# Irreducibility of some Quot schemes on Nodal curves

Parvez Rasul  Department of Mathematics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, Maharashtra, India<EMAIL_ADDRESS>
###### Abstract.

Let $C$ be an integral projective nodal curve over $\mathbb{C}$, of arithmetic
genus $g\geqslant 2$. Let $E$ be a vector bundle on $C$ of rank $r$ and degree
$e$. Let ${\rm Quot}_{C/\mathbb{C}}(E,k,d)$ denote the Quot scheme of
quotients of $E$ of rank $k$ and degree $d$. We show that ${\rm
Quot}_{C/\mathbb{C}}(E,k,d)$ is irreducible for $d\gg 0$.

###### Key words and phrases:

Quot scheme

###### 2010 Mathematics Subject Classification:

14H60

##  1\. Introduction

Let $C$ be a reduced and irreducible projective curve over $\mathbb{C}$ of
arithmetic genus $g$. For a coherent sheaf $F$ on $C$ of rank $k$, the degree
of $F$ is defined as

(1.1) $\deg F:=\chi(F)-k(1-g)\,.$

Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$. Let ${\rm
Quot}_{C/\mathbb{C}}(E,k,d)$ denote Grothendieck’s Quot scheme of coherent
quotients of $E$ of rank $k$ and degree $d$. We will use the simpler notation
$Q_{k,d}(E)$ to denote the quot scheme ${\rm Quot}_{C/\mathbb{C}}(E,k,d)$.
Quot schemes have played a central role in the study of vector bundles on
curves, in particular for constructing and understanding geometry of the
moduli spaces. Quot schemes also arise as a compactification of the space of
maps into Grassmannians.

When $C$ is a smooth curve, the quot scheme $Q_{0,d}(E)$ of torsion quotients
of $E$ is a smooth projective variety, which has been studied extensively. We
refer the reader to the recent works [BDH15],[BGS22], [OS23], [OP21] and
references therein. There are few results when $C$ is singular. In [Reg82],
the author shows that if $C$ is embeddable in a smooth surface then the quot
scheme $Q_{0,d}(\mathcal{O}^{\oplus r}_{C})$ is irreducible. It is also proved
that the compactification of moduli spaces of vector bundles on such curves is
irreducible. Some sort of converse is proved in [KK81], which shows that if
$C$ is not embeddable in a smooth surface then the quot scheme
$Q_{0,d}(\omega)$ is reducible, where $\omega$ is the dualizing sheaf on $C$.
The authors use this to show that the compactified Jacobian of such $C$ is
reducible. In [Ran05], the punctual Hilbert scheme parametrizing degree-$d$
subschemes of $C$ supported at the node is shown to be a chain of $d-1$
rational curves meeting transversely. It is also proved that the relative
Hilbert scheme $\textrm{Hilb}(X/B)$ parametrizing length-$m$ subschemes in the
fibres of the family $X/B$ is smooth, where $X$ is a 1-parameter smooth family
of curves over $B$.

We come to the case $k>0$ and when the curve $C$ is smooth. In genus 0 case,
when $E$ is the trivial bundle, Stromme proved that the Quot scheme
$Q_{k,d}(\mathcal{O}^{\oplus r}_{\mathbb{P}^{1}})$ is smooth and irreducible
and computed its Picard group in [Str87]. Let $C$ be smooth of genus $g>0$.
When $E$ is trivial, it is proved in [BDW96] that the Quot scheme
$Q_{k,d}(\mathcal{O}_{C}^{\oplus r})$ is irreducible and generically smooth
for $d\gg 0$. The authors also show that $Q_{k,d}(\mathcal{O}_{C}^{\oplus r})$
is a local complete intersection for large $d$. For any vector bundle $E$ on
$C$, it is proved in [PR03] that the quot scheme $Q_{k,d}(E)$ is irreducible
and generically smooth when $d\gg 0$. The Picard group of the Quot scheme
$Q_{k,d}(E)$ is computed in [GS22] for $d\gg 0$. The authors also show that if
$d\gg 0$ then the Quot scheme is an integral, normal variety which is a local
complete intersection and locally factorial.

In this article we extend some of the above results to the case when $C$ is a
nodal curve. Degeneration techniques have found many applications in moduli
problems and in enumerative geometry, for example, see [LW15], [Gol19] and
references therein. Consequently it is interesting to study quot schemes over
nodal curves. In [Gol19], the author defines a nice intersection theory on the
quot scheme over nodal curves by relating it to the Quot schemes on the
normalization. In order to do this, the author shows that, for a very general
vector bundle $E$ on $C$, the unique top dimensional component of the quot
scheme $Q_{k,d}(E)$ is generically smooth and contains all the points
corresponding to torsion-free quotients.

In this article we prove the following theorem.

######  Theorem (Theorem 6.15).

Let $C$ be a reduced irreducible projective curve over $\mathbb{C}$ of
arithmetic genus $g\geqslant 2$ with exactly one ordinary node and smooth
elsewhere. Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$. Let
$k$ be an integer such that $0<k<r$. Then there is a number $d_{Q}(E,k)$ such
that if $d\geqslant d_{Q}$ then the quot scheme $Q_{k,d}(E)$ is irreducible
and generically smooth of dimension $dr-ke+k(r-k)(1-g)$.

######  Remark 1.2.

Throughout the article and hence in Theorem 6.15, we have considered the curve
$C$ to have a single node. However we emphasize that all the results in the
article can be seen to generalise easily to the case when $C$ has more than
one nodes. So we have Theorem 6.15 for any integral projective nodal curve $C$
of arithmetic genus $g\geqslant 2$.

It would be interesting to know if a similar result holds when $C$ is an
integral projective curve. From [Vak06] it is known that, in general the
Hilbert schemes can be quite pathological. In [GS22], it is proved that for
large $d$, the quot scheme $Q_{k,d}(E)$ over a smooth curve is an integral
variety which is a local complete intersection and locally factorial. It is
interesting to know what kind of singularities occur in the quot schemes over
nodal curves.

We will discuss our strategy briefly. The strategy we use is similar to that
in [PR03, Section 6], however, some modifications and new inputs are required.
It will be convenient for the reader if we first discuss the strategy of
[PR03], where the theorem is proved for smooth curves.

  1. (1)

In Theorem 4.1 of [PR03], the authors compute an upper bound for the dimension
of the quot scheme $Q_{k,d}(E)$ which is as follows:

$\dim Q_{k,d}(E)\leq k(r-k)+(d-m(E,k))r,\quad\textrm{ for all }d\geqslant
m(E,k)\,.$

The number $m(E,k)$ is the minimal degree of a quotient of $E$ of rank $k$.

  2. (2)

The authors prove that the quot scheme always contains a stable locally free
quotient if $d$ is large enough. Then using the irreducibility of the moduli
space of stable locally free sheaves on $C$, it is concluded that there is a
unique component of the quot scheme whose general point corresponds to a
stable bundle quotient. Let us call this component $\mathcal{P}$. The
dimension of $\mathcal{P}$ is $dr-ke+k(r-k)(1-g)$.

  3. (3)

The authors prove that for large $d$, a general vector bundle quotient of $E$
can be deformed to a stable vector bundle quotient. A key ingredient for this
step is the upper bound for the dimension of the quot scheme given in step
(1). This proves the component $\mathcal{P}$ contains all the points
corresponding to locally free quotients.

  4. (4)

Let $Z$ denote the locus of points in the quot scheme which correspond to
quotients with torsion. The authors show that when $d$ is large, $\dim Z<dr-
ke+k(r-k)(1-g)$. There is a cohomological lower bound for the dimension of any
component of the quot scheme, which is $dr-ke+k(r-k)(1-g)$ in this case. This
proves that the points in $Z$ cannot be general in any component of the quot
scheme. This shows that a general point in any component corresponds to a
locally free quotient and so the component has to be $\mathcal{P}$. Thus, the
quot scheme $Q_{k,d}(E)$ is irreducible when $d$ is sufficiently large.

In the case of nodal curves, we proceed as follows. The first two steps are
very similar to that in [PR03]. However, dealing with quotients with torsion
is substantially harder in the nodal case.

  1. (a)

In a similar fashion, as in [PR03], we show that there is a unique component
of the quot scheme whose generic point corresponds to a stable locally free
quotient, when $d$ is large. Let us call this component $\mathcal{P}$. This
component has dimension $dr-ke+k(r-k)(1-g)$.

  2. (b)

Using Lemma 2.4 of [Bho06], we show that for large $d$, a general locally free
quotient in the quot scheme can be deformed to a stable locally free quotient.
This proves that the component $\mathcal{P}$ contains all the points
corresponding to locally free quotients, when $d$ is large. Similarly using
Lemma 2.3 of [Bho06], we show that for large $d$, a general torsion-free
quotient in the quot scheme can be deformed to a locally free quotient. This
proves that $\mathcal{P}$ contains all the points corresponding to torsion-
free quotients, when $d$ is large.

  3. (c)

It remains to deal with the locus of quotients with torsion. Let $Z$ denote
the locus of points in the quot scheme which correspond to quotients with
torsion. It is natural to try and follow step (4) to deal with this case. For
this we need a good cohomological lower bound. In the case of smooth curves,
any coherent sheaf is direct sum of a locally free sheaf and a torsion sheaf.
However, for nodal curves, this is not the case. As a result, computing a good
cohomological lower bound becomes difficult. Using [EL99, Lemma 2], we compute
the following lower bound

$\textrm{{cohomological lower bound}}=dr-ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

Unfortunately, unlike in the case of smooth curves (step (4)), the upper bound
on the dimension of $Z$ is not less than the cohomological lower bound. Thus,
the strategy in step (4) fails to generalize to the nodal case.

Instead, we show the following. Let $\mathcal{W}$ be a component whose general
point is in $Z$. We show that an inclusion $\ker q\hookrightarrow E$ coming
from a general point $[q]\in\mathcal{W}$ can be deformed to an inclusion
$S\hookrightarrow E$ whose cokernel is locally free. This implies that
$\mathcal{W}=\mathcal{P}$. From this we conclude the irreducibility of the
quot scheme when $d$ is large. This part is done in several steps and occupies
all of section 6.

Few points need to be highlighted. We use various results on the torsion-free
sheaves on nodal curves from [Bho06], in which the nodal curve is assumed to
have genus $g\geqslant 2$. For this reason, we impose the condition
$g\geqslant 2$ on the genus of the curve. To show the existence of
deformations mentioned in step (b) and step (c), some $H^{1}$ and
$\textnormal{Ext}^{1}$ vanishing results are required. These results rely on
dimension estimates of few subsets on the quot schemes. For example, the
codimension of the locus $Y_{\mu_{0}}=\\{[E\to F]\in Q_{k,d}^{{\rm
tf}}(E):\mu_{\min}(F)<\mu_{0}\\}$ increases linearly with $d$ (Lemma 3.21).
This is implicit in [PR03, Lemma 6.3], and is also used in the proof of [GS22,
Lemma 4.9].

The organization of the paper is as follows. In section 2 we go through some
preliminary lemmas regarding the sheaves on nodal curves and then we prove the
main lemma which estimates a $(\hom-\textnormal{ext}^{1})$ term. This lemma
directly gives us the cohomological lower bound for the quot scheme which is
mentioned in step $\eqref{step torsion locus}$. In section 3 we compute the
dimension estimates of a few subsets of the quot scheme which will help to
prove some $H^{1}$ and $\textnormal{Ext}^{1}$ vanishing results later. Section
4 is occupied by the step (a) mentioned above. This section is self-contained
and does not involve any result from the first two sections. In section 5 we
carry out the step (b). In section 6 we deal with the locus of points
corresponding to the quotients with torsion and carry out step (c) to prove
the main theorem 6.15.

.

Acknowledgements. I thank Usha Bhosle, Indranil Biswas and Chandranandan
Gangopadhyay for useful discussions. I thank my supervisor Ronnie Sebastian
for suggesting to work on this question and for several useful discussions.

##  2\. Preliminaries

######  Definition 2.1.

Let $G$ be a sheaf of rank $r$ on a projective scheme $X$ over $\mathbb{C}$.
For each rank $k<r$, define

$m(G,k)=\min_{\rank(F)=k}\left\\{\deg(F):F\textrm{ is a quotient of
}G\right\\}\,.$

######  Lemma 2.2.

Let $X$ and $Y$ be projective schemes over $\mathbb{C}$ and $\mathcal{G}$ be a
coherent sheaf on $X\times Y$ which is flat over $Y$. Then for a fixed
$k<\rank(\mathcal{G})$, the function $y\mapsto m(\mathcal{G}_{y},k)$ is lower
semicontinuous as a function from $Y$ to $\mathbb{Z}$. As a consequence the
set $\\{m(\mathcal{G}_{y},k)\\}_{y\in Y}$ is finite for fixed $k$.

###### Proof.

Let $l$ be an integer and define the set

$U=\\{y\in Y:m(\mathcal{G}_{y},k)>l\\}\,.$

We need to prove that $U$ is open in $Y$. Since the family of sheaves
$\\{\mathcal{G}_{y}\\}_{y\in Y}$ on $X$ is bounded, by [HL10, Lemma 1.7.6] we
have a sheaf $\mathcal{B}$ on $X$ such that each $\mathcal{G}_{y}$ is a
quotient of $\mathcal{B}$. Now a quotient of $\mathcal{G}_{y}$ is also a
quotient of $\mathcal{B}$. Hence $m(\mathcal{G}_{y},k)\geqslant
m(\mathcal{B},k)$ for all $y\in Y$. If $l<m(\mathcal{B},k)$ then $U=Y$ and we
are done. If $m(\mathcal{B},k)\leq l$, consider the relative Quot schemes
${\rm Quot}_{X\times Y/Y}(\mathcal{G},k,i)$ for each $i$ with
$m(\mathcal{B},k)\leq i\leq l$. Let $\sigma_{i}:{\rm Quot}_{X\times
Y/Y}(\mathcal{G},k,i)\to Y$ be the structure maps. Then

$U=Y\setminus\bigcup_{i=m(\mathcal{B},k)}^{l}\sigma_{i}({\rm Quot}_{X\times
Y/Y}(\mathcal{G},k,i))\,.$

Since each $\sigma_{i}$ is proper map, $U$ is open in $Y$. ∎

Let $C$ be a reduced irreducible, projective curve over $\mathbb{C}$ of
arithmetic genus $g\geqslant 2$ with exactly one singular point $x$. Further
we assume that the singularity at $x$ is an ordinary node. Let
$\mathcal{O}_{C,x}$ denote the local ring of $C$ at $x$ and $\mathfrak{m}_{x}$
denote the maximal ideal of $\mathcal{O}_{C,x}$. Let $\kappa(x)$ denote the
residue field of $\mathcal{O}_{C,x}$. We will also use $\kappa(x)$ to mean the
skyscraper sheaf on $C$ supported at the nodal point $x$.

######  Definition 2.3 (Type of a sheaf).

Let $F$ be a coherent sheaf on $C$ of rank $k$ such that the stalk $F_{x}$ at
the node $x$ is torsion-free. From [Ses82, Proposition 2, Part 8, page 168],
we know that there exists an integer $a$, $0\leq a\leq k$ such that the stalk
$F_{x}=\mathcal{O}_{C,x}^{\oplus k-a}\oplus\mathfrak{m}_{x}^{\oplus a}$. We
call such $F$ torsion free at node of type $a$. If $F$ is torsion free at node
of type $a$ then $\dim_{\mathbb{C}}(F\otimes\kappa(x))=k+a$.

######  Lemma 2.4.

Let $A$ and $B$ be sheaves on $C$ of rank $r_{A},r_{B}$ and degree
$d_{A},d_{B}$ respectively. Assume that $A$ and $B$ are torsion-free of type
$a$ and $b$ respectively. Then

$\hom(B,A)-\textnormal{ext}^{1}(B,A)=r_{B}d_{A}-r_{A}d_{B}+r_{A}r_{B}(1-g)-ab\,.$

###### Proof.

As $A$ and $B$ both are torsion-free, using [Bho06, Lemma 2.5(B)] we get that

$\textnormal{ext}^{1}(B,A)=\textrm{h}^{1}(C,\mathscr{H}om(B,A))+2ab\,.$

So we have

$\displaystyle\hom(B,A)-\textnormal{ext}^{1}(B,A)$
$\displaystyle=\hom(B,A)-\textrm{h}^{1}(C,\mathscr{H}om(B,A))-2ab$
$\displaystyle=\textrm{h}^{0}(C,\mathscr{H}om(B,A))-\textrm{h}^{1}(C,\mathscr{H}om(B,A))-2ab$
$\displaystyle=\chi(\mathscr{H}om(B,A))-2ab.$

Again [Bho06, Lemma 2.5(B)] says that the degree of the sheaf
$\mathscr{H}om(B,A)$ is

$r_{B}d_{A}-r_{A}d_{B}+ab\,.$

So using (1.1), we have

$\begin{split}\hom(B,A)-\textnormal{ext}^{1}(B,A)&=\chi(\mathscr{H}om(B,A))-2ab\\\
&=r_{B}d_{A}-r_{A}d_{B}+ab+r_{A}r_{B}(1-g)-2ab\\\
&=r_{B}d_{A}-r_{A}d_{B}+r_{A}r_{B}(1-g)-ab\,.\end{split}$

∎

######  Lemma 2.5.

Let

$0\to B\to E\to A\to 0$

be a short exact seqeunce of torsion-free sheaves and $E$ be locally free.
Then type of $A$ and type of $B$ are equal.

###### Proof.

Let the type of $A$ be $a$ and the type of $B$ be $b$. Tensoring the exact
sequence

$0\to B\to E\to A\to 0$

with $\kappa(x)$, we get a sequence

$0\to\textnormal{Tor}^{1}(A,\kappa(x))\to B|_{x}\to E|_{x}\to A|_{x}\to 0\,.$

Now $\dim_{\mathbb{C}}(A|_{x})=\text{rank}(A)+a$ and similarly
$\dim_{\mathbb{C}}(B|_{x})=\text{rank}(B)+b$. So
$b=\dim_{\mathbb{C}}\textnormal{Tor}^{1}(A,\kappa(x))-a$. Also
$\textnormal{Tor}^{1}(A,\kappa(x))=\textnormal{Tor}^{1}(\mathfrak{m}_{x}^{\oplus
a},\kappa(x))=2a$. It follows that $b=a$. ∎

For a torsion sheaf $\tau$ on $C$, we define the socle of $\tau_{x}$ as
$\textnormal{Soc}(\tau_{x}):=\textrm{Hom}_{C}(\kappa(x),\tau)$.

######  Lemma 2.6.

Let $0<k<r$. Let $A$ and $B$ be two torsion-free sheaves of rank $r-k$. Let
type of $A$ be $a$ and type of $B$ be $b$. Let $\tau$ be a torsion sheaf on
$C$ supported at the node $x$, which fits into a short exact sequence

(2.7) $0\to B\to A\to\tau\to 0\,.$

Then $\dim(\textnormal{Soc}(\tau))\leq r-k+2a+b$.

###### Proof.

Localizing the short exact sequence (2.7) at the node $x$, we get the short
exact sequence

$0\to B_{x}\to A_{x}\to\tau\to 0\,.$

As $A_{x}=\mathcal{O}_{C,x}^{\oplus r-k-a}\oplus\mathfrak{m}_{x}^{\oplus a}$,
so there is a surjection $\varphi:\mathcal{O}_{C,x}^{\oplus r-k+a}\to A_{x}$.
Let $\psi:\mathcal{O}_{C,x}^{\oplus r-k+a}\to\tau$ denote the composition of
$\varphi$ and the quotient $A\to\tau\to 0$. Let $K$ be the kernel of $\varphi$
and $M$ be the kernel $\psi$. So we have the following commutative diagram of
$\mathcal{O}_{C,x}$-modules, whose rows and columns are exact

(2.8) ${0}$${0}$${K}$${K}$${0}$${M}$${\mathcal{O}_{C,x}^{\oplus
r-k+a}}$${\tau}$${0}$${0}$${B_{x}}$${A_{x}}$${\tau}$${0}$${0}$${0\,.}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$

The sheaf $B$ is of rank $r-k$ and type $b$. So $B_{x}$ can be generated by
$r-k+b$ elements. By Lemma 2.5, $K$ is of rank $a$ and type $a$. So $K$ can be
generated by $2a$ elements. From the left vertical exact sequence it is clear
that the $M$ can be generated by $r-k+b+2a$ elements.

We note that the socle of $\tau$ depends only on the module structure over the
completion $\widehat{\mathcal{O}_{C,x}}$. We know that the completion
$\widehat{\mathcal{O}_{C,x}}$ is isomorphic to $\mathbb{C}[[X,Y]]/(XY)$. So
for the purpose of computing $\dim(\textnormal{Soc}(\tau))$, we can replace
$C$ by the curve $C^{\prime}$ defined by the vanishing of the polynomial
$\widetilde{X}\widetilde{Y}$ in the plane
$S:=\mathbb{P}^{2}_{\mathbb{C}}=\text{Proj}(\mathbb{C}[\widetilde{X},\widetilde{Y},\widetilde{Z}])$.
Thus we have a short exact sequence

$0\to\mathcal{O}_{S,x}\to\mathcal{O}_{S,x}\to\mathcal{O}_{C^{\prime},x}\to
0\,.$

Using this we can construct a commutative diagram whose rows and columns are
exact

(2.9) ${0}$${0}$${\mathcal{O}_{S,x}^{\oplus
r-k+a}}$${\mathcal{O}_{S,x}^{\oplus
r-k+a}}$${0}$${N}$${\mathcal{O}_{S,x}^{\oplus
r-k+a}}$${\tau}$${0}$${0}$${M}$${\mathcal{O}_{C^{\prime},x}^{\oplus
r-k+a}}$${\tau}$${0}$${0}$${0\,.}$$\scriptstyle{\psi}$

Similarly from this diagram we get that $N$ can be generated by
$(r-k+b+2a)+(r-k+a)=2(r-k)+3a+b$ elements. We view the surjective map
$\mathcal{O}_{S,x}^{\oplus r-k+a}\to\tau$ as a point in the Quot scheme of
torsion quotients of $\mathcal{O}_{S}^{\oplus r-k+a}$ on the surface $S$. Then
using [EL99, Lemma 2], we conclude that
$\dim(\textnormal{Soc}(\tau))\leq(2(r-k)+3a+b)-(r-k+a)=r-k+2a+b$. ∎

Recall that for a local ring $R$ and two $R$-modules $M$ and $N$, we have the
$R$-modules $\textnormal{Ext}^{i}_{R}(M,N)$. When $N$ has finite length, the
module $\textnormal{Ext}^{i}_{R}(M,N)$ has finite length. We denote the length
of $\textnormal{Ext}^{i}_{R}(M,N)$ by $\textnormal{ext}^{i}_{R}(M,N)$. Let us
take $R$ to be the local ring $\mathcal{O}_{C,x}$. Let $\tau$ be a torsion
sheaf over $C$ supported at the point $x$ and $B$ be any sheaf on $C$. Then we
have defined the numbers $\textnormal{ext}^{i}_{R}(B_{x},\tau)$. On the other
hand, $\textnormal{ext}^{i}(B,\tau)$ is the dimension of the
$\mathbb{C}$-vector space $\textnormal{Ext}^{i}_{C}(B,\tau)$. In the next
lemma we will use the fact that
$\textnormal{ext}^{i}(B,\tau)=\textnormal{ext}^{i}_{R}(B_{x},\tau)$ for all
$i$. This can be seen as follows. Consider the local-to-global spectral
sequence

$E_{2}^{p,q}=H^{p}(C,\mathscr{E}xt^{q}(B,\tau))\implies\textnormal{Ext}_{C}^{p+q}(B,\tau)\,.$

As $\tau$ is supported at the point $x$, the sheaves
$\mathscr{E}xt^{i}(B,\tau)$ are also supported at $x$. Consequently
$E_{2}^{p,q}=H^{p}(C,\mathscr{E}xt^{q}(B,\tau))=0$ for any $p\neq 0$ and for
any $q$. It follows that

$E_{\infty}^{p,q}=\begin{cases}H^{0}(C,\mathscr{E}xt^{q}(B,\tau))\quad&\text{if
}p=0\,,\\\ 0&\text{otherwise}\,.\end{cases}$

Now for any $i\geqslant 0$ it is clear that
$\textnormal{Ext}^{i}(B,\tau)=E^{0,i}_{\infty}=H^{0}(C,\mathscr{E}xt^{i}(B,\tau))$.
On the other hand, as $\mathscr{E}xt^{i}(B,\tau)$ is supported at the point
$x$, so
$H^{0}(C,\mathscr{E}xt^{i}(B,\tau))=\textnormal{Ext}^{i}_{R}(B_{x},\tau)$.
This proves that
$\textnormal{ext}^{i}(B,\tau)=\textnormal{ext}^{i}_{R}(B_{x},\tau)$ for any
$i\geqslant 0$.

######  Lemma 2.10.

Let $0<k<r$. Let $E$ be a locally free sheaf of rank $r$ and degree $e$ and
let $A$ be a sheaf of rank $k$ and degree $d$. Assume that $0\to B\to E\to
A\to 0$ is a short exact sequence of sheaves on $C$. Then we have

$\hom(B,A)-\textnormal{ext}^{1}(B,A)\geqslant dr-
ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

###### Proof.

Let $\tau$ be the torsion subsheaf of $A$ and $A^{\prime}$ be the torsion-free
quotient so that we have a short exact sequence

(2.11) $0\to\tau\to A\to A^{\prime}\to 0\,.$

Let the length of $\tau$ be $\delta$. We separate the torsion sheaf $\tau$ in
two parts. Let $\tau_{x}$ denote the torsion at the node $x$ and $\tau_{o}$
denote the torsion outside $x$. Thus, we have $\tau=\tau_{x}\oplus\tau_{o}$.
Let length$(\tau_{x})=\delta_{x}$ and length$(\tau_{o})=\delta_{o}$. So
$\delta_{x}+\delta_{o}=\delta$. Let $B^{\prime}$ be the kernel of the
surjection $E\to A^{\prime}\to 0$. By Lemma 2.5, type of $B^{\prime}$ is same
as type of $A^{\prime}$. It is easy to check that there is a short exact
sequence

(2.12) $0\to B\to B^{\prime}\to\tau\to 0\,.$

Applying $\textrm{Hom}(B,\\_)$ to the short exact sequence (2.11) we get,

(2.13) $\displaystyle\hom$ $\displaystyle(B,A)-\textnormal{ext}^{1}(B,A)$
$\displaystyle\geqslant\hom(B,A^{\prime})-\textnormal{ext}^{1}(B,A^{\prime})+\hom(B,\tau_{o})-\textnormal{ext}^{1}(B,\tau_{o})+\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x})\,.$

Next we compute $(\hom(B,A^{\prime})-\textnormal{ext}^{1}(B,A^{\prime}))$,
$(\hom(B,\tau_{o})-\textnormal{ext}^{1}(B,\tau_{o}))$ and
$(\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x}))$. The sheaves
$A^{\prime}$ and $B$ are torsion-free. Let the type of $A^{\prime}$ be
$a^{\prime}$ and let the type of $B$ be $b$. Degree of $A^{\prime}$ is
$d-\delta$ and degree of $B$ is $e-d$. By Lemma 2.4, we get that

(2.14) $\hom(B,A^{\prime})-\textnormal{ext}^{1}(B,A^{\prime})=dr-
ke+k(r-k)(1-g)-a^{\prime}b-\delta(r-k)\,.$

As $B$ is free on the support of $\tau_{o}$, so
$\hom(B,\tau_{o})=(r-k)\delta_{o}$ and $\textnormal{ext}^{1}(B,\tau_{o})=0$.
So we have

(2.15) $\hom(B,\tau_{o})-\textnormal{ext}^{1}(B,\tau_{o})=(r-k)\delta_{o}\,.$

Now we compute $\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x})$. Let $R$
denote the local ring $\mathcal{O}_{C,x}$ and $\mathfrak{m}_{x}$ denote the
maximal ideal of $R$. By the discussion before the lemma, we have
$\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x})=\hom_{R}(B_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(B_{x},\tau_{x})$.
Thus

(2.16) $\displaystyle\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x})$
$\displaystyle=\hom_{R}(B_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(B_{x},\tau_{x})$
$\displaystyle=\hom_{R}(R^{\oplus(r-k-b)}\oplus\mathfrak{m}^{\oplus
b},\tau_{x})-\textnormal{ext}^{1}_{R}(R^{\oplus(r-k-b)}\oplus\mathfrak{m}^{\oplus
b},\tau_{x})$
$\displaystyle=(r-k-b)[\hom_{R}(R,\tau_{x})-\textnormal{ext}^{1}_{R}(R,\tau_{x})]$
$\displaystyle\quad+b[\hom_{R}(\mathfrak{m}_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(\mathfrak{m}_{x},\tau_{x})]$
$\displaystyle=(r-k-b)\delta_{x}+b[\hom_{R}(\mathfrak{m}_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(\mathfrak{m}_{x},\tau_{x})]\,.$

Let $\widehat{R}$ denote the completion of $R$ with respect to the maximal
ideal $\mathfrak{m}_{x}$. As $\tau$ is also an $\widehat{R}$-module, so

$\hom_{R}(\mathfrak{m}_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(\mathfrak{m}_{x},\tau_{x})=\hom_{\widehat{R}}(\widehat{\mathfrak{m}}_{x},\tau_{x})-\textnormal{ext}^{1}_{\widehat{R}}(\widehat{\mathfrak{m}}_{x},\tau_{x})\,.$

We know that $\widehat{R}$ is isomorphic to $\mathbb{C}[[X,Y]]/(XY)$. So we
have a projective resolution of $\widehat{\mathfrak{m}}_{x}$,

$\dots\to\widehat{R}^{2}\xrightarrow{\begin{pmatrix}X&0\\\
0&Y\end{pmatrix}}\widehat{R}^{2}\xrightarrow{\begin{pmatrix}Y&0\\\
0&X\end{pmatrix}}\widehat{R}^{2}\longrightarrow\widehat{\mathfrak{m}}_{x}\to
0\,.$

Applying $\textrm{Hom}(\\_,\tau_{x})$ to the complex obtained by dropping
$\widehat{\mathfrak{m}}_{x}$, we get the complex

$\dots\xleftarrow{}\tau_{x}^{2}\xleftarrow{\phi_{2}=\begin{pmatrix}X&0\\\
0&Y\end{pmatrix}}\tau_{x}^{2}\xleftarrow{\phi_{1}=\begin{pmatrix}Y&0\\\
0&X\end{pmatrix}}\tau_{x}^{2}\xleftarrow{}0\,.$

It is easily checked that

$\hom_{\widehat{R}}(\widehat{\mathfrak{m}},\tau_{x})-\textnormal{ext}^{1}_{\widehat{R}}(\widehat{\mathfrak{m}},\tau_{x})=\dim(\text{im}\,(\phi_{2}))\,.$

Let $\Delta:\tau_{x}\to\tau_{x}^{\oplus 2}$ denote the diagonal map and let
$\phi:=\phi_{2}\circ\Delta$ denote the composite
$\tau_{x}\xrightarrow{\Delta}\tau_{x}^{\oplus
2}\xrightarrow{\phi_{2}}\tau_{x}^{\oplus 2}$. Then
$\dim(\text{im}\,(\phi_{2}))\geqslant\dim(\text{im}\,(\phi))$. The kernel of
$\phi$ is the subspace of $\tau_{x}$ which is annihilated by
$\widehat{\mathfrak{m}}_{x}$ and hence
$\ker(\phi)=\textnormal{Soc}(\tau_{x})$. As $\tau$ fits in the short exact
sequence (2.12), using Lemma 2.6 we get that
$\dim(\ker(\phi))=\dim(\textnormal{Soc}(\tau_{x}))\leq r-k+2a^{\prime}+b$, as
type of $B^{\prime}$ is $a^{\prime}$. So
$\dim(\text{im}\,(\phi))\geqslant\delta_{x}-(r-k+2a^{\prime}+b)$. Thus we have

$\hom_{\widehat{R}}(\widehat{\mathfrak{m}},\tau_{x})-\textnormal{ext}^{1}_{\widehat{R}}(\widehat{\mathfrak{m}},\tau_{x})=\dim(\text{im}\,(\phi_{2}))\geqslant\dim(\text{im}\,(\phi))\geqslant\delta_{x}-(r-k+2a^{\prime}+b)\,.$

Using (2.16), we get

(2.17) $\displaystyle\hom(B,\tau_{x})-\textnormal{ext}^{1}(B,\tau_{x})$
$\displaystyle=(r-k-b)\delta_{x}+b[\hom_{R}(\mathfrak{m}_{x},\tau_{x})-\textnormal{ext}^{1}_{R}(\mathfrak{m}_{x},\tau_{x})]$
$\displaystyle\geqslant(r-k-b)\delta_{x}+b(\delta_{x}-(r-k+2a^{\prime}+b))$
$\displaystyle\geqslant(r-k)\delta_{x}-b(r-k+2a^{\prime}+b)\,.$

Combining (2.13), (2.14), (2.15) and (2.16), we have

$\displaystyle\hom(B,A)-\textnormal{ext}^{1}(B,A)$ $\displaystyle\geqslant[dr-
ke+k(r-k)(1-g)-a^{\prime}b-\delta(r-k)]+[(r-k)\delta_{o}]+[(r-k)\delta_{x}-b(r-k+2a^{\prime}+b)]$
$\displaystyle=dr-ke+k(r-k)(1-g)-b(r-k+3a^{\prime}+b)\,.$

Note that $a^{\prime}\leq k$ and $b\leq(r-k)$. Thus we have

$\hom(B,A)-\textnormal{ext}^{1}(B,A)\geqslant dr-
ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

∎

##  3\. Dimensions of some subsets of the Quot scheme

Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$.

######  Proposition 3.1.

Any irreducible component of the quot scheme $Q_{k,d}(E)$ has dimension at
least

$dr-ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

###### Proof.

Let $[q:E\to F]$ be a closed point of $\mathcal{W}$ and $S_{F}$ denote the
kernel of $q$. Proposition 2.2.8 of [HL10] gives us the following bound for
the dimension of $Q_{k,d}(E)$ at the point $[q:E\to F]$,

(3.2)
$\dim_{[q]}Q_{k,d}(E)\geqslant\hom(S_{F},F)-\textnormal{ext}^{1}(S_{F},F)\,.$

Using Lemma 2.10, we have

$\dim_{[q]}Q_{k,d}(E)\geqslant\hom(S_{F},F)-\textnormal{ext}^{1}(S_{F},F)\geqslant
dr-ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

Since this is true for any closed point in $Q_{k,d}(E)$, it follows that any
irreducible component of $Q_{k,d}(E)$ has dimension at least $dr-
ke+k(r-k)(1-g)-(r-k)(2r+k)$. ∎

Let us define the following subsets of $Q_{k,d}(E)$.

$\displaystyle Q_{k,d}^{{\rm tf}}(E):=$ $\displaystyle\\{[q:E\to F]\in
Q_{k,d}(E):F\textrm{ is torsion-free }\\}\,,$ $\displaystyle Q_{k,d}^{{{\rm
tf}},a}(E):=$ $\displaystyle\\{[q:E\to F]\in Q_{k,d}(E):F\textrm{ is torsion-
free of type $a$}\\}$ $\displaystyle=$ $\displaystyle\\{[q:E\to F]\in
Q_{k,d}(E):F\textrm{ is torsion-free and
}\dim_{\mathbb{C}}(F\otimes\kappa(x))=k+a\\}\,.$

The set $Q_{k,d}^{{\rm tf}}(E)$ is open in $Q_{k,d}(E)$ by [HL10, Theorem
2.3.1]. The sets $Q_{k,d}^{{{\rm tf}},a}(E)$ are locally closed subsets of
$Q_{k,d}^{{{\rm tf}}}(E)$ by Lemma 3.3 below. We give the subsets
$Q_{k,d}^{{{\rm tf}},a}(E)$ the reduced subscheme structure. Consider the map

$\dim_{x}:Q_{k,d}^{{{\rm tf}}}(E)\to\mathbb{Z}$

which sends a closed point $[q:E\to F]$ to
$\dim_{\mathbb{C}}(F\otimes\kappa(x))$.

######  Lemma 3.3.

The map $\dim_{x}$ defined above is upper-semicontinuous.

###### Proof.

Let $\mathcal{F}$ be the restriction of the universal quotient sheaf on
$C\times Q_{k,d}^{{{\rm tf}}}(E)$. Consider the sheaf
$\mathcal{F}_{x}:=\mathcal{F}|_{\\{x\\}\times Q_{k,d}^{{{\rm tf}}}(E)}$ on
$Q_{k,d}^{{{\rm tf}}}(E)$. Then
$\dim_{\mathbb{C}}(F\otimes\kappa(x))=\dim_{\mathbb{C}}(\mathcal{F}_{x}|_{[q:E\to
F]})$. So the given map is upper-semicontinuous by [Har77, Example 12.7.2,
Chapter III]. ∎

Let $\pi:\widetilde{C}\to C$ be the normalization of $C$. Let $g$ and
$\widetilde{g}$ be the arithmetic genus of $C$ and $\widetilde{C}$
respectively. From the exact sequence of cohomology associated to the short
exact sequence

(3.4) $0\to\mathcal{O}_{C}\to\pi_{*}\mathcal{O}_{\widetilde{C}}\to\kappa(x)\to
0\,,$

one can see that $g=\widetilde{g}+1$. Let $x_{1}$ and $x_{2}$ be the two
points of $\widetilde{C}$ lying over the node $x$ of $C$.

######  Lemma 3.5.

Let $F$ be a torsion-free sheaf on $C$ of rank $k$ and type $a$. Then
$\deg(\pi^{*}F)=\deg(F)+a$. In particular, for any torsion-free sheaf $F$ on
$C$ of rank $k$, we have $\deg(\pi^{*}F)\leq\deg(F)+k$.

###### Proof.

Tensoring the short exact sequence
$0\to\mathcal{O}_{C}\to\pi_{*}\mathcal{O}_{\widetilde{C}}\to\kappa(x)\to 0$
with $F$, we get an exact sequence

$\textrm{Tor}^{1}(F,\kappa(x))\to F\to\pi_{*}\pi^{*}F\to F\otimes\kappa(x)\to
0\,.$

As $\textrm{Tor}^{1}(F,\kappa(x))$ is supported at the single point $x$ and
$F$ is torsion-free, we get a short exact sequence

(3.6) $0\to F\to\pi_{*}\pi^{*}F\to F\otimes\kappa(x)\to 0\,.$

This gives us $\chi(\pi_{*}\pi^{*}F)=\chi(F)+\chi(F\otimes\kappa(x))$. As
$\pi$ is a finite map,

(3.7) $\displaystyle\chi(\pi^{*}F)=\chi(\pi_{*}\pi^{*}F)$
$\displaystyle=\chi(F)+\chi(F\otimes\kappa(x))\,.$

Using (1.1) on both sides we have

$\deg(\pi^{*}F)+\text{rank}(F)(1-\widetilde{g})=\deg(F)+\text{rank}(F)(1-g)+k+a\,.$

Using $\widetilde{g}=g-1$, we get $\deg(\pi^{*}F)=\deg(F)+a$. ∎

######  Remark 3.8.

For any torsion-free sheaf $F$ on $C$, let $j:F\to\pi_{*}\pi^{*}F$ be the
inclusion in the short exact sequence (3.6). Restricting the map at $x$, we
have a map of vector spaces
$\iota:=j|_{x}:F|_{x}\to(\pi^{*}F|_{x_{1}}\oplus\pi^{*}F|_{x_{2}})$. As
$\text{coker}(j)$ is isomorphic to $F\otimes\kappa(x)$ which is supported
scheme theoretically at a single point $x$, so
$\text{coker}(j)=\text{coker}(j|_{x})$. So the short exact sequence (3.6) can
be rewritten as

(3.9) $0\to
F\xrightarrow{j}\pi_{*}\pi^{*}F\to(\pi^{*}F|_{x_{1}}\oplus\pi^{*}F|_{x_{2}})/\iota(F|_{x})\to
0\,.$

Let $\psi:F\to G$ be a map of torsion-free sheaves on $C$. Then the following
diagram is commutative which can be checked locally

(3.10)
${F}$${\pi_{*}\pi^{*}F}$${G}$${\pi_{*}\pi^{*}G\,.}$$\scriptstyle{J_{F}}$$\scriptstyle{\psi}$$\scriptstyle{\pi_{*}\pi^{*}\psi}$$\scriptstyle{j_{G}}$

######  Proposition 3.11.

For each type $a$, there exists a map

$f_{a}:Q_{k,d}^{{{\rm tf}},a}(E)\to{\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\pi^{*}E,k,d+a)$

which is injective on closed points.

###### Proof.

Let us fix a type $a$ with $0\leq a\leq k$. Let $p_{C}:C\times Q_{k,d}(E)\to
C$ be the projection map. There is a universal quotient
$p_{C}^{*}E\to\mathcal{F}\to 0$ on $C\times Q_{k,d}(E)$, $\mathcal{F}$ is flat
over $Q_{k,d}(E)$. Let $\mathcal{F}_{a}$ denote the restriction of
$\mathcal{F}$ on the subset $Q_{k,d}^{{{\rm tf}},a}(E)$. Let
$\widetilde{\pi}:\widetilde{C}\times Q_{k,d}^{{{\rm tf}},a}(E)\to C\times
Q_{k,d}^{{{\rm tf}},a}(E)$ denote the map
$\pi\times\textup{id}_{Q_{k,d}^{{{\rm tf}},a}(E)}$. Pulling back the quotient
via the map $\widetilde{\pi}$, we get a quotient on $\widetilde{C}\times
Q_{k,d}^{{{\rm tf}},a}(E)$

(3.12)
$\widetilde{\pi}^{*}p_{C}^{*}E\longrightarrow\widetilde{\pi}^{*}\mathcal{F}_{a}\longrightarrow
0\,.$

We aim to show that $\widetilde{\pi}^{*}\mathcal{F}_{a}$ is flat over
$Q_{k,d}^{{{\rm tf}},a}(E)$. Let $[\varphi:E\to F]\in Q_{k,d}^{{{\rm
tf}},a}(E)$ be a closed point. Then the fiber
$(\pi^{*}\mathcal{F}_{a})|_{[\varphi]}$ is the pullback sheaf $\pi^{*}F$ on
$\widetilde{C}$. As $F$ is torsion-free, we have the short exact sequence of
sheaves on $C$

$0\to F\to\pi_{*}\pi^{*}F\to F\otimes\kappa(x)\to 0\,.$

We easily see that the Hilbert polynomial of
$(\widetilde{\pi}^{*}\mathcal{F}_{a})_{[\varphi:E\to F]}=\pi^{*}F$ is constant
as a function of points $[\varphi:E\to F]$ in $Q_{k,d}^{{{\rm tf}},a}(E)$.
Using [HL10, Proposition 2.1.2], one concludes that
$\widetilde{\pi}^{*}\mathcal{F}_{a}$ is flat over $Q_{k,d}^{{{\rm tf}},a}(E)$.
So the quotient (3.12) induces a map

$f_{a}:Q_{k,d}^{{{\rm tf}},a}(E)\longrightarrow{\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\pi^{*}E,k,d+a)\,,$

as we have seen that $\deg(\pi^{*}F)=\deg(F)+a$ in Lemma 3.5.

Next we show that the map $f_{a}$ is injective on closed points. We start with
two closed points $\varphi_{1}:E\to F_{1}$ and $\varphi_{2}:E\to F_{2}$ in
$Q_{k,d}^{{{\rm tf}},a}(E)$ such that $\pi^{*}\varphi_{1}=\pi^{*}\varphi_{2}$.
So there is an isomoprhism $\psi:\pi^{*}F_{1}\to\pi^{*}F_{2}$ satisfying the
commutative diagram

${\pi^{*}E}$${\pi^{*}F_{1}}$${\pi^{*}E}$${\pi^{*}F_{2}\,.}$$\scriptstyle{\pi^{*}\varphi_{1}}$$\scriptstyle{\psi}$$\scriptstyle{\pi^{*}\varphi_{2}}$

Let $\iota_{i}$ and $j_{i}$ denote the maps $\iota_{F_{i}}$ and $j_{F_{i}}$
(see Remark 3.8 for the defintion of the maps). It is easy to see that

$(\psi|_{x_{1}}\oplus\psi|_{x_{2}})(\iota_{1}(F_{1}|_{x}))\subseteq\iota_{2}(F_{2}|_{x_{2}})$

using the following diagram in which the two squares and the two triangles
commute, and $\varphi_{1}|_{x}$ is surjective.

(3.13)
${F_{1}|_{x}}$${E|_{x}}$${\pi^{*}E|_{x_{1}}\oplus\pi^{*}E|_{x_{2}}}$${\pi^{*}F_{1}|_{x_{1}}\oplus\pi^{*}F_{1}|_{x_{2}}}$${E|_{x}}$${\pi^{*}E|_{x_{1}}\oplus\pi^{*}E|_{x_{2}}}$${\pi^{*}F_{1}|_{x_{1}}\oplus\pi^{*}F_{1}|_{x_{2}}}$${F_{2}|_{x}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\varphi_{1}|_{x}}$$\scriptstyle{\iota_{E}}$$\scriptstyle{\pi^{*}\varphi_{1}|_{x_{1}}\oplus\pi^{*}\varphi_{1}|_{x_{2}}}$$\scriptstyle{\psi|_{x_{1}}\oplus\psi|_{x_{2}}}$$\scriptstyle{\varphi_{2}|_{x}}$$\scriptstyle{\iota_{E}}$$\scriptstyle{\pi^{*}\varphi_{1}|_{x_{1}}\oplus\pi^{*}\varphi_{1}|_{x_{2}}}$$\scriptstyle{\iota_{2}}$

It follows that the map $\psi|_{x_{1}}\oplus\psi|_{x_{2}}$ induces a map

$\widetilde{\psi_{x}}:(\pi^{*}F_{1}|_{x_{1}}\oplus\pi^{*}F_{1}|_{x_{2}})/\iota_{1}(F_{1}|_{x})\longrightarrow(\pi^{*}F_{2}|_{x_{1}}\oplus\pi^{*}F_{2}|_{x_{2}})/\iota_{2}(F_{2}|_{x})\,.$

Consequently we get a map $\theta:F_{1}\to F_{2}$ given by the dashed arrow in
the following diagram

(3.14)
${0}$${F_{1}}$${\pi_{*}\pi^{*}F_{1}}$${(\pi^{*}F_{1}|_{x_{1}}\oplus\pi^{*}F_{2}|_{x_{2}})/\iota_{1}(F_{1}|_{x})}$${0}$${0}$${F_{2}}$${\pi_{*}\pi^{*}F_{2}}$${(\pi^{*}F_{2}|_{x_{1}}\oplus\pi^{*}F_{2}|_{x_{2}})/\iota_{2}(F_{2}|_{x})}$${0\,.}$$\scriptstyle{J_{1}}$$\scriptstyle{\theta}$$\scriptstyle{\pi_{*}\psi}$$\scriptstyle{\widetilde{\psi_{x}}}$$\scriptstyle{j_{2}}$

Considering the inverse of $\psi$ and proceeding similarly we get an inverse
of $\widetilde{\psi_{x}}$. As $\pi_{*}\psi$ and $\widetilde{\psi_{x}}$ both
are isomorphism, so is $\theta$. It remains to check that the following
diagram commutes

(3.15)
${E}$${F_{1}}$${E}$${F_{2}\,.}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\theta}$$\scriptstyle{\varphi_{2}}$

Since $\pi$ is an isomorphism outside the point $x$, it is clear that
$\theta=\pi^{*}\psi$ on $C\setminus\\{x\\}$. Now the set over which these two
maps agree is closed in $C$ containing $C\setminus\\{x\\}$, so it has to be
the whole of $C$. This proves that $\theta=\pi^{*}\psi$ on $C$. Hence
$\theta:F_{1}\to F_{2}$ is an isomorphism satisfying
$\theta\circ\varphi_{1}=\varphi_{2}$, which proves that $\varphi_{1}$ and
$\varphi_{2}$ corresponds to the same point in $Q_{k,d}^{{{\rm tf}},a}(E)$.
This proves injectivity of $f^{a}$ on closed points. ∎

We will use the map $f_{a}$ to bound the dimension of some subsets of the Quot
scheme $Q_{k,d}(E)$ using bounds of related subsets in the quot schemes ${\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\pi^{*}E,k,d^{\prime})$. We start by
computing dimension bound for a subset of ${\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\pi^{*}E,k,d^{\prime})$. The following lemma
is contained in the proof of [PR03, Lemma 6.3]. However we give a detailed
proof for the convenience of the reader. For ease of notation let us denote
$\pi^{*}E$ by $\widetilde{E}$ for rest of this section.

Let $d^{\prime}$, $d_{0}$ be integers and let $k_{0}$ be an integer such that
$0<k_{0}<k$. Let $\widetilde{Z}$ be the subset of ${\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d^{\prime})$ defined as
follows,

(3.16) $\widetilde{Z}=\\{[\widetilde{E}\to G]\in{\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d^{\prime}):\exists\textrm{ a
quotient }G\to G_{0}\textrm{ of rank }k_{0}\textrm{ and degree }d_{0}\\}\,.$

######  Lemma 3.17.

There exist constants $C_{1}(E,k,k_{0},d_{0})$ and
$\alpha_{1}(E,k,k_{0},d_{0})$ such that we have

$\dim\widetilde{Z}\leq d^{\prime}(r-k_{0})+C_{1}\quad\quad\text{ for any
}\quad d^{\prime}\geqslant\alpha_{1}\,.$

###### Proof.

Let $A$ denote the Quot scheme ${\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k_{0},d_{0})$. Let
$\mathscr{S}_{0}$ denote the universal kernel on $\widetilde{C}\times A$.
Consider the relative quot scheme

$D:={\rm Quot}_{\widetilde{C}\times
A/A}(\mathscr{S}_{0},k-k_{0},d^{\prime}-d_{0})\,.$

A closed point of $D$ corresponds to a pair $([q_{0}],[\varphi])$ where
$[q_{0}:\widetilde{E}\to F_{0}]\in A$ and $\varphi:\ker(q_{0})\to B$ is a
quotient of rank $k-k_{0}$ and degree $d^{\prime}-d_{0}$. Given such a point,
we can construct a quotient $q:\widetilde{E}\to F$ using the following push-
out diagram

${0}$${\ker(q_{0})}$${\widetilde{E}}$${F_{0}}$${0\phantom{\,.}}$${0}$${B}$${F}$${F_{0}}$${0\,.}$$\scriptstyle{\varphi}$$\scriptstyle{q_{0}}$$\scriptstyle{q}$

Clearly, the degree of $F$ is $d^{\prime}$ and so $q$ corresponds to a point
of ${\rm Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d^{\prime})$. It can
be checked that there is a map of schemes

$g:D\to{\rm Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d^{\prime})$

which sends the point $([q_{0}],[\varphi])$ to the point $[q]$. Given any
point $[q:\widetilde{E}\to F]$ in $\widetilde{Z}$, we have a quotient
$[q_{0}:\widetilde{E}\to F_{0}]\in A$ (by definition of the locus
$\widetilde{Z}$) and we get a quotient $\ker q_{0}\to B\to 0$ of rank
$k-k_{0}$ and degree $d^{\prime}-d_{0}$. This shows that $g$ maps onto the
subset $\widetilde{Z}$. Consequently, we have the following upper bound

(3.18) $\displaystyle\dim\widetilde{Z}$ $\displaystyle\leq\dim D\leq\dim
A+\max_{q_{0}\in A}\left\\{\dim\left({\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\ker
q_{0},d^{\prime}-d_{0},k-k_{0})\right)\right\\}\,.$

Using [PR03, Theorem 4.1] we have, for all $d^{\prime}-d_{0}\geqslant m(\ker
q_{0},k-k_{0}))$,

$\dim\left({\rm Quot}_{\widetilde{C}/\mathbb{C}}(\ker
q_{0},d^{\prime}-d_{0},k-k_{0})\right)\leqslant(k-k_{0})(r-k)+(d^{\prime}-d_{0}-m(\ker
q_{0},k-k_{0}))(r-k_{0})\,.$

As the collection $\\{\ker q_{0}:q_{0}\in A\\}$ is a bounded family, by Lemma
2.2, the set $\\{m(\ker q_{0},k-k_{0}):q_{0}\in A\\}$ is finite. So there
exist constants $M^{\prime}(E,k)$ and $M^{\prime\prime}(E,k)$ such that

$M^{\prime}\leqslant m(\ker q_{0},k-k_{0})\leqslant
M^{\prime\prime}\quad\textrm{ for all }[q_{0}]\in A.$

It follows that, for all $d^{\prime}\geqslant M^{\prime\prime}+d_{0}$, we have

$\displaystyle\dim\left({\rm Quot}_{\widetilde{C}/\mathbb{C}}(\ker
q_{0},d^{\prime}-d_{0},k-k_{0})\right)\leqslant(k-k_{0})(r-k)+$
$\displaystyle(d^{\prime}-d_{0}-M^{\prime})(r-k_{0})\,.$

Consequently, from (3.18) we have

$\displaystyle\dim\widetilde{Z}$ $\displaystyle\leqslant\dim A+\max_{q_{0}\in
A}\left\\{\dim\left({\rm Quot}_{\widetilde{C}/\mathbb{C}}(\ker
q_{0},d^{\prime}-d_{0},k-k_{0})\right)\right\\}$
$\displaystyle\leqslant\dim({\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k_{0},d_{0}))+(k-k_{0})(r-k)+(d^{\prime}-d_{0}-M^{\prime})(r-k_{0})$
$\displaystyle\leqslant d^{\prime}(r-k_{0})+\dim({\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k_{0},d_{0}))+(k-k_{0})(r-k)-(d_{0}+M^{\prime})(r-k_{0})\,.$

We define $\alpha_{1}:=M^{\prime\prime}+d_{0}$ and $C_{1}:=\dim({\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k_{0},d_{0}))+(k-k_{0})(r-k)-(d_{0}+M^{\prime})(r-k_{0})$.
This proves the lemma. ∎

######  Lemma 3.19.

Let $d_{0}$ and $k_{0}$ be integers such that $0<k_{0}<k$. Let
$Y_{d_{0},k_{0}}$ be a subset of $Q^{{\rm tf}}_{k,d}(E)$ defined as follows

(3.20) $Y_{d_{0},k_{0}}=\\{[E\to F]\in Q^{{\rm tf}}_{k,d}(E):\pi^{*}F\textrm{
has a quotient of rank $k_{0}$ and degree }d_{0}\\}\,.$

Then there exist constants $C_{2}(E,k,k_{0},d_{0})$ and
$\alpha_{2}(E,k,k_{0},d_{0})$ such that

$\dim Y_{d_{0},k_{0}}\leq d(r-k_{0})+C_{2}\quad\quad\text{ for any }\quad
d\geqslant\alpha_{2}\,.$

###### Proof.

For the sake of notation, we denote $Y_{d_{0},k_{0}}$ simply by $Y$ in this
proof. For any type $a$, let $Y^{a}:=Y\cap Q_{k,d}^{{{\rm tf}},a}(E)$. Then
$Y$ is disjoint union of the finitely many subsets $Y^{a}$ and so it is enough
to bound the dimension of $Y^{a}$. Fix a type $a$. Consider the map defined in
Proposition 3.11,

$f_{a}:Q_{k,d}^{{{\rm tf}},a}(E)\to{\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d+a)\,.$

Consider the subset $\widetilde{Z}$ of ${\rm
Quot}_{\widetilde{C}/\mathbb{C}}(\widetilde{E},k,d+a)$ as defined in (3.16),
taking $d^{\prime}=d+a$. It is clear that the image $f_{a}(Y^{a})$ is
contained in $\widetilde{Z}$. Using Lemma 3.17, we get constants $C_{1}$ and
$\alpha_{1}$ such that

$\dim\widetilde{Z}\leq(d+a)(r-k_{0})+C_{1}\quad\text{ for
}d+a\geqslant\alpha_{1}\,.$

As $f_{a}$ is injective on points, we have the following

$\dim
Y^{a}=\dim(f^{a}(Y^{a}))\leq\dim\widetilde{Z}\leq(d+a)(r-k_{0})+C_{1}\quad\text{
for }d\geqslant\alpha_{1}\,.$

Taking $\alpha_{2}:=\alpha_{1}$ and $C_{2}:=k(r-k_{0})+C_{1}$, we have

$\dim Y\leq d(r-k_{0})+C_{2}\text{ for }\quad d\geqslant\alpha_{2}\,.$

This proves the lemma. ∎

######  Lemma 3.21.

Given a number $\mu_{0}$, define the subset $Y_{\mu_{0}}$ of $Q^{{\rm
tf}}_{k,d}(E)$ as follows

$Y_{\mu_{0}}=\\{[E\to F]\in Q^{{\rm tf}}_{k,d}(E):\mu_{\min}(F)<\mu_{0}\\}\,.$

Then there exist constants $C_{3}(E,k,\mu_{0})$ and $\alpha_{3}(E,k,\mu_{0})$
such that

$\dim Y_{\mu_{0}}\leq d(r-1)+C_{3}\quad\quad\text{ for any }\quad
d\geqslant\alpha_{3}\,.$

###### Proof.

Let $[q:E\to F]\in Y_{\mu_{0}}$ be a closed point. Let $F\to F^{\prime}\to 0$
denote the minimal destabilizing quotient of $F$. Note that $F^{\prime}$ is
torsion-free and $\mu(F^{\prime})=\mu_{\min}(F)<\mu_{0}$. Again $F^{\prime}$
is a quotient of $E$, so $\mu(F^{\prime})\geqslant\mu_{\min}(E)$. Thus
$\mu_{\min}(E)\leq\mu(F^{\prime})<\mu_{0}$. As
$1\leq\text{rank}(F^{\prime})\leq k-1$, we have
$\mu_{\min}(E)\leq\deg(F^{\prime})<(k-1)\mu_{0}$. Using Lemma 3.5, we see that
$\mu_{\min}(E)\leq\deg(\pi^{*}F^{\prime})<(k-1)\mu_{0}+k$. In particular,
there are only finitely many choices for the degree of $\pi^{*}F^{\prime}$.
Recall the locus $Y_{d_{0},k_{0}}$ from (3.20). As $\pi^{*}F^{\prime}$ is a
quotient of $\pi^{*}F$, the point $[q:E\to F]$ is in the locus

$Y_{\deg(\pi^{*}F^{\prime}),\rank(\pi^{*}F^{\prime})}\,.$

Thus, we can write $Y_{\mu_{0}}$ as a union of finitely many subsets
$Y_{d_{0},k_{0}}$ with $1\leq k_{0}\leq k-1$. Now the lemma follows easily
using Lemma 3.19. ∎

Before going to the next lemma, let us recall the following result.

######  Lemma 3.22 (Corollary in Appendix, Page 87 of [OS79]).

Let $A\to B$ be a flat local homomorphism of noetherian local rings. Let $N$
be a $B$-module of finite type which is $A$-flat and satisfies
$\textnormal{Ext}^{1}(\overline{N},\overline{B})=0$, where
$\overline{B}=B\otimes_{A}k$ and $\overline{N}=N\otimes_{A}k$ for the residue
field $k$ of $A$. Then $\textrm{Hom}_{B}(N,B)$ is $A$-flat and
$\textrm{Hom}_{B}(N,B)\otimes_{A}k=\textrm{Hom}_{\overline{B}}(\overline{N},\overline{B})$.

######  Lemma 3.23.

For any degree $d$, there is a map of Quot schemes

$\mathcal{V}:Q_{k,d}^{{\rm tf}}(E)\to Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})$

which sends a closed point $[q:E\to F]$ to the point $[E^{\vee}\to(\ker
q)^{\vee}]$. Moreover, the map $\mathcal{V}$ is an isomorphism.

###### Proof.

Since $E,k$ and $d$ are fixed in this lemma, let us write $Q^{{\rm tf}}$
instead of $Q_{k,d}^{{\rm tf}}(E)$ in this proof. Let $p_{C}:C\times Q^{{\rm
tf}}\to C$ denote the projection map. Consider the universal short exact
sequence on $C\times Q^{{\rm tf}}$

(3.24) $0\to\mathscr{S}\to p_{C}^{*}E\to\mathcal{F}\to 0\,.$

The sheaves $\mathcal{F}$ and $\mathscr{S}$ are flat over $Q^{{\rm tf}}$.
Consider the duals

$\mathcal{F}^{\vee}:=\mathscr{H}om(\mathcal{F},\mathcal{O}_{C\times Q^{{\rm
tf}}})\quad\text{ and
}\quad\mathscr{S}^{\vee}:=\mathscr{H}om(\mathscr{S},\mathcal{O}_{C\times
Q^{{\rm tf}}})\,.$

We claim that the sheaf $\mathcal{F}^{\vee}$ is flat over $Q^{{\rm tf}}$.
Since flatness is a local property, it is enough to check that the stalk
$(\mathcal{F}^{\vee})_{(c,q)}$ is flat over the local ring
$\mathcal{O}_{Q^{{\rm tf}},q}$ for any closed point $(c,q)\in C\times Q^{{\rm
tf}}$. Let $(c,[q:E\to F])$ be a closed point of $C\times Q^{{\rm tf}}$.
Taking $A=\mathcal{O}_{Q^{{\rm tf}},q}$,
$B=\mathcal{O}_{C,c}\otimes\mathcal{O}_{Q^{{\rm tf}},q}$ and
$N=(\mathcal{F})_{(c,q)}$, we are in the situation of Lemma 3.22. Note that
$\overline{B}$ is the local ring $\mathcal{O}_{C,c}$ and $\overline{N}$ is the
stalk $F_{c}$. As $F$ is a torsion-free sheaf over $C$, so the stalk $F_{c}$
is a torsion-free $\overline{B}$-module. If $c$ is a non-singular point then
clearly $\textnormal{Ext}^{1}_{\mathcal{O}_{C,c}}(F_{c},\mathcal{O}_{C,c})=0$.
For the node $x$, we have
$\textnormal{Ext}^{1}_{\mathcal{O}_{C,x}}(\mathfrak{m}_{x},\mathcal{O}_{C,x})=0$
(for example see [Bho06, Lemma 2.1]). Hence,
$\textnormal{Ext}^{1}_{\mathcal{O}_{C,x}}(F_{x},\mathcal{O}_{C,x})=0$. This
gives us $Ext^{1}_{\overline{B}}(\overline{N},\overline{B})=0$. Using Lemma
3.22, we conclude that $(\mathcal{F}^{\vee})_{(c,q)}$ is $\mathcal{O}_{Q^{{\rm
tf}},q}$-flat, hence, $\mathcal{F}^{\vee}$ is $Q^{{\rm tf}}$-flat, and the
natural map $\mathcal{F}^{\vee}|_{C\times\\{q\\}}\longrightarrow F^{\vee}$ is
an isomorphism. Similarly, replacing $\mathcal{F}$ by $\mathscr{S}$ in the
above argument we get that $\mathscr{S}^{\vee}$ is $Q^{{\rm tf}}$-flat and the
natural map $\mathscr{S}^{\vee}|_{C\times\\{q\\}}\longrightarrow S^{\vee}$ is
an isomorphism.

Dualizing the short exact sequence (3.24) we get the exact sequence

(3.25)
$0\to\mathcal{F}^{\vee}\to(p_{C}^{*}E)^{\vee}\to\mathscr{S}^{\vee}\to\mathscr{E}xt^{1}(\mathcal{F},\mathcal{O}_{C\times
Q^{{\rm tf}}})\to 0\,.$

Restricting on the fiber $C\times\\{q\\}$, we get a sequence of maps

$\mathcal{F}^{\vee}|_{C\times\\{q\\}}\to(p_{C}^{*}E)^{\vee}|_{C\times\\{q\\}}\to\mathscr{S}^{\vee}|_{C\times\\{q\\}}\to\mathscr{E}xt^{1}(\mathcal{F},\mathcal{O}_{C\times
Q^{{\rm tf}}})|_{C\times\\{q\\}}$

such that the rightmost map is surjective. Restriction of the short exact
sequence (3.24) on the fiber $C\times\\{q\\}$ gives the short exact sequence

(3.26) $0\to S_{F}\to E\to F\to 0\,.$

Dualizing the short exact sequence (3.26), by Lemma 2.2(1) of [Bho06] we get
another short exact sequence

(3.27) $0\to F^{\vee}\to E^{\vee}\to(S_{F})^{\vee}\to 0\,.$

So we have the following diagram in which the vertical arrows are the natural
maps and the bottom sequence is exact

(3.28)
${\mathcal{F}^{\vee}|_{C\times\\{q\\}}}$${(p_{C}^{*}E)^{\vee}|_{C\times\\{q\\}}}$${\mathscr{S}^{\vee}|_{C\times\\{q\\}}}$${0}$${F^{\vee}}$${E^{\vee}}$${(S_{F})^{\vee}}$${0\,.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The squares commute by the naturality of vertical maps. It follows from the
diagram that the map
$(p_{C}^{*}E)^{\vee}|_{C\times\\{q\\}}\longrightarrow\mathscr{S}^{\vee}|_{C\times\\{q\\}}$
is surjective. Using Nakayama Lemma we conclude that the map
$(p_{C}^{*}E)^{\vee}\longrightarrow\mathscr{S}^{\vee}$ is surjective. So we
conclude that $\mathscr{E}xt^{1}(\mathcal{F},\mathcal{O}_{C\times Q^{{\rm
tf}}})=0$ and we have a short exact sequence

$0\to\mathcal{F}^{\vee}\to(p_{C}^{*}E)^{\vee}\to\mathscr{S}^{\vee}\to 0\,.$

The quotient $(p_{C}^{*}E)^{\vee}\to\mathscr{S}^{\vee}\to 0$ defines a map of
schemes

$\mathcal{V}:Q^{{\rm tf}}\to Q_{r-k,d-e}(E^{\vee})\,.$

which sends the closed point $[q:E\to F]$ to the point $E^{\vee}\to(\ker
q)^{\vee}\to 0$. It is clear that the image of $\mathcal{V}$ lies inside the
torsion-free locus $Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})$. Thus we have a map

$\mathcal{V}:Q_{k,d}^{{\rm tf}}(E)\longrightarrow Q^{{\rm
tf}}_{r-k,d-e}(E^{\vee})\,.$

Similarly we can get a map in the reverse direction

$\mathcal{V}^{\prime}:Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})\longrightarrow
Q_{k,d}^{{\rm tf}}(E)\,.$

To see that $\mathcal{V}^{\prime}\circ\mathcal{V}=\textnormal{Id}$, we need to
show that the natural map $\mathcal{F}\to\mathcal{F}^{\vee\vee}$ is an
isomorphism. Lemma 2.6(i) of [Bho06] says that for any torsion-free sheaf $G$
on $C$, the natural map $G\to G^{\vee\vee}$ is an isomorphism. Using this, it
is easy to see that the map $\mathcal{F}\to\mathcal{F}^{\vee\vee}$ is
surjective. Using flatness of $\mathcal{F}^{\vee\vee}$ over $Q_{k,d}^{{\rm
tf}}(E)$, one can check that the map $\mathcal{F}\to\mathcal{F}^{\vee\vee}$ is
also an inclusion and hence an isomorphism. Similarly one can prove that
$\mathcal{V}\circ\mathcal{V}^{\prime}=\textnormal{Id}$. It follows that
$\mathcal{V}$ is an isomorphism. ∎

######  Lemma 3.29.

Given a number $\mu_{0}$, define the subset $X_{\mu_{0}}$ of $Q^{{\rm
tf}}_{k,d}(E)$ as follows

$X_{\mu_{0}}=\\{[q:E\to F]\in Q^{{\rm tf}}_{k,d}(E):\mu_{\max}(\ker
q)>\mu_{0}\\}\,.$

Then there exist constants $C_{4}(E,k,\mu_{0})$ and $\alpha_{4}(E,k,\mu_{0})$
such that for $d\geqslant\alpha_{4}$,

$\dim X_{\mu_{0}}\leq d(r-1)+C_{4}\,.$

###### Proof.

Let $[q:E\to F]$ be a closed point in $X_{\mu_{0}}$ and $S_{F}$ denote the
kernel of $q$. Then we have $\mu_{\max}(S_{F})>\mu_{0}$ which implies that
$\mu_{\min}(S_{F}^{\vee})<-\mu_{0}$. We consider the map

$\mathcal{V}:Q_{k,d}^{{\rm tf}}(E)\to Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})$

which sends the point $[q:E\to F]$ to the point $[E^{\vee}\to S_{F}^{\vee}]$.
Replacing $E$ by $E^{\vee}$ in Lemma 3.21, we consider the subset
$Y_{-\mu_{0}}$ of $Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})$, which contains all the
points $[E^{\vee}\to G]$ such that $\mu_{\min}(G)<-\mu_{0}$. It follows that
the image $\mathcal{V}(X_{\mu_{0}})$ is same as the subset $Y_{-\mu_{0}}$ in
$Q^{{\rm tf}}_{r-k,d-e}(E^{\vee})$. By Lemma 3.21, we get constants
$C^{\prime}_{3}:=C_{3}(E^{\vee},k,-\mu_{0})$ and
$\alpha_{3}^{\prime}:=\alpha_{3}(E^{\vee},k,-\mu_{0})$ such that $\dim
Y_{-\mu_{0}}\leq(d-e)(r-1)+C^{\prime}_{3}$ for
$d\geqslant\alpha^{\prime}_{3}$. As $\mathcal{V}$ is bijective on points, we
get that for $d\geqslant\alpha^{\prime}_{3}$,

$\dim
X_{\mu_{0}}=\dim(\mathcal{V}(X_{\mu_{0}}))\leq(d-e)(r-1)+C^{\prime}_{3}\,.$

Taking $\alpha_{4}:=\alpha^{\prime}_{3}$ and $C_{4}:=-e(r-1)+C^{\prime}_{3}$,
the Lemma follows. ∎

##  4\. Irreducible components containing stable locally free quotients

In this section we will show that for large enough $d$, there is a unique
irreducible component of $Q_{k,d}(E)$ whose general point corresponds to a
stable locally free quotient. Before proving the theorem we prove few lemmas
which will be required in the proof of the theorem.

######  Lemma 4.1.

There exists an integer $d_{1}=d_{1}(E,k)$ such that for all stable locally
free sheaf $F$ of rank $k$ and degree $d\geqslant d_{1}$ on $C$, the sheaf
$E^{\vee}\otimes F$ is generated by global sections and $H^{1}(E^{\vee}\otimes
F)=0$.

###### Proof.

This can be proved exactly as it is proved in smooth curve case in [PR03,
Lemma 6.1]. ∎

######  Lemma 4.2.

If $d\geqslant d_{1}$, then for any quotient $E\to F\to 0$ where $F$ is a
stable locally free sheaf of rank $k$ and degree $d$, we have
$\textrm{h}^{1}(S_{F}^{\vee}\otimes F)=0$ where $S_{F}$ is the kernel of the
quotient $E\to F$. As a consequence we have

$\chi(S_{F}^{\vee}\otimes F)=h^{0}(S_{F}^{\vee}\otimes F)=dr-ke+k(r-k)(1-g)$

###### Proof.

As $d\geqslant d_{1}$, from lemma 4.1 we have $h^{1}(E^{\vee}\otimes F)=0$.
Using the long exact sequence of cohomology obtained from the short exact
sequence $0\to F^{\vee}\otimes F\to E^{\vee}\otimes F\to S_{F}^{\vee}\otimes
F\to 0$, we get that $h^{1}(S_{F}^{\vee}\otimes F)=0$. Using (1.1) we get
$h^{0}(S_{F}^{\vee}\otimes F)=\chi(S_{F}^{\vee}\otimes F)=dr-ke+k(r-k)(1-g)$.
∎

######  Lemma 4.3.

Let $F$ be a stable locally free sheaf of rank $k$ and degree $d$ such that
the sheaf $\mathscr{H}om(E,F)=E^{\vee}\otimes F$ is globally generated and
$H^{1}(E^{\vee}\otimes F)=0$. Then a general homomorphism from $E$ to $F$ is
surjective.

###### Proof.

The idea of the proof is taken from the proof of [GS22, Lemma 7.12]. Consider
the space $\mathbb{P}(\textrm{Hom}(E,F)^{\vee})$ which parametrizes non-zero
maps $E\to F$. Let $T$ be the subset of $\mathbb{P}(\textrm{Hom}(E,F)^{\vee})$
which contains all the maps $\varphi:E\to F$ such that $\varphi|_{c}=0$ for
atleast one point $c\in C$. Then $T$ is a closed subset and as a set

$T=\bigcup_{c\in
C}\mathbb{P}(\textrm{Hom}(E,F\otimes\mathfrak{m}_{c})^{\vee})$

where $\mathfrak{m}_{c}$ denotes the sheaf of ideals of the point $c$. As
$E^{\vee}\otimes F$ is globally generated and $H^{1}(E^{\vee}\otimes F)=0$, it
follows that $H^{1}(E^{\vee}\otimes F\otimes\mathfrak{m}_{c})=0$ for every
closed point $c\in C$. Using this one can check that

(4.4) $\dim(\mathbb{P}(\textrm{Hom}(E,F)^{\vee}))-\dim T\geqslant
rk-1\geqslant r-k.$

Let $V$ denote the open set $\mathbb{P}(\textrm{Hom}(E,F)^{\vee})\setminus T$.
One can define a map $\nu:C\times V\to\mathbb{P}(E\otimes F^{\vee})$ which
sends a closed point $(c,[\varphi])$ of $C\times V$ to the point
$(c,\varphi|_{c})$ in $\mathbb{P}(E\otimes F^{\vee})$. The map $\nu$ satisfies
the following diagram

${C\times V}$${\mathbb{P}(E\otimes
F^{\vee})}$${C}$$\scriptstyle{\nu}$$\scriptstyle{\eta}$

where $\eta$ is the structure map. It can be checked that, for any $c\in C$,
the restriction $\nu_{c}:V\to\mathbb{P}(E|_{c}\otimes F^{\vee}|_{c})$ is flat
(for details see page 17 of [GS22]). Now using [Stk, Tag 039E], it follows
that $\nu$ is flat. Consider the canonical map on $\mathbb{P}(E\otimes
F^{\vee})$,

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

and let $Z$ be the support of the cokernel of the map above. Then the set
$Z\cap\eta^{-1}(c)$ is the locus of the non-surjective maps in
$\mathbb{P}(E|_{c}\otimes F|_{c}^{\vee}))$. Using [ACGH85, Chapter II, section
2, page 67] we see that the codimension of $Z\cap\eta^{-1}(c)$ in
$\mathbb{P}(E|_{c}\otimes F|_{c}^{\vee}))$ is $r-k+1$. It follows that the
codimension of $Z$ in $\mathbb{P}(E\otimes F^{\vee})$ is $r-k+1$. Since $\nu$
has constant fiber dimension, the codimension of $\nu^{-1}(Z)$ in $C\times V$
is $r-k+1$. Let $p_{V}:C\times V\to V$ be the projection. Hence the
codimension of $p_{V}(\nu^{-1}(Z))$ in $V$ is $r-k$. So the locus of the
points in $\mathbb{P}(\textrm{Hom}(E,F)^{\vee})$ corresponding to non-
surjective maps is the set $T\cup p_{V}(\nu^{-1}(Z))$, which has codimension
at least $r-k\geqslant 1$. This proves the lemma. ∎

######  Lemma 4.5.

There exists a number $d_{2}:=d_{2}(E,k)$ such that if $d\geqslant d_{2}$ then
the locus of all stable locally free quotients in $Q_{k,d}(E)$ is irreducible.

###### Proof.

Using [New78, Lemma 5.2’] we get the number $d^{\prime}_{1}=k(2g-1+2k)$ such
that any stable locally free sheaf of rank $k$ and degree $d\geqslant
d^{\prime}_{1}$ is globally generated. Define

$d_{2}(E,k):=\max\\{d_{1},d^{\prime}_{1},1\\}\,.$

Assume $d\geqslant d_{2}$. Following the proof of irreduciblity of moduli
space of stable locally free sheaves (see [LP97, Theorem 8.5.2]), one can
construct a space $R$ which parametrizes pairs $(L,e_{L})$ where $L$ is a line
bundle of degree $d$ on $C$ and $e_{L}$ is an extension of the form

(4.6) $e_{L}:\quad 0\to\mathcal{O}_{C}^{k-1}\to F\to L\to 0\,.$

There is a universal sheaf $\mathcal{F}$ on $C\times R$. Let $R^{s}\subset R$
denote the open subset containing all the points such that $F$ is a stable
locally free sheaf. Let $M^{s}(k,d)$ denote the moduli space of stable locally
free sheaves on $C$ of rank $k$ and degree $d$. The restriction of the
universal sheaf $\mathcal{F}$ on $C\times R^{s}$ induces a morphism
$\Phi:R^{s}\to M^{s}_{C}(k,d)$ which sends a point
$(L,e_{L}:0\to\mathcal{O}_{C}^{k-1}\to F\to L\to 0)$ to the sheaf $F$. By the
choice of $d$, the space $R$ is irreducible and the map $\Phi$ is surjective.
In other words, the set $\\{\mathcal{F}_{r}:r\in R^{s}\\}$ is the set of all
the stable locally free sheaves on $C$ of rank $k$ and degree $d$.

Let $p_{C}:C\times R^{s}\to C$ and $p_{R}:C\times R^{s}\to R^{s}$ denote the
projections. The sheaf ${p_{R}}_{*}(p_{C}^{*}E^{\vee}\otimes\mathcal{F})$ is
locally free on $R^{s}$ using Lemma 4.1 and Guauert’s theorem. The
projectivization
$\mathbb{P}:=\mathbb{P}(({p_{R}}_{*}(p_{C}^{*}E^{\vee}\otimes\mathcal{F}))^{\vee})$
parametrizes pairs $(r,[q:E\to\mathcal{F}_{r}])$ where $r\in R^{s}$ and $q$ is
a non-zero map. The subset $U\subset\mathbb{P}$ containing all the points such
that $q$ is surjective, is open in $\mathbb{P}$. One can check that there is a
map $g:U\to Q_{k,d}(E)$ which sends the point $(r,[E\to\mathcal{F}_{r}])$ to
the point $[E\to\mathcal{F}_{r}]$. The image of $g$ is precisely the set of
all stable locally free quotients of $E$ of rank $k$ and degree $d$. As
$R^{s}$ is irreducible, so is $\mathbb{P}$ and $U$. Hence, we conclude that
the set parametrizing the stable locally free quotients in $Q_{k,d}(E)$ is
also irreducible. ∎

######  Theorem 4.7.

If $d\geqslant d_{2}(E,k)$ then there is a unique component of $Q_{k,d}(E)$
whose general point corresponds to a quotient $[E\to F]$ such that $F$ is
stable locally free. This component is generically smooth of dimension $dr-
ke+k(r-k)(1-g)$.

###### Proof.

Fix a stable bundle $F$ of rank $k$ and degree $d\geqslant d_{2}$. Using Lemma
4.1, we conclude that $\mathscr{H}om(E,F)$ is generated by global sections and
$H^{1}(E^{\vee}\otimes F)=0$. Lemma 4.3 says that a general morphism $E\to F$
is surjective. Such a quotient $\varphi:E\to F$ corresponds to a point in the
Quot scheme $Q_{k,d}(E)$. Let $\mathcal{W}$ be an irreducible component of
$Q_{k,d}(E)$ which contains the quotient $[\varphi:E\to F]$. Let $S_{F}$
denote the kernel of the quotient. By Lemma 4.2 we have
$\textrm{h}^{1}(S_{F}^{\vee}\otimes F)=0$ and $h^{0}(S_{F}^{\vee}\otimes
F)=dr-ke+k(r-k)(1-g)$. This implies that the point $[\varphi:E\to F]$ is a
smooth point of $\mathcal{W}$ and $\dim_{[\varphi]}\mathcal{W}=dr-
ke+k(r-k)(1-g)$. As the subset of stable locally free quotients is open in
$Q_{k,d}(E)$, a general point of $\mathcal{W}$ corresponds to stable locally
free quotient. This proves the existence of components whose general points
corresponds to stable locally free quotients and that any such component is
generically smooth and of dimension $dr-ke+k(r-k)(1-g)$. The uniqueness of
such component follows from the irreducibility of the locus of all stable
locally free quotients in $Q_{k,d}(E)$, which is proved in Lemma 4.5. ∎

######  Remark 4.8.

Let $\mathcal{P}$ denote the closure of the locus of stable locally free
quotients in $Q_{k,d}(E)$. If $d\geqslant d_{2}$, by Theorem 4.7, it follows
that $\mathcal{P}$ is an irreducible component of $Q_{k,d}(E)$. Moreover using
Lemma 4.5, we see that, if $\mathcal{W}\neq\mathcal{P}$ is another component
of $Q_{k,d}(E)$, then $\mathcal{W}$ cannot contain any stable locally free
quotient.

##  5\. Irreducible Component containing Torsion-free quotients

Let us first recall Serre duality for nodal curves. As $C$ is a projective
scheme over $\mathbb{C}$, by [Har77, Proposition 7.5, Chapter III], $C$ has a
dualizing sheaf $\omega^{\circ}_{C}$. As the nodal curve $C$ is a local
complete intersection, by [Har77, Theorem 7.11, Chapter III],
$\omega^{\circ}_{C}$ is locally free. Moreover, by [Har77, Theorem 7.6,
Chapter III], we have an isomorphism

(5.1)
$\textnormal{Ext}^{i}(G,\omega^{\circ}_{C})\xrightarrow{\sim}H^{1-i}(C,G)$

for any coherent sheaf $G$ on $C$ and $i\geqslant 0$.

######  Lemma 5.2.

There exists a number $d_{3}(E,k)$ such that if $d\geqslant d_{3}(E,k)$ then
we have the following. Let $\mathcal{W}$ be an irreducible component of
$Q_{k,d}(E)$ whose general point corresponds to a torsion-free quotient. Then
a general point $[E\to F]$ of $\mathcal{W}$ satisfies $H^{1}(E^{\vee}\otimes
F)=0$.

###### Proof.

First let us define $d_{3}$. Let
$\mu_{0}:=\mu_{\max}(E\otimes\omega^{\circ}_{C})+1$. For any $d$, consider the
set $Y_{\mu_{0}}$ as defined in Lemma 3.21,

$Y_{\mu_{0}}=\\{[E\to F]\in Q^{{\rm tf}}_{k,d}(E):\mu_{\min}(F)<\mu_{0}\\}\,.$

Using Lemma 3.21, we get constants $C_{3}(E,k,\mu_{0})$ and
$\alpha_{3}(E,k,\mu_{0})$ such that if $d\geqslant\alpha_{3}$ then

$\dim Y_{\mu_{0}}\leq d(r-1)+C_{3}\,.$

We define

$d_{3}:=\max\\{\alpha_{3},C_{3}-(-ke+k(r-k)(1-g)-(r-k)(2r+k))+1\\}\,.$

Thus, if $d\geqslant d_{3}$ then we have

(5.3) $\dim Y_{\mu_{0}}<dr-ke+k(r-k)(1-g)-(r-k)(2r+k)\,.$

Note that $\mu_{0}$ depends only on $E$ and hence the the number $d_{3}$
depends only on $E$ and $k$.

Now we assume $d\geqslant d_{3}$. Let $\mathcal{W}$ be an irreducible
component of $Q_{k,d}(E)$ whose general point corresponds to a torsion-free
quotient. Using Proposition 3.1, we have $\dim\mathcal{W}\geqslant[dr-
ke+k(r-k)(1-g)-(r-k)(2r+k)]$. As $d\geqslant d_{3}$, so (5.3) is satisfied and
hence $\dim Y_{\mu_{0}}<\dim\mathcal{W}$. It follows that a general torsion-
free quotient in $\mathcal{W}$ is not in $Y_{\mu_{0}}$, which translates to
the following. Let $[E\to F]$ be a general point of $\mathcal{W}$. By
assumption, $F$ is torsion-free. As
$\mu_{\min}(F)>\mu_{\max}(E\otimes\omega^{\circ}_{C})$, by [HL10, Lemma
1.3.3], it follows that $\textrm{Hom}(F,E\otimes\omega^{\circ}_{C})=0$. Using
Serre duality (5.1), we get $H^{1}(E^{\vee}\otimes F)=0$. This proves the
lemma. ∎

######  Theorem 5.4.

There is a number $d_{4}(E,k)$ such that if $d\geqslant d_{4}$, then there
exists a unique component of $Q_{k,d}(E)$ whose general point corresponds to a
locally free quotient.

###### Proof.

Define

$d_{4}:=\max\\{d_{2},d_{3}\\}\,.$

Assume $d\geqslant d_{4}$. As $d_{4}\geqslant d_{2}$, by Remark 4.8,
$Q_{k,d}(E)$ has a component $\mathcal{P}$ which contains all stable locally
free quotients. This proves the existence of a component whose general point
corresponds to a locally free quotient. We need to show the uniqueness of such
a component.

Let $\mathcal{W}$ be any component of $Q_{k,d}(E)$ whose general point
corresponds to a locally free quotient of $E$. We will show that $\mathcal{W}$
and $\mathcal{P}$ are the same components. Let $[\varphi:E\to
F]\in\mathcal{W}$ be a general locally free quotient in $\mathcal{W}$. By
Lemma 5.2, we have $H^{1}(E^{\vee}\otimes F)=0$.

By [Bho06, Lemma 2.4] there exists an irreducible variety $T$ and a $T$-flat
sheaf $\mathcal{F}$ on $C\times T$ such that $\mathcal{F}_{t}$ is a stable
vector bundle of rank $k$ and degree $d$ on $C$ for a general point $t\in T$
and $\mathcal{F}_{t_{0}}=F$ for some $t_{0}\in t$. Using $\mathcal{F}$ we want
to construct a family of quotients of $E$. Let $p_{C}:C\times T\to C$ and
$p_{T}:C\times T\to T$ be the projections. Consider the $T$-flat sheaf
$p_{C}^{*}E\otimes\mathcal{F}$ on $C\times T$. We replace $T$ by the open
subset containing points $t$ such that
$H^{1}(E^{\vee}\otimes\mathcal{F}_{t})=0$. Note that $T$ is non-empty as
$t_{0}\in T$. Using Grauert’s theorem, it follows that
${p_{T}}_{*}(p_{C}^{*}E^{\vee}\otimes\mathcal{F})$ is a locally free sheaf on
$T$. So the projectivization
$\mathbb{P}:=\mathbb{P}(({p_{T}}_{*}(p_{C}^{*}E^{\vee}\otimes\mathcal{F}))^{\vee})$
is irreducible. Let $\eta:\mathbb{P}\to T$ denote the natural projection. We
have the tautological quotient of sheaves on $\mathbb{P}$,

(5.5)
$\eta^{*}({p_{T}}_{*}(p_{C}^{*}E^{\vee}\otimes\mathcal{F}))^{\vee}\to\mathcal{O}_{\mathbb{P}}(1)\to
0\,.$

Let $\widetilde{\eta}$ and $\rho_{\mathbb{P}}$ denote the maps as shown in the
following Cartesian diagram

${C\times\mathbb{P}}$${C\times
T}$${\mathbb{P}}$${T}$$\scriptstyle{\widetilde{\eta}}$$\scriptstyle{\rho_{\mathbb{P}}}$$\scriptstyle{p_{T}}$$\scriptstyle{\eta}$

and let $\rho_{C}:C\times\mathbb{P}\longrightarrow C$ denote the projection.
From (5.5), one gets a map of sheaves on $C\times\mathbb{P}$,

(5.6)
$\theta:\rho_{C}^{*}E\to\rho_{\mathbb{P}}^{*}(\mathcal{O}_{\mathbb{P}}(1))\otimes\widetilde{\eta}^{*}\mathcal{F}\,.$

Let $U$ be the subset of $\mathbb{P}$ defined as follows

$U=\\{u\in\mathbb{P}:\theta|_{C\times\\{u\\}}:E\to\mathcal{F}_{\eta(u)}\textrm{
is surjective }\\}\,.$

The subset $U$ is open as it is the compliment of the closed set
$\rho_{\mathbb{P}}(\text{Supp}\,(Coker(\theta)))$ in $\mathbb{P}$. The
quotient $\varphi:E\to F$, we started with, corresponds to a point
$u_{0}\in\mathbb{P}$ such that $\eta(u_{0})=t_{0}$. Then the restriction of
the map $\theta$ on the fiber $C\times\\{u_{0}\\}$ is given by the quotient
$\varphi:E\to F$ itself. So $u_{0}$ is in $U$, which proves the non-emptiness
of $U$. Also note that $\mathbb{P}$ is irreducible and hence so is $U$. As
$\mathcal{F}$ is flat over $T$, so $\widetilde{\eta}^{*}\mathcal{F}$ is flat
over $\mathbb{P}$ and consequently
$\rho_{\mathbb{P}}^{*}(\mathcal{O}_{\mathbb{P}}(1))\otimes\widetilde{\eta}^{*}\mathcal{F}$
is flat over $\mathbb{P}$. So the quotient of sheaves $\theta$ restricted on
$C\times U$ gives a map

$g:U\to Q_{k,d}(E)\,.$

A closed point of $U$ corresponds to a pair $(t,\varphi_{t})$ where $t\in T$
and $\varphi_{t}:E\to\mathcal{F}_{t}$ is surjective. The map $g$ sends the
point $(t,\varphi_{t}:E\to\mathcal{F}_{t})$ to the point
$[\varphi_{t}:E\to\mathcal{F}_{t}]$ in $Q_{k,d}(E)$. Since a general point of
$T$ is a stable vector bundle and the projection $\eta:\mathbb{P}\to T$ is
surjective, so for a general point $(t,\varphi_{t}:E\to\mathcal{F}_{t})$ in
$\mathbb{P}$, $\mathcal{F}_{t}$ is a stable vector bundle. As $\mathbb{P}$ is
irreducible and $U$ is open in $\mathbb{P}$, we conclude that a general point
of $U$ corresponds to a stable locally free quotient. Let the corresponding
open set be denoted by $U^{\prime}$. Then $g(U^{\prime})\subset\mathcal{P}$.
This implies that $g(U)\subset
g(\overline{U^{\prime}})\subset\overline{\mathcal{P}}=\mathcal{P}$. So image
of $U$ is contained in $\mathcal{P}$ and hence $g(u_{0})=[\varphi:E\to
F]\in\mathcal{P}$. So a general element of $\mathcal{W}$ is in $\mathcal{P}$
which implies that $\mathcal{W}=\mathcal{P}$. This proves the theorem. ∎

######  Theorem 5.7.

If $d\geqslant d_{4}$, then there exists a unique component of $Q_{k,d}(E)$
whose general point corresponds to a torsion-free quotient of $E$.

###### Proof.

As $d\geqslant d_{4}$, the existence of such a component is already proved by
Theorem 5.4. Uniqueness of such a component can be proved by using the same
argument as in the proof of the Theorem 5.4 and using [Bho06, Lemma 2.3]
(instead of [Bho06, Lemma 2.4] which was used in the proof of Theorem 5.4) and
Theorem 5.4. ∎

######  Corollary 5.8.

Let $d\geqslant d_{4}$. Let $\mathcal{W}$ be an irreducible component of
$Q_{k,d}(E)$ which contains a torsion-free quotient. Then
$\mathcal{W}=\mathcal{P}$.

###### Proof.

Let $[E\to F]\in\mathcal{W}$ be a quotient such that $F$ is torsion-free. Let
$V=\\{[E\to G]\in\mathcal{W}:G\textrm{ is torsion-free\,}\\}$. By [HL10,
Proposition 2.3.1], $V$ is an open subset of $\mathcal{W}$. As $V$ is non-
empty, $\mathcal{W}$ is a component of $Q_{k,d}(E)$ whose general point
corresponds to a torsion-free quotient. Note that $\mathcal{P}$ is also a
component of $Q_{k,d}(E)$ whose general point corresponds to a torsion-free
quotient. By Theorem 5.7, it follows that $\mathcal{W}=\mathcal{P}$. ∎

######  Lemma 5.9.

Let $F$ be a torsion-free sheaf on $C$ of rank $k$, degree $d$ and type $a$.
Then there exists $n_{0}$ such that for any $n\geqslant n_{0}$ there is a
surjection $\mathcal{O}_{C}^{\oplus 2k+a}\to F(n)\to 0$.

###### Proof.

We take $n_{0}$ to be the integer such that if $n\geqslant n_{0}$ then $F(n)$
is generated by global sections. Let us fix $n\geqslant n_{0}$. So the natural
map

$\varphi:H^{0}(F(n))\otimes\mathcal{O}_{C}\longrightarrow F(n)$

is surjective. The vector space $(F(n))\otimes\kappa(x)$ is generated by
$(k+a)$ elements. So there is a subspace $V_{1}\subset H^{0}(F(n))$ of
dimension $k+a$ such that the natural map
$\varphi_{V_{1}}:V_{1}\otimes\mathcal{O}_{C}\to F(n)$ is surjective at the
node $x$. It follows that the map $\varphi_{V_{1}}$ is surjective on an open
set $U\subset C$ containing the node $x$. The complement of $U$ in $C$
consists of finitely many non-singular points, say $c_{1},\dots,c_{s}$. For
each $c_{i}$, the vector space $F(n)|_{c_{i}}$ has rank $k$. Let $G_{i}$
denote the set of all subspaces $W\subset H^{0}(F(n))$ of rank $k$ such that
the natural map $\varphi_{W}:W\otimes O_{C}\to F(n)$ is surjective at $c_{i}$.
Clearly, $G_{i}$ is non-empty as $\varphi$ is surjective. Moreover, $G_{i}$ is
an open subscheme of the Grassmannian scheme ${\rm Grass}(H^{0}(F(n)),k)$
which parametrises rank $k$ subspaces of $H^{0}(F(n))$. This implies that the
intersection $\cap_{i=1}^{s}G_{i}$ is non-empty. Hence we can find a subspace
$V_{2}\subset H^{0}(F(n))$ of rank $k$ such that the natural map
$\varphi_{V_{2}}:V_{2}\otimes O_{C}\to F(n)$ is surjective at every $c_{i}$.
Let $V:=V_{1}+V_{2}$. Then the natual map

$\varphi_{V}:V\otimes O_{C}\longrightarrow F(n)$

is surjective and rank of $V$ is at most $2k+a$. From this the lemma easily
follows. ∎

Combining Theorem 5.7 and Lemma 5.9, we get the Proposition 5.10.

######  Proposition 5.10.

Let $F$ be a torsion-free sheaf on $C$ of rank $k$ and degree $d$. Then there
exists a family $\mathcal{F}$ of torsion-free sheaves on $C$ of rank $k$ and
degree $d$, parametrized by an integral scheme $T$ such that for a general
point $t$ in $T$, the sheaf $\mathcal{F}_{t}$ is stable locally free and
$\mathcal{F}_{t_{0}}=F$ for some $t_{0}\in T$.

###### Proof.

Let the type of $F$ be $a$. For any degree $d^{\prime}$, consider the Quot
scheme $Q_{k,d^{\prime}}(\mathcal{O}_{C}^{\oplus 2k+a})$. Using Theorem 5.7,
we get a constant $d^{\prime}_{4}=d_{4}(\mathcal{O}_{C}^{\oplus 2k+a},k)$,
such that if $d^{\prime}\geqslant d^{\prime}_{4}$ then the torsion-free locus
$Q^{{\rm tf}}_{k,d^{\prime}}(\mathcal{O}_{C}^{\oplus 2k+a})$ is irreducible.
Using Lemma 5.9, we have $n_{0}$ such that if $n\geqslant n_{0}$ then there is
a surjection $q:\mathcal{O}_{C}^{\oplus 2k+a}\to F(n)\to 0$. Let us choose
$n\geqslant n_{0}$ such that $\deg(F(n))=d+nk\geqslant d^{\prime}_{4}$. Then
the surjection $q:\mathcal{O}_{C}^{\oplus 2k+a}\to F(n)$ corresponds to a
closed point in the Quot scheme $Q_{k,d+nk}(\mathcal{O}_{C}^{\oplus 2k+a})$.
Moreover $F$ is torsion-free, so $[q]$ is a point of $Q^{{\rm
tf}}_{k,d+nk}(\mathcal{O}_{C}^{\oplus 2k+a})$. We take $T$ to be this torsion-
free locus $Q^{{\rm tf}}_{k,d+nk}(\mathcal{O}_{C}^{\oplus 2k+a})$ with the
reduced induced scheme structure. As $d+nk\geqslant d^{\prime}_{4}$, $T$ is
irreducible. Let $\mathcal{F}^{\prime}$ denote the universal quotient sheaf on
$C\times T$. Then $\mathcal{F}^{\prime}_{q}=F(n)$. By Theorem 5.7 it follows
that $\mathcal{F}^{\prime}_{t}$ is stable and locally free for a general point
of $t\in T$. Taking $\mathcal{F}:=\mathcal{F}^{\prime}(-n)$ the propostion
follows. ∎

##  6\. Irreduciblity of the Quot scheme

Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$ and let $k$ be an
integer such that $0<k<r$. In the previous section we saw that for large $d$,
there is an irreducible component of $Q_{k,d}(E)$ which contains the locus of
torsion-free quotients. In this section we will show that for large $d$ the
quot scheme is irreducible. First we will prove the irreducibility for the
case when $k>\frac{r}{2}$. Next we will use this result to prove the
irreducibility for all $k$.

Let $Z$ denote the complement of torsion-free locus in the quot scheme
$Q_{k,d}(E)$. For a sheaf $F$, $\textnormal{Tor}(F)$ will denote the torsion
subsheaf of $F$. For $\delta>0$, let us define further subsets of $Z$,

$\displaystyle Z_{\delta}$ $\displaystyle:=\\{[E\to F]\in
Z:\quad\textnormal{length}(\textnormal{Tor}(F))=\delta\\}\,,$ $\displaystyle
Z^{\rm g}$ $\displaystyle:=\\{[E\to F]\in Z:\quad\text{the stalk $F_{x}$ is
free }\\}\,,$ $\displaystyle Z^{\rm g}_{\delta}$ $\displaystyle:=Z^{\rm g}\cap
Z_{\delta}\,.$

We emphasize that, for a point $[E\to F]\in Z$, $\textnormal{Tor}(F)$ is
supported away from the node and the stalk $F_{x}$ is free.

######  Lemma 6.1.

Assume $\frac{r}{2}<k<r$. For any $d$ and $\delta>0$, consider the subsets
$Z_{\delta}$ and $Z^{\rm g}_{\delta}$ in $Q_{k,d}(E)$. Then we have

$\displaystyle\dim Z_{\delta}$ $\displaystyle\leq\dim(Q_{k,d-\delta}^{{\rm
tf}}(E))+(r-1)\delta\,,$ $\displaystyle\dim Z^{\rm g}_{\delta}$
$\displaystyle\leq\dim(Q_{k,d-\delta}^{{\rm tf}}(E))+(r-k)\delta\,.$

Moreover $Z_{\delta}$ is non-empty only when $d-\delta\geqslant m(E,k)$.

###### Proof.

Let $[\varphi:E\to F]$ be a closed point in $Z_{\delta}$ and $\tau$ be the
torsion subsheaf of $F$ which has length $\delta$. Then we can construct a
torsion-free quotient $\varphi^{\prime}:E\to F^{\prime}$, and a torsion
quotient $\psi:S_{F^{\prime}}\to\tau$, which fit into the following
commutative diagram whose rows and columns are exact.

(6.2)
${0}$${0}$${0}$${S_{F}}$${S_{F^{\prime}}}$${\tau}$${0}$${0}$${S_{F}}$${E}$${F}$${0}$${F^{\prime}}$${F^{\prime}}$${0}$${0\,.}$$\scriptstyle{\psi}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\varphi}$

On the other hand, given the torsion-free quotient $\varphi^{\prime}:E\to
F^{\prime}$ and the torsion quotient $\psi:S_{F^{\prime}}\to\tau$ of length
$\delta$, one can get back the quotient $\varphi:E\to F$ using the following
push-out diagram

${0}$${S_{F^{\prime}}}$${E}$${F^{\prime}}$${0\phantom{\,.}}$${0}$${\tau}$${F}$${F^{\prime}}$${0\,.}$$\scriptstyle{\psi}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\varphi}$

This correspondence gives us

$\displaystyle\dim Z_{\delta}\leq$ $\displaystyle\dim(Q_{k,d-\delta}^{{\rm
tf}}(E))+\max_{q^{\prime}\in Q_{k,d-\delta}^{{\rm
tf}}(E)}\left\\{\dim(Q_{0,\delta}(S_{F^{\prime}})\right\\}\,.$

The above argument can be made precise by constructing a map from a relative
quot scheme onto the locus $Z_{\delta}$, as in the proof of Lemma 3.17.

If $S_{F^{\prime}}$ is of type $a$ then
$\dim(Q_{0,\delta}(S_{F^{\prime}}))\leq(r-k+a)\delta$. As $k>\frac{r}{2}$, the
rank of $S_{F^{\prime}}$ is $r-k$, which is less than $\frac{r}{2}$. As the
type of $S_{F^{\prime}}$ can be at most the rank of $S_{F^{\prime}}$, we get
that $a\leq r/2<k$. As $a\leq k-1$, we have
$\dim(Q_{0,\delta}(S_{F^{\prime}}))\leq(r-1)\delta$. Hence we have

$\dim Z_{\delta}\leq\dim(Q_{k,d-\delta}^{{\rm tf}}(E))+(r-1)\delta\,.$

Now for points in $Z^{\rm g}_{\delta}$, the sheaf $S_{F^{\prime}}$ is locally
free of rank $r-k$ and hence $\dim(Q_{0,\delta}(S_{F^{\prime}}))=(r-k)\delta$.
So we have

$\displaystyle\dim Z^{\rm g}_{\delta}\leq$
$\displaystyle\dim(Q_{k,d-\delta}^{{\rm tf}}(E))+(r-k)\delta\,.$

Observe that $Z_{\delta}$ is non-empty only when the quot scheme
$Q_{k,d-\delta}^{{\rm tf}}(E)$ is non-empty. Recall that $m(E,k)$ is the
minimal degree of a quotient of $E$ of rank $k$. So we must have
$d-\delta\geqslant m(E,k)$. This proves the lemma. ∎

######  Lemma 6.3.

There exists a number $d_{5}(E,k)$ such that if $d\geqslant d_{5}$ then there
is no component of $Q_{k,d}(E)$ whose general point is in $Z^{\rm g}$.

###### Proof.

First we define $d_{5}$. Recall that $m(E,k)$ is the minimal degree of a
quotient of $E$ of rank $k$. Also recall that by Theorem 5.7, if $d\geqslant
d_{4}(E,k)$ then the torsion-free locus $Q^{{\rm tf}}_{k,d}(E)$ is irreducible
of dimension $dr-ke+k(r-k)(1-g)$. Let us define

$M=\max_{m(E,k)\leq d^{\prime}<d_{4}}\left\\{\dim Q^{{\rm
tf}}_{k,d^{\prime}}(E)-[d^{\prime}r-ke+k(r-k)(1-g)]+d^{\prime}k\right\\}$

and

$d_{5}:=\max\\{d_{4},\frac{M}{k}+1\\}\,.$

Now we assume $d\geqslant d_{5}$. If possible, let $\mathcal{W}$ be a
component of $Q_{k,d}(E)$ whose general point corresponds to a quotient
$[\varphi:E\to F]$ in $Z^{\rm g}_{\delta}$, where $\delta>0$. Using Lemma 6.1,
we have $d-\delta\geqslant m(E,k)$. The stalk $F_{x}$ is free and so the
kernel $S_{F}$ of $\varphi$ is locally free. The dimension bound in (3.2)
gives

$\displaystyle\dim_{[\varphi]}Q_{k,d}(E)$
$\displaystyle\geqslant\hom(S_{F},F)-\textnormal{ext}^{1}(S_{F},F)$
$\displaystyle=\chi(S_{F}^{\vee}\otimes F)$ $\displaystyle=dr-
ke+k(r-k)(1-g)\,.$

So we have

(6.4) $\dim\mathcal{W}\geqslant dr-ke+k(r-k)(1-g)\,.$

Also from Lemma 6.1, we have

$\displaystyle\dim Z^{\rm g}_{\delta}\leq\dim(Q_{k,d-\delta}^{{\rm
tf}}(E))+(r-k)\delta\,.$

So

$\displaystyle\dim Z^{\rm g}_{\delta}-\dim\mathcal{W}$ $\displaystyle\leq\dim
Q^{{\rm tf}}_{k,d-\delta}(E)+(r-k)\delta-[dr-ke+k(r-k)(1-g)]$
$\displaystyle=\dim Q^{{\rm
tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]-\delta k\,.$

Observe that if $d-\delta\geqslant d_{4}$ then, by Theorem 5.7, $Q^{{\rm
tf}}_{k,d-\delta}(E)$ has dimension $[(d-\delta)r-ke+k(r-k)(1-g)]$. So the
right hand side is negative. Otherwise we have $m(E,k)\leq d-\delta<d_{4}$. As
$d>M/k$, in this case we have

$dk>M\geqslant\dim Q^{{\rm
tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]+(d-\delta)k$

which implies that

$\dim Q^{{\rm tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]-\delta k<0\,.$

So the difference $\dim Z^{\rm g}_{\delta}-\dim\mathcal{W}$ is negative. This
shows that the points of $Z^{\rm g}_{\delta}$ cannot be general in
$\mathcal{W}$. ∎

######  Proposition 6.5.

For $d\geqslant d_{5}(E,k)$, the locus $Z^{\rm g}$ in $Q_{k,d}(E)$ is
contained in the closure of the torsion-free locus $Q^{{\rm tf}}_{k,d}(E)$.

###### Proof.

Recall the irreducible component $\mathcal{P}$ defined in Remark 4.8, which
contains the locus of torsion-free quotients in $Q_{k,d}(E)$. We need to show
that $Z^{\rm g}$ is contained in $\mathcal{P}$. Let, if possible,
$[\varphi:E\to F]$ be a quotient in $Z^{\rm g}$ which is not in the component
$\mathcal{P}$. Let $\mathcal{W}$ be a component containing the point
$[\varphi:E\to F]$. Consider the open subset $V=\\{[q:E\to G]\in\mathcal{W}:$
the stalk $G_{x}$ is free}. Note that $[\varphi:E\to F]\in V$, which shows
that $V$ is non-empty. For any quotient $[q:E\to G]\in V$, either $G$ is
torsion-free or $[q:E\to G]\in Z^{\rm g}$. If $V$ contains a torsion-free
quotient then we must have $\mathcal{W}=\mathcal{P}$, by Corollary 5.8. This
is a contradiction. So $V$ cannot contain any torsion-free quotient. In other
words any point of $V$ is contained in $Z^{\rm g}$. But this is a
contradiction as the points of $Z^{\rm g}$ cannot be general in any component
by Lemma 6.3. So we conclude that the point $[\varphi:E\to F]$ is in
$\mathcal{P}$ and hence $Z^{\rm g}$ is contained in $\mathcal{P}$. ∎

For ease of notation, let us define the number

$\delta_{0}:=(r-k)(2r+k)\,.$

Recall that Proposition 3.1 says that any irreducible component of
$Q_{k,d}(E)$ has dimension at least $[dr-ke+k(r-k)(1-g)]-\delta_{0}$.

######  Lemma 6.6.

Assume $\frac{r}{2}<k<r$. There exists a number $d_{6}(E,k)$ such that if
$d\geqslant d_{6}$ then we have the following. If $\mathcal{W}$ is any
component of $Q_{k,d}(E)$ and $[E\to F]$ is a general point of $\mathcal{W}$
then length$(\textnormal{Tor}(F))\leq\delta_{0}$.

###### Proof.

We will use the same technique as in Lemma 6.3. First let us define
$d_{6}(E,k)$. Define

$M=\max_{m(E,k)\leq d^{\prime}<d_{4}}\left\\{\dim Q^{{\rm
tf}}_{k,d^{\prime}}(E)-(d^{\prime}r-ke+k(r-k)(1-g))+\delta_{0}+d^{\prime}\right\\}$

and

$d_{6}:=\max\\{d_{4},M+1\\}\,.$

Now we assume $d\geqslant d_{6}$ and $\delta>\delta_{0}$. If possible, let
$\mathcal{W}$ be a component of $Q_{k,d}(E)$ whose general point corresponds
to a quotient $[\varphi:E\to F]$ in $Z_{\delta}$. The dimension bound from
Proposition 3.1 gives

(6.7) $\dim\mathcal{W}\geqslant dr-ke+k(r-k)(1-g)-\delta_{0}\,.$

From Lemma 6.1, we have

$\displaystyle\dim Z_{\delta}\leq\dim(Q_{k,d-\delta}^{{\rm
tf}}(E))+(r-1)\delta\,.$

So

$\displaystyle\dim Z_{\delta}-\dim\mathcal{W}$ $\displaystyle\leq\dim Q^{{\rm
tf}}_{k,d-\delta}(E)+(r-1)\delta-[dr-ke+k(r-k)(1-g)]+\delta_{0}$
$\displaystyle=\dim Q^{{\rm
tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]-\delta+\delta_{0}\,.$

Observe that, if $d-\delta\geqslant d_{4}$ then by Theorem 5.7, $Q^{{\rm
tf}}_{k,d-\delta}(E)$ has dimension $[(d-\delta)r-ke+k(r-k)(1-g)]$. In this
case the right hand side is negative as $\delta>\delta_{0}$. Otherwise we have
$m(E,k)\leq d-\delta<d_{4}$. As $d>M$, in this case we have

$d>M\geqslant\dim Q^{{\rm
tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]+\delta_{0}+(d-\delta)$

which implies that

$\dim Q^{{\rm
tf}}_{k,d-\delta}(E)-[(d-\delta)r-ke+k(r-k)(1-g)]-\delta+\delta_{0}<0\,.$

So the difference $\dim Z_{\delta}-\dim\mathcal{W}$ is negative. This shows
that the points of $Z_{\delta}$ cannot be general in $\mathcal{W}$. ∎

######  Proposition 6.8.

Assume $\frac{r}{2}<k<r$. There exists a number $d_{7}(E,k)$ such that if
$d\geqslant d_{7}$ then the following holds. Let $\mathcal{W}$ be an
irreducible component of $Q_{k,d}(E)$ whose general point corresponds to a
quotient in $Z_{\delta}$, with $\delta>0$. Then a general point $[q:E\to F]$
of $\mathcal{W}$, satisfies $Ext^{1}(\ker q,E)=0$.

###### Proof.

First we define $d_{7}$. Let
$\mu_{0}:=\mu_{\min}(E\otimes(\omega^{\circ}_{C})^{\vee})+1$. Using Lemma
3.29, we have constants $C_{4}(E,k,\mu_{0})$ and $\alpha_{4}(E,k,\mu_{0})$.
Define

$d_{7}:=\max\\{d_{6},\alpha_{4}+\delta_{0},2\delta_{0}+C_{4}-[-ke+k(r-k)(1-g)]+1\\}\,.$

Consider the subset $X_{\mu_{0}}$ in $Q_{k,d-\delta}(E)$,

$X_{\mu_{0}}=\\{[q^{\prime}:E\to F^{\prime}]\in
Q_{k,d-\delta}(E):\mu_{\max}(\ker q^{\prime})>\mu_{0}\\}\,.$

By our choice of $d_{7}$ and Lemma 3.29, if $d\geqslant d_{7}$ then we have

(6.9) $\dim X_{\mu_{0}}<dr-ke+k(r-k)(1-g)-(r+1)\delta_{0}\,.$

Assume $d\geqslant d_{7}$. Let $\mathcal{W}$ be an irreducible component of
$Q_{k,d}(E)$ whose general point is in $Z_{\delta}$ for some $\delta>0$. As
$d\geqslant d_{6}$, using Lemma 6.6 we have $\delta\leq\delta_{0}$. Let $D$
denote the torsion-free locus of the quot scheme $Q_{k,d-\delta}(E)$ i.e.

$D:=Q^{{\rm tf}}_{k,d-\delta}(E)\,.$

Let $\mathscr{S}^{\prime}$ denote the universal kernel sheaf on $C\times D$.
Consider the relative quot scheme

$H:={\rm Quot}_{C\times D/D}(\mathscr{S}^{\prime},0,\delta)\,.$

A closed point of $H$ corresponds to the pair of quotients $(q^{\prime}:E\to
F^{\prime},\lambda:\ker q^{\prime}\to\tau)$. Given the point
$(q^{\prime},\lambda)$ in $H$, one can construct a quotient $q:E\to F$ using
the following push-out diagram

${0}$${\ker
q^{\prime}}$${E}$${F^{\prime}}$${0\phantom{\,.}}$${0}$${\tau}$${F}$${F^{\prime}}$${0\,.}$$\scriptstyle{\lambda}$$\scriptstyle{q^{\prime}}$$\scriptstyle{q}$

It is clear that the quotient $[q:E\to F]$ corresponds to a point in the quot
scheme $Q_{k,d}(E)$. One can check that there is a map of schemes

$g:H\to Q_{k,d}(E)$

which sends the point $(q^{\prime},\lambda)$ to the point $[q:E\to F]$.
Clearly, $H$ maps onto the subset $Z_{\delta}$. Thus, there is an irreducible
component, say $H^{\prime}$, of $H$ which maps to $\mathcal{W}$ and
$g(H^{\prime})$ is dense in $\mathcal{W}$. There is a structure map $\rho:H\to
D$ which sends the closed point $(q^{\prime},\lambda)$ to $[q^{\prime}]$. Let
$D^{\prime}:=\rho(H^{\prime})$. We have the following diagram where all the
schemes are irreducible,

${H^{\prime}}$${\mathcal{W}}$${D^{\prime}\,.}$$\scriptstyle{g}$$\scriptstyle{\rho}$

Let $[q^{\prime}:E\to F^{\prime}]$ be any closed point in $D^{\prime}$. The
fiber of $\rho$ over the point $[q^{\prime}]$ is contained in the quot scheme
$Q_{0,\delta}(\ker q^{\prime})$. It is easy to see that, if the torsion free
sheaf $\ker q^{\prime}$ has type $a$ then the quot scheme $Q_{0,\delta}(\ker
q^{\prime})$ has dimension at most $(r-k+a)\delta$. By Lemma 2.5, the type of
$F^{\prime}$ is also $a$ and so $a\leq k$. Consequently the fiber
$\rho^{-1}([q^{\prime}])$ has dimension at most $r\delta$. So we have

$\displaystyle\dim\mathcal{W}\leq\dim H^{\prime}\leq\dim
D^{\prime}+r\delta\leq\dim D^{\prime}+r\delta_{0}$

as $\delta\leq\delta_{0}$. In other words we have $\dim
D^{\prime}\geqslant\dim\mathcal{W}-r\delta_{0}$. Using the lower bound of
$\dim\mathcal{W}$ from Proposition 3.1, we get that

(6.10) $\dim D^{\prime}\geqslant dr-ke+k(r-k)(1-g)-(r+1)\delta_{0}\,.$

As $d\geqslant d_{7}$, it follows that $\dim D^{\prime}>\dim X_{\mu_{0}}$. So
a general point of $D^{\prime}$ is not in $X_{\mu_{0}}$. This implies that, a
general point $[q^{\prime}:E\to F^{\prime}]$ in $D^{\prime}$, satisfies the
inequality $\mu_{\max}(\ker q^{\prime})\leq\mu_{0}$. Let $[q^{\prime}:E\to
F^{\prime}]\in D^{\prime}$ be such a general point. Choose a point
$(q^{\prime},\lambda)$ in the fiber $\rho^{-1}([q^{\prime}])$ and let the
image $g(q^{\prime},\lambda)$ be $[q]$. It is easy to see from the description
of the map $g$ that $\ker q\subset\ker q^{\prime}$. Thus

$\mu_{\max}(\ker q)\leq\mu_{\max}(\ker q^{\prime})\leq\mu_{0}\,.$

This says that the point $[q]$ in $\mathcal{W}$, satisfies that
$\mu_{\max}(\ker q)<\mu_{\min}(E\otimes(\omega^{\circ}_{C})^{\vee})$ and hence
$\textrm{Hom}(E\otimes(\omega^{\circ}_{C})^{\vee},\ker q)=0$. Using Serre
duality (5.1) we get that $\textnormal{Ext}^{1}(\ker q,E)=0\,.$ Using [Lan83,
Theorem 1.4], we conclude the proof of the proposition. ∎

######  Lemma 6.11.

There exists a number $d_{8}(E,k)$ such that for any stable locally free sheaf
$S$ on $C$ of rank $r-k$ and degree $d\leq d_{8}$, the sheaf $S^{\vee}\otimes
E$ is generated by global sections and $H^{1}(S^{\vee}\otimes E)=0$. For such
a sheaf $S$ and a general homomorphism $\psi:S\to E$, the cokernel of $\psi$
is free at the node $x$.

###### Proof.

In Lemma 4.1, replacing $E$ by $E^{\vee}$ and $k$ by $r-k$, we get the number
$d^{\prime}_{1}:=d_{1}(E^{\vee},r-k)$ such that for any stable locally free
sheaf $G$ on $C$ of rank $r-k$ and degree $d\geqslant d^{\prime}_{1}$, the
sheaf $E\otimes G$ is globally generated and $H^{1}(E\otimes G)=0$. We take

$d_{8}:=-d^{\prime}_{1}\,.$

Let $S$ be any stable locally free sheaf on $C$ of rank $r-k$ and degree
$d\leq d_{8}$. Then $S^{\vee}$ is stable locally free of rank $r-k$ and degree
$-d\geqslant d^{\prime}_{1}$. So $S^{\vee}\otimes E$ is generated by global
sections and $H^{1}(S^{\vee}\otimes E)=0$. This proves the first assertion.

We prove the second assertion. A general homomorphism of the vector spaces
$S|_{x}\to E|_{x}$ is injective. Since $S^{\vee}\otimes E$ is globally
generated, it follows that for a general homomorphism $\psi:S\to E$, the
restriction $\psi|_{x}:S|_{x}\to E|_{x}$ is injective and consequently the
cokernel of $\psi$ is free at $x$. ∎

######  Lemma 6.12.

Let $Y$ be an irreducible Noetherian scheme and let $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ be coherent sheaves on $C\times Y$ which are flat over $Y$.
Let $\phi:\mathcal{G}_{1}\to\mathcal{G}_{2}$ be a map of sheaves. The locus of
points $y\in Y$, where $\phi_{y}$ is injective, is an open subset of $Y$.

###### Proof.

We will use the existence of flattening stratification, see [FGI+05, §5.4.2].
Let us assume that the set $y\in Y$ such the $\phi_{y}$ is injective is
nonempty, or else, there is nothing to prove. Let $\mathcal{G}$ denote the
cokernel of $\phi$. Let $I$ denote the finite set consisting of the Hilbert
polynomials of the sheaves $\\{\mathcal{G}_{y}:y\in Y\\}$. Let
$f_{1}<f_{2}<\ldots<f_{r}$ be the elements of $I$ with the usual total
ordering on polynomials ($f<g$ iff $f(n)<g(n)$ for all $n\gg 0$). Using
[FGI+05, Theorem 5.13], we get locally closed subschemes $Y_{f_{i}}$ where
$Y_{f_{i}}$ consists of all the points $y\in Y$ for which the Hilbert
polynomial of $\mathcal{G}_{y}$ is $f_{i}$. Moreover the subset $Y_{f_{1}}$ is
open. We have an exact sequence

$0\to\ker(\phi_{y})\to\mathcal{G}_{1,y}\xrightarrow{\phi_{y}}\mathcal{G}_{2,y}\to\mathcal{G}_{y}\to
0\,.$

We denote the Hilbert polynomial of a sheaf $F$ on $C$ by ${\rm Hilb}(F)$.
Hence we have

${\rm Hilb}(\mathcal{G}_{y})={\rm Hilb}(\mathcal{G}_{2,y})-{\rm
Hilb}(\mathcal{G}_{1,y})+{\rm Hilb}(\ker(\phi_{y}))\,.$

Let $y_{0}$ be a point for which $\phi_{y_{0}}$ is injective. Then

${\rm Hilb}(\mathcal{G}_{y_{0}})={\rm Hilb}(\mathcal{G}_{2,y_{0}})-{\rm
Hilb}(\mathcal{G}_{1,y_{0}})\,.$

Consequently, by flatness of $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$, we have

$\displaystyle{\rm Hilb}(\mathcal{G}_{y})$ $\displaystyle={\rm
Hilb}(\mathcal{G}_{2,y})-{\rm Hilb}(\mathcal{G}_{1,y})+{\rm
Hilb}(\ker(\phi_{y}))$ $\displaystyle={\rm Hilb}(\mathcal{G}_{2,y_{0}})-{\rm
Hilb}(\mathcal{G}_{1,y_{0}})+{\rm Hilb}(\ker(\phi_{y}))$ $\displaystyle={\rm
Hilb}(\mathcal{G}_{y_{0}})+{\rm Hilb}(\ker(\phi_{y}))\,.$

Thus for any point $y\in Y$ we have ${\rm Hilb}(\mathcal{G}_{y})\geqslant{\rm
Hilb}(\mathcal{G}_{y_{0}})$. Moreover, equality holds iff $\ker(\phi_{y})=0$.
This shows that ${\rm Hilb}(\mathcal{G}_{y_{0}})=f_{1}$ and that the set of
points for which $\phi_{y}$ is injective is precisely the set $Y_{f_{1}}$.
Hence the set of points $y$ for which $\phi_{y}$ is injective is open. ∎

######  Theorem 6.13.

Let $k$ be an integer such that $\frac{r}{2}<k<r$. Then there is a number
$d_{9}(E,k)$ such that if $d\geqslant d_{9}$ then the quot scheme $Q_{k,d}(E)$
is irreducible.

###### Proof.

We define

$d_{9}:=\max\\{d_{4},d_{5},d_{7},e-d_{8}\\}\,.$

Assume $d\geqslant d_{9}$. As $d\geqslant d_{4}$, recall that, by Theorem 5.7
or Corollary 5.8, there is a unique irreducible component $\mathcal{P}$ of
$Q_{k,d}(E)$ which contains the torsion-free locus $Q^{{\rm tf}}_{k,d}(E)$. We
want to show that there is no other component of the quot scheme $Q_{k,d}(E)$.

On the contrary, let us assume that $\mathcal{W}$ is an irreducible component
of $Q_{k,d}(E)$ different from $\mathcal{P}$. So any closed point in
$\mathcal{W}$ corresponds to a quotient with torsion. Let $[q:E\to F]$ be a
general point in $\mathcal{W}$. Let $S_{F}$ denote the kernel of $q$. Then
$S_{F}$ is a torsion-free sheaf on $C$ of rank $r-k$ and degree $e-d$. Using
Proposition 5.10, we get a family $\mathscr{S}$ of torsion-free sheaves on $C$
of rank $r-k$ and degree $e-d$, parametrized by an integral scheme $T$ such
that the sheaf $\mathscr{S}_{t}$ is stable locally free for a general point
$t\in T$ and $\mathscr{S}_{t_{0}}=S_{F}$ for some $t_{0}\in T$. Let
$p_{C}:C\times T\to T$ and $p_{T}:C\times T\to T$ denote the projections. We
consider the sheaf $\mathscr{H}om(\mathscr{S},p_{C}^{*}E)$ and let
$\mathcal{G}$ be its push-forward on $T$,

$\mathcal{G}:={p_{T}}_{*}(\mathscr{H}om(\mathscr{S},p_{C}^{*}E))\,.$

Proposition 6.8 shows that we can assume $\textnormal{Ext}^{1}(S_{F},E)=0$. We
replace $T$ by the open subset containing points $t$ such that
$\textnormal{Ext}^{1}(\mathscr{S}_{t},E)=0$. Clearly $t_{0}\in T$. Using
[Lan83, Theorem 1.4], we conclude that $G$ is a locally free sheaf on $T$. So
the projectivization $\mathbb{P}:=\mathbb{P}(\mathcal{G}^{\vee})$ is
irreducible. Let $\eta:\mathbb{P}\to T$ be the structure map. We have the
tautological quotient of sheaves on $\mathbb{P}$,

$\eta^{*}\mathcal{G}^{\vee}\longrightarrow\mathcal{O}_{\mathbb{P}}(1)\longrightarrow
0\,,$

that is,

(6.14)
$\eta^{*}({p_{T}}_{*}(\mathscr{H}om(\mathscr{S},p_{C}^{*}E)))^{\vee}\longrightarrow\mathcal{O}_{\mathbb{P}}(1)\longrightarrow
0\,.$

Let $\rho_{\mathbb{P}}$ and $\widetilde{\eta}$ denote the maps as shown in the
following Cartesian diagram and $\rho_{C}:C\times\mathbb{P}\to\mathbb{P}$
denote the projection

${C\times\mathbb{P}}$${C\times
T}$${\mathbb{P}}$${T\,.}$$\scriptstyle{\widetilde{\eta}}$$\scriptstyle{\rho_{\mathbb{P}}}$$\scriptstyle{p_{T}}$$\scriptstyle{\eta}$

Dualizing (6.14) and taking adjoint, we get a map of sheaves on
$C\times\mathbb{P}$,

$\rho_{\mathbb{P}}^{*}\mathcal{O}_{\mathbb{P}}(-1)\longrightarrow\widetilde{\eta}^{*}\mathscr{H}om(\mathscr{S},p_{C}^{*}E)\,.$

Tensoring with $\widetilde{\eta}^{*}\mathscr{S}$ and composing with the
natural map
$\widetilde{\eta}^{*}\textrm{Hom}(\mathscr{S},p_{C}^{*}E)\otimes\widetilde{\eta}^{*}\mathscr{S}\to\widetilde{\eta}^{*}p_{C}^{*}E$,
we get a map of sheaves on $C\times\mathbb{P}$,

$\Psi:\widetilde{\eta}^{*}\mathscr{S}\otimes\rho_{\mathbb{P}}^{*}\mathcal{O}_{\mathbb{P}}(-1)\longrightarrow\widetilde{\eta}^{*}p_{C}^{*}E\,.$

A closed point $u\in\mathbb{P}$ corresponds to a pair
$(t=\eta(u),\psi_{u}:\mathscr{S}_{t}\to E)$, where $\psi_{u}$ is a non-zero
map. Restriction of $\Psi$ on $C\times\\{u\\}$ is the map $\psi_{u}$ itself.
Let $U$ be the following subset of $\mathbb{P}$,

$U:=\\{u\in\mathbb{P}:\Psi|_{C\times\\{u\\}}:\mathscr{S}_{\eta(u)}\longrightarrow
E\textrm{ is injective }\\}\,.$

Lemma 6.12 shows that $U$ is an open subset of $\mathbb{P}$. There is a point
$u_{0}\in\mathbb{P}$ which corresponds to the pair
$(t_{0},S_{F}\hookrightarrow E)$. Clearly this point is in $U$, which shows
that $U$ is non-empty. Let $\mathcal{B}$ be the cokernel of $\Psi$. For any
$u\in U$, we have the short exact sequence $0\to\mathscr{S}_{\eta(u)}\to
E\to\mathcal{B}_{u}\to 0\,.$ It follows that the Hilbert polynomial of
$\mathcal{B}_{u}$ is constant as a function of $u$. Thus $\mathcal{B}$ is flat
over $U$. The surjection of sheaves
$\rho_{C}^{*}E\longrightarrow\mathcal{B}\longrightarrow 0$ on $C\times U$
gives a morphism of schemes

$g:U\longrightarrow Q_{k,d}(E)\,.$

The map $g$ sends a closed point $u\in U$ to the point $[E\to\mathcal{B}_{u}]$
in the Quot scheme. Recall that a general member of the family $\mathscr{S}$
is a stable locally free sheaf on $C$. So for a general point $u\in U$,
$\mathscr{S}_{\eta(u)}$ is a stable locally free sheaf on $C$ of rank $r-k$
and degree $e-d$. As $d\geqslant e-d_{8}$,
$\deg(\mathscr{S}_{\eta(u)})=e-d\leq d_{8}$. Using Lemma 6.11, we conclude
that for a general morphism $\alpha:\mathscr{S}_{\eta(u)}\longrightarrow E$,
the cokernel of $\alpha$ is free at the node $x\in C$. In other words, for a
general point $u\in U$, its image $g(u)=[E\to\mathcal{B}_{u}]$ is contained in
the locus $Z^{\rm g}$ of $Q_{k,d}(E)$. Lemma 6.5 shows that the locus $Z^{\rm
g}$ lies in the component $\mathcal{P}$. So image of a general point of $U$ is
in $\mathcal{P}$. Hence the image of whole $U$ is in $\mathcal{P}$. In
particular, the point we started with, that is, $[q:E\to F]=g(u_{0})$ is in
$\mathcal{P}$. As $[q:E\to F]$ is a general point in $\mathcal{W}$, this
proves that $\mathcal{W}=\mathcal{P}$, which is a contradiction. This proves
that the quot scheme $Q_{k,d}(E)$ is irreducible. ∎

######  Theorem 6.15.

Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$ and let $k$ be an
integer such that $0<k<r$. Then there is a number $d_{Q}(E,k)$ such that if
$d\geqslant d_{Q}$ then the quot scheme $Q_{k,d}(E)$ is irreducible. Moreover,
$Q_{k,d}(E)$ is generically smooth and has dimension $dr-ke+k(r-k)(1-g)$.

###### Proof.

If $k>\frac{r}{2}$ then we are done using Theorem 6.13, taking
$d_{Q}(E,k):=d_{9}(E,k)$. Otherwise we have $0<k\leq\frac{r}{2}$. In this
case, we will use Theorem 6.13 to prove irreduciblity of the quot scheme. Let
$t$ be the integer $t:=r-2k+1>0$ so that we have

$k+t>\frac{r+t}{2}\,.$

We define $r^{\prime}:=r+t$ and $k^{\prime}:=k+t$. So the new pair
$r^{\prime},k^{\prime}$ satisfies

$r^{\prime}>k^{\prime}>\frac{r^{\prime}}{2}\,.$

Let $E^{\prime}$ denote the vector bundle $E\oplus\mathcal{O}_{C}^{\oplus t}$
which has rank $r^{\prime}$ and degree $e$. Applying Theorem 6.13 for the
bundle $E^{\prime}$, we get a number
$d^{\prime}_{9}:=d_{9}(E^{\prime},k^{\prime})$ such that if $d\geqslant
d^{\prime}_{9}$ then the quot scheme $Q_{k^{\prime},d}(E^{\prime})$ is
irreducible. We will give a surjective map from an open subset of
$Q_{k^{\prime},d}(E^{\prime})$ to the quot scheme $Q_{k,d}(E)$. That will
prove irreducibility of $Q_{k,d}(E)$ when $d\geqslant d^{\prime}_{9}$.

We define $d_{Q}:=\max\\{d^{\prime}_{9},d_{2}(E,k)\\}$. Assume $d\geqslant
d_{Q}$. Let $Q^{\prime}$ denote the reduced scheme structure on the
irreducible quot scheme $Q_{k^{\prime},d}(E^{\prime})$. So $Q^{\prime}$ is
integral. There is a universal short exact sequence of sheaves on $C\times
Q^{\prime}$,

$0\to\mathscr{S}^{\prime}\to p_{C}^{*}E^{\prime}\to\mathcal{F}^{\prime}\to 0$

where $p_{C}:C\times Q^{\prime}\to Q^{\prime}$ is the projection. Let $\Psi$
denote the composition

$\mathscr{S}^{\prime}\hookrightarrow
p_{C}^{*}E^{\prime}=p_{C}^{*}(E\oplus\mathcal{O}_{C}^{\oplus
t})\twoheadrightarrow p_{C}^{*}E\,.$

The locus of points $q^{\prime}\in Q^{\prime}$ for which the map
$\Psi_{q^{\prime}}$ is injective is an open subset by lemma 6.12. Let $U$
denote this open locus. Since $U$ is integral, using the Hilbert polynomial,
it can be checked easily that the sheaf $\text{coker}(\Psi)|_{C\times U}$ is
flat over $U$. So the quotient of sheaves $p_{C}^{*}E^{\prime}|_{C\times
U}\to\text{coker}(\Psi)|_{C\times U}\to 0$ on $C\times U$ defines a map of
schemes

$g:U\longrightarrow Q_{k,d}(E)\,.$

Let us see the pointwise description of the map $g$. Let
$[q^{\prime}:E^{\prime}\to F^{\prime}]\in Q_{k^{\prime},d}(E^{\prime})$ be a
closed point in $U$. Let $S^{\prime}$ denote the kernel of $q^{\prime}$. As
$[q^{\prime}]$ is in $U$, so the composition $S^{\prime}\hookrightarrow
E^{\prime}\to E$ is injective. Let $B$ denote the cokernel of this inclusion
$S\hookrightarrow E$. Then $B$ is a sheaf of rank $k$ and degree $d$. The map
$g$ sends the point $[q^{\prime}:E^{\prime}\to F^{\prime}]$ to the point
$[E\to B]$. To see that the map $g$ is surjective on closed points, let
$[q:E\to F]$ be a closed point in the quot scheme $Q_{k,d}(E)$. Using $q$, we
get another quotient

$\widetilde{q}:=q\oplus Id:E^{\prime}=E\oplus\mathcal{O}_{C}^{\oplus
t}\longrightarrow F\oplus\mathcal{O}_{C}^{\oplus t}\,.$

Clearly $F\oplus\mathcal{O}_{C}^{\oplus t}$ is a sheaf of rank $k^{\prime}$
and degree $d$. So $\widetilde{q}$ corresponds to a closed point in
$Q^{\prime}$. Moreover, $\ker\widetilde{q}=\ker q\oplus 0\subset E^{\prime}$,
which implies that $[\widetilde{q}]$ is contained in the open set $U$. It is
easy to see that $g([\widetilde{q}])=[q]$. This shows that the map $g$ is
surjective on closed points. As $Q^{\prime}$ is irreducible, so the quot
scheme $Q_{k,d}(E)$ is also irreducible. This proves the theorem.

The second assertion follows from Theorem 4.7, as $d\geqslant d_{2}(E,k)$. ∎

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