# Realizability of matroid quotients

Alessio Borzì

###### Abstract

We characterize the realizability of a quotient of matroids, over an infinite
field $K$, in terms of the realizability over $K$ of a single matroid
associated to it, called the _Higgs major_. This result extends to
realizability of flag matroids. Further, we provide some applications to the
relative realizability problem for Bergman fans in tropical geometry.

Vita quieta, mente lieta, moderata dieta.

##  1 Introduction

Matroids are central objects in modern combinatorics. A matroid $M$ can be
seen as a combinatorial abstraction of a linear space. In a similar way, a
_matroid quotient_ $M\twoheadrightarrow N$ can be seen as a combinatorial
abstraction of an inclusion of two linear spaces. Even if $M$ and $N$ are
realizable matroids, this does not guarantee that the quotient as a whole is
realizable. An example of this was given in [4, §1.7.5. Example 7], and is as
follows. Let $P$ be the non-Pappus matroid, and let $e$ be one of the points
of the "non-line". Then, the contraction $P/e$ is a matroid quotient of the
deletion $P\setminus e$ and both matroids are realizable. However, the matroid
quotient $P\setminus e\twoheadrightarrow P/e$ is not realizable. It is
interesting to note that the non-realizability of the above quotient depends
on the non-realizability of the non-Pappus matroid.

12345678e$P$12345678$P\setminus e$14782,63,5$P/e$

The aim of this paper is to characterize the realizability of a matroid
quotient in terms of the realizability of a matroid associated to it. We will
do so by using the notions of _factorization_ and _major_ of a matroid
quotient, both appearing in a work of Kennedy [20]. The matroid we will
associate to a matroid quotient is called the _Higgs major_ (see also [21]).

######  Theorem A (Theorem 3.10).

Let $f:M_{1}\twoheadrightarrow M_{2}$ be a quotient of matroids, and let $K$
be an infinite field. The following statements are equivalent:

  1. 1.

$f$ is realizable over $K$,

  2. 2.

the Higgs factorization of $f$ is realizable over $K$,

  3. 3.

the Higgs major of $f$ is realizable over $K$.

A natural context in which to consider matroid quotients is flag varieties. A
sequence of matroid quotients $M_{1}\twoheadrightarrow
M_{2}\twoheadrightarrow\dots\twoheadrightarrow M_{n}$ is a _flag matroid_.
Flag matroids are related to flag varieties in the same natural way as
matroids are related to Grassmannians (see for instance [14] and [9]).

Tropical analogues of flag varieties were studied in [7, 5, 6, 15]. In
particular, in [7] a notion of _flag Dressian_ was introduced, and in order to
compare it with the tropical flag variety, it was proved that for ground sets
of cardinality up to five, all flag matroids are realizable. This result, in
the non-valuated case, can be now seen as an immediate consequence of the
above theorem (in the form of Corollary 3.11), combined with the fact that
matroids with ground set of small enough cardinality are realizable. A similar
reasoning might be applied accordingly for the linear degenerate flag Dressian
defined in [5].

Another natural application of our main theorem is about relative
realizability in tropical geometry (see for instance [19, Question C] and [13,
Question 1.1]).

######  Problem 1.1 (Relative realizability problem).

Given a pair of tropical varieties $\mathcal{Y}\subseteq\mathcal{X}$ and an
algebraic variety $X$ tropicalizing to $\mathcal{X}$, does there exist a
subvariety $Y\subseteq X$ tropicalizing to $\mathcal{Y}$?

There are many situations in which the above problem has a negative answer.
One of the most famously known is when considering tropical lines contained in
a tropical cubic surface. From the Cayley-Salmon theorem we know that a cubic
surface contains just $27$ lines, while there are examples of tropical cubic
surfaces that contain infinitely many tropical lines [25].

Even in the case when $\mathcal{X}$ and $\mathcal{Y}$ in Problem 1.1 are
linear, the answer might still be negative. An example of this was provided in
[22] in the context of tropical Fano schemes, and further studied in [19]. Our
main theorem offers another perspective on this phenomena (see Section 4.1 for
more details). Roughly speaking, in a similar way as a tropical linear space
has an underlying matroid, an inclusion of tropical linear spaces has an
underlying matroid quotient, and the corresponding relative realizability
problem is related to the realizability problem of the matroid quotient. We
also make some remarks about the nonlinear case in Section 4.3.

This paper is organized as follows. In Section 2 we recall some basic notions
about matroid quotients and review factorizations and majors, in Section 3 we
prove our main result (Theorem 3.10). In Section 4 we discuss some
applications to the relative realizability problem in tropical geometry, and
finally in Section 5 we list some questions.

#### Acknowledgements

I would like to thank Diane Maclagan, Bernd Sturmfels and Alex Fink for useful
discussions and email exchanges. A special thank to Alheydis Geiger for
insights about the relative realizability problem for lines on a cubic surface
and to Hamdi Dërvodeli for sharing his notes on the Veronese embedding.
Further thanks to Hannah Markwig and Victoria Schleis for fruitful exchanges
that partly inspired this work.

##  2 Preliminaries

We assume that the reader is familiar with the basics of matroid theory [24].

###  2.1 Morphisms, strong maps and quotients

Let $M_{1}$ and $M_{2}$ be matroids on the ground sets $E_{1}$ and $E_{2}$
respectively.

######  Definition 2.1.

A _morphism_ of matroids $f:M_{1}\rightarrow M_{2}$ is a function of sets
$f:{E_{1}}\rightarrow{E_{2}}$ such that for every flat $F$ of $M_{2}$,
$f^{-1}(F)$ is a flat of $M_{1}$.

For a set $E$ and an element $o$ not in $E$, we will denote $E\cup\\{o\\}$ by
$E_{o}$. Let $o$ be an element not in $E_{1}\cup E_{2}$, and let ${E_{i}}_{o}$
be the ground set of $M_{i}\oplus U_{0,1}$ for $i\in\\{1,2\\}$.

######  Definition 2.2.

A _strong map_ of matroids $f:M_{1}\rightarrow M_{2}$ is a morphism of
matroids $f_{o}:M_{1}\oplus U_{0,1}\rightarrow M_{2}\oplus U_{0,1}$ such that
$f_{o}(o)=o$.

From the definition, it is easy to see that the composition of two strong maps
is a strong map. Therefore, matroids together with strong maps form a
category. This category was studied in [17].

There are two natural examples of strong maps of matroids:

  * •

Extensions: if $M$ is a matroid on $E$, and $N$ is an extension of $M$ on
$E\cup S$, then the inclusion map $E_{o}\subseteq(E\cup S)_{o}$ is a strong
map from $M$ to $N$.

  * •

Contractions: if $I\subseteq E$, then the map $f:E_{o}\rightarrow
E_{o}\setminus I$ defined by $f(o)=o$ and for every $e\in E$

$f(e)=\begin{cases}e&\text{if }e\notin I,\\\ o&\text{if }e\in I,\end{cases}$

is a strong map from $M$ to $M/I$.

######  Definition 2.3.

If $M_{1}$ and $M_{2}$ have the same ground set $E$, and the identity on $E$
is a morphism of matroids, then $M_{2}$ is a _quotient_ of $M_{1}$, and the
identity map on $E_{o}$ is the _quotient map_ from $M_{1}$ to $M_{2}$.

Quotients will be denoted by $M_{1}\twoheadrightarrow M_{2}$ or by
$f:M_{1}\twoheadrightarrow M_{2}$ to emphasize the underlying quotient map.

As explained in [21, Section 8.1], in order to study strong maps of matroids,
often times one can essentially restrict to study matroids quotients.

###  2.2 Realizability

######  Definition 2.4.

A rank $r$ matroid $M$ on $E$ is _realizable_ (or _representable_) over a
field $K$ if there exists a $K$-vector space $V$ of dimension $r$ and a map
$\varphi:E\rightarrow V$ such that $A\subseteq E$ is independent on $M$ if and
only if the vectors $\varphi(A)$ are linearly independent.

Let $M$ be a matroid realizable over $K$, and let $\varphi_{i}:E\rightarrow
V_{i}$ for $i\in\\{1,2\\}$ be two maps realizing $M$. We say that
$\varphi_{1}$ and $\varphi_{2}$ are _equivalent_ if there exists an
isomorphism $\psi:V_{1}\rightarrow V_{2}$ such that
$\varphi_{2}=\psi\circ\varphi_{1}$. A _realization_ of $M$ over $K$ is an
equivalence class of such maps.

If $E$ has cardinality $n$, a map $\varphi:E\rightarrow V$ realizing $M$ is
often regarded as an $r\times n$ matrix $A$ with coefficients in $K$ once a
basis of $V$ is fixed. Thus, an equivalence class is obtained by multiplying
$A$ on the left by all invertible $r\times r$ matrices. Such an equivalence
class can be identified with the $r$-dimensional vector subspace of $K^{n}$
generated by the rows of $A$. This can be also thought as a point in the
Grassmannian $\operatorname{Gr}(r,n)$, which can be given by its Plücker
coordinates, via the Plücker embedding of $\operatorname{Gr}(r,n)$ into
$\mathbb{P}^{\binom{n}{r}-1}$.

######  Definition 2.5.

Let $M_{i}$ be a matroid on $E_{i}$ for $i\in\\{1,2\\}$. A morphism of
matroids $f:M_{1}\rightarrow M_{2}$ is _realizable_ over a field $K$ if both
$M_{1}$ and $M_{2}$ are realizable over $K$, and there exists a map
$\varphi_{i}:E_{i}\rightarrow V_{i}$ realizing $M_{i}$ for each
$i\in\\{1,2\\}$ and a linear map $g:V_{1}\rightarrow V_{2}$ such that the
following diagram is commutative:

${E_{1}}$${V_{1}}$${E_{2}}$${V_{2}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{g}$$\scriptstyle{\varphi_{2}}$

A strong map (or quotient) of matroids is _realizable_ over a field $K$ if so
is as a morphism of matroids.

Assume that the morphism of matroids $f:M_{1}\rightarrow M_{2}$ is realized
over $K$ by the following two commutative diagrams:

${E_{1}}$${V_{1}}$${E_{2}}$${V_{2}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{g}$$\scriptstyle{\varphi_{2}}$
${E_{1}}$${V_{1}^{\prime}}$${E_{2}}$${V_{2}^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\varphi_{2}^{\prime}}$

The above commutative diagrams realizing $f$ are _equivalent_ if there exist
two isomorphisms $\psi_{i}:V_{i}\rightarrow V_{i}^{\prime}$ for
$i\in\\{1,2\\}$ such that the following diagram is commutative:

${E_{1}}$${V_{1}}$${V_{1}^{\prime}}$${E_{2}}$${V_{2}}$${V_{2}^{\prime}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}^{\prime}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\sim}$$\scriptstyle{g}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{2}^{\prime}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\sim}$

A _realization_ of $f:M_{1}\rightarrow M_{2}$ over $K$ is an equivalence class
of such commutative diagrams.

For example, if $E_{1}$ and $E_{2}$ are the same set with cardinality $n$, and
the matroid $M_{i}$ has rank $r_{i}$ for $i\in\\{1,2\\}$, a realization of the
quotient $M_{1}\twoheadrightarrow M_{2}$ can be thought to be a realization of
$M_{i}$ given by an $r_{i}$-dimensional $K$-vector space $U_{i}\subseteq
K^{n}$ for $i\in\\{1,2\\}$ (in the sense we have described above) such that
$U_{2}\subseteq U_{1}$. Therefore, a realization of $M_{1}\twoheadrightarrow
M_{2}$ can be thought of as a point in the flag variety
$\operatorname{Fl}(r_{1},r_{2};n)$, which can be given by its Plücker
coordinates once we embed the flag variety in a product of Grassmannians
$\operatorname{Fl}(r_{1},r_{2};n)\subseteq\operatorname{Gr}(r_{1},n)\times\operatorname{Gr}(r_{2},n)\subseteq\mathbb{P}^{\binom{n}{r_{1}}-1}\times\mathbb{P}^{\binom{n}{r_{2}}-1}$.

######  Definition 2.6.

A _diagram of matroids_ consists of a set of matroids
$\\{M_{\lambda}\\}_{\lambda\in\Lambda}$ and a set of some morphisms between
them $\\{f:M_{\lambda}\rightarrow M_{\lambda^{\prime}}\\}$.

The previous definition can be recast in terms of category theory. Denote by
$\mathbf{Mat}$ the category of matroids with their morphisms. A diagram of
matroids can be thought as a functor $D:\mathbf{I}\rightarrow\mathbf{Mat}$,
where $\mathbf{I}$ is some index category.

######  Definition 2.7.

A diagram of matroids
$(\\{M_{\lambda}\\}_{\lambda\in\Lambda},\\{f:M_{\lambda}\rightarrow
M_{\lambda^{\prime}}\\})$ is _realizable_ over a field $K$ if for every
$\lambda\in\Lambda$ there exists a map
$\varphi_{\lambda}:E_{\lambda}\rightarrow V_{\lambda}$ that realizes
$M_{\lambda}$, and for every morphism $f:M_{\lambda}\rightarrow
M_{\lambda^{\prime}}$ in the diagram, there exists a linear map
$g:V_{\lambda}\rightarrow V_{\lambda^{\prime}}$ such that the following
diagram commutes:

${E_{\lambda}}$${V_{\lambda}}$${E_{\lambda^{\prime}}}$${V_{\lambda^{\prime}}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{\lambda}}$$\scriptstyle{g}$$\scriptstyle{\varphi_{\lambda^{\prime}}}$

Two such family of maps $(\\{\varphi_{i,\lambda}:E_{\lambda}\rightarrow
V_{i,\lambda}\\}_{\lambda\in\Lambda},\\{g_{i}:V_{i,\lambda}\rightarrow
V_{i,\lambda^{\prime}}\\})$ for $i\in\\{1,2\\}$ are _equivalent_ if for every
$\lambda\in\Lambda$ there exists an isomorphism
$\psi_{\lambda}:V_{1,\lambda}\rightarrow V_{2,\lambda}$ such that, for every
morphism $f:M_{\lambda}\rightarrow M_{\lambda^{\prime}}$ in the diagram of
matroids, the following diagram commutes:

${E_{\lambda}}$${V_{1,\lambda}}$${V_{2,\lambda}}$${E_{\lambda^{\prime}}}$${V_{1,\lambda^{\prime}}}$${V_{2,\lambda^{\prime}}}$$\scriptstyle{\varphi_{1,\lambda}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{2,\lambda}}$$\scriptstyle{\psi_{\lambda}}$$\scriptstyle{\sim}$$\scriptstyle{g_{1}}$$\scriptstyle{g_{2}}$$\scriptstyle{\varphi_{1,\lambda^{\prime}}}$$\scriptstyle{\varphi_{2,\lambda^{\prime}}}$$\scriptstyle{\psi_{\lambda^{\prime}}}$$\scriptstyle{\sim}$

A _realization_ of a diagram of matroids is an equivalence class of families
of maps $(\varphi_{i,\lambda},g_{i})$ as above.

The above definitions can be more compactly recast in terms of category
theory. We will follow a similar point of view taken in [10, Remark 2.1].

Let $\mathbf{Mat}(K)$ be the category in which an object is a map
$\varphi:E\rightarrow V$ realizing a matroid $M(\varphi)$ on the ground set
$E$ over a field $K$, and a morphism from $\varphi_{1}$ to $\varphi_{2}$ is a
commutative diagram of the form

${E_{1}}$${V_{1}}$${E_{2}}$${V_{2}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{g}$$\scriptstyle{\varphi_{2}}$

realizing a morphism $f:M(\varphi_{1})\rightarrow M(\varphi_{2})$. From the
definition, we have a functor
$\mathcal{F}:\mathbf{Mat}(K)\rightarrow\mathbf{Mat}$ defined by
$\varphi\mapsto M(\varphi)$.

Now let $D:\mathbf{I}\rightarrow\mathbf{Mat}$ be a diagram of matroids.
Without loss of generality, we can assume that $D$ is injective on objects.
Thus the image of $D$ is a subcategory $\operatorname{im}(D)$ of
$\mathbf{Mat}$. Then, the diagram $D$ is realized over a field $K$ by a
functor $F:\operatorname{im}(D)\rightarrow\mathbf{Mat}(K)$ such that
$\mathcal{F}\circ F$ is the identity on $\operatorname{im}(D)$. Two such
functors realizing $D$ are equivalent if they are naturally isomorphic, and a
realization of $D$ is an isomorphism class of such naturally isomorphic
functors.

######  Question 2.8.

Let $D:\mathbf{I}\rightarrow\mathbf{Mat}$ be a diagram of matroids. Does the
notion of realizability of Definition 2.7 give rise to a stratification of the
quiver Grassmannian corresponding to $D$ (defined as in [26])?

###  2.3 Modular cuts and weak maps

Following [21, Section 7.3], we now recall the notions of one element
extensions of a matroid and modular cuts.

An _extension_ of a matroid $M$ on $E$ is a matroid $N$ on $E\cup S$ such that
$N\setminus S=M$. If $S=\\{e\\}$ then $N$ is said a _one element extension_ of
$M$, denoted $M+e$. In order to define a one element extension $M+e$ of $M$,
we need just to specify, for each flat $F$ of $M$, whether or not $F$ and
$F\cup\\{e\\}$ are flats of $N$. Thus, we can partition the family
$\mathscr{F}(M)$ of flats of $M$ into three families:

$\displaystyle\mathscr{F}_{1}$
$\displaystyle=\\{F\in\mathscr{F}(M):F\in\mathscr{F}(M+e),F\cup\\{e\\}\in\mathscr{F}(M+e)\\},$
$\displaystyle\mathscr{F}_{2}$
$\displaystyle=\\{F\in\mathscr{F}(M):F\in\mathscr{F}(M+e),F\cup\\{e\\}\notin\mathscr{F}(M+e)\\},$
$\displaystyle\mathscr{F}_{3}$
$\displaystyle=\\{F\in\mathscr{F}(M):F\notin\mathscr{F}(M+e),F\cup\\{e\\}\in\mathscr{F}(M+e)\\}.$

Now $\mathscr{F}_{1}$ is downward closed, i.e. any flat contained in a flat in
$\mathscr{F}_{1}$ is in $\mathscr{F}_{1}$. On the other hand,
$\mathscr{F}_{3}$ is upward closed, i.e. any flat that contains a flat in
$\mathscr{F}_{3}$ is in $\mathscr{F}_{3}$. Finally, a flat is in
$\mathscr{F}_{2}$ if and only if it is not in $\mathscr{F}_{3}$ and it is
covered by a flat in $\mathscr{F}_{3}$. In light of the previous statements,
this partition is uniquely determined by $\mathscr{F}_{3}$. Families of flats
of type $\mathscr{F}_{3}$ are called _modular cuts_ , and they can be
characterized as follows. A family $\mathcal{M}$ of flats of $M$ is a modular
cut of $M$ if and only if

  * •

$\mathcal{M}$ is upward closed and nonempty,

  * •

if $A,B\in\mathcal{M}$ are such that
$\operatorname{rk}(A)+\operatorname{rk}(B)=\operatorname{rk}(A\cap
B)+\operatorname{rk}(A\cup B)$, then $A\cap B\in\mathcal{M}$.

It is now clear that, the one element extensions of $M$ are one to one with
modular cuts of $M$. We will denote by $M+_{\mathcal{M}}e$ the one element
extension of $M$ corresponding to the modular cut $\mathcal{M}$.

######  Definition 2.9.

Let $M_{i}$ be a matroid on $E_{i}$ for $i\in\\{1,2\\}$. A _weak map_ is a
function of sets $f:E_{1}\rightarrow E_{2}$ such that if $I$ is an independent
set of $M_{2}$, then $f^{-1}(I)$ is an independent set of $M_{1}$.

Weak maps provide the following partial order on matroids on a common ground
set $E$: if the identity on $E$ is a weak map from $M_{1}$ to $M_{2}$ then
$M_{1}\leq M_{2}$.

######  Lemma 2.10 ([20, Section 2.6]).

Let $M$ be a matroid on $E$, and let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$
be two modular cuts of $M$. The following are equivalent:

  1. 1.

$\mathcal{M}_{1}\subseteq\mathcal{M}_{2}$,

  2. 2.

$M+_{\mathcal{M}_{1}}e\leq M+_{\mathcal{M}_{2}}e$,

  3. 3.

$(M+_{\mathcal{M}_{1}}e)/e\leq(M+_{\mathcal{M}_{2}}e)/e$.

###  2.4 Factorizations and majors

In this section we recall the basics of factorizations and majors of matroid
quotients. Our main reference here is [20].

The _nullity_ of a morphism of matroids $f:M_{1}\rightarrow M_{2}$ (or strong
map, or quotient) is the integer
$n(f)=\operatorname{rk}(M_{1})-\operatorname{rk}(M_{2})$. Quotients
$M_{1}\twoheadrightarrow M_{2}$ of nullity one are called _elementary_. In
this case, $M_{2}$ is said to be an _elementary quotient_ of $M_{1}$, and
$M_{2}$ can be obtained from $M_{1}$ by a one element extension followed by a
contraction of the element added. Since, as we have seen in the previous
section, one element extensions correspond to modular cuts, so do elementary
quotients.

In the remainder of this section, if not stated otherwise,
$f:M_{1}\twoheadrightarrow M_{2}$ is a quotient of nullity $k$ of matroids
sharing a common ground set $E$ of cardinality $n$.

######  Definition 2.11.

A _factorization_ of a quotient $f:M_{1}\twoheadrightarrow M_{2}$ is a
sequence of matroids $N=(N_{0},N_{1},\dots,N_{k})$ such that we have a chain
of elementary quotients as follows:

$M_{1}=N_{0}\twoheadrightarrow N_{1}\twoheadrightarrow\dots\twoheadrightarrow
N_{k-1}\twoheadrightarrow N_{k}=M_{2}.$ (1)

The _length_ (or _nullity_) of a factorization is equal to the nullity of its
quotient.

The nullity of a subset $A\subseteq E$ is
$n(A)=\operatorname{rk}_{M_{1}}(A)-\operatorname{rk}_{M_{2}}(A)$. From [21,
Proposition 8.1.6] the nullity is an increasing set function. We will
frequently use this fact in the rest of the paper.

######  Proposition-Definition 2.12 (Higgs lift [21, Lemma 8.1.6]).

Given a quotient $f:M_{1}\twoheadrightarrow M_{2}$, the flats of $M_{2}$
together with the flats of $M_{1}$ with nullity strictly less than $i$ are the
flats of a matroid $L^{i}(f)$, called the _$i$ -th Higgs lift_ of $f$.

The Higgs lifts of $f$ form a chain of elementary quotients:

$M_{1}=L^{k}(f)\twoheadrightarrow
L^{k-1}(f)\twoheadrightarrow\dots\twoheadrightarrow L^{1}(f)\twoheadrightarrow
L^{0}(f)=M_{2}.$

Therefore, the sequence $(L^{i}(f))_{i}$ is a factorization of $f$, called the
_Higgs factorization_. Hence, every quotient has at least one factorization,
but in general factorizations are not unique (see [20]).

Factorizations are particular instances of diagrams of matroids. Thus we can
consider realizability and realizations of factorizations in the sense of
Definition 2.7.

######  Definition 2.13.

A _major_ of a quotient $f:M_{1}\twoheadrightarrow M_{2}$ of nullity $k$ is a
matroid $H$ on the ground set $E\cup K$, where $K$ is of cardinality $k$ and
disjoint from $E$, such that $M_{1}=H\setminus K$ and $M_{2}=H/K$.

In other words, if $H$ is a major of $f:M_{1}\twoheadrightarrow M_{2}$, then
$f$ can be factored into an extension followed by a contraction:
$M_{1}\rightarrow H\rightarrow M_{2}$.

######  Theorem 2.14 (Higgs [18]).

Every matroid quotient admits a major.

The major constructed by Higgs in the proof of the above theorem is called
_Higgs major_. The Higgs major of $f:M_{1}\twoheadrightarrow M_{2}$ is
constructed by taking the $k$-th Higgs lift of the quotient $M_{1}\oplus
U_{k,k}\twoheadrightarrow M_{2}\oplus U_{0,k}$. See [21, Theorem 8.2.7] for
more details.

In general, majors of a quotient are not unique. See [20, Example 3.14] for an
example of a quotient admitting two distinct majors.

######  Lemma 2.15 ( [20, Lemma 3.3-3.4]).

Let $M$ be a matroid on $E$, $e\in E$ and $f$ an element not in $E$.

  1. 1.

If $\mathcal{M}$ is a modular cut of $M/e$, then $\mathcal{M}\cup
e=\\{F\cup\\{e\\}:F\in\mathcal{M}\\}$ is a modular cut of $M$.

  2. 2.

$(M/e)+_{\mathcal{M}}f=(M+_{\mathcal{M}\cup e}f)/e$.

We now introduce some more notation: arrows of type
$\overset{e}{\underset{\mathcal{M}}{\longrightarrow}}$ denote a strong map
arising from a one element extension by an element $e$ with respect to the
modular cut $\mathcal{M}$; arrows of type $\overset{e}{\longrightarrow}$
denote the strong map arising from the quotient by the element $e$. With this
notation, the above lemma can be rephrased simply by saying that the following
diagram of matroids and strong maps, called the _standard extension-
contraction diamond_ in [20], commutes:

${M_{2}}$${M_{1}}$${M_{3}}$${M_{4}}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{\mathcal{M}\cup
e}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{\mathcal{M}}$ (2)

Now let $N=(N_{0},\dots,N_{k})$ be a factorization of a quotient
$f:M_{1}\twoheadrightarrow M_{2}$ of nullity $k$. Every elementary quotient
$N_{i-1}\twoheadrightarrow N_{i}$ can be factored into an extension by one
element $e_{i}$ with modular cut $\mathcal{M}_{i}$ followed by a contraction
by $e_{i}$. In particular, this means that we can go from $M_{1}=N_{0}$ to
$M_{2}=N_{k}$ by extending and then contracting $e_{i}$ for $i$ from $1$ to
$k$. By repeatedly applying Lemma 2.15, we can first extend $M_{1}$ by
$e_{1},e_{2},e_{3},\dots,e_{k}$ with respect to the modular cuts
$\mathcal{M}_{1},\mathcal{M}_{2}\cup e_{1},\mathcal{M}_{3}\cup e_{1}\cup
e_{2},\dots,\mathcal{M}_{k}\cup e_{1}\cup\dots\cup e_{k-1}$ respectively, and
then contract by $e_{1},\dots,e_{k}$ all at once. Thus, the matroid we obtain
after all such extensions is, by definition, a major of $f$. This major will
be denoted by $\mathfrak{M}(N)$, and clearly it depends on the factorization
$N$ we started with.

The above procedure can be better displayed by constructing a triangular
commutative diagram associated with the factorization. Starting with the
extension and contraction maps along the base of the triangle, each successive
level is constructed by applying Lemma 2.15. We illustrate this procedure with
the following triangular commutative diagram of a factorization of a quotient
of nullity $3$.

${N_{3,0}}$${N_{2,0}}$${N_{2,1}}$${N_{1,0}}$${N_{1,1}}$${N_{1,2}}$${N_{0,0}}$${N_{0,1}}$${N_{0,2}}$${N_{0,3}}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{3}}$$\scriptstyle{\mathcal{M}_{3}\cup
e_{1}\cup
e_{2}}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{2}}$$\scriptstyle{\mathcal{M}_{2}\cup
e_{1}}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{3}}$$\scriptstyle{\mathcal{M}_{3}\cup
e_{2}}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{3}}$$\scriptstyle{e_{1}}$$\scriptstyle{\mathcal{M}_{1}}$$\scriptstyle{e_{2}}$$\scriptstyle{\mathcal{M}_{2}}$$\scriptstyle{e_{3}}$$\scriptstyle{\mathcal{M}_{3}}$
(3)

We now wish to construct a factorization from a major. Let $H$ be a major of
$f:M_{1}\twoheadrightarrow M_{2}$ on the ground set $E\cup K$, where
$K=\\{e_{1},\dots,e_{k}\\}$. We define the factorization
$\mathfrak{F}(H)=(\mathfrak{F}(H)_{i})$ of $f$ by

$\mathfrak{F}(H)_{i}=\big{(}H/\\{e_{1},\dots,e_{i}\\}\big{)}\setminus\\{e_{i+1},\dots,e_{k}\\}.$

Let $\mathscr{F}_{f}$ denote the set of factorizations of the quotient $f$,
and let $\mathscr{M}_{f}$ denote the set of majors of $f$. Thus far, we have
defined the following two maps:

$\displaystyle\mathfrak{M}:\mathscr{F}_{f}\rightarrow\mathscr{M}_{f},$
$\displaystyle\mathfrak{F}:\mathscr{M}_{f}\rightarrow\mathscr{F}_{f}.$

######  Theorem 2.16 (Kennedy [20]).

  1. 1.

The map $\mathfrak{F}$ is surjective but not injective, whereas $\mathfrak{M}$
is injective but not surjective.

  2. 2.

The map $\mathfrak{F}\circ\mathfrak{M}$ is the identity on $\mathscr{F}_{f}$,
whereas $\mathfrak{M}\circ\mathfrak{F}$ is a coclosure operator on
$\mathscr{M}_{f}$ with the partial order given by the identity on the ground
set being a weak map.

  3. 3.

The map $\mathfrak{F}$ applied to the Higgs major gives us the Higgs
factorization and conversely $\mathfrak{M}$ applied to the Higgs factorization
gives us the Higgs major.

##  3 Realizability of quotients of matroids

######  Lemma 3.1.

Every elementary quotient $f:M_{1}\twoheadrightarrow M_{2}$ has a unique
major.

###### Proof.

Existence follows from Theorem 2.14. For uniqueness, let $H_{1}$ and $H_{2}$
be two majors of $f$. Then $H_{1}$ and $H_{2}$ are obtained by extending
$M_{1}$ with an element $e$ with respect to the modular cuts $\mathcal{M}_{1}$
and $\mathcal{M}_{2}$ respectively. From Lemma 2.10 we have that $H_{1}\leq
H_{2}$ if and only if $H_{1}/e\leq H_{2}/e$. Since $H_{1}/e=H_{2}/e=M_{2}$, it
follows that $H_{1}=H_{2}$. ∎

There are several results in the literature that are similar or implicitly
related to the next result. See for instance [7, Section 5.1] and [19, Section
2.2].

######  Proposition 3.2.

Let $f:M_{1}\twoheadrightarrow M_{2}$ be an elementary quotient. The
realizations of $f$ are in one-to-one correspondence with the realizations of
its unique major.

###### Proof.

Let $H$ be the major of $f$ on the ground set $E\cup e$, for some $e\notin E$.
Let $\varphi:E_{o}\cup e\rightarrow W$ be a realization of $H$, and let $U$ be
the space generated by $\varphi(e)$. Then a realization of $f$ is given by the
following commutative diagram

${E_{o}}$${W}$${E_{o}}$${W/U}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\varphi_{2}}$

where $\varphi_{1}$ is the restriction to $E_{o}$ of $\varphi$, and
$\varphi_{2}$ is the composition of $\varphi_{1}$ and the quotient map by $U$.
Conversely, consider a realization of $f$. We can assume that such a
realization is represented by a diagram as above, where again the space $U$ is
one dimensional, since $f$ is elementary. Thus, $U$ is generated by some
nonzero vector $v$, and the map $\varphi:E_{o}\cup e\rightarrow W$ defined by
$\varphi_{|E_{o}}=\varphi_{1}$ and $\varphi(e)=v$ is a realization of $H$.

Now suppose that $\psi_{i}:E_{o}\cup e\rightarrow W_{i}$ for $i\in\\{1,2\\}$
represent two realizations of $H$ such that they give rise to the same
realization of $f$. This means that we have a commutative diagram

${E_{o}}$${W_{1}}$${W_{2}}$${E_{o}}$${W_{1}/U_{1}}$${W_{2}/U_{2}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{h_{1}}$$\scriptstyle{\sim}$$\scriptstyle{/U_{1}}$$\scriptstyle{/U_{2}}$$\scriptstyle{/U_{1}\circ\varphi_{1}}$$\scriptstyle{/U_{2}\circ\varphi_{2}}$$\scriptstyle{h_{2}}$$\scriptstyle{\sim}$

where $U_{i}$ is the space generated by $\psi_{i}(e)$,
$\varphi_{i}={\psi_{i}}_{|E_{o}}$, and $h_{i}$ is an isomorphism, for
$i\in\\{1,2\\}$. Since the diagram is commutative, $h_{1}$ sends $U_{1}$ to
$U_{2}$. Further, $U_{1}$ and $U_{2}$ are one-dimensional, so $U_{i}$ is
generated by some nonzero vector $v_{i}$ for $i\in\\{1,2\\}$. Hence, $h_{1}$
sends $v_{1}$ to a scalar multiple of $v_{2}$, therefore $\psi_{1}$ and
$\psi_{2}$ represent the same realization of $H$. ∎

######  Lemma 3.3.

Consider the standard extension-contraction diamond as in (2) and let $K$ be
an infinite field. Then, $M_{2}$ is realizable over $K$ if and only if the
subdiagram
$M_{1}\xrightarrow{e}M_{4}\overset{f}{\underset{\mathcal{M}}{\longrightarrow}}M_{3}$
obtained from (2) is realizable over $K$.

###### Proof.

Since $M_{1}$, $M_{3}$ and $M_{4}$ are minors of $M_{2}$, if the latter is
realizable, then it is clear that the lower part of the diagram is realizable.

Conversely, assume the lower part of the diagram is realizable. Let $E$ be the
cardinality $n$ ground set of $M_{1}$, so that $E\cup f$, $E\cup f\setminus e$
and $E\setminus e$ are the ground sets of $M_{2}$, $M_{3}$ and $M_{4}$
respectively. If $e$ is a loop of $M_{1}$, then it is also a loop of $M_{2}$,
and contracting $e$ is the same as deleting $e$. Thus a realization of $M_{2}$
is given by extending the realization of $M_{3}$ by sending $e$ to the zero
vector. Therefore, we can assume that $e$ is not a loop of $M_{1}$ and part of
a basis. In addition, if $f$ is a coloop of $M_{3}$, this will imply
$\mathcal{M}=\emptyset$, so we will have $\mathcal{M}\cup e=\emptyset$ as
well, that is, $f$ will be a coloop also of $M_{2}$ and a realization of
$M_{2}$ can be constructed by extending a realization $\varphi:E\rightarrow V$
of $M_{1}$ to $E\cup f$ by sending $f$ to $\underline{0}\oplus 1\in V\oplus
K$. Therefore we can assume that $f$ is not a coloop of $M_{3}$. Thus, if
$M_{1}$ is of rank $r$ then $M_{2}$ has rank $r$ as well and $M_{3}$ and
$M_{4}$ have rank $r-1$.

Now, up to equivalence, the diagram
$M_{1}\xrightarrow{e}M_{4}\overset{f}{\underset{\mathcal{M}}{\longrightarrow}}M_{3}$
is realized by a commutative diagram of the following form:

${E_{o}}$${E_{o}\setminus e}$${E_{o}\cup f\setminus
e}$${K^{r}}$${K^{r-1}}$${K^{r-1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$$\scriptstyle{g}$$\scriptstyle{id_{K^{r-1}}}$

where $\varphi_{1}(e)=e_{1}\in K^{r}$, $g:K^{r}\rightarrow K^{r-1}$ is the
quotient map by $e_{1}$ and $\varphi_{3}(f)=(f_{2},\dots,f_{r})\in K^{r-1}$.
Now a realization of $M_{2}$ is given by a function $\varphi:E_{o}\cup
f\rightarrow K^{r}$ defined by $\varphi_{|E_{o}}=\varphi_{1}$ and
$\varphi(f)=(f_{1},f_{2},\dots,f_{r})\in K^{r}$, where the element $f_{1}\in
K$ is chosen such that the vector $\varphi(f)$ is in the subspace of $K^{r}$
generated by $\varphi_{1}(F)$, for $F$ a flat of $M_{1}$, if and only if
$F\in\mathcal{M}\cup e$. In the remainder of the proof, we will show that such
a choice is possible.

First, if $F$ is a flat of $M_{1}$ in $\mathcal{M}\cup e$, then $e\in F$ and
$F\setminus e\in\mathcal{M}$ by definition. Thus, $(f_{2},\dots,f_{r})$
belongs to the space generated by $\varphi_{2}(F\setminus e)$, and, since
$\varphi_{1}(e)=e_{1}$, this implies that
$(f_{1},\dots,f_{r})\in\langle\varphi_{1}(F)\rangle$ for every $f_{1}\in K$.
Now let $F$ be a flat of $M_{1}$ such that the space
$L=\langle\varphi_{1}(F)\rangle$ contains the affine line
$A=\\{(x,f_{2},\dots,f_{r})\in K^{r}:x\in K\\}$. This means that $L$ contains
$e_{1}$, so the flat $F$ contains $e$. In particular, the image of $L$ under
the quotient map $g$ contains the vector $(f_{2},\dots,f_{r})$, and this
implies that $F\setminus e\in\mathcal{M}$, that is $F\in\mathcal{M}\cup e$.
Thus, for any flat $F$ of $M_{1}$ not in $\mathcal{M}\cup e$, the space
$L=\langle\varphi_{1}(F)\rangle$ does not contain the affine line $A$, hence
it intersects $A$ in at most one point $p_{F}$. Now it is enough to chose
$f_{1}$ such that $\varphi(f)\in A\setminus\\{p_{F}:F\notin\mathcal{M}\cup
e\\}$. This last set is not empty since the points $p_{F}$ are finitely many,
as the flats of $M_{1}$ are finite, while the affine space $A$ contains
infinitely many points since $K$ is infinite. ∎

The above result is easily seen to be not true for finite fields, as the
following example shows.

######  Example 3.4.

Let $K=\\{0,1\\}$ be the field with two elements. Consider the following
standard extension-contraction diamond:

${U_{2,4}}$${U_{2,3}}$${U_{1,3}}$${U_{1,2}}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{\mathcal{M}\cup
e}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{\mathcal{M}}$

Up to permutations, the matroids $U_{2,3}$, $U_{1,3}$ and $U_{1,2}$ have each
a unique realization over $K$. These realizations constitute also a
realization of the diagram
$U_{2,3}\xrightarrow{e}U_{1,3}\overset{f}{\underset{\mathcal{M}}{\longrightarrow}}U_{1,2}$
over $K$, while $U_{2,4}$ is not realizable over $K$.

######  Corollary 3.5.

A factorization $N$ of a strong map $f:M_{1}\twoheadrightarrow M_{2}$ is
realizable over an infinite field $K$ if and only if so is its major
$\mathfrak{M}(N)$.

###### Proof.

Let $\Delta(N)$ be the triangular diagram induced by $N$, constructed by
following the procedure we illustrated in (3). Then, the statement follows by
applying Proposition 3.2 along the base of the triangle, and by repeatedly
applying Lemma 3.3 on all standard extension-contraction diamonds. ∎

######  Lemma 3.6.

Let $f:M_{1}\twoheadrightarrow M_{2}$ be a quotient of matroids with common
ground set $E$. A flat $F$ of $M_{1}$ is not closed in $M_{2}$ if and only if
it is covered by a flat of $M_{1}$ of nullity $n(F)+1$.

###### Proof.

Since from [21, Proposition 8.1.6] the nullity is a non-decreasing function,
any flat that covers $F$ has nullity greater or equal than $F$. Suppose that
$F$ is closed in $M_{2}$. Then, for every $a\in E\setminus F$ we have
$r_{M_{1}}(F\cup a)=r_{M_{1}}(F)+1$ and $r_{M_{2}}(F\cup a)=r_{M_{2}}(F)+1$,
thus $n(F\cup a)=n(F)$. Let $G$ be the closure of $F\cup a$ in $M_{1}$, then
$r_{M_{1}}(G)=r_{M_{1}}(F\cup a)$, also from [21, Proposition 8.1.6], $G$ is
contained in the closure of $F\cup a$ in $M_{2}$, so
$r_{M_{2}}(G)=r_{M_{2}}(F\cup a)$. Hence $n(G)=n(F)$, so all the flats of
$M_{1}$ that cover $F$ have the same nullity of $F$. On the other hand, if $F$
is not closed in $M_{2}$, then there exists $a\in E\setminus F$ such that
$r_{M_{2}}(F\cup a)=r_{M_{2}}(F)$, and for the same reasoning above, we have
$n(F\cup a)=n(F)+1$, so the closure of $F\cup a$ in $M_{1}$ is a flat of
$M_{1}$ of nullity $n(F)+1$ that covers $F$. ∎

######  Lemma 3.7.

Let $f:M_{1}\twoheadrightarrow M_{2}$ be a quotient of matroids of nullity $k$
and let $(L^{i}(f))_{i}$ be its Higgs factorization. Then, the modular cut of
$M_{1}$ associated to the elementary quotient $M_{1}\twoheadrightarrow
L^{k-1}(f)$ is given by all the nullity $k$ flats of $M_{1}$.

###### Proof.

From Lemma 3.6 all nullity $k$ flats of $M_{1}$ are closed in $M_{2}$.
Therefore, from Proposition-Definition 2.12, the flats of $L^{k-1}(f)$
consists of the flats of $M_{1}$ that are either closed in $M_{2}$ or do not
have nullity $k-1$. From Lemma 3.6, this means that we are removing from
$M_{1}$ all the flats covered by a nullity $k$ flat. This shows that the
modular cut of this elementary quotient is given by the nullity $k$ flats. ∎

As an application of the above lemma, we compute the Higgs major of quotients
of uniform matroids.

######  Example 3.8 (Higgs major of a quotient of uniform matroids).

Consider the quotient $f:U_{r+k,n}\twoheadrightarrow U_{r,n}$ of uniform
matroids on the common ground set $E$. The Higgs major of $f$ is
$U_{r+k,n+k}$. We can prove this by induction on $k$. In fact, from Lemma 3.7
the modular cut of the elementary quotient $U_{r+k,n}\twoheadrightarrow
L^{k-1}(f)$ is given by the nullity $k$ flats of $U_{r+k,n}$, which in this
case those consist of just the set $E$. This means that the corresponding one
element extension of $U_{r+k,n}$ give rise to the uniform matroid
$U_{r+k,n+1}$. Now, $L^{k-1}(f)$ is the contraction of $U_{r+k,n+1}$ by the
extended element, so we have $L^{k-1}(f)=U_{r+k-1,n}$. Now by induction, this
means that the Higgs factorization of $f$ is $U_{r+k,n}\twoheadrightarrow
U_{r+k-1,n}\twoheadrightarrow\dots\twoheadrightarrow
U_{r+1,n}\twoheadrightarrow U_{r,n}$, and the Higgs major is $U_{r+k,n+k}$.

######  Proposition 3.9.

Let $f:M_{1}\twoheadrightarrow M_{2}$ be a quotient of matroids of nullity
$k$. A realization of $f$ over an infinite field $K$ induces a realization of
the Higgs factorization $(L^{i}(f))_{i}$ over $K$.

###### Proof.

Fix a realization of $f$ given by the following commutative diagram

${E_{o}}$${V_{1}}$${E_{o}}$${V_{2}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\varphi_{2}}$

where $E$ is the common ground set of $M_{1}$ and $M_{2}$. Set $U=\ker g$, so
we have $\dim U=k$. Up to equivalence, we can assume that $V_{2}\simeq
V_{1}/U$ and $g$ is the quotient map by $U$. We proceed by induction on the
nullity $k$ of $f$. If $k=0$, then $M_{1}=M_{2}$ and there is nothing to
prove. Now assume $k>0$. By Lemma 3.7 the modular cut of $M_{1}$ associated to
the elementary quotient $M_{1}\twoheadrightarrow L^{k-1}(f)$ is given by the
family of flats of nullity $k$. Now, for every flat $F$ of $M_{1}$ we denote
by $L_{F}$ the subspace of $V_{1}$ generated by $\varphi_{1}(F)$. The nullity
of $F$ is equal to the dimension of $L_{F}\cap U$. In fact, we have

$n(F)=r_{M_{1}}(F)-r_{M_{2}}(F)=\dim(L_{F})-\dim(L_{F}/\big{(}L_{F}\cap
U)\big{)}=\dim(L_{F}\cap U).$

Therefore, if the nullity of $F$ is $k$ then $L_{F}$ contains $U$. Now the set
difference $W=U\setminus\bigcup_{n(F)=k-1}L_{F}$ is nonempty since the base
field $K$ is infinite. Thus we can choose a vector $v\in W$. The vector $v$ is
in $U$, thus it belongs to $L_{F}$ if $F$ is a flat of $M_{1}$ of nullity $k$,
but it does not if $F$ has nullity $k-1$. Hence, the quotient map by $\langle
v\rangle$ is a realization of $M\twoheadrightarrow L^{k-1}(f)$, and the
quotient map from $V_{1}/\langle v\rangle$ to $V_{2}$ by $U/\langle v\rangle$
is a realization of the quotient $L^{k-1}(f)\twoheadrightarrow M_{2}$. From
[20, Proposition 4.8] the sequence $(L^{k-1}(f),\dots,L^{0}(f))$ is the Higgs
factorization of $L^{k-1}(f)\twoheadrightarrow M_{2}$. Thus, we can apply
induction, so the realization of $L^{k-1}(f)\twoheadrightarrow M_{2}$ induces
a realization of the factorization starting from $k-1$ to $0$. Further, note
that in every step, we did not change the realizations of $M_{1}$ and $M_{2}$
we started with. Therefore, we can adjoin the realization of
$M_{1}\twoheadrightarrow L^{k-1}(f)$ with that of the rest of the
factorization. ∎

######  Theorem 3.10.

Let $f:M_{1}\twoheadrightarrow M_{2}$ be a quotient of matroids, and let $K$
be an infinite field. The following statements are equivalent:

  1. 1.

$f$ is realizable over $K$,

  2. 2.

the Higgs factorization of $f$ is realizable over $K$,

  3. 3.

the Higgs major of $f$ is realizable over $K$.

###### Proof.

The implication $(1)\Rightarrow(2)$ is Proposition 3.9 and $(2)\Rightarrow(1)$
is clear. Finally, $(2)\Leftrightarrow(3)$ follows from Corollary 3.5. ∎

The above theorem can be generalized to flag matroids. A _flag matroid_ of
length $l$ is a chain of matroid quotients of the form
$M_{1}\overset{f_{1}}{\twoheadrightarrow}M_{2}\overset{f_{2}}{\twoheadrightarrow}\dots\overset{f_{l}}{\twoheadrightarrow}M_{l+1}$,
and a flag matroid is realizable over a field $K$ if it is so as a diagram of
matroids. If we factor each quotient $f_{i}$ with the Higgs factorization, we
obtain a chain of elementary quotients that constitute a factorization of the
quotient $M_{1}\twoheadrightarrow M_{l+1}$ (note that this is not necessarily
equal to the Higgs factorization of this last quotient). We call such a
factorization the _Higgs factorization_ of the flag matroid, and the major of
this factorization the _Higgs major_ of the flag matroid.

######  Corollary 3.11.

Let
$M_{1}\overset{f_{1}}{\twoheadrightarrow}M_{2}\overset{f_{2}}{\twoheadrightarrow}\dots\overset{f_{l}}{\twoheadrightarrow}M_{l+1}$
be a length $l$ flag matroid, and let $K$ be an infinite field. The following
statements are equivalent:

  1. 1.

the flag matroid is realizable over $K$,

  2. 2.

the Higgs factorization of the flag matroid is realizable over $K$,

  3. 3.

the Higgs major of the flag matroid is realizable over $K$.

###### Proof.

For $(1)\Rightarrow(2)$ it is enough to apply Proposition 3.9 on each quotient
$f_{i}$. The rest of the proof is analogous to the proof of Theorem 3.10. ∎

Now we interpret Theorem 3.10 in terms of maps of realization spaces. Let
$[n]=\\{1,\dots,n\\}$, and denote by $\binom{[n]}{r}$ the subsets of $[n]$ of
cardinality $r$. Let $M$ be a rank $r$ matroid on $[n]$. The _realization
space_ of $M$ is the subset

$\mathcal{R}(M)=\left\\{(p_{B})_{B\in\binom{[n]}{r}}\in\operatorname{Gr}(r,n):p_{B}\neq
0\Leftrightarrow B\in\mathcal{B}(M)\right\\}\subseteq\operatorname{Gr}(r,n).$

If $N$ is a rank $s\leq r$ matroid on $[n]$ such that $M\twoheadrightarrow N$,
then the _realization space_ of this quotient is the subset

$\begin{split}\mathcal{R}(M\twoheadrightarrow
N)=\Big{\\{}(p_{B})_{B\in\binom{[n]}{r}\cup\binom{[n]}{s}}\in\operatorname{Fl}(s,r,n):p_{B}\neq
0\Leftrightarrow B\in\mathcal{B}(M)\cup\mathcal{B}(N)\Big{\\}}\\\
\subseteq\operatorname{Fl}(s,r;n)\subseteq\operatorname{Gr}(s,n)\times\operatorname{Gr}(r,n).\end{split}$

Now let $H$ be the Higgs major of $M\twoheadrightarrow N$ on the ground set
$[n]\cup[\overline{k}]$ where
$[\overline{k}]=\\{\overline{1},\overline{2},\dots,\overline{k}\\}$. Then,
there is a surjective map
$\varphi:\mathcal{R}(H)\twoheadrightarrow\mathcal{R}(M\twoheadrightarrow N)$
defined by

$\varphi\left((p_{B})_{B\in\binom{[n]\cup[\overline{k}]}{r}}\right)=(p_{B^{\prime}})_{B^{\prime}\in\binom{[n]}{r}}\times(p_{B^{\prime\prime}\cup[\overline{k}]})_{B^{\prime\prime}\in\binom{[n]}{s}}.$

In other words, $\varphi$ is projecting away the coordinates $p_{B}$ of
$\mathcal{R}(H)$ for all $B\in\binom{[n]\cup[\overline{k}]}{r}$ that are not
contained in $[n]$ and do not contain $[\overline{k}]$. Since
$\mathcal{R}(M\twoheadrightarrow
N)\subseteq\mathcal{R}(M)\times\mathcal{R}(N)$, a point in the realization
space of $M\twoheadrightarrow N$ can be viewed as a particular pair of
realizations of $M$ and $N$ respectively. Consider a realization of $H$,
corresponding to a point $(p_{B})_{B\in\binom{[n]\cup[\overline{k}]}{r}}$ in
$\mathcal{R}(H)$. The map $\varphi$ is sending this realization of $H$ to the
pair of realizations of $M$ and $N$ by identifying the realization of $M$ with
the point in $\mathcal{R}(M)$ with coordinates
$(p_{B^{\prime}})_{B^{\prime}\in\binom{[n]}{r}}$ and the realization of $N$
with the point in $\mathcal{R}(N)$ with coordinates
$(p_{B^{\prime\prime}\cup[\overline{k}]})_{B^{\prime\prime}\in\binom{[n]}{s}}$.

The fact that $\varphi$ is well defined can be verified directly from
comparing the Plücker relations of $\mathcal{R}(H)$ and the incidence Plücker
relations of $\mathcal{R}(M\twoheadrightarrow N)$. The surjectivity of
$\varphi$ is essentially a reformulation of $(1)\Rightarrow(3)$ of Theorem
3.10.

##  4 Applications and examples

In this section, we explain how quotients of matroids relate in a natural way
to relative realizability problems in tropical geometry.

###  4.1 Linear case

Let $M$ be a rank $r$ matroid with ground set $E=\\{0,1,\dots,n\\}$. We recall
that a _cycle_ of a matroid is a union of circuits. Denote by $\mathcal{C}(M)$
the set of circuits of $M$, and by $\mathcal{V}(M)$ the set of cycles of $M$.
A family $\mathcal{V}$ of subsets of $E$ is the set of cycles of a matroid if
and only if it satisfies the following axioms [8]:

  1. 1.

$\emptyset\in\mathcal{V}$,

  2. 2.

$\mathcal{V}$ is closed under taking unions,

  3. 3.

if $V_{1},V_{1}\in\mathcal{V}$ and $e\in V_{1}\cap V_{2}$ there exists
$V\in\mathcal{V}$ such that

$(V_{1}\cup V_{2})\setminus(V_{1}\cap V_{2})\subseteq V\subseteq(V_{1}\cup
V_{2})\setminus\\{e\\}.$

For a subset $A\subseteq E$ define $\operatorname{trop}(A)$ to be the set of
vectors $v=(v_{0},\dots,v_{n})\in\mathbb{R}^{n+1}$ such that the minimum of
the numbers $v_{i}$ is attained at least twice as $i$ ranges in $A$. Since if
$v\in\operatorname{trop}(A)$ then also
$v+\lambda\mathbf{1}\in\operatorname{trop}(A)$ for every
$\lambda\in\mathbb{R}$, where $\mathbf{1}=(1,\dots,1)\in\mathbb{R}^{n+1}$, we
regard $\operatorname{trop}(A)$ as a subset of the quotient space
$\mathbb{R}^{n+1}/\mathbf{1}\mathbb{R}\simeq\mathbb{R}^{n}$.

######  Proposition-Definition 4.1 (Tropical linear space).

Let $M$ be a matroid on $E=\\{0,1,\dots,n\\}$. The _tropical linear space_ of
$M$ is the set

$\operatorname{trop}(M)=\bigcap_{C\in\mathcal{C}(M)}\operatorname{trop}(C)=\bigcap_{V\in\mathcal{V}(M)}\operatorname{trop}(V)\subseteq\mathbb{R}^{n}.$

###### Proof.

We need to prove the second equality of the above equation. The inclusion
$\bigcap_{C\in\mathcal{C}(M)}\operatorname{trop}(C)\supseteq\bigcap_{V\in\mathcal{V}(M)}\operatorname{trop}(V)$
is obvious since every circuit is a cycle by definition. For the reverse
inclusion, let $v\in\bigcap_{C\in\mathcal{C}(M)}\operatorname{trop}(C)$ and
let $V$ be a cycle of $M$. It is enough to show that
$v\in\operatorname{trop}(V)$. Suppose that the minimum of the coordinates
$v_{i}$ of $v$, for $i$ ranging in $V$, is attained at $v_{j}$ for some $j\in
V$. Since $V$ is a union of circuits, there exists a circuit $C$ of $M$ that
contains $j$. Now, by the choice of $v$, we have $v\in\operatorname{trop}(C)$,
so the minimum of the numbers $v_{i}$ for $i$ ranging in $C$ is attained at
least twice. Since $C\subseteq V$ and $v_{j}$ was the minimum of the numbers
$v_{i}$ for $i\in V$, the minimum in $V$ is also attained at least twice,
hence $v\in\operatorname{trop}(V)$. ∎

We can give the set $\operatorname{trop}(M)$ various fan structures, and such
a fan is usually called _Bergman fan_ (see [23, Section 4.2] or [11] for more
information).

Let $M_{1}$ and $M_{2}$ be two matroids on the common ground set
$\\{0,1,\dots,n\\}$. From [27, Proposition 7.4.7] we have that
$M_{1}\twoheadleftarrow M_{2}$ if and only if every circuit of $M_{2}$ is a
union of circuits of $M_{1}$ (i.e. a cycle of $M_{1}$). It is easy to see that
this is equivalent to the condition that every cycle of $M_{2}$ is a cycle of
$M_{1}$. This observation, together with Proposition-Definition 4.1 imply the
following fact.

######  Corollary 4.2.

Let $M_{1}$ and $M_{2}$ be two matroids on a common ground set. We have

$M_{1}\twoheadleftarrow
M_{2}\Longleftrightarrow\operatorname{trop}(M_{1})\subseteq\operatorname{trop}(M_{2}).$

Let $I$ be a homogeneous ideal of $K[x_{0},\dots,x_{n}]$. We denote by
$\operatorname{trop}(V(I))$ the tropicalization of $V(I)\cap T^{n}$, where
$T^{n}$ is the algebraic torus of $\mathbb{P}^{n}$. If $K$ is infinite and $I$
is linear then the supports of the degree one polynomials in $I$ are the
cycles of a matroid $M$, called the underlying matroid of $I$ (see Lemma 4.6
below). In this situation we have the following (set-theoretic) equality:
$\operatorname{trop}(V(I))=\operatorname{trop}(M)$ (see [23, Chapter 4]).

In this paper, all tropicalizations are done with respect to the trivial
valuation. By a _tropical variety_ here we mean a pure-dimensional balanced
polyhedral complex obtained as the tropicalization of an algebraic variety,
that is, an object of the form $T=\operatorname{trop}(V(I))$ for some
homogeneous ideal $I$ of $K[x_{0},\dots,x_{n}]$. Let $T_{1}$ and $T_{2}$ be
two tropical varieties such that $T_{1}\subseteq T_{2}$. We say that such an
inclusion is _realizable_ over a field $K$ if there exists $I_{1}$ and $I_{2}$
two homogeneous ideals of $K[x_{0},\dots,x_{n}]$ such that
$T_{i}=\operatorname{trop}(V(I_{i}))$ for $i\in\\{1,2\\}$ and
$V(I_{1})\subseteq V(I_{2})$.

######  Proposition 4.3.

Let $T_{i}=\operatorname{trop}(V(I_{i}))$ for $i\in\\{1,2\\}$ be two tropical
varieties such that $T_{1}\subseteq T_{2}$, where $I_{1}$ and $I_{2}$ are
homogeneous linear ideals of $K[x_{0},\dots,x_{n}]$, and let $M_{1}$ and
$M_{2}$ be the underlying matroids of $I_{1}$ and $I_{2}$ respectively. The
inclusion $T_{1}\subseteq T_{2}$ is realizable over $K$ if and only if so is
the quotient of matroids $M_{1}\twoheadleftarrow M_{2}$.

###### Proof.

First assume that $T_{1}\subseteq T_{2}$ is realizable over $K$. This means
that there exist two homogeneous ideals $J_{1}$ and $J_{2}$ of
$K[x_{0},\dots,x_{n}]$ such that $T_{i}=\operatorname{trop}(V(J_{i}))$ for
$i\in\\{1,2\\}$ and $V(J_{1})\subseteq V(J_{2})$. Since $T_{1}$ and $T_{2}$
are linear tropical varieties, this implies that $J_{1}$ and $J_{2}$ are
linear ideals (see for instance [12, Theorem 4.7]). In particular, $J_{1}$ and
$J_{2}$ have the same underlying matroid of $I_{1}$ and $I_{2}$ respectively.
Now let $U_{i}$ be the vector space of polynomials of degree one in $J_{i}$,
for $i\in\\{1,2\\}$. Since $V(J_{1})\subseteq V(J_{2})$ and $J_{1}$ and
$J_{2}$ are linear, in particular radical, we have $J_{1}\supseteq J_{2}$, so
$U_{1}\supseteq U_{2}$, and this last inclusion is a realization of
$M_{1}\twoheadleftarrow M_{2}$ in the sense we discussed in Section 2.2.

Conversely, assume that $M_{1}\twoheadleftarrow M_{2}$ is realizable. Then
there exist two linear subspaces $U_{1}\supseteq U_{2}$ of $K^{n+1}$ that
realize this quotient. The vectors in $U_{i}$ can be thought as linear
polynomials in $K[x_{0},\dots,x_{n}]$, and we let $J_{i}$ be the linear
homogeneous ideal generated by these polynomials, for $i\in\\{1,2\\}$. By
construction $J_{1}\supseteq J_{2}$, so $V(J_{1})\subseteq V(J_{2})$, and
$T_{i}=\operatorname{trop}(V(J_{i}))$ as $J_{i}$ and $I_{i}$ have the same
underlying matroid, for $i\in\\{1,2\\}$. ∎

###  4.2 Tropical Fano schemes

In [22] Lamboglia studied _tropical Fano schemes_ and tropicalizations of Fano
schemes. The Fano scheme $F_{r}(X)$ of a projective variety $X$ is the fine
moduli space parametrizing linear spaces of dimension $r$ contained in $X$.
The tropicalization $\operatorname{trop}(F_{r}(X))$ of $F_{r}(X)$ inside
$\operatorname{trop}G(r+1,n+1)$ parametrizes the tropicalization of linear
spaces contained in $X$. On the other hand, the tropical Fano scheme
$F_{r}(\operatorname{trop}(X))$ of the tropical variety
$\operatorname{trop}(X)$ is a tropical prevariety parametrizing tropicalized
linear spaces of dimension $r$ contained in $\operatorname{trop}(X)$. From the
definition, it is immediate to observe that

$\operatorname{trop}(F_{r}(X))\subseteq F_{r}(\operatorname{trop}(X)).$

However, the above inclusion might be strict, already when $X$ is just a
linear space. In fact, [22, Theorem 3.1] describes a family of planes in
$\mathbb{P}^{n}$ with this property. Every plane $L$ of such family
tropicalizes to the standard tropical plane in
$\mathbb{R}^{n}\simeq\mathbb{R}^{n+1}/\mathbf{1}\mathbb{R}$. The
tropicalization thus contains the (realizable) tropical line $\Gamma$ whose
rays are $\operatorname{pos}(e_{i},e_{i+1})$ for $i$ ranging in all the even
numbers in $\\{0,1,\dots,n-1\\}$. However, no line contained in $L$
tropicalize to $\Gamma$.

Now the combination of Proposition 4.3 and Theorem 3.10 offers another
perspective on the above phenomenon. In fact, since $L$ tropicalize to the
standard tropical plane, its underlying matroid is $U_{3,n+1}$. Further, the
underlying matroid of $\Gamma$ is the matroid $M_{2}$ of rank $2$ on ground
set $E=\\{0,\dots,n\\}$ with no loops and parallel elements given by the
disjoint pairs of consecutive numbers: $\\{0,1\\}$, $\\{2,3\\}$ up to
$\\{n-1,n\\}$ or $\\{n-2,n-1\\}$ depending on the parity of $n$. Now since
$\Gamma\subseteq\operatorname{trop}(L)$, from Corollary 4.2 we have
$U_{3,n+1}\twoheadrightarrow M_{2}$. The Higgs major of this quotient is a
matroid $M_{3}=U_{3,n+1}+e$ of rank $3$ on the ground set
$\\{0,\dots,n\\}\cup\\{e\\}$ where the cardinality $3$ dependent subsets are
all of the form $\\{i,i+1,e\\}$ where $\\{i,i+1\\}$ are parallel elements in
$M_{2}$. Such a matroid is realizable over $\mathbb{C}$, and from Theorem 3.10
every realization of $M_{3}$ gives us a realization of the quotient
$U_{3,n+1}\twoheadrightarrow M_{2}$, which in turn, from Proposition 4.3,
provides us a pair of a line contained in a plane $\ell\subseteq L$ that
tropicalize to $\Gamma$ and the standard tropical plane respectively. The
crucial point here is that if we fix a realization of $U_{3,n+1}$ over
$\mathbb{C}$, it is not guaranteed that this realization can be extended to a
realization of $M_{3}$. This means that we cannot realize the quotient
$U_{3,n+1}\twoheadrightarrow M_{2}$ with this fixed realization of
$U_{3,n+1}$. We further illustrate this phenomenon with the examples discussed
in [22].

######  Example 4.4 ([22, Example 3.4]).

Let $L_{1}\subseteq\mathbb{P}^{5}$ be the plane spanned by the rows of the
following matrix

$\begin{bmatrix}1&3&0&1&5&7\\\ 0&0&1&3&-1&-1\\\
1&4&-1&-3&0&0\end{bmatrix}\in\mathbb{C}^{3,6}.$

The above matrix realizes $U_{3,6}$ over $\mathbb{C}$, therefore the
tropicalization of $L_{1}$ is the standard tropical plane. Now this
realization can be extended to a realization of $M_{3}$ given by the following
matrix, obtained by adding the column vector $(1,0,0)\in\mathbb{C}^{3}$:

$\begin{bmatrix}1&3&0&1&5&7&1\\\ 0&0&1&3&-1&-1&0\\\
1&4&-1&-3&0&0&0\end{bmatrix}\in\mathbb{C}^{3,7}.$

Now, by simply quotienting by the vector $(1,0,0)\in\mathbb{C}^{3}$
corresponding to the element $e$ of $M_{3}=U_{3,6}+e$, we obtain the following
realization of $M_{2}$:

$\begin{bmatrix}0&0&1&3&-1&-1\\\
1&4&-1&-3&0&0\end{bmatrix}\in\mathbb{C}^{2,6}.$

The rowspace of the above matrix is a $2$-dimensional vector space in
$\mathbb{C}^{6}$ corresponding to a line $\ell$ in $\mathbb{P}^{5}$ contained
in $L_{1}$, that tropicalizes to $\Gamma$.

######  Example 4.5 ([22, Example 3.3]).

Let $L_{2}\subseteq\mathbb{P}^{5}$ be the plane spanned by the rows of the
following matrix

$\begin{bmatrix}0&-271&-92&0&-13&-54\\\ 0&-18&-7&-1&0&-4\\\
-1&12293&4173&0&588&2450\end{bmatrix}\in\mathbb{C}^{3,6}.$

The above matrix realizes $U_{3,6}$ over $\mathbb{C}$, therefore the
tropicalization of $L_{1}$ is the standard tropical plane. Now this
realization cannot be extended to a realization of $M_{3}$. In fact, a
realization of $M_{3}$ would be given by a matrix of the following form

$A=\begin{bmatrix}0&-271&-92&0&-13&-54&x_{1}\\\ 0&-18&-7&-1&0&-4&x_{2}\\\
-1&12293&4173&0&588&2450&x_{3}\end{bmatrix}.$

In particular, the determinant of the square submatrices indexed by the
dependent subsets of cardinality $3$ should be zero. Therefore we have:

$\begin{array}[]{ccccc}\det(A_{0,1,e})=&-18x_{1}&+271x_{2}&&=0\\\
\det(A_{2,3,e})=&4173x_{1}&&+92x_{3}&=0\\\
\det(A_{4,5,e})=&2352x_{1}&+98x_{2}&+52x_{3}&=0\end{array}$

The above linear system has only one solution: $(x_{1},x_{2},x_{3})=(0,0,0)$,
but this is impossible since $M_{3}$ has no loops. Therefore, the realization
of $U_{3,6}$ given by $L_{2}$ cannot be extended to a realization of $M_{3}$.
Thus, by Theorem 3.10 no realization of the quotient
$U_{3,6}\twoheadrightarrow M_{2}$ can be constructed with such a realization
of $U_{3,6}$. From Proposition 4.3 and Theorem 3.10, this implies that there
is no line $\ell$ contained in $L_{2}$ tropicalizing to $\Gamma$.

###  4.3 Nonlinear case

Let $I$ be a homogeneous ideal of $K[x_{0},\dots,x_{n}]$. For every
$d\in\mathbb{N}$ denote by $I_{d}$ the the degree $d$ part of $I$, that is,
the set of polynomials of degree $d$ in $I$. Let $\operatorname{Mon}_{d}$ be
the set of monomials of degree $d$ in $K[x_{0},\dots,x_{d}]$. The _support_
$\operatorname{supp}(f)$ of a polynomial $f\in K[x_{0},\dots,x_{n}]$ is the
set of monomials with nonzero coefficients in $f$.

######  Lemma 4.6.

Let $K$ be an infinite field. For every $d\in\mathbb{N}$ the set of supports
of polynomials in $I_{d}$ is the set of cycles of a matroid $M(I_{d})$.

###### Proof.

We need to show that the set $\mathcal{V}$ of supports of $I_{d}$ satisfies
the axioms 4.1. The zero vector is in $I_{d}$ by convention, so
$\emptyset\in\mathcal{V}$ and $\mathcal{V}$ satisfies axiom $1$.

For axiom $2$, let $V_{1},V_{2}\in\mathcal{V}$, and let $f_{1},f_{2}\in I_{d}$
such that $\operatorname{supp}(f_{i})=V_{i}$ for $i\in\\{1,2\\}$. Let $a_{m}$
(resp. $b_{m}$) be the coefficient in $f_{1}$ (resp. $f_{2}$) of the monomial
$m\in\operatorname{Mon}_{d}$. Since $K$ is infinite, there exists $c\in K$
that is not equal to $-a_{m}/b_{m}$ for every $m\in\operatorname{Mon}_{d}$
such that $b_{m}\neq 0$. By construction, there are no cancellations in the
polynomial $f=f_{1}+cf_{2}\in I_{d}$, therefore
$V=\operatorname{supp}(f)=V_{1}\cup V_{2}\in\mathcal{V}$.

For axiom $3$, let $V_{1},V_{2}\in\mathcal{S}$ and $e\in V_{1}\cap V_{2}$,
then there exists $f_{1},f_{2}\in I_{d}$ such that
$V_{i}=\operatorname{supp}(f_{i})$ for $i\in\\{1,2\\}$. Now let $c_{i}\in K$
be the coefficient of the monomial $e$ in $f_{i}$ for $i\in\\{1,2\\}$. Then
the polynomial $f=f_{1}-(c_{1}/c_{2})f_{2}\in I_{d}$ has support
$V=\operatorname{supp}(f)\in\mathcal{V}$ such that $V\subseteq(V_{1}\cup
V_{2})\setminus\\{e\\}$ by construction, further $(V_{1}\cup
V_{2})\setminus(V_{1}\cap V_{2})\subseteq V$ as no cancellation can happen
outside $V_{1}\cap V_{2}$. ∎

Let $\nu_{d}:\mathbb{P}^{n}\rightarrow\mathbb{P}^{N}$ be the degree $d$
Veronese embedding, where $N=\binom{n+d}{d}-1$. Let
$U_{n,d}=\\{u\in\mathbb{N}^{n+1}:\sum_{i}u_{i}=d\\}$ and let $K[z_{u}:u\in
U_{n,d}]$ be the coordinate ring of $\mathbb{P}^{N}$. Now, let $I$ be a
homogeneous ideal of $K[x_{0},\dots,x_{n}]$ and choose $d\in\mathbb{N}$ such
that $V(I)=V(I_{d})$. We have

$\nu_{d}(V(I))=\nu_{d}(\mathbb{P}^{n})\cap L_{I,d}$ (4)

where $L_{I,d}$ is some linear subspace of $\mathbb{P}^{N}$ which linear
equations are obtained by substituting $x^{u}$ with $z_{u}$ for every $u\in
U_{n,d}$ in the polynomials $f\in I_{d}$ (see [16, Exercise 2.9]). Further,
note that, if $K$ is infinite, the underlying matroid of the linear space
$L_{I,d}$ is $M(I_{d})$, hence
$\operatorname{trop}(L_{I,d})=\operatorname{trop}(M(I_{d}))$.

Let $A\in\mathbb{R}^{N,n+1}$ be the matrix whose rows consists of the vectors
$u\in U_{n,d}$. Let $\ell_{A}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{N}$ be
the linear map given by the multiplication by $A$. Since
$\ell_{A}(\mathbf{1}_{n+1})=d\cdot\mathbf{1}_{N}$, the map $\ell_{A}$ induces
a linear map
$\operatorname{trop}(\nu_{d}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$
obtained by quotienting by $\mathbf{1}_{n+1}$ and identifying $\mathbb{R}^{n}$
with the quotient space $\mathbb{R}^{n+1}/\mathbf{1}\mathbb{R}$. Since the
Veronese embedding $\nu_{d}$ is a monomial map, from [23, Corollary 3.2.13] we
have the following result.

######  Lemma 4.7.

Let $I$ be an homogeneous ideal of $K[x_{0},\dots,x_{n}]$. For every
$d\in\mathbb{N}$ we have

$\operatorname{trop}(\nu_{d}(V(I)))=\operatorname{trop}(\nu_{d})(\operatorname{trop}(V(I)).$

######  Lemma 4.8.

Let $I$ be a homogeneous ideal of $K[x_{0},\dots,x_{n}]$ and let
$d\in\mathbb{N}$ such that $\operatorname{trop}(V(I))=\bigcap_{f\in
I_{d}}\operatorname{trop}(V(f))$ and $V(I)=V(I_{d})$. We have

$\operatorname{trop}(\nu_{d}(V(I))=\operatorname{trop}(\nu_{d}(\mathbb{P}^{n}))\cap\operatorname{trop}(L_{I,d}).$

###### Proof.

From (4) we have
$\operatorname{trop}(\nu_{d}(V(I))\subseteq\operatorname{trop}(\nu_{d}(\mathbb{P}^{n}))\cap\operatorname{trop}(L_{I,d})$.
For the reverse inclusion, let
$v\in\operatorname{trop}(\nu_{d}(\mathbb{P}^{n}))\cap\operatorname{trop}(L_{I,d})$.
From Lemma 4.7 we have that
$v\in\operatorname{trop}(\nu_{d}(\mathbb{P}^{n}))=\operatorname{trop}(\nu_{d})(\mathbb{R}^{n})$,
that is $v$ is in the image of $\operatorname{trop}(\nu_{d})$, so there exists
$w\in\mathbb{R}^{n+1}/\mathbf{1}\mathbb{R}\simeq\mathbb{R}^{n}$ such that
$v=\operatorname{trop}(\nu_{d})(w)$. Now
$v=\operatorname{trop}(\nu_{d})(w)\in\operatorname{trop}(L_{I,d})$ implies
that the minimum in $\operatorname{trop}(f)(w)$ is achieved at least twice for
every $f\in I_{d}$, that is $w\in\bigcap_{f\in
I_{d}}\operatorname{trop}(V(f))=\operatorname{trop}(V(I))$. Hence
$v=\operatorname{trop}(\nu_{d})(w)\in\operatorname{trop}(\nu_{d})(\operatorname{trop}(V(I)))=\operatorname{trop}(\nu_{d}(V(I)))$,
where in the last equality we applied Lemma 4.7. ∎

######  Proposition 4.9.

Let $K$ be an infinite field, and let $I$ and $J$ be two homogeneous ideals of
$K[x_{0},\dots,x_{n}]$. There exists $d_{0}\in\mathbb{N}$ such that for every
$d\geq d_{0}$ we have

$M(I_{d})\twoheadleftarrow
M(J_{d})\Longrightarrow\operatorname{trop}(V(I))\subseteq\operatorname{trop}(V(J)).$

###### Proof.

By applying [1, Theorem 3.7] we can choose $d_{0}\in\mathbb{N}$ sufficiently
large such that for every $d\geq d_{0}$ the hypothesis of Lemma 4.8 are
satisfied for both $I$ and $J$. Since the underlying matroid of $L_{I,d}$
(resp. $L_{J,d}$) is $M(I_{d})$ (resp. $M(J_{d})$), from Proposition-
Definition 4.1 we have
$\operatorname{trop}(L_{I,d})\subseteq\operatorname{trop}(L_{J,d})$. By
intersecting with $\operatorname{trop}(\nu_{d}(\mathbb{P}^{n}))$ and applying
Lemma 4.7 and Lemma 4.8 we obtain

$\operatorname{trop}(\nu_{d})(\operatorname{trop}(V(I)))\subseteq\operatorname{trop}(\nu_{d})(\operatorname{trop}(V(J)))$

which implies $\operatorname{trop}(V(I))\subseteq\operatorname{trop}(V(J))$
since $\operatorname{trop}(\nu_{d})$ is injective. ∎

The reverse implication of the above proposition is not true in general, as
the following example shows.

######  Example 4.10.

Let $I=(x_{0}+x_{1}+x_{2})$ and $J=(x_{0}^{p}+x_{1}^{p}+x_{2}^{p})$ be two
homogeneous ideals in $K[x_{0},x_{1},x_{2}]$, for some prime $p$. The
tropicalization of the varieties of these two ideals is (set-theoretically)
the same: the standard tropical line in $\mathbb{R}^{2}$. In this situation,
the number $d_{0}$ of Proposition 4.9 can be chosen equal to $p$, and we have
that, if $K$ has characteristic not equal to $p$, $M(I_{p})$ is not a quotient
of $M(J_{p})$.

######  Remark 4.11.

Another situation in which the reverse implication of Proposition 4.9 does not
hold concerns _symmetric powers_ of matroids. In fact, as explained in [2,
Remark 2.18], if a matrix $A$, with coefficients in some field $K$, realizes a
matroid $M$, the $d$-th Macaulay matrix of $A$ realizes a $d$-th symmetric
power of $M$. However, such a $d$-th symmetric power depends both on the
characteristic of $K$ and the matrix $A$ we started with (not just on the
matroid $M$). Thus, from two distinct realizations of $M$, we obtain two
distinct realizable symmetric powers of $M$. These two matroids will then be
the degree $d$ part of two ideals tropicalizing to the same tropical variety,
despite the fact that the two underlying matroids in degree $d$ are distinct
(so one is not the quotient of the other).

######  Remark 4.12.

While Proposition 4.3 essentially reduces the relative realizability problem
for tropical varieties of degree one to a problem of realizability of matroid
quotients, the same cannot be done directly for tropical varieties of higher
degrees. Example 4.10 is already an example of this. One could still try to
recover some non-realizability certificate from the non realizability of a
finite list of matroid quotients, although an algorithmic implementation of
such a procedure might not be feasible in practice.

##  5 Questions and future work

Theorem 3.10 and Corollary 3.11 characterize realizability of a (single)
quotient of matroids and a flag of matroids in terms of realizability of a
single matroid. It is natural to ask if something similar can be done more
generally for a diagram of matroids.

######  Question 5.1.

Can we characterize the realizability of a diagram of matroids in terms of the
realizability of a single matroid?

Here we restricted ourselves to (classical) matroids. In light of the
connection with relative realizability problems in tropical geometry, it is
natural to ask if a similar theory could be developed for valuated matroids.
Quotients of valuated matroids were studied in [7] and [5]. In particular, one
could try to define a _valuated_ Higgs lift, from which the notions of Higgs
factorization and major could be constructed.

######  Question 5.2.

Is there an analogue notion of Higgs lift for quotients of valuated matroids?

Let $\operatorname{Mon}_{d}$ be the set of monomials of degree $d$ in
$K[x_{0},\dots,x_{n}]$. Let $M_{1}$ and $M_{2}$ be two matroids on the ground
set $\operatorname{Mon}_{d}$ for some $d>0$. Denote by

$T_{i}=\operatorname{trop}(\nu_{d})^{-1}\big{(}\operatorname{trop}(M_{i})\cap\operatorname{trop}(\nu_{d}(\mathbb{P}^{n})\big{)}$

for $i\in\\{1,2\\}$. In light of Remark 4.11 it is natural to ask the
following question.

######  Question 5.3.

Under which conditions on $M_{1}$ and $M_{2}$ we have $T_{1}=T_{2}$?

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Alessio Borzì, MPI MiS Leipzig, Inselstraße 22, 04103 Leipzig, Germany
Email<EMAIL_ADDRESS>