# Nakajima’s quiver varieties and triangular bases of rank-2 cluster algebras

Li Li  Department of Mathematics and Statistics, Oakland University,
Rochester, MI 48309-4479, USA<EMAIL_ADDRESS>
###### Abstract.

Berenstein and Zelevinsky introduced quantum cluster algebras [3] and the
triangular bases [4]. The support conjecture in [12] asserts that the support
of a triangular basis element for a rank-2 cluster algebra is bounded by an
explicitly described region that is possibly concave. In this paper, we prove
the support conjecture for all skew-symmetric rank-2 cluster algebras.

###### 2010 Mathematics Subject Classification:

Primary 13F60; Secondary 14F06, 16G20, 32S60

###### Contents

  1. 1 Introduction
  2. 2 Quantum cluster algebras and triangular bases
    1. 2.1 Rank-2 quantum cluster algebras
    2. 2.2 Triangular bases
  3. 3 The case of real roots
  4. 4 Nakajima’s graded quiver varieties
  5. 5 Decomposition theorem
    1. 5.1 Algebraic Transversal Slice Theorem
    2. 5.2 The first proof of trivial local systems
  6. 6 Other desingularizations of $\mathbf{E}_{v,w}$
    1. 6.1 Variety $\tilde{\mathcal{F}}^{\rm swap}_{v,w}$
    2. 6.2 Varieties $\tilde{\mathcal{G}}_{v,w}$ and $\tilde{\mathcal{H}}_{v,w}$
  7. 7 Definition of dual canonical basis elements $L(w)$
    1. 7.1 Definition of $L(w)$ and $\mathbf{R}^{\mathrm{finite}}_{t}$
    2. 7.2 Definition of $M(w)$
    3. 7.3 Definition of $\chi$
    4. 7.4 $\\{\overline{\chi(M(w))}\\}$ and the standard monomial basis
    5. 7.5 $\\{\chi(L(w))\\}$ and the triangular basis
  8. 8 The proof of main results
    1. 8.1 Some facts on the BBDG Decomposition
    2. 8.2 $P_{-}(v,w)$
    3. 8.3 Proof of the main theorem (Theorem 1.2) and its corollaries
  9. 9 Analogue of Kazhdan-Lusztig polynomials
  10. 10 Examples
    1. 10.1 Two examples of triangular basis elements
    2. 10.2 An example to illustrate that $\\{\chi(L(w))\\}$ is identical with the triangular basis
    3. 10.3 An example to illustrate the proof of the main theorem
    4. 10.4 An example to Corollary 1.5
  11. 11 Some technical proofs
    1. 11.1 Proof of Lemma 4.6
    2. 11.2 Proof of Lemma 5.3
    3. 11.3 The second proof of trivial local systems

##  1\. Introduction

Cluster algebras and quantum cluster algebras were introduced by Fomin-
Zelevinsky [9] and Berenstein-Zelevinsky [3], respectively. A main goal of
introducing quantum cluster algebras is to understand good bases arising from
the representation theory of certain non-associative algebras. Mimicking the
construction of the dual canonical basis from a PBW basis [15], Berenstein and
Zelevinsky [4] constructed the triangular basis for acyclic quantum cluster
algebras. The triangular basis has many nice properties including
(conjecturally) the strong positivity, that is, all structure constants are
nonnegative. In this paper, we will focus on triangular bases for quantum
cluster algebras of rank 2, and will not discuss those of higher ranks; the
definition of rank-2 quantum cluster algebras and their triangular bases will
be reviewed in §2.

In [12], Lee, Rupel, Zelevinsky and the author proposed the following
conjecture. For $x\in\mathbb{R}$, denote $[x]_{+}=\max\\{x,0\\}$. Let $b,c$ be
positive integers, every triangular basis of the (coefficient-free) rank-2
quantum cluster algebra $\mathcal{A}_{\mathbf{v}}(b,c)$ is of the form

(1.1) $C[a_{1},a_{2}]=\sum_{p,q}e(p,q)X^{(bp-a_{1},cq-a_{2})}$

where $a_{1},a_{2},p,q\in\mathbb{Z}$, and $0\leq p\leq[a_{2}]_{+}$, $0\leq
q\leq[a_{1}]_{+}$. Define the set of positive imaginary roots as

$\Phi^{im}_{+}=\\{(a_{1},a_{2})\in\mathbb{Z}^{2}_{>0}:ca_{1}^{2}-bca_{1}a_{2}+ba_{2}^{2}\leq
0\\}.$

######  Conjecture 1.1.

[12, Conjecture 11] Let $b,c\in\mathbb{Z}_{>0}$,
$(a_{1},a_{2})\in\Phi^{im}_{+}$. Let $C[a_{1},a_{2}]$ be a triangular basis
element of the (coefficient-free) rank-2 quantum cluster algebra
$\mathcal{A}_{\mathbf{v}}(b,c)$. For integers $0\leq p\leq a_{2}$, $0\leq
q\leq a_{1}$, the coefficient $e(p,q)$ of $C[a_{1},a_{2}]$ is nonzero if and
only if $D(p,q)\geq 0$, where

(1.2) $D(p,q)=ca_{1}q+ba_{2}p-bp^{2}-bcpq-cq^{2}.$

To illustrate, the region of $\\{(p,q)\in\mathbb{R}^{2}\ |\ p,q\geq
0,D(p,q)\geq 0\\}$ is the “curved triangle” OAC in Figure 1 (6). The reader is
also referred to §10.1 for an example where $b=c=3$,
$(a_{1},a_{2})=(3,4)\in\Phi^{im}_{+}$, and an example where $b=c=3$,
$(a_{1},a_{2})=(2,8)\not\in\Phi^{im}_{+}$.

Our main theorem is to confirm the above conjecture in the skew-symmetric
case, that is, when $b=c$. Throughout the paper except §3, we assume that
$b=c=r$.

######  Theorem 1.2.

Assume $(b,c)=(r,r)$ where $r\in\mathbb{Z}_{>0}$,
$(a_{1},a_{2})\in\Phi^{im}_{+}$. For integers $0\leq p\leq a_{2}$, $0\leq
q\leq a_{1}$, let $D=D(p,q)=r(a_{1}q+a_{2}p-p^{2}-rpq-q^{2})$. Then the
following holds:

(i) $e(p,q)\neq 0$ if and only if $D\geq 0$.

(ii) Fix $(p,q)$ such that $D\geq 0$. Then $\displaystyle e(p,q)=\sum_{-D\leq
i\leq D}e_{i}\mathbf{v}^{i}$, where $e_{i}$’s are nonnegative integers
satisfying the following conditions:

$\bullet$ symmetry: $e_{i}=e_{-i}$ for every $i$;

$\bullet$ $e_{D}=e_{-D}=1$;

$\bullet$ $e_{i}\neq 0$ if and only if $i\in\\{-D,-D+2r,-D+4r,\dots,D\\}$;

$\bullet$ unimodality:

– if $2r|D$, then $e_{-D}\leq e_{-D+2r}\leq\cdots\leq e_{-2r}\leq e_{0}\geq
e_{2r}\geq\cdots\geq e_{D-2r}\geq e_{D}$;

– if $2r\nmid D$, then $e_{-D}\leq e_{-D+2r}\leq\cdots\leq e_{-3r}\leq
e_{-r}=e_{r}\geq e_{3r}\geq\cdots\geq e_{D-2r}\geq e_{D}$.

Next, as a corollary of Theorem 1.2 and a study of the case of real roots
(Theorem 3.1), we give a complete description for the support of any
triangular basis element. Define ${\bf d}_{n}=\begin{bmatrix}d^{\prime}_{n}\\\
d^{\prime\prime}_{n}\end{bmatrix}$ to be the denominator vector of the cluster
variable $X_{n}$. For $x,y\in\mathbb{R}$, define $\begin{bmatrix}x\\\
y\end{bmatrix}_{+}=\begin{bmatrix}[x]_{+}\\\ [y]_{+}\end{bmatrix}$. Define

$T_{n}=\textrm{ the convex hull of
}\\{(0,0),([d^{\prime\prime}_{n}]_{+},0),(0,[d^{\prime}_{n}]_{+})\\}.$

For $s\in\mathbb{R}_{\geq 0}$ and $T\subseteq\mathbb{R}^{2}$, define
$sT=\\{sz\ |\ z\in T\\}$.

######  Definition 1.3.

For $(a_{1},a_{2})\in\mathbb{Z}^{2}$, define a region
$R(a_{1},a_{2})\subseteq\mathbb{R}^{2}$ as follows (see Figure 1):

(a) If $(a_{1},a_{2})\notin\Phi^{im}_{+}$, we can write
$(a_{1},a_{2})=s_{1}{\bf d}_{n}+s_{2}{\bf d}_{n+1}$ for some $n\in\mathbb{Z}$,
$s_{1},s_{2}\in\mathbb{Z}_{\geq 0}$. Then define $R(a_{1},a_{2})$ be the
following Minkowski sum:111To verify the shape of $R(a_{1},a_{2})$ of case (5)
in Figure 1: assume $s_{1},s_{2}>0,n\neq 0,1,2$. Note that
$s_{1}T_{n}+s_{2}T_{n+1}$ is the convex hull of
$O=(0,0),A=(s_{1}d^{\prime\prime}_{n}+s_{2}d^{\prime\prime}_{n+1},0),B=(s_{2}d^{\prime\prime}_{n+1},s_{1}d^{\prime}_{n}),C=(0,s_{1}d^{\prime}_{n}+s_{2}d^{\prime}_{n+1}),D=(s_{1}d^{\prime\prime}_{n},s_{2}d^{\prime}_{n+1})$
We claim that $OABC$ is convex, and $D$ lies inside the triangle $OAC$. To
show this, it suffices to show that the vertices $A,B,C$ lie
counterclockwisely, and $A,C,D$ also lie counterclockwisely. The former
statement follows from the determinant computation:
$\begin{vmatrix}s_{1}d^{\prime\prime}_{n}+s_{2}d^{\prime\prime}_{n+1}&0&1\\\
s_{2}d^{\prime\prime}_{n+1}&s_{1}d^{\prime}_{n}&1\\\
0&s_{1}d^{\prime}_{n}+s_{2}d^{\prime}_{n+1}&1\\\
\end{vmatrix}=s_{1}s_{2}\begin{vmatrix}d^{\prime}_{n}&d^{\prime}_{n+1}\\\
d^{\prime\prime}_{n}&d^{\prime\prime}_{n+1}\\\ \end{vmatrix}=s_{1}s_{2}>0$.
The latter statement follows from the fact that $ABCD$ is a parallelogram.

$R(a_{1},a_{2})=s_{1}T_{n}+s_{2}T_{n+1}.$

(b) If $(a_{1},a_{2})\in\Phi^{im}_{+}$, then define

$R(a_{1},a_{2})=\\{(p,q)\in\mathbb{R}_{\geq 0}^{2}\ |\ D(p,q)\geq 0\\}.$

$p$$q$(1) $a_{1},a_{2}\leq 0$i.e. $(a_{1},a_{2})=s_{1}{\bf d}_{1}+s_{2}{\bf
d}_{2}$$p$$q$(2) $a_{1}\leq 0<a_{2}$i.e. $(a_{1},a_{2})=s_{1}{\bf
d}_{0}+s_{2}{\bf d}_{1}$$A$$p$$q$(3) $a_{2}\leq 0<a_{1}$i.e.
$(a_{1},a_{2})=s_{1}{\bf d}_{2}+s_{2}{\bf d}_{3}$$C$$sT_{n}$$p$$q$(4)
$(a_{1},a_{2})=s{\bf d}_{n}$,$n\neq
0,1,2,3$$C$$A$$s_{1}T_{n}+s_{2}T_{n+1}$$p$$q$(5) $(a_{1},a_{2})=s_{1}{\bf
d}_{n}+s_{2}{\bf d}_{n+1}$,$s_{1},s_{2}>0$, $n\neq 0,1,2$$C$$A$$B$$O$$A$$D\geq
0$$C$$p$$q$(6) $(a_{1},a_{2})\in\Phi^{im}_{+}$ Figure 1. The region
$R(a_{1},a_{2})$. The cases (1)–(5) are for real roots
$(a_{1},a_{2})\notin\Phi^{im}_{+}$, and $A=(a_{2},0)$, $C=(0,a_{1})$,
$B=(s_{2}d^{\prime\prime}_{n+1},s_{1}d^{\prime}_{n})$.

######  Corollary 1.4.

Let $(a_{1},a_{2})\in\mathbb{Z}^{2}$, let $\\{e(p,q)\\}$ be the coefficients
of the triangular basis $C[a_{1},a_{2}]$ as in (1.1). Then $e(p,q)\neq 0$ if
and only if $(p,q)\in R(a_{1},a_{2})$, and in this case we have

$\deg e(p,q)=D(p,q).$

Now we would like to put the theorem into a geometric context. For an
irreducible variety $X$, a local system $L$ on (an open dense subset of) $X$
and an integer $d$, denote by $IC_{X}(L)[d]$ the intersection cohomology
complex on $X$, with coefficients in $L$ shifted by $d$; for $d=0$,
$IC_{X}(L)[0]$ is just $IC_{X}(L)$; if $L=\mathbb{Q}$ is trivial, then we just
denote $IC_{X}(L)[d]$ by $IC_{X}[d]$. (Note that
$IC_{X}(L)=\mathcal{IC}_{X}(L)[\dim V]$ and in particular,
$IC_{X}=\mathbb{Q}[\dim X]$ when $X$ is nonsingular; please see [5, §1.6] for
the definition of $\mathcal{IC}$ and the difference between $IC$ and
$\mathcal{IC}$.) The famous BBDG Decomposition theorem [2] asserts that, for a
proper map of algebraic varieties $f:X\to Y$, the pushforward of the
intersection cohomology complex $IC_{X}$ can be decomposed (non-canonically)
into a direct sum of $IC_{V}(L)[d]$ for a collection of irreducible
subvarieties $V\subseteq Y$, local systems $L$ on (an open dense subset of)
$V$, and integers $d$. The support problem (that is, to determine which
subvarieties $V$ actually occur in the BBDG decomposition) can be quite
difficult, as stated in a comment in Ngo’s proof of the Fundamental Lemma of
Langlands theory: “…Their supports constitute an important topological
invariant of $f$. In general, it is difficult to explicitly determine this
invariant” (translated from French). Several recent studies are focusing on
the support problem of particular morphisms; the readers are referred to the
references [17, 16, 6, 18, 8].

In the above context, we have a corollary of the main theorem, which gives a
complete solution to the support problem for the natural projection from a
class of Nakajima’s nonsingular graded quiver varieties to the affine graded
quiver varieties.

First we fix some notations. Let $v=(v_{1},v_{2})\in\mathbb{Z}_{\geq
0}^{2},w=(0,w^{\prime}_{1},w_{2},0)\in\mathbb{Z}_{\geq 0}^{4}$. Let
${\tilde{\mathcal{F}}_{v,w}},\mathbf{E}_{v,w}$ be Nakajima’s nonsingular
graded quiver variety and affine graded quiver variety, respectively. (See §3
for more related definitions and notations). The condition $0\leq v_{1}\leq
w_{1}^{\prime}\textrm{ and }0\leq v_{2}\leq rv_{1}$ guarantees that that
${\tilde{\mathcal{F}}_{v,w}}$ is nonempty. There is a natural projection
$\pi:{\tilde{\mathcal{F}}_{v,w}}\to\mathbf{E}_{v,w}$, and the BBDG
decomposition takes the following form (and we will show that the local
systems are all trivial):

(1.3)
$\pi_{*}(IC_{\tilde{\mathcal{F}}_{v,w}})=\bigoplus_{v^{\prime}=(v^{\prime}_{1},v^{\prime}_{2})}\bigoplus_{d}a^{d}_{v,v^{\prime};w}IC_{\mathbf{E}_{v^{\prime},w}}[d].$

######  Corollary 1.5.

Let $v=(v_{1},v_{2})\in\mathbb{Z}_{\geq
0}^{2},w=(0,w^{\prime}_{1},w_{2},0)\in\mathbb{Z}_{\geq 0}^{4}$ such that
$0\leq v_{1}\leq w_{1}^{\prime}\textrm{ and }0\leq v_{2}\leq rv_{1}$. The
coefficients $a^{d}_{v,v^{\prime};w}$ in (1.3) are related to the triangular
basis as follows: let

(1.4)
$a_{1}=w^{\prime}_{1}-rv^{\prime}_{2},\;a_{2}=r[a_{1}]_{+}+rv^{\prime}_{1}-w_{2},\;p=v_{2}-v^{\prime}_{2},\;q=[a_{1}]_{+}-v_{1}+v^{\prime}_{1},$

and let $e(p,q)$ be determined by (1.1), then
$e(p,q)=\sum_{d}a_{v,v^{\prime};w}^{d}\mathbf{v}^{rd}.$

As a consequence, a subvariety $\mathbf{E}_{v^{\prime},w}$ appears in the BBDG
decomposition (1.3) if and only if “ $0\leq v^{\prime}_{1}\leq v_{1},\ 0\leq
v^{\prime}_{2}\leq v_{2}$, and $(p,q)\in R(a_{1},a_{2})$ for $a_{1},a_{2},p,q$
defined in (1.4)”.

The reader is referred to §10.4 for an example of the above corollary.

The paper is organized as follows. In §2 we recall the definition and
properties of quantum cluster algebra and triangular basis, and that
triangular basis coincides with dual canonical basis. In §3 we study the
support condition for cluster monomials. In §4 we recall some facts on
Nakajima’s quiver varieties. In §5 we prove an algebraic version of the
transversal slice theorem, and show that the local systems appeared in the
BBDG decomposition theorem are trivial. In §6 we introduce two other
desingularizations of $\mathbf{E}_{v,w}$. In §7 we study the triangular bases.
In §8 we prove the above main theorem (Theorem 1.2) and the two corollaries
(Corolloaries 1.4, 1.5). In §9 we discuss an analogue of the Kazhdan-Lusztig
polynomial that comes from the geometry of this paper. We end the paper with
some examples in §10.

Acknowledgments. he author learned a lot from the collaborated work with
Kyungyong Lee, Dylan Rupel, Andrei Zelevinsky, based on which the current work
is possible; he also gratefully acknowledges discussions with Mark de Cataldo
and Fan Qin; he thanks Pramod Achar for sharing a draft of his book, which is
now available in print [1]; he greatly appreciates the referee for carefully
reading through the paper and providing many helpful comments and suggestions.
Computer calculations were performed using SageMath [22].

##  2\. Quantum cluster algebras and triangular bases

In this section we recall the definition and some fundamental facts of rank-2
quantum cluster algebras. See [3, 4] for details.

###  2.1. Rank-2 quantum cluster algebras

First, let $\mathcal{T}$ be the quantum torus:

$\mathcal{T}=\mathbb{Z}[\mathbf{v}^{\pm 1}]\langle X_{1}^{\pm 1},X_{2}^{\pm
1}:X_{2}X_{1}=\mathbf{v}^{2}X_{1}X_{2}\rangle,$

let $\mathcal{F}$ be the skew-field of fractions of $\mathcal{T}$.

The _bar-involution_ is the $\mathbb{Z}$-linear anti-automorphism of
$\mathcal{T}$ determined by $\overline{f}(\mathbf{v})=f(\mathbf{v}^{-1})$ for
$f\in\mathbb{Z}[\mathbf{v}^{\pm 1}]$ and

$\overline{fX_{1}^{a_{1}}X_{2}^{a_{2}}}=\overline{f}X_{2}^{a_{2}}X_{1}^{a_{1}}=\mathbf{v}^{2a_{1}a_{2}}\overline{f}X_{1}^{a_{1}}X_{2}^{a_{2}}\quad\text{($a_{1},a_{2}\in\mathbb{Z}$)}.$

An element in $\mathcal{T}$ which is invariant under the bar-involution is
said to be _bar-invariant_.

The _rank-2 quantum cluster algebra_ $\mathcal{A}_{\mathbf{v}}(b,c)$ is the
$\mathbb{Z}[\mathbf{v}^{\pm 1}]$-subalgebra of $\mathcal{F}$ generated by the
_cluster variables_ $X_{m}$ ($m\in\mathbb{Z}$) defined recursively from the
_exchange relations_

$X_{m+1}X_{m-1}=\begin{cases}&\mathbf{v}^{b}X_{m}^{b}+1,\textrm{ if $m$ is
odd};\\\ &\mathbf{v}^{c}X_{m}^{c}+1,\textrm{ if $m$ is even};\\\ \end{cases}$

(In notations of [4], we take $\Lambda=\begin{bmatrix}0&-b\\\
c&0\end{bmatrix}$, $B=\begin{bmatrix}0&-b\\\ c&0\end{bmatrix}$,
$d_{1}=c,d_{2}=b$, with a linear order $1\vartriangleleft 2$.)

One can easily check that

(2.1) $X_{m+1}X_{m}=\mathbf{v}^{2}X_{m}X_{m+1}\quad(m\in\mathbb{Z}),$

and that all cluster variables are bar-invariant. Equation (2.1) implies that
each _cluster_ $\\{X_{m},X_{m+1}\\}$ generates a quantum torus
$\mathcal{T}_{m}=\mathbb{Z}[\mathbf{v}^{\pm 1}]\langle X_{m}^{\pm
1},X_{m+1}^{\pm 1}:X_{m+1}X_{m}=\mathbf{v}^{2}X_{m}X_{m+1}\rangle$. The (bar-
invariant) _quantum cluster monomials_ in the quantum torus $\mathcal{T}_{m}$
as

$X^{(a_{1},a_{2})}_{m}=\mathbf{v}^{a_{1}a_{2}}X_{m}^{a_{1}}X_{m+1}^{a_{2}}\quad(a_{1},a_{2}\in\mathbb{Z}_{\geq
0},\,\,m\in\mathbb{Z}).$

Denote $X^{(a_{1},a_{2})}=X^{(a_{1},a_{2})}_{1}$. Note that
$X^{(a_{1},a_{2})}X^{(b_{1},b_{2})}=\mathbf{v}^{a_{2}b_{1}-a_{1}b_{2}}X^{(a_{1}+b_{1},a_{2}+b_{2})}$.

###  2.2. Triangular bases

The construction of the triangular basis starts with the standard monomial
basis $\\{M[{\bf a}]\,:\,{\bf a}\in\mathbb{Z}^{2}\\}$. For every ${\bf
a}=(a_{1},a_{2})\in\mathbb{Z}^{2}$, the _standard monomial_ $M[{\bf a}]$
(which is denoted $E_{(-a_{1},-a_{2})}$ in [4]) is defined as follows, where
$X^{\prime}_{1}=X_{3}$, $X^{\prime}_{2}=X_{0}$, and $v({\bf a})\in\mathbb{Z}$
is determined by the condition that the leading term of $M({\bf a})$ is bar-
invariant):

(2.2) $M[{\bf a}]=\mathbf{v}^{v({\bf
a})}X^{([-a_{1}]_{+},[-a_{2}]_{+})}(X_{1}^{\prime})^{[a_{1}]_{+}}(X_{2}^{\prime})^{[a_{2}]_{+}}=\mathbf{v}^{a_{1}a_{2}}X_{3}^{[a_{1}]_{+}}X_{1}^{[-a_{1}]_{+}}X_{2}^{[-a_{2}]_{+}}X_{0}^{[a_{2}]_{+}}\
.$

It is known that the elements $M[{\bf a}]$ form a $\mathbb{Z}[\mathbf{v}^{\pm
1}]$-basis of the cluster algebra $\mathcal{A}_{\mathbf{v}}(b,c)$.

The standard monomials are not bar-invariant and do not contain all the
cluster monomials, moreover they are inherently dependent on the choice of an
initial cluster. These drawbacks provide a motivation to consider the
triangular basis (which is introduced in [4] and recalled below) constructed
from the standard monomial basis with a built-in bar-invariance property.

In this paper, for ${\bf a}^{\prime}=(a_{1}^{\prime},a_{2}^{\prime}),{\bf
a}=(a_{1},a_{2})\in\mathbb{Z}^{2}$, we denote

$\displaystyle{\bf a}^{\prime}\leq{\bf a}\ (\textrm{or }{\bf a}\geq{\bf
a}^{\prime}),\textrm{ if $a_{i}^{\prime}\leq a_{i}$ for $i=1,2$;}$
$\displaystyle{\bf a}^{\prime}<{\bf a}\ (\textrm{or }{\bf a}>{\bf
a}^{\prime}),\textrm{ if ${\bf a}^{\prime}\leq{\bf a}$ and ${\bf
a^{\prime}}\neq{\bf a}$.}$

######  Definition 2.1.

[4] The triangular basis $\\{C[{\bf a}]\,:\,{\bf a}\in\mathbb{Z}^{2}\\}$ is
the unique collection of elements in $\mathcal{A}_{\mathbf{v}}(b,c)$
satisfying:

  1. (P1) Each $C[{\bf a}]$ is bar-invariant.

  2. (P2) For each ${\bf a}\in\mathbb{Z}^{2}$, $\displaystyle C[{\bf a}]-M[{\bf a}]\in\bigoplus_{{\bf a}^{\prime}<{\bf a}}\mathbf{v}\mathbb{Z}[\mathbf{v}]M[{\bf a}^{\prime}].$

The triangular basis has the following nice property:

######  Theorem 2.2 ([4, Theorem 1.6]).

The triangular basis does not depend on the choice of initial cluster and it
contains all cluster monomials.

##  3\. The case of real roots

In this section, we study the support of a cluster monomial in a skew-
symmetrizable quantum cluster algebra $\mathcal{A}_{\mathbf{v}}(b,c)$, that
is, without assuming $b=c$. The denominator vectors $\\{{\bf d}_{n}\\}$ can be
determined recursively by

${\bf d}_{1}=\begin{bmatrix}-1\\\ 0\end{bmatrix},\quad{\bf
d}_{2}=\begin{bmatrix}0\\\ -1\end{bmatrix},\quad{\bf d}_{n+1}+{\bf
d}_{n-1}=\begin{cases}&b[{\bf d}_{n}]_{+},\textrm{ if $n$ is odd},\\\ &c[{\bf
d}_{n}]_{+},\textrm{ if $n$ is even}.\\\ \end{cases}$

Below are a few values of ${\bf d}_{n}$:

$n$ | $\cdots$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\cdots$
---|---|---|---|---|---|---|---|---|---|---|---|---
$d^{\prime}_{n}$ | $\cdots$ | $bc-1$ | $b$ | $1$ | $0$ | $-1$ | $0$ | $1$ | $b$ | $bc-1$ | $b^{2}c-2b$ | $\cdots$
$d^{\prime\prime}_{n}$ | $\cdots$ | $bc^{2}-2c$ | $bc-1$ | $c$ | $1$ | $0$ | $-1$ | $0$ | $1$ | $c$ | $bc-1$ | $\cdots$

######  Theorem 3.1.

Consider quantum cluster algebra $\mathcal{A}_{\mathbf{v}}(b,c)$ for arbitrary
$b,c\in\mathbb{Z}_{>0}$. Let
$(a_{1},a_{2})\in\mathbb{Z}^{2}\setminus\Phi^{im}_{+}$. (Thus
$(a_{1},a_{2})=s_{1}{\bf d}_{n}+s_{2}{\bf d}_{n+1}$ for some $n\in\mathbb{Z}$
and $s_{1},s_{2}\in\mathbb{Z}_{\geq 0}$, and
$C[a_{1},a_{2}]=\mathbf{v}^{s_{1}s_{2}}X_{n}^{s_{1}}X_{n+1}^{s_{2}}$ is a
cluster monomial.) Define $\\{e(p,q)\\}$ as in (1.1). Then $e(p,q)\neq 0$ if
and only if $(p,q)\in s_{1}T_{n}+s_{2}T_{n+1}=R(a_{1},a_{2})$, and for those
$(p,q)$, we have $\deg e(p,q)=D(p,q)$.

###### Proof.

We prove the statement by induction on $|n-1|$.

The base cases are $n=0,1,2$.

For $n=1$: $C[a_{1},a_{2}]=C[-s_{1},-s_{2}]=X^{(s_{1},s_{2})}$; $e(p,q)=1$ if
$(p,q)=(0,0)$, and $e(p,q)=0$ otherwise; $s_{1}T_{1}+s_{2}T_{2}=\\{(0,0)\\}$;
for $(p,q)=0$, $\deg e(0,0)=0=D(0,0)$. So the statement is true in this case.

For $n=2$:
$C[a_{1},a_{2}]=C[s_{2},-s_{1}]=X_{2}^{(s_{1},s_{2})}=\sum_{q=0}^{s_{2}}{s_{2}\brack
q}_{\mathbf{v}^{c}}X^{(-s_{2},s_{1}+cq)}$ (this computation appeared in the
proof of [13, Corollary 3.6]); $e(p,q)={s_{2}\brack q}_{\mathbf{v}^{c}}$ if
$p=0$ and $0\leq q\leq s_{2}$, and $e(p,q)=0$ otherwise;
$s_{1}T_{2}+s_{2}T_{3}=\\{(0,y)\ |\ 0\leq y\leq s_{2}\\}$; for $p=0$ and
$0\leq q\leq s_{2}$, $\deg e(0,q)=(s_{2}-q)qc=D(0,q)$. So the statement is
true in this case.

For $n=0$: similar as the $n=2$ case so we skip.

Recall in [13] we defined automorphisms $\sigma_{\ell}$ ($\ell=1,2$) on the
quantum torus such that $\sigma_{\ell}(X_{m})=X_{2\ell-m}$ and
$\sigma_{\ell}(\mathbf{v})=\mathbf{v}^{-1}$.

For the inductive step, we shall prove the case $n\geq 3$.

Without loss of generality we assume $s_{1}\neq 0$ (otherwise we can replace
$(n,s_{1},s_{2})$ by $(n-1,s_{2},0)$). Denote
$C[a^{\prime}_{1},a^{\prime}_{2}]=\sigma_{2}(C[a_{1},a_{2}])=v^{s_{2}s_{1}}X_{3-n}^{s_{2}}X_{4-n}^{s_{1}}$
(which is also a cluster monomial) and define
$e^{\prime}(p^{\prime},q^{\prime})$ by

$C[a_{1}^{\prime},a_{2}^{\prime}]=\sum_{p^{\prime},q^{\prime}}e^{\prime}(p^{\prime},q^{\prime})X^{(bp^{\prime}-a_{1}^{\prime},cq^{\prime}-a_{2}^{\prime})}.$

Then $a_{1}^{\prime}=ba_{2}-a_{1}$, $a_{2}^{\prime}=a_{2}$, and by inductive
hypothesis, the statement is true for $C[a^{\prime}_{1},a^{\prime}_{2}]$; that
is, $e^{\prime}(p^{\prime},q^{\prime})\neq 0$ if and only if $(p,q)\in
R(a^{\prime}_{1},a^{\prime}_{2})\ (=s_{2}T_{3-n}+s_{1}T_{4-n})$, and for those
$(p^{\prime},q^{\prime})$, $\deg
e^{\prime}(p^{\prime},q^{\prime})=D^{\prime}(p^{\prime},q^{\prime})$ where
$D^{\prime}(p^{\prime},q^{\prime})=ca^{\prime}_{1}q^{\prime}+ba^{\prime}_{2}p^{\prime}-bp^{\prime
2}-bcp^{\prime}q^{\prime}-cq^{\prime 2}$.

For a fixed $p$, denote $p^{\prime}=a_{2}-p$, there is the following relation
by [13, Proof of Lemma 2.2]:

(3.1)
$\sum_{q^{\prime}}e^{\prime}(p^{\prime},q^{\prime})t^{q^{\prime}}=\left(\sum_{\ell\geq
0}{{-a_{1}+bp}\brack{\ell}}_{\mathbf{v}^{c}}t^{\ell}\right)\left(\sum_{q}e(p,q)t^{q}\right)$

which we can also rewrite as

(3.2) $\left(\sum_{\ell\geq
0}{{a_{1}-bp}\brack{\ell}}_{\mathbf{v}^{c}}t^{\ell}\right)\left(\sum_{q^{\prime}}e^{\prime}(p^{\prime},q^{\prime})t^{q^{\prime}}\right)=\sum_{q}e(p,q)t^{q}$

By inductive hypothesis, $e^{\prime}(p^{\prime},q^{\prime})\neq
0\Rightarrow(p^{\prime},q^{\prime})\in R(a_{1}^{\prime},a_{2}^{\prime})$.
Considering the highest $t$-degree terms on both sides of (3.1) (if $a_{1}\leq
bp$) or (3.2) (if $a_{1}\geq bp$), we see that

(3.3) $a_{1}-bp+\max\\{q^{\prime}\ |\ e^{\prime}(p^{\prime},q^{\prime})\neq
0\\}=\max\\{q\ |\ e(p,q)\neq 0\\}$

We claim that the convex hull of $\\{(p,q)\ |\ e(p,q)\neq 0\\}$ is
$R(a_{1},a_{2})=s_{1}T_{n}+s_{2}T_{n+1}$. Indeed, by inductive hypothesis, the
convex hull of $\\{(p^{\prime},q^{\prime})\ |\
e^{\prime}(p^{\prime},q^{\prime})\neq 0\\}$ is
$R(a^{\prime}_{1},a^{\prime}_{2})=s_{2}T_{3-n}+s_{1}T_{4-n}$, a polygon with
vertices $(0,0)$ and
$(a^{\prime}_{2},0)=(s_{2}d^{\prime\prime}_{3-n}+s_{1}d^{\prime\prime}_{4-n},0)$,
$(s_{1}d^{\prime\prime}_{4-n},s_{2}d^{\prime}_{3-n})$,
$(0,a^{\prime}_{1})=(s_{2}d^{\prime}_{3-n}+s_{1}d^{\prime}_{4-n})$. Thus by
(3.3), the convex hull of $\\{(p,q)\ |\ e(p,q)\neq 0\\}$ has vertices $(0,0)$
and
$(0,a_{1}),(a_{2}-s_{1}d^{\prime\prime}_{4-n},a_{1}-b(a_{2}-s_{1}d^{\prime\prime}_{4-n})+s_{2}d^{\prime}_{3-n})=(s_{2}d^{\prime\prime}_{n+1},s_{1}d^{\prime}_{n}),(a_{2},a_{1}-b(a_{2}-0)+a^{\prime}_{1})=(a_{2},0)$;
so the convex hull is exactly $s_{1}T_{n}+s_{2}T_{n+1}$. Another way to see it
is to consider the map

(3.4) $\varphi_{R}:(x,y)\mapsto(a_{2}-x,y-(a_{1}-bp))$

which sends the top bound of $R(a_{1},a_{2})$ to the top bound of
$R(a^{\prime}_{1},a^{\prime}_{2})$.

Now we prove the theorem by considering two cases separately:

Case 1: if $a_{1}\geq bp$. Fix $q$ in (3.2), we get

$e(p,q)=\sum_{\ell+q^{\prime}=q}{{a_{1}-bp}\brack{\ell}}_{\mathbf{v}^{c}}e^{\prime}(p^{\prime},q^{\prime}).$

Assume $(p,q)\in R(a_{1},a_{2})$, we shall show that $\deg e(p,q)=D(p,q)$. It
suffices to show that
$\deg{{a_{1}-bp}\brack{\ell}}_{\mathbf{v}^{c}}e^{\prime}(p^{\prime},q^{\prime})\leq
D(p,q)$ and the equality holds exactly once. Indeed, using
$a^{\prime}_{1}=ba_{2}-a_{1},a^{\prime}_{2}=a_{2},p^{\prime}=a_{2}-p,q^{\prime}=q-\ell$,
we have

$\displaystyle
D(p,q)-\Big{(}\deg{{a_{1}-bp}\brack{\ell}}_{\mathbf{v}^{c}}\Big{)}e^{\prime}(p^{\prime},q^{\prime})=D(p,q)-(a_{1}-bp-\ell)\ell
c-D^{\prime}(p^{\prime},q^{\prime})$
$\displaystyle\quad=ca_{1}q+ba_{2}p-bp^{2}-bcpq-cq^{2}-(a_{1}-bp-\ell)\ell c$
$\displaystyle\quad\quad-c(b_{2}-a_{1})(q-\ell)-ba_{2}(a_{2}-p)+b(a_{2}-p)^{2}+bc(a_{2}-p)(q-\ell)+c(q-\ell)^{2}$
$\displaystyle\quad=2c(q-\ell)(a_{1}-bp-\ell)\geq 0$

since $\ell\leq\min\\{q,a_{1}-bp)$. Moreover, the equality can be obtained
exactly once, when $\ell=\min\\{q,a_{1}-bp\\}$; for this we need to verify
that the corresponding $(p^{\prime},q^{\prime})$ lies in
$R(a^{\prime}_{1},a^{\prime}_{2})$:

– if $q\leq a_{1}-bp$, then $\ell=q$, $q^{\prime}=0$, in which case
$(p^{\prime},q^{\prime})=(p^{\prime},0)\in R(a^{\prime}_{1},a^{\prime}_{2})$
(note that $(p,q)\in R(a_{1},a_{2})$ guarantees $0\leq p^{\prime}\leq
a^{\prime}_{2}$);

– if $q\geq a_{1}-bp$, then $\ell=a_{1}-bp$,
$(p^{\prime},q^{\prime})=(a_{2}-p,q-(a_{1}-bp))=\varphi(p,q)$ which is in
$R(a^{\prime}_{1},a^{\prime}_{2})$.

Case 2: if $a_{1}\leq bp$. By (3.1), for any fixed $q^{\prime}$ we have

(3.5)
$e^{\prime}(p^{\prime},q^{\prime})=\sum_{\ell+q=q^{\prime}}{{-a_{1}+bp}\brack{\ell}}_{\mathbf{v}^{c}}e(p,q).$

By a computation similar to Case 1, when $\ell+q=q^{\prime}$, we have

(3.6) $D(p^{\prime},q^{\prime})-(bp-a_{1}-\ell)\ell
c-D(p,q)=2c(q^{\prime}-\ell)(bp-a_{1}-\ell).$

We will prove that $\deg e(p,q)=D(p,q)$ for all $q$ with $(p,q)\in
R(a_{1},a_{2})$ by a downward induction.

As the base case, let $q_{\max}$ be the maximal $q$ satisfying $(p,q)\in
R(a_{1},a_{2})$ and let $\ell=bp-a_{1},q^{\prime}=\ell+q_{\max}$. Then
$(p^{\prime},q^{\prime})=(a_{2}-p,\ell+q_{\max})=\varphi_{R}(p,q_{\max})$ lies
in $R(a^{\prime}_{1},a^{\prime}_{2})$, thus $\deg
e^{\prime}(p^{\prime},q^{\prime})=D^{\prime}(p^{\prime},q^{\prime})$. Since
both $\ell$ and $q_{\max}$ are maximal, there is only one term in the right
side of (3.5), thus $\deg e(p,q_{\max})=(bp-a_{1}-\ell)\ell c+\deg
e(p,q_{\max})=D^{\prime}(p^{\prime},q^{\prime})=D(p,q_{\max})$.

For the inductive step, assume $\deg e(p,q^{*})=D(p,q^{*})$ for all $q^{*}>q$
with $(p,q^{*})\in R(a_{1},a_{2})$. Recall that we already proved
$e(p,q^{*})=0$ for $(p,q^{*})\notin R(a_{1},a_{2})$. Let $\ell=bp-
a_{1},q^{\prime}=\ell+q$. Then
$(p^{\prime},q^{\prime})=(a_{2}-p,\ell+q)=\varphi_{R}(p,q)$ lies in
$R(a^{\prime}_{1},a^{\prime}_{2})$, thus $\deg
e^{\prime}(p^{\prime},q^{\prime})=D^{\prime}(p^{\prime},q^{\prime})$.
Comparing the degrees of the two side of (3.5), we have

$D^{\prime}(p^{\prime},q^{\prime})\leq\max_{\stackrel{{\scriptstyle q\leq
q^{*}\leq q_{\max}}}{{\ell^{*}+q^{*}=q^{\prime}}}}\\{(bp-
a_{1}-\ell^{*})\ell^{*}c+\deg e(p,q^{*})\\}$

However, for $q^{*}$ satisfying $q+1\leq q^{*}\leq q_{\max}$, we already know
$\deg e(p,q^{*})=D(p,q^{*})$, so

$\displaystyle(bp-a_{1}-\ell^{*})\ell^{*}c+\deg e(p,q^{*})=(bp-
a_{1}-\ell^{*})\ell^{*}c+D(p,q^{*})$
$\displaystyle\stackrel{{\scriptstyle\text{by
}\eqref{eq:D'-D}}}{{=}}D^{\prime}(p^{\prime},q^{\prime})-2c(q^{\prime}-\ell^{*})(bp-
a_{1}-\ell^{*})=D^{\prime}(p^{\prime},q^{\prime})-2cq^{*}(q^{*}-q)<D^{\prime}(p^{\prime},q^{\prime}).$

So for $q^{*}=q$, $\ell^{*}=q^{\prime}-q^{*}=bp-a_{1}=\ell$, we must have

$D^{\prime}(p^{\prime},q^{\prime})=(bp-a_{1}-\ell^{*})\ell^{*}c+\deg
e(p,q^{*})$

It follows that

$\deg e(p,q)=D^{\prime}(p^{\prime},q^{\prime})-(bp-
a_{1}-\ell^{*})\ell^{*}c\stackrel{{\scriptstyle\text{by
}\eqref{eq:D'-D}}}{{=}}D(p,q)+2c(q^{\prime}-\ell^{*})(bp-
a_{1}-\ell^{*})=D(p,q).$

This completes the proof for the case $n\geq 3$. The case $n\leq-1$ can be
proved similarly by considering $\sigma_{1}$ instead of $\sigma_{2}$. ∎

In the rest of the paper we focus on skew-symmetric rank-2 cluster algebra
$\mathcal{A}_{\mathbf{v}}(r,r)$.

##  4\. Nakajima’s graded quiver varieties

In this section we recall Nakajima’s graded quiver varieties. For simplicity,
we focus to the special case corresponding to rank-2 cluster algebras. The
references for this section are [19, 21]. Many statements in this section are
known to hold in a much more general setting; but with a focus on the special
case, we can present an almost self-contained introduction by proving most of
the geometric facts without referring to the aforementioned papers.

Denote nonnegative integer tuples
$w=(w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime})\in\mathbb{Z}_{\geq 0}^{4}$ and
$v=(v_{1},v_{2})\in\mathbb{Z}_{\geq 0}^{2}$. Fix $\mathbb{C}$-vector spaces
$W_{1},W^{\prime}_{1},W_{2},W^{\prime}_{2},V_{1},V_{2}$ with dimensions
$w_{1},w^{\prime}_{1},w_{2},w^{\prime}_{2},v_{1},v_{2}$, respectively. Define

$W=W_{1}\oplus W_{1}^{\prime}\oplus W_{2}\oplus W_{2}^{\prime},\quad
V=V_{1}\oplus V_{2}.$

Consider $(x_{1},x_{2},y_{1},\dots,y_{r})$, where $x_{i}:W_{i}\to
W^{\prime}_{i}$ ($i=1,2$), $y_{h}:W_{2}\to W^{\prime}_{1}$ ($h=1,\dots,r$) are
linear maps.

(4.1)
$\textstyle{W_{2}^{\prime}}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{2}}$$\scriptstyle{\phantom{XX}y_{h}\;(h=1,\dots,r)}$$\textstyle{W_{1}^{\prime}}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{1}}$

Denote ${\bf x}=(x_{1},x_{2})$ and ${\bf y}=(y_{1},\dots,y_{r})$. Define two
linear maps as below:

$\displaystyle A=x_{1}+y_{1}+\cdots+y_{r}:$ $\displaystyle\;W_{1}\oplus
W_{2}^{\oplus r}\to W_{1}^{\prime},\quad(a_{1},b_{1},\dots,b_{r})\mapsto
x_{1}(a_{1})+y_{1}(b_{1})+\cdots+y_{r}(b_{r})$ $\displaystyle B=x_{2}\oplus
y_{1}\oplus\cdots\oplus y_{r}:$ $\displaystyle\;W_{2}\to
W_{2}^{\prime}\oplus(W_{1}^{\prime})^{\oplus r},\quad
b\mapsto(x_{2}(b),y_{1}(b),\dots,y_{r}(b))$

Alternatively, we can express these maps by matrices. Fix bases for $W_{i}$
and $W^{\prime}_{i}$ , and use $x_{1}$, or simply $x_{1}$ by abuse of
notation, to denote the matrix representing the linear map $x_{1}$. Similar
notations are used for $x_{2}$, $y_{h}$ ($h=1,\dots,r$). Then the above two
maps $A$ and $B$ are represented by the following block matrices of sizes
$w_{1}^{\prime}\times(w_{1}+rw_{2})$ and
$(w_{2}^{\prime}+rw_{1}^{\prime})\times w_{2}$, respectively (denote $p=({\bf
x},{\bf y})$):

(4.2) $\displaystyle A=A_{p}=A({\bf
x,y})=[x_{1},y_{1},\dots,y_{r}]_{\text{hor}}=\begin{bmatrix}\framebox{$x_{1}$}&\framebox{$y_{1}$}&\cdots&\framebox{$y_{r}$}\end{bmatrix},\quad$
$\displaystyle B=B_{p}=B({\bf
x,y})=[x_{2},y_{1},\dots,y_{r}]_{\text{vert}}=\begin{bmatrix}\,\framebox{$x_{2}$}\,\\\
\framebox{$y_{1}$}\\\ \vdots\\\ \framebox{$y_{r}$}\end{bmatrix}$

where we use the following notation for horizontally and vertically stacked
matrices:

$[M_{1},\dots,M_{n}]_{\text{hor}}=\begin{bmatrix}M_{1}&\cdots&M_{n}\\\
\end{bmatrix},\quad[M_{1},\dots,M_{n}]_{\text{vert}}=\begin{bmatrix}M_{1}\\\
\vdots\\\ M_{n}\\\ \end{bmatrix}$

Below is the list of sizes of various matrices:

matrix | $x_{1}$ | $x_{2}$ | $y_{h}$ | $A$ | $B$
---|---|---|---|---|---
size | $w_{1}^{\prime}\times w_{1}$ | $w_{2}^{\prime}\times w_{2}$ | $w_{1}^{\prime}\times w_{2}$ | $w^{\prime}_{1}\times(w_{1}+rw_{2})$ | $(w^{\prime}_{2}+rw^{\prime}_{1})\times w_{2}$

For an $m\times n$ matrix $A$, we denote

$A_{[I;J]}=[a_{ij}]_{i\in I,j\in J}=\text{ the submatrix of $A$ with row index
set $I$ and column index set $J$},$

denote

$A_{[-;J]}=A_{[1,\dots,m;J]},\quad A_{[I;-]}=A_{[I;1,\dots,n]}$

For a $\mathbb{C}$-vector space $E$ of dimension $e$, denote by $Gr(d,E)$ or
$Gr(d,e)$ the Grassmannian space parametrizing all $d$-dimensional linear
subspaces of $E$.

######  Definition 4.1.

[19, §4] Let $w=(w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime})\in\mathbb{Z}_{\geq
0}^{4}$ and $v=(v_{1},v_{2})\in\mathbb{Z}_{\geq 0}^{2}$.

(a) The variety $\mathbf{E}_{w}$ is defined as the space of quiver
representations of the quiver in (4.1):

$\mathbf{E}_{w}=\mathrm{Hom}(W_{1},W_{1}^{\prime})\oplus\mathrm{Hom}(W_{2},W_{2}^{\prime})\oplus\mathrm{Hom}(W_{2},W_{1}^{\prime})^{\oplus
r}\cong\mathbb{C}^{w_{1}w_{1}^{\prime}+w_{2}w_{2}^{\prime}+rw_{1}^{\prime}w_{2}}.$

We denote elements of $\mathbf{E}_{w}$ in the form $({\bf x},{\bf
y})=(x_{1},x_{2},y_{1},\dots,y_{r})$.

(b) The projective variety $\mathcal{F}_{v,w}$ is a subvariety of
$Gr(v_{1},W^{\prime}_{1})\times
Gr(v_{2},W^{\prime}_{2}\oplus{W^{\prime}_{1}}^{\oplus r})$ defined as

$\mathcal{F}_{v,w}=\\{(X_{1},X_{2})\;|\;\dim X_{i}=v_{i}\,(i=1,2),\quad
X_{1}\subseteq W^{\prime}_{1},\quad X_{2}\subseteq W_{2}^{\prime}\oplus
X_{1}^{\oplus r}\\}.$

(c) _Nakajima’s nonsingular graded quiver variety_ $\tilde{\mathcal{F}}_{v,w}$
is given by

$\tilde{\mathcal{F}}_{v,w}=\\{({\bf x},{\bf y},X_{1},X_{2})\;|\;({\bf x},{\bf
y})\in\mathbf{E}_{w},(X_{1},X_{2})\in\mathcal{F}_{v,w},\;{\mathrm{im}}A({\bf
x},{\bf y})\subseteq X_{1},\;{\mathrm{im}}B({\bf x},{\bf y})\subseteq
X_{2}\\}.$

where ${\mathrm{im}}A({\bf x},{\bf y})$ and ${\mathrm{im}}B({\bf x},{\bf y})$
mean the image of the corresponding linear maps.

(d) _Nakajima’s affine graded quiver variety_ $\mathbf{E}_{v,w}$ is given by

$\displaystyle\mathbf{E}_{v,w}$ $\displaystyle=\\{({\bf x},{\bf
y})\in\mathbf{E}_{w}\,|\,\mathrm{rank}A({\bf x,y})\leq
v_{1},\;\mathrm{rank}B({\bf x,y})\leq v_{2}\\}.$

######  Remark 4.2.

For readers who are used to notations in Nakajima’s paper [19], his notations
are related to ours in the following way: $W_{1}(q^{2})$, $W_{1}(1)$,
$W_{2}(q^{3})$, , $W_{2}(q)$, $\mathcal{F}(v,W)$, $\tilde{\mathcal{F}}(v,W)$,
$E_{W}$, $V_{1}(q)$, $V_{2}(q^{2})$ are our $W_{1}$, $W_{1}^{\prime}$,
$W_{2}$, $W_{2}^{\prime}$, $\mathcal{F}_{v,w}$, $\tilde{\mathcal{F}}_{v,w}$,
$\mathbf{E}_{w}$, $V_{1}$, $V_{2}$, respectively. The Cartan matrix
$\mathbf{C}=(c_{ij})=\begin{bmatrix}2&-r\\\ -r&2\end{bmatrix}$. The formula
[19, (3.2)] $C_{q}:(v_{i}(a))\mapsto(v^{\prime}_{i}(a)),\text{ where
}v^{\prime}_{i}(a)=v_{i}(aq)+v_{i}(aq^{-1})+\sum_{j\neq i}c_{ij}v_{j}(a)$
gives

$v^{\prime}_{1}(q^{2})=v_{1}-rv_{2},\;v^{\prime}_{1}(1)=v_{1},\;v^{\prime}_{2}(q^{3})=v_{2},\;v^{\prime}_{2}(q)=v_{2}-rv_{1}.$

In our notation, we write

(4.3) $C_{q}v=(v_{1}-rv_{2},v_{1},v_{2},v_{2}-rv_{1})$

The affine and graded quiver varieties are defined as algebro-geometric
quotient and GIT quotient, respectively:

(a) Affine graded quiver variety
${\mathfrak{M}}_{0}^{\bullet}(V,W)=\\{(\alpha_{i},\beta_{i},b_{j})\\}//\prod
GL(V_{i})$

(b) Nonsingular graded quiver variety
${\mathfrak{M}}^{\bullet}(V,W)=\\{(\alpha_{i},\beta_{i},b_{j})\\}^{s}/\prod
GL(V_{i})$

where $(\alpha_{i},\beta_{i},b_{j})$ are linear maps with domain and codomain
indicated in the following diagram:

(4.4)
$\textstyle{W_{2}^{\prime}}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{b_{1},\dots,b_{r}}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\textstyle{W_{1}^{\prime}}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$

The relation between $(\alpha_{i},\beta_{i},b_{j})$ and the maps
$(x_{1},x_{2},y_{h})$ in (4.1) is given by $x_{1}=\beta_{1}\alpha_{1}$,
$x_{2}=\beta_{2}\alpha_{2}$, $y_{h}=\beta_{1}b_{h}\alpha_{2}$ (for
$h=1,\dots,r$).

The following statement is proved by Nakajima [19]. For the reader’s
convenience, we give a proof here.

######  Lemma 4.3.

Let $w=(w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime})\in\mathbb{Z}_{\geq 0}^{4}$
and $v=(v_{1},v_{2})\in\mathbb{Z}_{\geq 0}^{2}$.

(a) The variety $\mathbf{E}_{v,w}\neq\emptyset$. The variety
$\mathcal{F}_{v,w}$ (thus $\tilde{\mathcal{F}}_{v,w}$) is nonempty if and only
if

(4.5) $0\leq v_{1}\leq w_{1}^{\prime}\textrm{ and }0\leq v_{2}\leq
w_{2}^{\prime}+rv_{1}.$

(b) If (4.5) holds, then $\tilde{\mathcal{F}}_{v,w}$ is (the total space of) a
vector bundle over $\mathcal{F}_{v,w}$ of rank $w_{1}v_{1}+w_{2}v_{2}$.
Meanwhile, the natural projection

(4.6) $\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w},\quad({\bf x},{\bf
y},X_{1},X_{2})\mapsto({\bf x},{\bf y})$

has the zero fiber $\pi^{-1}(0)\cong\mathcal{F}_{v,w}$.

###### Proof.

(a) and the second statement of (b) are obvious. For the first statement of
(b), consider the natural projection

$\tilde{\mathcal{F}}_{v,w}\to\mathcal{F}_{v,w},\quad({\bf x},{\bf
y},X_{1},X_{2})\mapsto(X_{1},X_{2}).$

We prove that it gives a vector bundle by identifying it with the pullback of
a vector bundle. Indeed, for fixed $(X_{1},X_{2})$, a tuple $({\bf x},{\bf
y},X_{1},X_{2})$ is in $\tilde{\mathcal{F}}_{v,w}$ if and only if the $w_{1}$
column vectors of the matrix $x_{1}$ are in $X_{1}$, and the $w_{2}$ column
vectors of the matrix $[x_{2},y_{1},\dots,y_{r}]_{\text{vert}}$ are in
$X_{2}$. (To show that these conditions implies $\mathrm{im}A({\bf x},{\bf
y})\subseteq X_{1}$, note that $X_{2}\subseteq W^{\prime}_{2}\oplus
X_{1}^{\oplus r}$, thus each ${\rm im}y_{i}$ is in $X_{1}$.) So

(4.7)
$\tilde{\mathcal{F}}_{v,w}\cong\iota^{*}\Big{(}S_{Gr(v_{1},W^{\prime}_{1})}^{\oplus
w_{1}}\bigoplus S_{Gr(v_{2},W^{\prime}_{2}\oplus{W^{\prime}_{1}}^{\oplus
r})}^{\oplus w_{2}}\Big{)}$

where $\iota$ is the natural embedding $\iota:\mathcal{F}_{v,w}\to
Gr(v_{1},W^{\prime}_{1})\times
Gr(v_{2},W^{\prime}_{2}\oplus{W^{\prime}_{1}}^{\oplus r})$, and $S_{Gr(d,e)}$
stands for the tautological subbundle (i.e., the universal subbundle) on the
Grassmannian variety $Gr(d,e)$. Since the pullback of a vector bundle is a
vector bundle of the same rank, it follows from (4.7) that
$\tilde{\mathcal{F}}_{v,w}$ is a vector bundle over $\mathcal{F}_{v,w}$ of
rank $v_{1}w_{1}+v_{2}w_{2}$. (In particular, it is locally trivial.) ∎

Now we study a stratification of $\pi$ given in (4.6). Consider a
stratification $\mathbf{E}_{w}=\bigcup_{v}\mathbf{E}^{\circ}_{v,w}$ where

(4.8) $\quad\mathbf{E}^{\circ}_{v,w}=\\{({\bf x},{\bf
y})\in\mathbf{E}_{w}\,|\,(\mathrm{rank}A({\bf x,y}),\mathrm{rank}B({\bf
x,y}))=(v_{1},v_{2})=v\\}.$

For $v^{\prime}=(v^{\prime}_{1},v^{\prime}_{2})$ and $v=(v_{1},v_{2})$, recall
that we denote $v^{\prime}\leq v$ if $v^{\prime}_{1}\leq v_{1}$ and
$v^{\prime}_{2}\leq v_{2}$. (Note that in [19] this is denoted oppositely as
$v^{\prime}\geq v$.) Recall the following definition of $l$-dominant condition
given in [19].

######  Definition 4.4.

Let $w=(w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime})\in\mathbb{Z}_{\geq 0}^{4}$
and $v=(v_{1},v_{2})\in\mathbb{Z}_{\geq 0}^{2}$. We say that $(v,w)$ is
$l$-_dominant_ if $w-C_{q}v\geq 0$; or equivalently, by (4.3):

(4.9) $w_{1}-v_{1}+rv_{2}\geq 0,\quad w_{1}^{\prime}-v_{1}\geq 0,\quad
w_{2}-v_{2}\geq 0,\quad w_{2}^{\prime}-v_{2}+rv_{1}\geq 0.$

For a fixed $w$, we define

$\mathrm{Dom}(w)=\\{v\in\mathbb{Z}_{\geq 0}^{2}\ |\ (v,w)\textrm{ is
$l$-dominant}\\}.$

We define

$\mathrm{Dom}=\\{(v,w)\in\mathbb{Z}_{\geq 0}^{2}\times\mathbb{Z}_{\geq 0}^{4}\
|\ (v,w)\textrm{ is $l$-dominant}\\}.$

Note that the condition (4.9) implies (4.5).

In the following lemma and proposition, we define $\bar{v}$ with the property
that $(\bar{v},w)$ is $l$-dominant and
$\mathbf{E}_{v,w}=\mathbf{E}_{\bar{v},w}$ (which will be proved in Proposition
4.7).

######  Lemma 4.5.

Let $w=(w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime})\in\mathbb{Z}_{\geq 0}^{4}$,
$v=(v_{1},v_{2})\in\mathbb{Z}_{\geq 0}^{2}$,
$V=\\{v^{\prime}\in\mathbb{Z}_{\geq 0}^{2}\ |\ v^{\prime}\leq v\\}$. Then the
intersection $V\cap\mathrm{Dom}(w)=\\{v^{\prime}\in\mathbb{Z}_{\geq 0}^{2}\ |\
v^{\prime}\leq v\textrm{ and $(v,w)$ is $l$-dominant}\\}$ has a unique maximal
element $\bar{v}$ in the sense that every $v^{\prime}\in V\cap\mathrm{Dom}(w)$
must satisfy $v^{\prime}\leq\bar{v}$. More explicitly,

(4.10)
$\bar{v}=\big{(}\min(v_{1},w^{\prime}_{1},w_{1}+rv_{2},w_{1}+rw_{2}),\min(v_{2},w_{2},w^{\prime}_{2}+rv_{1},w^{\prime}_{2}+rw^{\prime}_{1})\big{)}$

In particular, if $(v,w)$ satisfies (4.5), then

$\bar{v}=\big{(}\min(v_{1},w_{1}+rv_{2},w_{1}+rw_{2}),\min(v_{2},w_{2})\big{)};$

if $v\in\mathrm{Dom}(w)$, then $\bar{v}=v$.

###### Proof.

It is obvious that if $v\in\mathrm{Dom}(w)$, then $\bar{v}=v$. So in the rest
of the proof we assume $v\in\mathbb{Z}_{\geq 0}^{2}\setminus\mathrm{Dom}(w)$.

We want to show that the $\bar{v}$ given in (4.10) is the unique maximal
element in the intersection $V\cap\mathrm{Dom}(w)$.

We consider three cases, and only give a proof of Case 1, because the other
two cases can be proved similarly. See Figure 2 for the corresponding figures.

Case 1. $w_{1}-w_{1}^{\prime}+rw_{2}\geq 0$ and
$w_{2}^{\prime}-w_{2}+rw_{1}^{\prime}\geq 0$. In this case, (4.10) becomes

$\bar{v}=\begin{cases}(w_{1}^{\prime},w_{2}),&\textrm{ if $v_{1}\geq
w_{1}^{\prime}$ and $v_{2}\geq w_{2}$};\\\
\big{(}v_{1},\min(w_{2}^{\prime}+rv_{1},w_{2})\big{)},&\textrm{ if $v_{1}\leq
w_{1}^{\prime}$ and $v_{2}>\min(w_{2}^{\prime}+rv_{1},w_{2})$};\\\
\big{(}\min(w_{1}+rv_{2},w_{1}^{\prime}),v_{2}\big{)},&\textrm{ if
$v_{1}>\min(w_{1}+rv_{2},w_{1}^{\prime})$ and $v_{2}\leq w_{2}$}.\\\
\end{cases}$

For simplicity, we assume that $w_{1}\leq w^{\prime}_{1}$ and
$w^{\prime}_{2}\leq w_{2}$. (Other cases are similar). The set
$\mathrm{Dom}(w)$ is shown in Figure 2 Left. The line $w_{1}-x+ry=0$ has slope
$1/r$ whose $x$-intercept is $w_{1}\geq 0$; the line $w^{\prime}_{2}-y+rx=0$
has slope $r$ whose $y$-intercept is $w^{\prime}_{2}$. The convex hull of
$\mathrm{Dom}(w)$ is a (possibly degenerated) hexagon $H$ whose upper-right
corner is $(w^{\prime}_{1},w_{2})$.

The three regions of $v$ are illustrated in Figure 2 Left by different filling
patterns (diagonal stripes, vertical stripes, horizontal stripes), and there
are overlaps on the boundary. It is clear from the figure that each
$v\in\mathbb{Z}_{\geq 0}^{2}\setminus\mathrm{Dom}(w)$ belongs to at least one
of the three regions.

If $v_{1}\geq w_{1}^{\prime}$ and $v_{2}\geq w_{2}$, then $v$ is weakly to the
northeast of $(w^{\prime}_{1},w_{2})$, which implies $\mathrm{Dom}(w)\subseteq
V$, $V\cap\mathrm{Dom}(w)=\mathrm{Dom}(w)$, thus the unique maximal element in
$V\cap\mathrm{Dom}(w)$ is $(w^{\prime}_{1},w_{2})$.

If $v_{1}\leq w_{1}^{\prime}$ and $v_{2}>\min(w_{2}^{\prime}+rv_{1},w_{2})$,
then $v$ is above the hexagon $H$ and lies between the vertical lines $x=0$
and $x=w^{\prime}_{1}$. In this case, $\bar{v}$ is the intersection of the
vertical line $x=v_{1}$ with the upper boundary of $H$ (which consists of two
sides), which is the point $(v_{1},\min(w_{2}^{\prime}+rv_{1},w_{2}))$.

If $v_{1}>\min(x_{1}+rv_{2},w^{\prime}_{1})$ and $v_{2}<w_{2}$, then $v$ is to
the right the hexagon $H$ and lies between the horizontal lines $y=0$ and
$y=w_{2}$. In this case, $\bar{v}$ is the intersection of the horizontal line
$y=v_{2}$ with the right boundary of $H$ (which consists of two sides), which
is the point $(\min(w_{1}+rv_{2},w^{\prime}_{1}),v_{2})$.

This proves the formula (4.10) in this case.

The other two cases are proved similarly:

Case 2. If $w_{1}-w_{1}^{\prime}+rw_{2}<0$:

$\bar{v}=\begin{cases}(w_{1}+rw_{2},w_{2}),&\textrm{ if $v_{1}\geq
w_{1}+rw_{2}$ and $v_{2}\geq w_{2}$};\\\
\big{(}v_{1},\min(w_{2}^{\prime}+rv_{1},w_{2})\big{)},&\textrm{ if
$v_{2}>\min(w_{2}^{\prime}+rv_{1},w_{2})$ and $v_{1}\leq w_{1}+rw_{2}$};\\\
(w_{1}+rv_{2},v_{2}),&\textrm{ if $v_{1}>w_{1}+rv_{2}$ and $v_{2}\leq
w_{2}$}.\\\ \end{cases}$

Case 3. If $w_{2}^{\prime}-w_{2}+rw_{1}^{\prime}<0$:

$\bar{v}=\begin{cases}(w_{1}^{\prime},w_{2}^{\prime}+rw_{1}^{\prime}),&\textrm{
if $v_{1}\geq w_{1}^{\prime}$ and $v_{2}\geq
w_{2}^{\prime}+rw_{1}^{\prime}$};\\\ (v_{1},w_{2}^{\prime}+rv_{1}),&\textrm{
if $v_{1}\leq w_{1}$ and $v_{2}>w_{2}^{\prime}+rv_{1}$};\\\
\big{(}\min(w_{1}^{\prime},w_{1}+rv_{2}),v_{2}\big{)},&\textrm{ if
$v_{1}>\min(w_{1}^{\prime},w_{1}+rv_{2})$ and $v_{2}\leq
w_{2}^{\prime}+rw_{1}^{\prime}$}.\\\ \end{cases}$

Thus (4.10) holds for all cases. ∎

$w_{1}$$w^{\prime}_{1}$$x$$(0,0)$$w^{\prime}_{2}$$w_{2}$$y$$w_{1}-x+ry=0$$w^{\prime}_{2}-y+rx=0$$w_{1}$$w^{\prime}_{1}$$x$$(0,0)$$w^{\prime}_{2}$$w_{2}$$y$$w_{1}-x+ry=0$$w^{\prime}_{2}-y+rx=0$$w_{1}$$w^{\prime}_{1}$$x$$(0,0)$$w^{\prime}_{2}$$w_{2}$$y$$w_{1}-x+ry=0$$w^{\prime}_{2}-y+rx=0$
Figure 2. $l$-dominant lattices for Cases 1 (Left), 2 (Middle), and 3 (Right).

######  Lemma 4.6.

Fix $v\in\mathbb{Z}_{\geq 0}^{2}$ and $w\in\mathbb{Z}_{\geq 0}^{4}$. Let
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$ be the map defined in
(4.6).

(a) For each $v^{\prime}\leq v$, $\mathbf{E}^{\circ}_{v^{\prime},w}$, defined
in (4.8), is nonempty if and only if $(v^{\prime},w)$ is $l$-dominant. In
which case, it is nonsingular and locally closed in $\mathbf{E}_{w}$ (so is
also locally closed in $\mathbf{E}_{v,w}$), and is irreducible and rational.
Thus the variety $\mathbf{E}_{v,w}$ has a stratification

$\mathbf{E}_{v,w}=\bigcup_{v^{\prime}}\mathbf{E}^{\circ}_{v^{\prime},w}\,,\quad\textrm{where
$v^{\prime}\leq v$ and $(v^{\prime},w)$ are $l$-dominant}.$

(b) For each $v^{\prime}\leq v$ with $(v^{\prime},w)$ being $l$-dominant, the
restricted projection
$\pi^{-1}(\mathbf{E}^{\circ}_{v^{\prime},w})\to\mathbf{E}^{\circ}_{v^{\prime},w}$
is a Zariski locally trivial $\mathcal{M}$-bundle, where $\mathcal{M}$ itself
is a $Gr(v_{2}-v^{\prime}_{2},w_{2}^{\prime}+rv_{1}-v^{\prime}_{2})$-bundle
over $Gr(v_{1}-v^{\prime}_{1},w_{1}^{\prime}-v^{\prime}_{1})$ defined as

$\mathcal{M}=\Big{\\{}(X_{1}^{\prime},X_{2}^{\prime})\Big{|}X_{1}^{\prime}\in
Gr(v_{1}-v_{1}^{\prime},w_{1}^{\prime}-v_{1}^{\prime}),X_{2}^{\prime}\in
Gr\big{(}v_{2}-v_{2}^{\prime},\mathbb{C}^{w_{2}^{\prime}+rv_{1}^{\prime}-v_{2}^{\prime}}\oplus(X_{1}^{\prime})^{\oplus
r}\big{)}\Big{\\}}.$

In particular, $\mathcal{F}_{v,w}$ is a
$Gr(v_{2},w_{2}^{\prime}+rv_{1})$-bundle over $Gr(v_{1},w_{1}^{\prime})$.

We postpone its technical proof to the end of the paper (§11) so that it is
easier to follow the flow of arguments of the main results.

######  Proposition 4.7.

Fix $v\in\mathbb{Z}_{\geq 0}^{2}$ and $w\in\mathbb{Z}_{\geq 0}^{4}$. Let
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$ be the map defined in
(4.6). Let $\bar{v}$ be defined as in Lemma 4.5.

(a) $\mathbf{E}_{v,w}=\mathbf{E}_{\bar{v},w}$ is irreducible. Moreover,
$\mathbf{E}^{\circ}_{\bar{v},w}$ is its largest stratum; so
$\mathbf{E}_{v,w}=\overline{\mathbf{E}^{\circ}_{\bar{v},w}}$, the Zariski
closure of $\mathbf{E}^{\circ}_{\bar{v},w}$.

(b) Assume (4.5). Then $\tilde{\mathcal{F}}_{v,w}$ and $\mathcal{F}_{v,w}$ are
irreducible and nonsingular and $\pi$ is surjective. Moreover,

$\displaystyle\dim\mathcal{F}_{v,w}=d(v,w):=-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}w_{1}^{\prime}+v_{2}w_{2}^{\prime}$
$\displaystyle\dim\tilde{\mathcal{F}}_{v,w}=\tilde{d}(v,w):=\dim\mathcal{F}_{v,w}+w_{1}v_{1}+w_{2}v_{2}=-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}(w_{1}+w_{1}^{\prime})+v_{2}(w_{2}+w_{2}^{\prime})$
$\displaystyle\dim\mathbf{E}_{v,w}=\dim\mathbf{E}_{\bar{v},w}=\tilde{d}(\bar{v},w)=-\bar{v}_{1}^{2}+r\bar{v}_{1}\bar{v}_{2}-\bar{v}_{2}^{2}+\bar{v}_{1}(w_{1}+w_{1}^{\prime})+\bar{v}_{2}(w_{2}+w_{2}^{\prime})$

(c) Further assume that $(v,w)$ is $l$-dominant. Then $\pi$ is birational. In
this case, it restricts to an isomorphism $\pi^{-1}({\bf
E}^{\circ}_{v,w})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\bf
E}^{\circ}_{v,w}$.

###### Proof.

First, we prove the first statement of (b). The special case
$v^{\prime}=(0,0)$ of Lemma 4.6 (b) asserts that $\mathcal{F}_{v,w}$ is a
$Gr(v_{2},w_{2}^{\prime}+rv_{1})$-bundle over $Gr(v_{1},w_{1}^{\prime})$ and
therefore is irreducible and nonsingular. By Lemma 4.3 (b),
$\tilde{\mathcal{F}}_{v,w}$ is a vector bundle over $\mathcal{F}_{v,w}$, so is
also irreducible and nonsingular.

For any point in $\mathbf{E}_{v,w}$, it must be in a stratum
$\mathbf{E}^{\circ}_{v^{\prime},w}$ for some $v^{\prime}\leq v$ such that
$v^{\prime}\in\mathrm{Dom}(w)$ according to Lemma 4.6 (a). By Lemma 4.6 (b),
the preimage is $\mathcal{M}$, which is nonempty if $0\leq
v_{2}-v^{\prime}_{2}\leq w^{\prime}_{2}+rv_{1}-v^{\prime}_{2}$ and $0\leq
v_{1}-v^{\prime}_{1}\leq w^{\prime}_{1}-v^{\prime}_{1}$; this condition is
equivalent to (4.5). Therefore $\pi$ is surjective since all fibers are
nonempty.

(a) The equality $\mathbf{E}_{v,w}=\mathbf{E}_{\bar{v},w}$ follows from the
stratifications of both sides, and the fact that and $l$-dominant $v^{\prime}$
satisfying $v^{\prime}\leq v$ must also satisfy $v^{\prime}\leq\bar{v}$,
according to Lemma 4.5.

The projection $\tilde{\mathcal{F}}_{\bar{v},w}\to\mathbf{E}_{\bar{v},w}$ is
surjective and the source variety $\tilde{\mathcal{F}}_{\bar{v},w}$ is
irreducible by the first statement of (b). Therefore the target variety
$\mathbf{E}_{\bar{v},w}$ is also irreducible.

Since $\mathbf{E}_{\bar{v},w}$ is irreducible, it has a unique largest
stratum. Note
$\mathbf{E}_{\bar{v},w}=\bigcup_{v^{\prime}\leq\bar{v},v^{\prime}\in\mathrm{Dom}(w)}\mathbf{E}^{\circ}_{v^{\prime},w}$.
For each $v^{\prime}<v$ with $v^{\prime}\in\mathrm{Dom}(w)$, we have
$\mathbf{E}_{v^{\prime},w}=\bigcup_{v^{\prime\prime}\leq
v^{\prime},v^{\prime\prime}\in\mathrm{Dom}(w)}\mathbf{E}^{\circ}_{v^{\prime\prime},w}$
which is a proper subset of $\mathbf{E}_{v,w}$ because
$\mathbf{E}^{\circ}_{\bar{v},w}\neq\emptyset$, therefore

$\overline{\mathbf{E}^{\circ}_{v^{\prime},w}}\subseteq\mathbf{E}_{v^{\prime},w}\subsetneq\mathbf{E}_{v,w}$

so $\mathbf{E}^{\circ}_{v^{\prime},w}$ is not the largest stratum. Then
$\mathbf{E}^{\circ}_{\bar{v},w}$ is the largest stratum.

(c) It suffices to show that $\pi^{-1}({\bf
E}^{\circ}_{v,w})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\bf
E}^{\circ}_{v,w}$ is bijective, since a bijective morphism between nonsingular
complex algebraic varieties is an isomorphism. By Lemma 4.6 (b), each fiber is
a $Gr(v_{2}-v_{2},w^{\prime}_{2}+rv_{1}-v_{2})$-bundle over
$Gr(v_{1}-v_{1},w^{\prime}_{1}-v_{1})$, which is a point since both
Grassmannians are points. This implies the bijectivity.

Finally, we prove the second statement of (b). Since $\mathcal{F}_{v,w}$ is
isomorphic to the fiber over $0\in\mathbf{E}_{v,w}$, it is a
$Gr(v_{2},w_{2}^{\prime}+rv_{1})$-bundle over $Gr(v_{1},w_{1}^{\prime})$ by
Lemma 4.6 (b). Therefore
$\dim\mathcal{F}_{v,w}=v_{2}(w_{2}^{\prime}+rv_{1}-v_{2})+v_{1}(w_{1}^{\prime}-v_{1})=-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}w_{1}^{\prime}+v_{2}w_{2}^{\prime}$.

Then we obtain the formula of $\dim\tilde{\mathcal{F}}_{v,w}$ because
$\tilde{\mathcal{F}}_{v,w}\to\mathcal{F}_{v,w}$ is a vector bundle of rank
$w_{1}v_{1}+w_{2}v_{2}$ as observed in Lemma 4.3 (b).

The equality $\dim\mathbf{E}_{v,w}=\dim\mathbf{E}_{\bar{v},w}$ follows from
(a). The equality
$\dim\mathbf{E}_{\bar{v},w}=\dim\tilde{\mathcal{F}}_{\bar{v},w}=\tilde{d}(\bar{v},w)$
follows from (c). ∎

##  5\. Decomposition theorem

The following fact is proved by Nakajima in [20, Theorem 14.3.2] and
generalized by Qin in [21, Theorem 5.2.1]). Nakajima’s proof uses the
representation theory of quantum affine algebras in an essential way, and
focuses on type $ADE$; Qin’s proof uses a similar idea of Nakajima’s proof,
particularly the analytic transversal slice theorem, but does not provide
detailed explanations of why Nakajima’s representation-theoretic argument
still works in the more general setting. Instead, we will give an elementary
proof in our special case, without referring to representation theory.

######  Theorem 5.1.

Assume ${\tilde{\mathcal{F}}_{v,w}}\neq\emptyset$. The local system appeared
in the BBDG decomposition for $\pi_{*}IC_{\tilde{\mathcal{F}}_{v,w}}$ are all
trivial. Thus,

(5.1)
$\pi_{*}(IC_{\tilde{\mathcal{F}}_{v,w}})=\bigoplus_{v^{\prime}}\bigoplus_{d}a^{d}_{v,v^{\prime};w}IC_{\mathbf{E}_{v^{\prime},w}}[d]$

where $v^{\prime}\leq v$ satisfies the condition that $(v^{\prime},w)$ is
$l$-dominant, and $a^{d}_{v,v^{\prime};w}\in\mathbb{Z}_{\geq 0}$.

First, recall the Beilinson–Bernstein–Deligne–Gabber decomposition theorem
(also referred as BBDG or BBD decomposition Theorem).

######  Theorem 5.2 (BBDG Decomposition Theorem).

[2] Let $f:X\to Y$ be a proper algebraic morphism between complex algebraic
varieties. Then there is a finite list of triples $(Y_{a},L_{a},n_{a})$, where
for each $a$, $Y_{a}$ is locally closed smooth irreducible algebraic
subvariety of $Y$, $L_{a}$ is a semisimple local system on $Y_{a}$, $n_{a}$ is
an integer, such that:

(5.2) $f_{*}IC_{X}\cong\bigoplus_{a}IC_{\overline{Y_{a}}}(L_{a})[n_{a}]$

Moreover, even though the isomorphism is not necessarily canonical, the direct
summands appeared on the right hand side are canonical.

In the next two subsections, we shall provide two proofs of Theorem 5.1. The
first proof is based on the algebraic version of the Transversal Slice Theorem
(Lemma 5.3). The second proof is longer, but only based on the (weaker)
analytic version of the Transversal Slice Theorem proved by Nakajima [20].

###  5.1. Algebraic Transversal Slice Theorem

Nakajima proves an analytic transversal slice theorem [20, §3], and comments
that the technique used there “is based on a work of Sjamaar-Lerman in the
symplectic geometry”, so the transversal slice is analytic in nature and is
not algebraic; he further comments that “It is desirable to have a purely
algebraic construction of a transversal slice”. We shall prove an Algebraic
Transversal Slice Theorem for the varieties studied in this paper. In
contrast, by “Analytic Transversal Slice Theorem” we mean a weaker statement
than Lemma 5.3 where we replace “Zariski open” by “open in analytic topology”,
and replace the algebraic morphisms/isomorphisms by the analytic ones.

######  Lemma 5.3 (Algebraic Transversal Slice Theorem).

Let $p\in\mathbf{E}_{v,w}$ be a point in the stratum
$\mathbf{E}^{\circ}_{v^{0},w}$. So $v^{0}\leq v$. Define

$w^{\perp}=w-C_{q}v^{0}=(w_{1}-v_{1}^{0}+rv_{2}^{0},w_{1}^{\prime}-v_{1}^{0},w_{2}-v_{2}^{0},w_{2}^{\prime}-v_{2}^{0}+rv_{1}^{0}),\quad
v^{\perp}=v-v^{0}=(v_{1}-v_{1}^{0},v_{2}-v_{2}^{0}).$

Then there exist Zariski open neighborhoods $U\subseteq\mathbf{E}_{v,w}$ of
$p$, $U^{0}\subseteq\mathbf{E}^{\circ}_{v^{0},w}$ of $p$,
$U^{\perp}\subseteq\mathbf{E}^{\circ}(v^{\perp},w^{\perp})$ of $0$, and
isomorphisms $\varphi,\psi$ making the following diagram commute:

$\textstyle{\tilde{\mathcal{F}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\supseteq}$$\textstyle{\pi^{-1}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\scriptstyle{\cong}$$\textstyle{U^{0}\times\pi^{-1}(U^{\perp})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times\pi}$$\scriptstyle{\subseteq}$$\textstyle{\mathbf{E}^{\circ}_{v^{0},w}\times\tilde{\mathcal{F}}_{v^{\perp},w^{\perp}}}$$\textstyle{\mathbf{E}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\supseteq}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\cong}$$\textstyle{U^{0}\times
U^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbf{E}^{\circ}_{v^{0},w}\times\mathbf{E}_{v^{\perp},w^{\perp}}}$

Moreover, $\psi(p)=(p,0)$, the diagram is compatible with the stratifications,
in the sense that $\psi({\bf E}^{\circ}_{u,w}\cap U)=U_{0}\times({\bf
E}^{\circ}_{u^{\perp},w^{\perp}}\cap U^{\perp})$ for each $u$ satisfying
$v^{0}\leq u\leq v$, where we denote
$u^{\perp}=u-v^{0}=(u_{1}-v_{1}^{0},u_{2}-v_{2}^{0})$.

Its proof is delayed to §11.

###  5.2. The first proof of trivial local systems

In this subsection we give the first proof of Theorem 5.1. Another proof is
given in §11.3.

We need the following lemma describing the BBDG decomposition under pullpack.
It will be needed in the proof of Theorem 5.1.

######  Lemma 5.4.

Given $f:X\to Y$, $Y_{a},L_{a},n_{a}$ and the decomposition in Theorem 5.2:

$f_{*}IC_{X}\cong\bigoplus_{a}IC_{\overline{Y_{a}}}(L_{a})[n_{a}]$

Let $V$ be a nonsingular variety. Consider the following Cartesian diagram

$\textstyle{V\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\scriptstyle{1\times
f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y}$

where $p$ and $p^{\prime}$ are the natural projections. Define the pullback
$\tilde{L}_{a}=p^{*}L_{a}$, which is a semisimple local system on $V\times
Y_{a}$. Then

$(1\times f)_{*}IC_{V\times
X}\cong\bigoplus_{a}IC_{V\times\overline{Y_{a}}}(\tilde{L}_{a})[n_{a}]$

###### Proof.

Denote by $d$ the dimension of the fiber of $f$. Then

$(p^{\prime})^{*}[d]IC_{X}=IC_{V\times X},\quad
p^{*}[d]IC_{\overline{Y_{a}}}(L_{a})=IC_{V\times\overline{Y_{a}}}(\tilde{L}_{a})$

(for example, see [1, Lemma 3.6.3 and Corollary 3.6.9]). By proper base change
[7, §5.8], we have the isomorphism
$p^{*}f_{*}IC_{X}\stackrel{{\scriptstyle\sim}}{{\to}}(1\times
f)_{*}(p^{\prime})^{*}IC_{X}$. Thus,

$\displaystyle(1\times f)_{*}IC_{V\times X}$ $\displaystyle\cong(1\times
f)_{*}(p^{\prime})^{*}[d]IC_{X}\cong p^{*}[d](f_{*}IC_{X})\cong
p^{*}[d](\bigoplus_{a}IC_{\overline{Y_{a}}}(L_{a})[n_{a}])$
$\displaystyle\cong\bigoplus_{a}p^{*}[d]IC_{\overline{Y_{a}}}(L_{a})[n_{a}]\cong\bigoplus_{a}IC_{V\times\overline{Y_{a}}}(\tilde{L}_{a})[n_{a}].$

∎

###### The First Proof of Theorem 5.1.

Let $IC_{Z}(L)[n]$ be a direct summand that appears in the decomposition of
$\pi_{*}(IC_{\tilde{\mathcal{F}}_{v,w}})$. Assume a general point $p$ of $Z$
is in ${\bf E}^{\circ}_{v^{0},w}$ (thus $Z\subseteq{\bf E}_{v^{0},w}$), and we
adopt the notation in Lemma 5.3. In particular, $U^{0}$ is a Zariski open
subset of ${\bf E}^{\circ}_{v^{0},w}$.

By uniqueness of the BBDG Decomposition, the restriction $IC_{Z}(L)[n]|_{U}$
must coincide with a direct summand of the decomposition of
$\pi_{*}(IC_{\pi^{-1}U})$. Compare with Lemma 5.4 by setting $V=U^{0}$,
$X=\pi^{-1}(U^{\perp})$, $Y=U^{\perp}$, we conclude that
$IC_{Z}(L)[n]|_{U}\cong
IC_{U^{0}\times\overline{Y_{a}}}(\tilde{L}_{a})[n_{a}]$ for some $a$. Thus
$Z\cap U=U^{0}\times\overline{Y_{a}}$, $L\cong\tilde{L}_{a}$ on $Z\cap U$, and
$n=n_{a}$. But $Z\subseteq{\bf E}_{v^{0},w}$ implies $Z\cap U\subseteq
U^{0}\times\\{0\\}$. So we must have $Y_{a}=\\{0\\}$ and $Z\cap
U=U^{0}\times\\{0\\}$, thus $Z={\bf E}_{v^{0},w}$.

Moreover, $L_{a}$ is the trivial local system $\mathbb{Q}_{0}$. So the local
system $\tilde{L}_{a}$ on $U^{0}$ is also trivial since it is the pullback of
$L_{a}$ under the map $U^{0}\times\\{0\\}\to\\{0\\}$. Then $L$ is trivial on
$Z\cap U$, and we see that

$IC_{Z}(L)[n]\cong IC_{{\bf E}_{v^{0},w}}[n]$

Let $v^{\prime}=v^{0}$. Then $v^{\prime}\leq v$ as seen in Lemma 5.3, and
$(v^{\prime},w)$ is $l$-dominant because ${\bf E}^{\circ}_{v^{0},w}$ is
nonempty. ∎

######  Remark 5.5.

A key fact used in the above proof is that a local system on an irreducible
nonsingular variety $X$ is uniquely determined by its restriction on a Zariski
open dense subset $U$. This is true because a local system is determined by
the action of fundamental group on a stalk [1, Lemma 1.7.9], and the following
lemma modified from [1, Lemma 2.1.22].

######  Lemma 5.6.

Let $X$ be a smooth connected complex variety. For any Zariski open dense
subset $U\subseteq X$ and any point $x_{0}\in U$, the natural map
$\pi_{1}(U,x_{0})\to\pi_{1}(X,x_{0})$ is surjective. Moreover, the map is an
isomorphism if $X\setminus U$ has complex codimension at least 2.

###### Proof.

The first statement is proved in [1, Lemma 1.7.9]. The second statement can be
proved similarly, with the details given below. Recall the following fact.

Fact([10, p146, Theorem 2.3]): _if $M$ is a connected real smooth manifold
without boundary, and $N$ is a closed submanifold, $x$ be a point in
$M\setminus N$, then $\pi_{1}(M\setminus N,x)\to\pi_{1}(M,x)$ is surjective if
the real codimension of $N$ is at least 2, and the map is an isomorphism if
the real codimension of $N$ is at least 3._

Now assume $Z=X\setminus U$ has complex codimension at least 2. So its real
codimension is at least 4. Stratify $Z$ into locally closed smooth
subvarieties $Z=\cup_{i=1}^{k}Z_{i}$, indexed in decreasing dimensions. Then
$\cup_{i=1}^{j}Z_{i}$ is open in $Z$ for every $1\leq j\leq k$. Apply the Fact
to $(M,N)=(U\cup(\cup_{i=1}^{j}Z_{i}),Z_{j})$, we conclude that
$\pi_{1}(U\cup(\cup_{i=1}^{j-1}Z_{i}),x_{0})\to\pi_{1}(U\cup(\cup_{i=1}^{j}Z_{i}),x_{0})$
is an isomorphism. Thus the composition

$\pi_{1}(U,x_{0})\to\pi_{1}(U\cup
Z_{1},x_{0})\to\pi_{1}(U\cup(\cup_{i=1}^{2}Z_{i}),x_{0})\to\cdots\to\pi_{1}(U\cup(\cup_{i=1}^{k}Z_{i}),x_{0})=\pi_{1}(X,x_{0})$

is an isomorphism. ∎

##  6\. Other desingularizations of $\mathbf{E}_{v,w}$

In this section, we assume $(v,w)$ is $\ell$-dominant.

###  6.1. Variety $\tilde{\mathcal{F}}^{\rm swap}_{v,w}$

Given $(v,w)$, define

(6.1) $v^{\rm swap}=(v_{2},v_{1}),\ w^{\rm
swap}=(w^{\prime}_{2},w_{2},w_{1}^{\prime},w_{1}),\ \tilde{\mathcal{F}}^{\rm
swap}_{v,w}=\tilde{\mathcal{F}}_{v^{\rm swap},w^{\rm swap}}$

Note that $\iota:\mathbf{E}_{v,w}\to\mathbf{E}_{v^{\rm swap},w^{\rm swap}}$,
induced by $(A,B)\mapsto(B^{T},A^{T})$ is an isomorphism. The composition

$\pi^{\rm swap}:\ \tilde{\mathcal{F}}^{\rm swap}_{v,w}\to\mathbf{E}_{v^{\rm
swap},w^{\rm
swap}}\stackrel{{\scriptstyle\iota^{-1}}}{{\longrightarrow}}\mathbf{E}_{v,w}$

gives a desingularization of $\mathbf{E}_{v,w}$. Note that
$\tilde{\mathcal{F}}^{\rm swap}_{v,w}$ is not isomorphic to
$\tilde{\mathcal{F}}_{v,w}$ in general.

###  6.2. Varieties $\tilde{\mathcal{G}}_{v,w}$ and
$\tilde{\mathcal{H}}_{v,w}$

In this section we factor $\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$
through a possibly singular variety $\tilde{\mathcal{H}}_{v,w}$, and introduce
a nonsingular variety $\tilde{\mathcal{G}}_{v,w}$ such that there is a map
$\tilde{\mathcal{G}}_{v,w}\to\tilde{\mathcal{H}}_{v,w}$. See the following
diagram:

(6.2)
---
$\textstyle{\tilde{\mathcal{F}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi}$$\textstyle{\tilde{\mathcal{G}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\tilde{\mathcal{H}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{\mathbf{E}_{v,w}}$

Recall that a linear map $f:V\to W$ has a dual (or transpose, or adjoint)
$f^{T}:W^{*}\to V^{*}$, $\phi\mapsto\phi\circ f$. If $f$ is represented by the
matrix $A$ with respect to fixed bases of $V$ and $W$, then $f^{T}$ is
represented by the transpose matrix $A^{T}$ with respect to the dual bases.

######  Definition 6.1.

Define the quasi-projective variety

$\tilde{\mathcal{H}}_{v,w}=\\{({\bf x},{\bf y},X_{1})\;|\;({\bf x},{\bf
y})\in\mathbf{E}_{w},\ \dim X_{1}=v_{1},\ X_{1}\subseteq W^{\prime}_{1},\
{\mathrm{im}}A({\bf x},{\bf y})\subseteq X_{1},\ {\rm rank}B({\bf x},{\bf
y})\leq v_{2}\\}$

and define a stratification
$\\{U^{\prime}_{t}\subseteq\tilde{\mathcal{H}}_{v,w}\\}_{0\leq
t\leq\min(v_{2},w_{2})}$ as

(6.3) $U^{\prime}_{t}=\\{({\bf x},{\bf
y},X_{1})\in\tilde{\mathcal{H}}_{v,w}\;|\;{\rm rank}B({\bf x},{\bf y})=t\\}.$

Define the quasi-projective variety

$\displaystyle\tilde{\mathcal{G}}_{v,w}=$ $\displaystyle\\{({\bf x},{\bf
y},X_{1},X_{2}^{\prime})\;|\;\dim X_{1}=v_{1},\dim X_{2}^{\prime}=v_{2},$
$\displaystyle{\mathrm{im}}A({\bf x},{\bf y})\subseteq X_{1}\subseteq
W^{\prime}_{1},\;\mathrm{im}(B({\bf x},{\bf y})^{T})\subseteq
X_{2}^{\prime}\subseteq W_{2}^{*}\\}.$

######  Remark 6.2.

The strata $U_{t}^{\prime}$ is nonempty only if $0\leq
t\leq\min(v_{2},w_{2})$, because $\mathrm{rank}(B)\leq w_{2}$ (=the number of
columns of $B$) and $\mathrm{rank}(B)\leq v_{2}$ by the definition of
$\tilde{\mathcal{H}}_{v,w}$.

######  Proposition 6.3.

(a) Let $v_{1}\leq w^{\prime}_{1}$, $v_{2}\leq w_{2}$,
$\mathcal{G}_{v,w}=Gr(v_{1},w^{\prime}_{1})\times Gr(v_{2},w_{2})$. The
natural projection

$f:\tilde{\mathcal{G}}_{v,w}\to\mathcal{G}_{v,w},\quad({\bf x},{\bf
y},X_{1},X^{\prime}_{2})\mapsto(X_{1},X^{\prime}_{2})$

gives a vector bundle of rank $w_{1}v_{1}+w^{\prime}_{2}v_{2}+rv_{1}v_{2}$. So

$\dim\tilde{\mathcal{G}}_{v,w}=\dim\mathcal{G}_{v,w}+(w_{1}v_{1}+w^{\prime}_{2}v_{2}+rv_{1}v_{2})=v_{1}(w_{1}+w^{\prime}_{1}-v_{1})+v_{2}(w_{2}+w^{\prime}_{2}+rv_{1}-v_{2}).$

As a consequence, $\tilde{\mathcal{G}}_{v,w}$ is nonsingular.

(b) We can factorize the natural projection
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$ as follows:

$\displaystyle\tilde{\mathcal{F}}_{v,w}\stackrel{{\scriptstyle
p_{2}}}{{\longrightarrow}}\tilde{\mathcal{H}}_{v,w}\stackrel{{\scriptstyle
p_{1}}}{{\longrightarrow}}\mathbf{E}_{v,w}$ $\displaystyle({\bf x},{\bf
y},X_{1},X_{2})\mapsto({\bf x},{\bf y},X_{1})\mapsto({\bf x},{\bf y})$

Moreover, $p_{2}$ is stratified by $\\{U^{\prime}_{t}\\}$, and $p_{1}$ and
$\pi$ are both stratified by $\\{\mathbf{E}^{\circ}_{(s,t),w}\\}$ with
$((s,t),w)$ being $l$-dominant. The fibers of $p_{2}$ over $U^{\prime}_{t}$
are isomorphic to $Gr(v_{2}-t,w_{2}^{\prime}+rv_{1}-t)$. The fibers of $p_{1}$
over $U_{s,t}$ are isomorphic to $Gr(v_{1}-s,w_{1}^{\prime}-s)$.

(c) We can factorize the natural projection
$\pi^{\prime}:\tilde{\mathcal{G}}_{v,w}\to\mathbf{E}_{\bar{v},w}$ as follows:

$\displaystyle\tilde{\mathcal{G}}_{v,w}\stackrel{{\scriptstyle
p_{2}^{\prime}}}{{\longrightarrow}}\tilde{\mathcal{H}}_{v,w}\stackrel{{\scriptstyle
p_{1}}}{{\longrightarrow}}\mathbf{E}_{v,W}$ $\displaystyle({\bf x},{\bf
y},X_{1},X_{2}^{\prime})\mapsto({\bf x},{\bf y},X_{1})\mapsto({\bf x},{\bf
y})$

Moreover, $p_{2}^{\prime}$ is stratified by $\\{U^{\prime}_{t}\\}$, and
$\pi^{\prime}$ is stratified by $\\{U_{s,t}\\}$. The fibers of
$p_{2}^{\prime}$ over $U^{\prime}_{t}$ are isomorphic to
$Gr(v_{2}-t,w_{2}-t)$.

(d) $U^{\prime}_{t}=\bigcup_{s}\ p_{1}^{-1}(U_{s,t})$ is nonsingular.

###### Proof.

(a) Proposition 4.7 proves that $\tilde{\mathcal{F}}_{v,w}$ is nonsingular,
and we shall show that $\tilde{\mathcal{G}}_{v,w}$ is nonsingular by the same
method. We can think of the ambient vector bundle $E$ to be trivial bundle of
rank $(w_{1}w_{1}^{\prime}+w_{2}w_{2}^{\prime}+rw_{1}^{\prime}w_{2})$ over
$\mathcal{G}_{v,w}$ where each fiber is the vector space of all possible
$({\bf x},{\bf y})$ without restriction. Let $E_{1}$ be the subbundle of $E$
whose fiber over $(X_{1},X^{\prime}_{2})$ is

$\\{({\bf x},{\bf y})\ |\ \textrm{column vectors of $x_{1},y_{1},\dots,y_{r}$
are in }X_{1}\\}$

Let $E_{2}$ be the subbundle of $E$ whose fiber over $(X_{1},X^{\prime}_{2})$
is

$\\{({\bf x},{\bf y})\ |\ \textrm{row vectors of $x_{2},y_{1},\dots,y_{r}$ are
in }X_{2}^{\prime}\\}$

The fact that they are indeed (locally free) subbundles follows the
isomorphisms

$E_{1}\cong\pi_{1}^{*}S_{Gr(v_{1},w^{\prime}_{1})}^{\oplus(w_{1}+rw_{2})}\oplus\mathbb{C}^{w_{2}w^{\prime}_{2}},\quad
E_{2}\cong\pi_{2}^{*}S_{Gr(v_{2},w_{2})}^{\oplus(w^{\prime}_{2}+rw^{\prime}_{1})}\oplus\mathbb{C}^{w_{1}w_{1}^{\prime}}.$

where $\pi_{1},\pi_{2}$ are the natural projections from
$\mathcal{G}_{v,w}=Gr(v_{1},w^{\prime}_{1})\times Gr(v_{2},w_{2})$ to the
first and second factor, respectively, and $\mathbb{C}^{w_{2}w^{\prime}_{2}}$
denotes the trivial bundle whose fibers are the set of $x_{2}$,
$\mathbb{C}^{w_{1}w_{1}^{\prime}}$ denotes the trivial bundle whose fibers are
the set of $x_{1}$. It is well-known that the intersection of two subbundles
is a subbundle, provided that the dimensions of all fibers of the intersection
are equal. So in order to prove that the total space of
$\tilde{\mathcal{G}}_{v,w}=E_{1}\cap E_{2}$ is a subbundle, it suffices to
show that the dimensions of the fibers of $E_{1}\cap E_{2}$ are a constant
${w_{1}v_{1}+w_{2}^{\prime}{v}_{2}+rv_{1}{v}_{2}}$. Indeed, fix
$(X_{1},X_{2}^{\prime})$. For any nonnegative integer $n$, denote
$e_{1},\dots,e_{n}$ the standard basis of $\mathbb{C}^{n}$. Since
$\mathrm{rank}X_{1}=v_{1}$, there exists $P\in GL(w_{1}^{\prime})$ such that
$PX_{1}={\rm span}(e_{1},\dots,e_{v_{1}})$. Similarly, there exists $Q\in
GL(w_{2})$ such that $QX_{2}^{\prime}={\rm span}(e_{1},\dots,e_{{v}_{2}})$.
Define

$\displaystyle
Y=\\{(x_{1}^{\prime},x_{2}^{\prime},y_{1}^{\prime},\dots,y_{r}^{\prime})\;|$
$\displaystyle\mathrm{im}(x_{1}^{\prime}+y_{1}^{\prime}+\cdots+y_{r}^{\prime})\subseteq{\rm
span}(e_{1},\dots,e_{v_{1}}),$ $\displaystyle\mathrm{im}(x_{2}^{\prime
T}+y_{1}^{\prime T}+\cdots+y_{r}^{\prime T})\subseteq{\rm
span}(e_{1},\dots,e_{{v}_{2}})\;\\}.$

Then
$Y\cong\\{x_{1}^{\prime}\\}\times\\{x_{2}^{\prime}\\}\times\prod_{h=1}^{r}\\{y_{h}^{\prime}\\}\cong\mathbb{C}^{w_{1}v_{1}}\times\mathbb{C}^{w_{2}^{\prime}{v}_{2}}\times\prod_{h=1}^{r}\mathbb{C}^{v_{1}{v}_{2}}=\mathbb{C}^{w_{1}v_{1}+w_{2}^{\prime}{v}_{2}+rv_{1}{v}_{2}}$.
On the other hand, it is easy to see that the linear map

$f^{-1}(X_{1},X_{2}^{\prime})\to
Y,\quad(x_{1},x_{2},y_{1},\dots,y_{r})\mapsto(Px_{1},x_{2}Q^{T},Py_{1}Q^{T},\dots,Py_{r}Q^{T})$

is a well-defined isomorphism. This shows that $f^{-1}(X_{1},X_{2}^{\prime})$
is a vector space of dimension same as $\dim
Y=w_{1}v_{1}+w^{\prime}_{2}{v}_{2}+rv_{1}{v}_{2}$.

Now since $\tilde{\mathcal{G}}_{v,w}$ is the total space of a vector bundle
over a nonsingular variety $Gr(v_{1},w^{\prime}_{1})\times Gr({v}_{2},w_{2})$,
it must be nonsingular.

(b) It is easy to check $\pi=p_{1}\circ p_{2}$. The proof for the statement
that $p_{2}$ is stratified by $\\{U_{t}^{\prime}\\}$ is similar to Proposition
4.7(b). To prove that $p_{2}^{-1}(U^{\prime}_{t})\to U^{\prime}_{t}$ is a
locally trivial bundle, we show that it is a pullback of another locally
trivial bundle $\mathcal{X}\to\mathcal{F}_{(v_{1},t),w}$. Here is the diagram:

$\textstyle{p_{2}^{-1}(U_{t}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{U^{\prime}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathcal{F}_{(v_{1},t),w}}$

where

$\displaystyle\mathcal{X}=\\{(X_{1},X_{2},Y_{2})|(X_{1},Y_{2})\in\mathcal{F}_{(v_{1},t),w},Y_{2}\subseteq
X_{2}\subseteq W^{\prime}_{2}\oplus X_{1}^{\oplus r},\dim X_{2}=v_{2}\\}$
$\displaystyle f:({\bf x},{\bf
y},X_{1})\mapsto(X_{1},Y_{2})=(X_{1},\text{im}B({\bf x},{\bf y}))$
$\displaystyle q:(X_{1},X_{2},Y_{2})\mapsto(X_{1},Y_{2})$

We claim that $q$ gives a locally trivial fiber bundle with fibers isomorphic
to $\mathcal{M}$. Indeed, $\mathcal{F}_{(v_{1},t),w}$ can be covered by open
subsets over which the direct sum of universal subbundles
$S_{Gr(v_{1},W^{\prime}_{1})}\oplus
S_{Gr(t,W^{\prime}_{2}\oplus{W^{\prime}_{1}}^{\oplus r})}$ is trivial, so we
only need to discuss fiberwisely. For fixed $(X_{1},Y_{2})$, the set
$(X_{1},X_{2},Y_{2})$ satisfying $Y_{2}\subseteq X_{2}\subseteq
W^{\prime}_{2}\oplus X_{1}^{\oplus r}$ obviously form the variety

$Gr(\dim X_{2}-\dim Y_{2},\dim(W^{\prime}_{2}\oplus X_{1}^{\oplus r})-\dim
Y_{2})=Gr(v_{2}-t,w^{\prime}_{2}+rv_{1}-t)$

So $p_{2}$ is a locally trivial bundle with fibers isomorphic to
$Gr(v_{2}-t,w^{\prime}_{2}+rv_{1}-t)$.

Similarly, by considering the diagram

$\textstyle{p_{1}^{-1}(U_{s,t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{U_{s,t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Gr(s,W^{\prime}_{1})}$

where

$\displaystyle\mathcal{X}=\\{(X_{1},Y_{1})|Y_{1}\in
Gr(s,W^{\prime}_{1}),Y_{1}\subseteq X_{1}\\}$ $\displaystyle f:({\bf x},{\bf
y})\mapsto Y_{1}=\text{im}A({\bf x},{\bf y})$ $\displaystyle
q:(X_{1},Y_{1})\mapsto X_{1}$

we conclude that $p_{1}$ is a locally trivial bundle with fibers isomorphic to
$Gr(v_{1}-s,w_{1}^{\prime}-s).$

(c) can be proved similarly to (b). The diagram is

$\textstyle{p_{2}^{\prime-1}(U^{\prime}_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{2}}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{U^{\prime}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Gr(v_{1},W^{\prime}_{1})\times
Gr(t,W_{2})}$

where

$\displaystyle\mathcal{X}=\\{(X_{1},X^{\prime}_{2},Y^{\prime}_{2})|(X_{1},Y^{\prime}_{2})\in
Gr(v_{1},W^{\prime}_{1})\times Gr(t,W_{2}),Y^{\prime}_{2}\subseteq
X^{\prime}_{2}\subseteq W_{2},\dim X^{\prime}_{2}=v_{2}\\}$ $\displaystyle
f:({\bf x},{\bf y},X_{1})\mapsto(X_{1},Y^{\prime}_{2})=(X_{1},\text{im}(B({\bf
x},{\bf y})^{T}))$ $\displaystyle
q:(X_{1},X^{\prime}_{2},Y^{\prime}_{2})\mapsto(X_{1},Y^{\prime}_{2})$

(d) To prove that $U^{\prime}_{t}$ is nonsingular, we consider the following
open covering

$U^{\prime}_{t}=\bigcup U^{\prime}_{J}$

where $J\subseteq\\{1,\dots,w_{2}\\}$, $|J|=t$, $U^{\prime}_{J}\subseteq
U^{\prime}_{T}$ consists of those $({\bf x},\bf{y},X_{1})$ such that the
columns of $B({\bf x},{\bf y})$ indexed by $J$ are linearly independent. We
shall show that $U^{\prime}_{J}$ is nonsingular. Without loss of generality,
we assume $J=\\{1,\dots,t\\}$. Similar to the proof of Proposition 4.7 (b),
$U^{\prime}_{J}$ is isomorphic to the open subset of

$U=\mathbb{A}^{w^{\prime}_{2}t}\times\mathbb{A}^{(w_{2}-t)t}\times
S^{\oplus(w_{1}+rt)}_{Gr(v_{1},W^{\prime}_{1})}$

whose elements are denoted
$(Z^{\prime}_{2},N,x_{1},C^{\prime}_{1},\dots,C^{\prime}_{r},X_{1})$, where
$X_{1}\in Gr(v_{1},W^{\prime}_{1})$, and matrices:

$Z^{\prime}_{2}$ has size $w^{\prime}_{2}\times t$,

$N$ has size $t\times(w_{2}-t)$,

$x_{1}$ has size $w^{\prime}_{1}\times w_{1}$ and all column vectors are in
$X_{1}$,

$C^{\prime}_{i}$ has size $w^{\prime}_{1}\times t$ and all column vectors are
in $X_{1}$,

The isomorphism is sending
$(Z^{\prime}_{2},N,x_{1},C^{\prime}_{1},\dots,C^{\prime}_{r},X_{1})$ to
$(x_{1},x_{2},y_{1},\dots,y_{r},X_{1})$ with

$x_{2}=[Z^{\prime}_{2}\;\;Z^{\prime}_{2}N],\quad
y_{i}=[C^{\prime}_{i}\;\;C^{\prime}_{i}N]$

So

$A=\begin{bmatrix}x_{1}&C^{\prime}_{1}&C^{\prime}_{1}N&\cdots&C^{\prime}_{r}&C^{\prime}_{r}N\end{bmatrix},\quad
B=\begin{bmatrix}Z^{\prime}_{2}&Z^{\prime}_{2}N\\\
C^{\prime}_{1}&C^{\prime}_{1}N\\\ \vdots&\vdots\\\
C^{\prime}_{r}&C^{\prime}_{r}N\end{bmatrix}$

The open subset of $U^{\prime}_{J}$ is defined by the condition that
$[Z^{\prime}_{2},C^{\prime}_{1},\dots,C^{\prime}_{r}]_{\text{vert}}$ must have
full column rank $t$. ∎

To summarize the notation and computations we introduced so far:

$\textstyle{{\mathcal{F}}_{v,w}}$$\textstyle{\tilde{\mathcal{F}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi}$$\textstyle{U^{\prime}_{t}\;\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\mathcal{H}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{U_{s,t}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}_{\bar{v},w}}$
$\textstyle{{\mathcal{G}}_{v,w}}$$\textstyle{\tilde{\mathcal{G}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}_{2}}$$\scriptstyle{\pi^{\prime}}$$\textstyle{U^{\prime}_{t}\;\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\mathcal{H}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{U_{s,t}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}_{\bar{v},w}}$

space | dimension
---|---
$\mathcal{F}_{v,w}$ | $d_{v,w}=v_{2}(w_{2}^{\prime}+rv_{1}-v_{2})+v_{1}(w_{1}^{\prime}-v_{1})$
$\tilde{\mathcal{F}}_{v,w}$ | $\tilde{d}_{v,w}=v_{2}(w_{2}+w_{2}^{\prime}+rv_{1}-v_{2})+v_{1}(w_{1}+w_{1}^{\prime}-v_{1})$
$\mathbf{E}_{v,w}$ | $\bar{v}_{2}(w_{2}+w_{2}^{\prime}+r\bar{v}_{1}-\bar{v}_{2})+\bar{v}_{1}(w_{1}+w_{1}^{\prime}-\bar{v}_{1})$
$\mathcal{G}_{v,w}$ | $v_{1}(w^{\prime}_{1}-v_{1})+v_{2}(w_{2}-v_{2})$
$\tilde{\mathcal{G}}_{v,w}$ | $v_{1}(w_{1}+w^{\prime}_{1}-v_{1})+v_{2}(w_{2}+w^{\prime}_{2}+rv_{1}-v_{2})$
map | fiber
---|---
$\tilde{\mathcal{F}}_{v,w}\to\mathcal{F}_{v,w}$ | $\mathbb{C}^{w_{1}v_{1}+w_{2}v_{2}}$
$\tilde{\mathcal{G}}_{v,w}\to\mathcal{G}_{v,w}$ | $\mathbb{C}^{w_{1}v_{1}+w_{2}^{\prime}v_{2}+rv_{1}v_{2}}$
$p_{2}|_{U^{\prime}_{t}}$ | $Gr(v_{2}-t,w^{\prime}_{2}+rv_{1}-t)$
$p^{\prime}_{2}|_{U^{\prime}_{t}}$ | $Gr(v_{2}-t,w_{2}-t)$
$p_{1}|_{U_{s,t}}$ | $Gr(v_{1}-s,w^{\prime}_{1}-s)$

##  7\. Definition of dual canonical basis elements $L(w)$

The identification between dual canonical basis and triangular basis is
explained to us by Fan Qin, using results in [21, 14]. We will give a proof as
self-contained as possible after introducing the dual canonical basis.

###  7.1. Definition of $L(w)$ and $\mathbf{R}^{\mathrm{finite}}_{t}$

First, recall some definitions from [19, 21]: Define
$\mathcal{D}(\mathbf{E}_{w})$ to be the bounded derived category of
constructible sheaves of $\mathbb{Q}$-vector spaces on $\mathbf{E}_{w}$. (The
two references ) Define a set

$\mathcal{P}_{w}=\\{IC_{w}(v)=IC_{\mathbf{E}_{v,w}}\ |\
v\in\mathrm{Dom}(w)\\}$

Note that this is a finite set because $v$ must satisfy the condition (4.9).

Define a full subcategory $\mathcal{Q}_{w}$ of $\mathcal{D}(\mathbf{E}_{w})$
whose objects are finite direct sums of $IC_{w}(v)[k]$ for various
$v\in\mathrm{Dom}(w)$, $k\in\mathbb{Z}$.

Define an abelian group $\mathcal{K}_{w}$ to be generated by isomorphism
classes $(L)$ of objects of $\mathcal{Q}_{w}$ and quotient by the relations
$(L)+(L^{\prime})=(L^{\prime\prime})$ whenever $L\oplus L^{\prime}\cong
L^{\prime\prime}$. By abuse of notation we denote $(L)$ simply as $L$, as done
in [19]. We can view $\mathcal{K}_{w}$ as a free $\mathbb{Z}[t^{\pm}]$-module
with a basis $\mathcal{P}_{w}$, by defining $t^{i}L=L[i]$ for
$i\in\mathbb{Z}$. The duality on $\mathcal{D}(\mathbf{E}_{w})$ induces the bar
involution on $\mathcal{K}_{w}$ satisfying
$\overline{tL}=t^{-1}\overline{L},\;\overline{IC_{w}(v)}=IC_{w}(v)$. By the
BBDG decomposition and Theorem 5.1, for arbitrary $v,w$,

(7.1)
$\pi_{w}(v)=\pi_{*}(IC_{\widetilde{\mathcal{F}}_{v,w}})=\sum_{v^{\prime}\in\mathrm{Dom}(w)}\sum_{d}a^{d}_{v,v^{\prime};w}IC_{w}(v^{\prime})[d]=\sum_{v^{\prime}\in\mathrm{Dom}(w)}a_{v,v^{\prime};w}IC_{w}(v^{\prime})$

where $a_{v,v^{\prime};w}=\sum_{d}a^{d}_{v,v^{\prime};w}t^{d}$. It follows
from the (Algebraic or Analytic) Transversal Slice Theorem that

(7.2) $a_{v,v^{\prime};w}=a_{v^{\perp},v^{\prime\perp};w^{\perp}}\textrm{ for
any $v^{0}\in\mathrm{Dom}(w)$}.$

Note that $\\{\pi_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}$ is also a
$\mathbb{Z}[t^{\pm}]$-basis for $\mathcal{K}_{w}$, and for
$(v,w)\in\mathrm{Dom}$ we have $a_{v,v;w}=1$.

Define the dual $\mathcal{K}_{w}^{*}={\rm
Hom}_{\mathbb{Z}[t^{\pm}]}(\mathcal{K}_{w},\mathbb{Z}[t^{\pm}])$, which is a
free $\mathbb{Z}[t^{\pm}]$-module with a basis

$\\{L_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}=\textrm{the dual basis to
}\mathcal{P}_{w},$

that is, $\langle L_{w}(v),IC_{w}(v^{\prime})\rangle=\delta_{v,v^{\prime}}$.
The pairing
$\langle-,-\rangle:\mathcal{K}_{w}^{*}\times\mathcal{K}_{w}\to\mathbb{Z}[t^{\pm}]$
satisfies the condition

$\langle L,C[1]\rangle=\langle L[-1],C\rangle=t\langle
L,C\rangle,\quad\textrm{ for all
}L\in\mathcal{K}_{w}^{*},C\in\mathcal{K}_{w}.$

Note that $L_{w}(v)$ satisfies the property that

$\langle
L_{w}(v),IC_{w}(v^{\prime})\rangle=\delta_{v,v^{\prime}}=\delta_{v^{\perp},v^{\prime\perp}}=\langle
f_{w^{\perp}}(v^{\perp}),IC_{w^{\perp}}(v^{\prime\perp})\rangle\textrm{ for
any $v^{0}\leq v$},$

where $v^{0}$ is used to define
$w^{\perp}=w-C_{q}v^{0},v^{\perp}=v-v^{0},v^{\prime\perp}=v^{\prime}-v^{0}$.
Define

$\mathbf{R}_{t}=\\{(f_{w})\in\prod_{w}\mathcal{K}_{w}^{*}\ |\ \langle
f_{w},IC_{w}(v)\rangle=\langle
f_{w^{\perp}},IC_{w^{\perp}}(v^{\perp})\rangle\textrm{ whenever
}v\in\mathrm{Dom}(w),v^{0}\leq v\\}.$

For $(f_{w})\in\prod_{w}\mathcal{K}_{w}^{*}$, write
$f_{w}=\sum_{v\in\mathrm{Dom}(w)}c_{wv}L_{w}(v)$ where
$c_{wv}\in\mathbb{Z}[t^{\pm}]$. Then $c_{wv}=\langle f_{w},IC_{w}(v)\rangle$,
and

$(f_{w})\in\mathbf{R}_{t}\textrm{ if and only if
}c_{wv}=c_{w-C_{q}v,0}\textrm{ for every $l$-dominant pair $(w,v)$}.$

So an element in $\mathbf{R}_{t}$ is uniquely determined by
$\\{c_{w0}\\}_{w}$.

Define $L(w)\in\mathbf{R}_{t}$ to be induced by $L_{w}(0)$, that is, for any
$(v^{\prime},w^{\prime})\in\mathrm{Dom}$,

$\langle
L(w),IC_{w^{\prime}}(v^{\prime})\rangle=\delta_{w,w^{\prime}-C_{q}v^{\prime}}$

Define $\mathbf{R}^{\mathrm{finite}}_{t}$ to be the
$\mathbb{Z}[t^{\pm}]$-submodule of $\mathbf{R}_{t}$ spanned by
$\\{L(w)\\}_{w}$: 222Note that $\\{L(w)\\}$ is not a basis for
$\mathbf{R}_{t}$: for example, let all $c_{w0}=1$, that is,
$f_{w}=\sum_{v\in\mathrm{Dom}(w)}L_{w}(v)$; this gives an element in
$\mathbf{R}_{t}$ corresponding to the infinite sum $\sum_{w}L(w)$, which is
not in the span of $\\{L(w)\\}$. Nevertheless, each element in
$\mathbf{R}_{t}$ can be uniquely written as a possibly infinite linear
combination of $\\{L(w)\\}$. We can think of the projective limit
$\lim\limits_{\leftarrow}\mathcal{K}_{w}$ which identify $IC_{w}(v)$ with
$IC_{w^{\perp}}(v^{\perp})$ whenever $v\in\mathrm{Dom}(w),v^{0}\leq v$. Then
$\mathbf{R}_{t}$ is the dual of $\lim\limits_{\leftarrow}\mathcal{K}_{w}$, and
$\mathbf{R}^{\mathrm{finite}}_{t}$ consists of those linear functionals
induced from ones defined on finite rank.

$\displaystyle\mathbf{R}^{\mathrm{finite}}_{t}$ $\displaystyle=\textrm{ the
free $\mathbb{Z}[t^{\pm}]$-module with the basis $\\{L(w)\\}_{w}$ }$
$\displaystyle=\\{(f_{w})\in\mathbf{R}_{t}\ |\ \langle
f_{w},IC_{w}(0)\rangle=0\textrm{ for all but finitely many $w$}\\}$

###  7.2. Definition of $M(w)$

Next, define $\\{M_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}\in\mathcal{K}_{w}^{*}$ to
be the functional

$(L)\mapsto\sum_{k}t^{\dim{\bf E}^{\circ}_{v,w}-k}\dim
H^{k}(i^{!}_{x_{v,w}}L)$

where $x_{v,w}$ is a point in ${\bf E}^{\circ}_{v,w}$, and
$i_{x_{v,w}}:x_{v,w}\to{\bf E}_{w}$ is the natural embedding.

Define $M(w)\in\mathbf{R}_{t}$ to be induced by $M_{w}(0)$, that is, for any
$(v^{\prime},w^{\prime})\in\mathrm{Dom}$,

(7.3) $\displaystyle\langle M(w),IC_{w^{\prime}}(v^{\prime})\rangle$
$\displaystyle=\begin{cases}&\langle
M_{w}(0),IC_{w}(v)\rangle=\sum\limits_{k}t^{-k}\dim
H^{k}(i^{!}_{0}IC_{w}(v)),\textrm{ if $\exists v\in\mathbb{Z}_{\geq
0}^{2}:w-w^{\prime}=C_{q}(v-v^{\prime})$;}\\\ &0,\textrm{ otherwise}.\\\
\end{cases}$

where $i_{0}:\\{0\\}\to{\bf E}_{w}$ is the natural embedding. We claim that
$M(w)\in\mathbf{R}^{\mathrm{finite}}_{t}$. Write
$M(w)=\sum_{w^{\prime\prime}}b_{ww^{\prime\prime}}L(w^{\prime\prime})$, then
for any $(w^{\prime},v^{\prime})\in\mathrm{Dom}$:

$\langle
M(w),IC_{w^{\prime}}(v^{\prime})\rangle=\sum_{w^{\prime\prime}}b_{ww^{\prime\prime}}\delta_{w^{\prime\prime},w^{\prime}-C_{q}v^{\prime}}=b_{w,w^{\prime}-C_{q}v^{\prime}}$

Together with (7.3), we have

$b_{ww^{\prime\prime}}=\begin{cases}&b_{w,w-C_{q}v}=\sum\limits_{k}t^{-k}\dim
H^{k}(i^{!}_{0}IC_{w}(v)),\textrm{ if $w^{\prime\prime}=w-C_{q}v$ for some
$v\in\mathbb{Z}_{\geq 0}^{2}$};\\\ &0,\textrm{ otherwise}.\\\ \end{cases}$

As a consequence, $b_{w,w-C_{q}v}\neq 0\ \Rightarrow\ w-C_{q}v\geq 0\
\Rightarrow\ 0\leq v_{1}\leq w^{\prime}_{1}$ and $0\leq v_{2}\leq w_{2}$. So
for each fixed $w$, $b_{ww^{\prime\prime}}=0$ for all but finitely many
$w^{\prime\prime}$, thus $M(w)\in\mathbf{R}^{\mathrm{finite}}_{t}$.

We then claim that $\\{M(w)\\}$ is a basis of
$\mathbf{R}^{\mathrm{finite}}_{t}$. Indeed, denote a partial order

$w^{\prime}\leq_{\rm w}w\Longleftrightarrow w-w^{\prime}=C_{q}u$ for some
$u\geq 0$

and denote $w^{\prime}<_{\rm w}w$ if $w^{\prime}\leq_{\rm w}w$ and
$w^{\prime}\neq w$. Consider the $\mathbb{Z}[t^{\pm}]$-submodule $V$ of
$\mathbf{R}^{\mathrm{finite}}_{t}$ spanned by
$\\{L(w^{\prime})\\}_{w^{\prime}\leq_{\rm w}w}$. By the previous paragraph,
the transition matrix $B_{w}=[b_{w^{\prime}w^{\prime\prime}}]$ (where
$w^{\prime},w^{\prime\prime}\leq_{\rm w}w$) from
$\\{L(w^{\prime})\\}_{w^{\prime}\leq_{\rm w}w}$ to
$\\{M(w^{\prime})\\}_{w^{\prime}\leq_{\rm w}w}$ is a (finite) triangular
matrix. The diagonal entries $b_{w^{\prime}w^{\prime}}=\sum_{k}t^{-k}\dim
H^{k}(i^{!}_{0}IC_{w^{\prime}}(0))=1$ since
$IC_{w^{\prime}}(0)=\mathbb{Q}_{\\{0\\}}$ and

$H^{k}(i^{!}_{0}IC_{w^{\prime}}(0))=H^{k}_{\\{0\\}}({\bf
E}_{w^{\prime}},\mathbb{Q}_{\\{0\\}})=H^{k}({\bf
E}_{w^{\prime}},\mathbb{Q}_{\\{0\\}})=\begin{cases}&\mathbb{Q},\textrm{ if
$k=0$};\\\ &0,\textrm{ otherwise}.\\\ \end{cases}$

where the first equality uses the fact that if $f:X\to Y$ is a closed
embedding then $H^{k}(X,f^{!}K)=H_{X}(Y,K)$ ([7, 4.1.12]), the second equality
uses the long exact sequence of cohomology with supports, [11, III, Ex 2.3].
Thus the matrix $B_{w}$ is invertible (and we denote its inverse by
$B_{w}^{-1}=[b^{\prime}_{w^{\prime}w^{\prime\prime}}]$) and $L(w)$ is in the
span of $\\{M(w^{\prime})\\}_{w^{\prime}\leq_{\rm w}w}$. So $\\{M(w)\\}$ is a
basis of $\mathbf{R}^{\mathrm{finite}}_{t}$.

Moreover, by definition of the intersection cohomology complex
$IC_{w}(v)=IC_{{\bf E}_{v,w}}(\mathbb{Q})$, if $v>(0,0)$ and $\dim
H^{k}(i^{!}_{0}IC_{w}(v))\neq 0$, then $k\geq 1$ ([1, p142, Lemma 3.3.11]). So
$b_{ww^{\prime\prime}}\in t^{-1}\mathbb{Z}[t^{-1}]$, and

$M(w)=\sum_{w^{\prime\prime}\leq_{\rm
w}w}b_{ww^{\prime\prime}}L(w^{\prime\prime})\in
L(w)+\sum_{w^{\prime\prime}<_{\rm
w}w}t^{-1}\mathbb{Z}[t^{-1}]L(w^{\prime\prime}),$

which implies

(7.4) $L(w)=\sum_{w^{\prime\prime}\leq_{\rm
w}w}b^{\prime}_{ww^{\prime\prime}}M(w^{\prime\prime})\in
M(w)+\sum_{w^{\prime\prime}<_{\rm
w}w}t^{-1}\mathbb{Z}[t^{-1}]M(w^{\prime\prime}).$

###  7.3. Definition of $\chi$

Define a map $\phi:\mathbb{Z}^{4}\mapsto\mathbb{Z}^{2}:\
(u_{1},u_{1}^{\prime},u_{2},u^{\prime}_{2})\mapsto(u_{1}-u_{1}^{\prime},u_{2}-u_{2}^{\prime})$.
Define a map $\chi:\mathbf{R}^{\mathrm{finite}}_{t}\to\mathcal{T}$ by defining
it on the basis $\\{M(w)\\}$ (note that we do not require
$v\in\mathrm{Dom}(w)$):

(7.5) $\displaystyle\chi(M(w))$ $\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}\big{(}\langle
M(w),\pi_{w}(v)\rangle_{t\to\mathbf{v}^{r}}\big{)}X^{\phi(w-C_{q}v)}$
$\displaystyle=\sum_{v\in\mathbb{Z}_{\geq 0}^{2}}\sum_{k}\mathbf{v}^{-kr}\dim
H^{k}(i_{0}^{!}\pi_{w}(v))X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$

and extend it by the rules $\chi(L_{1}+L_{2})=\chi(L_{1})+\chi(L_{2})$ and
$\chi(tL)=\mathbf{v}^{r}\chi(L)$,

We claim the following is true: (again, we do not require
$v\in\mathrm{Dom}(w)$)

(7.6) $\chi(L(w))=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}a_{v,0;w}(\mathbf{v}^{r})X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$

Indeed,

$\displaystyle\chi(L(w))$ $\displaystyle=\sum_{w^{\prime\prime}\leq_{\rm
w}w}b^{\prime}_{ww^{\prime\prime}}\chi(M(w^{\prime\prime}))=\sum_{w^{\prime\prime}\leq_{\rm
w}w}b^{\prime}_{ww^{\prime\prime}}\sum_{v^{\prime\prime}}\big{(}\langle
M(w^{\prime\prime}),\pi_{w^{\prime\prime}}(v^{\prime\prime})\rangle_{t\to\mathbf{v}^{r}}\big{)}X^{\phi(w^{\prime\prime}-C_{q}v^{\prime\prime})}$
$\displaystyle=\sum_{w^{\prime\prime}\leq_{\rm
w}w}\sum_{v^{\prime}\in\mathrm{Dom}(w^{\prime\prime})}\sum_{v^{\prime\prime}}(a_{v^{\prime\prime},v^{\prime};w^{\prime\prime}}b^{\prime}_{ww^{\prime\prime}}b_{w^{\prime\prime},w^{\prime\prime}-C_{q}v^{\prime}})_{t\to\mathbf{v}^{r}}X^{\phi(w^{\prime\prime}-C_{q}v^{\prime\prime})}$

Now change the variables: let $u\in\mathbb{Z}_{\geq 0}^{2}$ satisfy
$w=w^{\prime\prime}+C_{q}u$, and let $v=v^{\prime\prime}+u$,
$u^{\prime}=v^{\prime}+u$. Then
$w-C_{q}=w^{\prime\prime}-C_{q}v^{\prime\prime}$,
$a_{v,u^{\prime};w}=a_{v^{\prime\prime},v^{\prime};w^{\prime\prime}}$, and

$\chi(L(w))=\sum_{u\in\mathrm{Dom}(w)}\sum_{\stackrel{{\scriptstyle
u^{\prime}\in\mathrm{Dom}(w)}}{{u^{\prime}\geq u}}}\sum_{v\geq
u}(a_{v,u^{\prime};w}b^{\prime}_{w,w-C_{q}u}b_{w-C_{q}u,w-C_{q}u^{\prime}})_{t\to\mathbf{v}^{r}}X^{\phi(w-C_{q}v)}$

In the above expression, all the conditions under the summation symbols can be
dropped: if $v\not\geq u$, then $v\not\geq u^{\prime}$, and thus
$a_{v,u^{\prime};w}=0$; if $u\notin\mathrm{Dom}(w)$ or
$u^{\prime}\notin\mathrm{Dom}(w)$ or $u^{\prime}\not\geq u$, then
$b_{w-C_{q}u,w-C_{q}u^{\prime}}=0$. So we drop the conditions and swap the
orders of the variables $u,u^{\prime},v$ in the iterated sums:

$\displaystyle\chi(L(w))$
$\displaystyle=\sum_{u}\sum_{u^{\prime}}\sum_{v}(a_{v,u^{\prime};w}b^{\prime}_{w,w-C_{q}u}b_{w-C_{q}u,w-C_{q}u^{\prime}})_{t\to\mathbf{v}^{r}}X^{\phi(w-C_{q}v)}$
$\displaystyle=\sum_{v}X^{\phi(w-C_{q}v)}\sum_{u^{\prime}}(a_{v,u^{\prime};w})_{t\to\mathbf{v}^{r}}\sum_{u}(b^{\prime}_{w,w-C_{q}u}b_{w-C_{q}u,w-C_{q}u^{\prime}})_{t\to\mathbf{v}^{r}}$

Since if $w^{\prime\prime}$ is not of the form $w-C_{q}u$ then
$b_{w-C_{q}u,w-C_{q}u^{\prime}}=0$, the last summation is

$\sum_{u}(b^{\prime}_{w,w-C_{q}u}b_{w-C_{q}u,w-C_{q}u^{\prime}})_{t\to\mathbf{v}^{r}}=\sum_{w^{\prime\prime}}(b^{\prime}_{w,w^{\prime\prime}}b_{w^{\prime\prime},w-C_{q}u^{\prime}})_{t\to\mathbf{v}^{r}}=\delta_{w,w-C_{q}u^{\prime}}$

thus $\chi(L(w))=\sum_{v}a_{v,0;w}(\mathbf{v}^{r})X^{\phi(w-C_{q}v)}$, (7.6)
is proved.

###  7.4. $\\{\overline{\chi(M(w))}\\}$ and the standard monomial basis

Denote $P_{t}(X)=\sum_{i}\dim H^{i}(X,\mathbb{Q})t^{i}$. Denote
$d_{v,w}=\dim\mathcal{F}_{v,w}$,
$\tilde{d}_{v,w}=\dim\widetilde{\mathcal{F}}_{v,w}$. We shall prove an
explicit formula for $\chi(M(w))$:

(7.7) $\displaystyle\chi(M(w))$ $\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}\mathbf{v}^{r(d_{v,w}-\tilde{d}_{v,w})}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{\mathbf{v}^{r}}{w^{\prime}_{1}\brack
v_{1}}_{\mathbf{v}^{r}}X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$
$\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}\mathbf{v}^{-r\dim\widetilde{\mathcal{F}}_{v,w}}P_{t}(\mathcal{F}_{v,w})X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$

Indeed, apply the base change isomorphism
$i_{0}^{!}\pi_{*}=\pi^{\prime}_{*}j^{!}$ for the following Cartesian square

${\mathcal{F}_{v,w}}$${\widetilde{\mathcal{F}}_{v,w}}$${\\{0\\}}$${\mathbf{E}_{v,w}}$$\scriptstyle{j}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{i_{0}}$

we have

$i_{0}^{!}\pi_{*}(IC_{\widetilde{\mathcal{F}}_{v,w}})=\pi^{\prime}_{*}j^{!}(\mathbb{Q}_{\widetilde{\mathcal{F}}_{v,w}}[\tilde{d}_{v,w}])=\pi^{\prime}_{*}j^{*}(\mathbb{Q}_{\widetilde{\mathcal{F}}_{v,w}}[\tilde{d}_{v,w}-2(\tilde{d}_{v,w}-d_{v,w})])=\pi^{\prime}_{*}(\mathbb{Q}_{{\mathcal{F}}_{v,w}}[2d_{v,w}-\tilde{d}_{v,w}]),$

where the second equality is because $j^{!}(C)=j^{*}[-2c](C)$ if $j$ is a
closed embedding of pure codimension $c$ transversal to all strata of a
stratification $\Sigma$ and the complex $C$ is $\Sigma$-constructible [7,
§4.1.12], here we just take the trivial stratification for
$\widetilde{\mathcal{F}}_{v,w}$. Then

(7.8) $\displaystyle\langle
M(w),\pi_{w}(v)\rangle=\sum_{k}t^{-k}H^{k}(i_{0}^{!}\pi_{*}(IC_{\widetilde{\mathcal{F}}_{v,w}}))$
$\displaystyle=\sum_{k}t^{-k}H^{k+2d-\tilde{d}_{v,w}}(\\{0\\},\pi^{\prime}_{*}(\mathbb{Q}_{{\mathcal{F}}_{v,w}}))=\sum_{k}\dim
t^{-k}H^{k+2d_{v,w}-\tilde{d}_{v,w}}({{\mathcal{F}}_{v,w}},\mathbb{Q})$
$\displaystyle=t^{2d_{v,w}-\tilde{d}_{v,w}}\sum_{k}\dim
t^{-k}H^{k}({{\mathcal{F}}_{v,w}},\mathbb{Q})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}t^{2d_{v,w}-\tilde{d}_{v,w}}\Big{(}t^{-\dim
Gr(v_{2},w_{2}^{\prime}+rv_{1})}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}\Big{)}\Big{(}t^{-\dim
Gr(v_{1},w_{1}^{\prime})}{w^{\prime}_{1}\brack v_{1}}_{t}\Big{)}$
$\displaystyle=t^{d_{v,w}-\tilde{d}_{v,w}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}{w^{\prime}_{1}\brack v_{1}}_{t}$

where (*) is because ${\mathcal{F}}_{v,w}$ is a
$Gr(v_{2},w_{2}^{\prime}+rv_{1})$-bundle over $Gr(v_{1},w_{1}^{\prime})$, and
in the last equality we use the fact that $\dim
Gr(v_{2},w_{2}^{\prime}+rv_{1})+\dim
Gr(v_{1},w_{1}^{\prime})=\dim{\mathcal{F}}_{v,w}=d_{v,w}$. Also note that

$P_{t}(\mathcal{F}_{v,w})=t^{d_{v,w}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}{w^{\prime}_{1}\brack v_{1}}_{t},$

thus we have proved (7.7).

Now we choose the initial seed to be $({\bf
X}^{\prime},B^{\prime})=\mu_{2}({\bf X},B)$. Thus
$B^{\prime}=\begin{bmatrix}0&r\\\ -r&0\end{bmatrix}$, the linear order is
$2\vartriangleleft 1$, and the initial cluster is $(X_{1},X_{0})$. The
standard monomial basis elements with respect to this initial seed is

$M^{\prime}[{\bf
b}]=\mathbf{v}^{b_{1}b_{2}}X_{2}^{[-b_{2}]_{+}}X_{0}^{[b_{2}]_{+}}X_{1}^{[b_{1}]_{+}}X_{-1}^{[-b_{1}]_{+}}$

More generally, define

$M^{*}[w]=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}}X_{-1}^{w^{\prime}_{1}}$

Each $w\in\mathbb{Z}_{\geq 0}^{4}$ can be uniquely written as
$w={}^{f}w+{}^{\phi}w$ where

${}^{f}w=\big{(}\min(w_{1},w^{\prime}_{1}),\min(w_{1},w^{\prime}_{1}),\min(w_{2},w^{\prime}_{2}),\min(w_{2},w^{\prime}_{2})\big{)}$

thus ${}^{\phi}w$ satisfies
${}^{\phi}w_{1}{}^{\phi}w^{\prime}_{1}={}^{\phi}w_{2}{}^{\phi}w^{\prime}_{2}=0$.

We have

(7.9) if $w={}^{\phi}w$ (that is,
$w_{1}w^{\prime}_{1}=w_{2}w^{\prime}_{2}=0$), then
$M^{*}[w]=M^{\prime}[w_{1}-w^{\prime}_{1},w_{2}^{\prime}-w_{2}]$.

######  Lemma 7.1.

For any $w\in\mathbb{Z}_{\geq 0}^{4}$, we have

$M^{*}[w]=\overline{\chi(M(w))}=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}\mathbf{v}^{r(-d_{v,w}+\tilde{d}_{v,w})}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{\mathbf{v}^{r}}{w^{\prime}_{1}\brack
v_{1}}_{\mathbf{v}^{r}}X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}.$

In particular, if $w={}^{\phi}w$, then $\overline{\chi(M(w))}$ is the standard
basis element $M^{\prime}[w_{1}-w^{\prime}_{1},w^{\prime}_{2}-w_{2}]$ with the
initial seed $\mu_{2}({\bf X},B)$.

###### Proof.

Note that $X_{0}=X^{(0,-1)}+X^{(r,-1)}$, $X_{-1}=X^{(-1,0)}+\sum_{i}{r\brack
i}_{\mathbf{v}^{r}}X^{(ri-1,-r)}=X_{0}^{(r,-1)}+X_{0}^{(0,-1)}$,
$X_{m-1}^{n}=\sum{n\brack i}_{\mathbf{v}^{r}}X_{m}^{(ri,-n)}=\sum{n\brack
i}_{\mathbf{v}^{r}}\mathbf{v}^{-rin}X_{m}^{ri}X_{m+1}^{-n}$. So

$\displaystyle
M^{*}[w]=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}}\sum_{v_{1}}{w^{\prime}_{1}\brack
v_{1}}\mathbf{v}^{-rv_{1}w^{\prime}_{1}}X_{0}^{rv_{1}}X_{1}^{-w^{\prime}_{1}}$
$\displaystyle=\sum_{v_{1}}\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})+2rv_{1}w_{1}}X_{2}^{w_{2}}{w^{\prime}_{1}\brack
v_{1}}\mathbf{v}^{-rv_{1}w^{\prime}_{1}}X_{0}^{w^{\prime}_{2}+rv_{1}}X_{1}^{w_{1}-w^{\prime}_{1}}$
$\displaystyle=\sum_{v_{1},v_{2}}\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})+2rv_{1}w_{1}-rv_{1}w^{\prime}_{1}-rv_{2}(w^{\prime}_{2}+rv_{1})}{w^{\prime}_{1}\brack
v_{1}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}X_{2}^{w_{2}}X_{1}^{rv_{2}}X_{2}^{-w^{\prime}_{2}-rv_{1}}X_{1}^{w_{1}-w^{\prime}_{1}}$

Since

$X_{2}^{w_{2}}X_{1}^{rv_{2}}X_{2}^{-w^{\prime}_{2}-rv_{1}}X_{1}^{w_{1}-w^{\prime}_{1}}=\mathbf{v}^{rv_{2}w_{2}+rv_{2}(w^{\prime}_{2}+rv_{1})+(w_{2}-w^{\prime}_{2}-rv_{1})(w_{1}-w^{\prime}_{1})}X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$

we have

$\displaystyle M^{*}[w]$
$\displaystyle=\mathbf{v}^{r(v_{1}w_{1}+v_{2}w_{2})}\sum_{v_{1},v_{2}}{w^{\prime}_{1}\brack
v_{1}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}X_{2}^{w_{2}}X_{1}^{rv_{2}}X_{2}^{-w^{\prime}_{2}-rv_{1}}X_{1}^{w_{1}-w^{\prime}_{1}}=\overline{\chi(M(w))}$

where the last equality is because
$v_{1}w_{1}+v_{2}w_{2}=\tilde{d}_{v,w}-d_{v,w}$. ∎

######  Lemma 7.2.

$M^{*}[w]\in M^{\prime}[w_{1}-w^{\prime}_{1},w^{\prime}_{2}-w_{2}]+\sum_{\bf
b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]$.

###### Proof.

If $\min(w_{2},w^{\prime}_{2})>0$, then

$\displaystyle
M^{*}[w]=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}-1}(X_{2}X_{0})X_{0}^{w_{2}^{\prime}-1}X_{1}^{w_{1}}X_{-1}^{w^{\prime}_{1}}$
$\displaystyle=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}-1}(1+\mathbf{v}^{r}X_{1}^{r})X_{0}^{w_{2}^{\prime}-1}X_{1}^{w_{1}}X_{-1}^{w^{\prime}_{1}}$
$\displaystyle=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}-1}X_{0}^{w_{2}^{\prime}-1}X_{1}^{w_{1}}X_{-1}^{w^{\prime}_{1}}+\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})+r}X_{2}^{w_{2}-1}X_{1}^{r}X_{0}^{w_{2}^{\prime}-1}X_{1}^{w_{1}}X_{-1}^{w^{\prime}_{1}}$
$\displaystyle=M^{*}[w_{1},w^{\prime}_{1},w_{2}-1,w^{\prime}_{2}-1]$
$\displaystyle\quad+\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})+r+2r(w^{\prime}_{2}-1)-(r+w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}M^{*}[r+w_{1},w^{\prime}_{1},w_{2}-1,w^{\prime}_{2}-1]$
$\displaystyle=M^{*}[w_{1},w^{\prime}_{1},w_{2}-1,w^{\prime}_{2}-1]+\mathbf{v}^{r(w_{2}+w^{\prime}_{2}-1)}M^{*}[r+w_{1},w^{\prime}_{1},w_{2}-1,w^{\prime}_{2}-1]$
$\displaystyle\in
M^{*}[w_{1},w^{\prime}_{1},w_{2}-1,w^{\prime}_{2}-1]+\bigoplus_{w^{\prime}}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{*}[w^{\prime}]$

Similarly, if $\min(w_{1},w^{\prime}_{1})>0$, then

$\displaystyle
M^{*}[w]=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}-1}(X_{1}X_{-1})X_{-1}^{w^{\prime}_{1}-1}$
$\displaystyle=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}-1}(1+\mathbf{v}^{r}X_{0}^{r})X_{-1}^{w^{\prime}_{1}-1}$
$\displaystyle=\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}-1}X_{-1}^{w^{\prime}_{1}-1}+\mathbf{v}^{(w_{1}-w^{\prime}_{1})(w^{\prime}_{2}-w_{2})+r}X_{2}^{w_{2}}X_{0}^{w_{2}^{\prime}}X_{1}^{w_{1}-1}X_{0}^{r}X_{-1}^{w^{\prime}_{1}-1}$
$\displaystyle=M^{*}[w_{1}-1,w^{\prime}_{1}-1,w_{2},w^{\prime}_{2}]+\mathbf{v}^{r(w_{1}+w^{\prime}_{1}-1)}M^{*}[w_{1}-1,w^{\prime}_{1}-1,w_{2},w^{\prime}_{2}+r]$
$\displaystyle\in
M^{*}[w_{1}-1,w^{\prime}_{1}-1,w_{2},w^{\prime}_{2}]+\bigoplus_{w^{\prime}}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{*}[w^{\prime}]$

Apply the above two formulas recursively, we obtain the result. ∎

###  7.5. $\\{\chi(L(w))\\}$ and the triangular basis

######  Lemma 7.3.

For any $w\in\mathbb{Z}_{\geq 0}^{4}$, we have
${\chi(L(w))}={C[w^{\prime}_{1}-w_{1},w_{2}^{\prime}-w_{2}+r[w^{\prime}_{1}-w_{1}]_{+}]}$,
which is a triangular basis element. In particular,
$\chi(L({}^{\phi}w))=\chi(L(w))$ (therefore $a_{v,0;w}=a_{v,0;{}^{\phi}w}$ for
any $v\in\mathbb{Z}_{\geq 0}^{2}$) and $\chi(L({}^{f}w))=1$.

###### Proof.

It follows from (7.6) that $\chi(L(w))$ is bar-invariant. It follows from
(7.4), Lemma 7.1 and Lemma 7.2 that

$\displaystyle\chi(L(w))-M^{\prime}[w_{1}-w^{\prime}_{1},w_{2}^{\prime}-w_{2}]\in\chi(L(w))-M^{*}[w]+\sum_{\bf
b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]\quad\text{ (by Lemma
\ref{M'' in M'}) }$
$\displaystyle\quad=\chi(L(w))-\overline{\chi(M(w))}+\sum_{\bf
b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]\quad\text{ (by Lemma
\ref{chiMbar=M''}) }$
$\displaystyle\quad\subseteq\overline{\chi(M(w))}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]\overline{\chi(M(w^{\prime\prime}))}-\overline{\chi(M(w))}+\sum_{\bf
b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]\quad\text{ (by
\eqref{L=sum M})}$
$\displaystyle\quad=\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{*}[w^{\prime\prime}]+\sum_{\bf
b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]\quad\text{ (by Lemma
\ref{chiMbar=M''})}$
$\displaystyle\quad\subseteq\sum\mathbf{v}\mathbb{Z}[\mathbf{v}](\sum\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf
b}^{\prime}])+\sum_{\bf b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf
b}]=\sum_{\bf b}\mathbf{v}\mathbb{Z}[\mathbf{v}]M^{\prime}[{\bf b}]$

By the definition of triangular basis, $\chi(L(w))$ is a triangular basis
element for the seed $({\bf X}^{\prime},B^{\prime})$. Since the triangular
basis does not depend on the chosen acyclic seed (proved in [4]), $\chi(L(w))$
is also a triangular basis element for the seed $({\bf X},B)$. Its denominator
vector is the same as the denominator vector ${\bf d}$ of
$M^{\prime}[w_{1}-w^{\prime}_{1},w^{\prime}_{2}-w_{2}]=M^{*}[{}^{\phi}w]$.
Since the denominator vectors of $X_{2},X_{0},X_{1},X_{-1}$ are
$(0,-1),((0,1),(-1,0),(1,r)$ respectively, we denote ${\bf
b}=(b_{1},b_{2})=(w_{1}-w^{\prime}_{1},w^{\prime}_{2}-w_{2})$ and get ${\bf
d}=[-b_{2}]_{+}(0,-1)+[b_{2}]_{+}(0,1)+[b_{1}]_{+}(-1,0)+[-b_{1}]_{+}(1,r)=(-b_{1},b_{2}+r[-b_{1}]_{+})=(w^{\prime}_{1}-w_{1},w^{\prime}_{2}-w_{2}+r[w^{\prime}_{1}-w_{1}]_{+})$.
So
$\chi(L(w))=C[w^{\prime}_{1}-w_{1},w^{\prime}_{2}-w_{2}+r[w^{\prime}_{1}-w_{1}]_{+}]$.
∎

##  8\. The proof of main results

This section is devoted to the proof of the main theorem and its corollaries.

###  8.1. Some facts on the BBDG Decomposition

We have the following two lemmas.

######  Lemma 8.1.

Let $f:Y\to X$ be a proper morphism between complex algebraic varieties, $Y$
be nonsingular, let $0$ be a point in $X$. Let $d=\dim Y$, $Y_{0}=f^{-1}(0)$
and $d_{0}=\dim Y_{0}$. Write the BBDG decomposition in the form

(8.1) $f_{*}IC_{Y}=\bigoplus_{b}IC_{0}^{\oplus
s_{0,b}}[b]\oplus\bigoplus_{V,L,b}IC_{V}^{\oplus s_{V,L,b}}(L)[b]$

where $V\neq 0$ are subvarieties of $X$, each $L$ is a local system on an open
dense subset of $V$, $s_{0,b},s_{V,L,b}\in\mathbb{Z}_{\geq 0}$ are
multiplicities of the corresponding $IC$-sheaves. Then $\\{s_{0,b}\\}$ satisfy
the following conditions:

i) $s_{0,b}=s_{0,-b}$ for every $b\in\mathbb{Z}$.

ii) $s_{0,b}\geq s_{0,b+2}$ for every $b\in\mathbb{Z}_{\geq 0}$.

iii) $s_{0,b}=0$ if $|b|>2d_{0}-d$. In particular, if $2d_{0}<d$, then
$s_{0,b}=0$ for all $b$.

###### Proof.

See [7] for i) Poincare duality and ii) Relative Hard Lefschez. For iii), let
$b\geq 0$. From (8.1) we get

$R^{d+b}f_{*}\mathbb{Q}_{Y}=R^{b}f_{*}\mathbb{Q}_{Y}[d]=\mathbb{Q}_{0}^{s_{0,-b}}\oplus\textrm{other
terms}$

Since the proper base change theorem implies
$(R^{d+b}f_{*}\mathbb{Q}_{Y})_{0}=H^{d+b}(Y_{0})$, the above implies (we
denote the cohomology to be $\mathbb{Q}$-coefficient)

$H^{d+b}(Y_{0})=\mathbb{Q}_{0}^{s_{0,-b}}\oplus\textrm{other terms}$

If $d+b>2\dim Y_{0}=2d_{0}$, then $H^{d+b}(Y_{0})=0$, and thus its direct
summand $\mathbb{Q}_{0}^{s_{0,-b}}$ must vanish. Thus $s_{0,b}=s_{0,-b}=0$ if
$b>2d_{0}-d$. ∎

######  Lemma 8.2.

Let $f:Y\to X$ be a birational proper morphism between complex algebraic
varieties, $Y$ be nonsingular, let $0$ be a point in $X$. Let $d=\dim Y=\dim
X$, $Y_{0}=f^{-1}(0)$ and $d_{0}=\dim Y_{0}$. Then

$H^{k}(i_{0}^{*}IC_{X})=0,\textrm{ if }k>2d_{0}-d$

###### Proof.

Since $f$ is birational, we can write the BBDG decomposition in the form

(8.2) $f_{*}IC_{Y}=IC_{X}\oplus\bigoplus_{V,L,b}IC_{V}^{s_{V,L,b}}(L)[b]$

where $V\subsetneq X$ are proper subvarieties and $L$ are local system on an
open dense subset of $V$. Similar to the proof of Lemma 8.1, we have

$H^{d+k}(Y_{0})=H^{k}(i_{0}^{*}IC_{X})\oplus\textrm{other terms}$

If $k>2d_{0}-d$, then $d+k>2d_{0}$, thus the left side vanishes, thus the
direct summand $H^{k}(i_{0}^{*}IC_{X})$ of the right side also vanishes. ∎

###  8.2. $P_{-}(v,w)$

We see in (7.1) that

$\pi_{*}(IC_{\widetilde{\mathcal{F}}_{v,w}})=\sum_{v^{\prime}\in\mathrm{Dom}(w)}a_{v,v^{\prime};w}IC_{w}(v^{\prime})$

Apply $\sum_{k}t^{-k}\dim H^{k}(i^{!}_{0}(-))$ to both sides and then apply
(7.8) to the left side, we obtain

(8.3) $t^{d_{v,w}-\tilde{d}_{v,w}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}{w^{\prime}_{1}\brack
v_{1}}_{t}=\sum_{v^{\prime}\in\mathrm{Dom}(w)}a_{v,v^{\prime};w}\sum_{k}t^{-k}\dim
H^{k}(i^{!}_{0}IC_{w}(v^{\prime})).$

######  Remark 8.3.

Note that $H^{k}(i_{0}^{!}IC_{w}(v))=H^{-k}(i_{0}^{*}IC_{w}(v))$. Indeed, let
$x$ be a point in $X$, $i:x\to X$ be the natural injective morphism. Let
$\mathbb{D}_{X}$ be the Verdier duality functor (see [1, §2.8]). Let
$\mathcal{F}\in D_{c}^{b}(X)$ be self-dual:
$\mathbb{D}_{X}(\mathcal{F})=\mathcal{F}$. (For example, the intersection
cohomology complex $\mathcal{F}=IC_{V}\in D_{c}^{b}(X)$ supported on $V$ with
a trivial local system satisfies the property; for a proof of this property,
see [1, Lemma 3.3.13]).) Then by [1, (2.8.3)],
$i^{!}\mathcal{F}=\mathbb{D}_{X}(i^{*}\mathbb{D}_{X}(\mathcal{F}))=\mathbb{D}_{X}(i^{*}\mathcal{F})$.
From here we see $H^{k}(i^{!}F)=H^{-k}(i^{*}F)$, thus

$\sum_{k}t^{-k}\dim H^{k}(i^{!}\mathcal{F})=\sum_{k}t^{k}\dim
H^{k}(i^{*}\mathcal{F})$

Based on this remark, we define a Laurent polynomial

$P_{-}(v,w)=\sum_{k}t^{k}\dim H^{k}(i_{0}^{*}IC_{w}(v))\in\mathbb{Z}[t^{\pm}]$

and rewrite (8.3) as

(8.4)
$\sum_{v^{\prime}\in\mathrm{Dom}(w)}a_{v,v^{\prime};w}P_{-}(v^{\prime},w)=t^{d_{v,w}-\tilde{d}_{v,w}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}{w^{\prime}_{1}\brack v_{1}}_{t}$

In the special case when $v\in\mathrm{Dom}(w)$, using the fact $a_{v,v;w}=1$
we get

(8.5) $P_{-}(v,w)=t^{d_{v,w}-\tilde{d}_{v,w}}{w^{\prime}_{2}+rv_{1}\brack
v_{2}}_{t}{w^{\prime}_{1}\brack
v_{1}}_{t}-\sum_{v^{\prime}\in\mathrm{Dom}(w)\setminus\\{v\\}}a_{v,v^{\prime};w}P_{-}(v^{\prime},w),\textrm{
if $v\in\mathrm{Dom}(w)$}.$

Recall $v^{\rm swap}=(v_{2},v_{1}),w^{\rm
swap}=(w^{\prime}_{2},w_{2},w_{1}^{\prime},w_{1})$. Define

$\displaystyle
f(v,w)=2d(v,w)-\tilde{d}(v,w)=-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}(w^{\prime}_{1}-w_{1})+v_{2}(w^{\prime}_{2}-w_{2})$
$\displaystyle f^{\rm swap}(v,w)=f(v^{\rm swap},w^{\rm
swap})=-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}(w_{1}-w^{\prime}_{1})+v_{2}(w_{2}-w^{\prime}_{2})$
$\displaystyle
g(v,w)=-v_{1}^{2}-rv_{1}v_{2}-v_{2}^{2}+v_{1}(w_{1}^{\prime}-w_{1})+v_{2}(w_{2}-w_{2}^{\prime})$

######  Lemma 8.4.

Assume $v\in\mathrm{Dom}(w)$.

If $\tilde{d}(v,w)=0$, then $P_{-}(v,w)=1$. This happens if and only if
$v=(0,0)$.

If $\tilde{d}(v,w)>0$, then the min-degree term of $P_{-}(v,w)$ is
$t^{-\tilde{d}(v,w)}$ (with coefficient 1); the max-degree of $P_{-}(v,w)$ is
$\leq\min\big{(}-1,f(v,w),f^{swap}(v,w),g(v,w)\big{)}$.

###### Proof.

First note that $\bar{v}=v$ by Lemma 4.5 and the assumption that $(v,w)$ is
$l$-dominant. So $\dim\mathbf{E}_{v,w}=\tilde{d}(v,w)$ by Proposition 4.7.

If $\tilde{d}(v,w)=0$, then $\mathbf{E}_{v,w}=\\{0\\}$ is a point,
$P_{-}(v,w)=\sum_{k}t^{k}\dim H^{k}(i_{0}^{*}\mathbb{Q}_{0})=1$. Solving
$\tilde{d}(v,w)=0$, that is,
$v_{1}(w^{\prime}_{1}-v_{1})+v_{2}(w_{2}+rv_{1}-v_{2})=0$, and using the
condition that $(v,w)$ is $l$-dominant, we must have $v=(0,0)$.

For the rest of the proof we assume $\tilde{d}(v,w)>0$.

For the min-degree of $P_{-}(v,w)$, [1, Exercise 3.10.2] asserts that for an
irreducible variety $X$ of dimension $d$,
$\tau^{\leq-d}IC_{X}\cong\mathbb{Q}_{X}[d]$; thus $H^{k}(IC_{X})=\tau^{\geq
k}\tau^{\leq k}IC_{X}=0$ for $k<-d$ and

$H^{-d}(IC_{X})[d]\cong\tau^{\geq-d}\tau^{\leq-d}IC_{X}\cong\tau^{\geq-d}(\mathbb{Q}_{X}[d])\cong\mathbb{Q}_{X}[d]$

Thus the min-degree of $P_{-}(v,w)$ is $-\tilde{d}(v,w)$, and since
$\mathbf{E}_{v,w}$ is irreducible, we get

$\text{`` The coefficient of $t^{-\tilde{d}(v,w)}$ in $P(v,w)$ '' $=\dim
H^{-d(v,w)}(i_{0}^{*}(IC_{w}(v))=\dim(i_{0}^{*}\mathbb{Q}_{\mathbf{E}_{v,w}})=1$}.$

For the max-degree of $P_{-}(v,w)$, it is $<0$ by the support condition of
intersection cohomology complex [7, §3.1 (20)].

Applying Lemma 8.2 to the morphism
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$, we conclude the max-degree
is no larger than $2d(v,w)-\tilde{d}(v,w)=f(v,w)$.

Applying Lemma 8.2 to the morphism $\pi^{\rm swap}:\tilde{\mathcal{F}}^{\rm
swap}_{v,w}\to\mathbf{E}_{v,w}$ introduced in §6.1, we conclude the max-degree
is no larger than $2d(v^{\rm swap},w^{\rm swap})-\tilde{d}(v^{\rm swap},w^{\rm
swap})=f^{\rm swap}(v,w)$.

Applying Lemma 8.2 to the morphism
$\pi^{\prime}:\tilde{\mathcal{G}}_{v,w}\to\mathbf{E}_{{v},w}$ introduced in
§6.2, we conclude the max-degree is no larger than
$2\dim\mathcal{G}_{v,w}-\dim\tilde{\mathcal{G}}(v,w)=2(v_{1}(w^{\prime}_{1}-v_{1})+v_{2}(w_{2}-v_{2}))-(v_{1}(w_{1}+w^{\prime}_{1}-v_{1})+v_{2}(w_{2}+w^{\prime}_{2}+rv_{1}-v_{2}))=g(v,w)$.
∎

###  8.3. Proof of the main theorem (Theorem 1.2) and its corollaries

Recall that in (1.1) we use $e(p,q)$ to denote the coefficients in the
triangular basis element $C[a_{1},a_{2}]=\sum e(p,q)X^{rp-a_{1},rq-a_{2}}$. We
can express $e(p,q)$ in terms of $a_{v,0;w}$ as follows.

######  Lemma 8.5.

Fix $(a_{1},a_{2})\in\mathbb{Z}^{2}$, $(p,q)\in\mathbb{Z}_{\geq 0}^{2}$.
Assume that $w,v$ satisfy

$w^{\prime}_{1}-w_{1}=a_{1},\;w^{\prime}_{2}-w_{2}=a_{2}-r[a_{1}]_{+},\;v=([a_{1}]_{+}-q,\,p)$

Then

$e(p,q)=a_{v,0;w}(\mathbf{v}^{r}).$

Moreover, if $a_{1}\geq 0$, then

$D(p,q)=rf(v,w)$

###### Proof.

By Lemma 7.3 and (7.6),

$\displaystyle
C[a_{1},a_{2}]=C[w^{\prime}_{1}-w_{1},w^{\prime}_{2}-w_{2}+r[w^{\prime}_{1}-w_{1}]_{+}]$
$\displaystyle=\chi(L(w))=\sum_{v\in\mathbb{Z}_{\geq
0}^{2}}a_{v,0;w}(\mathbf{v}^{r})X^{(w_{1}-w^{\prime}_{1}+rv_{2},w_{2}-w^{\prime}_{2}-rv_{1})}$

Therefore,

$\sum_{p,q}e(p,q)X^{(rp-a_{1},rq-
a_{2})}=\sum_{v_{1},v_{2}}a_{v,0;w}(\mathbf{v}^{r})X^{(rv_{2}-a_{1},r([a_{1}]_{+}-v_{1})-a_{2})}$

Comparing the coefficients gives us the first statement.

To prove the second statement: for $a_{1}\geq 0$,
$D(p,q)=D(v_{2},a_{1}-v_{1})=r(-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+a_{1}v_{1}+(a_{2}-ra_{1})v_{2})=r(-v_{1}^{2}+rv_{1}v_{2}-v_{2}^{2}+v_{1}(w^{\prime}_{1}-w_{1})+v_{2}(w^{\prime}_{2}-w_{2}))=rf(v,w)$.
∎

######  Lemma 8.6.

If $v_{1}>[w^{\prime}_{1}-w_{1}]_{+}$, then $a_{v,0;w}=0$.

###### Proof.

By Lemma 8.5, $a_{v,0;w}(\mathbf{v}^{r})=e(p,q)$ for
$v=([a_{1}]_{+}-q,p)=([w^{\prime}_{1}-w_{1}]_{+}-q,p)$. For
$v_{1}>[w^{\prime}_{1}-w_{1}]_{+}$, we have $q<0$, thus $e(p,q)=0$. ∎

If $r\geq 3$, we define

$r_{\rm small}=(r-\sqrt{r^{2}-4})/2,\quad r_{\rm big}=(r+\sqrt{r^{2}-4})/2$

to be the two roots of $x^{2}-rx+1=0$. Note that $1/r<r_{\rm small}<r_{\rm
big}<r$.

######  Lemma 8.7.

Let $r\geq 3$, $w=(0,w^{\prime}_{1},w_{2},0)$ where $r_{\rm
small}<w_{2}/w^{\prime}_{1}<r_{\rm big}$. Consider the curve
$C_{f}=\\{v\in\mathbb{R}_{\geq 0}^{2}\ |\ v_{1}\leq
w^{\prime}_{1},f(v,w)=0,v\neq(w^{\prime}_{1},0)\\}$. Then $C_{f}$ is concave
upward and with two endpoints $(0,0)$ and
$(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$. The slope of tangent of $C_{f}$ is
increasing as $v_{1}$ increases from $0$ to $w^{\prime}_{1}$; the slope at
$(0,0)$ is $w^{\prime}_{1}/w_{2}\ (>r_{\rm small})$, and the slope
$(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$ is
$r-\frac{w^{\prime}_{1}}{rw^{\prime}_{1}-w_{2}}\ (<r_{\rm big})$.

###### Proof.

The equation $\\{v\in\mathbb{R}^{2}\ |\ f(v,w)=0\\}$ is a hyperbola. Let
$x^{\prime}=(-v_{1}+v_{2})/\sqrt{2}$, $y^{\prime}=(v_{1}+v_{2})/\sqrt{2}$,
$x=x^{\prime}+\frac{w_{2}+w^{\prime}_{1}}{(2+r)\sqrt{2}}$,
$y=y^{\prime}+\frac{w_{2}-w^{\prime}_{1}}{(2-r)\sqrt{2}}$. We can rewrite the
equation $f(v,w)=0$ as

$(1+\frac{r}{2})x^{2}+(1-\frac{r}{2})y^{2}=\frac{1}{4}\bigg{(}\frac{(w_{2}+w^{\prime}_{1})^{2}}{r+2}-\frac{(w_{2}-w^{\prime}_{1})^{2}}{r-2}\bigg{)}$

It is easy to see that the right side is positive since $r_{\rm
small}<w_{2}/w^{\prime}_{1}<r_{\rm big}$. Thus the hyperbola is of the shape
in Figure 3, whose northwest branch passes $(0,0)$ and
$(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$.

$v_{1}$$v_{2}$$w^{\prime}_{1}$$rw^{\prime}_{1}-w_{2}$$f(v,w)\geq 0$$C_{f}$
Figure 3. $f(v,w)\geq 0$.

The slope of tangent at point $(v_{1},v_{2})$ is
$(-2v_{1}+rv_{2}+w^{\prime}_{1})/(-rv_{1}+2v_{2}+w_{2})$, so the value is
$w^{\prime}_{1}/w_{2}$ when $(v_{1},v_{2})=(0,0)$, is
$r-\frac{w^{\prime}_{1}}{rw^{\prime}_{1}-w_{2}}$ when
$(v_{1},v_{2})=(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$. ∎

######  Lemma 8.8.

Assume $w=(0,w^{\prime}_{1},w_{2},0)$,
$(w^{\prime}_{1},w_{2})\in\Phi^{im}_{+}$, $f(v,w)\geq 0$,
$v^{0}\in\mathrm{Dom}(w)\setminus\\{{\bf 0}\\}$ and $v^{0}\leq v$. Then

$\deg(a_{v,v^{0};w}P_{-}(v^{0},w))<f(v,w).$

###### Proof.

If $r=1$ then $\Phi^{im}_{+}$ is empty. So $r\geq 2$. The case $r=2$ is
degenerate (and much simpler than the $r\geq 3$ case) so we treat it at the
very end.

(I) Assume $r\geq 3$.

First we show that the inequality is true if $v^{0}=v$. Because in this case,
$a_{v,v^{\prime};w}=1$, $\deg P_{-}(v^{0},w)\leq 0$, so
$\deg(a_{v,v^{0};w}P_{-}(v^{0},w))<f(v,w)$, and the equality holds only if
$\deg P_{-}(v^{0},w)=0$, which is impossible since we assume $v^{0}\neq{\bf
0}$.

For the rest of the proof of (I), we assume $v^{0}<v$. As before, denote

$w^{\perp}=w-C_{q}v^{0}=(w_{1}-v_{1}^{0}+rv_{2}^{0},w_{1}^{\prime}-v_{1}^{0},w_{2}-v_{2}^{0},w_{2}^{\prime}-v_{2}^{0}+rv_{1}^{0}),\quad
v^{\perp}=v-v^{0}=(v_{1}-v_{1}^{0},v_{2}-v_{2}^{0}).$

We can assume $a_{v,v^{0};w}(=a_{v^{\perp},{\bf 0};w^{\perp}})$ is nonzero,
otherwise there is nothing to prove. Then $\deg
a_{v,v^{0};w}=f(v^{\perp};w^{\perp})$ if $a_{v,v^{0};w}\neq 0$. So it suffices
to prove that, assuming $f(v^{\perp},w^{\perp})\geq 0$, the following
inequality holds:

$f(v^{\perp},w^{\perp})+\deg P_{-}(v^{0},w)<f(v,w).$

By Lemma 8.4, it suffices to show the following claim:

Claim. If $0\leq v_{1}\leq w^{\prime}_{1}$, $w_{2}^{\prime}-v_{2}+rv_{1}\geq
0$,
$v^{0}\in\mathrm{Dom}(w)\setminus\\{(0,0),(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})\\}$,
$v^{0}<v$, $f(v,w)\geq 0,f(v^{\perp},w^{\perp})\geq 0$, then

$\min\big{(}-1,f(v^{0},w),f^{swap}(v^{0},w),g(v^{0},w)\big{)}<f(v,w)-f(v^{\perp},w^{\perp}).$

We allow them to be real numbers. Denote
$Q(v^{0})=Q(v^{0}_{1},v^{0}_{2})=(v^{0}_{1})^{2}-rv^{0}_{1}v^{0}_{2}+(v^{0}_{2})^{2}$.
Then

$\displaystyle
D_{1}=f(v,w)-f(v^{\perp},w^{\perp})=Q(v^{0})+(w^{\prime}_{1}-2v_{1})v^{0}_{1}+(2rv_{1}-2v_{2}-w_{2})v^{0}_{2}$
$\displaystyle
D_{2}=f(v,w)-f(v^{\perp},w^{\perp})-f(v^{0},w)=2\big{(}Q(v^{0})-v_{1}v^{0}_{1}+(rv_{1}-v_{2})v^{0}_{2}\big{)}$
$\displaystyle D_{3}=f(v,w)-f(v^{\perp},w^{\perp})-f^{\rm
swap}(v^{0},w)=2\big{(}Q(v^{0})+(w^{\prime}_{1}-v_{1})v^{0}_{1}+(rv_{1}-v_{2}-w_{2})v^{0}_{2}\big{)}$
$\displaystyle
D_{4}=f(v,w)-f(v^{\perp},w^{\perp})-g(v^{0},w)=2\big{(}(v^{0}_{1})^{2}+(v^{0}_{2})^{2}-v_{1}v^{0}_{1}+(rv_{1}-v_{2}-w_{2})v^{0}_{2}\big{)}$

To show the claim, it suffices to show that at least one of the following is
true: $D_{1}\geq 0$, $D_{2}>0$, $D_{3}>0$, or $D_{4}>0$.

Now fix $(v,w)$ and let $v^{0}=(at,bt)$ along a line with a fixed direction
$(a,b)\in\mathbb{R}_{\geq 0}^{2}\setminus\\{(0,0)\\}$ and parameter $t$.

The condition $v^{0}\in\mathrm{Dom}(w)\setminus\\{0\\}$ implies $rb-a\geq 0$,
$at\leq w^{\prime}_{1}$, $bt\leq w_{2}$, $ra-b\geq 0$. Thus $a,b>0$. (If
$a=0$, then $ra-b\geq 0$ forces $b=0$, contradicting to the assumption
$(a,b)\neq(0,0)$; if $b=0$, then $rb-a\geq 0$ forces $a=0$, contradicting to
the same assumption.) So we assume for the rest of the proof, that

$a=1,b\in\mathbb{Q}\textrm{ and }1/r\leq b\leq r.$

Denote

$\displaystyle Q_{0}=Q(a,b)=1-rb+b^{2},$ $\displaystyle
A_{2}=-v_{1}+(rv_{1}-v_{2})b,$ $\displaystyle
A_{3}=(w^{\prime}_{1}-v_{1})+(rv_{1}-v_{2}-w_{2})b.$ $\displaystyle
A_{1}=(w^{\prime}_{1}-2v_{1})+(2rv_{1}-2v_{2}-w_{2})b=A_{2}+A_{3}.$

Then

$D_{1}=Q_{0}t^{2}+A_{1}t,\quad D_{2}=2(Q_{0}t^{2}+A_{2}t),\quad
D_{3}=2(Q_{0}t^{2}+A_{3}t)$

Note that $A_{3}=A_{2}+w^{\prime}_{1}-w_{2}b$. Also note that the condition
$v^{0}\leq v$ implies $t\leq v_{1}$ and $t\leq v_{2}/b$. Then $t$ satisfies

$0<t\leq\min(v_{1},v_{2}/b,w^{\prime}_{1},w_{2}/b)$

Note that $Q_{0}\neq 0$, otherwise
$b=\frac{r\pm\sqrt{r^{2}-4}}{2}\notin\mathbb{Q}$ since we assume $r\geq 3$ in
part (I). Thus, we only need to consider two cases $Q_{0}>0$ (which consists
of two subcases 1a and 1b) and $Q_{0}<0$ (which consists of four subcases
2a–2d) below.

Case 1. If $Q_{0}>0$, we claim that either $A_{2}\geq 0$, or $A_{3}\geq 0$. It
then immediately follows that either $D_{2}>0$ for all $t>0$, or $D_{3}>0$ for
all $t>0$.

We consider two subcases: Cases 1a and 1b. See Figure 4.

$v_{1}$$v_{2}$$f(v,w)\geq 0$$A_{3}\leq
0$$w^{\prime}_{1}$$rw^{\prime}_{1}-w_{2}$$v_{1}$$v_{2}$$f(v,w)\geq
0$$A_{2}\leq 0$$w^{\prime}_{1}$$rw^{\prime}_{1}-w_{2}$ Figure 4. Left: Case
1a. Right: Case 1b.

Case 1a. $1/r\leq b<r_{\rm small}$. We claim that $A_{3}\geq 0$. Indeed, by
looking at Figure 4 (Left), it suffices to show that the slope of $A_{3}=0$ is
less than the slope of the diagonal, that is,

$r-1/b<r-w_{2}/w^{\prime}_{1}$

and the above inequality follows from $1/b>1/r_{\rm small}=r_{\rm
big}>w_{2}/w^{\prime}_{1}$. So there is no $(v_{1},v_{2})$ such that
$f(v,w)\geq 0$ and $A_{3}<0$ simultaneously.

Case 1b. $r_{\rm big}<b<r$. We claim that $A_{2}\geq 0$. Indeed, by looking at
Figure 4 (Right), it suffices to show that the slope of $A_{2}=0$ is greater
than the slope of the diagonal, that is,

$r-1/b>(rw^{\prime}_{1}-w_{2})/w^{\prime}_{1}$

and the above inequality follows from the inequality $1/b<1/r_{\rm big}=r_{\rm
small}<w_{2}/w^{\prime}_{1}$. So there is no $(v_{1},v_{2})$ such that
$f(v,w)\geq 0$ and $A_{2}<0$ simultaneously.

Case 2. If $Q_{0}<0$. Equivalently, if $r_{\rm small}<b<r_{\rm big}$. Then for
$i=1,2,3$, we have $D_{i}\geq 0$ for $0\leq t\leq-A_{i}/Q_{0}$. We shall
consider 4 subcases, Case 2a–2d.

For convenience, we introduce the following notation (and similar notation for
$\geq$ replaced by $\leq,>,<$):

For $i=1,2,3$, $H^{\geq 0}_{i}=\\{v\ |\ A_{i}\geq 0\\}$, $H_{i1}^{\geq
0}=\\{v\ |\ A_{i}/(-Q_{0})-v_{1}\geq 0\\}$, $H_{i2}^{\geq 0}=\\{v\ |\
A_{i}/(-Q_{0})-v_{2}/b\geq 0\\}$, $H_{i3}^{\geq 0}=\\{v\ |\
A_{i}/(-Q_{0})-(w^{\prime}_{1}-v_{1})/(rb-1)\geq 0\\}$. Define their boundary
lines by $\ell_{i},\ell_{i1},\ell_{i2},\ell_{i3}$, respectively.

Denote $P_{i,jb}=\ell_{i}\cap\ell_{jb}$. We collect some easy-to-be-verified
observations to be used later. For
$v=P_{2,33},P_{3,12},P_{3,22},P_{2,13},P_{12,13}$, we have $f(v,w)=0$, and the
coordinates of these points are explicitly computed below: (we use the
notation $P=(P^{x},P^{y})$ to denote the two coordinates for $P$)

$\displaystyle
P_{2,13}=(\frac{b(b(w^{\prime}_{1}-rw_{2})+w_{2})}{b^{2}-br+1},\frac{(br-1)(b(w^{\prime}_{1}-rw_{2})+w_{2})}{b^{2}-br+1})$
$\displaystyle
P_{2,33}=(P_{2,33}^{x},P_{2,33}^{y})=(\frac{b^{2}w^{\prime}_{1}+b(1-rb)w_{2}}{b^{2}-rb+1},\frac{(rb-1)(bw^{\prime}_{1}+(1-rb)w_{2})}{b^{2}-rb+1})$
$\displaystyle
P_{3,12}=(\frac{w^{\prime}_{1}-bw_{2}}{b^{2}-br+1},\frac{r(w^{\prime}_{1}-bw_{2})}{b^{2}-br+1})$
$\displaystyle
P_{3,22}=\big{(}(bw_{2}-w^{\prime}_{1})/(-Q_{0}),b(bw_{2}-w^{\prime}_{1})/(-Q_{0})\big{)}$
$\displaystyle
P_{12,13}=(\frac{b(bw^{\prime}_{1}+(br-1)w_{2})}{2b^{2}r^{2}+b^{2}-3br+1},\frac{b((2br-1)w^{\prime}_{1}-bw_{2})}{2b^{2}r^{2}+b^{2}-3br+1})$

$\bullet$ (slope of $\ell_{2}$) = (slope of $\ell_{3}=0$) $=r-\frac{1}{b}>b$.

$\bullet$ the line $\ell_{2}$ passes $O$.

$\bullet$ the line $\ell_{3}$ passes through the point
$(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$, and intersects with the north
boundary of $R_{f\geq 0}$ at the point
$\big{(}\frac{w^{\prime}_{1}-bw_{2}}{b^{2}-rb+1},\frac{b(w^{\prime}_{1}-bw_{2})}{b^{2}-rb+1}\big{)}$.

$\bullet$ (slope of $\ell_{13}$)
$=(2r^{2}b^{2}-3rb-b^{2}+1)/(2rb^{2}-2b)=r+(-b^{2}-rb+1)/(2rb^{2}-2b)>$ (slope
of $\ell_{3}$). So $\ell_{13}$ has positive slope.

Case 2a. If $A_{2}\geq A_{1},A_{3}$. We claim that $D_{2}>0$. See Figure 5.

Indeed, $A_{2}\geq A_{3}$ implies $w_{2}b\geq w^{\prime}_{1}$; $A_{2}\geq
A_{1}=A_{2}+A_{3}$ implies $A_{3}\leq 0$,

To prove $D_{2}\geq 0$, we need $Q_{0}t+A_{2}\geq 0$, that is, $t\leq
A_{2}/(-Q_{0})$. Since $t\leq v_{2}/b$, it suffices to show $v_{2}/b\leq
A_{2}/(-Q_{0})$, or equivalently $v\in H^{\geq 0}_{22}$, or equivalently

(8.6) $\frac{v_{2}}{b}\leq\frac{-v_{1}+(rv_{1}-v_{2})b}{-(1-rb+b^{2})}$

Note that $b>r_{\rm small}>1/r$, so after simplifying, (8.6) becomes

$v_{2}\leq v_{1}b.$

This is self-clear from Figure 5. The inequality $v_{2}\leq v_{1}b$ is
equivalent to saying that the intersection of the curved triangle $OCB$ with
the half plane $H^{\leq 0}_{3}$, which is the curved triangle $OP_{3,22}F$, is
below the line $\ell_{22}$.

If $P_{3,22}^{x}<w^{\prime}_{1}$, then the intersection of the region
$\\{A_{3}<0\\}$ with the region

$R_{f\geq 0}=\\{v|f(v,w)\geq 0,0\leq v_{1}\leq w^{\prime}_{1},v_{2}\geq 0\\}$

is below the line $\ell_{22}$ as shown in the figure.

If $P_{3,22}^{x}\geq w^{\prime}_{1}$, then the whole region $R_{f\geq 0}$ is
under the line $OE$.

In both cases, we get the expected inequality $v_{2}\leq v_{1}b$. Thus
$D_{2}\geq 0$.

Now we need to show that $D_{2}>0$. If not, then $D_{2}=0$. We inspect the
above computation and get $P_{3,22}=(v_{1},v_{2})=(v_{1},bv_{1})$,
$t=v_{1}^{0}=v_{1}=A_{2}/(-Q_{0})$, then $v^{0}_{2}=bt=bv_{1}^{0}=v_{2}$. Thus
$v^{0}=v$, contradicting to the assumption $v^{0}<v$.

$v_{1}$$v_{2}$$R_{f\geq
0}$$B=(w^{\prime}_{1},0)$$\ell_{22}$$rw^{\prime}_{1}-w_{2}$$H^{\leq
0}_{3}$$O$$\ell_{3}$$F$$P_{3,22}$$C$ Figure 5. Case 2a. If
$P_{3,22}^{x}<w^{\prime}_{1}$.

For the rest, since $a_{v^{\perp},{\bf 0};w^{\perp}}\neq 0$, by Lemma 8.6 we
have

(8.7)
$v_{1}-v^{0}_{1}=v^{\perp}_{1}\leq[(w^{\perp})^{\prime}_{1}-w^{\perp}_{1}]_{+}=[w^{\prime}_{1}-rv^{0}_{2}]_{+}$

For Case 2b and 2c below, we assume $w^{\prime}_{1}\geq rv^{0}_{2}$; for Case
2d, we assume $w^{\prime}_{1}<rv^{0}_{2}$.

If $w^{\prime}_{1}\geq rv^{0}_{2}$, then by (8.7),
$v^{0}_{1}-rv^{0}_{2}+w^{\prime}_{1}-v_{1}\geq 0$, $(rb-1)t\leq
w^{\prime}_{1}-v_{1}$, $t\leq(w^{\prime}_{1}-v_{1})/(rb-1)$. Therefore $t$
satisfies

(8.8)
$0<t\leq\min(v_{1},v_{2}/b,w^{\prime}_{1},w_{2}/b,(w^{\prime}_{1}-v_{1})/(rb-1))$

Case 2b. If $w^{\prime}_{1}\geq rv^{0}_{2}$, $A_{3}>A_{1}$, $A_{3}\geq A_{2}$.
We claim that $D_{3}>0$. See Figure 6.

$A_{3}>A_{1}$ implies $A_{2}<0$; $A_{3}\geq A_{2}$ implies $w_{2}b\leq
w^{\prime}_{1}$.

To prove $D_{3}\geq 0$, we need $Q_{0}t+A_{3}\geq 0$, that is, $t\leq
A_{3}/(-Q_{0})$. Since $t\leq(w^{\prime}_{1}-v_{1})/(rb-1)$, it suffices to
show $(w^{\prime}_{1}-v_{1})/(rb-1)\leq A_{3}/(-Q_{0})$, or equivalently $v\in
H^{\geq 0}_{33}$, or equivalently

(8.9)
$\frac{w^{\prime}_{1}-v_{1}}{rb-1}\leq\frac{(w^{\prime}_{1}-v_{1})+(rv_{1}-v_{2}-w_{2})b}{-(1-rb+b^{2})}$

After simplifying, (8.9) becomes

$(r+b-r^{2}b)(w^{\prime}_{1}-v_{1})+(rb-1)(rw^{\prime}_{1}-w_{2}-v_{2})\geq 0$

Note that $w^{\prime}_{1}-v_{1},rb-1,rw^{\prime}_{1}-w_{2}-v_{2}$ are all
nonnegative. So if $r+b-r^{2}b\geq 0$, then (8.9) holds. If $r+b-r^{2}b<0$,
then the above defines a half plane $H^{\geq 0}_{33}$ below the line

$\ell_{33}=\\{(v_{1},v_{2})\ |\
(r+b-r^{2}b)(w^{\prime}_{1}-v_{1})+(rb-1)(rw^{\prime}_{1}-w_{2}-v_{2})=0\\}$

which passes through $C=(w^{\prime}_{1},rw^{\prime}_{1}-w_{2})$ with slope
$(r^{2}b-r-b)/(rb-1)>0$.

Note that $A_{2}<0$ defines the upper half plane $H^{<0}_{2}$ with boundary
line $\ell_{2}$.

$\textrm{( slope of $\ell_{33}$ )}=r-\frac{b}{rb-1}>r-\frac{1}{b}=\textrm{(
slope of $\ell_{2}$ )}$

If $P_{2,33}^{x}\leq 0$, then $H^{<0}_{33}\cap R_{f\geq 0}=\emptyset$.

If $0\leq P_{2,33}^{x}\leq w^{\prime}_{1}$: let $F$ be the intersection of
$\ell_{2}$ with $BC$. Then

$H^{<0}_{2}\cap H^{<0}_{33}\cap R_{f\geq 0}=\emptyset$

If $P_{2,33}^{x}>w^{\prime}_{1}$, then

$H^{<0}_{2}\cap R_{f\geq 0}=\emptyset\subset H^{\geq 0}_{33}.$

In all cases, we get the expected inequality (8.9). Thus $D_{3}\geq 0$.

$v_{1}$$v_{2}$$R_{f\geq
0}$$B=(w^{\prime}_{1},0)$$\ell_{2}$$rw^{\prime}_{1}-w_{2}$$H^{<0}_{33}$$O$$\ell_{33}$$P_{2,33}$$C$$v_{1}$$v_{2}$$R_{f\geq
0}$$B=(w^{\prime}_{1},0)$$\ell_{2}$$rw^{\prime}_{1}-w_{2}$$H^{<0}_{2}\cap
H^{<0}_{33}$$O$$\ell_{33}$$F$$P_{2,33}$$C$$v_{1}$$v_{2}$$R_{f\geq
0}$$B=(w^{\prime}_{1},0)$$\ell_{2}$$rw^{\prime}_{1}-w_{2}$$H^{<0}_{2}$$O$$\ell_{33}$$P_{2,33}$$C$
Figure 6. Case 2b. Left: $P_{2,33}^{x}1\leq 0$. Middle: $0\leq
P_{2,33}^{x}\leq w^{\prime}_{1}$. Right: $P_{2,33}^{x}>w^{\prime}_{1}$.

Now we need to show that $D_{3}>0$. If not, then $D_{3}=0$. We inspect the
above computation and get that $v=C$ (if $P_{2,33}^{x}\leq w^{\prime}_{1}$) or
$v=P_{2,33}$ (if $0<P^{x}_{2,33}<w^{\prime}_{1}$).

If $v=C$, then by (8.8), $v^{0}_{1}=t\leq(w^{\prime}_{1}-v_{1})/(rb-1)=0$,
thus $v^{0}_{1}=0$, $v^{0}_{2}=0$, contradicting to the assumption
$v^{0}\neq{\bf 0}$.

If $v=P_{2,33}$, then $A_{2}=0$, $A_{3}=A_{1}\geq 0$. This is not Case 2b but
rather Case 2c, which will be discussed in the next case.

Case 2c. If $w^{\prime}_{1}\geq rv^{0}_{2}$ and $A_{1}\geq A_{2},A_{3}$. We
claim that $D_{1}\geq 0$.

In this case, $A_{2}\geq 0$, $A_{3}\geq 0$.

To prove $D_{1}\geq 0$, we need $Q_{0}t+A_{1}\geq 0$, that is, $t\leq
A_{1}/(-Q_{0})$.

Since $t\leq\min(v_{2}/b,(w^{\prime}_{1}-v_{1})/(rb-1))$, it suffices to show
“$v_{2}/b\leq A_{1}/(-Q_{0})$ or $(w^{\prime}_{1}-v_{1})/(rb-1)\leq
A_{1}/(-Q_{0})$”, or equivalently “$v\in H^{\geq 0}_{12}\cup H^{\geq
0}_{13}$”, or equivalently,

$\frac{v_{2}}{b}\leq\frac{(w^{\prime}_{1}-2v_{1})+(2rv_{1}-2v_{2}-w_{2})b}{-(1-rb+b^{2})}\textrm{
or
}\frac{w^{\prime}_{1}-v_{1}}{rb-1}\leq\frac{(w^{\prime}_{1}-2v_{1})+(2rv_{1}-2v_{2}-w_{2})b}{-(1-rb+b^{2})}.$

Note that $\ell_{13}$ intersects the curve $f=0$ at $P_{2,13}$ and
$P_{12,13}$, both of which are in $\mathbb{R}_{>0}^{2}$.

If $w_{1}^{\prime}\leq w_{2}b$, then $A_{3}\leq A_{2}$. The point $C$
satisfies $A_{13}\geq 0$ (because
$A_{13}(C)=(w_{2}b-w^{\prime}_{1})/(-b^{2}+rb-1)\geq 0$). See Figure 7.

(i) If $P_{3,12}$ is to the southwest of $P_{12,13}$, then

$H^{<0}_{12}\cap H^{<0}_{13}\cap H^{\geq 0}_{3}=\textrm{(curved
triangle)}P_{3,12}P_{12,13}P_{3,13},$

which has no intersection with $R_{f\geq 0}$ except $P_{3,12},P_{12,13}$.

(ii) If $P_{3,12}$ is to the northeast of $P_{12,13}$, then similarly,
$H^{<0}_{12}\cap H^{<0}_{13}\cap H^{\geq 0}_{3}=\emptyset$.

$v_{1}$$v_{2}$$H^{<0}_{12}\cap H^{<0}_{13}\cap
H^{>0}_{3}$$\ell_{12}$$\ell_{3}$$\ell_{13}$$B=(w^{\prime}_{1},0)$$O$$P_{3,12}$$P_{12,13}$$P_{3,13}$$C$$v_{1}$$v_{2}$$H^{<0}_{12}\cap
H^{<0}_{13}$$H^{>0}_{3}$$\ell_{12}$$\ell_{3}$$\ell_{13}$$B=(w^{\prime}_{1},0)$$O$$P_{12,13}$$P_{3,12}$$C$
Figure 7. Case 2c, $w_{1}^{\prime}\leq w_{2}b$. Left: $P_{3,12}^{x}\leq
P_{12,13}^{x}$. Right: $P_{3,12}^{x}>P_{12,13}^{x}$.

If $w_{1}^{\prime}>w_{2}b$, then $A_{3}>A_{2}$. The point $C$ satisfies
$A_{13}<0$, the origin $O$ satisfies $A_{12}>0$. See Figure 8.

(i) If $P_{12,13}$ is to the southwest of $P_{2,13}$, then $H^{<0}_{12}\cap
H^{<0}_{13}\cap H^{>0}_{2}\cap R_{f\geq 0}=\emptyset$.

(ii) If $P_{12,13}$ is to the northeast of $P_{2,13}$, then we even have
$H^{<0}_{12}\cap H^{<0}_{13}\cap H^{>0}_{2}=\emptyset$.

Thus $D_{1}\geq 0$.

$v_{1}$$v_{2}$$H^{<0}_{12}\cap H^{<0}_{13}\cap
H^{>0}_{2}$$\ell_{12}$$\ell_{2}$$\ell_{13}$$B=(w^{\prime}_{1},0)$$O$$P_{12,13}$$P_{2,13}$$C$$v_{1}$$v_{2}$$H^{<0}_{12}\cap
H^{<0}_{13}$$H^{>0}_{2}$$\ell_{12}$$\ell_{2}$$\ell_{13}$$B=(w^{\prime}_{1},0)$$O$$P_{2,13}$$P_{12,13}$$C$
Figure 8. Case 2c, $w_{1}^{\prime}>w_{2}b$. Left: $P_{12,13}^{x}\leq
P_{2,13}^{x}$. Right: $P_{12,13}^{x}>P_{2,13}^{x}$.

Case 2d. If $w^{\prime}_{1}<rv^{0}_{2}$. We assert that $D_{1}\geq 0$ or
$D_{4}>0$. Indeed, (8.7) forces $v_{1}^{0}=v_{1}$, so the assertion follows
from the following Claim.

Claim. If $r_{\rm small}\leq w_{2}/w^{\prime}_{1}\leq r_{\rm big}$, $0\leq
v^{0}_{2}\leq v_{2}$, $(v^{0},w)$ is $l$-dominant (that is, $rv^{0}_{2}\geq