When $M$ is finite-dimensional over $k$, the rank-nullity theorem implies

$\dim_{k}H^{0}(C_{M}^{\bullet})=2\dim_{k}\overline{W}_{M}.$ (6.23)

Taking the preimage of (6.22) with respect to the quotient map $\ker
A_{M}\twoheadrightarrow\ker A_{M}/\operatorname{im}A^{\prime}_{M}$, we define

$W^{0}(M):=\operatorname{im}A^{\prime}_{M}\subseteq
W^{1}(M):=T_{M}^{-1}(\operatorname{im}A^{\prime}_{M})\subseteq W^{2}(M):=\ker
A_{M}.$ (6.24)

Then (6.23) implies $\dim_{k}W^{1}(M)=(\dim_{k}W^{0}(M)+\dim_{k}W^{2}(M))/2$.
This middle-dimensionality is the only reason why we need Lemma 6.6.

We summarize the relations among relevant dimensions associated to
$C_{M}^{\bullet}$ and $T$ for a finite-dimensional $R$-module $M$. Recall
$a(M):=\dim_{k}\ker A_{M}$ and $b(M):=\dim_{k}\operatorname{im}A_{M}$, and we
have $a(M)+b(M)=2\dim_{k}M$ by the rank-nullity theorem.

######  Lemma 6.7.

For any $R$-module $M$ with $\dim_{k}M<\infty$, we have

  1. (a)

$\dim_{k}\ker A^{\prime}_{M}=\dim_{k}\ker A_{M}=\dim_{k}W^{0}(M)=a(M)$;

  2. (b)

$\dim_{k}\operatorname{im}A^{\prime}_{M}=\dim_{k}\operatorname{im}A_{M}=\dim_{k}W^{2}(M)=b(M)$.

  3. (c)

$\dim_{k}H^{0}(C_{M}^{\bullet})=\dim_{k}H^{1}(C_{M}^{\bullet})=a(M)-b(M)$.

  4. (d)

$\dim_{k}W^{1}(M)=\dfrac{a(M)+b(M)}{2}$.

###### Proof.

Identities (a)(b)(c) are immediate from tensoring Lemma 6.5 with $M$. The
identity (d) is restating the middle-dimensionality observation above. Note
that $\dim_{k}H^{0}(C_{M}^{\bullet})$ is _a priori_ $\dim_{k}\ker
A_{M}-\dim_{k}\operatorname{im}A^{\prime}_{M}$, so it is necessary to observe
(b) to express $\dim_{k}H^{0}(C_{M}^{\bullet})$ in terms of the rank and the
nullity of $A_{M}$. ∎

###  6.3. A stratification on $V_{d}$

In light of Lemma 6.7, we have a well-defined stratification on $V_{d}$ by

$V_{d}=\bigsqcup_{\begin{subarray}{c}a+b=2d\\\ a\geq b\geq
0\end{subarray}}V_{(a,b)},\quad V_{(a,b)}:=\\{M_{d}\in
V_{d}:a(M_{d})=a,b(M_{d})=b\\}.$ (6.25)

Recall that a point $(X_{d+1},Y_{d+1})$ of $V_{d+1}$ is determined by a point
$(X_{d},Y_{d})\in V_{d}$ and a vector $u:=\begin{bmatrix}z\\\
w\end{bmatrix}\in\ker A_{M_{d}}$, according to the rule (6.7). We then
determine the stratum that $(X_{d+1},Y_{d+1})$ belongs to in terms of $M_{d}$
and $u$.

######  Lemma 6.8.

In the notation above, suppose $M_{d}\in V_{(a,b)}$, then

$(X_{d+1},Y_{d+1})\in\begin{cases}V_{(a+2,b)},&\text{if }u\in W^{0}(M_{d});\\\
V_{(a+1,b+1)},&\text{if }u\in W^{1}(M_{d})\setminus W^{0}(M_{d});\\\
V_{(a,b+2)},&\text{if }u\in W^{2}(M_{d})\setminus W^{1}(M_{d}).\end{cases}$
(6.26)

Noting that the dimensions of $W^{0}(M_{d}),W^{1}(M_{d}),W^{2}(M_{d})$ are
$b,(a+b)/2,a$ respectively, Lemma 6.8 immediately implies the following
inductive formula for $V_{(a,b)}$, which concludes the proof of Theorem 6.2.

######  Corollary 6.9.

The motive of $V_{(a,b)}$ can be computed inductively by

$\begin{gathered}=1,[V_{(a,b)}]=0\text{ unless }a\geq b\geq 0,a\equiv
b\pmod{2};\\\
[V_{(a,b)}]=\mathbb{L}^{b}[V_{(a-2,b)}]+(\mathbb{L}^{\frac{a+b-2}{2}}-\mathbb{L}^{b-1})[V_{(a-1,b-1)}]+(\mathbb{L}^{a}-\mathbb{L}^{\frac{a+b-2}{2}})[V_{(a,b-2)}].\end{gathered}$
(6.27)

The motive of $V_{d}$ for $d\geq 0$ is given by

$[V_{d}]=\sum_{b=0}^{d}[V_{2d-b,b}].$ (6.28)

###### Proof of Lemma 6.8.

It suffices to determine $b(M_{d+1})$, and moreover, we shall use the
definition $b(M_{d+1})=\dim_{k}\operatorname{im}A^{\prime}_{M_{d+1}}$. The
matrix for $A^{\prime}_{M_{d+1}}$ on $M_{d+1}\oplus M_{d+1}=M_{d}\oplus
k\oplus M_{d}\oplus k$ is

$A^{\prime}_{M_{d+1}}=\begin{bmatrix}X_{d+1}&Y_{d+1}^{2}\\\
Y_{d+1}&X_{d+1}\end{bmatrix}=\begin{bmatrix}X_{d}&z&Y_{d}^{2}&Y_{d}w\\\
0&0&0&0\\\ Y_{d}&w&X_{d}&z\\\ 0&0&0&0\end{bmatrix}.$ (6.29)

Therefore, the image of $A^{\prime}_{M_{d+1}}$ actually lies in $M_{d}\oplus
M_{d}$, and is equal to

$\displaystyle\operatorname{im}A^{\prime}_{M_{d+1}}$
$\displaystyle=\operatorname{im}\begin{bmatrix}X_{d}&Y_{d}^{2}\\\
Y_{d}&X_{d}\end{bmatrix}+\operatorname{span}_{k}\left\\{\begin{bmatrix}z\\\
w\end{bmatrix},\begin{bmatrix}Y_{d}w\\\ z\end{bmatrix}\right\\}$ (6.30)
$\displaystyle=\operatorname{im}A^{\prime}_{M_{d}}+\operatorname{span}_{k}\\{u,T_{M_{d}}u\\}\in
M_{d}\subseteq M_{d}.$

We now work mod $\operatorname{im}A^{\prime}_{M_{d}}$. Since $u\in\ker
A_{M_{d}}$, we are working in $\ker
A_{M_{d}}/\operatorname{im}A^{\prime}_{M_{d}}=H^{0}(C_{M_{d}}^{\bullet})$.
Hence, the increase in the $b$-parameter is given by

$b(M_{d+1})-b(M_{d})=\dim_{k}\operatorname{span}_{k}\\{\overline{u},T^{0}_{M_{d}}\overline{u}\\}\subseteq
H^{0}(C_{M_{d}}^{\bullet}).$ (6.31)

We recall that $T^{0}_{M_{d}}$ is a square-zero linear map on
$H^{0}(C_{M_{d}}^{\bullet})$. By a standard fact about nilpotent linear maps
(Lemma 6.10), the dimension of
$\dim_{k}\operatorname{span}_{k}\\{\overline{u},T^{0}_{M_{d}}\overline{u}\\}$
is the number of nonzero elements in
$\\{\overline{u},T^{0}_{M_{d}}\overline{u}\\}$. This completes the proof. ∎

######  Lemma 6.10.

Let $V$ be a finite-dimensional vector space over a field $k$, and $T:V\to V$
be a nilpotent linear map. For any vector $v\in V$, consider the sequence
$v,Tv,T^{2}v,\dots$, which is eventually zero. Then the nonzero elements in
the sequence are linearly independent.

###### Proof.

Assume $T^{n-1}v\neq 0$ and $T^{n}v=0$. Let
$W=\operatorname{span}_{k}\\{v,Tv,\dots,T^{n-1}v\\}$, so $T$ is a nilpotent
endomorphism of $W$. We note that $T^{n-1}$ is not the zero map on $W$ because
$T^{n-1}v\neq 0$. Hence $\dim W>n-1$, so that $v,Tv,\dots,T^{n-1}v$ must be
linearly independent. ∎

######  Example 6.11.

Using Corollary 6.9, we compute the motive of $V_{d}$ for $d\leq 8$ in Table
1. The cases with $d\leq 3$ can be directly verified from the definition
(6.6); see §7.1.

$d$ | $[V_{d}]$
---|---
0 | 1
1 | 1
2 | $\mathbb{L}^{2}$
3 | $3\mathbb{L}^{4}-2\mathbb{L}^{3}$
4 | $2\mathbb{L}^{8}+3\mathbb{L}^{7}-5\mathbb{L}^{6}+\mathbb{L}^{5}$
5 | $10\mathbb{L}^{12}-5\mathbb{L}^{11}-9\mathbb{L}^{10}+5\mathbb{L}^{9}$
6 | $5\mathbb{L}^{18}+21\mathbb{L}^{17}-30\mathbb{L}^{16}-9\mathbb{L}^{15}+15\mathbb{L}^{14}-\mathbb{L}^{12}$
7 | $35\mathbb{L}^{24}+7\mathbb{L}^{23}-84\mathbb{L}^{22}+15\mathbb{L}^{21}+35\mathbb{L}^{20}-7\mathbb{L}^{18}$
8 | $14\mathbb{L}^{32}+112\mathbb{L}^{31}-112\mathbb{L}^{30}-162\mathbb{L}^{29}+113\mathbb{L}^{28}+70\mathbb{L}^{27}-7\mathbb{L}^{26}-28\mathbb{L}^{25}+\mathbb{L}^{22}$
Table 1. The motive of $[V_{d}]$ for $d\leq 8$

We observe from the table that as a polynomial in $\mathbb{L}$, the total
coefficient of $[V_{d}]$ is $1$. Indeed, if we substitute $\mathbb{L}\mapsto
1$ in Corollary 6.9, we have
$\left.{[V_{2d,0}]}\right\rvert_{\mathbb{L}\mapsto 1}=1$ and
$\left.{[V_{a,b}]}\right\rvert_{\mathbb{L}\mapsto 1}=0$ if $(a,b)\neq(2d,0)$,
so $\left.{[V_{d}]}\right\rvert_{\mathbb{L}\mapsto 1}=1$. Geometrically
speaking, this means that when $k={\mathbb{C}}$, the Euler characteristic of
$V_{d}$ in the analytic topology is $1$, which is expected because the obvious
${\mathbb{C}}^{\times}$-action $t\cdot(X,Y)=(tX,tY)$ on $V_{d}$ has only one
fixed point $(X,Y)=(0,0)$.

##  7\. Explicit computations in $d\leq 3$

We assume again that $R=k[[T^{2},T^{3}]]$ and $H_{d}(t):=H_{d,R}(t)$. We
compute $H_{d}(t)$ explicitly for $d\leq 3$ and prove the second part of
Theorem 1.8 using Theorem 5.14. As before, we present our proof in terms of
point counting over $k={{\mathbb{F}}_{q}}$. The same proof implies the motivic
formulas in Theorem 1.8 since the motivic version of Theorem 5.14 holds
according to Appendix B.

###  7.1. Pure-$K$ strata

When $d\leq 3$ and $\alpha$ is a pure-$K$ leading term datum, it is not hard
to describe the set $\operatorname{Hilb}^{0}(\alpha)$ (in fact the reduced
structure of the variety $V(\alpha)$ in (6.5)) explicitly. We give one typical
example.

######  Example 7.1.

Let $\alpha=(K(0),K(2),K(9))$. Then $V(\alpha)$ consists of $(X,Y)$ such that

$X=\begin{bmatrix}0&X_{12}&X_{13}\\\ 0&0&X_{23}\\\ 0&0&0\end{bmatrix},\quad
Y=\begin{bmatrix}0&Y_{12}&Y_{13}\\\ 0&0&Y_{23}\\\ 0&0&0\end{bmatrix},\quad
X_{12}=0$ (7.1)

and the matrix equations $X^{2}=Y^{3},XY=YX$ are equivalent to

$X_{12}X_{23}=0,\quad X_{12}Y_{23}=Y_{12}X_{23}.$ (7.2)

Since $X_{12}=0$, the only requirement is $Y_{12}X_{23}=0$. Thus
$[V(\alpha)]=(2\mathbb{L}-1)\mathbb{L}^{3}$, where $\mathbb{L}^{3}$ results
from the three freely chosen variables $X_{13},Y_{13},Y_{23}$.

According to (6.5), the variety $V(\alpha)$ only depends on the distance
matrix $\delta_{(ij)}(\alpha)$ of $\alpha$. Moreover, for each $i<j$, the
exact distance $\delta_{(ij)}(\alpha)$ does not matter; all that matters is
whether it is at most $1$ (labeled $1^{-}$), or $2$, or at least $3$ (labeled
$3^{+}$). For example, the above example corresponds to the case
$(\delta_{(12)}(\alpha),\delta_{(23)}(\alpha),\delta_{(13)}(\alpha))=(2,3^{+},3^{+})$.
We provide the motive of $V(\alpha)$ for every possible distance matrix in
$d\leq 3$. If $d=0,1$, then $V(\alpha)$ is just a point, so $[V(\alpha)]=1$.
If $d=2,3$, see Tables 2 and 3. Note that the last rows of Tables 2 and 3
verify the $d=2,3$ entries of Table 1.

$\delta_{(12)}(\alpha)$ | $[V(\alpha)]$
---|---
$1^{-}$ | $1$
2 | $\mathbb{L}$
$3^{+}$ | $\mathbb{L}^{2}$
Table 2. Pure-$K$ strata in terms of distances, $d=2$ $(\delta_{(12)}(\alpha),\delta_{(23)}(\alpha),\delta_{(13)}(\alpha))$ | $[V(\alpha)]$
---|---
$(1^{-},1^{-},1^{-})$ | $1$
$(1^{-},1^{-},2)$ | $\mathbb{L}$
$(1^{-},1^{-},3+)$ | $\mathbb{L}^{2}$
$(1^{-},2,2)$ | $\mathbb{L}^{2}$
$(1^{-},2,3+)$ | $\mathbb{L}^{3}$
$(1^{-},3+,3+)$ | $\mathbb{L}^{4}$
$(2,1^{-},2)$ | $\mathbb{L}^{2}$
$(2,1^{-},3^{+})$ | $\mathbb{L}^{3}$
$(2,2,3^{+})$ | $\mathbb{L}^{4}$
$(2,3^{+},3^{+})$ | $2\mathbb{L}^{4}-\mathbb{L}^{3}$
$(3^{+},1^{-},3^{+})$ | $\mathbb{L}^{4}$
$(3^{+},2,3^{+})$ | $2\mathbb{L}^{4}-\mathbb{L}^{3}$
$(3^{+},3^{+},3^{+})$ | $3\mathbb{L}^{4}-2\mathbb{L}^{3}$
Table 3. Pure-$K$ strata in terms of distances, $d=3$

###  7.2. Computation of $H_{d}(t)$ for $d\leq 3$

#### Case $d=0$

Clearly $H_{d}(t)=Q_{d}(t)=1$ if $d=0$.

For $d=1,2,3$, we shall compute $H_{d}(t)$ by first decomposing the set $\Xi$
of leading term data into a disjoint union of stable orbits, and then
computing the total contribution of each stable orbit using Lemma 5.12. The
existence of a decomposition is guaranteed by Theorem 5.14, but a more
efficient decomposition is often available. We will need to compute the
contents of finitely many leading term data along the way. Using (5.18), this
is always doable as long as we know $\operatorname{Hilb}^{0}(\alpha)$ for any
pure-$K$ leading term data up to rank $3$. From now on, we will provide the
quantities
$\lvert\operatorname{Hilb}(\alpha)\rvert,A(\alpha),B(\alpha),D(\alpha),\operatorname{Cont}(\alpha)$
as needed without proof.

All formulas in this section hold in the motivic sense as well if we replace
$q$ by $\mathbb{L}$, because the previous discussions show that $[V(\alpha)]$
is a polynomial in $\mathbb{L}$ if $\alpha$ is pure-$K$ of rank at most $3$.

#### Case $d=1$

The set $\Xi$ of leading term data is a disjoint union of two stable orbits:
$(K(0))$ stable under $\gamma_{(1)}$, and $(J(1))$ stable under
$\gamma_{(1)}$. We have $\operatorname{Cont}(K(0))=1$ and
$\operatorname{Cont}(J(1))=qt$. Hence

$H_{1}(t)=\frac{1}{1-t}+\frac{qt}{1-t}=\frac{1+qt}{1-t}.$ (7.3)

By (2.41), we have

$Q_{1}(t)=1+tH_{1}(t)=\frac{1+qt^{2}}{1-t}.$ (7.4)

This matches the well-known formula for the local Hilbert zeta function for
the cusp singularity; see for instance [31, Example 19, $A_{2}$].

#### Case $d=2$

We exhaust the elements of $\Xi$ using the four grids in Example 5.4. We say
an arrow is stable (marked as solid) if the leading term datum at the source
is stable under the spiral raising operator the arrow represents, and unstable
(marked as dashed) otherwise; see Figure 2. We introduce the notation
$\Xi_{c}$ to refer to the set of all $\alpha\in\Xi$ with color vector
$c(\alpha)=c$. From this grid, we can read that $\Xi_{(K,K)}$ is a disjoint
union of four stable orbits: $(K(0),K(0))$ under $\gamma_{(1)}$, $(K(0),K(1))$
under $\gamma_{(1)}$, $(K(0),K(2))$ under $\gamma_{(1)}$, and $(K(0),K(3))$
under $\gamma_{(1)}$, $\gamma_{(2)}$.

${{(K(0),K(0))}}$${{(K(0),K(1))}}$${{(K(0),K(2))}}$${{(K(0),K(3))}}$${\cdots}$${{(K(1),K(0))}}$${{(K(2),K(0))}}$${{(K(3),K(0))}}$${{(K(4),K(0))}}$${\cdots}$${{(K(1),K(1))}}$${{(K(1),K(2))}}$${{(K(1),K(3))}}$${{(K(1),K(4))}}$${\cdots}$${{(K(2),K(1))}}$${{(K(3),K(1))}}$${{(K(4),K(1))}}$${{(K(5),K(1))}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(2)}}$$\scriptstyle{\gamma_{(1)}}$
Figure 2. Stability in color vector $(K,K)$

We record the information in a simpler diagram Figure 3, where we only keep
the starting points of the orbits. We also omit the arrows $\gamma_{(1)}$,
with the understanding that every element of $\Xi$ is stable under
$\gamma_{(1)}$ (cf. Lemma 5.13). The stable arrows can thus be read from the
diagram as solid arrows plus $\gamma_{(1)}$. We also label the content of each
node.

${{\begin{matrix}(K(0),K(0))\\\
1\end{matrix}}}$${{\begin{matrix}(K(0),K(1))\\\
qt\end{matrix}}}$${{\begin{matrix}(K(0),K(2))\\\
q^{3}t^{2}\end{matrix}}}$${{\begin{matrix}(K(0),K(3))\\\
q^{5}t^{3}\end{matrix}}}$${\cdots}$$\scriptstyle{\gamma_{(2)}}$ Figure 3.
Stability and contents in $(K,K)$

From this diagram, we read

$\displaystyle\sum_{\alpha\in\Xi_{(K,K)}}\operatorname{Cont}(\alpha)$
$\displaystyle=\frac{1}{1-t}+\frac{qt}{1-t}+\frac{q^{3}t^{2}}{1-t}+\frac{q^{5}t^{3}}{(1-t)(1-qt)}$
(7.5)
$\displaystyle=\frac{1-q^{2}t^{2}+q^{3}t^{2}-q^{4}t^{3}+q^{5}t^{3}}{(1-t)(1-qt)}.$

Similar diagrams for other three color vectors are given in Figures 4, 5 and
6.

${\begin{matrix}(K(0),J(1))\\\
q^{3}t\end{matrix}}$${\cdots}$$\scriptstyle{\gamma_{(2)}}$

$\sum_{\alpha\in\Xi_{(K,J)}}\operatorname{Cont}(\alpha)=\frac{q^{3}t}{(1-t)(1-qt)}.$

Figure 4. Stability and contents in $(K,J)$

${\begin{matrix}(J(1),K(0))\\\
q^{2}t\end{matrix}}$${\begin{matrix}(J(1),K(1))\\\
q^{2}t^{2}\end{matrix}}$${\cdots}$$\scriptstyle{\gamma_{(2)}}$

$\sum_{\alpha\in\Xi_{(J,K)}}\operatorname{Cont}(\alpha)=\frac{q^{2}t}{1-t}+\frac{q^{2}t^{2}}{(1-t)(1-qt)}.$

Figure 5. Stability and contents in $(J,K)$

${\begin{matrix}(J(1),J(1))\\\
q^{4}t^{2}\end{matrix}}$${\begin{matrix}(J(1),J(2))\\\
q^{4}t^{3}\end{matrix}}$${\cdots}$$\scriptstyle{\gamma_{(2)}}$

$\sum_{\alpha\in\Xi_{(J,J)}}\operatorname{Cont}(\alpha)=\frac{q^{4}t^{2}}{1-t}+\frac{q^{4}t^{3}}{(1-t)(1-qt)}.$

Figure 6. Stability and contents in $(J,J)$

Summing up, we get

$H_{2}(t)=\sum_{\alpha\in\Xi}\operatorname{Cont}(\alpha)=\frac{1+q^{2}t+q^{3}t+q^{4}t^{2}}{(1-t)(1-qt)},$
(7.6)

where we note that all terms involving $t^{3}$ are cancelled.

By (2.41), we have

$Q_{2}(t)=1+(1+q)tH_{1}(qt)+t^{2}H_{2}(t)=\frac{1+q^{2}t^{2}+q^{3}t^{2}+q^{4}t^{4}}{(1-t)(1-qt)},$
(7.7)

where we note unexpectedly that the numerator of $Q_{2}(t)$ is the numerator
of $H_{2}(t)$ evaluated at $t\mapsto t^{2}$.

#### Case $d=3$

We now compute $H_{3}(t)$ by constructing similar diagrams for all eight color
vectors, see Figures 7 through 14. We put $\gamma_{(2)}$ on verticle arrows,
$\gamma_{(3)}$ on horizontal arrows, and omit $\gamma_{(1)}$ with the
understanding that every node is stable under $\gamma_{(1)}$. To save space,
we also omit the first component in the grid, which is always $K(0)$ when the
first component of the color vector is $K$, and $J(1)$ otherwise. For example,
in color $(J,K,J)$, the node $(J(1),J(2),K(0))$ is denoted by $(J(2),K(0))$.
For ease of verification, we label the content of each node in the form

$\operatorname{Cont}(\alpha)=A(\alpha)\cdot B(\alpha)\cdot
D(\alpha)t^{n(\alpha)}.$ (7.8)

We note that $A(\alpha)$ is generically unchanged by $\gamma_{(j)}$, unless
the distance between some $K$ components are stretched from $1$ to $2$ or from
$2$ to $3$. The factor $B(\alpha)$ stays the same on each grid, and it only
depends on the color vector (Lemma 5.5). The factor $D(\alpha)$ is generically
multiplied by $q^{j-1}$ under $\gamma_{(j)}$, unless some stretches are
obstructed (see Lemma 5.7). The exponent $n(\alpha)$ is always increased by
$1$ under $\gamma_{(j)}$. The top-left node $\mathbf{0}_{c}$ of each color
vector $c$ only consists of $K(0)$ and $J(1)$ components, and it has
$A(\mathbf{0}_{c})=1$, $D(\mathbf{0}_{c})=q^{d\\#_{J}(c)}=q^{3\\#_{J}(c)}$,
and $n(\mathbf{0}_{c})=\\#_{J}(c)$, where $\\#_{J}(c)$ is the number of $J$’s
in $c$. One can write down every factor but $A(\alpha)$ by starting from the
top-left node and tracing the arrows using the rules above. For $A(\alpha)$,
we consult Tables 2 and 3.

To obtain a rational formula for $H_{3}(t)$, we decompose each grid into a
disjoint union of stable orbits, marked by enclosing frames. Note that our
decomposition is different from the one in Theorem 5.14 because we exploit
stable arrows not guaranteed by Lemma 5.13; we only do so to reduce
computation. Note also that the choice of the “most efficient” decomposition
in Figure 12 is noncanonical. The contribution of each orbit to $H_{3}(t)$ is
computed by Lemma 5.12. For example, the top-right box of Figure 7 contributes
$\dfrac{q^{9}t^{4}}{(1-t)(1-q^{2}t)}$.

${{\begin{matrix}(J(1),J(1))\\\ 1\cdot 1\cdot
q^{9}t^{3}\end{matrix}}}$${{\begin{matrix}(J(1),J(2))\\\ 1\cdot 1\cdot
q^{9}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),J(1))\\\ 1\cdot
1\cdot q^{9}t^{4}\end{matrix}}}$${{\begin{matrix}(J(3),J(1))\\\ 1\cdot 1\cdot
q^{10}t^{5}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),J(2))\\\ 1\cdot
1\cdot q^{9}t^{5}\end{matrix}}}$${{\begin{matrix}(J(2),J(3))\\\ 1\cdot 1\cdot
q^{10}t^{6}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 7. Stability and contents in $(J,J,J)$ ${{\begin{matrix}(J(1),K(0))\\\
1\cdot 1\cdot q^{6}t^{2}\end{matrix}}}$${{\begin{matrix}(J(1),K(1))\\\ 1\cdot
1\cdot q^{6}t^{3}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),J(1))\\\
1\cdot 1\cdot q^{6}t^{3}\end{matrix}}}$${{\begin{matrix}(K(2),J(1))\\\ 1\cdot
1\cdot q^{7}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),K(1))\\\
1\cdot 1\cdot q^{6}t^{4}\end{matrix}}}$${{\begin{matrix}(J(2),K(2))\\\ 1\cdot
1\cdot
q^{7}t^{5}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 8. Stability and contents in $(J,J,K)$ ${{\begin{matrix}(K(0),J(1))\\\
1\cdot q\cdot q^{6}t^{2}\end{matrix}}}$${{\begin{matrix}(K(0),J(2))\\\ 1\cdot
q\cdot q^{7}t^{3}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),K(0))\\\
1\cdot q\cdot q^{6}t^{3}\end{matrix}}}$${{\begin{matrix}(J(3),K(0))\\\ 1\cdot
q\cdot q^{8}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),J(2))\\\
1\cdot q\cdot q^{6}t^{4}\end{matrix}}}$${{\begin{matrix}(K(1),J(3))\\\ 1\cdot
q\cdot
q^{8}t^{5}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 9. Stability and contents in $(J,K,J)$ ${{\begin{matrix}(K(0),K(0))\\\
1\cdot 1\cdot q^{3}t\end{matrix}}}$${{\begin{matrix}(K(0),K(1))\\\ 1\cdot
1\cdot q^{4}t^{2}\end{matrix}}}$${{\begin{matrix}(K(0),K(2))\\\ q\cdot 1\cdot
q^{6}t^{3}\end{matrix}}}$${{\begin{matrix}(K(0),K(3))\\\ q^{2}\cdot 1\cdot
q^{8}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),K(0))\\\ 1\cdot
1\cdot q^{3}t^{2}\end{matrix}}}$${{\begin{matrix}(K(2),K(0))\\\ 1\cdot 1\cdot
q^{5}t^{3}\end{matrix}}}$${{\begin{matrix}(K(3),K(0))\\\ q\cdot 1\cdot
q^{7}t^{4}\end{matrix}}}$${{\begin{matrix}(K(4),K(0))\\\ q^{2}\cdot 1\cdot
q^{9}t^{5}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),K(1))\\\ 1\cdot
1\cdot q^{3}t^{3}\end{matrix}}}$${{\begin{matrix}(K(1),K(2))\\\ 1\cdot 1\cdot
q^{5}t^{4}\end{matrix}}}$${{\begin{matrix}(K(1),K(3))\\\ q\cdot 1\cdot
q^{7}t^{5}\end{matrix}}}$${{\begin{matrix}(K(1),K(4))\\\ q^{2}\cdot 1\cdot
q^{9}t^{6}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 10. Stability and contents in $(J,K,K)$ ${{\begin{matrix}(J(1),J(1))\\\
1\cdot q^{2}\cdot q^{6}t^{2}\end{matrix}}}$${{\begin{matrix}(J(1),J(2))\\\
1\cdot q^{2}\cdot
q^{7}t^{3}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 11. Stability and contents in $(K,J,J)$ ${{\begin{matrix}(J(1),K(0))\\\
1\cdot q\cdot q^{3}t^{1}\end{matrix}}}$${{\begin{matrix}(J(1),K(1))\\\ 1\cdot
q\cdot q^{4}t^{2}\end{matrix}}}$${{\begin{matrix}(J(1),K(2))\\\ q\cdot q\cdot
q^{6}t^{3}\end{matrix}}}$${{\begin{matrix}(J(1),K(3))\\\ q^{2}\cdot q\cdot
q^{8}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),J(1))\\\ 1\cdot
q\cdot q^{4}t^{2}\end{matrix}}}$${{\begin{matrix}(K(2),J(1))\\\ q\cdot q\cdot
q^{5}t^{3}\end{matrix}}}$${{\begin{matrix}(K(3),J(1))\\\ q^{2}\cdot q\cdot
q^{7}t^{4}\end{matrix}}}$${{\begin{matrix}(K(4),J(1))\\\ q^{2}\cdot q\cdot
q^{9}t^{5}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),K(1))\\\ 1\cdot
q\cdot q^{5}t^{3}\end{matrix}}}$${{\begin{matrix}(J(2),K(2))\\\ q\cdot q\cdot
q^{6}t^{4}\end{matrix}}}$${{\begin{matrix}(J(2),K(3))\\\ q^{2}\cdot q\cdot
q^{8}t^{5}\end{matrix}}}$${{\begin{matrix}(J(2),K(4))\\\ q^{2}\cdot q\cdot
q^{10}t^{6}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(2),J(2))\\\ q\cdot
q\cdot q^{6}t^{4}\end{matrix}}}$${{\begin{matrix}(K(3),J(2))\\\ q^{2}\cdot
q\cdot q^{7}t^{5}\end{matrix}}}$${{\begin{matrix}(K(4),J(2))\\\ q^{2}\cdot
q\cdot q^{9}t^{6}\end{matrix}}}$${{\begin{matrix}(K(5),J(2))\\\ q^{2}\cdot
q\cdot q^{11}t^{7}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(3),K(2))\\\
q\cdot q\cdot q^{7}t^{5}\end{matrix}}}$${{\begin{matrix}(J(3),K(3))\\\
q^{2}\cdot q\cdot q^{8}t^{6}\end{matrix}}}$${{\begin{matrix}(J(3),K(4))\\\
q^{2}\cdot q\cdot q^{10}t^{7}\end{matrix}}}$${{\begin{matrix}(J(3),K(5))\\\
q^{2}\cdot q\cdot
q^{12}t^{8}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(3),J(3))\\\ q^{2}\cdot
q\cdot q^{8}t^{6}\end{matrix}}}$${{\begin{matrix}(K(4),J(3))\\\ q^{2}\cdot
q\cdot q^{9}t^{7}\end{matrix}}}$${{\begin{matrix}(K(5),J(3))\\\ q^{2}\cdot
q\cdot q^{11}t^{8}\end{matrix}}}$${{\begin{matrix}(K(6),J(3))\\\ q^{2}\cdot
q\cdot
q^{13}t^{9}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 12. Stability and contents in $(K,J,K)$ ${{\begin{matrix}(K(0),J(1))\\\
1\cdot q^{2}\cdot
q^{3}t\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(2),K(0))\\\ 1\cdot
q^{2}\cdot q^{4}t^{2}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),J(2))\\\
1\cdot q^{2}\cdot
q^{5}t^{3}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(3),K(1))\\\ 1\cdot
q^{2}\cdot q^{6}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(2),J(3))\\\
q\cdot q^{2}\cdot
q^{7}t^{5}\end{matrix}}}$${\cdots}$${{\begin{matrix}(J(4),K(2))\\\ q\cdot
q^{2}\cdot q^{8}t^{6}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(3),J(4))\\\
q^{2}\cdot q^{2}\cdot
q^{9}t^{7}\end{matrix}}}$${\cdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 13. Stability and contents in $(K,K,J)$ ${{\begin{matrix}(K(0),K(0))\\\
1\cdot 1\cdot 1\end{matrix}}}$${{\begin{matrix}(K(0),K(1))\\\ 1\cdot 1\cdot
q^{2}t\end{matrix}}}$${{\begin{matrix}(K(0),K(2))\\\ q^{2}\cdot 1\cdot
q^{4}t^{2}\end{matrix}}}$${{\begin{matrix}(K(0),K(3))\\\ q^{4}\cdot 1\cdot
q^{6}t^{3}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),K(0))\\\ 1\cdot
1\cdot qt\end{matrix}}}$${{\begin{matrix}(K(2),K(0))\\\ q\cdot 1\cdot
q^{3}t^{2}\end{matrix}}}$${{\begin{matrix}(K(3),K(0))\\\ q^{3}\cdot 1\cdot
q^{5}t^{3}\end{matrix}}}$${{\begin{matrix}(K(4),K(0))\\\ q^{4}\cdot 1\cdot
q^{7}t^{4}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(1),K(1))\\\ 1\cdot
1\cdot q^{2}t^{2}\end{matrix}}}$${{\begin{matrix}(K(1),K(2))\\\ q\cdot 1\cdot
q^{4}t^{3}\end{matrix}}}$${{\begin{matrix}(K(1),K(3))\\\ q^{3}\cdot 1\cdot
q^{6}t^{4}\end{matrix}}}$${{\begin{matrix}(K(1),K(4))\\\ q^{4}\cdot 1\cdot
q^{8}t^{5}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(2),K(1))\\\ q\cdot
1\cdot q^{3}t^{3}\end{matrix}}}$${{\begin{matrix}(K(3),K(1))\\\ q^{2}\cdot
1\cdot q^{5}t^{4}\end{matrix}}}$${{\begin{matrix}(K(4),K(1))\\\ q^{3}\cdot
1\cdot q^{7}t^{5}\end{matrix}}}$${{\begin{matrix}(K(5),K(1))\\\ q^{4}\cdot
1\cdot q^{9}t^{6}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(2),K(2))\\\
q^{2}\cdot 1\cdot q^{4}t^{4}\end{matrix}}}$${{\begin{matrix}(K(2),K(3))\\\
q^{3}\cdot 1\cdot q^{6}t^{5}\end{matrix}}}$${{\begin{matrix}(K(2),K(4))\\\
q^{4}\cdot 1\cdot q^{8}t^{6}\end{matrix}}}$${{\begin{matrix}(K(2),K(5))\\\
(2q^{4}-q^{3})\cdot 1\cdot
q^{10}t^{7}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(3),K(2))\\\ q^{3}\cdot
1\cdot q^{5}t^{5}\end{matrix}}}$${{\begin{matrix}(K(4),K(2))\\\ q^{3}\cdot
1\cdot q^{7}t^{6}\end{matrix}}}$${{\begin{matrix}(K(5),K(2))\\\ q^{4}\cdot
1\cdot q^{9}t^{7}\end{matrix}}}$${{\begin{matrix}(K(6),K(2))\\\
(2q^{4}-q^{3})\cdot 1\cdot
q^{11}t^{8}\end{matrix}}}$${\cdots}$${{\begin{matrix}(K(3),K(3))\\\ q^{4}\cdot
1\cdot q^{6}t^{6}\end{matrix}}}$${{\begin{matrix}(K(3),K(4))\\\ q^{4}\cdot
1\cdot q^{8}t^{7}\end{matrix}}}$${{\begin{matrix}(K(3),K(5))\\\
(2q^{4}-q^{3})\cdot 1\cdot
q^{10}t^{8}\end{matrix}}}$${{\begin{matrix}(K(3),K(6))\\\ (3q^{4}-2q^{3})\cdot
1\cdot
q^{12}t^{9}\end{matrix}}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\gamma_{(3)}}$$\scriptstyle{\gamma_{(2)}}$
Figure 14. Stability and contents in $(K,K,K)$

Summing up the contributions of 78 orbits in the eight diagrams, we obtain

$H_{3}(t)=\frac{1+q^{3}t+q^{4}t+q^{5}t+q^{6}t^{2}+q^{7}t^{2}+q^{8}t^{2}+q^{9}t^{3}}{(1-t)(1-qt)(1-q^{2}t)}$
(7.9)

and (2.41) gives

$Q_{3}(t)=\frac{1+q^{3}t^{2}+q^{4}t^{2}+q^{5}t^{2}+q^{6}t^{4}+q^{7}t^{4}+q^{8}t^{4}+q^{9}t^{6}}{(1-t)(1-qt)(1-q^{2}t)},$
(7.10)

where we note again that the numerator of $Q_{3}(t)$ is the numerator of
$H_{3}(t)$ evaluated at $t\mapsto t^{2}$.

The simplicity of the formula for $H_{3}(t)$ is striking. In terms of the
$t$-degrees of their contributions to the numerator of $H_{3}(t)$, individual
orbits have contributions up to $t^{9}$, and the total contribution of a color
vector has contribution up to $t^{7}$, but the $t$-degree of the numerator of
$H_{3}(t)$ is only $3$. See Table 4 for a breakdown of $H_{3}(t)$ in each
color, where we also mark the number of orbits used in our computation. We
multiply the content by $(t;q)_{3}=(1-t)(1-qt)(1-q^{2}t)$ to extract the
contribution to the numerator of $H_{3}(t)$.

$c$ | # orbits |  $(t;q)_{3}\sum_{\alpha\in\Xi_{c}}\operatorname{Cont}(\alpha)$
---|---|---
$(J,J,J)$ | 6 |  $q^{6}t^{2}+2q^{6}t^{3}-q^{7}t^{3}-q^{8}t^{3}+q^{6}t^{4}-q^{7}t^{4}-q^{8}t^{4}+q^{9}t^{4}+q^{7}t^{5}-2q^{8}t^{5}+q^{9}t^{5}$
$(J,J,K)$ | 6 |  $q^{9}t^{3}+2q^{9}t^{4}-q^{1}0t^{4}-q^{1}1t^{4}+q^{9}t^{5}-q^{10}t^{5}-q^{11}t^{5}+q^{12}t^{5}+q^{10}t^{6}-2q^{11}t^{6}+q^{12}t^{6}$
$(J,K,J)$ | 4 |  $q^{7}t^{2}+q^{7}t^{3}-q^{9}t^{3}+q^{7}t^{4}-q^{8}t^{4}-q^{9}t^{4}+q^{10}t^{4}$
$(J,K,K)$ | 12 |  $q^{3}t+q^{3}t^{2}-q^{5}t^{2}+q^{3}t^{3}-q^{4}t^{3}-q^{5}t^{3}+q^{7}t^{3}-q^{9}t^{4}+q^{10}t^{4}-q^{7}t^{5}+2q^{8}t^{5}-q^{9}t^{5}-q^{10}t^{6}+2q^{11}t^{6}-q^{12}t^{6}$
$(K,J,J)$ | 2 |  $q^{8}t^{2}+q^{9}t^{3}-q^{10}t^{3}$
$(K,J,K)$ | 13 |  $q^{4}t+q^{5}t^{2}-q^{6}t^{2}-q^{6}t^{3}+q^{8}t^{3}-q^{7}t^{4}+2q^{8}t^{4}-2q^{9}t^{4}+q^{11}t^{4}+q^{11}t^{5}-q^{12}t^{5}-q^{10}t^{6}+2q^{11}t^{6}-q^{12}t^{6}+q^{12}t^{7}-q^{13}t^{7}$
$(K,K,J)$ | 7 |  $q^{5}t-q^{9}t^{5}+q^{10}t^{5}-q^{12}t^{7}+q^{13}t^{7}$
$(K,K,K)$ | 28 |  $1-q^{3}t^{2}+q^{6}t^{2}-q^{3}t^{3}+q^{4}t^{3}+q^{5}t^{3}-q^{6}t^{3}-q^{7}t^{3}+q^{10}t^{3}-q^{6}t^{4}+q^{7}t^{4}+q^{9}t^{4}-q^{10}t^{4}+q^{10}t^{6}-2q^{11}t^{6}+q^{12}t^{6}$
Total | 78 |  $1+q^{3}t+q^{4}t+q^{5}t+q^{6}t^{2}+q^{7}t^{2}+q^{8}t^{2}+q^{9}t^{3}$
Table 4. $H_{3}(t)$ from each color

###  7.3. Conjectural formulas for general $d$

Following the notation of Theorem 1.8, let
$\mathit{NH}_{d}(t)=(t;q)_{d}H_{d}(t)$ and
$\mathit{NQ}_{d}(t)=(t;q)_{d}Q_{d}(t)$. They are now known to be polynomials
in $t$ by Theorem 5.14.

We translate Conjecture 1.4(b) together with the conjectural (1.25) into a
version for $\mathit{NH}_{d}(t)$ using (2.41).

######  Conjecture 7.2.

We have

  1. (a)

$\mathit{NH}_{d}(t^{2})=\sum_{r=0}^{d}t^{r}{d\brack
r}_{q}(t;q)_{d-r}\mathit{NH}_{r}(tq^{d-r})$;

  2. (b)

(Functional Equation)
$q^{d^{2}}t^{d}\mathit{NH}_{d}(q^{-2d}t^{-1})=\mathit{NH}_{d}(t)$.

We have just verified the above conjecture for $d\leq 3$. It turns out
Conjecture 7.2 is strong enough to _overdetermine_ $\mathit{NH}_{d}(t)$ for
all $d$. The following example for $d=4$ will demonstrate how to find
$\mathit{NH}_{d}(t)$ inductively and verify the consitency of Conjecture 7.2
for $d$ up to a given bound.

Assume Conjecture 7.2. Conjecture 7.2(a) then implies that
$\mathit{NH}_{d}(t)$ has constant term $1$. By Conjecture 7.2(b), the
$t$-leading term of $\mathit{NH}_{d}(t)$ is $q^{d^{2}}t^{d}$.

For any commutative ring $A$ with 1, let $A_{r}[t]$ be the $A$-module of
$A$-polynomials of degree at most $r$. Consider the $A$-linear map
$\Theta_{d}:A_{d}[t]\to A_{2d-1}[t]$ defined by

$\Theta_{d}(f(t)):=f(t^{2})-t^{d}f(t).$ (7.11)

Then Conjecture 7.2(a) can be rewritten as

$\Theta_{d}(\mathit{NH}_{d}(t))=\sum_{r=0}^{d-1}t^{r}{d\brack
r}_{q}(t;q)_{d-r}\mathit{NH}_{r}(tq^{d-r}).$ (7.12)

######  Example 7.3.

When $d=4$, based on the known expressions for $d\leq 3$, the equation (7.12)
becomes

$\begin{multlined}\Theta_{4}(\mathit{NH}_{4}(t))=1+(q^{4}+q^{5}+q^{6}+q^{7})t^{2}+(-1+q^{8}+q^{9}+2q^{10}+q^{11}+q^{12})t^{4}+\\\
(-q^{4}-q^{5}-q^{6}-q^{7})t^{5}+(-q^{8}-q^{9}-2q^{10}-q^{11}+q^{13}+q^{14}+q^{15})t^{6}+(-q^{12}-q^{13}-q^{14}-q^{15})t^{7}.\end{multlined}\Theta_{4}(\mathit{NH}_{4}(t))=1+(q^{4}+q^{5}+q^{6}+q^{7})t^{2}+(-1+q^{8}+q^{9}+2q^{10}+q^{11}+q^{12})t^{4}+\\\
(-q^{4}-q^{5}-q^{6}-q^{7})t^{5}+(-q^{8}-q^{9}-2q^{10}-q^{11}+q^{13}+q^{14}+q^{15})t^{6}+(-q^{12}-q^{13}-q^{14}-q^{15})t^{7}.$
(7.13)

Matching the $t^{4}$\- through $t^{7}$-coefficients of
$\Theta_{4}(\mathit{NH}_{4}(t))$, the only possibility for
$\mathit{NH}_{4}(t)$ is

$\mathit{NH}_{4}(t)=1+(q^{4}+q^{5}+q^{6}+q^{7})t+(q^{8}+q^{9}+2q^{10}+q^{11}+q^{12})t^{2}+(q^{12}+q^{13}+q^{14}+q^{15})t^{3}+q^{16}t^{4}.$
(7.14)

Here, the $t^{4}$-coefficient is determined by the functional equation, not by
(7.13). One can verify that $\mathit{NH}_{4}(t)$ indeed satisfies (7.13) and
Conjecture 7.2(b). We thus conclude that (7.14) is the unique
$\mathit{NH}_{4}(t)$ that makes Conjecture 7.2 consistent up to $d\leq 4$.

We also notice that $\mathit{NH}_{4}(t)$ is a polynomial in $q,t$, even if we
have not assumed so.

Based on computational evidence for $\mathit{NH}_{d}(t)$, $d\leq 30$, we
formulate the following conjecture independent of Conjecture 7.2.

######  Conjecture 7.4.

Assume $A$ is a commutative ring with 1, and $q$ is an element of $A$. Then
there are unique polynomials $\mathit{NH}_{d}(t)\in A[t]$ for $d\geq 0$ such
that

  1. (a)

$\mathit{NH}_{0}(t)=1$;

  2. (b)

$\mathit{NH}_{d}(t^{2})=\sum_{r=0}^{d}t^{r}{d\brack
r}_{q}(t;q)_{d-r}\mathit{NH}_{r}(tq^{d-r})$;

  3. (c)

(Functional Equation) $\mathit{NH}_{d}(t)=\sum_{i=0}^{d}a_{i}t^{i}$, with
$a_{d-i}=q^{d(d-2i)}a_{i}$ for $0\leq i\leq\lfloor d/2\rfloor$.

Moreover, $\mathit{NH}_{d}(t)$ is given by a polynomial $\mathit{NH}_{d}(t;q)$
in $t,q$ with nonnegative integer coefficients, defined by the following:

$\mathit{NH}_{d}(t;q):=\sum_{j=0}^{d}q^{{j+1\choose
2}+j(d-j)}c_{j}(q)t^{j},\quad\text{where
}\sum_{j=0}^{d}c_{j}(q)t^{j}=(-t;q)_{d}.$ (7.15)

In addition, let $H_{d}(t;q):=\mathit{NH}_{d}(t;q)/(t;q)_{d}$, and
$\widehat{Z}(t;q)=\sum_{d=0}^{\infty}\dfrac{q^{-d^{2}}t^{d}}{(q^{-1};q^{-1})_{d}}H_{d}(q^{-d}t;q)$
(compare (2.40)), then

$\widehat{Z}(t;q)=\frac{1}{(tq^{-1};q^{-1})_{\infty}}\sum_{n=0}^{\infty}\frac{q^{-n^{2}}t^{2n}}{(q^{-1};q^{-1})_{n}}.$
(7.16)

Conjecture 7.4 is purely combinatorial and is independent of the truth about
the motives of punctual Quot schemes of the cusp. Having the conjectured forms
of $\mathit{NH}_{d}(t;q)$ and $\widehat{Z}(t;q)$, verifying Conjecture 7.4
should be attainable in principle, but we have not made serious attempt.

######  Lemma 7.5.

Conjecture 7.4 and the motivic version of Conjecture 7.2 imply Conjecture 1.9.

###### Proof.

Apply Conjecture 7.4 with $A=K_{0}(\mathrm{Var}_{k})$ and $q=\mathbb{L}$. ∎

The above lemma suggests a “conceptual proof” of Conjecture 1.9 without
explicit computations and without knowing in advance whether the relevant
motives are polynomials of $\mathbb{L}$. Since Conjecture 1.9 implies
Conjecture 1.10, this provides a plan to attack Conjecture 1.10 without
computing the Gröbner strata and proving Conjecture 6.1. Of course, Conjecture
1.10 does not imply the stratum-wise polynomiality statement of Conjecture
6.1, so the latter conjecture would not be obsolete even if the former is
proved to be true.

We make a further remark of the formula (7.16). Let

$F(t;q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}t^{n}}{(q;q)_{n}}$ (7.17)

be a Rogers–Ramanujan series (see [4, Chapter 7.1]), then
$\widehat{Z}_{R}(t)=\frac{1}{(tq^{-1};q^{-1})_{\infty}}F(t^{2};q^{-1})$, so
the Rogers–Ramanujan identities would imply

$\displaystyle\widehat{Z}_{R}(1)$
$\displaystyle=\frac{1}{(q^{-1};q^{-1})_{\infty}(q^{-1};q^{-5})_{\infty}(q^{-4};q^{-5})_{\infty}};$
(7.18) $\displaystyle\widehat{Z}_{R}(-1)$
$\displaystyle=\frac{(q^{-1};q^{-2})_{\infty}}{(q^{-1};q^{-5})_{\infty}(q^{-4};q^{-5})_{\infty}}.$
(7.19)

If true, this would provide another evidence for an observation in [37] that
$\widehat{Z}_{R}(\pm 1)$ tend to have special values that admit infinite
product formulas, if $R$ is the local ring at a point on a curve. The
conjectured formula (7.16) also gives the first guess for the number of matrix
pairs
$\\{(A,B)\in\operatorname{Mat}_{n}({{\mathbb{F}}_{q}})^{2}:AB=BA,A^{2}=B^{3}\\}$.
Via (2.6) and (2.9), our guess (7.16) is equivalent to

$\big{\lvert}\\{(A,B)\in\operatorname{Mat}_{n}({{\mathbb{F}}_{q}})^{2}:AB=BA,A^{2}=B^{3}\\}\big{\rvert}=\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{j}q^{\frac{1}{2}\left(3j^{2}-j\right)+n(n-2j)}\frac{(q;q)_{n}}{(q;q)_{j}(q;q)_{n-2j}}.$
(7.20)

Unlike its nodal analog (which now has two proofs [24, 37] by counting
matrices $AB=BA=0$), the matrix counting problem (7.20) is not known to have a
direct approach.

We make another observation about $\mathit{NH}_{d}(t;q)$ defined in (7.15).

######  Proposition 7.6.

Let $\zeta_{r}$ be a primitive $r$-th root of unity in ${\mathbb{C}}$. Then

$\mathit{NH}_{d}(t;\zeta_{r})=(1+t^{r})^{d/r}=\mathit{NH}_{1}(t^{r};1)^{d/r}.$
(7.21)

if $r$ divides $d$. Equivalently, for any $i\in{\mathbb{Z}}$, let
$e=\gcd(i,d)$, then we have

$\mathit{NH}_{d}(t;\zeta_{d}^{i})=(1+t^{d/e})^{e}.$ (7.22)

###### Proof.

Assume $r>0$ divides $d$. Putting $q=\zeta_{r}$ in (7.15), we get

$\mathit{NH}_{d}(t;\zeta_{r})=\sum_{j=0}^{d}\zeta_{r}^{{j+1\choose
2}+j(d-j)}c_{j}(\zeta_{r})t^{j},$ (7.23)

where $\sum_{j=0}^{d}c_{j}(\zeta_{r})t^{j}=(-t;\zeta_{r})_{d}$. Since $r$
divides $d$, we have $\zeta_{r}^{d}=1$, so the factor $\zeta_{r}^{{j+1\choose
2}+j(d-j)}$ is just $\zeta_{r}^{-{j\choose 2}}$. Using

$(t;\zeta_{r})_{d}=((1-t)(1-\zeta_{r}t)\dots(1-\zeta_{r}^{r-1}t))^{d/r}=(1-t^{r})^{d/r}$
(7.24)

and substituting $t\mapsto-t$, we get $(-t;\zeta_{r})_{d}=(1-(-t)^{r})^{d/r}$,
so that $c_{j}(\zeta_{r})=0$ if $r\nmid j$, and

$c_{ir}(\zeta_{r})={d/r\choose i}(-(-1)^{r})^{i}={d/r\choose i}(-1)^{i(r-1)}$
(7.25)

for $0\leq i\leq d/r$. We have

$\displaystyle\mathit{NH}_{d}(t;\zeta_{r})$
$\displaystyle=\sum_{i=0}^{d/r}\zeta_{r}^{-{ir\choose 2}}{d/r\choose
i}(-1)^{i(r-1)}t^{ir}$ (7.26)
$\displaystyle=\sum_{i=0}^{d/r}\zeta_{2r}^{-2{ir\choose 2}}{d/r\choose
i}\zeta_{2r}^{r\cdot i(r-1)}t^{ir}$ (7.27)
$\displaystyle=\sum_{i=0}^{d/r}\zeta_{2r}^{-ir(ir-1)+ir(r-1)}{d/r\choose
i}t^{ir}$ (7.28)
$\displaystyle=\sum_{i=0}^{d/r}\zeta_{2r}^{-i(i-1)r^{2}}{d/r\choose i}t^{ir}.$
(7.29)

Since $i(i-1)$ is even, $\zeta_{2r}^{-i(i-1)r^{2}}=1$, so we have

$\mathit{NH}_{d}(t;\zeta_{r})=\sum_{i=0}^{d/r}{d/r\choose
i}t^{ir}=(1+t^{r})^{d/r}.$ (7.30)

∎

We note a consequence that has potential generalizations outside the cusp
case. Let $\mathit{NQ}_{d}(t;q)=\mathit{NH}_{d}(t^{2};q)$ and
$Q_{d}(t;q)=\mathit{NQ}_{d}(t;q)/(t;q)_{d}$. Then $\mathit{NQ}_{d}(t;q)$
clearly satisfies an analogous property
$\mathit{NQ}_{d}(t;\zeta_{r})=\mathit{NQ}_{1}(t^{r};1)^{d/r}$ for $r|d$.
Because the $q,t$-polynomial $(t;q)_{d}$ satisfies a similar property
$(t;\zeta_{r})_{d}=(1-t^{r})^{d/r}=(t^{r};1)_{1}^{d/r}$ when $r|d$, we have

$\displaystyle H_{d}(t;\zeta_{r})$ $\displaystyle=H_{1}(t^{r};1)^{d/r};$
(7.31) $\displaystyle Q_{d}(t;\zeta_{r})$ $\displaystyle=Q_{1}(t^{1};1)^{d/r}$

if $r$ divides $d$.

The analogous property holds for a smooth rational point on a curve, using
$H_{d,{{\mathbb{F}}_{q}}[[T]]}(t)=Q_{d,{{\mathbb{F}}_{q}}[[T]]}(t)=1/(t;q)_{d}$
and the property for $(t;q)_{d}$ above. If the formula in Conjecture 1.9 is
true, then the $r=1$ case of the proposition above would imply that if
$k={\mathbb{C}}$ and $X$ is a reduced curve with only cusp singularities, the
Euler characteristics of $\operatorname{Quot}_{d,n}(X)$ would satisfy

$\sum_{n=0}^{\infty}\chi(\operatorname{Quot}_{d,n}(X))t^{n}=\left\lparen\sum_{n=0}^{\infty}\chi(\operatorname{Hilb}_{n}(X))t^{n}\right\rparen^{d}.$
(7.32)

For a reduced curve $X$ with planar singularities, the right-hand side is
known in terms of the HOMFLY polynomial of the associated algebraic link.
Therefore, if (7.32) is true for $X$, it would give a formula for the Euler
characteristics of $\operatorname{Quot}_{d,n}(X)$. The truth of (7.32) for the
cusp singularity is unknown, and one may attempt it as another weak form of
Conjecture 1.9.

We conclude this section with a curious observation about factorizations of
$\mathit{NH}_{d}(t;q)$ over the rationals, verified for $d\leq 30$ and $t=\pm
1,\pm 2$. In the following statement, we say the constant polynomial $1$ is
irreducible as well.

######  Conjecture 7.7.

Define

$P_{d}(t;q):=\begin{cases}\mathit{NH}_{d}(t;q),&\text{ if $d$ is even;}\\\
\mathit{NH}_{d}(t;q)/(1+q^{d}t),&\text{ if $d$ is odd.}\end{cases}$ (7.33)

Then

  1. (a)

$P_{d}(t;q)$ is a polynomial in ${\mathbb{Z}}[t,q]$ with nonnegative
coefficients, and is irreducible (or $1$);

  2. (b)

For any integer $m\neq-1$, the polynomial $P_{d}(m;q)$ is irreducible in
${\mathbb{Z}}[q]$;

  3. (c)

The polynomial $P_{d}(-1;q)$ is of the form

$P_{d}(-1;q)=I_{d}(q)\prod_{\begin{subarray}{c}1\leq r\leq d\\\ r\text{
odd}\end{subarray}}\Phi_{r}(q)^{\lfloor\frac{d+r-1}{2r}\rfloor},$ (7.34)

where $I_{d}(q)$ is an irreducible polynomial in ${\mathbb{Z}}[q]$ and
$\Phi_{r}(q)$ is the cyclotomic polynomial of order $r$.

In other words, when $m$ is an integer different from $-1$, the polynomial
$\mathit{NH}_{d}(m;q)$ in $q$ only has the factorization dictated by the
generic factorization of $\mathit{NH}_{d}(t;q)$. If $m=-1$, then
$\mathit{NH}_{d}(m;q)$ has lots of cyclotomic factors of odd degrees, but
nothing else.

##  A. Appendix: Proofs in the Gröbner theory for monomial subrings

We prove the statements in §3 in the setting where $\Omega$ is a power series
ring over a field $k$. Note that the lower monomials are leading, i.e.,
$<\,=\,\prec$. Recall that $R\subseteq\Omega$ is a monomial subring, and
$F=Ru_{1}\oplus\dots\oplus Ru_{d}$ is a free $R$-module. We have a monomial
order $<$ on $F$.

######  Proposition A.1 (Proposition • ‣ 3.3).

Let $f,g_{1},\dots,g_{h}$ be elements of $F$. Then there is a (not necessarily
unique) expression

$f=r+\sum_{i=1}^{h}q_{i}g_{i}$ (A.1)

with $r\in F,q_{i}\in R$, such that

  * •

No term of $r$ is divisible by $\operatorname{LT}(g_{i})$ for any $i$.

  * •

$\operatorname{LT}(q_{i}g_{i})\succeq\operatorname{LT}(f)$ for any $i$,
namely, no $q_{i}g_{i}$ contains terms before (i.e., $\prec$)
$\operatorname{LT}(f)$.

###### Proof.

Assume $f,g_{1},\dots,g_{h}\in F-\\{0\\}$ without loss of generality. If no
term of $f$ is divisible by any $\operatorname{LT}(g_{i})$, when we are done
by taking $r=f,q_{i}=0$. Otherwise, we set $r=r^{(1)}=f,q_{i}=q_{i}^{(1)}=0$
to initialize, and we will modify the expression $f=r+\sum q_{i}g_{i}$ step by
step. The idea is to choose a $g_{j}$ and kill terms of $r$ _simultaneously_
using $\operatorname{LT}(g_{j})$, and we cycle around so that each $j$ will be
chosen infinitely often. In step $l$, let $j=j_{l}$ be such that $j\equiv
l\mod h$. Let $s$ denote the sum of all terms divisible by
$\operatorname{LT}(g_{j})$. We modify $q_{i}$ and $r$ according to

$\displaystyle q_{j}^{(l+1)}$
$\displaystyle=q_{i}^{(l)}+\frac{s}{\operatorname{LT}(g_{j})},$ (A.2)
$\displaystyle q_{i}^{(l+1)}$ $\displaystyle=q_{i}^{(l)},\quad i\neq j,$
$\displaystyle r^{(l+1)}$
$\displaystyle=r^{(l)}-\frac{s}{\operatorname{LT}(g_{j})}g_{j}.$

The construction preserves the property $f=r+\sum_{i=1}^{h}q_{i}g_{i}$ and
$\operatorname{LT}(q_{i}g_{i})\succeq\operatorname{LT}(f)$. If $s\neq 0$,
rewriting

$r^{(l+1)}=r^{(l)}-\frac{s}{\operatorname{LT}(g_{j})}\operatorname{LT}(g_{j})-\frac{s}{\operatorname{LT}(g_{j})}(g_{j}-\operatorname{LT}(g_{j})),$
(A.3)

we may interpret the process to go from $r^{(l)}$ to $r^{(l+1)}$ as, first
_kill_ the terms of $s$, and then _reintroduce_ some terms with
$-\frac{s}{\operatorname{LT}(g_{j})}(g_{j}-\operatorname{LT}(g_{j}))$. Some
reintroduced monomials may be equal to monomials appearing in $s$, but we
always have

$\operatorname{LT}\left\lparen-\frac{s}{\operatorname{LT}(g_{j})}(g_{j}-\operatorname{LT}(g_{j}))\right\rparen\succ\operatorname{LT}(s),$
(A.4)

by the defining property of a monomial order. Therefore, the term
$\operatorname{LT}(s)$ is killed but not reintroduced.

We claim that $q_{i}^{(l)}$ and $r^{(l)}$ converge (in the usual power series
topology in $\Omega$ or $\Omega^{d}$) as $l\to\infty$. If not, then there
exists a monomial $\mu\in F$ that is reintroduced (i.e., appears in
$-\frac{s}{\operatorname{LT}(g_{j})}(g_{j}-\operatorname{LT}(g_{j}))$)
infinitely many times. We may assume $\mu$ is the lowest with such property,
by well ordering. Reintroduction of $\mu$ must result from killing a term
involving a monomial $\nu$ of $r^{(l)}$ such that

$\mu=\frac{\nu}{\operatorname{LT}(g_{j})}\lambda,$ (A.5)

where $\lambda$ is a nonleading term of $\operatorname{LT}(g_{j})$. In
particular, $\nu$ precedes $\mu$, and $\nu$ equals
$\frac{\mu}{\lambda}\operatorname{LT}(g_{j})$. Let $u_{i_{0}}$ be the basis
vector involved in $\mu$. Then $\nu$ divides
$\frac{\mu}{u_{i_{0}}}\operatorname{LT}(g_{j})$. We note that there are only
finitely many monomials that divide a given monomial, so $\nu$ must belong to
a finite list of monomials, namely, monomials dividing one of
$\frac{\mu}{u_{i_{0}}}\operatorname{LT}(g_{1}),\dots,\frac{\mu}{u_{i_{0}}}\operatorname{LT}(g_{h})$.
In particular, there is a monomial $\nu\prec\mu$ that occurs in $r^{(l)}$ for
infinitely many $l$’s. By construction, every time $\nu$ occurs, it will be
killed immediately, so in order for $\nu$ to appear in $r^{(l)}$ infinitely
many times, $\nu$ must be reintroduced infinitely many times. Since $\nu$ is
lower than $\mu$, this contradicts the minimality of $\mu$.

Now, let the limits of $q_{i}^{(l)}$ and $r^{(l)}$ be $q_{i}^{(\infty)}$ and
$r^{(\infty)}$ respectively. Since $R$ is closed in $\Omega$, we have
$q_{i}^{(\infty)}\in R$ and $r^{(\infty)}\in F$. To show that
$f=r^{(\infty)}+\sum_{i=1}^{h}q_{i}^{(\infty)}g_{i}$ is a desired division
expression, it suffices to show that $r^{(\infty)}$ contains no term divisible
by any $\operatorname{LT}(g_{i})$. Suppose a monomial $\mu$ involved in
$r^{(\infty)}$ is divisible by $\operatorname{LT}(g_{j})$. In the above
discussion, we see that $\mu$ is only reintroduced finitely many times. After
the last time $\mu$ is reintroduced (if any), it will be killed when we deal
with $g_{j}$ next time (which will happen since we cycle through
$g_{1},\dots,g_{h}$), if not earlier, and $\mu$ will be never introduced
again. Hence $\mu$ does not appear in $r^{(\infty)}$, a contradiction. ∎

######  Lemma A.2 (Lemma 3.6).

Let $f,g_{1},\dots,g_{h}\in F$. If $G=\\{g_{1},\dots,g_{h}\\}$ is a Gröbner
basis, then the remainder of $f$ in a division expression by $G$ is unique
(even though the division expressions may not be unique). Moreover, the
remainder of $f$ by $G$ is zero if and only if $f\in(g_{1},\dots,g_{h})$.

###### Proof.

Let $r$ and $r^{\prime}$ be two possible remainders of $f$ divided by $G$.
Then $r-r^{\prime}$ lies in the submodule $M=(g_{1},\dots,g_{h})$. Moreover,
since no term of $r$ or $r^{\prime}$ is divisible by any of
$\operatorname{LT}(g_{i})$, no term of $r-r^{\prime}$ is divisible by any of
$\operatorname{LT}(g_{i})$. If $r-r^{\prime}$ is nonzero, then it has a
leading term $\operatorname{LT}(r-r^{\prime})$, which lies in
$\operatorname{LT}(M)$. Since $g_{1},\dots,g_{h}$ form a Gröbner basis,
$\operatorname{LT}(M)=(\operatorname{LT}(g_{1}),\dots,\operatorname{LT}(g_{h}))$.
As a property of monomial submodules, $\operatorname{LT}(r-r^{\prime})$ must
be divisible by one of $\operatorname{LT}(g_{i})$, a contradiction. Hence
$r-r^{\prime}=0$, proving the uniqueness of the remainder.

In the second assertion, we only need to prove the “if” direction. Let $f\in
M=(g_{1},\dots,g_{h})S$, and let $r$ be the remainder of $f$ divided by $G$.
Since $f\in M$, we have $r\in M$, so we may repeat the same argument above,
with $r-r^{\prime}$ replaced by $r$, and conclude that $r=0$. ∎

######  Lemma A.3.

Let $M\subseteq F$ be a submodule, and $g_{1},\dots,g_{h}\in M$. If
$\operatorname{LT}(M)=(\operatorname{LT}(g_{1}),\dots,\operatorname{LT}(g_{h}))$,
then we must have $M=(g_{1},\dots,g_{h})$, so that $\\{g_{1},\dots,g_{h}\\}$
is a Gröbner basis for $M$.

###### Proof.

Let $f\in M$, and we shall show that $f\in(g_{1},\dots,g_{h})$. Let

$f=\sum_{i=1}^{h}q_{i}g_{i}+r.$ (A.6)

be a division expression of $f$ divided by $g_{1},\dots,g_{h}$. It suffices to
show that $r=0$. If not, then $r$ has an initial term $\operatorname{LT}(r)$.
Since $r\in M$, we have
$\operatorname{LT}(r)\in\operatorname{LT}(M)=(\operatorname{LT}(g_{1}),\dots,\operatorname{LT}(g_{h}))$
by the definition of $\operatorname{LT}(M)$. As a result,
$\operatorname{LT}(r)$ is divisible by one of $\operatorname{LT}(g_{i})$, a
contradiction to the requirement of a division expression. ∎

######  Proposition A.4 (Proposition 3.7).

Every submodule $M$ of $F$ has a unique reduced Gröbner basis.

###### Proof.

Let $C(\operatorname{LT}(M))=\\{\mu_{1},\dots,\mu_{h}\\}$ be the set of
corners of $\operatorname{LT}(M)$.

We first prove the existence of a reduced Gröbner basis for $M$. By the
definition of $\operatorname{LT}(M)$, there are $g_{1},\dots,g_{h}\in M$ such
that $\operatorname{LT}(g_{i})=\mu_{i}$. By Lemma A.3, $M$ is generated by
$G:=\\{g_{1},\dots,g_{h}\\}$ and $G$ is a Gröbner basis for $M$.

We now modify $G$. For each $i$, divide $\mu_{i}$ by $G$, and call the
(unique) remainder $r_{i}$. Define

$g_{i}^{\prime}:=\mu_{i}-r_{i};$ (A.7)

note from the division expression that $g_{i}^{\prime}$ lies in $M$.

We claim that $g_{1}^{\prime},\dots,g_{h}^{\prime}$ form a reduced Gröbner
basis for $M$. Fix $i$. From the division algorithm, we observe that (1) no
term of $r_{i}$ is divisible by any $\mu_{j}$, and (2) no term of $r_{i}$ is
before $\mu_{i}$. In particular, $\mu_{i}$ is not involved in any term of
$r_{i}$, so every term of $r_{i}$ is strictly behind $\mu_{i}$. Therefore, we
have $\operatorname{LT}(g_{i}^{\prime})=\mu_{i}$, so
$g_{1}^{\prime},\dots,g_{h}^{\prime}$ form a Gröbner basis for $M$. It remains
to check the requirement (c) of Definition 3.4, namely, non nonleading term of
$g_{i}^{\prime}$ is divisible by any $\mu_{j}$. But this is just the
observation (1).

Finally, we prove the uniqueness of the reduced Gröbner basis. Say
$g_{1},\dots,g_{h}$ and $g_{1}^{\prime},\dots,g_{h}^{\prime}$ are two reduced
Gröbner bases for $M$ such that
$\operatorname{LT}(g_{i})=\operatorname{LT}(g_{i}^{\prime})=\mu_{i}$ for all
$i$. Fix $i$, and we claim that $g_{i}^{\prime}=g_{i}$. Consider
$r=g_{i}-g_{i}^{\prime}$. The leading terms of $g_{i}$ and $g_{i}^{\prime}$
cancel each other, and the nonleading terms of $g_{i}$ and $g_{i}^{\prime}$
are not divisible by any $\mu_{j}$ (by the requirement (c) of a reduced
Gröbner basis), so no term of $r$ is divisible by any $\mu_{j}$. If $r\neq 0$,
noting that $r$ is in $M$, we must have
$\operatorname{LT}(r)\in\operatorname{LT}(M)$, so $r$ has a term
$\operatorname{LT}(r)$ that is divisible by some $\mu_{j}$, a contradiction.
Hence $r=0$ and the proof is complete. ∎

######  Lemma A.5 (Lemma 3.1).

Let $M$ be a submodule of $F$. Then there is an isomorphism of $k$-vector
spaces

$\frac{F}{M}\cong\frac{F}{\operatorname{LT}(M)}.$ (A.8)

In fact, the set of monomials outside $\operatorname{LT}(M)$ is a $k$-basis
for $F/M$.

###### Proof.

Let $G=\\{g_{1},\dots,g_{h}\\}$ be a Gröbner basis for $M$, with
$\operatorname{LT}(g_{i})$ being monomials $\mu_{i}$. Let $\Delta$ be the set
of monomials of $F$ outside $\operatorname{LT}(M)$, which is equal to the set
of monomials not divisible by any $\mu_{i}$. Consider the map

$\displaystyle r_{G}:F$ $\displaystyle\to\operatorname{span}_{k}\Delta,$ (A.9)
$\displaystyle f$ $\displaystyle\mapsto(f\bmod G),$

where $(f\bmod G)$ denotes the remainder of $f$ divided by $G$, which is
unique by Lemma A.2. We claim that $r_{G}$ is $k$-linear and surjective with
kernel $M$, thus inducing a $k$-isomorphism

$\frac{F}{M}\overset{r_{G}}{\cong}\operatorname{span}_{k}\Delta=\frac{F}{\operatorname{LT}(M)}.$
(A.10)

By the second assertion of Lemma A.2, the kernel of $r_{G}$ is $M$. To prove
$r_{G}$ is surjective, let $f\in\operatorname{span}_{k}\Delta$. Then the
expression $f=f+\sum 0\cdot g_{i}$ is itself a division expression, so
$r_{G}(f)=f$.

It remains to show that $r_{G}$ is $k$-linear. It is obvious that
$r_{G}(cf)=cr_{G}(f)$ for any scalar $c\in k$. We shall show
$r_{G}(x)+r_{G}(y)+r_{G}(z)=0$ whenever $x+y+z=0$.

Without loss of generality, assume $x,y,z\neq 0$ and
$\operatorname{LT}(z)\preceq\operatorname{LT}(x),\operatorname{LT}(y)$. Let

$x=r_{G}(x)+\sum_{i=1}^{h}q^{x}_{i}g_{i}$ (A.11)

and

$y=r_{G}(y)+\sum_{i=1}^{h}q^{y}_{i}g_{i}$ (A.12)

be division expressions of $x$ and $y$ with repsect to $G$. Since
$\operatorname{LT}(q^{x}_{i}g_{i})\succeq\operatorname{LT}(x)\succeq\operatorname{LT}(z)$
and
$\operatorname{LT}(q^{y}_{i}g_{i})\succeq\operatorname{LT}(y)\succeq\operatorname{LT}(z)$,
the following expression

$-z=r_{G}(x)+r_{G}(y)+\sum_{i=1}^{h}(q^{x}_{i}+q^{y}_{i})g_{i}$ (A.13)

is a division expression of $z$ with respect to $G$. As a result,
$r_{G}(-z)=r_{G}(x)+r_{G}(y)$, so $r_{G}$ is $k$-linear. ∎

######  Lemma A.6 (Lemma 3.13).

Let
$G=\\{g_{1},\dots,g_{h}\\},G^{\prime}=\\{g_{1}^{\prime},\dots,g_{h}^{\prime}\\}$
be Gröbner bases for $M$ with
$\operatorname{LT}(g_{i})=\operatorname{LT}(g_{i}^{\prime})=\mu_{i}$, and
assume $G^{\prime}$ is reduced. If $g_{1}-\mu_{1}$ has no terms divisible by
any $\mu_{i}$, then we must have $g_{1}=g_{1}^{\prime}$.

###### Proof.

We use the algorithm (A.7) to reduce $G$ to a reduced Gröbner basis. In
particular, $g_{1}^{\prime}=\mu_{1}-r_{1}$, where $r_{1}$ is the remainder of
$\mu_{1}$ divided by $G$. However, if we let $r=\mu_{1}-g_{1}$, the expression

$\mu_{1}=1\cdot g_{1}+r$ (A.14)

is a division expression because $\operatorname{LT}(g_{1})=\mu_{1}$ and
$r=\mu_{1}-g_{1}$ has no terms divisible by any $\mu_{i}$ by our hypothesis.
Hence $r_{1}=r$ and $g_{1}^{\prime}=\mu_{1}-r=g_{1}$. ∎

###  A.1. Buchberger criterion

Let $M=(g_{1},\dots,g_{h})\subseteq F$ be a submodule, and assume
$\operatorname{LT}(g_{i})=\mu_{i}$ is a monomial. Denote
$G=\\{g_{1},\dots,g_{h}\\}$. We shall prove Proposition 3.11 that decides when
is $G$ a Gröbner basis. We shall divide it into two independent steps, Lemma
A.9 and Lemma A.10.

Recall that $\operatorname{LCM}(\mu_{i},\mu_{j})$ is the set of minimal common
multiples of $\mu_{i}$ and $\mu_{j}$, and the statement of Proposition 3.11
involves testing all relations of the form

$S^{\nu}_{ij}(g_{i},g_{j}):=\frac{\nu}{\mu_{i}}g_{i}-\frac{\nu}{\mu_{j}}g_{j},$
(A.15)

for all $i,j\in[h]$ and $\mu\in\operatorname{LCM}(\mu_{i},\mu_{j})$.

In fact, we will show that the only feature we need about the collection
$S^{\nu}_{ij}$ is that they generate the “module of relations” among monomials
$\mu_{1},\dots,\mu_{h}$. We now make the notion precise.

######  Definition A.7.

For any finite index set $I$, let $W_{I}:=\bigoplus_{i\in I}Rw_{i}$. We shall
always view elements of $W_{I}$ as an operator: if $f_{i},i\in I$ are elements
of any $R$-module $N$, and $\Theta=\sum_{i\in I}a_{i}w_{i}$ is an element of
$W_{I}$, then we write

$\Theta(f_{i}:i\in I):=\sum_{i\in I}a_{i}f_{i}.$ (A.16)

In particular, we have $\Theta=\Theta(w_{i}:i\in I)$.

If $I=[h]=\\{1,\dots,h\\}$, then we may abbreviate $W_{I}$ as $W_{h}$. If
$I\subseteq J$, then we shall view $W_{I}$ as a submodule of $W_{J}$ via the
natural inclusion sending $w_{i}$ to $w_{i}$.

For monomials $\underline{\mu}=\\{\mu_{i}:i\in I\\}$ in $F$, we define the
module of relations of $\underline{\mu}$ to be

$\operatorname{Rel}(\underline{\mu})=\left\\{\Theta\in
W_{I}:\Theta(\underline{\mu})=0\right\\}.$ (A.17)

We will need a grading on $W_{I}$ in the following proofs.

######  Definition A.8.

Let $\underline{\mu}=\\{\mu_{i}:i\in I\\}$ be monomials in $F$. Consider a
grading on $W_{I}$ with degrees indexed by monomials $F$, defined by

$\deg_{\underline{\mu}}(\tau w_{i}):=\tau\mu_{i}$ (A.18)

for any monomial $\tau\in R$ and index $i\in I$. We say
$\deg_{\underline{\mu}}$ is the grading on $W_{I}$ induced by
$\underline{\mu}$.

We note that the $R$-linear map $W_{I}\to F$,
$\Theta\mapsto\Theta(\underline{\mu})$ preserves the grading, where $F$ is
equipped with the natural grading by monomials in $F$.

We use the notation $[\cdot]_{\nu}$ to extract the degree-$\nu$ part. This
will applies to $W_{I}$ and $F$ graded by monomials in $F$, and $R$ graded by
monomials in $R$.

Technically speaking, the grading is not in the usual sense: given $\Theta$,
there may be infinitely many $\nu$ such that $[\Theta]_{\nu}$ is nonzero, just
like in the case of a power series ring. However, we do have that $\Theta=0$
if and only if $[\Theta]_{\nu}=0$ for all $\nu$.

######  Lemma A.9.

  1. (a)

For monomials $\mu_{1},\mu_{2}$ in $F$, the relation submodule
$\operatorname{Rel}(\mu_{1},\mu_{2})$ is generated by

$S^{\nu}:=\frac{\nu}{\mu_{1}}w_{1}-\frac{\nu}{\mu_{2}}w_{2}\in W_{\\{1,2\\}}$
(A.19)

for all $\nu\in\operatorname{LCM}(\mu_{1},\mu_{2})$.

  2. (b)

Let $\mu_{1},\dots,\mu_{h}$ be monomials in $F$. For each pair $1\leq i<j\leq
h$, let $\left\\{\Theta_{ij}^{\lambda}:\lambda\in\Lambda_{ij}\right\\}$ be a
generating set for $\operatorname{Rel}(\mu_{i},\mu_{j})$ in $W_{\\{i,j\\}}$.
Then the set

$\left\\{\Theta_{ij}^{\lambda}:i<j,\lambda\in\Lambda_{ij}\right\\}$ (A.20)

generates $\operatorname{Rel}(\mu_{1},\dots,\mu_{h})$ in $W_{h}$, where we
view $\Theta_{ij}^{\lambda}$ as an element of $W_{h}$ via the natural
inclusion $W_{\\{i,j\\}}\subseteq W_{h}$.

In consequence, elements of the form

$S_{ij}^{\nu}:=\frac{\nu}{\mu_{i}}w_{i}-\frac{\nu}{\mu_{j}}w_{j}$ (A.21)

for all $i,j\in[h],\nu\in\operatorname{LCM}(\mu_{i},\mu_{j})$ generate the
module of relations $\operatorname{Rel}(\mu_{1},\dots,\mu_{h})$ in $W_{h}$.

###### Proof.

  1. (a)

Let $\operatorname{LCM}(\mu_{1},\mu_{2})=\\{\nu^{1},\dots,\nu^{l}\\}$, and
consider an element $\Theta=a_{1}w_{1}+a_{2}w_{2}$ of
$\operatorname{Rel}(\mu_{1},\mu_{2})$, where $a_{i}\in R$. Thus
$a_{1}\mu_{1}+a_{2}\mu_{2}=0$ by definition. Use the grading on
$W_{\\{1,2\\}}$ induced by $\mu_{1},\mu_{2}$. Let $\nu$ be a monomial in $F$,
then it suffices to show that $[\Theta]_{\nu}$ is generated by
$S^{\nu^{\lambda}}$, $1\leq\lambda\leq l$. There are two cases:

    1. (i)

The monomial $\nu$ is divisible by both $\mu_{1}$ and $\mu_{2}$.

For $i=1,2$, write $\tau_{i}=\nu/\mu_{i}\in R$. Extracting the degree-$\nu$
part of both sides of $a_{1}\mu_{1}+a_{2}\mu_{2}=0$, we have

$[a_{1}]_{\tau_{1}}\mu_{1}+[a_{2}]_{\tau_{2}}\mu_{2}=0.$ (A.22)

Write $[a_{i}]_{\tau_{i}}=c_{i}\tau_{i}$ for $i=1,2$, where $c_{i}\in k$, then
we have

$0=c_{1}\tau_{1}\mu_{1}+c_{2}\tau_{2}\mu_{2}=(c_{1}+c_{2})\mu,$ (A.23)

and hence $c_{2}=-c_{1}$. Therefore,

$[\Theta]_{\nu}=c_{1}\tau_{1}w_{1}+c_{2}\tau_{2}w_{2}=c_{1}(\frac{\nu}{\mu_{1}}w_{1}-\frac{\nu}{\mu_{2}}w_{2}).$
(A.24)

By the definition of $\operatorname{LCM}(\mu_{1},\mu_{2})$, there exists
$\lambda$ such that $\nu^{\lambda}$ divides $\nu$. Thus

$[\Theta]_{\nu}=c_{1}\frac{\nu}{\nu^{\lambda}}\left\lparen\frac{\nu^{\lambda}}{\mu_{1}}w_{1}-\frac{\nu^{\lambda}}{\mu_{2}}w_{2}\right\rparen=a_{1}\frac{\nu}{\nu^{\lambda}}S^{\nu^{\lambda}},$
(A.25)

so one $S^{\nu^{\lambda}}$ is enough to generate $[\Theta]_{\nu}$.

    2. (ii)

The monomial $\nu$ is not divisible by at least one of $\mu_{1}$ or $\mu_{2}$.

In this case, $[\Theta]_{\nu}$ has at most one term, so no cancellation can
occur in the degree-$\nu$ part of $\Theta(\mu_{1},\mu_{2})$. As a result, in
order to have $[\Theta(\mu_{1},\mu_{2})]_{\nu}=0$, we must have
$[\Theta]_{\nu}=0$.

  2. (b)

Use the grading on $W_{h}$ induced by $\mu_{1},\dots,\mu_{h}$. Let $\Theta$ be
an element of $\operatorname{Rel}(\mu_{1},\dots,\mu_{h})$, and let $\nu$ be a
monomial in $F$. We claim that $[\Theta]_{\nu}$ is generated by
$\Theta_{ij}^{\lambda}$. Consider the degree-$\nu$ part of the equation
$\Theta(\mu_{1},\dots,\mu_{h})=0$. Let $A$ be the set of $i$ such that
$\mu_{i}$ divides $\nu$. If $\lvert A\rvert\leq 1$, then no cancellation
occurs in $[\Theta(\mu_{1},\dots,\mu_{h})]_{\nu}$, so we must have
$[\Theta]_{\nu}=0$. From now on, assume without loss of generality that
$A=\\{1,\dots,l\\}$ with $l\geq 2$.

Write $\tau_{i}=\nu/\mu_{i}\in R$, then $[\Theta]_{\nu}$ is of the form

$[\Theta]_{\nu}=c_{1}\tau_{1}w_{1}+\dots+c_{l}\tau_{l}w_{l},$ (A.26)

where $c_{i}\in k$. The equation $[\Theta(\mu_{1},\dots,\mu_{h})]_{\nu}=0$
then implies

$c_{1}\tau_{1}\mu_{1}+\dots+c_{l}\tau_{l}\mu_{l}=(c_{1}+\dots+c_{l})\tau=0,$
(A.27)

and thus $c_{1}+\dots+c_{l}=0$. It follows that

$\displaystyle[\Theta]_{\nu}$
$\displaystyle=-(c_{2}+\dots+c_{l})\tau_{1}w_{1}+c_{2}\tau_{2}w_{2}+\dots+c_{l}\tau_{l}w_{l}$
(A.28)
$\displaystyle=c_{2}(\tau_{2}w_{2}-\tau_{1}w_{1})+\dots+c_{l}(\tau_{l}w_{l}-\tau_{1}w_{1}).$
(A.29)

For $2\leq i\leq l$, we have
$\tau_{i}w_{i}-\tau_{1}w_{1}\in\operatorname{Rel}(\mu_{1},\mu_{i})$, since its
evaluation at the pair $(\mu_{1},\mu_{i})$ is
$\tau_{i}\mu_{i}-\tau_{1}\mu_{1}=\mu-\mu=0$. Because the relations
$\Theta_{1i}^{\lambda}\;(\lambda\in\Lambda_{1i})$ generate
$\operatorname{Rel}(\mu_{1},\mu_{i})$, we conclude that $[\Theta]_{\nu}$ is
generated by the elements $\Theta_{ij}^{\lambda}$ in $W_{h}$.

∎

We now use the above property for the collection $S_{ij}^{\nu}$ to prove the
Buchberger criterion. This is the only step that uses Assumption 3.9 that the
monomial order on $F$ is of order type $\omega$.

######  Lemma A.10.

Assume Assumption 3.9. Let $G=\\{g_{1},\dots,g_{h}\\}$ with $g_{i}\in F$ and
$\operatorname{LT}(g_{i})=\mu_{i}$. Let $\Theta^{\lambda}$,
$\lambda\in\Lambda$ be homogeneous elements of $W_{h}$ (*)with respect to the
grading induced by $\mu_{1},\dots,\mu_{h}$ that generate
$\operatorname{Rel}(\mu_{1},\dots,\mu_{h})$. If for each $\lambda\in\Lambda$,
a possible remainder of $\Theta^{\lambda}(g_{1},\dots,g_{h})$ divided by $G$
is zero, then $G$ is a Gröbner basis.

###### Proof.

Use the grading of $W_{h}$ induced by $\mu_{1},\dots,\mu_{h}$. In this proof,
we need to consider the notion of leading degree for any nonzero element
$\Theta$ of $W_{h}$:

$\deg(\Theta)=\mathrm{min}_{\prec}\\{\nu:[\Theta]_{\nu}\neq 0\\},$ (A.30)

the lowest monomial $\nu$ in $F$ such that $\Theta$ has a degree-$\nu$
component. Note that $\deg(\Theta)$ exists uniquely by well ordering.

Suppose $G$ is not Gröbner, then there exists an element $f$ of
$M=(g_{1},\dots,g_{h})$ such that $\operatorname{LT}(f)$ is not divisible by
any of $\mu_{i}$. We shall fix $f$ and derive a contradiction from here.

Since $f\in M$, there exists $P\in W_{h}$ such that $f=P(g_{1},\dots,g_{h})$.
Let $\nu=\deg P$. We note that $f$ is $[P]_{\nu}(\underline{\mu})$ (which is
of the form $c\nu,c\in k$) plus terms higher than $\nu$. Therefore,
$\operatorname{LM}(f)\succeq\nu$, and the inequality is strict if and only if
$[P]_{\nu}(\underline{\mu})=0$. On the other hand, we note that
$\operatorname{LM}(f)$ cannot be $\nu$, because $\nu$ must be divisible by one
of $\mu_{i}$ by the definition of the grading induced by $\underline{\mu}$,
while $\operatorname{LT}(f)$ is not. Hence $\operatorname{LM}(f)\succ\nu$, so
that $[P]_{\nu}(\underline{\mu})=0$.

We have concluded that any $P\in W_{h}$ such that $f=P(g_{1},\dots,g_{h})$
must satisfy $\deg P\prec\operatorname{LM}(f)$. By Assumption 3.9 that $\prec$
is of order type $\omega$, there are only finitely many monomials below
$\operatorname{LM}(f)$. Hence, we may assume $P$ is chosen such that $\nu=\deg
P$ is maximized.

We claim that there is $P^{\prime}\in W_{h}$ such that
$f=P^{\prime}(g_{1},\dots,g_{h})$ and $\deg P^{\prime}\succ\nu$, which yields
a contradiction. To prove the claim, recall that
$[P]_{\nu}(\underline{\mu})=0$, so $[P]_{\nu}$ lies in the module of relations
$\operatorname{Rel}(\mu_{1},\dots,\mu_{h})$. By our hypothesis, there exists
elements $a^{\lambda}\in R$ such that

$[P]_{\nu}=\sum_{\lambda\in\Lambda}a^{\lambda}\Theta^{\lambda}.$ (A.31)

Since $\Theta^{\lambda}$ is homogeneous (say, of degree $\nu^{\lambda}$), we
may assume $a^{\lambda}$ is homogeneous of degree $\nu/\nu^{\lambda}$ by
taking the degree-$\nu$ part of the right-hand side of (A.31). In other words,
we may assume $a^{\lambda}$ is of the form $c^{\lambda}\nu/\nu^{\lambda}$,
where $c^{\lambda}\in k$. (Here, we have assumed without loss of generality
that $\nu^{\lambda}$ divides $\nu$ for every $\lambda$ in the sum.)

By our hypothesis and the defining requirement of a division expression, for
each $\lambda$, there exists $Q^{\lambda}\in W_{h}$ such that

$Q^{\lambda}(g_{1},\dots,g_{h})=\Theta^{\lambda}(g_{1},\dots,g_{h})$ (A.32)

and $\deg
Q^{\lambda}\succeq\operatorname{LM}(\Theta^{\lambda}(g_{1},\dots,g_{h}))$. The
homogeneity of $\Theta^{\lambda}$ of degree $\nu^{\lambda}$ implies that

$\Theta^{\lambda}(g_{1},\dots,g_{h})=\Theta^{\lambda}(\underline{\mu})+(\text{terms
strictly higher than $\nu^{\lambda}$}).$ (A.33)

Since $\Theta^{\lambda}\in\operatorname{Rel}(\underline{\mu})$, we have
$\Theta^{\lambda}(\underline{\mu})=0$, so
$\operatorname{LM}(\Theta^{\lambda}(g_{1},\dots,g_{h}))\succ\nu^{\lambda}$.
Therefore, we reach an important conclusion

$\deg Q^{\lambda}\succ\nu^{\lambda}=\deg\Theta^{\lambda}.$ (A.34)

This allows us to replace the “leading terms” of $P$ by strictly higher terms,
which is what we want. To make it precise, let

$P^{\prime}=P+\sum_{\lambda\in\Lambda}a^{\lambda}\left\lparen
Q^{\lambda}-\Theta^{\lambda}\right\rparen.$ (A.35)

Then we have $P^{\prime}(g_{1},\dots,g_{h})=P(g_{1},\dots,g_{h})$. Moreover,
the terms $-\sum_{\lambda\in\Lambda}a^{\lambda}\Theta^{\lambda}$ precisely
cancel the degree-$\nu$ part of $P$, while the terms
$+\sum_{\lambda\in\Lambda}a^{\lambda}Q^{\lambda}$ only involve degrees higher
than $\nu$, because

$\deg(a^{\lambda}Q^{\lambda})\succ\deg(a^{\lambda}\nu^{\lambda})=\nu.$ (A.36)

It follows that $\deg P^{\prime}\succ\deg P$, so $P^{\prime}$ is constructed
as desired in the claim. We thus finish the proof of Lemma A.10. ∎

Combining Lemma A.9 and Lemma A.10, we complete the proof of the “if” part of
the Buchberger criterion (Proposition 3.11). The “only if” part is in fact
elementary: if $G=\\{g_{1},\dots,g_{h}\\}$ is a Gröbner basis for $M$, then
$S^{\nu}_{ij}(g_{i},g_{j})$ is in $M$, so its remainder in every division
expression by $G$ is zero by Lemma A.2. Hence the proof of every assertion in
§3 is complete.

##  B. Appendix: motivic considerations

We generalize some of our results to the Grothendieck ring of motives for
algebraic stacks (for an introduction, see [20]). To achieve this, we need
geometric considerations of moduli spaces. However, no much heavy machinery of
algebraic geometry is needed, since to work in the Grothendieck ring of
motives, we only need to care about reduced structures.

This section contains two main topics. The first is a proof of Theorem B.1 (or
Theorem 1.7), which occupies §B.1$\sim$ §B.4. As noted in Remark 2.6, the
proof of Proposition 2.3 does not naïvely generalize to the motivic setting.
In order to prove Theorem B.1, we will need to develop new machinaries. The
second is an exposition of Gröbner strata, which we present in §B.5. Abundant
results on Gröbner strata already exist in literature, for example, see [43].
The results in previous chapters are easily generalizable to the geometric
setting. Furthermore, Lemma 4.4 and Lemma 4.5 can be easily upgraded to their
motivic versions. So we actually get motivic decompositions of
$[\operatorname{Quot}_{d,n}^{d}(R)]$ with respect to Gröbner strata. We refer
the readers to §B.5.2 for the precise statements.

######  Theorem B.1 (A slight modification of Theorem 1.7).

Let $k$ be an arbitrary field and $R$ be a complete local ring of form
$k[[x_{1},...,x_{m}]]/I$. For any $n,d,r$ such that $0\leq
r\leq\min\\{n,d\\}$, the following identity holds in
${\widehat{K}_{0}(\mathrm{Var}_{k})}$:

$[\operatorname{Coh}_{n}^{r}(R)]=\frac{[\operatorname{Quot}_{d,n}^{r}(R)][\operatorname{GL}_{d-r}]}{\mathbb{L}^{d(n-d)+(d-r)^{2}}[\operatorname{GL}_{d}]}.$
(B.1)

The formula (B.1) recovers (1.21) by setting $R=\widehat{\mathcal{O}}_{X,p}$,
and substituting in

$[\operatorname{Gr}(d,r)]=[\operatorname{GL}_{d}]/\mathbb{L}^{r(d-r)}[\operatorname{GL}_{r}][\operatorname{GL}_{d-r}].$
(B.2)

In practice, we will only use the most important special case $r=d$.

####  B.0.1. Notation and assumptions

In order to keep notation clean, we will again use
$\operatorname{Coh}_{n}(R)$, $\operatorname{Coh}_{n}^{r}(R)$,
$\operatorname{Quot}_{d,n}(R)$, $\operatorname{Quot}_{d,n}^{r}(R)$,
$C_{n}(R)$, and $C_{n}^{r}(R)$ to denote the induced reduced structures of the
original schemes and stacks, see §B.1. In the whole chapter, the symbol $S$ is
reserved for denoting a reduced affine $k$-scheme, which will only be used in
the situation where we consider affine scheme points of various schemes and
stacks.

####  B.0.2. Outline of the proof for Theorem B.1

In §B.2.2, we will first observe a group action
$\operatorname{GL}[F]\curvearrowright\operatorname{Quot}_{d,n}^{r}(R)$ and
introduce an important group scheme $\mathscr{G}$ over
$\operatorname{Quot}_{d,n}^{r}(R)$, which is the “stablizer bundle” of the
$\operatorname{GL}[F]$-action. In §B.3, we will construct a Zariski
$\operatorname{GL}_{n}$-torsor ${\mathfrak{Q}}$ over
$\operatorname{Quot}_{d,n}^{r}(R)$ that sits in the following diagram

${{\mathfrak{Q}}}$${{\operatorname{Quot}_{d,n}^{r}(R)}}$${C_{n}^{r}(R)}$$\scriptstyle{\pi}$$\scriptstyle{\varpi}$
(B.3)

We will show that $\varpi$ admits Zariski local sections, i.e., we have a
Zariski cover $U\rightarrow C_{n}^{r}(R)$, and
$\sigma:U\rightarrow\mathfrak{Q}$ such that $\varpi\sigma=\text{Id}_{U}$. We
then show that $\operatorname{GL}[F]_{U}$ is an fppf $\mathscr{G}_{U}$-torsor
over $\mathfrak{Q}_{U}$, where $\mathscr{G}_{U}=\sigma^{*}\pi^{*}\mathscr{G}$
is a group scheme over $U$. We then show in Lemma B.6 that $U$ admits a
Zariski stratification over which $\mathscr{G}_{U}$ is special (in the sense
that any fppf torsor is Zariski locally trivial). We then use explicit
descriptions of $\operatorname{GL}[F]$ and $\mathscr{G}$ from Lemma B.2 and
Lemma B.3 to establish the formula (B.1).

###  B.1. Functor descriptions of several geometric objects

For later reference, we summarize the functor of points for various geometric
objects that we are considering. We remind the reader that $C_{n}(R)$ is the
commuting variety (§2.1). Fix $V_{n}$ to be a vector space of dimension $n$
over $k$. In the following, let $X$ be a reduced $k$-scheme.

$\operatorname{Coh}_{n}(R)(X)=\left\\{\begin{aligned} &\text{groupoid of
coherent sheaves }\mathcal{F}\text{ over }X_{R},\\\ &\text{such that
}\mathcal{F}\text{ is flat of rank }n\text{ over }X.\end{aligned}\right\\}.$
(B.4) $\operatorname{Coh}_{n}^{r}(R)(X)=\left\\{\begin{aligned}
&\text{groupoid of coherent sheaves }\mathcal{F}\text{ over }X_{R},\\\
&\text{such that }\mathcal{F}\text{ is flat of rank }n\text{ over }X\text{,
and}\\\ &\mathcal{F}/\mathfrak{m}\mathcal{F}\text{ is flat of rank }r\text{
over }X.\end{aligned}\right\\}.$ (B.5)
$\operatorname{Quot}_{d,n}(R)(X)=\left\\{N\subseteq\mathcal{O}_{X_{R}}^{d},\text{
such that
}\mathcal{O}_{X_{R}}^{d}/N\in\operatorname{Coh}_{n}(R)(X).\right\\}.$ (B.6)
$\operatorname{Quot}_{d,n}^{r}(R)(X)=\left\\{N\subseteq\mathcal{O}_{X_{R}}^{d},\text{
such that
}\mathcal{O}_{X_{R}}^{d}/N\in\operatorname{Coh}_{n}^{r}(R)(X).\right\\}.$
(B.7) $C_{n}(R)(X)=\left\\{\begin{aligned} &\text{equivalent classes of pairs
}(\mathcal{F},\iota)\text{ where
}\mathcal{F}\in\operatorname{Coh}_{n}(R)(X),\\\ &\text{ and
}\iota:\mathcal{O}_{X}\otimes V_{n}\simeq\mathcal{F}\text{ is an isomorphism
of }\mathcal{O}_{X}\text{-modules}.\end{aligned}\right\\}.$ (B.8)
$C_{n}^{r}(R)(X)=\left\\{\begin{aligned} &\text{equivalent classes of pairs
}(\mathcal{F},\iota)\text{ where
}\mathcal{F}\in\operatorname{Coh}_{n}^{r}(R)(X),\\\ &\text{ and
}\iota:\mathcal{O}_{X}\otimes V_{n}\simeq\mathcal{F}\text{ is an isomorphism
of }\mathcal{O}_{X}\text{-modules}.\end{aligned}\right\\}.$ (B.9)

Note that

$C_{n}^{r}(R)=C_{n}(R)\times_{\operatorname{Coh}_{n}(R)}\operatorname{Coh}_{n}^{r}(R),$
(B.10)

and $\operatorname{Coh}_{n}^{r}(R)=[C_{n}^{r}(R)/\operatorname{GL}_{n}]$.
Furthermore, when $n$ is fixed, there is a large Artin local quotient
$R^{\prime}$ of $R$ such that for all $d$ and $r$,
$\operatorname{Coh}_{n}(R)=\operatorname{Coh}_{n}(R^{\prime})$,
$\operatorname{Coh}_{n}^{r}(R)=\operatorname{Coh}_{n}^{r}(R^{\prime})$,
$\operatorname{Quot}_{d,n}(R)=\operatorname{Quot}_{d,n}(R^{\prime})$,
$\operatorname{Quot}_{d,n}(R)=\operatorname{Quot}_{d,n}(R^{\prime})$,
$\operatorname{Quot}_{d,n}^{r}(R)=\operatorname{Quot}_{d,n}^{r}(R^{\prime})$,
$C_{n}(R)=C_{n}(R^{\prime})$ and $C_{n}^{r}(R)=C_{n}^{r}(R^{\prime})$.

###  B.2. Geometry of $\operatorname{Quot}_{d,n}^{r}(R)$

By the last few sentences of §B.1, we may and will replace $R$ by a large
Artin quotient. Write $\dim_{k}R=b+1$ and
$F={R}^{d}=\bigoplus_{i=1}^{d}Ru_{i}$. If a submodule $N\subseteq F\otimes S$
lies in $\operatorname{Quot}_{d,n}^{r}(R)(S)$, then the quotient $(F\otimes
S)/N$ is a flat $S$-module of rank $n$, and $(\mathfrak{m}F\otimes S)\cap N$
is a flat $S$-module of rank $bd-n+r$, so we obtain a locally closed immersion

$\operatorname{Quot}_{d,n}^{r}(R)\hookrightarrow\operatorname{Gr}(F,bd+d-n)$
(B.11)

sending a submodule $N\in F\otimes S$ to the corresponding module in $F\otimes
S$. Indeed, $\operatorname{Quot}_{d,n}^{r}(R)$ admits a closed embedding into
the locally closed subvariety of $\operatorname{Gr}(F,bd+d-n)$ parametrizing
subspaces whose intersections with $\mathfrak{m}F$ have dimension exactly
$bd-n+r$. Note that when $d=r$, (B.11) is a closed embedding. Let

$0\rightarrow\mathscr{N}\rightarrow\underline{F}\rightarrow\mathscr{Q}\rightarrow
0$ (B.12)

be the restriction of the tautological sequence of universal bundles over
$\operatorname{Gr}(F,bd+d-n)$ to $\operatorname{Quot}_{d,n}^{r}(R)$, i.e.,
over a point $x\in\operatorname{Quot}_{d,n}^{r}(R)(S)$ corresponding to an
$N\subseteq F\otimes S$, the fiber $\mathscr{N}_{x}$ is $N$, while the fiber
of $\mathscr{Q}_{x}$ is $(F\otimes S)/N$. Furthermore, (B.12) is not only a
sequence of $\operatorname{Quot}_{d,n}^{r}(R)$-bundles, but also a sequence of
coherent $\operatorname{Quot}_{d,n}^{r}(R)\times\operatorname{Spec}R$ modules.
There is another related construction:

$\operatorname{Quot}_{d,n}^{r}(R)\rightarrow\operatorname{Quot}_{d,r}^{r}(R)=\operatorname{Gr}(F/\mathfrak{m}F,d-r)$
(B.13)

sending $N\subseteq F\otimes S$ to $N/(\mathfrak{m}F\otimes S)$. It is not
hard to see that this morphism is surjective with pointwise isomorphic fibers,
but we will not use this. The pullback of the tautological sequence of
universal bundles over $\operatorname{Gr}(F/\mathfrak{m}F,d-r)$ to
$\operatorname{Quot}_{d,n}^{r}(R)$ will be denoted by

$0\rightarrow\overline{\mathscr{N}}\rightarrow\underline{F}/\mathfrak{m}\underline{F}\rightarrow\overline{\mathscr{Q}}\rightarrow
0$ (B.14)

It is clear that
$\overline{\mathscr{N}}=\mathscr{N}/\mathfrak{m}\underline{F}$ and
$\overline{\mathscr{Q}}=\mathscr{Q}/\mathfrak{m}\mathscr{Q}$.

####  B.2.1. Group action

The most important feature of $\operatorname{Quot}_{d,n}^{r}(R)$ is that it is
equipped with an algebraic group action. For a finite $R$-module $M$, let
$\operatorname{GL}[M]$ be the group $\operatorname{Aut}_{R}(M)$, but viewed as
a linear algebraic group over $k$. Then $\operatorname{GL}[F]$ acts on
$\operatorname{Quot}_{d,n}^{r}(R)$, in the sense that any
$g\in\operatorname{GL}[F](S)$ acts on $\operatorname{Quot}_{d,n}^{r}(R)(S)$ by
sending a submodule $N\subseteq F\otimes S$ to $g\cdot N$.

There is a projection
$\operatorname{GL}[F]\rightarrow\operatorname{GL}[F/\mathfrak{m}F]$ sending
$g$ to $(g\bmod\mathfrak{m})$. Let $U(\mathfrak{m}F)$ be its kernel. The
canonical action of $\operatorname{GL}[F/\mathfrak{m}F]$ on
$\operatorname{Gr}(F/\mathfrak{m}F,d-r)$ is compatible with the action of
$\operatorname{GL}[F]$ on $\operatorname{Quot}_{d,n}^{r}(R)$ via (B.13). The
structure of $\operatorname{GL}[F]$ is easy to understand:

######  Lemma B.2.

Notation as above, the exact sequence

$1\rightarrow
U[\mathfrak{m}F]\rightarrow\operatorname{GL}[F]\rightarrow\operatorname{GL}[F/\mathfrak{m}F]\rightarrow
1,$ (B.15)

admits a canonical splitting. Furthermore,
$\operatorname{GL}[F/\mathfrak{m}F]\simeq\operatorname{GL}_{d}$ and
$U[\mathfrak{m}F]$ is a $k$-split unipotent group of dimension $bd^{2}$ (a
unipotent group is $k$-split if it is a successive extension of
$\mathbb{G}_{a}$ over $k$).

###### Proof.

It is clear that
$\operatorname{GL}[F/\mathfrak{m}F]\simeq\operatorname{GL}_{d}$. The morphism
$\pi:\operatorname{GL}[F]\rightarrow\operatorname{GL}[F/\mathfrak{m}F]$ has a
section
$\operatorname{GL}[F/\mathfrak{m}F]\hookrightarrow\operatorname{GL}[F]$
induced by the structural morphism $k\hookrightarrow R$. Note that
$U[\mathfrak{m}F](S)$ consists of $g$ of form

$g(u_{i})\in u_{i}+\mathfrak{m}F\otimes S,\,\,1\leq i\leq d.$ (B.16)

Therefore, $U[\mathfrak{m}F]$ is abstractly isomorphic to the affine space
$\mathbb{A}^{bd^{2}}$. It is unipotent since for any $g\in
U[\mathfrak{m}F](S)$, $(g-\text{Id})^{b}=0$. The fact that $U[\mathfrak{m}F]$
is $k$-split follows by induction on $b$ (the $k$-dimension of $R$): the
assertion is trivial for $b=1$, otherwise there is an element $0\neq v\in R$
such that $v\mathfrak{m}=0$. The subgroup $U[vF]\subseteq U[\mathfrak{m}F]$
whose $S$-points are elements $g$ with $g(u_{i})\in u_{i}+vF\otimes S$ lies in
the center of $U[\mathfrak{m}F]$, and is isomorphic to
$\mathbb{G}_{a}^{d^{2}}$. Let $F^{\prime}=(R/v)^{d}$. We have an exact
sequence

$1\rightarrow U[vF]\rightarrow U[\mathfrak{m}F]\rightarrow
U[\mathfrak{m}F^{\prime}]\rightarrow 1.$ (B.17)

By the induction hypothesis, $U[\mathfrak{m}F^{\prime}]$ is $k$-split. It
follows that $U[\mathfrak{m}F]$ is $k$-split. $\hfill\square$

####  B.2.2. Group schemes over $\operatorname{Quot}_{d,n}^{r}(R)$

The constructions in §B.2.1 can be globalized. They are even more important
group schemes over $\operatorname{Quot}_{d,n}^{r}(R)$. Define the following:

  * •

$\operatorname{GL}[\underline{F}]:=\operatorname{Aut}_{R}(\underline{F})$,
$\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]:=\operatorname{Aut}_{R}(\underline{F}/\mathfrak{m}\underline{F})$
and
$\operatorname{GL}[\overline{\mathscr{N}}]:=\operatorname{Aut}_{R}(\overline{\mathscr{N}})$.

  * •

$U(\mathfrak{m}\underline{F}):=\ker(\operatorname{GL}[\underline{F}]\rightarrow\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}])$.

  * •

$\mathscr{G}$ is the subgroup scheme of $\operatorname{GL}[\underline{F}]$
stabilizing $\mathscr{N}$ and reducing to identity on $\mathscr{Q}$.

  * •

$\operatorname{Fil}_{1}{\mathscr{G}}:=\ker(\mathscr{G}\rightarrow\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}])=U(\mathfrak{m}\underline{F})\cap\mathscr{G}$,
i.e., $\operatorname{Fil}_{1}{\mathscr{G}}$ is the subgroup scheme of
$\operatorname{GL}[\underline{F}]$ that reduces to identity on the whole
$\underline{F}/\mathfrak{m}\underline{F}$.

  * •

$\operatorname{Fil}_{2}\mathscr{G}$ is the subgroup scheme of
$\operatorname{GL}[\underline{F}]$ that reduces to identity on both
$\overline{\mathscr{N}}$ and $\overline{\mathscr{Q}}$.

Note that $\operatorname{GL}[\underline{F}]$ resp.
$\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]$ resp.
$U(\mathfrak{m}\underline{F})$ is just a trivial $\operatorname{GL}[F]$ resp.
$\operatorname{GL}[F/\mathfrak{m}F]$ resp. $U[\mathfrak{m}F]$ bundle over
$\operatorname{Quot}_{d,n}^{r}(R)$, and the split exact sequence (B.15) admits
an obvious global analogue. We will call $\mathscr{G}$ the stabilizer bundle,
in the sense that the fiber of $\mathscr{G}$ at an $S$-point is the subgroup
of $\operatorname{GL}[F](S)$ that acts trivially on the corresponding module.
It will play an important role in studying the action of
$\operatorname{GL}[F]$ on $\operatorname{Quot}_{d,n}^{r}(R)$. The group scheme
$\mathscr{G}$ admits a three step filtration of normal subgroups:

$\underline{1}\triangleleft\operatorname{Fil}_{1}\mathscr{G}\triangleleft\operatorname{Fil}_{2}\mathscr{G}\triangleleft\operatorname{Fil}_{3}\mathscr{G}=\mathscr{G}.$
(B.18)

Note that in the special case $d=r$, we have
$\mathscr{G}=\operatorname{Fil}_{2}\mathscr{G}$.

######  Lemma B.3.

The followings are true:

  1. (a)

$\operatorname{Fil}_{1}\mathscr{G}$ is a unipotent group scheme of rank
$d(bd-n+r)$.

  2. (b)

$\operatorname{gr}_{2}\mathscr{G}$ is a subgroup scheme of
$\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]$ which is Zariski
locally isomorphic to $\mathbb{G}_{a}^{r(d-r)}$.999By this we mean that there
is a Zariksi cover $U$ of $\operatorname{Quot}_{d,n}^{r}(R)$ such that
$\operatorname{gr}_{2}\mathscr{G}_{U}\simeq\mathbb{G}_{a,U}^{r(d-r)}$. We
adopt the same convention for (d).

  3. (c)

$\operatorname{Fil}_{2}\mathscr{G}$ is a unipotent group scheme of rank
$bd^{2}-d(n-d)-(d-r)^{2}$.

  4. (d)

$\operatorname{gr}_{3}\mathscr{G}\simeq\operatorname{GL}[\overline{\mathcal{N}}]$.
In particular, it is Zariski locally isomorphic to $\operatorname{GL}_{d-r}$.

###### Proof.

  1. (a)

We first describe the functor of points of
$\operatorname{Fil}_{1}\mathscr{G}$. Let
$x\in\operatorname{Quot}_{d,n}^{r}(R)(S)$ corresponds to a submodule
$N\subseteq F\otimes S$. Then the fiber of $\operatorname{Fil}_{1}\mathscr{G}$
over $x$ is

$\operatorname{Fil}_{1}\mathscr{G}|_{x}=\\{g\in\operatorname{GL}[F](S):\,\,g(u_{i})\in
u_{i}+N\cap(\mathfrak{m}F\otimes S),1\leq i\leq d\\}.$ (B.19)

This already implies that $\operatorname{Fil}_{1}\mathscr{G}$ is unipotent.
Furthermore, it implies that there is a vector bundle structure underlying
$\operatorname{Fil}_{1}\mathscr{G}$, which is isomorphic to
$(\mathscr{N}\cap\mathfrak{m}\underline{F})^{\oplus d}$. The assertion follows
from the fact that
$\operatorname{rk}(\mathscr{N}\cap\mathfrak{m}\underline{F})=bd-n+r$.

  2. (b)

Let $\mathscr{H}_{2}$ be the subgroup scheme of
$\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]$ fixing the
filtration (B.14) and reducing to identity on both $\overline{\mathscr{Q}}$
and $\overline{\mathscr{N}}$. Then
$\operatorname{gr}_{2}\mathscr{G}\subseteq\mathscr{H}_{2}$. The canonical
splitting of (B.15) gives rise to a morphism

$\mathscr{H}_{2}\hookrightarrow\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]\rightarrow\operatorname{GL}[\underline{F}]$
(B.20)

whose image lies in $\operatorname{Fil}_{2}\mathscr{G}$. This implies that
$\operatorname{gr}_{2}\mathscr{G}=\mathscr{H}_{2}$. Let $\operatorname{Spec}S$
be a Zariksi affine open of $\operatorname{Quot}_{d,n}^{r}(R)$ over which
$\overline{\mathscr{Q}}$ and $\overline{\mathscr{N}}$ are trivialized. We can
then pick a section of (B.14) and write
$(\underline{F}/\mathfrak{m}\underline{F})_{S}=\overline{\mathscr{N}}_{S}\oplus\overline{\mathscr{Q}}_{S}$.
Then the $S$-points of $\mathscr{H}_{2}|_{S}$ can be written as matrices

$\begin{bmatrix}\text{Id}_{\overline{\mathscr{N}}_{S}}&B\\\
0&\text{Id}_{\overline{\mathscr{Q}}_{S}}\end{bmatrix},\,\,B\in\mathcal{H}om(\overline{\mathscr{Q}},\overline{\mathscr{N}})(S).$
(B.21)

This particularly implies that
$\operatorname{gr}_{2}\mathscr{G}|_{S}=\mathbb{G}_{a,S}^{r(d-r)}$.

  3. (c)

This follows from (a) and (b).

  4. (d)

Let $\mathscr{H}$ be the subgroup scheme of
$\operatorname{GL}[\underline{F}/\mathfrak{m}\underline{F}]$ fixing the
filtration (B.14), and reducing to identity on $\overline{\mathscr{Q}}$. We
have $\mathscr{G}/\operatorname{Fil}_{1}\mathscr{G}\subseteq\mathscr{H}$. As
in (b), the canonical splitting of (B.15) again induces a morphism
$\mathscr{H}\rightarrow\operatorname{GL}[\underline{F}]$ whose image lies in
$\mathscr{G}$. This implies that
$\mathscr{G}/\operatorname{Fil}_{1}\mathscr{G}=\mathscr{H}$ and
$\operatorname{gr}_{3}\mathscr{G}=\mathscr{H}/\mathscr{H}_{2}$.

Let $\operatorname{Spec}S$ be a Zariksi affine open of
$\operatorname{Quot}_{d,n}^{r}(R)$ over which $\overline{\mathscr{Q}}$ and
$\overline{\mathscr{N}}$ are trivialized. Pick the same splitting of (B.14) as
in (b). Then the $S$-points of $\mathscr{H}|_{S}$ can be written into matrices

$\begin{bmatrix}A&B\\\
0&\text{Id}_{\overline{\mathscr{Q}}_{S}}\end{bmatrix},\,\,A\in\operatorname{GL}[\overline{\mathscr{N}}](S),\,\,B\in\mathcal{H}om(\overline{\mathscr{Q}},\overline{\mathscr{N}})(S).$
(B.22)

The assertion that
$\operatorname{gr}_{3}\mathscr{G}\simeq\operatorname{GL}[\overline{\mathcal{N}}]$
is clear if one takes quotient of this by (B.21).$\hfill\square$

###  B.3. Relating $\operatorname{Quot}_{d,n}^{r}(R)$ and $C_{n}^{r}(R)$

We start by constructing ${\mathfrak{Q}}$ in (B.3) as the
$\operatorname{GL}_{n}$-torsor corresponding to the bundle $\mathscr{Q}$. More
precisely, let $V_{n}$ be a fixed $k$-vector space of dimension $n$, and
$\underline{V_{n}}$ be the corresponding trivial bundle over
$\operatorname{Quot}_{d,n}^{r}(R)$, then

${\mathfrak{Q}}=\mathcal{I}som_{\operatorname{Quot}_{d,n}^{r}(R)}(\underline{V_{n}},{\mathscr{Q}}).$
(B.23)

Therefore we have a projection
$\pi:{\mathfrak{Q}}\rightarrow\operatorname{Quot}_{d,n}^{r}(R)$. On the other
hand, take $X={\mathfrak{Q}}$ and $\mathcal{F}=\pi^{*}{\mathscr{Q}}$ in (B.9),
since $\pi^{*}{\mathfrak{Q}}\rightarrow{\mathfrak{Q}}$ is a trivial torsor
(e.g., there is a global section which comes from the diagonal),
$\mathcal{F}\rightarrow{\mathfrak{Q}}$ is a trivial bundle, and is an object
in $\operatorname{Coh}_{n}^{r}(R)(X)$. Let

$\iota:\mathcal{O}_{X}\otimes_{k}V_{n}\simeq\mathcal{F}$ (B.24)

be the isomorphism of $\mathcal{O}_{X}$-modules induced by the evaluation map

$\underline{V_{n}}\times_{\operatorname{Quot}_{d,n}^{r}(R)}{\mathfrak{Q}}\rightarrow{\mathscr{Q}}.$
(B.25)

The pair $(\mathcal{F},\iota)$ gives rise to a morphism
$\varpi:{\mathfrak{Q}}\rightarrow C_{n}^{r}(R)$.

On the level of $S$-points, $\varpi$ can be easily understood as follows: let
$x$ be an $S$-point of ${\mathfrak{Q}}$ corresponding to a module $N\subseteq
F\otimes S$ with an $S$-isomorphism $\iota:V_{n}\otimes S\simeq(F\otimes
S)/N$, then $\varpi(x)=((F\otimes S)/N,\iota)\in C_{n}^{r}(R)(S)$. It is
easily seen that $\varpi$ is surjective, and is invariant under
$\operatorname{GL}[F]$-action. Indeed, let $g\in\operatorname{GL}[F](S)$, and
$x=(N,\iota)$ as above, we have $\varpi g(x)=((F\otimes
S)/gN,g\iota)\sim((F\otimes S)/N,\iota)$. This implies that $\varpi
g(x)=\varpi(x)$.

####  B.3.1. Local sections

There is a Zariski affine cover $U=\operatorname{Spec}D\rightarrow
C_{n}^{r}(R)$, such that the tautological point of $C_{n}^{r}(R)(U)$
corresponds to (the equivalent class of) a pair

$((F\otimes D)/N^{\text{taut}},\iota^{\text{taut}}).$ (B.26)

Fixing a pair (B.26) in the equivalence class, we get a $U$-point $\sigma$ of
${\mathfrak{Q}}$ such that $\varpi\sigma=\text{Id}_{U}$. Therefore, $\sigma$
is a section of $\varpi$ over $U$. Note that such sections are well defined up
to an action of $\operatorname{GL}[F](U)$. Let
${\mathfrak{Q}}_{U}={\mathfrak{Q}}\times_{C_{n}^{r}(R)}{U}$. Since $\varpi$ is
invariant under $\operatorname{GL}[F]$, the section $\sigma$ induces a
morphism of $U$-schemes

$p:\operatorname{GL}[{F}]_{U}=U\times\operatorname{GL}[F]\xrightarrow{\sigma\times\text{Id}}{\mathfrak{Q}}\times\operatorname{GL}[F]\xrightarrow{\cdot}{\mathfrak{Q}}.$
(B.27)

######  Lemma B.4.

Let $\mathscr{G}$ be the stabilizer bundle over
$\operatorname{Quot}_{d,n}^{r}(R)$ (§B.2.2) and let
$\mathscr{G}_{U}:=\sigma^{*}\pi^{*}\mathscr{G}$ be the pullback group scheme
over $U$. Then $p$ realizes ${\mathfrak{Q}}$ as the fppf quotient of
$\operatorname{GL}[{F}]_{U}$ by the fppf subgroup $\mathscr{G}_{U}$. In
particular, we have:

  1. (a)

$p$ is an fppf cover.

  2. (b)

$\operatorname{GL}[{F}]_{U}$ is an fppf $\mathscr{G}_{U}$-torsor over
${\mathfrak{Q}}_{U}$, trivialized under the covering $p$.

###### Proof.

First we check that ${\mathfrak{Q}}_{U}$ is the fppf quotient. Let $S$ be an
fppf affine over $U$ and $x$ be an $S$-point of $\sigma U$, corresponding to a
module $N\subseteq F\otimes S$ with an $S$-isomorphism $\iota:V_{n}\otimes
S\simeq(F\otimes S)/N$. For any $g\in\operatorname{GL}[F](S)$, we have
$p(x,g)=((F\otimes S)/gN,g\iota)$. Since $p(x,g)=p(x,g^{\prime})$ if and only
if $g^{\prime}g^{-1}$ is in the stabilizer of $x$, i.e., $g^{\prime}g^{-1}$ is
in the fiber of $\mathscr{G}_{S}$ over $x$, we obtain the fppf quotient
assertion. Once this is established, (a) and (b) are just formal properties of
fppf quotient, see [19, Proposition 4.31]. $\hfill\square$

###  B.4. The motivic formula

We say that a unipotent group scheme over a base $Y$ is $Y$-split, or simply
split, if it is a successive extension of $\mathbb{G}_{a,Y}$. We say a group
scheme $\mathcal{G}$ over $Y$ is special, if any fppf $\mathcal{G}$-torsor is
Zariski locally trivial. It is well-known that a split unipotent group scheme
is special, and an extension of special group schemes is special.

######  Lemma B.5.

Notation as in §B.2.2. There is a Zariski stratification

$\operatorname{Quot}_{d,n}^{r}(R)=\bigsqcup_{\beta\in\mathbf{B}}Z_{\beta},$
(B.28)

such that for each $\beta$, $\operatorname{Fil}_{2}\mathscr{G}_{Z_{\beta}}$ is
a split $Z_{\beta}$-unipotent group scheme.

###### Proof.

Note that it suffices to show that
$\operatorname{Fil}_{1}\mathscr{G}_{Z_{\beta}}$ is split for some Zariski
stratification $\\{Z_{\beta}\\}$, thanks to Lemma B.3(b). Take a nonempty
affine open $\operatorname{Spec}S\subseteq\operatorname{Quot}_{d,n}^{r}(R)$
and view it as an $S$-point of $\operatorname{Quot}_{d,n}^{r}(R)$. It then
corresponds to a submodule $N\subseteq F\otimes S$. Shrinking $S$, we can
assume that $N^{\prime}=N\cap(\mathfrak{m}F\otimes S)$ is free. The group
scheme $\operatorname{Fil}_{1}\mathscr{G}_{S}$ can be described as the functor

$T/S\rightarrow\\{g\in U[\mathfrak{m}F](T)|g(u_{i})\in u_{i}+N^{\prime}\otimes
T,1\leq i\leq d\\}.$ (B.29)

Let $\\{\omega_{i}\\}_{i=1}^{bd+d-n}$ be a basis of $N^{\prime}$. There is a
nonzero element $a\in N^{\prime}$ such that $\mathfrak{m}a=0$. Indeed, we know
that for any element $b\in N^{\prime}$, $\mathfrak{m}^{n}b=0$ for sufficiently
large $n$. So multiply $b$ by some suitable element $u\in\mathfrak{m}$, we
have $ub\neq 0$ but $\mathfrak{m}ub=0$. Let $a=ub$ and express
$a=\sum_{i}s_{i}\omega_{i}$. There is at least one $s_{i}$, say $s_{1}$, which
is nonzero. Since $S$ is reduced (recall that we have assumed everything is
reduced in §B.0.1), the localization $S[s_{1}^{-1}]$ is nonzero. Therefore,
$\\{a\\}\cup\\{\omega_{i}\\}_{i=2}^{bd+d-n}$ is a basis of
$N^{\prime}[s_{1}^{-1}]$. Therefore, after shrinking $S$, we can assume that
$N^{\prime}$ admits a basis $\\{\omega_{i}\\}_{i=1}^{bd+d-n}$ such that
$\mathfrak{m}\omega_{1}=0$.

Replace $N^{\prime}$ by $N^{\prime}/\omega_{1}$ and do the same trick. By
induction, we see that after shrinking $S$ sufficiently many times, there is a
basis $\\{\omega_{i}\\}_{i=1}^{bd+d-n}$ of $N^{\prime}$ such that
$\mathfrak{m}\omega_{j}\subseteq\operatorname{span}\\{\omega_{1},...,\omega_{j-1}\\}$.
Define a central series of unipotent group schemes

$0\triangleleft\mathscr{U}_{1,S}\triangleleft\mathscr{U}_{2,S}\triangleleft...\triangleleft\mathscr{U}_{bd-n+r,S}=\operatorname{Fil}_{1}\mathscr{G}_{S}$
(B.30)

by setting the functors of points as

$\mathscr{U}_{j,S}:T/S\rightarrow\\{g\in U[\mathfrak{m}F](T)|g(u_{i})\in
u_{i}+\operatorname{span}_{S}\\{\omega_{1},...,\omega_{j}\\}\otimes_{S}T,1\leq
i\leq d\\}.$ (B.31)

It follows easily from the construction that
$\mathscr{U}_{j,S}/\mathscr{U}_{j,S}\simeq\mathbb{G}_{a,S}^{d}$. Therefore
$\operatorname{Fil}_{1}\mathscr{G}_{S}$ is $S$-split.

Replace $\operatorname{Quot}_{d,n}^{r}(R)$ by the closed subscheme
$\operatorname{Quot}_{d,n}^{r}(R)-\operatorname{Spec}S$ and repeat the above
process. Since $\operatorname{Quot}_{d,n}^{r}(R)$ is finite type over $k$,
this ends in finitely many steps, and yields the desired stratification.
$\hfill\square$

######  Lemma B.6.

Notation as above. There is a Zariski stratification

$U=\bigsqcup_{\gamma\in\mathbf{C}}U_{\gamma},$ (B.32)

such that $\mathscr{G}_{U_{\gamma}}$ is a special $U_{\gamma}$-group scheme.

###### Proof.

From Lemma B.5 and Lemma B.3(d), we see that there is a stratification
$\\{U_{\gamma}\\}$ such that $\operatorname{Fil}_{2}\mathscr{G}_{U_{\gamma}}$
is split and $\operatorname{gr}_{3}\mathscr{G}_{U_{\gamma}}$ is isomorphic to
$\operatorname{GL}_{d-r,U_{\gamma}}$. It is classical that
$\operatorname{GL}_{d-r,U_{\gamma}}$ is special. Therefore
$\mathscr{G}_{U_{\gamma}}$, as an extension of special group schemes, is
special. $\hfill\square$
Proof of Theorem B.1. By Lemma B.6, there is a stratification (B.32) on $U$
such that the restriction of $\mathscr{G}_{U}$ to each stratum is special. It
follows from Lemma B.4 and the speciality of $\mathscr{G}_{U_{\gamma}}$,
together with Lemma B.3, that

$[U_{\gamma}][\operatorname{GL}[F]]=[{\mathfrak{Q}}_{U_{\gamma}}][\operatorname{GL}_{d-r}]\mathbb{L}^{bd^{2}-d(n-d)-(d-r)^{2}}.$
(B.33)

Since various ${\mathfrak{Q}}_{U_{\gamma}}$ also form a Zariski stratification
of ${\mathfrak{Q}}_{U}$, and $U$ is a Zariski cover of $C_{n}^{r}(R)$, we see
that

$[C_{n}^{r}(R)][\operatorname{GL}[F]]=[{\mathfrak{Q}}][\operatorname{GL}_{d-r}]\mathbb{L}^{bd^{2}-d(n-d)-(d-r)^{2}}.$
(B.34)

Since
$[{\mathfrak{Q}}]=[\operatorname{Quot}_{d,n}^{r}(R)][\operatorname{GL}_{n}]$,
$[C_{n}^{r}(R)]=[\operatorname{Coh}_{n}^{r}(R)][\operatorname{GL}_{n}]$ and
$[\operatorname{GL}[F]]=[\operatorname{GL}_{d}]\mathbb{L}^{bd^{2}}$ by Lemma
B.2, we obtain the formula (B.1). $\hfill\square$

###  B.5. Gröbner strata

As noted at the beginning, the theory of Gröbner strata in the Hilbert scheme
of points is well developed. Our case is slightly modified in the sense that
we work with (1) power series rings, and (2) Gröbner basis for modules rather
than ideals. Since we only classify submodules of $F:=R^{d}$ with finite
codimension, one shall expect that the Gröbner theory in our case bears no
much difference from the Gröbner theory in the usual polynomial ring case.
Moreover, one can even reduce the theory of Gröbner basis for modules to the
theory of Gröbner basis for ideals, in the sense that submodules $N\subseteq
F$ of rank $r$ and codimension $n$ can be regarded, via square-zero extension,
as ideals in the square-zero deformation $R[[u_{1},...,u_{r}]]/(u_{i}u_{j})$.
Therefore we can realize $\operatorname{Quot}_{d,n}^{d}(R)$ as certain
subscheme of the Hilbert scheme of points
$\operatorname{Hilb}_{n}(R[[u_{1},...,u_{d}]]/(u_{i}u_{j}))$. Ideally, the
Gröbner strata on the latter (the Hilbert scheme) shall restrict to Gröbner
strata on $\operatorname{Quot}_{d,n}^{d}(R)$. These observations and
intuitions suggest that the theory of Gröbner strata in our case is not too
different from the usual case.

Let $R$ be a monomial subring of a power series ring over a field $k$. In this
section, we establish a theory of Gröbner strata which is just adequate for
stratifying $\operatorname{Quot}_{d,n}^{d}(R)$. All treatments are similar and
parallel to [43], hence we will make them as concise as possible.

####  B.5.1. Gröbner basis for submodules of $F\otimes S$

Let $S$ be a finite type algebra over $k$. Then the ring $R\otimes S$ is
nothing other than the subring of a power series ring $S[[\underline{x}]]$
generated by certain monomials. A submodule $M\subseteq F\otimes S$ is called
monic, if $\operatorname{LT}(M)$ is a monomial submodule. (This notion is
exclusive to Gröbner theory over a ring, since over a field, any submodule is
monic.) A Gröbner basis of a submodule $M$ is a finite subset $G$ of $M$ such
$\operatorname{LT}(M)$ agrees with the submodule in $F\otimes S$ generated by
$\operatorname{LT}(g)$ for $g\in G$. Since $R\otimes S$ is Noetherian, every
submodule $M$ admits a Gröbner basis. Similar to §3, a Gröbner basis
$G=\\{g_{1},\dots,g_{h}\\}$ is reduced if

  1. (a)

$\operatorname{LT}(g_{1}),\dots,\operatorname{LT}(g_{h})$ are monomials and
are mutually indivisible;

  2. (b)

No nonleading term of $g_{i}$ is divisible by $\operatorname{LT}(g_{j})$, for
any $i,j$.

Similar to the classical setting (See [5, Theorem 2.11] and [58, Theorem 4]),
it is not hard to see that a submodule $M$ admits a reduced Gröbner basis if
and only if $M$ is monic. We also note that the Buchberger criterion still
holds in this case.

####  B.5.2. Gröbner subfunctors and the induced stratification

Following §4, let $\\{M_{\alpha}:\alpha\in\Xi\\}$ denote the set of finite-
codimensional monomial submodules of $\mathfrak{m}F$, and let $\Delta(\alpha)$
and $C(\alpha)$ be the set of monomials not contained in $M_{\alpha}$ and the
set of corners, respectively. Let
$n(\alpha):=\dim_{k}\mathfrak{m}F/M_{\alpha}$. Though §4 is specifically
written for the cusp, the notation above applies to any monomial subring $R$
of a power series ring over $k$.

We begin by defining the following two subfunctors of
$\operatorname{Hilb}_{d,n}(R)=\operatorname{Quot}_{d,n+d}^{d}(R)$. Let
$\preceq$ be a monomial order over $R$, for each $\alpha\in\Xi$ with
$n(\alpha)=n$, we define functors $\operatorname{Hilb}^{\Delta}(\alpha)$ and
$\operatorname{Hilb}(\alpha)$ by their points over affine $k$-algebras

$\operatorname{Hilb}^{\Delta}(\alpha)(S)=\left\\{\phi:F\otimes
S\twoheadrightarrow Q,\text{ such that $Q$ is free over }S\text{ with basis
}\phi(\Delta(\alpha))\right\\}/\sim$ (B.35)
$\operatorname{Hilb}(\alpha)(S)=\left.\left\\{\begin{aligned} &\phi:F\otimes
S\twoheadrightarrow Q,\text{ such that }\ker\phi\text{ has a reduced Gröbner
basis of form}\\\
&g_{\mu}=\mu+\sum_{{\nu}\in\Delta(\alpha),\mu\prec\nu}s_{\mu,\nu}\nu,\text{
where }\mu\in C(\alpha).\end{aligned}\right\\}\right/\sim$ (B.36)

Note that there is a bijection between $\Xi$ and the set of standard sets in
the sense of [43]. Therefore the two subfunctors defined here are the direct
analogues of the functors $\operatorname{Hilb}^{\Delta}$ and
$\operatorname{Hilb}^{\prec\Delta}$ defined in loc.cit. It is easy to check
that $\operatorname{Hilb}^{\Delta}(\alpha)$ and $\operatorname{Hilb}(\alpha)$
are sheaves in the big Zariski site over $k$. Note that the definition of
$\operatorname{Hilb}^{\Delta}(\alpha)$ does not require the monomial order.

######  Proposition B.7.

The followings are true:

  1. (a)

$\operatorname{Hilb}^{\Delta}(\alpha)$ are represented by open subschemes of
$\operatorname{Hilb}_{d,n}(R)$, and they form an open cover

$\operatorname{Hilb}_{d,n}(R)=\bigcup_{\begin{subarray}{c}\alpha\in\Xi\\\
n(\alpha)=n\end{subarray}}\operatorname{Hilb}^{\Delta}(\alpha).$ (B.37)

  2. (b)

$\operatorname{Hilb}(\alpha)$ is represented by a closed subscheme of
$\operatorname{Hilb}^{\Delta}(\alpha)$.

  3. (c)

$\operatorname{Hilb}_{d,n}(R)$ admits a Zariski stratification

$\operatorname{Hilb}_{d,n}(R)=\bigsqcup_{\begin{subarray}{c}\alpha\in\Xi\\\
n(\alpha)=n\end{subarray}}\operatorname{Hilb}(\alpha).$ (B.38)

As a result,
$[\operatorname{Hilb}_{d,n}(R)]=\sum_{\begin{subarray}{c}\alpha\in\Xi\\\
n(\alpha)=n\end{subarray}}[\operatorname{Hilb}(\alpha)]$ in
$K_{0}(\mathrm{Var}_{k})$.

###### Proof.

  1. (a)

It is clear from the definition that $\operatorname{Hilb}_{d,n}(R)$ is a union
of subfunctors
$\operatorname{Hilb}({\Delta}(\alpha)),\alpha\in\Xi,n(\alpha)=n$. We now prove
the openness.

Applying (B.11) with $r=d$ and $n_{\eqref{eq:embQGr}}=n+d$, we get a locally
closed immersion

$\operatorname{Hilb}_{d,n}(R)\hookrightarrow\operatorname{Gr}:=\operatorname{Gr}(F,bd-n).$
(B.39)

It is a closed embedding since $d=r$, as is observed before. Consider the
subspace $\operatorname{span}_{k}\Delta(\alpha)\subseteq\mathfrak{m}F$. Let
$\mathcal{V}(\alpha)\subseteq\operatorname{Gr}(k)$ be the subset parametrizing
subspaces that have trivial intersection with
$\operatorname{span}_{k}\Delta(\alpha)$. Since
$\dim\operatorname{span}_{k}\Delta(\alpha)=n$, the subspaces of
$\mathfrak{m}F$ of dimension $bd-n$ (i.e., of the complementary dimension to
$\operatorname{span}_{k}\Delta(\alpha)$) have generically trivial intersection
with $\operatorname{span}_{k}\Delta(\alpha)$. Therefore, $\mathcal{V}(\alpha)$
is open. Now we consider $\mathcal{V}(\alpha)$ as an open subscheme of
$\operatorname{Gr}$. Pulling $\mathcal{V}(\alpha)$ back to
$\operatorname{Hilb}_{d,n}(R)$ exactly gives
$\operatorname{Hilb}^{\Delta}(\alpha)$ which is, in particular, an open
subscheme of $\operatorname{Hilb}_{d,n}(R)$.

  2. (b)

The proof is almost identical to [43, Theorem 1]. Let
$\phi\in\operatorname{Hilb}^{\Delta}(\alpha)(S)$. For every $\mu\in C(\alpha)$
there is a unique polynomial of the form

$f_{\mu}=\mu+\sum_{\nu\in\Delta(\alpha)}d_{\mu,\nu}\nu\in\ker\phi.$ (B.40)

Let $J\subseteq S$ be the ideal generated by $d_{\mu,\nu}$ where $\mu$ runs
over $C(\alpha)$ and $\nu$ runs over elements of $\Delta$ such that
$\nu\prec\mu$. A morphism $\psi:S\rightarrow S^{\prime}$ in the category of
$k$-algebras factors through $S\rightarrow S/J$ if and only if
$\ker(\phi\otimes\psi)$ is monic with Gröbner basis $\\{\psi(f_{\mu})\\}$. The
same argument as in loc.cit shows that $\operatorname{Hilb}(\alpha)$ is
represented by a closed subscheme of $\operatorname{Hilb}^{\Delta}(\alpha)$.

  3. (c)

This assertion follows directly from (a) and (b), since every field-valued
point of $\operatorname{Hilb}_{d,n}(R)$ lies in exactly one stratum
$\operatorname{Hilb}(\alpha)$.

∎

######  Warning.

Our notion of stratification does not require that the closure of any stratum
is a disjoint union of strata. Since we are computing the motives but not the
intersection theory, our notion of Zariski stratification suffices. As is
pointed out in [43], the general theory of Gröbner stratification does not
guarantee that Proposition B.7(c) is a stratification in that stronger sense.

####  B.5.3. Motivic version of the counting results

Let $R=k[[T^{2},T^{3}]]$ and assume the setup of §4. Define a Zariski subsheaf
$\operatorname{Hilb}^{0}(\alpha)\subseteq\operatorname{Hilb}(\alpha)$ as

$\operatorname{Hilb}^{0}(\alpha)(S)=\left.\left\\{\begin{aligned}
&\phi:F\otimes S\twoheadrightarrow Q,\text{ such that }\ker\phi\text{ has a
reduced Gröbner basis of form}\\\
&g_{\mu}=\mu+\sum_{\mu\in\Delta(\alpha),\mu\prec\nu}s_{\mu,\nu}\nu,\text{
where }\mu\in C(\alpha),\text{ such that }s_{\mu,\nu}=0\text{ if
}\mu=\mu_{i}^{0}.\end{aligned}\right\\}\right/\sim$ (B.41)

This is the motivic version of the set $\operatorname{Hilb}^{0}(\alpha)$
defined in Lemma 4.4. Recall that we also defined a variety $V(\alpha)$ in
Conjecture 6.1 for a pure-$K$ leading term datum $\alpha$. The following is a
motivic upgrade of Lemma 4.4 and Lemma 4.5:

######  Theorem B.8.

$\operatorname{Hilb}^{0}(\alpha)$ is represented by a closed subscheme of
$\operatorname{Hilb}(\alpha)$. Furthermore, we have the following identities
in $K_{0}(\mathrm{Var}_{k})$:

$\displaystyle[\operatorname{Hilb}(\alpha)]=[\operatorname{Hilb}^{0}(\alpha)]\cdot\mathbb{L}^{\sum_{i}\big{\lvert}\Delta(\alpha)_{\succ\mu_{i}^{0}(\alpha)}\big{\rvert}}.$
(B.42)
$\displaystyle[\operatorname{Hilb}^{0}(\alpha)]=[V(\alpha|_{K_{\alpha}})]\cdot\mathbb{L}^{\sum_{i\in
K_{\alpha}}\left\lvert
C(\alpha|_{J_{\alpha}})_{\succ\mu_{i}^{0}}\right\rvert}.$ (B.43)

###### Proof.

To show that $\operatorname{Hilb}^{0}(\alpha)$ is a closed subscheme of
$\operatorname{Hilb}(\alpha)$, we simpy replace the ideal $J$ in the proof of
Lemma B.7(2) by an ideal $J_{0}$ generated by $J$ together with $d_{\mu,\nu}$
where $\mu=\mu_{i}^{0}$. A morphism $\psi:S\rightarrow S^{\prime}$ in the
category of $k$-algebras factors through $S\rightarrow S/J_{0}$ if and only if
$\ker(\phi\otimes\psi)$ is monic with Gröbner basis $\\{\psi(f_{\mu})\\}$,
such that $\psi(f_{\mu})=\psi(\mu)$ if $\mu=\mu_{i}^{0}$. It then follows that
$\operatorname{Hilb}^{0}(\alpha)$ is represented by a closed subscheme of
$\operatorname{Hilb}(\Delta(\alpha))$, hence a closed subscheme of
$\operatorname{Hilb}(\alpha)$. For the two formulas, one can simply upgrade
the proofs of Lemma 4.4 and Lemma 4.5. Indeed, one first establishes a
stronger structural result

$\displaystyle\operatorname{Hilb}(\alpha)\simeq\operatorname{Hilb}^{0}(\alpha)\times\mathbb{A}^{\sum_{i}\big{\lvert}\Delta(\alpha)_{\succ\mu_{i}^{0}(\alpha)}\big{\rvert}}.$
(B.44) $\displaystyle\operatorname{Hilb}^{0}(\alpha)\simeq
V(\alpha|_{K_{\alpha}})\times\mathbb{A}^{\sum_{i\in K_{\alpha}}\left\lvert
C(\alpha|_{J_{\alpha}})_{\succ\mu_{i}^{0}}\right\rvert}.$ (B.45)

by working with $S$-points (instead of $k$-points as we did) in the proofs of
above lemmas. It is not hard to see that the proofs of the two lemmas are
functorial in nature, and still carries over to $S$-points. We will leave the
details to the readers. $\hfill\square$

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