# Matroid stratifications of hypergraph varieties, their realization spaces,
and discrete conditional independence models

Oliver Clarke, Kevin Grace, Fatemeh Mohammadi, and Harshit J Motwani

###### Abstract

We study varieties associated to hypergraphs from the point of view of
projective geometry and matroid theory. We describe their decompositions into
matroid varieties, which may be reducible and can have arbitrary singularities
by the Mnëv–Sturmfels universality theorem. We focus on various families of
hypergraph varieties for which we explicitly compute an irredundant
irreducible decomposition. Our main findings in this direction are threefold:
(1) we describe minimal matroids of such hypergraphs; (2) we prove that the
varieties of these matroids are irreducible and their union is the hypergraph
variety; and (3) we show that every such matroid is realizable over real
numbers. As corollaries, we give conceptual decompositions of various,
previously-studied, varieties associated with graphs, hypergraphs, and
adjacent minors of generic matrices. In particular, our decomposition strategy
gives immediate matroid interpretations of the irreducible components of
multiple families of varieties associated to conditional independence (CI)
models in statistical theory and unravels their symmetric structures which
hugely simplifies the computations.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Matroid stratifications of hypergraph varieties
  4. 4 Irreducible point and line configurations
  5. 5 Combinatorial closures of configurations
  6. 6 Consecutive forest hypergraphs $\Delta_{G}$
  7. 7 Hypergraphs $\Delta^{s,t}$
  8. 8 Conditional independence models

##  1 Introduction

This work develops new connections between conditional independence (CI)
models in statistics [Stu05, DSS09, Sul18], projective geometry [RG11, LV13],
the theory of matroids [Oxl11, PW70] and their realization spaces [Mnë85,
Mnë88, Stu89], and determinantal varieties in commutative algebra [BC03]. In
particular, we study a family of varieties whose defining equations are
indexed by the edges of some hypergraphs, and show that their irreducible
decompositions lead to well-structured configurations of points and lines.
Moreover, we apply our results to classes of conditional independence
varieties which naturally arise in algebraic statistics. We also analyze well-
known results from the literature [HHH+10, Rau13, HS04, PS19, CMR20] and show,
in these cases, that the irreducible components of their concerned varieties
have concrete interpretations in terms of matroids.

Our main object of study is the determinantal hypergraph variety $V_{\Delta}$
which is associated to a hypergraph $\Delta$; see Definition 2.1. The defining
equations of $V_{\Delta}$ are the minors of a matrix $X$ of indeterminates and
the corresponding polynomial ideal $I_{\Delta}$ generalizes many familiar
families of determinantal ideals. For example, given integers $t\leq d\leq n$
and the $d\times n$ matrix $X=(x_{ij})$ of indeterminates, the classical ideal
$I_{t}(X)$ studied by Bruns and Conca [BC03] is the ideal of the hypergraph
$\Delta$ whose edges are all $t$-subsets of $[n]:=\\{1,\dots,n\\}$. These
ideals, along with many other families [Stu90, HHH+10, EHHM13, MR18, PS19],
are often studied using Gröbner bases. When it is possible to find them,
Gröbner bases are a powerful tool for understanding ideals. For instance, if
the initial ideal is square-free, then the original ideal is radical. However,
when the hypergraph $\Delta$ lies outside of one of the particularly nice
families in the list above, finding a Gröbner basis becomes exceedingly
difficult. In this paper, we develop a general framework called the
decomposition strategy for understanding the properties of arbitrary
hypergraph ideals, without computing their Gröbner bases, using matroid
varieties.

A matroid is a combinatorial axiomatization of linear dependence in a vector
space. We refer to [Oxl11] for a complete introduction to matroids, however,
it will now be useful for us to recall realizability and related notions.
Given a finite collection of vectors in a fixed vector space, the collection
of linearly dependent sets of vectors defines a matroid. If this process can
be reversed, i.e. for a given matroid $M$ we can find such a collection of
vectors, then we call these vectors a realization of $M$. We write
$\Gamma_{M}$ for the space of realizations of $M$. The matroid variety $V_{M}$
of $M$ is the Zariski closure of the realization space; see Definition 2.3.
Our main tool to study hypergraph varieties is the Decomposition Theorem which
provides a matroid stratification for $V_{\Delta}$. One can think of this
theorem as an altered version of the stratification of the Grassmannian by
matroids, which was studied in [GGMS87].

######  Theorem (Theorem 3.1).

The variety associated to the hypergraph $\Delta$ is the union of matroid
varieties. The union is taken over all realizable matroids $M$ whose dependent
sets contain the edges of $\Delta$.

Many of the matroid varieties appearing in the decomposition above are
redundant. So, to refine this decomposition, we introduce the notion of
combinatorial closures of matroids; see Definition 3.3. More precisely, the
combinatorial closure $V^{\mathrm{comb}}_{M}$ of a given matroid $M$ is the
hypergraph variety associated to its set of circuits.

We recall that a minimally dependent matroid for a given hypergraph $\Delta$
is a realizable matroid $M$ whose dependent sets contain $\Delta$ as a subset
and is minimal among all such matroids. In other words, there does not exist a
realizable matroid $N$ such that
$\Delta\subseteq\mathcal{D}(N)\subsetneq\mathcal{D}(M)$.

######  Proposition (Proposition 3.10).

The associated variety of a hypergraph $\Delta$ is the union of combinatorial
closures of matroid varieties, where the union is taken over all minimally
dependent matroids for $\Delta$.

We conjecture that combinatorial closures are sufficient to find the
irredundant irreducible decomposition of any hypergraph variety. In fact, we
show that this is the case for some of the well-known examples of hypergraph
varieties such as those corresponding to binomial edge ideals [HHH+10, Rau13],
conditional independence ideals with hidden variables [CMR20, CMM21] and
ideals of adjacent minors [HS04]. In all of these cases, each combinatorial
closure is equal to its central component which makes the computations easier.
One of the families of hypergraphs we study is the consecutive forest
hypergraphs; see Definition 6.1. We will see that the combinatorial closures
have non-central components. To compute the irreducible decomposition of
$V_{\Delta}$, we apply the following method and note the corresponding
sections for consecutive forest hypergraphs.

Decomposition Strategy §3.3.

  * (i)

Identify minimally dependent matroids for $\Delta$ (§6.1).

  * (ii)

For each minimally dependent matroid $M$, write the combinatorial closure
$V_{M}^{\mathrm{comb}}$ as a minimal union of matroid varieties (§5).

  * (iii)

For each matroid variety appearing in step (ii), show that it is irreducible
(§4).

  * (iv)

Determine redundancy of non-central components of combinatorial closures in
the resulting decomposition and show the matroids are realizable (§6.2 and
§6.3).

We are able to completely go through this strategy for various family of
hypergraphs in §4-§7. In particular, for Step (iii) above, we show that the
realization spaces of certain point and line configurations are irreducible;
see §4. These are a family of matroids whose varieties appear as irreducible
components of many examples of hypergraph varieties. We will prove Theorem 4.5
which allows us to inductively build up configurations with irreducible
varieties, and use this to show that all configurations with at most $6$ lines
have irreducible varieties.

######  Theorem (Theorems 4.2 and 4.5).

Suppose that $\mathcal{C}^{\prime}$ is a point and line configuration with
irreducible realization space $\Gamma_{\mathcal{C}^{\prime}}$. If
$\mathcal{C}$ is obtained from $\mathcal{C}^{\prime}$ by adding a single line
passing through at most two intersection points of $\mathcal{C}^{\prime}$,
then $\Gamma_{\mathcal{C}}$ is irreducible. In particular, the realization
spaces of configurations with at most $6$ lines are irreducible.

The conclusion of the decomposition strategy is the following characterization
of the minimal irreducible decomposition of the associated variety of each
consecutive forest hypergraph.

######  Theorem (Theorems 5.14 and 6.20).

The irredundant irreducible components of the consecutive forest variety
$V_{\Delta_{G}}$ are in one-to-one correspondence with the minimally dependent
matroids for the hypergraph $\Delta_{G}$. In particular, for each minimally
dependent matroid, the non-central components of its combinatorial closure are
completely characterized and are redundant in the irreducible decomposition.

Our decomposition strategy gives immediate interpretations of the prime
decompositions of families of ideals for which it is not feasible to calculate
an explicit Gröbner basis due to the time complexity of the algorithm and the
current available hardware. In [PS19], which is further explained in Example
3.12, the authors calculated the prime decomposition of a hypergraph ideal
which contains $16$ edges of size $3$. This computation is at the limit of
what is currently possible on available hardware. In this case, the prime
components of this ideal have a straight-forward interpretation in terms of
configurations of $12$ points in the projective plane.

Moreover, in §7 we describe the hypergraphs $\Delta^{s,t}$ which arise from
the study of conditional independence (CI) statements; see §8 and [Fin11,
CMR20, CMM21]. We give a summary of the known cases of $V_{\Delta^{s,t}}$ in
Remark 7.2 and Table 2. The family of varieties $V_{\Delta^{s,t}}$ for a fixed
$s,t$ can be studied effectively by understanding finite families of matroids.
In Table 1, we count the number of combinatorial types of dependent matroids
for $\Delta^{s,t}$. We show that as certain parameters increase, the number of
combinatorial types eventually stabilizes which simplifies the computational
task. The associated variety of $\Delta^{s,t}$ determines a statistical model
corresponding to a collection of CI statements with hidden variables. The
irreducible components of these varieties give information about additional
constraints satisfied by distributions within the given CI model. We describe
the irreducible components of these varieties in terms of so-called grid
matroids. The first step to understanding such decompositions is to find the
set of minimally dependent matroids for $\Delta^{s,t}$.

When the dimension of the ambient space $d$ is low enough, we show that:

######  Theorem (Theorem 7.5).

If $t\leq d\leq s+t-3$, then there is a unique minimal matroid for
$\Delta^{s,t}$.

In particular, the minimal matroid from the above theorem corresponds to the
special irreducible component of the CI varieties studied in [CMR20, CMM21],
which leads to the intersection axiom for CI models. This component is
particularly important for the inference problem in statistics, as it gives
information about additional constraints on distributions with full support
which satisfy the given CI statements.

When there is no bound on $d$, we show that as the parameters for this
hypergraph are allowed to become large, we obtain every matroid among the
minimally dependent matroids for $\Delta^{s,t}$, up to a mild equivalence.

######  Theorem (Theorem 7.7).

For every matroid $M$, there exists a hypergraph of the form $\Delta^{s,t}$
and a dependent matroid $M^{\prime}$ for $\Delta^{s,t}$ such that a
restriction of $M^{\prime}$ is isomorphic to $M$.

We prove this theorem using an algorithm from [MF14] to explicitly construct
the matroid $M^{\prime}$. This theorem shows that understanding the varieties
of a general CI model is very difficult since matroid varieties satisfy
various universality results. Similarly, determining the realizability of a CI
model is difficult as the matroids of its components might not be
$\mathbb{R}$-realizable. More generally, we propose the following
computational problem.

######  Question 1.1.

Find the irreducible components of $V_{\Delta^{s,t}}$ and determine whether
they contain a rational point.

We conclude by highlighting some of the difficulties arising in the study of
Question 1.1, in particular in determining the irreducibility of matroid
varieties. Following [BS89, RG99], consider a collection of vectors
$v_{1},\ldots,v_{s}$ in a finite dimensional vector space $V$ over a field
$\mathbb{K}$, which realizes a matroid $M$. They also define a hyperplane
arrangement $H_{v_{1}},\ldots,H_{v_{s}}\subset V^{*}$ in the dual vector
space. The combinatorial type of this hyperplane arrangement is defined by
$M$, thus, one can think of the realization space $\Gamma_{M}$ as a parameter
space of hyperplane arrangements of fixed combinatorial type. In [Ryb11],
Rybnikov constructs a pair of combinatorially equivalent hyperplane
arrangements whose complements have non-isomorphic fundamental groups. So, in
general, it is not possible to study matroid varieties by picking a single
generic point in the variety.

Unfortunately, very little can be said about the geometry of matroid varieties
in general. The Mnëv–Sturmfels Universality Theorem shows that matroid
varieties satisfy Murphy’s Law in Algebraic Geometry. Specifically, given any
singularity, appearing in a semi-algebraic set, there is a matroid variety
with the same singularity up to a mild equivalence; see [Mnë85, Mnë88, Stu89,
LV13]. In fact, point and line configurations already exhibit all
singularities. More precisely, the equivalence above is defined on pointed
schemes generated by $(X,p)\sim(Y,q)$ when there exists a smooth morphism
$(X,p)\to(Y,q)$. Then, the universality theorem states that point and line
configurations exhibit all equivalence classes of this relation called
singularity types.

We also note that the defining equations of matroid varieties are in general
very difficult to calculate. These equations often arise from geometric
constraints satisfied by the matroid. Some of them can be interpreted as rank
conditions on certain submatrices, however this is not true in general.
Example 3.6 shows the smallest matroid for which non-determinantal equations
appear. Proving that such a polynomial constraint holds is often achieved by
finding an equivalent condition in the Grassmann-Cayley algebra [SJS17,
STW21]. In the context of CI models, such conditions give further constraints
on distributions satisfying the given CI statements.

Outline of paper. In §2, we introduce the key concepts used throughout the
paper. In particular, we fix our notations for hypergraph, matroid varieties
and introduce the dependence order on matroids. In §3, we prove the
decomposition theorem and introduce the combinatorial closures of matroid
varieties. In §4, we define point and line configurations and prove that, for
certain families, their realization spaces are irreducible. In §5, we study
the family of forest-like point and line configurations. We give a complete
characterization of the irreducible components of their combinatorial closures
using perturbation arguments. In §6, we apply the decomposition strategy to
the family of consecutive forest hypergraphs. As a result, we prove Theorem
6.20 which gives an irredundant irreducible decomposition of the hypergraph
variety. In §7, we introduce grid matroids and apply them to describe the
irreducible components of the variety of $\Delta^{s,t}$; see Tables 1 and 2.
We then proceed to use algorithmic procedures to prove Theorem 7.7. In §8, we
show how the hypergraph $\Delta^{s,t}$ arises in algebraic statistics in the
study of CI statements with hidden variables. We conclude by explaining how
our results may shed light on a conjecture by Matúš in the context of CI
models.

##  2 Preliminaries

### 2.1 Hypergraph varieties.

Let $\mathbb{K}$ be a field, $d\leq n$ be two positive integers, $X=(x_{ij})$
be a $d\times n$ matrix of indeterminates and $R=\mathbb{K}[X]$ be the
polynomial ring over $\mathbb{K}$ in the indeterminates $x_{ij}$. It is often
convenient to write determinants of submatrices of $X$ as $[I|J]_{X}$ where
$I$ and $J$ are respectively the sets of rows and columns of the submatrix. If
$I=[d]$, that is the submatrix covers all rows of $X$, then we write $[J]$ for
$[I|J]_{X}$. We denote by $x_{i}$ the $i^{\rm th}$ column of $X$ and by
$X_{F}$ the submatrix of $X$ with columns indexed by $F\subseteq[n]$.

######  Definition 2.1.

A (simple) hypergraph $\Delta$ on the vertex set $[n]$ is a subset of the
power set $2^{[n]}$. We assume that no proper subset of an element of $\Delta$
is in $\Delta$. The elements of $\Delta$ are called (hyper)edges.

  * •

The _determinantal hypergraph ideal_ of $\Delta$ is

$I_{\Delta}=\big{\langle}[A|B]_{X}:A\subseteq[d],B\in\Delta,|A|=|B|\big{\rangle}\subset
R.$

  * •

The variety of $\Delta$ is the zero set of $I_{\Delta}$ which is given by

$V_{\Delta}=\\{X\in\mathbb{C}^{d\times n}:\ \textrm{Rank}(X_{F})<|F|\text{ for
each }F\text{ in }\Delta\\}.$

The ideal $I_{\Delta}$ and its variety $V_{\Delta}$ depend on the value $d$,
i.e. the dimension of the ambient space. However, to keep our notation
concise, we suppress $d$ from our notation. Unless otherwise stated, all
results for hypergraph ideals and their varieties hold for all $d$ as long as
$d\geq\max\\{|F|:F\in\Delta\\}$.

######  Problem 2.2.

Find an irredundant irreducible decomposition of $V_{\Delta}$ for any
hypergraph $\Delta$.

### 2.2 Matroid varieties.

In this subsection, we recall the definitions of the realization space of a
matroid and its associated variety. We refer the reader to [Oxl11] for basic
definitions concerning matroids.

######  Definition 2.3.

Let $M$ be a matroid on $[n]$ of rank $r$ and let $d\geq r$. If $\mathbb{K}$
is a field, a _realization_ of $M$ in $\mathbb{K}^{d}$ is a collection of
vectors $X=\\{x_{1},\dots,x_{n}\\}\subset\mathbb{K}^{d}$ such that

$\\{x_{i_{1}},\dots,x_{i_{p}}\\}\subset X\text{ is linearly
dependent}\iff\\{i_{1},\dots,i_{p}\\}\text{ is a dependent set of }M.$

If such a collection of vectors exists, we say that the matroid is
_realizable_ over $\mathbb{K}$. (Other words used interchangeably with
_realizable_ include _representable_ and _linear_.) In this paper, if
realizability is discussed without specifying $\mathbb{K}$, then
$\mathbb{K}=\mathbb{C}$. The _realization space_ of $M$ in $\mathbb{C}^{d}$ is

$\Gamma_{M}=\\{X\subset\mathbb{C}^{d}:X\text{ is a realization of }M\\}.$

So each element of $\Gamma_{M}$ is naturally identified with a $d\times n$
matrix over $\mathbb{C}$.

######  Definition 2.4.

The _matroid variety_ $V_{M}=\overline{\Gamma_{M}}$ is the Zariski closure of
the realization space of $M$. We denote $I_{M}=I(V_{M})\subseteq\mathbb{C}[X]$
for the corresponding ideal where $X=(x_{i,j})$ is a $d\times n$ matrix of
indeterminates.

Note that $I_{M}$ is a radical ideal. Similarly to hypergraph ideals and their
varieties, we will always use $d$ to denote the dimension of the ambient
space. In order to simplify our notation, we suppress $d$ from the notation of
the realization space of matroids and their ideals and varieties.

### 2.3 Minimal matroids.

We denote by $\leq$ the partial order on sets given by inclusion. We extend
this notion to matroids by identifying them with their collection of dependent
sets. We will denote the collection of dependent sets of a matroid $M$ by
$\mathcal{D}(M)$. So, given two matroids $M_{1}$ and $M_{2}$, if
$\mathcal{D}(M_{1})\subseteq\mathcal{D}(M_{2})$ then we write $M_{1}\leq
M_{2}$. This is the _dependency order_ on matroids. However, we caution the
reader that this is precisely opposite of the _weak order_ in the matroid
literature [Oxl11, BS89]. Suppose that $M$ is a matroid on ground set $E$. For
any given collection of subsets $\mathcal{D}$ of $E$, we say that $M$ is
_dependent_ for $\mathcal{D}$ if $\mathcal{D}\subseteq\mathcal{D}(M)$ and
write $\mathcal{D}\leq M$. If there does not exist a matroid $N$ on $E$ such
that $\mathcal{D}\subseteq\mathcal{D}(N)\subsetneq\mathcal{D}(M)$, then we say
that $M$ is _minimally dependent_ for $\mathcal{D}$, i.e. $M$ is a smallest
matroid such that $\mathcal{D}\leq M$.

######  Example 2.5.

Any matroid $M$ is minimally dependent for its collection of circuits
$\mathcal{C}$. In this case, $M$ is the unique such matroid. In general, there
can be many different minimally dependent matroids for a collection
$\mathcal{D}$. For example, if $E=[5]$, and $\mathcal{D}=\\{1234,1235\\}$,
then the uniform matroid $U^{3}_{5}$, whose circuits are all $4$-subsets of
$E$, and the matroid with a single circuit $123$ are both minimally dependent
for $\mathcal{D}$; see Figure 1.

Figure 1: Geometric representations for two minimally dependent matroids for
$\mathcal{D}=\\{1234,1235\\}$.

We will be interested in finding out the minimal matroids for a given
collection of sets. In general, this is difficult. However, we are able to do
this for some specific families; see §6.

######  Notation.

We denote the corresponding simple hypergraph of any collection $\mathcal{D}$
of subsets of $[n]$ as:

$\displaystyle\min(\mathcal{D})=\\{D\in\mathcal{D}:\textrm{ no set
}D^{\prime}\in\mathcal{D}\textrm{ exists such that }D^{\prime}\subsetneq
D\\}.$ (1)

######  Remark 2.6.

Let $\Delta^{\prime}$ be a simple hypergraph all of whose edges have size at
most $d$, and let $\Delta=\min(\Delta^{\prime}\cup\binom{[n]}{d+1})$. Note
that $I_{\Delta^{\prime}}=I_{\Delta}$ and that
$V_{\Delta^{\prime}}=V_{\Delta}$. Moreover, $\Gamma_{M}=\emptyset$ if $M$ has
rank greater than $d$. Therefore, finding the minimally dependent matroids of
rank at most $d$ for $\Delta$ is equivalent to finding those for
$\Delta^{\prime}$.

##  3 Matroid stratifications of hypergraph varieties

### 3.1 The general decomposition theorem.

We now use our notation from §2 to decompose any hypergraph variety into
certain matroid varieties. In subsequent sections, we will investigate
techniques which will allow us to strengthen this statement to an irredundant
irreducible decomposition. The following theorem may be interpreted from the
perspective of the stratification of Grassmannians by matroid varieties. See
Remark 3.2.

######  Theorem 3.1 (Decomposition Theorem).

Let $\Delta\subseteq 2^{[n]}$ be a hypergraph on $[n]$ and let
$\mathcal{M}(\Delta)$ be the collection of realizable matroids whose
collections of dependent sets contain $\Delta$. Then,

$V_{\Delta}=\bigcup_{M\in\mathcal{M}(\Delta)}V_{M}.$

In particular, we have
$\sqrt{I_{\Delta}}=\bigcap_{M\in\mathcal{M}(\Delta)}I_{M}$.

###### Proof.

Let $M\in\mathcal{M}(\Delta)$. We begin by showing that $I_{\Delta}\subseteq
I_{M}$. By definition, $I_{\Delta}$ is generated by minors, so let $f\in
I_{\Delta}$ be any such minor. We may assume without loss of generality that
$f$ is a maximal minor of the submatrix $X_{F}$ for some $F\in\Delta$. For
each matroid $M\in\mathcal{M}(\Delta)$ and each point $A$ in the realization
space $\Gamma_{M}$, the columns of $A_{F}$ are linearly dependent since $F$ is
a dependent set of $M$. Thus $f$ vanishes on $\Gamma_{M}$, and by definition,
$f$ vanishes on its Zariski closure which is $V_{M}$. Thus, we have that $f\in
I(V_{M})=I_{M}$, so $I_{\Delta}\subseteq I_{M}$. Hence, $V_{M}\subseteq
V_{\Delta}$. Let $A$ be any point in the variety $V_{\Delta}$. The matrix $A$
is a realization of a matroid, which we denote $M_{A}$, on $[n]$ in
$\mathbb{C}^{d}$. For each $F\in\Delta$, the ideal $I_{\Delta}$ contains all
$|F|$-minors of $X_{F}$, so all $|F|$-minors of $A_{F}$ vanish. Therefore, the
columns of $A_{F}$ are linearly dependent. Hence $F$ is a dependent set in
$M_{A}$, so $M_{A}\in\mathcal{M}(\Delta)$. ∎

### 3.2 Combinatorial closure.

In Theorem 3.1, we decomposed the hypergraph variety $V_{\Delta}$ into matroid
varieties. But this decomposition may have redundant components, because
$V_{M^{\prime}}\subset V_{M}$ for some $M,M^{\prime}\in\mathcal{M}(\Delta)$.
To solve Problem 2.2, we must determine the matroids that are necessary in the
decomposition. It is clear that all matroids which are minimal with respect to
the dependency order are necessary in the decomposition. One could conjecture
that these are enough. Unfortunately this is _not_ true, as we will see in
Example 3.6. Nevertheless, we will define an alternative decomposition for
which the minimal matroids correspond to irredundant parts of a decomposition.
We do this by introducing the _combinatorial closure_ of a matroid. This is a
generalization of the weak realization space of a matroid [BS89, page 70] to
an ambient space of dimension greater than the rank of the matroid.

In the following remark we connect our setting to that of the stratification
of the Grassmannian by matroids.

######  Remark 3.2.

The Grassmannian $\operatorname{Gr}(d,n+d)$ admits a stratification
$\operatorname{Gr}(d,n+d)=\bigcup_{\mathcal{M}}S_{\mathcal{M}}$ by matroids
[GGMS87, Stu89]. Here,
$S_{\mathcal{M}}:=\\{x\in\operatorname{Gr}(d,n+d)\mid\mathcal{M}_{x}=\mathcal{M}\\}$
is the matroid stratum of $\mathcal{M}$, where $\mathcal{M}_{x}$ is the
matroid associated to the point $x\in\operatorname{Gr}(d,n+d)$. Let us
restrict to the standard affine open patch $U_{0}$ of
$\operatorname{Gr}(d,n+d)$ where the matrices are of the form $[M\ I]$ such
that $M$ is a $d\times n$ matrix and $I$ is the $d\times d$ identity matrix.
We identify the ambient space $\mathbb{C}^{d\times n}$ with $U_{0}$ as they
are isomorphic. After this identification, we can see $V(I_{\Delta})$ as the
closed embedding of $U_{0}$ given by the vanishing of the corresponding
Plücker coordinates.

For example, take $\Delta=\\{12,134\\}$ and $d=3$. Now $V(I_{\Delta})$, is a
closed subset of the standard affine open patch $U_{0}$ of Grassmannian
$\operatorname{Gr}(3,7)$. The equations of $V(I_{\Delta})$ in terms of Plücker
coordinates are given by
$P_{123}=0,P_{124}=0,P_{125}=0,P_{126}=0,P_{127}=0,P_{134}=0$. So,
$V(I_{\Delta})$ can be written in the following form:

$V(I_{\Delta})\cap\operatorname{Gr}(d,n+d)\cap
U_{0}=\bigcup_{\mathcal{M}}S_{\mathcal{M}}\cap V(I_{\Delta})\cap U_{0}.$

In this setting, we may restate Theorem 3.1 as
$\bigcup_{\mathcal{M}}S_{\mathcal{M}}\cap V(I_{\Delta})\cap
U_{0}=\bigcup_{M\in\mathcal{M}(\Delta)}V_{M}.$

######  Definition 3.3.

We define the _combinatorial closure_ $V_{M}^{\mathrm{comb}}$ of a matroid $M$
to be the union of all matroid varieties $V_{M^{\prime}}$ for which
$M^{\prime}\geq M$. In other words,

$V_{M}^{\mathrm{comb}}=\bigcup_{M^{\prime}\geq M}V_{M^{\prime}}$

We will denote the ideal of the combinatorial closure by
$I_{M}^{\mathrm{comb}}$.

######  Remark 3.4.

  * (i)

Note that $V_{M}^{\mathrm{comb}}$ might not be a matroid variety itself. The
inclusion $V_{M}\subseteq V_{M}^{\mathrm{comb}}$ holds in general, however the
equality might not hold. (See Example 3.6 below.) We will call $V_{M}$ the
_central component_ of $V_{M}^{\mathrm{comb}}$ and matroid varieties which
intersect the complement will be called _non-central components_.

  * (ii)

The combinatorial closure is the closure of the union of realization spaces,
i.e. $V_{M}^{\mathrm{comb}}=\overline{\bigcup_{M^{\prime}\geq
M}\Gamma_{M^{\prime}}}$. This follows from the topological fact that a finite
union of closures coincides with the closure of a union of sets.

The ideal of the combinatorial closure of a matroid $M$ can be seen as
follows. The collection of circuits $\mathcal{C}(M)$ of $M$ can be considered
as a hypergraph. So the hypergraph ideal $I_{\mathcal{C}(M)}$ for a given $d$,
is defined as:

$I_{\mathcal{C}(M)}=\big{\langle}[A|B]_{X}:\
A\subseteq[d],B\in\mathcal{C}(M),|A|=|B|\big{\rangle}.$

######  Lemma 3.5.

For any matroid $M$ we have that

$V_{M}^{\mathrm{comb}}=V_{\mathcal{C}(M)}\quad\text{ or equivalently }\quad
I_{M}^{\mathrm{comb}}=\sqrt{I_{\mathcal{C}(M)}}\ .$

###### Proof.

Let $X\in V_{M}^{\mathrm{comb}}$ be a collection of vectors, then
$X\in\Gamma_{M^{\prime}}$ for some $M^{\prime}\geq M$. So, any
$B\in\mathcal{C}(M)$ is a dependent set of $M^{\prime}$, and therefore
$[A|B]_{X}=0$ for any $A\subseteq[d]$ with $|A|=|B|$. This shows that
$V_{M}^{\mathrm{comb}}\subseteq V_{\mathcal{C}(M)}$.

Similarly, let $X$ be a point in $V_{\mathcal{C}(M)}$, and let $M_{X}$ be the
matroid represented by $X$. For any $B\in\mathcal{C}(M)$ and any subset
$A\subseteq[d]$ with $|A|=|B|$, we have $[A|B]_{X}=0$. Therefore, any
$B\in\mathcal{C}(M)$ is also a dependent set of $M_{X}$, hence $M_{X}\geq M$.
This shows that $V_{\mathcal{C}(M)}\subseteq V_{M}^{\mathrm{comb}}$. ∎

In general, the combinatorial closures of matroid varieties are reducible.

Figure 2: (Left) A realization of $M$ from Example 3.6. (Center and Right)
Realizations of matroids corresponding to the prime components of $I_{\Delta}$
in Example 3.12.

######  Example 3.6.

Let $d=3$, $n=7$ and $\Delta=\min(\\{123,145,167\\}\cup\binom{[7]}{4})$. It is
easy to check that $\Delta$ is the collection of circuits for a matroid $M$,
hence $M$ is the unique minimal matroid for $\Delta$. By Lemma 3.5, the ideal
of the combinatorial closure is given by the radical ideal
$I_{\mathcal{C}(M)}=I_{\Delta}=\langle[123],[145],[167]\rangle\subseteq\mathbb{C}[x_{1,1},\dots,x_{3,7}]$.
Using Macaulay2 [GS], we find that $I_{\mathcal{C}(M)}$ has two prime
components given by

$I_{\mathcal{C}(M)}=I_{1}\cap I_{2}=\langle
x_{1,1},x_{2,1},x_{3,1}\rangle\cap\langle[123],[145],[167],[234][567]-[235][467]\rangle\
.$

By inspection, the ideal $I_{1}$ is the ideal of the matroid $M^{\prime}$ that
has a single circuit $1$. In this case, we say that $M^{\prime}$ is obtained
from $M$ by setting $1$ to be a loop. On the other hand, $I_{2}$ is the ideal
of the matroid $M$ and is called the central component of the combinatorial
closure. The generator $[234][567]-[235][467]$ of $I_{2}$ is a geometric
condition satisfied by realizations of $M$. More precisely, it is a condition
satisfied by six generic points which lie on three lines that intersect at a
common point, as shown in Figure 2.

In Lemma 5.13, we generalize this example to an arbitrary number of lines
meeting at a point.

######  Remark 3.7.

We note that Example 3.6 has been studied in [BS89, Theorem 4.28] to resolve a
problem of White, by showing that there exists a matroid $M$ and a dependent
matroid $N$ (Section 2.3) such that
$\overline{\Gamma_{M}}\cap\Gamma_{N}=\emptyset$. In Section 6, we provide a
family of matroids $M$ and dependent matroids $N$ such that
$\overline{\Gamma_{M}}\cap\Gamma_{N}\neq\emptyset$. In particular, the
perturbation procedure (Lemma 5.9) allows us to approximate realizations of
$N$ with realizations of $M$. This gives a way to generate families of
dependent matroids that give a positive answer to White’s problem.

We recall the definition of ideal quotients and saturation. See, e.g. [CLO15,
§4] for more details.

######  Lemma 3.8.

Consider a polynomial ring $R=\mathbb{K}[x_{1},\dots,x_{n}]$. Let $I\subset R$
be a radical ideal and $J\subset R$ any ideal. Then $I:J=I:J^{\infty}$, where

$I:J=\\{f\in R:fJ\subseteq I\\}\quad\text{ and }\quad
I:J^{\infty}=\bigcup_{i\geq 1}\,(I:J^{i}).$

For a matroid $M$, let us define its _bases ideal_ $J_{M}$ to be

$J_{M}=\sqrt{\prod_{B\text{ basis of }M}\big{\langle}[A|B]_{X}:\
A\subseteq[d],|A|=|B|\big{\rangle}}.$

If $M$ has rank $d$, then $J_{M}$ is generated by the product of all maximal
minors of $X$ whose columns are bases of $M$.

######  Proposition 3.9.

The ideal of the matroid variety can be obtained by saturating
$I(V_{M}^{\mathrm{comb}})$ with respect to $J_{M}$:

$I_{M}=I(V_{M}^{\mathrm{comb}}):J_{M}^{\infty}.$

###### Proof.

Geometrically, the saturation $I(V_{M}^{\mathrm{comb}}):J_{M}^{\infty}$
corresponds to the ideal of $\overline{V_{M}^{\mathrm{comb}}\setminus
V(J_{M})}$; see [CLO15, §4, Theorem 10(iii)]. By Lemma 3.8, if $I$ is a
radical ideal, then $I:J=I:J^{\infty}$. So it is enough to show that the
difference $V_{M}^{\mathrm{comb}}\setminus V(J_{M})$ is the realization space
$\Gamma_{M}$. For this, notice that the variety $V(J_{M})$ consists of
collections of vectors for which at least one of the bases $B$ of $M$ is
dependent. Therefore, the difference $V_{M}^{\mathrm{comb}}\setminus V(J_{M})$
consists of collections of vectors $V_{M}^{\mathrm{comb}}$ for which all bases
$B$ of $M$ are independent, which is by definition the realization space
$\Gamma_{M}$. ∎

The above observation is shown in [STW21, Proposition 2.1.3] under the
identification of $\mathbb{C}^{d\times n}$ with an open affine patch of the
Grassmannian, as explained in Remark 3.2.

######  Proposition 3.10.

Let $\Delta\subseteq 2^{[n]}$ be a hypergraph on $[n]$ and let
$\mathcal{M}^{\prime}(\Delta)$ be the collection of minimal realizable
matroids whose dependent sets contain $\Delta$. Then

$\sqrt{I_{\Delta}}=\bigcap_{M\in\mathcal{M}^{\prime}(\Delta)}I_{M}^{\mathrm{comb}}\quad\textrm{or
equivalently}\quad
V_{\Delta}=\bigcup_{M\in\mathcal{M}^{\prime}(\Delta)}V^{\mathrm{comb}}_{M}.$

###### Proof.

Following the notation of Theorem 3.1, let $\mathcal{M}(\Delta)$ be the
collection of realizable matroids whose collections of dependent sets contain
$\Delta$. We have that $\mathcal{M}^{\prime}(\Delta)$ consists of the minimal
matroids in $\mathcal{M}(\Delta)$. We also have that if $M\leq M^{\prime}$
then $I_{M}^{\mathrm{comb}}\subseteq I_{M^{\prime}}^{\mathrm{comb}}$. So it
follows that,

$\bigcap_{M\in\mathcal{M}^{\prime}(\Delta)}I_{M}^{\mathrm{comb}}=\bigcap_{M\in\mathcal{M}(\Delta)}I_{M}^{\mathrm{comb}}.$

And so the proof is complete by Theorem 3.1. ∎

### 3.3 Decomposition strategy.

We can now make precise the decomposition strategy detailed in §1. By
Proposition 3.10, we have a decomposition of the hypergraph variety
$V_{\Delta}$ into combinatorial closures of minimally dependent matroids for
$\Delta$. To upgrade this to an irreducible decomposition, there are several
remaining steps.

First, one has to find the minimal collection of matroid varieties which cover
$V_{M}^{\mathrm{comb}}$. In particular, one has to show that the matroids
appearing in this collection are realizable, since otherwise they are clearly
redundant. In general, these matroid varieties might not be irreducible, so
one has to either prove their irreducibility (which we will do in several
examples) or find their irreducible decomposition. By going through these
steps, we will obtain such decompositions for various families of hypergraphs.
It still might be the case that some of the components are redundant. So to
obtain the actual minimal decomposition, one has to check which ones are
necessary. We remark that step (ii) of the decomposition strategy, i.e.
writing $V^{\mathrm{comb}}_{M}$ as a minimal union of matroid varieties,
involves showing that the matroid varieties whose union is
$V_{M}^{\mathrm{comb}}$ are nonempty. In other words, we must show that the
matroids are realizable. In particular, if $V_{M}^{\mathrm{comb}}=V_{M}$ for
each minimally dependent matroid $M$ of $\Delta$, then step (ii) can be
accomplished by showing that all of the minimally dependent matroids for
$\Delta$ are realizable.

######  Example 3.11 (Fano plane).

Let $d=3$ and consider the hypergraph
$\Delta=\\{124,136,157,235,267,347,456\\}$. It is straightforward to check
that $\Delta$ is the collection of circuits for a matroid $M$, which is called
the Fano plane. The matroid $M$ is not realizable over $\mathbb{C}$ so its
realization space $\Gamma_{M}$ is empty. However its combinatorial closure is
non-empty and we can compute the associated prime ideals of $I_{\Delta}$ in
Macaulay2 [GS]. We find that $I_{\Delta}$ has $22$ associated primes which are
all matroid varieties of $4$ combinatorial types of point and line
configurations; see Figure 3. The configurations in the figure, from left to
right, are:

  * •

A single line with $7$ points. The ideal is a hypergraph ideal
$I_{\Delta_{0}}$ where $\Delta=\binom{[7]}{3}$.

  * •

A quadrilateral set, see [RG11, §8], together with a loop. There are $7$
associated primes of $I_{\Delta}$ with this combinatorial type which are
parametrised by the point of the Fano plane to be taken as the loop.

  * •

A line with $3$ points together with a free point with $4$-labels. There are
$7$ associated primes of $I_{\Delta}$ with this combinatorial type which are
parametrised by the lines of the Fano plane.

  * •

A line with $3$ points together with a free point. Each point on the line has
two labels. There are $7$ associated primes of $I_{\Delta}$ with this
combinatorial type which are parametrised by the point of the Fano plane to be
taken as the free point.

Figure 3: The Fano plane (left) and the configurations of the associated
primes of its combinatorial closure.

######  Example 3.12.

Let $d=3$ and $E=[9]\cup\\{\bar{1},\bar{2},\bar{3}\\}$. Let $\Delta$ be the
hypergraph

$\Delta=\\{123,456,789,\bar{1}\bar{2}\bar{3},147,\bar{1}14,\bar{1}47,\bar{1}17,258,\bar{2}58,\bar{2}28,\bar{2}25,369,\bar{3}69,\bar{3}39,\bar{3}36\\}$

depicted in Figure 2. It is shown in [PS19, Theorem 4.1] that $I_{\Delta}$ has
two prime components such that:

  * •

The first component is generated by all $3$-minors of a generic $3\times 12$
matrix. Hence, it corresponds to the matroid $M_{0}$ with dependent sets,

$\mathcal{C}(M_{0})=\\{\text{all $3$-subsets of $E$}\\}.$

  * •

The second component is generated by $44$ polynomials of which $16$ are the
original generators of $I_{\Delta}$ and the remaining $28$ generators are all
homogeneous of degree $12$.

Note that there exists a unique minimal matroid $M$ for $\Delta$. The circuits
of $M$ are elements of $\Delta$ and all subsets of size $4$ of $E$, which do
not contain an element of $\Delta$. (For the proof of a more general
statement, see Theorem 7.5.) Therefore, by Proposition 3.10, we have
$\sqrt{I_{\Delta}}=I_{M}^{\mathrm{comb}}$, so the result of [PS19] describes
the decomposition of the combinatorial closure of $M$. This provides a
geometric meaning of the second component of the decomposition. In particular,
it can be obtained by saturating $\sqrt{I_{\Delta}}$ as in Proposition 3.9;
see Figure 2.

Therefore, we can give a geometric interpretation of the second component
described above. More precisely, the $28$ non-determinantal generators are
analogous to the geometric constraints of quadrilateral sets; see [RG11, §8].
More precisely, given a generic collection of $12$ points arranged in a
$3\times 4$ grid in the projective plane, if we project this configuration
onto a line then there are constraints on the distances between the projected
points. The $28$ polynomials describe precisely these constraints.

##  4 Irreducible point and line configurations

In this section, we will introduce point and line configurations. These are a
family of matroids whose ideals appear as prime components of many examples of
hypergraph ideals. We will prove Theorem 4.5 which allows us to inductively
build up configurations with irreducible varieties. In particular, using this
theorem, we will show that all configurations with at most $6$ lines have
irreducible varieties; see Theorem 4.2.

######  Notation.

We write $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ for a
configuration of points and lines in the plane, or simply a configuration. The
set $\mathcal{P}=\\{p_{1},\dots,p_{n}\\}$ is the collection of _points_ ,
$\mathcal{L}=\\{\ell_{1},\dots,\ell_{k}\\}$ is the collection of _lines_ and
$\mathcal{I}\subseteq\mathcal{P}\times\mathcal{L}$ is the set of _incidences_
of points and lines. If the point $p$ lies on the line $\ell$ then
$(p,\ell)\in\mathcal{I}$ and we say that $p$ is incident to line $\ell$, or
$\ell$ passes through $p$. We assume that any two distinct lines meet in at
most one point.

For each point $p\in\mathcal{P}$, we write
$\mathcal{L}_{p}\subseteq\mathcal{L}$ for the set of lines which pass through
the point $p$. With a slight abuse of notation, we will often identify the
line $\ell\in\mathcal{L}$ with the set of points which lie on the line $\ell$.

######  Definition 4.1.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a configuration.
For a line $\ell\in\mathcal{L}$ we define
$\mathcal{C}\backslash\ell=(\mathcal{P}^{\prime},\mathcal{L}\backslash\ell,\mathcal{I}\cap(\mathcal{P}^{\prime}\times(\mathcal{L}\backslash\ell))$
to be the configuration where $\mathcal{P}^{\prime}$ is the collection of
points of $\mathcal{C}$ which do not lie solely on $\ell$.

For each configuration $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$,
there is a simple matroid of rank at most $3$ associated to it. The matroid
has ground set $\mathcal{P}$, and is defined such that a $3$-subset
$D\subseteq\mathcal{P}$ is dependent if and only if there exists a line
$\ell\in\mathcal{L}$ with $D\subseteq\ell$. The realization spaces for point
and line configurations, are the realization spaces for the matroid; see
Definition 2.3. In the projective plane, any two distinct points determine a
line and so any pair of points in $\mathcal{P}$ can be taken to lie on a line.
However, if there is not a third element of $\mathcal{P}$ on the line, then
this line does not give rise to any dependent set in the matroid. Therefore,
in the remainder of this section, we require that each line in $\mathcal{L}$
contain three or more points in $\mathcal{P}$.

We note that configurations $\mathcal{C}$ are purely combinatorial objects. In
particular, they need not be realizable, i.e. we may have
$\Gamma_{\mathcal{C}}=\emptyset$. For example, the Fano plane, see Example
3.11, is the smallest non-realizable configuration over $\mathbb{C}$. In the
following sections, the purpose of using configurations is to give a
parametrization of the irreducible components of certain hypergraph ideals via
their realization spaces $\Gamma_{\mathcal{C}}$. For these cases, we may
assume that the configurations are realizable. However, for families of
configurations that we study directly, such as forest-like configurations, we
will prove that they are indeed realizable.

######  Theorem 4.2.

For any configuration $\mathcal{C}$ with at most $6$ lines,
$\Gamma_{\mathcal{C}}$ is irreducible with respect to the Zariski topology.

We prove this theorem using the Grassmann-Cayley algebra. We review some of
its theory following [Stu08].

######  Definition 4.3.

Let $V$ be a vector space of dimension $d$ over $\mathbb{C}$ and let
$\bigwedge(V)$ be the exterior algebra of $V$. The Grassmann-Cayley algebra is
the vector space $\bigwedge(V)$ together with the two operations $\wedge$ and
$\vee$. We denote by $a_{1}\vee a_{2}\vee\dots\vee a_{k}\in\bigwedge(V)$ the
extensor of length $k$ which is also referred to as the join of
$a_{1},a_{2}\dots,a_{k}$. The meet operation $\wedge$ is a binary operation on
two extensors $A$ and $B$ of length $j,k$ with $j+k\geq d$ as:

$A\wedge B:=\sum_{\sigma}{\rm
sign}(\sigma)[a_{\sigma(1)},\dots,a_{\sigma(d-k)},b_{1},\dots,b_{k}]a_{\sigma(d-k+1)}\vee\cdots\vee
a_{\sigma(j)},$

where the sum is taken over all permutations $\sigma$ of $\\{1,2,\dots,j\\}$.

Any extensor $A=a_{1}\vee\dots\vee a_{k}$ has an associated vector space
$\overline{A}={\rm span}(a_{1},\dots,a_{k})$ with following properties.

######  Lemma 4.4.

Let $A$ and $B$ be two extensors. Then the following hold:

  * •

Any extensor $A$ is uniquely determined from $\overline{A}$ up to a scalar
multiple [Stu08, Section 3.3].

  * •

The meet of two extensors is again an extensor [Stu08, Theorem 3.3.2 (b)].

  * •

We have that $A\wedge B\neq 0$ if and only if $\overline{A}+\overline{B}=V$.
In this case $\overline{A\wedge B}=\overline{A}\cap\overline{B}$ [Stu08,
Theorem 3.3.2 (c)].

In order to prove Theorem 4.2, we will take cases on the possible
configurations. We begin by proving the following theorem which allows us to
build new irreducible configurations from old. The cases in the following
theorem are depicted in Figure 4 and explained further in Example 4.6.

######  Theorem 4.5.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a configuration and
$\ell$ a line in $\mathcal{L}$. Let $S=\\{p\in\ell:|\mathcal{L}_{p}|\geq 3\\}$
be set of the points on $\ell$ which lie on at least $2$ other lines. Suppose
that $|S|\leq 2$. If $\Gamma_{\mathcal{C}\backslash\ell}$ is irreducible then
so is $\Gamma_{\mathcal{C}}$.

###### Proof.

Let $n$ be the number of points in $\mathcal{C}\backslash\ell$ and
$\ell_{1},\dots,\ell_{m}$ be the lines of $\mathcal{C}\backslash\ell$ which
intersect $\ell$ in $\mathcal{L}$ at a unique point. That is, if $\ell_{i}$
intersects $\ell$ at the point $p$, then there are no other lines through $p$,
i.e. $\mathcal{L}_{p}=\\{\ell,\ell_{i}\\}$. For each line $\ell_{i}$ we fix
two distinct points $p_{i,1}$ and $p_{i,2}$ in $\mathcal{P}\backslash\ell$
which lie on $\ell_{i}$. For each realization
$\gamma\in\Gamma_{\mathcal{C}\backslash\ell}$ we write
$\gamma_{1},\dots,\gamma_{n}\in\mathbb{C}^{3}$ for the points of $\gamma$ and
$\gamma_{\ell_{1}},\dots,\gamma_{\ell_{m}}\subseteq\mathbb{C}^{3}$ for the
$2$-dimensional linear subspaces corresponding to the lines. Note that the
subspace $\gamma_{\ell_{i}}$ is the linear span of the points
$\gamma_{p_{i,1}}$ and $\gamma_{p_{i,2}}$. We denote by $r$ the number of
points of $\mathcal{C}$ which lie on $\ell$ and no other lines. We proceed by
taking cases on $|S|\in\\{0,1,2\\}$.

Case 1. Assume that $|S|=0$. We define the space

$X=\Gamma_{\mathcal{C}\backslash\ell}\times\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})\times(\mathbb{C}^{2})^{r}\times\mathbb{C}^{m},$

where $\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})$ denotes the
set of linear maps from $\mathbb{C}^{2}$ to $\mathbb{C}^{3}$. Here, we think
of a linear map
$\phi\in\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})$ as a
$3\times 2$ matrix with entries in $\mathbb{C}$. We define the subset
$X^{\prime}\subseteq X$ to be the collection of
$(\gamma,\phi,(y_{1},\dots,y_{r}),(\lambda_{1},\dots,\lambda_{m}))\in X$ such
that the following conditions hold:

  * (a)

$\operatorname{rank}(\phi)=2$,

  * (b)

for each $i\in[n]$, the point $\gamma_{i}$ does not lie in
$\phi(\mathbb{C}^{2})\subseteq\mathbb{C}^{3}$,

  * (c)

for each $i\in[r]$, the point $\phi(y_{i})$ does not lie in any of the
subspaces $\gamma_{\ell_{1}},\dots,\gamma_{\ell_{m}}$,

  * (d)

for any $i,j\in[r]$, if $\phi(y_{i})=\phi(y_{j})$ then $i=j$.

Since $\Gamma_{\mathcal{C}\backslash\ell}$, $(\mathbb{C}^{2})^{r}$ and
$\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})$ are irreducible, we
have that $X$ is a product of irreducible varieties over an algebraically
closed field. Hence, $X$ is also irreducible. By construction, each of the
conditions for the subset $X^{\prime}\subseteq X$ above is a rank constraint
on certain submatrices. Explicitly we have that:

  * •

$\operatorname{rank}(\phi)\neq 2$ if all of the $2$-minors of $\phi$ are zero,

  * •

Assuming that $\operatorname{rank}(\phi)=2$, a point $\gamma_{i}$ lies inside
$\phi(\mathbb{C}^{2})$ if the $3$-minor $[\gamma_{i},\phi(1,0),\phi(0,1)]$ is
zero,

  * •

A point $\phi(y_{i})$ lies inside the subspace $\gamma_{\ell_{i}}$ if the
$3$-minor $[\phi(y_{i}),\gamma_{p_{i,1}},\gamma_{p_{i,2}}]$ is zero,

  * •

If $\operatorname{rank}(\phi)=2$ then it follows that $\phi$ is injective.

Therefore $X^{\prime}\subseteq X$ is an open subset which implies that
$X^{\prime}$ is also irreducible.

Now, for each
$(\gamma,\phi,(y_{1},\dots,y_{r}),(\lambda_{1},\dots,\lambda_{m}))\in
X^{\prime}$ and for each $i\in[m]$ we fix a non-zero point
$q_{i}\in\mathbb{C}^{3}$ which lies in the intersection of $\gamma_{\ell_{i}}$
and $\phi(\mathbb{C}^{2})$. Note that the ambient space is $\mathbb{C}^{3}$,
hence any pair of $2$-dimensional linear subspaces intersect. By construction
of $X^{\prime}$, we have that $\phi(\mathbb{C}^{2})$ and $\gamma_{\ell_{i}}$
do not coincide. Hence, $\dim(\phi(\mathbb{C}^{2})\cap\gamma_{\ell_{i}})=1$
and $q_{i}$ is unique up to a non-zero scalar multiple. We give an explicit
formula for $q_{i}$ using the Grassmann-Cayley algebra as follows. Recall that
$\gamma_{\ell_{i}}$ is the span of $a_{1}:=\gamma_{p_{i,1}}$,
$a_{2}:=\gamma_{p_{i,2}}$ and $\phi(\mathbb{C}^{2})$ is the span of
$b_{1}:=\phi(1,0)$, $b_{2}:=\phi(0,1)$. Since the spaces $\gamma_{\ell_{i}}$
and $\phi(\mathbb{C}^{2})$ do not coincide, they must span the entire space
$\mathbb{C}^{3}$. Since $\dim(\phi(\mathbb{C}^{2})\cap\gamma_{\ell_{i}})=1$,
by Lemma 4.4, we have that
$q_{i}:=[a_{1},b_{1},b_{2}]a_{2}-[a_{2},b_{1},b_{2}]a_{1}\in\gamma_{\ell_{i}}\cap\phi(\mathbb{C}^{2})$
is a non-zero vector lying in the intersection. We define the map

$\psi:X^{\prime}\rightarrow(\mathbb{C}^{3})^{n+m+r}:(\gamma,\phi,(y_{1},\dots,y_{r}),(\lambda_{1},\dots,\lambda_{m}))\mapsto(\gamma,\lambda_{1}q_{1},\dots,\lambda_{m}q_{m},\phi(y_{1}),\dots,\phi(y_{r})).$

By construction, $\phi$ is a linear map and the coordinates of the points
$q_{i}\in\mathbb{C}^{3}$ are polynomials in the entries of $\gamma$. Thus
$\psi$ is a polynomial map and so it is continuous.

It remains to show that $\Gamma_{\mathcal{C}}$ is an open subset of the image
of $\psi$. It is easy to see that the image of $\psi$ contains
$\Gamma_{\mathcal{C}}$ since any realization of $\mathcal{C}$ can be viewed as
a realization of $\Gamma_{\mathcal{C}\backslash\ell}$ together with some
additional points on $\ell$. To show that $\Gamma_{\mathcal{C}}$ is open in
the image of $\psi$, we note that the image of $\psi$ is contained in the
combinatorial closure $V^{\mathrm{comb}}_{\mathcal{C}}$. So
$\Gamma_{\mathcal{C}}$ is obtained from the image of $\psi$ by removing the
vanishing locus of the bases ideal of $J_{\mathcal{C}}$; see Proposition 3.9.

Case 2. Assume that $|S|=1$ and write $S=\\{s\\}$ where $s$ is the
corresponding point of intersection in $\mathcal{C}$ and
$\mathcal{C}\backslash\ell$. For any $\gamma\in\Gamma_{\mathcal{C}}$ we denote
by $\gamma_{s}\in\mathbb{C}^{3}$ the coordinates of the point corresponding to
$s$. We define the space

$X=\\{(\gamma,\phi,(y_{1},\dots,y_{r}),(\lambda_{1},\dots,\lambda_{m})):\phi(1,0)=\gamma_{s}\\}\subseteq\Gamma_{\mathcal{C}\backslash\ell}\times\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})\times(\mathbb{C}^{2})^{r}\times\mathbb{C}^{m}$

where the condition $\phi(1,0)=\gamma_{s}$ on a matrix
$\phi\in\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})$ is
equivalent to the condition that the first column of $\phi$ is equal to
$\gamma_{s}$. So, we have that
$X\cong\Gamma_{\mathcal{C}}\times\mathbb{C}^{3}\times(\mathbb{C}^{2})^{r}\times\mathbb{C}^{m}$
which is irreducible. We define the subset $X^{\prime}\subseteq X$ identically
to Case 1, except that we allow $\gamma_{s}\in\phi(\mathbb{C}^{2})$ in
condition (b). Note that $\gamma_{s}\neq 0$ and so by the same argument,
$X^{\prime}\subseteq X$ is an open subset. The remainder of the argument from
Case 1, including the construction of the points
$q_{1},\dots,q_{m}\in\mathbb{C}^{3}$ and the map $\psi$, follows identically.

Case 3. Assume that $|S|=2$ and write $S=\\{s_{1},s_{2}\\}$ for the
corresponding points of intersection lying in both $\mathcal{C}$ and
$\mathcal{C}\backslash\ell$. For any $\gamma\in\Gamma_{\mathcal{C}}$ we denote
by $\gamma_{s_{1}},\gamma_{s_{2}}\in\mathbb{C}^{3}$ the coordinates of the
points corresponding to $s_{1}$ and $s_{2}$, respectively. We define the space

$X=\\{(\gamma,\phi,(y_{1},\dots,y_{r}),(\lambda_{1},\dots,\lambda_{m})):\phi(1,0)=\gamma_{s_{1}},\
\phi(0,1)=\gamma_{s_{2}}\\}\subseteq\Gamma_{\mathcal{C}\backslash\ell}\times\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})\times(\mathbb{C}^{2})^{r}\times\mathbb{C}^{m}$

where the conditions $\phi(1,0)=\gamma_{s_{1}}$ and $\phi(0,1)=\gamma_{s_{2}}$
on a matrix $\phi\in\textrm{Hom}_{\mathbb{C}}(\mathbb{C}^{2},\mathbb{C}^{3})$
are equivalent to the conditions that the first column of $\phi$ is equal to
$\gamma_{s_{1}}$ and the second column of $\phi$ is equal to $\gamma_{s_{2}}$,
respectively. Thus
$X\cong\Gamma_{\mathcal{C}}\times(\mathbb{C}^{2})^{r}\times\mathbb{C}^{m}$,
which is irreducible. We define the subset $X^{\prime}\subseteq X$ identically
to Case 1, except that we allow
$\gamma_{s_{1}},\gamma_{s_{2}}\in\phi(\mathbb{C}^{2})$ in condition (b). Note
that $s_{1}$ and $s_{2}$ are distinct points of $\mathcal{C}$ so
$\operatorname{rank}(\phi)=2$. By the same argument as in Case 1,
$X^{\prime}\subseteq X$ is an open subset. The remainder of the argument from
Case 1, including the construction of the points
$q_{1},\dots,q_{m}\in\mathbb{C}^{3}$ and the map $\psi$, follows identically.
∎

######  Example 4.6.

Let $\mathcal{C}$ be the configuration in Case 1 of Figure 4 and $\ell$ the
line indicated in the diagram. It is shown in [CMM21] that all configurations
with at most $4$ lines have irreducible realization spaces. In particular,
$\Gamma_{\mathcal{C}\backslash\ell}$ is irreducible. For each point $p$ on
$\ell$ we have that $|\mathcal{L}_{p}|=2$, thus $\Gamma_{\mathcal{C}}$ is
irreducible by Theorem 4.5. Note that, in the proof, the points
$p_{1,1},p_{1,2}$, as shown in the diagram, are used to give coordinates on
the line $\ell_{1}$, which allows us to give an explicit formula for the
intersection point $q_{1}$ of $\ell$ and $\ell_{1}$.

In the figure, the configurations labelled Case 2 and Case 3 show examples of
configurations, and the line $\ell$, corresponding to their respective cases
in the proof of Theorem 4.5. The points which lie in the set $S$ are labelled
by $s,s_{1},s_{2}$. In particular, each of the realization spaces for these
configurations is irreducible.

Figure 4: Configurations in Example 4.6. The left and middle figures are
obtained from the figure immediately to their right by removing the line
$\ell$.

The proof of Theorem 4.2 follows easily from the above theorem.

###### Proof of Theorem 4.2..

We show that $\Gamma_{\mathcal{C}}$ is irreducible by taking cases on the
number of lines in $\mathcal{C}$. If $\mathcal{C}$ contains at most $4$ lines
then it is irreducible by [CMM21, Corollary 4]. On the other hand, if
$\mathcal{C}$ contains $5$ or $6$ lines, then we will show that Theorem 4.5
applies, and so we reduce to a line configuration with fewer lines.

Let $\ell$ be any line of $\mathcal{C}$ and let
$S=\\{p\in\ell:|\mathcal{L}_{p}|\geq 3\\}$. We show that $|S|\leq 2$ by
contradiction. Assume that $|S|\geq 3$. So there are three distinct points
$p_{1},p_{2},p_{3}\in\ell$ such that $|\mathcal{L}_{p_{i}}|\geq 3$ for each
$i\in[3]$. Note that a pair of lines can intersect in at most one point. So if
$\ell^{\prime}\neq\ell$ is a line passing through $p_{1}$ then it does not
pass through $p_{2}$ or $p_{3}$. Since $|\mathcal{L}_{p_{i}}|\geq 3$ for each
$i$, the total number of distinct lines passing through $p_{1},p_{2}$ and
$p_{3}$ is at least $7$, a contradiction. By induction,
$\Gamma_{\mathcal{C}\backslash\ell}$ is irreducible. So by Theorem 4.5 we have
that $\Gamma_{\mathcal{C}}$ is irreducible. ∎

######  Remark 4.7.

The proof of Theorem 4.2 shows that Theorem 4.5 applies to all line
configurations with at most $6$ lines. However, for configurations with $7$
lines, consider the Fano plane depicted in Figure 3. This configuration
contains $7$ points each of which belongs to $3$ lines. Therefore, Theorem 4.5
does not apply.

##  5 Combinatorial closures of configurations

In this section, we will focus on point and line configurations whose
underlying graph is a forest. As in the previous section, if
$\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ is a point and line
configuration, then each line in $\mathcal{L}$ contains at least three points
in $\mathcal{P}$, unless stated otherwise.

We will give a complete description of their combinatorial closures by
describing their components. Our main tool for this section is the
perturbation argument which gives a way to determine when a particular point
in the combinatorial closure belongs to a specific realization space. Let us
begin by making precise our notation.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a point and line
configuration with the ordered set of points
$\mathcal{P}=\\{p_{1},\ldots,p_{n}\\}$. Let $G_{\mathcal{C}}$ be a graph with
vertex set $\mathcal{P}$ and edges $\\{p_{i},p_{j}\\}$ such that

  * •

both points lie on the same line $\ell\in\mathcal{L}$ (which is necessarily
unique) and

  * •

there is no point $p_{k}$ lying on $\ell$ with $i<k<j$ (or $j<k<i$).

Intuitively, $G_{\mathcal{C}}$ is obtained from $\mathcal{C}$ by making points
into vertices and lines into paths. Moreover, every edge of $G_{\mathcal{C}}$
is in exactly one of these paths. Each pair of paths intersects in at most one
vertex, and this occurs if and only if the corresponding lines intersect in a
point of $\mathcal{P}$.

######  Definition 5.1.

A configuration $\mathcal{C}$ is _forest-like_ if the corresponding graph
$G_{\mathcal{C}}$ is a forest, i.e. it has no cycle.

Note that the graph $G_{\mathcal{C}}$ depends on the choice of ordering of the
points in $\mathcal{P}$. However, the condition for $G_{\mathcal{C}}$ to be a
forest is independent of this choice. Essentially, swapping the order of two
points on a line corresponds to simply “flipping” an edge of
$G_{\mathcal{C}}$. We now make this more precise.

######  Lemma 5.2.

If $\mathcal{C}$ is forest-like, then any ordering of its points $\mathcal{P}$
makes $G_{\mathcal{C}}$ a forest.

###### Proof.

Let $\mathcal{P}=\\{p_{1},\ldots,p_{n}\\}$ be an ordering of $\mathcal{P}$
such that $G_{\mathcal{C}}$ is a forest. Inductively, it suffices to show that
the order $\\{p_{1},\ldots,p_{i-1},p_{i+1},p_{i},p_{i+2},\ldots,p_{n}\\}$
gives rise to a forest.

If $p_{i}$ and $p_{i+1}$ do not lie on the same line, then there is no change
to $G_{\mathcal{C}}$. So it suffices to consider the case where $p_{i}$ and
$p_{i+1}$ lie on a line $\ell\in\mathcal{L}$. Then $\\{p_{i},p_{i+1}\\}$ is an
edge in $G_{\mathcal{C}}$. Let us call these vertices of the graph $v$ and
$w$. Let $A(v)$ be the set of vertices of $G_{\mathcal{C}}$, adjacent to $v$,
whose corresponding points in $\mathcal{C}$ lie on some line
$\ell^{\prime}\neq\ell$, and define $A(w)$ similarly. Then swapping the order
of $i$ and $i+1$ corresponds to replacing the edges in $\\{\\{x,v\\}:x\in
A(v)\\}\cup\\{\\{x,w\\}:x\in A(w)\\}$ with the edges in $\\{\\{x,v\\}:x\in
A(w)\\}\cup\\{\\{x,w\\}:x\in A(v)\\}$. One can see that this resulting graph
is still a forest if $G_{\mathcal{C}}$ was a forest. ∎

Because of Lemma 5.2, Definition 5.1 is well-defined for point and line
configurations with unordered sets of vertices. So, we say that $\mathcal{C}$
is forest-like if there exists an ordering of its vertices such that the
resulting configuration is forest-like.

The next result will also be helpful.

######  Lemma 5.3.

If $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ is a forest-like
configuration with $\mathcal{L}\neq\emptyset$, then there exists a line $\ell$
which intersects $\mathcal{C}\setminus\ell$ in at most one point of
$\mathcal{P}$.

###### Proof.

Because two lines intersect in at most one point, it will suffice to show that
there exists a line $\ell$ which intersects $\mathcal{C}\setminus\ell$ (at a
point of $\mathcal{P}$) in at most one line. Suppose otherwise for a
contradiction. Then every line intersects with at least two other lines. Let
$\ell_{1},\ell_{2},\ldots$ be a sequence of lines in $\mathcal{C}$ such that
$\ell_{i}$ intersects both $\ell_{i-1}$ and $\ell_{i+1}$. Since there are only
finitely many lines, eventually there will be a repeated element $\ell_{k}$ of
the sequence.

Since these lines correspond to paths in $G_{\mathcal{C}}$, there is a cycle
in $G_{\mathcal{C}}$ whose set of edges includes at least one edge from each
path corresponding to the lines in the sequence
$\ell_{k},\ell_{k+1},\ldots,\ell_{k}$. This cycle contradicts the assumption
that $G_{\mathcal{C}}$ is a forest. ∎

It is perhaps worth noting that forest-likeness is a rather restrictive
condition for a matroid of rank $3$. However, forest-like configurations are
not regular (realizable over every field) matroids. For example, no line with
four or more points is a regular matroid. Each forest-like configuration is
realizable over all infinite fields. To show this, it will be useful to recall
the matroid-theoretic notion of _freely adding_ an element to a flat of a
matroid. Intuitively, this operation takes a new element of the ground set and
adds it to a flat as “freely” as possible, that is, we keep as many sets
independent as possible; see [Cra65] and [Oxl11, Section 7.2]. We will also
use this notion in Section 6.3.

######  Definition 5.4.

Let $F_{0}$ be a flat of a matroid $M$ with rank function $r_{M}$. We say that
$M^{\prime}$ is the single-element extension obtained by _freely adding_ $e$
to $F_{0}$ if the flats of $M^{\prime}$ fall into the following disjoint
classes:

  * •

flats $F$ of $M$ that do not contain $F_{0}$,

  * •

sets $F\cup e$ where $F$ is a flat of $M$ that contains $F_{0}$, and

  * •

sets $F\cup e$ where $F$ is a flat of $M$ that does not contain $F_{0}$, and
there is no flat $F^{\prime}$ of $M$ of rank $r_{M}(F)+1$ such that
$F\subseteq F^{\prime}$ and $F_{0}\subseteq F^{\prime}$.

The following results are fairly well-known. In particular, Lemma 5.5 follows
from a result of Piff and Welsh [PW70]; see also [Oxl11, Proposition 11.2.16].
We provide a more direct proof to keep the paper self-contained.

######  Lemma 5.5.

Let $\mathbb{F}$ be an infinite field, and let $M^{\prime}$ be a matroid
obtained by freely adding an element to a flat $F$ of a matroid $M$. Then
$M^{\prime}$ is $\mathbb{F}$-realizable if and only if $M$ is
$\mathbb{F}$-realizable.

###### Proof.

Realizability over a field is closed under deletion. Therefore, if
$M^{\prime}$ is $\mathbb{F}$-realizable, then so is $M$.

For the converse, let $M$ be $\mathbb{F}$-realizable, and let $r$ be the rank
function of $M$. A flat $F$ of $M$ corresponds to a linear subspace of
$\mathbb{F}^{r(M)}$ of dimension $r(F)$. To construct a representation of
$M^{\prime}$ we must find a point $p$ in the subspace corresponding to $F$
that is not contained in any subspace corresponding to a flat $F^{\prime}$ of
$M$ that does not contain $F$. For any such flat $F^{\prime}$, the rank of
$F\cap F^{\prime}$ is strictly less than $r(F)$. Therefore, since $\mathbb{F}$
is an infinite field, there are infinitely many such points $p$. ∎

######  Lemma 5.6.

Let $M^{\prime}$ be obtained by adding a coloop to an $\mathbb{F}$-realizable
matroid $M$. Then $M^{\prime}$ is $\mathbb{F}$-realizable.

###### Proof.

Let $A$ be a matrix whose columns form a realization of $M$. We construct a
matrix realizing $M^{\prime}$ by adding a row and column to $A$ such that all
entries of this row and column are $0$ except for the entry contained in both
the row and column. ∎

######  Proposition 5.7.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a forest-like point
and line configuration, and let $M$ be the simple matroid of rank at most $3$
associated with it. Then $M$ is realizable over all infinite fields.

###### Proof.

We proceed by induction on the number of lines in $\mathcal{L}$. If
$|\mathcal{L}|=0$, then $M$ is the uniform matroid $U_{r,|\mathcal{P}|}$,
where $r=\min\\{3,|\mathcal{P}|\\}$. For all positive integers $n$, it is
clear that $U_{n,n}$ is realizable over all fields. If $n>3$, then $U_{3,n}$
is obtained from $U_{3,3}$ by repeatedly freely adding elements to the
rank-$3$ flat of the matroid. Therefore, by Lemma 5.5, $U_{r,|\mathcal{P}|}$
is realizable over all infinite fields for all values of $n$.

Now, suppose that we have shown that the result holds for all forest-like
configurations with at most $k-1$ lines and we wish to prove the result for
$\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$, where $|\mathcal{L}|=k$.
By Lemma 5.3, there is a line $\ell$ in $\mathcal{L}$ that intersects with at
most one of the points in $\mathcal{C}\setminus\ell$. By the induction
hypothesis, the matroid associated with $\mathcal{C}\setminus\ell$ is
realizable over all infinite fields.

First, we consider the case where $\ell$ intersects with a point $v$ in
$\mathcal{C}\setminus\ell$. Let $w$ be an additional point in $\ell$, and let
$T=\ell\setminus\\{v,w\\}$. The matroid $M\backslash T$ is obtained from
$\mathcal{C}\setminus\ell$ either by adding $w$ as a coloop or by freely
adding $w$ to the unique rank-$3$ flat (depending on the rank of
$\mathcal{C}\setminus\ell$). Then $M$ is obtained from $M\backslash T$ by
freely adding the points in $T$ to the rank-$2$ flat defined by $v$ and $w$.

Now, we consider the case where $\ell$ and $\mathcal{C}\setminus\ell$ have no
points in common. Let $w_{1}$ and $w_{2}$ be points in $\ell$, and let
$T=\ell\setminus\\{w_{1},w_{2}\\}$. The matroid
$M\backslash(T\cup\\{w_{2}\\})$ is obtained from $\mathcal{C}\setminus\ell$
either by adding $w_{1}$ as a coloop or by freely adding $w_{1}$ to the unique
rank-$3$ flat (depending on the rank of $\mathcal{C}\setminus\ell$). Then
$M\backslash T$ is obtained from $M\backslash(T\cup\\{w_{2}\\})$ by freely
adding $w_{2}$ to the unique rank-$3$ flat. Then $M$ is obtained from
$M\backslash T$ by freely adding the points in $T$ to the rank-$2$ flat
defined by $w_{1}$ and $w_{2}$.

In either case, Lemmas 5.5 and 5.6 imply that $M$ is realizable over all
infinte fields. ∎

One can see that the last line $\ell$ to be added to a forest-like
configuration in the inductive process described in the proof of Proposition
5.7 has the property that $|\\{p\in\ell:|\mathcal{L}_{p}|\geq 3\\}|\leq 1$. In
particular, the only point $p\in\ell$ that might have the property that
$|\mathcal{L}_{p}|\geq 3$ is the one point that was on at least one of the
other $|\mathcal{L}|-1$ lines in $\mathcal{L}$.

So, we have the following straightforward corollary of Theorem 4.5.

######  Corollary 5.8.

The realization space of a forest-like configuration is irreducible with
respect to Zariski topology.

The following proof introduces the _perturbation procedure_ which we will use
throughout this section.

######  Lemma 5.9 (Perturbation procedure).

Let $\mathcal{C}$ be a forest-like configuration with realization space
$\Gamma_{\mathcal{C}}\subseteq\mathbb{C}^{d}$. Assume that
$|\mathcal{L}_{p}|\leq 2$ for every $p\in\mathcal{P}$. For every $\epsilon>0$
and for any $A\in
V^{\mathrm{comb}}_{\mathcal{C}}\backslash\Gamma_{\mathcal{C}}$ there exists
$A^{\prime}\in\Gamma_{\mathcal{C}}$ such that $||A-A^{\prime}||<\epsilon$,
where $||\cdot||$ is the Euclidean norm on $\mathbb{C}^{d\times n}$.

###### Proof.

We think of $A$ as a realization of a configuration $\mathcal{C}_{A}$. As
$A\in V^{\mathrm{comb}}_{\mathcal{C}}\backslash\Gamma_{\mathcal{C}}$ and the
rank of the corresponding matroid is at most $3$, the dependencies satisfied
by the configuration $\mathcal{C}_{A}$ that are not in $\mathcal{C}$ are the
following:

  * •

A point of $\mathcal{C}$ may be a loop in $\mathcal{C}_{A}$,

  * •

Two distinct points of $\mathcal{C}$ may coincide in $\mathcal{C}_{A}$,

  * •

Two distinct lines in $\mathcal{C}$ may coincide in $\mathcal{C}_{A}$,

  * •

A triple of non-collinear points in $\mathcal{C}$ may lie on a common line in
$\mathcal{C}_{A}$.

We now construct the realization $A^{\prime}$ by induction on the number of
lines. For the base case, assume that the configuration $\mathcal{C}_{A}$ has
no lines. Let us form $A^{\prime}$ by going through the points in order
$p_{1},\dots,p_{r}$ and perturbing the corresponding vector in $A$ by at most
$\epsilon/r$. For each $i\in[r]$, the vector corresponding to point $p_{i}$ is
perturbed such that it forms no dependencies, listed above, with subsets of
points from $\\{p_{j}:1\leq j\leq i\\}$.

For the induction step, let us assume that for any forest-like configuration
$\mathcal{C}^{\prime}=(\mathcal{P}^{\prime},\mathcal{L}^{\prime},\mathcal{I}^{\prime})$
with at most $n-1$ lines, we have that for all $\epsilon>0$ and any point
$x\in V^{\mathrm{comb}}_{\mathcal{C}^{\prime}}$ there exists
$x^{\prime}\in\Gamma_{\mathcal{C}^{\prime}}$ such that
$||x-x^{\prime}||<\epsilon$. Since the graph $G_{\mathcal{C}}$ is a forest, we
may take $\ell\in\mathcal{L}$ to be a line which intersects
$\mathcal{C}\backslash\ell$ in at most one point. By the inductive hypothesis,
we can perturb vectors in $A$ corresponding to the configuration
$\mathcal{C}\backslash\ell$ so that no additional dependencies are satisfied.
It remains to show that the points on the line $\ell$ may be perturbed to
remove all additional dependencies. We proceed by applying some or all of the
following steps in order. In particular, after each step we ensure that no new
dependencies are introduced which would be removed in a previous step.

  1. (a)

Suppose that the line $\ell$ intersects a point $p$ in $\mathcal{C}_{A}$ which
does not belong to $\ell$ in $\mathcal{C}$. We may perturb the line $\ell$ in
$A$ to produce $A^{\prime}$ so that $\ell$ does not pass through the point
$p$. In particular, if $\ell$ contains $k$ points which are not $p$, then we
perturb each point by at most $\epsilon/k$.

  2. (b)

Suppose that the line $\ell$ coincides with another line $\ell^{\prime}$ in
$\mathcal{C}_{A}$. Since $\ell$ intersects $\mathcal{C}\backslash\ell$ in at
most one point, we may rotate the line $\ell$ in $A$ (i.e. perturb each point
along the line) by small amount to obtain $A^{\prime}$ so that $\ell$ does not
coincide with any other line. In particular, if $\ell$ contains $k$ points
which do not lie on the intersection of $\ell$ and $\ell^{\prime}$ in
$\mathcal{C}$, then we perturb each point by at most $\epsilon/k$.

  3. (c)

Suppose that two points in $\ell$ coincide or some of the points are loops. We
may perturb these points away from each other so that they remain on the line
$\ell$. Similarly if any of the points are loops, then we can perturb these
points away from zero by the same method. In particular, if a point $p$ in
$\mathcal{C}_{A}$ is a loop then we recall that $p$ is incident to at most two
lines. If it is incident to exactly two lines $\ell,\ell^{\prime}$ in
$\mathcal{C}$, then we may perturb $p$ along $\ell\cap\ell^{\prime}$.
Otherwise if $p$ is incident only to $\ell$ in $\mathcal{C}$ then we may
perturb it to some non-zero point on $\ell$.

  4. (d)

Suppose that for some point $p_{i}$ in $\ell$, there are two distinct points
$p_{j},p_{k}$ in $\mathcal{C}\backslash\ell$ such that $p_{i},p_{j}$ and
$p_{k}$ lie on a line $\ell^{\prime}$ in $\mathcal{C}_{A}$ but do not lie on a
line in $\mathcal{C}$. By the step (a) of the procedure, we have that $\ell$
and $\ell^{\prime}$ are distinct lines. For this case, it is useful to
consider all lines of $\mathcal{C}_{A}$, including those which contain only
two points. The points of intersection between $\ell$ and all other distinct
lines of $\mathcal{C}_{A}$ is a finite set. Since we work over the infinite
field $\mathbb{C}$ and the line is homeomorphic to $\mathbb{C}^{1}$ with
respect to Euclidean topology, therefore we may perturb $p_{i}$ along $\ell$
so that it lies only on $\ell$ and no other line.

As a result of the above procedure, we have constructed a realization
$A^{\prime}$ of $\mathcal{C}$ with $||A-A^{\prime}||<\epsilon$. ∎

Figure 5: Depiction of the steps in the perturbation procedure: (a) line
$\ell$ is perturbed away from point $4$, (b) points $2,3\in\ell^{\prime}$ are
perturbed off $\ell$, (c) coincident points $1,2,3$ are perturbed away from
each other along $\ell$ and (d) point $1$ is perturbed along $\ell$ away from
the intersection.

######  Remark 5.10.

By perturbation procedure, the corresponding configurations are realizable
over any subfield of $\mathbb{C}$.

Let $\mathcal{C}$ be the configuration with two lines $123$ and $345$.
Consider the following points in $V^{\mathrm{comb}}_{\mathcal{C}}$:

$\displaystyle A=\begin{bmatrix}1&1&0&0&0\\\ 0&1&0&1&0\\\
0&0&0&1&1\end{bmatrix}\quad\text{and}\quad
A^{\prime}=\begin{bmatrix}1&1&0&0&0\\\ 0&1&\epsilon&1&0\\\
0&0&0&1&1\end{bmatrix}.$ (2)

Note that $A\notin\Gamma_{\mathcal{C}}$ is not a realization of $\mathcal{C}$
because $3$ is a loop in $\mathcal{C}_{A}$. See Figure 6. Following the
perturbation argument, we perturb $A$ to $A^{\prime}$ which corresponds to
moving point $3$ so that it is non-zero and lies on the intersection of the
planes spanned by $1$, $2$ and $4$, $5$.

Figure 6: On the right a configuration $\mathcal{C}$ and on the left the
configuration $\mathcal{C}_{A}$ for the matrix $A$ in (2), together with a
depiction of the perturbation of point $3$ taking $A\in
V^{\mathrm{comb}}_{\mathcal{C}}\backslash\Gamma_{\mathcal{C}}$ to
$A^{\prime}\in\Gamma_{\mathcal{C}}$. The loop $3$ in $\mathcal{C}_{A}$ is
shown in a square and the shaded planes in $\mathbb{C}^{3}$ correspond to the
lines in $\mathcal{C}_{A}$ and $\mathcal{C}$.

######  Theorem 5.11.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a forest-like
configuration with $|\mathcal{L}_{p}|\leq 2$ for every $p\in\mathcal{P}$. Then
the combinatorial closure $V^{\mathrm{comb}}_{\mathcal{C}}$ coincides with the
matroid variety $V_{\mathcal{C}}$ and in particular is irreducible.

###### Proof.

Recall that $V^{\mathrm{comb}}_{\mathcal{C}}=V_{C(\mathcal{C})}$ where
$C(\mathcal{C})$ denotes the circuits of the matroid corresponding to
$\mathcal{C}$. So it remains to show that
$V_{C(\mathcal{C})}=V_{\mathcal{C}}$. It is clear that
$V_{\mathcal{C}}\subseteq V_{C(\mathcal{C})}$. For the opposite inclusion, by
the perturbation procedure each $A\in V_{C(\mathcal{C})}$ is a limit point of
the configuration space $\Gamma_{\mathcal{C}}$ with respect to the Euclidean
topology. This implies $V_{C(\mathcal{C})}\subseteq V_{\mathcal{C}}$, since
the Zariski topology is coarser than the Euclidean topology. ∎

######  Definition 5.12.

Let $\mathcal{C}$ be a point and line configuration and $p$ a point. Let
$\mathcal{D}$ be the dependent sets of the matroid associated to $\mathcal{C}$
on ground set $\mathcal{P}$. By setting $P\subseteq\mathcal{P}$ to be loops,
we mean the matroid $M$ whose dependent sets are given by
$\mathcal{D}\cup\\{D\subseteq\mathcal{P}\colon P\cap D\neq\emptyset\\}$. In
other words the circuits of $M$ are: the circuits of the matroid associated to
$\mathcal{C}$ which do not meet $P$; and $P$ thought of as a set of loops.

Before proving an analogous result for a general forest-like point and line
configuration, we consider the following simple configuration which has at
most one point that lies on more than two lines.

######  Lemma 5.13.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a forest-like
configuration which has a unique point $p:=p_{i}$ such that
$|\mathcal{L}_{p}|>2$ and for all other points $p_{j}$ we have
$|\mathcal{L}_{p_{j}}|\leq 2$. Then the combinatorial closure
$V^{\mathrm{comb}}_{\mathcal{C}}$ has exactly two irreducible components: one
is the central component and the other arises by setting the intersection
point $p$ to be a loop.

###### Proof.

For each element $A\in V^{\mathrm{comb}}_{\mathcal{C}}$, we write
$A_{p}\in\mathbb{C}^{3}$ for the column vector corresponding to the
intersection point $p$. We have that $V_{\mathcal{C}}$ is the central
component of $V^{\mathrm{comb}}_{\mathcal{C}}$ and we write $V_{0}=\\{A\in
V^{\mathrm{comb}}_{\mathcal{C}}:A_{p}=\underline{0}\\}$ for the collection of
elements of the combinatorial closure where the intersection point is zero.
Let $\mathcal{C}_{0}$ be the configuration obtained from $\mathcal{C}$ by
setting the intersection point to be a loop. Clearly, we have
$V_{\mathcal{C}_{0}}\cup V_{\mathcal{C}}\subseteq V_{0}\cup
V_{\mathcal{C}}\subseteq V^{\mathrm{comb}}_{\mathcal{C}}$. Since
$\Gamma_{\mathcal{C}_{0}}$ and $\Gamma_{\mathcal{C}}$ are irreducible by
Corollary 5.8, the lemma follows from showing that $V_{\mathcal{C}_{0}}\cup
V_{C}=V^{\mathrm{comb}}_{\mathcal{C}}$ by proving the opposite inclusion.

Take any element $A\in V^{\mathrm{comb}}_{\mathcal{C}}$ and fix $\epsilon>0$.
We will show that there exists
$A^{\prime}\in\Gamma_{\mathcal{C}_{0}}\cup\Gamma_{\mathcal{C}}$ such that
$||A-A^{\prime}||<\epsilon$ by applying the perturbation procedure. Let us
take cases on whether $A_{p}$ is zero.

Case 1. Assume that $A_{p}=\underline{0}$. Since $\mathcal{C}_{0}$ is a
configuration that contains only points $p_{j}$ with
$|\mathcal{L}_{p_{j}}|\leq 2$, we may apply the perturbation procedure to
$\mathcal{C}_{0}\backslash p$. As a result we have
$A^{\prime}\in\Gamma_{\mathcal{C}_{0}}$.

Case 2. Assume that $A_{p}\neq\underline{0}$. We may now apply steps (a) to
(d) in the proof of the perturbation procedure to construct a point
$A^{\prime}\in\Gamma_{\mathcal{C}}$ with $||A-A^{\prime}||<\epsilon$. In the
procedure, by assumption we have that $A_{p}\neq\underline{0}$. This
assumption guarantees that if $p,p_{1},p_{2}$ are collinear points in
$\mathcal{C}$ then the corresponding points are collinear in
$\mathcal{C}_{A}$, and all lines arising in this way pass through the common
point $p$. ∎

We are now ready to state our main result in this section for a general
forest-like point and line configuration.

######  Theorem 5.14.

Let $\mathcal{C}=(\mathcal{P},\mathcal{L},\mathcal{I})$ be a forest-like
configuration. Let $S=\\{p\in\mathcal{P}:|\mathcal{L}_{p}|\geq 3\\}$ be the
collection of points contained in at least $3$ lines. Then the combinatorial
closure $V^{\mathrm{comb}}_{\mathcal{C}}$ has at most $2^{|S|}$ irredundant
irreducible components. Moreover, these components can be obtained from
$\mathcal{C}$ by setting a subset of $S$ to be loops.

###### Proof.

For each element $A\in V^{\mathrm{comb}}_{\mathcal{C}}$ and point
$p\in\mathcal{P}$, we write $A_{p}\in\mathbb{C}^{3}$ for the column vector
corresponding to the point $p$. For each subset $J\subseteq S$, we write
$\mathcal{C}_{J}$ for the configuration obtained from $\mathcal{C}$ by setting
the points in $J$ to be loops. We will show that $\bigcup_{J\subseteq
S}V_{\mathcal{C}_{J}}=V^{\mathrm{comb}}_{\mathcal{C}}$. Note that by Corollary
5.8, each variety $V_{\mathcal{C}_{J}}$ is irreducible. By construction, it is
immediately clear that $V_{\mathcal{C}_{J}}\subseteq
V^{\mathrm{comb}}_{\mathcal{C}}$ for each $J$. So, to prove the theorem, it
remains to show the opposite inclusion, i.e.
$V^{\mathrm{comb}}_{\mathcal{C}}\subseteq\bigcup_{J\subseteq
S}V_{\mathcal{C}_{J}}$. Fix $A\in V^{\mathrm{comb}}_{\mathcal{C}}$ and
$\epsilon>0$. We will construct $A^{\prime}\in\Gamma_{\mathcal{C}_{J}}$ for
some $J\subseteq S$ such that $||A-A^{\prime}||<\epsilon$.

We proceed by induction on $s:=|S|$. If $s=1$, then the result follows by
Lemma 5.13. So let us assume that $s>1$. Since $G_{\mathcal{C}}$ is a forest,
we can find a point $p\in S$ such that the path (if it exists) between any
other pair of points in $S$ does not pass through $p$. Let
$P\subseteq\mathcal{P}$ be the collection of points $q\in\mathcal{P}$ lying in
the same connected component of $G_{\mathcal{C}}$ as $p$, such that for all
$p^{\prime}\in S\backslash\\{p\\}$, if there is a path from $q$ to
$p^{\prime}$, then the path passes through $p$. By convention, we assume $p\in
P$.

Consider the configuration
$\mathcal{C}^{\prime}=(\mathcal{P}^{\prime},\mathcal{L}^{\prime},\mathcal{I}^{\prime})$
obtained from $\mathcal{C}$ by removing points $P$. We have
$S^{\prime}:=\\{p\in\mathcal{P}^{\prime}:|\mathcal{L}^{\prime}|\geq
3\\}=S\backslash\\{p\\}$. Clearly, $\mathcal{C}^{\prime}$ is a forest-like
configuration and $|S^{\prime}|=s-1$. So, by induction, we can perturb the
points in $\mathcal{P}^{\prime}$ so that they lie in a realization space
$\Gamma_{\mathcal{C}^{\prime}_{J^{\prime}}}$ for some $J^{\prime}\subseteq
S^{\prime}$. More precisely, there exists a subset $J^{\prime}\subseteq
S\backslash\\{p\\}$ and a realization
$A_{\mathcal{P}^{\prime}}^{\prime}\in\Gamma_{\mathcal{C}^{\prime}_{J^{\prime}}}$
such that
$||A_{\mathcal{P}^{\prime}}-A_{\mathcal{P}^{\prime}}^{\prime}||<\epsilon/2$,
where $A_{\mathcal{P}^{\prime}}$ is the set of vectors obtained from $A$ by
removing the points $P$. We proceed by taking cases on whether $A_{p}$ is
zero.

Case 1. Assume that $A_{p}=\underline{0}$. Let $J=J^{\prime}\cup\\{p\\}$, we
construct $A^{\prime}\in\Gamma_{\mathcal{C}_{J}}$ from
$A^{\prime}_{\mathcal{P}^{\prime}}$ by taking the vectors in $A$ for the
points in $P\backslash\\{p\\}$ and applying the perturbation procedure. By the
perturbation procedure we have ensured that
$A^{\prime}\in\Gamma_{\mathcal{C}_{J}}$ and by perturbing each point $p\in P$
by a distance of at most $\epsilon/2|P|$, we have that
$||A-A^{\prime}||\leq||A_{\mathcal{P}^{\prime}}-A^{\prime}_{\mathcal{P}^{\prime}}||+(|P|-1)(\epsilon/2|P|)<\epsilon$.

Case 2. Assume that $A_{p}\neq\underline{0}$. Let $J=J^{\prime}$ and construct
$A^{\prime}\in\Gamma_{\mathcal{C}_{J}}$ from
$A^{\prime}_{\mathcal{P}^{\prime}}$ by taking the vectors in $A$ for points in
$P$ and applying steps (a) to (d) in the proof of the perturbation procedure.
In the procedure, by assumption we have that $A_{p}\neq\underline{0}$. This
assumption guarantees that all lines of $\mathcal{C}$ passing through $p$ are
contained in a line of $\mathcal{C}_{A}$ passing through $p$. As a result we
obtain a realization $A^{\prime}\in\Gamma_{\mathcal{C}_{J}}$ and by perturbing
each point $p\in P$ by a distance of at most $\epsilon/2|P|$, we have that
$||A-A^{\prime}||\leq||A_{\mathcal{P}^{\prime}}-A^{\prime}_{\mathcal{P}^{\prime}}||+|P|(\epsilon/2|P|)<\epsilon$.
∎

######  Remark 5.15.

It is not hard (but a bit tedious) to classify all irreducible components of
$V^{\mathrm{comb}}_{\mathcal{C}}$ for forest-like configurations
$\mathcal{C}$. This classification follows from the same proof of Theorem 5.14
by taking into account that components may become redundant; see Example 5.16.

######  Example 5.16.

Consider the configuration $\mathcal{C}$ in Figure 7. Note that the points
$1,2$ and $3$ have degree $3$ in the graph $G_{\mathcal{C}}$. So by Theorem
5.14, $V^{\mathrm{comb}}_{\mathcal{C}}$ has at most $8$ irredundant
irreducible components which arise from setting the intersection points:
$1,2,$ and $3$ to be loops in some combination. When points are set to loops,
we remove the lines which contain at most two non-loop points, as they do not
contribute any new dependencies. The variety of $\mathcal{C}_{\\{1\\}}$
appears as an irredundant component of $V^{\mathrm{comb}}_{\mathcal{C}}$.
However $\mathcal{C}_{\\{1,2\\}}$ gives rise to a redundant component since
its variety is contained the variety of $\mathcal{C}_{\\{1\\}}$. The
irreducible decomposition of $V^{\mathrm{comb}}_{\mathcal{C}}$ has $4$
irredundant components which are: the central component $V_{\mathcal{C}}$, and
the non-central components
$V_{\mathcal{C}_{\\{1\\}}},V_{\mathcal{C}_{\\{2\\}}},V_{\mathcal{C}_{\\{3\\}}}$.

Figure 7: Depiction of $\mathcal{C}$, $\mathcal{C}_{\\{1\\}}$ and
$\mathcal{C}_{\\{1,2\\}}$ in Example 5.16. Shaded squares represent loops of
the configuration.

##  6 Consecutive forest hypergraphs $\Delta_{G}$

We fix a $d\times n$ matrix $X=(x_{i,j})$ of indeterminates and the polynomial
ring $R=\mathbb{C}[X]$. Throughout this section, we fix a forest $G$ (acyclic
graph) with vertices labeled $1,\dots,n$.

######  Definition 6.1.

We define the consecutive forest hypergraph of $G$ as

$\displaystyle\Delta_{G}=\min\left(P_{2}(G)\cup\binom{[n]}{4}\right)\quad\text{where}\quad
P_{2}(G)=\\{P\subseteq V(G):\ P\text{ is a $2$-path in $G$}\\}.$

We recall that a $k$-path $P$ in $G$ is a subgraph of $G$ whose vertex set is
a non-repeating sequence $p_{1},\dots,p_{k+1}\subseteq V(G)$ and whose edge
set is $\\{\\{p_{i},p_{i+1}\\}:1\leq i\leq k\\}$. We identify $P$ with its
vertex set, and simply write $C\subseteq P$ instead of $C\subseteq V(P)$. If
$Q$ is another path, then $P\cup Q$ is the set of all vertices of $G$ lying in
$P$ and $Q$.

######  Remark 6.2.

The hypergraph ideal $I_{\Delta_{G}}$ of the $n$-path $G$ is exactly the ideal
$I_{3n}(3)$ of adjacent minors studied in [HS04]. We note that our
construction generalises this particular family by allowing $G$ to be an
arbitrary graph. In [HS04], the ideals $I_{mn}(m)$ are studied for general
$m\geq 3$ which can be thought of as the consecutive forest hypergraph of the
$n$-path except with higher order minors. These ideals can be studied using
positroid varieties as described in [KLS13]. It is not too difficult to show
that each of the prime components of $I_{mn}(m)$ is a positroid variety. For
instance, one can use the characterisation of positroids via their excluded
minors [Oh09, Theorem 16] or the decomposition of the Grassmannian into
positroid varieties [KLS13, Section 5.2]. In particular, we note that the
defining ideals of positroid varieties are generated by minors [KLS13, Theorem
5.15]. However, this is no longer true for the consecutive forest hypergraph
varieties. More precisely, the varieties arising in the irreducible
decomposition of $V_{\Delta_{G}}$ are, in general, not positroid varieties;
for instance see Example 6.21.

### 6.1 Minimal matroids.

In this section, we will define so-called prime collections which are
collections of subsets of vertices of $G$. To each prime collection
$\mathcal{S}$ we will associate a unique matroid $M_{\mathcal{S}}$. We will
then see that such matroids are realizable over the real numbers and have
irreducible realization spaces. Moreover, we prove that such matroids appear
as minimal matroids for consecutive forest hypergraphs.

######  Definition 6.3.

Let $\mathcal{S}$ be a collection of singleton subsets of $[n]$. We say
$\mathcal{S}$ is a prime collection of singletons for $G$ if $\mathcal{S}$
satisfies the following inductive definition.

  * •

The empty set $\mathcal{S}=\emptyset$ is a prime collection of singletons.

  * •

If $|\mathcal{S}|\geq 1$, then
$\mathcal{S}=\\{\\{s_{1}\\},\dots,\\{s_{t}\\}\\}$ is a prime collection of
singletons if, for each natural number $i$ with $1\leq i\leq t$,
$\mathcal{S}\backslash\\{s_{i}\\}$ is a prime collection of singletons and
$s_{i}$ satisfies the following two rules:

    * 1.

$s_{i}$ is not a leaf or isolated vertex of $G^{\prime}$,

    * 2.

If $s_{i}$ is adjacent to a leaf of $G^{\prime}$ then it has degree at least
$3$ in $G^{\prime}$,

where $G^{\prime}$ is the induced subgraph of $G$ obtained by deleting the
vertices in the set $\\{s_{1},\dots,s_{i-1},s_{i+1},\dots,s_{t}\\}$.

For ease of notation and when it is clear, we denote a collection of
singletons $\mathcal{S}=\\{\\{s_{1}\\},\dots,\\{s_{t}\\}\\}$ as
$\\{s_{1},\dots,s_{t}\\}$.

######  Definition 6.4.

Let $\mathcal{S}$ be a collection of singleton subsets ${\rm
sing}(\mathcal{S})$ and $2$-subsets of $V(G)$. Let $G^{\prime}$ be the induced
subgraph of $G$ obtained by removing all vertices in ${\rm
sing}(\mathcal{S})$.

  * •

We say $\mathcal{S}$ is a prime collection for $G$ if:

    * –

The ${\rm sing}(\mathcal{S})$ is a prime collection of singletons for $G$.

    * –

The $2$-subsets in $\mathcal{S}$ are a subset of edges of $G^{\prime}$ such
that for every vertex $v\in V(G^{\prime})$ which is incident to an edge in
$\mathcal{S}$, there exists an edge $\\{v,w\\}$ of $G^{\prime}$ which is not
in $\mathcal{S}$.

  * •

We say $\mathcal{S}$ is a valid collection for $G$ if each $2$-subset is
disjoint from each singleton set.

Note that every prime collection is a valid collection, but the converse is
not true.

######  Example 6.5.

Consider the graph $G_{1}$ in Figure 8. Let
$\mathcal{S}_{1}=\\{1,23,24,45\\}$. We see that the singletons, i.e.
$\\{1\\}$, form a prime collection of singletons for $G_{1}$ because $1$ is
adjacent to a leaf of $G_{1}$ and has degree at least three. Then we consider
the induced subgraph $G^{\prime}$ of $G_{1}$ obtained by removing the vertex
$1$. The vertices which lie in some $2$-subset in $\mathcal{S}$ are shaded
yellow and vertices in $G^{\prime}$ which do not lie in any $2$-subset are
white. Since every yellow vertex is adjacent to a white vertex, we have that
the $2$-subsets form a prime collection of $2$-subsets for $G^{\prime}$.
Therefore, $\mathcal{S}_{2}$ is a prime collection for $G_{1}$.

On the other hand, in the graph $G_{2}$, the set
$\mathcal{S}_{2}=\\{1,2,34,45,67\\}$ is a valid collection since each
$2$-subset in $\mathcal{S}_{2}$ does not contain any of the singleton subsets.
However $\mathcal{S}$ is not a prime collection because $1$ is a leaf and so
the singletons do not form a prime collection of singletons. Additionally
$45\in\mathcal{S}$ is not an edge of $G_{2}$.

Figure 8: Graphs in Examples 6.5 and 6.7. (Left) $G_{1}$ with prime collection
$\mathcal{S}_{1}=\\{1,23,24,45\\}$ and clouds $234$ and $45$. (Right) $G_{2}$
has a valid collection $\mathcal{S}_{2}=\\{1,2,34,45,67\\}$ which is not
prime. The clouds are $345$ and $67$.

In the following, we show how a valid collection for $G$ can be extended to
the set of circuits of a dependent matroid for the consecutive forest
hypergraph $\Delta_{G}$.

######  Definition 6.6.

Let $\mathcal{S}$ be a valid collection.

  * •

Let $G(\mathcal{S})$ be the graph whose vertex set is given by the union of
all $2$-subsets of $\mathcal{S}$ and whose edges are the $2$-subsets of
$\mathcal{S}$. A _cloud of $\mathcal{S}$_ is the set of vertices of a
connected component of $G(\mathcal{S})$.

  * •

We say that a path $P:v_{0},v_{1},\dots,v_{t}$ in $G$ _crosses_ the cloud
containing $v_{j}$ and $v_{k}$ if there exists $i<j<k<\ell$ such that some
cloud of $\mathcal{S}$ contains $\\{v_{j},v_{k}\\}$ but no cloud of
$\mathcal{S}$ contains either $\\{v_{i},v_{j}\\}$ or $\\{v_{k},v_{\ell}\\}$.
(This may mean that $v_{i}$ or $v_{\ell}$ does not lie in any cloud.)

  * •

We say that the subset $A\subset V(G)$ is _blocked_ by $\mathcal{S}$ if there
exist $v,w\in A$ such that one of the following conditions holds:

    * (i)

The vertices $v$ and $w$ lie in different connected components of $G$.

    * (ii)

The vertices $v$ and $w$ lie in the same connected component of $G$ and the
unique path from $v$ to $w$ in $G$ crosses a cloud or contains a vertex $i\in
V(G)\backslash\\{v,w\\}$ such that $\\{i\\}$ is a singleton in $\mathcal{S}$.

We note that if a set $A\subseteq V(G)$ is not blocked by some valid set
$\mathcal{S}$ then it follows that $A$ is contained within a connected
component of $G$.

######  Example 6.7.

In Figure 8, we have illustrated a prime collection $\mathcal{S}_{1}$ for the
graph $G_{1}$ which contains two clouds $234$ and $45$. The path from $a$ to
$b$ crosses the cloud $234$ whereas the path from $c$ to $d$ does not cross
any cloud. Therefore any set of vertices containing $a$ and $b$ is blocked by
$\mathcal{S}$, but $\\{4,c,d\\}$ is not blocked by $\mathcal{S}$.

We also illustrate a non-prime valid collection $\mathcal{S}_{2}$ for the
graph $G_{2}$ which contains two clouds. The path from $a$ to $b$ crosses the
cloud $345$ since the path passes through both $3$ and $5$. However the path
from $c$ to $d$ does not cross any clouds of $\mathcal{S}_{2}$

We now give the construction of a matroid whose circuits include $\mathcal{S}$
for a given valid collection $\mathcal{S}$.

######  Proposition 6.8.

Let $\mathcal{S}$ be a valid collection. Let $\mathcal{C}$ be the collection
of subsets of $[n]$ consisting of the

  1. 1.

singletons in $\mathcal{S}$,

  2. 2.

$2$-subsets of clouds of $\mathcal{S}$,

  3. 3.

$3$-subsets of $[n]$ which are not blocked and do not contain any set in
$\mathcal{S}$, and

  4. 4.

$4$-subsets of $[n]$ containing none of the sets listed in 1, 2, or 3 above.

Then $\mathcal{C}$ is the collection of circuits of a matroid.

###### Proof.

We need to check that $\mathcal{C}$ satisfies the circuit elimination axiom.
That is, if $C_{1},C_{2}\in\mathcal{C}$ and $x\in C_{1}\cap C_{2}$, we must
check that there is a set in $\mathcal{C}$ contained in $(C_{1}\cup
C_{2})\backslash\\{x\\}$. This is clear if either $C_{1}$ or $C_{2}$ is a set
of size $1$ or $4$. If $C_{1}$ and $C_{2}$ both have size $2$, then $C_{1}$
and $C_{2}$ are contained in the same cloud. Thus, $(C_{1}\cup
C_{2})\backslash\\{x\\}$ is contained in the same cloud, implying that
$(C_{1}\cup C_{2})\backslash\\{x\\}\in\mathcal{C}$.

Suppose $|C_{1}|=2$ and $|C_{2}|=3$. Let $C_{1}=\\{v,x\\}$ and
$C_{2}=\\{x,y,z\\}$.

Since $C_{2}$ is not blocked, it is contained in a connected component of $G$.

Let $P_{1}$ be the path from $x$ to $y$; let $P_{2}$ be the path from $v$ to
$y$; and let $P_{3}$ be the path from $v$ to $x$. Since
$\\{x,y,z\\}\in\mathcal{C}$, we have $\\{x,y\\}\notin\mathcal{C}$. Therefore,
$y$ is not contained in the same cloud that contains $v$ and $x$. Since a
cloud contains every vertex on the path between two vertices in the cloud, we
see then that either $P_{3}\subseteq P_{1}$, or $P_{3}\subseteq P_{2}$, or
$P_{3}=P_{1}\cap P_{2}$. In either case, the fact that $\\{x,y\\}$ is not
blocked implies that $\\{v,y\\}$ is not blocked. Similarly, $\\{v,z\\}$ is not
blocked. We already know that $\\{y,z\\}$ is not blocked because
$\\{y,z\\}\subseteq\\{x,y,z\\}$. Therefore, we conclude that $\\{v,y,z\\}$ is
not blocked and contains a member of $\mathcal{C}$.

Finally, we consider the case where $|C_{1}|=|C_{2}|=3$. If $|C_{1}\cap
C_{2}|=1$, then $(C_{1}\cup C_{2})\backslash\\{x\\}$ has size $4$ and
therefore contains a member of $\mathcal{C}$. If $|C_{1}\cap C_{2}|=2$, let
$C_{1}=\\{v,x,y\\}$ and $C_{2}=\\{w,x,y\\}$. We will show that $(C_{1}\cup
C_{2})\backslash\\{x\\}=\\{v,w,y\\}$ contains a member of $\mathcal{C}$. Since
$\\{v,x,y\\}$ and $\\{w,x,y\\}$ are not blocked, the pairs $\\{v,y\\}$ and
$\\{w,y\\}$ are not blocked. Suppose for a contradiction that the pair
$\\{v,w\\}$ is blocked. Recalling that there is a unique path between any pair
of vertices in a connected component of a forest, one can see that either
$\\{v,x\\}$ or $\\{w,x\\}$ is blocked, a contradiction, or $x$ is contained in
the cloud that blocks $\\{v,w\\}$. So we deduce that $x$ is contained in the
cloud that blocks $\\{v,w\\}$. But then, either $\\{v,y\\}$ or $\\{w,y\\}$ is
blocked, or $x$ and $y$ are contained in the same cloud, a contradiction. ∎

######  Notation.

Given a valid collection $\mathcal{S}$, we denote by $M_{\mathcal{S}}$ its
corresponding matroid from Proposition 6.8.

######  Remark 6.9 (Flats of $M_{\mathcal{S}}$).

The set of loops $L$ of $M_{\mathcal{S}}$ is the set of singletons in
$\mathcal{S}$. Note that the loops of a matroid are contained in every flat of
the matroid. The flats of $M_{\mathcal{S}}$ of rank $1$ are either of the form
$B\cup L$, where $B$ is a cloud, or of the form $\\{x\\}\cup L$, where
$x\in[n]$ is a non-loop element contained in no cloud. The flats of rank $2$
are either of the form $F\cup L$, where $F$ is an inclusion-wise maximal set
that is not blocked, or of the form $F_{1}\cup F_{2}$, where $F_{1}$ and
$F_{2}$ are flats of rank $1$ such that $\\{x_{1},x_{2}\\}$ is blocked for
every pair $\\{x_{1},x_{2}\\}\in F_{1}\cup F_{2}$. Since, all $4$-subsets of
$[n]$ are dependent in $M_{\mathcal{S}}$, the matroid has rank at most $3$.

We now show that the matroids $M_{\mathcal{S}}$ associated to prime
collections $\mathcal{S}$ in Definition 6.4 are precisely the minimal matroids
for $\Delta_{G}$. Firstly, we observe that the simplification of
$M_{\mathcal{S}}$ is the matroid associated to a forest-like configuration.

######  Proposition 6.10.

Let $\mathcal{S}$ be a prime collection. Then the simplification of
$M_{\mathcal{S}}$ is the matroid of a forest-like configuration.

###### Proof.

By definition, the simplification $(M_{\mathcal{S}})^{s}$ is the matroid
obtained from $M_{\mathcal{S}}$ by deleting all its loops and deleting
elements from parallel classes such that each parallel class contains only one
element. Since $M_{\mathcal{S}}$ has rank at most $3$, it follows that
$(M_{\mathcal{S}})^{s}$ is completely determined by its dependent rank-$2$
flats (that is, the flats of rank $2$ with at least $3$ elements). We define
the point and line configuration $\mathcal{C}$ whose points are the elements
of the ground set of $(M_{\mathcal{S}})^{s}$ and whose lines are the dependent
rank-$2$ flats of $(M_{\mathcal{S}})^{s}$. Clearly, we have that the matroid
associated to $\mathcal{C}$ is equal to $(M_{\mathcal{S}})^{s}$. It remains to
show that $\mathcal{C}$ is forest-like.

We proceed by induction on $|\mathcal{S}|$. If $\mathcal{S}=\emptyset$, then
we have that the matroid associated to $\mathcal{C}$ is equal to
$M_{\mathcal{S}}$. For any labelling of the points in $\mathcal{C}$, we show
that $G_{\mathcal{C}}$ is a disjoint union of paths. By Definition 6.6, the
$3$-subsets of $V(G)$ contained in a connected component of $G$ are exactly
those which are not blocked by $\mathcal{S}$. So, by the definition of the
matroid $M_{\mathcal{S}}$, its dependent rank-$2$ flats are the connected
components of $G$. It follows immediately from the definition of the graph
$G_{\mathcal{C}}$ that its connected components are paths. Hence $\mathcal{C}$
is forest-like.

Assume that $|\mathcal{S}|\geq 1$. Then, either $\mathcal{S}$ contains a
singleton of $V(G)$ or an edge of $G$.

Case 1. Let $\\{v\\}\in\mathcal{S}$ be a singleton. By definition, we have
that $\mathcal{S}\backslash\\{v\\}$ is a prime collection. By induction, the
simplification $(M_{\mathcal{S}\backslash\\{v\\}})^{s}$ is the matroid of a
forest-like configuration $\mathcal{C}^{\prime}$. Since $v$ is not contained
in any cloud of $\mathcal{S}$, it follows that $v$ is contained in a unique
dependent rank-$2$ flat $F=\\{v,f_{1},\dots,f_{k}\\}$ of
$(M_{\mathcal{S}\backslash\\{v\\}})^{s}$. The dependent rank-$2$ flats of
$(M_{\mathcal{S}})^{s}$ are obtained from the dependent rank-$2$ flats of
$(M_{\mathcal{S}\backslash\\{v\\}})^{s}$ by replacing $F$ with a disjoint
collection of flats $F_{1},\dots,F_{d}$ which partitions
$\\{f_{1},\dots,f_{k}\\}$. Each flat $F_{i}$ corresponds to a neighbor of $v$
in $G$. To see this, let $A\subseteq V(G)\backslash v$ be a $3$-subset and
suppose that $A$ is not blocked by $\mathcal{S}\backslash\\{v\\}$. Then $A$ is
not blocked by $\mathcal{S}$ if and only if for any pair of vertices $x,y\in
A$ we have that the path from $x$ to $y$ in $G$ does not pass through $v$.
Therefore the point and line configuration $\mathcal{C}$ is obtained from
$\mathcal{C}^{\prime}$ by replacing the line containing $v$ with $d$ non-
intersecting lines. Since $\mathcal{C}^{\prime}$ is forest-like, it follows
that $\mathcal{C}$ is also forest-like.

Case 2. Let $\\{v,w\\}\in\mathcal{S}$ be an edge of $G$. Since $\mathcal{S}$
is a prime collection, the induced subgraph $G^{\prime}$ of $G$ on the
vertices of the cloud of $\mathcal{S}$ containing $\\{v,w\\}$ is connected. By
assumption $G$ is a forest, hence $G^{\prime}$ is forest. So, without loss of
generality, we may assume that $v$ is a leaf in $G^{\prime}$. By the
definition of a prime collection, it follows that
$\mathcal{S}\backslash\\{v,w\\}$ is a prime collection. By induction, the
simplification $(M_{\mathcal{S}\backslash\\{v,w\\}})^{s}$ is the matroid of a
forest-like configuration $\mathcal{C}^{\prime}$. Let $F$ be the unique
dependent flat of rank $2$ of $(M_{\mathcal{S}\backslash\\{v,w\\}})^{s}$ that
contains $v$. The dependent rank-$2$ flats of $(M_{\mathcal{S}})^{s}$ are
obtained from the dependent rank-$2$ flats of
$(M_{\mathcal{S}\backslash\\{v,w\\}})^{s}$ by replacing $F$ with a collection
of flats $F_{1},\dots,F_{d}$ such that $F_{i}\cap F_{j}=\\{p\\}$ where $p$ is
the element of the ground set of $(M_{\mathcal{S}})^{s}$ that corresponds to
the cloud of $\mathcal{S}$ containing $v$ and $w$. To see this, let
$A\subseteq V(G)$ be a $3$-subset and suppose that $A$ is not blocked by
$\mathcal{S}\backslash\\{v,w\\}$. Then $A$ is blocked by $\mathcal{S}$ if and
only if there exist $x,y\in A$ such that the path from $x$ to $y$ in $G$
crosses a cloud of $\mathcal{S}$ via the edge $\\{v,w\\}$. Therefore, the
point and line configuration $\mathcal{C}^{\prime}$ is obtained from
$\mathcal{C}$ by removing the line containing $v$ and replacing it with $d$
lines that pass through a common vertex. Since $\mathcal{C}^{\prime}$ is
forest-like, it follows that $\mathcal{C}$ is also forest-like. ∎

We now state and prove our main result, and then the auxiliary results used in
this proof.

######  Theorem 6.11.

The minimal matroids for $\Delta_{G}$ are
$\mathcal{M}_{G}=\\{M_{\mathcal{S}}:\mathcal{S}\text{ is a prime
collection}\\}.$

###### Proof.

On the one hand, if we take a minimal matroid $M$ for $\Delta_{G}$, then by
Lemma 6.17 we have that $M=M_{\mathcal{S}}$ for some prime collection
$\mathcal{S}$. On the other hand, let $\mathcal{S}$ be a prime collection.
First, we observe that, for every prime collection $\mathcal{S}$, every member
of $\Delta_{G}$ is indeed a dependent set in $M_{\mathcal{S}}$. More
precisely, $\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{S}})$ because every
path in $G$ with three vertices either contains a singleton set in
$\mathcal{S}$, contains two elements from a cloud of $\mathcal{S}$, or is not
blocked. Since $\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{S}})$, Lemma 6.17
implies that there is a prime collection $\mathcal{T}$ such that
$\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{T}})\subseteq\mathcal{D}(M_{\mathcal{S}})$
and such that $M_{\mathcal{T}}$ is a minimal matroid for $\Delta_{G}$. But
Lemma 6.18 implies that $\mathcal{S}=\mathcal{T}$. Therefore,
$M_{\mathcal{S}}$ is a minimal matroid for $\Delta_{G}$. ∎

######  Example 6.12.

Let $G$ be the graph on vertex set $[7]$ with edges
$E(G)=\\{12,23,34,45,56,47\\}$. The consecutive forest hypergraph for $G$ is
given by $\Delta=\\{123,234,345,347,456,457\\}$. In Figure 9, we write down
the prime collections $\mathcal{S}$ for $G$. For each $\mathcal{S}$, the
matroid $M_{\mathcal{S}}$ is the matroid of a point and line configuration
which we illustrate in the figure. For instance if $\mathcal{S}=\emptyset$,
then $M_{\mathcal{S}}$ is the matroid of the point and line configuration with
all seven points lying on a single line. If $\mathcal{S}=\\{34,45\\}$, then
the corresponding configuration has a line containing the points $1,2,3,4,5$
and two free points $6$ and $7$. The three points $3,4,5$ coincide since they
are a cloud of $\mathcal{S}$.

Figure 9: Prime collections $\mathcal{S}$ and their matroid $M_{\mathcal{S}}$
as a point and line configuration from Example 6.12. The dotted pairs of
vertices are the two subsets of $\mathcal{S}$ and the square points in the
configurations represent loops.

We devote the rest of this subsection to prove the lemmas used in the proof of
Theorem 6.11. We first prove a matroid result that we will need later.

######  Lemma 6.13.

Let $M$ be a matroid, and let $\\{a,b,c,d\\}$ be a subset of the ground set of
$M$ such that $\\{a,d\\}$ is an independent set. If $\\{a,b,d\\}$ and
$\\{a,c,d\\}$ are dependent sets, then so is $\\{a,b,c\\}$.

###### Proof.

Since $\\{a,b,d\\}$ is dependent, the rank of $\\{a,b,d\\}$ is at most $2$.
Since $\\{a,c,d\\}$ is dependent and $\\{a,d\\}$ is independent, $c$ is in the
closure of $\\{a,d\\}$. Therefore, $c$ is in the closure of $\\{a,b,d\\}$.
Since the rank of $\\{a,b,d\\}$ is at most $2$ and $c$ is in the closure of
$\\{a,b,d\\}$, it follows that the rank of $\\{a,b,c,d\\}$ is at most $2$.
Therefore, $\\{a,b,c\\}$ is a dependent set. ∎

Now we consider the loops of a minimal matroid and give an algorithmic
formulation of Definition 6.3. Given a collection $\mathcal{S}$ of singletons,
if $\mathcal{S}$ passes the following algorithm, it is a good candidate to be
a prime collection of singletons for $G$. While every prime collection of
singletons will pass Algorithm 1, not all sets which pass the algorithm are
prime collections of singletons. In order to guarantee that a set
$\mathcal{S}$ is a prime collection of singletons, we require that
$\mathcal{S}$ passes regardless of the ordering of its elements.

Input: $\mathcal{S}=(s_{1},\dots,s_{t})$ an ordered list of vertices of $G$

Output: pass or fail. (If fail then $\mathcal{S}$ is not a prime collection of
singletons for $G$.)

Initialize: $H\leftarrow G$

for  _$i\leftarrow 1$ to $t$_ do

if  _$s_{i}$ is a leaf or isolated vertex of $H$_ then

return _fail_

if  _$s_{i}$ is adjacent to a leaf of $H$ and $s_{i}$ has degree $2$ in $H$_
then

return _fail_

$H\leftarrow$ induced subgraph of $H$ obtained by deleting vertex $s_{i}$

return _pass_

Algorithm 1 Prime collection test

######  Proposition 6.14.

Let $\mathcal{S}=\\{s_{1},\dots,s_{t}\\}$ be a collection of singleton sets of
vertices of $G$. Then $\mathcal{S}$ is a prime collection of singletons for
$G$ if and only if $\mathcal{S}$ passes Algorithm 1 for any ordering of its
elements.

###### Proof.

Suppose that $\sigma\in S_{t}$ is a permutation such that the algorithm fails
with the input $(s_{\sigma(1)},\dots,s_{\sigma(t)})$. In particular, assume
that the algorithm fails at step $p\in\\{1,\dots,t\\}$. Then it follows that
$\\{s_{\sigma(1)},\dots,s_{\sigma(p)}\\}$ is not a prime collection of
singletons. So, by Definition 6.3, the set $\mathcal{S}$ is not a prime
collection of singletons. Conversely, if $\mathcal{S}$ is not a prime
collection of singletons then, by definition, there exists
$p\in\\{1,\dots,t\\}$ and $j\in\\{1,\dots,p\\}$ such that
$\\{s_{1},\dots,s_{p-1}\\}$ is a prime collection of singletons and either:
$s_{j}$ is a leaf or isolated vertex in $G^{\prime}$, or $s_{j}$ is adjacent
to a leaf of $G^{\prime}$ and has degree two, where $G^{\prime}$ is the
induced subgraph of $G$ obtained by deleting the vertices
$\\{s_{1},\dots,s_{j-1},s_{j+1},\dots,s_{p}\\}$. Now we order the elements of
$\mathcal{S}$ as:
$(s_{1},\dots,s_{j-1},s_{j+1},\dots,s_{p},s_{j},s_{p+1},\dots,s_{t}).$ By
construction, we have that Algorithm 1 fails in the for-loop when checking
vertex $s_{j}$. ∎

######  Lemma 6.15.

Let $M$ be a minimal matroid for $\Delta_{G}$. Then the loops of $M$ are a
prime collection of singletons for $G$.

###### Proof.

We follow the algorithmic description of the construction of a prime
collection of singletons. Let $\mathcal{S}$ be the loops of $M$. Suppose for a
contradiction that $\mathcal{S}$ fails to be a prime collection of singletons
for $G$. Then there exists an ordering of the loops, say
$\mathcal{S}=(s_{1},s_{2},\dots,s_{p})$, such that at step $i$ we are unable
to choose $s_{i}$ for the prime collection. Let $G^{\prime}$ be the induced
subgraph of $G$ obtained by removing vertices $s_{1},\dots,s_{i-1}$. In order
for us to be unable to choose $s_{i}$ for the prime collection there are two
cases:

  1. 1.

$s_{i}$ is a leaf or isolated vertex of $G^{\prime}$,

  2. 2.

$s_{i}$ is adjacent to a leaf and has degree two in $G^{\prime}$.

Case 1. We construct a new matroid $M^{\prime}$ on $[n]$ as follows. First
construct the matroid $M\backslash s_{i}$ by deleting $s_{i}$.

If $s_{i}$ is a leaf of $G^{\prime}$, denote by $x$ the unique vertex adjacent
to $s_{i}$. Then, let $M^{\prime}$ be the matroid obtained from $M\backslash
s_{i}$ by adding to the ground set the element $s_{i}$ in the same parallel
class as $x$.

If $s_{i}$ is an isolated vertex of $G^{\prime}$, note that every neighbor of
$s_{i}$ in $G$ is a loop of $M$. Let $M^{\prime}$ be the matroid obtained from
$M\backslash s_{i}$ by freely adding $s_{i}$ to the ground set of $M\backslash
s_{i}$.

In either case, note that the collection of dependent sets of $M^{\prime}$
contains $\Delta_{G}$ because every path with three vertices containing
$s_{i}$ contains either $x$ or some loop of $M^{\prime}$.

We now show that $M^{\prime}\lneq M$. Take any dependent set $D$ of
$M^{\prime}$. If $s_{i}\in D$, then clearly $D$ is a dependent set of $M$
since $s_{i}$ is a loop of $M$. On the other hand, if $s_{i}\not\in D$, then
$D$ is a dependent set of $M^{\prime}\backslash s_{i}=M\backslash s_{i}$.
Hence $D$ is a dependent set of $M$. To show that $M\neq M^{\prime}$ we simply
note that $s_{i}$ is a loop in $M$ but is not a loop in $M^{\prime}$. So we
have shown
$\Delta_{G}\subseteq\mathcal{D}(M^{\prime})\subsetneq\mathcal{D}(M)$. Hence
$M$ is not a minimal matroid for $\Delta_{G}$, a contradiction.

Case 2. Let $s_{i}$ be a vertex of degree two which is adjacent to a leaf of
$G^{\prime}$. Let $x$ and $y$ be the vertices adjacent to $s_{i}$, with $x$
being a leaf. We construct a new matroid $M^{\prime}$ by first deleting
$s_{i}$ to form $M\backslash s_{i}$. Then we define $M^{\prime}$ to be the
matroid obtained by adding $s_{i}$ to the same parallel class as $y$ in
$M\backslash s_{i}$. Note that $\mathcal{D}(M^{\prime})$ must contain
$\Delta_{G}$ because any path with three vertices containing $s_{i}$ (and not
containing any loop of $M^{\prime}$) must also contain $y$. To show that
$M^{\prime}\lneq M$ we use the same argument as Case 1, so we have that
$\Delta_{G}\subseteq\mathcal{D}(M^{\prime})\subsetneq\mathcal{D}(M)$.
Therefore $M$ is not a minimal matroid for $\Delta_{G}$, a contradiction. ∎

We now characterize matroids which are minimal among all the matroids whose
circuits of size one and two are exactly the members of $\mathcal{S}$. One can
think of this as a constrained notion of minimality. In the following lemma,
note that $M$ is fixed, and $\mathcal{S}$ is obtained from $M$. In a matroid,
every circuit of size two is disjoint from every circuit of size one.
Therefore, $\mathcal{S}$ is a valid collection for $G$, and $M_{\mathcal{S}}$
is well-defined.

######  Lemma 6.16 (Constrained minimal matroids).

Let $M$ be a matroid with $\Delta_{G}\subseteq\mathcal{D}(M)$ whose circuits
of size one and two are the members of $\mathcal{S}$. Then
$M_{\mathcal{S}}\leq M$.

###### Proof.

By assumption, all of the circuits of $M_{\mathcal{S}}$ of size one and two
are also circuits of $M$. All $4$-subsets of $[n]$ are dependent in both $M$
and $M_{\mathcal{S}}$. Therefore, to show that $M_{\mathcal{S}}\leq M$, it
suffices to show that all $3$-circuits of $M_{\mathcal{S}}$ are dependent in
$M$. Let $D=\\{a,b,c\\}$ be such a circuit of $M_{\mathcal{S}}$. Let
$G^{\prime}$ be the induced subgraph of $G$ obtained by removing those
vertices $v$ for which $\\{v\\}\in\mathcal{S}$. Now since $D$ is a circuit in
$M_{\mathcal{S}}$, we have that for any pair of elements $x,y\in D$, there is
a path in $G^{\prime}$ from $x$ to $y$ and it is not blocked by $\mathcal{S}$.
We denote by $\textrm{Conv}(x,y)$ the collection of vertices on the path from
$x$ to $y$. Similarly we define the convex hull of a set of vertices:

$\textrm{Conv}(A)=\bigcup_{x,y\in A}\textrm{Conv}(x,y).$

By abuse of notation, we identify $\textrm{Conv}(A)$ with the induced subgraph
of $G^{\prime}$ whose vertices are $\textrm{Conv}(A)$.

To complete the proof of the theorem, it suffices to prove the following
claim. Since, as a result we have that $\\{a,b,c\\}$ is a dependent set in
$M$. Therefore, every dependent set in $M_{\mathcal{S}}$ is also dependent in
$M$, as desired.

Claim. All $3$-subsets of ${\rm Conv}(a,b,c)$ are dependent in $M$.

We must show that every $3$-subset
$\\{a^{\prime},b^{\prime},c^{\prime}\\}\subseteq\textrm{Conv}(a,b,c)$ is
dependent in $M$. We proceed by induction on
$|\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})|$. For the base case,
suppose that $|\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})|=3$. Then
$\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})$ is a path in $G$ with three
vertices. By definition of $\Delta_{G}$, we have that
$\\{a^{\prime},b^{\prime},c^{\prime}\\}\in\Delta_{G}$. Since
$\Delta_{G}\subseteq\mathcal{D}(M)$, we have that
$\\{a^{\prime},b^{\prime},c^{\prime}\\}$ is dependent in $M$. For the
inductive step, assume that all $3$-subsets of ${\rm Conv}(a,b,c)$ whose
convex hull has size at most $k\geq 3$ are dependent and
$|\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})|=k+1$. Since
$|\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})|\geq 4$, there is a vertex
$d\in\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})\backslash\\{a^{\prime},b^{\prime},c^{\prime}\\}$.
Note that neither $a^{\prime}$, $b^{\prime}$, $c^{\prime}$, nor $d$ can be a
loop of $M$ because then it would be a loop of $M_{\mathcal{S}}$. This implies
either that $\\{a,b,c\\}$ is blocked or that an element of $\\{a,b,c\\}$ is a
loop of $M_{\mathcal{S}}$. This is impossible since $\\{a,b,c\\}$ is a circuit
of $M_{\mathcal{S}}$.

By induction, $\\{a^{\prime},b^{\prime},d\\}$,
$\\{a^{\prime},c^{\prime},d\\}$, and $\\{b^{\prime},c^{\prime},d\\}$, are all
dependent since their respective convex hulls lie strictly inside
$\textrm{Conv}(a^{\prime},b^{\prime},c^{\prime})$. Since
$\\{a^{\prime},b^{\prime},d\\}$ and $\\{a^{\prime},c^{\prime},d\\}$ are
dependent, Lemma 6.13 implies that either
$\\{a^{\prime},b^{\prime},c^{\prime}\\}$ or $\\{a^{\prime},d\\}$ is dependent.
Similarly, if $\\{a^{\prime},b^{\prime},c^{\prime}\\}$ is not dependent, then
$\\{b^{\prime},d\\}$ and $\\{c^{\prime},d\\}$ are dependent. Therefore, since
$d$ is not a loop, we may assume that the rank of
$\\{a^{\prime},b^{\prime},c^{\prime},d\\}$ is at most one, implying that
$\\{a^{\prime},b^{\prime},c^{\prime}\\}$ is dependent. This completes the
proof of the claim. ∎

######  Lemma 6.17.

If $M$ is a minimal matroid for $\Delta_{G}$, then $M=M_{\mathcal{S}}$ for
some prime collection $\mathcal{S}$.

###### Proof.

Let $\mathcal{S}$ be the collection of all circuits of $M$ of size one and
two. Since $\Delta_{G}\subseteq\mathcal{D}(M)$, Lemma 6.16 implies that
$M_{\mathcal{S}}\leq M$. However, $M$ is a minimal matroid. Since
$\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{S}})$, the minimality of $M$
implies that $M=M_{\mathcal{S}}$. So it suffices to show that $\mathcal{S}$ is
a prime collection.

By Lemma 6.15, we have that the loops of $M$ form a prime collection of
singletons for $G$. The collection of $2$-element circuits of a matroid does
not intersect the set of loops, so $\mathcal{S}$ is a valid collection of
subsets.

Let $G^{\prime}$ be the induced subgraph of $G$ obtained by deleting the
vertices that are singleton sets in $\mathcal{S}$. Suppose for a contradiction
that $\mathcal{S}$ is not a prime collection. Then it follows that there
exists a vertex $v$ in $G^{\prime}$ such that, for every vertex $w$ adjacent
to $v$ in $G^{\prime}$, we have that $\\{v,w\\}$ is a member of $\mathcal{S}$.

Let $w$ be a vertex adjacent to $v$, and consider the valid collection
$\mathcal{S}^{\prime}=\mathcal{S}\backslash\\{\\{v,w\\}\\}$. A set is a
singleton in $\mathcal{S}$ if and only if it is a singleton in
$\mathcal{S}^{\prime}$. Since every vertex adjacent to $v$ is in the same
cloud of $\mathcal{S}$ as $v$, a $3$-subset of $[n]$ is blocked by
$\mathcal{S}$ if and only if it is blocked by $\mathcal{S}^{\prime}$. We also
have $\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{S}}^{\prime})$.

Since the collection of $2$-subsets in $\mathcal{S}^{\prime}$ is properly
contained in the the collection of $2$-subsets in $\mathcal{S}$, we have that
$M_{\mathcal{S}}^{\prime}\lneq M_{\mathcal{S}}$, a contradiction. Therefore,
we conclude that $\mathcal{S}$ is a prime collection. ∎

The last ingredient in the proof of Theorem 6.11 is to show that
$M_{\mathcal{S}}$ is minimal for any prime collection $\mathcal{S}$.

######  Lemma 6.18.

If $\mathcal{S}$ and $\mathcal{T}$ are prime collections with
$M_{\mathcal{S}}\geq M_{\mathcal{T}}$, then $\mathcal{S}=\mathcal{T}$.

###### Proof.

Let us begin by considering the loops of $M_{\mathcal{S}}$ and
$M_{\mathcal{T}}$. Since $M_{\mathcal{S}}\geq M_{\mathcal{T}}$, we have that
every loop of $M_{\mathcal{T}}$ is a loop of $M_{\mathcal{S}}$. Suppose for a
contradiction that $M_{\mathcal{S}}$ has strictly more loops than
$M_{\mathcal{T}}$. Let $G_{\mathcal{S}}$ be the induced subgraph of $G$
obtained by removing the loops of $M_{\mathcal{S}}$. We define
$G_{\mathcal{T}}$ similarly. Pick any loop $v$ in $M_{\mathcal{S}}$ that is
not a loop in $M_{\mathcal{T}}$. We now show that $\deg(v)\geq 2$ in
$G_{\mathcal{T}}$. Suppose otherwise. Then all but at most one of the
neighbors of $v$ are loops in $M_{\mathcal{T}}$. Since $M_{\mathcal{S}}\geq
M_{\mathcal{T}}$, all but at most one of the neighbors of $v$ are loops in
$M_{\mathcal{S}}$ also. But this fact, with the assumption that $v$ is a loop
in $M_{\mathcal{S}}$, contradicts the assumption that $\mathcal{S}$ is a prime
collection. Therefore, $\deg(v)\geq 2$ in $G_{\mathcal{T}}$. Similar reasoning
shows that each vertex adjacent to $v$ in $G_{\mathcal{T}}$ must be adjacent
to a vertex in $G_{\mathcal{T}}$ that is not a loop in $M_{\mathcal{S}}$.

Since $\mathcal{T}$ is a prime collection, there is at least one vertex $x$
adjacent to $v$ in $G_{\mathcal{T}}$ such that $\\{x,v\\}\notin\mathcal{T}$.
Since $\deg(v)\geq 2$ in $G_{\mathcal{T}}$, let $y\neq x$ be another neighbor
of $v$ in $G_{\mathcal{T}}$. We have already observed that there must be a
vertex adjacent to $x$ in $G_{\mathcal{T}}$ that is not a loop in
$M_{\mathcal{S}}$. Call this vertex $x^{\prime}$. We may choose $x^{\prime}$
so that $\\{x,x^{\prime}\\}\notin\mathcal{S}$ because, if there is no such
vertex, then $x$ has no neighbor outside the cloud containing $x$ in
$G_{\mathcal{S}}$, which implies that $\mathcal{S}$ is not a prime collection.
Similarly, we let $y^{\prime}$ be any vertex adjacent to $y$ such that $y$ is
not a loop in $M_{\mathcal{S}}$ and such that
$\\{y,y^{\prime}\\}\notin\mathcal{S}$. For $X\subseteq[n]$, let $\text{cl}(X)$
denote the closure of $X$ in $M_{\mathcal{T}}$. Since
$\Delta_{G}\subseteq\mathcal{D}(M_{\mathcal{T}})$, we have that
$\\{x,x^{\prime},v\\}$ and $\\{x,y,v\\}$ are both dependent sets in
$M_{\mathcal{T}}$. By assumption, $\\{x,x^{\prime}\\}$ is independent in
$M_{\mathcal{S}}$, implying that it is independent in $M_{\mathcal{T}}$ also.
Since $\\{x,x^{\prime},v\\}$ is dependent, we have
$v\in\text{cl}(\\{x,x^{\prime}\\})$. Thus
$\text{cl}(\\{x,x^{\prime}\\})=\text{cl}(\\{x,x^{\prime},v\\})$. Similarly,
since $\\{x,v\\}$ is independent in $M_{\mathcal{T}}$ but $\\{x,y,v\\}$ is
dependent, we have that
$y\in\text{cl}(\\{x,v\\})\subseteq\text{cl}(\\{x,x^{\prime},v\\})=\text{cl}(\\{x,x^{\prime}\\})$.
Since $y\in\text{cl}(\\{x,x^{\prime}\\})$, we have that $\\{x,x^{\prime},y\\}$
is a dependent set in $M_{\mathcal{T}}$. However, by assumption, we have that
$\\{x,x^{\prime}\\}$ is independent in $M_{\mathcal{S}}$. Hence
$\\{x,x^{\prime},y\\}$ is independent in $M_{\mathcal{S}}$ because it is
blocked by $v$. So we have found a dependent set in $M_{\mathcal{T}}$ which is
independent in $M_{\mathcal{S}}$, a contradiction. Therefore, we deduce that
$M_{\mathcal{S}}$ and $M_{\mathcal{T}}$ have the same loops.

To show that $\mathcal{S}=\mathcal{T}$, it remains to show that the
$2$-subsets in $\mathcal{S}$ are the same as the $2$-subsets in $\mathcal{T}$.
Since $M_{\mathcal{S}}$ and $M_{\mathcal{T}}$ have the same loops and
$M_{\mathcal{S}}\geq M_{\mathcal{T}}$, it follows that every $2$-subset in
$\mathcal{T}$ is also in $\mathcal{S}$. Suppose for a contradiction that
$\mathcal{S}$ has strictly more $2$-subsets than $\mathcal{T}$. Therefore,
either $\mathcal{S}$ has more clouds than $\mathcal{T}$, or there is a cloud
of $\mathcal{T}$ that is properly contained in a cloud of $\mathcal{S}$. In
either case, since every vertex in the cloud is adjacent to a vertex not in
the cloud, there is a $3$-element set that is blocked by $\mathcal{S}$ but not
by $\mathcal{T}$. This contradicts the assumption that $M_{\mathcal{S}}\geq
M_{\mathcal{T}}$. ∎

### 6.2 Irreducible matroid varieties.

In this subsection, we combine the results of §6.1 to obtain a classification
of the irreducible components of $V_{\Delta_{G}}$. In particular, we will
prove Theorem 6.20. This gives a straightforward description of the
irreducible components.

######  Proposition 6.19.

Let $G$ be a forest and $\mathcal{S}$ be a prime collection for $G$ as in
Definition 6.4. Then for any non-central component $V_{N}$ of
$V_{M_{\mathcal{S}}}^{\mathrm{comb}}$ there exists another prime collection
$\mathcal{S}^{\prime}$ for $G$, such that $V_{N}\subseteq
V_{M_{\mathcal{S}^{\prime}}}$.

###### Proof.

Let $J$ be the ideal associated to the matroid $N$ and for notation we write
$N$ as $M_{J}$. By Proposition 6.10, the simplification of $M_{\mathcal{S}}$
is forest-like. So, by Theorem 5.14,

the matroid $M_{J}$ is obtained from $M_{\mathcal{S}}$ by setting all elements
inside certain clouds of $\mathcal{S}$ to be loops, see Definition 5.12.
Denote these clouds by $C_{1},\dots,C_{s}$. More precisely, the circuits of
$M_{J}$ are the singleton subsets of $C_{1}\cup\cdots\cup C_{s}$ along with
the collection of circuits of $M_{\mathcal{S}}\backslash(C_{1}\cup\cdots\cup
C_{s})$. Note that $M_{\mathcal{S}}<M_{J}$.

Let $(\mathcal{P},\mathcal{L},\mathcal{I})$ be the point and line
configuration whose matroid is the simplification of $M_{\mathcal{S}}$. By
Theorem 5.11, if a non-central component $J$ of
$I_{M_{\mathcal{S}}}^{\mathrm{comb}}$ exists, there is a point in
$\mathcal{P}$ contained in at least three lines in $\mathcal{L}$. This implies
that a cloud of $\mathcal{S}$ contains at least three elements.

Let $C$ be a cloud of $\mathcal{S}$ that is not one of $C_{1},\dots C_{s}$.
Let $p_{i}\in\mathcal{P}$ be the corresponding point in the simplification of
$M_{\mathcal{S}}$. We define the _star of $p_{i}$_ to be
$S(p_{i})=\bigcup_{\ell\in\mathcal{L}_{i}}\ell$, where $\ell$ is identified
with the set of points it passes through. By Lemma 5.13, we see that all
generators of $I_{M_{\mathcal{S}}}$ constructed using points in $S(p_{i})$ are
also generators of $J$. However, we also see that
$I_{M_{\mathcal{S}}}\nsubseteq J$ because the elements of $C_{1}\cup\cdots\cup
C_{s}$ are loops of $M_{J}$. Therefore, it is enough to show that there exists
a prime collection $\mathcal{S}^{\prime}$ such that:

  * •

$M_{\mathcal{S}^{\prime}}<M_{J}$ and

  * •

every cloud of $\mathcal{S}^{\prime}$ contained in any of $C_{1},\dots,C_{s}$
has size at most two.

We will construct $\mathcal{S}^{\prime}$ from $\mathcal{S}$ by an inductive
procedure modifying the clouds $C_{1},\dots,C_{s}$. Let
$\mathcal{S}_{0}=\mathcal{S}$. At each step of the procedure, we construct
$\mathcal{S}_{i+1}$ from $\mathcal{S}_{i}$ by modifying a cloud $C$ of
$\mathcal{S}_{i}$ with $|C|\geq 3$ such that $C\subseteq C_{1}\cup\cdots\cup
C_{s}$. By induction, we assume $\mathcal{S}_{i}$ is a prime collection. At
each step of this procedure, we construct a prime collection
$\mathcal{S}_{i+1}$ such that every set that is dependent in
$M_{\mathcal{S}_{i+1}}$ but not in $M_{\mathcal{S}_{i}}$ is dependent in
$M_{J}$. Since $M_{\mathcal{S}}<M_{J}$, this implies that
$M_{\mathcal{S}^{\prime}}<M_{J}$.

Let $G^{\prime}$ be the induced subgraph of $G$ obtained by deleting the
singletons in $\mathcal{S}_{i}$. Let $G[C]$ be the induced subgraph of $G$
with vertex set $C$, and let $\ell$ be a leaf of $G[C]$. Since $|C|\geq 3$,
the neighbor $x$ of $\ell$ in $G[C]$ is not a leaf of $G[C]$. In particular,
$\deg(x)\geq 3$ in $G^{\prime}$, so we can add $x$ to the collection of
singletons of $\mathcal{S}_{i}$ to result in a new prime collection of
singletons. Therefore, we modify $\mathcal{S}_{i}$ as follows. Make $x$ a
singleton set in $\mathcal{S}_{i+1}$ and remove all $2$-subsets in
$\mathcal{S}_{i}$ which contain $x$. We now consider the neighbors of $x$
which lie outside $C$. For each such neighbor $y$ of $x$, if $y$ belongs to a
cloud $C^{\prime}\neq C$ and $x$ is the only neighbor of $y$ not contained in
$C^{\prime}$, then we remove $y$ from the cloud $C^{\prime}$. This ensures
that the new collection is a prime collection. Let $\mathcal{S}_{i+1}$ be this
new prime collection. Note that this procedure does not add any new
$3$-circuits because all paths crossing $C^{\prime}$ that contain $y$
necessarily pass through $x$.

As a result of this procedure, the cloud $C$ no longer contains $x$ or $\ell$
and might also be split into multiple smaller clouds. Note that all other
clouds affected by this procedure have decreased in size. Thus, the only
subset of $[n]$ that is dependent in $\mathcal{S}_{i+1}$ but not in
$\mathcal{S}_{i}$ is the singleton set $\\{x\\}$, which is a loop of $M_{J}$.
Therefore, $M_{\mathcal{S}_{i+1}}<M_{J}$ as desired. We apply this procedure
inductively until we obtain a prime collection $\mathcal{S}^{\prime}$ such
that every cloud of $\mathcal{S}^{\prime}$ contained in $C_{1}\cup\cdots\cup
C_{s}$ has size $2$. ∎

######  Theorem 6.20.

Let $G$ be a forest and let $\Delta_{G}$ be the corresponding consecutive
forest hypergraph. Then,

$V_{\Delta_{G}}=\bigcup_{\mathcal{S}}V_{M_{\mathcal{S}}},$

is an irredundant irreducible decomposition of $V_{\Delta_{G}}$, where the
union is taken over all prime collections $\mathcal{S}$ of $G$. In particular,
a minimal prime decomposition of $\sqrt{I_{\Delta_{G}}}$ is given as
$\sqrt{I_{\Delta_{G}}}=\bigcap_{\mathcal{S}}I_{M_{\mathcal{S}}}.$

###### Proof.

By Theorem 6.11, the set of all minimal matroids of $\Delta_{G}$ is
$\\{M_{\mathcal{S}}:\mathcal{S}\text{ is a prime collection}\\}$. It follows,
by Proposition 3.10, that
$V_{\Delta_{G}}=\bigcup_{\mathcal{S}}V^{\mathrm{comb}}_{M_{\mathcal{S}}}.$ It
is easy to see that, the simplification of the matroid $M_{\mathcal{S}}$ is a
forest-like configuration. So, the irreducible components of
$V^{\mathrm{comb}}_{M_{\mathcal{S}}}$ are in one-to-one correspondence with
the irreducible components of the combinatorial closure of the matroid variety
of this configuration, which are given by Theorem 5.14. By Proposition 6.19,
for every non-central component $V_{M_{J}}$ of
$V^{\mathrm{comb}}_{M_{\mathcal{S}}}$, there exists a prime collection
$\mathcal{S}^{\prime}$ such that $V_{M_{J}}\subseteq
V_{M_{\mathcal{S}}^{\prime}}$. Moreover, by Theorem 6.22, each of the matroids
$M_{\mathcal{S}}$ is realizable. Hence, the irredundant irreducible
decomposition of $V_{\Delta_{G}}$ is given by
$V_{\Delta_{G}}=\bigcup_{\mathcal{S}}V_{M_{\mathcal{S}}}$, as desired. ∎

In the example below we show that the irreducible components of
$V_{\Delta_{G}}$ are not necessarily positroid varieties.

######  Example 6.21.

Let $G$ be the graph on the vertex set $\\{1,2,\dots,9\\}$ and edge set
$\\{12,23,14,45,56,47,78,89\\}$. Let $\mathcal{S}=\\{14,47\\}$ be a prime
collection. By Remark 6.9, the matroid $M_{\mathcal{S}}$ has three rank $2$
flats given by $12347,14567$ and $14789$. So the simplification of
$M_{\mathcal{S}}$ is the matroid of the configuration given by three lines
passing through a point. See Example 3.6. In particular, the ideal of
$M_{\mathcal{S}}$ is not generated by determinants. So by [KLS13, Theorem
5.15], the matroid $M_{\mathcal{S}}$ is not a positroid.

### 6.3 Realizability.

We are now ready to consider the minimal matroids for $\Delta_{G}$. Recall
that, if $\mathcal{S}$ is a valid collection for $G$, then the matroid
$M_{\mathcal{S}}$ arising from Proposition 6.8 is minimal with respect to
$\Delta_{G}$. Here, we show that $M_{\mathcal{S}}$ is realizable over the real
numbers hence realizable over $\mathbb{C}$.

######  Theorem 6.22.

If $\mathcal{S}$ is a valid collection, then $M_{\mathcal{S}}$ is
$\mathbb{R}$-realizable.

###### Proof.

Let $r$ be the rank function of $M_{\mathcal{S}}$. The elements of
$M_{\mathcal{S}}$ are the vertices of the forest $G$. Let $L$ be the set of