#
Hyperkähler varieties as Brill-Noether loci on curves

Soheyla Feyzbakhsh

###### Abstract.

Consider the moduli space $M_{C}(r;K_{C})$ of stable rank r vector bundles on
a curve $C$ with canonical determinant, and let $h$ be the maximum number of
linearly independent global sections of these bundles. If $C$ embeds in a K3
surface $X$ as a generator of $\operatorname{Pic}(X)$ and the genus of $C$ is
sufficiently high, we show the Brill-Noether locus
$\operatorname{BN}_{C}\subset M_{C}(r;K_{C})$ of bundles with $h$ global
sections is a smooth projective Hyperkähler manifold of dimension
$2g-2r\lfloor\frac{g}{r}\rfloor$, isomorphic to a moduli space of stable
vector bundles on $X$.

##  1\. Introduction

In this paper, we show that specific Brill-Noether loci of rank $r$-stable
vector bundles on K3 curves are isomorphic to moduli spaces of stable vector
bundles on K3 surfaces. This generalises Mukai’s program [Muk01, ABS14,
Fey20a, Fey20b] to Brill-Noether loci on K3 curves of dimension higher than
$2$.

Fix $(r,k)\in\mathbb{Z}_{>0}\oplus\mathbb{Z}_{>0}$ such that $\gcd(r,k)=1$ and
$k<r$. Then take $g\gg 0$. Let $(X,H)$ be a polarised K3 surface with
$\operatorname{Pic}(X)=\mathbb{Z}\cdot H$ and let $C$ be any curve in the
linear system $|H|$ of genus $g$. There is a unique $s\in\mathbb{Z}$ such that

(1) $-2\leq k^{2}H^{2}-2rs<-2+2r\,.$

We further assume $\gcd(s,k)=1$. Consider the Brill-Noether locus
$\operatorname{BN}_{C}(r,k(2g-2),r+s)$ of semistable111See Definition 2.1.
rank $r$-vector bundles on $C$ having degree $kH^{2}=k(2g-2)$ and possessing
at least $r+s$ linearly independent global sections. Also $M_{X}(\mathsf{v})$
denotes the moduli space of $H$-Gieseker semistable sheaves on $X$ with Mukai
vector $\mathsf{v}=(r,kH,s)$. It is a (non-empty) smooth quasi-projective
variety222See for instance [Huy16, Chapter 10, Corollary 2.1 & Theorem 2.7].
of dimension $\mathsf{v}^{2}+2=k^{2}H^{2}-2rs+2$.

######  Theorem 1.1.

There is an isomorphism

$\Psi\colon M_{X}(\mathsf{v})\rightarrow\operatorname{BN}_{C}(r,k(2g-2),r+s)$

which sends a vector bundle333Any $H$-Gieseker semistable sheaf of Mukai
vector $\mathsf{v}$ is locally free, see Lemma 3.5. $E$ on $X$ to its
restriction $E|_{C}$.

The above Theorem says that for any vector bundle $F$ on the curve $C$ in the
Brill-Noether locus $\operatorname{BN}_{C}(r,k(2g-2),r+s)$, there exists a
unique vector bundle on the surface $X$ whose restriction to $C$ is isomorphic
to $F$. In particular, we obtain the following:

######  Corollary 1.2.

Any vector bundle $F\in\operatorname{BN}_{C}(r,k(2g-2),r+s)$ is stable (cannot
be strictly semistable) and $\wedge^{r}F=\omega_{C}^{\otimes k}$ with
$h^{0}(C,F)=r+s$.

### Proof ideas

To prove Theorem 1.1, we go through the following steps:

Consider the embedding $\iota\colon C\hookrightarrow X$. The push-forward
$\iota_{*}F$ for any vector bundle $F\in\operatorname{BN}_{C}(r,kH^{2},r+s)$
is semistable in the large volume limit of the space of Bridgeland stability
conditions on $X$. We move down and study walls for objects of class
$\operatorname{ch}(\iota_{*}F)$ between the large volume limit and the Brill-
Noether wall that the structure sheaf $\mathcal{O}_{X}$ is making. The first
wall is made by stable sheaves with Mukai vector $\mathsf{v}=(r,kH,s)$. We
show $\iota_{*}F$ gets destabilised along this wall if and only if $F=E|_{C}$
for a sheaf $E\in M_{X}(\mathsf{v})$.

The assumption (1) implies that any stable coherent sheaf of Mukai vector
$\mathsf{v}$ is locally-free. We show that there is no wall for objects with
Mukai vector $\mathsf{v}$ between the large volume limit and the Brill-Noether
wall. That’s why we get $h^{0}(X,E)=r+s$ for any $E\in M_{X}(\mathsf{v})$.
Hence, there is a well-defined map

$\displaystyle\Psi\colon M_{X}(\mathsf{v})$
$\displaystyle\rightarrow\operatorname{BN}_{C}(r,kH^{2},r+s)$ $\displaystyle
E$ $\displaystyle\mapsto E|_{C}.$

Then a usual wall-crossing argument gives injectivity of $\Psi$ due to the
uniqueness of Harder-Narasimhan filtration.

To prove $\Psi$ is surjective we apply the technique developed in [Fey20a]
which bounds the number of global sections of sheaves on $X$ in terms of the
length of a convex polygon (which is the Harder-Narasimhan polygon in the
Brill-Noether region). This implies that if $h^{0}(X,\iota_{*}F)\geq r+s$,
then $F$ is the restriction of a vector bundle $E\in M_{X}(\mathsf{v})$ to the
curve $C$, so in particular, $\Psi$ is surjective.

Finally, any vector bundle $E\in M_{X}(\mathsf{v})$ is generated by global
sections and the kernel

$K_{E}\coloneqq\ker(\mathcal{O}_{X}^{\oplus r+s}\xrightarrow{\text{ev}}E).$

is a slope-stable vector bundle. By applying a wall-crossing argument, we show
that Hom$(K_{E},E(-H)[1])=0$. Then a usual deformation theory argument implies
that the derivative $d\Psi$ is surjetive, and so $\Psi$ is an isomorphism.

Note that in Theorem 1.1, the assumption $g\gg 0$ is necessary for all the
above steps. Remark 4.3 gives a list of inequalities that $g$ must satisfy to
get Theorem 1.1.

### Outlook

In this paper, we only work on curves on K3 surfaces of Picard rank one. More
generally one can consider a polarised K3 surfaces $(X,H)$ of arbitrary Picard
rank and pick a curve $C\in|H|$ of genus $g(C)\gg 0$. For any
$r,k\in\mathbb{Z}^{>0}$, we define444One can replace the structure sheaf
$\mathcal{O}_{C}$ with any other vector bundle on $C$ and consider the
corresponding Brill-Noether locus.

$h_{r,k}\coloneqq\max\left\\{h^{0}(C,F)\colon\text{rank $r$-stable vector
bundles $F$ on $C$ with $\wedge^{r}F=\omega_{C}^{\otimes k}$ }\right\\}.$

Consider the Brill-Noether locus
$\operatorname{BN}_{C}^{\text{st}}(r,\,\omega_{C}^{\otimes k},\,h_{r,k})$ of
stable rank $r$-vector bundles $F$ on $C$ of fixed determinant
$\omega_{C}^{\otimes k}$ and having $h_{r,k}$ linearly independent global
sections. Then the natural question is whether this Brill-Noether locus is a
Hyperkähler variety when $g(C)$ is high enough. I expect to answer this
question in the future by applying a more general wall-crossing argument.

Note that the assumption $g(C)\gg 0$ is necessary as there are examples of the
above Brill-Noether loci on K3 curves of genus $7$ [Muk01] and genus $12$
[Fey22] which are smooth Fano varieties. A similar technique as in this paper
can be applied to generalise Theorem 1.1 to

  1. (a)

polarised K3 surfaces $(X,H)$ such that $H^{2}$ divides $H.D$ for all curve
classes $D$ on $X$, and

  2. (b)

curves $C\hookrightarrow X$ which are not necessarily in the linear system
$|H|$.

### Plan

In Section 2, we review Bridgeland stability conditions on the bounded derived
category of coherent sheaves on a K3 surface $X$. Section 3 analyses walls for
the push-forward of vector bundles in our Brill-Noether locus to the K3
surface $X$ and as a result, proves Theorem 1.1. The computations for the
location of walls are all postponed to Section 4.

### Acknowledgement

I would like to thank Daniel Huybrechts who was the first person to encourage
me to think about higher-dimensional Hyperkähler varieties as Brill-Noether
loci on K3 curves. I would also like to thank Enrico Arbarello, Arend Bayer,
Ignacio Barros Reyes, Chunyi Li, Emanuele Macrì, Lenny Taelman, Richard
Thomas, and Yukinobu Toda for helpful discussions and comments about this
work. The author was supported by Marie Skłodowska–Curie individual
fellowships 887438 and EPSRC postdoctoral fellowship EP/T018658/1.

##  2\. Review: Bridgeland stability conditions on K3 surfaces

In this section, we review the description of a slice of the space of
stability conditions on the bounded derived category of coherent sheaves on a
K3 surface given in [Bri08, Section 1-7].

Let $X$ be a projective scheme over $\mathbb{C}$ of dimension $\dim(X)\geq 1$,
and let $H$ be an ample line bundle on $X$. The Hilbert polynomial $P(E,m)$ is
given by $m\mapsto\chi(\mathcal{O}_{X},E\otimes\mathcal{O}_{X}(mH))$. It can
be uniquely written in the form

$P(E,m)=\sum_{i=0}^{\dim(E)}\alpha_{i}(E)\frac{m^{i}}{i!}$

with integral coefficients $\alpha_{i}(E)$. The reduced Hilbert polynomial
$p(E,m)$ of a coherent sheaf $E$ of dimension $d$ is defined by

$p(E,m)=\frac{P(E,m)}{\alpha_{d}(E)}.$

######  Definition 2.1.

[HL10, Definition 1.2.4] A coherent sheaf $E$ of dimension $d$ is (semi)stable
if $E$ is pure and for any proper subsheaf $F\subset E$ one has
$p(F,m)\,(\leq)\,p(E,m)$ for $m\gg 0$. Here $(\leq)$ denotes $<$ for stability
and $\leq$ for semistability.

From now on, we always assume $(X,H)$ is a smooth polarized K3 surface over
$\mathbb{C}$ with Pic$(X)=\mathbb{Z}\cdot H$. Denote the bounded derived
category of coherent sheaves on $X$ by $\mathcal{D}(X)$ and its Grothendieck
group by $K(X):=K(\mathcal{D}(X))$. Given an object $E\in\mathcal{D}(X)$, we
write
$\operatorname{ch}(E)=(\operatorname{ch}_{0}(E),\operatorname{ch}_{1}(E),\operatorname{ch}_{2}(E))\in
H^{*}(X,\mathbb{Z})$ for its Chern characters. The Mukai vector of an object
$E\in\mathcal{D}(X)$ is an element of the numerical Grothendieck group
$\mathcal{N}(X)=\mathbb{Z}\oplus\text{NS}(X)\oplus\mathbb{Z}\cong\mathbb{Z}^{3}$
defined via

$v(E)\coloneqq\big{(}\operatorname{ch}_{0}(E),\,\operatorname{ch}_{1}(E),\,\operatorname{ch}_{0}(E)+\operatorname{ch}_{2}(E)\big{)}=\text{ch}(E)\sqrt{\text{td}(X)}\,.$

The Mukai bilinear form

$\left\langle
v(E),v(E^{\prime})\right\rangle=\operatorname{ch}_{1}(E)\operatorname{ch}_{1}(E^{\prime})+\operatorname{ch}_{0}(E)\left(\operatorname{ch}_{0}(E^{\prime})+\operatorname{ch}_{2}(E^{\prime})\right)+\operatorname{ch}_{0}(E^{\prime})\left(\operatorname{ch}_{0}(E)+\operatorname{ch}_{2}(E)\right)$

makes $\mathcal{N}(X)$ into a lattice of signature $(2,1)$. The Riemann-Roch
theorem implies that this form is the negative of the Euler form, defined as

$\chi(E,E^{\prime})=\sum_{i}(-1)^{i}\dim_{\mathbb{C}}\text{Hom}_{X}^{i}(E,E^{\prime})=-\left\langle
v(E),v(E^{\prime})\right\rangle.$

The slope of a coherent sheaf $E\in\operatorname{Coh}(X)$ is defined by

$\mu_{H}(E):=\begin{cases}\frac{H.\operatorname{ch}_{1}(E)}{H^{2}\operatorname{ch}_{0}(E)}&\text{if
$\operatorname{ch}_{0}(E)>0$}\\\ +\infty&\text{if
$\operatorname{ch}_{0}(E)=0$}.\end{cases}$

This leads to the usual notion of $\mu_{H}$-stability. Associated to it every
sheaf $E$ has a Harder-Narasimhan filtration. Its graded pieces have slopes
whose maximum we denote by $\mu_{H}^{+}(E)$ and minimum by $\mu_{H}^{-}(E)$.

For any $b\in\mathbb{R}$, let $\mathcal{A}(b)\subset\mathcal{D}(X)$ denote the
abelian category of complexes

(2) $\mathcal{A}(b)\ =\ \big{\\{}E^{-1}\xrightarrow{\,d\,}E^{0}\ \colon\
\mu_{H}^{+}(\ker d)\leq b\,,\ \mu_{H}^{-}(\operatorname{coker}d)>b\big{\\}}.$

Then $\mathcal{A}(b)$ is the heart of a bounded t-structure on
$\mathcal{D}(X)$ by [Bri08, Lemma 6.1]. For any pair $(b,w)\in\mathbb{R}^{2}$,
we define the group homomorphism $Z_{b,w}\colon K(X)\to\mathbb{C}$ by

(3)
$Z_{b,w}(E):=-\operatorname{ch}_{2}(E)+w\operatorname{ch}_{0}(E)+i\bigg{(}\frac{H\operatorname{ch}_{1}(E)}{H^{2}}-b\operatorname{ch}_{0}(E)\bigg{)}.$

Define the function $\Gamma\colon\mathbb{R}\rightarrow\mathbb{R}$ as

$\Gamma(b)\coloneqq\begin{cases}b^{2}\left(\frac{H^{2}}{2}+1\right)-1&\text{if
$b\neq 0$}\\\ 0&\text{if $b=0$}.\end{cases}$

By abuse of notations, we also denote the graph of $\Gamma$ by curve $\Gamma$
(see Figure 1). We define

$U\coloneqq\left\\{(b,w)\in\mathbb{R}^{2}\colon w>\Gamma(b)\right\\}.$

In figures, we will plot the $(b,w)$-plane simultaneously with the image of
the projection map

$\displaystyle\Pi\colon\
K(X)\setminus\big{\\{}E\colon\operatorname{ch}_{0}(E)=0\big{\\}}\\!$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{R}^{2},$ $\displaystyle
E$ $\displaystyle\shortmid\joinrel\relbar\joinrel\rightarrow$
$\displaystyle\\!\\!\bigg{(}\frac{\operatorname{ch}_{1}(E).H}{\operatorname{ch}_{0}(E)H^{2}}\,,\,\frac{\operatorname{ch}_{2}(E)}{\operatorname{ch}_{0}(E)}\bigg{)}.$

$b,\,\frac{\operatorname{ch}_{1}\\!.\hskip
0.6ptH}{\operatorname{ch}_{0}H^{2}}$$\Gamma$$w,\,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$\Pi(E)$$U$$\frac{\operatorname{ch}_{2}(E)}{\operatorname{ch}_{0}(E)}$$\frac{\operatorname{ch}_{1}(E).H}{\operatorname{ch}_{0}(E)H^{2}}$$(b,w)$
Figure 1. $(b,w)$-plane and the projection $\Pi(E)$

Consider the slope function

$\nu_{{b,w}}\colon\mathcal{A}(b)\to\mathbb{R}\cup\\{+\infty\\},\quad\nu_{b,w}(E)\coloneqq\begin{cases}-\frac{\text{Re}[Z_{b,w}(E)]}{\text{Im}[Z_{b,w}(E)]}&\text{if
$\text{Im}[Z_{b,w}(E)]>0$}\\\ +\infty&\text{if
$\text{Im}[Z_{b,w}(E)]=0$.}\end{cases}$

This defines our notion of stability in $\mathcal{A}(b)$:

######  Definition 2.2.

Fix $w>\Gamma(b)$. We say $E\in\mathcal{D}(X)$ is (semi)stable with respect to
the pair $\sigma_{b,w}\coloneqq\left(\mathcal{A}(b),\,Z_{b,w}\right)$ if and
only if

  * •

$E[k]\in\mathcal{A}(b)$ for some $k\in\mathbb{Z}$, and

  * •

$\nu_{b,w}(F)\,(\leq)\,\nu_{b,w}\big{(}E[k]/F\big{)}$ for all non-trivial
subobjects $F\hookrightarrow E[k]$ in $\mathcal{A}(b)$.

Let $E$ be a semistable coherent sheaf, or more generally a
$\sigma_{b,w}$-semistable object for some $(b,w)\in U$. By [FL21, Remark
2.3(a)]555Note that the curve $\Gamma$ defined in this paper lies above the
curve $\Gamma$ considered in [FL21]., the projection $\Pi(E)$ does not lie in
$U$. Thus the same argument as in [Bri08, Lemma 6.2] shows that the pair
$\sigma_{b,w}=\left(\mathcal{A}(b),\,Z_{b,w}\right)$ is a Bridgeland stability
condition on $\mathcal{D}(X)$, see [Bri07, Definition 1.1 & Proposition
5.3]666Up to the action of $\widetilde{\text{GL}}^{+}(2;\mathbb{R})$, the
stability conditions $\sigma_{b,w}$ are the same as the stability conditions
defined in [Bri08, section 6].. This in particular implies that every object
$E\in\mathcal{A}(b)$ admits a Harder–Narasimhan filtration which is a finite
sequence of objects in $\mathcal{A}(b)$:

$0=E_{0}\subset E_{1}\subset E_{2}\subset\dots\subset E_{k}=E$

whose factors $E_{i}/E_{i-1}$ are $\sigma_{b,w}$-semistable and
$\nu_{b,w}(E_{1})>\nu_{b,w}(E_{2})>\dots>\nu_{b,w}(E/E_{k-1})$. We denote
$\nu_{b,w}^{+}(E)\coloneqq\nu_{b,w}(E_{1})$ and
$\nu_{b,w}^{-}(E)\coloneqq\nu_{b,w}(E_{k})$. Summarizing, we have the
following:

######  Theorem 2.3 ([Bri08, Section 1]).

For any pair $(b,w)\in U$, the pair
$\sigma_{b,w}=\left(\mathcal{A}(b),Z_{b,w}\right)$ defines a stability
condition on $\mathcal{D}(X)$. Moreover, the map from $U$ to the space of
stability conditions $\mathop{\mathrm{Stab}}\nolimits(X)$ on $\mathcal{D}(X)$
given by $(b,w)\mapsto\sigma_{b,w}$ is continuous.

The second part of Theorem 2.3 implies that the two-dimensional family of
stability conditions $\sigma_{b,w}$ satisfies wall-crossing as $b$ and $w$
vary, see for instance [FT21, Proposition 4.1] or [FL21, Proposition 2.4].

######  Proposition 2.4 (Wall and chamber structure).

Fix a non-zero object $E\in\mathcal{D}(X)$. There exists a collection of line
segments (walls) $\mathcal{W}_{E}^{i}$ in $U$ (called “ _walls_ ”) which are
locally finite and satisfy

The extension of each line segment $\mathcal{W}_{E}^{i}$ passes through the
point $\Pi(E)$ if $\operatorname{ch}_{0}(E)\neq 0$, or has fixed slope
$\frac{\operatorname{ch}_{2}(E)H^{2}}{\operatorname{ch}_{1}(E).H}$ if
$\operatorname{ch}_{0}(E)=0$.

An endpoint of the segment $\mathcal{W}_{E}^{i}$ is either on the curve
$\Gamma$ or on the line segment through $(0,0)$ to $(0,-1)$.

The $\sigma_{b,w}$-(semi)stability of $E$ is unchanged as $(b,w)$ varies
within any connected component (called a “ _chamber_ ”) of
$U\setminus\bigcup_{i\in I}\mathcal{W}_{E}^{i}$.

For any wall $\mathcal{W}_{E}^{i}$ there is $k_{i}\in\mathbb{Z}$ and a map
$f\colon F\to E[k_{i}]$ in $\mathcal{D}(X)$ such that

  * •

for any $(b,w)\in\mathcal{W}_{E}^{i}$, the objects $E[k_{i}],\,F$ lies in the
heart $\mathcal{A}(b)$,

  * •

$E[k_{i}]$ is $\nu_{b,w}$-semistable with
$\nu_{b,w}(E)=\nu_{b,w}(F)=\,\mathrm{slope}\,(\mathcal{W}_{E}^{i})$ constant
on $\mathcal{W}_{E}^{i}$, and

  * •

$f$ is an injection $F\subset E[k_{i}]$ in $\mathcal{A}(b)$ which strictly
destabilises $E[k_{i}]$ for $(b,w)$ in one of the two chambers adjacent to the
wall $\mathcal{W}_{E}^{i}$.

$\mathcal{W}_{E}^{2}$$\mathcal{W}_{E}^{1}$$\operatorname{ch}_{0}(E)\neq
0$$\operatorname{ch}_{0}(E)=0$$b,\frac{\operatorname{ch}_{1}.H}{H^{2}\operatorname{ch}_{0}}$$w,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$b,\frac{\operatorname{ch}_{1}.H}{H^{2}\operatorname{ch}_{0}}$$w,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$U$$U$$\Pi(E)$$\mathcal{W}_{E}^{1}$$\mathcal{W}_{E}^{2}$
Figure 2. Walls $\mathcal{W}_{E}^{i}$ for $E$.

##  3\. Wall-crossing

Let $(X,H)$ be a smooth polarized K3 surface over $\mathbb{C}$ with
Pic$(X)=\mathbb{Z}\cdot H$, and let $C\in|H|$ be any curve of genus
$g=\frac{1}{2}H^{2}+1$. The Brill-Noether loci of vector bundles on $C$ when
genus $g$ is low have been highly studied, see e.g. [Muk96, Muk88, Muk02,
Muk01, Muk95]. But in this section, we analyse these loci when genus $g$ is
high.

As in the Introduction, fix $(r,k)\in\mathbb{Z}_{>0}\oplus\mathbb{Z}_{>0}$
such that $\gcd(r,k)=1$ and $k<r$. Then we take $g\gg 0$ and consider
polarised K3 surfaces $(X,H)$ of genus $g=\frac{1}{2}H^{2}+1$. Consider the
Mukai vector $\mathsf{v}\coloneqq(r,kH,s)$ such that $\gcd(s,k)=1$ and the
assumption (1) holds:

$-2\leq\mathsf{v}^{2}=k^{2}H^{2}-2rs<-2+2r\,.$

This implies

(4) $-1+\frac{1}{r}+\frac{k^{2}}{2r}H^{2}<\ s\
\leq\frac{1}{r}+\frac{k^{2}}{2r}H^{2}\,.$

We set Mukai vector $\alpha\coloneqq(s,-kH,r)$, so

$\Pi(\mathsf{v})=\left(\frac{k}{r},\,\frac{s-r}{r}\right)\
,\qquad\Pi(\alpha)=\left(-\frac{k}{s}\ ,\ \frac{r-s}{s}\right),$

and

$\Pi(\mathsf{v}(-H))=\Pi\left(r,(k-r)H,s-kH^{2}+\frac{r}{2}H^{2}\right)=\left(\frac{k-r}{r}\
,\ \frac{s-r}{r}-\frac{k}{r}H^{2}+\frac{1}{2}H^{2}\right).$

######  Definition 3.1.

Let $\ell^{*}$ be the lowest line of slope $H^{2}(\frac{k}{r}-\frac{1}{2})$
which intersects the curve $\Gamma$ at two points with $b$-values
$b_{1}^{*}<b_{2}^{*}$ satisfying

(5) $\max\left\\{b_{1}^{*}-\frac{k-r}{r}\,,\
\frac{k}{r}-b_{2}^{*}\right\\}\,\leq\,\frac{1}{r^{2}(r+1)}$

$b,\,\frac{\operatorname{ch}_{1}.H}{\operatorname{ch}_{0}H^{2}}$$\Gamma$$w,\,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$b_{2}^{\mathsf{v}}$$\tilde{b}_{2}$$\ell_{\mathsf{v}}$$\ell^{*}$$\tilde{\ell}$$b_{2}^{*}$$\Pi(\alpha)$$\ell_{\alpha}$$\Pi(\mathsf{v})$$\ell_{\mathsf{v}(-H)}$$b_{1}^{\mathsf{v}}$$b_{2}^{\alpha}$$b_{1}^{*}$$b_{1}^{\alpha}$$b_{1}^{\mathsf{v}(-H)}$$\tilde{b}_{1}$$\Pi(\mathsf{v}(-H))$
Figure 3. The lines $\ell^{*}$, $\ell_{\mathsf{v}}$, $\ell_{\mathsf{v}(-H)}$,
$\ell_{\alpha}$ and $\widetilde{\ell}$.

######  Proposition 3.2.

Suppose $H^{2}=2g-2\gg 0$.

The origin point $(0,0)$ lies below $\ell^{*}$ and

$\frac{k-r}{r}<b_{1}^{*}<b_{2}^{*}<\frac{k}{r}.$

The line $\widetilde{\ell}$ passing through $\Pi(\mathsf{v})$ and
$\Pi(\mathsf{v}(-H))$ is parallel to $\ell^{*}$ and lies above it. It
intersects the curve $\Gamma$ at two points with $b$-values
$\widetilde{b}_{1}<\widetilde{b}_{2}$ such that

$\frac{k-r}{r}\leq\widetilde{b}_{1}<b_{1}^{*}\qquad\text{and}\qquad
b_{2}^{*}<\widetilde{b}_{2}\leq\frac{k}{r}.$

The line $\ell_{\mathsf{v}}$ passing through $\Pi(\mathsf{v})$ and the origin
$(0,0)$ intersects $\Gamma$ at two points (except the origin) with $b$-values
$b_{1}^{\mathsf{v}}<b_{2}^{\mathsf{v}}$ satisfying

$b_{1}^{\mathsf{v}}<-\frac{k}{s}+\frac{1}{s(s-1)}\qquad\text{and}\qquad
b_{2}^{*}<b_{2}^{\mathsf{v}}\leq\frac{k}{r}.$

The line $\ell_{\alpha}$ passing through $\Pi(\mathsf{v}(-H))$ and
$\Pi(\alpha)$ intersects $\Gamma$ at two points with $b$-values
$b_{1}^{\alpha}<b_{2}^{\alpha}$ so that

$\frac{k-r}{r}\leq
b_{1}^{\alpha}<b_{1}^{*}\qquad\text{and}\qquad-\frac{k}{s}-\frac{1}{s(s-1)}<b_{2}^{\alpha}.$

The line $\ell_{\mathsf{v}(-H)}$ passing through $\Pi(\mathsf{v}(-H))$ and the
origin intersects the curve $\Gamma$ at two points (except the origin) with
$b$-values $b_{1}^{\mathsf{v}(-H)}<0<b_{2}^{\mathsf{v}(-H)}$ such that

$\frac{k-r}{r}\leq b_{1}^{\mathsf{v}(-H)}<b_{1}^{*}.$

######  Lemma 3.3.

Let $\ell^{*}$ intersects the vertical lines $b=\frac{k}{r}$ and
$b=\frac{k-r}{r}$ at $p_{1}=(\frac{k}{r},w_{1})$ and
$p_{2}=(\frac{k-r}{r},w_{2})$, respectively. The line $\ell_{1}$ that passes
through $p_{1}$ and the origin intersects the vertical line
$b=\frac{k}{r}-\frac{1}{r(r-1)}$ at a point inside $U$. Similarly, the line
$\ell_{2}$ through $p_{2}$ and the origin intersects
$b=\frac{k-r}{r}+\frac{1}{r(r-1)}$ at a point inside $U$.

We postpone the proof of Proposition 3.2 and Lemma 3.3 to Section 4. The next
Lemma gives vertical lines on which we can rule out walls of instability for
objects of certain classes.

######  Lemma 3.4.

Take an object $E\in\mathcal{D}(X)$ with $\operatorname{ch}_{\leq
1}(E)=(r^{\prime},k^{\prime}H)$ such that $\gcd(r^{\prime},k^{\prime})=1$.
There are unique $m^{\pm},n^{\pm}\in\mathbb{Z}$ such that $\left\lvert
n^{\pm}\right\rvert<r^{\prime}$, $n^{\pm}r^{\prime}>0$ and

$m^{-}r^{\prime}-n^{-}k^{\prime}=-1\qquad\text{and}\qquad
m^{+}r^{\prime}-n^{+}k^{\prime}=1.$

We set

$b_{E}^{\pm}\coloneqq\frac{m^{\pm}}{n^{\pm}}=\frac{k^{\prime}}{r^{\prime}}\pm\frac{1}{n^{\pm}r^{\prime}}\,,$

then there is no wall for $E$ crossing the vertical lines $b=b_{E}^{\pm}$,
i.e., if $E$ is $\sigma_{b=b_{E}^{\pm},w}$-semistable for some
$w>\Gamma(b_{E}^{\pm})$, then it is $\sigma_{b=b_{E}^{\pm},w}$-stable for any
$w>\Gamma(b_{E}^{\pm})$.

###### Proof.

The first part is trivial and the second part follows by the fact that the
imaginary part

$\left\lvert\text{Im}[Z_{b=b_{E}^{\pm},w}(E)]\right\rvert=\left\lvert
k^{\prime}-\frac{m_{i}}{n_{i}}r^{\prime}\right\rvert=\frac{1}{\left\lvert
n_{i}\right\rvert}$

is minimal, see [Fey21, Lemma 3.5] for more details. ∎

Let $M_{X}(\mathsf{v})$ be the moduli space of $H$-Gieseker semistable sheaves
on the surface $X$ with Mukai vector $\mathsf{v}=(r,kH,s)$.

######  Lemma 3.5.

Any coherent sheaf $E\in M_{X}(\mathsf{v})$ is a $\mu_{H}$-stable locally free
sheaf.

###### Proof.

Since $\gcd(r,k)=1$, any $H$-Gieseker-semistable sheaf of class $\mathsf{v}$
is $\mu_{H}$-stable. Assume $E$ is not locally-free, then taking its reflexive
hull gives the exact sequence

$E\hookrightarrow E^{\vee\vee}\twoheadrightarrow Q$

where $Q$ is a torsion sheaf supported in dimension zero. We know the
reflexive sheaf $E^{\vee\vee}$ is also slope-stable, so

$-2\leq
v(E^{\vee\vee})^{2}=(r,kH,s+\operatorname{ch}_{3}(Q))^{2}=k^{2}H^{2}-2r(s+\operatorname{ch}_{3}(Q))\leq
k^{2}H^{2}-2rs-2r$

which is not possible by our assumption (1). ∎

Define the object $K_{E}[1]\in\mathcal{D}(X)$ as the cone of the evaluation
map:

(6) $\mathcal{O}_{X}^{h^{0}(X,E)}\xrightarrow{\text{ev}}E\rightarrow
K_{E}[1].$

######  Proposition 3.6.

Let $E\in M_{X}(\mathsf{v})$ be a $\mu_{H}$-stable vector bundle on the
surface $X$. Then

  * (a)

$\operatorname{Hom}\big{(}E,E(-H)[1]\big{)}=0$.

  * (b)

For any curve $C\in|H|$, the restriction $E|_{C}$ is a stable777Stability on
the curve $C$ is defined as in Definition 2.1 vector bundle on $C$ and
$h^{0}(C,E|_{C})=r+s$.

  * (c)

The object $K_{E}$ is a $\mu_{H}$-stable locally free sheaf on $X$ of Mukai
vector $v(K_{E})=\alpha=(s,-kH,r)$ and
$\operatorname{Hom}\big{(}K_{E},E(-H)[1]\big{)}=0$.

###### Proof.

Step 1. We first do wall-crossing for the bundle $E$. By [Bri08, Proposition
14.2], $E$ is $\sigma_{b,w}$-stable for $b<\mu_{H}(E)$ and $w\gg 0$. As in
Lemma 3.4 consider the vertical line $b=b^{-}_{E}$. We know

$\frac{k}{r}-\frac{1}{r}\leq\,b_{E}^{-}\,\leq\frac{k}{r}-\frac{1}{r(r-1)}.$

By Proposition 3.2 (c), the line $\ell_{\mathsf{v}}$ intersects $\Gamma$ at a
point with positive $b$-value $b_{2}^{\mathsf{v}}$ satisfying

$\frac{k}{r}-\frac{1}{r^{2}(r+1)}\overset{\eqref{cond.for
ellstar}}{\leq}b_{2}^{*}<b_{2}^{\mathsf{v}}.$

Thus the line segment $\ell_{\mathsf{v}}\cap U$ intersects the vertical line
$b=b^{-}_{E}$ at a point inside $U$. Hence Lemma 3.4 implies that there is no
wall for $E$ above $\ell_{\mathsf{v}}$.

We have
$\operatorname{Hom}(\mathcal{O}_{X},E[2])=\operatorname{Hom}(E,\mathcal{O}_{X})^{*}=0$,
so

$\chi(\mathcal{O}_{X},E)=r+s=h^{0}(E)-h^{1}(E).$

Thus $h^{0}(E)\neq 0$ and $\mathcal{O}_{X}$ makes a wall for $E$. This wall is
indeed $\ell_{\mathsf{v}}\cap U$ for $b\in(b_{1}^{\mathsf{v}},0)$. Take a
point $(b,w)$ on this wall and consider the evaluation map

$\mathcal{O}_{X}^{h^{0}(E)}\xhookrightarrow{\text{ev}}E\twoheadrightarrow
K_{E}[1]$

We know $\mathcal{O}_{X}$ and $E$ are $\sigma_{b,w}$-semistable of the same
$\nu_{b,w}$-slope and $\mathcal{O}_{X}$ is strictly $\sigma_{b,w}$-stable.
Thus the map $\operatorname{ev}$ is injective in the abelian subcategory of
$\sigma_{b,w}$-semistable objects in $\mathcal{A}(b)$ of $\nu_{b,w}$-slope
equal to $\nu_{b,w}(\mathcal{O}_{X})$. Therefore the co-kernel $K_{E}[1]$ is
also $\sigma_{b,w}$-semistable.

By [Fey20a, Lemma 3.2], we know

$h^{0}(X,E)\leq\frac{r+s}{2}+\frac{\sqrt{(r-s)^{2}+k^{2}(2H^{2}+4)}}{2}\coloneqq
h$

We claim $h<r+s+1$, i.e.

$(r-s)^{2}+k^{2}(2H^{2}+4)<(r+s+2)^{2}.$

This is equivalent to $k^{2}H^{2}-2rs<2+2(r+s)-2k^{2}$. Since
$\mathsf{v}^{2}=k^{2}H^{2}-2rs<2r-2$ by (1), we only need to show

$2r-2\leq 2+2(r+s)-2k^{2}$

i.e. $-2+k^{2}\leq s$ which holds by (4) if $H^{2}>2r$. Therefore
$h^{0}(E)=r+s$, $h^{1}(E)=0$ and $v(K_{E}[1])=-\alpha=(-s,kH,-r)$.

Step 2. The next step is to examine walls for $K_{E}[1]$. By Proposition 3.2
(c), our Brill-Noether wall $\ell_{\mathsf{v}}\cap U$ intersects $\Gamma$ at a
point with $b$-value $b_{1}^{\mathsf{v}}<-\frac{k}{s}+\frac{1}{s(s-1)}$. We
know $\gcd(s,k)=1$, so if $k>1$, the value $b_{K_{E}}^{+}$ in Lemma 3.4
satisfies

$-\frac{k}{s}+\frac{1}{s(s-1)}\leq
b_{K_{E}}^{+}\leq-\frac{k}{s}+\frac{1}{s}<0.$

Thus the vertical line $b=b_{K_{E}}^{+}$ intersects $\ell_{\mathsf{v}}\cap U$
at a point inside $U$. Hence $\sigma_{b,w}$-semistability of $K_{E}[1]$ for
$(b,w)\in\ell_{\mathsf{v}}\cap U$ implies that it is
$\sigma_{b=b_{K_{E}}^{+},w}$-stable for any $w>\Gamma(b_{K_{E}}^{+})$ by Lemma
3.4.

If $k=1$, then we show directly that $K_{E}[1]$ is $\sigma_{b,w}$-stable for
$(b,w)\in\ell_{\mathsf{v}}\cap U$ when $b<0$. Assume otherwise, and let
$K_{1}$ be a $\sigma_{b,w}$-stable factor of $K_{E}$. There are
$t_{1},s_{1}\in\mathbb{Q}$ such that
$\operatorname{ch}(K_{1})=s_{1}\operatorname{ch}(\mathcal{O}_{X})+t_{1}\operatorname{ch}(E)$,
see for instance [Fey20a, Remark 2.5]. Taking $\operatorname{ch}_{1}$ implies
$t_{1}\in\mathbb{Z}$. Moving along the wall $\ell_{\mathsf{v}}$ towards the
origin implies that

$0\leq\lim\limits_{b\rightarrow
0}\text{Im}[Z_{(b,w)}(K_{1})]=t_{1}\leq\lim\limits_{b\rightarrow
0}\text{Im}[Z_{(b,w)}(K_{E}[1])]=1.$

Thus $t_{1}=0$ or $1$, so the structure sheaf $\mathcal{O}_{X}$ is a subobject
or a quotient of $K_{E}[1]$. By definition Hom$(\mathcal{O}_{X},K_{E}[1])=0$
and since Hom$(E,\mathcal{O}_{X})=0$, we have
Hom$(K_{E}[1],\mathcal{O}_{X})=0$, a contradiction. Therefore $K_{E}[1]$ is
$\sigma_{b,w}$-stable along the wall $\ell_{\mathsf{v}}\cap U$ for $b<0$.
Openness of stability implies that it is $\sigma_{b=0,w}$-stable for $0<w\ll
1$. Then Lemma 3.4 implies that $K_{E}[1]$ is $\sigma_{b=0,w}$-stable for any
$w>0$.

Hence in both cases, $K_{E}[1]$ is $\sigma_{b,w}$-stable for $b>\mu(K_{E})$
and $w\gg 0$. [MS17, Lemma 6.18] implies that $\mathcal{H}^{-1}(K_{E}[1])$ is
a $\mu_{H}$-semistable torsion-free sheaf and $\mathcal{H}^{0}(K_{E}[1])$ is
supported in dimension zero. Since $\gcd(s,k)=1$, $\mathcal{H}^{-1}(K_{E}[1])$
is $\mu_{H}$-stable. If $\mathcal{H}^{0}(K_{E}[1])$ is non-zero, then

$\displaystyle-2\leq v(\mathcal{H}^{-1}(K_{E}[1]))^{2}=\ $
$\displaystyle\big{(}s,-kH,r+\operatorname{ch}_{3}(\mathcal{H}^{0}(K_{E}[1]))\big{)}^{2}$
$\displaystyle=\ $ $\displaystyle
v(E)^{2}-2s\operatorname{ch}_{3}(\mathcal{H}^{0}(K_{E}[1]))$ (7)
$\displaystyle\overset{\eqref{assumption}}{<}\ $ $\displaystyle 2r-2-2s\,.$

This implies $s<r$, thus (4) gives

$-1+\frac{1}{r}+\frac{k^{2}}{2r}H^{2}<r$

which is not possible as $H^{2}>2r(r+1)$. Hence $K_{E}$ is a $\mu_{H}$-stable
sheaf, and so is $\sigma_{b,w}$-stable for $b>-\frac{k}{s}$ and $w\gg 0$ by
[Bri08, Proposition 14.2]. By taking the reflexive hull and doing the same
computations as in (3), one can easily check that $K_{E}$ is indeed locally-
free.

Step 3. The final step is to analyse walls for $K_{E}$ and $E(-H)[1]$ when
$b<\mu(K_{E})$. Consider the vertical line $b=b^{-}_{K_{E}}$ as in Lemma 3.4.
By Proposition 3.2 (d), the line $\ell_{\alpha}$ intersects $\Gamma$ at two
points with $b$-values $b_{1}^{\alpha}<b_{2}^{\alpha}$ such that

(8) $b_{1}^{\alpha}<-\frac{k}{s}-\frac{1}{s}\leq
b^{-}_{K_{E}}\leq-\frac{k}{s}-\frac{1}{s(s-1)}<b_{2}^{\alpha}.$

The first inequality follows from

$b_{1}^{\alpha}<b_{1}^{*}\overset{\eqref{cond.for
ellstar}}{\leq}\frac{k-r}{r}+\frac{1}{r^{2}(r+1)}$

and the point that for $g\gg 0$, we have

(9) $\displaystyle\frac{k}{r}-1+\frac{1}{r^{2}(r+1)}<\ $
$\displaystyle\frac{-(k+1)}{\frac{k^{2}}{2r}H^{2}-1}$
$\displaystyle\overset{\eqref{bound for s}}{<}$
$\displaystyle-\frac{k}{s}-\frac{1}{s}.$

Thus Lemma 3.4 shows that by moving down along the vertical line
$b=b_{K_{E}}^{-}$ we get $K_{E}$ is $\sigma_{b,w}$-stable for any
$(b,w)\in\ell_{\alpha}\cap U$.

On the other hand, $E$ is a $\mu_{H}$-stable vector bundle, thus [Fey21, Lemma
3.3] implies that $E(-H)[1]$ is $\sigma_{b,w}$-stable for $b>\frac{k-r}{r}$
and $w\gg 0$. If $k\neq r-1$, consider the vertical line $b=b^{+}_{E(-H)}$,
then by Proposition 3.2 (d),

$b_{1}^{\alpha}<b_{1}^{*}\overset{\eqref{cond.for
ellstar}}{\leq}\frac{k-r}{r}+\frac{1}{r(r-1)}\leq
b^{+}_{E(-H)}\leq\frac{k-r}{r}+\frac{1}{r}\leq-\frac{1}{r}.$

If $k=r-1$, then we consider the vertical line $b=-\frac{1}{r+1}$ and the same
argument as in Lemma 3.4 shows that there is no wall for $E(-H)[1]$ crossing
this vertical line. For $g\gg 0$, we have

(10) $\displaystyle-\frac{1}{r+1}<$
$\displaystyle\frac{-(k+1)}{\frac{k^{2}}{2r}H^{2}-1}$
$\displaystyle\overset{\eqref{bound for s}}{<}$
$\displaystyle-\frac{k+1}{s}<b_{2}^{\alpha},$

Hence Proposition 3.2 implies the lines $\widetilde{\ell}$,
$\ell_{\mathsf{v}(-H)}$ and $\ell_{\alpha}$ intersect the vertical line
$b=b^{+}_{E(-H)}$ if $k<r-1$ or $b=-\frac{1}{r+1}$ in case $k=r-1$ at points
inside $U$. Hence by Lemma 3.4, $E(-H)[1]$ is $\sigma_{b,w}$-stable along
these three lines and so we get the following:

(a) $E$ and $E(-H)[1]$ are stable of the same $\nu_{b,w}$-slope for
$(b,w)\in\widetilde{\ell}\,\cap\,U$, therefore
$\operatorname{Hom}(E,E(-H)[1])=0$ as claimed in part (a).

(b) We know $\iota_{*}E|_{C}$ lies in the exact sequence

$E\hookrightarrow\iota_{*}E|_{C}\twoheadrightarrow E(-H)[1]$

in $\mathcal{A}(b=0)$. Applying the same argument as in [Fey21, Corollary 4.3]
implies that $E|_{C}$ is stable. Since $E(-H)[1]$ and $\mathcal{O}_{X}$ are
$\sigma_{b,w}$-stable of the same $\nu_{b,w}$-slope for
$(b,w)\in\ell_{\mathsf{v}(-H)}\,\cap\,U$, we get
$\operatorname{Hom}(\mathcal{O}_{X},E(-H)[1])=0$. Thus
$h^{0}(C,E|_{C})=h^{0}(X,E)=r+s$. This completes the proof of (b).

(c) We have shown that $K_{E}$ is a $\mu_{H}$-stable locally free sheaf.
Moreover $E(-H)[1]$ and $K_{E}$ are $\sigma_{b,w}$-stable of the same slope
with respect to $(b,w)\in\ell_{\alpha}\,\cap\,U$, so
$\operatorname{Hom}(K_{E},E(-H)[1])=0$ as claimed in part (c). ∎

### Wall-crossing for the push-forward of vector bundles on curves

As before $C\xhookrightarrow{\iota}X$ is a curve in the linear system $|H|$.
Let $F$ be a semistable vector bundle on $C$ of rank $r$ and degree $kH^{2}$.
Then $\iota_{*}F$ is of Mukai vector

$v(\iota_{*}F)=\left(0,\ rH,\ kH^{2}-\frac{r}{2}H^{2}\right).$

Semistability of $F$ implies that $\iota_{*}F$ is $H$-Gieseker-semistable, so
it is $\sigma_{b,w}$-semistable for $w\gg 0$ [Bri08, Proposition 14.2]. The
walls of instability for $\iota_{*}F$ are parallel lines of slope
$H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)$.

######  Proposition 3.7.

Let $\ell$ be a wall for $\iota_{*}F$ which lies above or on the line
$\ell^{*}$ (see Definition 3.1). Let
$F_{1}\hookrightarrow\iota_{*}F\twoheadrightarrow F_{2}$ be a destabilising
sequence along $\ell$ with $\nu_{b,w}(F_{1})=\nu_{b,w}(\iota_{*}F)$ for
$(b,w)\in\ell\cap U$ and $\nu_{b,w^{-}}(F_{1})>\nu_{b,w^{-}}(\iota_{*}F)$ for
$(b,w^{-})\in U$ below $\ell\cap U$. Then

$F_{1}$ is a $\mu_{H}$-stable sheaf with $\operatorname{ch}_{\leq
1}(F_{1})=(r,kH)$.

$\mathcal{H}^{-1}(F_{2})$ is a $\mu_{H}$-stable sheaf with
$\operatorname{ch}_{\leq 1}(\mathcal{H}^{-1}(F_{2}))=(r,(k-r)H)$ and
$\mathcal{H}^{0}(F_{2})$ is either zero or a torsion sheaf supported in
dimension zero.

The sequence $F_{1}\hookrightarrow\iota_{*}F\twoheadrightarrow F_{2}$ is the
HN filtration of $\iota_{*}F$ with respect to $\sigma_{b=0,w}$ for $0<w\ll 1$.

The wall $\ell$ lies below or on $\widetilde{\ell}$. If it coincides with
$\widetilde{\ell}$, then $\operatorname{ch}_{2}(F_{1})$ is maximum (equal to
$s-r$) and $F=F_{1}|_{C}$.

###### Proof.

The first part of the argument is similar to [Fey20a, Proposition 4.2]; we add
it for completeness. Taking cohomology of the destabilising sequence gives a
long exact sequence of coherent sheaves

(11) $0\rightarrow\mathcal{H}^{-1}(F_{1})\rightarrow
0\rightarrow\mathcal{H}^{-1}(F_{2})\rightarrow\mathcal{H}^{0}(F_{1})\xrightarrow{d_{0}}\iota_{*}F\xrightarrow{d_{1}}\mathcal{H}^{0}(F_{2})\rightarrow
0.$

Thus $\mathcal{H}^{-1}(F_{1})=0$ and $\mathcal{H}^{0}(F_{1})\cong F_{1}$. Let
$v(F_{1})=\big{(}r^{\prime},k^{\prime}H,s^{\prime}\big{)}$. If $r^{\prime}=0$,
then $F_{1}$ and $i_{*}F$ have the same $\nu_{b,w}$-slope with respect to any
$(b,w)\in U$, so $F_{1}$ will not destabilise $\iota_{*}F$ below the wall.
Hence $r^{\prime}>0$. The first step is to show that $r^{\prime}=r$.

Let $T(F_{1})$ be the maximal torsion subsheaf of $F_{1}$ and $F_{1}/T(F_{1})$
be its torsion-free part. Let
$v\big{(}T(F_{1})\big{)}=(0,\tilde{r}H,\tilde{s})$. Right-exactness of the
underived pull-back $\iota^{*}$ applied to the short exact sequence

(12) $T(F_{1})\hookrightarrow F_{1}\twoheadrightarrow F_{1}/T(F_{1})$

implies that

(13)
$\text{rank}(\iota^{*}F_{1})\leq\text{rank}\big{(}\iota^{*}T(F_{1})\big{)}+\text{rank}\big{(}\iota^{*}\big{(}F_{1}/T(F_{1})\big{)}\big{)}.$

Take a point $(b,w)\in\ell\cap U$. By definition of $\mathcal{A}(b)$, the
sequence (12) is an exact triangle in $\mathcal{A}(b)$. Consider the
composition of injections $s\colon T(F_{1})\hookrightarrow
F_{1}\hookrightarrow\iota_{*}F$ in $\mathcal{A}(b)$. Then the cokernel $c(s)$
in $\mathcal{A}(b)$ is also a rank zero sheaf because if
$\mathcal{H}^{-1}(c(s))\neq 0$, the it must be a torsion-free sheaf which is
not possible. Therefore, $T(F_{1})$ is a subsheaf of $\iota_{*}F$. Since $F$
is a vector bundle on an irreducible curve $C$, we get
rank$(\iota^{*}T(F_{1}))=\tilde{r}$. Thus (13) gives
$\text{rank}(\iota^{*}F_{1})\leq\tilde{r}+r^{\prime}$. Let
$v\big{(}\mathcal{H}^{0}(F_{2})\big{)}=\big{(}0,k^{\prime\prime}H,s^{\prime\prime}\big{)}$.
The right-exactness of $\iota^{*}$ implies

(14) $r-k^{\prime\prime}=\text{rank}\big{(}\iota^{*}\ker
d_{1}\big{)}=\text{rank}\big{(}\iota^{*}\operatorname{im}d_{0}\big{)}\leq\text{rank}(\iota^{*}F_{1})\overset{\eqref{bound
for rank}}{\leq}\tilde{r}+r^{\prime}.$

Since $\ell$ lies above or on $\ell^{*}$, it intersects $\Gamma$ at two points
with $b$-values $b_{1}<b_{2}$ such that $b_{1}\leq b_{1}^{*}$ and
$b_{2}^{*}\leq b_{2}$. Note that the point $(0,0)$ lies below $\ell\cap U$ by
Proposition 3.2 (a). Therefore

(15) $\mu^{+}(\mathcal{H}^{-1}(F_{2}))\leq b_{1}^{*}\qquad\text{and}\qquad
b_{2}^{*}\leq\mu^{-}(F_{1}).$

This implies

$\displaystyle\dfrac{r-k^{\prime\prime}-\tilde{r}}{r^{\prime}}=\mu_{H}\big{(}F_{1}/T(F_{1})\big{)}-\mu_{H}\big{(}\mathcal{H}^{-1}(F_{2})\big{)}\geq
b_{2}^{*}-b_{1}^{*}$

and so

(16)
$r^{\prime}\leq\frac{r-k^{\prime\prime}-\tilde{r}}{b_{2}^{*}-b_{1}^{*}}<(r-k^{\prime\prime}-\tilde{r})+1.$

Here the right hand side inequality comes from

$\frac{r-k^{\prime\prime}-\tilde{r}}{r-k^{\prime\prime}-\tilde{r}+1}\leq\frac{r}{r+1}<1-\frac{1}{r^{2}(r+1)}\overset{\eqref{cond.for
ellstar}}{\leq}b_{2}^{*}-b_{1}^{*}\,.$

Comparing (14) with (16) implies $r^{\prime}=r-k^{\prime\prime}-\tilde{r}$.
Then (15) implies

(17) $b_{2}^{*}\leq\
\mu_{H}(F_{1}/T(F_{1}))=\frac{k^{\prime}-\tilde{r}}{r-k^{\prime\prime}-\tilde{r}}=\frac{\frac{\operatorname{ch}_{1}(\mathcal{H}^{-1}(F_{2})).H}{H^{2}}+r-k^{\prime\prime}-\tilde{r}}{r-k^{\prime\prime}-\tilde{r}}\
\leq b_{1}^{*}+1.$

Moreover, Proposition 3.2 (a) and (5) give

(18) $b_{2}^{*}<\frac{k}{r}<b_{1}^{*}+1\qquad\text{and}\qquad
b_{1}^{*}+1-b_{2}^{*}\leq\frac{2}{r^{2}(r+1)}.$

Thus comparing (17) and (18) shows

$\left\lvert\frac{k^{\prime}-\tilde{r}}{r-k^{\prime\prime}-\tilde{r}}-\frac{k}{r}\right\rvert<\frac{1}{r(r+1)}$

which is possible only if we have equality of two fractions
$\frac{k^{\prime}-\tilde{r}}{r-k^{\prime\prime}-\tilde{r}}=\frac{k}{r}$. Since
$\gcd(r,k)=1$, we get $k^{\prime\prime}=\tilde{r}=0$,
$(r^{\prime},k^{\prime})=(r,k)$ and $v(T(F_{1}))=(0,0,\tilde{s})$. However
$T(F_{1})$ cannot be a skyscraper sheaf because $T(F_{1})$ is a subsheaf of
$\iota_{*}F$ as explained above. Thus $T(F_{1})=0$ and $F_{1}$ is torsion-
free. Then (15) and (5) give

$\frac{k}{r}-\frac{1}{r^{2}(r+1)}\leq
b_{2}^{*}\leq\mu_{H}^{-}(F_{1})\leq\mu_{H}(F_{1})=\frac{k}{r}$

which implies $F_{1}$ is $\mu_{H}$-stable. This completes the proof of (a).

Since $k^{\prime\prime}=0$, we have $\operatorname{ch}_{\leq
1}(\mathcal{H}^{-1}(F_{2}))=(r,(k-r)H)$, so the first inequality in (15)
implies

$\frac{k-r}{r}\leq\mu_{H}^{+}(\mathcal{H}^{-1}(F_{2}))\leq
b_{1}^{*}\leq\frac{k-r}{r}+\frac{1}{r^{2}(r+1)}.$

Therefore $\mathcal{H}^{-1}(F_{2})$ is a $\mu_{H}$-stable sheaf as
$\gcd(r,k-r)=1$. This completes the proof of part (b).

Since $F_{1}$ is $\mu_{H}$-stable, we get

$v(F_{1})^{2}=k^{2}H^{2}-2r(r+\operatorname{ch}_{2}(F_{1}))\geq-2.$

Thus $\operatorname{ch}_{2}(F_{1})\leq s-r$. We know $\ell$ passes through
$\Pi(F_{1})$, so if $\operatorname{ch}_{2}(F_{1})=s-r$, then $\ell$ coincides
with $\widetilde{\ell}$ and if $\operatorname{ch}_{2}(F_{1})<s-r$, it lies
below $\widetilde{\ell}$.

We claim $F_{1}$ is $\sigma_{b=0,w}$-stable for any $w>0$. Consider the
vertical line $b=b_{F_{1}}^{-}$ as in Lemma 3.4; it satisfies

(19)
$0\leq\frac{k}{r}-\frac{1}{r}<b_{F_{1}}^{-}\leq\frac{k}{r}-\frac{1}{r(r-1)}\,.$

Let $\ell_{F_{1}}$ be the line passing through $\Pi(F_{1})$ and the origin.
The line segment $\ell_{F_{1}}\cap U$ lies above $\ell_{1}\cap U$ considered
in Lemma 3.3, see Figure 4. Thus combining (19) with Lemma 3.3 implies that
$\ell_{F_{1}}\cap U$ intersects the vertical line $b=b^{-}_{F_{1}}$ at a point
in the closure $\overline{U}$. We know the wall $\ell$ lies above
$\ell_{F_{1}}$ as the origin point $(0,0)$ lies below $\ell^{*}$ and so $\ell$
by Proposition 3.2(a). Thus $\ell$ intersects the vertical line
$b=b_{F_{1}}^{-}$ at a point $(b_{F_{1}}^{-},w)$ inside $U$, so semistability
of $F_{1}$ along the wall and Lemma 3.4 imply that $F_{1}$ is
$\sigma_{b=b_{F_{1}}^{-}}$-stable for any $w>\Gamma(b_{F_{1}}^{-})$. Hence the
structure of walls (which all pass through $\Pi(F_{1})$) implies that $F_{1}$
is $\sigma_{b,w}$-stable for any $(b,w)$ above $\ell_{F_{1}}$ when
$b<\frac{k}{r}$, so in particular, it is $\sigma_{b=0,w}$-stable for any
$w>0$.

$b,\,\frac{\operatorname{ch}_{1}\\!.\hskip
0.6ptH}{\operatorname{ch}_{0}H^{2}}$$\Gamma$$w,\,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$b=b_{F_{1}}^{-}$$\ell_{1}$$\ell_{F_{1}}$$\ell$$\ell^{*}$$\frac{k}{r}$$U$$\ell_{2}$$\ell_{F_{2}}$$\frac{k-r}{r}$$\Pi(F_{1})$$\Pi(F_{2})$$b=b_{F_{2}}^{+}$
Figure 4. The vertical lines $b=b_{F_{1}}^{-}$ and $b=b_{F_{2}}^{+}$

Similarly, we know

$\frac{k-r}{r}+\frac{1}{r(r-1)}\leq\ b_{F_{2}}^{+}\
\leq\frac{k-r}{r}+\frac{1}{r}\leq 0.$

Thus by the same argument as for $F_{1}$, Lemma 3.3 implies that the line
$\ell_{F_{2}}$ passing through $\Pi(F_{2})$ and the origin intersects the
vertical line $b=b_{F_{2}}^{+}$ at a point inside $U$, see Figure 4. Thus by
Lemma 3.4, $F_{2}$ is also $\sigma_{b=0,w}$-stable for any $w>0$. We know
$\nu_{b,w}$-slope of $F_{1}$ is bigger than $F_{2}$ for $(b,w)$ below the wall
$\ell$, thus the sequence $F_{1}\rightarrow\iota_{*}F\rightarrow F_{2}$ is the
HN filtration of $\iota_{*}F$ with respect to $\sigma_{0,w}$ for $0<w\ll 1$ as
claimed in part (c).

In case of equality $\operatorname{ch}_{3}(F_{1})=s-r$, Proposition 3.6
implies that $\iota_{*}F_{1}|_{C}$ is a stable sheaf. We know the non-zero
morphism $d_{0}$ in the long exact sequence (11) factors via the morphism
$d_{0}^{\prime}\colon i_{*}F_{1}|_{C}\rightarrow i_{*}F$. The sheaves
$i_{*}F_{1}|_{C}$ and $i_{*}F$ are both stable and have the same Mukai vector,
hence $d_{0}^{\prime}$ is an isomorphism. This completes the proof of part
(d). ∎

Proposition 3.7 only describes walls for class $v(\iota_{*}F)$ which are above
$\ell^{*}$. Instead of classifying walls below $\ell^{*}$, we jump over the
Brill-Noether region (neighbourhood of $\Pi(\mathcal{O}_{X})$) and find an
upper bound for the number of global sections of stable vector bundles on the
curve $C$ of rank $r$ and degree $kH^{2}$. We first recall definition of the
Harder-Narasimhan polygon.

######  Definition 3.8.

Given a stability condition $\sigma_{(b,w)}$ and an object
$E\in\mathcal{A}(b)$, the Harder-Narasimhan polygon of $E$ with respect to
$(b,w)$ is the convex hull of the points $Z_{b,w}(E^{\prime})$ for all
subobjects $E^{\prime}\subset E$ of $E$.

If the Harder-Narasimhan filtration of $E$ is the sequence

$0={E}_{0}\subset{E}_{1}\subset....\subset{E}_{n-1}\subset{E}_{n}=E,$

then the points $\left\\{p_{i}\coloneqq Z_{b,w}({E}_{i})\right\\}_{i=0}^{n}$
are the extremal points of the Harder-Narasimhan polygon of $E$ on the left
side of the line segment $\overline{oZ_{b,w}(E)}$, see Figure 5.

Re$[Z_{b,w}(-)]$Im$[Z_{b,w}(-)]$$o$$p_{1}$$p_{2}$$p_{3}$$p_{4}=Z_{b,w}(E)$
Figure 5. HN polygon

We want to consider the limit of HN polygon when $(b,w)$ is close to the
origin. Define the function $\overline{Z}\colon K(X)\rightarrow\mathbb{C}$ as

$\overline{Z}(E)\coloneqq\lim\limits_{w\rightarrow
0}Z_{b=0,w}(E)=-\operatorname{ch}_{2}(E)\,+\,i\,\frac{\operatorname{ch}_{1}(E).H}{H^{2}}.$

Take an object $E\in\mathcal{A}(b=0)$ which has no subobject
$E^{\prime}\subset E$ in $\mathcal{A}(0)$ with ch${}_{1}(E^{\prime})=0$, i.e.
$\nu_{b,w}^{+}(E)<+\infty$. [Fey20a, Proposition 3.4] implies that there
exists $w^{*}>0$ such that the Harder-Narasimhan filtration of $E$ is a fixed
sequence

(20) $0=E_{0}\subset E_{1}\subset....\subset E_{n-1}\subset E_{n}=E,$

with respect to all stability conditions $\sigma_{0,w}$ where $0<w<w^{*}$. We
define $\operatorname{P}_{E}$ to be the polygon with extremal points
$p_{i}=\overline{Z}(E_{i})$ and sides $\overline{p_{0}p_{n}}$ and
$\overline{p_{i}p_{i+1}}$ for $i\in[0,n-1]$. Since $P_{E}$ is the limit of HN
polygon when $w\rightarrow 0$, it is also a convex polygon.

######  Lemma 3.9.

Let $F$ be a rank $r$-semistable vector bundle on $C$ with degree $kH^{2}$ as
before. The polygon $\operatorname{P}_{\iota_{*}F}$ is contained in the
triangle $\triangle oz_{1}z_{2}$ where

$z_{1}\coloneqq\overline{Z}(\mathsf{v})=-s+r\ +i\,k\qquad\text{and}\qquad
z_{2}\coloneqq\overline{Z}(\iota_{*}F)=H^{2}\left(\frac{r}{2}-k\right)+\
i\,r.$

If $\operatorname{P}_{\iota_{*}F}$ coincides with $\triangle oz_{1}z_{2}$,
then $F=E|_{C}$ for a vector bundle $E\in M_{X}(\mathsf{v})$.

$o$Re$[\,\overline{Z}\,]=-\operatorname{ch}_{2}$Im$[\,\overline{Z}\,]=\frac{H\operatorname{ch}_{1}}{H^{2}}$$z_{1}$$z_{2}$$p_{1}$$p_{2}$
Figure 6. The polygon $P_{i_{*}F}$ is inside the triangle $T$

###### Proof.

Consider the HN filtration (20) for $E=\iota_{*}F$ with respect to
$\sigma_{0,w}$ where $0<w\ll 1$. Since $P_{\iota_{*}F}$ is a convex polygon,
to prove the first claim we only need to show

$-\dfrac{\text{Re}[\overline{Z}(E_{1})]}{\text{Im}[\overline{Z}(E_{1})]}\leq-\dfrac{\text{Re}[z_{1}]}{\text{Im}[z_{1}]}\qquad\text{and}\qquad-\dfrac{\text{Re}[z_{2}-z_{1}]}{\text{Im}[z_{2}-z_{1}]}\leq-\dfrac{\text{Re}[\overline{Z}(\iota_{*}F/E_{n-1})]}{\text{Im}[\overline{Z}(\iota_{*}F/E_{n-1})]},$

i.e.

(21)
$\frac{\operatorname{ch}_{2}(E_{1})H^{2}}{\operatorname{ch}_{1}(E_{1})H}\leq\frac{\operatorname{ch}_{2}(\mathsf{v})H^{2}}{\operatorname{ch}_{1}(\mathsf{v})H}\qquad\text{and}\qquad\frac{\operatorname{ch}_{2}(\mathsf{v}(-H))H^{2}}{\operatorname{ch}_{1}(\mathsf{v}(-H))H}\leq\frac{\operatorname{ch}_{2}(\iota_{*}F/E_{n-1})H^{2}}{\operatorname{ch}_{1}(\iota_{*}F/E_{n-1})H}.$

We first show that $\operatorname{ch}_{0}(E_{1})>0$ and
$\operatorname{ch}_{0}(\iota_{*}F/E_{n-1})<0$. Taking cohomology from the
exact sequence
$E_{1}\hookrightarrow\iota_{*}F\twoheadrightarrow\iota_{*}F/E_{1}$ in
$\mathcal{A}(b=0)$ implies that $E_{1}$ is a sheaf. If $E_{1}$ is of rank
zero, then $\mathcal{H}^{-1}(\iota_{*}F/E_{1})$ is zero as it is must be a
torsion-free sheaf, so $F_{1}$ is a subsheaf of $\iota_{*}F$ of bigger
$\nu_{0,w}$-slope which is not possible as $F$ is semistable888Recall that
$\nu_{b,w}$-slope of any rank zero sheaf $E$ is equal to
$\frac{\operatorname{ch}_{2}(E)H^{2}}{\operatorname{ch}_{1}(E)H^{2}}$ which is
independent of $(b,w)$., thus $\operatorname{ch}_{0}(E_{1})>0$. Similarly, the
exact triangle
$E_{n-1}\hookrightarrow\iota_{*}F\twoheadrightarrow\iota_{*}F/E_{n-1}$ implies
$\operatorname{ch}_{0}(E_{n-1})>0$ and so $\iota_{*}F/E_{n-1}$ is of negative
rank. Since $E_{1},\iota_{*}F/E_{n-1}\in\mathcal{A}(b=0)$ we get

$0\leq\frac{1}{\operatorname{ch}_{0}(E_{1})}\text{Im}\left[Z_{0,w}(E_{1})\right]=\mu(E_{1})\qquad\text{and}\qquad\frac{1}{\operatorname{ch}_{0}(\iota_{*}F/E_{n-1})}\text{Im}\left[Z_{0,w}(\iota_{*}F/E_{n-1})\right]\leq
0$

Therefore (21) is equivalent to the claim that $\Pi(E_{1})$ lies below or on
the line $\ell_{\mathsf{v}}$ and $\Pi(\iota_{*}F/E_{n-1})$ lies below or on
$\ell_{\mathsf{v}(-H)}$, see Figure 7.

$b,\,\frac{H\operatorname{ch}_{1}}{H^{2}\operatorname{ch}_{0}}$$\Gamma$$w,\,\frac{\operatorname{ch}_{2}}{\operatorname{ch}_{0}}$$\Pi(\mathsf{v})$$\Pi(\mathsf{v}(-H))$$\ell^{*}$$b_{2}^{*}$$b_{1}^{*}$$\ell_{\mathsf{v}}$$\ell_{\mathsf{v}(-H)}$
Figure 7. Walls for $\iota_{*}F$

If there exists a wall for $\iota_{*}F$ above or on $\ell^{*}$, Proposition
3.7 gives a complete description of the HN factors of $\iota_{*}F$ with
respect to $\sigma_{0,w}$ for $0<w\ll 1$ and so the claim follows. Thus we may
assume there is no wall above or on $\ell^{*}$ for $\iota_{*}F$ and so
$\iota_{*}F$ is $\sigma_{b,w}$-semistable for any $(b,w)\in U$ above
$\ell^{*}$.

Consider the line $\ell$ parallel to $\ell^{*}$ passing through $\Pi(E_{1})$.
We know $\iota_{*}F$ and $E_{1}$ have the same $\nu_{b,w}$-slope for
$(b,w)\in\ell\cap U$ and $E_{1}$ has bigger slope then $\iota_{*}F$ below the
line as the slope function $\nu_{b,w}(E_{1})$ is a decreasing function with
respect to $w$. Thus $\iota_{*}F$ is $\sigma_{b,w}$-unstable below $\ell$,
hence $\ell$ and so $\Pi(E_{1})$ lie below $\ell^{*}$. On the other hand,
$\sigma_{0,w}$-semistability of $E_{1}$ implies that $\Pi(E_{1})$ does not lie
above $\Gamma$, so $\Pi(E_{1})$ lies below the line passing through the origin
and $(b_{2}^{*},\Gamma(b_{2}^{*}))$. Then Proposition 3.2(c) implies
$\Pi(E_{1})$ lies below $\ell_{\mathsf{v}}$ as claimed. A similar argument
shows that $\Pi(\iota_{*}F/E_{n-1})$ lies below the line passing through the
origin and $(b_{1}^{*},\Gamma(b_{1}^{*}))$ and so it lies below
$\ell_{\mathsf{v}(-H)}$ by Proposition 3.2(e). This completes the proof of
(21) and so $P_{\iota_{*}F}$ is contained in the triangle $\triangle
oz_{1}z_{2}$.

If HN polygon coincides with the triangle $\triangle oz_{1}z_{2}$, then we
have equality in both inequalities of (21), i.e. $\Pi(E_{1})$ lies on
$\ell_{\mathsf{v}}$ and $\Pi(\iota_{*}F/E_{n-1})$ lies on
$\ell_{\mathsf{v}(-H)}$. Since $\Pi(E_{1})$ cannot lie inside $U$, it lies
above or on $\ell^{*}$. Thus there exits a wall for $\iota_{*}F$ above or on
$\ell^{*}$ and so Proposition 3.7 implies $v(F_{1})=\mathsf{v}$ and
$F=E_{1}|_{C}$ as claimed. ∎

Let $p_{i}=\overline{Z}(E_{i})$ be the extremal points of
$\operatorname{P}_{\iota_{*}F}$ where $E_{i}$’s are the factor in the HN
filtration of $E=\iota_{*}F$ as in (20). Then [Fey20a, Proposition 3.4]
implies that

(22)
$h^{0}(X,\iota_{*}F)\leq\dfrac{\chi(X,\iota_{*}F)}{2}+\dfrac{1}{2}\sum_{i=1}^{n}\lVert\overline{p_{i}p_{i-1}}\rVert$

where $\lVert.\rVert$ is the non-standard norm defined in [Fey20a, Equation
(3.2)], i.e. $\lVert x+iy\rVert=\sqrt{x^{2}+(2H^{2}+4)y^{2}}$.

######  Proposition 3.10.

Let $F$ be a rank $r$-semistable vector bundle on $C$ with degree $kH^{2}$ as
before. If the polygon $\operatorname{P}_{\iota_{*}F}$ lies strictly inside
$\triangle oz_{1}z_{2}$, then $h^{0}(C,F)<r+s$.

###### Proof.

Since the extremal points of $\operatorname{P}_{\iota_{*}F}$ have integral
coordinates, $\operatorname{P}_{\iota_{*}F}$ lies inside
$oz_{0}^{\prime}z_{1}^{\prime}z_{2}^{\prime}z_{2}$ where
$z_{0}^{\prime}=(-s+r)\frac{k-1}{k}+i\,(k-1)$, $z_{1}^{\prime}=-s+r+1+i\,k$
and

$z_{2}^{\prime}=(-kH^{2}+\frac{r}{2}H^{2}+s-r)\frac{1}{r-k}-s+r+i(k+1),$

see Figure 8.

$o$Re$[\,\overline{Z}\,]=-\operatorname{ch}_{2}$Im$[\,\overline{Z}\,]=\frac{H\operatorname{ch}_{1}}{H^{2}}$$z_{1}$$z_{1}^{\prime}$$z_{2}$$z_{2}^{\prime}$$z_{0}^{\prime}$
Figure 8. $\operatorname{P}_{i_{*}F}$ is inside the polygon
$oz_{0}^{\prime}z_{1}^{\prime}z_{2}^{\prime}z_{2}$

Convexity of $\operatorname{P}_{\iota_{*}F}$ and the polygon
$oz_{0}^{\prime}z_{1}^{\prime}z_{2}^{\prime}z_{2}$ gives

$\sum_{i=1}^{n}\lVert\overline{p_{i}p_{i-1}}\rVert\leq\lVert\overline{oz_{0}^{\prime}}\rVert+\lVert\overline{z_{0}^{\prime}z_{1}^{\prime}}\rVert+\lVert\overline{z_{1}^{\prime}z_{2}^{\prime}}\rVert+\lVert\overline{z_{2}^{\prime}z_{2}}\rVert\eqqcolon
Q_{\text{in}}.$

Let
$Q_{\text{out}}\coloneqq\lVert\overline{oz_{1}}\rVert+\lVert\overline{z_{1}z_{2}}\rVert$,
then

$Q_{\text{out}}-Q_{in}=\lVert\overline{z_{0}^{\prime}z_{1}}\rVert-\lVert\overline{z_{0}^{\prime}z^{\prime}_{1}}\rVert+\lVert\overline{z_{1}z_{2}^{\prime}}\rVert-\lVert\overline{z^{\prime}_{1}z^{\prime}_{2}}\rVert.$

Define
$\epsilon_{\text{out}}\coloneqq\frac{1}{2}\chi(\iota_{*}F)+\frac{1}{2}Q_{\text{out}}-(r+s)$.
Assume for a contradiction that $r+s\leq h^{0}(F)$, then (22) gives

$\dfrac{1}{2}\chi(\iota_{*}F)+\dfrac{1}{2}Q_{\text{out}}-\epsilon_{\text{out}}\hskip
0.6pt=\hskip 0.6ptr+s\hskip 0.6pt\leq
h^{0}(F)\leq\dfrac{\chi(\iota_{*}F)}{2}+\dfrac{1}{2}\sum_{i=1}^{n}\lVert\overline{p_{i}p_{i-1}}\rVert\leq\dfrac{1}{2}\chi(\iota_{*}F)+\dfrac{1}{2}Q_{\text{in}}.$

Thus $Q_{\text{out}}-Q_{\text{in}}\leq 2\,\epsilon_{\text{out}}$ but we show
for $g\gg 0$ it does not hold. We know

$\displaystyle 2\epsilon_{\text{out}}\coloneqq\ $ $\displaystyle
kH^{2}-\frac{r}{2}H^{2}+Q_{\text{out}}-2(r+s)$ $\displaystyle=\ $
$\displaystyle\sqrt{(s-r)^{2}+(2H^{2}+4)k^{2}}+\sqrt{\left(-kH^{2}+\frac{r}{2}H^{2}+s-r\right)^{2}+(2H^{2}+4)(r-k)^{2}}$
$\displaystyle-(s+r)-(-kH^{2}+\frac{r}{2}H^{2}+s+r)$ (23) $\displaystyle=\ $
$\displaystyle\frac{-4rs+(2H^{2}+4)k^{2}}{\sqrt{(s-r)^{2}+(2H^{2}+4)k^{2}}+(s+r)}\,+$
(24)
$\displaystyle\frac{-4r(-kH^{2}+\frac{r}{2}H^{2}+s)+(2H^{2}+4)(r-k)^{2}}{\sqrt{(-kH^{2}+\frac{r}{2}H^{2}+s-r)^{2}+(2H^{2}+4)(r-k)^{2}}+(-kH^{2}+\frac{r}{2}H^{2}+s+r)}\,.$

By our assumption (1), $2k^{2}H^{2}-4rs+4k^{2}\geq-4+4k^{2}\geq 0$ so

$(s+r)^{2}\leq(s+r)^{2}-4rs+k^{2}(2H^{2}+4)=(s-r)^{2}+k^{2}(2H^{2}+4).$

Thus $\eqref{aa.1}\leq\frac{-4rs+(2H^{2}+4)k^{2}}{2(r+s)}$.

The numerator of (24) is $2k^{2}H^{2}-4rs+4(r-k)^{2}\geq-4+4(r-k)^{2}\geq 0$
by (1), so

$\displaystyle\left(-kH^{2}+\frac{r}{2}H^{2}+s+r\right)^{2}\leq\ $
$\displaystyle\left(-kH^{2}+\frac{r}{2}H^{2}+s+r\right)^{2}+2k^{2}H^{2}-4rs+4(r-k)^{2}$
$\displaystyle=\ $
$\displaystyle\left(-kH^{2}+\frac{r}{2}H^{2}+s-r\right)^{2}+(2H^{2}+4)(r-k)^{2}.$

This implies
$\eqref{aa.2}\leq\frac{2k^{2}H^{2}-4rs+4(r-k)^{2}}{2(-kH^{2}+\frac{r}{2}H^{2}+s+r)}$,
thus

$\displaystyle
2\epsilon_{\text{out}}<\frac{k^{2}(H^{2}+2)-2rs}{r+s}+\frac{k^{2}H^{2}-2rs+2(r-k)^{2}}{-kH^{2}+\frac{r}{2}H^{2}+s+r}\eqqcolon
M_{1}$

When $H^{2}=2g-2\rightarrow+\infty$, we know
$s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$ by (4), so one gets
$M_{1}\rightarrow 0$. On the other hand,

(25) $\displaystyle Q_{\text{out}}-Q_{\text{in}}=\ $
$\displaystyle\sqrt{\left(\frac{1}{k}(s-r)\right)^{2}+(2H^{2}+4)}-\sqrt{\left(\frac{1}{k}(s-r)-1\right)^{2}+(2H^{2}+4)}\
\ +$
$\displaystyle\sqrt{\left(\frac{1}{r-k}\left(-kH^{2}+\frac{r}{2}H^{2}+s-r\right)\right)^{2}+(2H^{2}+4)}\
\ -$
$\displaystyle\sqrt{\left(\frac{1}{r-k}\left(-kH^{2}+\frac{r}{2}H^{2}+s-r\right)-1\right)^{2}+(2H^{2}+4)}$

By (4), when $H^{2}\rightarrow+\infty$, we have $\frac{1}{k}(s-r)>1$ and

$\displaystyle\frac{1}{r-k}\left(-kH^{2}+\frac{r}{2}H^{2}+s-r\right)\overset{\eqref{bound
for s}}{>}\ $
$\displaystyle\frac{1}{r-k}\left(-kH^{2}+\frac{r}{2}H^{2}+\frac{k^{2}}{2r}H^{2}-r-1+\frac{1}{r}\right)$
$\displaystyle=\ $
$\displaystyle\frac{r-k}{2r}H^{2}+\frac{1}{r-k}\left(-r-1+\frac{1}{r}\right)$
$\displaystyle>\ $ $\displaystyle 1\,.$

Thus (25) gives

$\displaystyle Q_{\text{out}}-Q_{\text{in}}>\ $
$\displaystyle\frac{\frac{s-r}{k}-\frac{1}{2}}{\sqrt{\frac{1}{k^{2}}(s-r)^{2}+(2H^{2}+4)}}+\frac{\frac{-kH^{2}+\frac{r}{2}H^{2}+s-r}{r-k}-\frac{1}{2}}{\sqrt{\frac{1}{(r-k)^{2}}(-kH^{2}+\frac{r}{2}H^{2}+s-r)^{2}+(2H^{2}+4)}}.$

When $s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$, the right hand
side goes to $2$, thus for $g\gg 0$, we obtain

(26) $2\epsilon_{\text{out}}<Q_{\text{out}}-Q_{\text{in}}$

as claimed. ∎

###### Proof of Theorem 1.1.

By Proposition 3.6(b), there is a well-defined map

$\displaystyle\Psi\colon M_{X}(\mathsf{v})$
$\displaystyle\rightarrow\operatorname{BN}_{C}(r,kH^{2},r+s)$ $\displaystyle
E$ $\displaystyle\mapsto E|_{C}.$

We know the exact sequence $E\rightarrow\iota_{*}E|_{C}\rightarrow E(-H)[1]$
in $\mathcal{A}(b=0)$ is the HN filtration of $\iota_{*}E|_{C}$ below the wall
$\widetilde{\ell}$, so the uniqueness of HN factors implies $\Psi$ is
injective. Combining Lemma 3.9 with Proposition 3.10 shows $\Psi$ is
surjective. Thus Proposition 3.6(b) implies that any vector bundle $F$ in the
Brill-Noether locus $\operatorname{BN}_{C}(r,\,k(2g-2),\,r+s)$ is stable and
$h^{0}(F)=r+s$. The Zariski tangent space to the Brill-Noether locus at the
point $[F]$ is the kernel of the map

$k_{1}\colon\text{Ext}^{1}(F,F)\rightarrow\text{Hom}\big{(}H^{0}(C,F),H^{1}(C,F)\big{)},$

where any $f\colon F\rightarrow
F[1]\in\text{Ext}^{1}(F,F)=\text{Hom}_{C}(F,F[1])$ goes to

$k_{1}(f)=H^{0}(f)\colon\text{Hom}_{C}(\mathcal{O}_{C},F)\rightarrow\text{Hom}_{C}(\mathcal{O}_{C},F[1]),$

see [BS13, Proposition 4.3] for more details. Note that the proof in [BS13] is
valid for any family of simple sheaves on a variety.

The moduli space $M_{X}(\mathsf{v})$ is a (non-empty) smooth quasi-projective
variety, see for instance [Huy16, Chapter 10, Corollary 2.1 & Theorem 2.7].
Hence to prove Theorem 1.1, we only need to show the derivative of the
restriction map

$d\Psi\colon T_{[E]}M_{X}(\mathsf{v})\rightarrow T_{[E|_{C}]}\mathcal{BN},$

is surjective. It sends any $f\colon E\rightarrow
E[1]\in\text{Hom}_{X}(E,E[1])$ to its restriction
$\iota^{*}f\colon\iota^{*}E\rightarrow\iota^{*}E[1]\in\ker(k_{1})$. By
Proposition 3.6(c), we can apply the same argument as in the proof of [Fey20a,
Theorem 1.2] to show surjectivity of $d\Psi$, hence $\Psi$ is an isomorphism.
∎

##  4\. location of the first wall

In this section, we prove Proposition 3.2 and Lemma 3.3 for $g\gg 0$. The
first step is to control the position of the lines $\ell_{\mathsf{v}}$ and
$\ell_{\alpha}$.

######  Proposition 4.1.

Fix $0<\epsilon,\epsilon^{\prime}<\frac{1}{2r}$. If $g\gg 0$, then

The line $\ell_{\mathsf{v}}$ passing through $\Pi(\mathsf{v})$ and the origin
intersects $\Gamma$ at two points (except the origin) with $b$-values
$b_{1}^{\mathsf{v}}<0<b_{2}^{\mathsf{v}}$ such that

$b_{1}^{\mathsf{v}}<-\frac{k}{s}+\frac{1}{s(s-1)}\qquad\text{and}\qquad\frac{k}{r}-\epsilon<b_{2}^{\mathsf{v}}\
.$

The line $\ell_{\alpha}$ passing through $\Pi(\mathsf{v}(-H))$ and
$\Pi(\alpha)$ intersects $\Gamma$ at two points with $b$-values
$b_{1}^{\alpha}<b_{2}^{\alpha}$ such that

$b_{1}^{\alpha}<\frac{k-r}{r}+\epsilon^{\prime}\qquad\text{and}\qquad-\frac{k}{s}-\frac{1}{s(s-1)}<b_{2}^{\alpha}.$

###### Proof.

The line $\ell_{\mathsf{v}}$ which is of equation $w=\frac{s-r}{k}b$ connects
the point $\Pi(\mathsf{v})$ which is outside or on $\Gamma$ to the point
$(0,0)$, so it intersects $\Gamma$ at two other points except the origin with
$b$-values $b_{1}^{\mathsf{v}}<b_{2}^{\mathsf{v}}$. The equation of $\Gamma$
for $b\neq 0$ is $\Gamma(b)=(1+\frac{H^{2}}{2})b^{2}-1$, thus

$b_{1}^{\mathsf{v}},b_{2}^{\mathsf{v}}=\frac{\frac{s-r}{k}\pm\sqrt{(\frac{s-r}{k})^{2}+2H^{2}+4}}{H^{2}+2}.$

We claim if $g\gg 0$, then

$b_{1}^{\mathsf{v}}<-\frac{k}{s}+\frac{1}{s(s-1)}$

i.e.

$\frac{s-r}{k}+\frac{\big{(}k(s-1)-1\big{)}(H^{2}+2)}{s(s-1)}<\sqrt{\left(\frac{s-r}{k}\right)^{2}+2H^{2}+4}\
.$

Taking the power 2 of both sides shows that we only need to prove

$\big{(}k(s-1)-1\big{)}\left(2\frac{s-r}{k}s(s-1)+\big{(}k(s-1)-1\big{)}(H^{2}+2)\right)<2s^{2}(s-1)^{2}$

which is equivalent to

(27)
$-\frac{2}{k}s(s-r)+(H^{2}+2)\left(k^{2}(s-1)+\frac{1}{s-1}-2k\right)<2rs(s-1).$

When $H^{2}\rightarrow+\infty$, then
$s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$ by (4), so the limit of
the left hand side is
$(H^{2})^{2}\,\frac{k^{3}}{2r}\left(k-\frac{1}{r}\right)$ but the limit of the
right hand side is $(H^{2})^{2}\,\frac{k^{4}}{2r}$, thus the claim follows.

The next step is to show for $g\gg 0$, we have

(28)
$b_{2}^{\mathsf{v}}=\frac{\frac{s-r}{k}+\sqrt{(\frac{s-r}{k})^{2}+2H^{2}+4}}{H^{2}+2}>\frac{k}{r}-\epsilon\,.$

This holds if

$\frac{1}{2}\left(\frac{k}{r}-\epsilon\right)^{2}(H^{2}+2)<\frac{s-r}{k}\left(\frac{k}{r}-\epsilon\right)+1\,.$

The limit of the right hand side when
$s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$ is
$\frac{k}{2r}\left(\frac{k}{r}-\epsilon\right)H^{2}$, so the above holds when
$g\gg 0$. This completes the proof of part (a).

The line $\ell_{\alpha}$ in part (b) is of equation $w=\theta b+\beta-1$ for

(29) $\theta\coloneqq\frac{s^{2}-r^{2}-skH^{2}+\frac{rs}{2}H^{2}}{s(k-r)+kr}\
,\qquad\beta\coloneqq\frac{r(k-r)+sk-k^{2}H^{2}+\frac{kr}{2}H^{2}}{s(k-r)+kr}.$

When $s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$, we know

(30) $\theta\ \rightarrow\ \frac{k-r}{2r}H^{2}\qquad\text{and}\qquad\beta\
\rightarrow\ \frac{k-r}{k}.$

Hence if $g\gg 0$, we have

(31) $\theta^{2}+2\beta(H^{2}+2)\geq 0.$

Therefore $\ell_{\alpha}$ intersects the parabola
$w=b^{2}(\frac{H^{2}}{2}+1)-1$ at two points with $b$-values

$b_{1}^{\alpha},\,b_{2}^{\alpha}=\frac{\theta\pm\sqrt{\theta^{2}+2\beta(H^{2}+2)}}{H^{2}+2}\,.$

We claim if $g\gg 0$, then

(32)
$-\frac{k}{s}-\frac{1}{s(s-1)}<b_{2}^{\alpha}=\frac{\theta+\sqrt{\theta^{2}+2\beta(H^{2}+2)}}{H^{2}+2}$

which holds if

(33)
$\frac{H^{2}+2}{2}\left(\frac{k}{s}+\frac{1}{s(s-1)}\right)^{2}<\beta-\theta\left(\frac{k}{s}+\frac{1}{s(s-1)}\right).$

The right hand side is equal to

$\beta-\theta\left(\frac{k}{s}+\frac{1}{s(s-1)}\right)=\frac{\frac{H^{2}}{s-1}\left(k-\frac{r}{2}\right)+r(k-r)-\frac{s}{s-1}+\frac{kr^{2}}{s}+\frac{r^{2}}{s(s-1)}}{s(k-r)+kr}$

thus its limit when $s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$ is
equal to
$\frac{1}{H^{2}}\left(2\frac{r^{2}}{k^{2}}+\frac{2r(r-k)}{k^{4}}\right)$.
Since the limit of the left hand side of (33) is equal to
$\frac{2r^{2}}{H^{2}k^{2}}$ the claim (33) follows.

The other intersection point of $\ell_{\alpha}$ with $\Gamma$ satisfies

$b_{1}^{\alpha}=\frac{\theta-\sqrt{\theta^{2}+2\beta(H^{2}+2)}}{H^{2}+2}<\frac{k-r}{r}+\epsilon^{\prime}$

if and only if

(34)
$\theta-(H^{2}+2)\left(\frac{k-r}{r}+\epsilon^{\prime}\right)<\sqrt{\theta^{2}+2\beta(H^{2}+2)}\,.$

Taking power 2 from both sides shows that (34) holds if

$\frac{H^{2}+2}{2}\left(\frac{r-k}{r}-\epsilon^{\prime}\right)^{2}<\beta-\theta\left(\frac{r-k}{r}-\epsilon^{\prime}\right)$

which is satisfied by (30) for $g\gg 0$ and $\epsilon^{\prime}<\frac{1}{2r}$.

∎

If $g\gg 0$, (30) implies

(35) $\beta-1<0$

where $\beta$ is defined as in (29). Thus the origin point $(0,0)$ and so the
line segment connecting $\Pi(\mathsf{v}(-H))$ to the origin lies above
$\ell_{\alpha}$. Thus it intersects the curve $\Gamma$ at a point with
$b$-value $b_{1}^{\mathsf{v}(-H)}$ satisfying

(36) $\frac{k-r}{r}\,\leq\,b_{1}^{\mathsf{v}(-H)}\,\leq\,b_{1}^{\alpha}\ .$

Consider the line $\widetilde{\ell}$ passing through $\Pi(\mathsf{v})$ and
$\Pi(\mathsf{v}(-H))$. It is of equation

$w=H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)b\,+\frac{s}{r}-H^{2}\frac{k}{r}\left(\frac{k}{r}-\frac{1}{2}\right)-1.$

When $s\rightarrow\frac{k^{2}}{2r}H^{2}\rightarrow+\infty$, we get

(37) $\frac{s}{r}-H^{2}\frac{k}{r}\left(\frac{k}{r}-\frac{1}{2}\right)-1\ >\
0.$

This implies the origin $(0,0)$ lies below $\widetilde{\ell}$. Therefore the
line segment connecting $\Pi(\mathsf{v})$ to $\Pi(\mathsf{v}(-H))$ (which is
part of $\widetilde{\ell}$) lies above $\ell_{\mathsf{v}}$ and
$\ell_{\mathsf{v}(-H)}$, so $\widetilde{\ell}$ intersects $\Gamma$ at two
points with $b$-values $\widetilde{b}_{1}<\widetilde{b}_{2}$ satisfying

$\frac{k-r}{r}\,\leq\,\widetilde{b}_{1}\,\leq\,b_{1}^{\mathsf{v}(-H)}\qquad\text{and}\qquad
b_{2}^{\mathsf{v}}\,\leq\,\widetilde{b}_{2}\,\leq\,\frac{k}{r}.$

Thus (36) together with Proposition 4.1 implies

(38)
$\frac{k-r}{r}\,\leq\,\widetilde{b}_{1}\,<\,\frac{k-r}{r}+\epsilon^{\prime}\qquad\text{and}\qquad\frac{k}{r}-\epsilon\,<\,\widetilde{b}_{2}\,\leq\,\frac{k}{r}\,.$

Thus if we assume $H^{2}$ is large enough, the line $\widetilde{\ell}$
intersects $\Gamma$ at two points which are arbitrary close to
$\Pi(\mathsf{v})$ and $\Pi(\mathsf{v}(-H))$. Consider the line $\ell^{*}$ as
in Definition 3.1. We set

(39)
$\epsilon^{\prime}=b_{1}^{*}-\frac{k-r}{r}\qquad\text{and}\qquad\epsilon=\frac{k}{r}-b_{2}^{*}\,.$

Then the line $\ell^{*}$ lies below the parallel line $\widetilde{\ell}$.

######  Lemma 4.2.

The origin point $(0,0)$ lies below $\ell^{*}$ and

(40) $\frac{k-r}{r}<b_{1}^{*}<0<b_{2}^{*}<\frac{k}{r}.$

###### Proof.

The equation of $\ell^{*}$ is given by
$w=H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)b+\alpha-1$ for some
$\alpha\in\mathbb{R}$. To prove the first claim we need to show $\alpha>1$. We
know the intersection points of $\ell^{*}$ with $\Gamma$ satisfies

$b_{2}^{*}-b_{1}^{*}=\frac{2\sqrt{(H^{2})^{2}\left(\frac{k}{r}-\frac{1}{2}\right)^{2}+2\alpha(H^{2}+2)}}{H^{2}+2}\geq\frac{k}{r}-\frac{1}{r^{2}(r+1)}-\frac{k-r}{r}-\frac{1}{r^{2}(r+1)}$

This implies for $g\gg 0$, we have

(41)
$2\alpha\geq(H^{2}+2)\left(\frac{1}{2}-\frac{1}{r^{2}(r+1)}\right)^{2}-\frac{(H^{2})^{2}}{H^{2}+2}\left(\frac{k}{r}-\frac{1}{2}\right)^{2}>2.$

The second claim (40) follows from (38) and the point that by the choice of
$\epsilon$ and $\epsilon^{\prime}$ as in (39), we know $\widetilde{\ell}$ lies
above $\ell^{*}$. ∎

The above arguments completes the proof of Proposition 3.2. The final step is
to prove Lemma 3.3.

###### Proof of Lemma 3.3.

The equation of the line $\ell^{*}$ is given by
$w=H^{2}(\frac{k}{r}-\frac{1}{2})b+\alpha-1$ where
$\alpha=(b_{2}^{*})^{2}\left(\frac{H^{2}}{2}+1\right)-H^{2}(\frac{k}{r}-\frac{1}{2})b_{2}^{*}$
and equation of the line $\ell_{1}$ is given by $w=\frac{r}{k}w_{1}b$. We know
$b_{2}^{*}=\frac{k}{r}-\epsilon$, so

$\alpha=\left(\frac{H^{2}}{2}+1\right)\left(\frac{k}{r}-\epsilon\right)^{2}-H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)\left(\frac{k}{r}-\epsilon\right).$

Thus when $H^{2}\rightarrow+\infty$, we know

$\alpha\ \rightarrow\
\frac{H^{2}}{2}\left(\frac{k}{r}-\epsilon\right)\left(-\frac{k}{r}-\epsilon+1\right)$

and so

(42) $\frac{r}{k}w_{1}\ \rightarrow\
H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)+\frac{H^{2}r}{2k}\left(\frac{k}{r}-\epsilon\right)\left(-\frac{k}{r}-\epsilon+1\right).$

The line $\ell_{1}$ intersects the vertical line
$b=\frac{k}{r}-\frac{1}{r(r-1)}$ at a point with
$w=\frac{r}{k}w_{1}(\frac{k}{r}-\frac{1}{r(r-1)})$. We claim if $g\gg 0$, this
point lies in $U$, i.e.

(43)
$\frac{r}{k}w_{1}>\Gamma\left(\frac{k}{r}-\frac{1}{r(r-1)}\right)\frac{1}{\frac{k}{r}-\frac{1}{r(r-1)}}\,.$

The limit of the right hand side when $H^{2}\rightarrow+\infty$ is
$\frac{H^{2}}{2}(\frac{k}{r}-\frac{1}{r(r-1)})$, so (42) shows that (43) holds
if

$\left(\frac{k}{r}-\epsilon\right)\left(-\frac{k}{r}-\epsilon+1\right)>\frac{k}{r}\left(1-\frac{k}{r}-\frac{1}{r(r-1)}\right).$

This is equivalent to

$\epsilon-\epsilon^{2}<\frac{k}{r^{2}(r-1)}$

which is satisfied by our assumption on $\epsilon=\frac{k}{r}-b_{2}^{*}$ in
Definition 3.1.

Similarly, we have

$w_{2}=H^{2}\frac{k-r}{r}\left(\frac{k}{r}-\frac{1}{2}\right)+\alpha-1$

We require

(44)
$\frac{r}{k-r}w_{2}<\Gamma\left(\frac{k-r}{r}+\frac{1}{r(r-1)}\right)\frac{1}{\frac{k-r}{r}+\frac{1}{r(r-1)}}\,.$

The limit of the left hand side is

$\displaystyle\lim\limits_{H^{2}\rightarrow+\infty}\frac{r}{k-r}w_{2}=\ $
$\displaystyle
H^{2}\left(\frac{k}{r}-\frac{1}{2}\right)+\frac{H^{2}r}{2(k-r)}\left(\frac{k}{r}-\epsilon\right)\left(-\frac{k}{r}-\epsilon+1\right)$

and the limit of the right hand side is
$\frac{H^{2}}{2}(\frac{k-r}{r}+\frac{1}{r(r-1)})$. Thus (44) holds for $g\gg
0$ if

$\displaystyle\frac{r}{k-r}\left(\frac{k}{r}-\epsilon\right)\left(-\frac{k}{r}-\epsilon+1\right)<\frac{1}{r(r-1)}-\frac{k}{r}.$

This is equivalent to

$\epsilon-\epsilon^{2}<\frac{r-k}{r^{2}(r-1)}$

which holds again by our assumptions on $\epsilon=\frac{k}{r}-b_{2}^{*}$. ∎

######  Remark 4.3.

The genus $g=\frac{H^{2}}{2}+1$ is large enough for Theorem 1.1 if
$H^{2}>2r(r+1)$ and inequalities (9), (10), (26), (27), (31), (32), (35),
(37), (41), (43), (44), and for the chosen values of
$\epsilon,\epsilon^{\prime}$ in (39), inequalities (28) and (34) are
satisfied.

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Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom