# Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects

Victoria Hoskins  V. Hoskins
IMAPP, Radboud University
PO Box 9010, 6500GL Nijmegen
The Netherlands<EMAIL_ADDRESS>and  Joshua Jackson  J. Jackson
Mathematics Department, Imperial College
London
SW7 2AZ
UK  Heilbronn Institute for Mathematical Research
Bristol
UK<EMAIL_ADDRESS>
###### Abstract.

Motivated by constructing moduli spaces of unstable objects, we use new ideas
in non-reductive GIT to construct quotients by parabolic group actions. For
moduli problems with semistable moduli spaces constructed by reductive GIT, we
consider associated instability (or HKKN) stratifications, which are often
closed related to Harder–Narasimhan stratifications, and construct quotients
of the unstable strata under various stabiliser assumptions by further
developing ideas of non-reductive GIT. Our approach is to construct parabolic
quotients in stages, in order for the required stabiliser assumptions to be
more readily verified. To illustrate these ideas, we construct moduli spaces
for certain sheaves of fixed Harder–Narasimhan type on a projective scheme in
cases where our stabiliser assumptions can be verified.

###### Contents

  1. 1 Introduction
  2. 2 Geometric invariant theory
  3. 3 Quotients of unstable HKKN strata
  4. 4 Quotienting-in-Stages for parabolics: Definitions and Results
  5. 5 Quotienting-in-stages for parabolics: Proof of the Main Theorem
  6. 6 An overview of applications

##  1\. Introduction

Many algebro-geometric moduli spaces are constructed using Mumford’s Geometric
Invariant Theory (GIT) [29]: given a reductive group $G$ acting linearly on a
projective scheme $Y$, one obtains a quotient of an open semistable locus
$Y^{ss}\subset Y$. Semistability is combinatorially determined via the
_Hilbert-Mumford criterion_ by calculating the weights of 1-parameters
subgroups (1-PS) of $G$. In good cases, this enables a moduli-theoretic
interpretation of GIT semistability from which one obtains a moduli space of
semistable objects; the paradigmatic example being slope semistability for
vector bundles on a smooth projective curve.

In this paper, we view the above as merely the beginning of the story and aim
to additionally construct moduli spaces of unstable objects. Instability in
reductive GIT is a structured phenomenon: for each unstable point of $Y$,
there is a _maximally destabilising_ 1-PS of $G$, unique up to conjugation
under a parabolic subgroup, which is most responsible111This notion depends on
a choice of norm on 1-PSs of $G$. for violating the Hilbert–Mumford criterion
[26]; this is akin to the Harder–Narasimhan filtration [19] of a vector bundle
being a maximally destabilising filtration. These maximally destabilising
1-PSs gives rise to an instability stratification (or HKKN stratification
following [20, 26, 27, 31]) of $Y$ into finitely many $G$-invariant locally
closed subschemes such that the lowest stratum $S_{0}$ is the semistable set
and the higher (unstable) strata $S_{\beta}$ are indexed by conjugacy classes
$\beta$ of rational 1-PSs of $G$ which are maximally destabilising for the
points in $S_{\beta}$. Recently, Halpern-Leistner greatly generalised this
idea to $\Theta$-stratifications of stacks [16]. The HKKN stratification
suggests that the instability types $\beta$ should be viewed as discrete
invariants and one should then look to construct moduli spaces for unstable
objects of instability type $\beta$. For the GIT construction of moduli of
vector bundles on a curve, GIT instability types correspond to
Harder–Narasimhan (HN) types [15, 23].

Our primary motivating examples come from moduli of objects in a linear
abelian category, where moduli spaces of semistable objects are constructed by
reductive GIT and the HKKN stratification is closely related to a Harder-
Narasimhan (or Shatz [34]) stratification; for example, this is the case for
moduli of bundles or sheaves [23, 22], Higgs sheaves [17] and quiver
representations [21]. In these cases, constructing quotients of the unstable
strata $S_{\beta}$ would give rise to moduli spaces of objects of fixed HN
type; this has been described for certain length $2$ HN filtrations in [10,
17, 25].

###  1.1. Moduli of Unstable Objects

The initial technical obstruction to constructing quotients of unstable strata
is clear enough: the strata consist by definition of GIT-unstable points, so
their quotients are empty. Since GIT semistability depends on a choice of
linearisation, a natural approach is to vary the linearisation $\mathcal{L}$
by twisting by a character to change the semistable locus, as is described by
variation of GIT [14, 36]. However, there may not exist a character
$\chi_{\beta}$ of $G$ which we can use to make all of $S_{\beta}$ semistable
(for example, if $G=\mathrm{SL}_{n}$, there are no non-trivial characters).
Fortunately, if we pick a representative 1-PS $\lambda_{\beta}$ of the
conjugacy class $\beta$, this determines a parabolic group
$P_{\beta}:=P(\lambda_{\beta})<G$ and locally closed subscheme
$Y_{\beta}^{ss}\subset S_{\beta}$ such that $S_{\beta}$ is the $G$-sweep of
$Y_{\beta}^{ss}$, and such that flowing under $\lambda_{\beta}$ induces a
retraction $p_{\beta}:Y_{\beta}^{ss}\rightarrow Z_{\beta}^{ss}$ [27].
Moreover, a categorical quotient of $G$ acting on $S_{\beta}$ is equivalent to
a categorical quotient of $P_{\beta}$ acting on $Y_{\beta}^{ss}$. The
parabolic subgroup $P_{\beta}$ has larger character group than $G$ and in
particular, there is a character $\chi_{\beta}$ such that the twisted
linearisation $\mathcal{L}_{\beta}$ (which we refer to as a borderline
linearisation below) produces a categorical $P_{\beta}$-quotient of
$Y_{\beta}^{ss}$ [23]. However, this quotient factors by the retraction
$p_{\beta}$ followed by the categorical quotient of $Z_{\beta}^{ss}$ under the
action of the Levi $L_{\beta}<P_{\beta}$. In particular, this categorical
quotient is far from being an orbit space. Often the moduli theoretic
interpretation of the retraction $p_{\beta}$ is given by taking the associated
graded of a Harder–Narasimhan filtration and so the obtained categorical
quotient is just a product of moduli spaces for the successive semistable
quotients in the HN filtration.

To avoid these identifications, one would like to remove the locus
$Z_{\beta}^{ss}\subset Y_{\beta}^{ss}$, which in many moduli theoretic
examples parametrises objects with completely degenerate HN filtrations. To do
this, we would like to further perturb the twisted linearisation
$\mathcal{L}_{\beta}$, so that $Z_{\beta}^{ss}$ becomes unstable. The
$P_{\beta}$-linearisation $\mathcal{L}_{\beta}$ on $Y_{\beta}^{ss}$ also
extends to an ample $P_{\beta}$-linearisation on the closure of
$Y_{\beta}^{ss}$ in $Y$, but it does not extend to an ample $G$-linearisation
on a natural compactification of $S_{\beta}$. For this reason, one must work
with the non-reductive group $P_{\beta}$ and so classical reductive GIT cannot
be applied.

Recently there has been great progress on developing non-reductive GIT (NRGIT)
for groups whose unipotent radical is ‘graded’ by a one-parameter subgroup [1,
2, 3, 4, 6]. The best results are obtained under certain strong stabiliser
assumptions by appropriately twisting the linearisation to make it ‘adapted’.
Even when these stabiliser assumptions fail, one can perform a blow-up
sequence to obtain these assumptions on the blow-up. Although there are
important applications in which the blow-up process is tractable (for example,
see [8]), in general explicitly determining the open set one obtains a
quotient of is often quite involved (see $\S$2.3). This theory has
applications to hyperbolicity questions [8] and the construction of moduli of
hypersurfaces in toric varieties [11], as well as providing tools for
cohomological computation via a symplectic description of the quotient [12];
but in this paper we will focus on its application to the construction of
moduli spaces of unstable objects, motivated by the desire to construct moduli
spaces of objects of fixed Harder–Narasimhan type. More precisely, given an
unstable stratum $S_{\beta}$ associated to a reductive GIT set-up, our aim is
to construct a non-reductive quotient of the action of the parabolic group
$P_{\beta}$ on $Y_{\beta}^{ss}$ (or rather its closure). This is an ideal
situation to apply non-reductive GIT, as the unipotent radical of $P_{\beta}$
is graded by the 1-PS $\lambda_{\beta}$. Consequently, we can apply NRGIT to
construct explicit quotients of unstable HKKN strata subject to certain
stabiliser assumptions denoted $[\widehat{U}]$,
$[\widehat{U};\overline{L}\text{-ss}]$ and $[\overline{L}]_{0}$ (see
Definition 2.20). The first main result of this paper is the following theorem
concerning quotients of unstable HKKN strata.

######  Theorem 1.1.

For a reductive group $G$ acting linearly on a projective scheme $Y$, let
$\beta$ be a HKKN index corresponding to an unstable stratum $S_{\beta}\subset
Y$. Consider the non-reductive group $P_{\beta}=U_{\beta}\rtimes L_{\beta}$
with unipotent radical $U_{\beta}$ internally graded by $\lambda_{\beta}$
acting on $X_{\beta}:=\overline{Y_{\beta}}$ with respect to the ample adapted
linearisation $\mathcal{L}_{\beta}^{\text{per}}$. Then the following
statements hold.

  1. i)

Assume $[\widehat{U}]$ holds for the action of
$\widehat{U}=\widehat{U}_{\beta}$; then the open stable set
$X_{\beta}^{\widehat{U}_{\beta}-s}:=Y_{\beta}-U_{\beta}Z_{\beta}$ admits a
geometric and projective $\widehat{U}_{\beta}$-quotient
$\overline{Y_{\beta}}^{\widehat{U}_{\beta}-s}/\widehat{U}_{\beta}$, given by
the projective spectrum of the $\widehat{U}$-invariants with respect to a
well-adapted twist of $\mathcal{L}_{\beta}^{\text{per}}$.

  2. ii)

Assume $[\widehat{U};\overline{L}\text{-ss}]$ and $[\overline{L}]_{0}$ hold
for the action of $P_{\beta}$; then the open stable set

$X_{\beta}^{P_{\beta}-s}:=Y_{\beta}^{ss}-UZ_{\beta}^{ss}=Y_{\beta}^{s}-UZ_{\beta}^{s}$

admits a geometric and projective $P_{\beta}$-quotient, given by the
projective spectrum of the $P_{\beta}$-invariants with respect to a well-
adapted twist of $\mathcal{L}_{\beta}^{\text{per}}$.

  3. iii)

Assuming $[\widehat{U}]$ holds, $U_{\beta}Z_{\beta}\subset Y_{\beta}$ is a
closed subscheme, the map $p_{\beta}:U_{\beta}Z_{\beta}\rightarrow Z_{\beta}$
is a geometric $\widehat{U}_{\beta}$-quotient and the reductive quotient
$Z_{\beta}/\\!/_{\mathcal{L}_{\beta}}L_{\beta}$ is a good $P_{\beta}$-quotient
of $U_{\beta}Z_{\beta}^{ss}$ (and also a geometric quotient if
$[\overline{L}]_{0}$ holds).

To apply this theorem, one needs to check whether these stabiliser assumptions
are satisfied. The assumptions $[\widehat{U}]$ and
$[\widehat{U};\overline{L}\text{-ss}]$ concern unipotent stabilisers, whereas
$[\overline{L}]_{0}$ concerns stabilisers for the reductive Levi factor and is
used to obtain an explicit description of the open ‘semistable’ set of which
we obtain a quotient. If these stabiliser assumptions fail, then NRGIT uses a
blow-up sequence to produce a (only quasi-projective) quotient of an open set,
which in practice can be hard to determine.

Unfortunately, for many unstable strata, the reductive stabiliser condition
$[\overline{L}]_{0}$ fails. More precisely, $[\overline{L}]_{0}$ asks for
semistability to coincide with stability for the action of the reductive Levi
factor $L_{\beta}$ on $Z_{\beta}$ (which is the minimal weight space in the
$\lambda_{\beta}$-fixed locus). Often in the case when our reductive GIT set-
up comes from moduli of objects in a linear abelian category, the subscheme
$Z_{\beta}$ parametrises the associated graded of Harder–Narasimhan
filtrations and the stabiliser group of such an object contains a torus of
dimension equal to the length of the HN filtration; this results in a torus
$T_{\beta}$ in the Levi $L_{\beta}$ acting trivially on $Z_{\beta}$ (but not
on $Y_{\beta}$) so that $[\overline{L}]_{0}$ fails except in low lengths. More
precisely, in the case of sheaves of HN type of length $l$, one obtains an
$(l-1)$-dimensional222Note that the dimension drops by one, as we are working
inside $\mathrm{SL}$ and not $\mathrm{GL}$. torus $T_{\beta}<L_{\beta}$ acting
trivially on $Z_{\beta}$ (see $\S$3.2.1). Hence, $[\overline{L}]_{0}$ fails
for HN filtrations of length $l>2$; this happens for moduli of (Higgs) sheaves
and quiver representations. For HN filtrations of length $l=2$, the problem of
constructing moduli spaces of fixed HN type for sheaves and Higgs bundles are
considered in [25] and [17].

###  1.2. Quotienting-in-Stages

The main result of this paper is motivated by the issue outlined above, namely
that in many applications of interest (both within ‘moduli spaces of unstable
objects’, and beyond) the stabiliser conditions of Theorem 1.1 are not met and
so we cannot obtain an explicit satisfactory quotient using the current
methods of NRGIT.

To overcome this, we prove a new theorem in NRGIT that is better suited to
this family of applications than Theorem 1.1. More precisely, we consider a
projective scheme $X$ with a linear action of a parabolic subgroup
$P=P(\lambda)=U\rtimes L<\mathrm{SL}_{N}$, defined by a one-parameter subgroup
$\lambda:\mathbb{G}_{m}\rightarrow\mathrm{SL}_{N}$. The basic idea is to break
the quotienting process up into stages, one for each step in the row
filtration of $P$ (one should consider these as the steps in the HN
filtration). At each stage, we take a NRGIT quotient of a maximal unipotent
group inside $U$ graded by a 1-PS coming from the central torus $T=Z(L)$. The
key challenges are determining an explicit description of the open locus we
get a quotient of at all these stages and determining the relevant stabiliser
conditions which are required for this procedure. To determine this open set
explicitly, we impose the so-called _Quotienting-in-Stages Assumption_ (QiS),
which are modelled on the behaviour of objects of fixed HN type in a linear
abelian category and are used to determine the relevant minimal weight spaces
at each stage in the quotient procedure. In place of the strong stabiliser
assumptions of Theorem 1.1 we introduce conditions on stabiliser groups that
are more likely to hold in this setting, which we call the _Weak Upstairs
Unipotent stabiliser Assumption_ (WUU) and the _Upstairs Stabiliser
Assumption_ (U). Under these assumptions, we obtain an explicit quotient as
follows.

######  Theorem 1.2.

Let $P<\mathrm{SL}_{N}$ be a parabolic group acting linearly on an irreducible
projective scheme $X$ satisfying (QiS). If (WUU) holds, then there is an open
set $X^{P-qs}$ with an explicit Hilbert-Mumford style description that admits
a geometric $P$-quotient given by a quasi-projective scheme with a natural
projective completion. If moreover, (U) holds, then this quotient of
$X^{P-qs}$ is projective.

The Upstairs Stabiliser Assumption (U) is chosen to induce the relevant
stabiliser conditions at each stage of the quotienting process, so that the
stronger results of NRGIT can inductively be applied at each stage. Crucially,
the reductive stabiliser assumptions in (U) do not concern the full Levi $L$,
but only its semisimple part and so the fact that the central torus $T=Z(L)$
acts trivially is no longer an issue. By performing a blow-up sequence, we can
weaken this to (WUU), which ensures we understand enough about the blow-up
procedure to explicitly describe the obtained quotient.

###  1.3. Applications

As mentioned above, the motivating family of applications for our results is
the construction of moduli spaces of objects of fixed HN type in a linear
abelian category for which a reductive GIT construction of the moduli space of
semistable objects is known. These applications, including sheaves, Higgs
sheaves, and quiver representations, will be dealt with in future papers [5,
18].

For ease of exposition, we illustrate our ideas by briefly analysing the
problem of constructing moduli spaces for sheaves of fixed HN type, concluding
with Theorem 6.2 which allows the construction of moduli spaces for certain
unstable sheaves in special situations. The open stable set $X^{P-qs}$
obtained by Theorem 1.2 in the case of sheaves of HN type $\tau$ admits a
moduli-theoretic interpretation which we call $\tau$-stability, see Definition
4.18: for a sheaf of HN type $\tau$ to lie in $X^{P-qs}$, each inclusion in
its HN filtration must be non-split and the successive HN quotients must be
stable rather than just semistable. In addition, one requires certain
unipotent stabiliser assumptions, which admit interpretations in terms of
certain filtered endomorphism groups of the sheaf. The semistability notion
given by NRGIT for sheaves of fixed HN type naturally fits with previous work
describing moduli of rank 2 bundles of fixed HN type [10] using a
projectivised Extension bundle associated to the universal families for the
two subquotients: this forces the HN filtration to be non-split and, for this
Extension sheaf to be a vector bundle, one requires constant dimensional
filtered endomorphism groups. More generally, for a length $l$ HN type,
provided one has universal families for the semistable subquotients, one can
iteratively take projectivised Extension bundles to obtain an analogous
construction. However, without the existence of such universal families, one
can instead apply the quotienting-in-stages procedure we develop.

However, the assumptions of Theorem 1.2 are very restrictive, due to the
necessity of condition (WUU), which at present we can only verify for sheaves
in slightly contrived examples (see $\S$ 6). A construction better adapted to
the case of sheaves in particular will be considered in [5]. On the other
hand, condition (WUU) is satisfied in far greater generality for Higgs
sheaves, and so the results of this paper are well suited to the construction
of moduli spaces of unstable Higgs sheaves; we will pursue this application in
[18].

###  1.4. Overview of the paper

In $\S$2, we recall the basic properties of reductive GIT and instability
stratifications. Moreover, in §2.3 we collect the results of NRGIT we shall
need and prove some slightly stronger results. We then apply these results in
$\S$3, leading to the proof of our first main result, Theorem 1.1 on
quotienting unstable strata.

The remainder of the paper is devoted to proving Theorem 1.2. Let us sketch
the structure; for the detailed road map, see $\S$4.5. In Section 4, we
explain the set-up and notation, including defining the quotienting-in-stages
stable locus $X^{P-qs}$ in Definition 4.11. We outline the quotienting-in-
stages process in $\S$4.3, which we can perform under so-called _Downstairs
Stabiliser Assumption_ (D) conditions. From $\S$4.4 onwards, we suppose the
_Quotienting-in-Stages Assumptions_ (QiS) hold. We then introduce certain
upstairs stabiliser assumptions (U) and (WUU), before moving onto the proof in
Section 5. Assuming (D), we first describe the open locus we obtain a quotient
of in $\S$5.1, and we show that (D) can be deduced from (U) in $\S$5.2. To
deduce the final version of Theorem 1.2 we perform certain blow-up procedures.
We conclude with a brief discussion of applications in $\S$6.

### Notation and Conventions

By a scheme, we mean a scheme of finite type over an algebraically closed
field of characteristic 0. By a sheaf over a scheme $X$, we mean a coherent
sheaf of $\mathcal{O}_{X}$-modules. For a linear algebraic group $G$ and a
subgroup $H\subset G$ acting on a quasi-projective scheme $Y$, the Borel
construction $G\times^{H}Y$ denotes the geometric quotient of $H$ acting on
$G\times Y$ diagonally.

### Acknowledgements

We are very grateful to F. Kirwan, G. Bérczi, D. Bunnett and E. Hamilton for
many helpful discussions concerning non-reductive GIT and the ideas in this
paper. The second named author is funded by the Heilbronn Institute for
Mathematical Research.

##  2\. Geometric invariant theory

For an action of a linear algebraic group on a scheme, the aim of geometric
invariant theory (GIT) is to provide a quotient of this action (potentially on
an invariant open subscheme) in the category of schemes. The theory splits
into two branches, according to whether the group acting is reductive or non-
reductive. The former was established in the 1960s by Mumford [29], building
on the results of 19th Century invariant theory. The latter has been studied
by several authors; in particular, the recently developed theory for non-
reductive groups with so-called graded unipotent radicals has many of the
favourable properties of reductive GIT [1]. In $\S$2.1, we survey the key
results of reductive GIT, including the Hilbert–Mumford criterion, and the
partial desingularisation blow-up procedure [28]; in $\S$2.2 we review the
instability (HKKN) stratifications arising from work of Hesselink [20] Kempf
[26], Kirwan [27] and Ness [31]. In $\S$2.3, we summarise the key results from
[1] which enable constructions of GIT quotients for actions of linear
algebraic groups with graded unipotent radicals.

###  2.1. Reductive GIT quotients

Let $G$ be a reductive group acting on a projective scheme $Y$ with an ample
linearisation of the action $\mathcal{L}$. One of the most well-known
properties of reductive groups actions is that the ring of $G$-invariants is
finitely generated; this is contrast to the counterexamples to Hilbert’s 14th
Problem [30] in the non-reductive case. In addition taking $G$-invariants is
exact, which makes the theory behave functorially with respect to
$G$-equivariant closed immersions. These properties were utilised by Hilbert
and Mumford in order to construct reductive GIT quotients. A standard
reference is [32].

The reductive GIT quotient $\pi:Y^{ss}\rightarrow Y/\\!/_{\\!\mathcal{L}}G$ is
a categorical and good quotient of the open subset
$Y^{ss}:=Y^{G-ss}(\mathcal{L})$ of GIT semistable points in $Y$ (with respect
to the $G$-linearisation $\mathcal{L}$) and was constructed by Mumford [29].
The quotient $Y/\\!/_{\\!\mathcal{L}}G$ is a projective scheme equal to the
projective spectrum of the graded ring of $G$-invariant sections of non-
negative powers of $\mathcal{L}$. In general, the GIT quotient is not an orbit
space: its closed points correspond to S-equivalence classes of semistable
orbits, where two semistable orbits are S-equivalent if their orbit closures
meet in the semistable set. However, the GIT quotient restricts to a geometric
quotient, which in particular is an orbit space, on the open subset
$Y^{s}(\mathcal{L})\subset Y^{ss}$ of stable points.

Stable points333Note that stable points in modern terminology were referred to
as properly stable points by Mumford [29]. can be described as semistable
points whose stabiliser groups are finite and whose orbits are closed in the
semistable set. Often when the linearisation and group is fixed, we will drop
the notation $\mathcal{L}$ and $G$ indicating the dependence of the semistable
set on this linearisation and $G$. Variation of GIT describes the birational
transformations between GIT quotients with respect to different
$G$-linearisations [14, 36].

The (semi)stable loci are defined in [29] using $G$-invariant sections of
positive powers of $\mathcal{L}$. Critical to the use of reductive GIT in
practice, though, is the ability to determine these loci without computing
ring of invariants, afforded by the Hilbert–Mumford criterion. This criterion
states that a closed point $y\in Y$ is semistable if and only if it is
semistable for each 1-parameter subgroup (1-PS) $\lambda$ of $G$. To state
this as a numerical criterion, we define the Hilbert–Mumford weight
$\mu(y,\lambda)$ to be minus the weight of the $\mathbb{G}_{m}$-action induced
by $\lambda$ on the fibre of $\mathcal{L}$ over the fixed point
$\lim_{t\rightarrow 0}\lambda(t)\cdot y$.

######  Proposition 2.1 (Hilbert–Mumford criterion).

A closed point $y\in Y$ is semistable if and only if $\mu(y,\lambda)\geq 0$
for all 1-PS $\lambda:\mathbb{G}_{m}\rightarrow G$.

Alternatively, by using the linearisation $\mathcal{L}$ to give an equivariant
embedding $Y\hookrightarrow\mathbb{P}^{n}$, the Hilbert–Mumford criterion can
be stated combinatorially using the weights of a maximal torus $T<G$ on
$\mathbb{P}^{n}$ (or rather the affine cone over this projective space). A
point is semistable for the $T$-action if its convex hull of $T$-weights
contains the origin [13, $\S$9.4]. Moreover, a point is semistable for the
$G$-action if all points in its $G$-orbit are $T$-semistable; thus we have

$Y^{G-ss}=\cap_{g\in G}\>gY^{T-ss}.$

####  2.1.1. Partial desingularisation via equivariant blow-ups

In this section, we summarise a partial desingularisation of the reductive GIT
quotient $Y/\\!/G$ that is constructed by performing a canonical sequence of
blow-ups along closed invariant subschemes determined by studying the
dimensions of reductive stabiliser groups of semistable points [28]. In [28]
this procedure was described when the stable locus was non-empty, in which
case one has that semistability and stability coincide on the final blow-up.
If one starts with a smooth projective scheme $Y$, the resulting blow-up
$\widetilde{Y}$ is also smooth and projective and its GIT quotient
$\widetilde{Y}/\\!/G$ has only orbifold singularities and thus provides a
partial desingularisation of $Y/\\!/G$. One can also perform this blow-up
procedure when then stable set is empty but the semistable set is non-empty;
in this case, the procedure can terminate in several different ways, as we
describe below.

To describe the blow-up sequence in [28] for irreducible $Y$ and also later
describe the non-reductive GIT blow-up sequences, we introduce the following
notation for the locus in a subscheme where the stabilisers of a certain group
have fixed dimension.

######  Definition 2.2.

Given a scheme $Y$ with an action by a linear algebraic group $G$, for any
subscheme $Y^{\prime}\subset Y$ and subgroup $G^{\prime}<G$ and any
$d\in\mathbb{N}$, we define the locally closed subscheme

$C(Y^{\prime},G^{\prime},d):=\\{y\in
Y^{\prime}:\dim\operatorname{Stab}_{G^{\prime}}(y)=d\\}\subset Y^{\prime}$

to be the locus in $Y^{\prime}$ where the $G^{\prime}$-stabilisers are
$d$-dimensional. We also write

$d_{\max}(Y^{\prime},G^{\prime}):=\max\\{\dim\operatorname{Stab}_{G^{\prime}}(y):y\in
Y^{\prime}\\}\quad\text{and}\quad
d_{\min}(Y^{\prime},G^{\prime}):=\min\\{\dim\operatorname{Stab}_{G^{\prime}}(y):y\in
Y^{\prime}\\}$

and
$C(Y^{\prime},G^{\prime}):=C(Y^{\prime},G^{\prime},d_{\max}(Y^{\prime},G^{\prime}))$
for the locus with maximal dimensional stabilisers.

Often for simplicity we will assume that $Y$ is irreducible (in the case when
$Y^{\prime}$ is not irreducible, for example if it is not connected, it may be
better to think of $d_{\max}(Y^{\prime},G^{\prime})$ as a tuple of dimensions
indexed by the irreducible components). Since the dimension of the stabiliser
group is upper semi-continuous, $C(Y^{\prime},G^{\prime})$ is a closed
subscheme of $Y^{\prime}$. If $Y^{\prime}$ is $G^{\prime}$-invariant, these
subschemes are $G^{\prime}$-invariant. If $Y^{\prime}$ is $G$-invariant and
$G^{\prime}\triangleleft G$ is normal then these subschemes are $G$-invariant,
so that the blow up of $Y$ along $Y^{\prime}$ is $G$-equivariant, and the
blown up space has a natural $G$-action.

Returning to the case where $G$ is a reductive group acting linearly on a
projective scheme $Y$, if $\emptyset\neq Y^{s}\subsetneq Y^{ss}$, then
$C(Y^{ss},G)$ is a proper closed $G$-invariant subscheme of $Y^{ss}$ which is
disjoint from $Y^{s}$. Assuming that $Y$ is irreducible for simplicity, the
first step444In fact, this first step in [28] is performed as a sequence of
blow-ups: one chooses representatives of all conjugacy classes of connected
reductive subgroups of $G$ of dimension $d_{\max}(Y^{ss},G)$ and for each such
subgroup $R$ one blows-up the closure of the $G$-sweep of the locus of
semistable points which are fixed by $R$. in the partial desingularisation
procedure in [28] is to blow-up $X$ along the closure of $C(Y^{ss},G)$ to
obtain $\varphi_{1}:\widetilde{Y}_{1}\rightarrow Y$. The $G$-action on $Y$
naturally lifts to $\widetilde{Y}_{1}$ and by pulling back the ample
linearisation $\mathcal{L}\rightarrow Y$ and twisting by a small multiple of
the exceptional divisor one obtains an ample $G$-linearisation
$\mathcal{L}_{1}\rightarrow\widetilde{Y}_{1}$ such that

$\varphi_{1}^{-1}(Y^{s})\subset\widetilde{Y}_{1}^{s}\subset\widetilde{Y}_{1}^{ss}\subset\varphi_{1}^{-1}(Y^{ss})$

and also

$d_{\max}(\widetilde{Y}^{ss}_{1},G)<d_{\max}(Y^{ss},G).$

By inductively blowing-up $\widetilde{Y}_{i}$ along the closure of
$C(\widetilde{Y}^{ss}_{i},G)$, we eventually obtain a projective scheme
$\widetilde{Y}$ for which semistability and stability coincide.

If the stable set is empty, the partial desingularisation procedure can
terminate in different ways; however, we will not be interested in these cases
in this paper, so we refer to [6] for an overview of the possible ways in
which this can terminate.

####  2.1.2. Example: moduli of semistable sheaves via reductive GIT

Let us recall the construction of moduli spaces of (Gieseker) semistable
sheaves with fixed Hilbert polynomial $P$ over a projective scheme $B$ with
fixed ample line bundle $\mathcal{O}_{B}(1)$ following Simpson [35]. As the
set of semistable sheaves with Hilbert polynomial $P$ is bounded, one can take
$n$ sufficiently large so they are all $n$-regular; thus for $\mathcal{E}$
semistable we have a surjective evaluation map

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

which with a choice of isomorphism $\mathbb{C}^{P(n)}\cong
H^{0}(\mathcal{E}(n))$ determines a point in the Quot scheme

$Q_{n}:=Q_{n}(P):=\operatorname{Quot}(\mathcal{O}_{B}(-n)^{\oplus P(n)},P)$

parametrising quotient sheaves of $\mathcal{O}_{B}(-n)^{\oplus P(n)}$ with
Hilbert polynomial $P$. Let $Q_{n}^{ss}:=Q^{ss}_{n}(P)$ be the open subscheme
of $Q_{n}$ that parametrises quotient sheaves $q:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ such that $\mathcal{E}$ is semistable and
$H^{0}(q(n))$ is an isomorphism; then the isomorphism classes of semistable
sheaves over $B$ with Hilbert polynomial $P$ are in bijective correspondence
with the orbits of the natural $\mathrm{GL}_{P(n)}$-action on $Q^{ss}_{n}$.
The moduli space of semistable sheaves is constructed as a reductive GIT
quotient of $\mathrm{SL}_{P(n)}$ acting on the closure $R_{n}:=R_{n}(P)$ of
$Q^{ss}_{n}$ in $Q_{n}$ with respect to the linearisation $\mathcal{L}_{n,m}$
for $m>\\!>n$ corresponding to Grothendieck’s embedding of the Quot scheme
$Q_{n}$ into a Grassmanninan of $P(m)$-dimensional quotients of
$H^{0}(\mathcal{O}_{B}(m-n))^{\oplus P(n)}$.

###  2.2. Instability stratifications

Associated to a linearised action of the reductive group $G$ on $Y$, there is
a GIT instability, or HKKN, stratification of $Y$ due to work of Kempf [26]
and Hesselink [20]. It also agrees with a symplectic Morse-theoretic
stratification associated to the norm square of the moment map for the action
of the maximal compact subgroup of $G$ [27, 31].

To define this stratification, we must also choose an invariant norm $||-||$
on the Lie algebra of $G$. More precisely, we mean a norm on
$\operatorname{Lie}G$ that is invariant under the adjoint action and such that
for a (or equivalently every) maximal torus $T\subset G$, the norm is integer
valued on the co-character lattice of $T$. In particular, this norm gives an
identification of the co-character lattice and character lattice of $T$ and of
the Lie algebra and co-Lie algebra of $T$.

The HKKN stratification of $Y$ is then a finite stratification by
$G$-invariant locally closed subschemes $S_{\beta}$ such that the lowest
stratum $S_{0}$ is $Y^{ss}$ and the unstable strata are indexed by conjugacy
classes of rational 1-PSs of $G$. The idea is to associate to each unstable
point $y\in Y-Y^{ss}$ the conjugacy class $\beta=[\lambda_{\beta}]$ of
rational 1-PSs of $G$ that are ‘most responsible’ for the instability of this
point (or rather its orbit), where most responsible means that a
representative minimises the normalised Hilbert–Mumford weight
$\mu(y,-)/||-||$.

The unstable strata $S_{\beta}$ can be inductively described following [27]:
first, pick a representative $\lambda_{\beta}$ of each conjugacy class $\beta$
(to do this canonically, fix a maximal torus $T$ and positive Weyl chamber
$\mathfrak{t}_{+}$ in the Lie algebra $\mathfrak{t}$ of $T$, and then pick the
representatives in $\mathfrak{t}_{+}$). For any 1-PS $\lambda$ of $G$, there
is a parabolic subgroup

(1) $P(\lambda):=\\{g\in G:\lim_{t\rightarrow
0}\lambda(t)\>g\>\lambda(t)^{-1}\mathrm{\>exists\>in\>}G\\},$

which is a semi-direct product of a reductive Levi group $L(\lambda)$ with a
unipotent group $U(\lambda)$. If $\lambda_{\beta}$ is the chosen
representative of the conjugacy class $\beta$, then we write
$P_{\beta}=P(\lambda_{\beta})=U_{\beta}\rtimes L_{\beta}$ and let
$q_{\beta}:P_{\beta}\rightarrow L_{\beta}$ denote the retraction onto the Levi
factor.

######  Definition 2.3.

Let $Z_{\beta}$ be the components of the fixed locus
$Y^{\lambda_{\beta}(\mathbb{G}_{m})}$ on which the Hilbert–Mumford weight
$\mu(-,\lambda_{\beta})$ takes the value $-||\lambda_{\beta}||^{2}$. Let
$Y_{\beta}$ be the locally closed subscheme of $Y$ whose closed points are
those $y\in Y$ such that $\lim_{t\rightarrow 0}\lambda_{\beta}(t)\cdot y\in
Z_{\beta}$.

There is a natural retraction $p_{\beta}:Y_{\beta}\rightarrow Z_{\beta}$.
Moreover, $Y_{\beta}$ is invariant under the $P_{\beta}$-action, $Z_{\beta}$
is invariant under the $L_{\beta}$-action and $p_{\beta}$ is equivariant with
respect to $q_{\beta}:P_{\beta}\rightarrow L_{\beta}$. Under the
identification between (rational) characters and co-characters given by
$||-||$, we let $\chi_{-\beta}:L_{\beta}\rightarrow\mathbb{G}_{m}$ denote the
rational character corresponding to the rational 1-PS
$\lambda_{\beta}^{-1}:\mathbb{G}_{m}\rightarrow Z(L_{\beta})$.

The subschemes $Y_{\beta}$ and $Z_{\beta}$ can be more explicitly defined in
terms of weights for the action of a maximal torus $T<G$ as follows. The
linearisation $\mathcal{L}$ (or some positive power) determines a
$G$-equivariant embedding $Y\hookrightarrow\mathbb{P}^{n}$ and we can choose
coordinates on $\mathbb{P}^{n}$ such that the action of the fixed maximal
torus $T\subset G$ is given by
$t\mapsto\operatorname{diag}(\alpha_{0}(t),\dots,\alpha_{n}(t))$ for
characters $\alpha_{i}:T\rightarrow\mathbb{G}_{m}$; then by [27], we have

$\displaystyle\overline{Y_{\beta}}$
$\displaystyle=Y\times_{\mathbb{P}^{n}}\\{[p_{0}:\dots:p_{n}]\in\mathbb{P}^{n}:p_{i}=0\text{
if }\alpha_{i}\cdot\beta<||\beta||^{2}\\}$ (2) $\displaystyle{Y_{\beta}}$
$\displaystyle=\overline{Y_{\beta}}\times_{\mathbb{P}^{n}}\\{[p_{0}:\dots:p_{n}]\in\mathbb{P}^{n}:\exists
i\text{ such that }p_{i}\neq 0\text{ and
}\alpha_{i}\cdot\beta=||\beta||^{2}\\}$ $\displaystyle Z_{\beta}$
$\displaystyle=Y\times_{\mathbb{P}^{n}}\\{[p_{0}:\dots:p_{n}]\in\mathbb{P}^{n}:p_{i}=0\text{
if }\alpha_{i}\cdot\beta\neq||\beta||^{2}\\}.$

######  Definition 2.4.

Let $\mathcal{L}_{\beta}$ denote the $L_{\beta}$-linearisation on $Z_{\beta}$
given by restricting the linearisation $\mathcal{L}$ to $Z_{\beta}$ and
twisting by the character $\chi_{-\beta}$. We define $Z_{\beta}^{ss}\subset
Z_{\beta}$ to be the open subset of GIT semistable points for the
$L_{\beta}$-action on $Z_{\beta}$ with respect to $\mathcal{L}_{\beta}$. Let
$Y_{\beta}^{ss}:=p_{\beta}^{-1}(Z_{\beta}^{ss})$.

Then the following theorem collects the key results concerning this
instability stratification, which we refer to as the Hesselink-Kempf-Kirwan-
Ness stratification (or HKKN stratification).

######  Theorem 2.5.

Let $G$ be a reductive group acting on a projective scheme $Y$ with respect to
an ample linearisation $\mathcal{L}$. For a choice of norm $||-||$ as above,
there is an associated stratification

$Y=\bigsqcup_{\beta\in\mathcal{B}}S_{\beta}$

into finitely many $G$-invariant locally closed subschemes $S_{\beta}$ with
the following properties.

  1. i)

If $Y^{ss}(\mathcal{L})\neq\emptyset$, then $\mathcal{B}$ has a minimal
element denoted $0$ such that $S_{0}=Y^{ss}(\mathcal{L})$.

  2. ii)

The closure of a stratum $S_{\beta}$ is contained in the union
$S_{\beta}\sqcup\bigsqcup_{\beta^{\prime}>\beta}S_{\beta^{\prime}}$.

  3. iii)

For $\beta\in\mathcal{B}$, we have $S_{\beta}:=G\cdot Y_{\beta}^{ss}\cong
G\times^{P_{\beta}}Y_{\beta}^{ss}$.

######  Remark 2.6.

The description of the unstable strata in [27] was originally only described
for varieties. For a scheme $Y$, one uses the linearisation $\mathcal{L}$ to
$G$-equivariantly embed $Y$ into $\mathbb{P}^{n}$ and then constructs a scheme
theoretic stratification of $Y$ by taking the fibre product inside
$\mathbb{P}^{n}$ of $Y$ with the strata in $\mathbb{P}^{n}$, which are locally
closed subvarieties of $\mathbb{P}^{n}$ (see [23, $\S$4]).

####  2.2.1. Example: HKKN stratification for sheaves

For a projective polarised scheme $(B,\mathcal{O}_{B}(1))$, recall from
$\S$2.1.2 that the moduli space of semistable sheaves on $B$ with Hilbert
polynomial $P$ is constructed as a GIT quotient of the
$\mathrm{SL}_{P(n)}$-action on the closed subscheme $R_{n}$ of the Quot scheme
$Q_{n}:=\operatorname{Quot}(\mathcal{O}_{B}(-n)^{\oplus P(n)},P)$ linearised
by $\mathcal{L}_{n,m}$, provided $m>\\!>n>\\!>0$.

Associated to the universal quotient sheaf $\mathcal{U}_{n}\rightarrow B\times
Q_{n}$, there is a schematic Harder–Narasimhan stratification [33]

(3) $Q_{n}=\bigsqcup_{\tau}Q^{\tau}_{n}.$

In [22, 23], this HN stratification was compared with the HKKN

$Q_{n}=\bigsqcup_{\beta\in\mathcal{B}_{n,m}}S_{\beta}$

associated to this $\mathrm{SL}_{P(n)}$-action on $Q_{n}$ linearised by
$\mathcal{L}_{n,m}$. The first step is to associate to a HN type $\tau$ a
candidate HKKN index $\beta$ (i.e. the conjugacy class of a rational 1-PS).

######  Definition 2.7.

For a tuple of Hilbert polynomials $\tau=(P_{1},\dots,P_{l})$ with
$\sum_{i=1}^{l}P_{i}=P$ and $(n,m)\in\mathbb{N}^{2}$, we let
$\beta_{n,m}(\tau)$ denote the conjugacy class of the following rational 1-PS
of $\mathrm{SL}_{P(n)}$

$\lambda_{\beta_{n,m}(\tau)}(t)=\operatorname{diag}(t^{\beta_{1}}I_{P_{1}(n)},\dots,t^{\beta_{l}}I_{P_{l}(n)})\>\>\text{
where }\>\beta_{i}:=\frac{P(m)}{P(n)}-\frac{P_{i}(m)}{P_{i}(n)}.$

Let $P_{\tau}:=P(\lambda_{\beta_{n,m}(\tau)})=U_{\tau}\rtimes
L_{\tau}<\mathrm{SL}_{P(n)}$ be the parabolic subgroup determined by
$\lambda_{\beta_{n,m}(\tau)}$.

######  Theorem 2.8 ([22, 23]).

Let $\tau=(P_{1},\dots,P_{l})$ be a HN type for sheaves on
$(B,\mathcal{O}_{B}(1))$ with Hilbert polynomial $P$; then, for
$m>\\!>n>\\!>0$, the following statements hold for
$\beta:=[\beta_{n,m}(\tau)]$.

  1. i)

All sheaves over $B$ with HN type $\tau$ are $n$-regular.

  2. ii)

If $S_{\tau}$ denotes the open subscheme of $Q_{n}^{\tau}$ consisting of
quotients $q:\mathcal{O}_{B}(-n)^{\oplus P(n)}\twoheadrightarrow\mathcal{E}$
such that $H^{0}(q(n))$ is an isomorphism, then the $SL_{P(n)}$-orbits of
closed points in $S_{\tau}$ are in bijection with the isomorphism classes of
sheaves over $B$ with HN type $\tau$.

  3. iii)

$\beta\in\mathcal{B}$ is a HKKN index for the $\mathrm{SL}_{P(n)}$-action on
$Q_{n}$ with respect to $\mathcal{L}_{n,m}$ and $S_{\tau}$ is a closed
subscheme of the corresponding HKKN stratum $S_{\beta}$.

  4. iv)

There is a closed $L_{\beta}$-invariant subscheme $Z_{\tau}^{ss}\subset
Z_{\beta}^{ss}$ such that if $Y_{\tau}^{ss}:=p_{\beta}^{-1}(Z_{\tau}^{ss})$;
then

$S_{\tau}=\mathrm{SL}_{P(n)}\cdot
Y_{\tau}^{ss}\cong\mathrm{SL}_{P(n)}\times^{P_{\tau}}Y_{\tau}^{ss}.$

Moreover, by [22, 23], we have an isomorphism

(4) $Z_{\tau}^{ss}\cong Q^{ss}_{n}(P_{1})\times\cdots\times
Q^{ss}_{n}(P_{l}),$

where $Q^{ss}_{n}(P_{i})$ is the open subscheme of
$Q_{n}(P_{i}):=\operatorname{Quot}(\mathcal{O}_{B}(-n)^{\oplus
P_{i}(n)},P_{i})$ consisting of quotient sheaves
$q_{i}:\mathcal{O}_{B}(-n)^{\oplus P_{i}(n)}\twoheadrightarrow\mathcal{E}_{i}$
such that $\mathcal{E}_{i}$ is semistable and $H^{0}(q_{i}(n))$ is an
isomorphism. In fact, if $R_{n}(P_{i})$ denotes the closure of
$Q^{ss}_{n}(P_{i})$ in $Q_{n}(P_{i})$; then $Z_{\tau}^{ss}$ is the open
subscheme of

(5) $Z_{\tau}\cong R_{n}(P_{1})\times\cdots\times R_{n}(P_{l})$

equal to the GIT semistable set for the action of the Levi subgroup
$L_{\tau}<P_{\tau}$ with respect to the canonical linearisation
$\mathcal{L}_{\tau}:=\mathcal{L}_{\beta}$. The closed points of
$Y_{\tau}^{ss}$ are quotients $q:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ such that $H^{0}(q(n))$ is an isomorphism,
$\mathcal{E}$ has HN type $\tau$ and the filtration on $\mathbb{C}^{P(n)}$
given by the 1-PS $\lambda_{\beta_{n,m}(\tau)}$ induces the HN filtration of
$\mathcal{E}$. The retraction $p_{\tau}:Y_{\tau}^{ss}\rightarrow
Z_{\tau}^{ss}$ sends a quotient sheaf $q:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ to the associated graded sheaf for its HN
filtration.

######  Remark 2.9.

If $B$ is a smooth projective curve, then for $n>\\!>0$, the Quot scheme
$Q^{ss}_{n}(P)$ is smooth (see [24, Proposition 2.2.8]). Consequently, for a
HN type $\tau$ of a vector bundle on a smooth projective curve, the HN stratum
$S_{\tau}$ is also smooth for $n>\\!>0$, as $Z_{\tau}^{ss}$ is isomorphic to a
product of smooth schemes by (4).

###  2.3. Non-reductive GIT quotients

The results of non-reductive GIT are strongest for linear algebraic groups
with so-called graded unipotent radical; for linear actions of such groups on
projective schemes, there are open (semi)stable sets with Hilbert–Mumford type
descriptions that admit quasi-projective good quotients with natural
projective completions [1, 3]. Thus the key features and computational
flexibility of reductive GIT carry over to this non-reductive setting. In this
subsection, we summarise and slightly expand on these results.

####  2.3.1. Linear algebraic groups with graded unipotent radical

Throughout this section, we let $H=U\rtimes L$ be a linear algebraic group,
where $U$ is the unipotent radical of $H$ and $L$ is a reductive Levi subgroup
(which is unique up to conjugation). We only consider linear algebraic groups
whose unipotent radical is graded in the following sense.

######  Definition 2.10.

An _internal grading of the unipotent radical_ $U$ of $H$ is a 1-PS
$\lambda:\mathbb{G}_{m}\rightarrow Z(L)$ such that $\lambda(\mathbb{G}_{m})$
acts via conjugation on the Lie algebra of $U$ with strictly positive weights.
If there is a central 1-PS $\lambda:\mathbb{G}_{m}\rightarrow Z(L)$ which
grades the unipotent radical $U$ of $H$ via its conjugation action, we say
that $\lambda$ _internally grades the unipotent radical_ $U$ of $H$, and
define the normal subgroup

$\widehat{U}:=U\rtimes\lambda(\mathbb{G}_{m}).$

Thus $\widehat{U}<H$ with quotient $\overline{L}:=L/\lambda(\mathbb{G}_{m})$.

For us, the key example of a group with internally graded unipotent radical is
a parabolic subgroup $P=U\rtimes L$ of a reductive group (see Lemma 3.4).

Note that the grading 1-PS is not necessarily unique. We will see that a
choice of grading 1-PS together with a choice of linearisation determines an
open subset admitting a quotient. The dependence of this open set on the
grading and linearisation is analogous to reductive Variation of GIT [14]
[36], and is explored in [7].

######  Remark 2.11.

In this paper, we will be primarily concerned with groups with internally
graded unipotent radical and so we describe the results of [3] in this
setting. There is however a more general notion of groups with _externally
graded unipotent radical_ , as follows. An _external grading of the unipotent
radical_ $U$ of $H$ is a 1-PS
$\lambda:\mathbb{G}_{m}\rightarrow\operatorname{Aut}(H)$ not coming from a
1-PS of $H$, which commutes with the action of $L$ and acts on $U$ with
strictly positive weights. One can then construct the associated _externally
graded group_ $\widehat{H}$ as the semi-direct product of $H$ and this 1-PS
$\lambda$

$\widehat{H}:=H\rtimes\lambda(\mathbb{G}_{m})=U\rtimes(L\times\lambda(\mathbb{G}_{m}))=\widehat{U}\rtimes
L\>\text{ where }\widehat{U}:=U\rtimes\lambda(\mathbb{G}_{m}).$

The general procedure for taking quotients using external gradings is
described in [3, 1].

Often in this paper we will choose a subgroup $L^{\prime}<L$ such that the
composition $L^{\prime}\rightarrow
L\rightarrow\overline{L}=L/\lambda(\mathbb{G}_{m})$ is surjective with finite
kernel, and then think of $\lambda$ as externally grading the subgroup
$H^{\prime}=U\rtimes L^{\prime}<H=U\rtimes L$; then the quotients of $Y$ by
the internally graded group $H$ and externally graded group
$\widehat{H}^{\prime}$ coincide.

####  2.3.2. Set-up and notation

######  Assumption 2.12.

Throughout $\S$2.3, we assume that $H=U\rtimes L$ is a linear algebraic group
which is internally graded by a fixed 1-PS $\lambda:\mathbb{G}_{m}\rightarrow
Z(L)$ and that $H$ acts on a projective scheme $Y$ with respect to a very
ample $H$-linearisation $\mathcal{L}\rightarrow Y$. We write
$\hat{U}:=U\rtimes\lambda(\mathbb{G}_{m})<H$ and
$\overline{L}:=L/\lambda(\mathbb{G}_{m})$ for the quotient of $L$ by this
central $\mathbb{G}_{m}$.

Following [3], we will first construct a quotient of the action of the graded
unipotent group $\widehat{U}$, and then use reductive GIT to construct the
residual quotient by $\overline{L}$. In $\S$2.3.5 and $\S$2.3.6, we explain
the central results in [3, 1], which give the construction of a quasi-
projective geometric quotient of the $H$-action on an open stable set of $Y$
admitting a Hilbert–Mumford type description together with a natural
projective completion. The first step is to show that one can construct a
projective quotient of the $H$-action if one assumes the stabilisers are
reasonably well-behaved (see Definition 2.20 below, which one can think of as
‘semistability coinciding with stability’ condition); we describe this case in
$\S$2.3.5 below (cf. Theorems 2.26 and 2.29). To prove the more general
statement without assuming this condition on the stabilisers, one performs a
sequence of equivariant blow-ups to arrange for this condition to hold on the
resulting blow-up and then we construct a quotient of an open set in $Y$ as an
open subset of the projective quotient of the blow-up; this is outlined in
$\S$2.3.6.

To describe the stable loci and the relevant stabiliser conditions, we need to
introduce some important notation.

######  Definition 2.13.

For the 1-PS $\lambda$ grading $U$, we let
$\omega_{\min}<\omega_{\min+1}<\cdots<\omega_{\max}$ denote the weights of the
$\lambda(\mathbb{G}_{m})$-action on $V:=H^{0}(Y,\mathcal{L})^{*}$ and we write
$V_{\min}$ for the weight space of $\omega_{\min}$.

  1. (1)

The _$\lambda$ -minimal weight space_, denoted $Z_{\min}=Z(Y,\lambda)_{\min}$
is the closed subscheme

$Z(Y,\lambda)_{\min}:=Y\cap\mathbb{P}(V_{\min})=\left\\{y\in
Y^{\lambda(\mathbb{G}_{m})}:\lambda(\mathbb{G}_{m})\text{ acts on
}\mathcal{L}^{*}|_{y}\text{ with weight }\omega_{\min}\right\\},$

  2. (2)

The _$\lambda$ -attracting open set_, denoted555In [1], the open scheme
$Y_{\min}$ is denoted $Y_{\min}^{0}$. $Y_{\min}=Y(\lambda)_{\min}$, is the
open subscheme

$Y(\lambda)_{\min}:=\\{y\in Y\mid p(y)\in Z_{\min}\\}\quad\mbox{and }\quad
p(y):=\lim_{\begin{subarray}{c}t\to 0\\\
t\in\mathbb{G}_{m}\end{subarray}}\lambda(t)\cdot y\quad\mbox{ for }y\in Y,$

  3. (3)

The _$\lambda$ -retraction_ is $p:Y_{\min}\rightarrow Z_{\min}$, given by
$p(y):=\lim_{t\rightarrow 0}\lambda(t)\cdot y$,

  4. (4)

The _(semi)stable minimal weight space_ is
$Z_{\min}^{(s)s}:=Z_{\min}^{\overline{L}-(s)s}$ and we let
$Y_{\min}^{(s)s}:=p^{-1}(Z_{\min}^{(s)s})$.

The subscheme $Z_{\min}\subset Y$ is closed and is contained in the fixed
locus $Y^{\lambda(\mathbb{G}_{m})}$. The subscheme $Y_{\min}$ is the open
stratum in the Białynicki-Birula decomposition for the downward flow of the
$\lambda(\mathbb{G}_{m})$-action on $Y$. The retraction $p:Y_{\min}\rightarrow
Z_{\min}$ is a Zariski locally trivial fibration in affine varieties (or even
affine spaces if $Y$ is smooth) [9]. We assume that there are at least two
distinct $\lambda(\mathbb{G}_{m})$-weights on $V$, as otherwise the $U$-action
on $Y$ is trivial, and in this case we could consider GIT for the action of
the reductive group $L=H/U$.

####  2.3.3. Adapted linearisations

The choice of linearisation in NRGIT is very important. In order to achieve
the best results, we will later require that the linearisation satisfies the
following _adaptedness_ condition.

######  Definition 2.14.

A very ample $H$-linearisation $\mathcal{L}$ over $Y$ is _adapted_ for the
action of $\widehat{U}<H$ if the $\lambda(\mathbb{G}_{m})$-weights on
$V:=H^{0}(Y,\mathcal{L})^{*}$ satisfy the following inequalities

$\omega_{\min}<0<\omega_{\min+1}<\cdots<\omega_{\max}.$

By twisting a $H$-linearisation $\mathcal{L}$ over $Y$ by a (rational)
character $\chi:H\rightarrow\mathbb{G}_{m}$, one effectively shifts the
$\lambda(\mathbb{G}_{m})$-weights on $V$ above, and thus one can arrange for
the twisted linearisation to be adapted. We note that the subschemes
$Z_{\min}$ and $Y_{\min}$ do not change under this twisting.

######  Remark 2.15.

In [1], there is also a notion of _well-adapted_ linearisation which one can
think of as requiring $0\in(\omega_{\min},\omega_{\min}+\epsilon)$ (see [1,
$\S$2] for the full definition).

For our major results we will always assume that the linearisation in question
is adapted. However, the ancillary notion of _borderline_
linearisations666Referred to as a canonical linearisation in [23, $\S$3.2]. is
also useful.

######  Definition 2.16.

A very ample $H$-linearisation $\mathcal{L}$ over $Y$ is _borderline_ for the
action of $\widehat{U}<H$ if the $\lambda(\mathbb{G}_{m})$-weights on
$V:=H^{0}(Y,\mathcal{L})^{*}$ satisfy the following inequalities

$\omega_{\min}=0<\omega_{\min+1}<\cdots<\omega_{\max}.$

The significance of a borderline linearisation is that it allows one to obtain
non-reductive GIT quotients without needing to impose any of the conditions on
stabiliser groups discussed in §2.3.4 below. Unfortunately, these quotients
are very far from being geometric, and so are of limited practical use: if one
works with such a borderline linearisation, one can obtain a categorical
$H$-quotient of a larger semistable set, but this $H$-quotient factors via the
retraction $p:Y_{\min}\rightarrow Z_{\min}$ given by flowing under the 1-PS
$\lambda$ and so contracts many orbits. More concretely, we have the following
result, whose proof follows analogously to the argument in [23, $\S$3.2].

######  Proposition 2.17.

Let $H=U\rtimes L$ be a linear algebraic group whose unipotent radical $U$ is
internally graded by $\lambda:\mathbb{G}_{m}\rightarrow L$, acting on a
projective scheme $Y$ with respect to a very ample borderline linearisation
$\mathcal{L}_{0}$. Then

$\bigoplus_{r\geq 0}H^{0}(Y,\mathcal{L}_{0}^{\otimes
r})^{H}\cong\bigoplus_{r\geq 0}H^{0}(Z_{\min},\mathcal{L}_{0}^{\otimes
r})^{L}$

and the associated projective scheme $Z_{\min}/\\!/_{\mathcal{L}_{0}}L$ is a
categorical quotient of

  1. (1)

the $L$-action on $Z_{\min}^{ss}$,

  2. (2)

the $L$-action on $Y_{\min}^{ss}$,

  3. (3)

the $H$-action on $Y_{\min}^{ss}$.

In particular, we have that

$Y_{\min}^{ss}=Y^{H-ss}(\mathcal{L}_{0}):=\\{y\in Y:\exists\sigma\in
H^{0}(Y,\mathcal{L}_{0}^{\otimes r})^{H}\text{ for }r>0\text{ with
}\sigma(y)\neq 0\\}$

and the morphism $Y_{\min}^{ss}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Z_{\min}^{ss}\rightarrow Z_{\min}/\\!/_{\mathcal{L}_{0}}L$
is a surjective categorical $H$-quotient that identifies every $y\in
Y_{\min}^{ss}$ with its limit $p(y)=\lim_{t\rightarrow 0}\lambda(t)\cdot y\in
Z_{\min}^{ss}$.

###### Proof.

This follows by adapting the arguments in [23, Lemma 3.1 and Proposition 3.2]
for the special case when $Y_{\min}=Y_{\beta}$ and $Z_{\min}=Z_{\beta}$ to the
retraction $p:Y_{\min}^{ss}\rightarrow Z_{\min}^{ss}$ which is equivariant
with respect to the surjection $q:H\twoheadrightarrow L$. ∎

Another use of borderline linearisations is that they enable us to obtain
information about adapated linearisations, via variation of GIT style
arguments. A well-adapted linearisation is a small perturbation of a
borderline linearisation $\mathcal{L}_{0}$; typically they are constructed by
twisting $\mathcal{L}_{0}$ by a small rational character. If the associated
ring of invariant sections for the perturbed linearisation are still finitely
generated, then the following result compares the associated ‘quotient
morphisms’ (which, for the perturbed linearisation, is not necessarily
surjective even when the invariant sections are finitely generated).

######  Proposition 2.18.

Let $H=U\rtimes L$ be a linear algebraic group with unipotent radical $U$
internally graded by $\lambda:\mathbb{G}_{m}\rightarrow L$ acting on a
projective scheme $Y$ with respect to a very ample borderline linearisation
$\mathcal{L}_{0}$. Given a (rational) character
$\chi:H\rightarrow\mathbb{G}_{m}$, we let $\mathcal{L}_{\chi}$ denote the
twist of $\mathcal{L}_{0}$ by the rational character $\epsilon\chi$ for
$\epsilon\in\mathbb{Q}_{>0}$ sufficiently small and let
$R(Y,\mathcal{L}_{\chi}):=\bigoplus_{r\geq
0}H^{0}(Y,\mathcal{L}_{\chi}^{\otimes r})$. If $R(Y,\mathcal{L}_{\chi})^{H}$
is finitely generated, then there is an induced, not necessarily surjective,
$H$-invariant morphism

$q_{H,\chi}:Y^{H-ss}(\mathcal{L}_{\chi})\longrightarrow
Y/\\!/_{\mathcal{L}_{\chi}}H:=\operatorname{Proj}(\bigoplus_{r\geq
0}H^{0}(Y,\mathcal{L}_{\chi}^{\otimes r})^{H}),$

where we define $Y^{H-ss}(\mathcal{M}):=\\{y\in Y:\exists\sigma\in
H^{0}(Y,\mathcal{M}^{\otimes r})^{H}\text{ for }r>0\text{ with }\sigma(y)\neq
0\\}$ for any linearisation, and we have

$Y^{H-ss}(\mathcal{L}_{\chi})\subset Y^{H-ss}(\mathcal{L}_{0})=Y_{\min}^{ss}.$

Furthermore, if $q_{H,\chi}$ is a categorical $H$-quotient, then there is an
induced morphism between the $H$-quotients with respect to
$\mathcal{L}_{\chi}$ and $\mathcal{L}_{0}$:

(6) $Y/\\!/_{\mathcal{L}_{\chi}}H\longrightarrow Y/\\!/_{\mathcal{L}_{0}}H.$

###### Proof.

The inclusions of invariant rings, which are by assumption finitely generated,

$R(Y,\mathcal{L}_{\chi})\supset R(Y,\mathcal{L}_{\chi})^{L}\supset
R(Y,\mathcal{L}_{\chi})^{H}$

induce rational morphisms of projective schemes

$q_{H,\chi}:Y\dashrightarrow Y/\\!/_{\mathcal{L}_{\chi}}L\dashrightarrow
Y/\\!/_{\mathcal{L}_{\chi}}H.$

Note that there is not a residual action of $U$ on
$Y/\\!/_{\mathcal{L}_{\chi}}L$. Nevertheless, we obtain that

$Y^{H-ss}(\mathcal{L}_{\chi})\subset Y^{L-ss}(\mathcal{L}_{\chi})\subset
Y^{L-ss}(\mathcal{L}_{0})=Y^{H-ss}(\mathcal{L}_{0})=Y^{ss}_{\min}$

where the first inclusion follows from the above factorisation of
$q_{H,\chi}$, the second inclusion follows by variation of reductive GIT
quotients (as $\mathcal{L}_{\chi}$ is a small perturbation of
$\mathcal{L}_{0}$), and the remaining equalities follow from Proposition 2.17.
Since the composition $Y^{H-ss}(\mathcal{L}_{\chi})\hookrightarrow
Y^{H-ss}(\mathcal{L}_{0})\rightarrow Y/\\!/_{\mathcal{L}_{0}}H$ is
$H$-invariant, provided $q_{H,\chi}$ is a categorical $H$-quotient, we obtain
via its universal property a morphism as in Equation (6). ∎

######  Remark 2.19.

In fact, in the above situation, we have $Y^{H-ss}(\mathcal{L}_{\chi})\subset
Y^{ss}_{\min}\setminus UZ^{ss}_{\min}$ provided the perturbation
$\mathcal{L}_{\chi}$ is non-trivial, as $Y^{H-ss}(\mathcal{L}_{\chi})\subset
Y^{\lambda(\mathbb{G}_{m})-ss}(\mathcal{L}_{\chi})$. The latter is either
equal to $Y_{\min}\setminus Z_{\min}$, if $\mathcal{L}_{\chi}$ is adapted, or
empty, if the borderline linearisation is perturbed in the other direction.

####  2.3.4. Stabiliser assumptions

Let us introduce some assumptions on the stabiliser groups, considered as
analogous to ‘semistability coinciding with (Mumford) stability for some
group’. For any $y\in Y$ and any subgroup $G<H$, we write
$\operatorname{Stab}_{G}(y)$ for the stabiliser group of $y$.

######  Definition 2.20.

For the $H$-action on $Y$ with respect to an ample adapted linearisation
$\mathcal{L}$, we define the following conditions:

  1. (1)

_Semistability coincides with stability for $\widehat{U}$_

$[\widehat{U}]_{0}$ $\dim\operatorname{Stab}_{U}(z)=0\>\text{ for all }z\in
Z_{\min},$

and _semistability coincides with stability for $\widehat{U}$ on the reductive
semistable locus_ if

$[\widehat{U};\overline{L}\text{-ss}]_{0}$
$\dim\operatorname{Stab}_{U}(z)=0\>\text{ for all }z\in Z_{\min}^{ss}.$

  2. (2)

If $U$ is abelian, we say _semistability coincides with Mumford stability for
$\widehat{U}$_ if

$[\widehat{U}]$ $\dim\operatorname{Stab}_{U}(-)\text{ is constant on
}Y_{\min},$

and _semistability coincides with Mumford stability for $\widehat{U}$ on the
reductive semistable locus_ if

$[\widehat{U};\overline{L}\text{-ss}]$ $\dim\operatorname{Stab}_{U}(-)\text{
is constant on }Y_{\min}^{ss}.$

  3. (3)

_Semistability coincides with stability for $\overline{L}$_

$[\overline{L}]_{0}$ $\dim\operatorname{Stab}_{\overline{L}}(z)=0\>\text{ for
all }z\in Z_{\min}^{ss}.$

  4. (4)

If $U$ is abelian, we say _the minimal weight space has minimal dimensional
$U$-stabilisers_ if

$[C(Z_{\min},U,d_{\min})\neq\emptyset]$
$\dim\operatorname{Stab}_{U}(z)=d_{\min}:=d_{\min}(Y_{\min},U)\text{ for
some}\>z\in Z_{\min},$

where $d_{\min}(Y_{\min},U)$ is as in Definition 2.2; this is equivalent to
$C(Z_{\min},U,d_{\min})\neq\emptyset$.

If $U$ is not abelian then (2) and (4) above are modified by choosing a series
of subgroups $\\{e\\}=U^{(0)}\leqslant U^{(1)}\cdots\leqslant U^{(r)}=U$
normalised by $H$ whose successive quotients are abelian and asking for these
conditions to hold for each $U^{(i)}$.

######  Remark 2.21.

Definitions (1) – (2) on the unipotent stabilisers appear in [1, 3], whereas
(3) is an additional condition that we use in this paper to determine the
stable locus explicitly. Condition (4) is a weakened version of (2): when (2)
fails, we will perform a blow-up sequence to arrange for (2) to hold, and (4)
ensures that we never blow up all of $Z_{\min}$ and so we can easily determine
the minimal weight space in the blow-up and obtain an explicit expression for
the stable set in this case.

Our primary motivation for constructing quotients by parabolic group actions
in stages is that conditions such as these may hold for the relevant groups at
each stage in the quotienting process, even when they fail to hold directly
for the initial $H$-action on $Y$ graded by $\lambda$. For example, when
looking at sheaves of fixed Harder–Narasimhan type of length $l>2$, Condition
$[\overline{L}]_{0}$ fails directly (see $\S$3.2.1) and our approach via
Quotienting-in-stages circumvents this problem by enabling us to set up a
sequence of quotients such that these types conditions may hold at each stage.

######  Remark 2.22.

  1. (1)

Condition $[\widehat{U}]_{0}$ is referred to as ‘semistability coincides with
stability’ in [3] and is denoted by $(ss=s\neq\emptyset[\widehat{U}])$ in [1].
As a unipotent group has no non-trivial finite subgroups, $[\widehat{U}]_{0}$
is equivalent to asking that all $U$-stabilisers on $Z_{\min}$ are trivial.
Furthermore, as the $\mathbb{G}_{m}$-action normalises the $U$-action,
$[\widehat{U}]_{0}$ is equivalent to $U$ acting freely on $Y_{\min}$ (see [1,
Remark 5.7]). This condition depends on the grading $\widehat{U}$ of $U$ by
$\lambda$, as the 1-PS $\lambda$ determines $Z_{\min}$.

  2. (2)

Condition $[\overline{L}]_{0}$ says that for $\overline{L}$ acting on
$Z_{\min}$, the GIT stable set coincides with the GIT semistable set.

  3. (3)

Conditions $[\widehat{U}]$ and $[\widehat{U};\overline{L}\text{-ss}]$ allow
for the presence of positive dimensional unipotent stabilisers, provided they
are of constant dimension for each unipotent subgroup in the chosen series.
See [1, Remark 7.12] and the proof of Theorem 2.26 below.

Let us finally include a result about stabilisers from [1] that we will need
later on.

######  Lemma 2.23.

[1, Lemma 5.2] For the $H$-action on $Y$ with respect to an ample adapted
linearisation $\mathcal{L}$, we have for $z\in Z_{\min}$ and $u\in U$ that
$uz\in Z_{\min}$ if and only if $u\in\operatorname{Stab}_{U}(z)$. Furthermore,

$\operatorname{Stab}_{H}(z)=\operatorname{Stab}_{U}(z)\rtimes\operatorname{Stab}_{L}(z)$

and, in particular,
$\dim\operatorname{Stab}_{H}(z)=\dim\operatorname{Stab}_{U}(z)+\dim\operatorname{Stab}_{L}(z)$.

###### Proof.

We extend the argument of [1, Lemma 5.2]. For the first statement, as $p$ is
$U$-invariant and restricts to the identity on $Z_{\min}$, if $uz\in
Z_{\min}$, we see $p(uz)=uz$ and also $p(uz)=z$, and so
$u\in\operatorname{Stab}_{U}(z)$. For the second statement, any $h\in H$ can
be written as $h=ul$ for $u\in U$ and $l\in L$. If $hz=z$ then $uz=l^{-1}z\in
Z_{\min}$ since $L$ preserves $Z_{\min}$ as the grading 1-PS $\lambda$ is
central in $L$. Thus any $h\in\operatorname{Stab}_{H}(z)$ can be written as a
product of $ul$ with both elements fixing $z$. Since
$\operatorname{Stab}_{U}(z)\cap\operatorname{Stab}_{L}(z)=\\{e\\}$ the result
follows. ∎

####  2.3.5. The construction of quotients under optimal stabiliser
assumptions

In this section, we assume we have an adapted linear action of $H=U\rtimes L$
on a projective scheme $Y$ with internally graded unipotent radical. Let us
define the stable loci in the case when the weaker assumption $[\widehat{U}]$
on unipotent stabilisers holds.

######  Definition 2.24.

For a $H$-action on $Y$ with respect to a very ample adapted linearisation
$\mathcal{L}$ satisfying $[\widehat{U}]$, we define the following stable sets.

  1. (1)

The $\widehat{U}$-stable set is defined to be

$Y^{\widehat{U}-s}:=Y_{\min}\setminus UZ_{\min}=\bigcap_{u\in
U}uY^{\lambda(\mathbb{G}_{m})-s}.$

  2. (2)

The $H$-stable set is defined to be

$Y^{H-s}:=Y^{s}_{\min}\setminus UZ_{\min}^{s}.$

######  Remark 2.25.

  1. (1)

The equality in the first definition holds as
$Y^{\lambda(\mathbb{G}_{m})-s}=Y^{\lambda(\mathbb{G}_{m})-ss}=Y_{\min}\setminus
Z_{\min}$ by the Hilbert–Mumford criterion for $\lambda(\mathbb{G}_{m})$ and
the assumption that the linearisation is adapted. This is an open set in $Y$,
as the assumption $[\widehat{U}]_{0}$ implies that $UZ_{\min}\subset
Y_{\min}^{0}$ is a closed subscheme (cf. [1, Lemma 3.4]).

  2. (2)

In [3], the $\widehat{U}$-stable set is denoted $Y^{\widehat{U}-s}_{\min+}$
rather than $Y^{\widehat{U}-s}$, but we have simplified the notation as we
will not introduce any other stable sets (see [3, 1] for a discussion of
various other notions of stability defined using invariant sections).

  3. (3)

Our definition of the $H$-stable set differs slightly from that given in [1].
However, under the assumption $[\overline{L}]_{0}$, both these notions
coincide (see Theorem 2.28 below).

  4. (4)

These definitions are functorial with respect to equivariant closed immersions
$Z\hookrightarrow Y$; that is, $Z^{H-s}=Z\times_{Y}Y^{H-s}$.

Let us state the main result concerning $\widehat{U}$-quotients from [3],
whose assumptions are weakened in [1].

######  Theorem 2.26.

[3, Theorem 0.2 and Corollary 0.6] Let $H=U\rtimes L$ be a linear algebraic
group with unipotent radical $U$ internally graded by
$\lambda:\mathbb{G}_{m}\rightarrow L$ acting on a projective scheme $Y$ with
respect to a very ample adapted linearisation $\mathcal{L}$. Assuming
condition $[\widehat{U}]$ holds, the following statements hold.

  1. _(i)_

The $\widehat{U}$-action on $Y^{\widehat{U}-s}=Y_{\min}\setminus UZ_{\min}$
has a projective geometric $\widehat{U}$-quotient

$q_{\widehat{U}}:Y^{\widehat{U}-s}\rightarrow
Y/\\!/\widehat{U}=\operatorname{Proj}R(Y,\mathcal{L})^{\widehat{U}}.$

  2. _(ii)_

By taking a reductive GIT quotient of the induced $\overline{L}$-action on the
projective geometric $\widehat{U}$-quotient $Y/\\!/\widehat{U}$

$\textstyle{q:Y^{\widehat{U}-s}=Y_{\min}\setminus
UZ_{\min}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\quad\quad
q_{\widehat{U}}}$$\textstyle{Y/\\!/\widehat{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\overline{L}}\quad\quad\quad}$$\textstyle{Y/\\!/H:=(Y/\\!/\widehat{U})/\\!/\overline{L}}$

one obtains a projective good $H$-quotient of an open set of
$Y^{\widehat{U}-s}$,

$\operatorname{dom}(q)\twoheadrightarrow
Y/\\!/H=\operatorname{Proj}R(Y,\mathcal{L})^{H}.$

###### Proof.

Under the stronger assumption $[\widehat{U}]_{0}$ this is [3, Theorem 0.2 and
Corollary 0.6]; let us describe how to weaken this assumption to
$[\widehat{U}]$ as outlined in [1, Remarks 2.8 and 7.12]. The idea is to
iteratively construct quotients by the abelian groups $U^{(j)}/U^{(j-1)}$ for
$j=1,\dots,r$ by locally choosing complementary subgroups to the stabiliser
groups. To illustrate the argument, we assume for simplicity that $U$ is
abelian; then by $[\widehat{U}]$, there exists $d$ such that
$\dim\operatorname{Stab}_{U}(y)=d$ for all $y\in Y_{\min}$. As $U$ is an
abelian unipotent group, for each stabiliser group
$\operatorname{Stab}_{U}(y)$ of $y\in Y_{\min}$, we can choose a complementary
subgroup $U^{\prime}$ with $\operatorname{Stab}_{U}(y)U^{\prime}=U$ and
$\operatorname{Stab}_{U}(y)\cap U=\\{1\\}$, by choosing a complementary
subspace in the Lie algebra. Taking the Lie algebra of the $U$-stabiliser
gives a $\widehat{U}$-invariant function
$Y_{\min}\rightarrow\mathrm{Gr}(d,\operatorname{Lie}(U))$ and being
complementary to a given $U^{\prime}<U$ is an open condition in this
Grassmannian. Therefore, we get a corresponding open $\widehat{U}$-invariant
set of $Y_{\min}$ on which all stabilisers have complementary subgroup
$U^{\prime}$ and thus quotienting by $U$ is equivalent to quotienting by
$U^{\prime}$. This open set of $Y_{\min}$ admits a geometric
$U^{\prime}$-quotient (and thus $U$-quotient) by [1, Proposition 7.4]. Then by
varying $U^{\prime}$ we can cover $Y_{\min}$ by open sets admitting geometric
$U$-quotients in order to obtain a geometric $U$-quotient of $Y_{\min}$. From
there one proceeds as in the proof of the case when $[\widehat{U}]_{0}$ holds.
∎

######  Remark 2.27.

Let us make a few comments about the construction of the quotients in the
above theorem.

  1. (1)

One first shows that there is a geometric $U$-quotient
$q_{U}:Y_{\min}\rightarrow Y_{\min}/U$ by gluing together affine geometric
quotients; this uses the grading 1-PS $\lambda$ and the fact that
$[\widehat{U}]_{0}$ holds (see [1, Proposition 7.4]).

  2. (2)

Then the grading 1-PS $\lambda$ is also used to give a projective
$\widehat{U}$-quotient as follows. First one constructs a
$\lambda(\mathbb{G}_{m})$-equivariant embedding of $Y_{\min}/U$ in a
projective space and we let $\overline{Y_{\min}/U}$ denote its projective
completion (see [1, Lemma 7.6]). Then the projective $\widehat{U}$-quotient is
constructed by taking a reductive quotient of $\overline{Y_{\min}/U}$ by
$\lambda(\mathbb{G}_{m})$, where one slightly perturbs the adapted
linearisation by twisting by a rational character to obtain a so-called well-
adapted linearisation (cf. [1, $\S$2]).

  3. (3)

If $[\widehat{U}]_{0}$ holds, then by [3, Theorem 0.2 (ii)],
$Y^{\widehat{U}-s}$ coincides with the _locally trivial stable locus_ (see [3,
Definition 1.5)], and it is shown that this geometric $\widehat{U}$-quotient
is projective and coincides with the _enveloping quotient_ (see [3, Definition
1.5]). Hence, the ring of $\widehat{U}$-invariant sections of a sufficiently
divisible tensor power of this well-adapted linearisation is finitely
generated and its projective spectrum is
$Y^{\widehat{U}-s}/\widehat{U}=Y/\\!/\widehat{U}$. In particular,
$q_{\widehat{U}}$ can be constructed by taking invariant sections for a slight
perturbation of the original linearisation.

  4. (4)

In general we will assume $UZ_{\min}\neq Y_{\min}$. If this does not hold then
the $\widehat{U}$-stable set $Y^{\widehat{U}-s}$ is empty and in this case, we
should define $Y_{\min}$ as a replacement stable set which admits a the
geometric $p:Y_{\min}\rightarrow Z_{\min}$ (see [1, Remark 2.5]). Moreover,
there is a good $H$-quotient $UZ_{\min}^{ss}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Z_{\min}^{ss}\rightarrow
Z_{\min}/\\!/_{\\!\mathcal{L}}\overline{L}$.

Determining the domain of the quotient $q$ in terms of the $H$-action on $Y$
is quite delicate, even though we can determine the domain of
$q_{\overline{L}}$ using the Hilbert–Mumford criterion for the reductive group
$\overline{L}$. If $T$ is a maximal torus of $L$ containing
$\lambda(\mathbb{G}_{m})$, one would like a description of
$\operatorname{dom}(q)$ in terms of $T$-weights on $Y$, but passing to the
$U$-quotient involves deleting some of these weights. Nevertheless, we can
sandwich the domain of definition between sets admitting Hilbert–Mumford type
descriptions in terms of weights as in Theorem 2.28 __(ii)__ below. Moreover,
under the assumption $[\overline{L}]_{0}$, we can explicitly determine the
domain of $q$. In [1, Lemma 7.8], a Hilbert–Mumford type statement is also
stated without assuming $[\overline{L}]_{0}$.

######  Theorem 2.28.

Let $H=U\rtimes L$ be a linear algebraic group with unipotent radical $U$
internally graded by $\lambda:\mathbb{G}_{m}\rightarrow L$ acting on a
projective scheme $Y$ with respect to a very ample adapted linearisation
$\mathcal{L}$ such that condition $[\widehat{U}]$ holds. Let
$q:Y\dashrightarrow Y/\\!/H$ be the non-reductive GIT quotient given by
Theorem 2.26.

  1. _(i)_

If $y\in Y^{s}_{\min}\setminus UZ^{s}_{\min}$, then $q_{U}(y)\in Y_{\min}/U$
is stable for the reductive group $L$ acting on $\mkern
1.5mu\overline{\mkern-1.5muY_{\min}/U\mkern-1.5mu}\mkern 1.5mu$.

  2. _(ii)_

The domain of $q$ sits between the open sets with Hilbert–Mumford type
descriptions

$Y^{H-s}=Y_{\min}^{s}\setminus
UZ_{\min}^{s}\subset\operatorname{dom}(q)\subset\bigcap_{h\in
H}hY^{T-ss}=\bigcap_{u\in U}uY^{L-ss}\subset Y_{\min}^{ss}\setminus
UZ_{\min}^{ss}.$

  3. _(iii)_

The open set $Y^{H-s}$ admits a quasi-projective geometric $H$-quotient by
restricting $q$

$q|_{Y^{H-s}}:Y^{H-s}=Y_{\min}^{s}\setminus UZ_{\min}^{s}\twoheadrightarrow
q(Y^{H-s})\subset Y/\\!/H,$

with a natural projective completion $Y/\\!/H$.

In particular, if one also assumes $[\overline{L}]_{0}$, then all the above
inclusions in _(ii)_ are equalities and so $q:Y^{H-s}\rightarrow Y/\\!/H$ is a
projective geometric $H$-quotient.

###### Proof.

After replacing the linearisation with a sufficiently high tensor power (which
does not change the notion of semistability), the quotient map $q_{U}$ is
constructed from a linear projection
$\mathbb{P}(V)\dashrightarrow\mathbb{P}(W)$ where $V=H^{0}(Y,\mathcal{L})^{*}$
and $W=(H^{0}(Y,\mathcal{L})^{U})^{*}$. More precisely, we obtain a locally
closed immersion $Y_{\min}/U\hookrightarrow\mathbb{P}(W)$ and we let $\mkern
1.5mu\overline{\mkern-1.5muY_{\min}/U\mkern-1.5mu}\mkern
1.5mu\subset\mathbb{P}(W)$ denote the projective completion. There is an
induced $L$-action on both of these projective schemes, whose (semi)stable
loci can be described in terms of the (reductive GIT) Hilbert–Mumford
criterion.

Let $T$ be a maximal torus in $L$ containing $\lambda(\mathbb{G}_{m})$ and
consider the $T$-weights on $V$ and $W$. For $y\in Y$ (or strictly speaking a
lift to the affine cone $V$), we need to compare the convex hull of
$T$-weights of $y$ and $q_{U}(y)$; since $q_{U}$ is induced by a linear
projection, it forgets some weights:

$\operatorname{Conv}_{T}(y)\supset\operatorname{Conv}_{T}(q_{U}(y)).$

However, the sections in $H^{0}(Y,\mathcal{L})$ with maximal $\lambda$-weights
(dually corresponding to the locus $V_{\min}\subset
V=H^{0}(Y,\mathcal{L})^{*}$ with minimal $\lambda$-weights) are all
$U$-invariant (see [1, $\S$7]) and thus these weights are preserved by
$q_{U}$. Let $H_{\min}\subset\mathfrak{t}^{*}$ be the hyperplane containing
the $T$-weights on $V$ which are $\lambda$-minimal (this hyperplane is
perpendicular to the ray corresponding to $\lambda$ when we identify
$\mathfrak{t}\cong\mathfrak{t}^{*}$ using our given norm); then we have

$\operatorname{Conv}_{T}(y)\cap H_{\min}=\operatorname{Conv}_{T}(q_{U}(y))\cap
H_{\min}.$

All the $T$-weights are contained in a closed half space bounded by this
hyperplane and, as the linearisation is twisted to be well-adapted, the origin
lies very close to this hyperplane. Points in $Y_{\min}$ have at least one
weight on $H_{\min}$ and points in $Z_{\min}$ have all their weights on this
hyperplane. The map $p:Y_{\min}\rightarrow Z_{\min}$ is the retraction to the
minimal $\lambda$-weight space $\mathbb{P}(V_{\min})$, i.e. it throws away all
the $T$-weights not lying on the hyperplane $H_{\min}$. If $y\in
Y_{\min}\setminus UZ_{\min}$, then the weight polytope
$\operatorname{Conv}_{T}(q_{U}(y))$ contains a weight outside of $H_{\min}$.
If $y\in Y^{s}_{\min}$, then $p(y)\in Z^{s}_{\min}$ is $\overline{L}$-stable;
that is, the projection of the origin to $H_{\min}$ is contained inside the
interior of $\operatorname{Conv}_{T}(y)\cap H_{\min}$.

For the proof of (i), if $y\in Y^{s}_{\min}\setminus UZ^{s}_{\min}$, then we
claim that the origin is contained in the interior of
$\operatorname{Conv}_{T}(q_{U}(y))$. Indeed, the origin lies very close to
$H_{\min}$, and since $y\in Y^{s}_{\min}$ this weight polytope meets
$H_{\min}$ in a codimension 1 polytope $\operatorname{Conv}_{T}(y)\cap
H_{\min}$ containing the projection of the origin. Since $y\notin UZ_{\min}$,
the weight polytope also contains one weight away from $H_{\min}$, so it is
full dimensional. Since $Y^{s}_{\min}\setminus UZ^{s}_{\min}$ is
$H$-invariant, the same is true for all points in the $H$-orbit of $y$, whose
image under $q_{U}$ is the $L$-orbit of $y$; hence, by the Hilbert–Mumford
criterion for the $L$-action on $\mathbb{P}(W)$, we conclude that $q_{U}(y)$
is $L$-stable.

For (ii), the first inclusion follows from (i), as
$\operatorname{dom}(q)=q_{U}^{-1}(\mkern
1.5mu\overline{\mkern-1.5muY_{\min}/U\mkern-1.5mu}\mkern 1.5mu^{L-ss})$. The
second inclusion follows as we can pullback invariant sections along $q_{U}$.
The middle equality is just the reductive Hilbert–Mumford criterion and the
right inclusion holds by a Hilbert–Mumford type argument involving $T$-weights
similar to that given in (i).

Finally (iii) follows from (i) and (ii), and the final statement follows as
then $Z_{\min}^{s}=Z_{\min}^{ss}$. ∎

Finally, let us explain how one can weaken $[\widehat{U}]$ to
$[\widehat{U};\overline{L}\text{-ss}]$ following [1, Remark 2.8]. In this
case, one minor difference is that although the $H$-quotient is projective,
the $\widehat{U}$-quotient is only quasi-projective.

######  Theorem 2.29.

For a $P$-action on $X$ with an adapted linearisation $\mathcal{L}$ satisfying
$[\widehat{U};\overline{L}\text{-ss}]$, there is a map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.57178pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q:Y^{ss}_{\min}\setminus
UZ^{ss}_{\min}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.01138pt\raise 5.95139pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.0625pt\hbox{$\scriptstyle{q_{\widehat{U}}\quad}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 79.57178pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.57178pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(Y^{ss}_{\min}\setminus
UZ^{ss}_{\min})/\widehat{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
149.1069pt\raise 5.6425pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.3714pt\hbox{$\scriptstyle{\quad\quad\quad
q_{\overline{L}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
198.11478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
198.11478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y/\\!/H}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

giving a projective good $H$-quotient $\operatorname{dom}(q)\twoheadrightarrow
Y/\\!/H$ of an open set of $Y^{ss}_{\min}\setminus UZ^{ss}_{\min}$, which
restricts to a quasi-projective geometric $H$-quotient of
$Y^{H-s}=Y_{\min}^{s}\setminus UZ_{\min}^{s}$. If additionally,
$[\overline{L}]_{0}$ holds, then $\operatorname{dom}(q)=Y^{H-s}$ and this has
a projective geometric quotient $Y/\\!/H$ given by taking the projective
spectrum of the ring of $H$-invariant sections for a twist of the
linearisation which is well-adapted.

###### Proof.

The argument is a minor modification of [1, §7]. For simplicity, let us
outline the proof in the case when $U$ is abelian and acts with zero
dimensional stabilisers on $Z^{ss}_{\min}$; if $U$ is not abelian, one needs
to successively take quotients by the abelian groups appearing as quotients in
a normal series and, if the stabilisers are positive dimensional, one needs to
choose complementary subgroups as in the proof of Theorem 2.26.

First one constructs a geometric $U$-quotient of $Y^{ss}_{\min}$ similarly to
the affine local construction of [1, Proposition 7.4]; however, one instead
takes a basis $\\{\sigma_{i}\\}_{i=1}^{r}$ of the $\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu$-invariant sections of
maximal $\lambda(\mathbb{G}_{m})$-weight $H^{0}(Y,\mathcal{L})^{\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}_{\max}$ (recall that
these sections are also $U$-invariant), then $Y_{\min}^{ss}$ is covered by the
open affine sets $Y_{\sigma_{i}}$ and each of these admits a trivial geometric
$U$-quotient given by taking rings of $U$-invariants (cf. [1, Lemmas 7.2 and
7.3]). Since these invariant rings are finitely generated, one can pick $s$ so
they are all generated in degree $s$. Let $W:=(H^{0}(Y,\mathcal{L}^{\otimes
s})^{U})^{*}$; then we have a natural $U$-invariant morphism
$\phi:Y^{ss}_{\min}\rightarrow\mathbb{P}:=\mathbb{P}(W)$, which thus factors
via the quotient $q_{U}:Y^{ss}_{\min}\rightarrow Y^{ss}_{\min}/U$ and a
locally closed immersion
$\overline{\phi}:Y^{ss}_{\min}/U\rightarrow\mathbb{P}$. Moreover, there is an
induced $L$-action on $\mathbb{P}$ such that $\phi$ is $L$-equivariant; we
denote the corresponding (semistable) $\lambda$-minimal weight space
$Z^{(ss)}_{\min}(\mathbb{P})$ and the corresponding open by
$\mathbb{P}_{\min}^{(ss)}$. After appropriately twisting the linearisation on
$\mathbb{P}$ so it is well-adapted, we have
$\mathbb{P}^{\lambda-(s)s}=\mathbb{P}_{\min}\setminus Z_{\min}(\mathbb{P})$ by
the Hilbert–Mumford criterion.

The same argument as in [1, Lemma 7.6] shows that the image of
$\overline{\phi}$ is contained in $\mathbb{P}_{\min}^{ss}$ as a closed
subscheme. Moreover, for $y\in Y_{\min}^{ss}$, one has $\phi(y)\in
Z^{ss}_{\min}(\mathbb{P})$ if and only if $y\in UZ_{\min}^{ss}$ analogously to
[1, Equation (3) of $\S$7]. Consequently, $\phi$ induces a closed immersion

$(Y_{\min}^{ss}\setminus
UZ_{\min}^{ss})/U\hookrightarrow\mathbb{P}_{\min}^{ss}\setminus
Z_{\min}^{ss}=\mathbb{P}_{\min}^{ss}\times_{\mathbb{P}}\mathbb{P}^{\lambda-s}.$

Thus one obtains a geometric quotient $(Y^{ss}_{\min}\setminus
UZ_{\min}^{ss})/\widehat{U}$ as a locally closed subscheme of the projective
scheme $\mathbb{P}/\\!/\lambda(\mathbb{G}_{m})$; in particular, this
$\widehat{U}$-quotient is not necessarily projective.

The reductive GIT quotient $q_{L}:\mathbb{P}\dashrightarrow\mathbb{P}/\\!/L$
factors via $\mathbb{P}/\\!/\lambda(\mathbb{G}_{m})$ and we will construct our
$H$-quotient as a closed subscheme of $\mathbb{P}/\\!/L$. A reductive
Hilbert–Mumford argument shows
$\mathbb{P}^{L-ss}\subset\mathbb{P}^{ss}_{\min}\setminus Z^{ss}_{\min}$, and
since $(Y_{\min}^{ss}\setminus UZ_{\min}^{ss})/U$ is closed in the latter, the
inclusion

$\mathbb{P}^{L-ss}\times_{\mathbb{P}}(Y_{\min}^{ss}\setminus
UZ_{\min}^{ss})/U\hookrightarrow\mathbb{P}^{L-ss}$

is closed and
$Y/\\!/H:=q_{L}(\mathbb{P}^{L-ss}\times_{\mathbb{P}}(Y_{\min}^{ss}\setminus
UZ_{\min}^{ss})/U)$ is a closed subscheme of $\mathbb{P}/\\!/L$. The
composition $Y_{\min}^{ss}\setminus UZ_{\min}^{ss}\rightarrow Y/\\!/H$ is thus
a projective good $H$-quotient.

If in addition, we assume that $[\overline{L}]_{0}$ holds, then
$\operatorname{dom}(q)=Y^{H-s}$ as in Theorem 2.28. To conclude that $Y/\\!/H$
is given by taking the projective spectrum of the ring of invariants for a
well-adapted perturbation of the linearisation, it suffices to show this
quotient coincides with the enveloping quotient by [2, Corollary 3.1.21]. To
prove this, one first shows that $\operatorname{dom}(q)$ is contained in the
locally trivial stable locus with respect to $H$ (see [2, Definition 3.3.2])
and then deduces that $Y/\\!/H$ coincides with the quotient of the locally
trivial stable locus and the enveloping quotient as in [1, $\S$7, Proof of
Theorem 2.16].∎

####  2.3.6. The construction of quotients in general via blow-up sequences

If condition $[\widehat{U}]$ does not hold for the linearised $H$-action on
$Y$, then as explained in [1, $\S$8] one can instead perform a sequence of
equivariant blow-ups in order to arrange that condition $[\widehat{U}]$ holds
on the blow-up. These blow-ups are similar to the blow up sequences used to
construct partial desingularisations of reductive GIT quotients (cf. $\S$2.1.1
and [28]), but where one now considers unipotent stabiliser groups instead.
Away from the exceptional locus, we can identify the semistable locus in the
blown up space $\widetilde{Y}$ with a certain open subset of $Y$; hence one
obtains a quasi-projective quotient of this open subset, and a
compactification given by the quotient of $\widetilde{Y}$. To identify this
open subset in $Y$, one has to describe the complement to the exceptional
divisor in the $\widehat{U}$-stable locus for $\widetilde{Y}$. This is more
difficult when all of $Z_{\min}$ is blown up (see [25] for further
discussion).

There are several cases to consider, of increasing complexity:

  1. (1)

If the generic dimension of $U$-stabilisers on $Z_{\min}$ is zero, then one
does not need to blow up all of $Z_{\min}$ and there is an explicit
description of the open subset admitting a quotient given by [1, Theorem
2.10].

  2. (2)

If $U$ is abelian, but the generic dimension of the $U$-stabilisers on
$Z_{\min}$ is positive, then either

    1. (a)

the generic dimension of the $U$-stabilisers on $Z_{\min}$ equals the generic
dimension of the $U$-stabilisers on $Y_{\min}$ or

    2. (b)

the generic dimension of the $U$-stabilisers on $Z_{\min}$ is bigger than the
generic dimension of the $U$-stabilisers on $Y_{\min}$.

In the first case, one does not need to blow up all of $Z_{\min}$ and it is
easier to describe the open subset that one obtains a quotient of; see
$\S$2.3.7 for a situation in which this is particularly straight-forward. In
the second case, one must blow-up all of $Z_{\min}$ and use jets to describe
the open subset admitting a quotient.

  3. (3)

For general $U$ such that the generic dimension of the $U$-stabilisers on
$Z_{\min}$ is positive, the results of the blow-up procedure is described by
[1, Theorem 2.20]. In this case one fixes a series of subgroups of $U$ which
are normal in $H$ and whose successive quotients are abelian with
$\lambda(\mathbb{G}_{m})$ acting by a single weight. One could then perform
quotients in stages doing blow-ups at each stage to arrange for the
stabilisers to be constant dimension on $Y_{\min}$ (cf. [1, Remarks 2.19 and
8.10]).

In this paper we will only consider the case where condition
$[C(Z_{\min},U,d_{\min})\neq\emptyset]$ holds, i.e. that there exists a point
in $Z_{\min}$ whose $U$-stabiliser dimension is minimal amongst all
$U$-stabilisers of points of $Y_{\min}$. This means we are at worst in case
$2$(a), and do not need to blow up all of $Z_{\min}$. Later on in $\S$4 we
will introduce condition (WUU), which guarantees this in the setting of
Quotienting-in-Stages. In the following subsection, we will explicitly
describe the sequence of blow-ups in the case when $U$ is abelian, as we will
only need this setting later in the paper.

######  Remark 2.30.

If the action does not satisfy $[\overline{L}]_{0}$, then one can perform
reductive blow-ups so that on the blow-up either $[\overline{L}]_{0}$ holds or
semistability equals Mumford stability for this reductive group action (i.e.
the dimensions of the $\overline{L}$-stabilisers of semistable points are
constant).

####  2.3.7. The blow-up procedure in a simple abelian setting

Throughout this section, we suppose that the unipotent radical $U$ is abelian,
and consider

$d_{\min}:=d_{\min}(Y_{\min}^{ss},U)\quad\text{and}\quad
d_{\max}:=d_{\max}(Y_{\min}^{ss},U).$

If $[\widehat{U};\overline{L}\text{-ss}]$ fails, then the $U$-stabilisers have
non-constant dimension on $Y_{\min}^{ss}$, so $d_{\min}<d_{\max}$. In this
case, we perform a sequence of $H$-equivariant blow-ups to arrange for
$[\widehat{U};\overline{L}\text{-ss}]$ to hold on the blow-up. The idea is to
start by considering points with maximal dimensional $U$-stabilisers. In fact,
there are several possible ways to perform this blow-up sequence: one can
blow-up the points in $Y_{\min}$ whose stabilisers have maximal $U$-dimension
or blow-up the $U$-sweep of the locus of points in $Z_{\min}$ whose
stabilisers have maximal $U$-dimension (the latter is equivalent to blowing-up
the locus of points in $Y_{\min}$ with maximal dimensional
$\widehat{U}$-stabiliser). In [1], the former approach is taken, but we will
take the latter approach. This approach involves blowing up a smaller locus
and it is easier to show this process terminates, as one is always blowing up
a locus of codimension at least 2.

Throughout this section, we will also assume that condition
$[C(Z_{\min},U,d_{\min})\neq\emptyset]$ holds. Let us introduce the centre of
each blow-up and construct the first blow-up in each sequence using the
notation of Definition 2.2.

######  Definition 2.31.

For a linearised $H$-action on $Y$ such that
$[\widehat{U};\overline{L}\text{-ss}]$ fails, we define

$D(Y):=C(Y_{\min}^{ss},\widehat{U}).$

We let $\pi_{(1)}:\widetilde{Y}_{(1)}\rightarrow Y$ denote the blow-up of $Y$
along the closure of $D(Y)$.

######  Remark 2.32.

By definition, $D(Y)$ is the closed subscheme of $Y^{ss}_{\min}$ on which the
dimension of the stabiliser for $\widehat{U}$ are maximal; equivalently this
is the $U$-sweep of $C(Z^{ss}_{\min},U)$. As $U<H$ is a normal subgroup, the
scheme $D(Y)$ and its closure in $Y$ are $H$-invariant. Therefore, $\pi_{(1)}$
is a $H$-equivariant blow-up and the induced $H$-action on
$\widetilde{Y}_{(1)}$ admits an ample linearisation given by pulling back the
linearisation on $Y$ and perturbing by a small multiple of the exceptional
divisor. The assumption $[C(Z_{\min},U,d_{\min})\neq\emptyset]$ ensures that
$Z_{\min}\not\subseteq D(Y)$. Consequently, the minimal weight space
$\widetilde{Z}_{(1),{\min}}$ in the blow-up $\widetilde{Y}_{(1)}$ is the
strict transform of the minimal weight space $Z_{\min}$ in $Y$. Assuming as we
always do that $UZ_{\min}\subsetneq Y_{\min}$ (as otherwise we proceed as in
Remark 2.27 (4)), the centre of this blow-up has codimension at least 2, so
the blown up space is not isomorphic to the space we started with.

Let us show that this first blow-up reduces the dimension of maximal
stabilisers; we will follow the argument [1, Proposition 8.8] even though
their blow-up procedure differs slights from ours.

######  Lemma 2.33.

Assume that $[\widehat{U};\overline{L}\text{-ss}]$ fails, but
$[C(Z_{\min},U,d_{\min})\neq\emptyset]$ holds. Then the dimensions of
$U$-stabilisers in the $\lambda$-minimal weight spaces drops after performing
the blow-up $\pi_{(1)}:\widetilde{Y}_{(1)}\rightarrow Y$

$d_{\max}(\widetilde{Z}_{(1),{\min}}^{ss},U)<d_{\max}(Z^{ss}_{\min},U)=d_{\max}$

where $\widetilde{Z}_{(1),{\min}}^{ss}$ denotes the semistable locus in the
minimal weight space for the blow-up $\widetilde{Y}_{(1)}$.

###### Proof.

This proof is an expanded version of the argument outlined in [1, Proposition
8.8]. By Lemma 2.34 below and the fact that $\widetilde{Y}_{(1)}\ncong Y$ (see
Remark 2.32), it suffices to show that if
$\tilde{z}\in\widetilde{Z}_{(1),{\min}}^{ss}$ lies over $z\in
C(Z^{ss}_{\min},U)$, then $\tilde{z}$ is not fixed by
$U^{\prime}:=\operatorname{Stab}_{U}(z)$. We have

$Z_{\min}^{U^{\prime}}\subset C(Z_{\min},U,d_{\max})\subset Z_{\min}.$

By embedding $Y$ in a projective space, we can assume without loss of
generality that $Y=\mathbb{P}^{m}$ and as $Z_{\min}^{U^{\prime}}$ and
$Z_{\min}$ are linear subspace we can assume we have taken coordinates so
there are $0\leq n\leq m^{\prime}\leq m$ such that $Z_{\min}^{U^{\prime}}$
(resp. $Z_{\min}$) is the vanishing locus of the first $m^{\prime}$ (resp. n)
coordinates and $p([x_{0}:\cdots:x_{m}])=[0:\cdots:0:x_{n}:\cdots:x_{m}]$. We
claim the $U^{\prime}$-action on $\mathbb{P}^{m}$ with respect to the blocks
$n\leq m^{\prime}\leq m$ has the form

$u^{\prime}\mapsto\left(\begin{array}[]{ccc}A(u^{\prime})&B(u^{\prime})&0\\\
0&I&0\\\ 0&0&I\end{array}\right).$

Indeed, the final column has this form, as $U^{\prime}$ acts trivially on
$Z_{\min}^{U^{\prime}}$. With respect to the two blocks given by $n\leq m$
(given by the minimal $\lambda$-weight), this matrix is block upper
triangular, as $\lambda$ grades $U$. Finally, the middle column has this form,
as $p(u\cdot x)=x$ for $x\in Z_{\min}$.

Since the exceptional divisor is given by the projectivised normal bundle,
$\tilde{z}$ is represented by some
$[\xi]\in\mathbb{P}(T_{z}Z_{\min}/T_{z}(Z_{\min}^{U^{\prime}}))=\pi_{(1)}^{-1}(z)$.
We can take coordinates so $z=[0:\cdots:0:1]$ and then take local coordinate
$x_{j}/x_{m}$ around $z$. From the above block form of the action, one sees
for any $\xi\in T_{z}Z_{\min}\setminus T_{z}(Z_{\min}^{U^{\prime}})$,

there is a $u^{\prime}\in U^{\prime}$ such that $u^{\prime}\xi-\xi\notin
T_{z}Z_{\min}$. Indeed, in the above representation of the
$U^{\prime}$-action, we have some $u^{\prime}$ where $B(u^{\prime})\neq 0$, as
$Z_{\min}^{U^{\prime}}\neq Z_{\min}$. Therefore $\tilde{z}$, which is
represented by some $[\xi]$, is not fixed by $U^{\prime}$. ∎

We used the following straight-forward lemma in the above proof.

######  Lemma 2.34.

Let $\pi:\widetilde{X}\rightarrow X$ be a $G$-equivariant blow-up for any
linear algebraic group $G$. Then for any subgroup $G^{\prime}<G$ and any
$x\in\widetilde{X}$, we have

$\dim\operatorname{Stab}_{G^{\prime}}(x)\leq\dim\operatorname{Stab}_{G^{\prime}}(\pi(x))$

with equality holding for all points outside the exceptional divisor.

We can finally prove that $[C(Z_{\min},U,d_{\min})\neq\emptyset]$ enables a
blow-up procedure that does not blow up all of $Z_{\min}$.

######  Proposition 2.35.

Let $H=U\rtimes L$ be a group with abelian unipotent radical internally graded
by $\lambda:\mathbb{G}_{m}\rightarrow L$. Let $Y$ be a projective $H$-scheme
with a very ample adapted linearisation $\mathcal{L}$ such that
$[\overline{L}]_{0}$ and $[C(Z_{\min},U,d_{\min})\neq\emptyset]$ hold. Then
there is a sequence of equivariant blow-ups along $H$-invariant closed
subschemes resulting in a projective scheme $\widetilde{Y}$ admitting a very
ample adapted linearisation888This linearisation is on a line bundle obtained
by tensoring the pullback of $\mathcal{L}$ with small multiples of the
exceptional divisors for each blow-up. such that on $\widetilde{Y}$ both
$[\overline{L}]_{0}$ and $[\widehat{U};\overline{L}\text{-ss}]$ hold.
Moreover, $[C(Z_{\min},U,d_{\min})\neq\emptyset]$ ensures that not all of
$Z_{\min}$ is blown up in this procedure.

###### Proof.

If $[\widehat{U};\overline{L}\text{-ss}]$ holds already for $Y$, then no blow-
ups are needed. Otherwise, let $\widetilde{Y}_{(0)}=Y$ and for $i=0,\dots$,
define $\widetilde{Y}_{(i+1)}$ to be the blow-up of $\widetilde{Y}_{(i)}$
along the closure of $D(\widetilde{Y}_{(i)})$. Since the dimensions of the
$U$-stabilisers on the semistable locus in the $\lambda$-minimal weight space
decreases at each stage, this procedure terminates with a scheme
$\widetilde{Y}:=\widetilde{Y}_{(n)}$ which has constant dimensional
$U$-stabilisers on the semistable locus in the $\lambda$-minimal weight space
equal to $d_{\min}$. We claim that $\dim\operatorname{Stab}_{U}$ is constant
and equal to $d_{\min}$ on all of $\widetilde{Y}_{\min}^{ss}$. If
$\tilde{y}\in\widetilde{Y}_{\min}^{ss}$ flows to
$\tilde{z}=\tilde{p}(\tilde{y})\in\widetilde{Z}^{ss}_{\min}$ under $\lambda$,
then

$d_{\min}\leq\dim\operatorname{Stab}_{U}(\tilde{y})\leq\dim\operatorname{Stab}_{U}(\tilde{z})=d_{\min}.$

Thus $[\widehat{U};\overline{L}\text{-ss}]$ holds for $\widetilde{Y}$, as does
$[\overline{L}]_{0}$ by Lemma 2.34, as it already held for $Y$. ∎

In this case, we define the following stable locus.

######  Definition 2.36.

Assume that $[\overline{L}]_{0}$ and $[C(Z_{\min},U,d_{\min})\neq\emptyset]$
holds, where $U$ is abelian. Then the _stable locus_ 999If additionally
$[\widehat{U};\overline{L}\text{-ss}]$ holds, then this coincides with the
stable locus in Definition 2.24. for the linearised $H$-action on $Y$ is

$Y^{H-s}:=\\{y\in Y^{s}_{\min}\setminus
UZ^{s}_{\min}:\>\dim\operatorname{Stab}_{U}(p(y))=d_{\min}\\}.$

The following result is a simplified version of [1, Theorems 2.10 and 2.20]
which shows this stable set admits a quasi-projective geometric $H$-quotient
(see also [25, $\S$3.3]).

######  Proposition 2.37.

Let $H=U\rtimes L$ be a group with abelian unipotent radical internally graded
by $\lambda:\mathbb{G}_{m}\rightarrow L$ acting on a projective scheme $Y$
with respect to a very ample adapted linearisation $\mathcal{L}$. Assume that
$[\overline{L}]_{0}$ and $[C(Z_{\min},U,d_{\min})\neq\emptyset]$ hold. If
$\pi:\widetilde{Y}\rightarrow Y$ is the $H$-equivariant blow-up of Proposition
2.35, then $\pi$ is an isomorphism over the open set $Y^{H-s}$ and thus via
$\pi$, the set $Y^{H-s}$ admits a quasi-projective geometric $H$-quotient with
natural projective completion $\widetilde{Y}/\\!/H$.

###### Proof.

We note that $Y^{H-s}$ is disjoint from the centre of each blow-up appearing
in Proposition 2.35 and so is contained in the open set over which $\pi$ is an
isomorphism. In fact, we claim $Y^{H-s}$ is contained in the intersection of
the open set over which $\pi$ is an isomorphism with the stable set
$\widetilde{Y}^{H-s}$ in the blow-up. For simplicity, suppose $H=\widehat{U}$.
Since not of all of $Z_{\min}$ is blown up, the minimal weight space in the
blow-up is the strict transform of $Z_{\min}$ and therefore
$Y^{\widehat{U}-s}$ is contained in the intersection of the open set over
which $\pi$ is an isomorphism with
$\widetilde{Y}^{\widehat{U}-s}=\widetilde{Y}_{\min}\setminus
U\widetilde{Z}_{\min}$. ∎

This stable set could potentially be empty; for example, when $H=\widehat{U}$,
this can happen if we have $UZ_{\min}\subsetneq X_{\min}$, but
$UC(Z_{\min},U,d_{\min})=p^{-1}(C(Z_{\min},U,d_{\min}))$ as in the next
example.

######  Example 2.38.

Let $H=\widehat{\mathbb{G}_{a}}=\mathbb{G}_{a}\rtimes\mathbb{G}_{m}$ act on
$Y:=\mathbb{P}^{2}$ with coordinates $[x:y:z]$ via the representation
$\mathbb{G}_{a}\rtimes\mathbb{G}_{m}\rightarrow\mathrm{GL}_{3}$

$(a,t)\mapsto\left(\begin{array}[]{ccc}t&a&0\\\ 0&t^{-1}&0\\\
0&0&t^{-1}\end{array}\right).$

Then $Z_{\min}=\\{x=0\\}\cong\mathbb{P}^{1}$ and $Y_{\min}\rightarrow
Z_{\min}$ is an $\mathbb{A}^{1}$-bundle. Furthermore, the only point in
$Z_{\min}$ with non-trivial $\mathbb{G}_{a}$-stabiliser is $y=[0:0:1]$ which
is fixed by $\mathbb{G}_{a}$; thus $d_{\min}=0$ and
$C:=C(Z_{\min},U,0)=Z_{\min}\setminus\\{y\\}$. Note that $UZ_{\min}\subsetneq
X_{\min}$, but $UC=p^{-1}(C)$ and so we have $Y^{H-s}=\emptyset$ using the
above definition. Since $[\widehat{U}]$ fails, we need to do blow-ups, but as
$[C(Z_{\min},U,d_{\min})\neq\emptyset]$ holds (and $[\overline{L}]_{0}$ holds
trivially) we can apply the above result. In this case, $\widetilde{Y}$ is
just the blow-up of $Y$ at $y$ and $\widetilde{Z}_{\min}=\mathbb{P}^{1}$ (as
it is the blow-up of $Z_{\min}$ at $y$) and one now has
$U\widetilde{Z}_{\min}=\widetilde{Y}_{\min}$, so also
$\widetilde{Y}^{H-s}=\emptyset$. In this case, one should instead define the
stable locus in $\widetilde{Y}$ to be $\widetilde{Y}_{\min}$ with $U$-quotient
$\widetilde{Y}_{\min}\rightarrow Z_{\min}$ as in Remark 2.27 (4), which under
$\pi$ would give a geometric quotient of the open set $p^{-1}(C)=UC$ in $Y$
via the map $p^{-1}(C)\rightarrow C$.

##  3\. Quotients of unstable HKKN strata

Let $G$ be a reductive group acting on a projective scheme $Y$ with respect to
an ample linearisation $\mathcal{L}$. As in $\S$2.2, associated to a choice of
norm $||-||$ on $\operatorname{Lie}G$ there is a HKKN stratification

$Y=\bigsqcup_{\beta\in\mathcal{B}}S_{\beta}$

into finitely many $G$-invariant locally closed subschemes $S_{\beta}\subset
Y$. If $Y^{ss}\neq\emptyset$, then this semistable locus is the minimal
stratum indexed by $0\in\mathcal{B}$ and admits a good quotient: the reductive
GIT quotient. If the stable locus is non-empty, this GIT quotient restricts to
a geometric quotient on the stable locus. The unstable HKKN strata
$S_{\beta}$, which correspond to non-zero indices $\beta\in\mathcal{B}$, admit
a description

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

for a parabolic subgroup $P_{\beta}\subset G$ and a locally closed subscheme
$Y_{\beta}^{ss}\subset Y$. Therefore, constructing a $G$-quotient of an open
subscheme of $S_{\beta}$ is equivalent to constructing a $P_{\beta}$-quotient
of an open subset of $Y_{\beta}^{ss}$. The goal of this section is to explain
how to construct such a quotient.

###  3.1. Categorical quotients from borderline linearisations

The problem of constructing a quotient of the $G$-action on an unstable
stratum $S_{\beta}\subset Y$, or equivalently the $P_{\beta}$-action on
$Y_{\beta}^{ss}$, was studied in [23, $\S$2]. As explained in $\S$1.1, it is
often preferable to consider the $P_{\beta}$-action on $Y_{\beta}^{ss}$ rather
than the $G$-action on $S_{\beta}$, as the parabolic group admits more
characters which can be used to twist the linearisation. The characters of
$P_{\beta}$ correspond to the characters of the Levi $L_{\beta}<P_{\beta}$,
and amongst these there is (after the choices we have made) a distinguished
character, $\chi_{-\beta}$ from which we obtain the canonical
$P_{\beta}$-linearisation $\mathcal{L}_{\beta}$ by twisting the original
linearisation $\mathcal{L}$ by the character
$\chi_{-\beta}:L_{\beta}\rightarrow\mathbb{G}_{m}$. This gives an ample
$P_{\beta}$-linearisation on the closure of $Y_{\beta}^{ss}$, but does not
extend to an ample $G$-linearisation on the closure of $S_{\beta}$, since the
character $\chi_{-\beta}$ may not extend to $G$.

As we will see in Lemma 3.4, the unipotent radical of $P_{\beta}$ is always
graded by $\lambda_{\beta}$, and with respect to this grading the canonical
linearisation is borderline, in the sense of Definition 2.16. Thus, this
canonical linearisation provides a categorical quotient of the unstable
stratum, via the following, which we generalised in Proposition 2.17.

######  Proposition 3.1.

[23, Proposition 3.2] The projective variety
$Z_{\beta}/\\!/_{\\!\mathcal{L}_{\beta}}L_{\beta}$ is a categorical quotient
for the $L_{\beta}$-action on $Z_{\beta}^{ss}$, the $P_{\beta}$-action on
$Y_{\beta}^{ss}$, and the $G$-action on $S_{\beta}$.

In general this categorical quotient of the $G$-action on $S_{\beta}$ and of
the $P_{\beta}$-action on $Y_{\beta}^{ss}$ is far from being an orbit space,
as it identifies each $y\in Y_{\beta}^{ss}$ with
$p_{\beta}(y):=\lim_{t\rightarrow 0}\lambda_{\beta}(t)\cdot y\in
Z_{\beta}^{ss}$, which in general only lies in the boundary of the orbit of
$y$.

######  Example 3.2.

For an unstable HN stratum $S_{\tau}$ for sheaves on a polarised projective
scheme, the map $p_{\tau}:Y_{\tau}^{ss}\rightarrow Z_{\tau}^{ss}$ corresponds
to sending a sheaf to its associated graded for the HN filtration. Hence the
above categorical quotient is just the product of moduli spaces of semistable
sheaves for each Hilbert polynomial $P_{i}$ in the HN type $\tau$.

######  Example 3.3.

Consider configurations of ordered tuples of $n\geq 4$ points on
$\mathbb{P}^{1}$ up to $\mathrm{SL}_{2}$-action, with the natural
linearisation and the Killing form on $\mathrm{SL}_{2}$. Such a configuration
is semistable as long as no more than $n/2$ of the points coincide anywhere,
and [27] shows that the HKKN stratification measures how many points do
coincide. In this case the above categorical quotient given above is most
unsatisfactory: for any $\beta$ it is simply a point; see [25] for more
details.

To avoid this type of identification, one would need to first remove
$P_{\beta}Z_{\beta}^{ss}=U_{\beta}Z_{\beta}^{ss}$, ideally by perturbing the
$P_{\beta}$-linearisation $\mathcal{L}_{\beta}$ on $Y_{\beta}^{ss}$ (or
strictly speaking its closure) by a sufficiently small rational character of
$P_{\beta}$, so that all points in $Z_{\beta}^{ss}$ are unstable for this
perturbed linearisation. In fact, we will construct a quotient of the action
of the non-reductive group $P_{\beta}$ on an open subset of semistable points
of $Y_{\beta}^{ss}$, which is disjoint from $Z_{\beta}^{ss}$, by using a
perturbation of the canonical linearisation $\mathcal{L}_{\beta}$ and recent
results from non-reductive GIT.

###  3.2. Applying non-reductive GIT to parabolic actions on unstable strata

In this section, we continue to assume we have a HKKN stratum $S_{\beta}\cong
G\times^{P_{\beta}}Y_{\beta}^{ss}$. In order to apply the above results from
non-reductive GIT to the $P_{\beta}$-action on $Y_{\beta}^{ss}$, we first need
to compactify $Y_{\beta}^{ss}$. Let $X_{\beta}:=\overline{Y_{\beta}}$ denote
the closure of the locally closed subscheme $Y_{\beta}\subset Y$. We recall
that we gave explicit descriptions (2.2) of $Y_{\beta}$ and $Z_{\beta}$ in
$\S$2.2 in terms of torus weights.

First, we check that the non-reductive group $P_{\beta}$ has graded unipotent
radical.

######  Lemma 3.4.

The parabolic subgroup $P_{\beta}=U_{\beta}\rtimes L_{\beta}$ has graded
unipotent radical $U_{\beta}$, where the grading $\mathbb{G}_{m}$ is given by
the central 1-PS $\lambda_{\beta}:\mathbb{G}_{m}\rightarrow L_{\beta}$.

###### Proof.

By definition $P_{\beta}=P(\lambda_{\beta})$ as in (1), it follows that the
weights of the conjugation action of $\lambda_{\beta}(\mathbb{G}_{m})$ on
$\operatorname{Lie}P_{\beta}$ are non-negative and are precisely zero on the
Levi factor $L_{\beta}$. Hence, all the weights of
$\lambda_{\beta}(\mathbb{G}_{m})$ on $\operatorname{Lie}U_{\beta}$ are
strictly positive. ∎

The associated minimal weight space and attracting open set admit the
following descriptions.

######  Proposition 3.5.

Let $\beta\in\mathcal{B}$ be a non-zero index of a HKKN stratum
$S_{\beta}\subset Y$. For the $P_{\beta}$-action on
$X_{\beta}=\overline{Y_{\beta}}$ with respect to the linearisation
$\mathcal{L}$ and the grading given by
$\lambda_{\beta}:\mathbb{G}_{m}\rightarrow L_{\beta}$ we have

$Z_{\beta,\min}:=Z(X_{\beta},\lambda_{\beta})_{\min}=Z_{\beta}\quad\text{and}\quad
X_{\beta,\min}:=X_{\beta}(\lambda_{\beta})_{\min}=Y_{\beta}$

and the associated retraction is $p_{\beta}:Y_{\beta}\rightarrow Z_{\beta}$.
Moreover, for $\epsilon\in\mathbb{Q}_{>0}$ sufficiently small, the twisted
linearisation $\mathcal{L}^{\chi_{-(1+\epsilon)\beta}}$ is adapted for the
graded unipotent group
$\widehat{U}_{\beta}=U_{\beta}\rtimes_{\lambda_{\beta}}\mathbb{G}_{m}$.

###### Proof.

Since for a $G$-equivariant closed immersion
$i:Y\hookrightarrow\mathbb{P}^{n}$ such that
$\mathcal{L}=i^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$, the HKKN strata on $Y$ are
the fibred products of the HKKN strata on $\mathbb{P}^{n}$ with $Y$ and
similarly for the fixed loci for $\lambda_{\beta}$, it suffices to prove that
$Z_{\beta,\min}=Z_{\beta}$ when $Y=\mathbb{P}^{n}$ and
$\mathcal{L}=\mathcal{O}_{\mathbb{P}^{n}}(1)$. We can choose a maximal torus
$T$ containing $\lambda_{\beta}(\mathbb{G}_{m})$ and coordinates on
$\mathbb{P}^{n}$ such that the action of the fixed maximal torus $T<G$ is
given by $t\mapsto\operatorname{diag}(\alpha_{0}(t),\dots,\alpha_{n}(t))$ for
characters $\alpha_{i}:T\rightarrow\mathbb{G}_{m}$.

Recall that we use an invariant inner product on $\mathfrak{t}$ to identify
characters and co-characters of $T$ and the associated norm is denoted
$||-||$. The weights of the $\lambda_{\beta}(\mathbb{G}_{m})$-action on
$H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(1))^{*}$ are given by the
projections $\frac{\alpha_{i}\cdot\beta}{||\beta||^{2}}\beta$ of the
$T$-weights $\alpha_{i}\in\mathfrak{t}$ to the line spanned by
$\beta\in\mathfrak{t}$. Thus to determine the minimum
$\lambda_{\beta}(\mathbb{G}_{m})$-weight $\omega_{\min}$, we want to minimise
$\alpha_{i}\cdot\beta$ over all weights $\alpha_{i}$ of points in $Y_{\beta}$.
As $\beta\in\mathcal{B}$, we have $Z_{\beta}\neq\emptyset$ and so from the
descriptions given in (2.2) of $\overline{Y_{\beta}}$ and $Z_{\beta}$, we see
that there is a point in $Z_{\beta}$ with at least one $T$-weight $\alpha_{i}$
satisfying $\alpha_{i}\cdot\beta=||\beta||^{2}$. The weights $\alpha_{j}$ of
points in $\overline{Y_{\beta}}$ all satisfy
$\alpha_{j}\cdot\beta\geq||\beta||^{2}$. Hence, the minimal
$\lambda_{\beta}(\mathbb{G}_{m})$-weight corresponds to $\beta\in\mathfrak{t}$
and by definition $Z_{\beta,\min}\subset X_{\beta}^{\lambda(\mathbb{G}_{m})}$
is the minimal $\lambda_{\beta}(\mathbb{G}_{m})$-weight space (cf. Definition
2.13); that is,

$Z_{\beta,\min}=\\{[p_{0}:\dots:p_{n}]:p_{i}=0\text{ if
}\alpha_{i}\cdot\beta\neq||\beta||^{2}\\},$

which is equal to $Z_{\beta}$ by (2.2). From the description in (2.2) of
$Y_{\beta}$, we see that $Y_{\beta}$ is the open subscheme of
$X_{\beta}=\overline{Y_{\beta}}$ consisting of points $y$ such that
$\lim_{t\rightarrow 0}\lambda_{\beta}(t)\cdot y\in Z_{\beta}$, and so it
follows that $X_{\beta,\min}=Y_{\beta}$ (cf. Definition 2.13). Note that if we
twist $\mathcal{L}$ by any scalar multiple of the character corresponding to
$\beta$, then the subsets $X_{\beta,\min}$ and $Z_{\beta,\min}$ remain
unchanged.

For the final statement, recall that the (rational) character
$\chi_{-(1+\epsilon)\beta}:P_{\beta}\rightarrow\mathbb{G}_{m}$ merely shifts
the $\lambda_{\beta}(\mathbb{G}_{m})$ weights by $-(1+\epsilon)||\beta||$. In
order for the $\widehat{U}_{\beta}$-action with respect to this twisted
linearisation to be adapted, we need the origin to separate the minimal weight
from all the other weights. Since the minimum weight for the original
linearisation $\mathcal{L}$ corresponds to $\beta$, we need to shift this
weight just beyond the origin so that this weight is negative and all the
other weights are positive. This is achieved by taking
$\epsilon\in\mathbb{Q}_{>0}$ sufficiently small. ∎

We fix $\epsilon\in\mathbb{Q}$ sufficiently small as above, so that the
perturbation
$\mathcal{L}_{\beta}^{\text{per}}:=\mathcal{L}^{\chi_{-(1+\epsilon)\beta}}$ of
the canonical linearisation $\mathcal{L}_{\beta}$ on $\overline{Y_{\beta}}$ is
adapted for the $\widehat{U}_{\beta}$-action. Now we can prove Theorem 1.1.

###### Proof of Theorem 1.1.

The proof follows from Theorem 2.29. For the second statement, we just need to
determine the semistable locus for the action of the reductive group
$\overline{L_{\beta}}:=L_{\beta}/\lambda_{\beta}(\mathbb{G}_{m})$ on
$Z_{\beta}$ with respect to the perturbed linearisation
$\mathcal{L}_{\beta}^{\text{per}}$, as by Definition 2.24 we have

$X_{\beta}^{P_{\beta}-s}=\\{y\in
X_{\beta}^{\widehat{U}_{\beta}-s}:p_{\beta}(y)\in
Z_{\beta}^{\overline{L_{\beta}}-s}(\mathcal{L}_{\beta}^{\text{per}})\\}.$

Since $[\overline{L}]_{0}$ holds, semistability coincides with stability for
the $\overline{L_{\beta}}$-action on $Z_{\beta}$ linearised by
$\mathcal{L}_{\beta}$. By variation of (reductive) GIT quotients, this
linearisation $\mathcal{L}_{\beta}$ lies inside a VGIT chamber and so the
stable locus is unaltered on passing to the small perturbation
$\mathcal{L}_{\beta}^{\text{per}}$; that is,

(7)
$Z_{\beta}^{\overline{L_{\beta}}-s}(\mathcal{L}_{\beta}^{\text{per}})=Z_{\beta}^{\overline{L_{\beta}}-s}(\mathcal{L}_{\beta})=Z_{\beta}^{\overline{L_{\beta}}-ss}(\mathcal{L}_{\beta})=Z_{\beta}^{{L_{\beta}}-ss}(\mathcal{L}_{\beta})=:Z_{\beta}^{ss}$

which completes the proof. ∎

######  Remark 3.6.

  1. (1)

As $Z_{\beta}$ is unstable for the $\lambda_{\beta}(\mathbb{G}_{m})$-action on
$\overline{Y_{\beta}}$ linearised by $\mathcal{L}^{\text{per}}_{\beta}$, we
see that a semistable point $y\in Y_{\beta}$ cannot be identified with its
limiting flow $p_{\beta}(y)\in Z_{\beta}$; hence the quotients constructed by
the above theorem avoid the collapsing described in Proposition 3.1. Indeed
they are geometric quotients and so are orbit spaces.

  2. (2)

The above theorem also holds on replacing condition $[\widehat{U}]_{0}$ with
condition $[\widehat{U}]$ (or $[\widehat{U};\overline{L}\text{-ss}]$). If this
condition fails, then one can construct a quotient of the $P_{\beta}$-action
on an open subset of $\overline{Y}_{\beta}$ by performing a sequence of blow-
ups as described in $\S$2.3.6.

####  3.2.1. Application to moduli of sheaves of fixed HN type

Let $\tau=(P_{1},\dots,P_{l})$ be a length $l\geq 2$ HN type for sheaves on a
polarised projective scheme $(B,\mathcal{O}_{B}(1))$. As explained in
$\S$2.2.1, there is a close relationship between the HN stratification on the
Quot schemes used for the construction of moduli of sheaves and the associated
HKKN stratification. In particular, constructing a moduli space of sheaves of
HN type $\tau$ is equivalent to constructing a quotient of the action of a
parabolic group $P_{\tau}$ on (an open subset of) $Y_{\tau}^{ss}$. We would
ideally like to apply Theorem 1.1 above to the $P_{\tau}$-action on
$X_{\tau}:=\overline{Y_{\tau}}$ linearised by
$\mathcal{L}_{\tau}^{\text{per}}$. However, for sheaves of fixed HN type the
various stabiliser conditions in Definition 2.20 do not hold in general.
Moreover, for length $l>2$, we have the following issues concerning both the
reductive and unipotent stabilisers (_cf._ [25]).

  1. (i)

The central torus $T_{\tau}:=Z(L_{\tau})\cong\mathbb{G}_{m}^{l-1}$, which
scales the pieces in the HN graded sheaf, acts trivially on $Z_{\tau}$ but
non-trivially on $X_{\tau}$. Thus the $T_{\tau}$-weight picture for $X_{\tau}$
linearised by $\mathcal{L}_{\tau}$ looks like a cone with a vertex at the
origin. Consequently semistability equals stability for the action of
$\overline{L_{\tau}}=L_{\tau}/\lambda(\mathbb{G}_{m})$ on $Z_{\tau}$ if and
only if $\tau$ is a coprime HN type (i.e. semistability coincides with
stability for all HN subquotients) of length $2$. Moreover, for $l>2$, the
$\overline{L_{\tau}}$-stable locus is empty, which implies that the $P$-stable
locus in Definition 2.24 is also empty.

  2. (ii)

In length $l>2$, the unipotent radical $U_{\tau}$ is no longer abelian.
Moreover, although the unipotent stabiliser groups often admit sheaf theoretic
interpretations, the dimension of these unipotent stabilisers can vary. In
particular, for $l>2$, the map $p_{\tau}:Y_{\tau}^{ss}\rightarrow
Z_{\tau}^{ss}$ does not preserve the dimensions of unipotent stabiliser groups
in general, thus the blow-up procedure in [1] is almost intractable as we may
blow-up all of $Z_{\tau}^{ss}$.

These statements are proved in Propositions 3.7 and 3.8 below.

These issues are not specific to sheaves of fixed HN type. For moduli of
objects in an abelian linear category (for example, moduli of quiver
representations, of Higgs sheaves or other decorated sheaves) one also
encounters these problems for HN types of length $l>2$. This is the main
motivation for the Quotienting-in-Stages procedure we develop later in this
paper.

Let us explain the issues mentioned above in the case of sheaves of HN type
$\tau$.

######  Proposition 3.7.

Consider the linearised action of $P_{\tau}=U_{\tau}\rtimes L_{\tau}$ on
$X_{\tau}=\overline{Y_{\tau}}$.

  1. _(i)_

The central torus $T_{\tau}:=Z(L_{\tau})\cong\mathbb{G}_{m}^{l-1}$ acts
trivially101010However, $T_{\tau}$ acts non-trivially on $Y_{\tau}$ if there
exists a sheaf $\mathcal{E}$ such that
$\mathcal{E}\ncong\operatorname{gr}(\mathcal{E})$. on $Z_{\tau}$.

  2. _(ii)_

If $l>2$, then $Z_{\tau}^{\overline{L_{\tau}}-s}=\emptyset$ and
$Z_{\tau}^{\overline{L_{\tau}}-ss}=Z_{\tau}^{ss}$. Thus semistability does not
coincide with stability for the $\overline{L_{\tau}}$-action on $Z_{\tau}$.

  3. _(iii)_

If $l=2$, then
$Z_{\tau}^{\overline{L_{\tau}}-s}=Z_{\tau}^{\overline{L_{\tau}}-ss}$ if and
only if $\tau$ is coprime.

###### Proof.

Equation (5) expresses $Z_{\tau}$ as a product of closed subschemes
$R_{n}(P_{i})$ of the Quot schemes $Q_{n}(P_{i})$. As
$L_{\tau}\cong(\prod_{i=1}^{l}\mathrm{GL}_{P_{i}(n)})\cap\mathrm{SL}_{P(n)}$
and the diagonal $\mathbb{G}_{m}$ in each $\mathrm{GL}_{P_{i}(n)}$ acts
trivially on each Quot scheme $Q_{n}(P_{i})$, we see that $T_{\tau}$ acts
trivially on $Z_{\tau}$. Since
$\overline{T_{\tau}}:=T_{\tau}/\lambda_{\tau}(\mathbb{G}_{m})\cong\mathbb{G}_{m}^{l-2}<\overline{L}_{\tau}$
acts trivially on $Z_{\tau}$, we see for $l>2$ that GIT semistability and
stability do not coincide for $\overline{L}_{\tau}$. For $l=2$, we have a
surjection
$R_{\tau}:=\mathrm{SL}_{P_{1}(n)}\times\mathrm{SL}_{P_{2}(n)}\twoheadrightarrow\overline{L_{\tau}}$
with finite kernel and so (semi)stability with respect to these groups
coincides. The final statement then follows, as GIT semistability coincides
with stability for the $\mathrm{SL}_{P_{i}(n)}$-action on $Q^{ss}(P_{i})$ if
and only if for sheaves with Hilbert polynomial $P_{i}$ semistability
coincides with stability. ∎

Let us now consider the various stabiliser conditions for the unipotent group
$U_{\tau}$. If $l>2$, then $U_{\tau}$ is not abelian and so we need to
consider stabilisers for the subgroups appearing in the lower central series
$U^{\\{l-1\\}}_{\tau}=\\{I\\}<U_{\tau}^{\\{l-2\\}}<\cdots<U_{\tau}^{\\{0\\}}=U_{\tau}$
(see Remark 4.2 below). The $U_{\tau}^{\\{l-2\\}}$-stabilisers of quotient
sheaves $q_{\mathcal{E}}:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ are related to certain filtered
endomorphism groups for the HN filtration
$0=\mathcal{E}^{(0)}\subsetneq\mathcal{E}^{(1)}\subsetneq\cdots\subsetneq\mathcal{E}^{(l)}=\mathcal{E}$.
Let $\operatorname{End}_{-j}(\mathcal{E})$ denote the global sections of the
sheaf $\mathcal{H}om_{-j}(\mathcal{E},\mathcal{E})$, whose sections over
$U\subset B$ are

$\mathcal{H}om_{-j}(\mathcal{E},\mathcal{E})(U):=\\{\varphi\in\mathcal{H}om(\mathcal{E},\mathcal{E})(U):\varphi(\mathcal{E}^{(i)}(U))\subset\mathcal{E}^{(i-j)}(U)\text{
for all }\>i\\}.$

######  Proposition 3.8.

For $q_{\mathcal{E}}:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ in $Y_{\tau}^{ss}$, the following
statements hold.

  1. _(i)_

We have
$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}}(q_{\mathcal{E}})\cong\operatorname{End}_{-1}(\mathcal{E})$
and, for $0\leq i\leq l-1$, we have

$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{\\{i\\}}}(q_{\mathcal{E}})\cong\operatorname{End}_{-(i+1)}(\mathcal{E}).$

  2. _(ii)_

If $q_{\mathcal{E}}\in Z_{\tau}^{ss}$, then
$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{\\{i\\}}}(q_{\mathcal{E}})=\oplus_{j>k+i}\operatorname{Hom}(\mathcal{E}_{j},\mathcal{E}_{k})$
for $0\leq i\leq l-1$.

In particular, the $P_{\tau}$-action on $\overline{Y}_{\tau}$ satisfies
condition $[\widehat{U};\overline{L}\text{-ss}]$ with respect to the lower
central series if and only if for $1\leq j<l$, the function
$\dim\operatorname{End}_{-j}(-)$ is constant on the set of all sheaves over
$X$ with HN type $\tau$.

###### Proof.

Statement (i) for $i=0$ follows as the stabiliser in $U_{\tau}$ of
$q_{\mathcal{E}}$ consists of all automorphisms of $\mathcal{E}$ of the form
$\text{Id}_{\mathcal{E}}+\alpha$, where
$\alpha\in\operatorname{End}_{-1}(\mathcal{E})$ for the HN filtration and the
version for higher $i$ follow similarly. Statement (ii) follows as for
$q_{\mathcal{E}}\in Z_{\tau}^{ss}$, we have
$\mathcal{E}=\operatorname{gr}(\mathcal{E})$.

∎

Thus, as explained in [25], even in HN length $l=2$ conditions
$[\widehat{U}]_{0}$, $[\widehat{U};\overline{L}\text{-ss}]_{0}$,
$[\widehat{U}]$ and $[\widehat{U};\overline{L}\text{-ss}]$ fail in general.
However, in length $l=2$ the above Proposition 3.8 simplifies as follows:

$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}}(q_{\mathcal{E}})\cong\operatorname{Hom}(\mathcal{E}_{2},\mathcal{E}_{1})\cong\operatorname{Lie}\operatorname{Stab}_{U_{\tau}}(p_{\tau}(q_{\mathcal{E}}))$

where $p_{\tau}(q_{\mathcal{E}})\in Z_{\tau}^{ss}$ is the quotient sheaf for
the associated graded
$\operatorname{gr}(\mathcal{E})=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$. This
means that $p_{\tau}$ preserves the dimension of $U_{\tau}$ stabilisers, which
is enough to ensure that we do not blow up the whole minimal weight space when
performing the blow-up process to obtain a quotient. In this way, [25]
constructs moduli spaces of sheaves of fixed HN type for length $l=2$.

However, in general this property of $p_{\tau}$ fails along with
$[\widehat{U}]_{0}$, $[\widehat{U};\overline{L}\text{-ss}]_{0}$,
$[\widehat{U}]$ and $[\widehat{U};\overline{L}\text{-ss}]$ for $l>2$. Let us
describe why, for $l=3$. In this case
$\operatorname{End}_{-2}(\mathcal{E})=\operatorname{Hom}(\mathcal{E}_{3},\mathcal{E}_{1})$
and $\operatorname{End}_{-1}(\mathcal{E})$ is given by a long exact sequence

$0\rightarrow\operatorname{End}_{-2}(\mathcal{E})\rightarrow\operatorname{End}_{-1}(\mathcal{E})\rightarrow\operatorname{Hom}(\mathcal{E}_{3},\mathcal{E}_{2})\oplus\operatorname{Hom}(\mathcal{E}_{2},\mathcal{E}_{1})\stackrel{{\scriptstyle\delta(\mathcal{E})}}{{\rightarrow}}\operatorname{Ext}^{1}_{-2}(\mathcal{E},\mathcal{E})\rightarrow\cdots$

where $\delta(\mathcal{E})\neq 0$ in general. For the associated graded, we
have $\delta(\operatorname{gr}(\mathcal{E}))=0$ and

$\operatorname{End}_{-1}(\operatorname{gr}(\mathcal{E}))=\operatorname{Hom}(\mathcal{E}_{3},\mathcal{E}_{1})\oplus\operatorname{Hom}(\mathcal{E}_{3},\mathcal{E}_{2})\oplus\operatorname{Hom}(\mathcal{E}_{2},\mathcal{E}_{1}).$

Hence, for length $l=3$, we have

$\dim\operatorname{Lie}\operatorname{Stab}_{U_{\tau}}(q_{\mathcal{E}})=\dim\operatorname{Lie}\operatorname{Stab}_{U_{\tau}}(p_{\tau}(q_{\mathcal{E}}))\iff\delta(\mathcal{E})=0$

and so $p_{\tau}$ will not in general preserve $U_{\tau}$-stabiliser
dimension, as $\delta(\mathcal{E})$ will be non-zero in general. As the length
increases, we need to consider $l-2$ long exact sequences to relate the
dimension of $\operatorname{End}_{-1}(\mathcal{E})$ with
$\operatorname{End}_{-1}(\operatorname{gr}(\mathcal{E}))=\oplus_{i>j}\operatorname{Hom}(\mathcal{E}_{i},\mathcal{E}_{j})$.
If the coboundary map in this long exact sequences is zero, then these groups
have the same dimension.

##  4\. Quotienting-in-Stages for parabolics: Definitions and Results

In this section, we introduce a new method for constructing quotients by
parabolic group actions in stages, where we use a sequence of different 1-PSs
which are central in the Levi subgroup to grade the unipotent radical of a
maximal parabolic at each stage. Our primary motivation is to construct
quotients of unstable HKKN strata $S_{\beta}\cong
G\times^{P_{\beta}}Y_{\beta}^{ss}$ representing moduli of objects of a fixed
HN type in a linear abelian category. Our construction will be suited to the
case when the centre of $L_{\beta}$ acts trivially on $Z_{\beta}$ and so one
cannot directly apply Theorem 1.1. We will focus on the case of a parabolic
subgroup $P$ of the special linear group $\mathrm{SL}_{N}$.

###  4.1. Parabolic subgroups of the special linear group

For concreteness, we will assume that our parabolic subgroup
$P<\mathrm{SL}_{N}$ is block upper triangular, since we can arrange this by
conjugating by an element in $\mathrm{SL}_{N}$. In particular, we assume that
$P$ is a parabolic subgroup associated to a diagonal (rational) 1-PS with
decreasing weights as follows.

######  Definition 4.1.

For a (rational) 1-PS $\lambda:\mathbb{G}_{m}\rightarrow\mathrm{SL}_{N}$ of
the form

(8) $\lambda(t)=\operatorname{diag}(t^{r_{1}},\dots,t^{r_{N}})\>\text{ with
}\>r_{1}\geq r_{2}\geq\cdots\geq r_{N}\>\text{ such that
}\sum_{i=1}^{N}r_{i}=0,$

we let $P=P(\lambda)$ be the associated parabolic subgroup and we write
$P=U\rtimes L$ where $U$ is the unipotent radical and $L$ is the Levi factor.
Let $l=l(\lambda)\geq 2$ denote the number of distinct weights of $\lambda$,
and write these weights as $\beta_{1}>\dots>\beta_{l}$ with multiplicities
$\underline{m}=\underline{m}(\lambda)=(m_{1},\dots,m_{l})$. We refer to $l$
and $\underline{m}$ as the _length_ and _multiplicities_ of $\lambda$ (or of
$P$). If $l=2$, then $P$ is a _maximal parabolic subgroup_ and $U$ is abelian,
and if $l>2$, then $U$ is non-abelian.

Note that the multiplicities sum to $N$ and $\sum_{i=1}^{l}m_{i}\beta_{i}=0$
as $\lambda$ is a 1-PS of $\mathrm{SL}_{N}$. Moreover, $l(\lambda)$ and
$\underline{m}(\lambda)$ only depend on the conjugacy class of $\lambda$. The
parabolic group $P$ is explicitly determined from $l$ and $\underline{m}$ as
follows.

######  Remark 4.2.

For $\lambda$ as in (8) with length $l=l(\lambda)$ and multiplicities
$(m_{1},\dots,m_{l})$, we have

$P=P(\lambda)=\left\\{\left(\begin{array}[]{ccccc}A_{11}&A_{12}&\cdots&\cdots&A_{1l}\\\
0&A_{22}&A_{23}&\cdots&A_{2l}\\\ \vdots&\ddots&\ddots&&\vdots\\\
\vdots&&0&\ddots&A_{l-1\>l}\\\
0&\cdots&\cdots&0&A_{ll}\end{array}\right)\in\mathrm{SL}_{N}:A_{ij}\in\operatorname{Mat}_{m_{i}\times
m_{j}}\right\\}.$

The Levi subgroup $L=\\{A\in P_{\beta}:A_{ij}=0\text{ for }1\leq i<j\leq
l\\}<P$ consists of block diagonal matrices and is the intersection of the
block diagonal embedding
$\prod_{i=1}^{l}\mathrm{GL}_{m_{i}}\hookrightarrow\mathrm{GL}_{N}$ with
$\mathrm{SL}_{N}$. The centre of the Levi is a rank $l-1$ torus, denoted

$T:=Z(L)=\\{\operatorname{diag}(t_{1}I_{m_{1}},\dots,t_{l}I_{m_{l}}):t_{i}\in\mathbb{G}_{m},\prod_{i=1}^{l}t_{i}^{m_{i}}=1\\}\cong\mathbb{G}_{m}^{l-1}.$

The unipotent radical $U=\\{A\in P_{\beta}:A_{ii}=I_{m_{i}}\text{ for }1\leq
i\leq l\\}<P$ has lower central series given by filtering by off-diagonals:

$U^{\\{l-1\\}}=\\{I\\}<U^{\\{l-2\\}}<\cdots<U^{\\{k\\}}:=\\{A\in
U:A_{ij}=0\text{ if }j-i\leq k\\}<\cdots<U^{\\{0\\}}=U.$

######  Definition 4.3.

For $P=P(\lambda)$ as above, we let $R:=\prod_{i=1}^{l}\mathrm{SL}_{m_{i}}$ be
the semisimple part of $L$. We define

$H:=U\rtimes R<P=U\rtimes L,\hskip 4.0pt\text{ and }\hskip
10.0pt\widehat{H}:=H\times T\cong H\times\mathbb{G}_{m}^{l-1}.$

Note that there is a surjection $R\rightarrow L/T$ with finite kernel.
Similarly, there is a surjection $H\rightarrow P/T$ with finite kernel.
Consequently, from the perspective of GIT, quotienting by $\widehat{H}$ and
$P$ are equivalent, since quotienting by finite groups poses no problem. Our
construction will take a quotient of the $\widehat{H}$-action rather than of
the $P$-action, in $l-1$ stages.

###  4.2. Important notation for the parabolic group action

Suppose our parabolic group $P<\mathrm{SL}_{N}$ acts on an irreducible
projective scheme $X$ with respect to an ample linearisation $\mathcal{L}$.
Before describing the Quotienting-in-Stages procedure for such an action, we
give some definitions of subgroups of $P$ and loci in $X$ that will play a
prominent role in the construction. After this in Definition 4.11 we define
the locus of which we will construct a quotient: the Quotienting-in-Stages
stable locus, denoted $X^{P-qs}$.

####  4.2.1. Subgroups of P

Instead of using the lower central series of Remark 4.2, which filters $U$
using the off-diagonals, we will consider a normal series for $U$ such that
the successive quotients are unipotent radicals of maximal parabolic groups;
that is, parabolics of length $l=2$. In fact, there are several ways in which
we could do this, depending on a choice of an ordering of the subgroups of $U$
which are unipotent radicals of maximal parabolic subgroups containing $P$.

######  Definition 4.4.

For $1\leq k\leq l-1$, define the subgroup

$U^{[k]}:=\\{A\in U:A_{ij}=0\text{ if }j\leq k\text{ or }i>k\\}<U.$

We let we $U^{(i)}:=\prod_{j\leq i}U^{[j]}$ to be the subgroup of $U$
generated by the first $i$ of these subgroups. Then we have the _row
filtration_ of $U$, a normal series filtering $U$ by its rows from top to
bottom

$\\{I\\}<U^{(1)}<\cdots<U^{(l-1)}=U$

whose successive quotients $U_{i}:=U^{(i)}/U^{(i-1)}\cong
U^{[\sigma(i)]}/U^{[i]}\cap U^{[i-1]}$ are abelian.

######  Remark 4.5.

Though we will not make use of this idea in the present paper, one could
instead take any permutation $\sigma\in S_{l-1}$, which determines a different
ordering $U^{[\sigma(1)]},\dots,U^{[\sigma(l-1)]}$ of these groups, and obtain
a filtration by $U_{\sigma}^{(i)}:=\prod_{j\leq i}U^{[\sigma(j)]}$ with
abelian subquotients. For example, the permutation $\sigma:i\mapsto l-i$
corresponds to filtering $U$ by its columns, rather than rows.

######  Definition 4.6.

Let us introduce some further subgroups and notation as follows. These groups
will feature heavily in the Quotienting-in-Stages procedure outlined in
$\S$4.3 below.

  1. (1)

For $1\leq i<l=l(\lambda)$, define 1-PSs $\lambda^{(i)}$ and
$\lambda^{[i]}:\mathbb{G}_{m}\rightarrow Z(L)$ by

$\quad\quad\lambda^{(i)}(t):=\operatorname{diag}(t^{\beta_{1}}I_{m_{1}},\dots,t^{\beta_{i}}I_{m_{i}},t^{\beta_{>i}}I_{m_{>i}})\quad\text{and}\quad\lambda^{[i]}(t):=\operatorname{diag}(t^{\beta_{\leq
i}}I_{m_{\leq i}},t^{\beta_{>i}}I_{m_{>i}})$

where $m_{>i}=\sum_{j>i}m_{j}$ and $m_{\leq i}=\sum_{j\leq i}m_{j}$, and the
exponents are defined by averaging:

$\beta_{>i}:=\frac{\sum_{j>i}\beta_{j}m_{j}}{m_{>i}}\quad\quad\beta_{\leq
i}:=\frac{\sum_{j\leq i}\beta_{j}m_{j}}{m_{\leq i}}.$

  2. (2)

The unipotent group $U^{[i]}<U$ is the unipotent radical of the maximal
parabolic subgroup $P^{[i]}:=P(\lambda^{[i]})>P$ associated to the 1-PS
$\lambda^{[i]}$. Consequently $\lambda^{[i]}$ grades $U^{[i]}$ and $U^{[i]}$
is normal in $P$, since it is normal in the larger group $P^{[i]}$.

  3. (3)

The group $U^{(i)}<U$ is the unipotent radical of the parabolic subgroup
$P(\lambda^{(i)})>P$ associated to the 1-PS $\lambda^{(i)}$ and thus
$\lambda^{(i)}$ grades $U^{(i)}$.

  4. (4)

We write $U^{[i-1,i]}:=U^{[i-1]}\cap U^{[i]}\leq U$.

  5. (5)

We obtain an increasing filtration $P^{(i)}:=U^{(i)}\rtimes L^{(i)}$ of $P$,
where for $i<l-1$,

$L^{(i)}:=\\{A\in L:A_{jj}=I_{m_{j}}\text{ if }j>i\\}$