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# Harder-Narasimhan stratification for the moduli stack of parabolic vector
bundles

Andres Fernandez Herrero

###### Abstract

For every set of parabolic weights, we construct a Harder-Narasimhan
stratification for the moduli stack of parabolic vector bundles on a curve. It
is based on the notion of parabolic slope, introduced by Mehta and Seshadri.
We also prove that the stratification is schematic, that each stratum is
complete, and establish an analogue of Behrend’s conjecture for parabolic
vector bundles. A comparison with recent $\Theta$-stratification approaches is
discussed.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
  2. 2 Parabolic Vector Bundles
    1. 2.1 Parabolic vector bundles on a curve
    2. 2.2 Parabolic weights and degree
    3. 2.3 Harder-Narasimhan filtrations
    4. 2.4 Parabolic vector bundles in families
  3. 3 Stacks of parabolic vector bundles
    1. 3.1 The group scheme $\text{GL}\left(\mathcal{V}\right)$
    2. 3.2 Moduli stacks of parabolic vector bundles
    3. 3.3 Harder-Narasimhan strata
  4. 4 Parabolic Quot Schemes
  5. 5 Openness of strata
    1. 5.1 Constructibility of strata
    2. 5.2 Openness of strata
  6. 6 Quasicompactness of strata
    1. 6.1 Harder-Narasimhan stratification for the moduli of vector bundles
    2. 6.2 Quasicompactness of parabolic Harder-Narasimhan strata
  7. 7 Completeness of strata
  8. 8 Harder-Narasimhan filtrations in families
  9. A Descent for finite covers
  10. B Comparison with previous work

##  1 Introduction

Let $C$ be a curve over a field $k$. Fix a finite set of $k$-points $x_{i}$ in
$C$. In this paper we investigate the geometry of moduli stacks
$\text{Bun}_{\mathcal{V}}$ of vector bundles over $C$ with the additional data
of a flag of vector spaces in the fiber at each $x_{i}$. The objects
parametrized by $\text{Bun}_{\mathcal{V}}$ are parabolic vector bundles, as in
[MS80].

The purpose of this article is to describe a stratification of
$\text{Bun}_{\mathcal{V}}$ analogous to the classical Harder-Narasimhan
stratification for the moduli of vector bundles [HN75]. It is the parabolic
analogue for $G=\text{GL}_{n}$ of the results of Gurjar and Nitsure [GN14]
when $\text{char}(k)=0$, and recently generalized in [GN18, GN16] to higher
dimensional varieties and general $k$. Our version of the Harder-Narasimhan
stratification is based on the notion of slope for a parabolic vector bundle.
This notion of slope depends on the choice of some stability parameters
$\overline{\lambda}$ known as parabolic weights. We use the slope to define a
parabolic version of the Harder-Narasimhan filtration. This filtration
provides us with an invariant of parabolic vector bundles, which we call the
HN datum. For each HN datum $P$, we define a stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ consisting of parabolic vector bundles
with HN datum bounded by $P$. We prove the following properties of the strata
in Theorems 5.11, 6.10, 7.6 and 8.6.

######  Main Theorem.

For every HN datum $P$, the following properties hold.

  1. (A)

The stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$ is a quasicompact open
substack of $\text{Bun}_{\mathcal{V}}$.

  2. (B)

The stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$ is complete.

  3. (C)

The locus $\text{Bun}_{\mathcal{V}}^{=P}$ of parabolic vector bundles with HN
datum $P$ can be naturally equipped with the structure of a locally closed
substack of $\text{Bun}_{\mathcal{V}}$. In fact,
$\text{Bun}_{\mathcal{V}}^{=P}\hookrightarrow\text{Bun}_{\mathcal{V}}^{\leq
P}$ is a closed immersion.

The article by Mehta and Seshadri [MS80] contains a definition of parabolic
degree similar to the one we give. They use the notion of parabolic degree to
build a moduli space of semistable parabolic vector bundles on a curve of
genus $g\geq 2$ over an algebraically closed field. They also prove a version
of our Theorem 5.11 in the special case of a semistable stratum by embedding
the semistable locus into a GIT problem. In contrast we deal with the whole
moduli stack, including the unstable locus. Our proofs work intrinsically with
the stack, without appealing to GIT.

Parabolic vector bundles can be alternatively seen as torsors for a parahoric
Bruhat-Tits group scheme, as in [PR10]. See Proposition 3.7 below for a proof
of this. It follows from [Hei10][Prop. 1] that $\text{Bun}_{\mathcal{V}}$ is a
smooth algebraic stack. There has been recent interest in developing
stratifications for moduli stacks of torsors for such group schemes. Gaitsgory
and Lurie [GL19] give an ad hoc construction of stratifications in the
semisimple case in order to derive a Grothendieck-Lefschetz trace formula for
the moduli stack of torsors. Balaji and Seshadri [BS15] described Harder-
Narasimhan stratifications for parahoric Bruhat-Tits group schemes under the
assumption that the generic fiber is split. They focus on the construction of
a moduli space for semistable torsors. They obtain results valid when $C$ is a
curve of genus $g\geq 2$ over a ground field of characteristic $0$.

Heinloth [Hei17] and Alper, Halpern-Leistner, Heinloth [AHLH18][§8] develop a
theory of $\Theta$-stratifications for the moduli of torsors of a parahoric
Bruhat-Tits group scheme in arbitrary characteristic, under some mild tameness
assumptions on the generic fiber. They use a modern variation of ideas from
GIT for stacks as in [HL14] in order to obtain a stratification associated to
a line bundle on the stack. In Appendix B we explain how to build certain
$\mathbb{R}$-line bundles on $\text{Bun}_{\mathcal{V}}$ associated to a
parabolic weight $\overline{\lambda}$. We explicitly describe the subset of
parabolic weights $\overline{\lambda}$ such that the corresponding line bundle
is admissible (cf. [Hei17]). Whenever this is the case, the line bundle
induces a $\Theta$-stratification that we show coincides with our parabolic
slope stratification. It should be remarked that this admissibility condition
on parabolic weights is restrictive and we do not impose it in this text.

Our techniques to establish the Main Theorem make contact with the classical
theory for vector bundles already present in [MS80]. In order to deal with
subbundles of parabolic vector bundles in families, we define an analogue of
the Quot scheme for parabolic vector bundles (Definition 4.9). We also
construct an iterated Quot scheme $Fil^{\alpha}_{\mathcal{W}}$ parametrizing
filtrations of a parabolic vector bundle with some given fixed set of
invariants (Definition 4.16).

We also prove certain properties of the stratification that do not hold in the
generality of [Hei17] when the ground field has positive characteristic. The
obstruction in the more general case is the failure of Behrend’s conjecture
for certain reductive groups in bad characteristics (established by Heinloth
[Hei08]). Recall that Behrend’s conjecture [Beh95] states that the canonical
parabolic reduction of a principal $G$-bundle over a curve with $G$-reductive
does not admit nontrivial deformations. We prove that an analogue of Behrend’s
conjecture holds in our context of parabolic vector bundles:

######  Proposition (= Proposition 2.31).

Let $k[\epsilon]\vcentcolon=k[T]\,/\,(T^{2})$ be the ring of dual numbers. We
denote by $\sigma:\text{Spec}(k)\rightarrow\text{Spec}\;k[\epsilon]$ the
unique section of the structure map
$p:\text{Spec}\;k[\epsilon]\rightarrow\text{Spec}(k)$. Let $\mathcal{V}$ be a
parabolic vector bundle over $C$. Suppose that we are given a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=p^{*}\mathcal{V}$
of $p^{*}\mathcal{V}$ by parabolic subbundles. Assume that the pulled-back
filtration
$0=\sigma^{*}\mathcal{W}_{0}\subset\sigma^{*}\mathcal{W}_{1}\subset\cdots\,\subset\sigma^{*}\mathcal{W}_{l-1}\subset\sigma^{*}\mathcal{W}_{l}=\mathcal{V}$
over the closed point is the Harder-Narasimhan filtration of $\mathcal{V}$.
Then $\mathcal{W}_{j}=p^{*}\sigma^{*}\mathcal{W}_{j}$ for all $1\leq j\leq
l-1$.

We use this result along with the construction of the filtration scheme
$Fil^{\alpha}_{\mathcal{W}}$ in order to develop a theory of schematic Harder-
Narasimhan filtrations. This is assertion (C) of the Main Theorem (see also
Theorem 8.6). It is the parabolic analogue of the work done by Gurjar and
Nitsure in [GN14], which deals with principal $G$-bundles over a curve when
the characteristic of the ground field is $0$.

Finally, we also prove in Theorem 7.6 that for every $P$ the stratum
$\text{Bun}^{\leq P}_{\mathcal{V}}$ satisfies a strong lifting criterion for
families over a discrete valuation ring. This is assertion (B) of the Main
Theorem. Heinloth [Hei17, Remark 3.20] noted that the weaker valuative
criterion [Sta, Tag 0CLK] holds for the semistable locus for more general
parahoric Bruhat-Tits group schemes. See Section 7 for more details on this.

We conclude this introduction with an outline of the paper.

In Section 2 we recall the notions of parabolic vector bundles (Definition
2.1), parabolic weights (Definition 2.5), degree and slope (Definition 2.7)
and the existence and uniqueness of the Harder-Narasimhan filtration
(Proposition 2.18). The section concludes with a discussion of parabolic
vector bundles in families (Subsection 2.4). This includes a proof of the
compatibility of the Harder-Narasimhan filtration with extensions of the
ground field (Lemma 2.30) and a proof of the analogue of Behrend’s conjecture
for parabolic vector bundles (Proposition 2.31). In Section 3 we define the
stack $\text{Bun}_{\mathcal{V}}$ of parabolic vector bundles of a given type
(Definition 3.6) and show that it is isomorphic to the stack of torsors for a
parahoric Bruhat-Tits group scheme over $C$ (Proposition 3.7). We define the
HN datum of a vector bundle (Definition 3.12) and the strata
$\text{Bun}_{\mathcal{V}}^{\leq P}$ (Definition 3.13).

In Section 4 we construct a parabolic version
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ of the Quot scheme (Definition
4.9). More generally, we consider nested versions
$\text{Fil}^{\alpha}_{\mathcal{V}}$ of parabolic Quot schemes (Definition
4.16). The main purpose of this section is proving that
$\text{Fil}^{\alpha}_{\mathcal{V}}$ is represented by a separated scheme of
finite type over the base (Proposition 4.17). This result is used in Sections
5.1 and 8. We recommend the reader to skip the proofs in Section 4 upon their
first read.

The purpose of Section 5 is to prove that each stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is represented by an open substack of
$\text{Bun}_{\mathcal{V}}$ (Theorem 5.11). This is achieved by first showing
that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is constructible (Proposition 5.1)
and then proving that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is closed under
generalization (Proposition 5.6). In Section 6 we prove that each stratum
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is quasicompact (Theorem 6.10). For this
purpose, we use the Harder-Narasimhan stratification of the moduli stack of
classical vector bundles, which we recall in Subsection 6.1. Our strategy is
to show that the forgetful morphism $\text{Bun}_{\mathcal{V}}^{\leq
P}\to\text{Bun}_{\text{GL}_{n}}(C)$ is quasicompact (Proposition 6.5), and
that the image lies in finitely many classical Harder-Narasimhan strata
(Corollary 6.9). In Section 7 we prove that each stratum is complete (see
Definition 7.1 and Theorem 7.6). We adapt the parabolic analogue of Langton’s
algorithm in Mehta-Seshadri [MS80] so that it applies to a general unstable
stratum $\text{Bun}_{\mathcal{V}}^{\leq P}$.

In Section 8 we define a notion of relative Harder-Narasimhan filtration in
families (Definition 8.4). We use this to equip each locally closed stratum
$\text{Bun}_{\mathcal{V}}^{=P}$ with the structure of a locally closed
substack of $\text{Bun}_{\mathcal{V}}$ (Theorem 8.6). The strategy of proof is
to show that the natural morphism
$\text{Bun}_{\mathcal{V}}^{=P}\to\text{Bun}_{\mathcal{V}}^{\leq P}$ is proper
(using the representability of $\text{Fil}^{\alpha}_{\mathcal{V}}$), radicial
(by the uniqueness of Harder-Narasimhan filtrations) and unramified (by the
parabolic analogue of Behrend’s conjecture in Proposition 2.31).

In Appendix A we provide an exposition of descent for finite morphisms, which
we adapt from [Gro95][B.3]. In Appendix B we discuss the relation of our
stratification with the $\Theta$-stratification for the moduli stack of
parahoric torsors constructed in [Hei17, AHLH18].

Acknowledgements The notion of slope employed here is a reinterpretation of
the definition suggested by J. Lurie in the outline of his Arizona Winter
School project [Lur19], and which we futher relate to [MS80]. Some of the
arguments greatly benefited from discussion with other members of the project
during the evening sessions. I would particularly like to thank Aaron
Landesman, David Yang and Bogdam Zavyalov for helpful discussions. I am happy
to thank my advisor Nicolas Templier for encouraging me to write down these
results and providing very valuable input for the redaction of the manuscript.
I thank Nitin Nitsure for helpful comments and David Mehrle for helping out
with LaTeX. I acknowledge support by NSF grants DMS-1454893 and DMS-2001071.

###  1.1 Notation

We work over a fixed ground field $k$. All schemes are understood to be
schemes over $k$. An undecorated product of $k$-schemes (e.g. $A\times B$)
should always be interpreted as a fiber product over $k$. If $R$ is a $k$
algebra and $S$ is a $k$-scheme, we may sometimes use the notation $S_{R}$ to
denote the fiber product $S\times\text{Spec}(R)$.

We write $s\in S$ to mean that $s$ is a (set theoretic) point of $S$.
Equivalently, $s$ is the Spec of the residue field of a topological point of
$S$. For such $s\in S$, we will write $\kappa(s)$ for the residue field.

We will often deal with pullbacks of quasicoherent sheaves on schemes. Let $X$
and $Y$ be schemes and let $\mathcal{Q}$ be quasicoherent sheaf on $Y$. It
will usually be the case that there is a clear choice of a morphism from
$f:X\rightarrow Y$. In such a situation we will not mention the choice of $f$
and write $\mathcal{Q}|_{X}$ to denote the pullback $f^{*}\mathcal{Q}$ of the
quasicoherent sheaf $\mathcal{Q}$ by $f$.

We fix once and for all a curve $\pi:C\rightarrow\text{Spec}(k)$ that is
smooth, projective and geometrically irreducible over $k$. We choose a finite
set $\\{x_{i}\\}_{i\in I}$ of $k$-points $x_{i}$ of $C$. We will often refer
to these $x_{i}$ as points of degeneration. For all $i\in I$, let us denote by
$q_{i}:x_{i}\rightarrow C$ the closed immersion of $x_{i}$ into $C$. We also
fix the choice of an ample line bundle $\mathcal{O}(1)$ of degree one on $C$.

##  2 Parabolic Vector Bundles

###  2.1 Parabolic vector bundles on a curve

######  Definition 2.1.

A parabolic vector bundle $\mathcal{V}$ over $C$ with parabolic structure at
$\\{x_{i}\\}_{i\in I}$ consists of the data of:

  1. (a)

A vector bundle $\mathcal{E}^{(0)}$ on $C$.

  2. (b)

For each $i\in I$, a chain of vector bundles
$\mathcal{E}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})$.

We define the rank of $\mathcal{V}$ to be
$\text{rank}\,\mathcal{V}\vcentcolon=\,\text{rank}\,\mathcal{E}^{(0)}$. For
each $i\in I$, we call $N_{i}$ the chain length at $i$.

For a parabolic vector bundle $\mathcal{V}$ as above, we will use the
convention $\mathcal{E}_{i}^{(0)}\vcentcolon=\mathcal{E}^{(0)}$ for all $i\in
I$. Note that for each $i\in I$ and $1\leq m\leq N_{i}$, the quotient
$\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}$ is the pushforward
$(q_{i})_{*}(V)$ of a finite dimensional $k$-vector space $V$. We will want to
keep track of the dimension of these vector spaces. For a parabolic vector
bundle $\mathcal{V}$, we write

$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$

to indicate that the $\mathcal{E}_{i}^{(m)}$s are the chains of vector bundles
appearing in Definition 2.1 and
$\text{dim}_{k}\left(\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}\right)\,=\,a_{i}^{(m)}$.

######  Definition 2.2.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(M_{i})}}{\subset}\,\mathcal{F}_{i}^{(M_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles. We say that $\mathcal{W}$ has type
$\mathcal{V}$ (or equivalently $\mathcal{V}$ has type $\mathcal{W}$) if

  1. (a)

$\text{rank}\,\mathcal{W}=\text{rank}\,\mathcal{V}$.

  2. (b)

For all $i\in I$, $M_{i}=N_{i}$.

  3. (c)

For all $i\in I$ and $1\leq m\leq N_{i}$, we have $a_{i}^{(m)}=b_{i}^{(m)}$.

The class of parabolic vector bundles with a given fixed set of chain lengths
forms a category. For the next two definitions, we let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles with the same chain length $N_{i}$ at each
$i\in I$.

######  Definition 2.3.

A morphism of parabolic vector bundles $f:\mathcal{W}\rightarrow\mathcal{V}$
consists of the data of a morphism of sheaves
$f_{i}^{(m)}:\mathcal{F}_{i}^{(m)}\rightarrow\mathcal{E}_{i}^{(m)}$ for each
$i\in I$ and $0\leq m\leq N_{i}$ satisfying the following two conditions

  1. (a)

$f_{i}^{(0)}=f_{j}^{(0)}$ for all $i,j\in I$.

  2. (b)

For all $i\in I$ and $1\leq m\leq N_{i}$, the following diagram commutes

${\mathcal{E}_{i}^{(m-1)}}$${\mathcal{E}_{i}^{(m)}}$${\mathcal{F}_{i}^{(m-1)}}$${\mathcal{F}_{i}^{(m)}}$$\subset$$\subset$$\scriptstyle{f_{i}^{(m-1)}}$$\scriptstyle{f_{i}^{(m)}}$

Recall that for a given vector bundle $\mathcal{E}$ on $C$, we say that a
subsheaf $\mathcal{F}\subset\mathcal{E}$ is a subbudle if $\mathcal{F}$ and
$\mathcal{E}\,/\,\mathcal{F}$ are themselves vector bundles. In this case
$\mathcal{E}\,/\,\mathcal{F}$ is called a quotient bundle.

######  Definition 2.4.

Let $f:\mathcal{W}\rightarrow\mathcal{V}$ be a morphism.

  1. (1)

We say that $\mathcal{V}$ is a parabolic quotient bundle of $\mathcal{W}$
(equivalently $f$ is an admissible epimorphism) if for each $i\in I$ and
$0\leq m\leq N_{i}$, the morphism $f_{i}^{(m)}$ witnesses
$\mathcal{E}_{i}^{(m)}$ as a quotient vector bundle of
$\mathcal{F}_{i}^{(m)}$.

  2. (2)

We say that $\mathcal{W}$ is a subbundle of $\mathcal{V}$ (equivalently $f$ is
an admissible monomorphism) if it satisfies the following two conditions

    1. (a)

For each $i\in I$ and $0\leq m\leq N_{i}$, the morphism $f_{i}^{(m)}$
witnesses $\mathcal{F}_{i}^{(m)}$ as a vector subbundle of
$\mathcal{E}_{i}^{(m)}$.

    2. (b)

For each $i\in I$ and $0\leq m\leq N_{i}$, we have
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$.

We can use these definitions to define the structure of an exact category on
the category of parabolic vector bundles over $C$ with fixed chain lengths
$N_{i}$. All morphisms in this category admit a kernel and an image. This
follows from the fact that subsheaves of vector bundles on $C$ are always
vector bundles (because $C$ is a regular curve).

###  2.2 Parabolic weights and degree

In order to stratify moduli stacks of parabolic bundles, we will define a
notion of slope. First we need the following definition.

######  Definition 2.5.

A set of parabolic weights $\overline{\lambda}$ consists of the data of a
tuple of real numbers
$\left(\lambda_{i}^{(1)},\lambda_{i}^{(2)},\,\cdots,\lambda_{i}^{(N_{i})}\right)$
for each $i\in I$ satisfying:

  1. (a)

$0<\lambda_{i}^{(m)}<1$ for all $i\in I$ and $1\leq m\leq N_{i}$.

  2. (b)

$\lambda_{i}^{(m-1)}<\lambda_{i}^{(m)}$ for all $i\in I$ and $2\leq m\leq
N_{i}$.

$N_{i}$ is called the chain length of $\overline{\lambda}$ at $i$.

######  Definition 2.6.

Let $\overline{\lambda}$ be a set of parabolic weights with chain length
$N_{i}$ at $i\in I$. We denote by $\text{Vect}_{\overline{\lambda}}$ the exact
category of parabolic vector bundles with chain length $N_{i}$ at each $i\in
I$.

For the rest of this paper, we fix a choice of parabolic weights
$\overline{\lambda}$ with chain length $N_{i}$ at $i\in I$. Unless otherwise
stated, all parabolic vector bundles will be in
$\text{Vect}_{\overline{\lambda}}$.

######  Definition 2.7.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic bundle. The degree of $\mathcal{V}$ is defined to be

$\text{deg}\;\mathcal{V}=\text{deg}\;\mathcal{E}^{(0)}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,a_{i}^{(j)}\right)$

The slope of $\mathcal{V}$ is defined by
$\mu\left(\mathcal{V}\right)\vcentcolon=\frac{\text{deg}\,\mathcal{V}}{\text{rank}\,\mathcal{V}}$.

######  Remark 2.8.

In [MS80], Mehta and Seshadri consider a slightly different notion of
parabolic vector bundle. We consider flags of vector spaces inside
$\mathcal{E}(x_{i})/\mathcal{E}$. [MS80] deals with flags in the fiber
$\mathcal{E}/\mathcal{E}(-x_{i})$. The inclusion
$\mathcal{O}_{C}(-x_{i})\hookrightarrow\mathcal{O}_{C}$ of the ideal sheaf of
$x_{i}$ induces an identification
$\mathcal{E}(x_{i})/\mathcal{E}\cong\mathcal{E}/\mathcal{E}(-x_{i})$.
Therefore our notion of parabolic vector bundle is in correspondence with the
one in Mehta-Seshadri. Our altered definition of degree reflects this change
of convention.

Let us state a couple lemmas that will come in handy in later sections. They
are immediate consequences of the definitions given above.

######  Lemma 2.9.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle of rank $n$. Then, we have

$deg\left(\mathcal{E}^{(0)}\right)\leq deg\left(\mathcal{V}\right)\leq
deg\left(\mathcal{E}^{(0)}\right)+n|I|$

∎

######  Lemma 2.10.

Let
$0\rightarrow\mathcal{W}\rightarrow\mathcal{V}\rightarrow\mathcal{Q}\rightarrow
0$ be a short exact sequence in $\text{Vect}_{\overline{\lambda}}$. Then, we
have
$\text{deg}\,\mathcal{V}\,=\,\text{deg}\,\mathcal{W}\,+\,\text{deg}\,\mathcal{Q}$.
∎

We end this section with a useful technical lemma.

######  Lemma 2.11.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ and
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be two parabolic vector bundles. Suppose that $\mathcal{W}$ is a suboject
of $\mathcal{V}$ (not necessarily a parabolic subbundle). Then,

  1. (a)

There exists a parabolic subbundle $\mathcal{W}^{sat}$ of $\mathcal{V}$
containing $\mathcal{W}$ as a subobject and satisfying
$\mu(\mathcal{W}^{sat})\geq\mu(\mathcal{W})$.

  2. (b)

If we have $\mu(\mathcal{W}^{sat})=\mu(\mathcal{W})$ in the construction for
part (a), then in fact $\mathcal{W}^{sat}=\mathcal{W}$.

  3. (c)

Given two parabolic subobjects
$\mathcal{W}_{1}\hookrightarrow\mathcal{W}_{2}\hookrightarrow\mathcal{V}$, we
have $\mathcal{W}_{1}^{sat}\subset\mathcal{W}_{2}^{sat}$.

###### Proof.

Set $n=\text{rank}\,\mathcal{W}$. For all $i\in I$ and $0\leq m\leq N_{i}$,
define
$\mathcal{P}_{i}^{(m)}\vcentcolon=\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{E}^{(m)}_{i}$.
Observe that $P_{i}^{(m)}\subset\mathcal{F}^{(0)}(x_{i})$ is a vector bundle,
because subsheaves of locally free sheaves in $C$ are locally free. Hence the
diagram of sheaves
$\mathcal{U}=\left[\;\mathcal{P}^{(0)}\,\overset{c_{i}^{(1)}}{\subset}\,\mathcal{P}_{i}^{(1)}\overset{c_{i}^{(2)}}{\subset}\cdots\;\overset{c_{i}^{(N_{i})}}{\subset}\,\mathcal{P}_{i}^{(N_{i})}=\,\mathcal{P}^{(0)}(x_{i})\;\right]_{i\in
I}$ is a parabolic vector bundle. We have that
$\mathcal{F}^{(m)}_{i}\subset\mathcal{P}^{(m)}_{i}$ and
$\mathcal{P}^{(0)}=\mathcal{F}^{(0)}$. Therefore $\mathcal{U}$ contains
$\mathcal{W}$ as a suboject. We now claim that
$\mu(\mathcal{U})\geq\mu(\mathcal{W})$. Since
$\text{rank}\,\mathcal{U}=\text{rank}\,\mathcal{W}=n$, the claim is equivalent
to $\text{deg}\,\mathcal{U}\geq\text{deg}\,\mathcal{W}$. Since
$\mathcal{F}^{(0)}=\mathcal{P}^{(0)}$, the inequality of degrees amounts to
showing that

$\sum_{i\in I}\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,c_{i}^{(j)}\,\leq\sum_{i\in
I}\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}$

We will actually show that for all $i\in I$ we have
$\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,c_{i}^{(j)}\,\leq\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,b_{i}^{(j)}$.
We can use summation by parts to rewrite both sides of the inequality. It
suffices to show

$n\lambda^{(N_{i})}\,+\,\sum_{j=1}^{N_{i}-1}(\lambda_{i}^{(j)}-\lambda^{(j+1)}_{i})\,\sum_{m=1}^{j}c_{i}^{(j)}\,\leq\,n\lambda^{(N_{i})}\,+\,\sum_{j=1}^{N_{i}-1}(\lambda_{i}^{(j)}-\lambda^{(j+1)}_{i})\,\sum_{m=1}^{j}b_{i}^{(j)}$

Since we have $\lambda_{i}^{(j)}<\lambda_{i}^{(j+1)}$ for all $1\leq j\leq
N_{i}-1$, it is sufficient to show that
$\sum_{m=1}^{j}c_{i}^{(m)}\,\geq\sum_{m=1}^{j}b_{i}^{(m)}$. By definition, we
have that
$\sum_{m=1}^{j}c_{i}^{(m)}=\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$.
Similarly, we have that
$\sum_{m=1}^{j}b_{i}^{(m)}=\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$.
Hence the inequality above is equivalent to

$\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}\geq\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$

This is now obvious, because
$\mathcal{P}_{i}^{(m)}\supset\mathcal{F}_{i}^{(m)}$. Suppose that
$\mu(\mathcal{U})=\mu(\mathcal{W})$. Then all of the inequalities above must
be equalities. In particular we have
$\text{dim}_{k}\,\mathcal{P}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}=\text{dim}_{k}\,\mathcal{F}_{i}^{(j)}\,/\,\mathcal{F}^{(0)}$
for all $i,j$. This implies that $\mathcal{U}=\mathcal{W}$. It is also clear
that inclusions of subobjects as in part (c) are preserved by this
construction.

Replacing $\mathcal{W}$ by $\mathcal{U}$ in the proposition, we can assume
from the beginning that we have
$\mathcal{F}_{i}^{(m)}=\mathcal{F}^{(0)}(x_{i})\,\cap\,\mathcal{E}^{(m)}_{i}$.
Let us define
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$.
This means that we have the following commutative diagram with exact columns:

${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$
Figure 1: Diagram 1

The problem is that the $\mathcal{Q}_{i}^{(m)}$s do not necessarily form a
parabolic bundle, since they could have nontrivial torsion. In order to solve
this, we kill the $C$-torsion. Let $\xi$ denote the generic point of $C$. Let
$\iota:\xi\longrightarrow C$ be the corresponding inclusion. Define

$K_{i}^{(m)}\vcentcolon=\text{Ker}\left(\mathcal{Q}_{i}^{(m)}\,\xrightarrow{unit}\,i_{*}i^{*}\,\mathcal{Q}_{i}^{(m)}\right)$
$\mathcal{Q}_{i}^{(m),\,sat}\vcentcolon=\text{Im}\left(\mathcal{Q}_{i}^{(m)}\,\xrightarrow{unit}\,i_{*}i^{*}\,\mathcal{Q}_{i}^{(m)}\right)$
$\mathcal{F}_{i}^{(m),\,sat}\vcentcolon=\text{Ker}\left(\mathcal{E}_{i}^{(m)}\,\twoheadrightarrow\,\mathcal{Q}_{i}^{(m)}\rightarrow\,\mathcal{Q}_{i}^{(m),\,sat}\right)$

There is a similar commutative diagram with exact columns:

${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0),\,sat}}$${\mathcal{Q}_{i}^{(1),\,sat}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2),\,sat}}$${\mathcal{Q}_{i}^{(N_{i}-1),\,sat}}$${\mathcal{Q}^{(0),\,sat}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0),\,sat}}$${\mathcal{F}_{i}^{(1),\,sat}}$${\cdots}$${{}^{s}\mathcal{F}_{i}^{(Ni-2),\,sat}}$${\mathcal{F}_{i}^{(N_{i}-1),\,sat}}$${\mathcal{F}^{(0),\,sat}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$

Figure 2: Diagram 2

We set
$\mathcal{W}^{sat}\vcentcolon=\left[\;\mathcal{F}^{(0),\,sat}\;\overset{c_{i}^{(1)}}{\subset}\;\mathcal{F}_{i}^{(1),\,sat}\;\overset{c_{i}^{(2)}}{\subset}\cdots\;\overset{c_{i}^{(N_{i})}}{\subset}\;\mathcal{F}_{i}^{(N_{i}),\,sat}=\,\mathcal{F}^{(0),\,sat}(x_{i})\;\right]_{i\in
I}$. By construction, $\mathcal{W}^{sat}$ is a parabolic subbundle of
$\mathcal{V}$ containing $\mathcal{W}$. It is clear that this construction
preserves inclusions of subobject as described in part (c). We are left to
show the claims about the slopes. By definition, we have the following
commutative diagram with exact rows:

${0}$${K_{i}^{(m-1)}}$${\mathcal{Q}_{i}^{(m-1)}}$${\mathcal{Q}_{i}^{(m-1),\,sat}}$${0}$${0}$${K_{i}^{(m)}}$${\mathcal{Q}_{i}^{(m)}}$${\mathcal{Q}_{i}^{(m),\,sat}}$${0}$

An application of the Snake Lemma yields a surjection
$\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\,\twoheadrightarrow\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}$.
Therefore, we have

$\text{dim}_{k}\,\left(\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}\right)\;\leq\;\text{dim}_{k}\,\left(\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\right)$

By applying the Snake Lemma to consecutive columns of Diagram 1 and Diagram 2
respectively, we obtain the two short exact sequences

$0\longrightarrow\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}\longrightarrow\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}\longrightarrow\mathcal{Q}_{i}^{(m)}\,/\,\mathcal{Q}_{i}^{(m-1)}\longrightarrow
0$
$0\longrightarrow\mathcal{F}_{i}^{(m),\,sat}\,/\,\mathcal{F}_{i}^{(m-1),\,sat}\longrightarrow\mathcal{E}_{i}^{(m),\,sat}\,/\,\mathcal{E}_{i}^{(m-1),\,sat}\longrightarrow\mathcal{Q}_{i}^{(m),\,sat}\,/\,\mathcal{Q}_{i}^{(m-1),\,sat}\longrightarrow
0$

We conclude that $c_{i}^{(m)}\geq b_{i}^{(m)}$. But note that for all $i\in I$
we have

$\sum_{m=1}^{N_{i}}c_{i}^{(m)}=\text{rank}\left(\mathcal{W}^{sat}\right)=\text{rank}\left(\mathcal{W}\right)=\sum_{m=1}^{N_{i}}b_{i}^{(m)}$

Therefore, we must have $c_{i}^{(m)}=b_{i}^{(m)}$ for all $i$ and $m$. By
construction we have $\mathcal{F}^{(0)}\subset\,\mathcal{F}^{(0),\,sat}$. In
particular, $deg\,\mathcal{F}^{(0)}\leq deg\,\mathcal{F}^{(0),\,sat}$ as
vector bundles. We therefore get a chain of (in)equalities:

$\displaystyle deg\,\mathcal{W}^{sat}$
$\displaystyle=\;deg\;\mathcal{F}^{(0),\,sat}+\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}^{sat}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,c_{i}^{(m)}\right)$
$\displaystyle=\;deg\;\mathcal{F}^{(0),\,sat}+\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}^{sat}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,b_{i}^{(m)}\right)$
$\displaystyle\geq\;deg\;\mathcal{F}^{(0)}\,+\;\sum_{i\in
I}\left(\text{rank}\,\mathcal{W}-\sum_{m=1}^{N_{i}}\lambda_{i}^{(m)}\,b_{i}^{(m)}\right)$
$\displaystyle=\;deg\,\mathcal{W}$

Part (a) of the lemma follows.

Suppose that $deg\,\mathcal{W}=deg\,\mathcal{W}^{sat}$. From the chain of
(in)equalities above we can conclude that
$deg\,\mathcal{F}^{(0)}=deg\,\mathcal{F}^{(0),\,sat}$. Hence we must have
$\mathcal{F}^{(0)}=\,\mathcal{F}^{(0),\,sat}$. We can use the fact that
$\mathcal{F}_{i}^{(m)}\subset\,\mathcal{F}_{i}^{(m),\,sat}$ and that

$\text{rank}\,\left(\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}^{(0)}\right)\,=\,\sum_{j=1}^{m}b_{i}^{(j)}\,=\,\text{rank}\,\left(\mathcal{F}_{i}^{(m),\,sat}\,/\,\mathcal{F}^{(0)}\right)$

in order to conclude that
$\mathcal{F}_{i}^{(m)}=\,\mathcal{F}_{i}^{(m),\,sat}$ for all $i$ and $m$.
Therefore, we have that $\mathcal{W}=\,\mathcal{W}^{sat}$, as stipulated in
part (b). ∎

###  2.3 Harder-Narasimhan filtrations

######  Definition 2.12.

A parabolic vector bundle $\mathcal{V}$ in $\text{Vect}_{\overline{\lambda}}$
is called semistable (with respect to $\overline{\lambda}$) if for all
parabolic subbundles $\mathcal{W}\subset\mathcal{V}$, we have
$\mu\left(\mathcal{W}\right)\,\leq\,\mu\left(\mathcal{V}\right)$.

######  Definition 2.13.

Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. A Harder-Narasimhan filtration of
$\mathcal{V}$ is a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by parabolic subbundles $\mathcal{W}_{j}$ satisfying the following two
conditions:

  1. (a)

For all $1\leq j\leq l$, the parabolic bundle
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ is semistable.

  2. (b)

For all $2\leq j\leq l$, we have
$\mu\left(\mathcal{W}_{j-1}\,/\,\mathcal{W}_{j-2}\right)\,>\,\mu\left(\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\right)$.

In order to prove the existence of such filtrations, we will need a few lemmas
along the way.

######  Lemma 2.14.

Let $\mathcal{V}$ be a parabolic bundle. There exists a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
by subbundles $\mathcal{W}_{j}$ such that
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ has rank 1 for all $1\leq j\leq l$.

###### Proof.

By induction on the rank of $\mathcal{V}$, it suffices to show that
$\mathcal{V}$ has a parabolic subbundle $\mathcal{W}$ of rank 1. Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is well known that $\mathcal{E}^{(0)}$ admits a vector subbundle
$\mathcal{F}^{(0)}$ of rank 1. Indeed, by Riemann-Roch there exists some $N>0$
such that $\mathcal{E}^{(0)}(N)$ has a nonzero global section
$\mathcal{O}_{C}\rightarrow\mathcal{E}^{(0)}(N)$. Tensoring by
$\mathcal{O}(-N)$ gives us a subsheaf
$\mathcal{O}(-N)\rightarrow\mathcal{E}^{(0)}$ that is a line bundle. Let
$j:\xi\rightarrow C$ denote the inclusion of the generic point $\xi$ into $C$.
Set
$\mathcal{F}^{(0)}\vcentcolon=\text{Ker}\left(\mathcal{E}^{(0)}\rightarrow\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)\xrightarrow{unit}j_{*}j^{*}\;\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)\right)$.
Then the quotient $\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}$ is torsion free,
because it is a subsheaf of
$j_{*}j^{*}\;\mathcal{E}^{(0)}\,/\,\mathcal{O}(-N)$. So $\mathcal{F}^{(0)}$ is
a subbundle of $\mathcal{E}^{(0)}$. On the other hand, we have that
$\mathcal{F}^{(0)}\,/\,\mathcal{O}(-N)$ is torsion. Therefore
$\text{rank}\,\mathcal{F}^{(0)}=\text{rank}\,\mathcal{O}(-N)=1$.

We can now set
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,\cap\,\mathcal{F}^{(0)}(x_{i})$
and define $\mathcal{W}$ to be the diagram of sheaves
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. It is clear that $\mathcal{W}$ is a parabolic subbundle of rank 1,
because all $\mathcal{F}_{i}^{(m)}\subset\mathcal{F}^{(0)}(x_{i})$ are line
bundles. ∎

######  Corollary 2.15.

Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$. Then,

  1. (a)

There is a constant $M$ such that any subbundle
$\mathcal{U}\subset\mathcal{V}$ satisfies $\text{deg}\,\mathcal{U}\leq M$.

  2. (b)

The set of all slopes of parabolic subbundles of $\mathcal{V}$ has a maximal
element.

###### Proof.

  1. (a)

Let
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
be a filtration as in Lemma 2.14. We set
$M=\underset{A\subset\\{1,2,\cdots,l\\}}{\text{max}}\,\left\\{\sum_{j\in
A}\text{deg}\,\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\right\\}$. Let
$\mathcal{U}$ be a parabolic subbundle of $\mathcal{V}$. We can take
intersections in the category of parabolic vector bundles, since
$\text{Vect}_{\overline{\lambda}}$ admits kernels. Consider the induced
filtration of $\mathcal{U}$ by parabolic subbundles

$0=\mathcal{W}_{0}\,\cap\mathcal{U}\,\subset\mathcal{W}_{1}\,\cap\mathcal{U}\,\subset\cdots\,\subset\mathcal{W}_{l-1}\,\cap\mathcal{U}\,\subset\mathcal{W}_{l}\,\cap\mathcal{U}\,=\mathcal{U}$

For all $1\leq j\leq l$ we have that
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)$
is isomorphic to a parabolic subbundle of
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$. Since
$\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$ has rank 1, we must have either
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\,=\,0$
or
$\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\;\cong\;\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}$.
Let

$A_{\mathcal{U}}\vcentcolon=\left\\{\;j\in\\{1,2,\cdots,l\\}\;\mid\;\left(\mathcal{W}_{j}\,\cap\,\mathcal{U}\right)\,/\,\left(\mathcal{W}_{j-1}\,\cap\,\mathcal{U}\right)\;\cong\;\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\;\right\\}$

A repeated application of Lemma 2.10 yields
$\text{deg}\,\mathcal{U}=\sum_{j\in
A_{\mathcal{U}}}\text{deg}\,\mathcal{W}_{j}\,/\,\mathcal{W}_{j-1}\leq M$.

  2. (b)

Fix a rank $k$ with $1\leq k\leq n$. We claim that the set
$\left\\{\;\mu(\mathcal{U})\;\mid\;\mathcal{U}\;\text{is a subbundle of
$\mathcal{V}$ of rank $k$}\;\right\\}$ has a maximal element. Once this claim
is proven, we can take the maximum among all possible ranks $1\leq k\leq n$ to
produce the maximal element as in part (b) of the corollary. We are left to
prove the claim.

Let $\mathcal{U}$ be any subbundle of rank $k$. By definition, we have
$\mu(\mathcal{U})=\frac{1}{k}\text{deg}\,\mathcal{U}$. So it suffices to
produce a subbundle $\mathcal{U}$ of maximal degree among those of rank $k$.
Consider the finite set
$S\vcentcolon=\left\\{A_{\mathcal{U}}\;\mid\;\mathcal{U}\,\text{is a subbundle
of $\mathcal{V}$ of rank $k$}\;\right\\}$, where $A_{\mathcal{U}}$ is defined
as in part (a). Let $A_{\mathcal{U}}$ be an element of $S$ maximizing the
quantity $\sum_{j\in A_{\mathcal{U}}}\text{deg}\,\mathcal{W}_{j}$. The same
argument as in part (a) shows that $\mathcal{U}$ is a subbundle of rank $k$
with maximal degree among subbundles of the same rank.

∎

######  Lemma 2.16.

Let $\mathcal{V}$, $\mathcal{W}$ be two semistable parabolic vector bundles.
Then,

  1. (a)

For any subobject $\mathcal{P}\hookrightarrow\mathcal{W}$ (not necessarily a
parabolic subbundle), we have $\mu(\mathcal{P})\leq\mu(\mathcal{W})$.

  2. (b)

If $\mathcal{Q}$ is a parabolic quotient bundle of $\mathcal{W}$, then
$\mu\left(\mathcal{Q}\right)\,\geq\,\mu\left(\mathcal{W}\right)$.

  3. (c)

If $\mu\left(\mathcal{W}\right)\,>\,\mu\left(\mathcal{V}\right)$, then
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{W},\,\mathcal{V}\right)=0$.

  4. (d)

Let $\mathcal{P}$ fit into an exact sequence
$0\rightarrow\mathcal{V}\rightarrow\mathcal{P}\rightarrow\mathcal{W}\rightarrow
0$. If $\mathcal{U}$ is a parabolic subbundle of $\mathcal{P}$, then
$\mu(U)\leq\text{max}\left\\{\mu(\mathcal{V}),\,\mu(\mathcal{W})\right\\}$

###### Proof.

  1. (a)

By Lemma 2.11, we have a subbundle $\mathcal{P}^{sat}$ of $\mathcal{W}$ with
$\mu(\mathcal{P}^{sat})\geq\mu(\mathcal{P})$. By semistability,
$\mu(\mathcal{W})\geq\mu(\mathcal{P}^{sat})$.

  2. (b)

Let $m=\text{rank}\,\mathcal{Q}$ and $n=\text{rank}\,\mathcal{W}$. Define
$\mathcal{K}$ to be the kernel of the admissible epimorphism
$\mathcal{W}\twoheadrightarrow\mathcal{Q}$. By Lemma 2.10, we have
$\text{deg}\,\mathcal{W}=\text{deg}\,\mathcal{K}\,+\,\text{deg}\,\mathcal{Q}$.
We can rewrite this as
$\mu(\mathcal{W})=\frac{n-m}{n}\mu(\mathcal{K})+\frac{m}{n}\mu(\mathcal{Q})$.
Since $\mathcal{W}$ is semistable, we have
$\mu(\mathcal{K})\leq\mu(\mathcal{W})$. Therefore

$\mu(\mathcal{Q})=\frac{n}{m}\left(\mu(\mathcal{W})-\frac{n-m}{n}\mu(\mathcal{K})\right)\geq\frac{n}{m}\left(\mu(\mathcal{W})-\frac{n-m}{n}\mu(\mathcal{W})\right)=\mu(\mathcal{W})$

  3. (c)

Suppose that there exists a nontrivial morphism
$f:\mathcal{W}\rightarrow\mathcal{V}$. Consider the nonzero image
$\mathcal{I}=\text{Im}\left(\,\mathcal{W}\xrightarrow{f}\mathcal{V}\,\right)$
in $\text{Vect}_{\overline{\lambda}}$. Then $\mathcal{I}$ is a parabolic
quotient bundle of $\mathcal{W}$. By part (b), we know that
$\mu(\mathcal{I})\geq\mu(\mathcal{W})$. Therefore
$\mu(\mathcal{I})>\mu(\mathcal{V})$. But $\mathcal{I}$ is a suboject of
$\mathcal{V}$. This contradicts semistability of $\mathcal{V}$ by part (a).

  4. (d)

Let
$M\vcentcolon=\text{max}\left\\{\mu(\mathcal{V}),\,\mu(\mathcal{W})\right\\}$.
Define $\mathcal{Q}$ to be the image of the composition
$\mathcal{U}\,\hookrightarrow\,\mathcal{P}\,\twoheadrightarrow\,\mathcal{W}$
in $\text{Vect}_{\overline{\lambda}}$. Then $\mathcal{Q}$ is a quotient bundle
of $\mathcal{U}$. We get a short exact sequence
$0\longrightarrow\mathcal{K}\longrightarrow\mathcal{U}\longrightarrow\mathcal{Q}\longrightarrow
0$, where $\mathcal{K}\hookrightarrow\mathcal{V}$ and
$\mathcal{Q}\hookrightarrow\mathcal{W}$. By part (a), we see that
$\mu(\mathcal{K})\leq\mu(\mathcal{V})\leq M$. Similarly,
$\mu(\mathcal{Q})\leq\mu(\mathcal{W})\leq M$. Therefore, we have

$\mu(\mathcal{U})=\frac{\text{rank}\,\mathcal{K}}{\text{rank}\,\mathcal{U}}\,\mu(\mathcal{K})\,+\,\frac{\text{rank}\,\mathcal{Q}}{\text{rank}\,\mathcal{U}}\,\mu(\mathcal{Q})\,\leq\,\frac{\text{rank}\,\mathcal{K}}{\text{rank}\,\mathcal{U}}\,M\,+\,\frac{\text{rank}\,\mathcal{Q}}{\text{rank}\,\mathcal{U}}\,M\,=\,M$

∎

######  Lemma 2.17.

Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Let $\mu$ denote the maximal element of
the set of all slopes of subbundles of $\mathcal{V}$, as in part (b) of
Corollary 2.15. Then,

  1. (a)

There exists a unique maximal semistable subbundle $\mathcal{U}$ of
$\mathcal{V}$ such that $\mu(\mathcal{U})=\mu$ and $\mathcal{U}$ contains all
other subbundles of slope $\mu$. We will call $\mathcal{U}$ the maximal
destabilizing subbundle of $\mathcal{V}$.

  2. (b)

If $\mathcal{U}$ is as in part (a), then all subobjects
$\overline{\mathcal{W}}\,\hookrightarrow\,\mathcal{V}\,/\,\mathcal{U}$ satisfy
$\mu(\overline{\mathcal{W}})<\mu$.

  3. (c)

For $\mathcal{U}$ as in part (a), we have
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{U},\;\mathcal{V}\,/\,\mathcal{U}\,\right)=0$.

###### Proof.

Uniqueness of such subbundle $\mathcal{U}$ is clear. It suffices to prove
existence. Let $\mathcal{U}$ be a parabolic subbundle of $\mathcal{V}$ with
$\mu(\mathcal{U})=\mu$ that has maximal rank among the subbundles with slope
$\mu$. By the definition of $\mu$, it is clear that $\mathcal{U}$ is
semistable.

We claim that $\mathcal{U}$ satisfies the statement in part (b) of the
proposition. Suppose that $\overline{\mathcal{W}}$ is a subobject of
$\mathcal{V}\,/\,\mathcal{U}$ with slope $\mu(\overline{\mathcal{W}})\geq\mu$.
By the axioms of exact categories, there is a subobject $\mathcal{W}$ of
$\mathcal{V}$ with $\mathcal{U}\subset\mathcal{W}$ and a short exact sequence
$0\longrightarrow\mathcal{U}\longrightarrow\mathcal{W}\longrightarrow\overline{\mathcal{W}}\rightarrow
0$.

By additivity of degree in short exact sequences (Lemma 2.10), we have

$\mu(\mathcal{W})=\frac{\text{rank}\,\mathcal{U}}{\text{rank}\,\mathcal{W}}\,\mu(\mathcal{U})\,+\,\frac{\text{rank}\,\overline{\mathcal{W}}}{\text{rank}\,\mathcal{W}}\,\mu(\overline{\mathcal{W}})\,\geq\,\mu$

contradicting maximality of the rank of $\mathcal{U}$.

Now let $\mathcal{P}$ be any parabolic subbundle of $\mathcal{V}$ with slope
$\mu(\mathcal{P})=\mu$. By the definition of $\mu$, $\mathcal{P}$ is
semistable. We claim that $\mathcal{P}\subset\mathcal{U}$. In order to see
this, it suffices to show that the composition
$\mathcal{P}\hookrightarrow\mathcal{V}\twoheadrightarrow\mathcal{V}\,/\,\mathcal{U}$
is 0. Let us show that
$\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{P},\;\mathcal{V}\,/\,\mathcal{U}\,\right)=0$.
This also implies part (c) as a special case.

For the sake of contradiction, suppose that
$\psi\in\text{Hom}_{\text{Vect}_{\overline{\lambda}}}\left(\mathcal{P},\;\mathcal{V}\,/\,\mathcal{U}\,\right)$
is nonzero. Let $\mathcal{I}$ be the nontrivial image $\text{Im}(\psi)$ of
$\mathcal{P}$ in $\mathcal{V}\,/\,\mathcal{U}$. $\mathcal{I}$ is a parabolic
quotient bundle of $\mathcal{P}$. By part (b) of Lemma 2.16 we have
$\mu(\mathcal{I})\geq\mu(\mathcal{P})$. Therefore $\mathcal{I}$ is a suboject
of $\mathcal{V}\,/\,\mathcal{U}$ with slope $\mu(\mathcal{I})\geq\mu$. We get
a contradiction, since we have proven in the paragraph above that this is not
possible. ∎

Now we are ready to prove the main proposition of this section.

######  Proposition 2.18.

Let $\mathcal{V}$ be a parabolic vector bundle in
$\text{Vec}_{\overline{\lambda}}$. Then $\mathcal{V}$ admits a unique Harder-
Narasimhan filtration.

###### Proof.

Let $\mathcal{U}_{1}$ be the maximal destabilizing subbundle of $\mathcal{V}$.
Set $\mu_{1}=\mu(\mathcal{U}_{1})$. We can iterate this process by recursion,
setting $\overline{\mathcal{U}_{j}}$ to be the maximal destabilizing subbundle
of $\mathcal{V}\,/\,\mathcal{U}_{j-1}$ and letting $\mathcal{U}_{j}$ be the
unique subbundle of $\mathcal{V}$ containing $\mathcal{U}_{j-1}$ such that
$\overline{\mathcal{U}_{j}}=\mathcal{U}_{j}\,/\,\mathcal{U}_{j-1}$. Define
$\mu_{j}\vcentcolon=\mu(\overline{\mathcal{U}_{j}})$. By part (b) of Lemma
2.11, we see that $\mu_{j}>\mu_{j-1}$. Since the rank of $\mathcal{V}$ is
finite, this process must terminate. Hence $\mathcal{U}_{l}=\mathcal{V}$ for
some $l$. We get a filtration

$0\subset\mathcal{U}_{1}\subset\cdots\,\subset\mathcal{U}_{l-1}\subset\mathcal{U}_{l}=\mathcal{V}$

which by construction is a Harder-Narasimhan filtration.

For uniqueness, we can induct on the length $l$ of the filtration above. We
are reduced to showing that for any Harder-Narasimhan filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{m-1}\subset\mathcal{W}_{m}=\mathcal{V}$
we have that $\mathcal{W}_{1}$ is the maximal destabilizing subsheaf
$\mathcal{U}_{1}$ of $\mathcal{V}$. By applying part (d) of Lemma 2.16 several
times, we conclude that we must have
$\mu(\mathcal{W}_{1})=\mu(\mathcal{U}_{1})=\mu_{1}$. Therefore, by maximality
of $\mathcal{U}_{1}$, we must have $\mathcal{U}_{1}\supset\mathcal{W}_{1}$.
Consider $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$. This is a subbundle of
$\mathcal{V}\,/\,\mathcal{W}_{1}$. Another sequence of applications of part
(d) of Lemma 2.16 shows that if $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$ is
nontrivial, then we must have
$\mu\left(\mathcal{U}_{1}\,/\,\mathcal{W}_{1}\right)<\mu_{1}$. This
contradicts part (b) of Lemma 2.16 applied to the semistable bundle
$\mathcal{U}_{1}$. Hence $\mathcal{U}_{1}\,/\,\mathcal{W}_{1}$ is trivial, and
therefore $\mathcal{U}_{1}=\mathcal{W}_{1}$. ∎

###  2.4 Parabolic vector bundles in families

In this subsection we extend the theory of parabolic vector bundles to the
relative setting.

######  Definition 2.19.

Let $S$ be a $k$-scheme. A parabolic vector bundle $\mathcal{V}$ of rank $n$
over $C\times S$ (with parabolic structure at $\\{x_{i}\\}$) consists of the
data of:

  1. (a)

A vector bundle $\mathcal{E}^{(0)}$ of rank $n$ on $C\times S$.

  2. (b)

For each $i\in I$, a chain of vector bundles
$\mathcal{E}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})$.

such that for all $i\in I$ and all $1\leq m\leq N_{i}$, the quotient
$\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}$ is $S$-flat and the
restriction $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times
S}$ is a vector bundle of constant rank on $x_{i}\times S$.

For each $i\in I$, the number $N_{i}$ will be called the chain length at $i$.

######  Remark 2.20.

Since $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times S}$ is
finitely presented and $S$-flat, it must be a vector bundle. The reason why we
require $\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times S}$
to have constant rank is to simplify notation. Note that this always true
locally on $S$.

For a parabolic vector bundle $\mathcal{V}$ over $C\times S$, we will write

$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$

to convey the information that
$a_{i}^{(m)}=\text{rank}\,\mathcal{E}^{(m)}_{i}\,/\,\mathcal{E}_{i}^{(m-1)}|_{x_{i}\times
S}$.

######  Definition 2.21.

Let $S$ be a $k$-scheme and let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$. For any $S$-scheme
$f:T\rightarrow S$, the pullback of $\mathcal{V}$ is defined to be the
parabolic vector bundle $f^{*}\mathcal{V}$ on $C\times T$ given by

$\displaystyle f^{*}\mathcal{V}=\left[\;(id_{C}\times
f)^{*}\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,(id_{C}\times
f)^{*}\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,(id_{C}\times
f)^{*}\mathcal{E}_{i}^{(N_{i})}=(id_{C}\times
f)^{*}\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in I}$

It will often be the case that the choice of a map $f$ is understood and
unambiguous, and then we will write $\mathcal{V}|_{C\times T}$ instead of
$f^{*}\mathcal{V}$.

######  Remark 2.22.

It is necessary to know that
$\mathcal{E}_{i}^{(m)}\,/\,\mathcal{E}_{i}^{(m-1)}$ is $S$-flat in order to be
able to conclude that the inclusion
$\mathcal{E}_{i}^{(m-1)}\subset\mathcal{E}_{i}^{(m)}$ remains a monomorphism
after pulling back by $f$. Note that the ranks $a_{i}^{(m)}$ are preserved by
pullbacks.

######  Definition 2.23.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic bundle over $C$. Let $S$ be a $k$-scheme and let
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(M_{i})}}{\subset}\,\mathcal{F}_{i}^{(M_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$. We say that $\mathcal{W}$
is of type $\mathcal{V}$ if all of the following conditions are satisfied

  1. (a)

$\text{rank}\,\mathcal{W}=\text{rank}\,\mathcal{V}$.

  2. (b)

For all $i\in I$, $N_{i}=M_{i}$ (they have the same chain lengths).

  3. (c)

For all $i\in I$ and $1\leq m\leq N_{i}$, we have $a_{i}^{(m)}=b_{i}^{(m)}$.

######  Remark 2.24.

The property of being equivalent to a given $\mathcal{V}$ is preserved by
pullbacks.

For the next two definitions, we fix a $k$-scheme $S$ and we let

$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$

be two parabolic vector bundles over $C\times S$ with the same chain length
$N_{i}$ at each $i\in I$.

######  Definition 2.25.

A morphism of parabolic Vector bundles $f:\mathcal{W}\rightarrow\mathcal{V}$
consists of of the data of a morphism of sheaves
$f_{i}^{(m)}:\mathcal{F}_{i}^{(m)}\rightarrow\mathcal{E}_{i}^{(m)}$ for each
$i\in I$ and $0\leq m\leq N_{i}$ satisfying the following two conditions:

  1. (a)

$f_{i}^{(0)}=f_{j}^{(0)}$ for all $i,j\in I$.

  2. (b)

For all $i\in I$ and $1\leq m\leq N_{i}$, the following diagram commutes

${\mathcal{E}_{i}^{(m-1)}}$${\mathcal{E}_{i}^{(m)}}$${\mathcal{F}_{i}^{(m-1)}}$${\mathcal{F}_{i}^{(m)}}$$\subset$$\subset$$\scriptstyle{f_{i}^{(m-1)}}$$\scriptstyle{f_{i}^{(m)}}$

######  Definition 2.26.

Let $f:\mathcal{W}\rightarrow\mathcal{V}$ be a morphism of parabolic bundles.

  1. 1.

We say that $\mathcal{V}$ is a parabolic quotient bundle of $\mathcal{W}$
(equivalently $f$ is an admissible epimorphism) if the morphism $f_{i}^{(m)}$
witnesses $\mathcal{E}_{i}^{(m)}$ as a quotient vector bundle of
$\mathcal{F}_{i}^{(m)}$ for all $i\in I$ and $0\leq m\leq N_{i}$.

  2. 2.

We say that $\mathcal{W}$ is a parabolic subbundle of $\mathcal{V}$
(equivalently $f$ is an admissible monomorphism) if it satisfies the following
two conditions:

    1. (a)

For each $i\in I$ and $0\leq m\leq N_{i}$, we have
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$.

    2. (b)

The naive quotient (as a diagrams of quasicoherent sheaves)

$\mathcal{Q}\vcentcolon=\left[\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}\,\subset\,\mathcal{E}_{i}^{(1)}\,/\,\mathcal{F}_{i}^{(1)}\,\subset\cdots\;\subset\,\mathcal{E}_{i}^{(N_{i})}\,/\,\mathcal{F}_{i}^{(N_{i})}\,=\,\left(\mathcal{E}^{(0)}\,/\,\mathcal{F}^{(0)}\right)(x_{i})\right]_{i\in
I}$

is a parabolic vector bundle over $C\times S$.

We can use this definitions to impose the structure of an exact category on
the class of parabolic vector bundles over $C\times S$ with a fixed set of
chain lengths the structure. In this case we don’t have kernels and images in
general, because it is no longer true that a subsheaf of a locally free sheaf
is locally free.

Let us fix a set of parabolic weights $\overline{\lambda}$ with chain length
$N_{i}$ at $i\in I$.

######  Definition 2.27.

Let $S$ be a $k$-scheme. We write $\text{Vect}_{\overline{\lambda}}(S)$ to
denote the exact category of parabolic vector bundles over $C\times S$ with
chain length $N_{i}$ for each $i\in I$.

######  Remark 2.28.

Since the pullbacks of parabolic vector bundles preserve exact sequences,
$\text{Vect}_{\overline{\lambda}}$ is a pseudofunctor from $k$-schemes into
the 2-category of exact categories (with exact functors as 1-morphisms).

######  Lemma 2.29.

Let $S$ be a scheme over $k$ and let $\mathcal{W}$ be a parabolic vector
bundle in $\text{Vect}_{\overline{\lambda}}(S)$. Then, the function
$f:S\longrightarrow\mathbb{R}$ given by
$f(s)=\text{deg}\;\mathcal{W}\,|_{C_{s}}$ is locally constant in the Zariski
topology.

###### Proof.

Suppose that
$\mathcal{W}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. By the formula for the degree in Definition 2.7, we have

$f(s)=\text{deg}\;\mathcal{E}^{(0)}\,|_{C_{s}}+\sum_{i\in
I}\left(n-\sum_{j=1}^{N_{i}}\lambda_{i}^{(j)}\,a_{i}^{(j)}\right)$

The corollary follows, because the degree of the fibers of $\mathcal{E}^{(0)}$
is know to be a locally constant function. ∎

The following lemma tells us that the Harder-Narasimhan filtration behaves
well under extension of scalars. The analogous result for classical vector
bundles was proven in [Lan75]. See also [HL97][Thm 1.3.7]. Our proof is a
modification of the argument found there.

######  Lemma 2.30.

Let $\mathcal{V}$ be a parabolic vector bundle over $C$ that belongs to
$\text{Vect}_{\overline{\lambda}}$. Let $K$ be a field extension of $k$. If
$\mathcal{V}$ is semistable, then $\mathcal{V}|_{C_{K}}$ is semistable.

In particular, the pullback of the Harder-Narasimhan filtration of
$\mathcal{V}$ to $C_{K}$ is the Harder-Narasimhan filtration of
$\mathcal{V}|_{C_{K}}$.

###### Proof.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a semistable parabolic vector bundle over $C$ as in the statement.
Suppose that $\mathcal{V}|_{C_{K}}$ is not semistable. Let $\mathcal{W}_{K}$
be the maximal destabilizing parabolic subbundle as in Lemma 2.17. It is clear
by the formula in Definition 2.7 that the slope of a parabolic vector bundle
is preserved under extension of ground fields. We will try to descend
$\mathcal{W}_{K}$ to a parabolic subbundle of $\mathcal{W}$ of $\mathcal{V}$
in order to contradict semistability of $\mathcal{V}$.

Suppose that $\mathcal{W}_{K}$ is given by
$\mathcal{W}_{K}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. The definition of parabolic subbundle implies
$\mathcal{F}_{i}^{(m)}=\mathcal{E}_{i}^{(m)}|_{C_{K}}\,\cap\,\mathcal{F}^{(0)}$.
Hence $\mathcal{W}_{K}$ is completely determined by the vector subbundle
$\mathcal{F}^{(0)}\subset\,\mathcal{E}^{(0)}|_{C_{K}}$. This vector subbundle
gives us a $K$-point in $\text{Quot}_{\mathcal{E}^{(0)}/C/k}$ (see [Nit05] or
Section 4 below). Since $\text{Quot}_{\mathcal{E}^{(0)}/C/k}$ is locally of
finite type, this actually comes from a $K^{\prime}$-point
$\mathcal{F}^{(0)}_{K^{\prime}}\subset\,\mathcal{E}^{(0)}|_{C_{K^{\prime}}}$,
where $K^{\prime}$ is a subextension of $K\supset k$ that is finitely
generated over $k$. Define
$\mathcal{F}_{i,\,K^{\prime}}^{(m)}\vcentcolon=\mathcal{F}^{(0)}_{K^{\prime}}\,\cap\,\mathcal{E}_{i}^{(m)}|_{C_{K^{\prime}}}$.

Set
$\mathcal{W}_{K^{\prime}}=\left[\;\mathcal{F}_{K^{\prime}}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i,\,K^{\prime}}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i,\,K^{\prime}}^{(N_{i})}=\,\mathcal{F}_{K^{\prime}}^{(0)}(x_{i})\;\right]_{i\in
I}$. This is a parabolic subbundle of $\mathcal{V}|_{C_{K^{\prime}}}$, because
$C_{K^{\prime}}$-flatness of the subsheaves and the quotients can be checked
fpqc-locally. $\mathcal{W}_{K^{\prime}}\subset\mathcal{V}|_{C_{K^{\prime}}}$
is the maximal parabolic subbundle with maximal slope. This is because there
are more subbundles to check over $C_{K}$ than over $C_{K^{\prime}}$. Hence we
can assume without loss of generality that $K\supset k$ is a finitely
generated field extension.

By enlarging the field if necessary, we can express $K\supset k$ as the
composition of a finitely generated purely transcendental extension, followed
by a finite Galois extension and a finite purely inseparable extension. We are
thus reduced to analyze each of these three cases.

  1. C 1

$K\supset k$ is a finite Galois extension Let $G=\text{Gal}(K\,/\,k)$. By the
discussion in Appendix A, it suffices to show that for all $\sigma\in G$ we
have $\sigma^{*}(\mathcal{W}_{K})=\mathcal{W}_{K}$ as parabolic subbundles of
$\mathcal{V}|_{C_{K}}$. From the formula for degree in Definition 2.7, it is
easy to see that parabolic degree is preserved by elements of the Galois
group. Since the rank is also preserved, $\sigma^{*}(\mathcal{W}_{K})$ is
still the unique maximal subbundle with maximal slope. Therefore
$\mathcal{W}_{K}$ descends.

  2. C 2

$K=k\left(a^{\frac{1}{p}}\right)$ for some $a\in k$ Let Der be the set of
$k$-derivations of $K$. For each $D\in\text{Der}$, we can extend $D$ to a
$\mathcal{O}_{C}$-linear derivation $D_{C}$ of $\mathcal{O}_{C_{K}}$. This
uniquely defines an $\mathcal{O}_{C}$-linear endomorphism of each
$\mathcal{E}_{i}^{(m)}|_{C_{K}}=\mathcal{E}_{i}^{(m)}\otimes\mathcal{O}_{C_{K}}$
by acting on the second component of the tensor product. This endomorphism
preserves the inclusion relations among the $\mathcal{E}_{i}^{(m)}$s, so it
gives a $\mathcal{O}_{C}$-linear endomorphism $D_{\mathcal{V}}$ of
$\mathcal{V}$. This describes the canonical descent data of
$\mathcal{V}|_{C_{K}}$ viewed as a diagram of quasicoherent sheaves, as
explained in Appendix A. By the discussion in Appendix A, it follows that
$\mathcal{W}_{K}\subset\mathcal{V}|_{C_{K}}$ descends to a parabolic subbundle
of $\mathcal{V}$ if and only if
$D_{\mathcal{V}}(\mathcal{W}_{K})\subset\mathcal{W}_{K}$ for all
$D\in\text{Der}$. This is equivalent to the composition
$\phi:\mathcal{W}_{K}\xrightarrow{D_{\mathcal{V}}}\mathcal{V}|_{C_{K}}\twoheadrightarrow\mathcal{V}|_{C_{K}}\,/\,\mathcal{W}_{K}$
being $0$. But, by the Leibniz rule, this composition is actually
$\mathcal{O}_{C_{K}}$-linear (because we are killing elements of
$\mathcal{W}_{K}$). Hence $\phi$ is a genuine morphism of parabolic vector
bundles. Since $\mathcal{W}_{K}$ is the maximal destabilizing sheaf, part (c)
of Lemma 2.17 implies that
$\text{Hom}\left(\mathcal{W}_{K},\,\mathcal{V}|_{C_{K}}\,/\;\mathcal{W}_{K}\right)=0$.
Therefore we must have $\phi=0$, as desired.

  3. C 3

$K=k(t)$ The same trick with the Quot scheme allows us to spread
$\mathcal{W}_{K}$ to a parabolic subbundle $\mathcal{W}_{U}$ of
$\mathcal{V}|_{C\times U}$, where $U$ is an open subset of
$\mathbb{A}^{1}_{k}$. We can first extend as a set of chains of vector
bundles. Then using opennes of the flat locus [Sta, Tag 0399] we can decrease
$U$ to ensure that all quotients in the chain are $U$-flat. Let $t\in U$ be a
closed point of $U$. Consider the parabolic subbundle
$\mathcal{W}_{U}|_{C_{t}}$ of $\mathcal{V}|_{C_{t}}$. By local constancy of
degree (Lemma 2.29), $\mathcal{W}_{U}|_{C_{t}}\subset\mathcal{V}|_{C_{t}}$
implies that $\mathcal{V}|_{C_{t}}$ is not semistable. Since the residue field
$\kappa(t)$ is finite over $k$, we are reduced to the case of finite field
extensions as in Cases 1,2 above.

For the second part of the proposition, suppose that
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=\mathcal{V}$
is the Harder-Narasimhan filtration of $\mathcal{V}$. The first part of the
proposition implies that
$\mathcal{W}_{j}|_{C_{K}}\,/\,\mathcal{W}_{j-1}|_{C_{K}}$ is semistable for
all $1\leq j\leq l$. Since slopes are preserved by field extensions, we
conclude that
$0=\mathcal{W}_{0}|_{C_{K}}\subset\mathcal{W}_{1}|_{C_{K}}\subset\cdots\,\subset\mathcal{W}_{l-1}|_{C_{K}}\subset\mathcal{W}_{l}|_{C_{K}}=\mathcal{V}|_{C_{K}}$
is a Harder-Narasimhan filtration of $\mathcal{V}|_{C_{K}}$. ∎

Next, we show that Harder-Narasimhan filtrations have no nontrivial first
order deformations. We denote by $k[\epsilon]\vcentcolon=k[T]\,/\,(T^{2})$ the
ring of dual numbers. We denote by
$\sigma:\text{Spec}(k)\rightarrow\text{Spec}\;k[\epsilon]$ the unique section
of the structure map $p:\text{Spec}\;k[\epsilon]\rightarrow\text{Spec}(k)$.

######  Proposition 2.31.

Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Suppose that we are
given a filtration
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l-1}\subset\mathcal{W}_{l}=p^{*}\mathcal{V}$
of $p^{*}\mathcal{V}$ by parabolic subbundles. Assume that the pulled-back
filtration
$0=\sigma^{*}\mathcal{W}_{0}\subset\sigma^{*}\mathcal{W}_{1}\subset\cdots\,\subset\sigma^{*}\mathcal{W}_{l-1}\subset\sigma^{*}\mathcal{W}_{l}=\mathcal{V}$
over the closed point is the Harder-Narasimhan filtration of $\mathcal{V}$.
Then $\mathcal{W}_{j}=p^{*}\sigma^{*}\mathcal{W}_{j}$ for all $1\leq j\leq
l-1$.

###### Proof.

By induction on the length $l$ of the filtration, we are reduced to showing
that $\mathcal{W}_{1}=p^{*}\sigma^{*}\,\mathcal{W}_{1}$. Let us show that
$\mathcal{W}_{1}\subset p^{*}\sigma^{*}\,\mathcal{W}_{1}$. Observe that
$p^{*}\sigma^{*}\,\mathcal{W}_{1}$ is a parabolic subbundle of
$p^{*}\mathcal{V}$. It suffices to prove that the composition
$f:\,\mathcal{W}_{1}\,\hookrightarrow\,p^{*}\mathcal{V}\,\twoheadrightarrow\,p^{*}\mathcal{V}\,/\,p^{*}\sigma^{*}\,\mathcal{W}_{1}$
is $0$. Restricting to the closed point gives us the map
$\sigma^{*}f:\,\sigma^{*}\mathcal{W}_{1}\,\hookrightarrow\,\mathcal{V}\,\twoheadrightarrow\,\mathcal{V}\,/\,\sigma^{*}\,\mathcal{W}_{1}$,
which is clearly $0$. Therefore, basic deformation theory tells us that
$f=\epsilon\,\psi$ for some homomorphism of parabolic bundles
$\psi\in\text{Hom}\left(\,\sigma^{*}\mathcal{W}_{1},\;\mathcal{V}\,/\,\sigma^{*}\mathcal{W}_{1}\,\right)$.
But note that, by assumption, $\sigma^{*}\mathcal{W}_{1}$ is the maximal
destabilizing parabolic subbundle of $\mathcal{V}$. Hence part (c) of Lemma
2.17 implies that
$\text{Hom}\left(\,\sigma^{*}\mathcal{W}_{1},\;\mathcal{V}\,/\,\sigma^{*}\mathcal{W}_{1}\,\right)=0$.
We conclude that $f=0$.

The same reasoning shows that
$p^{*}\sigma^{*}\,\mathcal{W}_{1}\subset\mathcal{W}_{1}$, thus concluding the
proof. ∎

##  3 Stacks of parabolic vector bundles

###  3.1 The group scheme $\text{GL}\left(\mathcal{V}\right)$

Let $J$ be a small category and let $X$ be a scheme. A diagram of sheaves on
$X$ of shape $J$ is a functor $F$ from $J$ into the abelian category of
quasicoherent sheaves $\text{QCoh}(X)$. Diagrams of sheaves on $X$ of shape
$J$ form an abelian category, where the morphisms are natural transformations.
Given a morphism of schemes $f:Y\rightarrow X$ and a diagram of sheaves
$F:J\rightarrow\text{QCoh}(X)$, the pullback $f^{*}F$ is defined to be the
diagram of sheaves on $Y$ obtained by postcomposing $F$ with the functor
$f^{*}:\text{QCoh}(X)\rightarrow\text{QCoh}(Y)$. As usual, we will write
$F|_{Y}$ instead of $f^{*}F$ when the morphism $f$ is implicitly understood.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C$. We can interpret $\mathcal{V}$ as a
diagram of sheaves on $C$. The source category $J_{\mathcal{V}}$ can be
described as follows. $J_{\mathcal{V}}$ has $1+\sum_{i\in I}N_{i}$ objects.
There is a initial object $X^{(0)}$, and the rest of the objects can be
labeled $X_{i}^{(m)}$ for $i\in I$ and $1\leq m\leq N_{i}$. For each $i\in I$
and $1\leq m_{1}\leq m_{2}\leq N_{i}$, there is a unique morphism
$X_{i}^{(m_{1})}\rightarrow X_{i}^{(m_{2})}$. These, along with the initial
morphims with source $X^{(0)}$, account for all morphisms in the category.
Here is a picture of $J_{\mathcal{V}}$ (we fix an enumeration
$I=\\{i_{1},i_{2},\cdots,i_{k}\\}$ for convenience):

${X_{i_{1}}^{(1)}}$${X_{i_{1}}^{(2)}}$${\cdots}$${X_{i_{1}}^{(N_{i_{1}}-1)}}$${X_{i_{1}}^{(N_{i_{1}})}}$${X^{(0)}}$${X_{i_{2}}^{(1)}}$${X_{i_{2}}^{(2)}}$${\cdots}$${X_{i_{1}}^{(N_{i_{2}}-1)}}$${X_{i_{2}}^{(N_{i_{2}})}}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${X_{i_{k}}^{(1)}}$${X_{i_{k}}^{(2)}}$${\cdots}$${X_{i_{k}}^{(N_{i_{k}}-1)}}$${X_{i_{k}}^{(N_{i_{k}})}}$

This description is compatible with some of the definitions given in Section
2. For example, morphisms of parabolic vector bundles are the same as
morphisms of diagrams of sheaves. Fixing the chain lengths amounts to fixing
the shape $J$ of the diagram.

######  Definition 3.1.

Let $\mathcal{V}$ a parabolic vector bundle as above. Denote by
$\text{GL}(\mathcal{V})$ the contravariant functor from the category of flat
$C$-schemes into Set given as follows. For $f:T\rightarrow C$ a flat
$C$-scheme, we set
$\text{GL}(\mathcal{V})\,(T)\vcentcolon=\text{Aut}(f^{*}\,\mathcal{V})$. Here
the pullbacks and automorphisms are to be interpreted in the category of
diagrams of sheaves of shape $J_{\mathcal{V}}$, as described in the paragraphs
above.

######  Proposition 3.2.

$\text{GL}(\mathcal{V})$ is represented by a smooth group scheme over $C$.

###### Proof.

By descent for morphisms of quasicoherent sheaves, we know that
$\text{GL}(\mathcal{V})$ is a sheaf in the Zariski topology. In particular, it
suffices to check the claim after passing to a Zariski cover of $C$. Let
$\bigsqcup_{j\in J}U_{j}\rightarrow C$ be an open cover of $C$ such that for
all $j\in J$ we have that
$\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus n}$. It suffices
to show that the automorphism functor $\text{GL}(\mathcal{V}|_{U_{j}})$ of the
diagram of sheaves $\mathcal{V}|_{U_{j}}$ is a smooth group scheme over
$U_{j}$ for any $j\in J$. If $U_{j}$ does not contain any of the points of
degeneration $\\{x_{i}\\}_{i\in I}$, then we have

$\text{GL}(\mathcal{V}|_{U_{j}})\,\cong\,\text{GL}(\mathcal{E}^{(0)}|_{U_{j}})\,\cong\,\text{GL}(\mathcal{O}_{U_{j}}^{\oplus
n})\,\cong\,\text{GL}_{n}\times U_{j},$

where $n=\text{rank}\,\mathcal{V}$. (Here we are abusing notation and writing
$\mathcal{E}^{(0)}$ to denote the constant diagram with identities as
morphisms). Hence we are done in this case.

Suppose that $U_{j}$ contains some of the points of degeneration
$\\{x_{i}\\}_{i\in I}$. By passing to a finer Zariski cover, we can assume
that $U_{j}$ contains only one such point of degeneration. Fix $i\in I$ such
that $x_{i}\in U_{j}$. For all indexes $s\neq i$, the inclusions in the $s$th
chain of vector bundles
$\mathcal{E}^{(0)}|_{U_{j}}\,\overset{a_{s}^{(1)}}{\subset}\,\mathcal{E}_{s}^{(1)}|_{U_{j}}\overset{a_{s}^{(2)}}{\subset}\cdots\;\overset{a_{s}^{(N_{s})}}{\subset}\,\mathcal{E}_{s}^{(N_{s})}|_{U_{j}}=\mathcal{E}^{(0)}(x_{s})|_{U_{j}}$
become isomorphisms. We conclude that the functor of automorphisms of the
diagram $\mathcal{V}|_{U_{j}}$ is the same of the functor of automorphisms
preserving the single chain of vector bundles

$\mathcal{E}^{(0)}|_{U_{j}}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{U_{j}}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{s})}|_{U_{j}}=\mathcal{E}^{(0)}(x_{i})|_{U_{j}}.$

By making $U_{j}$ smaller, we can assume that $U_{j}$ is the spectrum of a
Dedekind domain $R$ over $k$ and that the Cartier divisor $x_{i}$ is given by
the vanishing of a nonzerodivisor $t\in R$. We can also assume that
$\mathcal{E}_{i}^{(m)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus n}$ for all
$1\leq m\leq N_{i}$. Now $\mathcal{E}^{(0)}(x_{i})|_{U_{j}}$ is a free
$R$-module of rank $n$, let’s denote it by $F$. Similarly, for $1\leq m\leq
N_{i}-1$, let’s write $F_{m}$ for the free $R$-module
$\mathcal{E}^{(m)}_{i}|_{U_{j}}$. Our chain of vector bundles is given by a
set of inclusions
$t\,F\,\subset\,F_{1}\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\,\subset\,F$.

Observe that all maps in this chain become isomorphisms after inverting $t$.
Hence we get a collection of canonical isomorphisms $F_{l}|_{U_{j}\setminus
x_{i}}\xrightarrow{\sim}F|_{U_{j}\setminus x_{i}}$ for each $1\leq l\leq
N_{i}-1$. Consider the smooth group scheme $G$ over $R$ given by
$G\vcentcolon=\text{GL}(t\,F)\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F_{l})\,\times\text{GL}(F)$.
We have a diagonal morphism $\overline{\Delta}:\text{GL}(F)|_{U_{j}\setminus
x_{i}}\rightarrow G|_{U_{j}\setminus x_{i}}$ obtained by using the
aforementioned canonical isomorphisms $F_{l}|_{U_{j}\setminus
x_{i}}\xrightarrow{\sim}F|_{U_{j}\setminus x_{i}}$. To be precise,
$\overline{\Delta}$ is given by the composition

$\displaystyle\text{GL}(F)|_{U_{j}\setminus x_{i}}\,\xrightarrow{\Delta}$
$\displaystyle\;\text{GL}(F)|_{U_{j}\setminus
x_{i}}\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F)|_{U_{j}\setminus
x_{i}}\,\times\text{GL}(F)|_{U_{j}\setminus x_{i}}$
$\displaystyle\xrightarrow{\sim}$
$\displaystyle\;\text{GL}(t\,F)|_{U_{j}\setminus
x_{i}}\times\,\prod_{l=1}^{N_{i}-1}\text{GL}(F_{l})|_{U_{j}\setminus
x_{i}}\,\times\text{GL}(F)|_{U_{j}\setminus x_{i}}$

Let $S$ be a flat $R$-algebra. Automorphisms of the diagram of sheaves
$\mathcal{V}|_{\text{Spec}(S)}$ are the same as automorphisms the chain of
free modules $t\,F\otimes S\,\subset\,F_{1}\otimes
S\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\otimes S\,\subset\,F\otimes S$. By
$R$-flatness of $S$, we know that $t$ is still a nonzerodivisor in $S$. So the
map $S\rightarrow S[\frac{1}{t}]$ is injective. By unraveling the definition,
an automorphism of the chain of modules above is an element of $G(S)$ such
that its localization in $G(S[\frac{1}{t}])=G|_{U_{j}\setminus
x_{i}}(S[\frac{1}{t}])$ belongs to the image of the diagonal map
$\overline{\Delta}(S[\frac{1}{t}])$. These are exactly the $S$-points of the
scheme theoretic closure of the composition $D:\,\text{GL}(F)|_{U_{j}\setminus
x_{i}}\,\xrightarrow{\Delta}\,G|_{U_{j}\setminus x_{i}}\,\hookrightarrow\,G$.
This scheme theoretic closure $\text{Im}(D)$ must be $R$-flat, because it is
the Spec of a subalgebra of the torsion-free $R$-module
$\mathcal{O}_{\text{GL}(F)|_{U_{j}\setminus x_{i}}}$. We conclude that the
functor $\text{GL}(\mathcal{V})|_{U_{j}}$ is represented by an affine flat
$U_{j}$-scheme. We are left to show smoothness.

$\text{Im}(D)$ is of finite type, because it is a closed subscheme of $G$.
Also, we have seen that $\text{GL}(\mathcal{V})|_{U_{j}\setminus
x_{i}}\,\cong\,\text{GL}_{n}\times(U_{j}\setminus x_{i})$. We are left to show
that the fiber $\text{Im}(D)|_{x_{i}}$ is smooth. Since taking scheme
theoretic image of quasicompact maps commmutes with flat base-change, we can
pass to the completion of the local ring at $x_{i}$ without altering the fiber
$\text{Im}(D)|_{x_{i}}$. Therefore we can assume that $R$ is a complete
discrete valuation with residue field $k$. By the Cohen structure theorem, we
have $R=k[[{t}]]$.

Now we have put ourselves in a familiar situation, since the scheme theoretic
closure is by definition the parahoric group scheme of $\text{GL}_{n}$ defined
over $k[[{t}]]$ that preserves the chain of lattices
$t\,F\,\subset\,F_{1}\,\subset\,\cdots\,\subset\,F_{N_{i}-1}\,\,\subset\,F$.
This group scheme is known to be smooth over $R$, see [BT84b, §3]. ∎

######  Remark 3.3.

We have seen in the proof above that $\text{GL}(\mathcal{V})$ is generically
isomorphic to the constant group scheme $\text{GL}_{n}$ (via choosing a
trivialization of $\mathcal{E}^{(0)}$ at the generic point). At the finitely
many points of degeneration $\\{x_{i}\\}_{i\in I}$, the base-change to the
completion of the local ring
$\text{GL}(\mathcal{V})|_{\widehat{\mathcal{O}}_{C,\,x_{i}}}$ is a parahoric
group scheme of $\text{GL}_{n}$ in the sense of Bruhat-Tits [BT84a, Def.
5.2.6]. $\text{GL}(\mathcal{V})$ is a special case of what is sometimes called
a parahoric Bruhat-Tits group scheme in the literature, see [PR10], [BS15],
[Hei10].

We briefly recall the definition of the moduli stack of torsors for a group
scheme such as $\text{GL}(\mathcal{V})$ above. Let $X$ be a scheme. Let
$\mathcal{G}\rightarrow X$ be a relatively affine group scheme that is smooth
over $X$. Let $\mathcal{P}$ be a $X$-scheme. A right $\mathcal{G}$-action on
$\mathcal{P}$ consists of a map of $X$\- schemes
$a:\,\mathcal{P}\times_{X}\mathcal{G}\rightarrow\mathcal{P}$ such that for all
$X$-schemes $T$ the map induced on $T$-points
$a(T):\,\mathcal{P}(T)\times\mathcal{G}(T)\rightarrow\mathcal{P}(T)$ is a
right action of $\mathcal{G}(T)$ on the set $\mathcal{P}(T)$. A (right)
$\mathcal{G}$-torsor over $X$ is a scheme $\pi:\mathcal{P}\rightarrow X$
together with a right $\mathcal{G}$-action that makes $\mathcal{P}$ a
$\mathcal{G}$-torsor in the ètale topology [Sta, Tag 0497]. This means that
there is an ètale cover $T\rightarrow X$ such that the base-change $\pi\times
id_{T}:\mathcal{P}\times_{X}T\rightarrow T$ is $\mathcal{G}$-equivariantly
isomorphic to the trivial $\mathcal{G}$-torsor $\mathcal{G}\times_{X}T$. An
isomorphism between $\mathcal{G}$-torsors is an isomorphism of $X$-schemes
that intertwines the $\mathcal{G}$-actions.

Let $\mathcal{G}$ be a relatively affine smooth group scheme over $C$.

######  Definition 3.4.

The moduli stack of $\mathcal{G}$-torsors $\text{Bun}_{\mathcal{G}}(C)$ is
defined to be the pseudofunctor from $k$-schemes into groupoids given as
follows. For every $k$-scheme $T$, we let

$\text{Bun}_{\mathcal{G}}(C)\,(T)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of $\mathcal{G}\times T$-torsors over $C\times T$}\end{matrix}\right\\}$

See Proposition 1 in [Hei10] for a proof of the following using the Artin
criteria.

######  Proposition 3.5.

$\text{Bun}_{\mathcal{G}}$ is a smooth algebraic stack over $k$. ∎

###  3.2 Moduli stacks of parabolic vector bundles

Fix a parabolic vector bundle $\mathcal{V}$ over $C$. Let us define the stack
of parabolic vector bundles of type $\mathcal{V}$.

######  Definition 3.6.

We define the pseudofunctor $\text{Bun}_{\mathcal{V}}$ from $k$-schemes into
groupoids as follows. For any $k$-scheme $S$, we set

$\text{Bun}_{\mathcal{V}}(S)\;\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid
of parabolic vector bundles $\mathcal{W}$}\\\ \text{of type $\mathcal{V}$ over
$C\times S$}\end{matrix}\right\\}$

For any morphism $f:S\rightarrow T$ of $k$-schemes, we set
$\text{Bun}_{\mathcal{V}}(f)$ to be the pullback $f^{*}$ of parabolic vector
bundles.

For the next proposition, we write $\text{Bun}_{\text{GL}(\mathcal{V})}(C)$ to
denote the moduli stack of torsors of $\text{GL}(\mathcal{V})$ over $C$. See
Definition 3.4 and [Hei10] Example (2) pg. 500.

######  Proposition 3.7.

There is an equivalence of pseudofunctors
$\text{Bun}_{\text{GL}(\mathcal{V})}(C)\xrightarrow{\sim}\text{Bun}_{\mathcal{V}}$.

###### Proof.

Suppose that
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Recall that the group scheme $\text{GL}(\mathcal{V})$ acts on
$\mathcal{V}$ by automorphisms of diagrams of sheaves. Let $T$ be a
$k$-scheme. Let $\mathcal{P}$ be a $\text{GL}(\mathcal{V})\times T$-torsor
over $C\times T$. We can use the natural action of $\text{GL}(\mathcal{V})$ on
$\mathcal{V}$ in order to define an associated parabolic vector bundle
$\mathcal{P}\times_{\text{GL}(\mathcal{V})\times T}\,\mathcal{V}|_{C\times T}$
on $C\times T$. Let us describe how this is defined. Let $f:S\rightarrow
C\times T$ be an ètale cover of $C$ such that $f^{*}\mathcal{P}$ is isomorphic
to the trivial torsor. Choosing such an isomorphims, the descent datum for
$\mathcal{P}$ is given by an element
$h\in\text{GL}(\mathcal{V})(S\times_{C\times T}S)$. Here $h$ is viewed as an
automorphism of the trivial torsor $\text{GL}(\mathcal{V})|_{S}$ via left
multiplication. Now $h$ gives us an automorphims of
$\mathcal{V}|_{S\times_{C\times T}S}$, which we can interpret as a descent
datum for the diagram of sheaves $\mathcal{V}|_{S}$ relative to the cover
$f:S\rightarrow C\times T$. We can use effectivity of ètale descent for
quasicoherent sheaves in order to obtain a diagram of sheaves
$\mathcal{P}\times_{\text{GL}(\mathcal{V})\times T}\,\mathcal{V}|_{C\times T}$
such that its canonical descent datum is given by $h$. Recall that being a
vector bundle of a specified degree is a property of quasicoherent sheaves
that can be checked after passing to a flat cover. We can use this fact to
conclude that $\mathcal{P}\times_{\text{GL}(\mathcal{V})\times
T}\,\mathcal{V}|_{C\times T}$ is a parabolic vector bundle of type
$\mathcal{V}$ over $C\times T$.

Let us now define a functor
$F:\text{Bun}_{\text{GL}(\mathcal{V})}(C)\rightarrow\text{Bun}_{\mathcal{V}}$
as follows. For every $k$ scheme $T$ and for every
$\text{GL}(\mathcal{V})\times T$-torsor $\mathcal{P}$ on $C\times T$, we set
$F(T)\,(\mathcal{P})\vcentcolon=\mathcal{P}\times_{\text{GL}(\mathcal{V})\times
T}\mathcal{V}|_{C\times T}$.

We want to define a quasi-inverse functor $G$. Let $T$ be a $k$-scheme.
Suppose that we are given a parabolic vector bundle $\mathcal{W}$ of type
$\mathcal{V}$ over $C\times T$. We can define a presheaf of sets
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ on the category of flat $C\times
T$-schemes as follows. For every scheme $f:S\rightarrow C\times T$ that is
flat over $C\times T$, we define
$\text{Iso}(\mathcal{V},\,\mathcal{W})\,(S)\vcentcolon=\text{Iso}(f^{*}(\mathcal{V}|_{C\times
T}),\,f^{*}\mathcal{W})$. Here the isomorphisms are taken in the category of
diagrams of sheaves on $S$. It is clear by definition that
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ admits a right action of
$\text{GL}(\mathcal{V})$ given by precomposition.

We claim that $\text{Iso}(\mathcal{V},\,\mathcal{W})$ is in fact a
$\text{GL}(\mathcal{V})\times T$-torsor over $C\times T$. By ètale descent for
morphisms of quasicoherent sheaves, we know that
$\text{Iso}(\mathcal{V},\,\mathcal{W})$ is a sheaf in the ètale topology.
Therefore, in order to prove the claim it suffices to show that there is an
ètale cover $f:S\rightarrow C\times T$ such that $f^{*}\mathcal{W}\cong
f^{*}(\mathcal{V}|_{C\times T})$ as diagrams of sheaves.

Say
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $\bigsqcup_{j\in J}U_{j}\rightarrow C\times T$ be an open cover of
$C\times T$ such that for all $j\in J$ we have
$\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U}^{\oplus
n}\cong\mathcal{F}^{(0)}|_{U_{j}}$. By refining the cover, we can furthermore
assume that each $U_{j}$ intersects at most one of the divisors $x_{i}\times
T$ of $C\times T$.

Let’s focus our attention on a single open subset $U_{j}$. If $U_{j}$ does not
intersect any of the divisors $x_{i}\times T$, then we have a chain of
isomorphisms

$\mathcal{V}|_{U_{j}}\cong\mathcal{E}^{(0)}|_{U_{j}}\cong\mathcal{O}_{U_{j}}^{\oplus
n}\cong\mathcal{F}_{i}^{(0)}|_{U_{j}}\cong\mathcal{W}|_{U_{j}}$

(Here we are abusing notation for the intermediate terms; they are constant
diagrams of vector bundles with identities as morphisms). Hence we are done in
this case.

Suppose now that $U_{j}$ intersects one of the divisors $x_{i}\times T$. By
construction, it only intersects one such divisor. Fix $i\in I$ such that
$U_{j}\,\cap\,(x_{i}\times T)\neq\emptyset$. Let us denote by
$q_{i}:x_{i}\rightarrow C$ the closed immersion of the point $x_{i}$ into $C$.
Let $\pi:U_{j}\rightarrow T$ be the structure morphism. Let $W$ denote the
open subset of $T$ given by the inverse image of $U_{j}$ under the section
$q_{i}\times Id_{T}:\,T\rightarrow C\times T$. Define
$V\vcentcolon=\pi^{-1}(W)$. We can cover $U_{j}=V\,\bigcup\,(U_{j}\setminus
x_{i})$. Since $U_{j}\setminus x_{i}$ does not intersect any of the divisors
of degeneration, we have seen above that

$\mathcal{V}|_{U_{j}\setminus x_{i}}\cong\mathcal{E}^{(0)}|_{U_{j}\setminus
x_{i}}\cong\mathcal{O}_{U_{j}\setminus x_{i}}^{\oplus
n}\cong\mathcal{F}_{i}^{(0)}|_{U_{j}\setminus
x_{i}}\cong\mathcal{W}|_{U_{j}\setminus x_{i}}$

It remains to check the claim on $V$. Let $D\hookrightarrow V$ be the divisor
given by the intersection $V\,\cap\,(x_{i}\times T)$. By construction, we have
a structure morphims $\pi:V\rightarrow W$ and $D$ is given by a section
$s:W\rightarrow V$ of $\pi$. Since $V$ does not intersect any of the other
divisors of degeneration, for all $u\neq i$ the chain of inclusions with index
$u$ gives us canonical identifications
$\mathcal{E}_{u}^{(m)}|_{V}\cong\mathcal{E}^{(0)}|_{V}$ for all $m$.
Similarly, there are canonical identifications
$\mathcal{F}^{(m)}_{u}|_{V}\cong\mathcal{F}^{(0)}|_{V}$ for $u\neq i$. By
choosing isomorphisms
$\mathcal{E}^{(0)}|_{V}\cong\mathcal{F}^{(0)}|_{V}\cong\mathcal{O}_{V}^{\oplus
n}$ we get compatible isomorphisms
$\mathcal{E}_{u}^{(m)}|_{V}\cong\mathcal{F}_{u}^{(m)}|_{V}\cong\mathcal{O}_{V}^{\oplus
n}$ for all indexes $u\neq i$ and $m$ via the aforementioned canonical
identifications. Under these identifications, we see that $\mathcal{V}|_{V}$
and $\mathcal{W}|_{V}$ are determined by the following two chains of vector
bundles on $V$ respectively:

$\mathcal{O}_{V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{V}(D)^{\oplus
n}$ $\mathcal{O}_{V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}|_{V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{V}(D)^{\oplus
n}$

The claim is reduced to the problem of finding (maybe after passing to an
ètale cover of $V$) an automorphism $\text{GL}_{n}(V)$ of
$\mathcal{O}_{V}^{\oplus n}$ that sends one chain to the other. Consider the
corresponding flags of vector bundles on $W$

$0\,\overset{a_{i}^{(1)}}{\subset}\;s^{*}\,\left(\mathcal{E}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
$0\,\overset{a_{i}^{(1)}}{\subset}\;s^{*}\,\left(\mathcal{F}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{M_{l}}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$

These two flags represent two $W$-points of the partial flag variety
$\text{GL}_{n}\,/\,P$, where $P$ is the standard parabolic corresponding to
the partition $(a_{i}^{(m)})_{m}$ of $n$. Let us denote these points by
$p_{1},p_{2}\in(\text{GL}_{n}\,/\,P)(W)$. We know that the quotient
$\text{GL}_{n}\,/\,P$ represents the ètale sheafification of the naive coset
functor. So there is an ètale cover $w:Y\rightarrow W$ and an element
$g\in\text{GL}_{n}(Y)$ such that $g\cdot w^{*}p_{1}=w^{*}p_{2}$. In other
words, the element $g$ relates the following two flags:

$0\,\overset{a_{i}^{(1)}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{E}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$
$0\,\overset{a_{i}^{(1)}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{F}_{i}^{(1)}|_{V}\,/\,\mathcal{O}_{V}^{\oplus
n}\right)\;\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\;w^{*}\,s^{*}\,\left(\mathcal{O}_{V}(D)^{\oplus
n}\,/\,\mathcal{O}_{V}^{\oplus n}\right)$

The base-change $pr_{2}:Y\times_{W}V\rightarrow V$ is an ètale cover. We have
a structure morphism $pr_{1}:Y\times_{W}V\rightarrow Y$. By construction, the
automorphism $pr_{1}^{*}\,g$ of $\mathcal{O}_{Y\times_{W}V}^{\oplus n}$
relates the two chains of vector bundles over $Y\times_{W}V$

$\mathcal{O}_{Y\times_{W}V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}|_{Y\times_{W}V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{Y\times_{W}V}(D)^{\oplus
n}$ $\mathcal{O}_{Y\times_{W}V}^{\oplus
n}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}|_{Y\times_{W}V}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\mathcal{O}_{Y\times_{W}V}(D)^{\oplus
n}$

We conclude that
$\mathcal{W}|_{Y\times_{W}V}\cong\mathcal{V}|_{Y\times_{W}V}$, and the claim
is proven.

We can now define a quasi-inverse to $F$. We define
$G:\text{Bun}_{\mathcal{V}}\rightarrow\text{Bun}_{\text{GL}(\mathcal{V})}(C)$
as follows. For every $k$-scheme $T$ and every parabolic vector bundle
$\mathcal{W}$ of type $\mathcal{V}$ over $C\times T$, we let
$G(T)\,(\mathcal{W})\vcentcolon=\text{Iso}(\mathcal{V},\,\mathcal{W})$.

Let us prove that $F$ and $G$ are quasi-inverse functors. Fix a $k$-scheme
$T$. By construction, we have a canonical identification
$FG(T)\,(\mathcal{V}|_{C\times T})\cong\mathcal{V}|_{C\times T}$. It is also
clear by construction that under this canonical identification we have
$FG(T)\,(g)=g$ for any automorphism $g\in\text{Aut}(\mathcal{V}|_{C\times
T})$. This remains true for $\mathcal{V}|_{S}$, where $S$ is any ètale cover
of $C\times T$. Technically our functor $G$ is not defined for such $S$, but
the construction using the scheme
$\text{Iso}(\mathcal{V}|_{S},\,\mathcal{V}|_{S})$ still makes sense. So we
abuse notation and write $G$ for such construction.

Let $\mathcal{W}$ be any parabolic vector bundle of type $\mathcal{V}$ over
$C\times T$. We know that there exists an ètale cover $f:S\rightarrow C\times
T$ such that $f^{*}\mathcal{W}\cong f^{*}(\mathcal{V}|_{C\times T})$.
Therefore, $\mathcal{W}$ can be described by some ètale descent datum. In this
case this is just an element of
$\text{Aut}\left(\mathcal{V}|_{S\times_{C\times T}S}\right)$. We have seen
that we have a canonical isomorphism $FG\,(\mathcal{V}|_{S\times_{C\times
T}S})\cong\mathcal{V}|_{S\times_{C\times T}S}$. Under this canonical
identification, $FG(T)\,(\mathcal{W})$ is described by the same ètale descent
datum with respect to the cover $S\rightarrow C\times T$ (since we said that
$FG$ induces the identity on morphisms). So we can canonically identify
$\mathcal{W}$ and $FG(T)\,(\mathcal{W})$. The fact that $FG$ induces the
identity on automorphisms of $\mathcal{V}|_{C\times T}$ implies the same for
automorphims of $\mathcal{W}$ under the identification
$FG(T)\,(\mathcal{W})\,\cong\,\mathcal{W}$ we just constructed. The argument
for $GF(T)\cong id_{\text{Bun}_{\text{GL}(\mathcal{V})}(C)\,(T)}$ follows in a
similar manner. ∎

######  Corollary 3.8.

$\text{Bun}_{\mathcal{V}}$ is a smooth algebraic stack over $k$.

###### Proof.

This is an immediate consequence of Proposition 3.7 and Proposition 3.5. ∎

######  Remark 3.9.

It should be remarked that Corollary 3.8 can be done directly by hand. One can
describe a concrete atlas in terms of Quot schemes, just as is usually done
for the moduli stack of vector bundles. We wanted to prove the isomorphism in
Proposition 3.7 in order to show explicitly how the moduli of parabolic vector
bundles fits into the framework of moduli of torsors for Bruhat-Tits group
schemes over $C$.

###  3.3 Harder-Narasimhan strata

Fix a parabolic vector bundle $\mathcal{V}$ of rank $n$ over $C$. We will use
the existence of the Harder-Narasimhan filtration in order to attach
invariants to parabolic bundles of type $\mathcal{V}$. Let us first define the
kind of invariants we will be working with.

######  Definition 3.10.

A parabolic HN datum of rank $n$ is an $n$-tuple $(\mu_{l})_{l=1}^{n}$ of real
numbers $\mu_{l}$ such that $\mu_{l+1}\geq\mu_{l}$ for all $l$.

We can define a partial order on the set of Harder-Narasimhan data of a given
rank.

######  Definition 3.11.

Let $P_{1}=(\mu_{l})_{l=1}^{n}$ and $P_{2}=(\nu_{l})_{l=1}^{n}$ be two
parabolic HN data of rank $n$. We say $P_{1}\leq P_{2}$ if

  1. (1)

$\sum_{l=1}^{n}\mu_{l}=\sum_{l=1}^{n}\nu_{l}$.

  2. (2)

For all $1\leq k<n$, we have $\sum_{l=1}^{k}\mu_{l}\leq\sum_{l=1}^{k}\nu_{l}$.

For any parabolic bundle of type $\mathcal{V}$, we can associate a Harder-
Narasimhan datum.

######  Definition 3.12.

Let $\mathcal{W}$ be a parabolic vector bundle in
$\text{Vect}_{\overline{\lambda}}$. Suppose that it has Harder-Narasimhan
filtration given by
$0=\mathcal{W}_{0}\subset\mathcal{W}_{1}\subset\cdots\,\subset\mathcal{W}_{l}=\mathcal{W}$.
Then, we define the Harder-Narasimhan datum $HN(\mathcal{W})$ associated to
$\mathcal{W}$ to be the parabolic HN datum given by

$\displaystyle
HN(\mathcal{W})=(\;\underbrace{\mu(\mathcal{W}_{1}),\,\cdots,\,\mu(\mathcal{W}_{1})}_{\text{rank}\;\mathcal{W}_{1}/0\,\text{times}},\;\underbrace{\mu(\mathcal{W}_{2}/\,\mathcal{W}_{1}),\,\cdots,\,\mu(\mathcal{W}_{2}/\,\mathcal{W}_{1})}_{\text{rank}\;\mathcal{W}_{2}/\,\mathcal{W}_{1}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{W}_{l}/\,\mathcal{W}_{l-1}),\;\cdots,\;\mu(\mathcal{W}_{l}/\,\mathcal{W}_{l-1})}_{\text{rank}\;\mathcal{W}_{l}/\mathcal{W}_{l-1}\,\text{times}}\;)$

We are now ready to define some relevant subfunctors of
$\text{Bun}_{\mathcal{V}}$.

######  Definition 3.13.

Let $P$ be a HN-datum. We define $\text{Bun}_{\mathcal{V}}^{\leq P}$ to be the
pseudofunctor from $k$-schemes into groupoids given as follows. For every
$k$-scheme $S$,

$\text{Bun}_{\mathcal{V}}^{\leq
P}(S)\;\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of
}\mathcal{W}\in\text{Bun}_{\mathcal{V}}(S)\text{ such that}\\\
HN\left(\mathcal{W}\,|_{C_{s}}\right)\leq P\;\;\text{for all }s\in
S\end{matrix}\right\\}$

######  Remark 3.14.

The locus of semistable bundles of a given degree $d$ is
$\text{Bun}_{\mathcal{V}}^{\leq P}$, where $P$ is the the rank $n$ parabolic
HN datum given by $P=\left(\frac{d}{n},\frac{d}{n},\cdots,\frac{d}{n}\right)$.

Our main goal is to analyze the geometry of these subfunctors. It will turn
out that all $\text{Bun}_{\mathcal{V}}^{\leq P}$s are quasicompact open
substacks of $\text{Bun}_{\mathcal{V}}$. As we vary $P$, they form a
stratification of $\text{Bun}_{\mathcal{V}}$. We will refer to the subfunctors
$\text{Bun}_{\mathcal{V}}^{\leq P}$ collectively as Harder-Narasimhan strata.

##  4 Parabolic Quot Schemes

The main goal of this section is to prove the representability of the two
functors described in Definition 4.9 and Definition 4.16. These can be seen as
parabolic analogues of the Quot scheme and the iterated Quot scheme
respectively. The reader who is only interested in the main theorems described
in the introduction can safely skip this section and refer back to it when
needed.

######  Definition 4.1.

Let $\mathcal{Q}$ be a coherent sheaf on $C$. The Hilbert polynomial
$HP(\mathcal{Q})$ of $\mathcal{Q}$ with respect to $\mathcal{O}(1)$ is the
polynomial with rational coefficients whose value at an integer
$n\in\mathbb{Z}$ is given by
$HP(\mathcal{Q})(n)=\text{dim}_{k}\,H^{0}\left(\mathcal{Q}(n)\right)\,-\,\text{dim}_{k}\,H^{1}\left(\mathcal{Q}(n)\right)$.

######  Remark 4.2.

If $\mathcal{Q}$ is a vector bundle of rank $r$ and degree $d$, Riemann-Roch
tells us that $HP(\mathcal{Q})(n)=(r\cdot d)\,n+r\cdot\chi(\mathcal{O_{C}})$.
So the Hilbert polynomial contains precisely the information of the degree and
the rank of the vector bundle.

Let $S$ be a scheme of finite type over $k$. Let $\mathcal{E}$ be a coherent
sheaf on $C\times S$ that is flat over $S$. Choose a polynomial
$P\in\mathbb{Q}[T]$ with rational coefficients.

######  Definition 4.3.

We define $\text{Quot}_{\mathcal{E}/C\times S/S}^{P}$ to be the contravariant
functor from $S$-schemes to sets given as follows. For any $S$-scheme $T$,
$\text{Quot}_{\mathcal{E}/C\times S/S}^{P}(T)$ is the set of isomorphism
classes of quotients $\mathcal{Q}$ of $\mathcal{E}|_{C\times T}$ such that

  1. (a)

$\mathcal{Q}$ is flat over $T$

  2. (b)

Let $\mathcal{F}$ denote the kernel of $\mathcal{E}|_{C\times
T}\twoheadrightarrow\mathcal{Q}$. For all points $t\in T$, the Hilbert
polynomial of the fiber $\mathcal{F}|_{C_{t}}$ is $P$.

In this context, we have the following classical theorem of Grothendieck. See
[Nit05] for a proof.

######  Theorem 4.4.

$\text{Quot}_{\mathcal{E}/C\times S/S}^{P}$ is represented by a projective
$S$-scheme. ∎

We want to define a variation of the Quot scheme that will serve our needs in
latter sections. The set-up is as follows. We fix a finite set
$\\{x_{i}\\}_{i\in I}$ of $k$-points in $C$. Let $S$ be a scheme of finite
type over $k$. Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$ be a parabolic vector bundle over $C\times S$ with parabolic structure at
$\\{x_{i}\\}_{i\in I}$. We want to describe a functor that classifies
subbundles of $\mathcal{V}$ with some given invariants. Let us first describe
the invariants we will be looking at.

######  Definition 4.5.

A parabolic Quot datum of rank $k$ is a tuple $\left(P,\,b_{i}^{(m)}\right)$
where

  1. (a)

$P\in\mathbb{Q}[T]$ is a polynomial with rational coefficients.

  2. (b)

For each $i\in I$ and $1\leq m\leq N_{i}$, $b_{i}^{(m)}$ is a nonnegative
integer. These are required to satisfy $\sum_{m=1}^{N_{i}}b_{i}^{(m)}=k$ for
all $i\in I$.

We build our parabolic Quot scheme in stages. We start with the following
moduli problem:

######  Definition 4.6.

Let $S$ and $\mathcal{V}$ as above. Fix $P\in\mathbb{Q}[T]$. We denote by
$\text{Quot}_{\mathcal{V}}^{P}$ the subfunctor of
$\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$ given as follows. For a
$S$-scheme $T$, define $\text{Quot}_{\mathcal{V}}^{P}(T)$ to be the set of
isomorphism classes of quasicoherent quotients $\mathcal{Q}^{(0)}$ in
$\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}(T)$ such that for all $i\in
I$ and $0\leq m\leq N_{i}$, the sheaf $\mathcal{L}_{i}^{(m)}$ given as the
pushout of the following diagram is $T$-flat.

$\textstyle{\mathcal{Q}^{(0)}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{i}^{(k)}}$$\textstyle{\mathcal{E}^{(0)}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Remark 4.7.

Since $\mathcal{L}_{i}^{(m)}$ is given as a pushout, its formation commutes
with base-change. This is therefore a well defined functor.

######  Proposition 4.8.

$\text{Quot}_{\mathcal{V}}^{P}$ is a finite disjoint union of locally closed
subschemes of $\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$.

###### Proof.

Let
$\mathcal{Q}^{(0)}_{univ}\in\text{Coh}\left(C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\right)$ be the universal quotient of $\mathcal{E}^{(0)}$ on
$C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. Define
$\mathcal{L}_{i,\,univ}^{(k)}$ to be the pushout diagram:

$\textstyle{\mathcal{Q}^{(0)}_{univ}(x_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{i,\,univ}^{(m)}}$$\textstyle{\mathcal{E}^{(0)}(x_{i})|_{C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\right)|_{C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By definition,
$\text{Quot}_{\mathcal{V}}^{P}\longrightarrow\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}$ is the functor from $S$-schemes into set given as follows. For a
$S$-scheme $T$, we have

$\text{Quot}_{\mathcal{V}}^{P}(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\text{ such that}\\\ (id_{C}\times
f)^{*}\mathcal{L}_{i,\,univ}^{(m)}\;\text{is $T$-flat for all $i$ and
$m$}\end{matrix}\right\\}$

Now we can apply the existence of flattening stratifications (see [Nit05]) to
the projective map $pr_{2}:\,C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times
S/S}^{P}\;\longrightarrow\;\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$
and the coherent sheaves $\mathcal{L}_{i,\,univ}^{(m)}$ on
$C\times\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. We conclude that
this functor is represented by a finite disjoint union of locally closed
subschemes of $\text{Quot}_{\mathcal{E}^{(0)}/C\times S/S}^{P}$. ∎

Suppose that we have $S$, $\mathcal{V}$, $T$ and $\mathcal{Q}^{(0)}$ as in
Definition 4.6. We denote by $\mathcal{F}^{(0)}$ the kernel of the quotient
$\mathcal{E}^{(0)}\longrightarrow\mathcal{Q}^{(0)}$. Furthermore, define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\cap\mathcal{F}^{(0)}(x_{i})$
and
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$.
This means that we have the following commutative diagram with exact columns:

${0}$${0}$${\cdots}$${0}$${0}$${0}$${\mathcal{Q}^{(0)}}$${\mathcal{Q}_{i}^{(1)}}$${\cdots}$${\mathcal{Q}_{i}^{(Ni-2)}}$${\mathcal{Q}_{i}^{(N_{i}-1)}}$${\mathcal{Q}^{(0)}(x_{i})}$${\mathcal{E}^{(0)}}$${\mathcal{E}_{i}^{(1)}}$${\cdots}$${\mathcal{E}_{i}^{(Ni-2)}}$${\mathcal{E}_{i}^{(N_{i}-1)}}$${\mathcal{E}^{(0)}(x_{i})}$${\mathcal{F}^{(0)}}$${\mathcal{F}_{i}^{(1)}}$${\cdots}$${\mathcal{F}_{i}^{(Ni-2)}}$${\mathcal{F}_{i}^{(N_{i}-1)}}$${\mathcal{F}^{(0)}(x_{i})}$${0}$${0}$${\cdots}$${0}$${0}$${0}$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$

By construction, we have a short exact sequence
$0\longrightarrow\,\mathcal{Q}_{i}^{(m)}\,\longrightarrow\,\mathcal{Q}^{(0)}(x_{i})\,\longrightarrow\,\mathcal{L}_{i}^{(m)}\,\longrightarrow
0$. By the long exact sequence of Tors at each stalk, we conclude that
$\mathcal{Q}_{i}^{(m)}$ is $T$-flat. The same reasoning applied to the short
exact sequence
$0\longrightarrow\,\mathcal{F}_{i}^{(m)}\,\longrightarrow\,\mathcal{E}_{i}^{(m)}\,\longrightarrow\,\mathcal{Q}_{i}^{(m)}\,\longrightarrow
0$ shows that $\mathcal{F}_{i}^{(m)}$ is $T$-flat. In addition, the snake
lemma gives us a short exact sequence

$0\longrightarrow\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}\longrightarrow\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}\longrightarrow\mathcal{L}_{i}^{(m)}\longrightarrow
0$

By $T$-flatness of $\mathcal{E}^{(0)}(x_{i})\,/\,\mathcal{E}_{i}^{(m)}$ we
conclude that $\mathcal{F}^{(0)}(x_{i})\,/\,\mathcal{F}_{i}^{(m)}$ is
$T$-flat. Since all of these sheaves are $T$-flat, the formation of all of
$\mathcal{F}_{i}^{(m)}$, $\mathcal{Q}_{i}^{(m)}$,
$\mathcal{F}_{i}^{(m+1)}\,/\,\mathcal{F}_{i}^{(m)}$ and
$\mathcal{Q}_{i}^{(m+1)}\,/\,\mathcal{Q}_{i}^{(m)}$ commutes with base-change
on $T$. Hence the diagram above pulls back under base-change. This in turn
shows that the following is a well-defined subfunctor.

######  Definition 4.9.

Fix $S$ and $\mathcal{V}$ as above. Let $\left(P,\,b_{i}^{(m)}\right)$ a
parabolic Quot datum. We let $\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ be
the subfunctor of $\text{Quot}_{\mathcal{V}}^{P}$ defined as follows. For any
$S$-scheme $T$, $\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)$ is the set of
iso. classes of quotients $\mathcal{Q}^{(0)}\in\text{Quot}_{\mathcal{V}}^{P}$
satisfying

  1. (a)

For all $i\in I$ and $1\leq m\leq N_{i}$,
$\mathcal{F}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m-1)}|_{{x_{i}}\times T}$ is a
vector bundle of rank $b_{i}^{(m)}$.

  2. (b)

$\mathcal{Q}_{i}^{(m)}$ and $\mathcal{F}_{i}^{(m)}$ are vector bundles on
$C\times T$.

Here we define
$\mathcal{F}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times
T}\cap\mathcal{F}^{(0)}(x_{i})$ and
$\mathcal{Q}_{i}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}\,/\,\mathcal{F}_{i}^{(m)}$,
as explained in the paragraphs above.

######  Proposition 4.10.

$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ is an open subscheme of
$\text{Quot}_{\mathcal{V}}^{P}$. We call this the parabolic Quot scheme with
parabolic Quot datum $\left(P,b_{i}^{(m)}\right)$.

Before proving the proposition, let us state a lemma that will be used
repeatedly.

######  Lemma 4.11 (Fiberwise flatness criterion).

Let $B$ be a scheme and let $X$ be a scheme locally of finite presentation
over $B$. Let $\mathcal{F}\in\text{QCoh}(X)$ be a sheaf that is locally of
finite presentation as an $\mathcal{O}_{X}$-module. Suppose that $\mathcal{F}$
is flat over $B$. Let $x\in X$. Set $s=f(x)$. If the fiber
$\mathcal{F}|_{X_{s}}$ is $\mathcal{O}_{X_{s}}$-flat at $x$, then
$\mathcal{F}$ is $\mathcal{O}_{X}$-flat at $x$.

###### Proof.

This follows from [Sta, Tag 039C] by letting $X=Y$, $f=id_{X}$ and $B=S$. ∎

###### Proof of Proposition 4.10.

Let
$\mathcal{Q}^{(0)}_{univ}\in\text{Coh}(C\times\text{Quot}_{\mathcal{V}}^{P})$
be the universal quotient of $\mathcal{E}^{(0)}$ on
$C\times\text{Quot}_{\mathcal{V}}^{P}$. We denote by
$\mathcal{F}^{(0)}_{univ}$ the kernel of the quotient
$\mathcal{E}^{(0)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\longrightarrow\mathcal{Q}^{(0)}_{univ}$.
Furthermore, define
$\mathcal{F}_{i,\,univ}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\,\cap\,\mathcal{F}^{(0)}_{univ}(x_{i})$
and
$\mathcal{Q}_{i,\,univ}^{(m)}\vcentcolon=\mathcal{E}_{i}^{(m)}|_{C\times\text{Quot}_{\mathcal{V}}^{P}}\,/\,\mathcal{F}_{i,\,univ}^{(m)}$.
By definition, the functor
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow\text{Quot}_{\mathcal{V}}^{P}$
is given as follows. For every $S$-scheme $T$,

$\displaystyle\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such
that}\\\ \\\ (1)\quad\text{For all $i$ and $m$ we have that }\;(q_{i}\times
f)^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\\\
\text{is a vector bundle of rank $b_{i}^{(m)}$}\\\ \\\ (2)\quad\text{For all
$i$ and $m$, we have that both}\;(id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}\\\ \text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}\;\text{are $C\times
T$-flat}\end{matrix}\right\\}$

Let us define an intermediate functor $L$ given by imposing only the first
condition. In other words, we let $L$ be the functor from $S$-schemes $T$ into
sets given by

$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $i$, $m$ we have that}\\\ f^{*}\,(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\;\text{is
a vector bundle of rank $b_{i}^{(m)}$}\end{matrix}\right\\}$

By the discussion before Definition 4.9, we already know that $(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is a vector bundle. Hence for any morphism
$f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}$, we have that $f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is a vector bundle. Note that the condition for the rank can be checked at
each point $t\in T$. In other words, $L$ can be rewritten as

$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $i$, $m$ and $t\in T$ we have}\\\
\text{rank}_{\kappa(t)}\,\left(f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{t}\;=\;b_{i}^{(m)}\end{matrix}\right\\}$

But we know that

$\displaystyle\text{rank}_{\kappa(t)}\,\left(f^{*}(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{t}\;=\;\text{rank}_{\kappa(f(t))}\left((q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{f(t)}$

Denote by $U$ the set of points $s\in\text{Quot}_{\mathcal{V}}^{P}$ such that
$\text{rank}_{\kappa(s)}\left((q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}\right)|_{s}\,=\,b_{i}^{(m)}$
for all $i$ and $m$. Then $L$ is given by the set theoretic condition

$L(T)\;=\;\left\\{\begin{matrix}\text{morphisms of
$S$-schemes}\;f:T\rightarrow\text{Quot}_{\mathcal{V}}^{P}\text{ such that}\\\
\\\ (1)\quad\text{For all $t\in T$ we have $f(t)\in U$}\\\
\end{matrix}\right\\}$

Since $(q_{i}\times
id_{\text{Quot}_{\mathcal{V}}^{P}})^{*}\,\mathcal{F}_{i,\,univ}^{(m)}\,/\,\mathcal{F}_{i,\,univ}^{(m-1)}$
is locally free, the rank of the fibers is a locally constant function on
$\text{Quot}_{\mathcal{V}}^{P}$. In particular, we see that $U$ is an open
subset of $\text{Quot}_{\mathcal{V}}^{P}$. It follows that the functor $L$ is
represented by $U$ viewed as an open subscheme of
$\text{Quot}_{\mathcal{V}}^{P}$.

We can now go back to our functor
$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$. We see that it is given by

$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $i$ and $m$, we have that both}\\\ (id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}\;\text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}\;\text{are $C\times
T$-flat}\end{matrix}\right\\}$

By the discussion before Definition 4.9, the sheaves
$\mathcal{F}_{i,\,univ}^{(m)}$ and $\mathcal{Q}_{i,\,univ}^{(m)}$ are
$\text{Quot}_{\mathcal{V}}^{P}$-flat. So for any morphism $f:T\rightarrow U$
the sheaves $(id_{C}\times f)^{*}\mathcal{F}_{i,\,univ}^{(m)}$ and
$(id_{C}\times f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}$ are $T$ flat. Hence we can
apply Lemma 4.11 with $X=C\times T$ and $B=T$ to conclude that it suffices to
check flatness for the fibers $C_{t}$ at each $t\in T$. So our fuctor can be
described as:

$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $i$ and $m$, for every $t\in T$ we have that both}\;\\\ (id_{C}\times
f)^{*}\mathcal{F}_{i,\,univ}^{(m)}|_{C_{t}}\;\text{and}\;(id_{C}\times
f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{t}}\;\text{are $C_{t}$
flat}\end{matrix}\right\\}$

Since flatness can be checked after faithfully flat base-change, we see that
$(id_{C}\times f)^{*}\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{t}}$ is flat if and
only if $\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{f(t)}}$ is flat. The same reasoning
applies to $(id_{C}\times f)^{*}\mathcal{F}_{i,\,univ}^{(m)}|_{C_{t}}$. Let
$V$ be the set of points $s\in U$ such that
$\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{s}}$ and
$\mathcal{F}_{i,\,univ}^{(m)}|_{C_{s}}$ are $C_{s}$-flat for all $i$ and $m$.
Then we have just seen that the functor can described as

$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}(T)\;=\;\left\\{\begin{matrix}\text{morphisms
of $S$-schemes}\;f:T\rightarrow U\text{ such that}\\\ \\\ (2)\quad\text{For
all $t\in T$, we have that $f(t)\in V$}\end{matrix}\right\\}$

We are reduced to check that $V$ is an open subset of $U$. For a fixed choice
of indexes $i$ and $m$, let $V_{i,m}^{\mathcal{Q}}$ be the set of points $s\in
U$ such that $\mathcal{Q}_{i,\,univ}^{(m)}|_{C_{s}}$ is flat. We shall show
that $V_{i,m}^{\mathcal{Q}}$ is open.

Suppose that $s\in V_{i,m}^{\mathcal{Q}}$. The sheaf
$\mathcal{Q}_{i,\,univ}^{(m)}$ is $U$-flat (see the discussion before
Definition 4.9). Hence we can apply Lemma 4.11 to the morphism
$pr_{2}:\,C\times U\,\rightarrow U$ (here $X=C\times U$ and $B=U$) in order to
conclude that $\mathcal{Q}_{i,\,univ}^{(m)}$ is flat at all points $x\in
C\times U$ lying over $s$. Since the flatness locus of
$\mathcal{Q}_{i,\,univ}^{(m)}$ is open in $C\times U$ [Sta, Tag 0399], there
is an open $N\subset C\times U$ containing the fiber $C_{s}$ such that
$\mathcal{Q}_{i,\,univ}^{(m)}|_{N}$ is flat. Since the morphism
$pr_{2}:\,C\times U\,\rightarrow U$ is projective, $pr_{2}$ is in particular
closed. So we can find an open neighborhood $W$ of $s$ in $U$ such that
$(pr_{2})^{-1}(W)\subset N$. We conclude that $W\subset
V_{i,\,m}^{\mathcal{Q}}$, which shows that $V_{i,\,m}^{\mathcal{Q}}$ is open.

The same argument applies verbatim to the locus $V_{i\,m}^{\mathcal{F}}$ of
points $s\in U$ such that $\mathcal{F}_{i,\,univ}^{(m)}|_{C_{s}}$ is flat.
Since
$V=\bigcap_{i,\,m}\left(V_{i\,m}^{\mathcal{F}}\,\cap\,V_{i\,m}^{\mathcal{Q}}\right)$,
we conclude that $V$ is open. ∎

######  Corollary 4.12.

$\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$ is a separated scheme of finite
type over $S$.

###### Proof.

This is an immediate consequence of Theorem 4.4, Proposition 4.8 and
Proposition 4.10. ∎

######  Definition 4.13.

Let $\mathcal{V}$ is a parabolic vector bundle over $C$. Let $\mathcal{W}$ be
a parabolic subbundle of $\mathcal{V}$. There is a unique Quot datum
$\theta(\mathcal{W})$ such that $\mathcal{W}$ is a $k$-point of
$\text{Quot}_{\mathcal{V}}^{\theta(\mathcal{W})}$. We call
$\theta(\mathcal{W})$ the parabolic Quot datum associated to $\mathcal{W}$.
More explicitly, if we have
$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\,\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then
$\theta(\mathcal{W})=\left(HP(\mathcal{F}^{(0)}),\,b_{i}^{(m)}\right)$.

######  Remark 4.14.

  1. (i)

Let $\mathcal{W}$ be a parabolic vector bundle over $C$. Recall that the rank
and degree of a vector bundle can be recovered from its Hilbert polynomial via
Riemann-Roch. By the degree formula for parabolic vector bundles, it follows
that $\mu(\mathcal{W})$ is determined by $\theta(\mathcal{W})$.

  2. (ii)

Let $T$ be a connected $k$-scheme. Let $\mathcal{W}$ be a parabolic vector
bundle over $C\times T$. Since the degree and rank of a vector bundle are
locally constant on families, it follows that $\theta(\mathcal{W}|_{C_{t}})$
is the same for all $t\in T$.

We now want to iterate our construction in order to parametrize filtrations of
a parabolic vector bundle by parabolic subbundles with some given parabolic
Quot data.

######  Definition 4.15.

Let $S$ and $\mathcal{V}$ be as before. A filtration datum of length $l$ is a
pair of tuples
$\alpha=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}1\leq j\leq
l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{1\leq j\leq l-1}\right)$, where

  1. (a)

Each ${}_{j}b_{i}^{(m)}$ is a nonnegative integer.

  2. (b)

Each ${}_{j}P\in\mathbb{Q}[T]$ is a polynomial with rational coefficients.

######  Definition 4.16.

Let $S$ and $\mathcal{V}$ as usual. Fix a filtration datum $\alpha$ of length
$l$. Then, we define the functor $\text{Fil}_{\mathcal{V}}^{\alpha}$ over $S$
as follows. For any $S$-scheme $T$, $\text{Fil}^{\alpha}_{\mathcal{V}}(T)$
will denote the set of iso. classes of filtrations of $\mathcal{V}|_{T\times
C}$ by parabolic subbundles
$0=_{0}\mathcal{V}\,\subset\,_{1}\mathcal{V}\,\subset\,_{2}\mathcal{V}\,\subset\,\cdots\,\subset\,_{l}\mathcal{V}=\mathcal{V}$
such that if
${}_{j}\mathcal{V}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\subset\,_{j}\mathcal{F}_{i}^{(1)}\subset\cdots\;\subset\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then we have:

  1. (a)

For points $t\in T$ and all $1\leq j\leq l-1$, the Hilbert Polynomial of
${}_{j}\mathcal{F}^{(0)}\,|_{C_{t}}$ is ${}_{j}P$.

  2. (b)

For all $1\leq j\leq l-1$, $i\in I$ and $1\leq m\leq N_{i}$, we have that
${}_{j}\mathcal{F}_{i}^{(m)}\,/\,_{j}\mathcal{F}_{i}^{(m-1)}\,|_{T\times{x_{i}}}$
is a vector bundle of rank ${}_{j}b_{i}^{(m)}$.

######  Proposition 4.17.

$\text{Fil}^{\alpha}_{\mathcal{V}}$ is represented by a scheme that is
separated and of finite type over $S$.

###### Proof.

We induct on the length $l$ of the filtration datum. The case $l=1$ is just a
parabolic Quot scheme, so it follows from Corollary 4.12. Suppose that the
theorem has been proved for filtration data of size $l-1$. Let $\alpha$ be as
in the statement of the proposition. We can define
$\beta\vcentcolon=\left(\left({}_{j}b_{i}^{(m)}\right)_{\begin{subarray}{c}2\leq
j\leq l-1\\\ i\in I\\\ 1\leq m\leq
N_{i}\end{subarray}},\;\left({}_{j}P\right)_{2\leq j\leq l-1}\right)$, which
is a filtration datum of length $l-1$. By the induction hypothesis, we know
that $Fil_{\mathcal{V}}^{\beta}$ is a scheme that is separated and of finite
type over $S$.

There is a universal filtration
$0\subset\,\mathcal{W}_{2,\,univ}\,\subset\,\mathcal{W}_{3,\,univ}\,\subset\cdots\,\subset\,\mathcal{W}_{l,\,univ}=\mathcal{V}|_{C\times
Fil^{\beta}_{\mathcal{V}}}$ by parabolic subbundles. Here each
$\mathcal{W}_{j,\,univ}$ has the required associated parabolic data
$\theta(\mathcal{W}_{j,\,univ})=\left({}_{j}P,\,_{j}b_{i}^{(m)}\right)$ for
$2\leq j\leq l-1$. We can define a map $Fil^{\alpha}_{\mathcal{V}}\rightarrow
Fil^{\beta}_{\mathcal{V}}$ by forgetting the first element in the parabolic
filtration. By tracing back the definitions, the functor
$Fil^{\alpha}_{\mathcal{V}}$ is represented by the scheme
$\text{Quot}_{\mathcal{W}_{2,\,univ}}^{{}_{1}P,\,_{1}b_{i}^{(m)}}$. Since we
know that
$\text{Quot}_{\mathcal{W}_{2,\,univ}}^{{}_{1}P,\,_{1}b_{i}^{(m)}}\longrightarrow
Fil^{\beta}_{\mathcal{V}}$ is a separated morphism of finite type, we conclude
that the composition $Fil^{\alpha}_{\mathcal{V}}\longrightarrow
Fil^{\beta}_{\mathcal{V}}\longrightarrow S$is separated and of finite type. ∎

######  Definition 4.18.

Let $\mathcal{V}$ be a parabolic vector bundle over $C$. Suppose that we have
a filtration $\\{_{j}\mathcal{W}\\}$ of length $l$
$0=\,_{0}\mathcal{W}\subset\,_{1}\mathcal{W}\subset\cdots\,\subset\,_{l-1}\mathcal{W}\subset\,_{l}\mathcal{W}=\mathcal{V}$.
We know that such filtration is a $k$-point of
$\text{Fil}^{\,\psi(\mathcal{W}_{j})}_{\mathcal{V}}$ for some filtration datum
$\psi(\mathcal{W}_{j})$ of length $l$. We call this the filtration datum
associated to the filtration $\\{\mathcal{W}_{j}\\}$.

More explicitly, if we have
${}_{j}\mathcal{W}=\left[\;{}_{j}\mathcal{F}^{(0)}\,\overset{{}_{j}b_{i}^{(1)}}{\subset}\,_{j}\mathcal{F}_{i}^{(1)}\overset{{}_{j}b_{i}^{(2)}}{\subset}\cdots\;\overset{{}_{j}b_{i}^{(N_{i})}}{\subset}\,_{j}\mathcal{F}_{i}^{(N_{i})}=\,_{j}\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}$, then
$\psi(\mathcal{W}_{j})=\left({}_{j}b_{i}^{(m)},\,HP(_{j}\mathcal{F}^{(0)})\right)$.

##  5 Openness of strata

The goal of this section is to prove that each stratum
$\text{Bun}_{\mathcal{V}}(C)^{\leq P}$ is open (Theorem 5.11). This is
achieved by first proving that $\text{Bun}_{\mathcal{V}}^{\leq P}$ is
constructible (Proposition 5.1), and then proving that
$\text{Bun}_{\mathcal{V}}^{\leq P}$ is closed under generalization
(Proposition 5.9). A treatment of the analogous result in the case of
principal $G$-bundles over a curve of characteristic $0$ can be found in
[GN14][§2]. Our general approach is different from [GN14] in that we prove
constructibility of all strata directly without treating the openness of the
semistable locus first.

###  5.1 Constructibility of strata

The goal of this subsection is to prove the following proposition.

######  Proposition 5.1.

Let $T$ be a scheme of finite type over $k$. Let $\mathcal{V}$ be a parabolic
vector bundle of rank $n$ over $C\times T$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum. Then, the subset
of $T$ given by $T^{\leq P}=\left\\{t\in T\,\mid\,HN(\mathcal{V}|_{C_{t}})\leq
P\right\\}$ is constructible in the Zariski topology.

Before proving Proposition 5.1, we need a couple of lemmas.

######  Lemma 5.2.

Let $\mathcal{V}$ be a parabolic vector bundle of rank $n$ over $C$. Let
$P=(\mu_{1},\mu_{2},\cdots,\mu_{n})$ be a parabolic HN datum with
$\sum_{j=1}^{n}\mu_{j}=\text{deg}\,\mathcal{V}$. Suppose that
$HN(\mathcal{V})\nleq P$. Then, there exists a parabolic subbundle
$\mathcal{W}$ of $\mathcal{V}$ such that
$\mu(\mathcal{W})>\frac{1}{r}\sum_{l=1}^{r}\mu_{l}$, where $r$ is the rank of
$\mathcal{W}$.

###### Proof.

Suppose that $\mathcal{V}$ has Harder-Narasimhan filtration given by
$0\subset\mathcal{V}_{0}\subset\mathcal{V}_{1}\subset\cdots\,\subset\mathcal{V}_{l}=\mathcal{V}$.
Recall that

$HN(\mathcal{V})=(\;\underbrace{\mu(\mathcal{V}_{0}),\,\cdots,\,\mu(\mathcal{V}_{0})}_{\text{rank}\;\mathcal{V}_{0}/0\,\text{times}},\;\underbrace{\mu(\mathcal{V}_{1}/\,\mathcal{V}_{0}),\,\cdots,\,\mu(\mathcal{V}_{1}/\,\mathcal{V}_{0})}_{\text{rank}\;\mathcal{V}_{1}/\,\mathcal{V}_{0}\,\text{times}},\cdots,\,\underbrace{\mu(\mathcal{V}_{l}/\,\mathcal{V}_{l-1}),\;\cdots,\;\mu(\mathcal{V}_{l}/\,\mathcal{V}_{l-1})}_{\text{rank}\;\mathcal{V}_{l}/\mathcal{V}_{l-1}\,\text{times}}\;)$

Let us write $HN(\mathcal{V})=(\nu_{1},\nu_{2},\cdots,\nu_{n})$. Then we know
$\sum_{j=1}^{n}\nu_{j}=\text{deg}\,\mathcal{V}=\sum_{j=1}^{n}\mu_{j}$. By
definition, $HN(\mathcal{V})\nleq P$ implies that there exists $1\leq m<n$
such that $\sum_{j=1}^{m}\nu_{j}>\sum_{j=1}^{m}\mu_{j}$. Let $m_{0}$ the
maximal such $m$. Let $1\leq q<l$ be such that
$\text{rank}\,\mathcal{V}_{q}\leq m_{0}<\text{rank}\,\mathcal{V}_{q+1}$.

Define $r\vcentcolon=\text{rank}\,\mathcal{V}_{q}$. We claim that we have
$\sum_{j=1}^{r}\nu_{j}>\sum_{j=1}^{r}\mu_{j}$. Once we prove this claim we
will be done, because we can then set $\mathcal{W}=\mathcal{V}_{q}$.

In order to prove the claim, note that we have

$0<\sum_{j=1}^{m_{0}}\nu_{j}-\sum_{j=1}^{m_{0}}\mu_{j}=\left(\sum_{j=1}^{r}\nu_{j}-\sum_{j=1}^{r}\mu_{j}\right)+\sum_{j=r+1}^{m_{0}}(\nu_{j}-\mu_{j})$

So it suffices to show that $\nu_{j}\leq\mu_{j}$ for all $r+1\leq j\leq
m_{0}$. For all $r+1\leq j\leq m_{0}$ we have
$\nu_{j}=\mu(\mathcal{V}_{q+1})=\nu_{m_{0}+1}$. Using
$\mu_{j}\geq\mu_{m_{0}+1}$, we are reduced to showing that
$\nu_{m_{0}+1}\leq\mu_{m_{0}+1}$.

By maximality of $m_{0}$, we have
$\sum_{j=1}^{m_{0}}\nu_{j}>\sum_{j=1}^{m_{0}}\mu_{j}$ and
$\sum_{j=1}^{m_{0}+1}\nu_{j}\leq\sum_{j=1}^{m_{0}+1}\mu_{j}$. This means that
we must have $\nu_{m_{0}+1}<\mu_{m_{0}+1}$, as desired. ∎

######  Definition 5.3.

If $\mathcal{V}$, $P$ and $\mathcal{W}$ are as in Lemma 5.2, we say that
$\mathcal{W}$ is a $P$-destabilizing subbundle of $\mathcal{V}$.

For the next lemma, we need to talk about $P$-destabilizing subbundles for the
fibers of a family.

######  Definition 5.4.

Let $T$ be a $k$-scheme of finite type and let $\mathcal{V}$ be a parabolic
vector bundle over $T\times_{k}C$. Let $P$ be a HN datum. For any topological
point $t\in T$, we denote by $Dest_{t}^{P}(\mathcal{V})$ the set of
$P$-destabilizing subbundles of $\mathcal{V}|_{C_{t}}$.

######  Lemma 5.5.

Let $T$ , $\mathcal{V}$ and $P$ as in the definition above. The set of all
Hilbert polynomials

$\displaystyle\left\\{HP\left(\mathcal{F}^{(0)}\right)\;\mid\;\exists t\in
T,\;\exists\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\in Dest_{t}^{P}(\mathcal{V})\right\\}$

is finite.

###### Proof.

Let
$\mathcal{V}=\left[\;\mathcal{E}^{(0)}\,\overset{a_{i}^{(1)}}{\subset}\,\mathcal{E}_{i}^{(1)}\overset{a_{i}^{(2)}}{\subset}\cdots\;\overset{a_{i}^{(N_{i})}}{\subset}\,\mathcal{E}_{i}^{(N_{i})}=\mathcal{E}^{(0)}(x_{i})\;\right]_{i\in
I}$. Let $t\in T$. Recall (Remark 4.2) that the Hilbert polynomial of a vector
bundle on $C_{t}$ is completely determined by the degree and the rank of the
bundle. For any

$\mathcal{W}=\left[\;\mathcal{F}^{(0)}\,\overset{b_{i}^{(1)}}{\subset}\,\mathcal{F}_{i}^{(1)}\overset{b_{i}^{(2)}}{\subset}\cdots\;\overset{b_{i}^{(N_{i})}}{\subset}\,\mathcal{F}_{i}^{(N_{i})}=\mathcal{F}^{(0)}(x_{i})\;\right]_{i\in
I}\in Dest_{t}^{P}(\mathcal{V})$

we know that
$1\leq\text{rank}\,\mathcal{F}^{(0)}\leq\text{rank}\,\mathcal{V}$. So there
are only finitely many possible ranks for such $\mathcal{F}^{(0)}$. It
therefore suffices to fix some rank $r$ and show that there are finitely many
possible values of $\text{deg}\,\mathcal{F}^{(0)}$ for
$\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})$ of rank $r$ as above.

By definition of $P$-destabilizing bundle, we know that for such a
$\mathcal{W}$ we have $\text{deg}\,\mathcal{W}>\sum_{l=1}^{r}\mu_{l}$. By
Lemma 2.9 we know that $deg\,\mathcal{W}\,\leq
deg\left(\mathcal{F}^{(0)}\right)+r|I|$. Hence we must have
$\text{deg}\,\mathcal{F}^{(0)}>\sum_{l=1}^{r}\mu_{l}-r|I|$. It therefore
suffices to find a uniform upper bound for the possible values of
$\text{deg}\,\mathcal{F}^{(0)}$. We argue by contradiction. Suppose that the
degrees of all possible subbundles of $\mathcal{E}^{(0)}|_{C_{t}}$ for $t\in
T$ are not uniformly bounded above. Since
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}|_{C_{t}}\right)\supset
H^{0}\left(C_{t},\,\mathcal{F}^{(0)}\right)$, Riemann-Roch for
$\mathcal{F}^{(0)}$ implies that the dimension of
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}|_{C_{t}}\right)$ is not bounded above.
This in turn implies that for each $m\geq 0$ the dimension of
$H^{0}\left(C_{t},\,\mathcal{E}^{(0)}(m)|_{C_{t}}\right)$ is not bounded.
Serre vanishing and cohomology and base-change imply that there exists some
$m>>0$ such that $H^{0}\left(C_{t},\,\mathcal{E}^{(0)}(m)|_{C_{t}}\right)$ is
the fiber of the pushforward $(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ for all
$t\in T$. Since $\pi_{T}$ is a projective morphism, the sheaf
$(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ is coherent on the Noetherian scheme
$T$. This means that the dimension of the fibers of
$(\pi_{T})_{*}\,\mathcal{E}^{(0)}(m)$ are uniformly bounded, a contradition. ∎

We are now ready to prove the main proposition of this section.

###### Proof of Proposition 5.1.

The question is local, so we can restrict to each (open) connected component
of $T$. We can therefore assume that the degree of the fibers
$\mathcal{V}|_{C_{t}}$ is the same for all $t\in T$. Let us denote this
commond degree by $d$. We can assume without loss of generality that
$\sum_{j=1}^{n}\mu_{i}=d$, because otherwise we would have that $T^{\leq P}$
is empty. In order to prove constructibility, we shall show that the
complement $T^{\nleq P}\vcentcolon=|T|\setminus T^{\leq P}$ is constructible.

For each $t\in T$ and $\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})$, we can
associate a parabolic Quot datum $\theta(\mathcal{W})$ as in Definition 4.5.
Define
$\mathfrak{S}\vcentcolon=\left\\{\,\theta(\mathcal{W})\;\mid\;\mathcal{W}\in\text{Dest}_{t}^{P}(\mathcal{V})\;\text{for
some $t\in T$}\,\right\\}$. Lemma 5.5 tells us that the set of possible first
coordinates of tuples $\left(HP(\mathcal{F}^{(0)}),(b_{i}^{(m)})\right)$ in
$\mathfrak{S}$ is finite. But there are only finitely many possible choices of
nonnegative integers $b_{i}^{(m)}$, because
$\sum_{m=1}^{N_{i}}b_{i}^{(m)}=\text{rank}\,\mathcal{W}$ for all $i\in I$ and
the rank of $\mathcal{W}$ is always smaller than the rank of $\mathcal{V}$.
Hence $\mathfrak{S}$ is finite.

The $T$-scheme
$\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}$
is of finite type over $T$, because it is a finite disjoint union of schemes
of finite type over $T$. By Chevalley’s Theorem, the set theoretic image of
the structure morphism
$\pi:\underset{(P,\,b_{i}^{(m)})\in\mathfrak{S}}{\sqcup}\text{Quot}_{\mathcal{V}}^{P,\,b_{i}^{(m)}}\rightarrow
T$ is constructible. We claim that this image is precisely $T^{\nleq P}$.

Let $t\in T^{\nleq P}$. By Lemma 5.2, $\mathcal{V}|_{C_{t}}$ admits a
$P$-destabilizing subbundle $\mathcal{W}$ over $C_{t}$. By the definition of
the parabolic Quot scheme, $\mathcal{W}$ represents a $t$-point of
$\text{Bun}_{\mathcal{V}}^{\theta(\mathcal{W})}$. Therefore $t$ is in the
image of the structure morphism