Cluster Configuration Spaces of Finite Type

Cluster Configuration Spaces of Finite Type

Nima ARKANI-HAMED a, Song HE bcde and Thomas LAM f N. Arkani-Hamed, S. He and
T. Lam

a) School of Natural Sciences, Institute for Advanced Studies, Princeton, NJ,
08540, USA<EMAIL_ADDRESS>
b) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics,
b) Chinese Academy of Sciences, Beijing, 100190, China<EMAIL_ADDRESS>c)
School of Fundamental Physics and Mathematical Sciences,
c) Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China d)
ICTP-AP International Centre for Theoretical Physics Asia-Pacific,
d) Beijing/Hangzhou, China e) School of Physical Sciences, University of
Chinese Academy of Sciences,
e) No.19A Yuquan Road, Beijing 100049, China

f) Department of Mathematics, University of Michigan,
f) 530 Church St, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>http://math.lsa.umich.edu/~tfylam/

Received January 05, 2021, in final form October 04, 2021; Published online
October 16, 2021

For each Dynkin diagram $D$, we define a “cluster configuration space”
${\mathcal{M}}_{D}$ and a partial compactification
${\widetilde{\mathcal{M}}}_{D}$. For $D=A_{n-3}$, we have
${\mathcal{M}}_{A_{n-3}}={\mathcal{M}}_{0,n}$, the configuration space of $n$
points on ${\mathbb{P}}^{1}$, and the partial compactification
${\widetilde{\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The
space ${\widetilde{\mathcal{M}}}_{D}$ is a smooth affine algebraic variety
with a stratification in bijection with the faces of the
Chapoton–Fomin–Zelevinsky generalized associahedron. The regular functions on
${\widetilde{\mathcal{M}}}_{D}$ are generated by coordinates $u_{\gamma}$, in
bijection with the cluster variables of type $D$, and the relations are
described completely in terms of the compatibility degree function of the
cluster algebra. As an application, we define and study cluster algebra
analogues of tree-level open string amplitudes.

configuration space; cluster algebras; generalized associahedron; string
amplitudes 05E14; 13F60; 14N99; 81T30

##  1 Introduction

1.1. The configuration space ${\mathcal{M}}_{0,n}$ of $n$ distinct points on
${\mathbb{P}}^{1}$ is a smooth affine algebraic variety of dimension $n{-}3$,
and it has a very well-studied Deligne–Knudsen–Mumford compactification
${\overline{\mathcal{M}}}_{0,n}$, which is a smooth projective algebraic
variety. The boundary of ${\overline{\mathcal{M}}}_{0,n}$ consists of
$2^{n{-}1}{-}n{-}1$ divisors, satisfying factorization: each divisor is itself
a product
${\overline{\mathcal{M}}}_{0,n_{1}}\times{\overline{\mathcal{M}}}_{0,n_{2}}$,
where $n_{1}{+}n_{2}=n{+}2$.

The real points ${\mathcal{M}}_{0,n}({\mathbb{R}})$ have the structure of a
smooth real manifold with $(n{-}1)!/2$ connected components. Fixing once and
for all a (dihedral) ordering on $n$ points, we let
$({\mathcal{M}}_{0,n})_{>0}\subset{\mathcal{M}}_{0,n}({\mathbb{R}})$ denote
the connected component where the $n$ points are ordered on
${\mathbb{P}}^{1}({\mathbb{R}})=S^{1}$. The closure
$({\mathcal{M}}_{0,n})_{\geq 0}$ of $({\mathcal{M}}_{0,n})_{>0}$ in
${\overline{\mathcal{M}}}_{0,n}({\mathbb{R}})$ is a stratified space that is
homeomorphic to the face stratification of the associahedron polytope.

Let $W$ denote the union of those boundary divisors in
${\overline{\mathcal{M}}}_{0,n}$ whose intersection with
$({\mathcal{M}}_{0,n})_{\geq 0}$ is empty, and let
${\widetilde{\mathcal{M}}}_{0,n}:={\overline{\mathcal{M}}}_{0,n}\setminus W$.
The divisors that do intersect $({\mathcal{M}}_{0,n})_{\geq 0}$ correspond to
ways to divide $\\{1,2,\dots,n\\}$ into two cyclic intervals, each of size
greater than or equal to two. For example, ${\overline{\mathcal{M}}}_{0,5}$
has ten boundary divisors, and ${\widetilde{\mathcal{M}}}_{0,5}$ includes five
of them, corresponding to the five sides of the pentagon (the associahedron of
dimension two). Somewhat surprisingly, the partial compactification
${\widetilde{\mathcal{M}}}_{0,n}$ is an affine algebraic variety, and its ring
of regular functions has the following description. Let $u_{ij}$ be variables
labeled by the diagonals $(i,j)$ (not including sides) of a $n$-gon $P_{n}$.
Then ${\mathbb{C}}\big{[}{\widetilde{\mathcal{M}}}_{0,n}\big{]}$ is isomorphic
to the polynomial ring ${\mathbb{C}}[u_{ij}]$ modulo the relations

$\displaystyle R_{ij}:=u_{ij}+\Bigg{(}\prod_{(k,\ell)\text{ crossing
}(i,j)}u_{k\ell}\Bigg{)}-1,\qquad(i,j)\ \text{varying over all diagonals},$
(1.1)

and ${\mathcal{M}}_{0,n}\subset{\widetilde{\mathcal{M}}}_{0,n}$ is the locus
where $u_{ij}\neq 0$. The $u_{ij}$ are called _dihedral coordinates_. Brown
[11] describes the same space using a presentation with more relations (see
Section 10.1); the extra relations are implied by our smaller set. The
$u_{ij}$ are cross-ratios (see (7.1)) on ${\mathcal{M}}_{0,n}$ and appeared in
the study of scattering amplitudes in string theory and for the bi-adjoint
$\phi^{3}$-theory [1].

1.2. In this paper, we construct in an analogous manner two affine algebraic
varieties ${\mathcal{M}}_{D}\\!\subset\\!{\widetilde{\mathcal{M}}}_{D}$ for
each Dynkin diagram $D$ of finite type by considering the relations

$\displaystyle
R_{\gamma}:=u_{\gamma}+\prod_{\omega}u_{\omega}^{(\omega||\gamma)}-1.$ (1.2)

Here, $\gamma$ and $\omega$ denote mutable cluster variables of a cluster
algebra ${\mathcal{A}}$ of type $D$ [18], and $(\omega||\gamma)$ denotes the
compatibility degree. We call ${\mathcal{M}}_{D}$ the _cluster configuration
space of type $D$_. In the case $D=A_{n-3}$, we have
${\mathcal{M}}_{A_{n-3}}={\mathcal{M}}_{0,n}$ and
${\widetilde{\mathcal{M}}}_{A_{n-3}}={\widetilde{\mathcal{M}}}_{0,n}$. Amongst
many remarkable properties of these relations, let us immediately note that
$u_{\gamma}=0$ forces $u_{\omega}=1$ for all $\omega$ such that
$(\gamma||\omega)\neq 0$ (or equivalently, $(\omega||\gamma)\neq 0$). Thus,
factorization is manifest in (1.2). Some of the results of this work were
reported in [4], and ${\widetilde{\mathcal{M}}}_{D}$ is an example of the
notion of “binary geometry” discussed therein.

Whereas ${\widetilde{\mathcal{M}}}_{0,n}$ has a stratification indexed by the
faces of the associahedron, the space ${\widetilde{\mathcal{M}}}_{D}$ has a
stratification (Proposition 3.5) indexed by the faces of the
Chapoton–Fomin–Zelevinsky generalized associahedron for $D^{\vee}$ [12, 17].
We show (Theorem 3.3) that ${\mathcal{M}}_{D}$ and
${\widetilde{\mathcal{M}}}_{D}$ are smooth affine algebraic varieties and that
the boundary stratification of ${\widetilde{\mathcal{M}}}_{D}$ is simple
normal-crossing. These geometric properties depend on integrality properties
of the normal fan ${\mathcal{N}}(D^{\vee})$ of the generalized associahedron,
and an isomorphism (Theorem 5.3) between ${\widetilde{\mathcal{M}}}_{D}$ and
an affine open subset of the projective toric variety
$X_{{\mathcal{N}}(D^{\vee})}$ associated to ${\mathcal{N}}(D^{\vee})$. Like
$({\mathcal{M}}_{0,n})_{\geq 0}$, the variety ${\widetilde{\mathcal{M}}}_{D}$
contains a distinguished nonnegative part ${\mathcal{M}}_{D,\geq 0}$, which is
a stratified space homeomorphic to the face stratification of the generalized
associahedron (Theorem 8.1). The positive part
${\mathcal{M}}_{D,>0}\subset{\mathcal{M}}_{D}({\mathbb{R}})$ is a
distinguished connected component in ${\mathcal{M}}_{D}({\mathbb{R}})$, and is
cut out by the conditions $u_{\gamma}>0$. Though
${\widetilde{\mathcal{M}}}_{D}$ is not compact,
$\big{(}{\widetilde{\mathcal{M}}}_{D},({\mathcal{M}}_{D,\geq 0})_{\geq
0}\big{)}$ satisfies the other properties of a positive geometry in the sense
of [2].

1.3. The configuration space ${\mathcal{M}}_{0,n}$ is isomorphic to the
quotient of an open subset ${\mathring{{\rm Gr}}}(2,n)\subset{\rm Gr}(2,n)$ of
the Grassmannian of 2-planes by the diagonal torus
$T\subset\operatorname{SL}_{n}$ acting on ${\rm Gr}(2,n)$. Let ${\tilde{B}}$
be a full rank acyclic extended exchange matrix of type $D$. Let
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ be the corresponding cluster algebra,
$X\big{(}{\tilde{B}}\big{)}=\operatorname{Spec}\big{(}{\mathcal{A}}\big{(}{\tilde{B}}\big{)}\big{)}$
be the cluster variety, and let
${\mathring{X}}\big{(}{\tilde{B}}\big{)}\subset X\big{(}{\tilde{B}}\big{)}$
denote the locus where all cluster variables are non-vanishing. We show
(Theorem 4.2) that ${\mathcal{M}}_{D}$ is isomorphic to the (free) quotient of
${\mathring{X}}\big{(}{\tilde{B}}\big{)}$ by the cluster automorphism group
$T\big{(}{\tilde{B}}\big{)}$, generalizing the construction of
${\mathcal{M}}_{0,n}$ from ${\rm Gr}(2,n)$. The functions $u_{\gamma}$ are
particular $T\big{(}{\tilde{B}}\big{)}$-invariant rational functions on
$X\big{(}{\tilde{B}}\big{)}$. The $u_{\gamma}$ are related to some of the
“cluster $X$-coordinates” in the sense of Fock and Goncharov [16] by the
equation $u=X/(1+X)$. The cluster $X$-coordinates appearing here are exactly
those encountered in the Auslander–Reiten walk through cluster variables,
beginning from an acyclic quiver and mutating only on sources. It is important
to note that while the $u_{\gamma}$ are simply related to the cluster
$X$-variables in this way, they are actually in bijection with the cluster
$A$-variables, while in general there are more cluster $X$-variables than
cluster $A$-variables.

We do not have a good understanding of the relationship between
${\mathcal{M}}_{D}$ and cluster ${\mathcal{X}}$-varieties; for example,
${\mathcal{M}}_{D}$ does not contain a collection of (cluster) torus charts.

Our approach depends crucially on the flexibility in the choice of
${\tilde{B}}$. When ${\tilde{B}}={\tilde{B}}^{\operatorname{univ}}$ is the
extended exchange matrix for the universal coefficient cluster algebra [19,
27], the relation (1.2) is obtained from the primitive exchange relations of
${\mathcal{A}}\big{(}{\tilde{B}}^{\operatorname{univ}}\big{)}$ by setting all
mutable cluster variables to 1, and sending the universal frozen variables
$z_{\gamma}$ to $u_{\gamma}$. The non-primitive exchange relations give rise
to other relations of the form $U+U^{\prime}=1$, where $U$ and $U^{\prime}$
are monomials in the $u_{\gamma}$-s.

When ${\tilde{B}}={\tilde{B}}^{\operatorname{prin}}$ is the extended exchange
matrix for the principal coefficient cluster algebra, the functions
$u_{\gamma}$ become identified with certain ratios of the $F$-polynomials
$F_{\gamma}({\mathbf{y}})$. Bazier-Matte, Douville, Mousavand, Thomas, and
Yildrim have shown [8] in the case that $D$ is simply-laced that the Newton
polytope of $F_{\gamma}({\mathbf{y}})$ has normal fan a coarsening of the
${\mathbf{g}}$-vector fan ${\mathcal{N}}\big{(}D^{\vee}\big{)}$ of $D^{\vee}$,
and this result was extended to skew-symmetric cluster algebras by Fei [15].
We extend via folding this description to the case that $D$ is multiply-laced
finite type Dynkin diagram. The identification of
${\widetilde{\mathcal{M}}}_{D}$ with an open subset of the toric variety
$X_{{\mathcal{N}}(D^{\vee})}$ depends crucially on this analysis.

As an application of our results on quotients of cluster varieties and on
$F$-polynomials, we identify (Theorem 9.2) the positive tropicalization
$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ of the cluster configuration space
with the cluster fan ${\mathcal{N}}\big{(}D^{\vee}\big{)}$. In particular, we
resolve a conjecture of Speyer and Williams [29, Conjecture 8.1] on positive
tropicalizations of cluster varieties of finite type; see also [22].

1.4. Inspired by similar questions for ${\mathcal{M}}_{0,n}$, we proceed with
studying the topology of ${\mathcal{M}}_{D}({\mathbb{C}})$ and
${\mathcal{M}}_{D}({\mathbb{R}})$. We identify ${\mathcal{M}}_{B_{n}}$ with
the complement to the Shi-hyperplane arrangement and thereby compute point
counts over finite fields, and the Euler characteristics of
${\mathcal{M}}_{B_{n}}({\mathbb{R}})$ and
${\mathcal{M}}_{B_{n}}({\mathbb{C}})$. We give a configuration space style
description of ${\mathcal{M}}_{C_{n}}$ (Proposition 7.5) but were not able to
determine whether ${\mathcal{M}}_{C_{n}}$ is a hyperplane arrangement
complement. Nevertheless, we were able to compute the point count for
${\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q})$, and the number of connected
components of ${\mathcal{M}}_{C_{n}}({\mathbb{R}})$. We found numerically the
point counts for types $D_{4}$, $D_{5}$ and $G_{2}$, and obtained numerically
that the point count of ${\mathcal{M}}_{D_{4}}({\mathbb{F}}_{q})$ over a
finite field ${\mathbb{F}}_{q}$ is not a polynomial in $q$ but a quasi-
polynomial.

1.5. One of the main motivations for us are scattering amplitudes in string
theory. In [3], we introduced integral functions, called stringy canonical
forms,

$\displaystyle{\mathcal{I}}=\int_{{\mathbb{R}}_{>0}^{n}}\prod_{i}\frac{{\rm
d}x_{i}}{x_{i}}x_{i}^{\alpha^{\prime}X_{i}}\prod_{j}p_{j}({\mathbf{x}})^{-\alpha^{\prime}c_{j}},$
(1.3)

where $p_{j}({\mathbf{x}})$ is a positive Laurent polynomial. We showed in [3]
that the leading order $\lim_{\alpha^{\prime}\to
0}(\alpha^{\prime})^{n}{\mathcal{I}}$ is a rational function that for fixed
$c_{j}$-s coincides with the canonical rational function [2] of the Minkowski
sum of the Newton polytopes of $p_{j}({\mathbf{x}})$. Tree-level $n$-point
open superstring amplitudes are integrals on ${\mathcal{M}}_{0,n}$. It turns
out that for a suitable parametrization of ${\mathcal{M}}_{0,n}$, these
amplitudes can be written as an integral ${\mathcal{I}}_{A_{n-3}}$ in the form
(1.3), where the $p_{j}({\mathbf{x}})$ are the $F$-polynomials for the type
$A_{n-3}$ cluster algebra. The importance of the $u_{ij}$-variables appears in
the rewriting (see [3, Section 9] or [10, Section 3])

$\displaystyle{\mathcal{I}}_{A_{n-3}}=\int_{({\mathcal{M}}_{0,n})_{>0}}\Omega(({\mathcal{M}}_{0,n})_{>0})\prod_{(i,j)}u_{ij}^{\alpha^{\prime}X_{ij}}$

of the open-string amplitude. The poles of ${\mathcal{I}}_{A_{n-3}}$ are given
by $X_{ij}=0$, and at this pole, the factorization of
${\mathcal{I}}_{A_{n-3}}$ mimics the factorization of the equations (1.1). We
define the _cluster string amplitude_ (where $x_{\gamma}$, $x_{i}$ are cluster
variables of ${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ of type $D$)

$\displaystyle{\mathcal{I}}_{D}:=\int_{{\mathcal{M}}_{D,>0}}\Omega({\mathcal{M}}_{D,>0})\prod_{\gamma\in\Pi}x_{\gamma}^{\alpha^{\prime}s_{\gamma}}\prod_{i=n+1}^{n+m}x_{i}^{\alpha^{\prime}s_{i}}.$

The poles of ${\mathcal{I}}_{D}$ are made manifest by rewriting in terms of
the $u_{\gamma}$-s, and the leading order of ${\mathcal{I}}_{D}$ is controlled
by the combinatorics of the generalized associahedron of $D^{\vee}$.

##  2 Background on cluster algebras and generalized associahedra

In this section we review basic facts concerning cluster algebras. The most
important cluster algebra references for us are [8, 31]. For cluster
varieties, our conventions follow [23].

2.1. Let $D$ be a finite Dynkin diagram with vertex set $I$, and let
$A=(a_{ij})$ denote the $n\times n$ Cartan matrix of $D$, where $n=|I|$. Let
$B$ be a skew-symmetrizable exchange matrix, i.e., there exists a matrix $Z$
with positive diagonal entries such that $ZB$ is skew-symmetric. We say that
$B=(B_{ij})$ has type $D$ if

$\displaystyle a_{ij}=\begin{cases}2&\mbox{if}\quad i=j,\\\
-|B_{ij}|&\mbox{if}\quad i\neq j.\end{cases}$

In standard cluster algebra language, $B$ corresponds to an _acyclic_ initial
seed of a cluster algebra of finite type $D$. Given $D$, the possible exchange
matrices $B$ of type $D$ are in bijection with orientations of the underlying
tree of $D$: writing $i\to j$ for the directed edges of this orientation, we
have

$\displaystyle B_{ij}=\begin{cases}-a_{ij}&\mbox{if}\quad i\to j,\\\
a_{ij}&\mbox{if}\quad j\to i,\\\ 0&\mbox{otherwise.}\end{cases}$

For an $(n+m)\times n$ extended exchange matrix $\tilde{B}$ extending $B$, we
let ${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ denote the corresponding cluster
algebra of geometric type [18]. By convention,
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ is the ${\mathbb{C}}$-algebra
generated by all mutable cluster variables, all frozen variables, and the
inverses of all frozen variables. We let
$X\big{(}{\tilde{B}}\big{)}=\operatorname{Spec}{\mathcal{A}}\big{(}{\tilde{B}}\big{)}$
denote the _cluster variety_ [23]. This is a complex affine algebraic variety,
and in general it differs from the union of cluster tori, which is sometimes
called a cluster manifold.

2.2. We say that ${\tilde{B}}$ (or ${\mathcal{A}}$ or $X$) has _full rank_ if
${\tilde{B}}$ has rank $n$. We say that ${\tilde{B}}$ (or ${\mathcal{A}}$ or
$X$) has _really full rank_ if the rows of ${\tilde{B}}$ span
${\mathbb{Z}}^{n}$. If ${\tilde{B}}$ has full rank, then
$X\big{(}{\tilde{B}}\big{)}$ is a smooth affine algebraic variety [24, Theorem
7.7].

2.3. Let $\Pi=\Pi(B)$ be the indexing set for cluster variables, which depends
only on $B$. Set $r:=|\Pi|$. For $\gamma\in\Pi$, we let
$x_{\gamma}\in{\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ denote the corresponding
cluster variable. (Abusing terminology, sometimes we will refer to elements of
$\Pi$ as cluster variables.) We give $\Pi$ the structure of a simplicial
complex, called the _cluster complex_ , by declaring the maximal faces to be
the clusters $\\{\gamma_{1},\dots,\gamma_{n}\\}$.

The set $\Pi$ can be identified with the following set of pairs of integers:

$\displaystyle\Pi=\bigsqcup_{i\in I}\\{(s,i)\,|\,0\leq s\leq r_{i}\\},$ (2.1)

where $r_{i}$, $i\in I$ are some positive integers. The  _initial cluster_ is
$\\{(0,i)\,|\,i\in I\\}$. We let $\Pi^{+}\subset\Pi$ denote the subset of non-
initial cluster variables, i.e., those $\gamma=(t,j)$ with $t\neq 0$.

######  Remark 2.1.

In [31], the set $\Pi$ is identified with the set of weights
$\\{c^{s}\omega_{i}\,|\,0\leq s\leq r_{i}\\}$. We have chosen to index using
the pairs of integers $(s,i)$ instead. The choice $c$ of a Coxeter element in
[31] corresponds to our choice of an orientation of $D$ in determining the
exchange matrix $B$.

######  Remark 2.2.

Starting from the initial cluster $\\{x_{(0,i)}\,|\,i\in I\\}$, the cluster
$\\{x_{(1,i)}\,|\,i\in I\\}$ is obtained by mutating each vertex of $I$ once,
always mutating at sources. This process is repeated to obtain all the cluster
variables. In particular, the cluster variable $x_{(t,j)}$ is obtained by
mutation from $x_{(t-1,j)}$; see Proposition 2.5 for the exchange relation. We
refer the reader to [8] for an explanation of this _Auslander–Reiten walk_ ,
the relation to quiver representations, and many examples.

2.4. There is an involution ${}^{*}\colon I\to I$ sending $i$ to $i^{*}$
induced by the longest element of the Weyl group of the root system of $D$.
This involution is the identity in all types except for $A_{n}$, $D_{2n+1}$,
$E_{6}$, and in these types ${}^{*}\colon I\to I$ is the non-trivial
automorphism of $D$ (as a graph). We shall use the notation
$(-1,i):=(r_{i^{*}},i^{*})$; see [31, Proposition 1.3].

2.5. Each Dynkin diagram $D$ has a dual denoted $D^{\vee}$ defined by
requiring that the Cartan matrix of $D^{\vee}$ be transpose to that of $D$.
Note that $D$ and $D^{\vee}$ have the same underlying tree. If $B$ is an
exchange matrix of type $D$, then we let $B^{\vee}$ be the exchange matrix of
type $D^{\vee}$ associated to the same orientation of the underlying tree of
$D$ and $D^{\vee}$. For dual exchange matrices $B$ and $B^{\vee}$, the cluster
variables $\Pi(B)$ and $\Pi\big{(}B^{\vee}\big{)}$ are naturally in bijection
and under this bijection the cluster complexes are isomorphic.

2.6. For $\gamma,\omega\in\Pi$, we let $(\omega||\gamma)$ denote the
_compatibility degree_ , defined for example in [31, Proposition 5.1]. In
[31], the dependence of the compatibility degree on the choice of $c$
(equivalent to our choice of $B$) is made explicit, but we have suppressed
this dependence in our notation. By [31, Section 5], the compatibility degrees
for different choices of $B$ are equivalent under an appropriate renaming of
$\Pi$. Examples of the compatibility degree are given in Section 3.2.

We have $(\omega||\gamma)=0$ if and only if $(\gamma||\omega)=0$ and in this
case we say that $\omega$ and $\gamma$ are compatible. Otherwise, we call
$\omega$ and $\gamma$ incompatible. If $(\omega||\gamma)=(\gamma||\omega)=1$,
we say that $\omega$ and $\gamma$ are exchangeable. The faces of the cluster
complex consist of sets of cluster variables that are pairwise compatible.

2.7. Let
${\mathcal{A}}^{{\operatorname{prin}}}={\mathcal{A}}\big{(}{\tilde{B}}^{\operatorname{prin}}\big{)}$
denote the cluster algebra with principal coefficients [19]. Thus
${\tilde{B}}^{{\operatorname{prin}}}$ is a $2n\times n$ matrix whose top half
is equal to $B$ and bottom half is equal to the identity matrix. In this case,
the initial mutable variables are denoted $x_{1},x_{2},\dots,x_{n}$ and the
principal frozen variables are denoted $y_{1},y_{2},\dots,y_{n}$. We have a
${\mathbb{Z}}^{n}$-grading on the principal coefficient cluster algebra
${\mathcal{A}}^{\operatorname{prin}}$ given by

$\displaystyle\deg(x_{i})=e_{i}\qquad\text{and}\qquad\deg(y_{i})=-Be_{i}.$
(2.2)

Each mutable cluster variable is homogeneous with respect to this grading, and
we define the _${\mathbf{g}}$ -vector_ by
${\mathbf{g}}_{\gamma}:=\deg(x_{\gamma})$ for $\gamma\in\Pi$.

2.8. For $\gamma\in\Pi^{+}$, define the $F$-polynomial
$F_{\gamma}({\mathbf{y}})$ by setting the initial cluster variables to 1 in
the Laurent expansion of the cluster variable
$x^{\operatorname{prin}}_{\gamma}$ in the cluster algebra
${\mathcal{A}}^{\operatorname{prin}}$ with principal coefficients:

$\displaystyle
F_{\gamma}({\mathbf{y}}):=x^{\operatorname{prin}}_{\gamma}(x_{i}=1,y_{1},y_{2},\dots,y_{n}).$

By convention, we have $F_{\gamma}({\mathbf{y}})=1$ if $\gamma$ is initial.
Computations of ${\mathbf{g}}$-vectors and $F$-polynomials are given in
Examples 6.7 and 6.8. Further examples can be found in [8, 19].

2.9. The cluster fan ${\mathcal{N}}(B)$ is the collection of cones spanned by
$\\{{\mathbf{g}}_{\gamma_{1}},\dots,{\mathbf{g}}_{\gamma_{s}}\\}$ as
$\\{\gamma_{1},\dots,\gamma_{s}\\}$ varies over collections of cluster
variables that belong to the same cluster, called compatible cluster
variables. Recall that a cone $C$ is called simplicial if $\dim(C)$ is equal
to the number of extremal rays of $C$, and a fan is called simplicial if all
its cones are. A fan ${\mathcal{N}}$ in ${\mathbb{R}}^{n}$ is called smooth if
it is simplicial and for each maximal cone $C\in{\mathcal{N}}$ the primitive
integer vectors ${\mathbf{v}}_{1},\dots,{\mathbf{v}}_{n}$ spanning $C$ form an
integral basis for ${\mathbb{Z}}^{n}$.

######  Theorem 2.3 ([12, 17, 21]).

The collection of cones ${\mathcal{N}}(B)$ is a smooth, complete polyhedral
fan.

A _generalized associahedron of type $B$_ is any polytope whose normal fan is
equal to ${\mathcal{N}}(B)$. Often, we will say “generalized associahedron of
type $D$”, with the choice of $B$ of type $D$ understood.

2.10. For $\gamma\in\Pi$, let $P_{\gamma}$ denote the Newton polytope of the
$F$-polynomial $F_{\gamma}({\mathbf{y}})$. By convention, if $\gamma$ is
initial, we have set $F_{\gamma}({\mathbf{y}})=1$ and $P_{\gamma}=\\{0\\}$.
Let $F({\mathbf{y}})=\prod_{\gamma\in\Pi}F_{\gamma}({\mathbf{y}})$. Then the
Newton polytope $P$ of $F({\mathbf{y}})$ is the Minkowski sum
$\sum_{\gamma\in\Pi}P_{\gamma}$. The following result is established in [8]
when $D$ is simply-laced (and extended to not necessarily acyclic initial
seeds in [15]), and in Theorem 6.1 we extend the result to multiply-laced
finite type $D$ with acyclic initial seed.

######  Theorem 2.4.

The $($outer$)$ normal fan of the Minkowski sum
$\sum_{\gamma\in\Pi}P_{\gamma}$ is equal to ${\mathcal{N}}(B^{\vee})$.

2.11. Let $\tau\colon\Pi\to\Pi$ be the bijection defined by
$\tau(t,j)=(t-1,j)$ for $0\leq t\leq r_{j}$, denoted $\tau_{c}$ in [31]. Then
$(\tau\gamma||\tau\omega)=(\gamma||\omega)$ and $\tau$ induces an automorphism
of the cluster complex of $D$.

An exchange relation for ${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ is called
_primitive_ if it is of the form $x_{\gamma}x_{\omega}=M+M^{\prime}$, where
one of the two monomials $M$, $M^{\prime}$ does not contain any mutable
cluster variables. The primitive exchange relations are exactly the ones of
the form $x_{\tau\gamma}x_{\gamma}=M+M^{\prime}$.

2.12. Let
${\mathcal{A}}^{{\operatorname{univ}}}={\mathcal{A}}\big{(}{\tilde{B}}^{\operatorname{univ}}\big{)}$
denote the cluster algebra with universal coefficients, from [19, Theorem
12.4], [31, Section 5], and [27, Theorem 10.12 and Remark 10.13]. Thus
${\tilde{B}}^{{\operatorname{univ}}}$ is a $(n+r)\times n$ matrix whose top
part is equal to $B$ and whose bottom part has rows given by the
${\mathbf{g}}$-vectors of the cluster algebra with exchange matrix $B^{T}$,
see [27]. The bottom $r$ rows of ${\tilde{B}}^{\operatorname{univ}}$ are again
indexed by $\Pi$, and we denote the corresponding frozen variables by
$z_{\gamma}$, for $\gamma\in\Pi$.

######  Proposition 2.5 ([31, Proposition 5.6]).

The primitive exchange relations of
${\mathcal{A}}\big{(}{\tilde{B}}^{\operatorname{univ}}\big{)}$ are given by

$\displaystyle x_{(t-1,j)}x_{(t,j)}=z_{(t,j)}\prod_{i\to
j}x_{(t,i)}^{-a_{ij}}\prod_{j\to
i}x_{(t-1,i)}^{-a_{ij}}+\prod_{\omega}z_{\omega}^{(\omega||(t,j))}$ (2.3)

for $j\in I$ and $0\leq t\leq r_{j}$.

2.13. For $\gamma\in\Pi^{+}$, define the universal $F$-polynomial
$F^{\operatorname{univ}}_{\gamma}({\mathbf{z}})$ by setting the initial
cluster variables to 1 in the Laurent expansion of
$x^{\operatorname{univ}}_{\gamma}$:

$\displaystyle
F^{\operatorname{univ}}_{\gamma}({\mathbf{z}}):=x^{\operatorname{univ}}_{\gamma}(x_{i}=1,z_{\omega}).$

By convention, we have $F^{\operatorname{univ}}_{\gamma}({\mathbf{z}})=1$ if
$\gamma$ is initial.

2.14. Suppose $D$ is a multiply-laced Dynkin diagram whose underlying tree is
oriented. Then there exists a simply-laced Dynkin diagram ${\tilde{D}}$ such
that $D$ is obtained from ${\tilde{D}}$ by _folding_ [13], and the orientation
of $D$ is induced by the orientation of ${\tilde{D}}$. In this situation,
there is a finite group $\Gamma$ acting on $\tilde{I}$ and ${\tilde{\Pi}}$
such that $I$ and $\Pi$ are identified with the $\Gamma$-orbits on $\tilde{I}$
and ${\tilde{\Pi}}$. We obtain surjective quotient maps
$\nu\colon{\tilde{I}}\to I$ and $\nu\colon{\tilde{\Pi}}\to\Pi$. Abusing
notation, let $\nu\colon{\mathbb{R}}^{|{\tilde{\Pi}}|}\to{\mathbb{R}}^{|\Pi|}$
be given by $\nu(e_{{\tilde{\gamma}}})=e_{\nu({\tilde{\gamma}})}$, and
$\nu\colon{\mathbb{R}}^{|\tilde{I}|}\to{\mathbb{R}}^{|I|}$ be given by
$\nu(e_{\tilde{i}})=e_{\nu(\tilde{i})}$. Similarly, define
$\nu\colon{\mathbb{Z}}\big{[}y_{\tilde{i}}\,|\,\tilde{i}\in\tilde{I}\big{]}\to{\mathbb{Z}}[y_{i}\,|\,i\in
I]$ by $\nu(y_{\tilde{i}})=y_{\nu(\tilde{i})}$ and
$\nu\colon{\mathbb{Z}}\big{[}z_{{\tilde{\gamma}}}\,|\,{\tilde{\gamma}}\in{\tilde{\Pi}}\big{]}\to{\mathbb{Z}}[z_{\gamma}\,|\,\gamma\in\Pi]$
by $\nu(z_{{\tilde{\gamma}}})=z_{\nu({\tilde{\gamma}})}$. The following
results are a consequence of the definitions.

######  Proposition 2.6.

  1. $1.$

For ${\tilde{\gamma}},\tilde{\omega}\in{\tilde{\Pi}}$ and $g\in\Gamma$, we
have
$(g\cdot{\tilde{\gamma}}||g\cdot\tilde{\omega})=({\tilde{\gamma}}||\tilde{\omega})$.

  2. $2.$

For $\gamma,\omega\in\Pi$, we have
$(\gamma||\omega)_{D}=\sum_{{\tilde{\gamma}}\in\nu^{-1}(\gamma)}({\tilde{\gamma}}||\tilde{\omega})_{{\tilde{D}}}$
for any $\tilde{\omega}\in\bar{\omega}$.

  3. $3.$

For any ${\tilde{\gamma}}\in{\tilde{\Pi}}$, we have
$\nu(F_{{\tilde{\gamma}}}(\tilde{\mathbf{y}}))=F_{\nu({\tilde{\gamma}})}({\mathbf{y}})$
and
$\nu\big{(}F^{\operatorname{univ}}_{{\tilde{\gamma}}}(\tilde{\mathbf{z}})\big{)}=F^{\operatorname{univ}}_{\nu({\tilde{\gamma}})}({\mathbf{z}})$.

  4. $4.$

For any ${\tilde{\gamma}}\in{\tilde{\Pi}}$, we have
$\nu({\mathbf{g}}_{{\tilde{\gamma}}})={\mathbf{g}}_{\nu({\tilde{\gamma}})}$.

Examples of foldings are given in Section 3.2.

##  3 The cluster configuration space $\boldsymbol{{\mathcal{M}}_{D}}$

In this section, we define the cluster configuration space ${\mathcal{M}}_{D}$
and its partial compactification ${\widetilde{\mathcal{M}}}_{D}$, and we state
some geometric properties of these spaces. We also give examples of the
cluster compatibility degree appearing in the defining relations.

3.1. Let ${\mathbb{C}}[u]:={\mathbb{C}}[u_{\gamma}\,|\,\gamma\in\Pi]$ be the
polynomial ring with generators $u_{\gamma}$ and let ${\mathbb{C}}[u^{\pm 1}]$
denote the Laurent polynomial ring with the same generators.

######  Definition 3.1.

Let $I_{D}$ denote the ideal (in ${\mathbb{C}}[u]$ or
${\mathbb{C}}\big{[}u^{\pm 1}\big{]}$) generated by the elements

$\displaystyle
R_{\gamma}:=u_{\gamma}+\prod_{\omega}u_{\omega}^{(\omega||\gamma)}-1$ (3.1)

for $\gamma\in\Pi$, and $(\omega||\gamma)$ denotes the compatibility degree.

######  Definition 3.2.

Define the _cluster configuration space_ ${\mathcal{M}}_{D}$ and its partial
compactification ${\widetilde{\mathcal{M}}}_{D}$ by

$\displaystyle{\mathcal{M}}_{D}:=\operatorname{Spec}\big{(}{\mathbb{C}}\big{[}u^{\pm
1}\big{]}/I_{D}\big{)}\qquad\text{and}\qquad{\widetilde{\mathcal{M}}}_{D}:=\operatorname{Spec}\big{(}{\mathbb{C}}[u]/I_{D}\big{)}.$

The following result will be proved in Section 5.4.

######  Theorem 3.3.

The two schemes ${\mathcal{M}}_{D}$ and ${\widetilde{\mathcal{M}}}_{D}$ are
smooth, irreducible, affine algebraic varieties of dimension $n$. The boundary
divisor $\partial:={\widetilde{\mathcal{M}}}_{D}\setminus{\mathcal{M}}_{D}$ is
a simple normal-crossing divisor in ${\widetilde{\mathcal{M}}}_{D}$.

If $D=\varnothing$, we define
${\mathcal{M}}_{D}={\widetilde{\mathcal{M}}}_{D}=\operatorname{Spec}({\mathbb{C}})$
to be a point. If $D=A_{1}$, then
${\mathbb{C}}[u]/I_{D}={\mathbb{C}}[u,u^{\prime}]/(u+u^{\prime}=1)$ so
${\widetilde{\mathcal{M}}}_{A_{1}}={\mathbb{C}}$ and
${\mathcal{M}}_{A_{1}}={\mathbb{C}}\setminus\\{0,1\\}$.

######  Remark 3.4.

The definition of ${\mathcal{M}}_{D}$ and ${\widetilde{\mathcal{M}}}_{D}$
depends on the choice of exchange matrix $B$ of type $D$ only in the indexing
of the generators $u_{\gamma}$ by $\Pi$. For two different orientations of
$D$, there is a natural bijection between the two indexing sets $\Pi(B)$ that
arise, and a natural isomorphism between the resulting schemes
${\mathcal{M}}_{D}(B)$.

3.2. Let us give the relations $R_{\gamma}$ explicitly in types $A$, $B$, $C$,
$D$, $G$. In the following discussion, we use models for $\Pi$ involving
diagonals of a polygon; see [18] for further details. The precise
correspondence with (2.1) depends on the choice of initial cluster (for
example, a choice of triangulation of the polygon in type $A$), or
equivalently the choice of $B$, or equivalently the choice of orientation of
$D$.

3.2.1. Type $\boldsymbol{A_{n-3}}$. In this case, the set $\Pi$ can be
identified with the diagonals (not including sides!) $\\{(i,j)\\}$ of an
$n$-gon. The $R_{\gamma}$ are the equations (1.1). The compatibility degree is
given by the formula $((i,j)||(k,\ell))=1$ if $(i,j)$ and $(k,\ell)$ cross (in
the interior of the polygon) and $((i,j)||(k,\ell))=0$ if $(i,j)$ and
$(k,\ell)$ do not cross. The automorphism $\tau$ in this case corresponds to
the order $n$ rotation of the polygon. The clusters are exactly the maximal
sets of pairwise compatible diagonals. In other words, clusters are in
bijection with triangulations of the $n$-gon.

3.2.2. Type $\boldsymbol{C_{n-1}}$. Let $P_{2n}$ be the $2n$-gon with vertices
cyclically labeled $1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}$. The set $\Pi$
is identified with the union

$\displaystyle\Pi$
$\displaystyle=\Big{\\{}\big{[}i,\bar{i}\big{]}:=\big{(}i,\bar{i}\big{)}\,|\,1\leq
i\leq
n\Big{\\}}\cup\Big{\\{}[i,j]:=\big{(}(i,j),\big{(}\bar{i},\bar{j}\big{)}\big{)}\,|\,1\leq
i<j-1<n\Big{\\}}$ $\displaystyle{}\phantom{=\
}\cup\Big{\\{}\big{[}i,\bar{j}\big{]}:=\big{(}\big{(}i,\bar{j}\big{)},\big{(}j,\bar{i}\big{)}\big{)}\,|\,1\leq
i<j\leq n,\,(i,j)\neq(1,n)\Big{\\}}$ (3.2)

of the long diagonals, and pairs of centrally symmetric diagonals in $P_{2n}$.
In total we have $|\Pi|=n^{2}-n$.

The compatibility degree $(\gamma||\omega)$ is equal to the number of
crossings of one of the diagonals representing $\omega$ with the diagonals
representing $\gamma$. Thus for example
$\big{(}[1,\bar{1}]||[2,\bar{3}]\big{)}=1$ but
$\big{(}[2,\bar{3}]||[1,\bar{1}]\big{)}=2$. For example, for $C_{2}$,
Definition 3.1 gives the two equations

$\displaystyle
1=u_{[1\bar{1}]}+u_{[2\bar{2}]}u_{[3\bar{3}]}u_{[2\bar{3}]}^{2},$
$\displaystyle 1=u_{[2\bar{3}]}+u_{[1\bar{1}]}u_{[13]}u_{[1\bar{2}]}$ (3.3)

and the three cyclic rotations of each.

This case is obtained from $A_{2n-3}$ by folding. There is an action of the
two-element group $\Gamma$ on $P_{2n}$ mapping $i\leftrightarrow\bar{i}$. This
induces the natural map $\nu\colon\tilde{\Pi}\to\Pi$ sending diagonals of
$P_{2n}$ to $\Gamma$-orbits on the diagonals of $P_{2n}$. We may verify
Proposition 2.6(2): for example
$\big{(}[2,\bar{3}]||[1,\bar{1}]\big{)}_{C_{2}}=\big{(}(2,3)||(1,\bar{1})\big{)}_{A_{3}}+\big{(}(\bar{2},\bar{3})||(1,\bar{1})\big{)}_{A_{3}}=1+1=2$.
The automorphism $\tau$ is inherited from the rotation of the $2n$-gon
$P_{2n}$.

3.2.3. Type $\boldsymbol{D_{n}}$. Let $PP_{n}$ denote an $n$-gon $P_{n}$ with
vertices $1,2,\dots,n$ (in clockwise order) and an additional marked point $0$
in the middle. The set $\Pi$ consists of certain arcs in $PP_{n}$ connecting
vertices and $0$: ($a$) for $1\leq i\neq j\leq n$ and $i\neq j+1\mod n$ we
have an arc $(i,j)$ connecting $i$ to $j$ going counterclockwise around $0$,
and $(b)$ for each $1\leq i\leq n$ we have two arcs $[i]$ and $[\tilde{i}]$
connecting $i$ to $0$. We denote the corresponding $u$-variables by $u_{ij}$,
and $u_{i}$ and $u_{\tilde{i}}$. See Figure 1. (We caution the reader that the
notation $\tilde{i}$ here is unrelated to the notation ${\tilde{\gamma}}$ used
for foldings.) In this case, the automorphism $\tau$ is the composition of the
rotation of $P_{n}$ with “changing the tagging at 0” (i.e., switching from
$[i]$ to $[\tilde{i}]$ if the arc is incident to $0$).

The compatibility degree $(\gamma||\omega)$ is equal to the (minimal) number
of intersection points between the arc $\gamma$ and the arc $\omega$, if at
least one of $\gamma$ and $\omega$ do not connect to $0$. If both $\gamma$ and
$\omega$ connect to $0$, then we have
$([i]||[j])=\big{(}[\tilde{i}]||[\tilde{j}]\big{)}=\big{(}[i]||[\tilde{i}]\big{)}=0$
but $\big{(}[i]||[\tilde{j}]\big{)}=\big{(}[\tilde{j}]||[i]\big{)}=1$ if
$i\neq j$. For $D_{4}$, representative equations from Definition 3.1 are

$\displaystyle
1=u_{12}+u_{3}u_{\tilde{3}}u_{4}u_{\tilde{4}}u_{34}^{2}u_{23}u_{24}u_{41}u_{31},$
$\displaystyle 1=u_{13}+u_{4}u_{\tilde{4}}u_{41}u_{42}u_{24}u_{34},$
$\displaystyle
1=u_{1}+u_{\tilde{2}}u_{\tilde{3}}u_{\tilde{4}}u_{23}u_{34}u_{24}$ (3.4)

and we have $4$, $4$, $8$ equations of these types respectively, for a total
of $16$ equations.

$0$$1$$2$$3$$4$$u_{\tilde{1}}$$u_{1}$$u_{12}$$u_{13}$ Figure 1: The polygon
$PP_{4}$ and some $u$-variables. Note $u_{21}$ does not exist.

3.2.4. Type $\boldsymbol{B_{n-1}}$. The set $\Pi$ is the same as for $C_{n-1}$
(3.2). However, the compatibility degree $(\gamma||\omega)$ is equal to the
number of crossings of one of the diagonals representing $\gamma$ with the
diagonals representing $\omega$. Thus,
$\big{(}[1,\bar{1}]||[2,\bar{3}]\big{)}=2$ but
$\big{(}[2,\bar{3}]||[1,\bar{1}]\big{)}=1$. For $B_{2}$, Definition 3.1 gives
the two equations

$\displaystyle 1=u_{[1\bar{1}]}+u_{[2\bar{2}]}u_{[3\bar{3}]}u_{[2\bar{3}]},$
$\displaystyle 1=u_{[2\bar{3}]}+u_{[1\bar{1}]}^{2}u_{[13]}u_{[1\bar{2}]}$

and the three cyclic rotations of each. Note that ${\mathcal{M}}_{B_{2}}$ is
isomorphic to ${\mathcal{M}}_{C_{2}}$ under a non-trivial re-indexing of the
$u$-variables. However ${\mathcal{M}}_{B_{n}}$ and ${\mathcal{M}}_{C_{n}}$ are
not isomorphic for $n>2$.

Type $B_{n-1}$ can be obtained from type $D_{n}$ by folding. Let $\Gamma$ be
the two-element group acting on ${\tilde{\Pi}}=\Pi(D_{n})$ by sending
$[i]\leftrightarrow[\tilde{i}]$ and fixing all other $(i,j)$. The map
$\nu\colon{\tilde{\Pi}}\to\Pi$ sends $[i]$ and $[\tilde{i}]$ to $[i,\bar{i}]$,
and sends $(i,j)$ to $[i,\bar{j}]$ if $i<j$ and to $[j,i]$ if $i>j$. For
example, for $B_{3}$, the images of the equations from (3.4) are

$\displaystyle
1=u_{[1\bar{2}]}+u_{[3\bar{3}]}^{2}u_{[4\bar{4}]}^{2}u_{[3\bar{4}]}^{2}u_{[2\bar{3}]}u_{[2\bar{4}]}u_{[14]}u_{[13]},$
$\displaystyle
1=u_{[1\bar{3}]}+u_{[4\bar{4}]}^{2}u_{[14]}u_{[24]}u_{[2\bar{4}]}u_{[3\bar{4}]},$
$\displaystyle
1=u_{[1\bar{1}]}+u_{[2\bar{2}]}u_{[3\bar{3}]}u_{[4\bar{4}]}u_{[2\bar{3}]}u_{[3\bar{4}]}u_{[2\bar{4}]}.$
(3.5)

The automorphism $\tau$ is inherited from the rotation of the $n$-gon in type
$D_{n}$.

3.2.5. Type $\boldsymbol{G_{2}}$. In type $G_{2}$, we have $|\Pi|=8$ and we
denote $u$-variables by $a_{i}$, $b_{i}$ for $i=1,2,3,4$. The $u$-equations
for type $G_{2}$ are

$\displaystyle 1=a_{1}+a_{2}b_{2}a^{2}_{3}b_{3}a_{4},$ $\displaystyle
1=b_{1}+b_{2}a_{3}^{3}b_{3}^{2}a^{3}_{4}b_{4}$

and the cyclic rotations under the group ${\mathbb{Z}}/4{\mathbb{Z}}$. Type
$G_{2}$ can be obtained from $D_{4}$ by folding. (Though at present we only
consider foldings of simply-laced diagrams, $G_{2}$ can also be obtained from
$B_{3}$ by folding.)

3.3. Let $F$ be a face of the generalized associahedron, which we identify
with a pairwise compatible subset $\\{\gamma_{1},\dots,\gamma_{d}\\}$ of
$\Pi$. We define
${\widetilde{\mathcal{M}}}_{D}(F)\subset{\widetilde{\mathcal{M}}}_{D}$ to be
the closed subscheme cut out by the ideal
$(u_{\gamma_{1}},\dots,u_{\gamma_{d}})$. If $F=\varnothing$, then
${\widetilde{\mathcal{M}}}_{D}(\varnothing):={\widetilde{\mathcal{M}}}_{D}$.
We define ${\mathcal{M}}_{D}(F)\subset{\widetilde{\mathcal{M}}}_{D}(F)$ to be
the open subscheme of ${\widetilde{\mathcal{M}}}_{D}(F)$ where all the
variables $\\{u_{\gamma}\,|\,\gamma\notin F\\}$ are non-vanishing.

######  Proposition 3.5.

We have a natural stratification

$\displaystyle{\widetilde{\mathcal{M}}}_{D}=\bigsqcup_{F}{\mathcal{M}}_{D}(F),$
(3.6)

where $F$ varies over all the faces of the generalized associahedron.

###### Proof.

When $\omega$ and $\gamma$ are incompatible, the coordinate $u_{\omega}$ is
non-vanishing on ${\mathcal{M}}_{D}(\\{\gamma\\})$. This shows that the
subschemes ${\mathcal{M}}_{D}(F)$ cover ${\widetilde{\mathcal{M}}}_{D}$. ∎

3.4. Let us analyze ${\widetilde{\mathcal{M}}}_{D}(F)$ for $F=\\{\gamma\\}$.
Setting $u_{\gamma}=0$ in the equation

$\displaystyle u_{\omega}+\prod_{\tau}u_{\tau}^{(\tau||\omega)}=1$

we find that $u_{\omega}=1$ for all $\omega$ incompatible with $\gamma$. Let
$\Pi(\gamma)\subset\Pi$ be the subset of $\kappa\in\Pi$ that are compatible
with $\gamma$. Setting $u_{\omega}=1$ in $R_{\kappa}$ for
$\kappa\in\Pi(\gamma)$, we get

$\displaystyle
R^{\prime}_{\kappa}:=u_{\kappa}+\prod_{\tau\in\Pi(\gamma)}u_{\tau}^{(\tau||\kappa)}-1.$

It follows that the coordinate ring of ${\widetilde{\mathcal{M}}}_{D}(F)$ has
the following presentation

$\displaystyle{\mathbb{C}}[u_{\kappa}\,|\,\kappa\in\Pi(\gamma)]/(R^{\prime}_{\kappa}).$

######  Proposition 3.6.

Suppose $\gamma=(t,j)$ and removing $j$ from $D$ disconnects $D$ into
connected components $D_{1},\dots,D_{s}$. Then we have

$\displaystyle{\widetilde{\mathcal{M}}}_{D}(\\{\gamma\\})={\widetilde{\mathcal{M}}}_{D_{1}}\times{\widetilde{\mathcal{M}}}_{D_{2}}\times\cdots\times{\widetilde{\mathcal{M}}}_{D_{s}}$

and

$\displaystyle{\mathcal{M}}_{D}(\\{\gamma\\})={\mathcal{M}}_{D_{1}}\times{\mathcal{M}}_{D_{2}}\times\cdots\times{\mathcal{M}}_{D_{s}}.$

###### Proof.

Let $\gamma\in\Pi$. Since $\gamma$ defines a facet of the generalized
associahedron for $B$, it follows that the collection of clusters containing
$\gamma$ are connected by mutation, without mutating $\gamma$. It follows that
$\Pi(\gamma)$ is the cluster complex of a cluster algebra of finite type
associated to the disjoint union of $D_{1},D_{2},\dots,D_{s}$. ∎

Note that removing a vertex from a finite type Dynkin diagram produces at most
three components, so in Proposition 3.6 we have $s\leq 3$.

By applying Proposition 3.6 repeatedly, we have the following result.

######  Proposition 3.7.

Any ${\widetilde{\mathcal{M}}}_{D}(F)$ $($resp. ${\mathcal{M}}_{D}(F))$ is a
direct product of ${\widetilde{\mathcal{M}}}_{D^{\prime}}$ $($resp.
${\mathcal{M}}_{D^{\prime}})$ as $D^{\prime}$ varies over a finite set of
Dynkin diagrams obtained by removing some vertices from $D$.

3.5. Recall the bijection $\tau\colon\Pi\to\Pi$ of Section 2.11. We have
automorphisms $\tau\colon{\mathcal{M}}_{D}\to{\mathcal{M}}_{D}$ and
$\tau\colon{\widetilde{\mathcal{M}}}_{D}\to{\widetilde{\mathcal{M}}}_{D}$
induced by $u_{\gamma}\mapsto u_{\tau\gamma}$. The order of the automorphism
$\tau$ is either $h+2$ or $(h+2)/2$, where $h$ is the Coxeter number of $D$;
see for example [7]. It would be interesting to compute: (1) the group of
automorphisms of the variety ${\widetilde{\mathcal{M}}}_{D}$, (2) the group of
automorphisms of the variety ${\mathcal{M}}_{D}$, and (3) the group of
automorphisms of ${\mathcal{M}}_{D}$ that send the positive part
${\mathcal{M}}_{D,>0}$ (defined in Section 8.1) to itself.

In the case $D=A_{n-3}$, we have ${\mathcal{M}}_{D}={\mathcal{M}}_{0,n}$ which
has a natural action of $S_{n}$. The group $S_{n}$ acts transitively on the
connected components of ${\mathcal{M}}_{0,n}({\mathbb{R}})$. The positive part
${\mathcal{M}}_{D,>0}$ is one of the connected component of
${\mathcal{M}}_{0,n}({\mathbb{R}})$ and it is sent to itself by a dihedral
subgroup of $S_{n}$ of order $2$, and the number of connected components of
${\mathcal{M}}_{0,n}({\mathbb{R}})$ is equal to $n!/2n$, see Section 7.2.

3.6. Let $\gamma\in\Pi$. Then as in Proposition 3.6, we can uniquely associate
Dynkin diagrams $D_{1},\dots,D_{s(\gamma)}$ to $\gamma$, where $s\leq 3$.

######  Proposition 3.8.

Suppose that $\gamma\in\Pi$ and $s(\gamma)=1$. Then we have a natural morphism

$\displaystyle{\mathcal{M}}_{D}\longrightarrow{\mathcal{M}}_{D_{1}}.$ (3.7)

The proof of Proposition 3.8 is delayed to Section 4.7. The map of Proposition
3.8 corresponds to “forgetting a marked point” in the case of
${\mathcal{M}}_{A_{n-3}}={\mathcal{M}}_{0,n}$. We expect (3.7) to be a
fibration, similar to the ${\mathcal{M}}_{0,n}$ case.

3.7. Let $D$ be a folding of ${\tilde{D}}$ and let $\Gamma$ and
$\nu\colon{\tilde{\Pi}}\to\Pi$ be as in Section 2.14.

######  Proposition 3.9.

The quotient of ${\mathbb{C}}[u]/I_{{\tilde{D}}}$ by the ideal generated by
the equations $u_{{\tilde{\gamma}}}=u_{g\cdot{\tilde{\gamma}}}$ for
${\tilde{\gamma}}\in{\tilde{\Pi}}$ and $g\in\Gamma$ is canonically isomorphic
to ${\mathbb{C}}[u]/I_{D}$.

###### Proof.

Let
$\nu\colon{\mathbb{C}}\big{[}u_{{\tilde{\gamma}}}\,|\,{\tilde{\gamma}}\in{\tilde{\Pi}}\big{]}\to{\mathbb{C}}[u_{\gamma}\,|\,\gamma\in\Pi]$
be the ring homomorphism given by
$\nu(u_{{\tilde{\gamma}}})\allowbreak=u_{\nu({\tilde{\gamma}})}$. Then
applying Proposition 2.6(1),(2), we have
$\nu(R_{{\tilde{\gamma}}})=R_{\nu({\tilde{\gamma}})}$. The result follows. ∎

Thus ${\widetilde{\mathcal{M}}}_{D}$ can be identified with a closed subscheme
of ${\widetilde{\mathcal{M}}}_{{\tilde{D}}}$ and it is straightforward to see
that ${\mathcal{M}}_{D}$ is the intersection of
${\widetilde{\mathcal{M}}}_{D}\subset{\widetilde{\mathcal{M}}}_{{\tilde{D}}}$
with the open subset
${\mathcal{M}}_{{\tilde{D}}}\subset{\widetilde{\mathcal{M}}}_{{\tilde{D}}}$.

3.8. The significance of the following conjecture is unclear to us. We have
proved it by a direct, elementary calculation for $D=A_{n}$, $n\geq 2$.

######  Conjecture 3.10.

For $D$ not of type $A_{1}$, the ring ${\mathbb{C}}[u^{\pm 1}]/I_{D}$ is
generated by $u_{\gamma}^{-1}$, for $\gamma\in\Pi$.

##  4 $\boldsymbol{{\mathcal{M}}_{D}}$ as a quotient of a cluster variety

In this section we show that ${\mathcal{M}}_{D}$ can be obtained as a quotient
of an open subspace ${\mathring{X}}$ of the cluster variety
${\mathbf{X}}\big{(}{\tilde{B}}\big{)}$ by the action of the cluster
automorphism torus $T$ considered in [23]. An important role is played by
principal and universal coefficients, where
${\tilde{B}}={\tilde{B}}^{\operatorname{prin}}$ or
${\tilde{B}}={\tilde{B}}^{\operatorname{univ}}$. In particular, the defining
relations of ${\mathcal{M}}_{D}$ are obtained from the primitive exchange
relations of ${\mathbf{X}}\big{(}{\tilde{B}}^{\operatorname{univ}}\big{)}$.

4.1. Let ${\tilde{B}}$ be a full rank extended exchange matrix. Let
$T=T\big{(}{\tilde{B}}\big{)}$ be the cluster automorphism group [23] of
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$: this is the group of algebra
automorphisms
$\phi\colon{\mathcal{A}}\big{(}{\tilde{B}}\big{)}\to{\mathcal{A}}\big{(}{\tilde{B}}\big{)}$
such that for each (mutable or frozen) cluster variable $x$, we have
$\phi(x)=\zeta(x)x$ for $\zeta(x)\in{\mathbb{C}}^{\ast}$. Thus, $T$ acts on
any cluster torus of the cluster variety $X\big{(}{\tilde{B}}\big{)}$ by
scaling the coordinates. By [23, Proposition 5.1], we have

$\displaystyle
T=\operatorname{Hom}\big{(}{\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n},{\mathbb{C}}^{\ast}\big{)}.$
(4.1)

Since ${\tilde{B}}$ has full rank, the group $T$ is a (possibly disconnected)
abelian algebraic group of dimension $m$. The character group of $T$ is the
lattice ${\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n}$. By definition, the
torus $T$ acts on each cluster variable by a character, and we denote the
weight of the cluster variable $x\in{\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ by
$\operatorname{wt}(x)\in{\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n}$.

######  Lemma 4.1.

Let ${\tilde{B}}$ have full rank. Then ${\tilde{B}}$ has really full rank if
and only if ${\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n}$ has no torsion,
or equivalently, the group $T$ is connected, and thus a torus of dimension
$m$.

###### Proof.

The rows of ${\tilde{B}}$ span ${\mathbb{Z}}^{n}$ if and only if
${\tilde{B}}{\mathbb{Q}}^{n}\cap{\mathbb{Z}}^{n+m}={\tilde{B}}{\mathbb{Z}}^{n}$
if and only if ${\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n}$ has no
torsion. ∎

Let $X=X\big{(}{\tilde{B}}\big{)}$ be the cluster variety, which is a smooth
affine algebraic variety. Let ${\mathring{X}}\subset X$ be the locus where all
mutable cluster variables are non-vanishing. In terms of rings, we have

$\displaystyle{\mathring{X}}:=\operatorname{Spec}\big{(}{\mathcal{A}}[1/x\,|\,x\text{
is a mutable cluster variable}]\big{)}.$

Thus ${\mathring{X}}$ is a smooth affine subvariety of the initial (or any)
cluster subtorus of $X$, and it follows immediately from the definitions that
the action of $T$ preserves ${\mathring{X}}$, and furthermore the action of
$T$ is free on ${\mathring{X}}$. The geometric invariant theory quotient

$\displaystyle{\mathring{X}}\mathbin{/\mkern-6.0mu/}T:=\operatorname{Spec}\big{(}{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}\big{)}$

is again a smooth affine algebraic variety, and furthermore, there is a
bijection between closed points of ${\mathring{X}}\mathbin{/\mkern-6.0mu/}T$
and $T$-orbits on ${\mathring{X}}$. We thus simply denote
${\mathring{X}}\mathbin{/\mkern-6.0mu/}T$ by ${\mathring{X}}/T$. Explicitly,
the ring ${\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$ consists of all weight
zero Laurent polynomials in cluster variables.

4.2. Let $\tau\colon\Pi\to\Pi$ be the bijection defined by $\tau(t,j)=(t-1,j)$
for $0\leq t\leq r_{j}$. The primitive exchange relations are of the form

$\displaystyle x_{\tau\gamma}x_{\gamma}=M+M^{\prime},$

where $M^{\prime}$ only involves frozen variables. For each primitive exchange
relation, we define the rational function

$\displaystyle f_{\gamma}:=\frac{M}{x_{\tau\gamma}x_{\gamma}}.$

By definition, $f_{\gamma}\in{\mathbb{C}}\big{[}{\mathring{X}}\big{]}$, and it
is easy to see that $f_{\gamma}$ is $T$-invariant. Thus
$f_{\gamma}\in{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$.

######  Theorem 4.2.

Suppose that ${\tilde{B}}$ is a full rank extended exchange matrix, acyclic
and of finite type. Let $X=X\big{(}{\tilde{B}}\big{)}$. Then the quotient
${\mathring{X}}/T$ is a smooth affine variety isomorphic to
${\mathcal{M}}_{D}$, and the isomorphism
${\mathbb{C}}[{\mathcal{M}}_{D}]\cong{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$
is given by $u_{\gamma}\mapsto f_{\gamma}$.

######  Theorem 4.3.

Suppose that ${\tilde{B}}$ is a full rank extended exchange matrix, acyclic
and of finite type. Let $X=X^{\operatorname{prin}}\big{(}{\tilde{B}}\big{)}$
have principal coefficients. Then
${\mathcal{M}}_{D}\cong{\mathring{X}}^{\operatorname{prin}}/T^{\operatorname{prin}}$
is isomorphic to the locus $X^{\operatorname{prin}}(1)\subset
X^{\operatorname{prin}}$, where all initial mutable cluster variables have
been set to $1$. The coordinate ring ${\mathbb{C}}[{\mathcal{M}}_{D}]$ is
isomorphic to the subring of ${\mathbb{C}}(y_{1},y_{2},\dots,y_{n})$ generated
by $F_{\gamma}^{\pm 1}({\mathbf{y}})$ and $y_{i}^{\pm 1}$.

######  Theorem 4.4.

Suppose that ${\tilde{B}}$ is a full rank extended exchange matrix, acyclic
and of finite type. Let $X=X^{\operatorname{univ}}\big{(}{\tilde{B}}\big{)}$
have universal coefficients. Then
${\mathcal{M}}_{D}\cong{\mathring{X}}^{\operatorname{univ}}/T^{\operatorname{univ}}$
is isomorphic to the locus $X^{\operatorname{univ}}(1)\subset
X^{\operatorname{univ}}$, where all mutable cluster variables have been set to
$1$. The isomorphism
${\mathbb{C}}[{\mathcal{M}}_{D}]\cong{\mathbb{C}}[X^{\operatorname{univ}}(1)]$
is given by $u_{\gamma}\mapsto z_{\gamma}$.

4.3. The relations in the following corollary will be discussed in further
detail in Section 10.1.

######  Corollary 4.5.

The ideal $I_{D}$ has a natural set of generators of the form
$U+U^{\prime}-1$, given by the images of all exchange relations of
$X^{\operatorname{univ}}$.

The ideal $I_{D}$ also contains the $|\Pi|-n$ distinguished elements which are
images of $1-F^{\operatorname{univ}}_{\gamma}({\mathbf{z}})$.

4.4. Proof of Theorem 4.4. Recall that the mutable cluster variables of
${\mathcal{A}}^{\operatorname{univ}}$ are denoted $x_{\gamma}$ and the frozen
variables are denoted $z_{\gamma}$, where $\gamma\in\Pi$. Let
$X^{\operatorname{univ}}(1)\subset{\mathring{X}}^{\operatorname{univ}}\subset
X^{\operatorname{univ}}$ be the locus $\\{x_{\gamma}=1\\}$, where all mutable
cluster variables have been set to 1.

By Proposition 2.5, the primitive exchange relations are of the form

$\displaystyle
x_{\tau\gamma}x_{\gamma}=z_{\gamma}S+\prod_{\omega\in\Pi}z_{\omega}^{(\omega||\gamma)},$

where $S$ is a monomial in the mutable cluster variables. So,

$\displaystyle f_{\gamma}=\frac{z_{\gamma}S}{x_{\tau\gamma}x_{\gamma}}$

and thus on $X^{\operatorname{univ}}(1)$ we have
$(f_{\gamma})|_{X^{\operatorname{univ}}(1)}=z_{\gamma}$ and the relation

$\displaystyle
f_{\gamma}+\prod_{\omega\in\Pi}f_{\omega}^{(\omega||\gamma)}=1.$ (4.2)

We will now show that the multiplication map gives an isomorphism

$\displaystyle T^{\operatorname{univ}}\times
X^{\operatorname{univ}}(1)\cong{\mathring{X}}^{\operatorname{univ}},$

or equivalently, every $T^{\operatorname{univ}}$-orbit on
${\mathring{X}}^{\operatorname{univ}}$ intersects $X^{\operatorname{univ}}(1)$
in exactly one point. The character group of $T^{\operatorname{univ}}$ is
naturally isomorphic to ${\mathbb{Z}}^{n+r}/{\tilde{B}}{\mathbb{Z}}^{n}$,
which is a free abelian group of rank $r=|\Pi|$. Thus each cluster variable
$x_{\gamma}$ has a weight (or degree)
$\operatorname{wt}(x_{\gamma})\in{\mathbb{Z}}^{n+r}/{\tilde{B}}{\mathbb{Z}}^{n}$
(see (6.2) for the weight of initial and frozen variables). By Proposition
6.5, the set $\\{\operatorname{wt}(x_{\gamma})\,|\,\gamma\in\Pi\\}$ form a
basis of the lattice ${\mathbb{Z}}^{n+r}/{\tilde{B}}{\mathbb{Z}}^{n}$. Thus we
have a projection ${\mathring{X}}^{\operatorname{univ}}\mapsto
T^{\operatorname{univ}}$ given by sending
$x\in{\mathring{X}}^{\operatorname{univ}}$ to the coordinates
$(x_{\gamma})_{\gamma\in\Pi}$, and the fiber of this projection is
$X^{\operatorname{univ}}(1)$. This is an inverse to the multiplication map
$T^{\operatorname{univ}}\times
X^{\operatorname{univ}}(1)\to{\mathring{X}}^{\operatorname{univ}}$, and we
deduce that $T^{\operatorname{univ}}\times
X^{\operatorname{univ}}(1)\cong{\mathring{X}}^{\operatorname{univ}}$.

We conclude that
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}\\!\\!\cong{\mathbb{C}}[X^{\operatorname{univ}}(1)]$.
Now, any $T^{\operatorname{univ}}$-invariant function in
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}$ is a linear
combination of $T^{\operatorname{univ}}$-invariant Laurent monomials in
mutable and frozen variables. Each such Laurent monomial restricts to a
Laurent monomial in the $z_{\gamma}$-s on
${\mathbb{C}}[X^{\operatorname{univ}}(1)]$. It follows that the functions
$f_{\gamma}$ and their inverses generate
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}$,
and by (4.2) satisfy the same relations that
$u_{\gamma}\in{\mathbb{C}}[{\mathcal{M}}_{D}]$ satisfy. Finally, we check that
the generators $f_{\gamma}$ do not satisfy any further relations. Suppose we
have a polynomial identity $p(f_{\gamma})=0$ inside
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}$.
The equality $p(f_{\gamma})=0$ is equivalent to an equality
$q(x_{\gamma},z_{\gamma})=0$ inside
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}$, where
$q(x_{\gamma},z_{\gamma})$ is a Laurent polynomial. We claim that the
primitive exchange relations allow us to eliminate all the non-initial cluster
variables, i.e.,

$\displaystyle
q(x_{\gamma},z_{\gamma})=r(x_{1},x_{2},\dots,x_{n},z_{\gamma})\mod\mbox{ ideal
generated by primitive exchange relations},$

where $r(x_{1},x_{2},\dots,x_{n},z_{\gamma})$ is a Laurent polynomial and the
ideal is taken inside
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}$.
To see this, first note that $\deg(x_{1}),\dots,\deg(x_{n})$ and
$\deg(z_{\gamma})$, $\gamma\in\Pi$ together span ${\mathbb{Z}}^{n+r}$, and
thus we can always multiply $q(x_{\gamma},z_{\gamma})$ by a
$T^{\operatorname{univ}}$-invariant monomial so that the denominator involves
only initial $x_{i}$ and the $z_{\gamma}$. Next, we have

$\displaystyle x_{\gamma}=\frac{M+M^{\prime}}{x_{\tau\gamma}}-x_{\gamma}R,$

where $R=\frac{M+M^{\prime}}{x_{\tau\gamma}x_{\gamma}}-1$ is a primitive
exchange relation (divided by $x_{\tau\gamma}x_{\gamma}$). This allows us
(modulo the ideal) to replace $x_{\gamma}$ by an expression involving
$x_{\tau\gamma}$ and $M+M^{\prime}$. If $\gamma=(t,j)$, the mutable cluster
variables that appear in $M+M^{\prime}$ are either of the form $(t-1,i)$ or of
the form $(t,i)$, where $i\to j$ (see Proposition 2.5). It follows that
$x_{\gamma}$ will not appear again when this process is repeated. This proves
our claim.

But $r(x_{1},x_{2},\dots,x_{n},z_{\gamma})=0$ as an element of
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}\subset{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}\subset{\mathbb{C}}(X^{\operatorname{univ}})$
only if the polynomial $r$ is 0, since $x_{1},x_{2},\dots,x_{n},z_{\gamma}$
are algebraically independent. We conclude that $p(f_{\gamma})$ lies in the
ideal generated by primitive exchange relations. Thus the ideal of relations
satisfied by the $f_{\gamma}$ is generated by (4.2). We thus have an
isomorphism of rings

$\displaystyle{\mathbb{C}}[{\mathcal{M}}_{D}]\longrightarrow{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}},\qquad
u_{\gamma}\longmapsto f_{\gamma}$

and an isomorphism of varieties
$X^{\operatorname{univ}}(1)\cong{\mathcal{M}}_{D}$.

4.5. Proof of Theorem 4.2. By the defining property of universal coefficients,
we have a homomorphism of rings
$\phi\colon{\mathcal{A}}^{\operatorname{univ}}\to{\mathcal{A}}={\mathcal{A}}\big{(}{\tilde{B}}\big{)}$
such that $\phi(x_{\gamma}^{\operatorname{univ}})=x_{\gamma}$ and
$\phi(z_{\gamma})$ is a Laurent monomial in the frozen variables
$x_{n+1},\dots,x_{n+m}$ of ${\mathcal{A}}$. The homomorphism $\phi$ may not be
surjective, for example this would be the case if ${\tilde{B}}$ has rows equal
to 0, or rows that are repeated. However, the image
${\mathcal{A}}^{\prime}:=\phi({\mathcal{A}}^{\operatorname{univ}})$ is itself
a cluster algebra: it is generated by $\phi(x_{\gamma}^{\operatorname{univ}})$
and the monomials $\phi(z_{\gamma})$. The monomials $\phi(z_{\gamma})$ and
their inverses generate a Laurent polynomial subring
$S\subset{\mathbb{C}}\big{[}x_{n+1}^{\pm 1},\dots,x_{n+m}^{\pm 1}\big{]}$
which is the coefficient ring of ${\mathcal{A}}^{\prime}$. For any monomial
$M$ in $x_{n+1},\dots,x_{n+m}$, we can find $t\in T$ such that $t\cdot M$ is a
Laurent monomial in $S$. Thus, we have
${\mathcal{A}}^{T}={\mathcal{A}}^{\prime T^{\prime}}$. Since ${\mathcal{A}}$
has full rank, the quotient $X/T=X^{\prime}/T^{\prime}$ has dimension $n$, and
${\mathcal{A}}^{\prime}$ also has full rank.

Replacing ${\mathcal{A}}$ by ${\mathcal{A}}^{\prime}$, we now assume that
$\phi\colon{\mathcal{A}}^{\operatorname{univ}}\to{\mathcal{A}}$ is surjective,
and thus we have a closed immersion
$\psi\colon{\mathring{X}}\to{\mathring{X}}^{\operatorname{univ}}$. The
monomials $\phi(z_{\gamma})$, $\gamma\in\Pi$ together define a surjective
linear map $C^{\prime}\colon{\mathbb{Z}}^{r}\to{\mathbb{Z}}^{m}$. Extending by
the identity in the first $n$ coordinates, we get a linear map
${\mathbb{Z}}^{n+r}\to{\mathbb{Z}}^{n+m}$, represented by a matrix $C$
satisfying $C{\tilde{B}}^{\operatorname{univ}}={\tilde{B}}$. Suppose that
$t\in
T=\operatorname{Hom}\big{(}{\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n},{\mathbb{C}}^{\ast}\big{)}$.
Then composing $t$ with $C$, we get an element $t^{\prime}\in
T^{\operatorname{univ}}=\operatorname{Hom}\big{(}{\mathbb{Z}}^{n+r}/{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n},{\mathbb{C}}^{\ast}\big{)}$.
Since $C$ is surjective, the induced map $\psi\colon T\to
T^{\operatorname{univ}}$ is injective and thus the inclusion of a subgroup.

We need to show that
$\phi\colon{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}\to{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$
is an isomorphism. For surjectivity, suppose that
$f\in{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$. Then we may assume that
$f$ is a Laurent monomial in mutable and frozen variables. Let
$g\in{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}$ be such
that $\phi(g)=f$. It is immediate that $g$ is invariant under $T$, i.e., the
weight
$\operatorname{wt}(g)\in{\mathbb{Z}}^{n+r}/{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}$
of $g$ satisfies $C\operatorname{wt}(g)=0$ in
${\mathbb{Z}}^{n+m}/{\tilde{B}}{\mathbb{Z}}^{n}$. Thus, there exists
${\mathbf{u}}\in{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}$ such that
$C(\operatorname{wt}(g)+{\mathbf{u}})=0\in{\mathbb{Z}}^{n+m}$. The matrix $C$
is the identity in the first $n$-coordinates, so the first $n$ coordinates of
$\operatorname{wt}(g)+{\mathbf{u}}$ is 0. Let
$M=\prod_{\gamma\in\Pi}z_{\gamma}^{-a_{\gamma}}$, where $(a_{\gamma})$ are the
last $m$ coordinates of $\operatorname{wt}(g)$. Then by construction we have
$\operatorname{wt}(gM)+{\mathbf{u}}=0$, i.e.,
$\operatorname{wt}(gM)=0\in{\mathbb{Z}}^{n+r}/{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}$.
Furthermore, $\phi(M)=1$ and thus
$gM\in{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}$
satisfies $\phi(gM)=f$, proving surjectivity.

For injectivity, suppose that $\phi(g)=0$, where
$g\in{\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{univ}}\big{]}^{T^{\operatorname{univ}}}$
is nonzero. We have already shown that
$X^{\operatorname{univ}}/T^{\operatorname{univ}}$ is an irreducible affine
variety in Section 4.4. The affine variety
$\operatorname{Spec}\big{(}{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}\big{)}$
is thus identified with a subvariety of
$X^{\operatorname{univ}}/T^{\operatorname{univ}}$ of lower dimension. But this
is impossible, since
$\dim(X/T)=n=\dim(X^{\operatorname{univ}}/T^{\operatorname{univ}})$.

The isomorphism
${\mathbb{C}}[{\mathcal{M}}_{D}]\cong{\mathbb{C}}\big{[}{\mathring{X}}\big{]}^{T}$
given by $u_{\gamma}\mapsto f_{\gamma}$ now follows from Section 4.4.

4.6. Proof of Theorem 4.3. The group
${\mathbb{Z}}^{2n}/{\tilde{B}}^{\operatorname{prin}}{\mathbb{Z}}^{n}$ can be
naturally identified with the subgroup
${\mathbb{Z}}^{n}={\mathbb{Z}}^{[1,n]}\subset{\mathbb{Z}}^{[1,2n]}={\mathbb{Z}}^{2n}$
consisting of vectors which vanish in the last $n$-coordinates. Under this
identification, the torus $T^{\operatorname{prin}}$ has character lattice
${\mathbb{Z}}^{n}$, and the grading on ${\mathcal{A}}^{\operatorname{prin}}$
is given by (2.2). By Theorem 4.2, we have
${\mathcal{M}}_{D}\cong{\mathring{X}}^{\operatorname{prin}}/T^{\operatorname{prin}}$.
It follows from $\operatorname{wt}(x_{i})=e_{i}$ that
${\mathring{X}}^{\operatorname{prin}}/T^{\operatorname{prin}}$ is identified
with the subvariety
${\mathring{X}}^{\operatorname{prin}}(1)\subset{\mathring{X}}^{\operatorname{prin}}$,
where all initial cluster variables are set to 1.

The function $f^{\operatorname{prin}}_{\gamma}$ on
${\mathring{X}}^{\operatorname{prin}}(1)$ restricts to the rational function
in $y_{1},\dots,y_{n}$ given by (see [31, Theorem 1.5])

$\displaystyle f_{(t,j)}({\mathbf{y}})=\begin{cases}\frac{\prod_{i\to
j}F_{(t,i)}^{-a_{ij}}({\mathbf{y}})\prod_{j\to
i}F_{(t-1,i)}^{-a_{ij}}({\mathbf{y}})}{F_{(t-1,j)}({\mathbf{y}})F_{(t,j)}({\mathbf{y}})},&t\neq
0,\\\\[6.45831pt] \frac{y_{j}\prod_{i\to
j}F_{(t,i)}^{-a_{ij}}({\mathbf{y}})\prod_{j\to
i}F_{(t-1,i)}^{-a_{ij}}({\mathbf{y}})}{F_{(t-1,j)}({\mathbf{y}})F_{(t,j)}({\mathbf{y}})},&t=0\end{cases}$
(4.3)

for $\gamma=(t,j)$ with $0\leq t\leq r_{j}$. By the following result,
${\mathbb{C}}\big{[}{\mathring{X}}^{\operatorname{prin}}(1)]\simeq{\mathbb{C}}[{\mathcal{M}}_{D}\big{]}$
is isomorphic to the subring of ${\mathbb{C}}(y_{1},y_{2},\dots,y_{n})$
generated by $F_{\gamma}^{\pm 1}({\mathbf{y}})$ and $y_{i}^{\pm 1}$.

######  Proposition 4.6.

The rational functions $\\{f_{\gamma}({\mathbf{y}})\,|\,\gamma\in\Pi\\}$ and
$\\{y_{1},\dots,y_{n}\\}\cup\\{F_{\gamma}({\mathbf{y}})\,|\,\gamma\in\Pi^{+}\\}$
are related by an invertible monomial transformation.

The proof of Proposition 4.6 is delayed until Section 6.4.

4.7. Proof of Proposition 3.8. Using $\tau$, let us assume that $\gamma=(0,j)$
so that $x_{\gamma}=x_{j}$ is an initial mutable cluster variable. Let
${\tilde{B}}$ be full rank of type $D$ and let
${\mathcal{A}}\big{(}{\tilde{B}}_{j}\big{)}$ denote the cluster algebra of
type $D_{1}$ that is obtained by freezing the variable $x_{j}$ in
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$. The extended exchange matrix
${\tilde{B}}_{j}$ is obtained from ${\tilde{B}}$ by removing the $j$-th row
and we have
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}\big{[}x_{j}^{-1}\big{]}={\mathcal{A}}\big{(}{\tilde{B}}_{j}\big{)}$.
Thus
${\mathring{{\mathcal{A}}}}\big{(}{\tilde{B}}_{j}\big{)}\subset{\mathring{{\mathcal{A}}}}\big{(}{\tilde{B}}\big{)}$.
The action of the cluster automorphism group $T\big{(}{\tilde{B}}\big{)}$
extends to an action on ${\mathcal{A}}\big{(}{\tilde{B}}_{j}\big{)}$ and we
can identify $T\big{(}{\tilde{B}}\big{)}$ with a subgroup of
$T\big{(}{\tilde{B}}_{j}\big{)}$. The morphism
${\mathcal{M}}_{D}\to{\mathcal{M}}_{D_{1}}$ corresponds to the inclusion of
rings
${\mathring{{\mathcal{A}}}}\big{(}{\tilde{B}}_{j}\big{)}^{T({\tilde{B}}_{j})}\subset{\mathring{{\mathcal{A}}}}\big{(}{\tilde{B}}_{j}\big{)}^{T({\tilde{B}})}\subset{\mathring{{\mathcal{A}}}}\big{(}{\tilde{B}}\big{)}^{T({\tilde{B}})}$.

##  5 $\boldsymbol{{\widetilde{\mathcal{M}}}_{D}}$ as an affine open in a
projective toric variety

In this section, we show that the partial compactification
${\widetilde{\mathcal{M}}}_{D}$ is an affine open subspace of the projective
toric variety $X_{{\mathcal{N}}(B^{\vee})}$ associated to the cluster fan of
$B^{\vee}$. The stratification (Proposition 3.5) of
${\widetilde{\mathcal{M}}}_{D}$ is inherited from the natural stratification
of $X_{{\mathcal{N}}(B^{\vee})}$ by torus orbits. Our approach follows that of
[3].

5.1. Let ${\mathbb{C}}({\mathbf{y}})={\mathbb{C}}(y_{1},\dots,y_{n})$ denote
the field of rational functions. Recall that for $\gamma\in\Pi$, we have
defined $f_{\gamma}({\mathbf{y}})\in{\mathbb{C}}({\mathbf{y}})$ in (4.3). By
the proof of Theorem 4.3, ${\mathbb{C}}[{\mathcal{M}}_{D}]$ is isomorphic to
the subring of ${\mathbb{C}}({\mathbf{y}})$ generated by
$f_{\gamma}({\mathbf{y}})^{\pm 1}$. Define
$R_{B}\subset{\mathbb{C}}({\mathbf{y}})$ to be the subring generated by
$\\{f_{\gamma}({\mathbf{y}})\,|\,\gamma\in\Pi\\}$. Some examples of
$f_{\gamma}({\mathbf{y}})$ are computed in Examples 6.7 and 6.8.

######  Theorem 5.1.

The coordinate ring ${\mathbb{C}}\big{[}{\widetilde{\mathcal{M}}}_{D}\big{]}$
is isomorphic to $R_{B}$.

###### Proof.

There is a surjective ring homomorphism
$\varphi^{\prime}\colon{\mathbb{C}}[u]\to R_{B}$ given by $u_{\gamma}\mapsto
f_{\gamma}({\mathbf{y}})$. We already know that the kernel $K$ of $\varphi$
contains the ideal $I_{D}\subset{\mathbb{C}}[u]$. We need to show that the
homomorphism $\varphi\colon{\mathbb{C}}[u]/I_{D}\to R_{B}$ is an isomorphism.
From Theorem 4.3, we know this holds after inverting the
$\\{u_{\gamma}\,|\,\gamma\in\Pi\\}$ and $\\{f_{\gamma}\,|\,\gamma\in\Pi\\}$.

By definition, ${\widetilde{\mathcal{M}}}_{D}(\\{\gamma\\})$ is cut out of
${\widetilde{\mathcal{M}}}_{D}$ by the ideal $(u_{\gamma})$. Thus the ring
${\mathbb{C}}[u]/(I_{D}+(u_{\gamma}))$ is isomorphic to
${\mathbb{C}}\big{[}{\widetilde{\mathcal{M}}}_{D}(\\{\gamma\\})\big{]}$, and
by Proposition 3.6, we have
${\widetilde{\mathcal{M}}}_{D}(\\{\gamma\\})={\widetilde{\mathcal{M}}}_{D_{1}}\times{\widetilde{\mathcal{M}}}_{D_{2}}\times\cdots\allowbreak\times{\widetilde{\mathcal{M}}}_{D_{s}}$
for some Dynkin diagrams $D_{i}$. By induction on the rank of $D$, we have
that $\varphi_{i}\colon{\mathbb{C}}[u]/\allowbreak(I_{D}+(u_{\gamma}))\to
R_{B}/(f_{\gamma}({\mathbf{y}}))$ is an isomorphism. Applying Lemma A.1, we
conclude that $\varphi$ itself is an isomorphism. ∎

5.2. We give another description of $R_{B}\subset{\mathbb{C}}({\mathbf{y}})$.
Let
$R({\mathbf{y}})=P({\mathbf{y}})/Q({\mathbf{y}})\in{\mathbb{C}}({\mathbf{y}})$
be a rational function such that
$P({\mathbf{y}}),Q({\mathbf{y}})\in{\mathbb{Z}}[{\mathbf{y}}]$ have positive
integer coefficients. Then $\operatorname{Trop}(R({\mathbf{y}}))$ is the
piecewise-linear function on ${\mathbb{R}}^{n}$ given by the formal
substitution

$\displaystyle y_{i}\mapsto Y_{i},\qquad(+,\times,\div)\mapsto(\min,+,-).$

For example,
$\operatorname{Trop}\big{(}\big{(}3y_{1}^{2}y_{2}+y_{2}^{2}\big{)}/(y_{2}+6y_{3})\big{)}=\min(2Y_{1}+Y_{2},2Y_{2})-\min(Y_{2},Y_{3})$.
Note that the coefficients are unimportant since, for example,
$\operatorname{Trop}(2y)=\operatorname{Trop}(y+y)=\min(Y,Y)=Y$.

The domains of linearity of the piecewise-linear function
$L({\mathbf{Y}})=\operatorname{Trop}(R({\mathbf{y}}))$ define the structure of
a complete fan on ${\mathbb{R}}^{n}$. A piecewise-linear function
$L({\mathbf{Y}})\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ is called _nonnegative_
, denoted $L({\mathbf{Y}})\geq 0$, if it takes nonnegative values on
${\mathbb{R}}^{n}$.

######  Proposition 5.2.

The ring $R_{B}$ is equal to the subring of ${\mathbb{C}}({\mathbf{y}})$
generated by rational functions $R({\mathbf{y}})$ satisfying

  1. $(1)$

$R({\mathbf{y}})=\prod_{i=1}^{n}y_{i}^{a_{i}}\prod_{\gamma\in\Pi}F_{\gamma}({\mathbf{y}})^{a_{\gamma}}$
is a Laurent monomial in $y_{i}$ and $F_{\gamma}({\mathbf{y}})$,

  2. $(2)$

$\operatorname{Trop}(R({\mathbf{y}}))$ is nonnegative.

###### Proof.

Let $R({\mathbf{y}})$ be a Laurent monomial in
$\\{y_{i},F_{\gamma}({\mathbf{y}})\\}$. By Theorem 2.4, the domains of
linearity of the function
$L({\mathbf{Y}})=\operatorname{Trop}(R({\mathbf{y}}))$ is a coarsening of the
negative of the cluster fan $-{\mathcal{N}}(B^{\vee})$. Thus $L({\mathbf{Y}})$
is uniquely determined by $b_{\gamma}=L(-{\mathbf{g}}_{\gamma})$ as $\gamma$
varies over $\Pi$, and ${\mathbf{g}}_{\gamma}$ denotes a
${\mathbf{g}}$-vector. As in the proof of Proposition 4.6, we have
$R({\mathbf{y}})=\prod_{\gamma}f_{\gamma}({\mathbf{y}})^{-b_{\tau\gamma}}$.
The condition $L({\mathbf{Y}})\geq 0$ is equivalent to $b_{\gamma}\geq 0$ for
all $\gamma\in\Pi$. Thus the subring of rational functions $R({\mathbf{y}})$
satisfying (1) and (2) is exactly the subring $R_{D}$. ∎

5.3. The Laurent polynomial ring ${\mathbb{C}}\big{[}y_{1}^{\pm
1},\dots,y_{n}^{\pm 1}\big{]}$ is the coordinate ring of an $n$-dimensional
torus $T_{\mathbf{y}}$. Recall that
$F({\mathbf{y}})=\prod_{\gamma}F_{\gamma}({\mathbf{y}})$. The following result
is an application of [3, Section 10].

######  Theorem 5.3.

The affine scheme ${\widetilde{\mathcal{M}}}_{D}$ is isomorphic to the affine
open $\\{F({\mathbf{y}})\neq 0\\}$ in the projective toric variety
$X_{{\mathcal{N}}(B^{\vee})}$ associated to the complete fan
${\mathcal{N}}\big{(}B^{\vee}\big{)}$. The subvariety
${\mathcal{M}}_{D}\subset{\widetilde{\mathcal{M}}}_{D}$ is identified with the
intersection of $\\{F({\mathbf{y}})\neq 0\\}$ with the open torus orbit
$T_{\mathbf{y}}$ in $X_{{\mathcal{N}}(B^{\vee})}$.

###### Proof.

For any ${\mathbf{g}}\in{\mathbb{R}}^{n}$, the quantity
$\operatorname{Trop}(F({\mathbf{y}}))({\mathbf{g}})$ is equal to the minimum
value that the linear function
${\mathbf{Y}}\mapsto{\mathbf{Y}}\cdot{\mathbf{g}}$ takes on the Newton
polytope $P$ of $F({\mathbf{y}})$. Thus by Theorem 2.4, the outer normal fan
of $P$ is equal to ${\mathcal{N}}\big{(}B^{\vee}\big{)}$. Recall that a
lattice polytope $Q$ is called _very ample_ if for sufficiently large integers
$r>0$, every lattice point in $rQ$ is a sum of $r$ (not necessarily distinct)
lattice points in $Q$. For any lattice polytope $Q$, it is known that some
integer dilation $cQ$ is very ample. So let $c\in{\mathbb{Z}}_{>0}$ be such
that $cP$ is very ample and let
$\\{{\mathbf{v}}_{1},\dots,{\mathbf{v}}_{k}\\}=cP\cap{\mathbb{Z}}^{n}$ be the
set of all lattice points in $cP$. For ${\mathbf{v}}\in{\mathbb{Z}}^{n}$, let
${\mathbf{y}}^{\mathbf{v}}$ be the monomial with exponent vector
${\mathbf{v}}$. Then $X_{{\mathcal{N}}(B^{\vee})}$ can be explicitly realized
as the closure of the set of points

$\displaystyle\Big{\\{}\big{[}{\mathbf{y}}^{{\mathbf{v}}_{1}}:\cdots:{\mathbf{y}}^{{\mathbf{v}}_{k}}\big{]}\in{\mathbb{P}}^{k-1}\,|\,{\mathbf{y}}\in
T_{\mathbf{y}}\Big{\\}}$

inside the projective space ${\mathbb{P}}^{k-1}$. The polynomial
$F({\mathbf{y}})^{c}$ can be identified with a hyperplane section of
$X_{{\mathcal{N}}(B^{\vee})}$ in this projective embedding, and the affine
open $V:=\\{F({\mathbf{y}})\neq 0\\}$ is the complement of this hyperplane
section. The coordinate ring ${\mathbb{C}}[V]$ is generated by the functions
${\mathbf{y}}^{{\mathbf{v}}_{i}}/F({\mathbf{y}})^{c}$, $i=1,2,\dots,k$. Since
$\operatorname{Trop}\big{(}{\mathbf{y}}^{{\mathbf{v}}_{i}}/F({\mathbf{y}})^{c}\big{)}$
is nonnegative, by Proposition 5.2, we have ${\mathbb{C}}[V]\subset R_{B}$. It
is also not hard to see that $f_{\gamma}({\mathbf{y}})\in{\mathbb{C}}[V]$ (see
[3, Section 10]), and we have ${\mathbb{C}}[V]=R_{B}$ as subrings of
${\mathbb{C}}({\mathbf{y}})$. The theorem now follows from Theorem 5.1. ∎

######  Question 5.4.

Is $P$, the Newton polytope of
$F({\mathbf{y}})=\prod_{\gamma}F_{\gamma}({\mathbf{y}})$, very ample? Is $P$
normal?

######  Question 5.5.

Is the polynomial $F({\mathbf{y}})$ saturated?

######  Question 5.6.

Is every lattice point in $P$ a sum of lattice points in $P_{\gamma}$?

Fei [14] has shown that $F_{\gamma}({\mathbf{y}})$ is saturated in the simply-
laced case (and in more general situations). Thus Questions 5.5 and 5.6 are
equivalent in that case.

5.4. Proof of Theorem 3.3. By Theorem 2.3, the fan
${\mathcal{N}}\big{(}B^{\vee}\big{)}$ is a smooth, simplicial, polytopal,
complete fan. Thus $X_{{\mathcal{N}}(B^{\vee})}$ is a smooth projective toric
variety and the torus-orbit closure stratification of
$X_{{\mathcal{N}}(B^{\vee})}$ is simple normal-crossing.

##  6 Properties of $\boldsymbol{F}$-polynomials

We establish some technical properties of $F_{\gamma}({\mathbf{y}})$ and
$F^{\operatorname{univ}}_{\gamma}({\mathbf{z}})$, following the approach of
[8]. The statements are first established in the case of simply-laced $D$; the
multiply-laced case follows from folding. Another closely related approach is
that of [26], which would presumably avoid folding.

A key technical result is Theorem 6.6 which gives the values of the
tropicalization $\operatorname{Trop}(f_{\gamma}({\mathbf{y}}))$ on a (negated)
${\mathbf{g}}$-vector.

In this section, we will assume that $D$ is a finite type Dynkin diagram whose
underlying tree has been given an orientation, and we let $B$ denote the
corresponding exchange matrix. Recall that $D^{\vee}$ denotes the dual Dynkin
diagram, and we let $B^{\vee}$ denote the exchange matrix of type $D^{\vee}$,
satisfying the condition: $B_{ij}>0$ if and only if $B^{\vee}_{ij}>0$. Recall
that we write $i\to j$ if $B_{ij}>0$.

6.1. Let $B$ be the exchange matrix corresponding to the oriented Dynkin
diagram $D$. Let ${\mathbb{R}}^{\Pi}$ be the vector space with basis indexed
by $\Pi$, and write $(p_{\gamma})_{\gamma\in\Pi}$ for a typical vector in
${\mathbb{R}}^{\Pi}$. Define $\Pi^{+}:=\\{(s,i)\,|\,1\leq s\leq
r_{i}\\}\subset\Pi$ and let ${\mathbf{c}}=(c_{\gamma})_{\gamma\in\Pi^{+}}$
denote a typical vector in ${\mathbb{R}}^{\Pi^{+}}$. Following [8], we
consider the _${\mathbf{c}}$ -deformed mesh relations_

$\displaystyle p_{(t-1,j)}+p_{(t,j)}=c_{(t,j)}+\sum_{i\to
j}|B_{ij}|p_{(t,i)}+\sum_{j\to i}|B_{ij}|p_{(t-1,i)},$ (6.1)

where $(t,j)\in\Pi^{+}$. (Compare with (2.3), and note that if $i\to j$ then
$B_{ij}>0$, but if $j\to i$ then $B_{ij}<0$.) If ${\mathbf{c}}=0$, we call
(6.1) the ${\mathbf{0}}$-mesh relations.

For ${\mathbf{c}}=(c_{\gamma})\in{\mathbb{R}}^{\Pi^{+}}$, we let
${\mathbb{E}}_{\mathbf{c}}\subset{\mathbb{R}}^{\Pi}$ denote the solutions to
(6.1), and let
${\mathbb{U}}_{\mathbf{c}}:={\mathbb{E}}_{\mathbf{c}}\cap{\mathbb{R}}_{\geq
0}^{\Pi}$ denote the intersection of ${\mathbb{E}}_{\mathbf{c}}$ with the
positive orthant. Let $\pi\colon{\mathbb{R}}^{\Pi}\to{\mathbb{R}}^{n}$ denote
the projection onto the coordinates $p_{\gamma}$, where $\gamma$ varies over
$\\{(r_{i},i)\,|\,i=1,2,\dots,n\\}$. (Up to the action of $\tau^{-1}$, this is
the same as projection onto the initial cluster variables.)

We use the notation ${\mathbb{U}}(D)_{\mathbf{c}}$ and
${\mathbb{E}}(D)_{\mathbf{c}}$ (resp.
${\mathbb{U}}\big{(}D^{\vee}\big{)}_{\mathbf{c}}$ and
${\mathbb{E}}\big{(}D^{\vee}\big{)}_{\mathbf{c}}$) to denote these objects for
$B$ or $D$ (resp. $B^{\vee}$ or $D^{\vee}$). In the following, $e_{\gamma}$
denotes the unit basis vector in ${\mathbb{R}}_{>0}^{\Pi^{+}}$.

######  Theorem 6.1.

  1. $1.$

If ${\mathbf{c}}=(c_{\gamma})\in{\mathbb{R}}_{>0}^{\Pi^{+}}$, then the normal
fan of $\pi({\mathbb{U}}(D)_{{\mathbf{c}}})$ is equal to ${\mathcal{N}}(B)$.
If $(c_{\gamma})\in{\mathbb{R}}_{\geq 0}^{\Pi^{+}}$, then the normal fan of
$\pi({\mathbb{U}}(D)_{{\mathbf{c}}})$ is a coarsening of ${\mathcal{N}}(B)$.

  2. $2.$

For $\gamma\in\Pi^{+}$, the polytope
${\mathbb{U}}\big{(}D^{\vee}\big{)}_{e_{\gamma}}$ is the Newton polytope of
$F^{\operatorname{univ}}_{\gamma}({\mathbf{z}})$.

  3. $3.$

For $\gamma\in\Pi^{+}$, the polytope
$\pi\big{(}{\mathbb{U}}\big{(}D^{\vee}\big{)}_{e_{\gamma}}\big{)}$ is the
Newton polytope of $F_{\gamma}({\mathbf{y}})$.

###### Proof of Theorem 2.4.

By Theorem 6.1(3) the Newton polytope $P_{\gamma}$ of
$F_{\gamma}({\mathbf{y}})$ is
$\pi\big{(}{\mathbb{U}}\big{(}D^{\vee}\big{)}_{e_{\gamma}}\big{)}$. The Newton
polytope $P$ of $\prod_{\gamma}F_{\gamma}({\mathbf{y}})$ is the Minkowski sum
of the $P_{\gamma}$, and by Theorem 6.1(1), we conclude that $P$ is a
generalized associahedron. ∎

We let ${\mathbf{g}}^{\vee}_{\gamma}$ denote the ${\mathbf{g}}$-vector for
$B^{\vee}$ indexed by the element of $\Pi(B^{\vee})$ corresponding to $\gamma$
under the bijection of Section 2.5.

6.2. Proof of Theorem 6.1. For $D$ simply-laced, we have $D=D^{\vee}$ and
Theorem 6.1 is proven in [8]. We now prove it for multiply-laced $D$ via
folding.

Let $D$ be a folding of ${\tilde{D}}$ with folding group $\Gamma$, and
$\nu\colon{\tilde{\Pi}}\to\Pi$ the quotient map on cluster variables from
Section 2.14. Define
$\nu\colon{\mathbb{R}}^{{\tilde{\Pi}}}\to{\mathbb{R}}^{\Pi}$ by
$\nu(e_{\tilde{\gamma}})=e_{\nu({\tilde{\gamma}})}$, and
$\nu^{\vee}\colon{\mathbb{R}}^{{\tilde{\Pi}}}\to{\mathbb{R}}^{\Pi}$ by
$\nu^{\vee}(e_{\tilde{\gamma}})=\frac{1}{|\nu^{-1}(\nu({\tilde{\gamma}}))|}e_{\nu({\tilde{\gamma}})}$.
(The finite set $\nu^{-1}(\gamma)$ has cardinality one, two, or three.)
Similarly, we have
$\nu,\nu^{\vee}\colon{\mathbb{R}}^{{\tilde{\Pi}}^{+}}\to{\mathbb{R}}^{\Pi^{+}}$.

######  Lemma 6.2.

If
$(p_{{\tilde{\gamma}}})_{{\tilde{\gamma}}\in{\tilde{\Pi}}}\in{\mathbb{E}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}$
then
$\nu^{\vee}(p_{{\tilde{\gamma}}})\in{\mathbb{E}}(D)_{\nu^{\vee}({\tilde{\mathbf{c}}})}$
and
$\nu(p_{{\tilde{\gamma}}})\in{\mathbb{E}}\big{(}D^{\vee}\big{)}_{\nu({\tilde{\mathbf{c}}})}$.

The ${\mathbf{g}}$-vectors for ${\tilde{D}}$ are solutions to the $0$-mesh
relations in the following sense: for each $i=1,2,\dots,n$, the $i$-th
coordinates of ${\mathbf{g}}_{\gamma}$ give a vector ${\mathbf{g}}^{(i)}$ that
belongs to ${\mathbb{E}}_{0}$. This follows from [19, relation (6.13)], noting
that the sign-coherence conjecture [19, Conjecture 6.13] holds in our case.

The following follows from Lemma 6.2 and Proposition 2.6(3). (The appearance
of $\nu^{\vee}$ seems to contradict Lemma 6.2, but it is actually correct: the
$\nu^{\vee}$ in Lemma 6.2 acts on ${\mathbb{R}}^{|{\tilde{\Pi}}|}$ while the
$\nu^{\vee}$ below acts only on ${\mathbb{R}}^{{\tilde{I}}}$.)

######  Proposition 6.3.

We have
${\mathbf{g}}^{\vee}_{\gamma}=\nu^{\vee}\Big{(}\sum_{{\tilde{\gamma}}\in\nu^{-1}(\gamma)}{\mathbf{g}}_{\tilde{\gamma}}\Big{)}$.

We say that
$(p_{{\tilde{\gamma}}})_{{\tilde{\gamma}}\in{\tilde{\Pi}}}\in{\mathbb{R}}^{{\tilde{\Pi}}}$
is $\Gamma$-invariant and write
$(p_{{\tilde{\gamma}}})_{{\tilde{\gamma}}\in{\tilde{\Pi}}}\in\big{(}{\mathbb{R}}^{{\tilde{\Pi}}}\big{)}^{\Gamma}$
if for all $g\in\Gamma$, we have
$p_{{\tilde{\gamma}}}=p_{g\cdot{\tilde{\gamma}}}$. Similarly, we define
$\Gamma$-invariants
${\tilde{\mathbf{c}}}\in\big{(}{\mathbb{R}}^{{\tilde{\Pi}}^{+}}\big{)}^{\Gamma}$.
The following result follows from Lemma 6.2.

######  Proposition 6.4.

  1. $1.$

Suppose that
${\tilde{\mathbf{c}}}\in\big{(}{\mathbb{R}}^{{\tilde{\Pi}}^{+}}\big{)}^{\Gamma}$.
Then the linear map $\nu^{\vee}$ $($resp. $\nu)$ is a bijection between
${\mathbb{E}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\cap\big{(}{\mathbb{R}}^{{\tilde{\Pi}}}\big{)}^{\Gamma}$
and ${\mathbb{E}}(D)_{\nu^{\vee}({\tilde{\mathbf{c}}})}$ $\big{(}$resp.
${\mathbb{E}}\big{(}D^{\vee}\big{)}_{\nu({\tilde{\mathbf{c}}})}\big{)}$.

  2. $2.$

Suppose that ${\tilde{\mathbf{c}}}\in({\mathbb{R}}_{\geq
0}^{{\tilde{\Pi}}^{+}})^{\Gamma}$. Then the linear map $\nu^{\vee}$ $($resp.
$\nu)$ is a bijection between
${\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\cap\big{(}{\mathbb{R}}^{{\tilde{\Pi}}}\big{)}^{\Gamma}$
and ${\mathbb{U}}(D)_{\nu^{\vee}({\tilde{\mathbf{c}}})}$ $\big{(}$resp.
${\mathbb{U}}\big{(}D^{\vee}\big{)}_{\nu({\tilde{\mathbf{c}}})}\big{)}$.

###### Proof of Theorem 6.1.

In this proof we write ${\mathcal{N}}(D)$ for ${\mathcal{N}}(B)$ to avoid
conflict of notation. Let ${\tilde{D}}$ fold onto $D$. Let
${\mathbf{c}}\in{\mathbb{R}}_{>0}^{\Pi^{+}}$ and pick
${\tilde{\mathbf{c}}}\in\big{(}{\mathbb{R}}_{>0}^{{\tilde{\Pi}}^{+}}\big{)}^{\Gamma}$
satisfying ${\mathbf{c}}=\nu^{\vee}({\tilde{\mathbf{c}}})$. By Proposition
6.4(2), the map $\nu^{\vee}$ is a bijection between
${\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\cap\big{(}{\mathbb{R}}^{{\tilde{\Pi}}}\big{)}^{\Gamma}$
and ${\mathbb{U}}(D)_{{\mathbf{c}}}$. To prove Theorem 6.1(1) for $D$, it thus
suffices to show that
$\pi\big{(}{\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\cap\big{(}{\mathbb{R}}^{{\tilde{\Pi}}}\big{)}^{\Gamma}\big{)}=\pi\big{(}{\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\big{)}\cap\big{(}{\mathbb{R}}^{{\tilde{I}}}\big{)}^{\Gamma}\subset\big{(}{\mathbb{R}}^{{\tilde{I}}}\big{)}^{\Gamma}$
has normal fan ${\mathcal{N}}(D)$. By Theorem 6.1(1) for ${\tilde{D}}$, the
polytope
$\pi\big{(}{\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\big{)}$
has normal fan ${\mathcal{N}}\big{(}{\tilde{D}}\big{)}$, and by our choice of
${\tilde{\mathbf{c}}}$, it is $\Gamma$-invariant. The faces of
$\pi\big{(}{\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\big{)}$
that intersect $\big{(}{\mathbb{R}}^{{\tilde{I}}}\big{)}^{\Gamma}$ are exactly
those normal to the cones
$\\{{\tilde{\gamma}}_{1},\dots,{\tilde{\gamma}}_{a}\\}$ of
${\mathcal{N}}\big{(}{\tilde{D}}\big{)}$ consisting of $\Gamma$-invariant
pairwise compatible collections. Combining with Proposition 2.6(4), we deduce
that the normal fan of
$\pi\big{(}{\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{{\tilde{\mathbf{c}}}}\big{)}\cap\big{(}{\mathbb{R}}^{{\tilde{I}}}\big{)}^{\Gamma}$
is ${\mathcal{N}}(D)$. This proves the first statement of Theorem 6.1(1) for
$D$, and the second statement is similar.

Now, let $\nu({\tilde{\gamma}})=\gamma$. By Proposition 2.6(3), the Newton
polytope of $F_{\gamma}^{\operatorname{univ}}({\mathbf{z}})$ is the image of
the Newton polytope of
$F_{\tilde{\gamma}}^{\operatorname{univ}}({\mathbf{z}})$ under the map $\nu$.
By Proposition 6.4(2) and Theorem 6.1(2) for ${\tilde{D}}$, the Newton
polytope of
$\prod_{{\tilde{\gamma}}\in\nu^{-1}(\gamma)}F_{\tilde{\gamma}}^{\operatorname{univ}}({\mathbf{z}})$
is equal to
${\mathbb{U}}\big{(}{\tilde{D}}\big{)}_{\sum_{{\tilde{\gamma}}\in\nu^{-1}(\gamma)}e_{{\tilde{\gamma}}}}$.
Thus by Proposition 6.4(2), the Newton polytope of
$(F_{\gamma}^{\operatorname{univ}}({\mathbf{z}}))^{|\nu^{-1}(\gamma)|}$ is
equal to ${\mathbb{U}}\big{(}D^{\vee}\big{)}_{|\nu^{-1}(\gamma)|e_{\gamma}}$,
and Theorem 6.1(2) for $D$ follows. Finally, Theorem 6.1(3) follows from
Proposition 2.6(3). ∎

6.3. The character group of $T^{\operatorname{univ}}$ is
${\mathbb{Z}}^{n+r}/{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}$. Let
$\\{e_{1},\dots,e_{n}\\}\cup\\{e_{\gamma}\,|\,\gamma\in\Pi\\}$ be basis
vectors of ${\mathbb{Z}}^{n+r}$. We have

$\displaystyle\operatorname{wt}(x_{i})=e_{i}\quad\text{for}\quad
i=1,2,\dots,n,\qquad\text{and}\qquad\operatorname{wt}(z_{\gamma})=e_{\gamma}\quad\text{for}\quad\gamma\in\Pi.$
(6.2)

For $\gamma\in\Pi^{+}$, we have

$\displaystyle\operatorname{wt}(x_{\gamma})=\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}\mod{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}+\operatorname{span}({e_{1},\dots,e_{n}}).$

Note that all monomials in $F^{\operatorname{univ}}_{\gamma}$ have the same
weight modulo
${\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}+\operatorname{span}({e_{1},\dots,e_{n}})$.

######  Proposition 6.5.

The sets

$\displaystyle\big{\\{}\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}\,|\,\gamma\in\Pi^{+}\big{\\}}\qquad\text{and}\qquad\\{\operatorname{wt}(x_{\gamma})\,|\,\gamma\in\Pi\\}$

are bases of
${\mathbb{Z}}^{n+r}/\big{(}{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}+\operatorname{span}({e_{1},\dots,e_{n}})\big{)}$
and ${\mathbb{Z}}^{n+r}/{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}$
respectively.

###### Proof.

The first statement implies the second. By Theorem 6.1(2), for
$\gamma\in\Pi^{+}$, we have
$\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}\in{\mathbb{E}}(D^{\vee})_{e_{\gamma}}$.
The equations (6.1) define a linear map
$L\colon{\mathbb{R}}^{\Pi}\to{\mathbb{R}}^{\Pi^{+}}$, sending $(p_{\gamma})$
to $(c_{\gamma})$. Let $B^{\prime}$ be the last $r$ rows of
${\tilde{B}}^{\operatorname{univ}}$. By [27], see also [8, Section 8], the
matrix $B^{\prime}$ has rows given by $-{\mathbf{g}}^{\vee}_{\gamma}$. By [19,
relation (6.13)], the ${\mathbf{g}}$-vectors are solutions to the $0$-mesh
relations, and thus the kernel of $L$ is exactly $B^{\prime}{\mathbb{Z}}^{n}$.
We conclude that modulo $B^{\prime}{\mathbb{Z}}^{n}$, the last $r$ entries of
$\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}$ is equal to
the basis vector $e_{\gamma}$. Returning to the vector
$\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}\in{\mathbb{Z}}^{n+r}$,
we obtain

$\displaystyle\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}=e_{\gamma}\mod{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}+\operatorname{span}({e_{1},\dots,e_{n}}).$

Thus
$\big{\\{}\operatorname{wt}\big{(}F^{\operatorname{univ}}_{\gamma}\big{)}\,|\,\gamma\in\Pi^{+}\big{\\}}$
form a basis of
${\mathbb{Z}}^{n+r}/\big{(}{\tilde{B}}^{\operatorname{univ}}{\mathbb{Z}}^{n}+\operatorname{span}({e_{1},\dots,e_{n}})\big{)}$.
∎

6.4. Recall the definition of $f_{\gamma}({\mathbf{y}})$ from (4.3). Let
$A_{\gamma}({\mathbf{g}}):=\operatorname{Trop}(f_{\gamma}({\mathbf{y}}))$,
where we take $(g_{1},\dots,g_{n})$ as the tropicalization of
$y_{1},\dots,y_{n}$, for example
$\operatorname{Trop}(1+y_{1}+y_{1}y_{2})=\min(0,g_{1},\allowbreak
g_{1}+g_{2})$.

######  Theorem 6.6.

For $\gamma,\omega\in\Pi$, we have
$A_{\gamma}\big{(}{-}{\mathbf{g}}^{\vee}_{\omega}\big{)}=\delta_{\omega,\tau\gamma}$.

###### Proof.

First, assume that $D$ is simply-laced so that
${\mathbf{g}}_{\omega}={\mathbf{g}}^{\vee}_{\omega}$. For $\gamma\in\Pi$, let
$W_{\gamma}\in D^{b}({\rm rep}\,Q^{\rm op})$ be the object indexed by $\gamma$
in the bounded derived category of representations of the quiver $Q^{\rm op}$
corresponding to the reversed orientation of $D$, see [8, Section 3]. For any
${\mathbf{g}}\in{\mathbb{R}}^{n}$, the quantity
$\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))({\mathbf{g}})$ is equal to the
minimum value that the linear function
${\mathbf{Y}}\mapsto{\mathbf{Y}}\cdot{\mathbf{g}}$ takes on the Newton
polytope $P_{\gamma}$.

Now take $\gamma\in\Pi^{+}$. Let $G$ be the $n\times|\Pi|$ matrix whose
columns are ${\mathbf{g}}_{\omega}$. According to [8, Proof of Theorem 1], the
map ${\mathbf{Y}}\mapsto{\mathbf{Y}}\cdot(-G)+{\mathbf{v}}_{e_{\gamma}}$ is a
diffeomorphism between $P_{\gamma}$ and ${\mathbb{U}}_{e_{\gamma}}$, where
${\mathbf{v}}_{e_{\gamma}}\in{\mathbb{E}}_{e_{\gamma}}$ is the integer vector
given by
$({\mathbf{v}}_{e_{\gamma}})_{\omega}=\dim\operatorname{Hom}(W_{\omega},W_{\tau\gamma})$.
Here, $\operatorname{Hom}$ is taken within $D^{b}\big{(}{\rm rep}\,Q^{\rm
op}\big{)}$. Furthermore, it follows from [8] that ${\mathbb{U}}_{e_{\gamma}}$
has nonempty intersection with every coordinate hyperplane. Thus,

$\displaystyle\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))(-{\mathbf{g}}_{\omega})=-\dim\operatorname{Hom}(W_{\omega},W_{\tau\gamma}).$

Suppose $\tau\gamma=(t,i)\in\Pi^{+}$. Then we have an Auslander–Reiten
triangle in $D^{b}\big{(}{\rm rep}\,Q^{\rm op}\big{)}$

$\displaystyle W_{(t-1,i)}\to E\to W_{(t,i)}\to
W_{(t-1,i)}[1],\qquad\text{where}\quad E=\bigoplus_{i\to
j}W_{(t,j)}\oplus\bigoplus_{j\to i}W_{(t-1,j)}.$

We have an exact sequence

$\displaystyle
0\to\operatorname{Hom}\big{(}W_{\omega},W_{(t-1,i)}\big{)}\to\operatorname{Hom}(W_{\omega},E)\to\operatorname{Hom}\big{(}W_{\omega},W_{(t,i)}\big{)}$
$\displaystyle\hphantom{0}{}\to\operatorname{Hom}\big{(}W_{\omega},W_{(t-1,i)}[1]\big{)}\to\cdots.$

By the definition of Auslander–Reiten triangle, any map from $W_{\omega}$ to
$W_{(t,i)}$ which is not an isomorphism factors through $E$. Thus

$\displaystyle\big{(}\dim\operatorname{Hom}\big{(}W_{\omega},W_{(t-1,i)}\big{)}+\dim\operatorname{Hom}\big{(}W_{\omega},W_{(t,i)}\big{)}\big{)}-\dim\operatorname{Hom}(W_{\omega},E)=\delta_{\omega,(t,i)}$

and this is exactly $A_{\gamma}(-{\mathbf{g}}_{\omega})$. Now if
$\gamma\in\Pi^{+}$ but $\tau\gamma$ is initial, then we have an
Auslander–Reiten triangle of the form

$\displaystyle W_{(r_{i^{*}}-1,i^{*})}[-1]\to E\to W_{(0,i)}\to
W_{(r_{i^{*}}-1,i^{*})}$

and
$\operatorname{Hom}\big{(}W_{\omega},W_{(r_{i^{*}}-1,i^{*})}[-1]\big{)}=\operatorname{Ext}^{-1}\big{(}W_{\omega},W_{(r_{i^{*}}-1,i^{*})}\big{)}=0$
for all $\omega\in\Pi$, agreeing with our convention that
$F_{(0,i)}({\mathbf{y}})=1$, so again we have
$A_{\gamma}(-{\mathbf{g}}_{\omega})=\delta_{\omega,\tau\gamma}$. Finally,
suppose that $\gamma=(0,i)$ itself is initial. Then we have an
Auslander–Reiten triangle of the form

$\displaystyle W_{(r_{i^{*}}-1,i^{*})}\to E\to W_{(0,i)}[1]\to
W_{(r_{i^{*}}-1,i^{*})}[1].$

In this case, our formula for $f_{\gamma}({\mathbf{y}})$ includes a factor of
$y_{i}$, and $\operatorname{Trop}(y_{i})({\mathbf{g}}_{\omega})$ is simply the
$i$-th coordinate of ${\mathbf{g}}_{\omega}$. Our claim then follows from the
interpretation [8, Section 6] of ${\mathbf{g}}$-vectors as a change of basis
between the summands of a tilting object and the indecomposable projectives
$\big{\\{}W_{(0,1)},\dots,W_{(0,n)}\big{\\}}$ (see also [26, Theorem
3.23(ii)]).

Now, suppose that $D$ is multiply-laced and let ${\tilde{D}}$ be the simply-
laced diagram that folds to $D$. Note that for
${\mathbf{u}}\in{\mathbb{R}}^{|{\tilde{I}}|}$ and
${\mathbf{v}}\in\big{(}{\mathbb{R}}^{|{\tilde{I}}|}\big{)}^{\Gamma}$, we have
${\mathbf{u}}\cdot{\mathbf{v}}=\nu({\mathbf{u}})\cdot\nu^{\vee}({\mathbf{v}})$.
By Proposition 6.3,
$\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))(-{\mathbf{g}}^{\vee}_{\omega})$
is equal to the minimum value that the linear function
${\mathbf{Y}}\mapsto{\mathbf{Y}}\cdot\big{(}{-}\sum_{{\tilde{\omega}}\in\nu^{-1}(\omega)}{\mathbf{g}}_{\tilde{\omega}}\big{)}$
takes on the Newton polytope $P_{{\tilde{\gamma}}}$ (where
${\tilde{\gamma}}\in\nu^{-1}(\gamma)$), and we thus have
$\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))\big{(}{-}{\mathbf{g}}^{\vee}_{\omega}\big{)}=\sum_{{\tilde{\omega}}\in\nu^{-1}(\omega)}\operatorname{Trop}(F_{\tilde{\gamma}}({\mathbf{y}}))(-{\mathbf{g}}_{\tilde{\omega}})$.
It follows from the definitions that
$f_{\gamma}({\mathbf{y}})=\nu(f_{{\tilde{\gamma}}}({\mathbf{y}}))$ for any
${\tilde{\gamma}}\in\nu^{-1}(\gamma)$. The equality
$A_{\gamma}\big{(}{-}{\mathbf{g}}^{\vee}_{\omega}\big{)}=\delta_{\omega,\tau\gamma}$
for $D$ thus follows from the same equality for ${\tilde{D}}$. ∎

###### Proof of Proposition 4.6.

Let $m({\mathbf{y}})$ be a Laurent monomial in
$\\{y_{i},F_{\gamma}({\mathbf{y}})\\}$, and denote by
$G=G({\mathbf{g}}):=\operatorname{Trop}(m({\mathbf{y}}))$ the piecewise-linear
function that is the tropicalization of $m({\mathbf{y}})$. (Recall that by
convention the variables ${\mathbf{g}}$ are the tropicalizations of the
variables ${\mathbf{y}}$.) The domains of linearity of the function $G$ is a
coarsening of $-{\mathcal{N}}\big{(}B^{\vee}\big{)}$, so the function $G$ is
uniquely determined by the integer vector
$\big{(}G\big{(}{-}{\mathbf{g}}^{\vee}_{\gamma}\big{)}\,|\,\gamma\in\Pi\big{)}\in{\mathbb{Z}}^{|\Pi|}$.
By Theorem 6.6, any vector in ${\mathbb{Z}}^{|\Pi|}$ can arise in this way. It
follows that $m({\mathbf{y}})$ is uniquely determined by its tropicalization
$G$ by the formula
$m({\mathbf{y}})=\prod_{\gamma}f_{\gamma}({\mathbf{y}})^{G(-{\mathbf{g}}^{\vee}_{\tau\gamma})}$.
∎

######  Example 6.7.

We illustrate Theorem 6.6 for the exchange matrix

$\displaystyle B=\begin{bmatrix}0&1&0\\\ -1&0&-1\\\ 0&1&0\end{bmatrix}$

of type $A_{3}$. In this case, we have $\Pi=\\{0,1,2\\}\times\\{1,2,3\\}$ and
$i^{*}=4-i$. We tabulate ${\mathbf{g}}_{\gamma}$, $F_{\gamma}({\mathbf{y}})$,
and $f_{\gamma}({\mathbf{y}})$ below:

$\gamma$ | ${\mathbf{g}}_{\gamma}$ | $F_{\gamma}({\mathbf{y}})$ | $f_{\gamma}({\mathbf{y}})$
---|---|---|---
$(0,1)$ | $(1,0,0)$ | 1 | $\dfrac{y_{1}(1+y_{2})}{1+y_{1}+y_{1}y_{2}}\vphantom{\Bigg{(}}$
$(0,2)$ | $(0,1,0)$ | 1 | $\dfrac{y_{2}}{1+y_{2}}\vphantom{\Bigg{(}}$
$(0,3)$ | $(0,0,1)$ | 1 | $\dfrac{y_{3}(1+y_{2})}{1+y_{3}+y_{2}y_{3}}\vphantom{\Bigg{(}}$
$(1,1)$ | $(-1,1,0)$ | $1+y_{1}$ | $\dfrac{1}{1+y_{1}}\vphantom{\Bigg{(}}$
$(1,2)$ | $(-1,1,-1)$ | $1+y_{1}+y_{3}+y_{1}y_{3}+y_{1}y_{2}y_{3}$ | $\dfrac{(1+y_{1})(1+y_{3})}{1+y_{1}+y_{3}+y_{1}y_{3}+y_{1}y_{2}y_{3}}\vphantom{\Bigg{(}}$
$(1,3)$ | $(0,1,-1)$ | $1+y_{3}$ | $\dfrac{1}{1+y_{3}}\vphantom{\Bigg{(}}$
$(2,1)$ | $(0,0,-1)$ | $1+y_{3}+y_{2}y_{3}$ | $\dfrac{1+y_{1}+y_{3}+y_{1}y_{3}+y_{1}y_{2}y_{3}}{(1+y_{1})(1+y_{3}+y_{2}y_{3})}\vphantom{\Bigg{(}}$
$(2,2)$ | $(0,-1,0)$ | $1+y_{2}$ | $\dfrac{(1+y_{1}+y_{1}y_{2})(1+y_{3}+y_{2}y_{3})}{(1+y_{2})(1+y_{1}+y_{3}+y_{1}y_{3}+y_{1}y_{2}y_{3})}\vphantom{\Bigg{(}}$
$(2,3)$ | $(-1,0,0)$ | $1+y_{1}+y_{1}y_{2}$ | $\dfrac{1+y_{1}+y_{3}+y_{1}y_{3}+y_{1}y_{2}y_{3}}{(1+y_{3})(1+y_{1}+y_{1}y_{2})}\vphantom{\Bigg{(}}$

Taking $\gamma=(1,2)$ as an example, we have

$\displaystyle
A_{\gamma}({\mathbf{g}})=\min(0,g_{1})+\min(0,g_{3})-\min(0,g_{1},g_{3},g_{1}+g_{3},g_{1}+g_{2}+g_{3})$

and one can verify that it takes value 0 on all negatives of
${\mathbf{g}}$-vectors except for $-{\mathbf{g}}_{(0,2)}$, where it takes
value $1$.

######  Example 6.8.

Consider the following exchange matrix of type $B_{3}$:

$B=\begin{bmatrix}0&1&0\\\ -1&0&1\\\ 0&-2&0\end{bmatrix}\\!.$

We have $\Pi=\\{(t,j)\in[0,3]\times[1,3]\\}$ and $\tau(t,j)=((t-1)\mod 4,j)$.
We tabulate the ${\mathbf{g}}$-vectors, the ${\mathbf{g}}^{\vee}$-vectors, the
$F$-polynomials, and the polynomials $f_{\gamma}({\mathbf{y}})$ in Table 1.

Table 1: ${\mathbf{g}}$-vectors, $F$-polynomials, and $f_{\gamma}({\mathbf{y}})$ in type $B_{3}$. $\gamma$ | ${\mathbf{g}}_{\gamma}$ | ${\mathbf{g}}^{\vee}_{\gamma}$ | $F_{\gamma}({\mathbf{y}})$ | $f_{\gamma}({\mathbf{y}})$
---|---|---|---|---
$(0,1)$ | $\begin{bmatrix}1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 0\end{bmatrix}$ | $\begin{bmatrix}1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 0\end{bmatrix}$ | 1 | $\dfrac{y_{1}(1+y_{2}+2y_{2}y_{3}+y_{2}y_{3}^{2})}{1+y_{1}+y_{1}y_{2}+2y_{1}y_{2}y_{3}+y_{1}y_{2}y_{3}^{2}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(0,2)$ | $\begin{bmatrix}0\\\\[-4.30554pt] 1\\\\[-4.30554pt] 0\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] 1\\\\[-4.30554pt] 0\end{bmatrix}$ | 1 | $\dfrac{y_{2}(1+y_{3})^{2}}{1+y_{2}+2y_{2}y_{3}+y_{2}y_{3}^{2}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(0,3)$ | $\begin{bmatrix}0\\\\[-4.30554pt] 0\\\\[-4.30554pt] 1\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] 0\\\\[-4.30554pt] 1\end{bmatrix}$ | 1 | $\dfrac{y_{3}}{1+y_{3}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(1,1)$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 1\\\\[-4.30554pt] 0\end{bmatrix}$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 1\\\\[-4.30554pt] 0\end{bmatrix}$ | $1+y_{1}$ | $\dfrac{1}{1+y_{1}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(1,2)$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 2\end{bmatrix}$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 1\end{bmatrix}$ | $1+y_{1}+y_{1}y_{2}$ | $\dfrac{1+y_{1}}{1+y_{1}+y_{1}y_{2}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(1,3)$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 1\end{bmatrix}$ | $\begin{bmatrix}-2\\\\[-4.30554pt] 0\\\\[-4.30554pt] 1\end{bmatrix}$ | $1+y_{1}+y_{1}y_{2}+y_{1}y_{2}y_{3}$ | $\dfrac{1+y_{1}+y_{1}y_{2}}{1+y_{1}+y_{1}y_{2}+y_{1}y_{2}y_{3}}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(2,1)$ | $\begin{bmatrix}0\\\\[-4.30554pt] -1\\\\[-4.30554pt] 2\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] -1\\\\[-4.30554pt] 1\end{bmatrix}$ | $1+y_{2}$ | $\dfrac{1+y_{1}+y_{1}y_{2}}{(1+y_{1})(1+y_{2})}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(2,2)$ | $\begin{bmatrix}-1\\\\[-4.30554pt] -1\\\\[-4.30554pt] 2\end{bmatrix}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$ | $\begin{bmatrix}-1\\\\[-4.30554pt] -1\\\\[-4.30554pt] 1\end{bmatrix}$ |  | $1+y_{1}+y_{2}+2y_{1}y_{2}+y_{1}y_{2}^{2}+$
---
$2y_{1}y_{2}y_{3}+2y_{1}y_{2}^{2}y_{3}+y_{1}y_{2}^{2}y_{3}^{2}$
$\dfrac{(1+y_{2})(1+y_{1}+y_{1}y_{2}+y_{1}y_{2}y_{3})^{2}}{(1+y_{1}+y_{1}y_{2})f_{(2,2)({\mathbf{y}})}}$
$(2,3)$ | $\begin{bmatrix}0\\\\[-4.30554pt] -1\\\\[-4.30554pt] 1\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] -2\\\\[-4.30554pt] 1\end{bmatrix}$ | $1+y_{2}+y_{2}y_{3}$ | $\dfrac{F_{(2,2)}({\mathbf{y}})}{(1+y_{1}+y_{1}y_{2}+y_{1}y_{2}y_{3})(1+y_{2}+y_{2}y_{3})}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(3,1)$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 0\end{bmatrix}$ | $\begin{bmatrix}-1\\\\[-4.30554pt] 0\\\\[-4.30554pt] 0\end{bmatrix}$ | $1+y_{1}+y_{1}y_{2}+2y_{1}y_{2}y_{3}+y_{1}y_{2}y_{3}^{2}$ | $\dfrac{F_{(2,2)}({\mathbf{y}})}{(1+y_{2})f_{(3,1)}({\mathbf{y}})}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(3,2)$ | $\begin{bmatrix}0\\\\[-4.30554pt] -1\\\\[-4.30554pt] 0\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] -1\\\\[-4.30554pt] 0\end{bmatrix}$ | $1+y_{2}+2y_{2}y_{3}+y_{2}y_{3}^{2}$ | $\dfrac{(1+y_{2}+y_{2}y_{3})^{2}F_{(3,1)}({\mathbf{y}})}{F_{(2,2)}({\mathbf{y}})(1+y_{2}+2y_{2}y_{3}+y_{2}y_{3}^{2})}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$
$(3,3)$ | $\begin{bmatrix}0\\\\[-4.30554pt] 0\\\\[-4.30554pt] -1\end{bmatrix}$ | $\begin{bmatrix}0\\\\[-4.30554pt] 0\\\\[-4.30554pt] -1\end{bmatrix}$ | $1+y_{3}$ | $\dfrac{1+y_{2}+2y_{2}y_{3}+y_{2}y_{3}^{2}}{(1+y_{3})(1+y_{2}+y_{2}y_{3})}\vphantom{\begin{bmatrix}1\\\\[-2.15277pt] 0\\\\[-2.15277pt] 0\end{bmatrix}}$

##  7 Examples of $\boldsymbol{{\mathcal{M}}_{D}}$ as a configuration space

The space ${\mathcal{M}}_{A_{n-3}}$ can be identified with the configuration
space of $n$ distinct points on ${\mathbb{P}}^{1}$. In this section, we
investigate similar descriptions of ${\mathcal{M}}_{D}$ in the cases $D=B_{n}$
and $D=C_{n}$. We also consider the question of whether ${\mathcal{M}}_{D}$ is
a hyperplane arrangement complement.

So far we have considered ${\mathcal{M}}_{D}$ as a complex algebraic variety.
However, the equations (3.1) make sense over the integers, and we may also
consider ${\mathcal{M}}_{D}$ as a scheme over ${\mathbb{Z}}$. In particular,
in this section we will also consider ${\mathcal{M}}_{D}({\mathbb{F}}_{q})$,
the set of ${\mathbb{F}}_{q}$-points of ${\mathcal{M}}_{D}$, where
${\mathbb{F}}_{q}$ is a finite field.

Throughout this section, we use the description of $\Pi$ from Section 3.2.

7.1. Hyperplane arrangements. Let $H_{1},H_{2},\dots,H_{r}$ be (affine)
hyperplanes in ${\mathbb{C}}^{n}$, with the assumption that the hyperplanes
are defined over the integers. Let
$Z({\mathbb{C}}):={\mathbb{C}}^{n}-(H_{1}\cup H_{2}\cup\cdots\cup H_{r})$. By
our assumption $Z({\mathbb{R}})$ and $Z({\mathbb{F}}_{q})$ are also well-
defined. The following result is well-known [25, 30].

######  Theorem 7.1.

  1. $1.$

There exists a polynomial $\chi(t)$ so that $\chi(q)=\\#Z({\mathbb{F}}_{q})$,
where $q=p^{m}$ is a prime power with sufficiently large $p$.

  2. $2.$

The number of connected components $|\pi_{0}(Z({\mathbb{R}}))|$ of the real
hyperplane arrangement complement $Z({\mathbb{R}})$ is given by
$(-1)^{n}\chi(-1)$.

  3. $3.$

The cohomology ring $H^{*}(Z({\mathbb{C}}),{\mathbb{C}})$ is generated by the
classes of $\operatorname{dlog}f_{i}$, where $H_{i}=$ $\\{f_{i}=0\\}$, and we
have
$\sum_{i}\dim(H^{i}(Z({\mathbb{C}}),{\mathbb{C}}))\;t^{i}=(-t)^{n}\chi(-1/t)$.
Thus the Euler characteristic of $Z({\mathbb{C}})$ is equal to $\chi(1)$.

7.2. Type $\boldsymbol{A_{n}}$. Let $D=A_{n-3}$ with $n\geq 4$. Then $\Pi$ can
be identified with the diagonals of a $n$-gon $P_{n}$. We write $u_{ij}$ for
the $u$-variable indexed by a diagonal $(i,j)$. Then the relations defining
${\widetilde{\mathcal{M}}}_{A_{n-3}}$ are given by (1.1). These relations have
appeared a number of times in the literature, for example see [1, 11]. Let
${\mathcal{M}}_{0,n}$ denote the configuration space of $n$ (distinct) points
$(z_{1},z_{2},\dots,z_{n})$ on ${\mathbb{P}}^{1}$. Then the identification

$\displaystyle
u_{ij}=\frac{(z_{i}-z_{j+1})(z_{i+1}-z_{j})}{(z_{i}-z_{j})(z_{i+1}-z_{j+1})}$
(7.1)

of $u_{ij}$ with a cross ratio gives an isomorphism
${\mathcal{M}}_{A_{n-3}}\cong{\mathcal{M}}_{0,n}$. There is a well-studied
Deligne–Knudsen–Mumford compactification ${\overline{\mathcal{M}}}_{0,n}$, and
${\widetilde{\mathcal{M}}}_{0,n}:={\widetilde{\mathcal{M}}}_{A_{n-3}}$ is an
affine variety that sits between ${\mathcal{M}}_{0,n}$ and
${\overline{\mathcal{M}}}_{0,n}$, that is, we have open inclusions
${\mathcal{M}}_{0,n}\subset{\widetilde{\mathcal{M}}}_{0,n}\subset{\overline{\mathcal{M}}}_{0,n}$.

Let ${\rm Gr}(2,n)$ denote the Grassmannian of 2-planes in ${\mathbb{C}}^{n}$.
Let ${\mathring{\Pi}}(2,n)\subset{\rm Gr}(2,n)$ denote the open subset where
the adjacent cyclic minors $\Delta_{i,i+1}$ are non-vanishing. Then
${\mathring{\Pi}}(2,n)$ is a full rank cluster variety of type $A_{n-3}$, see
[18, Section 12.2]. Let ${\mathring{{\rm
Gr}}}(2,n)\subset{\mathring{\Pi}}(2,n)$ be the open subset where _all_ Plücker
coordinates $\Delta_{ij}$ are non-vanishing. This is the subset denoted
${\mathring{X}}$ in Section 4. Then ${\mathcal{M}}_{0,n}$ can be identified
with the quotient of ${\mathring{{\rm Gr}}}(2,n)$ by the diagonal torus $T$
sitting inside $\operatorname{GL}_{n}$ that acts on ${\rm Gr}(2,n)$. The
isomorphism ${\mathcal{M}}_{0,n}\cong{\mathring{{\rm Gr}}}(2,n)/T$ is an
instance of Theorem 4.2 for $D=A_{n-3}$. In the ${\rm Gr}(2,n)$ cluster
algebra, we have the primitive exchange relation

$\displaystyle\Delta_{i,j}\Delta_{i+1,j+1}=\Delta_{i,j+1}\Delta_{i+1,j}+\Delta_{i,i+1}\Delta_{j,j+1},$

where $\Delta_{i,i+1}$ and $\Delta_{j,j+1}$ are frozen variables. Thus (7.1),
or equivalently
$u_{ij}=\frac{\Delta_{i,j+1}\Delta_{i+1,j}}{\Delta_{i,j}\Delta_{i+1,j+1}}$,
agrees with the formula for $u_{\gamma}$ in Theorem 4.2.

The geometry and topology of ${\mathcal{M}}_{0,n}$ is very well-studied; see
for example [11]. We recall some basic facts in the context of Theorem 7.1.
Gauge-fixing $z_{1}$, $z_{n-1}$, $z_{n}$ to 0, 1, $\infty$, we have an
identification

$\displaystyle{\mathcal{M}}_{0,n}(k)=\big{\\{}(z_{2},z_{3},\dots,z_{n-2})\in
k^{n-3}\,|\,\mbox{$z_{i}\neq z_{j}$ and $z_{i}\notin\\{0,1\\}$}\big{\\}}$

for $k$ a field. In particular, ${\mathcal{M}}_{0,n}(k)$ is the complement in
$k^{n-3}$ of the hyperplane arrangement with hyperplanes

$\displaystyle z_{i}-z_{j}=0,\qquad z_{i}=0,\qquad 1-z_{i}=0.$

We may compute that
$\\#{\mathcal{M}}_{0,n}({\mathbb{F}}_{q})=(q-2)(q-3)\cdots(q-n+2).$ By Theorem
7.1, the number of connected components of ${\mathcal{M}}_{0,n}({\mathbb{R}})$
is given by

$\displaystyle|\pi_{0}({\mathcal{M}}_{0,n}({\mathbb{R}}))|=|(-3)\cdot(-4)\cdots(-n+1)|=(n-1)!/2.$

7.3. Type $\boldsymbol{B_{n}}$. By Theorem 4.2, ${\mathcal{M}}_{D}$ can be
identified with
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}/T\big{(}{\tilde{B}}\big{)}$ for any
full rank extended exchange matrix ${\tilde{B}}$ of type $D$. One such choice
of ${\tilde{B}}$, and thus of ${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ is
given in [18, Example 12.10]. Let ${\mathbb{C}}[X_{n+2}]$ be the ring
generated by the Plücker coordinates of the Grassmannian ${\rm Gr}(2,n+2)$.
Recall that $\Gamma$ is the two-element group whose non-trivial element maps
$i\leftrightarrow\bar{i}$. Consider the following functions in
${\mathbb{C}}[X_{n+2}]$, labeled by $\Gamma$-orbits of sides and diagonals in
the polygon $P_{2n+2}$ with vertices
$\big{\\{}1,2,\dots,n+1,\bar{1},\bar{2},\dots,\overline{n+1}\big{\\}}$:

$\displaystyle\big{[}a,\bar{a}\big{]}\mapsto\Delta_{a\bar{a}}=\Delta_{a,n+2}\qquad(1\leq
a\leq n+1),$
$\displaystyle\big{\\{}[a,b],\big{[}\bar{a},\bar{b}\big{]}\big{\\}}\mapsto\Delta_{ab}\qquad(1\leq
a<b\leq n+1),$
$\displaystyle\big{\\{}\big{[}a,\bar{b}\big{]},\big{[}\bar{a},b\big{]}\big{\\}}\mapsto\Delta_{a\bar{b}}=\Delta_{a,n+2}\Delta_{b,n+2}-\Delta_{ab}\qquad(1\leq
a<b\leq n+1).$

Let ${\mathring{V}}_{n}$ be the space of $2\times(n+2)$ matrices such that all
the above functions are non-vanishing, and let
$T_{n+1}\cong({\mathbb{C}}^{\times})^{n+1}$ act on ${\mathring{V}}_{n}$ by
scaling the first $n+1$ columns. Then the action of
$\operatorname{SL}_{2}\times T_{n+1}$ on ${\mathring{V}}_{n}$ is free.

######  Proposition 7.2.

We have an isomorphism
${\mathcal{M}}_{B_{n}}\cong\operatorname{SL}_{2}\backslash{\mathring{V}}_{n}/T_{n+1}$.

Using the action of $\operatorname{SL}_{2}$ we can gauge-fix the last column
of $M\in{\mathring{V}}_{n}$ to $[0,1]^{T}$, and using the action of $T_{n+1}$,
we may gauge-fix the first entry of columns $1,2,\dots,n+1$ of $M$ to $1$.
Thus modulo the action of $\operatorname{SL}_{2}\times T_{n+1}$, every point
in ${\mathring{V}}_{n}$ can be written in the form

$\displaystyle\begin{bmatrix}1&1&1&\cdots&1&0\\\
z_{1}&z_{2}&z_{3}&\cdots&z_{n+1}&1\end{bmatrix}\\!,$

where $z_{i}\in{\mathbb{C}}$, and two such matrices with parameters
${\mathbf{z}}=(z_{1},\dots,z_{n+1})$ and
${\mathbf{z}}^{\prime}=(z^{\prime}_{1},\dots,z^{\prime}_{n+1})$ are equivalent
if ${\mathbf{z}}^{\prime}-{\mathbf{z}}=c{\mathbf{1}}$, where ${\mathbf{1}}$ is
the all $1$-s vector. We may thus identify ${\mathcal{M}}_{B_{n}}$ with a
subspace of
${\mathbb{C}}^{n+1}/{\mathbb{C}}=(z_{1},\dots,z_{n+1})/{\mathbb{C}}\cdot{\mathbf{1}}$.
For these matrices, the cluster variables $\Delta_{a\bar{a}}$ are equal to
$1$, and we have

$\displaystyle\Delta_{ab}=z_{b}-z_{a}\qquad(1\leq a<b\leq n+1),$
$\displaystyle\Delta_{a\bar{b}}=1-z_{b}+z_{a}\qquad(1\leq a<b\leq n+1).$ (7.2)

We recognize the hyperplanes (7.2) as the _Shi arrangement_ [28].

######  Proposition 7.3.

${\mathcal{M}}_{B_{n}}$ is isomorphic to the complement in
${\mathbb{C}}^{n+1}/{\mathbb{C}}$ of the Shi arrangement.

Among many well-known properties, we obtain the following as immediate
consequences using Theorem 7.1: $(a)$
$\\#{\mathcal{M}}_{B_{n}}({\mathbb{F}}_{q})=(q-n-1)^{n}$, $(b)$ the number of
connected components
$|\pi_{0}({\mathcal{M}}_{B_{n}}({\mathbb{R}}))|=(-1)^{n}(\\#{\mathcal{M}}_{B_{n}}({\mathbb{F}}_{q})|_{q=-1})$
is equal to $(n+2)^{n}$, $(c)$ the cohomology
$H^{*}({\mathcal{M}}_{B_{n}},{\mathbb{C}})$ is generated in degree one by
$\operatorname{dlog}\Delta$, as $\Delta$ varies over the hyperplanes (7.2),
and $(d)$ $|\chi({\mathcal{M}}_{B_{n}})|\allowbreak=n^{n}$.

7.4. Type $\boldsymbol{C_{n}}$. By applying Theorem 4.2 to [18, Example
12.12], we obtain a description of ${\mathcal{M}}_{C_{n}}$. Recall that
$\Gamma$ is the two-element group whose non-trivial element maps
$i\leftrightarrow\bar{i}$. Consider the space ${\rm Mat}_{2,n+1}$ of
$2\times(n+1)$ matrices

$\displaystyle\begin{bmatrix}y_{11}&y_{12}&\cdots&y_{1,n+1}\\\
y_{21}&y_{22}&\cdots&y_{2,n+1}\end{bmatrix}$

and the cluster variables

$\displaystyle\big{\\{}[a,b],[\bar{a},\bar{b}]\big{\\}}\mapsto\Delta_{ab}=\frac{y_{1a}y_{2b}-y_{1b}y_{2a}}{2i}\qquad(1\leq
a<b\leq n+1),$
$\displaystyle\big{\\{}[a,\bar{b}],[\bar{a},b]\big{\\}}\mapsto\Delta_{a\bar{b}}=\frac{y_{1a}y_{2b}+y_{2a}y_{2b}}{2}\qquad(1\leq
a\leq b\leq n+1)$ (7.3)

labeled by $\Gamma$-orbits of sides and diagonals in the polygon $P_{2n+2}$
with vertices $\\{1,2,\dots,n+1,\bar{1},\bar{2},\dots,\overline{n+1}\\}$.
These functions generate the ring of invariant functions ${\mathbb{C}}[{\rm
Mat}_{2,n+1}]^{S}$, isomorphic to a cluster algebra
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ of type $C_{n}$ (when frozen
variables are inverted). Here, $S={\rm diag}(t,1/t)$ is the group of $2\times
2$ diagonal matrices with determinant 1. The cluster automorphism group is
$T=({\mathbb{C}}^{\times})^{n+1}\times({\mathbb{Z}}/2{\mathbb{Z}})$, where
$({\mathbb{C}}^{\times})^{n+1}\cong({\mathbb{C}}^{\times})^{n+1}/\langle(-1,-1,\dots,-1)\rangle$
acts on ${\rm Mat}_{2,n+1}$ by rescaling columns (with the element
$(-1,\dots,-1)$ acting trivially on $S\backslash{\rm Mat}_{2,n+1}$), and the
non-trivial element of the group $({\mathbb{Z}}/2{\mathbb{Z}})$ acts by
swapping the two rows.

######  Remark 7.4.

Note that in contrast to the $B_{n}$ case ([18, Example 12.10]), the
${\tilde{B}}$-matrix of [18, Example 12.12] is full rank but not really full
rank. For example, for $n=2$ we may choose an initial cluster so that we have

$\displaystyle{\tilde{B}}\quad\text{for type}\quad B_{2}=\begin{bmatrix}0&1\\\
-2&0\\\ 1&0\\\ 1&-1\\\
-1&0\end{bmatrix}\qquad\text{and}\qquad{\tilde{B}}\quad\text{for type}\quad
C_{2}=\begin{bmatrix}0&2\\\ -1&0\\\ 1&0\\\ 1&-2\\\ -1&0\end{bmatrix}$

respectively, where the rows are labelled by 13, $1\bar{1}$, 12, $1\bar{3}$,
23. This explains why the cluster automorphism group $T$ in our discussion is
disconnected.

On the locus where all cluster variables are non-vanishing, such
$2\times(n+1)$ matrices can be gauge-fixed, using $S$ and
$({\mathbb{C}}^{\times})^{n+1}\subset T$ to the form:

$\displaystyle\begin{bmatrix}1&1&\cdots&1&1\\\
z_{1}&z_{2}&\cdots&z_{n}&1\end{bmatrix}$ (7.4)

and the non-vanishing of the cluster variables is equivalent to the non-
vanishing of the linear forms

$\displaystyle z_{i}-z_{j},\qquad z_{i}+z_{j},\qquad 1-z_{i},\qquad
1+z_{i},\qquad z_{i}.$ (7.5)

Let ${\mathring{Z}}_{n}$ denote the space of matrices of the form (7.4), where
the linear forms (7.5) are non-vanishing. There is still a free action of
${\mathbb{Z}}/2{\mathbb{Z}}$ on ${\mathring{Z}}_{n}$, acting by swapping the
two rows, which induces $(z_{1},\dots,z_{n})\mapsto(1/z_{1},\dots,1/z_{n})$.
By Theorem 4.2, we obtain

######  Proposition 7.5.

We have an isomorphism
${\mathcal{M}}_{C_{n}}\cong{\mathring{Z}}_{n}/({\mathbb{Z}}/2{\mathbb{Z}})$.

This isomorphism is valid over the integers even though (7.3) involves the
scalars $1/2$ and $1/2i$: this is because the scalars cancel in any
$T$-invariant ratio of cluster variables.

######  Proposition 7.6.

We have

$\displaystyle\\#{\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q})=(q-n-1)(q-3)(q-5)\cdots(q-2n+1)$

for ${\rm char}(q)>2$ and

$\displaystyle|\pi_{0}({\mathcal{M}}_{C_{n}}({\mathbb{R}}))|=\frac{1}{2}\big{(}2^{n}(n+1)!+2^{n}n!\big{)}=2^{n-1}(n+2)n!.$

Note that
$|\pi_{0}({\mathcal{M}}_{C_{n}}({\mathbb{R}}))|=|(\\#{\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q}))|_{q=-1}|$,
agreeing with Theorem 7.1, even though we do not know whether
${\mathcal{M}}_{C_{n}}$ is isomorphic to a hyperplane arrangement complement.

###### Proof.

For a field $k$, the points of ${\mathcal{M}}_{C_{n}}(k)$ in general come from
${\mathring{Z}}_{n}(\bar{k})$, where $\bar{k}$ denotes the algebraic closure
of $k$.

First, suppose that $k={\mathbb{F}}_{q}$ with ${\rm char}(q)>2$. Let
${\bar{\mathbb{F}}}_{q}$ denote the algebraic closure of ${\mathbb{F}}_{q}$.
Then the Galois group is topologically generated by one generator $\sigma$,
called the _Frobenius automorphism_. It acts as the field automorphism
$\sigma(x)=x^{q}$, and furthermore, we have $\sigma(x)=x$ if and only if
$x\in{\mathbb{F}}_{q}$. The map
$\pi\colon{\mathring{Z}}_{n}({\bar{\mathbb{F}}}_{q})\to{\mathcal{M}}_{C_{n}}({\bar{\mathbb{F}}}_{q})$
commutes with the action of $\sigma$. Thus if
$\pi(z)=u\in{\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q})$, we have

$\displaystyle
u\in{\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q})\Leftrightarrow\sigma(u)=u\Leftrightarrow\sigma(\pi(z))=\pi(z)\Leftrightarrow\pi(\sigma(z))=\pi(z),$

and we have two possibilities: (1) $\sigma(z)=z$, or (2) $\sigma(z)=1/z$. For
case (1), we are just counting $\\#{\mathring{Z}}_{n}({\mathbb{F}}_{q})$.
Imposing the conditions (7.5), we get

$\displaystyle\\#{\mathring{Z}}_{n}({\mathbb{F}}_{q})=(q-3)(q-5)\cdots(q-2n-1).$

For case (2), the equation $\sigma(z)=1/z$ is equivalent to $z_{i}^{q+1}=1$
for $i=1,2,\dots,n$. There are $q+1$ solutions to the polynomial equation
$x^{q+1}-1$ in ${\bar{\mathbb{F}}}_{q}$. Two of the solutions are $x=\pm 1$.
The other $q-1$ solutions lie in ${\mathbb{F}}_{q^{2}}$, since
$x^{q+1}=1\implies x^{q^{2}-1}=1\implies x^{q^{2}}=x$. Thus in case (2), we
are counting $n$-tuples $(z_{1},z_{2},\dots,z_{n})\in{\mathbb{F}}_{q^{2}}$
satisfying the condition $x^{q+1}=1$ and the conditions imposed by (7.5). The
conditions $z_{i}\neq\pm 1$ are automatically satisfied, so we get

$\displaystyle(q-1)(q-3)\cdots(q-2n+1).$

In sum, taking into account the two-to-one covering
${\mathring{Z}}_{n}\to{\mathcal{M}}_{C_{n}}$, we have

$\displaystyle\\#{\mathcal{M}}_{C_{n}}({\mathbb{F}}_{q})$
$\displaystyle=\frac{1}{2}\left((2q-2n-2)(q-3)(q-5)\cdots(q-2n+1)\right)$
$\displaystyle=(q-n-1)(q-3)(q-5)\cdots(q-2n+1).$

(Note that this point count is also the same as the hyperplane arrangement
complement with hyperplanes $z_{i}+z_{j}$, $z_{i}-z_{j}$, $1+z_{i}$,
$1-z_{i}$, though we do not know an explanation for this.)

Next let us consider $k={\mathbb{R}}$, so $\bar{k}={\mathbb{C}}$. In this
case, $\sigma$ is replaced by complex conjugation. So the same argument says
that we should consider the two cases (1) $\bar{z}=z$, or (2) $\bar{z}=1/z$.
For case (1), we are looking at ${\mathring{Z}}_{n}({\mathbb{R}})$ and by
Theorem 7.1 we get

$\displaystyle|\pi_{0}({\mathring{Z}}_{n}({\mathbb{R}}))|=2^{n}(n+1)!$

since $\\#{\mathring{Z}}_{n}({\mathbb{F}}_{q})=(q-3)(q-5)\cdots(q-2n-1)$.

For case (2), we must have $z_{i}=\exp(i\theta_{i})$ on the unit circle, where
$\theta_{i}\in(\pi,\pi]$. The conditions (7.5) give

$\displaystyle\theta_{i}\neq\theta_{j},\qquad\theta_{i}\neq
0,\qquad\theta_{i}\neq\pi,\qquad\theta_{i}+\theta_{j}\neq 0.$

Thus the quantities $|\theta_{1}|,\dots,|\theta_{n}|$ lie in $(0,\pi)$ and
satisfy $|\theta_{i}|\neq|\theta_{j}|$. There are $n!$ regions in $|\theta|$
space, and $2^{n}$ sign choices going from $|\theta_{i}|$ to $\theta_{i}$,
giving

$\displaystyle\big{|}\pi_{0}\big{(}{\mathring{Z}}_{n}(S^{1})\big{)}\big{|}=2^{n}n!.$

In sum, taking into account the two-to-one covering
${\mathring{Z}}_{n}\to{\mathcal{M}}_{C_{n}}$, we have

$\displaystyle|\pi_{0}({\mathcal{M}}_{C_{n}}({\mathbb{R}}))|=\frac{1}{2}\left(2^{n}(n+1)!+2^{n}n!\right)=2^{n-1}(n+2)n!,$

as claimed. ∎

The variety ${\mathcal{M}}_{C_{n}}$ can be identified with a subvariety of
${\mathcal{M}}_{A_{2n-1}}$. Sending $(z_{1},\dots,z_{n})$ to

$\displaystyle\begin{bmatrix}1&1&\cdots&1&1&1&\cdots&1&1\\\
z_{1}&z_{2}&\cdots&z_{n}&1&-z_{1}&\cdots&-z_{n}&1\end{bmatrix}$ (7.6)

maps ${\mathcal{M}}_{C_{n}}$ into ${\mathcal{M}}_{A_{2n-1}}$. Note that the
non-vanishing of (7.5) is equivalent to the non-vanishing of all minors in
(7.6), and that $(z_{1},\dots,z_{n})$ and $(1/z_{1},\dots,1/z_{n})$ represent
the same point in ${\mathcal{M}}_{A_{2n-1}}$.

7.5. Type $\boldsymbol{D_{n}}$. We do not know a simple description of
${\mathcal{M}}_{D}$ in this case. Indeed, the point counts we have obtained
show that ${\mathcal{M}}_{D}$ cannot be a hyperplane arrangement complement.
In type $D_{4}$, numerical computations indicate that for $p\neq 2$, we have

$\displaystyle|{\mathcal{M}}_{D_{4}}({\mathbb{F}}_{p})|=\begin{cases}206-231p+93p^{2}-16p^{3}+p^{4}&\mbox{if}\quad
p=2\mod 3,\\\ 208-231p+93p^{2}-16p^{3}+p^{4}&\mbox{if}\quad p=1\mod
3.\end{cases}$ (7.7)

Substituting $p=-1$ in (7.7), we get $547$. We do not know for sure that
${\mathcal{M}}_{D_{4}}({\mathbb{R}})$ has $547$ connected components, but see
Section 11.4.

For type $D_{5}$, numerical computations indicate that for $p\neq 2,3$, we
have

$\displaystyle|{\mathcal{M}}_{D_{5}}({\mathbb{F}}_{p})|=-2318+2644p-1156p^{2}+244p^{3}-25p^{4}+p^{5}+(-36+5p)\delta_{3}(p)-\delta_{4}(p),$

where we define $\delta_{3}(p)=0$ for $p=2\mod 3$ or $2$ for $p=1\mod 3$ and
similarly $\delta_{4}(p)=0$ for $p=3\mod 4$ or $2$ for $p=1\mod 4$.

Substituting $p=-1$, we get $6388$. We do not know whether
${\mathcal{M}}_{D_{5}}({\mathbb{R}})$ has $6388$ connected components, but see
Section 11.4.

7.6. Type $\boldsymbol{G_{2}}$. Numerical computations give

$\displaystyle|{\mathcal{M}}_{G_{2}}({\mathbb{F}}_{p})|=\begin{cases}(p-4)^{2}&\mbox{if}\quad
p=2\mod 3,\\\ (p-4)^{2}+4&\mbox{if}\quad p=1\mod 3.\end{cases}$

Substituting $p=-1$, we get $25$, which we expect to be the number of
connected components of ${\mathcal{M}}_{G_{2}}({\mathbb{R}})$.

##  8 Positive part

In this section, we define the nonnegative subspace ${\mathcal{M}}_{D,\geq 0}$
of ${\mathcal{M}}_{D}({\mathbb{R}})$ and show that it is diffeomorphic to the
generalized associahedron of $D^{\vee}$.

8.1. We define the positive part
${\mathcal{M}}_{D,>0}\subset{\mathcal{M}}_{D}({\mathbb{R}})$ as the subspace

$\displaystyle{\mathcal{M}}_{D,>0}:=\big{\\{}x\in{\mathcal{M}}_{D}({\mathbb{R}})\,|\,u_{\gamma}(x)>0\text{
for all }\gamma\in\Pi\big{\\}}$

and the nonnegative part ${\mathcal{M}}_{D,\geq
0}\subset{\widetilde{\mathcal{M}}}_{D}({\mathbb{R}})$ by

$\displaystyle{\mathcal{M}}_{D,\geq
0}:=\overline{{\mathcal{M}}_{D,>0}}\subset{\widetilde{\mathcal{M}}}_{D}({\mathbb{R}}).$

Intersecting with the stratification (3.6), we obtain

$\displaystyle{\mathcal{M}}_{D,\geq
0}=\bigsqcup_{F}{\mathcal{M}}_{F,>0},\qquad\text{where}\quad{\mathcal{M}}_{F,>0}={\mathcal{M}}_{D,>0}\cap{\mathcal{M}}_{F}({\mathbb{R}}).$

Let $P$ be an integer polytope and ${\mathcal{N}}(P)$ denote its normal fan.
It is well-known [20, Chapter 4] that the nonnegative part
$X_{{\mathcal{N}}(P),\geq 0}$ of the projective toric variety
$X_{{\mathcal{N}}(P)}$ is diffeomorphic to the polytope $P$. The following
result thus follows from Theorem 5.3.

######  Theorem 8.1.

There is a face-preserving diffeomorphism between ${\mathcal{M}}_{D,\geq 0}$
and the generalized associahedron of $D^{\vee}$.

The positive part ${\mathcal{M}}_{D,>0}$ is identical to the positive part
$X_{{\mathcal{N}}(B^{\vee}),>0}$ of the ambient projective toric variety, and
${\mathcal{M}}_{D,>0}$ is equal to one of the connected components of the
smooth manifold ${\mathcal{M}}_{D}({\mathbb{R}})$.

8.2. The space ${\mathcal{M}}_{D}$ has a distinguished rational top-form
$\Omega({\mathcal{M}}_{D})$, called the _canonical form_ , that can be
described in a number of ways. Suppose ${\tilde{B}}$ is a full rank extended
exchange matrix of type $D$. The cluster algebra
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ has a natural top-form $\Omega$ which
in any cluster $(x_{1},x_{2},\dots,x_{n+m})$ can be written (up to sign):

$\displaystyle\Omega=\frac{{\rm d}x_{1}}{x_{1}}\wedge\cdots\wedge\frac{{\rm
d}x_{n+m}}{x_{n+m}},$

which is the natural top-form on the corresponding cluster torus
$({\mathbb{C}}^{\times})^{n+m}$. The cluster automorphism group
$T\big{(}{\tilde{B}}\big{)}$ can be identified with a subgroup of
$({\mathbb{C}}^{\times})^{n+m}$, and the quotient group
$({\mathbb{C}}^{\times})^{n+m}/T\big{(}{\tilde{B}}\big{)}$ is again an
algebraic torus $S=({\mathbb{C}}^{\times})^{n}$. The inclusion
$({\mathbb{C}}^{\times})^{n+m}\subset X\big{(}{\tilde{B}}\big{)}$ identifies
$S$ birationally with ${\mathcal{M}}_{D}$ (note that neither $S$ nor
${\mathcal{M}}_{D}$ contains the other, but the two share a common dense open
subset). The torus $S$ has a natural top-form, and the canonical form
$\Omega({\mathcal{M}}_{D})$ is the image of this form under the birational
isomorphism between $S$ and ${\mathcal{M}}_{D}$.

Another way to obtain the canonical form of ${\mathcal{M}}_{D}$ is via Theorem
5.3: any toric variety $X_{P}$ has a canonical rational top-form
$\Omega(X_{P})$, which is simply the (extension of the) natural top-form of
the dense algebraic torus in $X_{P}$. Restricting
$\Omega\big{(}X_{{\mathcal{N}}(B^{\vee})}\big{)}$ to ${\mathcal{M}}_{D}$ gives
a top-form on ${\mathcal{M}}_{D}$ which equals $\Omega({\mathcal{M}}_{D})$.

The pair $\big{(}{\widetilde{\mathcal{M}}}_{D},{\mathcal{M}}_{D,\geq
0}\big{)}$ is nearly a _positive geometry_ in the sense of [2], with
$\Omega\big{(}{\widetilde{\mathcal{M}}}_{D}\big{)}=\Omega({\mathcal{M}}_{D})$
as canonical form: the residue of $\Omega({\mathcal{M}}_{D})$ along a divisor
${\widetilde{\mathcal{M}}}_{D}(F)$ is equal to the canonical form
$\Omega\big{(}{\widetilde{\mathcal{M}}}_{D}(F)\big{)}$, which is simply the
product of canonical forms corresponding to the factorization of Proposition
3.6. This follows from the similar statement concerning
$X_{{\mathcal{N}}(B^{\vee})}$, proven in [2, Appendix G]. However,
${\widetilde{\mathcal{M}}}_{D}$ is an affine variety rather than a projective
variety, so it is not a positive geometry in the strict sense.

In the case $D=A_{n-3}$, we have ${\mathcal{M}}_{D}={\mathcal{M}}_{0,n}$, and
the canonical form can be written as

$\displaystyle\Omega\big{(}({\mathcal{M}}_{0,n})_{>0}\big{)}=\frac{{\rm
d}z_{2}\cdots{\rm d}z_{n-2}}{(z_{2}-z_{1})(z_{3}-z_{2})\cdots(z_{1}-z_{n})},$

where $(z_{1},z_{n-1},z_{n})=(0,1,\infty)$ as in Section 7.2, and denominator
factors equal to $\infty$ are understood to be omitted. This form is also
called a cell-form in [9] and the condition that
$\Omega(({\mathcal{M}}_{0,n})_{>0})$ only has poles along the boundary
divisors of ${\widetilde{\mathcal{M}}}_{0,n}$ (and not elsewhere in
${\overline{\mathcal{M}}}_{0,n}$) is [9, Proposition 2.7]. Combining with the
above discussion, we have

######  Proposition 8.2.

The pair $\big{(}{\overline{\mathcal{M}}}_{0,n},({\mathcal{M}}_{0,n})_{\geq
0}\big{)}$ is a positive geometry with canonical form
$\Omega\big{(}({\mathcal{M}}_{0,n})_{>0}\big{)}$.

##  9 Positive tropicalization

In this section, we consider the positive tropicalization of
${\mathcal{M}}_{D}$. We use our results to resolve a conjecture of Speyer and
Williams [29] on positive tropicalizations of cluster algebras of finite type.
We refer the reader to [29] for background on positive tropicalizations.

Let
${\mathcal{R}}=\bigcup_{n=1}^{\infty}{\mathbb{R}}\big{(}\big{(}t^{1/n}\big{)}\big{)}$
denote the field of Puiseux series over ${\mathbb{R}}$. We define ${\rm
val}\colon{\mathcal{R}}\to{\mathbb{R}}\cup\\{\infty\\}$ by ${\rm
val}(0)=\infty$ and ${\rm val}(x(t))=r$ if the lowest term of $x(t)$ is equal
to $\alpha t^{r}$, where $\alpha\in{\mathbb{R}}^{\times}$. We define
${\mathcal{R}}_{>0}\subset{\mathcal{R}}$ to be the semifield consisting of
Puiseux series $x(t)$ that are non-zero and such that coefficient of the
lowest term is a positive real number.

A point $u(t)\in{\mathcal{M}}_{D}({\mathcal{R}}_{>0})$ is a collection
$u(t)=\\{u_{\gamma}(t)\in{\mathcal{R}}_{>0},\gamma\in\Pi\\}$ of (positive)
Puiseux series satisfying the relations from Definition 3.1. We define the
_positive tropicalization_ $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ as the
closure of valuations

$\displaystyle\operatorname{Trop}_{>0}{\mathcal{M}}_{D}:=\overline{\big{\\{}{\rm
val}(u_{\gamma}(t))_{\gamma\in\Pi}\in{\mathbb{R}}^{\Pi}\,|\,u(t)\in{\mathcal{M}}_{D}({\mathcal{R}}_{>0})\big{\\}}}\subset{\mathbb{R}}^{\Pi}.$

######  Lemma 9.1.

The subspace $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ is a (not complete)
polyhedral fan inside the linear space ${\mathbb{R}}^{\Pi}$.

###### Proof.

We use the identification
${\mathcal{M}}_{D}\cong{\mathring{X}}^{\operatorname{prin}}/T^{\operatorname{prin}}$
from Theorem 4.3. According to Proposition 4.6, there is an invertible
monomial transformation between the functions $u_{\gamma}=f_{\gamma}$ and the
set of functions
$\\{y_{1},\dots,y_{n}\\}\cup\\{F_{\gamma}({\mathbf{y}})\,|\,\gamma\in\Pi^{+}\\}$.
Since each $F_{\gamma}({\mathbf{y}})$ is a positive Laurent polynomial in the
$y_{i}$-s, it follows that each $u_{\gamma}({\mathbf{y}})$ is a subtraction-
free rational function in the $y_{i}$-s. Thus the map
$(u_{\gamma})_{\gamma\in\Pi}\mapsto(y_{1},y_{2},\dots,y_{n})$ induces an
isomorphism
${\mathcal{M}}_{D}({\mathcal{R}}_{>0})\cong{\mathcal{R}}_{>0}^{n}$. It follows
that we have a homeomorphism
$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}\cong{\mathbb{R}}^{n}$. The
embedding of $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ in the (larger
dimensional) linear space ${\mathbb{R}}^{\Pi}$ endows it with the structure of
a polyhedral fan. ∎

######  Theorem 9.2.

The fan $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ is isomorphic to the
cluster fan ${\mathcal{N}}(B^{\vee})$.

###### Proof.

Under the isomorphism
$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}\cong{\mathbb{R}}^{n}$, the fan
structure of $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ gives a complete fan
in ${\mathbb{R}}^{n}$ whose maximal cones are the common domains of linearity
of the piecewise linear functions
$\operatorname{Trop}(u_{\gamma}({\mathbf{y}}))$. By Proposition 4.6, we can
equivalently take the common domains of linearity of the functions
$\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))$, $\gamma\in\Pi^{+}$. It is
well-known (see for example [6, Section 11.1]) that the resulting fan is the
normal fan of the Newton polytope of the Laurent polynomial
$\prod_{\gamma\in\Pi^{+}}F_{\gamma}({\mathbf{y}})$. By Theorem 2.4, we deduce
the isomorphism of fans
$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}\cong{\mathcal{N}}(B^{\vee})$. ∎

Now let ${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ denote a cluster algebra of
finite type, and $X\big{(}{\tilde{B}}\big{)}$ the corresponding cluster
variety. Using the set of cluster variables $x_{\gamma}$, $\gamma\in\Pi$ and
coefficient variables $x_{n+1},\dots,x_{n+m}$, we have an embedding
$X\big{(}{\tilde{B}}\big{)}\hookrightarrow{\mathbb{C}}^{|\Pi|+m}$. We define
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}\subset{\mathbb{R}}^{|\Pi|+m}$
as the closure of the image of
$X\big{(}{\tilde{B}}\big{)}({\mathcal{R}}_{>0})$ under the map ${\rm
val}\colon{\mathcal{R}}_{>0}\to{\mathbb{R}}$. Note that the tropicalization
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}$ depends only on the
cluster algebra ${\mathcal{A}}={\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ and not
on the choice of initial cluster.

The projection map $(x_{\gamma},x_{i})\mapsto(x_{1},x_{2},\dots,x_{n+m})$ from
${\mathbb{R}}^{|\Pi|+m}$ to ${\mathbb{R}}^{n+m}$ (onto the initial cluster
variables) identifies $\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}$
with a complete polyhedral fan in ${\mathbb{R}}^{n+m}$.

######  Proposition 9.3.

Suppose that ${\tilde{B}}$ is of full rank and has finite type $D$. Then
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}$ modulo its lineality
space $L$ is isomorphic to $\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$.

###### Proof.

The translation action of the lineality space $L$ on ${\mathbb{R}}^{n+m}$ is
simply the tropicalization of the action of the automorphism torus
$T\big{(}{\tilde{B}}\big{)}$ (4.1). Thus ${\tilde{B}}$ is of full rank if and
only if $L$ has dimension $m$ if and only if $T\big{(}{\tilde{B}}\big{)}$ is
of dimension $m$. In particular,
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}/L$ is a polyhedral fan of
dimension $n$.

Via the isomorphism of Theorem 4.2, each
$u_{\gamma}=f_{\gamma}\in{\mathbb{C}}[{\mathcal{M}}_{D}]$ can be identified
with a monomial in the cluster and coefficient variables $x_{\gamma}$,
$\gamma\in\Pi$ and $x_{n+1},\dots,x_{n+m}$. We thus have a linear projection
map $p\colon{\mathbb{R}}^{|\Pi|+m}\to{\mathbb{R}}^{|\Pi|}$ (the
tropicalization of the rational map sending $x$-s to $u$-s) mapping
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}$ surjectively to