\theta^{2}_{12}(z,w)\}$, $T(B_{9})=A_{5}$;

• $10)$

$F_{10}=\{u=\psi^{3}_{8}(z,w),\,\,v=\theta_{12}(z,w)\}$, $T(B_{{10}})=A_{3}$;

• $11)$

$F_{11}=\{u=\varphi_{6}^{4}(z,w),\,\,v=\psi^{3}_{8}(z,w)\}$, $T(B_{11})=D_{4}$;

• $12)$

$F_{12}=\{u=\varphi_{6}(z,w),\,\,v=\psi_{8}(z,w)\}$, $T(B_{12})=E_{6}$;

• $13)$

$F_{13}=\{u=\varphi^{2}_{6}(z,w),\,\,v=\theta_{12}(z,w)\}$, $T(B_{{13}})=E_{7}$;

• $14)$

$F_{14}=\{u=\varphi_{6}(z,w),\,\,v=\theta_{12}^{2}(z,w)\}$, $T(B_{{14}})=A_{7}$;

• $15)$

$F_{15}=\{u=\varphi^{2}_{6}(z,w),\,\,v=\theta_{12}^{2}(z,w)\}$, $T(B_{{15}})=D_{6}$;

• $16)$

$F_{16}=\{u=\psi_{20}(z,w),\,\,v=\theta_{30}(z,w)\}$, $T(B_{{16}})=A_{2}$;

• $17)$

$F_{17}=\{u=\psi_{20}(z,w),\,\,v=\theta_{30}^{2}(z,w)\}$, $T(B_{{17}})=A_{5}$;

• $18)$

$F_{18}=\{u=\psi_{20}^{3}(z,w),\,\,v=\theta_{30}(z,w)\}$, $T(B_{{18}})=A_{3}$;

• $19)$

$F_{19}=\{u=\varphi_{12}^{5}(z,w),\,\,v=\psi_{20}^{3}(z,w)\}$, $T(B_{19})=D_{4}$;

• $20)$

$F_{20}=\{u=\varphi_{12}(z,w),\,\,v=\theta_{30}(z,w)\}$, $T(B_{{20}})=A_{4}$;

• $21)$

$F_{21}=\{u=\varphi_{12}(z,w),\,\,v=\theta_{30}^{2}(z,w)\}$, $T(B_{{21}})=A_{3}$;

• $22)$

$F_{22}=\{u=\varphi_{12}(z,w),\,\,v=\psi_{20}(z,w)\}$, $T(B_{22})=E_{8}$;

In case $1)$ the monodromy group of $F_{1}$ is $G_{F_{1}}=\mathbb{S}_{3}$; in cases $2_{i})$, $i=1,2,3$, the groups $G_{F_{2_{i},pq,q}}=G(pq,q,2)$ (in notation used in [15]); in cases $j)$, $3\leqslant j\leqslant 22$, the group $G_{F_{j}}$ is the group in [15] with number $j$.

Proof.

By Cartan’s Lemma [3], the action of $G_{\widetilde{F}}$ on $\widetilde{U}$ can be linearized. Therefore $G_{\widetilde{F}}$ is a subgroup of $GL(2,\mathbb{C})$ generated by reflections. The list of germs of smooth Galois covers in Theorem 3 is in one-to-one correspondence with the list in [15] of finite subgroups of $GL(2,\mathbb{C})$ generated by reflections. The singularity types of the germs of the branch curves $B_{i}$ of $F_{i}$, $i=1,\dots,22$, are calculated in [9]. The rigidity of the covers $F_{i}$ easily follows from Cartan’s Lemma. ∎

3.3. Examples of finite groups which can (not) be realized as the monodromy groups of germs of finite covers.

The following two statements are a direct consequence of Theorem 3 and Proposition 13.

Corollary 4**.**

An abelian group $G$ can be realized as the monodromy group of the germ of a finite cover if and only if for each prime $p$, the number of cyclic $p$-primary factors entering in the presentation of $G$ as a direct product of cyclic groups is less than three.

The number of non-equivalent germs of finite covers $F:(U,o^{\prime})\to(V,o)$ whose monodrony group $G_{F}=Ab$, is equal to the number of non-isomorphic decompositions $Ab=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}$ (+ $1$, if $Ab$ is a cyclic group).

Proof.

There exists the unique relatively prime subgroup of each abelian group $G$, namely, the trivial subgroup. ∎

Corollary 5**.**

Groups $G=Q_{8}^{k}\times Ab$, $k\in\mathbb{N}$, where

[TABLE]

is the quaternion group and $Ab$ is any finite abelian group, are not realized as the monodromy groups of the germs of finite covers.

Proof.

There exists the unique relatively prime subgroup of each group $G=Q_{8}^{k}\times Ab$, namely, the trivial subgroup and the groups $G$ are not contained in the list of groups generated by reflections. ∎

Proposition 14**.**

The alternating group $\mathbb{A}_{4}$ can not be realized as the monodromy group of the germ of a finite cover.

Proof.

By Proposition 13, if $\mathbb{A}_{4}$ is the monodromy group of the germ $F:(U,o^{\prime})\to(V,o)$ of some finite cover, $\deg_{o^{\prime}}F=d$, then $d$ is equal either $4$, or $6$, or $12$. By Corollary 1, $d\neq 4$ and by Theorem 3, $d\neq 12$, since the group $\mathbb{A}_{4}$ is not a group generated by reflections.

Assume that there exists the germ $F$ of a finite cover of $\deg_{o^{\prime}}F=6$. Then the imbedding $G_{F}=\mathbb{A}_{4}\hookrightarrow\mathbb{S}_{6}$ corresponds to the action of $\mathbb{A}_{4}$ on the left cosets of a relatively simple subgroup $H_{F}\subset\mathbb{A}_{4}$ of order $2$. Consider the Galoisation $\widetilde{F}:(\widetilde{U},\widetilde{o})\to(V,o)$ of the cover $F$. The group $\mathbb{A}_{4}$ acts on $(\widetilde{U},\widetilde{o})$ such that $(\widetilde{U},\widetilde{o})/\mathbb{A}_{4}=(V,o)$ and $(\widetilde{U},\widetilde{o})/H_{1}=(U,o^{\prime})$, where $H_{1}=H_{F}$. The quotient cover $F_{H_{F}}:(\widetilde{U},\widetilde{o})\to(U,o^{\prime})$ is a two-sheeted germ of a Galois cover. Therefore it is branched along a germ of singular curve $B_{H_{F}}\subset U$, since $(\widetilde{U},\widetilde{o})$ is a germ of singular (normal) surface by Theorem 3. Hence, the ramification curve $R_{H_{F}}\subset(\widetilde{U},\widetilde{o})$ of $\widetilde{F}$ is the germ of a singular curve also.

Note that the curve germ $R_{H_{F}}$ can be defined as

[TABLE]

In the group $Kl_{4}\subset\mathbb{A}_{4}$ there exist three subgroups of order two, $H_{1}=H_{F}$, $H_{2}$, and $H_{3}$, conjugated in $\mathbb{A}_{4}$. Denote by $R_{i}=\{p\in\widetilde{U}\mid g(p)=p\,\,\,\text{for}\,\,g\in H_{i}\}$, $i=2,3$. Note that $R_{2}\neq R_{3}$ and $R_{i}$, $i=2,3$, are germs of singular curves.

The group $Kl_{4}\simeq H_{1}\times H_{2}$ acts on $(\widetilde{U},\widetilde{o})$. Consider the quotient germ $(\overline{U},\overline{o})=(\widetilde{U},\widetilde{o})/Kl_{4}$ and the quotient mapping $\overline{F}:(\widetilde{U},\widetilde{o})\to(\overline{U},\overline{o})$. We have $\overline{F}=F_{H_{2}}\circ F_{H_{F}}$, where $F_{H_{2}}:(U,o^{\prime})\to(U,o^{\prime})/H_{2}=(\overline{U},\overline{o})$, since $H_{2}$ is a normal subgroup of $Kl_{4}$ and therefore it acts on $(U,o^{\prime})$. The mapping $F_{H_{2}}$ is ramified along the curve germ $F_{H_{F}}(R_{2}\cup R_{3})$. The germ $F_{H_{F}}(R_{2}\cup R_{3})\subset U$ is also a singular curve germ, since either $F_{H_{F}}(R_{2})\neq F_{H_{1}}(R_{3})$ or if $F_{H_{F}}(R_{2})=F_{H_{1}}(R_{3})$ then $\deg F_{H_{1}\mid R_{i}}=1$, since $\deg_{\widetilde{o}}F_{H_{F}}=2$. On the other hand by Cartan’s Lemma, the action of $H_{2}$ on $(U,o^{\prime})$ can be linearized and the eigenvalues of the action of $H_{2}$ are $\pm 1$. If two eigenvalues of the action of $H_{2}$ are $-1$, then the ramification locus of $F_{H_{2}}$ is the point $o^{\prime}$, and if $H_{2}$ has the only one eigenvalue equals $-1$, when the ramification locus of $F_{H_{2}}$ is the germ of a non-singular curve. A contradiction. ∎

Note that the groups $\mathbb{S}_{2}\simeq\mathbb{Z}_{2}$ and $\mathbb{A}_{3}\simeq\mathbb{Z}_{3}$ are the monodromy groups of cyclic Galois covers of degree $2$ and $3$.

Proposition 15**.**

For $N\geqslant 3$ the symmetric group $\mathbb{S}_{N}$ and the alternating group $\mathbb{A}_{2N-1}$ are the monodromy groups of at least $[\frac{N-1}{2}]$ infinite series of rigid germs of finite covers.

Proof.

Consider the germ $F_{m,n,k}:(U,o^{\prime})\to(V,o)$ of a finite cover given by functions

[TABLE]

Obviously, $\deg_{o^{\prime}}F_{m,n,k}=m+n+1$ and $F_{m,n,k}$ is ramified with multiplicity $m+1$ along the curve germ $R_{1}$ given by $w=0$ and with multiplicity $n+1$ along the curve germ $R_{2}$ given by $w-z^{k}=0$. Therefore the branch locus of $F_{m,n,k}$ is $B=B_{1}\cup B_{2}$, where $B_{1}=F_{m,n,k}(R_{1})$ and $B_{2}=F_{m,n,k}(R_{2})$.

We have

[TABLE]

It follows from (54) that the restriction $F_{m,n,k|R_{i}}:R_{i}\to B_{i}$ of $F_{m,n,k}$ to $R_{i}$, $i=1,2$, is a biholomorphic mapping, $B_{1}$ is given by equation $v=0$ and $B_{2}$ is given by equation

[TABLE]

Therefore the singularity type of the curve germ $B$ is $T(B)=A_{2k(n+m+1)-1}$.

By Lemma 6, the group $\pi_{1}^{loc}(B,o)$ is generated by two bypasses $\gamma_{1}$ around $B_{1}$ and $\gamma_{2}$ around $B_{2}$, and it follows from the above discussion that $F_{*}(\gamma_{1})$ is a cycle of length $m+1$ and $F_{*}(\gamma_{2})$ is a cycle of length $n+1$ in the symmetric group $\mathbb{S}_{m+n+1}$. Therefore, up to conjugation in $\mathbb{S}_{m+n+1}$, we can assume that $F_{*}(\gamma_{1})=(1,2,\dots,m,m+n+1)$ and $F_{*}(\gamma_{2})=(m+1,m+2,\dots,m+n+1)$, since $F_{*}(\gamma_{1})$ and $F_{*}(\gamma_{2})$ generate a transitive subgroup $G_{F_{m,n,k}}$ of $\mathbb{S}_{m+n+1}$.

Lemma 17**.**

Let $G=\langle\tau,\sigma\rangle\subset\mathbb{S}_{m+n+1}$ be a subgroup generated by two cyclic permutations $\tau=(1,2,\dots,m,m+n+1)$ and $\sigma=(m+1,m+2,\dots,m+n+1)$, $m,n\in\mathbb{N}$. Then $\mathbb{A}_{m+n+1}\subset G$.

Proof.

Without loss of generality, we can assume that $m\geqslant n$. Let us show that $G$ contains the all cycles of length 3.

We have $\varrho=\tau\sigma=(1,2,\dots,m+n+1)\in G$,

[TABLE]

If $n=1$ then it follows from (55) that $G$ contains a transitive subgroup generated by transpositions $(j,m+1)$, $j=1,\dots,m$, and $\sigma$. Therefore $G=\mathbb{S}_{m+n+1}$. Therefore we can assume that $2\leqslant n\leqslant m$. If $m=2$, then direct check which we left to the reader shows that $\tau=(1,2,5)$ and $\sigma=(3,4,5)$ generate the group $\mathbb{A}_{5}$. Therefore we can assume that $m\geqslant 3$ and $m+n+1\geqslant 5$.

It is easy to check that

[TABLE]

Therefore

[TABLE]

for $1\leqslant j_{1}<j_{2}<j_{3}\leqslant m$ and $0\leqslant k\leqslant n+1$ and hence it suffices to prove that if the alternating groups $\mathbb{A}_{k}$ acting on the set $\{1,2,\dots,k\}$ and $\mathbb{A}_{3}$ acting on the set $\{k-1,k,k+1\}$ are subgroups of $G\subset\mathbb{S}_{m+n+1}$, then the alternating group $\mathbb{A}_{k+1}$ acting on the set $\{1,2,\dots,k,k+1\}$ is also a subgroup of $G$. But, it is obvious, since, first, the cycles $(j_{1},j_{2},j_{3})$ belong to $G$ for $\{j_{1},j_{2},j_{3}\}\subset\{1,2,\dots,k\}$; second, the cycles $(j,k-1,k+1)$ and $(j,k,k+1)$ belong to $G$ for $1\leqslant j\leqslant k-2$, since the cycles $(j,k-1,k)$ and $(j,k-1,k+1)$ belong to $G$; finally, the cycles $(j_{1},j_{2},k+1)$ belong to $G$ for $1\leqslant j_{1}<j_{2}\leqslant k-2$, since the cycles $(j_{1},k,k+1)$ and $(j_{2},k,k+1)$ belong to $G$. Therefore, to complete the proof of Lemma 17, it suffices to notice that the subgroup $H_{k+1}\subset G$ generated by all cycles $(j_{1},j_{2},j_{3})$ of length three, $1\leqslant j_{1}<j_{2}<j_{3}\leqslant k+1$, is a normal subgroup of the group $\mathbb{A}_{k+1}$ if $k\geqslant 4$ and hence, $H_{k+1}=\mathbb{A}_{k+1}$. ∎

It follows from Lemma 17 that if one of the cycles $(1,2,\dots,m,m+n+1)$ and $(m+1,m+2,\dots,m+n+1)$ is an odd permutation, then these cycles generate the symmetric group $\mathbb{S}_{m+n+1}$, and if the both of these two cycles are even permutations, then they generate the group $\mathbb{A}_{m+n+1}$. In the first case, if we put $N=m+n+1$, then we obtain that there are at least $[\frac{N-1}{2}]$ infinite series ($k\in\mathbb{N}$) of the germs $F_{m,n,k}$ ($m$ or $n$ is an odd number) of finite covers with the monodromy group $G_{F_{m,n,k}}=\mathbb{S}_{N}$. In the second case, if we put $2N=m+n+2$, then we obtain that there are at least $\frac{N-1}{2}$ infinite series ($k\in\mathbb{N}$) of the germs $F_{m,n,k}$ ($m$ and $n$ are even numbers) of finite covers with the monodromy group $G_{F_{m,n,k}}=\mathbb{A}_{2N-1}$.

The rigidity of the germs $F_{m,n,k}$ follows from the rigidity of the curve germs of singularity type $A_{2k(n+m+1)-1}$ and from Grauert - Remmert - Riemann - Stein Theorem, since the monodromy $F_{m,n,k*}$ is uniquely (up to conjugation in $\mathbb{S}_{m+n+1}$) determined by the triple $(m,n,k)$. ∎