Izvestiya: Mathematics
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

 Izvestiya: Mathematics, 2023, Volume 87, Issue 3, Pages 562–585DOI: https://doi.org/10.4213/im9357e (Mi im9357)

On the local fundamental group of the complement of a curve in a normal surface

Vik. S. Kulikov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

### Introduction

Let $(S,C)$ be a pair, where $S$ is a normal complex (not necessary compact) surface and $C\subset S$ is an effective reduced Weil divisor (possibly zero) on $S$ (hereinafter referred to as a curve lying in $S$) with the following properties:

(i) $C$ is a connected subset of $S$,

(ii) if $C$ is a zero divisor, then $C=o\in S$ is a point,

(iii) if an irreducible component of $C$ is not compact (in this case, such components will be called curve germs), then it is homeomorphic to the disc $\mathbb D_1=\{ z\in\mathbb C\mid |z|< 1\}$, has at most one singular point, and this point is the image of the centre of $\mathbb D_1$ (the image of the centre of $\mathbb D_1$ will be called the centre of this component),

(iv) each non-compact irreducible component of $C$ intersects other irreducible components of $C$ only at its centre.

Denote by $C_0$ the union of all compact irreducible components of $C$ (if $C$ has not a one-dimensional compact component, then $C_0$ is a point common to all irreducible components of $C$).

Let $U\subset S$ be an open neighbourhood (relative to the complex analytic topology) of the curve $C_0$. A proper holomorphic mapping $\nu\colon X\to U$ will be called the resolution of singularities of $C$ if

1) $X$ is a non-singular surface and $\nu\colon X\setminus \nu^{-1}(C)\to U\setminus C$ is a bi-holomorphic mapping,

2) $\widetilde C=\nu^{-1}(C)$ is a divisor with normal crossings in $X$,

3) each irreducible component of $\widetilde C$ is a non-singular curve,

4) any two different irreducible components of $\widetilde C$ can meet at most in a single point,

5) if $C=o$ is a point or all irreducible components of $C$ are not compact, then $\nu\colon X\to U$ is not a bi-holomorphic mapping.

We set $\widetilde C_0=\nu^{-1}(C_0)\subset X$. It follows from condition 5) that $\widetilde C_0$ is a non-empty union of compact curves, and by applying the Stein Factorization Theorem (see [1]), to the mapping $\nu$ and since $C$ is connected it follows that $\widetilde C$ and $\widetilde C_0$ are connected sets.

Based on existence of a polynomial real-valued function $\alpha \colon X\to \mathbb R$ such that $\alpha(x)\geqslant 0$ for $x\in X$ and $\alpha(\widetilde C_0)=0$, in [2] (see also [3]) a “tubular” neighbourhood of the curve $C_0\subset S$ was defined. In the present paper, we use a slightly modified definition of a set $\mathcal U_C$ of “tubular” neighbourhoods $U_{\varepsilon}\subset S$ of $C_0$ based on existence of good (with respect to the curve $\widetilde C$) Hermitian metrics (that is, positive definite Hermitian quadratic forms) $(ds)^2$ on $X$. The set $\mathcal U_C$ has the following properties:

– the set $\mathcal U_C$ is a base of open subsets of $S$ containing the curve $C_0$,

– for all $U_{\varepsilon}\in \mathcal U_C$, the fundamental groups $\pi_1(U_{\varepsilon}\setminus C)$ are naturally isomorphic.

We call $\pi_1^{\mathrm{loc}}(S,C):=\pi_1(U_{\varepsilon}\setminus C)$, $U_{\varepsilon}\in \mathcal U_C$, the local fundamental group of the complement of $C$ in $S$.

We say that a Hermitian metric $(ds)^2$ defined in a neighbourhood $\widetilde U\subset X$ of $\widetilde C_0$ is good (with respect to $\widetilde C$) if, for each singular point $p$ of $\widetilde C$, there is a neighbourhood $V_p\subset \widetilde U$ such that

(i) in each $V_p$, there are local coordinates $(z_1,z_2)$ such that $p=(0,0)$ and

$$\begin{equation*} V_p\simeq\mathbb B_2=\{(z_1,z_2)\in \mathbb C^2\mid \sqrt[2]{|z_1|^2+|z_2|^2}<2\}, \end{equation*} \notag$$

(ii) $z_1z_2=0$ is the equation of $\widetilde C\cap V_p$,

(iii) $(ds)^2$ is given in $V_p$ by $(ds)^2=dz_1\,d\overline z_1+dz_2\,d\overline z_2$,

(iv) $V_{p_1}\cap V_{p_2}=\varnothing$ for $p_1\neq p_2$.

To define the set of “tubular” neighbourhoods $\mathcal U_C$, consider a neighbourhood $U\subset S$ of $C_0$ such that: first, if a point $p\in \operatorname{Sing} S\cap U$, then $p\in C$, and, second, the pair $(U,U\cap C)$ satisfies conditions (i)–(iv). In § 1.1, it is proved the existence of good metrics in compactly imbedded in $X$ neighbourhoods $\widetilde U$ of $\widetilde C_0$, and, in § 2, it is shown that, for given good metric $(ds)^2$ in $\widetilde U\subset X$, there is a positive $\varepsilon_0:=\varepsilon_0((ds)^2)\ll 1$ such that the sets $\widetilde U_{\varepsilon_1}\setminus \widetilde C_0$ and $\widetilde U_{\varepsilon_2}\setminus \widetilde C_0$ are homeomorphic to each other if $\varepsilon_1,\varepsilon_2 <\varepsilon_0$, where

$$\begin{equation*} \widetilde U_{\varepsilon}= \{ p\in \widetilde U\mid \operatorname{dist}_{(ds)^2}(\widetilde C_0, p)<\varepsilon\}. \end{equation*} \notag$$

We denote by $\mathcal U_{\widetilde C,\nu, (ds)^2}:= \{ \widetilde U_{\varepsilon}\mid \varepsilon<\varepsilon_0((ds)^2)\}$ the set of all neighbourhoods $\widetilde U_{\varepsilon}$ of $\widetilde C_0$ with $\varepsilon<\varepsilon_0$; $\widetilde U_{\varepsilon}\in \mathcal U_{\widetilde C,\nu, (ds)^2}$ (see Definition 2 in § 1.2) will be called a tubular neighbourhood of $\widetilde C_0\subset X$ (defined via the good metric $(ds)^2$). Denote by $\mathcal U_{\widetilde C, \nu}:= \bigsqcup_{(\widetilde ds)^2}\mathcal U_{\widetilde C,\nu, (ds)^2}$ the disjunct union of sets $\mathcal U_{\widetilde C,\nu, (ds)^2}$ over all good metrics $(ds)^2$, by $\mathcal U_{\widetilde C}:=\bigsqcup_{\nu}\mathcal U_{\widetilde C,\nu}$ the disjunct union of the sets $\mathcal U_{\widetilde C,\nu}$ over all resolutions $\nu\colon X\to U$ of singularities of $C\subset S$. The open (in S) sets $U_{\varepsilon}$ lying in one of the sets $\mathcal U_{C,\nu}:=\{ U_{\varepsilon}= \nu(\widetilde U_{\varepsilon})\subset S\mid \widetilde U_{\varepsilon}\in \mathcal U_{\widetilde C,\nu}\}$ are called “tubularneighbourhoods of the curve $C_0\subset S$. Let $\mathcal U_{C}:=\bigsqcup_{\nu}\mathcal U_{C,\nu}$ and let ${\mathcal U}:=\bigsqcup_{C}\mathcal U_{\widetilde C}$ be the disjunct union of the sets $\mathcal U_{\widetilde C}$ over all pairs $(S,C)$.

The boundary $\partial U_{\varepsilon}\subset S$ of $U_{\varepsilon}\in \mathcal U_{C}$ is a compact connected three dimensional $C^0$-manifold without boundary. In § 2, we will prove the following result.

Theorem 1. Let $C\subset S$ be a compact curve in a normal complex surface $S$. Then

1) for $U_{\varepsilon} \in \mathcal U_{C,\nu, (ds)^2}$, there is a homeomorphism $\rho_{\varepsilon}\colon U_{\varepsilon}\setminus C\to \partial U_{\varepsilon}\times (0,\varepsilon)$ (here $(0,\varepsilon)= \{ t\in \mathbb R\mid 0< t< \varepsilon\})$ such that $\rho^{-1}_{\varepsilon}(\partial U_{\varepsilon}\times \{\varepsilon_1\}) = \partial U_{\varepsilon_1}\subset U_{\varepsilon}$ for $0<\varepsilon_1<\varepsilon$, and, in particular, $\pi_1(U_{\varepsilon}\setminus C)\simeq \pi_1(\partial U{_\varepsilon})$;

2) the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$.

Let $\Gamma(\widetilde C)$ be the dual partially bi-weighted graph of $\widetilde C=\nu^{-1}(C)= C_1\cup \dots\cup C_{m+k}$, where $C_1,\dots,C_m$ are the irreducible compact components of the graph $\widetilde C$, and $C_{m+1},\dots, C_{m+k}$ are the irreducible non-compact components. The vertex set of $\Gamma(\widetilde C)$ is $\{ v_{1,0},\dots, v_{m,0}\}\cup \{ v_{m+1,1},\dots, v_{m+k,1}\}$. The vertices $v_{j,0}$, $j=1,\dots, m$, of $\Gamma(\widetilde C)$ correspond to the compact components $C_j$, and their bi-weights are $(w_{1,j},w_{2,j})$, where $w_{1,j}=(C_{j}^2)_{X}$ is the self-intersection number and $w_{2,j}=g_j$ is the genus of $C_j$. The vertices $v_{j,1}$, $j=m+1,\dots, n=m+k$, correspond to the non-compact components of $\widetilde C$, and they have no bi-weights. Two vertices $v_{j_1,\delta_{j_1}}$ and $v_{j_2,\delta_{j_2}}$ are connected by the edge $e_{j_1,j_2}:=(v_{j_1,\delta_{j_1}},v_{j_2,\delta_{j_2}})$ if and only if $C_{j_1}\cap C_{j_2}\neq \varnothing$.

Note that $\Gamma(\widetilde C)$ is a connected graph, since so is $\widetilde C$.

Denote by $\mathcal G$ the set of all connected partially bi-weighted finite graphs $\Gamma$ with the following properties:

The following result can be proved quite easily (see § 2.1).

Proposition 1. The mapping $\gamma\colon (\widetilde U_{\varepsilon},\widetilde C) \in \mathcal U\mapsto \Gamma(\widetilde C)\in\mathcal G$ is a surjection.

Let $\Gamma_0$ be the subgraph of $\Gamma\in \mathcal G$ with vertex set $V(\Gamma_0)=\{ v_{1,0},\dots, v_{m,0}\}$ and the edge set formed of the edges of the graph $\Gamma$ that connect vertices from $V(\Gamma_0)$. The fundamental groups $\pi_1(\Gamma(\widetilde C),v_{1,0})$ and $\pi_1(\Gamma(\widetilde C_0),v_{1,0})$ are free groups of the same rank $r\geqslant 0$, since the valences $\operatorname{v}_j$ of the vertices $v_{j,1}$ are equal to $1$ for $j>m$.

We will consider a graph $\Gamma\in \mathcal G$ as a geometric graph. Let us choose $r$ edges $E_1,\dots, E_r$ of $\Gamma_0$ such that $\pi_1\bigl(\Gamma\setminus \bigcup_{l=1}^sE_{l},v_{1,0}\bigr)$ are free groups of rank $r-s$ for $s=1,\dots, r$. Let vertices $v_{j(E_s,1),0}$ and $v_{j(E_s,2),0}$ be connected by edges $E_s$. For each $s=1,\dots,r$, let us choose two points $v_{j(E_s,1),1}$ and $v_{j(E_s,2),1}$ in $E_s$ such that $v_{j(E_s,2),1}$ is met, first, when moving from $v_{j(E_s,1),0}$ towards $v_{j(E_s,2),0}$ along $E_s$. Let $\widetilde E=\{ E_1,\dots, E_r\}$ and ${\Gamma}_{\widetilde E}$ be the graph (called a tree of $\Gamma$) obtained from $\Gamma$ by adding the points $v_{j(E_s,1),1}$ and $v_{j(E_s,2),1}$, $s=1,\dots,r$, to the vertex set of $\Gamma$ and deleting the edges $(v_{j(E_s,2),1},v_{j(E_s,2),1})$ that connect the vertices $v_{j(E_s,2),1}$ and $v_{j(E_s,2),1}$ (see Figure 1). By definition, the vertices $v_{j(E_s,i),1}$ have no bi-weights for $i=1,2$.

For each pair of vertices $v_{i,\delta_i}$ and $v_{j,\delta_j}$ of ${\Gamma}_{\widetilde E}$, we define

$$\begin{equation*} \Delta_{(i,\delta_i),(j,\delta_j)}=\begin{cases} 1 &\text{if } v_{i,\delta_i}\text{ and }v_{j,\delta_j} \text{ are connected by an edge in } {\Gamma}_{\widetilde E}, \\ 0 &\text{if }v_{i,\delta_i}\text{ and }v_{j,\delta_j} \text{ are not connected by an edge in } {\Gamma}_{\widetilde E}, \\ 0 &\text{if }v_{i,\delta_i}=v_{j,\delta_j}. \end{cases} \end{equation*} \notag$$
Let $\operatorname{v}_{j_0}:=\operatorname{v}(v_{j_0,0})$ be the valence of a vertex $v_{j_0,0}\in {\Gamma}_{\widetilde E}$ and let
$$\begin{equation*} \Upsilon_{j_0}=\{ v_{i_1,\delta_{i_1}},\dots, v_{i_{\operatorname{v}_{j_0}},\delta_{i_{\operatorname{v}_{j_0}}}}\} \end{equation*} \notag$$
be the set of vertices $v_{j,\delta_j}\in {\Gamma}_{\widetilde E}$ connected with $v_{j_0,0}$ by edges. Let $I_{j_0}$ be the set of bi-indices $(j,\delta_j)$ of $v_{j,\delta_j}\in \Upsilon_{j_0}$. Consider the bijection
$$\begin{equation*} \overline o_{j_0}=(o_{j_0},d_{j_0})\colon \{ 1,\dots, \operatorname{v}_{j_0}\}\to I_{j_0}, \qquad (o_{j_0}(i),d_{j_0}(i))\in I_{j_0}. \end{equation*} \notag$$

With the vertices $v_{j,0}$, $j=1,\dots, m$, with bi-weights $(w_1,w_2)=(k_j,g_j)$, we associate the words

$$$$\mathcal W_j:= x_{j,0}^{k_j}\prod_{i=1}^{g_{j}}[\mu_{j,i},\lambda_{j,i}] \prod_{i=1}^{\operatorname{v}_{j}}x_{\overline{o}_{j}(i)}$$ \tag{1}$$
in the alphabet $\mu_{j,1},\lambda_{j,1},\dots, \mu_{j,g_{j}},\lambda_{j,g_{j}}$, $x_{j,0}$, $x_{\overline{o}_{j}(1)},\dots,x_{\overline{o}_{j}(\operatorname{v}_{j})}$, where $[\mu_{j,i},\lambda_{j,i}]= \mu_{j,i}\lambda_{j,i}\mu_{j,i}^{-1}\lambda_{j,i}^{-1}$.

In the following definition, we will use the above notation.

Definition 1. For a tree $\Gamma_{\widetilde E}$ of $\Gamma\in\mathcal G$, denote by $\pi^w_1(\Gamma_{\widetilde E})$ a group with the following presentation: it is generated by $m+k+3r+2\sum_{j=1}^mg_j$ elements

$(\mathrm{g}_1)$ $x_{j,0}$, $\mu_{j,i}$, $\lambda_{j,i}$, $1\leqslant j\leqslant m$, $1\leqslant i\leqslant g_j$,

$(\mathrm{g}_2)$ $x_{m+1,1},\dots, x_{m+k,1}$,

$(\mathrm{g}_3)$ $x_{j(E_s,1),1}$, $x_{j(E_s,2),1}$, $y_s$, $1\leqslant s\leqslant r$,

which are related by

$(\mathrm{r}_1)$ $\mathcal W_{j} =1$, $1\leqslant j \leqslant m$,

$(\mathrm{r}_2)$ $[x_{j,0}, \mu_{j,i}]=[x_{j,0}, \lambda_{j,i}] =1$, $1\leqslant j\leqslant m$, $1\leqslant i\leqslant g_j$,

$(\mathrm{r}_3)$ $[x_{j_1,\delta_1}, x_{j_2,\delta_2}] =1$, $\Delta_{(j_1,\delta_1),(j_2\delta_2)}=1$,

$(\mathrm{r}_4)$ $x_{j(E_s,1),0}^{-1}y_sx_{j(E_s,1),1}y_s^{-1} =1$, $1\leqslant s\leqslant r$,

$(\mathrm{r}_5)$ $x_{j(E_s,2),1}^{-1}y_s^{-1}x_{j(E_s,2),0}y_s =1$, $1\leqslant s\leqslant r$,

where the words $\mathcal W_{j}$ are defined in (1).

Theorem 2. Let $\Gamma_{\widetilde E_1}$ and $\Gamma_{\widetilde E_2}$ be two trees of $\Gamma\in\mathcal G$. Then $\pi^w_1(\Gamma_{\widetilde E_1})$ and $\pi^w_1(\Gamma_{\widetilde E_2})$ are isomorphic groups.

Let $\Gamma_{\widetilde E}$ be a tree of $\Gamma\,{\in}\,\mathcal G$. In view of Theorem 2, the group $\pi_1^w(\Gamma)\,{:=}\,\pi_1^w(\Gamma_{\widetilde E})$ will be called the fundamental group of the partially bi-weighted graph $\Gamma$, and the group $\pi_1(\Gamma,v_{1,0})$ will be simply called the fundamental group of the graph $\Gamma$.

Theorem 3. For $U_{\varepsilon}\in \mathcal U_{C,\nu}$ and a tree ${\Gamma}_{\widetilde E}(\widetilde{C})$ of $\Gamma(\widetilde C)$, and $\widetilde C=\nu^{-1}(C)$, the group $\pi_1(U_{\varepsilon}\setminus C)$ is isomorphic to $\pi_1^w(\Gamma_{\widetilde E})$.

The groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$.

In view of Proposition 1, it is easy to see that Theorem 2 is a direct consequence of Theorem 3.

In the case $C=o$ is a singular point of a normal surface $S$ and $\Gamma(\widetilde C)$ is a tree, Theorem 3 was proved by Mumford in [2] if $\widetilde C$ is a union of rational curves; Wagreich [4] proved the general case, that is, where the irreducible components of the curve $\widetilde C$ are not necessarily rational.1 A proof of Theorem 3 is also contained in [5] in the case $C$ is a compact curve; it is contained in [6] if $C_0$ is a singular point of a normal surface. Note also that Theorem 3 was formulated in the case when $C_0$ is a singular point of a normal surface, the graph $\Gamma(\widetilde C)$ is a tree, and the irreducible components of $\widetilde C_0$ are rational curves; this result was used in [7] and [8] in order to describe a connection between the set of rational Belyi pairs and the set of rigid germs of finite morphisms of smooth surfaces branched at curve germs with $\mathit{ADE}$ type singular points.

Let $\mathcal S$ be a subset of $\mathcal G$. Denote by $\Pi_w(\mathcal S)=\{\pi^w_1(\Gamma)\mid \Gamma\in \mathcal S\}$ the set of the fundamental groups of the partially bi-weighted graphs $\Gamma\in \mathcal S$ (considered up to isomorphisms of groups). The following theorem (a proof of which will be given in § 3) is a corollary to Theorem 3.

Theorem 4. 1. Let $\mathcal Ch^0_i\subset \mathcal G$, $i=0,1,2$, be the set of partially bi-weighted graphs that are chains with $i$ vertices without bi-weights, and the vertices $v_{j,0}$ of which have bi-weights $(k_j,0)$. Then

$$\begin{equation*} \Pi_w(\mathcal Ch^0_2)=\{ \mathbb Z\times \mathbb Z\}, \quad \Pi_w(\mathcal Ch^0_1)=\{ \mathbb Z\},\quad \Pi_w(\mathcal Ch^0_0)=\{ \mathbb Z/n\mathbb Z\mid n\geqslant 0\}. \end{equation*} \notag$$

2. Let $\mathcal L\subset \mathcal G$ be the set of partially bi-weighted graphs that are loops and whose vertices $v_{j,0}$ have bi-weights $(k_j,0)$. Then

$$\begin{equation*} \Pi_w(\mathcal L)= \{ \mathbb Z^2\ltimes_M\mathbb F_1\mid M\in \operatorname{SL}(2,\mathbb Z)\}, \end{equation*} \notag$$
where $\mathbb Z^2\ltimes_{M}\mathbb F_1=\langle (z_1,z_2),t\mid (z_1,z_2)\in \mathbb Z^2,\, t^{-1}(z_1,z_2)t=(z_1,z_2)M\rangle$ are the semidirect products of $\mathbb Z^2$ and $\mathbb F_1\simeq \mathbb Z$, and $M\in \operatorname{SL}(2,\mathbb Z)$ on $\mathbb Z^2$ from the right.

3. Let $\Gamma\in\mathcal G$ contain $n_i$ vertices $v_{j,0}$ whose bi-weights are $(k_j,i)$, let $i_0$ be a number such that $n_i=0$ for $i>i_0$, and let $\pi_1(\Gamma,v_{1,0})\simeq \mathbb F_r$ be the free group of rank $r$. Then there is an epimorphism

$$\begin{equation*} \pi^w_1(\Gamma)\to \biggl(\prod^{n_1}\mathcal R_1*\dots*\prod^{n_{i_0}} \mathcal R_{i_0}\biggr)*\mathbb F_r\to 1 \end{equation*} \notag$$
to the free product of the fundamental groups $\mathcal R_g=\pi_1(R_g)$ of Riemann surfaces $R_g$ of genus $g$ and $\mathbb F_r$.

Further we freely use the above notation.

### 1.1. Good metrics

Since the curve $\widetilde C_0$ is compact, we can choose a partition of unity $\{\rho_i\}$ subordinate to a finite open in $X$ cover $\{ W_i\}$ of $\widetilde C_0$ such that $\sum_i \rho_i(p)= 1$ for all $p\in\widetilde C_0$. Let $(z_{i,1},z_{i,2})$ be local coordinates in $W_i$. Then

$$\begin{equation*} (d\widetilde s)^2 =\sum_{i}\rho_i(dz_{i,1}\, d\overline z_{i,1}+ dz_{i,2}\, d\overline z_{i,2}) \end{equation*} \notag$$
is a Hermitian metric on a neighbourhood $\widetilde U\subset \bigcup_iW_i$ of $\widetilde C_0$.

Let $W\subset \widetilde U$ be a neighbourhood of a point $p\in \widetilde U$ with local coordinates $z_1$, $z_2$ such that $W$ is bi-holomorphic to the ball $\mathbb B_{r}=\{ (z_1,z_2)\in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}<r\}$ of radius $r$ and centred at $p$, $\mathbb B_{r}\simeq W$. Then $(d\widetilde s)^2$ is given in $W$ by

$$$$(d\widetilde s)^2=h_{1}\,dz_1\, d\overline z_1+h_{2}\, dz_2\, d\overline z_2+ (a+ib)\, dz_1\, d\overline z_2 +(a-ib)\, d\overline z_1\, dz_2,$$ \tag{2}$$
where $h_{1}:=h_{1}(z_1,z_2)$, $h_{2}:=h_{2}(z_1,z_2)$, $a:=a(z_1,z_2)$, and $b:=b(z_1,z_2)$ are real-valued functions in $W$. In addition, it follows from Sylvester’s criterion of positive definiteness of a quadratic form that the form given by (2) is positive definite if and only if the functions $h_{1}$, $h_{2}$, $a$, $b$ satisfy, ar each point of $W$,
$$$$h_{1}>0,\qquad h_{2}>0,\qquad h_{1}h_{2}-a^2-b^2>0.$$ \tag{3}$$

Without loss of generality, we can assume that $r=4$, and, for $\varepsilon\leqslant 4$, we consider the balls $\mathbb B_{\varepsilon}=\{ (z_1,z_2)\in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}<\varepsilon\}$ as open subsets of $W\simeq \mathbb B_{4}$.

Lemma 1. There exists a Hermitian metric $(ds_0)^2$ on $\widetilde U\subset X$ such that

$$\begin{equation*} (ds_0)^2= \begin{cases} dz_1\,d\overline z_1+dz_2\, d\overline z_2 &\textit{on }\mathbb B_{2} \subset W, \\ (d\widetilde s)^2 &\textit{on }\widetilde U\setminus W. \end{cases} \end{equation*} \notag$$

Proof. Let us choose monotone $C^{\infty}$-functions $f_1(t)$ and $g_1(t)$ such that
$$\begin{equation*} f_1(t)= \begin{cases} 2 &\text{if }t\leqslant 3, \\ 1 &\text{if }t\geqslant \dfrac{7}{2}, \end{cases} \qquad g_1(t)= \begin{cases} 0 &\textrm{if }t\leqslant 3, \\ 1 &\textrm{if }t\geqslant \dfrac{7}{2}. \end{cases} \end{equation*} \notag$$
Applying (3) it is easy to see that
$$\begin{equation*} (ds_1)^2= \begin{cases} f_1(\sqrt{|z_1|^2+|z_2|^2})(h_{1}\, dz_1\, d\overline z_2+h_{2}\, dz_2\, d\overline z_2) \\ \ +\,g_1(\sqrt{|z_1|^2+|z_2|^2})((a+ib)\, dz_1\, d\overline z_2+ (a-ib)\, d\overline z_1\, dz_2) &\text{on }\mathbb B_{7/2}\subset W, \\ (d\widetilde s)^2 &\text{on }\widetilde U\setminus \mathbb B_{7/2} \end{cases} \end{equation*} \notag$$
is a Hermitian metric on $\widetilde U$ such that $(ds_1)^2=2h_{1}\, dz_1\, d\overline z_1+2h_{2}\, dz_2\, d\overline z_2$ on $\mathbb B_{3}\,{\subset}\,W$.

Next, let us choose monotone $C^{\infty}$-functions $f_0(t)$ and $g_0(t)$ such that

$$\begin{equation*} f_0(t)= \begin{cases} 1 &\text{if }t\leqslant 2, \\ 0 &\textrm{if } t\geqslant \dfrac{5}{2}, \end{cases}\qquad g_0(t)= \begin{cases} 0 &\text{if }t\leqslant 2, \\ 1 &\text{if }t\geqslant \dfrac{5}{2}. \end{cases} \end{equation*} \notag$$
Then
$$\begin{equation*} (ds_0)^2= \begin{cases} \bigl(f_0(\sqrt{|z_1|^2+|z_2|^2})+2g_0(\sqrt{|z_1|^2+|z_2|^2}) h_{1}\bigr)\, dz_1\, d\overline z_2 \\ \ + \bigl(f_0(\sqrt{|z_1|^2+|z_2|^2})+2g_0(\sqrt{|z_1|^2+|z_2|^2}) h_{2}\bigr)\, dz_2\, d\overline dz_2 &\text{on }\mathbb B_{5/2}\subset W, \\ (ds_1)^2 &\textrm{on }\widetilde U\setminus \mathbb B_{5/2} \end{cases} \end{equation*} \notag$$
is a Hermitian metric on $\widetilde U$ such that $(d\widetilde s_0)^2=dz_1\, d\overline z_1+dz_2\, d\overline z_2$ on $\mathbb B_{2}\subset W$. Lemma 1 is proved.

For the points $p_{j_1,j_2}=C_{j_1}\cap C_{j_2}\subset \operatorname{Sing} \widetilde C$, $1\leqslant j_1 < j_2\leqslant n$, let us choose pairwise non-intersecting neighbourhoods $W_{j_1,j_2}\subset \widetilde U\subset X$ of the points $p_{j_1,j_2}$ bi-holomorphic to the ball $\mathbb B_{4}=\{ (z_1,z_2) \in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}< 4\}$, and such that, in the coordinates $(z_1,z_2)$ in $W_{j_1,j_2}$, the curve $W_{j_1}:=C_{j_1}\cap W_{j_1,j_2}$ (respectively, the curve $W_{j_2}:=C_{j_2}\cap W_{j_1,j_2}$) is given by the equation $z_1=0$ (respectively, $z_2=0$). Applying Lemma 1, in each neighbourhood $W_{j_1,j_2}$ we change the metric $(d\widetilde s)^2$ by $(ds_0)^2$ and obtain a good metric ($ds)^2$ (with respect to $\widetilde C$) on $\widetilde U\subset X$.

### 1.2. Tubular neighbourhoods

Let us fix $\widetilde U\subset X$ and a good metric $(ds)^2$ on $\widetilde U$. For positive $\varepsilon\ll 1$ and for $j=1,\dots,m$, denote by

$$\begin{equation*} U_{j,\varepsilon}= \{ p\in X\mid \operatorname{dist}_{(ds)^2}(C_j, p)<\varepsilon\} \end{equation*} \notag$$
the $\varepsilon$-neighbourhood of the compact curve $C_j$, let $\overline U_{j,\varepsilon}$ be its closure in $X$, and let
$$\begin{equation*} \partial U_{j,\varepsilon}= \{ p\in X\mid \operatorname{dist}_{(ds)^2}(C_j, p)=\varepsilon\} \end{equation*} \notag$$
be its boundary. Obviously, there is positive $\varepsilon_j\ll 1$ such that if $\varepsilon<\varepsilon_j$, then $\overline U_{j,\varepsilon}\subset \widetilde U$ and the set $\{ U_{j,\varepsilon}\}$ is a base of open subsets of $X$ containing $C_j$.

Consider the restriction $T_{X\mid C_j}$ of the tangent bundle $T_X$ of $X$ to $C_j$ as a bundle of four dimensional vector spaces over $\mathbb R$. Then $(ds)^2$ defines a splitting of $T_{X\mid C_j}$ into the direct sum $T_{C_j}\bigoplus N_{C_j,(ds)^2}$ of the tangent bundle $T_{C_j}$ of $C_j$ and the normal bundle

$$\begin{equation*} N_{C_j,(ds)^2}= \{ (p,v) \mid p\in C_j,\, v\in T_{X\mid C_j,p},\, v\perp T_{C_j,p}\} \end{equation*} \notag$$
on $C_j$ of the vector spaces transversal to $T_{C_j,p}$ in $T_{X\mid C_j,p}$ with respect to the scalar product
$$\begin{equation*} (v_1,v_2)=\frac{1}{2}[(ds)^2(v_1+v_2)-(ds)^2(v_1)-(ds)^2(v_2)]. \end{equation*} \notag$$
Denote by $\operatorname{pr}_j\colon N_{C_j,(ds)^2}\to C_j$ the projection of $N_{C_j,(ds)^2}$ to $C_j$ and denote by $S_{j,0}=\{ (p,0)\in N_{C_j,(ds)^2}\mid p\in C_j\}$ the zero section.

For $p\in C_j$, consider the restriction $\operatorname{Exp}_{p\mid N_{C_j,(ds)^2}}\colon N_{C_j,(ds)^2}\to \widetilde U\subset X$ to $N_{C_j,(ds)^2}$ of the exponential mapping $\operatorname{Exp}_p\colon T_{X,p}\to \widetilde U\subset X$ which sends segments of one dimensional vector spaces of $N_{C_j,(ds)^2}$ to geodesic lines perpendicular to $C_j$ at the point $p$. It is well-known that, for each point $p\in C_j$, there are neighbourhoods $W_p\subset C_j$ of $p$ and $V_p\subset \operatorname{pr}_j^{-1}(W_p)\subset N_{C_j,(ds)^2}$ of $S_{j,0}\cap \operatorname{pr}_j^{-1}(W_p)$ such that the mapping $\varphi_{j,p}\colon V_p \to \widetilde U$, as obtained at each point $(q,v)\in V_p$ from the exponential mapping $\operatorname{Exp}_{q\mid N_{C_j,(ds)^2}}\colon T_{X,q}\to \widetilde U$, is a diffeomorphism of $V_p$ and its image $\varphi_{j,p}(V_p)$. Since for $1\leqslant j\leqslant m$ the curves $C_j$ are compact, it is easy to see that there exist neighbourhoods $V_j\subset N_{C_j,(ds)^2}$ of the zero sections $S_{j,0}$ such that $\varphi_j\colon V_j\to \widetilde U$, as given by $\varphi_j((p,v))=\varphi_{j,p}((p,v))$, are diffeomorphisms of $V_j$ to their images. Without loss of generality, we can assume that $U_{j,\varepsilon}\subset \varphi_j(V_j)$ for $\varepsilon<\varepsilon_j$. We set $N_{C_j,(ds)^2}(\varepsilon):=\varphi_j^{-1}(U_{j,\varepsilon})$.

Below, we will identify $N_{C_j,(ds)^2}(\varepsilon)\subset N_{C_j,(ds)^2}$ with $U_{j,\varepsilon}\subset X$ if it does not lead to a misunderstanding. In particular, the projection $\operatorname{pr}_j$ defines a structure of a $C^{\infty}$-locally trivial bundle of closed discs of radius $\varepsilon$ on $\overline U_{j,\varepsilon}$ and defines a structure of a $C^{\infty}$-locally trivial bundle of circles of radius $\varepsilon$ on $\partial U_{j,\varepsilon}\subset \overline U_{j,\varepsilon}$. For $p\in C_j$, the set $\operatorname{pr}_j^{-1}(p)\cap \overline U_{j,\varepsilon}$, which is diffeomorphic to the disc $\mathbb D_{\varepsilon}=\{ z\in\mathbb C\mid |z|\leqslant\varepsilon\}$, is the union of segments of length $\varepsilon$ of geodesic lines lying in $\overline {U}_{j,\varepsilon}$ perpendicular to $C_j$ and emanating from the point $p\in C_j$.

We set $d_0=\min_{1\leqslant j\leqslant m} \varepsilon_j> 0$ and define

$$\begin{equation*} d_j = \operatorname{dist}_{ds^2}\biggl(C_j\setminus \biggl(\bigcup_{i=1}^{\operatorname{v}_j} V_{j,o_j(i)}\biggr), \widetilde C\setminus C_j\biggr)>0 \end{equation*} \notag$$
for $1\,{\leqslant}\, j\,{\leqslant}\, m$, where the neighbourhoods $V_{j,o_j(i)}:=V_{p_{j,o_j(i)}}$ are the neighbourhoods of the points $p_{j,o_j(i)}\,{=}\, C_j\cap C_{o_j(i)}$, involved in the definition of the good metric $(ds)^2$ (see the introduction). We set $\varepsilon_0=\frac{1}{2}\min(1,d_0,d_1,\dots,d_m)>0$.

Definition 2. The open subsets $\widetilde U_{\varepsilon}=\bigcup_{j=1}^mU_{j,\varepsilon}$ of $X$, $\varepsilon<\varepsilon_0$, are called the tubular neighbourhoods of $\widetilde C_0\subset X$.

Consider a curve $C_j$, $j\leqslant m$, and a curve $C_{o_j(i)}$ for some $i\leqslant \operatorname{v}_j$. By definition of the good metric $(ds)^2$, there is a neighbourhood $V:=V_{p_{j,o_j(i)}}$ of the point $p_{j,o_j(i)}=C_{j}\cap C_{o_j(i)}$ bi-holomorphic to the ball $\mathbb B_2\,{=}\,\{ (z_1,z_2)\,{\in}\,\mathbb C\mid \sqrt{|z_1|^2\,{+}\,|z_2|^2}\,{<}\,2\}$ and such that in the coordinates $(z_1,z_2)$ in $V_{p_{j,o_j(i)}}$ we have: $(ds)^2= dz_1\, d\overline z_1+dz_2\, d\overline z_2$, and, in addition, $C_{j}\cap V_{p_{j,o_j(i)}}$ is given by $z_1=0$ and $C_{o_j(i)}\cap V_{p_{j,o_j(i)}}$ is given by $z_2=0$.

Remark 1. Below, if $j_1=o_j(i)\leqslant m$, we will assume that $V\,{=}\,V_{p_{j_1,j}}\,{=}\,V_{p_{j,j_1}}\subset X$ and the identifications of the neighbourhood $V$ with the ball $\mathbb B_2=\{ (z_1,z_2)\in\mathbb C\mid \sqrt{|z_1|^2+|z_2|^2}<2\}$ define the bi-holomorphic automorphism of $\mathbb B_2$ sending $(z_1,z_2)$ to $(z_2,z_1)$.

It is easy to see that the following result holds.

Claim 1. For $1\leqslant j_i\leqslant m$, $i=1,2$, and for $\varepsilon<\varepsilon_0$, we have:

1) $U_{j_1,j_2,\varepsilon}:=U_{j_1,\varepsilon} \cap U_{j_2,\varepsilon}\subset V_{p_{j_1,j_2}}=V$,

$$\begin{equation*} U_{j_1,j_2,\varepsilon}\simeq \mathbb D^2_{\varepsilon}= \{\mathbb (z_1,z_2)\in \mathbb B_2\mid |z_1|<\varepsilon,\,\, |z_2|<\varepsilon\}\subset\mathbb B_2\simeq V_{p_{j_1,j_2}}, \end{equation*} \notag$$
and in the coordinates $(z_1,z_2)$ the projections $\operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}}\colon U_{j_1,j_2,\varepsilon}\to W_{j_i,\varepsilon}$, $i=1,2$, are given by $\operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}} \colon (z_1,z_2)\mapsto z_{\overline i}$, where $W_{j_i,\varepsilon}=C_{j_i}\cap U_{j_1,j_2,\varepsilon}$ and $\{ i,\overline i\}=\{1,2\}$, and, in particular, $U_{j_i,\varepsilon}\cap C_{j_{\overline i}}= \operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}}^{-1}(p_{j_1,j_2})$;

2) the segments of geodesic lines lying in $V_{p_{j_1,j_2}}$ perpendicular to $C_{j_1}$ (respectively, $C_{j_2})$ and emanating from a point $p=(0,z_{2,0})\in C_{j_1}$ (respectively, $p=(z_{1,0},0)\in C_{j_2}$) are

$$\begin{equation*} \begin{gathered} \, \{ (tz_1,z_{2,0})\mid 0\leqslant t< \sqrt{2-|z_{2,0}|^2}\} \\ \Bigl(\text{respectively,} \ \ \Bigl\{ (z_{1,0},tz_2)\Bigm| 0\leqslant t< \sqrt{2-|z_{1,0}|^2}\Bigr\}\Bigr), \end{gathered} \end{equation*} \notag$$
where $z_1\in\mathbb C$, $|z_1|=1$ (respectively, $z_2\in\mathbb C$, $|z_2|=1$).

### 1.3. On normal bundles of curves

Consider a curve $\mathcal C:=C_{j}$ corresponding to a vertex $v_{j,0}\in \Gamma(\widetilde C)$, $j=1,\dots ,m$. If $X$ and $\mathcal C$ are considered as complex manifolds, then their tangent bundles $T_X$ and $T_{\mathcal C}$ are vector bundles defined over $\mathbb C$ and the normal bundle $N_{\mathcal C}$ of $\mathcal C$ in $X$ is defined via the exact sequence

$$$$0\to T_{\mathcal C}\to T_{X \mid \mathcal C} \xrightarrow{\psi} N_{\mathcal C}\to 0.$$ \tag{4}$$
We set $\operatorname{v}:=\operatorname{v}_j$ and define $p_i:=p_{j,o_{j}(i)}=C_{j}\cap C_{o_{j}(i)}$. Let $M:=M_j$, $K:=K_j$ be two non-negative integers such that $(\mathcal C^2)_X=M-K$ and a divisor
$$\begin{equation*} D:=D_j=(p_{\operatorname{v}+1}+\dots + p_{\operatorname{v}+M})- (p_{\operatorname{v}+M+1}+\cdots+p_{\operatorname{v}+M+K})\in \operatorname{Pic}(\mathcal C) \end{equation*} \notag$$
is equivalent to the restriction $\mathcal C_{\mid \mathcal C}$ of the divisor $\mathcal C\in \operatorname{Pic}(X)$ to the curve $\mathcal C$. Then $N_{\mathcal C}=L_{\mathcal C,D}$, where $L_{\mathcal C,D}$ is the line bundle associated with the divisor $D$. We can assume that $p_{i}\notin \operatorname{Supp}(D)$ for $i=1,\dots,\operatorname{v}$.

Let us choose neighbourhoods $W_i\subset \mathcal C$ of the points $p_i$, $i=1,\dots,\operatorname{v}+M+K$, bi-holomorphic to the disc $\mathbb D_2=\{ w\in\mathcal C\mid |w|< 2\}$ and such that $W_{i_1}\cap W_{i_2}=\varnothing$ for $i_1\neq i_2$.

Let us identify $\mathcal C$ with the zero section $S_0$ of the bundle $L_{\mathcal C,D}$. We can compactify $L_{\mathcal C,D}$ by adding the section at “infinity” $S_{\infty}$, thereby obtaining a relatively minimal ruled surface $\overline L_{\mathcal C,D}$ over $\mathcal C$ with fibres $F_p=\operatorname{pr}^{-1}(p)\simeq\mathbb P^1$ over the points $p\in\mathcal C=S_0$ of the projection $\operatorname{pr}\colon \overline L_{\mathcal C,D}\to \mathcal C$.

In $\operatorname{Pic}(\overline L_{\mathcal C,D})$, we have

$$$$S_0= S_{\infty}+ \sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}}.$$ \tag{5}$$

For $\mathcal C=C_j$, let us add the imbedding $\iota_j\colon N_{C_j,(ds)^2}\to T_{X\mid C_j}$ to the exact sequence (4):

Note that $\psi_j\circ \iota_j\colon N_{C_j,(ds)^2}\to N_{C_j}$ is a $\mathbb R$-linear isomorphism of line bundles $\mathbb C$-linear over $W_i$ for $i=1,\dots,\operatorname{v}_j$. We set $\widetilde U_{j,\varepsilon}:= \psi_j\circ \iota_j(N_{C_j,(ds)^2}(\varepsilon))\subset N_{C_j}$. Obviously, the set $\widetilde{\mathcal U}_{j}= \{ \widetilde U_{j,\varepsilon}\mid \varepsilon< \varepsilon_0 \}$ is a base of open (in complex analytic topology) subsets of the normal bundle $N_{C_j}$ containing the zero section $S_{j,0}\subset N_{C_j}$. Applying Claim 1, we arrive at the following result.

Claim 2. The diffeomorphisms $\psi_j\circ \iota_j\circ \varphi_j^{-1}\colon U_{j,\varepsilon}\to \widetilde U_{j,\varepsilon}$ determine diffeomorphisms between $U_{j,\varepsilon}\setminus \widetilde C$ and $\widetilde U_{j,\varepsilon}\setminus \bigl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)$. In particular, we have isomorphisms of fundamental groups $\pi_1(U_{j,\varepsilon}\setminus \widetilde C)\simeq \pi_1\bigl(\widetilde U_{j,\varepsilon}\setminus \bigl(S_{j,0}\,\cup\, \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)\bigr)$.

### 1.4. Elementary transformations

For $p\in S_0$ (respectively, $\widetilde p=F_p\cap S_{\infty}$), we denote by $\operatorname{elm}_{p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,D'}$ (respectively, $\operatorname{elm}_{\widetilde p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,\widetilde D'}$) a bi-rational map (called the elementary transformation of $\overline L_{\mathcal C,D}$) consisting of the blowup of the point $p$ (respectively, $\widetilde p$) and after that the blowdown of the proper inverse image of the fibre $F_p$.

Lemma 2. Let $\operatorname{elm}_{p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,D'}$ (respectively, $\operatorname{elm}_{\widetilde p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,\widetilde D'}$) be an elementary transformation of $\overline L_{\mathcal C,D}$. Then $D'=D-p$ (respectively, $\widetilde D'=D+p$).

Proof. We will prove the lemma only in the case when the elementary transformation is $\operatorname{elm}_{p}$, since the proof in the second case is similar. Without loss of generality, we can assume that $p\not\in \operatorname{Supp} D$.

We have $\operatorname{elm}_{p}= \sigma\circ\sigma^{-1}_p\colon \overline L_{\mathcal C,D}\dashrightarrow \overline L_{\mathcal C,D'}$, where $\sigma_p\colon Y\to \overline L_{\mathcal C,D}$ is the $\sigma$-process with centre at the point $p= S_0\cap F_p$ and $\sigma\colon Y\to \overline L_{\mathcal C,D'}$ is the $\sigma$-process contracting the proper inverse image $\sigma^{-1}_p(F_p)$ of the fibre $F_p$. We set $F'_p=\sigma^{-1}_p(p)$ and denote the proper inverse images (and respectively, after that, the images under the mapping $\sigma$) of the sections $S_0$, $S_{\infty}$, and of the fibres $F_q$ over the points $q\in \operatorname{Supp} D$ by the same letters. We have $\sigma^*_p(S_0)=S_0+F'_p$. Therefore, by (5),

$$\begin{equation*} \sigma^*_p(S_0)=S_0+F'_p= S_{\infty}+\sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}} \end{equation*} \notag$$
in $\operatorname{Pic}(Y)$, and
$$\begin{equation*} \sigma_*(S_0)=S_0= S_{\infty}+\sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}}-F'_p \end{equation*} \notag$$
in $\operatorname{Pic}(\overline L_{\mathcal C,D'})$. Hence, $D'=D-p\in \operatorname{Pic}(\mathcal C)$. Lemma 2 is proved.

By Lemma 2, we have the bi-rational mapping

$$\begin{equation*} T=\operatorname{elm}_{p_{\operatorname{v}+1}}\cdots \operatorname{elm}_{p_{\operatorname{v}+M}} \operatorname{elm}_{\widetilde p_{\operatorname{v}+M+1}}\cdots \operatorname{elm}_{\widetilde p_{\operatorname{v}+M+K}}\colon \overline L_{\mathcal C,D}\dashrightarrow \overline L_{\mathcal C,0}\simeq \mathbb P^1\times \mathcal C \end{equation*} \notag$$
and the inverse mapping
$$$$T^{-1}=\operatorname{elm}_{\widetilde p_{\operatorname{v}+1}}\cdots \operatorname{elm}_{\widetilde p_{\operatorname{v}+M}} \operatorname{elm}_{p_{\operatorname{v}+M+1}}\cdots \operatorname{elm}_{p_{\operatorname{v}+M+K}}\colon \overline L_{\mathcal C,0}\dashrightarrow \overline L_{\mathcal C,D}.$$ \tag{6}$$

Consider a local case of elementary transformations, that is, the case

$$\begin{equation*} \operatorname{elm}_p\colon \mathbb P^1\times \mathbb D_2\dashrightarrow \mathbb P^1\times \mathbb D_2\quad (\text{respectively,} \operatorname{elm}_{\widetilde p}\colon \mathbb P^1\times \mathbb D_2 \dashrightarrow \mathbb P^1\times \mathbb D_2), \end{equation*} \notag$$
where $\mathbb D_2=\{ w\in \mathbb C\mid |w|< 2\}$ is the disc in $\mathbb C$ and $\mathbb P^1$ is the projective line. Let $(z_1:z_2)$ be homogeneous coordinates in $\mathbb P^1$ and $p=\{ (0,1)\}\times \{ w=0\}$ (respectively, $\widetilde p=\{ (1,0)\}\times \{ w=0\}$). As above, denote by $S_0=\{ z_1=0\}\times \mathbb D_2$ the zero section, and, by $S_{\infty}=\{ z_2=0\}\times \mathbb D_2$, the section at “infinity” of the projection to $\mathbb D_2$.

The mapping

$$\begin{equation*} \operatorname{elm}_p \text{ (respectively, } \operatorname{elm}_{\widetilde p}) \colon \mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p)\to \mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p) \end{equation*} \notag$$
is a bi-holomorphic mapping. Therefore, the mapping $\operatorname{elm}_p$ (respectively, $\operatorname{elm}_{\widetilde p}$) induces an isomorphism
\begin{equation*} \begin{aligned} \, &\operatorname{elm}_{p*} \text{ (respectively, } \operatorname{elm}_{\widetilde p*}) \colon \pi_1\bigl(\mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p)\bigr) \\ &\qquad\qquad\qquad\to \pi_1\bigl(\mathbb P^1\times \mathbb D_2 \setminus (S_0\cup S_{\infty}\cup F_p)\bigr). \end{aligned} \end{equation*} \notag

Denote by $z=z_1/z_2$ the coordinate in $\mathbb C=\{ (z_1:z_2)\in \mathbb P^1\mid z_2\neq 0\}$. The fundamental group $\pi_1(\mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p))= \pi_1(\mathbb C\times \mathbb D_2\setminus (S_0\cup F_p), q)$ is a free abelian group generated by two elements $x_0$, $x_1$, represented, respectively, by the loops

\begin{equation*} \begin{aligned} \, \gamma_0 &=\{ z=e^{2\pi\sqrt{-1}\, t},\, 0\leqslant t\leqslant 1 \}\times \{ w=1\}, \\ \gamma_1 &=\{ z=1 \}\times \{ w=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\}. \end{aligned} \end{equation*} \notag

Lemma 3. We have

\begin{equation*} \begin{alignedat}{2} \operatorname{elm}_{p*}(x_0) &=x_0, &\qquad \operatorname{elm}_{p*}(x_1) &=x_0^{-1}x_1, \\ \operatorname{elm}_{\widetilde p*}(x_0) &=x_0, &\qquad \operatorname{elm}_{\widetilde p*}(x_1) &=x_0x_1. \end{alignedat} \end{equation*} \notag

This result is a direct consequence of Lemma 7 in s[9].

### 1.5. Presentation of the fundamental group of an irreducible curve with punctures

We set $N:=\operatorname{v}+M+K$, and as above, let

$$\begin{equation*} W_i\simeq \mathbb D_2=\{ w_i\in \mathbb C\mid |w_i|<2\} \end{equation*} \notag$$
be pairwise non-intersecting $N$ neighbourhoods of the points $p_i=\{ w_i=0\}\in W_i$, $i=1,\dots, N$, in the curve $\mathcal C=C_j$.

The curve $\mathcal C$ of genus $g$, considered as a Riemann surface, is a sphere with $g$ handles. It is well-known that we can choose $g$ “meridians” $\widetilde{\mu}'_i:=\widetilde{\mu}'_{j,i}$ and $g$ “parallels” $\widetilde{\lambda}'_i:=\widetilde{\lambda}'_{j,i}$, $i=1,\dots, g$, that is, $\widetilde{\mu}'_1,\dots, \widetilde{\mu}'_g$, $\widetilde{\lambda}'_1,\dots, \widetilde{\lambda}'_g$ are oriented smooth loops intersecting only at a point $q'_0:=q'_{j,0}\in \mathcal C$ such that if we cut $\mathcal C$ along these loops, then we obtain a $2g$-polygon $P_{\widetilde{\mu}',\widetilde{\lambda}'}(\mathcal C)$. As is well known, the fundamental group $\pi_1\bigl(\mathcal C\setminus \bigl(\bigcup_{i=1}^N p_i\bigr),q'_0\bigr)$ of the Riemann surface $\mathcal C$ with $N$ punctures has the presentation

$$$$\pi_1\biggl(\mathcal C\setminus \biggl(\bigcup_{i=1}^N p_i\biggr),q'_0\biggr) = \biggl\langle\mu'_{1},\lambda'_{1},\dots,\mu'_g,\lambda'_{g},x'_{1}, \dots,x'_{N} \biggm| \prod_{i=1}^{g}[\mu'_{i},\lambda'_{i}] \prod_{i=1}^{N}x'_{i}=1 \biggr\rangle,$$ \tag{7}$$
where $\mu'_i$ and $\lambda'_i$ are represented by the loops $\widetilde{\mu'_i}$ and $\widetilde{\lambda}'_i$ and $x'_{i}$ are represented by the simple loops $\widetilde x'_i$ consisting of paths $l'_i$ from the point $q'_0$ to the points $q'_i=\{ w_i=1\}\in W_i$, the circles $\gamma'_i=\{ w_i=e^{2\pi\sqrt{-1}\, t},\, 0\leqslant t\leqslant 1\}\subset \mathbb D_2=\{ w_i\in \mathbb C\mid |w_i|<2\}\simeq W_i$ and the returns to the point $q'_0$ along the paths $l'_i$ (see Figure 2).

### 1.6. Presentation of the fundamental group of the complement of a union of zero section and fibres in the normal bundle of an irreducible curve

The surface $\overline L_{\mathcal C,0}\setminus \bigl(S_0 \cup S_{\infty} \cup \bigl( \bigcup_{i=1}^NF_{p_i}\bigr)\bigr)\subset \overline L_{\mathcal C,0}$ is isomorphic to $\mathbb C^*_{z}\times \bigl(\mathcal C\setminus \bigl(\bigcup_{i=1}^N p_i\bigr)\bigr)$, where $\mathbb C^*_{z}=\{ z\in\mathbb C \mid z\neq 0\}$. Therefore,

$$\begin{equation*} \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr)\simeq \pi_1(\mathbb C^*_{z},1)\times\pi_1\biggl(\mathcal C\setminus \biggl(\bigcup_{i=1}^N p_i\biggr),q'_0\biggr), \end{equation*} \notag$$
$q_0=\{z=1\}\times q'_0$, and the fundamental group $\pi_1\bigl(\overline L_{\mathcal C,0}\setminus \bigl(S_0 \cup S_{\infty}\cup \bigl(\bigcup_{i=1}^NF_{p_i}\bigr)\bigr),q_0\bigr)$ has the following presentation:
\begin{aligned} \, &\pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr) \nonumber \\ &\qquad= \biggl\langle \mu_{1},\lambda_{1},\dots,\mu_g,\lambda_{g},x_0, x_{1},\dots,x_{N} \biggm| \prod_{i=1}^{g}[\mu_{i},\lambda_{i}] \prod_{i=1}^{N}x_{i}=1, \nonumber \\ &\qquad\qquad [x_0,\mu_j]=[x_0,\lambda_j]=1 \text{ for } 1\leqslant j\leqslant g, \ [x_0,x_i]=1 \text{ for }1\leqslant i\leqslant N \biggr\rangle, \end{aligned} \tag{8}
where $x_0$ is represented by the circle $\widetilde x_0= \bigl\{ z=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\bigr\}\times q'_0$; $\mu_{i}$ and $\lambda_i$ are represented by the loops $\widetilde{\mu}_i=\{ z=1\}\times \widetilde{\mu}'_i$ and $\widetilde{\lambda}_{i}=\{ z=1\}\times \widetilde{\lambda}'_i$ obtained as a result of lifting in the section $S_1=\{ z=1\}\times \mathcal C\subset \overline L_{\mathcal C,0}$ of the loops representing the elements $\mu'_i$ and $\lambda'_{i}$; and $x_i$ are elements represented by simple loops $\widetilde x_i$ consisting of paths $l_i=\{ z=1\}\times l'_i$ from the point $q_0$ to the points $q_i=\{ z=1\}\times q'_i$, the circles $\gamma_i =\{ z=1\}\times \gamma'_i$ in $\{ z=1\}\times W_i$,
$$\begin{equation*} \gamma_i =\{ z=1\}\times \bigl\{ w_i= e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\bigr\}, \end{equation*} \notag$$
and the returns to the point $q_0$ along the paths $l_i$.

The bi-holomorphic mapping (see (6))

$$\begin{equation*} T^{-1}\colon \overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i}\biggr)\biggr)\to \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i})\biggr) \end{equation*} \notag$$
defines the isomorphism
\begin{equation*} \begin{aligned} \, &T^{-1}_*\colon \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr) \\ &\qquad \to \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr), \end{aligned} \end{equation*} \notag
where $\widetilde q_0=T^{-1}(q_0)$.

We set $\widetilde l_i=T^{-1}(l_i)$. In the notation of § 1.4, let us identify the neighbourhoods $W_i\subset\mathcal C$ with the neighbourhood $W$ considered in § 1.4 ($w_i:=w$) and denote by

$$\begin{equation*} y_i\in \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr),\qquad i=\operatorname{v}+1,\,\dots,\,\operatorname{v}+(M+K), \end{equation*} \notag$$
the elements represented by the loops consisting of the paths $\widetilde l_i=T^{-1}(l_i)$, the circles $\gamma_i =\{ z'=1\}\times \{ w_i=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\}$ in $\mathbb C_{z'}\times W_i=\mathbb C_{z'}\times W$, and the returns to the point $\widetilde q_0$ along the paths $\widetilde l_i$. We also denote by the same letters $\mu_1,\lambda_1,\dots,\mu_g,\lambda_g$ and $x_0,\dots, x_{\operatorname{v}}$ the images $T_*^{-1}(\mu_i)$, $T_*^{-1}(\lambda_i)$, $T_*^{-1}(x_i)$ of the elements
$$\begin{equation*} \mu_1,\lambda_1,\dots,\mu_g,\lambda_g,x_0,\dots, x_{\operatorname{v}}\in \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr). \end{equation*} \notag$$
Now by Lemma 3 $T_*^{-1}(x_i)=x_0y_i$ for $i=\operatorname{v}+1,\dots,\operatorname{v}+M$ and $T_*^{-1}(x_i)=x_0^{-1}y_i$ for $i=\operatorname{v}+M+1,\dots,\operatorname{v}+M+K$. In view of presentation (8), the fundamental group $\pi_1\bigl(\overline L_{\mathcal C,D}\setminus \bigl(S_0 \cup S_{\infty}\cup \bigl( \bigcup_{i=1}^NF_{p_i}\bigr)\bigr),\widetilde q_0\bigr)$ has the presentation, where $\omega= M-K$,
\begin{aligned} \, &\biggl\langle \mu_{1},\lambda_{1},\dots,\mu_g,\lambda_{g},x_0,x_{1}, \dots,x_{\operatorname{v}}, y_{\operatorname{v}+1},\dots, y_{\operatorname{v}+M+K} \biggm| \nonumber \\ &\qquad x_0^{\omega}\prod_{i=1}^{g}[\mu_{i},\lambda_{i}] \prod_{i=1}^{\operatorname{v}}x_{i}\prod_{i=1}^{M+K} y_{\operatorname{v}+i}=1,\ [x_0,\mu_j]=[x_0,\lambda_j]=1 \text{ for }1\leqslant j\leqslant g, \nonumber \\ &\qquad [x_0,x_i]=1 \text{ for } 1\leqslant i\leqslant \operatorname{v},\ [x_0,y_{\operatorname{v}+i}]=1 \text{ for } 1\leqslant i\leqslant M+K \biggr\rangle. \end{aligned} \tag{9}

The imbedding

$$\begin{equation*} i\colon \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr)\hookrightarrow \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^{\operatorname{v}}F_{p_i}\biggr)\biggr) \end{equation*} \notag$$
defines an epimorphism
$$\begin{equation*} i_*\colon \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr) \twoheadrightarrow \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^{\operatorname{v}}F_{p_i}\biggr) \biggr), \widetilde q_0\biggr). \end{equation*} \notag$$
It is obvious that the kernel of $i_*$ is the normal closure in
$$\begin{equation*} \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty} \cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr) \end{equation*} \notag$$
of the subgroup generated by elements $y_{\operatorname{v}+1},\dots, y_{\operatorname{v}+M+K}$. Therefore, by applying presentation (9) with $\mathcal C=C_j$, we obtain Proposition 2, in which we use the following notation: $x_{j,0}=i_*(x_0)$, $x_{o_j(l),1}=i_*(x_l)$ for $l=1,\dots, \operatorname{v}_j$, $\mu_{j,l}=i_*(\mu_l)$, and $\lambda_{j,l}=i_*(\lambda_l)$ for $l=1,\dots,g_j$.

Proposition 2. For $N_{C_j}=L_{C_j,D}$, the group

$$\begin{equation*} \pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0} \cup \biggl( \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr), \widetilde q_{j,0}\biggr) \end{equation*} \notag$$
has the presentation
\begin{aligned} \, &\pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0} \cup \biggl(\, \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr), \widetilde q_{j,0}\biggr) \nonumber \\ &\quad= \biggl\langle \mu_{j,1},\lambda_{j,1},\dots,\mu_{j,g_j} ,\lambda_{j,g_j}, x_{j,0},x_{o_j(1),1},\dots, x_{o_j(\operatorname{v}_j),1} \biggm| \nonumber \\ &\quad\qquad x_{j,0}^{k_{j}}\prod_{i=1}^{g_j}[\mu_{j,i},\lambda_{j,i}] \prod_{i=1}^{\operatorname{v}_j}x_{o_j(i),1} =1,\ [x_{j,0},\mu_{j,i}]= [x_{j,0},\lambda_{j,i}] =1 \textit{ for } 1\leqslant i\leqslant g_j, \nonumber \\ &\quad \qquad [x_{j,0},x_{o_j(i),1}] =1 \textit{ for } 1\leqslant i\leqslant \operatorname{v}_j \biggr\rangle,\quad \textit{where}\quad k_{j}= (C_j^2)_X. \end{aligned} \tag{10}

### 1.7. Presentation of the fundamental group of the complement of $\widetilde C$ in tubular neighbourhoods of its compact irreducible components

For $\varepsilon< \varepsilon_0$ the imbeddings $i_{j,\varepsilon}\colon \widetilde U_{j,\varepsilon}\hookrightarrow N_{C_j}$ (see § 1.3) define the homomorphisms

$$\begin{equation*} i_{j,\varepsilon*}\colon \pi_1\biggl(\widetilde U_{j,\varepsilon}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr)\to \pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr). \end{equation*} \notag$$

Proposition 3. For each $j$ and for each $\varepsilon<\varepsilon_0$, the homomorphism

$$\begin{equation*} i_{j,\varepsilon*}\colon \pi_1\biggl(\widetilde U_{j,\varepsilon}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j} F_{p_{j,o_j(i)}}\biggr)\biggr)\to\pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0} \cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr) \end{equation*} \notag$$
is an isomorphism.

Proof. Let us choose a finite covering $\{ W_{i}\}$ of the curve $C_j$, $C_j=\bigcup_iW_{i}$ such that the line bundle $N_{C_j}$ is trivial over each $W_{i}$. We identify $\operatorname{pr}_j^{-1}(W_{i})\subset N_{C_j}$ with $N_{C_j\mid W_{i}}=\mathbb C_z\times W_{i}$, and put $N_{C_j\mid W_{i}}(\delta):=\{ (z,p)\in \mathbb C_z\times W_{i}\mid |z|<\delta\}\subset N_{C_j}$. It is obvious that the set $\bigl\{ N_{C_j}(\delta):=\bigcup_iN_{C_j\mid W_{i}}(\delta)\bigr\}_{\delta>0}$ is a base of open subsets in $N_{C_j}$ containing the zero section $S_{j,0}$. Therefore, for each $\varepsilon<\varepsilon_0$, there is $\delta_{\varepsilon}>0$ such that $N_{C_j}(\delta)\subset \widetilde U_{j,\varepsilon}$ for each $\delta\leqslant\delta_{\varepsilon}$.

The automorphism group $\operatorname{Aut}(N_{C_j})$ contains the subgroup

$$\begin{equation*} A_j=\{ a_{j,r}\in \operatorname{Aut}(N_{C_j})\mid a_{j,r}(z,w)= (rz,w) \text{ for } (z,w)\in\mathbb C_z\times \widetilde W_{j,i}\}. \end{equation*} \notag$$
We have $a_{j,r}(N_{C_j}(\delta))=N_{C_j}(r\delta)$.

By (10), the fundamental group

$$\begin{equation*} \Pi_j:=\pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr),\widetilde q_0\biggr) \end{equation*} \notag$$
is finitely presented. Therefore, for each relation $\mathcal W(\overline{\mu}_j,\overline{\lambda}_j,\overline x_j)=1$ in (10), there is a continuous mapping $\theta_{\mathcal W}\colon \overline{\mathbb D}_1\to N_{C_j}\setminus \bigl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)$ of the closed disc $\overline{\mathbb D}_1=\{ z\in \mathbb C\mid |z|\leqslant 1\}$ such that the loop
$$$$\mathcal W\bigl(\widetilde{\mu}_{j,1}, \widetilde{\lambda}_{j,1},\dots, \widetilde{\mu}_{j,g_j}, \widetilde{\lambda}_{j,g_j}, \widetilde x_{j,0}, \widetilde x_{o_j(1),1},\dots, \widetilde x_{o_j(\operatorname{v}_j),1}\bigr) =\theta_{\mathcal W}(\partial \overline{\mathbb D}_1)$$ \tag{11}$$
is the image of the boundary $\partial \overline{\mathbb D}_1= \{ z=e^{2\pi \sqrt{-1}\, t}\mid 0\leqslant t\leqslant 1\}$ of the disc $\overline{\mathbb D}_1$ (here in the case $\mathcal C=C_j$ the loops $\widetilde{\mu}_{j,i}:=\widetilde{\mu}_i$, $\widetilde{\lambda}_{j,i}:=\widetilde{\lambda}_i$ for $i=1,\dots, g_j$, $\widetilde x_{j,0}:=\widetilde x_0$, and $\widetilde x_{o_j(i),1}:=\widetilde x_i$ for $i=1,\dots, \operatorname{v}_j$, where the loops $\widetilde{\mu}_{i}$, $\widetilde{\lambda}_{i}$, $\widetilde x_0$, and $\widetilde x_{i}$ from $\mathcal C$ are defined in § 1.6).

Since $\theta_{\mathcal W}(\overline{\mathbb D}_1)$ are compacts, there is a constant $r_{C_j}>0$ such that

$$$$\theta_{\mathcal W}(\overline{\mathbb D}_1)\subset N_{C_j}(r_{C_j})$$ \tag{12}$$
for all relations $\mathcal W$ in (10). Therefore, if $\delta<\delta_{\varepsilon}/r_{C_j}$, then
\begin{equation*} \begin{aligned} \, &a_{j,\delta}\bigl(\mathcal W\bigl(\widetilde{\mu}_{j,1}, \widetilde{\lambda}_{j,1},\dots, \widetilde{\mu}_{j,g_j}, \widetilde{\lambda}_{j,g_j}, \widetilde x_{j,0},\widetilde x_{o_j(1),1}, \dots, \widetilde x_{o_j(\operatorname{v}_j),1}\bigr)\bigr) \\ &\qquad\subset \widetilde U_{j,\varepsilon}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr), \end{aligned} \end{equation*} \notag
and it easily follows from (11) and (12) that $i_{\varepsilon*}$ is an isomorphism for each $\varepsilon<\varepsilon_0$. This proves Proposition 3.

### 1.8. Presentation of the fundamental group of the complement of $\widetilde C$ in a tubular neighbourhood of $\widetilde C_0$

Let $\widetilde{\mathcal U}_{\nu,\varepsilon}= \{ \widetilde U_{1,\varepsilon},\dots,\widetilde U_{m,\varepsilon}\}$ be the set of neighbourhoods $\widetilde U_{j,\varepsilon}\subset N_{C_j}$ of zero sections $S_{j,0}\subset N_{C_j}$. Denote by

$$\begin{equation*} E'=\{ (v_{j_1,0},v_{j_2,0}) \mid 1\leqslant j_1<j_2\leqslant m, \, \Delta_{(j_1,0),(j_2,0)}=1\} \end{equation*} \notag$$
the subset of the set of edges of the graph $\Gamma_{\widetilde E}(\widetilde C)$, and, for $(v_{j_1,0},v_{j_2,0})\in E'$, consider the neighbourhoods $U_{j_1,j_2,\varepsilon}\subset U_{j_1,\varepsilon}$ and $U_{j_2,j_1,\varepsilon}\subset U_{j_2,\varepsilon}$ defined in Claim 1. By Remark 1, $U_{j_1,j_2,\varepsilon}=U_{j_2,j_1,\varepsilon}$ (as subsets of $X)$. Therefore, applying the identifications of $U_{j,\varepsilon}$ with $\widetilde U_{j,\varepsilon}$ given by diffeomorphisms $\psi_j\circ\iota\circ\varphi_j^{-1}\colon U_{j,\varepsilon}\to \widetilde U_{j,\varepsilon}$, the neighbourhoods $\widetilde U_{j_1,\varepsilon}$ and $\widetilde U_{j_2,\varepsilon}$, $(v_{j_1,0},v_{j_2,0})\in E'$ can be glued along $U_{j_1,j_2,\varepsilon}\subset U_{j_1,\varepsilon}$ and $U_{j_2,j_1,\varepsilon}\subset U_{j_2,\varepsilon}$. As a result, we obtain a complex manifold $U'_{\varepsilon}$.

In the notation used in the proof of Proposition 3 (with identification of $\widetilde U_{j,\varepsilon}$ and $U_{j,\varepsilon}$), we set $Q_{j,0}:=a_{j,\epsilon}(\widetilde q_{j,0})$ and $Q_{j,o_j(i)}:=a_{j,\epsilon}(\widetilde q_{j,o_j(i)})\in U_{j,\varepsilon}\subset U'_{\varepsilon}$, $i= 1,\dots,\operatorname{v}_j$. In addition, without loss of generality, we can assume that $Q_{j_1,j_2}=Q_{j_2,j_1}\in U_{j_1,j_2,\varepsilon}$ if $C_{j_1}\cap C_{j_2}\neq\varnothing$. Also let $L_{j,o_j(i)}:=a_{j,\epsilon}(\widetilde l_{j,o_j(i)})$, $i=0,1,\dots,\operatorname{v}_j$, be the paths in $U_{j,\varepsilon}$ connecting the point $Q_{j,0}$ with the points $Q_{j,o_j(i)}$. Since $\Gamma_{\widetilde E}(\widetilde C)$ is a tree, we can assume that the paths $\widetilde l_{j,o_j(i)}$ are chosen so that

$$\begin{equation*} L=\bigcup_{j=1}^m\bigcup_{i=1}^{\operatorname{v}_j}L_{j,o_j(i)} \end{equation*} \notag$$
is a tree. For $j=1,\dots,m$ and $i=1,\dots, \operatorname{v}_j$, let $L_{0,j,o_j(i)}$ be the paths in $L$ from $Q_{1,0}$ to the points $Q_{j,o_j(i)}$.

Further, if this does not lead to a misunderstanding, we will denote again by $\mu_{j,i}$ and $\lambda_{j,i}$, and by $x_{j,0}$ for $j=1,\dots, m$ and $i=1,\dots,g_j$, and also we denote by $x_{o_j(i),1}$ for $i=1,\dots \operatorname{v}_j$, the elements in $\pi_1(U'_{\varepsilon}\setminus \widetilde C,Q_{1,0})$ represented by loops.

1. Go from $Q_{1,0}$ to $Q_{j,0}$ along $L_{0,j,0}$.

2. Go once along $a_{j,\epsilon}(\widetilde \mu_{j,i})$ (respectively, $a_{j,\epsilon}(\widetilde \lambda_{j,i})$, $a_{j,\epsilon}(\widetilde x_{j,0})$, and $a_{j,\epsilon}(\widetilde x_{j,o_j(i)}$).

3. Go back to $Q_{1,0}$ along $L_{0,j,0}$.

The choice of paths $L_{0,j,0}$ and the imbeddings $\alpha_j\colon U_{j,\varepsilon}\setminus \widetilde C\hookrightarrow U'_{\varepsilon}\setminus \widetilde C$ define the homomorphisms

$$\begin{equation*} \alpha_{j*}\colon \pi_1(U_{j,\varepsilon}\setminus \widetilde C,Q_{j,0}) \to \pi_1(U'_{\varepsilon}\setminus \widetilde C,Q_{1,0}) \end{equation*} \notag$$
and the equalities
$$\begin{equation*} \begin{gathered} \, \alpha_{j*}(a_{j,\epsilon}(\mu_{j,i}))=\mu_{j,i},\qquad \alpha_{j*}(a_{j,\epsilon}(\lambda_{ji}))=\lambda_{j,i},\qquad \alpha_{j*}(a_{j,\epsilon}(x_{j,0}))=x_{j,0}, \\ \alpha_{j*}(a_{j,\epsilon}(x_{j,o_j(i)}))=x_{o_j(i),1}. \end{gathered} \end{equation*} \notag$$
Note that, for $U_{j,o_j(i),\varepsilon}\subset U_{j,\varepsilon}$, the group $\alpha_{j*}(\pi_1(U_{j,o_j(i),\varepsilon}\setminus\widetilde C))$ is generated by $x_{j,0}=\alpha_{j*}(a_{j,\epsilon}(x_{j,0}))$ and $x_{o_j(i),1}=\alpha_{j*}(a_{j,\epsilon}(x_{j,o_j(i)}))$, and it is easy to see that, for $1\leqslant j_i\leqslant m$, $i=1,2$, such that $\Delta_{(j_1,0),(j_2,0)}=1$, we have
$$\begin{equation*} \alpha_{j_1*}\bigl(\pi_1(U_{j_1,j_2,\varepsilon}\setminus\widetilde C)\bigr)= \alpha_{j_2*}\bigl(\pi_1(U_{j_2,j_1,\varepsilon}\setminus\widetilde C)\bigr), \end{equation*} \notag$$
and $x_{j_1,0}=x_{j_2,1}$, $x_{j_2,0}=x_{j_1,1}$.

Since $L$ is a tree, using the paths $L_{0,j,o_j(i)}$, and applying Seifert–van Kampen Theorem (see [10], [11]) $m-1$ times, we obtain a presentation of the group $\pi_1(U'_{\varepsilon}\setminus \widetilde C)$ as the free product of the groups $\alpha_{j*}(\pi_1(U_{j_,\varepsilon}\setminus\widetilde C,Q_{j,0}))$ with amalgamation of the groups

$$\begin{equation*} \alpha_{j_1*}\bigl(\pi_1(U_{j_1,j_2,\varepsilon}\setminus \widetilde C, Q_{j_1,j_2})\bigr)=\alpha_{j_2*} \bigl(\pi_1(U_{j_2,j_1,\varepsilon}\setminus\widetilde C,Q_{j_2,j_1})\bigr), \end{equation*} \notag$$
$\Delta_{(j_1,0),(j_2,0)}=1$. As a result, we obtain the following fact.

Proposition 4. The group $\pi_1( U'_{\varepsilon}\setminus \widetilde C,Q_{1,0})$ has the following presentation: it is generated by $m+k+2\sum_{j=1}^mg_j+2r$ elements:

\begin{alignedat}{2} &x_{j,0}, \ \mu_{j,i},\ \lambda_{j,i}, &\qquad &1\leqslant j\leqslant m,\quad 1\leqslant i\leqslant g_j, \\ &x_{j(E_s,1),1},\ x_{j(E_s,2),1}, &\qquad &s=1,\dots, r, \\ &x_{m+l,1}, &\qquad &1\leqslant l\leqslant k, \end{alignedat} \tag{13}
and the defining relations are $(r_1)$–$(r_3)$ (here $x_{j,\delta}$ are in one-to-one correspondence with the vertices $v_{j,\delta}$ of the graph $\Gamma_{\widetilde E}(\widetilde C)$).

To obtain the surface $\widetilde U_{\varepsilon}$ from $U'_{\varepsilon}$, we should consequently glue $U_{j(E_s,1),\varepsilon}\subset U'_{\varepsilon}$ with $U_{j(E_s,2),\varepsilon}\subset U'_{\varepsilon}$ by their “intersection” $U_{j(E_s,1),j(E_s,2),\varepsilon}\,{=}\,U_{j(E_s,1),\varepsilon}\cap\, U_{j(E_s,2)\varepsilon}\subset \widetilde U_{\varepsilon}$, $s=1,\dots,r$. The paths $L_{0,j(E_s,1),j(E_s,2)}$ and $L_{0,j(E_s,2),j(E_s,1)}$ define two imbeddings

$$\begin{equation*} \alpha_{j(E_s,i),j(E_s,\overline i),\varepsilon} \colon \pi_1(U_{j(E_s,1),j(E_s,2),\varepsilon}\setminus \widetilde C,Q_{j(E_s,1),j(E_s,2)})\to \pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C, Q_{1,0}), \end{equation*} \notag$$
where $i=1,2$ and $\{ i,\overline i\}=\{ 1,2\}$. Therefore, to obtain a presentation of $\pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C)$, it suffices to apply $r$ times the $HNN$-extension of $\pi_1(U'_{\varepsilon}\setminus \widetilde C, Q_{1,0})$ relative to these imbeddings, that is, it suffices to add $r$ generators $y_1,\dots, y_r$ and relations $(i_4)$, $(r_5)$:
\begin{equation*} \begin{alignedat}{5} x_{j(E_s,1),0}^{-1}y_sx_{j(E_s,1),1}y_s^{-1} &=1 &\quad &\text{for} &\quad &1\leqslant s\leqslant r, \\ x_{j(E_s,2),0}^{-1}y_s^{-1}x_{j(E_s,2),1}y_s &=1 &\quad &\text{for} &\quad &1\leqslant s\leqslant r, \end{alignedat} \end{equation*} \notag
to the presentation claimed in Proposition 4 (here $y_s$ are elements of $\pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C)$, represented by the loops $L_{0,j(E_s,1),j(E_s,2)}\circ L^{-1}_{0,j(E_s,2),j(E_s,1)}$). As a result, we obtain the presentations claimed in Theorem 3. Therefore, to complete the proof of Theorem 3, it remains to verify that the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$.

### 1.9. Completion of the proof of Theorem 3

Note that, for all $U_{\varepsilon}\in \mathcal U_{C,\nu}$ and a selected tree ${\Gamma}_{\widetilde E}(\widetilde{C})$ of $\widetilde C=\nu^{-1}(C)$, the presentations of the fundamental groups $\pi_1(U_{\varepsilon}\setminus C)$ from Theorem 3 are independent of the choices of $\widetilde U\subset X$, of the good metrics $(ds)^2$ on $\widetilde U$, and of $\varepsilon<\varepsilon_0$. In addition, from above we have the following remark.

Remark 2. Two presentations, as obtained with two different choices of subsets $\widetilde E=\{ E_1,\dots,E_r\}$ (involved in the definition of the tree $\Gamma_{\widetilde E}(\widetilde C)$) in the edge set of $\Gamma(\widetilde C)$ (taking a part in the definition of a tree $\Gamma_{\widetilde E}(\widetilde C)$ ) are the presentations of the same group $\pi_1(U_{\varepsilon}\setminus C)$.

Let $\nu\colon X\to S$ be a resolution of singularities of $C$ let and $\sigma\colon X'\to X$ be the $\sigma$-process with centre at a point $p\in\widetilde C \subset X$. Then $\widetilde C'=(\nu\circ\sigma)^{-1}(C)= \sigma^{-1}(\widetilde C)\,\cup\, C_p$, where $C_p=\sigma^{-1}(p)$, and a tree $\Gamma_{\widetilde E'}(\widetilde C')$ can be obtained from $\Gamma_{\widetilde E}(\widetilde C)$ as follows. We add one more vertex $v_{p,0}$ to the set of vertices of $\Gamma_{\widetilde E}(\widetilde C)$. The bi-weight of $v_{p,0}$ is $(w_{p,1},w_{p,2})=(-1,0)$. Next, if $p\in C_{j_0}$ and $p\notin \operatorname{Sing}(\widetilde C)$, then we add one edge connecting the vertex $v_{j_0,0}\in\Gamma_{\widetilde E}(\widetilde C)$ with $v_{p,0}$, and change the bi-weight of $v_{j_0,0}$ by $(w_{j_0,1}-1,w_{j_0,2})$. If $p=p_{j_1,j_2}=C_{j_1}\cap C_{j_2}$, then by Remark 2 we can assume that $(v_{j_1,0},v_{j_2,0})\notin \{ E_1,\dots,E_r\}$, delete this edge, add two edges $(v_{j_1,0},v_{p,0})$ and $(v_{p,0},v_{j_2,0})$, and change the bi-weights of $v_{j_i,0}$ by $(w_{j_i,1}-1,w_{j_i,2})$, $i=1,2$.

Next, let $\langle \mathcal A \mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ (here $\mathcal A$ and $\mathcal A'$ are the alphabets, and $\mathcal R$ and $\mathcal R'$ are the sets of defining relations) be presentations of groups obtained via the graphs $\Gamma_{\widetilde E}(\widetilde C)$ and $\Gamma_{\widetilde E'}(\widetilde C')$. In the first case, $\mathcal A'=\mathcal A\cup\{x_{p,0}\}$, and the set of defining relations $\mathcal R'$ can be obtained from $\mathcal R$ as follows: in $\mathcal R$, we change the relation $x_{j_0,0}^{\omega_{j_0,2}}\mathcal W_{g_{j_0},\operatorname{v}_{j_0} }=1$ by $x_{j_0,0}^{\omega_{j_0,2}-1}\mathcal W_{g_{j_0}, \operatorname{v}_{j_0} }x_{p,0}=1$ and add two relations $x_{p,0}^{-1}x_{j_0,0}=[x_{j_0,0},x_{p,0}]\,{=}\,1$. Obviously, $\langle \mathcal A\mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ are isomorphic groups.

In the second case, $\mathcal A'=\mathcal A\cup\{x_{p,0}\}$ and the set of defining relations $\mathcal R'$ can be obtained from $\mathcal R$ as follows: in $\mathcal R$, we change three relations

$$$$x_{j_1,0}^{\omega_{j_1,2}}\mathcal W_{g_{j_1},\operatorname{v}_{j_1}} = x_{j_2,0}^{\omega_{j_2,2}}\mathcal W_{g_{j_2},\operatorname{v}_{j_2}} = [x_{j_1,0},x_{j_2,0}]=1$$ \tag{14}$$
by
$$$$x_{j_1,0}^{\omega_{j_i1,2}-1}\mathcal W'_{g_{j_1},\operatorname{v}_{j_1}} = x_{j_2,0}^{\omega_{j_2,2}-1}\mathcal W'_{g_{j_2},\operatorname{v}_{j_i}} = [x_{j_1,0},x_{p,0}]=[x_{j_2,0},x_{p,0}]=1,$$ \tag{15}$$
where the word $\mathcal W'_{g_{j_1},\operatorname{v}_{j_1} }$ is obtained from $\mathcal W_{g_{j_1},\operatorname{v}_{j_1} }$ by substituting $x_{p,0}$ for $x_{j_2,0}$, and $\mathcal W'_{g_{j_2},\operatorname{v}_{j_2} }$ is obtained from $\mathcal W_{g_{j_2},\operatorname{v}_{j_2} }$ by substituting $x_{p,0}$ for $x_{j_1,0}$; in addition, we add one more relation
$$$$x_{p,0}^{-1}x_{j_1,0}x_{j_2,0}=1.$$ \tag{16}$$

By (16) and (15), we have $x_{p,0}=x_{j_1,0}x_{j_2,0}=x_{j_2,0}x_{j_1,0}$ and it is easy to see that we obtain relations (14) from (15) and (16) if in (15) we substitute $x_{j_1,0}x_{j_2,0}$ in $W'_{g_{j_1},\operatorname{v}_{j_1} }$ and $\mathcal W'_{g_{j_2},\operatorname{v}_{j_2} }$ for $x_{p,0}$. Therefore, in the second case, $\langle \mathcal A\mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ are also isomorphic groups.

To complete the proof of Theorem 3, it suffices to note that, for any two resolutions of singularities $\nu_1\colon X_1\to S$ and $\nu_2\colon X_2\to S$, there exist $X$ and two sequences of blowups of points $\widetilde{\nu}_1\colon X\to X_1$ and $\widetilde{\nu}_2\colon X\to X_2$ such that $\nu_i\circ\widetilde{\nu}_i\colon X\to S$, $i=1,2$, are resolutions of singularities.

### 2.1. Proof of Proposition 1

Consider a graph $\Gamma\in \mathcal G$. Let

$$\begin{equation*} \{ v_{1,0},\dots, v_{m,0}\}\cup \{ v_{{m+1},1},\dots, v_{n,1}\} \end{equation*} \notag$$
be the vertex set and let $(k_j,g_j)$ be the bi-weights of $v_{j,0}$ for $j\leqslant m$.

For each vertex $v_{j,o}$, $j\leqslant m$, with bi-weight $(k_j,g_j)$, denote by $C_j$ the zero section (and the base) of a complex line bundle $\operatorname{pr}_j\colon L_j:=L_{C_j,D}\to C_j$ associated with a divisor $D$ of degree $k_j$ on a projective curve $C_j$ of genus $g_j$. We have $(C_j^2)_{L_j}=k_j$. Next, let us choose $\operatorname{v}_{v_{j,0}}$ fibres of the projection $\operatorname{pr}_j$, denote them by $F_{j,o_j(1)},\dots, F_{j,o_j(\operatorname{v}_j)}$, and choose a Hermitian metric $(ds_j)^2$ in $L_j$ which is good with respect to the curve $C_j\cup \bigl(\bigcup_{i=1}^{\operatorname{v}_j}F_{j,o_j(i)}\bigr)$. Let $\varepsilon _j$ be a positive number such that the sets

$$\begin{equation*} U_{j,\varepsilon}= \{ p\in L_j\mid \operatorname{dist}_{(ds_j)^2}(C_j, p)< \varepsilon\leqslant\varepsilon_j\} \end{equation*} \notag$$
are tubular neighbourhoods of $C_j$.

By definition of a good metric, for each $i=1,\dots, \operatorname{v}_j$, there is a neighbourhood $V_{j,o_j(i)}\subset U_{j,\varepsilon}$ of the point $p_{j,o_j(i)}=C_j\cap F_{o_j(i)}$ such that

(i) there are local coordinates $z_1,z_2$ in $V_{j,o_j(i)}$ such that

$$\begin{equation*} V_{j,o_j(i)}\simeq \{ (z_{j,1},z_{j,2})\in\mathbb C^2\mid |z_{j,1}|< \varepsilon,\, |z_{j,2}|<\varepsilon\}; \end{equation*} \notag$$

(ii) $z_{j,1}=0$ is the equation of $C_j\cap V_{j,o_j(i)}$ and $z_{j,2}=0$ is the equation of $F_{o_j(i)}\cap V_{j,o_j(i)}$;

(iii) the metric $(ds_j)^2$ in $V_{j,o_j(i)}$ is given by $(ds_j)^2=dz_{j,1}\,d\overline z_{j,1}+dz_{j,2}\,d\overline z_{j,2}$.

We set $\varepsilon_0=\min_{1\leqslant j\leqslant m} \varepsilon_j$ and define $C_{o_j(i)}=F_{o_j(i)}\cap V_{j,o_j(i)}\subset U_{j,\varepsilon_0}$ if $o_j(i) > m$. If, for $1\leqslant j_1\neq j_2\leqslant m$, there are $i_1$ and $i_2$ such that $o_{j_1}(i_2)=j_2$ and $o_{j_2}(i_1)=j_1$, then we glue $U_{j_1,\varepsilon_0}$ with $U_{j_2,\varepsilon_0}$ identifying $V_{j_1,j_2}\subset U_{j_1,\varepsilon_0}$ with $V_{j_2,j_1}\subset U_{j_2,\varepsilon_0}$ with the help of the bi-holomorphic isomorphism of $V_{j_1,j_2}$ and $V_{j_2,j_1}$ given by $z_{j_1,1}\leftrightarrow z_{j_2,2}$, $z_{j_1,2}\leftrightarrow z_{j_2,1}$.

As a result of all possible such glues, we obtain the tubular neighbourhood $\widetilde U_{\varepsilon_0}=\bigcup_{j=1}^mU_{j,\varepsilon_0}$ of the curve $\widetilde C=\bigcup_{j=1}^nC_j$. Obviously, the partially bi-weighted dual graph $\Gamma(\widetilde C)$ of the curve $\widetilde C\subset \widetilde U_{\varepsilon_0}$ and the partially bi-weighted graph $\Gamma\in\mathcal G$ are isomorphic as partially bi-weighted graphs.

### 2.2. Proof of Theorem 1

In the notation of § 1.2, for each point

\begin{equation*} \begin{aligned} \, p_{j_1,j_2} &=C_{j_1}\cap C_{j_2}\subset V_{p_{j_1,j_2}}\simeq \mathbb B_2 \\ &=\bigl\{ (z_1,z_2)\in \mathbb C^2\bigm| \sqrt{|z_1|^2+|z_2|^2}<2\bigr\}, \qquad 1\leqslant j_1,j_2\leqslant m, \end{aligned} \end{equation*} \notag
consider in $V_{p_{j_1,j_2}}\cap \partial U_{\varepsilon}$ the subset
$$\begin{equation*} \partial U_{j_1,j_2,\varepsilon}\simeq \{ (z_1,z_2)\mid |z_1|= \varepsilon,\, \varepsilon\leqslant |z_2|\leqslant 1\}\cup \{ (z_1,z_2)\mid \varepsilon \leqslant |z_1|\leqslant 1,\, |z_2|=\varepsilon\}. \end{equation*} \notag$$
Let $M_{j_1,j_2,\varepsilon}\,{=}\,\{ (|z_1|,|z_2|)\in \mathbb R^2\mid (z_1,z_2)\in\partial U_{j_1,j_2,\varepsilon}\} \subset \mathbb R^2$ be the set of modules of coordinates of points in $\partial U_{j_1,j_2,\varepsilon}$.

For $\varepsilon_1<\varepsilon_2<\varepsilon_0$, define the mapping $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$,

$$\begin{equation*} \rho_{\varepsilon_1,\varepsilon_2}\colon (z_1,z_2)\in \partial U_{j_1,j_2,\varepsilon_2}\mapsto \bigl(\rho_{\varepsilon_1,\varepsilon_2}(z_1), \rho_{\varepsilon_1,\varepsilon_2}(z_2)\bigr)\in \partial U_{j_1,j_2,\varepsilon_1}, \end{equation*} \notag$$
as follows: $\operatorname{arg}(\rho_{\varepsilon_1,\varepsilon_2}(z_i))= \operatorname{arg}(z_i)$ for $i=1,2$ and the induced by $\rho_{\varepsilon_1,\varepsilon_2}$ mapping $|\rho_{\varepsilon_1,\varepsilon_2}|\colon M_{j_1,j_2,\varepsilon_2}\to M_{j_1,j_2,\varepsilon_1}$ is the projection of $M_{j_1,j_2,\varepsilon_2}$ from the point $(1,1)\in\mathbb R^2$ to $M_{j_1,j_2,\varepsilon_1}$ (see Figure 3). It is easy to check that $\rho_{\varepsilon_1,\varepsilon_2}$ is given by
\begin{alignedat}{3} \rho_{\varepsilon_1,\varepsilon_2}(z_1,z_2) &= \biggl(\frac{(1-\varepsilon_1)(|z_1|-1)+ (1-\varepsilon_2)}{(1-\varepsilon_2)|z_1|}z_1, \, \frac{\varepsilon_1}{\varepsilon_2}z_2\biggr) &\quad &\text{if } |z_2|=\varepsilon_2, \\ \rho_{\varepsilon_1,\varepsilon_2}(z_1,z_2) &= \biggl(\frac{\varepsilon_1}{\varepsilon_2}z_1, \, \frac{(1-\varepsilon_1)(|z_2|-1)+(1-\varepsilon_2)} {(1-\varepsilon_2)|z_2|}z_2\biggr) &\quad &\text{if } |z_1|=\varepsilon_2. \end{alignedat} \tag{17}

Obviously, for $C_{j_1}\cap C_{j_2}\neq \varnothing$, the mappings $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ are homeomorphisms.

To extend the homeomorphisms $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ to a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial\widetilde U_{\varepsilon_2} \to \partial \widetilde U_{\varepsilon_1}$, we let $\bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon}$ denote the union over all $o_j(i)$ such that $\overline o_j(i)=(o_j(i),0)$. Applying Claim 1, it follows from (17) that, for the points $q=(z_1,z_2)\in \partial U_{j_1,j_2,\varepsilon_2}$ such that $|z_1|=1$ and $|z_2|=\varepsilon_2$ (respectively, $|z_1|=\varepsilon_2$ and $|z_2|=1$), the images $\rho_{\varepsilon_1,\varepsilon_2}(q)$ belong to the geodesic lines $\gamma\subset V_{j_1,j_2}$, transversal to $C_{j_2}$ (respectively, $C_{j_1}$) and passing through the points $q$. Therefore, the homeomorphisms $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ can be extended continuously to a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial\widetilde U_{\varepsilon_2} \to \partial \widetilde U_{\varepsilon_1}$ such that its restrictions

$$\begin{equation*} \rho_{j,\varepsilon_1,\varepsilon_2}\colon \partial U_{j,\varepsilon_2} \setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2} \to \partial U_{j,\varepsilon_1}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_1} \end{equation*} \notag$$
to $\partial U_{j,\varepsilon_2}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2}$ are defined as follows: the homeomorphisms $\rho_{j,\varepsilon_1,\varepsilon_2}$ send $q\in \partial U_{j,\varepsilon_2}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2}$ to $\rho_{j,\varepsilon_1,\varepsilon_2}(q)=\gamma_q\cap \bigl(\partial U_{j,\varepsilon_1}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_1}\bigr)$, where $\gamma_q$ are the geodesic lines transversal to $C_j$ and passing through the points $q$. As a result, we obtain a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{\varepsilon_2}\simeq \partial \widetilde U_{\varepsilon_2}\to \partial \widetilde U_{\varepsilon_1} \simeq \partial U_{\varepsilon_1}$.

For $q\in \widetilde U_{\varepsilon}$, denote by $d(q)=\operatorname{dist}_{(ds)^2}(q,\widetilde C)$ the distance between $q$ and the curve $\widetilde C$. We have $U_{\varepsilon}\setminus C\simeq \widetilde U_{\varepsilon}\setminus\widetilde C$ and it is easy to see that the mapping

$$\begin{equation*} \rho_{\varepsilon}\colon U_{\varepsilon}\setminus C\simeq \widetilde U_{\varepsilon}\setminus \widetilde C \to \partial \widetilde U_{\varepsilon}\times (0,\varepsilon)\simeq \partial U_{\varepsilon}\times (0,\varepsilon), \end{equation*} \notag$$
given by $\rho_{\varepsilon}(q)=(\rho^{-1}_{d(q),\varepsilon}(q),d(q))$ for $q\in \widetilde U_{\varepsilon}\setminus \widetilde C$, is an homeomorphism. Therefore, the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(\partial U_{\varepsilon})$ are isomorphic, and now assertions 1) and 2) of Theorem 1 follow from Theorem 3.

### 3.1. Extended Euclidean algorithm

Let $\mathbb Z[u_1,\dots,u_n,\dots]$ be the ring of polynomials in variables $u_1,\dots,u_n,\dots$ with coefficients in $\mathbb Z$, and let $\mathbb Q(u_1,\dots,u_n,\dots)$ be its field of quotients. Consider the rational functions

$$$$R_n(u_1,\dots, u_n)=u_n-\frac{1}{u_{n-1}-\cfrac{1}{u_{n-2}- \cfrac{1}{\dots-\cfrac{1}{u_1}}}}\in \mathcal R=\mathbb Q(u_1,\dots,u_n,\dots)$$ \tag{18}$$
and $R_0=P_0=1$. By induction, it is easy to check that $R_n(u_1,\dots,u_n)=P_n(u_1,\dots,u_n)/P_{n-1}(u_1,\dots,u_{n-1})$, where the polynomials $P_n(u_1,\dots,u_n)$ are given recursively by
$$$$\begin{gathered} \, P_0:=1,\qquad P_1:=P_1(u_1)=u_1, \nonumber \\ P_2:=P_2(u_1,u_2)= u_2P_1-P_0=u_1u_2-1, \nonumber \\ P_3:=P_3(u_1,u_2,u_3)=u_3P_2-P_1=u_1u_2u_3-u_1-u_3, \nonumber \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \nonumber \\ P_n:=P_n(u_1,\dots,u_n)=u_nP_{n-1}-P_{n-2}. \end{gathered}$$ \tag{19}$$

We set $Q_n:=P_{n-1}(u_2,\dots,u_n)$, $Q_{0}:=0$, and consider the set of $(2\times 2)$-matrices

$$\begin{equation*} \mathcal M=\biggl\{ M_m=(-1)^m\begin{pmatrix} -Q_{m-1} & Q_{m} \\ -P_{m-1} & P_m \end{pmatrix} \in \operatorname{Mat}(2,\mathcal R) \biggm| m\in\mathbb N \biggr\}. \end{equation*} \notag$$
We also define
$$\begin{equation*} M_1(u_m):=(-1)\begin{pmatrix} 0 & 1 \\ -1 & u_m \end{pmatrix} \in \operatorname{Mat}(2,\mathcal R),\qquad m\in\mathbb N; \end{equation*} \notag$$
in particular, $M_1=M_1(u_1)$.

Lemma 4. $\mathcal M=\{ M_m=M_1(u_1)\cdots M_1(u_m)\mid m\in \mathbb N\} \subset \operatorname{SL}(2,\mathcal R)$.

Proof. Using induction on $m$ and applying the equality
\begin{equation*} \begin{aligned} \, M_m M_1(u_{m+1}) &= (-1)^{m+1} \begin{pmatrix} -Q_{m-1} & Q_{m} \\ -P_{m-1} & P_m \end{pmatrix} \begin{pmatrix} 0 & 1 \\ -1 & u_{m+1} \end{pmatrix} \\ &=(-1)^{m+1} \begin{pmatrix} -Q_{m} & -Q_{m-1}+u_{m+1}Q_{m} \\ -P_{m} & -P_{m-1}+u_{m+1}P_m \end{pmatrix} \\ &\!\!\stackrel{(19)}= (-1)^{m+1} \begin{pmatrix} -Q_{m} & Q_{m+1} \\ -P_{m} & P_{m+1} \end{pmatrix}=M_{m+1}, \end{aligned} \end{equation*} \notag
we find that $M_m=M_1(u_1)\cdots M_1(u_m)$.

We have $M_m\in \operatorname{SL}(2,\mathcal R)$, since $\det M_1(u_n)=1$. Lemma 4 is proved.

Denote by $\overline k^{\,m}=(k_1,\dots,k_m)$, $k_j\in\mathbb Z$, the elements of $\mathbb Z^m$. The matrices $M_m\in \mathcal M$ define the mappings $\mu_m \colon \mathbb Z^m\to \operatorname{SL}(2,\mathbb Z)$ given by

\begin{equation*} \begin{aligned} \, &\mu_m \colon \overline k^{\,m}=(k_1,\dots,k_m) \\ &\qquad\qquad\mapsto M_{\overline k^{\,m}}=(-1)^m \begin{pmatrix} -Q_{m-1}(k_2,\dots, k_{m-1}) & Q_{m}(k_2,\dots, k_m) \\ -P_{m-1}(k_1,\dots,k_{m-1}) & P_m(k_1,\dots, k_m) \end{pmatrix}. \end{aligned} \end{equation*} \notag

Proposition 5. $\bigcup_{n=1}^{\infty}\mu_{4n}(\mathbb Z^{4n})= \bigcup_{n=1}^{\infty}\mu_n(\mathbb Z^{n})=\operatorname{SL}(2,\mathbb Z).$

Proof. It is well-known that $\operatorname{SL}(2,\mathbb Z)$ is generated by the matrices
$$\begin{equation*} T_{1,2}(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix},\qquad T_{2,1}(m)=\begin{pmatrix} 1 & 0 \\ m & 1 \end{pmatrix},\qquad m\in\mathbb Z, \end{equation*} \notag$$
that is, any matrix $A\in \operatorname{SL}(2,\mathbb Z)$ is a product $A=T_1(m_1)\cdots T_n(m_n)$ for some $n\in \mathbb N$, were each $T_i(m_i)$ is either $T_{1,2}(m_i)$, or $T_{2,1}(m_i)$ and if $T_i(m_i)=T_{1,2}(m_i)$, then $T_{i+1}(m_{i+1})=T_{2,1}(m_{i+1})$ and if $T_i(m_i)=T_{2,1}(m_i)$, then $T_{i+1}(m_{i+1})=T_{1,2}(m_{i+1})$. On the other hand, by Lemma 4, it is easy to check that
\begin{equation*} \begin{aligned} \, M_{(0,0,0,-k_4)} &=M_1(0)^3M_1(k_4)=T_{1,2}(k_4)\in \mu_4(\mathbb Z^4), \\ M_{(k_1,0,0,0)} &=M_1(k_1)M_1(0)^3=T_{2,1}(k_1)\in \mu_4(\mathbb Z^4). \end{aligned} \end{equation*} \notag
Therefore, by Lemma 4, the matrix $A=T_1(m_1)\cdots T_n(m_n)\in \mu_{4n}(\mathbb Z^{4n})$. This proves Proposition 5.

### 3.2. Proof of assertion 1) of Theorem 4

Consider a partially bi-weighted graph $\Gamma_{m,2}\in\mathcal Ch_2^0$ depicted in Figure 4 in which $(k_j,0)$ are bi-weights of vertices $v_{j,0}$.

By Theorem 2, the fundamental group $\pi_1^w(\Gamma_{m,2})$ has the presentation

\begin{aligned} \, \pi_1^w(\Gamma_{m,2}) &=\langle x_{1,0},\dots,x_{m,0},x_{m+1,1},x_{m+2,1} \mid x_{1,0}^{k_1}x_{m+1,1}x_{2,0}=x_{2,0}^{k_2}x_{1,0}x_{3,0} \nonumber \\ &\qquad\qquad =\dots=x_{m-1,0}^{k_{m-1}}x_{m-2,0}x_{m,0}= x_{m,0}^{k_m}x_{m-1,0}x_{m+2,1}=1, \nonumber \\ &\qquad\qquad [x_{1,0},x_{m+1,1}]=[x_{1,0},x_{2,0}]=[x_{2,0},x_{3,0}] \nonumber \\ &\qquad\qquad=\dots=[x_{m-1,0},x_{m,0}]=[x_{m,0},x_{m+2,1}]=1\rangle. \end{aligned} \tag{20}
By (20),
$$\begin{equation*} x_{2,0}=x_{m+1,1}^{-1}x_{1,0}^{-k_1}\quad\text{and}\quad [x_{2,0},x_{m+1,1}]=1, \end{equation*} \notag$$
since $[x_{1,0},x_{m+1,1}]=1$. Next,
$$\begin{equation*} x_{3,0}=x_{1,0}^{-1}x_{2,0}^{-k_2}=x_{m+1,1}^{k_2}x_{1,0}^{k_1,k_2-1}\quad \text{and}\quad [x_{1,0},x_{3,0}]=[x_{m+1,1},x_{3,0}]=1, \end{equation*} \notag$$
etc., and so
$$$$x_{j,0}=x_{m+1,1}^{(-1)^{j-1}Q_{j-1}}x_{1,0}^{(-1)^{j-1}P_{j-1}},\qquad j=2,\dots,m,$$ \tag{21}$$
$$$$x_{m+2,1}=x_{m+1,1}^{(-1)^{m}Q_{m}}x_{1,0}^{(-1)^{m}P_m},$$ \tag{22}$$
where $P_j=P_j(k_1,\dots,k_j)$ and $Q_{j}=P_{j-1}(k_2,\dots,k_j)$.

It follows from (21) and (22) that $\pi_1^w(\Gamma_{m,2})= \langle x_{1,0},x_{m+1,1}\mid [x_{1,0},x_{m+1,1}]=1\rangle$ is a free commutative group of rank $2$, and $\Pi_w(\mathcal Ch^0_2)=\{ \mathbb Z\times\mathbb Z\}$.

Let $\Gamma_{m,1}\in\mathcal Ch_1^0$ be a partially bi-weighted graph depicted in Figure 5.

To obtain a presentation of the group $\pi_1^w(\Gamma_{m,1})$, it suffices to add the relation $x_{m+2,1}=1$ to presentation (20). As a result,

$$$$\pi_1^w(\Gamma_{m,1})= \bigl\langle x_{1,0},x_{m+1,1}\bigm| [x_{1,0},x_{m+1,1}]= x_{m+1,1}^{Q_{m}}x_{1,0}^{P_m}=1\bigr\rangle.$$ \tag{23}$$
Therefore, $\pi_1^w(\Gamma_{m,1})=\mathbb Z$, since $Q_m$ and $P_m$ are coprime by Lemma 4.

Let $\Gamma_{m,0}\in\mathcal Ch_0^0$ be a partially bi-weighted graph depicted in Figure 6.

To obtain a presentation of the group $\pi_1^w(\Gamma_{m,}0)$, it suffices to add the relation $x_{m+1,1}=x_{m+2,1}=1$ to presentation (20). Hence

$$\begin{equation*} \pi_1^w(\Gamma_{m,0})=\bigl\langle x_{1,0}\bigm| x_{1,0}^{P_m}= 1\bigr\rangle \simeq \mathbb Z/|P_m|\mathbb Z \end{equation*} \notag$$
is a cyclic group. If $m=1$, then $P_1=k_1$, and hence, $\Pi_w(\mathcal Ch^0_0)=\{ \mathbb Z/n\mathbb Z\mid n\geqslant 0\}$.

### 3.3. Proof of assertion 2) of Theorem 4

Let $\Gamma_{m}\in\mathcal L$ be a partially bi-weighted loop whose vertex set $V(\Gamma_m)$ is $\{v_{1,0},\dots,v_{m,0}\}$ ($m\geqslant 3$ by condition (G4)). To give a presentation of the group $\pi_1^w(\Gamma_m)$ we need to choose a set of edges $\widetilde E$ and consider a tree $\Gamma_{m,\widetilde E}$ of $\Gamma_m$. We set $\widetilde E=\{ (v_{1,0},v_{m,0})\}$. The vertex set of the tree $\Gamma_{m,\widetilde E}$ is $\{ v_{1,0},\dots,v_{m,0}\}\cup \{v_{1,1},v_{m,1}\}$, and $\Gamma_{m,\widetilde E}$ coincides with the graph $\Gamma_{m,2}$, depicted in Figure 4, where $v_{1,1}=v_{m+2,1}$ and $v_{m,1}=v_{m+1,1}$. Therefore, to obtain a presentation of $\pi_1^w(\Gamma_m)$ it suffices to add an additional generator $y$ to the set of generators in presentation (20) and add two defining relations $y^{-1}x_{1,0}y=x_{m+2,1}$ and $y^{-1}x_{m+1,1}y=x_{m,0}$ to the set of defining relations in presentation (20). It follows from (21) and (22) that $\pi_1^w(\Gamma_m)$ has the presentation

\begin{equation*} \begin{aligned} \, \pi_1^w(\Gamma_m) &= \bigl\langle x_{1,0}, x_{m+1,1}, y\bigm| [x_{1,0},x_{m+1,1}]=1, \\ &\qquad\qquad y^{-1}x_{1,0}y =x_{1,0}^{(-1)^{m}P_m}x_{m+1,1}^{(-1)^{m}Q_{m}}, \\ &\qquad\qquad y^{-1}x_{m+1,1}y= x_{1,0}^{(-1)^{m-1}P_{m-1}}x_{m+1,1}^{(-1)^{m-1}Q_{m-1}}\bigr\rangle \end{aligned} \end{equation*} \notag
$$\begin{equation*} \pi_1^w(\Gamma_m)\simeq\mathbb Z^2\ltimes_{M^t_{\overline k^{\,m}}}\mathbb F_1 =\bigl\langle (z_1,z_2),t\bigm| (z_1,z_2)\in \mathbb Z^2,\, t^{-1}(z_1,z_2)t=(z_1,z_2)M^{\tau}_{\overline k^{\,m}}\bigr\rangle \end{equation*} \notag$$
is a semidirect product of $\mathbb Z^2$ and $\mathbb F_1\simeq \mathbb Z$, where (by Proposition 5)
$$\begin{equation*} M^{\tau}_{\overline k^{\,m}}= (-1)^m \begin{pmatrix} P_m & Q_{m} \\ -P_{m-1} & -Q_{m-1} \end{pmatrix} \end{equation*} \notag$$
is an arbitrary element of the group $\operatorname{SL}(2,\mathbb Z)$ acting on $\mathbb Z^2$ from the right.

### 3.4. Proof of assertion 3) of Theorem 4

Consider the presentation in Definition 1 of the fundamental group $\pi_1^w(\Gamma)$ of $\Gamma\in\mathcal G$. Let $N$ be the normal closure in $\pi_1^w(\Gamma)$ of the subgroup generated by

$$\begin{equation*} \! x_{1,0},\ \dots,\ x_{m,0},\ x_{m+1,1},\ \dots,\ x_{m+k,1},\ x_{j(E_1,1),1},\ x_{j(E_1,2),1},\ \dots,\ x_{j(E_r,1),1},\ x_{j(E_r,2),1}.\! \end{equation*} \notag$$
Then the quotient $\pi_1^w(\Gamma)/N$ has the presentation
\begin{equation*} \begin{aligned} \, \pi_1^w(\Gamma)/N &=\biggl\langle y_1,\dots, y_r, \mu_{j,i},\lambda_{j,i},\ 1\leqslant j\leqslant m,\, 1\leqslant i\leqslant g_j\biggm| \\ &\qquad\prod_{i=1}^{g_{j}}[\mu_{j,i},\lambda_{j,i}]=1,\ 1\leqslant j \leqslant m,\, 1\leqslant g_j \biggr\rangle, \end{aligned} \end{equation*} \notag
that is, $\pi_1^w(\Gamma)/N \simeq \bigl(\prod^{n_1}\mathcal R_1*\cdots* \prod^{n_{i_0}}\mathcal R_{i_0}\bigr)*\mathbb F_r$.

Acknowledgement. The author is grateful to the referee for valuable comments.

#### Bibliography

1. K. Stein, “Analytische Zerlegungen komplexer Räume”, Math. Ann., 132 (1956), 63–93
2. D. Mumford, “The topology of normal singularities of an algebraic surface and a criterion for simplicity”, Inst. Hautes Études Sci. Publ. Math., 9 (1961), 5–22      ; Russian transl. Matematika, 10:6 (1966), 3–24
3. A. H. Durfee, “Neighborhoods of algebraic sets”, Trans. Amer. Math. Soc., 276:2 (1983), 517–530
4. P. Wagreich, “Singularities of complex surfaces with solvable local fundamental group”, Topology, 11 (1972), 51–72
5. F. Catanese, Surface classification and local and global fundamental groups. I, arXiv: math/0602128v2
6. E. Artal Bartolo, J. I. Cogolludo-Augustín, and D. Matei, “Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links”, Eur. J. Math., 6:3 (2020), 624–645
7. Vik. S. Kulikov, “On rigid germs of finite morphisms of smooth surfaces”, Mat. Sb., 211:10 (2020), 3–31        ; English transl. Sb. Math., 211:10 (2020), 1354–1381
8. Vik. S. Kulikov, “Rigid germs of finite morphisms of smooth surfaces and rational Belyi pairs”, Mat. Sb., 212:9 (2021), 119–145        ; English transl. Sb. Math., 212:9 (2021), 1304–1328
9. Vik. S. Kulikov and E. I. Shustin, “On $G$-rigid surfaces”, Complex analysis and its applications, Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar, Trudy Mat. Inst. Steklova, 298, MAIK “Nauka/Interperiodika”, Moscow, 2017, 144–164        ; English transl. Proc. Steklov Inst. Math., 298 (2017), 133–151
10. E. R. van Kampen, “On the connection between the fundamental groups of some related spaces”, Amer. J. Math., 55 (1933), 261–267
11. H. Seifert, “Topologie dreidimensionaler gefaserter Räume”, Acta Math., 60:1 (1933), 147–238

 Statistics & downloads: Abstract page: 285 Russian version PDF: 13 English version PDF: 35 Russian version HTML: 79 English version HTML: 139 References: 16 First page: 11