# Free multiderivations of connected subgraph arrangements

Paul Mücksch  Institut für Algebra, Zahlentheorie und Diskrete Mathematik,
Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfen-
garten 1, D-30167 Hannover, Germany<EMAIL_ADDRESS>,  Gerhard
Röhrle  Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum,
Germany<EMAIL_ADDRESS>and  Sven Wiesner  Fakultät für Mathematik,
Ruhr-Universität Bochum, D-44780 Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

Cuntz and Kühne [CK24] introduced the class of connected subgraph arrangements
${\mathscr{A}}_{G}$, depending on a graph $G$, and classified all graphs $G$
such that the corresponding arrangement ${\mathscr{A}}_{G}$ is free. We extend
their result to the multiarrangement case and classify all graphs $G$ for
which the corresponding arrangement ${\mathscr{A}}_{G}$ supports some
multiplicity $\mu$ such that the multiarrangement $({\mathscr{A}}_{G},\mu)$ is
free.

###### Key words and phrases:

Free arrangement, Free multiarrangement, Connected subgraph arrangement

##  1\. Introduction

The study of modules of logarithmic vector fields tangent to hyperplane
arrangements (in the sequel also referred to as derivation modules) and in
particular the question about their freeness over the coordinate ring of the
ambient space is a classic topic at the crossroads of discrete geometry and
combinatorics, commutative algebra and algebraic geometry initiated in seminal
work by K. Saito [S80] and H. Terao [T80a]. Arrangements with free derivation
modules are simply called free arrangements. Subsequently, G. Ziegler [Z89]
extended the theory by also taking multiplicities into account. Such
multiarrangements naturally arise from arrangement constructions, i.e.
restrictions to hyperplanes, and the freeness of their mulitderivation modules
is closely linked to the freeness of the original simple arrangement due to
fundamental work by Ziegler and M. Yoshinaga [Y04].

To decide whether a given (multi-)arrangement is free is a hard problem in
general and at its core lies Terao’s conjecture, asserting that freeness only
depends on the underlying combinatorial structure provided by the intersection
lattice. To this day the conjecture is wide open. Therefore, a natural
approach in the investigation of such freeness questions is to restrict the
view to special classes of arrangements with additional structure. A pivotal
result along this line is a theorem of Terao stating that all (complex)
reflection arrangements are free [T80b]. Regarding multiarrangements, another
milestone is also due to Terao [T02] establishing the freeness of certain
natural multiplicities on Coxeter arrangements. This was subsequently used by
Yoshinaga in [Y04] to establish a conjecture of Edelman and Reiner about the
freeness of certain deformations of Weyl arrangements.

In the recent work [CK24], M. Cuntz and L. Kühne introduce a new class
stemming from simple graphs which are defined as follows.

######  Definition 1.1 ([CK24, Def. 1.1]).

Let $G=(N,E)$ be an undirected (simple) graph with a set of vertices
$N=\\{1,\dots,n\\}$ and a set of edges $E$. The _connected subgraph
arrangement_ ${\mathscr{A}}_{G}$ in $V=\mathbb{Q}^{n}$ associated to $G$ is
defined as

${\mathscr{A}}_{G}:=\\{H_{I}\mid\emptyset\neq I\subseteq N\text{ if
}G[I]\text{ is connected}\\},$

where $H_{I}$ is the hyperplane

$H_{I}=\ker\sum_{i\in I}x_{i}$

and $G[I]$ is the induced subgraph on the vertices $I\subseteq N$.

Cuntz and Kühne gave a complete characterization of free arrangements within
this class (see Theorem 2.23). In view of their work and because of the
importance of the broader perspective on freeness within the framework of
multiarrangements, the natural question arises if one can describe certain
free multiplicities extending their classification. This is the aim of this
paper. Our first main result is as follows.

######  Theorem 1.2 (Corollary 3.6).

Let $G$ be a connected graph. There exists a multiplicity $\mu$ such that the
connected subgraph multiarrangement $({\mathscr{A}}_{G},\mu)$ is free if and
only if $G$ is $G_{1}$, $G_{2}$, a path-graph, a cycle-graph, an almost-path-
graph, or a path-with-triangle-graph (see Definition 2.18 and Figure 1).

A natural and important question stemming from the study of free multi-Coxeter
arrangements by Terao [T02] is about the existence of constant free
multiplicities. Our next result gives the following answer for connected
subgraph arrangements.

######  Theorem 1.3 (Proposition 4.6 and Corollary 4.9).

Let $G$ be a connected graph with at least three vertices and $\mu\equiv k>1$
is a constant multiplicity on ${\mathscr{A}}_{G}$. Then
$({\mathscr{A}}_{G},\mu)$ is free if and only if $G$ is a path-graph $P_{n}$
or $G$ is the cycle-graph $C_{3}$ and $k=3$.

To classify free multiplicities in general is a notoriously hard problem.
Complete answers are known only in very few cases. E.g. characterizing the
free multiplicities even on braid arrangements of rank greater than $3$ is an
unresolved problem, cf. [DFMS20]. Nonetheless, in our study it turns out that
for a certain subarrangement ${\mathscr{D}}$ of ${\mathscr{A}}_{C_{3}}$ which
is obtained from ${\mathscr{A}}_{C_{3}}$ by deleting a single hyperplane, the
task is feasible. This constitutes our third theorem.

######  Theorem 1.4 (Theorem 5.9).

Let $(\mathscr{D},\mu)$ be the multiarrangement

$Q(\mathscr{D},\mu)=x_{1}^{a}x_{2}^{b}x_{3}^{c}(x_{1}+x_{2})^{d}(x_{1}+x_{3})^{e}(x_{2}+x_{3})^{f}$

with $a,b,c,d,e,f\geq 1$. Then $(\mathscr{D},\mu)$ is free if and only if
$\mu=(2k,2k,2k,1,1,1)$ for some $k\in\mathbb{N}_{\geq 1}$ where the order of
the hyperplanes is as above.

The classification of the free multiplicities on ${\mathscr{D}}$ was
previously obtained by M. DiPasquale and M. Wakefield in [DW18] by different
methods.

To conclude, our last chief result describes certain families of free
multiplicities on ${\mathscr{A}}_{C_{3}}$.

######  Theorem 1.5 (Proposition 5.13).

Let $({\mathscr{A}}_{C_{3}},\mu)$ be the multiarrangement

$Q({\mathscr{A}}_{C_{3}},\mu)=x_{1}^{a}x_{2}^{b}x_{3}^{c}(x_{1}+x_{2})^{d}(x_{1}+x_{3})^{e}(x_{2}+x_{3})^{f}(x_{1}+x_{2}+x_{3})^{g}$

with multiplicity $\mu=(a,b,c,d,e,f,g),(d\geq e\geq f)$. Suppose that either

  1. (i)

$\mu=(k,k,k,r,1,1,k)$, where $1\leq k\leq 3$ and $r\geq 1$, or

  2. (ii)

$\mu=(k,k,k,r,1,1,k)$, where $k>3$ and $r\geq 2k-5$.

Then $({\mathscr{A}}_{C_{3}},\mu)$ is free. Moreover, if $r\geq 2k$, then
$({\mathscr{A}}_{C_{3}},\mu)$ is inductively free with exponents
$(r,2k+1,2k+1)$. For $\mu=(3,3,3,1,1,1,3)$, $({\mathscr{A}}_{C_{3}},\mu)$ is
not inductively free.

For the notion of inducive freeness, see Definition 2.12.

The paper is organized as follows. In §2 we review the required notions from
the theory of free multiarrangements and connected subgraph arrangements. The
next section presents the arguments leading to a proof of Theorem 1.2. In §4,
we investigate the natural question which connected subgraph arrangements
support free constant multiplicities leading to our second main Theorem 1.3.
Along the way, in §4.1 we present a non-computational proof of the non-
freeness of the connected subgraph arrangements stemming from $G_{1}$ up to
$G_{8}$ from Figure 1. This was derived in [CK24] by means of elaborate
computer calculations. In our final section we study free multiplicities on
two special arrangements, culminating in Theorems 1.4 and 1.5. In the
appendices we list some tables containing data which we computed and utilised
for our proofs.

##  2\. Recollections and Preliminaries

###  2.1. Hyperplane arrangements, multiarrangements and their freeness

Let $V={\mathbb{K}}^{\ell}$ be an $\ell$-dimensional ${\mathbb{K}}$-vector
space. A _hyperplane arrangement_ is a pair $({\mathscr{A}},V)$, where
${\mathscr{A}}$ is a finite collection of hyperplanes in $V$. Usually, we
simply write ${\mathscr{A}}$ in place of $({\mathscr{A}},V)$. We write
$|{\mathscr{A}}|$ for the number of hyperplanes in ${\mathscr{A}}$. The empty
arrangement in $V$ is denoted by $\Phi_{\ell}$.

The _lattice_ $L({\mathscr{A}})$ of ${\mathscr{A}}$ is the set of subspaces of
$V$ of the form $H_{1}\cap\ldots\cap H_{i}$ where $\\{H_{1},\ldots,H_{i}\\}$
is a subset of ${\mathscr{A}}$. For $X\in L({\mathscr{A}})$, we have two
associated arrangements, firstly ${\mathscr{A}}_{X}:=\\{H\in{\mathscr{A}}\mid
X\subseteq H\\}\subseteq{\mathscr{A}}$, the _localization of ${\mathscr{A}}$
at $X$_, and secondly, the _restriction of ${\mathscr{A}}$ to $X$_,
$({\mathscr{A}}^{X},X)$, where ${\mathscr{A}}^{X}:=\\{X\cap H\mid
H\in{\mathscr{A}}\setminus{\mathscr{A}}_{X}\\}$. Note that $V$ belongs to
$L({\mathscr{A}})$ as the intersection of the empty collection of hyperplanes
and ${\mathscr{A}}^{V}={\mathscr{A}}$. The lattice $L({\mathscr{A}})$ is a
partially ordered set by reverse inclusion: $X\leq Y$ provided $Y\subseteq X$
for $X,Y\in L({\mathscr{A}})$. The rank function on $L({\mathscr{A}})$ is
given by $\operatorname{rank}(X)=\operatorname{codim}X$ for $X\in
L({\mathscr{A}})$ and we define $L({\mathscr{A}})_{k}=\\{X\in
L({\mathscr{A}})\mid\operatorname{rank}(X)=k\\}$.

Let $S=S[V]\cong{\mathbb{K}}[x_{1},\ldots,x_{\ell}]$ be the coordinate ring of
the ambient space with its usual grading by polynomial degree:
$S=\oplus_{p\in{\mathbb{Z}}}S_{p}$, where $S_{p}$ is the
${\mathbb{K}}$-subspace of polynomials of degree $p$ and $S_{p}=0$ in case
$p<0$.

A _multiarrangement_ is a pair $({\mathscr{A}},\mu)$ consisting of a
hyperplane arrangement ${\mathscr{A}}$ and a _multiplicity_ function
$\mu:{\mathscr{A}}\to{\mathbb{Z}}_{\geq 0}$ associating to each hyperplane $H$
in ${\mathscr{A}}$ a non-negative integer $\mu(H)$. The _order_ of the
multiarrangement $({\mathscr{A}},\mu)$ is defined by
$|\mu|:=\sum_{H\in{\mathscr{A}}}\mu(H)$. For a multiarrangement
$({\mathscr{A}},\mu)$, the underlying arrangement ${\mathscr{A}}$ is sometimes
called the associated _simple_ arrangement, and so $({\mathscr{A}},\mu)$
itself is simple if and only if $\mu(H)=1$ for each $H\in{\mathscr{A}}$. We
sometimes denote this simple multiplicity by $\mu=\mathbbm{1}$. For two
multiplicities $\mu,\mu^{\prime}$ on an arrangement ${\mathscr{A}}$ we
sometimes write $\mu\geq\mu^{\prime}$ if $\mu(H)\geq\mu^{\prime}(H)$ for all
$H\in{\mathscr{A}}$.

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $V$ and let $X\in
L({\mathscr{A}})$. The _localization of $({\mathscr{A}},\mu)$ at $X$_ is
defined to be $({\mathscr{A}}_{X},\mu_{X})$, where
$\mu_{X}=\mu|_{{\mathscr{A}}_{X}}$.

The _defining polynomial_ $Q({\mathscr{A}},\mu)$ of the multiarrangement
$({\mathscr{A}},\mu)$ is given by

$Q({\mathscr{A}},\mu):=\prod_{H\in{\mathscr{A}}}\alpha_{H}^{\mu(H)},$

a polynomial of degree $|\mu|$ in $S$.

The _module of ${\mathscr{A}}$-derivations_ of $({\mathscr{A}},\mu)$ is
defined as

$D({\mathscr{A}},\mu):=\\{\theta\in{\operatorname{Der}}(S)\mid\theta(\alpha_{H})\in\alpha_{H}^{\mu(H)}S\text{
for each }H\in{\mathscr{A}}\\}.$

We say that $({\mathscr{A}},\mu)$ is _free_ if $D({\mathscr{A}},\mu)$ is a
free $S$-module – a notion introduced by G. Ziegler [Z89, Def. 6], extending
K. Saito’s [S80] and H. Terao’s [T80a] freeness for simple arrangements.

The derivation module $D({\mathscr{A}},\mu)$ is a ${\mathbb{Z}}$-graded
$S$-module and thus, if $({\mathscr{A}},\mu)$ is free, there is a homogeneous
basis $\theta_{1},\ldots,\theta_{\ell}$ of $D({\mathscr{A}},\mu)$. The
multiset of the unique polynomial degrees $\operatorname{pdeg}\theta_{i}$
forms the set of _exponents_ of the free multiarrangement
$({\mathscr{A}},\mu)$ and is denoted by $\exp({\mathscr{A}},\mu)$. Freeness is
a local property for multiarrangements.

######  Theorem 2.1 ([ANN09, Prop. 1.7]).

For $U\subseteq V$ a subspace, the localization $({\mathscr{A}}_{U},\mu_{U})$
at $U$ is free provided $({\mathscr{A}},\mu)$ is free.

Let ${\mathscr{A}}$ be an arrangement and $H\in{\mathscr{A}}$. Define a
multiplicity $\kappa$ on ${\mathscr{A}}^{H}$ by

$\kappa(X)=|{\mathscr{A}}_{X}|-1,\quad X\in{\mathscr{A}}^{H}$

and call the pair $({\mathscr{A}}^{H},\kappa)$ the _Ziegler-restriction_ of
${\mathscr{A}}$ to $H$. Ziegler showed the following connection between the
freeness of ${\mathscr{A}}$ and the freeness of any of its Ziegler-
restrictions.

######  Theorem 2.2 ([Z89, Thm. 11]).

Suppose ${\mathscr{A}}$ is free with exponents $(1,e_{2},\dots,e_{n})$. Let
$H$ be a hyperplane in ${\mathscr{A}}$, then $({\mathscr{A}}^{H},\kappa)$ is
free with exponents $(e_{2},\dots,e_{n})$.

Ziegler also generalizes Saito’s criterion to the multiarrangement case.

######  Theorem 2.3 ([Z89, Thm. 8], Saito’s criterion).

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $\mathbb{K}^{n}$ and for
each $1\leq i\leq n$ let

$\theta_{i}=\sum_{j=1}^{n}p_{ij}\frac{\partial}{\partial x_{j}}\in
D({\mathscr{A}},\mu).$

Then $\\{\theta_{1},\theta_{2},\dots,\theta_{n}\\}$ is a basis for
$D({\mathscr{A}},\mu)$ if and only if $\det(p_{ij})=c\cdot
Q({\mathscr{A}},\mu)$ for some $c\in\mathbb{K}\setminus\\{0\\}$.

An arrangement ${\mathscr{A}}$ is called _locally free along_ $H$ for some
$H\in{\mathscr{A}}$, if ${\mathscr{A}}_{x}$ is a free arrangement for all
$x\in H\backslash\\{0\\}$. The following result is due to Yoshinaga.

######  Theorem 2.4 ([Y04, Thm. 2.2]).

Suppose $\ell\geq 3$. Then ${\mathscr{A}}$ is free with exponents
$(1,e_{1},\dots,e_{\ell})$ if and only if it is locally free along $H$ for
some $H\in{\mathscr{A}}$ and $({\mathscr{A}}^{H},\kappa)$ is free with
exponents $(e_{1},\dots,e_{\ell})$.

Theorem 2.4 readily implies the following.

######  Corollary 2.5.

Suppose $\ell\geq 3$. Let ${\mathscr{A}}$ be locally free along
$H\in{\mathscr{A}}$. If ${\mathscr{A}}_{X}$ fails to be free for some $X\in
L({\mathscr{A}})$ with $X\not\subset H$, then $({\mathscr{A}}^{H},\kappa)$ is
not free.

###### Proof.

Suppose $({\mathscr{A}}^{H},\kappa)$ is free. Then so is ${\mathscr{A}}$, by
Theorem 2.4, and so is ${\mathscr{A}}_{X}$ for any $X\in L({\mathscr{A}})$, by
Theorem 2.1 (with simple multiplicities), a contradiction. ∎

Recall that an $\ell$-arrangement ${\mathscr{A}}$ with
$\operatorname{rank}({\mathscr{A}})=r$ is called _generic_ if
$|{\mathscr{A}}_{X}|=k$ for all $X\in L({\mathscr{A}})_{k}$ and $1\leq k\leq
r-1$. An arrangement ${\mathscr{A}}$ is called _totally non-free_ if
$({\mathscr{A}},\mu)$ fails to be free for any multiplicity
$\mu\geq\mathbbm{1}$. The following result states that generic arrangements
are totally non-free.

######  Theorem 2.6 ([Y09, Prop. 4.1]).

Let ${\mathscr{A}}$ be a generic arrangement in $V$ with $\ell=\dim V\geq 3$.
If $|{\mathscr{A}}|>\ell$, then $({\mathscr{A}},\mu)$ is totally non-free.

The following construction from [ATW08] leads to an extension of Terao’s
seminal Addition-Deletion Theorem [T80a] for free simple arrangements to
multiarrangements.

######  Definition 2.7.

Let $({\mathscr{A}},\mu)\neq\Phi_{\ell}$ be a multiarrangement. Fix
$H_{0}=\ker(\alpha_{0})$ in ${\mathscr{A}}$. We define the _deletion_
$({\mathscr{A}}^{\prime},\mu^{\prime})$ and _restriction_
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ of $({\mathscr{A}},\mu)$ with respect
to $H_{0}$ as follows. If $\mu(H_{0})=1$, then set
${\mathscr{A}}^{\prime}={\mathscr{A}}\setminus\\{H_{0}\\}$ and define
$\mu^{\prime}(H)=\mu(H)$ for all $H\in{\mathscr{A}}^{\prime}$. If
$\mu(H_{0})>1$, then set ${\mathscr{A}}^{\prime}={\mathscr{A}}$ and define
$\mu^{\prime}(H_{0})=\mu(H_{0})-1$ and $\mu^{\prime}(H)=\mu(H)$ for all $H\neq
H_{0}$.

Let ${\mathscr{A}}^{\prime\prime}=\\{H\cap H_{0}\mid
H\in{\mathscr{A}}\setminus\\{H_{0}\\}\ \\}$. The _Euler multiplicity_
$\mu^{*}$ of ${\mathscr{A}}^{\prime\prime}$ is defined as follows. Let
$Y\in{\mathscr{A}}^{\prime\prime}$. Since the localization ${\mathscr{A}}_{Y}$
is of rank $2$, the multiarrangement $({\mathscr{A}}_{Y},\mu_{Y})$ is free,
[Z89, Cor. 7]. According to [ATW08, Prop. 2.1], the module of derivations
$D({\mathscr{A}}_{Y},\mu_{Y})$ admits a particular homogeneous basis
$\\{\theta_{Y},\psi_{Y},D_{3},\ldots,D_{\ell}\\}$, such that
$\theta_{Y}\notin\alpha_{0}{\operatorname{Der}}(S)$ and
$\psi_{Y}\in\alpha_{0}{\operatorname{Der}}(S)$, where $H_{0}=\ker\alpha_{0}$.
Then on $Y$ the Euler multiplicity $\mu^{*}$ is defined to be
$\mu^{*}(Y)=\operatorname{pdeg}\theta_{Y}$.

Often, $({\mathscr{A}},\mu),({\mathscr{A}}^{\prime},\mu^{\prime})$ and
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ is referred to as the _triple_ of
$({\mathscr{A}},\mu)$ with respect to $H_{0}$.

The following observation is immediate from Definition 2.7.

######  Remark 2.8.

Let $({\mathscr{A}},\mu)$ be a multiarrangement and $H_{0}\in{\mathscr{A}}$
and let $({\mathscr{A}}^{\prime\prime},\mu^{*})$ be the Euler multiplicity
corresponding to $H_{0}$. For $X\in{\mathscr{A}}^{\prime\prime}$ the value
$\mu^{*}(X)$ is simply the common value of the non-zero exponent of
$({\mathscr{A}}_{X},\mu_{X})$ and
$({\mathscr{A}}^{\prime}_{X},\mu_{X}^{\prime})$.

It is useful to be able to determine the Euler multiplicity explicitly in some
relevant instances. For that purpose, we recall parts of [ATW08, Prop. 4.1].

######  Proposition 2.9.

Let $({\mathscr{A}},\mu)$ be a multiarrangement. Let
$X\in{\mathscr{A}}^{H_{0}}$, $m_{0}=\mu(H_{0})$, $k=|{\mathscr{A}}_{X}|$, and
$m_{1}=\max\\{\mu(H)\mid H\in{\mathscr{A}}_{X}\backslash\\{H_{0}\\}\\}$.

  * (i)

If $k=2$, then $\mu^{*}(X)=m_{1}$.

  * (ii)

If $|\mu_{X}|\leq 2k-1$ and $m_{0}>1$, then $\mu^{*}(X)=k-1$.

  * (iii)

If $\mu_{X}\equiv 2$, then $\mu^{*}(X)=k$.

We can now state the general Addition-Deletion Theorem for free
multiarrangements.

######  Theorem 2.10 ([ATW08, Thm. 0.8] Addition-Deletion Theorem for
Multiarrangements).

Suppose that $({\mathscr{A}},\mu)\neq\Phi_{\ell}$. Fix $H_{0}$ in
${\mathscr{A}}$ and let
$({\mathscr{A}},\mu),({\mathscr{A}}^{\prime},\mu^{\prime})$ and
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ be the triple with respect to
$H_{0}$. Then any two of the following statements imply the third:

  * (i)

$({\mathscr{A}},\mu)$ is free with
$\exp({\mathscr{A}},\mu)=(b_{1},\ldots,b_{\ell-1},b_{\ell})$;

  * (ii)

$({\mathscr{A}}^{\prime},\mu^{\prime})$ is free with
$\exp({\mathscr{A}}^{\prime},\mu^{\prime})=(b_{1},\ldots,b_{\ell-1},b_{\ell}-1)$;

  * (iii)

$({\mathscr{A}}^{\prime\prime},\mu^{*})$ is free with
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})=(b_{1},\ldots,b_{\ell-1})$.

######  Remark 2.11.

If we restrict to simple multiplicities, Theorem 2.10 recovers Terao’s
original Addition-Deletion Theorem from [T80a]. As in the simple case if
$({\mathscr{A}},\mu)$ and $({\mathscr{A}}^{\prime},\mu^{\prime})$ are free,
then all three statements of Theorem 2.10 hold (see [ATW08, Thm. 0.4]).

Analogous to the simple case, Theorem 2.10 motivates the notion of inductive
freeness.

######  Definition 2.12 ([ATW08, Def. 0.9]).

The class ${\mathcal{IFM}}$ of _inductively free_ multiarrangements is the
smallest class of multiarrangements subject to

  * (i)

$\Phi_{\ell}\in{\mathcal{IFM}}$ for each $\ell\geq 0$;

  * (ii)

for a multiarrangement $({\mathscr{A}},\mu)$, if there exists a hyperplane
$H_{0}\in{\mathscr{A}}$ such that both $({\mathscr{A}}^{\prime},\mu^{\prime})$
and $({\mathscr{A}}^{\prime\prime},\mu^{*})$ belong to ${\mathcal{IFM}}$, and
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})\subseteq\exp({\mathscr{A}}^{\prime},\mu^{\prime})$,
then $({\mathscr{A}},\mu)$ also belongs to ${\mathcal{IFM}}$.

Similarly, we also have the notion of recursive freeness for
multiarrangements.

######  Definition 2.13 ([HRS17, Def. 2.21]).

The class ${\mathcal{RFM}}$ of _recursively free_ multiarrangements is the
smallest class of multiarrangements subject to

  * (i)

$\Phi_{\ell}\in{\mathcal{RFM}}$ for each $\ell\geq 0$;

  * (ii)

for a multiarrangement $({\mathscr{A}},\mu)$, if there exists a hyperplane
$H_{0}\in{\mathscr{A}}$ such that both $({\mathscr{A}}^{\prime},\mu^{\prime})$
and $({\mathscr{A}}^{\prime\prime},\mu^{*})$ belong to ${\mathcal{RFM}}$, and
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})\subseteq\exp({\mathscr{A}}^{\prime},\mu^{\prime})$,
then $({\mathscr{A}},\mu)$ also belongs to ${\mathcal{RFM}}$.

  * (iii)

for a multiarrangement $({\mathscr{A}},\mu)$, if there exists a hyperplane
$H_{0}\in{\mathscr{A}}$ such that both $({\mathscr{A}},\mu)$ and
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ belong to ${\mathcal{RFM}}$, and
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})\subseteq\exp({\mathscr{A}},\mu)$,
then $({\mathscr{A}}^{\prime},\mu^{\prime})$ also belongs to
${\mathcal{RFM}}$.

Moreover, we consider the following weaker notion.

######  Definition 2.14 ([HR24, Def. 3.3]).

The multiarrangement $({\mathscr{A}},\mu)$ is said to be _additively free_ if
there is a free filtration of multiplicities $\mu_{i}$ on ${\mathscr{A}}$,

$\mu_{0}<\mu_{1}<\cdots<\mu_{n}=\mu,$

i.e., where each $({\mathscr{A}},\mu_{i})$ is free with $|\mu_{i}|=i$. Denote
this class by ${\mathcal{AFM}}$. In particular,
$({\mathscr{A}},\mu_{0})=\Phi_{\ell}$.

######  Remark 2.15.

As for simple arrangements, if $r({\mathscr{A}})\leq 2$, then
$({\mathscr{A}},\mu)$ is inductively free, [Z89, Cor. 7]. Also,
${\mathcal{IFM}}\subseteq{\mathcal{RFM}}$ and
${\mathcal{IFM}}\subseteq{\mathcal{AFM}}$ and all inclusions are strict.

By Remark 2.11, restricting to the subclasses of simple arrangements within
the above defined classes, we recover the classes of inductively free
${\mathcal{IF}}$ (cf. [OT92, Def. 4.53]), additively free ${\mathcal{AF}}$
(cf. [A19, Def. 1.6]), and recursively free ${\mathcal{RF}}$ (cf. [OT92, Def.
4.60]) simple arrangements.

In general, it is a hard problem to compute the exponents of a free
multiarrangement even in the rank $2$ case. We require the following result by
Wakamiko.

######  Theorem 2.16 ([W07, Thm. 1.5]).

Let ${\mathscr{A}}=\\{H_{1},H_{2},H_{3}\\}$ be a $2$-arrangement of three
lines, $\mu$ a multiplicity on ${\mathscr{A}}$ with
$\mu(H_{i})=k_{i},(i=1,2,3)$ and $\exp({\mathscr{A}},\mu)=(d_{1},d_{2})$.
Assume that $k_{3}\geq\max\\{k_{1},k_{2}\\}$ and let $k=k_{1}+k_{2}+k_{3}$.

  * (i)

If $k_{3}<k_{1}+k_{2}-1$, then

$|d_{1}-d_{2}|=\begin{cases}0&\text{if $k$ is even,}\\\ 1&\text{if $k$ is
odd.}\\\ \end{cases}$

  * (ii)

If $k_{3}\geq k_{1}+k_{2}-1$, then
$\exp({\mathscr{A}},\mu)=(k_{1}+k_{2},k_{3})$.

###  2.2. Connected subgraph arrangements

We summarize the important definitions and results from [CK24]. We start by
recalling the definition of a connected subgraph arrangement. In [CK24] the
authors introduced this concept for an arbitrary underlying field. Here we
restrict ourselves to arrangements over ${\mathbb{Q}}$.

######  Definition 2.17 ([CK24, Def. 1.1]).

Let $G=(N,E)$ be an undirected (simple) graph with vertex set
$N=[n]=\\{1,\dots,n\\}$ and edges $E$. We define the _connected subgraph
arrangement_ ${\mathscr{A}}_{G}$ in ${\mathbb{Q}}^{n}$ as

${\mathscr{A}}_{G}:=\\{H_{I}\mid\emptyset\neq I\subseteq N\text{ if
}G[I]\text{ is connected}\\},$

where $H_{I}$ is the hyperplane

$H_{I}=\ker\sum_{i\in I}x_{i}$

and $G[I]$ is the induced subgraph on the vertices $I\subseteq N$.

By this definition, all defining linear forms of a connected subgraph
arrangement are $0/1$-vectors. The following families of graphs play a central
role in the investigation of free simple connected subgraph arrangements. They
are equally crucial for our study.

######  Definition 2.18 ([CK24]).

  * •

The _path-graph_ $P_{n}$ on $n$ vertices.

$P_{n}$:$1$$2$$n-1$$n$

  * •

The _almost-path-graph_ $A_{n,k}$ on $n+1$ vertices, where $1\leq k\leq n$.
Draw a path-graph on $n$ vertices, add an additional vertex $n+1$, and connect
the vertices $k$ and $n+1$ by an edge.

$A_{n,k}$:$1$$2$$k$$n$$n+1$

  * •

The _cycle-graph_ $C_{n}$ on $n$ vertices. Draw $n$ vertices and connect
vertex $1$ and $2$, $2$ and $3$, $\dots$, $n$ and $1$.

$C_{n}$:$1$$2$$n-1$$n$

  * •

The _path-with-triangle-graph_ $\Delta_{n,k}$, where $1\leq k\leq n-1$. Draw a
path-graph on $n$ vertices, add an additional vertex $n+1$, and connect the
vertices $k$ and $n+1$ by an edge as well as the vertices $k+1$ and $n+1$.

$\Delta_{n,k}$:$1$$2$$k$$k+1$$n$$n+1$

In [Y09], M. Yoshinaga investigated the freeness question of multiplicities
for a certain class of arrangements. His setting turns out to be important for
our study as well.

######  Definition 2.19 ([Y09, Def. 2.1]).

Let ${\mathscr{A}}$ be a hyperplane arrangement.

  * (i)

We call ${\mathscr{A}}$ _locally $A_{2}$_, if $|{\mathscr{A}}_{X}|\leq 3$ for
all $X\in L({\mathscr{A}})_{2}$.

  * (ii)

Fix a reduced system of defining forms $\Phi\subseteq V^{*}$ for
${\mathscr{A}}$, i.e., ${\mathscr{A}}=\\{\ker(\alpha)\mid\alpha\in\Phi\\}$ and
$|\Phi|=|{\mathscr{A}}|$. Then $\Phi$ is called a _positive system_ for the
locally $A_{2}$ arrangement ${\mathscr{A}}$ provided for each rank $2$
localization with three elements
${\mathscr{A}}_{X}=\\{\ker(\alpha),\ker(\beta),\ker(\gamma)\\}$
($\alpha,\beta,\gamma\in\Phi$) one of $\alpha=\beta+\gamma$,
$\beta=\alpha+\gamma$, $\gamma=\alpha+\beta$ holds.

Connected subgraph arrangements are locally $A_{2}$ and have positive systems.

######  Lemma 2.20 ([CK24, Lem. 2.10]).

For pairwise distinct nonempty subsets $A_{1},A_{2},A_{3}\subseteq[n]$ and
$n\geq 1$ the following two conditions are equivalent:

  * (i)

$\operatorname{rank}(H_{A_{1}}\cap H_{A_{2}}\cap H_{A_{3}})=2$ and

  * (ii)

$A_{i_{1}}\dot{\cup}A_{i_{2}}=A_{i_{3}}$ for some choice of pairwise distinct
indices $1\leq i_{1},i_{2},i_{3}\leq 3$ where $\dot{\cup}$ denotes a disjoint
set union.

######  Definition 2.21.

Let $G=(N,E)$ be a Graph and $e=\\{i,j\\}\in E$ an edge. We denote by $G/e$
the graph with contracted edge $e$, i.e. the graph in which the vertices $i,j$
are identified to one vertex $k$, each edge to $i$ or $j$ becomes an edge to
$k$, and multiple edges and loops are removed. For $S\subseteq N$ the _induced
subgraph_ of $G$ on $S$ we denote by $G[S]=(S,E\cap\binom{S}{2})$.

We have the following link between these operations and localizations of
${\mathscr{A}}_{G}$.

######  Lemma 2.22 ([CK24, Lem. 6.2, Lem. 6.4]).

Let $G=(N,E)$ be a graph. Then

  * (i)

for $S\subseteq N$ we have ${\mathscr{A}}_{G[S]}=({\mathscr{A}}_{G})_{X}$ for
some $X\in L({\mathscr{A}}_{G})$, and

  * (ii)

for $e\in E$, ${\mathscr{A}}_{G/{e}}=({\mathscr{A}}_{G})_{X}$ for some $X\in
L({\mathscr{A}}_{G})$.

In particular, if $P$ is a local arrangement property satisfied by
${\mathscr{A}}_{G}$, then both ${\mathscr{A}}_{G[S]}$ and
${\mathscr{A}}_{G/e}$ satisfy $P$ as well.

Cuntz and Kühne classified all graphs $G$ such that ${\mathscr{A}}_{G}$ is a
free arrangement. It turns out that ${\mathscr{A}}_{G}$ is free if and only if
$G$ belongs to one of the four families of graphs introduced in Definition
2.18.

######  Theorem 2.23 ([CK24, Thm. 1.6]).

Let $G$ be a connected graph. The connected subgraph arrangement
${\mathscr{A}}_{G}$ is free if and only if $G$ is a path-graph, a cycle-graph,
an almost-path-graph, or a path-with-triangle-graph.

In their proof, Cuntz and Kühne show that every graph that is not part of one
of the four families from Definition 2.18 can get modified using iterations of
the constructions from Definition 2.21 until we end up with one of the graphs
depicted in Figure 1. Note that $G_{2}$ is more commonly known as $K_{4}$.
Since the connected subgraph arrangements stemming from those eight graphs are
not free and freeness is a local property by Theorem 2.1, Lemma 2.22 finishes
the proof.

$1$$2$$3$$4$$G_{1}$

$1$$2$$3$$4$$G_{2}$

$1$$2$$3$$4$$5$$G_{3}$

$1$$2$$3$$4$$5$$G_{4}$

$1$$2$$3$$4$$5$$G_{5}$

$1$$2$$3$$4$$5$$G_{6}$

$1$$2$$3$$4$$5$$6$$G_{7}$

$1$$2$$3$$4$$5$$6$$7$$G_{8}$

Figure 1. The graphs $G_{1}$ up to $G_{8}$

Our aim in this note is to look at these classes within the framework of
multiarrangements and to extend this classification.

###  2.3. Extensions of a multiarrangement

The following definitions and results are due to Yoshinaga [Y09].

######  Definition 2.24.

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $\mathbb{C}^{\ell}$.

  * (i)

$({\mathscr{A}},\mu)$ is _is extendable_ if it can be obtained as a Ziegler-
restriction of a free arrangement in $\mathbb{C}^{\ell+1}$.

  * (ii)

Define an extension in $\mathbb{C}^{\ell+1}$ with
$(x_{1},x_{2},\dots,x_{\ell},x_{\ell+1})$ as a coordinate system by

${\mathscr{E}}({\mathscr{A}},\mu)=\\{\ker(x_{\ell+1})\\}\cup\left\\{\ker(\alpha_{H}-kx_{\ell+1})\
\middle|\ k\in\mathbb{Z},-\frac{\mu(H)-1}{2}\leq
k\leq\frac{\mu(H)}{2},H\in{\mathscr{A}}\right\\}.$

Note that
$({\mathscr{E}}({\mathscr{A}},\mu)^{\ker(x_{\ell+1})},\kappa)=({\mathscr{A}},\mu)$.

One of the main results in [Y09] is the following.

######  Theorem 2.25 ([Y09, Thm. 2.5]).

Let ${\mathscr{A}}$ be a locally $A_{2}$ arrangement with a positive system in
$V=\mathbb{C}^{\ell}$. We fix a positive system $\Phi^{+}=\\{\alpha_{H}\mid
H\in{\mathscr{A}}\\}\subset V^{*}$ of defining equations. Let
$\mu:{\mathscr{A}}\to\mathbb{Z}_{\geq 0}$ be a multiplicity. Assume the
following condition:

(2.26) Let ${\mathscr{A}}_{X}=\\{H_{i},H_{j},H_{k}\\}$ be a codimension two
localization with $\alpha_{i}=\alpha_{j}+\alpha_{k}$. If $\mu(H_{i})$ is odd,
then at least one of $\mu(H_{j})$ or $\mu(H_{k})$ is odd.

Then $({\mathscr{A}},\mu)$ is free, if and only if it is extendable. Indeed,
${\mathscr{E}}({\mathscr{A}},\mu)$ is a free arrangement.

###  2.4. Locally and globally mixed products

In [ATW08], a very useful tool is introduced to determine whether a given
multiarrangement $({\mathscr{A}},\mu)$ fails to be free.

######  Definition 2.27.

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $\mathbb{C}^{\ell}$. Let
$0\leq k\leq\ell$.

  * (i)

If the localization $({\mathscr{A}}_{X},\mu_{X})$ is free with exponents
$\exp({\mathscr{A}}_{X},\mu_{X})=(d_{1}^{X},d_{2}^{X},\dots,d_{k}^{X})$ for
each $X\in L_{k}$, then define the _$k$ -th local mixed product_ by

$\operatorname{LMP}(k)=\sum_{X\in L_{k}}d_{1}^{X}d_{2}^{X}\cdots d_{k}^{X}.$

  * (ii)

If $({\mathscr{A}},\mu)$ is free with exponents
$\exp({\mathscr{A}},\mu)=(d_{1},\dots,d_{\ell})$, define the _$k$ -th global
mixed product_ by

$\operatorname{GMP}(k)=\sum d_{i_{1}}d_{i_{2}}\cdots d_{i_{k}},$

where the sum is taken over all $k$-tuples from $\exp({\mathscr{A}},\mu)$ with
$1\leq i_{1}<\dots<i_{k}\leq\ell$.

Thanks to Remark 2.15, $\operatorname{LMP}(2)$ is always defined. These
invariants are rather useful for proving the non-freeness of a given
multiarrangement due to the following result.

######  Theorem 2.28 ([ATW07, Cor. 4.6]).

If $({\mathscr{A}},\mu)$ is free with
$\exp({\mathscr{A}},\mu)=(d_{1},\dots,d_{\ell})$ then for every $0\leq
k\leq\ell$, we have

$\operatorname{GMP}(k)=\operatorname{LMP}(k).$

######  Definition 2.29.

For $1\leq i\leq\ell$, let $d_{i}\in\mathbb{N}$. We use the shorthand
$(d_{1},\ldots,d_{\ell})_{\leq}\in\mathbb{N}^{\ell}$ to mean $d_{1}\leq
d_{2}\leq\ldots\leq d_{\ell}$. Then for the latter define the _$k$ -th global
mixed product_ by

$\operatorname{GMP}(k):=\sum d_{i_{1}}d_{i_{2}}\cdots d_{i_{k}},$

where the sum is taken over all $k$-tuples from
$(d_{1},\ldots,d_{\ell})_{\leq}$ with $1\leq i_{1}<\dots<i_{k}\leq\ell$.

######  Remark 2.30.

Following [ATW07], given two ordered sequences of non-negative integers
$(d_{1},d_{2},\dots,d_{\ell})_{\leq}$ and
$(e_{1},e_{2},\dots,e_{\ell})_{\leq}$ as above satisfying $\sum d_{i}=\sum
e_{i}$, we call $(d_{1},d_{2},\dots,d_{\ell})_{\leq}$ _more balanced_ than
$(e_{1},e_{2},\dots,e_{\ell})_{\leq}$ if $d_{\ell}-d_{1}<e_{\ell}-e_{1}$. It
follows from [ATW07, p.836] that if $(d_{1},d_{2},\dots,d_{\ell})_{\leq}$ is
more balanced than $(e_{1},e_{2},\dots,e_{\ell})_{\leq}$, then
$\operatorname{GMP}(k)$ is larger for $(d_{1},d_{2},\dots,d_{\ell})_{\leq}$
than for $(e_{1},e_{2},\dots,e_{\ell})_{\leq}$.

######  Example 2.31.

With notation as in Definition 2.29 let $S_{1}=(4,4,4,4)_{\leq}$ and
$S_{2}=(2,4,5,5)_{\leq}$. Then $S_{1}$ is more balanced than $S_{2}$ since
$4-4<5-2$. Among all tuples $(d_{1},d_{2},d_{3},d_{4})_{\leq}$ with $\sum
d_{i}=16$, we see that $S_{1}$ is _most balanced_.

The following is immediate from Theorem 2.28 and Remark 2.30.

######  Lemma 2.32.

Let ${\mathscr{A}}$ be a simple arrangement in $\mathbb{K}^{\ell}$.

  * (i)

Let $\mu$ be a multiplicity on ${\mathscr{A}}$ and let
$(e_{1},e_{2},\dots,e_{\ell})_{\leq}$ be the unique most balanced sequence of
integers with sum equal to $|\mu|$. If $\operatorname{GMP}(2)$ for
$(e_{1},\ldots,e_{\ell})_{\leq}$ is smaller than $\operatorname{LMP}(2)$ for
$({\mathscr{A}},\mu)$, then $({\mathscr{A}},\mu)$ is not free.

  * (ii)

If ${\mathscr{A}}$ is free, then $1\in\exp({\mathscr{A}})$. Let
$(e_{2},\dots,e_{\ell})_{\leq}$ be the unique most balanced sequence of
integers with sum equal to $|{\mathscr{A}}|-1$. If $\operatorname{GMP}(2)$ for
$(1,e_{2},\ldots,e_{\ell})_{\leq}$ is smaller than $\operatorname{LMP}(2)$ for
${\mathscr{A}}$, then ${\mathscr{A}}$ is not free.

##  3\. Free connected subgraph multiarrangements

The goal of this section is a generalization of Theorem 2.23, where we give a
list of all graphs $G$ for which there exists a multiplicity
$\mu:{\mathscr{A}}\to\mathbb{Z}_{>0}$ such that $({\mathscr{A}},\mu)$ is free.
As mentioned before, Cuntz and Kühne derive Theorem 2.23 by showing that any
graph not listed in Definition 2.18 can get manipulated by graph contractions
from Definition 2.21 such that the resulting graph possesses an induced
subgraph that affords a non-free arrangement. Unexpectedly, it turns out that
free multiplicities do exist for ${\mathscr{A}}_{G_{1}}$ and
${\mathscr{A}}_{G_{2}}$. Thus, in order to follow a similar line of argument,
as the one above, we need to expand the list of graphs we have to consider
which are shown in Figure 2.

$1$$2$$3$$4$$5$$G_{9}$

$1$$2$$3$$4$$5$$G_{10}$

$1$$2$$3$$4$$5$$G_{11}$

$1$$2$$3$$4$$5$$G_{12}$

$1$$2$$3$$4$$5$$G_{13}$

$1$$2$$3$$4$$5$$G_{14}$

$1$$2$$3$$4$$5$$G_{15}$

$1$$2$$3$$4$$5$$G_{16}$

$1$$2$$3$$4$$5$$G_{17}$

$1$$2$$3$$4$$5$$G_{18}$

$1$$2$$3$$4$$5$$G_{19}$

$1$$2$$3$$4$$5$$G_{20}$

Figure 2. Graphs $G_{9}$ to $G_{20}$

We begin with the following observation.

######  Proposition 3.1.

Let $G$ be one of the graphs $G_{3}$ to $G_{20}$ listed in Figures 1 and 2,
then the corresponding connected subgraph arrangement ${\mathscr{A}}_{G}$ is
totally non-free.

###### Proof.

Let $G$ be one of the graphs $G_{3}$ to $G_{20}$ listed in Figures 1 and 2.
For each such $G$ we present an element $X\in L({\mathscr{A}}_{G})$ in Table 1
such that the corresponding localization $({\mathscr{A}}_{G})_{X}$ is a
generic arrangement with
$\operatorname{rank}({\mathscr{A}}_{G})_{X}<|({\mathscr{A}}_{G})_{X}|$. By
Theorems 2.1 and 2.6 we can conclude that ${\mathscr{A}}_{G}$ is totally non-
free.

$G$ | $X$ | $\operatorname{rank}X$ | $({\mathscr{A}}_{G})_{X}$
---|---|---|---
$G_{3}$ | $H_{1}\cap H_{123}\cap H_{145}$ | $3$ | $\\{H_{1},H_{123},H_{145},H_{12345}\\}$
$G_{4}$ | $H_{1}\cap H_{124}\cap H_{135}$ | $3$ | $\\{H_{1},H_{124},H_{135},H_{12345}\\}$
$G_{5}$ | $H_{1}\cap H_{124}\cap H_{135}$ | $3$ | $\\{H_{1},H_{124},H_{135},H_{12345}\\}$
$G_{6}$ | $H_{12}\cap H_{14}\cap H_{1235}$ | $3$ | $\\{H_{12},H_{14},H_{1235},H_{1345}\\}$
$G_{7}$ | $H_{23}\cap H_{136}\cap H_{1235}\cap H_{1245}$ | $4$ | $\\{H_{23},H_{136},H_{1235},H_{1245},H_{12346}\\}$
$G_{8}$ | $H_{1}\cap H_{146}\cap H_{1245}\cap H_{12367}\cap H_{12467}$ | $5$ | $\\{H_{1},H_{146},H_{1245},H_{12367},H_{12467},H_{123456}\\}$
$G_{9}$ | $H_{1}\cap H_{124}\cap H_{135}$ | $3$ | $\\{H_{1},H_{124},H_{135},H_{12345}\\}$
$G_{10}$ | $H_{12}\cap H_{23}\cap H_{1245}$ | $3$ | $\\{H_{12},H_{23},H_{1245},H_{2345}\\}$
$G_{11}$ | $H_{13}\cap H_{23}\cap H_{1345}$ | $3$ | $\\{H_{13},H_{23},H_{1345},H_{2345}\\}$
$G_{12}$ | $H_{23}\cap H_{34}\cap H_{1235}$ | $3$ | $\\{H_{23},H_{34},H_{1235},H_{1345}\\}$
$G_{13}$ | $H_{15}\cap H_{35}\cap H_{1245}$ | $3$ | $\\{H_{15},H_{35},H_{1245},H_{2345}\\}$
$G_{14}$ | $H_{15}\cap H_{35}\cap H_{1245}$ | $3$ | $\\{H_{15},H_{35},H_{1245},H_{2345}\\}$
$G_{15}$ | $H_{125}\cap H_{135}\cap H_{245}$ | $3$ | $\\{H_{125},H_{135},H_{245},H_{345}\\}$
$G_{16}$ | $H_{15}\cap H_{35}\cap H_{1245}$ | $3$ | $\\{H_{15},H_{35},H_{1245},H_{2345}\\}$
$G_{17}$ | $H_{24}\cap H_{25}\cap H_{1234}$ | $3$ | $\\{H_{24},H_{25},H_{1234},H_{1235}\\}$
$G_{18}$ | $H_{13}\cap H_{14}\cap H_{1235}$ | $3$ | $\\{H_{13},H_{14},H_{1235},H_{1245}\\}$
$G_{19}$ | $H_{125}\cap H_{135}\cap H_{245}$ | $3$ | $\\{H_{125},H_{135},H_{245},H_{345}\\}$
$G_{20}$ | $H_{15}\cap H_{35}\cap H_{1245}$ | $3$ | $\\{H_{15},H_{35},H_{1245},H_{2345}\\}$
Table 1. Elements $X\in L({\mathscr{A}}_{G_{i}})$ such that
$({\mathscr{A}}_{G_{i}})_{X}$ is generic

∎

######  Corollary 3.2.

Let $G$ be a graph that has $G_{3}$ as a subgraph. Then ${\mathscr{A}}_{G}$ is
totally non-free.

###### Proof.

Let $v_{1},v_{2},v_{3},v_{4},v_{5}$ be the vertices of $G$ included in the
subgraph forming $G_{3}$. The induced subgraph of $G$ on
$\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ is isomorphic to $G_{3},$ $G_{4},$
$G_{5},$ $G_{9},$ $G_{11},$ $G_{13},$ $G_{14},$ $G_{15},$ $G_{16},$ $G_{18}$
or $G_{19}$. Thus, by Proposition 3.1, the arrangement of this induced
subgraph is totally non-free. It then follows from Lemma 2.22 and Theorem 2.1
that ${\mathscr{A}}_{G}$ is totally non-free as well. ∎

We are now able to address the main result of this section.

######  Theorem 3.3.

Let $G$ be a connected graph on at least $5$ vertices. Then there exists a
multiplicity $\mu\geq\mathbbm{1}$ such that the connected subgraph
multiarrangement $({\mathscr{A}}_{G},\mu)$ is free if and only if $G$ is a
path-graph, a cycle-graph, an almost-path-graph, or a path-with-triangle-
graph. In particular, all other connected graphs with at least $5$ vertices
yield totally non-free connected subgraph arrangements.

###### Proof.

By Theorem 2.23, $({\mathscr{A}}_{G},\mathbbm{1})$ is free for any graph in
the list above.

Conversely, assume that we have a graph $G=(E,V)$ and a multiplicity
$\mu\geq\mathbbm{1}$ on ${\mathscr{A}}_{G}$ such that
$({\mathscr{A}}_{G},\mu)$ is free.

1\. Assume a vertex of $G$ has at least $4$ neighbors.

Then $G_{3}$ is a subgraph of $G$ and ${\mathscr{A}}_{G}$ is totally non-free,
by Corollary 3.2. This implies that all the vertices of $G$ have degree at
most $3$.

2\. Assume that $G$ has at least $2$ cycles of length at least $3$ that share
an edge. Since we assume that $|V|\geq 5$, the graph cannot be equal to
$G_{1}$ or $G_{2}$. Let $C_{1},C_{2}$ be cycles of $G$ with a shared edge
$(v_{1},v_{2})$, choose $w_{1}\in C_{1},w_{2}\in C_{2}$ such that
$w_{1}\not\in C_{2},w_{2}\not\in C_{1}$. Assume first that both cycles have
length $3$. In this case the induced graph on $\\{v_{1},v_{2},w_{1},w_{2}\\}$
is of type $G_{1}$ or $G_{2}$. Since $|V|\geq 5$ and $G$ is connected there
exists a vertex $v$ that is a neighbor of at least one element of
$\\{v_{1},v_{2},w_{1},w_{2}\\}$, while not belonging to either $C_{1}$ or
$C_{2}$. Taking the induced subgraph on $\\{v_{1},v_{2},w_{1},w_{2},v\\}$
yields one of $G_{9}$ up to $G_{19}$. So ${\mathscr{A}}_{G}$ is again totally
non-free, thanks to Proposition 3.1. Assume that at least $C_{1}$ is of length
greater than $3$. Choose $v_{1},v_{2},w_{1},w_{2}$ as in the first part above
and an additional $w_{3}\in C_{1}$ and take the induced subgraph on
$\\{v_{1},v_{2},w_{1},w_{2},w_{3}\\}$. Now contract each edge at least one
vertex of which is not in $\\{v_{1},v_{2},w_{1},w_{2},w_{3}\\}$. We now arrive
at one of $G_{11},G_{12},G_{14},G_{15},G_{16},G_{17},G_{19},G_{20}$. So
${\mathscr{A}}_{G}$ is totally non-free, again by Proposition 3.1.

3\. Assume that $G$ has at least 2 cycles of length at least $3$ without any
shared edge. In this case we contract edges until we end up with $G_{3}$ as
subgraph and use Corollary 3.2.

4\. Assume $G$ has exactly one cycle $C$ of length at least $4$. Note that we
cannot have two vertices $v_{1},v_{2}\in C$ being connected by an edge $e$ not
belonging to $C$, since otherwise we could decompose $C$ into two cycles
$C_{1},C_{2}$ of length at least 3 with shared edge $e$, which was already
excluded in case 2. If $G$ is not a cycle graph, then we can choose a vertex
$v$ such that it is the neighbor of a vertex in $C$. After contracting edges
of $C$ until we arrive at $C_{4}$ and taking the subgraph induced by the
reduced cycle and $v$, we get one of $G_{6},G_{12},G_{15},G_{17},G_{20}$. So
${\mathscr{A}}_{G}$ it is totally non-free, again thanks to Proposition 3.1.

5\. Assume $G$ has exactly one cycle of length $3$. Let $v_{1},v_{2},v_{3}$ be
the unique cycle in $G$. Assume that all vertices $v_{1},v_{2},v_{3}$ are of
degree $3$. Taking the subgraph induced by $\\{v_{1},v_{2},v_{3}\\}$ together
with their respective third neighbors yields the graph $G_{7}$, since all
neighbors are distinct (($v_{1},v_{2},v_{3})$ is the only cycle in $G$). Using
Proposition 3.1, we see that ${\mathscr{A}}_{G_{7}}$ is totally non-free, so
we can assume that the vertex $v_{3}$ has degree $2$. If $G$ had a vertex $v$
of degree $3$ different from $v_{1}$ and $v_{2}$, then contracting all edges
between $v$ and $v_{1}$ (or $v$ and $v_{2}$, we choose whatever vertex can be
reached first by following the unique path) results in a graph with a vertex
of degree at least $4$. This is a contradiction as seen in case 1. Therefore,
all other vertices are of degree at most $2$ and as $v_{1},v_{2},v_{3}$
constitute its only cycle, $G$ must be a path-with-triangle-graph.

6\. Assume that $G$ is a tree. If there are two vertices of degree $3$, then
we contract edges until it becomes a vertex of at least degree $4$. This
connected subgraph arrangement is totally non-free, as seen in case 1. So
there is at most one vertex of degree $3$. If $G$ it is not a Path-Graph or an
Almost-Path-Graph, then it has $G_{8}$ as an induced subgraph and the
arrangement ${\mathscr{A}}_{G}$ is totally non-free, due to Proposition 3.1. ∎

The bound in Theorem 3.3 on the number of vertices is necessary since there
exist free multiplicities for ${\mathscr{A}}_{G_{1}}$ and
${\mathscr{A}}_{G_{2}}$. We present two of them in the following result. We
emphasize that the presented multiplicity on ${\mathscr{A}}_{G_{2}}$ is
recursively free but fails to be additively free. The given multiplicity on
${\mathscr{A}}_{G_{1}}$ in turn is inductively free.

######  Proposition 3.4.

Let ${\mathscr{A}}\in\\{{\mathscr{A}}_{G_{1}},{\mathscr{A}}_{G_{2}}\\}$. There
exists a multiplicity $\mu\neq\mathbbm{1}$ such that $({\mathscr{A}},\mu)$ is
free. In particular, there exists a multiplicity $\mu$ such that
$({\mathscr{A}}_{G_{2}},\mu)$ is recursively free but not additively free.

###### Proof.

We fix the following order on the hyperplanes of ${\mathscr{A}}_{G_{2}}$:

$H_{1},H_{2},H_{3},H_{4},H_{1234},H_{34},H_{23},H_{234},H_{14},H_{13},H_{134},H_{12},H_{124},H_{123},H_{24},$

and define the multiplicity $\mu_{2}=(2,2,2,2,2,1,1,1,1,1,1,1,1,1,1)$. Then
the multiarrangement $({\mathscr{A}}_{G_{2}},\mu_{2})$ is recursively free but
not additively free. It follows from Remark 2.11 that
$({\mathscr{A}}_{G_{2}},\mu_{2})$ fails to be additively free since we have
$|(({\mathscr{A}}_{G_{2}})^{H},(\mu_{2})^{*})|=14$ for an arbitrary
$H\in{\mathscr{A}}_{G_{2}}$.

For ${\mathscr{A}}_{G_{1}}$ we drop the last hyperplane $H_{24}$ from the list
above, but fix the same order on the rest of the hyperplanes and define
$\mu_{1}=(2,2,2,1,2,1,1,1,2,1,1,1,1,1)$. We claim that
$({\mathscr{A}}_{G_{1}},\mu_{1})$ is inductively free.

The facts that $({\mathscr{A}}_{G_{1}},\mu_{1})$ is inductively free and
$({\mathscr{A}}_{G_{2}},\mu_{2})$ is recursively free follow from the
corresponding induction tables, Table 2 and Table 4, respectively. ∎

######  Remark 3.5.

We note that ${\mathscr{A}}_{G_{2}}$ is the famous counterexample by Edelmann
and Reiner given in [ER93] to Orlik’s conjecture. In this context
${\mathscr{A}}_{G_{2}}$ appears as a non-free restriction of a free
arrangement ${\mathscr{B}}$. As a multiarrangement
$({\mathscr{A}}_{G_{2}},\mu_{2})=({\mathscr{B}}^{\prime\prime},\kappa)$ is the
Ziegler restriction of ${\mathscr{B}}$ (recall Theorem 2.2). This gives an
alternative proof of the freeness of $({\mathscr{A}}_{G_{2}},\mu_{2})$. Note
that ${\mathscr{A}}_{G_{2}}={\mathscr{A}}_{G_{1}}\cup\\{H_{24}\\}$. It turns
out that starting with the inductively free multiarrangement
$({\mathscr{A}}_{G_{1}},\mu_{1})$ and adding $H_{24}$ with multiplicity $1$
gives an inductively free multiplicity on ${\mathscr{A}}_{G_{2}}$.

Since every connected graph on four vertices is either
$G_{1},G_{2},P_{4},C_{4},A_{3,2}$ or $\Delta_{3,1}$ we can combine Theorem 3.3
and Proposition 3.4 to derive the following result.

######  Corollary 3.6.

Let $G$ be a connected graph. There exists a multiplicity $\mu$ such that the
connected subgraph multiarrangement $({\mathscr{A}}_{G},\mu)$ is free if and
only if $G$ is $G_{1}$, $G_{2}$, a path-graph, a cycle-graph, an almost-path-
graph, or a path-with-triangle-graph.

##  4\. Constant multiplicities on connected subgraph arrangements

###  4.1. Investigating $G_{1}$ and $G_{2}$

The proof in [CK24, §6.2] of the non-freeness of the connected subgraph
arrangements stemming from $G_{1}$ up to $G_{8}$ depends on tedious computer
calculations. In this section, we provide a non-computational, conceptional
proof giving a complete list of free constant multiplicities on the family of
graphs from Theorem 2.23, as well as giving a non-computational proof for the
non-freeness of ${\mathscr{A}}_{G_{1}}$ and ${\mathscr{A}}_{G_{2}}$ in the
process. Combined with Proposition 3.1 we obtain a proof for the non-freeness
of the simple connected subgraph arrangements stemming from $G_{1}$ up to
$G_{20}$ free of any machine calculations. We use the following method for
calculating $\operatorname{LMP}(2)$ of the connected subgraph arrangements.
Then we utilize Lemma 2.32.

######  Remark 4.1.

Algorithm to calculate $\operatorname{LMP}(2)$ for a simple
${\mathscr{A}}_{G}$:

  1. (1)

Determine the number of all types of connected induced subgraphs of $G$ with
at least $2$ vertices (i.e. path-graph, almost-path-graph, etc.).

  2. (2)

For every subgraph $G^{\prime}$ from step $1$ we determine the number of
triples $(G^{\prime},G[I],G[J])$, where $G[I],G[J]$ are connected induced
subgraphs of $G^{\prime}$ with $V(G[I])\cup V(G[J])=V(G^{\prime})$. Adding up
the number of all such triples for all connected induced subgraphs
$G^{\prime}$ from (1) provides an invariant $N_{1}$. Using Lemma 2.20 we see
that for a fixed graph $G^{\prime}$ this is just the number of rank $2$
localizations with $V(G^{\prime})=A_{i_{3}}$, with the notation as in the
lemma.

  3. (3)

Calculate $N_{2}:=\binom{|{\mathscr{A}}_{G}|}{2}$. If all rank $2$
localizations were to consist of exactly two elements, then $N_{2}$ would be
the number of all rank $2$ localizations.

  4. (4)

We have $\operatorname{LMP}(2)=N_{2}-N_{1}$. To see this, a rank $2$
localization with two elements is free with exponents $(1,1)$, while a rank
$2$ localization with three elements is free with exponents $(1,2)$. So if
instead of having three rank $2$ localizations with two elements we have one
rank $2$ localization with three elements, then $\operatorname{LMP}(2)$ gets
decreased by $1$ (since we assume for $N_{2}$ that all localizations contain
exactly two elements). Since $N_{1}$ is the number of rank $2$ localizations
with three elements, $N_{2}-N_{1}$ coincides with $\operatorname{LMP}(2)$.

We demonstrate how this algorithm works in the cases where
$G\in\\{G_{1},G_{2}\\}$.

######  Example 4.2.

We begin with $G=G_{1}$.

  * •

$5\times P_{2},2\times P_{3},2\times C_{3},1\times G_{1}$

  * •

$G^{\prime}=P_{2}:1\times(P_{2},P_{1},P_{1})$
$G^{\prime}=P_{3}:2\times(P_{3},P_{2},P_{1})$
$G^{\prime}=C_{3}:3\times(C_{3},P_{2},P_{1})$
$G^{\prime}=G_{1}:2\times(G_{1},C_{3},P_{1}),2\times(G_{1},P_{3},P_{1}),2\times(G_{1},P_{2},P_{2})$
$N_{1}=5\cdot 1+2\cdot 2+2\cdot 3+1\cdot 6=21$

  * •

$N_{2}=\binom{14}{2}=91$

  * •

$\operatorname{LMP}(2)=91-21=70$

For $G=G_{2}$ we get the following result.

  * •

$6\times P_{2},4\times C_{3},1\times G_{1}$

  * •

$G^{\prime}=P_{2}:1\times(P_{2},P_{1},P_{1})$
$G^{\prime}=C_{3}:3\times(C_{3},P_{2},P_{1})$
$G^{\prime}=G_{2}:4\times(G_{1},C_{3},P_{1}),3\times(G_{1},P_{2},P_{2})$
$N_{1}=6\cdot 1+4\cdot 3+1\cdot 7=25$

  * •

$N_{2}=\binom{15}{2}=105$

  * •

$\operatorname{LMP}(2)=105-25=80$

In view of Lemma 2.32, we derive the non-freeness of ${\mathscr{A}}_{G_{1}}$
and ${\mathscr{A}}_{G_{2}}$ without any machine calculations as follows.

######  Proposition 4.3.

Let $G\in\\{G_{1},G_{2}\\}$, then ${\mathscr{A}}_{G}$ is not free.

###### Proof.

Since $|{\mathscr{A}}_{G_{1}}|=14$, a tuple of potential exponents for
${\mathscr{A}}_{G_{1}}$ has to be $(1,4,4,5)_{\leq}$, or less balanced.
Calculating $\operatorname{GMP}(2)$ for this sequence gives
$\operatorname{GMP}(2)=69$. Since exponents that are less balanced afford a
smaller $\operatorname{GMP}(2)$ and since $\operatorname{LMP}(2)=70$, we have
$\operatorname{LMP}(2)\neq\operatorname{GMP}(2)$ and so
${\mathscr{A}}_{G_{1}}$ is not free, by Theorem 2.28.

Since $|{\mathscr{A}}_{G_{1}}|=15$, choose potential balanced exponents
$(1,4,5,5)_{\leq}$ for ${\mathscr{A}}_{G_{2}}$ to maximize
$\operatorname{GMP}(2)$. Using this tuple, we get $\operatorname{GMP}(2)=79$.
Since exponents that are less balanced yield a smaller value for
$\operatorname{GMP}(2)$ and since $\operatorname{LMP}(2)=80$, we have
$\operatorname{LMP}(2)\neq\operatorname{GMP}(2)$ and so
${\mathscr{A}}_{G_{2}}$ can’t be free, thanks to Theorem 2.28. ∎

######  Remark 4.4.

Propositions 3.1 and 4.3 combined give a non-computational, conceptional proof
for the non-freeness of the simple arrangements ${\mathscr{A}}_{G_{i}}$, for
$i\in\\{1,2,\dots,20\\}$.

Next we modify the algorithm from Remark 4.1 to calculate
$\operatorname{LMP}(2)$ when ${\mathscr{A}}_{G}$ gets equipped with a constant
multiplicity $\mu=k\mathbbm{1}$ for $k\neq 1$.

######  Remark 4.5.

Algorithm to calculate $\operatorname{LMP}(2)$ for a constant multiplicity
$\mu\equiv k$:

  1. (1)

Determine the number of all types of connected induced subgraphs of $G$ with
at least $2$ vertices (i.e. path-graph, almost-path-graph, etc.).

  2. (2)

For every subgraph $G^{\prime}$ from step (1) we determine the number of
triples $(G^{\prime},G[I],G[J])$, where $G[I],G[J]$ are connected induced
subgraphs of $G^{\prime}$ with $V(G[I])\cup V(G[J])=V(G^{\prime})$. Let
$N_{1}$ be the sum of the number of all such triples for all connected induced
subgraphs $G^{\prime}$ from part (1).

  3. (3)

Calculate $N_{2}:=\binom{|{\mathscr{A}}_{G}|}{2}$.

  4. (4)

Since $N_{1}$ is the number of rank $2$ localizations with three elements, the
number of rank $2$ localizations with two elements is equal to
$N_{3}:=N_{2}-3N_{1}$, because if there is one rank $2$ localization with
three elements, then the hyperplanes of this localization could have
potentially formed three rank $2$ localizations with two elements.

  5. (5)

Using Theorem 2.16 we derive the exponents of all rank $2$ localizations with
three elements (which depends on the parity of $k$), while localizations of
rank $2$ with two elements are free with exponents $(k,k)$. We get

$\operatorname{LMP}(2)=\begin{cases}N_{1}\cdot\left(\frac{3k}{2}\right)^{2}+N_{3}k^{2},&\text{
(if }k\text{ is even),}\\\
N_{1}\cdot\left(\frac{3k-1}{2}\cdot\frac{3k+1}{2}\right)+N_{3}k^{2},&\text{
(if }k\text{ is odd).}\end{cases}$

We now use Lemma 2.32 to extend Proposition 4.3 to arbitrary constant
multiplicities.

######  Proposition 4.6.

For $G\in\\{G_{1},G_{2}\\}$ and $\mu\equiv k\geq 1$ a constant multiplicity on
${\mathscr{A}}_{G}$, $({\mathscr{A}}_{G},\mu)$ is not free.

###### Proof.

The case $k=1$ is Proposition 4.3. So let $k\geq 2$. From the calculation in
the example we know that: $G_{1}:$ $N_{1}=21,N_{2}=91$ and $G_{2}:$
$N_{1}=25,N_{2}=105$, which gives $N_{3}=28$ for $G_{1}$ and $N_{3}=30$ for
$G_{2}$. So for $G_{1}$ we get

$\operatorname{LMP}(2)=\begin{cases}21\cdot\left(\frac{3k}{2}\right)^{2}+28k^{2}=75.25k^{2},&\text{if
$k$ is even.}\\\
21\cdot\left(\frac{3k-1}{2}\cdot\frac{3k+1}{2}\right)+28k^{2}=75.25k^{2}-5.25,&\text{if
$k$ is odd,}\end{cases}$

and for $G_{2}$ we get

$\operatorname{LMP}(2)=\begin{cases}25\cdot\left(\frac{3k}{2}\right)^{2}+30k^{2}=86.25k^{2},&\text{if
$k$ is even.}\\\
25\cdot\left(\frac{3k-1}{2}\cdot\frac{3k+1}{2}\right)+30k^{2})=86.25k^{2}-6.25,&\text{if
$k$ is odd.}\end{cases}$

If we assume perfectly balanced exponents for ${\mathscr{A}}_{G_{1}}$
(assuming $14k\equiv 0\mod 4$) we get the set of exponents
$(\frac{14k}{4},\frac{14k}{4},\frac{14k}{4},\frac{14k}{4})_{\leq}$ which
results in $\operatorname{GMP}(2)=73.5k^{2}$.
If we assume perfectly balanced exponents for ${\mathscr{A}}_{G_{2}}$
(assuming $15k\equiv 0\mod 4$) we get the set of exponents
$(\frac{15k}{4},\frac{15k}{4},\frac{15k}{4},\frac{15k}{4})_{\leq}$ which
results in $\operatorname{GMP}(2)=84.375k^{2}$.
Since $k\geq 2$, in any case, we have
$\operatorname{LMP}(2)>\operatorname{GMP}(2)$ and thus
$({\mathscr{A}}_{G_{2}},\mu)$ can’t be free, by Theorem 2.28. ∎

###  4.2. Constant multiplicities on a free ${\mathscr{A}}_{G}$

In this subsection, we inspect free constant multiplicities on a free
${\mathscr{A}}_{G}$. We give a complete list when $({\mathscr{A}}_{G},\mu)$ is
free for $G\in\\{P_{n},A_{n,k},\Delta_{n,k},C_{n}\\}$ and $\mu\equiv k\geq 1$.

First we consider the case $G=P_{n}$. Terao showed in [T02] that every
constant multiplicity on a Coxeter arrangement is free. Since
${\mathscr{A}}_{P_{n}}$ is the Coxeter arrangement of type $A_{n}$, we derive
the freeness of $({\mathscr{A}}_{P_{n}},\mu)$ for $\mu\equiv k\geq 1$.

Now suppose that $G\neq P_{n}$. We start with $G=C_{3}$ and follow the idea
described in Lemma 2.32 during every proof.

######  Proposition 4.7.

Let $G=C_{3}$ and $\mu\equiv k\geq 1$ is a constant multiplicity on
${\mathscr{A}}_{G}$. Then $({\mathscr{A}}_{G},\mu)$ is free if and only if
$k\in\\{1,3\\}$.

###### Proof.

For the forward implication, using the algorithm above, we get $N_{1}=1\cdot
3+3\cdot 1=6,N_{2}=\binom{7}{2}=21,N_{3}=21-3\cdot 6=3$. This gives

$\operatorname{LMP}(2)=\begin{cases}6\cdot\left(\frac{3k}{2}\right)^{2}+3k^{2}=16.5k^{2}&\text{
(if }k\text{ is even), or }\\\
6\cdot\left(\frac{3k-1}{2}\cdot\frac{3k+1}{2}\right)+3k^{2}=16.5k^{2}-1.5&\text{
(if }k\text{ is odd).}\end{cases}$

Assuming balanced exponents we get

$\operatorname{GMP}(2)=\begin{cases}16\frac{1}{3}k^{2},&\text{if }7k\equiv
0\mod 3\text{ or}\\\ 16\frac{1}{3}k^{2}-\frac{1}{3},&\text{else.}\end{cases}$

For $k\geq 4$ we get that $\operatorname{LMP}(2)>\operatorname{GMP}(2)$. For
$k=3$ we get equality and for $k=2$ we get $\operatorname{LMP}(2)=66$ and
depending on whether the assumed exponents are
$(3,5,6)_{\leq},(4,4,6)_{\leq}$, or $(4,5,5)_{\leq}$, a value of $63,64$, or
$65$, respectively for $\operatorname{GMP}(2)$. So if $k\not\in\\{1,3\\}$ we
have $\operatorname{LMP}(2)>\operatorname{GMP}(2)$ and thanks to Theorem 2.28,
we deduce non-freeness.

Next we address the reverse implication. The freeness of
$({\mathscr{A}}_{G},\mathbbm{1})$ follows from Theorem 2.23. In Table 3, we
present an induction table for $({\mathscr{A}}_{G},3\mathbbm{1})$ showing that
the latter is actually inductively free. ∎

######  Proposition 4.8.

Let $G\in\\{C_{n},\Delta_{n,k},A_{n,k}\\}$. Suppose
$\operatorname{rank}{\mathscr{A}}_{G}\geq 4$ and $\mu\equiv k\geq 1$ a
constant multiplicity on ${\mathscr{A}}_{G}$. Then $({\mathscr{A}}_{G},\mu)$
is free if and only if $k=1$.

###### Proof.

First let $G\in\\{C_{n},\Delta_{n,k}\\}$. Since $G$ either has $C_{3}$ as an
induced subgraph or becomes $C_{3}$ after contracting edges, we can use Lemma
2.22 and Proposition 4.7 to focus on the cases where $k\in\\{1,3\\}$. If $k=1$
we know that $({\mathscr{A}}_{G},\mu)$ is free. So let $\mu\equiv 3$ and
$G=C_{4}$.
Calculating $\operatorname{LMP}(2)$ we get $N_{1}=1\cdot(4+2)+4\cdot 2+4\cdot
1=18,N_{2}=\binom{13}{2}=78,N_{3}=78-3\cdot 18=24$ and so

$\operatorname{LMP}(2)=18\cdot\left(\frac{3\cdot 3-1}{2}\cdot\frac{3\cdot
3+1}{2}\right)+24\cdot 9=576.$

Since ${\mathscr{A}}_{C_{4}}$ consists of $13$ hyperplanes, assuming balanced
exponents $(9,10,10,10)_{\leq}$, we get $\operatorname{GMP}(2)=570$. So
$\operatorname{LMP}(2)>\operatorname{GMP}(2)$. Using Theorem 2.28, we see that
${\mathscr{A}}_{G}$ fails to be free. For a given $C_{n}$ (for $n>4$) we can
contract edges until we end up with $C_{4}$. The result follows.

Let $G=\Delta_{3,1}$ and note that every $\Delta_{n,k}\neq C_{3}$ can get
transformed into $G$ by contracting edges. If $\mu\equiv 3$ we get
$\operatorname{LMP}(2)=489$ and assuming exponents $(9,9,9,9)_{\leq}$ we get
$\operatorname{GMP}(2)=486$. So only the simple multiplicity can be free.

Let $G=A_{3,2}$. Let $\mu\equiv k$, then we get

$\operatorname{GMP}(2)=\begin{cases}45.375k^{2},&\text{if }11k\equiv 0\mod
4,\\\ 45.375k^{2}-\frac{1}{2},&\text{if }11k\equiv 2\mod 4,\\\
45.375k^{2}-\frac{3}{8},&\text{else.}\\\ \end{cases}$

Moreover, $\operatorname{LMP}(2)=46k^{2}$ (if $k$ is even) and $46k^{2}-3$ (if
$k$ is odd). Once more we see that
$\operatorname{LMP}(2)\neq\operatorname{GMP}(2)$ and so, by Theorem 2.28,
$({\mathscr{A}},\mu)$ fails to be free. Since every $A_{n,k}$ with $n\geq 4$
can be transformed into $A_{3,2}$ after contracting edges, the result follows
from Lemma 2.22. ∎

Combining the last two results we get the following.

######  Corollary 4.9.

Let $G\in\\{P_{n},A_{n,k},\Delta_{n,k},C_{n}\\}$ , $\mu\equiv k,(k>1)$. Then
$({\mathscr{A}}_{G},\mu)$ is free if and only if $G=P_{n}$ or $G=C_{3}$ and
$k=3$.

##  5\. Classifying free multiplicities on connected subgraph arrangements

Since connected subgraph arrangements are locally $A_{2}$ and admit a positive
system of defining equations, the extension techniques introduced by Yoshinaga
in [Y09] are well suited to attack the problem of classifying free
multiplicities on connected subgraph arrangements and their subarrangements.

We define a subarrangement of ${\mathscr{A}}_{C_{3}}$ for which we give a
complete list of free multiplicities. This classification was achieved
previously by DiPasquale and Wakefield in [DW18], where this arrangement
appeared in a different context, using homological algebra methods.

######  Definition 5.1.

We define the _deleted $C_{3}$ arrangement_ $\mathscr{D}$ as
$\mathscr{D}:={\mathscr{A}}_{C_{3}}\backslash\\{H_{123}\\}$. The defining
polynomial of $\mathscr{D}$ is given by

(5.2) $Q(\mathscr{D})=x_{1}x_{2}x_{3}(x_{1}+x_{2})(x_{1}+x_{3})(x_{2}+x_{3}).$

To prove the freeness of a certain family of multiplicities on $\mathscr{D}$
and to identify some free multiplicities on ${\mathscr{A}}_{C_{3}}$ at the end
of this section, we require the following lemma.

######  Lemma 5.3.

Let $({\mathscr{A}},\mu)=({\mathscr{A}}(B_{2}),\mu)$ be the multiarrangement

$Q({\mathscr{A}}(B_{2}),\mu)=x_{1}^{a}x_{2}^{b}(x_{1}+x_{2})^{c}(2x_{1}+x_{2})^{d}$

with multiplicity $\mu=(a,b,c,d)$.

  1. (1)

If $\mu=(2k,1,2k,1)$, then $({\mathscr{A}},\mu)$ is free with exponents
$(2k+1,2k+1)$.

  2. (2)

If $\mu=(2k+1,1,2k+1,1)$, then $({\mathscr{A}},\mu)$ is free with exponents
$(2k+1,2k+3)$.

  3. (3)

If $\mu=(k,1,k,2k-4)$ or $\mu=(k,2k-4,k,1)$, then $({\mathscr{A}},\mu)$ is
free with exponents $(2k-2,2k-1)$.

  4. (4)

If $\mu=(k+1,1,k+1,2k-4)$ or $\mu=(k+1,2k-4,k+1,1)$, then
$({\mathscr{A}},\mu)$ is free with exponents $(2k-1,2k)$.

###### Proof.

(1) and (2): We give an explicit basis for the module of derivations
$D({\mathscr{A}},\mu)$ in both cases. We utilize the linear isomorphism from
${\mathbb{C}}^{2}$ to itself, defined by $x_{1}\mapsto x_{2}$ and
$x_{2}\mapsto x_{1}-x_{2}$, sending
$\\{x_{1},x_{2},x_{1}+x_{2},2x_{1}+x_{2}\\}$ to
$\\{x_{1},x_{2},x_{1}+x_{2},x_{1}-x_{2}\\}$. Thus it is sufficient to
calculate a basis for $D(\mathscr{B},\nu)$, where
$Q(\mathscr{B},\nu)=x_{1}^{c}x_{2}^{a}(x_{1}+x_{2})^{d}(x_{1}-x_{2})^{b}$.
For (1), we have
$Q(\mathscr{B},\nu)=x_{1}^{2k}x_{2}^{2k}(x_{1}+x_{2})^{1}(x_{1}-x_{2})^{1}$
and a basis for $D(\mathscr{B},\nu)$ is given by

$\theta_{1}=x_{1}^{2k}x_{2}dx_{1}+x_{1}x_{2}^{2k}dx_{2}\quad\text{ and
}\quad\theta_{2}=x_{1}^{2k+1}dx_{1}+x_{2}^{2k+1}dx_{2}.$

For (2), we have
$Q(\mathscr{B},\nu)=x_{1}^{2k+1}x_{2}^{2k+1}(x_{1}+x_{2})^{1}(x_{1}-x_{2})^{1}$
and a basis for $D(\mathscr{B},\nu)$ is given by

$\theta_{1}=x_{1}^{2k+1}dx_{1}+x_{2}^{2k+1}dx_{2}\quad\text{ and
}\quad\theta_{2}=(x_{1}^{2}x_{2}^{2k+1}-x_{2}^{2k+3})dx_{2}.$

Verifying that $\theta_{1},\theta_{2}\in D(\mathscr{B},\nu)$ in both cases and
checking the determinant condition in Theorem 2.3 (which gives
$\det(p_{ij})=x_{1}^{2k}x_{2}^{2k+2}-x_{1}^{2k+2}x_{2}^{2k}=(x_{1}-x_{2})(x_{1}+x_{2})(-x_{1}^{2k})x_{2}^{2k}$
in the first case and
$\det(p_{ij})=(x_{1}-x_{2})(x_{1}+x_{2})x_{1}^{2k+1}x_{2}^{2k+1}$ in the
second case) shows that in both cases $\\{\theta_{1},\theta_{2}\\}$ is a basis
for $D(\mathscr{B},\nu)$, which completes the proof.
(3) and (4): Start with $\mu^{\prime}=(k,0,k,2k-4)$ resp.
$\mu^{\prime}=(k+1,0,k+1,2k-4)$ and use Theorem 2.16 to see that
$({\mathscr{A}},\mu^{\prime})$ is free with exponents $(2k-2,2k-2)$ resp.
$(2k-1,2k-1)$. So $({\mathscr{A}},\mu)$ has to be free with exponents
$(2k-2,2k-1)$ resp. $(2k-1,2k)$.
The argument is the same for the other two multiplicities, but we delete
$\ker(2x_{1}+x_{2})$ instead. ∎

###  5.1. Free multiplicities on $\mathscr{D}$.

We first introduce a special family of multiplicities $\mu$ such that
$(\mathscr{D},\mu)$ is a free multiarrangement. Then we proceed to show that
there are no other free multiplicities on $\mathscr{D}$.

######  Proposition 5.4.

Let $\mu=(2k,2k,2k,1,1,1)$ (for $k\in\mathbb{N}_{\geq 1}$) be a multiplicity
on $\mathscr{D}$ where the order on $\mathscr{D}$ is as in (5.2). Then
$(\mathscr{D},\mu)$ is free with exponents $(2k+1,2k+1,2k+1)$.

###### Proof.

The subarrangement $\mathscr{B}_{0}$ of $\mathscr{D}$ defined by
$Q(\mathscr{B}_{0},\mu_{0})=x_{1}^{2k}x_{2}^{2k}x_{3}^{2k}$ is boolean and
therefore free with exponents $(2k,2k,2k)$. Let
$Q(\mathscr{B}_{1},\mu_{1}):=x_{1}^{2k}x_{2}^{2k}x_{3}^{2k}(x_{1}+x_{2})$ and
$H_{1}=\ker(x_{1}+x_{2})$. Then we get
$\mathscr{B}_{1}^{H_{1}}=\\{\ker(x_{1}),\ker(x_{2})\\}$. Using Proposition
2.9, we get $\mu_{1}^{*}=(2k,2k)$ for the corresponding Euler-multiplicity.
Therefore, $(\mathscr{B}_{1}^{H_{1}},\mu_{1}^{*})$ is free with exponents
$\exp(\mathscr{B}_{1}^{H_{1}},\mu_{1}^{*})=(2k,2k)$. Using Theorem 2.10 we
deduce that $(\mathscr{B}_{1},\mu_{1})$ is free with exponents $(2k,2k,2k+1)$.
Now let
$Q(\mathscr{B}_{2},\mu_{2}):=x_{1}^{2k}x_{2}^{2k}x_{3}^{2k}(x_{1}+x_{2})(x_{1}+x_{3})$
and $H_{2}=\ker(x_{1}+x_{3})$, then we get
$\mathscr{B}_{2}^{H_{2}}=\\{\ker(x_{1}),\ker(x_{2}),\ker(x_{1}-x_{2})\\}$ and
Proposition 2.9 gives $\mu_{2}^{*}=(2k,2k,1)$. Therefore,
$(\mathscr{B}_{2}^{H_{2}},\mu_{2}^{*})$ is free with exponents
$\exp(\mathscr{B}_{2}^{H_{2}},\mu_{2}^{*})=(2k,2k+1)$, so
$(\mathscr{B}_{2},\mu_{2})$ is free with exponents $(2k,2k+1,2k+1)$. Finally,
restricting $(\mathscr{D},\mu)$ to $H_{3}=\ker(x_{2}+x_{3})$ gives
$\mathscr{D}^{H_{3}}=\\{x_{1},x_{2},x_{1}+x_{2},x_{1}-x_{2}\\}$ with Euler-
multiplicity $\mu^{*}=(2k,2k,1,1)$. In the proof of Lemma 5.3 we show that
$(\mathscr{D}^{H_{3}},\mu^{*})$ is free with exponents
$\exp(\mathscr{D}^{H_{3}},\mu^{*})=(2k+1,2k+1)$. Using Theorem 2.10 once more
completes the proof. ∎

The proof above actually shows that $(\mathscr{D},\mu)$ is inductively free
with $\mu$ as in the statement of the proposition. Next, we prove that there
are no other free multiplicities on $\mathscr{D}$.

###  5.2. Non-Free multiplicities on $\mathscr{D}$

The following Lemma shows that the non-trivial multiplicities as in
Proposition 5.4 all have to coincide.

######  Lemma 5.5.

Let
$Q(\mathscr{D},\mu)=x_{1}^{a}x_{2}^{b}x_{3}^{c}(x_{1}+x_{2})^{d}(x_{1}+x_{3})^{e}(x_{2}+x_{3})^{f}$
with multiplicity $\mu=(a,b,c,d,e,f),a\geq b\geq c$. If
$\mu=(even,even,even,1,1,1)$, then $(\mathscr{D},\mu)$ is free if and only if
$a=b=c$.

###### Proof.

The reverse implication follows from Proposition 5.4.

For the forward implication assume that $(\mathscr{D},\mu)$ is free. For some
$X\in L({\mathscr{A}})_{2}$ and the corresponding rank $2$ localization
${\mathscr{A}}_{X}=\\{H_{1},H_{2},H_{3}\\}$ or
${\mathscr{A}}_{X}=\\{H_{1},H_{2}\\}$ which is free with
$\exp({\mathscr{A}}_{X},\mu_{X})=(d_{1}^{X},d_{2}^{X})$, we write its summand
$d_{1}^{X}d_{2}^{X}$ in $\operatorname{LMP}(2)$ as
$[\mu(H_{1}),\mu(H_{2}),\mu(H_{3})]$ or $[\mu(H_{1}),\mu(H_{2})]$
respectively. Using this notation, $\operatorname{LMP}(2)$ of
$(\mathscr{D},\mu)$ equals

$\operatorname{LMP}(2)=[a,b,d]+[a,c,e]+[b,c,f]+[a,f]+[b,e]+[c,d]+[d,e]+[d,f]+[e,f].$

With our assumption $d=e=f=1$, we get

$\operatorname{LMP}(2)=[a,b,1]+[a,c,1]+[b,c,1]+a+b+c+3,$

which (because of $a\geq b\geq c$ and Theorem 2.16) equals

$a(b+1)+a(c+1)+b(c+1)+a+b+c+3=3a+2b+c+ab+ac+bc+3.$

Define the multiplicity $\mu_{1}=(a,b,c,0,0,0)$ on $\mathscr{D}$, then
$(\mathscr{D},\mu_{1})$ is free with exponents $(a,b,c)$ since it is boolean.
Assuming that $(\mathscr{D},\mu)$ is free, the arrangement has exponents equal
to one of $(a+3,b,c)_{\leq},(a+2,b+1,c)_{\leq},\dots,(a,b,c+3)_{\leq}$, thanks
to [HRS17, Rem. 2.8]. Writing $E:=ab+ac+bc$ this gives $\operatorname{GMP}(2)$
as follows:

Exponents | $(a+3,b,c)_{\leq}$ | $(a,b+3,c)_{\leq}$ | $(a,b,c+3)_{\leq}$ | $(a+2,b+1,c)_{\leq}$ | $(a+2,b,c+1)_{\leq}$
---|---|---|---|---|---
$\operatorname{GMP}(2)$ | $E+3b+3c$ | $E+3a+3c$ | $E+3a+3b$ | $E+a+2b+3c+2$ | $E+a+3b+2c+2$
Exponents | $(a+1,b+2,c)_{\leq}$ | $(a,b+2,c+1)_{\leq}$ | $(a+1,b,c+2)_{\leq}$ | $(a,b+1,c+2)_{\leq}$ | $(a+1,b+1,c+1)_{\leq}$
---|---|---|---|---|---
$\operatorname{GMP}(2)$ | $E+2a+b+3c+2$ | $E+3a+b+2c+2$ | $E+2a+3b+c+2$ | $E+3a+2b+c+2$ | $E+2a+2b+2c+3$

Since each of $a,b,c$ is even by assumption, we get that in all but the last
case (where $\exp(\mathscr{D},\mu)=(a+1,b+1,c+1)_{\leq}$),
$\operatorname{GMP}(2)$ is even. But since

$\operatorname{LMP}(2)=E+3a+2b+c+3$

is odd, we see that $\operatorname{LMP}(2)\neq\operatorname{GMP}(2)$ in those
cases. In particular, the arrangement cannot be free because of Theorem 2.28.
So let us assume $\exp(\mathscr{D},\mu)=(a+1,b+1,c+1)_{\leq}$ which gives
$\operatorname{GMP}(2)=E+2a+2b+2c+3$. But if we do not have $a=b=c$, then
$a>b$ or $b>c$. This implies $a>c$ and we have

$\operatorname{GMP}(2)=2a+2b+2c+E+3<3a+2b+c+E+3=\operatorname{LMP}(2).$

So using Theorem 2.28 again, we see that $(\mathscr{D},\mu)$ cannot be free
and we have $a=b=c$. ∎

Theorem 2.25 allows us to focus on the cases where $\mu$ does not satisfy
(2.26) as follows.

######  Corollary 5.6.

Let $\mu$ be a multiplicity on $\mathscr{D}$ that satisfies (2.26). Then
$(\mathscr{D},\mu)$ is not free. In particular, $\mathscr{D}$ is not free.

###### Proof.

By [CK24, Prop. 3.1], ${\mathscr{A}}_{C_{3}}$ is free with exponents
$(1,3,3)$. Let $H=H_{123}$ and note that ${\mathscr{A}}_{C_{3}}^{H}$ is free
with exponents $(1,2)$, since $|{\mathscr{A}}_{C_{3}}^{H}|=3$. According to
Theorem 2.10, ${\mathscr{A}}_{C_{3}}\backslash\\{H\\}=\mathscr{D}$ is not
free. By Theorem 2.25, $(\mathscr{D},\mu)$ is free if and only if
${\mathscr{E}}(\mathscr{D},\mu)$ free. However,
${\mathscr{E}}(\mathscr{D},\mu)_{X}=\mathscr{D}$ for $X=H_{1}\cap H_{2}\cap
H_{3}$. So ${\mathscr{E}}(\mathscr{D},\mu)$ cannot be free, by Theorem 2.1. ∎

We use a similar argument when (2.26) is not satisfied. For this, we define a
free extension for the multiarrangement $x^{a}y^{b}(x+y)^{c}$ when
$(a,b,c)=(even,even,odd)$. With this tool in hand, we construct an extension
of $(\mathscr{D},\mu)$ which is locally free along $H_{4}$ for an arbitrary
multiplicity $\mu$. This enables us to use Corollary 2.5 to show that
$(\mathscr{D},\mu)$ is not free.

Next we consider a class of extensions of a given multiarrangement
$({\mathscr{A}},\mu)$ which generalizes the one from Definition 2.24.

######  Definition 5.7.

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $\mathbb{K}^{\ell}$. Define
the extension ${\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)$ in
$\mathbb{K}^{\ell+1}$ depending on another fixed multiplicity
$\sigma:{\mathscr{A}}\to\mathbb{Z}$ as follows:

${\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)=\\{x_{\ell+1}=0\\}\cup\left\\{\alpha_{H}-kx_{\ell+1}=0\
\middle|\ k\in\mathbb{Z},-\frac{\mu(H)-1}{2}+\sigma(H)\leq
k\leq\frac{\mu(H)}{2}+\sigma(H)\right\\}.$

Note that
${\mathscr{E}}_{0}({\mathscr{A}},\mu)={\mathscr{E}}({\mathscr{A}},\mu)$ as in
Definition 2.24.

Analogous to multiplicities, we occasionally denote by
$\mu=(\mu(H_{1}),\mu(H_{2}),\dots,\mu(H_{n}))$, we write $\sigma$ as
$\sigma=(\sigma(H_{1}),\sigma(H_{2}),\dots,\sigma(H_{n}))$. We use this
notation in the following lemma.

######  Lemma 5.8.

Let $Q({\mathscr{A}},\mu)=x_{1}^{m_{1}}x_{2}^{m_{2}}(x_{1}+x_{2})^{m_{3}}$
with $m_{1}\leq m_{2}$. Let $\sigma=(0,0,1)$.

  * (i)

If $\mu=(m_{1},m_{2},m_{3})=(even,even,odd)$, then
${\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)$ is free.

  * (ii)

If $\mu=(m_{1},m_{2},m_{3})=(even,odd,odd)$ or $(odd,even,odd)$, then
${\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)$ is free.

###### Proof.

The idea of the proof is to increase the multiplicity of $H_{12}$ by $1$ which
results in a multiplicity $\mu_{1}=(m_{1},m_{2},m_{3}+1)$ such that (2.26) is
satisfied and thus ${\mathscr{E}}({\mathscr{A}},\mu_{1})$ is a free extension
of $({\mathscr{A}},\mu_{1})$. Now we remove a hyperplane $H$ so that
${\mathscr{E}}({\mathscr{A}},\mu_{1})\backslash\\{H\\}={\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)$
is free, thanks to Theorem 2.10. We demonstrate this in one case while the
other cases are handled analogously.

For (i) let $\mu_{1}=(m_{1},m_{2},m_{3}+1)=(even,even,even)$. The extended
arrangement ${\mathscr{A}}_{1}:={\mathscr{E}}_{0}({\mathscr{A}},\mu_{1})$ is
free, by Theorem 2.25. Note that for
$\sigma=(0,0,1),H=\ker(x_{1}+x_{2}+(\frac{m_{3}-1}{2})x_{3})$ and
${\mathscr{A}}_{2}:={\mathscr{E}}_{\sigma}({\mathscr{A}},\mu)$, we have
${\mathscr{A}}_{2}={\mathscr{A}}_{1}\backslash\\{H\\}$. So we start at the
free arrangement ${\mathscr{A}}_{1}$, fix the hyperplane $H$, and use Theorem
2.10 to derive the freeness of ${\mathscr{A}}_{2}$. Note that the exponents of
${\mathscr{A}}_{1}$ can be derived by using Theorems 2.16 and 2.2. The
following case by case investigation is necessary due to [Y09, Lem. 3.1].

Assume $m_{2}-m_{1}\leq m_{3}<m_{1}+m_{2}$. Then we have
$\exp({\mathscr{A}}_{1})=(1,\frac{m_{1}+m_{2}+m_{3}+1}{2},\frac{m_{1}+m_{2}+m_{3}+1}{2})$.
Note that it is sufficient to show that
$|{\mathscr{A}}_{1}^{H}|=1+\frac{m_{1}+m_{2}+m_{3}+1}{2}$, since
${\mathscr{A}}_{1}^{H}$ is a simple arrangement of rank $2$ and so is free
with exponents $(1,\frac{m_{1}+m_{2}+m_{3}+1}{2})$, satisfying the conditions
of Theorem 2.10. We count all elements $X\in L({\mathscr{A}}_{1})_{2}$ with
$X\subset H$ by listing all localizations ${\mathscr{A}}_{X}$. There is a
canonical bijection between the set containing those elements and the set of
hyperplanes of ${\mathscr{A}}_{1}^{H}$. We identify the hyperplanes of
${\mathscr{E}}({\mathscr{A}},\mu_{1})$ with their linear forms. We have a
single localization with more than $3$ elements:

$\left\\{H,x_{1}+x_{2}+\left(\frac{m_{3}-3}{2}\right)x_{3},\dots,x_{1}+x_{2}-\frac{m_{3}+1}{2}x_{3},x_{3}\right\\}$

and a total of $\frac{m_{1}+m_{2}-(m_{3}+1)}{2}$ localizations with $3$
elements

$\left\\{H,x_{1}+\left(\frac{m_{1}-2}{2}\right)x_{3},x_{2}-\left(\frac{m_{1}-(m_{3}+1)}{2}\right)x_{3}\right\\},\\\
\left\\{H,x_{1}+\left(\frac{m_{1}-4}{2}\right)x_{3},x_{2}-\left(\frac{m_{1}-(m_{3}+3)}{2}\right)x_{3}\right\\},\\\
\dots,\left\\{H,x_{1}-\left(\frac{m_{2}-(m_{3}+1)}{2}\right)x_{3},x_{2}+\left(\frac{m_{2}-2}{2}\right)x_{3}\right\\}.$

The rank $2$ localization with more than $3$ elements contains $m_{3}+1$
elements which are not equal to $H$ while all of the localizations with $3$
elements combined contain $m_{1}+m_{2}-m_{3}-1$ hyperplanes distinct from $H$.
Subtracting the number of hyperplanes included in these localizations from the
total number of hyperplanes in ${\mathscr{A}}_{1}$ shows that there are

$(m_{1}+m_{2}+(m_{3}+1))-(m_{3}+1)-(m_{1}+m_{2}-m_{3}-1)=m_{3}+1$

hyperplanes $H^{\prime}\in{\mathscr{A}}_{1}$ such that
$|({\mathscr{A}}_{1})_{H\cap H^{\prime}}|=2$. In conclusion we have

$1+\frac{m_{1}+m_{2}-(m_{3}+1)}{2}+(m_{3}+1)=1+\frac{m_{1}+m_{2}+m_{3}+1}{2}$

hyperplanes in $|{\mathscr{A}}_{1}^{H}|$. This implies that
$\exp({\mathscr{A}}_{1}^{H})=(1,\frac{m_{1}+m_{2}+m_{3}+1}{2})$. By Theorem
2.10, ${\mathscr{A}}_{2}$ is free with exponents
$\exp({\mathscr{A}}_{2})=(1,\frac{m_{1}+m_{2}+m_{3}-1}{2},\frac{m_{1}+m_{2}+m_{3}+1}{2})$.
The cases where $m_{3}+1<m_{2}-m_{1}+1$ and $m_{3}+1\geq m_{1}+m_{2}+1$ and
part (ii) are handled analogously. ∎

######  Theorem 5.9.

Let $(\mathscr{D},\mu)$ be the multiarrangement

$Q(\mathscr{D},\mu)=x_{1}^{a}x_{2}^{b}x_{3}^{c}(x_{1}+x_{2})^{d}(x_{1}+x_{3})^{e}(x_{2}+x_{3})^{f}$

with $a,b,c,d,e,f\geq 1$. Then $(\mathscr{D},\mu)$ is free if and only if
$\mu=(2k,2k,2k,1,1,1)$ for some $k\in\mathbb{N}_{\geq 1}$ where the order of
the hyperplanes is as above.

###### Proof.

The reverse implication follows from Proposition 5.4.
For the forward implication note that if at least two of the values $a,b,c$
are odd or all of $d,e,f$ are even, then $(\mathscr{D},\mu)$ is not free, by
Corollary 5.6. Consequently, we can assume (after possibly permuting some
basis vectors) that at least $d$ is odd and $a,b$ are even so (2.26) is not
satisfied. If $\mu=(even,even,even,1,1,1)$ and $\mu\neq(2k,2k,2k,1,1,1)$, then
we use Lemma 5.5 to see that $(\mathscr{D},\mu)$ is not free. For the
remaining multiplicities $\mu$ we give an extension of $(\mathscr{D},\mu)$
which is locally free along $H_{4}$, but has a localization isomorphic to
$\mathscr{D}$. So $(\mathscr{D},\mu)$ cannot be free, by Corollary 2.5. We now
give an alternative extension for all remaining multiplicities (up to a
permutation of coordinates) where $\mu\neq(2k,2k,2k,1,1,1)$. Note that the
local freeness along $H_{4}$ in the cases where (2.26) is not met is given by
Lemma 5.8. We write $H_{i4}=\ker(x_{i}-x_{4})$ and
$H_{ij4}=\ker(x_{i}+x_{j}-x_{4})$.

Let ${\mathscr{A}}={\mathscr{E}}_{\sigma}(\mathscr{D},\mu)$ be an extension of
$(\mathscr{D},\mu)$, where $\sigma$ is given as in the following table.

$a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $\sigma$ | Type
---|---|---|---|---|---|---|---
even | even | odd | odd | odd | $\geq 1$ | $(0,0,0,1,1,0)$ | $(a)$
even | even | odd | odd | even | $\geq 1$ | $(0,0,0,1,0,0)$ | $(a)$
even | even | even | $\geq 3$ | $\geq 1$ | $\geq 1$ |  $\sigma(H_{i})=0$ and $\sigma(H_{ij})=1\iff\mu(H_{ij})$ odd | $(c)$
even | even | even | $1$ | $\geq 2$ | $\geq 2$ |  $\sigma(H_{i})=0$ and $\sigma(H_{ij})=1\iff\mu(H_{ij})$ odd | $(b)$
even | even | even | $1$ | $1$ | $\geq 2$ |  $\sigma(H_{i})=0$ and $\sigma(H_{ij})=1\iff\mu(H_{ij})$ odd | $(a)$

The localizations of ${\mathscr{A}}={\mathscr{E}}_{\sigma}(\mathscr{D},\mu)$
isomorphic to $\mathscr{D}$ are:

  1. (a)

${\mathscr{A}}_{X}=\\{H_{2},H_{3},H_{14},H_{23},H_{124},H_{134}\\}$ for
$X=(H_{2}\cap H_{3}\cap H_{14})\in L({\mathscr{A}})$,

  2. (b)

${\mathscr{A}}_{X}=\\{H_{1},H_{3},H_{24},H_{13},H_{124},H_{234}\\}$ for
$X=(H_{1}\cap H_{3}\cap H_{24})\in L({\mathscr{A}})$,

  3. (c)

${\mathscr{A}}_{X}=\\{H_{1},H_{2},H_{34},H_{12},H_{134},H_{234}\\}$ for
$X=(H_{1}\cap H_{2}\cap H_{34})\in L({\mathscr{A}})$.

We now show how the table above can be applied to one of the remaining
multiplicities $\mu$ to derive the non-freeness for $(\mathscr{D},\mu)$. So
let $\mu=(2,4,4,1,1,3)$. First note that since in this case $a,b,c$ are even
and $d=e=1$, we use the last row of the table. The second to last column tells
us that $\sigma=(0,0,0,1,1,1)$, since all of the values $\mu(H_{ij})$ are odd.
Using the notation of Lemma 5.8 and choosing an arbitrary $X\in
L(\mathscr{D})_{2}$ with $|\mathscr{D}_{X}|=3$, we have parities
$(even,even,odd)$ on $\mathscr{D}_{X}$ and $\sigma|_{X}=(0,0,1)$. So Lemma 5.8
gives us the local freeness of ${\mathscr{E}}_{\sigma}(\mathscr{D},\mu)$ along
$H_{4}$. Since ${\mathscr{E}}_{\sigma}(\mathscr{D},\mu)$ has localization
$(a)$ with hyperplanes

$\\{\ker(x_{2}),\ker(x_{3}),\ker(x_{1}-x_{4}),\ker(x_{2}+x_{3}),\ker(x_{1}+x_{2}-x_{4}),\ker(x_{1}+x_{3}-x_{4})\\},$

(which is isomorphic to $\mathscr{D}$ and therefore non-free) we can apply
Corollary 2.5 to show that $(\mathscr{D},\mu)$ is not free. This completes the
proof of the theorem. ∎

Deleting a hyperplane from ${\mathscr{A}}_{C_{3}}$ results in either
${\mathscr{A}}_{P_{3}}={\mathscr{A}}(A_{3})$ or in $\mathscr{D}$. All free
multiplicities on the reflection arrangement ${\mathscr{A}}(A_{3})$ have been
classified, see [ATW08], [ANN09], [DFMS20]. Since all free multiplicities are
known on both types of deletions on ${\mathscr{A}}_{C_{3}}$, the next step is
to classify all free multiplicities on ${\mathscr{A}}_{C_{3}}$ itself.

###  5.3. Some free multiplicities on ${\mathscr{A}}_{C_{3}}$

In this final section we present some free multiplicities on
${\mathscr{A}}_{C_{3}}$ that we found by starting with a free multiplicity on
${\mathscr{A}}_{P_{3}}$ or $\mathscr{D}$ and adding a hyperplane via Theorem
2.10 to create a free multiplicity on ${\mathscr{A}}_{C_{3}}$. It turns out
that there is only one free multiplicity on $\mathscr{D}$ that extends to a
free multiplicity on ${\mathscr{A}}_{C_{3}}$ in such a way.

######  Corollary 5.10.

Let $H=H_{123}$ and

$({\mathscr{A}}_{C_{3}},\mu)=x_{1}^{2k}x_{2}^{2k}x_{3}^{2k}(x_{1}+x_{2})^{1}(x_{1}+x_{3})^{1}(x_{2}+x_{3})^{1}(x_{1}+x_{2}+x_{3})^{1}.$

Then $({\mathscr{A}}_{C_{3}},\mu)$ is free if and only if $k=1$.

###### Proof.

Let $H=H_{123}$. Owing to Theorem 5.9, we see that
$({\mathscr{A}}_{C_{3}}^{\prime},\mu^{\prime})$ is free with exponents
$(2k+1,2k+1,2k+1)$. Now restrict $({\mathscr{A}}_{C_{3}},\mu)$ to $H$ and
calculate $\mu^{*}$ by using Theorem 2.16 on the following rank $2$ flats:

$\\{H_{123},H_{12},H_{3}\\},\\{H_{123},H_{2},H_{13}\\},\\{H_{123},H_{1},H_{23}\\}.$

This gives $\mu^{*}=(2k,2k,2k)$ and therefore (again by Theorem 2.16)
$({\mathscr{A}}_{C_{3}}^{H},\mu^{*})$ is free with
$\exp({\mathscr{A}}_{C_{3}}^{H},\mu^{*})=(3k,3k)$. So by Remark 2.11
$({\mathscr{A}}_{C_{3}},\mu)$ is free if and only if $k=1$. ∎

######  Definition 5.11 ([ATW08, Def. 5.8]).

An arrangement ${\mathscr{A}}$ is supersolvable if there exists a filtration
${\mathscr{A}}={\mathscr{A}}_{r}\supset{\mathscr{A}}_{r-1}\supset\dots\supset{\mathscr{A}}_{2}\supset{\mathscr{A}}_{1}$
such that:

  * •

$\operatorname{rank}({\mathscr{A}}_{i})=i$ for $i=1,\dots,r$;

  * •

for any $H,H^{\prime}\in{\mathscr{A}}_{i}$, there exists some
$H^{\prime\prime}\in{\mathscr{A}}_{i-1}$ such that $H\cap H^{\prime}\subset
H^{\prime\prime}$.

######  Theorem 5.12 ([ATW08, Thm. 5.10]).

Suppose ${\mathscr{A}}$ is supersolvable with a filtration
${\mathscr{A}}={\mathscr{A}}_{r}\supset{\mathscr{A}}_{r-1}\supset\dots\supset{\mathscr{A}}_{2}\supset{\mathscr{A}}_{1}$
and $r\geq 2$ as in Definition 5.11. Let $\mu$ be a multiplicity on
${\mathscr{A}}$ and let $\mu_{i}=\mu|_{{\mathscr{A}}_{i}}$. Let
$\exp({\mathscr{A}}_{2},\mu_{2})=(d_{1},d_{2},0,\dots,0)$. Suppose that for
each $H^{\prime}\in{\mathscr{A}}_{d}\backslash{\mathscr{A}}_{d-1}$,
$H^{\prime\prime}\in{\mathscr{A}}_{d-1}$ $(d=3,\dots,r)$ and
$X:=H^{\prime}\cap H^{\prime\prime}$, we either have

${\mathscr{A}}_{X}=\\{H^{\prime},H^{\prime\prime}\\}$

or

$\mu(H^{\prime\prime})\geq\left(\sum_{X\subset
H\in{\mathscr{A}}_{d}\backslash{\mathscr{A}}_{d-1}}\mu(H)\right)-1.$

Then $({\mathscr{A}},\mu)$ is inductively free with
$\exp({\mathscr{A}},\mu)=(d_{1},d_{2},|\mu_{3}|-|\mu_{2}|,\dots,|\mu_{r}|-|\mu_{r-1}|,0,\dots,0)$.

We now use Theorem 5.12 to derive some classes of multiplicities on
${\mathscr{A}}_{C_{3}}$.

######  Proposition 5.13.

Let $({\mathscr{A}}_{C_{3}},\mu)$ be the multiarrangement

$Q({\mathscr{A}}_{C_{3}},\mu)=x_{1}^{a}x_{2}^{b}x_{3}^{c}(x_{1}+x_{2})^{d}(x_{1}+x_{3})^{e}(x_{2}+x_{3})^{f}(x_{1}+x_{2}+x_{3})^{g}$

with multiplicity $\mu=(a,b,c,d,e,f,g),(d\geq e\geq f)$. Suppose that

  1. (i)

$\mu=(k,k,k,r,1,1,k)$, where $1\leq k\leq 3$ arbitrary and $r\geq 1$, or

  2. (ii)

$\mu=(k,k,k,r,1,1,k)$, where $k>3$ arbitrary and $r\geq 2k-5$.

Then $({\mathscr{A}}_{C_{3}},\mu)$ is free. Moreover, if $r\geq 2k$, then
$({\mathscr{A}}_{C_{3}},\mu)$ is inductively free with exponents
$(r,2k+1,2k+1)$. For $\mu=(3,3,3,1,1,1,3)$, $({\mathscr{A}}_{C_{3}},\mu)$ is
not inductively free.

###### Proof.

Let $\mu(H_{1})=\mu(H_{2})=\mu(H_{3})=\mu(H_{123})=k$ and $\mu(H_{12})=r\geq
2k-1,\mu(H_{13})=\mu(H_{23})=1$ and fix $H=H_{23}$, so
$\mu^{\prime}=(k,k,k,r,1,0,k)$. We choose the following supersolvable
filtration for $({\mathscr{A}}_{C_{3}}^{\prime},\mu^{\prime})$:

$\\{H_{1}\\}\subset\\{H_{1},H_{2},H_{12}\\}\subset\\{H_{1},H_{2},H_{3},H_{12},H_{13},H_{123}\\}.$

This filtration satisfies the requirements of Theorem 5.12, because
$\mu^{\prime}(H_{1})\geq\mu^{\prime}(H_{3})+\mu^{\prime}(H_{13})-1\iff k\geq
k$ and
$\mu^{\prime}(H_{2})\geq\mu^{\prime}(H_{13})+\mu^{\prime}(H_{123})-1\iff k\geq
k$ and
$\mu^{\prime}(H_{12})\geq\mu^{\prime}(H_{3})+\mu^{\prime}(H_{123})-1\iff
2k-1\geq 2k-1$. So $({\mathscr{A}}_{C_{3}}^{\prime},\mu^{\prime})$ is
inductively free with exponents $(r,2k,2k+1)$. Restricting
$({\mathscr{A}}_{C_{3}},\mu)$ to $H_{23}$ gives hyperplanes in correspondence
to the rank $2$ flats

$\\{H,H_{2},H_{3}\\},\\{H,H_{1},H_{123}\\},\\{H,H_{12}\\},\\{H,H_{13}\\}$

with $\mu^{*}=(k,k,r,1)$. If $r\geq 2k$, then the restriction is free with
exponents $(2k+1,r)$ since deleting the hyperplane with multiplicity $1$ gives
a free arrangement with exponents $(2k,r)$ and using Theorem 2.10 shows that
$({\mathscr{A}}_{C_{3}},\mu)$ is inductively free with exponents
$(r,2k+1,2k+1)$. Deleting $H_{12}$ with Theorem 2.10 and Lemma 5.3 until
$r=2k-5$ (and we end up at exponents $(2k-1,2k-1,2k-1)$) shows the other
claim. In particular: If $r=3$, then $\mu=(3,3,3,1,1,1,3)$ and
$({\mathscr{A}}_{C_{3}},\mu)$ is free with exponents
$\exp({\mathscr{A}}_{C_{3}},\mu)=(5,5,5)$. Restricting to an arbitrary
$H\in{\mathscr{A}}_{C_{3}}$ results in $|({\mathscr{A}}_{C_{3}},\mu^{*})|=8$.
So Remark 2.11 shows that $({\mathscr{A}}_{C_{3}}^{\prime},\mu^{\prime})$ can
never be free and therefore $({\mathscr{A}}_{C_{3}},\mu)$ is neither
inductively nor additively free. ∎

It can be shown that $\mu=(3,3,3,1,1,1,3)$ is the minimal multiplicity (with
respect to $|\mu|$) on ${\mathscr{A}}_{C_{3}}$ that is free but not
inductively free.

Acknowledgments: The authors acknowledge the financial support of the DFG-
priority program SPP 2458 “Combinatorial Synergies” (Grant #RO 1072/25-1
(project number: 539865068) to G. Röhrle). We are grateful to M. DiPasquale,
T. Hoge, and M. Wakefield for some helpful comments.

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##  Appendix A Induction and Recursion tables

$\exp({\mathscr{A}},\mu)$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp({\mathscr{A}}_{G_{1}},\mu_{1})=(4,5,5,5)$ | $H_{123}$ | $(4,5,5)$
$(4,4,5,5)$ | $H_{34}$ | $(4,4,5)$
$(4,4,4,5)$ | $H_{12}$ | $(4,4,5)$
$(3,4,4,5)$ | $H_{13}$ | $(3,4,5)$
$(3,3,4,5)$ | $H_{134}$ | $(3,3,4)$
$(3,3,4,4)$ | $H_{234}$ | $(3,3,4)$
$(3,3,3,4)$ | $H_{3}$ | $(3,3,3)$
$(3,3,3,3)$ | $H_{3}$ | $(3,3,3)$
$(2,3,3,3)$ | $H_{1234}$ | $(2,3,3)$
$(2,2,3,3)$ | $H_{124}$ | $(2,2,3)$
$(2,2,2,3)$ | $H_{4}$ | $(2,2,2)$
$(2,2,2,2)$ | $H_{1234}$ | $(2,2,2)$
$(1,2,2,2)$ | $H_{2}$ | $(1,2,2)$
$(1,1,2,2)$ | $H_{14}$ | $(1,1,2)$
$(1,1,1,2)$ | $H_{2}$ | $(1,1,2)$
$(0,1,1,2)$ | $H_{23}$ | $(0,1,2)$
$(0,0,1,2)$ | $H_{1}$ | $(0,0,1)$
$(0,0,1,1)$ | $H_{1}$ | $(0,0,1)$
$(0,0,0,1)$ | $H_{14}$ | $(0,0,0)$
Table 2. Inductive chain for $({\mathscr{A}}_{G_{1}},\mu_{1})$ for $\mu=(2,2,2,1,2,1,1,1,2,1,1,1,1,1)$ $\exp({\mathscr{A}},\mu)$ | $H$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp({\mathscr{A}}_{C_{3}},\mu)=(7,7,7)$ | $H_{13}$ | $(7,7)$
$(6,7,7)$ | $H_{13}$ | $(7,7)$
$(5,7,7)$ | $H_{12}$ | $(5,7)$
$(5,6,7)$ | $H_{2}$ | $(5,6)$
$(5,6,6)$ | $H_{23}$ | $(5,6)$
$(5,5,6)$ | $H_{1}$ | $(5,5)$
$(5,5,5)$ | $H_{12}$ | $(5,5)$
$(4,5,5)$ | $H_{13}$ | $(4,5)$
$(4,4,5)$ | $H_{123}$ | $(4,4)$
$(4,4,4)$ | $H_{1}$ | $(4,4)$
$(3,4,4)$ | $H_{2}$ | $(3,4)$
$(3,3,4)$ | $H_{2}$ | $(3,3)$
$(3,3,3)$ | $H_{1}$ | $(3,3)$
$(2,3,3)$ | $H_{12}$ | $(2,3)$
$(2,2,3)$ | $H_{3}$ | $(2,2)$
$(2,2,2)$ | $H_{23}$ | $(2,2)$
$(1,2,2)$ | $H_{3}$ | $(1,2)$
$(1,1,2)$ | $H_{23}$ | $(1,2)$
$(0,1,2)$ | $H_{123}$ | $(0,1)$
$(0,1,1)$ | $H_{3}$ | $(0,1)$
$(0,0,1)$ | $H_{123}$ | $(0,0)$
Table 3. Inductive chain for $({\mathscr{A}}_{C_{3}},\mu)$ for $\mu\equiv 3$ Addition/Deletion | $\exp({\mathscr{A}},\mu)$ | $H$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---|---
Deletion | $\exp({\mathscr{A}}_{G_{2}},\mu_{2})=(5,5,5,5)$ | $H_{234}$ | $(5,5,5)$
Addition | $(5,5,5,6)$ | $H_{14}$ | $(5,5,5)$
Addition | $(5,5,5,5)$ | $H_{1}$ | $(5,5,5)$
Addition | $(4,5,5,5)$ | $H_{134}$ | $(4,5,5)$
Addition | $(4,4,5,5)$ | $H_{1234}$ | $(4,4,5)$
Addition | $(4,4,4,5)$ | $H_{4}$ | $(4,4,4)$
Addition | $(4,4,4,4)$ | $H_{124}$ | $(4,4,4)$
Addition | $(3,4,4,4)$ | $H_{2}$ | $(3,4,4)$
Addition | $(3,3,4,4)$ | $H_{23}$ | $(3,3,4)$
Addition | $(3,3,3,4)$ | $H_{1}$ | $(3,3,3)$
Addition | $(3,3,3,3)$ | $H_{3}$ | $(3,3,3)$
Addition | $(2,3,3,3)$ | $H_{13}$ | $(2,3,3)$
Addition | $(2,2,3,3)$ | $H_{123}$ | $(2,2,3)$
Addition | $(2,2,2,3)$ | $H_{234}$ | $(2,2,3)$
Addition | $(1,2,2,3)$ | $H_{3}$ | $(1,2,2)$
Addition | $(1,2,2,2)$ | $H_{12}$ | $(1,2,2)$
Addition | $(1,1,2,2)$ | $H_{34}$ | $(1,1,2)$
Addition | $(1,1,1,2)$ | $H_{2}$ | $(1,1,1)$
Addition | $(1,1,1,1)$ | $H_{24}$ | $(1,1,1)$
Addition | $(0,1,1,1)$ | $H_{4}$ | $(0,1,1)$
Addition | $(0,0,1,1)$ | $H_{1234}$ | $(0,0,1)$
Addition | $(0,0,0,1)$ | $H_{234}$ | $(0,0,0)$
Table 4. Recursive chain for $({\mathscr{A}}_{G_{2}},\mu_{2})$