# Homological mirror symmetry for the symmetric squares of punctured spheres

Yankı Lekili  and  Alexander Polishchuk  Imperial College London  University
of Oregon, National Research University Higher School of Economics, and Korea
Institute for Advanced Study

###### Abstract.

For an appropriate choice of a $\mathbb{Z}$-grading structure, we prove that
the wrapped Fukaya category of the symmetric square of a $(k+3)$-punctured
sphere, i.e. the Weinstein manifold given as the complement of $(k+3)$ generic
lines in $\mathbb{C}P^{2}$ is quasi-equivalent to the derived category of
coherent sheaves on a singular surface $\mathcal{Z}_{2,k}$ constructed as the
boundary of a toric Landau-Ginzburg model
$(\mathcal{X}_{2,k},{\mathbf{w}}_{2,k})$. We do this by first constructing a
quasi-equivalence between certain categorical resolutions of both sides and
then localising. We also provide a general homological mirror symmetry
conjecture concerning all the higher symmetric powers of punctured spheres.
The corresponding toric LG-models $(\mathcal{X}_{n,k},{\mathbf{w}}_{n,k})$ are
constructed from the combinatorics of curves on the punctured sphere and are
related to small toric resolutions of the singularity $x_{1}\ldots
x_{n+1}=v_{1}\ldots v_{k}$.

###### Key words and phrases:

Homological mirror symmetry, wrapped Fukaya category, symmetric products of
surfaces, matrix factorizations

###### 2010 Mathematics Subject Classification:

Primary 14J33; Secondary 14F05, 53D37

##  1\. Introduction

###  1.1. Motivating examples

We begin by considering the family

$\mathcal{Y}_{1,1}\coloneqq\mathrm{Spec}\,\mathbb{C}[x,y,V]/(xy-V)$

over $\mathbb{A}^{1}_{V}$ as a “toric degeneration” of the general fiber to
the singular variety $\mathcal{Z}_{1,1}\coloneqq\\{xy=0\\}$ which is a gluing
of two toric varieties $\mathbb{A}^{1}_{x}$ and $\mathbb{A}^{1}_{y}$ at their
toric boundary $0$. The symplectic mirror to this degeneration is well
understood [3]. Namely, we consider the cotangent bundle of the circle
$T^{*}\mathbb{S}^{1}$ as our symplectic manifold and $D_{1,1}\coloneqq\\{z\\}$
a marked point on $T^{*}\mathbb{S}^{1}$ as a symplectic divisor. Then, one has
an equivalence of pre-triangulated categories over $\mathbb{C}[V]$

$\displaystyle\mathcal{W}(T^{*}\mathbb{S}^{1},D_{1,1})\simeq
D^{b}\mathrm{Coh}(\mathcal{Y}_{1,1})$ (1.1)

where the left hand side is the wrapped Fukaya category of
$T^{*}\mathbb{S}^{1}$ relative to the divisor $D_{1,1}$ (cf. [21] [11] [25])
and the right hand side is the bounded derived category of $\mathcal{Y}_{1,1}$
over $\mathbb{A}^{1}_{V}$.

The equivalence (1.1) fits into the general framework of Gross-Siebert, Gross-
Hacking-Keel [6], Abouzaid-Auroux-Katzarkov [1], where the general fiber is a
log-Calabi-Yau variety and the degeneration given is a “large complex
structure limit” degeneration of such a log-Calabi-Yau variety to a union of
toric varieties. Even though in the case of this specific example both sides
can be computed rather easily, in general this is a harder problem. A viable
strategy to establish this equivalence is to first look at the special fiber
and prove the following equivalence

$\displaystyle\mathcal{W}(V_{1,1})\simeq D^{b}\mathrm{Coh}(\mathcal{Z}_{1,1})$
(1.2)

for $V_{1,1}\coloneqq T^{*}\mathbb{S}^{1}\setminus D_{1,1}$. Then one can try
to deduce the equivalence (1.1) via deformation theory (certain subcategories
of those in equivalence (1.2) deform to give rise to the equivalence (1.1)).

Note that $V_{1,1}$ is the pair-of-pants surface. To establish equivalences of
the form (1.2), we introduced a method via categorical partial
compactifications in [12], [14] (a similar idea was also applied in the log
Calabi-Yau setting in [8], [9]). On the A-side we introduce a stop
$\Lambda^{\circ}$, that is a symplectic hypersurface on the contact boundary
of $T^{*}\mathbb{S}^{1}\setminus D_{1,1}$ and consider the partially wrapped
Fukaya category, and on the B-side we partially compactify $\mathcal{Z}_{1,1}$
to $\overline{\mathcal{Z}}_{1,1}$ which is obtained by compactifying
$\mathbb{A}^{1}_{y}$ to $\mathbb{P}^{1}_{y}$ by adding a point $y=\infty$.
Thus, $\overline{\mathcal{Z}}_{1,1}$ is a union of $\mathbb{A}^{1}_{x}$ and
$\mathbb{P}^{1}_{y}$ which intersect transversely at a point. We establish
directly an easier equivalence

$\displaystyle\mathcal{W}(V_{1,1},\Lambda^{\circ})\simeq
D^{b}\mathrm{Coh}(\overline{\mathcal{Z}}_{1,1})$ (1.3)

and then show that the equivalence of (1.2) is obtained from (1.3) by
localization.

The Liouville domain $V_{1,1}$ with the stop $\Lambda^{\circ}$ is given by the
following picture.

$L_{0}$$L_{1}$$T_{1}$ Figure 1. Pair-of-pants

The equivalence (1.3) is proved by matching the generators $L_{0}$ and $L_{1}$
with $\mathcal{O}_{\mathbb{P}^{1}_{y}}$ and
$\mathcal{O}_{\mathbb{A}^{1}_{x}}$, respectively. To deduce the equivalence
(1.2), we localize at the subcategory generated by $T_{1}$ on the A-side and
by the skyscraper sheaf $\mathcal{O}_{y=\infty}$ on the $B$-side. The reason
that the equivalence (1.3) is easier to establish is because the endomorphism
algebra of our generators turns out to be formal which is not true before
compactification.

The example that we considered above and the proof of the equivalences (1.1),
(1.2),(1.3) can be generalized in a straightforward way to the case of
$(T^{*}\mathbb{S}^{1},D_{1,k})$ where $D_{1,k}$ is now $k$ points (see [12]
for a proof of (1.2), (1.3)) and the family $\mathcal{Y}_{1,k}$ is a
$k$-dimensional family over
$\mathbb{A}^{k}=\mathrm{Spec}\,\mathbb{C}[V_{1},V_{2},\ldots,V_{k}]$. The
general fiber is again isomorphic to $\mathbb{C}^{\times}$ and the special
fiber is a nodal chain given by the following figure:

$\mathbb{A}^{1}_{x}$$\mathbb{A}^{1}_{y}$ Figure 2. Mirror to $7$-punctured
sphere, $V_{1,5}$

The two components at the ends are $\mathbb{A}^{1}_{x}$ and
$\mathbb{A}^{1}_{y}$, and the “interior components” are isomorphic to
$\mathbb{P}^{1}$.

Another way to represent this singular curve is via the image under the moment
map induced by the toric action. This results in the following diagram where
the rays on the left and right represent $\mathbb{A}^{1}_{x}$ and
$\mathbb{A}^{1}_{y}$ and the interior intervals represent 4 components
isomorphic to $\mathbb{P}^{1}$.

Figure 3. Moment map picture of $\mathcal{Z}_{1,5}$ \- the mirror to $V_{1,5}$

The $A$-side in the above example has a natural generalization to higher
dimensions. Namely, for $n,k\geq 0$, let $\\{x_{1},x_{2},\ldots
x_{n+1}\\}\cup\\{v_{1},\ldots,v_{k}\\}$ be $(n+k+1)$ points in
$\mathbb{P}^{1}_{\mathbb{C}}$ that are mutually distinct. We consider the
exact symplectic manifold

$V_{n,k}=\mathrm{Sym}^{n}(\mathbb{P}_{\mathbb{C}}^{1}\setminus(\\{x_{1},x_{2},\ldots
x_{n+1}\\}\cup\\{v_{1},v_{2},\ldots,v_{k}\\}))$

with the standard Liouville structure which can alternatively be described as
the complement of $(n+k+1)$ generic hyperplanes in
$\mathbb{P}_{\mathbb{C}}^{n}$. Note that $V_{n,0}$ can be symplectically
identified with the cotangent bundle of the $n$-dimensional torus,

$\mathrm{Sym}^{n}(\mathbb{P}_{\mathbb{C}}^{1}\setminus(\\{x_{1},x_{2},\ldots
x_{n+1}\\})=T^{*}\mathbb{T}^{n},$

and we have $V_{n,k}=T^{*}\mathbb{T}^{n}\setminus D_{n,k}$ with

$D_{n,k}=\bigcup_{j=1}^{k}\\{v_{j}\\}\times\mathrm{Sym}^{n-1}(\mathbb{P}_{\mathbb{C}}^{1}\setminus(\\{x_{1},x_{2},\ldots
x_{n+1}\\})$

In our previous work [14], we studied (partially) wrapped Fukaya categories
associated with $V_{n,k}$. In the case $k=1$, i.e., the case of
$n$-dimensional pairs-of-pants, we proved a homological mirror symmetry
statement that provides an equivalence of $\mathbb{Z}$-graded pre-triangulated
categories over a regular commutative ring $\mathbf{k}$:

$\displaystyle\mathcal{W}(V_{n,1})\simeq\mathrm{D}^{b}\mathrm{Coh}(\mathcal{Z}_{n,1})\simeq\operatorname{MF}_{{\mathbb{G}}_{m}}(\mathcal{Y}_{n,1}\times\mathbb{A}^{1},\mathbf{w}_{n,1})$
(1.4)

where $\mathcal{W}(V_{n,1})$ is the fully wrapped Fukaya category of the
$n$-dimensional pair-of-pants $V_{n,1}$, the middle category is the derived
category of coherent sheaves on the singular affine variety

$\mathcal{Z}_{n,1}=\mathrm{Spec}\,\mathbf{k}[x_{1},\ldots,x_{n+1}]/(x_{1}x_{2}\ldots
x_{n+1})$

and the third category is the $\mathbb{G}_{m}$-equivariant derived category of
matrix factorizations on
$\mathcal{Y}_{n,1}\times\mathbb{A}^{1}_{U_{1}}\simeq\mathbb{A}^{n+2}$, where

$\mathcal{Y}_{n,1}\coloneqq\mathrm{Spec}\,\mathbb{C}[x_{1},x_{2},\ldots,x_{n+1},V_{1}]/(x_{1}x_{2}\ldots
x_{n+1}-V_{1})\simeq{\mathbb{A}}^{n+1}$

for the potential $\mathbf{w}_{n,1}=U_{1}V_{1}$.

###  1.2. Formulation of the main result

In this paper, we give an explicit candidate for the $B$-side for any $n$ and
$k$. To that end, inspired by the work of Abouzaid-Auroux-Katzarkov [1], we
introduce a smooth toric variety $\mathcal{Y}_{n,k}$ over
$\mathbb{A}^{k}=\mathrm{Spec}\,\mathbf{k}[V_{1},V_{2},\ldots,V_{k}]$ with the
general fiber isomorphic to the torus ${\mathbb{G}}_{m}^{n}$ and the special
fiber $\mathcal{Z}_{n,k}$ over $V_{1}=V_{2}=\ldots=V_{k}=0$ given by a union
of toric varieties glued along toric strata (see Sec. 1.3 for the
construction). In fact, ${\mathcal{Y}}_{n,k}$ is a small toric resolution of
the singularity $X_{1}\ldots X_{n+1}=V_{1}\ldots V_{k}$, in particular, it is
Calabi-Yau (see Sec. 2.4). We also consider a Landau-Ginzburg version by
considering the variety

$\mathcal{X}_{n,k}\coloneqq\mathcal{Y}_{n,k}\times\mathrm{Spec}\,\mathbf{k}[U_{1},U_{2},\ldots,U_{k}]$

with the potential

$\mathbf{w}_{n,k}=U_{1}V_{1}+U_{2}V_{2}+\ldots+U_{k}V_{k}$

There is a universal grading of the Fukaya category of $V_{n,k}$ with values
in an abelian group ${\mathbb{L}}\simeq\mathbb{Z}^{n+k+1}$ which is compatible
with a certain natural $\mathbb{Z}/2$-grading structure. In order to get a
relation with the B-side we identify ${\mathbb{L}}$ with a natural grading
group ${\mathbb{L}}_{B}$ on the B-side, which is a sublattice in the character
group of an algebraic torus acting on ${\mathcal{X}}_{n,k}$ in a way that
rescales $\mathbf{w}_{n,k}$ according to the given character
$l_{0}\in{\mathbb{L}}_{B}$.

The possible $\mathbb{Z}$-grading structures on $V_{n,k}$ is a torsor over
$H^{1}(V_{n,k})=\mathbb{Z}^{n+k}$. Among these there is a natural
$\mathbb{Z}$-grading structure $\eta_{0}$ on $V_{n,k}$ which roughly speaking
is determined by the property that the primitive chords around the $x_{i}$
have degree 0, and the primitive chords around $v_{i}$ have degree 2. This
corresponds to the unique subgroup ${\mathbb{G}}_{m}$ in the above torus, such
that all $x_{i}$ and $V_{j}$ have weight $0$ (and $U_{j}$ have weight $1$)
with respect to this subgroup.

We formulate a conjectural homological mirror symmetry statement that
generalizes the above result to arbitrary $n,k$.

######  Conjecture 1.2.1.

There exists a quasi-equivalence of $\mathbb{Z}$-graded pre-triangulated
categories

$\displaystyle\mathcal{W}(V_{n,k},\eta_{0})\simeq
D^{b}\mathrm{Coh}(\mathcal{Z}_{n,k}),$ (1.5)

and a quasi-equivalence of ${\mathbb{L}}$-graded pre-triangulated categories

$\displaystyle\mathcal{W}(V_{n,k})\simeq{\mathcal{B}}_{n,k},$ (1.6)

where ${\mathcal{B}}_{n,k}$ is the category of ${\mathbb{L}}_{B}$-graded
equivariant matrix factorizations of $\mathbf{w}_{n,k}$ on
${\mathcal{X}}_{n,k}$. Moreover, by considering deformations over
$\mathbb{A}^{k}$, there exists a quasi-equivalence

$\displaystyle\mathcal{W}(T^{*}\mathbb{T}^{n},D_{n,k},\eta_{0})\simeq
D^{b}\mathrm{Coh}(\mathcal{Y}_{n,k})$ (1.7)

We give a precise definition of the ${\mathbb{L}}$-graded dg-category of
equivariant matrix factorizations ${\mathcal{B}}_{n,k}$ in Sec. 2.3.3. Note
that (1.5) follows from (1.6) by choosing a specific $\mathbb{Z}$-grading and
applying localization theorems from [2] on the A-side and from [7] on the
B-side. Our main result is the proof of (1.5) and (1.6) when $n=2$. Along the
way, we also give a new proof for $n=1$.

######  Theorem 1.2.2.

Equivalence (1.5) holds for $n=1$ and $n=2$.

The equivalence (1.6) of ${\mathbb{L}}$-graded categories in the cases $n\leq
2$ should also be within reach. We prove its $\mathbb{Z}$-graded version, as
well as a version for the partially wrapped Fukaya category with one stop (see
Theorem 1.4.2 below). To deduce (1.6) one has to develop a setup for
localization of ${\mathbb{L}}$-graded $A_{\infty}$ (or dg-)categories.

Figure 4 describes the moment map picture of the singular variety
$\mathcal{Z}_{2,k}$ for $k=4$. This is a two-dimensional analogue of Figure 3.
The 3 components on the corners are all isomorphic to $\mathbb{A}^{2}$, 6
components isomorphic to $\mathbb{A}^{1}\times\mathbb{P}^{1}$, 3 components
isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and 3 components isomorphic
to a blow-up of $\mathbb{A}^{1}\times\mathbb{P}^{1}$. The markings $C_{ij}$
denote coordinates on one of the affine charts of each ${\mathbb{P}}^{1}$
along which the components are glued.

$x_{1}$$x_{1}$$x_{1}$$x_{1}$$x_{3}$$x_{3}$$x_{3}$$x_{3}$$x_{2}$$x_{2}$$x_{2}$$x_{2}$$C_{13}$$C_{12}$$C_{11}$$C_{13}$$C_{12}$$C_{13}$$C_{21}$$C_{22}$$C_{23}$$C_{21}$$C_{22}$$C_{21}$
Figure 4. Moment map picture of $\mathcal{Z}_{2,4}$ \- the mirror to $V_{2,4}$

###  1.3. Description of the toric variety

We now explicitly describe the toric variety $\mathcal{Y}_{n,k}$. Let us set

$\Sigma=\mathbb{P}^{1}_{\mathbb{C}}\setminus(\\{x_{1},x_{2},\ldots
x_{n+1}\\}\cup\\{v_{1},v_{2},\ldots,v_{k}\\}),$

the $(n+k+1)$ punctured surface whose boundary components are marked by
$x_{1},\ldots,x_{n+1}$ and $v_{1},\ldots,v_{n}$ as in Figure 5.

$C_{13}$$C_{12}$$C_{11}$$C_{23}$$C_{22}$$C_{21}$$a_{1}$$a_{2}$$b_{3}$$b_{2}$$b_{1}$$v_{4}$$v_{3}$$v_{2}$$v_{1}$$x_{1}$$x_{2}$$x_{3}$
Figure 5. The distinguished curves on the surface $\Sigma$; $n=2$, $k=4$.

Here we will describe the toric variety $\mathcal{Y}_{n,k}$ in terms of the
dual cones to cones forming the toric fan. These dual cones live in
$H\otimes\mathbb{R}$, where $H=H_{1}(\Sigma)$ is the homology lattice of the
punctured surface $\Sigma$ and are governed by the combinatorics of curves on
$\Sigma$. We denote by $T$ the algebraic torus with the character lattice $H$
($T$ will be the torus acting on ${\mathcal{Y}}_{n,k}$).

Let us equip each circle $x_{i}$ with the orientation induced by the
orientation of the surface, and we equip each circle $v_{j}$ with the opposite
orientation to the one induced by the orientation of the surface. We still
denote by $x_{i}$ and $v_{j}$ the corresponding classes in $H$. As in Figure
5, let us denote by $a_{i}$ the arc from $x_{i}$ to $x_{i+1}$ and $b_{j}$ the
arc from $v_{j}$ to $v_{j+1}$, and consider the circles $c_{ij}$ which
intersect $a_{i}$ and $b_{j}$ once. We equip $c_{ij}$ with the orientation
induced by the subsurface of $\Sigma$ to the left of $c_{ij}$. It is also
convenient to set

$c_{1k}=-x_{1}\ \text{ and }c_{n,0}=x_{n+1}.$

Note that we have the following relation in $H$:

$c_{ij}=v_{k}+v_{k-1}+\ldots+v_{j+1}-x_{1}-x_{2}-\ldots-x_{i}.$ (1.8)

We denote by $X_{i},V_{j},C_{ij}$ the invertible functions on $T$
corresponding to the elements $x_{i},v_{j},c_{ij}$ of the lattice $H$ (some of
these functions and their inverses will extend to regular functions forming
coordinates on affine charts of ${\mathcal{Y}}_{n,k}$).

We will often use the following relations in our lattice and the corresponding
relations between functions:

$\displaystyle c_{i,j}$ $\displaystyle=c_{i+1,j}+x_{i+1},\ \
C_{i,j}=C_{i+1,j}X_{i+1},$ (1.9) $\displaystyle c_{i,j}$
$\displaystyle=c_{i,j+1}+v_{j+1},\ \ C_{i,j}=C_{i,j+1}V_{j+1}.$ (1.10)

Now, we can define our cones in $H$. We start with the set $S\subset H$ given
by the classes $v_{1},\ldots,v_{k}$. The cones are numbered by nonincreasing
functions

$\phi:[1,n]\to[1,k].$

For each such function $\phi$, we will define a set of vectors $S_{\phi}$
obtained by removing some elements from $S$ and adding some new elements. The
vectors of $S_{\phi}$ will correspond to coordinates on an affine chart
${\mathcal{U}}_{\phi}\subset{\mathcal{Y}}_{n,k}$. Here is the recipe.

Let $[1,n]=I_{1}\sqcup\ldots\sqcup I_{r}$ be the partition of $[1,n]$ into the
level sets of the function $\phi$. For each interval $I=[i,p]\subset[1,n]$ and
an index $j$, $1\leq j\leq k$, we consider the subsurface $\Sigma(I,j)$
bounded by the circles

$-c_{i,j},-v_{j},c_{p,j-1},x_{i+1},\ldots,x_{p}$

Let us denote by $\partial^{\prime}\Sigma(I,j)$ the set of classes of all the
boundary components of $\Sigma(I,j)$ except for $-v_{j}$, so that $v_{j}$ is
equal to the sum of the classes in $\partial^{\prime}\Sigma(I,j)$. Now, for
our partition of the arcs $I_{1},\ldots,I_{r}$, we consider the subsurfaces
$\Sigma_{1},\ldots,\Sigma_{r}$, where $\Sigma_{m}=\Sigma(I_{m},\phi(I_{m}))$.
To get the set $S_{\phi}$ from $S$, we replace each $v_{\phi(I_{m})}$ for
$m=1,\ldots,r$ with the classes in $\partial^{\prime}\Sigma_{m}$. In other
words,

$S_{\phi}=\\{v_{j}|j\notin\mathrm{im}(\phi)\\}\sqcup\\{x_{i+1}|\phi(i)=\phi(i+1)\\}\sqcup$
$\\{-c_{i,\phi(i)}|\phi(i-1)>\phi(i)\text{ or
}i=1\\}\sqcup\\{c_{i,\phi(i)-1}|\phi(i+1)<\phi(i)\text{ or }i=n\\}$

Let $\sigma_{\phi}^{\vee}\subset H\otimes\mathbb{R}$ denote the cones
generated by $S_{\phi}$, and let $\sigma_{\phi}\subset
H^{\vee}\otimes\mathbb{R}$ be the dual cones. One can check that the cones
$\sigma_{\phi}$ form a smooth toric fan, and we define $\mathcal{Y}_{n,k}$ to
be the corresponding smooth toric scheme over the base ring $\mathbf{k}$. By
definition, ${\mathcal{Y}}_{n,k}$ is covered by the open affine charts
${\mathcal{U}}_{\phi}$, each isomorphic to the $(n+k)$-dimensional affine
space with the coordinates given by the multiplicative variables corresponding
to elements of $S_{\phi}$. Instead of checking that $(\sigma_{\phi})$ forms a
fan we will construct a projective morphism
${\mathcal{Y}}_{n,k}\to\mathbb{A}^{n+k+1}$, which proves that
${\mathcal{Y}}_{n,k}$ is separable (see Corollary 4.3.2).

Finally, we define

$\mathcal{X}_{n,k}\coloneqq\mathcal{Y}_{n,k}\times\mathbb{A}^{k}_{U_{1},\ldots,U_{k}}$

where $\mathbb{A}^{k}_{U_{1},\ldots,U_{k}}$ is the $k$-dimensional affine
space with coordinate functions $(U_{1},\ldots,U_{k})$.

The vectors $v_{1},\ldots,v_{k}$ belong to each of the cones
$\sigma_{\phi}^{\vee}$, so they give $n$ functions $V_{1},\ldots,V_{n}$ on
$\mathcal{Y}_{n,k}$. Now, we define the superpotential $\mathbf{w}_{n,k}$ on
the smooth toric variety $\mathcal{X}_{n,k}$ by

$\mathbf{w}_{n,k}\coloneqq U_{1}V_{1}+U_{2}V_{2}+\ldots+U_{k}V_{k}$

Furthermore, for each $\phi$ the vectors $v_{1},\ldots v_{k}$ are obtained by
taking sums of disjoint subsets of the bases $S_{\phi}$. Hence, the functions
$V_{1},\ldots,V_{k}$ form a regular sequence on $\mathcal{Y}_{n,k}$.

We define the singular variety

$\mathcal{Z}_{n,k}\coloneqq\\{V_{1}=V_{2}=\ldots,V_{k}=0\\}\subset\mathcal{Y}_{n,k}$

Let $T$ denote the torus (of dimension $n+k$) acting on
$\overline{{\mathcal{Y}}}_{n,k}$, so that the character lattice of $T$ is the
homology lattice $H$. With respect to this action the weight of the functions
$X_{i}$, $C_{ij}$ and $V_{i}$ are the corresponding homology classes $x_{i}$,
$c_{ij}$ and $v_{i}$. The torus $T\times{\mathbb{G}}_{m}^{k}$ acts naturally
on
${\mathcal{X}}_{n,k}={\mathcal{Y}}_{n,k}\times{\mathbb{A}}^{k}_{U_{1},\ldots,U_{k}}$,
where the second factor only rescales $U_{i}$’s. We restrict this action to
the subtorus $\widetilde{T}\subset T\times{\mathbb{G}}_{m}^{k}$ given by

$\widetilde{T}=\\{(t,\lambda_{1},\ldots,\lambda_{k})\ |\
v_{1}(t)\lambda_{1}=\ldots=v_{k}(t)\lambda_{k}\\}.$

We can identify the character lattice of $\widetilde{T}$ with

$\widetilde{H}:=(H\oplus(\bigoplus_{i}\mathbb{Z}\cdot
u_{i}))/(u_{i}+v_{i}-u_{j}-v_{j}\ |\ i<j),$

so that $u_{i}$ is the weight of the function $U_{i}$ on
${\mathcal{X}}_{n,k}$.

By definition, we have the character

$l_{0}:=u_{1}+v_{1}=\ldots=u_{k}+v_{k}\in\widetilde{H},$

so that the function ${\mathbf{w}}=\sum_{i}U_{i}V_{i}$ has weight $l_{0}$. Let
us consider the sublattice

${\mathbb{L}}_{B}:=2\widetilde{H}+\mathbb{Z}l_{0}\subset\widetilde{H}.$

The main category of interest for us is the ${\mathbb{L}}_{B}$-graded category
of $\widetilde{T}$-equivariant matrix factorizations of ${\mathbf{w}}_{n,k}$,

${\mathcal{B}}_{n,k}:=\operatorname{MF}_{\widetilde{T}}({\mathcal{X}}_{n,k},{\mathbf{w}}_{n,k})_{{\mathbb{L}}_{B}}$

(see Sec. 2.3.3 for the precise definition). The grading group
${\mathbb{L}}_{B}$ can be naturally identified with the grading group
${\mathbb{L}}$ appearing on the A-side.

######  Example 1.3.1.

Consider the case $n=2$, $k=4$. The homology classes of our distinguished
curves in this case are,
$x_{1},x_{2},x_{3},v_{1},v_{2},v_{3},v_{4},c_{11},c_{12},c_{13},c_{21},c_{22},c_{23}$.
We have the relations:

$\displaystyle v_{4}$ $\displaystyle=x_{1}+c_{13},$ $\displaystyle v_{3}$
$\displaystyle=c_{12}-c_{13}=c_{22}-c_{23},$ $\displaystyle v_{2}$
$\displaystyle=c_{11}-c_{12}=c_{21}-c_{22},$ $\displaystyle v_{1}$
$\displaystyle=x_{3}-c_{21},$ $\displaystyle c_{11}$
$\displaystyle=c_{21}+x_{2},\,c_{12}=c_{22}+x_{2},\,c_{13}=c_{23}+x_{2},$

For each such curve, we write the corresponding toric monomial with a capital
letter. The cones are numbered by $\phi:[1,2]\to[1,4]$. Let us write them as
$\sigma_{\phi(1)\phi(2)}\coloneqq\sigma_{\phi}$. We have 10 cones and the
corresponding monomials are given as follows:

$\displaystyle\sigma_{44}^{\vee}$ $\displaystyle=\langle
X_{1},X_{2},C_{23},V_{3},V_{2},V_{1}\rangle,$
$\displaystyle\sigma_{43}^{\vee}$ $\displaystyle=\langle
X_{1},C_{13},C^{-1}_{23},C_{22},V_{2},V_{1}\rangle,$
$\displaystyle\sigma_{42}^{\vee}$ $\displaystyle=\langle
X_{1},C_{13},V_{3},C_{22}^{-1},C_{21},V_{1}\rangle,$
$\displaystyle\sigma_{41}^{\vee}$ $\displaystyle=\langle
X_{1},C_{13},V_{3},V_{2},C_{21}^{-1},X_{3}\rangle$
$\displaystyle\sigma_{33}^{\vee}$ $\displaystyle=\langle
V_{4},C_{13}^{-1},X_{2},C_{22},V_{2},V_{1}\rangle,$
$\displaystyle\sigma_{32}^{\vee}$ $\displaystyle=\langle
V_{4},C_{13}^{-1},C_{12},C_{22}^{-1},C_{21},V_{1}\rangle$
$\displaystyle\sigma_{31}^{\vee}$ $\displaystyle=\langle
V_{4},C_{13}^{-1},C_{12},V_{2},C_{21}^{-1},X_{3}\rangle,$
$\displaystyle\sigma_{22}^{\vee}$ $\displaystyle=\langle
V_{4},V_{3},C_{12}^{-1},X_{2},C_{21},V_{1}\rangle$
$\displaystyle\sigma_{21}^{\vee}$ $\displaystyle=\langle
V_{4},V_{3},C_{12}^{-1},C_{11},C_{21}^{-1},X_{3}\rangle,$
$\displaystyle\sigma_{11}^{\vee}$ $\displaystyle=\langle
V_{4},V_{3},V_{2},C_{11}^{-1},X_{2},X_{3}\rangle$

Setting $V_{1}=V_{2}=V_{3}=V_{4}=0$ gives
$\mathbf{k}[\sigma_{ii}^{\vee}]/(V_{\bullet})\simeq\mathbb{C}[X,Y,Z]$ for all
$i$ and
$\mathbf{k}[\sigma_{ij}^{\vee}]/(V_{\bullet})\simeq\mathbb{C}[X,Y,Z,W]/(XY,ZW)$
for all $i>j$, which form the affine charts for $\mathcal{Z}_{2,4}$,
corresponding to the vertices of Figure 4. Note that these charts are mirror
to either the 2-dimensional pair of pants (i.e., the symmetric square of the
4-puncture sphere), or the product of two 1-dimensional pairs of pants. One
can see the corresponding 4-punctured spheres or embeddings of two disjoint
3-punctured spheres in the surface $\Sigma$ bounded by the corresponding
curves in Figure 5.

Via homological mirror symmetry, skyscraper sheaves on the components of
$\mathcal{Z}_{2,4}$ match with $\mathbb{Z}$-gradable Lagrangian 2-tori in
$V_{2,4}$ equipped with unitary local systems. Such Lagrangian tori can be
given as products of two disjoint circles on $\Sigma$. We describe this
correspondence without proof in Figure 6. Lagrangian tori appearing in the
same chamber are Hamiltonian isotopic ([16]). Any other products among our
distinguished curves not appearing in Figure 6 are non-gradable.

$C_{13}x_{3}$$C_{12}x_{3}$$C_{11}x_{3}$$C_{13}C_{21}$$C_{12}C_{21}$$C_{13}C_{22}$$x_{1}C_{21}$$x_{1}C_{22}$$x_{1}C_{23}$$x_{2}C_{11},x_{2}C_{21},C_{11}C_{21}$$x_{2}C_{23},x_{2}C_{13},C_{13}C_{23}$$x_{2}C_{12},x_{2}C_{22},C_{12}C_{22}$$x_{1}x_{2}$
$x_{1}x_{3}$ $x_{2}x_{3}$ Figure 6. Lagrangian tori corresponding to
skyscraper sheaves on the irreducible components of $\mathcal{Z}_{2,4}$. The
red line indicates the compactification divisor
$\overline{\mathcal{Z}}_{2,4}\setminus\mathcal{Z}_{2,4}$

###  1.4. Compactified LG model

It is easy to see that there is a natural locally closed toric embedding

${\mathcal{Y}}_{n,k}\to{\mathcal{Y}}_{n,k+1}$

whose image is the subscheme $V_{1}=0$ in the union of open charts of
${\mathcal{Y}}_{n,k+1}$ corresponding to $\phi:[1,n]\to[1,k+1]$ with
$\operatorname{im}(\phi)\subset[2,k+1]$. Under this embedding the functions
$V_{i+1}$ on ${\mathcal{Y}}_{n,k+1}$ restrict to $V_{i}$ on
${\mathcal{Y}}_{n,k}$, and function $X_{n+1}$ on ${\mathcal{Y}}_{n,k}$ gets
identified with the restriction of the function $C_{n,1}$ on these open charts
of ${\mathcal{Y}}_{n,k+1}$.

We define $\overline{{\mathcal{Y}}}_{n,k}$ (resp.,
$\overline{{\mathcal{Z}}}_{n,k}$) as the closure of ${\mathcal{Y}}_{n,k}$
(resp., ${\mathcal{Z}}_{n,k}$) under this embedding. The functions
$V_{1},\ldots,V_{k}$ extend regularly to $\overline{{\mathcal{Y}}}_{n,k}$, so
the potential ${\mathbf{w}}_{n,k}$ extends regularly to
$\overline{{\mathcal{X}}}_{n,k}=\overline{{\mathcal{Y}}}_{n,k}\times{\mathbb{A}}^{k}$.
We denote by $\overline{{\mathcal{B}}}_{n,k}$ the category of
${\mathbb{L}}_{B}$-graded equivariant matrix factorizations of
${\mathbf{w}}_{n,k}$ on $\overline{{\mathcal{X}}}_{n,k}$ (recall that
${\mathbb{L}}_{B}$ is naturally isomorphic to ${\mathbb{L}}$).

We have the following extensions of Conjecture 1.2.1 and Theorem 1.2.2. We
equip $V_{n,k}$ with one stop $\Lambda$ corresponding to a point on the
component marked by $x_{k}$ and consider the corresponding partially wrapped
Fukaya category.

######  Conjecture 1.4.1.

There exists a quasi-equivalence of $\mathbb{Z}$-graded pre-triangulated
categories

$\displaystyle\mathcal{W}(V_{n,k},\Lambda,\eta_{0})\simeq
D^{b}\mathrm{Coh}(\overline{{\mathcal{Z}}}_{n,k}),$ (1.11)

and a quasi-equivalence of ${\mathbb{L}}$-graded pre-triangulated categories

$\displaystyle\mathcal{W}(V_{n,k},\Lambda)\simeq\overline{{\mathcal{B}}}_{n,k},$
(1.12)

where $\overline{{\mathcal{B}}}_{n,k}$ is the category of
${\mathbb{L}}$-graded matrix factorizations of $\mathbf{w}_{n,k}$ on
$\overline{{\mathcal{X}}}_{n,k}$.

######  Theorem 1.4.2.

Equivalences (1.11) and (1.12) hold for $n=1$ and $n=2$.

Note that we derive Theorem 1.2.2 from Theorem 1.4.2 by passing to
localizations. The Landau-Ginzburg version is also crucial for our approach,
as it allows to prove formality of the $A_{\infty}$-endomorphism algebra of
the generators in $D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{n,k})$
(for $n\leq 2$). Namely, we observe that for a certain $\mathbb{Z}$-grading
the endomorphism algebra of the corresponding matrix factorizations is
concentrated in degree zero and derive from this formality of the
${\mathbb{L}}$-graded endomorphism algebra.

Conventions. For our main results we work over a ground field $\mathbf{k}$. In
some results $\mathbf{k}$ can be any regular Noetherian ring. Both for Fukaya
categories and for matrix factorization categories, we pass to a perfect
derived category of the original dg-category.

Acknowledgments. Y. L. would like to thank Denis Auroux and Nick Sheridan for
helpful correspondences. Y. L. was partially funded by the Royal Society
URF\R\180024. A. P. is partially supported by the NSF grant DMS-2001224, and
within the framework of the HSE University Basic Research Program and by the
Russian Academic Excellence Project ‘5-100’.

##  2\. Preliminaries

###  2.1. (Multi)gradings of $A_{\infty}$-algebras

Let ${\mathbb{L}}$ be an abelian group equipped with an element
$l_{0}\in{\mathbb{L}}$ and a homomorphism
${\mathbb{L}}\to\mathbb{Z}/2:x\mapsto|x|$, such that
$|l_{0}|=1\operatorname{mod}2$. We will use this homomorphism to collapse an
${\mathbb{L}}$-grading to a $\mathbb{Z}/2$-grading. The usual notion of an
$A_{\infty}$-algebra which is a $\mathbb{Z}$-graded vector space with some
operations has a natural ${\mathbb{L}}$-graded version.

######  Definition 2.1.1.

An ${\mathbb{L}}$-graded $A_{\infty}$-algebra is an ${\mathbb{L}}$-graded
vector space $A=\bigoplus_{l\in{\mathbb{L}}}A_{l}$ equipped with operations

$\mathfrak{m}_{n}:A^{\otimes n}\to A,\ n\geq 1,$

where the ${\mathbb{L}}$-degree of $\mathfrak{m}_{n}$ is $(2-n)l_{0}$. These
operations satisfy the usual $A_{\infty}$-axioms

$\sum_{m,n}(-1)^{|a_{1}|+\ldots+|a_{n}|-n}\mathfrak{m}_{d-m+1}(a_{d},\ldots,a_{n+m+1},\mathfrak{m}_{m}(a_{n+m},\ldots,a_{n+1}),a_{n},\ldots
a_{1})=0.$

where we use the induced $\mathbb{Z}/2$-grading on $A$. Similarly, we can
define the notion of an $A_{\infty}$-homomorphism between
${\mathbb{L}}$-graded $A_{\infty}$-algebras.

We will call a homomorphism $f:{\mathbb{L}}\to\mathbb{Z}$ admissible if
$f(l_{0})=1$ and $f(x)\equiv|x|\operatorname{mod}(2)$. Given an
${\mathbb{L}}$-graded $A_{\infty}$-algebra $A$ and an admissible homomorphism
$f:{\mathbb{L}}\to\mathbb{Z}$, we can collapse the ${\mathbb{L}}$-grading on
$A$ to a $\mathbb{Z}$-grading and get a usual $A_{\infty}$-algebra $A/f$. Note
that an admissible homomorphism $f$ induces a decomposition

${\mathbb{L}}=\mathbb{Z}\cdot l_{0}\oplus{\mathbb{L}}_{0},$

where ${\mathbb{L}}_{0}=\operatorname{ker}(f)$. Then the
${\mathbb{L}}$-grading on $A$ can be viewed as an ${\mathbb{L}}_{0}$-grading
on $A/f$ compatible with the $\mathbb{Z}$-grading, such that all
$\mathfrak{m}_{n}$ on $A/f$ preserve the ${\mathbb{L}}_{0}$-degree. The same
is true about $A_{\infty}$-homomorphisms (in particular, gauge equivalences)
between ${\mathbb{L}}$-graded $A_{\infty}$-algebras $A$ and $B$: they can be
viewed as usual $A_{\infty}$-homomorphisms between the usual
$A_{\infty}$-algebras $A/f$ and $B/f$, preserving the
${\mathbb{L}}_{0}$-degree.

Note that when ${\mathbb{L}}_{0}$ is finitely generated, an
${\mathbb{L}}_{0}$-graded vector space can be thought as an algebraic
representation of the corresponding commutative algebraic group $G$, such that
${\mathbb{L}}_{0}$ is the group of characters of $G$. Thus, in the presence of
a splitting ${\mathbb{L}}=\mathbb{Z}\oplus{\mathbb{L}}_{0}$, the above notion
of an ${\mathbb{L}}$-graded $A_{\infty}$-algebra is equivalent to that of a
$G$-equivariant $\mathbb{Z}$-graded $A_{\infty}$-algebra. Since for a
reductive group $G$ we can choose all the projections involved in the
homological perturbation theory to be $G$-equivariant, we have the following
statement.

######  Lemma 2.1.2.

Let $G$ be a linearly reductive algebraic group over a field $\mathbf{k}$. For
a dg-algebra $A$ over $\mathbf{k}$ equipped with an algebraic $G$-action
(respecting the dg-algebra structure) there is a $G$-equivariant version of
the homological perturbation producing a $G$-equivariant minimal
$A_{\infty}$-structure on $H^{*}(A)$.

We have the following obvious formality criterion for ${\mathbb{L}}$-graded
$A_{\infty}$-algebras.

######  Lemma 2.1.3.

Let $(A,\mathfrak{m}_{\bullet})$ be a minimal ${\mathbb{L}}$-graded
$A_{\infty}$-algebra such that for some admissible homomorphism
$f:{\mathbb{L}}\to\mathbb{Z}$, the algebra $A/f$ is concentrated in degree
$0$. Then $\mathfrak{m}_{n}=0$ for $n>2$.

###  2.2. Multigraded Fukaya categories

Multigradings appear naturally in the context of Fukaya categories: they have
been used in [23], [4] and studied extensively by Sheridan [24], [25]. Given a
symplectic manifold $V$, we consider the relative Lagrangian Grassmannian of
the tangent bundle $TV$,

$\mathcal{G}V\to V.$

The exact sequence associated to this fibration gives the following exact
sequence

$\mathbb{Z}=H_{1}(\mathcal{G}_{p}V)\to H_{1}(\mathcal{G}V)\to H_{1}(V)\to 0$

where ${\mathcal{G}}_{p}V$ is the Lagrangian Grassmannian of the tangent space
at a point $p\in V$, and the identification
$\mathbb{Z}=H_{1}(\mathcal{G}_{p}V)$ is given by the Maslov index. We set

${\mathbb{L}}:=H_{1}(\mathcal{G}V)$

and let $l_{0}$ be the image of $1\in\mathbb{Z}$ under the homomorphism
$H_{1}(\mathcal{G}_{p}V)\to H_{1}(\mathcal{G}V)$.

In addition, we choose a homomorphism

${\mathbb{L}}=H_{1}(\mathcal{G}V)\to\mathbb{Z}/2$

which sends $l_{0}$ to 1.

In [24] [25], a canonical choice of such a homomorphism is given by the
pairing with the first Stiefel-Whitney class $w_{1}(\mathcal{L})\in
H^{1}(\mathcal{G}V;\mathbb{Z}/2)$ of the universal bundle
$\mathcal{L}\to\mathcal{G}V$ whose fiber over a point is the Lagrangian
subspace at that point. This corresponds to the canonical
$\mathbb{Z}/2$-grading structure on Fukaya categories corresponding to the
fibrewise double cover of $\mathcal{G}V$ given by the oriented Lagrangian
Grassmannian of the tangent bundle $TV$.

To allow for other choices of $\mathbb{Z}/2$-gradings, we consider pairings
with

$w_{1}(\mathcal{L})+p^{*}\sigma\in H^{1}(\mathcal{G}V;\mathbb{Z}/2)$

for $\sigma\in H^{1}(V;\mathbb{Z}/2)$, where $p:\mathcal{G}V\to V$ is the
natural projection. By a slight abuse of notation, we denote the corresponding
homomorphism by

$\sigma:\mathbb{L}\to\mathbb{Z}/2$

Note that as we obtained this by pulling back a class from $V$, this
homomorphism still satisfies $\sigma(l_{0})=1$.

Sheridan explains in [24, Section 3] that the gluing formulae for index theory
of Cauchy-Riemann operators allow one to define an ${\mathbb{L}}$-graded
Fukaya category and in [25, Appendix B.3] provides careful elaborations of the
arguments by Seidel in [22, Sections 11 & 12 ]. The key topological
observation that makes this work is that if there is a holomorphic disk
contributing to some $A_{\infty}$-product, then its boundary is
nullhomologous. The proof given in [25, Appendix B.3] applies in the same way
if we use the more general $\mathbb{Z}/2$-gradings corresponding to
$\sigma:\mathbb{L}\to\mathbb{Z}/2$ after modifying the definition in [25,
Definition B.4] so that $\mathbb{Z}/2$ degree of a generator is computed with
respect to the $\mathbb{Z}/2$ grading associated to $\sigma$.

The objects of this Fukaya category are ${\mathbb{L}}$-graded Lagrangian
submanifolds of $V$. This means that, for a Lagrangian submanifold $L$, we
require a lift of its Gauss map $L\to\mathcal{G}V$ to the universal abelian
cover $\tilde{\mathcal{G}}V\to\mathcal{G}V$ and a Pin structure on $L$. If
$H_{1}(L)\to H_{1}(V)$ is trivial, then such a lift exists (and is unique up
to translation by ${\mathbb{L}}$ for connected $L$). A Pin structure on $L$
exists if and only if the second Stiefel-Whitney class vanishes, and the space
of Pin structures on $L$ is a torsor over $H^{1}(L;\mathbb{Z}_{2})$. In this
paper, we work only with contractible Lagrangians, so they will always be
gradable (uniquely up to translation by ${\mathbb{L}}$) and will have
canonical Pin structures on them.

In summary, for each $\mathbb{Z}/2$-grading structure $\sigma$ on $V$, which
can be prescribed by exhibiting a homomorphism
$\sigma:\mathbb{L}\to\mathbb{Z}_{2}$ with $\sigma(l_{0})=1$, we can construct
a $\mathbb{L}$-graded Fukaya category of $V$, where
$\mathbb{L}=H_{1}(\mathcal{G}V)$. There is an effective
$H^{1}(V;\mathbb{Z}_{2})$ worth of choices for the $\mathbb{Z}/2$-grading
structures, but once we fix such a $\mathbb{Z}/2$-grading, there is a
canonical $\mathbb{L}$-graded Fukaya category.

######  Remark 2.2.1.

i) Vanishing of $2c_{1}(V)$ guarantees that the morphism
$\mathbb{Z}=H_{1}(\mathcal{G}_{p}V)\to H_{1}(\mathcal{G}V)$ is injective, or
equivalently $l_{0}\in{\mathbb{L}}$ is non-torsion.
ii) Recall that the usual way of equipping Fukaya categories with a
$\mathbb{Z}$-grading goes via similar construction (see [20]) however instead
of using the universal abelian cover of $\mathcal{G}V$, one considers a
fibrewise universal cover of $\mathcal{G}V$. Such a cover exists if and only
if $2c_{1}(V)=0$. Moreover, if non-empty, the space of such covers is an
$H^{1}(V)$-torsor.

###  2.3. Categories of graded matrix factorizations

Let $X$ be a smooth space over $\mathbf{k}$ (possibly an algebraic stack) with
a regular function $W$. We are going to recall the definition of
$\mathbb{Z}$-graded categories of matrix factorizations following [17] and
explain how to generalize it to get multigraded versions of these categories.

####  2.3.1. $\mathbb{Z}$-graded categories of $G$-equivariant matrix
factorizations

Assume that we have an action of a reductive group $G$ (over $\mathbf{k}$) on
$X$, and the potential $W$ on $X$ satisfies

$W(gx)=l_{0}(g)W(x).$

for some character $l_{0}:G\to{\mathbb{G}}_{m}$. A $(G,l_{0})$-equivariant
matrix factorization of $W$ consists of $G$-equivariant vector bundles $E_{0}$
and $E_{1}$ and $G$-equivariant homomorphisms

$d_{1}:E_{1}\to E_{0},\ \ d_{0}:E_{0}\to E_{1}\otimes l_{0},$

such that $d_{1}d_{0}=d_{0}d_{1}=W$.

For a pair $(E,d_{E})$, $(F,d_{F})$ of $(G,l_{0})$-equivariant matrix
factorizations we define a $(G,l_{0})$-equivariant matrix factorization
$\underline{\operatorname{Hom}}=\underline{\operatorname{Hom}}((E,d_{E}),(F,d_{F}))$
of $0$, by setting 111This construction can be seen as a combination of two
operations on matrix factorizations, the tensor product and the duality (see
[18, Sec. 1]).

$\underline{\operatorname{Hom}}_{0}:=E_{0}^{\vee}\otimes F_{0}\oplus
E_{1}^{\vee}\otimes F_{1},\ \
\underline{\operatorname{Hom}}_{1}:=E_{0}^{\vee}\otimes F_{1}\oplus
E_{1}^{\vee}\otimes F_{0}\otimes l_{0}^{-1}.$

Now for any $(G,l_{0})$-equivariant matrix factorization $(H,d_{H})$ of $0$,
we can consider a $\mathbb{Z}$-graded complex of $G$-equivariant vector
bundles

$\operatorname{com}(H)=[\ldots H_{1}\to H_{0}\to H_{1}\otimes l_{0}\to
H_{0}\otimes l_{0}\ldots]$

with $H_{0}$ in degree $0$. Note that it is equipped with an isomorphism

$\alpha_{H}:\operatorname{com}(H)[2]\to\operatorname{com}(H)\otimes l_{0}.$
(2.1)

For a pair $E=(E,d_{E})$, $F=(F,d_{F})$ as above, the $\mathbb{Z}$-graded
morphism space is defined by

$\hom_{\mathbb{Z}}(E,F):=R\Gamma(X,\operatorname{com}(\underline{\operatorname{Hom}}(E,F)))^{G},$

where we use some functorial multiplicative resolutions to compute $R\Gamma$.
222Equivalently, one can first consider the naive dg-category, without
deriving the functor of global sections, and then pass to the quotient by the
subcategory of locally contractible objects (see [17]). We denote by
$\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}}$ the perfect derived category
of the $\mathbb{Z}$-graded dg-category of $(G,l_{0})$-equivariant matrix
factorizations of $W$. Note that if $W$ is not a zero then $l_{0}$ is uniquely
determined by $W$, so we will sometimes omit $l_{0}$ from the notation.

Note that due to isomorphism (2.1), for any $G$-equivariant matrix
factorization $E$ we have

$E\otimes l_{0}\simeq E[2].$

Assume that $W$ is not a zero divisor and consider the hypersurface
$X_{0}\subset X$ given by $W=0$. We denote by
$D_{\operatorname{sing}}([X_{0}/G])$ the singularity category of the stack
$[X_{0}/G]$, i.e., the quotient of the bounded derived category of
$G$-equivariant sheaves on $X_{0}$ by the subcategory perfect complexes.
Assuming that the stack $[X/G]$ is sufficiently nice (of finite cohomological
dimension and with a resolution property) one has a natural equivalence (see
[17, Thm. 3.14])

$\operatorname{MF}_{G}(X,W)_{\mathbb{Z}}\simeq
D_{\operatorname{sing}}([X_{0}/G]).$ (2.2)

####  2.3.2. ${\mathbb{G}}_{m}$-equivariant matrix factorizations of a
potential of degree $1$

An important case is when $G={\mathbb{G}}_{m}$ and
$l_{0}:{\mathbb{G}}_{m}\to{\mathbb{G}}_{m}$ is the identity character, so the
potential $W$ satisfies

$W(\lambda\cdot x)=\lambda\cdot W(x),$

where $\lambda\in{\mathbb{G}}_{m}$, $x\in X$. In this case the above
definition gives a definition of the dg category
$\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W):=\operatorname{MF}_{{\mathbb{G}}_{m},l_{0}}(X,W)_{\mathbb{Z}}$
of ${\mathbb{G}}_{m}$-equivariant matrix factorizations of $W$.

######  Remark 2.3.1.

1\. It can be shown that the above category
$\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W)$ is equivalent to the category of
$B$-branes defined as in Segal’s paper [19] for the doubled action of
${\mathbb{G}}_{m}$ on $X$.

2\. If $W$ is a potential of weight $d>0$ with respect to the
${\mathbb{G}}_{m}$-action, we can pass to the stack
${\mathcal{X}}:=[X/\mu_{d}]$ and equip it with the action of
${\mathbb{G}}_{m}/\mu_{d}\simeq{\mathbb{G}}_{m}$. Then $W$ descends to a
function on ${\mathcal{X}}$ of weight $1$ with respect to the
${\mathbb{G}}_{m}$-action. Then the above definition can be applied to the
stack ${\mathcal{X}}$ with the descended potential.

In the case when $W$ is not a zero divisor and the stack
$[X/{\mathbb{G}}_{m}]$ is sufficiently nice, we get from (2.2) an equivalence

$\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W)\simeq
D_{\operatorname{sing}}([X_{0}/{\mathbb{G}}_{m}]).$ (2.3)

Now assume that $V_{1},\ldots,V_{k}$ is a regular sequence of global functions
on a smooth scheme $Y$, and let $Z\subset Y$ be the zero locus
$Z(V_{1},\ldots,V_{k})$. Then we can consider the scheme
$X=Y\times{\mathbb{A}}^{k}_{U_{1},\ldots,U_{k}}$ with the potential
$W=U_{1}V_{1}+\ldots+U_{k}V_{k}$. We equip $X$ with the natural
${\mathbb{G}}_{m}$-action, trivial on $Y$ and such that each $U_{i}$ has
weight $1$, and consider the category of matrix factorizations
$\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W)$. Then Işık’s theorem [7] states
that there is an equivalence

$D^{b}\operatorname{Coh}(Z)\simeq\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W).$
(2.4)

####  2.3.3. Multigraded categories of matrix factorizations

Now we observe that in the context of Sec. 2.3.1, we can also define a
multigraded dg-category of matrix factorizations.

Namely, let $L$ denote the group of characters of $G$. We assume that
$l_{0}\not\in 2L$ and consider the sublattice

$L(l_{0}):=\mathbb{Z}\cdot l_{0}+2L\subset L.$

Note that we have a natural surjective homomorphism

$L(l_{0})\to\mathbb{Z}/2$

sending $l_{0}$ to $1\operatorname{mod}2$, and sending $2L$ to $0$.

For any $(G,l_{0})$-equivariant matrix factorization $(H,d_{H})$ of $0$ we can
define an $L(l_{0})$-graded complex of $G$-equivariant bundles
$\operatorname{com}_{L(l_{0})}(H)$ by

$\operatorname{com}_{L(l_{0})}(H)_{2l}:=H_{0}\otimes l,\ \
\operatorname{com}_{L(l_{0})}(H)_{2l+l_{0}}:=H_{1}\otimes l\otimes l_{0},$

with the differential of degree $l_{0}$ induced by $d_{H}$.

Now for a pair $E=(E,d_{E})$, $F=(F,d_{F})$ of $(G,l_{0})$-equivariant matrix
factorizations we define the $L(l_{0})$-graded space of morphisms

$\hom_{L(l_{0})}(E,F):=R\Gamma(X,\operatorname{com}_{L(l_{0})}(\underline{\operatorname{Hom}}(E,F)))^{G},$

where on the right we take the totalization of the $\mathbb{Z}\times
L(l_{0})$-graded complex with respect to the homomorphism $\mathbb{Z}\times
L(l_{0})\to L(l_{0}):(n,l)\mapsto nl_{0}+l$. We denote by
$\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}$ the corresponding
$L(l_{0})$-graded dg-category (sometimes we will omit $l_{0}$ from the
notation).

Note that we have a natural shift operation $(E,d_{E})\mapsto(E[1],d_{E[1]})$,
where

$E[1]_{0}=E_{1}\otimes l_{0},\ E[1]_{1}=E_{0}\otimes l_{0},$

and $d_{E[1]}$ is induced by $-d_{E}$. This shift operation is compatible with
the shift of degree by $l_{0}$ on the spaces of morphisms.

######  Remark 2.3.2.

It is clear from the definition that if a global function $f$ on $X$ has
weight $l$ with respect to the $G$-action, then the multiplication by $f$
gives an endomorphism of degree $2l$ of any object of
$\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}$. Thus, the natural $G$-weights
get doubled when passing to the endomorphisms in this category.

######  Lemma 2.3.3.

(i) There is an equivalence of $\mathbb{Z}$-graded dg-categories

$\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}l_{0}}\simeq\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}},$

where on the left we consider the subcategory corresponding to the
subcomplexes of $\hom$-spaces that have grading in $\mathbb{Z}l_{0}\subset
L(l_{0})$ (and identify the grading group $\mathbb{Z}l_{0}$ with
$\mathbb{Z}$).

(ii) Assume that $G$ is a split algebraic torus:
$G\simeq{\mathbb{G}}_{m}^{n}$. Then for a subgroup ${\mathbb{G}}_{m}\subset
G$, such that $l_{0}$ restricts to the identity character of
${\mathbb{G}}_{m}$, the natural forgetful functor

$\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}\to\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W)$

acts as identity on morphism spaces by collapsing the $L(l_{0})$-grading to
the $\mathbb{Z}$-grading via the homomorphism $L(l_{0})\hookrightarrow
L\to\mathbb{Z}$ dual to the embedding ${\mathbb{G}}_{m}\to G$.

###### Proof.

Part (i) follows immediately from the definitions. For part (ii), we first
observe that the homomorphism $p_{\mathbb{Z}}:L\to\mathbb{Z}$ induces a
splitting

$L=\mathbb{Z}l_{0}\oplus K,$

where $K=\operatorname{ker}(p_{\mathbb{Z}})$. From this we get an induced
splitting

$L(l_{0})=\mathbb{Z}l_{0}\oplus 2K.$

Next, let $(H,d_{H})=\underline{\operatorname{Hom}}(E,F)$, for some
$(G,l_{0})$-equivariant matrix factorizations $E$ and $F$. Then we observe
that

$\operatorname{com}_{L(l_{0})}(H)=\bigoplus_{k\in
K}\operatorname{com}(H)\otimes k.$

Due to our assumption on $G$, this implies that

$R\Gamma(X,\operatorname{com}_{L(l_{0})}(H))^{G}\simeq
R\Gamma(X,\operatorname{com}(H))^{{\mathbb{G}}_{m}}$

as claimed. ∎

From Lemma 2.3.3(i) we see that one can recover all morphisms in
$\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}$ from those in
$\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}}$ using the isomorphism of
morphism spaces in $\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}$,

$\operatorname{Hom}^{2l+ml_{0}}(E,F)\simeq\operatorname{Hom}^{ml_{0}}(E,F\otimes
l).$

Assume now that $W$ is not a zero divisor. Then using the equivalence (2.2) we
get an equivalence

$\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}l_{0}}\simeq\operatorname{MF}_{G,l_{0}}(X,W)_{\mathbb{Z}}\simeq
D_{\operatorname{sing}}([X_{0}/G]).$

Thus, given a $G$-equivariant coherent sheaf $F$ on $X_{0}$, we get from the
corresponding object of the $G$-equivariant singularity category an object of
$\operatorname{MF}_{G,l_{0}}(X,W)_{L(l_{0})}$.

####  2.3.4. Grothendieck-Serre duality and a vanishing criterion for
morphisms

Assume now that $G={\mathbb{G}}_{m}$ and $l_{0}$ is the identity character.
Recall that for a pair of ${\mathbb{G}}_{m}$-equivariant matrix factorizations
of $W$, $E$ and $F$, the cohomology of the morphism space are given by

$\operatorname{Hom}^{*}(E,F)\simeq
H^{*}(X,\operatorname{com}(\underline{\operatorname{Hom}}(E,F)))^{{\mathbb{G}}_{m}}.$

Now assume that we have a ${\mathbb{G}}_{m}$-equivariant proper morphism
$\pi:X\to S$, where $S=\operatorname{Spec}(A)$ is a regular Noetherian affine
scheme with a ${\mathbb{G}}_{m}$-action. Then we can define a
${\mathbb{G}}_{m}$-equivariant sheaf on $S$ by

$R\operatorname{Hom}(E,F)_{S}:=R\pi_{*}\operatorname{com}(\underline{\operatorname{Hom}}(E,F)).$

In addition, we will need the following construction from [17, Sec. 1.1]. For
a bounded complex $(C^{\bullet},\delta)$ of ${\mathbb{G}}_{m}$-equivariant
vector bundles on $X$, we define a ${\mathbb{G}}_{m}$-equivariant matrix
factorization $\operatorname{mf}(C)$ of $0$, by setting

$\operatorname{mf}(C)_{0}=\bigoplus_{n}C^{2n}\otimes l_{0}^{-n},\ \
\operatorname{mf}(C)_{1}=\bigoplus_{n}C^{2n-1}\otimes l_{0}^{-n},$

with the differential induced by $\delta$. It is easy to see that there is a
natural isomorphism of ${\mathbb{G}}_{m}$-equivariant complexes

$\operatorname{com}(\operatorname{mf}(C))=\bigoplus_{n}C\otimes
l_{0}^{n}[-2n].$

######  Lemma 2.3.4.

(i) In the above situation, for ${\mathbb{G}}_{m}$-equivariant matrix
factorizations $E$ and $F$, one has an isomorphism

$R\operatorname{Hom}(E,F)^{\vee}_{S}\simeq
R\operatorname{Hom}(F,E\otimes\omega_{X/S}[n])_{S},$

where $n$ is the relative dimension of $\pi$, and
$M^{\vee}=R\operatorname{Hom}(M,{\mathcal{O}}_{S})$, for $M\in
D_{{\mathbb{G}}_{m}}(S)$.

(ii) In the same situation assume that
$\underline{\operatorname{Hom}}(F,E)\simeq\operatorname{mf}(i_{*}H)$ for a
closed ${\mathbb{G}}_{m}$-invariant subscheme $i:Z\hookrightarrow X$ and $H\in
D_{{\mathbb{G}}_{m}}(Z)$, such that
$i^{*}\omega_{X/S}\simeq{\mathcal{O}}_{Z}\otimes l_{0}^{m}$ for some
$m\in\mathbb{Z}$. Then the vanishing of $\operatorname{Hom}^{*}(E,F)$ implies
the vanishing of $\operatorname{Hom}^{*}(F,E)$.

###### Proof.

(i) Since
$\operatorname{com}(\underline{\operatorname{Hom}}(F,E\otimes\omega_{X/S}))\simeq\operatorname{com}(\underline{\operatorname{Hom}}(E,F))^{\vee}\otimes\omega_{X/S}$,
this follows immediately from the usual (${\mathbb{G}}_{m}$-equivariant)
Grothendieck-Serre duality applied to
$\operatorname{com}(\underline{\operatorname{Hom}}(E,F))$.

(ii) Set ${\mathcal{H}}:=R\operatorname{Hom}(E,F)_{S}$. By assumption, we have
$R\Gamma(S,{\mathcal{H}})^{{\mathbb{G}}_{m}}=0$. But we also have

${\mathcal{H}}\otimes l_{0}^{n}\simeq{\mathcal{H}}[2n],$

hence, we get $R\Gamma(S,{\mathcal{H}}\otimes l_{0}^{n})^{{\mathbb{G}}_{m}}=0$
for every $n\in\mathbb{Z}$. Since $S$ is affine, this implies that
${\mathcal{H}}=0$. Therefore, by part (i), we deduce that

$R\pi_{*}\operatorname{com}(\underline{\operatorname{Hom}}(F,E\otimes\omega_{X/S}[n]))\simeq
R\operatorname{Hom}(F,E\otimes\omega_{X/S}[n])_{S}=0.$

By assumption, we have an isomorphism

$\operatorname{com}(\underline{\operatorname{Hom}}(F,E))\otimes\omega_{X/S}\simeq\bigoplus_{n}i_{*}H\otimes
l_{0}^{n}[-2n]\otimes\omega_{X/S}\simeq\bigoplus_{n}i_{*}H\otimes
l_{0}^{n+m}[-2n]\simeq\operatorname{com}(\underline{\operatorname{Hom}}(F,E))\otimes
l_{0}^{m}.$

Therefore, we deduce the vanishing of
$R\pi_{*}\operatorname{com}(\underline{\operatorname{Hom}}(F,E))$, and hence,
of $\operatorname{Hom}^{*}(F,E)$. ∎

####  2.3.5. Generation

We will use the standard generation result for the singularity categories (see
[15, Thm. 3.5]). However, we need to adapt this result to the multigraded
categories we work with.

Let us assume that we are in the setup of Sec. 2.3.3. Assume in addition that
$G$ is a split algebraic torus, and let us fix a subgroup
${\mathbb{G}}_{m}\subset G$, such that $l_{0}|_{{\mathbb{G}}_{m}}$ is the
identity character $\chi$.

######  Lemma 2.3.5.

Let $Z\subset X_{0}$ be a $G$-invariant closed subscheme containing the
singular locus of $X_{0}$, and let $(E_{i})_{i\in I}$ be a collection of
$G$-equivariant coherent sheaves on $Z$, which we view as objects of
$D_{\operatorname{sing}}([X_{0}/G])$ and as objects of
$\operatorname{MF}_{G}(X,W)_{L(l_{0})}$. Assume that
$(E_{i}\otimes\chi^{n})_{i\in I,n\in\mathbb{Z}}$ generate the
$D^{b}\operatorname{Coh}(Z/{\mathbb{G}}_{m})$. Then $(E_{i})$ generate
$\operatorname{MF}_{G}(X,W)_{L(l_{0})}$.

###### Proof.

The category $\operatorname{MF}:=\operatorname{MF}_{G}(X,W)_{L(l_{0})}$ has a
natural quasicoherent extension $\operatorname{MF}^{\operatorname{qcoh}}$,
such that the objects of $\operatorname{MF}$ are compact. Thus, to prove that
$(E_{i})$ generate $\operatorname{MF}$, it is enough to check that for
$X\in\operatorname{MF}^{\operatorname{qcoh}}$, the vanishing
$\operatorname{Hom}(E_{i},X)=0$ for all $i$ implies that $X=0$. It is enough
to know the same assertion after collapsing the $L(l_{0})$-grading to the
$\mathbb{Z}$-grading via some homomorphism $f:L(l_{0})\to\mathbb{Z}$ sending
$l_{0}$ to $1$. Hence, by Lemma 2.3.3(ii), we can replace $G$ with our fixed
subgroup ${\mathbb{G}}_{m}$. Thus, we are reduced to showing that $(E_{i})$
generate $\operatorname{MF}_{{\mathbb{G}}_{m}}(X,W)$. Note that $E_{i}\otimes
l_{0}^{n}\simeq E_{i}[2n]$ in this category. Hence, using equivalence with the
singularity category (2.3), it is enough to check that
$(E_{i}\otimes\chi^{n})$ generate
$D_{\operatorname{sing}}([X_{0}/{\mathbb{G}}_{m}])$. By the equivariant
version of [15, Thm. 3.5], it is enough to check that the subcategory of
$D^{b}\operatorname{Coh}(X_{0}/{\mathbb{G}}_{m})$ generated by
$(E_{i}\otimes\chi^{n})$ contains
$i_{*}D^{b}\operatorname{Coh}(\operatorname{Sing}X_{0})$, where
$i:\operatorname{Sing}X_{0}\hookrightarrow X_{0}$ is the singular locus. But
this follows from our assumption on $(E_{i})$. ∎

####  2.3.6. Elementary computations with Koszul matrix factorizations

Let us denote by $\\{a,b\\}$ the Koszul matrix factorization of $a\cdot b$,
where $a$ and $b$ are possibly sections of line bundles,

$E_{1}={\mathcal{O}}(-b)\rTo{b}E_{0}={\mathcal{O}}\rTo{a}{\mathcal{O}}(a)=E_{1}(ab)$

where $a$ (resp., $b$) is a section of a line bundle ${\mathcal{O}}(a)$
(resp., ${\mathcal{O}}(b)$). In the computations below we view coherent
sheaves as matrix factorizations of $0$ via the functor
$\operatorname{mf}(\cdot)$ described in Sec. 2.3.4.

######  Lemma 2.3.6.

(i) Assume $(a,b)$ is a regular sequence. Then one has natural
quasiisomorphisms

$\underline{\operatorname{End}}(\\{a,b\\})\simeq\\{a,b\\}^{\vee}|_{b=0}\simeq{\mathcal{O}}/(a,b).$

(ii) Assume $(a,b,c)$ is a regular sequence. Then one has natural
quasiisomorphisms

$\underline{\operatorname{Hom}}(\\{bc,a\\},\\{ca,b\\})\simeq\\{bc,a\\}^{\vee}|_{b=0}\simeq{\mathcal{O}}(a)/(a,b)[-1].$

The generating morphism comes from the exact triangle

${\mathcal{O}}(-a)/(b)\rTo{a}{\mathcal{O}}/(ab)\to{\mathcal{O}}/(a)\to{\mathcal{O}}(-a)/(b)[1].$

(iii) Assume $(a,b,c)$ is a regular sequence. Then one has natural
quasiisomorphisms

$\underline{\operatorname{Hom}}(\\{c,ab\\},\\{ca,b\\})\simeq\\{c,ab\\}^{\vee}|_{b=0}\simeq{\mathcal{O}}/(b,c),$
$\underline{\operatorname{Hom}}(\\{ca,b\\},\\{c,ab\\})\simeq\\{ca,b\\}^{\vee}|_{ab=0}\simeq{\mathcal{O}}(-a)/(b,c).$

The generating morphism for the former is the natural projection
${\mathcal{O}}/(ab)\to{\mathcal{O}}/(b)$, while the generating morphism for
the latter corresponds to the natural embedding

${\mathcal{O}}/(b)\rTo{a}{\mathcal{O}}(a)/(ab).$

###### Proof.

We use the standard quasiisomorphism of matrix factorizations of zero,

$\underline{\operatorname{Hom}}((E,d_{E}),\\{a,b\\})\simeq(E,d_{E})^{\vee}\otimes\\{a,b\\}\simeq(E,d_{E})^{\vee}|_{b=0}$

(see [18, Prop. 1.6.3(ii)]).

In (i) this leads to

${\mathcal{O}}(-a)/(b)\rTo{a}{\mathcal{O}}/(b)\rTo{0}\ldots$

which is quasiisomorphic to ${\mathcal{O}}/(a,b)$.

In (ii) we get

${\mathcal{O}}(-bc)/(b)\rTo{0}{\mathcal{O}}/(b)\rTo{a}{\mathcal{O}}(a)/(b)$

which is quasiisomorphic to ${\mathcal{O}}(a)/(a,b)[-1]$.

In (iii), for the first $\underline{\operatorname{Hom}}$ we get

${\mathcal{O}}(-c)/(b)\rTo{c}{\mathcal{O}}/(b)\rTo{0}\ldots$

which is quasiisomorphic to ${\mathcal{O}}/(b,c)$.

Finally, for second $\underline{\operatorname{Hom}}$ in (iii) we get

${\mathcal{O}}(-ca)/(ab)\rTo{ca}{\mathcal{O}}/(ab)\rTo{b}{\mathcal{O}}(b)/(ab)\to\ldots$

We observe that

$\operatorname{ker}(ca:{\mathcal{O}}/(ab)\to{\mathcal{O}}(ca)/(ab))=\operatorname{im}(b:{\mathcal{O}}(-b)/(ab)\to{\mathcal{O}}/(ab)),$
$\operatorname{ker}(b:{\mathcal{O}}/(ab)\to{\mathcal{O}}(b)/ab)=\operatorname{im}(a:{\mathcal{O}}(-a)/(ab)\to{\mathcal{O}}/(ab)).$

Hence, the above matrix factorization of $0$ is quasiisomorphic to

$\operatorname{coker}({\mathcal{O}}(-ca)/(ca)\rTo{ca}a{\mathcal{O}}(-a)/(ab))\simeq\operatorname{coker}({\mathcal{O}}(-ca)/(b)\rTo{c}{\mathcal{O}}(-a)/(b))\simeq{\mathcal{O}}(-a)/(b,c).$

∎

###  2.4. Toric geometry of ${\mathcal{Y}}_{n,k}$

Below we use the notation from the description of the dual cones of
${\mathcal{Y}}_{n,k}$ from Sec. 1.3. Let us denote by $\langle
S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$ the linear combinations of vectors in
$S_{\phi}$ with nonnegative integer coefficients.

######  Lemma 2.4.1.

Fix a nonincreasing map $\phi:[1,n]\to[1,k]$.

(i) Each subset $S_{\phi}\subset H$ is a basis of $H$.

(ii) One has $c_{i,j}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$
whenever $j<\phi(i)$ and $-c_{i,j}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq
0}}$ whenever $j\geq\phi(i)$.

(iii) For each $i=1,\ldots,k$ and each $j=1,\ldots,n+1$, the vectors $v_{i}$
and $x_{j}$ are in $\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$.

###### Proof.

(i) This follows immediately from the fact that the natural map of lattices

$H_{1}(\Sigma_{1})\oplus\ldots\oplus
H_{1}(\Sigma_{r})\oplus\bigoplus_{i\not\in\operatorname{im}(\phi)}\mathbb{Z}\cdot
v_{i}\to H$

is an isomorphism.

(ii) The relations $c_{i,j-1}=c_{i,j}+v_{j}$ and $-c_{i,j+1}=-c_{i,j}+v_{j+1}$
reduce this to

$c_{i,\phi(i)-1}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}},\ \
-c_{i,\phi(i)}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}.$

Let us prove the first inclusion—the proof of the second is similar. We use
descending induction on $i$. If $\phi(i+1)<\phi(i)$ or $i=n$ then
$c_{i,\phi(i)-1}\in S_{\phi}$ by the definition. If $\phi(i+1)=\phi(i)$ then
$x_{i+1}\in S_{\phi}$ and we can write
$c_{i,\phi(i)-1}=c_{i+1,\phi(i+1)-1}+x_{i+1}$, and the assertion follows from
the induction assumption.

(iii) The fact that $v_{i}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$
follows from the first definition of $S_{\phi}$ (to get $S_{\phi}$ from $S$ we
keep all $v_{i}$ for $i\not\in\operatorname{im}(\phi)$ and replace each
$v_{i}$ for $i\in\operatorname{im}(\phi)$ with a collection of vectors with
the sum $v_{i}$).

Next, let us prove that $x_{i}\in\langle S_{\phi}\rangle_{\mathbb{Z}_{\geq
0}}$. For $i=1$ this follows from the equality $x_{1}=-c_{1,k}$. For $i>1$ we
use the equality $x_{i}=c_{i-1,j}-c_{i,j}$. If $\phi(i)<\phi(i-1)$ then we set
$j=\phi(i)$ and note that $-c_{i,j}\in\langle
S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$ and $c_{i-1,j}\in\langle
S_{\phi}\rangle_{\mathbb{Z}_{\geq 0}}$ since $j<\phi(i-1)$. In the case
$\phi(i)=\phi(i-1)$ we have $x_{i}\in S_{\phi}$ by the definition. ∎

Let $\SS_{n,k}$ denote the toric hypersurface $X_{1}\ldots X_{n+1}=V_{1}\ldots
V_{k}$ in the affine space with coordinates
$(X_{1},\ldots,X_{n+1},V_{1},\ldots,V_{k})$. We have a natural toric
birational morphism

$\pi:{\mathcal{Y}}_{n,k}\to\SS_{n,k},$

which is a resolution of singularities since ${\mathcal{Y}}_{n,k}$ is smooth.

Note that $\SS_{n,k}$ is the affine toric variety associated with the cone
$C\subset{\mathbb{R}}H^{\vee}$ spanned by the vectors
$\omega_{ij}=e_{i}+f_{j}$, $i=1,\ldots,n+1$, $j=1,\ldots,k$. Here we use the
presentation

$H=(\mathbb{Z}x_{1}\oplus\ldots\oplus\mathbb{Z}x_{n+1}\oplus\mathbb{Z}v_{1}\oplus\ldots\oplus\mathbb{Z}v_{k})/(x_{1}+\ldots+x_{n+1}-v_{1}-\ldots-
v_{k}),$

which allows to identify ${\mathbb{R}}H^{\vee}$ with the hyperplane
$\alpha_{1}+\ldots+\alpha_{n+1}=\beta_{1}+\ldots+\beta_{k}$ in
${\mathbb{R}}^{n+1}\times{\mathbb{R}}^{k}$, and we denote by
$(e_{1},\ldots,e_{n+1})$ and $(f_{1},\ldots,f_{k})$ the standard bases in
${\mathbb{R}}^{n+1}$ and ${\mathbb{R}}^{k}$.

Note that the vectors $\omega_{ij}$ are precisely the vertices of the polytope

$\Pi:=\Delta_{n}\times\Delta_{k-1}\subset{\mathbb{R}}^{n+1}\times{\mathbb{R}}^{k},$

where $\Delta_{m}\subset{\mathbb{R}}^{m+1}$ is the standard simplex spanned by
the basis vectors. Thus, $C$ is the cone generated by this polytope. Now
taking cones over simplices of a standard triangulation of $\Pi$ (see e.g.,
[5, Ch. 7, Sec. 3D]) one gets a toric fan, and the corresponding toric variety
is a small resolution of $\SS_{n,k}$ (since we do not add new $1$-dimensional
cones). We claim that this precisely leads to our variety
${\mathcal{Y}}_{n,k}$.

######  Proposition 2.4.2.

The map $\pi$ is the small resolution, associated with the standard
triangulation of $\Pi$ corresponding to the usual ordering of the vertices of
$\Delta_{n}$ and the reverse ordering of the vertices of $\Delta_{k-1}$.

###### Proof.

We need to identify the dual simplices to $\sigma^{\vee}_{\phi}$, where $\phi$
runs through nonincreasing maps $[1,n]\to[1,k]$, with the simplices of the
standard triangulation of $\Pi$ described in [5, Ch. 7, Sec. 3D]. Note that
due to our ordering of the vertices of $\Delta_{k-1}$, the latter simplices
are numbered by lattice paths from $(1,k)$ to $(n+1,1)$, i.e., paths obtained
by starting at the point $(1,k)$ and moving either to the right or down along
the grid until reaching the point $(n+1,1)$. Alternatively, they can be
numbered by shuffles $w$ of the word $A_{1}A_{2}\ldots A_{n}B_{1}B_{2}\ldots
B_{k-1}$ (preserving the order of $A_{i}$’s and of $B_{j}$’s), where $A_{i}$
stands for the $i$th horizontal move and $B_{j}$ stands for the $j$th vertical
move.

The simplex $\Delta_{L}$ of the triangulation associated with the lattice path
$L$ is given inside $\Pi$ by the collection of inequalities:
$\alpha_{1}+\ldots+\alpha_{i}\leq\beta_{k}+\ldots+\beta_{j}$ whenever $A_{i}$
precedes $B_{j}$ in the corresponding shuffled word $w_{L}$, and
$\alpha_{1}+\ldots+\alpha_{i}\geq\beta_{k}+\ldots+\beta_{j}$ whenever $B_{j}$
precedes $A_{i}$.

We have a natural bijection between lattice paths and nonincreasing functions
$\phi:[1,n]\to[1,k]$: for a lattice path $L$ we let $\phi_{L}$ to be the
unique function such that $(i+1/2,\phi_{L}(i))$ lies on the path $L$ for each
$i=1,\ldots,n$. Now it is easy to see that $A_{i}$ precedes $B_{j}$ in the
shuffled word $w_{L}$ if and only if $\phi_{L}(i)\geq j$. Thus, due to formula
(1.8) the simplex $\Delta_{L}$ is given inside $\Pi$ by the inequalities
$\langle v,c_{i,j-1}\rangle\geq 0$ whenever $\phi_{L}(i)\geq j$, and $\langle
v,-c_{i,j-1}\rangle\geq 0$ whenever $\phi_{L}(i)<j$. Hence, the same
inequalities give the cone $\sigma_{L}$ generated by $\Delta_{L}$ inside the
cone $C$.

It remains to observe that the cone $C$ is dual to the cone generated by all
vectors $x_{i}$ and $v_{j}$ in $H$. Hence, Lemma 2.4.1 implies that the cone
$\sigma_{L}$ is dual to the cone $\sigma^{\vee}_{\phi_{L}}$. Thus,
${\mathcal{Y}}_{n,k}$ is precisely the toric variety associated with the fan
with the maximal cones $\sigma_{L}$. ∎

######  Corollary 2.4.3.

The canonical line bundle $\omega_{{\mathcal{Y}}_{n,k}}$ is trivial.

##  3\. A-side

###  3.1. Fukaya categories

####  3.1.1. Generalities

In [14], we studied the partially wrapped Fukaya category of
$\operatorname{Sym}^{n}(\Sigma)$ equipped with a stop of the form
$Z\times\mathrm{Sym}^{n-1}(\Sigma)$, where $\Sigma$ is a genus 0 surface with
boundary and $Z$ is either one point or two points lying on a single component
of $\partial\Sigma$. Our main tool was Auroux’s determination of generators
for this category [2] and the combinatorial description of their endomorphism
algebras via strand algebras from [10]. In the case where $Z$ is two points we
explicitly wrote out the endomorphism algebra of a particular generator of
these categories.

Here, we will concentrate in the case where $Z$ is a single point marked at
the outer boundary component as in Figure 7. We will use the same Lagrangians
as in [14]. Namely, they are given by products of $n$ disjoint arcs among the
Lagrangians drawn on $\Sigma$ in Figure 7 except that as we only consider the
case $Z$ is a single point, the last arc $L_{n+k}$ is not needed. Thus, our
generators are of the form

$L_{S}=L_{i_{1}}\times L_{i_{2}}\times\ldots\times L_{i_{n}}$

where $S$ is a subset of $[0,n+k-1]$ of size $n$. The computation given in
[14, Thm 3.2.5] readily gives the endomorphism algebras we need, and the
corresponding $A_{\infty}$-algebra is formal.

In order to avoid repetition, we have chosen to skip over many of the details;
the reader is referred to [14, Sections 2 & 3] for a complete background on
these calculations. We also note that in [14], $\Sigma$ was a
$(k+1)$-punctured surface of genus 0. Here, we let $\Sigma$ to be the genus 0
surface with $(n+k+1)$-punctures. This slight modification in notation turns
out to be more natural. Finally, we note that in [14] we worked with
$\mathbb{Z}$-graded categories (for all possible grading structures), whereas
here we consider the ${\mathbb{L}}$-graded categories, which in a sense is
equivalent to working with all the $\mathbb{Z}$-grading structures compatible
with a given $\mathbb{Z}/2$-grading structure. The latter structure is
characterized by the fact that all endomorphisms of our generators have even
grading (this choice is made in order to get an equivalence with the B-side).

$\ell_{1}$$\ell_{2}$$\ell_{k}$$\ell_{k+1}$$\ell_{n+k-1}$$r_{1}$$r_{2}$$r_{k}$$r_{k+1}$$r_{n+k-1}$$u_{1}$$u_{2}$$u_{k}$$x_{1}$$x_{n-1}$$x_{n}$$L_{0}$$L_{1}$$L_{k}$$L_{n+k-1}$$T$
Figure 7. Surface $\Sigma$ and generating Lagrangians

####  3.1.2. The universal grading group and its splitting

Let us set $V=\operatorname{Sym}^{n}(\Sigma)$. Note that there is an
isomorphism between $H_{1}(V)$ and $H_{1}(\Sigma)=\mathbb{Z}^{n+k}$ and
$c_{1}(V)=0$. Thus, the grading group ${\mathbb{L}}=H_{1}(\mathcal{G}V)$,
where ${\mathcal{G}}V$ is the bundle of Lagrangian subspaces in $TV$, fits
into an exact sequence

$0\to\mathbb{Z}\to{\mathbb{L}}\to H_{1}(\Sigma)\to 0$ (3.1)

In Lemma 3.1.1 below, we will see that there is a canonical splitting of this
sequence. Using this splitting, we can lift the boundary loops
$u_{1},\ldots,u_{k},x_{1},\ldots,x_{n}$ to homology classes in
${\mathbb{L}}=H_{1}(\mathcal{G}V)$ (denoted in the same way) and we choose the
unique $\mathbb{Z}/2$-grading structure

$\sigma:\mathbb{L}\to\mathbb{Z}/2$

which sends $u_{1},\ldots u_{k},x_{1},\ldots,x_{n}$ to zero (and
$\sigma(l_{0})=1$). This particular choice of $\mathbb{Z}/2$-grading structure
is made in order to match the $B$-side computations.

Following [24], the grading of a morphism in a partially wrapped Fukaya
category can be defined as follows. Suppose $L$ and $L^{\prime}$ are two
Lagrangians in $V$ equipped with grading structures, i.e., lifts of their
Gauss maps $L,L^{\prime}\to\mathcal{G}V$ to $\tilde{\mathcal{G}}V$, the
universal abelian cover of ${\mathcal{G}}V$. Note that the group
${\mathbb{L}}$ acts on $\tilde{\mathcal{G}}V$ by deck transformations. This
gives a simply transitive action of ${\mathbb{L}}$, on the set of grading
structures on a Lagrangian $L$, which we denote as $L\mapsto L(\ell)$, where
$\ell\in{\mathbb{L}}$. By definition, one has
$\hom^{\ell_{1}}(L,L^{\prime}(\ell_{2}))=\hom^{\ell_{1}+\ell_{2}}(L,L^{\prime})$,
so it is enough to characterize which morphisms have degree
$0\in{\mathbb{L}}$.

We have the following commutative diagram

${\tilde{\mathcal{G}}V}$${\tilde{V}}$${\mathcal{G}V}$${V}$

where $\tilde{V}$ is the universal abelian cover of $V$. Note that the top
arrow $\tilde{\mathcal{G}}V\to\tilde{V}$ also can be identified as the
(unique) fibre-wise universal cover of the Lagrangian Grassmannian
$\mathcal{G}\tilde{V}$. Thus, the symplectic manifold $\tilde{V}$ is equipped
with a (unique) $\mathbb{Z}$-grading structure.

Now assume that $\gamma\in\mathrm{hom}(L,L^{\prime})$ is a Reeb chord. The
grading structures on $L,L^{\prime}$ determine lifts $\tilde{L}$ and
$\tilde{L}^{\prime}$ of $L$ and $L^{\prime}$ to $\tilde{V}$. Furthermore,
$\tilde{L}$ and $\tilde{L}^{\prime}$ come equipped with $\mathbb{Z}$-grading
structures, so morphisms between $\tilde{L}$ and $\tilde{L}^{\prime}$ in the
partially wrapped Fukaya category of $\tilde{V}$ are equipped with
$\mathbb{Z}$-grading. The path $\gamma:L\to L^{\prime}$ lifts to a unique path
$\tilde{\gamma}:\tilde{L}\to\overline{L}^{\prime}$ where
$\overline{L}^{\prime}$ is a lift of $L^{\prime}$ to $\tilde{V}$, possibly
different from $\tilde{L}^{\prime}$. Now we say that $\gamma$ has degree
$0\in{\mathbb{L}}$ if $\overline{L}^{\prime}=\tilde{L}^{\prime}$ and the
$\mathbb{Z}$-degree of the morphism given by $\tilde{\gamma}$ is $0$.

Note that the classes of the boundary loops
$v_{1},\ldots,v_{k},x_{1},\ldots,x_{n}$ (see Figure 7) form a basis of
$H_{1}(\Sigma)$. Let us set $u_{i}$ the be the circle associated oriented with
the opposite orientation to $v_{i}$ so that its boundary orientation agrees
with the boundary orientation of the surface. We are going to use these
classes to define a splitting of the sequence (3.1).

######  Lemma 3.1.1.

There exist canonical lifts of the classes of the boundary loops
$u_{1},\ldots,u_{k}$, and $x_{1},\ldots,x_{n}$ to ${\mathbb{L}}$ (which we
will denote by the same letters), such that the ${\mathbb{L}}$-degree of any
nonzero endomorphism $u_{i}\in\operatorname{End}(L_{S})$ (resp.,
$x_{i}\in\operatorname{End}(L_{S})$) in the partially wrapped Fukaya category
of $V=\operatorname{Sym}^{n}(\Sigma)$ is given by this lift, i.e the
${\mathbb{L}}$-degree is independent of $S$.

###### Proof.

Recall that away from the big diagonal
$\Delta\subset\operatorname{Sym}^{n}(\Sigma)$, the Reeb flow along the contact
boundary is given by the product of the Reeb flow on the surface and this
agrees with the circular flow around the boundary components of $\Sigma$ in
the direction induced by the boundary orientations. Therefore, to an orbit
$\gamma$ going once around a boundary component of $\Sigma$, we can associate
a Reeb orbit of the form

$\tilde{\gamma}=p_{1}\times\ldots\times p_{n-1}\times\gamma$

in $\operatorname{Sym}^{n}(\Sigma)\setminus\Delta$, where
$p_{1},\ldots,p_{n-1}$ are fixed points.

Given a boundary component $\gamma$ of $\Sigma$, other than the one marked
with a stop, and an arc $L_{i}$ intersecting it, for any subset
$S=\\{i,i_{1},\ldots,i_{n-1}\\}$, we construct as above the Reeb orbit
$\tilde{\gamma}=p_{1}\times\ldots\times p_{n-1}\times\gamma$ with
$\tilde{\gamma}(0)=\tilde{\gamma}(1)\in L_{S}$, by fixing some points
$p_{1}\in L_{i_{1}},\ldots,p_{n-1}\in L_{i_{n-1}}$. We need to check that the
grading in ${\mathbb{L}}$ of the corresponding morphism in
$\mathrm{hom}(L_{S},L_{S})$ does not depend on $S$. Suppose we are given some
other $S^{\prime}$ containing $i$,
$S^{\prime}=\\{i,i^{\prime}_{1},\ldots,i^{\prime}_{n-1}\\}$ and some fixed
points $p^{\prime}_{1}\in L_{i^{\prime}_{1}},\ldots,p^{\prime}_{n-1}\in
L^{\prime}_{i^{\prime}_{n-1}}$, so that we have the corresponding Reeb chord
$\tilde{\gamma}^{\prime}=p^{\prime}_{1}\times\ldots\times
p^{\prime}_{n-1}\times\gamma$. Now we note that the definition of the grading
of $\tilde{\gamma}$ does not change if we replace $L_{S}$ by its open subset
$U_{S}=L_{i}\times U_{i_{1}}\times\ldots\times U_{i_{n-1}}$, where $U_{i_{r}}$
is an open neighborhood of $p_{i_{r}}$ in $L_{i_{r}}$. Similarly, we can
replace $L_{S^{\prime}}$ by an open subset $U_{S^{\prime}}$. Now we can deform
$(U_{S},\tilde{\gamma})$ to $(U_{S}^{\prime},\tilde{\gamma}^{\prime})$
continuously by moving the fixed points $p_{1},\ldots,p_{n-1}$ and their
neighborhoods in the arcs $L_{i_{1}},\ldots,L_{i_{n-1}}$ to the points
$p^{\prime}_{1},\ldots,p^{\prime}_{n-1}$ and their neighborhoods in the arcs
$L_{i^{\prime}_{1}},\ldots,L_{i^{\prime}_{n-1}}$. Clearly, such a continuous
deformation does not change the calculation of the grading.

Next, given another arc $L_{j}$ with $j=i\pm 1$, intersecting $\gamma$, and
indices $i_{1},\ldots,i_{n-1}$ distinct from $i$ and $j$, we have to check
that the grading of the chord from $L_{i}\times L_{i_{1}}\times\ldots\times
L_{i_{n-1}}$ to itself, induced by $\gamma$, is equal to the grading of the
similar chord from $L_{j}\times L_{i_{1}}\times\ldots L_{i_{n-1}}$. For this
we use a similar trick: we can replace $L_{i}$ and $L_{j}$ by neighborhoods of
the point of intersection with $\gamma$ and deform one into another.

Finally, for any indices $i_{1},\ldots,i_{n-2}$, we can check that the
gradings of the two chords from $L_{i}\times L_{j}\times
L_{i_{1}}\times\ldots\times L_{i_{n-2}}$ to itself, induced by $\gamma$ (which
we can stick either in the first or in the second factor) are the same: we
replace $L_{i}$ and $L_{j}$ by neighborhoods of the points of intersection
with $\gamma$ and move them along $\gamma$ until they swap. ∎

Lemma 3.1.1 allows us to determine the ${\mathbb{L}}$-grading on the
endomorphism algebra of our generators. Indeed, the ${\mathbb{L}}$-grading
structure can be computed as in [14, Thm. 3.2.5(iii)], which shows that up to
shifts of the generators, the grading is uniquely determined by the degrees of
$u_{r}$ and $x_{r}$ in ${\mathbb{L}}$.

###  3.2. Endomorphism algebra of the generating Lagrangians: case $n=1$

The generating objects are

$L_{0},L_{1},\ldots,L_{k}$

We can express the endomorphism algebra

$\mathcal{A}_{1,k}=\bigoplus_{i,j=0}^{k}\mathrm{hom}(L_{i},L_{j})$

via the following quiver

012(k-1)k$r_{1}$$\ell_{1}$$r_{2}$$\ell_{2}$$r_{k}$$\ell_{k}$$x_{1}$

with relations

$\displaystyle\ell_{i+1}\ell_{i}$ $\displaystyle=0=r_{i}r_{i+1}$
$\displaystyle x_{1}\ell_{k}$ $\displaystyle=0=r_{k}x_{1}$

Let us work with the $\mathbb{Z}$-graded category for which
$\deg(\ell_{i})=\deg(r_{i})=1$ for $i\leq k$ and $\deg(x_{1})=0$. To remove
the stop, we need to localize by the subcategory generated by an arc $T$
around the stop. It is easy to see that $T$ can be represented by the twisted
complex

$L_{0}[1]\xrightarrow{\ell_{1}}L_{1}[1]\xrightarrow{\ell_{2}}\ldots\xrightarrow{\ell_{k-1}}L_{k-1}[1]\xrightarrow{\ell_{k}}L_{k}[1]\xrightarrow{x_{1}}L_{k}\xrightarrow{r_{k}}L_{k-1}\ldots\xrightarrow{r_{2}}L_{1}\xrightarrow{r_{1}}L_{0}.$
(3.2)

###  3.3. Endomorphism algebra of the generating Lagrangians: case $n=2$

The generating objects are

$L_{i,j}\coloneqq L_{i}\times L_{j}\text{ for
}i,j\in\\{0,1,\ldots,k+1\\}\text{ with }i\neq j.$

The endomorphism algebra is defined by

$\mathcal{A}_{2,k}=\bigoplus_{S,S^{\prime}}\mathrm{hom}(L_{S},L_{S^{\prime}})$

where the sum is over distinct two element subsets of $\\{0,1\ldots,k+1\\}$.
Let us write $e_{ij}\in\mathrm{end}(L_{S})$ for the idempotent corresponding
to $S=\\{i,j\\}$.

$\mathcal{A}_{2,k}$ can be described via the following quiver over
$\mathbf{k}[u_{1},u_{2},\ldots,u_{k},x_{1},x_{2}]$

$01$$02$$03$$12$$13$$23$$k(k+1)$$r_{2}$$\ell_{2}$$r_{3}$$\ell_{3}$$\ell_{4}$$r_{4}$$r_{k+1}$$\ell_{k+1}$$r_{3}$$\ell_{3}$$r_{4}$$\ell_{4}$$r_{k+1}$$\ell_{k+1}$$r_{4}$$\ell_{4}$$r_{k+1}$$\ell_{k+1}$$r_{1}$$r_{1}$$r_{1}$$\ell_{1}$$\ell_{1}$$\ell_{1}$$r_{2}$$r_{2}$$\ell_{2}$$\ell_{2}$$r_{3}$$\ell_{3}$$r_{k}$$\ell_{k}$

with relations, using the convention that $x_{1}=u_{k+1},x_{2}=u_{k+2}$,

$\displaystyle u_{r}e_{ij}=0\text{ if }r\neq i,i+1,j,j+1,$
$\displaystyle\ell_{i+1}\ell_{i}=0=r_{i}r_{i+1}\text{ for all }i,$
$\displaystyle r_{i}r_{i+j}=r_{i+j}r_{i}\text{ for all }i\text{ and for all
}j\geq 2,$ $\displaystyle\ell_{i}\ell_{i+j}=\ell_{i+j}\ell_{i}\text{ for all
}i\text{ and for all }j\geq 2,$
$\displaystyle\ell_{i}r_{i+j}=r_{i+j}\ell_{i}\text{ for all }i\text{ and for
all }j\geq 2,$ $\displaystyle r_{i}\ell_{i+j}=\ell_{i+j}r_{i}\text{ for all
}i\text{ and for all }j\geq 2,$ $\displaystyle
r_{i}\ell_{i}=u_{i}=\ell_{i}r_{i}\text{ for }i=1,\ldots,k+1,$ $\displaystyle
x_{2}\ell_{k+1}=0=r_{k+1}x_{2}.$

To remove the stop, we need to localize by the subcategory generated by

$T\times L_{i},\text{ for }i=0,\ldots,k+1$

where $T$ is the arc around the stop. We will identify resolutions for these
objects in terms of the generators $(L_{i,j})$. Let us work in the
$\mathbb{Z}$-graded Fukaya category, where we use the gradings of $(L_{i,j})$
such that $\deg(\ell_{i})=\deg(r_{i})=1$ for $i\leq k$ and
$\deg(\ell_{k+1})=\deg(r_{k+1})=\deg(x_{2})=0$.

######  Proposition 3.3.1.

For $i\leq k$, $T\times L_{i}$ is represented by the twisted complex

$\begin{array}[]{l}L_{0,i}[1]\xrightarrow{\ell_{1}}\ldots\xrightarrow{\ell_{i-1}}L_{i-1,i}[1]\xrightarrow{\ell_{i}\times\ell_{i+1}}L_{i,i+1}[2]\xrightarrow{\ell_{i+2}}\ldots\xrightarrow{\ell_{k}}L_{i,k}[2]\xrightarrow{\ell_{k+1}}L_{i,k+1}[1]\xrightarrow{x_{2}}\\\
L_{i,k+1}\xrightarrow{r_{k+1}}L_{i,k}[-1]\xrightarrow{r_{k}}\ldots\xrightarrow{r_{i+2}}L_{i,i+1}[-1]\xrightarrow{r_{i}\times
r_{i+1}}L_{i-1,i}\xrightarrow{r_{i-1}}\ldots\xrightarrow{r_{1}}L_{0,i},\end{array}$
(3.3)

while $T\times L_{k+1}$ is represented by

$L_{0,k+1}[1]\xrightarrow{\ell_{1}}\ldots\xrightarrow{\ell_{k}}L_{k,k+1}[1]\xrightarrow{x_{1}}L_{k,k+1}\xrightarrow{r_{k}}\ldots\xrightarrow{r_{1}}L_{0,k+1}.$
(3.4)

###### Proof.

By considering the $n=1$ case, we already know that $T$ can be represented by
the twisted complex

$L_{0}[1]\xrightarrow{\ell_{1}}L_{1}[1]\xrightarrow{\ell_{2}}\ldots\xrightarrow{\ell_{k}}L_{k}[1]\xrightarrow{\ell_{k+1}}L_{k+1}[1]\xrightarrow{x_{2}}L_{k+1}\xrightarrow{r_{k+1}}L_{k}\ldots\xrightarrow{r_{2}}L_{1}\xrightarrow{r_{1}}L_{0}$

Since this description involves $L_{i}$, we cannot directly multiply this
complex with $L_{i}$ to get a description of $T\times L_{i}$. We instead will
argue in two steps by using the arcs $U$ and $D$ drawn in Figure 8. This
applies for $i\leq k$ and for $i=k+1$ we use the arc $E$ which plays a similar
role. Using these, we see that $T$ can be represented by

$\begin{array}[]{l}L_{0}[1]\xrightarrow{\ell_{1}}\cdots L_{i-2}[1]\to U\to
L_{i+2}[1]\xrightarrow{\ell_{i+3}}\cdots\xrightarrow{\ell_{k+1}}L_{k+1}[1]\xrightarrow{x_{2}}L_{k+1}\xrightarrow{r_{k+1}}\\\
L_{i+2}[-1]\to D\to L_{i-2}\cdots\xrightarrow{r_{1}}L_{0}\end{array}$ (3.5)

for $i\leq k$ and

$\begin{array}[]{l}L_{0}[1]\xrightarrow{\ell_{1}}\cdots\xrightarrow{\ell_{k-1}}L_{k-1}[1]\to
E\to L_{k-1}\xrightarrow{r_{k-1}}\cdots\xrightarrow{r_{1}}L_{0}\end{array}$
(3.6)

for $i=k+1$.

Since none of these complexes involve $L_{i}$, we can multiply every term by
$L_{i}$ to get a twisted complex describing $T\times L_{i}$.

Thus, it remains to describe $U\times L_{i}$, $D\times L_{i}$ for $i\leq k$
and $E\times L_{k+1}$ in terms of our generators. We will prove that $U\times
L_{i}$ is given by the cone

$L_{i-1}\times L_{i}\xrightarrow{\ell_{i}\times\ell_{i+1}}L_{i}\times L_{i+1}$

The other cases are similar. The existence of this triangle can be seen as
follows: First slide $L_{i-1}$ over $L_{i}$ to $\tilde{U}$. As this is a
Hamiltonian isotopy, this gives an equivalence $L_{i-1}\times
L_{i}\simeq\tilde{U}\times L_{i}$. Next, observe that sliding $\tilde{U}$ over
$L_{i+1}$ gives $U$, hence we have an exact $C$ is given as a cone
$\tilde{U}\to L_{i+1}$. Multiplying this with $L_{i}$, we conclude that
$L_{i-1}\times L_{i}\simeq\tilde{U}\times L_{i}\to L_{i}\times L_{i+1}$
describes the object $U\times L_{i}$.

Finally, since none of the slides involved lower part of the surface, the
morphism in the cone description has to be $\ell_{i}\times\ell_{i+1}$ as any
other morphism from $L_{i-1}\times L_{i}\to L_{i}\times L_{i+1}$ goes around
the holes. Indeed, one can see that this exact triangle in
${\mathbb{L}}$-graded setting which necessitates that the connecting morphism
to be $\ell_{i}\times\ell_{i+1}$. Similarly, note that there exists no
morphism from $L_{i-2}\times L_{i}$ to $L_{i}\times L_{i+1}$ (resp.
$L_{i-1}\times L_{i}$ to $L_{i}\times L_{i+2}$) and there exists a unique
morphism from $L_{i-2}\times L_{i}$ to $L_{i-1}\times L_{i}$ (resp.
$L_{i}\times L_{i+1}\to L_{i}\times L_{i+2}$), namely
$\ell_{i-1}\times\operatorname{id}_{{L}_{i}}$ (resp.
$\operatorname{id}_{{L}_{i}}\times\ell_{i+2}$), which does not go around the
holes. Hence, for reasons of ${\mathbb{L}}$-grading, it follows that the
morphisms in the twisted complex representation of the $T\times L_{i}$ in
terms of our generators must be those given in Equations 3.3 and 3.4. ∎

$L_{i-1}$$U$$D$$L_{i}$$\tilde{U}$$L_{i+1}$$L_{k}$$L_{k+1}$$E$ Figure 8.
Auxiliary Lagrangians

##  4\. B-side: case $n=1$

###  4.1. Compactified LG model

The variety $\overline{{\mathcal{Y}}}_{1,k}$ is covered by $k+1$ affine charts
${\mathcal{U}}_{0},\ldots,{\mathcal{U}}_{k}$, each isomorphic to the affine
space, where the coordinates on each chart are given as follows

$\begin{array}[]{l}{\mathcal{U}}_{0}:V_{k},\ldots,V_{2},V_{1},X_{2}^{-1},\\\
{\mathcal{U}}_{1}:V_{k},\ldots,V_{2},C_{1}^{-1},X_{2},\\\ \cdots\\\
{\mathcal{U}}_{k-1}:V_{k},C_{k-1}^{-1},C_{k-2},V_{k-2},\ldots,V_{1},\\\
{\mathcal{U}}_{k}:X_{1},C_{k-1},V_{k-1},\ldots,V_{1}.\end{array}$

where we abbreviate $C_{1i}$ as $C_{i}$. We have

$V_{k}=X_{1}C_{k-1},\ V_{k-1}=C_{k-1}^{-1}C_{k-2},\ \ldots,\
V_{1}=C_{1}^{-1}X_{2},$

and we consider
$\overline{{\mathcal{X}}}_{1,k}=\overline{{\mathcal{Y}}}_{1,k}\times{\mathbb{A}}^{k}$
with the potential

${\mathbf{w}}_{1,k}=U_{1}V_{1}+\ldots+U_{k}V_{k}.$

It is convenient to set $C_{k}=X_{1}^{-1}$, $C_{0}=X_{2}$.

The subvariety $\overline{{\mathcal{Z}}}_{1,k}$ given as the vanishing locus
of $(V_{1},\ldots,V_{k})$ is the nodal chain of rational curves
$\overline{R}_{0}\cup R_{1}\cup\ldots R_{k-1}\cup R_{k}$, where
$R_{k}={\mathbb{A}}^{1}$ with coordinate $X_{1}$; $R_{i}={\mathbb{P}}^{1}$ for
$0<i<k$ with $C_{i}$ and $C_{i}^{-1}$ being the coordinates on its affine
charts; $\overline{R}_{0}={\mathbb{P}}^{1}$ with $X_{2}$ and $X_{2}^{-1}$
being the coordinates on its affine charts. Note that the intersection point
$R_{i}\cap R_{i-1}$ lies on the nodal union of the affine lines
${\mathbb{A}}^{1}_{C_{i}^{-1}}\subset R_{i}$ and
${\mathbb{A}}^{1}_{C_{i-1}}\subset R_{i-1}$.

The critical locus
$\operatorname{crit}({\mathbf{w}}_{1,k})\subset\overline{{\mathcal{X}}}_{1,k}$
is the union of our nodal chain $\overline{{\mathcal{Z}}}_{1,k}$ with $k$
affine lines $L_{1},\ldots,L_{k}$, such that $L_{i}$ intersects the nodal
chain at the $i$th node.

######  Lemma 4.1.1.

(i) For every $i=0,\ldots,k-1$, there is a morphism
$p_{i}:\overline{{\mathcal{Y}}}_{1,k}\to{\mathbb{P}}^{1}$ such that
$p_{i}^{*}z=C_{i}$ for $i>0$ and $p_{0}^{*}z=X_{2}$, where $z$ is a coordinate
on ${\mathbb{A}}^{1}\subset{\mathbb{P}}^{1}$. Furthermore, the induced map

$f:\overline{{\mathcal{Y}}}_{1,k}\rTo{X_{1},V_{1},\ldots,V_{k},p_{0},\ldots,p_{k-1}}{\mathbb{A}}^{k+1}_{X_{1},V_{1},\ldots,V_{k}}\times({\mathbb{P}}^{1})^{k}$

is a closed embedding.

(ii) For $i=1,\ldots,k-1$, we have sections of $p_{i}^{*}{\mathcal{O}}(-1)$,
defined in an open neighborhood of $R_{i}$,

$s_{1}(i)=\frac{C_{i+1}^{-1}}{p_{i}^{*}z_{0}}=\frac{V_{i+1}}{p_{i}^{*}z_{1}},\
\ s_{2}(i)=\frac{C_{i-1}}{p_{i}^{*}z_{1}}=\frac{V_{i}}{p_{i}^{*}z_{0}}$ (4.1)

(where $(z_{0}:z_{1})$ are homogeneous coordinates on ${\mathbb{P}}^{1}$ such
that $z=z_{1}/z_{0}$), such that $R_{i}$ is the vanishing locus of

$V_{1},\ldots,V_{i-1},s_{1}(i),s_{2}(i),V_{i+2},\ldots,V_{k}.$

Similarly, we have a section of $p_{0}^{*}{\mathcal{O}}(-1)$, defined in a
neighborhood of $\overline{R}_{0}$

$s_{1}(0)=\frac{C_{1}^{-1}}{p_{0}^{*}z_{0}}=\frac{V_{1}}{p_{0}^{*}z_{1}},$

such that $\overline{R}_{k}$ is the vanishing locus
$s_{1}(0),V_{2},\ldots,V_{k}$. Finally, locally near $R_{k}$, it is given as
the vanishing locus of $V_{1},\ldots,V_{k-1},C_{k-1}$.

###### Proof.

(i) Let ${\mathbb{A}}_{0}=(z_{0}\neq 0)$ and ${\mathbb{A}}_{1}=(z_{1}\neq 0)$
be the standard open covering of ${\mathbb{P}}^{1}$ (where $(z_{0}:z_{1})$ are
the homogeneous coordinates such that $z=z_{1}/z_{0}$). Then we define $p_{i}$
as the unique map such that

$p_{i}^{-1}({\mathbb{A}}_{0})={\mathcal{U}}_{i+1}\cup\ldots\cup{\mathcal{U}}_{k},\
\
p_{i}^{-1}({\mathbb{A}}_{1})={\mathcal{U}}_{0}\cup\ldots\cup{\mathcal{U}}_{i},$

with $p_{i}^{*}z=C_{i}$ on
${\mathcal{U}}_{i+1}\cup\ldots\cup{\mathcal{U}}_{k}$ and
$p_{i}^{*}z^{-1}=C_{i}^{-1}$ on
${\mathcal{U}}_{0}\cup\ldots\cup{\mathcal{U}}_{i}$. The fact that $C_{i}$
(resp., $C_{i}^{-1}$) is indeed well defined on ${\mathcal{U}}_{j}$ with $j>i$
(resp., $j\leq i$) follows from the identity $C_{j}=C_{j+1}V_{j+1}$.

For a sequence $(i_{1},\ldots,i_{k})$ of $0$’s and $1$’s let us set

${\mathbb{A}}_{i_{1},\ldots,i_{k}}:={\mathbb{A}}_{i_{1}}\times\ldots\times{\mathbb{A}}_{i_{k}}\subset({\mathbb{P}}^{1})^{k}.$

To prove the second assertion we note that
$f^{-1}({\mathbb{A}}^{k+1}\times{\mathbb{A}}_{i_{1},\ldots,i_{k}})$ is
nonempty only when $(i_{1},\ldots,i_{k})=((0)^{i},(1)^{k-i})$ for some $0\leq
i\leq k$, and in this case

$f^{-1}({\mathbb{A}}^{k+1}\times{\mathbb{A}}_{(0)^{i},(1)^{k-i}})={\mathcal{U}}_{i}.$

It is easy to check that the induced map

${\mathcal{U}}_{i}\rTo{X_{1},V_{1},\ldots,V_{k},C_{0},\ldots,C_{i-1},C_{i}^{-1},\ldots,C_{k-1}}{\mathbb{A}}^{k+1}\times{\mathbb{A}}_{(0)^{i},(1)^{k-i}}$

is a closed embedding.

(ii) The fact that $s_{1}(i)$ (resp., $s_{2}(i)$) is well defined follows from
the identity $C_{i+1}^{-1}C_{i}=V_{i+1}$ (resp., $C_{i}^{-1}C_{i-1}=V_{i}$).
The fact that $R_{i}$ coincides with the claimed vanishing locus can be
checked locally. ∎

Let us set $\overline{R}_{i}=R_{i}$ for $i>0$, and consider the objects
$(P_{i})_{i=0,\ldots,k}$ of the multigraded category
$\overline{{\mathcal{B}}}_{1,k}$ of $\widetilde{T}$-equivariant matrix
factorization of ${\mathbf{w}}_{1,k}$ on $\overline{{\mathcal{X}}}_{1,k}$,
associated with the objects
${\mathcal{O}}_{\overline{R}_{i}\times{\mathbb{A}}^{k}}$ of the corresponding
singularity category. Note that the character group $\widetilde{H}$ of the
torus $\widetilde{T}$ is a free abelian group with the basis
$x_{1},u_{1},\ldots,u_{k},l_{0}$. Hence, the grading group
${\mathbb{L}}_{B}=2\widetilde{H}+\mathbb{Z}l_{0}$ is a free abelian group with
the basis $2x_{1},2u_{1},\ldots,2u_{k},l_{0}$, so that the grading of $X_{1}$
is $2x_{1}$ and the grading of $U_{i}$ is $2u_{i}$. We have
$v_{i}=l_{0}-u_{i}$, so the relations $v_{i}=-c_{i}+c_{i-1}$ allow to compute
the grading $c_{i}$ of $C_{i}$ (recall that $c_{k}=-x_{1}$).

######  Proposition 4.1.2.

The only nonzero morphisms between $(P_{i})$ in
$\overline{{\mathcal{B}}}_{1,k}$ are the following:

$\displaystyle\operatorname{End}(P_{k})=k[X_{1},U_{k}]/(X_{1}U_{k}),\
\operatorname{End}(P_{k-1})=k[U_{k},U_{k-1}]/(U_{k}U_{k-1}),\ \ldots,\
\operatorname{End}(P_{1})=k[U_{2},U_{1}]/(U_{2}U_{1}),$
$\displaystyle\operatorname{End}(P_{0})=k[U_{1}],$
$\operatorname{Hom}(P_{i},P_{i-1})=k[U_{i}]\cdot a_{i},\ \
\operatorname{Hom}(P_{i-1},P_{i})=k[U_{i}]\cdot b_{i},\ \text{ for
}i=1,\ldots,k,$

where

$\deg(a_{i})=l_{0}-2c_{i-1},\ \ \deg(b_{i})=l_{0}+2c_{i},$
$a_{i}b_{i}=b_{i}a_{i}=U_{i}.$

The proof uses presentations of $\overline{R}_{i}$ as complete intersections
and Lemma 2.3.6. We will give the details of similar calculations in the case
$n=2$.

There is a unique projection $|\cdot|:{\mathbb{L}}_{B}\to\mathbb{Z}$ sending
$l_{0}$ to $1$ and $x_{1}$ and $u_{i}$ to $0$. In the obtained
$\mathbb{Z}$-graded category the $\mathbb{Z}$-gradings are given by

$|X_{1}|=|U_{1}|=\ldots=|U_{k}|=0.$

With this choice of $\mathbb{Z}$-grading we have $|a_{i}|=1-2(k+1-j)$, and the
endomorphism algebra

$\operatorname{End}(P_{k}\oplus P_{k-1}[-1]\oplus
P_{k-2}[-4]\oplus\ldots\oplus P_{0}[-k^{2}])$

will be concentrated in degree $0$.

By Lemma 2.1.3, this implies formality of the ${\mathbb{L}}_{B}$-graded dg-
endomorphism algebra of $\bigoplus P_{i}$.

The following simple general result will help us to prove generation.

######  Lemma 4.1.3.

Suppose a scheme $X$ is the union of two closed subschemes: $X=Y\cup Z$. Set

$U=X\setminus Y=Z\setminus Y\cap Z.$

Assume we have a collection of coherent sheaves $(F_{i})_{i\in I}$ on $Y$,
generating $D^{b}\operatorname{Coh}(Y)$ and a collection of coherent sheaves
$(G_{j})_{j\in J}$ on $X$ such that $(G_{j}|_{U})_{j\in J}$ generate
$D^{b}\operatorname{Coh}(U)$. Then $(F_{i})_{i\in I}$ and $(G_{j})_{j\in J}$
together generate $D^{b}\operatorname{Coh}(X)$. The similar assertion holds
for categories of equivariant sheaves.

###### Proof.

Indeed, in this case $(F_{i})$ generate the subcategory in
$D^{b}\operatorname{Coh}(X)$ consisting of objects with cohomology supported
on $Y$. But latter subcategory is precisely the kernel of the restriction
functor

$D^{b}\operatorname{Coh}(X)\to D^{b}\operatorname{Coh}(U).$

The assertion follows from this. ∎

######  Lemma 4.1.4.

Let us work over a regular base ring $\mathbf{k}$. For any subgroup
${\mathbb{G}}_{m}\subset T$, the twists by $\chi^{n}$ of the equivariant
sheaves ${\mathcal{O}}_{\overline{R}_{0}},\ldots,{\mathcal{O}}_{R_{k}}$
(resp.,
${\mathcal{O}}_{R_{1}},\ldots,{\mathcal{O}}_{R_{k}},{\mathcal{O}}_{\overline{{\mathcal{Z}}}_{1,k}}$)
generate
$D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{1,k}/{\mathbb{G}}_{m})$.

###### Proof.

This can be proved by induction on $k$. By the induction assumption, the
objects ${\mathcal{O}}_{R_{1}},\ldots,{\mathcal{O}}_{R_{k}}$ and their twists
generate the subcategory of equivariant sheaves supported on
$R_{1}\cup\ldots\cup R_{k}$. Hence, by Lemma 4.1.3, it remains to check that
the twists of the restriction of ${\mathcal{O}}_{R_{0}}$ (resp.,
${\mathcal{O}}_{\overline{{\mathcal{Z}}}_{1,k}}$) to
$\overline{R}_{0}\setminus R_{1}$ generate
$D^{b}\operatorname{Coh}(\overline{R}_{0}\setminus R_{1})$. But this
restriction is the structure sheaf, and $\overline{R}_{0}\setminus R_{1}$ is
the affine line, so this is true. ∎

######  Proposition 4.1.5.

The objects $P_{0},\ldots,P_{k}$ generate the category
$\overline{{\mathcal{B}}}_{1,k}$.

###### Proof.

By Lemma 4.1.4 with the base ring being the polynomial ring in
$U_{1},\ldots,U_{k}$, we get that the twists by $\chi^{n}$ of the equivariant
coherent sheaves $({\mathcal{O}}_{\overline{R}_{i}\times{\mathbb{A}}^{k}})$
generate
$D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{1,k}/{\mathbb{G}}_{m}\times{\mathbb{A}}^{k})$.
It is easy to see that the critical locus
$\operatorname{crit}({\mathbf{w}}_{1,k})$ is contained in
$\overline{{\mathcal{Z}}}_{1,k}\times{\mathbb{A}}^{k}$, so the assertion
follows from Lemma 2.3.5. ∎

###  4.2. Localization for $n=1$

The complement $\overline{{\mathcal{Z}}}_{1,k}\setminus{\mathcal{Z}}_{1,k}$
consists of one point $p$, namely the infinite point of
$\overline{R}_{0}\simeq{\mathbb{P}}^{1}$. Furthermore, the critical locus of
${\mathbf{w}}_{1,k}$ intersects the complement
$\overline{{\mathcal{X}}}_{1,k}\setminus{\mathcal{X}}_{1,k}$ exactly at the
point $p$.

Let us denote by ${\mathcal{B}}_{1,k}^{\mathbb{Z}}$ (resp.,
$\overline{{\mathcal{B}}}_{1,k}^{\mathbb{Z}}$) the category of
$\mathbb{Z}$-graded matrix factorizations on ${\mathbf{w}}_{1,k}$ on
${\mathcal{X}}_{1,k}$ (resp., $\overline{{\mathcal{X}}}_{1,k}$), corresponding
to the grading

$|U_{i}|=2,\ |X_{i}|=|C_{i}|=|V_{i}|=0.$

By Isik’s theorem (see (2.4)), we have equivalences

${\mathcal{B}}_{1,k}^{\mathbb{Z}}\simeq
D^{b}\operatorname{Coh}({\mathcal{Z}}_{1,k}),\ \
\overline{{\mathcal{B}}}_{1,k}^{\mathbb{Z}}\simeq
D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{1,k}),$

compatible with the obvious restriction functors.

It is well known that the restriction functor induces an equivalence of
$D^{b}\operatorname{Coh}({\mathcal{Z}}_{1,k})$ with the quotient of
$D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{1,k})$ by the subcategory
of sheaves supported at
$\overline{{\mathcal{Z}}}_{1,k}\setminus{\mathcal{Z}}_{1,k}=\\{p\\}$. Since
the restriction functor has an obvious dg-enhancement, this is also true at
the level of dg-categories. Hence, we can identify
${\mathcal{B}}_{1,k}^{\mathbb{Z}}\simeq
D^{b}\operatorname{Coh}({\mathcal{Z}}_{1,k})$ with the quotient of
$\overline{{\mathcal{B}}}_{1,k}^{\mathbb{Z}}$ by the subcategory of matrix
factorizations supported at $p$. The latter category is generated by the
matrix factorization $E$ associated with the object
${\mathcal{O}}_{p\times{\mathbb{A}}^{k}}$ of the singularity category.

We have the following resolution of $E$ in terms of the generating objects
$P_{i}$. For an interval $[i,j]$, where $j<k$ and for integers
$m_{i},\ldots,m_{j}$, let us consider the object

${\mathcal{O}}_{R_{[i,j]}}(m_{i},\ldots,m_{j})=\bigotimes_{r=i}^{j}p_{r}^{*}{\mathcal{O}}(m_{r})|_{R_{[i,j]}},$

where $R_{[i,j]}=\overline{R}_{i}\cup\ldots\cup R_{j}$. Similarly, for $i<k$
we set

${\mathcal{O}}_{R_{[i,k]}}(m_{i},\ldots,m_{k-1})=\bigotimes_{r=i}^{k-1}p_{r}^{*}{\mathcal{O}}(m_{r})|_{R_{[i,k]}}.$

######  Lemma 4.2.1.

One has the following exact triangles

${\mathcal{O}}_{R_{[0,i]}}(1,\ldots,1,0)\rTo{p_{i}^{*}z_{1}}{\mathcal{O}}_{R_{[0,i+1]}}(1,\ldots,1,0)\to{\mathcal{O}}_{R_{i+1}}\to\ldots,\
\text{ for }i<k-1,$
${\mathcal{O}}_{R_{[0,k-1]}}(1,\ldots,1,0)\rTo{p_{k-1}^{*}z_{1}}{\mathcal{O}}_{R_{[0,k]}}(1,1,\ldots,1)\to{\mathcal{O}}_{R_{k}}\to\ldots,$
${\mathcal{O}}_{R_{i+1}}\rTo{p_{i+1}^{*}z_{0}}{\mathcal{O}}_{R_{[0,i+1]}}(0,1,\ldots,1)\to{\mathcal{O}}_{R_{[0,i]}}(0,1,\ldots,1)\to\ldots,\text{
for }0<i<k-1,$
${\mathcal{O}}_{R_{k}}\rTo{X_{1}}{\mathcal{O}}_{R_{[0,k]}}(0,1,\ldots,1)\to{\mathcal{O}}_{R_{[0,k-1]}}(0,1,\ldots,1)\to\ldots,$
${\mathcal{O}}_{R_{1}}\rTo{p_{1}^{*}z_{0}}{\mathcal{O}}_{R_{[0,1]}}(0,1)\to{\mathcal{O}}_{R_{0}}\to\ldots,$

which express ${\mathcal{O}}_{R_{[0,k]}}(1,\ldots,1)$ and
${\mathcal{O}}_{R_{[0,k]}}(0,1,\ldots,1)$ in terms of the generators. Finally,
there is an exact triangle

${\mathcal{O}}_{R_{[0,k]}}(0,1,\ldots,1)\rTo{X_{2}^{-1}}{\mathcal{O}}_{R_{[0,k]}}(1,\ldots,1)\to{\mathcal{O}}_{p}\to\ldots$

###### Proof.

These exact triangles come from the natural exact sequences of coherent
sheaves extending the surjections of the form $L|_{C}\to L|_{C^{\prime}}$,
where $L$ is a line bundle, $C$ is a union of components of the nodal curve,
$C^{\prime}$ is either a smaller union of components, or the point $p$. ∎

###  4.3. Toric maps ${\mathcal{Y}}_{n,k}\to{\mathcal{Y}}_{n,1}$

Here we will prove that ${\mathcal{Y}}_{n,k}$ is projective over
${\mathbb{A}}^{n+k+1}$, and hence it is a toric variety over $\mathbf{k}$. We
reduce this to the case $n=1$ using the following observation.

######  Proposition 4.3.1.

For every $a=1,\ldots,n$, there is a morphism

$f_{a}:{\mathcal{Y}}_{n,k}\to{\mathcal{Y}}_{1,k}$

such that
$f_{a}^{-1}{\mathcal{U}}_{i}=\cup_{\phi:\phi(a)=i}{\mathcal{U}}_{\phi}$, and

$f_{a}^{*}V_{i}=V_{i},\ f_{a}^{*}C_{i}=C_{ai},\ f_{a}^{*}X_{1}=X_{1}\ldots
X_{a},\ f_{a}^{*}X_{2}=X_{a+1}\ldots X_{n+1}.$

Furthermore, the morphism

${\mathcal{Y}}_{n,k}\rTo{(f_{1},\ldots,f_{n}),X_{2},\ldots,X_{n}}({\mathcal{Y}}_{1,k}\times_{A^{k}_{V_{1},\ldots,V_{k}}}\ldots\times_{{\mathbb{A}}^{k}_{V_{1},\ldots,V_{k}}}{\mathcal{Y}}_{1,k})\times{\mathbb{A}}^{n-1}_{X_{2},\ldots,X_{n}}$

is a closed embedding.

###### Proof.

The proof is a straightforward calculation in the affine charts. ∎

######  Corollary 4.3.2.

The morphism

${\mathcal{Y}}_{n,k}\rTo{V_{1},\ldots,V_{k},X_{1},\ldots,X_{n+1}}{\mathbb{A}}^{n+k+1}$

is projective.

###### Proof.

Combining Proposition 4.3.1 with Lemma 4.1.1 we see that the composition

${\mathcal{Y}}_{n,k}\to{\mathbb{A}}^{k}_{V_{1},\ldots,V_{k}}\times{\mathbb{A}}^{n+1}_{X_{1},\ldots,X_{n+1}}\rTo{\operatorname{id}\times\phi}{\mathbb{A}}^{k}_{V_{1},\ldots,V_{k}}\times{\mathbb{A}}^{3n-1}$

is projective, where

$\phi(X_{1},\ldots,X_{n+1})=(X_{1},X_{2}\ldots X_{n+1};X_{1}X_{2},X_{3}\ldots
X_{n+1};\ldots;X_{1}\ldots X_{n},X_{n+1};X_{2},\ldots,X_{n-1}).$

Since $\phi$ is a closed embedding, the morphism
${\mathcal{Y}}_{n,k}\to{\mathbb{A}}^{k+n+1}$ is also projective. ∎

##  5\. B-side: case $n=2$

###  5.1. Varieties ${\mathcal{Z}}_{2,k}$ and $\overline{{\mathcal{Z}}}_{2,k}$

The affine charts ${\mathcal{U}}_{ij}$ covering ${\mathcal{Y}}_{2,k}$ are
numbered by pairs $i\geq j$ in $[1,k]$, where $i=\phi(1)$, $j=\phi(2)$. For
$i>j$ the coordinates on ${\mathcal{U}}_{ij}$ are

$V_{k},\ldots,V_{i+1},C_{1,i}^{-1},C_{1,i-1},V_{i-1},\ldots,V_{j+1},C_{2,j}^{-1},C_{2,j-1},V_{j-1},\ldots,V_{1},$

while for $i=j$, the coordinates on ${\mathcal{U}}_{ii}$ are

$V_{k},\ldots,V_{i+1},C_{1,i}^{-1},X_{2},C_{2,i-1},V_{i-1},\ldots,V_{0}$

(recall that $C_{1k}^{-1}=X_{1}$, $C_{2,0}=X_{3}$).

The variety ${\mathcal{Z}}_{2,k}$ (given as the vanishing locus of
$V_{1},\ldots,V_{k}$) is $2$-dimensional and is a union of smooth irreducible
components $R_{ij}$, with $k\geq i\geq j\geq 0$. Here for $i<k$, $R_{ii}$ is
the closure of the locus $C_{1,i}^{-1}=0$ in
${\mathcal{U}}_{ii}\cap{\mathcal{Z}}_{2,k}$, while $R_{kk}$ is the locus
$C_{2,k-1}=0$ in ${\mathcal{U}}_{kk}\cap{\mathcal{Z}}_{2,k}$. For $k>i>j$,
$R_{ij}$ is the closure of the locus $C_{1,i}^{-1}=C_{2,j}^{-1}=0$ in
${\mathcal{U}}_{ij}\cap{\mathcal{Z}}_{2,k}$, while for $j<k-1$, $R_{kj}$ is
the closure of the locus $C_{1,k-1}=C_{2,j}^{-1}=0$ in
${\mathcal{U}}_{kj}\cap{\mathcal{Z}}_{2,k}$, and $R_{k,k-1}$ is the closure of
the locus $X_{2}=0$ in ${\mathcal{U}}_{kk}\cap{\mathcal{Z}}_{2,k}$.

Note that $R_{kk}$ is the affine plane with coordinates $X_{1},X_{2}$;
$R_{k0}$ is the affine plane with coordinates $X_{1},X_{3}$; and $R_{00}$ is
the affine plane with coordinates $X_{2},X_{3}$. The components $R_{kj}$ are
isomorphic to ${\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{1}}$, the components
$R_{i0}$ are isomorphic to ${\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{3}}$,
the components $R_{ii}$ are isomorphic to the blow up of the point $(0,0)$ in
${\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{2}}$, and the remaining components
are isomorphic to ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. In the case $k=4$,
the variety ${\mathcal{Z}}_{2,4}$ is schematically depicted in Figure 4, with
$R_{44}$ being the component at the top corner, $R_{40}$ at the bottom left
corner, and $R_{00}$ at the bottom right corner.

In the compactification $\overline{{\mathcal{Y}}}_{2,k}$ we have extra charts
${\mathcal{U}}_{i,0}$, where $i=0,\ldots,k$. For $i>1$ the coordinates on
${\mathcal{U}}_{i,0}$ are

$V_{k},\ldots,V_{i+1},C_{1,i}^{-1},C_{1,i-1},V_{i-1},\ldots,V_{1},X_{3}^{-1},$

while the coordinates on ${\mathcal{U}}_{1,0}$ are

$V_{k},\ldots,V_{2},C_{1,1}^{-1},X_{2}X_{3},X_{3}^{-1},$

and the coordinates on ${\mathcal{U}}_{0,0}$ are

$V_{k},\ldots,V_{1},X_{2}^{-1}X_{3}^{-1},X_{2}.$

The compactified variety $\overline{{\mathcal{Z}}}_{2,k}$ is the union of
irreducible components $\overline{R}_{ij}$, $k\geq i\geq j\geq 0$, where
$\overline{R}_{ij}=R_{ij}$ for $j>0$, while $R_{i0}$ get compactified as
follows. The affine plane $R_{k0}$ gets compactified to
${\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{1}}$. For $0<i<k$, the component
$R_{i0}={\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{3}}$ gets compactified to
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The component
$R_{00}={\mathbb{A}}^{2}_{X_{2},X_{3}}$ gets compactified to the blow up of a
point in ${\mathbb{P}}^{1}\times{\mathbb{A}}^{1}_{X_{2}}$.

Thus, the complement of ${\mathcal{Z}}_{2,k}$ in
$\overline{{\mathcal{Z}}}_{2,k}$ is the nodal chain

$D=D^{\prime}_{0}\cup D_{0}\cup D_{1}\cup\ldots\cup D_{k},$

where $D^{\prime}_{0}=D_{k}={\mathbb{A}}^{1}$, $D_{i}={\mathbb{P}}^{1}$ for
$i=0,\ldots,k-1$; $D_{i}=\overline{R}_{i0}\setminus R_{i0}$ for $i>0$,
$D^{\prime}_{0}\cup D_{0}=\overline{R}_{00}\setminus R_{00}$.

We have the following extension of Proposition 4.3.1 with a straightforward
proof.

######  Lemma 5.1.1.

(i) There are two natural morphisms

$f_{1}:\overline{{\mathcal{Y}}}_{2,k}\to{\mathcal{Y}}_{1,k},\ \
f_{2}:\overline{{\mathcal{Y}}}_{2,k}\to\overline{{\mathcal{Y}}}_{1,k}$

such that $f_{1}^{-1}{\mathcal{U}}_{i}=\cup_{j}{\mathcal{U}}_{ij}$,
$f_{2}^{-1}{\mathcal{U}}_{j}=\cup_{i}{\mathcal{U}}_{ij}$, and

$f_{1}^{*}X_{1}=X_{1},\ f_{1}^{*}X_{2}=X_{2}X_{3},\ f_{1}^{*}C_{i}=C_{1i},\
f_{1}^{*}V_{i}=V_{i}.$ $f_{2}^{*}X_{1}=X_{1}X_{2},\ f_{2}^{*}X_{2}=X_{3},\
f_{2}^{*}C_{i}=C_{2i},\ f_{2}^{*}V_{i}=V_{i},$

Furthermore, the morphism

$\overline{{\mathcal{Y}}}_{2,k}\rTo{(f_{1},f_{2}),X_{2}}({\mathcal{Y}}_{1,k}\times_{{\mathbb{A}}^{k}_{V_{1},\ldots,V_{k}}}\overline{{\mathcal{Y}}}_{1,k})\times{\mathbb{A}}^{1}_{X_{2}}$

is a closed embedding.

(ii) The schematic preimage $f_{2}^{-1}(\overline{R}_{j})$ is equal to the
union (with the reduced scheme structure)

$\overline{R}_{*j}:=\cup_{i}\overline{R}_{ij}$

(these are “rows” in Figure 4).

In the following result we determine equations that present
$\overline{R}_{ij}$ as a complete intersection in its open neighborhood in
$\overline{{\mathcal{Y}}}_{2,k}$. Note that for
$\overline{R}_{*j}=f_{2}^{-1}(\overline{R}_{j})$ we have such a description
due to Lemma 4.1.1.

######  Lemma 5.1.2.

(i) For $k>i>j>0$, such that $i-j>1$, $R_{ij}$ is the vanishing locus of

$V_{k},\ldots,V_{i+2},f_{1}^{*}s_{1}(i),f_{1}^{*}s_{2}(i),V_{i-1},\ldots,V_{j+2},f_{2}^{*}s_{1}(j),f_{2}^{*}s_{2}(j),V_{j-1},\ldots,V_{1},$

where $s_{1}(i)$ and $s_{2}(i)$ are defined by (4.1).

(ii) For $1\leq i<k$, there exists a section $s_{i}$ of
$f_{2}^{*}p_{i-1}^{*}{\mathcal{O}}(-1)\otimes
f_{1}^{*}p_{i}^{*}{\mathcal{O}}(-1)$, defined in a neighborhood of
$\overline{R}_{i,i-1}$, given by

$s_{i}:=\frac{f_{1}^{*}s_{2}(i)}{f_{2}^{*}p_{i-1}^{*}z_{1}}=\frac{f_{2}^{*}s_{1}(i-1)}{f_{1}^{*}p_{i}^{*}z_{0}}=\frac{X_{2}}{f_{2}^{*}p_{i-1}^{*}z_{0}\cdot
f_{1}^{*}p_{i}^{*}z_{1}}.$

For $1<i<k$, $R_{i,i-1}$ is given as the vanishing locus of

$V_{k},\ldots,V_{i+2},f_{1}^{*}s_{1}(i),s_{i},f_{2}^{*}s_{2}(i-1),V_{i-2},\ldots,V_{1},$

while $\overline{R}_{1,0}$ is the vanishing locus of

$V_{k},\ldots,V_{3},f_{1}^{*}s_{1}(1),s_{1}.$

On the other hand, there exists a section $s_{k}$ of
$f_{2}^{*}p_{1}^{*}{\mathcal{O}}(-1)$ defined locally near $R_{k,k-1}$, given
by

$s_{k}=\frac{X_{2}}{f_{2}^{*}p_{k-1}^{*}z_{0}}=\frac{C_{1,k-1}}{f_{2}^{*}p_{k-1}^{*}z_{1}},$

such that $R_{k,k-1}$ is the vanishing locus of

$s_{k},f_{2}^{*}s_{2}(k-1),V_{k-2},\ldots,V_{1}.$

(iii) For $0<j<k-1$, $\overline{R}_{kj}$ is cut out in its neighborhood as the
vanishing locus of

$C_{1,k-1},V_{k-1},\ldots,V_{j+2},f_{2}^{*}s_{1}(j),f_{2}^{*}s_{2}(j),V_{j-1},\ldots,V_{1}.$

For $1<i<k$, $\overline{R}_{i0}$ is the vanishing locus of

$V_{k},\ldots,V_{i+2},f_{1}^{*}s_{1}(i),f_{1}^{*}s_{2}(i),V_{i-1},\ldots,V_{2},f_{2}^{*}s_{1}(0).$

(iv) $\overline{R}_{k0}$ is cut out in its neighborhood as the vanishing locus
of

$C_{1,k-1},V_{k-1},\ldots,V_{2},f_{2}^{*}s_{1}(0).$

###### Proof.

(i) Note that for $i>j$, we have

$\overline{R}_{ij}=f_{1}^{-1}R_{i}\cap f_{2}^{-1}\overline{R_{j}}.$

Thus, our assertion follows from Lemma 4.1.1(ii), where we use the fact that

$V_{i+1}=s_{1}(i)\cdot p_{i}^{*}z_{1},\ V_{i}=s_{2}(i)\cdot p_{i}^{*}z_{0},$

so that $V_{i}$ and $V_{i+1}$ vanish on the zero locus of $s_{1}(i)$ and
$s_{2}(i)$.

(ii) The existence of $s_{j}$ follows from the claimed identities, which
themselves follow from

$f_{1}^{*}s_{2}(j)\cdot f_{1}^{*}p_{j}^{*}z_{0}=V_{j}=f_{2}^{*}s_{1}(j-1)\cdot
f_{2}^{*}p_{j-1}^{*}z_{1},$

which holds by the definition of $s_{1}(j-1)$ and $s_{2}(j)$, and from

$f_{2}^{*}s_{1}(j-1)\cdot f_{2}^{*}p_{j-1}^{*}z_{0}\cdot
f_{1}^{*}p_{j}^{*}(\frac{z_{1}}{z_{0}})=C_{2,j}^{-1}\cdot C_{1,j}=X_{2}.$

For the definition of $s_{k}$ we use the identity

$C_{1,k-1}\cdot\frac{f_{2}^{*}p_{k-1}^{*}z_{0}}{f_{2}^{*}p_{k-1}^{*}z_{1}}=C_{1,k-1}\cdot
C_{2,k-1}^{-1}=X_{2}.$

It is clear that the claimed $k$ sections vanish on $R_{j,j-1}$. To prove that
the subscheme they cut out is exactly $R_{j,j-1}$, we recall that by Lemma
4.1.1(ii), we know that for $1<j<k$, $\overline{R}_{j-1,j}$ is the vanishing
locus of

$V_{1},\ldots,V_{j+2},f_{1}^{*}s_{1}(j),\ f_{1}^{*}s_{2}(j),\
f_{2}^{*}s_{1}(j-1),\ f_{2}^{*}s_{2}(j-1),V_{j-2},\ldots,V_{k}.$

It remains to observe that $f_{1}^{*}s_{2}(j)$ and $f_{2}^{*}s_{1}(j-1)$ both
vanish on the zero locus of $s_{j}$.

In the case of $R_{k,k-1}$, from Lemma 4.1.1(ii) we get the description of
$R_{k,k-1}$ as the vanishing locus of

$C_{1,k-1},f_{2}^{*}s_{1}(k-1),f_{2}^{*}s_{2}(k-1),V_{k-2},\ldots,V_{1}.$

The fact that $C_{1,k-1}$ and $f_{2}^{*}s_{1}(k-1)$ vanish on the zero locus
of $s_{k}$ follows from the factorizations

$C_{1,k-1}=s_{k}\cdot f_{2}^{*}p_{k-1}^{*}z_{1},\ \
f_{2}^{*}s_{1}(k-1)=X_{1}\cdot s_{k}.$

(iii), (iv) The proofs are analogous, so we omit them. ∎

###  5.2. Generating matrix factorizations

For each $k\geq i>j\geq 0$, let $P_{ij}$ denote the matrix factorization of
${\mathbf{w}}_{2,k}$ on $\overline{{\mathcal{X}}}_{2,k}$ corresponding to the
object (denoted in the same way),

$P_{ij}:={\mathcal{O}}_{\overline{R}_{ij}\times{\mathbb{A}}^{k}}\otimes
f_{1}^{*}p_{i}^{*}{\mathcal{O}}(-1)$

of the singularity category, where our convention is that
$p_{k}^{*}{\mathcal{O}}(-1)$ is trivial. Here we equip
$f_{1}^{*}p_{i}^{*}{\mathcal{O}}(-1)$ with the $T$-equivariant structure by
letting $f_{1}^{*}p_{i}^{*}z_{0}$ to have weight $0$, and
$f_{1}^{*}p_{i}^{*}z_{1}$ to have weight $c_{1i}$.

We also define $Q_{j}$ to be the matrix factorization corresponding to the
object of the singularity category

$Q_{j}:={\mathcal{O}}_{\overline{R}_{*j}\times{\mathbb{A}}^{k}}.$

######  Lemma 5.2.1.

Let us work over a regular base ring. Then for any subgroup
${\mathbb{G}}_{m}\subset T$, the twists by $\chi^{n}$ of the equivariant
sheaves

$({\mathcal{O}}_{\overline{R}_{ij}}\otimes
f_{1}^{*}p_{i}^{*}{\mathcal{O}}(-1))_{0\geq i>j\geq k}\ \text{ and }\
({\mathcal{O}}_{\overline{R}_{*j}})_{0\leq j\leq k}$

generate
$D^{b}\operatorname{Coh}(\overline{{\mathcal{Z}}}_{2,k}/{\mathbb{G}}_{m})$.

###### Proof.

Let us prove by descending induction on $j_{0}\geq 0$ that the twists of the
sheaves $({\mathcal{O}}_{\overline{R}_{ij}}\otimes
f_{1}^{*}p_{i}^{*}{\mathcal{O}}(-1))_{j\geq j_{0}}$,
$({\mathcal{O}}_{\overline{R}_{*j}})_{j\geq j_{0}}$ generate the subcategory
of equivariant sheaves supported on

$\overline{R}_{*\geq
j_{0}}:=\overline{R}_{*k}\cup\ldots\cup\overline{R}_{*,j_{0}}.$

The base $j_{0}=k$ is clear as
$\overline{R}_{*k}=R_{kk}\simeq{\mathbb{A}}^{2}$, so the twists of
${\mathcal{O}}_{R_{kk}}$ generates the needed subcategory. Assume that
$j_{0}<k$ and the assertion holds for $j_{0}+1$. By Lemma 4.1.3, it is enough
to check generation after restricting to
$\overline{R}_{*j_{0}}\setminus\overline{R}_{*,j_{0}+1}$. Now we use the
further restriction

$D^{b}\operatorname{Coh}(\overline{R}_{*j_{0}}\setminus\overline{R}_{*,j_{0}+1})\to
D^{b}\operatorname{Coh}(\overline{R}_{j_{0}j_{0}}\setminus(\overline{R}_{j_{0}+1,j_{0}+1}\cup\overline{R}_{j_{0}+1,j_{0}})).$

We note that

$\overline{R}_{j_{0}j_{0}}\setminus(\overline{R}_{j_{0}+1,j_{0}+1}\cup\overline{R}_{j_{0}+1,j_{0}})\simeq{\mathbb{A}}^{2},$

so its derived category is generated by the twists of the structure sheaf
obtained as the restriction of ${\mathcal{O}}_{\overline{R}_{*k_{0}}}$. Hence,
it remains to show that the subcategory of
$D^{b}\operatorname{Coh}(\overline{R}_{*j_{0}}\setminus\overline{R}_{*,j_{0}+1})$
consisting of sheaves supported on
$(\overline{R}_{kj_{0}}\cup\ldots\overline{R}_{j_{0}+1,j_{0}})\setminus\overline{R}_{*,j_{0}+1}$
is generated by the restrictions of our objects. But we have

$(\overline{R}_{kj_{0}}\cup\ldots\overline{R}_{j_{0}+1,j_{0}})\setminus\overline{R}_{*,j_{0}+1}\simeq(R_{k}\cup\ldots\cup
R_{j_{0}+1})\times{\mathbb{A}}^{1},$

and our assertion follows from Lemma 4.1.4 (after tensoring with the
generators with the line bundle
$p_{k-1}^{*}{\mathcal{O}}(-1)\otimes\ldots\otimes
p_{j_{0}+1}^{*}{\mathcal{O}}(-1)$). ∎

######  Proposition 5.2.2.

The objects $(P_{ij})_{i>j}$ and $(Q_{j})$ generate
$\operatorname{MF}(\overline{{\mathcal{X}}}_{2,k},{\mathbf{w}}_{2,k})$.

###### Proof.

It is easy to see that the critical locus
$\operatorname{crit}({\mathbf{w}}_{2,k})$ is contained in
$\overline{{\mathcal{Z}}}_{2,k}\times{\mathbb{A}}^{k}$. Thus, as in
Proposition 4.1.5, the statement follows from Lemma 5.2.1 and Lemma 2.3.5. ∎

###  5.3. Computation of morphisms

For the computation of morphisms, we consider first the $\mathbb{Z}$-graded
category
$\operatorname{MF}_{{\mathbb{G}}_{m}}(\overline{{\mathcal{X}}}_{2,k},{\mathbf{w}}_{2,k})$,
where the subgroup ${\mathbb{G}}_{m}\subset\widetilde{T}$ corresponds to the
homomorphism $\widetilde{H}\to\mathbb{Z}$ sending
$x_{1}$,$x_{2}$,$u_{1},\ldots,u_{k}$ to zero and $l_{0}$ to $1$. This
corresponds to the following $\mathbb{Z}$-grading of the variables:

$|X_{1}|=|X_{2}|=|U_{i}|=0,\ \ |V_{i}|=2,\ \ |C_{1i}|=|C_{2i}|=2(k-i),\ \
|X_{3}|=2k.$ (5.1)

We will consider the case of an arbitrary grading later in Sec. 5.3.5.

####  5.3.1. Endomorphisms

Let us consider the critical locus
$\operatorname{crit}({\mathbf{w}}_{2,k})\subset\overline{{\mathcal{X}}}_{2,k}$.
It is easy to see that its projection to $\overline{{\mathcal{Y}}}_{2,k}$
factors through $\overline{{\mathcal{Z}}}_{2,k}$, so we have a natural
projection

$p:\operatorname{crit}({\mathbf{w}}_{2,k})\to\overline{{\mathcal{Z}}}_{2,k}.$

Recall that $\overline{{\mathcal{Z}}}_{2,k}$ is the union of the irreducible
components $\overline{R}_{ij}$. Let us introduce the notation for some other
strata in $\overline{{\mathcal{Z}}}_{2,k}$. For $k\geq i\geq j\geq 1$, set

$L^{h}_{ij}:=R_{i,j-1}\cap R_{i,j},$

Similarly, for $k\geq i\geq j\geq 1$, we set

$L^{v}_{ij}:=\overline{R}_{i,j-1}\cap\overline{R}_{i-1,j-1}.$

Also, for $i=1,\ldots,k$, set

$L^{d}_{i}:=R_{i-1,i-1}\cap R_{ii}.$

Finally, for $k\geq i>j\geq 1$, we set $p_{ij}:=L^{h}_{ij}\cap L^{v}_{ij}$. A
straightforward calculation gives the following description of the critical
locus of ${\mathbf{w}}_{2,k}$.

######  Lemma 5.3.1.

The locus $\operatorname{crit}({\mathbf{w}}_{2,k})$ is the union (with the
reduced scheme structure) of the following smooth two dimensional components:

  * •

$\overline{R}_{ij}$ for $k\geq i\geq j\geq 0$;

  * •

$L^{h}_{ij}\times{\mathbb{A}}^{1}_{U_{j}}$ for $k\geq i\geq j\geq 1$;

  * •

$L^{v}_{ij}\times{\mathbb{A}}^{1}_{U_{i}}$ for $k\geq i\geq j\geq 1$;

  * •

$L^{d}_{i}\times{\mathbb{A}}^{1}_{U_{i}}$ for $1\leq i\leq k$;

  * •

$p_{ij}\times{\mathbb{A}}^{2}_{U_{i},U_{j}}$ for $k\geq i>j\geq 1$.

The crucial step in calculating the morphism spaces is the following local
result.

######  Lemma 5.3.2.

For every $k\geq i>j\geq 0$, one has a natural quasiisomorphism

$\underline{\operatorname{End}}(P_{ij})\simeq{\mathcal{O}}_{p^{-1}\overline{R}_{ij}}.$

Let us set $L_{*j}=\cup_{i}L^{h}_{ij}\cup L^{d}_{j}$. Then for $0<j<k$ we have

$\underline{\operatorname{End}}(Q_{j})={\mathcal{O}}_{\overline{R}_{*j}\cup
L_{*j}\times{\mathbb{A}}^{1}_{U_{j}}\cup
L_{*j+1}\times{\mathbb{A}}^{1}_{U_{j+1}}},$

while

$\underline{\operatorname{End}}(Q_{k})={\mathcal{O}}_{p^{-1}R_{00}}={\mathcal{O}}_{R_{00}\cup
L_{*k}\times{\mathbb{A}}^{1}_{U_{k}}},$
$\underline{\operatorname{End}}(Q_{0})={\mathcal{O}}_{\overline{R}_{*0}\cup
L_{*1}\times{\mathbb{A}}^{1}_{U_{1}}}.$

###### Proof.

We will combine the description of $\overline{R}_{ij}$ as complete
intersections from Lemma 5.1.2 with the computation of Lemma 2.3.6(i). In the
case of $Q_{j}$ we use the description of
$\overline{R}_{*j}=f_{2}^{-1}(\overline{R}_{j})$ as a complete intersection.
Below we provide details of the calculation (in later calculations we will
skip over some similar steps).

Case of $\underline{\operatorname{End}}(P_{ij})$, for $k>i>j>0$.

Assume first that $i-j>1$. Then we can decompose $W$ as

$\displaystyle W=\sum_{r\neq i,i+1,j,j+1}V_{r}U_{r}+$ $\displaystyle
f_{1}^{*}s_{1}(i)\cdot f_{1}^{*}p_{i}^{*}z_{1}\cdot
U_{i+1}+f_{1}^{*}s_{2}(i)\cdot f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i}+f_{2}^{*}s_{1}(j)\cdot f_{2}^{*}p_{j}^{*}z_{1}\cdot
U_{j+1}+f_{2}^{*}s_{2}(j)\cdot f_{2}^{*}p_{j}^{*}z_{0}\cdot U_{j}.$ (5.2)

The description of $\overline{R}_{ij}$ as a complete intersection implies that
$P_{ij}\otimes f_{1}^{*}p_{i}^{*}{\mathcal{O}}(1)$ can be presented as the
tensor product of the following Koszul matrix factorizations:

  * •

for $r\neq i,i+1,j,j+1$, $\\{U_{r},V_{r}\\}$;

  * •

$\\{f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}s_{1}(i)\\},\
\\{f_{1}^{*}p_{i}^{*}z_{0}\cdot U_{i},f_{1}^{*}s_{2}(i)\\}$;

  * •

$\\{f_{2}^{*}p_{j}^{*}z_{1}\cdot U_{j+1},f_{2}^{*}s_{1}(j)\\},\
\\{f_{2}^{*}p_{j}^{*}z_{0}\cdot U_{j},f_{2}^{*}s_{2}(j)\\}$.

Hence, $\underline{\operatorname{End}}(P_{ij})$ is the tensor product of
endomorphism sheaves of these Koszul matrix factorizations. Applying Lemma
2.3.6(i), we get on $\overline{R}_{ij}\times{\mathbb{A}}^{k}$

$\displaystyle\underline{\operatorname{End}}(P_{ij})\simeq$
$\displaystyle{\mathcal{O}}/((U_{r}\ |\ r\neq
i,i+1,j,j+1),f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i},f_{2}^{*}p_{j}^{*}z_{1}\cdot U_{j+1},f_{2}^{*}p_{j}^{*}z_{0}\cdot
U_{j})={\mathcal{O}}_{p^{-1}\overline{R}_{ij}}.$

In the case $j=i-1$, $P_{i,i-1}\otimes f_{1}^{*}p_{i}^{*}{\mathcal{O}}(1)$ is
the tensor product of the Koszul matrix factorizations,

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq i-1,i,i+1$;

  * •

$\\{f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}s_{1}(i)\\},\
\\{f_{1}^{*}p_{i}^{*}z_{0}\cdot f_{2}^{*}p_{i-1}^{*}z_{1}\cdot
U_{i},s_{i}\\},\ \\{f_{2}^{*}p_{i-1}^{*}z_{0}\cdot
U_{i-1},f_{2}^{*}s_{2}(i-1)\\}.$

Hence, we get an isomorphism on $\overline{R}_{i,i-1}\times{\mathbb{A}}^{k}$,

$\displaystyle\underline{\operatorname{End}}(P_{i,i-1})\simeq$
$\displaystyle{\mathcal{O}}/((U_{r}\ |\ r\neq
i-1,i,i+1),f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}p_{i}^{*}z_{0}\cdot
f_{2}^{*}p_{i-1}^{*}z_{1}\cdot U_{i},f_{2}^{*}p_{i-1}^{*}z_{0}\cdot
U_{i-1})\simeq{\mathcal{O}}_{p^{-1}\overline{R}_{i,i-1}}.$

Case of $\underline{\operatorname{End}}(P_{i0})$, for $0<i<k$.

Assuming that $i>1$ we use the presentation of $P_{i0}$ as the tensor product
of

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq 0,i,i+1$;

  * •

$\\{f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}s_{1}(i)\\},\
\\{f_{1}^{*}p_{i}^{*}z_{0}\cdot U_{i},f_{1}^{*}s_{2}(i)\\},\
\\{f_{2}^{*}p_{0}^{*}z_{1}\cdot U_{1},f_{2}^{*}s_{1}(0)\\}.$

As before, from this we deduce an isomorphism on
$\overline{R}_{i0}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(P_{i0})\simeq{\mathcal{O}}/((U_{r}\ |\ r\neq
0,i,i+1),f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1},f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i},f_{2}^{*}p_{0}^{*}z_{1}\cdot
U_{1})\simeq{\mathcal{O}}_{p^{-1}\overline{R}_{i0}}.$

In the case $i=1$, we use the presentation of $P_{1,0}$ as the tensor product
of

  * •

$\\{U_{r},V_{r}\\}$, for $r>1$;

  * •

$\\{f_{1}^{*}p_{1}^{*}z_{1}\cdot U_{2},f_{1}^{*}s_{1}(1)\\},\
\\{f_{1}^{*}p_{1}^{*}z_{0}\cdot f_{2}^{*}p_{0}^{*}z_{1}\cdot U_{1},s_{1}\\}.$

From this we get an isomorphism on $\overline{R}_{1,0}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(P_{1,0})\simeq{\mathcal{O}}/((U_{r}\ |\
r>1),f_{1}^{*}p_{1}^{*}z_{1}\cdot U_{2},f_{1}^{*}p_{1}^{*}z_{0}\cdot
f_{2}^{*}p_{0}^{*}z_{1}\cdot
U_{1})\simeq{\mathcal{O}}_{p^{-1}\overline{R}_{1,0}}.$

Case of $\underline{\operatorname{End}}(P_{kj})$.

Assume first that $0<j<k-1$. Then in a neighborhood of
$R_{kj}\times{\mathbb{A}}^{k}$, $P_{kj}$ is obtained as the tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq j,j+1,k$;

  * •

$\\{X_{1}U_{k},C_{1,k-1}\\}$;

  * •

$\\{f_{2}^{*}p_{j}^{*}z_{1}\cdot U_{j+1},f_{2}^{*}s_{1}(j)\\},\
\\{f_{2}^{*}p_{j}^{*}z_{0}\cdot U_{j},f_{2}^{*}s_{2}(j)\\}$.

This leads to an isomorphism on $R_{kj}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(P_{kj})\simeq{\mathcal{O}}/((U_{r}\ |\ r\neq
j,j+1,k),X_{1}U_{k},f_{2}^{*}p_{j}^{*}z_{1}\cdot
U_{j+1},f_{2}^{*}p_{j}^{*}z_{0}\cdot
U_{j})\simeq{\mathcal{O}}_{p^{-1}R_{kj}}.$

In the case $j=k-1$, we present $P_{k,k-1}$ as the tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r<k-1$;

  * •

$\\{X_{1}U_{k}\cdot f_{2}^{*}p_{k-1}^{*}z_{1},s_{k}\\}$;

  * •

$\\{f_{2}^{*}p_{k-1}^{*}z_{0}\cdot U_{k-1},f_{2}^{*}s_{2}(k-1)\\}$,

which leads to an isomorphism on $R_{k,k-1}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(P_{k,k-1})\simeq{\mathcal{O}}/((U_{r}\ |\
r<k-1),X_{1}U_{k}\cdot
f_{2}^{*}p_{k-1}^{*}z_{1},f_{2}^{*}p_{k-1}^{*}z_{0}\cdot
U_{k-1})\simeq{\mathcal{O}}_{p^{-1}R_{k,k-1}}.$

In the case $j=0$, in a neighborhood of
$\overline{R}_{k0}\times{\mathbb{A}}^{k}$ we can present $P_{k0}$ as the
tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq 1,k$;

  * •

$\\{X_{1}U_{k},C_{1,k-1}\\}$;

  * •

$\\{f_{2}^{*}p_{0}^{*}z_{1}\cdot U_{1},f_{2}^{*}s_{1}(0)\\}$,

so we get an isomorphism on $\overline{R}_{k0}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(P_{k0})\simeq{\mathcal{O}}/((U_{r}\ |\ r\neq
1,k),X_{1}U_{k},f_{2}^{*}p_{0}^{*}z_{1}\cdot
U_{1})\simeq{\mathcal{O}}_{p^{-1}\overline{R}_{k0}}.$

Case of $\underline{\operatorname{End}}(Q_{j})$.

For $0<j<k$, we present $Q_{j}$ as the tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq j,j+1$;

  * •

$\\{f_{2}^{*}p_{j}^{*}z_{1}\cdot U_{j+1},f_{2}^{*}s_{1}(j)\\},\
\\{f_{2}^{*}p_{j}^{*}z_{0}\cdot U_{j},f_{2}^{*}s_{2}(j)\\}$.

From this we get an isomorphism on $R_{*j}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(Q_{j})\simeq{\mathcal{O}}/((U_{r}\ |\ r\neq
j,j+1),f_{2}^{*}p_{j}^{*}z_{1}\cdot U_{j+1},f_{2}^{*}p_{j}^{*}z_{0}\cdot
U_{j})\simeq{\mathcal{O}}_{Z},$

where $Z=\overline{R}_{*j}\cup L_{*j}\times{\mathbb{A}}^{1}_{U_{j}}\cup
L_{*j+1}\times{\mathbb{A}}^{1}_{U_{j+1}}$.

For $j=k$, in a neighborhood of $R_{kk}\times{\mathbb{A}}^{k}$, we can present
$Q_{k}$ as the tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r<k$;

  * •

$\\{X_{1}X_{2}\cdot U_{k},C_{2,k-1}\\}$,

which leads to an isomorphism on $R_{kk}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(Q_{k})\simeq{\mathcal{O}}/((U_{r}\ |\
r<k),X_{1}X_{2}\cdot U_{k})\simeq{\mathcal{O}}_{R_{kk}\cup
L_{*1}\times{\mathbb{A}}^{1}_{U_{1}}}.$

For $j=0$, we present $Q_{0}$ as the tensor product of

  * •

$\\{U_{r},V_{r}\\}$, for $r\neq 1$;

  * •

$\\{f_{2}^{*}p_{0}^{*}z_{1}\cdot U_{1},f_{2}^{*}s_{1}(0)\\}$,

which leads to an isomorphism on $\overline{R}_{*0}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{End}}(Q_{0})\simeq{\mathcal{O}}/((U_{r}\ |\ r\neq
1),f_{2}^{*}p_{0}^{*}z_{1}\cdot
U_{1})\simeq{\mathcal{O}}_{\overline{R}_{*0}\cup
L_{*1}\times{\mathbb{A}}^{1}_{U_{1}}}.$

∎

######  Lemma 5.3.3.

Let us set $U_{k+1}:=X_{1}$.

(i) For $k\geq i>j>0$, such that $i-j>1$, the natural embedding
$k[U_{i},U_{i+1},U_{j},U_{j+1}]\subset{\mathcal{O}}(\overline{{\mathcal{X}}}_{2,k})$
induces an isomorphism

$\operatorname{End}(P_{ij})\simeq
k[U_{i},U_{i+1},U_{j},U_{j+1}]/(U_{i}U_{i+1},U_{j}U_{j+1}).$

(ii) For $k\geq i>1$, there is a similar isomorphism

$\operatorname{End}(P_{i0})\simeq k[U_{1},U_{i},U_{i+1}]/(U_{i}U_{i+1}).$

(iii) For $i>1$, there is a similar isomorphism

$\operatorname{End}(P_{i,i-1})\simeq
k[U_{i-1},U_{i},U_{i+1}]/(U_{i-1}U_{i}U_{i+1}).$

(iv) There is a similar isomorphism

$\operatorname{End}(P_{1,0})\simeq k[U_{1},U_{2}].$

###### Proof.

By Lemma 5.3.2, we have to calculate
$H^{*}({\mathcal{O}}_{p^{-1}\overline{R}_{ij}})$. Let us consider the natural
projection

$\pi:\overline{{\mathcal{X}}}_{2,k}\to{\mathbb{A}}^{k+1}_{U_{1},\ldots,U_{k},U_{k+1}}.$

For part (i), it is enough to show the natural isomorphism

$R\pi_{*}{\mathcal{O}}_{p^{-1}\overline{R}_{ij}}\simeq{\mathcal{O}}/((U_{r})_{r\neq
i,i+1,j,j+1},U_{i}U_{i+1},U_{j}U_{j+1}),$

and similarly for parts (ii)–(iv).

Suppose that $k>i$, $j>0$ and $i-j>1$. Then

$p^{-1}R_{ij}\simeq C_{1}(U_{i},U_{i+1})\times C_{2}(U_{j},U_{j+1}),$

where $C_{1}(U_{i},U_{i+1})$ and $C_{2}(U_{j},U_{j+1})$ are nodal chains with
$3$ components:

$C_{1}(U_{i},U_{i+1})={\mathbb{A}}^{1}_{U_{i}}\cup{\mathbb{P}}^{1}\cup{\mathbb{A}}^{1}_{U_{i+1}},\
\
C_{2}(U_{j},U_{j+1})={\mathbb{A}}^{1}_{U_{j}}\cup{\mathbb{P}}^{1}\cup{\mathbb{A}}^{1}_{U_{j+1}},$

The morphism $C_{1}(U_{i},U_{i+1})\times
C_{2}(U_{j},U_{j+1})\to{\mathbb{A}}^{k}$ is the product of the natural maps
(contracting the ${\mathbb{P}}^{1}$ components)

$C_{1}(U_{i},U_{i+1})\to A^{2}_{U_{i},U_{i+1}},\ \
C_{2}(U_{j},U_{j+1})\to{\mathbb{A}}^{2}_{U_{j},U_{j+1}}$

composed with the coordinate embedding
${\mathbb{A}}^{4}_{U_{i},U_{i+1},U_{j},U_{j+1}}\hookrightarrow{\mathbb{A}}^{k+1}$.
This immediately implies the assertion.

In the case $i=k$ and $0<j<k-1$, we use a similar decomposition of
$p^{-1}R_{kj}$, where $C_{1}(U_{k},U_{k+1})$ is a nodal chain with the $2$
components:
$C_{1}(U_{k},U_{k+1})={\mathbb{A}}^{1}_{U_{k}}\cup{\mathbb{A}}^{1}_{U_{k+1}}$.
In the case $1<i<k$ and $j=0$, we use a similar decomposition

$p^{-1}\overline{R}_{i0}\simeq C_{1}(U_{i},U_{i+1})\times C_{2}(U_{1}),$

where $C_{2}(U_{1})$ is a nodal chain with $2$ components:
$C_{2}(U_{1})={\mathbb{A}}^{1}_{U_{1}}\cup{\mathbb{P}}^{1}$. In the case $i=k$
and $j=0$ we use

$p^{-1}\overline{R}_{k0}\simeq C_{1}(U_{1},U_{k+1})\times C_{2}(U_{1}).$

On the other hand, for $1<i<k$,

$p^{-1}R_{i,i-1}=Z\cup Z^{\prime},\ \ \text{ where }$ $Z\simeq
C_{1}(U_{i-1})\times C_{2}(U_{i},U_{i+1}),\ \ Z^{\prime}\simeq
C_{1}(U_{i-1},U_{i})\times C^{\prime}_{2}(U_{i+1}),$ $Z\cap Z^{\prime}\simeq
C_{1}(U_{i-1})\times C^{\prime}_{2}(U_{i+1}),$

where $C_{1}(U_{i-1})={\mathbb{A}}^{1}_{U_{i-1}}\cup{\mathbb{P}}^{1}$ and
$C^{\prime}_{2}(U_{i+1})={\mathbb{P}}^{1}\cup{\mathbb{A}}^{1}_{U_{i+1}}$. Now
the exact triangle

$R\pi_{*}{\mathcal{O}}_{p^{-1}R_{i,i-1}}\to R\pi_{*}{\mathcal{O}}_{Z}\oplus
R\pi_{*}{\mathcal{O}}_{Z^{\prime}}\to R\pi_{*}{\mathcal{O}}_{Z\cap
Z^{\prime}}\to\ldots$

easily gives the result.

Similarly for $i=1$ we use

$p^{-1}R_{1,0}=Z\cup Z^{\prime},\ \ \text{ where }$ $Z\simeq
C_{1}(U_{1})\times C_{2}(U_{0}),\ \ Z^{\prime}\simeq
C_{1}(U_{0},U_{1})\times{\mathbb{P}}^{1},$ $Z\cap Z^{\prime}\simeq
C_{1}(U_{1})\times{\mathbb{P}}^{1}.$

For $i=k$ we use

$p^{-1}R_{k,k-1}=Z\cup Z^{\prime},\ \ \text{ where }$
$Z\simeq{\mathbb{A}}^{1}_{U_{k+1}}\times C_{2}(U_{k-1},U_{k}),\ \
Z^{\prime}\simeq C_{1}(U_{k},U_{k+1})\times C^{\prime}_{2}(U_{k-1}),$ $Z\cap
Z^{\prime}\simeq{\mathbb{A}}^{1}_{U_{k+1}}\times C^{\prime}_{2}(U_{k-1}).$

∎

Similarly we compute endomorphisms of $Q_{j}$.

######  Lemma 5.3.4.

For $0<j<k$, the embedding
$k[X_{1},X_{2},U_{j},U_{j+1}]\subset{\mathcal{O}}(\overline{{\mathcal{X}}}_{2,k})$
induces an isomorphism

$\operatorname{End}(Q_{j})\simeq
k[X_{1},X_{2},U_{j},U_{j+1}]/(X_{1}X_{2},U_{j}U_{j+1}).$

Similarly,

$\operatorname{End}(Q_{k})\simeq k[X_{1},X_{2},U_{k}]/(U_{k}X_{1}X_{2}),$
$\operatorname{End}(Q_{0})\simeq k[U_{1},X_{1},X_{2}]/(X_{1}X_{2}).$

It follows from Lemmas 5.3.3 and 5.3.4 that for of our generators $G=P_{ij}$
or $G=Q_{j}$ the algebra $\operatorname{End}(G)$ is a quotient of the algebra
$A=\mathbf{k}[U_{1},\ldots,U_{k},X_{1},X_{2}]$. Let us set for any pair of our
generators $G,G^{\prime}$,

$\operatorname{End}(G,G^{\prime}):=\operatorname{End}(G)\otimes_{A}\operatorname{End}(G^{\prime}).$

The morphism spaces $\operatorname{Hom}(G,G^{\prime})$ and
$\operatorname{Hom}(G^{\prime},G)$ are
$\operatorname{End}(G,G^{\prime})$-modules, and we will show that they are
either $0$ or free of rank $1$ over $\operatorname{End}(G,G^{\prime})$.

####  5.3.2. Morphisms between $P_{ij}$’s

Looking at the supports we see that
$\operatorname{Hom}(P_{ij},P_{i^{\prime}j^{\prime}})=0$ unless
$|i^{\prime}-i|\leq 1$ and $|j^{\prime}-j|\leq 1$. The nonzero morphisms are
computed in the Lemma below.

######  Lemma 5.3.5.

(i) For $i>j$, we have generators

$\alpha_{ij}\in\operatorname{Hom}^{-1-2(k-j)}(P_{i,j+1},P_{i,j}),\
\beta_{ij}\in\operatorname{Hom}^{1+2(k-j)}(P_{i,j},P_{i,j+1})$

such that $\alpha_{ij}\beta_{ij}=\beta_{ij}\alpha_{ij}=U_{j+1}$, and such that
the spaces

$\operatorname{Hom}^{*}(P_{i,j+1},P_{i,j})=\operatorname{End}(P_{i,j},P_{i,j+1})\cdot\alpha_{ij},\
\
\operatorname{Hom}^{*}(P_{i,j},P_{i,j+1})=\operatorname{End}(P_{i,j},P_{i,j+1})\cdot\beta_{ij}$

are free $\operatorname{End}(P_{i,j},P_{i,j+1})$-modules. The generator
$\alpha_{ij}$ (resp., $\beta_{ij}$) comes from the exact triangle

${\mathcal{O}}_{R_{i,j}}(-2-2(k-j))\rTo{c_{2,j}}{\mathcal{O}}_{R_{i,j}\cup
R_{i,j+1}}\to{\mathcal{O}}_{R_{i,j+1}}\rTo{\alpha_{ij}}{\mathcal{O}}_{R_{i,j}}[1](-2-2(k-j))$

(resp., from the triangle

${\mathcal{O}}_{R_{i,j+1}}(2(k-j))\rTo{c_{2,j+1}^{-1}}{\mathcal{O}}_{R_{i,j}\cup
R_{i,j+1}}\to{\mathcal{O}}_{R_{i,j}}\rTo{\beta_{ij}}{\mathcal{O}}_{R_{i,j+1}}[1](2(k-j))\
),$

well defined near $R_{ij}\cap R_{i,j+1}$.

(ii) Similarly, for $i>j$, we have generators

$\gamma_{ij}\in\operatorname{Hom}^{-1}(P_{i+1,j},P_{i,j}),\
\delta_{ij}\in\operatorname{Hom}^{1}(P_{i,j},P_{i+1,j})$

such that $\gamma_{ij}\delta_{ij}=\delta_{ij}\gamma_{ij}=U_{i+1}$, and such
that the spaces

$\operatorname{Hom}^{*}(P_{i+1,j},P_{i,j})=\operatorname{End}(P_{i,j},P_{i+1,j})\cdot\gamma_{ij},\
\
\operatorname{Hom}^{*}(P_{i,j},P_{i+1,j})=\operatorname{End}(P_{i,j},P_{i+1,j})\cdot\delta_{ij}$

are free $\operatorname{End}(P_{i,j},P_{i+1,j})$-modules. For $i<k$, the
generator $\gamma_{ij}$ (resp., $\delta_{ij}$) has the form

$\gamma_{ij}=\gamma^{\prime}_{ij}\cdot\frac{f_{1}^{*}p_{i+1}^{*}z_{1}}{f_{1}^{*}p_{i}^{*}z_{0}},$
$(\text{ resp., }\
\delta_{ij}=\delta^{\prime}_{ij}\cdot\frac{f_{1}^{*}p_{i}^{*}z_{0}}{f_{1}^{*}p_{i+1}^{*}z_{1}}),$

where $\gamma^{\prime}_{ij}$ (resp., $\delta^{\prime}_{ij}$) comes from the
exact triangle

${\mathcal{O}}_{R_{i,j}}(-2-2(k-i))\rTo{c_{1,i}}{\mathcal{O}}_{R_{i,j}\cup
R_{i+1,j}}\to{\mathcal{O}}_{R_{i+1,j}}\rTo{\gamma^{\prime}_{ij}}{\mathcal{O}}_{R_{i,j}}[1](-2-2(k-i))$

(resp., from the triangle

${\mathcal{O}}_{R_{i+1,j}}(2(k-i))\rTo{c_{1,i+1}^{-1}}{\mathcal{O}}_{R_{i,j}\cup
R_{i+1,j}}\to{\mathcal{O}}_{R_{i,j}}\rTo{\delta^{\prime}_{ij}}{\mathcal{O}}_{R_{i+1,j}}[1](2(k-i))\
),$

well defined near $R_{i+1,j}\cap R_{i+1,j}$. For $i=k$, we have

$\gamma_{kj}=\gamma^{\prime}_{kj}\cdot\frac{1}{f_{1}^{*}p_{k-1}^{*}z_{0}},\ \
\delta_{kj}=\delta^{\prime}_{0j}\cdot f_{1}^{*}p_{k-1}^{*}z_{0},$

where $\gamma^{\prime}_{kj}$ and $\delta^{\prime}_{kj}$ come from the above
triangles with $i=k$.

(iii) For $i-j>1$, the space $\operatorname{Hom}(P_{i+1,j+1},P_{ij})$ (resp.,
$\operatorname{Hom}(P_{ij},P_{i+1,j+1})$) is a free
$\operatorname{End}(P_{ij},P_{i+1,j+1})$-module generated by
$\gamma_{i,j+1}\alpha_{ij}=\alpha_{i+1,j}\gamma_{ij}$ (resp.,
$\delta_{ij}\beta_{i+1,j}=\beta_{ij}\delta_{i,j+1}$). The similar result holds
for $\operatorname{Hom}(P_{i+1,j-1},P_{ij})$ (resp.,
$\operatorname{Hom}(P_{ij},P_{i+1,j-1})$). The space
$\operatorname{Hom}(P_{i+1,i+2},P_{i,i+1})$ (resp.,
$\operatorname{Hom}(P_{i,i+1},P_{i+1,i+2})$) is a free
$\operatorname{End}(P_{i,i+1},P_{i+1,i+2})$-module generated by
$\gamma_{i,i+2}\alpha_{i,i+1}$ (resp., $\beta_{i,i+1}\delta_{i,i+2}$).

###### Proof.

As before, we first compute the sheaves of homomorphisms using the complete
intersection equations for $\overline{R}_{ij}$ and Lemma 2.3.6. Furthermore,
when calculating $\operatorname{Hom}(P_{ij},P_{i^{\prime}j^{\prime}})$ we only
need to study the situation in a neighborhood of
$(\overline{R}_{ij}\cap\overline{R}_{i^{\prime}j^{\prime}})\times{\mathbb{A}}^{k}$.

For example, let us compute
$\underline{\operatorname{Hom}}(P_{i,j+1},P_{i,j})$, where $i<k$. We can work
near $(R_{i,j}\cap R_{i,j+1})\times{\mathbb{A}}^{k}$, so we can replace
$f_{2}^{*}s_{1}(j+1)$ with $V_{j+2}$ and $f_{2}^{*}s_{2}(j+1)$ with $C_{2,j}$
in the equations of $R_{i,j+1}$. Similarly, we can replace $f_{2}^{*}s_{1}(j)$
with $C_{2,j+1}^{-1}$ and $f_{2}^{*}s_{2}(j)$ with $V_{j}$ in the equations of
$R_{i,j}$. Thus, we split $W$ as

$W=\sum_{r\neq i,i+1,j+1}V_{r}U_{r}+f_{1}^{*}s_{1}(i)\cdot
f_{1}^{*}p_{i}^{*}z_{1}\cdot U_{i+1}+f_{1}^{*}s_{2}(i)\cdot
f_{1}^{*}p_{i}^{*}z_{0}\cdot U_{i}+C_{2,j+1}^{-1}C_{2,j}\cdot U_{j+1},$

and then take the tensor product of the following sheaves of homomorphisms
between Koszul matrix factorizations:

  * •

for $r\neq i,i+1,j+1$,
$\underline{\operatorname{End}}(\\{U_{r},V_{r}\\})\simeq{\mathcal{O}}/(U_{r},V_{r})$;

  * •

$\underline{\operatorname{End}}(\\{f_{1}^{*}p_{i}^{*}z_{1}\cdot
U_{i+1},f_{1}^{*}s_{1}(i)\\})\simeq{\mathcal{O}}/(f_{1}^{*}p_{i}^{*}z_{1}\cdot
U_{i+1},f_{1}^{*}s_{1}(i))$;

  * •

$\underline{\operatorname{End}}(\\{f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i},f_{1}^{*}s_{2}(i)\\})\simeq{\mathcal{O}}/(f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i},f_{1}^{*}s_{2}(i));$

  * •

$\underline{\operatorname{Hom}}(\\{C_{2,j+1}^{-1}\cdot
U_{j+1},C_{2,j}\\},\\{C_{2,j}\cdot
U_{j+1},C_{2,j+1}^{-1}\\})\simeq{\mathcal{O}}/(C_{2,j+1}^{-1},C_{2,j})[-1]$.

Thus, we get an isomorphism on
$\overline{R}_{ij}\cap\overline{R}_{i,j+1}\times{\mathbb{A}}^{k}$,

$\underline{\operatorname{Hom}}(P_{i,j+1},P_{i,j})\simeq{\mathcal{O}}/((U_{r}\
|\ r\neq i,i+1,j+1),f_{1}^{*}p_{i}^{*}z_{1}\cdot
U_{i+1},f_{1}^{*}p_{i}^{*}z_{0}\cdot
U_{i})[-1]\simeq{\mathcal{O}}_{C(U_{i},U_{i+1})\times{\mathbb{A}}^{1}_{U_{j+1}}}[-1],$

where $C(U_{i},U_{i+1})$ is the nodal chain with the components
${\mathbb{A}}^{1}_{U_{i}}$, ${\mathbb{P}}^{1}$, and
${\mathbb{A}}^{1}_{U_{i+1}}$. Thus, we get

$\operatorname{Hom}(P_{i,j+1},P_{i,j})\simeq\mathbf{k}[U_{i},U_{i+1},U_{j+1}]/(U_{i}U_{i+1})[-1]=\operatorname{End}(P_{i,j+1},P_{i,j})[-1].$

Chasing the generators from Lemma 2.3.6 we get the statement in this case.

In the case of $\underline{\operatorname{Hom}}(P_{i+1,j},P_{i,j})$ we take the
tensor product of the sheaves of homomorphisms,

  * •

for $r\neq i+1,j,j+1$,
$\underline{\operatorname{End}}(\\{U_{r},V_{r}\\})\simeq{\mathcal{O}}/(U_{r},V_{r})$;

  * •