# Connectivity and combinatorial interplay in the moduli space of line
arrangements

Benoît Guerville-Ballé  https://benoit-guervilleballe.com benoit.guerville-
<EMAIL_ADDRESS>and  Juan Viu-Sos  Dpto. de Matemáticas e Informática
(DMIAICN)
ETSI Caminos, Canales y Puertos
Universidad Politécnica de Madrid
C\Prof. Aranguren 3, 28040 Madrid, Spain https://jviusos.github.io/
<EMAIL_ADDRESS>
###### Abstract.

This paper aims to undertake an exploration of the behavior of the moduli
space of line arrangements while establishing its combinatorial interplay with
the incidence structure of the arrangement. In the first part, we investigate
combinatorial classes of arrangements whose moduli space is connected. We
unify the classes of simple and inductively connected arrangements appearing
in the literature. Then, we introduce the notion of arrangements with a rigid
pencil form. It ensures the connectivity of the moduli space and is less
restrictive that the class of $C_{3}$ arrangements of simple type. In the last
part, we obtain a combinatorial upper bound on the number of connected
components of the moduli space. Then, we exhibit examples with an arbitrarily
large number of connected components for which this upper bound is sharp.

###### 2010 Mathematics Subject Classification:

14H10, 14N20, 51A45, 14N10,

The first author was partially founded by RIMS. The second author was
supported by the Spanish grants PID2020-114750GB-C32 and MTM2016-76868-C2-2-P

###### Contents

  1. 1 Introduction
  2. 2 Moduli spaces of line arrangements
  3. 3 Families of arrangements with connected moduli space
  4. 4 On the number of connected components of $\mathcal{M}(\mathcal{A})$

##  1\. Introduction

Complex line arrangements are fundamental objects in algebraic geometry,
algebraic topology, and also combinatorics. The moduli space of a line
arrangement $\mathcal{A}$ is a geometrical object which describes isomorphism
classes, under the natural action of $\operatorname{PGL}_{3}(\mathbb{C})$, of
arrangements that are lattice-isomorphic to $\mathcal{A}$. The topological
study of the moduli space is extremely important in this context, e.g.
arrangements in the same connected component of the moduli space have
diffeomorphic complements [Ran89]. In this paper, we focus on a combinatorial
study of the moduli space of line arrangements and we explore how such a study
can provide fundamental information about its geometry and topology.

As proved by Mnëv in [Mnë88], the moduli space of a line arrangement can
behave as badly as one can imagine. Vakil qualifies such behavior as _Murphy’s
law_ [Vak06]. More precisely, Mnëv Universality Theorem states that every
singularity of finite type over $\mathbb{Z}$ appears in at least one moduli
space. Recently, it was shown in [CL23] that the realization space of line
arrangements is smooth for $|\mathcal{A}|\leq 11$ and then nodal singularities
appear for $|\mathcal{A}|=12$. In addition, there are several basic geometric
or topological aspects of the moduli space that can not be fully predicted by
a combinatorial study, as it is shown by the classical Pappus’ hexagon
theorem. Thus, a natural question follows: which property of the moduli space
can be combinatorially deduced?

We recall, in Section 2, the definitions of the realization space and moduli
space of a line arrangement. Also, we develop a systematic study of the
inductive construction of the moduli space of ordered arrangements by
introducing the notion of the _type_ (Definition 2.6). Roughly speaking, it
encodes the combinatorial complexity of the line-by-line construction of an
ordered arrangement by counting the multiple points contained in the new line
at each step.

Section 3 is dedicated to the investigation of several known combinatorial
classes of arrangements with connected moduli spaces: e.g. the class of _nice_
arrangements [JY94] and their generalization to _simple_ arrangements [WY05],
or the one of _inductively connected_ arrangements and the _$C_{3}$
arrangements of simple type_ both introduced in [NY12]. We prove in Theorem
3.9 that any generalized simple arrangement is inductively connected,
answering a question from Nazir and Yoshinaga (see the introduction of
[NY12]). In the last part of this section, we first introduce the
combinatorial class of _inductively rigid_ arrangements whose moduli space is
a unique point. Using this as a substructure, we define the class of
arrangements with a _rigid pencil form_ (Definition 3.17), which contains all
the $C_{3}$ arrangements of simple type. Likewise, any arrangement in this
class has a connected moduli space (Theorem 3.19).

In the latter part of this paper (Section 4), we introduce an upper bound on
the number of connected components of the moduli space of an arrangement. We
first define the notion of _$m$ -perturbation_ which transforms any
combinatorics into an inductively connected one (Definition 4.4). By the work
of Nazir and Yoshinaga [NY12], this allows to embed the moduli space of our
original arrangement in an affine space of a relatively small dimension
(Theorem 4.5). From this embedding, we derive an upper bound of the number of
connected components of the moduli space (Theorem 4.10) by inductively
bounding the degree of the $m$ polynomial conditions arising from the
$m$-perturbation. Finally, by providing explicit examples, we prove that the
upper bound obtained is sharp even for non-trivial cases for which the number
of connected components is arbitrarily large (Theorem 4.12).

##  2\. Moduli spaces of line arrangements

Throughout all the paper, a line arrangement $\mathcal{A}$ refers to a finite
set of lines $\\{\ell_{1},\dots,\ell_{n}\\}$ in the complex projective plane
$\mathbb{C}\mathbb{P}^{2}$. In this section, we will recall some basic
definitions about moduli space of line arrangements. We will also fix some
notations and define the type of an ordered arrangement and the naive
dimension of its moduli space.

###  2.1. Line combinatorics and line arrangements

######  Definition 2.1.

An _abstract line combinatorics_ is a pair
$\mathcal{C}=(\mathcal{L},\mathcal{P})$ where:

  * •

$\mathcal{L}$ is the data of an ordered set of $n$ _lines_ , codified by the
list $\mathcal{L}=\left\\{1,\ldots,n\right\\}$,

  * •

$\mathcal{P}$ is a finite subset of the power set of $\mathcal{L}$ codifying
the _singular points_ , satisfying:

    1. (1)

for all $P\in\mathcal{P}$, one has $|P|\geq 2$,

    2. (2)

for all $i,j\in\mathcal{L}$, there exists a unique $P\in\mathcal{P}$ such that
$i,j\in P$.

Two abstract line combinatorics $\mathcal{C}=(\mathcal{L},\mathcal{P})$ and
$\mathcal{C}^{\prime}=(\mathcal{L}^{\prime},\mathcal{P}^{\prime})$ are called
_equivalent_ , denoted $\mathcal{C}\simeq\mathcal{C}^{\prime}$, if there
exists a (non-necessarily order-preserving) bijection
$\Phi:\mathcal{L}\to\mathcal{L}^{\prime}$ such that
$\Phi(\mathcal{P})=\mathcal{P}^{\prime}$.

######  Remark 2.2.

Whenever there is no confusion, we will summarize an abstract line
combinatorics $\mathcal{C}$ by describing only the set $\mathcal{P}$.

Let $\operatorname{Arr}_{n}$ be the set of all the line arrangements of $n$
lines. Given $\mathcal{A}\in\operatorname{Arr}_{n}$, its abstract line
combinatorics $\mathcal{C}(\mathcal{A})$ (or shortly its _combinatorics_) is
given by $(\mathcal{A},\mathcal{P})$, with

$\mathcal{P}=\left\\{P\subset\mathcal{A}\,\left|\,\bigcap_{L\in
P}L\neq\emptyset\;\text{ and }\;\forall L^{\prime}\in\mathcal{A}\setminus
P:\;L^{\prime}\cap\bigcap_{L\in P}L=\emptyset\right.\right\\}.$

A _multiple point_ of $\mathcal{C}(\mathcal{A})$ is an element
$P\in\mathcal{P}$ such that $|P|\geq 3$. Singular points which are not
multiple points are called _double points_. An abstract line combinatorics
$\mathcal{C}$ is just called _line combinatorics_ if there exists an
arrangement $\mathcal{A}$ such that
$\mathcal{C}\simeq\mathcal{C}(\mathcal{A})$. By Pappus’ hexagon theorem, we
know there exist abstract line combinatorics which are not line combinatorics.

######  Example 2.3.

Let $\mathcal{A}$ be the Ceva arrangement defined by the following equations:

$\ell_{1}:x=0,\quad\ell_{2}:y=0,\quad\ell_{3}:z=0,\quad\ell_{4}:x-y=0,\quad\ell_{5}:x+z=0\quad\&\quad\ell_{6}:y+z=0.$

Its combinatorics is then:

$\mathcal{C}(\mathcal{A})=\left\\{\left\\{\ell_{1},\ell_{2},\ell_{4}\right\\},\,\left\\{\ell_{1},\ell_{3},\ell_{5}\right\\},\,\left\\{\ell_{1},\ell_{6}\right\\},\,\left\\{\ell_{2},\ell_{3},\ell_{6}\right\\},\,\left\\{\ell_{2},\ell_{5}\right\\},\,\left\\{\ell_{3},\ell_{4}\right\\},\left\\{\ell_{4},\ell_{5},\ell_{6}\right\\}\right\\}.$

In the following, we will consider two particular families of line
arrangements. Let $n$ be a positive natural number. We define:

  * •

the family of arrangements formed by a pencil of $n$ lines, denoted by
$\mathfrak{X}(n)\subset\operatorname{Arr}_{n}$,

  * •

the family of arrangements formed by a pencil of $n-1$ lines and a unique
generic line, denoted by
$\overline{\mathfrak{X}}(n)\subset\operatorname{Arr}_{n}$.

$\mathcal{A}\in\mathfrak{X}(4)$$\mathcal{A}^{\prime}\in\overline{\mathfrak{X}}(4)$
Figure 1. Arrangements in the classes $\mathfrak{X}(4)$ and
$\overline{\mathfrak{X}}(4)$.

###  2.2. Realization and moduli spaces

######  Definition 2.4.

Let $\mathcal{C}=(\mathcal{L},\mathcal{P})$ be an abstract line combinatorics,
and let $n$ be the cardinality of $\mathcal{L}$. The _realization space_ of
$\mathcal{C}$ is the set:

$\mathcal{R}(\mathcal{C})=\left\\{\mathcal{B}\in\operatorname{Arr}_{n}\mid\mathcal{C}(\mathcal{B})\simeq\mathcal{C}\right\\}.$

The _moduli space_ $\mathcal{M}(\mathcal{C})$ of the abstract line
combinatorics $\mathcal{C}$ is the quotient of the realization space
$\mathcal{R}(\mathcal{C})$ by the action of
$\operatorname{PGL}_{3}(\mathbb{C})$. For brevity sake, we set
$\mathcal{R}(\mathcal{A}):=\mathcal{R}(\mathcal{C}(\mathcal{A}))$ and
$\mathcal{M}(\mathcal{A}):=\mathcal{M}(\mathcal{C}(\mathcal{A}))$ for any
$\mathcal{A}\in\operatorname{Arr}_{n}$ with combinatorics
$\mathcal{C}(\mathcal{A})$.

A line $\ell:ax+by+cz=0$ in $\mathbb{C}\mathbb{P}^{2}$ can be consider as a
point $(a:b:c)$ in the dual $\mathbb{C}{\mathbb{P}^{\vee}}^{2}$, then an
element of $\operatorname{Arr}_{n}$ is naturally identified with a point in
$\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{n}$. Three different lines
$\ell_{i}=(a_{i}:b_{i}:c_{i})$, $\ell_{j}=(a_{j}:b_{j}:c_{j})$ and
$\ell_{k}=(a_{k}:b_{k}:c_{k})$ are _concurrent_ if and only if

$\Delta_{i,j,k}:=\det(\ell_{i},\ell_{j},\ell_{k})=\begin{vmatrix}a_{i}&a_{j}&a_{k}\\\
b_{i}&b_{j}&b_{k}\\\ c_{i}&c_{j}&c_{k}\end{vmatrix}=0.$

By definition, the realization space of a line combinatorics
$\mathcal{C}(\mathcal{A})$ with $\mathcal{A}\in\operatorname{Arr}_{n}$ can be
constructed as a subset of
$\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{n}$ as follows:

$\mathcal{R}(\mathcal{A})=\left\\{\begin{array}[]{c|ll}&\ell_{i}\neq\ell_{j},&\forall
i\neq j\\\
(\ell_{1},\dots,\ell_{n})\in\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{n}&\Delta_{i,j,k}=0,&\text{if
$\\{i,j,k\\}\subset P$, for a $P\in\mathcal{P}$}\\\ &\Delta_{i,j,k}\neq
0,&\text{otherwise}\\\ \end{array}\right\\}.$

If $\mathcal{A}$ is not in $\mathfrak{X}(n)$ or $\overline{\mathfrak{X}}(n)$,
then it contains four lines in general position. Up to a relabelling, one can
assume that these lines are $\ell_{1},\ldots,\ell_{4}$. Any isomorphism class
in $\mathcal{M}(\mathcal{A})$ admits a unique representative111Throughout this
paper any isomorphism class of $\mathcal{M}(\mathcal{A})$ is identified with
this unique representative. in $\mathcal{R}(\mathcal{A})$ such that
$\ell_{1},\ldots,\ell_{4}$ is the arrangement $S_{0}$ formed by the lines
$x=0$, $x-z=0$, $y=0$ and $y-z=0$. So the moduli space
$\mathcal{M}(\mathcal{A})$ can be considered in
$\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{n}$ as the trivial fibration
over the point $S_{0}\in\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{4}$
as follows:

$\mathcal{M}(\mathcal{A})=\\{S_{0}\\}\times\left\\{\begin{array}[]{c|ll}&\ell_{i}\neq\ell_{j},&\forall
i\neq j\in\\{1,\dots,n\\}\\\
(\ell_{5},\dots,\ell_{n})\in\big{(}\mathbb{C}{\mathbb{P}^{\vee}}^{2}\big{)}^{n-4}&\Delta_{i,j,k}=0,&\text{if
$\\{i,j,k\\}\subset P$, for a $P\in\mathcal{P}$}\\\ &\Delta_{i,j,k}\neq
0,&\text{otherwise}\\\ \end{array}\right\\}.$ (1)

It follows that both $\mathcal{R}(\mathcal{A})$ and $\mathcal{M}(\mathcal{A})$
are (quasi-projective, non-necessarily irreducible) algebraic varieties and so
the notions of _dimension_ and _irreducible components_ are well defined.

######  Remark 2.5.

Let $\mathcal{A}$ be in $\mathfrak{X}(n)$ (resp. in
$\overline{\mathfrak{X}}(n)$), the moduli space $\mathcal{M}(\mathcal{A})$ is
irreducible of dimension $\max(0,n-3)$ (resp. $\max(0,n-4)$). If $\mathcal{A}$
is neither in $\mathfrak{X}(n)$ nor in $\overline{\mathfrak{X}}(n)$, one has
that
$\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})=\dim_{\mathbb{C}}\mathcal{R}(\mathcal{A})-8$.

###  2.3. Type of an arrangement and naive dimension of the moduli space

An _ordered arrangement_ is a pair $(\mathcal{A},\omega)$, where
$\mathcal{A}\in\operatorname{Arr}_{n}$ and $\omega$ is a bijective map

$\omega:\mathcal{A}\longrightarrow\left\\{1,\ldots,n\right\\},$

called the _order_ on $\mathcal{A}$. For the sack of brevity, we will sometime
express an order arrangement as an ordered list
$\mathcal{A}=\left\\{\ell_{1},\ldots,\ell_{n}\right\\}$ where the indices of
the lines are chosen in a compatible way with the order, i.e. such that
$\omega(\ell_{i})=i$, for any $\ell_{i}\in\mathcal{A}$.

Let $(\mathcal{A},\omega)$ be an ordered arrangement and $\mathcal{B}$ be a
subarrangement of $\mathcal{A}$. There exists a unique order
$\omega_{\mathcal{B}}$ on $\mathcal{B}$ such that for any
$\ell\neq\ell^{\prime}\in\mathcal{B}$, one has that

$\omega_{\mathcal{B}}(\ell)<\omega_{\mathcal{B}}(\ell^{\prime})\iff\omega(\ell)<\omega(\ell^{\prime}).$

This order will be called the _order induced_ by $\omega$ on $\mathcal{B}$.

Let $\mathcal{A}_{i}=\omega^{-1}(\left\\{1,\dots,i\right\\})$, for
$i\in\left\\{1,\dots,n\right\\}$. When $\omega$ is the order given by the
indices, then $\mathcal{A}_{i}=\left\\{\ell_{1},\dots,\ell_{i}\right\\}$.
Consider the following ascending chain of (ordered) arrangements:

$\mathcal{A}_{1}=\left\\{\ell_{1}\right\\}\subsetneq\mathcal{A}_{2}\subsetneq\cdots\subsetneq\mathcal{A}_{i}\subsetneq\cdots\subsetneq\mathcal{A}_{n}=\mathcal{A},$
(2)

We denote by $\tau_{i}(\mathcal{A},\omega)$ (alternatively,
$\tau(\mathcal{A},\omega,\ell_{i})$ or $\tau_{i}$ depending on the context)
the cardinality of the intersection
$|\omega^{-1}(i)\cap\operatorname{Sing}(\mathcal{A}_{i-1})|$. Roughly
speaking, $\tau_{i}$ counts the number of multiple points of $\mathcal{A}_{i}$
contained in the $i$th line. Note that we also have:

$\tau_{i}=\sum_{P\in\mathcal{C}(\mathcal{A}_{i})}(|P|-2)\,-\sum_{Q\in\mathcal{C}(\mathcal{A}_{i-1})}(|Q|-2).$
(3)

######  Definition 2.6.

The _type_ of an ordered arrangement $(\mathcal{A},\omega)$ is the $n$-tuple
defined as

$\tau(\mathcal{A},\omega)=(\tau_{1},\dots,\tau_{n})\in\mathbb{N}^{n}.$

It is clear that the type depends on the choice of an order. In fact, both the
ascending chain (2) and the type are of combinatorial nature. So they can be
defined directly over an abstract line combinatorics $\mathcal{C}$. By an
extension of notation, we will also denote
$(\mathcal{C})_{1}\subsetneq\cdots\subsetneq(\mathcal{C})_{n}=\mathcal{C}$ and
$\tau(\mathcal{C},\omega)$.

######  Example 2.7.

Consider the arrangement $\mathcal{A}$ pictured in Figure 2 endowed with the
order $\omega$ induced by the indices. Its type is
$\tau(\mathcal{A},\omega)=(0,0,0,0,2,2,1,3)$.

$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\ell_{6}$$\ell_{7}$$\ell_{8}$
Figure 2. An arrangement with type
$\tau(\mathcal{A},\omega)=(0,0,0,0,2,2,1,3)$.

For two ordered arrangement $(\mathcal{A},\omega)$ and
$(\mathcal{A}^{\prime},\omega^{\prime})$ of $n$ and $n^{\prime}$ lines
respectively, and such that $\mathcal{A}\cap\mathcal{A}^{\prime}=\emptyset$
(i.e. without common line, but not necessarily with generic intersection), one
can define the operation
$(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})$
producing an ordered arrangements of $n+n^{\prime}$ lines with order:

$\begin{array}[]{rccl}\omega\oplus\omega^{\prime}:&\mathcal{A}\sqcup\mathcal{A}^{\prime}&\longrightarrow&\left\\{1,\ldots,n+n^{\prime}\right\\}\\\\[3.00003pt]
&\ell&\longmapsto&\begin{cases}\omega(\ell)&\text{if $\ell\in\mathcal{A}$},\\\
n+\omega^{\prime}(\ell)&\text{if
$\ell\in\mathcal{A}^{\prime}$}.\end{cases}\end{array}$

######  Remark 2.8.

If $\mathcal{A}\in\operatorname{Arr}_{n}$ and
$\mathcal{A}^{\prime}\in\operatorname{Arr}_{n^{\prime}}$ have no common lines,
for any orders $\omega$ and $\omega^{\prime}$ on $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ respectively, one has that:

$\tau_{i}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})=\tau_{i}(\mathcal{A},\omega)\quad\text{
and
}\quad\tau_{n+j}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})\geq\tau_{j}(\mathcal{A}^{\prime},\omega^{\prime}),$

for any $i\in\\{1,\dots,n\\}$ and $j\in\\{1,\dots,n^{\prime}\\}$. Furthermore,
for any other order $\overline{\omega}$ on $\mathcal{A}$, and for any
$j\in\\{1,\dots,n^{\prime}\\}$, we have that:

$\tau_{n+j}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})=\tau_{n+j}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\overline{\omega}\oplus\omega^{\prime}).$

######  Proposition 2.9.

Let $(\mathcal{A},\omega)$ be an ordered arrangement. The following equality
holds:

$\sum_{i=1}^{n}(2-\tau_{i})=2|\mathcal{A}|-\sum_{P\in\mathcal{C}(\mathcal{A})}(|P|-2).$

###### Proof.

First, if $|\mathcal{A}|=1$, then both sides of the equality are obviously
$0$.

Assume that the equality holds for any ordered arrangement of $n$ lines. Let
$\mathcal{A}$ be an ordered arrangement of $n+1$ lines and take
$\mathcal{A}^{\prime}=\mathcal{A}\setminus\\{\ell_{n+1}\\}$. By the induction
hypothesis and equation (3), one has:

$\sum_{i=1}^{n+1}(2-\tau_{i})=(2-\tau_{n+1})+2|\mathcal{A}^{\prime}|-\sum_{Q\in\mathcal{C}(\mathcal{A}^{\prime})}(|Q|-2)=2|\mathcal{A}|-\sum_{P\in\mathcal{C}(\mathcal{A})}(|P|-2).\qed$

The right-hand side of the equality in the preceding proposition can be seen
as a preliminary attempt to establish a combinatorial formula for the
dimension of the realization space $\mathcal{R}(\mathcal{A})$, despite its
naive nature. The part $2|\mathcal{A}|$ gives the number of required variables
in the construction, while the sum over $P\in\mathcal{C}(\mathcal{A})$
corresponds to the restriction imposed by the singular points. Since the
action of $\operatorname{PGL}_{3}(\mathbb{C})$ can fix four points in generic
position, this induces a reduction of $8$ to pass from the dimension of
$\mathcal{R}(\mathcal{A})$ to the dimension of $\mathcal{M}(\mathcal{A})$.

######  Definition 2.10.

Let $\mathcal{A}$ be a line arrangement. The _naive dimension_ of the moduli
space is:

$\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})=2|\mathcal{A}|-8-\sum_{P\in\mathcal{C}(\mathcal{A})}(|P|-2).$

######  Remark 2.11.

The naive dimension is fully determined by the combinatorics of $\mathcal{A}$,
so we can define it on any abstract line combinatorics. Recall that in
general, the naive dimension is not equal to the dimension of the moduli
space, by Pappus’ hexagon theorem.

From [DIM20, Remark 4.4], we have the following bound for the dimension of the
moduli space.

######  Proposition 2.12.

Let $\mathcal{A}$ be an arrangement. The following inequality holds:

$\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})\leq\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A}).$

##  3\. Families of arrangements with connected moduli space

In this section, we will study combinatorial classes of arrangements whose
moduli space is connected. The first one is the notion of inductively
connected arrangements introduced by Nazir and Yoshinaga in [NY12].It plays a
crucial role in all the following.

###  3.1. Inductively connected arrangement

The results of this section are reformulation and refinement of Nazir and
Yoshinaga’s [NY12]. First, using the notion of the type of an arrangement
introduced in Definition 2.6, we can reformulate the definition of inductively
connected arrangement.

######  Definition 3.1.

An arrangement $\mathcal{A}$ is _inductively connected_ if there exists an
order $\omega$ on $\mathcal{A}$ such that

$\max(\tau(\mathcal{A},\omega))\leq 2.$

######  Example 3.2.

The arrangement $\mathcal{A}$ pictured in Figure 3 endowed with the order
$\omega$ induced by the indices has type
$\tau(\mathcal{A},\omega)=(0,0,0,0,2,2,1)$. So it is an inductively connected
arrangement.

$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\ell_{6}$$\ell_{7}$ Figure
3. An inductively connected arrangement with type
$\tau(\mathcal{A},\omega)=(0,0,0,0,2,2,1)$.

######  Proposition 3.3.

Let $\mathcal{A}\in\operatorname{Arr}_{n}$ be an inductively connected
arrangement of $n\geq 4$ lines such that $\mathcal{A}$ is not in
$\mathfrak{X}(n)$ or $\overline{\mathfrak{X}}(n)$. There exists an order
$\omega$ on $\mathcal{A}$ verifying the following conditions:

  * •

$\max(\tau(\mathcal{A},\omega))\leq 2$,

  * •

the $4$th term $\mathcal{A}_{4}$ in the chain (2) is generic, i.e.
$\mathcal{A}_{4}\notin\mathfrak{X}(4)\cup\overline{\mathfrak{X}}(4)$.

###### Proof.

Since $\mathcal{A}$ is inductively connected, there exists an order
$\omega_{0}$ such that $\max(\tau(\mathcal{A},\omega_{0}))\leq 2$. We assume
that $\mathcal{A}_{4}$ is not generic, i.e. $\mathcal{A}_{4}$ is either
$\mathfrak{X}(4)$ or $\overline{\mathfrak{X}}(4)$ (otherwise, we choose
$\omega=\omega_{0}$), and also that $n\geq 5$. We denote by $P$ the unique
multiple point of $\mathcal{A}_{4}$.

Let $k$ be the smallest integer of $\\{5,\dots,n\\}$ such that
$\mathcal{A}_{k}$ is formed by lines in the pencil centered at $P$ and two
lines not passing through $P$. Note that $\mathcal{A}_{k}$ contains a generic
subarrangement $\mathcal{B}$ of $4$ lines. Let $\omega_{k}$ be any order on
$\mathcal{A}_{k}$ such that $\omega_{k}(\mathcal{B})=\\{1,2,3,4\\}$, and let
$\omega_{0}^{\prime}$ be the order induced by $\omega_{0}$ on
$\mathcal{A}\setminus\mathcal{A}_{k}$. By Remark 2.8, the order
$\omega=\omega_{k}\oplus\omega_{0}^{\prime}$ on
$\mathcal{A}=\mathcal{A}_{k}\sqcup(\mathcal{A}\setminus\mathcal{A}_{k})$
verifies the required conditions and the result holds. ∎

######  Proposition 3.4.

If $\mathcal{A}$ is an inductively connected arrangement such that
$\mathcal{A}\notin\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$, then
$\mathcal{M}(\mathcal{A})$ is isomorphic to a Zariski open subset of a complex
space of dimension $\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})$. As a
consequence, the moduli space $\mathcal{M}(\mathcal{A})$ is connected.

###### Proof.

In the proof of [NY12, Lemma 3.2], the authors show that
$\mathcal{M}(\mathcal{A}_{i+1})$ is isomorphic to either a
$\mathbb{C}\mathbb{P}^{1}$-fibration or a trivial
$\mathbb{C}\mathbb{P}^{2}$-fibration over $\mathcal{M}(\mathcal{A}_{i})$,
depending on the values of $\tau_{i}$.

When $\tau_{i}=1$, the $\mathbb{C}\mathbb{P}^{1}$-fibration describes the
choice of the new line $\ell_{i}$ in a pencil, which is supported by a
singular point of $\mathcal{A}_{i-1}$. Given that there is already a line in
$\mathcal{A}_{i-1}$ which belongs to this pencil and since $\ell_{i}$ should
be different from this one, we can thus describe the choice of $\ell_{i}$ as a
$\mathbb{C}$-fibration over $\mathcal{M}(\mathcal{A}_{i-1})$.

Similarly, when $\tau_{i}=0$ the $\mathbb{C}\mathbb{P}^{2}$-fibration
describes the choice of a generic line $\ell_{i}$. Fix
$P\in\operatorname{Sing}(\mathcal{A}_{i-1})$. Since $\ell_{i}$ should not pass
through $P$ and thus it does not belong to the pencil supported on $P$, one
can describe the choice of $\ell_{i}$ as a $\mathbb{C}^{2}$-fibration.

According to Proposition 3.3, we can assume that $\mathcal{A}_{4}$ is generic,
and it can be completely fixed by the action of $\operatorname{PGL}_{3}$, i.e.
$\mathcal{M}(\mathcal{A}_{4})=\left\\{\mathcal{A}_{4}\right\\}$. By induction,
we obtain that $\mathcal{M}(\mathcal{A})$ is a Zariski open subset of the
complex space of dimension $\sum_{i=5}^{n}(2-\tau_{i})$ and by Proposition 2.9
the result holds. ∎

In fact, an explicit parametrization of $\mathcal{M}(\mathcal{A})$ as an open
Zariski subset of an affine complex space is provided in Section 4.2 for any
arrangement $\mathcal{A}$.

######  Corollary 3.5.

If $\mathcal{A}\in\operatorname{Arr}_{n}$ is an inductively connected
arrangement such that
$\mathcal{A}\notin\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$, then

$\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})=\sum_{i=5}^{n}(2-\tau_{i})=\operatorname{d^{naive}}\mathcal{M}(\mathcal{A}).$

######  Remark 3.6.

In the same spirit as [NY12, Lemma 3.2], let $(\mathcal{A},\omega)$ and
$(\mathcal{A}^{\prime},\omega^{\prime})$ be two ordered arrangements with $n$
and $n^{\prime}$ lines respectively and such that
$\tau_{i}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})\leq
2$ for all $i\in\\{n+1,\dots,n+n^{\prime}\\}$. If $\mathcal{M}(\mathcal{A})$
is irreducible, then $\mathcal{M}(\mathcal{A}\sqcup\mathcal{A}^{\prime})$ is
irreducible thus connected and it has dimension

$\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A}\sqcup\mathcal{A}^{\prime})=\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})+\sum_{i=n+1}^{n+n^{\prime}}(2-\tau_{i}(\mathcal{A}\sqcup\mathcal{A}^{\prime},\omega\oplus\omega^{\prime})).$

Note that the reciprocal is not true in general, as it is shown in the
following example.

######  Example 3.7.

In [GBVS19, § 3.2] the authors present a line combinatorics $\mathcal{C}$ of
$13$ lines with a moduli space with two irreducible components of dimension
$1$. These components are geometrically characterized, e.g. there are
particular multiple points $P_{1}$, $P_{2}$, and $P_{3}$ which are collinear
in the first component but they are not in the second one (see [GBVS19, Prop.
4.6]). Consider the combinatorics $\mathcal{C}^{\prime}$ constructed on
$\mathcal{C}$ by the addition of a line $\ell_{14}$ passing through $P_{1}$
and $P_{2}$ but avoiding $P_{3}$. By construction, one has that
$\tau_{14}(\mathcal{C}^{\prime})=2$ and $\mathcal{M}(\mathcal{C}^{\prime})$ is
irreducible.

###  3.2. Nice, simple and generalized simple arrangements

In [JY94], Jiang and Yau defined the combinatorial class of _nice_
arrangements, which was generalized by Wang and Yau in [WY05] as the classes
of _simple_ and _generalized simple_ arrangements. In the introduction of
[NY12], Nazir and Yoshinaga ask about the relation between these combinatorial
classes and the class of inductively connected arrangements. The purpose of
this section is to prove that any generalized simple arrangement is actually
inductively connected. First, let us recall the definition of generalized
simple arrangements.

For any arrangement $\mathcal{A}$, we define $G_{\mathcal{A}}$ as the graph
whose vertices $v_{P}$ are in one-to-one correspondence with the multiple
points $P$ of $\mathcal{A}$ (i.e. the singular points whose multiplicity is at
least $3$), and two vertices $v_{P}$ and $v_{Q}$ are joined by an edge if and
only if the two corresponding points $P$ and $Q$ lie on the same line of
$\mathcal{A}$. Associated with this graph, one define the following notions:

  * •

A _star_ centered in $v_{P}$, and denoted $\operatorname{St}(v_{P})$, is the
subgraph of $G_{\mathcal{A}}$ generated by $v_{P}$ and any neighbor vertex of
$v_{P}$. The open star $\overset{\circ}{\operatorname{St}}(v_{P})$ is the
complex composed of $v_{P}$ and any adjacent edge (without the neighbor
vertices of $v_{P}$).

  * •

A _reduced circle_ of $G_{\mathcal{A}}$ is a tuple of $(v_{1},\dots,v_{k})$ of
vertices of $G_{\mathcal{A}}$, such that for all $i\in\\{1,\dots,k\\}$, one
has that $(v_{i},v_{i+1})$ is an edge of $G_{\mathcal{A}}$ and $v_{i-1}\neq
v_{i+1}$ (the indices are considered modulo $k$).

  * •

A _generalized free net_ based on a tuple $(B_{0},\dots,B_{m})$ of reduced
circles of $G_{\mathcal{A}}$ (where $B_{0}$ can also be a single vertex), is
the maximal subgraph $\operatorname{Net}(B_{0},\dots,B_{m})$ of
$G_{\mathcal{A}}$ whose vertices are at a distance of at most $1$ from at
least one of the circles $B_{i}$, and such that

    1. (N1)

for all $i\in\\{0,\dots,m\\}$, two vertices $v$ and $w$ of $B_{i}$ are
connected by an edge if and only if they are adjacent in $B_{i}$,

    2. (N2)

for all $i,j\in\\{0,\dots,m-2\\}$ and two vertices $v\in B_{i}$ and $w\in
B_{j}$, it does not exist $z\in B_{k}$ with $k>\max(i,j)$ which verifies that
$(v,z)$ and $(w,z)$ are both edges in $\operatorname{Net}(B_{0},\dots,B_{m})$,

    3. (N3)

for all pair of vertices $v,w\in\operatorname{Net}(B_{0},\dots,B_{m})$, it
does not exist $z\in
G_{\mathcal{A}}\setminus\operatorname{Net}(B_{0},\dots,B_{m})$ which verifies
that $(v,z)$ and $(w,z)$ are both edges in $G_{\mathcal{A}}$,

    4. (N4)

for all $i\in\\{1,\dots,n\\}$, there exists a vertex $v\in B_{i}$ which is not
connected by an edge to any reduced circle $B_{j}$ with $j<i$. The vertex $v$
is named the _free vertex of the circle $B_{i}$_.

The open net $\overset{\circ}{\operatorname{Net}}(B_{0},\dots,B_{m})$ is the
complex formed by the vertices of the $B_{i}$ and all the edges of
$\operatorname{Net}(B_{0},\dots,B_{m})$ (i.e. all the end vertices of the net
are removed).

######  Definition 3.8.

An arrangement $\mathcal{A}$ is _generalized simple_ if there are stars
$\operatorname{St}(v_{1}),\dots,\operatorname{St}(v_{k})$ and generalized free
nets $\operatorname{Net}_{1},\dots,\operatorname{Net}_{l}$ which are pairwise
disjoint in $G_{\mathcal{A}}$ and such that the graph

$G^{\prime}_{\mathcal{A}}=G_{\mathcal{A}}\setminus\left\\{\overset{\circ}{\operatorname{St}}(v_{1}),\dots,\overset{\circ}{\operatorname{St}}(v_{k}),\overset{\circ}{\operatorname{Net}}_{1},\dots,\overset{\circ}{\operatorname{Net}}_{l}\right\\}$

is a forest. When $l=0$, the arrangement $\mathcal{A}$ is _nice_ ; and it is
_simple_ when for all nets have $m=2$.

######  Theorem 3.9.

If $\mathcal{A}$ is a generalized simple arrangement, then $\mathcal{A}$ is
inductively connected.

In order to prove our result, the two following technical lemmas are required.
They allow to remove all the lines which pass through a multiple point with a
corresponding vertex of valence at most $2$ in the graph $G_{\mathcal{A}}$.

######  Lemma 3.10.

Let $\mathcal{A}\in\operatorname{Arr}_{n}$ be an arrangement and $v_{P}$ a
vertex of $G_{\mathcal{A}}$ of valence $m\leq 2$, and let
$\mathcal{A}_{P}=\\{L\in\mathcal{A}\mid P\in L\\}$. There exists an order
$\omega_{P}$ on $\mathcal{A}_{P}$ which verifies that for any order $\omega$
on $\mathcal{A}\setminus\mathcal{A}_{P}$, and for all
$\ell\in\mathcal{A}_{P}$, we have that
$\tau(\mathcal{A},\omega\oplus\omega_{P},\ell)\leq 2$.

###### Proof.

Let $\omega$ be a fixed order on $\mathcal{A}\setminus\mathcal{A}_{P}$. Note
that if $\omega_{P}$ is an order of $\mathcal{A}_{P}$ then
$\omega\oplus\omega_{P}$ is an order on $\mathcal{A}$. We study each case by
the possible values of the valence:

If $m=0$, then $v_{P}$ is an isolated vertex of $G_{\mathcal{A}}$. That is
equivalent to say that any line of $\mathcal{A}_{P}$ has only $P$ as multiple
point. So for any order $\omega_{P}$ on $\mathcal{A}_{P}$, one has
$\tau(\mathcal{A},\omega\oplus\omega_{P},\ell)\leq 1$, for all
$\ell\in\mathcal{A}_{P}$.

If $m=1$, we denote by $\ell_{P}$ the line corresponding to the unique edge of
$G_{\mathcal{A}}$ which ends at $v_{P}$. Remark that $\ell_{P}$ contains
exactly two multiple points, since otherwise $m$ should be at least $2$. As in
the previous case, any line of $\mathcal{A}_{P}\setminus\\{\ell_{P}\\}$
contains only $P$ as multiple point. Consider any order $\omega_{P}$ on
$\mathcal{A}_{P}$ such that $\omega_{P}(\ell_{P})=1$. For any line
$\ell\in\mathcal{A}_{P}\setminus\\{\ell_{P}\\}$, we have that
$\tau(\mathcal{A},\omega\oplus\omega_{P},\ell)\leq 1$. In
$\mathcal{A}\setminus\\{\ell\in\mathcal{A}_{P}\mid\ell\neq\ell_{P}\\}$, the
point $P$ is no longer a multiple point (neither a singular point). So, in the
previous arrangement, $\ell_{P}$ contains a unique multiple point. Then we
have $\tau(\mathcal{A},\omega\oplus\omega_{P},\ell_{P})=1$.

If $m=2$ and the two edges which join $v_{P}$ correspond to the same line
$\ell_{P}$, then $\ell_{P}$ contains at most $3$ multiple points. We consider
any order $\omega_{P}$ such that $\omega_{P}(\ell_{P})=1$, then end of the
proof is similar to the case $m=1$. The only difference is that in
$\mathcal{A}\setminus\\{\ell\in\mathcal{A}_{P}\mid\ell\neq\ell_{P}\\}$, the
line $\ell_{P}$ contains at most $2$ multiple points and thus
$\tau(\mathcal{A},\omega\oplus\omega_{P},\ell_{P})\leq 2$.

If $m=2$ and the two edges which join $v_{P}$ correspond to two different
lines $\ell_{P}$ and $\ell^{\prime}_{P}$, then each line contains exactly two
multiple points. So any line in $\mathcal{A}_{P}$ contains at most $2$
multiple points. This implies that for any order $\omega_{P}$ on
$\mathcal{A}_{P}$, one has $\tau(\mathcal{A},\omega\oplus\omega_{P},\ell)\leq
2$, for all $\ell\in\mathcal{A}_{P}$. ∎

######  Lemma 3.11.

Let $\mathcal{A}$ be a line arrangement, let $v_{P}$ be a vertex of
$G_{\mathcal{A}}$ with valence at most $2$ and let
$\mathcal{A}_{P}=\\{\ell\in\mathcal{A}\mid P\in\ell\\}$. If
$\mathcal{A}\setminus\mathcal{A}_{P}$ is inductively connected then so is
$\mathcal{A}$.

###### Proof.

Assume that $\mathcal{A}\setminus\mathcal{A}_{P}$ inductively connected, then
there exists an order $\omega_{0}$ on $\mathcal{A}\setminus\mathcal{A}_{P}$
such that $\tau(\mathcal{A}\setminus\mathcal{A}_{P},\omega_{0},\ell)\leq 2$,
for all $\ell\in\mathcal{A}\setminus\mathcal{A}_{P}$. By Lemma 3.10, there
exists an order $\omega_{P}$ on $\mathcal{A}_{P}$ such that
$\tau(\mathcal{A},\omega_{0}\oplus\omega_{P},\ell)\leq 2$, for all
$\ell\in\mathcal{A}_{P}$. By Remark 2.8, we deduce that
$\max(\tau(\mathcal{A},\omega_{0}\oplus\omega_{P}))\leq 2$, and so that
$\mathcal{A}$ is inductively connected. ∎

###### Proof of Theorem 3.9.

For any multiple point $P$ of $\mathcal{A}$, we denote by $\mathcal{A}_{P}$
the arrangement $\\{\ell\in\mathcal{A}\mid P\in\mathcal{A}\\}$. Note that
$G_{\mathcal{A}\setminus\mathcal{A}_{P}}$ is a (non necessarily strict)
subgraph of
$G_{\mathcal{A}}\setminus\overset{\circ}{\operatorname{St}}(v_{P})$. The
strategy of the proof is to remove one-by-one all the vertices of
$G_{\mathcal{A}}$ in such a way that, at each step, the removed vertex has
valence at most $2$.

Step 1. The vertices of $G^{\prime}_{\mathcal{A}}$.

By hypothesis, $G^{\prime}_{\mathcal{A}}$ is a forest. Let $v_{P}$ be an end-
point of $G^{\prime}_{\mathcal{A}}$, that is there is at most one edge in
$G^{\prime}_{\mathcal{A}}$ which meets $v_{P}$. Then, we know that the stars
and the generalized net are disjoint. Furthermore, condition (N3) implies that
there are no two edges in the same generalized net that meet $v_{P}$. This
implies that there is at most $1$ edge in the union of the stars and of the
generalized nets which meet $v_{P}$. So the valence of $v_{P}$ is at most $2$.
So by Lemma 3.11, if $\mathcal{A}\setminus\mathcal{A}_{P}$ is inductively
connected then so is $\mathcal{A}$.
As noted at the beginning of the proof,
$G_{\mathcal{A}\setminus\mathcal{A}_{P}}$ is a subgraph of
$G_{\mathcal{A}}\setminus\overset{\circ}{\operatorname{St}}(v_{P})$. The
remaining part of $G^{\prime}_{\mathcal{A}}$ in
$G_{\mathcal{A}\setminus\mathcal{A}_{P}}$ is also a forest (as a subgraph of a
forest) and then we can apply the same argument on a new end-point. By
successive applications of this argument, we obtain that if
$\mathcal{B}=\mathcal{A}\setminus\bigcup_{v_{P}\in
G^{\prime}_{\mathcal{A}}}\mathcal{A}_{P}$ is inductively connected then so is
$\mathcal{A}$.

Step 2. The vertices of $\operatorname{St}(v_{P})$.

The graph $G_{\mathcal{B}}$ is contained in the disjoint union of the stars
and of the generalized nets. Remark that the union of the stars is also a
forest. So we can apply exactly the same argument as in step 1 to remove all
the vertices included in any star. Thus, we obtain that if
$\mathcal{D}=\mathcal{B}\setminus\bigcup_{v_{P}\in\cup\operatorname{St}(v_{i})}\mathcal{A}_{P}$
is inductively connected then so is $\mathcal{B}$.

Step 3. The vertices of $\operatorname{Net}_{i}$.

The graph $G_{\mathcal{D}}$ is contained in the disjoint union of the
generalized nets. By definition, we have that
$\operatorname{Net}_{l}=\operatorname{Net}(B_{0},\dots,B_{m})$ for a fixed
$m$. Remark that all the vertices outside the reduced circles $B_{i}$ are not
in $G_{\mathcal{D}}$ since they have been removed at Step 1. Let $v_{P}$ be
the free vertex of $B_{m}$, and let $v_{Q}$ be one of its two neighbors in
$B_{m}$. According to conditions (N1) and (N4), the valence of $v_{f}$ is
exactly $2$. By Lemma 3.11, if $\mathcal{D}\setminus\mathcal{A}_{P}$ is
inductively connected then so is $\mathcal{D}$. It follows from conditions
(N1) and (N2), that in $G_{\mathcal{D}\setminus\mathcal{A}_{P}}$ the vertex
$v_{Q}$ as valence at most $2$. So we can inductively apply this argument on
all the vertices of $B_{m}$, and then on each reduced circle $B_{i}$ of
$\operatorname{Net}_{l}$. At the end, we obtain that if
$\mathcal{D}\setminus\bigcup_{i=0}^{m}\bigcup_{v_{P}\in B_{i}}\mathcal{A}_{P}$
is inductively connected then so is $\mathcal{D}$.

Applying the latter on each generalized net
$\operatorname{Net}(B^{j}_{0},\dots,B^{j}_{m_{j}})$, we obtain that if the
arrangement
$\mathcal{E}=\mathcal{D}\setminus\bigcup_{j=0}^{l}\bigcup_{i=0}^{m_{j}}\bigcup_{v_{P}\in
B^{j}_{i}}\mathcal{A}_{P}$ is inductively connected, then so is $\mathcal{D}$.

As a consequence of the previous steps, the obtained graph $G_{\mathcal{E}}$
is empty. Thus, the arrangement $\mathcal{E}$ is either empty or generic. In
both cases, we have that $\mathcal{E}$ is inductively connected. According to
the previous steps, this implies that $\mathcal{A}$ is also inductively
connected. ∎

The fact that the diffeomorphism type of the complement of both nice and
generalized simple arrangements is combinatorially determined is well-known
and forms the main theorems of [JY94] and [WY05], respectively. This can also
be deduced from Theorem 3.9, Proposition 3.4 and Randell’s Lattice–Isotopy
Theorem [Ran89].

###  3.3. Inductively rigid arrangements

Following a similar approach as for inductively connected arrangements, we
introduce another combinatorial class of arrangements with connected moduli
space.

######  Definition 3.12.

An arrangement $\mathcal{A}\in\operatorname{Arr}_{n}$ is _inductively rigid_
if there exists an order $\omega$ such that for any arrangement
$\mathcal{A}_{i}$ in the ascending chain (2), we have
$\mathcal{M}(\mathcal{A}_{i})=\\{\mathcal{A}_{i}\\}$.

Note that if an arrangement $\mathcal{A}$ is inductively rigid, then any
$\mathcal{A}_{i}$ in the induced ordered chain (2) is also inductively rigid.
As a consequence, any arrangement $\mathcal{A}$ with $|\mathcal{A}|\leq 3$ is
inductively rigid. In addition, no arrangement in the classes
$\mathfrak{X}(n)$ and $\overline{\mathfrak{X}}(n+1)$ is inductively rigid for
$n\geq 4$, by Remark 2.5.

######  Proposition 3.13.

An arrangement $\mathcal{A}\in\operatorname{Arr}_{n}$ is inductively rigid if
and only if either $n\leq 4$ and $\mathcal{A}\notin\mathfrak{X}(4)$, or $n\geq
5$ and there exists an order $\omega$ on $\mathcal{A}$ such that:

  1. (A1)

$\mathcal{A}_{4}$ is generic, i.e. for all $i\in\\{1,2,3,4\\}$, one has
$\tau_{i}(\mathcal{A},\omega)=0$,

  2. (A2)

for all $i\in\\{5,\dots,n\\}$, one has $\tau_{i}(\mathcal{A},\omega)\geq 2$.

###### Proof.

Any arrangement of at most three lines is inductively rigid. Whenever $n=4$,
either $\mathcal{A}\in\mathfrak{X}(4)$ or $\mathcal{A}$ is fixed by the
$\operatorname{PGL}_{3}$-action and thus
$\mathcal{M}(\mathcal{A})=\left\\{\mathcal{A}\right\\}$. The first case is not
inductively rigid while the second one is. Assume that $n\geq 5$, one has
that:

  * •

If for any order $\omega$ the condition (A1) does not hold, then
$\mathcal{A}\in\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$ with $n\geq 5$
and thus, due to Remark 2.5, $\mathcal{A}$ is not inductively rigid.

  * •

If for any order $\omega$, the condition (A2) does not hold, then there exists
$i\geq 5$ such that $\tau_{i}(\mathcal{A},\omega)<2$. This implies that
$\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A}_{i})>0$ and thus $\mathcal{A}$ is
not inductively rigid.

  * •

If there exists an order $\omega$ over $\mathcal{A}$ such that (A1) and (A2)
hold, then by a recursive application of [NY12, Lemma 3.2], we deduce that
$\mathcal{M}(\mathcal{A}_{i})=\\{\mathcal{A}_{i}\\}$ for all
$i\in\\{5,\dots,n\\}$. Then, $\mathcal{A}$ is inductively rigid.

Since we have covered all possible cases, we obtain the stated equivalence. ∎

As an important consequence of the previous proposition, we have that the
class of inductively rigid arrangements is combinatorial.

######  Example 3.14.

Consider the arrangement $\mathcal{B}$ given in Figure 4, with order $\omega$
induced by the line numbering. It is inductively rigid since its type is
$\tau(\mathcal{B},\omega)=(0,0,0,0,2,2,2,2,2,3)$. However, it is not
inductively connected since all the lines contain at least $3$ multiple
points. Furthermore, one can also verify that it is not $C_{3}$ of simple type
(see Section 3.4 for the definition).

$\ell_{1}$$\ell_{2}$$\ell_{3}$$\ell_{4}$$\ell_{5}$$\infty$$\ell_{6}$$\ell_{7}$$\ell_{8}$$\ell_{9}$$\ell_{10}$
Figure 4. An inductively rigid arrangement which is not inductively connected
nor $C_{3}$ of simple type.

###  3.4. Arrangements with a rigid pencil form

An arrangement $\mathcal{A}$ is $C_{k}$ if $k$ is the minimal integer such
that there exists a subarrangement $\mathcal{D}_{k}\subset\mathcal{A}$ of $k$
lines such that $\operatorname{Sing}_{\geq 3}(\mathcal{A})$ is contained in
$\mathcal{D}_{k}$. Nazir and Yoshinaga proved that if an arrangement is
$C_{0}$, $C_{1}$ or $C_{2}$ then it is inductively connected, and so that its
moduli space is connected, see [NY12, Theorem. 3.11]. They also introduced the
combinatorial class $C_{3}$ of simple222The term simple used here is not
related to the one of Section 3.2. type.

######  Definition 3.15.

An arrangement $\mathcal{A}$ of class $C_{3}$ is of _simple type_ if one of
following conditions holds:

  1. (i)

the three lines of $\mathcal{D}_{3}$ are in generic position, and one of them
contains a unique multiple point,

  2. (ii)

the three lines of $\mathcal{D}_{3}$ are concurrent.

######  Theorem 3.16 ([NY12, Thm. 3.15]).

If $\mathcal{A}$ is a $C_{3}$ arrangement of simple type then its moduli space
$\mathcal{M}(\mathcal{A})$ is connected.

The main argument to prove that $C_{3}$ arrangement of simple type with
concurrent lines is irreducible use the fact that one can fix the three lines
using the action of $\operatorname{PGL}_{3}(\mathbb{C})$. Using the
combinatorial notion of inductively rigid arrangements, one can define a
larger class of arrangements with a connected moduli space.

######  Definition 3.17.

An arrangement $\mathcal{A}$ has a _rigid pencil form_ if it contains an
inductively rigid subarrangement $\mathcal{A}^{\prime}$ with a singular point
$P_{0}$ such that for any multiple point $P\in\operatorname{Sing}_{\geq
3}(\mathcal{A})$ one of the following conditions holds:

  1. (RP1)

$P$ is a singular point of $\mathcal{A}^{\prime}$,

  2. (RP2)

the line $(P,P_{0})$ is in $\mathcal{A}^{\prime}$.

######  Proposition 3.18.

Any $C_{3}$ arrangement of simple type has a rigid pencil form.

###### Proof.

Let $\mathcal{A}$ be a $C_{3}$ arrangement of simple type. Assume that
$\mathcal{A}$ verifies condition (i) of Definition 3.15. Let
$\mathcal{D}_{3}=\\{\ell,\ell^{\prime},\ell^{\prime\prime}\\}$ be a generic
subarrangement that contains all the multiple points of $\mathcal{A}$, and
such that $\ell^{\prime\prime}$ contains a unique multiple point $Q_{0}$.
There exist two lines $\ell_{Q_{0}},\ell^{\prime}_{Q_{0}}\in\mathcal{A}$
passing through $Q_{0}$ which do not contained $\ell\cap\ell^{\prime}$. The
arrangement
$\mathcal{A}^{\prime}=\\{\ell,\ell^{\prime},\ell_{Q_{0}},\ell^{\prime}_{Q_{0}}\\}$
is inductively rigid. The multiple point $Q_{0}$ is contained in
$\ell_{Q_{0}}$ and $\ell^{\prime}_{Q_{0}}$, and all the other multiple points
of $\mathcal{A}$ are contained in $\ell\cup\ell^{\prime}$. So $\mathcal{A}$
has a rigid pencil form.

Assume that $\mathcal{A}$ verifies condition (ii) of Definition 3.15. Then any
multiple point is in the $3$-pencil $\mathcal{D}_{3}$. Since $\mathcal{D}_{3}$
is in $\mathfrak{X}(3)$, then it is inductively rigid. So $\mathcal{A}$ has a
rigid pencil form. ∎

######  Theorem 3.19.

If $\mathcal{A}$ has a rigid pencil form then its moduli space
$\mathcal{M}(\mathcal{A})$ is connected.

###### Proof.

The idea of the proof is similar to the one of [NY12, Proof of Theorem 3.15,
case (ii)]. One can assume first that $\mathcal{A}^{\prime}$ is maximal, in
the sense that for all $\ell\in\mathcal{A}\setminus\mathcal{A}^{\prime}$ one
has $|\ell\cap\operatorname{Sing}(\mathcal{A}^{\prime})|<2$. Up to a
projective transformation, one can also assume that $P_{0}$ has coordinates
$(0:1:0)$ and that any line $\ell_{i}\in\mathcal{A}$ which pass through
$P_{0}$ is defined by the equation $x-\lambda_{i}z=0$, for a fixed
$\lambda_{i}\in\mathbb{C}$. Denote by $P_{i}$ the multiple points of
$\mathcal{A}$ which are not singular points of $\mathcal{A}^{\prime}$. By
Condition (RP2), each $P_{i}$ has coordinates $(\lambda_{i}:\alpha_{i}:1)$,
for some $\alpha_{i}\in\mathbb{C}$. The moduli space is then defined by some
collinearity relations between the points $P_{i}$ and the singular points of
$\mathcal{A}^{\prime}$. More precisely, there are two types:

  * •

The collinearity between three points $(P_{i},P_{j},P_{k})$ given by the
linear equation on $\alpha_{i}$, $\alpha_{j}$ and $\alpha_{k}$:

$\alpha_{i}(\lambda_{k}-\lambda_{j})+\alpha_{j}(\lambda_{i}-\lambda_{k})+\alpha_{k}(\lambda_{j}-\lambda_{k})=0.$

  * •

The collinearity between a point
$Q\in\operatorname{Sing}(\mathcal{A}^{\prime})$ and two points $P_{i}$ and
$P_{j}$ given by the affine equation on $\alpha_{i}$ and $\alpha_{j}$:

$\alpha_{i}(r_{Q}-\lambda_{j}t_{Q})+\alpha_{j}(\lambda_{i}t_{Q}-r_{Q})+s(\lambda_{i}-\lambda_{j})=0.$

where $Q=(r_{Q}:s_{Q}:t_{Q})\in\mathbb{C}\mathbb{P}^{2}$.

Due to the assumption that $\mathcal{A}^{\prime}$ is maximal, there is no
other type of relations. Since
$\mathcal{M}(\mathcal{A}^{\prime})=\left\\{\mathcal{A}^{\prime}\right\\}$ by
hypothesis, the moduli space $\mathcal{M}(\mathcal{A})$ is described by the
previous equations on the $\alpha_{i}$’s with fixed values
$\left\\{\lambda_{i}\right\\}_{\ell_{i}\in\mathcal{A}^{\prime}}$ and
$\left\\{(r_{Q}:s_{Q}:t_{Q})\right\\}_{Q\in\operatorname{Sing}(\mathcal{A}^{\prime})}$.
Thus, $\mathcal{M}(\mathcal{A})$ is an open Zariski subset of the kernel of an
affine map defined by the previous equations. As a consequence, it is
irreducible and then connected. ∎

######  Example 3.20.

Consider the arrangement $\mathcal{A}\in\operatorname{Arr}_{10}$ pictured in
Figure 5 and defined by the equations:

$\begin{array}[]{cccc}\ell_{1}:x=0,&\quad\ell_{2}:x-z=0,&\quad\ell_{3}:y=0,&\quad\ell_{4}:y-z=0,\\\
\ell_{5}:z=0,&\quad\ell_{6}:-x+y=0,&\quad\ell_{7}:x+y-z=0,&\quad\ell_{8}:-2x+4y-z=0,\\\
\ell_{9}:2x-3y+z=0,&\quad\ell_{10}:-4x+6y=0.\end{array}$

Its combinatorics is given by:

$\left\\{\begin{array}[]{c}\left\\{\ell_{1},\ell_{2},\ell_{5}\right\\},\left\\{\ell_{1},\ell_{3},\ell_{6},\ell_{10}\right\\},\left\\{\ell_{1},\ell_{4},\ell_{7}\right\\},\left\\{\ell_{1},\ell_{8}\right\\},\left\\{\ell_{1},\ell_{9}\right\\},\left\\{\ell_{2},\ell_{3},\ell_{7}\right\\}\\\\[3.00003pt]
\left\\{\ell_{2},\ell_{4},\ell_{6},\ell_{9}\right\\},\left\\{\ell_{2},\ell_{8}\right\\},\left\\{\ell_{2},\ell_{10}\right\\},\left\\{\ell_{3},\ell_{4},\ell_{5}\right\\},\left\\{\ell_{3},\ell_{8},\ell_{9}\right\\},\left\\{\ell_{4},\ell_{8},\ell_{10}\right\\}\\\\[3.00003pt]
\left\\{\ell_{5},\ell_{6}\right\\},\left\\{\ell_{5},\ell_{7}\right\\},\left\\{\ell_{5},\ell_{8}\right\\},\left\\{\ell_{5},\ell_{9},\ell_{10}\right\\},\left\\{\ell_{6},\ell_{7},\ell_{8}\right\\},\left\\{\ell_{7},\ell_{9}\right\\},\left\\{\ell_{7},\ell_{10}\right\\}\end{array}\right\\}.$

Let $\mathcal{A}^{\prime}=\\{\ell_{1},\dots,\ell_{7}\\}$ be the subarrangement
of $\mathcal{A}$ with combinatorics:

$\left\\{\left\\{\ell_{1},\ell_{2},\ell_{5}\right\\},\left\\{\ell_{1},\ell_{3},\ell_{6}\right\\},\left\\{\ell_{1},\ell_{4},\ell_{7}\right\\},\left\\{\ell_{2},\ell_{3},\ell_{7}\right\\},\left\\{\ell_{2},\ell_{4},\ell_{6}\right\\},\left\\{\ell_{3},\ell_{4},\ell_{5}\right\\},\left\\{\ell_{5},\ell_{6}\right\\},\left\\{\ell_{5},\ell_{7}\right\\},\left\\{\ell_{6},\ell_{7}\right\\}\right\\}.$

Once again, if we consider the order $\omega$ induced by the indices, we have
$\tau(\mathcal{A}^{\prime},\omega)=(0,0,0,0,2,2,2)$ and thus
$\mathcal{A}^{\prime}$ is an inductively rigid arrangement. Furthermore, all
multiple points of $\mathcal{A}$ are contained in the three concurrent lines
$\ell_{3}$, $\ell_{4}$ and $\ell_{5}$, except for the point
$\\{\ell_{6},\ell_{7},\ell_{8}\\}$ which contains the lines $\ell_{6}$ and
$\ell_{7}$ of $\mathcal{A}^{\prime}$. Thus $\mathcal{A}$ has a rigid pencil
form. By Theorem 3.19, we deduce that $\mathcal{M}(\mathcal{A})$ is connected.

It is worth noticing that all multiple points of $\mathcal{A}$ are also
contained in the $4$-pencil $\\{\ell_{1},\ell_{3},\ell_{6},\ell_{10}\\}$ or
$\\{\ell_{2},\ell_{4},\ell_{6},\ell_{9}\\}$. This implies that $\mathcal{A}$
is a $C_{4}$ arrangement. Nevertheless, any of these two $4$-pencils is
contained in an inductively rigid subarrangement of $\mathcal{A}$. So
$\mathcal{A}$ does not admit a rigid pencil form.

$\ell_{3}$$\ell_{4}$$\ell_{2}$$\ell_{1}$$\ell_{5}$$\infty$$\ell_{7}$$\ell_{6}$$\ell_{9}$$\ell_{10}$$\ell_{8}$
Figure 5. An arrangement with rigid pencil form.

######  Remark 3.21.

The statement of Theorem 3.19 still holds if the subarrangement
$\mathcal{A}^{\prime}$ verifies that
$\mathcal{M}(\mathcal{A}^{\prime})=\\{\mathcal{A}^{\prime}\\}$ instead of
being inductively rigid. Nevertheless, one should notice that this is a
geometric but not combinatorial condition.

##  4\. On the number of connected components of $\mathcal{M}(\mathcal{A})$

The objective of this section is to introduce a strategy to inductively build
an upper bound on the number of connected components of
$\mathcal{M}(\mathcal{A})$ by means of certain “tower of modifications” in the
combinatorics.

###  4.1. Perturbations of a combinatorics

######  Definition 4.1.

Let $\mathcal{C}=(\mathcal{L},\mathcal{P})$ be an abstract line combinatorics.
Let $P_{0}\in\mathcal{P}$ be a fixed multiple point of $\mathcal{C}$ and let
$\ell\in P_{0}$. An _elementary perturbation_ of $\mathcal{C}$ at
$(\ell,P_{0})$ is an abstract combinatorics
$\widetilde{\mathcal{C}}=(\mathcal{L},\widetilde{\mathcal{P}})$ such that:

  1. (1)

for all $P\in\mathcal{P}$, $P\neq P_{0}$, one has
$P\in\widetilde{\mathcal{P}}$,

  2. (2)

$\widetilde{P}_{0}=P_{0}\setminus\\{\ell\\}$ is in $\widetilde{\mathcal{P}}$.

Such a relation is denoted $\widetilde{\mathcal{C}}\prec\mathcal{C}$. For any
multiple point $P\in\mathcal{P}$, we call the _parent of $P$_ in
$\widetilde{\mathcal{C}}$ the unique element
$\widetilde{P}\in\widetilde{\mathcal{P}}$ such that either $\widetilde{P}=P$
or $\widetilde{P}=\widetilde{P}_{0}$.

######  Example 4.2.

Let
$\mathcal{C}=\\{\\{\ell_{1},\ell_{2},\ell_{3},\ell_{4}\\},\\{\ell_{1},\ell_{5}\\},\\{\ell_{2},\ell_{5}\\},\\{\ell_{3},\ell_{5}\\},\\{\ell_{4},\ell_{5}\\}\\}$
be the combinatorics of an arrangement in $\overline{\mathfrak{X}}(5)$. The
elementary perturbation of $\mathcal{C}$ at $(P_{0},\ell_{4})$, for
$P_{0}=\\{\ell_{1},\ell_{2},\ell_{3},\ell_{4}\\}$, is:

$\\{\\{\ell_{1},\ell_{2},\ell_{3}\\},\\{\ell_{1},\ell_{4}\\},\\{\ell_{1},\ell_{5}\\},\\{\ell_{2},\ell_{4}\\},\\{\ell_{2},\ell_{5}\\},\\{\ell_{3},\ell_{5}\\},\\{\ell_{4},\ell_{5}\\},\\{\ell_{4},\ell_{5}\\}\\},$

and the parent of $P_{0}=\\{\ell_{1},\ell_{2},\ell_{3},\ell_{4}\\}$ is
$\\{\ell_{1},\ell_{2},\ell_{3}\\}$.

######  Lemma 4.3.

If $\widetilde{C}$ is an elementary perturbation of $\mathcal{C}$ then
$\sum_{P\in\widetilde{\mathcal{P}}}(|P|-2)=\sum_{P\in\mathcal{P}}(|P|-2)-1$.

###### Proof.

Assume that $\widetilde{\mathcal{C}}$ is an elementary perturbation of
$\mathcal{C}$ at $(\ell,P_{0})$. By Definition 4.1-(1), all the terms in the
right-sided sum, except for that depending on $P_{0}$, are also in the left-
sided sum. Since $\widetilde{P}_{0}$ is the parent of $P_{0}$, one has
$(|\widetilde{P}_{0}|-2)=(|P_{0}|-2)+1$. Finally, notice that all remaining
terms of the left-sided sum correspond to double points of
$\widetilde{\mathcal{C}}$ and thus they do not contribute to the sum. ∎

Note that, according to Definition 2.10, the previous lemma implies that

$1+\operatorname{d^{naive}}\mathcal{M}(\mathcal{C})=\operatorname{d^{naive}}\mathcal{M}(\widetilde{\mathcal{C}}).$

######  Definition 4.4.

A _$m$ -perturbation_ of $\mathcal{C}=(\mathcal{L},\mathcal{P})$ is a sequence
of $m$ elementary perturbations such that

$\mathcal{C}_{0}\prec\mathcal{C}_{1}\prec\cdots\prec\mathcal{C}_{m}=\mathcal{C},$

and $\mathcal{C}_{0}$ is inductively connected with associated order
$\omega_{0}$. Since $\mathcal{C}_{0}$ and $\mathcal{C}$ have the same set of
lines, then $\omega_{0}$ is also an order on $\mathcal{C}$, and it will be
called the _perturbation order_. For any multiple point $P$ in
$\mathcal{P}_{i}$, there is a unique chain of parents coming from each
elementary perturbation:

$\widetilde{P}=P_{0}\subset P_{1}\subset\cdots\subset P_{i}=P.$

The multiple point $\widetilde{P}$ in $\mathcal{C}_{0}$ is called the
_ancestor of $P$_.

For brevity sake, a $m$-perturbation
$\mathcal{C}_{0}\prec\mathcal{C}_{1}\prec\cdots\prec\mathcal{C}_{m}=\mathcal{C}$
will be denoted by $\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}$. Any
combinatorics can be reduce to the generic combinatorics by enough successive
elementary perturbations. Since the generic combinatorics is inductively
connected then any combinatorics admits a $m$-perturbation for some $m$.

###  4.2. Parametrization of the moduli space.

The purpose of this section is to give a constructible proof of the following
result.

######  Theorem 4.5.

Let $\mathcal{A}\in\operatorname{Arr}_{n}$ be an arrangement, with
$\mathcal{A}\notin\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$, which admits
a $m$-perturbation
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$, and denote
$d_{0}=\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})$. There exists an
open Zariski subset $W_{0}$ of the affine space of dimension $d_{0}$ and $m$
polynomials $\Delta_{1},\ldots,\Delta_{m}$ such that

$\mathcal{M}(\mathcal{A})\simeq\left\\{(v_{1},\dots,v_{d_{0}})\in
W_{0}\mid\Delta_{1}=\dots=\Delta_{m}=0\right\\}.$ (4)

Remark that we may have that some of the $\Delta_{i}$’s are always trivial. It
is for example the case in Pappus arrangement. From the previous result, we
can deduce the following corollary. The left-hand inequality corresponds to
the one of Proposition 2.12.

######  Corollary 4.6.

Let $\mathcal{A}\in\operatorname{Arr}_{n}$ be a line arrangement. If
$\mathcal{C}(\mathcal{A})$ admits a $m$-perturbation then

$0\leq\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})-\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})\leq
m.$

###### Proof.

According to Theorem 4.5, we can describe $\mathcal{M}(\mathcal{A})$ as the
intersection of at most $m$ hypersurfaces in some non-empty open Zariski
subset of the affine space of dimension
$\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})$. So, using classical
arguments on the codimension of the intersection of hypersurfaces, we deduce
that
$\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})\geq\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})\geq\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})-m$.

By successive applications of Lemma 4.3, we have that
$\sum_{P\in\widetilde{\mathcal{P}}_{0}}(|P|-2)=\sum_{P\in\mathcal{P}}(|P|-2)-m$.
This can be reformulated as
$\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})=\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})+m$.
Then, we substitute $\operatorname{d^{naive}}\mathcal{M}(\mathcal{C}_{0})$ by
$\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})+m$ in the previous
inequalities, and we obtain:

$\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})\leq\dim_{\mathbb{C}}\mathcal{M}(\mathcal{A})\leq\operatorname{d^{naive}}\mathcal{M}(\mathcal{A})+m.\qed$

The strategy of the proof of Theorem 4.5 is the following. The goal is to
express the moduli space $\mathcal{M}(\mathcal{A})$ described in (1) from the
following data:

  * •

An open Zariski subset $W_{0}$ of an affine linear space, codifying the open
conditions in (1).

  * •

A map $\Psi$ over $W_{0}$ which parametrizes the arrangements in
$\operatorname{Arr}_{n}$ verifying the Zariski-closed conditions of
$\mathcal{M}(\mathcal{C}_{0})$, e.g those of type $\Delta_{i,j,k}=0$ with
$\left\\{i,j,k\right\\}\subset\widetilde{P}$ for a multiple point
$\widetilde{P}$ in $\mathcal{C}_{0}$.

  * •

A list $\Delta_{1},\ldots,\Delta_{m}$ of $m$ polynomials, i.e. the remaining
Zariski-closed conditions coming from multiple points $P$ in
$\mathcal{C}(\mathcal{A})$ which are not in $\mathcal{C}_{0}$, where each
polynomial represents a step of the $m$-perturbation
$C_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$.

We start by constructing an expression of the map $\Psi$, then we define the
polynomials $\Delta_{1},\ldots,\Delta_{m}$ and finally the existence of
$W_{0}$ is discussed.

######  Remark 4.7.

Let $P=\left\\{\ell_{i_{1}},\ldots,\ell_{i_{m}}\right\\}$ be a multiple point
in $\mathcal{C}$. The set of Zariski-closed conditions
$\left\\{\Delta_{i,j,k}=0\mid\left\\{i,j,k\right\\}\subset P\right\\}$ can be
reduced to $\left\\{\Delta_{i_{1},i_{2},i_{k}}=0\mid k=3,\ldots,m\right\\}$.

Let $\operatorname{\mathbf{v}}=(v_{1},\dots,v_{d_{0}})$ be a system of
coordinates in $V_{0}=\mathbb{C}^{d_{0}}$. For each line $\ell_{i}$, we
associate three polynomials
$a_{i},b_{i},c_{i}\in\mathbb{C}[v_{1},\dots,v_{d_{0}}]$ such that

$\ell_{i}:a_{i}(\operatorname{\mathbf{v}})x+b_{i}(\operatorname{\mathbf{v}})y+c_{i}(\operatorname{\mathbf{v}})z=0.$

In this way, one can define a map called _parametrization_ of
$\mathcal{M}(\mathcal{A})$:

$\Psi:\operatorname{\mathbf{v}}\in
W_{0}\longmapsto\left(\left(\begin{array}[]{c}a_{1}(\operatorname{\mathbf{v}})\\\
b_{1}(\operatorname{\mathbf{v}})\\\
c_{1}(\operatorname{\mathbf{v}})\end{array}\right),\ldots,\left(\begin{array}[]{c}a_{n}(\operatorname{\mathbf{v}})\\\
b_{n}(\operatorname{\mathbf{v}})\\\
c_{n}(\operatorname{\mathbf{v}})\end{array}\right)\right)\in(\mathbb{C}^{3})^{n}.$
(5)

In addition to $\Psi$, one can construct another map $\Phi$ that parameterize
the multiple points333In the following, it may happen that we define the map
$\Phi$ for a double point of $\mathcal{C}(\mathcal{A})$. This allows to
lighten the explanation and it has no incidence on the result. of the elements
in $\mathcal{M}(\mathcal{A})$ as follows. Let $P$ be a multiple point in the
combinatorics $\mathcal{C}(\mathcal{A})$, and let $\widetilde{P}$ be the
ancestor of $P$ in $\mathcal{C}_{0}$. Take $\ell_{i}$ and $\ell_{j}$ the two
lines in $\widetilde{P}$ which are minimal with respect to the order
$\omega_{0}$. We define

$\Phi_{P}=(b_{i}c_{j}-b_{j}c_{i},a_{j}c_{i}-a_{i}c_{j},a_{i}b_{j}-a_{j}b_{i})\in\mathbb{C}[v_{1},\dots,v_{d_{0}}]^{3},$
(6)

If an arrangement $\mathcal{A}_{0}\in\mathcal{M}(\mathcal{A})$ is given by
$\Psi(\operatorname{\mathbf{v}})$ then for any multiple point $P$ of
$\mathcal{C}(\mathcal{A})$, the vector
$\Phi_{P}(\operatorname{\mathbf{v}})\in\mathbb{C}^{3}$ express the homogeneous
coordinates of $P$ in $\mathbb{C}\mathbb{P}^{2}$.

Let us describe in detail how the polynomials $a_{i}$, $b_{i}$ and $c_{i}$ are
inductively constructed. Since
$\mathcal{A}\notin\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$, one can
follow similar arguments as in Proposition 3.3, and assume that the
perturbation order $\omega_{0}$ is such that the lines $\ell_{1}$, $\ell_{2}$,
$\ell_{3}$ and $\ell_{4}$ are in generic position in both $\mathcal{C}_{0}$
and $\mathcal{C}(\mathcal{A})$. Using the action of
$\operatorname{PGL}_{3}(\mathbb{C})$, we fix them as $x=0$, $x-z=0$, $y=0$ and
$y-z=0$, respectively. In other words, for all $\operatorname{\mathbf{v}}\in
V_{0}$, we define

$\Psi(\operatorname{\mathbf{v}})_{1}=(1,0,0),\quad\Psi(\operatorname{\mathbf{v}})_{2}=(1,0,-1),\quad\Psi(\operatorname{\mathbf{v}})_{3}=(0,1,0),\quad\Psi(\operatorname{\mathbf{v}})_{4}=(0,1,-1).$
(7)

It follows from equation (6) that the parametrization $\Phi$ of the singular
points of these four lines are given for any $\operatorname{\mathbf{v}}\in
V_{0}$ by:

$\begin{array}[]{ccccc}\Phi_{\langle\ell_{1},\ell_{2}\rangle}(\operatorname{\mathbf{v}})=(0,1,0),&&\Phi_{\langle\ell_{1},\ell_{3}\rangle}(\operatorname{\mathbf{v}})=(0,0,1),&&\Phi_{\langle\ell_{1},\ell_{4}\rangle}(\operatorname{\mathbf{v}})=(0,1,1),\\\
\Phi_{\langle\ell_{2},\ell_{3}\rangle}(\operatorname{\mathbf{v}})=(1,0,1),&&\Phi_{\langle\ell_{2},\ell_{4}\rangle}(\operatorname{\mathbf{v}})=(1,1,1),&&\Phi_{\langle\ell_{3},\ell_{4}\rangle}(\operatorname{\mathbf{v}})=(1,0,0).\end{array}$
(8)

where $\langle\ell_{i_{1}},\ldots,\ell_{i_{k}}\rangle$ with $k\geq 2$ is the
unique point $P$ in $\mathcal{C}(\mathcal{A})$ verifying that
$\left\\{\ell_{i_{1}},\ldots,\ell_{i_{k}}\right\\}\subset P$, if it exists.

The induction process goes as follows. Assume that the maps $\Psi$ and $\Phi$
parametrize $\mathcal{A}_{i-1}$ in the chain (2) with respect to the
perturbation order $\omega_{0}$. Note that the number of parameters used in
this parametrization is
$d_{i}=\sum_{j=5}^{i-1}\big{(}2-\tau_{j}(\mathcal{C}_{0},\omega_{0}))$. The
next step is then to extend these parametrization maps to $\mathcal{A}_{i}$ .
They are determined by the values $\tau_{i}(\mathcal{C}_{0},\omega_{0})$:

  * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=0$, i.e. the line $\ell_{i}$ is
generic in $(\mathcal{C}_{0})_{i}$. Since
$\ell_{i}\not\in\left\langle\ell_{1},\ell_{2}\right\rangle$ or
$\ell_{i}\not\in\left\langle\ell_{1},\ell_{3}\right\rangle$ in
$\mathcal{C}(\mathcal{A})$, we can parametrize $\ell_{i}$ using only two
complex parameters $v_{d_{i}}$ and $v_{d_{i}+1}$,

$\Psi(\operatorname{\mathbf{v}})_{i}=\begin{cases}(v_{d_{i}},1,v_{d_{i}+1})&\text{if
$\ell_{i}\not\in\left\langle\ell_{1},\ell_{2}\right\rangle$},\\\
(1,v_{d_{i}},v_{d_{i}+1})&\text{if
$\ell_{i}\not\in\left\langle\ell_{1},\ell_{3}\right\rangle$}.\end{cases}$ (9)

  * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=1$, i.e. the line $\ell_{i}$ passes
through a unique multiple point $P_{0}$ in $(\mathcal{C}_{0})_{i}$. Take
$\ell_{j}$ and $\ell_{k}$ the two lines in $P_{0}$ which are minimal with
respect to the order $\omega_{0}$. One define:

$\Psi(\operatorname{\mathbf{v}})_{i}=\Psi(\operatorname{\mathbf{v}})_{j}+v_{d_{i}}\cdot\Psi(\operatorname{\mathbf{v}})_{k}.$
(10)

  * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=2$, i.e. the line $\ell_{i}$ passes
through two multiple points $P_{0}$ and $Q_{0}$ in $(\mathcal{C}_{0})_{i}$.
The line $\ell_{i}$ is then parametrized by

$\Psi(\operatorname{\mathbf{v}})_{i}=\Phi_{\left\langle
P_{0}\right\rangle}(\operatorname{\mathbf{v}})\times\Phi_{\left\langle
Q_{0}\right\rangle}(\operatorname{\mathbf{v}}),$ (11)

where “$\times$” stands for the usual cross product.

The next step is to construct the polynomials $\Delta_{1},\ldots,\Delta_{m}$.
Assume that the $i$th elementary perturbation
$\mathcal{C}_{i-1}\prec\mathcal{C}_{i}$ in
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$ is at
$(\ell_{j},P)$ and let $\widetilde{P}$ be the parent of $P$ in
$\mathcal{C}_{i-1}$. Take two distinct elements
$\ell_{i_{1}},\ell_{i_{2}}\in\widetilde{P}$ and define:

$\Delta_{i}(\operatorname{\mathbf{v}})=\det\left(\Psi(\operatorname{\mathbf{v}})_{i_{1}},\Psi(\operatorname{\mathbf{v}})_{i_{2}},\Psi(\operatorname{\mathbf{v}})_{j}\right)\in\mathbb{C}[v_{1},\ldots,v_{d_{0}}].$
(12)

It is worth noticing that another choice of
$\ell_{i_{1}},\ell_{i_{2}}\in\widetilde{P}$ will lead to an equivalent
construction of $\mathcal{M}(\mathcal{A})$, due to Remark 4.7.

The Zariski open subset $W_{0}$ can be expressed in a similar way as

$W_{0}=\left\\{\operatorname{\mathbf{v}}\in\mathbb{C}^{d_{0}}\mid\det\left(\Psi(\operatorname{\mathbf{v}})_{i},\Psi(\operatorname{\mathbf{v}})_{j},\Psi(\operatorname{\mathbf{v}})_{k}\right)\neq
0,\,\text{if $\not\exists P$ in $\mathcal{C}(\mathcal{A})$ such that
$\left\\{i,j,k\right\\}\subset P$}\right\\}.$

Since $\mathcal{A}\not\in\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$ by
hypothesis, the above polynomial inequalities imply that
$\ell_{i}\neq\ell_{j}$, for any $i\neq j\in\left\\{1,\ldots,n\right\\}$.

###  4.3. Inductive upper bound

The purpose of this section is to compute an upper bound for
$\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}))$, the number of connected
components of $\mathcal{M}(\mathcal{A})$. To achieve this goal, we construct
an upper bound on the number of irreducible components of
$\mathcal{M}(\mathcal{A})$, and then we apply the following classical lemma,
e.g. [Sha13, Sec. 7.2].

######  Lemma 4.8.

Let $\mathcal{A}$ be a line arrangement. We have the following inequality:

$\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}))\leq\\#\operatorname{Irr}(\mathcal{M}(\mathcal{A})),$

where $\operatorname{Irr}(\mathcal{M}(\mathcal{A}))$ is the set of irreducible
components of $\mathcal{M}(\mathcal{A})$.

When $\mathcal{A}$ is in $\mathfrak{X}(n)$ or $\overline{\mathfrak{X}}(n)$, we
know by Remark 2.5 that its moduli space $\mathcal{M}(\mathcal{A})$ is
irreducible and so connected. Among this section, we assume that
$\mathcal{A}\not\in\mathfrak{X}(n)\cup\overline{\mathfrak{X}}(n)$.

Let $\mathcal{A}=\\{\ell_{1},\dots,\ell_{n}\\}$ be an arrangement with a
$m$-perturbation
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})=(\mathcal{A},\mathcal{P})$
and perturbation order $\omega_{0}$. Up to relabelling, we assume that
$\omega_{0}(\ell_{i})=i$. Furthermore, we can also assume that $\ell_{1}$,
$\ell_{2}$, $\ell_{3}$ and $\ell_{4}$ are in generic position in
$\mathcal{C}(\mathcal{A})$. We define recursively two applications
$\Lambda:\mathcal{A}\rightarrow\mathbb{N}$ and
$\Theta:\mathcal{P}\rightarrow\mathbb{N}$ as follows.

  1. (R1)

For $i\in\\{1,2,3,4\\}$, we fix $\Lambda(\ell_{i})=0$.

  2. (R2)

For any $P\in\mathcal{C}(\mathcal{A}_{4})$, we fix $\Theta(\left\langle
P\right\rangle)=0$.

  3. (R3)

For $i\in\\{5,\dots,n\\}$, the expression of $\Lambda(\ell_{i})$ depends on
the value $\tau_{i}(\mathcal{C}_{0},\omega_{0})$, following the ideas of the
proof in Section 4.2:

     * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=0$, we fix

$\Lambda(\ell_{i})=1.$

     * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=1$, i.e. the line $\ell_{i}$ passes
through a unique singular point $P_{0}$ in $(\mathcal{C}_{0})_{i}$, we define:

$\Lambda(\ell_{i})=\left\\{\begin{array}[]{ll}1+\Lambda(\ell_{j})&\text{ if
}\Lambda(\ell_{j})=\Lambda(\ell_{k}),\\\
\max(\Lambda(\ell_{j}),\Lambda(\ell_{k}))&\text{ otherwise.
}\end{array}\right.$

where $\ell_{j}$ and $\ell_{k}$ are two lines in the parent of $P_{0}$.

     * •

If $\tau_{i}(\mathcal{C}_{0},\omega_{0})=2$, i.e. the line $\ell_{i}$ passes
through two singular points $P_{0}$ and $Q_{0}$ in $(\mathcal{C}_{0})_{i}$, we
define

$\Lambda(\ell_{i})=\Theta(\left\langle P_{0}\right\rangle)+\Theta(\left\langle
Q_{0}\right\rangle).$

  4. (R4)

For $i\in\\{5,\dots,n\\}$ and for any double point $P=\\{\ell_{i},\ell_{j}\\}$
in $(\mathcal{C}_{0})_{i}$, we define

$\Theta(\left\langle P\right\rangle)=\Lambda(\ell_{i})+\Lambda(\ell_{j}).$

######  Lemma 4.9.

Let $\mathcal{A}$ be an arrangement and let
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$ be a
$m$-perturbation. There exists a parametrization $\Psi$ of
$\mathcal{M}(\mathcal{A})$ as in (5) such that for all $j\in\\{1,\dots,n\\}$:

$\max(\deg a_{j},\deg b_{j},\deg c_{j})\leq\Lambda(\ell_{j}).$

Moreover, consider the description of $\mathcal{M}(\mathcal{A})$ induced by
$\Psi$ as in (4). If $\mathcal{C}_{i-1}\prec\mathcal{C}_{i}$ is the elementary
perturbation at $(\ell_{j},P)$, and if $\ell_{j_{1}}$ and $\ell_{j_{2}}$ are
two lines passing through the parent of $P$ in $\mathcal{C}_{i-1}$, then:

$\deg\Delta_{i}\leq\Lambda(\ell_{j_{1}})+\Lambda(\ell_{j_{2}})+\Lambda(\ell_{j}).$

###### Proof.

The first inequality holds straightforward by construction of $\Lambda$ and
$\Theta$. More precisely, the rules (R1), (R2) and (R4) control the behaviour
of the degrees of the polynomials in equations (7), (8) and (6), respectively,
meanwhile (R3) controls the one of (9), (10) and (11). Furthermore, by
Equation (12),
$\Delta_{i}=\det(\Psi(\operatorname{\mathbf{v}})_{j_{1}},\Psi(\operatorname{\mathbf{v}})_{j_{2}},\Psi(\operatorname{\mathbf{v}})_{j})$,
then the first inequality induces the second part of the lemma. ∎

######  Theorem 4.10.

Let $\mathcal{A}$ be a line arrangement and assume that
$\mathcal{C}(\mathcal{A})$ admits a $m$-perturbation
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$. The following
inequality holds:

$\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}))\leq\prod_{i=1}^{m}\big{(}\Lambda(\ell_{i_{1}})+\Lambda(\ell_{i_{2}})+\Lambda(\ell_{j})\big{)}.$

where $\mathcal{C}_{i-1}\prec\mathcal{C}_{i}$ is the elementary perturbation
at $(\ell_{j},P)$, and $\ell_{j_{1}}$, $\ell_{j_{2}}$ are two lines contained
in the parent of $P$ in $\mathcal{C}_{i-1}$.

###### Proof.

In Theorem 4.5, we have obtain a description of $\mathcal{M}(\mathcal{A})$ as
the intersection of at most $m$ proper algebraic hypersurfaces
$\left\\{V_{i}:\Delta_{i}=0\right\\}_{i=1}^{m}$, in a non-empty Zariski open
subset of an affine linear space. By [Ful98, Ex. 8.3.6], this implies that
$\\#\operatorname{Irr}(\mathcal{M}(\mathcal{A}))\leq\prod_{i=1}^{m}\deg\Delta_{i}$.
By Lemmas 4.8 and 4.9, we obtain the result. ∎

######  Remark 4.11.

There are three types of choices in the construction of the upper bound in the
previous theorem: the first one is obviously the choice of the
$m$-perturbation
$\mathcal{C}_{0}\prec\mkern-5.0mu\prec\mathcal{C}(\mathcal{A})$, the second
one is in rule (R3) (the choice of the lines $\ell_{i_{1}}$ and $\ell_{i_{2}}$
when $\tau_{i}(\mathcal{C}_{0},\omega_{0})=1$), and the third one is the
choice of the lines $\ell_{j_{1}}$ and $\ell_{j_{2}}$ in the elementary
perturbation $\mathcal{C}_{i-1}\prec\mathcal{C}_{i}$. Note that the value of
the upper bound depends on the previous choices.
Heuristically, we can optimize the upper bound by taking the lines
$\ell_{i_{1}},\ell_{i_{2}}$ and $\ell_{j_{1}},\ell_{j_{2}}$ which minimize the
value of their images by the map $\Lambda$.

###  4.4. Sharpness of the upper bound

By construction, the inequality in Theorem 4.10 becomes an equality for any
inductively connected or inductively rigid arrangement. The question is then,
is this inequality still sharp in non-trivial cases?

######  Theorem 4.12.

For any $N\in\mathbb{N}_{\geq 2}$, there exists an arrangement $\mathcal{A}$
such that $\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}))\geq N$, and

$\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}))=\prod_{i=1}^{m}\big{(}\Lambda(\ell_{i_{1}})+\Lambda(\ell_{i_{2}})+\Lambda(\ell_{j})\big{)},$

with the notation of Theorem 4.10.

###### Proof.

Let $p$ be a prime number such that $p\geq N$. Fix a primitive $p$-root of
unity $\zeta$. Consider the arrangement
$\mathcal{A}_{p}=\left\\{\ell_{1},\ldots,\ell_{2p+2}\right\\}$ with lines:

$\ell_{1}\colon x-y=0,\quad\ell_{2}\colon x-\zeta y=0,\quad\ell_{2i+1}\colon
x+\zeta^{i}z=0\quad\text{ and }\quad\ell_{2i+2}:\zeta^{-i}y+z=0,$

for $i\in\\{1,\dots,p\\}$. The multiple points of $\mathcal{A}_{p}$ are the
two points of multiplicity $p$:

$\\{\ell_{3},\ell_{5},\dots,\ell_{2p+1}\\}\quad\text{and}\quad\\{\ell_{4},\ell_{6},\dots,\ell_{2p+2}\\},$

and the $2p$ triple points given by:

$\\{\ell_{1},\ell_{2i+1},\ell_{2i+2}\\}\quad\text{and}\quad\\{\ell_{2},\ell_{2i+2},\ell_{[2i]+3}\\}$

for $i\in\\{1,\dots,p\\}$, where $[a]$ is the value of $a$ modulo $2p$ such
that $0\leq[a]<2p$.

Consider the elementary perturbation
$\widetilde{\mathcal{C}}(\mathcal{A}_{p})$ on $\mathcal{C}(\mathcal{A}_{p})$
at $(\ell_{2p+2},\\{\ell_{2},\ell_{3},\ell_{2p+2}\\})$. Let $\omega_{0}$ be
the following order on $\widetilde{\mathcal{C}}(\mathcal{A}_{p})$:

$(\ell_{1},\dots,\ell_{2p+2})\longmapsto(5,6,1,2,3,4,7,8,\dots,(2p+2)).$

We have that
$\tau(\widetilde{\mathcal{C}}(\mathcal{A}_{p}),\omega_{0})=(0,0,0,0,2,1,2,2\dots,2,2)$.
So $\widetilde{\mathcal{C}}(\mathcal{A}_{p})$ is inductively connected, and
thus it is a $1$-perturbation of $\mathcal{C}(\mathcal{A}_{p})$. Using the
rules (R1)-(R2)-(R3)-(R4), we can compute that:

$\Lambda_{p}:(\ell_{1},\dots,\ell_{2p+2})\longmapsto(0,1,0,0,0,0,1,1,2,2,\dots,p-2,p-2).$

By Theorem 4.10, we obtain that $\mathcal{M}(\mathcal{A}_{p})$ has at most
$\Lambda_{p}(\ell_{2})+\Lambda_{p}(\ell_{3})+\Lambda_{p}(\ell_{2p+2})=1+0+(p-2)=p-1$
connected components. On the other hand, the computation of the moduli space
using Theorem 4.5 shows that the dimension is zero. Furthermore, it contains
$\mathcal{A}_{p}$ for any choice of primitive $p$-root of unity $\zeta$. We
conclude that $\\#\operatorname{CC}(\mathcal{M}(\mathcal{A}_{p}))=p-1$. ∎

Note that the arrangement $\mathcal{A}_{2}$ corresponds to the Ceva
arrangement, while the arrangements $\mathcal{A}_{3}$ are the MacLane
arrangements [Mac36]. This family of arrangements appears in [CS18] in the
context of Zariski pairs as subarrangements of the reflection arrangements
associated with $G(N,N,3)$.

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