# Picard Groups of Some Quot Schemes

Chandranandan Gangopadhyay  Department of Mathematics, Indian Institute of
Science Education and Research, Pune, 411008, Maharashtra, India.
<EMAIL_ADDRESS>and  Ronnie Sebastian  Department of
Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076,
Maharashtra, India<EMAIL_ADDRESS>
###### Abstract.

Let $C$ be a smooth projective curve over the field of complex numbers
$\mathbb{C}$ of genus $g(C)>0$. Let $E$ be a locally free sheaf on $C$ of rank
$r$ and degree $e$. Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(E,k,d)$ denote
the Quot scheme of quotients of $E$ of rank $k$ and degree $d$. For $k>0$ and
$d\gg 0$ we compute the Picard group of $\mathcal{Q}$.

###### Key words and phrases:

Quot Scheme

###### 2010 Mathematics Subject Classification:

14J60

##  1\. Introduction

Let $C$ be a smooth projective curve over the field of complex numbers
$\mathbb{C}$. We shall denote the genus of $C$ by $g(C)$. Throughout this
article we shall assume that $g(C)\geqslant 1$. Let $E$ be a locally free
sheaf on $C$ of rank $r$ and degree $e$. Throughout this article

(1.1) $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(E,k,d)$

will denote the Quot scheme of quotients of $E$ of rank $k$ and degree $d$.

Stromme proved that
$\mathcal{Q}_{\mathbb{P}^{1}/\mathbb{C}}(\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n},k,d)$ is a smooth projective variety, computed its Picard group and nef
cone. In [Jow12], the author computes the effective cone of
$\mathcal{Q}_{\mathbb{P}^{1}/\mathbb{C}}(\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n},k,d)$. In [Ito17], the author studies the birational geometry of
$\mathcal{Q}_{\mathbb{P}^{1}/\mathbb{C}}(\mathcal{O}_{\mathbb{P}^{1}}^{\oplus
n},k,d)$. When $E$ is trivial and $g(C)\geqslant 1$, the space $\mathcal{Q}$
is studied in [BDW96] and it is proved that when $d\gg 0$ it is irreducible
and generically smooth. For $g(C)\geqslant 1$ and $E$ trivial, the divisor
class group of $\mathcal{Q}$ was computed in [HO10] under the assumption $d\gg
0$.

When $g(C)\geqslant 1$, it was proved in [PR03] that $\mathcal{Q}$ is
irreducible and generically smooth when $d\gg 0$. See also [Gol19], [CCH21],
[CCH22] for similar results on other variations of this Quot scheme. We use
this as a starting point to further investigate the space $\mathcal{Q}$ when
$d\gg 0$ and compute its Picard group. In the case when $k=r-1$ we have that
$\mathcal{Q}$ is a projective bundle over the Jacobian of $C$ for $d\gg 0$
(Theorem 3.3) and as a result its Picard group can be computed easily
(Corollary 3.5). In Theorem 6.3 we show that if $d\gg 0$ then $\mathcal{Q}$ is
an integral variety which is a local complete intersection and locally
factorial. We compute the Picard group of $\mathcal{Q}$ in the following
cases.

######  Theorem 1.2 (Theorem 7.16).

Let $k\leqslant r-2$. Assume one of the following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Then for $d\gg 0$ we have

${\rm Pic}(\mathcal{Q})\cong{\rm Pic}({\rm
Pic}^{0}(C))\times\mathbb{Z}\times\mathbb{Z}\,.$

Note that we have a natural determinant map

${\rm det}:\mathcal{Q}\longrightarrow{\rm Pic}^{d}(C)$

which sends a quotient $[E\longrightarrow F]\mapsto{\rm det}(F)$. In Theorem
6.3 we show that ${\rm det}$ is a flat map when $d\gg 0$. For $[L]\in{\rm
Pic}^{d}(C)$ let $\mathcal{Q}_{L}$ be the scheme theoretic fiber of det over
$[L]$. We prove the following analogous results for $\mathcal{Q}_{L}$.

######  Theorem 1.3 (Theorem 8.7).

Let $k\geqslant 2,g(C)\geqslant 2$. Let $d\gg 0$. Then $\mathcal{Q}_{L}$ is a
local complete intersection, integral, normal and locally factorial scheme.

######  Theorem 1.4 (Theorem 8.9).

Let $k\leqslant r-2$. Assume one of the following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Let $d\gg 0$. Then ${\rm
Pic}(\mathcal{Q}_{L})\cong\mathbb{Z}\times\mathbb{Z}$.

When $k=1$ the above results can be improved to the case $g(C)\geqslant 1$. In
Theorem 9.1 we show that if $d\gg 0$ then ${\rm Pic}(\mathcal{Q})\cong{\rm
Pic}({\rm Pic}^{0}(C))\times\mathbb{Z}\times\mathbb{Z}$ and ${\rm
Pic}(\mathcal{Q}_{L})\cong\mathbb{Z}\times\mathbb{Z}$.

We say a few words about how the above results are proved. By a very large
open subset we mean an open set whose complement has codimension $\geqslant
2$. When $d\gg 0$ the Quot scheme $\mathcal{Q}$ is a local complete
intersection. This follows easily using [HL10, Proposition 2.2.8] and is the
content of Lemma 6.1. Using dimension bounds from [PR03] we show that the
locus of singular points in $\mathcal{Q}$ has large codimension. These are
used to prove Theorem 6.3. To compute the Picard group, we first show that the
locus of quotients $[E\longrightarrow F]$ with $F$ stable is a very large open
subset. Let $\mathcal{Q}^{s}$ denote this locus. We consider a map
$\mathcal{Q}^{s}\longrightarrow M^{s}$, to a moduli space of stable bundles of
rank $k$ and suitable degree, see (7.9). After base change by a principal
${\rm PGL}(N)$-bundle, the domain becomes a very large open subset of a
projective bundle associated to a vector bundle. From this we compute the
Picard group of $\mathcal{Q}^{s}$ in terms of the Picard group of $M^{s}$. The
assertions about $\mathcal{Q}_{L}$ follow in a similar manner using the
assertions about $\mathcal{Q}$ and the flat map ${\rm
det}:\mathcal{Q}\longrightarrow{\rm Pic}^{d}(C)$.

##  2\. Preliminaries

For a locally closed subset $Z\subset X$ we shall refer to ${\rm dim}(X)-{\rm
dim}(Z)$ as the codimension of $Z$ in $X$. For a morphism $f:X\longrightarrow
Y$ and a closed point $y\in Y$ we denote by $X_{y}$ the fiber over $Y$.

######  Lemma 2.1.

Let $f:X\longrightarrow Y$ be a dominant morphism of integral schemes of
finite type over a field $k$. Let $U\subset X$ be an open subset such that
nonempty fibers of $f|_{U}$ have constant dimension. Let $Z:=X\setminus U$.

  1. (1)

If ${\rm dim}(X)-{\rm dim}(Z)>{\rm dim}(Y)$ then the dimension of nonempty
fibers of $f$ is constant.

  2. (2)

Let $t_{0}\geqslant 0$ be an integer and assume ${\rm dim}(X)-{\rm
dim}(Z)>{\rm dim}(Y)+t_{0}$. Let $y\in Y$ be a closed point such that $Z_{y}$
is nonempty. Then ${\rm dim}(X_{y})-{\rm dim}(Z_{y})>t_{0}$.

###### Proof.

Let $y\in Y$ be a closed point such that $X_{y}$ is nonempty. Note that
$X_{y}=U_{y}\bigsqcup Z_{y}$. If $U_{y}$ is empty then

${\rm dim}(Z)\geqslant{\rm dim}(Z_{y})={\rm dim}(X_{y})\geqslant{\rm
dim}(X)-{\rm dim}(Y)\,.$

This contradicts the hypothesis that ${\rm dim}(X)-{\rm dim}(Z)>{\rm dim}(Y)$.
Thus, $U_{y}$ is nonempty. Since $f|_{U}$ has constant fiber dimension, it
follows that ${\rm dim}(U_{y})={\rm dim}(X)-{\rm dim}(Y)$, see [Har77, Chapter
2, Exercise 3.22(b)]. As ${\rm dim}(X)-{\rm dim}(Z_{y})\geqslant{\rm
dim}(X)-{\rm dim}(Z)>{\rm dim}(Y)$ it follows that ${\rm dim}(Z_{y})<{\rm
dim}(X)-{\rm dim}(Y)$. It follows that

${\rm dim}(X_{y})={\rm max}\\{{\rm dim}(U_{y}),{\rm dim}(Z_{y})\\}={\rm
dim}(X)-{\rm dim}(Y)\,.$

This proves (1).

Let $y\in Y$ be a closed point such that $Z_{y}$ is nonempty. Then $X_{y}$ is
nonempty and so by the previous part we get that ${\rm dim}(X_{y})={\rm
dim}(X)-{\rm dim}(Y)$. As ${\rm dim}(X)-{\rm dim}(Z_{y})\geqslant{\rm
dim}(X)-{\rm dim}(Z)>{\rm dim}(Y)+t_{0}$ it follows that ${\rm
dim}(Z_{y})<{\rm dim}(X)-{\rm dim}(Y)-t_{0}={\rm dim}(X_{y})-t_{0}$. This
proves (2) and completes the proof of the Lemma. ∎

######  Lemma 2.2.

Let $f:X\longrightarrow Y$ be a morphism of irreducible schemes of finite type
over a field $k$ which is surjective on closed points. Let $Y^{\prime}\subset
Y$ be a closed subset. Then ${\rm dim}(X)-{\rm
dim}(f^{-1}(Y^{\prime}))\leqslant{\rm dim}(Y)-{\rm dim}(Y^{\prime})$.

###### Proof.

Since it suffices to consider reduced schemes, we look at the map $f_{\rm
red}:X_{\rm red}\longrightarrow Y_{\rm red}$. Thus, we may assume that $X$ and
$Y$ are integral schemes. Let $Y^{\prime\prime}\subset Y^{\prime}$ be an
irreducible component such that ${\rm dim}(Y^{\prime\prime})={\rm
dim}(Y^{\prime})$. Let $Z^{\prime\prime}$ be an irreducible component of
$f^{-1}(Y^{\prime\prime})$ which surjects onto $Y^{\prime\prime}$. By [Har77,
Chapter 2, Exercise 3.22(a)] we have ${\rm dim}(X)-{\rm
dim}(Z^{\prime\prime})\leqslant{\rm dim}(Y)-{\rm dim}(Y^{\prime\prime})$. As
$Z^{\prime\prime}\subset f^{-1}(Y^{\prime})$ it follows that

${\rm dim}(X)-{\rm dim}(f^{-1}(Y^{\prime}))\leqslant{\rm dim}(X)-{\rm
dim}(Z^{\prime\prime})\leqslant{\rm dim}(Y)-{\rm dim}(Y^{\prime\prime})={\rm
dim}(Y)-{\rm dim}(Y^{\prime})\,.$

This completes the proof of the Lemma. ∎

Recall the space $\mathcal{Q}$ from (1.1). Let

(2.3) $p_{1}:C\times\mathcal{Q}\longrightarrow C\qquad
p_{2}:C\times\mathcal{Q}\longrightarrow\mathcal{Q}$

denote the projections. Let

(2.4) $0\longrightarrow\mathcal{K}\longrightarrow
p_{1}^{*}E\longrightarrow\mathcal{F}\longrightarrow 0$

denote the universal quotient on $C\times\mathcal{Q}$. The sheaf $\mathcal{K}$
is locally free and so ${\rm det}(\mathcal{K}):=\wedge^{r-k}\mathcal{K}$ is a
line bundle on $C\times\mathcal{Q}$ which is flat over $\mathcal{Q}$. Using
this we define the determinant map as the following composite

(2.5) ${\rm det}:\mathcal{Q}\longrightarrow{\rm
Pic}^{e-d}(C)\longrightarrow{\rm Pic}^{d}C\,,$

which has the following pointwise description. Let $[q:E\longrightarrow
F]\in\mathcal{Q}$ be a closed point. We denote the kernel of $q$ by $K$, so
that there is a short exact sequence

(2.6) $0\longrightarrow K\longrightarrow E\stackrel{{\scriptstyle
q}}{{\longrightarrow}}F\longrightarrow 0\,.$

Then

${\rm det}(q):={\rm det}(E)\otimes{\rm det}(K)^{-1}={\rm det}(F)\,.$

Next we describe the differential of this map ${\rm det}$.

######  Lemma 2.7.

The differential of the map det (2.5) at the point $q$ is the composite

${\rm Hom}(K,F)\stackrel{{\scriptstyle-\delta}}{{\longrightarrow}}{\rm
Ext}^{1}(F,F)\stackrel{{\scriptstyle
tr}}{{\longrightarrow}}H^{1}(C,\mathcal{O}_{C})\,,$

where the first map is obtained by applying ${\rm Hom}(-,F)$ to (2.6) and the
second map is the trace.

###### Proof.

Given a vector $v\in{\rm Hom}(K,F)$ it corresponds to an element in the
Zariski tangent space at $q\in\mathcal{Q}$, and so it corresponds to a short
exact sequence on $C\times{\rm Spec}\,(k[\epsilon]/(\epsilon^{2}))$

$0\longrightarrow\tilde{K}\longrightarrow
p_{1}^{*}E\longrightarrow\tilde{F}\longrightarrow 0\,,$

whose restriction to $C$ gives the sequence (2.6). Moreover, $\tilde{F}$ is
flat over ${\rm Spec}\,(k[\epsilon]/(\epsilon^{2}))$. Consider the line bundle
${\rm det}(\tilde{F})$ on $C\times{\rm Spec}\,(k[\epsilon]/(\epsilon^{2}))$.
Tensoring this line bundle with the short exact sequence

(2.8) $0\longrightarrow(\epsilon)\longrightarrow
k[\epsilon]/(\epsilon^{2})\longrightarrow k\longrightarrow 0$

gives the short exact sequence

(2.9) $0\longrightarrow{\rm det}(F)\longrightarrow{\rm
det}(\tilde{F})\longrightarrow{\rm det}(F)\longrightarrow 0\,.$

Using the definition of the differential of the map ${\rm det}$ it is clear
that

(2.10) $d\,{\rm det}_{q}(v)=\text{extension class of \eqref{ddet_q}}\in
H^{1}(C,\mathcal{O}_{C})\,.$

Tensoring (2.8) with $\tilde{F}$ gives a short exact sequence

(2.11) $0\longrightarrow F\longrightarrow\tilde{F}\longrightarrow
F\longrightarrow 0\,.$

One checks easily, using the discussion before [HL10, Lemma 2.2.6], that the
above extension, and in particular the sheaf $\tilde{F}$, is obtained by
taking the pushout of the sequence (2.6) along the map $-v$. That is, the
extension class of (2.11) is precisely $-\delta(v)$.

For a coherent sheaf $G$, consider the trace map $tr:{\rm
Ext}^{1}(G,G)\longrightarrow H^{1}(C,\mathcal{O}_{C})$. An element $v\in{\rm
Ext}^{1}(G,G)$ corresponds to a short exact sequence

$0\longrightarrow G\longrightarrow\tilde{G}\longrightarrow G\longrightarrow
0\,,$

on $C\times{\rm Spec}\,(k[\epsilon]/(\epsilon^{2}))$ such that $\tilde{G}$ is
flat over ${\rm Spec}\,(k[\epsilon]/(\epsilon^{2}))$. The image $tr(v)$ in
$H^{1}(C,\mathcal{O}_{C})$ corresponds to the extension class obtained by
tensoring (2.8) with the line bundle ${\rm det}(\tilde{G})$ on $C\times{\rm
Spec}\,(k[\epsilon]/(\epsilon^{2}))$. When $G$ is locally free this can be
seen using a Cech description, for example, see [Nit09]. The general case
reduces to the locally free case using the discussion in [HL10, §10.1.2]. In
particular, we can apply this discussion by taking $G=F$. We get that
$tr(-\delta(v))$ is the extension class obtained by tensoring ${\rm
det}(\tilde{F})$ in (2.11) with (2.8). But note that we obtained (2.9) also be
tensoring ${\rm det}(\tilde{F})$ with (2.8). This proves that

$d\,{\rm det}_{q}(v)=tr(-\delta(v))$

and completes the proof of the Lemma. ∎

##  3\. Quot Schemes ${\rm Quot}_{C/\mathbb{C}}(E,r-1,d)$

In this section we describe the Quot scheme ${\rm
Quot}_{C/\mathbb{C}}(E,r-1,d)$ which parametrizes quotients of $E$ of rank
$(r-1)$ and degree $d>2g-2+e-\mu_{\rm min}(E)$. Let

(3.1) $\rho_{1}:C\times{\rm Pic}^{e-d}(C)\longrightarrow
C\,,\quad\quad\rho_{2}:C\times{\rm Pic}^{e-d}(C)\longrightarrow{\rm
Pic}^{e-d}(C)$

be the projections. Let $\mathcal{L}$ be a Poincare bundle on $C\times{\rm
Pic}^{e-d}(C)$. Define

$\mathcal{E}:=\rho_{2*}[\rho_{1}^{*}E\otimes\mathcal{L}^{\vee}]\,.$

######  Lemma 3.2.

Assume $d>2g-2+e-\mu_{\rm min}(E)$. Then $\mathcal{E}$ is a vector bundle on
${\rm Pic}^{e-d}(C)$ of rank $rd-(r-1)e-r(g-1)$.

###### Proof.

Let $K_{C}$ denote the canonical bundle of $C$. For any $L\in{\rm
Pic}^{e-d}(C)$, we claim

$H^{1}(C,E\otimes L^{\vee})=H^{0}(C,E^{\vee}\otimes L\otimes
K_{C})^{\vee}=0\,.$

This is because a nonzero section of $H^{0}(C,E^{\vee}\otimes L\otimes K_{C})$
corresponds to a nonzero map $E\longrightarrow L\otimes K_{C}$ which cannot
exist since by assumption $\mu_{\rm min}(E)>{\rm deg}(L\otimes
K_{C})=e-d+2g-2$. Therefore by Grauert’s theorem $\mathcal{E}$ is a vector
bundle of rank $h^{0}(C,E\otimes L^{\vee})$ which by Riemann-Roch is
$rd-(r-1)e-r(g-1)$. ∎

Let $\pi:\mathbb{P}(\mathcal{E}^{\vee})\longrightarrow{\rm Pic}^{e-d}(C)$ be
the projective bundle associated to $\mathcal{E}^{\vee}$. Recall that we have
the map ${\rm det}:\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)\longrightarrow{\rm
Pic}^{e-d}(C)$ which sends a quotient $[E\longrightarrow F\longrightarrow 0]$
to its kernel.

######  Theorem 3.3.

Assume $d>2g-2+e-\mu_{\rm min}(E)$. We have an isomorphism of schemes over
${\rm Pic}^{e-d}(C)$

$\mathbb{P}(\mathcal{E}^{\vee})\xrightarrow{\sim}\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)\,.$

###### Proof.

Let

$\sigma_{1}:C\times\mathbb{P}(\mathcal{E})\longrightarrow
C\,,\quad\quad\sigma_{2}:C\times\mathbb{P}(\mathcal{E})\longrightarrow{\rm
Pic}^{e-d}(C)$

be the projections. We define the map
$\mathbb{P}(\mathcal{E}^{\vee})\longrightarrow\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)$
by producing a quotient on $C\times\mathbb{P}(\mathcal{E}^{\vee})$.

Recall the maps $\rho_{i}$ from (3.1). By adjunction we have a natural map on
$C\times{\rm Pic}^{e-d}(C)$

$\rho^{*}_{2}\mathcal{E}\otimes\mathcal{L}\longrightarrow\rho^{*}_{1}E\,.$

Pulling this morphism to $C\times\mathbb{P}(\mathcal{E}^{\vee})$ we get a map

$({\rm
Id}_{C}\times\pi)^{*}[\rho_{2}^{*}\mathcal{E}\otimes\mathcal{L}]=(\pi\circ\sigma_{2})^{*}\mathcal{E}\otimes({\rm
Id}_{C}\times\pi)^{*}\mathcal{L}\longrightarrow\sigma_{1}^{*}E\,.$

We also have the morphism of sheaves on $\mathbb{P}(\mathcal{E}^{\vee})$

$\mathcal{O}(-1)\hookrightarrow\pi^{*}\mathcal{E}\,.$

Pulling this back to $C\times\mathbb{P}(\mathcal{E}^{\vee})$ we get a
composition of sheaves on $C\times\mathbb{P}(\mathcal{E}^{\vee})$

(3.4) $\sigma_{2}^{*}\mathcal{O}(-1)\otimes({\rm
Id}_{C}\times\pi)^{*}\mathcal{L}\longrightarrow(\pi\circ\sigma_{2})^{*}\mathcal{E}\otimes({\rm
Id}_{C}\times\pi)^{*}\mathcal{L}\longrightarrow\sigma_{1}^{*}E\,.$

As $\sigma_{2}^{*}\mathcal{O}(-1)\otimes({\rm
Id}_{C}\times\pi)^{*}\mathcal{L}$ is a line bundle and
$C\times\mathbb{P}(\mathcal{E}^{\vee})$ is smooth, it easily follows that
(3.4) is an inclusion as it is nonzero. By the previous lemma, a point
$x\in\mathbb{P}(\mathcal{E}^{\vee})$ corresponds to a pair
$(L,\phi:L\longrightarrow E)$ where $L$ is a line bundle of degree $e-d$ and
$\phi$ is a nonzero homomorphism of sheaves. The inclusion (3.4) restricted to
$C\times x$ is nothing but the nonzero homomorphism $\phi$. Therefore we get
that the cokernel of $(\ref{eqn universal kernel})$, which we denote
$\mathcal{F}$, is flat over $\mathbb{P}(\mathcal{E}^{\vee})$, and the
restriction $\mathcal{F}|_{C\times x}$ has rank $r-1$ and degree $d$. Thus,
$\mathcal{F}$ defines a map
$\phi:\mathbb{P}(\mathcal{E}^{\vee})\longrightarrow\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)$.
It is easily checked that this map is bijective on closed points.

Let point $x=[E\longrightarrow F\longrightarrow
0]\in\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)$. Let $L$ be the kernel. Then we have
an exact sequence

${\rm Ext}^{1}(L,L)\longrightarrow{\rm Ext}^{1}(L,E)\longrightarrow{\rm
Ext}^{1}(L,F)\longrightarrow 0\,.$

From the proof of Lemma 3.2 it follows that ${\rm Ext}^{1}(L,E)=0$. Hence
${\rm Ext}^{1}(L,F)=0$. Therefore $\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)$ is
smooth at $x$ [HL10, Proposition 2.2.8]. As $\phi$ is bijective on closed
points, it follows it is an isomorphism. ∎

######  Corollary 3.5.

Assume $d>2g-2+e-\mu_{\rm min}(E)$. Then $\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d)$
is a smooth projective variety and ${\rm
Pic}(\mathcal{Q}_{C/\mathbb{C}}(E,r-1,d))\cong{\rm Pic}({\rm
Pic}^{0}(C))\times\mathbb{Z}$.

When $E$ is the trivial bundle the above result is proved in [BDW96, Corollary
4.23].

##  4\. The good locus for torsion free quotients

The following Lemma is an easy consequence of [PR03, Lemma 6.1].

######  Lemma 4.1.

Let $k$ be an integer. There is a number $\mu_{0}(E,k)$, which depends only on
$E$ and $k$, such that for all torsion free sheaves $F$ with ${\rm
rk}(F)\leqslant k$ and $\mu_{\rm min}(F)\geqslant\mu_{0}(E,k)$ we have
$H^{1}(E^{\vee}\otimes F)=0$.

###### Proof.

When $F$ is stable and ${\rm rk}(F)\leqslant k$, it follows using [PR03, Lemma
6.1], that there is $\mu_{0}(E,k)$ such that if ${\rm rk}(F)\leqslant k$ and
$\mu(F)\geqslant\mu_{0}(E,k)$ then $H^{1}(E^{\vee}\otimes F)=0$.

Next let $F$ be semistable. Take a Jordan-Holder filtration for $F$ and let
$G$ be a graded piece of this filtration. As ${\rm rk}(G)\leqslant k$ and
$\mu(G)=\mu(F)\geqslant\mu_{0}(E,k)$ it follows from the stable case that
$H^{1}(E^{\vee}\otimes G)=0$. From this it easily follows that if $F$ is
semistable, ${\rm rk}(F)\leqslant k$ and $\mu(F)\geqslant\mu_{0}(E,k)$ then
$H^{1}(E^{\vee}\otimes F)=0$.

Now let $F$ be a locally free sheaf with ${\rm rk}(F)\leqslant k$ and let

$0=F_{0}\subsetneq F_{1}\subsetneq\ldots\subsetneq F_{l}=F$

be its Harder-Narasimhan filtration. Each graded piece is semistable with
slope

$\mu(F_{i}/F_{i-1})\geqslant\mu(F_{l}/F_{l-1})=\mu_{\rm min}(F)\,.$

Thus, if $\mu_{\rm min}(F)\geqslant\mu_{0}(E,k)$ then $\mu_{\rm
min}(F_{i}/F_{i-1})\geqslant\mu_{0}(E,k)$ and so from the semistable case it
follows that $H^{1}(E^{\vee}\otimes(F_{i}/F_{i-1}))$. Again it follows that
$H^{1}(E^{\vee}\otimes F)=0$. This proves the lemma. ∎

Let $G$ be a locally free sheaf on $C$ and let $k$ be an integer. Define

(4.2) $d_{k}(G):={\rm min}\\{d\,|\,\text{$\exists$ quotient $G\longrightarrow
F$ with deg($F$)=$d$, rk$(F)=k$}\\}.$

######  Remark 4.3.

We recall some results from [PR03] (see [PR03, Lemma 6.1, Theorem 6.4] and the
remarks following these). There is an integer $\alpha(E,k)$ such that when
$d\geqslant\alpha(E,k)$, the following three assertions hold:

  1. (1)

If $F$ is a stable bundle of rank $k$ and degree $d$, then $E^{\vee}\otimes F$
is globally generated.

  2. (2)

$\mathcal{Q}$ is irreducible and generically smooth of dimension $rd-
ek-k(r-k)(g-1)$.

  3. (3)

For the general quotient $E\longrightarrow F$, with $F$ having rank $k$ and
degree $d$, we have $F$ is torsion free and stable. ∎

######  Definition 4.4.

Let $a,b$ be integers. Let ${\rm Quot}_{C/\mathbb{C}}(E,a,b)$ be the Quot
scheme parametrizing quotients of $E$ of rank $a$ and degree $b$. For a
locally closed subset $A\subset{\rm Quot}_{C/\mathbb{C}}(E,a,b)$ define the
following locally closed subsets of $A$.

(4.5) $\displaystyle A_{\rm g}$ $\displaystyle:=\\{[E\longrightarrow F]\in
A\,|\,H^{1}(E^{\vee}\otimes F)=0\\}$ $\displaystyle A_{\rm b}$
$\displaystyle:=A\setminus A_{\rm g}$ $\displaystyle A^{\rm tf}$
$\displaystyle:=\\{[E\longrightarrow F]\in A\,|\,\text{$F$ is torsion
free}\\}$ $\displaystyle A^{\rm tf}_{\rm g}$ $\displaystyle:=A^{\rm tf}\cap
A_{\rm g}$ $\displaystyle A^{\rm tf}_{\rm b}$ $\displaystyle:=A^{\rm tf}\cap
A_{\rm b}$

In particular, we get a subsets $\mathcal{Q}^{\rm tf}_{\rm g}$,
$\mathcal{Q}^{\rm tf}_{\rm b}$.

For integers $0<k^{\prime\prime}<k<r$ define constants

(4.6) $\displaystyle C_{1}(E,k,k^{\prime\prime})$
$\displaystyle:=k^{\prime\prime}(r-k^{\prime\prime})-d_{k^{\prime\prime}}(E)r+(k-k^{\prime\prime})(r-k)-d_{k}(E)(r-k^{\prime\prime})$
$\displaystyle C_{2}(E,k,k^{\prime\prime})$
$\displaystyle:=-ek-k(r-k)(g-1)-C_{1}(E,k,k^{\prime\prime})$ $\displaystyle
C_{3}(E,k)$ $\displaystyle:=\underset{k^{\prime\prime}<k}{\rm
min}\\{C_{2}(E,k^{\prime\prime},k)\\}\,.$

Let $t_{0}$ be a positive integer. Define

(4.7) $\beta(E,k,t_{0}):={\rm
max}\\{(r-1)\mu_{0}(E,r-1),r^{2}\mu_{0}(E,r-1)+t_{0}-C_{3}(E,k),\alpha(E,k)\\}\,.$

######  Remark 4.8.

From the definition it is clear that $\beta(E,k,t_{0})\geqslant\alpha(E,k)$
for all integers $t_{0}\geqslant 1$ and if $t_{1}\geqslant t_{0}\geqslant 1$
then $\beta(E,k,t_{1})\geqslant\beta(E,k,t_{0})$.

######  Lemma 4.9.

Fix a positive integers $t_{0}$ and $k$ such that $k<r$. Let
$d\geqslant\beta(E,k,t_{0})$. Let $S$ be an irreducible component of
$\mathcal{Q}^{\rm tf}_{\rm b}$. Then ${\rm dim}(\mathcal{Q})-{\rm
dim}(S)>t_{0}$ and so also ${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}^{\rm
tf}_{\rm b})>t_{0}$.

###### Proof.

We give $S$ the reduced subscheme structure so that $S$ is an integral scheme.
Let $q\in S$ be a closed point corresponding to a quotient $E\longrightarrow
F$. If $F$ is semistable, then using
$d\geqslant\beta(E,k,t_{0})\geqslant(r-1)\mu_{0}(E,r-1)$ we get

$\mu(F)=\mu_{\rm
min}(F)=\frac{d}{k}\geqslant\frac{d}{r-1}\geqslant\mu_{0}(E,r-1)\,.$

It follows from Lemma 4.1 that $q\in\mathcal{Q}^{\rm tf}_{\rm g}$, which is a
contradiction as $q\in\mathcal{Q}^{\rm tf}_{\rm b}$. Thus, $F$ is not
semistable.

Let $p_{1}:C\times S\longrightarrow C$ denote the projection. Consider the
pullback of the universal quotient from $C\times\mathcal{Q}$ to $C\times S$
and denote it

$p_{1}^{*}E\longrightarrow\mathcal{F}\,.$

From [HL10, Theorem 2.3.2] (existence of relative Harder-Narasimhan
filtration) it follows that there is a dense open subset $U\subset S$ and a
filtration

$0=\mathcal{F}_{0}\subsetneq\mathcal{F}_{1}\subsetneq\ldots\subsetneq\mathcal{F}_{l}=\mathcal{F}$

such that $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ is flat over $U$ and for each
closed point $u\in U$, the sheaf $\mathcal{F}_{i,u}/\mathcal{F}_{i-1,u}$ is
semistable. Consider the quotient
$p_{1}^{*}E\longrightarrow\mathcal{F}_{l}\longrightarrow\mathcal{F}_{l}/\mathcal{F}_{l-1}$.
Denote the kernel by $\mathcal{S}$ so that we have an exact sequence

$0\longrightarrow\mathcal{S}\longrightarrow
p_{1}^{*}E\longrightarrow\mathcal{F}_{l}/\mathcal{F}_{l-1}\longrightarrow 0$

on $C\times U$. Let us denote

$\mathcal{F}^{\prime\prime}:=\mathcal{F}_{l}/\mathcal{F}_{l-1}\,,\qquad\mathcal{F}^{\prime}:=\mathcal{F}_{l-1}\,.$

With this notation we have a commutative diagram of short exact sequences on
$C\times U$

(4.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p_{1}^{*}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\phantom{\,.}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,.}$

In particular, we observe that the map $E\longrightarrow\mathcal{F}_{u}$ can
be obtained as a pushout of the short exact sequence
$0\longrightarrow\mathcal{S}_{u}\longrightarrow
E\longrightarrow\mathcal{F}^{\prime\prime}_{u}\longrightarrow 0$ along the map
$\mathcal{S}_{u}\longrightarrow\mathcal{F}^{\prime}_{u}$.

For a closed point $u\in U$ define

$k^{\prime\prime}:={\rm rk}(\mathcal{F}^{\prime\prime}_{u})\,,\qquad
d^{\prime\prime}:={\rm deg}(\mathcal{F}^{\prime\prime}_{u})\,.$

Then

${\rm rk}(\mathcal{F}^{\prime}_{u})=k-k^{\prime\prime}\,,\qquad{\rm
deg}(\mathcal{F}^{\prime}_{u})=d-d^{\prime\prime}\,.$

The top row of (4.10) defines a map

$\theta:U\longrightarrow{\rm
Quot}_{C/\mathbb{C}}(E,k^{\prime\prime},d^{\prime\prime})\,.$

For ease of notation let us denote $A:={\rm
Quot}_{C/\mathbb{C}}(E,k^{\prime\prime},d^{\prime\prime})$. Let
$\mathcal{S}_{1}$ denote the universal kernel bundle on $C\times A$. Then
$({\rm Id}_{C}\times\theta)^{*}\mathcal{S}_{1}=\mathcal{S}$. The left vertical
arrow of (4.10) defines a map to the relative Quot scheme

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\tilde{\theta}}$$\textstyle{{\rm
Quot}_{C\times
A/A}(\mathcal{S}_{1},k-k^{\prime\prime},d-d^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A}$

One easily checks (using the observation after (4.10)) that the map
$\tilde{\theta}$ is injective on closed points.

Let us compute the dimension of ${\rm Quot}_{C\times
A/A}(\mathcal{S}_{1},k-k^{\prime\prime},d-d^{\prime\prime})$. Consider a
quotient $[E\longrightarrow F^{\prime\prime}]$ which corresponds to a closed
point in $A$. Let $S_{F^{\prime\prime}}$ denote the kernel. It has rank
$r-k^{\prime\prime}$. The fiber of $\pi$ over $[E\longrightarrow
F^{\prime\prime}]$ is the Quot scheme ${\rm
Quot}_{C/\mathbb{C}}(S_{F^{\prime\prime}},k-k^{\prime\prime},d-d^{\prime\prime})$.
Recall from (4.2) the integer $d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})$,
which is the smallest possible degree among all quotients of
$S_{F^{\prime\prime}}$ of rank $k-k^{\prime\prime}$. Thus, if the fiber is
nonempty then we have that

$d-d^{\prime\prime}\geqslant d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})\,.$

By [PR03, Theorem 4.1] it follows that, if the fiber is nonempty then

(4.11) $\displaystyle{\rm dim}({\rm
Quot}_{C/\mathbb{C}}(S_{F^{\prime\prime}},k-k^{\prime\prime},d-d^{\prime\prime}))$
$\displaystyle\leqslant(k-k^{\prime\prime})(r-k)+$
$\displaystyle(d-d^{\prime\prime}-d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}}))(r-k^{\prime\prime})\,.$

We will find a lower bound for $d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})$.
Let $S_{F^{\prime\prime}}\longrightarrow G$ be a quotient such that ${\rm
deg}(G)=d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})$. Then we can form the
pushout $\tilde{G}$ which sits in the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{F^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\phantom{\,.}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,.}$

Since $\tilde{G}$ is a quotient of $E$ of rank $k$, it follows that

${\rm
deg}(\tilde{G})=d^{\prime\prime}+d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})\geqslant
d_{k}(E)\,.$

This shows that $d_{k-k^{\prime\prime}}(S_{F^{\prime\prime}})\geqslant
d_{k}(E)-d^{\prime\prime}$. Combining this with (4.11) yields

(4.12) $\displaystyle{\rm dim}({\rm
Quot}_{C/\mathbb{C}}(S_{F^{\prime\prime}},k-k^{\prime\prime},d-d^{\prime\prime}))\leqslant(k-k^{\prime\prime})(r-k)+(d-d_{k}(E))(r-k^{\prime\prime})\,.$

Again using [PR03, Theorem 4.1] it follows that

(4.13) ${\rm dim}(A)={\rm dim}({\rm
Quot}_{C/\mathbb{C}}(E,k^{\prime\prime},d^{\prime\prime}))\leqslant
k^{\prime\prime}(r-k^{\prime\prime})+(d^{\prime\prime}-d_{k^{\prime\prime}}(E))r\,.$

Combining (4.12) and (4.13), and using (4.6) we get that

$\displaystyle{\rm dim}(U)$ $\displaystyle\leqslant{\rm Quot}_{C\times
A/A}(\mathcal{S}_{1},k-k^{\prime\prime},d-d^{\prime\prime})$
$\displaystyle\leqslant
k^{\prime\prime}(r-k^{\prime\prime})+(d^{\prime\prime}-d_{k^{\prime\prime}}(E))r+(k-k^{\prime\prime})(r-k)+(d-d_{k}(E))(r-k^{\prime\prime})$
$\displaystyle=C_{1}(E,k,k^{\prime\prime})+d(r-k^{\prime\prime})+d^{\prime\prime}r$

From this, Remark 4.3(2) and (4.6) it follows that

(4.14) $\displaystyle{\rm dim}(\mathcal{Q})-{\rm dim}(U)\geqslant
C_{2}(E,k,k^{\prime\prime})+dk^{\prime\prime}-d^{\prime\prime}r\,.$

We claim that
$C_{2}(E,k,k^{\prime\prime})+dk^{\prime\prime}-d^{\prime\prime}r>t_{0}$. If
not, then we have

$\frac{C_{2}(E,k,k^{\prime\prime})+dk^{\prime\prime}-t_{0}}{r}\leqslant
d^{\prime\prime}\,.$

But this yields

(4.15)
$\frac{C_{3}(E,k)+d-t_{0}}{r^{2}}\leqslant\frac{C_{2}(E,k,k^{\prime\prime})+d-t_{0}}{r^{2}}<\frac{C_{2}(E,k,k^{\prime\prime})+dk^{\prime\prime}-t_{0}}{rk^{\prime\prime}}\leqslant\frac{d^{\prime\prime}}{k^{\prime\prime}}\,.$

Let $u\in U$ be a closed point. Then $\mu_{\rm
min}(\mathcal{F}_{u})=d^{\prime\prime}/k^{\prime\prime}$. By the assumption on
$d$ we have that

$d\geqslant\beta(E,k,t_{0})\geqslant r^{2}\mu_{0}(E,r-1)+t_{0}-C_{3}(E,k)\,.$

Using this and (4.15) gives

$\mu_{0}(E,r-1)\leqslant\frac{C_{3}(E,k)+d-t_{0}}{r^{2}}<\frac{d^{\prime\prime}}{k^{\prime\prime}}=\mu_{\rm
min}(\mathcal{F}_{u})\,.$

It follows from Lemma 4.1 that $H^{1}(E^{\vee}\otimes\mathcal{F}_{u})=0$, that
is, $u\in\mathcal{Q}^{\rm tf}_{\rm g}$. But this is a contradiction as
$U\subset\mathcal{Q}^{\rm tf}_{\rm b}$. Thus, it follows from (4.14) that

${\rm dim}(\mathcal{Q})-{\rm dim}(S)\geqslant
C_{2}(E,k,k^{\prime\prime})+dk^{\prime\prime}-d^{\prime\prime}r>t_{0}\,.$

This completes the proof of the Lemma. ∎

##  5\. Locus of quotients which are not torsion free

For a sheaf $F$, denote the torsion subsheaf of $F$ by ${\rm Tor}(F)$. For an
integer $i\geqslant 1$ define the locally closed subset

(5.1) $Z_{i}:=\\{[q:E\longrightarrow F]\in\mathcal{Q}\,|\,{\rm deg}({\rm
Tor}(F))=i\\}\,.$

We now estimate the dimension of $Z_{i}$ and $(Z_{i})_{b}$ (recall the
definition of $(Z_{i})_{b}$ from (4.5)).

######  Lemma 5.2.

With notation as above we have

  1. (1)

Assume that $d-i\geqslant\alpha(E,k)$ (see Remark 4.3). Then $Z_{i}$ is
irreducible and ${\rm dim}(Z_{i})={\rm dim}(\mathcal{Q})-ki$. Moreover,
$\bar{Z}_{i}\supset\bigcup_{j\geqslant i}Z_{j}$.

  2. (2)

Let $t_{1}$ be a positive integer. If $d-i\geqslant\beta(E,k,t_{1})$ (see
(4.7) for definition of $\beta$) then ${\rm dim}(Z_{i})-{\rm dim}((Z_{i})_{\rm
b})>t_{1}$.

  3. (3)

If $d-i\geqslant\beta(E,k,t_{1})$ then ${\rm dim}(\mathcal{Q})-{\rm
dim}((Z_{i})_{\rm b})>t_{1}+ki$.

###### Proof.

Consider the Quot scheme ${\rm Quot}_{C/\mathbb{C}}(E,k,d-i)$. For ease of
notation we denote $A={\rm Quot}_{C/\mathbb{C}}(E,k,d-i)$. Let

$0\longrightarrow\mathscr{S}\longrightarrow
p_{1}^{*}E\longrightarrow\mathscr{F}\longrightarrow 0$

be the universal quotient on $C\times A$. Consider the relative Quot scheme

(5.3) ${\rm Quot}_{C\times
A/A}(\mathscr{S},0,i)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}A\,.$

There is a map

(5.4) ${\rm Quot}_{C\times
A/A}(\mathscr{S},0,i)\stackrel{{\scriptstyle\pi^{\prime}}}{{\longrightarrow}}\mathcal{Q}$

whose image consists of precisely those quotients $[E\longrightarrow F]$ for
which ${\rm deg}({\rm Tor}(F))\geqslant i$. Recall the locus $A^{\rm tf}$ from
(4.5). One checks easily that

(5.5) $\pi^{\prime-1}(Z_{i})=\pi^{-1}(A^{\rm tf})\,.$

In fact, one easily checks that $\pi^{\prime}:\pi^{-1}(A^{\rm
tf})\longrightarrow Z_{i}$ is a bijection on points and so they have the same
dimension. As $d-i\geqslant\alpha(E,k)$, by Remark 4.3(2), it follows that $A$
is irreducible of dimension

${\rm dim}(A)=r(d-i)-ek-k(r-k)(g-1)\,.$

By Remark 4.3(3), it follows that $A^{\rm tf}$ is a dense open subset of $A$.
If $[E\longrightarrow F]\in A$ is a quotient, let $S_{F}$ denote the kernel.
The fiber of $\pi$ over this point is the Quot scheme ${\rm
Quot}_{C/\mathbb{C}}(S_{F},0,i)$, which is irreducible and has dimension
$(r-k)i$. From this it follows that ${\rm Quot}_{C\times
A/A}(\mathscr{S},0,i)$ is irreducible of dimension ${\rm
dim}(\mathcal{Q})-ki$. Thus, the open set $\pi^{-1}(A^{\rm tf})$ also has the
same dimension and is irreducible. As this open subset dominates $Z_{i}$, the
claim about the irreducibility and dimension of $Z_{i}$ follows. We have
already observed that the image of $\pi^{\prime}$ is the locus
$\bigcup_{j\geqslant i}Z_{j}$. As $\pi^{-1}(A^{\rm tf})$ is dense in ${\rm
Quot}_{C\times A/A}(\mathscr{S},0,i)$, the proof of (1) is complete.

To prove the second assertion, note that

$H^{1}(E^{\vee}\otimes F)=H^{1}(E^{\vee}\otimes(F/{\rm Tor}(F)))\,.$

One checks easily that

(5.6) $\pi^{\prime-1}((Z_{i})_{\rm b})=\pi^{-1}(A^{\rm tf}_{\rm b})\,.$

As $\pi$ has constant fiber dimension, we see

${\rm dim}(A^{\rm tf})-{\rm dim}(A^{\rm tf}_{\rm b})={\rm dim}(\pi^{-1}(A^{\rm
tf}))-{\rm dim}(\pi^{-1}(A^{\rm tf}_{\rm b}))\,.$

By applying Lemma 2.2 to the map $\pi^{\prime}$, and using (5.5) and (5.6), we
get

$\displaystyle{\rm dim}(A^{\rm tf})-{\rm dim}(A^{\rm tf}_{\rm b})$
$\displaystyle={\rm dim}(\pi^{-1}(A^{\rm tf}))-{\rm dim}(\pi^{-1}(A^{\rm
tf}_{\rm b}))$ $\displaystyle={\rm dim}(\pi^{\prime-1}(Z_{i}))-{\rm
dim}(\pi^{\prime-1}((Z_{i})_{\rm b}))\leqslant{\rm dim}(Z_{i})-{\rm
dim}((Z_{i})_{\rm b})\,.$

As $d-i\geqslant\beta(E,k,t_{1})\geqslant\alpha(E,k)$ it follows from Remark
4.3(2) that ${\rm Quot}_{C/\mathbb{C}}(E,k,d-i)$ is irreducible and so ${\rm
dim}({\rm Quot}_{C/\mathbb{C}}(E,k,d-i))={\rm dim}({\rm
Quot}_{C/\mathbb{C}}(E,k,d-i)^{\rm tf})$. By Lemma 4.9 it follows that

${\rm dim}(A^{\rm tf})-{\rm dim}(A^{\rm tf}_{\rm b})={\rm dim}({\rm
Quot}_{C/\mathbb{C}}(E,k,d-i)^{\rm tf})-{\rm dim}({\rm
Quot}_{C/\mathbb{C}}(E,k,d-i)^{\rm tf}_{\rm b})>t_{1}\,.$

This proves that ${\rm dim}(Z_{i})-{\rm dim}((Z_{i})_{\rm b})>t_{1}$. This
proves (2).

Assertion (3) of the Lemma follows easily using the first two. ∎

##  6\. Flatness of det

We begin by showing that when $d\gg 0$, $\mathcal{Q}$ is a local complete
intersection. This result seems well known to experts (see [BDW96, Theorem
1.6] and the para following it), however, we include it as we could not find a
precise reference.

######  Lemma 6.1.

Let $d\geqslant\alpha(E,k)$. Then $\mathcal{Q}$ is a local complete
intersection scheme. In particular, it is Cohen-Macaulay.

###### Proof.

By Remark 4.3(2), $\mathcal{Q}$ is irreducible and so ${\rm
dim}_{q}(\mathcal{Q})$ is independent of the closed point $q\in\mathcal{Q}$.
Let $\mathcal{F}$ denote the universal quotient and let $\mathcal{K}$ denote
the universal kernel on $C\times\mathcal{Q}$. For a closed point
$q\in\mathcal{Q}$ we shall denote the restrictions of these sheaves to
$C\times q$ by $\mathcal{K}_{q}$ and $\mathcal{F}_{q}$. The sheaf
$\mathcal{K}$ is locally free on $C\times\mathcal{Q}$. It follows that
$\mathcal{K}^{\vee}\otimes\mathcal{F}$ is flat over $\mathcal{Q}$, and so the
Euler characteristic of $\mathcal{K}^{\vee}_{q}\otimes\mathcal{F}_{q}$ is
constant, call it $\chi$. As $\mathcal{Q}^{\rm tf}_{\rm g}$ is nonempty, let
$q\in\mathcal{Q}^{\rm tf}_{\rm g}$ be a closed point. As
$h^{1}(\mathcal{K}^{\vee}_{q}\otimes\mathcal{F}_{q})=0$, it follows from
[HL10, Proposition 2.2.8] that

${\rm
dim}_{q}(\mathcal{Q})=h^{0}(\mathcal{K}^{\vee}_{q}\otimes\mathcal{F}_{q})=h^{0}(\mathcal{K}^{\vee}_{q}\otimes\mathcal{F}_{q})-h^{1}(\mathcal{K}^{\vee}_{q}\otimes\mathcal{F}_{q})=\chi\,.$

Let $t\in\mathcal{Q}$ be a closed point. We already observed that ${\rm
dim}_{t}(\mathcal{Q})$ is independent of the closed point $t\in\mathcal{Q}$
and so is equal to $\chi$. It follows that for all closed points
$t\in\mathcal{Q}$ we have

${\rm
dim}_{t}(\mathcal{Q})=\chi=h^{0}(\mathcal{K}^{\vee}_{t}\otimes\mathcal{F}_{t})-h^{1}(\mathcal{K}^{\vee}_{t}\otimes\mathcal{F}_{t})\,.$

By [HL10, Proposition 2.2.8] it follows that the space $\mathcal{Q}$ is a
local complete intersection at any closed point and so is also Cohen-Macaulay.
∎

######  Lemma 6.2.

Fix a positive integer $t_{0}$. Let $i_{0}$ be the smallest integer such that
$ki_{0}>g(C)+t_{0}$. If $d\geqslant\beta(E,k,g(C)+t_{0})+i_{0}$ then ${\rm
dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}_{b})>g(C)+t_{0}$.

###### Proof.

First observe that we can write

$\mathcal{Q}=\mathcal{Q}^{\rm tf}\bigsqcup_{i\geqslant 1}Z_{i}\,.$

Only finitely many indices $i$ appear. In fact, $i$ can be at most
$d-d_{k}(E)$, see (4.2). In view of this we get

$\mathcal{Q}_{\rm b}=\mathcal{Q}^{\rm tf}_{\rm b}\bigsqcup_{i\geqslant
1}(Z_{i})_{\rm b}\,.$

By Lemma 4.9, since $d\geqslant\beta(E,k,g(C)+t_{0})$ we have

${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}^{\rm tf}_{\rm b})>g(C)+t_{0}\,.$

If $1\leqslant i\leqslant i_{0}$ then $d-i\geqslant
d-i_{0}\geqslant\beta(E,k,g(C)+t_{0})$, and so by Lemma 5.2(3) we get

${\rm dim}(\mathcal{Q})-{\rm dim}((Z_{i})_{\rm b})>g(C)+t_{0}+ki\,.$

By Lemma 5.2(1) we also get that $\bar{Z}_{i_{0}}\supset\bigcup_{j\geqslant
i_{0}}Z_{j}$. For $j\geqslant i_{0}$,

${\rm dim}((Z_{j})_{\rm b})\leqslant{\rm dim}(Z_{j})\leqslant{\rm
dim}(Z_{i_{0}})\leqslant{\rm dim}(\mathcal{Q})-ki_{0}\,.$

This shows that for $j\geqslant i_{0}$ we have

${\rm dim}(\mathcal{Q})-{\rm dim}((Z_{j})_{\rm b})\geqslant
ki_{0}>g(C)+t_{0}\,.$

Combining these shows that ${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}_{\rm
b})>g(C)+t_{0}$. This completes the proof of the Lemma. ∎

######  Theorem 6.3.

Recall the map det defined in (2.5).

  1. (1)

Let $n_{0}$ be the smallest integer such that $kn_{0}>g(C)+1$. Let
$d\geqslant\beta(E,k,g(C)+1)+n_{0}$. Then ${\rm
det}:\mathcal{Q}\longrightarrow{\rm Pic}^{d}(C)$ is a flat map. Further,
$\mathcal{Q}$ is an integral and normal variety.

  2. (2)

Let $n_{1}$ be the smallest integer such that $kn_{1}>g(C)+3$. Let
$d\geqslant\beta(E,k,g(C)+3)+n_{1}$. Then $\mathcal{Q}$ is locally factorial.

###### Proof.

Let $q\in\mathcal{Q}$ be a closed point and let $K$ denote the kernel of the
quotient $q$. Then we have a short exact sequence

$0\longrightarrow K\longrightarrow E\longrightarrow F\longrightarrow 0\,.$

Applying ${\rm Hom}(-,F)$ and using Lemma 2.7 we get the following diagram, in
which the top row is exact.

(6.4) |  |
---|---|---
$\textstyle{{\rm
Hom}(K,F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d({\rm
det})_{q}}$$\textstyle{{\rm
Ext}^{1}(F,F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{tr}$$\textstyle{{\rm
Ext}^{1}(E,F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Ext}^{1}(K,F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H^{1}(C,\mathcal{O}_{C})}$

If $H^{1}(E^{\vee}\otimes F)=0$ then we make the following two observations.
First observe that it follows $H^{1}(K^{\vee}\otimes F)=0$, which shows that
$\mathcal{Q}_{\rm g}$ is contained in the smooth locus of $\mathcal{Q}$, by
[HL10, Proposition 2.2.8]. Second observe that the map ${\rm
Hom}(K,F)\longrightarrow{\rm Ext}^{1}(F,F)$ will be surjective. As ${\rm
Ext}^{1}(F,F)\longrightarrow H^{1}(C,\mathcal{O}_{C})$ is surjective, it
follows that if $H^{1}(E^{\vee}\otimes F)=0$ then the diagonal map in the
above diagram is surjective. However, the diagonal map is precisely the
differential of det at the point $q$. As ${\mathcal{Q}}_{\rm g}$ and ${\rm
Pic}^{d}(C)$ are smooth, it follows that the restriction of det to
${\mathcal{Q}}_{\rm g}$ is a smooth morphism and so also flat.

In view Lemma 6.1 and [Mat86, Theorem 23.1] or [Stk, Tag 00R4], to prove the
first assertion of the theorem, it suffices to show that the fibers of det
have constant dimension. To show this we will use Lemma 2.1. Assume
$d\geqslant\beta(E,k,g(C)+1)+n_{0}$. Applying Lemma 6.2 we get

${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}_{\rm b})>g(C)+1\,.$

Applying Lemma 2.1(1) we get that det is flat. We observed in Lemma 6.1 that
$\mathcal{Q}$ is a Cohen-Macaulay scheme and so it satisfies Serre’s condition
$S_{2}$. The open subset $\mathcal{Q}_{\rm g}$ is smooth. As $\mathcal{Q}_{\rm
b}=\mathcal{Q}\setminus\mathcal{Q}_{\rm g}$, it follows that $\mathcal{Q}$
satisfies Serre’s conditions $R_{1}$. Thus, $\mathcal{Q}$ is an integral and
normal variety. This proves (1).

Now we prove (2). Assume $d\geqslant\beta(E,k,g(C)+3)+n_{1}$. Applying Lemma
6.2 we get

${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}_{\rm b})>g(C)+3\,.$

This implies that the singular locus has codimension 4 or more. Now we use a
result of Grothendieck which states that if $R$ is a local ring that is a
complete intersection in which the singular locus has codimension 4 or more,
then $R$ is a UFD. We refer the reader to [Gro05], [Cal94], [AH20, Theorem
1.4]. This implies that $\mathcal{Q}$ is locally factorial. The proof of the
theorem is now complete. ∎

##  7\. Locus of stable quotients and Picard group of $\mathcal{Q}$

### 7.1.

We begin by explaining a result from [Bho99] that we need. Assume one of the
following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Let $d\geqslant\alpha(E,k)$. Fix a closed point $P\in C$. For a closed point
$q\in\mathcal{Q}$, let $[E\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\mathcal{F}_{q}]$ denote the quotient corresponding to
this closed point. We may choose $n\gg 0$ such that for all
$q\in\mathcal{Q}^{\rm tf}_{\rm g}$ we have $H^{1}(C,\mathcal{F}_{q}(nP))=0$
and $\mathcal{F}_{q}(nP)$ is globally generated. As $d\geqslant\alpha(E,k)$,
by Remark 4.3, it follows that $\mathcal{Q}^{\rm tf}_{\rm g}$ is irreducible,
and so $h^{0}(C,\mathcal{F}_{q}(nP))$ is independent of $q$. Let

(7.1) $N:=h^{0}(C,\mathcal{F}_{q}(nP))$

and consider the Quot scheme ${\rm
Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus N},k,d+kn)$. Let
$\mathcal{G}^{\prime}$ denote the universal quotient on $C\times{\rm
Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus N},k,d+kn)$. Let $R\subset{\rm
Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus N},k,d+kn)$ be the open subset
containing closed points $[x:\mathcal{O}_{C}^{\oplus
N}\longrightarrow\mathcal{G}^{\prime}_{x}]$ such that
$\mathcal{G}^{\prime}_{x}$ is torsion free,
$H^{1}(C,\mathcal{G}^{\prime}_{x})=0$ and the quotient map
$\mathcal{O}_{C}^{\oplus N}\longrightarrow\mathcal{G}^{\prime}_{x}$ induces an
isomorphism
$\mathbb{C}^{N}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(C,\mathcal{G}^{\prime}_{x})$.
This is the space $R$ in [Bho99, page 246, Proposition 1.2], see [Bho99, page
246, Notation 1.1]. The space $R$ is a smooth equidimensional scheme. Let
$R^{s}$ denote the open subset of $R$ consisting of closed points $x$ for
which $\mathcal{G}^{\prime}_{x}$ is stable. In [Bho99, page 246, Proposition
1.2] it is proved that ${\rm dim}(R)-{\rm dim}(R\setminus R^{s})\geqslant 2$.

Let

$\displaystyle p_{1}:C\times{\rm Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus
N},k,d+kn)$ $\displaystyle\longrightarrow C$ $\displaystyle p_{2}:C\times{\rm
Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus N},k,d+kn)$
$\displaystyle\longrightarrow{\rm Quot}_{C/\mathbb{C}}(\mathcal{O}_{C}^{\oplus
N},k,d+kn)$

denote the projections. Let

$\mathcal{G}:=\mathcal{G}^{\prime}\otimes p_{1}^{*}(\mathcal{O}_{C}(-np))\,.$

Let $R^{\prime}\subset R$ be the open subset containing closed points $x$ for
which $H^{1}(C,E^{\vee}\otimes\mathcal{G}_{x})=0$. By Cohomology and Base
change theorem it follows that $p_{2*}(p_{1}^{*}E^{\vee}\otimes\mathcal{G})$
is locally free on $R^{\prime}$. The fiber over a point $x\in R^{\prime}$ is
isomorphic to the vector space ${\rm Hom}(E,\mathcal{G}_{x})$. Consider the
projective bundle

(7.2)
$\mathbb{P}(p_{2*}(p_{1}^{*}E^{\vee}\otimes\mathcal{G})^{\vee})\stackrel{{\scriptstyle\Theta}}{{\longrightarrow}}R^{\prime}\,.$

The fiber of $\Theta$ over a point $x\in R^{\prime}$ is the space of lines in
the vector space ${\rm Hom}(E,\mathcal{G}_{x})$. For ease of notation we
denote $\mathbb{P}(p_{2*}(p_{1}^{*}E^{\vee}\otimes\mathcal{G})^{\vee})$ by
$\mathbb{P}$. Denote the projection maps from $C\times\mathbb{P}$ by

$p^{\prime}_{1}:C\times\mathbb{P}\longrightarrow C\,,\qquad
p^{\prime}_{2}:C\times\mathbb{P}\longrightarrow\mathbb{P}\,.$

Consider the following Cartesian square

$\textstyle{C\times\mathbb{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Theta}}$$\scriptstyle{p^{\prime}_{2}}$$\textstyle{C\times
R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathbb{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{R^{\prime}}$

Let $\mathcal{O}(1)$ denote the tautological line bundle on $\mathbb{P}$. Then
we have a map of sheaves on $C\times\mathbb{P}$

(7.3) $p^{\prime*}_{1}E\longrightarrow\tilde{\Theta}^{*}\mathcal{G}\otimes
p^{\prime*}_{2}\mathcal{O}(1)\,.$

A closed point $v\in\mathbb{P}$ corresponds to the closed point $\Theta(v)\in
R^{\prime}$ and a line in the vector space $w_{v}\in{\rm
Hom}(E,\mathcal{G}_{\Theta(v)})$. The restriction of (7.3) to $C\times v$
gives the map $w_{v}:E\longrightarrow\mathcal{G}_{\Theta(v)}$. Let
$\mathbb{U}\subset\mathbb{P}$ denote the open subset parametrizing points $v$
such that $w_{v}$ is surjective. On $C\times\mathbb{U}$ we have a surjection

(7.4) $p^{\prime*}_{1}E\longrightarrow\tilde{\Theta}^{*}\mathcal{G}\otimes
p^{\prime*}_{2}\mathcal{O}(1)\,.$

This defines a morphism

(7.5) $\Psi:\mathbb{U}\longrightarrow\mathcal{Q}^{\rm tf}_{\rm g}\,.$

######  Lemma 7.6.

$\Psi$ is surjective on closed points.

###### Proof.

Let $[q:E\longrightarrow\mathcal{F}_{q}]\in\mathcal{Q}^{\rm tf}_{\rm g}$ be a
closed point. By our choice of $n$ and $N$ (see (7.1)), we have that
$\mathcal{F}_{q}(nP)$ is globally generated and
$N=h^{0}(C,\mathcal{F}_{q}(nP))$. A choice of basis for
$H^{0}(C,\mathcal{F}_{q}(nP))$ and the canonical map
$[H^{0}(C,\mathcal{F}_{q}(nP))\otimes\mathcal{O}_{C}\longrightarrow\mathcal{F}_{q}(nP)]$
corresponds to a closed point $x\in R$. This point is in fact contained in the
subset $R^{\prime}$. It is clear that the fiber $\Theta^{-1}(x)$ contains the
line corresponding to $q\in{\rm Hom}(E,\mathcal{F}_{q})$. It follows that
$\Psi$ is surjective on closed points. ∎

Now further assume $d\geqslant{\rm max}\\{\alpha(E,k),k\mu_{0}(E,k)\\}$. By
Lemma 4.1 we have $H^{1}(C,E^{\vee}\otimes\mathcal{G}_{q})=0$ for $q\in
R^{s}$. Thus, we have inclusions of open sets $R^{s}\subset R^{\prime}\subset
R$. Let $\mathbb{P}^{s}\subset\mathbb{P}$ denote the inverse image of $R^{s}$
under the map $\Theta$. Similarly, let $\mathbb{U}^{s}\subset\mathbb{U}$
denote the inverse image of $R^{s}$ under the restriction of $\Theta$ to
$\mathbb{U}$. Let

(7.7) $\mathcal{Q}^{\rm s}:=\\{[E\longrightarrow
F]\in\mathcal{Q}\,|\,\text{$F$ is stable}\\}.$

As $d\geqslant k\mu_{0}(E,k)$, by Lemma 4.1 we have $H^{1}(C,E^{\vee}\otimes
F)=0$ for $[E\longrightarrow F]\in\mathcal{Q}^{s}$. It follows that
$\mathcal{Q}^{s}\subset\mathcal{Q}^{\rm tf}_{\rm g}$. It is easily checked
that

(7.8) $\Psi^{-1}(\mathcal{Q}^{s})=\mathbb{U}^{s}\,.$

The group ${\rm PGL}(N)$ acts freely on $\mathbb{P}^{s}$ and leaves the open
subset $\mathbb{U}^{s}$ invariant. Consider the trivial action of ${\rm
PGL}(N)$ on $\mathcal{Q}^{s}$. Then the restriction
$\Psi:\mathbb{U}^{s}\longrightarrow\mathcal{Q}^{s}$ is ${\rm
PGL}(N)$-equivariant. It is clear that the restriction of the map
$\Theta:\mathbb{P}^{s}\longrightarrow R^{s}$ is also ${\rm
PGL}(N)$-equivariant. Let $M^{s}_{k,d+kn}$ denote the moduli space of stable
bundles of rank $k$ and degree $d+kn$. Then $M^{s}_{k,d+kn}$ is the GIT
quotient

$\psi:R^{s}{\longrightarrow}R^{s}/\\!\\!/{\rm PGL}(N)=M^{s}_{k,d+kn}\,.$

Let $p_{C}:C\times\mathcal{Q}\longrightarrow C$ denote the projection and let
$p_{1}^{*}E\longrightarrow\mathcal{F}$ denote the universal quotient on
$C\times\mathcal{Q}$. The sheaf
$p_{C}^{*}\mathcal{O}_{C}(nP)\otimes\mathcal{F}$ on $C\times\mathcal{Q}^{s}$
defines a morphism
$\mathcal{Q}^{s}\stackrel{{\scriptstyle\theta}}{{\longrightarrow}}M^{s}_{k,d+kn}$.
One easily checks that we have the following commutative diagram, in which all
arrows are surjective on closed points

(7.9)
$\textstyle{\mathbb{U}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Theta}$$\textstyle{\mathcal{Q}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{R^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{M^{s}_{k,d+kn}\,.}$

The map $\psi$ is a principal ${\rm PGL}(N)$-bundle. For a closed point $x\in
R^{s}$, the points in the fiber $\Theta^{-1}(x)$ are in bijection with the
points in the fiber $\theta^{-1}(\psi(x))$. The natural map from
$\mathbb{U}^{s}$ to the Cartesian product of $\psi$ and $\theta$ is a
bijective map of smooth varieties and hence an isomorphism. This shows that
the above diagram is Cartesian.

In this section we shall compute the Picard group of $\mathcal{Q}$ when $d\gg
0$. As we saw in Theorem 6.3, $\mathcal{Q}$ is locally factorial and so the
Picard group is isomorphic to the divisor class group. Let
$CH^{1}(\mathcal{Q})$ denote the divisor class group of $\mathcal{Q}$. We
shall first show that
$CH^{1}(\mathcal{Q})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}CH^{1}(\mathcal{Q}^{s})$
and then use the diagram (7.9) to compute $CH^{1}(\mathcal{Q}^{s})$.

In the following Lemma we shall use the fact that $R$ is irreducible.

######  Lemma 7.10.

Assume one of the following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Also assume $d\geqslant{\rm
max}\\{\alpha(E,k)+1,k\mu_{0}(E,k),\beta(E,k,1)\\}$. Then the map
$CH^{1}(\mathcal{Q}){\longrightarrow}CH^{1}(\mathcal{Q}^{s})$ is an
isomorphism.

###### Proof.

Recall the definition of $Z_{1}$ from (5.1) and observe that $\mathcal{Q}^{\rm
tf}=\mathcal{Q}\setminus\bar{Z}_{1}$. Taking $i=1$ in Lemma 5.2(1) we get
${\rm dim}(\mathcal{Q})-{\rm dim}(\bar{Z}_{1})\geqslant k$. Since $k\geqslant
2$, it follows that $CH^{1}(\mathcal{Q})=CH^{1}(\mathcal{Q}^{\rm tf})$.

By Lemma 4.9 it follows that

${\rm dim}(\mathcal{Q}^{\rm tf})-{\rm dim}(\mathcal{Q}^{\rm tf}_{\rm b})={\rm
dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}^{\rm tf}_{\rm b})>1\,.$

Observe that $\mathcal{Q}^{\rm tf}_{\rm g}=\mathcal{Q}^{\rm
tf}\setminus\mathcal{Q}^{\rm tf}_{\rm b}$. It follows that
$CH^{1}(\mathcal{Q}^{\rm tf})=CH^{1}(\mathcal{Q}^{\rm tf}_{\rm g})$.

We had observed earlier that $\mathcal{Q}^{s}\subset\mathcal{Q}^{\rm tf}_{\rm
g}$. To prove the Lemma it suffices to show that

${\rm dim}(\mathcal{Q}^{\rm tf}_{\rm g})-{\rm dim}(\mathcal{Q}^{\rm tf}_{\rm
g}\setminus\mathcal{Q}^{s})>1\,.$

We will now show this.

As $d\geqslant k\mu_{0}(E,k)$, by Lemma 4.1 we have
$H^{1}(C,E^{\vee}\otimes\mathcal{G}_{q})=0$ for $q\in R^{s}$. We have already
checked above, see (7.8), that $\Psi^{-1}(\mathcal{Q}^{s})=\mathbb{U}^{s}$. By
[Bho99, Proposition 1.2] we have ${\rm dim}(R)-{\rm dim}(R\setminus
R^{s})\geqslant 2$ and so also ${\rm dim}(R^{\prime})-{\rm
dim}(R^{\prime}\setminus R^{s})\geqslant 2$. As $R^{\prime}$ is integral, so
is $\mathbb{U}$. As the map $\Theta$ is flat, it follows using Lemma 2.2
(applied to the map $\Psi:\mathbb{U}\longrightarrow\mathcal{Q}^{\rm tf}_{\rm
g}$) that

$2\leqslant{\rm dim}(R^{\prime})-{\rm dim}(R^{\prime}\setminus R^{s})={\rm
dim}(\mathbb{U})-{\rm dim}(\mathbb{U}\setminus\mathbb{U}^{s})\leqslant{\rm
dim}(\mathcal{Q}^{\rm tf}_{\rm g})-{\rm dim}(\mathcal{Q}^{\rm tf}_{\rm
g}\setminus\mathcal{Q}^{s})\,.$

This completes the proof of the Lemma. ∎

######  Lemma 7.11.

Let $r-k\geqslant 2$. Let $d\geqslant{\rm
max}\\{\alpha(E,k),k\mu_{0}(E,k)+k\\}$. The natural map
$CH^{1}(\mathbb{P}^{s})\longrightarrow CH^{1}(\mathbb{U}^{s})$ is an
isomorphism.

###### Proof.

It suffices to show that ${\rm dim}(\mathbb{P}^{s})-{\rm
dim}(\mathbb{P}^{s}\setminus\mathbb{U}^{s})\geqslant 2$. Let
$[x:\mathcal{O}_{C}^{\oplus N}\longrightarrow F]$ be a quotient corresponding
to a closed point $x\in R^{s}$. It suffices to show that ${\rm
dim}(\Theta^{-1}(x))-{\rm dim}(\Theta^{-1}(x)\setminus\mathbb{U}^{s})\geqslant
2$ for every closed point $x\in R^{s}$. We now show this.

The space $\Theta^{-1}(x)$ is the space $\mathbb{P}({\rm Hom}(E,F)^{\vee})$
parametrizing lines in the vector space ${\rm Hom}(E,F)$. Let $c\in C$ be a
closed point. As $F$ is stable, note $\mu_{\rm min}(F(-c))=\mu(F)-1$. As
$d\geqslant k\mu_{0}(E,k)+k$, it follows that

$\mu_{\rm min}(F(-c))=\mu(F)-1=\frac{d-k}{k}\geqslant\mu_{0}(E,k)\,.$

Let $p_{i}$ denote the projections from $C\times C$. Let $\Delta$ denote the
diagonal in $C\times C$. Consider the short exact sequence of sheaves on
$C\times C$ given by

$0\longrightarrow p_{1}^{*}(E^{\vee}\otimes F)(-\Delta)\longrightarrow
p_{1}^{*}(E^{\vee}\otimes F)\longrightarrow\Delta_{*}(E^{\vee}\otimes
F)\longrightarrow 0\,.$

By Lemma 4.1 we have $H^{1}(E^{\vee}\otimes F(-c))=0$. Applying $p_{2*}$ to
the above, we get that the sheaf

$\mathcal{V}:=p_{2*}(p_{1}^{*}(E^{\vee}\otimes F)(-\Delta))\,,$

which is locally free on $C$, sits in a short exact sequence

$0\longrightarrow\mathcal{V}\longrightarrow{\rm
Hom}(E,F)\otimes\mathcal{O}_{C}\longrightarrow E^{\vee}\otimes
F\longrightarrow 0\,.$

The restriction of the above sequence to a closed point $c\in C$ gives the
short exact sequence of vector spaces

(7.12) $0\longrightarrow{\rm Hom}(E,F(-c))\longrightarrow{\rm
Hom}(E,F)\longrightarrow{\rm Hom}(E|_{c},F|_{c})\longrightarrow 0\,.$

Consider the closed subset
$\mathbb{P}(\mathcal{V}^{\vee})\subset\mathbb{P}({\rm Hom}(E,F)^{\vee})\times
C$. Let $T\subset\mathbb{P}({\rm Hom}(E,F)^{\vee})$ denote the image of
$\mathbb{P}(\mathcal{V}^{\vee})$ under the projection map

$\mathbb{P}({\rm Hom}(E,F)^{\vee})\times C\longrightarrow\mathbb{P}({\rm
Hom}(E,F)^{\vee})\,.$

Then $T$ is a closed subset and set theoretically it is the union

$T=\bigcup_{c\in C}\mathbb{P}({\rm Hom}(E,F(-c))^{\vee})\,.$

As $r-k\geqslant 2$ we have $rk\geqslant(k+2)k>2$. Therefore,

(7.13) ${\rm dim}(\mathbb{P}({\rm Hom}(E,F)^{\vee}))-{\rm dim}(T)\geqslant{\rm
dim}(\mathbb{P}({\rm Hom}(E,F)^{\vee}))-{\rm
dim}(\mathbb{P}(\mathcal{V}^{\vee}))=rk-1\geqslant 2\,.$

Let $V$ denote the open set $\mathbb{P}({\rm Hom}(E,F)^{\vee})\setminus T$.
Let $\mathcal{O}(1)$ denote the restriction of the tautological bundle on
$\mathbb{P}({\rm Hom}(E,F)^{\vee})$ to $V$. Let $p_{C}$ denote the projection
from $C\times V$ to $C$ and let $p_{V}$ denote the projection to $V$. Consider
the canonical map of sheaves on $C\times V$

(7.14) $p_{C}^{*}(E\otimes F^{\vee})\longrightarrow{\rm
Hom}(E,F)^{\vee}\otimes\mathcal{O}_{C\times V}\longrightarrow
p_{V}^{*}\mathcal{O}(1)\,.$

Let $\varphi\neq 0$ be an element in ${\rm Hom}(E,F)$ such that the line
$[\varphi]$ it defines is in $V$. For a point $c\in C$, the map (7.14)
restricted to $(c,[\varphi])$ is adjoint to the map
$E|_{c}\stackrel{{\scriptstyle\varphi|_{c}}}{{\longrightarrow}}F|_{c}$. As
$[\varphi]\in V$, it follows that the map
$E|_{c}\stackrel{{\scriptstyle\varphi|_{c}}}{{\longrightarrow}}F|_{c}$ is
nonzero, and so it follows that the restriction of (7.14) to $(c,[\varphi])$
is nonzero, that is, $E|_{c}\otimes F|_{c}^{\vee}\longrightarrow\mathbb{C}$ is
nonzero and hence surjective. This proves that the map (7.14) is surjective.
This defines a map $C\times
V\stackrel{{\scriptstyle\kappa}}{{\longrightarrow}}\mathbb{P}(E\otimes
F^{\vee})$ which sits in a commutative diagram

$\textstyle{C\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{\mathbb{P}(E\otimes
F^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C}$

The restriction of the map $\kappa$ over a point $c\in C$ is the composite map
below, where the second arrow is obtained using (7.12)

$V\longrightarrow\mathbb{P}({\rm Hom}(E,F)^{\vee})\setminus\mathbb{P}({\rm
Hom}(E,F(-c))^{\vee})\longrightarrow\mathbb{P}({\rm
Hom}(E|_{c},F|_{c})^{\vee})\,.$

The second arrow is a surjective flat map and the first arrow is an open
immersion. It follows that the composite is a flat map and hence has constant
fiber dimension. It follows that the map $\kappa$ has constant fiber
dimension, and so using [Mat86, Theorem 23.1] or [Stk, Tag 00R4] we see that
$\kappa$ is a flat map. Consider the canonical map

$\pi^{*}E\longrightarrow\pi^{*}F\otimes\mathcal{O}_{\mathbb{P}(E\otimes
F^{\vee})}(1)$

on $\mathbb{P}(E\otimes F^{\vee})$ and let $Z$ denote the support of the
cokernel. The set $Z\cap\pi^{-1}(c)$ is precisely the locus of non-surjective
maps in $\mathbb{P}(E|_{c}\otimes F|_{c}^{\vee})$. By [ACGH85, Chapter II, §2,
page 67] we have that codimension of $Z\cap\pi^{-1}(c)$ in
$\mathbb{P}(E|_{c}\otimes F|_{c}^{\vee})$ is $r-k+1$. It follows that
codimension of $Z$ in $\mathbb{P}(E\otimes F^{\vee})$ is $r-k+1$. It follows
that the codimension of $\kappa^{-1}(Z)$ in $C\times V$ is $r-k+1$ and the
codimension of $p_{V}(\kappa^{-1}(Z))$ in $V$ is at least $r-k\geqslant 2$.
The set $V\setminus p_{V}(\kappa^{-1}(Z))$ is precisely the locus of points in
$\mathbb{P}({\rm Hom}(E,F)^{\vee})$ corresponding to maps which are
surjective. The locus of points in $\mathbb{P}({\rm Hom}(E,F)^{\vee})$
corresponding to non-surjective maps $E\longrightarrow F$ is the set $T\bigcup
p_{V}(\kappa^{-1}(Z))$, which has codimension at least 2. This proves that
${\rm dim}(\Theta^{-1}(q))-{\rm
dim}(\Theta^{-1}(q)\setminus\mathbb{U}^{s})\geqslant 2$, which completes the
proof of the Lemma. ∎

######  Remark 7.15.

The proof of Lemma 7.11 also shows the following. Let $k=1$ and $r\geqslant 3$
so that $k\leqslant r-2$. Let $d\geqslant{\rm
max}\\{\alpha(E,1),\mu_{0}(E,1)+1\\}$. Let $L$ be a line bundle on $C$ of
degree $d$. Then the closed subset in $\mathbb{P}({\rm Hom}(E,L)^{\vee})$
consisting of non-surjective maps has codimension $\geqslant 2$.

######  Theorem 7.16.

Let $r-k\geqslant 2$. Assume one of the following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Let $n_{1}$ be the smallest integer such that $kn_{1}>g(C)+3$. Assume

$d\geqslant{\rm
max}\\{\alpha(E,k)+1,k\mu_{0}(E,k)+k,\beta(E,k,g(C)+3)+n_{1}\\}\,.$

Then

${\rm Pic}(\mathcal{Q})\cong{\rm Pic}(M^{s}_{k,d+kn})\times\mathbb{Z}\cong{\rm
Pic}({\rm Pic}^{0}(C))\times\mathbb{Z}\times\mathbb{Z}\,.$

###### Proof.

We saw in Theorem 6.3 that $\mathcal{Q}$ is an integral variety which is
normal and locally factorial. So the Picard group is isomorphic to the divisor
class group. By Lemma 7.10 it is enough to show that

${\rm Pic}(\mathcal{Q}^{s})\cong{\rm Pic}(M^{s}_{k,d+kn})\times\mathbb{Z}\,.$

Recall that we have the following diagram (7.9), which we checked is
Cartesian:

$\textstyle{\mathbb{U}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Theta}$$\textstyle{\mathcal{Q}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{R^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{M^{s}_{k,d+kn}\,.}$

Recall from §7.1 that we had fixed a closed point $P\in C$. Note that for any
$[q:\mathcal{O}^{N}_{C}\longrightarrow F(nP)]\in R^{s}$, the fibre
$\Theta^{-1}(q)\cong\theta^{-1}([F])$. Now from the proof of Lemma 7.11 it
follows that

${\rm Pic}(\Theta^{-1}(q))=\theta^{-1}([F])\cong\mathbb{Z}\,.$

Therefore we have the restriction map

${\rm res}:{\rm Pic}(\mathcal{Q})\cong{\rm
Pic}(\mathcal{Q}^{s})\longrightarrow{\rm
Pic}(\theta^{-1}([F]))\cong\mathbb{Z}\,.$

We claim this map is nontrivial. Let $\mathcal{L}$ be a very ample line bundle
on $\mathcal{Q}$. If ${\rm res}(\mathcal{L})$ were trivial, it follows that
the restriction of $\mathcal{L}$ to $\theta^{-1}([F])=\Theta^{-1}(q)$ would be
trivial and very ample, which is a contradiction as $\Theta^{-1}(q)$ is an
open subset of a projective space whose complement has codimension $\geqslant
2$. Thus, the image of ${\rm res}$ is isomorphic to a copy of $\mathbb{Z}$. We
will show that the kernel of ${\rm res}$ is isomorphic to ${\rm
Pic}(M^{s}_{k,d+kn})$.

Let $L\in{\rm Pic}(\mathcal{Q}^{s})$ be such that ${\rm res}(L)$ restricted to
$\theta^{-1}([F])$ is trivial. We need to show that $L$ is isomorphic to the
pullback of some line bundle on ${\rm Pic}(M^{s}_{k,d+kn})$. Consider the
pullback $\Psi^{*}L$. Since $\Psi$ is ${\rm PGL}(N)$-invariant, this line
bundle carries a ${\rm PGL}(N)$-linearization. By Lemma 7.11 the codimension
of complement of $\mathbb{U}^{s}$ in $\mathbb{P}^{s}$ has codimension
$\geqslant 2$. Therefore, both $L$ and this ${\rm PGL}(N)$-linearization
extends uniquely to $\mathbb{P}^{s}$. Let us denote this extension of
$\Psi^{*}L$ to $\mathbb{P}^{s}$ by $L^{\prime}$ and the linearization on ${\rm
PGL}(N)\times\mathbb{P}^{s}$ by
$\alpha^{\prime}:m^{*}_{\mathbb{P}^{s}}L^{\prime}\longrightarrow
p_{\mathbb{P}^{s}}^{*}L$, where $m_{\mathbb{P}^{s}}:{\rm
PGL}(N)\times\mathbb{P}^{s}\longrightarrow\mathbb{P}^{s}$ is the
multiplication map and $p_{\mathbb{P}^{s}}:{\rm
PGL}(N)\times\mathbb{P}^{s}\longrightarrow\mathbb{P}^{s}$ is the second
projection. Since $\Theta:\mathbb{P}^{s}\longrightarrow R^{s}$ is a projective
bundle, $L^{\prime}\cong\mathcal{O}(n)\otimes\Theta^{*}L^{\prime\prime}$ for
some $L^{\prime}\in{\rm Pic}(R^{s})$. However, since the fibers of $\Theta$
and $\theta$ are isomorphic, the condition ${\rm res}(L)$ is trivial implies
that $n=0$, that is, $L^{\prime}\cong\Theta^{*}L^{\prime\prime}$. Now note
that since the map $\mathbb{P}^{s}\longrightarrow R^{s}$ is ${\rm
PGL}(N)$-equivariant we have a commutative diagram

${{\rm PGL}(N)\times\mathbb{P}^{s}}$${\mathbb{P}^{s}}$${{\rm PGL}(N)\times
R^{s}}$${R^{s}}$$\scriptstyle{m_{\mathbb{P}^{s}}}$$\scriptstyle{{\rm
Id}\times\Theta}$$\scriptstyle{\Theta}$$\scriptstyle{m_{R^{s}}}$

From this diagram it follows that we have an isomorphism of sheaves

$({\rm Id}\times\Theta_{\mathbb{P}^{s}})^{*}m^{*}_{R^{s}}L^{\prime\prime}\cong
m^{*}_{\mathbb{P}^{s}}\Theta^{*}L^{\prime\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}p_{\mathbb{P}^{s}}^{*}\Theta^{*}L^{\prime\prime}\cong({\rm
Id}\times\Theta)^{*}p_{R^{s}}^{*}L^{\prime\prime}\,.$

where the middle isomorphism is given by the linearization $\alpha^{\prime}$.
Since ${\rm Id}\times\Theta$ is a projective bundle, applying $({\rm
Id}\times\Theta)_{*}$ to this composition of isomorphism we get a
linearization

$\alpha^{\prime\prime}:m^{*}_{R^{s}}L^{\prime\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}p_{R^{s}}^{*}L^{\prime\prime}$

of $L^{\prime\prime}$ such that $({\rm
Id}\times\Theta)^{*}\alpha^{\prime\prime}=\alpha^{\prime}$. Now recall that
the map $\psi$ is a principal ${\rm PGL}(N)$-bundle. By [HL10, Theorem 4.2.14]
we get that there exists $L^{\prime\prime\prime}\in{\rm Pic}(M^{s}_{k,d+kn})$
such that $\psi^{*}L^{\prime\prime\prime}\cong L^{\prime\prime}$ and the
induced ${\rm PGL}(N)$ linearization is $\alpha^{\prime\prime}$. Therefore we
get that

$\Psi^{*}\theta^{*}L^{\prime\prime\prime}\cong\Theta^{*}\psi^{*}L^{\prime\prime\prime}\cong\Psi^{*}L$

and also the induced ${\rm PGL}(N)$-linearizations are also the same. Since
the diagram (7.9) is Cartesian, the map $\Psi$ is a principal ${\rm
PGL}(N)$-bundle. Hence by [HL10, Theorem 4.2.14] we get that
$\theta^{*}L^{\prime\prime\prime}\cong L$. This completes the proof of the
first equality in the statement of the Theorem. The second equality follows
from [DN89, Theorem A, Theorem C] and from the fact that

${\rm dim}(M_{k,d+kn})-{\rm dim}(M_{k,d+kn}\setminus M^{s}_{k,d+kn})\geqslant
2\,.$

∎

##  8\. Fibers of ${\rm det}$

Let $L$ be a line bundle on $C$ of degree $d$ and let $\mathcal{Q}_{L}$ denote
the scheme theoretic fiber ${\rm det}^{-1}(L)$. As a corollary of Theorem 6.3
we have the following Proposition.

######  Proposition 8.1.

Let $n_{1}$ be the smallest integer such that $kn_{1}>g(C)+3$. Let
$d\geqslant\beta(E,k,g(C)+3)+n_{1}$. Then $\mathcal{Q}_{L}$ is a local
complete intersection scheme which is equidimensional, normal and locally
factorial.

###### Proof.

We use Theorem 6.3 and Lemma 6.1. As $\mathcal{Q}$ is a local complete
intersection scheme, ${\rm Pic}^{d}(C)$ is smooth and the map ${\rm det}$ is
flat, it follows using [Avr77, (1.9.2)] (see also [BH93, Remark 2.3.5] and
[Stk, Tag 09Q2]) that $\mathcal{Q}_{L}$ is a local complete intersection
scheme and so also Cohen-Macaulay. As $\mathcal{Q}$ is irreducible, flatness
of ${\rm det}$ also implies that $\mathcal{Q}_{L}$ is equidimensional.

We observed in the proof of Theorem 6.3 that the restriction of ${\rm det}$ to
the open subset $\mathcal{Q}_{\rm g}$ is a smooth morphism. It follows that
$\mathcal{Q}_{L}\cap\mathcal{Q}_{\rm g}$ is contained in the smooth locus of
$\mathcal{Q}_{L}$. The singular locus of $\mathcal{Q}_{L}$ is thus contained
in $\mathcal{Q}_{L}\cap\mathcal{Q}_{\rm b}$. As
$d\geqslant\beta(E,k,g(C)+3)+n_{1}$, applying Lemma 6.2 we get

${\rm dim}(\mathcal{Q})-{\rm dim}(\mathcal{Q}_{\rm b})>g(C)+3\,.$

By Lemma 2.1(2) it follows that

(8.2) ${\rm dim}(\mathcal{Q}_{L})-{\rm
dim}(\mathcal{Q}_{L}\cap\mathcal{Q}_{\rm b})>3\,.$

It follows that the singular locus of $\mathcal{Q}_{L}$ has codimension 4 or
more. This proves that $\mathcal{Q}_{L}$ is normal, that is, it is the
disjoint union of finitely many normal varieties, all of the same dimension.
Using Grothendieck’s theorem (see [AH20, Theorem 1.4]) it follows that
$\mathcal{Q}_{L}$ is locally factorial. ∎

Next we want to find conditions under which $\mathcal{Q}_{L}$ becomes
irreducible. We use the notation used in Lemma 5.2.

######  Lemma 8.3.

Let $n_{0}$ be the smallest integer such that $kn_{0}>g(C)+1$. Let $n_{1}$ be
the smallest integer such that $kn_{1}>g(C)+3$. Let

$d\geqslant{\rm
max}\\{\beta(E,k,g(C)+1)+n_{0}+1,\beta(E,k,g(C)+3)+n_{1}\\}\,.$

Then $\mathcal{Q}^{\rm tf}_{L}$ is dense in $\mathcal{Q}_{L}$.

###### Proof.

Recall the relative Quot scheme in equation (5.3). We are interested in the
case $i=1$, that is, the relative Quot scheme ${\rm Quot}_{C\times
A/A}(\mathscr{S},0,1)$, where $A$ is the Quot scheme ${\rm
Quot}_{C/\mathbb{C}}(E,k,d-1)$. For ease of notation we denote by $B$ the
scheme ${\rm Quot}_{C\times A/A}(\mathscr{S},0,1)$. Recall the map
$\pi:B\longrightarrow A$ from (5.3). On $C\times B$ we have a quotient

(8.4) $({\rm Id_{C}}\times\pi)^{*}\mathscr{S}\longrightarrow\mathscr{T}\,,$

such that $\mathscr{T}$ is flat over $B$. Using $\mathscr{T}$ we get the
determinant map

${\rm det}_{B}:B\longrightarrow{\rm Pic}^{1}(C)\,.$

This map has the following pointwise description. A closed point $b\in B$
gives rise to the closed point $\pi(b)\in A$, which corresponds to a short
exact sequence on $C$

$0\longrightarrow S_{F}\longrightarrow E\longrightarrow F\longrightarrow 0\,,$

where $F$ is of rank $k$ and degree $d-1$ on $C$. The restriction of the
universal quotient (8.4) to the point $b$ is a torsion quotient on $C$

$S_{F}\longrightarrow M\,,$

such that ${\rm length}(M)=1$. Let $c={\rm Supp}(M)$. Then ${\rm
det}_{B}(b)=\mathcal{O}_{C}(c)$. Consider the natural embedding (recall that
$g(C)>0$) $\iota:C\hookrightarrow{\rm Pic}^{1}(C)$ given by
$c\mapsto\mathcal{O}_{C}(c)$. It is clear that the image of $B$ is the image
of $\iota$. Next we want to show that $B$ is an integral scheme.

As $d-1\geqslant\alpha(E,k)$, it follows from Lemma 6.1 that $A$ is a local
complete intersection. By Theorem 6.3(1) it follows that $A$ is integral. As
$i=1$, it is easily checked that the map $\pi:B\longrightarrow A$, see (5.3),
is the projective bundle $\mathbb{P}(\mathscr{S})\longrightarrow A$. It
follows that $B$ is integral and a local complete intersection and so Cohen-
Macaulay. As $B$ is integral, the map ${\rm det}_{B}$ factors through the map
$\iota$, that is, we have a commutative diagram

---
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
det}_{B}}$$\scriptstyle{{\rm det}_{T}}$$\textstyle{{\rm
Pic}^{1}(C)}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

Let ${\rm det}_{A}:A\longrightarrow{\rm Pic}^{d-1}(C)$ denote the determinant
map for the Quot scheme $A$. This is flat due to Theorem 6.3(1). Consider the
map

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\rm
det}_{T},\,{\rm det}_{A}\circ\pi)}$$\scriptstyle{{\rm
det}_{B}}$$\textstyle{C\times{\rm
Pic}^{d-1}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Pic}^{d}(C)\,.}$

The second map is given by $(c,M)\mapsto M\otimes\mathcal{O}_{C}(c)$. It is
easily checked that both maps have constant fiber dimension. In view of
[Mat86, Theorem 23.1] it follows that both maps are flat and so the composite
${\rm det}_{B}$ is also flat. Recall the map $\pi^{\prime}$ from (5.4). It is
clear that we have a commutative diagram

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{{\rm
det}_{B}}$$\textstyle{\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
det}}$$\textstyle{{\rm Pic}^{d}(C)\,.}$

Recall the definition of $Z_{1}$, see (5.1). We saw in the proof of Lemma 5.2
that $\pi^{\prime}(B)=\bar{Z}_{1}$. Let

$\bar{Z}_{1,L}:=\\{[q:E\longrightarrow F]\in\bar{Z}_{1}\,|\,{\rm
det}(F)=L\\}\,.$

Let $B_{L}:={\rm det}_{B}^{-1}(L)$ denote the scheme theoretic fiber over $L$.
Then it is clear that $\pi^{\prime}(B_{L})=\bar{Z}_{1,L}$. In the proof of
Lemma 5.2 we had remarked that there is an open set $U\subset B$ such that
$\pi^{\prime}$ is injective on points of $U$. Thus, $\pi^{\prime}$ is also
injective on the subset $U\cap B_{L}$. It follows that

(8.5) ${\rm dim}(\bar{Z}_{1,L})={\rm dim}(B_{L})={\rm
dim}(\mathcal{Q})-k-g={\rm dim}(\mathcal{Q}_{L})-k\,.$

As $k\geqslant 1$, and all irreducible components of $\mathcal{Q}_{L}$ have
the same dimension, it follows that
$\mathcal{Q}_{L}\setminus\bar{Z}_{1,L}=\mathcal{Q}^{\rm tf}_{L}$ is dense in
$\mathcal{Q}_{L}$. ∎

The above Lemma implies that irreducibility of $\mathcal{Q}_{L}$ is equivalent
to the irreducibility of the open subset $\mathcal{Q}_{L}^{\rm tf}$. Let

$\mathcal{Q}^{\rm tf}_{g,L}:=\mathcal{Q}^{\rm tf}_{g}\cap\mathcal{Q}_{L}\,.$

Combining Proposition 8.1 and Lemma 8.3 we get the following.

######  Lemma 8.6.

Let $n_{0}$ be the smallest integer such that $kn_{0}>g(C)+1$. Let $n_{1}$ be
the smallest integer such that $kn_{1}>g(C)+3$. Let

$d\geqslant{\rm
max}\\{\beta(E,k,g(C)+1)+n_{0}+1,\beta(E,k,g(C)+3)+n_{1}\\}\,.$

Then $\mathcal{Q}^{\rm tf}_{{\rm g},L}$ is dense in $\mathcal{Q}^{\rm
tf}_{L}$.

###### Proof.

As all components of $\mathcal{Q}_{L}$ have the same dimension, the same holds
for the open subset $\mathcal{Q}^{\rm tf}_{L}$. Note that

$\mathcal{Q}^{\rm tf}_{L}\setminus\mathcal{Q}^{\rm tf}_{{\rm
g},L}=\mathcal{Q}^{\rm tf}_{L}\cap\mathcal{Q}_{\rm b}\,.$

The Lemma follows using (8.2). ∎

Combining the above results we have the following.

######  Theorem 8.7.

Let $k\geqslant 2,g(C)\geqslant 2$. Let $n_{0}$ be the smallest integer such
that $kn_{0}>g(C)+1$. Let $n_{1}$ be the smallest integer such that
$kn_{1}>g(C)+3$. Let

$d\geqslant{\rm
max}\\{\beta(E,k,g(C)+1)+n_{0}+1,\beta(E,k,g(C)+3)+n_{1}\\}\,.$

Then $\mathcal{Q}_{L}$ is a local complete intersection scheme which is also
integral, normal and locally factorial.

###### Proof.

The Theorem follows using Proposition 8.1 once we show that $\mathcal{Q}_{L}$
is irreducible. In view of Lemma 8.3 and Lemma 8.6, it suffices to show that
$\mathcal{Q}^{\rm tf}_{{\rm g},L}$ is irreducible.

Recall the notation from §7, in particular, the map $\Psi$ from (7.5). This
sits in the following commutative diagram whose maps we describe next.

(8.8) ${\mathbb{U}}$${\mathcal{Q}^{\rm tf}_{\rm g}}$${R^{\prime}}$${{\rm
Pic}^{d+kn}(C)}$$\scriptstyle{\Psi}$

The bottom horizontal map sends a closed point $[x:\mathcal{O}_{C}^{\oplus
N}\longrightarrow F]\in R^{\prime}$ to ${\rm det}(F)$. The right vertical map
sends a closed point $[q:E\longrightarrow F]\in\mathcal{Q}^{\rm tf}_{\rm g}$
to ${\rm det}(F)\otimes\mathcal{O}_{C}(knP)$. Let
$L^{\prime}:=L\otimes\mathcal{O}_{C}(knP)$.

The bottom horizontal map in (8.8) is a smooth morphism. This follows using
Lemma 2.7 and the reason explained after (6.4) applied to the space
$R^{\prime}$. In particular, the morphism $R^{\prime}\longrightarrow{\rm
Pic}^{d+kn}(C)$ is flat. Thus, $R^{\prime}_{L^{\prime}}$ is a smooth
equidimensional scheme. Using [Bho99, Corollary 1.3] we easily see that
$R^{\prime}_{L^{\prime}}$ is irreducible. Taking “fiber” of (8.8) over the
point $[L^{\prime}]\in{\rm Pic}^{d+kn}(C)$ we get the following commutative
diagram

${\mathbb{U}_{L^{\prime}}}$${\mathcal{Q}^{\rm tf}_{{\rm
g},L}}$${R^{\prime}_{L^{\prime}}}$${{[L^{\prime}]}}$$\scriptstyle{\Psi_{L^{\prime}}}$$\scriptstyle{\Theta_{L^{\prime}}}$

It follows that $\mathbb{U}_{L^{\prime}}$ is irreducible. Using $\Psi$ is
surjective on closed points we get that $\Psi_{L^{\prime}}$ is also surjective
on closed points. It follows that $\mathcal{Q}^{\rm tf}_{{\rm g},L}$ is
irreducible. This completes the proof of the Theorem. ∎

######  Theorem 8.9.

Let $r-k\geqslant 2$. Assume one of the following two holds

  * •

$k\geqslant 2$ and $g(C)\geqslant 3$, or

  * •

$k\geqslant 3$ and $g(C)=2$.

Let $n_{0}$ be the smallest integer such that $kn_{0}>g(C)+1$. Let $n_{1}$ be
the smallest integer such that $kn_{1}>g(C)+3$. Let

$d\geqslant{\rm
max}\\{k\mu_{0}(E,k)+k,\beta(E,k,g(C)+1)+n_{0}+1,\beta(E,k,g(C)+3)+n_{1}\\}\,.$

We have isomorphisms

${\rm Pic}(\mathcal{Q}_{L})\cong{\rm
Pic}(M^{s}_{k,L})\times\mathbb{Z}\cong\mathbb{Z}\times\mathbb{Z}\,.$

###### Proof.

The proof is similar to Theorem 7.16 and so we only sketch it. From (8.2) and
(8.5) it follows that

${\rm dim}(\mathcal{Q}_{L})-{\rm dim}(\mathcal{Q}_{L}\setminus\mathcal{Q}^{\rm
tf}_{{\rm g},L})\geqslant 2\,.$

Now consider the diagram

${\mathbb{U}_{L^{\prime}}}$${\mathcal{Q}^{\rm tf}_{{\rm
g},L}}$${R^{\prime}_{L^{\prime}}}$${{[L^{\prime}]}}$$\scriptstyle{\Psi_{L^{\prime}}}$$\scriptstyle{\Theta_{L^{\prime}}}$

Just as in Lemma 7.10, using [Bho99, Corollary 1.3], and Lemma 2.2 we have

${\rm dim}(\mathcal{Q}^{\rm tf}_{{\rm g},L})-{\rm dim}(\mathcal{Q}^{\rm
tf}_{{\rm g},L}\setminus\mathcal{Q}^{s}_{L})\geqslant 2\,.$

Therefore we get that

${\rm dim}(\mathcal{Q}_{L})-{\rm
dim}(\mathcal{Q}_{L}\setminus\mathcal{Q}^{s}_{L})\geqslant 2\,.$

Since $\mathcal{Q}_{L}$ is locally factorial we have

${\rm Pic}(\mathcal{Q}_{L})\cong{\rm Pic}(\mathcal{Q}^{s}_{L})\,.$

Now we have the cartesian diagram

(8.10)
$\textstyle{\mathbb{U}^{s}_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Theta_{L}}$$\textstyle{\mathcal{Q}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{L}}$$\textstyle{R^{s}_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{M^{s}_{k,L}\,.}$

we get by taking fiber over $[L]$ of the diagram (7.9). The rest of the proof
is same as the proof of Theorem 7.16, by considering this diagram instead of
(7.9). The second equality follows from [DN89, Theorem B]. ∎

##  9\. Quot Schemes ${\rm Quot}_{C/\mathbb{C}}(E,1,d)$

In this section we consider the case $k=1$. We only sketch the proofs as they
are similar to the earlier cases considered.

######  Theorem 9.1.

Let $k=1$. Let $d\geqslant{\rm
max}\\{\mu_{0}(E,1)+1,\beta(E,1,g(C)+3)+g(C)+4\\}$. Then

${\rm Pic}(\mathcal{Q})\cong{\rm Pic}({\rm
Pic}^{d}(C))\times\mathbb{Z}\times\mathbb{Z}\,,\quad\quad{\rm
Pic}(\mathcal{Q}_{L})\cong\mathbb{Z}\times\mathbb{Z}\,.$

###### Proof.

We can apply Theorem 6.3 to conclude that $\mathcal{Q}$ is integral, normal
and locally factorial. Let

(9.2) $\rho_{1}:C\times{\rm Pic}^{d}(C)\longrightarrow
C\,,\quad\quad\rho_{2}:C\times{\rm Pic}^{d}(C)\longrightarrow{\rm Pic}^{d}(C)$

be the projections. Let $\mathcal{L}$ be a Picard bundle on $C\times{\rm
Pic}^{d}(C)$. Define

$\mathcal{E}:=\rho_{2*}[\rho_{1}^{*}E^{\vee}\otimes\mathcal{L}]\,.$

Using Lemma 4.1 and cohomology and base change we easily conclude that
$\mathcal{E}$ is a locally free sheaf on ${\rm Pic}^{d}(C)$ such that the
fibre over the point $[L]\in{\rm Pic}^{d}(C)$ is isomorphic to ${\rm
Hom}(E,L)$. Let $\mathbb{W}\subset\mathbb{P}(\mathcal{E}^{\vee})$ be the open
subset consisting of points parametrizing surjective maps. Both $\mathbb{W}$
and $\mathcal{Q}^{\rm tf}$ are smooth. There is a map
$\mathbb{W}\longrightarrow{\mathcal{Q}}^{\rm tf}$ which is bijective on points
(and hence an isomorphism as both are smooth) and sits in a commutative
diagram

$\textstyle{\mathbb{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathcal{Q}^{\rm
tf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
det}$$\textstyle{{\rm Pic}^{d}(C)}$

Using Remark 7.15 it follows that ${\rm
dim}(\mathbb{P}(\mathcal{E}^{\vee}))-{\rm
dim}(\mathbb{P}(\mathcal{E}^{\vee})\setminus\mathbb{W})\geqslant 2$. Thus, is
follows that ${\rm Pic}(\mathcal{Q}^{\rm tf})\cong{\rm
Pic}(\mathbb{W})\cong{\rm Pic}(\mathbb{P}(\mathcal{E}^{\vee}))\cong{\rm
Pic}({\rm Pic}^{d}(C))\times\mathbb{Z}$. By Lemma 5.2,
$\mathcal{Q}\setminus\mathcal{Q}^{\rm tf}=\bar{Z}_{1}$ is irreducible of
codimension 1 and so we have an exact sequence

$0\longrightarrow\mathbb{Z}\longrightarrow{\rm
Pic}(\mathcal{Q})\longrightarrow{\rm Pic}(\mathcal{Q}^{\rm tf})\longrightarrow
0\,.$

It easily follows that we have an isomorphism

${\rm Pic}(\mathcal{Q})\cong{\rm Pic}({\rm
Pic}^{d}(C))\times\mathbb{Z}\times\mathbb{Z}\,.$

For $\mathcal{Q}_{L}$, we first show that $\mathcal{Q}_{L}$ is integral,
normal and locally factorial. This is easily done using Proposition 8.1, Lemma
8.3 and using the fact that $\mathcal{Q}^{\rm tf}_{L}\cong\mathbb{W}_{L}$. The
rest of the proof follows in the same way as that of $\mathcal{Q}$, once we
use the irreducibility of $\bar{Z}_{1,L}$ and the fact that it is of
codimension 1, see (8.5). We remark that when $k=1$, unlike in Theorem 8.7, we
do not need to use [Bho99] and hence do not need the hypothesis that
$g(C)\geqslant 2$. ∎

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