# Moduli Spaces of Unstable Objects:
Sheaves of Harder-Narasimhan length 2

Joshua Jackson  J. Jackson
Mathematics Department, Imperial College
London
SW7 2AZ
UK  Heilbronn Institute for Mathematical Research
Bristol
UK<EMAIL_ADDRESS>
###### Abstract.

Given a moduli problem posed using Geometric Invariant Theory, one can use
Non-Reductive Geometric Invariant Theory to quotient unstable HKKN strata and
construct ‘moduli spaces of unstable objects’, extending the usual moduli
classifications. After giving a self-contained account of how to do this, we
apply this method to construct moduli spaces for certain unstable coherent
sheaves of HN length 2 on a projective scheme, which we call $\tau$-stable
sheaves. This extends a previous result of Brambila-Paz and Mata-Gutiérrez for
rank two vector bundles on a curve.

## 1\. Introduction

For more than fifty years, Geometric Invariant Theory (GIT) has been a central
tool in moduli theory, and many important algebro-geometric moduli spaces can
be constructed this way. Constructions generally all follow the same strategy.
One starts by producing a scheme $X$ which parameterises the objects of
interest, together with a redundancy given by the action of some _reductive_
linear algebraic group $G$. After describing the GIT-semistability condition
$X^{ss}\subset X$ for this set-up in an object-intrinsic way, Mumford’s GIT
[MFK94], [New78] then guarantees a projective quotient $X^{ss}\rightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G$
of the semistable locus, which is a good moduli space for these semistable
objects.

One way of viewing such constructions, then, is as picking out a certain
subclass of ‘semistable’ objects whose members can be parameterised by a well-
behaved space; the projectivity of the moduli space then corresponds to the
assertion that this subclass is maximal, in the sense that no more members can
be added to it. The remaining objects are called _unstable_ ,111The convention
that ‘unstable’ means ‘not semistable’ rather than ‘not stable’ is in some
ways unfortunate, but by now far too well embedded to be avoided. and, with
few exceptions, moduli spaces parameterising them have yet to be constructed.

First, we provide a self-contained account of results of [HJ] explaining how,
subject to some fairly mild assumptions, moduli spaces for unstable objects
can indeed be constructed. Secondly, to apply this approach to construct
moduli spaces for unstable coherent sheaves of Harder-Narasimhan length two,
generalising a result of [BPMG13]. The main technical tools come from a new
form of Geometric Invariant Theory for Non-Reductive Groups, developed in
[DK07] [BDHK20] [BDHK] [BK17] [BHKD18][BDK][BJK18]

### 1.1. Moduli of unstable objects

In the classical reductive GIT set-up described above, the moduli
classification thus obtained says nothing at all about those objects whose
corresponding points were in $X\setminus X^{ss}$, that is, _unstable_. While
there are good geometric reasons not to try to include these objects in the
same moduli space as the semistable ones, there is usually no good reason to
ignore them altogether. We take the view that this is unsatisfactory, and
that, rather than simply throw away the unstable points, the more natural
thing to do is to regard the classical quotient
$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G$
(the moduli space of semistable objects) as merely the first step in the
solution of the moduli problem.

Just as classical GIT tells us which objects we can include in the moduli
space of semistables, it also tells us how we should divide the problem
further in order to make sense of the unstable part. This information is
provided by the Hesselink-Kempf-Kirwan-Ness (HKKN) stratification, also called
the instability stratification [Hes78] [Kem78] [Kir84] [NM84]. As explained in
§2.2, this stratification may be thought of as measuring how unstable points
in $X\setminus X^{ss}$ are. Given a choice of invariant inner on product on
$\operatorname{Lie}G$, one obtains a $G$-invariant locally closed
stratification

$X=X^{ss}\sqcup\bigsqcup_{\beta\in\mathscr{B}}S_{\beta},$

which, roughly speaking, uses a normalised version of the Hilbert-Mumford
function, building on work of Kempf [Kem78], to associate to each unstable
point $x\in X\setminus X^{ss}$ a cocharacter of $G$ that is most responsible
for the instability of $x$. In other words, one uses the fact that instability
in GIT is structured phenomenon. One can then try to find, for every
instability index $\beta\in\mathscr{B}$, a quotient of the stratum $S_{\beta}$
by the $G$ action, and hence obtain a moduli space of objects with that
instability index. A suitable slogan might be: we treat instability itself as
a discrete invariant for the moduli problem.

The classical GIT quotients of $S_{\beta}$ by $G$ are of course empty, at
least with respect to the linearisation we started with, however due to work
of Kirwan [Kir84] (see Theorem 2.2.10), we may view the problem as equivalent
to that of finding a quotient for the action of a certain parabolic group,
$P_{\beta}\leqslant G$, on a subvariety $Y_{\beta}^{ss}\subset S_{\beta}$.
Considering this latter question instead allows a greater range of
linearisations to be brought into play, via twisting by characters. The trade-
off for this is that _$P_{\beta}$ is in general non-reductive_, meaning that
Mumford’s GIT cannot be used to construct a quotient. Hence, we will need to
make use of the recent progress [BDHK20] [BK17] [BHKD18][BDK][BJK18] [BDHK],
[BDHK20] in producing a version of Mumford’s GIT that works for non-reductive
groups.

Non-reductive GIT encounters considerable _prima facie_ obstacles, the most
famous of which is the non-finite generation of invariants seen in the Nagata
counterexample [Nag60] to Hilbert’s 14th Problem. Nevertheless, the results of
[BDHK20] provide a powerful machine that allows many of the good properties of
the reductive case to be recovered, under mild technical assumptions. We will
refer to these results collectively as the different versions of the $\hat{U}$
Theorem, summarised in §3. Taking this as a starting point, in [HJ] we prove
the following. (See Theorem 4.1.2 for a precise statement.)

######  Theorem 1.1.1.

[HJ] Let $X$ be a projective scheme, acted on by $G$ a reductive linear
algebraic group. Suppose that the $G$ action is linearised with respect to an
ample line bundle. Choose an invariant inner product $\|\cdot\|$ on
$\operatorname{Lie}G$, and consider the associated HKKN stratification.

Then:

  1. (1)

For each $\beta\in\mathscr{B}$ such that a certain ‘semistability coincides
with stability’ condition holds for the unipotent radical of $P_{\beta}$, ‘
_ss=s for_ $\hat{U}$’, there is a projective good quotient by the action of
the non-reductive group $P_{\beta}$,

$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta,\varepsilon}}P_{\beta}$

of a certain open locus in $Y_{\beta}^{ss}$, defined in the same way as in
[BDHK20].

  2. (2)

Further, this quotient map restricts to give a quasi-projective geometric
quotient of the _totally stable_ locus $Y_{\beta}^{ts}\subseteq\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}^{P_{\beta}-ss}$,
which we define at Definition 4.1.1 and is given by a Hilbert-Mumford type
criterion (§3.2).

  3. (3)

Moreoever, if a certain ‘semistability coincides with stability’ condition ‘
_ss=s for $\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’
holds for the reductive part of $P_{\beta}$, then $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}=Y_{\beta}^{ts}$, so that this geometric quotient
of the totally stable locus is projective.

Thus, one might summarise the strategy of moduli of unstable objects in the
following way:

  1. I)

Pose the given moduli problem as a GIT problem.222Of course, the sets of
objects in which we are interested will usually come in unbounded families,
and in any given GIT-setup we will only see finitely many strata. So strictly
speaking what we really need is an infinite family of GIT problems that
somehow approximate our moduli problems asymptotically, so that we can work
with an asymptotic instability stratification, in the manner of [Hos18].

  2. II)

Interpret the instability stratification associated to this GIT problem in an
intrinsic way. That is: convert the purely geometric-invariant-theoretic
notion of HKKN type into a moduli-theoretic instability type. This gives a
discrete classification, analogous to fixing the Hilbert polynomial or similar
invariant.

  3. III)

Use non-reductive GIT as set out in [BDHK20] to quotient the unstable strata,
obtaining moduli spaces of objects of fixed instability type.

The output of this is a moduli problem that has been split into (usually an
infinite number of) manageable pieces, each with its own moduli space. Each
isomorphism (or equivalence) class of object should appear in one, and only
one, of these moduli spaces.

### 1.2. Unstable coherent sheaves

The remainder of this paper deals with our main application. Let
$(X,\mathscr{O}_{X}(1))$ be a projective scheme with an ample line bundle, and
suppose we wish to study the moduli of coherent sheaves on $X$. For sheaves
that are _semistable_ (Definition 5.1.1), this is a very well studied
question: [HL10] is an excellent introduction to the vast literature on this
topic. Unstable sheaves, by contrast, have received almost no attention: to
our knowledge the only result is [BPMG13]. The latter result constructs a
moduli space for rank two unstable vector bundles on a smooth projective
curve, and proves a Torelli-type theorem in this context. The existence of
this moduli space may be recovered from Theorem 1.2.1 below. Our aim here is
to try to understand unstable sheaves in a systematic way, using the strategy
given above.

The now-standard GIT construction of the moduli space of semistable sheaves is
due to Simpson [Sim94], and it is this construction that we take as our
starting point, giving a brief recap in §5.1. The HKKN stratification coming
from Simpson’s construction is also well understood. As outlined in §6,
Nitsure and Shatz have shown [Nit11] that there is a second stratification on
the parameter space, by Harder-Narasimhan (HN) type (Definition 6.1.1). As
both the HN and HKKN stratifications both in some sense measure the failure of
a sheaf to be semistable, it is reasonable to suppose that they might
coincide, and due to work of Hoskins and Kirwan [HK12] [Hos18], we know that
this is essentially the case (Theorem 6.2.5). Thus, in the case of unstable
sheaves, parts (I) and (II) of the above strategy have already been completed,
and we give a partial solution to (III), using Theorem 1.1.1.

However, the quotient problem (III) for sheaves cannot be solved by a
straightforward application of Thereoem 1.1.1. Indeed, we show in §7.2 that
the condition ‘ _ss=s for_ $\hat{U}$’ essentially never holds for unstable
sheaves, so that instead we must use a different version of the $\hat{U}$
Theorem: following [BDHK20] we employ a kind of non-reductive ‘partial
desingularisation’ process (the analogy is with [Kir85]) to achieve a state of
affairs in which the quotient may be performed. The output of this blow-up
process may be thought of as a refinement of the HKKN stratification in the
manner of [BHK18], with moduli spaces constructed for each refined HKKN-type.
Thus, in the case of sheaves, one obtains some refinement of the HN type,
which must be interpreted sheaf-theoretically. For sheaves of HN length two,
we show in §7 that this refined invariant corresponds to fixing the dimension
of the endomorphism group of the sheaf.

Whilst one may in principle perform this non-reductive partial
desingularisation process for sheaves of arbitrary HN length, the sheaf-
theoretic interpretation of the resultant refined HKKN stratification as a
refinement of HN type does not appear to be tractable for sheaves of Harder-
Narasimhan lengths greater than two, for technical reasons explained at §3.3
and in Remark 7.2.7. In addition, it is only for sheaves of Harder-Narasimhan
length 2 that the totally stable locus $Y^{ts}_{\beta}$ is non-empty. Thus, we
restrict our attention to these sheaves, and leave the general case to be
dealt with in a later work [BHJK], via a different non-reductive GIT
construction.

After restricting to sheaves of HN length 2 with fixed HN type $\tau$, we
introduce the condition of _$\tau$ -stability_ (Definition 7.1.2), which
captures the NRGIT property of being totally stable in a sheaf-theoretic way,
just as (semi)stability for sheaves captures GIT (semi)stability. For HN types
of length 2 that are coprime, in the sense of Definition 6.1.1,
$\tau$-stability is equivalent to indecomposability. Thus, our main result is:

######  Theorem 1.2.1.

Let $(X,\mathscr{O}_{X}(1))$ be a projective scheme with an ample line bundle,
and let $\tau=(P_{1},P_{2})$ be a Harder-Narasimhan type of length 2. Then

  1. (1)

There is a quasi-projective moduli space $M_{\tau,d}$ for those $\tau$-stable
(Definition 7.1.2) coherent sheaves such that $\dim\operatorname{End}(E)=d$.

  2. (2)

This moduli space has a canonical projective completion
$M_{\tau,d}\subset\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{\tau,d}$, constructed via the nonreductive partial desingularisation
procedure of §3.3, such that the map

$M_{\tau,d}\rightarrow M^{ss}_{P_{1}}\times M^{ss}_{P_{2}},$

to the product of two moduli spaces of semistable sheaves, which takes a sheaf
to the associated graded of its HN filtration, extends to $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{\tau,d}$.

  3. (3)

Furthermore, if $\tau$ is coprime in the sense of Definition 7.1.1, then
$M_{\tau,d}$ is the moduli space of indecomposable sheaves of type $\tau$ such
that $\dim\operatorname{End}(E)=d$.

### 1.3. Relation to previous work

The basic idea of using non-reductive GIT to construct moduli spaces of
unstable objects goes back at least as far as [Hos11] [HK12]. However, the
necessary results in Non-reductive GIT have only recently established. A
thorough, if technical, account is given in [BHK18], where an inductive
procedure is used to show that there exists a refinement of the HKKN
stratification such that each refined stratum has a quasi-projective geometric
quotient (that is, the quotient stack has a coarse moduli space). Our
contribution is to give a closed-form expression for the open stratum of this
construction, in the case covered by Theorem 1.1.1, and to interpret the GIT
condition in a moduli-theoretic way, in the case of sheaves of HN length 2.

To our knowledge, the only previous construction of a moduli space for sheaves
of fixed HN type is [BPMG13], where the base scheme is a smooth projective
curve, and the length two HN type is the type of a rank two vector bundle. The
argument of [BPMG13] does not use GIT at all, either reductive or non-
reductive, but rather constructs the vector bundle directly using a suitable
sheaf of extensions. It is therefore of interest that the refinement of HN
type thus obtained should coincide with ours.

As explained elsewhere in this paper, the case of HN length 2 enjoys certain
special properties making it amenable to the methods we develop here, which do
not hold in higher HN length. The construction of moduli spaces for unstable
sheaves of HN length greater than two will be given in [BHJK], via a different
NRGIT set-up.

### 1.4. Plan of the paper

In §2 we recap the basics of reductive GIT and the HKKN stratification,
concluding in §2.3 with a review of previous attempts to quotient unstable
strata using reductive GIT . This motivates the use of Non-reductive GIT. In
§3 we give an introduction to the results and terms from Non-reductive GIT
that we will use later, including an extend discussion of the non-reductive
partial desingularisation process, in §3.3. In §4.1 we give a precise
statement and proof of Theorem 1.1.1, and illustrate this with a simple
example, in §4.2.

We then turn to moduli of sheaves. We give the outline of Simpson’s GIT
construction for semistable sheaves in §5, and in §6 we review the work of
Hoskins and Kirwan on comparing the HKKN and HN stratifications. This done, we
conclude in §7 by giving an interpretation of the refined HKKN/HN
stratification for length two sheaves, and proving Theorem 1.2.1.

Acknowledgements

The author is grateful to Gergely Bérczi and Frances Kirwan for their very
valuable help. Special thanks are due to Victoria Hoskins, whose thesis
inspired this work, and for many helpful conversations.

### Notation

Throughout we work over an algebraically closed field $k$ of characteristic
$0$. Unless otherwise stated, we will always use ‘sheaf’ to mean coherent
sheaf and ‘point’ to mean closed point. Unless otherwise specified, following
a helpful convention introduced by [DK07], $G$ will always be a reductive
linear algebraic group, and $H$ an arbitrary (i.e. possibly non-reductive)
linear algebraic group.

## 2\. Reductive GIT and Instability Stratifications

### 2.1. Recap of Reductive GIT

Given a projective scheme $X$ equipped with the action of a linear algebraic
group, the question of whether, or in what sense, a quotient variety exists
can be very subtle. If the group $G$ is reductive, the answer to this question
was provided by Mumford in the 1960’s [MFK94], and may be summarised as
follows. More details may be found in the approachable introduction, [New78].

Embedding $X\subseteq\mathbb{P}^{n}$ using a very ample line bundle $L$, we
choose a linearisation of the $G$ action, i.e. a lift of the action to the
total space of $L$. This extends the action to
$G\curvearrowright\mathbb{P}^{n}$ via a representation
$G\rightarrow\mathrm{GL}_{n+1}$. We say the action is linear, and denote by
$\mathscr{L}$ the line bundle $L$ together with this choice of extra data.
This induces an action on the sections of all non-negative tensor powers of
$L$, and hence an action on the graded ring,

$G\curvearrowright\bigoplus_{i=0}^{\infty}H^{0}(X,\mathscr{L}^{i}),$

where $H^{0}(X,\mathscr{L}^{i})$ is just $H^{0}(X,L^{i})$ equipped with the
structure of a $G$-representation. This in turn allows us to speak of the sub-
algebra

$\bigoplus_{i=0}^{\infty}H^{0}(X,\mathscr{L}^{i})^{G}\subseteq\bigoplus_{i=0}^{\infty}H^{0}(X,\mathscr{L}^{i})$

of invariant sections of non-negative powers of $L$. Reductivity of $G$
ensures that this algebra is finitely generated, meaning that the projective
spectrum of this algebra is a projective scheme. We then define the
(reductive) GIT quotient,

$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}G\coloneqq\operatorname{Proj}\left(\bigoplus_{i=0}^{\infty}H^{0}(X,\mathscr{L}^{i})^{G}\right).$

One obtains a rational map $X\dashrightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}G$
induced via the $\operatorname{Proj}$ functor by the inclusion of the
invariants into the whole algebra, whose domain of regularity is the open set
$X^{ss}\subseteq X$, called the semistable locus333We will write e.g.
$X^{G-ss}$ if we wish to make explicit the dependence on the group.. This
rational map restricts to give a good, and in particular categorical, quotient
$X^{ss}\rightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}G$,
which, locally on affine opens, is given by taking Spec of invariants. It
restricts further to give a geometric quotient of the stable locus,
$X^{s}\subseteq X^{ss}\subset X$, which in good cases will coincide with the
semistable locus.

As a topological space, we have

$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}G\cong
X^{ss}/\sim_{S}$

where the relation, called $S$-equivalence, is given by

$x\sim_{S}y\iff\mkern 1.5mu\overline{\mkern-1.5muGx\mkern-1.5mu}\mkern
1.5mu\cap\mkern 1.5mu\overline{\mkern-1.5muGy\mkern-1.5mu}\mkern 1.5mu\cap
X^{ss}\neq\emptyset.$

Hence to understand the quotient it suffices to understand the semistable
locus. Calculating a generating set of invariants by hand, however, is seldom
possible in practice. It is therefore crucial to the utility of this theory
that one has the Hilbert-Mumford criterion, as in [MFK94], Chapter 2.1. This
allows one to calculate $X^{ss}$, and hence understand the quotient, by
understanding semistability with respect to cocharacters
$\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow G$,
i.e. without having to calculate the subalgebra of invariants.

Given a cocharacter $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow G$ and $x\in X$, the valuative criterion
of properness guarantees that the map $\lambda(t)\cdot
x\mathrel{\mathop{\mathchar
58\relax}}\mathbb{A}_{t}^{1}\setminus\\{0\\}\rightarrow X$ can be extended
uniquely over the origin. We denote the image of the origin by the limit444If
$k=\mathbb{C}$ this is the limit in the analytic sense. $\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern
1.5mu\coloneqq\lim_{t\rightarrow 0}\lambda(t)x$. This point is a fixed point
for the $\lambda(\mathbb{G}_{m})$ action on $X$, and hence the action of
$\lambda(\mathbb{G}_{m})$ restricts to an action on the fibre $L_{\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu}$.

######  Definition 2.1.1.

We define the value of the _Hilbert-Mumford function_
$\mu(\lambda,x)\in\mathbb{Z}$ to be the weight of the $\mathbb{G}_{m}$ action
on $L_{\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu}$ If we
wish to make the dependence on the linearisation explicit, we will write this
as $\mu_{\mathscr{L}}(\lambda,x)$.

Then the _Hilbert-Mumford criterion_ tells us that

$X^{ss}=\\{x\in X\mid\mu(\lambda,x)\geqslant 0\text{ for all cocharacters
}\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow G\\}$
$X^{s}=\\{x\in X\mid\mu(\lambda,x)>0\text{ for all cocharacters
}\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow G\\}.$

Rather than checking cocharacters of $G$ one at a time, we can work with a
whole maximal torus $T\leqslant G$ at once. Let $\mathfrak{t}^{+}$ be a
positive Weyl chamber of $\mathfrak{t}=\operatorname{Lie}(T)$. Choosing
coordinates $x_{0},\dots x_{n}$ on $\mathbb{P}^{n}$ that diagonalise the $T$
action, the induced representation $T\rightarrow\mathrm{GL}_{n+1}$ is
determined by the weights
$w_{i}\coloneqq\text{wt}_{T}(x_{i})\in\mathfrak{t}^{*}$, for $i=0,\dots,n$.

######  Definition 2.1.2.

We associate to $x=[a_{0}\mathrel{\mathop{\mathchar 58\relax}}\dots a_{n}]\in
X\subseteq\mathbb{P}^{n}$ the convex polytope

$\text{conv}_{T}(x)\coloneqq\text{conv}(w_{i}\mid a_{i}\neq
0)\subset\mathfrak{t}^{*}$

i.e. the convex hull of the set of weights $w_{i}$ whose corresponding
coordinates $a_{i}$ are non-zero.

It is not hard to show ([Dol03]) that for tori the Hilbert-Mumford criterion
of semistability can be reformulated to say:

$x\in X^{T-ss}\iff 0\in\text{conv}_{T}(x),$ $x\in X^{T-s}\iff
0\in\text{Interior}(\text{conv}_{T}(x)).$

and that a point $x$ is (semi)stable for $G$ if and only if all points of its
$G$-orbit are (semi)stable for $T$. Taken together, all of this reduces the
calculation of $G$-(semi)stability to a finite problem, and gives a hint as to
how we might measure instability.

######  Remark 2.1.3.

A point that is not usually emphasised, but will be important later, is that
the sense of ‘interior’ meant above is _with respect to the ambient space,
$\mathfrak{t}^{*}$_. That is to say, if the polytope is not full-dimensional,
then it is considered to have empty interior.

### 2.2. HKKN stratification

In fact, we can obtain more from reductive GIT than just a moduli space of
semistables, via the Hesselink-Kempf-Kirwan-Ness (HKKN) stratification, or
instability stratification, which we now recall. This stratification was used
in [Kir84], where it was used to calculate the cohomology of reductive GIT
quotients (see e.g. [KL89]) Our use for the stratification, however, is
different: we regard it as a way to break the ‘full moduli problem’ into
manageable pieces.

Take $G\curvearrowright X\subseteq\mathbb{P}^{n}$, a reductive group acting
linearly on a projective variety, and notation as in the previous section. The
stratification also depends on piece of additional data: we fix a norm $\|-\|$
on the set of cocharacters of $G$. That is, we take a norm on
$\operatorname{Lie}G$ that is invariant under the action of the Weyl
group.555In our main application, as well as many others, $G=\mathrm{SL}_{r}$,
so the Killing form is the only such choice up to scaling.

######  Remark 2.2.1.

If $X$ is smooth, the HKKN stratification also admits a symplectic description
in terms of the Morse stratification associated to the norm square of a
suitable moment map. We will not use this description here. Details may be
found in [Kir84].

######  Definition 2.2.2.

The index set $\mathscr{B}$ for the HKKN stratification is the set of points
$\beta\in\mathfrak{t}^{*}$ that are the closest points to the origin of the
convex hull of some subset of the $T$-weights $w_{i}$.

The rough idea of the instability stratification is that it indexes points
$x\in X\setminus X^{ss}$ according to how unstable they are, by associating a
cocharacter that is in a sense most responsible for the instability of $x$. To
make this idea precise: Kempf showed [Kem78] that for each $x\in X$ there is a
cocharacter that achieves the minimum of the normalised Hilbert-Mumford
function

$M(\lambda,x)\coloneqq\frac{\mu(\lambda,x)}{\|\lambda\|}$

######  Definition 2.2.3.

We say that a cocharacter $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow G$ is _optimal_ for a point $x\in X$ if it
achieves the minimum $M(-,x)$ over all cocharacters of $G$.

Kempf shows that optimal cocharacters are unique up to conjugacy by a certain
parabolic group $P_{x}$ associated to $x$, and each has a unique
representative in $\mathfrak{t}^{+}$, a fixed positive Weyl chamber. This
parabolic is given by

$P_{x}=P_{\lambda}\coloneqq\\{g\in G\mid\lim_{t\rightarrow
0}\lambda^{-1}(t)\cdot g\cdot\lambda(t)\text{ exists in $G$}\\}$

where $\lambda$ is any optimal cocharacter.

######  Remark 2.2.4.

In the case we consider later, where $G=\mathrm{SL}(V)$, a cocharacter
$\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow G$
gives a splitting $V=\oplus_{r\in\mathbb{Z}}V_{r}$ of $V$ into weight spaces
for the $\mathbb{G}_{m}$ action. Choosing a basis such that
$\lambda(t)=\operatorname{diag}(t^{a_{0}},\dots,t^{a_{n}})$ with
$a_{i}>a_{i+1}$, the parabolic $P_{\lambda}$ is the group of block upper
triangular matrices preserving the filtration of $V$ whose pieces are
$V^{i}\coloneqq\oplus_{r\geqslant i}V_{r}$. More generally, if
$G\leqslant\mathrm{GL}_{n}$ for some $n$, then $P_{\lambda}$ is the
intersection with $G$ of the standard parabolic of $\mathrm{GL}_{n}$
consisting of elements preserving the weight filtration of $\lambda$.

Next we need some notation that will be important later.

######  Definition 2.2.5.

[Kir84]

  1. i)

Using the inner product $\|.\|$, identify $\mathfrak{t}\cong\mathfrak{t}^{*}$.
Then given $\beta\in\mathfrak{t}^{*}$, we define

$\lambda_{\beta}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow
G$

to be the cocharacter corresponding $\beta$ under this identification.

  2. ii)

We define $Z_{\beta}$ to be the closed subscheme of the fixed point locus
$X^{\beta(\mathbb{G}_{m})}$ consisting of the points for which
$\mu(x,\lambda_{\beta})=-\|\beta\|^{2}$. Considering $\beta\in\mathfrak{t}$,
we see that $Z_{\beta}$ is those points of $X$ whose $T$-weights lie on the
hyperplane $H_{\beta}$ perpendicular to $\beta$ with respect to our chosen
inner product.

  3. iii)

We define $p_{\beta}\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow
X^{\beta(\mathbb{G}_{m})}$ to be the retraction

$p_{\beta}(x)\coloneqq\lim_{t\rightarrow 0}\lambda_{\beta}(t)\cdot x.$

  4. iv)

We define the locally closed subscheme $Y_{\beta}=p_{\beta}^{-1}(Z_{\beta})$.
Equivalently, $Y_{\beta}$ is the set of points in $X$ whose $T$-weights all
lie on the half-space

$H_{\beta}^{+}=\\{w\in\mathfrak{t}\mid\langle
v,\beta\rangle\geqslant\|\beta\|^{2}\\}$

and at least one of whose weight lies on $H_{\beta}$.

  5. v)

We further define

$Y_{\beta}^{ss}=\\{x\in Y_{\beta}\mid\lambda_{\beta}\text{ is optimal for
}x\\},$ $Z_{\beta}^{ss}=\\{x\in Z_{\beta}\mid\lambda_{\beta}\text{ is optimal
for }x\\},$ $S_{\beta}=G\cdot Y_{\beta}^{ss}.$

  6. vi)

Finally we define $\mkern
1.5mu\overline{\mkern-1.5muY_{\beta}\mkern-1.5mu}\mkern 1.5mu$ to be the
closure of $Y_{\beta}$ (or equivalently $Y_{\beta}^{ss}$) in $X$.

######  Definition 2.2.6.

Denote by $\operatorname{Stab}\beta$ the stabiliser of $\beta$ under the
coadjoint action of $G$, and note that $Z_{\beta}$ is invariant under the
action of $\operatorname{Stab}\beta$. We restrict the linearisation to the
action of $\operatorname{Stab}\beta$ on $Z_{\beta}$, and twist it by the
character $-\beta$; this has the effect of shifting the weights by this
vector. We denote this linearisation by $\mathscr{L}_{\beta}$, and we will
call it the _canonical linearisation_.

######  Remark 2.2.7.

This character need not exist as a character of $G$, but the centrality of
$\lambda_{\beta}$ in $\operatorname{Stab}\beta$ ensures that it exists as a
character of $\operatorname{Stab}\beta$, and, consequently, in $P_{\beta}$.
This extra choice of character is one benefit of working with the non-
reductive GIT problem we will consider later.

######  Remark 2.2.8.

The subscheme $Z_{\beta}^{ss}$ is exactly the semistable locus for the action
of $\operatorname{Stab}\beta$ on $Z_{\beta}$ with respect to the twisted
linearisation, and $Y_{\beta}^{ss}=p_{\beta}^{-1}(Z_{\beta}^{ss})$ ([Kir84],
12.21).

######  Definition 2.2.9.

Given $\beta\in\mathscr{B}$, denote by $P_{\beta}$ the parabolic subgroup of
$G$ associated to $\lambda_{\beta}$.

We can now summarise the relevant results of [Kir84] that give the
stratification.

######  Theorem 2.2.10.

[Kir84] Let $X$ be a projective scheme, acted on by a reductive linear
algebraic group $G$, linearly with respect to an ample line bundle
$\mathscr{L}$. Then, with notation as above, we have:

  1. (1)

There is a stratification of $X$ into disjoint locally closed $G$-invariant
subschemes,

$X=\bigsqcup_{\beta\in\mathscr{B}}S_{\beta},$

where the open stratum, $S_{0}$, is $X^{ss}$, and the closure of a stratum
$S_{\beta}$ satisfies

$\mkern 1.5mu\overline{\mkern-1.5muS_{\beta}\mkern-1.5mu}\mkern
1.5mu\subset\bigcup_{\begin{subarray}{c}\gamma\text{ s.t.}\\\
\|\gamma\|\geqslant\|\beta\|\end{subarray}}S_{\gamma}.$

  2. (2)

There is, for every $\beta\in\mathscr{B}$, a $G$-equivariant isomorphism

$Y_{\beta}^{ss}\times^{P_{\beta}}G\cong S_{\beta},$

where $Y_{\beta}^{ss}\times^{P_{\beta}}G$ is the quotient of
$Y_{\beta}^{ss}\times G$ by the free action of $P_{\beta}$ given by

$h\cdot(g,x)\coloneqq(gh^{-1},hx).$

It is therefore natural to try to find, for every $\beta\in\mathscr{B}$, a
quotient of the stratum $S_{\beta}$ by the $G$ action. By the second part of
the theorem above, this is equivalent to the problem of finding a quotient for
the action of $P_{\beta}$ on $Y_{\beta}^{ss}$. Thus, we are naturally led from
a reductive GIT problem to a non-reductive one.

### 2.3. Reductive quotients of the unstable strata

By way of motivation for the use of the $\hat{U}$ Theorem in this setting, we
illustrate what happens if we try to quotient the unstable strata using
reductive GIT. This was the approach taken in [Hos11], before an adequate non-
reductive GIT was available. Recall that in the best setting for GIT we have a
reductive group $G$ acting on a projective scheme $X$ with ample linearisation
$\mathscr{L}$, where the quotient morphism

$X^{ss}\rightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G$

is induced via the Proj functor by the inclusion of rings

$\bigoplus\limits_{r\geqslant 0}H^{0}(X,\mathscr{L}^{\otimes
r})^{G}\hookrightarrow\bigoplus\limits_{r\geqslant
0}H^{0}(X,\mathscr{L}^{\otimes r}).$

Outside of this setting, say when $L$ is not ample or $G$ is not reductive, we
can still consider the morphism of schemes resulting from this inclusion, but
this will not necessarily coincide with the usual GIT quotient in the sense of
Mumford. Moroever, if the invariants are not finitely generated, the scheme
obtained thereby will not be of finite type. A more tractable quotient may be
obtained, following [Hos11], by taking a finitely generated subalgebra

$R\subseteq\bigoplus\limits_{r\geqslant 0}H^{0}(X,\mathscr{L}^{\otimes
r})^{G}$

and then defining the ‘generalised GIT quotient’ to be the variety

$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L},R}G\coloneqq\operatorname{Proj}R.$

This comes equipped with a natural rational map from the scheme $X$, defined
by the corresponding inclusion of subrings.

Let $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}$ be
the closure of $Y_{\beta}$ inside the ambient projective space, and define

$\hat{S}_{\beta}=G\times^{P_{\beta}}\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}.$

In [Hos11], Hoskins constructs a linearisation on $\hat{S}_{\beta}$, which is
also denoted by $\mathscr{L}_{\beta}$ and also called the canonical
linearisation (cf. Definition 2.2.6). This has the property that there are
isomorphisms of algebras

$\bigoplus\limits_{n\geqslant
0}H^{0}(S_{\beta},\mathscr{L}_{\beta}^{\otimes{n}})^{G}\cong\bigoplus\limits_{n\geqslant
0}H^{0}(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta},\mathscr{L}_{\beta}^{\otimes{n}})^{P_{\beta}}\cong\bigoplus\limits_{n\geqslant
0}H^{0}(Z_{\beta},\mathscr{L}_{\beta}^{\otimes{n}})^{\text{Stab}\beta}.$

By the reductivity of Stab$\beta$, we know that the invariant sections on
$\hat{S}_{\beta}$ are finitely generated. The canonical linearisation on
$\hat{S}_{\beta}$ is therefore particularly amenable to the strategy of
‘generalised GIT’– we can take $R$ to be, in each case, the whole subalgebra
of invariants (Let us abuse notation and denote them all by $R$, since they
are isomorphic.) On the other hand, this linearisation has the drawback that
it is not usually ample.666See [Hos11], Example 2.2.5 The result of this
approach is summarised as follows.

######  Theorem 2.3.1.

([Hos11], $2.2$)

  1. 1)

The semistable locus for the $P_{\beta}$ action on $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}$ is
$Y_{\beta}^{ss}$, and we have isomorphisms of generalised GIT quotients

$\hat{S}_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{(\mathscr{L}_{\beta},R)}G\cong\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{(\mathscr{L}_{\beta},R)}P_{\beta}\cong
Z_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta}}\text{Stab}\beta.$

  2. 2)

The quotient map

$Y_{\beta}^{ss}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{(\mathscr{L}_{\beta},R)}P_{\beta}$

factors through the morphism $p_{\beta}\mathrel{\mathop{\mathchar
58\relax}}Y_{\beta}\rightarrow Z_{\beta}$.

  3. 3)

The morphism $S_{\beta}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{(\mathscr{L}_{\beta},R)}P_{\beta}$
is a categorical quotient for the action of $G$ on $S_{\beta}$.

The first point tell us the good news: in taking a quotient with respect to
the canonical linearisation on $S_{\beta}$ we can essentially treat the
problem as if it were classical GIT. That is, the quotient we are looking for
is equivalent to a quotient of the action of the reductive group Stab$\beta$
on $Z_{\beta}$ with respect to the ample canonical linearisation.

However, the second point shows that each point of $Y_{\beta}^{ss}$ is
identified in the quotient with its image under the retraction
$p_{\beta}\mathrel{\mathop{\mathchar 58\relax}}Y_{\beta}\rightarrow
Z_{\beta}$. This is generally undesirable, and leads to GIT quotients being of
much smaller dimension than they ought to be. For example, in the setting of
moduli of unstable sheaves, this results in a sheaf being identified with the
associated graded of its Harder-Narasimhan filtration; this means that the
moduli spaces of unstable sheaves constructed in this way would be isomorphic
to products of moduli spaces of semistable sheaves, with no information about
the extensions that glue them together. This is plainly not satisfactory. A
more straightforward illustration, where the moduli spaces resulting from this
method of construction are just points, is given in §4.2, and we will see in
§4 how we can use non-reductive GIT to improve on this.

In §4 we follow an idea due to Hoskins [Hos11], first executed in [HJ]. If we
are to find a quotient morphism that does not factor through $p_{\beta}$, we
must somehow make $Z_{\beta}$ unstable, otherwise we will have the problem
that too many orbit closures meet there. Hence, a good approach would be to
perturb the canonical linearisation, thus splitting each stratum into two
pieces: a piece corresponding to $Z_{\beta}$, whose associated moduli space is
the reductive quotient by $\operatorname{Stab}\beta$ above, and another, open,
piece777This will be $Y_{\beta}^{ts}=Y_{\beta}^{s}\setminus
U_{\beta}Z_{\beta}^{s}$; compare to $X^{0}_{\min}\setminus UZ_{\min}$ in §3.
which we will quotient using non-reductive GIT with respect to this
perturbation of the canonical linearisation on $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}$. The problem
arises, however, that perturbing the linearisation may potentially destroy the
resemblance to a classical GIT problem noted in the theorem above, and creates
a need for the methods of non-reductive GIT.

## 3\. Non-reductive GIT: The $\hat{U}$ Theorem

Let us return to the situation above, in which a projective variety
$X\subseteq\mathbb{P}^{n}$ is acted on linearly by a linear algebraic group
$H$, now _no longer assumed to be reductive_.

### 3.1. Non-Reductive GIT: The $\hat{U}$ Theorems

Suppose one attempts to generalise Mumford’s theory in the most naive way, by
proceeding exactly as in the reductive case: that is, declaring the quotient
map to be the rational map from $X$ to the projective spectrum of the algebra
of invariants, and taking $X^{ss}$ to be the locus of regularity of this map.
Unfortunately, several things go wrong straight away.

  * •

The subalgebra of invariants need not be finitely generated, so that this
quotient may fail to be a scheme of finite type. This was first demonstrated
by the Nagata counterexamples to Hilbert’s 14th Problem, and simpler examples
we later given by Roberts. [Nag60] [Rob90]

  * •

Even when the subalgebra of invariants does happen to be finitely generated,
there is no guarantee that the morphism as defined above have the topological
properties we desire in a quotient map. In particular, the morphism can fail
to be surjective. [DK07]

  * •

Unlike in the reductive case, taking invariants is not right exact.
Consequently, non-reductive GIT quotients can in general behave badly with
respect to $H$-equivariant closed embeddings.

  * •

One way to interpret the Hilbert-Mumford criterion is as saying a reductive
group $G$ has ‘enough’ cocharacters to detect $G$-semistability. There is
obviously no hope that an unmodified Hilbert-Mumford criterion can continue to
hold for general non-reductive groups: for example, unipotent groups have no
cocharacters at all.

Nevertheless, all of these problems can be resolved under remarkably weak
hypotheses. This generalisation of GIT relies on two crucial technical
concepts, both of which we will show later are present in the cases of
interest to us. We sketch it briefly here, and refer the reader in the first
instance to [BDHK20], as well as to [BK17] [BDHK], for a more detailed
account.

######  Definition 3.1.1.

Let $H$ be an arbitrary linear algebraic group, and let $U$ denote its
unipotent radical (i.e. its maximal connected normal unipotent subgroup).
Since we are in characteristic zero, we can choose a reductive Levi factor
$R\leqslant H$ and write $H=U\rtimes R$. Let
$\mathfrak{u}=\operatorname{Lie}(U)$. We say that a cocharacter
$\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow H$
_grades $U$_, or that $U$ is _positively graded_ if it has trivial conjugation
action on $H/U\cong R$ (i.e. its image lies in the centre of $R$) and the
weights for its conjugation action on $\mathfrak{u}$ are all strictly
positive.

######  Definition 3.1.2.

We say that a linearisation is _adapted_ with respect to some choice of
grading cocharacter $\lambda$, if the lowest weight of the
$\lambda(\mathbb{G}_{m})$ action on $X$ is strictly negative, and the rest are
strictly positive.

######  Remark 3.1.3.

If the linearisation we are working with is not adapted, we can twist it by a
suitable character to make it so. In the setting of moduli of unstable
objects, we can always find a grading cocharacter, and adapted linearisation.
Namely, if we wish to quotient $Y_{\beta}^{ss}$, we can use $\lambda_{\beta}$
itself to grade the unipotent radical of $P_{\beta}$, and twist the
linearisation by $(1+\varepsilon)(-\beta)$, where $0<\varepsilon\ll 1$ and
$\beta$ is the character dual to $\lambda_{\beta}$ under our choice of
invariant inner product.

Before we can state the result from NRGIT that we will use, we need a few more
pieces of notation, which we adopt in order to be consistent with [BDHK20].

######  Definition 3.1.4.

Fix a grading $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow H$. Let $Z_{\min}$ be the closed subscheme
of the fixed point locus $X^{\lambda(\mathbb{G}_{m})}$ consisting of those
points whose unique $\lambda$-weight is the minimal one. Then define
$X^{0}_{\min}=p_{\lambda}^{-1}(Z_{\min})$, where $p_{\lambda}$ is as defined
at Definition 2.2.5.

We now quote the so-called $\hat{U}$ Theorem. The theorem get its name from a
piece of notation: for a linear algebraic group $H$ with unipotent radical $U$
graded by a cocharacter $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow H$, we form the semidirect product
$U\rtimes_{\lambda}\mathbb{G}_{m}$ and denote it by $\hat{U}$. There are
various versions of the theorem that apply in different settings, but all of
them require the following as a starting point. We will call these the
$\hat{U}$ conditions.

######  Condition 3.1.5.

Let $X$ be a projective scheme, with an action of a linear algebraic group
$H$. Let $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow H$ be a grading cocharacter. Suppose that
the action is linear with respect to an ample linearisation, and suppose that
this linearisation is adapted in the sense of Definition 3.1.2.

######  Theorem 3.1.6.

_( $\hat{U}$ Theorem), [BDHK20] _ Suppose the $\hat{U}$ conditions hold in the
sense of 3.1.5. Then with notation as in Definition 3.1.1, suppose further
that _semistability coincides with stability for the unipotent radical, $U$,
of $H$_. That is, suppose we have888Requiring this condition for
$X^{0}_{\min}$ is equivalent requiring it for $Z_{\min}$: since $U$ is
normalised by $\mathbb{G}_{m}$, the $U$-stabiliser dimensions cannot decrease
along the retraction $p\mathrel{\mathop{\mathchar
58\relax}}X^{0}_{\min}\rightarrow Z_{\min}$.

[‘ _ss=s for_ $\hat{U}$’] $\operatorname{Stab}_{U}(x)=\\{e\\}\text{ for all
}x\in Z_{\min}.$

Then

  1. (1)

The $\hat{U}$-invariants are finitely generated, and the inclusion of the
$\hat{U}$-invariant algebra induces a projective geometric quotient

$X^{\hat{U}-s}\rightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}\hat{U}.$

of an open subscheme of $X$ which we call the $\hat{U}$-stable locus:

$X^{\hat{U}-s}\mathrel{\mathop{\mathchar 58\relax}}=X^{0}_{\min}\setminus
UZ_{\min}.$

  2. (2)

Consequently the $H$-invariants are finitely generated, and the inclusion of
the $H$-invariant subalgebra induces a good quotient of an open subscheme of
$X$ (the $H$-semistable locus):

$X^{H-ss}\rightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}H,$

where
$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}H$
is the GIT quotient of
$X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}\hat{U}$
by the induced action of the reductive group $H/\hat{U}\cong R/\mathbb{G}_{m}$
with respect to the induced linearisation. This good quotient of $X^{H-ss}$
restricts to a geometric quotient

$X^{H-s}\rightarrow X^{H-s}/H\subseteq
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}}H$

of the $H$-stable locus $X^{H-s}$.

######  Remark 3.1.7.

The definitions of $X^{H-ss}$ and $X^{H-s}$ may be found in [BDHK20]; for our
present purposes we need not determine these sets precisely, as it will be
enough to work with a certain subset of $X^{H-s}$, which we explicitly
describe in §3.2.

For our main application, and others of interest, we will not have condition
(‘ _ss=s for_ $\hat{U}$’). Fortunately, there are other versions of the
$\hat{U}$ Theorem that can be used in such cases. These other versions follow
a process that may be considered as analogous to the partial desingularisation
procedure used in [Kir85] for reductive groups. An important caveat in this
analogy is that, unlike in the reductive case, we must perform these blow-ups
in order to obtain a quotient at all.

The version of the $\hat{U}$ Theorem that we will use in the construction of
moduli spaces for sheaves of Harder-Narasimhan length 2 is the following.

######  Theorem 3.1.8.

[BDHK20] Take the assumptions of 3.1.5, without assuming (‘ _ss=s for_
$\hat{U}$’), and suppose that the unipotent radical $U$ is abelian.

Then there is an $H$-equivariant series of blow-ups of $X$, resulting in a
variety $\pi\mathrel{\mathop{\mathchar 58\relax}}\widehat{X}\rightarrow X$,
with an $H$ action lifting the action on $X$, carrying an $H$-linearisation
that is some small perturbation of $\pi^{*}\mathscr{L}$, such that 3.1.5 holds
and in addition we have

[$\star_{d}$] $\exists d\in\mathbb{N}\text{ such that
}\dim\operatorname{Stab}_{U}(x)=d\text{ for all }x\in\widehat{X}^{0}_{\min}.$

Furthermore, the conclusions (1) and (2) of 3.1.6 then hold for $\widehat{X}$.

######  Remark 3.1.9.

As remarked in [BDHK20], the above theorem can be combined with a certain
quotienting-in-stages procedure, to yield the stronger version of the
$\hat{U}$ Theorem that may be applied whenever we conditions of 3.1.5 are
satisfied. For this we first must choose the a subnormal series

$0\trianglelefteq U^{1}\trianglelefteq\dots\trianglelefteq U^{\ell}=U,$

satisfying the condition that each successive quotient $U^{j+1}/U^{j}$ is
abelian, and that the adjoint action of the distinguished cocharacter
$\lambda(\mathbb{G}_{m})\leqslant H$ on $\operatorname{Lie}U^{j+1}/U^{j}$
consists of a single weight space. Such a subnormal series can always be
found: indeed, in the situation of moduli of unstable objects discussed in §4,
where we use the Killing form as our invariant inner product, we can use the
derived series of $U$. One can then perform the quotient in stages, by first
performing the blowups for the abelian group $U^{1}$ as in 3.1.8, and then
following an iterative procedure for the remainder of $U$.

The utility of 3.1.8, and its generalisation described above, lies in the fact
that the map $\pi\mathrel{\mathop{\mathchar 58\relax}}\widehat{X}\rightarrow
X$ restricts away from the centres of the blow-ups to give an $H$-equivariant
isomorphism of some nonempty open subset of $X$ with a nonempty open subset of
$X$. Because being a good quotient is a local property, this set then has a
good quotient by $H$. The difference in outcome between theorems 3.1.6 and
3.1.8 is that in the former case we obtain an open subset of $X$ with a
projective quotient, whereas in the latter case we obtain an open subset of
$X$ whose quotient is only quasi-projective, but has a natural projective
completion given by including orbits from $\widehat{X}$.

######  Remark 3.1.10.

A final important property of this generalisation of reductive GIT, which we
will not use here, is that Variation of GIT as in [DH98] [Tha96], can be shown
to have an avatar in this setting [BJK18].

### 3.2. Hilbert-Mumford Criterion

As we have observed, it is essential to have a tractable way of computing the
stable and semistable loci if non-reductive GIT is to be useful in practice.
Fortunately, subject to the conditions of the $\hat{U}$ Theorem, the Hilbert-
Mumford criterion holds in exactly the sense we would like. A proof is given
in full generality in [BDHK20]; here we will give a slightly simplified proof
which is enough to establish the result in the case we will need it.

Continuing to use the notation as in theorem 3.1.6, write $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu=R/\lambda(\mathbb{G}_{m})$ for the quotient of the Levi by the grading
cocharacter. Since this cocharacter acts trivially on $Z_{\min}$ by
construction, we get an induced action $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu\curvearrowright
Z_{\min}$. Since the image of $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow H$ lies in the centre of $R$, the weight
$w_{\min}$ with which $\lambda(\mathbb{G}_{m})$ acts on $Z_{\min}$, _a priori_
only a character of $\lambda(\mathbb{G}_{m})$, in fact corresponds to a
character of $R$. Hence, twisting $\mathscr{L}$ by this character, we obtain a
linearisation $\mkern
1.5mu\overline{\mkern-1.5mu\mathscr{L}\mkern-1.5mu}\mkern 1.5mu$ of the action
of $\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$ on
$Z_{\min}$. With respect to this linearisation, we consider the following
condition:

######  Definition 3.2.1.

We define the _totally $H$-stable locus_ to be

$X^{H-ts}\mathrel{\mathop{\mathchar 58\relax}}=\\{x\in X^{0}_{\min}\setminus
UZ_{\min}\mid\lim_{t\rightarrow 0}\lambda(t)x\mathrel{\mathop{\mathchar
58\relax}}=p_{\lambda}(x)\in Z_{\min}^{\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu-s}\\}$

Since $HZ_{\min}=UZ_{\min}$, this property and its negation are invariant
under the action of $H$.

######  Remark 3.2.2.

We will see in §7 that this condition arises naturally when one considers
unstable strata in the standard GIT construction of moduli of sheaves. Indeed,
the HN-stable sheaves are exactly the sheaves satisfying this condition.

######  Theorem 3.2.3.

(Hilbert-Mumford Criterion) (cf. [BDHK20], [HJ])

Take the assumptions of Theorem 3.1.6. Then

  1. (1)

All totally $H$-stable points are $H$-stable. That is, we have
$X^{H-ts}\subseteq X^{H-s}$, and the totally $H$-stable locus therefore has a
quasi-projective geometric quotient by $H$.

  2. (2)

If we have the following condition

[‘ _ss=s for $\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’]
$Z_{\min}^{\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu-s}=Z_{\min}^{\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu-ss}$

then there are equalities

$X^{H-ts}=X^{H-s}=X^{H-ss},$

so that the locus on the left has a projective geometric $H$-quotient.

  3. (3)

Furthermore, (1) and (2) hold also in the situation of Theorem 3.1.8. That is
to say, if we do not have condition (‘ _ss=s for_ $\hat{U}$’), but the
unipotent radical is abelian, we obtain results as in (1) and (2) with
$\widehat{X}$ replacing $X$.

###### Proof.

We show how the introduction of the assumption (‘ _ss=s for $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’) allows one to give
an easier proof of this Hilbert-Mumford criterion, by simplifying the proof of
Lemma 7.8 of [BDHK20]. We give the proof for the situation of Theorem 3.1.6;
the proof for Theorem 3.1.8 is the same, following Remark 7.12 of [BDHK20].

Adopting notation as in [BDHK20] §7.4, where Proposition 7.4 gives a locally
trivial $U$-quotient

$q_{U}\mathrel{\mathop{\mathchar 58\relax}}X^{0}_{\min}\rightarrow
X^{0}_{\min}/U.$

We choose $s>0$ sufficiently large as determined by [BDHK20], and define

$V=H^{0}(X,L^{\otimes s})^{*}$ $V^{U}=(H^{0}(X,L^{\otimes s})^{U})^{*}.$

We then obtain a locally closed immersion

$\iota\mathrel{\mathop{\mathchar
58\relax}}X^{0}_{\min}/U\hookrightarrow\mathbb{P}(V^{U}).$

There is an induced $\lambda(\mathbb{G}_{m})$ action on $\mathbb{P}(V^{U})$;
we write $\mathbb{P}(V^{U})_{\min}$ for its minimal weight space and $\mkern
1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern
1.5mu\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(V^{U})_{\min}^{0}\rightarrow\mathbb{P}(V^{U})_{\min}$ for
the induced retraction. Now $\mathbb{P}(V^{U})$ has an action of the maximal
torus $T\leqslant H$, which is reductive and hence enjoys a classical Hilbert-
Mumford criterion. The idea of the proof in [BDHK20] is to bootstrap this
Hilbert-Mumford criterion from $\mathbb{P}(W^{*})$ to $X^{0}_{\min}$ via
$X^{0}_{\min}/U$. Recall from §2.1 the formulation of the Hilbert-Mumford
criterion which says that a point is stable for a torus action exactly when
its weight polytope contains the origin in its interior.

We must now compare, for a point $x\in X^{0}_{\min}$, the two weight polytopes
$\text{conv}_{T}(x)$ and $\text{conv}_{T}(q_{U}(x))$. As is standard, we may
replace $L$ with a positive tensor power, since the property that the weight
polytope contain (or not contain) the origin is unaffected by this. Hence we
may assume that the composition of the $U$-quotient map with $\iota$
corresponds to the inclusion of the $U$-invariant sections of
$H^{0}(X,L^{\otimes s})$, and thus comes from a linear projection of the
associated projective spaces $\mathbb{P}(V)\dashrightarrow\mathbb{P}(V^{U})$.
Thus, we have

$\text{conv}_{T}(q_{U}(x))\subseteq\text{conv}_{T}(x),$

and _a priori_ this could be any subset. However, as observed in [BDHK20] the
condition of $\lambda(\mathbb{G}_{m})$ grading $U$ guarantees that
$H^{0}(X,L^{\otimes s})_{\max}\subset H^{0}(X,L^{\otimes s})^{U}$ , so all
weights of $\text{conv}_{T}(x)$ that are minimal for $\lambda$ are still
present in $\text{conv}_{T}(q_{U}(x))$. Hence if $p(x)\in Z_{\min}^{\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu-s}$, then the same will
be true after the $U$-quotient, i.e. $\mkern
1.5mu\overline{\mkern-1.5mup\mkern-1.5mu}\mkern 1.5mu\circ
q_{U}(x)\in\mathbb{P}(V^{U})_{\min}^{\mkern
1.5mu\overline{\mkern-1.5muR-s\mkern-1.5mu}\mkern 1.5mu}$. Crucially, the
weight polytope of the latter point will be of codimension one in
$\mathfrak{t}^{*}$. Furthermore, we have $x\notin UZ_{\min}$, from which it
follows that $q_{U}(x)\notin\mathbb{P}(V^{U})_{\min}$, since the $U$-quotient
is geometric. Hence, $\text{conv}_{T}(x)$ contains at least one weight that is
off the minimal weight space, and the same is true of
$\text{conv}_{T}(q_{U}(x))$. The combination of these facts with well-
adaptedness of the linearisation ensures that $q_{U}(x)$ satisfies the
Hilbert-Mumford criterion for $T$-stability. Moreover, the same will be true
for every point in the $H$-orbit of $x$, since having stable limit in
$Z_{\min}$ is an $H$-invariant property. Hence, $x\in X^{H-s}$.

∎

### 3.3. The blow-up process in more detail

We have seen that, in quotienting by non-reductive groups, the dimensions of
the stabilisers in the unipotent radical play an important role. For unstable
sheaves of Harder-Narasimhan length 2, we will interpret these stabilisers in
§7.2, and see that we do not have condition ‘ _ss=s for $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’. Consequently, we
will need the full force of Theorem 3.1.8, and so we will need to understand
the blow-up process used to prove that theorem. We adopt the assumptions of
Theorem 3.1.8.

In order for the blow-up process described in [BDHK20] to work, it is
necessary that the centre of the blow-up is always of codimension at least
two, or else the blow-up will be an equivariant isomorphism and the induction
will stall. To prevent this problem from arising, here and in [HJ], we adopt a
different approach, following an earlier arXiv version of [BDHK20]. In this
approach, we blow up according to the dimension of $\hat{U}$-stabilisers
rather than $U$-stabilisers - or equivalently, blowing up the locus in
$UZ_{\min}$ of maximal $U$-stabiliser dimension. This implies that the centre
of the blow-up is always of codimension at least two; that is unless we have
$UZ_{\min}=X^{0}_{\min}$, in which case the reductive quotient
$Z\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}R$
is the appropriate $H$-quotient of $X$.

Now let us examine this blow-up process. Proofs of the below statements, as
far as they do not simply follow from the theorems quoted above, may be found
in [BDHK20] and [HJ].

######  Definition 3.3.1.

We will use the following notation for the blow-ups’ centres. For
$d\in\mathbb{N}$, let

$C_{d}(X)=\\{x\in X^{0}_{\min}\mid\dim\operatorname{Stab}_{\hat{U}}(x)=d\\}$

and define

$d_{\max}(X)=\max\\{\dim\operatorname{Stab}_{\hat{U}}(x)\mid x\in
X^{0}_{\min}\\}.$

With this notation, we perform a series of blow-ups to achieve condition
($\star_{d}$) for $U$\- that is, to have a constant-dimensional $U$-stabiliser
across all of $X^{0}_{\min}$. We begin by blowing up along the closure of the
locus $C_{d_{\max}}(X)$. This gives us a new space
$\pi^{(1)}\mathrel{\mathop{\mathchar 58\relax}}X^{(1)}\rightarrow X$, with
exceptional locus $E^{(1)}$. Because $\hat{U}$ is normal, the centre of the
blow-up was $H$-invariant, so the new space has an $H$ action lifting the
action on $X$, which we linearise by defining

$\mathscr{L}^{(1)}\coloneqq(\pi^{(1)})^{*}\mathscr{L}^{r}\otimes\mathcal{O}_{X}(-E^{(1)}).$

We choose $r\gg 0$, so that, since replacing a linearisation with a positive
multiple makes no difference to (semi)stability, we may think of this
linearisation as a perturbation of the original linearisation of the action on
$X$ by a sufficiently small multiple of the exceptional divisor.

We denote the new minimal weight space for the distinguished
$\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow H$ by
$Z_{\min}(X^{(1)})$. Then let

$(X^{(1)})^{0}_{\min}=\\{x\in X^{(1)}\mid\lim_{t\rightarrow 0}\lambda(t)x\in
Z_{\min}(X^{(1)})\\}.$

The arguments of [BDHK20] (see also [HJ]§2.3.6-2.3.7) tell us that we now have

$d^{1}_{\max}(X^{(1)})\coloneqq\max\\{\dim\operatorname{Stab}_{\hat{U}}(x)\mid
x\in X^{(1)}\\}<d^{1}_{\max}.$

So we repeat the process: we blow up $X^{(1)}$ along the closure of
$C_{d_{\max}}(X^{(1)})$, obtaining $X^{(2)}$ with another action lifting that
of $X$, which we again perturb by a small multiple of the exceptional divisor
$E^{(2)}$.

Inductively, we see that this process will terminate with a space
$\widehat{X}\mathrel{\mathop{\mathchar 58\relax}}=X^{(l)}$ and a blow-down map
$\pi\mathrel{\mathop{\mathchar 58\relax}}\widehat{X}\rightarrow X$ such that
the action of $U$ satisfies condition ($\star_{d}$). At this point we apply
Theorem 3.1.8.

######  Remark 3.3.2.

Let us pause at this point to illustrate the properties of the linearisation
on the blow-up. For simplicity, assume that we have a $\mathbb{G}_{m}$ action
on $\mathbb{P}^{n}$, with weights
$w_{i}=\operatorname{wt}_{\mathbb{G}_{m}}(x_{i})$, and that we are blowing up
along a linear subspace $\mathbb{P}^{k}\subset\mathbb{P}^{n}$. Then the blow-
up is

$\operatorname{Bl}_{\mathbb{P}^{k}}\mathbb{P}^{n}=\\{([x_{0}\mathrel{\mathop{\mathchar
58\relax}}\dots\mathrel{\mathop{\mathchar
58\relax}}x_{n}],[\tilde{x}_{k+1}\mathrel{\mathop{\mathchar
58\relax}}\dots\tilde{x}_{n}]\in\mathbb{P}^{n}\times\mathbb{P}^{n-k-1}\mid
x_{i}\tilde{x}_{j}=x_{j}\tilde{x}_{i}\\}.$

With respect to the choice of linearisation given above, weights of the
$\mathbb{G}_{m}$ action on the blow-up will be given by summing $r$ of the
weights $w_{0},\dots,w_{n}$ and then perturbing by one of the weights
$w_{k+1},\dots w_{n}$: that is, we replace each weight $w_{i}$ by a small
cluster of weights, and each weight in the cluster corresponds to a weight in
the projectivised normal bundle of the centre of the blow-up.

In general, we will be blowing up a scheme $X$ that may be singular, non-
reduced, etc, along a possibly singular locus. However, our choice of ample
linearisation gives an embedding $X\subset\mathbb{P}^{n}$, and we may perform
the blow-ups along the relevant loci considered in this space, which, by the
arguments of [BDHK20], will be smooth because $\mathbb{P}^{n}$ is. This means
that the identification of the exceptional divisor in
$\operatorname{Bl}_{\mathbb{P}^{k}}\mathbb{P}^{n}$ with the projectivised
normal bundle to the centre of the blow-up holds, and we may consider the
exceptional divisor in the blow-up of $X$ to be the intersection of the proper
transform of $X$ with this, which will therefore have weights as described
above.

######  Remark 3.3.3.

For non-abelian $U$, as in the situation of Remark 3.1.9, the idea is
essentially the same. We first perform the blow-ups described above, with the
abelian group $U^{1}$ in place of $U$. Having achieved condition ($\star_{d}$)
for $U^{1}$, we then repeat the whole process for $U^{2}$, blowing up along
closures of loci of the above form, and then inductively replacing $U^{j}$
with $U^{j-1}$ until we have a blow-up of $X$ satifying ($\star_{d}$) for the
full unipotent radical $U$.

Of course, what one is really interested in is obtaining a quotient of an open
subset of $X$. In the situation that will be of greatest concern to us, when
we have posed a particular moduli problems using GIT, the $H$-orbits in $X$
will have some moduli-theoretic interpretation as equivalence classes of
objects to be classified, whereas the points of $\widehat{X}$ may not readily
admit such an interpretation. As a result, we will mainly use Theorem 3.1.8 in
the following form.

######  Corollary 3.3.4.

The blow down map $\pi\mathrel{\mathop{\mathchar
58\relax}}\widehat{X}\rightarrow X$ restricts to the complement of the
exceptional divisor $E$, to give an $H$-equivariant identification
$X\setminus\pi(E)\cong\widehat{X}\setminus E$. Consequently, we may define in
$X$ an open subset,

$X^{H-\widehat{ss}}\coloneqq\pi(\widehat{X}^{H-ss}\setminus
E)\cong\widehat{X}^{H-ss}\setminus E.$

This open set admits a quasi-projective good quotient, which we denote

$X^{H-\widehat{ss}}\rightarrow
X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!_{\mathscr{L}}H,$

of which
$\widehat{X}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}H$
is a projective completion.

Hence, the price that we must pay for the construction of moduli spaces when
we do not have (‘ _ss=s for_ $\hat{U}$’) or ($\star_{d}$) for the unipotent
radical is that the resultant spaces will not be compact; that is, they will
be quasi-projective rather than projective, and the projective completions
yielded by the above process will not necessarily readily admit modular
interpretation.

In order to determine $X^{H-\widehat{ss}}$, we must keep track of the minimal
weight space and its basin of attraction at each stage of the process. To
simplify notation slightly, let’s suppose that we are at the very beginning of
the process, i.e. that we are working with $X$ and have yet to do any blow-
ups. We blow up along the closure of $C_{d_{\max}}(X)$ to get
$\pi^{(1)}\mathrel{\mathop{\mathchar 58\relax}}X^{(1)}\rightarrow X$ with
exceptional locus $E^{(1)}$. Now there are two cases.

  1. 1)

If $Z_{\min}\not\subseteq C_{d_{\max}(X)}$, then the minimal weight space in
the blow-up, $Z_{\min}(X^{(1)})$ is simply the proper transform of the old
one, and $\pi^{(1)}$ gives an identification

$(X^{(1)})^{0}_{\min}\setminus E\cong\pi^{(1)}((X^{(1)})^{0}_{\min}\setminus
E)=\\{x\in X^{0}_{\min}\mid\dim\operatorname{Stab}_{U}^{(}(p(x))<d_{\max}\\},$

where we recall that $p\mathrel{\mathop{\mathchar
58\relax}}X^{0}_{\min}\rightarrow Z_{\min}$ is the map given by

$x\longmapsto\lim_{t\rightarrow 0}\lambda(t)x.$

  2. 2)

If $Z_{\min}\subseteq C_{d_{\max}}(X)$, the situation is more complicated.
Roughly speaking, since we are blowing up the whole minimal weight space,
there will be nothing left of it, and consequently its place will be taken by
that part of the exceptional divisor which lies over the old minimal weight
space and corresponds to the lowest weight space that we haven’t blown up
entirely. More precisely, for $r\in\mathbb{Z}$, define $Z_{r}$ to be the
$\lambda$ weight space of weight $r$, and then let

$W^{0}_{r}=\\{x\in X^{0}_{\min}\mid\lim_{t\rightarrow 0}\lambda^{-1}(t)x\in
Z_{r}\\}.$

That is to say, $W^{0}_{r}$ is those points of $X^{0}_{\min}$ whose highest
weight is $r$. Now define

$r_{\min}=\min\\{r\in\mathbb{Z}\mid W^{0}_{r}\not\subseteq
C_{d_{\max}}(X)\\}.$

Then the new minimal weight space is that subset of
$(\pi^{(1)})^{-1}(Z_{\min})$ given by

$p^{(1)}\circ(\pi^{(1)})^{-1}(W^{0}_{r}\setminus C_{d_{\max}}(X)),$

where $p^{(1)}$ is the map induced by $p$ on the blow-up.

It is clear from the above that, although Corollary 3.3.4 gives us an open
subset of $X$ that has a categorical quotient, determining this locus in
practice can be non-trivial - especially if $U$ is not abelian. Furthermore,
if we are to find an interpretation of the categorical quotient
$X^{H-\widehat{ss}}\rightarrow
X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!_{\mathscr{L}}H$
as a moduli space for certain objects in a moduli problem, we must understand
the whole blow-up process in terms of the objects parameterised by $X$, and
such conditions as being in a certain $C_{d}$ or $W^{0}_{r}$ should have some
meaning intrinsic to the objects themselves.

Fortunately, as we will see in §7 in the blow-up process the sheaves of HN
length $\ell=2$ one can show that the following additional condition holds.

######  Definition 3.3.5.

We say that _$p$ preserves $U$-stabiliser dimension if_

[$\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x))$] $\forall
x\in
X^{0}_{\min}\quad\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x)).$

The condition
$\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x))$ simplifies
the blow-up considerably, as one can easily show that when this condition
holds the blow-up process never leaves case (1) above. This will be crucial to
our application to sheaves.

On the other hand, for $\ell>2$ this condition does not hold, as observed in
[HJ], and so we will sooner or later find ourselves in case (2). Interpreting
the various loci of the form $C_{d}$ and $W^{0}_{r}$ that arise in the blow-up
process for general length involves complicated nested Brill-Noether type
conditions, and as such seems rather intractable. This, as well as the failure
of condition ‘ _ss=s for $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’, is the reason that
we confine ourselves to the case of Harder-Narasimhan length 2 in the present
article. The general case will be dealt with in [BHJK].

######  Remark 3.3.6.

Although we will not consider the external case in detail here, one may use
all of the $\hat{U}$ Theorems to perform the quotient of $X$ by $H$ using an
external $\mathbb{G}_{m}$, in the following sense. We apply one of the
theorems above with $H$ replaced by $\hat{H}=H\rtimes\mathbb{G}_{m}$, and $X$
replaced by $X\times\mathbb{P}^{1}$, linearising the action by tensoring
$\mathscr{L}$ with $\mathscr{O}_{\mathbb{P}^{1}}(N)$ for $N\gg 0$. This yields
a projective scheme, which if semistability equals stability for the unipotent
radical is just
$(X\times\mathbb{P}^{1})\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}\hat{H}$,
containing an open subscheme which is a geometric quotient of a certain
$H$-invariant open subset $X^{H-\hat{s}}\subset X$ by $H$. This approach is of
course particularly useful if no admissible internal $\mathbb{G}_{m}$ exists,
for example if $H$ is unipotent.

## 4\. Moduli Spaces of Unstable Objects

Having presented the essentials of Non-reductive GIT in the form of the
$\hat{U}$ Theorem and its consequences, we turn to the question of
constructing moduli spaces of unstable objects, outlined at §1.1. Recalling
the set-up of §2.2, assume we have some moduli problem of interest posed using
reductive GIT. That is to say, we have an action of a reductive group $G$ on a
projective scheme $X$, such that orbits in $X$ correspond to isomorphism (or
equivalence) classes of the objects we wish to classify. Assume also that the
action is linearised with respect to some very ample line bundle, and that we
have made a choice of invariant inner product $\|.\|$, so that we may consider
the associated HKKN stratification of $X$. Then classical reductive GIT gives
us a ‘moduli space of semistables’ -a quotient of the open HKKN stratum- and
our goal here is to find reasonable quotients for the unstable HKKN strata,
and hence classify unstable objects too, by constructing for them their own
moduli spaces. The $\hat{U}$ Theorem turns out to be ideally suited to this
purpose.

### 4.1. Non-Reductive Quotients of unstable strata

Recall the notation of §2.2, and in particular of Definition 2.2.5. Since we
are over an algebraically closed field of characteristic zero, we can write
$P_{\beta}=U_{\beta}\rtimes\operatorname{Stab}\beta$ where $U_{\beta}$ is the
unipotent radical of $P_{\beta}$, and $\operatorname{Stab}\beta$ is as defined
in §2.2. We will also need one further piece of notation.

######  Definition 4.1.1.

Recalling Remark 2.2.8, we define $Z_{\beta}^{s}$ by analogy to be the stable
locus for the action of $\operatorname{Stab}\beta$ on $Z_{\beta}$, and
$Y_{\beta}^{s}=p_{\beta}^{-1}(Z_{\beta}^{s})$. We then define by analogy with
Definition 3.2.1 the _totally stable locus_ :

$Y_{\beta}^{ts}\mathrel{\mathop{\mathchar 58\relax}}=Y_{\beta}^{s}\setminus
U_{\beta}Z_{\beta}^{s}.$

Let us now give a precise statement and proof of Theorem 1.1.1.

######  Theorem 4.1.2.

[HJ] Let $G$ be a reductive group acting linearly on a projective scheme
$X\subseteq\mathbb{P}^{n}$. Let $\|.\|$ be a Weyl-invariant inner product on
$\operatorname{Lie}G$, and consider the associated HKKN stratification

$X=\bigsqcup\limits_{\beta\in\mathscr{B}}S_{\beta}.$

Suppose that for some $\beta\in\mathscr{B}\setminus\\{0\\}$ such that we have
condition (‘ _ss=s for_ $\hat{U}$’) as in Theorem 3.1.6, i.e. that

$\operatorname{Stab}_{U_{\beta}}(x)=\\{e\\}\quad\forall x\in Z_{\beta}.$

Then the following holds:

  1. 1)

The locus $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}$ as in Theorem 3.1.6 satisfies $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}\subseteq Y_{\beta}^{ss}$, and has a projective
good quotient

$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta,\varepsilon}}P_{\beta}$

  2. 2)

We have, for $0<\varepsilon\ll 1$,

$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta,\varepsilon}}P_{\beta}=\operatorname{Proj}\bigoplus_{i=0}^{\infty}H^{0}(\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta},\mathscr{L}^{\otimes i}_{\beta,\varepsilon}),$

where the quotient map is defined by the inclusion of the invariant
subalgebra, which is finitely generated. The quotient is independent of
$\varepsilon$, taken to be strictly positive and sufficiently small.

  3. 3)

This quotient restricts to give a quasi-projective geometric quotient of the
locus

$Y_{\beta}^{ts}=Y_{\beta}^{s}\setminus
UZ_{\beta}^{s}\rightarrow(Y_{\beta}^{s}\setminus
UZ_{\beta}^{s})/P_{\beta}\subseteq\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta,\varepsilon}}P_{\beta}$

  4. 4)

Furthermore, suppose condition (‘ _ss=s for $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’) holds, i.e. that we
have

$Z_{\beta}^{\operatorname{Stab}\beta/\lambda_{\beta}(\mathbb{G}_{m})-s}=Z_{\beta}^{\operatorname{Stab}\beta/\lambda_{\beta}(\mathbb{G}_{m})-ss}$

for the action of $\operatorname{Stab}\beta/\lambda_{\beta}(\mathbb{G}_{m})$
with respect to the linearisation $\mathscr{L}_{\beta}$. In this case, we have
$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}^{P_{\beta}-ss}=Y^{ts}_{\beta}$ so that these geometric and
projective quotients coincide.

###### Proof.

We use Theorem 3.1.6 and the Hilbert-Mumford criterion as stated in Theorem
3.2.3. We fix $\beta\in\mathscr{B}\setminus\\{0\\}$, and aim to provide a
quotient of $Y_{\beta}^{ss}$ by its action of $P_{\beta}$, which will be our
$H$. Since we have a linearisation of the action of $G\curvearrowright X$, we
have in particular an action of $P_{\beta}\leqslant G$ on the ambient
projective space, and by the standard properties that an orbit closures is
union of orbits and hence setwise invariant (see e.g.[Bri] 1.11), we see that
$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\beta}$ is
setwise invariant under this $P_{\beta}$ action. So $P_{\beta}$ acts on the
projective variety $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\beta}$, which will play the role of $X$.

Up to conjugacy, there is a natural choice of distinguished cocharacter,
namely an optimal cocharacter for $Y_{\beta}^{ss}$. To get a conjugacy class
representative, we take $T$ a maximal torus of $G$, and intersect with
$P_{\beta}$ to get a maximal torus $T_{\beta}$ of $P_{\beta}$; that is, a
maximal torus $T_{\beta}\leqslant\operatorname{Stab}\beta$. Let
$\lambda_{\beta}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{G}_{m}\rightarrow
P_{\beta}$ be, amongst the $P_{\beta}$-conjugacy class denoted by $\beta$, the
unique representative that lies in $T_{\beta}$. This is the cocharacter we
will use to grade $U_{\beta}$. Indeed, since $P_{\beta}=P(\lambda_{\beta})$ as
defined in §2.2, all the weights of the adjoint action of $\lambda_{\beta}$ on
$\operatorname{Lie}P_{\beta}$ must be non-negative. This establishes that the
unipotent radical $U_{\beta}\leqslant P_{\beta}$ is graded by
$\lambda_{\beta}$.

For the reductive part, observe that $\lambda_{\beta}(\mathbb{G}_{m})$ is by
definition central in $\operatorname{Stab}\beta$, so there is a character
$\chi_{\beta}$ of $P_{\beta}$, given by the composition

$\chi_{\beta}\mathrel{\mathop{\mathchar 58\relax}}P_{\beta}\rightarrow
L_{\beta}\rightarrow\mathbb{G}_{m}.$

The first map in the composition is just the quotient map with kernel
$U_{\beta}$, and the second is obtained by extending to
$\operatorname{Stab}\beta$ the character of the torus $T_{\beta}$ dual to
$\lambda_{\beta}$ with respect to our chosen inner product; this is possible
because $\lambda_{\beta}$ is central in $L_{\beta}$. We then define the
linearisation $\mathscr{L}_{\beta,\varepsilon}$ to be $\mathscr{L}$ twisted by
the character $(1+\varepsilon)(-\chi_{\beta})$. By construction, for
$0<\varepsilon\ll 1$ this is adapted, since the minimal weight of
$\lambda_{\beta}$ is $\beta$ and all other weights are strictly greater. In
fact, by the arguments of [BDHK20] the choice of $\varepsilon$ will not matter
so long as it is taken sufficiently small and strictly positive.

With all of this in place, the theorem follows from the first two statements
in Theorem 3.2.3.

∎

######  Remark 4.1.3.

Our construction of moduli spaces for unstable sheaves is, morally speaking, a
matter of applying Theorem 4.1.2, but it not quite a straightforward
application of this theorem, because we do not have condition (‘ _ss=s for_
$\hat{U}$’). However, for length two sheaves we will use Theorem 3.1.8
instead, essentially proving a variant of Theorem 4.1.2 that holds in this
setting.

### 4.2. Toy Example: Unordered points on $\mathbb{P}^{1}$

Now that the ideas have been introduced, the general situation of quotienting
unstable strata is best clarified by working all of this out in detail for a
simple example. Our description of the HKKN stratification here follows
[Kir84]. Take $G\coloneqq\mathrm{SL}(2)$ and consider the natural action on

$X\coloneqq\text{Sym}^{n}(\mathbb{P}^{1})\cong\mathbb{P}^{n}$

induced by the action on $\mathbb{P}^{1}$. We think of $X$ as the space of
degree $n$ homogeneous polynomials $F(x,y)$, with the zeros of the polynomial
giving the choice of points. This comes equipped with the action of $G$
induced by the dual action on $\mathbb{P}^{1}$. That is,

$(g\cdot F)(x,y)=F(g^{-1}\cdot(x,y)).$

The linearisation is that induced from the standard
$\mathscr{O}(1)$-linearisation on $\mathbb{P}^{1}$: in representation-
theoretic terms, this is $\operatorname{Sym}^{n}V$, where $V$ is the two-
dimensional irreducible representation. Our chosen maximal torus $T$ is
identified with $\mathbb{G}_{m}$ via the isomorphism

$T\rightarrow\mathbb{G}_{m}$ $\begin{pmatrix}t&0\\\
0&t^{-1}\end{pmatrix}\longmapsto t.$

This torus acts on $X$ diagonally with respect to the basis formed by
monomials in $x$ and $y$, and hence the weights of $T$ are

$\\{n-2i\in\mathbb{Z}\mid i=0,...n\\}\subset\mathbb{Z}.$

We observe that if more than half of the points are at zero, there will be no
negative weight, and if more that half are at infinity, there will be no
positive weight. We therefore find, as is well known, that the first level of
our stratification, the $G$-semistable locus, is:

$S_{0}=X^{ss}=\\{x\in X\mid\text{not more than half of the $n$ points coincide
anywhere}\\}.$

We choose the positive Weyl chamber
$\mathfrak{t}^{+}\cong\mathbb{R}_{\geqslant 0}$, and so the indexing set is:

$\mathscr{B}=\Big{\\{}0\Big{\\}}\cup\Big{\\{}2i-n\mid\frac{n}{2}<i\leqslant
n\Big{\\}}.$

Immediately from the definition we compute that the parabolics and stabilisers
are the same for every member of the indexing set, so that for all $i$ we
have:

$P_{n-2i}=P\coloneqq\Big{\\{}\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix}\mid
a\in k^{*},b\in k\Big{\\}}$

That is, the standard Borel subgroup in $\mathrm{SL}_{2}$. We also compute

$\operatorname{Stab}(n-2i)=T.$

The $Z_{\beta}$’s each consist of a Stab$\beta$-invariant single point:

$Z_{n-2i}=\\{x\in X\mid\text{ Exactly }i\text{ of the points are at $\infty$
and the rest are at $0$}\\},$

and we find:

$Y_{n-2i}=\\{x\in X\mid\text{ Exactly }i\text{ of the points are at
$\infty$}\\},$

which we can check directly is $P$-invariant.

As a consequence, the reductive quotient of each stratum as described in §2.3
is just a point, which is clearly unsatisfactory. Let us see how using non-
reductive GIT can improve on this.

We consider the shifted linearisation $\mathscr{L}_{\beta}$ on a specific
stratum, as used in [Hos11]. For a fixed $i$ we have the $T$-character

$\chi_{n-2i}\mathrel{\mathop{\mathchar 58\relax}}T\rightarrow\mathbb{G}_{m}$
$\begin{pmatrix}t&0\\\ 0&t^{-1}\end{pmatrix}\longmapsto t^{n-2i}$

and we define $\mathscr{L}_{n-2i}$ to be the linearisation of the $G$ action
given by twisting $\mathscr{L}$ by $\chi_{n-2i}$. The condition (‘ _ss=s for
$\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$_’) holds
trivially, since $\operatorname{Stab}\beta=\lambda_{\beta}(\mathbb{G}_{m})$.
The (semi)stable locus of $Z_{\beta}$ with respect to $\mathscr{L}_{n-2i}$ is

$Z_{n-2i}^{s}=Z_{n-2i}^{ss}=Z_{n-2i}=\\{\text{pt}\\}.$

Hence also, for each $\beta_{i}=(n-2i)\in\mathscr{B}$, we have

$Y_{n-2i}^{s}=Y_{n-2i}^{ss}=Y_{n-2i}.$

Taking the projective closure yields:

$\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{n-2i}=\\{x\in
X\mid\text{At least }i\text{ of the points are at
$\infty$}\\}\cong\mathbb{P}^{n-i},$

so, taking the $G$-sweep of the $Y_{n-2i}^{ss}$, the stratification is:

$S_{n-2i}=\\{x\in X\mid\text{Exactly }i\text{ of the points coincide}.\\}$

Now we can apply the Theorem 4.1.2 to find quotients, once we have verified
the remaining remaining hypotheses. The subgroup $P$ fixes the point
$\infty\in\mathbb{P}^{1}$, so $P$ acts on $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu^{ss}_{n-2i}$ as stated
above. Indeed, $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu^{ss}_{n-2i}\cong\mathbb{P}^{n-i}$ and the action is the natural one, and
indeed it extends to an SL$(2)$ action on each $\mathbb{P}^{n-i}$. Now take

$\begin{pmatrix}0&\eta\\\ 0&0\end{pmatrix}\in\text{Lie}(\mathbb{G}_{a})\cong
k$ $g_{\lambda}=\begin{pmatrix}\lambda&0\\\ 0&\lambda^{-1}\end{pmatrix}.$

We see adjoint action of $\mathbb{G}_{m}$ on the Lie algebra of
$\mathbb{G}_{a}$ is via

$g_{\lambda}\cdot X\cdot g_{\lambda}^{-1}=\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}\begin{pmatrix}0&\eta\\\
0&0\end{pmatrix}\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}^{-1}=\begin{pmatrix}0&\lambda^{2}\eta\\\
0&0\end{pmatrix}.$

So the $\mathbb{G}_{m}$ in the Levi decomposition of $P$ acts on the Lie
algebra Lie$(\mathbb{G}_{a})$ of the $\mathbb{G}_{a}$ with sole weight $2>0$,
in agreeement with the above.

Now we turn to weights of the $\mathbb{G}_{m}$ action from our linearisation.
We began with the linearisation $\mathscr{L}$, with respect to which the
weights on $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{n-2i}$ are $\\{(2i-n),...n\\}$, and twist by the rational character
associated to

$(1+\varepsilon)(n-2i),$

where $\varepsilon>0$ is a small rational number.

It remain to verify the final hypothesis, to determine whether (‘ _ss=s for_
$\hat{U}$’) holds. For this it suffices to note that no point of $Z_{n-2i}$ is
fixed under the translating action of the $\mathbb{G}_{a}$, unless $i=n$ and
we are considering the case where all points coincide at $\infty$. This
$Y_{\beta}$ corresponds to the $S_{\beta}$ stratum where all points coincide,
and it is clear in this case that the quotient ought to be a point. At all
other levels of the stratification, the hypotheses are verified and we can
apply the Theorem 4.1.2.

The theorem tells us that we have

$Y_{n-2i}^{ts}=(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{n-2i})\setminus(\mathbb{G}_{a}\cdot Z_{n-2i}).$

Now we recognise the $\mathbb{G}_{a}$-sweep of $Z_{n-2i}$ as being the locus
where $i$ points are at $\infty$ and the remaining points are affine and all
coincide somewhere. So:

$Y_{n-2i}^{ts}=\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{n-2i}^{P-ss}=\\{\text{Exactly $i$ points at $\infty$ and the remaining
points don't all coincide}\\}.$

Since Theorem 4.1.2 gives us a geometric quotient of this locus, we see that
using non-reductive GIT gives a far better answer than the methods of §2.3.

## 5\. Moduli of Semistable Sheaves

For the remainder of this article, we will be concerned with the moduli of
coherent sheaves on a projective scheme. Our ultimate goal is to apply the
results of §4 to quotient unstable strata in Simpson’s reductive GIT
construction of the moduli space of semistable sheaves, to obtain moduli
spaces of unstable sheaves of fixed Harder-Narasimhan type, in the case of
Harder-Narasimhan length 2. In this section we give a very brief review of
semistable sheaves and Simpson’s moduli construction [Sim94]. The standard
reference for this classical theory is [HL10], and a very nice account of its
historical development is given in [New78], §5.1.

Let $X$ be a projective scheme over $k$ with a fixed ample line bundle
$\mathcal{O}_{X}(1)$. For a coherent sheaf $F$ on $X$, we let
$P(F,t)\in\mathbb{Q}[t]$ denote the Hilbert polynomial of $F$ with respect to
$\mathcal{O}_{X}(1)$. As the Hilbert polynomial of such a sheaf is locally
constant in flat families, from a moduli perspective it is natural to fix this
discrete invariant. Hence, we fix $P\in\mathbb{Q}[t]$, and from now on, unless
specified otherwise, all coherent sheaves will have this Hilbert polynomial.

### 5.1. Semistable sheaves

We recall the definition of semistability for sheaves. We define a partial
ordering $\preccurlyeq$ on $\mathbb{Q}[t]$ by saying that $P\preccurlyeq Q$
whenever

$\frac{P(n)}{P(m)}\leqslant\frac{Q(n)}{Q(m)}$

for $m\gg n\gg 0$. We define $\prec$ analogously by replacing $\leqslant$ with
$<$.

######  Definition 5.1.1.

We say $F$ is semistable if for all non-zero proper subsheaves
$F^{\prime}\lneq F$, we have $P(F^{\prime})\preccurlyeq P(F)$. If we have
$P(F^{\prime})\prec P(F)$ for all non-zero proper subsheaves, we say $F$ is
stable.

######  Remark 5.1.2.

The definition of semistability above is due to Rudakov [Rud97], and is
equivalent to the notion of Gieseker-semistability used by Simpson. In
particular, to be semistable it is necessary that a sheaf be pure; that is,
the dimensions of the supports of all nonzero subsheaves must be equal. Hence,
if $X$ is a smooth projective curve, the above is also equivalent to the
notion of slope semistability for vector bundles on a curve. Our reason for
using this formulation of semistability is that it allows us to assign Harder-
Narasimhan filtrations to any coherent sheaf, not just the pure ones that
would be allowed under Gieseker’s definition.

Every semistable sheaf has a Jordan-Hölder filtration ([HL10] I§1.5) that is,
a filtration such that every successive subquotient is stable, and all
subquotients have the same Hilbert polynomial as $F$. This filtration, unlike
the Harder-Narasimhan filtration we will consider later, is not canonical.
However, its associated graded sheaf is determined up to isomorphism by $F$.

######  Definition 5.1.3.

We say two semistable sheaves are _S-equivalent_ if the associated graded
sheaves from their Jordan-Hölder filtrations are isomorphic.

######  Remark 5.1.4.

The moduli-theoretic motivation for this definition is as follows. If a
semistable sheaf $F$ can be given as an extension

$0\rightarrow F_{1}\rightarrow F\rightarrow F_{2}\rightarrow 0$

of two other sheaves, $F_{1}$, $F_{2}$ also semistable, then once can
construct a family $\mathscr{F}$ of sheaves over $X\times\mathbb{A}^{1}$ \-
parameterised for example by a suitable line in
$\operatorname{Ext}^{1}(F_{2},F_{1})$ \- such that $\mathscr{F}_{0}\cong
F_{1}\oplus F_{2}$, and for $t\neq 0$ we have $\mathscr{F}_{t}\cong F$. This
is an example of so-called jump phenomena, and shows us that any separated
moduli space including all semistable sheaves must identify $F$ and
$F_{1}\oplus F_{2}$. In particular, it shows that any such moduli space must
identify all sheaves in the same S-equivalence class.

### 5.2. The GIT problem

By the Le Potier estimates ([Sim94] Theorem 1.1), the family of semistable
sheaves over $X$ with Hilbert polynomial $P$ is bounded. Hence, for $n\gg 0$,
all such sheaves are $n$-regular in the sense of Castelnuovo-Mumford ([Mum66]
Lecture 14), i.e. we have $H^{i}(X,F(n-i))=0$ for all $i>0$. As observed by
Mumford (_loc. cit._), $n$-regularity implies that

  * •

The sheaf $F(n)$ is generated by global sections, so that the evaluation map

$H^{0}(F(n))\otimes\mathcal{O}_{X}(-n)\rightarrow F$

is surjective, by right exactness of tensor product;

  * •

$H^{i}(F(n))=0$ for $i>0$, so $\dim H^{0}(F(n))=P(F,n)$.

This means that, if we fix a $k$-vector space $V$ of dimension $P(n)$, and are
prepared to make a choice of isomorphism $H^{0}(F(n))\simeq V$, we can
instantiate any $n$-regular sheaf $F$ with Hilbert polynomial $P$ as the image
of a quotient map

$q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F.$

Hence we can associate to any such sheaf a point in the Quot scheme

$[q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F]\in\operatorname{Quot}_{n}\coloneqq\operatorname{Quot}(V\otimes\mathcal{O}_{X}(-n),P)$

parameterising quotient sheaves of $V\otimes\mathcal{O}_{X}(-n)$ with Hilbert
polynomial $P$. By a quotient sheaf we mean nothing more than an equivalence
class of surjections $q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow F$, where two such maps
$q_{i}\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow F_{i}$, $i=1,2$, are regarded
as equivalent, and hence correspond to the same point in the Quot scheme, if
they have the same kernel; or, equivalently, if there exists an isomorphism of
sheaves $\varphi\mathrel{\mathop{\mathchar 58\relax}}F_{1}\rightarrow F_{2}$
making the below diagram commute:

[1]
${V\otimes\mathcal{O}_{X}(-n)}$${V\otimes\mathcal{O}_{X}(-n)}$${F_{1}}$${F_{2}.}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{2}}$$\scriptstyle{\varphi}$

It will be helpful to the reader to bear in mind that the Quot scheme is
projective but not irreducible. The points associated to $n$-regular sheaves
will all lie in the open subscheme

$Q=Q_{n}\subset\operatorname{Quot}_{n}$

defined to be that locus of quotient $q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow F$ such that $H^{0}(q(n))$ is
an isomorphism.

The point of $Q$ associated to a sheaf is not canonical: it depends on the
choice of isomorphism $H^{0}(F(n))\simeq V$ above. In other words, a point of
the Quot scheme corresponds to a sheaf $F$, together with a choice of basis of
$H^{0}(F(n))$. This gives rise to a natural action of $\mathrm{GL}(V)$ on
$\operatorname{Quot}_{n}$, induced by the action on $V$, which preserves $Q$.
For

$[q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F]\in\operatorname{Quot}_{n}$

and $g\in SL(V)$, we define $g\cdot[q]$ to be the quotient map given by the
composition

${V\otimes\mathcal{O}_{X}(-n)}$${V\otimes\mathcal{O}_{X}(-n)}$${F.}$$\scriptstyle{g^{-1}\otimes\text{id}}$$\scriptstyle{q}$

For later convenience, we note the following properties of this action on $Q$.

######  Lemma 5.2.1.

([HL10], §4.3)

  1. 1)

Orbits of closed points of $Q$ under the $\mathrm{GL}(V)$ action correspond to
isomorphism classes of sheaves.

  2. 2)

For a closed point $[q]\in Q$ corresponding to a globally generated sheaf $F$,
there is an isomorphism
$\operatorname{Stab}_{\mathrm{GL}(V)}(q)\cong\operatorname{Aut}(F)$.

###### Proof.

The first statement is evident from the discussion above. For the second
statement we note that any automorphism $\varphi\mathrel{\mathop{\mathchar
58\relax}}F\rightarrow F$ is determined by the map of vector spaces

$H^{0}(\varphi(n))\mathrel{\mathop{\mathchar 58\relax}}H^{0}(F(n))\rightarrow
H^{0}(F(n)).$

Conversely, the isomorphism $H^{0}(q(n))\mathrel{\mathop{\mathchar
58\relax}}V\rightarrow H^{0}(F(n))$ coming from the point $q\in Q$ allows us
to view an element of $g\in\mathrm{GL}(V)$ as an isomorphism

$H^{0}(q(n))\circ g\circ H^{0}(q(n))^{-1}\mathrel{\mathop{\mathchar
58\relax}}H^{0}(F(n))\rightarrow H^{0}(F(n)).$

By the definition of equivalence of points in the Quot scheme given by the
diagram above, this isomorphism of twisted global sections induces an
isomorphism of $F$ if and only if $g\in\operatorname{Stab}(q)$. ∎

######  Remark 5.2.2.

  1. 1)

From the above proposition, it is clear that the centre
$\mathcal{Z}(\mathrm{GL}(V))$ of $\mathrm{GL}(V)$ acts trivially, and so in
our GIT set-up we will pass to considering an action of $SL(V)$. Up to
replacing the linearisation with an appropriate tensor power, which we are
certainly prepared to do, this is equivalent to considering the action of
$\mathrm{GL}(V)/\mathcal{Z}(\mathrm{GL}(V))=\mathrm{PGL}(V)$. See [HL10] 4.3
for more discussion of this.

  2. 2)

The above properties may be interpreted stack-theoretically, as saying that we
have a presentation of $\mathcal{C}oh_{X,P}$, the stack of coherent sheaves on
$X$ with Hilbert Polynomial $P$, as an infinite increasing union of quotient
stacks. This approach is taken, for example, in [Hos18], [BHK18]. For this it
is necessary that we work with $\mathrm{GL}(V)$, since we must record all
automorphisms of the sheaves, including the $\mathbb{G}_{m}$ common to all.

The final ingredient necessary to turn this set-up into a reductive GIT
problem is a linearisation. Take $m>n$ and write $H\coloneqq
H^{0}(\mathscr{O}_{X}(m-n)),$ following [HK12]. Now, we have a map

$\operatorname{Quot}_{n}\rightarrow\text{Gr}(V\otimes H,P(m))$
$[q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathscr{O}_{X}(-n)\rightarrow
F]\longmapsto[V\otimes H\twoheadrightarrow H^{0}(F(m))].$

to the Grassmannian of $P(m)$-dimensional vector space quotients

$V\otimes H\rightarrow W.$

Furthermore, because Ker$(q)$ will be $m$-regular for $m$ large enough, if we
take $m\gg n\gg 0$, this map is an embedding ([HL10], 2.2.4). Following this
up with the usual Plücker embedding of the Grassmannian

$\text{Gr}(V\otimes
H,P(m))\hookrightarrow\mathbb{P}\big{(}\bigwedge^{P(m)}(V\otimes
H)^{*}\big{)}\eqqcolon\mathbb{P}^{N}$

we obtain an equivariant embedding $i_{m,n}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Quot}_{n}\hookrightarrow\mathbb{P}^{N}$, which we may
restrict to $Q$. Denote by $\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu$ the closure of $Q$ in
this projective space. Since $\operatorname{Quot}_{n}$ is projective ( see
e.g. [FGI+05]), we have $\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu\subset\operatorname{Quot}_{n}$. Let us denote by $\mathscr{L}_{m,n}$ the
very ample line bundle on $\operatorname{Quot}_{n}$ obtained from pulling back
$\mathscr{O}_{\mathbb{P}^{N}}(1)$ along this embedding.

We are now in the best setting for reductive GIT: we have a projective scheme
$\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu$ carrying an
action of a reductive $G$, and the very ample line bundle corresponding to
this embedding has a natural linearisation of the $G$ action. This is the set-
up that Simpson uses to prove the following theorem.

######  Theorem 5.2.3.

[Sim94] Let $X$ be a projective scheme, and fix a Hilbert Polynomial
$P\in\mathbb{Q}[t]$. Then

  1. 1)

If $m\gg n\gg 0$, the semistable locus for the GIT problem above is

$\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu^{ss}=\\{[q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathscr{O}_{X}(-n)\rightarrow F]\in Q\mid F\text{ is a
semistable sheaf in the sense of \ref{ss sheaf defn}}\\}.$

Similarly, the GIT stable locus consists of those points of $Q$ corresponding
to stable sheaves.

  2. 2)

The GIT quotient $M_{P,ss}\coloneqq\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{m,n}}G$
is a coarse moduli space for $S$-equivalence classes of semistable coherent
sheaves on $X$ with Hilbert polynomial $P$.

This moduli space, the natural successor to the moduli space of semistable
vector bundles on a smooth curve, is by now a central object in algebraic
geometry, and is very well studied from many different perspectives. From our
point of view, however, it is just a starting point. Our eventual goal is to
extend the classification of coherent sheaves on $X$ to include the unstable
ones as well, so that Simpson’s moduli space may be thought of as a special
case: that of Harder-Narasimhan length one.

## 6\. Stratifications for Sheaves

Theorem 5.2.3 gives a very satisfactory tool for understanding semistable
sheaves and their moduli, but says nothing about sheaves that are _unstable_ ,
i.e. do not satisfy Definition 5.1.1. In understanding those, the following is
the natural preliminary step.

### 6.1. Harder-Narasimhan Stratification

To any sheaf, one may associate a canonical filtration by subsheaves, first
introduced by Harder and Narasimhan in [HN75]. A proof of its existence and
uniqueness in our setting may be found in [HL10].

######  Definition 6.1.1.

Let $F$ be a sheaf on $X$. The _Harder-Narasimhan filtration_ (or _HN
filtration_) of $F$ is the unique filtration by subsheaves

$0=F^{0}\lneq F^{1}\lneq\dots\lneq F^{\ell}=F$

such that the subquotients $F_{i}\coloneqq F^{i}/F^{i-1}$ are semistable with
strictly decreasing Hilbert polynomials $P(F_{1})\succ P(F_{2})\succ\dots\succ
P(F_{\ell})$. The _associated graded_ of $F$ is
$\mathrm{gr}(F)\coloneqq\oplus_{i=1}^{\ell}F_{i}.$ We say that $F$ has
_Harder-Narasimhan type_ $\tau=(P(F_{1}),\dots,P(F_{\ell}))$, and Harder-
Narasimhan length $\ell$.

######  Notation 6.1.2.

_From now on, to prevent needless repetition, given a sheaf $F$ the terms in
the Harder-Narasimhan filtration of $F$ will always be denoted by $F^{i}$, and
the subquotients will always be denoted by $F_{i}$._

######  Remark 6.1.3.

By additivity of Hilbert Polynomials in exact sequences, if $F$ is of type
$\tau=(P_{1},\dots,P_{\ell})$, then $\sum_{i}P_{i}=P$.

By results of Shatz and Nitsure ([Sha77, Nit11]), there is an upper semi-
continuous ‘HN type function’, which, given any family of sheaves
$\mathscr{F}\rightarrow S$, and a point $s\in S$, gives the Harder-Narasimhan
type of $\mathscr{F}_{s}$. Hence, defining

$\operatorname{Quot}_{n}^{\tau}\coloneqq\\{[q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F]\in\operatorname{Quot}_{n}\mid F\text{ has HN type $\tau$}\\},$

one has a HN stratification999Sometimes also called the Shatz stratification.
into locally closed subschemes

$\operatorname{Quot}_{n}=\bigsqcup_{\tau}\operatorname{Quot}^{\tau}_{n}.$

This in turn gives a stratification

$Q=\bigsqcup_{\tau}Q_{\tau}$

defined in the same way.

######  Definition 6.1.4.

We let $S_{\tau}$ denote the open subscheme of $Q_{\tau}$ consisting of those
quotients $[q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow F]$ such that $H^{0}(q(n))$
is an isomorphism.

######  Remark 6.1.5.

All of the above can be translated into the language of stacks. It was shown
in [Hos18] that there is a Harder-Narasimhan type stratification on the stack,
which may be thought of, in an appropriate sense, as a kind of colimit of the
Harder-Narasimhan stratifications of the Quot schemes, just as the stack
itself is in some sense their colimit.

### 6.2. Instability stratification for sheaves

In addition to the stratification by Harder-Narasimhan type one also has,
according to the general principles described in §2.2, an HKKN instability
stratification of the Quot scheme,

$Q=\bigsqcup_{\beta\in\mathcal{B}(n,m)}S_{\beta}$

coming from Simpson’s GIT set-up.101010This stratification depends in a non-
trivial way upon the parameters $m$ and $n$, so strictly speaking the first
question one should ask is whether there is any meaningful sense in which
these stratifications can be said to stabilise. Fortunately, as we shall see,
there is. The natural thing to do, then, is to compare the two. Indeed, since
they both measure the failure of a sheaf to be semistable, one might hope that
they would coincide. The relationship between these stratifications was
studied by Hoskins and Kirwan. First, in [HK12], it was shown that for a pure
HN type111111That is, a HN type of a pure sheaf. $\tau$ and for $m\gg n\gg 0$
depending on $\tau$, there is a conjugacy class $\beta_{n,m}(\tau)$ of
rational cocharacters of $G$ such that the HN stratum $Q^{\tau}$ is contained
(set theoretically) in the instability stratum $S_{\beta_{n,m}}(\tau)$. In
[Hos18], this result was extended to a scheme theoretic statement for all HN
types of coherent sheaves. Theorem 6.2.5 summarises these results. First, give
the definition of the HKKN index $\beta_{n,m}(\tau)$ corresponding to a HN
type $\tau$.

######  Definition 6.2.1.

[Hos18] For a Harder-Narasimhan type $\tau=(P_{1},\dots,P_{l})$, we define a
rational $\mathrm{SL}_{P(n)}$-cocharacter
$\lambda_{\beta(n,m,\tau)}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\mathrm{SL}_{P(n)}$, such that
$\lambda_{\beta(n,m,\tau)}(t)$ is diagonal with entries

$(t^{\beta_{1}},\dots t^{\beta_{1}},t^{\beta_{2}},\dots
t^{\beta_{2}},\dots,t^{\beta_{\ell}},\dots,t^{\beta_{\ell}})$

where each

$\beta_{i}\mathrel{\mathop{\mathchar
58\relax}}=\frac{P(m)}{P(n)}-\frac{P_{i}(m)}{P_{i}(n)}$

is repeated $P_{i}(n)$ times. We write $\beta(n,m,\tau)$ for the conjugacy
class of this cocharacter under conjugation by its associated parabolic
$P_{\lambda_{\beta(n,m,\tau)}}$, as defined in §2.2.

######  Remark 6.2.2.

Instead of working with rational cocharacters -that is, cocharacters with
rational weights- we could instead define $\lambda_{\beta(n,m,\tau)}(t)$ to be
the unique indivisible cocharacter obtained as a positive rational multiple of
the cocharacter defined above.

######  Remark 6.2.3.

Because two values will not always uniquely specify a polynomial, even for
$m\gg n\gg 0$ the map $\tau\mapsto\beta(n,m,\tau)$ will not be injective if
$\dim X>1$. However, in [Hos18] Hoskins and Kirwan have shown that there is an
asymptotic HKKN stratification on the stack of coherent sheaves, and this
coincides with the stratification by HN type.

A cocharacter $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow\mathrm{SL}(V)$ is the same thing as the
data of its weight filtration

$\\{0\\}\subseteq\dots\subseteq V^{i}\subseteq V^{i-1}\subseteq\dots\subseteq
V$

where $V^{i}$ denotes the span of those vectors on which $\lambda$ acts with
weight at least $i$, so that only finitely many of the
$V_{i}\mathrel{\mathop{\mathchar 58\relax}}=\faktor{V^{i}}{V^{i-1}}$ are non-
zero. Thus, given a point

$[q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F]\in\operatorname{Quot}_{n},$

such a weight filtration induces a filtration of $F$, by defining

$F^{i}\coloneqq q(V^{i}\otimes\mathscr{O}_{X}(-n)).$

Limits under cocharacters are easy to compute.

######  Lemma 6.2.4.

([HL10] Lemma 4.4.3) We have

$\lim_{t\rightarrow 0}\lambda(t)[q]=[\oplus_{i}q_{i}\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow\oplus_{i}F_{i}]\in\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu$

where for each $i$,

$q_{i}\mathrel{\mathop{\mathchar
58\relax}}V_{i}\otimes\mathcal{O}_{X}(-n)\rightarrow F_{i}$

is the surjection induced from $q$. In words: $F$ is mapped to the associated
graded of the filtration determined by $\lambda$.

The origin of the cocharacter $\lambda_{\beta(n,m,\tau)}$, then, is not so
mysterious. In looking for optimal cocharacters for $F$, a sensible initial
guess is that this filtration should be $F$’s Harder-Narasimhan filtration. As
in [Hos11], one then obtains $\lambda_{\beta(n,m,\tau)}$, by maximising (with
Lagrange multipliers) the normalised Hilbert-Mumford function over all weight
filtrations that induce the Harder-Narasimhan filtration of $F$.

The fixed point locus $\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu^{\lambda_{\beta}}$ thus consists of all quotients

$[\oplus_{i}q_{i}\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow\oplus_{i}F_{i}]\in\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu$

that split with respect to the weight filtration of $\lambda_{\beta}$. Because
the Hilbert polynomials of the summand image sheaves must be locally constant,
there is a component $F$ of $\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu^{\lambda_{\beta}}$ where
for each $i$, the sheaf $F_{i}$ has Hilbert polynomial $P_{i}$. And so,
writing,

$\operatorname{Quot}_{n,i}=\operatorname{Quot}(V_{i}\otimes\mathcal{O}_{X}(-n),P_{i})$

one has:

$F\cong\operatorname{Quot}_{n,1}\times\dots\times\operatorname{Quot}_{n,\ell}.$

As above, we let $Q_{i}\subset\operatorname{Quot}_{n,i}$ be the open locus of
pure dimension $d$ sheaves such that $H^{0}(q_{i}(n))$ is an isomorphism. This
allows one to define:

$Z_{\tau}=\\{\oplus_{i}q_{i}\in F\mid q_{i}\in Q_{i}\\}.$

Thus $Z_{\tau}$ is open in $F$, but its closure is a proper subscheme. Now let
$Q^{ss}\subset Q_{i}$ be the open locus consisting of those sheaves which are
in addition semistable, and define

$Z^{ss}_{\tau}=\\{\oplus_{i}q_{i}\in F\mid q_{i}\in Q^{ss}_{i}\\}.$

Note that all sheaves in $Z^{ss}_{\tau}$ are of Harder-Narasimhan type $\tau$,
and isomorphic to their Harder-Narasimhan graded. We now have the notation we
need to state the results of [HK12] [Hos18].

######  Theorem 6.2.5 ([HK12, Hos18]).

Let $\tau=(P_{1},\dots,P_{l})$ be a HN type for sheaves on $X$ with Hilbert
polynomial $P$; then, for $m\gg>n\gg 0$, the following statements hold for
$\beta\mathrel{\mathop{\mathchar 58\relax}}=\beta(n,m,\tau)$.

  1. i)

Isomorphism classes of sheaves of Harder-Narasimhan type $\tau$ are in
bijection with $G$-orbits of points in $S_{\tau}$.

  2. ii)

The conjugacy class $\beta\in\mathcal{B}$ is an instability index for the $G$
action on $Q$ with respect to $\mathscr{L}_{n,m}$ and $S_{\tau}$ is a closed
subscheme of the corresponding instability stratum $S_{\beta}$.

  3. iii)

We have $Z_{\tau}^{ss}\subset Z_{\beta}^{ss}$ as a
$\operatorname{Stab}\beta$-invariant closed subscheme.

  4. iv)

If $Y_{\tau}^{ss}\mathrel{\mathop{\mathchar
58\relax}}=p_{\beta}^{-1}(Z_{\tau}^{ss})$, then $S_{\tau}=G\cdot
Y_{\tau}^{ss}\cong G\times^{P_{\beta}}Y_{\tau}^{ss}.$

  5. v)

The subcheme $Y_{\tau}^{ss}$ is precisely the locus of points
$[q\mathrel{\mathop{\mathchar 58\relax}}V\otimes\mathcal{O}_{X}(-n)\rightarrow
F]\in\operatorname{Quot}$ such that $H^{0}(q(n))$ is an isomorphism, $F$ has
Harder-Narasimhan type $\tau$, and the weight filtration of $\lambda_{\beta}$
induces the Harder-Narasimhan filtration of $F$. The map
$p_{\beta}\mathrel{\mathop{\mathchar 58\relax}}Y_{\tau}^{ss}\rightarrow
Z_{\tau}^{ss}$ takes these sheaves to their associated graded.

We can give explicit GIT characterisations of the subschemes $Y_{\tau}^{ss}$
and $Z_{\tau}^{ss}$, in the manner usual for instability stratifications. A
detailed account is in [Hos11]; we give an overview here to fix notation for
later.

######  Remark 6.2.6.

We recall from Remark 2.2.4 that $P_{\beta}=P_{\lambda_{\beta}}$ is the
subgroup of $\mathrm{SL}(V)$ consisting of elements that respect the weight
filtration of $\lambda_{\beta(n,m,\tau)}$, i.e. block upper triangular
matrices. Thus $P_{\lambda_{\beta}}$ splits as a semidirect product of a
reductive part, $\operatorname{Stab}\beta$, which consists of block diagonal
matrices, and its unipotent radical $U_{\beta}$, which consists of block upper
triangular matrices with identity matrices on the diagonal blocks. Thus one
has

$\operatorname{Stab}\beta\cong\left(\prod_{i}\mathrm{GL}(V_{i})\right)\cap\mathrm{SL}(V),$

and the action on $Z_{\tau}$ is via each $\mathrm{GL}(V_{i})$ acting on
$Q_{i}$ in the way described at §5.2.

To analyse the GIT semistability with respect to $\operatorname{Stab}\beta$,
[HK12] considers the subgroup

$\prod_{i=1}^{\ell}\mathrm{SL}_{P_{i}}\leqslant\operatorname{Stab}\beta.$

This group acts on

$Z_{\tau}\subset Q_{1}\times\dots\times Q_{\ell}$

diagonally via the natural action of $\mathrm{SL}_{P_{i}}$ on $Q_{i}$,
discussed in §5.2. Furthermore, the linearisation induced for this action by
the restriction of the linearisation $\mathscr{L}=\mathscr{L}_{m,n}$, is
nothing but

$\mathscr{L}_{\pi}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{1}^{*}\mathscr{L}_{1}\otimes\dots\otimes\pi_{\ell}^{*}\mathscr{L}_{\ell}$

where $\mathscr{L}_{i}$ is the linearisation given at §5.2, and
$\pi_{i}\mathrel{\mathop{\mathchar 58\relax}}\prod_{j}Q_{j}\rightarrow Q_{i}$
is the projection.

If $Z(\operatorname{Stab}\beta)\cong\mathbb{G}_{m}^{\ell-1}$ denotes the
centre of $\operatorname{Stab}\beta$, then one has a surjection with finite
kernel

$\prod_{i=1}^{\ell}\mathrm{SL}_{P_{i}}\times
Z(\operatorname{Stab}\beta)\rightarrow\operatorname{Stab}\beta.$

Twisting the linearisation by the character $\beta$ ensures that the centre
$Z(\operatorname{Stab}\beta)$ acts trivially on the canonical linearisation
$\mathscr{L}_{\beta}$ of Definition 2.2.6. Hence, since GIT (semi)stability is
unaffected by finite groups, one has:

$Z_{\tau}^{\operatorname{Stab}\beta-
ss}(\mathscr{L}_{\beta})=Z_{\tau}^{\prod\mathrm{SL}(V_{i})-ss}(\mathscr{L}_{\pi}),$

i.e. the semistable loci for $\operatorname{Stab}\beta$ with respect to
$\mathscr{L}_{\beta}$, and $\prod_{i}\mathrm{SL}_{P_{i}}$ with respect to
$\mathscr{L}_{\pi}$, coincide. From the form of the linearisation we have, for
any cocharacter

$\lambda=(\lambda_{1},\dots\lambda_{\ell})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow\prod_{i}\mathrm{SL}_{P_{i}}$

and any point $q=(q_{1},\dots,q_{\ell})\in Z_{\tau}$, that

$\mu_{\mathscr{L}_{\pi}}(q,\lambda)=\sum_{i=1}^{\ell}\mu_{\mathscr{L}_{i}}(q_{i},\lambda_{i}).$

From this one readily deduces:

######  Proposition 6.2.7.

[HK12, Hos18] The subscheme $Z_{\tau}^{ss}\subset Z_{\tau}$ is exactly the
semistable locus for the $\operatorname{Stab}\beta$ action on $Z_{\tau}$ with
respect to the linearisation $\mathscr{L}_{\beta}$.

Thus, following §2.3 it is possible to obtain categorical quotients of
$Y_{\tau}^{ss}$ using only reductive GIT, by taking the $P_{\beta}$-quotient
of $Y_{\tau}^{ss}$ using the canonical linearisation $\mathscr{L}_{\beta}$.
However, §2.3 tells us that if we do this we will learn nothing new: we
identify each sheaf with its accociated Harder-Narasimhan graded, and hence
the resultant ‘moduli spaces of unstable sheaves’ are isomorphic to products
of moduli spaces of semistable sheaves. We now turn to non-reductive GIT and
the methods of §4.1, to see if this situation can be improved.

## 7\. Moduli of $\tau$-stable Sheaves of Harder-Narasimhan length 2

Having set up all of the machinery for constructing moduli spaces of unstable
objects via non-reductive GIT, we now turn to our main application: sheaves of
Harder-Narasimhan length two. Throughout we fix a Harder-Narasimhan type
$\tau=(P_{1},P_{2})$, of length $\ell=2$. Fix $m\gg n\gg 0$ and let
$\beta=\beta(n,m,\tau)$ be the associated HKKN index.

### 7.1. On $\tau$-stable sheaves

######  Definition 7.1.1.

We say that a Harder-Narasimhan type $\tau=(P_{1},\dots,P_{\ell})$ is
coprime121212The reason for this choice of term is that coprimality
generalises the situation of semistability and stability coinciding for vector
bundles on a curve, which happens when the rank and degree, two positive
integers, are coprime in the usual sense. if, for all $i=1,\dots\ell$, all
semistable sheaves on $X$ of Hilbert Polynomial $P_{i}$ are stable.

We now define the sheaves for which we will construct moduli spaces.

######  Definition 7.1.2.

We say that a sheaf $F$ of HN length 2 is $\tau$-stable if it is of Harder-
Narasimhan type $\tau$, we have $F\ncong\mathrm{gr}(F)$, and the HN-
subquotients $F_{i}$ in its associated graded
$\mathrm{gr}(F)\cong\oplus_{i=1}^{\ell}F_{i}$, are stable.

Later on we will give a GIT interpretation of such sheaves in the HKKN picture
for Simpson’s construction. This will allows us to prove our main result,
Theorem 1.2.1, which we repeat here for the reader’s convenience.

######  Theorem 7.1.3.

Let $(X,\mathscr{O}_{X}(1))$ be a projective scheme with an ample line bundle,
and let $\tau=(P_{1},P_{2})$ be a Harder-Narasimhan type of length 2. Then

  1. (1)

There is a quasi-projective moduli space $M_{\tau,d}$ for those $\tau$-stable
(Definition 7.1.2) coherent sheaves such that $\dim\operatorname{End}(E)=d$.

  2. (2)

This moduli space has a canonical projective completion
$M_{\tau,d}\subset\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{\tau,d}$, constructed via the nonreductive partial desingularisation
procedure of §3.3, such that the map

$M_{\tau,d}\rightarrow M^{ss}_{P_{1}}\times M^{ss}_{P_{2}},$

to the product of two moduli spaces of semistable sheaves, which takes a sheaf
to the associated graded of its HN filtration, extends to $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{\tau,d}$.

  3. (3)

Furthermore, if $\tau$ is coprime in the sense of Definition 7.1.1, then
$M_{\tau,d}$ is the moduli space of indecomposable sheaves of type $\tau$ such
that $\dim\operatorname{End}(E)=d$.

We pause to make the following elementary observation.

######  Lemma 7.1.4.

A sheaf of Harder-Narasimhan length 2 is indecomposable if and only if
$\mathrm{gr}F\ncong F$. In particular, $\tau$-stable sheaves are
indecomposable. If $\tau$ is coprime, the converse holds as well.

###### Proof.

Of the first statement, one implication is clear. For the other, recall that
every stable sheaf $E$ is simple, (see Corollary 1.2.8 of [HL10]) i.e. it
satisfies $\operatorname{End}(E)\cong k$, so in particular is indecomposable.
Hence if there were a splitting $F\cong F^{\prime}\oplus F^{\prime\prime}$,
then $F^{1}$ would be entirely contained within one factor, say $F^{\prime}$.
But then $F_{2}\cong F^{\prime}/F^{1}\oplus F^{\prime}$, contradicting the
simplicity, and hence the stability, of $F_{2}$.

For the last statement, any indecomposable sheaf of type $\tau$ must have non-
split Harder-Narasimhan filtration, and the condition that the subquotients
$F_{i}$ are stable holds trivially because $\tau$ is coprime. ∎

As might be expected, there is a GIT characterisation of $\tau$-stability,
analogously to §6.2.

######  Definition 7.1.5.

Write $Z_{\tau}^{s}$ for the subscheme of $Z_{\tau}^{ss}$ consisting of HN-
graded sheaves whose HN-summands are all stable.

The following is proved in the same way as Proposition 6.2.7.

######  Proposition 7.1.6.

The locus $Z_{\tau}^{s}$ is precisely the stable locus for the action of
$\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu_{\beta}=\faktor{\operatorname{Stab}\beta}{\lambda_{\beta}(\mathbb{G}_{m})}$
on $Z_{\tau}$, with respect to the linearisation $\mathscr{L}_{\beta}$.

###### Proof.

Recall the discussion preceding Proposition 6.2.7, and in particular the
surjection

$\prod_{i=1}^{\ell}\mathrm{SL}_{P_{i}}\times
Z(\operatorname{Stab}\beta)\rightarrow\operatorname{Stab}\beta.$

Since we are in Harder-Narasimhan length 2, we have
$\lambda_{\beta}(\mathbb{G}_{m})\cong Z(\operatorname{Stab}\beta)$, so we get
a surjection

$\prod_{i}\mathrm{SL}_{P_{i}}\rightarrow\faktor{\operatorname{Stab}\beta}{\lambda_{\beta}(\mathbb{G}_{m})}=\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu_{\beta}$

which also has finite kernel. Since GIT stability is also unaffected by finite
group actions, we have an equality of stable loci,

$Z_{\tau}^{\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu_{\beta}-s}(\mathscr{L}_{\beta})=Z_{\tau}^{\prod_{i}\mathrm{SL}_{P_{i}}-s}(\mathscr{L}_{\beta}).$

Hence recalling the Hilbert-Mumford criterion for stability from §2.1, it
suffices to show that for all nontrivial

$\lambda=(\lambda_{1},\dots\lambda_{\ell})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{m}\rightarrow\prod_{i}\mathrm{SL}_{P_{i}}$

and any point $q=(q_{1},\dots,q_{\ell})\in Z_{\tau}$, that
$\mu_{\mathscr{L}_{\beta}}(q,\lambda)>0.$ By the decomposition

$\mu_{\mathscr{L}_{\pi}}(q,\lambda)=\sum_{i=1}^{\ell}\mu_{\mathscr{L}_{i}}(q_{i},\lambda_{i}),$

this will hold if and only if $\mu_{\mathscr{L}_{i}}(q_{i},\lambda_{i})>0$ for
each $i$. And this is the case if and only if each $q_{i}$ is a stable sheaf-
that is, if and only if $q\in Z_{\tau}^{s}$.

∎

This allows us to deduce a GIT characterisation of the sheaves for which we
will construct a moduli space.

######  Corollary 7.1.7.

If we define $Y^{s}_{\tau}\mathrel{\mathop{\mathchar
58\relax}}=p_{\beta}^{-1}(Z_{\tau}^{s})$, then $P_{\beta}$-orbits in
$Y^{s}_{\tau}\setminus U_{\beta}Z_{\tau}^{s}$ are in one-to-one correspondence
with isomorphism classes of $\tau$-stable sheaves.

######  Remark 7.1.8.

A word of warning is in order here. It might seem that the above proposition
is almost a tautology, and should therefore be true in for any HN length.
However, this is not the case, and the reason is somewhat subtle. Consider a
point $q\in Z_{\tau}^{s}$, where now $\tau$ has length $\ell$. The
corresponding HN-graded sheaf $F\cong\mathrm{gr}(F)=\oplus_{i=1}^{\ell}F_{i}$
of Harder-Narasimhan length $\ell$ has a copy of $\mathbb{G}_{m}^{\ell}$ in
its automorphism group (and hence, by Lemma 5.2.1 its
$\mathrm{GL}(V)$-stabiliser in the Quot scheme) which comes from scaling each
factor.

Passing to $\mathrm{SL}(V)$ removes the diagonal $\mathbb{G}_{m}$ that is
common to all sheaves, and so we are left with a copy of
$\mathbb{G}_{m}^{\ell-1}$ inside the $\operatorname{Stab}\beta$-stabiliser of
such a sheaf. Upon quotienting by $\lambda_{\beta}(\mathbb{G}_{m})$, this
reduces to a $\mathbb{G}_{m}^{\ell-2}$-dimensional torus inside the $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu_{\beta}$-stabiliser of
$q\in Z_{\tau}^{s}$, meaning that such points cannot be $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu_{\beta}$-stable if
$\ell>2$. Put another way, in the language of the convex hull formulation of
the Hilbert-Mumford criterion, the origin is contained in the interior of the
weight polytope when the interior is taken with respect induced topology on
the polytope itself, but not when the interior is taken with respect to the
topology on the codimension-one subspace of $\mathfrak{t}^{*}$ corresponding
to the Lie algebra of a maximal torus of $\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu_{\beta}$. This is one
reason for the alternative construction pursued in [HJ], which constructs
moduli spaces for certain sheaves of Harder-Narasimhan length $\ell>2$.

### 7.2. Automorphism groups of sheaves of Harder-Narasimhan length 2

We are now almost ready to prove Theorem 7.1.3.

Following §3, we will need to understand the unipotent radicals of stabilisers
of points in the Quot scheme corresponding to sheaves Harder-Narasimhan length
2. By Lemma 5.2.1, the $\mathrm{GL}$-stabiliser of such a point is isomorphic
to the automorphism group of the sheaf, and so we collect some observations
about automorphism groups of unstable sheaves. We begin with the following,
doubtless well known, which shows that there is a natural parabolic structure
on these automorphism groups.

######  Lemma 7.2.1.

Let $F$ be a coherent sheaf on $X$. Then all automorphisms of $F$ preserve the
Harder-Narasimhan filtration of $F$.

###### Proof.

This follows directly from the uniqueness of the Harder-Narasimhan filtration.
∎

######  Remark 7.2.2.

The above lemma may be seen as an instance of a more general phenomenon
occurring in the setting of moduli spaces of unstable objects. In view of the
GIT construction in §5 and §6, the statement for automorphisms follows from
[Kem78], so long as we are careful to use the action of $\mathrm{GL}(V)$
rather than $\mathrm{SL}(V)$ for the GIT setup, i.e. to have the right stack-
theoretic parameter space.

Next we need some notation for versions in $\mathrm{GL}(V)$ of the groups we
have encountered.

######  Notation 7.2.3.

We write $\widetilde{P}_{\beta}\cong P_{\beta}\times Z(\mathrm{GL}(V))$ for
the subgroup of $\mathrm{GL}(V)$ generated by $P_{\beta}$ and the diagonal
$\mathbb{G}_{m}$. Thus,
$\widetilde{P}_{\beta}\cong\mathrm{U}_{\beta}\rtimes\widetilde{\operatorname{Stab}}\beta$,
where $\widetilde{\operatorname{Stab}}\beta$ is the group generated by
$\operatorname{Stab}\beta$ and the central $\mathbb{G}_{m}$ in
$\mathrm{GL}(V)$. In other words, we let $\widetilde{P}_{\beta}$ be the
parabolic group defined by the weight filtration of
$\lambda_{\beta}(\mathbb{G}_{m})$, taken in $\mathrm{GL}(V)$.

######  Remark 7.2.4.

Note that, since the centre of $\mathrm{GL}(V)$ acts trivially on the Quot
scheme, the orbits for these groups are the same as for their analogues in
$\mathrm{SL}(V)$. The only difference is the action of $Z(\mathbb{G}_{m})$ on
the linearisation, which is by the determinantal character of
$\mathrm{GL}(V)$. Thus, we could just as well set up the GIT problem using
these groups, so long as we twist the linearisation we are using by that
character. Indeed, as we have seen, from a stack-theoretic perspective this
contruction is more natural, since it gives the correct automorphism groups.

Now fix a sheaf $F$ of type $\tau=(P_{1},P_{2})$, represented by a choice of
point in the Quot scheme $[q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathscr{O}_{X}(-n)\rightarrow F]\in Y_{\tau}^{ss}$. This
choice allows us to make free use of the identification

$H^{0}(q(n))\mathrel{\mathop{\mathchar 58\relax}}V\rightarrow H^{0}(F(n)),$

and gives us a faithful representation
$\operatorname{Aut}(F)\hookrightarrow\mathrm{GL}(V)$, restricting to an
isomorphism

$\operatorname{Stab}_{\mathrm{GL}(V)}(q)\cong\operatorname{Aut}(F)$

as in Lemma 5.2.1.

We write

$\operatorname{Aut}(F)\cong\operatorname{Aut}_{U}(F)\rtimes\operatorname{Aut}_{R}(F),$

where $\operatorname{Aut}_{U}(F)\trianglelefteq\operatorname{Aut}(F)$ is the
unipotent radical, and $\operatorname{Aut}_{R}(F)$ is a chosen Levi factor. By
the above Lemma 7.2.1, in fact
$\operatorname{Aut}(F)\leqslant\widetilde{P}_{\beta}$, from which it follows
that $\operatorname{Aut}_{U}(F)=\operatorname{Aut}(F)\cap U_{\beta}$, and we
can choose
$\operatorname{Aut}_{R}(F)=\operatorname{Aut}(F)\cap\widetilde{\operatorname{Stab}}\beta$.

######  Proposition 7.2.5.

(_cf.[BPMG13]_) Let $F$ be a sheaf of Harder-Narasimhan length $\ell=2$. Then
we have

$\operatorname{Aut}_{U}(F)\cong\operatorname{id}_{F}+\operatorname{Hom}(F_{2},F_{1}),$

as groups, with composition being the operation on the right. Furthermore, if
$F$ is $\tau$-stable then we have $\operatorname{End}(F)\cong
k\oplus\operatorname{Hom}(F_{2},F_{1}).$

###### Proof.

Using the representation of $\operatorname{Aut}(F)$ above, from
$\operatorname{Aut}_{U}(F)=\operatorname{Aut}(F)\cap U_{\beta}$ we see

$\operatorname{Aut}_{U}(F)\subseteq\\{\operatorname{id}_{F}+\varphi\in\operatorname{End}(F)\mid\varphi\in\operatorname{Hom}(F,F_{1})\\}.$

In fact, since such an element must have $F_{1}$ in its kernel by its block
form, we see that $\varphi\in\operatorname{Hom}(F_{2},F_{1})$, where we use
the natural injection
$\operatorname{Hom}(F_{2},F_{1})\hookrightarrow\operatorname{Hom}(F,F_{1})$
which comes from applying $\operatorname{Hom}(-,F_{1})$ to the short exact
sequence $0\rightarrow F_{1}\rightarrow F\rightarrow F_{2}\rightarrow 0.$
Conversely any $\varphi\in\operatorname{Hom}(F_{2},F_{1})$ yields an element
$\operatorname{id}_{F}+\varphi\in\operatorname{Aut}_{U}(F)$ in this manner.

Now suppose that $F$ is $\tau$-stable By Lemma 7.1.4 it is then
indecomposable, and by [Ati56] this means that we have

$\operatorname{End}(F)\cong k\oplus\operatorname{Nil}(E)$

where $\operatorname{Nil}(E)$ is the nilpotent endomorphisms of $E$. There is
an injective map

$\operatorname{Hom}(E_{2},E_{1})\rightarrow\operatorname{Nil}(E)$

sending $\varphi\in\operatorname{Hom}(E_{2},E_{1})$ to the composition

${E}$${E_{2}}$${E_{1}}$${E.}$$\scriptstyle{\varphi}$

Now for surjectivity. Take $\alpha\in\operatorname{Nil}(E)$. We observe that
since all automorphisms of $F$ preserve the HN filtration, by considering
$\operatorname{id}_{F}+\alpha$, so must $\alpha$. Thus we get
$\alpha|_{E^{1}}\mathrel{\mathop{\mathchar 58\relax}}E^{1}\rightarrow E^{1}$,
which must be zero since $E^{1}$ is simple and $\alpha$ is nilpotent. By the
same reasoning, the induced map $\mkern
1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern
1.5mu\mathrel{\mathop{\mathchar 58\relax}}E_{2}\rightarrow E_{2}$ must be
zero, which shows that $\alpha\in\operatorname{Hom}(E_{2},E_{1})$. ∎

Putting the pieces together, we find the following important corollary, which
greatly simplifies the blow-up process, and gives a sheaf-theoretic intrinsic
description of the resultant refined Harder-Narasimhan type.

######  Corollary 7.2.6.

  1. (1)

For any $q\in Y_{\tau}^{ss}$ corresponding to a sheaf $F$ of type $\tau$, we
have

$\dim\operatorname{Stab}_{U_{\beta}}(q)=\dim\operatorname{Hom}(F_{2},F_{1})$

and hence

$\dim\operatorname{Stab}_{U_{\beta}}(q)=\dim\operatorname{Stab}_{U_{\beta}}(p_{\beta}(q)),$

so condition
$\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x))$ holds.

  2. (2)

If in addition $F$ is $\tau$-stable, we have

$\dim\operatorname{End}(F)=\dim\operatorname{Stab}_{U_{\beta}}(q)+1.$

######  Remark 7.2.7.

The failure of this Corollary for higher length Harder-Narasimhan types, and