# On the birational geometry of matroids

###### Abstract.

This paper investigates isomorphisms of Bergman fans of matroids respecting
different fan structures, which we regard as matroid analogs of birational
maps. We show that isomorphisms respecting the fine fan structure are induced
by matroid isomorphisms.

We introduce Cremona automorphisms of the coarse structure of Bergman fans,
which are not induced by matroid automorphisms. We show that the automorphism
group of the coarse fan structure is generated by matroid automorphisms and
Cremona maps in the case of rank $3$ matroids which are not parallel
connections and for modularly complemented matroids.

Kris Shaw and Annette Werner

MSC(2020): 14T20, 52B40, 14E07

##  1\. Introduction

This paper investigates the following question: What are the isomorphisms of
Bergman fans of matroids with some choice of fan structures? Some, but not
all, isomorphisms of Bergman fans of matroids arise from isomorphisms of
matroids. For matroids which are realizable as an essential hyperplane
arrangement in projective space, birational morphisms which are regular on the
arrangement complement give rise to invertible linear maps between supports of
the corresponding Bergman fans. Therefore we regard isomorphisms of Bergman
fans as examples of matroid analogs of birational maps.

Recently, there have been exciting breakthroughs in matroid theory motivated
by algebraic geometry. In [2], [6], [4] a framework of Hodge theory for
matroids was developed with applications to deep open combinatorial problems.
For an overview see [5]. One key ingredient in these results is the study of
Bergman fans and the Chow rings of matroids. These tools also play a decisive
role in the present paper.

To be precise, let $M$ and $M^{\prime}$ be matroids on ground sets $E$ and
$E^{\prime}$ respectively. For $k\in E$, let $v_{k}\in{\mathbb{Z}}^{E}$ be the
canonical basis vector associated to $k$. The Bergman fan of $M$ is denoted
$B(M)$ and is viewed as a subset of
${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$, where $\mathbbm{1}=\sum_{k\in
E}v_{k}$. We use the notation $B_{\ast}(M)$ to denote a chosen fan structure,
see Section 2. We mostly investigate the coarse and fine fan structures,
denoted $B_{c}(M)$ and $B_{f}(M)$ respectively, as defined in Section 2.

An isomorphism of Bergman fans $B_{\ast}(M)$ and $B_{\ast}(M^{\prime})$, where
$\ast$ denotes choices of fan structures, is defined as a linear map
$\phi:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}\to{\mathbb{R}}^{E^{\prime}}/{\mathbb{R}}\mathbbm{1}$
derived from a lattice isomorphism
${\mathbb{Z}}^{E}\to{\mathbb{Z}}^{E^{\prime}}$, such that $\phi$ restricts to
an isomorphism of fans $B_{\ast}(M)\to B_{\ast}(M^{\prime})$, see Definition
2.5.

Note that for simple matroids $M_{1}$, $M_{2}$ of rank $2$, every isomorphism
between their Bergman fans (where all fan structures coincide) is obviously
induced by a matroid isomorphism. Our first main theorem establishes the
analogous claim for simple matroids of all ranks, if we consider the fine
structure:

######  Theorem (6.3).

Let $M_{1}$ and $M_{2}$ be simple matroids such that neither is totally
disconnected. If $\phi:B_{f}(M_{1})\to B_{f}(M_{2})$ is an isomorphism of
fans, then $\phi$ is induced by a matroid isomorphism.

If $M$ is not a simple matroid on the ground set $E$, let $\hat{M}$ denote its
simplification on the ground set $\hat{E}$. Note that the affine Bergman fan
of $M$ lives in a vector subspace of ${\mathbb{R}}^{E}$ which is determined by
the parallel elements of $M$ and which is isomorphic to
${\mathbb{R}}^{\hat{E}}$. Moreover, the coordinate projection
${\mathbb{R}}^{E}\to{\mathbb{R}}^{\hat{E}}$ restricts to an isomorphism on
this subspace and gives an isomorphism of the Bergman fan of $M$ with the
Bergman fan of its simplification. Therefore, in order for the above theorem
to hold, we need to assume that $M_{1}$ and $M_{2}$ are simple. Moreover, the
connectedness assumption in Theorem 6.3 is essential as the following example
shows.

######  Example 1.1.

If $M$ is totally disconnected and simple, then $M=U_{n+1,n+1}$, so that the
lattice of flats is the Boolean lattice on $n+1$ elements. Let
$\phi:{\mathbb{R}}^{E}\to{\mathbb{R}}^{E}$ be the linear map sending
$v\mapsto-v$. This map descends to a fan automorphism $\phi:B_{f}(M)\to
B_{f}(M)$ which does not arise from a matroid automorphism.

The proof of the Theorem 6.3 is based on the observation that $\phi$ induces
an isomorphism of Chow rings. In Section 4, we review the Chow ring of fans
and the degree map in the case of matroid fans, see [2] and more generally
[4]. We show that an isomorphism of Bergman fans of matroids with arbitrary
fan structures induces an isomorphism of the respective Chow rings which is
compatible with the degree map, see Proposition 4.4. This isomorphism and
Eur’s degree formula [16] are the main tools used to prove Theorem 6.3.

In Section 5, we show that the existence of isomorphisms of Bergman fans
implies that the underlying matroids have the same characteristic polynomials.
It is an open question, which non-isomorphic matroids have the same
characteristic polynomials. As an example, the operation of parallel
connection discussed in Section 2 produces examples of non-isomorphic matroids
with the same characteristic polynomials [15]. In Section 5, we also conclude
that the Chern-Schwartz-MacPherson classes of matroids from [25] are preserved
under matroid fan isomorphisms, see Proposition 5.5.

We then consider our main question for coarse fan structures. In rank $3$ we
show that there are no isomorphisms $\phi:B_{c}(M_{1})\to B_{c}(M_{2})$ for
non-isomorphic matroids $M_{1}$ and $M_{2}$ , see Theorem 9.1. However, the
situation is different from rank $4$ on. In Section 7, we show that there are
non-isomorphic matroids arising from parallel connections with isomorphic
coarse Bergman fans, see Corollary 7.3. The hyperplane arrangement analogue of
the operation of parallel connection produces non-isomorphic arrangements with
diffeomorphic complements [15].

We then switch perspective to automorphisms of a single Bergman fan and ask:
What can be said of the automorphism group of a Bergman fan of a matroid with
a fixed fan structure? Inspired by birational geometry, we introduce and study
Cremona maps for matroids. These are automorphisms of the coarse Bergman fan
not induced by matroid automorphisms. The map $\phi$ from Example 1.1 is the
first simple example of a Cremona map, and is the tropicalization of the
standard Cremona map on projective space. For connected matroids, we provide
an intrinsic definition of Cremona maps. As an example, we describe in Section
3 the automorphism group of the coarse Bergman fan of the braid arrangement
matroid. Note that the Bergman fan of the braid arrangement in dimension $n$
is the moduli space of tropical genus zero curves with $(n+2)$-marked points.
It was shown in [1], that the automorphism group of this tropical moduli space
is isomorphic to the symmetric group $S_{n+2}$. We show in Proposition 3.2,
that this implies that the automorphism group of the coarse Bergman fan of the
braid arrangement matroid is generated by the matroid automorphisms and a
single Cremona map. Theorem 6.3 implies that in this example the automorphism
group of the coarse fan structure is strictly larger than the automorphism
group of the fine fan structure.

In general, we define Cremona maps as follows. If $M$ is a simple matroid on
the ground set $E$ with basis $b=\\{b_{0},\dots,b_{d}\\}$, we define the flat
$B_{j}$ as the closure of $b\backslash\\{b_{j}\\}$ and put
$v_{B_{j}}=\sum_{k\in B_{j}}v_{k}\in{\mathbb{R}}^{E}$. Then we define the
${\mathbb{Z}}$-linear map
$\operatorname{\text{Crem}}_{b}\colon{\mathbb{R}}^{E}\to{\mathbb{R}}^{E}$ by
$v_{b_{j}}\mapsto v_{B_{j}}$ for the basis elements $b_{j}$, and $v_{k}\mapsto
v_{k}$ for all $k\in E\backslash b$. If $\operatorname{\text{Crem}}_{b}$ maps
the line ${\mathbb{R}}\mathbbm{1}$ to itself, we also write
$\operatorname{\text{Crem}}_{b}:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}{\rightarrow}{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$
for the quotient map. We then prove the following necessary and sufficient
criterion for the map $\operatorname{\text{Crem}}_{b}$ to be an automorphism
of the coarse fan structure of the Bergman fan.

######  Theorem (8.3).

Let $b$ be a basis of a simple connected matroid $M$. For any pair of elements
$i,j\in E$, let $F_{ij}$ be the rank $2$ flat which is the closure of $i$ and
$j$. The map $\operatorname{\text{Crem}}_{b}$ descends to a linear map
$\operatorname{\text{Crem}}_{b}:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}\rightarrow{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$
mapping $B(M)$ to itself, if only if the sets
$\\{F_{ij}\backslash\\{i,j\\}\\}_{i,j\in b}$ partition the set $E\backslash b$
into pairwise disjoint subsets.

We then focus on the case of rank three matroids in Section 9. In particular,
Theorem 9.1 states that if there is an isomorphism of coarse Bergman fans of
rank three matroids, then the underlying matroids are isomorphic. From this we
can also deduce the following corollary.

######  Corollary (9.5).

The automorphism group of $B_{c}(M)$ for a rank $3$ matroid $M$ which is not a
parallel connection is generated by matroid automorphisms and Cremona maps.

We conclude the paper by studying modularly complemented matroids. A simple
modularly complemented matroid of rank at least $4$ is either a sufficiently
big submatroid of projective space or a Dowling matroid associated to a group
$G$. For the modularly complemented submatroids of projective space we prove
in Section 10 that the automorphism group of $B_{c}(M)$ is equal to the group
of matroid automorphisms. For Dowling matroids however, we find a Cremona
transformation in the automorphism group of the coarse Bergman fan. This leads
to the following theorem, combining Proposition 10.2 and Proposition 10.3.

######  Theorem 1.2.

For every simple modularly complemented matroid $M$, the automorphism group of
$B_{c}(M)$ is generated by Cremona maps and matroid automorphisms.

Based on our results, one might ask: under which conditions is the
automorphism group of the coarse Bergman fan of a simple matroid $M$ generated
by matroid automorphisms and Cremona maps?

Acknowledgements:  The research of the first author was supported by the Trond
Mohn Foundation project “Algebraic and topological cycles in tropical and
complex geometry”. The second author was partially supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and
Arithmetic of Uniformized Structures, project number 444845124.

##  2\. Matroids and Bergman fans

A matroid is a pair $M=(E,r)$ where $E$ is a finite set and
$r:\mathcal{P}(E)\to{\mathbb{Z}}_{\geq 0}$ is a rank function satisfying the
axioms

  1. (1)

$r(A)\geq 0$ for all $A\subseteq E$ and $r(\emptyset)=0$,

  2. (2)

For $A,B\subseteq E$ we have $r(A\cup B)+r(A\cap B)\leq r(A)+r(B)$,

  3. (3)

For any $i\in E$ and $A\subseteq E$, we have $r(A)\leq r(A\cup\\{i\\})\leq
r(A)+1$.

The rank of a matroid $M$ is $r(E)$ and is denoted $r(M)$. A _flat_ of a
matroid $M$ is a subset $F\subseteq E$ which is closed under the rank
function, which means that for all $i\notin F$ the rank of $F\cup\\{i\\}$ is
strictly greater than the rank of $F$. The flats are ordered by inclusion and
form a lattice, in the sense of partially ordered sets. For every subset $S$
of $E$ we write $\operatorname{\text{cl}}(S)$ for the inclusion minimal flat
containing $S$.

A _circuit_ of a matroid is a minimally dependent subset in the sense of
inclusion. A _loop_ of a matroid $M=(E,r)$ is a element $i\in E$ such
$r(\\{i\\})=0$. A pair of elements $i,j\in E$ are _parallel_ if neither $i$
nor $j$ is a loop and $r(\\{i,j\\})=1$. A matroid is _simple_ if it contains
no loops or parallel elements. A _connected component_ of a matroid is an
equivalence class under the relation defined by $i\sim j$ if $i$ and $j$ are
contained in a circuit. Each connected component forms a matroid and a matroid
decomposes into a direct sum of its connected components. A matroid $M$ is
_totally disconnected_ if all of its connected components have rank less than
or equal to one. The _parallel connection_ $P_{p}(M_{1},M_{2})$ of two
matroids $M_{1}$ and $M_{2}$ such that $E(M_{1})\cap E(M_{2})=\\{p\\}$ is a
matroid on the set $E(M_{1})\cup E(M_{2})$, see [29], Section 7.1. We say that
$M$ is a non-trivial parallel connection along $p\in E(M)$ if there exist
matroids $M_{1}$ and $M_{2}$, which are both of rank at least two, so that $M$
is isomorphic to $P_{p}(M_{1},M_{2})$. We say that $M$ is a _non-trivial
parallel connection_ , if there exists some $p\in E(M)$ such that $M$ is a
non-trivial parallel connection along $p$.

######  Definition 2.1.

Let $M_{1}=(E_{1},r_{1})$ and $M_{2}=(E_{2},r_{2})$ be two matroids. A matroid
isomorphism $f\colon M_{1}\to M_{2}$ is a bijection $f\colon E_{1}\to E_{2}$
such that for every subset $I\subseteq E_{1}$ we have $r_{2}(f(I))=r_{1}(I)$.

Since we are interested in Bergman fans, we will generally assume that our
matroids are loopfree. Suppose $M=(E,r)$ is a matroid containing no loops. For
every subset $F\subseteq E$, define a vector $v_{F}:=\sum_{i\in
F}v_{i}\in{\mathbb{R}}^{E}$. We also write $v_{E}=\mathbbm{1}$.

The (affine) _Bergman fan_ $\tilde{B}(M)$ associated to a loopfree matroid $M$
is defined as the subset of ${\mathbb{R}}^{E}$ consisting of all vectors
$\sum_{i}a_{i}v_{i}$ such that for each circuit $C$ of $M$ the minimum of the
set $\\{a_{i}:i\in C\\}$ is attained twice, see e.g. [17]. It is invariant
under scaling by the vector $\mathbbm{1}$. We will often work with the
_projective Bergman fan_ $B(M)$, which is defined as the image of the affine
Bergman fan under the quotient map ${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$.

######  Remark 2.2.

Given a polyhedral fan $\Sigma$ in a vector space ${\mathbb{R}}^{n}$ we may
ask if $\Sigma$ is equal to the Bergman fan of a matroid with some fan
structure up to a transformation in $\text{GL}_{n}({\mathbb{Z}})$. Studying
the collection of such transformations in $\text{GL}_{n}({\mathbb{Z}})$
provides an alternative point of view to our main question. If
$A,B\in\text{GL}_{n}({\mathbb{Z}})$ are such that $A(\Sigma)$ and $B(\Sigma)$
are both Bergman fans of matroids with some fan structures, then there is an
isomorphism of Bergman fans $\phi:A(\Sigma)\to B(\Sigma).$ The underlying
matroids of these two fans need not be the same.

Tropical manifolds are locally modelled on matroid fans up to coordinate
changes in $\text{GL}_{n}({\mathbb{Z}})$ [27]. Therefore, answering our main
question also has applications to tropical geometry.

The set $B(M)$ carries several interesting fan structures which have been
investigated by Feichtner and Sturmfels [17]:

i) We denote by $B_{c}(M)$ the natural fan structure as a subfan of the normal
fan of the matroid polytope [17], Proposition 2.5. We refer to it as the
coarse structure, since it is the coarsest fan structure on the set $B(M)$,
see [26], Section 4.2 or [21], Proposition 3.4.1.

ii) The minimal nested set structure, which we denote by $B_{m}(M)$. The rays
of the minimal nested set fan structure on $B(M)$ correspond to flats of $M$
which are _connected_. A flat is connected if $M|F$ is connected. The cones of
$B_{m}(M)$ correspond to _nested_ collections of connected flats. A collection
$\mathcal{S}$ of connected flats is called nested if for any subcollection of
incomparable flats $F_{1},\dots,F_{t}\in\mathcal{S}$ the join is not
connected.

iii) The fine structure which we denote by $B_{f}(M)$. This subdivision was
first described in [7]. Here the building set in the sense of [17] consists of
all flats, i.e. it is maximal. We can describe the cones in the fine structure
as follows: They are induced by flag of flats

$\mathcal{F}:=\emptyset\subsetneq F_{1}\subsetneq
F_{2}\subsetneq\dots\subsetneq\dots F_{k}\subsetneq E$

as the rational cones

$\rho_{\mathcal{F}}:=\langle
v_{F_{1}},v_{F_{2}},\dots,v_{F_{k}}\rangle_{\mathbb{R}_{\geq 0}}.$

iv) We write $B_{\ast}(M)$ for an arbitrary unimodular fan structure. In this
case, the rays of $B_{\ast}(M)$ need not correspond to flats of $M$. These fan
structures are studied in [4]. Another example on such a fan structure is the
conormal variety of a matroid from [6].

Note that by [18, Proposition 2], the Bergman fan of a matroid from any
building set is a unimodular fan. This implies that $B_{m}(M)$ and $B_{f}(M)$
are both unimodular fans. The following result by Feichtner and Sturmfels will
be useful to study isomorphisms of Bergman fans with the coarse structure.

######  Lemma 2.3 ([17], Theorem 5.3).

The minimal nested set fan structure $B_{m}(M)$ coincides with the coarse fan
structure $B_{c}(M)$ on the Bergman fan, if the matroid $M|G/F$ is connected
for all flats $F,G$ such that $G$ is connected and contains $F$.

For a polyhedral fan $\Sigma$ in ${\mathbb{R}}^{n}$ and a point $w\in|\Sigma|$
we define the star of $\Sigma$ at $w$ to be the subset of

$\operatorname{\text{star}}_{w}(\Sigma)=\\{x\in{\mathbb{R}}^{n}\ |\
\exists\epsilon>0\ s.t.\ \forall 0<\delta<\epsilon,\ w+\delta
x\in|\Sigma|\\}.$

If the fan is an affine Bergman fan of a matroid $\tilde{B}(M)$, and
$v_{\mathcal{F}}$ is the indicator of a flag of flats
$\mathcal{F}=\\{\emptyset=F_{0}\subsetneq F_{1}\subsetneq\dots\subsetneq
F_{k}\subsetneq F_{k+1}=E\\}$ of $M$ then by [19, Lemma 2.2] and [7,
Proposition 2.2] the star is the support of another affine Bergman fan, namely

$\operatorname{\text{star}}_{v_{\mathcal{F}}}(\tilde{B}(M))=\tilde{B}(M|F_{1}/F_{0})\times\dots\times\tilde{B}(M|F_{k+1}/F_{k}).$
(2.1)

It follows from this that the star of the projective Bergman fan
$\operatorname{\text{star}}_{v_{\mathcal{F}}}(B(M))$ is then also a matroid
fan $B\left(\bigoplus_{i=1}^{k+1}M|F_{i}/F_{i-1}\right).$

######  Lemma 2.4.

Let $M$ be a simple matroid, and let $i$ be an element of $E=E(M)$. The ray
$\langle v_{i}\rangle_{\mathbb{R}_{\geq 0}}$ is not a cone of some fan
structure $B_{\ast}(M)$ if and only if $M$ is a non-trivial parallel
connection along $i$.

###### Proof.

It suffices to consider the coarse fan structure. Suppose that $\langle
v_{i}\rangle_{\mathbb{R}_{\geq 0}}$ is not a cone in $B_{c}(M)$. Then the star
fan of $B_{c}(M)$ in direction $v_{i}$ has lineality space of dimension at
least two. By Formula (2.1), the star of $B(M)$ at $v_{i}$ is the matroid fan
$B(M/i\oplus M|i)$. By [19, Lemma 2.3], the dimension of the lineality space
is one less than the number of connected components of the matroid $M/i\oplus
M|i$. The matroid $M|i$ is a single coloop and hence connected. Therefore, the
ray $\langle v_{i}\rangle_{\mathbb{R}_{\geq 0}}$ is not in $B_{c}(M)$ if and
only if $M/i$ is disconnected. By [29, Proposition 7.1.15 iii) and 7.1.16
ii)], the matroid $M/i$ is disconnected if and only if $M$ is a non-trivial
parallel connection along $i$. This completes the proof. ∎

Note that for simple $M$, every $i\in E$ gives rise to a ray in $B_{m}(M)$ and
$B_{f}(M)$. If $B_{\ast}(M)$ comes from a building set, then all rays of the
fan come from flats of $M$. In this case, we define the rank of a ray in
$B_{\ast}(M)$ as the rank of the associated flat. We use the same terminology
for rays in the coarse structure $B_{c}(M)$.

######  Definition 2.5.

Let $N_{1}$ and $N_{2}$ be lattices and $\Sigma_{1}$ and $\Sigma_{2}$ be
rational polyhedral fans in $N_{1}\otimes{\mathbb{R}}$ and
$N_{2}\otimes{\mathbb{R}}$, respectively. A morphism of rational polyhedral
fans $\phi:\Sigma_{1}\to\Sigma_{2}$ is the restriction of a linear map induced
by a lattice morphism $N_{1}\to N_{2}$, which sends cones of $\Sigma_{1}$ to
cones of $\Sigma_{2}$. A morphism of fans is an isomorphism if it is a
bijective morphism whose inverse map is also a fan morphism.

We write $\mathrm{Aut}(\Sigma)$ for the group of automorphisms of $\Sigma$ in
this sense.

Let $M_{1},M_{2}$ be loopfree matroids on ground sets $E_{1}$ and $E_{2}$,
respectively. Note that any isomorphism of matroids $f:M_{1}\to M_{2}$ gives
rise to a ${\mathbb{Z}}$-linear map
$\phi_{f}:{\mathbb{R}}^{E_{1}}\to{\mathbb{R}}^{E_{2}}$ satisfying
$\phi_{f}(v_{i})=v_{f(i)}$. The isomorphism descends to the quotients, and the
map $\phi_{f}$ induces an isomorphism of the fine Bergman fans
$\phi_{f}:B_{f}(M_{1})\to B_{f}(M_{2})$ and also of the coarse Bergman fans
$\phi_{f}:B_{c}(M_{1})\to B_{c}(M_{2})$. In particular, there is a map

$\text{Iso}(M_{1},M_{2})\to\text{Iso}(B_{\ast}(M_{1}),B_{\ast}(M_{2}))$

for $\ast$ denoting either the coarse or fine fan structure. If both matroids
are simple, the map is injective.

More generally, if there is a matroid isomorphism $f:M_{1}\to M_{2}$, then
given any fan structure $B_{\ast}(M_{1})$ on $B(M_{1})$ there is a compatible
fan structure $B_{\ast}(M_{2})$ on $B(M_{2})$ such that the induced map
$\phi_{f}:B_{\ast}(M_{1})\to B_{\ast}(M_{2})$ is a fan isomorphism.

##  3\. Realizable matroids

If the matroid $M$ arises from an arrangement of hyperplanes $\mathcal{A}$
with trivial intersection in projective space over some field, its Bergman fan
is given by the tropicalization of the associated hyperplane complement
$\Omega_{\mathcal{A}}$ (over a trivially valued ground field), which is a very
affine variety sitting in the intrinsic torus. In this case, every fan
structure on the Bergman fan gives rise to a compactification of
$\Omega_{\mathcal{A}}$ by taking the closure in the toric variety associated
to the fan, see [34], Proposition 2.3. In this way, the minimal nested set fan
induces the (minimal) wonderful compactification introduced by de Concini and
Procesi [14]. It is shown in [24] that for essential and connected hyperplane
arrangements every dominant morphism
$f:\Omega_{\mathcal{A}}\rightarrow\Omega_{\mathcal{A}}$ extends to an
endomorphism of the visible contour compactification, which is induced by the
coarse structure on the Bergman fan. Hence it extends to an endomorphism of
the (minimal) wonderful compactification whenever the criterion in Lemma 2.3
by Feichtner and Sturmfels is satisfied. In fact, the proof of [24], Theorem
5.1 shows that every dominant endomorphism $f$ of $\Omega_{\mathcal{A}}$
extends to a endomorphism of the intrinsic torus which is a homomorphism up to
translation. Hence by passing to the tropicalization we get a linear map
preserving the Bergman fan, i.e. an endomorphism of $B_{c}(M)$ in the sense of
Definition 2.5. Therefore we regard linear maps preserving the different fan
structures on the Bergman fan as examples of analogs of birational maps
between matroids.

We will now look at an interesting example of a realizable matroid, namely the
matroid given by the $A_{n}$ type root system:

######  Example 3.1.

The braid arrangement. The root hyperplanes of the root system of type $A_{n}$
for $n\geq 2$ induce (after passing to the quotient by their intersection) the
essential arrangement in $\mathbb{P}^{n}_{K}$ (where $K$ is any algebraically
closed field) given by the hyperplanes $V(x_{i})$ for $i=0\ldots,n-1$ and
$V(x_{i}-x_{j})$ for $i<j$ in $\\{0,\ldots,n-1\\}$. This arrangement
$\mathcal{A}$ is called the braid arrangement. Mapping the class of a point
$(p_{1},\ldots p_{n-1},1)\in\mathbb{G}_{m,K}^{n}$ to the points
$p_{1},\ldots,p_{n-1},1,0,\infty$ in $\mathbb{P}^{1}_{K}$ induces an
isomorphism between the associated hyperplane complement
$\Omega_{\mathcal{A}}$ in $\mathbb{G}_{m,K}^{n}$ and the moduli space
$M_{0,n+2}$ of $n+2$-pointed curves of genus zero. The minimal wonderful
compactification of $\Omega_{\mathcal{A}}$ is isomorphic to the Deligne-
Mumford compactification $\overline{M}_{0,n+2}$. It is shown in [11] that the
automorphism group of $\overline{M}_{0,n+2}$ is isomorphic to $S_{n+2}$ and
hence equal to the automorphism group of $\Omega_{\mathcal{A}}$. The new
transpositions in the symmetric group are realized by Cremona automorphisms.

The matroid $M(\mathcal{A})$ induced by this arrangement is isomorphic to the
graphic matroid given by the complete graph $K_{n+1}$. This implies that the
automorphism group of $M(\mathcal{A})$ is equal to the symmetric group
$S_{n+1}$, i.e. the Weyl group of the root system. The Bergman fan of the
matroid $M(\mathcal{A})$ can be identified with the moduli space
$M_{0,n+2}^{trop}$ of tropical genus zero curves with $n+2$ marked points, see
[7], chapter 4 and [19], Example 7.2. In [1] it is shown that the automorphism
group of $M_{0,n+2}^{trop}$ (and also the automorphism group of its
compactification) is equal to $S_{n+2}$, i.e. it is the same as the
automorphism group in the algebraic case. This implies that
$Aut(B_{c}(M(\mathcal{A})))$ is $S_{n+2}$.

######  Proposition 3.2.

Let $M(\mathcal{A})$ be the matroid given by the braid arrangement. Then the
automorphism group of the coarse Bergman fan $B_{c}(M(\mathcal{A}))$ is
generated by the matroid automorphisms and a combinatorial Cremona map.

###### Proof.

We have to make some of the identifications discussed above explicit. Let
$M=M(\mathcal{A})$ be the matroid given by the braid arrangement for $A_{n}$.
First of all, let us identify $E=E(M)$ with the set
$[\binom{n}{2}]=\\{1,\ldots,\binom{n}{2}\\}$ (where we write two-element
subsets as ordered pairs) by the following map:

$H_{ij}\mapsto(i+1,j+1)\mbox{ for all }i<j\mbox{ in }\\{0,\ldots,n-1\\}$

and

$H_{i}\mapsto(i+1,n+1)\mbox{ for all }i\mbox{ in }\\{0,\ldots,n-1\\}.$

The elements $b_{i}=(i,n+1)$ for $i=1,\ldots n$ form a basis of $M$, which
satisfies the condition from Theorem 8.3. Hence
$\operatorname{\text{Crem}}_{b}$ is an automorphism of $B_{c}(M)$.

The identification of $B_{c}(M)$ with $M_{0,n+2}^{trop}$ (e.g. spelled out in
[12], Section 2.1.2) now shows that $\operatorname{\text{Crem}}_{b}$ induces
the transposition of $n+1$ and $n+2$ in the natural action of $S_{n+2}$ on
$M_{0,n+2}^{trop}$. Moreover, the automorphism group $S_{n+1}$ of $M$ acts on
$M_{0,n+2}^{trop}$ as permutation of the first $n+1$ markings. This implies
our claim. ∎

######  Example 3.3.

Notice that in general not all automorphisms of $B(M({\mathcal{A}}))$ lift to
automorphisms of $\Omega_{{\mathcal{A}}}$. Consider for example the
arrangement ${\mathcal{A}}=\\{0,1,\infty,p\\}$ on $\mathbb{P}^{1}$. The
automorphism group of $B(M({\mathcal{A}}))$ is the symmetric group $S_{4}$. On
the other hand, any automorphism of $\Omega_{{\mathcal{A}}}$ extends to an
automorphism of $\mathbb{P}^{1}$ and is therefore contained in
$\text{PGL}_{2}$ and thus preserves the cross ratio of $(0,1,\infty,p)$. On
the other hand, for appropriate choices of permutations in $S_{4}$ and points
$p$ the cross ratio will not be preserved. Hence automorphisms of Bergman fans
do not always arise from automorphisms of complements.

##  4\. Chow rings of matroids

Let $\Sigma$ be a rational polyhedral fan in
$N_{{\mathbb{R}}}=N\otimes{\mathbb{R}}$ where $N$ is a rank $n$ lattice. Let
$N^{*}$ denote the dual lattice.

######  Definition 4.1.

The combinatorial Chow ring of $\Sigma$ is the ring

$A^{*}(\Sigma)=\frac{\mathbb{Z}[x_{\rho}\ |\ \rho\in\Sigma^{1}]}{I+J},$

where $I$ is the ideal generated by squarefree monomials coming from non-faces
of $\Sigma$, i.e.

$I=(x_{\rho_{1}}\dots x_{\rho_{k}}\ |\
\rho_{1},\dots,\rho_{k}\in\Sigma^{1}\text{ pairwise distinct, not spanning a
cone })$

and $J$ is the ideal

$J=\big{(}\sum_{\rho\in\Sigma^{1}}m(v_{\rho})x_{\rho}\ |\ m\in N^{*}\big{)},$

where $v_{\rho}$ is the primitive integer vector in direction $\rho$.

An analogous ring was introduced by Danilov in [13] as the Chow ring of a
complete non-singular toric variety. The combinatorial Chow ring is the Chow
ring of a non-singular toric variety of a unimodular fan $\Sigma$ even when
the fan is not complete by [10, Section 3.1]. When the fan $\Sigma$ is
unimodular, quasi-projective, and has support $B(M)$ the ring $A^{*}(\Sigma)$
has Poincaré duality, Hard Lefschetz, and satisfies the Hodge Riemann bilinear
relations [4].

######  Proposition 4.2.

An isomorphism of rational polyhedral fans $\phi:\Sigma_{1}\to\Sigma_{2}$
induces an isomorphism of graded rings $\phi_{*}:A^{*}(\Sigma_{1})\to
A^{*}(\Sigma_{2})$.

###### Proof.

Denote the Chow rings of $\Sigma_{i}$ by

$A^{*}(\Sigma_{i})=\frac{\mathbb{Z}[x_{\rho}\ |\
\rho\in\Sigma_{i}^{1}]}{I_{i}+J_{i}},$

where the ideals $I_{i}$ and $J_{i}$ are described above.

The map $\phi$ maps rays of $\Sigma_{1}$ bijectively to rays of $\Sigma_{2}$
which induces a bijection between the generators of the two Chow rings.
Moreover, the ideal $I_{1}$ from the non-faces of $\Sigma_{1}$ is sent exactly
to the corresponding ideal $I_{2}$ of $\Sigma_{2}$. To show that $J_{1}$ maps
to $J_{2}$ consider a relation
$\sum_{\rho\in\Sigma_{1}^{1}}m(v_{\rho})x_{\rho}$ for some $m\in N^{*}$. Then

$\sum_{\rho\in\Sigma_{1}^{1}}m(v_{\rho})x_{\phi(\rho)}=\sum_{\rho^{\prime}\in\Sigma^{1}_{2}}m(\phi^{-1}v_{\rho^{\prime}})x_{\rho^{\prime}}=\sum_{\rho^{\prime}\in\Sigma^{1}_{2}}\phi^{*}m(v_{\rho^{\prime}})x_{\rho^{\prime}},$

therefore $\phi(J_{1})\subseteq J_{2}$. Reversing the argument shows that
$\phi(J_{1})=J_{2}$. Hence the map on the Chow rings induced by $\phi$ is an
isomorphism. ∎

Let us now consider the fine fan structure on a Bergman fan of a loopfree
matroid $M$. For brevity we will call the Chow ring of $B_{f}(M)$ simply the
Chow ring of the matroid $M$ and denote it by $A^{*}(M)$. In terms of
generators and relations this ring is

$A^{*}(M)=\frac{\langle x_{F}\ |\ F\text{ proper flat of
}M\rangle_{\mathbb{Z}}}{I+J},$

where $I$ is the ideal generated by the monomials $x_{F}x_{G}$ for
$F\not\subseteq G$ and $G\not\subseteq F$.

Set $E=\\{0,\dots,n\\}$, then a basis for $N$ is $\\{v_{1},\dots,v_{n}\\}$,
where $v_{k}\in N$ is the image of the standard basis vector of
${\mathbb{Z}}^{E}$ under the quotient. We claim the ideal $J$ is generated by
the degree $1$ polynomials

$\sum_{i\in F}x_{F}-\sum_{0\in G}x_{G},$

for all $i\in E\backslash 0$. Recall that the matroid fan $B_{f}(M)$ is
defined with respect to the lattice
$N={\mathbb{Z}}^{E}/{\mathbb{Z}}\mathbbm{1}$. A collection of generators of
the ideal $J$ is obtained by considering the relations in Definition 4.1
coming from a basis of $N^{*}$. The relation above is obtained by taking
$v_{k}^{*}\in N^{*}$ in Definition 4.1.

This ring is the Chow ring of the _maximal wonderful compactification_ of the
complement of a hyperplane arrangement in projective space when $M$ is
representable over ${\mathbb{C}}$, see [14], [18]. The maximal wonderful
compactification is obtained from projective space by blowing up all
intersections of the hyperplanes in ${\mathcal{A}}$ starting from the
intersections which are points and proceeding by dimension.

If $M$ is a simple matroid of rank $d+1$, then by [2, Proposition 5.10] there
is an isomorphism

$\deg:A^{d}(M)\to\mathbb{Z}$

such that $\deg(x_{F_{1}}\dots x_{F_{d}})=1$ whenever
$F_{1}\subsetneq\dots\subsetneq F_{d}$ is a maximal flag of flats.

Moreover, there is the following formula for the degree map on monomials from
[16] in terms of the coefficients of the characteristic polynomial of a
matroid. We write the reduced characteristic polynomial of a loopfree matroid
$M$ as

$\tilde{\chi}_{M}(t)=\mu^{0}(M)t^{rankM-1}-\mu^{1}(M)t^{rankN-2}+\dots\pm\mu^{rankM-1}(M),$

denoting its $k$-th unsigned coefficient by $\mu^{k}(M)$. The reduced
characteristic polynomial is always monic, hence $\mu_{0}=1$.

######  Theorem 4.3.

[16, Theorem 3.2] Let $F_{1}\subsetneq F_{2}\subsetneq\dots\subsetneq F_{k}$
denote a flag of proper flats of $M$ where $r_{i}=r(F_{i})$ then for all
$d_{i}\geq 0$ with $r(M)-1=d=\sum_{i=1}^{k}d_{i}$

$\deg(x_{F_{1}}^{d_{1}}\dots
x_{F_{k}}^{d_{k}})=(-1)^{d-k}\prod_{i=1}^{k}\binom{d_{i}-1}{\tilde{d}_{i}-r_{i}}\mu^{\tilde{d}_{i}-r_{i}}(M|F_{i+1}/F_{i})$
(4.1)

where $\tilde{d_{i}}=\sum_{j=0}^{i}d_{j}$

For more general unimodular fan structures $\Sigma$ on $B(M)$, Amini and
Piquerez show that the Chow ring of $\Sigma$ still satisfies
$A^{d}(\Sigma)\cong\mathbb{Z}$ where $d+1$ is the rank of $M$ [4, Theorem
1.2]. Moreover, the isomorphism is given by the degree map

$\deg:A^{d}(\Sigma)\to\mathbb{Z}$

sending $x_{\rho_{1}}x_{\rho_{2}}\dots x_{\rho_{d}}$ to $1$, whenever
$\rho_{1},\dots,\rho_{d}\in\Sigma^{1}$ are rays of a $d$-dimensional face of
$\Sigma$, see [4, Section 3.4].

######  Proposition 4.4.

Let $M_{1}$ and $M_{2}$ be loopfree matroids, and let $\Sigma_{1},\Sigma_{2}$
be unimodular fans of dimension $d$ whose supports are respectively $B(M_{1})$
and $B(M_{2})$. If $\phi:\Sigma_{1}\to\Sigma_{2}$ is an isomorphism of fans
then there exists an isomorphism of Chow rings
$\phi_{\ast}:A^{*}(\Sigma_{1})\to A^{*}(\Sigma_{2})$ which is compatible with
the degree maps. More precisely, for any $\alpha\in A^{d}(\Sigma_{1})$ we have
$\deg(\alpha)=\deg(\phi(\alpha)).$

###### Proof.

By Proposition 4.2, the isomorphism $\phi$ of fans induces an isomorphism
$\phi_{\ast}:A^{*}(\Sigma_{1})\to A^{*}(\Sigma_{2})$ of Chow groups. The map
$\phi_{\ast}$ is compatible with the degree map since rays
$\rho_{1},\dots,\rho_{d}$ of $\Sigma_{1}$ generate a $d$-dimensional face of
$\Sigma_{1}$ if and only if the rays $\phi(\rho_{1}),\dots,\phi(\rho_{d})$
generate a $d$-dimensional face of $\Sigma_{2}$. ∎

##  5\. Orlik-Solomon algebra and the characteristic polynomials of matroids

From the previous section and the intersection formula in Equation 4.3, we see
that the reduced characteristic polynomials of minors of $M$ determine the
intersection numbers in $A_{\ast}(M)$. By Proposition 4.2, these intersection
numbers must be preserved under an isomorphism of fine fan structures. In
fact, we show in this section that if an isomorphism between any fan
structures exists the matroids must have the same characteristic polynomial.

The following proposition describes the coefficients of the reduced
characteristic polynomial in terms of the support of the Bergman fan of $M$.
We define

$\mathcal{F}_{p}(B(M))=\sum_{\sigma\subset
B_{\ast}(M)}\bigwedge^{p}\langle\sigma\rangle\subset\bigwedge^{p}({\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}),$

where $\langle\sigma\rangle\subset{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$
denotes the linear span of the cone $\sigma$, and $B_{\ast}(M)$ denotes the
choice of any fan structure on $B(M)$. We set
$\mathcal{F}_{0}(B(M))={\mathbb{R}}$.

######  Proposition 5.1.

If $M$ is a loopfree matroid, then

$\mu^{p}(M)=\dim\mathcal{F}_{p}(B(M)).$

###### Proof.

By [35, Theorem 4], the dual of $\mathcal{F}_{p}(B(M))$ is isomorphic to
$OS^{p}(M)$ where $OS^{*}(M)$ denotes the Orlik-Solomon algebra of $M$ over
${\mathbb{R}}$. Moreover, by [28] the reduced characteristic polynomial of $M$
is

$\tilde{\chi}_{M}(t)=\sum_{p=0}^{r(M)-1}(-1)^{r(M)-p-1}\dim
OS^{p}(M)t^{r(M)-p-1}.$

This completes the proof. ∎

######  Example 5.2.

We can use Proposition 5.1 to determine the coefficient $\mu^{1}(M)$ for any
simple matroid $M$. Since $M$ has no parallel elements, then the vectors
$v_{i}$ are contained in $\tilde{B}(M)$ for any $i\in E$. Upon taking the
projective Bergman fan we have $\dim{\mathcal{F}}_{1}(M)=|E|-1.$ Therefore, we
have $\mu^{1}(M)=|E|-1$.

######  Proposition 5.3.

If $\phi:B(M_{1})\to B(M_{2})$ is an isomorphism of supports of Bergman fans,
then $\dim\mathcal{F}_{p}(B(M_{1}))=\dim\mathcal{F}_{p}(B(M_{2}))$ for all
$p$, and $M_{1}$ and $M_{2}$ have the same reduced characteristic polynomial.

###### Proof.

The map $\phi$ induces isomorphisms on the exterior powers
$\phi_{\ast}:\bigwedge^{p}{\mathbb{R}}^{E_{1}}\to\bigwedge^{p}{\mathbb{R}}^{E_{2}}$.
Moreover, since $\phi$ preserves the supports of the fans it sends the
generators of $\mathcal{F}_{p}(B(M_{1}))$ to the generators of
$\mathcal{F}_{p}(B(M_{2}))$ for all $p$. This proves the claim about the
dimensions. This claim regarding the characteristic polynomial follows from
Proposition 5.1. ∎

Non-isomorphic matroids with the same characteristic polynomials have been
studied in [15]. There are known examples of combinatorially equivalent
arrangements of lines in ${\mathbb{C}}P^{2}$ with non-homotopic complements
[31]. Since they are combinatorially equivalent, the arrangements define the
same matroid, and hence have the same characteristic polynomial. The above
proposition implies that classifying matroids up to isomorphisms of their fans
is a priori a finer notion of equivalence than up to equality of their
characteristic polynomials.

The Chern-Schwartz-MacPherson (CSM) cycles of a matroid were introduced in
[25]. Here they are defined as Minkowski weights supported on the matroid fan
equipped with the fine structure. Recall that a $k$-dimensional Minkowski
weight on a fan $\Sigma$ is an assignment $w:\Sigma^{k}\to{\mathbb{Z}}$
satisfying a _balancing condition_ on all faces of codimension one [20]. In
[25], the CSM cycles of matroids are defined as Minkowski weights on the fine
fan structure of Bergman fans. Moreover, the weights of the cones are
described in terms of invariants of the underlying matroid in the following
way. Let $\mathcal{F}=\\{\emptyset\subsetneq F_{1}\subsetneq\dots\subsetneq
F_{k}\subsetneq E\\}$ be a flag of flats of a rank $d+1$ matroid $M$, then the
$k$-th CSM cycle of $M$ is the weight function
$w_{{\text{csm}}_{k}(M)}:\Sigma^{k}\to{\mathbb{Z}}$ given by

$w_{{\text{csm}}_{k}(M)}(\sigma_{\mathcal{F}})=(-1)^{d-k}\prod_{i=1}^{k}\beta(M|F_{i}/F_{i-1}),$

where $\beta$ denotes the beta invariant of $M$, namely
$\beta(M)=(-1)^{d}\tilde{\chi}_{M}(1)$. The above formula determines the
underlying matroid $M$ and a priori the weight of a face may be different in
the presence of matroid fan isomorphisms. Moreover, the above definition of
the CSM cycle is only a Minkowski weight for the fine fan structure.

The next lemma provides an alternative formula for the weights of the CSM
cycles, and shows they can be recovered from the support of the fan. Hence
this recipe produces CSM cycles as Minkowski weights on $B(M)$ with any fan
structure. This recipe also allows us to consider the CSM cycles of matroids
as fan tropical cycles. Tropical cycles are equivalence classes of Minkowski
weights [3]. Two Minkowski weights $w_{1}:\Sigma_{1}^{k}\to{\mathbb{Z}}$ and
$w_{2}:\Sigma_{2}^{k}\to{\mathbb{Z}}$ are equivalent and hence represent the
same tropical cycle if there exists a fan $\Sigma_{3}$ which refines both
$\Sigma_{1}$ and $\Sigma_{2}$ and a Minkowski weight
$w_{3}:\Sigma_{3}\to{\mathbb{Z}}$ such that
$w_{1}(\sigma_{1})=w_{2}(\sigma_{2})=w_{3}(\sigma_{3})$ if both $\sigma_{1}$
and $\sigma_{2}$ lie in a common face $\sigma_{3}$ of $\Sigma_{3}$.

To alternatively define the CSM cycles, first set

$\tilde{\chi}_{\sigma}(t)=\sum_{i=0}^{d}(-1)^{d-k}\dim\mathcal{F}_{p}(\Sigma_{M}(\sigma))$

where
$\mathcal{F}_{p}(\Sigma_{M}(\sigma))=\sum_{\sigma^{\prime}\supset\sigma}\bigwedge^{p}\langle\sigma^{\prime}\rangle$.
This differs from the definition of the vector space
$\mathcal{F}_{p}(\Sigma_{M})$ in that it only takes into account faces
containing $\sigma$. Notice again that the vector spaces
$\mathcal{F}_{p}(\Sigma_{M}(\sigma))$ are not dependent on the fan structure
chosen and can be defined using any point
$x\in\operatorname{\text{int}}(\sigma)$.

Now let $\Sigma_{M}$ be an arbitrary fan structure on $B(M)$ for some loopfree
matroid $M$. Then set

$w_{{\text{csm}}_{k}(\Sigma_{M})}(\sigma)=\frac{\tilde{\chi}_{\sigma}(t)}{(1-t)^{k}}|_{t=1}.$
(5.1)

######  Lemma 5.4.

Let $M$ be a loopfree matroid and $B_{f}(M)$ be the Bergman fan with fine fan
structure. Then

$w_{{\text{csm}}_{k}(M)}(\sigma_{{\mathcal{F}}})=w_{{\text{csm}}_{k}(B_{f}(M))}(\sigma_{{\mathcal{F}}})$

for all flags of flats ${\mathcal{F}}$ of $M$.

Moreover, for another fan structure $\Sigma_{M}^{\prime}$ on $B(M)$ the
Minkowski weights $w_{{\text{csm}}_{k}(\Sigma_{M})}$ and
$w_{{\text{csm}}_{k}(\Sigma_{M}^{\prime})}$ define the same tropical cycles.

###### Proof.

The first statement follows from the fact that the beta invariant of a matroid
is equal to $\beta(M)=(-1)^{d}\tilde{\chi}_{M}(1)$ and
$\chi_{M_{1}}(t)\chi_{M_{2}}(t)=\chi_{M_{1}\oplus M_{2}}(t)$, where
$\chi_{M}(t)$ denotes the non-reduced characteristic polynomial, that is
$(1-t)\tilde{\chi}_{M}(t)=\chi_{M}(t)$.

For the second statement, if $\Sigma_{M}$ and $\Sigma_{M}^{\prime}$ are two
fan structures on $B(M)$ then there exists a common refinement
$\Sigma^{\prime\prime}_{M}$. Since the recipe for the weight
$w_{{\text{csm}}_{k}(\Sigma)}(\sigma)$ depends only on the support of $\Sigma$
locally about $\sigma$ the CSM Minkowski weights of all three fans are
equivalent. ∎

######  Proposition 5.5.

If $\phi:B_{\ast}(M_{1})\to B_{\ast}(M_{2})$ is an isomorphism of fans where
$\ast$ denotes arbitrary fan structures, then $\phi$ sends
${\text{csm}}_{k}(M_{1})$ to ${\text{csm}}_{k}(M_{2})$ for all $k$.

###### Proof.

To shorten notation let $\Sigma_{i}=B_{\ast}(M_{i})$. Since $\phi$ is an
isomorphism of fans it sends cones of dimension $k$ of $\Sigma_{1}$ to cones
of dimension $k$ to $\Sigma_{2}$. It suffices to prove that the weight of a
face $\sigma\in\Sigma_{1}^{k}$ in ${\text{csm}}_{k}(M_{1})$ is the same as the
weight of $\phi(\sigma)\in\Sigma^{k}_{2}$ in ${\text{csm}}_{k}(M_{2})$. By the
same proof as for Proposition 5.3, for all $p$ we have
$\dim{\mathcal{F}}_{p}(\Sigma_{1}(\sigma))=\dim{\mathcal{F}}_{p}(\Sigma_{2}(\phi(\sigma)))$.
The statement now follows from the formula for the weights of faces in CSM
classes in Equation 5.1. ∎

##  6\. Isomorphisms preserving the fine structure

The goal of this section is to prove Theorem 6.3 which states that every
isomorphism between Bergman fans preserving the fine structure is induced by a
matroid isomorphism if the matroids are not totally disconnected.

######  Lemma 6.1.

If $M$ is a loopfree matroid, then the following are equivalent:

  1. (1)

$M$ is totally disconnected.

  2. (2)

$M$ does not contain a circuit of size greater than or equal to $3$.

  3. (3)

$M/F$ is totally disconnected for every rank $1$ flat $F$.

  4. (4)

$B(M)$ is a linear space of dimension $r(M)-1$.

  5. (5)

the unsigned constant term of the (reduced) characteristic polynomial
$\mu_{d}(M)$ is $1$.

###### Proof.

We first show the equivalence of statements $1)$ and $2)$. The circuits of a
matroid which is totally disconnected are all of size less than or equal to
$2$. Conversely, if $M$ is not totally disconnected then there must be a
connected component of $M$ which contains a circuit of size at least $3$. This
circuit is also a circuit of $M$.

Statement $1)$ implies $3)$ directly. Suppose statement $3)$ holds and let $C$
be a circuit of $M$. If $C$ is contained in a rank $1$ flat of $M$, then
$|C|=2$. Therefore, suppose $C=\\{i_{1},\dots,i_{k}\\}$ with $i_{j}\in F_{j}$
where $F_{j}$ are distinct rank $1$ flats of $M$ and $k\geq 3$. Then
$r_{M/F_{k}}(\\{i_{1},\dots,i_{k-1}\\})=k-1$ since $M/F_{k}$ is totally
disconnected and $F_{k}$ is of rank $1$. This implies that
$r(C)=r_{M/F_{k}}(\\{i_{1},\dots i_{k-1}\\})+r(F_{k})=k$ contradicting the
fact that $C$ is a circuit. Therefore, all circuits of $M$ have size less than
or equal to $2$ so $3)$ implies $2)$.

The equivalence of $1)$ and $4)$ follows from the fact that the dimension of
the lineality space of a Bergman fan is one less than the number of connected
components of the corresponding matroid, Proposition [19, Lemma 2.3]. The
equivalence of $4)$ and $5)$ follows from Proposition 5.1. This completes the
proof. ∎

######  Lemma 6.2.

Let $M_{1}$ and $M_{2}$ be simple matroids, and let $i$ be in $E(M_{1})$.
Suppose $\phi:B_{f}(M_{1})\to B_{f}(M_{2})$ is a isomorphism mapping the ray
associated to the flat $\\{i\\}$ to a ray of a corank one flat in $M_{2}$.
Then $M_{1}/\\{i\\}$ is totally disconnected.

###### Proof.

Let $d+1$ be the rank of $M_{1}$ and $M_{2}$. By Formula 4.3, we can compute

$\deg x_{i}^{d}=(-1)^{d-1}\mu^{d-1}(M_{1}/\\{i\\})\text{ and }\deg
x_{F}^{d}=(-1)^{d-1}\mu^{0}(M_{2}/F)=(-1)^{d-1},$

where $\operatorname{\text{rank}}F=d$. By Proposition 4.4 the isomorphism
$\phi$ induces an isomorphism of Chow rings which is compatible with the
degree map. Therefore, we must have $\mu^{d-1}(M_{1}/\\{i\\})=1$. It follows
from Lemma 6.1 part 5), that the matroid $M_{1}/\\{i\\}$ is totally
disconnected. ∎

######  Theorem 6.3.

Let $M_{1}$ and $M_{2}$ be simple matroids such that neither is totally
disconnected. If $\phi:B_{f}(M_{1})\to B_{f}(M_{2})$ is an isomorphism of
fans, then $\phi$ is induced by a matroid isomorphism.

###### Proof.

Note that if there exists an isomorphism of their Bergman fans, then $M_{1}$
and $M_{2}$ must have the same rank, which we denote by $d+1$. If $M_{1}$ and
$M_{2}$ are of rank $2$, then the simplicity assumption implies that both are
isomorphic to $U_{2,n}$ for some $n$. Hence we may assume that $d\geq 2$.

Since $\phi$ maps rays to rays, we get a bijective correspondence $\phi$
between the flats of $M_{1}$ and the flats of $M_{2}$ preserving adjacency,
i.e. if $F_{1}\subsetneq F_{2}$ is a flag of flats in $M_{1}$, we find that
either $\phi(F_{1})\subsetneq\phi(F_{2})$ or
$\phi(F_{2})\subsetneq\phi(F_{1})$.

We begin by showing that flats of rank $d$ of $M_{1}$ must be mapped to flats
of rank $d$ of $M_{2}$. Let $F$ be a flat of rank $d$ in $M_{1}$. As in the
proof of Lemma 6.2, we deduce from Theorem 4.3 that we have
$\deg(x_{F}^{d})=(-1)^{d-1}$ in the Chow ring of $M_{1}$. If $F^{\prime}$ is a
flat of rank $k$ in $M_{2}$, find again by Theorem 4.3 that

$\deg(x_{F^{\prime}}^{d})=(-1)^{d-1}\binom{d-1}{d-k}\mu^{d-k}(M_{2}/F^{\prime}).$

By Proposition 4.4, the isomorphism $\phi$ induces an isomorphism of Chow
rings compatible with the degree map. If $\phi(F)$ is a flat of rank $k$, we
deduce that $\deg(x_{\phi(F)}^{d})=(-1)^{d-1}$, which implies
$\binom{d-1}{d-k}\mu^{d-k}(M_{2}/\phi(F))=1$. In particular, $k$ must be equal
to $d$ or to $1$.

Assume that $k=1$, so that $\phi(F)=\\{i\\}$ for some $i\in E(M_{2})$. In this
case we find $\mu^{d-1}(M_{2}/\\{i\\})=1$, which implies by Lemma 6.2 that the
matroid $M_{2}/\\{i\\}$ is totally disconnected.

Consider any $j\in F$, then by Theorem 4.3 we have
$\deg(x_{j}x_{F}^{d-1})=(-1)^{d-2}$. Suppose that $j$ is sent to the flat $G$
of $M_{2}$. Then we must have $\deg(x_{i}^{d-1}x_{G})=(-1)^{d-2}$. However,
the degree of the product $x_{i}^{d-1}x_{G}$ contains the factor
$\mu^{d-2}(M_{2}|G/i)$. The rank of the matroid $M_{2}|G/i$ is
$\operatorname{\text{rank}}(G)-1$, since $i$ is not a loop. So if
$\operatorname{\text{rank}}(G)<d$, then $\mu^{d-2}(M_{2}|G/i)=0$, hence the
degree of $x_{i}^{d-1}x_{G}$ would be zero, see also [16, Proposition 3.12].
Hence every $j\in F$ must be sent to a flat of rank $d$ of $M_{2}$. Carrying
on, we see that all rank $d$ flats of $M_{1}$ which contain any $j\in F$ must
be sent to rank $1$ flats of $M_{2}$.

Now take any rank $d$ flat $F^{\prime\prime}$ in $M_{1}$. We can always find a
rank $d$ flat $F^{\prime}$ in $M_{1}$ such that $F\cap F^{\prime}$ and
$F^{\prime}\cap F^{\prime\prime}$ are non-empty, which implies that
$\phi(F^{\prime\prime})$ also has to have rank one. Reasoning as above we find
that every rank $d$ flat in $M_{1}$ is mapped to a rank one flat in $M_{2}$.
Note that conversely every rank one flat in $M_{2}$ is in fact the image of a
rank $d$ flat in $M_{1}$: Any rank one flat $G$ in $M_{2}$ is of the form
$\phi(F)$ for some flat $F$ in $M_{1}$. If $F$ were contained in two different
rank $d$ flats, then $G$ would be adjacent to two different lines, which is
impossible. Therefore we find by Lemma 6.2 applied to $\phi^{-1}$ that
$M_{2}/\\{i\\}$ is totally disconnected for all $i\in E(M_{2})$ and by Lemma
6.1 that $M_{2}$ is totally disconnected which contradicts our hypothesis.
Therefore we conclude that all flats of rank $d$ in $M_{1}$ are sent to flats
of rank $d$ in $M_{2}$.

Now we show by downward induction that for all $k=2,\ldots,d$ the map $\phi$
maps flats of rank $k$ to flats of rank $k$. Assume that the claim is true for
all $l>k$, where $k$ is between $2$ and $d-1$. Then $\phi$ induces a map
between the set of flats of $M_{1}$ and $M_{2}$ of rank less than or equal to
$k$. In particular, it gives a map between the lattice of flats of the
truncation matroids $\operatorname{\text{tr}}_{k+1}(M_{1})$ and
$\operatorname{\text{tr}}_{k+1}(M_{2})$. Recall that the $k+1$-truncation of
the matroid $M$ has rank function

$\operatorname{\text{tr}}_{k+1}r_{M}(I)=\min\\{r(I),k+1\\}.$

Assume that $\phi$ maps a flat of rank $k$ to a flat of rank strictly less
than $k$. We apply the argument above to the truncated matroids. Replacing $d$
by $k$ we find that $\operatorname{\text{tr}}_{k+1}M_{2}/\\{i\\}$ is totally
disconnected for all $i\in E(M_{2})$. So again by Lemma 6.1, the truncation
$\operatorname{\text{tr}}_{k+1}M_{2}$ is totally disconnected and all circuits
are of size less than or equal to $2$.

We claim that this implies that $M_{2}$ is totally disconnected for $k\geq 2$.
Suppose that $M_{2}$ has a circuit $C$ with $|C|\geq 3$. If
$\operatorname{\text{tr}}_{k+1}r_{M_{2}}(C)=\min\\{r_{M_{2}}(C),k+1\\}=r_{M_{2}}(C)=|C|-1$,
then $C$ is still a circuit in $\operatorname{\text{tr}}_{k+1}M_{2}$ which is
a contradiction. Otherwise we have $|C|>k+2$. If $C^{\prime}\subsetneq C$ and
$|C^{\prime}|=k+2$, then $C^{\prime}$ is a circuit of
$\operatorname{\text{tr}}_{k+1}M_{2}$ since
$\min\\{r_{M_{2}}(C^{\prime}),k+1\\}=|C^{\prime}|-1$ and every subset
$I\subsetneq C^{\prime}$ satisfies $\min\\{r_{M_{2}}(I),k+1\\}=|I|$. However,
$k+2\geq 3$ which contradicts that $\operatorname{\text{tr}}_{k+1}M_{2}$ is
totally disconnected. ∎

######  Corollary 6.4.

Let $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ be the maximal wonderful
compactifications of the complements $\Omega_{\mathcal{A}_{1}}$ and
$\Omega_{\mathcal{A}_{2}}$ of two essential and connected hyperplane
arrangements $\mathcal{A}_{1}$, $\mathcal{A}_{2}$. Any isomorphism
$\Omega_{\mathcal{A}_{1}}\to\Omega_{\mathcal{A}_{2}}$ extending to the maximal
wonderful compactifications gives rise to a map of Bergman fans which is
induced by an isomorphism of matroids $M(\mathcal{A}_{1})\to
M(\mathcal{A}_{2})$.

###### Proof.

As in the proof of [24, Theorem 5.1] we see that an isomorphism
$f:\Omega_{\mathcal{A}_{1}}\to\Omega_{\mathcal{A}_{2}}$ induces an isomorphism
between the intrinsic tori and hence via tropicalization a $\mathbb{Z}$-linear
isomorphism mapping $B(M(\mathcal{A}_{1}))$ to $B(M(\mathcal{A}_{2}))$.

If $f:\Omega_{\mathcal{A}_{1}}\to\Omega_{\mathcal{A}_{2}}$ extends to the
maximal wonderful compactifications, it induces an isomorphism between the
boundaries. The boundary of the maximal wonderful compactification of
$\Omega_{\mathcal{A}_{i}}$ is the union of divisors associated to the flats in
the fine structure of the Bergman fan associated to $M(\mathcal{A}_{i})$.
Since two divisors meet if and only if the associated flats are nested, $f$
induces an isomorphism between the two Bergman fans with their fine structure.
Hence our claim follows from Theorem 6.3. ∎

######  Corollary 6.5.

Every finite group $G$ occurs as the automorphism group of the Bergman fan
$B_{f}(M)$ for some simple matroid $M$ of rank $3$.

###### Proof.

This follow from Theorem 6.3 together with the main statement of [8] which
shows that every finite group is the automorphism group of a simple rank $3$
matroid. ∎

##  7\. Isomorphisms of coarse fan structures

In this section we investigate isomorphisms between Bergman fans endowed with
the coarse structure for loopfree matroids of higher rank. The following
trivial example shows that we need to make some connectedness assumptions to
get interesting results.

######  Example 7.1.

For the totally disconnected matroid $M=U_{n,n}$ on $n\geq 3$ elements the
Bergman fan $B(M)$ is simply ${\mathbb{R}}^{n}/{\mathbb{R}}\mathbbm{1}$. Hence
the automorphism group of $B_{c}(M)$ is isomorphic to $GL_{n-1}(\mathbb{Z})$.

If Bergman fans decompose in a product of fans, automorphisms can be defined
component-wise. We will show now that such a product decomposition only happen
for matroids which are non-trivial parallel connections.

######  Theorem 7.2.

Let $M$ be a loopfree matroid of rank at least $3$. If $M$ is a non-trivial
parallel connection of matroids $M_{1}$ and $M_{2}$, then the Bergman fan
$B_{c}(M)$ is isomorphic to the product $B_{c}(M_{1})\times B_{c}(M_{2})$. On
the other hand, if $B_{c}(M)$ is isomorphic to a product $B_{c}(M_{1})\times
B_{c}(M_{2})$ of two Bergman fans for matroids $M_{i}$ of rank at least two,
then $M$ is a non-trivial parallel connection.

###### Proof.

We begin with the first statement. Consider two matroids $M_{1}$ and $M_{2}$
on the ground sets $E_{1}$ and $E_{2}$, and suppose that $M$ is the parallel
connection of $M_{1}$ and $M_{2}$ along $p$. Hence $E=E(M)=E_{1}\cup E_{2}$
and $E_{1}\cap E_{2}=\\{p\\}$.

If $p$ is a coloop of either $M_{1}$ or $M_{2}$, then the parallel connection
along $p$ is a disconnected matroid and the first statement follows from [19,
Lemma 2.1]. Hence we may assume that $p$ is neither a loop nor a coloop in
either $M_{1}$ or $M_{2}$. We will show that the support of the fan
${B}_{c}(M)$ can be mapped bijectively to

$B_{c}(M_{1})\times
B_{c}(M_{2})\subset{\mathbb{R}}^{E_{1}}/{\mathbb{R}}\mathbbm{1}_{E_{1}}\times{\mathbb{R}}^{E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{2}}.$

We use the notation $w_{i}$ for the canonical basis element of
${\mathbb{Z}}^{E_{1}}$ or ${\mathbb{Z}}^{E_{2}}$ when $i$ is in $E_{1}$ or
$E_{2}$. The element $p$ in the intersection $E_{1}\cap E_{2}$ gives rise to
vectors $w_{p_{E_{1}}}\in{\mathbb{R}}^{E_{1}}$ and
$w_{p_{E_{2}}}\in{\mathbb{R}}^{E_{2}}$.

Now consider the map

$\phi_{M}:{\mathbb{R}}^{E_{1}\cup E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{1}\cup
E_{2}}\to{\mathbb{R}}^{E_{1}}/{\mathbb{R}}\mathbbm{1}_{E_{1}}\times{\mathbb{R}}^{E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{2}}$
(7.1)

given by $v_{i}\mapsto w_{i}$ for $i\neq p$ and $v_{p}\mapsto
w_{p_{E_{1}}}+w_{p_{E_{2}}}.$ This map is an isomorphism of vector spaces, and
the inverse is given by $w_{i}\mapsto v_{i}$ for $i\in E_{1}\cup E_{2}$,
$i\neq p$ and $w_{p_{E_{i}}}\mapsto-v_{E_{i}\backslash\\{p\\}}.$

Recall that the affine Bergman fan $\tilde{B}(M)$ consists of points $\sum
a_{i}v_{i}\in{\mathbb{R}}^{E}$ such that $\max_{i\in C}a_{i}$ is attained at
least twice for all circuits $C$ of $M$. Since $p$ is not a loop or coloop in
either $M_{1}$ or $M_{2}$, we find by [29], Proposition 7.1.4, that the
circuits of the parallel connection $M$ of $M_{1}$ and $M_{2}$ along $p$ are
of the form $C=C_{i}$ where $C_{i}$ is a circuit of $M_{i}$ or
$C=C_{1}\backslash\\{p\\}\cup C_{2}\backslash\\{p\\}$ where $C_{i}$ is a
circuit of $M_{i}$ containing $p$. A direct check shows that $x\in B_{c}(M)$
if and only if $\phi_{M}(x)\in B_{c}(M_{1})\times B_{c}(M_{2}).$

Conversely, suppose that $B_{c}(M)$ is isomorphic to the product of two
Bergman fans $B_{c}(M_{1})$ and $B_{c}(M_{2})$ via a linear isomorphism
$\phi:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}_{E}\to{\mathbb{R}}^{E_{1}}/{\mathbb{R}}\mathbbm{1}_{E_{1}}\times{\mathbb{R}}^{E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{2}}$,
where $M_{i}$ is a matroid of rank $d_{i}+1$ on a ground set $E_{i}$ with
$|E_{i}|=n_{i}$. Suppose $|E|=n+1$ and $M$ has rank $d$. Then $d=d_{1}+d_{2}$
and $n+1=n_{1}+n_{2}-1$. By passing to the simplification of $M$ if necessary,
we may suppose that $M$ has no parallel elements. Assume that every $i\in E$
gives rise to a ray in the coarse fan structure. Since $B_{c}(M)$ is a
product, each of its rays corresponds to a ray of either $B_{c}(M_{1})$ or
$B_{c}(M_{2})$. This produces a partition of the rank $1$ rays of $B_{c}(M)$,
which equivalently defines a partition of $E=R_{1}\sqcup R_{2}.$

Again let $v_{k}$ denote the standard basis vector of ${\mathbb{R}}^{E}$
corresponding to $k\in E$. The map $\phi$ sends $\phi(v_{i})$ to
${\mathbb{R}}^{E_{1}}/{\mathbb{R}}\mathbbm{1}_{E_{1}}\times 0$ for $i\in
R_{1}$ and $\phi(v_{j})$ to
$0\times{\mathbb{R}}^{E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{2}}$. Since
$n+1=n_{1}+n_{2}-1$, either $|R_{1}|\geq n_{1}$ or $|R_{2}|\geq n_{2}$, so
that either the vectors $\\{\phi(v_{i})\ |\ i\in R_{1}\\}$ are linearly
dependent in ${\mathbb{R}}^{E_{1}}/{\mathbb{R}}\mathbbm{1}_{E_{1}}$, or the
vectors $\\{\phi(v_{j})\ |\ j\in R_{2}\\}$ are linearly dependent in
${\mathbb{R}}^{E_{2}}/{\mathbb{R}}\mathbbm{1}_{E_{2}}$. However, there is only
one linear dependency among the vectors $v_{k}$ for $k\in E$, namely
$\sum_{k\in E}v_{k}=0$ and the map $\phi$ is an isomorphism. Therefore, our
assumption that every $i\in E$ gives rise to a ray in the coarse fan structure
is false. It follows from Lemma 2.4, that the matroid $M$ must be a parallel
connection along the element $i$. ∎

The next result shows that Bergman fans of non-isomorphic parallel connection
matroids may be isomorphic.

######  Corollary 7.3.

Let $M$ and $M^{\prime}$ be two loopfree matroids on a set $E$ of size $n+1$
obtained as the non-trivial parallel connection of $M_{1}$ and $M_{2}$ along
possibly different pairs of elements. Then there exists an isomorphism from
$B_{c}(M)$ to $B_{c}(M^{\prime})$.

###### Proof.

It suffices to compose $\phi_{M}$ and $\phi^{-1}_{M^{\prime}}$ from Equation
7.1 for two parallel connections $M$ and $M^{\prime}$ of the same matroids
$M_{1}$, $M_{2}$ along different elements. ∎

We will now show that for matroids with a certain connectedness property,
automorphisms of the coarse Bergman fan can map rays to rank $k$ only to rays
of rank or corank $k$.

######  Theorem 7.4.

Suppose $\phi:B_{c}(M)\to B_{c}(M^{\prime})$ is an isomorphism and let $F$ be
a flat of $M$ of rank $k$ giving rise to a ray in the coarse structure.
Suppose neither $M|F$ nor $M/F$ are non-trivial parallel connections, then
$\phi$ sends the ray of $B_{c}(M)$ in direction $F$ to a ray of rank $k$ or
corank $k$ in $B_{c}(M^{\prime})$.

###### Proof.

By Formula 2.1, for any flat $F$ any matroid $M$, we have

$\operatorname{\text{star}}_{v_{F}}(\tilde{B}(M))=\tilde{B}(M|{F})\times\tilde{B}(M/{F}).$

The above fan has a two dimensional lineality space spanned by
$\mathbbm{1}_{E}$ and $v_{F}$. Therefore, taking the quotient of the star of
the projective Bergman fan by $v_{F}$ gives,

$\frac{\operatorname{\text{star}}_{v_{F}}(B(M))}{\langle v_{F}\rangle}\cong
B(M|{F})\times B(M/{F}),$

where $\dim B(M|F)=\operatorname{\text{rank}}(F)-1=k-1$ as well as
$\dim{B}(M/F)=\operatorname{\text{rank}}(M)-\operatorname{\text{rank}}(F)-1=\operatorname{\text{rank}}(M)-k.$
If neither $M|F$ nor $M/F$ are non-trivial parallel connections, then by
Theorem 7.2 neither $B(M|F)$ nor ${B}(M/F)$ splits as a non-trivial product of
Bergman fans.

Suppose the ray generated by $v_{F}$ is a ray in the coarse structure, which
is mapped by $\phi$ to the ray generated by $v_{F^{\prime}}$ for a flat
$F^{\prime}$ of $M^{\prime}$. Then the above quotient of the star fan of
$B_{c}(M)$ at $v_{F}$ must be sent to the corresponding quotient of the star
fan of $B_{c}(M^{\prime})$ at $v_{F^{\prime}}$. Hence the star of
$B(M^{\prime})$ at $v_{F^{\prime}}$ must be a product of Bergman fans of the
same dimensions. Hence $F^{\prime}$ is of rank or corank $k$. ∎

In Section 10, we will apply this result in the following situation.

######  Corollary 7.5.

Let $\phi$ be an automorphism of the coarse Bergman fan $B_{c}(M)$ of a
loopfree matroid $M$ with ground set $E$. Suppose that for every element of
the ground set $i\in E$ the matroid $M/i$ is not a non-trivial parallel
connection. Then $\phi$ maps rays of rank $1$ to rays of rank $1$ or of corank
$1$.

###### Proof.

The statement is an immediate specialisation of Theorem 7.4 in the case $k=1$.
∎

##  8\. Cremona maps

Let $M$ be a matroid of rank $d+1$ on a ground set $E$ of $n+1$ elements. As
in Section 2, we write $v_{F}=\sum_{i\in F}v_{i}$ or every subset $F\subseteq
E$. Recall that $\mathbbm{1}=v_{E}$.

######  Definition 8.1.

Consider a basis $b=\\{b_{0},\dots,b_{d}\\}$ of a simple matroid $M$. For each
basis element $b_{j}$ in $b$ we define a flat $B_{j}$ by

$B_{j}:=\operatorname{\text{cl}}\\{b_{0},\dots,\widehat{b_{j}},\dots,b_{d}\\},$

where $\widehat{b_{j}}$ indicates that $b_{j}$ is left out.

We define the ${\mathbb{Z}}$-linear map
$\operatorname{\text{Crem}}_{b}\colon{\mathbb{R}}^{E}\to{\mathbb{R}}^{E}$ by
$v_{b_{j}}\mapsto v_{B_{j}}$ for the basis elements $b_{j}$, and $v_{k}\mapsto
v_{k}$ for all $k\in E\backslash b$.

If $\operatorname{\text{Crem}}_{b}$ maps the line ${\mathbb{R}}\mathbbm{1}$ to
itself, we also write
$\operatorname{\text{Crem}}_{b}:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}{\rightarrow}{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$
for the quotient map.

We wish to know under what conditions on the matroid $M$ the map
$\operatorname{\text{Crem}}_{b}$ preserves the coarse Bergman fan $B_{c}(M)$.
This is the case if and only if $\operatorname{\text{Crem}}_{b}$ preserves the
support of the (projective) Bergman fan, which we denote by $B(M)$.

Recall the Cremona map $\operatorname{\text{Crem}}_{E}\colon B(U_{n+1,n+1})\to
B(U_{n+1,n+1})$ from Example 1.1. This map is defined by $v_{i}\to-v_{i}$ for
all $i\in E$. We can consider the action of this map
$\operatorname{\text{Crem}}_{E}$ restricted to Bergman fans $B(M)$ for any
matroid $M$ on the ground set $E$.

######  Lemma 8.2.

Let $M$ be a matroid of rank $d+1$ on the ground set $E$. The image of $B(M)$
under $\operatorname{\text{Crem}}_{E}$ is the support of a Bergman fan
$B(M^{\prime})$ if and only if $M$ is totally disconnected.

###### Proof.

Suppose $|E|=n+1$. We use the description of the reduced characteristic
polynomial of Huh and Katz [22] in terms of intersection theory of Minkowski
weights on the $n$-dimensional permutahedral toric variety. This is the toric
variety defined by the complete fan $B(U_{n+1,n+1})$. The map
$\operatorname{\text{Crem}}_{E}$ is an automorphism of the fan
$B(U_{n+1,n+1})$ and hence produces an automorphism of its Chow ring
compatible with the degree map by Proposition 4.4. Hence we also have an
automorphism of the ring of Minkowski weights.

By [22, Lemma 6.1], there is a Minkowski weight $\alpha$ such that for all
$i$, we have
$\mu^{i}(M)=\alpha^{d-i}\cup\operatorname{\text{Crem}}_{E}(\alpha)^{i}\cup
B(M)$, where $\cup$ denotes the intersection product and $\mu^{i}(M)$ is the
$i$-th unsigned coefficient of the reduced characteristic polynomial of $M$.
Since the map $\operatorname{\text{Crem}}_{E}$ gives an automorphism of the
ring of Minkowski weights we have

$\alpha^{d}\cup B(M)=\operatorname{\text{Crem}}_{E}(\alpha)^{d}\cup B(M).$

Now the left hand side is $\mu^{0}(M)=1$ and the right hand side is equal to
$\mu^{d}(M)$. By Lemma 6.1, the coefficient $\mu^{d}(M)$ is equal to one if
and only if $M$ is totally disconnected. ∎

We will now prove a criterion about the existence of Cremona maps (with
respect to a suitable basis) acting as automorphisms of the Bergman fan with
the coarse structure.

######  Theorem 8.3.

Let $b$ be a basis of a simple connected matroid $M$. For any pair of elements
$i,j\in E$, let $F_{ij}$ be the rank $2$ flat which is the closure of $i$ and
$j$. The map $\operatorname{\text{Crem}}_{b}$ descends to a linear map
$\operatorname{\text{Crem}}_{b}:{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}\rightarrow{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$
mapping $B(M)$ to itself, if only if the sets
$\\{F_{ij}\backslash\\{i,j\\}\\}_{i,j\in b}$ partition the set $E\backslash b$
into pairwise disjoint subsets.

We will need the following lemma in the proof.

######  Lemma 8.4.

Assume that $b$ is a basis of a simple matroid $M$, such that the sets
$\\{F_{ij}\backslash\\{i,j\\}\\}_{i,j\in b}$ partition $E\backslash b$. Then
$\operatorname{\text{Crem}}_{b}$ descends to a linear automorphism on the
quotient ${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$.

For all subsets $a\subset b$ we define an associated flat
$F_{a}=\operatorname{\text{cl}}\\{b_{i}:b_{i}\in a\\}$. Then each flat $G$ of
$M$ decomposes into the disjoint union of the two flats $F_{b\cap G}$ and
$G\cap F_{b\backslash G}$. Moreover,

$\operatorname{\text{Crem}}_{b}(v_{G\cap F_{b\backslash G}})=v_{G\cap
F_{b\backslash
G}}\qquad\text{and}\qquad\operatorname{\text{Crem}}_{b}(v_{F_{b\cap
G}})=v_{F_{b\backslash G}}\in{\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}.$

###### Proof.

Note that the partition hypothesis implies that every element $a\in
E\backslash b$ is contained in precisely $d-1$ of the sets
$B_{j}:=\operatorname{\text{cl}}\\{b_{0},\dots,\hat{b_{j}},\dots,b_{d}\\}$.
Therefore the Cremona automorphism $\operatorname{\text{Crem}}_{b}$ of
${\mathbb{R}}^{E}$ maps $\mathbbm{1}$ to
$\sum_{i=0}^{d}v_{B_{i}}+\sum_{a\notin b}v_{a}=d\mathbbm{1}$, hence it
descends to the quotient space ${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$ as
the endomorphism $\operatorname{\text{Crem}}_{b}$ from Definition 8.1.

Let $G$ be any flat in $M$ and consider the subflat $F_{b\cap G}\subset G$.
Let $h\in G\backslash F_{b\cap G}$. By assumption,
$h\in\operatorname{\text{cl}}\\{b_{i},b_{j}\\}$ for some $i,j$. The basis
vector $b_{i}$ cannot be contained in $G$, since this would imply
$b_{j}\in\operatorname{\text{cl}}\\{h,b_{i}\\}\subset G$, and therefore $h\in
F_{b\cap G}$. For the same reason the basis vector $b_{j}$ cannot be contained
in $G$. Hence $h\in F_{b\backslash G}$. This implies that $G\backslash
F_{b\cap G}=G\cap F_{b\backslash G}$. Therefore $G$ is the disjoint union of
the two flats $F_{b\cap G}$ and $G\cap F_{b\backslash G}$.

To prove the remaining statements, note that $G\cap F_{b\backslash G}$ does
not intersect $b$. Therefore, the map $\operatorname{\text{Crem}}_{b}$ fixes
the vector $v_{G\cap F_{b\backslash G}}$. Next put $g=|b\cap G|$ and consider
the action of Cremona:

$\operatorname{\text{Crem}}_{b}(v_{F_{b\cap G}})=\sum_{b_{i}\in b\cap
G}\operatorname{\text{Crem}}_{b}(v_{b_{i}})+v_{F_{b\cap G}\backslash
b}=\sum_{b_{i}\in b\cap G}v_{B_{i}}+v_{F_{b\cap G}\backslash b}.$

We write $C_{pq}=F_{b_{p},b_{q}}\backslash\\{b_{p},b_{q}\\}$. By our
assumption on the matroid $M$ the set $F_{b\cap G}\backslash b$ is the union
of all $C_{pq}$ for $b_{p}$ and $b_{q}$ in $b\cap G$, so that $v_{F_{b\cap
G}\backslash b}=\sum_{b_{p},b_{q}\in b\cap G}v_{C_{pq}}$. Similarly,
$B_{i}\backslash b$ is the union of all $C_{pq}$ for all $b_{p}$ and $b_{q}$
both not equal to $b_{i}$, so that $v_{B_{i}\backslash b}=\sum_{b_{p}\neq
b_{i},b_{q}\neq b_{i}}v_{C_{pq}}$.

Hence we can calculate

$\sum_{b_{i}\in b\cap G}v_{B_{i}}=g\sum_{b_{j}\in b\backslash
G}v_{b_{j}}+(g-1)\sum_{b_{j}\in b\cap G}v_{b_{j}}+$ $(g-2)\sum_{b_{p},b_{q}\in
G}v_{C_{pq}}+(g-1)\sum_{\mbox{either }b_{p}\mbox{ or }b_{q}\in
G}v_{C_{pq}}+g\sum_{b_{p},b_{q}\notin G}v_{C_{pq}},$

so that

$\operatorname{\text{Crem}}(v_{F_{b\cap G}})=v_{F_{b\backslash
G}}+(g-1)\mathbbm{1}.$

This proves the lemma. ∎

###### Proof of Theorem 8.3 .

Let $b$ be a basis such that $\operatorname{\text{Crem}}_{b}$ descends to
${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$ and preserves $B(M)$. Put
$C=E\backslash b$. We consider the matroid $M|b$ which is the same as the
matroid $M\backslash C$ and isomorphic to the uniform matroid $U_{d+1,d+1}$
for $d+1=rk(M)$. For each $k\in C$, consider the rank $d+1$ matroid
$M\backslash\\{C\backslash k\\}$. This matroid is loopfree since $M$ is
assumed to be loopfree. Therefore, $M\backslash\\{C\backslash k\\}/k$ is a
matroid of rank $d$ on the ground set $b$.

Since $M$ is simple, an element $i\in b$ and $k$ cannot be parallel in $M$, so
that $i$ is not a loop of $M\backslash\\{C\backslash k\\}/k$, and we can
consider its Bergman fan in ${\mathbb{R}}^{b}/\mathbbm{1}_{b}$. It must be
preserved under $\operatorname{\text{Crem}}_{b}$ acting on
${\mathbb{R}}^{b}/\mathbbm{1}_{b}$, since the Cremona map commutes with
coordinate projections and the Bergman fans of matroid minors of $M$ can be
determined from the coordinate projections [32, Proposition 2.22].

By Lemma 8.2, the matroid $M/k\backslash\\{C\backslash k\\}$ must be totally
disconnected for all $k$. Since it is a rank $d$ loopfree matroid on the $d+1$
elements of $b$, by the pigeon hole principle there must exist a unique pair
of elements in $b$, say $\\{i,j\\}$, which are parallel elements in
$M/k\backslash\\{C\backslash k\\}$. This implies that $k\in
F_{i,j}\backslash\\{i,j\\}$ and we have a partition of $E\backslash b$ given
by the sets $F_{i,j}\backslash\\{i,j\\}$ for $i,j\in b$.

For the other direction assume that $b$ is a basis of $M$ such that the sets
$\\{F_{ij}\backslash\\{i,j\\}\\}_{i,j\in b}$ partition $E\backslash b$. For
simplicity we write again $C_{ij}:=F_{ij}\backslash\\{i,j\\}$. Then
$C=\bigcup_{p,q}C_{pq}$.

Note that if $k\in E\backslash b$, then $v_{k}$ is fixed by
$\operatorname{\text{Crem}}_{b}$. If $b_{i}\in b$, we have

$\displaystyle\operatorname{\text{Crem}}_{b}^{2}(v_{b_{i}})$ $\displaystyle=$
$\displaystyle\operatorname{\text{Crem}}_{b}(v_{B_{i}})=\sum_{j\neq
i}v_{B_{j}}+\sum_{p\neq i,q\neq i}v_{C_{pq}}$ $\displaystyle=$ $\displaystyle
dv_{b_{i}}+(d-1)\sum_{j\neq i}v_{b_{j}}+(d-1)\sum_{k\in C}v_{k},$

so that under the above hypothesis
$\operatorname{\text{Crem}}_{b}^{2}(v_{b_{i}})=v_{b_{i}}+(d-1)\mathbbm{1}.$
Therefore, $\operatorname{\text{Crem}}_{b}$ is an involution on
${\mathbb{R}}^{E}/{\mathbb{R}}\mathbbm{1}$.

We have to show that $\operatorname{\text{Crem}}_{b}$ preserves the support
$B(M)$ of the Bergman fan. Since we are only interested in the support of the
fan, the fan structure we consider on it does not matter. Therefore it
suffices to show that for every maximal cone $\gamma$ in $B_{f}(M)$ (with
respect to the the fine fan structure), the image
$\operatorname{\text{Crem}}_{b}(\gamma)$ is contained in a cone in $B_{m}(M)$
(with respect to the minimal nested set structure). Namely this implies
$\operatorname{\text{Crem}}_{b}(B(M)\subset{B}(M)$, and since
$\operatorname{\text{Crem}}_{b}$ is an involution, equality of sets follows.

Again write $F_{a}=\operatorname{\text{cl}}\\{b_{i}:b_{i}\in a\\}$ for all
subsets $a$ of our basis $b$. Hence $F_{ij}=F_{\\{b_{i},b_{j}\\}}$. Let
$\mathcal{G}:=\emptyset\subsetneq G_{1}\subsetneq\ldots\subsetneq G_{d}$ be a
flag of pairwise different flats in $E$, giving rise to the maximal cone
$\rho_{\mathcal{G}}:=\langle
v_{G_{1}},v_{G_{2}},\dots,v_{G_{d}}\rangle_{\mathbb{R}_{\geq 0}}$ in the fine
subdivision of $B(M)$.

For every $j\in\\{1,\ldots,r\\}$ we write $G_{j}$ as the union of the two
flats $F_{b\cap G_{j}}$ and $G_{j}\cap F_{b\backslash G_{j}}$ as in the proof
of Lemma 8.4. The sets $b\backslash G_{j}$ give a chain of pairwise different
subsets $\emptyset\subsetneq I_{1}\subsetneq...\subsetneq I_{r}$ of $b$.
Moreover, we decompose every flat $G_{j}\cap F_{b\backslash G_{j}}$ into its
connected components. All those components for all $G_{j}$ form a finite set
of connected flats $K_{1},\ldots,K_{s}$ of $M$, such that each $K_{i}$ are
contained in $E\backslash b$.

We claim that the set of flats
$\mathcal{F}=\\{F_{I_{1}},\ldots,F_{I_{r}},K_{1},\ldots,K_{s}\\}$ is nested
for the minimal nested set structure of $M$. Consider a subset $\mathcal{S}$
of $\mathcal{F}$ consisting of at least two pairwise incomparable elements.
This implies that at most one of those elements is of the form $F_{I_{k}}$.
The other elements of $\mathcal{S}$ are given by pairwise incomparable flats
$K_{i}$ for $i\in I$, where $I$ is a subset of $\\{1,\ldots,s\\}$. We have to
show that the flat $\operatorname{\text{cl}}(\bigcup_{S\in\mathcal{S}}S)$ is
disconnected.

Assume first that $\mathcal{S}$ only contains flats of the form $K_{i}$. Each
$K_{i}$ is contained in some $G_{j(i)}$. We can find some $G_{j}$ containing
all of them, so that at least one of the $K_{i}$ is contained in $G_{j}\cap
F_{b\backslash G_{j}}$. Since $G_{j}$ decomposes into the two flats $F_{b\cap
G_{j}}$ and $G_{j}\cap F_{b\backslash G_{j}}$, we find

$\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}=(\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}\cap
F_{b\cap G_{j}})\cup(\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}\cap
G_{j}\cap F_{b\backslash G_{j}})$

If at least one of the elements $K_{i}$ in $\mathcal{S}$ is contained in the
first set, then this decomposition shows that the flat
$\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}$ is disconnected. If all
$K_{i}$ in $\mathcal{S}$ are contained in $G_{j}\cap F_{b\backslash G_{j}}$
which is the disjoint union of some of the flats $K_{q}$ by construction, we
find that $\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}$ is a disconnected
subflat of $G_{j}\cap F_{b\backslash G_{j}}$.

Let us now assume that $\mathcal{S}$ contains $F_{I_{k}}$, and let $G_{j}$ be
an element in our flat $\mathcal{G}$ such that $I_{k}=b\backslash G_{j}$. For
every $K_{i}$ in $\mathcal{S}$ there exists some $G_{j(i)}$ containing
$K_{i}$, i.e. $K_{i}\subset G_{j(i)}\cap F_{b\backslash G_{j(i)}}$. Since
$K_{i}$ and $F_{I_{k}}$ are incomparable, this implies that $j>j(i)$. Hence
all $K_{i}$ are contained in the flat $G_{j}$, which is the disjoint union of
$F_{b\cap G_{j}}$ and $G_{j}\cap F_{b\backslash G_{j}}$. Since $K_{i}$ and
$F_{G_{j}\backslash b}$ are disjoint, we deduce that all $K_{i}$ are contained
in fact in $F_{b\cap G_{j}}$. Therefore we can decompose

$\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}=(\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}\cap
F_{b\cap G_{j}})\cup(\operatorname{\text{cl}}\\{S:S\in\mathcal{S}\\}\cap
F_{b\backslash G_{j}})$

into two disjoint non-empty flats, so that the flat on the left hand side is
indeed disconnected.

This concludes the proof that the set of flats
$\mathcal{F}=\\{F_{I_{1}},\ldots,F_{I_{r}},K_{1},\ldots,K_{s}\\}$ is nested
for the minimal nested set structure. Hence the cone

$\rho_{\mathcal{F}}=\langle v_{F_{I_{1}}},\dots,v_{F_{I_{r}}},v_{K_{1}},\ldots
v_{K_{r}}\rangle_{\mathbb{R}_{\geq 0}}+{\mathbb{R}}\mathbbm{1}$

is a cone in the minimal nested set structure of the Bergman fan.

By Lemma 8.4, we have $v_{G_{j}}=v_{F_{b\cap G_{j}}}+v_{G_{j}\cap
F_{b\backslash G_{j}}}$, and

$\operatorname{\text{Crem}}_{b}(v_{G_{j}})=v_{F_{b\backslash
G_{j}}}+v_{G_{j}\cap F_{b\backslash G_{j}}}+{\mathbb{R}}\mathbbm{1}.$

Hence we deduce that the image of the cone $\rho_{\mathcal{G}}$ under
$\operatorname{\text{Crem}}_{b}$ is contained in $\rho_{\mathcal{F}}$, and
hence in $B(M)$. ∎

##  9\. The Cremona group of a matroid in rank $3$

For matroids of rank $3$, we have the following strong result:

######  Theorem 9.1.

[33, Theorem 2.8] If $M_{1}$ and $M_{2}$ are simple matroids of rank $3$ and
there exists an isomorphism $\phi:B_{c}(M_{1})\to B_{c}(M_{2})$, then $M_{1}$
and $M_{2}$ are isomorphic.

The statement of the above proposition asserts that $M_{1}$ and $M_{2}$ are
isomorphic, however this does not mean that $\phi$ is necessarily induced by a
matroid isomorphism. We have seen in the last section, that there are Cremona
automorphisms preserving Bergman fans and these do not come from matroid
automorphism. A concrete example to keep in mind is the matroid $M$ of the
braid arrangement $A_{3}$, see Example 3.1. In this case, we have
$\text{Aut}(M)\subsetneq\text{Aut}(B(M))=S_{5}$.

We will prove Theorem 9.1 by considering the case of parallel connections
separately.

######  Lemma 9.2.

Let $M_{1}$ and $M_{2}$ be simple matroids of rank $3$. If one of them is a
non-trivial parallel connection, and there exists an isomorphism
$\phi:B_{c}(M_{1})\to B_{c}(M_{2})$, then $M_{1}$ and $M_{2}$ are isomorphic.

Moreover the automorphism group of $B_{c}(M)$ when $M$ is a non-trivial
parallel connection of rank $3$ is the automorphism group of a complete
bipartite graph.

###### Proof.

Firstly, if one of the matroids $M_{1}$, $M_{2}$ is a non-trivial parallel
connection and there exists an isomorphism of their Bergman fans as in our
claim, then both matroids are non-trivial parallel connections by Proposition
7.2. Since $B_{c}(M_{i})$ is two dimensional, the matroids $M_{i}$ are non-
trivial parallel connections if and only if their fans are a product of two
one dimensional projective Bergman fans. The number of rays and faces of the
fans determines the number of rays in each of these one dimensional Bergman
fans, which in turn determines the two rank $2$ matroids involved in the
parallel connections. Hence the matroids are isomorphic.

Since the fan $B_{c}(M)$ of a rank $3$ parallel connection of two rank $2$
matroids is a product of two one dimensional fans, the fan is the cone over a
complete bipartite graph. Hence any automorphism of the fan induces an
automorphism of the complete bipartite graph. Conversely, any automorphism of
this complete bipartite graph yields an integer linear map which preserves the
Bergman fan of the matroid. ∎

######  Theorem 9.3.

Let $M_{1}$ and $M_{2}$ be simple matroids of rank $3$ which are not non-
trivial parallel connections. Any isomorphism $\phi:B_{c}(M_{1})\to
B_{c}(M_{2})$ is either induced by a matroid isomorphism or the composition of
a Cremona map and matroid isomorphism.

To prove the above lemma we will require some calculations in the Chow ring of
the coarse fan structure on $B(M_{i})$. Note that for a rank $3$ matroid $M$
which is not a non-trivial parallel connection, the coarse fan structure and
the minimal nested set structure of the Bergman fan coincide by Lemma 2.3.

######  Lemma 9.4.

Let $A_{c}^{*}(M)$ denote the Chow ring of the coarse subdivision of the fan
of a rank $3$ simple matroid $M$ which is not a parallel connection of two
rank $2$ matroids. Let $F,F^{\prime}$ be flats of rank $2$ of $M$ which
produce rays in $B_{c}(M)$. Let $\deg$ denote the isomorphism
$A^{2}_{c}(M)\to{\mathbb{Z}}$. Then

  1. (1)

$\deg x_{k}x_{F}=1$ if $k\in F$, for all $k\in E$,

  2. (2)

$\deg x_{k}^{2}=1-|\\{F\ |\ k\in F,|F|>\operatorname{\text{rank}}F=2\\}|$,

  3. (3)

$\deg x_{F}^{2}=-1$,

  4. (4)

$\deg x_{F}x_{F}^{\prime}=0$ for $F\neq F^{\prime}$.

###### Proof.

To simplify notation let $\Sigma$ denote $B_{c}(M)$. We recall that the degree
map is defined on $A^{2}_{c}(M)$ by $\deg x_{k}x_{F}=1$ if $k\in F$ [4].
Recall from Section 4 that

$\sum_{k\in F}x_{F}-\sum_{l\in G}x_{G}\in J,$ (9.1)

where $F$ and $G$ denote connected flats of rank $2$. Thus this sum is $0$ in
$A^{1}_{c}(M)$. Therefore, for any $l\in E$ not equal to $k$ we have

$x_{k}^{2}=x_{k}(x_{l}+\sum_{\begin{subarray}{c}l\in G\\\ G\neq
l\end{subarray}}x_{G}-\sum_{\begin{subarray}{c}k\in F\\\ F\neq
k\end{subarray}}x_{F}).$ (9.2)

There are two cases to consider, either
$\operatorname{\text{cl}}\\{l,k\\}=\\{l,k\\}$ or not. If
$\operatorname{\text{cl}}\\{l,k\\}=\\{l,k\\}$, then the rays $\rho_{l}$,
$\rho_{k}$, corresponding to $l$ and $k$ respectively, span a face of $\Sigma$
and there is no face spanned by $\rho_{k}$ and $\rho_{G}$ for rank $2$
connected flats $G$ with $l\in G$. Then the product becomes

$\deg x_{k}^{2}=1-|\\{F\ |\ k\in F,|F|>\operatorname{\text{rank}}F=2\\}|,$

as claimed. If $F_{lk}=\operatorname{\text{cl}}\\{l,k\\}\supsetneq\\{l,k\\}$,
then there is a ray for $F$ in the coarse subdivision. Moreover, we have the
products $\deg x_{l}x_{k}=0$ and $\deg x_{k}x_{F_{lk}}=1$. Therefore the
product in (9.2) still has degree

$1-|\\{F\ |\ k\in F,|F|>\operatorname{\text{rank}}F=2\\}|.$

Using the relation (9.1), we obtain

$x_{F}^{2}=x_{F}[x_{l}+\sum_{\\{l\\}\subsetneq G}x_{G}-(x_{k}+\sum_{k\in
F^{\prime},F^{\prime}\neq F}x_{F^{\prime}})],$

without loss of generality we can assume that $l\not\in F$, in which case,
$x_{F}^{2}=-x_{F}x_{k}$ which has degree $-1$.

Lastly, if $F\neq F^{\prime}$ and both are of rank $2$, then $F$ and
$F^{\prime}$ are not nested hence their corresponding rays do not span a cone
of $B_{c}(M)$. This implies that $\deg x_{F}x_{F}^{\prime}=0$. ∎

###### Proof of Theorem 9.3.

Recall from Example 5.2, a simple matroid $M$ with ground set $E$ has
$\mu_{1}(M)=\dim\mathcal{F}_{1}(M)=|E|-1$, and the dimension of this vector
space is preserved by a fan isomorphism by Proposition 5.3. Hence, if $E_{1}$
and $E_{2}$ are the ground sets of $M_{1}$ and $M_{2}$, respectively, then we
must have $|E_{1}|=|E_{2}|$. By Lemma 2.4, every ray of rank $1$ is in
$B_{c}(M_{i})$. Any connected flat of rank $2$ of $M_{i}$ induces a ray of
$B_{m}(M_{i})$ and hence of $B_{c}(M_{i})$. If $P_{i}$ is the set of connected
flats of $M_{i}$ then we must also have $|P_{1}|=|P_{2}|$.

Now if $\phi$ is not induced by a matroid automorphism, then there is a ray of
rank $1$ in $B_{c}(M_{1})$ which is sent to a ray of rank $2$ in
$B_{c}(M_{2})$ by $\phi$. Let $\tilde{E}_{i},\tilde{P}_{i}$ be the subsets of
the ground set $E_{i}$, and rank two connected flats $P_{i}$, respectively,
whose ranks are changed by $\phi$ or $\phi^{-1}$. In particular, the map
$\phi$ also induces bijections between $\tilde{E}_{i}$ and $\tilde{P}_{j}$ for
$\\{i,j\\}=\\{1,2\\}$. Moreover, since the number of rank one flats sent to
rank one flats is the same for $\phi$ and $\phi^{-1}$, we have
$|E_{1}|-|\tilde{E}_{1}|=|E_{2}|-|\tilde{E}_{2}|$. Therefore, all of
$\tilde{E}_{1},\tilde{E}_{2},\tilde{P}_{1},$ and $\tilde{P}_{2}$ have the same
size.

Let $G$ be the graph with vertex set $V=\tilde{E}_{1}\cup\tilde{P}_{1}$ and
with edges between two vertices if and only if the corresponding rays span a
face of $B_{c}(M_{1})$. We will prove that this is a bipartite cycle graph
with $6$ vertices and with vertex partition given by
$V=\tilde{E}_{1}\cup\tilde{P}_{1}$.

Claim $1$: If $k\in\tilde{E}_{1}$ then every connected rank $2$ flat $F\ni k$
must be in $\tilde{P}_{1}$. Otherwise if $\phi_{\ast}(x_{k})=x_{F^{\prime}}$
and $\phi_{\ast}(x_{F})=x_{F^{\prime\prime}}$ then $\deg(x_{k}x_{F})=1$ yet
$\deg(x_{F^{\prime}}x_{F^{\prime\prime}})=0$ by Lemma 9.4. This contradicts
that $\phi$ induces a ring isomorphism $A_{c}^{*}(M_{1})\to A_{c}^{*}(M_{2})$
compatible with the degree map.

Claim $2$: If $k\in\tilde{E}_{1}$ then $|\\{F\ |\ k\in
F,|F|>\operatorname{\text{rank}}F=2\\}|=2$. By Proposition 4.4 the map $\phi$
induces a Chow ring isomorphism compatible with the degree map. If
$\phi_{\ast}(x_{k})=x_{F^{\prime}}$ then
$\deg(x_{k}^{2})=\deg(x_{F^{\prime}}^{2})=-1$ and by Lemma 9.4 we have $|\\{F\
|\ k\in F,|F|>\operatorname{\text{rank}}F=2\\}|=2$.

Claim 1 and 2 together imply that every vertex in $\tilde{E}_{1}$ has valency
$2$ in $G$. Repeating the argument for the inverse map of $\phi$ proves the
same assertion for $\tilde{P}_{1}$. Notice that this also implies that
$|\tilde{P}_{1}|\geq 2$ and hence that $|\tilde{E}_{1}|\geq 2$.

Claim $3$: If $k,l\in\tilde{E}_{1}$ then
$F=\operatorname{\text{cl}}{\\{k,l\\}}\in\tilde{P}_{1}$. In particular,
$\operatorname{\text{cl}}{\\{k,l\\}}$ must be a ray in the minimal
subdivision. This proves that the graph $G$ is bipartite and connected.

Therefore, the graph $G$ is a connected 2-regular bipartite graph with $2m$
vertices, where $m=|\tilde{E}_{1}|=|\tilde{P}_{1}|$. Hence it must be the
cycle graph $C_{2m}$. If the graph is $C_{2m}$, then there are $m$ elements in
$\tilde{E}_{1}$ and by Claim $3$ a vertex in $\tilde{E}_{1}$ would have
valency $m-1$. Since the graph is $2$-regular we have $m=3$. Moreover, this
shows that the map $\phi$ sends $j,k,l$ to rank $2$ flats of $M_{2}$ where
$\tilde{E}_{1}=\\{j,k,l\\}$. All other elements of $E_{1}$ are sent to
elements of $E_{2}$.

First notice that $\operatorname{\text{rank}}(\tilde{E}_{1})=3$, therefore
this set is a basis. We prove that the Cremona map
$\operatorname{\text{Crem}}_{\tilde{E}_{1}}$ preserves the fan by using
Theorem 8.3.

To apply the theorem we must show that every $p\in E_{1}\backslash\\{j,k,l\\}$
is contained in one of the flats
$\operatorname{\text{cl}}{\\{k,l\\}},\operatorname{\text{cl}}{\\{j,l\\}},\operatorname{\text{cl}}{\\{k,j\\}}$.
Suppose otherwise, then the ray corresponding to $p\in E_{1}$ spans two
dimensional cones with the rays of $j$, $k$, and $l$. This is because the
closure of $p$ and any one of $j,k,$ or $l$ must be disconnected, as there are
only two connected rank $2$ flats containing any one of $j,k,l$ and they are
among
$\operatorname{\text{cl}}{\\{k,l\\}},\operatorname{\text{cl}}{\\{j,l\\}},\operatorname{\text{cl}}{\\{k,j\\}}$.

By Lemma 9.4, for any singleton $i$ in a rank $3$ matroid $M$ we have in
$A_{c}(M)$,

$\deg(x_{i}^{2})=1-|\\{F\ |\ F\text{ flat of }M,F\ni
i,|F|>\operatorname{\text{rank}}F=2\\}|.$

Yet since $p\not\in\tilde{E}_{1}$ its ray is sent to a rank one ray in the fan
of $M_{2}$ corresponding to $p^{\prime}\in E_{2}$. However, the elements
$j,k,l$ are sent to connected rank $2$ flats of $M_{2}$ containing
$p^{\prime}$, so that there are three more rank two flats contributing to
$\deg(x_{p^{\prime}}^{2})$ than to $\deg(x_{p}^{2})$, which contradicts
Proposition 4.4.

Hence every $p\in E_{1}\backslash\\{j,k,l\\}$ is contained in one of
$\operatorname{\text{cl}}{\\{k,l\\}},\operatorname{\text{cl}}{\\{j,l\\}},\operatorname{\text{cl}}{\\{k,j\\}}$
and the Cremona map $\operatorname{\text{Crem}}_{\tilde{E}_{1}}$ preserves the
fan. Moreover, the composition of $\operatorname{\text{Crem}}_{\tilde{E}_{1}}$
and $\phi$ is sends all rank 1 rays to rank 1 rays and hence is induced by a
matroid automorphism. Therefore, the map $\phi$ is the composition of a
Cremona map and a matroid automorphism. ∎

######  Corollary 9.5.

The automorphism group of $B_{c}(M)$ for a simple rank $3$ matroid $M$ which
is not a parallel connection is generated by matroid automorphisms and Cremona
maps.

###### Proof of Theorem 9.1.

The statement follows immediately from Theorem 9.3. ∎

It is an interesting question under which hypotheses an analogous statement is
true for matroids of higher rank. In Section 10, we prove an analogous
statement for modularly complemented matroids.

######  Example 9.6.

Consider $M$ the parallel connection of $U_{2,n}$ and $U_{2,m}$ along a common
element $0$ contained in both ground sets. By Theorem 7.2, this rank $3$
matroid is the product of two $1$-dimensional Bergman fans. Therefore the
automorphism group of $B_{c}(M)$ is isomorphic to the automorphism group of a
complete bipartite graph $K_{m,n}$ where $m$ and $n$ are the number of rays of
the two $1$-dimensional fans in the product. We assume that $m$ and $n$ are at
least $3$. By Lemma 2.4, the element $0\in E$ is such that $v_{0}$ is not a
ray of the coarse fan structure.

The parallel connection $M$ has exactly two connected flats of rank $2$. These
are the only connected rank $2$ flats containing $0$. These produce two rays
in the coarse fan structure $B_{c}(M)$ of rank $2$. We claim that an
automorphism of $K_{m,n}$ which fixes the first vertex set, yet acts freely on
the other vertex set cannot be obtained by composing matroid automorphisms and
Cremona maps. Notice that such an isomorphism necessarily fixes one of the two
connected rank $2$ rays and sends the other rank $2$ ray to a ray of rank $1$.
A matroid isomorphism preserves the ranks of rays, thus such an isomorphism
either leaves the two rays of rank $2$ fixed or it may swap them if we are in
the case $n=m$. Any basis $b$ of $M$ containing $0$ gives a Cremona map
$\operatorname{\text{Crem}}_{b}$ which is an automorphism of the coarse
Bergman fan of $M$ and hence of $K_{m,n}$. These are the only bases for which
Cremona maps are possible. Applying the Cremona map swaps the two rays of rank
$2$ for two rays of rank $1$. Thus it is not possible to obtain all of
$\mathrm{Aut}(K_{m,n})$ from Cremona maps and matroid isomorphisms.

##  10\. Modularly complemented matroids

For matroids $M_{1}$ and $M_{2}$ of rank bigger than $3$, the determination of
the fan isomorphisms between $B_{c}(M_{1})$ and $B_{c}(M_{2})$ is much more
involved. In the present section we will determine the automorphism groups of
$B_{c}(M)$ for modularly complemented matroids of rank at least $4$.

Let us recall some definitions from [29], Section 6.9. Two flats $X$ and $Y$
in a matroid $M$ form a _modular pair_ if

$r(X)+r(Y)=r(X\cup Y)+r(X\cap Y).$

A flat $X$ is called modular, if and only if for all flats $Y$ the pair
$(X,Y)$ is a modular pair. A matroid is _modular_ , if every flat is modular.
This is a very restrictive condition. In fact, a connected simple modular
matroid is either a free matroid $U_{n,n}$ or a finite projective geometry
[29], Proposition 6.9.1. Recall from Example 7.1 that the coarse Bergman fan
of the totally disconnected matroid $U_{n,n}$ is a linear space with
automorphism group $GL_{n-1}(\mathbb{Z})$.

We will now deal with the case that $M$ is a finite projective geometry of
rank at least $3$, i.e. $M=PG(n,q)$ for some $n\geq 2$ is the matroid on the
set $E(M)$ of all one-dimensional subspaces of the vector space
$\mathbb{F}_{q}^{n+1}$, with rank function given as the dimension of the
linear span of a subset of $E$. Hence the flats in $M$ correspond to the
linear subspaces of $\mathbb{F}_{q}^{n+1}$. The matroid $M=PG(n,q)$ is
isomorphic to the matroid represented by the hyperplane arrangement
$\mathcal{A}$ in $\mathbb{P}_{\mathbb{F}_{q}}^{n}$ consisting of all
$\mathbb{F}_{q}$-rational hyperplanes. It is shown in [30] that every
automorphism of the complement $\Omega_{\mathcal{A}}$ extends to
$\mathbb{P}_{\mathbb{F}_{q}}^{n}$, so that the automorphism group of
$\Omega_{\mathcal{A}}$ is $PGL(n+1,\mathbb{F}_{q})$.

######  Proposition 10.1.

Let $M=PG(n,q)$ be a finite projective geometry with $n\geq 2$. Then every
isomorphism of $B_{c}(M)$ is induced by a matroid automorphism.

###### Proof.

Note that for all connected non-nested flats $F_{1}$ and $F_{2}$ in $M$, the
closure of $F_{1}\cup F_{2}$ is strictly bigger than $F_{1}\cup F_{2}$, hence
it is not disconnected. Therefore the minimal nested set structure and the
fine structure on the Bergman fan coincide. Moreover, by Lemma 2.3 the minimal
nested set structure is the coarsest fan structure on $B(M)$. Hence our claim
follows from Theorem 6.3. ∎

Let us next investigate matroids which satisfy the weaker property of being
modularly complemented. A matroid $M$ is _modularly complemented_ if and only
if every flat $X$ in $M$ has a modular complement $Y$, i.e. $Y$ is a modular
flat such that $r(X\cup Y)=d+1$ and $X\cap Y=\emptyset$.

Note that in a connected modularly complemented matroid $M$ of rank $d$, there
exist $d$ modular flats $H_{1},\ldots,H_{d}$ of rank $d-1$ such that
$H_{1}\cap\ldots\cap H_{d}=\emptyset$, and such that the intersection of any
$d-1$ of them is a element of $E(M)$. Let us write

$\\{b_{i}\\}=H_{1}\cap\ldots\cap H_{i-1}\cap H_{i+1}\cap\ldots\cap H_{d}.$

Then $b_{1},\ldots,b_{d}$ is a basis of $M$, and every flat generated by a
subset of this basis is modular (see [23], p.244).

For matroids of rank at least $4$ there is the following classification of
modularly complemented matroids: By [23], a connected simple modularly
complemented matroid of rank at least $4$ is either a Dowling matroid
$Q_{d}(G)$ or a certain submatroid of the projective geometry $PG(d,q)$.

We will deal with Dowling matroids below. Let us first investigate the case
that $M$ is a submatroid of some $PG(d,q)$ for $d\geq 3$ such that the ground
set $E$ contains all internal elements of $PG(d,q)$, i.e. all lines in
$\mathbb{F}_{q}^{d+1}$ contained in one of the coordinate hyperplanes
$\\{x_{i}=0\\}$, where $(x_{i})_{i}$ is the dual basis of the canonical basis
of $\mathbb{F}_{q}^{d+1}$.

######  Proposition 10.2.

Let $M$ be a submatroid of $PG(d,q)$ for $d\geq 3$ containing all internal
elements. Then every automorphism of $B_{c}(M)$ is induced from a matroid
automorphism.

###### Proof.

Every flat of rank at least $2$ in $PG(d,q)$ corresponds to a subspace $W$ of
$\mathbb{F}_{q}^{d+1}$ of dimension at least $2$. A straightforward argument
shows that $W$ has a basis consisting only of vectors in
$\mathbb{F}_{q}^{d+1}$ which lie in one of the coordinate hyperplanes. Hence
every flat of rank at least $2$ in $PG(d,q)$ is also a flat in $M$.

Applying Lemma 2.3, we find that the minimal nested set structure on $B(M)$
coincides with the coarse structure. Let $\varphi$ be an automorphism of
$B_{c}(M)$. By looking at circuits of size $3$, one can check that $M/i$ is
not a non-trivial parallel connection for all $i$. Hence by Corollary 7.5 rays
of rank or corank $1$ are mapped by $\varphi$ to rays of rank or corank $1$
(i.e. rank $d-1$).

Let us assume that $\varphi$ sends the corank $1$ ray associated to a
coordinate hyperplanes $H_{i}=\\{x_{i}=0\\}$ to the rank one ray associated to
the element $l$. To shorten notation, call this hyperplane $H$. It contains
$\frac{q^{d-1}-1}{q-1}$ many one-dimensional subspaces, which all give rise to
internal elements of $PG(d,q)$. Hence there are $\frac{q^{d-1}-1}{q-1}$ many
rays of rank one in $B_{m}(M)$ which are neighbors of the ray associated to
$H$ in the minimal nested set fan structure. Applying $\varphi$, we find that
the ray associated to $l$ has precisely $\frac{q^{d-1}-1}{q-1}$ many neighbor
rays of rank $1$ or corank $1$ in the minimal nested set structure. On the
other hand, the one-dimensional subspace given by $l$ in
$\mathbb{F}_{q}^{d+1}$ is contained in $\frac{q^{d-1}-1}{q-1}$ hyperplanes.
Therefore its ray cannot have any rank $1$ neighbor in the minimal nested set
structure.

Every element of $E(M)$ contained in $H$ gives therefore rise to a ray which
is mapped to a ray of corank $1$ under $\varphi$. This in turn implies that
all corank $1$ rays are mapped to rank $1$ rays by $\varphi$, since every
hyperplane intersects $H$ non-trivially. Now $M$ contains all corank $1$ flats
of $PG(d,q)$. Since all associated rays map to rank one rays, $M$ must contain
all points from $PG(d,q)$. Since this contradicts Proposition 10.1, our
assumption must be false, which means that $\varphi$ maps the corank $1$ rays
associated to $H_{1},\ldots,H_{d+1}$ to corank $1$ rays.

This implies that all rays corresponding to internal points are mapped to rank
$1$ rays. It suffices to show that the rays associated to the non-internal
points in $E(M)$ are also mapped to rank $1$ rays, since this implies that
$\varphi$ is induced by a matroid automorphism.

If $l$ is any element in $E(M)$, then we find a corank $1$ flat $F$ containing
both $l$ and an internal point $p$. If the ray associated to $l$ is mapped to
a corank $1$ ray, the ray associated to $F$ is mapped to a rank one ray given
by the element $m$. Applying the previous discussion to $\varphi^{-1}$, we
find that $m$ is not an internal point. Now the ray associated to $p$ is
mapped by $\varphi$ to a rank $1$ ray given by a point $q$. Since $m$ is not
internal, the linear span of $q$ and $m$ contains an additional internal
vector, hence the rays associated to $m$ and $q$ cannot be neighbors in the
minimal nested set structure, contradicting our assumption that the ray for
$l$ is sent to a corank one ray. Therefore $\varphi$ is indeed induced by a
matroid automorphism. ∎

We will now discuss the second class of modularly complemented matroids,
namely the Dowling matroids associated to arbitrary finite groups $G$. Let
$d\geq 4$, and put $[d]=\\{1,\ldots,d\\}$. We recall the definition of the
Dowling matroid $Q_{d}(G)$, which is a simple matroid of rank $d$ on
$E(Q_{d}(G))=B\cup\Gamma$ where $B=\\{b_{1},\ldots,b_{d}\\}$ is the set of
coordinate points, and $\Gamma=\\{g_{ij}:g\in G,1\leq i<j\leq d\\}$ is the set
of non-coordinate points. For simplicity, let us write $x_{1}\vee\ldots\vee
x_{d}$ for the closure of $\\{x_{1},\ldots,x_{d}\\}$ in $M$.

Putting $g_{ij}=g_{ji}$ whenever $i>j$, we can list the flats of rank $2$ of
$Q_{d}(G)$ as follows: The rank $2$ flats are precisely the coordinate flats
of the form

$b_{i}\vee b_{j}=\\{b_{i},b_{j}\\}\cup\\{g_{ij}:g\in G\\}\mbox{ for $i\neq j$
}$

and the non-coordinate flats of the form

$\\{g_{ij},h_{jk},(gh)_{ik}\\}\mbox{ for $g,h,gh\in G$ and $i,j,k$ pairwise
different in $[d]$}.$

A subset $F$ of $E(Q_{d}(G))$ is a flat if and only if it is line-closed,
which means that for every two different elements $x,y$ in $F$ the flat $x\vee
y$ is also contained in $F$.

Note that a Dowling matroid $Q_{d}(G)$ is realizable over a field $F$ if and
only if $G$ is isomorphic to a subgroup of $F^{\ast}$, see [29, Theorem
6.10.10]. The matroid automorphisms of $Q_{d}(G)$ are determined in [9].

Let us now determine the automorphism group the Bergman fan of a Dowling
geometry $Q_{d}(G)$ equipped with the coarse fan structure. In order to
determine the minimal nested set structure, we have to study the decomposition
of flats into connected components. Note that every flat $F$ of $Q_{d}(G)$ is
a product of flats in graphic matroids associated to complete graphs and at
most one Dowling matroid on a subset of $B$, see [29, Proposition 6.10.18]. In
particular the only connected corank $1$ flats in $Q_{d}(G)$ are the
coordinate flats of the form $b_{1}\vee\ldots\vee\widehat{b}_{i}\vee\ldots
b_{d}$, i.e. the closure of $B\backslash\\{b_{i}\\}$, and the non-coordinate
flats of the form
$g^{(1)}_{i1}\vee\ldots\vee\widehat{g}^{(i)}_{ii}\vee\ldots\vee g^{(n)}_{in}$
for some $i\in[d]$ and some elements $g^{(1)},\ldots,g^{(d)}$ of the group
$G$.

######  Proposition 10.3.

Let $\varphi:B_{c}(Q_{d}(M))\rightarrow B_{c}(Q_{d}(M))$ be a linear
automorphism of the Bergman fan of a Dowling matroid of rank at least $4$ with
its coarse structure. Then $\varphi$ is either induced by a matroid
automorphism of $Q_{d}(M)$ or $\varphi$ is of the form
$\varphi=\operatorname{\text{Crem}}_{B}\circ\mu$, where $\mu$ is a matroid
automorphism $Q_{d}(M)$ and $\operatorname{\text{Crem}}_{B}$ is the Cremona
map for the basis $B=\\{b_{1},\ldots,b_{d}\\}$.

###### Proof.

First of all note that by Theorem 8.3 the Cremona map
$\operatorname{\text{Crem}}_{B}$ preserves indeed the support of the Bergman
fan.

The matroid $Q_{d}(G)$ is obviously not a non-trivial parallel connection, and
by [29, Proposition 6.10.18 (i)] we find that for every $x\in E(Q_{d}(G))$ the
simplification of $Q_{d}(G)/x$ is again a Dowling matroid of rank at least
$3$. Hence $Q_{d}(G)/x$ is not a non-trivial parallel connection, so that we
can apply Corollary 7.5 to deduce that $\varphi$ maps any ray of rank $1$ to a
ray of rank or corank $1$.

Let $F$ be a connected flat in $Q_{d}(G)$. By [29, Proposition 6.10.18 (ii)],
the restriction $Q_{d}(G)|{F}$ is either isomorphic to $Q_{r}(G)$ for some
$r\leq d$ or to the cycle matroid $M(K_{s})$ of a complete graph $K_{s}$.
Moreover, if $F$ is any flat of rank $k$ in $Q_{r}(G)$, by [29, Section 6.10,
exercise 1], the simplification of $Q_{r}(G)/F$ is isomorphic to $Q_{r-k}(G)$
and hence connected. It is easy to see that for every flat $F$ in $M(K_{s})$
the simplification of $M(K_{s})/F$ is the cycle matroid associated to a
complete graph and hence also connected. We deduce that the matroid $Q_{d}(G)$
satisfies the criterion in Lemma 2.3, so that the coarse structure on the
Bergman fan of $Q_{d}(G)$ coincides with the minimal nested set structure. We
consider the following two cases.

Case 1: $\varphi$ maps all rays of rank $1$ to rays of rank $1$. Then it is
induced by a matroid automorphism of $Q_{d}(G)$.

Case 2: There exists a rank $1$ ray mapping to a corank $1$ ray. Let $F$ be
such a flat of rank $1$ such that the associated ray is mapped to the ray
associated to the corank $1$ flat $F^{\prime}$. We claim that $F$ must be a
coordinate flat $\\{b_{i}\\}$. Let us first consider the case that $|G|=1$.
Then there exists precisely one non-coordinate corank $1$ flat in $Q_{d}(G)$,
which contains every non-coordinate rank $1$ flat. Such a non-coordinate rank
$1$ ray is furthermore adjacent to $d-1\choose 2$ rays of rank $1$. If $F$ was
a non-coordinate rank $1$ flat, we deduce that $F^{\prime}$ would contain
${d-1\choose 2}+1$ flats of rank $1$. This is neither satisfied for a
coordinate corank $1$ flat nor for the non-coordinate corank $1$ flat. This
proves our claim for trivial $G$. Let us now assume that $|G|\geq 2$. If $F$
was a non-coordinate rank $1$ flat, the ray of any non-coordinate corank $1$
flat containing it would be mapped to a rank $1$ ray by $\phi$. Every
connected non-coordinate corank $1$ flat is of the form
$g^{(1)}_{i1}\vee\ldots\vee\widehat{g}^{(i)}_{ii}\vee\ldots\vee g^{(d)}_{id}$
for some $i$ and some elements $g^{(1)},\ldots,g^{(d)}$ of the group $G$. This
flat contains ${d\choose 2}$ rank $1$ flats, and hence this number counts the
neighbor rays of rank $1$ or corank $1$ in the minimal nested set structure.
Since $|G|\geq 2$, a straightforward counting argument shows that for every
rank $1$ ray in $Q_{d}(G)$ the number of neighbor rays of either rank or
corank equal to $1$ is strictly bigger than ${d\choose 2}$. This proves our
claim for non-trivial $G$.

Therefore every rank $1$ ray mapping to a corank $1$ ray is indeed associated
to some $\\{b_{i}\\}$, whereas $\phi$ maps non-coordinate rank $1$ rays to
rank $1$ rays. Let $H$ be a coordinate hyperplane containing $b_{i}$. The ray
associated to $H$ is mapped by $\phi$ to a coordinate rank $1$ ray associated
to some ${b_{k}}$. Now $H$ contains ${d-1\choose 2}|G|$ non-coordinate rank
$1$ flats. Their rays are all mapped to rank $1$ neighbors of the ray given by
${b_{k}}$. Since this ray has precisely ${d-1\choose 2}|G|$ many rank $1$
neighbors, the rays of all coordinate vectors in $H$ must map to corank $1$
rays.

Hence we deduce that $\varphi$ maps every coordinate rank $1$ ray to a
coordinate corank $1$ ray.

Since every permutation on $[d]$ induces a matroid automorphism of $Q_{d}(G)$,
we deduce that there exists a matroid automorphism $\mu_{1}$ such that
$\varphi\circ\mu_{1}$ maps each rank $1$ flat $\\{b_{i}\\}$ to the corank $1$
flat $b_{1}\vee\ldots\vee\widehat{b}_{i}\vee\ldots b_{d}$. Hence
$\operatorname{\text{Crem}}_{B}\circ\varphi\circ\mu_{1}$ maps every ray of
rank $1$ to itself. Therefore it is induced by a matroid automorphism
$\mu_{2}$, and our claim follows. ∎

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Kris Shaw,

Annette Werner,