# Multi-Rees Algebras of Strongly Stable Ideals

Selvi Kara  Department of Mathematics and Statistics, University of South
Alabama, 411 University Boulevard North, Mobile, AL 36688, USA
<EMAIL_ADDRESS>,  Kuei-Nuan Lin  Department of Mathematics, Penn State
University, Greater Allegheny campus
4000 University Dr, McKeesport, PA 15132, USA<EMAIL_ADDRESS>and  Gabriel Sosa
Department of Mathematics, Colgate University, 3 Oak Drive Hamilton, NY 13346,
USA<EMAIL_ADDRESS>
###### Abstract.

We prove that the multi-Rees algebra $\mathcal{R}(I_{1}\oplus\cdots\oplus
I_{r})$ of a collection of strongly stable ideals $I_{1},\ldots,I_{r}$ is of
fiber type. In particular, we provide a Gröbner basis for its defining ideal
as a union of a Gröbner basis for its special fiber and binomial syzygies. We
also study the Koszulness of $\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$
based on parameters associated to the collection. Furthermore, we establish a
quadratic Gröbner basis of the defining ideal of $\mathcal{R}(I_{1}\oplus
I_{2})$ where each of the strongly stable ideals has two quadric Borel
generators. As a consequence, we conclude that this multi-Rees algebra is
Koszul.

###### Key words and phrases:

Rees algebra, special fiber ring, toric ring, Koszul algebra, Gröbner basis,
strongly stable ideals, fiber graph

2020 Mathematics Subject Classification. Primary 13A30, 13P10; Secondary
05E40, 13C05, 13F55, 05C20

## Introduction

Rees algebras are central and important objects in commutative algebra and
algebraic geometry (see [23]) with numerous appearances in various other
fields including elimination theory, geometric modeling and chemical reaction
networks (see [5], [8] and [9]). From a geometrical point of view, the Rees
algebra $\mathcal{R}(I)$ of an ideal $I$ in a polynomial ring
$S=\mathbb{K}[x_{1},\ldots,x_{n}]$ is the homogeneous coordinate ring of two
fundamental objects: the blowup of a projective space along the subscheme
defined by $I$ and the graph of a rational map between projective spaces
defined by the generators of $I$. Naturally, the multi-Rees algebra of a
collection of ideals $I_{1},\ldots,I_{r}$ of $S$ is the homogeneous coordinate
ring of the blowup along the subschemes defined by the ideals. The multi-Rees
algebra of $I_{1},\ldots,I_{r}$ is also the Rees algebra of the module
$I_{1}\oplus\cdots\oplus I_{r}$ and it is defined as the multigraded
$S$-algebra:

$\mathcal{R}(I_{1}\oplus\cdots\oplus
I_{r})=\bigoplus_{(a_{1},\ldots,a_{r})\in\mathbb{Z}_{\geq
0}^{n}}I_{1}^{a_{1}}\cdots I_{r}^{a_{r}}t_{1}^{a_{1}}\cdots
t_{r}^{a_{r}}\subseteq S[t_{1},\ldots,t_{r}]$

for auxiliary variables $t_{1},\ldots,t_{r}$. A related object is the special
fiber ring which is associated to the image of the blowup map. We denote the
special fiber ring by $\mathcal{F}(I_{1}\oplus\cdots\oplus I_{r})$ and it is
equal to $\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})\otimes_{S}\mathbb{K}$.

A fundamental problem in the study of Rees and multi-Rees algebras is finding
the implicit equations of the defining ideals $\mathcal{L}$ and $\mathcal{K}$
of the multi-Rees algebra and its special fiber, respectively, such that
$\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})\cong
S[T_{1},\ldots,T_{N}]/\mathcal{L}$ and $\mathcal{F}(I_{1}\oplus\cdots\oplus
I_{r})\cong\mathbb{K}[T_{1},\ldots,T_{N}]/\mathcal{K}$ where $N$ is the total
number of minimal generators of $I_{1},\ldots,I_{r}$. This challenging problem
has been studied extensively for Rees algebras ($r=1$), however, it is still
open for many classes of ideals. In recent years, there has been some progress
toward addressing this problem for the multi-Rees algebras [12, 18, 19, 21]
albeit the increased complexity of this setting. In the light of all the
established cases, a complete solution can be provided if the ideals in
question have specific structures.

In this paper, we study the multi-Rees algebras of strongly stable ideals
(also called Borel ideals in literature). These ideals are of special
importance in computational commutative algebra due to their nice
combinatorial structure; in characteristic zero, these ideals coincide with
Borel-fixed ideals and they occur as generic initial ideals ([1], [15]).

In addition to finding defining equations of the multi-Rees algebras of
strongly stable ideals and their special fiber rings, we are interested in
detecting when these algebras are Koszul. An arbitrary graded ring $R$ over a
field $R_{0}=\mathbb{K}$ is Koszul if the residue field
$R/R_{+}\cong\mathbb{K}$ has a linear resolution over $R$. A common approach
to prove an algebra $R$ is Koszul is to show that it is $G$-quadratic, that
is, $R\cong S/J$ is the quotient of a polynomial ring $S$ by an ideal
$J\subset S$ such that $J$ has a Gröbner basis of quadratics with respect to
some monomial order (see [13, Theorem 6.7], [14]). Many of the classically
studied rings in commutative algebra and algebraic geometry are Koszul and
these algebras have good homological features; as the authors of [7] put “a
homological life is worth living in a Koszul algebra.” It is well-known that
Koszulness of $\mathcal{R}(I)$ implies that $I$ has linear powers, i.e., every
power of $I$ has a linear resolution [3, Corollary 3.6]. The multigraded
version of this result is proved by Bruns and Conca in [4, Theorem 3.4]; when
$\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$ is Koszul, then products of
$I_{1},\ldots,I_{r}$ have linear resolutions.

Both problems of interest, implicit equations for the defining ideals and
Koszulness, can be answered simultaneously by identifying an explicit
quadratic Gröbner basis for the defining ideal of the multi-Rees algebra or
its special fiber ring. However, it is difficult to find such Gröbner basis in
general. One of the earlier works on the multi-Rees algebras of strongly
stable ideals addressing both problems is due to Lin and Polini [19] in which
they consider powers of maximal ideals. In particular, they provide quadratic
Gröbner bases of $\mathcal{L}$ and $\mathcal{K}$ in [19, Theorem 2.4] and
further show that the associated multi-Rees algebra and its special fiber ring
are both Koszul domain. Their work is extended to a more general class of
principal strongly stable ideals by Sosa in [21, Theorem 2.2].

A closer look into the literature on the Rees algebras of strongly stable
ideals suggest a systematic approach for the multi-Rees setting. Namely, one
can study the multi-Rees algebras of strongly stable ideals
$I_{1},\ldots,I_{r}$ by considering three parameters: the number of ideals
$r$, the number of _Borel generators_ of each ideal, and the degrees of Borel
generators. Note that when the first parameter $r$ is equal to one, the
objects are the Rees algebras of strongly stable ideals and the Koszulness of
these objects are studied by controlling the remaining two parameters. Authors
of [4], [10], and [11] investigate the Koszulness of the Rees algebras of
strongly stable ideals (see Section 4 for details).

In the investigation of quadratic Gröbner bases of $\mathcal{L}$ and
$\mathcal{K}$ in the multi-Rees setting, it is reasonable to start by
considering classes of ideals whose Rees algebras or special fiber rings are
$G$-quadratic. Emergence of this natural approach can be observed in [4]. In
one such instance, based on De Negri’s result on the Koszulness of
$\mathcal{F}(I)$ for a principal strongly stable ideal $I$ from [10], Bruns
and Conca suggest that it is “very likely” that
$\mathcal{F}(I_{1}\oplus\cdots\oplus I_{r})$ is defined by a Gröbner basis of
quadratics when each $I_{i}$ is principal strongly stable (see [4, Page 3]).
Very recently, DiPasquale and Nezhad confirm an extended version of Bruns and
Conca’s _suggestion_ for $\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$ when
each $I_{i}$ is principal strongly stable [12, Corollary 6.4].

In the study of Rees algebras of strongly stable ideals, it is possible to
deduce the Koszulness of the Rees algebra from its special fiber thanks to a
result of Herzog, Hibi and Vladoiu [17, Theorem 5.1] which states that
_$\mathcal{R}(I)$ is of fiber type for a strongly stable ideal $I$_. The
multi-Rees version of Herzog, Hibi and Vladoiu’s result would be quite useful
in the multi-Rees world. We generalize their result to the multi-Rees algebra
case and show that the multi-Rees algebra of strongly stable ideals is of
fiber type, that is, the ideal $\mathcal{L}\subseteq S[T_{1},\ldots,T_{N}]$ is
generated by linear relations in $T_{1},\ldots,T_{N}$ variables and the
generators of $\mathcal{K}$. In particular, we prove that a Gröbner basis of
$\mathcal{L}$ is the union of a Gröbner basis of $\mathcal{K}$ and a set of
quadratic binomials associated to first syzygies of $I_{1}\oplus\cdots\oplus
I_{r}$ (see Theorem 3.2). It is worth pointing that we allow ideals
$I_{1},\ldots,I_{r}$ to be generated in different degrees while each Borel
generator of $I_{i}$ is of the same degree for $i\in\\{1,\ldots,r\\}$. An
immediate application of Theorem 3.2 is that the multi-Rees algebra of
strongly stable ideals is $G$-quadratic whenever its special fiber ring is.
Therefore, Koszulness of the multi-Rees algebra of strongly stable ideals can
be obtained from that of its special fiber ring.

Following the footsteps of Conca and De Negri, we provide a collection of
examples of strongly stable ideals whose multi-Rees algebras are not Koszul
(see Section 4). Hence, if $\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$ is
always Koszul for a fixed arrangement of the three sets of parameters
introduced earlier (i.e. number of ideals, number of Borel generators of each
ideal and degree of each Borel generator), we identify all possible values of
the parameters in the arrangement (see Proposition 4.4).

In [11, Question 6.4], the authors conclude their paper with a question asking
whether the multi-Rees algebra of strongly stable ideals $I_{1},\ldots,I_{r}$
is Koszul where each $I_{i}$ has either one or two Borel generators. In this
case, by Proposition 4.4, at most two of the ideals have two Borel generators.
Furthermore, if exactly two of the ideals have two Borel generators then these
generators must be quadric or cubic. In the second half of the paper, we focus
on the multi-Rees algebra of strongly stable ideals $I_{1}$ and $I_{2}$ with
two quadric Borel generators. In particular, we prove in Section 6 that
$\mathcal{F}(I_{1}\oplus I_{2})$ has a quadratic Gröbner basis. We achieve it
by obtaining an explicit quadratic Gröbner basis for the toric ideal of the
special fiber ring with respect to _head and tail order_ (see Theorem 6.9).
Hence Koszulness of $\mathcal{R}(I_{1}\oplus I_{2})$ follows from Theorem 3.2.

Our paper is structured as follows. In Section 1, we collect the necessary
terminology to be used throughout the paper. In Section 2, we introduce
_directed graph of a monomial_ and use these objects to determine when a given
collection of binomials (with a marked term) form a Gröbner basis (see Theorem
2.5). We prove our first main result, Theorem 3.2, in Section 3. In Section 4,
we investigate Koszulness of multi-Rees algebras of strongly stable ideals
through examples and identify large classes of ideals whose multi-Rees
algebras are not necessarily Koszul, and present our second main result,
Proposition 4.4. In Sections 5 and 6, we study multi-Rees algebras of strongly
stable ideals with two quadratic Borel generators. In particular, we prove our
third main result, Theorem 6.9, and its corollary, Corollary 6.18, in Section
6.

##  1\. Preliminaries

Let $S={\mathbb{K}}[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$
variables over a field ${\mathbb{K}}$ and suppose $S$ is equipped with the
standard grading as a ${\mathbb{K}}$-algebra.

######  Definition 1.1.

  1. (a)

Let $\succ_{rlex}$ denote the reverse lexicographic order with the variable
order $x_{1}>x_{2}>\cdots>x_{n}$ on $S$. We write $[n]=\\{1,\ldots,n\\}$.

  2. (b)

A monomial ideal $I\subseteq S$ is called _strongly stable_ if for each
monomial $m\in I$, we have $x_{j}\frac{m}{x_{i}}\in I$ whenever $x_{i}$
divides $m$ and $j<i$.

  3. (c)

Fix a monomial $m\in S$, and suppose that $x_{i}$ divides $m.$ Then for any
$j<i$, the monomial $m^{\prime}=x_{j}\frac{m}{x_{i}}$ is called a _one step
strongly stable reduction_ of $m$.

  4. (d)

Fix a degree $d.$ Then one can define a partial order $<$, called the
_strongly stable order_ , on the degree $d$ monomials of $S$ by setting
$m^{\prime}<m$ whenever $m^{\prime}$ can be obtained from $m$ by a sequence of
one step strongly stable reductions. In this case, we say $m^{\prime}$ comes
before $m$ in the strongly stable order.

  5. (e)

Let $\mathcal{M}=\\{m_{1},\ldots,m_{k}\\}$ be a collection of degree $d$
monomials. Then the smallest strongly stable ideal containing $\mathcal{M}$ is
called the _strongly stable ideal generated by_ $\mathcal{M}$ and denoted by
${\mathcal{B}}(\mathcal{M})$. If $I={\mathcal{B}}(\mathcal{M})$, the set
$\mathcal{M}$ is called a strongly stable generating set for $I$. The strongly
stable ideal $I$ has a unique minimal strongly stable generating set
corresponding to the latest monomial generators in the strongly stable order
and these monomials are called its _strongly stable generators_ or _Borel
generators_. For the remainder of the paper, we shall use the term Borel
generators. If $I={\mathcal{B}}(m)$ has only one Borel generator, we say $I$
is a prinpical strongly stable ideal.

######  Remark 1.2.

Strongly stable ideals are _Borel-fixed_ (i.e., fixed under the action of the
Borel subgroup). In characteristic zero, the notion of strongly stable
coincide with Borel-fixed.

In this paper, our main objects are the multi-Rees algebras of strongly stable
ideals. We recall the definition of a multi-Rees algebra and its special fiber
ring below.

######  Definition 1.3.

Let $I_{1},\dots,I_{r}$ be a collection of ideals in $S$. The _multi-Rees
algebra of_ $I_{1},\dots,I_{r}$ is the multi-graded $S$-algebra

$\mathcal{R}(I_{1}\oplus\dots\oplus
I_{r})=\bigoplus_{(a_{1},\ldots,a_{r})\in\mathbb{Z}_{\geq
0}^{n}}I_{1}^{a_{1}}\cdots I_{r}^{a_{r}}t_{1}^{a_{1}}\cdots
t_{r}^{a_{r}}\subseteq S[t_{1},\ldots,t_{r}]$

where $t_{1},\ldots,t_{r}$ are indeterminates over $S$. In particular, in the
case $r=1$, it is the classical Rees algebra of an ideal $I$. The multi-Rees
algebra $\mathcal{R}(I_{1}\oplus\dots\oplus I_{r})$ is also the Rees algebra
of the module $I_{1}\oplus\cdots\oplus I_{r}$. The _special fiber ring of_
$I_{1},\dots,I_{r}$ is defined as

$\mathcal{F}(I_{1}\oplus\dots\oplus I_{r})=\mathcal{R}(I_{1}\oplus\dots\oplus
I_{r})\otimes_{S}{\mathbb{K}}$

where we regard ${\mathbb{K}}$ as an $S$-algebra via ${\mathbb{K}}\cong
S/\langle x_{1},\dots,x_{n}\rangle$.

Let $I_{i}$ be a monomial ideal and $\\{u_{i,j}:1\leq j\leq s_{i}\\}$ be the
minimal generating set of $I_{i}$ for each $i\in[r]$. We can construct the
presentation of $\mathcal{R}(I_{1}\oplus\dots\oplus I_{r})$ as follows.
Consider the $S$-algebra homomorphism given by

$\displaystyle\varphi:R=S[T_{i,j}~{}:~{}1\leq i\leq r,1\leq j\leq s_{i}]$
$\displaystyle\longrightarrow S[t_{1},\dots,t_{r}]$ $\displaystyle T_{i,j}$
$\displaystyle\longmapsto u_{i,j}t_{i}$

and extended algebraically. The ideal $\mathcal{I}=\ker(\varphi)$ is called
the _defining ideal_ of the multi-Rees algebra and the minimal generators of
$\mathcal{I}$ are called the _defining equations_ of
$\mathcal{R}(I_{1}\oplus\dots\oplus I_{r})$. The multi-Rees algebra of
$I_{1},\ldots,I_{r}$ is the quotient

$\mathcal{R}(I_{1}\oplus\dots\oplus I_{r})=S[T_{i,j}~{}:~{}1\leq i\leq r,1\leq
j\leq s_{i}]/\ker(\varphi)\cong\operatorname{Im}\varphi.$

The map $\varphi$ induces a surjective ${\mathbb{K}}$-algebra homomorphism
$\varphi^{\prime}$ such that

$\displaystyle\varphi^{\prime}:R^{\prime}={\mathbb{K}}[T_{i,j}~{}:~{}1\leq
i\leq r,1\leq j\leq s_{i}]$
$\displaystyle\longrightarrow{\mathbb{K}}[u_{i,j}t_{i}~{}:~{}1\leq i\leq
r,1\leq j\leq s_{i}]$ $\displaystyle T_{i,j}$ $\displaystyle\longmapsto
u_{i,j}t_{i}.$

The special fiber ring, $\mathcal{F}(I_{1}\oplus\dots\oplus I_{r})$, is
isomorphic to the image of $\varphi^{\prime}$ and the ideal
$\ker(\varphi^{\prime})$ is called the _defining ideal_ of the special fiber
ring. Note that $\mathcal{F}(I_{1}\oplus\dots\oplus I_{r})$ is a toric ring
because $\ker(\varphi^{\prime})$ is a toric ideal associated to the monomial
map $\varphi^{\prime}$. We often use the notation $T(I_{1}\oplus\dots\oplus
I_{r})$ to denote $\ker(\varphi^{\prime})$.

In what follows, we provide a quick overview of marked polynomials and
Noetherian reductions in the context of Gröbner bases theory.

######  Definition 1.4.

A _marked polynomial_ is a polynomial $g\in S$ together with a “marked”
initial term $\text{in}(g)$, say $g=c\underline{u}+h$ where $u$ is a monomial
with $c\in{\mathbb{K}}^{*}$ and $h$ is a polynomial such that
$u\notin\operatorname{supp}(h)$ and $\text{in}(g)=u$. Note that any of the
terms appearing in $g$ can be chosen as $\text{in}(g)$.

Let $\mathcal{G}=\\{g_{i}=c_{i}\underline{u_{i}}+h_{i}~{}:~{}1\leq i\leq s\\}$
be a collection of marked polynomials. In the following two examples, we
investigate the existence of a monomial order $<_{\tau}$ such that
$\mathcal{G}$ is marked coherently, namely, one has
$\operatorname{in}_{<_{\tau}}(g_{i})=\underline{u_{i}}$ for each
$g_{i}\in\mathcal{G}$.

######  Example 1.5.

Consider the polynomials $f=\underline{xy}-yz$ and $g=\underline{z^{2}}+x^{2}$
with the marked terms as the leading ones. If there exists a monomial order
$<_{\tau}$ such that $\operatorname{in}_{<_{\tau}}(f)=\underline{xy}$ and
$\operatorname{in}_{<_{\tau}}(g)=\underline{z^{2}}$, then $yz<_{\tau}xy$
implies $z<_{\tau}x$ and $x^{2}<_{\tau}z^{2}$ implies that $x<_{\tau}z$ which
is not possible. Hence, there is no such monomial order. In fact, for the
marked initial terms, there is no monomial order that makes $\\{f,g\\}$ a
Gröbner basis for $\langle f,g\rangle$ unless $\text{char}(k)\neq 2$.

######  Example 1.6.

Consider the ideal $I$ generated by the $2\times 2$ minors of the following
matrix.

$\left[\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\\
x_{4}&x_{5}&x_{6}\end{array}\right]$

The ideal $I$ is generated by $f=\underline{x_{1}x_{5}}-x_{2}x_{4}$,
$g=\underline{x_{2}x_{6}}-x_{3}x_{5}$ and
$h=\underline{x_{3}x_{4}}-x_{1}x_{6}$. As in the previous example, there is no
monomial order $<_{\tau}$ such that
$\operatorname{in}_{<_{\tau}}(f)=\underline{x_{1}x_{5}}$,
$\operatorname{in}_{<_{\tau}}(g)=\underline{x_{2}x_{6}},$ and
$\operatorname{in}_{<_{\tau}}(h)=\underline{x_{3}x_{4}}$. If such an order
exists, then $x_{2}x_{4}<_{\tau}x_{1}x_{5}$, $x_{3}x_{5}<_{\tau}x_{2}x_{6}$
and $x_{1}x_{6}<_{\tau}x_{3}x_{4}$. It implies that
$(x_{2}x_{4})(x_{3}x_{5})(x_{1}x_{6})<_{\tau}(x_{1}x_{5})(x_{2}x_{6})(x_{3}x_{4})$
which is not possible.

As we see from the above examples, given a small number of marked polynomials,
it is rather straightforward to show there is no monomial order $<_{\tau}$
such that marked terms are initial terms with respect to $<_{\tau}$. If there
are more marked polynomials, it is more difficult to figure out whether
$\mathcal{G}$ is marked coherently. Naturally, one might ask whether there is
a way to determine when $\mathcal{G}$ is marked coherently. We could further
question when $\mathcal{G}$ is a Gröbner basis for the ideal generated by the
collection with respect to $<_{\tau}$. These questions are answered in [22] in
terms of reduction relations. We conclude this section by collecting several
necessary definitions and recalling the related answer from [22].

######  Definition 1.7.

Let $\mathcal{G}=\\{g_{1},\dots,g_{s}\\}$ be a collection of marked
polynomials with initial terms $\\{u_{1},\dots,u_{s}\\}$ where
$\operatorname{in}(g_{i})=u_{i}$ for each $i$. Let $f$ be a polynomial in $S$.
If there exists a monomial $u\in\operatorname{supp}(f)$ such that $u_{i}$
divides $u$ for some $1\leq i\leq s$, let $\displaystyle
f^{\prime}:=f-\frac{cu}{c_{i}u_{i}}g_{i}$ where $f=cu+h$ such that
$u\notin\operatorname{supp}(h)$. We call the polynomial $f^{\prime}$ a _one
step reduction_ of $f$ with respect to $\mathcal{G}$. We denote this reduction
relation by $f\longrightarrow_{\mathcal{G}}f^{\prime}$.

A collection of marked polynomials $\mathcal{G}=\\{g_{1},\dots,g_{s}\\}$ is
said to define a _Noetherian reduction relation_ if the number of successive
one step reductions of any polynomial is finite, i.e., for a polynomial $f$
and a chain of reduction relations
$f\longrightarrow_{\mathcal{G}}f^{\prime}\longrightarrow_{\mathcal{G}}\cdots\longrightarrow_{\mathcal{G}}f^{\prime\prime}$,
there is no possible one step reduction of $f^{\prime\prime}$ with respect to
$\mathcal{G}$. Noetherian

######  Theorem 1.8.

[22, Theorem 3.12] Given a collection of marked polynomials
$\mathcal{G}=\\{g_{1},\dots,g_{s}\\}$, there exists a monomial order,
$<_{\tau}$, such that $\operatorname{in}_{<_{\tau}}(g_{i})=u_{i}$ and
$\mathcal{G}$ is a Gröbner basis for $\langle g_{1},\dots,g_{s}\rangle$ with
respect to $<_{\tau}$ if and only if the reduction relation
$\longrightarrow_{\mathcal{G}}$ is Noetherian.

##  2\. Directed graphs of monomials via reductions

In this section, we consider collections of marked polynomials consisting
exclusively of homogeneous binomials of the same degree, i.e.,
$\mathcal{G}=\\{g_{1},\dots,g_{s}\\}$ such that
$g_{i}=\underline{u_{i,1}}-u_{i,2}$ where $u_{i,1}$ and $u_{i,2}$ are both
monomials of degree $d$ for all $i$. We first introduce a new notion called
the _directed graph_ of a monomial in $S$ and use its combinatorial structure
to detect when $\mathcal{G}$ is a Gröbner basis with respect to some monomial
order in Theorem 2.5.

######  Definition 2.1.

Let $\mu$ be a monomial in $S$ and $\mathcal{G}$ be a collection of marked
binomials. The _directed graph_ of $\mu$ with respect to $\mathcal{G},$
denoted by $\Gamma_{\mu}(\mathcal{G})$, is defined as follows:

  * •

The vertices are the monomials that can be obtained from $\mu$ by a sequence
of one step reductions with respect to $\mathcal{G}$.

  * •

For two vertices $v$ and $w$, there is a directed edge from $v$ to $w$ if
$v\longrightarrow_{\mathcal{G}}w$, i.e., $w$ is a one step reduction of $v$
with respect to $\mathcal{G}$.

######  Example 2.2.

Consider the collection of marked polynomials
$\mathcal{G}=\\{\underline{x_{1}x_{3}}-x_{2}^{2},\underline{x_{1}x_{2}}-x_{3}^{2}\\}$
and the monomial $x_{1}x_{2}x_{3}$. Let
$g_{1}=\underline{x_{1}x_{3}}-x_{2}^{2}$ and
$g_{2}=\underline{x_{1}x_{2}}-x_{3}^{2}$. Then the directed graph of
$x_{1}x_{2}x_{3}$ with respect to $\mathcal{G}$ is given below. Notice that
this directed graph has two different sinks.

$x_{1}x_{2}x_{3}$$x_{3}^{3}$$x_{2}^{3}$

One can see that the marked monomials are the leading terms for the collection
$\mathcal{G}$ with respect to the (degree) lexicographic order where
$x_{1}>x_{2}>x_{3}$. However, $\mathcal{G}$ is not a Gröbner basis because the
$S$-polynomial of $g_{1}$ and $g_{2}$ given below

$S(g_{1},g_{2})=x_{2}g_{1}-x_{3}g_{2}=x_{3}^{3}-x_{2}^{3}$

has a non-zero remainder when divided by $\mathcal{G}$.

######  Example 2.3.

Consider
$\mathcal{G}=\\{\underline{x_{1}x_{4}}-x_{2}x_{5},\underline{x_{2}x_{3}}-x_{4}^{2}\\}$
and the monomial $x_{1}x_{2}x_{3}x_{4}$. The directed graph of
$x_{1}x_{2}x_{3}x_{4}$ with respect to $\mathcal{G}$ is given as follows.

$x_{1}x_{2}x_{3}x_{4}$$x_{1}x_{4}^{3}$$x_{2}^{2}x_{3}x_{5}$$x_{2}x_{4}^{2}x_{5}$

Note that the directed graph possesses a unique sink and has no (directed)
cycles. Additionally, observe that $\mathcal{G}$ is a Gröbner basis with
respect to the monomial order $\succ_{rlex}$.

######  Example 2.4.

Consider the monomial $x_{1}x_{3}x_{5}x_{6}$ and
$\mathcal{G}=\\{\underline{x_{1}x_{5}}-x_{2}x_{4},\underline{x_{2}x_{6}}-x_{3}x_{5},\underline{x_{3}x_{4}}-x_{1}x_{6}\\}$
from Example 1.6. Below is the directed graph of $x_{1}x_{3}x_{5}x_{6}$ with
respect to $\mathcal{G}$.

$x_{1}x_{3}x_{5}x_{6}$$x_{2}x_{3}x_{4}x_{6}$$x_{3}^{2}x_{4}x_{5}$$x_{1}x_{2}x_{6}^{2}$

It is clear that the directed graph has no sinks and it has two cycles.
Additionally, recall from Example 1.6 that $\mathcal{G}$ is not a Gröbner
basis.

In the light of the previous examples, one can ask whether
$\Gamma_{\mu}(\mathcal{G})$ possesses a unique sink and has no cycles if there
exists a monomial order $<_{\tau}$ such that the collection of marked
binomials $\mathcal{G}$ is a Gröbner basis with respect to $<_{\tau}$ and the
initial terms are the marked monomials. In the next theorem, we answer this
question and further show that the converse is true.

######  Theorem 2.5.

Let $\mathcal{G}$ be a collection of marked binomials and $\mu$ be a monomial
in $S$. The directed graph of $\mu$ with respect to $\mathcal{G}$ possesses a
unique sink and has no cycles if and only if there exists a monomial order
$<_{\tau}$ such that $\mathcal{G}$ is a Gröbner basis with the marked
monomials as the initial terms.

###### Proof.

If there exists a monomial $\mu\in S$ such that $\Gamma_{\mu}(\mathcal{G})$
has a cycle, then there is an infinite sequence of one step reductions with
respect to $\mathcal{G}$ starting at $\mu$. Thus $\mathcal{G}$ fails to define
a Noetherian reduction relation. It follows from Theorem 1.8 that there is no
monomial order $<_{\tau}$ such that $\mathcal{G}$ is a Gröbner basis with the
marked monomials as the initial terms.

If there exists a monomial $\mu\in S$ such that $\Gamma_{\mu}(\mathcal{G})$
has more than one sink, then division of $\mu$ by $\mathcal{G}$ does not have
a unique remainder since each sink is the remainder of $\mu$ when it is
divided by $\mathcal{G}$. Thus, $\mathcal{G}$ can not be a Gröbner basis with
respect to a monomial order $<_{\tau}$ such that the marked monomials as the
initial terms with respect to $<_{\tau}$. Therefore, it is necessary that the
directed graph has a unique sink and no cycles for the existence of a monomial
order $<_{\tau}$ such that $\mathcal{G}$ is a Gröbner basis with leading terms
as the marked monomials.

Let $\mu$ be a monomial in $S$. Suppose that $\Gamma_{\mu}(\mathcal{G})$ has a
unique sink and no cycles. Our goal is to prove that there exists a monomial
order $<_{\tau}$ such that $\mathcal{G}$ is a Gröbner basis with the marked
monomials as the initial terms. It suffices to show that $\mathcal{G}$ defines
a Noetherian reduction relation by Theorem 1.8. For this purpose, we define an
invariant called the longest path length, denoted by $\ell_{\max}(\mu)$, where

$\ell_{\max}(\mu)=\\{\text{the longest distance from }\mu\text{ to the unique
sink in }\Gamma_{\mu}(\mathcal{G})\\}.$

The longest path length $\ell_{\max}(\mu)$ is well-defined because there can
be only a finite number of directed paths starting at $\mu$ and ending at the
unique sink in $\Gamma_{\mu}(\mathcal{G})$. Additionally, the longest path
length is a nonnegative integer, i.e., $\ell_{\max}(\mu)\geq 0$. We define a
related invariant for a polynomial $f$ in $S$ and it is given by

$\displaystyle\ell_{\max}(f):=\sum_{\mu\in\text{supp}(f)}\ell_{\max}(\mu).$

If $f$ has no one step reductions with respect to $\mathcal{G}$, we can choose
another polynomial in $S$ until that polynomial has a one step reduction.
Without loss of generality, we may assume that there exists a polynomial
$f^{\prime}$ such that $f\longrightarrow_{\mathcal{G}}f^{\prime}$. Then there
exists a monomial $v\in\operatorname{supp}(f)$ such that $u_{i,1}$ divides $v$
for some $1\leq i\leq s$ and

$f^{\prime}=f-\frac{c_{v}v}{u_{i,1}}g_{i}=(f-c_{v}v)+c_{v}\frac{v}{u_{i,1}}u_{i,2}$

where $f=c_{v}v+h$ such that $v\notin\operatorname{supp}(h)$.

In order to prove $\mathcal{G}$ defines a Noetherian reduction relation, it
suffices to show $\ell_{\max}(f^{\prime})<\ell_{\max}(f)$. Let $\displaystyle
v^{\prime}=\frac{v}{u_{i,1}}u_{i,2}$. If
$v^{\prime}\in\operatorname{supp}(f)$, then

$\ell_{\max}(f^{\prime})\leq\left(\sum_{\mu\in\operatorname{supp}(f)}\ell_{\max}(\mu)\right)-\ell_{\max}(v)<\sum_{\mu\in\operatorname{supp}(f)}\ell_{\max}(\mu)=\ell_{\max}(f).$

The last inequality is due to the fact that $\ell_{\max}(v)>0$ because $v$ can
not be the unique sink in $\Gamma_{v}(\mathcal{G})$ as $\displaystyle
v\longrightarrow_{\mathcal{G}}v^{\prime}$. If
$v^{\prime}\notin\operatorname{supp}(f)$, then

$\ell_{\max}(f^{\prime})=\displaystyle\left(\sum_{\mu\in\operatorname{supp}(f)}\ell_{\max}(\mu)\right)-\ell_{\max}(v)+\ell_{\max}(v^{\prime})<\sum_{\mu\in\operatorname{supp}(f)}\ell_{\max}(\mu)=\ell_{\max}(f).$

The strict inequality follows from the fact
$\ell_{\max}(v)>\ell_{\max}(v^{\prime})$ which is proved as follows. Since
$v\longrightarrow_{\mathcal{G}}v^{\prime}$, the fiber graph of $v^{\prime}$
with respect to $\mathcal{G}$ is a subgraph of $\Gamma_{v}(\mathcal{G})$;
moreover, both fiber graphs share the same sink. Thus, existence of a directed
edge from $v$ to $v^{\prime}$ guarantees that
$\ell_{\max}(v)>\ell_{\max}(v^{\prime})$. ∎

######  Remark 2.6.

Let $\mathcal{G}$ be a Gröbner basis consisting exclusively of binomials with
leading terms as the marked monomials. Then for any polynomial $f\in S$, there
is always a unique $\tilde{f}$ obtained through a sequence of one step
reductions with respect to $\mathcal{G}$ such that $\ell_{max}(\tilde{f})=0$.
It is also worth noting that $\tilde{f}$ is the remainder of $f$ when it is
divided with respect to $\mathcal{G}$.

##  3\. Strongly Stable Ideals and their multi-Rees Algebras

In [17], Herzog, Hibi and Vladoiu determine the defining equations of the Rees
algebra of a strongly stable ideal as the union of a Gröbner basis for the
toric ideal of its special fiber and a set of first syzygies, proving that
Rees algebras of strongly stable ideals are of fiber type. The main result of
this section (Theorem 3.2) generalizes their result to the case of multi-Rees
algebras.

######  Notation 3.1.

Let $I_{1},\dots,I_{r}$ be a collection of strongly stable ideals in $S$ and
$\\{u_{i,1},\dots,u_{i,s_{i}}\\}$ be the set of minimal monomial generators of
$I_{i}$ for each $i$ such that $\deg(u_{i,j})=d_{i}$ for all $1\leq j\leq
s_{i}$ and

$u_{i,1}\succ_{rlex}u_{i,2}\succ_{rlex}\dots\succ_{rlex}u_{i,s_{i}}.$

Recall that the defining ideals of the multi-Rees algebra
$\mathcal{R}(I_{1}\oplus\dots\oplus I_{r})$ and its special fiber
$\mathcal{F}(I_{1}\oplus\dots\oplus I_{r})$ are denoted by
$\mathcal{L}\subseteq R$ and $T(I_{1}\oplus\dots\oplus I_{r})\subseteq
R^{\prime}$, respectively, where

$R=S[T_{i,j}:1\leq i\leq r,~{}1\leq j\leq s_{i}]\text{ and
}R^{\prime}={\mathbb{K}}[T_{i,j}:1\leq i\leq r,~{}1\leq j\leq s_{i}].$

In what follows, we present the main result of this section and establish its
proof through two auxiliary lemmas. Explicit statements and proofs of these
lemmas are provided in the following steps.

######  Theorem 3.2.

If $\mathcal{G}$ is a Gröbner basis for the toric ideal
$T(I_{1}\oplus\dots\oplus I_{r})$, then

$\mathcal{G}^{\prime}:=\left\\{x_{i}T_{l,k}-x_{j}T_{l,k^{\prime}}~{}:~{}i<j\text{
and }x_{i}u_{l,k}=x_{j}u_{l,k^{\prime}}\right\\}\cup\mathcal{G}$

is a Gröbner basis for the defining ideal of the multi-Rees algebra
$\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$.

###### Proof.

In Lemma 3.3, we show that the collection of binomials $\mathcal{G}^{\prime}$
defines a Noetherian reduction relation. It then follows from Theorem 1.8 that
there exists a monomial term order $<_{\tau^{\prime}}$ such that
$\mathcal{G}^{\prime}$ is a Gröbner basis for the ideal generated by
$\mathcal{G}^{\prime}$ with respect to $<_{\tau^{\prime}}$. Finally, in Lemma
3.4, we prove that the initial ideal of $\langle\mathcal{G}^{\prime}\rangle$
and the initial ideal of $\mathcal{L}$ with respect to $<_{\tau^{\prime}}$ are
equal. Thus, the collection $\mathcal{G}^{\prime}$ is a Gröbner basis for
$\mathcal{L}$. ∎

######  Lemma 3.3.

Let $\mathcal{G}$ be a Gröbner basis for the toric ideal
$T(I_{1}\oplus\dots\oplus I_{r})$. The collection of marked binomials

$\mathcal{G}^{\prime}=\left\\{\underline{x_{i}T_{l,k}}-x_{j}T_{l,k^{\prime}}~{}:~{}i<j\text{
and }x_{i}u_{l,k}=x_{j}u_{l,k^{\prime}}\right\\}\cup\mathcal{G}$

defines a Noetherian reduction relation in $R$.

###### Proof.

Let $u$ be a monomial in $R^{\prime}$ which is presented by

$u=\prod_{i=1}^{r}\prod_{j=1}^{c_{i}}T_{i,i_{j}}=\Big{(}T_{1,1_{1}}\cdots
T_{1,1_{c_{1}}}\Big{)}\cdots\Big{(}T_{r,r_{1}}\cdots T_{r,r_{c_{r}}}\Big{)}$

where $1\leq i_{1}\leq\cdots\leq i_{c_{i}}\leq s_{i}$ for all $1\leq i\leq r$.
We define the _content_ of $u$ as the monomial in $S$, denoted by
$\textbf{x}^{\alpha_{u}}$, and given by

$\textbf{x}^{\alpha_{u}}:=x_{1}^{\alpha_{u,1}}\cdots
x_{n}^{\alpha_{u,n}}=\prod_{i=1}^{r}\prod_{j=1}^{c_{i}}u_{i,i_{j}}.$

Let $m$ be a monomial in $S$ expressed as $\displaystyle
m=\prod_{j=1}^{c}x_{i_{j}}$ where $i_{j}\in\\{1,\ldots,n\\}$.

In the remainder of the proof, we use the following order for the elements of
${\mathbb{N}}^{2}.$ Given $\mathbf{a},\mathbf{b}\in\mathbb{N}^{2}$, we say
$\mathbf{a}<_{lex,2}\mathbf{b}$ if the first nonzero entry of $\mathbf{b-a}$
is positive.

In order to show that $\mathcal{G}^{\prime}$ defines a Noetherian reduction
relation in $R$, we first prove that it is true for all monomials in $R$. For
this purpose, we introduce an invariant for a monomial $mu$ in $R$ by
associating it to an ordered pair $(o_{mu},l_{mu})$ where

$\displaystyle
o_{mu}=\sum_{q=1}^{c}\left(\sum_{t=i_{q}+1}^{n}\alpha_{u,t}\right)\text{ and
}l_{mu}=\ell_{\max}(u)$

such that $l_{mu}$ is the longest path length invariant defined in Theorem 2.5
and $o_{mu}$ can be interpreted as the iterated cumulative degree of the
content of $u$ with respect to $m$. Thus, it suffices to show the following
claim to prove $\longrightarrow_{\mathcal{G}^{\prime}}$ is Noetherian for
monomials in $R$.

$\text{ If }mu\longrightarrow_{\mathcal{G}^{\prime}}m^{\prime}u^{\prime}\text{
then
}(o_{m^{\prime}u^{\prime}},l_{m^{\prime}u^{\prime}})<_{lex,2}(o_{mu},l_{mu}).$

If there exists a reduction relation
$mu\longrightarrow_{\mathcal{G}^{\prime}}m^{\prime}u^{\prime}$, it must be
associated to an element of $\mathcal{G}$ or
$\mathcal{G}^{\prime}\setminus\mathcal{G}$. We call this element a reduction
element. If the reduction element is in $\mathcal{G},$ it follows from Theorem
2.5 that $\ell_{\max}(u^{\prime})<\ell_{\max}(u)$. Since $m=m^{\prime}$ and
the contents of $u$ and $u^{\prime}$ are the same, it is immediate that
$o_{mu}=o_{m^{\prime}u^{\prime}}$. Thus,
$(o_{m^{\prime}u^{\prime}},l_{m^{\prime}u^{\prime}})<_{lex,2}(o_{mu},l_{mu})$,
as desired.

If the reduction element is in $\mathcal{G}^{\prime}\setminus\mathcal{G}$, it
is of the form $\underline{x_{i}T_{l,k}}-x_{j}T_{l,k^{\prime}}$ where $i<j$
and $x_{i}u_{l,k}=x_{j}u_{l,k^{\prime}}$. Then, there exists a variable in the
support of $m$ such that $x_{i_{p}}=x_{i}$ for some $p\in\\{1,\ldots,c\\}$ and

$\displaystyle
m^{\prime}=x_{j}\frac{m}{x_{i}}=\prod_{q=1}^{c}x_{i^{\prime}_{q}}\text{ and
}u^{\prime}=T_{l,k^{\prime}}\frac{u}{T_{l,k}}$

where $x_{i^{\prime}_{p}}=x_{j}$ and $x_{i^{\prime}_{q}}=x_{i_{q}}$ for all
the remaining $q\in\\{1,\ldots,c\\}.$ Since
$x_{i}u_{l,k}=x_{j}u_{l,k^{\prime}}$, contents of $u$ and $u^{\prime}$ only
differ at the variables $x_{i}$ and $x_{j}$. In particular,
$\alpha_{u,t}=\alpha_{u^{\prime},t}$ for all $t\neq i,j$ while
$\alpha_{u,i}=\alpha_{u^{\prime},i}-1$ and
$\alpha_{u,j}=\alpha_{u^{\prime},j}+1.$ By making use of these equalities, we
obtain $o_{mu}>o_{m^{\prime}u^{\prime}}$ through the following term by term
comparison.

$\displaystyle o_{mu}$ $\displaystyle=\sum_{\begin{subarray}{c}t=i+1\\\
(q=p)\end{subarray}}^{n}\alpha_{u,t}+\sum_{\begin{subarray}{c}t=i_{q}+1\\\
q\neq p\text{ and
}i_{q}<i\end{subarray}}^{n}\alpha_{u,t}+\sum_{\begin{subarray}{c}t=i_{q}+1\\\
q\neq p\text{ and }i_{q}\geq i\end{subarray}}^{n}\alpha_{u,t}$
$\displaystyle>\sum_{\begin{subarray}{c}t=j+1\\\
(q=p)\end{subarray}}^{n}\alpha_{u,t}+\sum_{\begin{subarray}{c}t=i_{q}+1\\\
q\neq p\text{ and
}i_{q}<i\end{subarray}}^{n}\alpha_{u,t}+\left(\sum_{\begin{subarray}{c}t=i_{q}+1\\\
q\neq p\text{ and }i_{q}\geq i\end{subarray}}^{n}\alpha_{u,t}-1\right)$
$\displaystyle=\sum_{\begin{subarray}{c}t=j+1\\\
(q=p)\end{subarray}}^{n}\alpha_{u^{\prime},t}+\sum_{\begin{subarray}{c}t=i^{\prime}_{q}+1\\\
q\neq p\text{ and
}i_{q}<i\end{subarray}}^{n}\alpha_{u^{\prime},t}+\sum_{\begin{subarray}{c}t=i^{\prime}_{q}+1\\\
q\neq p\text{ and }i_{q}\geq
i\end{subarray}}^{n}\alpha_{u^{\prime},t}=o_{m^{\prime}u^{\prime}}$

Note that the transition from the second to the third line is achieved via
term by term equality. Hence,
$(o_{m^{\prime}u^{\prime}},l_{m^{\prime}u^{\prime}})<_{lex,2}(o_{mu},l_{mu})$
which implies that $\mathcal{G}^{\prime}$ defines a Noetherian reduction
relation for all monomials in $R$.

As in the proof of Theorem 2.5, for a polynomial $f$ in $R,$ one can define an
ordered pair $(o_{f},l_{f})$ as the sum of all the ordered pairs of the
monomials in the support of $f$. By repeating the arguments employed in the
proof of Theorem 2.5, it can be shown that $\mathcal{G}^{\prime}$ defines a
Noetherian reduction relation in $R$. ∎

######  Lemma 3.4.

Let $\mathcal{G}^{\prime}$ be the collection given in Lemma 3.3. Then there
exists a term order $<_{\tau^{\prime}}$ such that
$\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle=\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{L})\rangle$.

###### Proof.

Recall from Lemma 3.3 that $\mathcal{G}^{\prime}$ defines a Noetherian
reduction relation. Then, by Theorem 1.8, there exists a term order
$<_{\tau^{\prime}}$ such that $\mathcal{G}^{\prime}$ is a Gröbner bases with
respect to $<_{\tau^{\prime}}$ where the leading terms are the marked
monomials of $\mathcal{G}^{\prime}$. It is clear that
$\langle\mathcal{G}^{\prime}\rangle\subseteq\mathcal{L}$. Our goal is to show
$\langle\text{in}_{\tau^{\prime}}(\mathcal{L})\rangle\subseteq\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$.

Let $u,m,o_{mu}$ and $\ell_{mu}$ be given as in the proof of Lemma 3.3. In
what follows, we observe that if $mu$ does not belong to
$\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$, then
$o_{mu}=0=l_{mu}$. Suppose $o_{mu}>0$. It follows from the definition of
$o_{mu}$ that there exists a variable $x_{i}$ dividing $m$ such that
$\alpha_{u,j}>0$ for some $j\geq i+1$; moreover, $x_{j}$ divides $u_{l,k}$ for
some $l,k$ and $T_{lk}$ divides $u$ where $l\in[r]$ and
$k\in\\{1,\ldots,s_{l}\\}$. Since $u_{l,k}\in I_{l}$ and $I_{l}$ is a strongly
stable ideal, there exists $k^{\prime}>k$ such that
$x_{i}u_{l,k}=x_{j}u_{l,k^{\prime}}$. It implies that
$\underline{x_{i}T_{l,k}}-x_{j}T_{l,k^{\prime}}\in\mathcal{G}^{\prime}$ while
$x_{i}T_{lk}$ divides $mu$. Thus
$mu\in\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$.

If $l_{mu}>0,$ then $u$ can not be the sink of the fiber graph
$\Gamma_{u}(\mathcal{G})$ which implies that there is an element
$g=\underline{g_{1}}-g_{2}$ in $\mathcal{G}$ such that
$g_{1}=\operatorname{in}_{\tau^{\prime}}(g)$ divides $u$. Hence, $mu$ is
divisible by $\operatorname{in}_{\tau^{\prime}}(g)$, proving that
$mu\in\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$. This
completes the proof of the observation.

Proceed by contradiction and assume that there is a binomial generator of
$\mathcal{L}$, say $h$, such that
$\operatorname{in}_{\tau^{\prime}}(h)\notin\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$.
Let $h^{\prime}$ be the unique remainder of $h$ when it is divided by
$\mathcal{G}^{\prime}$ and let $h^{\prime}=mu-m^{\prime}u^{\prime}$ where

$m=\prod_{q=1}^{c_{0}}x_{i_{q}},~{}~{}m^{\prime}=\prod_{q=1}^{b_{0}}x_{j_{q}},~{}~{}u=\prod_{i=1}^{r}\prod_{j=1}^{c_{i}}T_{i,k_{j}},~{}~{}u^{\prime}=\prod_{i=1}^{r}\prod_{j=1}^{b_{i}}T_{i,k^{\prime}_{j}}$

such that the monomials $mu,m^{\prime}u^{\prime}$ do not belong to
$\langle\text{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$. Since
$h^{\prime}\in\mathcal{L}=\ker(\varphi),$ we have the following equality which
in turn implies that $b_{i}=c_{i}$ for all $i\in[r]$.

$\displaystyle\left(\prod_{q=1}^{c_{0}}x_{i_{q}}\right)\underbrace{\left(x_{1}^{\alpha_{u,1}}\cdots
x_{n}^{\alpha_{u,n}}\right)}_{\text{content of }u}\left(t_{1}^{c_{1}}\cdots
t_{r}^{c_{r}}\right)$
$\displaystyle=\left(\prod_{q=1}^{b_{0}}x_{j_{q}}\right)\underbrace{\left(x_{1}^{\alpha_{u^{\prime},1}}\cdots
x_{n}^{\alpha_{u^{\prime},n}}\right)}_{\text{content of
}u^{\prime}}\left(t_{1}^{b_{1}}\cdots t_{r}^{b_{r}}\right)$ (3.1)

Since $\deg(u_{i,j})=d_{i}$ for all $j\in\\{1,\ldots,s_{i}\\}$, one can
conclude that total degrees of contents of both $u$ and $u^{\prime}$ are
equal; moreover,

$\sum_{i=1}^{n}\alpha_{u,i}=\sum_{i=1}^{n}\alpha_{u^{\prime},i}=\sum_{i=1}^{r}c_{i}d_{i},$

implying $b_{0}=c_{0}$. Let $c=\max\\{k~{}:~{}\alpha_{u,k}\neq 0\\}$ and
$d=\max\\{k~{}:~{}\alpha_{u^{\prime},k}\neq 0\\}$. It follows from the
observation that $o_{mu}=0=o_{m^{\prime}u^{\prime}}$. Then one can obtain the
refined expressions of $m$ and $m^{\prime}$ given below.

$m=\prod_{i=c}^{n}x_{i}^{\gamma_{i}},~{}~{}~{}~{}~{}m^{\prime}=\prod_{i=d}^{n}x_{i}^{\delta_{i}}$

As a result, Equation 3.1 can be rewritten as

$\prod_{i=1}^{c}x_{i}^{\alpha_{u,i}}\prod_{i=c}^{n}x_{i}^{\gamma_{i}}=\prod_{i=1}^{d}x_{i}^{\alpha_{u^{\prime},i}}\prod_{i=d}^{n}x_{i}^{\delta_{i}}$
(3.2)

We may assume $c\leq d$. Then Equation 3.2 implies that
$\alpha_{u,i}=\alpha_{u^{\prime},i}$ when $i<c$ and $\gamma_{i}=\delta_{i}$
when $d<i$. Note that $\displaystyle\sum_{c\leq i\leq d}\gamma_{i}=\delta_{d}$
since $\deg(m)=\deg(m^{\prime})$ and $\gamma_{i}=\delta_{i}$ when $d<i$.

One can further show that $c=d$. Otherwise, Equation 3.2 results with the
following set of equalities.

$\gamma_{c}=\alpha_{u^{\prime},c}-\alpha_{u,c},~{}~{}\gamma_{i}=\alpha_{u^{\prime},i}\text{
for }c<i<d,~{}~{}\gamma_{d}=\alpha_{u^{\prime},d}+\delta_{d}$

Subtituting the above equalities in $\displaystyle\sum_{c\leq i\leq
d}\gamma_{i}=\delta_{d}$ yields to

$\alpha_{u,c}-\alpha_{u^{\prime},c}=\sum_{c<i\leq
d}\alpha_{u^{\prime},c}=-\gamma_{c}\leq 0,$

a contradiction because $\alpha_{u^{\prime},d}>0$ by definition. Since $c=d$,
one must have $\gamma_{c}=\delta_{c}$. In addition, we also have
$\alpha_{u,c}=\alpha_{u^{\prime},c}$ from Equation 3.2. Thus
$\gamma_{i}=\delta_{i}$ and $\alpha_{u,i}=\alpha_{u^{\prime},i}$ for all
$1\leq i\leq c$.

As a result, we conclude $m=m^{\prime}$ and contents of $u$ and $u^{\prime}$
are equal. Hence, $h^{\prime}=m(u-u^{\prime})$ where
$u-u^{\prime}\in\ker(\varphi^{\prime})=\langle\mathcal{G}\rangle$. Then
$\operatorname{in}_{\tau^{\prime}}(h^{\prime})=m\overline{u}$ where
$\overline{u}$ is in the ideal generated by the leading terms of $\mathcal{G}$
which is a subset of
$\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$. Thus
$\operatorname{in}_{\tau^{\prime}}(h^{\prime})=m\overline{u}\in\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle,$
a contradiction because none of the terms of $h^{\prime}$ belongs to
$\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle$.
Therefore,
$\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{G}^{\prime})\rangle=\langle\operatorname{in}_{\tau^{\prime}}(\mathcal{L})\rangle$.
∎

##  4\. Koszulness of the Multi-Rees Algebras of Strongly Stable Ideals

In this section, we provide a collection of examples of strongly stable ideals
whose multi-Rees algebras are not necessarily Koszul. One can systematically
study the multi-Rees algebras of strongly stable ideals $I_{1},\ldots,I_{r}$
by considering the following parameters.

  * •

$r:$ the number of ideals,

  * •

$g_{i}:$ the number of Borel generators of $I_{i}$ where
$I_{i}=\mathcal{B}(m_{i,1},\ldots,m_{i,g_{i}})$,

  * •

$d_{i}:$ the degree of Borel generators of $I_{i}$ where $\deg(m_{i,j})=d_{i}$
for $1\leq j\leq g_{i}$.

If the first parameter value is set equal to one ($r=1$), we are in the world
of Rees algebras of strongly stable ideals. A closer look into the literature
on these objects support our systematic approach. In particular, in [10], De
Negri proves that the special fiber ring $\mathcal{F}(I)$ of the Rees algebra
$\mathcal{R}(I)$ is Koszul when $I$ is a principal strongly stable ideal
($g_{1}=1$). In this case, one can further conclude that $\mathcal{R}(I)$ is
Koszul thanks to [17, Theorem 5.1], a result of Herzog, Hibi and Vladoiu
stating that the Rees algebras of strongly stable ideals are of fiber type. In
[4, Example 1.3], Bruns and Conca present an example of a strongly stable
ideal with three Borel generators ($g_{1}=3$) such that the toric ideal has a
minimal cubic generator. Thus, the special fiber ring or the Rees algebra are
not necessarily Koszul when $g_{1}\geq 3$. Motivated by this example, authors
of [11] studies the Rees algebra of a strongly stable ideal $I$ with two Borel
generators ($g_{1}=2$). Furthermore, they complete the general study on
Koszulness of the Rees algebras of strongly stable ideals by proving
$\mathcal{R}(I)$ is always Koszul when $g_{1}=2$.

In the multi-Rees setting, it is natural to consider the classes of strongly
stable ideals whose Rees algebras are always Koszul, in other words,
$g_{i}\in\\{1,2\\}$ for each $1\leq i\leq r$. In fact, authors of [11]
conclude their paper with the question asking whether the multi-Rees algebras
of strongly stable ideals $I_{1},\ldots,I_{r}$ is Koszul when $1\leq g_{i}\leq
2$ for each $i$. In a recent paper [12], it is proved that the multi-Rees
algebra of principal strongly stable ideals is Koszul, answering the question
for the case $g_{i}=1$ for each $i.$ Note that there are no restrictions on
the last set of parameters $d_{1},\ldots,d_{r}$.

In what follows, we provide several examples to illustrate some conditions on
$r,g_{i},d_{i}$ that diminish the possibility of a multi-Rees algebra of
strongly stable ideals to be Koszul. As a natural consequence of these
examples, we collect the possible conditions on $r,g_{i},d_{i}$ allowing
Koszulness.

######  Example 4.1.

Consider the following strongly stable ideals with two Borel generators

$I_{1}={\mathcal{B}}(x_{3}^{2}x_{6}^{a},x_{1}x_{5}x_{6}^{a}),~{}~{}I_{2}={\mathcal{B}}(x_{3}^{2}x_{6}^{b},x_{2}x_{4}x_{6}^{b}),~{}~{}I_{3}={\mathcal{B}}(x_{2}x_{4}x_{6}^{c},x_{1}x_{5}x_{6}^{c}),$

where $a,b,c$ non-negative integers. Then the syzygy

$(x_{1}x_{5}x_{6}^{a})(x_{3}^{2}x_{6}^{b})(x_{2}x_{4}x_{6}^{c})=(x_{3}^{2}x_{6}^{a})(x_{2}x_{4}x_{6}^{b})(x_{1}x_{5}x_{6}^{c})$

is minimal and it corresponds to a cubic minimal generator of the toric ideal
$T(I_{1}\oplus I_{2}\oplus I_{3})$.

This example shows that the multi-Rees algebra of three or more strongly
stable ideals (none of which are principal) is not necessarily Koszul ($r\geq
3$, $g_{i}\geq 2$, and $d_{i}\geq 2$ for each $1\leq i\leq r$).

######  Example 4.2.

Consider the following strongly stable ideals with two Borel generators

$I_{1}={\mathcal{B}}(x_{1}^{2}x_{3}^{2}x_{4}^{a},x_{1}x_{2}^{2}x_{3}x_{4}^{a}),~{}~{}I_{2}={\mathcal{B}}(x_{1}^{2}x_{3}^{2}x_{4}^{b},x_{2}^{4}x_{4}^{b})$

where $a,b,c$ non-negative integers. Then the syzygy

$(x_{1}^{2}x_{3}^{2}x_{4}^{a})^{2}(x_{2}^{4}x_{4}^{b})=(x_{1}x_{2}^{2}x_{3}x_{4}^{a})^{2}(x_{1}^{2}x_{3}^{2}x_{4}^{b})$

is minimal and it corresponds to a cubic minimal generator of the toric ideal
$T(I_{1}\oplus I_{2})$.

This example shows that the multi-Rees algebra of two strongly stable ideals
with two Borel generators of degree four or higher is not necessarily Koszul
($r=2$, $g_{1}=g_{2}=2,$ and $d_{1},d_{2}\geq 4$).

######  Example 4.3.

Consider the following strongly stable ideals with two Borel generators

$I_{1}={\mathcal{B}}(x_{1}x_{3},x_{2}^{2}),~{}~{}I_{2}={\mathcal{B}}(x_{1}^{2}x_{3}^{2}x_{4}^{a},x_{2}^{4}x_{4}^{a})$

where $a$ is a non-negative integer. Then the syzygy

$(x_{1}x_{3})^{2}(x_{2}^{4}x_{4}^{a})=(x_{2}^{2})^{2}(x_{1}^{2}x_{3}^{2}x_{4}^{a})$

is minimal and it corresponds to a cubic minimal generator of the toric ideal
$T(I_{1}\oplus I_{2})$.

This example shows that the multi-Rees algebra of two strongly stable ideals
with two Borel generators such that exactly one of the ideals is generated in
degree four or higher is not necessarily Koszul ($r=2$, $g_{1}=g_{2}=2,$ and
$d_{1}=2,d_{2}\geq 4$).

######  Proposition 4.4.

Consider two collections of natural numbers $1\leq g_{1}\leq
g_{2}\leq\dots\leq g_{r}$ and $d_{1},\dots,d_{r}$ satisfying $1\leq d_{i}\leq
d_{j}$ whenever both $i<j$ and $g_{i}=g_{j}$.

If the multi-Rees algebra of $I_{1},\dots,I_{r}$ is Koszul for any collection
of strongly stable ideals $I_{1},\dots,I_{r}$ where each $I_{i}$ possesses
exactly $g_{i}$ Borel generators of degree $d_{i}$, then one of the following
must be true.

  1. (a)

$r=2$, $g_{1}=g_{2}=2$ and $2\leq d_{1}\leq d_{2}\leq 3$

  2. (b)

$r>2$, $g_{1}=g_{2}=\ldots=g_{r-2}=1$, $g_{r-1}=g_{r}=2$ and $2\leq
d_{r-1}\leq d_{r}\leq 3$

  3. (c)

$r\geq 2$, $g_{1}=g_{2}=\ldots=g_{r-1}=1$ and $g_{r}\leq 2$

###### Proof.

Combination of the observations from Examples 4.1, 4.2, and 4.3 leads to the
conclusion. ∎

In the remainder of the paper, we obtain a partial converse of Proposition 4.4
concerning a subcase of (a) that is $r=2$, $g_{1}=g_{2}=2$ and
$d_{1}=d_{2}=2$. In particular, we show that the multi-Rees algebra of two
strongly stable ideals with two quadric Borel generators is Koszul.

##  5\. Strongly Stable Ideals and Their Fiber Graphs

In this section, we focus on strongly stable ideals with two quadric Borel
generators. In particular, we investigate the toric ideal of the special fiber
$\mathcal{F}(I)$ associated to the ideal $I={\mathcal{B}}(M,N)$ where
$M=x_{a}x_{b}$ and $N=x_{c}x_{d}$ such that $c<a\leq b<d$. Our main objective
in this section is to study _fiber graphs_ of these ideals to obtain a
quadratic Gröbner basis for their toric ideals with respect to two different
monomial orders. Results of this section forms the foundation of the next
section in which we investigate the Koszulness of the multi-Rees algebras of
strongly stable ideals with two quadric Borel generators via fiber graphs.

In Section 2, we provided a combinatorial method to investigate whether a
collection of marked binomials $\mathcal{G}=\\{g_{1},\dots,g_{s}\\}$ form a
Gröbner basis. More specifically, we introduced the notion of directed graph
of a monomial with respect to $\mathcal{G}$ and showed that $\mathcal{G}$ is a
Gröbner basis if and only if the directed graph of any monomial has a unique
sink and has no cycles. One could further investigate whether $\mathcal{G}$ is
a Gröbner basis of a particular ideal through related combinatorial objects
called fiber graphs. Fiber graphs are first introduced in [2] to study Gröbner
basis of toric ideals of graphic matroids and further developed in [11, 12,
20] for different classes of toric ideals.

######  Definition 5.1.

Let $\\{u_{i}:i\in\mathcal{P}\\}\subseteq S$ be a finite collection of
monomials of the same degree in the polynomial ring
$S={\mathbb{K}}[x_{1},\dots,x_{n}]$. Let $I$ be the ideal generated by the
given collection. Consider the toric map

$\begin{array}[]{rcl}\phi_{I}:{\mathbb{K}}[T_{u_{i}}:i\in\mathcal{P}]&\rightarrow&{\mathbb{K}}[u_{i}:i\in\mathcal{P}]\\\
T_{u_{i}}&\rightarrow&u_{i}\end{array}$

and extended algebraically. The kernel of the toric map $\phi_{I}$ is the
toric ideal of $I$ which we denote by $T(I)$. Note that the ring
$R={\mathbb{K}}[T_{u_{i}}:i\in\mathcal{P}]$ inherits the multigrading from $S$
where $R=\bigoplus_{\mu}R_{\mu}$ with $\mu$ ranging over monomials of $S$ and
$R_{\mu}$ being the ${\mathbb{K}}$-vectors space described as

$R_{\mu}=\operatorname{span}_{{\mathbb{K}}}\\{T=\prod T_{u_{i}}^{a_{u_{i}}}\in
R~{}:~{}\phi_{I}(T)=\mu\\}.$

For the remainder of the paper, we abuse notation by referring to monomials of
$S$ as multidegrees.

In what follows, we recall the fiber graph construction from [2, 20] (see also
[12, 11]). These objects are defined in terms of fibers of a multidegree under
the toric map and a collection of marked binomials in $R$.

######  Definition 5.2.

Let $\mu\in S$ be a multidegree and $\mathcal{G}$ be a collection of marked
binomials in $R$ such that the marked monomial is the initial term of the
binomial with respect to a given term order. The _fiber graph of $I$ at $\mu$
with respect to $\mathcal{G}$_, denoted by $\Gamma_{\mu}(I)$, is defined as
follows: the vertices of $\Gamma_{\mu}(I)$ correspond to monomials in
$R_{\mu}$, i.e., a monomial $T$ is a vertex if and only if $\phi_{I}(T)=\mu$.
For each pair of vertices $T$ and $T^{\prime}$, there is a directed edge from
$T$ to $T^{\prime}$ whenever $T\rightarrow_{\mathcal{G}}T^{\prime}$. For the
sake of simplicity, we suppress the collection of marked binomials
$\mathcal{G}$ in the notation for $\Gamma_{\mu}(I)$ as our choice of
$\mathcal{G}$ can be understood from context.

######  Remark 5.3.

It follows from the construction of fiber graphs that for each
$T\rightarrow_{\mathcal{G}}T^{\prime}$, the monomial $T$ comes before the
monomial $T^{\prime}$ with respect to a term order $\succ_{\tau}$ such that
the marked monomials of $\mathcal{G}$ are the initial terms with respect to
$\succ_{\tau}$. Thus, none of the fiber graphs possesses cycles.

Fiber graphs are significant tools to study Gröbner bases of $T(I)$ and this
importance is due to the following corollary which is a direct consequence of
Theorem 2.5 and the definition of fiber graphs (see also [11, Proposition 4.5]
and [12, Proposition 2.3]).

######  Corollary 5.4.

The collection of marked binomials $\mathcal{G}\subseteq R$ is a Gröbner basis
for $T(I)$ if and only if the fiber graph $\Gamma_{\mu}(I)$ is either empty or
has a unique sink for every multidegree $\mu\in S$.

######  Remark 5.5.

Notice that the fiber graph construction given in Definition 5.2 is associated
to a toric map $\phi_{I}$. Thus one can define fiber graphs for direct sum of
ideals $I_{1}\oplus\dots\oplus I_{r}$ via the toric map $\varphi^{\prime}$
from Definition 1.3. As a result, we can extend Corollary 5.4 to show that a
collection of marked binomials form a Gröbner basis for
$T(I_{1}\oplus\dots\oplus I_{r})$.

######  Notation 5.6.

For the remainder of this section, we use the notation $I={\mathcal{B}}(M,N)$
to denote a strongly stable ideal with two quadric Borel generators $M$ and
$N$ where $M=x_{a}x_{b}$, $N=x_{c}x_{d}$ such that $c<a\leq b<d$. Furthermore,
we denote the collection of the minimal monomial generators of
${\mathcal{B}}(M)$ by ${\mathcal{B}}_{M}$ and collection of the minimal
monomial generators in ${\mathcal{B}}(N)\setminus{\mathcal{B}}(M)$ by
${\mathcal{B}}_{N}$.

In certain cases, one can classify vertices of a fiber graph based on the
factorization of a vertex. In such situations, the structure of one vertex
determines the structure of all other vertices of a fiber graph. Before we
provide this classification, we introduce the following notation.

######  Definition 5.7.

A vertex $T$ in $\Gamma_{\mu}(I)$ is called a _type $M$ vertex_ if
$T=\prod_{m\in{\mathcal{B}}_{M}}T_{m}$. Similarly, if
$T=\prod_{n\in{\mathcal{B}}_{N}}T_{n}$, we say $T$ is a _type $N$ vertex_.

A fiber graph may have vertices which are neither type $M$ nor type $N$.
However, in the existence of a type $M$ vertex, it forces all other vertices
to be type $M$. Same situation occurs when a fiber graph has a vertex of type
$N$.

######  Lemma 5.8.

Let $\mu$ be a multidegree in $S$ and $I={\mathcal{B}}(M,N)$.

  1. (a)

If $\Gamma_{\mu}(I)$ has a vertex of type $M$, then all other vertices of
$\Gamma_{\mu}(I)$ are of the same type.

  2. (b)

If $\Gamma_{\mu}(I)$ has a vertex of type $N$, then all other vertices of
$\Gamma_{\mu}(I)$ are of the same type.

###### Proof.

(a) Let $T=T_{m_{1}}\cdots T_{m_{k}}$ where
$m_{i}=x_{s_{i}}x_{t_{i}}\in{\mathcal{B}}_{M}$. Since $T$ is a vertex in
$\Gamma_{\mu}(I)$, we have $\mu=m_{1}\cdots
m_{k}=\prod_{i=1}^{k}x_{s_{i}}x_{t_{i}}$ where $s_{i}\leq t_{i}\leq b$ for
each $i$. Note that there exists no $x_{q}$ dividing $\mu$ such that $q>b$.

Suppose there exists another vertex $T^{\prime}$ of $\Gamma_{\mu}(I)$ such
that
$T^{\prime}=\Big{(}\prod_{i=1}^{k^{\prime}}T_{f_{i}}\Big{)}\Big{(}\prod_{j=1}^{l}T_{n_{j}}\Big{)}$
where $f_{i}\in{\mathcal{B}}_{M}$ for each $i\in[k^{\prime}]$ and
$n_{j}\in{\mathcal{B}}_{N}$ for each $j\in[l]$. Let $f_{i}=x_{a_{i}}x_{b_{i}}$
and $n_{j}=x_{p_{j}}x_{q_{j}}$ for each $i$ and $j$. Then

$\mu=\Big{(}\prod_{i=1}^{k^{\prime}}x_{a_{i}}x_{b_{i}}\Big{)}\Big{(}\prod_{j=1}^{l}x_{p_{j}}x_{q_{j}}\Big{)}$

where $a_{i}\leq b_{i}\leq b$ and $p_{j}\leq c<b<q_{j}$ for each $i$ and $j$.
Since $\mu$ does not have any factor $x_{q}$ with $q>b$, we must have $l=0$.
Thus $k=k^{\prime}$ and $T^{\prime}$ is a vertex of type $M$.

(b) Let $T=T_{n_{1}}\cdots T_{n_{l}}$ where
$n_{i}=x_{p_{i}}x_{q_{i}}\in{\mathcal{B}}_{N}$ such that $p_{i}\leq c<b<q_{i}$
for each $i$. Since $T$ is a vertex in $\Gamma_{\mu}(I)$, we have
$\mu=n_{1}\cdots n_{l}$. Note that there exists no factor of $\mu$ other than
$x_{q_{1}}\cdots x_{q_{l}}$ such that index of each of the variables in the
factor is greater than $b$.

Suppose there exists another vertex $T^{\prime}$ of $\Gamma_{\mu}(I)$ such
that
$T^{\prime}=\Big{(}\prod_{i=1}^{k}T_{m_{i}}\Big{)}\Big{(}\prod_{j=1}^{l^{\prime}}T_{g_{j}}\Big{)}$
where $m_{i}\in{\mathcal{B}}_{M}$ for each $i\in[k]$ and
$g_{j}\in{\mathcal{B}}_{N}$ for each $j\in[l^{\prime}]$. Let
$m_{i}=x_{s_{i}}x_{t_{i}}$ and $q_{j}=x_{c_{j}}x_{d_{j}}$ for each $i$ and
$j$. Then

$\mu=\Big{(}\prod_{i=1}^{k}x_{s_{i}}x_{t_{i}}\Big{)}\Big{(}\prod_{j=1}^{l^{\prime}}x_{c_{j}}x_{d_{j}}\Big{)}$

where $s_{i}\leq t_{i}\leq b$ and $c_{j}\leq c<b<d_{j}$ for each $i$ and $j$.
Then we must have $x_{q_{1}}\cdots x_{q_{l}}=x_{d_{1}}\cdots
x_{d_{l^{\prime}}}$ which implies that $l=l^{\prime}$. Thus $k=0$ and
$T^{\prime}$ is a vertex of type $N$. ∎

######  Notation 5.9.

For the sake of simplicity, we often use the notation $T_{ij}$ to refer to
$T_{x_{i}x_{j}}$ for $i\leq j$. It should also be noted that
$T_{ij}=T_{x_{i}x_{j}}=T_{x_{j}x_{i}}=T_{ji}$. We set $T_{ij}$ as our standard
notation where $i\leq j$.

###  5.1. The revlex Gröbner basis

In this subsection, we describe a Gröbner basis for $T(I)$ where
$I={\mathcal{B}}(M,N)$ with respect to reverse lexicographic order by using
fiber graphs.

######  Definition 5.10.

We say $T_{ij}>_{rlex}T_{i^{\prime}j^{\prime}}$ if
$x_{i}x_{j}\succ_{rlex}x_{i}^{\prime}x_{j}^{\prime}$ in
$S=K[x_{1},\dots,x_{n}]$. We use $\succ_{rlex}$ to also denote the reverse
lexicographic order in $R$ induced by $>_{rlex}$ on the $T$ variables.

The following result was first proved in [6, Theorem 1.3] for symmetric ladder
ideals (see Remark 5.12). Our proof differs from [6] as it employs fiber
graphs to obtain a given collection as a Gröbner basis. Additionally, the
ideas used in the proof of the following lemma is fundamental to establish our
last main result (Theorem 6.9).

######  Theorem 5.11.

Let
$\mathcal{G}_{1}=\\{\underline{T_{u}T_{v}}-T_{u^{\prime}}T_{v^{\prime}}:uv=u^{\prime}v^{\prime}\text{
and }u,v\succ_{rlex}v^{\prime}\\}$. Then $\mathcal{G}_{1}$ is a Gröbner basis
of $T(I)$ with respect to the term order $\succ_{rlex}$.

###### Proof.

It suffices to prove that, for each multidegree $\mu\in S$ with respect to
$\mathcal{G}_{1}$, the fiber graph $\Gamma_{\mu}(I)$ is empty or has a unique
sink by Corollary 5.4. Thus our goal is to prove the claim: “If $\mu$ is a
multidegree in $S$, then $\Gamma_{\mu}(I)$ is empty or possesses a unique
sink.” If the degree of $\mu$ is odd, then $\Gamma_{\mu}(I)$ must be empty.
Suppose the degree of $\mu$ is even. If $\Gamma_{\mu}(I)$ is empty, we are
done. If not, the proof follows from Corollary 5.17. ∎

$c$$a$$b$$d$$c$$a$$b$$d$ Figure 1. The symmetric ladder diagram for
$I=\mathcal{B}(x_{a}x_{b},x_{c}x_{d})$ with $c<a\leq b<d$

######  Remark 5.12.

The set $\mathcal{G}_{1}$ corresponds to the collection of $2\times 2$ minors
of the symmetric ladder diagram given in Figure 1. In this case, the leading
(marked) monomial is chosen according to the term order $\succ_{rlex}$,
defined in Definition 5.10. The variables of $R$ are ordered in the following
way.

$\displaystyle
T_{11}>_{rlex}T_{12}>_{rlex}T_{22}>_{rlex}\dots>_{rlex}T_{1b}>_{rlex}T_{2b}>_{rlex}\dots>_{rlex}T_{ab}>_{rlex}$
$\displaystyle
T_{1(b+1)}>\dots>_{rlex}T_{c(b+1)}>_{rlex}\dots>_{rlex}T_{1d}>_{rlex}\dots>_{rlex}T_{cd}$

In order to study fiber graphs in a systemic way, we set the following
standard factorization for a monomial in $R$ with respect to the
$\succ_{rlex}$ order.

######  Notation 5.13.

Let $T$ be a monomial in $R$. We express $T$ as $T=(T_{p_{1}}\cdots
T_{p_{k}})(T_{p_{k+1}}\cdots T_{p_{k+l}})$ such that

  * •

$p_{i}\in{\mathcal{B}}_{M}$ for each $i\in\\{1,\ldots,k\\}$ and
$p_{k+j}\in{\mathcal{B}}_{N}$ for each $j\in\\{1,\ldots,l\\}$,

  * •

$p_{1}\succeq_{rlex}\dots\succeq_{rlex}p_{k}\succeq_{rlex}p_{k+1}\succeq_{rlex}\dots\succeq_{rlex}p_{k+l}$.

In the above expression, we denote the latest variable factor of $T$ in
${\mathcal{B}}_{M}$ by $L_{M}(T)$ and the latest variable factor of $T$ in
${\mathcal{B}}_{N}$ by $L_{N}(T)$. Based on the above expression of $T$, one
has $L_{M}(T)=T_{p_{k}}$ and $L_{N}(T)=T_{p_{k+l}}$ unless $k$ or $l$ is equal
to zero.

Given a multidegree $\mu$ in $S$, we use $M^{\prime}$ and $N^{\prime}$ to
denote the smallest elements in ${\mathcal{B}}_{M}$ and ${\mathcal{B}}_{N}$,
respectively, with respect to $\succ_{rlex}$ dividing $\mu$.

######  Example 5.14.

Let $I=\mathcal{B}(x_{3}^{2},x_{2}x_{5})$ and let
$\mu=x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}x_{5}.$ Then the fiber graph
$\Gamma_{\mu}(I)$ is given in Figure 2 and vertices of $\Gamma_{\mu}(I)$ are
labeled based on 5.13.

$T_{23}^{2}T_{14}T_{15}$$T_{13}T_{23}T_{24}T_{15}$$T_{22}T_{33}T_{14}T_{15}$$T_{13}T_{23}T_{14}T_{25}$$T_{12}T_{33}T_{24}T_{15}$$T_{12}T_{33}T_{14}T_{25}$$T_{13}^{2}T_{24}T_{25}$$T_{11}T_{33}T_{24}T_{25}$
Figure 2. The fiber graph $\Gamma_{x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}x_{5}}(I)$
for $I=\mathcal{B}(x_{3}^{2},x_{2}x_{5})$.

Note that the fiber graph given in Figure 2 has no cycles and has a unique
sink $T=T_{11}T_{33}T_{24}T_{25}$ where $L_{M}(T)=T_{33}$ and
$L_{N}(T)=T_{25}$. In addition, both Borel generators $M=x_{3}^{2}$ and
$N=x_{2}x_{5}$ divides $\mu$, so $M=M^{\prime}$ and $N=N^{\prime}$. As we
shall see in the following lemma, it is not a coincidence that
$L_{N}(T)=T_{25}=T_{N}$ for the sink vertex $T$.

######  Lemma 5.15.

Let $\mu$ be a multidegree and $T$ be a vertex of $\Gamma_{\mu}(I)$.

  1. (a)

If $L_{N}(T)$ exists and $L_{N}(T)\neq T_{N^{\prime}}$, then there is an edge
from $T$ to a vertex $T^{\prime}$ of $\Gamma_{\mu}(I)$ such that

$L_{N}(T)>_{rlex}L_{N}(T^{\prime})\geq_{rlex}T_{N^{\prime}}.$

  2. (b)

Suppose $L_{N}(T)$ does not exist. If $L_{M}(T)$ exists and $L_{M}(T)\neq
T_{M^{\prime}}$ then there is an edge from $T$ to $T^{\prime}$ such that

$L_{M}(T)>_{rlex}L_{M}(T^{\prime})\geq_{rlex}T_{M^{\prime}}.$

###### Proof.

(a) Let $L_{N}(T)=T_{ij}$ and $N^{\prime}=x_{i^{\prime}}x_{j^{\prime}}$ where
$x_{i}x_{j}$ and $x_{i^{\prime}}x_{j^{\prime}}$ are both in
${\mathcal{B}}_{N}$. It follows from our assumption $L_{N}(T)\neq
T_{N^{\prime}}$ that $i<i^{\prime}$ or $j<j^{\prime}$. Suppose $i<i^{\prime}.$
Since $N^{\prime}$ divides $\mu$, there exists a monomial $u$ divisible by
$x_{i^{\prime}}$ such that $T_{u}$ is in $\operatorname{supp}(T),$ the support
of $T$. We claim that $T_{u}T_{ij}$ divides $T$. In order to prove the claim,
it suffices to show $T_{u}\neq T_{ij}$. If $u=x_{i}x_{j}$, we must have
$i^{\prime}=j$ as $u$ is divisible by $x_{i^{\prime}}$ where $i<i^{\prime}$.
Additionally, since $N^{\prime}$ divides $\mu$, there exists a monomial $v$
divisible by $x_{j^{\prime}}$ such that $T_{v}$ is in
$\operatorname{supp}(T)$. Note that $T_{v}\geq_{rlex}T_{ii^{\prime}}$ as
$L_{N}(T)=T_{ii^{\prime}}$ is the last variable in the standard factorization
of $T$. Thus, we must have $i^{\prime}=j^{\prime}$ which is a contradiction
because $x_{i^{\prime}}x_{j^{\prime}}\in{\mathcal{B}}_{N}$ implies
$i^{\prime}<j^{\prime}$. Therefore, $T_{u}\neq T_{ij}$.

Recall that $u\in\mathcal{B}(M,N)$. It follows from the definition of strongly
stable ideals that $u^{\prime}:=x_{i}(u/x_{i^{\prime}})$ belongs to
${\mathcal{B}}(M,N)$ as $i<i^{\prime}$. Furthermore,
$x_{i^{\prime}}x_{j}\in{\mathcal{B}}_{N}$ because
$x_{i^{\prime}}x_{j^{\prime}}\in{\mathcal{B}}_{N}$ where $j\leq j^{\prime}$.
Thus, $T_{u}T_{ij}-T_{u^{\prime}}T_{i^{\prime}j}\in\mathcal{G}_{1}$ which
implies that $T\rightarrow_{\mathcal{G}_{1}}T^{\prime}$ where

$\displaystyle
T^{\prime}=T_{u^{\prime}}T_{i^{\prime}j}\Big{(}\frac{T}{T_{u}T_{ij}}\Big{)}$

is a vertex in $\Gamma_{\mu}(I)$. Note that
$L_{N}(T^{\prime})=T_{i^{\prime}j}$ where
$T_{u}>_{rlex}L_{N}(T)>_{rlex}L_{N}(T^{\prime})\geq_{rlex}T_{N^{\prime}}$, and
the equality is achieved when $j=j^{\prime}$. If $i=i^{\prime},$ then
$j<j^{\prime}$. The proof of this case follows from similar arguments used in
$i<i^{\prime}$ case.

(b) Let $L_{M}(T)=T_{ij}$ and $M^{\prime}=x_{i^{\prime}}x_{j^{\prime}}$ where
$x_{i}x_{j}$ and $x_{i^{\prime}}x_{j^{\prime}}$ are both in
${\mathcal{B}}_{M}$. It follows from our assumption $L_{M}(T)\neq
T_{M^{\prime}}$ that $i<i^{\prime}$ or $j<j^{\prime}$. Suppose $j<j^{\prime}$.
Since $M^{\prime}$ divides $\mu$, there exists $T_{u}$ in
$\operatorname{supp}(T)$ such that $u$ is divisible by $x_{j^{\prime}}.$
Observe that $T_{u}\neq T_{ij}$. Otherwise, if $u=x_{i}x_{j}$, we must have
$j^{\prime}=i$ since $x_{j^{\prime}}$ divides $u$ and $j<j^{\prime}$. We then
conclude that $j^{\prime}=i\leq j$, a contradiction. Thus, $T_{u}\neq T_{ij}$
and $T_{u}T_{ij}$ divides $T$. Recall that $L_{N}(T)$ does not exists. Then
$u\in{\mathcal{B}}_{M}$. Since $x_{j^{\prime}}$ divides $u$ where
$j<j^{\prime},$ we must have
$u^{\prime}:=x_{j}(u/x_{j^{\prime}})\in{\mathcal{B}}_{M}$. Furthermore,
$x_{i}x_{j^{\prime}}\in{\mathcal{B}}_{M}$ because
$x_{i^{\prime}}x_{j^{\prime}}\in{\mathcal{B}}_{M}$ where $i\leq i^{\prime}$.
Thus, $T_{u}T_{ij}-T_{u^{\prime}}T_{ij^{\prime}}\in\mathcal{G}_{1}$ which
implies that $T\rightarrow_{\mathcal{G}_{1}}T^{\prime}$ where

$\displaystyle
T^{\prime}=T_{u^{\prime}}T_{ij^{\prime}}\Big{(}\frac{T}{T_{u}T_{ij}}\Big{)}$

is a vertex in $\Gamma_{\mu}(I)$. Note that
$L_{M}(T^{\prime})=T_{ij^{\prime}}$ where
$T_{u}>L_{M}(T)>_{rlex}L_{M}(T^{\prime})\geq_{rlex}T_{M^{\prime}}$, and the
equality is achieved when $i=i^{\prime}$.

Suppose $j=j^{\prime}$. Then $i<i^{\prime}$. Similar to the previous
paragraphs, we claim that there exists a $u\in{\mathcal{B}}_{M}$ such that it
is divisible by $x_{i^{\prime}}$ and $T_{u}T_{ij}$ divides $T$. Since
$M^{\prime}$ divides $\mu$, there exists a $T_{u}$ in $\operatorname{supp}(T)$
such that $u$ is divisible by $x_{i^{\prime}}$. If $T_{u}\neq T_{ij},$ then
$T_{u}T_{ij}$ divides $T$. If $T_{u}=T_{ij}$, then $u=x_{i}x_{j}$. Since
$i<i^{\prime}$ and $u$ is divisible by $x_{i^{\prime}}$, we must have
$i^{\prime}=j=j^{\prime}$ which implies that $M^{\prime}=x_{i^{\prime}}^{2}$
and $T_{u}=L_{M}(T)=T_{ii^{\prime}}$. Since $M^{\prime}$ divides $\mu,$ there
exists another $T_{v}$ in the support of $T$ such that $v$ is divisible by
$x_{i^{\prime}}$. Then $T_{v}T_{ij}$ divides $T$. Since $L_{N}(T)$ does not
exist, $u,v$ must be both in ${\mathcal{B}}_{M}$ and the claim holds. Since
$u\in{\mathcal{B}}_{M}$ and $x_{i^{\prime}}$ divides $u$ where $i<i^{\prime}$,
the monomial $u^{\prime}:=x_{i}(u/x_{i^{\prime}})$ is in ${\mathcal{B}}_{M}.$
Recall also that $M^{\prime}=x_{i^{\prime}}x_{j}\in{\mathcal{B}}_{M}$. Then,
$T_{u}T_{ij}-T_{u^{\prime}}T_{i^{\prime}j}\in\mathcal{G}_{1}$ which implies
that $T\rightarrow_{\mathcal{G}_{1}}T^{\prime}$ where

$T^{\prime}=T_{u^{\prime}}T_{i^{\prime}j}\Big{(}\frac{T}{T_{u}T_{ij}}\Big{)}$

is a vertex in $\Gamma_{\mu}(I)$. Note that
$L_{M}(T^{\prime})=T_{i^{\prime}j}$ where
$L_{M}(T)>_{rlex}L_{M}(T^{\prime})=T_{M^{\prime}}$. ∎

######  Lemma 5.16.

Let $\mu$ be a multidegree such that $\Gamma_{\mu}(I)$ is nonempty. If the
fiber graph $\Gamma_{\mu}(I)$ has a vertex of type $M$, then any sink of
$\Gamma_{\mu}(I)$ is of type $M$. In particular, every sink of
$\Gamma_{\mu}(I)$ is of the form $T_{M^{\prime}}Z_{M^{\prime}}$ where
$Z_{M^{\prime}}$ is a sink of $\Gamma_{\frac{\mu}{M^{\prime}}}(I)$. Otherwise,
each sink of $\Gamma_{\mu}(I)$ is of the form $T_{N^{\prime}}Z_{N^{\prime}}$
where $Z_{N^{\prime}}$ is a sink of $\Gamma_{\frac{\mu}{N^{\prime}}}(I)$.

###### Proof.

Recall that whenever there is a directed edge from $T$ to $T^{\prime}$, we
have $T\succ_{rlex}T^{\prime}$. Additionally, observe that a sink of
$\Gamma_{\mu}(I)$ must be reached eventually because there is no infinite
descending chain of monomials with respect to a monomial order (see [16, Lemma
2.1.7]).

If there is a vertex $T$ of type $M$ in $\Gamma_{\mu}(I)$, then every vertex
of $\Gamma_{\mu}(I)$ must be of type $M$ by Lemma 5.8 (a). Thus any sink is of
type $M$. Furthermore, every sink of $\Gamma_{\mu}(I)$ must be divisible by
$T_{M^{\prime}}$. Otherwise, there is an edge directed from a sink by Lemma
5.15 (b), a contradiction. Hence each sink is of the form
$T_{M^{\prime}}Z_{M^{\prime}}$ where $Z_{M^{\prime}}$ is a vertex in
$\Gamma_{\frac{\mu}{M^{\prime}}}(I)$. Finally, the monomial $Z_{M^{\prime}}$
must be a sink in $\Gamma_{\frac{\mu}{M^{\prime}}}(I)$. Otherwise, there must
be an edge directed from $Z_{M^{\prime}}$ to another vertex in
$\Gamma_{\frac{\mu}{M^{\prime}}}(I)$, say $Z^{\prime}$. Then,
$Z_{M^{\prime}}\rightarrow_{\mathcal{G}_{1}}Z^{\prime}$ which in turn implies
that
$T_{M^{\prime}}Z_{M^{\prime}}\rightarrow_{\mathcal{G}_{1}}T_{M^{\prime}}Z^{\prime}$,
contradicting the fact that $T_{M^{\prime}}Z_{M^{\prime}}$ is a sink in
$\Gamma_{\mu}(I)$.

If $\Gamma_{\mu}(I)$ has no vertices of type $M$, then $L_{N}(T)$ exists for
each vertex $T$ of $\Gamma_{\mu}(I)$. Furthermore, every sink of
$\Gamma_{\mu}(I)$ must be divisible by $T_{N^{\prime}}$. Otherwise, it could
not be a sink due to Lemma 5.15 (a). Therefore, each sink is of the form
$T_{N^{\prime}}Z_{N^{\prime}}$ where $Z_{N^{\prime}}$ is a vertex in
$\Gamma_{\frac{\mu}{N^{\prime}}}(I)$. As in the previous paragraph, one can
show that $Z_{N^{\prime}}$ must be a sink in
$\Gamma_{\frac{\mu}{N^{\prime}}}(I)$. ∎

######  Corollary 5.17.

Every nonempty $\Gamma_{\mu}(I)$ with respect to $\mathcal{G}_{1}$ has a
unique sink.

###### Proof.

Let $\mu$ be a multidegree. We induct on the degree of $\mu$. If the degree of
$\mu$ is odd, then $\Gamma_{\mu}(I)$ must be empty. Thus, the degree of $\mu$
must be even. If the degree of $\mu$ is two, then $\Gamma_{\mu}(I)$ is either
an empty graph or it consists of exactly one vertex, thus it has a unique
sink. Induction step follows immediately from Lemma 5.16. ∎

We conclude this subsection by presenting an explicit description of the
relations between the indices of the variables that are involved in the
standard factorization of a sink. These relations will play a central role in
the proof of the main result in Section 6. Furthermore, they can be used to
count the number of different fiber graphs for a given multidegree.

######  Lemma 5.18.

Let $\Gamma_{\mu}(I)$ be a nonempty fiber graph and $T=T_{p_{1}}\cdots
T_{p_{r}}$ be its unique sink. Let $p_{k}=x_{i}x_{j}$ and
$p_{l}=x_{i^{\prime}}x_{j^{\prime}}$ where $k<l$.

  1. (a)

If $p_{k}$ and $p_{l}$ are both in ${\mathcal{B}}_{M}$ or ${\mathcal{B}}_{N}$,
then $i\leq i^{\prime}$ and $j\leq j^{\prime}$.

  2. (b)

If $p_{k}\in{\mathcal{B}}_{M}$ and $p_{l}\in{\mathcal{B}}_{N}$, then $i\leq
i^{\prime}$ or $c<i$. In particular, we must have $i=j$ or $c<j$ when
$i=i^{\prime}$.

###### Proof.

(a) The last inequality $j\leq j^{\prime}$ follows from our assumption that
$p_{k}\succeq_{rlex}p_{l}$. For the first inequality, by contradiction,
suppose $i>i^{\prime}$. Observe that we must have $j<j^{\prime}$ as
$p_{k}\succ_{rlex}p_{l}$. Before we proceed further recall that $i\leq j$ and
$i^{\prime}\leq j^{\prime}$. If $i^{\prime}=j^{\prime}$, then
$p_{l}\in{\mathcal{B}}_{M}$ as the equality of two indices is not possible for
monomials in ${\mathcal{B}}_{N}$. Then we obtain $i^{\prime}<i\leq j\leq
j^{\prime}$ which implies that $i^{\prime}<j^{\prime}$, a contradiction. Thus,
we must have $i\leq i^{\prime}$ in this case. For the remainder of the proof,
we may assume that $i^{\prime}<j^{\prime}$. Note that $x_{i^{\prime}}x_{j}$
and $x_{i}x_{j^{\prime}}$ are both in ${\mathcal{B}}_{M}$ if
$p_{k},p_{l}\in{\mathcal{B}}_{M}$. Similarly, they are both in
${\mathcal{B}}_{N}$ if $p_{k},p_{l}\in{\mathcal{B}}_{N}$. Furthermore,
$p_{k},p_{l}\succ_{rlex}x_{i}x_{j^{\prime}}$. Then
$T_{p_{k}}T_{p_{l}}-T_{i^{\prime}j}T_{ij^{\prime}}\in\mathcal{G}_{1}$, which
contradicts to $T$ being a sink. Therefore, $i\leq i^{\prime}$.

(b) By contradiction, suppose $i^{\prime}<i\leq c.$ It follows from the
definition of strongly stable ideals that
$x_{i^{\prime}}x_{j}\in{\mathcal{B}}_{M}$ as $x_{i}x_{j}\in{\mathcal{B}}_{M}$
where $i^{\prime}<i$. Furthermore, we have
$x_{i}x_{j^{\prime}}\in{\mathcal{B}}_{N}$ as $i\leq c$ and
$p_{k},p_{l}\succ_{rlex}x_{i}x_{j^{\prime}}$. Then
$T_{p_{k}}T_{p_{l}}-T_{i^{\prime}j}T_{ij^{\prime}}\in\mathcal{G}_{1}$, a
contradiction.

For the last statement, assume that $i=i^{\prime}$. Since
$p_{k}\succ_{rlex}p_{l},$ it is immediate that $j<j^{\prime}$. On the
contrary, suppose $i<j\leq c$. Then, we have
$x_{j}x_{j^{\prime}}\in{\mathcal{B}}_{N}$ as $j\leq c$. Moreover,
$x_{i}^{2}\in{\mathcal{B}}_{M}$ since $x_{i}x_{j}\in{\mathcal{B}}_{M}$ while
$i<j$. Note that $p_{k},p_{l}\succ_{rlex}x_{j}x_{j^{\prime}}$. Thus
$T_{p_{k}}T_{p_{l}}-T_{ii}T_{jj^{\prime}}\in\mathcal{G}_{1}$, a contradiction.
∎

###  5.2. The mixed revlex Gröbner basis

Our goal in this subsection is to describe a Gröbner basis of $T(I)$ with
respect to a new order called _mixed reverse lexicographic order_. We use
similar approaches to the ones employed in the previous subsection and depict
the differences in the structure of fiber graphs with respect to a new
collection $\mathcal{G}_{2}$.

######  Definition 5.19.

Consider all the monomials in ${\mathcal{B}}_{M}\cup{\mathcal{B}}_{N}.$ We say
$m\succ_{mrlex}n$ if one of the followings hold:

  1. (i)

$m\succ_{rlex}n$ and $m,n\in{\mathcal{B}}_{M}$,

  2. (ii)

$m\succ_{rlex}n$ where $m\in{\mathcal{B}}_{N}$ and $n\in{\mathcal{B}}_{M}$,

  3. (iii)

$m\succ_{rlex}n$ and $m,n\in{\mathcal{B}}_{N}$.

We say $T_{m}>_{mrlex}T_{n}$ if and only if $m\succ_{mrlex}n$. The $T$
variables are ordered as follows:

$\displaystyle
T_{1(b+1)}>_{mrlex}\dots>_{mrlex}T_{c(b+1)}>_{mrlex}\dots>_{mrlex}T_{1d}>_{mrlex}\dots>_{mrlex}T_{cd}>_{mrlex}$
$\displaystyle
T_{11}>_{mrlex}T_{12}>_{mrlex}T_{22}>_{mrlex}\dots>_{mrlex}T_{1b}>_{mrlex}T_{2b}>_{mrlex}\dots>_{mrlex}T_{ab}.$

We use the notation $\succ_{mrlex}$ to denote the reverse lexicographic order
induced by $>_{mrlex}$ variable order on $R$. We call this new term order the
_mixed reverse lexicographic order_.

Similar to 5.13, we set a standard factorization to express each monomial in
$R$ with respect to mixed reverse lexicographic order.

######  Notation 5.20.

Let $T$ be a monomial in $R$. We express $T$ as $T=(T_{p_{1}}\cdots
T_{p_{k}})(T_{p_{k+1}}\cdots T_{p_{k+l}})$ such that

  * •

$p_{i}\in{\mathcal{B}}_{N}$ for each $i\in\\{1,\ldots,k\\}$ and
$p_{k+j}\in{\mathcal{B}}_{M}$ for each $j\in\\{1,\ldots,l\\}$,

  * •

$p_{1}\succeq_{mrlex}\dots\succeq_{mrlex}p_{k}\succ_{mrlex}p_{k+1}\succeq_{mrlex}\dots\succeq_{mrlex}p_{k+l}$.

We still denote the latest variable factor of $T$ in ${\mathcal{B}}_{M}$ by
$L_{M}(T)$ and the latest variable factor of $T$ in ${\mathcal{B}}_{N}$ by
$L_{N}(T)$. Based on the above expression of $T$, one has $L_{N}(T)=T_{p_{k}}$
and $L_{M}(T)=T_{p_{k+l}}$ unless $k$ or $l$ is equal to zero.

Given a multidegree $\mu$ in $S,$ we use $M^{\prime}$ and $N^{\prime}$ to
denote the smallest elements in ${\mathcal{B}}_{M}$ and ${\mathcal{B}}_{N}$,
respectively, with respect to $\succ_{mrlex}$ order dividing $\mu$.

$x_{3}^{2}$$x_{2}x_{3}$$x_{2}^{2}$$x_{1}x_{3}$$x_{1}x_{2}$$x_{1}^{2}$$x_{2}x_{4}$$x_{1}x_{4}$$x_{1}x_{5}$$x_{2}x_{5}$$T_{4}$$T_{2}$$T_{3}$$T_{1}$$T_{0}$$T_{5}$$T_{7}$$T_{6}$$T_{8}$$T_{9}$$M$$N$
Figure 3.  $\succ_{rlex}$ order

$x_{3}^{2}$$x_{2}x_{3}$$x_{2}^{2}$$x_{1}x_{3}$$x_{1}x_{2}$$x_{1}^{2}$$x_{2}x_{4}$$x_{1}x_{4}$$x_{1}x_{5}$$x_{2}x_{5}$$T_{8}$$T_{6}$$T_{7}$$T_{5}$$T_{4}$$T_{9}$$T_{1}$$T_{0}$$T_{2}$$T_{3}$$M$$N$
Figure 4. $\succ_{mrlex}$ order

The two Hasse diagrams given above in Figure 4 and Figure 4 highlight the
differences between the $\succ_{rlex}$ order and $\succ_{mrlex}$ order.

The main result of this section describes a Gröbner basis of $T(I)$ with
respect to the mixed reverse lexicographic order, $\succ_{mrlex}$, on $R$.

######  Theorem 5.21.

Let
$\mathcal{G}_{2}=\\{\underline{T_{u}T_{v}}-T_{u^{\prime}}T_{v^{\prime}}:uv=u^{\prime}v^{\prime}\text{
and }u,v\succ_{mrlex}v^{\prime}\\}$. Then $\mathcal{G}_{2}$ is a Gröbner basis
of $T(I)$ with respect to the mixed reverse lexicographic order
$\succ_{mrlex}$.

###### Proof.

Analogous to the proof of Theorem 5.11, it suffices to show that, for each
multidegree $\mu$, the fiber graph $\Gamma_{\mu}(I)$ with respect to
$\mathcal{G}_{2}$ is empty or it has a unique sink by Corollary 5.4. As
discussed in the proof of Theorem 5.11, when $\mu$ has an odd degree, its
fiber graph is empty. It is possible that the fiber graph of $\mu$ is empty
when its degree is even. In these two cases, there is nothing to prove. The
remaining situation is proved in Corollary 5.24. ∎

The following lemmas are analogous versions of Lemma 5.15 and Lemma 5.16 in
the mixed reverse lexicographic order.

######  Lemma 5.22.

Let $\mu$ be a multidegree and $T$ be a vertex of $\Gamma_{\mu}(I)$.

  1. (a)

If $L_{M}(T)$ exists and $L_{M}(T)\neq T_{M^{\prime}}$ then there is an edge
from $T$ to to a vertex of $\Gamma_{\mu}(I)$, $T^{\prime}$, such that

$L_{M}(T)>_{mrlex}L_{M}(T^{\prime})\geq_{mrlex}T_{M^{\prime}}.$

  2. (b)

Suppose $L_{M}(T)$ does not exist. If $L_{N}(T)$ exists and $L_{N}(T)\neq
T_{N^{\prime}}$, then there is an edge from $T$ to $T^{\prime}$ such that

$L_{N}(T)>_{mrlex}L_{N}(T^{\prime})\geq_{mrlex}T_{N^{\prime}}.$

###### Proof.

The proof of this lemma is analogous to that of Lemma 5.15 with the following
difference. For the proof of part (b), one can find a monomial $u$ such that
$T_{u}L_{N}(T)$ divides $T$. This monomial $u$ has to be in
${\mathcal{B}}_{N}$ since $L_{M}(T)$ does not exists. Depending on our
conditions, we can create a monomial $u^{\prime}$ as in the proof of Lemma
5.15 and this monomial $u^{\prime}$ must be in ${\mathcal{B}}_{N}$. ∎

######  Lemma 5.23.

Let $\mu$ be a multidegree such that $\Gamma_{\mu}(I)$ is not empty. If
$\Gamma_{\mu}(I)$ has a vertex of type $N$, then any sink of $\Gamma_{\mu}(I)$
is of type $N$. In particular, each sink is of the form
$T_{N^{\prime}}Z_{N^{\prime}}$ where $Z_{N^{\prime}}$ is a sink of
$\Gamma_{\frac{\mu}{N^{\prime}}}(I)$. Otherwise, every sink of
$\Gamma_{\mu}(I)$ is of the form $T_{M^{\prime}}Z_{M^{\prime}}$ where
$Z_{M^{\prime}}$ is a sink of $\Gamma_{\frac{\mu}{M^{\prime}}}(I)$.

The proof of Lemma 5.23 is completely analogous to Lemma 5.16. Similar to
Corollary 5.17, we conclude the following.

######  Corollary 5.24.

Every nonempty $\Gamma_{\mu}(I)$ with respect to $\mathcal{G}_{2}$ has a
unique sink.

As in the previous subsection, we conclude the current subsection by
presenting an explicit description of the relations between the indices of the
variables that are involved in the standard factorization of a sink with
respect to mixed reverse lexicographic order.

######  Lemma 5.25.

Let $\Gamma_{\mu}(I)$ be a nonempty fiber graph and $T=T_{p_{1}}\cdots
T_{p_{r}}$ be its unique sink. Let $p_{k}=x_{i}x_{j}$ and
$p_{l}=x_{i^{\prime}}x_{j^{\prime}}$ where $k<l$.

  1. (a)

If $p_{k}$ and $p_{l}$ are both in ${\mathcal{B}}_{M}$ or ${\mathcal{B}}_{N}$,
then $i\leq i^{\prime}$ and $j\leq j^{\prime}$.

  2. (b)

If $p_{k}\in{\mathcal{B}}_{N}$ and $p_{l}\in{\mathcal{B}}_{M}$, then $i\leq
i^{\prime}$.

###### Proof.

(a) Notice that when $p_{k}$ and $p_{l}$ are both in ${\mathcal{B}}_{N}$ or
${\mathcal{B}}_{M}$, $\succ_{rlex}$ and $\succ_{mrlex}$ orders coincide. Thus
the proof is obtained from Lemma 5.18.

(b) Suppose $p_{k}\in{\mathcal{B}}_{N}$ and $p_{l}\in{\mathcal{B}}_{M}$. If
$i>i^{\prime}$, then $x_{i}x_{j^{\prime}}\in{\mathcal{B}}_{M}$ and
$x_{i^{\prime}}x_{j}\in{\mathcal{B}}_{N}$ which can be easily verified by
checking the indices. Note that $p_{k},p_{l}\succ_{mrlex}x_{i}x_{j^{\prime}}$.
Then $T_{p_{k}}T_{p_{l}}-T_{i^{\prime}j}T_{ij^{\prime}}\in\mathcal{G}_{2}$
implying that $T$ is not a sink, a contradiction. ∎

##  6\. Gröbner Bases of the Multi-Rees Algebras of Strongly Stable Ideals

In this section, we provide a quadratic Gröbner basis for the defining ideal
of the multi-Rees algebra $\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$ when
$r=2$ and each ideal is a strongly stable ideal with two quadric Borel
generators. As a result of Theorem 3.2, it suffices to find a quadratic
Gröbner basis for the toric ideal associated to its special fiber. Similar to
Section 5, we follow an approach that involves fiber graphs of multidegrees
with respect to a collection of marked binomials.

We begin by setting our notation and introducing our objects. Throughout this
section, we set $I_{1}={\mathcal{B}}(M_{1},N_{1})$ and
$I_{2}={\mathcal{B}}(M_{2},N_{2})$ where $M_{i}=x_{a_{i}}x_{b_{i}}$ and
$N_{i}=x_{c_{i}}x_{d_{i}}$ for $i=1,2$. Note that $c_{i}<a_{i}\leq
b_{i}<d_{i}$ for each $i$. Additionally, we assume $d_{1}\leq d_{2}$. As in
the previous section, we denote the list of the minimal monomial generators of
${\mathcal{B}}(M_{i})$ by ${\mathcal{B}}_{M_{i}}$ and the list of the minimal
monomial generators of ${\mathcal{B}}(N_{i})\setminus{\mathcal{B}}(M_{i})$ by
${\mathcal{B}}_{N_{i}}$ for each $i$.

Recall that the defining ideal of $\mathcal{R}(I_{1}\oplus I_{2})$ is the
kernel of the following $S$-algebra homomorphism given by

$\begin{array}[]{rcl}\varphi:S[T_{u},Z_{v}:u\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}},v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}]&\rightarrow&S[t,z]=\mathbb{K}[x_{1},\dots,x_{n},t,z]\\\
T_{u}&\rightarrow&ut\\\ Z_{v}&\rightarrow&vz,\end{array}$

and extended algebraically. The defining ideal of the special fiber
$\mathcal{F}(I_{1}\oplus I_{2})$ is the kernel of $\varphi^{\prime}$, the
induced $\mathbb{K}$-algebra homomorphism of $\varphi$, and it is denoted by
$T(I_{1}\oplus I_{2})$.

Note that the ring
$R=\mathbb{K}[T_{u},Z_{v}:u\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}},v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}]$
inherits the multigrading from $S[t,z]$ where $R=\bigoplus_{\mu}R_{\mu}$ with
$\mu$ ranging over monomials of $S[t,z]$ and $R_{\mu}$ being the
${\mathbb{K}}$-vector space described as

$R_{\mu}=\operatorname{span}_{\mathbb{K}}\\{V=\prod
T_{u}^{a_{u}}Z_{v}^{b_{v}}\in R~{}:~{}\varphi(V)=\mu\\}.$

In what follows, we introduce a collection of marked binomials $\mathcal{G}$
to study fiber graphs of multidegrees with respect to $\mathcal{G}$. In
particular, we prove that $\mathcal{G}$ is a Gröbner basis of $T(I_{1}\oplus
I_{2})$ with respect to the following monomial order on $R$.

######  Definition 6.1.

We define the _head and tail order_ , denoted by $\succ_{ht}$, on $R$ as
follows. We first introduce the following variable order, denoted by $>_{ht}$,
on $R$.

  * •

If $u,v\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}}$, and $u\succ_{rlex}v$,
then $T_{u}>_{ht}T_{v}$.

  * •

If $u\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}}$ and
$v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}},$ then $T_{u}>_{ht}Z_{v}$.

  * •

If $u,v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}$, and
$u\succ_{mrlex}v$, then $Z_{u}>_{ht}Z_{v}$.

In particular, we order the variables of $R$ in the following way.

$\displaystyle
T_{11}>_{ht}T_{12}>_{ht}T_{22}>_{ht}\dots>_{ht}T_{1b_{1}}>_{ht}T_{2b_{1}}>_{ht}\dots>T_{a_{1}b_{1}}>_{ht}$
$\displaystyle
T_{1(b_{1}+1)}>_{ht}\dots>_{ht}T_{c_{1}(b_{1}+1)}>_{ht}\dots>_{ht}T_{1d_{1}}>_{ht}\dots>_{ht}T_{c_{1}d_{1}}>_{ht}$
$\displaystyle
Z_{1(b_{2}+1)}>_{ht}\dots>_{ht}Z_{c_{2}(b_{2}+1)}>_{ht}\dots>_{ht}Z_{1d_{2}}>_{ht}\dots>_{ht}Z_{c_{2}d_{2}}>_{ht}$
$\displaystyle
Z_{11}>_{ht}Z_{12}>_{ht}Z_{22}>_{ht}\dots>_{ht}Z_{1b_{2}}>_{ht}Z_{2b_{2}}>_{ht}\dots>_{ht}Z_{a_{2}b_{2}}.$

Finally, the head and tail order, $\succ_{ht}$, is the reverse lexicographical
order on $R$ induced by $>_{ht}$.

######  Remark 6.2.

The head and tail order, $>_{ht}$, is actually the direct product order of
$>_{rlex}$ and $>_{mrlex}$ on
$\mathbb{K}[T_{u}:u\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}}]$ and
$\mathbb{K}[Z_{v}:v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}]$ defined
in [13, Section 2.1.2].

######  Notation 6.3.

Let $\mathcal{G}$ be the union of the following collection of marked binomials
where

$\displaystyle\mathcal{G}_{1}$
$\displaystyle=\\{\underline{T_{u}T_{v}}-T_{u^{\prime}}T_{v^{\prime}}:uv=u^{\prime}v^{\prime}\text{
and }u,v\succ_{rlex}v^{\prime}\\},$ $\displaystyle\mathcal{G}_{2}$
$\displaystyle=\\{\underline{Z_{u}Z_{v}}-Z_{u^{\prime}}Z_{v^{\prime}}:uv=u^{\prime}v^{\prime}\text{
and }u,v\succ_{mrlex}v^{\prime}\\},$ $\displaystyle\mathcal{G}_{3}$
$\displaystyle=\\{\underline{T_{u}Z_{v}}-T_{u^{\prime}}Z_{v^{\prime}}:uv=u^{\prime}v^{\prime}\text{
and }v\succ_{mrlex}v^{\prime}\\}.$

Recall that $\mathcal{G}_{1}$ is the Gröbner basis of $I_{1}$ with respect to
term order $\succ_{rlex}$ on
$\mathbb{K}[T_{u}:u\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}}]$ and
$\mathcal{G}_{2}$ is the Gröbner basis of $I_{2}$ with respect to the mixed
reverse lexicographical order, $\succ_{mrlex}$, on
$\mathbb{K}[Z_{v}:v\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}]$ by
Theorem 5.11 and Theorem 5.21. Marked terms of each $\mathcal{G}_{i}$ for
$i=1,2,3$ are leading terms with respect to the head and tail order,
$\succ_{ht}$, on $R$.

######  Remark 6.4.

One can also view $\mathcal{G}$ as the collection of $2\times 2$ minors of the
symmetric ladders where $\mathcal{G}_{i}$ corresponds to the collection of
$2\times 2$ minors of symmetric ladders associated to $I_{i}$ for $i=1,2$.
Since $\mathcal{G}_{3}$ is expressed in terms of $T$ and $Z$ variables, we can
view it as the collection of $2\times 2$ minors of “stacked” symmetric
ladders. Here we use the term stacked to imply that the minors can be read
from two different levels of symmetric ladders where the $i$th level is
associated to $I_{i}$ for $i=1,2$ such that the two levels are aligned at
their lower left corners based on Figure 1.

As we have seen in 5.13 and 5.20, it is useful to set a standard factorization
to express each monomial in $R$ with respect to the head and tail order.

######  Definition 6.5.

Let $V$ be a monomial in $R$. We express $V$ as $V=(T_{m_{1}}\cdots
T_{m_{p}})(Z_{n_{1}}\cdots Z_{n_{q}})$ such that
$T_{m_{1}}\geq_{ht}\dots\geq_{ht}T_{m_{p}}>_{ht}Z_{n_{1}}\geq_{ht}\dots\geq_{ht}Z_{n_{q}}$.
Note that $m_{i}\in\mathcal{B}_{M_{1}}\cup{\mathcal{B}}_{N_{1}}$ and
$n_{j}\in\mathcal{B}_{M_{2}}\cup{\mathcal{B}}_{N_{2}}$ for each $i\in[p]$ and
$j\in[q]$.

One can define fiber graphs of $I_{1}\oplus I_{2}$ at multidegrees as in
Definition 5.2.

######  Definition 6.6.

Given a multidegree $\mu\in S[t,z]$, we denote the fiber graph of $I_{1}\oplus
I_{2}$ at $\mu$ with respect to $\mathcal{G}$ by $\Gamma_{\mu}(I_{1},I_{2})$.
The vertices of the fiber graph are the monomials of $R_{\mu}$. Given two
vertices $V$ and $V^{\prime}$, there is a directed edge from $V$ to
$V^{\prime}$ whenever $V\rightarrow_{\mathcal{G}}V^{\prime}$.

######  Remark 6.7.

Let $V=(T_{m_{1}}\cdots T_{m_{p}})(Z_{n_{1}}\cdots Z_{n_{q}})$ be a vertex in
$\Gamma_{\mu}(I_{1},I_{2})$. If $p=0$, then $V$ is a vertex in
$\displaystyle\Gamma_{\mu/t^{p}}(I_{2})$ with respect to $\mathcal{G}_{2}$. If
$q=0$, then $V$ is a vertex in $\Gamma_{\mu/z^{q}}(I_{1})$ with respect to
$\mathcal{G}_{1}$.

$T_{25}T_{44}Z_{36}$$T_{26}T_{44}Z_{35}$$T_{35}T_{44}Z_{26}$$T_{24}T_{45}Z_{36}$$T_{26}T_{35}Z_{44}$$T_{34}T_{45}Z_{26}$$T_{26}T_{45}Z_{34}$sink$T_{26}Z_{35}\rightarrow_{\mathcal{G}}T_{25}Z_{36}$$T_{44}Z_{35}\rightarrow_{\mathcal{G}}T_{35}Z_{44}$$T_{44}T_{25}\rightarrow_{\mathcal{G}}T_{24}T_{45}$$T_{35}Z_{26}\rightarrow_{\mathcal{G}}T_{25}Z_{36}$$T_{44}T_{35}\rightarrow_{\mathcal{G}}T_{34}T_{45}$$T_{24}Z_{36}\rightarrow_{\mathcal{G}}T_{26}Z_{34}$$T_{34}Z_{26}\rightarrow_{\mathcal{G}}T_{26}Z_{34}$$T_{45}Z_{34}\rightarrow_{\mathcal{G}}T_{35}Z_{44}$
Figure 5. The fiber graph of $I_{1}\oplus I_{2}$ at the multidegree
$x_{2}x_{3}x_{4}^{2}x_{5}x_{6}t^{2}z$

######  Example 6.8.

Let
$I_{1}={\mathcal{B}}(x_{4}x_{5},x_{2}x_{6}),I_{2}={\mathcal{B}}(x_{4}^{2},x_{3}x_{6})$
and $\mu=x_{2}x_{3}x_{4}^{2}x_{5}x_{6}t^{2}z\in S[t,z].$ Note that all
elements of $R_{\mu}$ are the vertices of $\Gamma_{\mu}(I_{1},I_{2})$. In
Figure 5, we label each edge of the fiber graph with the related reduction
producing it.

We are now ready to state the main result of this section.

######  Theorem 6.9.

The collection $\mathcal{G}$ is a Gröbner basis of $T(I_{1}\oplus I_{2})$ with
respect to the head and tail order, $\succ_{ht}$, on $R$.

###### Proof.

Analogous to the proofs of Theorem 5.11 and Theorem 5.21, it suffices to show
that for a multidegree $\mu\in S[t,z]$, the fiber graph
$\Gamma_{\mu}(I_{1},I_{2})$ is empty or has a unique sink by Corollary 5.4 and
Remark 5.5. If the fiber graph is empty, we are done. If not, this statement
is proved in Proposition 6.17. ∎

In comparison to fiber graphs of $I_{1}$ and $I_{2}$, studying sinks of fiber
graphs of $I_{1}\oplus I_{2}$ is much more complex because one needs to
consider reductions associated to both ideals via the collection
$\mathcal{G}_{3}$ given in terms of $T$ and $Z$ variables. In what follows, we
carefully set our notation and establish several auxiliary lemmas to utilize
in our proof of Proposition 6.17.

######  Notation 6.10.

Let $\mu$ be a mutlidegree such that $\Gamma_{\mu}(I_{1},I_{2})$ is nonempty
and $p,q\geq 1$. Let $V$ and $V^{\prime}$ be two vertices of
$\Gamma_{\mu}(I_{1},I_{2})$ such that

$\displaystyle V=\underbrace{(T_{m_{1}}\cdots
T_{m_{p}})}_{V_{1}}\underbrace{(Z_{n_{1}}\cdots Z_{n_{q}})}_{V_{2}}\text{ and
}V^{\prime}=\underbrace{(T_{m^{\prime}_{1}}\cdots
T_{m^{\prime}_{p}})}_{V^{\prime}_{1}}\underbrace{(Z_{n^{\prime}_{1}}\cdots
Z_{n^{\prime}_{q}})}_{V^{\prime}_{2}}.$

Notice that $\mu=\mu_{1}\mu_{2}=\mu^{\prime}_{1}\mu^{\prime}_{2}$ where
$\mu_{1}=m_{1}\cdots m_{p}$, $\mu_{2}=n_{1}\cdots n_{q}$,
$\mu^{\prime}_{1}=m^{\prime}_{1}\cdots m^{\prime}_{p}$, and
$\mu^{\prime}_{2}=n^{\prime}_{1}\cdots n^{\prime}_{q}$. We use
$\textbf{i}_{k},\textbf{j}_{k},\textbf{i}^{\prime}_{k},\textbf{j}^{\prime}_{k}$
for the indices associated to $m_{k}$ and $m^{\prime}_{k}$, and
$i_{l},j_{l},i^{\prime}_{l},j^{\prime}_{l}$ for the indices associated to
$n_{l}$ and $n^{\prime}_{l}$. These two lists of indices should not be
confused. More explicitly, for each $k\in[p]$ and $l\in[q]$, we set

$\displaystyle m_{k}=x_{\textbf{i}_{k}}x_{\textbf{j}_{k}}\text{ and
}n_{l}=x_{i_{l}}x_{j_{l}}$ $\displaystyle
m^{\prime}_{k}=x_{\textbf{i}^{\prime}_{k}}x_{\textbf{j}^{\prime}_{k}}\text{
and }n^{\prime}_{l}=x_{i^{\prime}_{l}}x_{j^{\prime}_{l}}.$

We further introduce the following lists in which an element can appear more
than once.

$\displaystyle\mathcal{V}=\mathcal{V}_{1}\cup\mathcal{V}_{2}\text{ where }$
$\displaystyle\mathcal{V}_{1}=\\{\textbf{i}_{k},\textbf{j}_{k}:k\in[p]\\}\text{
and }\mathcal{V}_{2}=\\{i_{l},j_{l}:l\in[q]\\}$
$\displaystyle\mathcal{V}^{\prime}=\mathcal{V}^{\prime}_{1}\cup\mathcal{V}^{\prime}_{2}\text{
where }$
$\displaystyle\mathcal{V}^{\prime}_{1}=\\{\textbf{i}^{\prime}_{k},\textbf{j}^{\prime}_{k}:k\in[p]\\}\text{
and }\mathcal{V}^{\prime}_{2}=\\{i^{\prime}_{l},j^{\prime}_{l}:l\in[q]\\}$

Note that the two lists $\mathcal{V}$ and $\mathcal{V}^{\prime}$ coincide as
$\mu_{1}\mu_{2}=\mu=\mu^{\prime}_{1}\mu^{\prime}_{2}$. In addition, we denote
the first and second half of the indices associated to $n_{1},\ldots,n_{q}$ by

$\mathcal{I}=\\{i_{1},\ldots,i_{q}\\}\text{ and
}\mathcal{J}=\\{j_{1},\ldots,j_{q}\\}.$

It is clear that $\mathcal{V}_{2}=\mathcal{I}\cup\mathcal{J}$. For the
remainder of this section, we adopt 6.10.

######  Remark 6.11.

If $V$ is a sink of $\Gamma_{\mu}(I_{1},I_{2})$ with respect to $\mathcal{G}$,
then $V_{1}$ must be the unique sink of $\Gamma_{\mu_{1}}(I_{1})$ with respect
to $\mathcal{G}_{1}$ and $V_{2}$ must be the unique sink of
$\Gamma_{\mu_{2}}(I_{2})$ with respect to $\mathcal{G}_{2}$.

Given a vertex $V$ as in 6.10, its associated monomials $m_{1},\ldots m_{p}$
belong to ${\mathcal{B}}_{M_{1}}$ region or ${\mathcal{B}}_{N_{1}}$ region and
monomials $n_{1},\ldots,n_{q}$ belong ${\mathcal{B}}_{M_{2}}$ region or
${\mathcal{B}}_{N_{2}}$ region. If $V$ is a sink, by Lemma 5.18 and Lemma
5.25, we can deduce several relations between the elements of
$\mathcal{V}_{i}$ based on regions of the associated monomials of $V_{i}$ for
each $i=1,2$. In the following lemmas, we describe several relations between
the elements $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ for a sink $V$ by taking
into account the regions of certain monomials.

######  Lemma 6.12.

Suppose $V$ is a sink of $\Gamma_{\mu}(I_{1},I_{2})$ such that
$n_{q}\in{\mathcal{B}}_{N_{2}}$. Then

  1. (a)

each of $n_{1},\ldots,n_{q}$ is contained in ${\mathcal{B}}_{N_{2}}$ and

$\displaystyle i_{1}\leq i_{2}\leq\cdots\leq i_{q}\leq c_{2}<b_{2}<j_{1}\leq
j_{2}\leq\cdots\leq j_{q}\leq d_{2},$ (6.1)

  2. (b)

each _$\textbf{j}_{k}$_ is at most $j_{1}$ for $k\in[p]$,

  3. (c)

each _$\textbf{i}_{k}$_ is either at most $i_{1}$ or greater than $c_{2}$ for
$k\in[p]$.

###### Proof.

(a) Recall that $Z_{n_{q}}$ is the latest variable among
$Z_{n_{1}},\ldots,Z_{n_{q}}$ with respect to the head and tail order. Then,
each $n_{l}$ is contained in ${\mathcal{B}}_{N_{2}}$ for $l\in[q]$ and we have
(6.1) from Lemma 5.25.

(b) Note that $\textbf{j}_{k}\leq\textbf{j}_{p}$ for each $k\in[p]$. Thus, it
suffices to show that $\textbf{j}_{p}\leq j_{1}$. On the contrary, suppose
$j_{1}<\textbf{j}_{p}$. Then, $x_{\textbf{i}_{p}}x_{j_{1}}$ is contained in
${\mathcal{B}}(M_{1},N_{1})$ because
$m_{p}=x_{\textbf{i}_{p}}x_{\textbf{j}_{p}}\in{\mathcal{B}}(M_{1},N_{1})$
while $j_{1}<\textbf{j}_{p}$. In addition, observe that
$x_{i_{1}}x_{\textbf{j}_{p}}$ is in ${\mathcal{B}}_{N_{2}}$ by (6.1) and our
assumption $d_{1}\leq d_{2}$. Note that
$n_{1}\succ_{mrlex}x_{i_{1}}x_{\textbf{j}_{p}}$. Hence, the binomial
$T_{m_{p}}Z_{n_{1}}-T_{\textbf{i}_{p}j_{1}}Z_{i_{1}\textbf{j}_{p}}$ belongs to
$\mathcal{G}$, which contradicts to our assumption on $V$ being a sink.
Therefore, index $\textbf{j}_{p}$ is at most $j_{1}$.

(c) Suppose $i_{1}<\textbf{i}_{k}\leq c_{2}$ for some $k\in[p]$. Since
$m_{k}=x_{\textbf{i}_{k}}x_{\textbf{j}_{k}}\in{\mathcal{B}}(M_{1},N_{1})$ and
$i_{1}<\textbf{i}_{k}$, we must have
$x_{i_{1}}x_{\textbf{j}_{k}}\in{\mathcal{B}}(M_{1},N_{1})$. Furthermore, the
monomial $x_{\textbf{i}_{k}}x_{j_{1}}$ is contained in ${\mathcal{B}}_{N_{2}}$
and $n_{1}\succ_{mrlex}x_{\textbf{i}_{k}}x_{j_{1}}$. Thus,
$T_{m_{k}}Z_{n_{1}}-T_{i_{1}\textbf{j}_{k}}Z_{\textbf{i}_{k}j_{1}}\in\mathcal{G}$,
a contradiction. ∎

Under certain assumptions, we can relate $V$ and $V^{\prime}$ through the
lists given in 6.10 and counting arguments. In the following set of
observations, we provide complementary facts which will be quite useful in our
study of sinks of $\Gamma_{\mu}(I_{1},I_{2})$.

######  Observation 6.13.

Suppose $n_{q}\in{\mathcal{B}}_{N_{2}}$ and
$n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$. One can decompose
$\mathcal{V}_{2}^{\prime}$ into the following disjoint lists.

$\mathcal{N}=\\{v\in\mathcal{V}_{2}^{\prime}:v>b_{2}\\}\text{ and
}\mathcal{N}^{c}=\\{v\in\mathcal{V}_{2}^{\prime}:v\leq b_{2}\\}$

  1. (i)

Observe that
$\mathcal{N}\subseteq\\{j^{\prime}_{1},\ldots,j^{\prime}_{q-1}\\}\text{ and
}\\{i^{\prime}_{1},\ldots,i^{\prime}_{q},j^{\prime}_{q}\\}\subseteq\mathcal{N}^{c}$
since $j^{\prime}_{q}\leq b_{2}$ and $i^{\prime}_{l}\leq a_{2}$ for each
$l\in[q]$. Then $\mathcal{N}$ has at most $q-1$ elements and $\mathcal{N}^{c}$
has at least $q+1$ elements.

  2. (ii)

Let $j_{r}$ be the smallest element in $\mathcal{J}$ such that $j_{r}$ is not
contained in $\mathcal{V}^{\prime}_{2}$. Existence of such an element is
guaranteed by observing that $\mathcal{V}^{\prime}_{2}$ can contain at most
$q-1$ of the elements from $\mathcal{J}$. Then, there exists $h\in[p]$ such
that $x_{j_{r}}$ divides $m^{\prime}_{h}$. As a result, we have $j_{1}\leq
j_{r}\leq d_{1}$ where the first inequality is due to (6.1).

  3. (iii)

Since at most $q$ many of the elements of $\mathcal{N}^{c}$ can be contained
in $\mathcal{I}$ and none of them is in $\mathcal{J}$, there exists $f\in[p]$
such that $m_{f}$ is divisible by $x_{e}$ for $e\in\mathcal{N}^{c}$, i.e.,
$e\leq b_{2}$. For the remainder of this section, we shall denote
$m_{f}=x_{e}x_{e^{\prime}}$ by avoiding our standard notation as it is
possible to have $e^{\prime}<e$.

In the existence of another sink vertex, we can provide further relations
between the elements of $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ with the
additional region assumption. These relations will be used in our proof of
Proposition 6.17.

######  Lemma 6.14.

Suppose $V$ and $V^{\prime}$ are sinks of $\Gamma_{\mu}(I_{1},I_{2})$ such
that $n_{q}\in{\mathcal{B}}_{N_{2}}$ and
$n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$.

  1. (a)

None of the monomials among $m_{1},\ldots,m_{p}$ is contained in
${\mathcal{B}}(x_{c_{1}}x_{b_{2}})$.

  2. (b)

If the index _$\textbf{j}_{1}$_ is greater than $b_{2}$, then
_$\textbf{j}_{1}$_ is at most $a_{1}$.

  3. (c)

The index $j_{1}$ is greater than $b_{1}$.

###### Proof.

(a) On the contrary, suppose there exists $k\in[p]$ such that
$m_{k}=x_{\textbf{i}_{k}}x_{\textbf{j}_{k}}\in{\mathcal{B}}(x_{c_{1}}x_{b_{2}})$.
Note that the monomial $x_{c_{1}}x_{b_{2}}$ is not written according to our
standard notation as it is possible to have $b_{2}<c_{1}$ while we set
$\textbf{i}_{k}\leq c_{1}$ and $\textbf{j}_{k}\leq b_{2}$. Consider the index
$r$ from 6.13 (ii) such that $n_{r}=x_{i_{r}}x_{j_{r}}$ where $i_{r}\leq
c_{2}$ and $b_{2}<j_{r}\leq d_{1}$. Then
$x_{i_{r}}x_{\textbf{j}_{k}}\in{\mathcal{B}}_{M_{2}}$ and
$x_{\textbf{i}_{k}}x_{j_{r}}\in{\mathcal{B}}(M_{1},N_{1})$ which can be
verified by checking the indexes. Note that
$n_{r}\succ_{mrlex}x_{i_{r}}x_{\textbf{j}_{k}}$. Thus, the binomial
$T_{m_{k}}Z_{n_{r}}-T_{\textbf{i}_{k}j_{r}}Z_{i_{r}\textbf{j}_{k}}$ is
contained in $\mathcal{G}$, a contradiction.

(b) Suppose $a_{1}<\textbf{j}_{1}$ and let
$\mathcal{J}^{\prime}=\\{\textbf{j}_{1},\ldots,\textbf{j}_{p}\\}$. Observe
that each element of $\mathcal{J}\cup\mathcal{J}^{\prime}\subset\mathcal{V}$
is greater than $a_{1}$ and $b_{2}$ by Lemma 6.12 and (6.1). Thus
$\mathcal{V}$ has at least $p+q$ such elements. Recall from 6.13 (i) that
$\mathcal{N}$ is the list of all such elements in $\mathcal{V}_{2}^{\prime}$
and it has at most $q-1$ elements. Since
$\mathcal{V}=\mathcal{V}^{\prime}=\mathcal{V}^{\prime}_{1}\cup\mathcal{V}^{\prime}_{2}$,
at least $p+1$ elements of $\mathcal{J}\cup\mathcal{J}^{\prime}$ must be
contained in $\mathcal{V}^{\prime}_{1}$. It follows from the pigeonhole
principle that there exists $k\in[p]$ such that
$m^{\prime}_{k}=x_{\textbf{i}^{\prime}_{k}}x_{\textbf{j}^{\prime}_{k}}$ where
$\textbf{i}^{\prime}_{k},\textbf{j}^{\prime}_{k}\in\mathcal{J}\cup\mathcal{J}^{\prime}$.
Then $a_{1}<\textbf{i}^{\prime}_{k}\leq\textbf{j}^{\prime}_{k}$ which implies
that $m^{\prime}_{k}\notin{\mathcal{B}}(M_{1},N_{1})$, a contradiction.

(c) On the contrary, suppose $j_{1}\leq b_{1}$. We first claim that
$\textbf{j}_{k}>b_{2}$ for all $k\in[p]$. Otherwise, suppose there exists
$k\in[p]$ such that $\textbf{j}_{k}\leq b_{2}<j_{1}$. Then
$x_{i_{1}}x_{\textbf{j}_{k}}\in{\mathcal{B}}_{M_{2}}$ as
$n_{1}=x_{i_{1}}x_{j_{1}}\in{\mathcal{B}}_{N_{2}}$. Note that
$x_{i_{1}}x_{\textbf{j}_{k}}$ comes later than $n_{q}$ with respect to
$\succ_{mrlex}$. Furthermore, we have
$x_{\textbf{i}_{k}}x_{j_{1}}\in{\mathcal{B}}_{M_{1}}$ since $j_{1}\leq b_{1}$.
Thus,
$T_{m_{k}}Z_{n_{1}}-T_{\textbf{i}_{k}j_{1}}Z_{i_{1}\textbf{j}_{k}}\in\mathcal{G}$,
a contradiction. Thus, the claim holds.

It follows from the claim that the index $e$ from 6.13 (iii) must belong to
the list $\\{\textbf{i}_{1},\ldots,\textbf{i}_{p}\\}$ and, as a result, we
have $\textbf{i}_{1}\leq b_{2}<j_{1}$. In addition, $\textbf{j}_{1}\leq a_{1}$
by part (b) of this lemma. Then, the monomial $x_{i_{1}}x_{\textbf{i}_{1}}$ is
in ${\mathcal{B}}_{M_{2}}$ because $i_{1}\leq c_{2}$ and $\textbf{i}_{1}\leq
b_{2}$; moreover, $x_{\textbf{j}_{1}}x_{j_{1}}\in{\mathcal{B}}_{M_{1}}$ as
$\textbf{j}_{1}\leq a_{1}$ and $j_{1}\leq b_{1}$. Thus, the binomial
$T_{m_{1}}Z_{n_{1}}-T_{\textbf{j}_{1}j_{1}}Z_{i_{1}\textbf{i}_{1}}$ belongs to
$\mathcal{G}$, a contradiction. ∎

######  Remark 6.15.

Suppose $V$ and $V^{\prime}$ are sinks of $\Gamma_{\mu}(I_{1},I_{2})$ such
that $n_{q}\in{\mathcal{B}}_{N_{2}}$ and
$n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$. In the light of Lemma 6.14, we can
better understand the indices of the monomials from 6.13 (ii) and (iii).

  1. (a)

It follows from Lemma 6.14 (a) that $c_{1}<e^{\prime}$ for the monomial
$m_{f}=x_{e}x_{e^{\prime}}$ from 6.13 (iii). In addition, if
$b_{2}<\textbf{j}_{1}$, we have $e^{\prime}=\textbf{j}_{f}$ and
$e=\textbf{i}_{f}\leq b_{2}$.

  2. (b)

Recall from 6.13 (ii) that $j_{r}$ belongs to
$\\{\textbf{i}^{\prime}_{h},\textbf{j}^{\prime}_{h}\\}$. Note that
$\textbf{i}^{\prime}_{h}$ can not be greater than $c_{1}$. If
$c_{1}<\textbf{i}^{\prime}_{h}$, then the monomial $m^{\prime}_{h}$ is
contained in ${\mathcal{B}}_{M_{1}}\setminus{\mathcal{B}}_{N_{1}}$ such that
$\textbf{i}^{\prime}_{h}\leq\textbf{j}^{\prime}_{h}\leq b_{1}$. Since
$b_{1}<j_{1}\leq j_{r}$ by Lemma 6.14 (c) and (6.1), we conclude that
$j_{r}\notin\\{\textbf{i}^{\prime}_{h},\textbf{j}^{\prime}_{h}\\}$, a
contradiction. Therefore, $\textbf{i}^{\prime}_{h}\leq c_{1}$ and
$j_{r}=\textbf{j}^{\prime}_{h}$.

Before the proof of Proposition 6.17, we present our last auxiliary lemma.

######  Lemma 6.16.

Suppose $V$ and $V^{\prime}$ are sinks of $\Gamma_{\mu}(I_{1},I_{2})$ with
$n_{q}\in{\mathcal{B}}_{N_{2}}$ and $n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$.
Then _$\textbf{i}_{1}$_ is at most $c_{1}$ or one of the monomials from
$m_{1},\ldots,m_{p}$ is in ${\mathcal{B}}_{N_{1}}$.

###### Proof.

On the contrary, suppose $c_{1}<\textbf{i}_{1}$ and
$m_{k}\in{\mathcal{B}}_{M_{1}}$ for each $k\in[p]$. It follows from Lemma 5.18
(a) that

$\displaystyle
c_{1}<\textbf{i}_{1}\leq\textbf{i}_{2}\leq\cdots\leq\textbf{i}_{p}\text{ and
}\textbf{j}_{1}\leq\textbf{j}_{2}\leq\cdots\leq\textbf{j}_{p}\leq b_{1}\text{
where }\textbf{i}_{k}\leq\textbf{j}_{k}\text{ for }k\in[p].$ (6.2)

In order to get a contradiction, our first step is to show that
$n_{1}\in{\mathcal{B}}_{N_{1}}$. For this purpose, recall that $e\leq b_{2}$
where $m_{f}=x_{e}x_{e^{\prime}}$ is the monomial from 6.13 (iii). By using
the chain of inequalities (6.2), we conclude that $c_{1}<b_{2}$ since
$e\in\\{\textbf{i}_{f},\textbf{j}_{f}\\}$. Additionally, it is clear that the
multidegree $\mu=\mu_{1}\mu_{2}$ is divisible by $x_{\textbf{i}^{\prime}_{h}}$
where $m^{\prime}_{h}=x_{\textbf{i}^{\prime}_{h}}x_{\textbf{j}^{\prime}_{h}}$
such that $\textbf{i}^{\prime}_{h}\leq c_{1}$ and
$\textbf{j}^{\prime}_{h}=j_{r}$ based on Remark 6.15 (b). Then
$\textbf{i}^{\prime}_{h}$ must be in $\mathcal{V}_{1}\cup\mathcal{V}_{2}$
where $\mathcal{V}_{2}=\mathcal{I}\cup\mathcal{J}$. Note that
$\textbf{i}^{\prime}_{h}$ is not contained in
$\mathcal{V}_{1}=\\{\textbf{i}_{k},\textbf{j}_{k}:k\in[p]\\}$ as each element
of this list is greater than $c_{1}$ by (6.2) whereas
$\textbf{i}^{\prime}_{h}\leq c_{1}<b_{2}$; furthermore, it can not be in
$\mathcal{J}$ as all elements of $\mathcal{J}$ are greater than $b_{2}$. Then
the only option is $\textbf{i}^{\prime}_{h}\in\mathcal{I}$ which implies that
$i_{1}\leq\textbf{i}^{\prime}_{h}\leq c_{1}$. Thus,
$n_{1}=x_{i_{1}}x_{j_{1}}\in{\mathcal{B}}_{N_{1}}$ by Lemma 6.14 (c) and 6.13
(ii).

Our next step is to show that $b_{2}<\textbf{j}_{1}\leq a_{1}.$ It suffices to
show that $b_{2}<\textbf{j}_{1}$ by Lemma 6.14 (b). First observe that there
exist no $k\in[p]$ such that $m_{k}\in{\mathcal{B}}_{M_{2}}$. Otherwise,
$T_{m_{k}}Z_{n_{1}}-T_{n_{1}}Z_{m_{k}}\in\mathcal{G}$, a contradiction. Note
that $\mathcal{V}^{\prime}_{2}$ has at least $q$ many elements which are at
most $a_{2}$ while $\mathcal{V}_{2}\setminus\\{\textbf{i}^{\prime}_{h}\\}$ has
$q-1$ many such elements. Since $\mathcal{V}=\mathcal{V}^{\prime}$, there
exists $\textbf{i}_{k}\in\mathcal{V}_{1}$ such that
$\textbf{i}^{\prime}_{h}\neq\textbf{i}_{k}\leq a_{2}$ which implies that
$\textbf{i}_{1}\leq a_{2}$. Thus, $\textbf{j}_{1}>b_{2}$. If not, the monomial
$m_{1}=x_{\textbf{i}_{1}}x_{\textbf{j}_{1}}$ is contained in
${\mathcal{B}}_{M_{2}}$, a contradiction.

In the final step of the proof, recall from 6.13 (i) that
$\mathcal{V}^{\prime}_{2}$ has at least $q+1$ elements which are at most
$b_{2}$ and $\mathcal{N}^{c}$ is the list containing all such elements. Since
$\mathcal{V}=\mathcal{V}^{\prime}$ and
$\mathcal{V}_{2}\setminus\\{\textbf{i}^{\prime}_{h}\\}$ has $q-1$ many such
elements, the list
$\mathcal{V}_{1}=\mathcal{I}^{\prime}\cup\mathcal{J}^{\prime}$ must contain at
least two elements from $\mathcal{N}^{c}$ where
$\mathcal{I}^{\prime}=\\{\textbf{i}_{1},\ldots,\textbf{i}_{p}\\}$ and
$\mathcal{J}^{\prime}=\\{\textbf{j}_{1},\ldots,\textbf{j}_{p}\\}$. In
particular, both such elements of $\mathcal{V}_{1}$ must be in
$\mathcal{I}^{\prime}$ because each element of $\mathcal{J}^{\prime}$ is
greater than $b_{2}$. Since $e=\textbf{i}_{f}$ is already one of these two
elements, the other element must be $\textbf{i}_{g}$ for some $g\in[p]$. Let
$L=\max\\{f,g\\}$. Then
$x_{\textbf{i}_{1}}x_{\textbf{i}_{L}}\in{\mathcal{B}}_{M_{1}}$ as
$m_{1}\in{\mathcal{B}}_{M_{1}}$ where $\textbf{i}_{L}\leq
b_{2}<\textbf{j}_{1}$. Furthermore,
$x_{\textbf{j}_{1}}x_{\textbf{j}_{L}}\in{\mathcal{B}}_{M_{1}}$ since
$\textbf{j}_{L}\leq b_{1}$ and $\textbf{j}_{1}\leq a_{1}$. Thus,
$m_{1}\succeq_{rlex}m_{L}=x_{\textbf{i}_{L}}x_{\textbf{j}_{L}}\succ_{rlex}x_{\textbf{j}_{1}}x_{\textbf{j}_{L}}$
and
$T_{m_{1}}T_{m_{L}}-T_{\textbf{i}_{1}\textbf{i}_{L}}T_{\textbf{j}_{1}\textbf{j}_{L}}\in\mathcal{G},$
contradiction. ∎

######  Proposition 6.17.

Every nonempty $\Gamma_{\mu}(I_{1},I_{2})$ with respect to $\mathcal{G}$ has a
unique sink.

###### Proof.

Let $\mu$ be a multidegree in $S[t,z]$ such that $\Gamma_{\mu}(I_{1},I_{2})$
is nonempty. Let $V$ be a vertex in $\Gamma_{\mu}(I_{1},I_{2})$ of the form
$V=(T_{m_{1}}\cdots T_{m_{p}})(Z_{n_{1}}\cdots Z_{n_{q}})$. If $p=0$ or $q=0,$
the fiber graph $\Gamma_{\mu}(I_{1},I_{2})$ becomes a fiber graph of $I_{1}$
or $I_{2}$ as discussed in Remark 6.7. It follows from Corollary 5.17 and
Corollary 5.24 that both fiber graphs have unique sinks. For the remainder of
the proof, we may assume that $p,q\geq 1$.

We proceed by contradiction. Suppose $\Gamma_{\mu}(I_{1},I_{2})$ has two
different sinks $V$ and $V^{\prime}$. It is immediate that
$\mu_{2}\neq\mu^{\prime}_{2}$ following 6.10. Otherwise, we must have
$\mu_{1}=\mu^{\prime}_{1}$ which in turn implies that $V=V^{\prime}$, a
contradiction. Furthermore, we may assume that $n_{q}\neq n^{\prime}_{q}$.
Otherwise, we can consider the multidegree $\mu/n_{q}$; if
$n_{q-1}=n^{\prime}_{q-1}$, consider the multidegree $\mu/n_{q}n_{q-1}$; and
so on. Without loss of generality, let $n_{q}\succ_{mrlex}n^{\prime}_{q}$.
Rest of the proof is established by considering all possible regions where
$n_{q}$ and $n^{\prime}_{q}$ belong to in ${\mathcal{B}}(M_{2},N_{2})$. In
particular, we collect all possibilities under two cases and show that neither
of them is possible. Hence, the fiber graph $\Gamma_{\mu}(I_{1},I_{2})$ must
have a unique sink.

Case I: _Suppose $n_{q}=x_{i}x_{j}$ and
$n^{\prime}_{q}=x_{i^{\prime}}x_{j^{\prime}}$ are both in
${\mathcal{B}}_{M_{2}}$ or ${\mathcal{B}}_{N_{2}}$._

We first consider the case when they are both in ${\mathcal{B}}_{N_{2}}$. It
follows from our assumption $n_{q}\succ_{mrlex}n^{\prime}_{q}$ that
$j<j^{\prime}$ or $i<i^{\prime}$. Suppose $j<j^{\prime}$. It is clear that
$x_{j^{\prime}}$ does not divide $n_{q}$. Moreover, there exists no
$Z_{n_{l}}$ in the support of $V$ such that $x_{j^{\prime}}$ divides $n_{l}$
for $l\in[q-1]$. Otherwise, $n_{q}\succ_{mrlex}n_{l}$, which is not possible.
Thus, $\mu_{2}$ is not divisible by $x_{j^{\prime}}$ while $\mu$ is. Then
there exists $T_{m_{l}}$ in the support of $V$ such that $x_{j^{\prime}}$
divides $m_{l}$. Since $m_{l}\in{\mathcal{B}}(M_{1},N_{1})$ is divisible by
$x_{j^{\prime}}$ while $j<j^{\prime}$, the monomial
$m^{\prime}:=x_{j}(m_{l}/x_{j^{\prime}})\in{\mathcal{B}}(M_{1},N_{1})$. In
addition, it is immediate that $x_{i}x_{j^{\prime}}\in{\mathcal{B}}_{N_{2}}$
and $n_{q}\succ_{mrlex}x_{i}x_{j^{\prime}}$. Hence,
$T_{m_{l}}Z_{n_{q}}-T_{m^{\prime}}Z_{ij^{\prime}}\in\mathcal{G}$, a
contradiction. Then, we may assume that $j=j^{\prime}$ by Lemma 5.25.

Suppose $i<i^{\prime}$. Note that $n_{l}\in{\mathcal{B}}_{N_{2}}$ for each
$n_{1},\ldots,n_{q}$ from Item (a) (a). Then by Lemma 5.25 and the fact that
$i<i^{\prime}\leq c_{2}$, there exists no $n_{l}$ such that $x_{i^{\prime}}$
divides $n_{l}$. Thus, $\mu_{2}$ is not divisible by $x_{i^{\prime}}$. Since
$\mu$ is divisible by $x_{i^{\prime}}$, there exists $T_{m_{l}}$ in the
support of $V$ such that $x_{i^{\prime}}$ divides $m_{l}$. As in the previous
paragraph, by letting $m^{\prime}:=x_{i}(m_{l}/x_{i^{\prime}})$, we have
$m^{\prime}\in{\mathcal{B}}(M_{1},N_{1})$. Hence,
$T_{m_{l}}Z_{n_{q}}-T_{m^{\prime}}Z_{n^{\prime}_{q}}\in\mathcal{G}$, a
contradiction. Therefore, this case is not possible.

Investigation of the remaining case yields to a contradiction, as well.
Arguments of this step is very similar to the previous one and we omit the
details.

Case II: _Suppose $n_{q}\in{\mathcal{B}}_{N_{2}}$ and
$n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$._

Based on Lemma 6.16, one needs to consider two subcases to investigate whether
the fiber graph can have two sinks. By employing our auxiliary lemmas, we
provide a contradiction for each of the two cases and it completes the proof.

Subcase 1: _Suppose_ $\textbf{i}_{1}$ _is at most_ $c_{1}$.

First note that $\textbf{j}_{1}>b_{2}$ because
$m_{1}=x_{\textbf{i}_{1}}x_{\textbf{j}_{1}}$ is not in
${\mathcal{B}}(x_{c_{1}}x_{b_{2}})$ by Lemma 6.14 (a). Then
$\textbf{j}_{k}>b_{2}$ for each $k\in[p]$ since $\textbf{j}_{1}$ is the
smallest element in $\\{\textbf{j}_{1},\ldots,\textbf{j}_{p}\\}.$ An immediate
implication of this lower bound is that $e^{\prime}=\textbf{j}_{f}$ and
$e=\textbf{i}_{f}\leq a_{1}$ where $m_{f}$ is the monomial from 6.13 (iii).
Additionally, the monomial $m_{1}$ is in ${\mathcal{B}}_{M_{1}}$ because
$\textbf{j}_{1}\leq a_{1}$ by Lemma 6.14 (b).

In what follows, we conclude that $c_{1}<\textbf{j}_{1}$. On the contrary,
suppose $\textbf{j}_{1}\leq c_{1}$. Then, we have
$x_{\textbf{i}_{1}}x_{\textbf{i}_{f}}\in{\mathcal{B}}_{M_{1}}$ and
$x_{\textbf{j}_{1}}x_{\textbf{j}_{f}}\in{\mathcal{B}}(M_{1},N_{1})$ by
checking the indices; moreover,
$m_{1}\succ_{rlex}m_{f}\succ_{rlex}x_{\textbf{j}_{1}}x_{\textbf{j}_{f}}$
because $e=\textbf{i}_{f}\leq b_{2}<\textbf{j}_{1}$. Thus,
$T_{m_{1}}T_{m_{f}}-T_{\textbf{i}_{1}\textbf{i}_{f}}T_{\textbf{j}_{1}\textbf{j}_{f}}\in\mathcal{G}$,
a contradiction.

In this step, we show that if $\textbf{i}_{k}\leq c_{1}$ for some
$\textbf{i}_{k}$ in $\\{\textbf{i}_{2},\ldots,\textbf{i}_{p}\\}$, then
$m_{k}\in{\mathcal{B}}_{N_{1}}$. Otherwise, if
$m_{k}\in{\mathcal{B}}_{M_{1}}$, one can easily verify that
$x_{\textbf{j}_{1}}x_{\textbf{j}_{k}}\in{\mathcal{B}}_{M_{1}}$. Additionally,
$x_{\textbf{i}_{1}}x_{\textbf{i}_{k}}\in{\mathcal{B}}_{M_{1}}$ since
$m_{k}\in{\mathcal{B}}_{M_{1}}$ where $\textbf{i}_{k}<\textbf{j}_{1}$. Since
$m_{1}\succ_{rlex}m_{k}\succ_{rlex}x_{\textbf{j}_{1}}x_{\textbf{j}_{k}}$, we
have
$T_{m_{1}}T_{m_{k}}-T_{\textbf{i}_{1}\textbf{i}_{k}}T_{\textbf{j}_{1}\textbf{j}_{k}}\in\mathcal{G}$,
a contradiction.

In the next step, our goal is to show that there exists $t>1$ such that
$m_{t}$ in ${\mathcal{B}}_{M_{1}}$ when $p\geq 2$. On the contrary, suppose
all the monomials in $\\{m_{2},\ldots,m_{p}\\}$ are in
${\mathcal{B}}_{N_{1}}$. Let
$\mathcal{M}^{\prime}=\\{\textbf{j}_{2},\ldots,\textbf{j}_{p},j_{1},\ldots,j_{q}\\}\subseteq\mathcal{V}_{1}\cup\mathcal{V}_{2}$.
Notice that each element of this list is greater than $b_{1}$ by our
assumption that each $m_{k}\in{\mathcal{B}}_{N_{1}}$ and Lemma 6.14 (c).
Furthermore, each of its elements is greater than $b_{2}$ since
$b_{2}<\textbf{j}_{1}$ by the first step. Note that $\mathcal{V}^{\prime}_{1}$
can have at most $p$ elements which are greater than $b_{1}$. In addition,
$\mathcal{V}^{\prime}_{2}$ can have at most $q-1$ such elements which are
greater than $b_{2}$ by 6.13 (i). Since $\mathcal{M}^{\prime}$ have $p+q-1$
elements and $\mathcal{V}=\mathcal{V}^{\prime}$, the list
$\mathcal{V}^{\prime}_{1}$ has exactly $p$ elements from
$\mathcal{M}^{\prime}$ and the remaining $q-1$ elements of
$\mathcal{M}^{\prime}$ must be in $\mathcal{V}^{\prime}_{2}$. In particular,
it means
$\mathcal{M}^{\prime}=\\{\textbf{j}^{\prime}_{1},\ldots,\textbf{j}^{\prime}_{p},j^{\prime}_{1},\ldots,j^{\prime}_{q-1}\\}$
since $n^{\prime}_{q}\in{\mathcal{B}}_{M_{2}}$. This implies that each
$m^{\prime}_{k}\in{\mathcal{B}}_{N_{1}}$ and each $n^{\prime}_{l}$ belongs to
${\mathcal{B}}_{N_{2}}$ for $k\in[p]$ and $l\in[q-1]$. Then
$\mathcal{V}^{\prime}\setminus\mathcal{M}^{\prime}$ contains elements which
are either at most $c_{1}$ or $b_{2}$. Moreover, this list must contain
$\textbf{j}_{1}$ since $\mathcal{V}=\mathcal{V}^{\prime}$. This is not
possible because $\textbf{j}_{1}$ is greater than both $c_{1}$ and $b_{2}$
from the first two steps of this subcase. Therefore, there exists $t>1$ such
that $m_{t}\in{\mathcal{B}}_{M_{1}}$ when $p\geq 2$.

It is worth noting that $\textbf{j}_{1}\leq\textbf{i}_{t}$ for each
$m_{t}\in{\mathcal{B}}_{M_{1}}$ where $t>1$. Otherwise, both monomials
$x_{\textbf{i}_{1}}x_{\textbf{i}_{t}}$ and
$x_{\textbf{j}_{1}}x_{\textbf{j}_{t}}$ are contained in
${\mathcal{B}}_{M_{1}}$. In addition,
$m_{1}\succ_{rlex}m_{t}\succ_{rlex}x_{\textbf{j}_{1}}x_{\textbf{j}_{t}}$.
Thus, the binomial
$T_{m_{1}}T_{m_{t}}-T_{\textbf{i}_{1}\textbf{i}_{t}}T_{\textbf{j}_{1}\textbf{j}_{t}}$
belongs to $\mathcal{G}$, a contradiction.

Let $m_{1},\ldots,m_{s}\in{\mathcal{B}}_{M_{1}}$ and
$m_{s+1},\ldots,m_{p}\in{\mathcal{B}}_{N_{1}}$. Then, by making use of Lemma
5.18, Lemma 6.12, and the main conclusions of each step of this subcase, we
obtain the following chain of inequalities.

$\displaystyle\textbf{i}_{1}\leq$
$\displaystyle\textbf{i}_{s+1}\leq\cdots\leq\textbf{i}_{p}\leq
c_{1}<\textbf{j}_{1}\leq\textbf{i}_{2}\leq\cdots\leq\textbf{i}_{s}\leq a_{1}$
$\displaystyle c_{1},b_{2}<\textbf{j}_{1}$
$\displaystyle\leq\cdots\leq\textbf{j}_{s}\leq
b_{1}<\textbf{j}_{s+1}\leq\cdots\leq\textbf{j}_{p}\leq j_{1}\leq\cdots\leq
j_{q}$

Consider the list
$\mathcal{O}=\\{\textbf{i}_{2},\ldots,\textbf{i}_{s},\textbf{j}_{1},\ldots,\textbf{j}_{p},j_{1},\ldots,j_{q}\\}\subseteq\mathcal{V}=\mathcal{V}^{\prime}_{1}\cup\mathcal{V}^{\prime}_{2}$
with $p+q+s-1$ elements. Note that each element of $\mathcal{O}$ is greater
than both $c_{1}$ and $b_{2}$. Then, $\mathcal{V}^{\prime}_{1}$ must contain
at least $p+s$ many elements from $\mathcal{O}$ since
$\mathcal{V}^{\prime}_{2}$ can have at most $q-1$ elements from $\mathcal{O}$
by 6.13 (i). It follows from the pigeonhole principle that there exists
$m^{\prime}_{l_{1}}\succ_{rlex}\cdots\succ_{rlex}m^{\prime}_{l_{s}}$ such that
$\textbf{i}^{\prime}_{l_{k}},\textbf{j}^{\prime}_{l_{k}}\in\mathcal{O}$ for
each $k\in[s]$. Since each $\textbf{i}^{\prime}_{l_{k}}\in\mathcal{O}$ is
greater than $c_{1}$, each monomial $m^{\prime}_{l_{k}}$ must be contained in
${\mathcal{B}}(M_{1})\setminus{\mathcal{B}}(N_{1})$, in other words,
$\textbf{i}^{\prime}_{l_{k}}\leq\textbf{j}^{\prime}_{l_{k}}\leq b_{1}$ for
$k\in[s]$. As a result, the list $\mathcal{O}$ has at least $2s$ elements
which are at most $b_{1}$. However, the only such elements of $\mathcal{O}$
are
$\\{\textbf{i}_{2},\ldots,\textbf{i}_{s},\textbf{j}_{1},\ldots,\textbf{j}_{s}\\}$,
a contradiction. As a result, Subcase 1 is eliminated.

Subcase 2: _Suppose_ $\textbf{i}_{1}>c_{1}$ _and one of the monomials from_
$m_{2},\ldots,m_{p}$ _is in_ ${\mathcal{B}}_{N_{1}}$.

Let $m_{1},\ldots,m_{s}\in{\mathcal{B}}_{M_{1}}$ and
$m_{s+1},\ldots,m_{p}\in{\mathcal{B}}_{N_{1}}$ where $s\geq 1$. Then, we
obtain the following chain of inequalities by making use of Lemma 5.18, Item
(a) (a) and (b), Item (a) (a) and (c).

$\displaystyle\textbf{i}_{s+1}\leq\cdots\leq\textbf{i}_{p}\leq
c_{1}<\textbf{i}_{1}\leq\cdots\leq\textbf{i}_{s}\leq a_{1}$
$\displaystyle\text{ and }c_{1}<\textbf{j}_{1}\leq\cdots\leq\textbf{j}_{s}\leq
b_{1}$ $\displaystyle
b_{1},b_{2}<\textbf{j}_{s+1}\leq\cdots\leq\textbf{j}_{p}\leq
j_{1}\leq\cdots\leq j_{q}$ $\displaystyle\text{ and }i_{1}\leq\cdots\leq
i_{q}\leq c_{2}$ (6.3)

Consider the list
$\mathcal{O}=\\{\textbf{j}_{s+1},\ldots,\textbf{j}_{p},j_{1},\ldots,j_{q}\\}\subseteq\mathcal{V}$
with $p+q-s$ elements. Note that $\mathcal{O}$ is the collection of all
elements of $\mathcal{V}$ which are greater than both $b_{1}$ and $b_{2}$.
Recall from 6.13 (i) that $\mathcal{V}^{\prime}_{2}$ can have at most $q-1$
elements from $\mathcal{O}$. Thus, the list $\mathcal{V}^{\prime}_{1}$ must
contain at least $p-s+1$ many elements from $\mathcal{O}$. Suppose
$\mathcal{V}^{\prime}_{1}$ contains exactly $\tilde{p}:=p+t-s+1$ many of the
elements of $\mathcal{O}$ and the remaining $\tilde{q}:=q-t-1$ elements of
$\mathcal{O}$ are contained in $\mathcal{V}^{\prime}_{2}$ where $0\leq t\leq
q-1$. Then there exists
$\textbf{i}^{\prime}_{l_{k}},\textbf{j}^{\prime}_{l_{k}}\in\mathcal{V}^{\prime}_{1}$
such that each $\textbf{j}^{\prime}_{l_{k}}\in\mathcal{O}$ for
$k\in[\tilde{p}]$. Since elements of $\mathcal{O}$ are greater than $b_{1}$,
we must have $\textbf{i}^{\prime}_{l_{k}}\leq c_{1}$ for each
$k\in[\tilde{p}]$. It means that $\mathcal{V}$ has at least $p-s+1$ many
elements which are less than $c_{1}$. In what follows, we show that
$\mathcal{V}$ has only $p-s$ such elements and it yields to a contradiction.
Hence, Subcase 2 is eliminated and, as a result, the fiber graph can not have
two sinks satisfying the statement of Case II.

Let $\mathcal{C}$ be the collection of all elements of $\mathcal{V}$ which are
less than $c_{1}$. It follows from the chain of inequalities given in (6) that
$\mathcal{C}\subseteq\\{\textbf{i}_{s+1},\ldots,\textbf{i}_{p},i_{1},\ldots,i_{q}\\}$.
We claim that $\mathcal{C}=\\{\textbf{i}_{s+1},\ldots,\textbf{i}_{p}\\}$, in
other words, $\mathcal{C}$ has $p-s$ elements. In order to prove the claim, it
suffices to show that $c_{1}<i_{1}$.

Let
$\mathcal{S}=\\{\textbf{i}_{k},\textbf{j}_{k}:k\in[s]\\}\subseteq\mathcal{V}=\mathcal{V}^{\prime}_{1}\cup\mathcal{V}^{\prime}_{2}$
and set $\mathcal{S}_{i}=\\{u\in\mathcal{V}^{\prime}_{i}:u\in\mathcal{S}\\}$
for $i=1,2$. Note that none of the elements of $\mathcal{S}_{1}$ is equal to
$\textbf{i}^{\prime}_{l_{k}}$ or $\textbf{j}^{\prime}_{l_{k}}$ because
$\textbf{i}^{\prime}_{l_{k}}\leq c_{1}$ and
$b_{1}<\textbf{j}^{\prime}_{l_{k}}$ for $k\in[\tilde{p}]$. Thus, the list
$\mathcal{S}_{1}$ is contained in
$\mathcal{V}^{\prime}_{1}\setminus\\{\textbf{i}^{\prime}_{l_{k}},\textbf{j}^{\prime}_{l_{k}}:k\in[\tilde{p}]\\}$.
This means $\mathcal{S}_{1}$ has at most $2s-2t-2$ elements while
$\mathcal{S}_{2}$ has at least $2t+2$ elements. For each
$u\in\mathcal{S}_{2}$, we have either $u\leq c_{2}$ or $u>c_{2}$. If $u\leq
c_{2}$ for some $u\in\mathcal{S}_{2},$ then $\textbf{i}_{1}\leq u\leq c_{2}$.
It follows from Lemma 6.12 (c) that $c_{1}<\textbf{i}_{1}\leq i_{1}$, proving
the claim.

Suppose each elements of $\mathcal{S}_{2}$ are greater than $c_{2}$. Recall
that $\mathcal{V}^{\prime}_{2}$ has exactly $\tilde{q}$ elements from
$\mathcal{O}$; there exists
$i^{\prime}_{k_{l}},j^{\prime}_{k_{l}}\in\mathcal{V}^{\prime}_{2}$ such that
each $j^{\prime}_{k_{l}}\in\mathcal{O}$ for $l\in[\tilde{q}]$. Since each
element of $\mathcal{O}$ is greater than $b_{2}$, each $i^{\prime}_{k_{l}}\leq
c_{2}$ for $l\in[\tilde{q}]$ which implies that each
$n^{\prime}_{k_{l}}\in{\mathcal{B}}_{N_{2}}$. Note that none of the elements
of $\mathcal{S}_{2}$ is equal to $i^{\prime}_{k_{l}}$ or $j^{\prime}_{k_{l}}$
for $l\in[\tilde{q}]$. Thus, $\mathcal{S}_{2}$ is contained in
$\mathcal{V}^{\prime}_{2}\setminus\\{i^{\prime}_{k_{l}},j^{\prime}_{k_{l}}:l\in[\tilde{q}]\\}$.
Since $\mathcal{S}_{2}$ has at least $2t+2$ elements while the latter set has
exactly $2t+2$ elements, we must have

$\displaystyle\mathcal{S}_{2}=\mathcal{V}^{\prime}_{2}\setminus\\{i^{\prime}_{k_{l}},j^{\prime}_{k_{l}}:l\in[\tilde{q}]\\}\text{
and
}\mathcal{S}_{1}=\mathcal{V}^{\prime}_{1}\setminus\\{\textbf{i}^{\prime}_{l_{k}},\textbf{j}^{\prime}_{l_{k}}:k\in[\tilde{p}]\\}$

where the second equality follows from observing $\mathcal{S}_{1}$ must have
$2s-2t-2$ elements. In the light of the first equality and the fact that each
element of $\mathcal{S}_{2}$ is greater than $c_{2}$, the list
$\mathcal{S}_{2}$ contains pairs $i^{\prime}_{l},j^{\prime}_{l}$ where the
associated monomial
$n^{\prime}_{l}\in{\mathcal{B}}(M_{2})\setminus{\mathcal{B}}(N_{2})$. Recall
that for the pairs that are not in $\mathcal{S}_{2}$, their associated
monomials are in ${\mathcal{B}}_{N_{2}}$. Since we use the $>_{mrlex}$ order
on the $Z$ variables, we must have
$\mathcal{S}_{2}=\\{i^{\prime}_{l},j^{\prime}_{l}:l\in\\{\tilde{q}+1,\dots,q\\}\\}$
where $i^{\prime}_{l}\leq a_{2}$ and $j^{\prime}_{l}\leq b_{2}.$

In the final step, our goal is to show that $m_{1}\in{\mathcal{B}}_{M_{2}}$.
Notice that if $m_{1}\in{\mathcal{B}}_{M_{2}}$, then $c_{1}<i_{1}$. Otherwise,
we have $n_{1}\in{\mathcal{B}}_{N_{1}}$ by (6). Then $n_{1}\succ_{mrlex}m_{1}$
and $T_{m_{1}}Z_{n_{1}}-T_{n_{1}}Z_{m_{1}}\in\mathcal{G}$, a contradiction.
Consider the list $\mathcal{S}_{2}$. Since $\textbf{i}_{1}$ is the smallest
element in $\mathcal{S},$ it is clear that $\textbf{i}_{1}\leq a_{2}$ because
$i^{\prime}_{l}\leq a_{2}$ for $i^{\prime}_{l}\in\mathcal{S}_{2}$. Similarly,
since $\textbf{j}_{1}$ is the smallest among
$\textbf{j}_{1},\ldots,\textbf{j}_{s}$, if $\mathcal{S}_{2}$ contains at least
one $\textbf{j}_{k}$ for $k\in[s]$, then $\textbf{j}_{1}\leq b_{2}.$ So the
only case that we need to consider is when $\mathcal{S}_{2}$ does not contain
any such element. Note that it happens when $\textbf{j}_{k}\in\mathcal{S}_{1}$
for each $k\in[s]$. By the pigeonhole principle, there exists
$\textbf{i}^{\prime}_{k},\textbf{j}^{\prime}_{k}\in\mathcal{S}_{1}$ such that
$\textbf{i}^{\prime}_{k}=\textbf{j}_{u}$ and
$\textbf{j}^{\prime}_{k}=\textbf{j}_{v}$ for some $u,v\in[s]$ and it implies
$\textbf{j}_{1}\leq\textbf{j}_{u}\leq a_{1}$. Observe that
$\textbf{j}_{1}\leq\textbf{i}_{2}.$ Otherwise,
$m_{1}\succ_{rlex}m_{2}\succ_{rlex}x_{\textbf{j}_{1}}x_{\textbf{j}_{2}}$ and
$T_{m_{1}}T_{m_{2}}-T_{\textbf{i}_{1}\textbf{i}_{2}}T_{\textbf{j}_{1}\textbf{j}_{2}}\in\mathcal{G},$
a contradiction. Since $\mathcal{S}_{2}$ has at least two elements, there
exists $k>1$ such that $\textbf{i}_{k}\in\mathcal{S}_{2}$. Thus
$\textbf{j}_{1}\leq\textbf{i}_{2}\leq b_{2}.$ Therefore,
$m_{1}\in{\mathcal{B}}_{M_{2}}$ and it concludes the proof. ∎

As an immediate corollary of Theorem 6.9, we obtain the following result on
the Koszulness of multi-Rees algebras.

######  Corollary 6.18.

The multi-Rees algebra, $\mathcal{R}(I_{1}\oplus I_{2})$, of strongly stable
ideals $I_{1}$ and $I_{2}$ with two quadric Borel generators is Koszul. In
particular, products of powers of $I_{1}$ and $I_{2}$ have linear resolutions.

###### Proof.

Recall from Theorem 6.9 that the toric ideal $T(I_{1}\oplus I_{2})$ has a
quadratic Gröbner basis with respect to the head and tail order. It then
follows from Theorem 3.2 that the defining ideal of $\mathcal{R}(I_{1}\oplus
I_{2})$ has a quadratic Gröbner basis. Thus $\mathcal{R}(I_{1}\oplus I_{2})$
is Koszul by [13, Theorem 6.7]. The second statement follows from [4, Theorem
3.4]. ∎

We close the paper with the following question inviting interested researchers
to study Koszulness of the multi-Rees algebras of strongly stable ideals.

######  Question 6.19.

Let $I_{1},\ldots,I_{r}$ be a collection of strongly stable ideals satisfying
one of the conditions given in Proposition 4.4. Is the multi-Rees algebra
$\mathcal{R}(I_{1}\oplus\cdots\oplus I_{r})$ Koszul?

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