# A Categorical Quantum Toroidal Action On The Hilbert Schemes

Yu Zhao

###### Abstract.

We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$
and their infinitely many counterparts in the quantum toroidal algebra
$U_{q_{1},q_{2}}(\ddot{gl_{1}})$ acting on the Grothendieck groups of Hilbert
schemes from [11, 34, 26, 29]. By combining our result with [29], one obtains
a geometric categorical $U_{q_{1},q_{2}}(\ddot{gl_{1}})$ action on the derived
category of Hilbert schemes. Our main technical tool is a detailed geometric
study of certain nested Hilbert schemes of triples and quadruples, through the
lens of the minimal model program, by showing that these nested Hilbert
schemes are either canonical or semi-divisorial log terminal singularities.

To Andrei, Lucas and Linyuan with gratitude

##  1\. Introduction

###  1.1. The Description of Our Result

The quantum toroidal algebra $U_{q_{1},q_{2}}(\ddot{gl}_{1})$ (Definition 2.5)
is an affinization of the quantum Heisenberg algebra, studied by Ding-Iohara-
Miki [7, 24], and has been realized in several contexts:

  * •

the elliptic Hall algebra in [2, 33],

  * •

the double shuffle algebra in [9, 10, 27],

  * •

the trace of the deformed Khovanov Heisenberg category in [4](when
$q_{1}=q_{2}$).

Given a smooth quasi-projective surface $S$ over $k=\mathbb{C}$, let

$\mathcal{M}=\bigsqcup_{n=0}^{\infty}S^{[n]}$

be the Hilbert schemes of points on $S$. Schiffmann-Vasserot [34], Feigin-
Tsymbaliuk [11] and Neguţ [29] constructed the
$U_{q_{1},q_{2}}(\ddot{gl}_{1})$ action on the Grothendieck group of
$\mathcal{M}$. It generalizes the action of

  * •

the Heisenberg algebra (Nakajima [25] and Grojnowski [12])

  * •

the $W$ algebra (Li-Qin-Wang [22])

on the cohomology of Hilbert schemes. There are already some applications of
this action in algebraic geometry, like the Beauville-Voison conjecture for
the Hilbert schemes of points on $K3$ surfaces [23].

The purpose of this paper is to categorify the above quantum toroidal algebra
action. We prove that

######  Theorem 1.1 (Theorem 6.2).

Consider the correspondences $e_{k},f_{k}:D^{b}(\mathcal{M})\to
D^{b}(\mathcal{M}\times S)$ induced from:

$\displaystyle e_{k}:=\mathcal{L}^{k}\mathcal{O}_{S^{[n,n+1]}}\in
D^{b}(S^{[n]}\times S^{[n+1]}\times S)$ $\displaystyle
f_{k}:=\mathcal{L}^{k-1}\mathcal{O}_{S^{[n,n+1]}}[1]\in D^{b}(S^{[n+1]}\times
S^{[n]}\times S).$

where $S^{[n,n+1]}$ is the nested Hilbert scheme

$S^{[n,n+1]}:=\\{(\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in S^{[n]}\times
S^{[n+1]}\times
S|\mathcal{I}_{n+1}\subset\mathcal{I}_{n},\mathcal{I}_{n}/\mathcal{I}_{n+1}=k_{x}\\}$

and the line bundle $\mathcal{L}$ on $S^{[n,n+1]}$ has fibers equal to
$\mathcal{I}_{n}/\mathcal{I}_{n+1}$. Then

  1. (1)

For every two integers $m$ and $r$, there exists natural transformations

(1.1) $\begin{cases}f_{r}e_{m-r}\to e_{m-r}f_{r}&\text{ if }m>0\\\
e_{m-r}f_{r}\to f_{r}e_{m-r}&\text{ if }m<0\\\
f_{r}e_{-r}=e_{r}f_{-r}\oplus\mathcal{O}_{\Delta}[1]\end{cases}$

where $\Delta$ is the diagonal of $\mathcal{M}\times\mathcal{M}\times S\times
S$.

  2. (2)

When $m\neq 0$, the cone of the natural transformations in 1.1 has a
filtration with associated graded object

$\begin{cases}\bigoplus\limits_{k=0}^{m-1}R\Delta_{*}(h_{m,k}^{+})&\text{ if
}m>0\\\ \bigoplus\limits_{k=m+1}^{0}R\Delta_{*}(h_{m,k}^{-})&\text{ if
}m<0\end{cases}$

where $h_{m,k}^{+},h_{m,k}^{-}\in D^{b}(\mathcal{M}\times S)$ are defined in
Definition 3.1 and are complexes of universal sheaves on $\mathcal{M}\times
S$.

  3. (3)

(Theorem 3.2) At the level of Grothendieck groups, we have the formula:

$\displaystyle(1-[\omega_{S}])\sum_{k=0}^{m-1}[h_{m,k}^{+}]=h_{m}^{+}$
$\displaystyle\qquad m>0$
$\displaystyle(1-[\omega_{S}])\sum_{k=m+1}^{0}[h_{m,k}^{-}]=h_{-m}^{-}$
$\displaystyle\qquad m<0$

where $\omega_{S}$ is the canonical line bundle of $S$ and $h_{m}^{\pm}$ is
defined in [26] (also see Definition 2.8) .

Note that the non-triviality of the extension is a feature of the derived
category statement, which is not visible at the level of Grothendieck groups.
Proposition 7.1 provides a precise extension formula.

The positive part of the quantum toroidal algebra action was already
categorified in [29]. Theorem 1.1 categorifies the commutation of the positive
part and the negative part, and thus accounts for the action of the whole
quantum toroidal algebra.

######  Remark 1.2.

We have not categorified all the relations defining the elliptic Hall algebra
of [2] (see [29] for more discussions).

######  Remark 1.3.

Our categorification is a geometric categorification in the sense of [3]. The
$2$-categorification of the quantum toroidal algebra is still unclear to us.

###  1.2. Outline of The Proof

Let us first review the categorification of the commutation of $e_{k}$ and
$e_{l}$ for different $k$ and $l$ in [29]. Consider the moduli space
$\mathfrak{Z}_{2},\mathfrak{Z}_{2}^{\prime}$ which parameterize diagrams

(1.2)
${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}}$${\mathcal{I}_{n+1}}$$\scriptstyle{x}$$\scriptstyle{y}$

(1.3)
${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}^{\prime}}$${\mathcal{I}_{n+1}}$$\scriptstyle{y}$$\scriptstyle{x}$

respectively, of ideal sheaves where each successive inclusion is colength $1$
and supported at the point indicated on the diagrams. Then $e_{k}e_{l}$ and
$e_{l}e_{k}$ are the derived pushfoward of line bundles on $\mathfrak{Z}_{2}$
and $\mathfrak{Z}_{2}^{\prime}$ to $S^{[n-1]}\times S^{[n+1]}\times S\times S$
respectively.

In order to compare $e_{k}e_{l}$ and $e_{l}e_{k}$, [29] introduced the
quadruple moduli space $\mathfrak{Y}$ which parameterize diagrams

(1.4)
${\mathcal{I}_{n}}$${\mathcal{I}_{n+1}}$${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{y}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{x}$

of ideal sheaves where each successive inclusion is colength $1$ and supported
at the point indicated on the diagrams. $\mathfrak{Z}_{+}$ is smooth and
induces resolutions of $\mathfrak{Z}_{2}$ and $\mathfrak{Z}_{2}^{\prime}$.
Proposition 2.30 of [29] proved that $\mathfrak{Z}_{2}$ and
$\mathfrak{Z}_{2}^{\prime}$ are rational singularities, based on the fact that
any fiber of the resolution has dimension $\leq 1$. Thus $e_{k}e_{l}$ and
$e_{l}e_{k}$ could be compared through line bundles on $\mathfrak{Y}$.

Now in order to compare $f_{r}e_{m-r}$ and $e_{m-r}f_{r}$, we introduce the
triple moduli spaces $\mathfrak{Z}_{+},\mathfrak{Z}_{-}$ which parameterize
diagrams

(1.5)
${\mathcal{I}_{n}}$${\mathcal{I}_{n+1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{x}$$\scriptstyle{y}$

(1.6)
${\mathcal{I}_{n}}$${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{y}$$\scriptstyle{x}$

Then $f_{r}e_{m-r}$ and $e_{m-r}f_{r}$ are the derived push forward of line
bundles on $\mathfrak{Z}_{+},\mathfrak{Z}_{-}$ respectively. The quadruple
moduli space $\mathfrak{Y}$ still induces resolutions of $\mathfrak{Z}_{-}$,
but $\mathfrak{Z}_{+}$ have two irreducible components. One irreducible
component is $S^{[n,n+1]}$, denoted by $W_{1}$ and the other irreducible
component is denoted by $W_{0}$ in Section 7. $\mathfrak{Y}$ induces a
resolution of $W_{0}$.

In order to compare $f_{r}e_{m-r}$ and $e_{m-r}f_{r}$ through line bundles on
$\mathfrak{Y}$, one must prove that $\mathfrak{Z}_{-}$ and $W_{0}$ are
rational singularities. The approach in [29] did not work here, as the fiber
could have pretty large dimensions. Instead, we study the singularity
structure of $\mathfrak{Z}_{+}$ and $\mathfrak{Z}_{-}$ through the viewpoint
of the minimal model program (MMP) [19, 18]. We prove that

######  Proposition 1.4 (Proposition 5.6 and Proposition 5.13).

The pair $(\mathfrak{Z}_{+},0)$ is semi-dlt. $\mathcal{Z}_{-}$ and $W_{0}$ are
canonical singularities.

We prove Proposition 1.4 by explicitly computing the discrepancy (see Section
7 for the definitions of semi-dlt, canonical singularities and the
discrepancy). Canonical singularities are always rational singularities by
Theorem 4.7.

###  1.3. Higher Rank Stable Sheaves and the Deformed $\mathcal{W}$-algebra

The quantum toroidal algebra also acts on the Grothendieck group of higher
rank stable sheaves [29]. The action factors through the deformed $W$-algebra
[28], which leads to the AGT correspondences for algebraic surfaces.

We will pursue the categorification of the above algebra action in the future.
Different to the case of Hilbert schemes, $\mathfrak{Z}_{+}$ is no longer
equi-dimensional and the obstruction bundle has to be accounted for.

###  1.4. Categorical Heisenberg Actions

Khovanov [17] defined the Heisenberg category through graphical calculus.
Cautis-Licata [5] constructed a categorical Heisenberg action on the derived
category of Hilbert schemes of points of the minimal resolution of the type
ADE singularities. Krug [21] constructed the weak categorical Heisenberg
action on the derived category of Hilbert schemes of points on smooth
surfaces. Our categorification is different from those above, as it is given
in terms of explicit correspondences and independent of the derived McKay
correspondence.

Although the higher Nakajima operators were categorified by the objects
$e_{(0,...,0)}$ of [29], the relations between them (as well as the morphisms
between them in Khovanov’s Heisenberg category) are still unclear to us.

###  1.5. Double Categorified Hall Algebra

The study of Cohomological Hall algebra was initiated by Kontsevich-Soibelman
[20] and Schiffmann-Vasserot [32]. Kapranov-Vasserot [16] and the author [39]
constructed the $K$-theoretic Hall algebra on surfaces, which was categorified
by Porta-Sala [30]. It also categorified the positive half of
$U_{q}(\ddot{gl}_{1})$ when $S=\mathbb{A}^{2}$. The relation between the
categorified Hall algebra of minimal resolution of type A singularities and
quivers was studied by Diaconescu-Porta-Sala [6]. On the other hand, the
Drinfeld double of the categorified Hall algebra is still mysterious. As an
attempt to understand the action of the “double Categorified Hall algebra”, it
is natural to expect that our approach could be generalized to
categorifications in other settings, like those of Toda [38] and Rapcak-
Soibelman-Yang-Zhao [31].

###  1.6. Other Related Work

Recently, Addington-Takahashi [37] studied certain sequences of moduli spaces
of sheaves on $K3$ surfaces and show that these sequences can be given the
structure of a geometric categorical $\mathfrak{sl}_{2}$ action in the sense
of [3]. It would be interesting to explore the interactions between their
action and ours.

Another related work is Jiang-Leung’s projectivization formula [15]. Through
this formula, they obtained a semi-orthogonal decomposition of the derived
category of the nested Hilbert schemes.

###  1.7. The Organization of The Paper

The proof of the main theorem is in Section 6 and the extension formula is in
Section 7. The other sections are organized as follows:

Section 2:

we review the action of $U_{q_{1},q_{2}}(\ddot{gl}_{1})$ on the Grothendieck
group of Hilbert schemes [11, 34, 26, 29];

Section 3:

we define $h_{m,k}^{+}\in D^{b}(\mathcal{M}\times S)$ and prove the third part
of Theorem 1.1;

Section 4:

we make a review of the singularity of the minimal model program.

Section 5:

we study the singularity structures of $\mathfrak{Z}_{-}$ and
$\mathfrak{Z}_{+}$ through the singularity theory of the minimal model
program.

###  1.8. Acknowledgment

The author is grateful to Jihao Liu and Ziquan Zhuang for their help in
understanding the minimal model program and Yuchen Liu for checking the
Section 4. The author would like to thank Qingyuan Jiang, Huachen Chen,
Xiaolei Zhao, Shizhuo Zhang, Andreas Krug and Sabin Cautis for many
interesting discussions on the subject. The author is supported by the NSF
Career grant DMS-1845034.

##  2\. The Quantum Toroidal Algebra $U_{q_{1},q_{2}}(\ddot{gl}_{1})$ and the
$K$-theory of Hilbert scheme of points on surfaces

In this section, we will review the action of $U_{q_{1},q_{2}}(\ddot{gl}_{1})$
on the $K$-theory on Hilbert scheme of points on surfaces from [11, 34, 26,
29]. It will be formulated in Theorem 2.10.

In this paper, we will denote $K(X)$ the Grothendieck group of coherent
sheaves on $X$ for a scheme $X$.

###  2.1. Hilbert Schemes and Nested Hilbert Schemes

Given an integer $n>0$ and a smooth quasi-projective surface $S$ over
$k=\mathbb{C}$, let

$S^{[n]}:=\\{\mathcal{I}_{n}\subset\mathcal{O}|\mathcal{O}/\mathcal{I}_{n}\text{
is dimension }0\text{ and length }n\\}$

be the Hilbert scheme of $n$ points on $S$. There is a universal ideal sheaf
on $S^{[n]}\times S$, still denoted by $\mathcal{I}_{n}$, and a universal
closed subscheme $\mathcal{Z}_{n}\subset S^{[n]}\times S$. ’

######  Proposition 2.1 (Proposition 2.11 of [29]).

There exists a resolution of $\mathcal{I}_{n}$ by

(2.1) $0\to W_{n}\xrightarrow{s}V_{n}\to\mathcal{I}_{n}\to 0$

where $W_{n}$ and $V_{n}$ are locally free coherent sheaves with the same
determinant. Let $w_{n}$ and $v_{n}$ be the rank of $W_{n}$ and $V_{n}$,
respectively. Then $v_{n}-w_{n}=1$.

######  Definition 2.2.

The nested Hilbert scheme $S^{[n,n+1]}$ is defined to be

$S^{[n,n+1]}:=\\{(\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in S^{[n]}\times
S^{[n+1]}\times
S|\mathcal{I}_{n+1}\subset\mathcal{I}_{n},\mathcal{I}_{n}/\mathcal{I}_{n+1}=k_{x}\\}$

with natural projection maps

(2.2)
${S^{[n,n+1]}}$${S^{[n]}}$${S}$${S^{[n+1]}}$$\scriptstyle{p_{n}^{-}}$$\scriptstyle{p_{n}^{+}}$$\scriptstyle{\pi_{n}}$

(2.3)
${(\mathcal{I}_{n},\mathcal{I}_{n+1},x)}$${\mathcal{I}_{n}}$${x}$${\mathcal{I}_{n+1}}$$\scriptstyle{p_{n}^{-}}$$\scriptstyle{p_{n}^{+}}$$\scriptstyle{\pi_{n}}$

and let

$\displaystyle p_{n}:=(p_{n}^{+},\pi_{n}):S^{[n,n+1]}\to S^{[n]}\times S.$

The tautological line bundle $\mathcal{L}$ is the line bundle whose fibers are
$\mathcal{I}_{n}/\mathcal{I}_{n+1}$.

######  Definition 2.3.

Define the nested Hilbert scheme $S^{[n-1,n,n+1]}$ by the Cartesian diagram:

(2.4) ${S^{[n-1,n,n+1]}}$${S^{[n,n+1]}}$${S^{[n-1,n]}}$${S^{[n]}\times
S}$$\scriptstyle{q_{n}}$$\scriptstyle{p_{n}}$$\scriptstyle{(p_{n-1}^{-},\pi_{n-1})}$

which consists of

$\\{(\mathcal{I}_{n-1},\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in S^{[n-1]}\times
S^{[n]}\times S^{[n+1]}\times
S|\mathcal{I}_{n-1}/\mathcal{I}_{n}=k_{x},\mathcal{I}_{n}/\mathcal{I}_{n+1}=k_{x}\\}.$

There are two tautological line bundles $\mathcal{L}_{1},\mathcal{L}_{2}$ on
$S^{[n-1,n,n+1]}$ whose fibers are
$\mathcal{I}_{n}/\mathcal{I}_{n+1},\mathcal{I}_{n-1}/\mathcal{I}_{n}$
respectively. We denote

$q_{n}:S^{[n-1,n,n+1]}\to S^{[n,n+1]}$

the projection morphism.

We define

$\mathcal{M}:=\bigsqcup_{n=0}^{\infty}S^{[n]}$

to be the Hilbert schemes of points on $S$.

######  Example 2.4.

Let $\Delta_{S}:S\to S\times S$ be the diagonal embedding and
$\mathcal{I}_{\Delta_{S}}$ be the ideal sheaf of the diagonal. Then

$\displaystyle S^{[1,2]}=Bl_{\Delta_{S}}(S\times S)=\mathbb{P}_{S\times
S}(\mathcal{I}_{\Delta_{S}})\qquad S^{[2]}=Bl_{\Delta_{S}}(S\times
S)/\mathbb{Z}_{2}$

where the $\mathbb{Z}_{2}$ action on $Bl_{S\times
S}(\mathcal{I}_{\Delta_{S}})$ is induced by the $\mathbb{Z}_{2}$ action

$i:S\times S\to S\times S\quad i(x,y)=(y,x).$

By [35], the projection morphism

$(p_{2}^{-},\pi_{2}):S^{[1,2]}\to S^{[2]}\times
S\quad(\mathcal{I}_{1},\mathcal{I}_{2},x)\to(\mathcal{I}_{2},x)$

has image $\mathcal{Z}_{2}$ and is an isomorphism when restricting to
$\mathcal{Z}_{2}$. By 2.4, $S^{[1,2,3]}\cong p_{2}^{-1}(\mathcal{Z}_{2})$.

###  2.2. The Quantum Toroidal algebra $U_{q_{1},q_{2}}(\ddot{gl}_{1})$

We follow the notation of [29] for the definition of the quantum toroidal
algebra $U_{q_{1},q_{2}}(\ddot{gl}_{1})$. Given two formal parameters $q_{1}$
and $q_{2}$, let $q=q_{1}q_{2}$. Let

$\mathbb{K}=\mathbb{Z}[q_{1}^{\pm 1},q_{2}^{\pm
1}]^{Sym}_{([1],[2],[3],\cdots)}$

where $Sym$ refers to polynomials which are symmetric in $q_{1}$ and $q_{2}$.

######  Definition 2.5.

The quantum toroidal algebra $U_{q_{1},q_{2}}(\ddot{gl_{1}})$ is the
$\mathbb{K}$-algebra with generators:

$\\{E_{k},F_{k},H_{l}^{\pm}\\}_{k\in\mathbb{Z},l\in\mathbb{N}}$

modulo the following relations:

(2.5) $(z-wq_{1})(z-wq_{2})(z-\frac{w}{q})E(z)E(w)=\\\
=(z-\frac{w}{q_{1}})(z-\frac{w}{q_{2}})(z-wq)E(w)E(z)$ (2.6)
$(z-wq_{1})(z-wq_{2})(z-\frac{w}{q})E(z)H^{\pm}(w)=\\\
=(z-\frac{w}{q_{1}})(z-\frac{w}{q_{2}})(z-wq)H^{\pm}(w)E(z)$ (2.7)
$[[E_{k+1},E_{k-1}],E_{k}]=0\quad\forall k\in\mathbb{Z}$

together with the opposite relations for $F(z)$ instead of $E(z)$, as well as:

(2.8)
$[E(z),F(w)]=\delta(\frac{z}{w})(1-q_{1})(1-q_{2})(\frac{H^{+}(z)-H^{-}(w)}{1-q})$

where

(2.9) $E(z)=\sum_{k\in\mathbb{Z}}\frac{E_{k}}{z^{k}},\quad
F(z)=\sum_{k\in\mathbb{Z}}\frac{F_{k}}{z^{k}},\quad
H^{\pm}(z)=\sum_{l\in\mathbb{N}\cup\\{0\\}}\frac{H_{l}^{\pm}}{z^{\pm l}}$

where

$\delta(z)=\sum_{n\in\mathbb{Z}}z^{n}\in\mathbb{Q}\\{\\{z\\}\\}.$

We will set $H_{0}^{+}=q$ and $H_{0}^{-}=1$.

###  2.3. The Quantum Toroidal Algebra Action on the $K$-theory of Hilbert
Schemes

We will write $S_{1},S_{2}$ for two copies of $S$, in order to emphasize the
factors of $S\times S$.

######  Definition 2.6 (Definition 4.10 and Definition 4.11 of [29]).

For any group homomorphisms $x,y:K(\mathcal{M})\to K(\mathcal{M}\times S)$, we
define:

$xy|_{\Delta_{S}}=\\{K(\mathcal{M})\xrightarrow{y}K(\mathcal{M}\times
S)\xrightarrow{x\times Id_{S}}K(\mathcal{M}\times S\times
S)\xrightarrow{Id_{\mathcal{M}}\times\Delta_{S}^{*}}K(\mathcal{M}\times S)\\}$

where $\Delta_{S}:S\to S\times S$ is the diagonal embedding. Also define:

$[x,y]=\\{K(\mathcal{M})\xrightarrow{y}K(\mathcal{M}\times
S_{2})\xrightarrow{s\times Id_{S_{2}}}K(\mathcal{M}\times S_{1}\times
S_{2})\\}\\\ -\\{K(\mathcal{M})\xrightarrow{x}K(\mathcal{M}\times
S_{1})\xrightarrow{y\times Id_{S_{1}}}K(\mathcal{M}\times S_{1}\times
S_{2})\\}$

We define

$[x,y]_{red}=z$

for a group homomorphism $z:K(\mathcal{M})\to K(\mathcal{M\times S})$ if

$[x,y]=\Delta_{S*}(z).$

The definition is unambiguous, since $\Delta_{S*}:K(S)\to K(S\times S)$ is
injective, and so $z$ is unique.

######  Definition 2.7.

For a two term complex of locally free sheaves

$U:=\\{W\xrightarrow{\mathfrak{u}}V\\}$

we define

$[\wedge^{k}(U)]:=\sum_{i=0}^{k}(-1)^{i}[S^{i}W][\wedge^{k-i}V]\quad[S^{k}(U)]:=\sum_{i=0}^{k}(-1)^{i}[\wedge^{i}W][S^{k-i}V]$

as elements in $K(X)$ and define $det(U):=\frac{det(V)}{det(W)}$. We define
$U^{\vee}$ to be the two term complex

$\\{V^{\vee}\xrightarrow{\mathfrak{u}^{\vee}}W^{\vee}\\}.$

######  Definition 2.8.

We define $h_{0}^{+}:=[\omega_{S}]$, $h_{0}^{-}:=1$ and when $m>0$

(2.10) $\displaystyle
h_{m}^{+}:=(1-\omega_{S})\sum_{j=0}^{m-1}[\omega_{S}^{-j}]\sum_{i=0}^{j}(-1)^{i}[S^{m-i}\mathcal{I}_{n}][\wedge^{i}\mathcal{I}_{n}]$
(2.11) $\displaystyle
h_{m}^{-}:=(1-\omega_{S})\sum_{j=0}^{m-1}(-1)^{j}[\omega_{S}^{j}]\sum_{i=0}^{j}(-1)^{i}[\wedge^{m-i}\mathcal{I}_{n}^{\vee}][S^{i}\mathcal{I}_{n}^{\vee}]$

as elements in $K(S^{[n]}\times S).$ Here we abuse the notation to denote

$\mathcal{I}_{n}:=\\{W_{n}\xrightarrow{s}V_{n}\\}$

in the short exact sequence 2.1.

######  Remark 2.9.

Definition 2.8 is equivalent to the definition of $h_{m}^{\pm}$ in [26]. We
will prove it in Appendix A.

######  Theorem 2.10 (Theorem 1.2 of [26]).

Let $T^{*}S$ be the cotangent bundle of $S$ and $\omega_{S}$ be the canonical
bundle of $S$. The morphism:

$q_{1}+q_{2}\to[T^{*}S]\quad q=q_{1}q_{2}\to[\omega_{S}]$

induces a homomorphism:

$\mathbb{K}\to K_{S}.$

Regarding

$\displaystyle\tilde{e}_{i}:=[\mathcal{L}^{i}\mathcal{O}_{S^{[n,n+1]}}]\in
K(S^{[n]}\times S^{[n+1]}\times S),$
$\displaystyle\tilde{f}_{i}:=-[\mathcal{L}^{i-1}\mathcal{O}_{S^{[n,n+1]}}]\in
K(S^{[n+1]}\times S^{[n]}\times S),$
$\displaystyle\tilde{h}_{i}^{\pm}:=[h_{i}^{\pm}\mathcal{O}_{S^{[n]}\times
S}]\in K(S^{[n]}\times S^{[n]}\times S)$

as operators $K(\mathcal{M})\to K(\mathcal{M}\times S)$ through the
$K$-theoretic correspondences, where we regard $S^{[n]}\times S$ as a closed
subscheme of $S^{[n]}\times S^{[n]}\times S$ through the diagonal embedding,
then there exists a unique $\mathbb{K}$-homomorphism

$\Phi:U_{q_{1},q_{2}}(\ddot{gl}_{1})\to
Hom(K_{\mathcal{M}},K_{\mathcal{M}\times S})$

such that

  1. (1)

$\Phi(E_{i})=\tilde{e}_{i},\quad\Phi(F_{i})=\tilde{f}_{i},\quad\Phi(H_{i}^{\pm})=\tilde{h}_{i}^{\pm}$

  2. (2)

For all $x,y\in U_{q_{1},q_{2}}(\ddot{gl}_{1})$, we have

$\displaystyle\Phi(xy)=\Phi(x)\Phi(y)|_{\Delta_{S}}$
$\displaystyle[\Phi(x),\Phi(y)]|_{red}=\Phi(\frac{[x,y]}{(1-q_{1})(1-q_{2})}).$

The right-hand side is well defined due to the fact that all commutators in
$U_{q_{1},q_{2}}(\ddot{gl}_{1})$ are multiples of $(1-q_{1})(1-q_{2})$(see
Theorem 2.4 of [28]).

##  3\. Nested Hilbert Schemes and $h_{m,k}^{\pm}$

Given any scheme $X$, let $D^{b}(X)$ be the bounded derived category of
coherent sheaves on $X$. In this section, we will define objects

$h_{m,k}^{\pm}\in D^{b}(S^{[n]}\times S)$

such that at the level of Grothendieck groups

(3.1)
$\begin{cases}(1-[\omega_{S}])\sum\limits_{k=0}^{m-1}[h_{m,k}^{+}]=h_{m}^{+}&\qquad
m>0\\\ (1-[\omega_{S}])\sum\limits_{k=m+1}^{0}[h_{m,k}^{-}]=h_{-m}^{-}&\qquad
m<0.\end{cases}$

######  Definition 3.1.

For two integers $k,m$ such that $m>k\geq 0$, we define $h_{m,k}^{+}\in
D^{b}(S^{[n]}\times S)$ by

(3.2) $h_{m,k}^{+}:=\begin{cases}R(p_{n}\circ
q_{n})_{*}(\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k})[1]&k>0\\\
Rp_{n*}(\mathcal{L}^{m})[2]&k=0\end{cases}$

and if $m<k\leq 0$, we define $h_{m,k}^{-}\in D^{b}(S^{[n]}\times S)$ by

(3.3) $h_{m,k}^{-}:=\begin{cases}R(p_{n}\circ
q_{n})_{*}(\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k})[1]&k<0\\\
Rp_{n*}(\mathcal{L}^{-m-1})[1]&k=0.\end{cases}$

######  Theorem 3.2.

At the level of Grothendieck groups,

$\displaystyle[h_{m,k}^{+}]=[\omega_{S}^{-k}]\sum_{i=0}^{k}(-1)^{i}[S^{m-i}\mathcal{I}_{n}][\wedge^{i}\mathcal{I}_{n}]$
$\displaystyle[h_{m,k}^{-}]=(-1)^{k}[\omega_{S}^{k}]\sum_{i=0}^{k}(-1)^{i}[\wedge^{m-i}\mathcal{I}_{n}^{\vee}][S^{i}\mathcal{I}_{n}^{\vee}]$

and 3.1 holds.

Theorem 3.2 will be proved later in this section.

###  3.1. Projectivization and A Categorical Projection Lemma

In this paper, for any complex

$\\{\cdots\to C_{-1}\to C_{0}\to 0\\}$

we will assume that $C_{0}$ has cohomological degree $0$ unless explicitly
pointing out the cohomological degree.

######  Definition 3.3.

Given a two term complex of locally free sheaves $U:=\\{W\xrightarrow{s}V\\}$,
we define the symmetric product and the wedge product complexes:

$\displaystyle S^{k}(U):=\\{\wedge^{k}W\cdots\to\cdots\to W\otimes S^{k-1}V\to
S^{k}(V)\\}$
$\displaystyle\wedge^{k}(U):=\\{S^{k}W\to\cdots\to\wedge^{k-1}(V)\otimes
W\to\wedge^{k}(V)\\}$

and $S^{k}(U)=\wedge^{k}(U)=0$ when $k<0$.

######  Definition 3.4.

Let

$U:=\\{W\xrightarrow{s}V\\}$

be a two term complex of locally free sheaves over a scheme $X$ such that $W$
has rank $w$ and $V$ has rank $v$. Let $Z\subset\mathbb{P}_{X}(V)$ be the
closed subscheme such that $\mathcal{O}_{Z}$ is the cokernel of the
composition of morphisms

$\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-1)\xrightarrow{\rho^{*}(s)}\rho^{*}V\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-1)\xrightarrow{taut}\mathcal{O}_{\mathbb{P}_{X}(V)}$

where $\rho:\mathbb{P}_{X}(V)\to X$ is the projection morphism. We define $Z$
to be the projectivization of $U$ over $X$, denoted by

$Z=\mathbb{P}_{X}(U)$

if $\mathcal{O}_{Z}$ is resolved by the Koszul complex:

$0\to\wedge^{w}\rho^{*}(W)\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-w)\to\cdots\to\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-1)\to\mathcal{O}_{\mathbb{P}_{X}(V)}\to\mathcal{O}_{Z}\to
0.$

When $Z$ is a projectivization of $U$ over $X$, we have a categorical
projection lemma for $R\rho_{*}(\mathcal{O}_{Z}(k))$:

######  Lemma 3.5 (Categorical Projection Lemma).

If $Z$ is the projectivization of $U$ over $X$ in Definition 3.4, then the
tensor contraction

(3.4)
$det(U)^{-1}\wedge^{w-v-k}W^{\vee}=\wedge^{v}(V^{\vee})\wedge^{k+v}(W)\to\wedge^{k}(W)$

induces a morphism of complexes

(3.5) $det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[-k]\to S^{k}(U)$

and $R\rho_{*}(\mathcal{O}_{Z}(k))$ is quasi-isomorphic to its cone.

###### Proof.

$\mathcal{O}_{Z}(k)$ is quasi-isomorphic to the complex

$\\{\cdots\to\wedge^{j}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-j+k)\to\cdots\to\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(k-1)\to\mathcal{O}_{\mathbb{P}_{X}(V)}(k)\to
0\\}.$

Consider the following two complexes

$\displaystyle
F_{0}=\\{\cdots\to\wedge^{k+v+1}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-v-1)\to\wedge^{k+v}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-v)\\}[-v-k+1]$
$\displaystyle
F_{1}=\\{\wedge^{k+v-1}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-v+1)\cdots\to\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(k-1)\to\mathcal{O}_{\mathbb{P}_{X}(V)}(k)\\}.$

Then the morphism
$\wedge^{k+v}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-v)\to\wedge^{k+v-1}\rho^{*}W\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(-v+1)$
induces a morphism of $F_{0}\to F_{1}$ with the cone quasi-isomorphic to
$\mathcal{O}_{Z}(k)$.

By Exercise III.8.4 of Hartshorne [13],

(3.6) $R\rho_{*}(\mathcal{O}(j))=\begin{cases}S^{j}(V)&j\geq 0\\\ 0&-v<j<0\\\
detV^{-1}\otimes S^{-j-v}V^{\vee}[v-1]&j\leq-v.\end{cases}$

and thus

$R\rho_{*}F_{0}\cong det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[-k],\quad
R\rho_{*}F_{1}\cong S^{k}(U).$

Hence $R\rho_{*}(\mathcal{O}_{Z}(k))$ is quasi-isomorphic to the cone

$det(U)^{-1}\wedge^{w-v-k}(U^{\vee})[-k]\to S^{k}(U).$

∎

###  3.2. Nested Hilbert Schemes as Projectivization

Recall the short exact sequence 2.1:

$0\to W_{n}\xrightarrow{s_{n}}V_{n}\to\mathcal{I}_{n}\to 0.$

Nested Hilbert schemes can be realized as projectivizations, as in the
following Propositions:

######  Proposition 3.6 (Proposition 2.2 of [8]).

The nested Hilbert scheme $S^{[n,n+1]}$ is the blow up of $\mathcal{Z}_{n}$ in
$S^{[n]}\times S$:

$S^{[n,n+1]}\cong\mathbb{P}_{S^{[n]}\times S}(\mathcal{I}_{n})$

and is smooth of dimension $2n+2$. Moreover, $S^{[n,n+1]}$ is the
projectivization of

$W_{n}\xrightarrow{s}V_{n}$

over $S^{[n]}\times S$. The tautological line bundle $\mathcal{L}$ is the
restriction of $\mathcal{O}_{\mathbb{P}_{S^{[n]}\times S}(V_{n})}(1)$ to
$S^{[n,n+1]}$.

######  Corollary 3.7.

$\mathcal{L}$ is the exceptional divisor of $S^{[n,n+1]}$ as the blow up of
$\mathcal{Z}_{n}$, i.e. we have the short exact sequence:

(3.7) $0\to\mathcal{L}\to\mathcal{O}_{S^{[n,n+1]}}\to
p_{n}^{-1}\mathcal{O}_{\mathcal{Z}_{n}}\to 0.$

###### Proof.

It is obvious from the Proposition 7.13 of [13]. ∎

Let $\overline{V_{n}}$ be the kernel of the surjective morphism

$p_{n}^{*}V_{n}\to\mathcal{O}_{S^{[n,n+1]}}(1)=\mathcal{L}.$

Then $\overline{V_{n}}$ is also locally free. The morphism
$p_{n}^{*}(W_{n})\to p_{n}^{*}(V_{n})$ factors through $\overline{V_{n}}$ and
induces a morphsim

$\overline{V_{n}}^{\vee}\otimes\omega_{S}\to
p_{n}^{*}(W_{n}^{\vee})\otimes\omega_{S}.$

######  Proposition 3.8 (Proposition 4.15 of [28]).

The scheme $S^{[n-1,n,n+1]}$ is smooth of dimension $2n+1$. Moreover, it is
the projectivization of

$\overline{V_{n}}^{\vee}\otimes\omega_{S}\to
p_{n}^{*}(W_{n}^{\vee})\otimes\omega_{S}$

over $S^{[n,n+1]}$. For the two tautological line bundles $\mathcal{L}_{1}$,
$\mathcal{L}_{2}$, $\mathcal{L}_{1}=q_{n}^{*}(\mathcal{O}_{S^{[n,n+1]}}(1))$
and $\mathcal{L}_{2}$ is the restriction of
$\mathcal{O}_{\mathbb{P}_{S^{[n,n+1]}}(p_{n}^{*}(W_{n}^{\vee})\otimes\omega_{S})}(-1)$
in $S^{[n,n+1]}$.

###  3.3. The derived push-forward of line bundles on $S^{[n-1,n,n+1]}$

Given complexes $\\{C_{i}|i\in\mathbb{Z}\\}$ with morphisms $d_{i}:C_{i}\to
C_{i+1}$ such that $d_{i}\circ d_{i+1}=0$, we will write

$\\{\cdots\to C_{i+1}\to C_{i}\to\cdots\\}$

for the total complex of the double complex $C_{\bullet}$. In this subsection,
we still abuse the notation to denote

$\mathcal{I}_{n}:=\\{W_{n}\to
V_{n}\\},\qquad\mathcal{I}_{n}^{\vee}:=\\{V_{n}^{\vee}\to W_{n}^{\vee}\\}.$

######  Lemma 3.9.

We have the following formula for the derived push-forward
$Rp_{n*}\mathcal{L}^{j}$:

(3.8) $Rp_{n*}(\mathcal{L}^{j})=\begin{cases}S^{j}(\mathcal{I}_{n})&j\geq 0\\\
\wedge^{-j-1}(\mathcal{I}_{n}^{\vee})[-j-1]&j<0.\end{cases}$

###### Proof.

By Proposition 3.6, $S^{[n,n+1]}$ is the projectivization of

$W_{n}\xrightarrow{s_{n}}V_{n}.$

Thus 3.8 follows from Lemma 3.5. ∎

######  Lemma 3.10.

We have the following formula for $Rq_{n*}\mathcal{L}_{2}^{k}$:

(3.9)
$Rq_{n*}(\mathcal{L}_{2}^{k})=\begin{cases}\\{\mathcal{L}\to\mathcal{O}_{S^{[n,n+1]}}\\}&k=0\\\
\omega_{S}^{-k}\otimes\\{Lp_{n}^{*}(\wedge^{k}\mathcal{I}_{n})\mathcal{L}\to\cdots\to
Lp_{n}^{*}(\mathcal{I}_{n})\otimes\mathcal{L}^{k}\to\mathcal{L}^{k+1}\\}[2k-1]&k>0\\\
\omega_{S}^{-k}\otimes\\{\cdots\to\mathcal{L}^{-1}Lp_{n}^{*}(S^{-k-1}\mathcal{I}_{n}^{\vee})[-1]\to
Lp_{n}^{*}(S^{-k}\mathcal{I}_{n}^{\vee})\\}&k<0\end{cases}$

where $Lp_{n}^{*}:D^{b}(S^{[n]}\times S)\to D^{b}(S^{[n,n+1]})$ is the derived
pull back morphism.

###### Proof.

By Proposition 3.8, $S^{[n-1,n,n+1]}$ is the projectivization of

$\overline{V_{n}}^{\vee}\otimes\omega_{S}\to
p_{n}^{*}(W_{n}^{\vee})\otimes\omega_{S}$

over $S^{[n,n+1]}$. We notice that $p_{n}^{*}(W_{n}^{\vee})\otimes\omega_{S}$
and $\overline{V_{n}}^{\vee}\otimes\omega_{S}$ have the same rank and

$\displaystyle\frac{det(p_{n}^{*}(W_{n})\otimes\omega_{S})}{det(\overline{V_{n}})\otimes\omega_{S}}=\frac{det(p_{n}^{*}(W_{n}))}{det(\overline{V_{n}})}=\frac{det(p_{n}^{*}(V_{n}))}{det(p_{n}^{*}(V_{n}))\mathcal{L}^{-1}}=\mathcal{L}.$

By Lemma 3.5, we have

  * •

When $k=0$,

(3.10)
$Rq_{n*}(\mathcal{O}_{S^{[n-1,n,n+1]}})=\\{\mathcal{L}\to\mathcal{O}_{S^{[n,n+1]}}\\},$

  * •

When $k>0$, we have

$Rq_{n*}\mathcal{L}_{2}^{k}=\mathcal{L}\otimes\omega_{S}^{-k}\otimes\\{S^{k}(p_{n}^{*}W_{n})\to\overline{V_{n}}\otimes
S^{k-1}(p_{n}^{*}W_{n})\cdots\to\wedge^{k}(\overline{V_{n}})\\}[k-1].$

By the resolution of $\wedge^{k}(\overline{V_{n}})$:

$0\to\wedge^{k}(\overline{V_{n}})\to\wedge^{k}(V_{n})\to\wedge^{k-1}(V_{n})\otimes\mathcal{L}\to\cdots\to\mathcal{L}^{k}\to
0,$

we have

$Rq_{n*}(\mathcal{L}_{2}^{k})=\omega_{S}^{-k}\otimes\\{Lp_{n}^{*}(\wedge^{k}\mathcal{I}_{n})\mathcal{L}\to\cdots\to
Lp_{n}^{*}(\mathcal{I}_{n})\otimes\mathcal{L}^{k}\to\mathcal{L}^{k+1}\\}[2k-1].$

  * •

When $k<0$, we have

$\displaystyle
Rq_{n*}(\mathcal{L}_{2}^{k})=\omega_{S}^{-k}\otimes\\{\cdots\to\overline{V_{n}}^{\vee}\otimes
S^{-k-1}(p_{n}^{*}W_{n}^{\vee})\to S^{-k}(p_{n}^{*}W_{n}^{\vee})\\}$

By the resolution of $\wedge^{k}(\overline{V_{n}}^{\vee})$:

$0\to\mathcal{L}^{k}\to\cdots\to\wedge^{-k-1}(V_{n}^{\vee})\mathcal{L}^{-1}\to\wedge^{k}(V_{n}^{\vee})\to\wedge^{k}(\overline{V_{n}}^{\vee})\to
0,$

we have

(3.11) $\displaystyle Rq_{n*}(\mathcal{L}_{2}^{k})$
$\displaystyle=\omega_{S}^{-k}\otimes\\{\cdots\to\mathcal{L}^{-1}Lp_{n}^{*}(S^{-k-1}\mathcal{I}_{n}^{\vee})[-1]\to
Lp_{n}^{*}(S^{-k}\mathcal{I}_{n}^{\vee})\\}$

∎

By Lemma 3.9 and Lemma 3.10, we have

######  Corollary 3.11.

We have the following formula for $R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k}$:

(3.12) $R(p_{n}\circ
q_{n})_{*}(\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k})=\\\
\begin{cases}\omega_{S}^{-k}\otimes\\{\wedge^{-m}(\mathcal{I}_{n}^{\vee})\to\cdots\to\wedge^{-m+k}(\mathcal{I}_{n}^{\vee})S^{-k}(\mathcal{I}_{n}^{\vee})\\}[-m+k]&m\leq
k\leq-1\\\
\omega_{S}^{-k}\otimes\\{\wedge^{k}(\mathcal{I}_{n})S^{m-k}(\mathcal{I}_{n})\to\cdots\to\mathcal{I}_{n}\otimes
S^{m-1}(\mathcal{I}_{n})\to S^{m}(\mathcal{I}_{n})\\}[2k-1]&1\leq k\leq m.\\\
\end{cases}$

######  Remark 3.12.

When $m=k<0$, same as the computation in the Grothendieck group, the complex

$\\{\wedge^{-k}(\mathcal{I}_{n}^{\vee})\to\cdots\to
S^{-k}(\mathcal{I}_{n}^{\vee})\\}$

is quasi-isomorphic to $0$ and thus we have

(3.13) $R(p_{n}\circ q_{n})_{*}\mathcal{L}^{-1}_{1}\mathcal{L}_{2}^{k}=0.$

###### Proof of Theorem 3.2.

It follows from Definition 2.8, Lemma 3.9 and Corollary 3.11. ∎

##  4\. Singularity of the Minimal Model Program

In this section, we will review the singularities in the minimal model program
from [19, 18]. We use the notation $D$ to replace $\Delta$ in [19, 18], as
$\Delta$ is already used to denote the diagonal embedding in our paper. For a
normal variety, we will denote $K_{X}$ the canonical Weil divisor. We will
denote by $\omega_{X}$ the dualizing sheaf when $X$ is Cohen-Macaulay. They
coincide when $X$ is Gorenstein.

###  4.1. Discrepancy and Classifacation of Singularities

######  Definition 4.1 (Definition 2.25 of [19] or Definition 2.4 of [18],
Discrepancy).

Let $(X,D)$ be a pair where $X$ is a normal variety and $D=\sum
a_{i}D_{i},a_{i}\in\mathbb{Q}$ is a sum of distinct prime divisors. Assume
that $m(K_{X}+D)$ is Cartier for some $m>0$. Suppose $f:Y\to X$ is a
birational morphism from a normal variety $Y$. Let $E\subset Y$ denote the
exceptional locus of $f$ and $E_{i}\subset E$ the irreducible exceptional
divisors. The two line bundles

$\mathcal{O}(m(K_{Y}+f^{-1}D))|_{Y-E}\text{ and
}f^{*}\mathcal{O}_{X}(m(K_{X}+D))|_{Y-E}$

are naturally isomorphic. Thus there are rational numbers $a(E_{i},X,D)$ such
that $ma(E_{i},X,D)$ are integers, and

$\mathcal{O}_{Y}(m(K_{Y}+f^{-1}D))\cong
f^{*}\mathcal{O}_{X}(m(K_{X}+D))\otimes\mathcal{O}_{Y}(\sum_{i}ma(E_{i},X,D)E_{i}).$

$a(E,X,D)$ is called the discrepancy of $E$ with respect to $(X,D)$. We define
the center of $E$ in $X$ by

$center_{X}(E):=f(E).$

When $D=0$, then $a(E_{i},X,D)$ depends only on $E_{i}$ but not on $f$.

######  Lemma 4.2 (Lemma 2.29 of [19]).

Let $X$ be a smooth variety and $D=\sum a_{i}D_{i}$ a sum of distinct prime
divisors. Let $Z\subset X$ be a closed subvariety of codimension $k$. Let
$p:Bl_{Z}X\to X$ be the blow up of $Z$ and $E\subset Bl_{Z}X$ the irreducible
component of the exceptional divisor which dominates $Z$, (if $Z$ is smooth,
then this is the only component). Then

$a(E,X,D)=k-1-\sum_{i}a_{i}mult_{Z}D_{i}$

where $mult_{Z}D_{i}$ is the multiplicity of $D_{i}$ in $Z$.

######  Definition 4.3 (Definition 2.34 and 2.37 of [19], or Definition 2.8 of
[18]).

Let $(X,D)$ be a pair where $X$ is a normal variety and $D=\sum a_{i}D_{i}$ is
a sum of distinct prime divisors where $a_{i}\in\mathbb{Q}$ and $a_{i}\leq 1$.
Assume that $m(K_{X}+D)$ is Cartier for some $m>0$. We say that $(X,D)$ is

$\begin{cases}terminal\\\ canonical\\\ klt\\\ plt\\\ dlt\\\ lc\\\ \end{cases}$

$\text{if }a(E,X,D)\text{ is }\begin{cases}>0,\text{ for every exceptional
}E\\\ \geq 0,\text{ for every exceptional }E\\\ >-1,\text{ for every }E\\\
>-1,\text{ for every exceptional }E\\\ >-1,\text{ if
}center_{X}E\subset\text{non-snc}(X,D)\\\ \geq-1.\text{ for every }E\\\
\end{cases}$

Here klt is short for “Kawamata log terminal”, plt for “pure log terminal” and
lc for “log canonical”.Above, non-snc$(X,D)$ denotes the set of points where
$(X,D)$ is not simple normal crossing(snc for short).

We say that $X$ is terminal (canonical,etc) if and only if $(X,0)$ is terminal
(canonical,etc).

Each class contains the previous one, except canonical does not imply klt if
$D$ contains a divisor with coefficient $1$.

######  Theorem 4.4 (Theorem 5.50 of [19], or Theorem 4.9 of [18], Inversion
of adjunction).

Let $X$ be normal and $S\subset X$ a normal Weil divisor which is Cartier in
codimension $2$. Let $B$ be an effective $\mathbb{Q}$-divisor and assume that
$K_{X}+S+B$ is $\mathbb{Q}$-Cartier. Then $(X,S+B)$ is plt near $S$ iff
$(S,B|_{S})$ is klt.

###  4.2. Rational Singularities

######  Definition 4.5 (Definition 5.8 of [19]).

Let $X$ be a variety of a field of characteristic $0$. We say that $X$ is a
rational singularity if there exists one resolution of singularities $f:Y\to
X$,

  1. (1)

$f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$(equivalently, $X$ is normal), and

  2. (2)

$R^{i}f_{*}\mathcal{O}_{Y}=0$ for $i>0$.

######  Remark 4.6.

By Theorem 5.10 of [19], if $X$ is a rational singularity, then for all
resolution of singularities $f:Y\to X$,

  1. (1)

$f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$,

  2. (2)

$R^{i}f_{*}\mathcal{O}_{Y}=0$ for $i>0$.

######  Theorem 4.7 (Theorem 5.22 of [19]).

Let $X$ be a normal variety over a field of characteristic $0$. If $X$ is a
canonical singularity, then $X$ is a rational singularity. If $X$ is
Gorenstein, then $X$ is a canonical singularity if $X$ is a rational
singularity.

Here are also some examples of rational singularities:

######  Example 4.8.

By [35], the universal closed subscheme $\mathcal{Z}_{n}$ is a rational
singularity.

######  Example 4.9.

The morphism $q_{n}:S^{[n-1,n,n+1]}\to S^{[n,n+1]}$ factors through
$p_{n}^{-1}\mathcal{Z}_{n}$ and induces a resolution of
$p_{n}^{-1}\mathcal{Z}_{n}$. By Lemma 3.10 and Corollary 3.7,
$p_{n}^{-1}\mathcal{Z}_{n}$ is a Gorenstein rational singularity and thus is a
canonical singularity by Theorem 4.7.

###  4.3. Semi-dlt Pairs

######  Definition 4.10 (Definition 1.10 of [18], Semi-snc pairs).

Let $W$ be a regular subscheme and $\sum_{i\in I}E_{i}$ a snc divisor on $W$.
Write $I=I_{Y}\cup I_{D}$ as a disjoint union. Set $Y:=\sum_{i\in I_{V}}E_{i}$
as a subscheme and $D_{Y}:=\sum_{i\in I_{D}}a_{i}E_{i}|_{Y}$ as a divisor on
$Y$ for some $a_{i}\in\mathbb{Q}$. We call $(Y,D_{Y})$ an embedded semi-snc
pair. A pair $(X,D)$ is called semi-snc if it is Zariski locally isomorphic to
an embedded semi-snc pair.

######  Example 4.11.

We have the following three examples of semi-snc pairs $(X,D)$:

  1. (1)

$X=\\{z=0\\}\subset\mathbb{A}^{3}$ and $D=a_{x}(x|_{X}=0)+a_{y}(y|_{X}=0)$.

  2. (2)

$X=\\{yz=0\\}\subset\mathbb{A}^{3}$ and $D=a_{x}(x|_{X}=0)$.

  3. (3)

$X=\\{xyz=0\\}\subset\mathbb{A}^{3}$ and $D=0$.

######  Definition 4.12 (Definition 5.1 of [18], Demi-normal schemes).

A scheme $X$ is called demi-normal if it is $S_{2}$ and codimension $1$ points
are either regular points or nodes. Here we say a scheme $X$ has a node at a
point $x\in X$ if its local ring $\mathcal{O}_{x,X}$ can be written as $R/(f)$
where $(R,m)$ is a regular local ring of dimension $2$, $f\in m^{2}$ and $f$
is not a square in $m^{2}/m^{3}$.

######  Definition 4.13 (Section 5.2 of [18], Conductor).

Let $X$ be a reduced scheme and $\pi:\bar{X}\to X$ its normalization. The
conductor ideal

$cond_{X}:=\mathcal{H}om(\pi_{*}\mathcal{O}_{X},\mathcal{O}_{X})\subset\mathcal{O}_{X}$

is the largest ideal sheaf on $X$ that is also an ideal sheaf on $\bar{X}$. We
write it as $cond_{\bar{X}}$ when we view the conductor as an ideal sheaf on
$\bar{X}$. The conductor subschemes are defined as

$T:=Spec_{X}(\mathcal{O}_{X}/cond_{X})\qquad\bar{T}:=Spec_{\bar{X}}(\mathcal{O}_{\bar{X}}/cond_{\bar{X}}).$

######  Definition 4.14 (Definition-Lemma 5.10 of [18]).

Let $X$ be a demi-normal scheme with normalization $\pi:\bar{X}\to X$ and
conductors $T\subset X$ and $\bar{T}\subset\bar{X}$. Let $D$ be an effective
$\mathbb{Q}$-divisor whose support does not contain any irreducible components
of $T$ and $\bar{D}$ the divisorial part of $\pi^{-1}(D)$. The pair $(X,D)$ is
called semi log canonical or slc if

  1. (1)

$K_{X}+D$ is $\mathbb{Q}$-Cartier, and

  2. (2)

$(\bar{X},\bar{D}+\bar{T})$ is lc.

######  Definition 4.15 (Definition 5.19 of [18]).

An slc pair $(X,D)$ is semi-divisorial log terminal or semi-dlt if
$a(E,X,D)>-1$ for every exceptional divisor $E$ over $X$ such that $(X,D)$ is
not snc at the generic point of $center_{X}E$.

######  Example 4.16.

A semi-snc pair $(X,D)$ is always semi-dlt.

######  Proposition 4.17 (Proposition 5.20 of [18]).

Let $(X,D)$ be a demi-normal pair over a field of characteristic $0$. Assume
that the normalization $(\bar{X},\bar{T}+\bar{D})$ is dlt and there is a
codimension $3$ set $W\subset X$ such that $(X\backslash W,D|_{X\backslash
W})$ is semi-dlt. Then

  1. (1)

the irreducible components of $X$ are normal,

  2. (2)

$K_{X}+D$ is $\mathbb{Q}$-Cartier and

  3. (3)

$(X,D)$ is semi-dlt.

##  5\. The Quadrulple/Triple Moduli Spaces and the Minimal Model Program

In this section we introduce the triple moduli spaces
$\mathfrak{Z}_{+},\mathfrak{Z}_{-}$ and the quadruple moduli space
$\mathfrak{Y}$ which parameterize diagrams:

(5.1)
${\mathcal{I}_{n}}$${\mathcal{I}_{n+1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{x}$$\scriptstyle{y}$

(5.2)
${\mathcal{I}_{n}}$${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{y}$$\scriptstyle{x}$

(5.3)
${\mathcal{I}_{n}}$${\mathcal{I}_{n+1}}$${\mathcal{I}_{n-1}}$${\mathcal{I}_{n}^{\prime}}$$\scriptstyle{y}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{x}$

respectively, of ideal sheaves where each successive inclusion is colength $1$
and supported at the point indicated on the diagrams. We consider line bundles
$\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{1}^{\prime},\mathcal{L}_{2}^{\prime}$
over triple/quadruple moduli spaces with fiber
$\mathcal{I}_{n+1}/\mathcal{I}_{n},\mathcal{I}_{n}/\mathcal{I}_{n-1},\mathcal{I}_{n+1}/\mathcal{I}_{n}^{\prime},\mathcal{I}_{n}^{\prime}/\mathcal{I}_{n-1}$
respectively.

We consider the Cartesian diagram:

(5.4) ${\mathfrak{Y}}$${\mathfrak{Z}_{+}}$${\mathfrak{Z}_{-}}$${S^{[n]}\times
S^{[n]}\times S\times
S}$$\scriptstyle{\alpha_{+}}$$\scriptstyle{\alpha_{-}}$$\scriptstyle{\theta}$$\scriptstyle{\beta_{+}}$$\scriptstyle{\beta_{-}}$
${(\mathcal{I}_{n-1},\mathcal{I}_{n},\mathcal{I}_{n}^{\prime},\mathcal{I}_{n+1},x,y)}$${(\mathcal{I}_{n},\mathcal{I}_{n}^{\prime},\mathcal{I}_{n+1},x,y)}$${(\mathcal{I}_{n-1},\mathcal{I}_{n},\mathcal{I}_{n}^{\prime},x,y)}$${(\mathcal{I}_{n},\mathcal{I}_{n}^{\prime},x,y).}$$\scriptstyle{\alpha_{+}}$$\scriptstyle{\alpha_{-}}$$\scriptstyle{\theta}$$\scriptstyle{\beta_{+}}$$\scriptstyle{\beta_{-}}$

2.4 is the restriction of 5.4 to the diagonal $\Delta:S^{[n]}\times S\to
S^{[n]}\times S^{[n]}\times S\times S$.

######  Example 5.1.

When $n=1$, then $\mathfrak{Y}=Bl_{\Delta_{S}}(S\times S)$, where
$\Delta_{S}:S\to S\times S$ is the diagonal embedding.
$\mathfrak{Z}_{-}=S\times S$ and $\mathfrak{Z}_{+}$ is induced by the
Cartesian diagram

(5.5) ${\mathfrak{Z}_{+}}$${Bl_{\Delta_{S}}(S\times
S)}$${Bl_{\Delta_{S}}(S\times S)}$${Bl_{\Delta_{S}}(S\times
S)/\mathbb{Z}_{2}.}$

$\mathfrak{Z}_{+}$ has two irreducible components and each one is isomorphic
to $Bl_{\Delta}(S\times S)$.

The main purpose of this section is to compute
$R\alpha_{+*}\mathcal{O}_{\mathfrak{Y}}$ and
$R\alpha_{-*}\mathcal{O}_{\mathfrak{Y}}$ explicitly.

######  Proposition 5.2.

We have the formula

(5.6)
$R\alpha_{-*}\mathcal{O}_{\mathfrak{Y}}=\mathcal{O}_{\mathfrak{Z}_{-}},\quad
R\alpha_{+*}\mathcal{O}_{\mathfrak{Y}}=\mathcal{O}_{W_{0}}$

where $W_{0}$ will be defined in Section 5.3.

Proposition 5.2 follows from Corollary 5.7 and Corollary 5.14, which will be
proved later in this section.

###  5.1. The Geometry of $\mathfrak{Y}$

######  Theorem 5.3 (Proposition 2.28 and Proposition 5.28 of [29]).

The scheme $\mathfrak{Y}$ is smooth of dimension $2n+2$. The closed embedding:

$\Delta_{\mathfrak{Y}}:S^{[n-1,n,n+1]}\to\mathfrak{Y}\quad(\mathcal{I}_{n-1},\mathcal{I}_{n},\mathcal{I}_{n+1},x)\to(\mathcal{I}_{n-1},\mathcal{I}_{n},\mathcal{I}_{n},\mathcal{I}_{n+1},x,x)$

is a regular closed subscheme of codimension $1$. If we abuse the notation to
denote $S^{[n-1,n,n+1]}$ by $\Delta_{\mathfrak{Y}}$, then the morphism of
coherent sheaves over $\mathfrak{Y}\times S$:

$\mathcal{I}_{n}/\mathcal{I}_{n+1}\to\mathcal{I}_{n-1}/\mathcal{I}_{n+1}\to\mathcal{I}_{n-1}/\mathcal{I}_{n}^{\prime},\quad\mathcal{I}_{n}^{\prime}/\mathcal{I}_{n+1}\to\mathcal{I}_{n-1}/\mathcal{I}_{n+1}\to\mathcal{I}_{n-1}/\mathcal{I}_{n}$

induce short exact sequences

$\displaystyle
0\to\mathcal{L}_{1}\to\mathcal{L}_{2}^{\prime}\to\mathcal{L}_{2}^{\prime}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to
0$ $\displaystyle
0\to\mathcal{L}_{1}^{\prime}\to\mathcal{L}_{2}\to\mathcal{L}_{2}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to
0.$

Moreover,
$\mathcal{L}_{1}\mathcal{L}_{2}^{\prime-1}=\mathcal{L}_{1}^{\prime}\mathcal{L}_{2}^{-1}=\mathcal{O}(-\Delta_{\mathfrak{Y}})$.
The normal bundle
$N_{\mathfrak{Y}/\Delta_{\mathfrak{Y}}}=\mathcal{L}_{1}^{-1}\mathcal{L}_{2}$.

######  Remark 5.4.

From now on we will abuse the notation to denote $S^{[n-1,n,n+1]}$ by
$\Delta_{\mathfrak{Y}}$.

######  Lemma 5.5 (Claim 3.8 of [26]).

Let $U$ be the complement of $\Delta$. Then $\alpha_{+}$ and $\alpha_{-}$ in
5.4 are isomorphisms when restricting to $U$.

###  5.2. The Geometry of $\mathfrak{Z}_{-}$

######  Proposition 5.6.

The scheme $\mathfrak{Z}_{-}$ is an irreducible $2n+2$ dimensional locally
complete intersection scheme and is a canonical singularity.

###### Proof.

By Lemma 5.5, $\alpha^{-1}_{-}(U)\cong\theta^{-1}(U)$ is a $2n+2$ dimensional
smooth open subscheme and the complement is the $2n$ dimensional closed
subscheme $S^{[n-1,n]}$ by 5.4. Hence $\mathfrak{Z}_{-}$ is a $2n+2$
dimensional $R_{1}$ scheme in the sense of Serre’s condition.

On the other hand,

$\mathfrak{Z}_{-}=S^{[n-1,n]}\times_{S^{[n-1]}}S^{[n-1,n]}$

has the expected dimension also equal to $2n+2$. Thus $\mathfrak{Z}_{-}$ is a
locally complete intersection scheme and thus is Cohen-Macaulay and normal.

Now we prove that $\mathfrak{Z}_{-}$ is a canonical singularity. The
complement of $\theta^{-1}(U)$ in $\mathfrak{Y}$ is
$\Delta_{\mathfrak{Y}}=S^{[n-1,n,n+1]}$ and thus there exists $a\in\mathbb{Q}$
such that

$\alpha_{-}^{*}K_{\mathfrak{Z}_{-}}=K_{\mathfrak{Y}}+a\mathcal{O}(\Delta_{\mathfrak{Y}})$

We denote $\mathfrak{Y}_{1}$ the quadruple moduli space $\mathfrak{Y}$ when
$n=1$. Let

$V_{1}:=\\{(\mathcal{I}_{n-1},(\mathcal{I}_{1},\mathcal{I}_{1}^{\prime},\mathcal{I}_{2},x,y))\in
S^{[n-1]}\times\mathfrak{Y}_{1}|(\mathcal{I}_{n-1},x)\notin\mathcal{Z}_{n-1}\text{
and }(\mathcal{I}_{n-1},y)\notin\mathcal{Z}_{n-1}\\}$

where $\mathcal{Z}_{n-1}$ is the universal closed subscheme of
$S^{[n-1]}\times S$. The morphism

$(\mathcal{I}_{n-1},(\mathcal{I}_{1},\mathcal{I}_{1}^{\prime},\mathcal{I}_{2},x,y))\to(\mathcal{I}_{n-1},\mathcal{I}_{n-1}\cap\mathcal{I}_{1},\mathcal{I}_{n-1}\cap\mathcal{I}_{1}^{\prime},\mathcal{I}_{n-1}\cap\mathcal{I}_{2},x,y)$

induces an open embedding $V_{1}\subset\mathfrak{Y}$.

We define

$V_{2}:=\\{(\mathcal{I}_{n-1},x,y)\in S^{[n]}\times S\times
S|(\mathcal{I}_{n-1},x)\notin\mathcal{Z}_{n-1}\text{ and
}(\mathcal{I}_{n-1},y)\notin\mathcal{Z}_{n-1}\\},$

and the morphism

$(\mathcal{I}_{n-1},x,y)\to(\mathcal{I}_{n-1},\mathcal{I}_{n-1}\cap\mathcal{I}_{x},\mathcal{I}_{n-1}\cap\mathcal{I}_{y})$

induces an open embedding $V_{2}\subset\mathfrak{Y}_{-}$.
$\alpha_{-}^{-1}(V_{2})=V_{1}$ and we have the Cartesian diagram:

${V_{1}}$${\mathfrak{Y}_{1}}$${V_{2}}$${S\times S}$

where all horizontal morphisms are natural projections. By Example 5.1,
$\mathfrak{Y}_{2}=Bl_{\Delta_{S}}(S\times S)$. Thus by Lemma 4.2, $a=1$ and
$\mathfrak{Z}_{-}$ is a canonical singularity. ∎

######  Corollary 5.7.

We have the formula

$R\alpha_{-*}(\mathcal{O}_{\mathfrak{Y}})=\mathcal{O}_{\mathfrak{Z}_{-}}.$

###  5.3. The Geometry of $\mathfrak{Y}_{+}$

The geometry of $\mathfrak{Y}_{+}$ is more complicated, as it is no longer
irreducible. We define $W_{0}$ as a closed subscheme of $\mathfrak{Z}_{+}$ by

$W_{0}:=\\{(\mathcal{I}_{n},\mathcal{I}_{n}^{\prime},\mathcal{I}_{n+1},x,y)\in\mathfrak{Z}_{+}|(\mathcal{I}_{n}^{\prime},y)\in\mathcal{Z}_{n}\text{
and }(\mathcal{I}_{n},x)\in\mathcal{Z}_{n}\\}.$

$W_{0}$ is the closure of $\beta_{+}^{-1}(U)$ in $\mathfrak{Z}_{+}$. We define
$W_{1}:=S^{[n,n+1]}$.

######  Proposition 5.8.

The scheme $\mathfrak{Z}_{+}$ is a locally complete intersection scheme (and
hence Cohen-Macaulay) of dimension $2n+2$, with two irreducible components
$W_{0}$ and $W_{1}$. $W_{0}\cap W_{1}=p_{n}^{-1}\mathcal{Z}_{n}$.

###### Proof.

$W_{1}$ is $2n+2$ dimensional and the complement of $W_{1}$ in
$\mathfrak{Z}_{+}$ is $\beta_{+}^{-1}(U)=\theta^{-1}(U)$, which is also $2n+2$
dimensional by Lemma 5.5. Thus $\mathfrak{Z}_{+}$ is $2n+2$ dimensional and

$\mathfrak{Z}_{+}=S^{[n,n+1]}\times_{S^{[n+1]}}S^{[n,n+1]}$

has expected dimension also equal to $2n+2$. Hence $\mathfrak{Z}_{+}$ is a
locally complete intersection scheme.

Any closed point of $W_{0}\cap W_{1}$ corresponds to
$(\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in S^{[n,n+1]}$ such that $x$ has
length $\geq 1$ in $\mathcal{O}/\mathcal{I}_{n}$ and thus is in
$p_{n}^{-1}\mathcal{Z}_{n}$. ∎

######  Example 5.9.

When $n=2$, by Example 2.4, $\mathcal{Z}_{2}=S^{[1,2]}$. Thus for any point
$(\mathcal{I}_{2}^{\prime},\mathcal{I}_{2},\mathcal{I}_{3},x,y)\in W_{0}$,
there exists a unique ideal sheaf $\mathcal{I}_{1}$ with two short exact
sequences

$0\to\mathcal{I}_{2}^{\prime}\to\mathcal{I}_{1}\to k_{y}\to 0\quad
0\to\mathcal{I}_{2}\to\mathcal{I}_{1}\to k_{x}\to 0.$

Thus $W_{0}\cong\mathfrak{Y}$.

$W_{0}\cap W_{1}=S^{[1,2,3]}$ by Example 2.4 and is smooth. So
$(\mathfrak{Z}_{+},0)$ is a semi-snc pair.

Consider the closed subscheme $W_{2}\subset p_{n}^{-1}\mathcal{Z}_{n}$

$W_{2}:=\\{(\mathcal{I}_{n},\mathcal{I}_{n+1},x)\in W_{1}|\text{ the length of
}k_{x}\text{ in }\mathcal{O}/\mathcal{I}_{n}\geq 3.\\}.$

######  Lemma 5.10.

The scheme $q_{n}^{-1}(W_{2})$ and $W_{2}$ have dimension less or equal to
$2n-1$.

###### Proof.

The scheme

$S^{[n-2,n-1,n,n+1]}:=S^{[n-2,n-1,n]}\times_{S^{[n-1,n]}}S^{[n-1,n,n+1]}$

has dimension $2n-1$ by (5.21) of [29]. The image of the projection morphism

$S^{[n-2,n-2,n,n+1]}\to S^{[n,n+1]}$

is $W_{2}$ and the image of the projecton morphism

$S^{[n-2,n-2,n,n+1]}\to S^{[n-1,n,n+1]}$

is $q_{n}^{-1}(W_{2})$. Hence $q_{n}^{-1}(W_{2})$ and $W_{2}$ have dimension
less or equal to $2n-1$. ∎

As the morphism $\alpha_{+}:\mathfrak{Y}\to\mathfrak{Z}_{+}$ factors through
$W_{0}$, we will abuse the notation to denote the morphism
$\alpha_{+}:\mathfrak{Y}\to W_{0}$.

######  Lemma 5.11.

The morphism $\alpha_{+}:\mathfrak{Y}\to W_{0}$ is an isomorphism when
restricting to $W_{0}-W_{2}$.

###### Proof.

We denote $\mathfrak{Y}_{2}$ the quadruple moduli space when $n=2$. Let

$V_{3}:=\\{(\mathcal{I}_{n-2},(\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{2}^{\prime},\mathcal{I}_{3},x,y))\in
S^{[n-1]}\times\mathfrak{Y}_{2}|\mathcal{I}_{n-2}+\mathcal{I}_{3}=\mathcal{O}\\}$

and $V_{4}:=\alpha_{+}V_{3}$. The morphism

$(\mathcal{I}_{n-2},(\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{2}^{\prime},\mathcal{I}_{3},x,y))\to(\mathcal{I}_{n-2}\cap\mathcal{I}_{1},\mathcal{I}_{n-2}\cap\mathcal{I}_{2},\mathcal{I}_{n-2}\cap\mathcal{I}_{2}^{\prime},\mathcal{I}_{n-2}\cap\mathcal{I}_{3},x,y)$

induces an open emebedding $V_{3}\subset\mathfrak{Y}$. By Example 5.9,

$V_{3}\cong V_{4}$

By Lemma 5.5, $\alpha_{+}$ is also an isomorphism when restricting to
$\beta_{+}^{-1}(U)$. $W_{2}$ is the complment of $\beta_{+}^{-1}(U)\cup V_{4}$
in $W_{0}$. ∎

Let $\overline{W_{0}}$ be the normalization of $W_{0}$ and
$\overline{W_{0}\cap W_{1}},\overline{W_{2}}$ the preimage of $W_{0}\cap
W_{1}$ and $W_{2}$ in the normalization. The morphism
$\alpha_{+}:\mathfrak{Y}\to W_{0}$ will factor through
$\overline{\alpha_{+}}:\mathfrak{Y}\to\overline{W_{0}}$

######  Lemma 5.12.

The scheme $\overline{W}_{0}$ is a canonical singularity and the pair
$(\overline{W_{0}},\overline{W_{0}\cap W_{1}})$ is plt.

###### Proof.

The codimension of $\overline{W_{2}}$ in $\overline{W_{0}}$ is $3$ and
$\overline{\alpha_{+}}$ is an isomorphism outside of $\overline{W_{2}}$. The
preimage of $\overline{W_{0}\cap W_{1}}$ in $\mathfrak{Y}$ is
$\Delta_{\mathfrak{Y}}$, which is a smooth divisor of $\mathfrak{Y}$. Hence
$\overline{W}_{0}$ is a canonical singularity and the pair
$(\overline{W_{0}},\overline{W_{0}\cap W_{1}})$ is plt. ∎

######  Proposition 5.13.

The pair $(\mathfrak{Z}_{+},0)$ is semi-dlt and $W_{0}$ is normal.

###### Proof.

Let

$V_{5}:=\mathcal{I}_{n-2},(\mathcal{I}_{2},\mathcal{I}_{2}^{\prime},\mathcal{I}_{3},x,y))\in
S^{[n-1]}\times\mathfrak{Z}_{2}^{+}|\mathcal{I}_{n-2}+\mathcal{I}_{3}=\mathcal{O}\\}$

where $\mathfrak{Z}_{2}^{+}=S^{[2,3]}\times_{S^{[2]}}S^{[2,3]}$ is the triple
moduli space when $n=2$. $V_{5}$ is an open subscheme of $\mathfrak{Z}_{+}$.
The pair$(V_{5},0)$ is a semi-snc by Example 5.9 and so is
$(V_{5}\cup\beta_{+}^{-1}(U),0)$. $W_{2}$ is the complement of
$V_{5}\cup\beta_{+}^{-1}(U)$ in $\mathfrak{Z}_{+}$ and

$codim_{W_{2}}\mathfrak{Z}_{+}=3.$

By Definition 4.13, $W_{0}\cap W_{1}$ is the conductor subscheme of
$\mathfrak{Z}_{+}$. By Example 4.9, $W_{0}\cap W_{1}$ is a canonical
singularity. By the inversion of adjunction theorem Theorem 4.4, the pair
$(W_{1},W_{0}\cap W_{1})$ is plt and thus dlt.

By Lemma 5.12, $(\overline{W_{0}},\overline{W_{0}\cap W_{1}})$ is also a plt
pair and thus a dlt pair.

By Proposition 4.17, $(\mathfrak{Z}_{+},0)$ is a semi-dlt pair, and $W_{0}$ is
normal. ∎

######  Corollary 5.14.

We have the formula

$R\alpha_{+*}(\mathcal{O}_{\mathfrak{Y}})=\mathcal{O}_{W_{0}}.$

##  6\. The Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

######  Definition 6.1.

We will write $S_{1},S_{2}$ for two copies of $S$, in order to emphasize the
factors of $S\times S$ and write $\mathcal{M}_{1}$, $\mathcal{M}_{2}$ and
$\mathcal{M}_{3}$ for three copies of $\mathcal{M}$, in order to emphasize the
factors of $\mathcal{M}\times\mathcal{M}\times\mathcal{M}$. Given $P\in
D^{b}(\mathcal{M}_{1}\times\mathcal{M}_{2}\times S_{1})$ and $Q\in
D^{b}(\mathcal{M}_{2}\times\mathcal{M}_{3}\times S_{2})$, we define the
composition $QP\in D^{b}(\mathcal{M}_{1}\times\mathcal{M}_{3}\times
S_{1}\times S_{2})$ by

$QP:=R\pi_{13_{*}}(L\pi_{12}^{*}P\otimes L\pi_{23}^{*}Q),$

where $\pi_{12},\pi_{23}$ and $\pi_{13}$ are the projections from
$\mathcal{M}_{1}\times\mathcal{M}_{2}\times\mathcal{M}_{3}\times S_{1}\times
S_{2}$ to $\mathcal{M}_{1}\times\mathcal{M}_{2}\times S_{1}$,
$\mathcal{M}_{2}\times\mathcal{M}_{3}\times S_{2}$ and
$\mathcal{M}_{1}\times\mathcal{M}_{3}\times S_{1}\times S_{2}$ respectively.

######  Theorem 6.2.

Given two integers $m$ and $r$ and let

$\displaystyle e_{m-r}=\mathcal{L}_{1}^{m-r}\mathcal{O}_{S^{[n,n+1]}}\in
D^{b}(S^{[n]}\times S^{[n+1]}\times S)$ $\displaystyle
f_{r}=\mathcal{L}_{1}^{k-1}\mathcal{O}_{S^{[n,n+1]}}[1]\in
D^{b}(S^{[n+1]}\times S^{[n]}\times S).$

Then

  1. (1)

If $m>0$, then there are $b_{m,r}^{k}\in D^{b}(S^{[n]}\times S^{[n]}\times
S\times S)$ for $0\leq k\leq m$ such that $b_{m,r}^{m}=e_{m-r}f_{r}$ and
$b_{m,r}^{0}=f_{r}e_{m-r}$, with exact triangles

$\mathfrak{B}_{m,r}^{k}:R\Delta_{*}(h_{m,k}^{+})[-1]\to b_{m,r}^{k}\to
b_{m,r}^{k+1}\to R\Delta_{*}(h_{m,k}^{+})$

  2. (2)

If $m<0$, then there are $c_{m,r}^{r}\in D^{b}(S^{[n]}\times S^{[n]}\times
S\times S)$ for $m\leq k\leq 0$ such that $c_{m,r}^{m}=e_{m-r}f_{r}$ and
$c_{m,r}^{0}=f_{r}e_{m-r}$, and exact triangles

$\mathfrak{C}_{m,r}^{k}:R\Delta_{*}(h_{m,k}^{-})[-1]\to c_{m,r}^{k-1}\to
c_{m,r}^{k}\to R\Delta_{*}(h_{m,k}^{-})$

  3. (3)

If $m=0$, there is an explicit isomorphism
$f_{r}e_{-r}=e_{r}f_{-r}\oplus\mathcal{O}_{\Delta}[1]$.

Theorem 1.1 follows from Theorem 6.2 and Theorem 3.2. We will prove Theorem
6.2 later in this section.

###  6.1. $e_{m-r}f_{r}$ and $f_{r}e_{m-r}$ Revisited

As $\mathfrak{Z}_{-}$ and $\mathfrak{Z}_{+}$ are both Cohen-Macaulay of
expected dimension, we have the following formula:

$\displaystyle e_{m-r}f_{r}=R\beta_{-*}(\mathcal{L}_{2}^{\prime
m-r}\mathcal{L}_{2}^{r-1}\mathcal{O}_{\mathfrak{Z}_{-}})[1]\in
D^{b}(S^{[n]}\times S^{[n]}\times S\times S)$ $\displaystyle
f_{r}e_{m-r}=R\beta_{+*}(\mathcal{L}_{1}^{m-r}\mathcal{L}_{1}^{\prime
r-1}\mathcal{O}_{\mathfrak{Z}_{+}})[1]\in D^{b}(S^{[n]}\times S^{[n]}\times
S\times S)$

By Proposition 5.6,
$R\alpha_{-*}\mathcal{O}_{\mathfrak{Y}}=\mathcal{O}_{\mathfrak{Z}_{-}}$ and by
Theorem 5.3

$\displaystyle e_{m-r}f_{r}$
$\displaystyle=R\theta_{*}(\mathcal{L}_{2}^{\prime
m-r}\mathcal{L}_{2}^{r-1}\mathcal{O}_{\mathfrak{Y}})[1]$ (6.1)
$\displaystyle=R\theta_{*}(\mathcal{L}_{1}^{\prime
m-r}\mathcal{L}_{1}^{r-1}\mathcal{O}((m-1)\Delta_{\mathfrak{Y}}))[1].$

###  6.2. $a_{m,r}^{k}$ and $\mathfrak{A}_{m,r}^{k}$

We recall the short exact sequence

(6.2)
$0\to\mathcal{O}_{\mathfrak{Y}}(-\Delta_{\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{Y}}\to\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to
0.$

######  Definition 6.3.

For any integer $r$, we define

$a_{m,r}^{k}=R\theta_{*}(\mathcal{L}_{1}^{\prime
m-r}\mathcal{L}_{1}^{r-1}\mathcal{O}(k\Delta_{\mathfrak{Y}}))[1].$

We define the exact traingles

(6.3) $\mathfrak{A}_{m,r}^{k}:a_{m,r}^{k-1}\to a_{m,r}^{k}\to
R\Delta_{*}(R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k})[1]\to a_{m,r}^{k-1}[1]$

by applying the functor $R\theta_{*}(-\otimes\mathcal{L}_{1}^{\prime
m-r}\mathcal{L}_{1}^{r-1}\mathcal{O}(k\Delta_{\mathfrak{Y}}))[1]$ to 6.2.

By 6.1, $e_{m-r}f_{r}\cong a_{m,r}^{m-1}$. Moreover, when $m<0$, by 3.13,
$R(p_{n}\circ q_{n})_{*}\mathcal{L}_{1}^{-1}\mathcal{L}_{2}^{m}=0$ and
$e_{m-r}f_{r}\cong a_{m,r}^{m-1}\cong a_{m,r}^{m}$.

###  6.3. $\mathfrak{B}_{m,r}^{\prime 0}$ and $\mathfrak{C}_{m,r}^{\prime 0}$

By Example 4.9,
$R\alpha_{+*}\mathcal{O}_{\Delta_{\mathfrak{Y}}}=\mathcal{O}_{W_{0}\cap
W_{1}}$. Taking the $R\alpha_{+*}$ functor to the short exact sequence 6.2, we
have the short exact sequence

$0\to
R\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})\to\mathcal{O}_{W_{0}}\to\mathcal{O}_{W_{0}\cap
W_{1}}\to 0.$

Recalling the short exact sequence in Corollary 3.7:

$0\to\mathcal{L}\otimes\mathcal{O}_{W_{1}}\to\mathcal{O}_{W_{1}}\to\mathcal{O}_{W_{0}\cap
W_{1}}\to 0.$

We have the following commutative diagram:

${0}$${0}$${R\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})}$${R\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})}$${0}$${\mathcal{L}\otimes\mathcal{O}_{W_{1}}}$${\mathcal{O}_{\mathfrak{Z}_{+}}}$${\mathcal{O}_{W_{0}}}$${0}$${0}$${\mathcal{L}\otimes\mathcal{O}_{W_{1}}}$${\mathcal{O}_{W_{1}}}$${\mathcal{O}_{W_{0}\cap
W_{1}}}$${0}$${0}$${0}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where all rows and columns are short exact sequences. Thus we have short exact
sequences:

(6.4) $\displaystyle 0\to
R\alpha_{+*}\mathcal{O}(-\Delta_{\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{Z}_{+}}\to\mathcal{O}_{W_{1}}\to
0$ (6.5) $\displaystyle
0\to\mathcal{L}_{1}\mathcal{O}_{S^{[n,n+1]}}\to\mathcal{O}_{\mathfrak{Z}_{+}}\to\mathcal{O}_{W_{0}}\to
0.$

Taking the functor $R\beta_{+*}(-\otimes\mathcal{L}_{1}^{\prime
m-r}\mathcal{L}_{1}^{r-1})[1]$ to 6.4 and 6.5 respectively, we get the exact
triangles

(6.6) $\displaystyle\mathfrak{C}_{m,r}^{\prime 0}:a_{m,r}^{-1}\to
f_{r}e_{m-r}\to R\Delta_{*}(Rp_{n*}\mathcal{L}^{m-1})[1]\to a_{m,r}^{-1}[1]$
(6.7) $\displaystyle\mathfrak{B}_{m,r}^{\prime
0}:R\Delta_{*}(Rp_{n*}\mathcal{L}^{m})[1]\to f_{r}e_{m-r}\to a_{m,r}^{0}\to
R\Delta_{*}(Rp_{n*}\mathcal{L}^{m})[2].$

###  6.4. $\mathfrak{B}_{m,r}^{k}$ and $\mathfrak{C}_{m,r}^{k}$

######  Definition 6.4.

When $m>0$, we define $b_{m.r}^{k}\in D^{b}(S^{[n]}\times S^{[n]}\times
S\times S)$ and natural transforms $\mathfrak{B}_{m,r}^{k}:b_{m,r}^{k}\to
b_{m,r}^{k+1}$ by

(6.8) $b_{m,r}^{k}:=\begin{cases}f_{r}e_{m-r}&k=0\\\ a_{m,r}^{k-1}&1\leq k\leq
m\end{cases}$

(6.9) $\mathfrak{B}_{m,r}^{k}:=\begin{cases}\mathfrak{B}_{m,r}^{\prime
0}&k=0\\\ \mathfrak{A}_{m,r}^{k}&1\leq k\leq m-1.\end{cases}$

We have $b_{m,r}^{0}=f_{r}e_{m-r}$ and $b_{m,r}^{m}=e_{m-r}f_{r}$.

When $m<0$, we define $c_{m,r}^{k}\in D^{b}(S^{[n]}\times S^{[n]}\times
S\times S)$ and $\mathfrak{C}_{m,r}^{k}:c_{m,r}^{k-1}\to c_{m,r}^{k}$ by

(6.10) $c_{m,r}^{k}=\begin{cases}f_{r}e_{m-r}&k=0\\\ a_{m,r}^{k}&m\leq
k\leq-1\end{cases}$

(6.11) $\mathfrak{C}_{m,r}^{k}=\begin{cases}\mathfrak{C}_{m,r}^{\prime
0}&k=0\\\ \mathfrak{A}_{m,r}^{k}&m+1\leq k\leq-1\end{cases}$

We have $f_{r}e_{m-r}=c_{m,r}^{0}$ and $e_{m-r}f_{r}=c_{m,r}^{m}$.

###### Proof of Theorem 6.2.

When $m>0$, we only need to prove that the cone of $\mathfrak{B}_{m,r}^{k}$ is
$R\Delta_{*}(h_{m,k}^{+})$ and the cone of $\mathfrak{C}_{m,r}^{k}$ is
$R\Delta_{*}(h_{m,k}^{-})$. It follows from 6.3, 6.6 and 6.7.

When $m=0$,

$\displaystyle
e_{-r}f_{r}=R\beta_{+*}(R\alpha_{+*}(\mathcal{L}_{1}^{\prime-r}\mathcal{L}_{1}^{r-1}\mathcal{O}(-\Delta_{\mathfrak{Y}}))).$

We have the short exact sequence:

(6.12)
$0\to\mathcal{L}_{1}^{\prime-r}\mathcal{L}_{1}^{r-1}\mathcal{O}_{\mathfrak{Z}_{+}}\to\mathcal{L}_{1}^{\prime-r}\mathcal{L}_{1}^{r-1}(\mathcal{O}_{W_{0}}\oplus\mathcal{O}_{W_{1}})\to\mathcal{L}^{-1}\mathcal{O}_{W_{0}\cap
W_{1}}\to 0.$

and

$\displaystyle R\beta_{+*}(\mathcal{L}^{-1}\mathcal{O}_{W_{0}\cap W_{1}})$
$\displaystyle=\\{R\beta_{+*}(\mathcal{O}_{W_{1}}\to\mathcal{L}^{-1}\mathcal{O}_{W_{1}})\\}$
$\displaystyle=R\Delta_{*}\\{\mathcal{O}_{S^{[n]}\times
S}\to\mathcal{O}_{S^{[n]}\times S}\\}\text{ by \lx@cref{creftype~refnum}{L1}
}$ $\displaystyle=0$

By 6.4 and 6.12, we have isomorphisms

$\displaystyle f_{r}e_{-r}\cong
R\beta_{+*}\mathcal{L}_{1}^{\prime-r}\mathcal{L}_{1}^{r-1}(\mathcal{O}_{W_{0}})[1]\oplus\mathcal{O}_{\Delta}[1]$
$\displaystyle e_{-r}f_{r}\cong
R\beta_{+*}\mathcal{L}_{1}^{\prime-r}\mathcal{L}_{1}^{r-1}(\mathcal{O}_{W_{0}})[1]$

∎

##  7\. Extension Classes between $h_{m,k}^{\pm}$

When $m>0$, the composition of $\mathfrak{B}_{m,r}^{k}$ and
$\mathfrak{B}_{m,r}^{k+1}$ in Theorem 6.2 induces an extension of
$R\Delta_{*}h_{m,k}^{+}$ and $R\Delta_{*}h_{m,k+1}^{+}$, i.e. an extension
class of

$Hom(R\Delta_{*}h_{m,k+1}^{+},R\Delta_{*}h_{m,k}^{+}[1]).$

By the Hochschild-Kostant-Rosenberg theorem ([36], also Remark 7.4),

$Hom(R\Delta_{*}h_{m,k+1}^{+},R\Delta_{*}h_{m,k}^{+}[1])=\bigoplus_{j=0}^{\infty}Hom(h_{m,k+1}^{+}\otimes\wedge^{j}T^{*}(S^{[n]}\times
S),h_{m,k}^{+}[1-j]).$

When $m<0$, there is also be an extension class of

$Hom(R\Delta_{*}h_{m,k}^{-},R\Delta_{*}h_{m,k-1}^{-}[1])=\bigoplus_{j=0}^{\infty}Hom(h_{m,k}^{-}\otimes\wedge^{j}T^{*}(S^{[n]}\times
S),h_{m,k-1}^{-}[1-j]).$

In this section, we will explicitly computing the above extension classes and
prove that

######  Proposition 7.1.

The extension class in
$Hom(R\Delta_{*}h_{m,k+1}^{+},R\Delta_{*}h_{m,k}^{+}[1])$ is

$(0,\mathfrak{H}_{m,k}^{+},0,0,\cdots)$

and the extension class in
$Hom(R\Delta_{*}h_{m,k}^{-},R\Delta_{*}h_{m,k-1}^{-}[1])$ is

$(0,\mathfrak{H}_{m,k}^{-},0,0,\cdots)$

where $\mathfrak{H}_{m,k}^{-}$ and $\mathfrak{H}_{m,k}^{-}$ will be defined in
Definition 7.6.

Proposition 7.1 will be proved later in this section.

###  7.1. A Splitting Formula and HKR isomorphism

Let $j:Y\subset X$ be a regular embedding of smooth schemes and $N_{Y/X}$ be
the normal bundle. In a first step, we will suppose that $Y$ is given as the
zero locus of a regular section $s\in H^{0}(X,\mathcal{E})$ of a locally free
sheaf $\mathcal{E}$ of rank $r$. In this case, the structure sheaf
$\mathcal{O}_{Y}$ can be resolved by the Koszul complex:

(7.1)
$0\to\wedge^{r}\mathcal{E}^{\vee}\to\cdots\to\mathcal{E}^{\vee}\to\mathcal{O}_{X}\to
j_{*}\mathcal{O}_{Y}\to 0,$

with isomorphism given by contraction with $s$. The normal bundle $N_{Y/X}$ is
$\mathcal{E}|_{Y}$.

######  Proposition 7.2 (Proposition 11.1 of [14]).

There exists a canonical isomorphism:

$Lj^{*}Rj_{*}\mathcal{O}_{Y}\cong\bigoplus_{k}\wedge^{k}N_{X/Y}^{\vee}[k].$

A stronger version of Proposition 7.2 was studied in [1]:

######  Theorem 7.3 (Theorem 1.8, Lemma 3.1 of [1]).

Let $j:Y\to X$ be a closed embedding of smooth varieties. A choice of
splitting of

(7.2) $0\to T_{Y}\to T_{X}|_{Y}\to N_{X/Y}\to 0$

determines an isomorphism:

$Lj^{*}Rj_{*}(\mathcal{O}_{Y})\cong\bigoplus_{k}\wedge^{k}N_{X/Y}^{\vee}[k].$

######  Remark 7.4.

Let $X$ be a smooth variety and $\Delta_{X}:X\to X\times X$ the diagonal
embedding. Then $T_{X\times X}|_{X}=T_{X}\oplus T_{X}$ has a canonical
splitting and thus

$L\Delta_{X}^{*}R\Delta_{X*}(\mathcal{O}_{X})=\bigoplus_{k=0}^{dim(X)}\wedge^{k}T^{*}X[k]$

which is the HKR isomorphism in [36].

###  7.2. A morphism $s:\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ
q_{n})^{*}T(S^{n}\times S)$

In this subsection, we first construct two canonical morphisms:

(7.3) $\displaystyle s_{1}:\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ
q_{n})^{*}TS^{[n]}\quad
s_{2}:\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ q_{n})^{*}(TS)$

Let $\pi:S^{[n-1,n,n+1]}\times S\to S^{[n-1,n,n+1]}$ be the projection
morphism and $\Gamma$ be the graph of the projection map to $S$. Then there
exists short exact sequences:

$0\to\mathcal{I}_{n+1}\to\mathcal{I}_{n}\to\pi^{*}\mathcal{L}_{1}\otimes\mathcal{O}_{\Gamma}\to
0\quad
0\to\mathcal{I}_{n}\to\mathcal{I}_{n-1}\to\pi^{*}\mathcal{L}_{2}\otimes\mathcal{O}_{\Gamma}\to
0$

which induces two global sections:

$r_{1}:\mathcal{L}_{1}^{-1}\to\mathcal{H}om_{\pi}(\mathcal{I}_{n},\mathcal{O}_{\Gamma}),\quad
r_{2}:\mathcal{L}_{2}\to\mathcal{E}xt^{1}_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_{n})$

Let $\text{proj}_{n}:S^{[n]}\times S\to S^{[n]}$ be the projection map and
$\Gamma_{n}$ be the graph of the projection map from $S^{[n]}\times S\to S$.
Then

$\displaystyle(p_{n}\circ q_{n})^{*}TS^{[n]}$ $\displaystyle=(p_{n}\circ
q_{n})^{*}\mathcal{E}xt_{\text{proj}_{n}}^{1}(\mathcal{I}_{n},\mathcal{I}_{n})$
$\displaystyle=\mathcal{E}xt^{1}_{\pi}(\mathcal{I}_{n},\mathcal{I}_{n})$
$\displaystyle(p_{n}\circ q_{n})^{*}TS$
$\displaystyle=\mathcal{E}xt^{1}_{\pi}(\mathcal{O}_{\Gamma},\mathcal{O}_{\Gamma})$

by the flat base change theorem. With the composition of the following two
natural homomorphisms:

$\displaystyle\mathcal{E}xt^{1}_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_{n})\otimes\mathcal{H}om_{\pi}(\mathcal{I}_{n},\mathcal{O}_{\Gamma})\to\mathcal{E}xt^{1}_{\pi}(\mathcal{I}_{n},\mathcal{I}_{n})=(p_{n}\circ
q_{n})^{*}TS^{[n]}$
$\displaystyle\mathcal{H}om_{\pi}(\mathcal{I}_{n},\mathcal{O}_{\Gamma})\otimes\mathcal{E}xt^{1}_{\pi}(\mathcal{O}_{\Gamma},\mathcal{I}_{n})\to\mathcal{E}xt^{1}_{\pi}(\mathcal{O}_{\Gamma},\mathcal{O}_{\Gamma})=(p_{n}\circ
q_{n})^{*}TS,$

we get the two canonical morphisms in 7.3

$\displaystyle s_{1}:\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ
q_{n})^{*}TS^{[n]}\quad
s_{2}:\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ q_{n})^{*}(TS).$

Let

$s=(r_{1},r_{2}):\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ
q_{n})^{*}T(S^{[n]}\times S).$

Another way of inducing the morphism is restricting $\theta:\mathfrak{Y}\to
S^{[n]}\times S^{[n]}\times S\times S$ to the diagonal of $S^{[n]}\times
S^{[n]}\times S\times S$. We get the Cartesian diagram:

(7.4) ${S^{[n-1,n,n+1]}}$${\mathfrak{Y}}$${S^{[n]}\times S}$${S^{[n]}\times
S^{[n]}\times S\times S}$$\scriptstyle{p_{n}\circ
q_{n}}$$\scriptstyle{\Delta_{\mathfrak{Y}}}$$\scriptstyle{\theta}$$\scriptstyle{\Delta}$

and the normal bundle of $S^{[n]}\times S$ in $S^{[n]}\times S^{[n]}\times
S\times S$ is $T(S^{[n]}\times S)$. Thus the diagram 7.4 induces a morphism

(7.5)
$s^{\prime}:N_{\mathfrak{Y}/\Delta_{\mathfrak{Y}}}=\mathcal{L}_{1}^{-1}\mathcal{L}_{2}\to(p_{n}\circ
q_{n})_{*}T(S^{[n]}\times S).$

######  Proposition 7.5.

The morphism $s^{\prime}$ coincides with $s$ in 7.3.

###### Proof.

A closed point $t$ on $S^{[n-1,n,n+1]}$ corresponds to two short exact
sequences of coherent sheaves on $S$:

$\displaystyle
0\to\mathcal{I}_{n}\xrightarrow{i_{n}}\mathcal{I}_{n-1}\xrightarrow{j_{n}}k_{x}\to
0$ $\displaystyle\quad
0\to\mathcal{I}_{n+1}\xrightarrow{i_{n+1}}\mathcal{I}_{n}\xrightarrow{j_{n+1}}k_{x}\to
0.$

which induces $r_{1}\in Hom(\mathcal{I}_{n},k_{x})$ and $r_{2}\in
Ext^{1}(k_{x},\mathcal{I}_{n})$. Let $V$ be the vector space of pairs
$\\{(w_{0},w_{1})\in Ext^{1}(\mathcal{I}_{n-1},\mathcal{I}_{n-1})\oplus
Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\\}$ such that $w_{0},w_{1}$ map to
the same element of $Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n-1})$. By the proof
of Proposition 5.28 of [29], elements in $V$ are in 1-1 correspondence with
the commutative diagrams of short exact sequences:

${0}$${0}$${0}$${0}$${\mathcal{I}_{n}}$${\mathcal{A}}$${\mathcal{I}_{n}}$${0}$${0}$${\mathcal{I}_{n-1}}$${\mathcal{B}}$${\mathcal{I}_{n-1}}$${0}$${0}$${k_{x}}$${\mathcal{C}}$${k_{x}}$${0}$${0}$${0}$${0}$$\scriptstyle{i_{n}}$$\scriptstyle{i_{n}}$$\scriptstyle{j_{n}}$$\scriptstyle{j_{n}}$

and thus induce a morphism $dp_{S}^{1}:V\to Ext^{1}(k_{x},k_{x})$. Let
$V^{\prime}$ be the vector space of pairs

$\\{(w_{1}^{\prime},w_{2})\in Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\oplus
Ext^{1}(\mathcal{I}_{n+1},\mathcal{I}_{n+1})\\}$

auch that $w_{1}^{\prime},w_{2}$ map to the same element of
$Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n+1})$. Then elements in $V^{\prime}$
are in 1-1 correspondence with the commutative diagrams of short exact
sequences:

${0}$${0}$${0}$${0}$${\mathcal{I}_{n+1}}$${\mathcal{A}^{\prime}}$${\mathcal{I}_{n+1}}$${0}$${0}$${\mathcal{I}_{n}}$${\mathcal{B}^{\prime}}$${\mathcal{I}_{n}}$${0}$${0}$${k_{x}}$${\mathcal{C}^{\prime}}$${k_{x}}$${0}$${0}$${0}$${0}$

and thus also induces a morphism $dp_{S}^{2}:V^{\prime}\to
Ext^{1}(k_{x},k_{x})$.

Consider the natural morphism:

$\displaystyle\hat{s}_{1}:Hom(\mathcal{I}_{n},k_{x})\otimes
Ext^{1}(k_{x},\mathcal{I}_{n})\to Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})$
$\displaystyle\hat{s}_{2}:Hom(\mathcal{I}_{n},k_{x})\otimes
Ext^{1}(k_{x},\mathcal{I}_{n})\to Ext^{1}(k_{x},k_{x}).$

and the short exact sequences

$\displaystyle Hom(\mathcal{I}_{n},k_{x})\xrightarrow{\hat{s}_{1}(-\otimes
r_{2})}Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\to
Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n-1})$ $\displaystyle
Ext^{1}(k_{x},\mathcal{I}_{n})\xrightarrow{\hat{s}_{1}(r_{1}\otimes-)}Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\to
Ext^{1}(\mathcal{I}_{n+1},\mathcal{I}_{n}).$

For any element $u\in Hom(\mathcal{I}_{n},k_{x})$, $(\hat{s}_{1}(u\otimes
r_{2}),0)\in V$. Moreover, by diagram chasing (we left it to interested
readers)

(7.6) $dp_{S}^{1}(\hat{s}_{1}(u\otimes r_{2}),0)=\hat{s}_{2}(u\otimes r_{2}).$

Simlilarly, for any element $v\in Ext^{1}(k_{x},\mathcal{I}_{n})$,
$(0,\hat{s}_{1}(r_{1}\otimes v))\in V^{\prime}$ and

(7.7) $dp_{S}^{2}(0,\hat{s}_{1}(r_{1}\otimes v))=\hat{s}_{2}(r_{1}\otimes v).$

By Proposition 5.28 of [29], the tangent space $T_{t}\mathfrak{Y}$ is the
space of $(w_{0},w_{1},w_{1}^{\prime},w_{2},u,v)$ in

$Ext^{1}(\mathcal{I}_{n-1},\mathcal{I}_{n-1})\oplus
Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\oplus
Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\oplus
Ext^{1}(\mathcal{I}_{n+1},\mathcal{I}_{n+1})\oplus Ext^{1}(k_{x},k_{x})\oplus
Ext^{1}(k_{x},k_{x})$

such that

  1. (1)

$(w_{0},w_{1})$ and $(w_{0},w_{1}^{\prime})$ are in $A$.

  2. (2)

$(w_{1},w_{2})$ and $(w_{1}^{\prime},w_{2})$ are in $A^{\prime}$.

  3. (3)

$dp_{S}^{1}(w_{1},w_{0})=dp_{S}^{2}(w_{2},w_{1}^{\prime})=u$,

  4. (4)

$dp_{S}^{1}(w_{1}^{\prime},w_{0})=dp_{S}^{2}(w_{2},w_{1})=v$.

If $(w_{0},w_{1},w_{1}^{\prime},w_{2},u,v)$ is in the tangent space, then so
is

$\frac{(2w_{0},w_{1}+w_{1}^{\prime},w_{1}+w_{1}^{\prime},2w_{2},u+v,u+v)}{2}\text{
and }(0,w_{1}^{\prime}-w_{1},w_{1}^{\prime}-w_{1},0,u-v,v-u).$

Hence the tangent space $T_{t}\mathfrak{Y}$ decompose into a direct sum of two
subspaces: the subspace $w_{1}=w_{1}^{\prime}$ which is $T_{t}S^{[n-1,n,n+1]}$
and the subspace $N$ which consists of elements $(w_{1},u)\in
Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n})\oplus Ext^{1}(k_{x},k_{x})$ such that

  1. (1)

$w_{1}$ maps to $0$ in $Ext^{1}(\mathcal{I}_{n},\mathcal{I}_{n-1})$ and
$Ext^{1}(\mathcal{I}_{n+1},\mathcal{I}_{n}),$

  2. (2)

$dp_{S}^{1}(w_{1},0)=dp_{S}^{2}(0,w_{1})=u.$

The image of $s$ is $(\hat{s}_{1}(r_{1}\otimes r_{2}),\hat{s}_{2}(r_{1}\otimes
r_{2}))$ and we only need to prove that it is in $N$. It follows from the fact
that

$dp_{S}^{1}(\hat{s}_{1}(r_{1}\otimes
r_{2}),0)=dp_{S}^{2}(0,\hat{s}_{1}(r_{1}\otimes
r_{2}))=\hat{s}_{2}(r_{1}\otimes r_{2})$

by 7.6, 7.7.

∎

###  7.3. The definition of $\mathfrak{H}_{m,k}^{\pm}$

Consider the dual of $s$

$s^{\vee}:(p_{n}\circ q_{n})^{*}T^{*}(S^{[n]}\times
S)\to\mathcal{L}_{1}\mathcal{L}_{2}^{-1}.$

Since $L(p_{n}\circ q_{n})^{*}$ is the left adjoint functor of $R(p_{n}\circ
q_{n})_{*}$, $s^{\vee}$ induces another morphism

$\mathfrak{h}:T^{*}(S^{[n]}\times S)\to R(p_{n}\circ
q_{n})_{*}(\mathcal{L}_{1}\mathcal{L}_{2}^{-1}).$

The identity

$\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k}\otimes\mathcal{L}_{1}\mathcal{L}_{2}^{-1}\cong\mathcal{L}_{1}^{m-k}\mathcal{L}_{2}^{k-1}$

induces morphisms

(7.8) $R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k}\otimes^{L}R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}\mathcal{L}_{2}^{-1}\to R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}^{m-k}\mathcal{L}_{2}^{k-1}$

and we define

(7.9) $\tau_{m}^{k}:R(p_{n}\circ
q_{n})_{*}\mathcal{L}_{1}^{m-1-k}\mathcal{L}_{2}^{k}\otimes^{L}T^{*}(S^{[n]}\times
S)\to R(p_{n}\circ q_{n})_{*}\mathcal{L}_{1}^{m-k}\mathcal{L}_{2}^{k-1}.$

to be the composition of $\mathfrak{h}$ and 7.8.

Still by Lemma 3.10,

$Rq_{n*}(\mathcal{L}_{1}^{m-1})=\\{\mathcal{L}^{m}\to\mathcal{L}^{m-1}\\}$

is the cone of $\mathcal{L}^{m}$ to $\mathcal{L}^{m-1}$. Hence we have exact
triangles

(7.10) $Rp_{n*}\mathcal{L}^{m}\to
Rp_{n*}\mathcal{L}^{m-1}\xrightarrow{\mu_{m}}R(p_{n}\circ
q_{n})_{*}(\mathcal{L}_{1}^{m-1})\xrightarrow{\nu_{m}}Rp_{n*}\mathcal{L}^{m}[1].$

######  Definition 7.6.

When $m>k\geq 0$, we define

$\mathfrak{H}_{m,k}^{+}:h_{m,k+1}^{+}\otimes T^{*}(S^{[n]}\times S)\to
h_{m,k}^{+}$

by

(7.11) $\mathfrak{H}_{m,k}^{+}=\begin{cases}\tau_{m,k}&k>0\\\
\nu_{m}\circ\tau_{m,0}&k=0\end{cases}$

and when $m<k\leq 0$, we define

$\mathfrak{H}_{m,k}^{-}:h_{m,k}^{-}\otimes T^{*}(S^{[n]}\times S)\to
h_{m,k-1}^{+}$

by

(7.12) $\mathfrak{H}_{m,k}^{+}=\begin{cases}\tau_{m,k}&k<0\\\
\tau_{m,0}\circ\mu_{m}&k=0\end{cases}$

###### Proof of Proposition 7.1.

We only compute the postive part of the extension and only consider the $k>0$
and the case $k=0$ follows from the 6.5. The extension class is induced from
the short exact sequence on $\mathfrak{Y}$:

$0\to\mathcal{L}_{1}^{m-k-1}\mathcal{L}_{2}^{k}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to\mathcal{L}_{1}^{m-k-2}\mathcal{L}_{2}^{k+1}\mathcal{O}_{2\Delta_{\mathfrak{Y}}}\to\mathcal{L}_{1}^{m-k-2}\mathcal{L}_{2}^{k+1}\mathcal{O}_{\Delta_{\mathfrak{Y}}}\to
0$

where $\mathcal{O}_{2\Delta_{\mathfrak{Y}}}$ is the cokernel of
$\mathcal{O}(-2\Delta_{\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{Y}}$. It
induces an extension class in

$\displaystyle
Ext^{1}_{\mathfrak{Y}}(\mathcal{L}_{1}^{m-k-2}\mathcal{L}_{2}^{k+1}\mathcal{O}_{\Delta_{\mathfrak{Y}}},\mathcal{L}_{1}^{m-k-1}\mathcal{L}_{2}^{k}\mathcal{O}_{\Delta_{\mathfrak{Y}}})$
$\displaystyle=$ $\displaystyle
Ext^{1}_{\Delta_{\mathfrak{Y}}}(\mathcal{L}_{1}^{m-k-2}\mathcal{L}_{2}^{k+1},\mathcal{L}_{1}^{m-k-1}\mathcal{L}_{2}^{k})\oplus
Hom_{\Delta{\mathfrak{Y}}}(\mathcal{L}_{1}^{m-k-1}\mathcal{L}_{2}^{k},\mathcal{L}_{1}^{m-k-1}\mathcal{L}_{2}^{k}).$

by Theorem 7.3 and (3) of Theorem 5.3. Still by Theorem 7.3, the extension
class is $(0,id)$ and thus would map to
$(0,\mathfrak{H}_{m,k}^{+},0,0,\cdots)$ by the functorial property. ∎

##  Appendix A  Exterior Powers and Formal Series

######  Definition A.1.

Let $V$ be a locally free sheaf over $X$, we define the exterior powers of $V$
by

$\wedge^{\bullet}(xV)=\sum_{i=0}^{\infty}(-x)^{i}[\wedge^{i}V],\quad\wedge^{\bullet}(-xV)=\sum_{i=0}^{\infty}x^{i}[S^{i}V]$

as elements in $K(X)[[x]]$.

######  Definition A.2.

For a two term complex of locally free sheaves

$U:=\\{W\xrightarrow{\mathfrak{u}}V\\}$

we define

$\wedge^{\bullet}(-xU)=\wedge^{\bullet}(xW)\wedge^{\bullet}(-xV)\qquad\wedge^{\bullet}(xU)=\wedge^{\bullet}(xV)\wedge^{\bullet}(-xW).n$

$[S^{k}(U)]$ and $[\wedge^{k}(U)]$ are the $x^{k}$ coefficients of
$\wedge^{\bullet}(-xU)$ and $\wedge^{\bullet}(xU)$ respectively.

######  Lemma A.3 (Exercise II.5.16 of [13]).

For a two term complex of locally free sheaves

$U:=\\{W\xrightarrow{\mathfrak{u}}V\\},$

we have

(A.1) $\wedge^{\bullet}(xU)\wedge^{\bullet}(-xU)=1.$

######  Definition A.4 (Section 3.8 of [26]).

For any non-negative integer $m$, we define $h_{m}^{\pm}\in K(S^{[n]}\times
S)$

(A.2) $\displaystyle
h_{m}^{+}:=[\omega_{S}]\sum_{i=0}^{m}(-1)^{m-i}[\omega_{S}^{-i}][S^{i}\mathcal{I}_{n}][\wedge^{m-i}\mathcal{I}_{n}]$
(A.3) $\displaystyle
h_{m}^{-}:=\sum_{i=0}^{m}(-1)^{i}[\omega_{S}^{i}][\wedge^{i}\mathcal{I}_{n}^{\vee}][S^{m-i}\mathcal{I}_{n}^{\vee}]$

where we abuse the notation to denote

$\mathcal{I}_{n}:=\\{W_{n}\xrightarrow{s}V_{n}\\}$

in the short exact sequence 2.1.

######  Lemma A.5.

Definition A.4 and Definition 2.8 are equivalent.

###### Proof.

First note that

(A.4) $\sum_{i=0}^{m}[\wedge^{i}\mathcal{I}_{n}][S^{m-i}\mathcal{I}_{n}]=0$

by A.1. Thus

$\displaystyle[\omega_{S}]\sum_{i=0}^{m}(-1)^{m-i}[\omega_{S}^{-i}][S^{i}\mathcal{I}_{n}][\wedge^{m-i}\mathcal{I}_{n}]$
$\displaystyle=[\omega_{S}]\sum_{i=1}^{m}(-1)^{m-i}([\omega_{S}^{-i}]-1)[S^{i}\mathcal{I}_{n}][\wedge^{m-i}\mathcal{I}_{n}]$
by A.4
$\displaystyle=([\omega_{S}]-1)\sum_{i=1}^{m}(-1)^{m-i}[S^{i}\mathcal{I}_{n}][\wedge^{m-i}\mathcal{I}_{n}]\sum_{j=1}^{i}[\omega_{S}^{-j+1}]$
$\displaystyle=([\omega_{S}]-1)\sum_{j=1}^{m}[\omega_{S}^{-j+1}]\sum_{i=j}^{m}(-1)^{m-i}[S^{i}\mathcal{I}_{n}][\wedge^{m-i}\mathcal{I}_{n}]$
$\displaystyle=(1-[\omega_{S}])\sum_{j=0}^{m-1}[\omega_{S}^{-j}]\sum_{i=0}^{j}(-1)^{i}[S^{m-i}\mathcal{I}_{n}][\wedge^{i}\mathcal{I}_{n}].$
by A.4

Hence $h_{m}^{+}$ are equivalent in two defintions.

Then we note that

(A.5)
$\sum_{i=0}^{m}(-1)^{i}[S^{i}\mathcal{I}_{n}^{\vee}][\wedge^{m-i}\mathcal{I}_{n}^{\vee}]=0$

by A.1. Thus

$\displaystyle\sum_{i=0}^{m}(-1)^{i}[\omega_{S}^{i}][\wedge^{i}\mathcal{I}_{n}^{\vee}][S^{m-i}\mathcal{I}_{n}^{\vee}]$
$\displaystyle=\sum_{i=1}^{m}(-1)^{i}([\omega_{S}^{i}]-1)[\wedge^{i}\mathcal{I}_{n}^{\vee}][S^{m-i}\mathcal{I}_{n}^{\vee}]$
by A.5
$\displaystyle=([\omega_{S}]-1)\sum_{i=1}^{m}(-1)^{i}[\wedge^{i}\mathcal{I}_{n}^{\vee}][S^{m-i}\mathcal{I}_{n}^{\vee}]\sum_{j=1}^{i}[\omega_{S}^{j-1}]$
$\displaystyle=([\omega_{S}]-1)\sum_{j=1}^{m}[\omega_{S}^{j-1}]\sum_{i=j}^{m}(-1)^{i}[\wedge^{i}\mathcal{I}_{n}^{\vee}][S^{m-i}\mathcal{I}_{n}^{\vee}]$
$\displaystyle=(1-[\omega_{S}])\sum_{j=0}^{m-1}(-1)^{j}[\omega_{S}^{j}]\sum_{i=0}^{j}(-1)^{i}[\wedge^{m-i}\mathcal{I}_{n}^{\vee}][S^{i}\mathcal{I}_{n}^{\vee}]$
by A.5

Hence $h_{m}^{-}$ are equivalent in two defintions. ∎

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