# Braid and Phantom

Jenia Tevelev

###### Abstract.

Let $N$ be the moduli space of stable rank $2$ vector bundles on a smooth
projective curve of genus $g\geq 2$ with fixed odd determinant. In [TT], we
previously found a semi-orthogonal decomposition of the bounded derived
category of $N$ into bounded derived categories of symmetric powers of the
curve and, possibly, a phantom block. In this work, we employ the theory of
weaving patterns to eliminate the possibility of a phantom, completing the
proof of the decomposition conjectured by Narasimhan and, independently, by
Belmans–Galkin–Mukhopadhyay.

##  1\. Introduction

Let $N$ be the moduli space of stable rank $2$ vector bundles on a smooth
projective curve of genus $g\geq 2$ with a fixed odd determinant. Let $\theta$
be an ample generator of $\operatorname{Pic}N=\mathbb{Z}$ and let
$\operatorname{\mathcal{E}}$ be the Poincaré vector bundle on $C\times N$
normalized so that $\det\operatorname{\mathcal{E}}|_{\\{p\\}\times N}=\theta$
for every $p\in C$. The tensor vector bundles
$\operatorname{\mathcal{E}}^{\boxtimes k}$ on $\operatorname{Sym}^{k}C\times
N$ of rank $2^{k}$ were previously studied in [TT], where it was proved that
the Fourier–Mukai functor
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{E}}^{\boxtimes
k}}:\,D^{b}(\operatorname{Sym}^{k}C)\to D^{b}(N)$ is fully faithful for $k\leq
g-1$. In general, the image of a fully faithful Fourier–Mukai functor
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{A}}}:\,D^{b}(X)\to
D^{b}(Y)$ between derived categories of smooth projective varieties with
kernel $\operatorname{\mathcal{A}}\in D^{b}(X\times Y)$, is an admissible
subcategory of $D^{b}(Y)$, which we denote by
$\langle\operatorname{\mathcal{A}}\rangle$. For example, the Beilinson semi-
orthogonal decomposition can be written as
$D^{b}(\operatorname{{\mathbb{P}}}^{n})=\langle\operatorname{\mathcal{O}},\ldots,\operatorname{\mathcal{O}}(n)\rangle$.
With this notation, our goal is to finish the proof of the following theorem.

######  Theorem 1.1.

$D^{b}(N)$ has a semi-orthogonal decomposition into
$D^{b}(\operatorname{Sym}^{k}C)$ (two blocks for $0\leq k\leq g-2$ and one
block for $k=g-1$), arranged into four mega-blocks, and embedded by the
following Fourier–Mukai functors:

$\Bigl{\langle}\bigl{\langle}{\theta}^{1-g+k}\otimes\operatorname{\mathcal{E}}^{\boxtimes
g-2-2k}\bigr{\rangle}_{0\leq k\leq\lfloor{g-2\over
2}\rfloor},\quad\bigl{\langle}{\theta}^{2-g+k}\otimes\operatorname{\mathcal{E}}^{\boxtimes
g-3-2k}\bigr{\rangle}_{0\leq k\leq\lfloor{g-3\over 2}\rfloor},\quad{}$
${}\quad\bigl{\langle}{\theta}^{2-g+k}\otimes\operatorname{\mathcal{E}}^{\boxtimes
g-2-2k}\bigr{\rangle}_{0\leq k\leq\lfloor{g-2\over
2}\rfloor},\quad\bigl{\langle}{\theta}^{2-g+k}\otimes\operatorname{\mathcal{E}}^{\boxtimes
g-1-2k}\bigr{\rangle}_{0\leq k\leq\lfloor{g-1\over 2}\rfloor}\Bigr{\rangle}.$

Within the mega-blocks, the blocks are arranged in decreasing order of $k$.

We refer to [TT] for the history of the problem and the proof of semi-
orthogonality. For a slightly different decomposition used there, see Theorem
6.14. As in [TT], we work on another Fano variety, the moduli space $M$ of
stable pairs $(F,s)$. Here $F$ is a stable rank $2$ vector bundle with a fixed
odd determinant $\Lambda$ of degree $2g-1$, and $s\in H^{0}(C,F)$ is a non-
zero section.

We play the two ray game. The variety $M$ admits a forgetful birational
morphism $\zeta:\,M\to N$, $\zeta(F,s)=F$, and a birational map
$\psi:\,M\dashrightarrow\operatorname{{\mathbb{P}}}^{3g-3}$, which sends a
general stable pair $(F,s)\in M$ to the point of
$\operatorname{{\mathbb{P}}}(\operatorname{Ext}^{1}(\Lambda,\operatorname{\mathcal{O}}))$
representing the unique extension $0\to\operatorname{\mathcal{O}}\to
F\to\Lambda\to 0$ with $s\in H^{0}(C,\operatorname{\mathcal{O}})$. The map
$\psi$ is factored into flips followed by a divisorial contraction [thaddeus]:

(1.1) $M=M_{g-1}\dashrightarrow M_{g-2}\dashrightarrow\ldots\dashrightarrow
M_{1}\to M_{0}=\operatorname{{\mathbb{P}}}^{3g-3}.$

On each step $M_{k}\dashrightarrow M_{k-1}$, one projective bundle over
$\operatorname{Sym}^{k}C$ is flipped into another, of smaller rank. By [TT,
Proposition 3.18], this gives a semi-orthogonal decomposition of $D^{b}(M)$
with $3g-2-3k$ blocks isomorphic to $D^{b}(\operatorname{Sym}^{k}C)$ for
$k=0,\ldots g-1$ and supported on the exceptional locus of the contraction
$\psi:\,M\dashrightarrow\operatorname{{\mathbb{P}}}^{3g-3}$. The goal of this
paper is to construct an element of the braid group on ${g(3g-1)\over 2}$
strands that mutates this semi-orthogonal decomposition into the decomposition
compatible with the morphism $\zeta:\,M\to N$, which contracts a divisor
$Z\subset M$ to the Brill–Noether locus $B=\\{F\,|\,h^{0}(C,F)>1\\}\subset N$.
Some blocks go into the blocks of Theorem 1.1 (pulled back to $M$), and the
rest go into the blocks of the perpendicular subcategory
$\\{X\,|\,R\zeta_{*}(X)=0\\}\subset D^{b}(M)$, proving Theorem 1.1.

To construct the mutation, we combine vanishing theorems from [TT] with an
analysis of “weaving patterns” related to various geometric constructions. The
result is illustrated in Figure 1 (for genus 5). We use the term “weaving” as
opposed to “braiding” to emphasize the importance of invisible wefts, which
break the mutation into a sequence of standard steps. In contrast to real-
world weaving, vertical warps are not parallel. When they interlace, the warp
that stays above connects an admissible subcategory to itself, while the warp
that goes under connects it to a differently embedded (but equivalent)
subcategory. Crossing strands correspond to mutually perpendicular
subcategories that both remain the same after the mutation.

### Notational quirks

In complicated formulas, we follow [thaddeus] and drop the tensor product
symbol, for example between line bundles. We often mix notation for line
bundles and Cartier divisors, derived and underived functors (when they are
the same), and denote pull-backs of vector bundles by the same letters. For
example, we denote $\zeta^{*}\theta$ simply by $\theta$. Standard facts about
Fourier–Mukai functors from [huybrechts] are used without much ado.

### Acknowledgments

This project would have been impossible without our collaboration with
Sebastián Torres [TT]. I am grateful to Alex Perry for helpful discussions and
to Elias Sink for helping to fix a mistake in the earlier version of the
paper. Ideas have been borrowed from the papers [myself] on Bott vanishing on
GIT quotients, [castravet1, castravet2] on derived category of
$\overline{M}_{0,n}$, and [toda, kosekitoda] on d-critical flips of stable
pair moduli spaces. The paper was written during a visit to Imperial College
in January 2023, and I thank Paolo Cascini, Alessio Corti, and Yanki Lekili
for sharing an inspirational environment. The research was partially supported
by the NSF grant DMS-2101726. Graphics were created by
www.plainformstudio.com.

Figure 1. All weaving patterns in genus $5$

##  2\. Farey Twill

The Farey Twill, illustrated in Figure 2, produces an improved version of the
semi-orthogonal decomposition of $D^{b}(M)$ given in [TT, Prop. 3.18]. Its
blocks are isomorphic to $D^{b}(\operatorname{Sym}^{k}C)$ for $k=0,\ldots,g-1$
and are supported on the exceptional locus of the contraction
$\psi:\,M\dashrightarrow\operatorname{{\mathbb{P}}}^{3g-3}$.

Figure 2. Farey Twill in genus $5$

The goal of this section is to prove Theorem 2.2 below, except for several
lemmas, which will be proved in Section 3. We first introduce some notation,
mostly from [TT] and [thaddeus], that will be used throughout the paper.

######  Notation 2.1.

Let $(\operatorname{\mathcal{F}},\Sigma)$ be the universal stable pair on
$C\times M$. Then

$\operatorname{\mathcal{F}}=\operatorname{\mathcal{E}}(-Z),\quad\det\operatorname{\mathcal{F}}=\Lambda\boxtimes\operatorname{{\boldsymbol{\Lambda}}},\quad\hbox{\rm
and}\quad\theta=\operatorname{{\boldsymbol{\Lambda}}}(2Z),$

where $Z$ is the exceptional divisor of the contraction $M\to N$. Let
$\operatorname{\mathcal{D}}^{k}$ be the structure sheaf of a reduced subscheme

$D^{k}=\\{(D,F,s):\,s|_{D}=0\\}\subset\operatorname{Sym}^{k}C\times M,$

where we view $D\in\operatorname{Sym}^{k}C$ as a closed subscheme of $C$.

######  Theorem 2.2.

The Fourier–Mukai functor
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}}:\,D^{b}(\operatorname{Sym}^{k}C)\to
D^{b}(M)$ is fully faithful for $k\leq g-1$ and $D^{b}(M)$ has a semi-
orthogonal decomposition into admissible subcategories arranged into three
mega-blocks, as follows:

$\Bigl{\langle}\bigl{\langle}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{j}\otimes\operatorname{\mathcal{D}}^{k}\bigr{\rangle}_{j+k\leq
g-2\atop j,k\geq 0},\
\bigl{\langle}(Z\operatorname{{\boldsymbol{\Lambda}}}){\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{j}\otimes\operatorname{\mathcal{D}}^{k}\bigr{\rangle}_{j+k\leq
g-2\atop j,k\geq 0},\
\bigl{\langle}(\theta\operatorname{{\boldsymbol{\Lambda}}}){\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{j}\otimes\otimes\operatorname{\mathcal{D}}^{k}\bigr{\rangle}_{j+k\leq
g-1\atop j,k\geq 0}\Bigr{\rangle}.$

Within each of the three mega-blocks, the blocks are first arranged by $j+k$
(in the increasing order) and, for a fixed $j+k$, by $j$ (in the increasing
order).

######  Notation 2.3.

Let $E$ be the exceptional divisor of the birational morphism $M_{1}\to
M_{0}=\operatorname{{\mathbb{P}}}^{3g-3}$ and let $H$ be the pullback of the
hyperplane divisor. The varieties $M_{1},\ldots,M_{g-1}=M$ appearing in the
sequence of flips (1.1) are isomorphic in codimension $1$. This allows us to
use the same notation for their line bundles, including an important line
bundle [thaddeus]

$\operatorname{\mathcal{O}}(m,n):=(m+n)H-nE.$

We use throughout that $M_{i}$ is the moduli space of stable pairs (for
varying stability parameter) as in [thaddeus]. Mnemonically, if $(F,s)\in
M_{i}$ then the scheme of zeros of $s$ in $C$ has degree at most $i$. For
$0\leq k\leq i$, let $\operatorname{\mathcal{D}}_{i}^{k}$ be the structure
sheaf of a reduced subscheme

$D_{i}^{k}=\\{(D,F,s):\,s|_{D}=0\\}\subset\operatorname{Sym}^{k}C\times
M_{i}.$

This agrees with the definition of $D^{k}$ and
$\operatorname{\mathcal{D}}^{k}$ above when $i=g-1$.

The following lemma, as well as several others, will be proved in Section 3.

######  Lemma 2.4.

The Fourier–Mukai functor
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}:\,D^{b}(\operatorname{Sym}^{k}C)\to
D^{b}(M_{i})$ is fully faithful for $0\leq k\leq i\leq g-1$.

######  Definition 2.5.

We introduce admissible (by Lemma 2.4) subcategories

$\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle\subset D^{b}(M_{\lfloor
t\rfloor})$

for integer parameters $0\leq k\leq g-1$, $0\leq s\leq 3g-3-3k$, and a real
parameter $k<t<g$. Namely,
$\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle$ is the image of
$\operatorname{Sym}^{k}C$ under the Fourier–Mukai functor with the kernel
$\operatorname{\mathcal{D}}^{k,s}_{t}=\operatorname{\mathcal{D}}^{k}_{\lfloor
t\rfloor}\otimes L^{k,s}_{t}$, where

(2.1)
$L^{k,s}_{t}=\begin{cases}\operatorname{\mathcal{O}}\left(\left\lfloor{s\over
t-k}\right\rfloor,s+\left\lfloor{s\over
t-k}\right\rfloor(k-1)\right)&\hbox{\rm if}\quad{\lfloor
t\rfloor}>k,\cr\operatorname{\mathcal{O}}(s,sk)&\hbox{\rm if}\quad{\lfloor
t\rfloor}=k.\cr\end{cases}$

Here $\lfloor x\rfloor$ denotes the round-down of a real number $x$.

We interpret the variable $t$ as time. In the Farey Twill, named after the
Farey fractions, the admissible subcategory
$\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle$ “moves” in the
$(x,t)$-plane along the trajectory

(2.2) $x_{k,s}(t)={s\over t-k},$

with the $t$-axis pointing down and the $x$-axis to the right. On the level
$t$, the blocks $\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle\subset
D^{b}(M_{{\lfloor t\rfloor}})$ are ordered by the value of the function
$x_{k,s}(t)$. When the trajectories cross, the blocks mutate as will be
described below. The paths of the blocks with $s=0$ have to be modified to
allow them to participate in the mutations on the integer levels $t={\lfloor
t\rfloor}$. The paths (2.2) (and modified paths of blocks with $s=0$) are
plotted in Figure 3 in genus $5$.

Figure 3. Genuine paths of blocks in the Farey Twill (genus $5$)

######  Example 2.6.

We analyze the Farey Twill for small values of $t$, see Figure 4. In this and
other illustrations, we apply an $(x,t)$-plane transformation to better
visualize intersections of trajectories. For examples, paths of the line
bundles $H,2H,3H,\ldots$ should come from infinity (see the top of Figure 3),
but in Figures 1, 2, and 4 we draw them as parallel and vertical.

Figure 4. Starting the Farey Twill in genus $5$

$0<t<1$. $\operatorname{\mathcal{D}}^{0,s}_{t}=\operatorname{\mathcal{O}}(sH)$
and we start with the Beilinson decomposition:

$D^{b}(M_{0})=D^{b}(\operatorname{{\mathbb{P}}}^{3g-3})=\langle\operatorname{\mathcal{O}},H,\ldots,(3g-3)H\rangle.$

$t=1$.
$\operatorname{\mathcal{D}}^{0,s}_{1}=\operatorname{\mathcal{O}}(s,0)=\operatorname{\mathcal{O}}(sH)$
but pulled back to $M_{1}$. This gives an admissible subcategory
$\psi^{*}D^{b}(\operatorname{{\mathbb{P}}}^{3g-3})=\langle\operatorname{\mathcal{O}},H,\ldots,(3g-3)H\rangle\subset
D^{b}(M_{1})$.

$1<t<2$. By [thaddeus],
$M_{1}\simeq\operatorname{Bl}_{C}\operatorname{{\mathbb{P}}}^{3g-3}$ (where
the curve $C$ is embedded in $\operatorname{{\mathbb{P}}}^{3g-3}$ by the
linear system $|K_{C}+\Lambda|$) and the locus

$D^{1}_{1}=\\{(p,x)\,|\,\psi(x)=p\\}\subset C\times M_{1}$

is a projective bundle over $C$ isomorphic to the exceptional divisor
$E\subset M_{1}$ via the second projection. The line bundles
$\operatorname{\mathcal{O}}(s,s)$ restrict to $\operatorname{\mathcal{O}}(s)$
on the fibers $\operatorname{{\mathbb{P}}}^{3g-5}$ of the projective bundle.
So the sheaves
$\operatorname{\mathcal{D}}^{1,s}_{t}=\operatorname{\mathcal{D}}^{1}_{1}(s,s)$
are the Fourier–Mukai kernels that appear in the Orlov decomposition [orlov]

(2.3)
$D^{b}(M_{1})=D^{b}(\operatorname{Bl}_{C}\operatorname{{\mathbb{P}}}^{3g-3})=\langle\operatorname{\mathcal{O}},H,\ldots,(3g-3)H,\operatorname{\mathcal{D}}^{1,0}_{t},\ldots,\operatorname{\mathcal{D}}^{1,3g-6}_{t}\rangle.$

The paths (2.2) of the new blocks $\operatorname{\mathcal{D}}^{1,s}_{t}$ come
from infinity, corresponding to the fact that these blocks appear on the right
in (2.3).

The sheaves $\operatorname{\mathcal{D}}^{1,s}_{t}$ don’t change when $1<t<2$
but the line bundles in (2.3) will undergo mutations as $t$ increases from $1$
to $2$. Note that
$\operatorname{\mathcal{D}}^{0,s}_{1+\epsilon}=\operatorname{\mathcal{O}}(s-1,1)=\operatorname{\mathcal{O}}(sH-E)$
for $0<\epsilon\ll 1$, so these blocks have already changed compared to $t=1$.
The corresponding mutations are encoded in the convention that paths of blocks
with $s=0$ have to be modified to give the asymptotes of the hyperbolas (2.2).
Namely, we add the horizontal line $t=1$ to the trajectory of the block
$\operatorname{\mathcal{D}}^{1,0}_{t}$ and get a sequence of mutations on
level $t=1$ induced by the standard short exact sequences (for $s>0$),

(2.4)
$0\to\operatorname{\mathcal{O}}(sH-E)\to\operatorname{\mathcal{O}}(sH)\to\operatorname{\mathcal{O}}_{E}(sH)\to
0.$

This gives a semi-orthogonal decomposition of $D^{b}(M_{1})$ on the level
$t=1+\epsilon$:

$\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{D}}^{1,0}_{t},H-E,2H-E,\ldots,(3g-3)H-E,\operatorname{\mathcal{D}}^{1,1}_{t},\ldots,\operatorname{\mathcal{D}}^{1,3g-6}_{t}\rangle.$

Note that we keep the block $\operatorname{\mathcal{D}}^{1,0}_{t}$ to the
right of $\operatorname{\mathcal{O}}$. Formally speaking, we modify the
vertical part of its path as well (make it $x=\epsilon$ for $0<\epsilon\ll
1$).

The next block to start crossing paths in the Farey Twill is
$\operatorname{\mathcal{D}}^{1,1}_{t}$, then
$\operatorname{\mathcal{D}}^{1,2}_{t}$, etc. Mutations are given by the line
bundle twists of (2.4), eventually giving a semi-orthogonal decomposition of
$D^{b}(M_{1})$ on the level $t=2-\epsilon$:

(2.5)
$\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{D}}^{1,0}_{t},H-E,2H-E,\operatorname{\mathcal{D}}^{1,1}_{t},3H-2E,4H-2E,\operatorname{\mathcal{D}}^{1,2}_{t},\ldots,\operatorname{\mathcal{D}}^{1,3g-6}_{t}\rangle.$

with many blocks $\operatorname{\mathcal{D}}^{1,s}_{t}$ left at the end of the
decomposition.

What is the advantage of (2.5) compared to (2.3)? By [TT, Section 3], we have
a “windows” embedding $\iota:\,D^{b}(M_{1})\hookrightarrow D^{b}(M_{2})$ that
factors as $D^{b}(M_{1})\cong\mathbb{G}_{w}\subset
D^{b}(\operatorname{\mathcal{M}})\mathop{\to}\limits^{r}D^{b}(M_{2})$, where
$\operatorname{\mathcal{M}}$ is an appropriate quotient stack that contains
$M_{1}$ and $M_{2}$ as open substacks, $r$ is the restriction, and the windows
subcategory $\mathbb{G}_{w}\subset D^{b}(\operatorname{\mathcal{M}})$ consists
of complexes in equivariant derived category with cohomology sheaves having
weights in the range $[0,1]$ for the wall crossing from $M_{1}$ to $M_{2}$
(see [TT, Proposition 3.18]).

The line bundles $\operatorname{\mathcal{O}},H-E,2H-E,3H-2E,4H-2E,\ldots$
appearing in (2.5) have weights $0,1,0,1,0,\ldots$ (we refer to [TT, Section
3] for the calculation of weights of all standard vector bundles), and so the
windows embedding $\iota$ maps them to the same line bundles on $M_{2}$.
Furthermore, $\iota$ takes the decomposition (2.5) into the following
admissible subcategory:

$t=2$.

$\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{D}}^{1,0}_{2},H-E,2H-E,\operatorname{\mathcal{D}}^{1,1}_{2},3H-2E,4H-2E,\operatorname{\mathcal{D}}^{1,2}_{2},\ldots,\operatorname{\mathcal{D}}^{1,3g-6}_{2}\rangle\subset
D^{b}(M_{2}).$

$2<t<3$. We complete this admissible subcategory of $D^{b}(M_{2})$ to a semi-
orthogonal decomposition of $D^{b}(M_{2})$ by adding blocks
$\operatorname{\mathcal{D}}^{2,0}_{2+\epsilon},\ldots,\operatorname{\mathcal{D}}^{2,3g-9}_{2+\epsilon}$
at the end and continue with mutations encoded in intersections of paths.

To realize this program in general, we consider the “windows” embedding from
[TT, Proposition 3.18],

(2.6) $\iota:\,D^{b}(M_{i-1})\cong\mathbb{G}_{w}\subset
D^{b}(\operatorname{\mathcal{M}})\mathop{\to}\limits^{r}D^{b}(M_{i}),$

where $\operatorname{\mathcal{M}}$ is an appropriate quotient stack that
contains $M_{i-1}$ and $M_{i}$ as open substacks, $r$ is the restriction, and
$\mathbb{G}_{w}\subset D^{b}(\operatorname{\mathcal{M}})$ is a full
subcategory of complexes in the equivariant derived category with cohomology
sheaves having weights in the range $[0,i-1]$ for the wall crossing
$M_{i-1}\dashrightarrow M_{i}$.

######  Lemma 2.7.

For $0\leq k<i$,
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}_{i}}=\iota\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}_{i-1}}$.
Furthermore, objects in the subcategory
$\langle\operatorname{\mathcal{D}}^{k}_{i-1}\rangle\subset
D^{b}(M_{i-1})\cong\mathbb{G}_{w}$ have weights in the range $[0,\ldots,k]$.

Given Lemma 2.7, we claim that all subcategories
$\langle\operatorname{\mathcal{D}}^{k,s}_{i-\epsilon}\rangle\subset
D^{b}(M_{i-1})$ for $0<\epsilon\ll 1$ and $k<i$ map to the subcategories
$\langle\operatorname{\mathcal{D}}^{k,s}_{i}\rangle\subset D^{b}(M_{i})$ under
the windows embedding. More precisely, we have
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k,s}_{i}}=\iota\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k,s}_{i-\epsilon}}$.
Indeed,
$\operatorname{\mathcal{D}}^{k,s}_{i-\epsilon}=\operatorname{\mathcal{D}}^{k}_{i-1}\otimes
L^{k,s}_{i}$ and
$\operatorname{\mathcal{D}}^{k,s}_{i}=\operatorname{\mathcal{D}}^{k}_{i}\otimes
L^{k,s}_{i}$, where

$L^{k,s}_{i}=\begin{cases}\operatorname{\mathcal{O}}\left(\left\lfloor{s\over
i-k}\right\rfloor,s+\left\lfloor{s\over
i-k}\right\rfloor(k-1)\right)&\hbox{\rm if}\quad
i-1>k,\cr\operatorname{\mathcal{O}}(s,s(i-1))&\hbox{\rm if}\quad
i-1=k.\cr\end{cases}$

The weight of $L^{k,s}_{i}$ is equal to $s\mod(i-k)$ if $i-k>1$ and 0
otherwise. In either case, the weight is less than $i-k$. So we have

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k,s}_{i}}=\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}_{i}}\otimes
L^{k,s}_{i}=(\iota\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}_{i-1}})\otimes
L^{k,s}_{i}=\iota\circ(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k}_{i-1}}\otimes
L^{k,s}_{i})=\iota\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k,s}_{i-\epsilon}}.$

This analysis shows that the Farey Twill is compatible with the windows
embeddings $\iota:\,D^{b}(M_{i-1})\hookrightarrow D^{b}(M_{i})$.

Next, we analyze what happens when the blocks
$\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle$ and
$\langle\operatorname{\mathcal{D}}^{k^{\prime},s^{\prime}}_{t}\rangle$ cross
trajectories at level $t$. This happens when ${s\over t-k}={s^{\prime}\over
t-k^{\prime}}$ with one exception: as explained above, we have to modify the
paths of blocks $\langle\operatorname{\mathcal{D}}^{k,0}_{t}\rangle$ to be the
horizontal line $t=k$ followed by the vertical line $x=k\epsilon$, $t>k$ for
$0<\epsilon\ll 1$.

######  Lemma 2.8.

If ${s\over t-k}={s^{\prime}\over
t-k^{\prime}}\not\in\operatorname{{\mathbb{Z}}}$ then
$\langle\operatorname{\mathcal{D}}^{k,s}_{t}\rangle$ and
$\langle\operatorname{\mathcal{D}}^{k^{\prime},s^{\prime}}_{t}\rangle$ are
mutually orthogonal. Therefore, the Farey Twill at level $t$ only reorders
these blocks.

Next we analyze the crossings for integer values of $x_{k,s}={s\over
t-k}=n\in\operatorname{{\mathbb{Z}}}$. We have

(2.7) $n={s\over t-k}={s-n\over t-(k+1)}={s-2n\over
t-(k+2)}=\ldots={s-({\lfloor t\rfloor}-k)n\over t-{\lfloor t\rfloor}}.$

So all the blocks
$\operatorname{\mathcal{D}}^{k,s}_{t},\operatorname{\mathcal{D}}^{k+1,s-n}_{t},\operatorname{\mathcal{D}}^{k+2,s-2n}_{t},\operatorname{\mathcal{D}}^{{\lfloor
t\rfloor},s-({\lfloor t\rfloor}-k)n}_{t}$ are crossing paths. If $k\geq 1$ and
$s+n\leq 3g-3-3(k-1)$ then we also have $n={s+n\over t-(k-1)}$. So, without
loss of generality, we may assume that (2.7) starts with the smallest possible
$k$. It follows that, on the level $t$, the blocks of the subcategory

(2.8)
$\langle\operatorname{\mathcal{D}}^{k,s}_{t},\operatorname{\mathcal{D}}^{k+1,s-n}_{t},\operatorname{\mathcal{D}}^{k+2,s-2n}_{t},\ldots,\operatorname{\mathcal{D}}^{{\lfloor
t\rfloor},s-({\lfloor t\rfloor}-k)n}_{t}\rangle$

will undergo mutation that will result in the blocks

(2.9) $\langle\operatorname{\mathcal{D}}^{{\lfloor t\rfloor},s-({\lfloor
t\rfloor}-k)n}_{t+\epsilon},\ldots,\operatorname{\mathcal{D}}^{k+2,s-2n}_{t+\epsilon},\operatorname{\mathcal{D}}^{k+1,s-n}_{t+\epsilon},\operatorname{\mathcal{D}}^{k,s}_{t+\epsilon}\rangle.$

Note the blocks are arranged in the opposite order (according to the slopes of
the paths (2.2)). The twisting line bundle
$\operatorname{\mathcal{O}}\left(\left\lfloor{s\over
t-k}\right\rfloor,s+\left\lfloor{s\over t-k}\right\rfloor(k-1)\right)$ changes
precisely when ${s\over t-k}$ decreases and passes through an integer value
$n$. However, the twisting line bundle of the last block
$\operatorname{\mathcal{D}}^{{\lfloor t\rfloor},s-({\lfloor
t\rfloor}-k)n}_{t}$ is $\operatorname{\mathcal{O}}(s,sk)$, which is the same
$\operatorname{\mathcal{D}}^{{\lfloor t\rfloor},s-({\lfloor
t\rfloor}-k)n}_{t+\epsilon}$. So this block doesn’t change.

######  Lemma 2.9.

The mutation in $D^{b}(M_{{\lfloor t\rfloor}})$ from the semi-orthogonal
decomposition (2.8) to the decomposition (2.9) is given by the following
Figure 5:

Figure 5. Basic Farey Twill Mutation

In this analysis, the case $t\in\operatorname{{\mathbb{Z}}}$ is special: the
last fraction in (2.7) is $0\over 0$. Nevertheless, we still include the block
$\operatorname{\mathcal{D}}^{{\lfloor t\rfloor},s-({\lfloor
t\rfloor}-k)n}_{t}=\operatorname{\mathcal{D}}^{t,0}_{t}$ in the mutation
according to our convention that the path of this block contains the
horizontal line $t=k$. On the integer levels $t=k$, several mutations happen.
Blocks participating in these mutations form several subsequences (2.8) that
are disjoint, except for the last block $\operatorname{\mathcal{D}}^{k,0}_{k}$
that participates in all these mutations consecutively, starting with the
rightmost subsequence of the blocks.

With these results, we can continue the Farey Twill to the level
$t=g-\epsilon$ for $0<\epsilon\ll 1$, proving the following semi-orthogonal
decomposition:

######  Corollary 2.10.

$D^{b}(M_{g-1})$ admits a semi-orthogonal decomposition

$\left\langle\operatorname{\mathcal{D}}^{k}_{g-1}\left(\left\lfloor{s\over
g-k}\right\rfloor,s+\left\lfloor{s\over
g-k}\right\rfloor(k-1)\right)\right\rangle_{0\leq k\leq g-1\atop 0\leq s\leq
3g-3-3k}.$

The blocks are arranged in the increasing order by the function ${s\over
g-\epsilon-k}$.The blocks with $s=0$ are additionally ordered by $k$.

This decomposition is different from the decomposition of Theorem 2.2. To
finish the proof of Theorem 2.2, we have to “undo” some of the mutations of
the Farey Twill. Equivalently, we have to stop it earlier (at times that
depend on the blocks). To explain the algorithm, we first employ the Farey
Twill until the level $t=g-1$ (and not $t=g-\epsilon$ as in Corollary 2.10).
This gives a semi-orthogonal decomposition of $D^{b}(M_{g-1})$ with blocks
$\langle\operatorname{\mathcal{D}}^{k}_{g-1}\otimes L^{k,s}_{g-1}\rangle$ for
$0\leq k\leq g-1$, $0\leq s\leq 3g-3-3k$, where the line bundle

$L^{k,s}_{g-1}=\begin{cases}\operatorname{\mathcal{O}}\left(\left\lfloor{s\over
g-1-k}\right\rfloor,s+\left\lfloor{s\over
g-1-k}\right\rfloor(k-1)\right)&\hbox{\rm if}\quad
k<g-1,\cr\operatorname{\mathcal{O}}&\hbox{\rm if}\quad k=g-1.\cr\end{cases}$

Note that the block $\operatorname{\mathcal{D}}^{g-1,0}_{g-1}$ has not
participated in the Farey Twill yet. On the level $t=g-1$, we do a single
additional chain mutation of Lemma 2.9, from the sequence of blocks
$\langle\operatorname{\mathcal{D}}^{0,3g-3}_{g-1},\operatorname{\mathcal{D}}^{1,3g-6}_{g-1},\ldots,\operatorname{\mathcal{D}}^{g-2,3}_{g-1},\operatorname{\mathcal{D}}^{g-1,0}_{g-1}\rangle$
to the sequence of blocks
$\langle\operatorname{\mathcal{D}}^{g-1,0}_{g-1},\operatorname{\mathcal{D}}^{g-2,3}_{g-1+\epsilon},\ldots,\operatorname{\mathcal{D}}^{1,3g-6}_{g-1+\epsilon},\operatorname{\mathcal{D}}^{0,3g-3}_{g-1+\epsilon}\rangle$.
This gives precisely the very last blocks
$\bigl{\langle}\operatorname{\mathcal{D}}^{k}(2,2g-4+j)\bigr{\rangle}_{j+k=g-1\atop
j,k\geq 0}$ of the third mega-block of Theorem 2.2, in the correct order.

It remains to explain how to obtain the remaining blocks in Theorem 2.2. These
blocks are arranged into three mega-blocks

$\Bigl{\langle}\bigl{\langle}\operatorname{\mathcal{D}}^{k}(0,j)\bigr{\rangle},\quad\bigl{\langle}\operatorname{\mathcal{D}}^{k}(1,g-2+j\bigr{\rangle},\quad\bigl{\langle}\operatorname{\mathcal{D}}^{k}(2,2g-4+j)\bigr{\rangle}\Bigr{\rangle}$

that appear on the left of the semi-orthogonal decomposition. In each of the
three mega-blocks, $j,k\geq 0$, $j+k\leq g-2$, and the blocks are first
arranged by $j+k$ and then by $j$ (both in the increasing order).

These blocks will come from the blocks
$\langle\operatorname{\mathcal{D}}^{k,s}_{g-1}\rangle$ of the Farey Twill with
$0\leq k\leq g-2$, $0\leq s\leq 3g-4-3k$. The idea is to stop the Farey Twill
for the block $\langle\operatorname{\mathcal{D}}^{k,s}_{g-1}\rangle$ shortly
after $\left\lfloor{s\over g-1-k}\right\rfloor$ reaches its minimum, since
afterwards this block no longer changes. We will also ensure that it will no
longer participate in mutations of other blocks. Concretely, we choose the
stopping time $t=t(s,k)$ as follows:

${s\over t-k}=\left\lfloor{s\over g-1-k}\right\rfloor+{j+k\over
g-1},\quad\hbox{\rm where}\quad j=s-\left\lfloor{s\over
g-1-k}\right\rfloor(g-1-k).$

After this time $t(s,k)$, the Farey Twill path is contained in the vertical
strip ${s\over t-k}\in[0,1)$, $[1,2)$, or $[2,3)$, depending on the value of
$\left\lfloor{s\over g-1-k}\right\rfloor=0,1$, or $2$. So the blocks no longer
mutate in the Farey Twill, only permute. Instead of doing these permutations,
we just order the blocks by the value of ${s\over t(s,k)-k}$, or equivalently
by $j+k$, within each of the three mega-blocks. When $j+k$ is the same, the
values of the stopping time $t(s,k)$ are different, with the larger value of
$j$ corresponding to the larger value of $t$. It follows that the blocks with
the same $j+k$ are ordered by $j$ in the increasing order.

##  3\. Cross Warp

In this section we will study a rather striking weaving pattern in $D^{b}(M)$,
which is illustrated in Figure 6 (in genus $5$). We call it the Cross Warp.

Figure 6. Cross Warp in genus $5$

Theorem 3.1 below provides a semi-orthogonal decomposition of $D^{b}(M)$ with
blocks embedded by Fourier–Mukai functors with the kernels given by the tensor
vector bundles $\operatorname{\mathcal{F}}^{*\boxtimes k}$ on
$\operatorname{Sym}^{k}C\times M$ (twisted by line bundles). We refer to [TT,
Section 2] for basic properties of tensor vector bundles. The semi-orthogonal
decompositions of Theorem 2.2 and Theorem 3.1 are connected by the Cross Warp
pattern illustrated in Figure 6 (in genus $5$).

######  Theorem 3.1.

$D^{b}(M)$ has a semi-orthogonal decomposition into admissible subcategories
arranged into three mega-blocks, as follows:

$\Bigl{\langle}\bigl{\langle}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}{\operatorname{\mathcal{F}}^{*\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-2\atop j,k\geq 0},\
\bigl{\langle}{(Z\operatorname{{\boldsymbol{\Lambda}}})\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}{\operatorname{\mathcal{F}}^{*\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-2\atop j,k\geq 0},\
\bigl{\langle}{(\theta\operatorname{{\boldsymbol{\Lambda}}})\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}{\operatorname{\mathcal{F}}^{*\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-1\atop j,k\geq 0}\Bigr{\rangle}.$

Within each of the three mega-blocks, the blocks are arranged first by $k$ (in
the decreasing order) and then, for a fixed $k$, by $j$ (in the decreasing
order).

The proof is a simple corollary of the following theorem, which will also
imply several lemmas that were needed in Section 2.

######  Theorem 3.2.

The Fourier–Mukai functors
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}:\,D^{b}(\operatorname{Sym}^{k}C)\to
D^{b}(M_{i})$ introduced in Section 2 have the following properties:

  1. (a)

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}$ is fully
faithful for $0\leq k\leq i\leq g-1$.

  2. (b)

For $k\leq i-1$,
$\iota\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i-1}^{k}}=\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}$,
where $\iota$ is the embedding (2.6).

  3. (c)

The Fourier–Mukai functors
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}},\
\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*\boxtimes
k}}:\,D^{b}(\operatorname{Sym}^{k}C)\to D^{b}(M_{i})$ are related via the
Cross Warp mutation illustrated in Figure 7.

Figure 7. Basic Cross Warp Mutation

We will first prove various corollaries of Theorem 3.2 and introduce
additional notation, before proving the theorem itself in the next section.

###### Proof of Theorem 3.1.

Each of the three mega-blocks of the semi-orthogonal decomposition of Theorem
2.2 mutates into the corresponding mega-block of the decomposition of Theorem
3.1, without any interaction between the mega-blocks. Furthermore, the second
(resp., third) mega-block differ from the first one by a twist with the line
bundle $Z\otimes\operatorname{{\boldsymbol{\Lambda}}}$ (resp.,
$\theta\otimes\operatorname{{\boldsymbol{\Lambda}}}$). In addition, the third
mega-block is larger (it includes the kernel with
$\operatorname{\mathcal{D}}^{g-1}$ whereas the first two mega-blocks continue
only up to $\operatorname{\mathcal{D}}^{g-2}$). So it is enough to describe
the mutation for the first mega-block, which is illustrated in Figure 8.

Figure 8. First mega-block of the Cross Warp in genus $5$

The Cross Warp mutation of Theorem 3.2 (c) is designed so that these mutations
stack on top of each other. More precisely, the mutation with
$\operatorname{\mathcal{D}}^{k}$ at the top in the middle connects to the
mutation with $\operatorname{\mathcal{D}}^{k-1}$ on the top left and with the
mutation with
$\operatorname{\mathcal{D}}^{k-1}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*}$
on the bottom left. This gives the Cross Warp pattern, where we indicate the
top middles of each block:

$\begin{matrix}\operatorname{\mathcal{O}}\cr&\ldots&\cr\ldots&&\operatorname{\mathcal{D}}^{g-5}&&&\cr&\ldots&&\operatorname{\mathcal{D}}^{g-4}&&\cr\ldots&&\operatorname{\mathcal{D}}^{g-5}\operatorname{{\boldsymbol{\Lambda}}}^{*}&&\operatorname{\mathcal{D}}^{g-3}&\cr&\ldots&&\operatorname{\mathcal{D}}^{g-4}\operatorname{{\boldsymbol{\Lambda}}}^{*}&&\operatorname{\mathcal{D}}^{g-2}\cr\ldots&&\operatorname{\mathcal{D}}^{g-5}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{2}&&\operatorname{\mathcal{D}}^{g-3}\operatorname{{\boldsymbol{\Lambda}}}^{*}&\cr&\ldots&&\operatorname{\mathcal{D}}^{g-4}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{2}&&\cr\ldots&&\operatorname{\mathcal{D}}^{g-5}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{3}&&&\cr&\ldots&\cr{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-2}\cr\end{matrix}$

Altogether this gives the mutation from the semi-orthogonal decomposition

$\langle\operatorname{\mathcal{O}}\rangle,\langle\operatorname{\mathcal{D}}^{1},\operatorname{{\boldsymbol{\Lambda}}}^{*}\rangle,\langle\operatorname{\mathcal{D}}^{2},\operatorname{\mathcal{D}}^{1}\operatorname{{\boldsymbol{\Lambda}}}^{*},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{2}\rangle,\ldots,\langle\operatorname{\mathcal{D}}^{g-2},\operatorname{\mathcal{D}}^{g-3}{\operatorname{{\boldsymbol{\Lambda}}}^{*}},\ldots,\operatorname{\mathcal{D}}^{1}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-3},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-2}\rangle$

of Theorem 2.2 to the semi-orthogonal decomposition

$\langle{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-2}\rangle,\langle\operatorname{\mathcal{F}}^{*}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-3},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{g-3}\rangle,,\ldots,\langle{\operatorname{\mathcal{F}}^{*\boxtimes
g-2}},{\operatorname{\mathcal{F}}^{*\boxtimes
g-3}},\ldots,\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$

of Theorem 3.1. The other two mega-blocks differ by line bundle twists. ∎

###### Proof of Lemma 2.4.

This is a reformulation of Theorem 3.2 (a). ∎

###### Proof of Lemma 2.7.

The first part is a reformulation of Theorem 3.2 (b). The second part follows
from Theorem 3.2 (c) and induction on $k$, since the weights of the vector
bundles $\operatorname{\mathcal{F}}^{*\boxtimes j}$ are in $[0,j]$ by [TT,
Section 2]. ∎

######  Notation 3.3.

In addition to $\operatorname{Sym}^{k}C$, we work on $C^{k}$. The
$S_{k}$-quotient

$\tau:\,C^{k}\times M_{i}\to\operatorname{Sym}^{k}C\times M_{i}$

commutes with arbitrary base change [TT, Section 2]. To avoid confusing
objects on $\operatorname{Sym}^{k}C\times M_{i}$ and $C^{k}\times M_{i}$, we
adorn the latter with the hat.For example,
$\hat{\operatorname{\mathcal{F}}}^{*\boxtimes
k}=\pi_{1}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\otimes\pi_{k}^{*}\operatorname{\mathcal{F}}^{*}$
and $\operatorname{\mathcal{F}}^{*\boxtimes
k}=\tau^{S_{k}}_{*}\left(\hat{\operatorname{\mathcal{F}}}^{{}^{*}\boxtimes
k}\right)$. Let

$\bar{\operatorname{\mathcal{F}}}^{\boxtimes
k}:=\tau^{S_{k}}_{*}\bigl{(}({\hat{\operatorname{\mathcal{F}}}}^{\boxtimes
k})\otimes\operatorname{sgn}\bigr{)},$

where $\operatorname{sgn}$ is the sign representation of $S_{k}$

######  Lemma 3.4.

We have the following formulas:

$\operatorname{\mathcal{F}}^{*\boxtimes k}\simeq(\Lambda^{*\boxtimes
k}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*k})\otimes\operatorname{\mathcal{F}}^{\boxtimes
k},$ $(\operatorname{\mathcal{F}}^{*\boxtimes
k})^{*}\simeq\bar{\operatorname{\mathcal{F}}}^{\boxtimes k}(\Delta/2),$

where
$\operatorname{\mathcal{O}}(-\Delta/2)=\tau_{*}^{S_{k}}(\operatorname{\mathcal{O}}\otimes\operatorname{sgn})$
is a line bundle on $\operatorname{Sym}^{k}C$ such that
$\operatorname{\mathcal{O}}(-\Delta/2)^{\otimes
2}\simeq\operatorname{\mathcal{O}}(-\Delta)$, where
$\Delta\subset\operatorname{Sym}^{k}C$ is the diagonal divisor.

###### Proof.

By [DN, Theorem 2.3], $\Lambda^{*\boxtimes k}$ is a descent of
$\hat{\Lambda}^{*\boxtimes k}$. So we have
$\operatorname{\mathcal{F}}^{*\boxtimes
k}=\tau^{S_{k}}_{*}\left((\hat{\Lambda}^{*\boxtimes
k}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*k})\otimes\hat{\operatorname{\mathcal{F}}}^{\boxtimes
k}\right)\simeq(\Lambda^{*\boxtimes
k}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*k})\otimes\tau^{S_{k}}_{*}\left(\hat{\operatorname{\mathcal{F}}}\right)^{\boxtimes
k}$ by projection formula. The latter expression is equal to
$(\Lambda^{*\boxtimes
k}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*k})\otimes\operatorname{\mathcal{F}}^{\boxtimes
k}$.

The morphism $\tau$ is ramified along $B=\tau^{-1}(\Delta)$ generically of
order $2$. So $\operatorname{\mathcal{O}}(B)$ is a relative dualizing sheaf
for $\tau$. The equivariant structure on $\operatorname{\mathcal{O}}(B)$ is
dual to the equivariant structure of the ideal sheaf
$\operatorname{\mathcal{O}}(-B)\subset\operatorname{\mathcal{O}}$. Since the
local equation of $B$ is anti-invariant,
$\operatorname{\mathcal{O}}(B)\simeq\tau^{*}\operatorname{\mathcal{O}}(\Delta/2)\otimes\operatorname{sgn}$.
By duality,

$(\operatorname{\mathcal{F}}^{*\boxtimes
k})^{*}\simeq\tau^{S_{k}}_{*}\left(\hat{\operatorname{\mathcal{F}}}^{\boxtimes
k}(B)\right)\simeq\tau^{S_{k}}_{*}\left(\hat{\operatorname{\mathcal{F}}}^{\boxtimes
k}\otimes\operatorname{sgn}\right)(\Delta/2)\simeq\bar{\operatorname{\mathcal{F}}}^{\boxtimes
k}(\Delta/2).$

This proves the lemma. ∎

######  Remark 3.5.

It is worth mentioning that $\operatorname{\mathcal{F}}^{\boxtimes k}$ is not
a descent of $\hat{\operatorname{\mathcal{F}}}^{\boxtimes k}$. Indeed, the
stabilizer of a general point $(p,p,\ldots)$ of the diagonal in $C^{k}$ does
not act trivially on the fiber
$\operatorname{\mathcal{F}}_{p}\otimes\operatorname{\mathcal{F}}_{p}\otimes\ldots$
of $\hat{\operatorname{\mathcal{F}}}^{\boxtimes k}$ at that point.

######  Corollary 3.6.

The Fourier–Mukai functors $D^{b}(\operatorname{Sym}^{k}C)\to D^{b}(M)$ with
kernels $(\operatorname{\mathcal{F}}^{*\boxtimes k})^{*}$ and
$(\bar{\operatorname{\mathcal{F}}}^{\boxtimes k})$ differ by an
autoequivalence of $D^{b}(\operatorname{Sym}^{k}C)$ given by tensoring with a
line bundle on $\operatorname{Sym}^{k}C$. In particular, we have the following
isomorphisms of evaluation vector bundles on $M$:

$(\operatorname{\mathcal{F}}_{D}^{*\boxtimes
k})^{*}\simeq(\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
k})\quad\hbox{\rm for every $D\in\operatorname{Sym}^{k}C$.}$

######  Notation 3.7.

To analyze loci $D_{i}^{k}$ inductively, we need moduli spaces
$M_{i}(\Lambda^{\prime})$ of stable pairs with determinant $\Lambda^{\prime}$
of degree $1\leq d\leq 2g-1$. We denote them by $M_{i}(d)$ when
$\Lambda^{\prime}$ is not important. We refer to [thaddeus, TT] for the
detailed treatment. As in degree $2g-1$ studied so far, there is a sequence of
flips followed by a divisorial contraction

$M_{\lfloor{d-1\over 2}\rfloor}(d)\dashrightarrow\ldots\dashrightarrow
M_{1}(d)\to M_{0}(d)=\operatorname{{\mathbb{P}}}^{d+g-2}.$

We may drop the subscript in $M_{\lfloor{d-1\over
2}\rfloor}(\Lambda^{\prime})$ and denote it simply by $M(\Lambda^{\prime})$.
The projection morphism $D_{i}^{k}\hookrightarrow\operatorname{Sym}^{k}C\times
M_{i}\to\operatorname{Sym}^{k}C$ is smooth, with the fiber over
$D\in\operatorname{Sym}^{k}C$ embedded in $M_{i}$ and isomorphic to
$M_{i-k}(\Lambda(-2D))$. For example, $D_{k}^{k}$ is a projective bundle over
$\operatorname{Sym}^{k}C$. The line bundle $\operatorname{\mathcal{O}}(m,n)$
on $M_{i}$ restricts to the line bundle $\operatorname{\mathcal{O}}(m,n-km)$
on $M_{i-k}(\Lambda(-2D))$, where we use the same notation for line bundles on
$M_{i}(d)$ as we did for $M_{i}$.

We use vanishing theorems for tensor vector bundles from [TT], which we state
here for ease of reference.

######  Theorem 3.8 ([TT, Theorem 7.1]).

Suppose $2<d\leq 2g+1$, and $1\leq j\leq\lfloor{d-1\over 2}\rfloor$. Let
$D\in\operatorname{Sym}^{a}C$ and $D^{\prime}\in\operatorname{Sym}^{b}C$ with
$a,\ b\leq d+g-2j-1$, and let $t$ be an integer satisfying

$a-j-1<t<d+g-2j-1-b.$

Suppose, further, that $t\notin[0,a]$. Then we have

(3.1) $R\Gamma_{M_{j}(d)}({{\bar{\operatorname{\mathcal{F}}}^{\boxtimes
a}_{D}}}{}^{*}\otimes\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t})=0$

and, if $D=\sum\alpha_{i}x_{i}$,

(3.2)
$R\Gamma_{M_{j}(d)}(\bigotimes_{i}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}^{\otimes\alpha_{i}}\otimes\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t})=0.$

######  Theorem 3.9 ([TT, Lemma 7.3, Theorem 7.4]).

Let $d>2$ and $1\leq j\leq\lfloor{d-1\over 2}\rfloor$. Let
$D\in\operatorname{Sym}^{a}C$ and $D^{\prime}\in\operatorname{Sym}^{b}C$, and
let $t$ be an integer satisfying

$a<t<d+g-2j-1-b.$

Then

(3.3) $R\Gamma_{M_{j}(d)}({\bar{\operatorname{\mathcal{F}}}^{\boxtimes
a}_{D}}{}^{*}\otimes{\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t})=R\Gamma_{M_{j}(d)}({\operatorname{\mathcal{F}}^{\boxtimes
a}_{D}}^{*}\otimes\operatorname{\mathcal{F}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t})=0.$

Moreover, the same vanishing holds for $d>0$ and $j=0$.

######  Theorem 3.10 ([TT, Theorem 9.6]).

Suppose $2<d\leq 2g+1$ and $1\leq j\leq\lfloor{d-1\over 2}\rfloor$. Let
$D\in\operatorname{Sym}^{a}C$ and $D^{\prime}\in\operatorname{Sym}^{b}C$ with
$a\leq j$, $b<d+g-2j-1$. Suppose $D\not\leq D^{\prime}$ (for example, $a>b$).
Then

(3.4) $R\Gamma_{M_{j}(d)}({\bar{\operatorname{\mathcal{F}}}^{\boxtimes
a}_{D}}{}^{*}\otimes{\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}})=0.$

###### Proof of Lemma 2.8.

Let $r={s\over t-k}={s^{\prime}\over
t-k^{\prime}}\not\in\operatorname{{\mathbb{Z}}}$ and $k^{\prime}>k$. Clearly,
$k^{\prime}\geq k+2$. Let $i={\lfloor t\rfloor}$. Then
$\operatorname{\mathcal{D}}^{k,s}_{t}=\operatorname{\mathcal{D}}^{k}_{i}\left({\lfloor
r\rfloor},s+{\lfloor r\rfloor}(k-1)\right)$ and
$\operatorname{\mathcal{D}}^{k^{\prime},s^{\prime}}_{t}=\operatorname{\mathcal{D}}^{k^{\prime}}_{i}\otimes
L$, where

$L=\begin{cases}\operatorname{\mathcal{O}}\left({\lfloor
r\rfloor},s^{\prime}+{\lfloor r\rfloor}(k^{\prime}-1)\right)&\hbox{\rm
if}\quad
k^{\prime}<i,\cr\operatorname{\mathcal{O}}(s^{\prime},s^{\prime}k^{\prime})&\hbox{\rm
if}\quad k^{\prime}=i.\cr\end{cases}$

We claim that
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k,s}_{t}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k^{\prime},s^{\prime}}_{t}}(X^{\prime}))=0$
for any $X\in D^{b}(\operatorname{Sym}^{k}C)$, $X^{\prime}\in
D^{b}(\operatorname{Sym}^{k^{\prime}}C)$. By Theorem 3.2 (c), it suffices to
show that

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}{\operatorname{\mathcal{F}}^{*\boxtimes
m}}}(X)\left({\lfloor r\rfloor},s+{\lfloor
r\rfloor}(k-1)\right),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}^{k^{\prime},s^{\prime}}_{t}}(X^{\prime}))=0,$

where $X\in D^{b}(\operatorname{Sym}^{m}C)$, $X^{\prime}\in
D^{b}(\operatorname{Sym}^{k^{\prime}}C)$ and $l+m\leq k$, $l,m\geq 0$.
Furthermore, we can assume that $X$ (resp., $X^{\prime}$) is the skyscraper
sheaf of a point $D\in\operatorname{Sym}^{m}C$ (resp.,
$D^{\prime}\in\operatorname{Sym}^{k^{\prime}}C$), and so we need to check that

$\operatorname{RHom}({\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}{\operatorname{\mathcal{F}}_{D}^{*\boxtimes
m}}\left({\lfloor r\rfloor},s+{\lfloor
r\rfloor}(k-1)\right),\operatorname{\mathcal{O}}_{M_{i-k^{\prime}}(\Lambda(-2D^{\prime}))}\otimes
L)=0.$

We first consider the case when $k^{\prime}=i$, so that
$M_{i-k^{\prime}}(\Lambda(-2D^{\prime}))=\operatorname{{\mathbb{P}}}^{3g-3-2k^{\prime}}$
and we need to show that

(3.5) $R\Gamma(\operatorname{{\mathbb{P}}}^{3g-3-2k^{\prime}},\
{\operatorname{{\boldsymbol{\Lambda}}}}^{l}({\operatorname{\mathcal{F}}_{D}^{*\boxtimes
m}})^{*}\operatorname{\mathcal{O}}(s^{\prime}-s-{\lfloor r\rfloor}k+{\lfloor
r\rfloor}k^{\prime}))=$
$R\Gamma(\operatorname{{\mathbb{P}}}^{3g-3-2k^{\prime}},\
{({\operatorname{\mathcal{F}}_{D}^{\boxtimes
m}})^{*}\operatorname{{\boldsymbol{\Lambda}}}}^{l+m-(s^{\prime}-s-{\lfloor
r\rfloor}k+{\lfloor r\rfloor}k^{\prime})})=0.$

But $s^{\prime}-s-{\lfloor r\rfloor}k+{\lfloor
r\rfloor}k^{\prime}=\\{r\\}(k-k^{\prime})<0$ since $r$ is not an integer. On
the other hand, $l+m+\\{r\\}(k^{\prime}-k)\leq k^{\prime}\leq
3g-3-2k^{\prime}$. So we have (3.5) by Theorem 3.9 with $j=0$,
$d=2g-1-2k^{\prime}$, $a=m$, $b=0$, $t=l+m+\\{r\\}(k^{\prime}-k)$.

Now suppose $k^{\prime}<i$. Arguing as above, we need to show that

$R\Gamma(M_{i-k^{\prime}}(\Lambda(-2D^{\prime})),\
{\operatorname{\mathcal{F}}_{D}^{\boxtimes
m}}^{*}{\operatorname{{\boldsymbol{\Lambda}}}}^{t})=0,$

where $t=l+m+\\{r\\}(k^{\prime}-k)$. This follows from Theorem 3.9 with
parameters $a=m$, $b=0$, $j=i-k^{\prime}$, $d=2g-1-2k^{\prime}$,
$t=l+m+\\{r\\}(k^{\prime}-k)$. ∎

###### Proof of Lemma 2.9.

We consider the following divisor in $D_{i}^{k}$:

$E_{i}^{k,1}=\\{(D,F,s)\in D_{i}^{k}\,|\,\deg Z(s)\geq k+1\\}.$

Let $\operatorname{\mathcal{E}}_{i}^{k,1}$ be its structure sheaf. Over
$D\in\operatorname{Sym}^{k}C$, the fiber of $D_{i}^{k}$ is
$M_{i-k}(\Lambda(-2D))$ and the fiber of $E_{i}^{k,1}$ is the divisor
$E\subset M_{i-k}(\Lambda(-2D))$. So we have a short exact sequence of sheaves
on $\operatorname{Sym}^{k}C\times M$

$0\to\operatorname{\mathcal{D}}_{i}^{k}(-1,-(k-1))\to\operatorname{\mathcal{D}}_{i}^{k}\to\operatorname{\mathcal{E}}_{i}^{k,1}\to
0$

and, for every $X\in D^{b}(\operatorname{Sym}^{k}C)$, an exact triangle in
$D^{b}(M_{i})$,

(3.6)
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X)(-1,-(k-1))\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X)\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{E}}_{i}^{k,1}}(X)\to.$

######  Claim 3.11.

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{E}}_{i}^{k,1}}(X)\in\langle\operatorname{\mathcal{D}}_{i}^{k+1},\ldots,\operatorname{\mathcal{D}}_{i}^{i}\rangle$.

We will prove Claim 3.11 by induction.

######  Definition 3.12.

For inductive purposes, we define

$D_{i}^{k_{1},\ldots,k_{r}}=\\{(D_{1},\ldots,D_{r},F,s)\ |\
s_{D_{1}+\ldots+D_{r}}=0\\}\subset\qquad\qquad\qquad\qquad\qquad\qquad$
$\qquad\qquad\qquad\qquad\qquad\qquad\subset\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r}}C\times
M_{i},$ $E_{i}^{k_{1},\ldots,k_{r}}=\\{(D_{1},\ldots,D_{r-1},F,s)\in
D_{i}^{k_{1},\ldots,k_{r-1}}\ |\ \deg Z(s)\geq k_{1}+\ldots+k_{r}\\}.$

Let $\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}$,
$\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r}}$ be the structure sheaves
of these loci.

The locus $D_{i}^{k_{1},\ldots,k_{r}}$ is smooth and

(3.7) $\nu:\,D_{i}^{k_{1},\ldots,k_{r}}\to
E_{i}^{k_{1},\ldots,k_{r}},\quad(D_{1},\ldots,D_{r},F,s)\mapsto(D_{1},\ldots,D_{r-1},F,s)$

is a normalization morphism. Furthermore, by [TT, Lemma 6.5] (applied
relatively over
$\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r-1}}C$),
we have a commutative diagram

(3.8)
${0}$${\nu_{*}\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}(-E_{i}^{k_{1},\ldots,k_{r},1})}$${\nu_{*}\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}}$${\nu_{*}\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r},1}}$${0}$${0}$${\mathcal{I}_{E_{i}^{k_{1},\ldots,k_{r-1},k_{r}+1}}}$${\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r}}}$${\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r-1},k_{r}+1}}$${0}$$\scriptstyle{\sim}$

where
$\mathcal{I}_{E_{i}^{k_{1},\ldots,k_{r-1},k_{r}+1}}\simeq\nu_{*}\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}(-E_{i}^{k_{1},\ldots,k_{r},1})$
is the conductor sheaf of the normalization (3.7) and
$E_{i}^{k_{1},\ldots,k_{r},1}$ (resp. $E_{i}^{k_{1},\ldots,k_{r-1},k_{r}+1}$)
is a conductor subscheme in $D_{i}^{k_{1},\ldots,k_{r}}$ (resp.
$E_{i}^{k_{1},\ldots,k_{r}}$).

Instead of Claim 3.11, we will prove a slightly stronger

######  Claim 3.13.

For every $X\in
D^{b}(\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r-1}}C)$,
we have

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r}}}(X)\in\langle\operatorname{\mathcal{D}}_{i}^{k_{1}+\ldots+k_{r}},\ldots,\operatorname{\mathcal{D}}_{i}^{i}\rangle.$

###### Proof of the claim.

We argue by downward induction on $k_{1}+\ldots+k_{r}$. By inductive
assumption and (3.8), it suffices to prove the same statement for the
Fourier–Mukai functors with kernels
$\nu_{*}\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}$ and
$\nu_{*}\operatorname{\mathcal{E}}_{i}^{k_{1},\ldots,k_{r},1}$. By projection
formula and inductive assumption, it suffices to prove that
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k_{1},\ldots,k_{r}}}(X)\in\langle\operatorname{\mathcal{D}}_{i}^{k_{1}+\ldots+k_{r}},\ldots,\operatorname{\mathcal{D}}_{i}^{i}\rangle$
for $X\in
D^{b}(\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r}}C)$.

In fact, consider projections
$\pi_{1}:\,D_{i}^{k_{1},\ldots,k_{r}}\to\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r}}C$
and $\pi_{2}:\,D_{i}^{k_{1},\ldots,k_{r}}\to M_{i}$. It suffices to show that
$R{\pi_{2}}_{*}(L\pi_{1}^{*}X)\in\langle\operatorname{\mathcal{D}}_{i}^{k_{1}+\ldots+k_{r}}\rangle$.
We have a Cartesian diagram

${D_{i}^{k_{1},\ldots,k_{r}}}$${D_{i}^{k_{1}+\ldots+k_{r}}}$${\operatorname{Sym}^{k_{1}}C\times\ldots\times\operatorname{Sym}^{k_{r}}C}$${\operatorname{Sym}^{k_{1}+\ldots+k_{r}}C}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\mu}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\mu}$

where $\mu$ is the addition map $(D_{1},\ldots,D_{r})\mapsto
D_{1}+\ldots+D_{r}$. Furthermore,
$R{\pi_{2}}_{*}(L\pi_{1}^{*}X)=R{\pi_{2}}_{*}(R\mu_{*}L\pi_{1}^{*}X)=R{\pi_{2}}_{*}(L\pi_{1}^{*}(R\mu_{*}X))$
by cohomology and base change. But the latter is contained in
$\langle\operatorname{\mathcal{D}}_{i}^{k_{1}+\ldots+k_{r}}\rangle$ by
definition. ∎

To finish the proof that
$\langle\operatorname{\mathcal{D}}_{i}^{k}(-1,-(k-1))\rangle$ is the mutation
of $\langle\operatorname{\mathcal{D}}_{i}^{k}\rangle$, we need to show that
$\langle\operatorname{\mathcal{D}}_{i}^{k}(-1,-(k-1))\rangle\subset{}^{\perp}\langle\operatorname{\mathcal{D}}_{i}^{k+1},\ldots,\operatorname{\mathcal{D}}_{i}^{i}\rangle$.
It suffices to show that, for $X\in D^{b}(\operatorname{Sym}^{k}C)$,
$X^{\prime}\in D^{b}(\operatorname{Sym}^{\alpha}C)$, $k<\alpha\leq i$, we have
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X)(-1,-(k-1)),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{\alpha}}(X^{\prime}))=0$.
By (3.6) and (3.11) (both applied to $k=\alpha$) and the downward induction on
$\alpha$, it suffices to prove that
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X)(-1,-(k-1)),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{\alpha}}(X^{\prime})(-1,-(\alpha-1)))=0$.

By Theorem 3.2 (c), we can prove instead that

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}{\operatorname{\mathcal{F}}^{*\boxtimes
m}}}(X)(-1,-(k-1)),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{\alpha}}(X^{\prime})(-1,-(\alpha-1)))=0$

for $l+m\leq k$, $l,m\geq 0$. Furthermore, we can assume that $X$ (resp.,
$X^{\prime}$) is a skyscraper sheaf of a point $D\in\operatorname{Sym}^{m}C$
(resp., $D^{\prime}\in\operatorname{Sym}^{\alpha}C$). This is equivalent to
proving that $R\Gamma(M_{i-\alpha}(\Lambda(-2D^{\prime})),\
{\operatorname{{\boldsymbol{\Lambda}}}}^{l}{(\operatorname{\mathcal{F}}_{D}^{*\boxtimes
m})^{*}}(0,k-i))=$$R\Gamma(M_{i-\alpha}(\Lambda(-2D^{\prime})),\
{{(\operatorname{\mathcal{F}}_{D}^{\boxtimes
m})^{*}}\operatorname{{\boldsymbol{\Lambda}}}}^{l+m+i-k})=0$, which follows
from Theorem 3.9 with $j=i-\alpha$, $d=2g-1-2\alpha$, $a=m$, $b=0$,
$t=l+m+i-k$: If $i-\alpha>0$, then $d\geq 2g-1-2(g-2)=3$. Since $i>k$ and
$l\geq 0$, we have $a=m<t$. Since $l+m\leq k$ and $i\leq g-1$, we have $t\leq
i<3g-2-2i=d+g-1-2j-b$. ∎

##  4\. Analysis of the basic Cross Warp mutation

The goal of this section is to prove Theorem 3.2. We continue to use notation
of the previous section and begin with establishing semi-orthogonality of
subcategories appearing in the basic Cross Warp mutation (see Figure 7) given
by Fourier–Mukai functors with the kernels given by vector bundles.

######  Lemma 4.1.

For $1\leq k\leq i$, $D^{b}(M_{i})$ admits admissible subcategories

$\langle{\operatorname{\mathcal{F}}^{*\boxtimes
k-1}},\ldots,{\operatorname{\mathcal{F}}}^{*},\operatorname{\mathcal{O}},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}\rangle\quad\hbox{\rm
and}\quad\langle{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k},{\operatorname{\mathcal{F}}^{*\boxtimes
k}},{\operatorname{\mathcal{F}}^{*\boxtimes
k-1}},\ldots,{\operatorname{\mathcal{F}}}^{*},\operatorname{\mathcal{O}}\rangle.$

###### Proof.

Fourier–Mukai functors
$\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes k}}}$are
fully faithful [TT, Theorem 9.2]. Semi-orthogonality follows from the
vanishing theorems in Section 3.We have
$R\Gamma({\operatorname{\mathcal{F}}_{D}^{*\boxtimes
a}}\otimes({\operatorname{\mathcal{F}}_{D^{\prime}}^{*\boxtimes
b}})^{*})=R\Gamma(\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
a*}\otimes{\bar{\operatorname{\mathcal{F}}}_{D^{\prime}}^{\boxtimes b}})=0$
for $b<a<k$ by Theorem 3.10. Next,
$R\Gamma({\operatorname{\mathcal{F}}_{D}^{*\boxtimes
l}}^{*}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k})=R\Gamma({\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
l}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}}^{-k})=0$ for $l<k$ by
Theorem 3.8 with $j=i$, $d=2g-1$, $a=0$, $b=l$, $t=-k$, as we have
$0,l<i<3g-2-2i$ and $-i-1<-k<-l<3g-2-2i-l$. Finally,
$R\Gamma({\operatorname{\mathcal{F}}_{D}^{*\boxtimes
l}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}}^{k})=0$ by Theorem 3.9 with
$j=i$, $d=2g-1$, $a=l$, $b=0$, $t=k$, as $l<k<i<3g-2-2i$. ∎

We prove Theorem 3.2 by induction on $k$. We can start with an obvious case
$k=0$, but, to introduce notation and a few ideas, we begin with $k=1$.

The universal section $\Sigma$ of the universal stable pair
$(\operatorname{\mathcal{F}},\Sigma)$ on $C\times M_{i}$ vanishes on the locus
$D_{i}^{1}$, which has codimension $2$. It follows that we have an exact
Koszul complex on $C\times M_{i}$,

(4.1)
$0\to\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}\to\operatorname{\mathcal{F}}^{*}\mathop{\longrightarrow}\limits^{\Sigma}\operatorname{\mathcal{O}}\to\operatorname{\mathcal{D}}_{i}^{1}\to
0.$

######  Definition 4.2.

We define the complex
$\operatorname{\mathcal{F}}^{\bullet}=[\operatorname{\mathcal{F}}^{*}\mathop{\longrightarrow}\limits^{\Sigma}\operatorname{\mathcal{O}}]$
in $D^{b}(C\times M_{i})$, normalized so that $\operatorname{\mathcal{O}}$ is
placed in cohomological degree $0$.

By exactness of (4.1), we compute

(4.2)
$\operatorname{\mathcal{H}}^{0}(\operatorname{\mathcal{F}}^{\bullet})=\operatorname{\mathcal{D}}_{i}^{1}\quad\hbox{\rm
and}\quad\operatorname{\mathcal{H}}^{-1}(\operatorname{\mathcal{F}}^{\bullet})=\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}.$

This gives two exact triangles in $D^{b}(C\times M_{i})$,

(4.3)
$\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}[1]\to\operatorname{\mathcal{F}}^{\bullet}\to\operatorname{\mathcal{D}}_{i}^{1}\to\qquad\hbox{\rm
and}\qquad\operatorname{\mathcal{O}}\to\operatorname{\mathcal{F}}^{\bullet}\to\operatorname{\mathcal{F}}^{*}[1]\to.$

Applying the Fourier–Mukai functors to any $X\in D^{b}(C)$ gives exact
triangles

(4.4)
$\operatorname{\mathcal{P}}_{\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(X)[1]\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X)\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(X)\to$

and

(4.5)
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}}(X)\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X)\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(X)[1]\to.$

######  Lemma 4.3.

The Fourier–Mukai functor
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}$ is fully
faithful. Furthermore, there is a mutation of admissible subcategories
$\langle\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle\to\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{F}}^{\bullet}\rangle$.

###### Proof.

We first claim that

(4.6)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}}(Y))=0\quad\hbox{\rm
for all}\quad X,Y\in D^{b}(C).$

By (4.4), it suffices to demonstrate that

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(X),\operatorname{\mathcal{O}})=\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(X),\operatorname{\mathcal{O}})=0.$

For brevity, in the following formulas we omit the shift in the Serre duality:

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(X),\operatorname{\mathcal{O}})^{*}[\ldots]=R\Gamma(M_{i},\operatorname{\mathcal{P}}_{\Lambda^{*}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(X)\otimes\omega_{M_{i}})=$
$R\Gamma(C\times
M_{i},(X\otimes\Lambda^{*})\boxtimes(\operatorname{{\boldsymbol{\Lambda}}}^{*}\otimes\omega_{M_{i}}))=R\Gamma(C,X\otimes\Lambda^{*}\otimes^{L}{R\pi_{1}}_{*}(\operatorname{{\boldsymbol{\Lambda}}}^{*}\otimes\omega_{M_{i}}))=0.$

Indeed, by cohomology and base change it suffices to demonstrate that
$R\Gamma(M_{i},\operatorname{{\boldsymbol{\Lambda}}}^{*}\otimes\omega_{M_{i}})=0$,
but this follows from Serre duality and Theorem 3.9 applied with $j=i$,
$d=2g-1$, $t=1$, $a=b=0$, as $0<1<3g-2-2i$. Likewise, we have

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(X),\operatorname{\mathcal{O}})^{*}[\ldots]=R\Gamma(M_{i},\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(X)\otimes\omega_{M_{i}})=$
$R\Gamma(C\times
M_{i},X\otimes^{L}\operatorname{\mathcal{D}}_{i}^{1}\otimes\omega_{M_{i}})=R\Gamma(C,X\otimes^{L}R{\pi_{1}}_{*}(\operatorname{\mathcal{D}}_{i}^{1}\otimes\omega_{M_{i}}))=0.$

Indeed, by cohomology and base change it suffices to prove that, for $p\in C$,

$R\Gamma(M_{i-1}(\Lambda(-2p)),\omega_{M_{i}}|_{M_{i-1}(\Lambda(-2p))})=0.$

Since $\omega_{M_{i}(d)}=\operatorname{\mathcal{O}}(-3,-(d+g-4))$, see [TT,
Section 3], the latter is equal to
$R\Gamma(M_{i-1}(\Lambda(-2p)),\omega_{M_{i-1}(\Lambda(-2p))}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*})$,
which is equal to $0$ by Serre duality and Theorem 3.9 with $j=i-1$, $d=2g-3$,
$t=1$, $a=b=0$.

The formula (4.6) follows. We also have
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(Y))=0$
by Lemma 4.1. From (4.5), it follows that

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(Y))=\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(Y)[1])$
$=\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(Y))=\operatorname{RHom}(X,Y)$

by Lemma 4.1, which shows that
$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}$ is fully
faithful. ∎

######  Lemma 4.4.

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}$ is fully
faithful. There is a mutation
$\langle\operatorname{{\boldsymbol{\Lambda}}}^{*},\operatorname{\mathcal{F}}^{\bullet}\rangle\to\langle\operatorname{\mathcal{D}}_{i}^{1},\operatorname{{\boldsymbol{\Lambda}}}^{*}\rangle$.

###### Proof.

Note that we, obviously, have
$\langle\operatorname{{\boldsymbol{\Lambda}}}^{*}\rangle=\langle\Lambda^{*}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*}\rangle$.
We first claim that

(4.7)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\Lambda^{*}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(Y))=0\quad\hbox{\rm
for all}\quad X,Y\in D^{b}(C).$

By (4.5), this follows from

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}}(X),\operatorname{{\boldsymbol{\Lambda}}}^{*})=R\Gamma(C,X)\otimes
R\Gamma(M_{i},\operatorname{{\boldsymbol{\Lambda}}})=0$

(Theorem 3.9 with $j=i$, $d=2g-1$, $t=1$, $a=b=0$) and

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(X),\operatorname{{\boldsymbol{\Lambda}}}^{*})^{*}[\ldots]=\operatorname{RHom}(\operatorname{{\boldsymbol{\Lambda}}}^{*},\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{*}}(X)\otimes\omega_{M_{i}})=$
$R\Gamma(C\times
M_{i},(X\boxtimes(\operatorname{{\boldsymbol{\Lambda}}}\otimes\omega_{M_{i}}))\otimes\operatorname{\mathcal{F}}^{*})=R\Gamma(C,X\otimes^{L}{R\pi_{1}}_{*}(\operatorname{\mathcal{F}}^{*}\otimes\operatorname{{\boldsymbol{\Lambda}}}\otimes\omega_{M_{i}}))=0$

by cohomology and base change and Serre duality. Indeed, for any $p\in C$,
$R\Gamma(M_{i},\operatorname{\mathcal{F}}_{p}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*})=0$:
since
$\operatorname{\mathcal{F}}_{p}\simeq\bar{\operatorname{\mathcal{F}}}_{p}$ and
$-i-1<-1<3g-3-2i$, this follows from Theorem 3.8 with $j=i$, $d=2g-1$, $t=-1$,
$a=0$, $b=1$. In addition to (4.7), we also have

(4.8)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\Lambda^{*}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(Y))=0\quad\hbox{\rm
for all}\quad X,Y\in D^{b}(C).$

Indeed, it suffice to show that
$\operatorname{RHom}(\operatorname{{\boldsymbol{\Lambda}}}^{*},\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(Y))=0$.
But this is equal to

$R\Gamma(M_{i},\operatorname{{\boldsymbol{\Lambda}}}\otimes\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(Y))=R\Gamma(M_{i},\operatorname{{\boldsymbol{\Lambda}}}\otimes{R\pi_{2}}_{*}(L\pi_{1}^{*}(Y)\otimes^{L}\operatorname{\mathcal{D}}_{i}^{1}))=$
$R\Gamma(C\times
M_{i},(\pi_{1}^{*}(Y)\boxtimes\operatorname{{\boldsymbol{\Lambda}}})\otimes^{L}\operatorname{\mathcal{D}}_{i}^{1})=R\Gamma(C,Y\otimes^{L}{R\pi_{1}}_{*}(\operatorname{\mathcal{D}}_{i}^{1}\otimes\operatorname{{\boldsymbol{\Lambda}}}))=0$

by cohomology and base change. Indeed,
$R\Gamma(M_{i-1}(\Lambda(-2p)),\operatorname{{\boldsymbol{\Lambda}}})=0$ for
any $p\in C$ by Theorem 3.9 with $j=i-1$, $d=2g-3$, $t=1$, $a=b=0$.

Using (4.4), (4.7) and (4.8), we compute

$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(Y))=\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{1}}(Y))=$
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{F}}^{\bullet}}(Y))=\operatorname{RHom}(X,Y)$

by Lemma 4.3. This proves the lemma. ∎

###### Proof of Theorem 3.2 for $k=1$.

Part (a) was proved in Lemma 4.4. Furthermore, part (c) follows from Lemmas
4.1, 4.3, 4.4. Finally, part (b) follows from part (c). Indeed, we can perform
the basic Cross Warp mutation
$\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{D}}_{i}^{1},\Lambda^{*}\rangle\to\langle\Lambda^{*},\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$
in $D^{b}(M_{i-1})$, then embed
$\langle\Lambda^{*},\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$
by the windows embedding into itself [TT, Section 3] in $D^{b}(M_{i})$, then
undo the basic Cross Warp mutation
$\langle\Lambda^{*},\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle\to\langle\operatorname{\mathcal{O}},\operatorname{\mathcal{D}}_{i}^{1},\Lambda^{*}\rangle$
in $D^{b}(M_{i})$. ∎

To scale up our method to handle $k>1$, we introduce further notation.

######  Definition 4.5.

We define

$\hat{D}_{i}^{k}=\\{(p_{1},\ldots,p_{k},F,s)\,|\,s_{p_{1}+\ldots+p_{k}}=0\\}\subset
C^{k}\times M_{i}$

and let $\hat{\operatorname{\mathcal{D}}}_{i}^{k}$ be its structure sheaf.
This is the same as $D_{i}^{1,\ldots,1}$ and
$\operatorname{\mathcal{D}}_{i}^{1,\ldots,1}$ (with $k$ ones in the
superscript) in the notation of Definition 3.12.

Like $D_{i}^{k}=\tau(\hat{D}_{i}^{k})$, the scheme $\hat{D}_{i}^{k}$ is
regular, of codimension $2k$. It is the main component of the intersection
scheme $\pi_{1}^{-1}D_{i}^{1}\cap\ldots\cap\pi_{k}^{-1}D_{i}^{1}$, which
contains other irreducible components, of smaller codimension. So, while the
intersection scheme is the zero locus of the section
$(\pi_{1}^{*}\Sigma,\ldots\pi_{k}^{*}\Sigma)$ of the tautological bundle
$\pi_{1}^{*}\operatorname{\mathcal{F}}\oplus\ldots\oplus\pi_{k}^{*}\operatorname{\mathcal{F}}$,
its Koszul complex is not exact. We will analyze cohomology sheaves of a
related but simpler complex.

######  Definition 4.6.

We define

${\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}=\tau_{*}^{S_{k}}\left({\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}}\otimes\operatorname{sgn}\right),\quad\hbox{\rm
where}\quad{\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}}=L\pi_{1}^{*}\operatorname{\mathcal{F}}^{\bullet}\otimes^{L}\ldots\otimes^{L}L\pi_{k}^{*}\operatorname{\mathcal{F}}^{\bullet}.$

Recall that the symmetric group acts on
${\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes k}}$ in such a way that
an adjacent transposition $(t,t+1)$, in addition to permuting the
corresponding factors of a homogeneous tensor $g_{1}\otimes\ldots\otimes
g_{k}$, also multiplies it by $(-1)^{\deg(g_{t})\deg(g_{t+1})}$.

The following Lemmas 4.7, 4.9, and 4.12 will be proved along with Theorem 3.2,
by the same induction on $k$. In their proofs, we assume that they hold for
smaller values of $k$. In Lemma 4.7, we analyze naive truncations of the
complex ${\operatorname{\mathcal{F}}^{\bullet\boxtimes k}}$, while in Lemma
4.9, we study its cohomology sheaves.

######  Lemma 4.7.

We have the following exact triangle in $D^{b}(\operatorname{Sym}^{k}C\times
M_{i})$:

(4.9) $G_{k}\to{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}\to{\operatorname{\mathcal{F}}^{*\boxtimes k}}[k]\to$

with
$\operatorname{\mathcal{P}}_{G_{k}}(X)\in\langle{\operatorname{\mathcal{F}}^{*\boxtimes
k-1}},\ldots,\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$
for every $X\in D^{b}(\operatorname{Sym}^{k}C)$.

###### Proof.

Since the action of $S_{k}$ on
${\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes k}}$ is signed, the term
of the complex ${\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}=\tau^{S_{k}}_{*}({\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}}\otimes\operatorname{sgn})$ in degree $-k$ is given by
$\tau^{S_{k}}_{*}\bigl{(}({\hat{\operatorname{\mathcal{F}}}}^{*\boxtimes
k})\otimes\operatorname{sgn}\otimes\operatorname{sgn}\bigr{)}=\tau^{S_{k}}_{*}\bigl{(}({\hat{\operatorname{\mathcal{F}}}}^{*\boxtimes
k})\bigr{)}=(\operatorname{\mathcal{F}}^{*\boxtimes k})$, which gives the
truncation triangle (4.9).

######  Claim 4.8.

For $l<k$, the degree $-l$ term of the complex
${\operatorname{\mathcal{F}}^{\bullet\boxtimes k}}$ is given by

${R(\pi_{\operatorname{Sym}^{k}C\times
M_{i}}})_{*}\left(L\pi_{\operatorname{Sym}^{k}C\times\operatorname{Sym}^{l}C}^{*}\operatorname{\mathcal{O}}_{W}(-B/2)\otimes
L\pi_{\operatorname{Sym}^{l}C\times
M_{i}}^{*}{\operatorname{\mathcal{F}}^{*\boxtimes l}}\right),$

where projections are from the triple product
$\operatorname{Sym}^{k}C\times\operatorname{Sym}^{l}C\times M_{i}$,$W$ is the
correspondence $\\{(D,D^{\prime})\,|\,D\geq
D^{\prime}\\}\subset\operatorname{Sym}^{k}C\times\operatorname{Sym}^{l}C$, and
$B$ is the pullback of the diagonal in $\operatorname{Sym}^{k-l}C$ with
respect to the subtraction map $W\to\operatorname{Sym}^{k-l}C$,
$(D,D^{\prime})\mapsto D-D^{\prime}$.

###### Proof of the claim.

The degree $-l$ term of ${\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}}\otimes\operatorname{sgn}$ is given by

$K=\bigoplus\limits_{1\leq i_{1}<\ldots<i_{l}\leq
k}\pi_{i_{1}}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\pi_{i_{l}}^{*}\operatorname{\mathcal{F}}^{*}.$

The action of the symmetric group $S_{k}$ is induced from the action of
$S_{l}\times S_{k-l}$ on
$\pi_{i_{1}}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\pi_{i_{l}}^{*}\operatorname{\mathcal{F}}^{*}$,
where $S_{l}$ acts by permuting tensor factors (and the corresponding factors
of $C^{k}$), while $S_{k-l}$ acts by permuting the remaining factors of
$C^{k}$ and tensored with the sign representation. Consider the triple product
$C^{k}\times C^{l}\times M_{i}$, where, in addition to the action of the
symmetric group $S_{k}$ on $C^{k}$, we also have the action of $S_{l}$ on
$C^{l}$. For a sequence $i_{1},\ldots,i_{l}$ of different (but not necessarily
increasing) elements of the set $\\{1,\ldots,k\\}$, let
$\operatorname{\mathcal{O}}_{i_{1},\ldots,i_{l}}$ be the structure sheaf of
the graph of the morphism $C^{k}\times M_{i}\to C^{l}$ that sends
$(p_{1},\ldots,p_{k},F,s)$ to $(p_{i_{1}},\ldots,p_{i_{l}})$. Then

$K=R({\pi_{C^{k}\times
M_{i}}})^{S_{l}}_{*}\Bigl{(}\bigoplus_{i_{1},\ldots,i_{l}}\operatorname{\mathcal{O}}_{i_{1},\ldots,i_{l}}\otimes
L\pi^{*}_{{C^{l}\times
M_{i}}}\Bigl{(}\pi_{1}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\otimes\pi_{l}^{*}\operatorname{\mathcal{F}}^{*}\Bigr{)}\Bigr{)}.$

Here
$\pi_{1}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\pi_{l}^{*}\operatorname{\mathcal{F}}^{*}$
is a sheaf on $C^{l}\times M_{i}$, $S_{l}$ acts on it by permuting factors of
the tensor product (and $C^{l}$), while the action of $S_{k}$ on the sheaf
$\bigoplus_{i_{1},\ldots,i_{l}}\operatorname{\mathcal{O}}_{i_{1},\ldots,i_{l}}$
is induced from the action of $S_{k-l}$ on
$\operatorname{\mathcal{O}}_{i_{1},\ldots,i_{l}}$ tensored with the sign
representation. Interchanging commuting $\tau_{*}^{S_{k}}$ and
$R{\pi^{S_{l}}_{13}}_{*}$ gives

$\tau_{*}^{S_{k}}K=R({\pi_{\operatorname{Sym}^{k}C\times
M_{i}}})^{S_{l}}_{*}\left(\operatorname{\mathcal{O}}_{\hat{W}}(-\hat{B}/2)\otimes
L\pi^{*}_{{C^{l}\times
M_{i}}}\left(\pi_{1}^{*}\operatorname{\mathcal{F}}^{*}\otimes\ldots\otimes\pi_{l}^{*}\operatorname{\mathcal{F}}^{*}\right)\right),$

where $\hat{W}=\\{(D,p_{1},\ldots,p_{l})\,|\,D\geq
p_{1}+\ldots+p_{l}\\}\subset\operatorname{Sym}^{k}C\times C^{l}$ and $\hat{B}$
is the pullback of the diagonal in $\operatorname{Sym}^{k-l}C$ with respect to
the subtraction map $\hat{W}\to\operatorname{Sym}^{k-l}C$,
$(D,p_{1},\ldots,p_{l})\mapsto D-p_{1}-\ldots-p_{l}$. The projection
$\pi_{\operatorname{Sym}^{k}C\times M_{i}}$ is the composition of the
$S_{l}$-quotient morphism $\operatorname{Sym}^{k}C\times C^{l}\times
M_{i}\to\operatorname{Sym}^{k}C\times\operatorname{Sym}^{l}C\times M_{i}$ and
the projection $\pi_{\operatorname{Sym}^{k}C\times M_{i}}$. This gives
$\tau_{*}^{S_{k}}K=R{\pi_{\operatorname{Sym}^{k}C\times
M_{i}}}_{*}\left(\operatorname{\mathcal{O}}_{W}(-B/2)\otimes{\operatorname{\mathcal{F}}^{*\boxtimes
l}}\right),$ as claimed. ∎

By Claim 4.8, the Fourier–Mukai functor with the kernel given by the degree
$-l$ term of the complex ${\operatorname{\mathcal{F}}^{\bullet\boxtimes k}}$
is the composition of the Fourier–Mukai functors
$\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
l}}}\circ\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}_{W}(-B/2)}$.
So its image belongs to the subcategory
$\langle{\operatorname{\mathcal{F}}^{*\boxtimes l}}\rangle$. It follows that
$\operatorname{\mathcal{P}}_{G_{k}}(X)\in\langle{\operatorname{\mathcal{F}}^{*\boxtimes
k-1}},\ldots,\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$
for $X\in D^{b}(\operatorname{Sym}^{k}C)$. ∎

######  Lemma 4.9.

The Fourier–Mukai functor $D^{b}(\operatorname{Sym}^{k}C)\to D^{b}(M_{i})$
with the kernel
$\operatorname{\mathcal{H}}^{-l}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})$ is a composition of the Fourier–Mukai functor

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{O}}_{W}(-B/2)\otimes(\Lambda^{*\boxtimes
k}\boxtimes\Lambda^{\boxtimes k-l})}:\,D^{b}(\operatorname{Sym}^{k}C)\to
D^{b}(\operatorname{Sym}^{k-l}C)$

and the Fourier–Mukai functor

$\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{D}}^{k-l}_{i}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}}:\,D^{b}(\operatorname{Sym}^{k-l}C)\to
D^{b}(M_{i}).$

Here $W=\\{(D,D^{\prime})\,|\,D\geq
D^{\prime}\\}\subset\operatorname{Sym}^{k}C\times\operatorname{Sym}^{k-l}C$
and $B\subset\operatorname{Sym}^{k-l}C$ is the diagonal.

###### Proof.

The complex $\operatorname{\mathcal{F}}^{\bullet\boxtimes k}$ is the direct
summand of $S_{k}$-anti-invariants of the complex
$\tau_{*}\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes k}$, where
$\tau:\,C^{k}\times M_{i}\to\operatorname{Sym}^{k}C\times M_{i}$ is the
$S_{k}$-quotient morphism. Via the $S_{k}$-equivariant projection
$\tau_{*}\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}\to\operatorname{\mathcal{F}}^{\bullet\boxtimes k}$, the complex
$\operatorname{\mathcal{F}}^{\bullet\boxtimes k}$ is also isomorphic to the
complex of $S_{k}$-anti-coinvariants, namely the quotient complex of
$\tau_{*}\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes k}$ by a
subcomplex generated by local sections of the form
$a+\operatorname{sgn}(\sigma)\sigma(a)$ for $\sigma\in S_{k}$. Since $\tau$ is
a finite morphism, the same properties hold for the cohomology sheaf
$\operatorname{\mathcal{H}}^{-l}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})$: it is the direct summand of $S_{k}$-anti-invariants of the sheaf
$\tau_{*}\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$ and it is canonically isomorphic to the quotient sheaf of $S_{k}$-anti-
coinvariants of
$\tau_{*}\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$.

To prove Lemma 4.9, we are going to prove that

$\operatorname{\mathcal{H}}^{-l}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})\simeq\tau_{*}^{S_{k}}\Bigl{(}\bigoplus_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\Bigl{(}\pi^{*}_{L}\hat{\operatorname{\mathcal{D}}}^{k-l}_{i}\otimes\bigotimes_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\otimes\operatorname{sgn}^{\prime}\Bigr{)}\Bigr{)}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l},$

where the action of $S_{k}$ is induced from the permutation action of
$S_{l}\times S_{k-l}$ and $\operatorname{sgn}^{\prime}$ is the sign
representation of $S_{k-l}$.

We write
$\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})=\operatorname{Ker}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})/\operatorname{Im}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$. By (4.2), the degree $-l$ component of
$\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes k}$, the sheaf
$\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\operatorname{\mathcal{F}}^{*}$, contains a subsheaf
$\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}$,
which in fact is contained in the sheaf
$\operatorname{Ker}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$. This induces an $S_{k}$-equivariant injection

$\phi:\,\operatorname{\mathcal{B}}^{-l}:=\bigoplus_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\bigotimes_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\Big{/}\phi^{-1}(\operatorname{Im}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}))\hookrightarrow\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}).$

On the other hand, also by (4.2), we have an $S_{k}$-equivariant surjection

(4.10)
$\psi:\,\operatorname{\mathcal{B}}^{-l}\twoheadrightarrow\operatorname{\mathcal{C}}^{-l}:=\bigoplus_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\pi^{*}_{L}\hat{\operatorname{\mathcal{D}}}^{k-l}_{i}\otimes\bigotimes_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}.$

In other words, $\operatorname{\mathcal{C}}^{-l}$ is a quotient-sub-sheaf of
$\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$.

The plan is to show that both morphisms $\phi$ and $\psi$ become isomorphisms
after applying the functor $\tau^{S_{k}}(\bullet\otimes\operatorname{sgn})$,
where the action of $S_{k}$ is induced from the permutation action of
$S_{l}\times S_{k-l}$ tensored with the sign representation of $S_{k-l}$. In
addition to our claim about cohomology sheaves
$\operatorname{\mathcal{H}}^{-l}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})$ of the complex $\operatorname{\mathcal{F}}^{\bullet\boxtimes k}$, this
claim also proves the following statement about the kernel of its
differential, which will be used in the last section:

######  Corollary 4.10.

We have

$\operatorname{Ker}^{-l}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})\simeq\tau_{*}^{S_{k}}\Bigl{(}\bigoplus_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l}\Bigl{(}\bigotimes_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\otimes\operatorname{sgn}^{\prime}\Bigr{)}\Bigr{)}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}.$

In particular, the image of the Fourier-Mukai functor
$D^{b}(\operatorname{Sym}^{k}C)\to D^{b}(M)$ with this kernel belongs to the
subcategory $\langle{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\rangle$.

To proceed with the plan, we notice that, for every
$\alpha\in\\{1,\ldots,k\\}$, the $2$-step decreasing filtration on the
$2$-term complex $\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{\bullet}$
induces a $2$-step decreasing filtration on
$\hat{\operatorname{\mathcal{F}}}^{\bullet k}$ and
$\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})$. These filtrations are permuted by the action of $S_{k}$ and are
compatible with the morphisms $\phi$ and $\psi$. We have injections
$G_{\alpha}^{p}(\operatorname{\mathcal{B}}^{-l})\hookrightarrow
G_{\alpha}^{p}(\operatorname{\mathcal{H}}^{-l})$ and surjections
$G_{\alpha}^{p}(\operatorname{\mathcal{B}}^{-l})\twoheadrightarrow
G_{\alpha}^{p}(\operatorname{\mathcal{C}}^{-l})$ for $p=0,-1$, where we denote
associated graded sheaves by $G_{\alpha}^{0}$ and $G_{\alpha}^{-1}$.
Furthermore, $\operatorname{\mathcal{C}}^{-l}$ is a direct sum
$G_{\alpha}^{0}(\operatorname{\mathcal{C}}^{-l})\oplus
G_{\alpha}^{-1}(\operatorname{\mathcal{C}}^{-l})$. Namely, the terms in
$G_{\alpha}^{0}(\operatorname{\mathcal{C}}^{-l})$ are direct summands in
(4.10) with $\alpha\in L$, while the terms in
$G_{\alpha}^{-1}(\operatorname{\mathcal{C}}^{-l})$ are the ones with
$\alpha\not\in L$.

We first apply the functor
$\tau_{*}^{S_{k-1}}(\bullet\otimes\operatorname{sgn})$, where $S_{k-1}$ is the
stabilizer of $\alpha\in\\{1,\ldots,k\\}$. We can decompose $\tau$ as follows:

$C^{k}\times
M_{i}\mathop{\longrightarrow}\limits^{\eta}\operatorname{Sym}^{k-1}C\times
C\times
M_{i}\mathop{\longrightarrow}\limits^{\rho}\operatorname{Sym}^{k}C\times
M_{i},$

where $\eta$ is the quotient by $S_{k-1}$-action. The complex
$\tau_{*}^{S_{k-1}}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k}\otimes\operatorname{sgn})$ is equal to
$\rho_{*}\left(\eta_{*}^{S_{k-1}}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k-1}\otimes\operatorname{sgn})\otimes\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{\bullet}\right)=\rho_{*}\left(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k-1}\otimes\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{\bullet}\right)$. Thus,

$\tau_{*}^{S_{k-1}}\left(\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})\otimes\operatorname{sgn}\right)=\rho_{*}\operatorname{\mathcal{H}}^{-l}\left(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k-1}\otimes\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{\bullet}\right).$

The associated graded components of
$\tau_{*}^{S_{k-1}}\left(\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})\otimes\operatorname{sgn}\right)$ can be computed using the $2$-row
spectral sequence. By the inductive assumption on cohomology of
$\operatorname{\mathcal{F}}^{\bullet\boxtimes k-1}$, the $E_{1}$ term has the
$(-l)$-th column given by

$\tau_{*}^{S_{k-1}}\Bigl{(}\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l,\
\alpha\in L}K_{L}\otimes\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\otimes\operatorname{sgn}^{\prime}\Bigr{)},$

where $K_{L}$ is the complex
$\Bigl{[}\pi^{*}_{L\setminus\alpha}\hat{\operatorname{\mathcal{D}}}^{k-1-l}_{i}\otimes\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{*}\mathop{\longrightarrow}\limits^{\pi_{\alpha}^{*}\Sigma}\pi^{*}_{L\setminus\alpha}\hat{\operatorname{\mathcal{D}}}^{k-1-l}_{i}\Bigr{]}$.
We consider the locus $Y_{L}=\pi_{L\setminus\alpha}^{-1}(\hat{D}^{k-1-l}_{i})$
and rewrite this complex as

$K_{L}=\left[\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{*}|_{Y_{L}}\mathop{\longrightarrow}\limits^{\pi_{\alpha}^{*}\Sigma}\operatorname{\mathcal{O}}_{Y_{L}}\right].$

######  Claim 4.11.

We have the following isomorphisms:

$\operatorname{\mathcal{H}}^{-1}(K_{L})\simeq\pi_{\alpha}^{*}\Lambda^{*}|_{Y_{L}}(\Delta_{L})\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*},\quad\operatorname{\mathcal{H}}^{0}(K_{L})\simeq\operatorname{\mathcal{O}}_{\Delta_{L}\cup
W_{L}},\quad\hbox{\rm where}$ $\Delta_{L}=\\{(p_{1},\ldots,p_{k},F,s)\in
Y_{L}\ |\ p_{j}=p_{\alpha}\quad\hbox{\rm for some}\quad j\in
L\setminus\\{\alpha\\}\\}$

is a divisor in $Y_{L}$ and $W_{L}=\pi_{L}^{-1}(\hat{D}^{k-l}_{i})$ has
codimension $2$ in $Y_{L}$. The loci $\Delta_{L}$ and $W_{L}$ intersect
transversally in $Y_{L}$.

###### Proof of the claim.

The section $\pi_{\alpha}^{*}\Sigma$ of the vector bundle
$\pi_{\alpha}^{*}\operatorname{\mathcal{F}}|_{Y_{L}}$ vanishes along the union
of subvarieties $\Delta_{L}\cup W_{L}\subset Y_{L}$, which intersect
transversally. It follows that the induced section of
$\pi_{\alpha}^{*}\operatorname{\mathcal{F}}|_{Y_{L}}(-\Delta_{L})$ vanishes
along $W_{L}$ and that the Koszul complex
$\Bigl{[}\pi_{\alpha}^{*}\Lambda^{*}(2\Delta_{L})|_{Y_{L}}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}\to\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{*}|_{Y_{L}}(\Delta_{L})\to\operatorname{\mathcal{O}}_{Y_{L}}\Bigr{]}$
resolves $\operatorname{\mathcal{O}}_{W_{L}}$. We twist the Koszul complex by
$\operatorname{\mathcal{O}}_{Y_{L}}(-\Delta_{L})$ and truncate it to the
complex
$K_{L}^{\prime}=\Bigl{[}\pi_{\alpha}^{*}\operatorname{\mathcal{F}}^{*}|_{Y_{L}}\mathop{\longrightarrow}\limits\operatorname{\mathcal{O}}_{Y_{L}}(-\Delta_{L})\Bigr{]}$,
which is a subcomplex of $K_{L}$. The isomorphisms of the Claim follow by
Snake Lemma by comparing cohomology sheaves of these two complexes, since
cohomology sheaves of $K_{L}^{\prime}$ are given by
$\operatorname{\mathcal{H}}^{-1}=\pi_{\alpha}^{*}\Lambda^{*}(\Delta_{L})|_{Y_{L}}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*}$
and
$\operatorname{\mathcal{H}}^{0}=\operatorname{\mathcal{O}}_{W_{L}}(-\Delta_{L})$.
∎

By Claim 4.11 and our spectral sequence,
$\tau_{*}^{S_{k-1}}G^{0}_{\alpha}(\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})\otimes\operatorname{sgn})$ is isomorphic to

$\tau_{*}^{S_{k-1}}\Bigl{(}\\!\\!\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l,\
\alpha\in L}\operatorname{\mathcal{O}}_{\Delta_{L}\cup
W_{L}}\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\otimes\operatorname{sgn}^{\prime}\Bigr{)},$

which we claim has the same image in the quotient-sheaf of $S_{k}$-anti-co-
invariants as its quotient-subsheaf
$\tau_{*}^{S_{k-1}}G^{0}_{\alpha}(\operatorname{\mathcal{C}}^{-l}\otimes\operatorname{sgn})$,
which is given by

$\tau_{*}^{S_{k-1}}\Bigl{(}\\!\\!\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l,\
\alpha\in L}\operatorname{\mathcal{O}}_{W_{L}}\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\otimes\operatorname{sgn}^{\prime}\Bigr{)}.$

Indeed, $\Delta_{L}$ has irreducible components given by diagonals
$\Delta_{j\alpha}$ for $j\in L$. Local sections of the sheaf
$\operatorname{\mathcal{O}}_{\Delta_{j\alpha}}$ are invariant under the
transposition $(j\alpha)$, and therefore go to $0$ in the quotient sheaf of
$S_{k}$-anti-coinvariants.

Now let $l\geq 1$. The sheaf
$\tau_{*}^{S_{k-1}}G^{-1}_{\alpha}(\operatorname{\mathcal{H}}^{-l}(\hat{\operatorname{\mathcal{F}}}^{\bullet\boxtimes
k})\otimes\operatorname{sgn})$ is given by

(4.11)
$\tau_{*}^{S_{k-1}}\Bigl{(}\\!\\!\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l,\
\alpha\not\in
L}\pi_{L}^{*}\hat{\operatorname{\mathcal{D}}}_{i}^{k-l}(\Delta_{L})\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\otimes\operatorname{sgn}^{\prime}\Bigr{)},$

which we claim has the same image in the quotient-sheaf of $S_{k}$-anti-co-
invariants as its quotient-subsheaf
$\tau_{*}^{S_{k-1}}G^{-1}_{\alpha}(\operatorname{\mathcal{C}}^{-l}\otimes\operatorname{sgn})$,
which is given by

(4.12)
$\tau_{*}^{S_{k-1}}\Bigl{(}\\!\\!\bigoplus\limits_{L\subset\\{1,\ldots,k\\}\atop|L|=k-l,\
\alpha\not\in
L}\pi_{L}^{*}\hat{\operatorname{\mathcal{D}}}_{i}^{k-l}\bigotimes\limits_{j\not\in
L}\pi_{j}^{*}\Lambda^{*}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l}\otimes\operatorname{sgn}^{\prime}\Bigr{)}.$

As in the proof of the Claim 4.8, we compute (4.11) as the pushforward by
$({\pi_{\operatorname{Sym}^{k-1}C\times C\times M_{i}}})_{*}$ of the sheaf

$\pi^{*}_{\operatorname{Sym}^{k-1}C\times C\times
M_{i}}(\operatorname{\mathcal{O}}_{W}(\Delta_{p})\boxtimes\Lambda^{*\boxtimes
k}\boxtimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l})\otimes\pi^{*}_{\operatorname{Sym}^{k-l}C\times
M_{i}}(\operatorname{\mathcal{D}}_{i}^{k-l}(-B/2)\otimes\Lambda^{\boxtimes
k-l})$

from the product $\operatorname{Sym}^{k-1}C\times
C\times\operatorname{Sym}^{k-l}C\times M_{i}$. Here we use the following
notation: $W=\\{(D,D^{\prime})\ |\ D\geq
D^{\prime}\\}\subset\operatorname{Sym}^{k-1}C\times\operatorname{Sym}^{k-l}C$,
$B\subset\operatorname{Sym}^{k-l}$ is the diagonal, and
$\Delta_{p}\subset\operatorname{Sym}^{k-1}C\times C$ is the locus $\\{(D,p)\
|\ p\in D\\}$. For (4.12), the formula is the same except that it has
$\operatorname{\mathcal{O}}_{W}$ instead of
$\operatorname{\mathcal{O}}_{W}(\Delta_{p})$.

Next, apply the morphism $\rho:\,\operatorname{Sym}^{k-1}C\times C\times
M_{i}\to\operatorname{Sym}^{k}C\times M_{i}$. Interchanging $\rho_{*}$ and
$({\pi_{\operatorname{Sym}^{k-1}C\times C\times M_{i}}})_{*}$, we need to
prove that $\bar{\rho}_{*}(\operatorname{\mathcal{O}}_{W}(\Delta_{p}))$ and
$\bar{\rho}_{*}(\operatorname{\mathcal{O}}_{W})$ have the same image in the
quotient-sheaf of $S_{k}$-co-invariants. where $\bar{\rho}$ is a morphism
$\bar{\rho}:\,\operatorname{Sym}^{k-1}\times
C\times\operatorname{Sym}^{k-l}C\to\operatorname{Sym}^{k}C\times\operatorname{Sym}^{k-l}C$.

We claim that this image is the structure sheaf of the correspondence
$\overline{W}=\bar{\rho}(W)=\\{(D,D^{\prime}))\ |\ D\geq D^{\prime}\\}$ in
$\operatorname{Sym}^{k}C\times\operatorname{Sym}^{k-l}C$. Indeed, the
$S_{k}$-orbit of the preimage of $W$ in $C^{k}\times\operatorname{Sym}^{k-l}C$
is the locus

$\hat{W}=\\{(p_{1},\ldots,p_{k},D^{\prime}))\ |\ p_{1}+\ldots+p_{k}\geq
D^{\prime}\\}\subset C^{k}\times\operatorname{Sym}^{k-l}C.$

The morphism $R:\,\hat{W}\to\overline{W}$ is the categorical quotient, so
$R^{S_{k}}_{*}(\operatorname{\mathcal{O}}_{\hat{W}})\simeq\operatorname{\mathcal{O}}_{\overline{W}}$.
On the other hand, by duality,
$R^{S_{k}}_{*}(\operatorname{\mathcal{O}}_{\hat{W}}(\Delta))\simeq(R^{S_{k}}_{*}(\operatorname{\mathcal{O}}_{\hat{W}}))^{*}\simeq\operatorname{\mathcal{O}}_{\overline{W}}^{*}\simeq\operatorname{\mathcal{O}}_{\overline{W}}$
because $\operatorname{\mathcal{O}}_{\hat{W}}(\Delta)$ is the relative
dualizing sheaf for $R_{*}$. ∎

######  Lemma 4.12.

We have the following exact triangle in $D^{b}(\operatorname{Sym}^{k}C\times
M_{i})$:

(4.13) $H_{k}\to{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}\to\operatorname{\mathcal{D}}_{i}^{k}(-B/2)\to$

with
$\operatorname{\mathcal{P}}_{H_{k}}(X)\in\langle\operatorname{\mathcal{D}}_{i}^{k-1}\operatorname{{\boldsymbol{\Lambda}}}^{*},\ldots,\operatorname{\mathcal{D}}_{i}^{1}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-1},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}\rangle$
for every $X\in D^{b}(\operatorname{Sym}^{k}C)$, where
$B\subset\operatorname{Sym}^{k}C$ is the diagonal.

###### Proof.

By Lemma 4.9,
$\operatorname{\mathcal{H}}^{0}(\operatorname{\mathcal{F}}^{\bullet\boxtimes
k})\simeq\operatorname{\mathcal{D}}_{i}^{k}(-B/2)$. So we have a morphism
${\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}\to\operatorname{\mathcal{D}}_{i}^{k}(-B/2)$ in
$D^{b}(\operatorname{Sym}^{k}C\times M_{i})$, which gives the exact triangle
(4.13). Continuing with the smart truncations of the complex
${\operatorname{\mathcal{F}}^{\bullet\boxtimes k}}$ and using Lemma 4.9 gives
the remaining statements. ∎

######  Corollary 4.13.

Applying the Fourier–Mukai functors to the exact triangles (4.9) and (4.13)
gives exact triangles

(4.14)
$\operatorname{\mathcal{P}}_{G_{k}}(X)\to\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X)\to\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
k}}}(X)[k]\to$

and

(4.15)
$\operatorname{\mathcal{P}}_{H_{k}}(X)\to\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X)\to\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X(-B/2))\to$

for every $X\in D^{b}(\operatorname{Sym}^{k}C)$. Here
$\operatorname{\mathcal{P}}_{G_{k}}(X)\in\langle{\operatorname{\mathcal{F}}^{*\boxtimes
k-1}},\ldots,\operatorname{\mathcal{F}}^{*},\operatorname{\mathcal{O}}\rangle$
and
$\operatorname{\mathcal{P}}_{H_{k}}(X)\in\langle\operatorname{\mathcal{D}}_{i}^{k-1}\operatorname{{\boldsymbol{\Lambda}}}^{*},\ldots,\operatorname{\mathcal{D}}_{i}^{1}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-1},{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}\rangle$.

To finish the proof of Theorem 3.2], we need Lemma 4.14 and Lemma 4.15, which
break the required mutation in two steps, as illustrated in Figure 9.

Figure 9. A fleeting glimpse of the subcategory
$\langle\operatorname{\mathcal{F}}^{\bullet\boxtimes k}\rangle$.

######  Lemma 4.14.

$\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}$ is fully faithful and there is a mutation

$\langle{\operatorname{\mathcal{F}}^{*\boxtimes
k}},\operatorname{\mathcal{F}}^{*\boxtimes
k-1},\ldots,\operatorname{\mathcal{O}}\rangle\to\langle\operatorname{\mathcal{F}}^{*\boxtimes
k-1},\ldots,\operatorname{\mathcal{O}},{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}\rangle.$

###### Proof.

We first claim that

(4.16)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
m}}}(Y))=0$

for $X\in D^{b}(\operatorname{Sym}^{k}C)$, $Y\in
D^{b}(\operatorname{Sym}^{m}C)$, $m<k$. By (4.15), it suffices to show
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{l}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}}(X),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
m}}}(Y))=0$ for $l\leq k$. By Serre duality, it suffices to prove that
$R\Gamma(M_{i},\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{l}{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}}(X)\otimes^{L}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes{m}}}}(Y))^{*}\otimes\omega_{M_{i}})=0$.
Without loss of generality, we take $X$ and $Y$ to be skyscraper sheaves. Then
$R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{M_{i-l}(\Lambda(-2D))}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}(\operatorname{\mathcal{F}}^{*\boxtimes
m}_{D^{\prime}})^{*}\omega_{M_{i}})=R\Gamma_{M_{i-l}(\Lambda(-2D))}({\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}\bar{\operatorname{\mathcal{F}}}^{\boxtimes
m}_{D^{\prime}}\otimes\omega_{M_{i-l}(\Lambda(-2D))}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{l})=R\Gamma(M_{i-l}(\Lambda(-2D)),{\operatorname{{\boldsymbol{\Lambda}}}}^{k}\otimes\bar{\operatorname{\mathcal{F}}}^{\boxtimes
m*}_{D^{\prime}})^{*}[\ldots]=0$ by Theorem 3.9 with $j=i-l$, $d=2g-1-2l$,
$t=k$, $a=m$, $b=0$ (we have $2g-1-2l>2$ if $l<i$, and $m<k\leq i<2g-3-2i$).

The formula (4.16) follows. Combining with Lemma 4.1 and (4.14), it follows
that
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(Y))=\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
k}}}(Y)[k])=\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
k}}}(X),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
k}}}(Y))=\operatorname{RHom}(X,Y)$ by Lemma 4.1, which shows that
$\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes k}}}$ is
fully faithful and we have a required mutation. ∎

######  Lemma 4.15.

$\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}$ is fully
faithful. There is a mutation

$\langle\operatorname{\mathcal{D}}_{i}^{k-1}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*},\ldots,{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k},\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}\rangle\to\langle\operatorname{\mathcal{D}}_{i}^{k},\operatorname{\mathcal{D}}_{i}^{k-1}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{*},\ldots,{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k}\rangle.$

###### Proof.

We first claim that

(4.17)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{l}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}}(Y))=0$

for all $X\in D^{b}(\operatorname{Sym}^{k}C)$, $Y\in
D^{b}(\operatorname{Sym}^{l}C)$, $l<k$. By (4.14), it suffices to prove that
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
m}}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{l}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}}(Y))=0$
for $m\leq k$. Without loss of generality, we take $X$ and $Y$ to be
skyscraper sheaves. Then
$\operatorname{RHom}({{\operatorname{\mathcal{F}}_{D}^{*\boxtimes
m}},\operatorname{\mathcal{O}}_{M_{i-l}(\Lambda(-2D^{\prime}))}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}})=R\Gamma_{M_{i-l}(\Lambda(-2D^{\prime}))}({{(\operatorname{\mathcal{F}}_{D}^{*\boxtimes
m})^{*}}{\operatorname{{\boldsymbol{\Lambda}}}}^{l-k}})=R\Gamma_{M_{i-l}(\Lambda(-2D^{\prime}))}({{\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
m}}{\operatorname{{\boldsymbol{\Lambda}}}}^{l-k}})=0$ by Theorem 3.8 with
$j=i-l$, $d=2g-1-2l$, $t=l+m-k$, $a=m$, $b=0$ (since $0<l<k\leq i\leq g-1$, we
have $1\leq i-l\leq g-1$ and $2<2g-1-2l\leq 2g+1$; also,
$m-(i-l)-1<l+m-k<i<2g-3-2i$ and $l+m-k>m$). In addition to (4.17), we have

(4.18)
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{l}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{k-l}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(Y))=0$

for $X\in D^{b}(\operatorname{Sym}^{l}C)$, $Y\in
D^{b}(\operatorname{Sym}^{k}C)$, $l<k$. Indeed, by inductive assumption in
Theorem 3.2, it suffices to check instead that, for $l<k$, $l+m\leq k$, $m>0$,
we have
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{*\boxtimes
l}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{m}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(Y))=0$.
Without loss of generality, we take $X$ and $Y$ to be skyscraper sheaves. Then
we compute $\operatorname{RHom}({\operatorname{\mathcal{F}}_{D}^{*\boxtimes
l}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}^{*}}^{m},\operatorname{\mathcal{O}}_{M_{i-k}(\Lambda(-2D^{\prime}))})=R\Gamma(M_{i-k}(\Lambda(-2D^{\prime})),{(\operatorname{\mathcal{F}}_{D}^{*\boxtimes
l})^{*}}\otimes{\operatorname{{\boldsymbol{\Lambda}}}}^{m})=R\Gamma_{M_{i-k}(\Lambda(-2D^{\prime}))}({{\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
l}}{\operatorname{{\boldsymbol{\Lambda}}}}^{m}})=0$ by Theorem 3.9 with
$j=i-k$, $d=2g-1-2k$, $t=m$, $a=0$, $b=l$ (we have $d>2$ if $j>0$, and
$0<m\leq i<3g-2-2i$).

Combining (4.15), (4.17), (4.18), we see that
$\operatorname{RHom}(\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(X),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(Y))\simeq\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X({B\over
2})),\operatorname{\mathcal{P}}_{\operatorname{\mathcal{D}}_{i}^{k}}(Y))\simeq\operatorname{RHom}(\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(X({B\over
2})),\operatorname{\mathcal{P}}_{{\operatorname{\mathcal{F}}^{\bullet\boxtimes
k}}}(Y({B\over 2})))\simeq\operatorname{RHom}(X({B\over 2}),Y({B\over 2}))$
for $X,Y\in D^{b}(\operatorname{Sym}^{k}C)$ by Lemma 4.14. But the latter
group is isomorphic to $\operatorname{RHom}(X,Y)$. This proves the lemma. ∎

###### Proof of Theorem 3.2.

Part (a) is a part of Lemma 4.15, Part (b) follows from part (c) and
induction. Indeed, we can perform the basic Cross Warp mutation Theorem 3.2
(c) in $D^{b}(M_{i-1})$. By inductive assumption and [TT, Section 3], the
resulting admissible subcategory maps into itself in $D^{b}(M_{i})$ by the
windows embedding $\iota$. It remains to perform the inverse of the basic
Cross Warp mutation Theorem 3.2 (c) in $D^{b}(M_{i})$.

Finally, part (c) is a combination of Lemmas 4.1, 4.14, 4.15, and inductive
assumption, as illustrated in Figure 9. ∎

##  5\. Broken Loom

In this section we perform a transition from the semi-orthogonal
decompositions of $M=M_{g-1}$ with three mega-blocks associated with the
birational contraction
$\psi:\,M\dashrightarrow\operatorname{{\mathbb{P}}}^{3g-3}$ studied in the
previous sections to decompositions with four mega-blocks associated with the
birational contraction $\zeta:\,M\to N$. Recall that this morphism is the
forgetful map $(F,s)\mapsto F$.

######  Lemma 5.1.

$D^{b}(M)$ has a semi-orthogonal decomposition into admissible subcategories
arranged into three mega-blocks, as follows:

$\Bigl{\langle}\bigl{\langle}Z^{3+j+2k-g}{\theta^{*}}^{k+j+1}{\operatorname{\mathcal{E}}^{\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-2\atop j,k\geq 0},\qquad\qquad\qquad\qquad\qquad$
(5.1)
$\bigl{\langle}Z^{2+j+2k-g}{\theta^{*}}^{k+j}{\operatorname{\mathcal{E}}^{\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-2\atop j,k\geq 0},\ $
$\qquad\qquad\qquad\qquad\qquad\bigl{\langle}Z^{1+j+2k-g}{\theta^{*}}^{k+j-1}{\operatorname{\mathcal{E}}^{\boxtimes
j}}\bigr{\rangle}_{j+k\leq g-1\atop j,k\geq 0}\Bigr{\rangle}.$

Within each of the three mega-blocks, the blocks are arranged first by $k$ (in
the decreasing order) and then, for a fixed $k$, by $j$ (in the decreasing
order).

Figure 10. Repeated helix mutation in genus $5$

###### Proof.

We tensor the semi-orthogonal decomposition of Theorem 3.1 with the line
bundle
$\omega_{M}^{3-g}=Z^{g-3}\theta^{g-3}\operatorname{{\boldsymbol{\Lambda}}}^{g-3}$,
which corresponds to the full rotation of the helix mutation repeated $g-3$
times, see Figure 10 for $g=5$.

We use the formulas $\operatorname{\mathcal{F}}^{*\boxtimes
j}\simeq\operatorname{\mathcal{F}}^{\boxtimes j}\otimes(\Lambda^{*\boxtimes
j}\boxtimes\operatorname{{\boldsymbol{\Lambda}}}^{*j})\simeq\operatorname{\mathcal{E}}^{\boxtimes
j}\otimes(\Lambda^{*\boxtimes j}\boxtimes
Z^{-j}\operatorname{{\boldsymbol{\Lambda}}}^{*j})$ to rewrite the
decomposition using Fourier–Mukai functors associated with tensor bundles of
$\operatorname{\mathcal{E}}$ rather than $\operatorname{\mathcal{F}}$. We use
the formula $\operatorname{{\boldsymbol{\Lambda}}}=\theta Z^{-2}$ to eliminate
$\operatorname{{\boldsymbol{\Lambda}}}$. Finally, we twist with the line
bundle ${\theta^{*}}^{2g-5}$. ∎

######  Remark 5.2.

The twist by the line bundle $\omega_{M}^{3-g}$ is required to create blocks
of the semi-orthogonal decomposition compatible with the contraction $\zeta$.
In contrast, the twist by the line bundle ${\theta^{*}}^{2g-5}$, which is
pulled back from $N$, is not necessary and we only do it to simplify the
formulas. In the illustrations of the braid, we ignore this twist even though
we still use it to label the blocks, for consistency.

The next step is crucial. As it turns out, the blocks within each mega-block
of (5.1) can be reordered differently. This is illustrated in Figure 11.

Figure 11. Reordering of the blocks in genus $5$

######  Theorem 5.3.

$D^{b}(M)$ has a semi-orthogonal decomposition into admissible subcategories
arranged into three mega-blocks, as follows:

$\Bigl{\langle}\bigl{\langle}Z^{3+\lambda-g}{\theta^{*}}^{\lambda-k+1}{\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}}\bigr{\rangle}_{\lambda-k\leq
g-2\atop\lambda-2k,k\geq 0},\qquad\qquad\qquad\qquad\qquad$ (5.2)
$\bigl{\langle}Z^{2+\lambda-g}{\theta^{*}}^{\lambda-k}{\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}}\bigr{\rangle}_{\lambda-k\leq
g-2\atop\lambda-2k,k\geq 0},\ $
$\qquad\qquad\qquad\qquad\qquad\bigl{\langle}Z^{1+\lambda-g}{\theta^{*}}^{\lambda-k-1}{\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}}\bigr{\rangle}_{\lambda-k\leq
g-1\atop\lambda-2k,k\geq 0}\Bigr{\rangle}.$

Within each of the three mega-blocks, the blocks are arranged in decreasing
order of $\lambda$. Blocks with the same $\lambda$ are arranged in decreasing
order of $k$.

We start with a lemma.

######  Lemma 5.4.

Suppose $2<d\leq 2g+1$ and $1\leq j\leq\lfloor{d-1\over 2}\rfloor$. Let
$D\in\operatorname{Sym}^{a}C$ and $D^{\prime}\in\operatorname{Sym}^{b}C$ with
$a,\ b\leq\min(d+g-2j-1,j)$, and let $t$ be an integer satisfying

$a-j-1<t<d+g-2j-1-b.$

Suppose, further, that $2t<a-b$. Then

(5.3) $R\Gamma_{M_{j}(d)}({\operatorname{\mathcal{F}}^{\boxtimes
a}_{D}}^{*}\otimes\operatorname{\mathcal{F}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t})=0$

###### Proof.

We write $D=\sum\alpha_{i}x_{i}$ and $D^{\prime}=\sum\beta_{i}y_{i}$, where
$\sum\alpha_{i}=a$, $\sum\beta_{i}=b$, and $x_{i}\neq x_{j}$, $y_{i}\neq
y_{j}$ for $i\neq j$. It suffices to prove that, under the assumptions of the
Lemma,

(5.4)
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}^{\otimes\alpha_{i}}\otimes\bigotimes_{i}\operatorname{\mathcal{F}}_{y_{i}}^{\otimes\beta_{i}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right)=0.$

Indeed, (5.4) implies (5.3) (see also Remark 5.7) since vector bundles in
(5.3) are deformations of vector bundles in (5.4) over
$\operatorname{{\mathbb{A}}}^{1}$ by [TT, Corollary 2.9].

######  Claim 5.5.

It suffices to prove that, under the assumptions of the Lemma,

(5.5)
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}\otimes\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right)=0.$

Indeed, for every $i$, we have a direct sum decomposition,

(5.6)
$\operatorname{\mathcal{F}}_{y_{i}}^{\otimes\beta_{i}}=\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}\oplus[\operatorname{{\boldsymbol{\Lambda}}}\otimes\operatorname{Sym}^{\beta_{i}-2}\operatorname{\mathcal{F}}_{y_{i}}]^{\oplus\ldots}\oplus[\operatorname{{\boldsymbol{\Lambda}}}^{2}\otimes\operatorname{Sym}^{\beta_{i}-4}\operatorname{\mathcal{F}}_{y_{i}}]^{\oplus\ldots}\oplus\ldots,$

which is valid for any rank $2$ vector bundle on any scheme, where the number
of copies of each summand is not important for us. We also have the same
decompositions for $\operatorname{\mathcal{F}}_{x_{i}}^{\otimes\alpha_{i}}$.
Tensored together, these decompositions give a direct sum decomposition of
$\bigotimes_{i}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}^{\otimes\alpha_{i}}\otimes\bigotimes_{i}\operatorname{\mathcal{F}}_{y_{i}}^{\otimes\beta_{i}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}$,
where all summands have form (5.5) but for different parameters $\alpha_{i}$,
$\beta_{i}$, $t$. Concretely, we have to change $a\to a-2u$, $b\to b-2v$,
$t\to t-u+v$ for some $u,v\geq 0$. On the other hand, all inequalities in
Lemma 5.4 remain valid under this change of parameters. Therefore, if (5.5) is
valid for the range of inequalities in the lemma then (5.4) is valid as well.
This proves Claim 5.5.

Furthermore, we can assume, without loss of generality, that $b\geq a$ since
otherwise we can swap the bundles:
$\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}\otimes\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\simeq\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}_{x_{i}}}\otimes\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}^{*}_{y_{i}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t-a+b}$
and inequalities in the statement of the lemma still hold: $2(t-a+b)<b-a$,
$b-j-1<t-a+b<d+g-2j-1-a$. Note that if $b\geq a$ then $t<0$ by our assumptions
and we have vanishing (3.2) by Theorem 3.8. We will reduce (5.5) (for $b\geq
a$) to (3.2).

We prove (5.5) by induction on $a+b$. The base case $a=b=0$ is (3.2) for
$a=b=0$. As above, we can assume that $b\geq a$ (and so $t<0$).

######  Claim 5.6.

We have a short exact sequence

(5.7)
$0\to\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}\to\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}\to Q\to 0.$

Under the induction assumption,
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}\otimes
Q\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right)=0$.

Given the claim, we note that
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}\otimes\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right)$
vanishes as a direct summand of
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}^{\otimes\alpha_{i}}\otimes\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right),$ which
is equal to $0$ by (3.2). Therefore, (5.5) follows from the claim.

It remains to prove Claim 5.6. We use technique and notation from the proof of
[TT, Lemma 7.7]. Let $D_{\beta}=\mathbb{C}[t]/(t^{\beta})$ and
$B_{\beta}=\frac{\mathbb{C}[t_{1},\ldots,t_{\beta}]}{(\sigma_{1},\ldots,\sigma_{\beta})}$,
where $\sigma_{1},\ldots,\sigma_{\beta}$ are elementary symmetric polynomials.
Let $\operatorname{{\mathbb{D}}}_{\beta}=\operatorname{Spec}D_{\beta}$ and
$\operatorname{{\mathbb{B}}}_{\beta}=\operatorname{Spec}B_{\beta}$. Write the
indexing set $\\{1,\ldots,b\\}$ as a disjoint union of sets $X_{k}$ of
cardinality $\beta_{k}$ for $k=1,\ldots,s$, and denote
$B=B_{\beta_{1}}\otimes\ldots\otimes B_{\beta_{s}}$. Let $M=M_{j}(d)$. For
every $j\in X_{k}$, we have a commutative diagram [TT, (7.6)]

(5.8)
${\mathbb{B}_{\beta_{1}}\times\ldots\times\operatorname{{\mathbb{B}}}_{\beta_{s}}\times
M}$${\mathbb{D}_{\beta_{k}}\times M}$${C\times
M}$${M}$$\scriptstyle{\pi_{j}}$$\scriptstyle{\tau}$$\scriptstyle{q_{k}}$

We have $\bar{\operatorname{\mathcal{F}}}^{\boxtimes
b}_{D^{\prime}}=\tau_{*}^{S_{\beta_{1}}\times\ldots\times
S_{\beta_{s}}}\left(\bigotimes_{j\in
X_{k}}\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\otimes\operatorname{sgn}\right)$,
where $\operatorname{\mathcal{F}}_{k}=q_{k}^{*}{\operatorname{\mathcal{F}}}$.

The vector bundle
$\tau_{*}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$ has
a filtration by vector subbundles $\tau_{*}\left(B_{\geq
d}\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$, where $B_{\geq
d}\subset B$ is the ideal of monomials of degree $\geq d$. In particular,
$\tau_{*}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$ has
a vector subbundle
$\tau_{*}\left(\Delta_{\beta_{1}}\ldots\Delta_{\beta_{s}}\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$,
where $\Delta_{\beta}\in B_{\beta}$ is the top degree monomial (the
Vandermonde determinant). The associated graded bundle is
$\bigotimes_{k}\operatorname{\mathcal{F}}_{x_{k}}^{\otimes\beta_{k}}\otimes_{\mathcal{O}_{M}}B$.
For example, the subbundle
$\tau_{*}\left(\Delta_{\beta_{1}}\ldots\Delta_{\beta_{s}}\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)\subset\tau_{*}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$
is isomorphic to
$\bigotimes_{k}\operatorname{\mathcal{F}}_{x_{k}}^{\otimes\beta_{k}}$.

The group $S_{\beta_{1}}\times\ldots\times S_{\beta_{s}}$ acts on
$\bigotimes_{k}\operatorname{\mathcal{F}}_{x_{k}}^{\otimes\beta_{k}}$ by
permuting tensor factors within each subset $X_{k}$. The action on $B$ is by
the regular representation. The direct sum decomposition of
$\bigotimes_{k}\operatorname{\mathcal{F}}_{x_{k}}^{\otimes\beta_{k}}$ induced
by (5.6) is equivariant under $S_{\beta_{1}}\times\ldots\times S_{\beta_{s}}$.
By Frobenius reciprocity, the vector bundle
$\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}$
appears in
$\left(\bigotimes_{k}\operatorname{\mathcal{F}}_{x_{k}}^{\otimes\beta_{k}}\otimes_{\mathcal{O}_{M}}B\otimes\operatorname{sgn}\right)^{S_{\beta_{1}}\times\ldots\times
S_{\beta_{s}}}$ only once and corresponds to the subbundle
$\tau_{*}\left(\Delta_{\beta_{1}}\ldots\Delta_{\beta_{s}}\bigotimes_{i}\operatorname{Sym}^{\beta_{i}}\operatorname{\mathcal{F}}_{y_{i}}\right)\subset\tau_{*}\left(\Delta_{\beta_{1}}\ldots\Delta_{\beta_{s}}\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)\subset\tau_{*}(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k})$.
It follows that we have a short exact sequence (5.7), where the quotient
bundle $Q$ is a direct summand (of skew-invariants) in the bundle
$\tilde{Q}=\tau_{*}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)/\tau_{*}\left(\Delta_{\beta_{1}}\ldots\Delta_{\beta_{s}}\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$.
By the above, $\tilde{Q}$ has a $(S_{\beta_{1}}\times\ldots\times
S_{\beta_{s}})$-equivariant filtration by vector bundles with subquotients
given by direct sums of vector bundles of the form
$\operatorname{{\boldsymbol{\Lambda}}}^{\sum
v_{i}}\bigotimes_{i}\operatorname{Sym}^{\beta_{i}-2v_{i}}\operatorname{\mathcal{F}}_{y_{i}}$
with $\sum v_{i}\geq 0$. Furthermore, vector bundles with $\sum v_{i}=0$ have
a trivial component of skew-invariants. It follows that
$R\Gamma_{M_{j}(d)}\left(\bigotimes_{i}\operatorname{Sym}^{\alpha_{i}}{\operatorname{\mathcal{F}}^{*}_{x_{i}}}\otimes
Q\otimes\operatorname{{\boldsymbol{\Lambda}}}^{t}\right)=0$ by the induction
assumption. ∎

######  Remark 5.7.

The proof of Lemma 5.4 also shows that (5.3) holds with
$\operatorname{\mathcal{F}}^{\boxtimes a}_{D}$ replaced with
$\bar{\operatorname{\mathcal{F}}}^{\boxtimes a}_{D}$ and with
$\operatorname{\mathcal{F}}^{\boxtimes b}_{D^{\prime}}$ replaced with
$\bar{\operatorname{\mathcal{F}}}^{\boxtimes b}_{D^{\prime}}$.

###### Proof of Theorem 5.3.

The mega-blocks are the same as in Lemma 5.1, we just do the change of
variables $\lambda=2k+j$. The three mega-blocks differ by line bundle twists,
in addition, the last mega-block is larger. So it suffices to prove the
statement for the last mega-block. We need to check that, whenever
$\lambda_{1}<\lambda_{2}$ or $\lambda_{1}=\lambda_{2},k_{1}<k_{2}$,

$\operatorname{RHom}(Z^{1+\lambda_{1}-g}{\theta^{*}}^{\lambda_{1}-k_{1}-1}{\operatorname{\mathcal{E}}_{D}^{\boxtimes\lambda_{1}-2k_{1}}},Z^{1+\lambda_{2}-g}{\theta^{*}}^{\lambda_{2}-k_{2}-1}{\operatorname{\mathcal{E}}_{D^{\prime}}^{\boxtimes\lambda_{2}-2k_{2}}})=0$

for every $D\in\operatorname{Sym}^{\lambda_{1}-2k_{1}}C$,
$D^{\prime}\in\operatorname{Sym}^{\lambda_{1}-2k_{1}}C$. This
$\operatorname{RHom}$ is equal to
$R\Gamma\Bigl{(}\Lambda^{t}\otimes(\operatorname{\mathcal{F}}_{D}^{\boxtimes
j_{1}})^{*}\otimes\operatorname{\mathcal{F}}^{\boxtimes
j_{2}}_{D^{\prime}}\Bigr{)}$, where $t=(k_{1}+j_{1})-(k_{2}+j_{2})$ and, for
$i=1,2$, $j_{i}=\lambda_{i}-2k_{i}$. Suppose first that
$\lambda_{1}<\lambda_{2}$. We have $j_{1},j_{2}\leq\min(g,g-1)$, so by Lemma
5.4, it suffices to check two numerical conditions:

(5.9) $j_{1}-g<t<g-j_{2},$ (5.10) $2t<j_{1}-j_{2}.$

Inequalities (5.9) hold for all blocks within the mega-block because, for
$i=1,2$, we have that $j_{i}\leq j_{i}+k_{i}$ and $j_{i}+k_{i}\leq g-1<g$.
When $\lambda_{1}<\lambda_{2}$, we have

$2t=2(\lambda_{1}-\lambda_{2})-2(k_{1}-k_{2})<(\lambda_{1}-\lambda_{2})-2(k_{1}-k_{2})=j_{1}-j_{2},$

so (5.10) holds. On the other hand, when $\lambda_{1}=\lambda_{2}$ and
$k_{1}<k_{2}$, we have

$t=k_{2}-k_{1}<2(k_{2}-k_{1})=j_{1}-j_{2},$

so the result follows from Theorem 3.8 (3.2) since

$\Lambda^{t}\otimes(\operatorname{\mathcal{F}}_{D}^{\boxtimes
j_{1}})^{*}\otimes\operatorname{\mathcal{F}}^{\boxtimes
j_{2}}_{D^{\prime}}\cong\Lambda^{t+j_{2}-j_{1}}\otimes(\bar{\operatorname{\mathcal{F}}}_{D}^{\boxtimes
j_{1}})\otimes{\operatorname{\mathcal{F}}^{*\boxtimes j_{2}}_{D^{\prime}}}$

by Lemma 3.4. ∎

Finally, we prove the following theorem:

######  Theorem 5.8.

$D^{b}(M)$ has a semi-orthogonal decomposition with semi-orthogonal blocks
arranged into the following four mega-blocks:

$\bigl{\langle}Z^{\lambda+2-g}{\theta^{*}}^{\lambda-k+1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}\bigr{\rangle}_{0\leq\lambda\leq
g-2\atop 0\leq k\leq\lfloor{\lambda\over
2}\rfloor},\bigl{\langle}Z^{\lambda+3-g}{\theta^{*}}^{\lambda-k+1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}\bigr{\rangle}_{0\leq\lambda\leq
2(g-2)\atop 0\leq k\leq\lfloor{\lambda\over 2}\rfloor,\lambda-k\leq g-2},$
$\bigl{\langle}Z^{\lambda+2-g}{\theta^{*}}^{\lambda-k}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}\bigr{\rangle}\\!\\!\\!{}_{0\leq\lambda\leq
2(g-2)\atop 0\leq k\leq\lfloor{\lambda\over 2}\rfloor,\lambda-k\leq
g-2}\\!\\!\\!\\!,\bigl{\langle}Z^{\lambda+1-g}{\theta^{*}}^{\lambda-k-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2k}\bigr{\rangle}\\!\\!\\!\\!{}_{g-1\leq\lambda\leq
2(g-1)\atop 0\leq k\leq\lfloor{\lambda\over 2}\rfloor,\lambda-k\leq
g-1}\\!\\!\\!\\!.$

Within each mega-block, the blocks are arranged in decreasing order of
$\lambda$ and those with identical $\lambda$ are further arranged by
decreasing $k$.

###### Proof.

We split the last mega-block of the semi-orthogonal decomposition of Theorem
5.3 in half: we keep the blocks with $g-1\leq\lambda\leq 2(g-1)$ and tensor
the blocks with $0\leq\lambda\leq g-2$ by $\omega_{M}=Z\otimes\theta^{-2}$,
which corresponds to mutating them in front of the decomposition as in Figure
12. ∎

Figure 12. Creation of the fourth mega-block in genus $5$

##  6\. Plain Weave

In the last section we prove Theorem 1.1. We start with the semi-orthogonal
decomposition of Theorem 5.8 and observe that the blocks with the trivial
power of $Z$ (they correspond to $\lambda=g-2$, $g-3$, $g-2$, and $g-1$ for
mega-blocks I, II, III, and IV, respectively,) are pulled-back from the semi-
orthogonal blocks of $D^{b}(N)$ of Theorem 1.1. The Plain Weave pattern,
illustrated in Figure 13, moves the remaining blocks out of the way.

Figure 13. Plain Weave in genus $5$

######  Notation 6.1.

In view of the semi-orthogonal decomposition

$D^{b}(M)=\langle\operatorname{\mathcal{A}},\
L\zeta^{*}D^{b}(N)\rangle,\quad\hbox{\rm where}\
\operatorname{\mathcal{A}}=\\{T\ :\ R\zeta_{*}(T)=0\\}\subset D^{b}(M),$

it suffices to prove that the blocks from Theorem 5.8 that are not pulled back
from $N$ can be mutated into the subcategory $\operatorname{\mathcal{A}}$. To
preserve symmetry, we actually mutate some of the blocks that are not pulled
back from $N$ to the left of the subcategory generated by the pulled back
blocks, and prove that they go into the subcategory
$\operatorname{\mathcal{A}}$, and mutate the remaining blocks that are not
pulled back from $N$ to the right of the subcategory generated by the pulled
back blocks, and prove that they end up in the subcategory
$\operatorname{\mathcal{A}}^{\prime}=\\{T\ :\
R\zeta_{*}(T\otimes\omega_{M})=0\\}$. This is illustrated in Figure 13. Once
this is done, we can tensor the blocks contained in
$\operatorname{\mathcal{A}}^{\prime}$ with $\omega_{M}$, which mutates them to
the left of the semi-orthogonal decomposition and into the subcategory
$\operatorname{\mathcal{A}}$. This will prove Theorem 1.1.

######  Remark 6.2.

In contrast to the previous sections of the paper, we don’t have to study the
kernels of the Fourier–Mukai functors that embed the left-over blocks into the
subcategory $\operatorname{\mathcal{A}}$. By the main result of [kosekitoda],
the category $\operatorname{\mathcal{A}}$ is isomorphic to the derived
category of the moduli space of stable pairs with fixed determinant
$K_{C}\otimes\Lambda^{*}$ of degree $2g-3$. The latter category has a semi-
orthogonal decomposition obtained using the tower of flips similar to Theorem
2.2. There likely exists a mutation that connects blocks of this semi-
orthogonal decomposition to our left-over blocks in $D^{b}(M)$.

We need the following theorem, proved later in this section. Recall that
$\operatorname{\mathcal{D}}^{l}$ is the structure sheaf of
$D^{l}=\\{(D,F,s):\,s|_{D}=0\\}\subset\operatorname{Sym}^{l}C\times M$.

######  Theorem 6.3.

Let $0\leq 2l\leq k\leq g-1$. Let
$\Phi:\,D^{b}(M)\to{}^{\perp}\langle\operatorname{\mathcal{E}}^{\boxtimes
k}\rangle$ be the projection functor. Then
$\Phi(Z^{k}\otimes\langle(\operatorname{\mathcal{D}}^{l})^{\vee}\rangle)\subset\operatorname{\mathcal{A}}^{\prime}$.

We will first finish the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

We process blocks of the semi-orthogonal decomposition of Theorem 5.8, mega-
block by mega-block. We also split mega-blocks II and III into half-mega-
blocks, II=IIa+IIb and III=IIIa+IIIb. Recall that, within each mega-block, the
blocks are arranged by $\lambda$ in the decreasing order. The reader may wish
to inspect Figure 13 as we go along.

On the right side of Figure 13, Mega-block IV contains the blocks

(6.1)
$\bigl{\langle}Z^{\lambda+1-g}{\theta^{*}}^{\lambda-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\bigr{\rangle}_{g-1\leq\lambda\leq
2(g-1)\atop 0\leq m\leq\lfloor{\lambda\over 2}\rfloor,\lambda-m\leq g-1}$

and ends with the blocks with $\lambda=g-1$,

(6.2)
$\bigl{\langle}{\theta^{*}}^{(g-1)-k-1}\operatorname{\mathcal{E}}^{\boxtimes(g-1)-2k}\bigr{\rangle}_{0\leq
k\leq\lfloor{g-1\over 2}\rfloor},$

which are pulled back from $N$ and are ordered by $k$, in decreasing order. We
process the remaining blocks in the increasing order of $\lambda$. Take one of
the blocks $\operatorname{\mathcal{C}}$ from (6.1) with $\lambda\geq g$. We
will take some blocks (one or two) from (6.2), temporarily move them in front
of all other blocks in (6.2), and call the subcategory generated by them
$\operatorname{\mathcal{B}}$. Some of the remaining blocks in (6.1) will
mutate in the process but will remain in the subcategory pulled back from $N$.
We then mutate the block $\operatorname{\mathcal{C}}$ to the right of
$\operatorname{\mathcal{B}}$ (into the subcategory
${}^{\perp}\operatorname{\mathcal{B}}$) and show that the mutated block
$\operatorname{\mathcal{C}}^{\prime}$ is contained in the subcategory
$\operatorname{\mathcal{A}}^{\prime}$. Therefore, the mutated block
$\operatorname{\mathcal{C}}^{\prime}$ becomes orthogonal to the remaining
blocks in (6.2), some of them mutated, since all of them are pulled back from
$N$. So we can move the mutated block $\operatorname{\mathcal{C}}^{\prime}$,
unchanged, to the right of all the blocks in (6.2). Finally, we move the
blocks from $\operatorname{\mathcal{B}}$ back into their position in the
sequence (6.2). To realize this program, we need the following lemma:

######  Lemma 6.4.

Suppose $g\leq\lambda\leq 2(g-1)$, $0\leq m\leq\lfloor{\lambda\over
2}\rfloor$, and $\lambda-m\leq g-1$. The orthogonal projector onto
${}^{\perp}\operatorname{\mathcal{B}}$ takes the block
$\bigl{\langle}Z^{\lambda+1-g}{\theta^{*}}^{\lambda-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\bigr{\rangle}$
into $\operatorname{\mathcal{A}}^{\prime}$, where $\operatorname{\mathcal{B}}$
is given by two blocks from (6.2), namely one block with $k=g-1-\lambda+m$ and
another with $k=g-\lambda+m$, unless $\lambda=2m$, in which case we take only
one block, namely the block with $k=g-1-\lambda+m$.

###### Proof.

We rewrite the kernel vector bundle as

$Z^{\lambda+1-g}{\theta^{*}}^{\lambda-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\simeq
Z^{2m+1-g}{\theta^{*}}^{m-1}\operatorname{\mathcal{F}}^{*\boxtimes\lambda-2m}.$

Take the complex
$Z^{2m+1-g}{\theta^{*}}^{m-1}\operatorname{\mathcal{F}}^{\bullet\lambda-2m}$
from Definition 4.6. If $\lambda\neq 2m$, let
$Z^{2m+1-g}{\theta^{*}}^{m-1}\otimes[\operatorname{\mathcal{F}}^{*\boxtimes\lambda-2m}\to\operatorname{Ker}^{1-(\lambda-2m)}]$
be its $2$-step smart truncation. By Lemma 4.9 and Corollary 4.10, it suffices
to prove that the subcategory $\langle
Z^{2m+1-g}{\theta^{*}}^{m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m}\rangle$,
and subcategories $\langle
Z^{2m+1-g}{\theta^{*}}^{m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m-1}\rangle$
and $\langle
Z^{2m+1-g}{\theta^{*}}^{m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m}[\operatorname{\mathcal{D}}^{1}]^{\vee}\rangle$,
which are necessary only if $\lambda\neq 2m$, are moved into
$\operatorname{\mathcal{A}}^{\prime}$ by the projector onto
${}^{\perp}\operatorname{\mathcal{B}}$ from the lemma. But these subcategories
are the same as $\langle
Z^{1-g+2\lambda-2m}{\theta^{*}}^{\lambda-m-1}\rangle$, $\langle
Z^{-1-g+2\lambda-2m}{\theta^{*}}^{\lambda-m-2}\rangle$, and $\langle
Z^{1-g+2\lambda-2m}{\theta^{*}}^{\lambda-m-1}[\operatorname{\mathcal{D}}^{1}]^{\vee}\rangle$,
so the claim follows from Theorem 6.3.

We only need to check that the block with $k=g-1-\lambda+m$ (and also with
$k=g-\lambda+m$ if $\lambda\neq 2m$) is among the blocks in (6.2), i.e. that
$0\leq k\leq\lfloor{g-1\over 2}\rfloor$. The first inequality follows from the
inequality $\lambda-m\leq g-1$. To show that $g-\lambda+m\leq\lfloor{g-1\over
2}\rfloor$, or equivalently that $\lambda-m\geq\lceil{g+1\over 2}\rceil$ we
use that $\lambda-m\geq\lambda-\lfloor{\lambda\over
2}\rfloor=\lceil{\lambda\over 2}\rceil\geq\lceil{g\over 2}\rceil$. This shows
that $\lambda-m\geq\lceil{g+1\over 2}\rceil$ unless $\lambda=2m$, in which
case a weaker inequality $\lambda-m\geq\lceil{g-1\over 2}\rceil$ still holds.
∎

At the end of the process, the blocks from (6.2) move to the left side of the
mega-block IV unchanged and all other blocks from (6.1) mutate to the right
side and into the subcategory $\operatorname{\mathcal{A}}^{\prime}$.

On the left side of Figure 13, Mega-block I contains the blocks

(6.3)
$\bigl{\langle}Z^{\lambda-(g-2)}{\theta^{*}}^{\lambda-m+1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\bigr{\rangle}_{0\leq\lambda\leq
g-2\atop 0\leq m\leq\lfloor{\lambda\over 2}\rfloor}$

and starts with the blocks with $\lambda=g-2$,

(6.4)
$\bigl{\langle}{\theta^{*}}^{(g-2)-k+1}\operatorname{\mathcal{E}}^{\boxtimes(g-2)-2k}\bigr{\rangle}_{0\leq
k\leq\lfloor{g-2\over 2}\rfloor},$

which are pulled back from $N$ and are ordered by decreasing $k$. We process
the remaining blocks in (6.3) inductively, in the decreasing order of
$\lambda$. The processed blocks will mutate to the left of the blocks (6.4)
and into the subcategory $\operatorname{\mathcal{A}}$. Equivalently, we can
mutate the blocks dual to (6.3),

(6.5)
$\bigl{\langle}Z^{-\lambda+(g-2)}{\theta^{*}}^{-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\bigr{\rangle}_{0\leq\lambda\leq
g-2\atop 0\leq m\leq\lfloor{\lambda\over 2}\rfloor},$

into $\operatorname{\mathcal{A}}^{\prime}$ under the blocks dual to (6.4),
namely

(6.6)
$\bigl{\langle}{\theta^{*}}^{-k-1}\operatorname{\mathcal{E}}^{\boxtimes(g-2)-2k}\bigr{\rangle}_{0\leq
k\leq\lfloor{g-2\over 2}\rfloor}.$

Take the next block to process from (6.5). Arguing as in the case of mega-
block IV, it suffices to prove the following lemma:

######  Lemma 6.5.

Suppose $0\leq\lambda\leq g-2$, $0\leq m\leq\lfloor{\lambda\over 2}\rfloor$.
The orthogonal projector onto ${}^{\perp}\operatorname{\mathcal{B}}$ takes the
block
$\bigl{\langle}Z^{-\lambda+(g-2)}{\theta^{*}}^{-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\bigr{\rangle}$
into $\operatorname{\mathcal{A}}^{\prime}$, where $\operatorname{\mathcal{B}}$
is given by two blocks from (6.4) with $k=m$ and $m+1$, unless $\lambda=2m$,
in which case we take only one block, namely the block with $k=m$.

###### Proof.

We rewrite the kernel vector bundle as

$Z^{-\lambda+(g-2)}{\theta^{*}}^{-m-1}\operatorname{\mathcal{E}}^{\boxtimes\lambda-2m}\simeq
Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\operatorname{\mathcal{F}}^{*\boxtimes\lambda-2m}.$

Take the complex
$Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\operatorname{\mathcal{F}}^{\bullet\lambda-2m}$
from Definition 4.6.If $\lambda\neq 2m$, let
$Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\otimes[\operatorname{\mathcal{F}}^{*\boxtimes\lambda-2m}\to\operatorname{Ker}^{1-(\lambda-2m)}]$
be its $2$-step smart truncation. By Lemma 4.9 and Corollary 4.10, it suffices
to prove that the subcategory $\langle
Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m}\rangle$,
and subcategories $\langle
Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m-1}\rangle$,
$\langle
Z^{(g-2)+2m-2\lambda}{\theta^{*}}^{-\lambda+m-1}\operatorname{{\boldsymbol{\Lambda}}}^{*\lambda-2m}[\operatorname{\mathcal{D}}^{1}]^{\vee}\rangle$
if $\lambda\neq 2m$, are moved into $\operatorname{\mathcal{A}}^{\prime}$ by
the projector from the lemma. These subcategories are the same as $\langle
Z^{(g-2)-2m}{\theta^{*}}^{-m-1}\rangle$, $\langle
Z^{(g-2)-2m-2}{\theta^{*}}^{-m-2}\rangle$, $\langle
Z^{(g-2)-2m}{\theta^{*}}^{-m-1}[\operatorname{\mathcal{D}}^{1}]^{\vee}\rangle$.
So the claim follows from Theorem 6.3. ∎

Mega-block IIa contains the blocks

(6.7)