Families of Gröbner Degenerations, Grassmannians and Universal Cluster
Algebras

Families of Gröbner Degenerations, Grassmannians
and Universal Cluster Algebras

Lara BOSSINGER a, Fatemeh MOHAMMADI bc and Alfredo NÁJERA CHÁVEZ ad

L. Bossinger, F. Mohammadi and A. Nájera Chávez

a) Instituto de Matemáticas UNAM Unidad Oaxaca,
a) León 2, altos, Oaxaca de Juárez, Centro Histórico, 68000 Oaxaca, Mexico
<EMAIL_ADDRESS><EMAIL_ADDRESS>https://www.matem.unam.mx/~lara/,
https://www.matem.unam.mx/~najera/

b) Department of Mathematics: Algebra and Geometry, Ghent University, 9000
Gent, Belgium<EMAIL_ADDRESS>https://www.fatemehmohammadi.com

c) Department of Mathematics and Statistics, UiT – The Arctic University of
Norway,
c) 9037 Tromsø, Norway

d) Consejo Nacional de Ciencia y Tecnología, Insurgentes Sur 1582,
d) Alcaldía Benito Juárez, 03940 CDMX, Mexico

Received October 21, 2020, in final form May 29, 2021; Published online June
10, 2021

Let $V$ be the weighted projective variety defined by a weighted homogeneous
ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We
construct a flat family over $\mathbb{A}^{m}$ that assembles the Gröbner
degenerations of $V$ associated with all faces of $C$. This is a multi-
parameter generalization of the classical one-parameter Gröbner degeneration
associated to a weight. We explain how our family can be constructed from
Kaveh–Manon’s recent work on the classification of toric flat families over
toric varieties: it is the pull-back of a toric family defined by a Rees
algebra with base $X_{C}$ (the toric variety associated to $C$) along the
universal torsor $\mathbb{A}^{m}\to X_{C}$. We apply this construction to the
Grassmannians ${\rm Gr}(2,\mathbb{C}^{n})$ with their Plücker embeddings and
the Grassmannian ${\rm Gr}\big{(}3,\mathbb{C}^{6}\big{)}$ with its cluster
embedding. In each case, there exists a unique maximal Gröbner cone whose
associated initial ideal is the Stanley–Reisner ideal of the cluster complex.
We show that the corresponding cluster algebra with universal coefficients
arises as the algebra defining the flat family associated to this cone.
Further, for ${\rm Gr}(2,\mathbb{C}^{n})$ we show how Escobar–Harada’s
mutation of Newton–Okounkov bodies can be recovered as tropicalized cluster
mutation.

cluster algebras; Gröbner basis; Gröbner fan; Grassmannians; flat
degenerations; Newton–Okounkov bodies

13F60; 14D06; 14M25; 14M15; 13P10

##  1 Introduction

The theory of Gröbner fans, introduced by Mora and Robbiano [40], is a popular
tool in commutative algebra, and one of its modern applications is
degenerating ideals into simpler ones such as monomial, binomial or toric
ideals. More precisely, let $\mathbb{K}$ be an algebraically closed field and
$J\subseteq\mathbb{K}[x_{1},\dots,x_{n}]$ a ${\bf d}$-weighted homogeneous
ideal for some ${\bf d}\in\mathbb{Z}_{>0}^{n}$. The Gröbner fan of $J$ is a
complete fan in $\mathbb{R}^{n}$ whose elements represent weight vectors on
the variables $x_{1},\dots,x_{n}$. Two weight vectors lie in the same open
cone if and only if they give rise to the same initial ideal of $J$, see
def:init. The ideal $J$ defines a weighted projective variety $V$ inside the
weighted projective space $\mathbb{P}({\bf d})$. Every open cone in the
Gröbner fan gives rise to a one-parameter flat family degenerating $V$ to the
variety defined by the associated initial ideal of $J$. This construction is
realized by choosing a weight in the relative interior of the cone.

We modify the classical one-parameter construction as follows: for a maximal
cone $C$ in the Gröbner fan of $J$ we choose integral generators of its rays
$r_{1},\dots,r_{m}$ and denote by ${\bf r}$ the $(m\times n)$-matrix whose
rows are $r_{1},\dots,r_{m}$. For an element
$f=\sum_{\alpha\in\mathbb{Z}_{\geq 0}^{n}}c_{\alpha}{\bf x}^{\alpha}$ of $J$,
we define $\mu(f)\in\mathbb{Z}^{m}$ as the vector whose $i$-th entry is
$\min_{c_{\alpha}\not=0}\\{r_{i}\cdot\alpha\\}$. Then the lift of $f$ is

$\tilde{f}:=f\big{(}{\bf t}^{{\bf r}\cdot e_{1}}x_{1},\dots,{\bf t}^{{\bf
r}\cdot e_{n}}x_{n}\big{)}{\bf t}^{-\mu(f)},$

where ${\bf t}^{\bf a}$ for ${\bf a}\in\mathbb{Z}^{m}$ denotes the monomial
$\prod_{i=1}^{m}t_{i}^{a_{i}}$. The lifted ideal
$\tilde{J}\subseteq\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]$ is the
ideal generated by the lifts of all polynomials in $J$. We prove that
$\tilde{J}$ is generated by the lifts of elements of the reduced Gröbner basis
for $J$ and $C$, see prop:lifted generators. The lifts of these elements are
independent of the choice of ${\bf r}$ and homogeneous with respect to the
${\bf d}$-grading on $x_{i}$’s, see prop:change rays. Consequently,
$\tilde{J}$ is independent of the choice of ${\bf r}$ and it defines a variety
inside $\mathbb{P}({\bf d})\times\mathbb{A}^{m}$. Our first main result is the
following.

######  Theorem 1.1 (thm:family).

Let $J$ be a weighted homogeneous ideal, $C$ a maximal cone in the Gröbner fan
of $J$ and ${\bf r}$ an $(m\times n)$-matrix whose rows are integral ray
generators of $C$. Then the algebra
$\tilde{A}:=\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]/\tilde{J}$ is a
free $\mathbb{K}[t_{1},\dots,t_{m}]$-algebra. It defines a flat family

${{\rm Proj}\big{(}\tilde{A}\big{)}}$${\mathbb{P}({\bf
d})\times\mathbb{A}^{m}}$${\mathbb{A}^{m}}$$\scriptstyle{\pi}$

such that for every face $\tau$ of $C$ there exists ${\bf
a}_{\tau}\in\mathbb{A}^{m}$ with fiber $\pi^{-1}({\bf a}_{\tau})$ isomorphic
to the variety defined by the initial ideal associated to $\tau$. In
particular, generic fibers are isomorphic to ${\rm Proj}(A)$, where
$A=\mathbb{K}[x_{1},\dots,x_{n}]/J$, and there exist special fibers for every
proper face $\tau\subset C$.

Next, we explain how the algebra $\tilde{A}$ arises in Kaveh–Manon’s recent
work on the classification of affine toric flat families of finite type over
toric varieties [33]. Consider a fan $\Sigma$ defining a toric variety
$X_{\Sigma}$ which contains a dense torus $T$. Then a toric family is a
$T$-equivariant flat sheaf $\mathcal{A}$ of positively graded algebras of
finite type over $X_{\Sigma}$ such that: $(i)$ the relative spectrum of
$\mathcal{A}$ has reduced fibers and $(ii)$ its generic fibers are isomorphic
to the spectrum of some positively graded $\mathbb{K}$-algebra $A$. Such
families are classified by so-called PL-quasivaluations on $A$ whose codomain
is the semifield of piecewise linear functions on the intersection of $\Sigma$
with the cocharacter lattice of $T$ (see [33, Section 1.1] or Section 3.1
below). Given a PL-quasivaluation, Kaveh–Manon construct a sheaf of Rees
algebras on $X_{\Sigma}$ that is a toric family. In the special case when
$\Sigma$ is a cone, their construction yields a single Rees algebra rather
than a sheaf. We prove the following result:

######  Theorem 1.2 (thm:KM max cone).

Let $J\subseteq\mathbb{K}[{x_{1},\dots,x_{n}}]$ be a weighted homogeneous
ideal and $C$ a maximal cone in the Gröbner fan of $J$. Let
$\mathcal{R}_{C}(A)$ be the Rees algebra associated to the PL-quasivaluation
on $A=\mathbb{K}[{x_{1},\dots,x_{n}}]/J$ defined by $C$ and let
$\psi\colon\operatorname{Spec}(\mathcal{R}_{C}(A))\to X_{C}$ be the
corresponding toric family. The morphism
$\pi\colon\operatorname{Spec}\big{(}\tilde{A}\big{)}\to\mathbb{A}^{m}$ fits
into a pull-back diagram as follows:

${\operatorname{Spec}(\mathcal{R}_{C}(A))}$${\operatorname{Spec}\big{(}\tilde{A}\big{)}}$${X_{C}}$${\mathbb{A}^{m}.}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$$\scriptstyle{p_{C}}$

Here $p_{C}\colon\mathbb{A}^{m}\to X_{C}$ is the universal torsor of $X_{C}$
$($see (3.9) for details$)$ and $m$ is the number of rays of $C$. Both flat
families, the one defined by $\psi$ and the one defined by $\pi$, are direct
sums of line bundles indexed by the standard monomial basis associated to the
maximal cone $C$.

Toric degenerations are a particular class of flat families with a single
special fiber that is a toric variety. They are of significant interest and
have been widely studied in recent years. Among the faces of the maximal cone
$C$ one may find binomial prime initial ideals that hence define toric Gröbner
degenerations of $V$. Such faces lie in the tropicalization of $J$ denoted
$\operatorname{Trop}(J)$, which by definition is the subfan of the Gröbner fan
consisting of cones whose associated initial ideal does not contain monomials.
Let $\Sigma$ be the intersection of $C$ with the tropicalization of $J$. Then
the affine space $\mathbb{A}^{m}$ contains the universal torsor
$\mathcal{T}_{\Sigma}$ of the toric variety $X_{\Sigma}$ and all fibers of
$\pi^{-1}(\mathcal{T}_{\Sigma})\to\mathcal{T}_{\Sigma}$ correspond to initial
ideals of cones in $\Sigma$. In thm:KM vs grobner we show how the family
defined by $\pi|_{\pi^{-1}(\mathcal{T}_{\Sigma})}$ also arises as a pull-back
from a toric family defined by a sheaf of Rees algebras on $X_{\Sigma}$.

Grassmannians and cluster algebras. The interest of studying toric
degenerations in the context of cluster algebras has grown in the last years
(see for example [9, 28]). Therefore, we would like to understand the
framework introduced above from the perspective of cluster algebras. As a
first step in this direction, we analyze in depth the situation for the
Grassmannians $\operatorname{Gr}(2,\mathbb{C}^{n})$ and
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$ whose coordinate rings are
cluster algebras.

For $\operatorname{Gr}(2,\mathbb{C}^{n})$ we choose its Plücker embedding and
obtain the homogeneous coordinate ring $A_{2,n}$ as a quotient of the
polynomial ring in Plücker variables $\mathbb{C}[p_{ij}\colon 1\leq i<j\leq
n]$ by the Plücker ideal $I_{2,n}$. It was shown in [23] that $A_{2,n}$ is a
cluster algebra in which the cluster variables are in one-to-one
correspondence with Plücker coordinates $\bar{p}_{ij}\in A_{2,n}$ and the
seeds are in one-to-one correspondence with triangulations of the $n$-gon.
Every triangulation of the $n$-gon gives rise to a toric degeneration of
$\operatorname{Gr}(2,\mathbb{C}^{n})$ obtained by adding principal
coefficients to $A_{2,n}$ at the corresponding seed [28]. Principal
coefficients were introduced by Fomin and Zelevinsky in [25] and are a core
concept in the theory of cluster algebras. A central piece of the construction
is endowing every cluster variable (i.e., every Plücker coordinate) with a so-
called ${\bf g}$-vector depending on the fixed seed. We want to understand the
toric degenerations coming from principal coefficients in the context of
Gröbner theory: to achieve this first fix a triangulation $T$. We use ${\bf
g}$-vectors for Plücker coordinates to construct a weight vector ${\bf
w}_{T}$, see def:wt of T. In thm:toric wt ideals we prove that the initial
ideal with respect to ${\bf w}_{T}$ is binomial and prime, hence ${\bf w}_{T}$
lies in the tropicalization of $I_{2,n}$. Moreover, the central fiber of the
Gröbner degeneration induced by ${\bf w}_{T}$ is isomorphic to the central
fiber of the toric degeneration induced by endowing $A_{2,n}$ with principal
coefficients at the seed determined by $T$, see cor:central fibre.

There is a single object in cluster theory that simultaneously encodes
principal coefficients at all seeds and thus the ${\bf g}$-vectors of all
Plücker coordinates with respect to all triangulations. Namely, the cluster
algebra with universal coefficients $A^{\rm univ}_{2,n}$ associated to
$A_{2,n}$, see def:Plucker univ. In spirit, this algebra is very similar to
the algebra defining the flat family associated to a maximal cone in the
Gröbner fan of $I_{2,n}$ as it encodes various toric degenerations of
$\operatorname{Gr}(2,\mathbb{C}^{n})$ at the same time. It is therefore
natural to ask if the cluster algebra with universal coefficients fits into
the above framework. The following result answers this question for
$\operatorname{Gr}(2,\mathbb{C}^{n})$:

######  Theorem 1.3.

There exists a maximal cone $C$ in the Gröbner fan of $I_{2,n}$ whose rays are
in bijection with diagonals of the $n$-gon. Moreover, the unique cone $C$ has
the following properties:

  * $(i)$

The standard monomial basis for $A_{2,n}$ associated with $C$ coincides with
the basis of cluster monomials $($cluster vs SM basis$)$.

  * $(ii)$

For every triangulation $T$ of the $n$-gon the weight vector ${\bf w}_{T}$
lies in the boundary of $C$ and the intersection of $C$ with the
tropicalization of $I_{2,n}$ is the totally positive part of
$\operatorname{Trop}(I_{2,n})$ $($prop:unique mono ideal and thm:trop+$)$.

  * $(iii)$

Let $\tilde{I}_{2,n}$ be the lift of $I_{2,n}$ with respect to $C$ and denote
by $\tilde{A}_{2,n}$ the quotient $\mathbb{C}[t_{ij}\colon 2\leq i+1<j\leq
n][p_{ij}\colon 1\leq i<j\leq n]/\tilde{I}_{2,n}$. Then there exists a
canonical isomorphism between the cluster algebra with universal coefficients
$A_{2,n}^{\rm univ}$ and $\tilde{A}_{2,n}$ $($thm:A univ as quotient$)$.

  * $(iv)$

The monomial initial ideal of $I_{2,n}$ with respect to $C$ is squarefree and
coincides with the Stanley–Reisner ideal of the cluster complex
$($cor:Stanley-Reisner$)$.

Similarly, we describe the cluster algebra with universal coefficients for
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$ from the viewpoint of
Gröbner theory. In this case, we fix its cluster embedding obtained as
follows: consider the Plücker embedding and let $A_{3,6}$ be the homogeneous
coordinate ring. As a cluster algebra of type $\mathtt{D}_{4}$, $A_{3,6}$ has
22 cluster variables, 20 of which are the Plücker coordinates. The additional
two cluster variables are homogeneous binomials in Plücker coordinates of
degree 2. Hence, we can present $A_{3,6}$ as the quotient of a polynomial ring
in 22 variables by a weighted homogeneous ideal denoted by $I^{\rm ex}$. This
yields an embedding of $\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$ in
the weighted projective space $\mathbb{P}(1,\dots,1,2,2)$, where the Plücker
coordinates correspond to the coordinates of weight one, and the additional
cluster variables to those of weight two, see Section 4.4. Our main result is
the following:

######  Theorem 1.4.

There exists a unique maximal cone $C$ in the Gröbner fan of $I^{\rm ex}$ such
that

  * $(i)$

The algebra $\tilde{A}_{3,6}$ is canonically isomorphic to the cluster algebra
with universal coefficients $A^{\rm univ}_{3,6}$, where rays of $C$ are
identified with mutable cluster variables of $A_{3,6}$ $($Theorem
$\ref{thm:Gr36}(ii))$.

  * $(ii)$

For every seed of $A_{3,6}$ there exists a face of $C$ whose associated
initial ideal is a totally positive binomial prime ideal. More precisely, the
intersection of $C$ with $\operatorname{Trop}(I^{\rm ex})$ is the totally
positive part of $\operatorname{Trop}(I^{\rm ex})$ $($thm:Gr36$(iii))$.

  * $(iii)$

The monomial initial ideal $\operatorname{in}_{C}(I^{\rm ex})$ associated to
$C$ is squarefree and coincides with the Stanley–Reisner ideal of the cluster
complex. In particular, the basis of standard monomials associated to $C$
coincides with the basis of cluster monomials $($cor:Stanley-Reisner$)$.

As mentioned before, the toric fibers of the above families are of particular
interest. They arise from maximal cones in the tropicalization whose initial
ideal is binomial and prime; such cones are called maximal prime cones. The
corresponding projective toric varieties (respectively, their normalizations)
have associated polytopes. Following [34] these polytopes can be realized as
Newton–Okounkov bodies of full rank valuations constructed from maximal prime
cones. In the recent preprint [19] Escobar and Harada study wall-crossing
phenomena of Newton–Okounkov bodies associated to maximal prime cones that
intersect in a facet. They give piecewise linear maps called flip and shift
that relate the Newton–Okounkov bodies. For cluster algebras like $A_{3,6}$
and $A_{2,n}$ it has been shown in several cases that Newton–Okounkov bodies
can equivalently be obtained from the cluster structure (see, e.g., [6, 8, 26,
46]). Hence, one might wonder how Escobar–Harada’s wall-crossing formulas can
be understood in the context of cluster algebras. In the particular case of
$A_{2,n}$ each Newton–Okounkov body arising from
$\operatorname{Trop}(I_{2,n})$ is unimodularly equivalent to the convex hull
of g-vectors of Plücker coordinates for some triangulation. More precisely, we
obtain the following result:

######  Corollary 1.5 (cor:mutate NObodies).

Let $\sigma_{1}$ and $\sigma_{2}$ be two maximal prime cones in
$\operatorname{Trop}(I_{2,n})$ that intersect in a facet. Then their
associated Newton–Okounkov bodies are $($up to unimodular equivalence$)$
related by a shear map obtained from tropicalizing a cluster mutation.

In combination with Escobar–Harada’s results about the flip map for
$\operatorname{Gr}(2,\mathbb{C}^{n})$ this corollary implies that it is of
cluster nature (for details see Section 4.6).

We would like to remark that this paper is not the first to make a connection
between cluster algebras and Gröbner theory. In [41] Muller, Rajchgot and
Zykoski obtained presentations for lower bounds of cluster algebras using
Gröbner theory. Further, it is worth noticing that the theory of universal
coefficients for cluster algebras is particularly well-developed for finite
and surface type cluster algebras, see [44]. We believe that the above results
can be extended to projective varieties containing a cluster variety of finite
type. It is further an interesting and challenging problem to extend the
results of this paper to cluster algebras of (infinite) surface type as it
would involve Gröbner theory of non-Noetherian algebras.

Structure of the paper. In Section 2 we recall the background on weighted
projective varieties and Gröbner basis theory. In Section 3 we introduce the
construction of the flat families and prove the main theorem. We relate to
Kaveh–Manon’s work in Section 3.1. In Section 4 we turn to the Grassmannians
$\operatorname{Gr}(2,\mathbb{C}^{n})$ and
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$. We recall the background on
cluster algebras and universal coefficients in Section 4.1. We explain in
detail how toric degenerations from cluster algebras arise as Gröbner
degenerations for $\operatorname{Gr}(2,\mathbb{C}^{n})$ in Section 4.2. Then
we apply the main construction to $\operatorname{Gr}(2,\mathbb{C}^{n})$ in
Section 4.3 and afterwards to
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$ in Section 4.4. We explore
further connections to Escobar–Harada’s work in Section 4.6. Finally, in the
Appendix A we present computational results used for the application to
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$.

##  2 Preliminaries

We first fix our notation throughout the note. Let $\mathbb{K}$ be an
algebraically closed field. We are mainly interested in the case when
$\mathbb{K}=\mathbb{C}$. In the polynomial ring
$\mathbb{K}[x_{1},\dots,x_{n}]$, we fix the notation ${\bf x}^{\alpha}$ with
$\alpha=(a_{1},\ldots,a_{n})\in\mathbb{Z}_{\geq 0}^{n}$ denoting the monomial
$x_{1}^{a_{1}}\dots x_{n}^{a_{n}}$. Throughout when we write
$f=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf x}^{\alpha}$ for
$f\in\mathbb{K}[x_{1},\dots,x_{n}]$ we refer to the expression of $f$ in the
basis of monomials.

###  2.1 Weighted projective varieties

Fix a vector ${\bf d}=(d_{1},\dots,d_{n})\in\mathbb{R}^{n}$. Let
$\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ be the polynomial ring
$\mathbb{K}[x_{1},\dots,x_{n}]$ endowed with the $\mathbb{R}$-grading
determined by setting $\text{deg}(x_{i})=d_{i}$ for $i=1,\dots,n$.

######  Definition 2.1.

def:d homog For $c\in\mathbb{K}^{\ast}$ and
$\alpha=(a_{1},\dots,a_{n})\in\mathbb{Z}_{\geq 0}^{n}$ the weight of the
monomial $c{\bf x}^{\alpha}\in\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ is ${\bf
d}\cdot\alpha=\sum_{i=1}^{n}d_{i}a_{i}$. We extend this definition and say
$f\in\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ is d-homogeneous if $f$ can be
expressed as a sum of monomials of the same weight. Similarly, an ideal
$J\subseteq\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ is d-homogeneous if it is
generated by ${\bf d}$-homogeneous elements.

Assume ${\bf d}\in\mathbb{Z}^{n}_{>0}$. In this case $\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]$ is $\mathbb{Z}$-graded and we can define the weighted
projective space $\mathbb{P}(d_{1},\dots,d_{n})$ as the quotient of
$\mathbb{K}^{n}$ under the equivalence relation

$(x_{1},\dots,x_{n})\sim\big{(}\lambda^{d_{1}}x_{1},\dots,\lambda^{d_{n}}x_{n}\big{)}\qquad\text{for}\quad\lambda\in\mathbb{K}^{\ast}.$

For $\mathbb{K}=\mathbb{C}$ it is well-known that
$\mathbb{P}(d_{1},\dots,d_{n})$ is a projective toric variety [15]. Moreover,
as a scheme $\mathbb{P}(d_{1},\dots,d_{n})$ is the homogeneous spectrum (or
Proj) of $\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ [17, Notation 1.1]. For
simplicity, we denote $\mathbb{P}(d_{1},\dots,d_{n})$ by $\mathbb{P}({\bf d})$
and let $[x_{1}:\dots:x_{n}]$ denote the class of $(x_{1},\dots,x_{n})$ in
$\mathbb{P}({\bf d})$. Given a d-homogeneous ideal $J\subseteq\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]$ its set of zeros is

$V(J)=\big{\\{}[x_{1}:\dots:x_{n}]\in\mathbb{P}({\bf d})\colon
f(x_{1},\dots,x_{n})=0\text{ for all }f\in J\big{\\}}.$

Conversely, given a subset $X\subseteq\mathbb{P}(d_{1},\dots,d_{n})$, its
associated ideal $I(X)$ is generated by polynomials:

$\big{\\{}f\in\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]\colon f\text{ is }{\bf
d}\text{-homogeneous and }f(p)=0\text{ for all }p\in X\big{\\}}.$

Subsets of $\mathbb{P}({\bf d})$ of the form $V(J)$ for some weighted
projective ideal $J$ are called weighted projective varieties. These sets are
the closed sets of a Zariski-type topology on $\mathbb{P}({\bf d})$. The
theory of weighted projective varieties is very similar to the theory of usual
projective varieties and background on this theory can be found for example in
[17, 30, 45]. In particular, the projective space $\mathbb{P}^{n-1}$ can be
realized as $\mathbb{P}^{n-1}=\mathbb{P}({\bf 1})$, where ${\bf
1}=(1,\dots,1)\in\mathbb{Z}^{n}$. So every projective variety can be
considered as a weighted projective variety of $\mathbb{P}({\bf 1})$.

######  Definition 2.2.

The weighted homogeneous coordinate ring of a weighted projective variety
$X\subseteq\mathbb{P}({\bf d})$ is defined as

$S(X):=\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]/I(X).$

By construction, $S(X)$ is a positively graded ring. Moreover, the weighted
homogeneous coordinate ring of a projective variety (considered as a weighted
projective variety) coincides with its homogeneous coordinate ring. There is a
natural notion of morphism between weighted projective varieties which we will
not need in this work. It will be sufficient to recall that if
$X\subseteq\mathbb{P}(d_{1},\dots,d_{n})$ and
$Y\subseteq\mathbb{P}(d^{\prime}_{1},\dots,d^{\prime}_{m})$ are weighted
projective varieties, then an isomorphism between the graded rings $S(X)$ and
$S(Y)$ induces an isomorphism of the weighted projective varieties $X$ and
$Y$.

###  2.2 Gröbner basis theory

We first review some results in Gröbner basis theory in order to fix our
notation and keep the paper self-contained. Most of the material here is well-
known and we refer to Buchberger’s thesis [11] and standard books (e.g., [1,
29, 37, 51]) for proofs and more details.

######  Definition 2.3.

def:init Let $f=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}\in\mathbb{K}[x_{1},\dots,x_{n}]$. Given a weight vector
$w\in\mathbb{R}^{n}$ the initial form of $f$ with respect to $w$ is defined as

$\operatorname{in}_{w}(f):=\sum_{\alpha\colon w\cdot\alpha=a}c_{\alpha}{\bf
x}^{\alpha},$

where $a=\min\\{w\cdot\alpha\colon c_{\alpha}\not=0\\}$. For an ideal
$J\subseteq\mathbb{K}[x_{1},\dots,x_{n}]$ its initial ideal with respect to
$w$ is defined as
$\operatorname{in}_{w}(J)\colon=\langle\operatorname{in}_{w}(f)\colon f\in
J\rangle$. A finite set $\mathcal{G}=\\{g_{1},\ldots,g_{r}\\}\subseteq J$ is
called a Gröbner basis for $J$ with respect to $w$ if
$\operatorname{in}_{w}(J)=\langle\operatorname{in}_{w}(g_{1}),\ldots,\operatorname{in}_{w}(g_{r})\rangle$.

######  Definition 2.4.

def:mono order A monomial term order on $\mathbb{K}[x_{1},\dots,x_{n}]$ is a
total order $<$ on the set of monic monomials in
$\mathbb{K}[x_{1},\dots,x_{n}]$ such that for every $\alpha,\beta,\gamma$ in
$\mathbb{Z}_{\geq 0}^{n}$ we have that

$(i)\quad 1\leq{\bf
x}^{\alpha},\qquad\text{and}\qquad(ii)\quad\text{if}\quad{\bf x}^{\alpha}<{\bf
x}^{\beta},\quad\text{then}\quad{\bf x}^{\alpha+\gamma}<{\bf
x}^{\beta+\gamma}.$

The initial monomial of an element $f=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq
0}}c_{\alpha}{\bf x}^{\alpha}\in\mathbb{K}[x_{1},\dots,x_{n}]$ with respect to
$<$ is $\operatorname{in}_{<}(f):=c_{\beta}{\bf x}^{\beta}$, where ${\bf
x}^{\beta}=\max_{<}\\{{\bf x}^{\alpha}\colon c_{\alpha}\not=0\\}$. The initial
ideal of an ideal $J\subseteq\mathbb{K}[x_{1},\dots,x_{n}]$ with respect to
$<$ is defined as
$\operatorname{in}_{<}(J):=\langle\operatorname{in}_{<}(f)\colon f\in
J\rangle$.

Recall that given an ideal $J\subseteq\mathbb{K}[x_{1},\dots,x_{n}]$ and a
monomial term order $<$ there always exists a weight vector
$-w\in\mathbb{Z}_{\geq 0}^{n}$ such that
$\operatorname{in}_{w}(J)=\operatorname{in}_{<}(J)$, see, e.g., [29, Theorem
3.1.2] (Note the sign switch here, this is due to our $\min$-convention for
initial ideals with respect to weight vectors and $\max$-convention for
initial ideals with respect to monomial term orders.) On the other hand, if
$\operatorname{in}_{w}(J)$ is generated by monomials and there exists a
monomial term order $<$ such that
$\operatorname{in}_{w}(J)=\operatorname{in}_{<}(J)$, then we say that $w$ is
compatible with $<$.

######  Definition 2.5.

def:SMT Let $\operatorname{in}_{<}(J)$ be a monomial initial ideal of the
ideal $J$ for some monomial term order $<$ on
$\mathbb{K}[x_{1},\ldots,x_{n}]$. Then the set $\mathbb{B}_{<}:=\\{\bar{\bf
x}^{\alpha}\colon{\bf x}^{\alpha}\not\in\operatorname{in}_{<}(J)\\}$ is a
vector space basis of $\mathbb{K}[x_{1},\dots,x_{n}]/J$ (and
$\mathbb{K}[x_{1},\dots,x_{n}]/\operatorname{in}_{<}(J)$) called standard
monomial basis.

######  Theorem 2.6 (the division algorithm).

division alg Fix a monomial term order $<$ on $\mathbb{K}[x_{1},\dots,x_{n}]$
and let $g_{1},\dots,g_{s}$ be non-zero polynomials in
$\mathbb{K}[x_{1},\dots,x_{n}]$. Given
$0\not=f\in\mathbb{K}[x_{1},\dots,x_{n}]$ there exist
$f_{1},\dots,f_{s},f^{\prime}\in\mathbb{K}[x_{1},\dots,x_{n}]$ with
$f=f_{1}g_{1}+\dots+f_{s}g_{s}+f^{\prime}$, such that

  * $(i)$

if $f^{\prime}\neq 0$ then no monomial of $f^{\prime}$ is divisible by any of
the monomials
$\operatorname{in}_{<}(g_{1}),\dots,\operatorname{in}_{<}(g_{s});$

  * $(ii)$

if $f_{i}\not=0$ then
$\operatorname{in}_{<}(f)\geq\operatorname{in}_{<}(f_{i}g_{i})$ for all $i$.

We say that $f$ reduces to $f^{\prime}$ with respect to
$\\{g_{1},\dots,g_{s}\\}$.

######  Definition 2.7.

Let $<$ be a monomial term order on $\mathbb{K}[x_{1},\dots,x_{n}]$ and
$\mathcal{G}=\\{g_{1},\ldots,g_{s}\\}$ a finite generating set of an ideal
$J$. Then the $S$-polynomial of $g_{i}$ and $g_{j}$ is defined as

$S(g_{i},g_{j}):=\frac{{\rm
lcm}\big{(}\operatorname{in}_{<}(g_{i}),\operatorname{in}_{<}(g_{j})\big{)}}{\operatorname{in}_{<}(g_{i})}g_{i}-\frac{{\rm
lcm}\big{(}\operatorname{in}_{<}(g_{i}),\operatorname{in}_{<}(g_{j})\big{)}}{\operatorname{in}_{<}(g_{j})}g_{j}.$

We say that Buchberger’s criterion holds if for all $1\leq i<j\leq s$, the
$S$-pairs reduce to zero with respect to $\\{g_{1},\dots,g_{s}\\}$. If
Buchberger’s criterion holds, then $\mathcal{G}$ forms a Gröbner basis for $J$
with respect to $<$. Moreover, a Gröbner basis $\mathcal{G}$ for $J$ with
respect to $<$ is reduced if

  * $(i)$

$\operatorname{in}_{<}(g_{i})$ is monic for all $1\leq i\leq s$, and

  * $(ii)$

for $i\neq j$ no monomial in $g_{i}$ is divisible by
$\operatorname{in}_{<}(g_{j})$.

A reduced Gröbner basis for $J$ with respect to $<$ always exists and is
unique, see, e.g., [29, Theorem 2.2.7]. We let $\mathcal{G}_{<}(J)$ denote the
reduced Gröbner basis for $J$ with respect to $<$.

Studying all possible initial ideals of a given ideal leads to the notion of
Gröbner fan (see, e.g., [51, Proposition 2.4]):

######  Definition 2.8.

def:Gfan Let $J\subseteq\mathbb{K}[x_{1},\dots,x_{n}]$ be an ideal. The
Gröbner region of $J$ denoted by $\text{GR}(J)$ is the set of
$w\in\mathbb{R}^{n}$ such that there exists a monomial term order $<$ with
$\operatorname{in}_{<}(\operatorname{in}_{w}(J))=\operatorname{in}_{<}(J)$.
The Gröbner fan of $J$ denoted by $\operatorname{GF}(J)$ is a fan with support
$\text{GR}(J)$ in which a pair of elements $u,w\in\mathbb{R}^{n}$ lie in the
relative interior of the same cone $C\subset\mathbb{R}^{n}$ (denoted by
$C^{\circ}$) if and only if
$\operatorname{in}_{u}(J)=\operatorname{in}_{w}(J)$. We introduce the notation
$\operatorname{in}_{C}(J):=\operatorname{in}_{w}(J)$ for any $w\in C^{\circ}$.
By definition of $\text{GR}(J)$, for every full-dimensional cone $C$, there
exists a monomial term order $<$ such that $C$ is the topological closure of
$\\{w\in\mathbb{R}^{n}\colon\operatorname{in}_{w}(J)=\operatorname{in}_{<}(J)\\}$.
Moreover, we define the lineality space $\mathcal{L}(J)$ as the linear
subspace of $\operatorname{GF}(J)$ that contains all elements $u$ for which
$\operatorname{in}_{u}(J)=J$.

Integral weight vectors in $\operatorname{GF}(J)$ can be seen as points in a
lattice $N=\mathbb{Z}^{n}$ whose dual lattice $M=(\mathbb{Z}^{n})^{*}$
contains exponent vectors of monomials in $\mathbb{K}[x_{1},\dots,x_{n}]$.
Consequently, for $w\in N$ and $\alpha\in M$ we denote by $w\cdot\alpha$ the
pairing between the two lattices.

######  Remark 2.9.

Proposition 15.16 in [18] gives a criterion for whether a weight vector is
compatible with a monomial term order $<$ for $J$, or not. Namely, a weight
$w$ is compatible with a monomial term order $<$ if and only if
$\operatorname{in}_{w}(g)=\operatorname{in}_{<}(g)$ for every element of
$\mathcal{G}_{<}(J)$. Let $C$ be the topological closure of the corresponding
Gröbner cone of $<$, then $w\in C$ if and only if
$\operatorname{in}_{<}(g)=\operatorname{in}_{<}(\operatorname{in}_{w}(g))$ for
every $g\in\mathcal{G}_{<}(J)$.

######  Lemma 2.10.

lem:wt homog Let ${\bf d}\in\mathbb{Z}_{>0}^{n}$ and
$J\subseteq\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ be a ${\bf d}$-homogeneous
ideal. Then ${\bf d}\in\mathcal{L}(J)$. More generally, $\mathcal{L}(J)$
contains all elements ${\bf u}\in\mathbb{R}^{n}$ such that $J$ is ${\bf
u}$-homogeneous.

###### Proof.

Observe that for every ${\bf d}$-homogeneous polynomial $f\in\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]$ the equality $\operatorname{in}_{\bf d}(f)=f$ holds.
Next, every $f\in J$ can be uniquely written as a finite sum
$f=\sum_{i=0}^{d}f_{i}$ for some $d\in\mathbb{Z}_{\geq 0}$ and $f_{i}\in J$ of
d-degree $i$ (see [30, Lemma 3.0.7]). Then $\operatorname{in}_{\bf
d}(f)=f_{j}$, where $j=\min\\{i\colon f_{i}\neq 0\\}$. In particular, for any
$f\in J$ we have that $\operatorname{in}_{\bf d}(f)\in J$. Therefore,
$\operatorname{in}_{\bf d}(J)=J$. ∎

######  Lemma 2.11.

lem:init ell Fix an arbitrary monomial term order $<$. Then for every
$\ell\in\mathcal{L}(J)$ and every $g\in\mathcal{G}_{<}(J)$ we have
$\operatorname{in}_{\ell}(g)=g$. In particular, if $J$ is ${\bf
d}$-homogeneous then so is every $g\in\mathcal{G}_{<}(J)$.

###### Proof.

Consider a Gröbner basis $F:=\\{f_{1},\dots,f_{t}\\}$ of $J$ with respect to a
fixed element $\ell\in\mathcal{L}(J)$. Then $J=\operatorname{in}_{\ell}(J)$ is
generated by
$\operatorname{in}(F):=\\{\operatorname{in}_{\ell}(f_{1}),\dots,\operatorname{in}_{\ell}(f_{t})\\}$.
In particular, we have
$\operatorname{in}_{\ell}(\operatorname{in}_{\ell}(f))=\operatorname{in}_{\ell}(f)$
for all $f\in F$. From the set $F$ we can construct the reduced Gröbner basis
$\mathcal{G}_{<}(J)$ by doing the following three steps: First, extend $F$ to
a Gröbner basis $H$ with respect to $<$ by adding $S$-pairs using Buchberger’s
criterion. Then if necessary eliminate elements from $H$ until the outcome is
a minimal Gröbner basis $G$ (see [1, Corollary 1.8.3]). Finally, reduce all
elements in $G$ to obtains $\mathcal{G}_{<}(J)$ (see [1, Corollary 1.8.6]). We
invite the reader to verify that the property $\operatorname{in}_{\ell}(g)=g$
holds for all $g\in H$, hence for all $g\in G$ and finally for all
$g\in\mathcal{G}_{<}(J)$. ∎

Recall from [40, Corollary 5.7] that for a ${\bf d}$-homogeneous ideal
$J\subseteq\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ the support of its Gröbner
fan, i.e., $\text{GR}(J)$, is $\mathbb{R}^{n}$. In other words, there exists a
compatible monomial term order for every maximal cone
$C\in\operatorname{GF}(J)$.

######  Lemma 2.12.

lem:facet Let $J\subseteq\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ be a ${\bf
d}$-homogeneous ideal, $C\in\operatorname{GF}(J)$ a maximal cone, and $<$ a
compatible monomial term order of $C$. Consider $v\in C\setminus C^{\circ}$
and $u\in\mathbb{R}^{n}$ such that $w:=u+v\in C^{\circ}$. Then for every
$g\in\mathcal{G}_{<}(J)$, we have
$\operatorname{in}_{<}(g)=\operatorname{in}_{w}(g)=\operatorname{in}_{u}(\operatorname{in}_{v}(g))$.

###### Proof.

Consider an element $g$ in $\mathcal{G}_{<}(J)$. Since $w\in C^{\circ}$ we
have $\operatorname{in}_{<}(g)=\operatorname{in}_{w}(g)$. On the other hand,
since $v\in C$, by Remark 2.9 we have that
$\operatorname{in}_{<}(g)=\operatorname{in}_{<}(\operatorname{in}_{v}(g))$.
This implies that $\operatorname{in}_{w}(g)$ is a refinement of
$\operatorname{in}_{v}(g)$. In other words, $\operatorname{in}_{v}(g)$
contains the monomial $\operatorname{in}_{w}(g)$ with possibly some extra
terms which will disappear after taking its initial form with respect to $u$.
∎

We note that lem:facet does not hold for arbitrary elements of $J$. See
exp:lem facet.

######  Lemma 2.13.

lem:strictly_convex For every cone
$C\in\operatorname{GF}(J)\subseteq\mathbb{R}^{n}$, the cone
$\overline{C}:=C/\mathcal{L}(J)\subseteq\mathbb{R}^{n}/\mathcal{L}(J)$ is
strongly convex, that is $\overline{C}\cap(-\overline{C})=\\{0\\}$.

###### Proof.

As we have chosen $J$ to be ${\bf d}$-homogeneous every cone in
$\operatorname{GF}(J)$ is a face of a full-dimensional cone. Hence, we may
assume without loss of generality that $C$ is a cone of dimension $n$. Let $<$
be the corresponding monomial term order and $g\in\mathcal{G}_{<}(J)$. Let $v$
and $-v$ be in $C\setminus C^{\circ}$ and choose
$u,u^{\prime}\in\mathbb{R}^{n}$ such that $u+v,u^{\prime}-v\in C^{\circ}$. By
lem:facet we have that
$\operatorname{in}_{<}(g)=\operatorname{in}_{u}(\operatorname{in}_{v}(g))=\operatorname{in}_{u^{\prime}}(\operatorname{in}_{-v}(g))$.
However, $\operatorname{in}_{<}(g)$ cannot simultaneously be a monomial in
$\operatorname{in}_{v}(g)$ and $\operatorname{in}_{-v}(g)$ unless
$v\in\mathcal{L}(J)$. ∎

######  Definition 2.14.

def:trop For a ${\bf d}$-homogeneous ideal $J\subseteq\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]$ we define its tropicalization

$\operatorname{Trop}(J):=\big{\\{}w\in\mathbb{R}^{n}\colon\operatorname{in}_{w}(J)\text{
does not contain any monomial}\big{\\}}.$

In fact, $\operatorname{Trop}(J)$ is a $d$-dimensional subfan of
$\operatorname{GF}(J)$, where $d$ is the Krull-dimension of $\mathbb{K}_{\bf
d}[x_{1},\dots,\linebreak x_{n}]/J$.

######  Definition 2.15.

def:trop+ An ideal $J\subseteq\mathbb{R}[x_{1},\dots,x_{n}]$ is called totally
positive if it does not contain any non-zero element of $\mathbb{R}_{\geq
0}[x_{1},\dots,x_{n}]$. For a ${\bf d}$-homogeneous ideal
$J\subseteq\mathbb{R}[x_{1},\dots,x_{n}]$ the totally positive part of its
tropicalization is defined as

$\operatorname{Trop}^{+}(J):=\\{w\in\operatorname{Trop}(J)\colon\operatorname{in}_{w}(J)\text{
is totally positive}\\}.$

Due to [50] $\operatorname{Trop}^{+}(J)$ is a closed subfan of
$\operatorname{Trop}(J)$ (that may be empty).

##  3 Families of Gröbner degenerations

In this section, we introduce the main construction of the paper. Let
$J\subseteq\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]$ be a d-homogeneous ideal,
$C$ a maximal cone in $\operatorname{GF}(J)$ and $A$ the quotient
$\mathbb{K}_{\bf d}[x_{1},\dots,x_{n}]/J$. Our aim is to construct a flat
family of degenerations of ${\rm Proj}(A)$ that contains as fibers the
varieties corresponding to the various initial ideals associated to the
interior of the faces of $C$. To achieve this we generalize the following
classical construction: take $w\in C^{\circ}$ and consider the ideal

$\hat{J}_{w}:=\bigg{\langle}\sum_{\alpha\in\mathbb{Z}^{n}_{\geq
0}}c_{\alpha}{\bf x}^{\alpha}t^{w\cdot\alpha-\min\\{w\cdot\beta\colon
c_{\beta}\not=0\\}}\colon\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}\in J\bigg{\rangle}\subseteq\mathbb{K}[t][x_{1},\dots,x_{n}].$
(3.1)

######  Remark 3.1.

The ideal $\hat{J}_{w}$ induces a flat family
$\operatorname{Spec}\big{(}\mathbb{K}[t][x_{1},\dots,x_{n}]/\hat{J}_{w}\big{)}\to\operatorname{Spec}(\mathbb{K}[t])$
whose fiber over the closed point $(t)$ is isomorphic to
$\operatorname{Spec}(\mathbb{K}[x_{1},\dots,x_{n}]/\operatorname{in}_{w}(J))$
and the fiber over any non-zero closed point $(t-a)$ is isomorphic to
$\operatorname{Spec}(A)$, see [18, Theorem 15.17]. In fact, (3.1) and [18,
Theorem 15.17] hold more generally for arbitrary cones in
$\operatorname{GF}(J)$. However, for the following generalization we focus on
maximal cones for simplicity.

To generalize the construction (3.1) we fix vectors $r_{1},\dots,r_{m}\in C$
such that $\\{\overline{r}_{1},\dots,\overline{r}_{m}\\}$ is the set of
primitive ray generators for $\overline{C}$, which is possible due to
lem:strictly_convex. Using lem:wt homog we may assume if necessary that
$r_{1},\dots,r_{m}$ are non-negative or positive vectors. We denote by ${\bf
r}$ the $(m\times n)$-matrix whose rows are $r_{1},\dots,r_{m}$. Additionally,
we write $<$ for a monomial term order compatible with $C$ and denote by
$\mathcal{G}$ the associated reduced Gröbner basis.

######  Definition 3.2.

def:lift For $f=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}\in J$ set ${\mu}_{\bf
r}(f):=\big{(}\min_{c_{\alpha}\not=0}\\{r_{i}\cdot\alpha\\}\big{)}_{i=1,\dots,m}\in\mathbb{Z}^{m\times
1}$, that is we think of $\mu_{\bf r}(f)$ as a column vector with $m$ entries.
We define the lift of $f$ to be the polynomial $\tilde{f}_{\bf
r}\in\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]$ given by the following
formula

$\tilde{f}_{\bf r}:=\tilde{f}_{\bf r}({\bf t},{\bf x}):=f({\bf t}^{{\bf
r}\cdot e_{1}}x_{1},\dots,{\bf t}^{{\bf r}\cdot e_{n}}x_{n}){\bf t}^{-\mu_{\bf
r}(f)}=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf x}^{\alpha}{\bf
t}^{{\bf r}\cdot\alpha-\mu_{\bf r}(f)}.$

Similarly, we define the lifted ideal as

$\tilde{J}_{\bf r}:=\big{\langle}\tilde{f}_{\bf r}\colon f\in
J\big{\rangle}\subseteq\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}],$

and the lifted algebra as the quotient

$\tilde{A}_{\bf
r}:=\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]/\tilde{J}_{\bf r}.$ (3.2)

We proceed by showing that the lifted algebra $\tilde{A}_{\bf r}$ is
independent of the choice of vectors $r_{1},\dots,r_{m}\in C$ that represent
the primitive ray generators $\overline{r}_{1},\dots,\overline{r}_{m}$ of
$\overline{C}$. For this, we identify generators of the ideal $\tilde{J}_{\bf
r}$ which is a crucial matter for applications.

######  Proposition 3.3.

prop:change rays Suppose $r^{\prime}_{1},\dots,r^{\prime}_{m}\in C$ are such
that $\overline{r}^{\prime}_{i}=\overline{r}_{i}$ in
$\mathbb{R}^{n}/\mathcal{L}(J)$. Let ${\bf r^{\prime}}$ be the $(m\times
n)$-matrix whose rows are $r^{\prime}_{1},\dots,r^{\prime}_{m}$. Then for
$g\in\mathcal{G}$ we have that $\tilde{g}_{\bf r}=\tilde{g}_{\bf r^{\prime}}$.

###### Proof.

We have that $r^{\prime}_{i}=r_{i}+l_{i}$ for some $l_{i}\in\mathcal{L}(J)$.
Let $L$ be the $(m\times n)$-matrix whose rows are $l_{1},\dots,l_{m}$. In
particular, ${\bf r^{\prime}}={\bf r}+L$. Write
$g=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf x}^{\alpha}$. Since
$g\in\mathcal{G}$ we have by lem:init ell that the value $l_{i}\cdot\alpha$ is
the same for all $\alpha$ with $c_{\alpha}\neq 0$. Let $a_{i}$ be this common
value. Observe that $\min_{c_{\alpha}\neq
0}\\{r^{\prime}_{i}\cdot\alpha\\}=\min_{c_{\alpha}\neq
0}\\{r_{i}\cdot\alpha\\}+a_{i}$. In particular, we have that the column vector
$(a_{1},\dots,a_{m})$ is equal to $L\cdot\alpha$ for all $\alpha$ with
$c_{\alpha}\neq 0$ and therefore $\mu_{\bf r^{\prime}}(f)=\mu_{\bf
r}(f)+L\cdot\alpha$. Finally, we compute

$\tilde{g}_{\bf r^{\prime}}=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq
0}}c_{\alpha}{\bf x}^{\alpha}{\bf t}^{{\bf r}^{\prime}\cdot\alpha-\mu_{\bf
r^{\prime}}(g)}=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}{\bf t}^{{\bf r}\cdot\alpha+L\cdot\alpha-(\mu_{\bf
r}(g)+L\cdot\alpha)}=\tilde{g}_{\bf r}.$

In the following, when there is no risk of confusion, we write $\tilde{J}$ for
$\tilde{J}_{\bf r},\tilde{f}$ for $\tilde{f}_{\bf r}$ and $\tilde{A}$ for
$\tilde{A}_{\bf r}$.

######  Notation 3.4.

notation For ${\bf a}=(a_{1},\dots,a_{m})\in\mathbb{K}^{m}$ denote by $({\bf
t}-{\bf a})$ the maximal ideal $\langle t_{1}-a_{1},\dots$,
$t_{m}-a_{m}\rangle$ of $\mathbb{K}[t_{1},\dots,t_{m}]$. Then we denote by
$\tilde{A}|_{{\bf t}={\bf a}}$ the tensor product
$\tilde{A}\otimes_{\mathbb{K}[t_{1},\dots,t_{m}]}\mathbb{K}[t_{1},\dots,t_{m}]$
$/({\bf t}-{\bf a})$. Additionally, for
$f\in\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]$ let $f|_{{\bf t}={\bf
a}}\in\mathbb{K}[x_{1},\dots,x_{n}]$ be the evaluation of $f$ at $t_{i}=a_{i}$
for $i\in[m]$. Similarly, denote by $\tilde{J}|_{{\bf t}={\bf a}}$ the ideal
of $\mathbb{K}[x_{1},\dots,x_{n}]$ generated by the set $\\{\tilde{f}|_{{\bf
t}={\bf a}}\colon f\in J\\}$. For a set
$B\subset\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]$ let $B|_{{\bf
t}={\bf a}}:=\\{b|_{{\bf t}={\bf a}}\colon b\in B\\}$. Recall the isomorphism
$\mathbb{K}[t_{1},\dots,t_{m}]/({\bf t}-{\bf a})\to\mathbb{K}$ that sends an
element $\bar{f}$ to $f|_{{\bf t}={\bf a}}$. It extends to an isomorphism
$\tilde{A}|_{{\bf t}={\bf
a}}\cong\mathbb{K}[x_{1},\dots,x_{n}]/\tilde{J}|_{{\bf t}={\bf a}}$, which
explains our choice of notation.

Denote ${\bf 1}:=(1,\dots,1)\in\mathbb{Z}^{m}$.

######  Lemma 3.5.

lem:w’homogeneous Let $w=(w_{1},\dots,w_{n})=r_{1}+\cdots+r_{m}$, and
$w^{\prime}=(-1,\ldots,-1,w_{1},\ldots,w_{n})\in\mathbb{Z}^{m+n}$, where
$w_{i}$ is the weight of $x_{i}$ and $-1$ is the weight of $t_{j}$ for
$i=1,\ldots,n$ and $j=1,\ldots,m$. Then for every $f\in J$ its lift
$\tilde{f}$ is $w^{\prime}$-homogeneous. In particular, the ideal $\tilde{J}$
is ${w^{\prime}}$-homogeneous.

###### Proof.

Let $f=\sum_{\alpha\in\mathbb{Z}_{\geq 0}^{n}}c_{\alpha}{\bf x}^{\alpha}$.
Then the lift of $f$ is by definition $\tilde{f}=\sum c_{\alpha}{\bf
x}^{\alpha}{\bf t}^{{\bf r}\cdot\alpha-\mu(f)}$. Notice that the
$w^{\prime}$-weight of a monomial $c_{\alpha}{\bf x}^{\alpha}{\bf t}^{{\bf
r}\cdot\alpha-\mu(f)}$ in $\tilde{f}$ is

$w\cdot\alpha-{\bf 1}\cdot({\bf
r}\cdot\alpha-\mu(f))=\sum_{i=1}^{m}r_{i}\cdot\alpha-\sum_{i=1}^{m}r_{i}\cdot\alpha+\sum_{i=1}^{m}\min_{c_{\beta}\not=0}(r_{i}\cdot\beta)=\sum_{i=1}^{m}\min_{c_{\beta}\not=0}(r_{i}\cdot\beta).$

This implies the first claim. The second claim is immediate as
$\tilde{J}=\langle\tilde{f}\colon f\in J\rangle$. ∎

######  Remark 3.6.

rmk:homogeneous weights In lem:w’homogeneous more generally we can choose
$v=c_{1}r_{1}+\dots+c_{m}r_{m}\in C^{\circ}$ with $c_{i}\in\mathbb{R}_{\geq
0}$ and $v^{\prime}=(-c_{1},\dots,-c_{m},v_{1},\dots,v_{n}).$ Then $\tilde{J}$
is $v^{\prime}$-homogeneous.

######  Lemma 3.7.

lem:homogeneous Let $h\in\tilde{J}$ be $w^{\prime}$-homogeneous with
$w^{\prime}$ as in lem:w’homogeneous. Then $h={\bf t}^{\bf u}\tilde{f}$ for
some $f\in J$ and ${\bf u}\in\mathbb{Z}_{\geq 0}^{m}$.

###### Proof.

Let $h=\sum_{i=1}^{s}{p_{i}}\tilde{f}_{i}$ with $f_{i}\in J$ and
$p_{i}\in\mathbb{K}[t_{1},\ldots,t_{m}][x_{1},\ldots,x_{n}]$
$w^{\prime}$-homogeneous. Then $h|_{{\bf t}={\bf
1}}=\sum_{i=1}^{s}p_{i}|_{{\bf t}={\bf 1}}f_{i}\in J$. Furthermore, since $h$
is $w^{\prime}$-homogeneous, $h={\bf t}^{\bf u}\widetilde{h|_{{\bf t}={\bf
1}}}$ for some ${\bf u}\in\mathbb{Z}_{\geq 0}^{m}$. Hence, we can take
$h|_{{\bf t}={\bf 1}}$ as $f$. ∎

######  Lemma 3.8.

lem:lifts Consider $f=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}\in J$ with a unique monomial $c_{\beta}{\bf
x}^{\beta}\in\operatorname{in}_{<}(J)$. That is, for every $\alpha\neq\beta$
with $c_{\alpha}\neq 0$, there exists no $g^{\prime}\in\mathcal{G}$ such that
${\bf x}^{\alpha}$ is divisible by $\operatorname{in}_{<}(g^{\prime})$. Then

$\tilde{f}=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf
x}^{\alpha}{\bf t}^{{\bf r}\cdot(\alpha-\beta)}=c_{\beta}{\bf
x}^{\beta}+\sum_{\alpha\not=\beta}c_{\alpha}{\bf x}^{\alpha}{\bf t}^{{\bf
r}\cdot(\alpha-\beta)},$ (3.2)

contains a unique monomial with coefficient in $\mathbb{K}$. In particular,
this is the case for elements of the reduced Gröbner basis $\mathcal{G}$.

###### Proof.

The assumption that only the monomial $c_{\beta}{\bf x}^{\beta}$ of $f$ is
contained in $\operatorname{in}_{<}(J)$ ensures that $c_{\beta}{\bf
x}^{\beta}$ is also a monomial appearing in $\operatorname{in}_{r_{i}}(f)$ for
all $i$. To see this, assume there is a ray $r_{i}$ such that $c_{\beta}{\bf
x}^{\beta}$ is not a monomial in $\operatorname{in}_{r_{i}}(f)$ and let
$w^{\prime}=\sum_{j\not=i}r_{j}$. By [38, Lemma 2.4.5] we can find an
$\epsilon$ so that $\operatorname{in}_{\epsilon
w^{\prime}+r_{i}}(f)=\operatorname{in}_{w^{\prime}}(\operatorname{in}_{r_{i}}(f))$.
As we assumed $c_{\beta}{\bf x}^{\beta}$ is not a monomial in
$\operatorname{in}_{r_{i}}(f)$, it is also not a monomial in
$\operatorname{in}_{\epsilon w^{\prime}+r_{i}}(f)$. But $\epsilon
w^{\prime}+r_{i}\in C^{\circ}$ and so $\operatorname{in}_{\epsilon
w^{\prime}+r_{i}}(f)\in\operatorname{in}_{\epsilon
w^{\prime}+r_{i}}(J)=\operatorname{in}_{<}(J)$. This however is a
contradiction as we assumed that $c_{\beta}{\bf x}^{\beta}$ is the only
monomial in $f$ that is contained in $\operatorname{in}_{<}(J)$. Therefore,
$r_{i}\cdot\beta=\min_{c_{\alpha}\not=0}\\{r_{i}\cdot\alpha\\}$ for all $i$
and ${\bf r}\cdot\beta=\mu_{\bf r}(f)$. Assume on the contrary that there
exists another monomial $c_{\gamma}{\bf x}^{\gamma}$ in $f$ with ${\bf
r}\cdot\gamma={\bf r}\cdot\beta$. Then $c_{\gamma}{\bf x}^{\gamma}$ is also a
monomial in $\operatorname{in}_{r_{i}}(f)$ for all $i$. Moreover, for
$w=r_{1}+\dots+r_{m}\in C^{\circ}$ the initial form $\operatorname{in}_{w}(f)$
contains both monomials $c_{\beta}{\bf x}^{\beta}$ and $c_{\gamma}{\bf
x}^{\gamma}$. But as $\operatorname{in}_{w}(J)=\operatorname{in}_{<}(J)$ this
implies that $c_{\gamma}{\bf x}^{\gamma}\in\operatorname{in}_{<}(J)$, a
contradiction. Hence, $c_{\beta}{\bf x}^{\beta}$ is the unique monomial in $f$
with $\mu_{\bf r}(f)={\bf r}\cdot\beta$ and we obtain (3.2) as ${\bf
r}\cdot(\alpha-\beta)\not=0$ for all $\alpha\not=\beta$. This completes the
proof of the first claim.

To prove the second part, assume that $g\in\mathcal{G}$ has a monomial term
$c_{\gamma}{\bf x}^{\gamma}\in\operatorname{in}_{<}(J)$ with $c_{\gamma}{\bf
x}^{\gamma}\neq\operatorname{in}_{<}(g)$. Then by the definition of the
reduced Gröbner basis, $c_{\gamma}{\bf x}^{\gamma}$ is not divisible by
$\operatorname{in}_{<}(g^{\prime})$ for any
$g^{\prime}\in\mathcal{G}\setminus\\{g\\}$. Hence, it should be divisible by
$\operatorname{in}_{<}(g)$, but this cannot happen as $g$ is ${\bf
d}$-homogeneous by lem:init ell. ∎

We extend the monomial term order $<$ on $\mathbb{K}[x_{1},\dots,x_{n}]$ to a
monomial term order $\ll$ on the polynomial ring
$\mathbb{K}[t_{1},\dots,t_{m},x_{1},\dots,x_{n}]$ in such a way that $t_{i}\ll
x_{j}$ for all $1\leq i\leq m$ and $1\leq j\leq n$, and

$\displaystyle{\bf x}^{\alpha}{\bf t}^{\lambda}\ll{\bf x}^{\beta}{\bf
t}^{\mu}\quad\text{if and only if}\quad(i)\quad{\bf x}^{\alpha}<{\bf
x}^{\beta}\quad\text{or}\quad(ii)\quad{\bf x}^{\alpha}={\bf
x}^{\beta}\quad\text{and}\quad{\bf t}^{\lambda}<_{\rm lex}{\bf t}^{\mu}.$
(3.3)

######  Proposition 3.9.

prop:lifted generators The set $\tilde{\mathcal{G}}=\\{\tilde{g}\colon
g\in\mathcal{G}\\}$ is a Gröbner basis for $\tilde{J}$ with respect to $\ll$.
In particular, $\tilde{\mathcal{G}}$ is a generating set for $\tilde{J}$.

###### Proof.

Since $\tilde{J}$ is $w^{\prime}$-homogeneous by Lemma
LABEL:lem:w'homogeneous, it is enough to show that for every
$w^{\prime}$-homogeneous polynomial $h\in\tilde{J}$ there exists some
$\tilde{g}_{i}\in\tilde{\mathcal{G}}$ whose initial term divides
$\operatorname{in}_{\ll}(h)$. Let $h\in\tilde{J}$ be $w^{\prime}$-homogeneous.
By Lemma LABEL:lem:homogeneous there exist $f\in J$ and ${\bf
u}\in\mathbb{Z}_{\geq 0}^{m}$ such that $h={\bf t}^{\bf u}\tilde{f}$. We
compute

$\displaystyle\operatorname{in}_{\ll}(h)={\bf t}^{\bf
u}\operatorname{in}_{\ll}(\tilde{f})\overset{\text{Def. }\ll}{=}{\bf t}^{{\bf
u}+{\bf v}}\operatorname{in}_{<}(f)\overset{\exists\ g\in\mathcal{G}}{=}{\bf
t}^{{\bf u}+{\bf v}}c_{\alpha}{\bf
x}^{\alpha}\operatorname{in}_{<}(g)\overset{\text{\thref{lem:lifts}}}{=}{\bf
t}^{{\bf u}+{\bf v}}c_{\alpha}{\bf
x}^{\alpha}\operatorname{in}_{\ll}(\tilde{g}),$

where $g$ is a suitable element of the Gröbner basis $\mathcal{G}$ for $J$ and
${\bf v}\in\mathbb{Z}_{\geq 0}^{m}$. As $\tilde{g}\in\tilde{\mathcal{G}}$,
this shows that $\tilde{\mathcal{G}}$ is a Gröbner basis for $\tilde{J}$.
Hence, by [29, Theorem 2.1.8] $\tilde{\mathcal{G}}$ is a generating set for
$\tilde{J}$. ∎

As a direct consequence of prop:change rays and prop:lifted generators we
obtain that the lifted algebra is independent of the choice of vectors
$r_{1},\dots,r_{m}$.

######  Corollary 3.10.

For ${\bf r}$ and ${\bf r}^{\prime}$ as in prop:change rays we have that
$\tilde{A}_{\bf r}=\tilde{A}_{\bf r^{\prime}}$.

The algebras $\tilde{A}|_{{\bf t}={\bf a}}$ constitute the fibers of a flat
family introduced below. The following result leads to fibers over different
points being isomorphic if they have zero entries in the same positions.

######  Proposition 3.11.

prop:generic fibers Consider ${\bf a}=(a_{1},\dots,a_{m})$ and ${\bf
b}=(b_{1},\dots,b_{m})$ in $\mathbb{K}^{m}$ with the property that $a_{i}=0$
if and only if $b_{i}=0$. Then the algebras $\tilde{A}|_{{\bf t}={\bf a}}$ and
$\tilde{A}|_{{\bf t}={\bf b}}$ are isomorphic.

###### Proof.

Let $\varphi$ be the $\mathbb{K}$-algebra automorphism of
$\mathbb{K}[t_{1},\dots,t_{m}]$ defined by
$\varphi(t_{i})=\frac{a_{i}}{b_{i}}t_{i}$ if $b_{i}\not=0$ and
$\varphi(t_{i})=t_{i}$ if $b_{i}=0$. Then $\varphi({\bf t}-{\bf a})=({\bf
t}-{\bf b})$, so the claim follows by notation. ∎

######  Remark 3.12.

rmk:non primitive rays For computational reasons it might be desirable to work
with ray generators for $C$ that are not representatives of primitive ray
generators for $\overline{C}$. Let ${\bf r}$ and ${\bf r^{\prime}}$ be two
choices of ray matrices whose rows satisfy
$\overline{r^{\prime}}_{j}=q_{j}\overline{r}_{j}$ with $q_{j}\in\mathbb{Q}$
for all $j$. Then we still have an isomorphism between $\tilde{A}_{\bf
r}|_{{\bf t}={\bf a}}$ and $\tilde{A}_{\bf r^{\prime}}|_{{\bf t}={\bf a}}$ for
all ${\bf a}\in\mathbb{K}^{m}$. For the proof it is necessary to extend the
polynomial ring $\mathbb{K}[t_{1},\dots,t_{m}]$ to a ring
$\mathbb{K}\big{[}t_{1}^{\mathbb{Q}},\dots,t_{m}^{\mathbb{Q}}\big{]}$, where
the $t_{i}$’s are allowed to have rational exponents. The automorphism of
$\mathbb{K}\big{[}t_{1}^{\mathbb{Q}},\dots,t_{m}^{\mathbb{Q}}\big{]}$ defined
by $h(t_{i})=t_{i}^{q_{i}}$ extends to an automorphism of
$\mathbb{K}\big{[}t_{1}^{\mathbb{Q}},\dots,t_{m}^{\mathbb{Q}}\big{]}[x_{1},\dots,x_{n}]$
with the property $h\big{(}\tilde{f}_{\bf r}\big{)}=\tilde{f}_{\bf
r^{\prime}}$ for all ${\bf d}$-homogeneous polynomials $f\in J$. The rest
follows from prop:generic fibers.

Before presenting our main result we explain how the ideal $\tilde{J}_{\bf r}$
is related to the ideal $\hat{J}_{w}$ in (3.1).

######  Proposition 3.13.

prop:special fibers Consider a face $\tau$ of $C$ spanned by a subset
$\\{r_{i_{1}},\dots,r_{i_{s}}\\}$ of $\\{r_{1},\dots,r_{m}\\}$ and the
lineality space $\mathcal{L}(J)$. We define ${\bf
t}_{\tau}\in\mathbb{K}[t]^{m}$ by

$({\bf t}_{\tau})_{k}:=\begin{cases}t&\text{if}\quad
k\in\\{i_{1},\ldots,i_{s}\\},\\\
1&\text{otherwise,}\end{cases}\qquad\text{for}\quad k=1,\ldots,m.$

Then for $w_{\tau}=r_{i_{1}}+\dots+r_{i_{s}}$ we have $\tilde{J}|_{{\bf
t}={\bf t}_{\tau}}=\hat{J}_{w_{\tau}}$.

###### Proof.

Let $<$ be the monomial term order compatible with $C$. Consider an element
$g=\sum_{\alpha\in\mathbb{Z}^{n}_{\geq 0}}c_{\alpha}{\bf x}^{\alpha}$ in
$\mathcal{G}$ with $\operatorname{in}_{<}(g)=c_{\beta}{\bf x}^{\beta}$. Since
the vectors $w_{\tau}$, $w:=w_{\tau}-r_{i_{s}}$ and $r_{i_{s}}$ are all in
$C$, by Remark 2.9:

$\operatorname{in}_{<}(\operatorname{in}_{w_{\tau}}(g))=\operatorname{in}_{<}(\operatorname{in}_{w}(g))=\operatorname{in}_{<}(\operatorname{in}_{r_{i_{s}}}(g))=\operatorname{in}_{<}(g)=c_{\beta}{\bf
x}^{\beta}.$

This implies that the initial forms of $\operatorname{in}_{w_{\tau}}(g)$,
$\operatorname{in}_{w}(g)$ and $\operatorname{in}_{r_{i_{s}}}(g)$ contain
$c_{\beta}{\bf x}^{\beta}$. In other words,

$\min_{c_{\alpha}\not=0}\\{w_{\tau}\cdot\alpha\\}=w_{\tau}\cdot\beta=(w+r_{i_{s}})\cdot\beta,\qquad\min_{c_{\alpha}\not=0}\\{w\cdot\alpha\\}=w\cdot\beta,\qquad\min_{c_{\alpha}\not=0}\\{r_{i_{s}}\cdot\alpha\\}=r_{i_{s}}\cdot\beta.$

Therefore,

$(w+r_{i_{s}})\cdot\beta=\min_{c_{\alpha}\not=0}\\{(w+r_{i_{s}})\cdot\alpha\\}=\min_{c_{\alpha}\not=0}\\{w\cdot\alpha\\}+\min_{c_{\alpha}\not=0}\\{r_{i_{s}}\cdot\alpha\\}.$

Using the same argument multiple times we obtain that

$\min_{c_{\alpha}\not=0}\Bigg{\\{}\bigg{(}\sum_{j=1}^{s}r_{i_{j}}\bigg{)}\cdot\alpha\Bigg{\\}}=\sum_{j=1}^{s}\min_{c_{\alpha}\not=0}\\{r_{i_{j}}\cdot\alpha\\}.$

Now the claim follows by comparing the generating sets of $\hat{J}_{w_{\tau}}$
and $\tilde{J}|_{{\bf t}={\bf t}_{\tau}}$. ∎

We are now prepared to present our main theorem:

######  Theorem 3.14.

thm:family Let $J$ be a ${\bf d}$-homogeneous ideal in $\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]$, $A=\mathbb{K}_{\bf d}[x_{1},\ldots,x_{n}]/J$, $C$ a
maximal cone in $\operatorname{GF}(J)$ with compatible monomial term order $<$
and ${\bf r}$ an $(m\times n)$-matrix whose rows are representatives of the
primitive ray generators of
$\overline{C}\subset\mathbb{R}^{n}/\mathcal{L}(J)$. Then:

  * $(i)$

The algebra $\tilde{A}_{\bf r}$ is a free
$\mathbb{K}[t_{1},\dots,t_{m}]$-module with basis $\mathbb{B}_{<}$, the
standard monomial basis of $A$ with respect to $\operatorname{in}_{<}(J)$. In
particular, we have a flat family

${{\rm Proj}(\tilde{A})}$${\mathbb{P}({\bf
d})\times\mathbb{A}^{m}}$${\mathbb{A}^{m}.}$$\scriptstyle{\pi}$

  * $(ii)$

For every face $\tau$ of $C$ there exists ${\bf a}_{\tau}\in\mathbb{A}^{m}$
such that $\pi^{-1}({\bf a}_{\tau})={\rm Proj}(\mathbb{K}_{\bf
d}[x_{1},\dots,x_{n}]/$ $\operatorname{in}_{\tau}(J))$. In particular, generic
fibers are isomorphic to ${\rm Proj}(A)$ and there exist special fibers for
every proper face $\tau\subset C$.

###### Proof.

$(i)$ We extend $<$ to a monomial term order $\ll$ on
$\mathbb{K}[t_{1},\dots,t_{m},x_{1},\dots,x_{n}]$ as in (3.3). Let
$\mathbb{B}_{\ll}$ be the standard monomial basis for $\tilde{A}_{\bf r}$
induced by $\operatorname{in}_{\ll}\big{(}\tilde{J}_{\bf r}\big{)}$. Then
$\mathbb{B}_{\ll}$ is a basis for $\tilde{A}_{\bf r}$ as a $\mathbb{K}$-vector
space and a generating set for $\tilde{A}_{\bf r}$ as a
$\mathbb{K}[t_{1},\dots,t_{m}]$-module. By prop:lifted generators and [29,
Proposition 1.1.5] we have

$\displaystyle\mathbb{B}_{\ll}=\big{\\{}\bar{\bf t}^{\beta}\bar{\bf
x}^{\alpha}\colon\operatorname{in}_{\ll}(\tilde{g})\not|\ {\bf t}^{\beta}{\bf
x}^{\alpha}\text{ for all
}g\in\mathcal{G}\big{\\}},\qquad\mathbb{B}_{<}=\big{\\{}\bar{\bf
x}^{\alpha}\colon\operatorname{in}_{<}(g)\not|\ {\bf x}^{\alpha}\text{ for all
}g\in\mathcal{G}\big{\\}}.$

Observe that $\mathbb{B}_{\ll}$ can be reduced to a
$\mathbb{K}[t_{1},\dots,t_{m}]$-basis of $\tilde{A}_{\bf r}$ by defining

$\mathbb{B}^{\prime}_{\ll}:=\mathbb{B}_{\ll}\setminus\big{\\{}\bar{\bf
t}^{\beta}\bar{\bf x}^{\alpha}\colon\exists\ \bar{\bf t}^{\gamma}\bar{\bf
x}^{\alpha}\in\mathbb{B}_{\ll}\text{ with }\gamma<\beta\big{\\}},$

where $\gamma<\beta$ means that $\gamma_{i}\leq\beta_{i}$ for all $i$ and
$\gamma_{j}<\beta_{j}$ for some $j$. Note that
$\mathbb{B}_{<}\subseteq\mathbb{B}^{\prime}_{\ll}$. Now assume there is a
monomial $\bar{\bf t}^{\beta}\bar{\bf
x}^{\alpha}\in\mathbb{B}^{\prime}_{\ll}\setminus\mathbb{B}_{<}$. By
prop:lifted generators we have that
$\operatorname{in}_{\ll}\big{(}\tilde{J}\big{)}|_{{\bf t}={\bf
1}}=\operatorname{in}_{<}(J)$. Hence, $\mathbb{B}^{\prime}_{\ll}|_{{\bf
t}={\bf 1}}=\mathbb{B}_{\ll}|_{{\bf t}={\bf 1}}=\mathbb{B}_{<}$. In
particular, this implies that $\bar{\bf x}^{\alpha}\in\mathbb{B}_{<}$. But as
$\mathbb{B}_{<}\subset\mathbb{B}_{\ll}$ this is a contradiction to the
definition of $\mathbb{B}^{\prime}_{\ll}$.

For the second claim, let $\mathbb{K}[t_{1},\dots,t_{m}]_{\bf
d}[x_{1},\dots,x_{n}]$ denote the polynomial ring in $x_{1},\dots,x_{n}$ with
coefficients in $\mathbb{K}[t_{1},\dots,t_{m}]$ and grading induced by ${\bf
d}$. Then by lem:lifts and prop:lifted generators
$\tilde{J}\subseteq\mathbb{K}[t_{1},\dots,t_{m}]_{\bf d}[x_{1},\dots,x_{n}]$
is ${\bf d}$-homogeneous. This yields the embedding ${\rm
Proj}\big{(}\tilde{A}\big{)}\hookrightarrow\mathbb{P}({\bf
d})\times\mathbb{A}^{m}$. The projection $\mathbb{P}({\bf
d})\times\mathbb{A}^{m}\twoheadrightarrow\mathbb{A}^{m}$ induces the flat
morphism $\pi\colon{\rm Proj}(\tilde{A})\to\mathbb{A}^{m}$ as $\tilde{A}$ has
a $\mathbb{K}[t_{1},\dots,t_{m}]$-basis by the first claim.

$(ii)$ Every face $\tau$ of $C$ induces a face $\overline{\tau}$ of
$\overline{C}$ with primitive ray generators
$\overline{r}_{i_{1}},\dots,\overline{r}_{i_{s}}$ for some
$i_{1},\dots,i_{s}\in[m]$. We define

$({\bf a}_{\tau})_{k}:=\begin{cases}0&\text{if}\quad
k\in\\{i_{1},\dots,i_{s}\\},\\\
1&\text{otherwise},\end{cases}\qquad\text{for}\quad k=1,\ldots,m.$

Then by prop:special fibers we have $\tilde{J}|_{{\bf t}={\bf
a}_{\tau}}=\hat{J}_{w_{\tau}}|_{t=0}$ which is equal to
$\operatorname{in}_{\tau}(J)$ by Remark 3.1. ∎

###  3.1 Torus equivariant families

We explain how the above results are related to Kaveh–Manon’s recent work [33]
on the classification of torus equivariant families.

Consider the lattice $N=\mathbb{Z}^{n}$, its dual lattice $M=N^{*}$ and a fan
$\Sigma\subset N\otimes_{\mathbb{Z}}\mathbb{R}$. We write $X_{\Sigma}$ for the
toric variety associated to $\Sigma$. Furthermore, we define $\mathcal{O}_{N}$
to be the semifield of piecewise linear functions on $N$ and
$\mathcal{O}_{\Sigma}$ the semifield of piecewise linear functions on
$|\Sigma|\cap N$. For $a,b\in\mathcal{O}_{\Sigma}$ we have $a\otimes b:=a+b$
and $a\oplus b:=\min\\{a,b\\}$, where the minimum is taken pointwise. In
particular, $\mathcal{O}_{\Sigma}$ is partially ordered: $a\geq b$ if $a\oplus
b=b$. A PL-quasivaluation on an algebra $A$ is a map
$\operatorname{\mathfrak{v}}\colon A\to\mathcal{O}_{\Sigma}$ that satisfies:
$(i)$
$\operatorname{\mathfrak{v}}(fg)\geq\operatorname{\mathfrak{v}}(f)+\operatorname{\mathfrak{v}}(g)$
and $(ii)$
$\operatorname{\mathfrak{v}}(f+g)\geq\min\\{\operatorname{\mathfrak{v}}(f),\operatorname{\mathfrak{v}}(g)\\}$,
where the minimum is taken pointwise in $\mathcal{O}_{\Sigma}$. If $(i)$ is an
equality, $\operatorname{\mathfrak{v}}$ is called a PL-valuation. If $A$ is a
graded algebra $A=\bigoplus_{n\in\mathbb{Z}}A_{n}$ then
$\operatorname{\mathfrak{v}}$ is called homogeneous if it is compatible with
the grading, i.e., for $f=\sum_{n\in\mathbb{Z}}c_{n}g_{n}$ with $g_{n}\in
A_{n}$ and coefficients $c_{n}$ we have
$\operatorname{\mathfrak{v}}(f)=\operatorname{\mathfrak{v}}(g_{k})$ for the
smallest $k$ with $c_{k}\not=0$.

Given a sheaf of algebras $\mathcal{A}$ on $X_{\Sigma}$ its relative spectrum
denoted by $\mathbf{Spec}(\mathcal{A})$ is the scheme obtained from gluing
affine schemes $\operatorname{Spec}(\mathcal{A}(U_{i}))$, where
$\bigcup_{i}U_{i}$ is an open cover of $X_{\Sigma}$ and $\mathcal{A}(U_{i})$
is the corresponding section of $\mathcal{A}$.

######  Theorem 3.15 ([33, Theorem 1.1]).

Let $A=\bigoplus_{n\geq 0}A_{n}$ be a positively graded algebra and a domain
over an algebraically closed field of characteristic zero and $\Sigma$ a fan
in $N\otimes_{\mathbb{Z}}\mathbb{R}$. Then the following are equivalent
information:

  * $(i)$

a $T_{N}$-equivariant flat sheaf $\mathcal{A}$ of positively graded algebras
of finite type over $X_{\Sigma}$ such that $\mathbf{Spec}(\mathcal{A})$ has
reduced and irreducible fibers and its generic fibers are isomorphic to
$\operatorname{Spec}(A);$

  * $(ii)$

a homogeneous PL-valuation $\operatorname{\mathfrak{v}}\colon
A\to\mathcal{O}_{\Sigma}$ with finite Khovanskii basis. $($Given
$\operatorname{\mathfrak{v}}\colon A\to\mathcal{O}_{\Sigma}$ for every ray
$\rho\in\Sigma$ one constructs a quasivaluation
$\operatorname{\mathfrak{v}}_{\rho}\colon A\to\mathbb{Z}\cup\\{\infty\\}$ with
associated graded algebra ${\rm gr}_{\rho}(A)$, see [33, Section 4.1]. A
Khovanskii basis for $\operatorname{\mathfrak{v}}$ is a subset
$\mathcal{B}\subset A$ such that for every ray $\rho\in\Sigma$ the image of
$\mathcal{B}$ in ${\rm gr}_{\rho}(A)$ is a set of algebra generators.$)$

Consider a presentation of $A$, i.e., a surjection ${\rm
pr}\colon\mathbb{K}[x_{1},\dots,x_{n}]\to A$ whose kernel is a weighted
homogeneous prime ideal $J$ and so it induces an isomorphism of graded
algebras $A\cong\mathbb{K}[x_{1},\dots,x_{n}]/J$. Take a cone $\sigma$ in
$\operatorname{GF}(J)$. Then $\sigma$ defines PL-quasivaluation
$\operatorname{\mathfrak{v}}_{\sigma}\colon A\to\mathcal{O}_{\sigma}$ as
follows: let $C$ be a maximal cone in $\operatorname{GF}(J)$ with face
$\sigma$ and denote by $\mathbb{B}_{<}$ the standard monomial basis associated
to $C$ (here $<$ is the compatible monomial term order). If
$\operatorname{in}_{\sigma}(J)$ is prime, then
$\operatorname{\mathfrak{v}}_{\sigma}$ is a PL-valuation. Then for $\bar{\bf
x}^{\alpha}\in\mathbb{B}_{<}$ we have
$\operatorname{\mathfrak{v}}_{\sigma}(\bar{\bf
x}^{\alpha}):=-\cdot\alpha\colon\sigma\to\mathbb{Z}$, where $-\cdot-$ is the
pairing between the lattices $N$ and $M$, as above. For an arbitrary element
$f\in A$ write $f=\sum_{\bar{\bf
x}^{\alpha}\in\mathbb{B}_{<}}c_{\alpha}\bar{\bf x}^{\alpha}$. Then
$\operatorname{\mathfrak{v}}_{\sigma}(f):=\min_{c_{\alpha}\not=0}\\{\operatorname{\mathfrak{v}}_{\sigma}(\bar{\bf
x}^{\alpha})\\}$.

######  Definition 3.16 ([33]).

def:Rees The PL-quasivaluation $\operatorname{\mathfrak{v}}_{\sigma}\colon
A\to\mathcal{O}_{\sigma}$ defines a filtration on $A$ with filtered pieces
$F_{m}(\operatorname{\mathfrak{v}}_{\sigma}):=\\{f\in
A\colon\operatorname{\mathfrak{v}}_{\sigma}(f)\geq-\cdot m\\}$ for $m\in M$.
The Rees algebra of the PL-valuation $\operatorname{\mathfrak{v}}_{\sigma}$ is
$\mathcal{R}_{\sigma}(A):=\bigoplus_{m\in
M}F_{m}(\operatorname{\mathfrak{v}}_{\sigma})t^{m}$, where we think of $t^{m}$
as a character of the torus $T_{N}=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$.

More generally, a PL-quasivaluation can be obtained from a collection of
equidimensional cones $\sigma_{1},\dots,\sigma_{k}$ in
$\operatorname{Trop}(J)$ which are faces of the same maximal cone $C$ in
$\operatorname{GF}(J)$. Let $\Sigma\subset\operatorname{Trop}(J)$ be the
subfan whose maximal cones are $\sigma_{1},\dots,\sigma_{k}$ and
$\operatorname{\mathfrak{v}}_{\Sigma}\colon A\to\mathcal{O}_{N}$ be the
corresponding PL-valuation. By Kaveh–Manon’s classification of toric families
this yields a flat family

$\psi\colon\ \mathbf{Spec}(\mathcal{R}_{\Sigma}(A))\to X_{\Sigma}.$ (3.4)

Here $\mathcal{R}_{\Sigma}(A)$ is the $T_{N}$-equivariant flat sheaf of Rees
algebras on $X_{\Sigma}$. The scheme ${\bf Spec}(\mathcal{R}_{\Sigma}(A))$ is
glued from $\operatorname{Spec}(\mathcal{R}_{\sigma})$ for $\sigma\in\Sigma$.
For $p\in T_{N}\subset X_{\Sigma}$ we have
$\psi^{-1}(p)\cong\operatorname{Spec}(A)$. Moreover, $\psi_{\Sigma}$ has a
special fiber over every torus fixed point of $X$. More precisely, let
$p_{\sigma_{i}}$ be the torus fixed point corresponding to the (maximal) cone
$\sigma_{i}\in\Sigma$. Then

$\psi^{-1}(p_{\sigma_{i}})\cong\operatorname{Spec}\big{(}\mathbb{K}[x_{1},\dots,x_{n}]/\operatorname{in}_{\sigma_{i}}(J)\big{)}.$
(3.5)

Note that we can apply the construction of def:lift to the ideal $J$ and the
maximal cone $C\in\operatorname{GF}(J)$. In what follows we explore the
relation between the flat families from (3.4) and thm:family. Before stating
our results (thm:KM max cone and thm:KM vs grobner), we recall necessary
background from [33]. We fix a maximal cone $C$ in $\operatorname{GF}(J)$,
denote by $\mathcal{G}$ the associated reduced Gröbner basis and let
$\mathbb{B}_{<}$ be the standard monomial basis for the compatible monomial
term order $<$.

######  Definition 3.17 ([33]).

The PL-quasivaluation
$\mathfrak{w}_{C}\colon\mathbb{K}[x_{1},\dots,x_{n}]\to\mathcal{O}_{C}$
associated to $C$ is defined by $\mathfrak{w}_{C}({\bf
x}^{\alpha})=-\cdot\alpha\colon C\to\mathbb{Z}$. For a polynomial $f=\sum
c_{\alpha}{\bf x}^{\alpha}$ we have
$\mathfrak{w}_{C}(f):=\min_{c_{\alpha}\not=0}\\{-\cdot\alpha\\}$. The
associated Rees algebra is
$\mathcal{R}_{C}(\mathbb{K}[x_{1},\dots,x_{n}])=\bigoplus_{m\in
M}F_{m}(\mathfrak{w}_{C})t^{m}$, where $F_{m}(\mathfrak{w}_{C})$ is defined
analogous to $F_{m}(\operatorname{\mathfrak{v}}_{\sigma})$ in def:Rees.

The PL-quasivaluation $\operatorname{\mathfrak{v}}_{C}\colon
A\to\mathcal{O}_{C}$ is obtained as a pushforward of the PL-quasivaluation
$\mathfrak{w}_{C}\colon\mathbb{K}[x_{1},\dots,x_{n}]\to\mathcal{O}_{C}$ along
the morphism ${\rm pr}\colon\mathbb{K}[x_{1},\dots,x_{n}]\to A$. An adapted
basis for $\operatorname{\mathfrak{v}}_{C}$ is a vector space basis
$\mathbb{B}$ for $A$ such that $\mathbb{B}\cap
F_{m}(\operatorname{\mathfrak{v}}_{\sigma})$ is a vector space basis of
$F_{m}(\operatorname{\mathfrak{v}}_{\sigma})$ for every $m\in M$. By [33] the
standard monomial basis $\mathbb{B}_{<}$ is an adapted basis for
$\operatorname{\mathfrak{v}}_{C}$. Similarly, the monomial basis of
$\mathbb{K}[x_{1},\dots,x_{n}]$ is adapted to $\mathfrak{w}_{C}$. Hence,
$\mathcal{R}_{C}(A)$ is a free $\mathbb{K}[S_{C}]$-algebra with basis
$\mathbb{B}_{<}$ and $\mathcal{R}_{C}(\mathbb{K}[x_{1},\dots,x_{n}])$ is a
free $\mathbb{K}[S_{C}]$-algebra whose basis is the monomial basis, where
$S_{C}:=-C^{\vee}\cap\mathbb{Z}^{n}$ due to our $\min$-convention. In
particular:

$\mathcal{R}_{C}\big{(}\mathbb{K}[x_{1},\dots,x_{n}]\big{)}\cong\mathbb{K}[S_{C}][x_{1},\dots,x_{n}].$
(3.6)

Manon explained the proof of the following proposition to us. It follows from
results in [33].

######  Proposition 3.18.

prop:Rees quotient For any maximal cone $C\in\operatorname{GF}(J)$, the Rees
algebra $\mathcal{R}_{C}(A)$ has an explicit presentation of form
$\mathbb{K}[S_{C}][x_{1},\dots,x_{n}]/\hat{J}$. More precisely, the ideal
$\hat{J}$ is generated by homogenizations of elements in the reduced Gröbner
basis $\mathcal{G}$: let $g={\bf x}^{\gamma}+\sum c_{\alpha}{\bf
x}^{\alpha}\in\mathcal{G}$ and ${\bf x}^{\gamma}=\operatorname{in}_{C}(g)$,
then the homogenization of $g$ is

$\hat{g}={\bf x}^{\gamma}+\sum c_{\alpha}{\bf
x}^{\alpha}t^{\beta}\qquad\text{with}\quad\gamma-\alpha=\beta\in S_{C}.$ (3.7)

###### Proof.

Consider $f\in\mathbb{K}[x_{1},\dots,x_{n}]$. It can be written as
$f=\sum_{{\bf x}^{\alpha}\in\mathbb{B}_{<}}c_{\alpha}{\bf
x}^{\alpha}+\sum_{{\bf x}^{\beta}\not\in\mathbb{B}_{<}}c_{\beta}{\bf
x}^{\beta}$. In particular, if $\mathfrak{w}_{C}(f)\geq-\cdot m$ in
$\mathcal{O}_{C}$, then
$\operatorname{\mathfrak{v}}_{C}(\text{pr}(f))\geq-\cdot m$. So the morphism
$\text{pr}\colon\mathbb{K}[x_{1},\dots,x_{n}]$ $\to A$ induces a map
$\text{pr}_{m}\colon F_{m}(\mathfrak{w}_{C})\to
F_{m}(\operatorname{\mathfrak{v}}_{C})$ for all $m$. Further, every element in
$\mathbb{B}_{<}\cap F_{m}(\operatorname{\mathfrak{v}}_{C})$ is the image of a
monomial in $F_{m}(\mathfrak{w}_{C})$. Hence, $\text{pr}_{m}$ is surjective
and we obtain:

$\widehat{\text{pr}}:=\bigoplus_{m\in(\mathbb{Z}^{n})^{*}}\text{pr}_{m}\colon\
\mathcal{R}_{C}(\mathbb{K}[x_{1},\dots,x_{n}])\to\mathcal{R}_{C}(A).$

By (3.6) this implies the first claim. Now let $\hat{J}$ be the kernel of
$\widehat{\text{pr}}$. Then $\hat{J}=\bigoplus_{m\in M}\hat{J}_{m}$, where
$\hat{J}_{m}=\ker(\text{pr}_{m})\subset F_{m}(\mathfrak{w}_{C})$. Now consider
an element $g\in\mathcal{G}$ of form ${\bf x}^{\gamma}+\sum c_{\alpha}{\bf
x}^{\alpha}$ with ${\bf x}^{\gamma}=\operatorname{in}_{C}(g)$. We want to show
that the element $\hat{g}$ defined in (3.7) lies in $\hat{J}$ for all
$g\in\mathcal{G}$. Recall that by the proof of prop:special fibers, $\gamma$
is such that $w\cdot\gamma\leq w\cdot\alpha$ for all $w\in C$ and all $\alpha$
with $c_{\alpha}\not=0$ in $g$. Hence, $w\cdot(\gamma-\alpha)\leq 0$ and so
$\beta=\gamma-\alpha\in-C^{\vee}$. In particular, this implies that
$\hat{g}\in\hat{J}_{\gamma}\subset F_{\gamma}(\mathfrak{w}_{\sigma})$.

Lastly, we need to show that $\hat{J}$ is generated by $\\{\hat{g}\colon
g\in\mathcal{G}\\}$. Consider $h\in\hat{J}$. As $\hat{J}=\bigoplus_{m\in
M}\hat{J}_{m}$, we may assume that $h\in\hat{J}_{m}$. Further, as
$\hat{J}_{m}\subset
F_{m}(\mathfrak{w}_{C})\subset\mathbb{K}[x_{1},\dots,x_{n}]$ we can think of
$h$ as an element of $\mathbb{K}[x_{1},\dots,x_{n}]$ as well as an element of
$\mathcal{R}_{C}(\mathbb{K}[x_{1},\dots,x_{n}])$. Then
$\operatorname{in}_{<}(h)={\bf x}^{\mu}\operatorname{in}_{<}(g)$ for some
$g\in\mathcal{G}$ of form $g={\bf x}^{\gamma}+\sum c_{\alpha}{\bf x}^{\alpha}$
with ${\bf x}^{\gamma}=\operatorname{in}_{<}(g)$. In particular, similar to
the above the exponent of $\operatorname{in}_{<}({\bf x}^{\mu}g)={\bf
x}^{\mu+\gamma}$ has the property that $w\cdot(\mu+\gamma)\leq
w\cdot(\mu+\alpha)$ for all $w\in C$ and all $\alpha$ with $c_{\alpha}\not=0$
in $g$. So, $h-{\bf x}^{\mu}g\in F_{m}(\mathfrak{w}_{C})$. In particular, the
division algorithm with respect to $\mathcal{G}$ takes place inside
$F_{m}(\mathfrak{w}_{C})$ and yields an expression of $h$ in terms of
$\\{\hat{g}\colon g\in\mathcal{G}\\}$. ∎

We now recall the Cox construction [14] for toric varieties and refer to [15,
Section 5] for more details. Let $\Sigma=C\cap\operatorname{Trop}(J)$ and $l$
denote the dimension of $\mathcal{L}(J)$. Fix an $(m\times n)$-matrix ${\bf
r}$ whose rows $r_{1},\dots,r_{m}$ are representatives of the primitive ray
generators of $\overline{C}\subset\mathbb{R}^{n}/\mathcal{L}(J)$. The rays of
$\Sigma$, denoted by $\Sigma(1)$, form a subset of
$\\{{r}_{1},\dots,{r}_{m}\\}$. Recall the definition of the lifted algebra
$\tilde{A}_{\bf r}$ from (3.2). We can apply the Cox construction in two ways,
to $X_{C}$ and to $X_{\Sigma}$: First, let $X_{C}$ be the affine toric variety
associated to $C$. Recall that a quasitorus is a product of a torus with a
finite abelian group. Then $X_{C}$ is isomorphic to the almost geometric
quotient $\mathbb{A}^{m}/\\!/G$ (see, e.g., [15, Theorem 5.1.11]), where the
group $G$ by [15, Lemma 5.1.1(b)] is the quasitorus

$G=\\{(t_{1},\dots,t_{m})\in(\mathbb{C}^{*})^{m}\colon t_{1}^{r_{1}\cdot
e_{i}}\cdots t_{m}^{r_{m}\cdot e_{i}}=1\,\forall\,i\in[n]\\}.$ (3.8)

One can easily check that if $p,q\in\mathbb{A}^{m}$ lie in the same $G$-orbit,
then $\pi^{-1}(p)\cong\pi^{-1}(q)$ by prop:generic fibers. Hence, the flat
family $\pi\colon\operatorname{Spec}(\tilde{A}_{\bf r})\to\mathbb{A}^{m}$
induces the commutative diagram

${\operatorname{Spec}(\tilde{A}_{\bf
r})}$${X_{C}}$${\mathbb{A}^{m}.}$$\scriptstyle{\pi}$$\scriptstyle{p_{C}\circ\pi}$$\scriptstyle{p_{C}}$
(3.9)

If $p,q\in X_{C}$ lie in the same torus orbit, then
$(p_{C}\circ\pi)^{-1}(p)\cong(p_{C}\circ\pi)^{-1}(q)$ by prop:generic fibers.
Note that $p_{C}\colon\mathbb{A}^{m}\to X_{C}$ is indeed a morphism. Namely,
it is the universal torsor for $X_{C}$.

Similarly, we can apply the Cox construction to the toric variety
$X_{\Sigma}$. For simplicity we assume that
$\Sigma(1)=\\{{r}_{1},\dots,{r}_{m}\\}$. In this case the construction has two
steps: first we remove the locus of $\mathbb{A}^{m}$ that does not correspond
to torus orbits in $X_{\Sigma}$, denoted by $Z(\Sigma)$. Recall that
$\mathcal{C}\subset\\{{r}_{1},\dots,{r}_{m}\\}$ is a primitive collection for
$\Sigma$, if $(i)$ $\mathcal{C}\not\subset\sigma(1)$ for all
$\sigma\in\Sigma$, and $(ii)$ for every
$\mathcal{C}^{\prime}\subsetneq\mathcal{C}$ there exists $\sigma\in\Sigma$
with $\mathcal{C}^{\prime}\subset\sigma(1)$. Then by [15, Proposition 5.1.6]

$Z(\Sigma)=\bigcup_{\mathcal{C}\text{ primitive collection }}V(t_{i}\colon
r_{i}\in\mathcal{C})\subset\mathbb{A}^{m}.$ (3.10)

Now $X_{\Sigma}$ is isomorphic to the almost geometric quotient
$(\mathbb{A}^{m}-Z(\Sigma))/\\!/G$, where $G$ is as in (3.8) (see, e.g., [15,
Theorem 5.1.11]). In contrast to the morphism $p_{C}\colon\mathbb{A}^{m}\to
X_{C}$ from (3.9), we obtain a rational map
$p_{\Sigma}\colon\mathbb{A}^{m}\dashrightarrow X_{\Sigma}$ as $p_{\Sigma}$ is
only defined on $\mathcal{T}_{\Sigma}:=\mathbb{A}^{m}-Z(\Sigma)$. Moreover,
$p_{\Sigma}\colon\mathcal{T}_{\Sigma}\to X_{\Sigma}$ is the universal torsor
of $X_{\Sigma}$. Similarly to the first case, this gives the diagram:

${\pi^{-1}(\mathcal{T}_{\Sigma})}$${\operatorname{Spec}(\tilde{A}_{\bf
r})}$${X_{\Sigma}}$${\mathcal{T}_{\Sigma}}$${\mathbb{A}^{m}}$$\scriptstyle{\pi|_{\pi^{-1}(\mathcal{T}_{\Sigma})}}$$\scriptstyle{p_{\Sigma}\circ\pi}$$\scriptstyle{\pi}$$\scriptstyle{p_{\Sigma}}$
(3.11)

which is commutative. The fibers of the family defined by
$p_{\Sigma}\circ\pi|_{\pi^{-1}(\mathcal{T}_{\Sigma})}$ coincide with the
fibers of $\psi$ defined in (3.4):
$(p_{\Sigma}\circ\pi)^{-1}(x)\cong\psi^{-1}(x)$ for every $x\in X_{\Sigma}$,
see (3.5).

In the following, we explain how the flat family of thm:family is related to
Kaveh–Manon’s flat family associated to a maximal cone in the Gröbner fan.

######  Theorem 3.19.

thm:KM max cone Let $J\subseteq\mathbb{K}_{\bf d}[{x_{1},\dots,x_{n}}]$ be a
d-homogeneous ideal and $C$ a maximal cone in $\operatorname{GF}(J)$. Let
${\bf r}$ be an $(m\times n)$-matrix whose rows are representatives of
primitive ray generators for $\overline{C}$. Then the morphism
$\pi\colon\operatorname{Spec}\big{(}\tilde{A}_{\bf r}\big{)}\to\mathbb{A}^{m}$
fits into a pull-back diagram

${\operatorname{Spec}(\mathcal{R}_{C})}$${\operatorname{Spec}\big{(}\tilde{A}_{\bf
r}\big{)}}$${X_{C}}$${\mathbb{A}^{m}.}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$$\scriptstyle{p_{C}}$

Here $p_{C}\colon\mathbb{A}^{m}\to X_{C}$ is the universal torsor of $X_{C}$
obtained from the Cox construction as in (3.9).

###### Proof.

We prove equivalently that the corresponding diagram between the algebras is a
push-out. Hence, it suffices to show that

$\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]\cong\tilde{A}.$

Note that the map $p_{C}\colon\mathbb{A}^{m}\to X_{C}$ corresponds to
$p_{C}^{\\#}\colon\mathbb{K}[S_{C}]{\hookrightarrow}\mathbb{K}[t_{1},\dots,t_{m}]$.
In particular, for a character $t^{-r_{i}^{*}}\in\mathbb{K}[S_{C}]$ with
$r_{i}^{*}\in C^{\vee}\subset M$ the dual of $r_{i}\in C\subset N$ we have
$p_{C}^{\\#}(t^{-r_{i}^{*}})$ $=t_{i}\in\mathbb{K}[t_{1},\dots,t_{m}]$. Then
under the extension of scalars (i.e., applying the functor
$-\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]$) the
homogenization $\hat{g}$ of an element $g\in\mathcal{G}$ is sent to the lift
$\tilde{g}$. This can be seen as follows: by lem:lifts we have $\tilde{g}={\bf
x}^{\gamma}+\sum c_{\alpha}{\bf x}^{\alpha}{\bf t}^{{\bf
r}\cdot(\alpha-\gamma)}$. By lem:init ell for every $\ell\in\mathcal{L}(J)$ we
have $\operatorname{in}_{\ell}(g)=g$. In particular,
$\ell\cdot\alpha=\ell\cdot\gamma$ and so $\alpha-\gamma\in(\mathcal{L}(J)\cap
N)^{\perp}$. Let $\\{\ell_{1},\dots,\ell_{k}\\}$ be a basis for
$\mathcal{L}(J)\cap N$. Then $\\{r_{1},\dots,r_{m},\ell_{1},\dots,\ell_{k}\\}$
spans $N$ and $\\{r_{1}^{*},\dots,r_{m}^{*}\\}$ spans $(\mathcal{L}(J)\cap
N)^{\perp}\subset M$. So every $\alpha-\gamma$ has an expression of form
$\alpha-\gamma=\sum c_{i}r_{i}^{*}$. We compute

$t^{-(\alpha-\gamma)}=t^{-c_{1}r_{1}^{*}}\cdots
t^{-c_{m}r_{m}^{*}}\overset{p_{C}^{\\#}}{\longmapsto}t_{1}^{c_{1}}\cdots
t_{m}^{c_{m}}={\bf t}^{{\bf r}\cdot(\alpha-\gamma)}.$ (3.12)

Note that $\gamma$ has the property that
$r_{i}\cdot\gamma=\min_{c_{\alpha}\not=0}\\{r_{i}\cdot\alpha\\}$. So,
$r_{i}\cdot(\alpha-\gamma)\geq 0$ and therefore $\alpha-\gamma\in C^{\vee}$.
Hence, we have

$\displaystyle\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]\overset{\text{{Proposition}~{}\ref{prop:Rees
quotient}}}{\cong}\big{(}\mathbb{K}[S_{C}][x_{1},\dots,x_{n}]/\hat{J}\big{)}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]$
$\displaystyle\hphantom{\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]Pro}{}\overset{\eqref{eq:t's}}{\cong}\mathbb{K}[t_{1},\dots,t_{m}][x_{1},\dots,x_{n}]/\tilde{J}=\tilde{A}.$
∎

######  Corollary 3.20.

thm:KM vs grobner Let $J\subseteq\mathbb{K}[{x_{1},\dots,x_{n}}]$ be a
weighted homogeneous ideal as in Definition LABEL:def:d_homog and $C$ a
maximal cone in $\operatorname{GF}(J)$. Let ${\bf r}$ be the matrix whose rows
are representatives of primitive ray generators for $\overline{C}$ and denote
$\tilde{A}_{\bf r}$ by $\tilde{A}$. Let $\Sigma$ be the intersection of $C$
with $\operatorname{Trop}(J)$. Then the restriction of morphism
$\pi\colon\operatorname{Spec}\big{(}\tilde{A}\big{)}\to\mathbb{A}^{m}$ to
$\pi^{-1}(\mathcal{T}_{\Sigma})=\operatorname{Spec}\big{(}\tilde{A}\big{)}-\pi^{-1}(Z(\Sigma))$
fits into a pull-back diagram

${\mathbf{Spec}(\mathcal{R}_{\Sigma})}$${\pi^{-1}(\mathcal{T}_{\Sigma})}$${X_{\Sigma}}$${\mathcal{T}_{\Sigma}.}$$\scriptstyle{\psi}$$\scriptstyle{\pi|_{\pi^{-1}(\mathcal{T}_{\Sigma})}}$$\scriptstyle{p_{\Sigma}}$

Here $p_{\Sigma}\colon\mathcal{T}_{\Sigma}\to X_{\Sigma}$ is the universal
torsor of $X_{\Sigma}$ obtained from the Cox construction as in (3.11).

###### Proof.

The scheme $X_{\Sigma}$ and the sheaf $\mathcal{R}_{\Sigma}$ are defined
locally for affine pieces
$U_{\sigma}=\operatorname{Spec}(\mathbb{K}[S_{\sigma}])$ $\subset X_{\Sigma}$
for $\sigma\in\Sigma$. Algebraically, for every $\sigma\in\Sigma$ the pull-
back of $\psi$ and $p_{\Sigma}$ corresponds to the following push-out diagram:

${\mathcal{R}_{\sigma}}$${\mathcal{R}_{\sigma}\otimes_{\mathbb{K}[S_{\sigma}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon
r_{i}\not\in\sigma\rangle}}$${\mathbb{K}[S_{\sigma}]}$${\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon r_{i}\not\in\sigma\rangle.}}$

Observe that by (3.10) the localization
$\mathbb{K}[t_{1},\dots,t_{m}]_{\langle t_{i}\colon
r_{i}\not\in\sigma\rangle}$ corresponds to an affine piece in $\mathbb{A}^{m}$
that does not intersect $Z(\Sigma)$. So it is in fact an affine piece of
$\mathcal{T}_{\Sigma}$. We have to show that

$\mathcal{R}_{\sigma}\otimes_{\mathbb{K}[S_{\sigma}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon r_{i}\not\in\sigma\rangle}\cong\tilde{A}_{\langle t_{i}\colon
r_{i}\not\in\sigma\rangle}.$

As $\sigma$ is a face of the maximal cone $C$ by [33, Proposition 3.13] we
have
$\mathcal{R}_{\sigma}\cong\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[S_{\sigma}]$.
Using thm:KM max cone we compute

$\displaystyle\mathcal{R}_{\sigma}\otimes_{\mathbb{K}[S_{\sigma}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon
r_{i}\not\in\sigma\rangle}\cong\big{(}\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[S_{\sigma}]\big{)}\otimes_{\mathbb{K}[S_{\sigma}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon r_{i}\not\in\sigma\rangle}$
$\displaystyle\hphantom{\mathcal{R}_{\sigma}\otimes_{\mathbb{K}[S_{\sigma}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon
r_{i}\not\in\sigma\rangle}}{}\cong\mathcal{R}_{C}\otimes_{\mathbb{K}[S_{C}]}\mathbb{K}[t_{1},\dots,t_{m}]_{\langle
t_{i}\colon r_{i}\not\in\sigma\rangle}\cong\tilde{A}_{\langle t_{i}\colon
r_{i}\not\in\sigma\rangle}.$ ∎

##  4 Grassmannians and cluster algebras

We now apply the machinery developed in Section 3 to the Grassmannians
$\operatorname{Gr}(2,\mathbb{C}^{n})$ and
$\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$. Our aim is to make this
paper as self-contained as possible and accessible to readers of broad
mathematical background. Therefore, we recall background on tropical
Grassmannians below, on cluster algebras and the cluster structure of
$\operatorname{Gr}(2,\mathbb{C}^{n})$ in Section 4.1 and then in Section 4.2
we translate between toric degenerations obtained from the tropicalizations
and from the cluster structure. In Section 4.3, we turn to the application of
the flat families introduced in Section 3 and show how to recover the cluster
algebra with universal coefficients in this case. Similarly, in Section 4.4 we
treat the case of $\operatorname{Gr}\big{(}3,\mathbb{C}^{6}\big{)}$.

We first fix our notation for $\operatorname{Gr}(2,\mathbb{C}^{n})$. Denote by
$S$ the polynomial ring in Plücker variables $\mathbb{C}[p_{ij}\colon 1\leq
i<j\leq n]$. Here, Plücker variables are associated with arcs in an $n$-gon
whose vertices are labeled by $[n]$ in clockwise order. Hence, we think of
$[n]:=\\{1,\ldots,n\\}$ as a cyclically ordered set. In particular, $i<j<k<l$
indicates that in clockwise order starting at the vertex $i$ the other
vertices appear in order $j$, $k$, $l$. Note that in general we might not have
$1\leq i<j<k<l\leq n$. We set the convention $p_{ij}:=-p_{ji}$. In $S$ we
define for every $i<j<k<l$ the Plücker relation:

$R_{ijkl}:=p_{ij}p_{kl}-p_{ik}p_{jl}+p_{il}p_{jk}\in S.$

The ideal $I_{2,n}\subset S$ generated by all Plücker relations $R_{ijkl}$ is
called the Plücker ideal. The quotient $A_{2,n}:=S/I_{2,n}$ is the Plücker
algebra which is the homogeneous coordinate ring of the Grassmannian
$\operatorname{Gr}(2,\mathbb{C}^{n})$ with respect to its Plücker embedding
into $\mathbb{P}^{\binom{n}{2}-1}$. The cosets of Plücker variables in the
quotient are denoted by $\bar{p}_{ij}\in A_{2,n}$ and are called Plücker
coordinates. Denote by $d:=2(n-2)$ the dimension of
$\operatorname{Gr}(2,\mathbb{C}^{n})$, so $d+1$ is the Krull-dimension of
$A_{2,n}$.

In $S$, we fix the notation ${\bf p}^{m}$ with
$m=(m_{ij})_{ij}\in\mathbb{Z}_{\geq 0}^{\binom{n}{2}}$ denoting the monomial
$\prod_{1\leq i<j\leq n}p_{ij}^{m_{ij}}$. The maximal cones in the Gröbner fan
$\operatorname{GF}(I_{2,n})$ are those of full dimension in the ambient space
$\mathbb{R}^{\binom{n}{2}}$ with associated monomial initial ideals. The
lineality space
$\mathcal{L}_{2,n}:=\mathcal{L}(I_{2,n})\subset\operatorname{GF}(I_{2,n})$ is
$n$-dimensional and spanned by the elements $\ell_{s}$ for $s\in[n]$ which are
defined by

$(\ell_{s})_{ij}:=\begin{cases}1&\text{if}\quad s=i\quad\text{or}\quad s=j,\\\
0&\text{otherwise}.\end{cases}$

Note that ${\bf 1}:=(1,\dots,1)\in\mathbb{R}^{\binom{n}{2}}$ lies in
$\mathcal{L}_{2,n}$. The Gröbner fan of the Plücker ideal has a subfan that is
particularly interesting when studying toric degenerations of
$\operatorname{Gr}(2,\mathbb{C}^{n})$.

######  Definition 4.1.

def:tropGr The tropical Grassmannian is a simplicial $(d+1)$-dimensional
subfan of $\operatorname{GF}(I_{2,n})$ defined as
$\operatorname{Trop}(I_{2,n})$.

Maximal cones in $\operatorname{Trop}(I_{2,n})$ are called tropical maximal
cones. They are in correspondence with trivalent trees with $n$ leaves as
shown by Speyer and Sturmfels in [49]. To a trivalent tree $\Upsilon$ one
associates a weight vector in the relative interior of the corresponding cone
of $\operatorname{Trop}(I_{2,n})$ by

$\displaystyle\underline{w}^{\Upsilon}(p_{ij}):=-\\#\\{\text{edges on the
unique path }i\to j\text{ in }\Upsilon\\}.$ (4.1)

To simplify notation we denote
$\operatorname{in}_{\Upsilon}(I):=\operatorname{in}_{\underline{w}^{\Upsilon}}(I)$.

######  Theorem 4.2 ([49]).

thm:SS For every trivalent tree $\Upsilon$ with $n$ leaves the ideal
$\operatorname{in}_{{\Upsilon}}(I)$ is toric, i.e., binomial and prime.
Moreover, it is generated by $\operatorname{in}_{{\Upsilon}}(R_{ijkl})$ for
all $1\leq i<j<k<l\leq n$.

######  Definition 4.3.

def:trop+Gr The totally positive tropical Grassmannian is
$\operatorname{Trop}^{+}(I_{2,n})$.

In [50, Section 5], it was shown that maximal cones in
$\operatorname{Trop}^{+}(I_{2,n})$ are in bijection with planar trivalent
trees, i.e., trivalent trees whose vertices are labeled $1,\dots,n$ in a
clockwise manner.

###  4.1 Preliminaries on cluster algebras

The Plücker algebra $A_{2,n}$ is in fact a cluster algebra (see Definition
4.4) whose cluster structure is closely related to the combinatorics governing
the tropical Grassmannian. To exhibit this connection, we recall some facts
about skew-symmetric cluster algebras following Fomin and Zelevinsky [23, 25].
Afterwards, we define finite type cluster algebras with frozen variables and
universal coefficients.

Let $m,f\in\mathbb{N}$ be positive integers and
$\mathcal{F}=\mathbb{C}(u_{1},\dots,u_{m+f})$ be the field of rational
functions on $m+f$ variables. Here $m$ stands for mutable and $f$ for frozen.
A seed in $\mathcal{F}$ is a pair $\big{(}\widetilde{{\bf
x}},\widetilde{B}\big{)}$, where $\widetilde{{\bf
x}}=\\{x_{1},\dots,x_{m+f}\\}$ is a free generating set of $\mathcal{F}$ and
$\widetilde{B}=(b_{ij})$ is an $(m+f)\times m$ rectangular matrix with the
following property: the top square submatrix of $\widetilde{B}$ (that is the
$m\times m$ submatrix of $\widetilde{B}$ formed by its first $m$ rows) is
skew-symmetric. We call $\widetilde{B}$ an extended exchange matrix and refer
to its top square submatrix $B$ as an exchange matrix. We call ${\bf
x}=\\{x_{1},\dots,x_{m}\\}$ a cluster, $\\{x_{m+1},\dots,x_{m+f}\\}$ the set
of frozen variables and $\widetilde{{\bf x}}$ an extended cluster.

Given $k\in\\{1,\dots,m\\}$, the mutation in direction $k$ of a seed
$\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$ is the new seed
$\mu_{k}\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}=\big{(}\widetilde{{\bf
x}}^{\prime},\widetilde{B}^{\prime}\big{)}$, defined as follows:

  * •

The extended cluster $\widetilde{{\bf x}}^{\prime}=\widetilde{{\bf
x}}\setminus\\{x_{k}\\}\cup\\{x^{\prime}_{k}\\}$, where

$x_{k}x^{\prime}_{k}=\prod_{i\colon b_{ik}>0}x^{b_{ik}}_{i}+\prod_{i\colon
b_{ik}<0}x^{-b_{ik}}_{i}.$ (4.2)

We call an expression of the form (4.2) an exchange relation and the monomial
$x_{k}x_{k}^{\prime}$ an exchange monomial.

  * •

The extended exchange matrix $\widetilde{B}^{\prime}=(b^{\prime}_{ij})$ is
defined by the following formula

$b^{\prime}_{ij}:=\begin{cases}-b_{ij}\qquad\text{if}\quad
i=k\quad\text{or}\quad j=k,\\\ b_{ij}+\mathop{\rm sgn}(b_{ik})\mathop{\rm
max}(b_{ik}b_{kj},0)\qquad\text{else},\end{cases}$

where

$\mathop{\rm sgn}(b_{ik}):=\begin{cases}-1&\text{if}\quad b_{ik}<0,\\\
0&\text{if}\quad b_{ik}=0,\\\ 1&\text{if}\quad b_{ik}>0.\end{cases}$

######  Definition 4.4.

The cluster algebra $A\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$
associated to a seed $\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$ is the
$\mathbb{C}$-subalgebra of $\mathcal{F}$ generated by the elements of _all_
the extended clusters that can be obtained from the initial seed
$\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$ by a finite sequence of
mutations. The elements of the clusters constructed in this way are called the
cluster variables. If $f\neq 0$ we say that the cluster algebra
$A\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$ has frozen variables,
namely, $x_{m+1},\dots,x_{m+f}$.

######  Remark 4.5.

It can be easily verified that, up to a canonical isomorphism,
$A\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$ is independent of
$\widetilde{{\bf x}}$. Therefore, we write $A\big{(}\widetilde{B}\big{)}$
instead of $A\big{(}\widetilde{{\bf x}},\widetilde{B}\big{)}$.

######  Remark 4.6.

Fomin and Zelevinsky usually define cluster algebras over $\mathbb{Q}$. We
recover the construction described above applying tensor product with
$\mathbb{C}$ to Fomin and Zelevinsky’s construction.

######  Theorem 4.7 ([25, Laurent phenomenon]).

thm:SLaurent The cluster algebra $A\big{(}\widetilde{B}\big{)}$ is a
$\mathbb{C}[x_{m+1},\dots,x_{m+f}]$-subalgebra of $\mathbb{C}\big{[}x_{1}^{\pm
1},\dots,x_{m}^{\pm 1},x_{m+1},\dots,x_{m+f}\big{]}$.

######  Definition 4.8.

def:prin The cluster algebra with principal coefficients associated to an
$m\times m$ skew-symmetric matrix $B$ is the cluster algebra
$A(B^{\rm{prin}})$, where $B^{\rm{prin}}$ is the $2m\times m$ matrix whose top
$m\times m$ square submatrix is $B$ and whose lower $m\times m$ square
submatrix is the identity matrix $\mathbbm{1}_{m}$.

The notion of cluster algebras with principal coefficients is central in
cluster theory. Its relevance in the framework of toric degenerations of
cluster varieties was highlighted in [28]. In particular, in loc. cit. the
authors work implicitly with cluster algebras with frozen variables and
principal coefficients. This is precisely the notion we need here.

######  Definition 4.9.

Let $\widetilde{B}$ be an extended exchange matrix. Let $\widehat{B}$ be the
$(m+f)\times(m+f)$ skew-symmetric square matrix whose left $(m+f)\times m$
submatrix is $\widetilde{B}$ and whose bottom right $f\times f$ square
submatrix is the zero matrix (this information fully determines $\widehat{B}$
since it is skew-symmetric). Denote by $t_{1},\dots,t_{m+f}$ the variables of
$A\big{(}\widehat{B}^{\rm prin}\big{)}$ associated to the last $m+f$
directions, which in this case are the frozen directions. Then the cluster
algebra with principal coefficients and frozen variables
$A\big{(}\widetilde{B}^{\rm prin}\big{)}$ associated to $\widetilde{B}$ is the
$\mathbb{C}[t_{1},\dots,t_{m+f}]$-subalgebra of $A\big{(}\widehat{B}^{\rm
prin}\big{)}$ spanned by the cluster variables that can be obtained from the
initial extended cluster by iterated mutation in the first $m$ mutable
directions.

######  Theorem 4.10 ([16, Theorem 1.7]).

thm:g-vecs Let $X$ be a cluster variable associated to
$A\big{(}\widetilde{B}^{\rm{prin}}\big{)}$. By Theorem LABEL:thm:SLaurent we
know that $X\in\mathbb{C}[t_{1},\dots,t_{m+f}]\big{[}x_{1}^{\pm
1},\dots,x_{m}^{\pm 1},x_{m+1},\dots,x_{m+f}\big{]}$. Let $X|_{{\bf t}=0}$ be
the Laurent polynomial in $\mathbb{C}\big{[}x_{1}^{\pm 1},\dots,x_{m}^{\pm
1},x_{m+1},\dots,x_{m+f}\big{]}$ obtained after the evaluation of $X$ at
$t_{1}=0,\dots,t_{m+f}=0$. Then $X|_{{\bf t}=0}$ is a non-constant Laurent
monomial with coefficient $1$. In other words, for a non-zero vector ${\bf
g}(X)=\big{(}g_{1}^{X},\dots,g_{m+f}^{X}\big{)}\in\mathbb{Z}^{m+f}$ we have
that

$X|_{{\bf t}=0}=\prod_{i=1}^{m+f}x_{i}^{g^{X}_{i}}.$

Moreover, if $X$ and $X^{\prime}$ are different cluster variables then ${\bf
g}(X)\neq{\bf g}(X^{\prime})$.

######  Definition 4.11.

def:g-vectors Let $X$ be a cluster variable associated to
$A\big{(}\widetilde{B}^{\rm{prin}}\big{)}$. The vector ${\bf
g}(X)\in\mathbb{Z}^{m+f}$ introduced in thm:g-vecs is the g-vector associated
to $X$. We refer to the vector
$\big{(}g_{1}^{X},\dots,g_{m}^{X}\big{)}\in\mathbb{Z}^{m}$ as the truncated
g-vector of $X$. The set of (truncated) g-vectors associated to
$\widetilde{B}$ is the set of (truncated) g-vectors of all the cluster
variables in $A\big{(}\widetilde{B}^{\rm{prin}}\big{)}$.

######  Remark 4.12.

Let $X|_{{\bf t}=1}$ be the Laurent polynomial in $\mathbb{C}\big{[}x_{1}^{\pm
1},\dots,x_{m}^{\pm 1},x_{m+1},\dots,x_{m+f}\big{]}$ obtained after the
evaluation of $X$ at $t_{1}=1,\dots,t_{m+f}=1$. Then $X|_{{\bf t}=1}$ is a
cluster variable of $A\big{(}\widetilde{B}\big{)}$. By a slight abuse of
terminology we say ${\bf g}(X)$ is the ${\bf g}$-vector of the cluster
variable $X|_{{\bf t}=1}$.

Universal coefficients. We now turn to the definition of finite type cluster
algebras _with_ frozen variables _and_ universal coefficients. This notion is
a natural (and slight) generalization of the usual notion of universal
coefficients for cluster algebras _without_ frozen variables introduced by
Fomin and Zelevinsky in [25, Section 12]. The difference between these notions
arises from the distinction we make between coefficients and frozen variables,
notions that are usually identified. This distinction was first suggested in
[9] and is of particular importance in the study of toric degenerations of
cluster varieties. It was shown in [25] that finite type cluster algebras can
be endowed with universal coefficients. This construction was categorified in
[42] and the categorification was used to perform various computations in the
following sections. To further clarify the preceding discussion let us first
recall the notion of finite type cluster algebras with universal coefficients.

######  Definition 4.13.

ice quiver/matrix An ice quiver is a pair $(Q,F)$, where $Q$ is a finite
quiver with the vertex set $V(Q)$ and $F\subset V(Q)$ is a set of frozen
vertices. The vertices in $V(Q)\backslash F$ are called mutable. If $(Q,F)$ is
an ice quiver then the corresponding $|V(Q)|\times|V(Q)\setminus F|$ matrix is
$B_{(Q,F)}:=(b_{ij})$, where

$b_{ij}:=\\#\\{\text{arrows }i\to j\text{ in }Q\\}-\\#\\{\text{arrows }j\to
i\text{ in }Q\\}$ (4.3)

for $i\in V(Q)$ and $j\in V(Q)\setminus F$. Note that
$(\ref{eq:matrix_quiver})$ can be similarly used to recover $(Q,F)$ from
$B_{(Q,F)}$. The quiver mutation is the operation induced by matrix mutation
at the level of quivers. If $F=\emptyset$ we set $B_{Q}:=B_{(Q,\emptyset)}$.

######  Theorem-Definition 4.14 ([43, Corollary 8.15]).

def:univ_coeff_R Let $Q$ be a quiver of finite cluster type, that is $Q$ is
mutation equivalent to an orientation of a Dynkin diagram. The cluster algebra
with universal coefficients associated to $Q$ is $A\big{(}B_{Q}^{{\rm
univ}}\big{)}$, where

$B^{{\rm univ}}_{Q}=\left(\begin{array}[]{c}B_{Q}\\\ \hdashline
U_{Q}\end{array}\right)$

and $U_{Q}$ is the rectangular matrix whose rows are the g-vectors of the
opposite quiver $Q^{\rm op}$.

Observe that we have described the rows of $U_{Q}$ but did not specify in
which order they appear. Any order we choose provides a realization of
$A\big{(}B_{Q}^{{\rm univ}}\big{)}$ since a rearrangement of the rows of
$U_{Q}$ amounts to reindexing the corresponding coefficients. So any choice
gives rise to an isomorphic algebra. It can be verified that the cluster
algebra with universal coefficients associated to $Q$ is an invariant of the
mutation class of $Q$. In other words, if $Q^{\prime}$ is mutation equivalent
to $Q$ then $A\big{(}B_{Q}^{{\rm univ}}\big{)}$ is canonically isomorphic to
$A\big{(}B_{Q^{\prime}}^{{\rm univ}}\big{)}$. The canonical isomorphism is
completely determined by sending the elements of the initial cluster of
$B_{Q}^{{\rm univ}}$ to those of $B_{Q^{\prime}}^{{\rm univ}}$.

######  Remark 4.15.

rem:univ_property Cluster algebras with universal coefficients as treated by
Fomin and Zelevinsky are defined by a universal property [25, Definition
12.1]. We have chosen Reading’s equivalent definition since we will not use
the universal property here.

If $Q$ is a bipartite orientation of a Dynkin diagram $\Delta$ then the matrix
$B^{\rm univ}_{Q}$ has a particularly nice and explicit description. More
precisely, let $\Phi^{\vee}$ be the root system dual to the root system
associated to $\Delta$. Fix $\Phi^{\vee}_{+}\subset\Phi^{\vee}$ a subset of
positive coroots with
$\\{\alpha_{1},\dots,\alpha_{n}\\}\subset\Phi^{\vee}_{+}$ the set of simple
coroots. The almost positive coroots are the coroots in the set
$\Phi^{\vee}_{\geq-1}:=\Phi^{\vee}_{+}\cup\\{-\alpha_{1},\dots,-\alpha_{n}\\}$.
The cardinality of $\Phi^{\vee}_{\geq-1}$ coincides with the number of cluster
variables in $A\big{(}B^{\rm univ}_{Q}\big{)}$ which is the number of
coefficients for $A\big{(}B^{\rm univ}_{Q}\big{)}$. Hence, we may consider
$\big{\\{}y_{\alpha}\colon\alpha\in\Phi^{\vee}_{\geq-1}\big{\\}}$ as the set
of coefficients for $A\big{(}B^{\rm univ}_{Q}\big{)}$. Next, for a coroot
$\beta=\sum_{i=1}^{n}c_{i}\alpha_{i}$ we define $[\beta:\alpha_{i}]:=c_{i}$.
With this notation we have the following description of $B^{\rm univ}_{Q}$
which is a reformulation of [25, Theorem 12.4].

######  Proposition 4.16.

prop:bipartite_univ_Q The entry in $B_{Q}^{\rm univ}$ in the column $i$ and
the row corresponding to $y_{\beta}$ is $[\beta:\alpha_{i}]$ if $i$ is a
source and $-[\beta:\alpha_{i}]$ if $i$ is a sink.

######  Definition 4.17.

def:univ frozen Let $(Q,F)$ be an ice quiver whose mutable part is a quiver
$Q^{\rm mut}$ of finite cluster type. The associated cluster algebra with
universal coefficients is $A\big{(}B_{(Q,F)}^{{\rm univ}}\big{)}$, where

$B^{{\rm univ}}_{(Q,F)}=\left(\begin{array}[]{c}B_{(Q,F)}\\\ \hdashline
U_{Q^{\rm mut}}\end{array}\right)$

and $U_{Q^{\rm mut}}$ is the matrix whose rows are the truncated g-vectors of
the opposite quiver $(Q^{\rm mut})^{\rm op}$.

######  Remark 4.18.

The algebra $A\big{(}B_{(Q,F)}^{{\rm univ}}\big{)}$ satisfies a universal
property which is a straight forward generalization of the universal property
of a cluster algebra without frozen directions and with universal
coefficients, see rem:univ_property.

The Grassmannian cluster algebra $\boldsymbol{A_{2,n}}$. We now turn to the
cluster algebra that is of most interest to us, and use the combinatorics
governing the cluster structure of $A_{2,n}$: triangulations of an $n$-gon.
The vertices of the $n$-gon are labeled by $[n]$ in the clockwise order. An
arc connecting two vertices $i$ and $j$ is denoted by $\overline{ij}$ and we
associate to it the Plücker coordinate $\bar{p}_{ij}\in A_{2,n}$.

######  Definition 4.19.

def:triang A triangulation of the $n$-gon is a maximal collection of non-
crossing arcs dividing the $n$-gon into $n-2$ triangles. To a triangulation
$T$ we associate a collection of Plücker coordinates ${\bf
x}_{T}:=\big{\\{}\bar{p}_{ij}\colon\overline{ij}\text{ is an arc in
}T\big{\\}}$ which is the associated cluster. The elements of ${\bf x}_{T}$
are the cluster variables. The monomials in Plücker coordinates which are all
part of one cluster are the cluster monomials.

Note that every triangulation $T$ contains the boundary edges
$\overline{12},\overline{23},\dots,\overline{n1}$. The corresponding cluster
variables $\bar{p}_{12},\bar{p}_{23},\dots,\bar{p}_{1n}$ are frozen cluster
variables. All other variables $\bar{p}_{ij}$ with $i\not=j\pm 1$ are mutable
cluster variables. To every triangulation $T$ we associated an ice quiver.

######  Definition 4.20.

def:quiver For a triangulation $T$, we define its associated ice quiver
$\big{(}Q_{T},F_{T}\big{)}$: the vertices $V(Q_{T})$ are in correspondence
with arcs and boundary edges $\overline{ij}\in T$; Two vertices
$v_{\overline{ij}}$ and $v_{\overline{kl}}$ are connected by an arrow, if they
correspond to arcs in one triangle. Inside every triangle, we orient the
arrows clockwise. The set $F_{T}$ consists of vertices corresponding to the
boundary edges of the $n$-gon, hence $F$ does not depend on $T$. We neglect
arrows between vertices in $F$, see Figure 1.

123456781$1^{\prime}$2$2^{\prime}$3$3^{\prime}$4$4^{\prime}$5$5^{\prime}$6$6^{\prime}$7$7^{\prime}$8$8^{\prime}$12345678
Figure 1: On the right a triangulation $T$ of the $8$-gon and its
corresponding quiver $Q_{T}$. On the left its corresponding extended tree
$\widehat{D}_{T}$ dual to $T$ which is a trivalent tree with $16$ leaves.

We first recall some classic results due to Fomin–Zelevinsky and
Fomin–Shapiro–Thurston.

######  Theorem 4.21 ([23]).

For all $n\geq 3$ the ring $A_{2,n}$ is a cluster algebra with frozen
variables. More precisely, if $\big{(}Q_{T},F_{T}\big{)}$ is the ice quiver
associated to a triangulation $T$ of the $n$-gon then $A_{2,n}\cong
A\big{(}B_{(Q_{T},F_{T})}\big{)}$ as $\mathbb{C}$-algebras. Moreover, the
cluster variables associated to $B_{(Q_{T},F_{T})}$ are precisely the Plücker
coordinates and the exchange relations are precisely the Plücker relations.

######  Theorem 4.22 ([21]).

thm:cluster mono basis The set of all cluster monomials is a
$\mathbb{C}$-vector space basis for the algebra $A_{2,n}$ called the basis of
cluster monomials.

We endow $A_{2,n}$ with universal coefficients associated to its mutable part.
This is an example of a cluster algebra with frozen directions and
coefficients in the sense of [9].

######  Notation 4.23.

Let $\big{(}Q_{T},F_{T}\big{)}$ be the quiver associated to a triangulation of
the $n$-gon. We denote by $Q_{T}^{\rm{mut}}$ the full subquiver of $Q_{T}$
supported in the mutable vertices $V(Q_{T})\setminus F_{T}$. In particular,
$Q_{T}^{\rm{mut}}$ is mutation equivalent to an orientation of a type
$\mathtt{A}$ Dynkin diagram.

######  Definition 4.24.

def:Plucker univ Let $T$ be a triangulation of the $n$-gon. The Plücker
algebra with universal coefficients $A_{2,n}^{\rm{univ}}$ is the cluster
algebra defined by the extended exchange matrix

$B_{(Q_{T},F_{T})}^{\rm{univ}}=\left(\begin{array}[]{c}B_{(Q_{T},F_{T})}\\\
\hdashline U_{Q^{\rm{mut}}_{T}}\end{array}\right)\\!.$

Up to canonical isomorphism,
$A_{2,n}^{\rm{univ}}=A\big{(}B_{(Q_{T},F_{T})}^{\rm{univ}}\big{)}$ is
independent of $T$.

The quivers $Q_{T}^{\rm mut}$ and $\big{(}Q_{T}^{\rm mut}\big{)}^{\rm op}$
define canonically isomorphic cluster algebras whose cluster variables
(denoted by $\bar{p}_{ij}$) are both in bijection with arcs $\overline{ij}$ of
the $n$-gon for which $2\leq i+1<j\leq n$. Hence, the rows of $U_{Q_{T}^{\rm
mut}}$ are also in bijection with these arcs. We write $y_{ij}$ for the
coefficient variable of $A_{2,n}^{\rm univ}$ corresponding to the arc
$\overline{ij}$.

######  Example 4.25.

Let $\big{(}Q_{T},F_{T}\big{)}$ be the ice quiver associated to the
triangulation of the pentagon depicted in Figure 2. The corresponding
rectangular matrix for universal coefficients can be found on the right side
of Figure 2. Incidentally, in this case the rows corresponding to the
universal coefficients coincide with the rows corresponding to the frozen part
of $\big{(}Q_{T},F_{T}\big{)}$. However, for polygons with more than 5 sides
this will not be the case.

12345$B^{\rm univ}_{(Q_{T},F_{T})}=\left(\scriptsize{\begin{array}[]{rr}0&1\\\
-1&0\\\ 1&0\\\ 0&-1\\\ 0&1\\\ 1&-1\\\ -1&0\\\ \hdashline 1&0\\\ 0&-1\\\ 0&1\\\
1&-1\\\ -1&0\end{array}}\right)=\left(\begin{array}[]{c}B_{(Q_{T},F_{T})}\\\
\hdashline U_{Q_{T}^{\rm mut}}\end{array}\right)$ Figure 2: A triangulation of
the $5$-gon and the associated matrix $B_{(Q_{T},F_{T})}^{\rm univ}$.

To get a better understanding of the cluster algebra $A_{2,n}^{\rm univ}$ we
now turn to the combinatorial gadgets that govern its definition: g-vectors of
Plücker coordinates. Their description involves two more combinatorial objects
associated to a triangulation $T$, namely its dual graph $D_{T}$ and its
extended dual graph $\widehat{D}_{T}$. Recall that non-interior vertices of a
tree graph are called leaves. Moreover, two leaves that are connected to the
same interior vertex form a so-called cherry.

######  Definition 4.26.

def:trees Fix a triangulation $T$ of the $n$-gon. The trivalent tree $D_{T}$
is the dual graph or ribbon graph of $T$. It has $n$ leaves and its interior
vertices correspond to triangles in $T$. Two interior vertices are connected
by an edge if their corresponding triangles share an arc. Vertices
corresponding to triangles involving a boundary edge $\overline{i-1,i}$ are
connected to the leaf $i$. The trivalent tree $\widehat{D}_{T}$ is obtained
from $D_{T}$ by replacing every leaf $i$ by a cherry with leaves $i$ and
$i^{\prime}$, where (in the clockwise order) $i^{\prime}$ labels the first
leaf. We call $\widehat{D}_{T}$ the extended tree dual to $T$.

Note that the tree $D_{T}$ is by definition planar. See Figure 1 for an
example. Moreover, the extended tree $\widehat{D}_{T}$ can alternatively be
defined as the dual graph of a triangulation $\widehat{T}$ of the $2n$-gon
with vertices $1^{\prime},1,2^{\prime},2,\dots,n^{\prime},n$ in the clockwise
order. Here $\widehat{T}$ is obtained from $T$ by replacing every boundary
edge $\overline{i-1,i}$ by a triangle with a new vertex labeled by
$i^{\prime}$. Then $\widehat{D}_{T}=D_{\widehat{T}}$.

We now focus on giving a combinatorial definition of the ${\bf g}$-vector
associated to a Plücker coordinate with respect to a triangulation $T$. The
reader can find an equivalent definition above in def:g-vectors. The ${\bf
g}$-vectors can be read from the extended tree $\widehat{D}_{T}$. First,
observe that the interior edges of $\widehat{D}_{T}$ correspond to the arcs
and the boundary edges in $T$ and, therefore to the cluster variables of the
associated seed.

Sign conventions for ${\bf g}$-vectors. Consider a path in the tree
$\widehat{D}_{T}$ with the end points in $\\{1,\dots,n\\}$ (not in
$\\{1^{\prime},\dots,n^{\prime}\\}$). Say the path goes from $i$ to $j$. For
simplicity, we fix an orientation of the path and denote it by
$i\rightsquigarrow j$ (what follows is independent of the choice of
orientation). As $\widehat{D}_{T}$ is trivalent, at every interior vertex the
path can either turn _right_ or _left_. Denote by $e_{\overline{ab}}$ the edge
in $\widehat{D}_{T}$ corresponding to an arc $\overline{ab}\in T$ and let
$v_{abc}$ and $v_{abd}$ be the vertices in $\widehat{D}_{T}$ adjacent to
$e_{\overline{ab}}$. Given the path $i\rightsquigarrow j$ we associate signs
$\sigma_{i\rightsquigarrow j}^{ab}\in\\{-1,0,+1\\}$ to every $\overline{ab}\in
T$ as follows:

  1. 1.

If the path $i\rightsquigarrow j$ either does not pass through
$e_{\overline{ab}}$, or it passes through $e_{\overline{ab}}$ by turning
_right_ at $v_{abc}$ and turning _right_ at $v_{abd}$, or it passes through
$e_{\overline{ab}}$ by turning _left_ at $v_{abc}$ and turning _left_ at
$v_{abd}$: then $\sigma_{i\rightsquigarrow j}^{{ab}}=0$.

  2. 2.

If the path $i\rightsquigarrow j$ passes through the $e_{\overline{ab}}$ by
turning _left_ at $v_{abc}$ and turning _right_ at $v_{abd}$: then
$\sigma_{i\rightsquigarrow j}^{{ab}}=+1$.

  3. 3.

If the path $i\rightsquigarrow j$ passes through the $e_{\overline{ab}}$ by
turning _right_ at $v_{abc}$ and turning _left_ at $v_{abd}$: then
$\sigma_{i\rightsquigarrow j}^{{ab}}=-1$.

Cases 2 and 3 are depicted in Figure 3. We leave it to the reader to verify
that $\sigma_{i\rightsquigarrow j}^{{ab}}=\sigma_{j\rightsquigarrow
i}^{{ab}}$. Later we replace the end points $i$ and $j$ by the interior
vertices of $\widehat{D}_{T}$ and use the same symbol.

$-$$+$ Figure 3: Sign conventions to compute ${\bf g}$-vectors.

######  Definition 4.27.

def:g Let $\\{f_{{ab}}\\}_{\overline{ab}\in T}$ be the standard basis of
$\mathbb{Z}^{2n-3}$. For the Plücker coordinate $\bar{p}_{ij}$ consider the
path from $i$ to $j$ in $\widehat{D}_{T}$. The ${\bf g}$-vector of
$\bar{p}_{ij}$ with respect to $T$ is defined as

${\bf g}^{\widehat{T}}_{ij}:=\sum_{\overline{ab}\in
T}\sigma_{i\rightsquigarrow j}^{ab}f_{ab}.$

Similarly, we define the g-vector of a cluster monomial as the sum of the
g-vectors of its factors.

Note that for $\overline{ij}\in T$ we have ${\bf g}_{ij}=f_{ij}$. When there
is no ambiguity, we write ${\bf g}_{ij}$ instead of ${\bf
g}^{\widehat{T}}_{ij}$.

Figure 4: Combinatorial translation between laminations on triangulations (as
in [22]) and paths on trees (as in def:g) to compute truncated ${\bf
g}$-vectors. The red lines corresponds to the simple lamination associated to
$\overline{ij}$.

######  Lemma 4.28.

lem:g are gv Let $T$ be a triangulation of the $n$-gon. The ${\bf g}$-vector
${\bf g}^{\widehat{T}}_{ij}$ of a Plücker coordinate $\overline{p}_{ij}$ with
respect to $T$ introduced in def:g coincides with the ${\bf g}$-vector ${\bf
g}(\bar{p}_{ij})$ $($from Definition $\ref{def:g-vectors})$ of the cluster
variable $\bar{p}_{ij}$ of $A\big{(}B_{(Q_{T},F_{T})}\big{)}$.

###### Proof.

We use some of the terminology of cluster algebras associated to surfaces
developed in [22]. The algebra $A_{2,n}$ is a cluster algebra arising from an
orientable surface with marked points, in this case a disc with $n$ marked
points in its boundary. As such, the truncated ${\bf g}$-vectors of its
cluster variables can be computed using laminations in the surface as
explained in [22, Proposition 17.3]. To be more precise, the truncated
g-vector of $\bar{p}_{ij}$ is the vector containing the shear coordinates of
the internal arcs of $T$ with respect to the simple lamination associated to
the arc $\overline{ij}$. The key observation is that the combinatorial rule
defining shear coordinates translates to our combinatorial rule to compute
${\bf g}^{\widehat{T}}_{ij}$. In Figure 4 we visualize the translation from
our combinatorial rule to the one defining shear coordinates from [22,
Definition 12.2, Figure 34]. Shear coordinates are only associated to internal
arcs and, therefore, can only compute truncated ${\bf g}$-vectors. However,
the extended exchange matrix $\widehat{B}_{(Q_{T},F_{T})}$ is the submatrix of
$B_{Q_{\widehat{T}},F_{\widehat{T}}}$ associated to the internal arcs of the
triangulation $\widehat{T}$ of the $2n$-gon. As a consequence ${\bf
g}(\bar{p}_{ij})={\bf g}^{\widehat{T}}_{ij}$. ∎

######  Example 4.29.

Consider the triangulation $T$ depicted in Figure 1 and its corresponding tree
$\widehat{D}_{T}$. Then ${\bf g}_{ij}=f_{ij}$ for $\overline{ij}\in T$. We
compute the ${\bf g}$-vectors of the remaining Plücker coordinates:

$\begin{split}{\bf g}_{15}&=f_{17}-f_{47}+f_{45},\\\ {\bf
g}_{24}&=f_{12}-f_{13}+f_{34},\\\ {\bf g}_{25}&=f_{12}-f_{14}+f_{45},\\\ {\bf
g}_{46}&=f_{47}-f_{57}+f_{56},\\\ {\bf g}_{58}&=f_{78}-f_{47}+f_{45},\\\
\end{split}\quad\quad\begin{split}{\bf g}_{16}&=f_{17}-f_{57}+f_{56},\\\ {\bf
g}_{27}&=f_{12}-f_{14}+f_{47},\\\ {\bf g}_{28}&=f_{12}-f_{17}+f_{78},\\\ {\bf
g}_{48}&=f_{14}-f_{17}+f_{78},\\\ {\bf g}_{68}&=f_{56}-f_{57}+f_{78},\\\
\end{split}\quad\quad\begin{split}{\bf g}_{37}&=f_{47}-f_{14},\\\ {\bf
g}_{35}&=f_{45}-f_{14},\\\ {\bf g}_{38}&=f_{78}-f_{17},\\\ {\bf
g}_{26}&=f_{12}-f_{14}+f_{47}-f_{57}+f_{56},\\\ {\bf
g}_{36}&=-f_{14}+f_{47}-f_{57}+f_{56}.\\\ \end{split}$

###  4.2 Toric degenerations

In this section, we use the ${\bf g}$-vectors from def:g to obtain a weight
vector for every triangulation of the $n$-gon. The initial ideals of $I_{2,n}$
arising this way are toric. Throughout this section we fix a triangulation
$T$.

######  Definition 4.30.

def:monomial ideal Consider a quadrilateral with vertices $i$, $j$, $k$, $l$
inside the $n$-gon. The arcs $\overline{ik}$ and $\overline{jl}$ are called
crossing arcs, if when drawn inside the $n$-gon the corresponding arcs cross
each other. Let $M_{2,n}$ be the set of crossing monomials $p_{ik}p_{jl}$,
where in the quadrilateral with vertices $i$, $j$, $k$, $l$ the arcs
$\overline{ik}$ and $\overline{jl}$ are crossing. Let
$\mathcal{M}_{2,n}\subset S$ be the monomial ideal generated by $M_{2,n}$.

######  Lemma 4.31.

lem:g-v on relations Given a Plücker relation
$p_{ij}p_{kl}-p_{ik}p_{jl}+p_{il}p_{jk}$ with $i<j<k<l$ the multidegree
induced by the ${\bf g}$-vectors for the triangulation $T$ on the monomials
agrees for exactly two terms. Moreover, one of those terms is the crossing
monomial $p_{ik}p_{jl}$.

$i$$v_{i}$$l$$v_{l}$$a_{1}$$b_{1}$$c_{1}$$v^{\prime}$$v$$j$$v_{j}$$k$$v_{k}$$a_{2}$$b_{2}$$c_{2}$$i^{\prime}$$v_{i^{\prime}}$$d$
Figure 5: The full subtree of $\widehat{D}_{T}$ with leaves $i$, $i^{\prime}$,
$j$, $k$, $l$ and labeling as in the proof of lem:g-v on relations (ignoring
$i^{\prime}$, $d$, $v^{\prime}$) and prop:wt map. Blue marks refer to certain
paths in the tree and the labels $v_{i}$, $v_{j}$, $v_{k}$, $v_{l}$, $v$,
$v^{\prime}$ to the corresponding elements of $\mathbb{Z}^{2n-3}$. We follow
the convention in Figure 3.

###### Proof.

Consider the extended tree $\widehat{D}_{T}$ and restrict it to the full
subtree with leaves $i$, $j$, $k$, $l$. By definition, the subtree has four
leaves and edges of valency three or less. Without loss of generality we may
assume the leaves are arranged as in Figure 5. Two vertices of the subtree are
trivalent, namely the one where the paths $i\rightsquigarrow j$ and
$l\rightsquigarrow j$ meet, respectively where the paths $j\rightsquigarrow i$
and $k\rightsquigarrow i$ meet. Assume these vertices come from triangles in
$T$ labeled by $a_{1}$, $b_{1}$, $c_{1}$, respectively $a_{2}$, $b_{2}$,
$c_{2}$. Denote the vertex of $\widehat{D}_{T}$ corresponding to the triangle
$a_{1}$, $b_{1}$, $c_{1}$ (respectively $a_{2}$, $b_{2}$, $c_{2}$) by
$\Delta_{1}$ (respectively $\Delta_{2}$). We compute the ${\bf g}$-vectors for
all Plücker coordinates whose indices are a subset of $\\{i,j,k,l\\}$. To
simplify the notation in the computation we use the following abbreviations

$\displaystyle v_{i}:=\sum_{\overline{pq}\in
T}\sigma_{i\rightsquigarrow\Delta_{1}}^{pq}f_{pq},\quad
v_{j}:=\sum_{\overline{pq}\in
T}\sigma_{j\rightsquigarrow\Delta_{2}}^{pq}f_{pq},\quad
v_{l}:=\sum_{\overline{pq}\in
T}\sigma_{l\rightsquigarrow\Delta_{1}}^{pq}f_{pq},\quad
v_{k}:=\sum_{\overline{pq}\in
T}\sigma_{k\rightsquigarrow\Delta_{2}}^{pq}f_{pq}.$

So $v_{i}$ is the weighted sum (with respect to the sign convention as in
Figure 3) of the edges on the path from $i$ to the vertex $\Delta_{1}$
excluding $f_{a_{1}c_{1}}$, etc. Further, let $v:=\sum_{\overline{pq}\in
T}\sigma_{\Delta_{1}\rightsquigarrow\Delta_{2}}^{pq}f_{pq}$. Then

$\displaystyle{\bf g}_{ij}=v_{i}+\sigma_{i\rightsquigarrow
j}^{a_{1}c_{1}}f_{a_{1}c_{1}}+\sigma_{i\rightsquigarrow
j}^{a_{1}b_{1}}f_{a_{1}b_{1}}+v+\sigma_{i\rightsquigarrow
j}^{a_{2}c_{2}}f_{a_{2}c_{2}}+\sigma_{i\rightsquigarrow
j}^{a_{2}b_{2}}f_{ab}+v_{j},$ $\displaystyle{\bf
g}_{ik}=v_{i}+\sigma_{i\rightsquigarrow
j}^{a_{1}c_{1}}f_{a_{1}c_{1}}+\sigma_{i\rightsquigarrow
j}^{a_{1}b_{1}}f_{a_{1}b_{1}}+v+\big{(}1+\sigma_{i\rightsquigarrow
j}^{a_{2}c_{2}}\big{)}f_{a_{2}c_{2}}+\sigma_{i\rightsquigarrow
k}^{b_{2}c_{2}}f_{b_{2}c_{2}}+v_{k},$ $\displaystyle{\bf
g}_{jl}=v_{l}+\sigma_{j\rightsquigarrow
l}^{b_{1}c_{1}}f_{b_{1}c_{1}}-\big{(}1-\sigma_{i\rightsquigarrow
j}^{a_{1}b_{1}}\big{)}f_{a_{1}b_{1}}+v+\sigma_{i\rightsquigarrow
j}^{a_{2}c_{2}}f_{a_{2}c_{2}}+\sigma_{i\rightsquigarrow
j}^{a_{2}b_{2}}f_{a_{2}b_{2}}+v_{j},$ $\displaystyle{\bf
g}_{kl}=v_{l}+\sigma_{j\rightsquigarrow
l}^{b_{1}c_{1}}f_{b_{1}c_{1}}-\big{(}1-\sigma_{i\rightsquigarrow
j}^{a_{1}b_{1}}\big{)}f_{a_{1}b_{1}}+v+\big{(}1+\sigma_{i\rightsquigarrow
j}^{a_{2}c_{2}}\big{)}f_{a_{2}c_{2}}+\sigma_{i\rightsquigarrow
k}^{b_{2}c_{2}}f_{b_{2}c_{2}}+v_{k},$ $\displaystyle{\bf
g}_{il}=v_{i}+\big{(}1+\sigma_{i\rightsquigarrow
j}^{a_{1}c_{1}}\big{)}f_{a_{1}c_{1}}-\big{(}1-\sigma_{j\rightsquigarrow
l}^{b_{1}c_{1}}\big{)}f_{b_{1}c_{1}}+v_{l},$ $\displaystyle{\bf
g}_{jk}=v_{j}-\big{(}1+\sigma_{i\rightsquigarrow
j}^{a_{2}b_{2}}\big{)}f_{a_{2}b_{2}}+\big{(}1+\sigma_{i\rightsquigarrow
k}^{b_{2}c_{2}}\big{)}f_{b_{2}c_{2}}+v_{k}.$

In particular, we have ${\bf g}_{ij}+{\bf g}_{kl}={\bf g}_{ik}+{\bf
g}_{jl}\not={\bf g}_{il}+{\bf g}_{jk}$. ∎

Recall from (4.1) how to obtain a weight vector from a trivalent tree with $n$
leaves in $\text{Trop}(I_{2,n})$. The following corollary is a direct
consequence of the proof of lem:g-v on relations.

######  Corollary 4.32.

cor:trop and g Without loss of generality by lem:g-v on relations assume that
for $i<j<k<l$, ${\bf g}_{ij}+{\bf g}_{kl}={\bf g}_{ik}+{\bf g}_{jl}\not={\bf
g}_{il}+{\bf g}_{jk}$. Then
$\operatorname{in}_{D_{T}}(p_{ij}p_{kl}-p_{ik}p_{jl}+p_{il}p_{jk})=p_{ij}p_{kl}-p_{ik}p_{jl}$.

######  Definition 4.33.

def:wt map A quadrilateral weight map for a triangulation $T$ is a linear map
$w_{T}:\mathbb{R}^{2n-3}$ $\to\mathbb{R}$, such that for every quadrilateral
$\overline{abcd}$ with diagonal $\overline{ac}$ in $T$ we have:

$w_{T}(f_{ad}+f_{bc})>w_{T}(f_{ab}+f_{cd}).$ (4.4)

Note that (4.4) is equivalent to saying that for every mutable vertex
$\overline{ac}$ of the quiver $Q_{T}$ we require
$w_{T}\big{(}\sum_{\overline{pq}\to\overline{ac}}{\bf
g}_{pq}\big{)}>w_{T}\big{(}\sum_{\overline{ac}\to\overline{pq}}{\bf
g}_{pq}\big{)}$. Before we show that quadrilateral weight maps do in fact
exist (in lem:ex quad, using Algorithm 1) we proceed by showing how they can
be used to construct weight vectors in $\operatorname{Trop}(I_{2,n})$ from
${\bf g}$-vectors.

######  Proposition 4.34.

prop:wt map Consider any $i<j<k<l$ with ${\bf g}_{ij}+{\bf g}_{kl}={\bf
g}_{ik}+{\bf g}_{jl}\not={\bf g}_{il}+{\bf g}_{jk}$ and any quadrilateral
linear map $w_{T}\colon\mathbb{R}^{2n-3}\to\mathbb{R}$. Then $w_{T}({\bf
g}_{ij}+{\bf g}_{kl})=w_{T}({\bf g}_{ik}+{\bf g}_{jl})<w_{T}({\bf g}_{il}+{\bf
g}_{jk})$.

###### Proof.

Without loss of generality assume that the full subtree of $\widehat{D}_{T}$
with leaves $i$, $j$, $k$, $l$ is of form as in Figure 5. For $s=1,2$ let
$\Delta_{s}$ be the vertex of $\widehat{D}_{T}$ corresponding to the triangle
$a_{s}$, $b_{s}$, $c_{s}$ in $T$. We proceed by induction on the number of
edges along the unique path from $\Delta_{2}$ to $\Delta_{1}$, denote it by
$p$.

$p=1$: In this case we have $a_{1}=d=a_{2}$ and $b_{1}={c_{2}}$ in Figure 5.
Note that every quadrilateral linear map $w_{T}$ satisfies
$w_{T}({f}_{a_{2}c_{1}}+{f}_{b_{2}c_{2}})>w_{T}({f}_{a_{2}b_{2}}+{f}_{c_{2}c_{1}})$.
Using lem:g-v on relations we compute

$w_{T}({\bf g}_{il}+{\bf g}_{jk})=w_{T}({\bf g}_{ij}+{\bf
g}_{kl}+f_{a_{2}c_{1}}+f_{b_{2}c_{2}}-f_{a_{2}b_{2}}-f_{c_{2}c_{1}})>w_{T}({\bf
g}_{ij}+{\bf g}_{kl}).$

$p\geq 1$: Assume that the result holds for $p$. When the number of edges
along the path from $v_{a_{2}b_{2}c_{2}}$ to $v_{a_{1}b_{1}c_{1}}$ is $p+1$ we
know there exists at least one leaf of form either $i^{\prime}$ with
$i<i^{\prime}<j$ or $k^{\prime}$ with $k<k^{\prime}<l$. We treat the first
case, where $i^{\prime}$ exists depicted in Figure 5. The proof of the second
case is similar. All expressions of ${\bf g}$-vectors for Plücker variables in
the relation $R_{ijkl}$ appear in the proof of lem:g-v on relations. The ${\bf
g}$-vectors involving $i^{\prime}$ can be computed similarly. By induction we
know that for the relation $R_{ii^{\prime}jl}$ we have $w_{T}({\bf
g}_{ij}+{\bf g}_{i^{\prime}l})<w_{T}({\bf g}_{il}+{\bf g}_{i^{\prime}j})$.
Further, by assumption on $w_{T}$ we have
$w_{T}({f}_{a_{2}d}+{f}_{b_{2}c_{2}})>w_{T}({f}_{a_{2}b_{2}}+{f}_{c_{2}d})$.
One verifies by direct computation that

$\displaystyle w_{T}({\bf g}_{il}+{\bf g}_{jk})$ $\displaystyle=w_{T}({\bf
g}_{il}+{\bf g}_{i^{\prime}j})+w_{T}({\bf g}_{jk}-{\bf g}_{i^{\prime}j})$
$\displaystyle>w_{T}({\bf g}_{ij}+{\bf g}_{kl})+w_{T}({\bf g}_{jk}-{\bf
g}_{i^{\prime}j}+{\bf g}_{i^{\prime}l}-{\bf g}_{kl})$
$\displaystyle=w_{T}({\bf g}_{ij}+{\bf
g}_{kl})+w_{T}({f}_{a_{2}d}+{f}_{b_{2}c_{2}}-{f}_{a_{2}b_{2}}-{f}_{c_{2}d})>w_{T}({\bf
g}_{ij}+{\bf g}_{kl}).\\!\\!\\!\\!\\!\\!$ ∎

######  Definition 4.35.

def:wt of T Fix a quadrilateral linear map
$w_{T}\colon\mathbb{R}^{2n-3}\to\mathbb{R}$. We define ${\bf
w}_{T}\in\mathbb{R}^{\binom{n}{2}}$ the weight vector associated to $T$ and
$w_{T}$ as $({\bf w}_{T})_{ij}:=w_{T}({\bf g}_{ij})$.

######  Proposition 4.36.

thm:toric wt ideals For each quadrilateral weight map
$w_{T}\colon\mathbb{R}^{2n-3}\to\mathbb{R}$, the initial ideal
$\operatorname{in}_{{\bf w}_{T}}(I_{2,n})$ is toric. Moreover, every binomial
in the minimal generating set of $\operatorname{in}_{{\bf w}_{T}}(I_{2,n})$
corresponds to a quadrilateral and it contains the monomial associated to the
crossing in that quadrilateral.

###### Proof.

By prop:wt map and Corollary LABEL:cor:trop_and_g we have
$\operatorname{in}_{{D_{T}}}(R_{ijkl})=\operatorname{in}_{{\bf
w}_{T}}(R_{ijkl})$ for every Plücker relation $R_{ijkl}$. By lem:g-v on
relations these initial forms are binomials and one monomial corresponds to
the crossing in the quadrilateral $\overline{ijkl}$. By [49, Proof of Theorem
3.4] we know that $\operatorname{in}_{{D_{T}}}(I_{2,n})$ is generated by
$\operatorname{in}_{{D_{T}}}(R_{ijkl})$ for all $1\leq i<j<k<l\leq n$.
Moreover, by [49, Corollary 4.4] the ideal
$\operatorname{in}_{{D_{T}}}(I_{2,n})$ is binomial and prime. ∎

######  Remark 4.37.

rmk:toric degen As shown in thm:toric wt ideals, each triangulation of the
$n$-gon gives rise to a Gröbner toric degeneration of
$\operatorname{Gr}(2,\mathbb{C}^{n})$. For general Grassmannians, there are
other examples of combinatorial objects leading to toric degenerations such as
plabic graphs [7, 46] and matching fields [12, 13, 39]. All such degenerations
can be realized as Gröbner degenerations, nevertheless, this is not true in
general; see, e.g., [32] for a family of toric degenerations arising from
graphs that cannot be identified as Gröbner degenerations.

Existence of quadrilateral weight maps: For every triangulation $T$ we give a
weight map obtained from a partition of the cluster ${\bf x}_{T}$, which is
the output of Algorithm 1. We use it to define a linear map
$v_{T}\colon\mathbb{R}^{2n-3}\to\mathbb{R}$ and show in lem:ex quad that it is
quadrilateral for $T$.

Input: An initial cluster ${\bf x}_{T}$ with its corresponding triangulation
$T$ and its quiver $Q_{T}$.

Output: A partitioning of the initial seed indices.

Initialization: $i\leftarrow 0$;

$T\leftarrow$ the corresponding triangulation of ${\bf x}_{T}$;

$Q\leftarrow Q_{T}$;

while  _$T$ is not empty_ do

$V_{i}:=\\{v\colon v\text{ is a frozen vertex in }Q\text{ and a sink}\\}$;

foreach  _triangle $t$ in $T$_ do

if  _$t$ has an edge whose corresponding vertex in $Q$ is in $V_{i}$_ then

$T\leftarrow T\setminus\\{t\\}$ (Remove the triangle $t$ from $T$);

$Q\leftarrow Q_{T}$;

end if

end foreach

$i\leftarrow i+1$;

end while

$V_{i}\leftarrow{\bf x}_{T}\setminus(V_{1}\cup\cdots\cup V_{i-1})$;

return _$(V_{0},\ldots,V_{i})$_ ;

Algorithm 1 Partitioning the cluster ${\bf x}_{T}$ based on its triangulation
$T$ (union of triangles).

Description of Algorithm 1. The input of the algorithm is a triangulation $T$
of the $n$-gon. The output is an ordered partition of ${\bf x}_{T}$. We
repeatedly shrink $T$ to obtain a quiver associated to smaller triangulations.
More precisely, in step $i$ we identify the frozen vertices of $Q_{T}$ that
are sinks, i.e., they have no outgoing arrows. Then we add the corresponding
arcs to the set $V_{i}$ and remove the corresponding triangles from $T$. Note
that in this process, we might remove edges whose corresponding vertices are
not in $V_{i}$, but they are part of a triangle with an edge in $V_{i}$.

Let $V_{0},\dots,V_{i}$ be the partition of the initial seed indices obtained
by applying Algorithm 1 to the triangulation $T$ associated to the initial
seed ${\bf x}_{T}$. Refining the output of the algorithm yields an ordering on
the variables of our initial seed, or equivalently the arcs of our initial
triangulation.

######  Example 4.38.

We continue with our running example and compute a partition of the initial
seed indices as $V_{0}=\\{56,12,78\\}$, $V_{1}=\\{13,45\\}$,
$V_{2}=\\{17,14,47\\}$ and $V_{3}=\\{57,34,23,67,18\\}$.

######  Lemma 4.39.

lem:ex quad Given a triangulation $T$ consider the partition of ${\bf
x}_{T}=V_{0}\cup\dots\cup V_{s}$ obtained from Algorithm 1. Let
$v_{T}\colon\mathbb{R}^{2n-3}\to\mathbb{R}$ be the linear map defined by
sending basis elements ${f}_{{ij}}$ with $\overline{ij}\in T$ to $q$, where
$\overline{ij}\in V_{q}$. Then $v_{T}$ is a quadrilateral linear map for $T$.

###### Proof.

Consider a quadrilateral $i<j<k<l$ in $T$ with $\overline{ij}$,
$\overline{jk}$, $\overline{kl}$, $\overline{li}$ and $\overline{ik}\in T$. As
the algorithm ends when there is only one triangle or one arc left, there are
two possibilities: either one of the triangles $\overline{ijk}$,
$\overline{kli}$ is cut off first, or both triangles are cut off at the same
time. In the first case we may assume the triangle $\overline{ijk}$ is removed
first. Then $\overline{ij}\in V_{p}$, $\overline{lk}\in V_{q}$ and
$\overline{jk},\overline{il}\in V_{s}$ for some $p\leq q\leq s$ with $p<s$. In
the first case $p$ and $q$ are different, while in the second they are equal.
So in either case, we have
$v_{T}({f}_{ij}{+f}_{kl})=p+q<s+s=v_{T}({f}_{jk}+{f}_{il})$. ∎

###  4.3 A distinguished maximal Gröbner cone

In order to apply the construction from Section 3 in this section we identify
a particular maximal cone $C$ in $\operatorname{GF}(I_{2,n})$. We analyze the
cone $C$ and apply thm:family. In the following section we will show how this
construction is related to adding universal coefficients to the cluster
algebra $A_{2,n}$.

######  Definition 4.40.

def:mono wt Let $u\in\mathbb{Q}^{\binom{n}{2}}$ be the weight vector such that
the weight of $p_{ij}$ is:

$u(p_{ij}):=-\left(j-i-\frac{n}{2}\right)^{2}\qquad\text{for}\quad 1\leq
i<j\leq n.$

######  Example 4.41.

exp:mono wt For $n=8$, the Plücker variables associated to boundary arcs
receive the lowest weight, which is $-9$. Mutable Plücker variables have the
following weights:

$u(p_{ij})=\begin{cases}0,&\text{if}\quad\min\\{|j-i|,|i-j|\\}=4,\\\
-1,&\text{if}\quad\min\\{|j-i|,|i-j|\\}=3,\\\
-4,&\text{if}\quad\min\\{|j-i|,|i-j|\\}=2.\end{cases}$

######  Lemma 4.42.

lem: mono ideal With $u$ as in def:mono wt we have
$\operatorname{in}_{u}(I_{2,n})=\mathcal{M}_{2,n}$ from def:monomial ideal.

###### Proof.

Consider the quadrilateral with vertices $i<j<k<l$ and its corresponding
Plücker relation $R_{ijkl}=p_{ik}p_{jl}-p_{ij}p_{kl}-p_{il}p_{jk}$. The arcs
$\overline{ik}$ and $\overline{jl}$ are crossing, hence we have to show that
($a$) $\operatorname{in}_{u}(R_{ijkl})=p_{ik}p_{jl}$ and ($b$) the crossing
monomials generate $\operatorname{in}_{u}(I_{2,n})$. To prove ($a$) we verify
that

$\displaystyle u(p_{ik}p_{jl})=u(p_{il}p_{jk})+2(j-i)(l-k)>u(p_{il}p_{jk}),$
$\displaystyle u(p_{ik}p_{jl})=u(p_{ij}p_{kl})+2(k-j)(n+i-l)>u(p_{ij}p_{kl}).$

To establish ($b$), we apply Buchberger’s criterion and show that all the
$S$-pairs $S(R_{ijkl},R_{abcd})$ reduce to $0$ modulo $\\{R_{ijkl}\\}_{1\leq
i<j<k<l\leq n}$. If $\operatorname{in}_{u}(R_{ijkl})$ and
$\operatorname{in}_{u}(R_{abcd})$ have no common factor, then
$S(R_{ijkl},R_{abcd})$ reduces to zero (see, for example, [29, Lemma 2.3.1]).
Assuming that $\operatorname{in}_{u}(R_{ijkl})$ and
$\operatorname{in}_{u}(R_{abcd})$ have a common factor we distinguish two
cases, in each we underline the crossing monomials. Consider $R_{ijls}$ and
$R_{iklr}$, where without loss of generality $i<j<k<l<r<s$:

$\displaystyle\begin{split}&S(R_{ijls},R_{iklr})=\underline{p_{kr}p_{ls}}p_{ij}+\underline{p_{kr}p_{jl}}p_{is}-\underline{p_{js}p_{ik}}p_{lr}-\underline{p_{js}p_{ir}}p_{kl}\\\
&\hphantom{S(R_{ijls},R_{iklr})}{}=-p_{ij}R_{klrs}-p_{is}R_{jklr}+p_{lr}R_{ijks}+p_{kl}R_{ijrs}.\end{split}$

The second case for $R_{ijkl}$ and $R_{jklm}$ with $i<j<k<l<m$ is similar. ∎

Let $C\in\operatorname{GF}(I_{2,n})$ be the maximal cone corresponding to the
monomial initial ideal $\mathcal{M}_{2,n}$ and let $<$ denote the compatible
monomial term order on $S$. By definition the standard monomial basis
$\mathbb{B}_{<}$ (see def:SMT) is the set of monomials that are not contained
in the ideal generated by _crossing monomials_. So we call it the basis of
non-crossing monomials.

######  Proposition 4.43.