# Moduli Spaces of One-Line Extensions of $(10_{3})$ Configurations

Moshe Cohen  Mathematics Department, State University of New York at New
Paltz, New Paltz, New York, USA<EMAIL_ADDRESS>and  Baian Liu
Department of Mathematics, The Ohio State University, Columbus, Ohio
<EMAIL_ADDRESS>
###### Abstract.

Two line arrangements in $\mathbb{CP}^{2}$ can have different topological
properties even if they are combinatorially isomorphic. Results by Dan Cohen
and Suciu and by Randell show that a reducible moduli space under complex
conjugation is a necessary condition.

We present a method to produce many examples of combinatorial line
arrangements with a reducible moduli space obtained from a set of examples
with irreducible moduli spaces.

In this paper, we determine the reducibility of the moduli spaces of a family
of arrangements of 11 lines constructed by adding a line to one of the ten
$(10_{3})$ configurations. Out of the four hundred ninety-five combinatorial
line arrangements in this family, ninety-five have a reducible moduli space,
seventy-six of which are still reducible after the quotient by complex
conjugation.

###### Key words and phrases:

$(n_{3})$ configuration, geometric matroid, extension by an element

###### 2020 Mathematics Subject Classification:

52C35, 32S22, 14N20, 14H37, 14Q05

##  1\. Introduction

Hyperplane arrangements are not only classically interesting [OS80] but have
been studied more recently [Dim17a]. These objects have interesting
combinatorial structures [BLVSWZ99, HOV15] and even have applications to
database searches [SBA15]. Interesting questions concerning hyperplane
arrangements include those regarding whether the combinatorics determine
topological properties [DIM17b].

We focus on hyperplane arrangements in two dimensions: line arrangements. We
define a combinatorial line arrangement
$\mathcal{A}=(\mathcal{P},\mathcal{L})$ to be a finite set of points
$\mathcal{P}$ and a finite set of lines $\mathcal{L}$, which are subsets of
$\mathcal{P}$. We also require that the intersection of two distinct lines be
one point or empty. Given a combinatorial line arrangement $\mathcal{A}$, we
consider its moduli space $\mathcal{M}_{\mathcal{A}}$, its set of all
geometric realizations in $\mathbb{CP}^{2}$. Elements of the same moduli space
are combinatorially equivalent.

The number of components of a moduli space gives us information about the
topology of the arrangement. In an irreducible moduli space or in a single
connected component, Randell’s Isotopy Theorem [Ran89] states that any two
arrangements $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ in 1-parameter family of
combinatorially equivalent arrangements have diffeomorphic complements and
that ($\mathbb{CP}^{2}$,$\mathcal{A}_{1}$) is homeomorphic to
($\mathbb{CP}^{2}$,$\mathcal{A}_{2}$). Furthermore, a result by Dan Cohen and
Suciu [CS97, Theorem 3.9] states that complex conjugate arrangements have
equivalent braid monodromies and so also have diffeomorphic complements.

Thus we seek combinatorial line arrangements with a reducible moduli space
$\mathcal{M}_{\mathcal{A}}$ and a reducible moduli space modulo complex
conjugation $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$. We construct numerous
such examples of combinatorial line arrangements using one-line extensions. We
say that a one-line extension of a combinatorial line arrangement
$\mathcal{A}$ is $\mathcal{A}$ together with an additional line. More commonly
in the literature, matroids are extended by single elements; according to
Oxley, this “can be fraught with difficulty” [Oxl19]. Our method is a specific
type of single-element extension, in a dual sense. For more on matroids, see
Oxley’s textbook [Oxl92]. Kocay [Koc16] uses a method of one-point extensions
in order to find coordinatizations of arrangements.

We apply one-line extensions to $(10_{3})$ configurations. More generally, an
($\bf n_{3}$) configuration is a combinatorial line arrangement with $n$ lines
and $n$ triple points, which are points that lie in exactly three lines or, in
other words, have multiplicity three. Such arrangements are not only
classically interesting [Mar87, DvS03, Gro90, SW90] but have been studied more
recently [Grü09, AAB14, EP13, Koc16, Koc21].

The reason we apply one-line extension to $(10_{3})$ configurations is to find
combinatorial line arrangements with 11 lines and reducible moduli space.
Combinatorial line arrangements with 10 or fewer lines and a reducible moduli
space have been classified. In 1997, Fan showed that there are no
combinatorial line arrangements with a reducible moduli space of up to and
including 6 lines [Fan97]. Garber, Teicher, and Vishne showed that there are
combinatorial line arrangements with a reducible moduli space of up to 8 lines
that are realizable in $\mathbb{RP}^{2}$ [GTV03]. Nazir and Yoshinaga verified
that for combinatorial line arrangements of up to and including 9 lines, those
with a reducible moduli space must contain one of three combinatorial line
arrangements as a subarrangement: the MacLane arrangement, also known as the
Möbius-Kantor arrangement or the unique ($8_{3}$) configuration [Kan81, Mö28,
Rey82, Sch89, Mac36]; the Nazir-Yoshinaga arrangement [NY12]; and the Falk-
Sturmfels arrangement [CS97, cited as unpublished]. We say an arrangement is
exceptional if it contains one of these three arrangements as a subarrangement
and unexceptional otherwise.

Suppose a combinatorial line arrangement $\mathcal{A}$ contains a line $\ell$
that passes through at most two points of multiplicity three or greater. Form
another combinatorial line arrangement $\mathcal{A}\backslash\ell$ by deleting
$\ell$ from $\mathcal{A}$. Nazir and Yoshinaga show that if the moduli space
of $\mathcal{A}\backslash\ell$ is irreducible, then the moduli space of
$\mathcal{A}$ is irreducible [NY12, Lemma 3.2]. We say a combinatorial line
arrangement is reductive if it contains a line that passes through at most two
points of multiplicity three or greater and non-reductive otherwise.

The first author with Amram, Teicher, and Ye completed the classification of
irreducibility of the moduli space of non-reductive, unexceptional
arrangements of 10 lines in [ATY13] and [ACTY13], producing eighteen examples.
Further analysis of these authors together with Sun and Zarkh reduced the
number of candidates from eighteen to fifteen [ACSTYZ15]. Motivating this
work, two of the nine combinatorial line arrangements from [ACTY13] with a
reducible moduli space are on the list of the eleven one-line extensions of
($9_{3}$) configurations.

Amram, Gong, Teicher, and Xu classify the moduli spaces of non-reductive
arrangements of 11 lines with at least one point of multiplicity at least five
in [AGTX15], identifying thirty-eight arrangements that satisfy the necessary
moduli space condition. Further interesting examples of arrangements of 11
lines from the literature include a reductive, exceptional example by Artal
Bartolo, Carmona Ruber, Cogolludo-Agustín, and Marco Buzunáriz and twenty-nine
examples by Guerville-Ballé [GB22].

In this current work, we describe the one-line extension construction. While
it does not produce reducible moduli spaces all of the time, it can
empirically produce reducible moduli spaces a lot of the time. We use this to
continue the classification of moduli spaces of non-reductive combinatorial
line arrangements of 11 lines.

Main Results.  Given a $(10_{3})$ configuration, we determine the number of
possible ways to add an eleventh line through some number of the double
points: at least three so that the arrangement is not reductive; and at most
five because a line through six existing doubles must belong to an arrangement
of at least 13 lines. Over the ten $(10_{3})$ configurations, this gives: a
subtotal of three hundred thirty-six arrangements, fifteen of which appear
twice, for three double points; a subtotal of one hundred eighty-eight
arrangements, thirty-seven of which appear twice, for four double points; and
a total of twenty-three arrangements for five double points. We then classify
the moduli spaces of these arrangements. A summary of these results can be
found in Tables 1, 2, and 3.

######  Theorem 1.1.

Out of the three hundred twenty-one distinct arrangements obtained by adding
an eleventh line through three double points in one of the ten $(10_{3})$
configurations, just one of them has a reducible moduli space modulo complex
conjugation: $(10_{3})_{7}.ADO$ as discussed in Example 1.6.

######  Theorem 1.2.

Out of the one hundred fifty-one distinct arrangements obtained by adding an
eleventh line through four double points in one of the ten $(10_{3})$
configurations, seventy-four of them have a reducible moduli space modulo
complex conjugation. These are listed in Table 6 at the end of the
Introduction.

######  Theorem 1.3.

Out of the twenty-three distinct arrangements obtained by adding an eleventh
line through five double points in one of the ten $(10_{3})$ configurations,
just one of them has a reducible moduli space modulo complex conjugation:
$(10_{3})_{1}.AEIKO$ as discussed in Example 1.7.

$j$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Subtotal | Total
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ arrangements constructed from $(10_{3})_{j}$ | 4 | 17 | 42 | 11 | 76 | 30 | 50 | 50 | 39 | 17 | 336 | 321
$\\#$ with irreducible, non-empty moduli space | 3 | 12 | 34 | 0 | 73 | 25 | 43 | 48 | 36 | 17 | 291 | 282
$\\#$ with empty moduli space | 1 | 5 | 7 | 11 | 3 | 4 | 6 | 2 | 3 | 0 | 42 | 36
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}$ but irreducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | 2
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1

Table 1. Classification of the moduli space of arrangements obtained by adding
a line through _three_ double points of a $(10_{3})$ configuration

$j$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Subtotal | Total
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ arrangements constructed from $(10_{3})_{j}$ | 2 | 8 | 21 | 5 | 45 | 16 | 25 | 30 | 24 | 12 | 188 | 151
$\\#$ with irreducible, non-empty moduli space | 1 | 2 | 3 | 0 | 0 | 2 | 3 | 1 | 0 | 1 | 13 | 10
$\\#$ with empty moduli space | 0 | 4 | 8 | 5 | 13 | 8 | 8 | 9 | 11 | 1 | 67 | 50
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}$ but irreducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 0 | 1 | 4 | 0 | 5 | 2 | 1 | 4 | 2 | 3 | 22 | 17
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 1 | 1 | 6 | 0 | 27 | 4 | 13 | 16 | 11 | 7 | 86 | 74

Table 2. Classification of the moduli space of arrangements obtained by adding
a line through _four_ double points of a $(10_{3})$ configuration

$j$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total
---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ arrangements constructed from $(10_{3})_{j}$ | 1 | 1 | 2 | 1 | 5 | 2 | 2 | 3 | 3 | 3 | 23
$\\#$ with irreducible, non-empty moduli space | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1
$\\#$ with empty moduli space | 0 | 1 | 2 | 1 | 5 | 2 | 2 | 3 | 3 | 2 | 21
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}$ but irreducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\\#$ with reducible $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1

Table 3. Classification of the moduli space of arrangements obtained by adding
a line through _five_ double points of a $(10_{3})$ configuration

######  Remark 1.4.

All of the $(10_{3})$ configurations have a 2-dimensional moduli space, except
for $(10_{3})_{1}$ and $(10_{3})_{4}$. The configuration $(10_{3})_{1}$ has a
3-dimensional moduli space, and $(10_{3})_{4}$ has an empty moduli space.
Introducing a new line through 4 double points gives 2 restraints on the
moduli space. If neither of these restraints are already present, then the
moduli space is 0-dimensional or empty, and a 0-dimensional moduli space is
reducible as long as it contains more than one point. This is empirically why
one-line extensions of $(10_{3})$ configurations through 4 double points yield
a high proportion of reducible moduli spaces.

######  Remark 1.5.

Our example $(10_{3})_{6}.AFIO$ appears as $C_{28}$ in a recent work by
Guerville-Ballé [GB22]; it is the only such overlap.

######  Example 1.6.

Consider the $(10_{3})$ configuration $(10_{3})_{7}$ with an eleventh line
passing through the intersections $L_{1}\cap L_{5}$, $L_{2}\cap L_{4}$, and
$L_{9}\cap L_{10}$. We call this arrangement $(10_{3})_{7}.ADO$. Its
arrangement table is given in Table 4, and two of its geometric realizations
from the two different irreducible components of its moduli space are given in
Figure 1.

$L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 4 | 6 | 5 | 3 | 7 | 2 | $A$
2 | 4 | 6 | 8 | 8 | 9 | 7 | 5 | 3 | 4 | $D$
3 | 5 | 7 | 9 | 0 | 0 | 8 | 9 | 0 | 6 | $O$
$A$ | $D$ |  | $D$ | $A$ |  |  |  | $O$ | $O$ |
Table 4. The arrangement table for $(10_{3})_{7}.ADO$, which has a Galois
conjugate moduli space

$L_{10}$$L_{1}$$L_{2}$$L_{3}$$L_{4}$$L_{5}$$L_{6}$$L_{7}$$L_{8}$$L_{9}$$ADO$

$L_{10}$$L_{1}$$L_{2}$$L_{3}$$L_{4}$$L_{5}$$L_{6}$$L_{7}$$L_{8}$$L_{9}$$ADO$

Figure 1. Arrangement $(10_{3})_{7}.ADO$ with geometric realizations from the
two different irreducible components of its Galois conjugate moduli space. The
realization on the left corresponds to the values of $a=3$ and
$c=\frac{3}{4}(1+\sqrt{5})$, and the realization on the right corresponds to
the values $a=3$ and $c=\frac{3}{4}(1-\sqrt{5})$

The geometric realizations of $(10_{3})_{7}.ADO$ in $\mathbb{CP}^{2}$ with
coordinates $[x:y:z]$, up to a projective transformation, can be described by
the equation

(1)
$\displaystyle\begin{split}(y)(\sqrt{3}x+y)(\sqrt{3}x-y)(y-z)(\sqrt{3}(b+c)x+(b+c)y+(2\sqrt{3}bc-c)z)&\\\
(\sqrt{3}bx-
by+(a+2b+\sqrt{3}ab)z)(\sqrt{3}(b+c)x-(c-b)y+2\sqrt{3}bcz)(-\sqrt{3}bx-(a+b)y+\sqrt{3}abz)&\\\
(\sqrt{3}cx-(a+c)y-\sqrt{3}acz)(z)(\sqrt{3}(b+c)x-(2\sqrt{3}bc-b-3c)y+(2\sqrt{3}bc-2c)z)&=0,\end{split}$

where $a$ is a complex number with a finite number of exceptions,
$b=-\tfrac{4c^{2}+ac}{a+2c-2\sqrt{3}c^{2}}$, and
$c^{\pm}=\tfrac{a}{4}(1\pm\sqrt{5})$. This shows that we have two irreducible
components in the moduli space: one corresponding to
$\tfrac{a}{4}(1+\sqrt{5})$ and another corresponding to
$\tfrac{a}{4}(1-\sqrt{5})$. Since we have one free parameter, each of the
irreducible components is one-dimensional.

######  Example 1.7.

The arrangement $\mathcal{A}=(10_{3})_{1}.AEIKO$ has arrangement table given
in Table 5. Its moduli space can be parameterized by $a$ and $b$ satisfying
$a^{\pm}=(\frac{3\pm\sqrt{5}}{2})b$, where $b$ is a complex number with a
finite number of exceptions.We also know that
$\operatorname{Aut}((10_{3})_{1}.AEIKO)\cong
F_{20}\cong\langle(12345),(1243)\rangle$, the Frobenius group of order 20.

$L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 3 | 4 | 5 | $A$
2 | 4 | 6 | 9 | 4 | 5 | 6 | 7 | 6 | 7 | $E$
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | $I$
$A$ | $E$ | $I$ | $A$ | $K$ | $I$ | $E$ | $O$ | $O$ | $K$ | $K$
|  |  |  |  |  |  |  |  |  | $O$
Table 5. The arrangement table for $(10_{3})_{1}.AEIKO$, whose moduli space is
two Galois conjugate points

Organization.  Section 2 describes in more detail the objects with which we
work, including the moduli space of a combinatorial line arrangement.

Section 3 details the one-line extension construction.

Section 4 applies the one-line extension construction to all ten $(10_{3})$
configurations. We calculate that, up to isomorphism, there are three hundred
twenty-one one-line extensions of $(10_{3})$ configurations through three
double points, one hundred fifty-one one-line extensions of $(10_{3})$
configurations through four double points, and twenty-three one-line
extensions of $(10_{3})$ configurations through five double points.

Section 5 discusses how the moduli space is calculated and the algebraic
techniques to determine its irreducibility. We conclude that there are
seventy-six one-line extensions of $(10_{3})$ configurations with a reducible
moduli space up to complex conjugation.

Section 6 offers minor corrections to the work by the first author with Amram,
Teicher, and Ye [ACTY13].

Acknowledgements.  The authors would like to thank the Undergraduate Research
Summer Institute at Vassar College, an in-house research experience for
undergraduates, for their funding of a portion of this research while the
second author was a rising senior at Vassar during the summer of 2017.

Arrangement $\mathcal{A}$ | $\lvert\mathcal{M}_{\mathcal{A}}\rvert$ | $\lvert\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}\rvert$ | $\operatorname{Aut}(\mathcal{A})$
---|---|---|---
1.AEIK | $\infty^{1}$ | $\infty^{1}$ | ${\mathbb{Z}}/4{\mathbb{Z}}$
2.AENO $\cong$ 7.ADLO | 2 | 2 |
3.BDHL $\cong$ 9.BDIM | 3 | 2 |
3.BDIK $\cong$ 6.AENO | 3 | 2 |
3.BDIL | 3 | 2 |
3.BDKL $\cong$ 9.BDMN | 3 | 3 |
3.BFJM | 3 | 2 |
3.DHLO | 3 | 2 |
5.AEIK | 5 | 3 |
5.AEJO | 3 | 2 |
5.AEKN | 5 | 3 |
5.AENO | 4 | 3 |
5.AFIK | 4 | 2 |
5.AFIO | 4 | 3 | ${\mathbb{Z}}/2{\mathbb{Z}}$
5.AFKL | 4 | 3 |
5.AFLO | 4 | 4 |
5.BDHL $\cong$ 5.BFIK | 4 | 2 |
5.BDHN $\cong$ 7.AIJL | 3 | 2 |
5.BDIK $\cong$ 5.BDKL $\cong$ 10.AEIJ | 6 | 3 |
5.BDJL $\cong$ 5.BEIK | 3 | 2 |
5.BDKN $\cong$ 10.ADKM | 5 | 3 |
5.BEGJ | 5 | 3 |
5.BEGK | 5 | 4 |
5.BEGN | 4 | 2 |
5.BEKN | 4 | 2 |
5.BFGK | 4 | 3 |
5.BFKL | 3 | 2 |
5.BGJL | 3 | 2 |
5.BGKL | 5 | 3 |
5.BGKN | 4 | 3 |
5.DHLO | 3 | 3 |
5.DHMN $\cong$ 7.AILO | 3 | 3 |
5.DHNO $\cong$ 7.AFIL | 2 | 2 |
5.DIKM | 4 | 2 |
5.DKMN | 2 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$
1 Two 1-dimensional components |  |  |
Arrangement $\mathcal{A}$ | $\lvert\mathcal{M}_{\mathcal{A}}\rvert$ | $\lvert\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}\rvert$ | $\operatorname{Aut}(\mathcal{A})$
---|---|---|---
6.AEHO | 4 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$
6.AFIJ | 3 | 2 |
6.AFIO | 4 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$
7.ADIL | 3 | 2 |
7.ADIM $\cong$ 7.AFIM | 2 | 2 |
7.AEIJ | 3 | 2 |
7.AEIM | 2 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$
7.AEIO | 2 | 2 |
7.AEJM | 2 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$
7.AEJN | 2 | 2 |
7.AIJM | 4 | 2 |
7.BIJM | 3 | 2 |
8.ADIM | 4 | 3 |
8.AEGM | 4 | 2 |
8.AEIM | 7 | 4 |
8.AFIJ $\cong$ 9.ADIM | 5 | 3 |
8.AFIL | 5 | 3 |
8.AFJO | 5 | 3 |
8.AIJM | 5 | 3 |
8.AILM | 5 | 4 |
8.BDIM | 4 | 2 |
8.BFIJ | 5 | 3 |
8.BFIK $\cong$ 9.ADLN | 5 | 4 |
8.BFJO | 4 | 2 |
8.BFKO | 3 | 2 |
8.BIJM | 4 | 2 |
8.BIKM | 6 | 4 |
8.CFIJ $\cong$ 9.AEIM | 4 | 3 | ${\mathbb{Z}}/2{\mathbb{Z}}$
9.ADIL | 5 | 3 |
9.ADMN | 5 | 3 |
9.AEGM | 3 | 2 |
9.BDHM | 3 | 2 |
9.BDHO | 3 | 2 |
9.BGMN | 3 | 2 |
10.ADHM | 4 | 3 |
10.ADHN | 4 | 3 | ${\mathbb{Z}}/2{\mathbb{Z}}$
10.ADIN | 5 | 4 |
10.AEGM | 6 | 4 |
10.AEGN | 2 | 2 | ${\mathbb{Z}}/2{\mathbb{Z}}$

Table 6.  The list of 74 arrangements with reducible moduli space modulo
complex conjugation obtained from one-line extensions of $(10_{3})$
arrangements appearing in Theorem 1.2, whose automorphism groups are trivial
unless otherwise noted

##  2\. Background

We can describe a combinatorial line arrangement as a collection of “points”
and “lines” along with incidence relations between them. The following
definition has been adapted from the definition of combinatorial configuration
in Grünbaum’s textbook _Configurations of Points and Lines_ [Grü09].

######  Definition 2.1.

A combinatorial line arrangement $\mathcal{A}=(\mathcal{P},\mathcal{L})$
consists of a finite set of points $\mathcal{P}$ and a finite set of lines
$\mathcal{L}$, which are subsets of $\mathcal{P}$. We also require that the
intersection of two lines is at most one point. Our convention in constructing
non-reductive arrangements also requires that each point in $\mathcal{P}$
appears in at least three elements of $\mathcal{L}$. If a point
$P\in\mathcal{P}$ is on the line $L\in\mathcal{L}$, we say that $P$ is
incident to $L$ or that $L$ is incident to $P$.

We refer to the intersection of exactly two lines as a double point, and we
define the set of double points of $\mathcal{A}$ to be

$\operatorname{Doubles}(\mathcal{A})=\left\\{\\{L,L^{\prime}\\}\in\binom{\mathcal{L}}{2}\mid
L\cap L^{\prime}=\emptyset\right\\}.$

The reason why these 2-tuples are called double points is that we will be
attempting to realize these arrangements in projective space, where all pairs
of lines intersect exactly once. Using our convention that each point in
$\mathcal{P}$ appears in at least three elements of $\mathcal{L}$, we see that
if $L\cap L^{\prime}=\emptyset$, then the intersection of $L$ and
$L^{\prime}$, in the realization in projective space, is not a point that
appears in at least three elements of $\mathcal{L}$, so this intersection
shall be named a double point since $L$ and $L^{\prime}$ are the only two
lines passing through this intersection.

On the same note, We define points of higher multiplicity as the elements of
$\mathcal{P}$. A triple point is an element in $\mathcal{P}$ that appears in
exactly three elements of $\mathcal{L}$.

These arrangements can be presented in an arrangement table, in which the
headers are the names of the lines and the columns contain the names of the
points incident to each line.

######  Example 2.2.

The well-known Fano arrangement is a combinatorial line arrangement whose
arrangement table can be found in Table 7.

$L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$
---|---|---|---|---|---|---
$P_{1}$ | $P_{1}$ | $P_{1}$ | $P_{2}$ | $P_{2}$ | $P_{3}$ | $P_{3}$
$P_{2}$ | $P_{4}$ | $P_{6}$ | $P_{4}$ | $P_{5}$ | $P_{4}$ | $P_{5}$
$P_{3}$ | $P_{5}$ | $P_{7}$ | $P_{6}$ | $P_{7}$ | $P_{7}$ | $P_{6}$
Table 7. An arrangement table for the Fano arrangement

We see that the arrangement table presentation of the Fano arrangement
corresponds to the usual geometric presentation given in Figure 2.

$P_{5}$$P_{1}$$P_{3}$$P_{7}$$L_{1}$$L_{3}$$L_{6}$$L_{2}$$L_{7}$$L_{5}$$P_{2}$$P_{6}$$P_{4}$
Figure 2. The Fano arrangement with the middle circle as the combinatorial
“line” $L_{4}$

###  2.1. Moduli space of a combinatorial line arrangement

With this in mind, we are able to manipulate these combinatorial line
arrangements by adding or removing a line.

######  Definition 2.3.

Let $\mathcal{A}=(\mathcal{P},\mathcal{L})$ be a combinatorial line
arrangement with $\mathcal{L}=\\{L_{1},L_{2},\dots,L_{n}\\}$, and let
$L\subseteq\operatorname{Doubles}(\mathcal{A})$ be a subset of the set of
double points. We define the one-line extension of $\mathcal{A}$ by the line
$L$, written $\mathcal{A}\cup L$, to be the line arrangement $(\mathcal{P}\cup
L,\mathcal{L}^{\prime})$, where
$\mathcal{L}^{\prime}=\\{L_{1}^{\prime},L_{2}^{\prime},\dots,L_{n}^{\prime},L\\}$
and $L_{i}^{\prime}=L_{i}\cup D$ if there exists $D\in L$ such that $L_{i}\in
D$ and $L_{i}^{\prime}=L_{i}$ otherwise.

Intuitively, this is adding the line $L$ through its specified points and
attaching double points in $\mathcal{A}$ that have now turned into triple
points to the appropriate lines.

A similar construction involves removing a line from an arrangement.

######  Definition 2.4.

Let $\mathcal{A}=(\mathcal{P},\mathcal{L})$ be a combinatorial line
arrangement and $L\in\mathcal{L}$. We define $\mathcal{A}$ minus $L$, written
$\mathcal{A}\setminus L$, to be

$\mathcal{A}\setminus L=(\mathcal{P},\mathcal{L}\setminus\\{L\\})$

with the convention that double points are omitted.

Intuitively, this is removing the line $L$ from $\mathcal{A}$ while removing
triple points that are now double points from $\mathcal{P}$ and the
corresponding lines.

The following notion of an isomorphism for combinatorial line arrangements is
used to identify those we deem to have the same combinatorial information.

######  Definition 2.5.

Let $\mathcal{A}=(\mathcal{P},\mathcal{L})$ and
$\mathcal{A}^{\prime}=(\mathcal{P}^{\prime},\mathcal{L}^{\prime})$ be two
combinatorial line arrangements. We say that $\mathcal{A}$ and
$\mathcal{A}^{\prime}$ are isomorphic, denoted by
$\mathcal{A}\cong\mathcal{A}^{\prime}$, if there exists function
$\varphi:\mathcal{P}\cup\mathcal{L}\to\mathcal{P}^{\prime}\cup\mathcal{L}^{\prime}$
such that $\varphi|_{\mathcal{P}}:\mathcal{P}\to\mathcal{P}^{\prime}$ and
$\varphi|_{\mathcal{L}}:\mathcal{L}\to\mathcal{L}^{\prime}$ are both
bijections. Additionally, for all $L\in\mathcal{L}$ and $P,Q\in L$, we have
that $\varphi(P),\varphi(Q)\in\varphi(L)$. Also, if
$L,L^{\prime}\in\mathcal{L}$ are such that $L\cap L^{\prime}=\\{P\\}$ for some
point $P\in\mathcal{P}$, then
$\varphi(L)\cap\varphi(L^{\prime})=\\{\varphi(P)\\}$. Such a function
$\varphi$ is called an isomorphism. Also denote by
$\operatorname{Aut}(\mathcal{A})=\\{\varphi:\mathcal{A}\to\mathcal{A}\mid\text{$\varphi$
is an isomorphism}\\}$ the automorphism group of $\mathcal{A}$.

######  Example 2.6.

It is known that the automorphism group for the Fano arrangement is
$\operatorname{GL}(3,\mathbb{F}_{2})$.

The main tool we use to analyze combinatorial line arrangements is the moduli
space. The moduli space $\mathcal{M}_{\mathcal{A}}$ is the space of all
geometric realizations of the combinatorial line arrangement $\mathcal{A}$ in
$\mathbb{CP}^{2}$. In order to define the moduli space, we need to define a
geometric realization for a combinatorial line arrangement.

######  Definition 2.7.

Let $\mathcal{A}$ be a combinatorial line arrangement with lines
$L_{1},L_{2},\ldots,L_{n}$. A geometric realization of $\bf\mathcal{A}$ in
$\mathbb{CP}^{2}$ is a collection of lines $\ell_{1},\ell_{2},\ldots,\ell_{n}$
in $\mathbb{CP}^{2}$ such that for any subset of $S\subseteq\\{1,2,\dots,n\\}$
with $\lvert S\rvert\geq 3$, we have $\bigcap_{i\in S}\ell_{i}$ is nonempty if
and only if $\bigcap_{i\in S}L_{i}$ is nonempty. We say that $\mathcal{A}$ is
geometrically realizable in $\mathbb{CP}^{2}$ if there exists a geometric
realization of $\mathcal{A}$ in $\mathbb{CP}^{2}$.

Consider the complex projective line $\ell$ with equation $ax+by+cz=0$. Then
the complex projective point dual to this line in $\mathbb{CP}^{2}$ is
$\ell^{*}=[a:b:c]\in(\mathbb{CP}^{2})^{*}$. This is a convenient way to
represent lines as points for the sake of notation.

Now we are ready to introduce the moduli space.

######  Definition 2.8.

The ordered moduli space of a combinatorial line arrangement $\mathcal{A}$ is
defined to be

$\mathcal{M}_{\mathcal{A}}=\\{(\ell_{1}^{*},\ell_{2}^{*},\ldots,\ell_{n}^{*})\in((\mathbb{CP}^{2})^{*})^{n}\mid\text{$(\ell_{1},\ell_{2},\ldots,\ell_{n})$
is a geometic realization of $\mathcal{A}$ in
$\mathbb{CP}^{2}$}\\}/\operatorname{PGL}(3,\mathbb{C}),$

which are all geometric realizations of $\mathcal{A}$ in $\mathbb{CP}^{2}$, up
to a projective transformation.

We refer to this throughout as simply the moduli space of an arrangement,
noting that there exists a related notion of the unordered moduli space,
obtain by a quotient by the automorphism group. The moduli space is endowed
the topology induced by the Zariski topology on $((\mathbb{CP}^{2})^{*})^{n}$.
We will also sometimes endow the moduli space with the finer Euclidean
topology. An important quotient of the moduli space is
$\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$, which is defined to be the quotient
of $\mathcal{M}_{\mathcal{A}}$ under complex conjugation.

######  Example 2.9.

The Fano arrangement is not geometrically realizable in $\mathbb{CP}^{2}$; its
moduli space is empty. These notions are equivalent in general.

Given a combinatorial line arrangement $\mathcal{A}$, it is a necessary
condition for its moduli space $\mathcal{M}_{\mathcal{A}}$ to be reducible for
two geometric realizations of $\mathcal{A}$ to be different topologically. If
$\mathcal{M}_{\mathcal{A}}$ is irreducible, then we can apply Randell’s
Isotopy Theorem to show that all of the geometric realizations of
$\mathcal{A}$ are the same topologically.

######  Theorem 2.10.

(Randell’s Isotopy Theorem [Ran89]) Two combinatorially isomorphic
arrangements $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ connected by a
1-parameter family of isomorphic arrangements have complements in
$\mathbb{CP}^{2}$ that are diffeomorphic. Furthermore,
$(\mathbb{CP}^{2},\mathcal{R}_{1})$ and $(\mathbb{CP}^{2},\mathcal{R}_{2})$
are of the same topological type.

We also consider the moduli space modulo complex conjugation due the following
result from Cohen and Suciu.

######  Theorem 2.11.

(Cohen-Suciu [CS97, Theorem 3.9]) The braid monodromies of complex conjugated
curves are equivalent.

For the purpose of classifying irreducibility of moduli spaces, it turns out
that we only need to analyze the moduli spaces of non-reductive combinatorial
line arrangements. This is a consequence of the following result, which shows
that we can remove a line incident to at most two points of higher
multiplicity and use the irreducibility of the moduli space of the smaller
arrangement to deduce the irreducibility of the moduli space of the original,
larger arrangement.

######  Theorem 2.12.

(Nazir-Yoshinaga [NY12, reframed Lemma 3.2]) Let
$\mathcal{A}=(\mathcal{P},\\{L_{1},L_{2},\ldots,L_{n}\\})$ be a combinatorial
line arrangement with $L_{n}$ being incident to at most two points of higher
multiplicity, and let $\mathcal{A}^{\prime}=\mathcal{A}\setminus L_{n}$. Then
$\mathcal{M}_{\mathcal{A}}$ is irreducible if
$\mathcal{M}_{\mathcal{A}^{\prime}}$ is irreducible.

We use the one-line extension construction, which we describe in the next
section, to provide examples of combinatorial line arrangements with a
reducible moduli space. These come from known arrangements which we discuss in
the next subsection.

###  2.2. $(n_{3})$ configurations

In anticipation of our later construction, we introduce these examples from
the literature.

######  Definition 2.13.

(see for example [Grü09]) An $(n_{k})$ configuration is a combinatorial line
arrangement with $n$ lines and $n$ points where each line is incident to $k$
points and each point is incident to $k$ lines.

We are specifically interested in $(n_{3})$ configurations. Table 8 provides a
non-exhaustive enumeration of $(n_{3})$ configurations as found in Grünbaum’s
textbook [Grü09, Theorem 2.2.1], in which he cites others [Mar87, DvS03,
Gro90]. It is worth noting that since $(n_{3})$ configurations can be seen as
$3$-regular, $3$-uniform hypergraphs of girth $\geq 3$, any finite
automorphism group is achievable with an $(n_{3})$ configuration [Vog86].

$n$ | $\leq 6$ | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---
number of $(n_{3})$ configurations | 0 | 1 | 1 | 3 | 10 | 31 | 229
Table 8. Number of $(n_{3})$ configurations as found in [Grü09, Theorem 2.2.1]

The construction in [ACTY13] begins with one of the three $(9_{3})$
configurations and considers the set of all double points. Then all subsets of
cardinality three are considered, and this set is quotiented out by the
automorphism group of the $(9_{3})$ configuration. This gives the possible
combinatorial line arrangements of 10 lines obtained by adding a line passing
through three double points.

Table 20 at the end of Section 6 gives the eleven arrangements that result
from this construction. Following the naming convention in [ACTY13], the
numbers correspond to points in the original $(9_{3})$ configuration, and the
capital letters correspond to double points in the original $(9_{3})$
configuration that have become triple points after the tenth line has been
added.

######  Remark 2.14.

We note here that in [ACTY13] there are fourteen arrangements given, but in
Section 6 we show that three of these are redundant due to arrangement
$(9_{3})_{1}$ having a larger automorphism group than acknowledged in that
work.

Table 9 lists all ten $(10_{3})$ arrangements. For ease of naming new
combinatorial line arrangements, we name each of the 15 double points with the
upper case letters $A$ through $O$ for each of the $(10_{3})$ configurations
$\mathcal{A}$. These are given explicitly in Table 10.

$\text{Aut}((10_{3})_{1})\cong S_{5}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 3 4 5 2 4 6 9 4 5 6 7 6 7 3 5 7 0 8 8 9 9 0 0 |  $\text{Aut}((10_{3})_{2})\cong D_{12}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 3 4 5 2 4 6 9 4 7 6 5 6 7 3 5 7 0 8 8 9 9 0 0
---|---
$\text{Aut}((10_{3})_{3})\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 3 4 5 2 4 6 9 4 6 7 5 6 7 3 5 7 0 8 8 9 9 0 0 |  $\text{Aut}((10_{3})_{4})\cong S_{4}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 3 4 5 2 4 6 9 4 6 5 7 6 7 3 5 7 0 8 8 9 9 0 0
$\text{Aut}((10_{3})_{5})\cong{\mathbb{Z}}/2{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 4 3 5 2 4 6 9 4 7 5 6 6 7 3 5 7 0 8 8 9 9 0 0 |  $\text{Aut}((10_{3})_{6})\cong S_{3}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 2 3 2 5 3 4 2 4 6 9 4 7 6 7 5 6 3 5 7 0 8 8 9 9 0 0
$\text{Aut}((10_{3})_{7})\cong{\mathbb{Z}}/3{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 2 4 6 5 3 7 2 2 4 6 8 8 9 7 5 3 4 3 5 7 9 0 0 8 9 0 6 |  $\text{Aut}((10_{3})_{8})\cong{\mathbb{Z}}/3{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 3 5 7 2 6 4 2 2 4 6 8 8 9 7 5 3 4 3 5 7 9 0 0 8 9 0 6
$\text{Aut}((10_{3})_{9})\cong{\mathbb{Z}}/4{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 2 4 6 5 3 2 3 2 4 6 8 8 9 7 5 7 4 3 5 7 9 0 0 8 9 0 6 |  $\text{Aut}((10_{3})_{10})\cong{\mathbb{Z}}/10{\mathbb{Z}}$. $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 3 2 7 5 6 4 2 2 4 6 8 8 9 7 5 3 4 3 5 7 9 0 0 8 9 0 6
Table 9. The ten $(10_{3})$ configurations with their arrangement tables and
automorphism groups

Double | $(10_{3})_{1}$ | $(10_{3})_{2}$ | $(10_{3})_{3}$ | $(10_{3})_{4}$ | $(10_{3})_{5}$ | $(10_{3})_{6}$ | $(10_{3})_{7}$ | $(10_{3})_{8}$ | $(10_{3})_{9}$ | $(10_{3})_{10}$
---|---|---|---|---|---|---|---|---|---|---
A | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{4}$ | $L_{1}\cap L_{5}$ | $L_{1}\cap L_{5}$ | $L_{1}\cap L_{5}$ | $L_{1}\cap L_{6}$
B | $L_{1}\cap L_{9}$ | $L_{1}\cap L_{9}$ | $L_{1}\cap L_{9}$ | $L_{1}\cap L_{9}$ | $L_{1}\cap L_{8}$ | $L_{1}\cap L_{8}$ | $L_{1}\cap L_{6}$ | $L_{1}\cap L_{6}$ | $L_{1}\cap L_{6}$ | $L_{1}\cap L_{7}$
C | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{10}$ | $L_{1}\cap L_{7}$ | $L_{1}\cap L_{8}$ | $L_{1}\cap L_{7}$ | $L_{1}\cap L_{8}$
D | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$ | $L_{2}\cap L_{4}$
E | $L_{2}\cap L_{7}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{6}$ | $L_{2}\cap L_{5}$
F | $L_{2}\cap L_{8}$ | $L_{2}\cap L_{7}$ | $L_{2}\cap L_{7}$ | $L_{2}\cap L_{8}$ | $L_{2}\cap L_{9}$ | $L_{2}\cap L_{7}$ | $L_{2}\cap L_{9}$ | $L_{2}\cap L_{7}$ | $L_{2}\cap L_{9}$ | $L_{2}\cap L_{6}$
G | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$ | $L_{3}\cap L_{4}$
H | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$ | $L_{3}\cap L_{5}$
I | $L_{3}\cap L_{6}$ | $L_{3}\cap L_{8}$ | $L_{3}\cap L_{8}$ | $L_{3}\cap L_{7}$ | $L_{3}\cap L_{7}$ | $L_{3}\cap L_{9}$ | $L_{3}\cap L_{8}$ | $L_{3}\cap L_{9}$ | $L_{3}\cap L_{8}$ | $L_{3}\cap L_{9}$
J | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{9}$ | $L_{5}\cap L_{8}$ | $L_{4}\cap L_{9}$ | $L_{4}\cap L_{10}$ | $L_{4}\cap L_{10}$ | $L_{4}\cap L_{10}$
K | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{9}$ | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{10}$ | $L_{5}\cap L_{8}$ | $L_{5}\cap L_{8}$
L | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L{10}$ | $L_{6}\cap L_{7}$ | $L_{6}\cap L_{10}$
M | $L_{6}\cap L_{9}$ | $L_{6}\cap L_{9}$ | $L_{6}\cap L_{10}$ | $L_{6}\cap L_{10}$ | $L_{6}\cap L_{8}$ | $L_{6}\cap L_{10}$ | $L_{7}\cap L_{10}$ | $L_{7}\cap L_{8}$ | $L_{7}\cap L_{10}$ | $L_{7}\cap L_{9}$
N | $L_{7}\cap L_{10}$ | $L_{7}\cap L_{10}$ | $L_{7}\cap L_{9}$ | $L_{7}\cap L_{9}$ | $L_{7}\cap L_{9}$ | $L_{7}\cap L_{9}$ | $L_{8}\cap L_{10}$ | $L_{7}\cap L_{9}$ | $L_{8}\cap L_{9}$ | $L_{7}\cap L_{10}$
O | $L_{8}\cap L_{9}$ | $L_{8}\cap L_{9}$ | $L_{8}\cap L_{9}$ | $L_{8}\cap L_{9}$ | $L_{8}\cap L_{10}$ | $L_{8}\cap L_{10}$ | $L_{9}\cap L_{10}$ | $L_{8}\cap L_{9}$ | $L_{9}\cap L_{10}$ | $L_{8}\cap L_{9}$

Table 10.  Labels of the double points in the $(10_{3})$ arrangements

We highlight two geometric pictures of arrangements that better display the
symmetry of their respective automorphism group: the arrangements
$(10_{3})_{7}$ and $(10_{3})_{10}$ in Figure 3.

2463579801$L_{1}$$L_{2}$$L_{3}$$L_{7}$$L_{8}$$L_{9}$$L_{4}$$L_{6}$$L_{5}$$L_{10}$

0529174863$L_{1}$$L_{2}$$L_{3}$$L_{4}$$L_{5}$$L_{6}$$L_{7}$$L_{8}$$L_{9}$$L_{10}$

Figure 3. A geometric picture of $(10_{3})_{7}$ on the left showing its
$\mathbb{Z}/3{\mathbb{Z}}$ automorphism group, and a geometric picture of
$(10_{3})_{10}$ on the right showing the ${\mathbb{Z}}/5{\mathbb{Z}}$ subgroup
of its $\mathbb{Z}/10{\mathbb{Z}}$ automorphism group

##  3\. The One-Line Extension Construction

We first describe the one-line extension construction in full generality. Then
we describe the one-line extension construction through a specified number of
double points. Lastly, we discuss the effects of this construction on the
moduli spaces.

######  Definition 3.1.

Let $\mathcal{A}=(\mathcal{P},\mathcal{L})$ be a combinatorial line
arrangement. We define the set of one-line extensions of $\mathcal{A}$ as

$\operatorname{OL}(\mathcal{A})=\left\\{\mathcal{A}\cup
L\,\middle|\,L\in\binom{\mathcal{P}\cup\operatorname{Doubles}(\mathcal{A})}{k},k\in{\mathbb{Z}},k>0,\mathcal{A}\cup
L\text{ is a line arrangement}\right\\}/\sim,$

where $A\cup L\sim A\cup L^{\prime}$ if the two arrangements are
combinatorially isomorphic.

In other words, $\operatorname{OL}(\mathcal{A})$ is the set of all
combinatorial line arrangements with $\left|\mathcal{L}\right|+1$ lines
containing $\mathcal{A}$ as a subarrangement, up to isomorphism. The
requirement that $\mathcal{A}\cup L$ is a line arrangement ensures that every
pair of lines intersect no more than once.

An interesting family of one-line extensions are extensions purely through
double points. Some, but not all, isomorphisms between different extensions
are induced by the automorphisms of $\mathcal{A}$.

######  Definition 3.2.

Let $\mathcal{A}$ be a combinatorial line arrangement. We define the  set of
extension lines of $\mathcal{A}$ by $k$ double points as

$\operatorname{OLExt}(k,\mathcal{A})=\left\\{L\in\binom{\operatorname{Doubles}(\mathcal{A})}{k}\,\middle|\,\text{$\mathcal{A}\cup
L$ is a line arrangement}\right\\}/\operatorname{Aut}(\mathcal{A}).$

We take $k\geq 3$ to avoid producing reductive arrangements.

Again, to make sure that $\mathcal{A}\cup L$ is a line arrangement, we have to
make sure that every pair of lines intersect no more than once.

######  Definition 3.3.

The following set contains all the possible combinatorial line arrangements
constructed by adding a line through $k$ double points in $\mathcal{A}$ up to
isomorphism:

$\operatorname{OLExtArrs}(k,\mathcal{A})=\\{\mathcal{A}\cup
L\mid[L]\in\operatorname{OLExt}(k,\mathcal{A})\\}/\sim,$

where $A\cup L\sim A\cup L^{\prime}$ if the two arrangements are
combinatorially isomorphic. We call this set one line extensions of
$\mathcal{A}$ by $k$ double points. Each element is well-defined since for
each $[L]\in\operatorname{OLExt}(k,\mathcal{A})$, if $[L]=[L^{\prime}]$, the
automorphism of $\mathcal{A}$ induces an automorphism between $A\cup L$ and
$A\cup L^{\prime}$.

######  Remark 3.4.

Note that it is possible that $[L]\neq[L^{\prime}]$ as elements of
$\operatorname{OLExt}(k,\mathcal{A})$ yet $\mathcal{A}\cup
L\cong\mathcal{A}\cup L^{\prime}$.

######  Example 3.5.

Table 11 gives an example of a pair of isomorphic arrangements constructed
using different classes of $\operatorname{OLExt}(3,\mathcal{A})$, where
$\mathcal{A}$ is a $(10_{3})$ configuration.

$(10_{3})_{5}.BDL$ | $(10_{3})_{5}.BIK$
---|---
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 4 | 3 | 5 | B
2 | 4 | 6 | 9 | 4 | 7 | 5 | 6 | 6 | 7 | D
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | L
B | D |  | D |  | L | L | B |  |  |
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 4 | 3 | 5 | B
2 | 4 | 6 | 9 | 4 | 7 | 5 | 6 | 6 | 7 | I
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | K
B |  | I |  | K |  | I | B |  | K |
Table 11. Two isomorphic arrangements in
$\operatorname{OLExtArrs}(3,(10_{3})_{5})$

An isomorphism $\varphi:(10_{3})_{5}.BDL\to(10_{3})_{5}.BIK$ given in Table
12. Note that
$\varphi(\\{L_{1},L_{2},\dots,L_{10}\\})\neq\\{L_{1},L_{2},\dots,L_{10}\\}$
or, equivalently, $\varphi(L_{11})\neq L_{11}$, so $(10_{3})_{5}.BDL$ and
$(10_{3})_{5}.BIK$ would not have been identified via the quotient of
$\operatorname{OLExt}(3,\mathcal{A})$ by $\operatorname{Aut}((10_{3})_{5})$.
However, the two arrangements are identified since they are combinatorially
isomorphic.

Point $p$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | B | D | L
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\varphi(p)$ | 8 | 6 | I | 7 | B | 9 | 3 | 0 | K | 2 | 1 | 4 | 5
Line $L$ | $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$ |  |
$\varphi(L)$ | $L_{3}$ | $L_{8}$ | $L_{9}$ | $L_{5}$ | $L_{11}$ | $L_{10}$ | $L_{7}$ | $L_{1}$ | $L_{6}$ | $L_{4}$ | $L_{2}$ |  |
Table 12.  The explicit isomorphism between $(10_{3})_{5}.BDL$ and
$(10_{3})_{5}.BIK$

To detect this phenomenon, we rely on the following lemma:

######  Lemma 3.6.

Fix $k\in{\mathbb{Z}}$ such that $k\geq 3$. Let
$\mathcal{A}=(\mathcal{P},\mathcal{L})$ be a combinatorial line arrangement,
and let $[L],[L^{\prime}]\in\operatorname{OLExt}(k,\mathcal{A})$ such that
$[L]\neq[L^{\prime}]$. If there exists an isomorphism $\varphi:\mathcal{A}\cup
L\to\mathcal{A}\cup L^{\prime}$, then there exists a line $\ell\in\mathcal{L}$
such that $(\mathcal{A}\cup L^{\prime})\setminus\ell\cong\mathcal{A}$.

###### Proof.

Since $[L]\neq[L^{\prime}]$, we know that $\varphi(L)\neq L^{\prime}$, meaning
$\varphi(L)=\ell$ for some $\ell\in\mathcal{L}$, so
$\mathcal{A}=(\mathcal{A}\cup L)\setminus L\cong(\mathcal{A}\cup
L^{\prime})\setminus\ell$. ∎

When considering one-line extensions of different combinatorial arrangements,
it is also possible to have one-line extensions of non-isomorphic arrangements
be isomorphic. In other words, it is possible for two non-isomorphic
combinatorial line arrangements $\mathcal{A}_{1}\not\cong\mathcal{A}_{2}$ to
have $\mathcal{A}_{1}^{\prime}\in\operatorname{OLExtArrs}(k,\mathcal{A}_{1})$
and
$\mathcal{A}_{2}^{\prime}\in\operatorname{OLExtArrs}(k^{\prime},\mathcal{C}_{2})$
such that $\mathcal{A}_{1}^{\prime}\cong\mathcal{A}_{2}^{\prime}$.

######  Example 3.7.

The following is an example of a pair of isomorphic arrangements: one comes
from some $\operatorname{OLExtArrs}(3,\mathcal{A})$ and the other comes from
some $\operatorname{OLExtArrs}(3,\mathcal{B})$, where $\mathcal{A}$ and
$\mathcal{B}$ are distinct, non-isomorphic $(10_{3})$ configurations:

$(10_{3})_{1}.AEM$ | $(10_{3})_{6}.KLO$
---|---
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 3 | 4 | 5 | A
2 | 4 | 6 | 9 | 4 | 5 | 6 | 7 | 6 | 7 | E
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | M
A | E |  | A |  | M | E |  | M |  |
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 5 | 3 | 4 | K
2 | 4 | 6 | 9 | 4 | 7 | 6 | 7 | 5 | 6 | L
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | O
|  |  |  | K | L | L | O | K | L |
Table 13. Two isomorphic arrangements: one comes from
$\operatorname{OLExtArrs}(3,(10_{3})_{1})$ and the other comes from
$\operatorname{OLExtArrs}(3,(10_{3})_{6})$

An isomorphism $\varphi:(10_{3})_{1}.AEM\to(10_{3})_{6}.KLO$ is given in Table
14.

Point $p$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | E | M
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\varphi(p)$ | 2 | 5 | O | 7 | K | 3 | 4 | 1 | L | 6 | 9 | 0 | 8
Line $L$ | $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$ |  |
$\varphi(L)$ | $L_{8}$ | $L_{9}$ | $L_{2}$ | $L_{7}$ | $L_{11}$ | $L_{6}$ | $L_{7}$ | $L_{3}$ | $L_{5}$ | $L_{1}$ | $L_{4}$ |  |
Table 14.  The explicit isomorphism between $(10_{3})_{1}.AEM$ and
$(10_{3})_{6}.KLO$

To detect when one-line extensions from different arrangements are isomorphic,
we have the following tool.

######  Lemma 3.8.

Let $\mathcal{A}_{1}=(\mathcal{P}_{1},\mathcal{L}_{1})$ and
$\mathcal{A}_{2}=(\mathcal{P}_{2},\mathcal{L}_{2})$ be two arrangements such
that $\mathcal{A}_{1}\not\cong\mathcal{A}_{2}$, and let
$[L]\in\operatorname{OLExt}(k,\mathcal{A}_{1})$ and
$[L^{\prime}]\in\operatorname{OLExt}(k,\mathcal{A}_{2})$. If there exists an
isomorphism $\varphi:\mathcal{A}_{1}\cup L\to\mathcal{A}_{2}\cup L^{\prime}$,
then there exists a line $\ell\in\mathcal{L}_{2}$ such that
$\mathcal{A}_{2}\cup L^{\prime}\setminus\ell\cong\mathcal{A}_{1}$.

###### Proof.

If $\varphi(L)=L^{\prime}$, then it must be the case that
$\varphi(\mathcal{A}_{1})=\mathcal{A}_{2}$, which goes against our hypothesis
that $\mathcal{A}_{1}\not\cong\mathcal{A}_{2}$. We may then assume that
$\varphi(L)\neq L^{\prime}$ and $\varphi(L)=\ell$ for some
$\ell\in\mathcal{L}_{2}$. Then $\mathcal{A}_{1}=(\mathcal{A}_{1}\cup
L)\setminus L\cong(\mathcal{A}_{2}\cup L^{\prime})\setminus\ell$ by $\varphi$.
∎

Empirically, the one-line extension construction is effective since the
construction generally reduces the dimension of the moduli space and a zero
dimensional space is reducible as long as it is not empty or a singleton. The
dimension of the moduli space is reduced after a one-line extension since
requiring three points to be collinear is an additional constraint on the
moduli space unless those three points are already collinear. Requiring four
points to be collinear gives two additional constraints in the generic case,
and requiring fives points to be collinear gives three additional constraints
in the generic case.

##  4\. One-Line Extensions of $(10_{3})$ Configurations

Out of the eleven possible one-line extensions of $(9_{3})$ configurations by
three double points, six have a reducible moduli space. With this as
motivation, we apply one-line extensions to all $(10_{3})$ configurations in
hopes of producing combinatorial line arrangements of 11 lines with a
reducible moduli space.

We now apply the one-line extension construction through $k$ double points to
all of the $(10_{3})$ configurations, for $k=3,4,5$. Note that $k\geq 6$ is
impossible since 6 double points require 12 other lines. The construction is
detailed in Algorithm 1.

Algorithm 1 Enumeration Algorithm

0: A list $\mathfrak{C}$ of all $(10_{3})$ configurations up to isomorphism
and their automorphism groups, and a value for $k$

0: A list $\mathfrak{A}$ of combinatorial line arrangements of $11$ lines that
can be constructed by adding a line through $k$ double points in a
configuration in $\mathfrak{C}$, up to isomorphism

Initialize $\mathfrak{A}:=\emptyset$

Initialize $\mathfrak{F}:=\emptyset$, the arrangements needing additional
testing

for $\mathcal{C}=(\mathcal{P},\mathcal{L})\in\mathfrak{C}$ do

Calculate $\operatorname{OLExt}(k,\mathcal{C})$

for $L\in\operatorname{OLExt}(k,\mathcal{C})$ do

Set $\mathfrak{A}:=\mathfrak{A}\cup\\{\mathcal{C}\cup L\\}$

for $L^{*}\in\mathcal{L}$ do

if $\mathcal{C}\cup L\setminus L^{*}$ is a $(10_{3})$ configuration then

$\mathfrak{F}:=\mathfrak{F}\cup\\{\mathcal{C}\cup L\\}$

break

end if

end for

end for

end for

Check for isomorphisms among the arrangements in $\mathfrak{F}$, throwing away
isomorphic copies in $\mathfrak{A}$

return $\mathfrak{A}$

In order to start implementing this construction, we need the automorphism
groups of the $(10_{3})$ configurations. The generators for the automorphism
groups were also provided in [Mar87]. A correspondence between the naming of
the configurations in [Mar87] and [Grü09] is provided in [Grü09]. We indicate
the automorphism groups in Table 9.

Now we enumerate the one-line extensions of $(10_{3})$ configurations through
3, 4, or 5 double points up to isomorphism utilizing Lemmas 3.6 and 3.8. Fix
$k=3,4,5$. We can calculate $\operatorname{OLExt}(k,(10_{3})_{i})$ for
$i=1,\dots,10$ directly. Then we form the set

$\bigcup_{i=1}^{10}\\{(10_{3})_{i}\cup
L\mid[L]\in\operatorname{OLExt}(k,(10_{3})_{i}\\},$

which has all of the possible one-line extensions of $(10_{3})$ configurations
through $k$ double points, but some isomorphism classes might be represented
more than once. Lemmas 3.6 and 3.8 imply that such an isomorphism class can
only come from arrangements of the form $(10_{3})_{i}\cup L$ such that there
exists a line $\ell$ of $(10_{3})_{i}$ where $(10_{3})_{i}\cup L\setminus\ell$
is a $(10_{3})$ configuration. This creates a much smaller set to check for
pairwise isomorphisms. This enumeration process is detailed in Algorithm 1.

Using the automorphism groups of all the $(10_{3})$ configurations from Table
9, we can apply Algorithm 1. For $k=3$, the first block of code results in
three hundred thirty-seven arrangements. Lemma 3.6 identifies a pair of
isomorphic arrangements both constructed from $(10_{3})_{5}$, leaving us with
a subtotal of three hundred thirty-six arrangements. Lemma 3.8 identifies
fifteen pairs of isomorphic line arrangements, and we are able to conclude
that the remaining three hundred twenty-one arrangements are pairwise non-
isomorphic. See Table 1 for more details. For $k=4$ and $k=5$, the results of
the construction are summarized in Tables 2 and 3. The results in those tables
about the reducibility of various notions of the moduli space are obtained by
using tools from the next section.

##  5\. Irreducibility of Moduli Spaces

In order to use results from Section 2, we must be able to calculate the
moduli space of a combinatorial line arrangement. We follow Algorithm 2 in
[BP15] to achieve this.

Given a combinatorial line arrangement $\mathcal{A}$, we fix a projective
basis of four points (or lines), accounting for the quotient by
$\operatorname{PGL}(3,\mathbb{C})$. Then we add in a line or point from
$\mathcal{A}$ one at a time, parameterizing when needed. We are then left with
parameters $v_{1},v_{2},\dots,v_{r}$ and polynomials
$f_{1},f_{2},\dots,f_{s}\in{\mathbb{C}}[v_{1},v_{2},\dots,v_{r}]$ such that
$f_{i}(v_{1},v_{2},\dots,v_{r})=0$ for all $i=1,2,\dots,s$ to ensure the
collinearity relations. There are also
$g_{1},g_{2},\dots,g_{t}\in{\mathbb{C}}[v_{1},v_{2},\dots,v_{r}]$ such that
$g_{j}(a_{1},a_{2},\dots,a_{r})\neq 0$ for all $j=1,2,\dots,t$, since
$g_{i}(v_{1},v_{2},\dots,v_{r})=0$ would correspond to a degenerate
realizations of the arrangement.

######  Example 5.1.

The construction of $(10_{3})_{5}$ in [ACTY13] required the introduction of 3
parameters named $a,b,c$ and the constraint on the parameters is

$-a^{2}b^{2}c+a^{2}b^{2}+a^{2}bc-a^{2}c+2abc^{2}-3abc-ac^{2}+2ac-
bc^{2}+bc+c^{2}-c=0.$

One of the new arrangements is obtained by adding a line through the double
points $A,N,O$ and has the arrangement table given in Table 15.

$L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ | $L_{11}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 8 | 2 | 3 | 2 | 4 | 3 | 5 | A
2 | 4 | 6 | 9 | 4 | 7 | 5 | 6 | 6 | 7 | N
3 | 5 | 7 | 0 | 8 | 8 | 9 | 9 | 0 | 0 | O
A |  |  | A |  |  | N | O | N | O |
Table 15. An arrangement table for the arrangement $(10_{3})_{5}.ANO$

The additional line requires the additional constraint that

$\displaystyle
2a^{2}b^{2}c-a^{2}b^{2}-2a^{2}bc^{2}+a^{2}c^{2}+ab^{2}c^{3}-2ab^{2}c+ab^{2}-2abc^{3}$
$\displaystyle+abc^{2}+abc+2ac^{3}-2ac^{2}-bc^{4}+bc^{3}+bc^{2}-bc+c^{4}-2c^{3}+c^{2}$
$\displaystyle=0.$

If the system $\\{f_{i}=0\mid i=1,2,\dots s\\}\cup\\{g_{j}\neq 0\mid
j=1,2,\dots,t\\}$ has no solutions, then the combinatorial line arrangement is
not geometrically realizable. If the system does have a solution, then we
analyze the irreducibility of the moduli space. In order to look at the
irreducibility of the moduli space, we eliminate the factors of each $f_{i}$
that contradict the system $\\{g_{j}\neq 0\mid j=1,2,\dots,t\\}$. We end up
with polynomials
$h_{1},h_{2},\dots,h_{m}\in{\mathbb{C}}[v_{1},v_{2},\dots,v_{r}]$ so that the
moduli space is the variety $V(h_{1},h_{2},\dots,h_{m})\setminus
V(g_{1},g_{2},\dots,g_{t})$ and its closure is $V(h_{1},h_{2},\dots,h_{m})$.
Then the irreducible components of the moduli space correspond to the
irreducible components of $V(h_{1},h_{2},\dots,h_{m})$.

For all of our arrangements, $V(h_{1},h_{2},\dots,h_{m})$ is a finite set or
there exists a parameterization such that $m=1$. This is achieved by selecting
a projective basis that minimizes $m$. If $V(h_{1},h_{2},\dots,h_{m})$ is a
finite set, then $\mathcal{M}_{\mathcal{A}}$ is a finite set and the moduli
space $\mathcal{M}_{\mathcal{A}}$ is reducible if it contains at least two
points. If $m=1$ and $h_{1}$ is irreducible over ${\mathbb{C}}$, then
$V(h_{1})$ and therefore $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ are
irreducible. We will discuss how to decide whether $h_{1}$ is irreducible over
${\mathbb{C}}$ later in this section.

######  Example 5.2.

In Example 5.1, the moduli space is a variety that is described by two
polynomials. It would be easier to determine irreducibility if we can
reparamaterize the moduli space so that it is described by a singular
polynomial. Indeed, we can reparametrize the moduli space by starting with the
following four points and coordinates:

$9:\;[1:0:0]\hskip 28.45274pt2:\;[0:1:0]\hskip 28.45274pt0:\;[0:0:1]\hskip
28.45274pt12:\;[1:1:1]$

Then the points and lines were constructed the following order:

$9,2,0,O,L_{10},L_{8},L_{7},L_{4},5,L_{1},A,L_{11},N,L_{9},3,6,L_{2},1,4,L_{3},L_{5},8,7,L_{6}.$

This leads to a new parametrization with only two variables and the single
constraint that

$a^{4}b^{2}+a^{4}b-3a^{3}b^{2}-3a^{3}b+a^{2}b^{2}+2a^{2}b-2ab-a+1=0,$

which is easier to interpret, since we will see that this is an irreducible
polynomial over ${\mathbb{C}}$ by Theorem 5.8 and Lemma 5.9 so the moduli
space is irreducible.

Suppose $m=1$ and $h_{1}$ is reducible over ${\mathbb{C}}$. Here we look at
the intersections of the irreducible components of $V(h_{1})$ to see if they
are in $\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$. If an intersection point of
the irreducible components of $V(h_{1})$ is in
$\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$, the irreducible components
containing this intersection point are in the same Euclidean-connected
component.

As we have discussed earlier, for many of our arrangements, we can translate
the problem of determining the irreducibility of the moduli space into a
problem of determining the irreducibliity of the polynomial describing the
moduli space.

######  Definition 5.3.

A polynomial $f$ over a field $K$ is absolutely irreducible if $f$ is
irreducible over any field extension of $K$, or equivalently, $f$ is
irreducible over $\overline{K}$, the algebraic closure of $K$.

The problem of determining whether $f\in K[x_{1},x_{2},\ldots,x_{n}]$ is
absolutely irreducible can be translated into a problem about the Newton
polytope of $f$, as we are able to put a monoidal operation on Newton
polytopes that mimics the multiplication of polynomials.

######  Definition 5.4.

Let $K$ be a field and let $f\in K[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial.
Then the Newton polytope of $f$ is defined to be the convex hull of a set of
points in ${\mathbb{R}}^{n}$:

$P_{f}=\textup{Conv}\left\\{(u_{1},u_{2},\ldots,u_{n})\,\middle|\,\alpha_{u_{1},u_{2},\ldots,u_{n}}\neq
0,f=\sum_{j_{1},j_{2},\ldots,j_{n}}\alpha_{j_{1},j_{2},\ldots,j_{n}}x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots
x_{n}^{j_{n}}\right\\}.$

The Newton polytope is defined this way so that the points in $P_{f}$ provide
a geometric interpretation of the combinations of exponents in each term in
$f$. Now we want an operation on Newton polytopes that provides an analogy for
polynomial multiplication in this geometric interpretation.

######  Definition 5.5.

(see for example Bertone-Chèze-Galligo [BCG10, Definition 5]) If $A_{1}$ and
$A_{2}$ are two subsets of ${\mathbb{R}}^{n}$, then we define their Minkowski
sum as

$A_{1}+A_{2}=\\{a_{1}+a_{2}\mid a_{1}\in A_{1},a_{2}\in A_{2}\\}.$

In fact, if we view $(K[x_{1},x_{2},\ldots,x_{n}],\cdot)$ as a monoid and
$(\textup{{NewP}}_{n},+)$, where $\textup{{NewP}}_{n}$ is the set of all
convex polytopes with nonnegative integer coordinates in ${\mathbb{R}}^{n}$,
also as a monoid, then $P:K[x_{1},x_{2},\ldots,x_{n}]\to\textup{{NewP}}_{n}$
defined by $f\mapsto P_{f}$ is a monoid homomorphism by the following
proposition.

######  Proposition 5.6.

(Ostrowski [Ost75], as found in Bertone-Chèze-Galligo [BCG10, Lemma 6]) Let
$f,g\in K[x_{1},x_{2},\ldots,x_{n}]$. Then $P_{fg}=P_{f}+P_{g}$.

Using the language of Newton polytopes, the following proposition leads to a
useful criterion for detecting the absolute irreducibility of polynomials.

######  Proposition 5.7.

([Rup04]) If $f\in K[x_{1},x_{2},\ldots,x_{n}]$ is absolutely reducible and
$f=f_{1}f_{2}\cdots f_{s}$, then $P_{f}=P_{f_{1}}+P_{f_{2}}+\cdots
P_{f_{s}}=\underbrace{P_{f_{1}}+\cdots P_{f_{1}}}_{\text{$s$ times}}$

###### Proof.

The factors $f_{1},f_{2},\ldots,f_{s}$ are conjugates over $K$ so the
corresponding Newton polytopes are the same. ∎

This leads to the following reformulation of a result by [Gao01] as found in
[BCG10].

######  Theorem 5.8.

(Gao [Gao01], as found in Bertone-Chèze-Galligo [BCG10]) Let
$f(x_{1},x_{2},\ldots,x_{n})$ be an irreducible polynomial in
$K[x_{1},x_{2},\ldots,x_{n}]$, where $K$ is a field. If the Newton polytope
has the following convex hull:

$P_{f}=\\{(x_{1}^{(1)},x_{2}^{(1)},\dots,x_{n}^{(1)}),(x_{1}^{(2)},x_{2}^{(2)},\dots,x_{n}^{(2)}),\ldots,(x_{1}^{(k)},x_{2}^{(k)},\dots,x_{n}^{(k)})\\}$

and the coordinates of the points on the convex hull are coprime, i.e.

(2)
$\gcd(x_{1}^{(1)},x_{2}^{(1)},\dots,x_{n}^{(1)},x_{1}^{(2)},x_{2}^{(2)},\dots,x_{n}^{(2)},\ldots,x_{1}^{(k)},x_{2}^{(k)},\dots,x_{n}^{(k)})=1,$

then $f$ is absolutely irreducible.

It is easy to check algorithmically the $\gcd$ condition of Equation 2 in
Theorem 5.8. However, we require a little bit more theory to see whether $f$
is indeed irreducible over $K$, in order to satisfy the first hypothesis of
Theorem 5.8. For the purpose of this paper, we are only looking at polynomials
over $K=\mathbb{Q}$. In fact, the polynomials we are considering have
coefficients in ${\mathbb{Z}}$, and due to Gauss’s Lemma, a primitive
polynomial $f$ irreducibe over ${\mathbb{Z}}$ is an irreducible polynomial
viewed as an element in ${\mathbb{Q}}[x_{1},x_{2},\dots,x_{n}]$ so we need a
criteria for a polynomial to be irreducible over ${\mathbb{Z}}$, such as the
following.

######  Lemma 5.9.

Let $f\in{\mathbb{Z}}[x_{1},x_{2},\ldots,x_{n}]$. Now let
$g(x_{1})=f(x_{1},t_{2},t_{3}\dots,t_{n})$ for some particular values
$t_{2},t_{3},\dots,t_{n}\in{\mathbb{Z}}$. Furthermore, let $p$ be a prime
number and define $h(x_{1})\in{\mathbb{Z}}/p{\mathbb{Z}}[x_{1}]$ to be
$g(x_{1})$ with its coefficients reduced modulo $p$. If both $h$ and $f$ have
the same degree in the variable $x_{1}$ and $h$ is irreducible over
${\mathbb{Z}}/p{\mathbb{Z}}$, then $f$ is irreducible over ${\mathbb{Z}}$.

Since there are only a small number of irreducible univariate polynomials over
${\mathbb{Z}}/p{\mathbb{Z}}$ of a certain degree for a small prime $p$, it is
easy to check if $h$ is irreducible over ${\mathbb{Z}}/p{\mathbb{Z}}$ in the
above lemma.

######  Example 5.10.

The polynomial in Example 5.2 is
$f(a,b):=a^{4}b^{2}+a^{4}b-3a^{3}b^{2}-3a^{3}b+a^{2}b^{2}+2a^{2}b-2ab-a+1$,
whose Newton polytope has vertices $(0,0),(1,0),(2,2),(4,1),(4,2)$. The
greatest common divisor of the coordinates of these vertices is 1, so Theorem
5.8 implies that $f(a,b)$ is absolutely irreducible as long as $f(a,b)$ is
irreducible over ${\mathbb{Q}}$.

We consider $f(-1,b)=5b^{2}+8b+2$. We see that $f(-1,b)$ has no roots modulo
7, so $f(-1,b)$ is irreducible over ${\mathbb{Z}}$ and therefore $f(a,b)$ is
irreducible over ${\mathbb{Z}}$ by Lemma 5.9. Then Gauss’ Lemma implies that
$f(a,b)$ is irreducible over ${\mathbb{Q}}$.

Theorem 5.8 is easy to check and suffices to show absolute irreducibility for
the majority of the polynomials with which we are concerned. The following
theorem handles the rest of the polynomials we have by removing some points
from the Newton Polytope and then applying Theorem 5.8.

######  Theorem 5.11.

(Bertone-Chèze-Galligo [BCG10, Proposition 9], Kaltofen [Kal95]) Let
$f\in{\mathbb{Z}}[x_{1},x_{2},\dots,x_{n}]$ and let $\overline{f}$ be $f$ with
its coefficients reduced modulo $p$ for some prime $p$. If
$\deg(f)=\deg(\overline{f})$ and $\overline{f}$ is absolutely irreducible,
then $f$ is absolutely irreducible.

Now Theorems 5.8 and 5.11 provide us with criteria to test the irreducibility
of the polynomial constraints on the parameters of the moduli spaces. This
then allows us to determine the number of connected components the moduli
space has. The results are shown in Theorems 1.1, 1.2, and 1.3, showing the
number of arrangements in each family, classified by their moduli spaces. We
break down the tables by which $(10_{3})$ configuration each arrangement is a
one-line extension of and also by whether the moduli space is irreducible,
empty, $\mathcal{M}_{\mathcal{A}}$ being reducible and
$\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ being irreducible, or
$\mathcal{M}_{\mathcal{A}}^{\mathbb{C}}$ being reducible. The subtotal simply
adds all of the columns together, whereas the total accounts for
identifications up to isomorphism.

##  6\. Corrections

We now remedy a small error in previous work by the first author with Amram,
Teicher, and Ye that leads to three arrangements being counted twice.

######  Proposition 6.1.

There are only two distinct line arrangements with ten lines obtained by
adding a tenth line through three double points in the configuration
$(9_{3})_{1}$.

This replaces the following now-incorrect result:

######  Proposition 6.2.

(Amram-Cohen-Teicher-Ye [ACTY13, Lemma 8.2]) There are five line arrangements
with ten lines obtained by adding a tenth line through three double points in
the configuration $(9_{3})_{1}$.

The argument in [ACTY13] relies on the automorphism group of the configuration
$(9_{3})_{1}$ to identify isomorphic arrangements with ten lines constructed
from the configuration $(9_{3})_{1}$ as described. It appears that [ACTY13]
has depicted the configuration $(9_{3})_{1}$ as having an automorphism group
isomorphic to $D_{6}$, the dihedral group of order twelve. However, the
automorphism group of the configuration $(9_{3})_{1}$ has order larger than
twelve.

######  Proposition 6.3.

(Coxeter [Cox77]) The automorphism group of the configuration $(9_{3})_{1}$ is
isomorphic to $PG(2,3)$, a group of order one hundred eight.

A larger automorphism group would mean that we could potentially identify some
of the arrangements with ten lines constructed from $(9_{3})_{1}$ as
isomorphic. Indeed, this is the case.

We conclude with a proof of Proposition 6.1.

###### Proof.

In [ACTY13], five line arrangements with 10 lines are said to be generated as
described, but we show explicitly that some of the new line arrangements are
isomorphic.

Using the names of the arrangements in [ACTY13], we first show
$(9_{3})_{1}.CDI\cong(9_{3})_{1}.CFH$ , whose arrangement tables are given in
Table 16.

$(9_{3})_{1}.CDI$ | $(9_{3})_{1}.CFH$
---|---
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$
---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 3 | 5 | 0 | 0 | 0 | C
2 | 4 | 6 | 4 | 6 | 7 | 4 | 2 | 3 | D
3 | 5 | 7 | 8 | 8 | 8 | 6 | 7 | 5 | I
C | I |  | D |  | C |  | I | D |
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$
---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 3 | 5 | 0 | 0 | 0 | C
2 | 4 | 6 | 4 | 6 | 7 | 4 | 2 | 3 | F
3 | 5 | 7 | 8 | 8 | 8 | 6 | 7 | 5 | H
C | H | F | F | H | C |  |  |  |
Table 16.  Arrangement tables for two isomorphic arrangements
$(9_{3})_{1}.CDI$ and $(9_{3})_{1}.CFH$

The arrangements $(9_{3})_{1}.CDI$ and $(9_{3})_{1}.CFH$ are isomorphic due to
the following isomorphism $\varphi$. The isomorphism $\varphi$ also induces an
automorphism of the lines. See Table 17.

Point $p$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | C | D | I
---|---|---|---|---|---|---|---|---|---|---|---|---
$\varphi(p)$ | 4 | 3 | 1 | 2 | 6 | 8 | 0 | 5 | 7 | C | F | H
Line $L$ | $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ |  |
$\varphi(L)$ | $L_{1}$ | $L_{5}$ | $L_{9}$ | $L_{3}$ | $L_{8}$ | $L_{6}$ | $L_{7}$ | $L_{2}$ | $L_{4}$ | $L_{10}$ |  |
Table 17.  An explicit isomorphism $\varphi:(9_{3})_{1}.CDI\to(9_{3})_{1}.CFH$

Now we show that
$(9_{3})_{1}.CDG\cong_{\varphi_{1}}(9_{3})_{1}.CDH\cong_{\varphi_{2}}(9_{3})_{1}.CFG$
, whose arrangement tables are given in Table 18.

$(9_{3})_{1}.CDG$ | $(9_{3})_{1}.CDH$ | $(9_{3})_{1}.CFG$
---|---|---
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$
---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 3 | 5 | 0 | 0 | 0 | C
2 | 4 | 6 | 4 | 6 | 7 | 4 | 2 | 3 | D
3 | 5 | 7 | 8 | 8 | 8 | 6 | 7 | 5 | G
C |  |  | D | G | C |  | G | D |
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$
---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 3 | 5 | 0 | 0 | 0 | C
2 | 4 | 6 | 4 | 6 | 7 | 4 | 2 | 3 | D
3 | 5 | 7 | 8 | 8 | 8 | 6 | 7 | 5 | H
C | H |  | D | H | C |  |  | D |
| $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$
---|---|---|---|---|---|---|---|---|---
1 | 1 | 1 | 2 | 3 | 5 | 0 | 0 | 0 | C
2 | 4 | 6 | 4 | 6 | 7 | 4 | 2 | 3 | F
3 | 5 | 7 | 8 | 8 | 8 | 6 | 7 | 5 | G
C |  | F | F | G | C |  | G |  |

Table 18.  Arrangement tables for three isomorphic arrangements
$(9_{3})_{1}.CDG,(9_{3})_{1}.CDH$ and $(9_{3})_{1}.CFG$

All three arrangements are isomorphic due to the isomorphisms $\varphi_{1}$
and $\varphi_{2}$. The isomorphisms $\varphi_{1},\varphi_{2}$ also induce an
automorphisms of the lines. See Table 19. ∎

Point $P$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | C | D | G
---|---|---|---|---|---|---|---|---|---|---|---|---
$\varphi_{1}(P)$ | 4 | 7 | 5 | 8 | 0 | 2 | 6 | 1 | 3 | C | D | H
$\varphi_{2}(\varphi_{1}(P))$ | 6 | 5 | 8 | 7 | 4 | 1 | 0 | 3 | 2 | C | F | G
Line $L$ | $L_{1}$ | $L_{2}$ | $L_{3}$ | $L_{4}$ | $L_{5}$ | $L_{6}$ | $L_{7}$ | $L_{8}$ | $L_{9}$ | $L_{10}$ |  |
$\varphi_{1}(L)$ | $L_{6}$ | $L_{8}$ | $L_{3}$ | $L_{9}$ | $L_{5}$ | $L_{1}$ | $L_{7}$ | $L_{2}$ | $L_{4}$ | $L_{10}$ |  |
$\varphi_{2}(\varphi_{1}(L))$ | $L_{6}$ | $L_{2}$ | $L_{9}$ | $L_{4}$ | $L_{8}$ | $L_{1}$ | $L_{7}$ | $L_{5}$ | $L_{3}$ | $L_{10}$ |  |
Table 19.  Explicit isomorphisms for $(9_{3})_{1}.CDG\cong_{\varphi_{1}}(9_{3})_{1}.CDH\cong_{\varphi_{2}}(9_{3})_{1}.CFG$ $(9_{3})_{1}.CFH$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 2 3 5 0 0 0 C 2 4 6 4 6 7 4 2 3 F 3 5 7 8 8 8 6 7 5 H C H F F H C |  $(9_{3})_{3}.BDF$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 9 9 9 B 2 4 6 2 5 3 2 4 3 D 3 5 7 4 6 7 5 7 6 F F B B D D F
---|---
$(9_{3})_{1}.CDG$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 2 3 5 0 0 0 C 2 4 6 4 6 7 4 2 3 D 3 5 7 8 8 8 6 7 5 G C D G C G D |  $(9_{3})_{3}.ACG$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 9 9 9 A 2 4 6 2 5 3 2 4 3 C 3 5 7 4 6 7 5 7 6 G G C A G C A
$(9_{3})_{2}.DFI$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 4 3 2 D 2 4 6 4 2 3 7 5 5 F 3 5 7 6 7 9 9 6 9 I I D F I D F |  $(9_{3})_{3}.AEG$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 9 9 9 A 2 4 6 2 5 3 2 4 3 E 3 5 7 4 6 7 5 7 6 G G E A G E A
$(9_{3})_{2}.CFI$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 4 3 2 C 2 4 6 4 2 3 7 5 5 F 3 5 7 6 7 9 9 6 9 I I F I C F C |  $(9_{3})_{3}.ADG$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 9 9 9 A 2 4 6 2 5 3 2 4 3 D 3 5 7 4 6 7 5 7 6 G G A G D D A
$(9_{3})_{2}.DFA$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 4 3 2 D 2 4 6 4 2 3 7 5 5 F 3 5 7 6 7 9 9 6 9 A A D F D F A |  $(9_{3})_{3}.BEG$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 9 9 9 A 2 4 6 2 5 3 2 4 3 D 3 5 7 4 6 7 5 7 6 G G A G D D A
$(9_{3})_{2}.DFH$ $L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$ $L_{5}$ $L_{6}$ $L_{7}$ $L_{8}$ $L_{9}$ $L_{10}$ 1 1 1 8 8 8 4 3 2 D 2 4 6 4 2 3 7 5 5 F 3 5 7 6 7 9 9 6 9 H D F H D F H |
Table 20. The eleven one-line extensions of $(9_{3})$ configurations found in
[ACTY13]

## References

  * [AAB14] Abdullah Al-Azemi and Dieter Betten, _The configurations $(12_{3})$ revisited_, J. Geom. 105 (2014), 391–417.
  * [ACSTYZ15] Meirav Amram, Moshe Cohen, Hao Sun, Mina Teicher, Fei Ye, and Anna Zarkh, _Combinatorial symmetry of line arrangements and applications_ , Topology Appl. 193 (2015), 226–247.
  * [ACTY13] Meirav Amram, Moshe Cohen, Mina Teicher, and Fei Ye, _Moduli spaces of ten-line arrangements with double and triple points_ , arXiv:1306.6105, 2013.
  * [AGTX15] Meirav Amram, Cheng Gong, Mina Teicher, and Wan-Yuan Xu, _Moduli spaces of arrangements of 11 projective lines with a quintuple point_ , Turk. J. Math. 39 (2015), 618–644.
  * [ATY13] Meirav Amram, Mina Teicher, and Fei Ye, _Moduli spaces of arrangements of 10 projective lines with quadruple points_ , Adv. Appl. Math. 51 (2013), 392–418.
  * [BCG10] Cristina Bertone, Guillaume Chèze, and André Galligo, _Modular Las Vegas algorithms for polynomial absolute factorization_ , J. Symbolic Comput. (2010), 1280–1295.
  * [BLVSWZ99] Anders Björner, Michael Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, _Oriented matroids_ , Cambridge University Press, 1999\.
  * [BP15] Jürgen Bokowski and Vincent Pilaud, _On topological and geometric $(19_{4})$ configurations_, Eur. J. Combin. 50 (2015), 4–17.
  * [Cox77] Harold Scott MacDonald Coxeter, _The pappus configuration and its groups_ , Pi Mu Epsilon J. 6 (1977), 331–336.
  * [CS97] Daniel Cohen and Alexander Suciu, _The braid monodromy of plane algebraic curves and hyperplane arrangements_ , Comment. Math. Helv. 72 (1997), 285–315.
  * [Dim17a] Alexandru Dimca, _Hyperplane arrangements: An introduction_ , Springer, 2017\.
  * [DIM17b] Alexandru Dimca, Denis Ibadula, and Anca Macinic, _Freeness and near freeness are combinatorial for line arrangements in small degrees_ , arXiv:1712.04400, 2017.
  * [DvS03] Robert Daublebsky von Sterneck, _Über die zu den configurationen $12_{3}$ zugehörigen gruppen von substitutionen_, Monatsh. Math. 14 (1903), 254–260.
  * [EP13] Eric Ens and Ranganathan Padmanabhan, _Group embedding of the projective plane ${\rm PG}(2,3)$_, Automated reasoning and mathematics, Lecture Notes in Comput. Sci., vol. 7788, Springer, Heidelberg, 2013, pp. 131–138.
  * [Fan97] Kwai-Man Fan, _Direct product of free groups as the fundamental group of the complement of a union of lines_ , Michigan Math. 44 (1997), 283–291.
  * [Gao01] Shuhong Gao, _Absolute irreducibility of polynomials via newton polytopes_ , J. Algebra (2001), 501–520.
  * [GB22] Benoît Guerville-Ballé, _The loop-linking number of line arrangements_ , Math. Z. 301 (2022), no. 2, 1821–1850.
  * [Gro90] H. Gropp, _On the existence and nonexistence of configurations $n_{k}$_, J. Comb. Inf. Syst. Sci. 15 (1990), 34–48.
  * [Grü09] Branko Grünbaum, _Configurations of points and lines_ , American Mathematical Society, 2009.
  * [GTV03] David Garber, Mina Teicher, and Uzi Vishne, _$\pi_{1}$ -classification of real arrangements with up to eight lines_, Topology 42 (2003), 265–289.
  * [HOV15] Jessica Hauschild, Jazmin Ortiz, and Oscar Vega, _On the levi graph of point-line configurations_ , Involve 8 (2015), no. 5, 893–900.
  * [Kal95] E. Kaltofen, _Effective noether irreducibility forms and applications_ , Journal of Computer and System Sciences 50 (1995), no. 2, 274 – 295\.
  * [Kan81] S. Kantor, _Ueber die configurationen (3,3) mit den Indices 8,9 und ihren Zusammenhang mit den Curven dritter Ordnung_ , Wien. Ber. (1881), 915–932.
  * [Koc16] William L. Kocay, _One-point extensions in $n_{3}$ configurations_, Ars Math. Contemp. 10 (2016), 291–322.
  * [Koc21] by same author, _The configurations $(13_{3})$_, Art Discrete Appl. Math. 4 (2021), no. 3, Paper No. 3.15, 16.
  * [Mö28] A. F. Möbius, _Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heißen?_ , J. Reine Angew. Math. 3 (1828), 273–278.
  * [Mac36] Saunders MacLane, _Some interpretations of abstract linear dependence in terms of projective geometry_ , Amer. J. Math. 58 (1936), no. 1, 236–240.
  * [Mar87] Vittorio Martinetti, _Sulle configurazioni piane $\mu_{3}$_, Ann. Mat. Pura Appl. 2 (1887), 1–26.
  * [NY12] Shaheen Nazir and Masahiko Yoshinaga, _On the connectivity of the realization spaces of line arrangements_ , Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012), 921–937.
  * [OS80] Peter Orlik and Louis Solomon, _Combinatorics and topology of complements of hyperplanes_ , Invent. Math. 56 (1980), 167–189.
  * [Ost75] Alexander Markowich Ostrowski, _On multiplication and factorization of polynomials, i. lexicographic orderings and extreme aggregates of terms_ , Aequ. Math. 13 (1975), no. 3, 201–228.
  * [Oxl92] James G. Oxley, _Matroid theory_ , Oxford University Press, 1992.
  * [Oxl19] James Oxley, _A matroid extension result_ , SIAM J. Discrete Math. 33 (2019), no. 1, 138–152.
  * [Ran89] Richard Randell, _Lattice-isotopic arrangements are topologically isomorphic_ , Proc. Amer. Math. Soc. 107 (1989), 555–559.
  * [Rey82] Th. Reye, _Das Problem der Configurationen_ , Acta Math. 1 (1882), no. 1, 93–96.
  * [Rup04] David Rupprecht, _Semi-numerical absolute factorization of polynomials with integer coefficients_ , Journal of Symbolic Computation 37 (2004), no. 5, 557–574.
  * [SBA15] Klara Stokes and Maria Bras-Amorós, _A survey on the use of combinatorial configurations for anonymous database search_ , Adv. Res. Data Privacy 567 (2015), 407–419.
  * [Sch89] H. Schröter, _über die Bildungsweise und geometrische Construction der Configurationen $10_{3}$_, Nachr. Ges. Wiss Göittingen (1889), 239–253.
  * [SW90] Bernd Sturmfels and Neil White, _Rational realizations of $11_{3}$\- and $12_{3}$-configurations_, Aequationes Math. 39 (1990), 92–123.
  * [Vog86] Walter Vogler, _Representing groups by graphs with constant link and hypergraphs_ , Journal of Graph Theory 10 (1986), no. 4, 461–475.
  * [Ye13] Fei Ye, _Classification of moduli spaces of arrangements of 9 projective lines_ , Pacific J. of Math. 265 (2013), 243–256.