# The BGMN conjecture via stable pairs

Jenia Tevelev  and  Sebastián Torres  In memory of M. S. Narasimhan

###### Abstract.

Let $C$ be a smooth projective curve of genus $g\geq 2$ and let $N$ be the
moduli space of stable rank $2$ vector bundles on $C$ and fixed determinant of
odd degree. We construct a semi-orthogonal decomposition of $D^{b}(N)$
conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two
blocks for each $i$-th symmetric power of $C$ for $i=0,\ldots,g-2$ and one
block for the $(g-1)$-st symmetric power. We conjecture that the subcategory
generated by our blocks has a trivial semi-orthogonal complement, proving the
full BGMN conjecture. Our proof is based on an analysis of wall-crossing
between moduli spaces of stable pairs, combining classical vector bundles
techniques with the method of windows.

##  1\. Introduction

Let $C$ be a smooth complex projective curve of genus $g\geq 2$. Let
$N=M_{C}(2,\Lambda)$ be the moduli space of stable vector bundles on $C$ of
rank $2$ and fixed determinant $\Lambda$ of odd degree. It is a smooth Fano
variety of index $2$, let $\theta$ be an ample generator of
$\operatorname{Pic}N=\mathbb{Z}$.

######  Theorem 1.1.

$D^{b}(N)$ has a semi-orthogonal decomposition
$\langle\operatorname{\mathcal{P}},\operatorname{\mathcal{A}}\rangle$, where

(1.1)
$\begin{matrix}\hskip-6.0pt\operatorname{\mathcal{A}}=\langle&\theta^{*}\otimes\operatorname{\mathcal{G}}_{0},&(\theta^{*})^{2}\otimes\operatorname{\mathcal{G}}_{2},&(\theta^{*})^{3}\otimes\operatorname{\mathcal{G}}_{4},&(\theta^{*})^{4}\otimes\operatorname{\mathcal{G}}_{6},&\ldots&,\\\
&\theta^{*}\otimes\operatorname{\mathcal{G}}_{1},&(\theta^{*})^{2}\otimes\operatorname{\mathcal{G}}_{3},&(\theta^{*})^{3}\otimes\operatorname{\mathcal{G}}_{5},&(\theta^{*})^{4}\otimes\operatorname{\mathcal{G}}_{7},&\ldots&,\\\
&\operatorname{\mathcal{G}}_{0},&\theta^{*}\otimes\operatorname{\mathcal{G}}_{2},&(\theta^{*})^{2}\otimes\operatorname{\mathcal{G}}_{4},&(\theta^{*})^{3}\otimes\operatorname{\mathcal{G}}_{6},&\ldots&,\\\
&\operatorname{\mathcal{G}}_{1},&\theta^{*}\otimes\operatorname{\mathcal{G}}_{3},&(\theta^{*})^{2}\otimes\operatorname{\mathcal{G}}_{5},&(\theta^{*})^{3}\otimes\operatorname{\mathcal{G}}_{7},&\ldots&\rangle\\\
\end{matrix}$

Each subcategory $\operatorname{\mathcal{G}}_{i}\cong
D^{b}(\operatorname{Sym}^{i}C)$ is embedded in $D^{b}(N)$ by a fully faithful
Fourier-Mukai functor with kernel given by the $i$-th tensor bundle
$\operatorname{\mathcal{E}}^{\boxtimes i}$ (see Section 2) of the Poincaré
bundle $\operatorname{\mathcal{E}}$ on $C\times N$ normalized so that
$\det\operatorname{\mathcal{E}}_{x}\cong\theta$ for every $x\in C$. There are
two blocks isomorphic to $D^{b}(\operatorname{Sym}^{i}C)$ for each
$i=0,\ldots,g-2$ and one block isomorphic to
$D^{b}(\operatorname{Sym}^{g-1}C)$, which appears on the $1$st or $2$nd line
of (1.1), depending on parity of $g$.

The blocks appearing in (1.1) cannot be further decomposed [lin], but there is
some flexibility in their order, most importantly blocks within each of the
four lines are mutually orthogonal (see Corollary 9.8). This is remarkably
compatible with the results of Muñoz [munoz1, munoz2, munoz3] (cf. [belmans21,
Prop. 6.4.2]), that the operator of the quantum multiplication by $c_{1}(N)$
on the quantum cohomology $QH^{\bullet}(N)$ has eigenvalues $8\lambda$, where

$\lambda=(1-g),\ (2-g)\sqrt{-1},\ (3-g),\ldots,\ (g-3),\ (g-2)\sqrt{-1},\
(g-1)$

and the eigenspace of $8\lambda$ is isomorphic to
$H^{\bullet}(\operatorname{Sym}^{g-1-|\lambda|}C)$. The four lines of (1.1)
correspond to the four coordinate rays of the complex plane. There are many
other results, e.g. [delbano, lee], on cohomology and motivic decomposition of
$N$ compatible with (1.1). This provides an ample evidence towards the
expectation that $\operatorname{\mathcal{P}}=0$. We hope to address this
question in the future. We also hope that our methods will be useful in
studying properties of analogous Fourier–Mukai functors for moduli spaces of
vector bundles of higher rank on curves and for moduli spaces of sheaves with
$1$-dimensional support on K3 surfaces.

Partial results towards Theorem 1.1 have appeared in the literature. The case
$g=2$ is a classical theorem of Bondal and Orlov [bondal-orlov, Theorem 2.9],
who also proved that $\operatorname{\mathcal{P}}=0$ in that case. Fonarev and
Kuznetsov [fonarev] proved that $D^{b}(C)\hookrightarrow D^{b}(N)$ if $C$ is a
hyperelliptic curve using an explicit description of $N$ due to Desale and
Ramanan [desale]. They also proved that $D^{b}(C)\hookrightarrow D^{b}(N)$ for
a general curve $C$ by a deformation argument. Narasimhan proved that
$D^{b}(C)\hookrightarrow D^{b}(N)$ for all curves [narasimhan1, narasimhan2]
using Hecke correspondences. He also showed that one can add the line bundles
$\operatorname{\mathcal{O}}$ and $\theta^{*}$ to $D^{b}(C)$ to start a semi-
orthogonal decomposition of $D^{b}(N)$.

In [belmans], Belmans and Mukhopadhyay work with the moduli space
$M_{C}(r,\Lambda)$ of vector bundles of rank $r$ and determinant $\Lambda$,
where $r\geq 2$ and $(r,\deg\Lambda)=1$. They show that there is a fully
faithful embedding $D^{b}(C)\hookrightarrow D^{b}(M_{C}(r,\Lambda))$ provided
the genus is sufficiently high. Moreover, they use this embedding to find the
start of a semi-orthogonal decomposition of $D^{b}(M_{C}(r,\Lambda))$ of the
form $\theta^{*}$, $D^{b}(C)$, $\operatorname{\mathcal{O}}$,
$\theta^{*}\otimes D^{b}(C)$, this way extending the decomposition on
$N=M_{C}(2,\Lambda)$ present in [narasimhan2]. Belmans, Galkin and
Mukhopadhyay have conjectured, independently of Narasimhan, that $D^{b}(N)$
should have a semi-orthogonal decomposition with blocks
$D^{b}(\operatorname{Sym}^{i}C)$ (see [oberwolfach, lee]), and have collected
additional evidence towards this conjecture in [belmans21]. Lee and Narasimhan
[lee-narasimhan] proved using Hecke correspondences that, if $C$ is non-
hyperelliptic and $g\geq 16$, there is a fully faithful functor
$D^{b}(\operatorname{Sym}^{2}C)\hookrightarrow D^{b}(N)$ whose image is left
semi-orthogonal to the copy of $D^{b}(C)$ obtained earlier. They also
introduced tensor bundles $\operatorname{\mathcal{E}}^{\boxtimes i}$ of the
Poincaré bundle (see Section 2), which we discovered independently. If
$D\in\operatorname{Sym}^{i}C$ is a reduced sum of points $x_{1}+\ldots+x_{i}$,
the fiber $(\operatorname{\mathcal{E}}^{\boxtimes i})_{D}$ is a vector bundle
on $N$ isomorphic to the tensor product
$\operatorname{\mathcal{E}}_{x_{1}}\otimes\ldots\otimes\operatorname{\mathcal{E}}_{x_{1}}$.
If the points have multiplicities, $(\operatorname{\mathcal{E}}^{\boxtimes
i})_{D}$ is a deformation of the tensor product over
$\operatorname{{\mathbb{A}}}^{1}$ (see Corollary 2.10).

Instead of using Hecke correspondences (although they do make a guest
appearance in Section 6), we prove Theorem 1.1 by analyzing Fourier–Mukai
functors given by tensor bundles $F^{\boxtimes i}$ of the universal bundle $F$
on the moduli space of stable pairs $(E,\phi)$, where $E$ is a rank-two vector
bundle on $C$ with fixed odd determinant line bundle of degree $d$ and
$\phi\in H^{0}(E)$ is a non-zero section. Stability condition on these spaces
depends on a parameter, and we use extensively results of Thaddeus [thaddeus]
on wall-crossing. If $d=2g-1$ then there is a well-known diagram of flips

(1.2)
${\tilde{M}_{2}}$${\tilde{M}_{3}}$${\tilde{M}_{g-1}}$${M_{1}}$${M_{2}}$${\cdots}$${M_{g-1}}$${M_{0}}$${N}$

where $M_{0}={\mathbb{P}}^{3g-3}$, $M_{1}$ is the blow up of $M_{0}$ in $C$,
the rational map $M_{i-1}\dashrightarrow M_{i}$ is a standard flip of
projective bundles over $\operatorname{Sym}^{i}C$, and $\xi:\,M_{g-1}\to N$ is
a birational Abel–Jacobi map with fiber $\mathbb{P}H^{0}(E)$ over a stable
vector bundle $E$. Accordingly, $D^{b}(M_{i})$ has a semi-orthogonal
decomposition into $D^{b}(M_{i-1})$ and several blocks equivalent to
$D^{b}(\operatorname{Sym}^{i}C)$ with torsion supports (see Proposition 3.15
or [bfr]). While these decompositions do not descend to $N$ and are not
associated with the universal bundle, they are useful. Philosophically, tensor
bundles on $\operatorname{Sym}^{i}C\times N$ should be related to exterior
powers of the tautological bundle of the universal bundle, which appear in the
Koszul complex of the tautological section that vanishes on the flipped locus.
One can try to connect two Fourier–Mukai functors via mutations. In practice,
this Koszul complex is difficult to analyze except for $M_{1}$ (see Section 5)
and we followed a less direct strategy towards proving Theorem 1.1.

In order to prove semi-orthogonality in (1.1) and full faithfulness of the
Fourier–Mukai functors via the Bondal–Orlov criterion, we had to investigate
coherent cohomology for a large class of vector bundles. The main difficulty
in this kind of analysis is to find a priori numerical bounds on the class of
acyclic vector bundles to get the induction going.

######  Definition 1.2.

For an object $\operatorname{\mathcal{F}}$ in the derived category of a scheme
$M$, we say that $\operatorname{\mathcal{F}}$ is $\Gamma$-acyclic if
$R\Gamma(\operatorname{\mathcal{F}})=0$. That is, for us $\Gamma$-acyclicity
will mean vanishing of _all_ cohomology groups, including
$H^{0}(\operatorname{\mathcal{F}})$. Other authors have used the term
_immaculate_ for this property (cf. [immaculate]).

Theorem 1.1 then requires the proof of $\Gamma$-acyclicity for several vector
bundles. It is worth emphasizing that the moduli space $N$ depends on the
complex structure of the curve $C$ by a classical theorem of Mumford and
Newstead [mumford-newstead] later extended by Narasimhan and Ramanan
[narasimhan-ramanan75]. The uniform shape of Theorem 1.1 is thus a
surprisingly strong statement about coherent cohomology of vector bundles on
$N$ that does not involve any conditions of the Brill–Noether type. Our
approach utilizes the method of windows into derived categories of GIT
quotients of Teleman, Halpern–Leistner, and Ballard–Favero–Katzarkov [teleman,
dhl, katzarkov] to systematically predict behavior of coherent cohomology
under wall-crossing. This dramatically reduces otherwise unwieldy
cohomological computations to a few key cases, which can be analyzed using
other techniques. Rather unexpectedly, one of the difficult ingredients in the
proof is acyclicity of certain line bundles (see Section 6). While cohomology
of line bundles on the space of stable pairs was extensively studied in
[thaddeus] in order to prove the Verlinde formula, the line bundle that we
need is outside of the scope of that paper.

Analogous recent applications of windows to moduli spaces include the proof of
the Manin–Orlov conjecture on $\bar{M}_{0,n}$ by Castravet and Tevelev
[castravet1, castravet1.1, castravet2] and analysis of Bott vanishing on GIT
quotients by Torres [myself].

We are grateful to Pieter Belmans for bringing papers [oberwolfach, belmans21]
to our attention and to the anonymous referee for sending us a long list of
thoughtful comments and suggestions. J.T. was supported by the NSF grant
DMS-2101726. Some results of the paper have first appeared in the PhD thesis
[mythesis] of S.T.

##  2\. Tensor vector bundles

Let $C$ be a smooth projective curve over $\mathbb{C}$. For integers
$\alpha\geq 1$ and $1\leq j\leq\alpha$, let $\pi_{j}:C^{\alpha}\to C$ be the
$j$-th projection and $\tau:C^{\alpha}\to\operatorname{Sym}^{\alpha}C$ the
categorical $S_{\alpha}$-quotient, where $S_{\alpha}$ is the symmetric group.
Since $C^{\alpha}$ is Cohen–Macaulay (in fact smooth),
$\operatorname{Sym}^{\alpha}C$ is smooth, and $\tau$ is equi-dimensional, we
conclude that $\tau$ is flat by miracle flatness. Therefore, any base change
$\tau:\,C^{\alpha}\times M\to\operatorname{Sym}^{\alpha}C\times M$ is also a
finite and flat categorical $S_{\alpha}$-quotient, where $M$ is any scheme
over $\mathbb{C}$. The constructions in this section are functorial in $M$. In
the following sections, $M$ will be one of the moduli spaces we consider.

######  Notation 2.1.

For an $S_{\alpha}$-equivariant vector bundle $\operatorname{\mathcal{E}}$ on
$C^{\alpha}\times M$, we will denote by
$\tau_{*}^{S_{\alpha}}\operatorname{\mathcal{E}}$ the $S_{\alpha}$-invariant
part of the pushforward $\tau_{*}\operatorname{\mathcal{E}}$.

######  Lemma 2.2.

Let $\operatorname{\mathcal{E}}$ be an $S_{\alpha}$-equivariant locally free
sheaf on $C^{\alpha}\times M$. Then $\tau_{*}\operatorname{\mathcal{E}}$ and
$\tau_{*}^{S_{\alpha}}\operatorname{\mathcal{E}}$ are locally free sheaves on
$\operatorname{Sym}^{\alpha}C\times M$.

###### Proof.

The scheme $C^{\alpha}\times M$ is covered by $S_{\alpha}$-equivariant affine
charts $\operatorname{Spec}R$ and $\tau^{*}$ is given by the inclusion of
invariants $R^{S_{\alpha}}\subset R$. Since $R$ is a finitely generated and
flat $R^{G}$-module, it is also a projective $R^{G}$-module. Let
$E=H^{0}(\operatorname{Spec}R,\operatorname{\mathcal{E}})$. Since $E$ is a
projective $R$-module, it is a direct summand of $R^{s}$ for some $s$. It
follows that $E$ is a projective $R^{G}$-module, i.e.
$\tau_{*}\operatorname{\mathcal{E}}$ is locally free. Since $E^{S_{\alpha}}$
is a direct summand of $E$ as an $R^{G}$-module, it is also a projective
$R^{G}$-module. Therefore, $\tau_{*}^{S_{\alpha}}\operatorname{\mathcal{E}}$
is a locally free sheaf as well. ∎

######  Definition 2.3.

For any vector bundle $\mathcal{F}$ on $C\times M$, we define the following
tensor vector bundles on $\operatorname{Sym}^{\alpha}C\times M$,

$\mathcal{F}^{\boxtimes\alpha}=\tau_{*}^{S_{\alpha}}\left(\bigotimes_{j=1}^{\alpha}\pi_{j}^{*}\mathcal{F}\right)\quad\hbox{\rm
and}\quad\overline{\mathcal{F}}^{\boxtimes\alpha}=\tau_{*}^{S_{\alpha}}\left(\bigotimes_{j=1}^{\alpha}\pi_{j}^{*}\mathcal{F}\otimes\operatorname{sgn}\right),$

where $S_{\alpha}$ acts on $C^{\alpha}$ and also permutes the factors of the
corresponding vector bundle on $C^{\alpha}$. Here $\operatorname{sgn}$ is the
sign representation of $S_{\alpha}$.

######  Lemma 2.4.

The formation of tensor vector bundles is functorial in $M$, that is, given a
morphism $f:\,M^{\prime}\to M$ and its base changes $C\times M^{\prime}\to
C\times M$ and $\operatorname{Sym}^{\alpha}C\times
M^{\prime}\to\operatorname{Sym}^{\alpha}C\times M$, which we also denote by
$f$, we have

$f^{*}(\mathcal{F}^{\boxtimes\alpha})=(f^{*}\mathcal{F})^{\boxtimes\alpha}\quad\hbox{\rm
and}\quad
f^{*}(\overline{\mathcal{F}}^{\boxtimes\alpha})=\overline{(f^{*}\mathcal{F})}^{\boxtimes\alpha}.$

###### Proof.

Since $\tau$ is flat, this follows from cohomology and base change. ∎

######  Remark 2.5.

Similarly, one can define bundles
$\mathcal{F}^{\boxplus\alpha}=\tau_{*}^{S_{\alpha}}\left(\bigoplus_{j=1}^{\alpha}\pi_{j}^{*}\mathcal{F}\right)$
and
$\overline{\mathcal{F}}^{\boxplus\alpha}=\tau_{*}^{S_{\alpha}}\left(\bigoplus_{j=1}^{\alpha}\pi_{j}^{*}\mathcal{F}\otimes\operatorname{sgn}\right)$.

######  Remark 2.6.

Since $\operatorname{\mathcal{O}}_{C^{\alpha}\times
M}\otimes\operatorname{sgn}$ is the equivariant dualizing sheaf for $\tau$
(see [ks, Lemma 5.8]), by Grothendieck duality we have
$\left(\mathcal{F}^{\boxtimes\alpha}\right)^{\vee}=\overline{\mathcal{F}^{\vee}}^{\boxtimes\alpha}.$

For a divisor $D\in\operatorname{Sym}^{\alpha}C$ and a vector bundle
$\mathcal{G}$ on $\operatorname{Sym}^{\alpha}C\times M$, let us denote by
$\mathcal{G}_{D}=\mathcal{G}|_{\\{D\\}\times M}$. We usually view
$\mathcal{G}_{D}$ as a vector bundle on $M$.

######  Lemma 2.7.

If $D=\sum\alpha_{k}x_{k}$ with $x_{k}\neq x_{l}$ for $k\neq l$, then we have

(2.1) $\displaystyle\begin{aligned}
\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}=\bigotimes\left(\mathcal{F}^{\boxtimes\alpha_{k}}\right)_{\alpha_{k}x_{k}},\qquad\left(\overline{\mathcal{F}}^{\boxtimes\alpha}\right)_{D}=\bigotimes\left(\overline{\mathcal{F}}^{\boxtimes\alpha_{k}}\right)_{\alpha_{k}x_{k}}.\end{aligned}$

###### Proof.

Indeed, the quotient $\tau:\,C^{\alpha}\to\operatorname{Sym}^{\alpha}C$ is
étale-locally near $D\in\operatorname{Sym}^{\alpha}C$ isomorphic to the
product of quotients $\prod
C^{\alpha_{k}}\to\prod\operatorname{Sym}^{\alpha_{k}}C$. Moreover,
$\operatorname{sgn}$ restricts to the tensor product of sign representations
of $\prod S_{\alpha_{k}}$. ∎

Consider the non-reduced scheme
$\mathbb{D}_{\alpha}=\operatorname{Spec}\mathbb{C}[t]/t^{\alpha}$, with maps
$\operatorname{pt}\xhookrightarrow{\imath}\mathbb{D}_{\alpha}\xrightarrow{\rho}\operatorname{pt}$
given by the obvious pullbacks
$\mathbb{C}\xrightarrow{\rho^{*}}\mathbb{C}[t]/t^{\alpha}\xrightarrow{\imath^{*}}\mathbb{C}$.
We still write $\imath$ and $\rho$ for the base changes to $M$ of these
morphisms, that is, $M\xrightarrow{\imath}\mathbb{D}_{\alpha}\times
M\xrightarrow{\rho}M$. For a locally free sheaf $\mathcal{F}$ on
$\mathbb{D}_{\alpha}\times M$, we denote by
$\mathcal{F}_{0}=\imath^{*}\mathcal{F}$ its restriction to $M$.

######  Definition 2.8.

For two vector bundles $\operatorname{\mathcal{F}}$,
$\operatorname{\mathcal{G}}$ on a scheme $M$, we will say that
$\operatorname{\mathcal{F}}$ is a _deformation of $\operatorname{\mathcal{G}}$
over $\operatorname{{\mathbb{A}}}^{1}$_ if there is a coherent sheaf
$\tilde{\operatorname{\mathcal{F}}}$ on
$M\times\operatorname{{\mathbb{A}}}^{1}$, flat over
$\operatorname{{\mathbb{A}}}^{1}$, with
$\tilde{\operatorname{\mathcal{F}}}|_{t}\cong\operatorname{\mathcal{F}}$ for
$t\neq 0$, while
$\tilde{\operatorname{\mathcal{F}}}|_{0}\cong\operatorname{\mathcal{G}}$.

######  Lemma 2.9.

Every locally free sheaf $\mathcal{F}$ on $\mathbb{D}_{\alpha}\times M$ is a
deformation of $\rho^{*}\mathcal{F}_{0}$ over $\mathbb{A}^{1}$. In particular,
$\rho_{*}\mathcal{F}$ is a deformation of $\mathcal{F}_{0}^{\oplus\alpha}$
over $\mathbb{A}^{1}$.

###### Proof.

Let
$\lambda:\mathbb{A}^{1}_{s}\times\mathbb{D}_{\alpha}\to\mathbb{D}_{\alpha}$ be
the map defined by its pullback $\lambda^{*}:t\mapsto ts$, and also denote by
$\lambda$ its base change to $M$. We claim that the locally free sheaf
$\lambda^{*}\mathcal{F}$ gives the required deformation. Indeed, the
restriction of $\lambda^{*}\mathcal{F}$ to $\\{s_{0}\\}\in\mathbb{A}^{1}_{s}$
is the pullback of $\mathcal{F}$ along the composition $b_{s_{0}}=\lambda\circ
j_{s_{0}}$

$\mathbb{D}_{\alpha}\times
M\xhookrightarrow{j_{s_{0}}}\mathbb{D}_{\alpha}\times\mathbb{A}^{1}_{s}\times
M\xrightarrow{\lambda}\mathbb{D}_{\alpha}\times M$

determined by its pullback $b_{s_{0}}^{*}:t\mapsto s_{0}t$. When $s_{0}\neq
0$, $b_{s_{0}}^{*}\mathcal{F}\cong\mathcal{F}$. On the other hand, when
$s_{0}=0$, the map $b_{0}$ factors as the composition

$\mathbb{D}_{\alpha}\times
M\xrightarrow{\rho}M\xrightarrow{\imath}\mathbb{D}_{\alpha}\times M$

so
$b_{0}^{*}\mathcal{F}=\rho^{*}\imath^{*}\mathcal{F}=\rho^{*}\mathcal{F}_{0}$,
as desired. The last statement follows from projection formula and the fact
that $\rho_{*}\rho^{*}\mathcal{O}_{M}\cong\mathcal{O}_{M}^{\oplus\alpha}$. ∎

Suppose $D=\alpha x$ is a fat point, i.e. a divisor given by a single point
$x$ with multiplicity $\alpha$, and let $t$ be a local parameter on $C$ at
$x$. Note that the notation $\operatorname{\mathcal{O}}_{D}$ is unfortunately
ambiguous, because it can denote both the structure sheaf of the subscheme
$D\subset C$ and the skyscraper sheaf of the point
$\\{D\\}\in\operatorname{Sym}^{\alpha}C$. When confusion is possible, we
denote the latter sheaf by $\operatorname{\mathcal{O}}_{\\{D\\}}$. Then

(2.2)
$\displaystyle\tau^{*}\mathcal{O}_{\\{D\\}}\cong\frac{\mathbb{C}[t_{1},\ldots,t_{\alpha}]}{(\sigma_{1},\ldots,\sigma_{\alpha})}$

is the so-called covariant algebra, where $\sigma_{1},\ldots,\sigma_{\alpha}$
are the elementary symmetric functions in variables $t_{j}=\pi_{j}^{*}(t)$.
Call $\mathbb{B}_{\alpha}=\operatorname{Spec}\tau^{*}\mathcal{O}_{\\{D\\}}$.
By the Newton formulas, $t_{j}^{\alpha}=0$ for every $j=1,\ldots,\alpha$, and
in particular, every map $\pi_{j}:\,\mathbb{B}_{\alpha}\to C$ factors through
$\mathbb{D}_{\alpha}$. By abuse of notation, we have a diagram of morphisms

(2.3) ${\mathbb{B}_{\alpha}\times M}$${\mathbb{D}_{\alpha}\times M}$${C\times
M}$${M}$$\scriptstyle{\pi_{j}}$$\scriptstyle{\tau}$$\scriptstyle{q}$$\scriptstyle{\rho}$$\scriptstyle{\imath}$

######  Corollary 2.10.

Let $D=x_{1}+\ldots+x_{\alpha}$ (possibly with repetitions). Then both
$\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ and
$\left(\overline{\mathcal{F}}^{\boxtimes\alpha}\right)_{D}$ are deformations
of $\mathcal{F}_{x_{1}}\otimes\ldots\otimes\mathcal{F}_{x_{\alpha}}$ over
$\mathbb{A}^{1}$.

###### Proof.

By (2.1), it suffices to consider the case when $D=\alpha x$. Using the
notation as in the diagram (2.3), the restriction
$\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ can be written as
$\tau_{*}^{S_{\alpha}}\left(\bigotimes\pi_{j}^{*}q^{*}\mathcal{F}\right)$, by
flatness of $\tau$. The construction of Lemma 2.9 commutes with the
$S_{\alpha}$-action, so
$\tau_{*}^{S_{\alpha}}\left(\bigotimes\pi_{j}^{*}q^{*}\mathcal{F}\right)$ is a
deformation of
$\tau_{*}^{S_{\alpha}}\left(\bigotimes\pi_{j}^{*}\rho^{*}\mathcal{F}_{x}\right)$
over $\operatorname{{\mathbb{A}}}^{1}$, since
$(q^{*}\mathcal{F})_{0}=\mathcal{F}_{x}=\left.\mathcal{F}\right|_{\\{x\\}\times
M}$. Note $\pi_{j}^{*}\rho^{*}=\tau^{*}$, so using the projection formula, we
get that $\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ is a deformation of
$\left(\bigotimes_{j=1}^{\alpha}\mathcal{F}_{x}\right)\otimes\tau_{*}^{S_{\alpha}}(\mathcal{O}_{\mathbb{B}_{\alpha}\times
M})$, and similarly,
$\left(\overline{\mathcal{F}}^{\boxtimes\alpha}\right)_{D}$ is a deformation
of
$\left(\bigotimes_{j=1}^{\alpha}\mathcal{F}_{x}\right)\otimes\tau_{*}^{S_{\alpha}}(\mathcal{O}_{\mathbb{B}_{\alpha}\times
M}\otimes\operatorname{sgn})$. By flatness of the quotient
$C^{\alpha}\to\operatorname{Sym}^{\alpha}C$, the covariant algebra
$\mathcal{O}_{\mathbb{B}_{\alpha}}$ (2.2) is the regular representation
$\mathbb{C}[S_{\alpha}]$ of $S_{\alpha}$. It follows that it contains the
trivial and the sign representations each with multiplicity $1$, and therefore
$\tau_{*}^{S_{\alpha}}(\mathcal{O}_{\mathbb{B}_{\alpha}\times
M})=\tau_{*}^{S_{\alpha}}(\mathcal{O}_{\mathbb{B}_{\alpha}\times
M}\otimes\operatorname{sgn})=\mathcal{O}_{M}$. This concludes the proof. ∎

######  Remark 2.11.

If we have a $G$-action on $M$ and a $G$-equivariant bundle
$\operatorname{\mathcal{F}}$, then the deformation of Corollary 2.10 can also
be chosen to be $G$-equivariant, i.e. given by a $G$-equivariant bundle on
$\operatorname{{\mathbb{A}}}^{1}\times M$. This is because the deformation in
the proof of Lemma 2.9 can be chosen to be $G$-equivariant.

######  Definition 2.12.

A vector bundle $\mathcal{F}$ on a scheme $M$ is said to be a stable
deformation of a vector bundle $\mathcal{G}$ _over $\mathbb{A}^{1}$_ if there
is some vector bundle $\mathcal{K}$ such that $\mathcal{F}\oplus\mathcal{K}$
is a deformation of a direct sum $\mathcal{G}^{\oplus r}$ for some $r>0$.

######  Proposition 2.13.

Let $D=x+\tilde{D}$. Then the vector bundle
$\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ is a stable deformation of
the vector bundle
$\mathcal{F}_{x}\otimes\left(\mathcal{F}^{\boxtimes(\alpha-1)}\right)_{\tilde{D}}$
over $\operatorname{{\mathbb{A}}}^{1}$.

###### Proof.

By Lemma 2.7, it suffices to consider the case $D=\alpha x$. Let
$W_{\alpha}=\mathbb{C}^{\alpha}$ be the tautological representation of
$S_{\alpha}$, which splits as a sum of the trivial and the standard
representations, $W_{\alpha}=\mathbb{C}\oplus V_{\alpha}$. For any
$S_{\alpha}$-equivariant vector bundle $\mathcal{E}$ on
$\mathbb{B}_{\alpha}\times M$, we have

(2.4) $\displaystyle\tau_{*}^{S_{\alpha}}(\mathcal{E}\otimes
W_{\alpha})=\tau_{*}^{S_{\alpha}}(\mathcal{E})\oplus\tau_{*}^{S_{\alpha}}(\mathcal{E}\otimes
V_{\alpha}).$

On the other hand, we have $W_{\alpha}=\mathbb{C}[S_{\alpha}/S_{\alpha-1}]$,
where $S_{\alpha-1}\hookrightarrow S_{\alpha}$ is the inclusion given by
fixing the $\alpha$-th element. Then, by Frobenius reciprocity,
$\tau_{*}^{S_{\alpha}}(\mathcal{E}\otimes
W_{\alpha})=\tau_{*}^{S_{\alpha-1}}(\mathcal{E})=\rho_{*}\circ(\pi_{\alpha})^{{S_{\alpha-1}}}_{*}(\mathcal{E})$,
where $\pi_{\alpha}$ is the $\alpha$-th projection. By Lemma 2.9 this bundle
is a deformation of
$\left((\pi_{\alpha})^{S_{\alpha-1}}_{*}\mathcal{E}\right)_{0}^{\oplus\alpha}$
over $\operatorname{{\mathbb{A}}}^{1}$. Now let $\mathcal{E}$ be
$\bigotimes\pi_{j}^{*}q^{*}\mathcal{F}$. Then
$\tau_{*}^{S_{\alpha}}(\mathcal{E})$ is precisely
$\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ and, by projection formula,

$\displaystyle\left((\pi_{\alpha})^{S_{\alpha-1}}_{*}\mathcal{E}\right)_{0}$
$\displaystyle=\mathcal{F}_{x}\otimes\left((\pi_{\alpha})_{*}^{S_{\alpha-1}}\left(\bigotimes_{j=1}^{\alpha-1}\pi_{j}^{*}q^{*}\mathcal{F}\right)\right)_{0}$
$\displaystyle=\mathcal{F}_{x}\otimes(\pi_{\alpha})_{*}^{S_{\alpha-1}}\bigotimes_{j=1}^{\alpha-1}(\pi_{j}^{*}q^{*}\mathcal{F})|_{t_{\alpha}=0}=\mathcal{F}_{x}\otimes\left(\mathcal{F}^{\boxtimes(\alpha-1)}\right)_{(\alpha-1)x}$

since the subscheme $(t_{\alpha}=0)\subset\mathbb{B}_{\alpha}$ is isomorphic
to $\mathbb{B}_{\alpha-1}$ and the restriction of $\pi_{\alpha}$ to it is
isomorphic to the quotient $\tau$ (for the group $S_{\alpha-1}$). ∎

######  Remark 2.14.

We will use stable deformations for semi-continuity arguments. If
$\mathcal{F}$ is a stable deformation of $\mathcal{G}$, $M$ is proper and
$H^{p}(\mathcal{G})=0$, then, by the semi-continuity theorem,
$H^{p}(\mathcal{F})=0$, too. In particular, if $\mathcal{G}$ is
$\Gamma$-acyclic, then so is $\mathcal{F}$.

######  Remark 2.15.

Let $D=x_{1}+\tilde{D}$, $\tilde{D}=x_{2}+\ldots+x_{\alpha}$ (possibly with
repetitions). Suppose $M$ is proper. Since
$\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D}$ and
$\mathcal{F}_{x_{1}}\otimes\left(\mathcal{F}^{\boxtimes(\alpha-1)}\right)_{\tilde{D}}$
are both deformations of
$\mathcal{F}_{x_{1}}\otimes\ldots\otimes\mathcal{F}_{x_{\alpha}}$ over
$\operatorname{{\mathbb{A}}}^{1}$ by Corollary 2.10, they have the same Euler
characteristic. Combining this with Remark 2.14, if
$H^{p}(\mathcal{F}_{x}\otimes\left(\mathcal{F}^{\boxtimes(\alpha-1)}\right)_{\tilde{D}})=0$
for $p>0$ then both $H^{p}(\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D})=0$
for $p>0$ and
$H^{0}(\left(\mathcal{F}^{\boxtimes\alpha}\right)_{D})=H^{0}(\mathcal{F}_{x}\otimes\left(\mathcal{F}^{\boxtimes(\alpha-1)}\right)_{\tilde{D}})$.
The same results hold for
$\left(\overline{\mathcal{F}}^{\boxtimes\alpha}\right)_{D}$ and
$\mathcal{F}_{x}\otimes\left(\overline{\mathcal{F}}^{\boxtimes(\alpha-1)}\right)_{\tilde{D}}$.

##  3\. Wall-crossing on moduli spaces of stable pairs

Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb{C}$. In
[thaddeus], Thaddeus studies moduli spaces of pairs $(E,\phi)$, where $E$ is a
rank-two vector bundle on $C$ with fixed determinant line bundle $\Lambda$ and
$\phi\in H^{0}(E)$ is a non-zero section. We use these results extensively and
so, for ease of reference, try to follow the notation in [thaddeus] as closely
as possible. We always assume that $d=\deg E>0$. For a given choice of a
parameter $\sigma\in\mathbb{Q}$ the following stability condition is imposed:
for every line subbundle $L\subset E$, one must have

$\deg L\leq\begin{cases}\frac{d}{2}-\sigma\quad&\text{ if }\phi\in
H^{0}(L),\\\ \frac{d}{2}+\sigma\quad&\text{ if }\phi\notin
H^{0}(L).\end{cases}$

Semi-stable pairs exist whenever $\sigma\in(0,d/2]$ [thaddeus, 1.3], which we
will assume. The next lemma follows the ideas of [thaddeus, 2.1]:

######  Lemma 3.1.

For a given line bundle $\Lambda$ of degree $d$, the moduli stack
$\operatorname{\mathcal{M}}_{\sigma}(\Lambda)$ of semi-stable pairs is a
smooth algebraic stack.

###### Proof.

$\operatorname{\mathcal{M}}_{\sigma}(\Lambda)$ is a fiber of the morphism
$\operatorname{\mathcal{M}}_{\sigma}^{d}\to\operatorname{Pic}^{d}(C)$,
$(E,\phi)\mapsto\det E$, from the stack of semi-stable pairs $(E,\phi)$, where
$E$ is a degree $d$ vector bundle. We first show that
$\operatorname{\mathcal{M}}_{\sigma}^{d}$ is smooth. Obstructions to
deformations of a morphism of sheaves $\phi$ from a fixed source
$\operatorname{\mathcal{O}}_{C}$ to a varying target $E$ lie in
$\operatorname{Ext}^{1}([\operatorname{\mathcal{O}}_{C}\mathop{\to}\limits^{\phi}E],E)$.
The truncation exact triangle of the complex
$[\operatorname{\mathcal{O}}_{C}\mathop{\to}\limits^{\phi}E]$ yields an exact
sequence

$\operatorname{Ext}^{1}(E,E)\mathop{\to}\limits^{\phi}\operatorname{Ext}^{1}(\operatorname{\mathcal{O}}_{C},E)\to\operatorname{Ext}^{1}([\operatorname{\mathcal{O}}_{C}\mathop{\to}\limits^{\phi}E],E)\to
0.$

We claim that the first map is surjective, so obstructions vanish. By Serre
duality, it suffices to prove injectivity of the map of sheaves
$E^{*}(K_{C})\mathop{\to}\limits^{\phi}E^{*}\otimes E(K_{C})$ and this follows
from $\phi\neq 0$ (cf. the proof of [thaddeus, 2.1]). Next we consider
obstructions to deformations of $(E,\phi)$ fixing the determinant, which
amounts to studying the map
$\operatorname{Ext}^{1}(E,E)_{0}\mathop{\to}\limits^{\phi}\operatorname{Ext}^{1}(\operatorname{\mathcal{O}}_{C},E)$,
where $\operatorname{Ext}^{1}(E,E)_{0}$ denotes traceless endomorphisms.
However, this map is also surjective because the Serre-dual map is induced by
the map of sheaves
$E^{*}(K_{C})\mathop{\to}\limits^{\phi}\operatorname{\mathcal{E}}\\!nd(E,E)_{0}(K_{C})$,
which is still injective - a non-zero scalar matrix cannot have rank $1$. ∎

The moduli space $M_{\sigma}(\Lambda)$ of $S$-equivalence classes of stable
pairs exists as a projective variety and, in the case there is no strictly
semi-stable locus, it is smooth, isomorphic to the stack
$\operatorname{\mathcal{M}}_{\sigma}(\Lambda)$ and carries a universal bundle
$F$ with a universal section $\tilde{\phi}:\operatorname{\mathcal{O}}_{C\times
M_{\sigma}(\Lambda)}\to F$. A salient point is that stable pairs, unlike
stable vector bundles, don’t have non-trivial automorphisms [thaddeus, 1.6].

The spaces $M_{\sigma}(\Lambda)$ can be obtained as GIT quotients as follows.
Let $\chi=\chi(E)=d+2-2g$. If $d\gg 0$, a bundle $E$ is generated by global
sections, and $\chi=h^{0}(E)$. Then $M_{\sigma}(\Lambda)$ is a GIT quotient of
$U\times\mathbb{PC}^{\chi}$ by $SL_{\chi}$, where
$U\subset\operatorname{Quot}$ is the locally closed subscheme of the Quot
scheme [groth] corresponding to locally free quotients
$\mathcal{O}_{C}^{\chi}\twoheadrightarrow E$ inducing an isomorphism
$s:\mathbb{C}^{\chi}\xrightarrow{\sim}H^{0}(E)$ and such that
$\Lambda^{2}E=\Lambda$. The isomorphism $s$ induces a map
$\Lambda^{2}\mathbb{C}^{\chi}\to H^{0}(\Lambda)$, and we get an inclusion
$U\times\mathbb{PC}^{\chi}\hookrightarrow\mathbb{P}\operatorname{Hom}\times\mathbb{PC}^{\chi}$,
where we write $\mathbb{P}\operatorname{Hom}$ for
$\mathbb{P}\operatorname{Hom}(\Lambda^{2}\mathbb{C}^{\chi},H^{0}(\Lambda))$,
and a quotient $s:\mathcal{O}_{C}^{\chi}\twoheadrightarrow E$ on the left is
sent to the induced map in the first coordinate. Then $M_{\sigma}(\Lambda)$
can be seen as the GIT quotient of a closed subset of
$\mathbb{P}\operatorname{Hom}\times\mathbb{PC}^{\chi}$ by $SL_{\chi}$, where
the linearization is given by $\mathcal{O}(\chi+2\sigma,4\sigma)$.

For arbitrary $d$, we pick any effective divisor $D$ on $C$ with $\deg D\gg
0$, and $M_{\sigma}(\Lambda)$ can be seen as the closed subset of
$M_{\sigma}(\Lambda(2D))$ consisting of pairs $(E,\phi)$ such that
$\phi|_{D}=0$. This way, $M_{\sigma}(\Lambda)$ is a GIT quotient by
$SL_{\chi^{\prime}}$, with $\chi^{\prime}=d+2-2g+2\deg D$, of the closed
subset $X\subset U^{\prime}\times\mathbb{PC}^{\chi^{\prime}}$ determined by
the condition that $\phi$ vanishes along $D$ [thaddeus, §1]. Regardless of the
GIT, the embedding $M_{\sigma}(\Lambda)\subset M_{\sigma}(\Lambda(2D))$ will
play an important role.

######  Remark 3.2.

Scalar matrices in $SL_{\chi^{\prime}}$ act trivially on
$U\times\mathbb{PC}^{\chi^{\prime}}$, so the action factors through the
quotient $SL_{\chi^{\prime}}\to PGL_{\chi^{\prime}}$. If we replace
$\mathcal{O}(\chi^{\prime}+2\sigma,4\sigma)$ by its $\chi^{\prime}$-th power,
this line bundle carries a $PGL_{\chi^{\prime}}$-linearization and
$M_{\sigma}(\Lambda)$ can also be written as a GIT quotient
$X\operatorname{/\\!\\!/}PGL_{\chi^{\prime}}$. Moreover, the moduli stack
$\operatorname{\mathcal{M}}_{\sigma}(\Lambda)$ is isomorphic to the
corresponding GIT quotient stack $[X^{ss}/PGL_{\chi^{\prime}}]$.

For fixed $\Lambda$, the spaces $M_{\sigma}(\Lambda)$ are all GIT quotients of
the same scheme, with different stability conditions. The GIT walls occur when
$\sigma\in d/2+\mathbb{Z}$, and for $0\leq i\leq v=\lfloor(d-1)/2\rfloor$ we
have different GIT chambers with moduli spaces $M_{0},M_{1},\ldots,M_{v}$,
where $M_{i}=M_{i}(\Lambda)=M_{\sigma}(\Lambda)$ for
$\sigma\in(\max(0,d/2-i-1),d/2-i)$. These $M_{i}$ are smooth projective
rational varieties of dimension $d+g-2$. Indeed,
$M_{0}=\mathbb{P}H^{1}(C,\Lambda^{-1})$ is a projective space, $M_{1}$ is a
blow-up of $M_{0}$ along a copy of $C$ embedded by the complete linear system
of $\omega_{C}\otimes\Lambda$, and the remaining ones are small modifications
of $M_{1}$. More precisely, for each $0\leq i\leq v=\lfloor(d-1)/2\rfloor$
there are projective bundles $\mathbb{P}W_{i}^{+}$ and $\mathbb{P}W_{i}^{-}$
over the symmetric product $\operatorname{Sym}^{i}C$, of (projective) ranks
$d+g-2i-2$, $i-1$, respectively, with embeddings
$\mathbb{P}W_{i}^{+}\hookrightarrow M_{i}$ and
$\mathbb{P}W_{i}^{-}\hookrightarrow M_{i-1}$, and such that
$\mathbb{P}W_{i}^{+}$ parametrizes the pairs $(E,\phi)$ appearing in $M_{i}$
but not in $M_{i-1}$, while $\mathbb{P}W_{i}^{-}$ parametrizes those appearing
in $M_{i-1}$ but not in $M_{i}$.

We have a diagram of flips (3.1), where $\tilde{M}_{i}$ is the blow-up of
$M_{i-1}$ along $\mathbb{P}W_{i}^{-}$ and also the blow-up of $M_{i}$ along
$\mathbb{P}W_{i}^{+}$. Here $N$ is the moduli space of ordinary slope-
semistable vector bundles as in the Introduction and the map $M_{v}\to N$ is
an “Abel-Jacobi” map with fiber $\mathbb{P}H^{0}(C,E)$ over a vector bundle
$E$. If $d\geq 2g-1$ the Abel-Jacobi map is surjective, and if $d=2g-1$ it is
a birational morphism (see [thaddeus, §3] for details).

(3.1)
${\tilde{M}_{2}}$${\tilde{M}_{3}}$${\tilde{M}_{v}}$${M_{1}}$${M_{2}}$${\cdots}$${M_{v}}$${M_{0}}$${N}$

######  Notation 3.3.

By abuse of notation, we will sometimes write $M_{i}(d)$ to denote the moduli
space $M_{i}=M_{i}(\Lambda)$, where $d=\deg\Lambda$.

######  Notation 3.4.

In what follows, $v$ will always denote $\lfloor(d-1)/2\rfloor$.

The Picard group of $M_{1}=\operatorname{Bl}_{C}M_{0}$ is generated by a
hyperplane section $H$ in $M_{0}=\mathbb{P}^{d+g-2}$ and the exceptional
divisor $E_{1}$ of the morphism $M_{1}\to M_{0}$. Since the maps
$M_{i}\dashrightarrow M_{i+1}$ are small birational modifications for each
$i\geq 1$, there are natural isomorphisms
$\operatorname{Pic}M_{1}\cong\operatorname{Pic}M_{i}$, $i\geq 1$. The
following notation is taken from [thaddeus, §5].

######  Definition 3.5.

For each $m$, $n$, we denote the line bundle
$\mathcal{O}_{M_{1}}((m+n)H-nE_{1})$ by $\mathcal{O}_{1}(m,n)$, while
$\mathcal{O}_{i}(m,n)$ will denote the image of $\mathcal{O}_{M_{1}}(m,n)$
under the isomorphism $\operatorname{Pic}M_{1}\cong\operatorname{Pic}M_{i}$.

######  Remark 3.6.

By [thaddeus, 5.3], the ample cone of $M_{i}$ is bounded by
$\operatorname{\mathcal{O}}_{i}(1,i-1)$ and
$\operatorname{\mathcal{O}}_{i}(1,i)$ for $0<i<v$, while the ample cone of
$M_{v}$ is bounded below by $\operatorname{\mathcal{O}}_{v}(1,v-1)$ and
contains the cone bounded on the other side by
$\operatorname{\mathcal{O}}_{v}(2,d-2)$.

######  Remark 3.7.

For any effective divisor $D$ on $C$ of $\deg D=\alpha$, we have a closed
immersion $M_{i-\alpha}(\Lambda(-2D))\hookrightarrow M_{i}(\Lambda)$, as the
locus of pairs $(E,\phi)$ where the section $\phi$ vanishes along $D$
[thaddeus, 1.9]. The restriction of $\operatorname{\mathcal{O}}_{i}(m,n)$ to
$M_{i-\alpha}(\Lambda(-2D))$ is
$\operatorname{\mathcal{O}}_{i-\alpha}(m,n-m\alpha)$ [thaddeus, 5.7].

Suppose $d\gg 0$. Then the universal bundle $F$ on $M_{i}$ is the descent from
the equivariant vector bundle $\mathcal{F}(1)$ on $X\times C\subset
U\times\mathbb{PC}^{\chi}\times C$, where
$\mathcal{O}^{\chi}\twoheadrightarrow\mathcal{F}$ is the universal quotient
bundle over $U\times C$, and the universal section $\tilde{\phi}$ descends
from the universal section of $\mathcal{F}(1)$ [thaddeus, 1.19]. Let
$\pi:C\times M_{i}\to M_{i}$ be the projection. For every $i\geq 1$, the
determinant of cohomology line bundle $\det\pi_{!}F$ (cf. [det-coh]) descends
from $\mathcal{O}(0,\chi)$ on
$\mathbb{P}\operatorname{Hom}\times\mathbb{PC}^{\chi}$ [thaddeus, 5.4 and
5.14]. On $M_{1}$, $\det\pi_{!}F$ corresponds to
$\mathcal{O}_{M_{1}}((g-d-1)H-(g-d)E_{1})=\mathcal{O}_{1}(-1,g-d)$. For $x\in
C$, call $F_{x}=F|_{\\{x\\}\times M}$. The line bundle $\det
F_{x}=\Lambda^{2}F_{x}$ does not depend on $x$, and it is the descent of
$\mathcal{O}(1,2)$ on $\mathbb{P}\operatorname{Hom}\times\mathbb{PC}^{\chi}$.
It corresponds to $\mathcal{O}_{M_{1}}(E_{1}-H)=\mathcal{O}_{i}(0,-1)$
[thaddeus, §5].

For arbitrary $d$, consider an embedding $\imath:M_{i}\hookrightarrow
M^{\prime}=M_{\sigma}(\Lambda(2D))$, $\deg D\gg 0$, as above, and let
$F^{\prime}$ be the universal bundle on $M^{\prime}$. Then we have a short
exact sequence [thaddeus, 1.19]

(3.2) $\displaystyle 0\to
F\to\imath^{*}F^{\prime}\to\imath^{*}F^{\prime}|_{D\times M_{i}}\to 0.$

In particular, $F$ is the descent from an object on $X\times C\subset
U^{\prime}\times\mathbb{PC}^{\chi^{\prime}}\times C$. The same is true for
$\det\pi_{!}F$ and $\Lambda^{2}F_{x}$.

######  Lemma 3.8.

$F_{x}\cong\imath^{*}F^{\prime}_{x}$ for every $x\in C$.

###### Proof.

We tensor (3.2) with $\operatorname{\mathcal{O}}_{\\{x\\}\times M_{i}}$, which
gives an exact sequence

$0\to{\mathcal{T}or}^{1}_{C\times M_{i}}(\imath^{*}F^{\prime}|_{D\times
M_{i}},\operatorname{\mathcal{O}}_{\\{x\\}\times M_{i}})\to
F_{x}\to\imath^{*}F^{\prime}_{x}\to\qquad\qquad\qquad$
$\qquad\qquad\qquad\to\imath^{*}F^{\prime}|_{D\times M_{i}}\otimes_{C\times
M_{i}}\operatorname{\mathcal{O}}_{\\{x\\}\times M_{i}}\to 0.$

If $x\not\in D$ then ${\mathcal{T}or}^{1}(\imath^{*}F^{\prime}|_{D\times
M_{i}},\operatorname{\mathcal{O}}_{\\{x\\}\times
M_{i}})=\imath^{*}F^{\prime}|_{D\times
M_{i}}\otimes\operatorname{\mathcal{O}}_{\\{x\\}\times M_{i}}=0$ and we get
$F_{x}\cong\imath^{*}F^{\prime}_{x}$. If $x\in D$ then
${\mathcal{T}or}^{1}_{C}(\operatorname{\mathcal{O}}_{D},\operatorname{\mathcal{O}}_{x})\cong\operatorname{\mathcal{O}}_{D}\otimes_{C}\operatorname{\mathcal{O}}_{x}\cong\operatorname{\mathcal{O}}_{x}$,
and the sequence splits into two isomorphisms,
$F_{x}\cong\imath^{*}F^{\prime}_{x}$ and
$\imath^{*}F^{\prime}_{x}\cong\imath^{*}F^{\prime}_{x}$. ∎

######  Definition 3.9.

We introduce important line bundles

$\psi^{-1}:=\det\pi_{!}F=\mathcal{O}_{i}(-1,g-d),$
$\Lambda_{M}:=\Lambda^{2}F_{x}=\mathcal{O}_{i}(0,-1),$
$\zeta:=\psi\otimes\Lambda_{M}^{d-2g+1}=\operatorname{\mathcal{O}}_{i}(1,g-1)$

and

$\theta:=\psi^{2}\otimes\Lambda_{M}^{\chi}=\mathcal{O}_{i}(2,d-2),$

where $\chi=d+2-2g$ (cf. [narasimhan1, Proposition 2.1]).

######  Lemma 3.10.

For a point $x\in C$ and every $i\geq 1$, we have exact sequences

(3.3) $\displaystyle 0\to\Lambda_{M}^{-1}\to
F_{x}^{\vee}\to\mathcal{O}_{M_{i}(\Lambda)}\to\mathcal{O}_{M_{i-1}(\Lambda(-2x))}\to
0$

and

(3.4) $\displaystyle 0\to\mathcal{O}_{M_{i}(\Lambda)}\to
F_{x}\to\Lambda_{M}\to\left.\Lambda_{M}\right|_{M_{i-1}(\Lambda(-2x))}\to 0.$

###### Proof.

By Remark 3.7, the zero locus of the section $\phi_{x}$ of $F_{x}$ is smooth
and has codimension $2$. Therefore, the Koszul complex and the dual Koszul
complex of $(F_{x},\phi_{x})$ are exact. ∎

######  Definition 3.11.

Let $M=M_{i}(\Lambda)$ be a moduli space in the interior of a GIT chamber, as
above, and let $F$ be the universal bundle on $C\times M$. We apply the
constructions of Section 2 to $F$. In particular, for a divisor
$D\in\operatorname{Sym}^{\alpha}C$, we will denote

$G_{D}=\left(F^{\boxtimes\alpha}\right)_{D}\quad\hbox{\rm and}\quad
G_{D}^{\vee}=\left(\overline{F^{\vee}}^{\boxtimes\alpha}\right)_{D}.$

Consider again the diagram (3.1). The wall between two consecutive chambers
$M_{i-1}$ and $M_{i}$ occurs at $\sigma=d/2-i$. The birational transformation
$M_{i-1}\dashrightarrow M_{i}$ is an isomorphism outside of the loci
$\mathbb{P}W_{i}^{-}\subset M_{i-1}$, $\mathbb{P}W_{i}^{+}\subset M_{i}$,
where $W_{i}^{-}$ and $W_{i}^{+}$ are vector bundles over the symmetric
product $\operatorname{Sym}^{i}C$ of rank $i$ and $d+g-1-2i$, respectively. We
have a diagram

${\tilde{M}}$${M_{i-1}=M_{\sigma+\epsilon}}$${M_{\sigma-\epsilon}=M_{i}}$${M_{\sigma}}$

where $\tilde{M}$ is both the blow-up of $M_{\sigma+\epsilon}=M_{i-1}$ along
$\mathbb{P}W_{i}^{-}$ and the blow-up of $M_{\sigma-\epsilon}=M_{i}$ along
$\mathbb{P}W_{i}^{+}$. The variety $M_{\sigma}$ is singular, obtained from the
contraction to $\operatorname{Sym}^{i}C$ of the exceptional locus
$\mathbb{P}W_{i}^{-}\times_{\operatorname{Sym}^{i}C}\mathbb{P}W_{i}^{+}\subset\tilde{M}$.

When $d\gg 0$, $M_{\sigma\pm\epsilon}(\Lambda)$ and $M_{\sigma}(\Lambda)$ are
obtained as GIT quotients of $U\times\mathbb{PC}^{\chi}$, with $\chi=d+2-2g$.
When $d$ is arbitrary, take an effective divisor $D^{\prime}$ of large degree,
so that $M_{\sigma}\hookrightarrow
M^{\prime}_{\sigma}:=M_{\sigma}(\Lambda(2D^{\prime}))$, where
$M^{\prime}_{\sigma}$ is a GIT quotient with a semi-stable locus
$X^{\prime}\subset U^{\prime}\times\mathbb{PC}^{\chi^{\prime}}$,
$\chi^{\prime}=d+2-2g+2\deg D^{\prime}$. The spaces
$M_{\sigma\pm\epsilon}(\Lambda)$ and $M_{\sigma}(\Lambda)$ are then GIT
quotients by $SL_{\chi^{\prime}}$ of a closed subset of
$U^{\prime}\times\mathbb{PC}^{\chi^{\prime}}$ determined by the condition that
in the pair $(E^{\prime},\phi^{\prime})$, the section $\phi^{\prime}$ vanishes
along $D^{\prime}$. If we call $\mathcal{L}_{\pm}$, $\mathcal{L}_{0}$ the
corresponding linearizations, we can write $X\subset X^{\prime}$, the semi-
stable locus of $\operatorname{\mathcal{L}}_{0}$, as the union
$X=X^{ss}(\mathcal{L}_{+})\cup X^{ss}(\mathcal{L}_{-})\sqcup Z$, where the
locus $Z=X^{u}(\mathcal{L}_{+})\cap X^{u}(\mathcal{L}_{-})$ corresponds to
pairs $(E^{\prime},\phi^{\prime})$, such that $E^{\prime}$ splits as

$E^{\prime}=L^{\prime}\oplus K^{\prime},$

with $\deg L^{\prime}=i+\deg D^{\prime}$, $\deg K^{\prime}=d-i+\deg
D^{\prime}$, and $\phi^{\prime}\in H^{0}(L^{\prime})$ vanishes along
$D^{\prime}$ (see [thaddeus, 1.4]). The map
$\mathcal{O}_{C}^{\chi^{\prime}}\twoheadrightarrow E^{\prime}$ is then given
by a block-diagonal matrix $(\mathcal{O}_{C}^{a}\twoheadrightarrow
L^{\prime})\oplus(\mathcal{O}_{C}^{b}\twoheadrightarrow K^{\prime})$, where
$a=h^{0}(L^{\prime})$, $b=h^{0}(K^{\prime})$ and $a+b=h^{0}(L^{\prime}\oplus
K^{\prime})=\chi^{\prime}$. The strictly semi-stable locus
$X^{sss}(\operatorname{\mathcal{L}}_{0})=X^{u}(\mathcal{L}_{+})\cup
X^{u}(\mathcal{L}_{-})$ consists of the orbits whose closure intersects $Z$
(cf. [potashnik, Remark 7.4]).

Using techniques from [dhl] and [katzarkov], we compare the derived categories
on either side of the wall $M_{\sigma}$. We write
$M_{\sigma\pm\epsilon}=X\operatorname{/\\!\\!/}_{\mathcal{L}_{\pm}}PGL_{\chi^{\prime}}$
(cf. Remark 3.2) and take Kempf–Ness stratifications of the unstable loci
$X^{u}(\mathcal{L}_{\pm})$ with strata $S^{j}_{\pm}$ determined by pairs
$(Z^{j},\lambda^{j}_{\pm})$, where
$\lambda^{j}_{-}(t)=\lambda^{j}_{+}(t)^{-1}$ are one-parameter subgroups and
$Z^{j}$ is the fixed locus of $\lambda^{j}=\lambda^{j}_{+}$ (see [dhl, §2.1]
for details).

######  Remark 3.12.

From the discussion above, it follows that in this case the KN stratification
of the unstable locus in $X$ with respect to $\mathcal{L}_{\pm}$ has only one
stratum $S_{\pm}$, consisting of the vector bundle $W_{i}^{\pm}$ over the
locus $Z$ of split bundles described above. In the notation of [dhl, §2], the
stratum $S_{\pm}$ is determined by the pair $(Z,\lambda)$, where
$\lambda=\lambda_{+}=\mathbb{G}_{m}$ is the stabilizer of $Z$, acting on a
split bundle $E^{\prime}=L^{\prime}\oplus K^{\prime}$ by $(t^{b},t^{-a})$.

######  Remark 3.13.

Let $\mathfrak{Z}$ be the stack $[Z/\mathbb{L}]$, where $\mathbb{L}$ is the
Levi subgroup, i.e. the centralizer of $\lambda$ in $PGL_{\chi^{\prime}}$. We
have a short exact sequence of groups $1\to\mathbb{G}_{m}\to\mathbb{L}\to
PGL_{a}\times PGL_{b}\to 1$ with $\mathbb{G}_{m}=\lambda$ acting on $Z$
trivially and $[Z/PGL_{a}\times PGL_{b}]\cong\operatorname{Sym}^{i}C$. Indeed,
the action of $PGL_{a}\times PGL_{b}$ on $Z$ is free, and each orbit is
determined by a divisor $D\in\operatorname{Sym}^{i}C$, where $D+D^{\prime}$ is
the zero locus of the section $\phi^{\prime}\in H^{0}(L^{\prime})$. Therefore
$\mathfrak{Z}\cong[\operatorname{Sym}^{i}C/\mathbb{G}_{m}]$, with the trivial
action of $\mathbb{G}_{m}$.

For $\sigma=d/2-i$ with $1<i\leq v$, $M_{\sigma\pm\epsilon}$ ($\epsilon>0$) is
isomorphic to the corresponding quotient stack, since the action of
$PGL_{\chi^{\prime}}$ is free on the stable locus by [thaddeus, 1.6]. Let
$\eta_{\pm}=\operatorname{weight}_{\lambda_{\pm}}\left.\det\mathcal{N}^{\vee}_{S_{\pm}/X}\right|_{Z}$.
For any choice of an integer $w$, $D^{b}(M_{\sigma\pm\epsilon})$ is equivalent
to the window subcategory $G_{w}^{\pm}\subset D^{b}([X/PGL_{\chi^{\prime}}])$
determined by objects having $\lambda_{\pm}$-weights in the range $[w,\
w+\eta_{\pm})$ for the unique stratum $S_{\pm}$ (see [dhl, Theorem 2.10]). If
$\operatorname{weight}_{\lambda}\omega_{X}|_{Z}=\eta_{-}-\eta_{+}>0$, we get
an embedding $D^{b}(M_{\sigma+\epsilon})\subset D^{b}(M_{\sigma-\epsilon})$
(see [dhl, Proposition 4.5] and the Remark following it).

######  Lemma 3.14.

In the wall-crossing between the spaces $M_{\sigma+\epsilon}(\Lambda)=M_{i-1}$
and $M_{\sigma-\epsilon}(\Lambda)=M_{i}$, the window has width $\eta_{+}=i$,
$\eta_{-}=d+g-1-2i$ after the standard rescaling (see the proof).

###### Proof.

We use the notation as in the discussion above, with
$M_{\sigma}\hookrightarrow
M^{\prime}_{\sigma}:=M_{\sigma}(\Lambda(2D^{\prime}))$, $D^{\prime}$ effective
with $\deg D^{\prime}\gg 0$. For $\mathcal{L}_{\pm}$, there is no strictly
semi-stable locus and in fact $PGL_{\chi^{\prime}}$ acts freely on the semi-
stable locus [thaddeus, 1.6], so
$M_{i-1}=M_{\sigma+\epsilon}(\Lambda)=X\operatorname{/\\!\\!/}_{\mathcal{L}_{+}}SL_{\chi^{\prime}}$
and
$M_{i}=M_{\sigma-\epsilon}(\Lambda)=X\operatorname{/\\!\\!/}_{\mathcal{L}_{-}}SL_{\chi^{\prime}}$
are isomorphic to the quotient stacks
$[{X}^{ss}(\mathcal{L}_{\pm})/PGL_{\chi^{\prime}}]$ (cf. Remark 3.2). By Lemma
3.1, both $[X/PGL_{\chi^{\prime}}]$ and $[X^{\prime}/PGL_{\chi^{\prime}}]$ are
smooth quotient stacks of dimension $d+g-2$ and $d+g-2+2\deg D^{\prime}$,
respectively, and thus $X$ and $X^{\prime}$ are both smooth and $X\subset
X^{\prime}$ is a local complete intersection cut out precisely by the $2\deg
D^{\prime}$ conditions imposed by the vanishing of a section along
$D^{\prime}$.

Recall that the unique KN stratum of $X^{u}(\operatorname{\mathcal{L}}_{\pm})$
is determined by $(Z,\lambda)$ (cf. Remark 3.12), where for a pair
$(E^{\prime},\phi^{\prime})\in Z$, the bundle $E^{\prime}=L^{\prime}\oplus
K^{\prime}$ is acted on by $\lambda=\mathbb{G}_{m}$ by $(t^{b},t^{-a})$. By
[potashnik, Lemma 7.6] and its proof, the $\lambda$-weights of
$\mathcal{N}^{\vee}_{S/X^{\prime}}$ on $Z$ are all $\pm(a+b)=\pm\chi^{\prime}$
or $0$. Then the weights of $\mathcal{N}_{S_{\pm}/X}^{\vee}$ are all
$\pm\chi^{\prime}$, and
$\eta_{\pm}=\operatorname{weight}_{\lambda_{\pm}}\left.\det\mathcal{N}^{\vee}_{S_{\pm}/X}\right|_{Z}$
is just the codimension of $S_{\pm}\subset X$. Since $S_{\pm}$ is the bundle
$W_{i}^{\pm}$ on $Z$, we have $\operatorname{codim}(S_{\pm}\subset
X)=\operatorname{rk}W_{i}^{\mp}$, so that $\eta_{+}=i\chi^{\prime}$ and
$\eta_{-}=(d+g-1-2i)\chi^{\prime}$. We get the formula above after the
standard rescaling of all weights by $1/\chi^{\prime}$. ∎

Using this we obtain the following result.

######  Proposition 3.15.

For $1\leq i\leq\frac{d+g-1}{3}$ (resp., $i\geq\frac{d+g-1}{3}$) there is an
admissible embedding $D^{b}(M_{i-1})\hookrightarrow D^{b}(M_{i})$ (resp.,
$D^{b}(M_{i})\hookrightarrow D^{b}(M_{i-1})$). When $1<i\leq\frac{d+g-1}{3}$,
the admissible embedding can be chosen to be the window subcategory
$G_{0}^{+}\subset D^{b}(M_{i})$ determined by the range of weights
$[0,i)\subset[0,d+g-1-2i)$ (cf. [dhl]) and moreover there is a semi-orthogonal
decomposition

(3.5) $\displaystyle D^{b}(M_{i})=\langle
D^{b}(M_{i-1}),D^{b}(\operatorname{Sym}^{i}C),\ldots,D^{b}(\operatorname{Sym}^{i}C)\rangle$

with $\mu=d+g-3i-1$ copies of $D^{b}(\operatorname{Sym}^{i}C)$ given by the
fully faithful images of functors
$Rj_{*}(L\pi^{*}(\cdot)\otimes^{L}\mathcal{O}_{\pi}(l)):D^{b}(\operatorname{Sym}^{i}C)\to
D^{b}(M_{i})$ for $l=0,\ldots,\mu-1$, where
$\pi:\mathbb{P}W_{i}^{+}\to\operatorname{Sym}^{i}C$ is the projection and
$j:\mathbb{P}W_{i}^{+}\hookrightarrow M_{i}$ the inclusion.

The semi-orthogonal decomposition (3.5) follows from [bfr], as the birational
transformation between $M_{i-1}$ and $M_{i}$ is a standard flip of projective
bundles over $\operatorname{Sym}^{i}C$. Here we provide an alternative proof
for this case. We also note that [potashnik, Corollary 8.1] shows the
admissible embeddings $D^{b}(M_{i-1})\hookrightarrow D^{b}(M_{i})$ when $i$ is
in the specified range.

As explained in the introduction, Proposition 3.15 does not provide a semi-
orthogonal decomposition with Fourier-Mukai functors associated with Poincaré
bundles and it is not used in our paper. However, we find this result
relevant.

###### Proof.

If $i=1$, this follows from Orlov’s blow-up formula [orlov]. Let $i>1$. From
Lemma 3.14,
$\operatorname{weight}_{\lambda}\omega_{X}|_{Z}=\eta_{-}-\eta_{+}=(d+g-1-3i)$.
By [dhl, Proposition 4.5 and Remark 4.6], and since
$M_{\sigma\pm\epsilon}\cong[{X}^{ss}(\mathcal{L}_{\pm})/PGL_{\chi^{\prime}}]$,
we get a window embedding $D^{b}(M_{\sigma+\epsilon})\subset
D^{b}(M_{\sigma-\epsilon})$ if $\eta_{+}\leq\eta_{-}$ and the other way around
if $\eta_{+}\geq\eta_{-}$. Moreover, if $G_{w}^{+}=D^{b}(M_{\sigma+\epsilon})$
is a window, determined by the range of weights $[w,\ w+\eta_{+})\subset[w,\
w+\eta_{-})$, then [dhl, Theorem 2.10] and [katzarkov, Theorem 1] give semi-
orthogonal blocks $D^{b}(\mathfrak{Z})_{k}$, so that

(3.6) $\displaystyle D^{b}(M_{\sigma-\epsilon})=\langle
G_{w}^{+},D^{b}(\mathfrak{Z})_{w},\ldots,D^{b}(\mathfrak{Z})_{w+\mu-1}\rangle,$

where $\mu\chi^{\prime}=\eta_{-}-\eta_{+}$ and $\mathfrak{Z}=[Z/\mathbb{L}]$
is the quotient stack by the Levi subgroup. By Remark 3.13,
$D^{b}(\mathfrak{Z})=D_{\mathbb{G}_{m}}^{b}(\operatorname{Sym}^{i}C)$, so the
blocks in (3.6) are given by the fully faithful images of
$Rj_{*}(L\pi^{*}(\cdot)\otimes^{L}\mathcal{O}_{\pi}(l)):D^{b}(\operatorname{Sym}^{i}C)\to
D^{b}(M_{i})$ for $l\in[w,\ w+\mu)$, where
$\pi:\mathbb{P}W_{i}^{+}\to\operatorname{Sym}^{i}C$ is the projection and
$j:\mathbb{P}W_{i}^{+}\hookrightarrow M_{i}$ the inclusion. Taking $w=0$ gives
the claim. ∎

######  Corollary 3.16.

If $d\leq 2g-1$, then $D^{b}(M_{i-1})\subset D^{b}(M_{i})$ for any $1\leq
i\leq v$.

###### Proof.

In this case $i\leq(d-1)/2\leq g-1$, so the inequality $i<(d+g-1)/3$ holds for
every $i$. ∎

Consider an object $G$ in $D^{b}([X/PGL_{\chi^{\prime}}])$ descending to some
objects on $D^{b}(M_{i-1})$ and $D^{b}(M_{i})$. We can use windows to
determine when such object can “cross the wall”. Namely, if the weights of $G$
are in the required range, cohomology groups will be the same on either side.
By abuse of notation, we often denote in the same way both the object on
$D^{b}([X/PGL_{\chi^{\prime}}])$ and the objects it descends to in
$M_{\sigma\pm\epsilon}(\Lambda)$.

######  Theorem 3.17.

Let $\sigma=d/2-i$, $1<i\leq v$. If $A$, $B$ are objects in
$D^{b}([X/PGL_{\chi^{\prime}}])$, with $\lambda=\lambda_{+}$-weights
satisfying the inequalities

(3.7) $\displaystyle
1+2i-d-g<\operatorname{weight}_{\lambda}B|_{Z}-\operatorname{weight}_{\lambda}A|_{Z}<i$

then
$R\operatorname{Hom}_{M_{\sigma+\epsilon}}(A,B)=R\operatorname{Hom}_{M_{\sigma-\epsilon}}(A,B)$.
In particular, if $1+2i-d-g<\operatorname{weight}_{\lambda}B|_{Z}<i$ then
$R\Gamma_{M_{i-1}}(B)=R\Gamma_{M_{i}}(B)$.

###### Proof.

By Lemma 3.14, (3.7) is equivalent to inequalities

$-\eta_{-}<\operatorname{weight}_{\lambda}B|_{Z}-\operatorname{weight}_{\lambda}A|_{Z}<\eta_{+},$

so the Quantization Theorem [dhl, Theorem 3.29] implies that

$\displaystyle
R\operatorname{Hom}_{M_{\sigma+\epsilon}}(A,B)=R\operatorname{Hom}_{[X/PGL_{\chi^{\prime}}]}(A,B)=R\operatorname{Hom}_{M_{\sigma-\epsilon}}(A,B).$

Indeed, the first equality follows directly from [dhl, Theorem 3.29] applied
on $M_{\sigma+\epsilon}$, while the second is the same theorem applied on
$M_{\sigma-\epsilon}$, using the fact that
$\operatorname{weight}_{\lambda_{-}}B|_{Z}-\operatorname{weight}_{\lambda_{-}}A|_{Z}=-(\operatorname{weight}_{\lambda}B|_{Z}-\operatorname{weight}_{\lambda}A|_{Z})$.
∎

We finish this section with the computation of all weights that we need in
order to construct the semi-orthogonal decompositions.

######  Theorem 3.18.

The objects of the form $F_{x}$, $\Lambda_{M}$, $\psi$, $\zeta$, $G_{D}$ on
both $M_{i-1}$ and $M_{i}$ are the descents of objects $\tilde{F}_{x}$,
$\tilde{\Lambda}_{M}$, $\tilde{\psi}$, $\tilde{\zeta}$, $\tilde{G}_{D}$ on
$D^{b}([X/PGL_{\chi^{\prime}}])$ that, after the standard rescaling (see the
proof of Lemma 3.14), have $\lambda$-weights

$\displaystyle\operatorname{weights}_{\lambda}\tilde{F}_{x}|_{Z}=\\{0,-1\\}$
$\displaystyle\operatorname{weight}_{\lambda}\tilde{\Lambda}_{M}|_{Z}=-1$
$\displaystyle\operatorname{weight}_{\lambda}\tilde{\psi}|_{Z}=d+1-g-i$
$\displaystyle\operatorname{weight}_{\lambda}\tilde{\zeta}|_{Z}=g-i$
$\displaystyle\operatorname{weights}_{\lambda}\tilde{G}_{D}|_{Z}=\\{0,-1,\ldots,-\deg
D\\}.$

###### Proof.

Let $\sigma=d/2-i$ and embed $\imath:M_{\sigma}(\Lambda)\hookrightarrow
M^{\prime}_{\sigma}=M_{\sigma}(\Lambda(2D^{\prime}))$ for an effective divisor
$D^{\prime}$, $\deg D^{\prime}\gg 0$, as usual. Recall that the universal
bundle $F^{\prime}$ on $C\times M^{\prime}_{\sigma\pm\epsilon}$ is the descent
of $\mathcal{F}^{\prime}(1)$ on $C\times X^{\prime}\subset C\times
U^{\prime}\times\mathbb{PC}^{\chi^{\prime}}$, where $\mathcal{F}^{\prime}$ is
the universal family on $C\times U^{\prime}$ [thaddeus, 1.19]. Let us compute
the $\lambda$-weights of $\mathcal{F}^{\prime}_{x}(1)$ on the
$\sigma$-strictly semi-stable locus, for a point $x\in C$. The fiber of
$\mathcal{F}^{\prime}_{x}$ over $L^{\prime}\oplus K^{\prime}$ is
$L^{\prime}_{x}\oplus K^{\prime}_{x}$, which is acted on with weights $b$ in
the first component and $-a$ in the second. Since the $\lambda$-weight of
$\mathcal{O}_{\mathbb{PC}^{\chi^{\prime}}}(1)$ over the section
$(\phi^{\prime},0)$ is $-b$, the weights of $\mathcal{F}^{\prime}_{x}(1)$ are
$0$ and $-a-b=-\chi^{\prime}$. By Lemma 3.8, we have
$F_{x}\cong\imath^{*}F^{\prime}_{x}$. Hence, $F_{x}$ also is the descent of an
object with weights $0$ and $-\chi^{\prime}$.

The bundle $\det{\pi_{!}}F^{\prime}$ descends from
$\det{\pi_{!}}\mathcal{F}^{\prime}(1)$. On the fiber of
$\pi_{!}\mathcal{F}^{\prime}$ over $L^{\prime}\oplus K^{\prime}$, $\lambda$
acts on $H^{0}(L^{\prime})\oplus H^{0}(K^{\prime})$ with weights $b$ and $-a$,
with multiplicities $h^{0}(L^{\prime})=a$ and $h^{0}(K^{\prime})=b$,
respectively. Taking tensor product with
$\mathcal{O}_{\mathbb{PC}^{\chi^{\prime}}}(1)$ shifts each weight by $-b$, and
then taking the determinant we get
$\operatorname{weight}_{\lambda}\det{\pi_{!}}\mathcal{F}^{\prime}(1)|_{Z^{\prime}}=0\cdot
a+(-a-b)\cdot b=-b\chi^{\prime}$. For $\det F^{\prime}_{x}$, which is the
descent of $\det\mathcal{F}^{\prime}_{x}(1)$, we see that $\lambda$ acts with
weights $b,-a$ on $L^{\prime}_{x}\oplus K^{\prime}_{x}$ and then shifting by
$-b$ and taking determinants we get
$\operatorname{weight}_{\lambda}\det\mathcal{F}^{\prime}_{x}(1)|_{Z^{\prime}}=-a-b=-\chi^{\prime}$.

Now for the universal bundle $F$ on $C\times M_{\sigma\pm\epsilon}(\Lambda)$,
we use the short exact sequence (3.2). From this we see that $\Lambda_{M}=\det
F_{x}\cong\det F^{\prime}_{x}$ is the descent of an object with
$\lambda$-weight equal to $-\chi^{\prime}$. Also, since
$\det\pi_{!}F^{\prime}|_{D^{\prime}\times
M_{\sigma\pm\epsilon}}=\det\bigoplus_{x\in D^{\prime}}F^{\prime}_{x}=(\det
F^{\prime}_{x})^{\deg D^{\prime}}$, we obtain that
$\psi^{-1}=\det\pi_{!}F=\det{\pi_{!}}F^{\prime}\otimes(\det
F^{\prime}_{x})^{-\deg D^{\prime}}$ is the descent of an object with
$\lambda$-weight equal to $-b\chi^{\prime}+\deg D^{\prime}\chi^{\prime}$.
Recall $\deg L^{\prime}=i+\deg D^{\prime}$, $\deg K^{\prime}=d-i+\deg
D^{\prime}$ (see the discussion before Remark 3.12), so by Riemann-Roch
$b=h^{0}(K^{\prime})=d-i+\deg D^{\prime}+1-g$ and the weight of ${\psi}$ is
$-\chi^{\prime}(\deg D^{\prime}-b)=\chi^{\prime}(d+1-g-i)$. As for
$\zeta=\psi\otimes\Lambda_{M}^{d-2g+1}$, the weights must be
$(d+1-g-i-(d-2g+1))\chi^{\prime}=(g-i)\chi^{\prime}$. Rescaling everything by
$1/\chi^{\prime}$ as in Lemma 3.14, we get the weights as in the statement.

Finally, we consider $G_{D}$. Let $D=x_{1}+\ldots+x_{\alpha}$. Since by
construction tensor bundles are functorial in $M$, the bundle $G_{D}$ is the
descent of a vector bundle
$(\operatorname{\mathcal{E}}^{\boxtimes\alpha})_{D}$ on $X$, where
$M=X\operatorname{/\\!\\!/}SL_{\chi^{\prime}}$ and
$\operatorname{\mathcal{E}}$ descends to $F$. By Lemma 2.10,
$(\operatorname{\mathcal{E}}^{\boxtimes\alpha})_{D}$ is a deformation of
$\operatorname{\mathcal{E}}_{x_{1}}\otimes\ldots\otimes\operatorname{\mathcal{E}}_{x_{\alpha}}$,
and the deformation can be chosen to be $SL_{\chi^{\prime}}$-equivariant (see
Remark 2.11). Therefore, $(\operatorname{\mathcal{E}}^{\boxtimes\alpha})_{D}$
has the same weights as the tensor product
$\operatorname{\mathcal{E}}_{x_{1}}\otimes\ldots\otimes\operatorname{\mathcal{E}}_{x_{\alpha}}$,
i.e. $0,-1,\ldots,-\alpha$ (after rescaling). ∎

######  Remark 3.19.

Observe that
$\operatorname{\mathcal{O}}_{i}(1,0)=\psi\otimes\Lambda_{M}^{d-g}$ and
$\operatorname{\mathcal{O}}_{i}(0,1)=\Lambda_{M}^{-1}$, so we can use the
previous theorem to see that in general, a line bundle
$\operatorname{\mathcal{O}}_{i}(m,n)$ is the descent on both $M_{i-1}$ and
$M_{i}$ of an object having $\lambda$-weight $m(1-i)+n$ on the strictly semi-
stable locus of the wall.

##  4\. Acyclic vector bundles on $M_{i}$ – easy cases

In order to prove Theorem 1.1, we will first construct fully faithful functors
$\Phi_{\alpha}^{i}:D^{b}(\operatorname{Sym}^{\alpha}C)\hookrightarrow
D^{b}(M_{i})$ for $1\leq\alpha\leq i$ and show that, after suitable twists,
the essential images of these functors are semi-orthogonal to each other in
the required way (see Theorem 9.2, Definition 9.3 and Theorem 9.6 below). By
means of Bondal-Orlov’s criterion [bondal-orlov], this reduces to the
computation of $R\Gamma$ for a large class of vector bundles on $M_{i}$. In
particular, we will need to prove $\Gamma$-acyclicity for several of these
vector bundles.

######  Theorem 4.1.

Let $D$, $D^{\prime}$ be effective divisors on $C$. Let $d>2$ and $1\leq i\leq
v$. Suppose

$\deg D-g<t<d-\deg D^{\prime}-i-1.$

Then $R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1})=0$.

We start with a lemma.

######  Lemma 4.2.

$R\Gamma_{M_{1}(d)}(\mathcal{O}_{M_{1}(d)}(-kH+lE_{1}))=0$ for $0<k\leq d+g-2$
and $0\leq l\leq d+g-4$. In particular,
$R\Gamma_{M_{1}(d)}(\Lambda_{M}^{t})=0$ for $t=1,\ldots,d+g-4$.

###### Proof.

Consider the short exact sequence

(4.1) $\displaystyle
0\to\mathcal{O}_{M_{1}(d)}\to\mathcal{O}_{M_{1}(d)}(E_{1})\to\mathcal{O}_{\pi}(-1)\to
0,$

where $E_{1}=\mathbb{P}W_{1}^{+}$ and $\pi:E_{1}\to C$ is the
$\mathbb{P}^{r}$-bundle, $r=d+g-4$. $\mathcal{O}_{M_{1}(d)}(-kH)$ is
$\Gamma$-acyclic provided $0<k\leq d+g-2=\dim M_{1}(d)$. Then twisting (4.1)
by $\mathcal{O}_{M_{1}(d)}(-kH)$ and taking a long exact sequence in
cohomology gives $\Gamma$-acyclicity of $\mathcal{O}_{M_{1}(d)}(-kH+E_{1})$
for such $k$. Similarly, twisting by powers of $\mathcal{O}_{M_{1}(d)}(E_{1})$
and using induction, we get that
$R\Gamma_{M_{1}(d)}(\mathcal{O}_{M_{1}(d)}(-kH+lE))=0$ as well, since
$\mathcal{O}_{\pi}(-l)$ is $\Gamma$-acyclic for $0<l\leq d+g-4$. ∎

Next we prove Theorem 4.1 for the case $i=1$.

######  Lemma 4.3.

The statement of Theorem 4.1 holds for $i=1$.

###### Proof.

Let $\alpha=\deg D$, $\beta=\deg D^{\prime}$. We are given that
$\alpha-g<t<d-\beta-2$. We do induction on $\alpha+\beta$. If
$\alpha=\beta=0$, we have to check that
$\Lambda_{M}^{t}\otimes\zeta^{-1}=-(t+g)H+(g+t-1)E_{1}$ is $\Gamma$-acyclic on
$M_{1}(d)$. By Lemma 4.2, this holds provided $0<t+g\leq d+g-2$ and $0\leq
g+t-1\leq d+g-4$, which is true by hypothesis.

If $\alpha>0$, we can write $D=x+\tilde{D}$ for some point $x\in C$. Then by
Proposition 2.13 $G_{D}^{\vee}$ is a stable deformation of
$F_{x}^{\vee}\otimes G_{\tilde{D}}^{\vee}$ over
$\operatorname{{\mathbb{A}}}^{1}$, so it suffices by semi-continuity to show
$\Gamma$-acyclicity of $F_{x}^{\vee}\otimes G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$. Consider the exact
sequence (3.3) from Lemma 3.10,

$\displaystyle 0\to\Lambda_{M}^{-1}\to
F_{x}^{\vee}\to\mathcal{O}_{M_{1}(d)}\to\mathcal{O}_{M_{0}(d-2)}\to 0,$

and twist it by $G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$. The restriction of
$F_{y}$ to $M_{0}(d-2)=\mathbb{P}^{r}$, $r=d+g-4$, is equal to
$\mathcal{O}_{\mathbb{P}^{r}}\oplus\mathcal{O}_{\mathbb{P}^{r}}(-1)$ (see the
proof of [thaddeus, 3.2], with $i=1$) so, by Corollary 2.10, we have that
$\left.G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}\right|_{M_{0}(d-2)}$ is
a deformation over $\operatorname{{\mathbb{A}}}^{1}$ of a sum of bundles
$\bigoplus\mathcal{O}_{\mathbb{P}^{r}}(s_{j})\otimes\mathcal{O}_{\mathbb{P}^{r}}(1-t-g)$,
with $-\beta\leq s_{j}\leq\alpha-1$. These are all $\Gamma$-acyclic on
$\mathbb{P}^{d+g-4}$, since by hypothesis

(4.2) $\displaystyle\alpha-t-g<0,\quad-\beta+1-t-g\geq-(d+g-4).$

Then $\left.G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}\right|_{M_{0}(d-2)}$ is
$\Gamma$-acyclic too, by semi-continuity.

The other two terms from the sequence (3.3) are $G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$ and
$G_{\tilde{D}}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t-1}\otimes\zeta^{-1}$. Observe they both
satisfy the inequalities of the hypothesis, so by induction they are
$\Gamma$-acyclic on $M_{1}(d)$, and we conclude that so is
$G_{D}^{\vee}\otimes G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$,
as desired.

Similarly, if $\beta>0$ we write $D^{\prime}=y+\tilde{D^{\prime}}$ and use the
exact sequence

$\displaystyle 0\to\mathcal{O}_{M_{1}(d)}\to
F_{y}\to\Lambda_{M}\to\left.\Lambda_{M}\right|_{M_{0}(d-2)}\to 0,$

twisted with $G_{D}^{\vee}\otimes
G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$. The resulting
term on the right is a deformation over $\operatorname{{\mathbb{A}}}^{1}$ of a
sum
$\bigoplus\mathcal{O}_{\mathbb{P}^{r}}(s_{j})\otimes\mathcal{O}_{\mathbb{P}^{r}}(-t-g)$,
with $-\beta+1\leq s_{j}\leq\alpha$, and it is again $\Gamma$-acyclic by the
same inequalities (4.2). Finally, the remaining two terms $G_{D}^{\vee}\otimes
G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$ and
$G_{D}^{\vee}\otimes
G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t+1}\otimes\zeta^{-1}$ are
$\Gamma$-acyclic by induction, and we conclude that
$R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1})=0$. ∎

###### Proof of Theorem 4.1.

Let $\alpha=\deg D$ and $\beta=\deg D^{\prime}$. We do induction on $i$. If
$i=1$, this is Lemma 4.3. Suppose $i>1$. By Lemma 4.3 we still have

$R\Gamma_{M_{1}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1})=0.$

Let $1<j\leq i$ and consider the wall-crossing between $M_{j-1}$ and $M_{j}$.
Here, the bundle $G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1}$ descends from an object
with weights

$\operatorname{weights}_{\lambda}\tilde{G}_{D}^{\vee}\otimes\tilde{G}_{D^{\prime}}\otimes\tilde{\Lambda}_{M}^{t}\otimes\tilde{\zeta}^{-1}=\\{-\beta-t+j-g,\ldots,\alpha-t+j-g\\}$

(see Theorem 3.18). Our hypothesis guarantees that $\alpha-t+j-g<j=\eta_{+}$
and $-\beta-t+j-g>1+2j-d-g=-\eta_{-}$, that is, all these weights live in the
range $(-\eta_{-},\ \eta_{+})$ in the respective wall-crossings. By Theorem
3.17 this implies

$R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1})=R\Gamma_{M_{1}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}\otimes\zeta^{-1})=0.$

This proves the theorem. ∎

##  5\. A fully faithful embedding $D^{b}(C)\subset D^{b}(M_{1})$

The following Theorem 5.1 is a special case of Theorem 9.2, and will be needed
for our proof of the latter. Namely, the result of Theorem 5.1 will be used in
Sections 7 and 8, which are necessary for Theorem 9.2. While Theorem 5.1 could
be avoided by including it as a step of a more complicated inductive proof, we
find it more convenient to prove it first, both to make the inductions less
cumbersome and to introduce some ideas that will help understand the general
picture.

We assume that $v\geq 1$, i.e. $d\geq 3$. As before, let $E_{1}\subset M_{1}$
be the exceptional locus of the blow-up $M_{1}\to M_{0}$ along $C\subset
M_{0}$. By Orlov’s blow-up formula [orlov], we have a fully faithful functor
$\Psi:D^{b}(C)\hookrightarrow D^{b}(M_{1})$, corresponding to the Fourier-
Mukai transform given by $\mathcal{O}_{Z}(E_{1})$, where $Z=C\times_{C}E_{1}$.
Now consider the Fourier-Mukai transform

$\Phi_{F}=Rp_{*}(Lq^{*}(\cdot)\otimes^{L}F):D^{b}(C)\to D^{b}(M_{1})$

determined by the universal bundle $F$ on $C\times M_{1}$.

######  Theorem 5.1.

The functor $\Phi_{F}$ is fully faithful.

We need a few constructions and lemmas first. Observe that
$Z=C\times_{C}E_{1}$ is supported precisely on the zero locus of the universal
section $\tilde{\phi}:\mathcal{O}_{C\times M_{1}}\to F$. Indeed, pairs
$(E,\phi)$ in $\mathbb{P}W_{1}^{+}=E_{1}$ parametrize extensions

$\displaystyle 0\to\mathcal{O}_{C}(x)\to E\to\Lambda(-x)\to 0$

with the canonical section $\phi\in H^{0}(C,\mathcal{O}_{C}(x))$ vanishing on
$x\in C$ [thaddeus, 3.2], and in fact $\tilde{\phi}$ has no zeros outside this
locus, since $M_{1}\backslash E_{1}$ consists of extensions
$0\to\mathcal{O}_{C}\to E\to\Lambda\to 0$ together with a (constant) section
$\phi\in H^{0}(C,\mathcal{O}_{C})$ [thaddeus, 3.1]. Since $Z$ has codimension
$2$, we have a Koszul resolution

(5.1) $\displaystyle\left[\Lambda^{2}F^{\vee}\to
F^{\vee}\xrightarrow{\tilde{\phi}}\mathcal{O}_{C\times
M_{1}}\right]\xrightarrow{\sim}\mathcal{O}_{Z}.$

######  Lemma 5.2.

$R\Gamma_{M_{1}}(\Lambda_{M}^{-1})=0$.

###### Proof.

Recall $\Lambda_{M}^{-1}=\mathcal{O}_{M_{1}}(H-E_{1})$. Since the embedding

$C\xhookrightarrow{|\omega_{C}\otimes\Lambda|}M_{0}=\mathbb{P}^{d+g-2}$

is given by a complete linear system [thaddeus, 3.4], the image of $C$ is not
contained in any hyperplane and thus
$H^{0}(M_{1},\mathcal{O}_{M_{1}}(H-E_{1}))=0$. Now use the exact sequence

(5.2) $\displaystyle
0\to\mathcal{O}_{M_{1}}(H-E_{1})\to\mathcal{O}_{M_{1}}(H)\to\mathcal{O}_{E_{1}}(H)\to
0.$

Observe that
$H^{p}(M_{1},\mathcal{O}_{E_{1}}(H))=H^{p}(C,\omega_{C}\otimes\Lambda)$,
because $C\hookrightarrow M_{0}$ is given by $|\omega_{C}\otimes\Lambda|$.
Since $\deg\omega_{C}\otimes\Lambda>\deg\omega_{C}$, we have
$H^{1}(C,\omega_{C}\otimes\Lambda)=0$. On the other hand,
$H^{0}(M_{1},\mathcal{O}_{M_{1}}(H))=\mathbb{C}^{d+g-1}$ and
$H^{>0}(M_{1},\mathcal{O}_{M_{1}}(H))=0$. Taking a long exact sequence in
cohomology from (5.2) we get

(5.3) $\displaystyle 0\to H^{0}(M_{1},\mathcal{O}_{M_{1}}(H))\to
H^{0}(C,\omega_{C}\otimes\Lambda)\to H^{1}(M_{1},\Lambda_{M}^{-1})\to 0$

and $H^{p}(M_{1},\Lambda_{M}^{-1})=0$ for $p\neq 1$. By Riemann-Roch,
$H^{0}(C,\omega_{C}\otimes\Lambda)=\mathbb{C}^{d+g-1}$, so the first map in
(5.3) is an isomorphism and $H^{1}(M_{1},\Lambda_{M}^{-1})=0$ as well. ∎

######  Lemma 5.3.

Let $x\in C$. Then $R\Gamma_{M_{1}}(F_{x}^{\vee})=0$, while
$R\Gamma_{M_{1}}(F_{x})=\mathbb{C}$, with $H^{0}(M_{1},F_{x})=\mathbb{C}$
given by restriction of the universal section $\tilde{\phi}$ of $F$ to
$\\{x\\}\times M_{1}$.

###### Proof.

Consider the resolution (5.1) and restrict to $\\{x\\}\times M_{1}$ to get

(5.4) $\displaystyle\left[\Lambda_{M}^{-1}\to
F_{x}^{\vee}\to\mathcal{O}_{M_{1}}\right]\xrightarrow{\sim}\mathcal{O}_{\mathbb{P}_{x}^{r}}$

where $\mathbb{P}_{x}^{r}=M_{0}(\Lambda(-2x))$ is the fiber over $x\in
C\subset M_{0}$ along the blow-up $\pi:M_{1}\to M_{0}$. We twist by
$\Lambda_{M}=\mathcal{O}_{M_{1}}(E_{1}-H)$ to get

(5.5)
$\displaystyle\left[\mathcal{O}_{M_{1}}\xrightarrow{\tilde{\phi}}F_{x}\to\Lambda_{M}\right]\xrightarrow{\sim}\mathcal{O}_{\mathbb{P}_{x}^{r}}(-1),$

using that $F_{x}^{\vee}\otimes\Lambda_{M}=F_{x}$ and that
$\mathcal{O}_{M_{1}}(H)$ restricts trivially to
$\mathcal{O}_{\mathbb{P}_{x}^{r}}$ (see Lemma 3.10 for a generalization of
(5.4) and (5.5)). It is well-known that
$R\Gamma(\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}^{r}_{x}}(-1))=0$.
By Lemma 4.2, we also have $R\Gamma(\Lambda_{M})=0$. Hence, by (5.5),
$\tilde{\phi}$ induces an isomorphism
$R\Gamma(\operatorname{\mathcal{O}}_{M_{1}})\cong R\Gamma(F_{x})$. As $M_{1}$
is a blow up of a projective space along a smooth center, we get
$R\Gamma(F_{x})\cong
R\Gamma(\operatorname{\mathcal{O}}_{M_{1}})\cong\mathbb{C}$, with
$H^{0}(M_{1},F_{x})=\mathbb{C}$ given by restriction of $\tilde{\phi}$ to
$\\{x\\}\times M_{1}$.

To show that $R\Gamma_{M_{1}}(F_{x}^{\vee})=0$, we apply $R\Gamma$ to (5.4).
We know that
$R\Gamma_{M_{1}}(\mathcal{O}_{M_{1}})=R\Gamma_{\mathbb{P}^{r}}(\mathcal{O}_{\mathbb{P}^{r}})=\mathbb{C}$,
while $R\Gamma_{M_{1}}(\Lambda_{M}^{-1})=0$ by Lemma 5.2. Then the claim will
be proved if we show that $H^{0}(M_{1},F_{x}^{\vee})=0$, as this would also
imply that $H^{p}(M_{1},F_{x}^{\vee})$ vanishes for $p>0$. Any global section
$s\in H^{0}(M_{1},F_{x}^{\vee})$, composed with
$F_{x}^{\vee}\to\mathcal{O}_{M_{1}}$ gives a constant section
$\mathcal{O}_{M_{1}}\to\mathcal{O}_{M_{1}}$ vanishing along the locus $Z$,
hence identically $0$. By exactness of $0\to\Lambda_{M}^{-1}\to
F_{x}^{\vee}\to\mathcal{O}_{M_{1}}$, the section $s:\mathcal{O}_{M_{1}}\to
F_{x}^{\vee}$ must lift to a section $\mathcal{O}_{M_{1}}\to\Lambda_{M}^{-1}$.
But $\Lambda_{M}^{-1}$ has no global sections by Lemma 5.2. ∎

###### Proof of Theorem 5.1.

By Bondal-Orlov’s criterion [bondal-orlov], in order to show full faithfulness
of $\Phi_{F}$ we only need to consider the sheaves
$\Phi_{F}(\mathcal{O}_{x})=F_{x}$ for closed points $x\in C$. On the other
hand, consider the functor $\Psi$ from Orlov’s blow-up formula, with Fourier-
Mukai kernel $\mathcal{O}_{Z}(E_{1})$, $Z=C\times_{C}E_{1}$. We can compute
$\Psi(\mathcal{O}_{x})=\Phi_{\mathcal{O}_{Z}(E_{1})}(\mathcal{O}_{x})$ for a
point $x\in C$ using (5.1) as follows. As before, let
$\mathbb{P}_{x}^{r}=M_{0}(\Lambda(-2x))$ denote the fiber over $x\in C\subset
M_{0}$ along the blow-up. The fact that $\mathcal{O}_{M_{1}}(H)$ restricts
trivially to this fiber implies that both $\Lambda_{M}$ and
$\mathcal{O}_{M_{1}}(E_{1})$ restrict to
$\mathcal{O}_{\mathbb{P}_{x}^{r}}(-1)$ there. Now we restrict (5.1) to
$\\{x\\}\times M_{1}$ and twist it by $\Lambda_{M}$ to get
$\Phi_{\mathcal{O}_{Z}(E_{1})}(\mathcal{O}_{x})\cong\left[\mathcal{O}_{M_{1}}\to
F_{x}\to\Lambda_{M}\right]\cong\mathcal{O}_{\mathbb{P}_{x}^{r}}(-1)$, as in
(5.5). Since we already know that $\Psi$ is fully faithful, we have

(5.6)
$\displaystyle\operatorname{Hom}_{D^{b}(M_{1})}(\Psi(\mathcal{O}_{x}),\Psi(\mathcal{O}_{y})[k])=\begin{cases}0\quad&\text{if
}x\neq y\\\ 0\quad&\text{if }x=y\text{ and }k\neq 0,1\\\ \mathbb{C}&\text{if
}x=y\text{ and }k=0,1.\end{cases}$

But
$R\operatorname{Hom}_{D^{b}(M_{1})}(\Psi(\mathcal{O}_{x}),\Psi(\mathcal{O}_{y}))\cong
R\Gamma\circ R\operatorname{\mathscr{H}\text{\kern-3.0pt{\calligra\large
om}}\,}(\Psi(\mathcal{O}_{x}),\Psi(\mathcal{O}_{y}))$ can also be obtained as
follows: take $R\operatorname{\mathscr{H}\text{\kern-3.0pt{\calligra\large
om}}\,}(\Psi(\mathcal{O}_{x}),\Psi(\mathcal{O}_{y}))\cong\Psi(\mathcal{O}_{x})^{\vee}\otimes^{L}\Psi(\mathcal{O}_{y})$
as an inner tensor product obtained from the double complex

(5.7)
${\mathcal{O}_{M_{1}}}$${F_{x}^{\vee}\otimes\Lambda_{M}}$${\Lambda_{M}}$${\Lambda_{M}^{-1}\otimes
F_{y}}$${F_{x}^{\vee}\otimes
F_{y}}$${F_{y}}$${\Lambda_{M}^{-1}}$${F_{x}^{\vee}}$${\mathcal{O}_{M_{1}}}$

which produces the total complex

(5.8) $\displaystyle\begin{aligned} &\left[\Lambda_{M}^{-1}\to
F_{x}^{\vee}\oplus F_{y}^{\vee}\to\mathcal{O}_{M_{1}}^{\oplus
2}\oplus(F_{x}^{\vee}\otimes F_{y})\to F_{x}\oplus
F_{y}\to\Lambda_{M}\right]\\\
\cong&\phantom{[}\Psi(\mathcal{O}_{x})^{\vee}\otimes^{L}\Psi(\mathcal{O}_{y}),\end{aligned}$

again using $F_{x}=F_{x}^{\vee}\otimes\Lambda_{M}$. Recall that our
descriptions of $\Psi(\mathcal{O}_{y})$ and $\Psi(\mathcal{O}_{x})^{\vee}$
were obtained from the Koszul resolution (5.5) and its dual. In particular,
the maps $\operatorname{\mathcal{O}}_{M_{1}}\to
F_{x}^{\vee}\otimes\Lambda_{M}=F_{x}$ and
$\operatorname{\mathcal{O}}_{M_{1}}\to F_{y}$ appearing in (5.7) correspond to
the restriction of the universal section $\tilde{\phi}$ to $\\{x\\}\times
M_{1}$ and $\\{y\\}\times M_{1}$, respectively.

The hypercohomology $R\Gamma$ of (5.8) can be computed by taking the spectral
sequence with first page
$E_{1}^{p,q}=H^{q}(X,\operatorname{\mathcal{F}}^{p})\Rightarrow
H^{p+q}(X,\operatorname{\mathcal{F}}^{\bullet})$. On the other hand, we know
that $R\Gamma$ of this complex is given by (5.6). We will combine these to
show that

(5.9) $\displaystyle R\Gamma(F_{x}^{\vee}\otimes
F_{y})=\begin{cases}0\quad&\text{if }x\neq y\\\
\mathbb{C}\oplus\mathbb{C}[-1]&\text{if }x=y.\end{cases}$

By Lemma 4.2, $R\Gamma(\Lambda_{M})=0$, and by Lemma 5.2
$R\Gamma(\Lambda_{M}^{-1})=0$. Also, Lemma 5.3 computes hypercohomology of
both $F_{x}$ and $F_{x}^{\vee}$. Summing up, applying $R\Gamma$ to (5.8)
yields a spectral sequence $E_{1}^{p,q}$ of the form

${\vdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots\phantom{.}}$${0}$${0}$${H^{1}(F_{x}^{\vee}\otimes
F_{y})}$${0}$${0\phantom{.}}$${0}$${0}$${H^{0}(\mathcal{O}_{M_{1}})^{\oplus
2}\oplus H^{0}(F_{x}^{\vee}\otimes F_{y})}$${H^{0}(F_{x})\oplus
H^{0}(F_{y})}$${0,}$

where the map $H^{0}(\mathcal{O}_{M_{1}})^{\oplus 2}\to H^{0}(F_{x})\oplus
H^{0}(F_{y})$ is the isomorphism
$\mathbb{C}^{2}\xrightarrow{\sim}\mathbb{C}^{2}$ given by the universal
section in each coordinate, by Lemma 5.3 and the discussion above. Since this
spectral sequence converges to (5.6), we obtain (5.9). ∎

##  6\. Acyclicity of powers of $\Lambda_{M}^{\vee}$

The goal of the present section is to prove the following generalization of
Lemma 5.2:

######  Theorem 6.1.

Suppose $2<d\leq 2g+1$ and $1\leq k\leq l\leq v$. Then

$R\Gamma_{M_{l}(d)}(\Lambda_{M}^{-k})=0.$

$\Gamma$-acyclicity of these negative powers of $\Lambda_{M}$ will be crucial
for the cohomology computations in the upcoming sections.

######  Lemma 6.2.

Under the assumptions of Theorem 6.1, $H^{0}(M_{l}(d),\Lambda_{M}^{-k})=0.$

###### Proof.

Since $M_{l}$ is isomorphic to $M_{1}$ in codimension $1$, it suffices to
prove that $H^{0}(M_{1},\Lambda_{M}^{-k})=H^{0}(M_{1},kH-kE_{1})=0.$ Recall
that $M_{1}$ is the blow-up of $\operatorname{{\mathbb{P}}}^{r}$ in $C$
embedded by a complete linear system of $K_{C}+\Lambda$, $r=d+g-2$, $E_{1}$ is
the exceptional divisor and $H$ is a hyperplane divisor. The claim is that
there is no hypersurface $D\subset\operatorname{{\mathbb{P}}}^{r}$ of degree
$k$ that vanishes along $C$ with multiplicity $\geq k$. We argue by
contradiction. Choose $r+1$ points $p_{1},\ldots,p_{r+1}\in C$ in linearly
general position. Then $D$ vanishes at these points with multiplicity $\geq
k$. Let $R$ be a rational normal curve passing through $p_{1},\ldots,p_{r+1}$.
Let $\tilde{R}$ and $\tilde{D}$ be the proper transforms of $R$ and $D$ in
$\operatorname{Bl}_{p_{1},\ldots,p_{r+1}}\operatorname{{\mathbb{P}}}^{r}$.
Then $\tilde{D}\cdot\tilde{R}\leq kr-k(r+1)<0.$ It follows that $R\subset D$.
But we can choose $R$ passing through a general point of
$\operatorname{{\mathbb{P}}}^{r}$, which is a contradiction. ∎

######  Lemma 6.3.

Under the assumptions of Theorem 6.1, if
$R\Gamma_{M_{k}(d)}(\Lambda_{M}^{-k})=0.$ then
$R\Gamma_{M_{l}(d)}(\Lambda_{M}^{-k})=0.$

###### Proof.

By Theorem 3.18, in the wall between $M_{l-1}$ and $M_{l}$, $\Lambda_{M}^{-k}$
descends from an object of weight $k$, with $-\eta_{-}<k<\eta_{+}$ when
$k<l\leq v$, that is, $1+2l-d-g<k<l$ for $l$ in that range. This way,
$0=R\Gamma_{M_{k}}(\Lambda_{M}^{-k})=R\Gamma_{M_{l}}(\Lambda_{M}^{-k})$ for
$l\geq k$ by Theorem 3.17 ∎

######  Definition 6.4.

For $0\leq\alpha\leq i$, we introduce the following loci:

$\displaystyle E_{i}^{\alpha}$ $\displaystyle:=\\{(E,s)\mid Z(s)\subset
C\text{ has degree }\geq\alpha\\}\subset M_{i},$
$\displaystyle\operatorname{\mathcal{D}}_{i}^{\alpha}$
$\displaystyle:=\\{(D,E,s)\mid
s|_{D}=0\\}\subset\operatorname{Sym}^{\alpha}C\times M_{i},$ $\displaystyle
R_{i}^{\alpha}$ $\displaystyle:=\\{(D,E,s)\mid s|_{D}=0\text{ and }Z(s)\text{
has degree }\geq\alpha+1\\}\subset\operatorname{\mathcal{D}}_{i}^{\alpha},$

where $Z(s)$ denotes the zero locus subscheme of the section $s$.

Note that $E_{i}^{i}$ is precisely $\mathbb{P}W_{i}^{+}$, while
$E_{i}^{1}=E_{i}$ is the proper transform of $E_{1}$ under the birational
equivalence given by (3.1). Recall
$\operatorname{\mathcal{O}}(E_{i})=\operatorname{\mathcal{O}}_{i}(1,-1)$
according to Definition 3.5. For a divisor $D\in\operatorname{Sym}^{\alpha}C$,
we observe that the fiber $(\operatorname{\mathcal{D}}_{i}^{\alpha})_{D}$
along the projection $\operatorname{Sym}^{\alpha}C\times
M_{i}\to\operatorname{Sym}^{\alpha}C$ is isomorphic to
$M_{i-\alpha}(\Lambda(-2D))$, see Remark 3.7 or [thaddeus, 1.9]. Similarly,
$(R_{i}^{\alpha})_{D}\cong E_{i-\alpha}(\Lambda(-2D))$. In particular,
$\operatorname{\mathcal{D}}_{i}^{\alpha}$ is smooth, and the projection

(6.1)
${R_{i}^{\alpha}}$${\operatorname{\mathcal{D}}_{i}^{\alpha}}$${E_{i}^{\alpha}}$$\scriptstyle{\nu}$

is the normalization morphism.

######  Lemma 6.5.

We have the following commutative diagram

(6.2)
${0}$${\nu_{*}\operatorname{\mathcal{O}}_{D_{i}^{\alpha}}(-R_{i}^{\alpha})}$${\nu_{*}\operatorname{\mathcal{O}}_{D_{i}^{\alpha}}}$${\nu_{*}\operatorname{\mathcal{O}}_{R_{i}^{\alpha}}}$${0}$${0}$${\mathcal{I}_{E_{i}^{\alpha+1}}}$${\operatorname{\mathcal{O}}_{E_{i}^{\alpha}}}$${\operatorname{\mathcal{O}}_{E_{i}^{\alpha+1}}}$${0}$$\scriptstyle{\sim}$$\scriptstyle{\beta}$

where
$\mathcal{I}_{E_{i}^{\alpha+1}}\cong\nu_{*}\operatorname{\mathcal{O}}_{D_{i}^{\alpha}}(-R_{i}^{\alpha})$
is the conductor sheaf of the normalization (6.1) and $R_{i}^{\alpha}$ (resp.
$E_{i}^{\alpha+1}$) is a conductor subscheme in
$\operatorname{\mathcal{D}}_{i}^{\alpha}$ (resp. $E_{i}^{\alpha}$).

###### Proof.

From the flipping diagram (3.1), $E_{\alpha}^{\alpha}\subset M_{\alpha}$ is
the projective bundle $\operatorname{{\mathbb{P}}}W_{\alpha}^{+}$ and
$E_{\alpha+1}^{\alpha}\subset M_{\alpha+1}$ is isomorphic to
$E_{\alpha}^{\alpha}$ away from
$E_{\alpha+1}^{\alpha+1}\cong\operatorname{{\mathbb{P}}}W_{\alpha+1}^{+}$.

######  Claim 6.6.

$E_{\alpha+1}^{\alpha}$ has a multicross singularity generically along
$E_{\alpha+1}^{\alpha+1}$ (concretely, this means that a general section of
$E_{\alpha+1}^{\alpha}$ that intersects $E_{\alpha+1}^{\alpha+1}$ in a point
is étale locally isomorphic to the union of coordinate axes in
$\operatorname{{\mathbb{A}}}^{\alpha+1}$).

Given the claim, and since multicross singularities are semi-normal [LV],
$E_{\alpha+1}^{\alpha}$ has semi-normal singularities in codimension $1$. For
$i>\alpha+1$, $E_{i}^{\alpha}$ is isomorphic to $E_{\alpha+1}^{\alpha}$ in
codimension $2$, and so also has semi-normal singularities in codimension $1$.
Next we argue by induction on $\alpha$ that $E_{i}^{\alpha}$ is semi-normal
and Cohen–Macaulay with reduced conductor subschemes $E_{i}^{\alpha+1}\subset
E_{i}^{\alpha}$ and
$R_{i}^{\alpha}\subset\operatorname{\mathcal{D}}_{i}^{\alpha}$, and in
particular that we have a commutative diagram (6.2).

Indeed, $E^{1}_{i}\subset M_{i}$ is Cohen–Macaulay as a hypersurface in a
smooth variety. Suppose $E_{i}^{\alpha}$ is Cohen–Macaulay. Since it is semi-
normal in codimension $1$ by the above, it is semi-normal everywhere [greco,
2.7]. Therefore, its conductor subschemes in $E_{i}^{\alpha}$ and
$\operatorname{\mathcal{D}}_{i}^{\alpha}$ are both reduced [traverso, 1.3] and
all of their associated primes have height $1$ in $E_{i}^{\alpha}$ and
$\operatorname{\mathcal{D}}_{i}^{\alpha}$, respectively [greco, 7.4]. It
follows that these conductor subschemes are equal to $E_{i}^{\alpha+1}$ and
$R_{i}^{\alpha}$, respectively. Finally,
$R_{i}^{\alpha}\subset\operatorname{\mathcal{D}}_{i}^{\alpha}$ is
Cohen–Macaulay as a hypersurface in a smooth variety and therefore
$E_{i}^{\alpha+1}\subset E_{i}^{\alpha}$ is also Cohen–Macaulay [roberts,
2.2], and we can proceed with induction.

It remains to prove the claim. We analyze the flipping diagram (3.1) between
the spaces $M_{\alpha}$ and $M_{\alpha+1}$, where $M_{\alpha}$ contains
projective bundles $\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-}$ (over
$\operatorname{Sym}^{\alpha+1}C$) and
$\operatorname{{\mathbb{P}}}W_{\alpha}^{+}\cong E_{\alpha}^{\alpha}$ (over
$\operatorname{Sym}^{\alpha}C$) of dimensions $2\alpha+1$ and $d+g-2-\alpha$,
respectively. What is their intersection over a point
$D^{\prime}\in\operatorname{Sym}^{\alpha+1}C$, for simplicity a reduced sum of
points? By [thaddeus, 3.3], $\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-}$
parametrizes pairs $(E,\phi)$ that appear in extensions

$0\to L\to E\to\Lambda\otimes L^{-1}\to 0$

with $\deg L=d-\alpha-1$ and $\phi\not\in H^{0}(L)$. Projecting $\phi$ to
$\Lambda\otimes L^{-1}$ gives a non-zero vector $\gamma\in
H^{0}(\Lambda\otimes L^{-1})$ so that $\Lambda\otimes
L^{-1}=\operatorname{\mathcal{O}}(D^{\prime})$, where $\deg
D^{\prime}=\alpha+1$ (this gives the map from
$\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-}$ to
$\operatorname{Sym}^{\alpha+1}C$). Moreover, at $D^{\prime}$ the section lifts
to a section of $\operatorname{\mathcal{O}}_{D^{\prime}}\otimes
L\cong\operatorname{\mathcal{O}}_{D^{\prime}}\otimes\Lambda(-D^{\prime})$, and
this vector $p\in
H^{0}(\operatorname{\mathcal{O}}_{D^{\prime}}\otimes\Lambda(-D^{\prime}))$
(determined uniquely up to a scalar) determines $(E,\phi)$ uniquely [thaddeus,
3.3].

The same pair $(E,\phi)$ belongs to
$\operatorname{{\mathbb{P}}}W_{\alpha}^{+}$ if it can be given by an extension

$0\to\operatorname{\mathcal{O}}(D)\to E\to\Lambda(-D)\to 0$

with $\phi\in H^{0}(\operatorname{\mathcal{O}}(D))$ and $\deg D=\alpha$
[thaddeus, 3.2]. Since $\phi$ vanishes at $D$ and its image in
$\operatorname{\mathcal{O}}(D^{\prime})$ vanishes at $D^{\prime}$, we have
$D\subset D^{\prime}$. Since we assume that $D^{\prime}$ is a reduced divisor,
there are exactly $\alpha+1$ choices for $D$. Since $p$ has to vanish at
points of $D\subset D^{\prime}$, for each of these choices of $D$, there is
exactly one vector $p\in
H^{0}(\operatorname{\mathcal{O}}_{D^{\prime}}\otimes\Lambda(-D^{\prime}))$ (up
to a multiple) that works. Moreover, in this way we get a basis of
$H^{0}(\operatorname{\mathcal{O}}_{D^{\prime}}\otimes\Lambda(-D^{\prime}))\cong\mathbb{C}^{\alpha+1}$.
It follows that, over $D^{\prime}\in\operatorname{Sym}^{\alpha+1}C$,
$\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-}$ and
$\operatorname{{\mathbb{P}}}W_{\alpha}^{+}\cong E_{\alpha}^{\alpha}$ intersect
in $\alpha+1$ reduced points which form a basis of the projective space
$(\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-})_{D^{\prime}}\cong\operatorname{{\mathbb{P}}}^{\alpha}$.

The strict transform of $\operatorname{{\mathbb{P}}}W_{\alpha}^{+}$ in
$M_{\alpha+1}$ is $E_{\alpha+1}^{\alpha}$, which contains the bundle
$\operatorname{{\mathbb{P}}}W_{\alpha+1}^{+}$ of dimension $d+g-3-\alpha$ (the
flipped locus). After the flip, linearly independent intersection points in
$(\operatorname{{\mathbb{P}}}W_{\alpha+1}^{-})_{D^{\prime}}\cap\operatorname{{\mathbb{P}}}W_{\alpha}^{+}$
become linearly independent normal directions of branches of
$E_{\alpha+1}^{\alpha}$ along $\operatorname{{\mathbb{P}}}W_{\alpha+1}^{+}$,
i.e. $E_{\alpha+1}^{\alpha}$ has a multicross singularity in codimension $1$,
as claimed. We illustrate the geometry of $M_{\alpha}$, $M_{\alpha+1}$ and the
common resolution $\tilde{M}_{\alpha+1}$ in Figure 1. ∎

Figure 1. Common resolution $\tilde{M}_{\alpha+1}$ of $M_{\alpha}$ and
$M_{\alpha+1}$.

######  Corollary 6.7.

If the claim of Theorem 6.1 is proved for $1\leq k=l\leq i-1$, then, for
$1\leq\alpha\leq i-1$,
$R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{E_{i}^{\alpha}}(1,i-1))\cong
R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{E_{i}^{\alpha+1}}(1,i-1))$ via
$R\Gamma(\beta)$.

###### Proof.

Twisting by $\operatorname{\mathcal{O}}_{i}(1,i-1)$ and applying $R\Gamma$ to
the bottom sequence in (6.2), we see that it suffices to show
$\mathcal{I}_{E_{i}^{\alpha+1}}(1,i-1)\cong\nu_{*}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{D}}_{i}^{\alpha}}(-R_{i}^{\alpha})(1,i-1)$
is $\Gamma$-acyclic. But $\nu$ is a finite map, so this is equivalent to
$\Gamma$-acyclicity of
$\operatorname{\mathcal{O}}_{\operatorname{\mathcal{D}}_{i}^{\alpha}}(-R_{i}^{\alpha})(1,i-1)$.
Using the Leray spectral sequence for the fibration
$p:\operatorname{\mathcal{D}}_{i}^{\alpha}\to\operatorname{Sym}^{\alpha}C$, it
suffices to prove that
$R\Gamma(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{D}}_{i}^{\alpha},D}(-R_{i,D}^{\alpha})(1,i-1))=0.$
Under the isomorphism $(\operatorname{\mathcal{D}}_{i}^{\alpha})_{D}\cong
M_{i-\alpha}(\Lambda(-2D))$,
$R_{i}^{\alpha}\subset\operatorname{Sym}^{\alpha}C\times M_{i}$ restricts to
$E_{i-\alpha}^{1}$ on $M_{i-\alpha}(\Lambda(-2D))$, while
$\operatorname{\mathcal{O}}_{i}(m,n)$ on $M_{i}(\Lambda)$ restricts to
$\operatorname{\mathcal{O}}(m,n-m\alpha)$ on $M_{i-\alpha}(\Lambda(-2D))$ (cf.
Remark 3.7). Therefore,

$R\Gamma_{M_{i}(d)}(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{D}}_{i}^{\alpha},D}(-R_{i,D}^{\alpha})(1,i-1))=R\Gamma_{M_{i-\alpha}(d-2\alpha)}(\Lambda_{M}^{\alpha-i})$

which is zero by hypothesis. ∎

######  Lemma 6.8.

Suppose $d\leq 2g+1$. Then for $1\leq i\leq d+1-g$, $i\leq v$ we have
$H^{p}(M_{i}(d),\operatorname{\mathcal{O}}_{i}(1,i-1))=0$ for any $p>0$.

###### Proof.

First, we see that there is some $k\geq i$ such that $k\leq v$ and
$\operatorname{\mathcal{O}}_{M_{k}}(1,i-1)\otimes\omega_{M_{k}}^{-1}$ is big
and nef on $M_{k}(d)$. Recall
$\omega_{M_{k}}=\operatorname{\mathcal{O}}_{M_{k}}(-3,4-d-g)$ (see [thaddeus,
6.1]), so
$\operatorname{\mathcal{O}}_{M_{k}}(1,i-1)\otimes\omega_{M_{k}}^{-1}=\operatorname{\mathcal{O}}_{M_{k}}(4,d+g+i-5)$
and all we need to check is that this is in the cone bounded below by
$\operatorname{\mathcal{O}}_{M_{k}}(1,i-1)$ and above by
$\operatorname{\mathcal{O}}_{M_{k}}(2,d-2)$ (cf. Remark 3.6). This is
equivalent to $i-1\leq\frac{d+g+i-5}{4}\leq\frac{d-2}{2}.$ The inequality on
the left is equivalent to $3i\leq d+g-1$, which is guaranteed by the fact that
$i\leq v=\lfloor(d-1)/2\rfloor$ and $d\leq 2g+1$. The other inequality is
equivalent to $i\leq d+1-g$, which is given as a hypothesis. Therefore, there
is some $k\geq i$, $k\leq v$ such that
$\operatorname{\mathcal{O}}_{M_{k}}(1,i-1)\otimes\omega_{M_{k}}^{-1}$ is big
and nef. By the Kawamata–Viehweg vanishing theorem,
$H^{p}(M_{k},\operatorname{\mathcal{O}}_{k}(1,i-1))=0$ for $p>0$.

Now, we claim that in fact

(6.3)
$R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{i}(1,i-1))=R\Gamma_{M_{i+1}}(\operatorname{\mathcal{O}}_{i+1}(1,i-1))=\ldots=R\Gamma_{M_{k}}(\operatorname{\mathcal{O}}_{k}(1,i-1)).$

Indeed, in the wall-crossing between $M_{l-1}$ and $M_{l}$, there are windows
of width $\eta_{+}=l$ and $\eta_{-}=d+g-1-2l$ and
$\operatorname{\mathcal{O}}_{l}(1,i-1)$,
$\operatorname{\mathcal{O}}_{l-1}(1,i-1)$ both descend from the same object,
that has $\lambda$-weight $i-l$ (see Proposition 3.18 and Remark 3.19). By
Theorem 3.17, we will have
$R\Gamma_{M_{l-1}}(\operatorname{\mathcal{O}}_{l-1}(1,i-1))=R\Gamma_{M_{l}}(\operatorname{\mathcal{O}}_{l}(1,i-1))$
whenever

(6.4) $1+2l-d-g<i-l<l.$

But (6.4) holds for any $i<l\leq k$, because then $i<2l$, while $3l\leq
3(d-1)/2<i+d+g-1$ provided $d\leq 2g+1$. Therefore, (6.3) holds and in
particular $H^{p}(M_{i},\operatorname{\mathcal{O}}_{i}(1,i-1))=0$ for $p>0$. ∎

######  Remark 6.9.

Suppose that $d\leq 2g+1$. Then (6.4) holds for $l\in(i/2,v]$, and the same
reasoning shows that
$R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{i}(1,i-1))=R\Gamma_{M_{l}}(\operatorname{\mathcal{O}}_{l}(1,i-1))$
for every $\lfloor i/2\rfloor\leq l\leq v$. In particular, under the same
hypotheses of Lemma 6.8, $\operatorname{\mathcal{O}}_{l}(1,i-1)$ has no higher
cohomology whenever $\lfloor i/2\rfloor\leq l\leq v$.

######  Definition 6.10.

Let $L_{i}$ be the line bundle on $\operatorname{Sym}^{i}C$ defined by

(6.5)
$L_{i}=\det\nolimits^{-1}\pi_{!}\Lambda(-\Delta)\otimes\det\nolimits^{-1}\pi_{!}\operatorname{\mathcal{O}}(\Delta),$

where $\Delta\subset\operatorname{Sym}^{i}C\times C$ is the universal divisor,
cf. [thaddeus, 6.5]. To emphasize the degree $d$, sometimes we denote this
line bundle by $L_{i}(d)$.

######  Lemma 6.11.

$H^{p}(\operatorname{Sym}^{i}C,L_{i}(d))=0$ if $p>0$, $1\leq i\leq d-g$.

###### Proof.

By [thaddeus, 7.5] (see also [macdonald]), and mixing notation for line
bundles and divisors,

(6.6) $L_{i}(d)=(d-2i)\eta+2\sigma\quad\hbox{\rm and}\quad
K_{\operatorname{Sym}^{i}C}=(g-i-1)\eta+\sigma,$

where $\eta=p_{0}+\operatorname{Sym}^{i-1}C\subset\operatorname{Sym}^{i}C$ is
an ample divisor and $\sigma\subset\operatorname{Sym}^{i}C$ is a pull-back of
a theta-divisor via the Abel–Jacobi map, in particular $\sigma$ is nef. It
follows that $L_{i}(d)-K_{\operatorname{Sym}^{i}C}=(d-i-g+1)\eta+\sigma$ is
ample if $i\leq d-g$ and the result follows by Kodaira vanishing theorem. ∎

######  Lemma 6.12.

Suppose $i+g\leq d\leq 2g+1$. Then
$\chi(M_{i}(d),\operatorname{\mathcal{O}}_{i}(1,i-1))=\chi(\operatorname{Sym}^{i}C,L_{i}(d)).$

###### Proof.

Since $i\leq d-g$, we can use [thaddeus, 6.2] together with [thaddeus, 7.8] to
compute $\chi(\operatorname{\mathcal{O}}_{i}(1,i-1))=$

(6.7) $\displaystyle\begin{aligned}
=&\underset{t=0}{\operatorname{Res}}\left\\{\frac{(1-t^{3})^{2i-d-1}(1-t^{2})^{2d+1-2i-2g}}{t^{i+1}(1-t)^{d+g-1}}(1-5(1-t)t^{2}-t^{5})^{g}dt\right\\}\\\
=&\underset{t=0}{\operatorname{Res}}\left\\{\frac{(1+t)^{2d+1-2i-2g}(t^{2}+3t+1)^{g}(t-1)}{t^{i+1}(1+t+t^{2})^{d+1-2i}}dt\right\\}.\end{aligned}$

On the other hand, we use Hirzebruch–Riemann–Roch theorem to compute, using
the formulas

$\operatorname{ch}(L_{i})=e^{(d-2i)\eta+\sigma},\quad\operatorname{td}(\operatorname{Sym}^{i}C)=\left(\frac{\eta}{1-e^{-\eta}}\right)^{i-g+1}\exp\left(\frac{\sigma}{e^{\eta}-1}-\frac{\sigma}{\eta}\right)$

(see [thaddeus, §7]) and notation from the proof of Lemma 6.11, that

$\chi(L_{i})=\underset{\eta=0}{\operatorname{Res}}\left\\{\frac{e^{\eta(d-2i)}}{(1-e^{-\eta})^{i-g+1}}\left(2+\frac{1}{e^{\eta}-1}\right)^{g}d\eta\right\\}.$

If we let $u(\eta)=e^{\eta}-1$, then $u$ is biholomorphic near $\eta=0$, with
$u(0)=0$, $u^{\prime}(0)=1$, so we can do a change of variables
$u=e^{\eta}-1$, $du=e^{\eta}d\eta$ to obtain

(6.8)
$\displaystyle\chi(L_{i})=\underset{u=0}{\operatorname{Res}}\left\\{\frac{(1+u)^{d-i-g}(2u+1)^{g}}{u^{i+1}}du\right\\}.$

Next, we apply an ad hoc change of variables

$u=\frac{t}{t^{2}+t+1},\quad du=\frac{1-t^{2}}{(t^{2}+t+1)^{2}}dt$

to (6.8) and we get precisely (6.7) after some algebraic manipulations. ∎

For what follows we need some geometric constructions. Fix a point $p_{0}\in
C$ and consider a subvariety $M_{i-1}(d-1)\subset M_{i}(d+1)$ of codimension
$2$ as in Remark 3.7, with $D=p_{0}$. Let $B$ be the blow-up of $M_{i}(d+1)$
in $M_{i-1}(d-1)$ with exceptional divisor $\operatorname{\mathcal{E}}$.

Consider the $\operatorname{{\mathbb{P}}}^{1}$-bundle
$\operatorname{{\mathbb{P}}}F_{p_{0}}$ over $M_{i}(d+1)$ that parametrizes
triples $(E,\phi,l)$, where $\phi$ is a non-zero section of $E$ and $l\subset
E_{p_{0}}$ is a line, subject to the usual stability condition (see Section 3)
that for every line subbundle $L\subset E$, one must have

(6.9) $\deg L\leq\begin{cases}i+\frac{1}{2}\quad&\text{ if }\phi\in
H^{0}(L),\\\ d-i+\frac{1}{2}\quad&\text{ if }\phi\notin H^{0}(L).\end{cases}$

######  Lemma 6.13.

With the notation as above, the blow-up $B$ of $M_{i}(d+1)$ in $M_{i-1}(d-1)$
is isomorphic to the following locus:

$Z=\\{(E,\phi,l)\,:\,\phi(p_{0})\in
l\\}\subset\operatorname{{\mathbb{P}}}F_{p_{0}}.$

###### Proof.

Indeed, the projection of $Z$ onto $M_{i}(d+1)$ is clearly an isomorphism
outside of $M_{i-1}(d-1)$, since the latter is precisely the locus where
$\phi(p_{0})=0$. Over $M_{i-1}(d-1)$, the fiber of this projection is
$\operatorname{{\mathbb{P}}}^{1}$. By the universal property of the blow-up,
it suffices to check that $Z$ is the blow-up of $M_{i}(d+1)$ in $M_{i-1}(d-1)$
locally near $(E,\phi)\in M_{i-1}(d-1)$, where we can trivialize
$F_{p_{0}}\cong\operatorname{\mathcal{O}}\oplus\operatorname{\mathcal{O}}$.
Its universal section can be written as $s=(a,b)$, where
$a,b\in\operatorname{\mathcal{O}}$ is a regular sequence (its vanishing locus
is $M_{i-1}(d-1)$ locally near $(E,\phi)$). Then $Z$ is locally given by the
equation $ay-bx=0$, where $[x:y]$ are homogeneous coordinates of the
$\operatorname{{\mathbb{P}}}^{1}$-bundle
$\operatorname{{\mathbb{P}}}F_{p_{0}}$ given by the trivialization
$F_{p_{0}}\cong\operatorname{\mathcal{O}}\oplus\operatorname{\mathcal{O}}$.
Thus $Z$ is indeed isomorphic to the blow-up $B$. ∎

Now we can prove the main result of this section.

###### Proof of Theorem 6.1.

We proceed by induction on $k$. Suppose $R\Gamma_{M_{k}}(\Lambda^{-k})$ is
zero for $1\leq k<i$. By Lemma 6.3, it suffices to show that
$R\Gamma_{M_{i}}(\Lambda^{-i})=0$. Twist the tautological short exact sequence
for $E_{i}\subset M_{i}$ by $\operatorname{\mathcal{O}}_{i}(1,i-1)$ to get

$0\to\Lambda_{M}^{-i}\to\operatorname{\mathcal{O}}_{i}(1,i-1)\mathop{\to}^{\gamma}\operatorname{\mathcal{O}}_{E_{i}}(1,i-1)\to
0.$

It suffices to prove that
$R\Gamma_{M_{i}}(\operatorname{\mathcal{O}}_{i}(1,i-1))\cong
R\Gamma_{E_{i}}(\operatorname{\mathcal{O}}_{E_{i}}(1,i-1))$ via $R\gamma$. By
the induction hypothesis, we can apply Corollary 6.7 to see that

$R\Gamma(\operatorname{\mathcal{O}}_{E_{i}}(1,i-1))\cong\ldots\cong
R\Gamma(\operatorname{\mathcal{O}}_{E_{i}^{i}}(1,i-1))=R\Gamma(\operatorname{\mathcal{O}}_{\mathbb{P}W_{i}^{+}}(1,i-1)).$

But $\mathcal{O}_{\mathbb{P}W_{i}^{+}}(1,i-1)$ restricts trivially to each
fiber of $\mathbb{P}W_{i}^{+}$ and is a pull-back of $L_{i}$ (6.5) on
$\operatorname{Sym}^{i}C$ [thaddeus, §5], which implies
$R\Gamma(\operatorname{\mathcal{O}}_{\mathbb{P}W_{i}^{+}}(1,i-1))\cong
R\Gamma(\operatorname{Sym}^{i}C,L_{i})$. Therefore, it suffices to show that

(6.10) $R\Gamma_{M_{i}(d)}(\operatorname{\mathcal{O}}_{i}(1,i-1))\cong
R\Gamma_{\operatorname{Sym}^{i}C}(L_{i}(d))$

via the composition of functors as above.

######  Claim 6.14.

If $d\geq i+g$, then (6.10) holds.

###### Proof.

In this case
$H^{p}(M_{i},\operatorname{\mathcal{O}}_{i}(1,i-1))=H^{p}(\operatorname{Sym}^{i}C,L_{i})=0$
for $p>0$ by Lemmas 6.8 and 6.11. Using this together with the fact that
$\Lambda_{M}^{-i}=\operatorname{\mathcal{O}}_{i}(0,i)$ has no global sections
by Lemma 6.2, it suffices to prove that
$h^{0}(M_{i},\operatorname{\mathcal{O}}_{i}(1,i-1))=h^{0}(\operatorname{Sym}^{i}C,L_{i})$
or, equivalently, that
$\chi(M_{i},\operatorname{\mathcal{O}}_{i}(1,i-1))=\chi(\operatorname{Sym}^{i}C,L_{i}).$
Thus, Lemma 6.12 proves the Claim. ∎

We now proceed by a downward induction on $d$, starting with any $d$ such that
$d\geq i+g$. For such $d$, we have the result by the Claim above.

Next we perform a step of the downward induction assuming the theorem holds
for degree $d+1$. As above, we fix a point $p_{0}\in C$ and consider the
subvariety $M_{i-1}(d-1)\subset M_{i}(d+1)$ of codimension $2$ determined by
Remark 3.7. Let
$\operatorname{\mathcal{I}}\subset\operatorname{\mathcal{O}}_{M_{i}(d+1)}$ be
its ideal sheaf. As in the proof of Lemma 6.11, we denote the divisor
$p_{0}+\operatorname{Sym}^{i-1}C\subset\operatorname{Sym}^{i}C$ by $\eta$ and,
by abuse of notation, we denote its pull-back to the projective bundle
$\operatorname{{\mathbb{P}}}W_{i}^{+}$, which as a locus in $M_{i-1}(d-1)$ is
nothing but $\operatorname{{\mathbb{P}}}W_{i-1}^{+}$, by $\eta$ as well. To
summarize, we have a commutative diagram of sheaves on $M_{i}(d+1)$ with exact
rows, where we suppress closed embeddings from notation.

(6.11)
${0}$${\operatorname{\mathcal{I}}}$${\operatorname{\mathcal{O}}_{M_{i}(d+1)}}$${\operatorname{\mathcal{O}}_{M_{i-1}(d-1)}}$${0}$${0}$${\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}W_{i}^{+}}(-\eta)}$${\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}W_{i}^{+}}}$${\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}W_{i-1}^{+}}}$${0}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

We tensor (6.11) with $\operatorname{\mathcal{O}}(1,i-1)$. Recall that the
restriction of $\operatorname{\mathcal{O}}(1,i-1)$ to $M_{i-1}(d-1)$ is
$\operatorname{\mathcal{O}}(1,i-2)$, to $\operatorname{{\mathbb{P}}}W_{i}^{+}$
is the pull-back of $L_{i}(d+1)$ from $\operatorname{Sym}^{i}C$, and to
$\operatorname{{\mathbb{P}}}W_{i-1}^{+}$ is the pull-back of $L_{i-1}(d-1)$
from $\operatorname{Sym}^{i-1}C$. By inductive hypothesis and Claim 6.14, it
follows that the arrows $\beta$ and $\gamma$ in (6.11) give an isomorphism in
cohomology after tensoring with $\operatorname{\mathcal{O}}(1,i-1)$. By the
5-lemma, we conclude that we have an isomorphism

(6.12) $R\Gamma(\operatorname{\mathcal{I}}(1,i-1))\cong
R\Gamma(\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}W_{i}^{+}}(-\eta)(1,i-1)).$

Note that
$\operatorname{\mathcal{O}}_{\operatorname{{\mathbb{P}}}W_{i}^{+}}(-\eta)(1,i-1)$
is precisely the pull-back of $L_{i}(d)$ from $\operatorname{Sym}^{i}C$ to the
projective bundle (see (6.6)). On the other hand, (6.12) gives

(6.13)
$R\Gamma_{B}(\operatorname{\mathcal{O}}_{B}(1,i-1)(-\operatorname{\mathcal{E}}))\cong
R\Gamma_{\operatorname{Sym}^{i}C}(L_{i}(d)),$

where $B$ is the blow-up of $M_{i}(d+1)$ in $M_{i-1}(d-1)$ and
$\operatorname{\mathcal{E}}$ its exceptional divisor.

Recall that the goal is to prove (6.10). We can do one extra simplification.
Let $\sigma=\frac{d}{2}-i$ be the slope on the wall between the moduli spaces
$M_{i}(d)$ and $M_{i-1}(d)$ and let $M_{\sigma}(d)$ be the corresponding
(singular) GIT quotient. The birational morphism $M_{i}(d)\to M_{\sigma}(d)$
contracts the projective bundle $\operatorname{{\mathbb{P}}}W_{i}^{+}$ to its
base $\operatorname{Sym}^{i}C$, and in particular proving (6.10) is equivalent
to proving that

(6.14) $R\Gamma_{M_{\sigma}(d)}(\operatorname{\mathcal{O}}_{i}(1,i-1))\cong
R\Gamma_{\operatorname{Sym}^{i}C}(L_{i}(d))$

by projection formula and Boutot theorem [boutot]. To show how (6.13) implies
(6.14), we need a geometric construction, a variant of the Hecke
correspondence, relating $B$ to $M_{\sigma}(d)$.

By Lemma 6.13, $B$ carries a family of parabolic (at $p_{0}\in C$) rank $2$
vector bundles $E$ with a section $\phi$. The parabolic line at $p_{0}$
defines a quotient $E\to\operatorname{\mathcal{O}}_{p_{0}}$, and we define a
rank $2$ vector bundle $E^{\prime}$ as an elementary transformation, by the
formula

(6.15) $0\to E^{\prime}\to E\to\operatorname{\mathcal{O}}_{p_{0}}\to 0.$

Our condition $\phi(p_{0})\in l$ implies that the section $\phi$ lifts to a
section $\phi^{\prime}$ of $E^{\prime}$. Elementary transformation is well-
known to be a functorial construction [NR78], in fact we claim that
$(E^{\prime},\phi^{\prime})$ is a $\sigma$-semistable pair, i.e. we have a
morphism

$h:\,B\to M_{\sigma}(d),\quad(E,\phi,l)\mapsto(E^{\prime},\phi^{\prime}).$

Indeed, we need to check that

$\displaystyle\deg L^{\prime}\leq\begin{cases}i\quad&\text{ if
}\phi^{\prime}\in H^{0}(L^{\prime}),\\\ d-i\quad&\text{ if
}\phi^{\prime}\notin H^{0}(L^{\prime}).\end{cases}$

for every line subbundle $L^{\prime}\subset E^{\prime}$, which follows from
(6.9) applied to $L^{\prime}$.

By the Kollár vanishing theorem [kollar, Theorem 7.1],
$Rh_{*}\operatorname{\mathcal{O}}_{B}=\operatorname{\mathcal{O}}_{M_{\sigma}(d)}$.
Indeed, $B$ is smooth, $M_{\sigma}(d)$ has rational singularities and a
general geometric fiber of $h$ is isomorphic to
$\operatorname{{\mathbb{P}}}^{1}$ (given by extensions (6.15) with fixed
$E^{\prime}$). By projection formula, (6.13) implies (6.14) if we can show
that

$h^{*}\operatorname{\mathcal{O}}_{i}(1,i-1)\cong\operatorname{\mathcal{O}}_{B}(1,i-1)(-\operatorname{\mathcal{E}}).$

Outside of $\operatorname{\mathcal{E}}$ and for any $q\in C$, the bundle
$F_{q}$ over the stack of the $\sigma$-semistable pairs (resp. its determinant
$\Lambda^{\prime}$), pulls back to the bundle $F_{q}$ over
$B\setminus\operatorname{\mathcal{E}}$ (resp. its determinant $\Lambda$), by
(6.15). On the other hand, the divisor $E^{\prime}_{i}$ of $\sigma$-semistable
stable pairs $(E^{\prime},\phi^{\prime})$ such that $\phi^{\prime}$ has a
zero, pulls back to the analogous divisor $E_{i}$ of
$B\setminus\operatorname{\mathcal{E}}$, because the section $\phi$ of $E$ is
the same as the section $\phi^{\prime}$ of $E^{\prime}$. Since $E$ and
$\Lambda$ generate the Picard group of $B\setminus\operatorname{\mathcal{E}}$,
it follows that
$h^{*}\operatorname{\mathcal{O}}_{i}(1,i-1)\cong\operatorname{\mathcal{O}}_{B}(1,i-1)(-c\operatorname{\mathcal{E}})$
for some integer $c$. It remains to show that $c=1$. To this end, we re-
examine the diagram (6.11). Note that the proper transform
$\tilde{\operatorname{{\mathbb{P}}}}$ of
$\operatorname{{\mathbb{P}}}W_{i}^{+}$ in $B$ is isomorphic to its blow-up in
$\operatorname{{\mathbb{P}}}W_{i-1}^{+}$, which is a Cartier divisor $\eta$.
Therefore,
$\tilde{\operatorname{{\mathbb{P}}}}\cong\operatorname{{\mathbb{P}}}W_{i}^{+}$.
However, the restriction
$h^{*}\operatorname{\mathcal{O}}_{i}(1,i-1)|_{\tilde{\operatorname{{\mathbb{P}}}}}$
is isomorphic to the pull-back of $L_{i}^{d}$ from $\operatorname{Sym}^{i}C$,
while the restriction
$\operatorname{\mathcal{O}}_{B}(1,i-1)|_{\tilde{\operatorname{{\mathbb{P}}}}}$
is isomorphic to the pull-back of $L_{i}(d+1)$. Since $L_{i}(d)\cong
L_{i}(d+1)(-\eta)$, and $\operatorname{\mathcal{E}}$ restricts to
$\tilde{\operatorname{{\mathbb{P}}}}$ as $\eta$, the claim follows. ∎

##  7\. Acyclic vector bundles on $M_{i}$ – hard cases

The main goal of the present section is to prove the following results.

######  Theorem 7.1.

Suppose $2<d\leq 2g-1$ and $1\leq i\leq v$. Let $D=x_{1}+\ldots+x_{\alpha}$,
$D^{\prime}=y_{1}+\ldots+y_{\beta}$ (possibly with repetitions) of degrees
$\alpha,\ \beta\leq d+g-2i-1$, and let $t$ be an integer satisfying

(7.1) $\displaystyle\deg D-i-1<t<d+g-2i-1-\deg D^{\prime}.$

We have:

  1. (a)

If $t\notin[0,\ \deg D]$, then
$R\Gamma_{M_{i}(d)}((\bigotimes_{k=1}^{\alpha}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$.

  2. (b)

If $\deg D\notin[t,\ t+\deg D^{\prime}]$, then
$R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes(\bigotimes_{k=1}^{\beta}F_{y_{k}})\otimes\Lambda_{M}^{t})=0$.

In particular, $R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$ in either of these cases.

######  Proposition 7.2.

Suppose $0<d\leq 2g-1$ and $0\leq i\leq v$. Let $D=x_{1}+\ldots+x_{\alpha}$
(possibly with repetitions), with $\alpha=\deg D<d+g-2i-1$. Then

(7.2)
$R\Gamma_{M_{i}}\left(\bigotimes_{k=1}^{\alpha}F_{x_{k}}\right)=R\Gamma_{M_{i}}(G_{D})=R\Gamma_{M_{i}}(G_{D}^{\vee}\otimes\Lambda_{M}^{\alpha})=\mathbb{C}.$

Moreover, the unique (up to a scalar) global section of these bundles vanishes
precisely along the union of codimension $2$ loci $M_{i-1}(\Lambda(-2x_{k}))$,
for $k\in\\{1,\ldots,\alpha\\}$.

These computations will allow us to verify both the Bondal-Orlov conditions
for the fully faithful embeddings of $D^{b}(\operatorname{Sym}^{\alpha}C)$
into $D^{b}(M_{i})$, for $\alpha\leq i$, as well as the vanishings needed in
order to show semi-orthogonality between the corresponding subcategories of
$D^{b}(M_{i})$ thus defined.

######  Remark 7.3.

If $\alpha=1$ then $F_{x_{1}}=G_{D}=G_{D}^{\vee}\otimes\Lambda_{M}$. If
$\alpha>1$ then the precise relationship between vector bundles $G_{D}$ and
$G_{D}^{\vee}\otimes\Lambda_{M}^{\alpha}$ is not clear, however both are
deformations of $\bigotimes_{k=1}^{\alpha}F_{x_{k}}$ by Corollary 2.10, see
also the proof of Lemma 7.5.

We will prove both Theorem 7.1 and Proposition 7.2 simultaneously, by a
combined induction on the degrees of the divisors.

######  Lemma 7.4.

With the notation as in Theorem 7.1, let $0<d\leq 2g-1$, $i=0$ and suppose the
stronger inequality $\deg D<t<d+g-1-\deg D^{\prime}$ holds. Then
$R\Gamma_{M_{0}(d)}(G_{D}^{\vee}\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t})=R\Gamma_{M_{0}(d)}((\bigotimes_{k=1}^{\alpha}F_{x_{k}}^{\vee})\otimes(\bigotimes_{k=1}^{\beta}F_{y_{k}})\otimes\Lambda_{M}^{t})=0$.

###### Proof.

The vector bundle $\left.G_{D}^{\vee}\otimes
G_{\tilde{D}}\otimes\Lambda_{M}^{t}\right|_{M_{0}}$ is a deformation over
$\operatorname{{\mathbb{A}}}^{1}$ of
$\left.(\bigotimes_{k=1}^{\alpha}F_{x_{k}}^{\vee})\otimes(\bigotimes_{k=1}^{\beta}F_{y_{k}})\otimes\Lambda_{M}^{t}\right|_{M_{0}}$,
which has the form
$\bigoplus\operatorname{\mathcal{O}}_{\mathbb{P}^{d+g-2}}(s_{j}-t)$ on
$M_{0}=\mathbb{P}^{d+g-2}$, where $-\beta\leq s_{j}\leq\alpha$. By hypothesis,
$\alpha-t<0$ and $-\beta-t\geq-(d+g-2)$, so this bundle is $\Gamma$-acyclic. ∎

######  Lemma 7.5.

Suppose $2<d\leq 2g-1$ and $1\leq i\leq v$. Let $D$ be an effective divisor on
$C$ and suppose that $\deg D\leq d+g-2i-1$. Assume the statement of
Proposition 7.2 holds for all effective divisors $\tilde{D}$ with
$\deg\tilde{D}<\deg D$. Then

(7.3) $R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes\Lambda_{M}^{\deg
D-1})=R\Gamma_{M_{i}(d)}(G_{D}\otimes\Lambda_{M}^{-1})=0.$

######  Remark 7.6.

Note that (7.3) is a special case of Theorem 7.1.

###### Proof.

Note that for $1<j\leq i$, the bundles $G_{D}^{\vee}\otimes\Lambda_{M}^{\deg
D-1}$ and $G_{{D}}\otimes\Lambda_{M}^{-1}$ descend from objects with weights
within $[-\deg D+1,\ 1]$, where $1<j$ and $-\deg D+1>1+2j-d-g$ by hypothesis.
By Theorem 3.17, it suffices to show (7.3) when $i=1$. We write
$D=\alpha_{1}x_{1}+\ldots+\alpha_{s}x_{s}$ with $x_{k}\neq x_{j}$. If $\deg
D=0$ then we are done by Theorem 6.1 and if, say, $\alpha_{1}=1$ then we are
done by induction on $\deg D$ by writing $D=\tilde{D}+x_{1}$ and using the
exact sequences obtained from Lemma 3.10,

(7.4) $\displaystyle\begin{aligned} 0\to
G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{\deg\tilde{D}-1}\to
G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D-1}\to\qquad\qquad\qquad\qquad\qquad\\\
\qquad\qquad\qquad\qquad\qquad\to
G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{\deg\tilde{D}}\to
G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{\deg\tilde{D}}|_{M_{0}}\to
0\end{aligned}$

and

(7.5) $\displaystyle 0\to G_{\tilde{D}}\otimes\Lambda_{M}^{-1}\to
G_{D}\otimes\Lambda_{M}^{-1}\to G_{\tilde{D}}\to G_{\tilde{D}}|_{M_{0}}\to 0,$

where $M_{0}=M_{0}(\Lambda(-2x_{1}))$. By Proposition 7.2 (applied to
$\tilde{D}$), the last two terms in each sequence have vanishing higher
cohomology and $H^{0}=\mathbb{C}$ with a global section that does not vanish
along $M_{0}(\Lambda(-2x_{1}))$. Thus

$R\Gamma_{M_{1}(d)}(G_{D}^{\vee}\otimes\Lambda_{M}^{\deg
D-1})=R\Gamma_{M_{1}(d)}(G_{D}\otimes\Lambda_{M}^{-1})=0$

by the hypercohomology spectral sequence
$E_{1}^{p,q}=H^{q}(X,\operatorname{\mathcal{F}}^{p})$. So we can assume that
$\alpha_{k}>1$ for all $k$. By the same reasoning as above, using the
sequences (7.4)–(7.5) and the same hypercohomology spectral sequence, we
conclude that $F_{x_{1}}^{\vee}\otimes
G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{\deg D-1}$ and $F_{x_{1}}\otimes
G_{\tilde{D}}\otimes\Lambda_{M}^{-1}$ (and thus their stable deformations
$G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D-1}$ and
$G_{D}\otimes\Lambda_{M}^{-1}$, see Remark 2.15), have the following
cohomology: $h^{p}=0$ for $p\geq 2$ and $h^{0}=h^{1}$. So it suffices to show
that

(7.6)
$H^{0}(M_{1}(d),G_{D}\otimes\Lambda_{M}^{-1})=H^{0}(M_{1}(d),G_{D}^{\vee}\otimes\Lambda_{M}^{\deg
D-1})=0.$

We recall the construction of $G_{D}$ from the proof of Corollary 2.10 adapted
to our case when $D$ is not necessarily a fat point. Let $M=M_{1}(d)$.

Let
$\operatorname{{\mathbb{B}}}_{\alpha}=\operatorname{Spec}\frac{\mathbb{C}[t_{1},\ldots,t_{\alpha}]}{(\sigma_{1},\ldots,\sigma_{\alpha})}$
be the spectrum of the covariant algebra. Write the indexing set
$\\{1,\ldots,\alpha\\}$ as a disjoint union of sets $A_{k}$ of cardinality
$\alpha_{k}$ for $k=1,\ldots,s$. For every $j\in A_{k}$, we have a diagram of
morphisms as in (2.3),

(7.7)
${\mathbb{B}_{\alpha_{1}}\times\ldots\times\operatorname{{\mathbb{B}}}_{\alpha_{s}}\times
M}$${\mathbb{D}_{\alpha_{k}}\times M}$${C\times
M}$${M}$$\scriptstyle{\pi_{j}}$$\scriptstyle{\tau}$$\scriptstyle{q_{k}}$$\scriptstyle{\rho}$$\scriptstyle{\imath}$

We let $\operatorname{\mathcal{F}}_{k}=q_{k}^{*}{F}$, where $F$ is the
universal bundle, and therefore
$G_{D}=\tau_{*}^{S_{\alpha_{1}}\times\ldots\times
S_{\alpha_{s}}}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\right)$
and $G_{D}^{\vee}\otimes\Lambda_{M}^{\deg
D}=\tau_{*}^{S_{\alpha_{1}}\times\ldots\times
S_{\alpha_{s}}}\left(\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}\otimes\operatorname{sgn}\right)$.
Thus (7.6) is equivalent to the following:
$\Lambda_{M}^{-1}\otimes\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}$
does not have invariant or skew-invariant global sections (with respect to
each factor of $S_{\alpha_{1}}\times\ldots\times S_{\alpha_{s}}$).

The restriction of
$\Lambda_{M}^{-1}\otimes\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{i}$
to the special fiber $M$ is $\Lambda_{M}^{-1}\otimes\bigotimes
F_{x_{k}}^{\otimes\alpha_{k}}$. While the group
$S_{\alpha_{1}}\times\ldots\times S_{\alpha_{s}}$ acts trivially on the
special fiber, the action on the vector bundle is still non-trivial (the
action permutes tensor factors within each block).

######  Claim 7.7.

As a representation of $S_{\alpha_{1}}\times\ldots\times S_{\alpha_{s}}$, the
space $H^{0}(M,\Lambda_{M}^{-1}\otimes\bigotimes
F_{x_{k}}^{\otimes\alpha_{k}})$ is isomorphic to the direct sum
$V_{\alpha_{1}}\oplus\ldots\oplus V_{\alpha_{s}}$ of irreducible
representations, where each $V_{\alpha_{k}}$ is the standard
$(\alpha_{k}-1)$-dimensional irreducible representation of $S_{\alpha_{k}}$
and the other factors $S_{\alpha_{l}}$, $l\neq k$, act on $V_{\alpha_{k}}$
trivially. If we realize the representation $V_{\alpha_{k}}$ as $\\{\sum
a_{j}e_{j}\,|\,\sum a_{j}=0\\}\subset\mathbb{C}^{\alpha_{k}}$ then the vector
$e_{j^{\prime}}-e_{j^{\prime\prime}}\in V_{\alpha_{k}}$ corresponds to the
global section $s_{j^{\prime}j^{\prime\prime}}$ of
$\Lambda_{M}^{-1}\otimes\bigotimes F_{x_{k}}^{\otimes\alpha_{k}}$ that can be
written as a tensor product of universal sections $s_{k}$ of $F_{x_{k}}$ in
positions $j\neq j^{\prime},j^{\prime\prime}$ and the section of
$\Lambda_{M}^{-1}\otimes F_{x_{k}}\otimes F_{x_{k}}$ (in positions
$j^{\prime}$, $j^{\prime\prime}$) given by wedging (recall that $\Lambda_{M}$
is the determinant of $F_{x_{k}}$).

###### Proof of the Claim.

The sections $s_{j^{\prime}j^{\prime\prime}}$ satisfy the same linear
relations as difference vectors $e_{j^{\prime}}-e_{j^{\prime\prime}}$, namely
that $s_{j_{1}j_{2}}+s_{j_{2}j_{3}}+\ldots+s_{j_{r-1}j_{r}}+s_{j_{r}j_{1}}=0$
for $j_{1},\ldots,j_{r}\in A_{k}$. Indeed, choose a basis $\\{f_{1},f_{2}\\}$
in a fiber of the rank $2$ bundle $F_{x_{k}}$ so that the universal section is
equal to $f_{2}$ and the determinant is given by $f_{1}\wedge f_{2}$. After
reordering of $j_{1},\ldots,j_{r}$, we have

$\displaystyle s_{12}+s_{23}+\ldots+s_{r1}=$ $\displaystyle(f_{1}\otimes
f_{2})\otimes f_{2}\otimes\ldots\otimes f_{2}-(f_{2}\otimes f_{1})\otimes
f_{2}\otimes\ldots\otimes f_{2}+$ $\displaystyle f_{2}\otimes(f_{1}\otimes
f_{2})\otimes\ldots\otimes f_{2}-f_{2}\otimes(f_{2}\otimes
f_{1})\otimes\ldots\otimes f_{2}+$ $\displaystyle\ldots=0.$

Let $j_{k}\in A_{k}$ be the first index of this subset for $k=1,\ldots,s$. It
suffices to prove that the sections $s_{j_{k}j}$ for $k=1,\ldots,s$ and $j\in
A_{k}\setminus\\{j_{k}\\}$, form a basis of
$H^{0}(M,\Lambda_{M}^{-1}\otimes\bigotimes F_{x_{k}}^{\otimes\alpha_{k}})$. We
prove this by induction on $\alpha$. Let
$\tilde{F}=F_{x_{1}}^{\otimes\alpha_{1}}\otimes\ldots\otimes
F_{x_{s}}^{\otimes(\alpha_{s}-1)}$. We have the usual exact sequence obtained
from Lemma 3.10:

(7.8) $\displaystyle
0\to\Lambda_{M}^{-1}\otimes\tilde{F}\to\Lambda_{M}^{-1}\otimes\tilde{F}\otimes
F_{x_{s}}\to\tilde{F}\to\tilde{F}|_{M_{0}}\to 0,$

where $M_{0}=M_{0}(\Lambda(-2x_{s}))$. By Proposition 7.2, the last two terms
have vanishing higher cohomology and $H^{0}=\mathbb{C}$. If $\alpha$ is the
first index in its group $A_{s}$, i.e. if $\alpha=j_{s}$, then the global
section of $\tilde{F}$ does not vanish along $M_{0}$ and therefore
$H^{0}(\Lambda_{M}^{-1}\otimes\tilde{F})=H^{0}(\Lambda_{M}^{-1}\otimes\tilde{F}\otimes
F_{x_{s}})$ by the corresponding hypercohomology spectral sequence, and the
basis stays the same. On the other hand, if $\alpha\neq j_{s}$ then the global
section of $\tilde{F}$ (the tensor product of universal sections) vanishes
along $M_{0}$ inducing the zero map $H^{0}(\tilde{F})\to
H^{0}(\tilde{F}|_{M_{0}})$. Moreover, the section $s_{j_{s}\alpha}\in
H^{0}(\Lambda_{M}^{-1}\otimes\tilde{F}\otimes F_{x_{s}})$ maps onto the global
section of $\tilde{F}$. Thus the claim also follows from the hypercohomology
spectral sequence. ∎

If $\alpha_{k}>2$ then $V_{\alpha_{k}}$ is neither trivial nor the sign
representation of $S_{\alpha_{k}}$. On the other hand, if $\alpha_{k}=2$ but
$\alpha>2$ then $V_{\alpha_{k}}$ is the sign representation of
$S_{\alpha_{k}}$ and the trivial representation for other factors
$S_{\alpha_{l}}$, $l\neq k$. Therefore, in either of these cases, no invariant
or skew-invariant global sections of
$\Lambda_{M}^{-1}\otimes\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}$
can specialize to $V_{\alpha_{k}}$, in particular the lemma is proved in these
cases. It follows that the only case that we need to consider is
$\alpha=\alpha_{1}=2$. Let $x=x_{1}$. Let $\nu:\,\mathbb{B}_{\alpha}\times
M\to M$ be the degree $2$ projection map. It suffices to show that the (skew-
symmetric) section $s_{12}$ of $\Lambda_{M}^{-1}\otimes F_{x_{1}}^{\otimes 2}$
on $M$ does not deform to a global section of
$\Lambda_{M}^{-1}\otimes\pi_{1}^{*}\operatorname{\mathcal{F}}_{1}\otimes\pi_{2}^{*}\operatorname{\mathcal{F}}_{1}$
on $\mathbb{B}_{\alpha}\times M$. Equivalently, since
$\Lambda_{M}^{-1}\otimes\pi_{1}^{*}\operatorname{\mathcal{F}}_{1}\cong\pi_{1}^{*}\operatorname{\mathcal{F}}_{1}^{\vee}$,
we claim that the morphism
$\operatorname{Id}\in\operatorname{Hom}(F_{x_{1}},F_{x_{1}})$ does not deform
to a morphism
$\widetilde{\operatorname{Id}}\in\operatorname{Hom}(\pi_{1}^{*}\operatorname{\mathcal{F}}_{1},\pi_{2}^{*}\operatorname{\mathcal{F}}_{1})$.
We argue by contradiction and suppose that $\widetilde{\operatorname{Id}}$
exists.

In the diagram (2.3), let $\operatorname{\mathcal{F}}=q^{*}F$ and
$\bar{F}=\rho_{*}\operatorname{\mathcal{F}}$. Note that
$\operatorname{{\mathbb{B}}}_{2}\cong\operatorname{{\mathbb{D}}}_{2}\cong\mathbb{C}[\epsilon]/(\epsilon^{2})$,
so $\bar{F}$ is a rank $4$ vector bundle on $M$, which is an extension of
$F_{x}$ by $F_{x}$. We denote the subbundle by $F_{x}^{\prime}$ and the
quotient bundle by $F_{x}^{\prime\prime}$. In fact
$F_{x}^{\prime}=\rho_{*}(\epsilon\operatorname{\mathcal{F}})$ and
$F_{x}^{\prime\prime}=\rho_{*}(\operatorname{\mathcal{F}}/\epsilon\operatorname{\mathcal{F}})$
is isomorphic to the restriction of $\operatorname{\mathcal{F}}$ to $M$.
Pushing forward $\widetilde{\operatorname{Id}}$ to $M$ by $\nu$ gives a
morphism, which we also denote by

$\widetilde{\operatorname{Id}}:\,\bar{F}\to\bar{F}.$

Note that $\widetilde{\operatorname{Id}}$ restricts to $-\operatorname{Id}$ on
the subbundle $F_{x}^{\prime}$ (the action of $S_{2}$ on
$\operatorname{{\mathbb{B}}}_{2}$ amounts to $\epsilon\mapsto-\epsilon$) and
induces identity on the quotient bundle $F_{x}^{\prime\prime}$. It follows
that the eigenspace decomposition of the endomorphism
$\widetilde{\operatorname{Id}}$ of $\bar{F}$ is a splitting $\bar{F}\cong
F_{x_{1}}^{\prime}\oplus F_{x_{1}}^{\prime\prime}$. Let
$\Phi_{F}:\,D^{b}(C)\to D^{b}(M)$ be the Fourier–Mukai transform with kernel
$F$. Then $\bar{F}=\Phi_{F}(\operatorname{\mathcal{O}}_{2x})$ and since
$\Phi_{F}$ is fully faithful (Theorem 5.1), $\bar{F}$ is indecomposable. This
gives a contradiction. ∎

######  Lemma 7.8.

Suppose Theorem 7.1 holds for all divisors of $\deg D,\deg
D^{\prime}<a<d+g-2i-1$. Then Proposition 7.2 holds for all divisors with
degree $\alpha\leq a$.

###### Proof.

When $i=0$,
$F_{x_{k}}=\operatorname{\mathcal{O}}_{\mathbb{P}^{r}}\oplus\operatorname{\mathcal{O}}_{\mathbb{P}^{r}}(-1)$
on $M_{0}=\mathbb{P}^{r}$, $r=d+g-2$, and $\bigotimes F_{x_{k}}$ splits as a
sum of line bundles
$\bigoplus\operatorname{\mathcal{O}}_{\mathbb{P}^{r}}(s_{j})$, where
$-\alpha\leq s_{j}\leq 0$ and exactly one of the summands is
$\operatorname{\mathcal{O}}_{\mathbb{P}^{r}}$. Since $\alpha\leq d+g-2$,
$R\Gamma_{M_{i}}(\bigotimes_{k=1}^{\alpha}F_{x_{k}})=\mathbb{C}$ in this case.
Since $G_{D}$ and $G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D}$ are deformations
of $\bigotimes_{k=1}^{\alpha}F_{x_{k}}$ over
$\operatorname{{\mathbb{A}}}^{1}$, we have (7.2) by semi-continuity and
equality of the Euler characteristic.

Let $i\geq 1$. We see that, using Theorem 3.17, it suffices to prove (7.2) on
$M_{1}(d)$. In fact, by Theorem 3.18, $\bigotimes_{k=1}^{\alpha}F_{x_{k}}$
descends from an object with weights within $[-\alpha,\ 0]$, all of which live
in the window $(1+2j-d-g,\ j)$ for $1<j\leq i$, since $1+2j-d-g\leq
1+2i-d-g<-\alpha$ by hypothesis. This way we get
$R\Gamma_{M_{i}}(\bigotimes_{k=1}^{\alpha}F_{x_{k}})=R\Gamma_{M_{1}}(\bigotimes_{k=1}^{\alpha}F_{x_{k}})$.
Similarly, $R\Gamma_{M_{i}}(G_{D})=R\Gamma_{M_{1}}(G_{D})$ and
$R\Gamma_{M_{i}}(G_{D}^{\vee}\otimes\Lambda_{M}^{\alpha})=R\Gamma_{M_{1}}(G_{D}^{\vee}\otimes\Lambda_{M}^{\alpha})$.

Hence, we take $i=1$ and $\alpha<d+g-3$. In this case $d>2$. Let us show the
result for $\bigotimes F_{x_{k}}$ first. We do induction on $\alpha$. If
$D=0$, the result is trivial. Otherwise, use the sequence (3.4) on
$F_{x_{\alpha}}$ to obtain an exact sequence

$\displaystyle
0\to\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\to\bigotimes_{k=1}^{\alpha}F_{x_{k}}\to\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\otimes\Lambda_{M}\to\left.\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\otimes\Lambda_{M}\right|_{M_{0}(d-2)}\to
0.$

Of these terms, we get
$R\Gamma_{M_{1}(d)}(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\otimes\Lambda_{M})=0$
from the vanishing in Theorem 7.1, which by hypothesis holds in this case.
Also, we have
$R\Gamma_{M_{0}(d-2)}(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\otimes\Lambda_{M})=0$
from Lemma 7.4, given that $t=1\neq 0$ and $0<1<d+g-3-(\alpha-1)$. Using the
hypercohomology spectral sequence
$E_{1}^{p,q}=H^{q}(X,\operatorname{\mathcal{F}}^{p})$ and induction, we obtain

$\displaystyle
R\Gamma_{M_{1}}\left(\bigotimes_{k=1}^{\alpha}F_{x_{k}}\right)=R\Gamma_{M_{1}}\left(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}\right)=\mathbb{C}.$

Finally, by Corollary 2.10 both $G_{D}$ and
$G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D}$ are deformations over
$\operatorname{{\mathbb{A}}}^{1}$ of $\bigotimes_{k=1}^{\alpha}F_{x_{k}}$, so
we have (7.2) by semi-continuity and equality of the Euler characteristic. It
also follows that the global section of $G_{D}$ (resp.,
$G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D}$) is a deformation of the global
section of $\bigotimes_{k=1}^{\alpha}F_{x_{k}}$ over
$\operatorname{{\mathbb{A}}}^{1}$, which does not vanish outside of the union
of loci $M_{i-1}(\Lambda(-2x_{k}))$ for $k=1,\ldots,\alpha$. On the other
hand, tautological sections of these bundles (the descent of the tensor
product of tautological sections of
$\bigotimes\pi_{j}^{*}\operatorname{\mathcal{F}}_{k}$ (resp., this tensor
product tensored with the sign representation) for $G_{D}$ (resp.,
$G_{D}^{\vee}\otimes\Lambda_{M}^{\deg D}$)) vanish precisely along these loci.
∎

Now we can prove Theorem 7.1. Observe that by Lemma 7.8, this will also
complete the proof of Proposition 7.2.

###### Proof of Theorem 7.1.

We consider cases (a) and (b) separately.

The case $t\notin[0,\ \deg D]$. We first suppose $D=0$ and do induction on
$\deg D^{\prime}$. If $D=D^{\prime}=0$, we need to show that for $t\neq 0$
with $-i-1<t<d+g-2i-1$ we have $R\Gamma_{M_{i}(d)}(\Lambda_{M}^{t})=0$. If
$t>0$, Lemma 4.2 ensures $R\Gamma_{M_{1}(d)}(\Lambda_{M}^{t})=0$, since $i\geq
1$ and so $t\leq d+g-4$. But also for every $1<j\leq i$ we have
$1+2j-d-g<-t<0<j$, that is, the weight of $\Lambda_{M}^{t}$ lives in the
window between $M_{j-1}$ and $M_{j}$, so we conclude
$R\Gamma_{M_{i}(d)}(\Lambda_{M}^{t})=R\Gamma_{M_{1}(d)}(\Lambda_{M}^{t})=0$ by
Theorem 3.17. Suppose now $t<0$, so that $-i\leq t<0$. By Theorem 6.1,
$R\Gamma_{M_{i}(d)}(\Lambda_{M}^{t})=0$.

Let $D=0$ and $\deg D^{\prime}\geq 1$. Assume the result holds for divisors
$\tilde{D^{\prime}}$ with $\deg\tilde{D^{\prime}}<\deg D^{\prime}$. In this
case (7.3) holds for $D^{\prime}$ by Lemma 7.8 and Lemma 7.5, since $\deg
D^{\prime}\leq d+g-2i-1\leq d+g-3$. We need to show that this implies
$R\Gamma_{M_{i}(d)}(G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$ for
$-i-1<t<d+g-2i-1-\deg D^{\prime}$ and $t\neq 0$. If $t=-1$, then
$R\Gamma_{M_{i}(d)}(G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$ by (7.3). If
$t\neq-1$, we write $D^{\prime}=\tilde{D^{\prime}}+y$ and use the fact that
$G_{D^{\prime}}$ is a stable deformation of $F_{y}\otimes
G_{\tilde{D^{\prime}}}$ over $\operatorname{{\mathbb{A}}}^{1}$ (see
Proposition 2.13). If we take the sequence (3.4) twisted by
$G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}$, we get an exact sequence

$\displaystyle 0\to G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}\to
F_{y}\otimes G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}\to
G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t+1}\to\left.G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t+1}\right|_{M_{i-1}}\to
0.$

Observe that this is an acyclic chain complex involving $F_{y}\otimes
G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t}$ and where the remaining three
terms satisfy the corresponding inequalities from (7.1):
$-i-1<t<d+g-2i-1-\deg\tilde{D^{\prime}}$,
$-i-1<t+1<d+g-2i-1-\deg\tilde{D^{\prime}}$,
$-(i-1)-1<t+1<d-2+g-2(i-1)-1-\deg\tilde{D^{\prime}}$. Notice that the
inequality $\deg\tilde{D^{\prime}}\leq(d-2)+g-2(i-1)-1$ is preserved too.
Given that $t\neq-1$, we have both $t,\ t+1\neq 0$ so by induction we see that
$R\Gamma_{M_{i}(d)}(G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t})=R\Gamma_{M_{i}(d)}(G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t+1})=0$.
On the other hand, we obtain
$R\Gamma_{M_{i-1}(d-2)}(G_{\tilde{D^{\prime}}}\otimes\Lambda_{M}^{t+1})=0$
either by induction if $i>1$, or from Lemma 7.4 if $i=1$. Therefore we get the
desired vanishing from the corresponding hypercohomology spectral sequence and
semi-continuity.

Next we do induction on $\alpha=\deg D$. If $\alpha\geq 1$, we write
$D=\tilde{D}+x_{\alpha}$ and take the sequence (3.3) with
$F_{x_{\alpha}}^{\vee}$, twisted by
$(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}$. This way we get an exact sequence
involving $(\bigotimes_{k=1}^{\alpha}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}$, and where the remaining terms are
$(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t-1}$ and
$(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}$ on $M_{i}(d)$, and
$(\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t}$ on $M_{i-1}(d-2)$. All three still
satisfy the inequalities (7.1): $\deg\tilde{D}-i-1<t-1<d+g-2i-1-\deg
D^{\prime}$, $\deg\tilde{D}-i-1<t<d+g-2i-1-\deg D^{\prime}$,
$\deg\tilde{D}-(i-1)-1<t<d-2+g-2(i-1)-1-\deg D^{\prime}$. Further, $t,\
t-1\notin[0,\ \deg\tilde{D}]$ so by induction
$R\Gamma_{M_{i}(d)}((\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t-1})=R\Gamma_{M_{i}(d)}((\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$, while
$R\Gamma_{M_{i-1}(d-2)}((\bigotimes_{k=1}^{\alpha-1}F_{x_{k}}^{\vee})\otimes
G_{D^{\prime}}\otimes\Lambda_{M}^{t})=0$ either by induction when $i>1$ or by
Lemma 7.4 when $i=1$ (observe that when $i=1$ we must have $t>\deg\tilde{D}$).
By looking at the corresponding hypercohomology spectral sequence we obtain
the vanishing
$R\Gamma_{M_{i}(d)}((\bigotimes_{k=1}^{\alpha}F_{x_{k}}^{\vee})\otimes
G_{{D^{\prime}}}\otimes\Lambda_{M}^{t})=0$.

The case $\deg D\notin[t,\ t+\deg D^{\prime}]$. The proof is similar. If
$D=D^{\prime}=0$, then $t\neq 0$ and the vanishing has already been shown
above. Let $D^{\prime}=0$ and $\deg D\geq 1$. If $t=\deg D-1$, the vanishing
follows from induction and Lemma 7.5. If $t\neq\deg D-1$, we write
$D=\tilde{D}+x$. Twisting (3.3) by
$G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{t}$ yields an exact sequence
involving $F_{x}^{\vee}\otimes G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{t}$ and
where the remaining terms are $G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{t-1}$
and $G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{t}$ on $M_{i}(d)$ and
$G_{\tilde{D}}^{\vee}\otimes\Lambda_{M}^{t}$ on $M_{i-1}(d-2)$. Inequalities
(7.1) are preserved in all of them. Further, $t\neq\deg D-1$ implies that both
$\deg\tilde{D}\neq t,\ t-1$, so by induction and Lemma 7.4 they are all
$\Gamma$-acyclic, and semi-continuity provides
$R\Gamma_{M_{i}(d)}(G_{D}^{\vee}\otimes\Lambda_{M}^{t})=0$.

Finally, we do induction on $\beta=\deg D^{\prime}\geq 1$. We write
$D^{\prime}=\tilde{D^{\prime}}+y_{\beta}$ and use (3.4) twisted by
$G_{D}^{\vee}\otimes(\bigotimes_{k=1}^{\beta-1}F_{y_{k}})\otimes\Lambda_{M}^{t}$.
The resulting terms other than
$G_{D}^{\vee}\otimes(\bigotimes_{k=1}^{\beta}F_{y_{k}})\otimes\Lambda_{M}^{t}$
all satisfy (7.1), and the fact that $\deg{D}\notin[t,\ t+\deg
D^{\prime}]=[t,\ t+1+\deg\tilde{D^{\prime}}]$, together with Lemma 7.4 when
$i=1$, guarantee they are all $\Gamma$-acyclic. Again, the hypercohomology
spectral sequence yields the required vanishing. The final statement is a
direct consequence of Corollary 2.10 and semi-continuity. ∎

##  8\. Computation of $R\operatorname{Hom}(G_{D},G_{D^{\prime}})$