# Restriction theorems for semistable sheaves

Mihai Pavel

###### Abstract

In this paper we prove restriction theorems for torsion-free sheaves that are
(semi)stable with respect to the truncated Hilbert polynomial over a smooth
projective variety. Our results apply in particular to Gieseker-semistable
sheaves and generalize the well-known restriction theorems of Mehta and
Ramanathan. As an application we construct a moduli space of sheaves in higher
dimensions.

##  1 Introduction

In the moduli theory of sheaves, the so-called restriction theorems for slope
semistability play a particularly important role. In many cases they provide a
useful instrument to reduce the study of sheaves to lower dimensions. More
exactly, they ensure that slope-semistability (or slope-stability) is
preserved by restriction to a general divisor of sufficiently large degree on
a smooth projective variety. For example, they were used to show Bogomolov’s
inequality in higher dimensions (see [10, Thm. 7.3.1]). Another noteworthy
application was in the construction of moduli spaces of slope-semistable
sheaves over surfaces by Le Potier [14] and Jun Li [17].

At this point there are several different proofs of the restriction theorems
in the literature. The first general results were given by Mehta and
Ramanathan [20, 21]. In zero characteristic, Flenner later proved in [4] an
effective restriction theorem but only for slope-semistability. He found a
bound on the degree of the divisor that depends only on the numerical
invariants of the sheaf. Finally, by a different approach, Langer gave in [11]
stronger effective restriction theorems that work also in positive
characteristic.

######  Main results.

Let $X$ be a smooth projective variety over an algebraically closed field $k$,
and fix $\mathcal{O}_{X}(1)$ a very ample line bundle. In this paper we prove
restriction theorems for torsion-free sheaves on $X$ that are semistable with
respect to the truncated Hilbert polynomial. We call such sheaves
$\ell$-semistable (see Definition 2.1), where $\ell$ marks the level of
truncation. In particular one recovers the notion of slope-(semi)stability for
$\ell=1$ and of Gieseker-(semi)stability for $\ell=\dim(X)$. Our aim is to
prove the following two theorems, thus answering in the affirmative a
conjecture posed by Langer in [12, Conj. 3.13].

######  Theorem 1.1.

Let $\ell<\dim(X)$ and $E$ be an $\ell$-semistable torsion-free sheaf on $X$.
The restriction $E|_{D}$ to a general divisor $D\in|\mathcal{O}_{X}(a)|$
remains $\ell$-semistable for $a\gg 0$.

######  Theorem 1.2.

Let $\ell<\dim(X)$ and $E$ be an $\ell$-stable torsion-free sheaf on $X$. The
restriction $E|_{D}$ to a general divisor $D\in|\mathcal{O}_{X}(a)|$ remains
$\ell$-stable for $a\gg 0$.

As a consequence of Theorem 1.1, we also obtain a restriction theorem for
Gieseker-semistable sheaves. However the range for $\ell$ is sharp in Theorem
1.2. Indeed, the result is false for a Gieseker-stable but not slope-stable
sheaf on a surface. For a different example see also [2, Exam. 3.1].

Our proof is by induction on $\ell$ and follows the strategy applied by Mehta
and Ramanathan [20, 21] in the case of slope-semistability. However, these
methods give no effective bound on the degree of the divisor from which the
restriction remains $\ell$-semistable. Finding such effective bounds may be an
incentive for future studies. One may also ask whether the theorems hold for
pure sheaves supported in positive codimension. In this regard, we proved in
[23] an effective restriction theorem for slope-(semi)stable pure sheaves, by
following a different approach due to Langer [11]. We expect the result to
hold more generally for $\ell$-semistable pure sheaves:

######  Conjecture.

Let $E$ be an $\ell$-(semi)stable pure sheaf of dimension $d$ on $X$ with
$\ell<d$. The restriction $E|_{D}$ to a general divisor
$D\in|\mathcal{O}_{X}(a)|$ remains $\ell$-(semi)stable for $a\gg 0$.

Our restriction theorems may find applications in the moduli theory of
sheaves. To give an example already envisioned by Le Potier [14], one may use
the restriction theorems to construct moduli spaces of $\ell$-semistable
sheaves on $X$. In this respect, here we give the construction when
$\ell=\dim(X)-1$, to obtain a moduli space of sheaves in higher dimensions
over the field of complex numbers. We recover a result due to Huybrechts–Lehn
[10, Sect. 8] when $X$ is a surface. By similar methods, Greb and Toma [7]
constructed moduli spaces of slope-semistable sheaves in higher dimensions,
but they had to work over the category of weakly normal varieties. In contrast
to their result, we need not to impose such restrictive assumptions on schemes
(compare Theorem 4.2 with [7, Main Thm.]), but on the other hand we have a
stronger stability condition on sheaves. Furthermore, in our case we are able
to give a complete description of the geometric points of the moduli space
(see Theorem 4.3).

The moduli spaces of slope-semistable sheaves constructed by Huybrechts–Lehn
and Greb–Toma are closely related to the Donaldson–Uhlenbeck compactification
of Hermitian Yang–Mills (HYM) connections (see [17] and [6]). This is partly
due to the so-called Kobayashi–Hitchin correspondence, which says that vector
bundles over compact complex manifolds admit HYM connections (or
Hermite–Einstein metrics) if and only if they are slope-polystable. There is
an analogue correspondence between Gieseker-stable sheaves and almost
Hermite–Einstein metrics (see [16]). For this reason, we expect that the
moduli space we construct here may find applications in the context of gauge
theory.

######  Aknowledgments.

I would like to thank my advisor Matei Toma for his patient guidance,
insightful suggestions and encouragement during my PhD studies.

##  2 Preliminaries

In this section we set up notation and terminology. First we introduce the
notion of $\ell$-(semi)stability and recall some of its basic properties,
following [10, Sect. 1].

Throughout this paper, let $X$ be a smooth integral projective scheme of
dimension $n$ over a field $k$, and $\mathcal{O}_{X}(1)$ a very ample line
bundle on $X$. For the moment $k$ is a field of arbitrary characteristic,
which is not necessarily algebraically closed.

Let $E$ be a pure sheaf of dimension $d$ on $X$, i.e. any non-zero subsheaf
$F\subset E$ has dimension $d$. The Hilbert polynomial of $E$ is given by its
Euler characteristic $P(E,m)\coloneqq\chi(X,E\otimes\mathcal{O}_{X}(m))$ and
has the form

$\displaystyle P(E,m)=\sum_{j=0}^{d}\alpha_{d-j}(E)\frac{m^{d-j}}{(d-j)!},$

with rational coefficients $\alpha_{d-j}(E)$. In what follows, a pure sheaf on
$X$ of dimension $n=\dim(X)$ will be called torsion-free.

######  Definition 2.1.

Let $E$ be a coherent sheaf of dimension $d$ on $X$ and take $1\leq\ell\leq
d$. Then $E$ is called $\ell$-semistable (resp. $\ell$-stable) if it is pure
and

$\displaystyle\left(\frac{\alpha_{d-1}(F)}{\alpha_{d}(F)},\ldots,\frac{\alpha_{d-\ell}(F)}{\alpha_{d}(F)}\right)\leq\left(\frac{\alpha_{d-1}(E)}{\alpha_{d}(E)},\ldots,\frac{\alpha_{d-\ell}(E)}{\alpha_{d}(E)}\right)\quad(\textnormal{resp.\leavevmode\nobreak\
}<)$

for all subsheaves $F\subset E$ such that $0<\alpha_{d}(F)<\alpha_{d}(E)$,
where $\leq$ is the lexicographic order.

With this definition we recover the notion of slope-(semi)stability for
$\ell=1$, and of Gieseker-(semi)stability for $\ell=d$. More generally, one
may define $\ell$-stability in the quotient category
$\operatorname{Coh}_{d}(X)/\operatorname{Coh}_{d-\ell-1}(X)$ (see [10, Sect.
1.6]), where $\operatorname{Coh}_{d}(X)$ is the category of coherent sheaves
of dimension $\leq d$. However, for the limited purpose of this paper, we
restrict to work within the category $\operatorname{Coh}(X)$.

Sometimes it is convenient to formulate the $\ell$-semistability conditions in
terms of truncated Hilbert polynomials. If $\mathbb{Q}[T]_{j}$ denotes the
vector subspace of polynomials of degree $\leq j$ in $\mathbb{Q}[T]$, let
$P_{\ell}(E)\in\mathbb{Q}[T]_{d}/\mathbb{Q}[T]_{d-\ell-1}$ be the
$\ell$-truncation of the Hilbert polynomial of $E$ and $p_{\ell}(E)\coloneqq
P_{\ell}(E)/\alpha_{d}(E)$ its reduced form. Then, clearly, $E$ is
$\ell$-semistable (resp. $\ell$-stable) if and only if

$\displaystyle p_{\ell}(F)\leq
p_{\ell}(E)\quad(\text{resp.}\leavevmode\nobreak\ <)$

for all subsheaves $F\subset E$ such that $0<\alpha_{d}(F)<\alpha_{d}(E)$.
Above we compare the polynomials with respect to the natural lexicographic
order of their coefficients.

### Properties of $\ell$-semistability

Most of the basic properties of Gieseker-(semi)stability, such as the ones
treated in [10, Sect. 1], hold also for $\ell$-(semi)stability with almost
identical proofs. Furthermore, one can show by using known techniques that
families of $\ell$-semistable sheaves satisfy the following properties:
boundedness, openness and properness, which are particularly important for
studying moduli of sheaves.

######  Lemma 2.1.

Assume that $k$ is algebraically closed. The family of $\ell$-semistable
sheaves of fixed Hilbert polynomial on $X$ is bounded.

###### Proof.

See [11, Thm. 4.4]. ∎

######  Lemma 2.2.

The property of being $\ell$-semistable, resp. simple, geometrically
$\ell$-stable, is open in flat families of sheaves.

###### Proof.

The proof is adapted from [10, Prop. 2.3.1]. The property of being simple,
i.e. $\dim_{k(s)}(\operatorname{End}(\mathcal{F}_{s}))=1$, is open by the
semicontinuity properties of the relative Ext sheaves [1, Satz 3]. We treat
next the case of $\ell$-semistability, see the cited reference for the
remaining case.

Let $S$ be a $k$-scheme of finite type and $\mathcal{F}$ an $S$-flat family of
sheaves on $X$ with Hilbert polynomial $P$. Since the property of being pure
is open in flat families, we may assume the fibers $\mathcal{F}_{s}$ are pure
of dimension $d\coloneqq\deg(P)$. We first show that all destabilizing
quotients $\mathcal{F}_{s}\to F^{\prime}$ fit inside a proper Quot scheme.
Consider the following set

$\displaystyle H=\\{$ $\displaystyle
P(F^{\prime})\in\mathbb{Q}[T]:\mathcal{F}_{s}\to F^{\prime}\textnormal{ is a
pure quotient of dimension $d$}$ $\displaystyle\text{for a closed point $s\in
S$ such that $p_{\ell}(F^{\prime})<p_{\ell}$}\\}.$

where $p_{\ell}$ denotes the reduced $\ell$-truncation of $P$. By a
boundedness result due to Grothendieck (see [10, Lem. 1.7.9]) we know that $H$
is finite. Consider the relative Quot scheme
$\varphi:Q\coloneqq\operatorname{Quot}(\mathcal{F},H)\to S$ of quotients
$\mathcal{F}_{s}\to F^{\prime}$ with $P(F^{\prime})\in H$. Since $Q$ is
proper, its scheme-theoretic image $\varphi(Q)$ is a closed subset of $S$.
Then a fiber $\mathcal{F}_{s}$ is $\ell$-semistable if and only if $s$ is not
contained in $\varphi(Q)$. ∎

######  Lemma 2.3.

Assume that $k$ is algebraically closed. Let $A$ be a discrete valuation ring
with residue field $k$ and quotient field $K$. Let $E$ be an $A$-flat family
of $d$-dimensional sheaves on $X$ such that $E_{K}=E\otimes_{A}K$ is
$\ell$-semistable on $X_{K}$. Then there exists a subsheaf $F\subset E$ such
that $F_{K}=E_{K}$ and $F_{k}$ is $\ell$-semistable on $X$.

###### Proof.

In the slope-semistable case this result is due to Langton [13]. In fact her
proof can be adapted to work also for $\ell$-semistability (see [10, Thm.
2.B.1]). ∎

######  Definition 2.2.

Let $E$ be a coherent sheaf on $X$. Then $E$ is called $\ell$-simple if
$\operatorname{End}(E|_{X\setminus Y})\cong k$ for every closed subset
$Y\subset X$ such that $\operatorname{codim}(\operatorname{Supp}(E)\cap
Y,\operatorname{Supp}(E))\geq\ell+1$.

######  Remark 2.1.

Clearly any $\ell$-simple sheaf is also $(\ell+1)$-simple. Thus any
$\ell$-simple sheaf $E$ is in particular simple, i.e.
$\operatorname{End}(E)\cong k$. Also, one easily checks that any $\ell$-stable
sheaf is $\ell$-simple.

######  Lemma 2.4.

Let $E$ be a torsion-free sheaf on $X$. Then there exists a unique coherent
sheaf $E\subset G$ on $X$ such that

  1. (1)

$E$ and $G$ are isomorphic in codimension $\ell$,

  2. (2)

$G$ satisfies

$\displaystyle\operatorname{depth}_{x}(G)\geq 2\text{ for every }x\in X\text{
such that }\dim(\mathcal{O}_{X,x})\geq\ell+1$ ($\star$)

  3. (3)

for every closed subset $Y\subset X$ such that
$\operatorname{codim}(Y,X)\geq\ell+1$,

$\displaystyle\operatorname{Hom}(G,G)\to\operatorname{Hom}(G|_{X\setminus
Y},G|_{X\setminus Y})$

is an isomorphism.

###### Proof.

Let $F\coloneqq E^{\vee\vee}$ be the double dual of $E$ and set $Q\coloneqq
F/E$. Denote by $T$ the torsion part of $Q$ supported in codimension
$\geq\ell+1$, and take $G\subset F$ such that $G/E\cong T$. Then
$\operatorname{depth}_{x}(Q/T)\geq 1$ for every $x\in X$ such that
$\dim(\mathcal{O}_{X,x})\geq\ell+1$. From the short exact sequence

$\displaystyle 0\to G\to F\to Q/T\to 0,$

we have by [25, Tag 00LX]

$\displaystyle\operatorname{depth}_{x}(G)\geq\min\\{\operatorname{depth}_{x}(F),\operatorname{depth}_{x}(Q/T)+1\\}$

for every point $x\in X$. But $F$ is reflexive, and so
$\operatorname{depth}_{x}(F)\geq 2$ whenever $\dim(\mathcal{O}_{X,x})\geq 2$.
Hence $G$ satisfies ($\star$ ‣ 2). Also, note that $E$ and $G$ are isomorphic
in codimension $\ell$, outside the support of $T$.

It remains to prove $(3)$. Let $Y\subset X$ be a closed subset as in the
statement. As $\operatorname{depth}_{Y}(G)\geq 2$, by a standard result in
local cohomology (see [8, Thm. 3.8]) we have
$\operatorname{\mathcal{E}\\!\mathit{xt}}_{Y}^{i}(G,G)=0$ for $i=0,1$. This
further implies

$\displaystyle\operatorname{Hom}_{Y}(G,G)=\operatorname{Ext}_{Y}^{1}(G,G)=0,$

by using the spectral sequence

$\displaystyle
E_{2}^{pq}=H^{p}(X,\operatorname{\mathcal{E}\\!\mathit{xt}}_{Y}^{q}(G,G))\Rightarrow\operatorname{Ext}_{Y}^{p+q}(G,G)$

given in [9, VI, Thm. 1.6]. From the exact sequence (see [9, VI, Cor. 1.9])

$\displaystyle\operatorname{Hom}_{Y}(G,G)\to\operatorname{Hom}(G,G)\to\operatorname{Hom}(G|_{X\setminus
Y},G|_{X\setminus Y})\to\operatorname{Ext}_{Y}^{1}(G,G)$

it follows that

$\displaystyle\operatorname{Hom}(G,G)\to\operatorname{Hom}(G|_{X\setminus
Y},G|_{X\setminus Y})$

is an isomorphism. ∎

######  Lemma 2.5.

Let $S$ be an integral scheme of finite type over $k$ and $\xi\in S$ its
generic point. Let $f:Z\to S$ be a smooth projective morphism and
$\mathcal{F}$ an $S$-flat family of torsion-free sheaves such that
$\mathcal{F}_{s}$ is $\ell$-simple for every closed point $s\in S$. Then
$\mathcal{F}_{\xi}$ is also $\ell$-simple.

###### Proof.

Let $\iota:\mathcal{F}_{\xi}\to\mathcal{G}_{\xi}$ be the inclusion given by
Lemma 2.4, in particular $\mathcal{F}_{\xi}$ and $G_{\xi}$ are isomorphic in
codimension $\ell$. By shrinking $S$ if necessary, we can extend $\iota$ to an
inclusion $\mathcal{F}\subset\mathcal{G}$ such that $\mathcal{G}$ is an
$S$-flat family and moreover, $\mathcal{F}_{s}$ and $\mathcal{G}_{s}$ are
isomorphic in codimension $\ell$ for every $s\in S$. Note that
$\mathcal{F}_{s}$ is $\ell$-simple if and only if $\mathcal{G}_{s}$ is
$\ell$-simple, hence it is enough to prove the statement for the family
$\mathcal{G}$.

Let $Y_{\xi}\subset Z_{\xi}$ be a closed subset of codimension $\geq\ell+1$ in
$Z_{\xi}$. By Lemma 2.4 there is an isomorphism

$\displaystyle\operatorname{Hom}(\mathcal{G}_{\xi},\mathcal{G}_{\xi})\to\operatorname{Hom}(\mathcal{G}_{\xi}|_{Z_{\xi}\setminus
Y_{\xi}},\mathcal{G}_{\xi}|_{Z_{\xi}\setminus Y_{\xi}}).$

By the semicontinuity properties of the relative Ext sheaves [1, Satz 3], it
follows that $\mathcal{G}_{\xi}$ is simple (see also Lemma 2.2). We conclude
that $\mathcal{G}_{\xi}$ is $\ell$-simple. ∎

We next recall the existence of the Harder-Narasimhan filtration. The result
is standard and we omit its proof.

######  Lemma 2.6.

If $E$ is a pure sheaf on $X$, then $E$ has a unique $\ell$-Harder-Narasimhan
filtration

$\displaystyle 0=E_{0}\subset E_{1}\subset\ldots\subset E_{m}=E$

such that its factors $E_{i}/E_{i-1}$ are $\ell$-semistable and satisfy

$\displaystyle
p_{\ell}(E_{1})>p_{\ell}(E_{2}/E_{1})>\ldots>p_{\ell}(E/E_{m-1}).$

We call $E/E_{m-1}$ the minimal $\ell$-destabilizing quotient of $E$.

We also have a relative version of the Harder-Narasimhan filtration, which
will provide a key ingredient in proving Theorem 1.1.

######  Lemma 2.7.

Let $S$ be an integral scheme of finite type over $k$ and $\mathcal{F}$ an
$S$-flat family of $\ell$-semistable sheaves on $X$. Then there exists a dense
open subset $U\subset S$ and a filtration

$\displaystyle
0=\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\ldots\subset\mathcal{F}_{m}=\mathcal{F}$

such that

  1. 1.

its factors $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ are flat over U,

  2. 2.

for each closed point $s\in U$, the restriction
$0=\mathcal{F}_{0,s}\subset\mathcal{F}_{1,s}\subset\ldots\subset\mathcal{F}_{m,s}=\mathcal{F}_{s}$
is the $\ell$-Harder-Narasimhan filtration of $\mathcal{F}_{s}$.

###### Proof.

The proof is very similar to that in [10, Thm. 2.3.2]. ∎

If $E$ is already $\ell$-semistable, then $E$ has a so-called Jordan-Hölder
filtration

$\displaystyle 0=E_{0}\subset E_{1}\subset\ldots\subset E_{m}=E$

such that its factors $E_{i}/E_{i-1}$ are $\ell$-stable with
$p_{\ell}(E_{i}/E_{i-1})=p_{\ell}(E)$. This filtration is not unique in
general, but nevertheless its graded sheaf
$\operatorname{gr}_{\ell}(E)=\oplus_{i}{E_{i}/E_{i-1}}$ is so in codimension
$\ell$, i.e. outside a closed subset of codimension $\geq\ell+1$ in
$\operatorname{Supp}(E)$.

As in the case of Gieseker-semistability, $E$ has a unique extended socle,
that we describe precisely below. The following is adapted from [10, Sect.
1.5] and treats the case of $\ell$-stability. A similar discussion for slope-
stability can be found in [5, Sect. 6]. We define the $\ell$-socle
$\operatorname{Soc}(E)$ of $E$ as the saturation of the sum of all
$\ell$-stable subsheaves $F\subset E$ with $p_{\ell}(F)=p_{\ell}(E)$.

######  Definition 2.3.

The extended $\ell$-socle of an $\ell$-semistable sheaf $E$ is the maximal
subsheaf $F\subset E$ with $p_{\ell}(F)=p_{\ell}(E)$ such that every graded
factor of $\operatorname{gr}_{\ell}(F)$ is isomorphic with a graded factor of
$\operatorname{gr}_{\ell}(\operatorname{Soc}(E))$ in codimension $\ell$.

######  Lemma 2.8.

Let $E$ be an $\ell$-stable sheaf on $X$ and $k\subset K$ the algebraic
closure. Then $E_{K}=E\otimes_{k}K$ equals its extended $\ell$-socle.

###### Proof.

The proof is identical to that of [5, Lem. 6.2]. ∎

######  Lemma 2.9.

Let $E$ be a coherent sheaf on $X$. If $E$ is $\ell$-simple, $\ell$-semistable
and equals its extended $\ell$-socle, then $E$ is $\ell$-stable.

###### Proof.

Suppose that $E$ is not $\ell$-stable, so there exists an $\ell$-stable
quotient $E\to F$ with $p_{\ell}(F)=p_{\ell}(E)$. As $E$ equals its extended
$\ell$-socle, there is an open subset $U\subset\operatorname{Supp}(E)$ with
$\operatorname{codim}(\operatorname{Supp}(E)\setminus
U,\operatorname{Supp}(E))\geq\ell+1$ such that $F|_{U}\subset E|_{U}$. This
gives a non-trivial composition $E|_{U}\to F|_{U}\to E|_{U}$. But $E$ is
$\ell$-simple, so $F|_{U}\cong E|_{U}$, which yields a contradiction. ∎

######  Lemma 2.10.

If $E$ is $\ell$-simple, then $E$ is $\ell$-stable if only if $E$ is
geometrically $\ell$-stable.

###### Proof.

This follows immediately from Lemma 2.8 and Lemma 2.9. ∎

### The Hilbert polynomial

Below we relate the coefficients of the Hilbert polynomial of a sheaf $E$ to
those of its restriction $E|_{D}$ to a divisor. For $a>0$, let
$\Pi_{a}\coloneqq|\mathcal{O}_{X}(a)|$ be the linear system of hypersurfaces
of degree $a$ on $X$. For a pure sheaf $E$ of dimension $d$ on $X$, consider
the short exact sequence

$\displaystyle 0\to E(-a)\to E\to E|_{D}\to 0,$

for a divisor $D\in\Pi_{a}$ avoiding the associated points of $E$. By the
additivity of the Hilbert polynomial in short exact sequences we obtain

$\displaystyle\alpha_{d-j}(E|_{D})={}$
$\displaystyle\alpha_{d-j}(E)-\alpha_{d-j}(E(-a))$ $\displaystyle={}$
$\displaystyle\sum_{l=1}^{j}\frac{(-1)^{l-1}a^{l}}{l!}\alpha_{d-j+l}(E)$ (2.1)

for $1\leq j\leq d$.

######  Lemma 2.11.

Let $E$ and $F$ be pure sheaves of dimension $d$ on $X$ and $D\in\Pi_{a}$ a
divisor such that $E|_{D}$ and $F|_{D}$ are still pure. If $\ell<d$, then
$p_{\ell}(E|_{D})\geq p_{\ell}(F|_{D})$ (resp. $>$) if and only if
$p_{\ell}(E)\geq p_{\ell}(F)$ (resp. $>$).

###### Proof.

The inequalities follow in an iterative manner from (2). ∎

Note that (2) gives us no information about the free coefficient
$\alpha_{0}(E)$, which explains the upper bound $\ell<d$ in the statement
above.

For $1\leq j\leq d$ and $D\in\Pi_{a}$, set

$\displaystyle\beta_{d-j}(E|_{D})\coloneqq\alpha_{d-j}(E|_{D})+\frac{a}{2}\alpha_{d-j+1}(E|_{D})+\sum_{l=1}^{j-1}\frac{(-1)^{l-1}B_{2l}}{(2l)!}a^{2l}\alpha_{d-j+1+l}(E|_{D}),$

where $B_{2l}$ denotes the $l$-th Bernoulli number. Recall that the Bernoulli
numbers can be defined by the generating power series

$\displaystyle Q(x)=\frac{x}{1-e^{-x}}=1+\frac{x}{2}+\sum_{l\geq
1}\frac{(-1)^{l-1}B_{2l}}{(2l)!}x^{2l}=1+\frac{x}{2}+\frac{x^{2}}{12}-\frac{x^{4}}{720}+\ldots$

By using its inverse

$\displaystyle Q(x)^{-1}=\frac{1-e^{-x}}{x}=\sum_{l\geq
1}\frac{(-1)^{l-1}x^{l-1}}{l!},$

one can derive

$\displaystyle\alpha_{d-j}(E|_{D})=\sum_{l=1}^{j}\frac{(-1)^{l-1}a^{l-1}}{l!}\beta_{d-j+l-1}(E|_{D}).$
(2.2)

A direct comparison of (2) and (2.2) shows that
$\beta_{d-j}(E|_{D})/a=\alpha_{d-j+1}(E)$ whenever $D$ avoids the associated
points of $E$.

### Determinant line bundles

The following discussion will serve only in Section 4, so the reader may skip
this part and come back when needed.

Fix a numerical Grothendieck class $c\in K(X)_{\text{num}}$. Let $S$ be a
scheme over $k$ and $\mathcal{E}$ an $S$-flat family of coherent sheaves of
class $c$ on $X$. Consider the map
$\lambda_{\mathcal{E}}:K(X)\to\operatorname{Pic}(S)$ given by the composition:

$\displaystyle K(X)\xrightarrow{p_{2}^{*}}K^{0}(S\times
X)\xrightarrow{\cdot[\mathcal{E}]}K^{0}(S\times
X)\xrightarrow{{p_{1}}_{!}}K^{0}(S)\xrightarrow{\det}\operatorname{Pic}(S),$
(2.3)

which associates to a class $u\in K(X)$ its corresponding determinant line
bundle $\lambda_{\mathcal{E}}(u)$ over $S$. One notices that this construction
behaves well with respect to base change. We refer the reader to [15, 14] for
an account of the properties of $\lambda_{\mathcal{E}}$.

The Grothendieck group $K(X)$ is endowed with a natural quadratic form given
by the Euler characteristic $\chi$. If $A\subset K(X)$ is a subset, let
$A^{\perp}\subset K(X)$ be the orthogonal complement of $A$ with respect to
$\chi$. Let $h$ denote the class of $\mathcal{O}_{X}(1)$ in $K(X)$. If $u\in
K(X)$ is a class such that $u\in
c^{\perp}\cap\\{1,h,\ldots,h^{n}\\}^{\perp\perp}$ then there is a natural
determinant line bundle $\lambda(u)$ over the moduli space $M^{Gss}(c)$ of
Gieseker-semistable sheaves of class $c$ on $X$ (see [10, Thm. 8.1.5]).

###  $G$-properness and Iitaka $G$-fibration

In this part we work over the category $(\text{Sch}^{ft}/\mathbb{C})$ of
schemes of finite type over $\mathbb{C}$. Fix $G$ a connected algebraic group
over $\mathbb{C}$.

######  Definition 2.4.

Let $R$ be a $G$-scheme and $f:R\to S$ a $G$-invariant separated morphism. We
say that $f$ is $G$-proper if for every commutative diagram

${\operatorname{Spec}(K)}$${R}$${\operatorname{Spec}(A)}$${S}$$\scriptstyle{j}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{i}$

where $A$ is a discrete valuation ring over $\mathbb{C}$ of quotient field
$K$, there exists a morphism $i:\operatorname{Spec}(A)\to R$ such that $i\circ
j$ and $g$ differ by a group element in $G(K)$.

Let $R$ be a $G$-proper scheme over $\mathbb{C}$ endowed with a
$G$-equivariant line bundle $\mathcal{L}$ that is $G$-semiample, i.e.
$\mathcal{L}$ is globally generated by $G$-invariant sections. If $G$ is
reductive, then one can form the Iitaka $G$-fibration of $(R,\mathcal{L})$ as
in [23, Sect. 5], which is given by a Proj scheme

$\displaystyle\textup{Proj}\bigoplus_{k\geq 0}H^{0}(R,\mathcal{L}^{ke})^{G}$

for some $e>0$. In Section 4, we will use the Iitaka $G$-fibration to
construct a moduli space of sheaves.

##  3 Restriction theorems for $\ell$-semistability

In this section we assume that the base field $k$ is algebraically closed of
arbitrary characteristic. We will show both restriction theorems by following
the methods of Mehta and Ramanathan. For this reason, in some situations the
proofs follow a similar pattern. Whenever this happens we only sketch the
arguments and refer the reader to [20, 21] or [10, Sect. 7] for further
details. We also recommend the lecture notes in [5] for a detailed exposition
of the slope-(semi)stable case.

We first show Theorem 1.1. The basic idea of the proof is to extend a
destabilizing quotient $E|_{D}\to F_{D}$ over $X$, since then Lemma 2.11 will
give a contradiction. We argue by induction on $\ell$. We already know by the
Mehta–Ramanathan restriction theorem that the result holds for $\ell=1$, so it
remains to prove the induction step.

Fix $1<\ell<n$ and let $E$ be an $\ell$-semistable torsion-free sheaf on $X$.
We want to vary the minimal $\ell$-destabilizing quotient of $E|_{D}$ in a
flat family over $\Pi_{a}$. For this consider the incidence variety
$Z_{a}=\\{([D],x)\in\Pi_{a}\times X|x\in D\\}$ with its natural projections:

${Z_{a}}$${X}$${\Pi_{a}}$$\scriptstyle{p}$$\scriptstyle{q}$

######  Remark 3.1.

If $\mathcal{K}$ denotes the kernel of the evaluation map

$\displaystyle
H^{0}(X,\mathcal{O}_{X}(a))\otimes\mathcal{O}_{X}\to\mathcal{O}_{X}(a),$

then $q:Z_{a}\to X$ is in fact isomorphic to the projective bundle
$\mathbb{P}(\mathcal{K}^{\vee})\to X$. Also, there exists an isomorphism
$\operatorname{Pic}(Z_{a})\cong\operatorname{Pic}(X)\times\operatorname{Pic}(\Pi_{a})$
such that $\mathcal{O}_{\mathbb{P}(\mathcal{K}^{\vee})}(1)\cong
p^{*}(\mathcal{O}_{\Pi_{a}}(1))$ (see the proof in [20, Prop. 2.1]).

As a matter of notation, we will denote by $[D]\in\Pi_{a}$, or simply by
$D\in\Pi_{a}$, the closed point of $\Pi_{a}$ coressponding to an effective
divisor $D\subset X$. Also we will often identify the fibre $p^{-1}([D])$ with
$D\subset X$ via $q$.

By Lemma 2.7 there exists a relative $\ell$-Harder-Narasimhan filtration of
$q^{*}E$, which one can use to prove the following result:

######  Lemma 3.1.

For $a>0$ there exists a dense open subset $U_{a}\subset\Pi_{a}$ and a
quotient $q^{*}E\to\mathcal{F}_{a}$ such that

  1. (1)

each divisor $D\in U_{a}$ is smooth and integral,

  2. (2)

$\mathcal{F}_{a}$ is flat over $U_{a}$,

  3. (3)

for $[D]\in U_{a}$, $E|_{D}\to\mathcal{F}_{a}|_{D}$ is the minimal
$\ell$-destabilizing quotient of $E|_{D}$,

  4. (4)

for $a\gg 0$, $E|_{D}$ is $(\ell-1)$-semistable.

###### Proof.

$(2)$ and $(3)$ are direct consequences of Lemma 2.7. We may also assume by
induction that $E|_{D}$ is $(\ell-1)$-semistable for general $[D]\in\Pi_{a}$
of sufficiently large degree. Finally $(1)$ holds by Bertini’s Theorem. ∎

Since $\mathcal{F}_{a}$ is flat over $U_{a}$, the Hilbert polynomial of
$\mathcal{F}_{a}|_{D}$ does not vary with $D\in U_{a}$. Hence we set
$P(a)\coloneqq P(\mathcal{F}_{a}|_{D})$,
$\operatorname{rk}(a)\coloneqq\operatorname{rk}(\mathcal{F}_{a}|_{D})$,
$\alpha_{n-j}(a)\coloneqq\alpha_{n-j}(\mathcal{F}_{a}|_{D})$ and so forth.

######  Remark 3.2.

Since $E|_{D}$ is already $(\ell-1)$-semistable, we have
$p_{\ell-1}(a)=p_{\ell-1}(E|_{D})$ for $a\gg 0$. Then

$\displaystyle\frac{\beta_{n-j}(a)}{a\operatorname{rk}(a)}=\frac{\beta_{n-j}(E|_{D})}{a\operatorname{rk}(E|_{D})}=\frac{\alpha_{n-j+1}(E)}{\operatorname{rk}(E)}$

is constant for any $1\leq j\leq\ell$ and $a\gg 0$.

Let $D=D_{1}+D_{2}$ be a simple normal crossing divisor of degree
$a=a_{1}+a_{2}$ with $[D_{1}]\in U_{a_{1}}$ and $[D_{2}]\in U_{a_{2}}$. Then
there exists a smooth (locally closed) curve $C\subset\Pi_{a}$ passing through
$[D]$ such that $C\setminus[D]\subset U_{a}$ and $Z_{C}\coloneqq
Z_{a}\times_{\Pi_{a}}C$ is smooth in codimension 2 (see [10, Lem. 7.2.3]).
Restricting $q^{*}E\to\mathcal{F}_{a}$ to $Z_{C\setminus[D]}$ gives a quotient
$q^{*}E|_{Z_{C\setminus[D]}}\to\mathcal{F}_{a}|_{Z_{C\setminus[D]}}$, which
induces a morphism of schemes

$\displaystyle
C\setminus[D]\to\operatorname{Quot}_{Z_{a}/\Pi_{a}}(q^{*}E,P(a))$

As $C$ is a smooth curve and the Quot scheme is projective, we can extend the
above morphism over $[D]$ and get a quotient
$q^{*}E|_{Z_{C}}\to\mathcal{F}_{C}$.

In what follows $i=1,2$. Let
$F\coloneqq(\mathcal{F}_{C}|_{D})/T(\mathcal{F}_{C}|_{D})$, $F_{i}\coloneqq
F|_{D_{i}}$ and denote by $T_{i}\coloneqq T(F_{i})$ the torsion of $F_{i}$.
For $1\leq j\leq\ell+1$, consider the following property $(P_{j})$:

  1. 1.

for $a\gg 0$, $T(\mathcal{F}_{C}|_{D})$ is supported in codimension $\geq j$
in $D$ and

$\displaystyle\frac{\alpha_{n-j}(F)}{\operatorname{rk}(F)}=\frac{\alpha_{n-j}(a)}{\operatorname{rk}(a)},$

  2. 2.

for $a_{i}\gg 0$, $T_{i}$ is supported in codimension $\geq j$ in $D_{i}$ and

$\displaystyle\frac{\alpha_{n-j}(F_{i}/T_{i})}{\operatorname{rk}(F_{i}/T_{i})}=\frac{\alpha_{n-j}(a_{i})}{\operatorname{rk}(a_{i})}.$

We will show by induction that $(P_{j})$ holds true for $1\leq j\leq\ell+1$.
First note that
$\alpha_{n-1}(F)=\alpha_{n-1}(\mathcal{F}_{C}|_{D})=\alpha_{n-1}(a)$ by the
flatness of $\mathcal{F}_{C}$. The second part of $(P_{1})$ is also clearly
satisfied. We next show the induction step, whose proof we divided in a
sequence of lemmas. In what follows, let $j>1$ and suppose that
$(P_{1}),\ldots,(P_{j-1})$ hold true.

######  Lemma 3.2.

For $a_{1},a_{2}\gg 0$ we have

$\displaystyle\frac{\alpha_{n-j}(a)}{\operatorname{rk}(a)}\geq{}$
$\displaystyle\frac{\alpha_{n-j}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}+\frac{\alpha_{n-j}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}-$
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2l!}\left(a_{2}^{l}\frac{\alpha_{n-j+l}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}+a_{1}^{l}\frac{\alpha_{n-j+l}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}\right).$
(3.1)

###### Proof.

Consider the short exact sequence

$\displaystyle
0\to\mathcal{O}_{D}\to\mathcal{O}_{D_{1}}\oplus\mathcal{O}_{D_{2}}\to\mathcal{O}_{D_{1}\cap
D_{2}}\to 0$

corresponding to the normal crossing divisor $D=D_{1}+D_{2}$. After tensoring
with the pure sheaf $F$ the sequence

$\displaystyle 0\to F\to F_{1}\oplus F_{2}\to F|_{D_{1}\cap D_{2}}\to 0$ (3.2)

remains exact. We get by additivity

$\displaystyle\alpha_{n-j}(F)=\alpha_{n-j}(F_{1})+\alpha_{n-j}(F_{2})-\alpha_{n-j}(F|_{D_{1}\cap
D_{2}}).$

From the short exact sequence

$\displaystyle 0\to T_{i}\to F_{i}\to F_{i}/T_{i}\to 0,$ (3.3)

we also have

$\displaystyle\alpha_{n-j}(F_{i})=\alpha_{n-j}(T_{i})+\alpha_{n-j}(F_{i}/T_{i}).$

As $F_{i}/T_{i}$ has no torsion over $D_{1}\cap D_{2}$, the restriction of
(3.3) to $D_{1}\cap D_{2}$ gives the short exact sequence

$\displaystyle 0\to T_{i}|_{D_{1}\cap D_{2}}\to F|_{D_{1}\cap
D_{2}}\to(F_{i}/T_{i})|_{D_{1}\cap D_{2}}\to 0.$

Thus

$\displaystyle\alpha_{n-j}(F|_{D_{1}\cap
D_{2}})=\alpha_{n-j}(T_{i}|_{D_{1}\cap
D_{2}})+\alpha_{n-j}((F_{i}/T_{i})|_{D_{1}\cap D_{2}}).$

Putting together the above identities and using
$\alpha_{n-j}(T_{i})\geq\alpha_{n-j}(T_{i}|_{D_{1}\cap D_{2}})$, we get

$\displaystyle\alpha_{n-j}(F)\geq{}$
$\displaystyle\alpha_{n-j}(F_{1})+\alpha_{n-j}(F_{2}/T_{2})-\alpha_{n-j}((F_{2}/T_{2})|_{D_{1}\cap
D_{2}})$ $\displaystyle\geq{}$
$\displaystyle\alpha_{n-j}(F_{1}/T_{1})+\alpha_{n-j}(F_{2}/T_{2})-\alpha_{n-j}((F_{2}/T_{2})|_{D_{1}\cap
D_{2}}),$ (3.4)

where the last inequality holds since the torsion is supported in codimension
$\geq j-1$ by the induction hypothesis.

On the other hand, from the short exact sequence

$\displaystyle 0\to(F_{2}/T_{2})(-D_{1})\to
F_{2}/T_{2}\to(F_{2}/T_{2})|_{D_{1}\cap D_{2}}\to 0,$

it follows by formula (2) that

$\displaystyle\alpha_{n-j}((F_{2}/T_{2})|_{D_{1}\cap
D_{2}})=\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{1}^{l}}{l!}\alpha_{n-j+l}(F_{2}/T_{2}).$

From this we can rewrite (3) as

$\displaystyle\alpha_{n-j}(F)\geq\alpha_{n-j}(F_{1}/T_{1})+\alpha_{n-j}(F_{2}/T_{2})-\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{1}^{l}}{l!}\alpha_{n-j+l}(F_{2}/T_{2}).$

By a similar computation, we also have

$\displaystyle\alpha_{n-j}(F)\geq\alpha_{n-j}(F_{1}/T_{1})+\alpha_{n-j}(F_{2}/T_{2})-\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{2}^{l}}{l!}\alpha_{n-j+l}(F_{1}/T_{1}).$

Adding the last two inequalities and dividing by 2 we obtain

$\displaystyle\alpha_{n-j}(F)\geq{}$
$\displaystyle\alpha_{n-j}(F_{1}/T_{1})+\alpha_{n-j}(F_{2}/T_{2})-$
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2l!}(a_{2}^{l}\alpha_{n-j+l}(F_{1}/T_{1})+a_{1}^{l}\alpha_{n-j+l}(F_{2}/T_{2})).$
(3.5)

By the flatness of $\mathcal{F}_{C}$ over $C$, we have
$\alpha_{n-j}(\mathcal{F}_{C}|_{D})=\alpha_{n-j}(a)$. Moreover, since
$\operatorname{Supp}(T(\mathcal{F}_{C}|_{D}))$ has codimension $\geq j-1$ in
$D$ by $(P_{j-1})$, we get

$\displaystyle\alpha_{n-j}(a)=\alpha_{n-j}(\mathcal{F}_{C}|_{D})\geq\alpha_{n-j}(F).$
(3.6)

We also have
$\operatorname{rk}(F_{i}/T_{i})=\operatorname{rk}(F)=\operatorname{rk}(a)$
(see [5, Lem. 2.12]). From these considerations, dividing (3) by
$\operatorname{rk}(a)$ we finally deduce (3.2). ∎

######  Lemma 3.3.

For $a_{1},a_{2}\gg 0$ we have

$\displaystyle\frac{\beta_{n-j}(a)}{\operatorname{rk}(a)}\geq\frac{\beta_{n-j}(a_{1})}{\operatorname{rk}(a_{1})}+\frac{\beta_{n-j}(a_{2})}{\operatorname{rk}(a_{2})}.$

###### Proof.

The basic idea is to use (2.2) in order to replace the $\alpha$-terms in (3.2)
by $\beta$-terms. This way the LHS of (3.2) can be rewritten as

$\displaystyle\frac{\alpha_{n-j}(a)}{\operatorname{rk}(a)}=\frac{\beta_{n-j}(a)}{\operatorname{rk}(a)}+\sum_{l=1}^{j-1}\frac{(-1)^{l}}{(l+1)!}a^{l}\frac{\beta_{n-j+l}(a)}{\operatorname{rk}(a)}.$

We are now concerned with the RHS of (3.2). A straightforward computation
gives

$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{2}^{l}}{2l!}\frac{\alpha_{n-j+l}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}=$
$\displaystyle={}$
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{2}^{l}}{2l!}\sum_{s=1}^{j-l}\frac{(-1)^{s-1}a_{1}^{s-1}}{s!}\frac{\beta_{n-j+l+s-1}(a_{1})}{\operatorname{rk}(a_{1})}$
(by (2.2) and $(P_{<j}.2$)) $\displaystyle={}$
$\displaystyle\sum_{p=1}^{j-1}\frac{\beta_{n-j+p}(a_{1})}{\operatorname{rk}(a_{1})}\sum_{l=1}^{p}\frac{(-1)^{l-1}a_{2}^{l}}{2l!}\frac{(-1)^{p-l}a_{1}^{p-l}}{(p-l+1)!}$
(by the change of variables $p=s+l-1$) $\displaystyle={}$
$\displaystyle\sum_{p=1}^{j-1}\frac{\beta_{n-j+p}(a_{1})}{a_{1}\operatorname{rk}(a_{1})}\frac{(-1)^{p-1}}{2(p+1)!}((a_{1}+a_{2})^{p+1}-a_{1}^{p+1}-a_{2}^{p+1})$
(by binomial formula) $\displaystyle={}$
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2(l+1)!}\frac{\beta_{n-j+l}(a_{1})}{a_{1}\operatorname{rk}(a_{1})}(a^{l+1}-a_{1}^{l+1}-a_{2}^{l+1}).$
(by the change of variables $l=p$)

In a similar fashion,

$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}a_{1}^{l}}{2l!}\frac{\alpha_{n-j+l}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}=\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2(l+1)!}\frac{\beta_{n-j+l}(a_{2})}{a_{2}\operatorname{rk}(a_{2})}(a^{l+1}-a_{2}^{l+1}-a_{1}^{l+1}).$

Adding the last two identities, for $a_{1},a_{2}\gg 0$ we obtain by Remark 3.2
that

$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2l!}(a_{2}^{l}\frac{\alpha_{n-j+l}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}+a_{1}^{l}\frac{\alpha_{n-j+l}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}=$
$\displaystyle={}$
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{(l+1)!}\left(\frac{\beta_{n-j+l}(a)}{a\operatorname{rk}(a)}a^{l+1}-\frac{\beta_{n-j+l}(a_{1})}{a_{1}\operatorname{rk}(a_{1})}a_{1}^{l+1}-\frac{\beta_{n-j+l}(a_{2})}{a_{2}\operatorname{rk}(a_{2})}a_{2}^{l+1}\right).$
(3.7)

As $F_{i}/T_{i}$ is a torsion-free quotient of $E|_{D_{i}}$, it must satisfy

$\displaystyle\frac{\alpha_{n-j}(F_{i}/T_{i})}{\operatorname{rk}(F_{i}/T_{i})}\geq\frac{\alpha_{n-j}(\mathcal{F}_{a_{i}}|_{D_{i}})}{\operatorname{rk}(\mathcal{F}_{a_{i}}|_{D_{i}})}=\frac{\alpha_{n-j}(a_{i})}{\operatorname{rk}(a_{i})}.$
(3.8)

Hence the RHS of $\eqref{eq:gammaineq}$ satisfies

$\displaystyle\frac{\alpha_{n-j}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}+\frac{\alpha_{n-j}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}-\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{2l!}\left(a_{2}^{l}\frac{\alpha_{n-j+l}(F_{1}/T_{1})}{\operatorname{rk}(F_{1}/T_{1})}+a_{1}^{l}\frac{\alpha_{n-j+l}(F_{2}/T_{2})}{\operatorname{rk}(F_{2}/T_{2})}\right)$
$\displaystyle\geq{}$
$\displaystyle\frac{\alpha_{n-j}(a_{1})}{\operatorname{rk}(a_{1})}+\frac{\alpha_{n-j}(a_{2})}{\operatorname{rk}(a_{2})}-$
(by (3) and (3.8))
$\displaystyle\sum_{l=1}^{j-1}\frac{(-1)^{l-1}}{(l+1)!}\left(\frac{\beta_{n-j+l}(a)}{a\operatorname{rk}(a)}a^{l+1}-\frac{\beta_{n-j+l}(a_{1})}{a_{1}\operatorname{rk}(a_{1})}a_{1}^{l+1}-\frac{\beta_{n-j+l}(a_{2})}{a_{2}\operatorname{rk}(a_{2})}a_{2}^{l+1}\right)$
$\displaystyle\geq{}$
$\displaystyle\frac{\beta_{n-j}(a_{1})}{\operatorname{rk}(a_{1})}+\frac{\beta_{n-j}(a_{2})}{\operatorname{rk}(a_{2})}+\sum_{l=1}^{j-1}\frac{(-1)^{l}}{(l+1)!}a^{l}\frac{\beta_{n-j+l}(a)}{\operatorname{rk}(a)}.$
(by (2.2))

Comparing both sides of (3.2), we conclude that for $a_{1},a_{2}\gg 0$

$\displaystyle\frac{\beta_{n-j}(a)}{\operatorname{rk}(a)}\geq\frac{\beta_{n-j}(a_{1})}{\operatorname{rk}(a_{1})}+\frac{\beta_{n-j}(a_{2})}{\operatorname{rk}(a_{2})}.$

∎

######  Lemma 3.4.

$\frac{\beta_{n-j}(a)}{a\operatorname{rk}(a)}$ is constant for $a\gg 0$.

###### Proof.

We already know the result holds true for $1\leq j\leq\ell$ by Remark 3.2, so
it only remains to prove it when $j=\ell+1$.

First, we show that
$\\{\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}:a>0\\}$ is a discrete
subset of $\mathbb{Q}$. Recall that $q:Z_{a}\to X$ is isomorphic to the
projective bundle $\mathbb{P}(\mathcal{K}^{\vee})\to X$ and, via this
isomorphism, we have $\mathcal{O}_{\mathbb{P}(\mathcal{K}^{\vee})}(1)\cong
p^{*}(\mathcal{O}_{\Pi_{a}}(1))$ (see Remark 3.1). Denote by $\xi$ the class
of $\mathcal{O}_{\Pi_{a}}(1)$ in the Grothendieck group $K(\Pi_{a})$ and set
$r\coloneqq\operatorname{rk}(\mathcal{K})$. We have the following isomorphism
of $K(X)$-algebras (see [18, Thm. 4.5])

$\displaystyle K(Z_{a})\cong K(\mathbb{P}(\mathcal{K}^{\vee}))\cong
K(X)[T]/(\sum_{i=0}^{r}(-1)^{i}cl(\Lambda^{r-i}(\mathcal{K}^{\vee}))T^{i}),$

where $cl(-)$ denotes the corresponding Grothendieck class. Moreover

$\displaystyle T\text{ mod
}(\sum_{i=0}^{r}(-1)^{i}cl(\Lambda^{r-i}(\mathcal{K}^{\vee}))T^{i})=cl(p^{*}\xi)\in
K(Z_{a}).$

Using this isomorphism we can write

$\displaystyle cl(\mathcal{F}_{a})$
$\displaystyle=\sum_{i=0}^{r-1}cl(G_{i})cl(p^{*}\xi)^{i}$ (3.9)

for $G_{i}$ coherent sheaves on $X$. As $X$ is smooth and projective, any
$G_{i}$ has a finite free resolution. Thus, there is a finite sum

$\displaystyle cl(G_{i})=\sum_{p}n_{i,p}cl(G_{i,p})$

in $K(X)$ with $n_{i,p}\in\mathbb{Z}$ and $G_{i,p}$ locally free sheaves on
$X$. Restricting (3.9) to a fiber over $[D]\in U_{a}$, we obtain

$\displaystyle
i^{*}cl(\mathcal{F}_{a})=\sum_{i=0}^{r-1}\sum_{p}n_{i,p}cl(G_{i,p}|_{D})$

in $K(Z_{[D]})\cong K(D)$, where $i:Z_{[D]}\to Z$ is the natural immersion.
For $1\leq j\leq\ell+1$, we get by additivity

$\displaystyle\frac{\alpha_{n-j}(a)}{a\operatorname{rk}(a)}$
$\displaystyle=\sum_{i=0}^{r-1}\sum_{p}n_{i,p}\frac{\alpha_{n-j}(G_{i,p}|_{D})}{a\operatorname{rk}(a)}$
$\displaystyle=\sum_{i=0}^{r-1}\sum_{p}n_{i,p}\sum_{l=1}^{j}\frac{(-1)^{l-1}a^{l-1}}{l!}\frac{\alpha_{n-j+l}(G_{i,p})}{\operatorname{rk}(a)}\in\frac{\mathbb{Z}}{(\ell+1)!\operatorname{rk}(E)!}.$

from which we obtain that
$\\{\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}:a>0\\}$ is discrete in
$\mathbb{Q}$.

Second, let us show that the function
$a\mapsto\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}$ is bounded from
above. By Remark 3.2 we have

$\displaystyle\frac{\beta_{n-j}(a)}{a\operatorname{rk}(a)}=\frac{\alpha_{n-j+1}(E)}{\operatorname{rk}(E)}$
(3.10)

for $1\leq j\leq\ell$. Then

$\displaystyle\frac{\alpha_{n-1-\ell}(E|_{D})}{a\operatorname{rk}(E|_{D})}={}$
$\displaystyle\sum_{l=1}^{\ell+1}\frac{(-1)^{l-1}a^{l}}{l!}\frac{\alpha_{n-1-\ell+l}(E)}{a\operatorname{rk}(E|_{D})}$
(by (2)) $\displaystyle={}$
$\displaystyle\frac{\alpha_{n-\ell}(E)}{\operatorname{rk}(E|_{D})}+\sum_{l=2}^{\ell+1}\frac{(-1)^{l-1}a^{l-1}}{l!}\frac{\beta_{n-1-\ell+l}(a)}{a\operatorname{rk}(a)}.$
(by (3.10))

On the other hand, as $E|_{D}\to\mathcal{F}_{a}|_{D}$ is an
$\ell$-destabilizing quotient, we obtain

$\displaystyle\frac{\alpha_{n-1-\ell}(E|_{D})}{a\operatorname{rk}(E|_{D})}\geq{}$
$\displaystyle\frac{\alpha_{n-1-\ell}(a)}{a\operatorname{rk}(a)}$
$\displaystyle\geq{}$
$\displaystyle\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}+\sum_{l=2}^{\ell+1}\frac{(-1)^{l-1}a^{l-1}}{l!}\frac{\beta_{n-1-\ell+l}(a)}{a\operatorname{rk}(a)}.$
(by (2.2))

Then

$\displaystyle\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}\leq\frac{\alpha_{n-\ell}(E)}{\operatorname{rk}(E)}.$

Finally, we prove that $\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}$ is
constant for $a\gg 0$. Since the map is bounded from above and discrete, its
maximum is attained at an integer $b>0$. Consider the set

$\displaystyle
B=\\{\frac{\beta_{n-1-\ell}(b^{\prime})}{b^{\prime}\operatorname{rk}(b^{\prime})}:b^{\prime}\text{
and }b\text{ are relatively prime}\\}.$

There exists $b^{\prime}\in\mathbb{N}$ such that
$\frac{\beta_{n-1-\ell}(b^{\prime})}{b^{\prime}\operatorname{rk}(b^{\prime})}$
is the maximum of $B$. Suppose that $a=cb+c^{\prime}b^{\prime}$ for some
integers $c,c^{\prime}>0$. By Lemma (3.3) there is an integer $c_{0}>0$ such
that for $c,c^{\prime}\geq c_{0}$

$\displaystyle\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}\geq{}$
$\displaystyle\frac{c}{a}\frac{\beta_{n-1-\ell}(b)}{\operatorname{rk}(b)}+\frac{c^{\prime}}{a}\frac{\beta_{n-1-\ell}(b^{\prime})}{\operatorname{rk}(b^{\prime})}$
$\displaystyle\geq{}$
$\displaystyle\frac{cb}{a}\frac{\beta_{n-1-\ell}(b)}{b\operatorname{rk}(b)}+\frac{c^{\prime}b^{\prime}}{a}\frac{\beta_{n-1-\ell}(b)}{b^{\prime}\operatorname{rk}(b^{\prime})}$
$\displaystyle\geq{}$
$\displaystyle\frac{\beta_{n-1-\ell}(b^{\prime})}{b^{\prime}\operatorname{rk}(b^{\prime})}.$

But $a$ and $b^{\prime}$ are coprime, hence we have equality above. In
particular

$\displaystyle\frac{\beta_{n-1-\ell}(b)}{b\operatorname{rk}(b)}=\frac{\beta_{n-1-\ell}(b^{\prime})}{b^{\prime}\operatorname{rk}(b^{\prime})},$

which further gives

$\displaystyle\frac{\beta_{n-1-\ell}(a)}{a\operatorname{rk}(a)}=\frac{\beta_{n-1-\ell}(b)}{b\operatorname{rk}(b)}.$

As $b$ and $b^{\prime}$ are coprime, for $a\gg 0$ there are always integers
$c,c^{\prime}\geq c_{0}$ such that $a=cb+c^{\prime}b^{\prime}$, which
concludes the proof. ∎

######  Corollary 3.1.

$(P_{j})$ holds true.

###### Proof.

By Lemma 3.4 and Lemma 3.3 we get

$\displaystyle\frac{\beta_{n-j}(a)}{\operatorname{rk}(a)}=\frac{\beta_{n-j}(a_{1})}{\operatorname{rk}(a_{1})}+\frac{\beta_{n-j}(a_{2})}{\operatorname{rk}(a_{2})}$

for $a_{1},a_{2}\gg 0$. It follows that all inequalities employed so far in
the proof are in fact equalities, which proves $(P_{j})$. ∎

######  Corollary 3.2.

$\operatorname{rk}(a)$ is constant for $a\gg 0$.

###### Proof.

For $a_{i}\gg 0$, since $(P_{j})$ is true for $1\leq j\leq\ell+1$, and so
$p_{\ell}(F_{i}/T_{i})=p_{\ell}(a_{i})=p_{\ell}(\mathcal{F}_{a_{i}}|_{D_{i}})$.
Moreover, as $\mathcal{F}_{a_{i}}|_{D_{i}}$ is the minimal
$\ell$-destabilizing quotient of $E|_{D_{i}}$, we have
$\operatorname{rk}(F_{i}/T_{i})=\operatorname{rk}(a)\leq\operatorname{rk}(a_{i})$.
Then there is an integer $a_{0}>0$ such that
$\operatorname{rk}(a)\leq\min\\{\operatorname{rk}(a_{1}),\operatorname{rk}(a_{2})\\}$
for all $a_{1},a_{2}\geq a_{0}$. Let $R\coloneqq\\{\operatorname{rk}(a):a\geq
a_{0}\\}$. Since $R$ is discrete and bounded, there exists $b\geq a_{0}$ such
that $\operatorname{rk}(b)$ is the minimum of $R$. Then, for $a\geq b+a_{0}$
we have
$\operatorname{rk}(a)\leq\min\\{\operatorname{rk}(b),\operatorname{rk}(a-b)\\}=\operatorname{rk}(b)$.
This implies that $\operatorname{rk}(a)=\operatorname{rk}(b)$ and
consequently, $\operatorname{rk}(a)$ is constant for $a\gg 0$. ∎

Denote by $\mathcal{L}_{a}\coloneqq\Lambda^{r(a)}(\mathcal{F}_{a})^{\vee\vee}$
the determinant of $\mathcal{F}_{a}$. By Remark 3.1
$\operatorname{Pic}(Z_{a})\cong\operatorname{Pic}(X)\times\operatorname{Pic}(\Pi_{a})$,
thus $\mathcal{L}_{a}\cong p^{*}M_{a}\otimes q^{*}L_{a}$ for some
$L_{a}\in\operatorname{Pic}(X)$ and $M_{a}\in\operatorname{Pic}(\Pi_{a})$. In
particular $\det(\mathcal{F}_{a}|_{D})\cong L_{a}|_{D}$ for $[D]\in U_{a}$.

######  Lemma 3.5.

There is a line bundle $L\in\operatorname{Pic}(X)$ such that $L\cong L_{a}$
for $a\gg 0$.

###### Proof.

We have proved that
$p_{\ell}(F_{i}/T_{i})=p_{\ell}(a_{i})=p_{\ell}(\mathcal{F}_{a_{i}}|_{D_{i}})$.
Then $F_{i}/T_{i}\cong\mathcal{F}_{a_{i}}|_{D_{i}}$ since
$\mathcal{F}_{a_{i}}|_{D_{i}}$ is the (unique) minimal $\ell$-destabilizing
quotient of $E|_{D_{i}}$ and has the same rank as $F_{i}/T_{i}$. In particular
they have the same determinant line bundle. This is enough to prove the
result, which now follows exactly the same as in [10, Lem. 7.2.7]. ∎

By using Lemma 3.5, one can extend the minimal destabilizing quotient of
$E|_{D}$ over $X$. Precisely, we prove the following result:

######  Proposition 3.1.

There is a torsion-free quotient $E\to F$ on $X$ that extends
$E|_{D}\to\mathcal{F}_{a}|_{D}$ for $[D]\in U_{a}$ and $a\gg 0$.

###### Sketch of proof.

For a more detailed exposition of the argument below we refer the reader to
[10, p. 201]. Let $[D]\in U_{a}$ and consider the destabilizing quotient
$E|_{D}\to\mathcal{F}_{a}|_{D}$. Clearly, there is an open subset
$D^{\prime}\subset D$ such that $E|_{D^{\prime}}$ and
$\mathcal{F}_{a}|_{D^{\prime}}$ are both locally free sheaves and
$\operatorname{codim}(D\setminus D^{\prime},D)\geq 2$. If
$r\coloneqq\operatorname{rk}(a)$, then by taking the $r$-th exterior power of
$E|_{D}\to\mathcal{F}_{a}|_{D}$ we obtain a morphism
$\sigma_{D}:\Lambda^{r}(E|_{D})\to L|_{D}$, which is surjective over
$D^{\prime}$. Note that here we used the crucial fact that the determinant
line bundles $L_{a}$ are isomorphic for $a\gg 0$ by Lemma 3.5. Now,
$\sigma_{D}$ induces a map

$\displaystyle
D^{\prime}\to\operatorname{Grass}(E,r)\subset\mathbb{P}(\Lambda^{r}E),$

where the last inclusion is given by the Plucker embedding. By a standard
argument using Serre duality, for $a\gg 0$ one can uniquely extend
$\sigma_{D}$ to a morphism $\sigma:\Lambda^{r}E\to L$, which is surjective
over an open subset $X^{\prime}\subset X$ such that
$\operatorname{codim}(X\setminus X^{\prime},X)\geq 2$. This induces a map
$X^{\prime}\to\mathbb{P}(\Lambda^{r}E)$ which actually factorizes through
$\operatorname{Grass}(E,r)$. So we get a quotient $E|_{X^{\prime}}\to
F^{\prime}$ that extends $E|_{D^{\prime}}\to\mathcal{F}_{a}|_{D^{\prime}}$.
Let $G^{\prime}$ be its kernel and take
$G\coloneqq\iota^{-1}(i_{*}G^{\prime})$, where $i:X^{\prime}\to X$ is the
inclusion and $\iota:E\to E^{\vee\vee}$ is the canonical injection of $E$ into
its double dual. Then we have a short exact sequence

$\displaystyle 0\to G\to E\to F\to 0$

on $X$ such that $F|_{X^{\prime}}=F^{\prime}$. Restricting it to $D$, we get a
short exact sequence

$\displaystyle 0\to G|_{D}\to E|_{D}\to F|_{D}\to 0.$

On the other side, if $G_{a}$ denotes the kernel of
$E|_{D}\to\mathcal{F}_{a}|_{D}$, then we have another short exact sequence

$\displaystyle 0\to G_{a}\to E|_{D}\to\mathcal{F}_{a}|_{D}\to 0.$

Note that, by construction, $G|_{D^{\prime}}$ is isomorphic to
$G_{a}|_{D^{\prime}}$. Since both $G|_{D}$ and $G_{a}$ are saturated in
$E|_{D}$ and have the same double dual, it follows by the lemma below that
they are in fact isomorphic over $D$. Consequently, $F|_{D}$ is isomorphic to
$\mathcal{F}_{a}|_{D}$. ∎

######  Lemma 3.6.

Let $M$ be a torsion-free sheaf and $N\subset M$ a subsheaf. Let $\iota:M\to
M^{\vee\vee}$ be the canonical injection. Then the saturation of $N$ in $M$ is
equal to $\iota^{-1}(N^{\vee\vee})$.

###### Proof.

Denote by $S$ the saturation of $N$ in $M$ and set
$N^{\prime}\coloneqq\iota^{-1}(N^{\vee\vee})$. Then we have an injection
$M/N^{\prime}\to M^{\vee\vee}/N^{\vee\vee}$ such that
$M^{\vee\vee}/N^{\vee\vee}$ is torsion-free, and so $M/N^{\prime}$ is also
torsion-free. Hence $S\subset N^{\prime}$. We get a diagram as follows

where $\alpha$ and $\beta$ are the canonical maps. By applying Snake’s Lemma,
we obtain an injection $N^{\prime}/S\to\operatorname{Coker}(\alpha)$. But
$N^{\prime}/S\subset E/S$ is torsion-free and $\operatorname{Coker}(\alpha)$
is supported in positive codimension, thus $N^{\prime}/S=0$. ∎

We can complete now the proof of Theorem 1.1. Suppose the theorem were false.
Then $E|_{D}$ is not $\ell$-semistable and admits an $\ell$-destabilizing
quotient $E|_{D}\to\mathcal{F}_{a}|_{D}$ for $D\in\Pi_{a}$ general and $a\gg
0$. By Proposition 3.1 we can extend this quotient over $X$ for $D\in U_{a}$
of sufficiently large degree, which yields a contradiction by Lemma 2.11.

######  Corollary 3.3.

If $E$ is a Gieseker-semistable (i.e. $n$-semistable) sheaf on $X$, then the
restriction $E|_{D}$ is still Gieseker-semistable for a general divisor
$D\in\Pi_{a}$ of sufficiently large degree.

Clearly the result of Corollary 3.3 does not hold in general for Gieseker-
stability, as we have already pointed out in the introduction.

### The case of $\ell$-stability

We next show Theorem 1.2. Fix $\ell<n$ and let $E$ be an $\ell$-stable
torsion-free sheaf on $X$. Recall that any $\ell$-stable sheaf is in
particular $\ell$-simple. Therefore a necessary condition for the theorem to
hold is that the restriction $E|_{D}$ to a general divisor of sufficiently
large degree remains $\ell$-simple. We show below that this is indeed the case
whenever $\ell<n$.

######  Lemma 3.7.

Under the same assumptions as above, the restriction $E|_{D}$ to a general
divisor $D\in\Pi_{a}$ remains $\ell$-simple and torsion-free for $a\gg 0$.

###### Proof.

By Lemma 2.4 there exists a sheaf $G$ on $X$ which is isomorphic with $E$ in
codimension $\ell$ and moreover satisfies ($\star$ ‣ 2). In particular $G$ is
also $\ell$-simple. In this case it is enough to show that the lemma holds for
$G$.

As $\operatorname{depth}_{x}(G)\geq 2$ for any closed point $x\in X$, we have
by Auslander-Buchsbaum formula

$\displaystyle\operatorname{hd}_{x}(E)=\dim(X)-\operatorname{depth}_{x}(E)\leq
n-2,$

which further gives $\operatorname{\mathcal{E}\\!\mathit{xt}}^{n-1}(G,G)=0$.

Applying the functor $\operatorname{Hom}(G,-)$ to the short exact sequence

$\displaystyle 0\to G(-D)\to G\to G|_{D}\to 0,$

we obtain a long exact sequence

$\displaystyle\operatorname{End}(G)\to\operatorname{End}(G|_{D})\to\operatorname{Ext}^{1}(G,G(-D))\to\ldots.$
(3.11)

By using Serre duality and the local-to-global Ext spectral sequence, one
obtains

$\displaystyle\operatorname{Ext}^{1}(G,G(-D))\cong
H^{0}(X,\operatorname{\mathcal{E}\\!\mathit{xt}}^{n-1}(G,G)\otimes\omega_{X}(D))=0$

for $a\gg 0$, where $\omega_{X}$ denotes the dualizing line bundle on $X$. As
$G$ is $\ell$-simple, it is in particular simple. It follows by (3.11) that
$G|_{D}$ is also simple for $a\gg 0$. Moreover, for $[D]\in\Pi_{a}$ general we
may further assume by [10, Lem. 1.1.13] that $G|_{D}$ is a torsion-free sheaf
satisfying ($\star$ ‣ 2).

Let $Y\subset D$ be a closed subset such that
$\operatorname{codim}(Y,D)\geq\ell+1$. As $G|_{D}$ satisfies ($\star$ ‣ 2), we
have by Lemma 2.4 that

$\displaystyle\operatorname{Hom}(G|_{D},G|_{D})\to\operatorname{Hom}(G|_{D\setminus
Y},G|_{D\setminus Y})$

is an isomorphism, which proves that $G|_{D}$ is also $\ell$-simple. ∎

We proceed with the proof of Theorem 1.2. One main difference compared to the
case of $\ell$-semistability lies in the fact that the Harder-Narasimhan
filtration of an $\ell$-stable sheaf is trivial. We will use instead the so-
called extended $\ell$-socle (see Definition 2.3) to construct a flat family
of destabilizing quotients. This approach is due to Mehta and Ramanathan [21].

In what follows we suppose that the theorem is false, i.e. for any integer
$a_{0}>0$ there exists $a\geq a_{0}$ such that $E|_{D}$ is $\ell$-unstable for
general $D\in\Pi_{a}$.

######  Lemma 3.8.

Let $a>0$ be an integer such that $E|_{D}$ is $\ell$-unstable and
$\ell$-simple for general $[D]\in\Pi_{a}$. Then there exists a dense open
subset $W_{a}\subset\Pi_{a}$ and a quotient $q^{*}E\to\mathcal{G}_{a}$ such
that

  1. (1)

each divisor $D\in W_{a}$ is smooth and integral,

  2. (2)

$\mathcal{G}_{a}$ is flat over $W_{a}$,

  3. (3)

for $[D]\in W_{a}$, $E|_{D}\to\mathcal{G}_{a}|_{D}$ is $\ell$-destabilizing,
i.e. $p_{\ell}(E|_{D})=p_{\ell}(\mathcal{G}_{a}|_{D})$,

  4. (4)

for $a\gg 0$, $E|_{D}$ is $\ell$-semistable.

###### Proof.

Since $k$ is algebraically closed, $E|_{D}$ is geometrically $\ell$-unstable.
Recall that the property of being geometrically $\ell$-stable is open in flat
families, cf. Lemma 2.2. Thus, if $D_{\xi}\in\Pi_{a}$ denotes the generic
point of $\Pi_{a}$, then $E|_{D_{\xi}}\coloneqq q^{*}E_{D_{\xi}}$ must be
geometrically $\ell$-unstable on $Z_{D_{\xi}}$. On the other side, from Lemma
2.5 we obtain that $E|_{D_{\xi}}$ is also $\ell$-simple. Hence, according to
Lemma 2.10, $E|_{D_{\xi}}$ is $\ell$-unstable.

Now consider the quotient $G_{\xi}$ of $E|_{D_{\xi}}$ by its $\ell$-extended
socle (see Definition 2.3). Extend $G_{\xi}$ to a quotient
$q^{*}E\to\mathcal{G}_{a}$ over $Z_{a}$. By generic flatness there is an open
subset $W_{a}\subset\Pi_{a}$ such that $\mathcal{G}_{a}$ is flat over $W_{a}$
and $E|_{D}\to\mathcal{G}_{a}|_{D}$ is $\ell$-destabilizing for all $[D]\in
W_{a}$. Moreover, by Bertini’s Theorem and shrinking $W_{a}$ if necessary, we
may assume that each divisor $D\in W_{a}$ is smooth and integral. Also note
that (4) holds true by Theorem 1.1. ∎

######  Lemma 3.9.

For $a\gg 0$ the restriction $E|_{D}$ to a general divisor $D\in\Pi_{a}$ is
$\ell$-unstable.

###### Sketch of proof.

We argue by contradiction. Let $a_{1}>0$ be an integer such that $E|_{D_{1}}$
is $\ell$-stable for every divisor $D_{1}$ in some dense open subset
$W_{a_{1}}\subset\Pi_{a_{1}}$. There exists $a\geq a_{1}$ such that $E|_{D}$
is $\ell$-unstable and $\ell$-simple (cf. Lemma 3.7) for general
$[D]\in\Pi_{a}$. Then we have a flat quotient $q^{*}E\to\mathcal{G}_{a}$ over
$W_{a}\subset\Pi_{a}$ as given by Lemma 3.8.

Set $a_{2}\coloneqq a-a_{1}$. By Theorem 1.1, there exists a non-empty open
subset $W_{a_{2}}\subset\Pi_{a_{2}}$ such that $E|_{D_{2}}$ is
$\ell$-semistable for every $D_{2}\in W_{a_{2}}$. Consider a normal crossing
divisor $D=D_{1}+D_{2}$ with $D_{1}\in W_{a_{1}}$ and $D_{2}\in W_{a_{2}}$.
Similarly to the discussion below Remark 3.2, there exists a smooth curve
$C\subset\Pi_{a}$ passing through $[D]$ such that $C\setminus[D]\subset
W_{a}$, and there is a flat quotient $q^{*}E\to\mathcal{G}_{C}$ over $Z_{C}$
obtained by restricting $q^{*}E\to\mathcal{G}_{a}$.

Set $G\coloneqq(\mathcal{G}_{C}|_{D})/T(\mathcal{G}_{C}|_{D})$ and
$G_{i}\coloneqq G|_{D_{i}}$ for $i=1,2$. As $E|_{D_{i}}$ is $\ell$-semistable
by assumption, we have

$\displaystyle p_{\ell}(G_{i}/T(G_{i}))\geq p_{\ell}(E|_{D_{i}}).$

Then, by similar arguments to that used in the $\ell$-semistable case, one
shows that in fact we have equality above. In particular

$\displaystyle p_{\ell}(G_{1}/T(G_{1}))=p_{\ell}(E|_{D_{1}}),$

which further implies that $G_{1}/T(G_{1})\cong E|_{D_{1}}$ as $E|_{D_{1}}$ is
$\ell$-stable. However, we also have
$\operatorname{rk}(G_{1}/T(G_{1}))=\operatorname{rk}(\mathcal{G}_{a})<\operatorname{rk}(E)$
since $\mathcal{G}_{a}$ is a family of (proper) destabilizing quotients. This
yields a contradiction. ∎

We have shown above how to construct a flat family $q^{*}E\to\mathcal{G}_{a}$
of (proper) destabilizing quotients for every $a\gg 0$. As before, there are
line bundles $L_{a}$ on $X$ such that $\det(\mathcal{G}_{a}|_{D})=L_{a}|_{D}$
for $[D]\in W_{a}$.

######  Lemma 3.10.

There is an infinite subset $N\subset\mathbb{N}$ and a line bundle
$L\in\operatorname{Pic}(X)$ such that $L\cong L_{a}$ for every $a\in N$.

###### Proof.

The result follows as in the slope-stable case. We refer the reader to [10,
Sect. 7] or [5, Sect. 8] for a complete proof. ∎

Now a similar argument as done for Proposition 3.1 shows that, for a
sufficiently large integer $a\in N$, $E|_{D}\to\mathcal{G}_{a}|_{D}$ extends
to an $\ell$-destabilizing quotient $E\to G_{a}$ on $X$. This will give a
contradiction and complete the proof of Theorem 1.2.

##  4 Application: the moduli space of $(n-1)$-semistable sheaves

In this section we work over the field of complex numbers $\mathbb{C}$. Choose
a numerical Grothendieck class $c\in K(X)_{\text{num}}$ of dimension
$n\coloneqq\dim(X)>1$, which will fix the topological type of sheaves. We will
use the restriction theorems to construct a moduli space of $(n-1)$-semistable
sheaves of class $c$ on $X$. Afterwards, we will describe precisely its
geometric points.

### The construction

Since the family of $(n-1)$-semistable sheaves of class $c$ on $X$ is bounded,
cf. Lemma 2.1, for sufficiently large $m>0$ every such semistable sheaf $E$ is
$m$-regular (in the sense of Castelnuovo–Mumford) and has Hilbert polynomial
$P\coloneqq P(c)$. In particular $E(m)$ is globally generated with
$P(m)=h^{0}(E(m))$. Thus the evaluation map induces a quotient

$\displaystyle H^{0}(E(m))\otimes\mathcal{O}_{X}(-m)\to E\to 0.$

Let $V\coloneqq\mathbb{C}^{P(m)}$ and consider the subset
$R^{ss}\subset\operatorname{Quot}(V\otimes\mathcal{O}_{X}(-m),P)$ of quotients
$[q:V\otimes\mathcal{O}_{X}(-m)\to E]$ on $X$ such that

  1. nosep

$E$ is $(n-1)$-semistable of class $c$,

  2. nosep

$q$ induces an isomorphism $V\to H^{0}(E(m))$.

Then $R^{ss}$ is a locally closed subscheme that parametrizes
$(n-1)$-semistable sheaves of class $c$ on $X$. However, note that any
semistable sheaf is uniquely determined in $R^{ss}$ up to a base change of
$V$. In this respect, $R^{ss}$ is endowed with a natural
$G\coloneqq\textup{SL}(V)$ action.

For the following result, see Definition 2.4 regarding $G$-properness.

######  Lemma 4.1.

$R^{ss}$ is a $G$-proper scheme over $\mathbb{C}$.

###### Proof.

Let $A$ be a discrete valuation ring over $\mathbb{C}$ of quotient field $K$.
Consider a commutative diagram as follows

${\operatorname{Spec}(K)}$${R^{ss}}$${\operatorname{Spec}(A)}$${\operatorname{Spec}(\mathbb{C})}$$\scriptstyle{j}$$\scriptstyle{g}$

The map $\operatorname{Spec}(K)\to R^{ss}$ corresponds to a quotient
$[q_{K}:V\otimes K\to E_{K}]\in R^{ss}$, with $E_{K}$ $\ell$-semistable. By
Lemma 2.3, we can extend $E_{K}$ to an $A$-flat family $E$ of
$\ell$-semistable sheaves on $X$. This induces a quotient $[q_{A}:V\otimes
A\to E]\in R^{ss}$, or equivalently an $A$-point $i:\operatorname{Spec}(A)\to
R^{ss}$. As $E\otimes_{A}K\cong E_{K}$, $g$ and $i\circ j$ differ by a group
element in $G(K)$, which completes the proof. ∎

Next we want to define a $G$-semiample line bundle over $R^{ss}$, i.e. a
$G$-equivariant line bundle globally generated by $G$-invariant sections. Let
$\mathcal{F}$ denote the universal family of quotients over $R^{ss}$ with its
natural $G$-action induced from $R^{ss}$. Choose a smooth divisor
$D\in\Pi_{a}$. Since $\mathcal{F}_{s}(-D)\to\mathcal{F}_{s}$ is injective for
every $s\in R^{ss}$, we have a short exact sequence

$\displaystyle 0\to\mathcal{F}(-D)\to\mathcal{F}\to\mathcal{F}|_{R^{ss}\times
D}\to 0$ (4.1)

such that $\mathcal{F}|_{R^{ss}\times D}$ remains $R^{ss}$-flat, cf. [19, Thm.
22.5]. Furthermore, for $m^{\prime}$ sufficiently large we may also assume
that each fiber of $\mathcal{F}|_{R^{ss}\times D}$ over $R^{ss}$ is
$m^{\prime}$-regular. In particular
$R^{i}{p_{1}}_{*}(\mathcal{F}|_{R^{ss}\times D}(m^{\prime}))=0$ for $i>0$ and
${p_{1}}_{*}(\mathcal{F}|_{R^{ss}\times D}(m^{\prime}))$ is a locally free
$G$-equivariant sheaf on $R^{ss}$ of rank $P^{\prime}(m^{\prime})\coloneqq
P(c|_{D}(m^{\prime}))$.

Denote by $\mathcal{R}$ the projective frame bundle corresponding to
${p_{1}}_{*}(\mathcal{F}|_{R^{ss}\times D}(m^{\prime}))$ and let
$\pi:\mathcal{R}\to R^{ss}$ be the natural projection. Then there exists a
natural quotient

$\displaystyle\mathcal{O}_{\mathcal{R}}\otimes\mathcal{O}_{D}(-m^{\prime})^{P^{\prime}(m^{\prime})}\to\pi^{*}(\mathcal{F}|_{R^{ss}\times
D})\otimes\mathcal{O}_{\pi}(1)\to 0.$ (4.2)

Set $V^{\prime}\coloneqq\mathbb{C}^{P^{\prime}(m^{\prime})}$ and consider the
Quot scheme
$\operatorname{Quot}_{D}\coloneqq\operatorname{Quot}(V^{\prime}\otimes\mathcal{O}_{D}(-m^{\prime}),P^{\prime})$
over $D$. Then the quotient map (4.2) induces a natural morphism
$\Phi:\mathcal{R}\to\operatorname{Quot}_{D}$ and we have a diagram

${\mathcal{R}}$${\operatorname{Quot}_{D}}$${R^{ss}}$${M_{D}}$$\scriptstyle{\pi}$$\scriptstyle{\Phi}$$\scriptstyle{\varphi^{\prime}}$

where $M_{D}$ denotes the Simpson moduli space of Gieseker-semistable sheaves
of Hilbert polynomial $P^{\prime}$ on $D$. Several remarks are in order:

######  Remark 4.1.

  1. wide

Let $\mathcal{F}^{\prime}$ denote the universal family of quotients over
$\operatorname{Quot}_{D}$. For $l^{\prime}>0$, the Grothendieck class
$[\mathcal{O}_{D}(l^{\prime})]\in K(X)$ gives the following determinant line
bundle

$\displaystyle
H_{l^{\prime}}\coloneqq\lambda_{\mathcal{F}^{\prime}}([\mathcal{O}_{D}(l^{\prime})])$

over $\operatorname{Quot}_{D}$, see (2.3). In fact $H_{l^{\prime}}$ is very
ample for $l^{\prime}\gg 0$, cf. [10, Prop. 2.2.5]. Moreover,
$\operatorname{Quot}_{D}$ is endowed with a natural
$G^{\prime}\coloneqq\textup{SL}(V^{\prime})$ action, which further induces a
$G^{\prime}$-linearization of $H_{l^{\prime}}$. Now consider the $G$-stable
open subscheme $R_{D}\subset\operatorname{Quot}_{D}$ consisting of
$m^{\prime}$-regular Gieseker-semistable quotients
$[q:V^{\prime}\otimes\mathcal{O}_{D}(-m^{\prime})\to F]$ such that the induced
map $V^{\prime}\to H^{0}(F(m^{\prime}))$ is an isomorphism. By Simpson’s
construction [24], for $l^{\prime}\gg m^{\prime}\gg 0$, $R_{D}$ is the locus
of GIT-semistable points of the closure
$\overline{R_{D}}\subset\operatorname{Quot}_{D}$ with respect to
$H_{l^{\prime}}$. Furthermore, the rational map $\varphi^{\prime}$ is well-
defined over $R_{D}$ and $R_{D}\to M_{D}$ is a good GIT quotient, in the sense
of Mumford [22].

  2. wiide

Note that the image of $\Phi$ is contained in the open subscheme
$Q\subset\operatorname{Quot}_{D}$ consisting of quotients
$[q:V^{\prime}\otimes\mathcal{O}_{D}(-m^{\prime})\to F]$ such that

    1. (i)

$H^{i}(F(m^{\prime}))=0$ for all $i>0$,

    2. (ii)

the induced map $V^{\prime}\to H^{0}(F(m^{\prime}))$ is an isomorphism.

Clearly $R_{D}$ is an open subset of $Q$.

  3. wiiide

The given $G$-action on $R^{ss}$ lifts to a natural $G$-action on
$\mathcal{R}$. There is also a natural $G^{\prime}$-action on $\mathcal{R}$
induced by ${p_{1}}_{*}(\mathcal{F}|_{R^{ss}\times D}(m^{\prime}))$ and which
is compatible with its $G$-action. Then $\Phi$ is a $G\times
G^{\prime}$-equivariant morphism.

Consider the following Grothendieck class in $K(X)$

$\displaystyle w\coloneqq\chi(c(m^{\prime})\cdot
h)[\mathcal{O}_{X}(l^{\prime})]-\chi(c(l^{\prime})\cdot
h)[\mathcal{O}_{X}(m^{\prime})],$

with $h\coloneqq[O_{D}]\in K(X)$, which we use to form the determinant line
bundle

$\displaystyle\mathcal{L}\coloneqq\lambda_{\mathcal{F}}(w\cdot h)$

over $R^{ss}$, see (2.3). Note that $\mathcal{L}$ does not depend on the
chosen divisor $D$ in $\Pi_{a}$, but only on the degree $a$.

######  Theorem 4.1.

Under the same assumptions as above, for $l^{\prime}\gg m^{\prime}\gg 0$ and
$a\gg 0$, there exists an integer $\nu>0$ such that $\mathcal{L}^{\nu}$ is
globally generated by $G$-invariant sections over $R^{ss}$.

###### Proof.

Let $s\in R^{ss}$ be a closed point corresponding to an $(n-1)$-semistable
quotient sheaf $E$ in $R^{ss}$. By Theorem 1.1, the restriction $E|_{D}$ to a
general divisor $D\in\Pi_{a}$ is Gieseker-semistable for $a\gg 0$. Then, by
Simpson’s construction, there exists a $G^{\prime}$-invariant section
$\sigma\in H^{0}(\overline{R_{D}},H_{l^{\prime}}^{\nu^{\prime}})^{G^{\prime}}$
non-vanishing at $[E|_{D}]$ for some $\nu^{\prime}>0$. By [23, Lem. 4.1],
there exists a $G^{\prime}$-equivariant isomorphism
$H_{l^{\prime}}^{\nu^{\prime}}\cong\lambda_{\mathcal{F}^{\prime}}(w|_{D})^{\nu}$
over $Q$ for some $\nu>0$. The restriction of $\sigma$ to $Q$ induces a
$G^{\prime}$-invariant section $\sigma_{Q}$ in
$H^{0}(Q,\lambda_{\mathcal{F^{\prime}}}(w|_{D})^{\nu})^{G^{\prime}}$. Then its
pull-back $\Phi^{*}(\sigma_{Q})$ is a global $G\times G^{\prime}$-invariant
section, which descends to a $G$-invariant section in
$H^{0}(R^{ss},\lambda_{\mathcal{F}|_{R^{ss}\times D}}(w|_{D})^{\nu})^{G}$, as
$\pi$ is a good quotient. Applying the determinant map $\lambda_{(-)}(w)$ to
the exact sequence (4.1), we obtain a $G$-equivariant isomorphism

$\displaystyle\mathcal{L}=\lambda_{\mathcal{F}}(w\cdot
h)\cong\lambda_{\mathcal{F}|_{R^{ss}\times D}}(w|_{D})$

over $R^{ss}$. From the above considerations, we have constructed a linear map

$\displaystyle\Gamma:H^{0}(\overline{R_{D}},H_{l^{\prime}}^{\nu^{\prime}})^{G^{\prime}}\to
H^{0}(R^{ss},\mathcal{L}^{\nu})^{G}$

such that $\Gamma(\sigma)$ is a $G$-invariant section of $\mathcal{L}^{\nu}$
non-vanishing at $s$. ∎

We have shown above that $R^{ss}$ is a $G$-proper scheme over $\mathbb{C}$
endowed with a $G$-semiample line bundle $\mathcal{L}$. Since $G$ is
reductive, one can form the Iitaka $G$-fibration corresponding to
$(R^{ss},\mathcal{L})$ as in [23, Thm. 5.3], which will define our moduli
space $M$ of $(n-1)$-semistable sheaves of class $c$ on $X$.

Consider the moduli functor

$\displaystyle\mathcal{M}:(\text{Sch}^{ft}/\mathbb{C})^{\textnormal{op}}\to(\textnormal{Sets})$

that associates to any scheme $S$ of finite type over $\mathbb{C}$ the set of
all equivalence classes of $S$-flat families of $(n-1)$-semistable sheaves of
class $c$ on $X$. Unfortunately our moduli space $M$ does not (co)represent
$\mathcal{M}$. Nevertheless, we have the following functorial property of $M$:

######  Theorem 4.2.

There exists a unique triple $(M,\mathcal{A},e)$ formed of a projective scheme
$M$ over $\mathbb{C}$ endowed with an ample line bundle $\mathcal{A}$ and a
natural number $e>0$ such that there is a natural transformation
$\Psi:\mathcal{M}\to\operatorname{Hom}(-,M)$, that associates to any
$\mathbb{C}$-scheme of finite type $S$ and any $S$-flat family $\mathcal{E}$
of $(n-1)$-semistable sheaves of class $c$ on $X$ a classifying morphism
$\Psi_{\mathcal{E}}:S\to M$, satisfying the following properties:

  1. (1)

For any $S$-flat family $\mathcal{E}$ of $(n-1)$-semistable sheaves of class
$c$ on $X$, the classifying morphism $\Psi_{\mathcal{E}}$ satisfies

$\displaystyle\Psi_{\mathcal{E}}^{*}(\mathcal{A})\cong\lambda_{\mathcal{E}}(w\cdot
h)^{e},$

where $\lambda_{\mathcal{E}}(w\cdot h)$ is the determinant line bundle on $S$
defined by (2.3).

  2. (2)

For any other triple $(M^{\prime},\mathcal{A}^{\prime},e^{\prime})$, with
$M^{\prime}$ a projective scheme over $\mathbb{C}$, $\mathcal{A}^{\prime}$ an
ample line bundle on $M^{\prime}$ and $e^{\prime}$ a natural number satisfying
property (1), we have $e|e^{\prime}$ and there exists a unique morphism
$\phi:M\to M^{\prime}$ such that
$\phi^{*}\mathcal{A}^{\prime}\cong\mathcal{A}^{(e^{\prime}/e)}$.

###### Proof.

The proof follows exactly as in [7, Prop. 4.5] or [23, Main Thm.]. We omit it.
∎

### Geometric points of the moduli space

We end this section by studying the geometric points of $M$. If $E$ is an
$(n-1)$-semistable sheaf on $X$, then there exists a Jordan-Hölder filtration

$\displaystyle 0=E_{0}\subset E_{1}\subset\ldots\subset E_{l}=E$

such that its factors $E_{i}/E_{i-1}$ are $(n-1)$-stable with
$p_{n-1}(E_{i}/E_{i-1})=p_{n-1}(E)$. We denote by
$\operatorname{gr}(E)\coloneqq\oplus_{i}E_{i}/E_{i-1}$ its corresponding
graded torsion-free module. We say that $E$ is $(n-1)$-polystable if $E$ is
isomorphic to a direct sum of $(n-1)$-stable sheaves.

######  Lemma 4.2.

Let $E$ be an $(n-1)$-semistable sheaf on $X$ and $\operatorname{gr}(E)$ the
graded module corresponding to a Jordan-Hölder filtration of $E$. Then $E$ and
$\operatorname{gr}(E)$ give the same point in $M$.

###### Proof.

It is not difficult to construct an $\mathbb{A}^{1}$-flat family $\mathcal{E}$
of $(n-1)$-semistable sheaves on $X$ such that
$\mathcal{E}_{0}\cong\operatorname{gr}(E)$ and $\mathcal{E}_{t}\cong E$ for
all $t\neq 0$. Then the classifying morphism
$\Psi_{\mathcal{E}}:\mathbb{A}^{1}\to M$ corresponding to $\mathcal{E}$ is
constant, which shows that $E$ and $\operatorname{gr}(E)$ define the same
point in $M$. ∎

Thus, it will suffice to analyze the separation of polystable sheaves. If $E$
is $(n-1)$-polystable, we denote by $E^{[n-1]}$ the corresponding torsion-free
sheaf satisfying property $(\star)$ given by Lemma 2.4. Note that $E^{[n-1]}$
is also $(n-1)$-polystable as it differs from $E$ only in a finite number of
points. We let $l_{E}$ denote the zero-cycle associated to $E^{[n-1]}/E$.

Next we aim to prove the following result:

######  Theorem 4.3.

Let $E_{1}$ and $E_{2}$ be two $(n-1)$-polystable sheaves of class $c$ on $X$.
Then $E_{1}$ and $E_{2}$ define the same point in $M$ if and only if
$E_{1}^{[n-1]}\cong E_{2}^{[n-1]}$ and $l_{E_{1}}=l_{E_{2}}$.

######  Remark 4.2.

Let $E$ be an $(n-1)$-semistable sheaf on $X$ and $\operatorname{gr}(E)$ the
graded module corresponding to a Jordan-Hölder filtration of $E$. Note that
$\operatorname{gr}(E)$ is uniquely determined only in codimension $n-1$.
However, one can use the result above to show that the pair
$(\operatorname{gr}(E)^{[n-1]},l_{\operatorname{gr}(E)})$ is in fact uniquely
determined by $E$. Indeed, suppose we have two graded modules
$\operatorname{gr}_{1}(E)$ and $\operatorname{gr}_{2}(E)$ corresponding to two
different Jordan-Hölder filtrations of $E$. Then, by Lemma 4.2 and Theorem
4.3, we must have
$\operatorname{gr}_{1}(E)^{[n-1]}\cong\operatorname{gr}_{2}(E)^{[n-1]}$ and
$l_{\operatorname{gr}_{1}(E)}=l_{\operatorname{gr}_{2}(E)}$.

We divided the proof of Theorem 4.3 in a sequence of lemmas.

######  Lemma 4.3.

Let $E_{1}$ and $E_{2}$ be two $(n-1)$-polystable sheaves of class $c$ on $X$
such that $E_{1}^{[n-1]}\ncong E_{2}^{[n-1]}$. Then $E_{1}$ and $E_{2}$ give
distinct points in $M$.

###### Proof.

By Theorem 1.2 and [10, Lem. 1.1.13], we can choose a general divisor
$D\in\Pi_{a}$, with $a\gg 0$, such that

  1. nosep

$E_{1}|_{D}$ and $E_{2}|_{D}$ remain polystable,

  2. nosep

$E_{1}|_{D}=E_{1}^{[n-1]}|_{D}$ and $E_{2}|_{D}=E_{2}^{[n-1]}|_{D}$ are
torsion-free and satisfy $(\star)$.

As in the proof of Lemma 3.7, one can show that

$\displaystyle\operatorname{Hom}(E_{1}^{[n-1]},E_{2}^{[n-1]})\to\operatorname{Hom}(E_{1}^{[n-1]}|_{D},E_{2}^{[n-1]}|_{D})$

is an isomorphism for $a\gg 0$. By assumption $E_{1}^{[n-1]}\ncong
E_{2}^{[n-1]}$, and thus $E_{1}|_{D}$ and $E_{2}|_{D}$ are non-isomorphic
polystable sheaves on $D$. Then $E_{1}|_{D}$ and $E_{2}|_{D}$ give distinct
points in the Simpson moduli space $M_{D}$ of Gieseker-semistable sheaves on
$D$. Hence there exists a section $\sigma\in
H^{0}(\overline{R}_{D},H_{l^{\prime}}^{\nu^{\prime}})^{G^{\prime}}$, for some
$\nu^{\prime}>0$, that separates $E_{1}|_{D}$ and $E_{2}|_{D}$ (see Remark 4.1
for notation). Following the proof of Theorem 4.1, we find a section
$\Gamma(\sigma)\in H^{0}(R^{ss},\mathcal{L}^{\nu})^{G}$, for $\nu>0$, that
separates $E_{1}$ and $E_{2}$ in $R^{ss}$. This implies that $E_{1}$ and
$E_{2}$ give distinct points in $M$. ∎

######  Lemma 4.4.

Let $E_{1}$ and $E_{2}$ be two $(n-1)$-polystable sheaves of class $c$ on $X$
such that $l_{E_{1}}\neq l_{E_{2}}$. Then $E_{1}$ and $E_{2}$ give distinct
points in $M$.

###### Proof.

The proof goes as in [17, Lem. 3.6], we omit it. ∎

######  Lemma 4.5.

Let $E_{1}$ and $E_{2}$ be two $(n-1)$-polystable sheaves of class $c$ on $X$
such that $E_{1}^{[n-1]}\cong E_{2}^{[n-1]}$ and $l_{E_{1}}=l_{E_{2}}$. Then
$E_{1}$ and $E_{2}$ define the same point in $M$.

###### Proof.

Set $F\coloneqq E_{1}^{[n-1]}\cong E_{2}^{[n-1]}$, $\xi\coloneqq
l_{E_{1}}=l_{E_{2}}$ and $l\coloneqq\operatorname{length}(\xi)$. Then $E_{1}$
and $E_{2}$ correspond to two closed points $y_{1}$ and $y_{2}$ respectively
in the Quot scheme $\operatorname{Quot}(F,l)$. Denote by
$\rho:\operatorname{Quot}(F,l)\to S^{l}(X)$ the Quot-to-Chow morphism that
associates to any zero-dimensional quotient of $F$ of length $l$ its
corresponding zero-cycle. By assumption $y_{1}$ and $y_{2}$ lie inside the
fiber $\rho^{-1}(\xi)$.

Suppose that $y_{1}$ and $y_{2}$ lie in the same connected component $C$ of
$\rho^{-1}(\xi)$. Let $\mathcal{U}$ be the universal family of quotients over
$\operatorname{Quot}(F,l)$, which fits in a short exact sequence

$\displaystyle 0\to\mathcal{K}\to\mathcal{O}_{\operatorname{Quot}}\otimes
F\to\mathcal{U}\to 0$

such that $\mathcal{K}$ is flat over $\operatorname{Quot}(F,l)$. As $F$ is
$(n-1)$-semistable, it follows that $\mathcal{K}|_{C}$ is a $C$-flat family of
$(n-1)$-semistable sheaves of class $c$ on $X$. Choose a general divisor
$D\in\Pi_{a}$ avoiding the support of $\xi$ such that $F|_{D}$ is Gieseker-
polystable, which exists for $a\gg 0$ by Theorem 1.2. Then there exists a
short exact sequence

$\displaystyle
0\to\mathcal{K}|_{C}(-D)\to\mathcal{K}|_{C}\to\mathcal{K}|_{C\times D}\to 0$
(4.3)

such that $\mathcal{K}|_{C\times D}$ is a $C$-flat family of fiber $F|_{D}$,
cf. [19, Thm. 22.5]. This induces a constant morphism $\Psi_{\mathcal{K}}:C\to
M_{D}$, where $M_{D}$ denotes the Simpson moduli space of Gieseker-semistable
sheaves of Hilbert polynomial $P(F|_{D})$ on $D$. As $w|_{D}\in
c^{\perp}\cap\\{1,h|_{D},\ldots,h|_{D}^{n-1}\\}^{\perp\perp}$, there exists a
natural determinant line bundle $\lambda(w|_{D})$ over $M_{D}$ such that

$\displaystyle\Psi_{\mathcal{K}}^{*}(\lambda(w|_{D}))\cong\lambda_{\mathcal{K}|_{C\times
D}}(w|_{D})\cong\lambda_{\mathcal{K}|_{C}}(w\cdot h),$

see [10, Thm. 8.1.5]. The last isomorphism above follows by applying the map
$\lambda_{(-)}(w)$ to the short exact sequence (4.3) and using [10, Lem.
8.1.2, i)]. But since $\Psi_{\mathcal{K}}$ is constant, we obtain that
$\lambda_{\mathcal{K|_{C}}}(w\cdot h)$ is trivial. Therefore, the classifying
morphism $C\to M$ corresponding to the family $\mathcal{K}|_{C}$, as given by
Theorem 4.2, is constant. This shows that $E_{1}$ and $E_{2}$ define the same
point in $M$ if they lie in the same connected component of $\rho^{-1}(\xi)$.
Below we show that all fibers of $\rho$ are in fact connected, which will
complete the proof. ∎

The proof of the following result is inspired by [3].

######  Theorem 4.4.

Let $F$ be a torsion-free sheaf on $X$ and $\xi\in S^{l}(X)$ a zero-cycle of
length $l$. Then the fiber
$\operatorname{Quot}_{\xi}(F,l)\coloneqq\rho^{-1}(\xi)$ over $\xi$ of the
Quot-to-Chow morphism $\rho:\operatorname{Quot}(F,l)\to S^{l}(X)$ is
connected.

###### Proof.

The proof is by induction on $l$. It is enough to prove the statement when
$\xi$ is supported at a single closed point of $X$, say $x\in X$. If $l=1$,
then $\xi$ corresponds to the closed point $x\in X$, and so
$\operatorname{Quot}_{\xi}(F,1)$ is isomorphic to the projective space
$\mathbb{P}(F(x))$, with $F(x)=F_{x}\otimes k(x)$, which is clearly connected.

Now suppose that $\operatorname{Quot}_{\xi}(F,l)$ is connected for some $l>0$.
Let $\mathcal{U}$ be the universal family of quotients over
$\operatorname{Quot}_{\xi}(F,l)$ and consider the short exact sequence

$\displaystyle 0\to\mathcal{K}\to\mathcal{O}_{\operatorname{Quot}}\otimes
F\to\mathcal{U}\to 0$

over $\operatorname{Quot}_{\xi}(F,l)$. Then
$Z\coloneqq\mathbb{P}(\mathcal{K})$ is a projective space over
$\operatorname{Quot}_{\xi}(F,l)\times X$. Note that $Z$ is connected, since
$\operatorname{Quot}_{\xi}(F,l)\times X$ is so.

Next we want to construct a surjective morphism
$\varphi:Z\to\operatorname{Quot}_{\xi}(F,l+1)$. Let us first describe this map
set-theoretically, at the level of geometric points. By construction, a closed
point of $Z$ corresponds to a quotient $[\mu:K\to k(x)]\in\mathbb{P}(K)$,
where $K$ is the kernel of some closed point $[F\to
Q]\in\operatorname{Quot}_{\xi}(F,l)$. If
$K^{\prime}\coloneqq\operatorname{Ker}(\mu)$, then $F/K^{\prime}$ is a zero-
dimensional quotient of $F$ of length $l+1$ on $X$. We send $[K\to k(x)]\in Z$
through $\varphi$ to the quotient $[F\to
F/K^{\prime}]\in\operatorname{Quot}_{\xi}(F,l+1)$. In fact, one can do this
construction in families, as described in [3, Prop. 5], to define an algebraic
morphism $\varphi:Z\to\operatorname{Quot}_{\xi}(F,l+1)$. It remains to show
that $\varphi$ is surjective. Let $[q:F\to
T]\in\operatorname{Quot}_{\xi}(F,l+1)$ be a closed point and denote by
$K\coloneqq\operatorname{Ker}(q)$ its kernel. Let $M\subset T_{x}$ be the
subsheaf of all elements in $T_{x}$ annihilated by the maximal ideal in
$\mathcal{O}_{X,x}$ and choose a point $[\lambda:k(x)\to
M]\in\mathbb{P}(M^{\vee})$. This induces a diagram as follows

${0}$${0}$${0}$${k(x)}$${T}$${Q}$${0}$${0}$${K}$${F}$${Q}$${0}$$\scriptstyle{\lambda}$$\scriptstyle{\mu}$$\scriptstyle{q}$

such that $[F\to Q]$ is a zero-dimensional quotient of length $l$ on $X$. Then
$[\mu:K\to k(x)]\in Z$ is mapped to $[q:F\to T]$ by $\varphi$, which shows
that $\varphi$ is surjective. As $Z$ is connected, we conclude that
$\operatorname{Quot}_{\xi}(F,l+1)$ must also be connected. ∎

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