# Fundamental groups of partial compactifications of arrangements and homology
planes

Rodolfo Aguilar Aguilar

###### Abstract.

Following our previous work, we develop an algorithm to compute a presentation
of the fundamental group of certain partial compactifications of the
complement of a complex arrangement of lines in the projective plane. It
applies, in particular, to homology planes arising from arrangements of lines.
This permits to determine the infinitude of the fundamental group of homology
planes of log-general type that may be the first examples of this type.

Partially supported by ANR project Hodgefun ANR-16-CE40-0011-01 and by the
National program “Young scientists and Postdoctoral students”, Funded by the
Ministry of Education and Science of Bulgaria

##  1\. Introduction

The computation of the fundamental group of smooth quasi-projective complex
varieties is in general a difficult task. There exist, however, certain
classes of these type of varieties where specific methods have been developed
in order to obtain at least, a presentation for these groups. For example, for
the complement $\mathbb{P}^{2}\setminus\mathscr{A}$ of an arrangement of lines
$\mathscr{A}$ in $\mathbb{P}^{2}$, a presentation for
$\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ is obtained in [Ran85], [Arv92],
[CS97], [FGBMB15].

In [AA20], by modifying the method in [Ran85], we have developed a method to
obtain a presentation for the fundamental group of certain partial
compactifications of the complement $\mathbb{P}^{2}\setminus\mathscr{A}$ of an
arrangement of lines $\mathscr{A}\subset\mathbb{P}^{2}$ under the hypothesis
that the lines in $\mathscr{A}$ are defined by real linear forms.

The purpose of this note is to generalize our method in two directions:

  * •

to admit a general arrangement $\mathscr{A}\subset\mathbb{P}^{2}$ defined by
complex linear forms and

  * •

to admit a more general class $M(\mathscr{A},I,P)$ of partial
compactifications of $\mathbb{P}^{2}\setminus\mathscr{A}$. See 2.3.1 for a
precise definition.

In particular, some of these partial compactifications give rise to homology
planes, see below. It also contains some partial compactifications constructed
by Fowler in the study of rational homology disks in [Fow13], see also [Wah20,
BW21].

We proceed in two different ways: firstly, following [Arv92] and [CS97], whose
work generalize [Ran85] to complex arrangements, we define a wiring diagram
$\mathcal{W}$ that encodes some over or under-crossing of the lines in
$\mathscr{A}$ arising by the complex nature of the forms defining them. The
graph $\mathcal{W}$ encodes enough information to obtain a presentation of
$\pi_{1}(M(\mathscr{A},I,P))$.

######  Theorem 1.1.

A presentation for $\pi_{1}(M(\mathscr{A},I,P))$ can be obtained from
$\mathcal{W}$. The set of generators are in correspondence with the set of
lines in $\mathscr{A}$ and the set of relations has two types of them:

  * •

those relations $R_{p}$ coming from a singular point $p$ of $\mathscr{A}$.
These relations already appeared in a presentation of
$\pi_{1}(M(\mathscr{A}))$ and

  * •

for each element $\iota$ either in $I$ or in $P$, a relation $R_{\iota}$ which
is a product of conjugates of some generators depending on $\iota$.

As in [AA20], a main step in the proof of Theorem 1.1 consists in the explicit
computation of an expression for the meridians around certain exceptional
divisors, obtained by blowing-up $\mathscr{A}$ in some singular points, in
terms of the generators. We also obtain in Theorem 3.17, similar to loc. cit.,
a presentation for the fundamental group of an orbifold whose moduli space
$\operatorname{Bl}_{P_{0}}\mathbb{P}^{2}$ is a blow-up of $\mathbb{P}^{2}$ at
some singular points $P_{0}$ of the arrangement $\mathscr{A}$ and the non-
trivial isotropy groups lie in a divisor $D$ contained in the total transform
of $\mathscr{A}$ in $\operatorname{Bl}_{P_{0}}\mathbb{P}^{2}$.

Secondly, let $U$ denote a closed regular tubular neighborhood of
$\mathscr{A}$ in $\mathbb{P}^{2}$. We call $\partial U$ the boundary manifold
of $\mathscr{A}$. In [FGBMB15], a presentation for $\pi_{1}(M(\mathscr{A}))$
is obtained from a presentation of $\pi_{1}(\partial U)$ by studying the map
$\pi_{1}(\partial U)\to\pi_{1}(M(\mathscr{A}))$ induced by the inclusion
$\partial U\hookrightarrow M(\mathscr{A})$. For the homological version see
[GB13].

It turns out that their methods can also be applied to determine a
presentation for the fundamental group of some partial compactifications
$M(\mathscr{A},I,P)$. However, in order to study the boundary manifolds
$\partial U_{D}$ of strict transforms $D$ of $\mathscr{A}$ in some birational
model of $\mathbb{P}^{2}$, we start from a different presentation for the
boundary manifold $\partial U$ of $\mathscr{A}$ than that used in [FGBMB15].

Indeed, when $D=\sum D_{i}$ is a connected, simple normal crossing divisor
such that $\pi_{1}(D)$ is trivial, Mumford gave a presentation for
$\pi_{1}(\partial U_{D})$ in [Mum61]. This, together with the graph-manifold
structure in the sense of Waldhausen [Wal67], permitted Westlund to give a
presentation of $\pi_{1}(\partial U)$ in [Wes67] (see also [CS08]). Here, by a
choice of a surface birational to $\mathbb{P}^{2}$ where the strict transform
of $\mathscr{A}$ satisfies the hypothesis for the presentation of Mumford, we
obtain the same presentation of Westlund. See Theorem 4.1. Following this
construction, we are able to give a presentation for the fundamental group of
a boundary manifold $\partial U_{D}$ of a divisor $D$ lying in a surface
$\bar{X}$ obtained by successive blows-up of $\mathbb{P}^{2}$ such that
$M(\mathscr{A},I,P)=\bar{X}\setminus D$.

We obtain in Theorem 4.3 a presentation for $\pi_{1}(M(\mathscr{A}))$ by
studying the map $i_{*}:\pi_{1}(\partial U)\to\pi_{1}(M(\mathscr{A}))$.
Moreover, as the construction for $\pi_{1}(\partial U)$ depends of several
choices, we can make them in such a way that the image under $i$ of the
meridians of the lines in $\mathscr{A}$ lying in $\partial U$, whose homotopy
class are part of the generators of $\pi_{1}(\partial U)$, lie in the same
homotopy class as the meridians constructed for Theorem 1.1. From this, we do
not only obtain that the presentation of Theorem 4.3 and 1.1 are equivalent,
but that the image of the set of relation in the presentation of
$\pi_{1}(M(\mathscr{A}))$ coincides with the relations as in Theorem 1.1. From
this, we can obtain a presentation for partial compactifications
$\pi_{1}(M(\mathscr{A},I,P)$, see Theorem 4.14.

The presentation of Theorem 1.1 applies in particular for homology planes,
affine smooth surfaces with trivial reduced integral homology, arising from
arrangement of lines. To the knowledge of the author, no general algorithm for
computing a presentation for their fundamental groups were known.

The arrangements of lines giving rise to homology planes were classified by T.
Tom Dieck and T. Petri, see [TDP89, TDP91, TD90] . There exists one infinite
family $L(1,n+1)$ and six arrangement $L(2),\ldots,L(7)$ of at most ten lines.

As in algebraic surfaces, the study of homology planes is usually divided by
its log-Kodaira dimension: the only homology plane $X$ with
$\bar{k}(X)=-\infty$ satisfies $X\cong\mathbb{C}^{2}$ [Fuj79, Theorem 3.2],
there are no homology planes with $\bar{k}(X)=0$ and all the homology planes
with $\bar{k}(X)=1$ can be obtained from $L(1,n+1)$, see [Miy00].

For homology planes $X$ with $\bar{k}(X)=2$, Tom Dieck and Petri gave examples
arising from $L(2)$ in [TDP93] as well as a general algorithm to obtain
homology planes out of arrangement of lines. As the fundamental group
$\pi_{1}(\mathbb{P}^{2}\setminus L(2))$ is abelian, all homology planes
arising from $L(2)$ are contractible. In [Zai93], Zaidenberg gave a countable
number of contractible homology planes arising from the arrangement $L(3)$.
Here we can obtain the following result.

######  Theorem 1.2.

There exists homology planes of log-general type with infinite fundamental
group.

These homology planes arise as in [TDP93]. They may be the first examples of
this type. The infinitude was obtained at first, using the presentation of
Theorem 1.1, but the nature of the groups suggested the following more general
construction: using certain pencils of lines associated to the
(sub)-arrangements $L(3),L(4),L(5)$, as presented by Suciu in [Suc01], we can
construct homology planes $X$ having a fibration that induces a surjective
homomorphism to the fundamental group of an orbicurve which is infinite. In
fact, all homology planes with infinite fundamental group that we obtain are
of this form. We present examples where the general fiber is $\mathbb{C}$, see
examples 5.3, 5.5 and others where the general fiber is
$\mathbb{P}^{1}\setminus\\{3-\text{points}\\}$, see examples 5.2, 5.4. This
suggests the following question:

######  Question 1.

Let $X$ be a homology plane with infinite fundamental group. Does $\pi_{1}(X)$
always admit an infinite quotient isomorphic to the fundamental group of an
orbicurve?

This is the case for the homology planes $X$ with $\bar{k}(X)=1$, see [Miy00].
Adapting these arguments, we show in Proposition 5.6, that this is also the
case for the homology planes arising from $L(3)$.

Finally, by using again Theorem 1.1, we give examples of contractible homology
planes arising from $L(6)$ and $L(7)$. Following the arguments of Zaidenberg
in [Zai93], we can show that these examples are not isomorphic to those
obtained from $L(1,n+1)$ and $L(3)$. These arguments also show that these
homology planes are of log-general type.

Acknowledgments. I would like to thank my advisor P. Eyssidieux for a careful
reading and valuable comments of this note. I would also like to thank M.
Zaidenberg who introduced me to homology planes and the referees of my thesis
E. Artal Bartolo and M. Teicher.

##  2\. Preliminaries

###  2.1. Notations

We will denote by $\mathbb{P}^{2}$ the complex projective plane.

Let $\mathscr{A}=\\{L_{1},\ldots,L_{n+1}\\}$ be an arrangement of $n+1$ lines
in $\mathbb{P}^{2}$. The complement of the arrangement will be sometimes
denoted by $M(\mathscr{A}):=\mathbb{P}^{2}\setminus\mathscr{A}$.

Let $X$ be a complex manifold, for $p\in X$ we denote by
$\pi:\operatorname{Bl}_{p}X\to X$ the blow up of $X$ at $p$. If $D\subset X$
is a divisor, we denote by $\left|D\right|$ the reduced divisor with the same
support as $D$ and by $\operatorname{Sing}D$ the set of singular points of
$D$.

We will denote by $a^{b}=b^{-1}ab$ if $a,b\in G$ with $G$ a group. If $a\in G$
and $b\in\mathbb{Z}$, we denote as well by $a^{b}$ the $b$-power of $a$.

###  2.2. Meridians

Let $X$ be a complex manifold and $H\subset X$ a hypersurface. Let $p\in H$ be
a smooth point and $\Delta$ a disc cutting transversaly $H$ at $p$. A loop
$\gamma$ in $\pi_{1}(X\setminus H)$ freely homotopic to the boundary of
$\Delta$ with the natural orientation is called a _meridian_.

The following Proposition is well-known, for a proof see [Shi07].

######  Proposition 2.1.

Let $X$ be a complex manifold and $D=\sum D_{i}$ a divisor such that each
irreducible reduced component $\left|D_{i}\right|$ of $D$ is smooth. Let
$\gamma_{i}$ be a meridian of $\left|D_{i}\right|$, then every other meridian
of $\left|D_{i}\right|$ is a conjugate of $\gamma_{i}$ in $\pi_{1}(X\setminus
D)$ and the kernel of the map $\pi_{1}(X\setminus D)\to\pi_{1}(X)$ is the
normal subgroup generated by the meridians of its irreducible components.

###  2.3. Dual graph of a divisor and partial compactifications of its
complement

Let $\bar{X}$ be a projective smooth surface and let
$D=\sum_{i=1}^{N}D_{i}\subset\bar{X}$ be a reduced simple normal crossing
divisor with the $D_{i}$ being the irreducible components of $D$ and denote by
$w_{i}=D_{i}\cdot D_{i}$ the self-intersection number of $D_{i}$. Let $\Delta$
be the unoriented graph where the vertices
$V(\Delta):=\\{v_{1},\ldots,v_{N}\\}$ are in correspondence with the
irreducible components $D_{i}$ of $D$ and the edges $\mathcal{E}(\Delta)$
correspond with the intersection of the irreducible components of $D$, this
is, there is an unoriented edge joining $v_{i}$ and $v_{j}$ for each point in
$D_{i}\cap D_{j}$. Denote by $X:=\bar{X}\setminus D$.

We want to define some partial compactifications of $X$. The idea goes as
follows: we choose a subset of irreducible components of $D$ indexed by $I$
which are not to be removed from $\bar{X}$, we then select a subset $P$ of
points in $\operatorname{Sing}\sum_{i\not\in I}D_{i}$ to be blown-up and
remove the _strict_ transform of $\sum_{i\not\in I}D_{i}$ in
$\operatorname{Bl}_{P}\bar{X}$.

More precisely, let $I\subset\\{1,\ldots,N\\}$,
$P=\\{p_{1},\ldots,p_{s_{1}}\\}\subset\operatorname{Sing}(\sum_{i\not\in
I}D_{i})$ and denote by $\pi:\operatorname{Bl}_{P}\bar{X}\to\bar{X}$ the
composite of the blow-ups at the points in $P$. Denote by
$\pi^{*}D=\sum_{i=1}^{N+s_{1}}D_{i}^{\prime}$ the total transform of the
divisor $D$ in $\operatorname{Bl}_{P}\bar{X}$, suppose that for
$i=1,\ldots,N$, we have that $D_{i}^{\prime}$ is a strict transform of $D_{i}$
and for $j=1,\ldots,s_{1}$, the $D_{N+j}^{\prime}$ are exceptional divisors.
Define the divisor

$D^{\prime}(I,P)=\pi^{*}D-\sum_{i\in
I}D_{i}^{\prime}-\sum_{N<j}D_{j}^{\prime}.$

Note that $\operatorname{Bl}_{P}\bar{X}\setminus\pi^{*}D\hookrightarrow
X^{\prime}(I,P):=\operatorname{Bl}_{P}\bar{X}\setminus D^{\prime}(I,P)$. By
restricting $\pi$, we obtain an isomorphism
$\operatorname{Bl}_{P}\bar{X}\setminus\pi^{*}D\overset{\sim}{\to}X$. We call
$X^{\prime}(I,P)$ a _partial compactification_ of $X=\bar{X}\setminus D$. By
Proposition 2.1, the induced homomorphism
$\pi_{1}(X)\to\pi_{1}(X^{\prime}(P,I))$ is surjective.

We comment on the effects of this construction in the dual graph. Denote by
$\Delta^{\prime}(I,P)$ the dual graph of $D^{\prime}(I,P)$. It is obtained
from $\Delta$ by deleting the following vertices and edges: for the set $I$,
we have a subset $V(I)\subset V(\Delta)$ of vertices corresponding to the
lines $D_{i}$ for $i\in I$, remove these vertices from $\Delta$, together with
all edges in $\mathcal{E}(\Delta)$ having an endpoint in $V(I)$. We also
remove the edges corresponding to $P$: let $p_{j}\in P$, there exists
$j_{1},j_{2}\in\\{1,\ldots,N\\}$ such that $p_{j}=D_{j_{1}}\cap D_{j_{2}}$. In
the dual graph of $\pi^{*}D$ the edge corresponding to $p_{j}$ in $\Delta$ has
been divided in two, with a vertex in between corresponding to the exceptional
divisor coming from $p_{j}$.

####  2.3.1. Partial compactifications for an arrangement of lines

We can carry the above construction for a divisor $D\subset\bar{X}$ coming
from an arrangement of lines
$\mathscr{A}=\\{L_{1},\ldots,L_{n+1}\\}\subset\mathbb{P}^{2}$. In fact, this
will be the only case we will be interested in.

Let $\mathscr{A}\subset\mathbb{P}^{2}$ be an arrangement of lines. Denote by
$P_{0}:=\\{p_{1},\ldots,p_{s_{0}}\\}\subset\operatorname{Sing}\mathscr{A}$ the
points with multiplicity strictly bigger that $2$. Define
$\pi:\bar{X}:=\operatorname{Bl}_{P_{0}}\mathbb{P}^{2}\to\mathbb{P}^{2}$ and
denote by $D=\left|\pi^{*}\mathscr{A}\right|=\sum_{i=1}^{n+1+s_{0}}D_{i}$ the
reduced total transform of $\mathscr{A}$ in $\bar{X}$. Note that $D$ is simple
normal crossing. For a divisor $D$ where the irreducible components are smooth
rational curves, the set of edges $\mathcal{E}(\Delta)$ of the dual graph
$\Delta$ can be described as
$\mathcal{E}(\Delta)=\\{(i,j)\in\\{1,\ldots,n+1+s_{0}\\}^{2}\mid D_{i}\cap
D_{j}\not=\varnothing,i<j\\}$ once the irreducible components of $D$ are
numbered. We assume that $D_{i}$ is the strict transform of $L_{i}$.

Let $I\subset\\{1,\ldots,N=n+1+s_{0}\\}$ and
$P=\\{p_{1}^{\prime},\ldots,p_{s_{1}}^{\prime}\\}\subset\operatorname{Sing}\sum_{i\not\in
I}D_{i}$. Consider $\pi^{\prime}:\operatorname{Bl}_{P}\bar{X}\to\bar{X}$ and
let $D^{\prime}=\pi^{\prime*}(D)-\sum_{i\in
I}D_{i}^{\prime}-\sum_{N<j}D_{j}^{\prime}$ as above. We write
$M(\mathscr{A},I,P):=X^{\prime}(I,P)=\operatorname{Bl}_{P}\bar{X}\setminus
D^{\prime}$ for a _partial compactification_ of the complement of an
arrangement $M(\mathscr{A})=\mathbb{P}^{2}\setminus\mathscr{A}$.

We can iterate this construction in the following way; consider a sequence of
blow-ups:

$\operatorname{Bl}_{P_{k},\ldots,P_{1}}\bar{X}\overset{\pi^{(k)}}{\to}\operatorname{Bl}_{P_{k-1},\ldots,P_{1}}\bar{X}\overset{\pi^{(k-1)}}{\to}\ldots\overset{\pi^{(2)}}{\to}\operatorname{Bl}_{P_{1}}\bar{X}\overset{\pi^{(1)}}{\to}\bar{X}\overset{\pi^{(0)}}{\to}\mathbb{P}^{2}$

with
$P_{l}\subset\operatorname{Sing}((\pi^{(0)}\circ\pi^{(1)}\circ\cdots\circ\pi^{(l-1)})^{*}\mathscr{A})$
for $l=1,\ldots,k$ and
$\pi^{(l)}:\operatorname{Bl}_{P_{l},\ldots,P_{1}}\bar{X}\to\operatorname{Bl}_{P_{l-1},\ldots,P_{1}}\bar{X}$
denoting the blow-up of $\operatorname{Bl}_{P_{l-1},\ldots,P_{1}}\bar{X}$ at
$P_{l}$. We can suppose that the irreducible components of the reduced divisor

$D^{\prime}:=\left|(\pi^{(0)}\circ\cdots\circ\pi^{(k)})^{*}\mathscr{A}\right|=\sum_{1}^{\left|\mathscr{A}\right|}D_{i}^{\prime}+\sum_{\left|\mathscr{A}\right|}^{\left|\mathscr{A}\right|+\left|P_{0}\right|}D_{j}^{\prime}+\ldots+\sum_{\left|\mathscr{A}\right|+\left|P_{0}\right|+\ldots+\left|P_{k-1}\right|}^{\left|\mathscr{A}\right|+\left|P_{0}\right|+\ldots+\left|P_{k}\right|}D_{l}^{\prime},$

where $\left|P\right|$ denotes the cardinality of the set $P$, are ordered in
such a way that $\pi^{(l)}\circ\cdots\circ\pi^{(k)}$ contracts the curves
$D_{i}^{\prime}$ with $i>\left|\mathscr{A}\right|+\ldots+\left|P_{l-1}\right|$
for $l=1,\ldots,k$. Let
$I\subset\\{1,\ldots,\left|\mathscr{A}\right|+\ldots+\left|P_{k}\right|\\}$
and define
$M(\mathscr{A},I,P_{1},\ldots,P_{k}):=\operatorname{Bl}_{P_{k},\ldots,P_{1}}\bar{X}\setminus
D^{\prime}-\sum_{i\in I}D_{i}^{\prime}$ as an _iterated partial
compactification_ of $M(\mathscr{A})$.

######  Lemma 2.2.

Let $(\bar{X}^{\prime},D^{\prime})$ be a smooth projective surface such that

  1. (1)

the divisor $D^{\prime}$ is a simple normal crossing divisor,

  2. (2)

there is a birational morphism $\bar{X}^{\prime}\overset{\psi}{\to}\bar{X}$,

  3. (3)

we have that $\psi^{*}D\supset D^{\prime}$

then there exists an iterated partial compactification
$(\bar{X}^{\prime\prime},D^{\prime\prime})$ and a proper birational morphism
$\bar{X}^{\prime}\overset{\psi^{\prime\prime}}{\to}\bar{X}^{\prime\prime}$
such that $(\psi^{\prime\prime})^{-1}D^{\prime\prime}\supset D^{\prime}$ and
$\pi_{1}(\bar{X}^{\prime\prime}\setminus
D^{\prime\prime})\overset{\sim}{\leftarrow}\pi_{1}(\bar{X}^{\prime}\setminus
D^{\prime})$ is an isomorphism.

Here we will restrict the study to $M(\mathscr{A},I,P)$ unless otherwise
stated. The results are easily generalized to the above more general setting
of iterated partial compactifications.

######  Remark 2.3.

We have that $\pi_{1}(\bar{X}^{\prime}\setminus D^{\prime})$ is a quotient
group of $\pi_{1}(M(\mathscr{A}))$ by proposition 2.1.

###  2.4. Boundary manifolds

Let $\bar{X}$ be a projective smooth surface and
$D=\sum_{i=1}^{k}D_{i}\subset\bar{X}$ be a connected divisor. We can construct
a regular tubular neighborhood $U$ of $D$ in $\bar{X}$ which comes with a
surjective continuous retraction $\varphi:U\to D$ such that
$\varphi|_{D}=id_{D}$. The boundary $\partial U$ of $U$ is an oriented,
connected, closed $3$-manifold (see [Mum61]). We call the $3$-manifold
$\partial U$ the _boundary manifold_ of $D$ and denote by $\psi:\partial U\to
D$ the restriction of $\varphi$ to $\partial U$.

Suppose now that $(X,D)$ is simple normal crossing and assume that:

  * •

the divisor $D$ is connected,

  * •

the irreducible components $D_{i}$ of $D$ are rational curves ,

  * •

the dual graph of $D$ has no cycles, in particular $\\#D_{i}\cap D_{j}=0\text{
or }1\text{ if }i\not=j$. This dual graph is a tree that we denote by
$\mathcal{T}$.

For such a pair, a presentation of $\pi_{1}(\partial U)$ is given in [Mum61,
p. 235] (See also [Hir64]). As we shall need the notations, let us describe
it.

Fix a base point $Q_{i}\in D_{i}\setminus\cup_{i\not=m}D_{m}$ in every
rational curve $i=1,\ldots,k$. Denote by $P_{im}^{\prime}$ the unique point in
$D_{i}\cap D_{m}$, if any. Select a simple contractible oriented curve
$l_{i}\subset D_{i}$ containing $Q_{i}$ and passing through every point
$P_{im}^{\prime}\in D_{i}$ as in figure 1(a) and denote by $l=\cup
l_{i}\subset D$. We can construct a continuous map $h:l\to\partial U$ such
that $\psi\circ h|_{l_{i}}=id_{l_{i}}$ and $h(l_{i})\cap
h(l_{m})\not=\varnothing$ if $P_{im}^{\prime}=D_{i}\cap D_{m}=l_{i}\cap
l_{m}\not=\varnothing$.

$\partial\Delta(P_{i{1}})$$\partial\Delta(P_{i2})$$\partial\Delta(P_{i{k_{1}}})$$P_{i{1}}$$P_{i2}$$P_{i{k_{i}}}$$Q_{i}$$l_{i}$$D_{i}$••••
(a) Mumford generators

$P_{i{1}}$$P_{i2}$$P_{i{k_{1}}}$$Q_{i}$$l_{i}^{\prime}$$\beta_{i1}^{\prime}$$D_{i}$••••
(b) Paths in $D_{i}^{*}$

Figure 1. Generators

It is easy to see that $l$ is a homeomorphic image of a tree and deformation
retracts to a point.

Label the points $P_{im}^{\prime}\in D_{i}$ by the order they intersect
$l_{i}$ as $P_{i1},\ldots,P_{i{k_{i}}}$, see Figure 1(a). Denote by
$\psi_{i}:\partial U_{i}\to D_{i}$ the boundary manifold of $D_{i}$. Let
$D_{i}^{*}=D_{i}\setminus\cup_{m=1}^{k_{i}}\Delta(P_{im})$ with
$\Delta(P_{im})$ a small open disk around $P_{im}$ in $D_{i}$. Define
$\partial U_{i}^{*}:=\psi_{i}^{-1}(D_{i}^{*})$. We may suppose that $\partial
U\cap\partial U_{i}=\partial U_{i}^{*}$.

We may also assume that $Q_{i}\in D_{i}^{*}$. Define another contractible path
$l_{i}^{\prime}\subset D_{i}^{*}$ as follows: join every two connected
components of $l_{i}\cap D_{i}^{*}$ touching the boundary of a disk
$\partial\Delta(P_{im})$, by the segment of $\partial\Delta(P_{im})$ that
connects these two points when traveled in the natural orientation, see figure
1(b). We assume $l_{i}$ and $\partial\Delta(P_{im})$ intersect transversally
at two points for all $m=1,\ldots,k_{i}-1$.

Consider the circle $\partial\Delta(P_{im})$ traveled in the natural
orientation and connect it to $Q_{i}$ via a segment of $l_{i}^{\prime}$. We
obtain a path $\beta_{im}^{\prime}\in\pi_{1}(D_{i}^{*},Q_{i})$, for
$m=1,\ldots,k_{i}$, see figure 1(b). Note that
$\beta_{i1}^{\prime}\cdots\beta_{ik_{i}}^{\prime}=1$ in $\pi_{1}(D_{i}^{*})$.

We can construct continuous maps
$h_{i}:\cup_{m=1}^{k_{i}}\beta_{im}^{\prime}\to\partial U_{i}^{*}$ such that
$\psi_{i}\circ h_{i}|_{\beta_{im}^{\prime}}=id_{\beta_{im}^{\prime}}$ for
every $i=1,\ldots,k$. Let $h_{i}(Q_{i})$ be a base point in $\partial
U_{i}^{*}$, denote by $\gamma_{im}^{\prime}=h_{i}(\beta_{im}^{\prime})$ and
let $\gamma_{i}^{\prime}$ be a fiber $S^{1}$ at $Q_{i}$ of $\partial
U_{i}^{*}$ traveled in the natural orientation.

By using the long homotopy sequence of a fiber bundle, Mumford obtained the
following presentation in [Mum61]. See also [Hir64].

######  Lemma 2.4.

[Mum61, p. 236-237] The fundamental group of $\partial U_{i}^{*}$ is given by
the following presentation

(1)
$\left\langle\gamma_{i1}^{\prime},\ldots,\gamma_{i{k_{i}}}^{\prime},\gamma_{i}^{\prime}\mid[\gamma_{im}^{\prime},\gamma_{i}^{\prime}]\
m=1,\ldots,k_{i},\gamma_{i}^{\prime-
w_{i}}=\gamma_{i1}^{\prime}\cdots\gamma_{ik_{i}}^{\prime}\right\rangle$

with $w_{i}=D_{i}\cdot D_{i}$ the self-intersection number of $D_{i}$.

######  Remark 2.5.

Note that $\partial U_{i}^{*}$ is non canonically homeomorphic to the trivial
bundle $S^{1}\times D_{i}^{*}$, but the image of the paths
$\gamma_{im}^{\prime}$ are not longer identified with a path freely homotopic
to one of the form $\\{\text{point}\\}\times\partial\Delta(P_{i_{m}})$. In
fact, we need to twist this image by a multiple of $\gamma_{i}^{\prime}$ for
it to be of such form. See [Mum61, p. 235].

Now, to globalize this construction to $\partial U$, we can use
$h(l)\subset\partial U$ as a skeleton to define paths generating
$\pi_{1}(\partial U)$. Let $\gamma_{i}$ be the loop based at $h(Q_{1})$
constructed as follows. Join $h(Q_{1})$ to $h(Q_{i})$ by a segment $\lambda$
of $h(l)$, follow $\gamma_{i}^{\prime}$ and come back by $\lambda^{-1}$. Then
it is homotopic to the canonical representative of $\gamma_{i}^{\prime}$ in
$\pi_{1}(\partial U_{i}^{*}\cup h(l),h(Q_{1}))$ using the natural isomorphism
$\pi_{1}(\partial U_{i}^{*}\cup h(l),h(Q_{1}))\to\pi_{1}(\partial
U_{i}^{*},Q_{i})$ thus obtained. Define similarly $\gamma_{im}$ for $1\leq
m\leq k_{i}$. Then $\gamma_{im}=\gamma_{j_{\mathcal{T}}(i,m)}$ for some
injective map $m\mapsto j_{\mathcal{T}}(i,m)$ from $\\{1,\ldots,k_{i}\\}$ to
$\\{1,\ldots,k\\}$.

By gluing the $\partial U_{i}^{*}$ together and by using van Kampen theorem,
Mumford obtained the following presentation for $\pi_{1}(\partial U)$.

######  Theorem 2.6 ([Mum61]).

With the notations and assumptions as above, a presentation for
$\pi_{1}(\partial U)$ is given by:

$\pi_{1}(\partial
U)=\left\langle\gamma_{1},\ldots,\gamma_{k}\mid[\gamma_{i},\gamma_{j_{\mathcal{T}}(i,m)}],\
m=1,\ldots,k_{i},\gamma_{i}^{-w_{i}}=\prod_{m=1}^{k_{i}}\gamma_{j_{\mathcal{T}}(i,m)},1\leq
i\leq k\right\rangle.$

where $w_{i}=D_{i}\cdot D_{i}$, $[a,b]=aba^{-1}b^{-1}$, $a^{0}=1$, the
identity of the group.

##  3\. Wiring diagrams and a first presentation of the fundamental group of a
partial compactification

We will describe the construction of a diagram permitting to express some
meridians around different points in the lines of $\mathscr{A}$, lying in a
pencil of lines passing through a base point
$R\in\mathbb{P}^{2}\setminus\mathscr{A}$, in terms of a fixed set of meridians
lying in a fixed fiber of the pencil.

As an application we obtain a first presentation for the fundamental group of
a partial compactification $M(\mathscr{A},I,P)$. To do that we will use a
modification of the presentation of the fundamental group of $M(\mathscr{A})$
given in [Arv92] and [CS97].

This diagram will also carry the information to compute the image of the
cycles in the boundary manifolds of $\mathscr{A}$ into $M(\mathscr{A})$. This
will be done in section 4

###  3.1. Wiring diagram associated to a complex arrangement

Consider an arrangement of lines $\mathscr{A}$ in $\mathbb{P}^{2}$. Let us fix
a base point $R\in\mathbb{P}^{2}\setminus\mathscr{A}$ and denote by
$\pi_{R}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{2}$ the blow-up at
$R$. Let $\bar{f}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$ be the
morphism defined by the pencil of lines passing through $R$. In what follows,
we assume that we have chosen $R$ in such a way that
$\bar{f}|_{\operatorname{Sing}\mathscr{A}}:\operatorname{Sing}\mathscr{A}\to\mathbb{P}^{1}$
is injective.

Let $\ast\in\mathbb{P}^{1}$, consider a simple piece-wise linear path
$\beta:([0,1],0)\to(\mathbb{P}^{1},\ast)$ starting at $\ast$ and passing
through every point $\bar{f}(p)$ for all $p\in\operatorname{Sing}\mathscr{A}$,
being locally linear around these points.

By abuse of notation let us denote by $\mathscr{A}$ the union of the lines of
arrangement in $\mathbb{P}^{2}$.

######  Definition 3.1.

The _wiring diagram_ of $\mathscr{A}$ with respect to $\beta$ is
$\mathcal{W}=\bigcup_{t\in[0,1]}(\mathscr{A}\cap\bar{f}^{-1}{(\beta(t)}))\subset\operatorname{Bl}_{R}\mathbb{P}^{2}$.
The $i$-wire $W_{i}$ is $L_{i}\cap\mathcal{W}$. Here, we view
$\mathscr{A},L_{i}$ as subvarieties of $\operatorname{Bl}_{R}\mathbb{P}^{2}$
since $R\not\in\mathscr{A}$.

By the choice of $\beta$, as it passes through the points $\bar{f}(p)$ for
$p\in\operatorname{Sing}\mathscr{A}$, we have that
$\operatorname{Sing}\mathscr{A}\subset\mathcal{W}$.

######  Lemma 3.1.

Every wire is a piece-wise linear simple curve.

###### Proof.

As no line in $\mathscr{A}$ passes through $R$, every $L_{i}\in\mathscr{A}$
induces a section of $\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$
which is in fact an isomorphism. By the choice of $\beta$ the result follows.
∎

####  3.1.1. Planar representation of the wiring diagram

By considering the pullback $\beta^{\ast}(\mathcal{W})$ and a trivialization
$\beta^{*}\operatorname{Bl}_{R}\mathbb{P}^{2}\cong[0,1]\times\mathbb{P}^{1}$,
we can view $\beta^{*}(\mathcal{W})$ as a closed graph embedded in
$[0,1]\times\mathbb{P}^{1}$. Sometimes we will continuing writing
$\mathcal{W}$ for $\beta^{*}(\mathcal{W})$. Moreover we can remove the
exceptional divisor $\pi_{R}^{-1}(R)$ from $[0,1]\times\mathbb{P}^{1}$ and we
can view $\mathcal{W}$ as a closed graph embedded in $[0,1]\times\mathbb{C}$
via a piece-wise linear isomorphism.

There exists a complex coordinate $z$ in $\mathbb{C}$ such that the projection
$(p:[0,1]\times\mathbb{C}\to[0,1]\times\mathbb{R},(t,z)\mapsto(t,\Re(z)))$ is
generic, in the sense that the extra crossings in $p(\mathcal{W})$ arise as
transversal intersection of only two wires $p(W_{i})$ and $p(W_{k})$ for
certain $t\in[0,1]$ and wires $W_{i},W_{k}$ that do not intersect in
$\bar{f}^{-1}{(\beta(t))}$. We call these crossings virtual vertices. We
obtain a planar diagram which can be represented as in the figure 2.

We assume that the order of the lines $L_{1},\ldots,L_{n}$ is such that, at
the very right of the planar representation of $\mathcal{W}$, the wire $W_{1}$
is at the bottom of $\mathcal{W}$, above it is the wire $W_{2}$ and then
$W_{3}$, continuing in this way until $W_{n}$.

######  Definition 3.2.

Consider coordinates $(t,x,y)$ in $\mathbb{R}^{3}$. We say that a wire $W_{i}$
passes above $W_{k}$ at a point $t^{\prime}\in[0,1]$ if
$(t^{\prime},x,y_{i})\in W_{i},(t^{\prime},x,y_{k})\in W_{k}$ and
$y_{i}<y_{k}$.

In order to distinguish the virtual vertices arising in the projection we mark
the projection $p(W_{i})\cap p(W_{k})$ to indicate if the wires over or under
crossed in $\beta^{\ast}\mathcal{W}$ as in Figure 5. We call the first a
positive braiding (or positive virtual vertex) and the second a negative
braiding (or negative virtual vertex).

######  Remark 3.2.

As in the [CS97], we read the wiring diagram from right to left.

$W_{1}$$W_{1}$$W_{2}$$W_{2}$$W_{3}$$W_{3}$$W_{4}$$W_{4}$$W_{5}$$W_{5}$$\downarrow$$1$$0$
Figure 2. Wiring Diagram

######  Example 3.1.

Let $(z_{1}:z_{2}:z_{3})$ be homogeneous coordinates of $\mathbb{P}^{2}$.
Consider the arrangement consisting of two transverse pairs of parallel lines
in $\mathbb{C}^{2}\cong\mathbb{P}^{2}\setminus\\{z_{3}=0\\}$, defined by the
equation
$(z_{2}-z_{1})(z_{2}-z_{1}+z_{3})(z_{2}+z_{1})(z_{2}+z_{1}-z_{3})z_{3}=0$. The
wiring diagram associated to this arrangement is shown in Figure 2. There are
no virtual vertices since the arrangement is real and $\beta$ is a real
segment.

######  Remark 3.3.

When no under or overcrossing is marked in a wiring diagram $\mathcal{W}$, it
coincides with the notion of wiring diagram in [BLVS+99]. They are in
correspondence with arrangement of ”pseudo-lines”, in particular there exists
a wiring diagram of $9$ wires that does not comes from an arrangement of lines
(the so called non-Pappus arrangement, see [BLVS+99, Proposition 8.3.1]),
however for $8$-wires or less they are in correspondence with the real
arrangement of lines [BLVS+99, Thm 6.3.1].

###  3.2. Using the diagram to obtain presentations

####  3.2.1. Algorithm for computing a presentation of the fundamental group
of $M(\mathscr{A})$

We will use the following well-known Lemma.

######  Lemma 3.4.

Let $Z\subset X$ be an algebraic subvariety of an algebraic smooth surface
$X$. Fix a point $R\in X\setminus Z$. Denote by
$\pi_{R}:\operatorname{Bl}_{R}X\to X$, then $\pi_{1}(X\setminus
Z)\cong\pi_{1}(\operatorname{Bl}_{R}X\setminus\pi_{R}^{*}Z)$.

This allow us to compute $\pi_{1}(M(\mathscr{A}))$ in the total space of the
fiber bundle $\bar{f}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$.
We will find suitable subspaces of the total space of this fiber bundle to
apply the van-Kampen Theorem.

Let $\mathcal{W}\subset\operatorname{Bl}_{R}\mathbb{P}^{2}$ be a wiring
diagram. Let $\beta^{*}(\mathcal{W})\subset[0,1]\times\mathbb{P}^{1}$ be as in
3.1.1. Every vertical line $t\times\mathbb{P}^{1}$ in
$[0,1]\times\mathbb{P}^{1}$ corresponds to the fiber
$\bar{f}^{-1}{(\beta(t))}$. Recall that if $p\in\bar{f}^{-1}{(\beta(t_{p}))}$
for $p\in\operatorname{Sing}\mathscr{A}$ and $t_{p}\in[0,1]$, then no other
point in $\operatorname{Sing}\mathscr{A}$ lies in the same fiber. Suppose that
there are $s$ points $t_{p_{1}},\ldots,t_{p_{s}}$ corresponding to
$p_{1},\ldots,p_{s}$ in $\operatorname{Sing}\mathscr{A}$.

By fixing a planar representation $p(\beta^{*}(\mathcal{W}))$ of
$\beta^{*}(\mathcal{W})$ as in 3.1.1, some under or over-crossing can arise.
As the projection is generic, they correspond to a finite number
$t_{1}^{\prime},\ldots,t_{\nu}^{\prime}$ of elements of $[0,1]$ distinct from
the $t_{p_{r}}$.

Order the set
$\\{t_{p_{1}},\ldots,t_{p_{s}},t_{1}^{\prime},\ldots,t_{\nu}^{\prime}\\}$ by
increasing order and relabel them by $t_{\kappa}$ for $\kappa=1,\ldots,\nu+s$.
Let $B_{\kappa}\subset\mathbb{P}^{1}$ be a neighborhood of $\beta(t_{\kappa})$
homeomorphic to a disk in $\mathbb{C}$ such that $B_{\kappa}\cap
B_{j}=\varnothing$ if $\left|\kappa-j\right|>1$ and $B_{\kappa}\cap
B_{\kappa+1}$ is homeomorphic to a disk. Consider
$M_{\kappa}:=\bar{f}^{-1}(B_{\kappa})\subset\operatorname{Bl}_{R}(\mathbb{P}^{2})$
for $\kappa=1,\ldots,\nu+s$ and denote by
$M_{\kappa}(\mathscr{A}):=M_{\kappa}\setminus
M_{\kappa}\cap\pi_{R}^{*}\mathscr{A}$.

######  Lemma 3.5.

We have that

$\pi_{1}(M_{\kappa}(\mathscr{A})\cap M_{\kappa+1}(\mathscr{A}))\cong
F_{n}\text{ for }\kappa=1,\ldots,\nu+s-1$

with $F_{n}$ the free group in $n$ generators.

###### Proof.

First note that as $B_{\kappa}\cap
B_{\kappa+1}\subset\mathbb{P}^{1}\setminus\\{\bar{f}(p)\mid
p\in\operatorname{Sing}\mathscr{A}\\}$ we have that
$M_{\kappa}(\mathscr{A})\cap
M_{\kappa+1}(\mathscr{A})=\bar{f}^{-1}(B_{\kappa}\cap B_{\kappa+1})$ is the
restriction of a fiber bundle to a contractible base. The fundamental group of
any fiber in $B_{\kappa}\cap B_{\kappa+1}$ is a free group in $n$ generators.
∎

######  Proposition 3.6.

We have that

$\pi_{1}(M(\mathscr{A}))\cong\pi_{1}(M_{1}(\mathscr{A}))\underset{\pi_{1}(M_{1}(\mathscr{A})\cap
M_{2}(\mathscr{A}))}{*}\cdots\underset{\pi_{1}(M_{\nu+s-1}(\mathscr{A})\cap
M_{\nu+s}(\mathscr{A}))}{*}\pi_{1}(M_{\nu+s}(\mathscr{A})).$

###### Proof.

By Lemma 3.4, we have that the morphism
$\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}\to
M(\mathscr{A})=\mathbb{P}^{2}\setminus\mathscr{A}$ induces an isomorphism in
the fundamental groups.

Denote the restriction of $\bar{f}$ to
$\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}$ by
$f:\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}\to\mathbb{P}^{1}$.
Let $\infty\in\mathbb{P}^{1}\setminus\cup_{\kappa=1}^{\nu+s}B_{\kappa}$ and
note that $f^{-1}(\mathbb{P}^{1}\setminus\\{\infty\\})$ is the complement in
$\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}$ of a
smooth irreducible divisor $D_{\infty}$ that is the restriction to
$\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}$ of the
strict transform of a line in $\mathbb{P}^{2}$ passing through $R$ .

By Proposition 2.1, we have that

$\pi_{1}(\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A})=\pi_{1}\left(f^{-1}(\mathbb{P}^{1}\setminus\\{\infty\\})\right)/\langle\langle\gamma_{D_{\infty}}\rangle\rangle$

where $\gamma_{D_{\infty}}$ is a meridian around $D_{\infty}$.

Note that, as $R\in\mathbb{P}^{2}\setminus\mathscr{A}$, we have that
$\pi_{R}^{-1}(R)\subset\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\pi_{R}^{*}\mathscr{A}$
and its restriction to $f^{-1}(\mathbb{P}^{1}\setminus\\{\infty\\})$ is
isomorphic to $\mathbb{C}$. The meridian $\gamma_{D_{\infty}}$ can be chosen
to lie inside this restriction and therefore $\gamma_{D_{\infty}}=1$. We
obtain that
$\pi_{1}(f^{-1}(\mathbb{P}^{1}\setminus\\{\infty\\}))\cong\pi_{1}(M(\mathscr{A}))$.

Observe that $\cup_{\kappa=1}^{\nu+s}M_{\kappa}(\mathscr{A})$ has the same
homotopy as
$(\operatorname{Bl}_{R}\mathbb{P}^{2}\setminus\mathscr{A})\setminus
f^{-1}(\infty)$. We conclude by successive applications of the van-Kampen
Theorem: by construction $B_{1}\cap B_{\nu+s}=\varnothing$, we obtain that
$\pi_{1}(\cup_{\kappa=1}^{\nu+s}M_{\kappa}(\mathscr{A}))$ is isomorphic to

$\pi_{1}(M_{1}(\mathscr{A}))\underset{\pi_{1}(M_{1}(\mathscr{A})\cap
M_{2}(\mathscr{A}))}{*}\cdots\underset{\pi_{1}(M_{\nu+s-1}(\mathscr{A})\cap
M_{\nu+s}(\mathscr{A}))}{*}\pi_{1}(M_{\nu+s}(\mathscr{A}))$

∎

We want to compute now $\pi_{1}(M_{\kappa}(\mathscr{A}))$ for
$\kappa=1,\ldots,\nu+s$ and the morphisms of amalgamation
$\pi_{1}(M_{\kappa}(\mathscr{A}))\leftarrow\pi_{1}(M_{\kappa}(\mathscr{A})\cap
M_{\kappa+1}(\mathscr{A}))\to\pi_{1}(M_{\kappa+1}(\mathscr{A}))$. In fact, if
no point of $\operatorname{Sing}\mathscr{A}$ lies in $M_{\kappa}(\mathscr{A})$
we will have that $\pi_{1}(M_{\kappa}(\mathscr{A}))\cong F_{n}$. However, some
conjugations may arise in the meridians due to braiding of the wires in
$\mathcal{W}$.

We have to distinguish $3$ cases depending in the nature of $M_{\kappa}$:
$M_{\kappa}$ contains a point of $\operatorname{Sing}\mathscr{A}$, it contains
a _positive braiding_ of $\mathcal{W}$ or it contains a _negative braiding_.

Let $\theta_{\kappa}<t_{\kappa}$ be sufficiently close so that
$\beta(\theta_{\kappa})\in B_{\kappa}\cap B_{\kappa-1}$ and denote by
$x_{1}^{(\kappa)},\ldots,x_{n+1}^{(\kappa)}$ the set of points in the planar
representation $p(\beta^{*}(\mathcal{W}))$ of the wiring diagram $\mathcal{W}$
labeled from bottom to top corresponding to the points in
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})\cap\mathscr{A}$.

######  Definition 3.3.

A _geometric generating set_
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
of the group
$\pi_{1}(\bar{f}^{-1}{(\beta(\theta_{\kappa})})\setminus(\bar{f}^{-1}{(\beta(\theta_{\kappa})})\cap\mathscr{A}),q_{\kappa})$
with $q_{\kappa}=\pi_{R}^{-1}(R)\cap\bar{f}^{-1}{(\beta(\theta_{\kappa})})$ is
the datum of $\lambda_{1}^{(\kappa)},\ldots,$ $\lambda_{n+1}^{(\kappa)}$
meridians around $x_{1}^{(\kappa)},\ldots,x_{n+1}^{(\kappa)}$ respectively,
all of them based at $q_{\kappa}$ such that
$\lambda_{n+1}^{(\kappa)}\cdots\lambda_{1}^{(\kappa)}$ is nullhomotopic in
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})\setminus\\{x_{1}^{(\kappa)},\ldots,x_{n+1}^{(\kappa)}\\}\cong\mathbb{P}^{1}\setminus\\{(n+1)-\text{points}\\}$.

######  Remark 3.7.

A geometric generating set
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
induces a geometric base
${\Gamma^{(\kappa)}}^{\prime}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n}^{(\kappa)}\\}$
of
$\pi_{1}(\mathbb{C}\setminus\\{x_{1}^{(\kappa)},\ldots,x_{n}^{(\kappa)}\\},q_{\kappa})$.

We consider here the geometric generating set
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
as in figure 3(a). As $\pi_{1}(\pi_{R}^{-1}(R))$ is trivial, we can fix a
point $q\in\pi_{R}^{-1}(R)$ as a global base point for all the geometric
generating set $\Gamma^{(\kappa)}$ with $\kappa=1,\ldots,\nu+s$ by joining
$q_{\kappa}$ to $q$ by a simple path in $\pi_{R}^{-1}(R)$.

$\lambda_{j}^{(\kappa)}$$\lambda_{j+1}^{(\kappa)}$$\lambda_{m}^{(\kappa)}$•••
(a) $\Gamma^{(\kappa)}$ in $\bar{f}^{-1}{(\beta(\theta_{\kappa})})$

$\lambda_{m}^{(\kappa)}$$\lambda_{j+1}^{(\kappa)}$$\lambda_{j}^{(\kappa)}$•••
(b) $\Gamma^{(\kappa)}$ in $\bar{f}^{-1}{(\beta(\theta_{\kappa+1})})$ .

Figure 3. Geometric generating set in different fibers

We describe how the meridians change when we move the generators of
$\Gamma^{(\kappa)}$ to the fiber $\bar{f}^{-1}{(\beta(\theta_{\kappa+1})})$
and express them in the generators $\Gamma^{(\kappa+1)}$, see figure 3(b). We
record as well the relations arising in between.

Suppose that $p\in\operatorname{Sing}\mathscr{A}\cap M_{\kappa}$, and let
$\Gamma^{(\kappa)}$ be as above. Denote by $j$ the first index of the
meridians of $\Gamma^{(\kappa)}$ corresponding to a line passing through $p$,
and by $m$ the last. We have that
$\lambda_{k}^{(\kappa+1)}=\lambda_{k}^{(\kappa)}$ for $k<j$ and $k>m$ as we
can deform continuously $\lambda_{k}^{(\kappa)}$ to $\lambda_{k}^{(i+1)}$
having the same homotopy type in $\pi_{1}(M_{\kappa}(\mathscr{A}))$.

Let
$R_{\kappa}=[\lambda_{m}^{(\kappa)},\lambda_{m-1}^{(\kappa)},\ldots,\lambda_{j}^{(\kappa)}]$
denote the set of equations of the form
$\lambda_{m}^{(\kappa)}\cdot\lambda_{m-1}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}=\lambda_{\sigma(m)}^{(\kappa)}\lambda_{\sigma(m-1)}^{(\kappa)}\cdots\lambda_{\sigma(j)}^{(\kappa)}$
where $\sigma$ varies in the set of cyclic permutations in $m-j+1$ elements.

$p$$a$$a$$b^{a}$$b$$c^{ba}$$c$$d^{cba}$$d$$\Gamma^{(\kappa)}$$\Gamma^{(\kappa+1)}$
Figure 4. Actual vertex

######  Lemma 3.8.

Let $p\in\operatorname{Sing}\mathscr{A}\cap M_{\kappa}$. Then
$\pi_{1}(M_{\kappa}(\mathscr{A}),q_{\kappa})$ is generated by the elements of
$\Gamma^{(\kappa)}$ and $\Gamma^{(\kappa+1)}$ together with the relations
$R_{\kappa}$, $\lambda_{k}^{(\kappa+1)}=\lambda_{k}^{(\kappa)}$ for $k<j$ or
$m<k$, $\lambda_{n+1}^{(\kappa)}\cdots\lambda_{1}^{(\kappa)}=1$ and

$\displaystyle\lambda_{m}^{(\kappa+1)}=\lambda_{j}^{(\kappa)},$
$\displaystyle\lambda_{m-1}^{(\kappa+1)}={\lambda_{j+1}^{(\kappa)}}^{{\lambda_{j}}^{(\kappa)}},$
$\displaystyle\lambda_{m-2}^{(\kappa+1)}={\lambda_{j+2}^{(\kappa)}}^{{\lambda_{j+1}}^{(\kappa)}{\lambda_{j}}^{(\kappa)}},$
$\displaystyle\vdots$
$\displaystyle\lambda_{j}^{(\kappa+1)}={\lambda_{m}^{(\kappa)}}^{\lambda_{m-1}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}}.$

(see figures 3 and 4.)

###### Proof.

Let $V_{p}$ be a neighborhood around $p$ homeomorphic to a product
$B_{\kappa}\times D$ with $D$ a disk not intersecting $L_{k}\in\mathscr{A}$
with $k<j$ or $k>m$. The local fundamental
$\pi_{1}(V_{p}\setminus\mathscr{A})$ equals the fundamental group of the link
associated to the singularity $p$ which is a Hopf link of $m-j+1$ circles (see
[OT92, Lemma 5.75]).

For the complement $M_{\kappa}(\mathscr{A})\setminus V_{p}$ we have
$\pi_{1}(M_{\kappa}(\mathscr{A})\setminus V_{p})\cong F_{n-(m-j)}$ and if
$V^{\prime}$ is a small neighborhood of $V_{p}$ we have that as
$V_{p}\setminus\mathscr{A}$ retracts to $\partial V_{p}\setminus\mathscr{A}$
then $\pi_{1}((M_{\kappa}(\mathscr{A})\setminus V_{p})\cap
V^{\prime})\cong\pi_{1}(V_{p}\setminus\mathscr{A})$. By van-Kampen we obtain
the relation $\lambda_{n+1}^{(\kappa)}\cdots\lambda_{1}^{(\kappa)}=1$. ∎

######  Lemma 3.9.

Suppose that there is a _positive braiding_ of the wires $j$ and $j+1$ in
$M_{\kappa}(\mathscr{A})$. Then the group
$\pi_{1}(M_{\kappa}(\mathscr{A}),q_{\kappa})$ admits the presentation

$\left\langle\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)},\lambda_{j}^{(\kappa+1)},\lambda_{j+1}^{(\kappa+1)}\mid\lambda_{j+1}^{(\kappa+1)}=\lambda_{j}^{(\kappa)},\lambda_{j}^{(\kappa+1)}={\lambda_{j+1}^{(\kappa)}}^{\lambda_{j}^{(\kappa)}}\right\rangle$

(See fig. 5(a).)

###### Proof.

As in lemma 3.8, we have that we can deform $\lambda_{k}^{(\kappa+1)}$ to
$\lambda_{k}^{(\kappa)}$ for $k<j$ or $j+1<k$ without changing the homotopy
type.

The result follows from the Wirtinger presentation of a braid interchanging
the $j$ and the $j+1$ wire: consider the meridians
$\lambda_{j}^{(\kappa)},\lambda_{j+1}^{(\kappa)}$ in
$\bar{f}^{-1}(\beta(\theta_{\kappa+1}))$ as in figure 3(b). Note that in
$\pi_{1}(\bar{f}^{-1}(\beta(\theta_{\kappa+1})),q_{\kappa+1})$ these meridians
satisfy the relations:

$\lambda_{j+1}^{(\kappa+1)}=\lambda_{j}^{(\kappa)},\quad\lambda_{j}^{(\kappa+1)}={\lambda_{j+1}^{(\kappa)}}^{\lambda_{j+1}^{(\kappa+1)}}={\lambda_{j+1}^{(\kappa)}}^{\lambda_{j}^{(\kappa)}}.$

This can be seen directly from Figure 3(b). (C.f. [OT92, Lemmas 5.73, 5.74].)
∎

$t$$x$$y$${\lambda_{j}^{(\kappa)}}$$\lambda_{j}^{(\kappa)}$${\lambda_{j+1}^{(\kappa)}}^{\lambda_{j}^{(\kappa)}}$$\lambda_{j+1}^{(\kappa)}$
(a) Positive braiding

${\lambda_{j}^{(\kappa)}}^{{\lambda_{j+1}^{(\kappa)}}^{-1}}$$\lambda_{j}^{(\kappa)}$${\lambda_{j+1}^{(\kappa)}}$$\lambda_{j+1}^{(\kappa)}$
(b) Negative braiding

Figure 5. Braiding in $\mathcal{W}$

######  Lemma 3.10.

Suppose that there is a _negative braiding_ in $M_{\kappa}(\mathscr{A})$, then
the group $\pi_{1}(M_{\kappa}(\mathscr{A}),q_{\kappa})$ admits the
presentation

$\left\langle\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)},\lambda_{j}^{(\kappa+1)},\lambda_{j+1}^{(\kappa+1)}\mid\lambda_{j+1}^{(\kappa+1)}={\lambda_{j}^{(\kappa)}}^{{\lambda_{j+1}^{(\kappa)}}^{-1}},\lambda_{j}^{(\kappa+1)}=\lambda_{j+1}^{(\kappa)}\right\rangle$

(See fig. 5(b).)

We can summarize the information carried by a wiring diagram $\mathcal{W}$ and
the changes in the geometric sets $\Gamma^{(\kappa)}$ as they cross a vertex
in $\mathcal{W}$ as follows.

For every $t_{\kappa}\in\\{t_{1},\ldots,t_{\nu+s}\\}$ there exists a crossing
$p_{\kappa}$ in the planar representation of $\mathcal{W}$, let
$\Pi^{(\kappa)}=\\{\sigma^{(\kappa)}(1)<\ldots<\sigma^{(\kappa)}(n+1)\\}$ be
an ordered set, with $\sigma^{(\kappa)}$ a permutation of $\\{1,\ldots,n+1\\}$
such that the $k$-th element $\sigma^{(\kappa)}(k)$ records the position of
the wire $W_{\sigma^{(\kappa)}(k)}$ in the fiber
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})$, when $\mathcal{W}$ is read from
bottom to top, with $\theta_{\kappa}$ as in Definition 3.3. This is,
$x_{k}^{(\kappa)}\in W_{\sigma^{(\kappa)}(k)}$ for $k=1,\ldots,n+1$. Note that
$\sigma^{(1)}=id$.

The order in $\Pi^{(\kappa)}$ records the _local position_ of the wires of
$\mathcal{W}$ in $\bar{f}^{-1}{(\beta(\theta_{\kappa})})$, while the order
$\\{1,\ldots,n+1\\}$ induced from the order of the lines in $\mathscr{A}$ is a
_global order_. For a wire $W_{k}$ of $\mathcal{W}$, we write
${\sigma^{(\kappa)}}^{-1}(k)$ to indicate that the wire $W_{k}$ is in the
${\sigma^{(\kappa)}}^{-1}(k)$ position in the fiber
$\bar{f}(\beta(\theta_{\kappa}))$.

Consider the free group $F_{n+1}^{(\kappa)}$ generated by the meridians in
$\Gamma^{(\kappa)}$ and let $\tau^{(\kappa)}:\\{1,\ldots,n+1\\}\to
F_{n+1}^{(\kappa)}$ defined as follows:

Suppose that the crossing $p_{\kappa}$ corresponding to $t_{\kappa}$ satisfies
$p_{\kappa}=W_{\sigma^{(\kappa)}(j)}\cap
W_{\sigma^{(\kappa)}(j+1)}\cap\ldots\cap W_{\sigma^{(\kappa)}(m)}$, then

$\tau^{(\kappa)}(k)=\left\\{\begin{array}[]{ll}e&\text{ for
}k=1,\ldots,j,m+1,\ldots,n+1,\\\
\lambda_{k-1}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}&\text{ for }j<k\leq
m,\par\end{array}\right.$

if $t_{\kappa}$ is an actual vertex,

$\tau^{(\kappa)}(k)=\left\\{\begin{array}[]{ll}e&\text{ for
}k=1,\ldots,j,j+2,\ldots,n+1,\\\ \lambda_{j}^{(\kappa)}&\text{ for
}k=j+1,\par\end{array}\right.$

if $t_{\kappa}$ is a positive virtual vertex, and

$\tau^{(\kappa)}(k)=\left\\{\begin{array}[]{ll}e&\text{ for
}k=1,\ldots,j-1,j+1,\ldots,n+1,\\\ (\lambda_{j+1}^{(\kappa)})^{-1}&\text{ for
}k=j,\par\end{array}\right.$

if $t_{\kappa}$ is a negative virtual vertex.

The Lemmas 3.8, 3.9 and 3.10 imply the following proposition.

######  Proposition 3.11.

Let
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\},\Gamma^{(\kappa+1)}=\\{\lambda_{1}^{(\kappa+1)},\ldots,\lambda_{n+1}^{(\kappa+1)}\\}$
be geometric generating set as in 3.2.1 and suppose that $p_{\kappa}\in
M_{\kappa}$. Then we have that in
$\pi_{1}(M_{\kappa}(\mathscr{A}),q_{\kappa})$:

$\lambda_{{\sigma^{(\kappa+1)}}^{-1}(\sigma^{(\kappa)}(k))}^{(\kappa+1)}=(\lambda_{k}^{(\kappa)})^{\tau^{(\kappa)}(k)}\quad\text{
for }k=1,\ldots,n+1,$

or equivalently,

$\lambda_{k}^{(\kappa+1)}=(\lambda_{{\sigma^{(\kappa)}}^{-1}(\sigma^{(\kappa)}(k))}^{(\kappa)})^{\tau^{(\kappa)}\left({\sigma^{(\kappa)}}^{-1}(\sigma^{(\kappa+1)}(k))\right)}\quad\text{
for }k=1,\ldots,n+1.$

Note that if $p_{\kappa}=W_{\sigma^{(\kappa)}(j)}\cap\ldots\cap
W_{\sigma^{(\kappa)}(m)}$ we have that

${\sigma^{(\kappa+1)}}^{-1}(\sigma^{(\kappa)}(k))=\left\\{\begin{array}[]{ll}k&\text{for
}k=1,\ldots,j-1,m+1,\ldots,n+1,\\\ m-\iota&\text{for }k=j+\iota\text{ and
}\iota=0,\ldots,m-j.\end{array}\right.$

As the fundamental group of $M(\mathscr{A})$ is generated by the meridians
around each line, we fix the geometric generating set
$\Gamma^{(1)}=\\{\lambda_{1}^{(1)},\ldots,\lambda_{n+1}^{(1)}\\}=\\{\lambda_{1},\ldots,\lambda_{n+1}\\}\subset
M_{1}(\mathscr{A})$.

######  Theorem 3.12.

Let $\mathscr{A}=\\{L_{1},\ldots,L_{n+1}\\}$ be a complex arrangement of lines
in $\mathbb{P}^{2}$ and let $\Gamma^{(1)}$ be a geometric generating set as
above. A presentation for the fundamental group of $M(\mathscr{A})$ is given
by

$\pi_{1}(M(\mathscr{A}),q)=\left\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\bigcup_{\kappa}R_{\kappa},\lambda_{n+1}\cdots\lambda_{1}\right\rangle$

with $R_{\kappa}$ as Lemma 3.8 and each $\kappa$ corresponding to a point
$p_{\kappa}\in\operatorname{Sing}\mathscr{A}$.

######  Remark 3.13.

The relations $R_{\kappa}$ are expressed in terms of the geometric generating
set $\Gamma^{(1)}$ by substituting $\lambda_{k}^{(\kappa)}$ by a conjugate of
$\lambda_{\sigma^{(\kappa)}(k)}^{(1)}$ by elements of $\Gamma^{(1)}$ by
repeated applications of Proposition 3.11.

###### Proof.

From Proposition 3.6 we know that
$\pi_{1}(M(\mathscr{A}))=\pi_{1}(M_{1}(\mathscr{A}))*_{F_{n}}\cdots*_{F_{n}}\pi_{1}(M_{\nu+s}(\mathscr{A}))$.
Now, the groups $\pi_{1}(M_{\kappa}(\mathscr{A}))$ are presented in generators
$\Gamma^{(\kappa)}$ and $\Gamma^{(\kappa+1)}$, and relations which are words
in these letters (see Lemmas 3.8, 3.9, 3.10). The geometric generating set
$\Gamma^{(\kappa)}$ is chosen in such a way that it lies in a fiber over a
point of $B_{\kappa-1}\cap B_{\kappa}$, and therefore, we can assume that the
amalgamation
$\pi_{1}(M_{\kappa-1}(\mathscr{A}))*_{F_{n}}\pi_{1}(M_{\kappa}(\mathscr{A}))$
permits to see $\Gamma^{(\kappa)}$ in $M_{\kappa}(\mathscr{A})$ and
$M_{\kappa-1}(\mathscr{A})$ simultaneously.

Note that
$\lambda_{m}^{(\kappa+1)}\cdots\lambda_{j}^{(\kappa+1)}=\lambda_{m}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}$,
hence
$\lambda_{n+1}^{(\kappa+1)}\ldots\lambda_{1}^{(\kappa+1)}=\lambda_{n+1}^{(\kappa)}\cdots\lambda_{1}^{(\kappa)}$
for every $\kappa=1,\ldots,\nu+s-1$.

The relations in $\pi_{1}(M_{\kappa}(\mathscr{A}))$ when there is a positive
or virtual vertex in $M_{\kappa}(\mathscr{A})$, can be omitted in the
presentation of $\pi_{1}(M(\mathscr{A}))$ by writing every meridian of
$\Gamma^{(\kappa+1)}$ in terms of $\Gamma^{(\kappa)}$ as in Lemmas 3.9, 3.10.

When there is an actual vertex in $M_{\kappa}(\mathscr{A})$, the relation
$R_{\kappa}=[\lambda_{m}^{(\kappa)},\lambda_{m-1}^{(\kappa)},\ldots,\lambda_{j}^{(\kappa)}]$
will appear in the presentation of $\pi_{1}(M(\mathscr{A}))$. This relation
can be expressed in terms of $\Gamma^{(1)}$ in a recursive way, by expressing
$\Gamma^{(\kappa)}$ in terms of $\Gamma^{(\kappa-1)}$ by using the
amalgamation of $\pi_{1}(M_{\kappa-1}(\mathscr{A}))$ and
$\pi_{1}(M_{\kappa}(\mathscr{A}))$ over $B_{\kappa-1}\cap B_{\kappa}$ and the
presentation of $M_{\kappa-1}(\mathscr{A})$ given by proposition 3.11. More
precisely, we have that

$\lambda_{k}^{(\kappa)}=\lambda_{\sigma^{(\kappa)}(k)}^{\tau^{(1)}\left(\sigma^{(\kappa)}(k)\right)\cdot\tau^{(2)}\left({\sigma^{(2)}}^{-1}(\sigma^{(\kappa)}(k)\right)\cdots\tau^{(\kappa-1)}\left({\sigma^{(\kappa-1)}}^{-1}(\sigma^{(\kappa)}(k))\right)}\text{
for }k=1,\ldots,n+1,$

and every $\tau^{(r)}\left({\sigma^{(r)}}^{-1}(\sigma^{(\kappa)}(k))\right)$
can be expressed in terms of $\Gamma^{(1)}$ in a recursive way for
$r=1,\ldots,i-1$. ∎

####  3.2.2. Algorithm for determining the presentation for a partial
compactification $M(\mathscr{A},I,P)$

Let $\mathcal{W}$ be a wiring diagram and
$\left\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\cup_{k}R_{k},\lambda_{n+1}\cdots\lambda_{1}\right\rangle$
a presentation of $\pi_{1}(M(\mathscr{A}))$ as in Theorem 3.12.

Consider a partial compactification $M(\mathscr{A},I,P)$ of
$\mathbb{P}^{2}\setminus\mathscr{A}$ as in 2.3.1. Here, we let
$P_{0}=\\{p_{1},\ldots,p_{s_{0}}\\}\subset\operatorname{Sing}\mathscr{A}$
denote the points of multiplicity strictly bigger than two, consider
$\pi:\bar{X}=\operatorname{Bl}_{P_{0}}\mathbb{P}^{2}\to\mathbb{P}^{2}$ and
denote by $D=\sum_{i=1}^{n+1+s_{0}}D_{i}=\pi^{*}\mathscr{A}$. Select
$I\subset\\{1,\ldots,n+1+s_{0}\\}$ and
$P=\\{p_{1}^{\prime},\ldots,p_{s_{1}}^{\prime}\\}\subset\operatorname{Sing}\sum_{i\not\in
I}D_{i}$. Consider another blow-up
$\pi^{\prime}:\operatorname{Bl}_{P}\bar{X}\to\bar{X}$ and write
${\pi^{\prime}}^{*}D=\sum_{i=1}^{n+1+s_{0}+s_{1}}D_{i}^{\prime}$. Define
$D^{\prime}={\pi^{\prime}}^{*}(D)-\sum_{i\in
I}D_{i}^{\prime}-\sum_{i>n+1+s_{0}}D_{i}^{\prime}$ and
$M(\mathscr{A},I,P)=\operatorname{Bl}_{P_{1}}\bar{X}\setminus D^{\prime}$.

From Proposition 2.1, we have that a presentation for the fundamental group
$\pi_{1}(M(\mathscr{A},I,P))$ can be obtained from
$\left\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\cup_{k}R_{k},\lambda_{n+1}\cdots\lambda_{1}\right\rangle$
by adding as relations certain words $\lambda(D_{i}^{\prime})$ representing
some meridians around the irreducible components $D_{i}^{\prime}$ with either
$i\in I$ or $n+s_{0}+1<i$. In order to do so, we have to distinguish four
cases for these irreducible components $D_{i}^{\prime}$ of
${\pi^{\prime}}^{*}D$:

  1. (1)

$D_{i}^{\prime}$ is the strict transform of a line in $\mathscr{A}$. In this
case $i\leq n+1$.

  2. (2)

$D_{i}^{\prime}$ is the strict transform of an exceptional divisor $D_{i}$ in
$\bar{X}$. In this case $n+1<i\leq n+1+s_{0}$,

  3. (3)

$D_{p}^{\prime}$ is an exceptional divisor coming from a double point $p$ in
$\operatorname{Sing}\mathscr{A}$.

  4. (4)

$D_{p}^{\prime}$ is an exceptional divisor obtained by blowing-up a point
$p=D_{r}\cap D_{k}$ with $r\leq n+1$ and $n+1<k\leq n+1+s_{0}$.

For the lines as in (1) we let $\lambda(D_{i}^{\prime})=\lambda_{i}$.

For the lines as in (2), suppose that $D_{i}\subset\bar{X}$ is an exceptional
divisor coming from a point $p\in\operatorname{Sing}\mathscr{A}$ and suppose
that $p\in M_{\kappa}$, this is, $t_{p}$ is the $\kappa$-element in the
ordered set of vertices $t_{1},\ldots,t_{\nu+s}$ of a planar representation of
$\mathcal{W}$ as in 3.2.1. In other words $t_{p}=t_{\kappa}\in[0,1]$ satisfies
$\beta(t_{\kappa})=\bar{f}(p)$ and consider
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
the geometric generating set of
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})\setminus(x_{1}^{(\kappa)},\ldots,x_{n+1}^{(\kappa)})\subset
M_{\kappa}(\mathscr{A})$ and suppose that $p=W_{\sigma^{(\kappa)}(j)}\cap
W_{\sigma^{(\kappa)}(j+1)}\cap\ldots\cap W_{\sigma^{(\kappa)}(m)}$ with the
local index
$\Pi^{(\kappa)}=\\{\sigma^{(\kappa)}(1)<\ldots<\sigma^{(\kappa)}(n+1)\\}$ as
in 3.2.1. Associate to $D_{i}$ and to its strict transform $D_{i}^{\prime}$,
the word
$\lambda(D_{i})=\lambda(D_{i}^{\prime})=\lambda_{m}^{(\kappa)}\cdot\lambda_{m-1}^{(\kappa)}\cdots\lambda_{j+1}^{(\kappa)}\cdot\lambda_{j}^{(\kappa)}.$

######  Lemma 3.14.

Let $D_{i}^{\prime}$ be a line as in (2). Then
$\lambda(D_{i}^{\prime})=\lambda_{m}^{(\kappa)}\cdot\lambda_{m-1}^{(\kappa)}\cdots\lambda_{j+1}^{(\kappa)}\cdot\lambda_{j}^{(\kappa)}$
represents a meridian around $D_{i}$, and by pull-back, also around
$D_{i}^{\prime}$.

###### Proof.

Let $\psi:U\to D$ and $\psi_{i}:U_{i}\to D_{i}$ be the boundary manifolds of
$D$ and $D_{i}$ in $\bar{X}$ respectively. Note that we can use the meridians
$\lambda_{j}^{(\kappa)},\ldots,\lambda_{m}^{(\kappa)}$ to give a presentation
of $\pi_{1}(\partial U^{*})$, with $\partial U_{i}^{*}=\partial U\cap\partial
U_{i}$ as in 2.4, as follows: the projection $\pi(\partial U_{i})$ to
$\mathbb{P}^{2}$ can be seen as the boundary of a $4$-real ball $B_{p}$
centered at $p$. There exists $R_{p}\in\partial B_{p}$ such that for each
$j\leq k\leq m$ the loop $\alpha_{k}:=\lambda_{k}^{(\kappa)}$ is homotopic to
a product $\alpha_{k_{1}}^{\alpha_{k_{2}}}$ with

  * •

The loop $\alpha_{k_{1}}$ starting at $R_{p}$, lying completely in $\partial
B_{p}$ and surrounding the line $L_{\sigma^{(\kappa)}(k)}$.

  * •

The loop $\alpha_{k_{2}}$ is a simple path connecting $R_{p}$ and the point
$R\in\mathbb{P}^{2}\setminus\mathscr{A}$.

By pulling-back the meridians $\alpha_{j_{1}},\ldots,\alpha_{m_{1}}$ to
$\bar{X}$ we can see them as lying in $\partial U$. By construction of the
geometric generating set $\Gamma^{(\kappa)}$, the product
$\alpha_{m_{1}}\cdots\alpha_{j_{1}}$ is homotopic to a path encircling the
lines $L_{\sigma^{(\kappa)}(j)},\ldots,L_{\sigma^{(\kappa)}(m)}$ and therefore
the projection ${\psi_{i}}_{*}(\alpha_{m_{1}}\cdots\alpha_{j_{1}})=e$ in
$\pi_{1}(D_{i}^{*},\psi_{i}(R_{p}))$. We can construct a continuous map
$h_{i}:\cup_{k=j}^{m}\psi_{i}(\alpha_{k_{1}})\to\cup_{k=j}^{m}\alpha_{k_{1}}$
such that $h_{i}(\psi(\alpha_{k_{1}}))=\alpha_{k_{1}}$ and therefore the loops
$\alpha_{j_{1}},\ldots,\alpha_{m_{1}}$ together with a fiber $\alpha_{i}$ of
$\partial U_{i}$ generate the group $\pi_{1}(\partial U_{i}^{*})$ as in Lemma
2.4. Moreover, as $D_{i}\cdot D_{i}=-1$, we have the relation
$\alpha_{i}=\alpha_{m_{1}}\cdots\alpha_{j_{1}}$ in $\pi_{1}(\partial
U_{i}^{*},R_{p})$.

By construction of $\Gamma^{(\kappa)}$, we have that every two
$\alpha_{k_{2}}$ and $\alpha_{k_{2}^{\prime}}$ with $j\leq k,k^{\prime}\leq m$
are homotopic. Therefore, by connecting $\alpha_{i}$ to $R$ via
$\alpha_{j_{2}}$, we obtain the relation
$\alpha_{i}^{\alpha_{j_{2}}}=\lambda_{m}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}$
in $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$.

By pulling-back $\alpha_{i}^{\alpha_{j_{2}}}$ to $\operatorname{Bl}_{P_{1}}$
we obtain that it is homotopic to a meridian around $D_{i}^{\prime}$. ∎

For the lines $D_{p}^{\prime}$ as in (3), suppose that $p=D_{r}\cap D_{k}$
with $r,k\leq n+1$. Consider the unique index $1\leq\kappa\leq\nu+s$ such that
$p\in M_{\kappa}$ and let
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
be a geometric generating set of $\pi_{1}(M_{\kappa}(\mathscr{A}))$. We denote

$\lambda(D_{p}^{\prime}):=\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}\lambda_{{\sigma^{(\kappa)}}^{-1}(k)}^{(\kappa)}.$

Recall that ${\sigma^{(\kappa)}}^{-1}(r)$ and ${\sigma^{(\kappa)}}^{-1}(k)$
record the local position of the wires $W_{r},W_{k}$ respectively, in the
local order of the wires of $\mathcal{W}$ in
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})$ given by
$\Pi^{(\kappa)}=\\{\sigma^{(\kappa)}(1)<\ldots<\sigma^{(\kappa)}(n+1)\\}$.

Finally, let $D_{p}^{\prime}$ be as in (4) with $p\in P$. We have that
$p=D_{r}\cap D_{k}$ with $r\leq n+1$ and $D_{k}$ an exceptional divisor coming
from a point $p(k)\in P_{0}$. Let us suppose that $p(k)\in M_{\kappa}$. Denote
by
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}\subset
M_{\kappa}(\mathscr{A})$ the geometric generating set as above. We can suppose
that $p(k)=W_{\sigma^{(\kappa)}(j)}\cap\ldots\cap W_{\sigma^{(\kappa)}(m)}$.
As $n+1\leq k\leq n+1+s_{0}$, we can consider the word
$\lambda(D_{k})=\lambda_{m}^{(\kappa)}\cdots\lambda_{j}^{(\kappa)}$ as in
Lemma 3.14 above.

######  Lemma 3.15.

A meridian of $D_{p}^{\prime}$ is given by
$\lambda(D_{p}^{\prime})=\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}\lambda(D_{k})$.
Moreover, $\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}$ commutes with
$\lambda(D_{k})$.

###### Proof.

Recall that by construction,
$\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}$ is the meridian of $L_{r}$
lying in the geometric generating set $\Gamma^{(\kappa)}$.

Let $\psi_{D_{k}}:\partial U_{D_{k}}\to D_{k}$ be the boundary manifold of
$D_{k}$ in $\bar{X}$. For $k^{\prime}=j,\ldots,m$, let us decompose the loops
$\alpha_{k^{\prime}}=\lambda_{k^{\prime}}^{(\kappa)}$ in two parts
$\alpha_{k_{1}^{\prime}},\alpha_{k_{2}^{\prime}}$, as in the proof of the
Lemma 3.14, such that $\alpha_{k^{\prime}}$ is homotopic to
$\alpha_{k_{1}^{\prime}}^{\alpha_{k_{2}^{\prime}}}$. The proof of the same
Lemma and 2.4 give us that

$\pi_{1}(\partial
U_{D_{k}}^{*},R_{k})=\langle\alpha_{j_{1}},\ldots,\alpha_{m_{1}},\alpha_{k}\mid[\alpha_{k},\alpha_{k_{1}^{\prime}}],\alpha_{k}=\alpha_{m_{1}}\cdots\alpha_{j_{1}}\rangle.$

for a point $R_{k}\in\partial U_{D_{k}}^{*}$ and $\alpha_{k}$ a fiber of
$\partial U_{D_{k}}^{*}$. We can globalize the relations in this presentation
by considering $\alpha_{k_{1}^{\prime}}^{\alpha_{k_{2}^{\prime}}}$ and obtain
that $\lambda(D_{k})$ commutes with $\lambda_{k^{\prime}}^{(\kappa)}$ for
$k^{\prime}=j,\ldots,m$, in particular as $D_{r}$ intersect $D_{k}$, we have
that $\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}$ commutes with
$\lambda(D_{k})$.

Furthermore, the point $R_{k}$ can be chosen to lie in the boundary $\partial
B_{p}$ of a ball $B_{p}\subset\bar{X}$ around $p$. Let
$\psi_{D_{p}^{\prime}}:\partial U_{D_{p}^{\prime}}\to D_{p}^{\prime}$ be the
boundary manifold of $D_{p}^{\prime}$ in $\operatorname{Bl}_{P}\bar{X}$ and
$\Delta_{1},\Delta_{2}$ a pair of disks about the points $D_{p}^{\prime}\cap
D_{r}^{\prime}$ and $D_{p}^{\prime}\cap D_{k}^{\prime}$ respectively. Denote
$\partial
U_{D_{p}^{\prime}}^{*}=\psi_{D_{p}^{\prime}}^{-1}(D_{p}^{\prime}\setminus(\Delta_{1}\cup\Delta_{2}))$.
By working in local coordinates, it can be seen that
$\alpha_{k},\alpha_{{\sigma^{(\kappa)}}^{-1}(r)_{1}}$ and a fiber $\alpha_{p}$
of $\partial U_{D_{p}^{\prime}}$ at $R_{k}$ generate the group
$\pi_{1}(\partial U_{D_{p}^{\prime}}^{*})$ and that

$\pi_{1}(\partial
U_{D_{p}^{\prime}}^{*})=\left\langle\alpha_{k},\alpha_{{\sigma^{(\kappa)}}^{-1}(r)_{1}},\alpha_{p}\left\lvert\begin{array}[]{l}[\alpha_{k},\alpha_{p}],\
[\alpha_{{\sigma^{(\kappa)}}^{-1}(r)_{1}},\alpha_{p}],\\\
\alpha_{p}=\alpha_{k}\cdot\alpha_{{\sigma^{(\kappa)}}^{-1}(r)_{1}}\end{array}\right.\right\rangle$

by Lemma 2.4 and because $D_{p}^{\prime}\cdot D_{p}^{\prime}=-1$. ∎

######  Theorem 3.16.

Let $\mathscr{A}\subset\mathbb{P}^{2}$ be an arrangement of lines,
$\mathcal{W}$ a wiring diagram and $M(\mathscr{A},I,P)$ a partial
compactification. Then

$\pi_{1}(M(\mathscr{A},I,P),q)=\left\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\bigcup_{k}R_{k},\lambda_{n+1}\cdots\lambda_{1},\bigcup_{i\in
I}\lambda(D_{i}^{\prime}),\bigcup_{p\in
P}\lambda(D_{p}^{\prime}),\right\rangle$

is a presentation for the fundamental group of the partial compactification.

###### Proof.

We only have to justify the expression for those meridians around lines as in
(1) and (3). For the meridians of lines as in (2) and (4), the expression
$\lambda(D_{i}^{\prime})$ and $\lambda(D_{p}^{\prime})$ is explained by Lemmas
3.14 and 3.15 respectively. We will conclude by Proposition 2.1.

For the meridians around lines as in (1), it is immediate by the
biholomorphism property of the blow-up outside the exceptional divisor.

Consider a line $D_{i}$ as in (3) and suppose that it comes from a point
$p=D_{r}\cap D_{k}$ with $r,k\leq n+1$. Note that there is essentially no
difference with a line as in (2) besides the change of local indexation to a
global one, and therefore, we can proceed as in the proof of Lemma 3.14 to
obtain that
$\lambda_{{\sigma^{(\kappa)}}^{-1}(r)}^{(\kappa)}\lambda_{{\sigma^{(\kappa)}}^{-1}(k)}^{(\kappa)}$
is homotopic to a fiber of $\partial U_{i}^{*}$ connected to the global base
point $R$.

∎

As $D=\sum_{i=1}^{N}D_{i}$ is a simple normal crossing divisor with
$N=n+1+s_{0}$, we can consider an orbifold structure in
$(\operatorname{Bl}_{P_{0}}\mathbb{P}^{2},D)$ (see [Eys17] for the notation)
by choosing weights
$r=(r_{1},\ldots,r_{N})\in(\mathbb{N}^{*}\cup\\{+\infty\\})^{N}$.

######  Theorem 3.17.

Let $\mathscr{A}$ be a complex arrangement of lines, $\mathcal{W}$ a wiring
diagram and consider the weights $r$ of $D$ as above. The fundamental group
$\pi_{1}(\mathcal{X}(\operatorname{Bl}_{P_{0}}\mathbb{P}^{2},D,r))$ of the
orbifold $\mathcal{X}(\operatorname{Bl}_{P_{0}}\mathbb{P}^{2},D,r)$ admits the
following presentation:

$\left\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\bigcup_{k}R_{k},\lambda_{n+1}\cdots\lambda_{1},\bigcup_{i=1}^{N}\lambda(D_{i})^{r_{i}}\right\rangle$

where the relation $\lambda(D_{i})^{r_{i}}$ is omitted if $r_{i}=+\infty$.

##  4\. Boundary Manifolds methods

In this Section we use the results of Mumford as stated in 2.4 in order to
study the fundamental group of the boundary manifold $\partial U$ of an
arrangement of lines $\mathscr{A}$.

The notion of wiring diagram defined in the previous section will play an
important role, a presentation of $\pi_{1}(M(\mathscr{A},I,P))$ will be
obtained as a quotient of the presentation of $\pi_{1}(\partial U)$ and
compared with Theorem 3.16.

###  4.1. Boundary manifold of an arrangement of lines

####  4.1.1. Fundamental group of the boundary manifold of an arrangement of
lines

Let $\mathscr{A}=\\{L_{1},\ldots,L_{n+1}\\}\subset\mathbb{P}^{2}$ be an
arrangement of lines and denote by $\pi:\bar{X}\to\mathbb{P}^{2}$ the blow-up
of the projective plane at the $s_{0}$ points of
$\operatorname{Sing}\mathscr{A}$ of multiplicity equal or higher than $3$ as
in 2.3.1. Recall that $D=\left|\pi^{*}D\right|=\sum_{i=1}^{n+s_{0}+1}D_{i}$ is
the reduced total transform of $\mathscr{A}$ in $\bar{X}$ and let
$\psi:\partial U\to D$ be its boundary manifold.

Using the description of Mumford (Theorem 2.6) and that of a weighted graph,
Westlund gave a presentation of the fundamental group $\pi_{1}(\partial U)$ of
$\partial U$ [Wes67] (see also [CS08]). Let us describe this presentation.

Denote by $\Delta$ the dual graph of $D$ and by $\mathcal{E}$ the set of edges
of $\Delta$ as in 2.3 above. Associate to each vertex $v_{i}$ a weight $w_{i}$
corresponding with the self-intersection number of the associated line $D_{i}$
in $\bar{X}$.

Let $\mathcal{T}$ be a maximal tree of $\Delta$ (a subgraph of $\Delta$
containing no cycles and all the vertices of $\Delta$) and denote by
$\mathcal{C}=\Delta\setminus\mathcal{T}$. Note that
$g=\left|\mathcal{C}\right|=b_{1}(\Delta)$ equals the number of independent
cycles in $\Delta$.

The edges in $\mathcal{C}$ correspond to $g$ points $\\{p_{1},\ldots,p_{g}\\}$
in $\operatorname{Sing}D$. Let us denote by
$\pi^{(1)}:\operatorname{Bl}_{p_{1},\ldots,p_{g}}\bar{X}\to\bar{X}$ the blow-
up at these points. Denote by
$D^{\prime}=\sum_{i=1}^{n+s_{0}+1}D_{i}^{\prime}$ the strict transform of $D$
in $\operatorname{Bl}_{p_{1},\ldots,p_{g}}\bar{X}$ and let
$\psi^{\prime}:\partial U^{\prime}\to D^{\prime}$ be the boundary manifold of
$D^{\prime}$. Note that the dual graph of $D^{\prime}$ is a tree that can be
identified with $\mathcal{T}$ by removing from $\Delta$ the edges in
$\mathcal{C}$. In particular, $D^{\prime}$ and $\partial U^{\prime}$ are
connected. Let ${\pi^{(1)}}^{*}(D)=D^{\prime}+\sum_{k=1}^{g}E_{k}$ be the
total transform of $D$ with $E_{1},\ldots,E_{k}$ exceptional divisors.

Now, if $(i,j)\in\mathcal{C}$ corresponds to the point $p_{k}$ for some $1\leq
k\leq g$, there exists an exceptional divisor
$E_{k}\in\operatorname{Bl}_{p_{1},\ldots,p_{g}}\bar{X}$ and
$D_{i}^{\prime},D_{j}^{\prime}$ strict transforms of irreducible components
$D_{i},D_{j}$ of $D$ respectively such that $E_{k}\cap
D_{i}^{\prime}\not=\varnothing$, $E_{k}\cap D_{j}^{\prime}\not=\varnothing$
and $D_{i}\cap D_{j}=p_{k}$. Denote its boundary manifold by
$\psi_{E_{k}}:\partial U_{E_{k}}\to E_{k},\psi_{i}^{\prime}:\partial
U_{i}^{\prime}\to D_{i}^{\prime},\psi_{j}^{\prime}:\partial U_{j}^{\prime}\to
D_{j}^{\prime}$.

Select a base point $Q_{i}\in
D_{i}^{\prime}\setminus(\cup_{j\not=i}D_{j}^{\prime}\bigcup\cup_{k=1}^{g}E_{k})$
as in 2.4 and a simple curve $l_{i}\subset D_{i}^{\prime}$ containing $Q_{i}$
and every intersection of the form:

  1. (1)

$D_{i}^{\prime}\cap D_{j}^{\prime},\text{ with }(i,j)\text{ an edge in
}\mathcal{T}$,

  2. (2)

$D_{i}^{\prime}\cap E_{k}$, with $E_{k}$ coming from a point $p_{k}$
corresponding to an edge $(i,j)$ in $\mathcal{C}$.

Let us label these points by the order they intersect $l_{i}$ as
$P_{i1},\ldots,P_{i{k_{i}^{\prime}}}$. Note that for every $P_{im}$ there
corresponds a unique edge $(i,j_{\Delta}(i,m))$ in $\Delta$. This defines an
injective function $m\mapsto j_{\Delta}(i,m)$ from
$\\{1,\ldots,k_{i}^{\prime}\\}$ to $\\{1,\ldots,n+s_{0}+1\\}$.

We also label only the points as in (1) by the order they intersect $l_{i}$ as
$P_{i1}^{\prime},\ldots,P_{i{k_{i}}}^{\prime}$ and define a function $m\mapsto
j_{\mathcal{T}}(i,m)$ from $\\{1,\ldots,k_{i}\\}$ to
$\\{1,\ldots,n+s_{0}+1\\}$ as in 2.4.

Let $l=\cup l_{k}\subset D^{\prime}$ and $h^{\prime}:l\to\partial U^{\prime}$
be a continuous function such that $\psi^{\prime}\circ h^{\prime}=id_{l}$. For
an exceptional divisor $E_{k}$ corresponding to an edge $(i,j)$ in
$\mathcal{C}$, we let $l_{E_{k}}\subset E_{k}$ be a simple path connecting
$E_{k}\cap D_{i}^{\prime}$ to $E_{k}\cap D_{j}^{\prime}$ and
$h_{E_{k}}:l_{E_{k}}\to\partial U_{E_{k}}$ such that $\psi_{E_{k}}\circ
h_{E_{k}}=id_{l_{E_{k}}}$, $h_{E_{k}}(l_{E_{k}})\cap
h^{\prime}(l_{i})\not=\varnothing$ and $h_{E_{k}}(l_{E_{k}})\cap
h^{\prime}(l_{j})\not=\varnothing$. This create a cycle $c_{k}=c_{ij}$ in the
boundary manifold of ${\pi^{(1)}}^{*}(D)$, which we orient passing first by
$h^{\prime}(l_{i})$, following $h_{E_{k}}(l_{E_{k}})$ and coming back by
$h^{\prime}(l_{j})$. We denote by $\gamma_{1},\ldots,\gamma_{n+1+s_{0}}$ the
meridians around $D_{1}^{\prime},\ldots,D_{n+1+s_{0}}^{\prime}$ obtained as in
2.4 using $h^{\prime}(l)$.

######  Theorem 4.1 (Westlund).

A presentation for $\pi_{1}(\partial U)$ is given by

$\pi_{1}(\partial
U)=\left\langle\begin{array}[]{llll}\gamma_{1},\ldots,\gamma_{n+s_{0}+1}&\big{\lvert}[\gamma_{i},\gamma_{j}^{s_{ij}}],&(i,j)\in\mathcal{E}\\\
c_{1},\ldots,c_{g}&\bigg{\lvert}\gamma_{i}^{-w_{i}}=\prod_{m=1}^{k_{i}^{\prime}}\gamma_{j_{\Delta}(i,m)}^{s_{ij_{\Delta}(i,m)}}&1\leq
i\leq n+s_{0}+1\end{array}\right\rangle$

where

$s_{ij}=\left\\{\begin{array}[]{ll}c_{k}^{-1}&\text{ if }(i,j)\text{ equals
the $k$-th element in }\mathcal{C},\\\ c_{k}&\text{ if }(j,i)\text{ equals the
$k$-th element in }\mathcal{C},\\\ 1&\text{ if }(i,j)\text{ is an edge of
}\mathcal{T}.\\\ \end{array}\right.$

###### Proof.

From Theorem 2.6 we know that

$\pi_{1}(\partial
U^{\prime})=\left\langle\gamma_{1},\ldots\gamma_{n+s_{0}+1}\bigg{\lvert}[\gamma_{i},\gamma_{j_{\mathcal{T}(i,m)}}]\
m=1,\ldots,k_{i},\gamma_{i}^{-w_{i}^{\prime}}=\prod_{m=1}^{k_{i}}\gamma_{j_{\mathcal{T}(i,m)}}\right\rangle$

where $w_{i}^{\prime}$ is the intersection number of the strict transform
$D_{i}^{\prime}$ of $D_{i}$ in
$\operatorname{Bl}_{p_{1},\ldots,p_{g}}\bar{X}$. Note that $(i,l)$ is an edge
of $\mathcal{T}$ if and only if $l=j_{\mathcal{T}}(i,m)$ for some
$m\in\\{1,\ldots,k_{i}\\}$ and therefore the set of relations
$A=\\{[\gamma_{i},\gamma_{j_{\mathcal{T}}(i,m)}]\mid
m=1,\ldots,k_{i},i=1,\ldots,n+s_{0}+1\\}$ is the same as
$B=\\{[\gamma_{i},\gamma_{l}]\mid(i,l)\text{ an edge of }\mathcal{T}\\}$.

Let $E_{k}$ be an exceptional divisor corresponding to an edge $(i,j)$ in
$\mathcal{C}$ as above. We can remove two disks $\Delta_{1}^{\prime}\subset
D_{i}^{\prime},\Delta_{2}^{\prime}\subset D_{j}^{\prime}$ in $D^{\prime}$
around the points $E_{k}\cap D_{i}^{\prime}$ and $E_{k}\cap D_{j}^{\prime}$
respectively, and obtain a pair of torus $T_{i}^{\prime},T_{j}^{\prime}$ as
boundary from $\partial
U^{\prime\circ}=\psi^{\prime-1}(D^{\prime}\setminus\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime})$.
Let $\gamma(E_{k})_{i}^{\prime},\gamma_{i}$ and
$\gamma(E_{k})_{j}^{\prime},\gamma_{j}$ be generators of
$\pi_{1}(T_{i}^{\prime})$ and $\pi_{1}(T_{j}^{\prime})$ with
$\gamma(E_{k})_{i}^{\prime},\gamma(E_{k})_{j}^{\prime}$ constructed from
$\partial\Delta_{1}^{\prime}$, $\partial\Delta_{2}^{\prime}$ as in 2.4. We
obtain the following presentation for $\pi_{1}(\partial U^{\prime\circ})$:

$\left\langle\gamma_{1},\ldots\gamma_{n+s_{0}+1},\gamma(E_{k})_{i}^{\prime},\gamma(E_{k})_{j}^{\prime}\Bigg{\lvert}\begin{array}[]{l}A,{[\gamma_{i},\gamma(E_{k})_{i}^{\prime}],[\gamma_{j},\gamma(E_{k})_{j}^{\prime}]}\\\
\gamma_{l}^{-w_{l}^{\prime}}=\prod_{m=1}^{k_{l}}\gamma_{j_{\mathcal{T}}(l,m)}\text{
for }l\not=i,j,\\\
\gamma_{i}^{-w_{i}^{\prime}}=\gamma_{j_{\mathcal{T}}(i,1)}\cdots\gamma(E_{k})_{i}^{\prime}\cdots\gamma_{j_{\mathcal{T}}(i,k_{i})},\\\
\gamma_{j}^{-w_{j}^{\prime}}=\gamma_{j_{\mathcal{T}}(j,1)}\cdots\gamma(E_{k})_{j}^{\prime}\cdots\gamma_{j_{\mathcal{T}}(i,k_{j})}\end{array}\right\rangle$

where the products in the lowest row of the relations are taken in such a way
that
$1=\psi_{i}(\gamma_{i1}^{\prime})\cdots\psi_{i}(\gamma(E_{k})_{i}^{\prime})\cdots\psi_{i}(\gamma_{i{k_{i}}}^{\prime})$
holds in $\pi_{1}({D_{i}^{\prime}}^{*}\setminus\Delta_{1})$ and similarly
$1=\psi_{j}(\gamma_{j1}^{\prime})\cdots\psi_{j}(\gamma(E_{k})_{j}^{\prime})\cdots\psi_{j}(\gamma_{j{k_{j}}}^{\prime})$
in $\pi_{1}({D_{j}^{\prime}}^{*}\setminus\Delta_{2})$ for generators
$\gamma_{r},\gamma_{r1}^{\prime},\ldots,\gamma_{r{k_{r}}}^{\prime}$ generators
of $\pi_{1}(\partial{U_{r}^{\prime}}^{*})$ for $r=i,j$ as in 2.4.

Let $E_{k}^{*}$ denote the submanifold of $E_{k}$ obtained by removing another
pair of disks $\Delta_{1},\Delta_{2}$ of $E_{k}$ about the points $E_{k}\cap
D_{i}^{\prime}$ and $E_{k}\cap D_{j}^{\prime}$ as in 2.4. Write $\partial
U_{E_{k}}^{*}$ for $\psi_{E_{k}}^{-1}(E_{k}^{*})$. Note that the boundary of
$\partial U_{E_{k}}^{*}$ consists also of a pair of torus $T_{i},T_{j}$
corresponding to $\Delta_{1}$ and $\Delta_{2}$ respectively. Let
$\gamma_{i}^{\prime},\gamma(E_{k})$ and $\gamma_{j}^{\prime},\gamma(E_{k})$ be
generators of $\pi_{1}(T_{i})$ and $\pi_{1}(T_{j})$ respectively. By (1), we
have that $\pi_{1}(\partial
U_{E_{k}}^{*})=\langle\gamma(E_{k}),\gamma_{i}^{\prime},\gamma_{j}^{\prime}\mid[\gamma(E_{k}),\gamma_{i}^{\prime}],[\gamma(E_{k}),\gamma_{j}^{\prime}],\gamma(E_{k})=\gamma_{i}^{\prime}\gamma_{j}^{\prime}\rangle\cong\langle\gamma_{i}^{\prime},\gamma_{j}^{\prime}\mid[\gamma_{i}^{\prime},\gamma_{j}^{\prime}]\rangle\cong\mathbb{Z}^{2}$
because $E_{k}\cdot E_{k}=-1$.

We can glue $\partial U^{\prime\circ}$ to $\partial U_{E_{k}}^{*}$ by first
gluing $T_{i}$ to $T_{i}^{\prime}$ by a longitude-to-meridian orientation-
preserving attaching map $f$, and similarly $T_{j}$ to $T_{j}^{\prime}$ by a
map $g$.

First, by the van Kampen Theorem we obtain that
$\gamma_{i}=\gamma_{i}^{\prime}$ and
$\gamma(E_{k})=\gamma(E_{k})_{i}^{\prime}$. Then, from HNN extension we get
$\gamma_{j}=c_{k}^{-1}\gamma_{j}^{\prime}c_{k}$ and
$\gamma(E_{k})_{j}^{\prime}=c_{k}^{-1}\gamma(E_{k})c_{k}$.

We obtain the following presentation of $\pi_{1}(\partial
U^{\prime\circ}\cup_{f,g}\partial U_{E_{k}}^{*})$ by replacing
$\gamma_{i}^{\prime}=\gamma_{i},\gamma_{j}^{\prime}=\gamma_{j}^{c_{k}^{-1}},\gamma(E_{k})=\gamma_{i}\gamma_{j}^{c_{k}^{-1}},$
$\gamma(E_{k})_{i}^{\prime}=\gamma_{i}\gamma_{j}^{c_{k}^{-1}},\gamma(E_{k})_{j}^{\prime}=\gamma_{i}^{c_{k}}\gamma_{j}$
in terms of $\gamma_{i},\gamma_{j},c_{k}$

$\left\langle\gamma_{1},\ldots\gamma_{n+s_{0}+1},c_{k}\bigg{\lvert}\begin{array}[]{l}{[\gamma_{i},\gamma_{j}^{c_{k}^{-1}}]},[\gamma_{i},\gamma_{l}]\text{
with }(i,l)\in\mathcal{T}\\\
\gamma_{l}^{-w_{l}^{\prime}}=\prod_{m=1}^{k_{l}}\gamma_{j_{\mathcal{T}}(l,m)}\text{
for }l\not=i,j,\\\
\gamma_{i}^{-w_{i}^{\prime}}=\gamma_{j_{\mathcal{T}}(i,1)}\cdots\gamma_{i}\gamma_{j}^{c_{k}^{-1}}\cdots\gamma_{j_{\mathcal{T}(i,k_{i})}},\\\
\gamma_{j}^{-w_{j}^{\prime}}=\gamma_{j_{\mathcal{T}(j,1)}}\cdots\gamma_{i}^{c_{k}}\gamma_{j}\cdots\gamma_{j_{\mathcal{T}(j,k_{j})}}\end{array}\right\rangle$

Note that the row of the relations corresponding to $i$ can be simplified to

(2)
$\gamma_{i}^{-(w_{i}^{\prime}+1)}=\gamma_{j_{\mathcal{T}}(i,1)}\cdots\gamma_{j}^{c_{k}^{-1}}\cdots\gamma_{j_{\mathcal{T}(i,k_{i})}},$

as $\gamma_{i}$ commutes with every $\gamma_{j_{\mathcal{T}}(i,m)}$. A similar
simplification can be made for the relation corresponding to $j$.

We repeat the above process for every $E_{k}$ with $k=1,\ldots,g$. After this,
the order for the product as in (2), is given by the function $m\mapsto
j_{\Delta}(i,m)$ and the conjugations $s_{ij}$ as in the statement of the
Theorem. We get that
$\gamma_{i}^{-(w_{i}^{\prime}+(k_{i}^{\prime}-k_{i}))}=\prod_{m=1}^{k_{i}^{\prime}}\gamma_{j_{\Delta}(i,m)}^{s_{ij_{\Delta}(i,m)}}$.
Note that $k_{i}^{\prime}-k_{i}$ equals the number of points in
$\\{p_{1},\ldots,p_{g}\\}\cap D_{i}$, and therefore
$w_{i}^{\prime}+(k_{i}^{\prime}-k_{i})=w_{i}$.

This gives a presentation for the fundamental group of the boundary manifold
of the total transform of $D$, which is homeomorphic to $\partial U$. ∎

A central computation in our work is the expression of the meridians around
the exceptional divisors $D_{n+2},\ldots,D_{n+s_{0}+1}$ in
$D=\sum_{i=1}^{n+s_{0}+1}D_{i}$ in terms of meridians of the lines in
$\mathscr{A}$. As a partial result we obtain an expression in the following
corollary. The cycles $c_{k}$ will be expressed in terms of meridians of the
lines in 4.2.

######  Corollary 4.2.

For $r=n+2,\ldots,n+s_{0}+1$, we have that in $\pi_{1}(\partial U)$,

$\gamma_{r}=\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}\text{
with }s_{rj}=\left\\{\begin{array}[]{cc}c_{k}^{-1}&\text{ if }(r,j)\text{
equals the $k$-th element in }\mathcal{C},\\\ c_{k}&\text{ if }(j,r)\text{
equals the $k$-th element in }\mathcal{C},\\\ 1&\text{ if }(r,j)\text{ is an
edge in }\mathcal{T}.\\\ \end{array}\right.$

###### Proof.

It follows from the relation
$\gamma_{r}^{-w_{r}}=\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(i,m)}^{s_{rj_{\Delta}(r,m)}}$
in the presentation of $\pi_{1}(\partial U)$ in Theorem 4.1, the fact that
$w_{r}=-1$ because $D_{r}$ is an exceptional divisor and hence
$j_{\Delta}(r,m)\in\\{1,\ldots,n+1\\}$. ∎

####  4.1.2. Choice of a maximal tree

In what follows, we will define a maximal tree $\mathcal{T}^{\prime}$ of the
dual graph $\Delta$ of $D$ as defined in [CS08, Section 3.3].

In the arrangement $\mathscr{A}=\\{L_{1},\ldots,L_{n+1}\\}$, we will fix the
line $L_{n+1}$ as the line at infinity, recall that we denote by $D_{i}$ the
strict transform by $L_{i}$ for $i\leq n+1$ in $D=\sum D_{i}\subset\bar{X}$.

Consider the following subset of edges
$\mathcal{E}^{\prime}\subset\mathcal{E}$ which defines a maximal tree
$\mathcal{T}^{\prime}\subset\Delta$ of the dual graph $\Delta$ of $D$:

  1. (1)

Let $(j,n+1),(n+1,j)\in\mathcal{E}^{\prime}$ if $D_{n+1}\cap
D_{j}\not=\varnothing$. This is, all the edges having as an endpoint the
vertex corresponding to $D_{n+1}$.

  2. (2)

Let $(i,j)\in\mathcal{E}^{\prime}$ if $n+1<j$ ($D_{j}$ is an exceptional
divisor) with either

     * •

$D_{j}\cap D_{n+1}=\varnothing$ and $i=\min\\{l\mid D_{l}\cap
D_{j}\not=\varnothing\\}$. Note that $D_{j}$ comes from a point in
$\operatorname{Sing}\mathscr{A}\setminus L_{n+1}$.

     * •

or $D_{j}\cap D_{n+1}\not=\varnothing$ and $D_{i}\cap D_{j}\not=\varnothing$.
The line $D_{i}$ corresponds then to a line $L_{i}$ touching $L_{n+1}$ in a
point of multiplicity $>2$.

Note that $\mathcal{E}\setminus\mathcal{E}^{\prime}$ consists either:

  * •

of edges corresponding to double points $L_{i}\cap L_{j}$ with $i,j<n+1$,

  * •

or, if $p=L_{i_{1}}\cap\ldots\cap L_{i_{l}}$ with $i_{1}<\ldots<i_{l}<n+1$,
$2<l$, and $E_{j}$ denotes the exceptional divisor obtained by blowing up at
$p$, of edges of the form $(i_{r},j)$ with $r=2,\ldots,l$.

Let us consider the presentation of $\pi_{1}(\partial U)$ as in Theorem 4.1.
If $(i,j)$ equals the $k$-th element in $\Delta\setminus\mathcal{T}^{\prime}$
as in the first point above, we denote the cycle $c_{k}$ by $c_{i,j}$. Recall
that if $i<j$, we pass first through $h^{\prime}(l_{i})$ and then through
$h^{\prime}(l_{j})$.

For the cycles created by the edges in the second point, let us suppose that
the irreducible component of $D$ are ordered in such a way that $D_{n+1}\cap
D_{j}\not=\varnothing$ for $j=n+2,\ldots,s^{\prime}$ and $D_{n+1}\cap
D_{k}=\varnothing$ for $k>s^{\prime}$.

For $s^{\prime}<\iota\leq n+1+s_{0}$, we have that, as $D_{\iota}$ is an
exceptional divisor, $1\leq j_{\Delta}(\iota,m)\leq n$ for $1\leq m\leq
k_{i}^{\prime}$, and
$\gamma_{\iota}^{-w_{\iota}}=\prod_{m=1}^{k_{\iota}^{\prime}}\gamma_{j_{\Delta}(\iota,m)}^{s_{\iota
j_{\Delta}(\iota,m)}}$ holds as in Theorem 4.1. Note that if
$(j_{\Delta}(\iota,m),\iota)$ equals the $k$-th element in
$\Delta\setminus\mathcal{T}^{\prime}$, we have that $s_{\iota
j_{\Delta}(\iota,m)}=c_{k}$. In this case, we denote $c_{k}$ by
$c_{j_{\Delta}(\iota,m),\iota}$. As $\mathcal{T}^{\prime}$ is a maximal tree,
the edges corresponding to $(j_{\Delta}(\iota,m),\iota)$ for $1<m\leq
k_{i}^{\prime}$, give rise to $k_{i}^{\prime}-1$ independent cycles
$c_{j_{\Delta}(\iota,m),\iota}$ in $\Delta$.

Using the tree $\mathcal{T}^{\prime}$ and corollary 4.2, we can express the
meridian around an exceptional divisor in terms of the meridians of the lines
and the cycles $c_{s_{t}^{\prime\prime},s^{\prime\prime}}$:

(3)
$\gamma_{\iota}=\gamma_{j_{\Delta}(\iota,1)}^{c_{j_{\Delta}(\iota,1),\iota}}\gamma_{j_{\Delta}(\iota,2)}^{c_{j_{\Delta}(\iota,2),\iota}}\cdots\gamma_{j_{\Delta}(\iota,k_{\iota}^{\prime})}^{c_{j_{\Delta}(\iota,k_{\iota}^{\prime}),\iota}}\quad\text{
for }s^{\prime}<\iota\leq n+1+s_{0}$

with $c_{j_{\Delta}(\iota,r),\iota}=1$ if $r=\min\\{j_{\Delta}(\iota,m)\mid
m=1,\ldots,k_{i}^{\prime}\\}$.

###  4.2. From a presentation for the boundary manifold of an arrangement of
lines to a presentation of its complement

Let $\mathscr{A}\subset\mathbb{P}^{2}$ be an arrangement of lines and
$\partial U$ its boundary manifold. We identify $\partial U$ with the boundary
manifold of the total transform $D$ of $\mathscr{A}$ in
$\pi:\bar{X}\to\mathbb{P}^{2}$, the blow-up of $\mathbb{P}^{2}$ at the points
of $\operatorname{Sing}\mathscr{A}$ of multiplicity higher than two. Denote by
$i:\partial U\hookrightarrow\mathbb{P}^{2}\setminus\mathscr{A}$ the inclusion
map and by $i_{*}:\pi_{1}(\partial
U)\to\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ the induced homomorphism.

Consider the presentations
$\langle\gamma_{1},\ldots,\gamma_{n+1},c_{1},\ldots,c_{g}\mid
R^{\prime}\rangle$ of $\pi_{1}(\partial U)$ with $R^{\prime}$ the set of
relations as in Theorem 4.1 and
$\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\cup
R_{k},\lambda_{n+1}\cdots\lambda_{1}\rangle$ of
$\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ as in Theorem 3.12.

Recall that the construction of the meridian $\gamma_{k}$ around the
irreducible component $D_{k}$ of $D=\sum_{k=1}^{n+1+s_{0}}D_{k}$ depends on a
choice of a maximal tree $\mathcal{T}$ of the dual graph $\Delta$ of $D$,
contractible paths $l_{k}\subset D_{k}$, and a section $h:l=\cup
l_{k}\to\partial U$, see 2.4. We choose the maximal tree
$\mathcal{T}^{\prime}$ constructed at 4.1.2. For $p=L_{\eta_{1}}\cap\ldots\cap
L_{\eta_{r}}\in\operatorname{Sing}\mathscr{A}\setminus L_{n+1}$, we have a
unique cycle $c_{\eta_{1},\eta_{2}}$ if $r=2$ and $r-1$ cycles if $r>2$, in
this case let us denote by $D_{\iota}$ the corresponding exceptional divisor
in $\bar{X}$, therefore we have the cycles
$c_{\eta_{2},\iota},\ldots,c_{\eta_{r},\iota}$. See 4.1.2.

Consider a wiring diagram $\mathcal{W}$ of $\mathscr{A}$ as in 3.1. There
exists $\kappa\in\mathbb{N}^{*}$ such that $p\in M_{\kappa}$. Consider the
geometric generating set
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$.
Recall that, as in remark 3.13, there exists a word $\xi_{j}^{(\kappa)}$ in
$\lambda_{1}^{(1)},\ldots,\lambda_{n+1}^{(1)}$ (see also 4.2.2), such that

$\lambda_{{\sigma^{(\kappa)}}^{-1}(j)}^{(\kappa)}={\lambda_{j}^{(1)}}^{\xi_{j}^{(\kappa)}}\quad\text{
for }j=1,\ldots,n+1.$

The main objective of this subsection is to prove the following Theorem.

######  Theorem 4.3.

The paths $l_{1},\ldots,l_{n+1+s_{0}}$, the map $h:l\to\partial U$ and the
wiring diagram $\mathcal{W}$ of $\mathscr{A}$ can be chosen in such a way that

  1. (1)

The generator $\gamma_{k}$ of $\pi_{1}(\partial U)$ lies in the same homotopy
class as $\lambda_{k}$ in $\mathbb{P}^{2}\setminus\mathscr{A}$ for
$k=1,\ldots,n+1$.

  2. (2)

If $p=L_{\eta_{1}}\cap\ldots\cap
L_{\eta_{r}}\in\operatorname{Sing}\mathscr{A}\setminus L_{n+1}$ and $p\in
M_{\kappa}$ as above, then

     * •

if $r=2$, the cycle $c_{\eta_{1},\eta_{2}}$ is homotopic in
$\mathbb{P}^{2}\setminus\mathscr{A}$ to
$\xi_{\eta_{1}}^{(\kappa)}(\xi_{\eta_{2}}^{(\kappa)})^{-1}$ and

     * •

if $r>2$, the cycle $c_{\eta_{a},\iota}$ is homotopic to
$\xi_{\eta_{a}}^{(\kappa)}{\xi_{\eta_{1}}^{(\kappa)}}^{-1}$, for
$a=2,\ldots,r$.

By the point (1), we can also consider each $\xi_{\eta_{a}}^{(\kappa)}$ as a
word in $\gamma_{1},\ldots,\gamma_{n+1}$.

  3. (3)

If $p=L_{\eta_{1}}\cap\ldots\cap
L_{\eta_{r}}\in\operatorname{Sing}\mathscr{A}\setminus L_{n+1}$, denote by
$R^{\prime}(p)$ the set of relations:

     * •

$\\{c_{\eta_{1},\eta_{2}}^{-1}\xi_{\eta_{1}}^{(\kappa)}(\xi_{\eta_{2}}^{(\kappa)})^{-1}\\}$
if $r=2$ , or

     * •

$\\{c_{\eta_{a},\iota}^{-1}\xi_{\eta_{a}}^{(\kappa)}(\xi_{\eta_{1}}^{(\kappa)})^{-1}\mid
a=2,\ldots,r\\}$ if $r>2$.

We have that $\langle\gamma_{1},\ldots,\gamma_{n+1},c_{1},\ldots,c_{g}\mid
R^{\prime},\cup_{p\in\operatorname{Sing}\mathscr{A}\setminus
L_{n+1}}R^{\prime}(p)\rangle$ and
$\langle\lambda_{1},\ldots,\lambda_{n+1}\mid\cup
R_{k},\lambda_{n+1}\cdots\lambda_{1}\rangle$ are Tietze-equivalent
presentations of $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$.

By using a different presentation of $\pi_{1}(\partial U)$ and different
techniques, the image of the generators of $\pi_{1}(\partial U)$ under the map
$i_{*}$ was computed in [FGBMB15] (See Proposition 2.13 and Theorem 4.5 of
loc. cit.). The proof of Theorem 4.3 is inspired by the ideas of [FGBMB15].

####  4.2.1. Constructing equivalent generators

Let us choose the point $R\in\mathbb{P}^{2}\setminus\mathscr{A}$ close to
$L_{n+1}$, consider the blow-up
$\pi_{R}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{2}$ and denote by
$\bar{f}:Bl_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$ the associated pencil as in
3.1.

Let $\beta:[0,1]\to\mathbb{P}^{1}$ be as in 3.1 such that it passes first
through the projection of the points $\operatorname{Sing}\mathscr{A}\cap
L_{n+1}$ to $\mathbb{P}^{1}$ via $\bar{f}$. Take its associated wiring diagram
$\mathcal{W}$ corresponding to the arrangement $\mathscr{A}$ and fix a planar
representation $p(\beta^{*}\mathcal{W})$ as in 3.1.1.

Let us order the representation of all the singular points
$\operatorname{Sing}\mathscr{A}=\\{p_{1},\ldots,p_{s}\\}$ in
$p(\beta^{*}\mathcal{W})$ together with the virtual vertices
$\\{p_{1}^{\prime},\ldots,p_{\nu}^{\prime}\\}\in p(\beta^{*}\mathcal{W})$, by
the order they are crossed by the fiber $p(\beta^{*}\mathcal{W})|_{t}$ with
$t$ increasing in $[0,1]$, and let $t_{1},\ldots,t_{s+\nu}\in(0,1)$ be such
that either an actual or a virtual vertex lies in
$p(\beta^{*}(\mathcal{W}))|_{t_{\kappa}}$, for all $\kappa=1,\ldots,s+\nu$. By
abuse of notation we will also denote by $t_{\kappa}$ the crossings in
$p(\beta^{*}\mathcal{W})$ at the fiber
$p(\beta^{*}(\mathcal{W}))|_{t_{\kappa}}$ and we will write $\mathcal{W}$ for
$p(\beta^{*}\mathcal{W})$. Let
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
be the geometric generating set defined in 3.2.1, for $\kappa=1,\ldots,s+\nu$.

Recall that we have assumed that the order of the lines $L_{1},\ldots,L_{n}$
is such that, at the very right of the planar representation of $\mathcal{W}$,
the wire $W_{1}$ is at the bottom of $\mathcal{W}$, above it is the wire
$W_{2}$ and then $W_{3}$, continuing in this way until $W_{n}$.

For an irreducible component $D_{k}$ of $D\subset\bar{X}$, denote its boundary
manifold by $\psi_{k}:\partial U_{k}\to D_{k}$ and recall that we can consider
$\psi_{k}|_{\partial U_{k}^{*}}:\partial U_{k}^{*}=\partial U_{k}\cap\partial
U\to D_{k}^{*}$ (see 2.4). A set of generators for $\pi_{1}(\partial
U_{k}^{*})$ was constructed by fixing a base point $Q_{k}\in D_{k}^{*}$,
simple paths $l_{k}\subset D_{k}$ from which we obtain paths
$l_{k}^{\prime}\subset D_{k}^{*}$ (see figure 1) and $h:\cup l_{k}\to\partial
U$ as in 2.4. The generators $\gamma_{1},\ldots,\gamma_{n+1+s}$ were
constructed by joining the different generators of $\pi_{1}(\partial
U_{k}^{*})$ to a common base point $Q$ via the contractible path $h(\cup
l_{k})$ in $\partial U$.

Recall that the first $n+1$ irreducible components $D_{1},\ldots,D_{n+1}$ of
$D$ correspond to the lines $L_{1},\ldots,L_{n+1}$ respectively and that, as
in the end of 4.1.1, there exists $s^{\prime}$ such that for
$j=n+2,\ldots,s^{\prime}$, we have that $D_{n+1}\cap D_{j}\not=\varnothing$
and for $j>s^{\prime}$, we have $D_{n+1}\cap D_{j}=\varnothing$.

######  Lemma 4.4.

For $k=n+1,\ldots,s^{\prime}$, we can choose $l_{k}^{\prime}\subset
D_{k}^{*}$, a continuous map $h_{k}^{\prime}:l_{k}^{\prime}\to\partial
U_{k}^{*}$ and a base point $Q\in h_{n+1}(l_{n+1}^{\prime})$ for the
fundamental group $\pi_{1}(\partial U)$ in such a way that $i_{*}(\gamma_{r})$
lies in the same homotopy class as $\lambda_{r}^{(1)}$ for $r=1,\ldots,n+1$.

###### Proof.

We begin by defining those $l_{k}^{\prime}$ for $k=n+2,\ldots,s^{\prime}$.
Essentially, we arrange the choices in an appropriate way to obtain the stated
in the lemma.

More precisely, let $D_{k}$ be an exceptional divisor corresponding to a point
$p=p(k)\in\operatorname{Sing}\mathscr{A}\cap L_{n+1}$ with multiplicity higher
or equal to three. Suppose that $p=L_{j}\cap L_{j+1}\cap\ldots\cap L_{m-1}\cap
L_{m}\cap L_{n+1}$ (which can be written in this way by the order of the lines
chosen above). Consider the boundary manifold $\psi_{k}:\partial U_{k}\to
D_{k}$ in $\bar{X}$. We will also write $\partial U_{p}$ for the image
$\pi(\partial U_{k})\subset\mathbb{P}^{2}$ under the map
$\pi:\bar{X}\to\mathbb{P}^{2}$. For $r=j,\ldots,m$, each meridian
$\lambda_{r}^{(1)}$ (see figure 3(a)) is homotopic to a meridian
${\lambda_{r}^{\prime}}^{(1)}$ (see figure 6) that can be decomposed in the
following way:
${\lambda_{r}^{\prime}}^{(1)}=\lambda_{p}\lambda_{r_{1}}\lambda_{p}^{-1}$ with
$\lambda_{r_{1}}\subset\partial U_{p}$ a meridian of $L_{r}$ based at a point
$q_{p}\in\partial U_{p}$ and $\lambda_{p}$ a path connecting $R$ and $q_{p}$.

We can further decompose each $\lambda_{r_{1}}$ as the boundary of a disk
$\Delta$ around a point in $L_{r}$ and a path $\lambda_{r_{2}}$ connecting the
point $q_{p}$ to $\partial\Delta$. Define the path $l_{k}^{\prime}$ in
$D_{k}^{*}$ as the projection $\psi_{k}(\cup_{r=j}^{m}\lambda_{r_{2}})$. We
define $h|_{l_{k}^{\prime}}$ such that
$h|_{l_{k}^{\prime}}\circ\psi_{k}|_{\cup_{r=j}^{m}\lambda_{r_{2}}}=id|_{\cup_{r=j}^{m}\lambda_{r_{2}}}$.

${\lambda_{j}^{\prime}}^{(1)}$${\lambda_{j+1}^{\prime}}^{(1)}$${\lambda_{m}^{\prime}}^{(1)}$$\partial\Delta$${\lambda_{m_{2}}}$${\lambda_{j_{2}}}$${\lambda_{p}}$$R$$q_{p}$••••
Figure 6. Decomposing a meridian

Note that, up to a slight change in $R$, the paths $\lambda_{p}$ are homotopic
to paths $\lambda_{p}^{\prime}$ lying in $\partial U_{n+1}^{*}$.

Now, let $p=L_{j}\cap L_{n+1}$ be a double point in
$\operatorname{Sing}\mathscr{A}\cap L_{n+1}$. The meridian $\lambda_{j}^{(1)}$
can be decomposed as
$\lambda_{j}^{\prime}\cdot\partial\Delta_{j}\cdot\lambda_{j}^{\prime-1}$, with
$\partial\Delta_{j}$ a fiber of $\partial U_{j}^{*}$ and
$\lambda_{j}^{\prime}\subset\partial U_{n+1}^{*}$ a path starting at $R$ and
finishing at point $q_{j}\in\partial U_{j}^{*}\cap\partial U_{n+1}^{*}$.

Finally, for $k=n+1$ we define $l_{k}\in D_{k}$ as the image of $\beta([0,1])$
under the section of the pencil
$\bar{f}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$ with range
$D_{k}$. By construction $l_{k}$ passes over all the points in
$\operatorname{Sing}\mathscr{A}\cap L_{n+1}$. We let $h|_{l_{n+1}}$ be a
continuous function such that $\psi_{n+1}|_{h|_{l_{n+1}}(l_{n+1})}\circ
h|_{l_{n+1}}=id_{l_{n+1}}$, $h|_{l_{n+1}}(l_{n+1})$ is a simple path passing
through each $q_{j}$ with $L_{j}\cap L_{n+1}$ a double point, touching each
$\lambda_{(n+1)_{2}}(p)$ for each point
$p\in\operatorname{Sing}\mathscr{A}\cap L_{n+1}$ of multiplicity greater or
equal to two, and such that each $\lambda_{p}\cdot\lambda_{j_{2}}$ is
homotopic to a segment of $h|_{l_{n+1}}(l_{n+1})$.

By the construction of the maximal tree $\mathcal{T}^{\prime}$, these paths
are sufficient to construct $\gamma_{j}$ for $j=1,\ldots,n+1$ and by
construction, they lie in the same homotopy class as $\lambda_{j}^{(1)}$. ∎

######  Corollary 4.5.

The morphism $i_{*}:\pi_{1}(\partial
U)\to\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ is surjective.

###### Proof.

The group $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ is generated by the
elements $\Gamma^{(1)}=\\{\lambda_{1}^{(1)},\ldots,\lambda_{n+1}^{(1)}\\}$, as
$i_{*}(\gamma_{k})=\lambda_{k}^{(1)}$ for $k=1,\ldots,n+1$ the result follows.
∎

Suppose that $p\in M_{\kappa}\cap(\operatorname{Sing}\mathscr{A}\setminus
L_{n+1})$ is of multiplicity higher or equal to three and that
$p=L_{\sigma^{(\kappa)}(j)}\cap L_{\sigma^{(\kappa)}(j+1)}\cap\ldots\cap
L_{\sigma^{(\kappa)}(m)}$. Denote by $\psi_{p}:\partial U_{p}\to E_{p}$ the
boundary manifold of the exceptional divisor $E_{p}\subset\bar{X}$ obtained by
blowing-up $p$. We select $l_{p}^{\prime}\subset E_{p}^{*}$ in a similar way
as in the proof of the precedent Lemma for a point of multiplicity higher than
two lying in $L_{n+1}$: decompose each
$\gamma_{j}^{(\kappa)},\ldots,\gamma_{m}^{(\kappa)}$ into a path
$\lambda_{p}^{(\kappa)}$ connecting $R\subset\mathbb{P}^{2}$ and a point
$q_{p}\in\partial U_{p}$, and ${\lambda_{r_{1}}}^{(\kappa)}$ with
$r\in\\{j,\ldots,m\\}$ based at $q_{p}$ and generating $\pi_{1}(\partial
U_{p}^{*})$ as in figure 6. Decompose further ${\lambda_{r_{1}}}^{(\kappa)}$
into a boundary of a disk $\partial\Delta_{r}$ around a point of the line
$L_{\sigma^{(\kappa)}(r)}$ and a path ${\lambda_{r_{2}}}^{(\kappa)}$
connecting $q_{p}$ and $\partial\Delta_{r}$. We take
$l_{p}^{\prime}=\psi_{p}(\cup{\lambda_{r_{2}}}^{(\kappa)})$ and define
$h|_{l_{p}^{\prime}}$ such that
$h|_{l_{p}^{\prime}}\circ\psi_{p}|_{\cup{\lambda_{r_{2}}}^{(\kappa)}}=id|_{\cup{\lambda_{r_{2}}}^{(\kappa)}}$.

For every $k=1,\ldots,n$, we define $l_{k}\subset D_{k}$ as the image of
$\beta([0,1])$ under the section of
$\bar{f}:\operatorname{Bl}_{R}\mathbb{P}^{2}\to\mathbb{P}^{1}$ that has as
range $D_{k}$. We define $h|_{l_{k}}$ such that it is continuous,
$\psi_{k}|_{h|_{l_{k}}(l_{k})}\circ h_{l_{k}}=id_{l_{k}}$, $h_{l_{k}}(l_{k})$
intersects $\cup\lambda_{r_{2}}^{(\kappa)}$ in a point if $p\in
M_{\kappa}\cap(\operatorname{Sing}\mathscr{A}\cap L_{k})$ with the notations
as in the paragraph above, $h_{l_{k}}(l_{k})\cap
h_{l_{k^{\prime}}}(l_{k^{\prime}})\not=\varnothing$ if $L_{k}\cap
L_{k^{\prime}}\not=\varnothing$ is a double point and $h_{l_{k}}(l_{k})$ is
not homotopic to a multiple of a fiber $S^{1}$ of $\partial U_{k}$.

####  4.2.2. Expressing the cycles in terms of the meridians

Let $t_{\eta}\in\mathcal{W}$ be an actual vertex and suppose that
$t_{\eta}=W_{\eta_{1}}\cap W_{\eta_{2}}\cap\ldots\cap W_{\eta_{r}}$ with the
global order of the wires of $\mathcal{W}$ such that
$\eta_{1}<\eta_{2}<\ldots<{\eta_{r}}<n+1$. By definition of the maximal tree
$\mathcal{T}^{\prime}$, to each $\eta_{a}$, with $a>1$, corresponds a cycle
$c_{\eta_{a},t_{\eta}}$ which is a generator of $\pi_{1}(\partial U)$, see
4.1.2. This cycle is constructed by connecting
$h|_{\eta_{a}}(l_{\eta_{a}})\cdot h|_{t_{\eta}}(l_{t_{\eta}}^{\prime})\cdot
h|_{\eta_{1}}(l_{\eta_{1}})$ to $R$ if $r>2$ and by connecting
$h|_{\eta_{1}}(l_{\eta_{1}})\cdot h|_{\eta_{a}}(l_{\eta_{a}})$ to $R$ if
$r=2$.

For every $\kappa\leq\eta$, consider the geometric generating set
$\Gamma^{(\kappa)}=\\{\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)}\\}$
as in 3.2.1 and recall the construction of the functions
$\tau^{(\kappa)}:\\{1,\ldots,n+1\\}\to F_{n+1}^{(\kappa)}$ as defined before
Proposition 3.11. For $1\leq a\leq r$, denote by

(4)
$\xi_{\eta_{a}}^{(\kappa)}=\tau^{(1)}(\eta_{a})\cdot\tau^{(2)}\left({\sigma^{(2)}}^{-1}(\eta_{a})\right)\ldots\tau^{(\kappa-1)}\left({\sigma^{(\kappa-1)}}^{-1}(\eta_{a})\right).$

######  Proposition 4.6.

Let $1<a\leq r$. The image of the cycle $c_{t_{\eta},\eta_{a}}$ under the map
$i_{*}$ equals $\xi_{\eta_{1}}^{(\eta)}(\xi_{\eta_{a}}^{(\eta)})^{-1}$ if
$r=2$ or $\xi_{\eta_{a}}^{(\eta)}(\xi_{\eta_{1}}^{(\eta)})^{-1}$ if $r>2$.

We consider the points $\theta_{\kappa}<t_{\kappa}$ very close to $t_{\kappa}$
as before Definition 3.3.

######  Lemma 4.7.

Let
$\Gamma^{(\kappa)}=(\lambda_{1}^{(\kappa)},\ldots,\lambda_{n+1}^{(\kappa)})$
be a generating set as above. Then, for $\iota=1,\ldots,n+1$ we have that
$(\lambda_{\iota}^{(\kappa)})^{{\xi_{\eta_{a}}^{(\kappa)}}^{-1}}$ is homotopic
to a meridian of $L_{\sigma^{(\kappa)}(\iota)}$ at the point
$x_{\iota}^{(\kappa)}=\bar{f}^{-1}{(\beta(\theta_{\kappa})})\cap
L_{\sigma^{(\kappa)}(\iota)}$ constructed by:

  1. (1)

following $h|_{\eta_{a}}(l_{\eta_{a}})$ until
$\bar{f}^{-1}{(\beta(\theta_{\kappa})})$,

  2. (2)

then joining it to a circle in $\bar{f}^{-1}{(\beta(\theta_{\kappa})})$ about
$x_{\iota}^{(\kappa)}$ and,

  3. (3)

coming back via $h|_{\eta_{a}}(l_{\eta_{a}})$.

See figure 7.

$(\gamma_{\iota}^{(2)})^{{\tau^{(1)}(\eta_{a})}^{-1}}$$\vdots$$x_{{\sigma^{(2)}}^{-1}(\eta_{a})}^{(2)}$$x_{\iota}^{(2)}$$x_{\iota-1}^{(2)}$•••
Figure 7. A meridian follows another boundary manifold

###### Proof.

Note that if $\sigma^{(\kappa)}(\iota)=\eta_{a}$ and as
$\lambda_{\iota}^{(\kappa)}=\lambda_{\eta_{a}}^{\xi_{\eta_{a}}^{(\kappa)}}$,
by successive applications of Proposition 3.11, we can choose a meridian in
the homotopy class of
$(\lambda_{\iota}^{(\kappa)})^{{\xi_{\eta_{a}}^{(\kappa)}}^{-1}}=\lambda_{\eta_{a}}$
that satisfies the properties stated in the Lemma (see figure 3(b)).

Now, if $\sigma^{(\kappa)}(\iota)\not=\eta_{a}$, we proceed by induction. Let
$t_{1}=W_{j}\cap W_{j+1}\cap\ldots\cap W_{m}$ and consider

$\xi_{\eta_{a}}^{(2)}=\tau^{(1)}(\eta_{a})=\left\\{\begin{array}[]{ll}\lambda_{\eta_{a}-1}^{(1)}\cdots\lambda_{j}^{(1)}&\text{if
}n_{a}\in\\{j+1,\ldots,m\\},\\\ 1&\text{if
}\eta_{a}\not\in\\{j+1,\ldots,m\\}.\end{array}\right.$

By construction, for $k\in\\{j+1,\ldots,m\\}$ the product
$\lambda_{k}^{(1)}\ldots\lambda_{j}^{(1)}$ is freely homotopic to a circle
containing the points $x_{j}^{(1)},\ldots,x_{k}^{(1)}$.

Now, note that the paths in $\Gamma^{(1)}$ are homotopic to paths in the fiber
$\bar{f}^{-1}({\beta(\theta_{2})})$ as in Figure 3(b). Such representative of
the homotopy class of $\lambda_{\iota}^{(1)}$ can be seen as lying in the
boundary manifold $\partial U_{\iota}^{*}$.

By considering $(\lambda_{\iota}^{(2)})^{{\xi_{\eta_{a}}^{(2)}}^{-1}}$ we
obtain a path as in figure 7 if $\tau^{(1)}(\eta_{a})\not=1$. This meridian
can be decomposed as stated.

For a general $\Gamma^{(\kappa+1)}$, note that as
$\xi_{\eta_{a}}^{(\kappa+1)}=\xi_{\eta_{a}}^{(\kappa)}\cdot\tau^{(\kappa)}(\sigma^{{(\kappa)}^{-1}}(\eta_{a}))$
and by repeating the above procedure, we can decompose
$(\lambda_{\iota}^{(\kappa+1)})^{{\tau^{(\kappa)}}^{-1}({\sigma^{(\kappa)}}^{-1}(\eta_{a}))}$
as a meridian of $L_{\sigma^{(\kappa+1)}(\iota)}$ that follows
$l_{\eta_{a}}^{\prime}$ between $\bar{f}^{-1}{(\beta(\theta_{\kappa})})$ and
$\bar{f}^{-1}({\beta(\theta_{\kappa+1})})$ (see figure 7). By applying
induction, we obtain that
$(\lambda_{\iota}^{(\kappa+1)})^{{\xi_{\eta_{a}}^{(\kappa+1)}}^{-1}}$ can be
decomposed as stated in the lemma. ∎

###### Proof of Proposition 4.6.

Note that we have that

$\displaystyle\xi_{\eta_{a}}^{(\eta)}$
$\displaystyle=\tau^{(1)}(\eta_{a})\cdots\tau^{(\eta-1)}({\sigma^{(\eta-1)}}^{-1}(\eta_{a}))$
$\displaystyle=\tau^{(\eta-1)}\left({\sigma^{(\eta-1)}}^{-1}(\eta_{a})\right)^{{\xi_{\eta_{a}}^{(\eta-1)}}^{-1}}\cdots\tau^{(2)}\left({\sigma^{(2)}}^{-1}(\eta_{a})\right)^{{\xi_{\eta_{a}}^{(2)}}^{-1}}\tau^{(1)}(\eta_{a}).$

by using
$\xi_{\eta_{a}}^{(\kappa)}=\xi_{\eta_{a}}^{(\kappa-1)}\cdot\tau^{(\kappa-1)}\left({\sigma^{(\kappa-1)}}^{-1}(\eta_{a}))\right)$
for $\kappa=2,\ldots,\eta$.

Now, if $\tau^{(\kappa)}({\sigma^{(\kappa)}}^{-1}(\eta_{a}))\not=1$, it is
homotopic to a path in $\bar{f}^{-1}{(\beta(\theta_{\kappa})})$ encircling the
points $x_{j}^{(\kappa)},\ldots,$
$x_{{\sigma^{(\kappa)}}^{-1}(\eta_{a})}^{(\kappa)}$ and by applying Lemma 4.7
to each factor of
$\tau^{(\kappa)}({\sigma^{(\kappa)}}^{-1}(\eta_{a}))=\lambda_{{\sigma^{(\kappa)}}^{-1}(\eta_{a})}^{(\kappa)}\ldots\lambda_{j}^{(\kappa)}$
we obtain that $\tau^{(\kappa)}({\sigma^{(\kappa)}}^{-1}(\eta_{a}))$ can be
decomposed in three parts as in Lemma 4.7.

Recall that we have constructed the $l_{\eta_{a}}\subset D_{\eta_{a}}$ from a
section of the map $\bar{f}$. By the choice of $h|_{\eta_{a}}$ we can suppose
that $h_{\eta_{a}}\subset\bar{f}^{-1}(\beta[0,1])\cap\partial U_{\eta_{a}}$.

By considering
$Y=\bar{f}^{-1}(\beta([0,1]))\setminus\pi_{R}^{-1}(R)\subset\operatorname{Bl}_{R}\mathbb{P}^{2}$,
we can see the cycles $c_{t_{\eta},\eta_{a}}\subset Y\subset\mathbb{R}^{3}$.
Moreover, we can choose coordinates in $\mathbb{R}^{3}$ and define that
$h|_{\eta_{a}}(l_{\eta_{a}})$ passes above $h|_{l_{k}}(l_{k})$ (or
$h|_{l_{k}}(l_{k})$ passes below $h|_{l_{\eta_{a}}}(l_{\eta_{a}})$) in some
fiber $\bar{f}^{-1}({\beta(\theta^{\prime})})$ with
$\theta^{\prime}\in[t_{\kappa}-\varepsilon,t_{\kappa}+\varepsilon]$ with
$\varepsilon>0$ sufficiently small, if the wires $W_{\eta_{a}}\cap
W_{k}\not=\emptyset$ in a planar representation of the fiber
$\bar{f}^{-1}({\beta(t_{\kappa})})$ and
${\sigma^{(\kappa)}}^{-1}(\eta_{a})<{\sigma^{(\kappa)}}^{-1}(k)$.

We can see then
$(\tau^{(\kappa)}({\sigma^{(\kappa)}}(\eta_{a})))^{{\xi_{\eta_{a}}^{(\kappa)}}^{-1}}$
as a path encircling the lines $L_{k}$ corresponding to those
$h|_{l_{k}}(l_{k})$ passing below $h|_{l_{\eta_{a}}}(l_{\eta_{a}})$ in some
fiber $\bar{f}^{-1}({\beta^{\prime}(\theta^{\prime})})$ with
$\theta^{\prime}\in[t_{\kappa}-\varepsilon,t_{\kappa}+\varepsilon]$. By
construction, $\xi_{\eta_{a}}^{(\eta)}$ is homotopic to a path encircling all
the lines $L_{k}$ such that $h|_{l_{k}}(l_{k})$ lies below
$h_{l_{\eta_{a}}}(l_{\eta_{a}})$ at some point in $\beta([0,t_{\eta}])$.

Therefore, we can decompose $\xi_{\eta_{a}}^{(\eta)}$ in three parts:

  1. (1)

The first path starting at $Q\in h(l_{n+1})$ and following $h(l_{\eta_{a}})$
until $\bar{f}^{-1}({\beta(\theta_{\eta})})$. Then,

  2. (2)

a simple path starting at
$h(l_{\eta_{a}})\cap\bar{f}^{-1}({\beta(\theta_{\eta})})$, lying completely in
$\bar{f}^{-1}({\beta(\theta_{\eta})})$ and finishing at
$\bar{f}^{-1}({\beta(\theta_{\eta})})\cap\pi_{R}^{-1}(R)$, and

  3. (3)

a path connecting $\pi_{R}^{-1}(R)\cap\bar{f}^{-1}({\beta(\theta_{\eta})})$ to
$Q\subset\pi_{R}^{-1}(R)$.

By decomposing in a similar fashion $\xi_{\eta_{1}}^{(\eta)}$, it follows that
the cycle $c_{t_{\eta},\eta_{a}}$ is homotopic in
$\mathbb{P}^{2}\setminus\mathscr{A}$ to
$\xi_{\eta_{1}}^{(\eta)}(\xi_{\eta_{a}}^{(\eta)})^{-1}$ if $r=2$ and to
$\xi_{\eta_{a}}^{(\eta)}(\xi_{\eta_{1}}^{(\eta)})^{-1}$ if $r>2$. ∎

####  4.2.3. Expressing the relations in terms of the generators

For every $r=n+2,\ldots,n+s_{0}+1$, we let $R_{r}^{\prime}$ be the subset of
the set of relations $R^{\prime}$ of the presentation of $\pi_{1}(\partial U)$
as in Theorem 4.1 such that

$R_{r}^{\prime}=\\{[\gamma_{k},\gamma_{r}^{s_{kr}}],\gamma_{r}^{-w_{r}}=\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}\mid(k,r)\in\mathcal{E}\\}$

with

$s_{kr}=\left\\{\begin{array}[]{ll}c_{\iota}^{-1}&\text{ if }(k,r)\text{
equals the $\iota$-th element in }\mathcal{C},\\\ c_{\iota}&\text{ if
}(r,k)\text{ equals the $\iota$-th element in }\mathcal{C},\\\ 1&\text{ if
}(k,r)\text{ is an edge of }\mathcal{T}^{\prime}.\\\ \end{array}\right.$

######  Proposition 4.8.

Consider an exceptional divisor $E_{\kappa}=D_{r}\subset\bar{X}$ coming from a
singular point $t_{\kappa}\in\operatorname{Sing}\mathscr{A}\cap M_{\kappa}$ of
multiplicity higher or equal to $3$. The image of the set of relations
$R_{r}^{\prime}$ as above, under the map $i_{*}$, equals the set of relations
$R_{\kappa}=[\lambda_{m}^{(\kappa)},\ldots,\lambda_{j}^{(\kappa)}]$ as in
Lemma 3.8.

###### Proof.

Let $t_{\kappa}=W_{\sigma^{(\kappa)}(j)}\cap
W_{\sigma^{(\kappa)}(j+1)}\cap\ldots\cap W_{\sigma^{(\kappa)}(m)}$ with the
local order given by
$\Pi^{(\kappa)}=\\{\sigma^{(\kappa)}(1)<\ldots<\sigma^{(\kappa)}(n+1)\\}$. As
$w_{r}=-1$ and by the local order of the wires we have that
$\gamma_{r}=\gamma_{\sigma^{(\kappa)}(m)}^{c_{\sigma^{(\kappa)}(m),t_{\kappa}}}\cdots\gamma_{\sigma^{(\kappa)}(j)}^{c_{\sigma^{(\kappa)}(j),t_{\kappa}}}$,
that $[\gamma_{r},\gamma_{\sigma(\kappa)}^{c_{\sigma(\kappa),t_{\kappa}}}]$
and that
$c_{\sigma^{(\kappa)}(k),t_{\kappa}}=c_{t_{\kappa},\sigma^{(\kappa)}(k)}^{-1}$.

Let us omit the superscript $\lambda_{\kappa}=\lambda_{\kappa}^{(1)}$ for the
elements in $\Gamma^{(1)}$.

By considering the image under $i_{*}$ of the elements in $R_{r}^{\prime}$, we
have by Lemma 4.4 and by Proposition 4.6 that

$i_{*}(\gamma_{r})=\lambda_{\sigma^{(\kappa)}(m)}^{\xi_{\sigma^{(\kappa)}(m)}^{(\kappa)}{\xi_{\iota}^{(\kappa)}}^{-1}}\cdots\lambda_{\sigma^{(\kappa)}(j+1)}^{\xi_{\sigma^{(\kappa)}(j+1)}^{(\kappa)}{\xi_{\iota}^{(\kappa)}}^{-1}}\lambda_{\sigma^{(\kappa)}(j)}^{\xi_{\sigma^{(\kappa)}(j)}^{(\kappa)}{\xi_{\iota}^{(\kappa)}}^{-1}},\quad[i_{*}(\gamma_{r}),\lambda_{\sigma^{(\kappa)}(k)}^{\xi_{\sigma^{(\kappa)}(k)}^{(\kappa)}{\xi_{\iota}^{(\kappa)}}^{-1}}].$

with
$\iota=\min\\{\sigma^{(\kappa)}(j),\sigma^{(\kappa)}(j+1),\ldots,\sigma^{(\kappa)}(m)\\}$.

The commutators can also be written as
$[i_{*}(\gamma_{r})^{\xi_{\iota}^{(\kappa)}},\lambda_{\sigma^{(\kappa)}(k)}^{\xi_{\sigma^{(\kappa)}(k)}^{(\kappa)}}]$
. But as
$i_{*}(\gamma_{r})^{\xi_{\iota}^{(\kappa)}}=\lambda_{\sigma^{(\kappa)}(m)}^{\xi_{\sigma^{(\kappa)}(m)}^{(\kappa)}}\cdots\lambda_{\sigma^{(\kappa)}(j+1)}^{\xi_{\sigma^{(\kappa)}(j+1)}^{(\kappa)}}\lambda_{\sigma^{(\kappa)}(j)}^{\xi_{\sigma^{(\kappa)}(j)}^{(\kappa)}},$
we have that the relations
$[i_{*}(\gamma_{r})^{\xi_{\iota}^{(\kappa)}},\lambda_{\sigma^{(\kappa)}(k)}^{\xi_{\sigma^{(\kappa)}(k)}^{(\kappa)}}]$
can be condensed as
$[\lambda_{\sigma^{(\kappa)}(m)}^{\xi_{\sigma^{(\kappa)}(m)}^{(\kappa)}},\cdots,\lambda_{\sigma^{(\kappa)}(j+1)}^{\xi_{\sigma^{(\kappa)}(j+1)}^{(\kappa)}},\lambda_{\sigma^{(\kappa)}(j)}^{\xi_{\sigma^{(\kappa)}(j)}^{(\kappa)}}]$.
Now, if
$R_{\kappa}=[\lambda_{m}^{(\kappa)},\ldots,\lambda_{j+1}^{(\kappa)},\lambda_{j}^{(\kappa)}]$
denotes the relation given in Theorem 3.12 for the point $t_{\kappa}$ as in
Lemma 3.8, recall that we have the equality
$\lambda_{k}^{(\kappa)}=\lambda_{\sigma^{(\kappa)}(k)}^{\xi_{\sigma^{(\kappa)}(k)}}$.
By replacing it in the commutators above, the result follows. ∎

######  Proposition 4.9.

For $r=1,\ldots,n+1$, we have the equality

$i_{*}(\gamma_{r}^{w_{r}}\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}})=\gamma_{n+1}\cdots\gamma_{1}$

in $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ with $s_{rj}$ as above.

###### Proof.

Fix $L_{r}\in\mathscr{A}$. Let
$\\{p_{1},\ldots,p_{b}\\}\subset\operatorname{Sing}\mathscr{A}\cap L_{r}$ be
the singular points of the arrangement lying in $L_{r}$. Note that
$b=k_{r}^{\prime}$. Indeed, we can find a partition $A\cup
B=\\{1,\ldots,k_{r}^{\prime}\\}$, with $A$ a set indexing the double points of
$\operatorname{Sing}\mathscr{A}\cap L_{r}$, and $B$ indexing the points of
multiplicity strictly bigger than two. Let $\pi:\bar{X}\to\mathbb{P}^{2}$ be
the blow-up of $\mathbb{P}^{2}$ at the points of
$\operatorname{Sing}\mathscr{A}$ of multiplicity strictly bigger than two and
let $D_{r}$ denote the strict transform of $L_{r}$ in $\bar{X}$. We have that
$A$ also indexes all the strict transforms of lines in $\mathscr{A}$ which
have no empty intersection with $D_{r}$, and $B$ the exceptional divisors of
$\bar{X}$ crossing $D_{r}$. It is clear then that $b=k_{r}^{\prime}$.

It follows that $\gamma_{j_{\Delta}(r,m)}$ is a meridian of an irreducible
component $D_{j_{\Delta}(r,m)}$ of $D=\pi^{*}D$ for
$m=1,\ldots,k_{r}^{\prime}$. Recall that $\gamma_{r}$ commutes with
$\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}$ and note that the self-
intersection number $w_{r}$ of $D_{r}$ is $1-\left|B\right|$.

Let us study the geometric meaning of the product
$\gamma_{r}^{-1}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}$ with $m\in
B$. Let us write $\iota=j_{\Delta}(r,m)$, denote by
$D_{\iota}=D_{j_{\Delta}(r,m)}$ the exceptional divisor that
$\gamma_{j_{\Delta}(r,m)}$ surrounds, and let
$D_{\iota_{1}},\ldots,D_{\iota_{k}}$ be the irreducibles components of
$D=\pi^{*}\mathscr{A}$ that intersect $D_{\iota}$ ordered in such a way that,
if we denote by $\gamma_{\iota_{j}}$ the meridians around $D_{\iota_{j}}$ used
for the presentation of $\pi_{1}(\partial U)$,
$\gamma_{\iota}^{-w_{\iota}}=\gamma_{\iota_{1}}^{s_{\iota\iota_{1}}}\cdots\gamma_{\iota_{k}}^{s_{\iota\iota_{k}}}$
holds. As $D_{\iota}$ is an exceptional divisor, we have that $w_{\iota}=-1$.
By Theorem 4.1, we have that
$[\gamma_{\iota},\gamma_{\iota_{j}}^{s_{\iota\iota_{j}}}]$ for
$j=1,\ldots,k=k(\iota)$.

Replacing the expression $\gamma_{\iota}$ as above in
$[\gamma_{\iota},\gamma_{\iota_{j}}^{s_{\iota\iota_{j}}}]$, we can show that
these commutators relations are equivalent to

$\gamma_{\iota_{1}}^{s_{\iota\iota_{1}}}\gamma_{\iota_{2}}^{s_{\iota\iota_{2}}}\cdots\gamma_{\iota_{k}}^{s_{\iota\iota_{k}}}=\gamma_{\sigma(\iota_{1})}^{s_{\iota\sigma(\iota_{1})}}\gamma_{\sigma(\iota_{2})}^{s_{\iota\sigma(\iota_{2})}}\cdots\gamma_{\sigma(\iota_{k})}^{s_{\iota\sigma(\iota_{k})}}$

where $\sigma$ runs over the cyclic permutations of the elements
$\\{\iota_{1},\ldots,\iota_{k}\\}$. Hence there exists some cyclic permutation
$\sigma^{\prime}$ such that $\sigma^{\prime}(\iota_{1})=r$ because $D_{r}$
intersects $D_{j_{\Delta}(r,m)}=D_{\iota}$. Note that $s_{r\iota}=s_{\iota
r}^{-1}=s_{\iota\sigma^{\prime}(\iota_{1})}^{-1}$ and hence
$\gamma_{r}^{-1}\gamma_{\iota}^{s_{r\iota}}=(\gamma_{\sigma^{\prime}(\iota_{2})}^{s_{\iota\sigma^{\prime}(\iota_{2})}}\cdots\gamma_{\sigma^{\prime}(\iota_{k})}^{s_{\iota\sigma^{\prime}(\iota_{k})}})^{s_{r\iota}}$
represents a loop which surrounds the lines
$L_{\sigma^{\prime}(\iota_{2})},\ldots,L_{\sigma^{\prime}(\iota_{k})}$
following $l_{r}^{\prime}$ by construction of the cycle $s_{r\iota}$.

Now, the product
$\gamma_{r}^{w_{r}}\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}$
can be written as $\gamma_{r}\prod_{m=1}^{k_{r}^{\prime}}\Upsilon_{m}$ with

$\Upsilon_{m}=\left\\{\begin{array}[]{ll}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}&\text{if
}m\in A,\\\
\gamma^{-1}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}&\text{if }m\in
B.\end{array}\right.$

by commuting $\gamma_{r}$ with
$\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}$. Note that, for
$\Upsilon_{m}$ with $m\in A$, the path $\Upsilon_{m}$ is a meridian around the
other line that intersects $D_{r}$ in the double point corresponding to $m\in
A$. Hence, by the precedent paragraph,
$\gamma_{r}\prod_{m=1}^{k_{r}^{\prime}}\Upsilon_{m}$ is a product of the
meridians of all the lines in $\mathscr{A}$ ordered in the way they intersect
$L_{r}$.

Now, by choosing a line $L$ sufficiently close to $L_{r}$ we have that the
product
$\gamma_{r}^{w_{r}}\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}}$
is a path encircling $L\setminus(L\cap\mathscr{A})$ and therefore it is
equivalent to $\lambda_{n+1}\cdots\lambda_{1}$ in
$\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$. ∎

####  4.2.4. End of proof of the Theorem 4.3

The point (1) of the Theorem is obtained by Lemma 4.4.

The point (2) follows from Proposition 4.8.

For the point (3), recall that $R^{\prime}$ denotes the set of relations for
the presentation of $\pi_{1}(\partial U)$ as in Theorem 4.1. Using the
notation of 4.2.3 we have that

$R^{\prime}\setminus\cup
R_{k}^{\prime}=\\{\gamma_{r}^{w_{r}}\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}},[\gamma_{k},\gamma_{r}^{s_{kr}}]\mid
r=1,\ldots,n+1,D_{k}\cap D_{r}\in\operatorname{Sing}\mathscr{A}\setminus
P_{0}\\}$

this is, $D_{k}\cap D_{r}$ is a double point.

By Proposition 4.9, we have that
$i_{*}(\gamma_{r}^{w_{r}}\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(r,m)}})=\lambda_{n+1}\ldots\lambda_{1}$.

By proceeding as in Proposition 4.8, it can be seen that for a double point
$p_{\kappa}=D_{k}\cap D_{r}$, the relation $[\gamma_{k},\gamma_{r}^{s_{kr}}]$
correspond to the relation $R_{\kappa}$ as in Theorem 3.12.

Hence, in $\langle\gamma_{1},\ldots,\gamma_{n+1},c_{1},\ldots,c_{g}\mid
R^{\prime},\cup_{p\in\operatorname{Sing}\mathscr{A}\setminus
L_{n+1}}R^{\prime}(p)\rangle$ the set of relations $R^{\prime}$ is equivalent
to the set of relations $\cup
R_{\kappa}\cup\\{\lambda_{n+1}\cdots\lambda_{1}\\}$.

This concludes the proof of Theorem 4.3.

####  4.2.5. Independence of the maximal tree

Let $D=\sum_{k=1}^{n+1+s_{0}}D_{k}$ the total transform of the arrangement
$\mathscr{A}$ in $\bar{X}$ and denote by $\Delta$ the dual graph of $D$ as
above.

Let $\mathcal{T}\subset\Delta$ be an arbitrary maximal tree and denote by
$G(\mathcal{T})=\\{\gamma_{1}(\mathcal{T}),\ldots,$
$\gamma_{n+1+s_{0}}(\mathcal{T}),c_{1}(\mathcal{T}),\cdots,c_{g}(\mathcal{T})\\}$,
the set of generators of $\pi_{1}(\partial U)$ as in Theorem 4.1. Recall that
these are constructed using $\mathcal{T}$. Denote by $R(\mathcal{T})$ the set
of relations given in the same Theorem.

Consider also the maximal tree $\mathcal{T}^{\prime}$ defined as in 4.1.2 and
denote by $\gamma_{1},\ldots,\gamma_{n+1+s_{0}},$ $c_{1},\ldots,c_{g}$ the
generators of $\pi_{1}(\partial U)$ as in Theorem 4.3 and by $R$ the set of
relations.

Consider the inclusion $i:\partial
U\hookrightarrow\mathbb{P}^{2}\setminus\mathscr{A}$ and fix
$i_{*}(\gamma_{1})=\lambda_{1},\ldots,i_{*}(\gamma_{n+1+s_{0}})=\lambda_{n+1+s_{0}}$
as a set of generators for $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ with
$\Gamma^{(1)}=\\{\lambda_{1},\ldots,\lambda_{n+1}\\}$ as in Theorem 4.3. For
$\iota=1,\ldots,n+1+s_{0}$, we have that $i_{*}(\gamma_{\iota}(\mathcal{T}))$
and $\lambda_{\iota}$ are meridians of the same smooth curve $D_{\iota}$,
therefore, we can express $i_{*}(\gamma_{\iota}(\mathcal{T}))$ as a conjugate
of $\lambda_{\iota}$ by elements in $\lambda_{1},\ldots,\lambda_{n+1}$. We let
$\delta_{\iota}$ denote the word in
$\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ representing $i_{*}(c_{\iota})$
in the letters $\lambda_{1},\ldots,\lambda_{n+1+s_{0}}$ and by
$\delta_{\iota}^{\prime}$ the same word in the letters
$\gamma_{1},\ldots,\gamma_{n+1+s_{0}}$ as in Theorem 4.3.

Reciprocally, by fixing
$i_{*}(\gamma_{1}(\mathcal{T})),\ldots,i_{*}(\gamma_{n+1+s_{0}}(\mathcal{T}))$
as generators of $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$, we can express
$\lambda_{\iota}$ as a conjugate of $i_{*}(\gamma_{\iota}(\mathcal{T}))$ by
elements in $i_{*}(\gamma_{1}(\mathcal{T})),\ldots,$
$i_{*}(\gamma_{n+1+s_{0}}(\mathcal{T}))$ for $\iota=1,\ldots,n+1$. The image
$i_{*}(c_{\iota}(\mathcal{T}))$ of the cycle $c_{\iota}(\mathcal{T})$ can be
expressed in terms of
$i_{*}(\gamma_{1}(\mathcal{T})),\ldots,i_{*}(\gamma_{n+1+s_{0}}(\mathcal{T}))$
for $\iota=1,\ldots,g$. We let $\delta_{\iota}(\mathcal{T})$ be this
expression when it is written in terms of
$\gamma_{1}(\mathcal{T}),\ldots,\gamma_{n+1+s_{0}}(\mathcal{T})$ such that
$\delta_{\iota}(\mathcal{T})\in\langle G(\mathcal{T})\mid
R(\mathcal{T})\rangle$.

######  Proposition 4.10.

A presentation of $\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})$ can be
obtained as follows

$\pi_{1}(\mathbb{P}^{2}\setminus\mathscr{A})\cong\langle G(\mathcal{T})\mid
R(\mathcal{T}),c_{1}(\mathcal{T})\cdot\delta_{1}(\mathcal{T})^{-1},\cdots,c_{g}(\mathcal{T})\cdot\delta_{g}(\mathcal{T}))^{-1}\rangle,$

###### Proof.

The presentations $\langle G(\mathcal{T})\mid R(\mathcal{T})\rangle$ and
$\langle\gamma_{1},\ldots,\gamma_{n+1+s_{0}},c_{1},\ldots,c_{g}\mid R\rangle$
of $\pi_{1}(\partial U)$ as in Theorem 4.1 can also be obtained as graphs of
groups (see [Hir00]). These graphs of groups are constructed over $\Delta$ as
follows: the vertices groups are given as in Lemma 2.4, the edges groups are
$\mathbb{Z}^{2}$. To each tree of $\Delta$ there correspond a presentation and
the presentations are Tietze-equivalent.

Let us fix $v_{n+1}$, the vertex corresponding to $D_{n+1}$ as a base point
for $\pi_{1}(\Delta)$ and $c_{1},\ldots,c_{g}$ a generating set. Every cycle
$c_{\iota}(\mathcal{T})\in\pi_{1}(\Delta,v_{n+1})$ can be expressed as
$c_{\iota}(\mathcal{T})=c_{\iota 1}\cdots c_{\iota r_{\iota}}$ where $c_{\iota
m}\in\\{c_{1},\ldots,c_{g}\\}$ with $m=1,\ldots,r_{\iota}$ and
$\iota=1,\ldots,g$. Therefore $i_{*}(c_{\iota}(\mathcal{T}))=i_{*}(c_{\iota
1})\cdots i_{*}(c_{\iota r_{\iota}})=\delta_{\iota 1}\cdots\delta_{\iota
r_{\iota}}$. Let us show that

$c_{\iota}(\mathcal{T})\cdot\delta_{\iota}(\mathcal{T})^{-1}=c_{\iota 1}\cdots
c_{\iota r_{\iota}}{\delta_{\iota
r_{\iota}}^{\prime}}^{-1}\cdots{\delta_{\iota
1}^{\prime}}^{-1}\in\langle\langle c_{1}\cdot{\delta_{\iota
1}^{\prime}}^{-1},\ldots,c_{g}\cdot{\delta_{g}^{\prime}}^{-1}\rangle\rangle.$

Note that

$\displaystyle(c_{\iota 1}{\delta_{\iota 1}^{\prime}}^{-1})(c_{\iota
2}{\delta_{\iota 2}^{\prime}}^{-1})^{\delta_{\iota 1}^{-1}}$
$\displaystyle=c_{\iota 1}c_{\iota 2}{\delta_{\iota
2}^{\prime}}^{-1}{\delta_{\iota 1}^{\prime}}^{-1}$ $\displaystyle\ \
\\!\vdots$ $\displaystyle(c_{\iota 1}{\delta_{\iota
1}^{\prime}}^{-1})\cdots(c_{\iota r_{\iota}}{\delta_{\iota
r_{\iota}}^{\prime}}^{-1})^{{\delta_{\iota
r_{\iota}}^{\prime}}^{-1}\cdots{\delta_{\iota 1}^{\prime}}^{-1}}$
$\displaystyle=c_{\iota 1}\cdots c_{\iota\iota_{r}}{\delta_{\iota
r_{\iota}}^{\prime}}^{-1}\cdots{\delta_{\iota 1}^{\prime}}^{-1}$

In a similar way we can prove that
$c_{\iota}\cdot{\delta_{\iota}^{\prime}}^{-1}\in\langle\langle
c_{1}(\mathcal{T})\cdot\delta_{1}(\mathcal{T})^{-1},\ldots,c_{g}(\mathcal{T})\cdot\delta_{g}(\mathcal{T})^{-1}\rangle\rangle$.
This proves that the presentations $\langle G(\mathcal{T})\mid
R(\mathcal{T}),c_{1}(\mathcal{T})\delta_{1}(\mathcal{T})^{-1},\cdots,c_{g}(\mathcal{T})\delta_{g}(\mathcal{T}))^{-1}\rangle,$
and $\langle\gamma_{1},\ldots,\gamma_{n+1+s_{0}},c_{1},\ldots,c_{g}\mid
R,c_{1}{\delta_{1}^{\prime}}^{-1},\ldots,c_{g}{\delta_{g}^{\prime}}^{-1}\rangle$
are equivalent. We conclude by Theorem 4.3. ∎

###  4.3. Boundary manifold of a partial compactification

Here we will present another presentation for the fundamental group of certain
partial compactifications $M(\mathscr{A},I,P)$, where $M(\mathscr{A},I,P)$ is
as in 2.3.1, but the lines of $D$ indexed by $I$ correspond only to
exceptional divisors, this is, $I\subset\\{n+2,\ldots,n+1+s_{0}\\}$

####  4.3.1. Inclusion of the boundary of a partial compactification

Let us recall the notation of section 2.3.1.

Let $\mathscr{A}\subset\mathbb{P}^{2}$ be an arrangement of lines and
$\bar{X}$ the blow-up at the points $P_{0}=\\{p_{1},\ldots,p_{s_{0}}\\}$ of
$\operatorname{Sing}\mathscr{A}$ with multiplicity strictly higher than two
and let $D=\sum_{k=1}^{n+1+s_{0}}D_{k}$ be the reduced total transform of
$\mathscr{A}$ in $\bar{X}$.

Here, _we suppose that_ $I\subset\\{n+2,\ldots,n+1+s_{0}\\}$ and let
$P=\\{p_{1}^{\prime},\ldots,p_{s_{1}}^{\prime}\\}\subset\operatorname{Sing}\sum_{k\not\in
I}D_{k}$. Denote by $\pi^{\prime}:\operatorname{Bl}_{P}\bar{X}\to\bar{X}$ the
blow-up map and the dual graph of $\left|\pi^{\prime*}D\right|$ by $\Delta$.
Note that in the previous section $\Delta$ denoted instead the dual graph of
$D$. Consider the divisor $D^{\prime}\subset\operatorname{Bl}_{P}\bar{X}$ as
in 2.3.1 and denote by $\Delta^{\prime}$ the dual graph of $D^{\prime}$.
Recall that $\Delta^{\prime}$ is obtained from $\Delta$ by removing some
vertices and the corresponding adjacent edges.

In 2.3.1 we defined the partial compactification $M(\mathscr{A},I,P)$ of
$M(\mathscr{A})$ as $\operatorname{Bl}_{P}\bar{X}\setminus D^{\prime}$.

Let us assume that $D^{\prime}$ _is connected_ , which is equivalent to
$\Delta^{\prime}$ being connected. Therefore, there exists a maximal tree
$\mathcal{T}_{\Delta^{\prime}}\subset\Delta^{\prime}$. Note that every cycle
in $\Delta^{\prime}$ can be seen as a cycle in $\Delta$.

######  Lemma 4.11.

Any maximal tree $\mathcal{T}_{\Delta^{\prime}}$ can be completed to a maximal
tree $\mathcal{T}_{\Delta^{\prime},\Delta}$ in $\Delta$.

###### Proof.

Let $\\{v_{1},\ldots,v_{k}\\}$ be the vertices of $\Delta$ which are to be
removed along with its adjacent edges in order to obtain $\Delta^{\prime}$.

As $I\subset\\{n+2,\ldots,n+1+s_{0}\\}$ and
$P\subset\operatorname{Sing}\sum_{\iota\not\in I}D_{\iota}$, we have that all
the vertices in $\\{v_{1},\ldots,v_{k}\\}$ correspond to exceptional divisors
in $\operatorname{Bl}_{P}\bar{X}$, therefore there is no edge connecting
$v_{\iota}$ and $v_{j}$ for $\iota\not=j$ and to complete
$\mathcal{T}_{\Delta^{\prime}}$ to a maximal tree of $\Delta$ it suffices to
take no matter what edge connecting a vertex in
$\mathcal{T}_{\Delta^{\prime}}$ and $v_{\iota}$ for $\iota=1,\ldots,k$ because
no cycle will be created in this way. ∎

######  Corollary 4.12.

Let $g$ denote the number of independent cycles in $\Delta$. Let
$c_{1}(\mathcal{T}_{\Delta^{\prime}}),$
$\ldots,c_{g^{\prime}}(\mathcal{T}_{\Delta^{\prime}})$ be independent cycles
in $\Delta^{\prime}$ each one formed by adjoining one edge in
$\Delta^{\prime}$ to the maximal tree $\mathcal{T}_{\Delta^{\prime}}$. There
exists
$c_{g^{\prime}+1}(\mathcal{T}_{\Delta^{\prime},\Delta}),\ldots,c_{g}(\mathcal{T}_{\Delta^{\prime},\Delta})$
cycles in $\Delta$ that together with
$c_{1}(\mathcal{T}_{\Delta^{\prime},\Delta})=c_{1}(\mathcal{T}_{\Delta^{\prime}}),\ldots,c_{g^{\prime}}(\mathcal{T}_{\Delta^{\prime},\Delta})=c_{g^{\prime}}(\mathcal{T}_{\Delta^{\prime}})$
complete a generating set of $\pi_{1}(\Delta,v_{n+1})$.

Let us denote by $\partial U$ the boundary manifold of the total transform of
$D$ in $\operatorname{Bl}_{P}\bar{X}$. By proceeding as in the proof of
Theorem 4.1, we have that a presentation for $\pi_{1}(\partial U)$, by using
the maximal tree $\mathcal{T}_{\Delta^{\prime},\Delta}$, has generators
$\gamma_{1}=\gamma_{1}(\mathcal{T}_{\Delta^{\prime},\Delta}),\ldots,$
$\gamma_{n+1+s_{0}+s_{1}}=\gamma_{n+1+s_{0}+s_{1}}(\mathcal{T}_{\Delta^{\prime},\Delta}),c_{1}=c_{1}(\mathcal{T}_{\Delta^{\prime},\Delta}),\ldots,c_{g}=c_{g}(\mathcal{T}_{\Delta^{\prime},\Delta})$
and a set of relations

(5)
$R=\left\\{\begin{array}[]{ll}[\gamma_{r},\gamma_{j}^{s_{rj}}],&(r,j)\in\mathcal{E}(\Delta)\\\
\gamma_{r}^{-w_{r}^{\prime}}=\prod_{m=1}^{k_{r}^{\prime}}\gamma_{j_{\Delta}(r,m)}^{s_{rj_{\Delta}(i,m)}}&1\leq
r\leq n+1+s_{0}+s_{1}\end{array}\right\\}$

where $w_{r}^{\prime}=D_{r}^{\prime}\cdot D_{r}^{\prime}$, for an irreducible
component $D_{r}^{\prime}$ of $\pi^{\prime*}D$, we denoted by $k_{r}^{\prime}$
the number of points in $\operatorname{Sing}\pi^{\prime*}D\cap D_{r}^{\prime}$
(see the proof of Proposition 4.9), and

$s_{rj}=\left\\{\begin{array}[]{ll}c_{k}^{-1}&\text{ if }(r,j)\text{ equals
the $k$-th element in }\Delta\setminus\mathcal{T}_{\Delta^{\prime},\Delta},\\\
c_{k}&\text{ if }(j,r)\text{ equals the $k$-th element in
}\Delta\setminus\mathcal{T}_{\Delta^{\prime},\Delta},\\\ 1&\text{ if
}(r,j)\text{ is an edge of }\mathcal{T}_{\Delta^{\prime},\Delta}.\\\
\end{array}\right.$

Moreover, let $\partial U^{\prime}$ denote the boundary manifold of
$D^{\prime}\subset\operatorname{Bl}_{P}\bar{X}$. Here, if $r\not\in I$ let us
denote by $k_{r}^{\prime\prime}$ the number of points in
$(D_{k}\cap\sum_{\iota\not\in I}D_{\iota})\setminus P$ or equivalently, in
$D_{k}^{\prime}\cap D^{\prime}$. By using the maximal tree
$\mathcal{T}_{\Delta^{\prime}}$ of $\Delta^{\prime}$ and proceeding as in the
proof of Theorem 4.1, we obtain the following Proposition.

######  Proposition 4.13.

A presentation for $\pi_{1}(\partial U^{\prime})$ is given by

$\left\langle\begin{array}[]{llll}\gamma_{\iota},\quad\iota\in
J&\big{\lvert}[\gamma_{r},\gamma_{j}^{s_{rj}}],&(r,j)\in\mathcal{E}(\Delta^{\prime})\\\
c_{1},\ldots,c_{g^{\prime}}&\bigg{\lvert}\gamma_{r}^{-w_{r}^{\prime}}=\prod_{m=1}^{k_{r}^{\prime\prime}}\gamma_{j_{\Delta^{\prime}}(r,m)}^{s_{rj_{\Delta^{\prime}}(r,m)}}&r\in
J\end{array}\right\rangle$

where $J=\\{1,\ldots,n+1+s_{0}\\}\setminus I$, $\mathcal{E}(\Delta^{\prime})$
denotes the set of edges of $\Delta^{\prime}$, $w_{r}^{\prime}$ the self-
intersection number of the strict transform $D_{r}^{\prime}$ of $D_{r}$ in
$\operatorname{Bl}_{P}\bar{X}$ and

$s_{rj}=\left\\{\begin{array}[]{ll}c_{k}^{-1}&\text{ if }(r,j)\text{ equals
the $k$-th element in
}\Delta^{\prime}\setminus\mathcal{T}_{\Delta^{\prime}},\\\ c_{k}&\text{ if
}(j,r)\text{ equals the $k$-th element in
}\Delta^{\prime}\setminus\mathcal{T}_{\Delta^{\prime}},\\\ 1&\text{ if
}(r,j)\text{ is an edge of }\mathcal{T}_{\Delta^{\prime}}.\\\
\end{array}\right.$

For every $\iota\in I$, we have that, as $D_{\iota}^{\prime}$ is an
exceptional divisor, the following relation is in $R$:

(6)
$\gamma_{\iota}=\prod_{m=1}^{k_{\iota}^{\prime}}\gamma_{j_{\Delta^{\prime},\Delta}(\iota,m)}^{s_{\iota
j_{\Delta^{\prime},\Delta}(\iota,m)}}.$

Analogously, if $p=p_{\iota}\in P$, by abuse of notation we will write
$p={n+1+s_{0}+\iota}$. We have that if $p=D_{r}\cap D_{j}$:

(7) $\gamma_{p}=\gamma_{r}^{s_{pr}}\gamma_{j}^{s_{pj}}$

By using the map $i:\partial U\to
M(\mathscr{A})=\mathbb{P}^{2}\setminus\mathscr{A}$ as in 4.2.5, we can express
the image of the cycles $i_{*}(c_{r})$ as a word in the letters
$i_{*}(\gamma_{1}),\ldots,i_{*}(\gamma_{n+1})$, for $r=1,\ldots,g$. Let us
denote by $\delta_{r}$ the word obtained by replacing the letters
$i_{*}(\gamma_{1}),\ldots,i_{*}(\gamma_{n+1})$ by
$\gamma_{1},\ldots,\gamma_{n+1}$ in this precedent word associated to
$i_{*}(c_{r})$.

By using $\delta_{1},\ldots,\delta_{g}$ and replacing
$i_{*}(\gamma_{1}),\ldots,i_{*}(\gamma_{n+1})$ by
$\gamma_{1},\ldots,\gamma_{n+1}$, we can express the words
$i_{*}(\prod_{m=1}^{k_{\iota}^{\prime}}\gamma_{j_{\Delta^{\prime},\Delta}(\iota,m)}^{s_{\iota
j_{\Delta^{\prime},\Delta}(\iota,m)}})$ and
$i_{*}(\gamma_{r}^{s_{pr}}\gamma_{j}^{s_{pj}})$ with $\iota\in I$ and $p\in P$
as words $\gamma(\iota),\gamma(p)\in\pi_{1}(\partial U^{\prime})$
respectively.

Let us denote by $R^{\prime}$ the set of relations in the presentation given
by Proposition 4.13.

######  Theorem 4.14.

A presentation of $\pi_{1}(M(\mathscr{A},I,P))$ is given by

$\langle c_{1},\ldots,c_{g^{\prime}},\gamma_{\iota}\quad\iota\in J\mid
R^{\prime},c_{1}\delta_{1}^{-1},\ldots,c_{g^{\prime}}\delta_{g^{\prime}}^{-1},\cup_{\iota\in
I}\gamma(\iota),\cup_{p\in P}\gamma(p)\rangle$

with $J=\\{1,\ldots,n+1+s_{0}\\}\setminus I$.

###### Proof.

Consider the following diagram:

${\pi_{1}(\partial U)}$${\pi_{1}(M(\mathscr{A}))}$${\pi_{1}(\partial
U)/{\langle\langle
c_{1}\delta_{l}^{-1},\ldots,c_{g}\delta_{g}^{-1}\rangle\rangle}}$${\pi_{1}(\partial
U^{\prime})}$${\pi_{1}(M(\mathscr{A},I,P))}$${{\pi_{1}(M(\mathscr{A}))}/{\langle\langle
i_{*}(\gamma_{\iota}),i_{*}(\gamma_{p})\rangle\rangle}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Where the isomorphism in the right of the first row comes from Theorem 4.3 and
Proposition 4.10.

From the rightest column we obtain that

(8) $\pi_{1}(\partial U)/{\langle\langle
c_{1}\delta_{1}^{-1},\ldots,c_{g}\delta_{g}^{-1},\gamma(\iota),\gamma(p)\rangle\rangle}\cong\pi_{1}(M(\mathscr{A},I,P)).$

We will see that this presentation is equivalent to

(9) $\pi_{1}(\partial