In this section we recall concepts about hyperplane arrangements, Fan’s graph, characteristic varieties and resonance varieties. Each subsection concludes with several lemmas that are used to prove the main theorems of this paper.

#### 2.3. Graphs of Fan Type

In this paper, we will use a graph defined by Fan in [

7]. We recall the definition of the graph, and then give lemmas that will be useful later.

**Definition 2.1.** Let ${\mathcal{A}}^{*}=\{{L}_{1},{L}_{2},\cdots ,{L}_{n}\}$ be an arrangement of n distinct projective lines in ${\mathbb{CP}}^{2}$. Let M denote the set of points in ${\mathbb{CP}}^{2}$ where two or more lines from the arrangement ${\mathcal{A}}^{*}$ intersect and let ${M}_{3}$ be the subset of M consisting of points with multiplicity greater than or equal to three.

Define a graph denoted by

$F\left(\mathcal{A}\right)$ called a

**graph of Fan type** of the arrangement

${\mathcal{A}}^{*}$ as follows.

Let the set of points ${M}_{3}$ be the vertices of $F\left(\mathcal{A}\right)$.

For each line ${L}_{i}\in {\mathcal{A}}^{*}$, let ${S}_{i}={M}_{3}\cap {L}_{i}$. If the set ${S}_{i}$ is not empty, then choose an ordering of the points in ${S}_{i}$ given by ${S}_{i}=\{{p}_{1},{p}_{2},\cdots ,{p}_{m}\}$. For each $j\in \{1,\cdots ,m-1\}$, choose a simple arc ${a}_{j}$ in ${L}_{i}$ that connects ${p}_{j}$ to ${p}_{j+1}$, avoids all points in M, and avoids all arcs previously chosen. Let ${A}_{i}$ be the set of simple arcs chosen for the line ${L}_{i}$. The edges of $F\left(\mathcal{A}\right)$ will consist of the set of arcs ${A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n}$.

The vertices of graphs of Fan type are uniquely defined. However, the edges are not uniquely defined since any line containing more than two multiple points admits many orderings of those points that leads to different sets of edges. This situation motivates the following definition.

**Definition 2.2.** Let $\mathcal{F}\left({\mathcal{A}}^{*}\right)$ denote the set of all possible graphs of Fan type for the arrangement ${\mathcal{A}}^{*}.$

We introduce the following definition for the sake of brevity.

**Definition 2.1.** If an arrangement $\mathcal{A}$ in ${\mathbb{C}}^{2}$ is the union of two nontrivial subarrangements ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ that intersect in exactly $|{\mathcal{A}}_{1}|\xb7|{\mathcal{A}}_{2}|$ points of multiplicity two, then we say that $\mathcal{A}={\mathcal{A}}_{1}\cup {\mathcal{A}}_{2}$ is a **general position partition** of the arrangement.

We use the phrase “general position” since two distinct lines in the complex plane intersect transversely in a point of multiplicity two or have no point of intersection. We collect some useful properties of graphs of Fan type of an arrangement ${\mathcal{A}}^{*}$ that imply ${\mathcal{A}}^{*}$ has a decone with a general position partition.

**Lemma 2.3.** Let ${\mathcal{A}}^{*}$ be a projective line arrangement in ${\mathbb{CP}}^{2}$. If a graph of Fan type $F\in \mathcal{F}\left({\mathcal{A}}^{*}\right)$ is disconnected, then ${\mathcal{A}}^{*}$ has a decone with a general position partition.

Proof. Let C be a connected component of the graph F and let $\mathcal{C}$ denote the set of all lines in ${\mathcal{A}}^{*}$ that contain a vertex in C. Let $\mathcal{D}={\mathcal{A}}^{*}\setminus \mathcal{C}$. The set $\mathcal{D}$ is non-empty as otherwise the graph F would be connected.

Let ${H}_{c}\in \mathcal{C}$ and ${H}_{D}\in \mathcal{D}$. Suppose that ${H}_{C}$ and ${H}_{D}$ intersect in a point of multiplicity greater than two. (As the lines are in projective space, the multiplicity of their intersection is at least two.) As ${H}_{C}\in \mathcal{C}$, then by definition the line ${H}_{C}$ contains a vertex in the connected component C of the graph F. By definition of graphs of Fan type, this means either ${H}_{C}\cap {H}_{D}$ is a vertex of the component C or there is a path from another vertex in C contained in ${H}_{C}$ to ${H}_{C}\cap {H}_{D}$. In either case, this implies that ${H}_{D}$ contains a vertex in the connected component C; therefore, ${H}_{D}$ is in the set $\mathcal{C}$, which contradicts the fact that ${H}_{D}\in \mathcal{D}$. Therefore, ${H}_{C}$ and ${H}_{D}$ intersect in a point of multiplicity two.

Let L be an arbitrary line in the arrangement $\mathcal{C}$. Then the decone ${\mathbf{d}}_{L}\mathcal{A}$ has two components ${\mathbf{d}}_{L}\mathcal{C}$ and ${\mathbf{d}}_{L}\mathcal{D}$, the images of $\mathcal{C}$ and $\mathcal{D}$ under the decone operation. As the arrangements $\mathcal{C}$ and $\mathcal{D}$ were in general position in projective space, their images are in general position in affine space after the decone. Therefore, ${\mathcal{A}}^{*}$ has a decone with a general position partition. ☐

**Lemma 2.4.** Let ${\mathcal{A}}^{*}$ be a projective arrangement of lines in ${\mathbb{CP}}^{2}$ containing at least three lines. If there is a line H in ${\mathcal{A}}^{*}$ such that all multiple points contained in H are points of multiplicity two, then ${\mathcal{A}}^{*}$ has a decone with a general position partition.

Proof. Let L denote any line in ${\mathcal{A}}^{*}$ that is not H. Then the decone ${\mathbf{d}}_{L}\mathcal{A}$ may be written as two subarrangements $\mathcal{H}=\left\{{\mathbf{d}}_{L}H\right\}$ and ${\mathbf{d}}_{L}\mathcal{A}\setminus \mathcal{H}$. As all lines in ${\mathcal{A}}^{*}$ intersect H in double points, the image of these lines will also intersect the image of H in double points in ${\mathbb{C}}^{2}$. Therefore, $\mathcal{H}$ and ${\mathbf{d}}_{L}\mathcal{A}\setminus \mathcal{H}$ are in general position. Whereby, we conclude that ${\mathcal{A}}^{*}$ has a decone with a general position partition. ☐

**Lemma 2.5.** Suppose that all graphs in $\mathcal{F}\left({\mathcal{A}}^{*}\right)$ are connected and every hyperplane in ${\mathcal{A}}^{*}$ contains a point of multiplicity at least three. If there is a graph $F\in \mathcal{F}\left({\mathcal{A}}^{*}\right)$ such that F has an edge that is not part of a simple circuit, then ${\mathcal{A}}^{*}$ has a decone with a general position partition.

Proof. Recall that a simple circuit is a path in a graph such that the first and last vertices of the path are the same, no vertices are repeated (except for the first vertex as the last vertex) and no edges are repeated.

Let L be the line in ${\mathcal{A}}^{*}$ containing the edge e that is not part of a simple circuit in F. Let v and w denote the vertices of the edge e . The graph $F\setminus \left\{e\right\}$ has two connected components. If not, then there is a simple path P from v to w. Combining the path P with the edge e would create a simple circuit containing e, which is a contradiction.

Denote the components of $F\setminus \left\{e\right\}$ by ${F}_{v}$ and ${F}_{w}$ where ${F}_{v}$ is the component containing v and ${F}_{w}$ is the component containing w. Let ${\mathcal{B}}_{v}$ denote the set of lines in ${\mathcal{A}}^{*}$ containing vertices in ${F}_{v}$ except for L. Likewise let ${\mathcal{B}}_{w}$ denote the set of lines in ${\mathcal{A}}^{*}$ containing vertices in ${F}_{w}$ except for L. Since every line in ${\mathcal{A}}^{*}$ contains a higher order multiple point, each line besides L must be in either ${\mathcal{B}}_{v}$ or ${\mathcal{B}}_{w}$. Therefore, ${\mathcal{A}}^{*}={\mathcal{B}}_{w}\phantom{\rule{3.33333pt}{0ex}}\dot{\cup}\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{v}\phantom{\rule{3.33333pt}{0ex}}\dot{\cup}\phantom{\rule{3.33333pt}{0ex}}\left\{L\right\}$. The sets of lines ${\mathcal{B}}_{v}$ and ${\mathcal{B}}_{w}$ are disjoint. If they have a line, H, in common, then H would contain a vertex that is connected to v in ${F}_{v}$ and a vertex that is connected to w in ${F}_{w}$. By definition of graphs of Fan type, there is a path between these two vertices, hence the vertices v and w are connected, which is a contradiction.

Let ${H}_{v}\in {\mathcal{B}}_{v}$ and ${H}_{w}\in {\mathcal{B}}_{w}$ be chosen arbitrarily. Suppose that ${H}_{v}$ and ${H}_{w}$ intersect in a point of multiplicity greater than two in ${\mathcal{A}}^{*}$. (We know the lines intersect in a point of at least multiplicity two as they are in ${\mathbb{CP}}^{2}$.) Then $z={H}_{v}\cap {H}_{w}$ will be a vertex in all graphs of Fan type. As ${H}_{v}\in {\mathcal{B}}_{v}$, by definition there is a path from the point v to z in the graph $F\setminus \left\{e\right\}$. Likewise, as ${H}_{w}\in {\mathcal{B}}_{w}$ there is a path from the point z to the w in $F\setminus \left\{e\right\}.$ Combining these paths creates a path from v to w. This yields a contradiction as v and w are in distinct connected components. Therefore ${H}_{v}$ and ${H}_{w}$ intersect in a point of multiplicity two. As these lines were chosen arbitrarily, we see that ${\mathcal{B}}_{v}$ and ${\mathcal{B}}_{w}$ intersect in general position in ${\mathbb{CP}}^{2}$.

Therefore, the arrangement ${\mathbf{d}}_{L}\mathcal{A}$ has two subarrangements $\mathbf{d}{\mathcal{B}}_{v}$ and $\mathbf{d}{\mathcal{B}}_{w}$ the images of ${\mathcal{B}}_{v}$ and ${\mathcal{B}}_{w}$ respectively. These arrangements intersect in general position, therefore ${\mathcal{A}}^{*}$ has a decone with a general position partition. ☐

**Corollary 2.1.** If ${\mathcal{A}}^{*}$ does not have a decone with a general position partition, then for every $F\in \mathcal{F}\left({\mathcal{A}}^{*}\right)$ it follows that F is connected, every edge in F is contained in a simple circuit and every hyperplane in ${\mathcal{A}}^{*}$ contains a vertex in F.

**Theorem 2.1.** Let ${\mathcal{A}}^{*}$ be an arrangement of projective lines. If there is an edge e on a line L such that for all $F\in \mathcal{F}\left({\mathcal{A}}^{*}\right)$ with $e\in F$ every simple circuit containing e has at least two edges contained in the line L, then ${\mathcal{A}}^{*}$ has a decone with a general position partition.

Proof. From Lemma 2.3, we know that if any choice of Fan’s graph is disconnected, then the conclusion follows. Therefore, we assume that any choice of graph of Fan type from the collection $\mathcal{F}\left({\mathcal{A}}^{*}\right)$ is connected. Let $F\in \mathcal{F}\left({\mathcal{A}}^{*}\right)$ be any choice of graph of Fan type that contains the edge e as described in the statement of the theorem. Let v and w denote the vertices of the edge e.

Let E denote the set of edges in F that are contained in the line L. Then, ${F}^{*}=F\setminus E$ is a subgraph of F . If ${F}^{*}$ is connected, then there is a simple path from v to w. However, combining this path with the edge e would create a simple circuit in the graph F, but this contradicts the hypothesis that every simple circuit containing the edge e in the graph F has at least two edges contained in the line L since the path comes from ${F}^{*}.$ Therefore the graph ${F}^{*}$ has at least two connected components and w and v are in different components.

Let ${\mathcal{B}}_{v}$ denote the set of lines in ${\mathcal{A}}^{*}\setminus \left\{L\right\}$ that contain vertices with paths to v in ${F}^{*}$. Let ${\mathcal{B}}_{w}=\left({\mathcal{A}}^{*}\setminus \left\{L\right\}\right)\setminus {\mathcal{B}}_{v}$. Both of these arrangements are non-empty as v and w represent points of multiplicity greater than three. Let ${H}_{v}\in {\mathcal{B}}_{v}$ and ${H}_{w}\in {\mathcal{B}}_{w}$ be chosen arbitrarily. Suppose ${H}_{v}\cap {H}_{w}$ is a point of multiplicity greater than two. Then by definition of ${H}_{v}\in {\mathcal{B}}_{v}$ there is a path from the vertex ${H}_{v}\cap {H}_{w}$ to v. Therefore, by definition of the set ${\mathcal{B}}_{v}$, we have ${H}_{w}\in {\mathcal{B}}_{v}$. However, this is a contradiction as ${H}_{w}\in {\mathcal{B}}_{w}$. Therefore, the lines ${H}_{v}$ and ${H}_{w}$ intersect in a point of multiplicity two in the arrangement ${\mathcal{A}}^{*}$. As these lines were chosen arbitrarily, the arrangements ${\mathcal{B}}_{v}$ and ${\mathcal{B}}_{v}$ are in general position. Using L as the line at infinity will result in the arrangement ${\mathbf{d}}_{L}{\mathcal{A}}^{*}$ that has a general position partition given by ${\mathbf{d}}_{L}{\mathcal{B}}_{v}\cup {\mathbf{d}}_{L}{\mathcal{B}}_{w}$. ☐

#### 2.4. Characteristic Varieties

The characteristic varieties can be defined for any space that is homotopy equivalent to CW-complex with finitely many cells in each dimension [

10]. In this paper, we restrict out attention to spaces that are complements of hyperplane arrangements.

Let $\mathcal{A}=\{{H}_{1},\cdots ,{H}_{n}\}$ be an arrangement of hyperplanes in ${\mathbb{C}}^{l}$. As the fundamental group ${\pi}_{1}\left(M\left(\mathcal{A}\right)\right)$ has torsion-free abelianization with rank n, the character variety $\text{Hom}({\pi}_{1}\left(M\left(\mathcal{A}\right)\right),{\mathbb{C}}^{*})$ is identified with ${\left({\mathbb{C}}^{*}\right)}^{n}$. A generating set for a presentation of the fundamental group ${\pi}_{1}\left(M\left(\mathcal{A}\right)\right)$ is given by $\{{\gamma}_{1},\cdots ,{\gamma}_{n}\}$ where each ${\gamma}_{i}$ is a meridional loop around ${H}_{i}$ whose orientation is given by the complex structure.

**Definition 2.3** ([

10]). The

**characteristic varieties** of the arrangement

$\mathcal{A}$ are the cohomology jumping loci of

$M\left(\mathcal{A}\right)$ with coefficients in the rank 1 local systems over

${\mathbb{C}}^{*}$:

where

$\mathbf{t}=\{{t}_{1},\cdots ,{t}_{n}\}$ determines a representation

${\pi}_{1}\left(M\left(\mathcal{A}\right)\right)\to {\mathbb{C}}^{*}$,

${\gamma}_{i}\mapsto {t}_{i}$ that induces a rank one local system

${\mathbb{C}}_{\mathbf{t}}$.

In this paper, we shall only be concerned with the varieties where

$i=1$ and

$d=1$. The characteristic varieties

${V}_{1}^{1}\left(\mathcal{A}\right)$ depend (up to a monomial automorphism of the algebraic torus

${\left({\mathbb{C}}^{*}\right)}^{n}$) only on the fundamental group

$G={\pi}_{1}\left(M\left(\mathcal{A}\right)\right)$ (Subsection 2.5, [

10]), therefore we will also use the notation

${V}_{1}^{1}\left(G\right)$.

#### 2.4.1. Direct Products of Groups

Let $\mathcal{A}$ denote an arrangement n of hyperplanes and let $G={\pi}_{1}\left(M\left(\mathcal{A}\right)\right)$ denote the fundamental group of the complement of the arrangement. Further suppose that $G\cong {G}_{1}\times {G}_{2}$ where ${G}_{1}$ and ${G}_{2}$ have ranks ${n}_{1}\ge 1$ and ${n}_{2}\ge 1$ respectively. As G may be finitely presented with rank equal to the number of hyperplanes in the arrangement, it follows that ${G}_{1}$ and ${G}_{2}$ may also be finitely presented and $n={n}_{1}+{n}_{2}$. The characteristic variety ${V}_{1}^{1}\left({G}_{i}\right)$ is a subset of the algebraic torus ${\left({\mathbb{C}}^{*}\right)}^{{n}_{i}}$.

**Theorem 2.2.** If $G\cong {G}_{1}\times {G}_{2}$ is the fundamental group of an arrangement $\mathcal{A}$ of n hyperplanes, then the characteristic varieties ${V}_{1}^{1}\left(\mathcal{A}\right)$ are isomorphic to Proof. Let

${M}_{1}$ and

${M}_{2}$ be the canonical CW complexes generated by finite presentations for

${G}_{1}$ and

${G}_{2}$ respectively. By Theorem 3.2 in [

11],

As the characteristic varieties depend only on the group, there are monomial automorphisms of the algebraic torus

${\left({\mathbb{C}}^{*}\right)}^{n}$ such that

Thus, the varieties are isomorphic as desired. ☐

**Corollary 2.1.** Let ${\mathcal{A}}^{*}$ be a projective arrangement of lines in ${\mathbb{CP}}^{2}$.

If ${\pi}_{1}\left(M\left({\mathcal{A}}^{*}\right)\right)\cong {G}_{1}\times {G}_{2}$ then ${V}_{1}^{1}\left(M\left(\mathbf{c}\mathcal{A}\right)\right)$ is isomorphic to Proof. From Lemma 2.2, we conclude that ${\pi}_{1}\left(M\left(\mathbf{c}\mathcal{A}\right)\right)\cong {\pi}_{1}\left(M\left({\mathcal{A}}^{*}\right)\right)\times \mathbb{Z}\cong {G}_{1}\times {G}_{2}\times \mathbb{Z}$. Also, ${V}_{1}^{1}\left(\mathbb{Z}\right)=\left\{1\right\}$. Therefore, using Theorem 2.2 twice, the conclusion follows. ☐

#### 2.5. Resonance Varieties

Let

$\mathcal{A}$ be an arrangement in

${\mathbb{C}}^{l}$. Then, the cone of the arrangement

$\mathbf{c}\mathcal{A}$ is an arrangement in

${\mathbb{C}}^{l+1}$. We denote the projectivization of the arrangement in

${\mathbb{CP}}^{l}$ by

${\mathcal{A}}^{*}$. The intersection poset is the set of non-empty intersections of hyperplanes in the arrangement and is denoted by

The rank of an element

$X\in L\left(\mathcal{A}\right)$ is equal to the codimension of the space

X in

${\mathbb{C}}^{l}$. The rank

n elements are denoted by

${L}_{n}\left(\mathcal{A}\right)$.

Falk introduced the resonance varieties associated to an arrangement in [

12]. We recall the basic notation and ideas here and direct the interested reader to Falk’s original work for more information. Let

$A=A\left(\mathcal{A}\right)$ be the graded Orlik–Solomon algebra associated to an arrangement generated by

$\{{a}_{1},\cdots ,{a}_{n}\}$ where

${a}_{i}$ is associated to

${H}_{i}\in \mathcal{A}$. If we fix an element

$\omega \in {A}^{1}$, then the map

${d}_{\omega}:{A}^{p}\to {A}^{p+1}$ defined by left multiplication creates a complex

$(A,{d}_{\omega})$ as

${d}_{\omega}\circ {d}_{\omega}=0$. Notice that

$\omega ={\displaystyle \sum _{i=1}^{n}}{\lambda}_{i}{a}_{i}$ where

${\lambda}_{i}\in \mathbb{C}$. Therefore, we associate each

ω with a vector

$\lambda \in {\mathbb{C}}^{n}$.

As

$(A,{d}_{\omega})$ is a complex, we denote the cohomology of the complex by

${H}^{p}(A,\omega )={H}^{p}(A,\lambda )$. Finally, we define the

**resonance varieties** associated to the arrangement by

The following definition follows from Lemma 3.14 in [

12]:

**Definition 2.4.** Let

$\mathcal{A}={\left\{{H}_{i}\right\}}_{i=1}^{n}\in {\mathbb{C}}^{d}$ be a central arrangement of hyperplanes. For each

$X\in {L}_{2}\left(\mathcal{A}\right)$ with

X contained in at least three hyperplanes in

$\mathcal{A}$, the

**local component of X** in

${R}_{1}\left(\mathcal{A}\right)$ is given by

We will denote this component by the simpler notation

${R}_{1}^{loc}\left(X\right)$ when there is no confusion about the arrangement.

The next lemma follows easily from the definition as each vertex corresponds to a point in ${\mathcal{A}}^{*}$ of multiplicity at least three, hence a rank two component of $L\left(\mathbf{c}\mathcal{A}\right)$.

**Lemma 2.6.** Let ${\mathcal{A}}^{*}$ be a projective arrangement of lines in ${\mathbb{CP}}^{2}.$ Each vertex in a graph of Fan type of ${\mathcal{A}}^{*}$ induces a non-trivial local component of the resonance varieties in ${R}_{1}\left(\mathbf{c}\mathcal{A}\right)$.

**Example 2.1.** Consider the arrangement

$\mathcal{A}$ in

${\mathbb{C}}^{3}$ defined by the polynomial

$Q(x,y,z)=xyz(x+y)(y+z)(x+z)$. This is the rank 3 braid arrangement with associated matroid

$M\left({K}_{4}\right)$. We may depict the real part of the projectivization of this arrangement in

Figure 1. Using the labeling of the lines as in the figure, we have four local components in

${R}_{1}\left(\mathcal{A}\right)$. Let

$\{i,j,k\}$ denote the point in

${\mathcal{A}}^{*}$where the lines labelled by

$i,j,k$ intersect. The four local components are

Let

$\{{f}_{1},\cdots ,{f}_{6}\}$ be the canonical set of basis vectors for

${\mathbb{C}}^{6}$. Associate each hyperplane

${H}_{i}$ with the vector

${f}_{i}$. Then we may write

${R}_{1}(\mathcal{A},\{1,2,6\})$ as the span of the vectors

${f}_{1}-{f}_{2}$ and

${f}_{2}-{f}_{6}$.

**Notation 2.1.** Let $V\subseteq {\mathbb{C}}^{n}$ be any variety that is the union of linear subspaces. Then, each point $v\in V$ may be regarded as a vector $\mathbf{v}\in {\mathbb{C}}^{n}$. Denote by $\mathrm{Span}\left\{V\right\}$ the linear subspace of ${\mathbb{C}}^{n}$ spanned by the vectors $\mathbf{v}\in V.$

**Theorem 2.3.** Let $\mathcal{A}$ be a central arrangement of n hyperplanes such thatwhere $n={n}_{1}+{n}_{2}$.

Then, the resonance variety ${R}_{1}\left(\mathcal{A}\right)$ decomposes into a union of varieties ${R}_{1}\left(\mathcal{A}\right)={R}_{1}\left({M}_{1}\right)\cup {R}_{1}\left({M}_{2}\right)$ such that the intersection of $\mathrm{Span}\left\{{R}_{1}\left({M}_{1}\right)\right\}$ and $\mathrm{Span}\left\{{R}_{1}\left({M}_{2}\right)\right\}$ is the trivial vector. Proof. By Theorem 5.2 in [

11], the tangent cone

${\mathcal{V}}_{1}\left(\mathcal{A}\right)$ of the characteristic variety

${V}_{1}^{1}\left(\mathcal{A}\right)$ at the point

$\mathbf{1}$ coincides with the resonance variety

${R}_{1}\left(\mathcal{A}\right)$. More explicitly, there is a linear isomorphism

$\varphi $ from the tangent space of

${\mathbb{C}}^{n}$ at

$\mathbf{1}$ to

${\mathbb{C}}^{n}$ such that

$\varphi \left({\mathcal{V}}_{k}\left(\mathcal{A}\right)\right)={R}_{1}\left(\mathcal{A}\right)$.

Let $\mathbb{C}[{t}_{1},\cdots ,{t}_{{n}_{1}},{t}_{{n}_{1}+1},\cdots ,{t}_{{n}_{1}+{n}_{2}}]$ be a coordinate ring for $V=({V}_{1}^{1}\left({M}_{1}\right)\times {\mathbf{1}}^{{n}_{2}})\cup ({\mathbf{1}}^{{n}_{1}}\times {V}_{1}^{1}\left({M}_{2}\right))$ and let $\mathbb{C}[{z}_{1},\cdots ,{z}_{{n}_{1}},{z}_{{n}_{1}+1},\cdots ,{z}_{{n}_{1}+{n}_{2}}]$ be a coordinate ring for the tangent space of $\mathbf{1}$ in ${\mathbb{C}}^{n}.$ Then the variety ${V}_{1}^{1}\left({M}_{1}\right)\times {\mathbf{1}}^{{n}_{2}}$ is defined by an ideal ${I}_{1}\in \mathbb{C}[{t}_{1},\cdots ,{t}_{{n}_{1}},{t}_{{n}_{1}+1},\cdots ,{t}_{{n}_{1}+{n}_{2}}]$ generated by the union of an ideal ${I}_{1}^{\prime}\in \mathbb{C}[{t}_{1},\cdots ,{t}_{{n}_{1}}]$ and the set $\{{t}_{{n}_{1}+1}-1,\cdots ,{t}_{{n}_{1}+{n}_{2}}-1\}$. Therefore, the tangent cone $\mathcal{V}\left({M}_{1}\right)$ of the variety ${V}_{1}^{1}\left({M}_{1}\right)\times {\mathbf{1}}^{{n}_{2}}$ at $\mathbf{1}$ is defined by the ideal generated by an ideal ${J}_{1}\in \mathbb{C}[{z}_{1},\cdots ,{z}_{{n}_{1}}]$ and the set $\{{z}_{{n}_{1}+1},\cdots ,{z}_{{n}_{1}+{n}_{2}}\}$. In a similar manner, the tangent cone of ${\mathbf{1}}^{{n}_{1}}\times {V}_{1}^{1}\left({M}_{2}\right)$ at $\mathbf{1}$ denoted by $\mathcal{V}\left({M}_{2}\right)$ is defined by the ideal generated by an ideal ${J}_{2}\in \mathbb{C}[{z}_{{n}_{1}+1},\cdots ,{z}_{{n}_{1}+{n}_{2}}]$ and the set $\{{z}_{1},\cdots ,{z}_{{n}_{1}}\}$. Therefore, the tangent cone of V at $\mathbf{1}$ is given by $\mathcal{V}\left({M}_{1}\right)\cup \mathcal{V}\left({M}_{2}\right)$. From the definition of the ideals generating $\mathcal{V}\left({M}_{1}\right)$ and $\mathcal{V}\left({M}_{2}\right)$, the varieties are orthogonal. Therefore, the subspaces $\mathrm{Span}\left\{\mathcal{V}\left({M}_{1}\right)\right\}$ and $\mathrm{Span}\left\{\mathcal{V}\left({M}_{2}\right)\right\}$ intersect only at the origin.

As ${V}_{1}^{1}\left(\mathcal{A}\right)\cong V$, there is a monomial automorphism g of ${\mathbb{C}}^{n}$ inducing a linear isomorphism ${g}^{*}$ of tangent cones such that ${g}^{*}(\mathcal{V}\left({M}_{1}\right)\cup \mathcal{V}\left({M}_{2}\right))={\mathcal{V}}_{1}\left(\mathcal{A}\right)$. Combined with the map $\varphi $, we have ${R}_{1}\left(\mathcal{A}\right)=\varphi \left({g}^{*}(\mathcal{V}\left({M}_{1}\right)\cup \mathcal{V}\left({M}_{2}\right))\right)$. Define ${R}_{1}\left({M}_{i}\right)=\varphi \left({g}^{*}\left(\mathcal{V}\left({M}_{i}\right)\right)\right)$. Then, as linear isomorphisms preserve unions and intersections of linear subspaces, the conclusion of the theorem follows. ☐

**Theorem 2.4.** Let ${\mathcal{A}}^{*}$ be an arrangement of projective lines in ${\mathbb{CP}}^{2}$ such that>where the varieties ${V}_{1}^{1}\left({M}_{i}\right)$ are not trivial. Let ${R}_{1}\left(\mathcal{A}\right)={R}_{1}\left({M}_{1}\right)\cup {R}_{1}\left({M}_{2}\right)$ be the decomposition of the resonance variety from Theorem 2.3. If either ${R}_{1}\left({M}_{1}\right)$ or ${R}_{1}\left({M}_{2}\right)$ does not have local components, then ${\mathcal{A}}^{*}$ has a decone with a general position partition. Proof. Suppose that both ${R}_{1}\left({M}_{1}\right)$ and ${R}_{1}\left({M}_{2}\right)$ do not have local components. Then there are no multiple points in the arrangement ${\mathcal{A}}^{*}$. Deconing ${\mathcal{A}}^{*}$ with respect to any line will result in an arrangement of lines in general position. Thus an arrangement that has a general position partition.

Without loss of generality, assume

${R}_{1}\left({M}_{1}\right)$ has local components and

${R}_{1}\left({M}_{2}\right)$ does not have local components. Let

${\mathcal{B}}_{1}$ be the set of lines in

${\mathcal{A}}^{*}$ containing points that induce local components in

${R}_{1}\left({M}_{1}\right)$ and let

${\mathcal{B}}_{2}={\mathcal{A}}^{*}\setminus {\mathcal{B}}_{1}$. If

${\mathcal{B}}_{1}={\mathcal{A}}^{*}$, then every line in the arrangement induces a local component contained in

${R}_{1}\left({M}_{1}\right)$. Then

$\mathrm{Span}\left\{{R}_{1}\left({M}_{1}\right)\right\}=\u25b5=\left\{\lambda \in {\mathbb{C}}^{n}|{\displaystyle \sum _{i=1}^{n}}{\lambda}_{i}=0\right\}$. However, as

${R}_{1}\left(\mathbf{c}\mathcal{A}\right)\subseteq \u25b5$ [

12],

${R}_{1}\left({M}_{2}\right)$ must be the trivial subspace. By hypothesis,

${V}_{1}^{1}\left({M}_{2}\right)$ is not trivial, therefore its tangent cone and the variety

${R}_{1}\left({M}_{2}\right)$ are not trivial subspaces. Therefore we have a contradiction and may conclude that

${\mathcal{B}}_{1}$ and

${\mathcal{B}}_{2}$ are not empty. In fact,

$|{\mathcal{B}}_{1}|\ge 3$ as it contains at least one local component.

Any line ${L}_{1}\in {\mathcal{B}}_{1}$ and ${L}_{2}\in {\mathcal{B}}_{2}$ must intersect in a point of multiplicity two. If they intersect in a higher order point, the intersection induces a local component, but the lines in ${\mathcal{B}}_{2}$ do not induce local components. Therefore, ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{2}$ intersect in general position in ${\mathbb{CP}}^{2}.$

Pick any line $L\in {\mathcal{B}}_{1}$ and consider ${\mathbf{d}}_{L}{\mathcal{A}}^{*}$. The images of ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{2}$ intersect transversely in ${\mathbb{C}}^{2}$, thus ${\mathcal{A}}^{*}$ has a decone with a general position partition. ☐